ROBUST MODEL PREDICTIVE CONTROL OF INTEGRATING AND ... · sem precedentes. Aqui preciso honrá-la,...

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MÁRCIO ANDRÉ FERNANDES MARTINS ROBUST MODEL PREDICTIVE CONTROL OF INTEGRATING AND UNSTABLE TIME DELAY PROCESSES Tese apresentada à Escola Politécnica da Universidade de São Paulo para obtenção do Título de Doutor em En- genharia. São Paulo 2014

Transcript of ROBUST MODEL PREDICTIVE CONTROL OF INTEGRATING AND ... · sem precedentes. Aqui preciso honrá-la,...

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MÁRCIO ANDRÉ FERNANDES MARTINS

ROBUST MODEL PREDICTIVE CONTROL OF INTEGRATING ANDUNSTABLE TIME DELAY PROCESSES

Tese apresentada à Escola Politécnicada Universidade de São Paulo paraobtenção do Título de Doutor em En-genharia.

São Paulo2014

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MÁRCIO ANDRÉ FERNANDES MARTINS

ROBUST MODEL PREDICTIVE CONTROL OF INTEGRATING AND

UNSTABLE TIME DELAY PROCESSES

Tese apresentada à Escola Politécnicada Universidade de São Paulo paraobtenção do Título de Doutor em En-genharia.

Área de Concentração:Engenharia Química

Orientador:Prof. Dr. Darci Odloak

São Paulo2014

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Este exemplar foi revisado e alterado em relação à versão original,sob responsabilidade única do autor e com a anuência de seu orien-tador.

São Paulo, 05 de novembro de 2014.

Assinatura do autor

Assinatura do orientador

CATALOGAÇÃO-NA-PUBLICAÇÃO

Martins, Márcio André Fernandes.Robust model predictive control of integrating and unstable

time delay processes / M.A.F. Martins. – versão corr.– SãoPaulo, 2014.

121 p.

Tese (Doutorado) – Escola Politécnica da Universidade deSão Paulo. Departamento de Engenharia Química.

1. Controle preditivo 2. Processos químicos I. Universi-dade de São Paulo. Escola Politécnica. Departamento deEngenharia Química. II. t.

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Ao estado de graça amado Bahia. Junto-me a João Gilberto quando ele afirma e canta: “Na Bahiaque é meu lugar. Tem meu ão, tem meu céu, tem meu mar”.

Ao grande amor da minha vida, àquela que me tem em verso, prosa e pœsia – minha querida esposaLucimar. Não há outra pessoa que mereça tanto a partilha deste sonho quanto ela. Ao falar de Lulu,sou tomado de uma explosão de bons sentimentos, companheirismo, lealdade, unidade, carinho, afeto e amorsem precedentes. Aqui preciso honrá-la, pela sua abnegação e sacrifício para suportar minhas ausênciasnos inúmeros momentos de elaboração da tese. Em tempo, gostaria de agradecer a Deus pela maravilhosavida que vivemos até aqui e, antecipadamente, àquela melhor que nos aguarda. Decerto, os valores quenos cingem, assim como nosso terno e sincero amor, me levam a acreditar que somos e seremos umafamília feliz. Encerro esta dedicatória a você, meu amor, com os seguintes versos:

De repente...Percebi que deveria mesmo me apaixonar...Que felicidade a minha ter sido Lucimar:A mulher que me foi destinada a amar.

Que eu seja o teu baluarte,Tua bebida salutar,Tua substancial comida, eTodo infinito amor desta vida.

A cada anoitecerQuero ser o teu seguro acolherNas noites quentesSejamos ardentes amantesEm noite friaEstejamos unidos por simpatiaE que em cada amanhecerPossamos cada vez mais nos conhecer.

São muitas as afinidades existentesIsso atrai-nos ferozmentePor isso roubo as palavras de um pœta que enfatiza:Tu és o máximo denominador comum da minha vida.

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Acknowledgments (in Portuguese)

É natural que o desenvolvimento de um trabalho árduo, longo emeticuloso quanto este rogue

esforços de outras pessoas, as quais registro aqui que elas contribuíram incondicionalmente

com seu conhecimento, experiência, disponibilidade, paciência e incentivos. Nas circunstâncias

presentes, tenho o dever de restringir-me às influências mais significativas que até mesmo uma

memória falha (como a minha) não poderia esquecer por completo. Embora, deixo, desde já,

minha profunda gratidão a cada pessoa que pode, justificadamente, encontrar sinais de suas

influências nesta obra.

Inicialmente, devoto merecidos agradecimentos ao maior e mais enigmático dos seres sobrena-

turais: DEUS, o criador; a (in)certeza da sua onipresença, onipotência e onisciência conforta-me

nas inúmeras crises existenciais que me assombram constantemente.

Transmito minha eterna gratidão ao Prof. Darci Odloak por sua valiosa orientação e amizade.

Sua engenhosidade e sagacidade se fizeram presentes em demasia nesta obra, o que coube a

mim foi apenas seguir diligentemente seus ensinamentos e direcionamentos. Tenho a grata sa-

tisfação em confessar que minha admiração e respeito por suacompetência e brilhante trajetória

acadêmica só aumentaram ao longo do desenvolvimento desta tese.

Necessito registrar também as contribuições dos membros dabanca examinadora, professores

Dr. Claudio Garcia, Dr. Marcelo Embiruçu, Dr. Oscar Sotomayor Zanabria e o engenheiro Dr.

Antônio Zanin, que, devidamente acatadas e incorporadas aotexto, tornaram a tese melhor do

que aquela que eu poderia fazê-la sozinho.

Não poderia deixar de enfatizar o agradável convívio com todos os colegas, professores e

funcionários do PQI, assim como os companheiros do Semi-Industrial, Aldo, Capron, David,

Diego, Ricardo, Zé e, em especial, ao Bruno e André, que dividiram comigo as alegrias, desco-

bertas, indagações e angústias desta laboriosa e gratificante jornada... Não me esquecerei dos

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excelentes momentos de entretenimento vívidos: nas corridas e pedaladas, nos churrascos do

bloco 21, nos almoços de bandejão, na taça carabina,...

Gostaria de registrar meus sinceros agradecimentos ao Prof. Ricardo Kalid, quem primeiro me

introduziu neste fascinante mundo da pesquisa científica; seus inúmeros incentivos, conselhos

e confiança depositados em mim foram fundamentais e decisivos para que eu pudesse enxergar

esta vocação profissional. Valeu muito a pena te ouvir meu amigo!

Meus agradecimentos finais são creditados aos meus familiares. Às pessoas sem as quais este

trabalho não poderia ter se concretizado, meus queridos e amados pais, Manoel e Antonia;

a eles reafirmo meus leais votos de ternura e amor sem precedentes. A contribuição deles à

materialização desta obra iniciou-se no momento em que elesentenderam e permitiram que

minha vocação fosse levada adiante desde o fim da graduação. Aos meus irmãos, Marcelo,

Érica e Bruna, agradeço toda amizade e fraternidade experimentadas e a serem vívidas; a vocês

tenho a declarar: tragam navalha e nem sangue lhes faltará. Àminha amada esposa, reconheço

minha infindável dívida para com sua paciência e compreensãodurante os três longos anos em

que me ocupei deste trabalho. Não sei como agradecer-lhe meuamor! Ao meu sogro, Pedro,

e minha sogra, Laurentina, que me abriram a porta de casa e convidaram-me a fazer parte de

mais outro equilibrado seio familiar. Muito obrigado!

Procedendo-se às formalidades de praxe, encerro esta seçãoagradecendo os suportes financeiros

dispensados pela CAPES, no primeiro ano de pesquisa, e pela FAPESP (processo 2011/22313-

5), nos dois anos subsequentes.

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“Utopia [. . . ] está en el horizonte. Me acerco dos pasos, ellase aleja dos pasos. Camino diez

pasos y el horizonte se corre diez pasos más allá. Por mucho que yo camine, nunca la

alcanzaré. Para que sirve la utopia? Para eso sirve: para caminar.”

(Eduardo Galeano, escritor e jornalista uruguaio.)

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Abstract

The design of stable model predictive control (MPC) strategies that explicitly incorporate the

model uncertainty into the control formulation still remains an open issue, although a rich theory

has been developed to the synthesis of robustly stabilizingMPC schemes. In fact, the existing

solutions to the robust MPC problem seem far from an acceptable stage of practical imple-

mentations, chiefly when the process system is composed of integrating and unstable poles, as

well as time delays between its input and output variables. Within this perspective, the ultimate

goal of this thesis is to develop a new framework for robust MPC synthesis which guarantees

closed-loop stability of integrating and unstable time delay processes. On this subject, three

different robust MPC strategies are developed. The two firstconcerns on integrating time delay

processes; the former is based on a two-step control formulation, whereas the latter is posed

as a one-step control optimization problem and state-spacemodel description is more general

than that adopted in the former formulation. The third proposed strategy focuses on one-step

control formulation-based unstable time delay processes.Aiming at practical implementation

purposes, the controllers proposed herein comprise the following aspects: (i) the offset free

control laws are obtained without the need to include an additional steady-state calculation op-

timization layer due to the enclosure of proper state-spacemodels in the incremental form of

the inputs, which are derived of analytical expressions of step response of the process system;

(ii) the uncertainty of all model parameters, e.g. gains, time constants, time delays and so on,

is considered in the problem formulation; (iii) the proofs of robust Lyapunov stability are easily

carried out of an intuitive way by imposing terminal equality constraints and cost-contracting

constraints; (iv) the suitable inclusion of slack variables, which does not commit the stabil-

ity properties of the controllers, ensure that the proposedoptimization problems are always

feasible; (v) stable integration with real-time optimization layer, seeing as the controllers are

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designed to work in the optimum target tracking scheme wherethey should drive the process

to the optimum operating point, while maintaining the remaining inputs and outputs inside pre-

defined zones instead of fixed set-points. Simulation examples typical of the process industry

are exploited to illustrate the helpfulness of the proposedcontrol methods and demonstrate that

they can be implemented in real applications.

Keywords: Model Predictive Control. Infinite Horizon. Robust Stability.

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Resumo

O projeto de estratégias de controle preditivo (MPC) com estabilidade garantida, que incorpora

explicitamente a incerteza de modelo na formulação de controle, ainda permanece uma questão

em aberto na literatura, embora uma ampla teoria já tenha sido desenvolvida para a síntese de

algoritmos MPC robustamente estáveis. Em verdade, as soluções existentes para o problema

de MPC robusto estão longe de uma etapa aceitável de implementação prática, principalmente

se o sistema de processo é composto de modos integradores ou instáveis, e também apresenta

atrasos de tempo (tempos mortos) entre suas variáveis de entrada e saída. Sob esta perspec-

tiva, o objetivo principal desta tese é desenvolver uma estrutura de síntese de controladores

MPC com estabilidade robusta garantida para sistemas de processo com as características in-

tegradoras ou instáveis, assim como tempos mortos entre as variáveis. Particularmente, três

diferentes estratégias de MPC robusto são desenvolvidas neste trabalho. As duas primeiras

referem-se a sistemas integradores com tempos mortos: o primeiro algoritmo é baseado em

uma formulação de controle em dois passos, enquanto o segundo é posto como um problema

de otimização de controle em um passo e a representação de modelo em variáveis de estado

é mais geral do que aquela adotada na formulação do primeiro método. A terceira estraté-

gia proposta focaliza os sistemas instáveis com tempos mortos através de uma formulação de

controle em um passo. Ademais, visando o caso de implementação prática, os controladores

desenvolvidos compreende os seguintes aspectos: (i) as leis de controle livre de erro perma-

nente são obtidas sem a necessidade de incluir uma camada de otimização adicional de cálculo

de estados estacionários, devido à formulação adequada de modelos em espaço de estados na

forma incremental das entradas, os quais são derivados de expressões analíticas de resposta ao

degrau do sistema de processo; (ii) a incerteza de todos os parâmetros do modelo, e.g. gan-

hos, constantes de tempo, atrasos de tempo, é considerada naformulação do problema; (iii) as

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provas de estabilidade robusta segundo Lyapunov são realizadas de uma forma intuitiva através

da imposição de restrições terminais de igualdade e restrições de contração de custo; (iv) a in-

clusão adequada de variáveis de folga, que não comprometem as propriedades estabilizantes

dos controladores, assegura que os problemas de otimizaçãosão sempre viáveis; (v) integração

estável com camada de otimização em tempo real, visto que os controladores são projetados de

tal forma a rastrear targets ótimos para algumas entradas e saídas do processo, mantendo as var-

iáveis remanescentes dentro de faixas pré-definidas, ao invés de set-points fixos. Exemplos de

simulação típicos da indústria de processo são explorados para ilustrar as potenciais utilidades

dos métodos propostos e demonstrar que eles podem ser aplicados em casos reais.

Palavras-chave:Controle Preditivo. Horizonte Infinito. Estabilidade Robusta.

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Contents

1 Introduction and literature review 33

1.1 Advanced control in the process industry . . . . . . . . . . . . .. . . . . . . . 33

1.2 A survey of robust model predictive control strategies .. . . . . . . . . . . . . 36

1.3 Research objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 42

1.4 Outline of the work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .43

2 Two-step formulation of robust MPC for integrating time delay processes 45

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .45

2.2 The proposed model formulation . . . . . . . . . . . . . . . . . . . . .. . . . 46

2.3 The robust MPC formulation . . . . . . . . . . . . . . . . . . . . . . . . .. . 48

2.4 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 57

2.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .66

3 One-step formulation of robust MPC for integrating time delay processes 69

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .69

3.2 Development of the proposed robust MPC . . . . . . . . . . . . . . .. . . . . 70

3.3 Application of the controller to the partial combustionFCC converter . . . . . 75

3.3.1 The control problem of the FCC systems . . . . . . . . . . . . . . .. 75

3.3.2 The control structure of the FCC unit . . . . . . . . . . . . . . . .. . 77

3.3.3 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

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4 One-step formulation of robust MPC for unstable time delay processes 91

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .91

4.2 Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 92

4.3 The formulation of the nominal MPC . . . . . . . . . . . . . . . . . . .. . . 95

4.4 The formulation of the robust MPC . . . . . . . . . . . . . . . . . . . .. . . . 100

4.5 Application to an unstable reactor system . . . . . . . . . . . .. . . . . . . . 103

4.6 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .109

5 Conclusions and future research directions 111

5.1 Summary of contributions . . . . . . . . . . . . . . . . . . . . . . . . . .. . 111

5.2 Suggestions for future work . . . . . . . . . . . . . . . . . . . . . . . .. . . . 112

References 115

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List of Figures

2.1 Controlled outputs of the ethylene oxide reactor system .. . . . . . . . . . . . 60

2.2 Manipulated inputs of the ethylene oxide reactor system. . . . . . . . . . . . 60

2.3 Control costs for the ethylene oxide reactor system . . . . .. . . . . . . . . . 61

2.4 Controlled inputs of the CSTR system . . . . . . . . . . . . . . . . . . .. . . 65

2.5 Manipulated inputs of the CSTR system . . . . . . . . . . . . . . . . .. . . . 66

3.1 Schematic representation of the FCC converter . . . . . . . . .. . . . . . . . 78

3.2 Controlled outputs of the FCC process system in Case 1 . . . . . .. . . . . . . 85

3.3 Manipulated inputs of the FCC process system in Case 1 . . . . .. . . . . . . 86

3.4 Controlled outputs of the FCC process system in Case 2 . . . . . .. . . . . . . 87

3.5 Manipulated inputs of the FCC process system in Case 2 . . . . .. . . . . . . 88

3.6 Control cost of the robust MPC for the FCC process system . . .. . . . . . . . 89

4.1 Controlled outputs of the unstable reactor system . . . . . .. . . . . . . . . . 106

4.2 Manipulated inputs of the unstable reactor system . . . . .. . . . . . . . . . . 107

4.3 Controlled outputs of the unstable reactor system for thetracking case . . . . . 108

4.4 Manipulated inputs of the unstable reactor system for the tracking case . . . . . 109

4.5 Control costs for the unstable reactor system . . . . . . . . . .. . . . . . . . . 109

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List of Tables

2.1 Controller bounds for the manipulated inputs of the ethylene oxide system . . . 59

2.2 Nominal parameters of the CSTR system . . . . . . . . . . . . . . . . .. . . 62

2.3 Different steady state conditions for the CSTR system . . .. . . . . . . . . . . 63

2.4 Controller bounds for the manipulated inputs of the CSTR system . . . . . . . 63

3.1 Process model corresponding to the nominal operating condition . . . . . . . . 79

3.2 Process model corresponding to the operating conditionM1 . . . . . . . . . . 80

3.3 Process model corresponding to the operating conditionM2 . . . . . . . . . . 80

3.4 Operational bounds for the manipulated inputs of the FCC process system . . . 81

3.5 Operational bounds for the controlled outputs of the FCC process system . . . 81

3.6 Output zones of the FCC process system . . . . . . . . . . . . . . . . .. . . . 84

3.7 Modified output zones of the FCC process system . . . . . . . . . .. . . . . . 85

4.1 CSTR model parameters values . . . . . . . . . . . . . . . . . . . . . . . .. . 104

4.2 Different equilibrium points for the unstable reactor system . . . . . . . . . . . 104

4.3 Control zones for the unstable reactor system . . . . . . . . . .. . . . . . . . 105

4.4 Controller bounds for the manipulated inputs of the unstable reactor system . . 106

4.5 New control zones for the unstable reactor system . . . . . .. . . . . . . . . . 107

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Nomenclature

Scalars

A, B Chemical species.

cA Reactant concentration inside CSTR.

cA,in Reactant concentration in the feed stream of CSTR.

Cp Heat capacity of the reaction mixture.

d0i,j Gain of the transfer functionGi,j(s).

dii,j Coeficient corresponding to the integrating pole of the step response ofGi,j(s) obtained from the partial fractions expansion.

dsti,j,k k-th coeficient of step response of the stable poles ofGi,j(s) obtainedfrom the partial fractions expansion.

duni,j,k k-th coeficient of step response of the unstable poles ofGi,j(s) obtainedfrom the partial fractions expansion.

E/R Activation energy/universal gas constant.

Fin Inlet flow rate of CSTR.

Fout Outlet flow rate of CSTR.

Gi,j(s) Transfer function of the outputyi and inputuj.

h Liquid level of CSTR.

k Current time step.

k0 Pre-exponential factor.

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L Number of models that define the multi-plant uncertainty.

m Control horizon length of the MPC controllers.

na Number of distinct stable poles ofGi,j(s).

nc Number of distinct unstable poles ofGi,j(s).

nst Total number of stable poles in the multivariable system, and it is de-fined by product (nst = ny.nu.na).

nu Number of process system inputs.

nun Total number of unstable poles in the multivariable system,and it isdefined by product (nun = ny.nu.nc).

nx Dimension of the state vectorx.

ny Number of process system outputs.

p Maximum time delay between the input and output variables ofthe pro-cess system.

q Dimensionless feed flow rate to the reactor.

qc Dimensionless jacket feed flow rate of the reactor.

r Radius of CSTR.

rsti,j,k k-th stable pole ofGi,j(s).

runi,j,k k-th unstable pole ofGi,j(s).

Si,j(s) Step response ofGi,j(s).

T Reaction temperature inside CSTR.

Tc Cooling fluid temperature of CSTR.

Tin Temperature of the feed stream of CSTR.

U Overall heat transfer coefficient.

Vk Cost function at time stepk.

x1 Dimensionless reactor concentration.

x1f Dimensionless reactor feed concentration.

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x3 Dimensionless reactor temperature.

x2f Dimensionless reactor feed temperature.

x3 Dimensionless cooling jacket temperature.

x3f Dimensionless cooling jacket feed temperature.

β Dimensionless enthapy of reaction.

γ Dimensionless activation energy.

γi,j Time delay between the outputyi and inputuj of the process system.

δ Dimensionless heat transfer coeficient.

δ1 Reactor to cooling jacket volume ratio.

δ2 Reactor to cooling jacket density heat capacity ratio.

∆H Enthalpy of reaction.

κ Dimensionless Arrhenius reaction rate non-linearity.

λj Parameter specifying the linear combination for(Aj,Bj ,Cj) necessaryto form (A,B,C) ∈ Co, for

∑L

j=1 λj = 1, ∀λj ≥ 0 .

ρ Density of the reaction mixture.

τ Dimensionless time.

φ Nominal Damköhler number based on the reaction feed.

Vectors

u Input of the process system.

u(0) Initial value of the input.

u(k + j|k) Input at time stepj in the control horizon with measurement up to timestepk.

udes Input target.

umin,umax Minimum and maximum input constraints.

x State of the process system.

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xi Integrating component of the state of the state-space models defined inchapters 2 and 3.

xs Integrated state component produced by the incremental form of thestate-space models defined in chapters 2, 3 and 4.

xst Stable component of the state of the state-space models defined in chap-ters 2, 3 and 4.

xun Unstable component of the state of the state-space model defined inchapter 4.

y Output of the process system.

y(0) Initial value of the output.

y(k + j|k) Output at time stepj in the prediction horizon with measurement up totime stepk.

ymin,ymax Lower and upper zone limits.

ysp,k Set-point for the output computed at time stepk.

δi Slack variable for the integrating state of the state-spacemodels definedin chapters 2 and 3.

δu Slack variable for the process system inputs.

δun Slack variable for the unstable state of the state-space model defined inChapter 4.

δy Slack variable for the process system outputs.

∆t Sampling time.

∆u(k) Single input moves at time stepk.

∆uk Vector of input moves from time stepk to k +m− 1.

∆umax Maximum move of the process system inputs.

Matrices

0 Matrix of appropriate dimension with entries equal to zero.

A,B,C State-space model matrices in discrete time.

Bil Matrix that relates control actions with components of the statexi.

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Bsl Matrix that relates control actions with components of the statexs.

Bstl Matrix that relates control actions with components of the statexst.

Bunl Matrix that relates control actions with components of the statexun.

Co Convex hull ofΩ.

Dst Diagonal matrix that concentrates all the coeficientsdsti,j,k.

Dun Diagonal matrix that concentrates all the coeficientsduni,j,k.

Fst Diagonal matrix associated with all the stable poles of the process sys-tem.

Fun Diagonal matrix associated with all the unstable poles of the processsystem.

In Identity matrix with dimensionn.

I∗ny Diagonal matrix of dimensionny whose entries are 1 for the integratingoutputs and 0 for the stable outputs.

Jst, Jun Auxiliary matrices composed of zeros and ones used in the formulationof the proposed state-space models.

Ns(Ni,Nun) Auxiliary matrix used to extract the component of statexs (cf. xi, xun).

Nst(Nun) Auxiliary matrix that relates the control actions with the components ofBst

l (cf. Bunl ).

Qu Weight of the input target in the cost function of MPC.

Qy Weight of the output prediction error in the cost function ofMPC.

Q Terminal weight of the infinite horizon MPC.

R Weight that penalizes the input moves in the cost function ofMPC.

Si Weight of the slack for the integrating state in the cost function of MPC.

Su Weight of the input slack in the cost function of MPC.

Sun Weight of the slack for the unstable state in the cost function of MPC.

Sy Weight of the output slack in the cost function of MPC.

W Auxiliary matrix associated with the controllability matrix of the pro-cess system.

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Θ Set of parameter matrices of the state-space models for which the multi-plant uncertainty is defined.

Θn Set of parameter matrices of the state-space models for the modeln inthe multi-plant uncertainty description.

ΘN Set of parameter matrices of the state-space models for the nominalplant model in the multi-plant uncertainty description.

ΘT Set of parameter matrices of the state-space models for the true plantmodel in the multi-plant uncertainty description.

Ψst(Ψun) Matrix that relates the process system outputs with components of thestatexst (cf. xun).

Ω Set of models that defines the vertices of the multi-plant uncertaintydescription.Ω = Θ1, . . . ,ΘL.

Mathematical symbols

∥∥∥ •

∥∥∥

2

PDenotes the weighted Euclidean norm of•, beingP ≥ 0 a symetricmatrix.

Cn Set of complex numbers inn-space.

N Set of natural numbers.

Rn Set of real numbers inn-space.

U Set of constrained subspace for the input vector.

Superscript

i Refers to the integrating states of the process system.

s Refers to the integrating states produced by the incrementalform of thestate-space models.

st Refers to the stable states of the process system.

un Refers to the unstable states of the process system.

⊤ Denotes the transpose of a matrix (vector).

∗ Refers to the optimal solution.

˜ Refers to a feasible solution.

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Subscript

i, j, k Indexes.

l Time delay index of the process system variables.

max Maximum value.

min Minimum value.

n Model index in the multi-plant uncertainty description.

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Acronyms

ANN Artificial Neural Network.

CSTR Continuous Stirred Tank Reactor.

CLF Control Lyapunov Function.

FCC Fluidized-Bed Catalytic Cracking.

GA Genetic Algorithm.

LMI Linear Matrix Inequality.

LQR Linear Quadratic Regulator.

MIMO Multi-Input, Multi-Output.

MPC Model Predictive Control.

NLP Non-Linear Programming.

NMPC Nonlinear Model Predictive Control.

PSO Particle Swarm Optimization.

QP Quadratic Programming.

RMPC Robust Model Predictive Control.

RPI Robust Positively Invariant.

RTO Real-Time Optimizer.

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List of publications

• Journal papers:

1. Márcio A.F. Martins, André S. Yamashita, Bruno F. Santoro,Darci Odloak. Robust

model predictive control of integrating time delay processes.Journal of Process Con-

trol, 23 (5), 917–932, 2013. (Qualis A1 of CAPES – Engineering II).

2. Márcio A.F. Martins, Antonio C. Zanin, Darci Odloak. Robustmodel predictive

control of an industrial partial combustion fluidized-bed catalytic cracking converter.

Chemical Engineering Research and Design, 92(5), 917-930, 2014. (Qualis A1 of

CAPES – Engineering II).

3. Márcio A.F. Martins, Darci Odloak. A robustly stabilizing model predictive control

strategy of stable and unstable time delay processes. It will be submitted to the journal

Industrial and Engineering Chemistry Research. (Qualis A1 of CAPES – Engineering

II).

• Conference papers:

1. André S. Yamashita, Márcio A.F. Martins, Bruno F. Santoro,Darci Odloak. A stable

MPC for integrating systems with dead-time. 20th International Congress of Chemi-

cal and Process Engineering (CHISA), Czech Republic, 2012.

2. Márcio A.F. Martins, Antonio C. Zanin, Darci Odloak. Infinite Horizon MPC applied

to an industrial FCC converter. 9th Asian Control Conference (ASCC), Istanbul, 2013.

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CHAPTER 1

Introduction and literature review

This chapter concerns the motivation and objectives of the present thesis, as well as the in-

troduction of the research work developed here. First of all, it is presented the robust model

predictive control (RMPC) as a quite useful tool for controlling perturbed, uncertain and con-

strained systems from the process industry. Then, the majorissues related to the contributions

of this work are highlighted through a survey of the researchworks that have been recently pub-

lished in the field of model predictive control (MPC). Finally, the typical problems for which

this work proposes a solution will be presented.

1.1 Advanced control in the process industry

In their vast majority, process systems are characterized as having very complex dynamics,

in which there are strong interactions between their process variables (multivariable systems),

and unpredictable market disturbances with respect to demand and/or supply of their products.

Moreover, issues concerning safety, product quality, and compliance with environmental laws

are still part of daily routine of these industrial processes, and therefore they should, a priori,

strictly adhere to. From the industrial point of view, it is broadly recognized that the most prof-

itable operating point of a process plant typically lies at the intersections of the aforementioned

conditions (constraints) (Prett and Gillette, 1979). Then, in order to be successful, a control

system for the industrial plant must be designed in such a wayas to keep the process as close

as possible to constraints without violating them (Garcíaet al., 1989).

In the classical control structure, in which the usual approach is the PID-based control tech-

33

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34

nique, the topics exposed previously are addressed in an ad hoc manner, i.e., there is not an

integrated control solution to systematically deal with the multivariable interactions and numer-

ous constraints which are imposed to the industrial processes. As a general rule, the classical

control solution relies upon a combination of PID controllers accomplished by using different

control strategies, such as feed forward, override, split-range, delay compensations, decou-

pling, and so on. In view of the fact that the design of such control strategies varies greatly

from application to application, it is clear that the effective use of all these approaches does

not yield a systematic solution, which makes their implementation fairly restrictive. In or-

der to circumvent these shortcomings, model predictive control appears to be the most suit-

able control solution. In fact, MPC has been thoroughly studied since its introduction in the

late 1970s and nowadays, it is a well-established technology which has become the standard

advanced control strategy for tackling typical multivariable constrained control problems of

the process industries. A considerable amount of literature has been published on theoret-

ical and practical issues associated with MPC technology, including comprehensive reviews

(Rawlings, 2000; Qin and Badgwell, 2003; Darby and Nikolaou, 2012) and some recent books

(Camacho and Bordons, 2007; Rawlings and Mayne, 2009; Zheng, 2012).

MPC refers to a class of control algorithms that explicitly makes use of a process model to

forecast the future behavior of the controlled process. It computes a suitable control sequence

of input moves (control actions) through an optimization problem, which may be formulated

for any combination of constraints on the process variables. Even though, at a given time step,

the optimal sequence of the control actions is obtained, only the first action is injected into

the plant, and the entire calculation is repeated at subsequent time steps using updated mea-

surements from the plant. Clearly, the model is a key element for successful implementations

of MPC algorithms; however, it is well-known that no model isa perfect forecaster, either

due to poor design of test signals or invalid assumptions in data analysis, or both, thus giving

rise to its associated uncertainty. Within an industrial context, several other sources of uncer-

tainty may contribute considerably to differences betweenthe process model and the real plant,

among them are: unknown disturbances, measurement noises,changes in the plant operating

conditions, neglected dynamics and time delays. The MPC controllers that assume the process

model to be exactly known (nominal case) can result in a quitepoor control performance or

even worse, cause closed-loop instability. On the other hand, MPC schemes which plainly take

into account the model uncertainty in their formulation, the so-called robust MPC, are able to

provide a better control performance than the corresponding ones based solely on the nominal

model.

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35

In the study of robust control there have been many differentmethods to characterize the

model uncertainty; the interested reader on this subject isreferred to the thesis of Wang (2002)

and the references therein. Nevertheless, in the robust MPCliterature, the model uncertainty has

been tackled considering certain relevant classes of uncertainty description, namely: polytopic

system, multi-plant system and bounded exogenous disturbance.

In the polytopic representation of model uncertainty, the key purpose is to produce a con-

vex linear combination of a set of linear process models in state-space form defined byΩ =

Co(A1,B1,C1), , . . . , (AL,BL,CL), in whichCo denotes the convex hull. In this case, the

real plant model is not known exactly but it lies in the polytope (i.e. it belongs toΩ), if

(A,B,C

)=

L∑

j=1

λj

(Aj,Bj ,Cj

),

L∑

j=1

λj = 1, ∀λj ≥ 0. (1.1)

With this uncertainty description, the plant can vary with time (it does need to be fixed), so long

as it remains within the polytope. Thus, a polytopic system usually arises when the process is

represented by a linear time-variant system with the state-space matrices defined in (1.1).

With regard to the multi-plant uncertainty description, the state-space matrices(A,B,C

)

that correspond to the real plant are also unknown but lie within a set

Ω =Aj ,Bj ,Cj

, j = 1, . . . , L, (1.2)

in which eachj corresponds to a possible model of the process system. Note that this uncer-

tainty representation is a particular case of the polytopicuncertainty description.

From the practical point of view, it is noteworthy that the family of linear models used to

construct the setsΩ for both the polytopic and multi-plant uncertainty representations can be

viewed as a result of the linearization of the nonlinear process model at different operating

conditions. Therefore, it is sought a controller design that guarantees at least some kind of

performance, which can include closed-loop stability, forthe whole family of the possible plant

representations.

Many robust MPC formulations have dealt with bounded exogenous disturbance (w) as a

result of the model uncertainty. This disturbance is usually norm-bounded or lies in a compact

and convex polyhedron, besides it is commonly associated with the system state, or

x(k + 1) = Ax(k) +Bu(k) +w(k)y(k) = Cx(k).

(1.3)

Although it has been presented in the linear equation (1.3),this uncertainty description may

also be used in process systems represented by nonlinear models without loss of generality.

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36

It should be stressed here that, opposite to what happens with the polytopic or multi-plant

uncertainty descriptions, which are based on model parameter uncertainties, the bounded distur-

bance based uncertainty representation is only used to globally describe the model uncertainty

and it characterizes any kind of model prediction mismatch caused by either external distur-

bance or measurement noise.

From the preceding discussion on the uncertainty descriptions employed in MPC formula-

tions, this work will focus on solutions to the robust MPC problem of process systems described

by polytopic or multi-plant uncertainties, or both. In whatfollows other arguments, which also

corroborate with this decision, are addressed in light of the different robust MPC formulations.

1.2 A survey of robust model predictive control strategies

A major issue to be required in the application of MPC algorithms to industrial processes is its

stability when the system is in closed-loop. Such a condition is always a desired property be-

cause the resulting MPC control law guarantees that the closed-loop system is stable regardless

of the controller tuning parameters, which will be selectedconsidering solely the improvement

on the system performance. By and large, considerable progress has been made in analyzing

closed-loop stability of MPC schemes for the case in which the plant model is perfect, the so-

called nominal stability problem. One class of approaches is to impose stability constraints in

the MPC optimization problem such as terminal state constraints (Keerthi and Gilbert, 1988;

Mayne and Michalska, 1990; Gilbert and Tan, 1991; Polak and Yang, 1993), or terminal set

constraints (Michalska and Mayne, 1993; Scokaertet al., 1999). Another class of approaches

(more popular) for obtaining a nominally stable MPC is to adopt an infinite prediction horizon

(Rawlings and Muske, 1993; Muske and Rawlings, 1993; Meadowset al., 1995). For open-

loop stable systems, this approach consists in reducing theinfinite-horizon cost function of the

controller to a finite-horizon cost by defining a well-designed terminal penalty term (termi-

nal cost), which is obtained from the solution of a Lyapunov equation. On the other hand,

in the case where the system also comprises integrating and/or unstable modes, it requires

that additional sets of terminal equality constraints mustbe added to the control problem in

such a way as to keep the infinite-horizon cost bounded, (e.g.Rodrigues and Odloak, 2003a;

Carrapiço and Odloak, 2005).

As mentioned earlier, the process model used in the MPC framework is never perfect and so

nominal stability results are not rigorously applicable. Within this context, the design of sta-

bilizing MPC algorithms that explicitly takes into accountthe model uncertainty (the so-called

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37

Robust MPC or RMPC) is therefore crucial to the success of MPC applications. The search

for robustly stabilizing controllers has received plentiful attention in the control literature, so

that the several approaches that have been developed so far can be grouped into four main cate-

gories: (i) Min-Max control formulations; (ii) Stabilizing state-feedback schemes; (iii) Control

Lyapunov functions; (iv) Cost contracting constraints.

The first approach aims to achieve robust stability by minimizing the objective function

for the worst possible realization of the uncertainty description. In the middle of the stud-

ies available on this topic, the main research work that makes use of such an approach is

the one that was reported by Lee and Yu (1997). They proposed robustly stabilizing infinite-

horizon MPC algorithms for open-loop stable systems with norm-bounded time-invariant or

time-varying parametric uncertainties. In their method, it is also shown that the control cost

is a Lyapunov function to the dynamic system when the objective is calculated considering all

possible combinations of system models along the control horizon. Following the same line,

Lee and Cooley (2000) extended the method to systems with integrating poles. There, in order

to turn the infinite-horizon control cost bounded, the integrating states of the system at the end

of the control horizon are forced to be minimized through a new optimization problem of the

worst-case cost. Then the solution of this proposed optimization problem is properly transferred

to the original min-max control optimization problem as an equally constraint, which guarantees

the convergence properties of the controller. With the aim of producing a highly structured con-

vex optimization problem for the Lee and Cooley’s control formulation, Ralhan and Badgwell

(2000) added cost contracting constraints associated withthe uncertainty description to the

original control problem, allowing for more efficient numerical solutions. A major limitation

behind the aforementioned approaches is the assumption that the plant steady-state is always

known (set-point at the system origin), i.e. the min-max predictive controllers are simply regu-

lators, and consequently as soon as a disturbance enters theprocess or the set-point changes to

values away from the origin, the control law cannot eliminate the resulting offset. This short-

coming is overcome in the work by Odloak (2004) that proposeda min-max robust MPC to the

general case where the system equilibrium point is unknown (output tracking case).

Even so the min-max control theories suffer serious limitations, namely: the main barrier is

the fact that the application of these strategies results ina very conservative control performance

under the conditions in which there are guarantee of stability; other disadvantage of the min-max

predictive control formulations is its computational intensity since their resulting optimization

problems are very expensive to solve on-line. Therefore, from the point of view of the process

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38

industry such a control approach becomes infeasible for practical implementation purposes.

One of the most heavily studied robust stability methods is the one based on the state-

feedback scheme. The core of these control strategies consists of incorporating the state feed-

back control law into the RMPC problem formulation. The state-feedback RMPC schemes

involve either adding contracting constraints to all possible system states along the infinite hori-

zon, or applying the dual-mode control structure, which is composed of two distinct control

modes: in the first mode, the RMPC drives the uncertain system over a finite horizon from the

initial state towards a point inside a given robust positively invariant (RPI) set, while in the sec-

ond control mode a local controller having the form of a state-feedback takes over and holds the

state within this set for all admissible uncertainties. Following this approach, the first RMPC ap-

peared in the work by Kothareet al. (1996), where the authors proposed a robust MPC strategy

for polytopic uncertain models, which is based on an infinite-horizon linear quadratic regulator

(LQR). The controller was extended to the constrained case, through the inclusion of conserva-

tive linear matrix inequality (LMI) constraints on the inputs and outputs. Even though, with this

method, stability can be achieved for stable, unstable or integrating systems with some classes

of model uncertainty, it yields a quite conservative control law due to the fact that the control

actions are obtained by a fixed state-feedback gain throughout the infinite prediction horizon.

Besides, it may cause feasibility problems because of the hard state contraction constraints.

There have been attempts in the literature to improve the feasibility issues of the method, by

applying a parameter-dependent Lyapunov function (Cuzzolaet al., 2002; Mao, 2003), by intro-

ducing relaxation matrices in the robust control optimization problem (Leeet al., 2008), or by

providing LMIs as approximations to the state contracting constraints (Jiaet al., 2005). With re-

gard to reducing conservatism, the method has been extendedby several works so as to become

more general by providing extra degrees of freedom to the controller through the dual-mode

prediction paradigm. That is, instead of parameterizing the control actions as a single linear

state-feedback law for the entire infinite horizon, one addsa set of free control inputs along

a finite horizon, which are now considered as decision variables in the optimization problem,

and subsequently one applies the state-feedback law in the terminal robustly stabilizable set

(Casavolaet al., 2000; Lu and Arkun, 2000; Schuurmans and Rossiter, 2000; Ding et al., 2004;

Tahir and Jaimoukha, 2013; Ding and Zou, 2014).

The state-feedback RMPC schemes for the polytopic uncertainmodel cited above have the

drawback of huge on-line computational burdens stemming from the combinatorial nature of

the resulting optimization problem. Furthermore, the rigorous stability and feasibility of these

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39

methods are guaranteed only if the desired reference valuesfor the system inputs and states are

fixed equilibrium points (typically the origin). Within thedual-mode prediction approach the

finite (control) horizon must be large enough such that the states at the end of the horizon lie in

the RPI set and, in addition, the off-line calculation concerning the parameters determination of

the RPI set is not trivial. A possible way, proposed by Limonet al. (2010), remedies in part the

aforementioned drawbacks by pointing out a robust tube based dual-mode MPC formulation.

In their research work, the reference tracking case is solved by considering the steady-state

conditions of the system as decision variables (artificial reference) of a single RMPC optimiza-

tion problem, whose recursive feasibility and enlarged domain of attraction of the controller are

achieved because the terminal constraint (state-feedbacklaw) that is a RPI set accommodates

any equilibrium point (non-zero set-points). The computational burden is drastically reduced as

the optimization problem is posed as a standard quadratic programming (QP) problem, which

can be easily solved by available commercial package routines. However, such a computational

improvement is efficiently attained because the system uncertainty is restricted to the bounded

additive disturbance case.

Alternatively, the Control Lyapunov Function (CLF)-based MPC schemes are motivated by

the fact that is possible to explicitly characterize the stability region, guaranteed feasibility and

closed-loop stability of the controller without the need ofimposing an infinite prediction horizon

(Mhaskaret al., 2005, 2006); please see the work of Christofideset al. (2013) for a survey of

this approach. Regarding the applications of this approach,it has been successfully applied

in the field of control actuator fault (Mhaskar, 2006) and switched systems (Mhaskaret al.,

2008). Although the CLF-based RMPC strategies are promising,two significant barriers must

be overcome before these results can be implemented in practice. The first is that the problem

formulation can only be applied to uncertain systems represented by exogenous-type bounded

disturbances. The other serious limitation of this body of research concerns on the ways of

obtaining CLFs, which may not be a trivial task, and the authors of this area give no clue as to

how to obtain it generically.

An interesting approach to the problem of robust MPC, which seems already in an accept-

able level of development for practical implementation, isthe one based on the cost-contraction

formulation, proposed in the seminal work of Badgwell (1997). This method was developed

for the regulator operation of open-loop stable systems with multi-plant uncertainty, and its

stability is achieved by adding constraints to the control optimization problem that prevent the

controller cost functions to increase at successive sampling steps, in conjunction with the con-

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40

sideration of an infinite prediction horizon (Rawlings and Muske, 1993). For the same sort of

model uncertainty, Odloak (2004) extended that method to the output tracking case of systems

with unmeasured disturbances. This MPC controller is offset free as the adopted state-space

model is an incremental form of the system inputs and, therefore, there is no need to include an

intermediary layer in the control structure to compute a feasible steady state as proposed e.g.

in Kassmannet al. (2000). Nonetheless, in the case where the system also comprises integrat-

ing and/or unstable modes the cost-contracting approach requires the inclusion of an additional

set of terminal equality constraints into the control optimization problem. The keystone of

imposing these constraints is to zero at the end of the control horizon both the system integrat-

ing and unstable modes, thus maintaining the infinite-horizon cost bounded. To work around

this, Cano and Odloak (2003) put forward a robust MPC for pure integrating systems in which

good performance to the output-tracking case was reached, even when the system was subject

to unknown disturbances. That controller includes a set of suitable slack variables to soften

the hard terminal constraints related to the integrating modes, which implies that the proposed

control optimization problem will be always feasible. After a few years, the method of Cano

and Odloak was extended by Gonzálezet al. (2007) to systems composed of both stable and

integrating modes, considering a suitable extension of theanalytical step response state-space

model previously proposed by Odloak (2004).

On the other hand, chemical and petrochemical processes usually show significant time de-

lays associated with the transport of fluids, and uncertainty in these time delays may also be a

consequence of changes in operating conditions. Time delaycompensation does not constitute

an obstacle to the implementation of nominally stable MPC strategies because it is always possi-

ble to consider an augmented state in the representation of the system (Normey-Rico and Camacho,

2007; Santoro, 2011). Conversely, in the robust MPC context,the available solutions seem still

far from the practical application stage, as will be presented hereafter. For controllers that work

as regulators, Lombardiet al. (2012) proposed a predictive control strategy for linear time in-

variant systems with variable time delay. Their approach produces an explicit solution to the

control problem obtained via multi-parametric programming. Ding and Huang (2007) proposed

a min–max robust MPC for open-loop stable time delay systemswith polytopic uncertainty. For

the integrating system, the approach needs the consideration of free control moves before the

extended state-feedback control law. Thus, the method was extended in Dinget al. (2008) to

polytopic systems with time varying delays and to quasi linear parameter varying systems with

bounded disturbances in Ding (2010). More recently, Santoset al. (2012) describe a robust

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41

tube-based MPC strategy that is capable of explicitly handling dead-time compensation using

a filtered Smith predictor scheme when the system is represented by first-order and integrating

models. Although such a method is applicable to the output tracking case, the uncertainty of

the process system is restricted to a bounded additive disturbance.

Another important issue that should be addressed in the practical implementation of MPC

controllers is that they are usually implemented as an intermediary layer of the multilevel control

structure. Above the MPC layer, a real time optimizer (RTO) or a static economic layer defines

optimal targets for some of the inputs and outputs (Kassmannet al., 2000). Furthermore, in

these process systems where the operation is optimized, theaim of the MPC layer is merely to

maintain some controlled outputs within appropriate ranges or zones, rather than at a fixed set-

point or desired value; this strategy is the so-called zone control (Maciejowski, 2002). Within a

perspective of the process industry, the zone control strategy may be used in two major distinct

modes of application. In the first mode, when there are highlycorrelated outputs to be controlled

and so there are not enough inputs to control them. In the second mode, when one desires to

use the surge capacity of tanks to smooth out the operation ofa process unit, in this case it is

somewhat opportune to let the level of the tanks to float between appropriate limits so as to

buffer the effect of disturbances entering in the upstream sections of the process plant.

Published research focusing on practical solutions of RMPC schemes with guarantee of sta-

bility and based on a layered control structure is still a bitscarce. In this sense, it is worth

mentioning the following research works. In the work of Ferramoscaet al. (2012) is developed

a stabilizing state-feedback based RMPC strategy for tracking both target sets and zone regions.

The method consists of an extension of the control strategy proposed by Limonet al. (2010),

whose key purpose was to incorporate a term to the original cost function (denoted as offset

cost function) for assuring that the targeted variables aresteered to their optimum targets while

the non-targeted outputs are controlled within a predefinedregion; the offset cost function is

related to the deviation between the artificial references (decision variables of the optimization

problem) and the targets of some outputs and inputs, as well as the terminal set (polyhedron)

that defines the zone region. This method preserves all the good stability properties (e.g. recur-

sive feasibility and local optimality) of the original method, but it also inherits its disadvantages

such as trade-off between feasibility and performance due to the off-line design of terminal RPI

set and the uncertainty description is simply a bounded additive disturbance.

Following the cost-contracting approach for open-loop stable systems described by multi-

plant uncertainty, Alvarez and Odloak (2010) extended the method of Odloak (2004) to the

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42

tracking case of optimizing targets and zone control. In their approach, it is included in the

infinite-horizon MPC cost function a term that penalizes thedistance between the desired values

for the targeted inputs, which are produced by the RTO layer,and their predicted values along

the control horizon. In addition, the output set point is posed as an additional vector of decision

variables that can assume any value inside the output zone. Also in this approach, new slack

variables are added to the optimization problem in order to soft the hard constraints that force

the deviation between the input values predicted at the end of the control horizon and their

targets to be zeroed. This slack assures that the resulting control law is always made feasible to

a large class of unknown disturbances and changes of the optimizing targets and zone control.

Such an approach was extended recently in the work of González and Odloak (2011) to the time

delayed systems on the basis of a suitable modification of theaugmented state-space model.

Next, reformulating the optimization problem of this latter method as a LMI control problem,

Capronet al. (2012) have enlarged the robustness of the controller to thecase in which the

process gains can be represented by polytopic uncertainty.The limitation of these methods

arises when the proposed model description is extended to the integrating time delay systems,

since it will be controllable and detectable if and only if the dead-times associated with an

integrating output is the same for all the inputs (Santoro and Odloak, 2012). This means that

such an approach cannot be applied in practice.

1.3 Research objectives

From the survey presented in section 1.2, it is noteworthy that the design of RMPC schemes

with guarantee of stability remains an open field of research, mostly when integrating and un-

stable time delay processes shall be considered into the problem formulation. In an attempt to

circumvent some of the remaining challenges associated with the RMPC problem, as already

explained earlier, it is intended in this thesis the development of robustly stabilizing MPC strate-

gies for integrating and unstable time delay processes, which can be directly implemented in

real systems. Particularly, with the purpose of accommodating the practical case, the proposed

algorithms have been structured in such a way that:

(i) The uncertainty associated with all model parameters isconsidered in the problem formu-

lation (gains, time constants and time delays). That is, it will be considered the multi-plant

and/or polytopic uncertainty descriptions;

(ii) The robust stability proof is made considerably easier. For this purpose one adopts cost-

contracting constraints and terminal constraints that aresufficient to cancel the effect of

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43

both integrating (real and artificial) and unstable modes onthe output prediction at time

steps beyond the delayed control horizon;

(iii) The feasibility of the control optimization problem will be always guaranteed. Here, slack

variables are properly included in the optimization problem, in such a way that they will

not affect the stability properties of the controllers;

(iv) The RTO layer, which defines the optimizing targets for the process variables, is duly han-

dled by the RMPC controller. The optimizing targets for some of the process system inputs

and outputs, and interval tracking (zone control) for the remaining controlled outputs, are

explicitly incorporated in the formulation of the control strategies.

1.4 Outline of the work

The text is organized into five chapters, including this introductory chapter. The outline of each

chapter is presented as follows.

Chapter 2 derives a robust model predictive control of stableand integrating time delay pro-

cesses with guaranteed stability obtained from the sequential solution of two optimization prob-

lems (two-stage formulation). In this approach, a state-space model based on the analytical step

response model, proposed previously by Gonzálezet al.(2007), is suitably extended to the case

of time delay systems.

Chapter 3 presents a new proposal for a robustly stable model predictive control of stable

and integrating time delay processes. The method uses a moregeneral state-space model than

the corresponding one of Chapter 2 and, in addition, the sketch of the stability proof is obtained

for a one-step formulation.

Chapter 4 wraps up the development of a closed-loop stable MPCfor stable and unstable

time delay processes with model uncertainty. This proposedcontroller is much more appealing

than the algorithms existing in the literature, particularly as it achieves an offset-free control

law and proves the Lyapunov stability of the closed-loop system through a one-step control

formulation.

Finally, Chapter 5 summarizes the contributions done by thisthesis and outlines directions

for future research.

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44

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CHAPTER 2

Two-step formulation of robust MPC forintegrating time delay processes

2.1 Introduction

This chapter focuses on the solution to the problem of model predictive control of time delay

processes with both stable and integrating modes and model uncertainty. In this case, the objec-

tive is to extend the work previously presented in Gonzálezet al.(2007) to time delay processes

so as to produce a robustly stabilizing MPC algorithm for themulti-plant uncertainty. The model

extension proposed here is such that the additional states that represent the time delays do not

depend upon the model parameters, and consequently, they donot affect the model uncertainty.

Such a strategy is quite useful in the development of the end constraint that guarantees the sta-

bility for the multi-plant system in the proposed two-step control formulation. Also, aiming at

the practical case, the control problems are formulated to accommodate optimizing targets for

some of the process system inputs and outputs, and interval tracking for the remaining outputs.

The chapter is organized as follows. The next section presents the extension of the analytical

step response model to represent the time delay system with stable and integrating modes. Then,

the robust MPC with guaranteed stability is developed for the model proposed here. Finally,

simulation results of an ethylene oxide reactor and a nonlinear CSTR are presented and followed

by appropriate conclusions arising from this study.

45

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46

2.2 The proposed model formulation

Gonzálezet al. (2007) presented a state-space model, which is equivalent to the analytical rep-

resentation of the step response of systems with stable and integrating modes. Here, this model

is extended to the time delay system. For this purpose, consider a MIMO (Multi-Input, Multi-

Output) system withnu inputs andny outputs, where the transfer function relating inputuj to

outputyi is

Gi,j(s) =bi,j,0 + bi,j,1s+ . . .+ bi,j,nbs

nb

s(s− rsti,j,1) . . . (s− rsti,j,na)e−γi,j ·s, (2.1)

wherena, nb ∈ N|nb < na andrsti,j,1, . . . , rsti,j,na are the distinct stable poles of the system.

Assume, initially that the time delay is equal to zero (γi,j = 0). Then, for a sampling period∆t,

the corresponding step response at time stepk can be computed by the expression:

Si,j(k) = d0i,j +na∑

l=1

dsti,j,lersti,j,l

·k·∆t + dii,jk∆t, (2.2)

in which coefficientsd0i,j, dsti,j,1, . . . , d

sti,j,na anddii,j are obtained from the partial fractions expan-

sion of the functionGi,j/s. The step response model is equivalent to a state-space model, which

is suitable for offset free MPC, and can be written in the following form:

x(k + 1) = Ax(k) + B∆u(k)

y(k) = Cx(k),(2.3)

where the statex(k) is defined as follows:

x(k) =[xs(k)⊤ xst(k)⊤ xi(k)⊤

]⊤, x ∈ C

nx, xs ∈ Rny, xst ∈ C

nst, xi ∈ Rnu,

nst = ny.nu.max(na), nx′ = ny + nst+ nu,

and

A =

Iny 0 ∆tDi

0 Fst 0

0 0 Inu

∈ Cnx′×nx′

, B =

Bs

Bst

Inu

∈ Cnx′×nu,

C =[Iny Ψst 0

]∈ R

ny×nx′

,

in which:

Bs = D0+∆tDi, Bst = DstFstNst, Di =

di1,1 · · · di1,nu...

. . ....

diny,1 · · · diny,nu

, D0 =

d01,1 · · · d01,nu...

. .....

d0ny,1 · · · d0ny,nu

,

Fst = diag(e∆t·rst

1,1,1 · · · e∆t·rst1,1,na · · · e∆t·rstny,nu,1 · · · e∆t·rstny,nu,na

)∈ C

nst×nst,

Dst = diag(dst1,1,1 · · · d

st1,1,na · · · d

stny,nu,1 · · · d

stny,nu,na

)∈ C

nst×nst,

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47

Nst =

Jst

Jst

...Jst

ny

, Jst =

na

1 0 · · · 0...

.... . .

...1 0 · · · 0

na

0 1 · · · 0...

.... . .

...0 1 · · · 0

...

na

0 0 · · · 1...

.... . .

...0 0 · · · 1

, Ψst =

nu.na︷ ︸︸ ︷

1 1 · · · 10 0 · · · 0...

..... .

...0 0 · · · 0

· · ·

nu.na︷ ︸︸ ︷

0 0 · · · 00 0 · · · 0...

..... .

...1 1 · · · 1

,

Nst ∈ Rnst×nu, Jst ∈ R

nu.na×nu, Ψst ∈ Rny×nst, I and0 are the identity and null matrices

with appropriate dimensions, respectively.

In such a model,xs(k) represents the prediction of the system output at steady-state and

is related to the integrating modes introduced by the incremental form of the inputs,xst(k)

corresponds to the stable states andxi(k) stands for the integrating states of the system.

Remark 2.1 If the rank ofDi is equal to the number of the system inputs, then the model

defined in (2.3) is completely state controllable and observable. However, if the number of

integrated inputs (ni) is less than the total number of inputs, only the ni components of state

xi will be controllable and observable. Therefore, in order to assure the controllability and

observability properties of the model, the remaining (nu − ni) components are fixed at the

origin (Gonzálezet al., 2007)

In the model defined above, the integrating state corresponds to the total increment (sum) of

past input moves, or

xi(k) =k−1∑

j=0

∆u(j). (2.4)

To extend the above model to the time delay system, the statexi should be augmented to ac-

commodate the maximum time delay between the system inputs and outputs. Then, considering

(2.4), the augmented integrating state is defined through the following sequence:

xi0(k + 1)

xi1(k + 1)

xi2(k + 1)

...xip−1(k + 1)xip(k + 1)

=

Inu 0 · · · 0 0

Inu 0 · · · 0 0

0 Inu · · · 0 0...

.... ..

......

0 0 · · · 0 0

0 0 · · · Inu 0

xi0(k)

xi1(k)

xi2(k)...

xip−1(k)xip(k)

+

Inu0

0...0

0

∆u(k), (2.5)

wherep = maxi,j

γi,j.

Then, combining (2.5) and (2.3), and after some simple algebraic manipulations, the ex-

tended incremental state-space model that is equivalent tothe analytical step response takes the

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48

form

x(k + 1) = Ax(k) +B∆u(k)y(k) = Cx(k),

(2.6)

in which:

x(k) =[xs(k)⊤ xst(k)⊤ xi

0(k)⊤ xi

1(k)⊤ · · · xi

p(k)⊤]⊤

∈ Cnx,

nx = ny + nst+ (p+ 1).nu,

A =

Iny 0 ∆tDi0 +Bs

1 ∆tDi1 +Bs

2 −Bs1 · · · ∆tDi

p−1 +Bsp −Bs

p−1 ∆tDip −Bs

p

0 Fst Bst1 Bst

2 −Bst1 · · · Bst

p −Bstp−1 −Bst

p

0 0 Inu 0 · · · 0 0

0 0 Inu 0 · · · 0 0

0 0 0 Inu · · · 0 0...

......

.... . .

......

0 0 0 0 · · · 0 0

0 0 0 0 · · · Inu 0

,

A ∈ Cnx×nx, B =

Bs0

Bst0

Inu0...0

0

∈ Cnx×nu, C =

[

Iny Ψst

p+1︷ ︸︸ ︷

0 · · · 0

]

∈ Rny×nx.

In the model defined in (2.6), the statexil stands for the integrating state associated with the

time delayl. The matrices that actually relate the system inputs to the states, for each given time

delayl = 0, . . . , p, areBsl , B

stl andDi

l. MatricesBstl andDi

l, whose dimensions areny × nu,

are defined as follows:

• If l = γi,j, then[Bsl ]i,j = d0i,j +∆tdii,j, and[Di

l]i,j = dii,j;

• Else,[Bsl ]i,j = 0 and[Di

l]i,j = 0.

Matrix Bstl is a copy of matrixDstFstNst, but with zeros in the positions corresponding to the

transfer functions with time delays that are not equal tol.

This proposed model preserves the main characteristic of the previous model, proposed by

Gonzálezet al. (2007), or more specifically, the integrating states do not depend of the param-

eters related with the integrating outputs. This is a step further towards the development of a

MPC strategy that is robust to model uncertainty and can be implemented in practice.

2.3 The robust MPC formulation

Process system models are usually affected by parameter uncertainty, i.e. the model param-

eters that are not exactly known. With the model formulationdefined in (2.6), model uncer-

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49

tainty would be concentrated on matricesA andB, which are related to the uncertainty in

matricesFst, Bsl , B

stl andDi

l, as well as on the time delayγi,j. Then, one can define the

set of possible plants asΩ = Θ1, . . . ,ΘL, where each particular plantΘn is defined as

Θn = Fstn ,B

sl,n,B

stl,n,D

il,n, γi,j,n, for n = 1, . . . , L andl = 0, . . . , p. Also, it is convenient to

designate the true plant asΘT and the nominal or most probable plant asΘN .

The multi-plant uncertainty considered here corresponds to a scenario in which it is assumed

that the process system can operate at several different operating conditions. At each operating

point, one can obtain through plant test a linear modelΘn that includes the time delay. Once

the system reaches an operating point, it will remain near this point for a large period of time.

The definition of setΩ is more the result of a decision to be taken during the design phase of

the process system and depends on the operating policy of theprocess plant, than a decision to

be taken during the design and implementation of the controlstructure, where, one has only to

define which are the linear models that constitute the setΩ.

Now, for each modelΘn, one can define the control cost as follows:

V1,k(Θn) =∞∑

j=0

∥∥∥yn(k + j|k)− ysp,k(Θn)− δy,k(Θn)− (j −m− p)∆tDi(Θn)δi,k

∥∥∥

2

Qy

+∞∑

j=0

∥∥∥u(k + j|k)− udes,k − δu,k

∥∥∥

2

Qu

+m−1∑

j=0

∥∥∥∆u(k + j|k)

∥∥∥

2

R(2.7)

+∥∥∥δy,k(Θn)

∥∥∥

2

Sy

+∥∥∥δu,k

∥∥∥

2

Su

+∥∥∥δi,k

∥∥∥

2

Si

,

where:Qy ∈ Rny×ny is positive definite;Qu ∈ R

nu×nu andR ∈ Rnu×nu are positive semidef-

inite; yn(k + j|k) is the output prediction corresponding to each modelΘn at time stepk + j

computed at time stepk including the effects of the future control actions;ysp,k is the computed

output reference;∆u(k+ j|k) is the input move (beyond the control horizonm the input moves

are assumed here to be equal to zero);udes,k is the input target; and the matrix of the coefficients

corresponding to the integrating poles associated with each particular modelΘn (Di ∈ Rny×nu)

is obtained by[Di]i,j(Θn) = dii,j(Θn).

Furthermore, in order to enlarge the region where the controller is feasible, slack variables

δy,k ∈ Rny, δu,k ∈ R

nu andδi,k ∈ Rnu are introduced in the control problem, whileSy ∈

Rny×ny, Su ∈ R

nu×nu andSi ∈ Rnu×nu are positive definite weight matrices associated with

these slacks.

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50

The control cost defined in (2.7) can also be written in the following form:

V1,k(Θn) =

m+p∑

j=0

∥∥∥yn(k + j|k)− ysp,k(Θn)− δy,k(Θn)− (j −m− p)∆tDi(Θn)δi,k

∥∥∥

2

Qy

+

∞∑

j=1

∥∥∥yn(k +m+ p+ j|k)− ysp,k(Θn)− δy,k(Θn)− j∆tDi(Θn)δi,k

∥∥∥

2

Qy

+

m−1∑

j=0

∥∥∥u(k + j|k)− udes,k − δu,k

∥∥∥

2

Qu

+

m−1∑

j=0

∥∥∥∆u(k + j|k)

∥∥∥

2

R(2.8)

+∞∑

j=0

∥∥∥u(k +m+ j|k)− udes,k − δu,k

∥∥∥

2

Qu

+∥∥∥δy,k(Θn)

∥∥∥

2

Sy

+∥∥∥δu,k

∥∥∥

2

Su

+∥∥∥δi,k

∥∥∥

2

Si

.

From the state-space model defined in (2.6), and with the assumption that no control actions

are considered after the time stepm, it is easy to show that

yn(k +m+ p+ j|k) = Cxn(k +m+ p+ j|k)

= xsn(k +m+ p|k) +Ψst(Fst)j(Θn)x

dn(k +m+ p|k) (2.9)

+j∆tDi(Θn)xi0(k +m+ p|k).

So, substituting (2.9) in (2.8), the control cost becomes

V1,k(Θn) =

m+p∑

j=0

∥∥∥yn(k + j|k)− ysp,k(Θn)− δy,k(Θn)− (j −m− p)∆tDi(Θn)δi,k

∥∥∥

2

Qy

+∞∑

j=1

∥∥∥x

sn(k +m+ p+ j|k)− ysp,k(Θn)− δy,k(Θn)

∥∥∥

2

Qy

+∞∑

j=1

∥∥∥Ψ

st(Fst)j(Θn)xdn(k +m+ p|k) + j∆tDi(Θn)(x

i0(k +m+ p|k)− δi,k)

∥∥∥

2

Qy

+m−1∑

j=0

∥∥∥u(k + j|k)− udes,k − δu,k

∥∥∥

2

Qu

+m−1∑

j=0

∥∥∥∆u(k + j|k)

∥∥∥

2

R(2.10)

+∞∑

j=0

∥∥∥u(k +m+ j|k)− udes,k − δu,k

∥∥∥

2

Qu

+∥∥∥δy,k(Θn)

∥∥∥

2

Sy

+∥∥∥δu,k

∥∥∥

2

Su

+∥∥∥δi,k

∥∥∥

2

Si

.

In order to keep the control cost bounded, the following equality constraints must be added to

the control problem:

xsn(k +m+ p|k)− ysp,k(Θn)− δy,k(Θn) = 0, n = 1, . . . , L (2.11)

u(k +m− 1|k)− udes,k − δu,k = 0, (2.12)

xi0(k +m+ p|k)− δi,k = 0. (2.13)

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51

It should be noted here that that constraint (2.13) needs to be imposed only for the state com-

ponents related with the integrated inputs. The remaining components corresponding to the

non-integrated inputs are arbitrarily placed at the originso that the model formulation is both

controllable and observable, and consequently, the valuesof the slacksδi,k associated with the

non-integrated inputs do not affect the control problem andcan be assumed null. Therefore,

the proposed controller is applicable for systems in which the number of integrating poles as-

sociated with each output is smaller than the number of inputs, thus assuring the existence of

enough degrees of freedom to steer the output towards the intended goal.

Using the state components defined in (2.6), equations (2.11) to (2.13) are equivalent to the

following equations, respectively,

Ns

(

Am+p(Θn)x(k)−Ap(Θn)W(Θn)∆uk

)

− ysp,k(Θn)− δy,k(Θn) = 0,

n = 1, . . . , L (2.14)

I⊤nu∆uk + u(k − 1)− udes,k − δu,k = 0, (2.15)

xi0(k) + I⊤nu∆uk − δi,k = 0, (2.16)

where:

Ns =

[

Iny 0

p+1︷ ︸︸ ︷

0 · · · 0

]

,

W(Θn) =[Am−1(Θn)B(Θn) Am−2(Θn)B(Θn) · · · A(Θn)B(Θn) B(Θn)

],

∆uk =[∆u(k|k)⊤ · · · ∆u(k +m− 1|k)⊤

]⊤, I⊤nu =

[Inu · · · Inu

].

Now, considering the constraints defined in (2.14) to (2.16), the control cost defined in (2.10)

becomes

V1,k(Θn) =

m+p∑

j=0

∥∥∥yn(k + j|k)− ysp,k(Θn)− δy,k(Θn)− (j −m− p)∆tDi(Θn)δi,k

∥∥∥

2

Qy

+∞∑

j=1

∥∥∥Ψ

st(Fst)j(Θn)xdn(k +m+ p|k)

∥∥∥

2

Qy

+m−1∑

j=0

∥∥∥∆u(k + j|k)

∥∥∥

2

R(2.17)

+m−1∑

j=0

∥∥∥u(k + j|k)− udes,k − δu,k

∥∥∥

2

Qu

+∥∥∥δy,k(Θn)

∥∥∥

2

Sy

+∥∥∥δu,k

∥∥∥

2

Su

+∥∥∥δi,k

∥∥∥

2

Si

.

Following the standard procedure in the MPC literature, e.g. see the book of Maciejowski

(2002), the infinite sum contained in (2.17) is replaced witha terminal weight matrixQ(Θn),

which is obtained from the solution to the following Lyapunov equation:

Q(Θn) = (Fst)⊤(Θn)(Ψst)⊤QyF

st(Θn)Ψst + (Fst)⊤(Θn)Q(Θn)F

st(Θn), n = 1, . . . , L.

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52

Therefore, the robust MPC for time delayed processes proposed here results from the solu-

tion to the following optimization problem:

Problem2–1

min∆uk,ysp,k(Θn=1,··· ,L),δy,k(Θn=1,··· ,L),δu,k,δi,k

V1,k(ΘN),

V1,k(ΘN) =

m+p∑

j=0

∥∥∥yN(k + j|k)− ysp,k(ΘN)− δy,k(ΘN)− (j −m− p)∆tDi(ΘN)δi,k

∥∥∥

2

Qy

+m−1∑

j=0

∥∥∥∆u(k + j|k)

∥∥∥

2

R+∥∥∥u(k + j|k)− udes,k − δu,k

∥∥∥

2

Qu

(2.18)

+∥∥∥x

dN(k +m+ p|k)

∥∥∥

2

Q(ΘN )+∥∥∥δy,k(ΘN)

∥∥∥

2

Sy

+∥∥∥δu,k

∥∥∥

2

Su

+∥∥∥δi,k

∥∥∥

2

Si

,

subject to (2.14), (2.15), (2.16) and

∆u(k + j|k) ∈ U, j = 0, . . . ,m− 1. (2.19)

U =

−∆umax ≤ ∆u(k + j|k) ≤ ∆umax

∆u(k + j|k) = 0, j ≥ m

umin ≤ u(k − 1) +∑j

i=0 ∆u(k + i|k) ≤ umax

ymin ≤ ysp,k(Θn) ≤ ymax, n = 1, . . . , L (2.20)

V1,k(Θn) ≤ V1,k(Θn), n = 1, . . . , L (2.21)

where, considering that(

∆u∗k−1,y

∗sp,k−1(Θn), δ

∗y,k−1(Θn), δ

∗u,k−1, δ

∗i,k−1

)

is the optimal solu-

tion to Problem 2–1 at time stepk-1, one defines:

∆uk =[∆u∗(k|k − 1)⊤ · · · ∆u∗(k +m− 2|k − 1)⊤ 0⊤

]⊤,

ysp,k(Θn) = y∗sp,k−1(Θn), n = 1, . . . , L,

and, the slacksδy,k(Θn), δu,k andδi,k are such that

Ns

(

Am+p(Θn)x(k)−Ap(Θn)W(Θn)∆uk

)

− ysp,k(Θn)− δy,k(Θn) = 0, n = 1, . . . , L,

I⊤nu∆uk + u(k − 1)− udes,k − δu,k = 0,

xi0(k) + I⊤nu∆uk − δi,k = 0.

Then,V1,k(Θn) is the control cost corresponding to the variables defined above, assuming that

the system still starts from the current state.

Remark 2.2 A simple approach to produce a robust MPC for the multi-plantuncertainty is to

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53

consider constraints that force the cost related to each plant lying inΩ to decrease at successive

time steps. In the control problem defined above, these constraints are defined in (2.21).

Remark 2.3 With the model representation considered here, the slacks associated with the

integrating states and system inputs,δi,k andδu,k, are the same for all the models lying inΩ.

Remark 2.4 As in González and Odloak (2009), the zone control strategy adopted in the con-

troller proposed here considers that the output targetsysp,k(Θn=1,...,L) are additional decision

variables of the control problem instead of fixed set-points.

Although Problem 2–1 is always feasible due to the slack variables included in the controller

formulation, one can verify (through simple simulation examples) that the solution of this con-

trol problem at successive time steps does not necessarily produce an asymptotically decreasing

cost function, while the slack variables associated with the integrating states is not zeroed. How-

ever, a simple strategy to force the convergence of this control cost is to split Problem 2–1 into

two sub-problems as follows (Santoro and Odloak, 2012):

Problem2–2a

min∆ua,k,δi,k

V2a,k =∥∥∥δi,k

∥∥∥

2

Si

,

subject to:

∆ua(k + j|k) ∈ U, j = 0, . . . ,m− 1, (2.22)

and

xi0(k) + I⊤nu∆ua,k − δi,k = 0, (2.23)

where∆ua,k =[∆ua(k|k)

⊤ · · · ∆ua(k +m− 1|k)⊤]⊤

.

The optimal solution to Problem 2–2a is denoted(

∆u∗a,k, δ

∗i,k

)

and the total input increment

corresponding to this optimal solution is given by

u∗a(k +m− 1|k)− u(k − 1) =

m−1∑

j=0

∆u∗a(k + j|k).

Then, the total input increment is imposed as a constraint toa second problem that is solved

within the same time step, and is defined as follows:

Problem2–2b

min∆ub,k,ysp,k(Θn=1,··· ,L),δy,k(Θn=1,··· ,L),δu,k

V2b,k(ΘN),

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54

V2b,k(ΘN) =

m+p∑

j=0

∥∥∥yN(k + j|k)− ysp,k(ΘN)− δy,k(ΘN)

∥∥∥

2

Qy

+m−1∑

j=0

∥∥∥∆u(k + j|k)

∥∥∥

2

R+∥∥∥u(k + j|k)− udes,k − δu,k

∥∥∥

2

Qu

(2.24)

+∥∥∥x

dN(k +m+ p|k)

∥∥∥

2

Q(ΘN )+∥∥∥δy,k(ΘN)

∥∥∥

2

Sy

+∥∥∥δu,k

∥∥∥

2

Su

,

subject to (2.14), (2.20) and

∆ub(k + j|k) ∈ U, j = 0, . . . ,m− 1, (2.25)

I⊤nu∆ub,k + u(k − 1)− udes,k − δu,k = 0, (2.26)

ub(k +m− 1|k) = u∗a(k +m− 1|k), (2.27)

V2b,k(Θn) ≤ V2b,k(Θn), n = 1, . . . , L, (2.28)

where, following the same reasoning as the one that led to theformulation of constraint (2.21),

that is, assuming that(

∆u∗k−1,y

∗sp,k−1(Θn), δ

∗y,k−1(Θn), δ

∗u,k−1

)

is the optimal solution to Prob-

lem 2–2b at time stepk-1, thenV2b,k(Θn) is computed with:

∆ub,k =[∆u∗

b(k|k − 1)⊤ · · · ∆u∗b(k +m− 2|k − 1)⊤ 0⊤

]⊤, (2.29)

ysp,k(Θn) = y∗sp,k−1(Θn), n = 1, . . . , L, (2.30)

and, the slacksδy,k(Θn) andδu,k are obtained from

Ns

(

Am+p(Θn)x(k)−Ap(Θn)W(Θn)∆ub,k

)

− ysp,k(Θn)− δy,k(Θn) = 0,

n = 1, . . . , L, (2.31)

and

I⊤nu∆ub,k + u(k − 1)− udes,k − δu,k = 0. (2.32)

Remark 2.5 While the slackδi,k has not been zeroed, the solution proposed in (2.29) to (2.32)

is not necessarily feasible. For instance, consider that the optimal control sequence of Problem

2–2a is∆u∗a,k =

[∆u⊤

max · · · ∆u⊤max

]⊤, then, because of (2.27), the only possible solution

to Problem 2–2b would be∆u∗b,k = ∆u∗

a,k, which may not satisfy (2.28). An alternative to

overcome this limitation is to apply the following algorithm, which makes Problem 2-2b feasible

in a finite number of time steps:

• At any time stepk solve Problem 2–2a, which provides the control sequence∆u∗a,k.

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55

• Try to solve Problem 2–2b, if it results infeasible because of a conflict between constraints

(2.27) and (2.28), assume∆u∗b,k = ∆u∗

a,k and inject the first control move into the real

process.

The convergence of the closed-loop system with the robust controller resulting from the

sequential solution to Problems 2–2a and 2–2b is guaranteedby the following theorems:

Theorem 2.1 For an undisturbed stabilizable system at time stepk, the sequential solution

of Problems 2–2a and 2–2b results in the convergence of the slack vectorδi,k to zero and,

consequently, the convergence of the integrating states atthe end of the delayed control horizon,

xi0(k +m+ p|k), to zero, after a finite number of time steps.

Proof. Consider that at time stepk the optimal solution to Problem 2–2a is given by(

∆u∗a,k =

[∆u∗

a(k|k)⊤ · · · ∆u∗

a(k +m− 1|k)⊤]⊤

, δ∗i,k

)

,

and let the corresponding optimal control cost beV ∗2a,k. Then, the corresponding total input

increment is fed, through the equality constraint (2.27), to Problem 2–2b, which is solved at the

same time step and let the optimal control sequence corresponding to the solution to Problem

2–2b be represented as

∆u∗b,k =

[∆u∗

b(k|k)⊤ · · · ∆u∗

b(k +m− 1|k)⊤]⊤

,

from which the first control move∆u∗b(k|k) is injected into the true process. Now, at time step

k+1, consider a solution to Problem 2–2a given by

∆ua,k+1 =[∆u∗

b(k + 1|k)⊤ · · · ∆u∗b(k +m− 1|k)⊤ 0⊤

]⊤, (2.33)

δi,k+1 = δ∗i,k. (2.34)

To show that this solution is feasible, observe that the integrating states at the end of the de-

layed control horizon computed at timek+1 can be related to the proposed control sequence as

follows:

xi0(k +m+ p+ 1|k + 1) = xi

0(k + 1|k + 1) + I⊤nu∆ua,k+1

= xi0(k + 1|k + 1) + ∆u∗

b(k + 1|k) + · · ·+∆u∗b(k +m− 1|k) + 0.

For the undisturbed system, since the integrating statexi0 is the same to all the modelsΘn=1,...,L,

then one hasxi0(k + 1|k + 1) = xi

0(k + 1|k), and from the model defined in (2.6), it results

xi0(k + 1|k + 1) = xi

0(k) + ∆u∗(k|k). Hence,

xi0(k +m+ p+ 1|k + 1) = xi

0(k) + ∆u∗b(k|k) + · · ·+∆u∗

b(k +m− 1|k),

= xi∗

0 (k +m+ p|k). (2.35)

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56

Now, let V2a,k+1 be the control cost of Problem 2–2a associated with the solution proposed in

(2.33) to (2.34). From (2.35), it is clear thatV2a,k+1 = V ∗2a,k, and consequently the optimal

solution to Problem 2–2a at time stepk+1 will correspond toV2a,k+1 ≤ V ∗2a,k, which shows that

this control cost will converge to zero if the system is controllable. It is easy to see that the

convergence will happen in a finite number of steps. For instance, if the input increment is not

limiting, the control sequence∆u(k + j|k) = −xi0(k)/m, j = 0, . . . ,m − 1, will reduce the

integrating state to zero inm steps.

From Theorem 2.1, after convergence of the integrating state to zero, the solution to Problem

2–2a becomes equivalent to the following trivial constraint:

xi0(k) + I⊤nu∆ua,k = 0. (2.36)

This implies that the sequential solution of Problems 2–2a and 2–2b becomes equivalent to the

solution to the following optimization problem:

Problem2–3

min∆uk,ysp,k(Θn=1,··· ,L),δy,k(Θn=1,··· ,L),δu,k

V3,k(ΘN) ≡ V2b,k(ΘN),

subject to (2.14), (2.15), (2.19), (2.20), (2.28) and (2.36).

The asymptotic convergence of the controller resulting from the solution to the optimization

problem defined above can be stated by the following theorem:

Theorem 2.2 Consider a time delay process with stable and integrating modes whose true

model is unknown but lies withinΩ. Also, assume that this process is stabilizable at the desired

input targets and output zones and the prediction of the integrating states at the end of the de-

layed control horizon is zeroed. Then, the control actions obtained from the solution to Problem

2-3 will drive the true plant to its input targets and output zones.

Proof. The full proof of this theorem follows the same steps as in Gonzálezet al. (2007),

adapted to the time delay system. Here, it is just shown that for the undisturbed system, the ex-

tended cost function of the proposed controller is non-increasing and tends to zero if the desired

equilibrium point is reachable. Thus, assume that at time stepk, Problem 2–3 is solved and its

optimal solution is represented by(

∆u∗k,y

∗sp,k(Θn), δ

∗y,k(Θn), δ

∗u,k

)

, such that the control cost

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57

associated with the true plant is given by

V ∗3,k(ΘT ) =

m+p∑

j=0

∥∥∥yT (k + j|k)− y∗

sp,k(ΘT )− δ∗y,k(ΘT )

∥∥∥

2

Qy

+m−1∑

j=0

∥∥∥∆u∗(k + j|k)

∥∥∥

2

R+∥∥∥u

∗(k + j|k)− udes,k − δ∗u,k

∥∥∥

2

Qu

+∥∥∥x

dT (k +m+ p|k)

∥∥∥

2

Q(ΘT )+∥∥∥δ

∗y,k(ΘT )

∥∥∥

2

Sy

+∥∥∥δ

∗u,k

∥∥∥

2

Su

.

Now, one injects the first control move∆u∗(k|k) into the real process and one moves to time

stepk+1. At this time step, consider the following solution:(

∆uk+1, ysp,k+1(Θn), δy,k+1(Θn), δu,k+1

)

,

where:

∆uk+1 =[∆u∗(k + 1|k)⊤ · · · ∆u∗(k +m− 1|k)⊤ 0⊤

]⊤, (2.37)

ysp,k+1(Θn) = y∗sp,k(Θn), n = 1, . . . , L, (2.38)

and, the slacksδy,k+1(Θn) andδu,k+1 are such that

Ns

(

Am+p(Θn)x(k)−Ap(Θn)W(Θn)∆uk+1

)

− ysp,k+1(Θn)− δy,k+1(Θn) = 0,

n = 1, . . . , L, (2.39)

I⊤nu∆uk+1 + u(k|k)− udes,k − δu,k+1 = 0. (2.40)

It is easy to verify that this is a feasible solution to Problem 2–3 and, in addition, one can prove

that for the undisturbed true plantδy,k+1 = δ∗y,k andδu,k+1 = δ

∗u,k. Hence, the control cost for

the true plant corresponding to this feasible solution is

V3,k+1(ΘT ) = V ∗3,k(ΘT )−

∥∥∥y(k)− y∗

sp,k(ΘT )− δ∗y,k(ΘT )

∥∥∥

2

Qy

−∥∥∥u

∗(k|k)− udes,k − δ∗u,k

∥∥∥

2

Qu

−∥∥∥∆u∗(k|k)

∥∥∥

2

R,

and, consequentlyV3,k+1(ΘT ) ≤ V ∗3,k(ΘT ). From (2.28), it is clear thatV ∗

3,k+1(ΘT ) ≤ V3,k+1(ΘT ),

and soV ∗3,k+1(ΘT ) ≤ V ∗

3,k(ΘT ).

Therefore, if the system is controllable at the desired equilibrium point, the sequence of

optimal costs for the true plant is decreasing and convergesto zero, even thoughV3,k(Θn)Θn 6=ΘT

is not necessarily decreasing.

2.4 Simulation results

In this section, it is simulated the application of the proposed robust MPC to two process sys-

tems: a subsystem of an industrial ethylene oxide reactor and a continuous stirred tank reactor

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58

(CSTR). In the first case, it is assumed that the system is linearbut the true model is unknown.

In the second case the true system is nonlinear and a set of linear models is used to approximate

the true model.

Example 1. The first system has been previously studied in Rodrigues and Odloak (2003b).

This is a typical example of the chemical process industry that exhibits stable and integrating

modes as well as significant time delays. The results obtained with the robust MPC are com-

pared to the corresponding ones with the nominal MPC, which isobtained through the solution

to the same control problem but considering only the nominalmodel.

The analysis of this case study is concerned with the tracking of output intervals and input

target. Furthermore, uncertainties on time delays were included here in order to take into ac-

count a potential degradation of the process model along an operation campaign. It is assumed

that setΩ includes three models as follows:

Θ1 =

−10−4(−9s+ 1)e−s

32.16s2 + 4.65s+ 1

−2.3× 10−3

s

−3.2× 10−3(−s+ 1)e−2s

64.55s2 + 8.83s+ 1

−1.69× 10−4e−3s

s

2.1× 10−4e−8s

s

−1.9× 10−3(1.47s+ 1)

9.67s2 + 13.55s+ 1

,

Θ2 =

−1.5× 10−4(−27s+ 1)e−4s

64.32s2 + 9.03s+ 1

−3.5× 10−3e−s

s

−4.8× 10−3(−3s+ 1)e−8s

129.10s2 + 17.66s+ 1

−2.5× 10−4e−6s

s

3.2× 10−4

s

−6.1× 10−3(−4.41s+ 1)e−5s

19.34s2 + 27.10s+ 1

,

Θ3 =

−5× 10−4(−3.6s+ 1)e−8s

19.30s2 + 2.81s+ 1

−1.2× 10−3e−2s

s

−1.6× 10−3(−0.4s+ 1)e−3s

38.73s2 + 5.31s+ 1

−8.5× 10−5e−5s

s

1.1× 10−4e−2s

s

−9.5× 10−4(0.58s+ 1)e−4s

5.8s2 + 8.13s+ 1

.

In this reduced system, the manipulated inputs are: the oxygen flow rate to the reactor (u1), the

ethylene feed flow rate (u2) and the valve opening of the cooling oil (u3). The controlled outputs

are the molar fraction of oxygen in the recycle gas (y1) and the molar fraction of ethylene in the

recycle gas (y2).

The simulation results obtained with the proposed robust MPC, defined through Problems 2–

2a and 2–2b, are compared to the results corresponding to thenominal MPC. In the simulations

presented here, modelΘ2 is assumed to be the nominal model, which is used in the control cost

that is minimized in the controller, whereas modelΘ3 represents the true process. The sampling

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59

time is∆t = 1 min and the control horizon ism = 3, the remaining tuning parameters are:

Qy = diag(1, 0.3), Qu = diag(0, 0, 1), R = diag(0.1, 0.1, 0.01), Sy = diag(105, 7 × 104),

Su = diag(0, 0, 10), andSi = diag(10, 10, 10). The upper and lower bounds related to the

manipulated inputs are presented in Table 2.1.

Table 2.1: Controller bounds for the manipulated inputs of the ethylene oxide reactor system.Inputs umin umax ∆umax Unitu1 5700 6900 25 kg/hu2 4500 5700 25 kg/hu3 0 100 2 %

The simulation results of both robust and nominal controllers are shown in figures 2.1 and

2.2, for the case in which the system starts from the following initial operating point:

u(0) =[6357 (kg/h) 5280 (kg/h) 82 (%)

]⊤andy(0) =

[6.2 (%) 15.7 (%)

]⊤.

Note that at this starting point the outputs lie outside their control zones. From the starting

point until time stepk = 399, the optimum target related to the nonintegrated input (u3) is

u3,des = 82% and the output zone limits are:

ymin =[5.8 (%) 16.0 (%)

]⊤andymax =

[6 (%) 16.2 (%)

]⊤.

At time stepk = 400, when the system has reached a new steady state, the input target is moved

to u3,des = 45% and the output zone limits are:

ymin =[6 (%) 16.0 (%)

]⊤andymax =

[6.2 (%) 16.2 (%)

]⊤.

Finally, after time stepk = 550, once the system has already stabilized, the true process is

switched to the model represented byΘ1 and the control of outputy2 is reduced to a fixed

set-point. In this case, zone limits become

ymin =[6 (%) 15.9 (%)

]⊤andymax =

[6.2 (%) 15.9 (%)

]⊤.

From the simulation results, one can observe that both controllers can drive inputu3 to its

target, but only the robust controller is capable of bringing all outputs to their control zones and

maintaining them inside these zones throughout the simulation. The lack of guaranteed stability

of the nominal MPC, where the control law is based only on modelΘ2, results in a rather poor

performance of the closed-loop system. Meanwhile, the proposed robust MPC, which is based

on the three models, can perform the required tasks more efficiently.

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Figure 2.1: Controlled outputs of the ethylene oxide reactor system. Nominal MPC (−−), robust MPC (—),upper lower limits (− · −).

Figure 2.2: Manipulated inputs of the ethylene oxide reactor system. Nominal MPC (−−), robust MPC (—),optimum target (− · −).

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61

In accordance with Theorem 2-2, the control cost of the robust MPC associated to the true

process must be strictly decreasing, and this is shown in Figure 2.3. Note also that although

there is no guarantee that the control cost is strictly decreasing for the nominal MPC, in this

particular case, the nominal cost also decreases but with a slower rate than the true cost in the

robust controller.

Figure 2.3: Control costs for the ethylene oxide reactor system. Nominal MPC (−−), and robust MPC (—).

Example 2. The aim of this simulation example is to test the performanceof the proposed

robust controller based on the multi-plant representationwhen applied to a nonlinear process

system and to compare with the MPC based on the nominal model.For this purpose, it is

studied the control of a CSTR with cooling jacket (Pannocchiaand Rawlings, 2003) where an

elementary irreversible reaction takes place A→ B. It is assumed that the physical properties

and heat transfer coefficient are constant and the model, which represents the true plant is given

by the following set of nonlinear equations:

dh(t)

dt=

Fin(t)− Fout(t)

πr2,

dcA(t)

dt=

[cA,in − cA(t)]Fin(t)

πr2h(t)− k0 exp

[

−E

RT (t)

]

cA(t), (2.41)

dT (t)

dt=

[Tin − T (t)]Fin(t)

πr2h(t)+

(−∆H)

ρCp

k0 exp

[

−E

RT (t)

]

cA(t) +2U

ρrCp

[Tc(t)− T (t)].

The parameters associated with this system are given in Table 2.2. In the3× 3 control struc-

ture considered here, the controlled outputs are the liquidlevel in the reactory1 [h (m)], the

reactant concentrationy2 [cA (kmol/m3)] and the reaction temperaturey3 [T (K)]. The manipu-

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62

lated inputs are the inlet flow rateu1 [Fin (m3/min)], the outlet flow rateu2 [Fout (m3/min)] and

the cooling fluid temperatureu3 [Tc (K)].

Table 2.2: Nominal parameters of the CSTR system.Description ValuescA,in(reactant concentration in the feed stream) 1.0 kmol·m−3

Tin (temperature of the feed stream) 350 Kr (radius of the reactor) 0.219 mk0 (pre-exponential factor) 7.2× 1010 min−1

E/R(activation energy/universal gas constant) 8,750 KU (overall heat transfer coefficient) 915.6 W·m−2·K−1

ρ (density of the reaction mixture) 103 kg·m−3

Cp (heat capacity of the reaction mixture) 239 J·kg−1·K−1

∆H (enthalpy of reaction) -5× 107 J·kmol−1

To illustrate the application of the method, setΩ, on which the robust MPC is based, is

constituted of three linear models that approximate the model defined in (2.41) at three differ-

ent operating points. The operating conditions corresponding to these equilibrium points are

represented in Table 2.3. The transfer functions that correspond to the linear models are the

following:

Θ1 =

8.194e−s

s

−8.194

s0

(4.375s2 + 0.364s− 7.314)e−3s

s(s2 + 3.719s+ 3.276)

(5.478s+ 7.314)e−4s

s(s2 + 3.719s+ 3.276)

−0.013

s2 + 3.719s+ 3.276

(803.8s2 + 137.7− 1433)e−s

s(s2 + 3.719s+ 3.276)

(1006s+ 1433)

s(s2 + 3.719s+ 3.276)

(2.332s+ 3.107)e−s

s2 + 3.719s+ 3.276

,

Θ2 =

5.485e−3s

s

−5.485e−s

s0

(4.316s2 − 19.15s+ 9.272)e−4s

s(s2 + 5.153s+ 21.4)

(2.387s− 9.272)e−2s

s(s2 + 5.153s+ 21.4)

−0.071e−2s

s2 + 5.153s+ 21.4

(−162.7s2 + 7313s− 3995)

s(s2 + 5.153s+ 21.4)

(−89.99s+ 3995)

s(s2 + 5.153s+ 21.4)

(1.908s+ 19.91)e−2s

s2 + 5.153s+ 21.4

,

Θ3 =

6.637e−4s

s

−6.637e−s

s0

(7.073s2 − 32.6s+ 25.62)e−4s

s(s2 + 60.99s+ 201.1)

(7.109s− 25.62)e−s

s(s2 + 60.99s+ 201.1)

−0.103e−3s

s2 + 60.99s+ 201.1

(−508.3s2 + 6519s− 6500)

s(s2 + 60.99s+ 201.1)

(−510.8s+ 6500)

s(s2 + 60.99s+ 201.1)

(2.099s+ 143)e−3s

s2 + 60.99s+ 201.1

.

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63

It can be noted that time delays were included in the transferfunction model in order to simulate

the more realistic condition for which the robust controller was developed. Also, uncertainty

was considered for the parameters associated with the integrating output, which can be inter-

preted as uncertainty in the measurement of the radius of thereactor. In all the simulations

carried out here, it is assumed that the nonlinear model defined in (2.41) represents the real

plant, the robust MPC is based on the three models defined above while the nominal MPC uses

only modelΘ3 to predict the process system behavior.

Table 2.3: Different steady state conditions for the CSTR system.Variables Steady state 1 Steady state 2 Steady state 3Fin /(m3·min−1) 0.05 0.12 0.14Fout/(m3·min−1) 0.05 0.12 0.14Tc /(K) 295. 350. 380.h /(m) 0.327 1.190 0.925cA /(kmol·m−3) 0.825 0.0635 0.0147T /(K) 317.9 385.3 420.8r /(m) 0.197 0.233 0.219

In the application of the controllers, one adopts a samplingtime∆t = 0.5 min and a con-

trol horizon (m) equal to 7. The remaining tuning parameters are:Qy = diag(8 × 104, 5 ×

102, 1), Qu = diag(0, 0, 10−2), R = diag(103, 103, 10−2), Sy = diag(8 × 105, 104, 10),

Si = diag(102, 10, 1), andSu = diag(0, 0, 0.5). The constraints related to the manipulated

inputs are summarized in Table 2.4.

Table 2.4: Controller bounds for the manipulated inputs of the CSTR system.Inputs umin umax ∆umax Unitu1 0.01 0.4 0.015 m3/minu2 0.01 0.4 0.015 m3/minu3 290. 410. 10. K

Furthermore, in the simulation considered here, the systemstarts from the initial condition:

u(0) =[0.05 (m3/min) 0.05 (m3/min) 275 (K)

]⊤and

y(0) =[

0.3272 (m) 0.8253 (kmol/m3) 317.9 (K)]⊤

;

the initial output zone limits are:

ymin =[

0.75 (m) 0 (kmol/m3) 420 (K)]⊤

and

ymax =[

1.0 (m) 0.05 (kmol/m3) 440 (K)]⊤

,

and the target for the only nonintegrated manipulated inputu3 is set to 395 K. Note that all

outputs and the inputu3 start from outside their control zones and optimum target, respectively.

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64

Therefore, it was expected that initially the controllers would drive the outputs to their desired

zones. From figures 2.4 and 2.5, one can conclude that both themulti-model MPC and nominal

MPC performed satisfactorily. The similarity between the responses of the two controllers can

be easily understood if one observes that the operating point defined by the initial output zones

and input target is near to the equilibrium point where modelΘ3 was obtained. This means that

the nominal model is very close to the true plant model.

After 40 simulation steps, the reactor is forced to a new operating point that corresponds to

the output control zones:

ymin =[

1.0 (m) 0.01 (kmol/m3) 375 (K)]⊤

and

ymax =[

1.2 (m) 0.10 (kmol/m3) 400 (K)]⊤

,

and the following optimizing targetu3,des = 350 K. This new operating point is close to the

equilibrium point where modelΘ2 was obtained and since this point is not far from the previous

operating point, the performance of the nominal MPC is stillreasonable and not too different

from the performance of the robust MPC. Only for outputy1, the response of the nominal

controller is slightly deteriorated. Finally, at time stepk = 80, the reactor is forced to a third

operating region defined through the output zone limits

ymin =[

0.3 (m) 0.70 (kmol/m3) 300 (K)]⊤

and

ymax =[

0.65 (m) 0.95 (kmol/m3) 330 (K)]⊤

,

and a target foru3 equal to 305 K. It is easy to see that this new operating condition is close to the

equilibrium point where modelΘ1 was obtained and is quite far from the point corresponding to

modelΘ3. This justifies the poor performance of the nominal controller while the performance

of the robust MPC is almost unaffected.

From the foregoing results, one can arrive to the conclusionthat, although the robust MPC

proposed in this work does not ensure stability for the nonlinear plant, it can drive the process

system to the desired goals without a significant deterioration of the performance, mainly if the

results are compared to the corresponding results of the nominal MPC.

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65

Figure 2.4: Controlled inputs of the CSTR system. Nominal MPC (−−), robust MPC (—), lower and upperbounds (− · −).

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66

Figure 2.5: Manipulated inputs of the CSTR system. Nominal MPC (−−), robust MPC (—), optimum target(− · −).

2.5 Concluding remarks

In this chapter, it is proposed an extension of a controller that was presented in previous work

(Gonzálezet al., 2007) in order to produce a convergent robust MPC with guaranteed stability

for stable and integrating systems. This extension is related with the following aspects of the

robust controller:

(i) The state-space model adopted in the controller is generalized to consider the time delay

system. This model is based on the step responses associatedwith the transfer function

models of the process;

(ii) Model uncertainty is considered not only on the usual state-space matrices, but also in the

time delays;

(iii) Focusing on the practical implementation, the robustcontroller considers output control

zone and optimizing input targets that are usually related to economic targets.

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67

The simulation results of typical industrial examples presented here show that when there is

model uncertainty, the performance of the proposed robust approach can be significantly better

than the performance of the corresponding nominal infinite horizon MPC. These results also

show that the industrial implementation of the robust MPC becomes a competitive alternative

when compared to the implementation of conventional MPC.

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CHAPTER 3

One-step formulation of robust MPC forintegrating time delay processes

3.1 Introduction

A novel robustly stable model predictive control strategy of integrating time delay processes is

developed in this chapter. The approach proposed here generalizes the robust MPC controller

presented in Chapter 2, which still has some shortcomings to be circumvented, even though

it has numerous advantages over the conventional MPC approach. In particular there are two

limitations to that robust control strategy, and they are summarized as follows.

First, in the two-step controller proposed in Chapter 2, at any time stepk, Problem 2-2a

is solved first and then one solves Problem 2-2b, and the two problems interact. However,

because Problem 2-2a does not consider the dynamic behaviorof the process system, one might

speculate that the performance of the closed-loop system may not be satisfactory for some

simulated scenarios. The second inadequacy is related withthe state-space model formulation,

insofar as it is only completely controllable when the number of integrating poles associated

with any controlled output is less than number of available manipulated inputs. Although such

a system representation is applicable to many cases of the process industry, there are practical

cases where all the system inputs are integrated by the controlled output, e.g. liquid level

in process tanks, component concentration in systems containing recycle streams, and some

operating modes of fluidized-bed catalytic cracking (FCC) systems. In this way, the features

described earlier encourage and justify the development ofa new (more general) robust MPC

69

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70

strategy.

With regard to the general model formulation, Santoro (2011) developed a state-space model

based on the analytical representation of the step responsemodel of the system, and that can be

applied to the process systems whose outputs can be integrating process variables with respect

to all system inputs. This proposed model formulation was used to synthesize a nominally stable

MPC for stable and integrating time delay systems (Santoro and Odloak, 2012). Based on this

model description, the main scope of this chapter is to develop a robustly stable MPC within a

one-step optimization formulation, thus overcoming the interaction disadvantage inherited from

the two-step formulation of the controller proposed in Chapter 2.

The text of this chapter has been organized in the following way. In Section 3.2 the more

general robust MPC of integrating time delay processes is developed considering the aforemen-

tioned state-space model description. Section 3.3 deals with the application of the proposed

control strategy for different disturbance scenarios and plant/model mismatch of an industrial

FCC converter unit, and the results are favorably compared tothose from a conventional MPC

existing in the real process. Finally, Section 3.4 concludes the chapter.

3.2 Development of the proposed robust MPC

The robust MPC proposed here is based on a minimal order state-space model, which is equiv-

alent to the step response model corresponding to the systemtransfer function, developed by

Santoro (2011). Here, this model representation is summarized.

Consider a process withnu inputs andny outputs, assume also that the poles related with

any inputuj to any outputyi are non-repeated and can be stable and/or integrating, and the

corresponding time delay is designated asγi,j, with p = maxi,j

γi,j. Then, the state-space model

considered here takes the following form:

x(k + 1) = Ax(k) +B∆u(k)y(k) = Cx(k),

(3.1)

where: x(k) =[xs(k)⊤ xst(k)⊤ xi(k)⊤ z1(k)

⊤ · · · zp(k)⊤]⊤

, x ∈ Cnx, xs ∈ R

ny,

xst ∈ Cnst, xi ∈ R

ny, z1 · · · zp ∈ Rnu, nx = ny + nst+ ny + p.nu,

A =

Iny 0 ∆tI∗ny Bs1 Bs

2 · · · Bsp−1 Bs

p

0 Fst 0 Bst1 Bst

2 · · · Bstp−1 Bst

p

0 0 I∗ny Bi1 Bi

2 · · · Bip−1 Bi

p

0 0 0 0 0 · · · 0 0

0 0 0 Inu 0 · · · 0 0...

......

......

.. ....

...0 0 0 0 0 · · · Inu 0

∈ Cnx×nx, B =

Bs0

Bst0

Bi0

Inu0...0

∈ Cnx×nu,

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71

C =

[

Iny Ψst 0

p︷ ︸︸ ︷

0 · · · 0

]

∈ Rny×nx.

∆t is the sampling time considered in the time discretization,I∗ny is a diagonal matrix of dimen-

sionny × ny whose entries are 1 for the integrating outputs and 0 for the stable outputs. The

more detailed rules to obtain the matricesFst, Ψst, Bsl , B

stl andBi

l, for l = 1, . . . , p, can be

found in the original work of Santoro (2011).

Remark 3.1 In the model defined in (3.1),xs(k) represents the integrating states produced

by the incremental form of inputs and corresponds to the predicted output steady-state,xst(k)

corresponds to the stable states,xi(k) stands for the true integrating states of the process and

each state componentzl(k) corresponds to past input moves associated with the time delay l,

wherel varies from 1 top.

Remark 3.2Notice also that the state-space model is written in the incremental form of the in-

puts, which implies an offset free MPC and allows the elimination of the target calculation layer

that is frequently adopted to prevent offset in MPC implementations (Pannocchia and Rawlings,

2003; Muske and Badgwell, 2002).

In the sequel to this study, one recasts the multi-model robust MPC into the framework

of the state-space model described above. With respect to the model structure defined in

(3.1), its uncertainty as a whole is directly related to uncertainty in matricesFst, Bsl , Bst

l

and Bil, as well as on the time delayγi,j. So, with the multi-plant uncertainty represen-

tation, setΩ is defined asΩ = (Θ1, . . . ,ΘL), where eachΘn denotes a particular plant

Θn = Fstn ,B

sl,n,B

stl,n,B

il,n, γi,j,n, for n = 1, . . . , L and l = 0, . . . , p. For convenience of

notation, the nominal (most likely) plant and the true plantare designated here asΘN andΘT ,

respectively. Thus, for each modelΘn, at time stepk the control cost is defined as follows:

V1,k(Θn) =∞∑

j=0

∥∥∥yn(k + j|k)− ysp,k(Θn)− δy,k(Θn)− (j −m− p)∆tδi,k(Θn)

∥∥∥

2

Qy

+∞∑

j=0

∥∥∥u(k + j|k)− udes,k − δu,k

∥∥∥

2

Qu

+m−1∑

j=0

∥∥∥∆u(k + j|k)

∥∥∥

2

R(3.2)

+∥∥∥δy,k(Θn)

∥∥∥

2

Sy

+∥∥∥δu,k

∥∥∥

2

Su

+∥∥∥δi,k(Θn)

∥∥∥

2

Si

,

in which∆u(k + j|k) ∈ Rnu is the input move computed at time stepk to be applied at time

stepk + j; m is the control horizon where the inputs are allowed to move;yn(k + j|k) ∈ Rny

is the prediction of the output of modelΘn at time stepk + j including the effect of the future

input moves;udes,k ∈ Rnu is the input target;ysp,k ∈ R

ny is output reference (set-point), which

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72

is also considered a constrained variable of the control problem.

In the control objective defined in (3.2), it is also includedthe unconstrained slack variables

δy,k(Θn) ∈ Rny (associated with the predicted steady-state),δi,k(Θn) ∈ R

ny (associated with

the integrating outputs) andδu,k ∈ Rnu (associated with the inputs). The key purpose of these

slacks is to provide extra degrees of freedom to the control problem and to widen the feasible

region of the proposed controller, as will be shown later. The weighting matricesQy, Qu, R,

Sy, Su andSi are tuning parameters of controller considered here.

Once an infinite prediction horizon is adopted, it can be shown that the control cost defined

in (3.2) will be bounded only if the following terminal constraints are included in the control

problem:

xsn(k +m+ p|k)− ysp,k(Θn)− δy,k(Θn) = 0, n = 1, . . . , L, (3.3)

xin(k +m+ p|k)− δi,k(Θn) = 0, n = 1, . . . , L, (3.4)

u(k +m− 1|k)− udes,k − δu,k = 0. (3.5)

If the state predictions and the future input values are written in terms of the input moves,

equations (3.3) to (3.5) can be written as follows:

Ns

(

Am+p(Θn)x(k)−Ap(Θn)W(Θn)∆uk

)

− ysp,k(Θn)− δy,k(Θn) = 0, n = 1, . . . , L,

(3.6)

Ni

(

Am+p(Θn)x(k)−Ap(Θn)W(Θn)∆uk

)

− δi,k(Θn) = 0, n = 1, . . . , L, (3.7)

I⊤nu∆uk + u(k − 1)− udes,k − δu,k = 0, (3.8)

where:

Ns =

[

Iny 0ny×nst 0ny

p︷ ︸︸ ︷

0ny×nu · · · 0ny×nu

]

,

Ni =

[

0ny 0ny×nst Iny

p︷ ︸︸ ︷

0ny×nu · · · 0ny×nu

]

,

W(Θn) =[Am−1(Θn)B(Θn) Am−2(Θn)B(Θn) · · · A(Θn)B(Θn) B(Θn)

],

∆uk =[∆u(k|k)⊤ · · · ∆u(k +m− 1|k)⊤

]⊤, I⊤nu =

[Inu · · · Inu

].

Now, if the equality constraints (3.6) to (3.8) are includedin the control problem, then the

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73

control cost defined in (3.2) can be rewritten as follows:

V1,k(Θn) =

m+p∑

j=0

∥∥∥yn(k + j|k)− ysp,k(Θn)− δy,k(Θn)− (j −m− p)∆tδi,k(Θn)

∥∥∥

2

Qy

+∞∑

j=1

∥∥∥Ψ

st(Fst)j(Θn)xstn (k +m+ p|k)

∥∥∥

2

Qy

+m−1∑

j=0

∥∥∥∆u(k + j|k)

∥∥∥

2

R(3.9)

+m−1∑

j=0

∥∥∥u(k + j|k)− udes,k − δu,k

∥∥∥

2

Qu

+∥∥∥δy,k(Θn)

∥∥∥

2

Sy

+∥∥∥δu,k

∥∥∥

2

Su

+∥∥∥δi,k(Θn)

∥∥∥

2

Si

.

Since the infinite sum contained in (3.9) depends only on the stable modes that appear in

Fst(Θn=1,...,L) , this infinite sum can be replaced by a quadratic norm of the stable compo-

nent of the state penalized by weighting matrixQ, which is obtained from the iterative solution

of the discrete Lyapunov equation (Maciejowski, 2002)

Q(Θn) = (Fst)⊤(Θn)(Ψst)⊤QyF

st(Θn)Ψst + (Fst)⊤(Θn)Q(Θn)F

st(Θn), n = 1, . . . , L.

Therefore, the control problem of the proposed MPC reduces to the solution to the following

optimization problem:

Problem3–1

min∆uk,ysp,k(Θn=1,...,L),δy,k(Θn=1,...,L),δu,k,δi,k(Θn=1,...,L)

V1,k(ΘN),

V1,k(ΘN) =

m+p∑

j=0

∥∥∥yN(k + j|k)− ysp,k(ΘN)− δy,k(ΘN)− (j −m− p)∆tδi,k(ΘN)

∥∥∥

2

Qy

+m−1∑

j=0

∥∥∥u(k + j|k)− udes,k − δu,k

∥∥∥

2

Qu

+m−1∑

j=0

∥∥∥∆u(k + j|k)

∥∥∥

2

R

+∥∥∥x

stN(k +m+ p|k)

∥∥∥

2

Q(ΘN )+∥∥∥δy,k(ΘN)

∥∥∥

2

Sy

+∥∥∥δu,k

∥∥∥

2

Su

+∥∥∥δi,k(ΘN)

∥∥∥

2

Si

,

subject to (3.6), (3.7), (3.8) and

∆u(k + j|k) ∈ U, j = 0, . . . ,m− 1. (3.10)

U =

−∆umax ≤ ∆u(k + j|k) ≤ ∆umax

∆u(k + j|k) = 0, j ≥ m

umin ≤ u(k − 1) +∑j

i=0 ∆u(k + i|k) ≤ umax

, (3.11)

ymin ≤ ysp,k(Θn) ≤ ymax, n = 1, . . . , L, (3.12)

V1,k(Θn) ≤ V1,k(Θn), n = 1, . . . , L, (3.13)

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74

where the calculation of the control costV1,k(Θn) is based on the optimal solution to Problem

3–1 at the time stepk − 1. This is done by defining the following variables:

∆uk =[∆u∗(k|k − 1)⊤ · · · ∆u∗(k +m− 2|k − 1)⊤ 0⊤

]⊤,

ysp,k(Θn=1,...,L) = y∗sp,k−1(Θn=1,...,L).

and the pseudo-slack variablesδy,k(Θn=1,...,L), δi,k(Θn=1,...,L) andδu,k are such that

Ns

(

Am+p(Θn)x(k)−Ap(Θn)W(Θn)∆uk

)

− ysp,k(Θn)− δy,k(Θn) = 0, n = 1, . . . , L,

Ni

(

Am+p(Θn)x(k)−Ap(Θn)W(Θn)∆uk

)

− δi,k(Θn) = 0, n = 1, . . . , L,

I⊤nu∆uk + u(k − 1)− udes,k − δu,k = 0.

There are some important points about this optimization problem that should be noted here:

(i) The state of the process at time stepk is assumed to be known;

(ii) The introduction of the pseudo-slack variablesδy,k(Θn=1,...,L), δi,k(Θn=1,...,L) andδu,k is

necessary in order to accommodate the feedback from the current outputy(k) to all the

plants belonging to setΩ;

(iii) For the adopted zone control strategy, the desired values for the output of each plant

ysp,k(Θn=1,...,L) become additional constrained decision variables to the control optimiza-

tion problem. It is worth noting that when one has a target fora controlled output, the

corresponding output zone is settled such that the upper bound equals the lower bound.

(iv) Problem 3–1 is no longer a quadratic programming (QP) optimization problem due to

the presence of the nonlinear cost-contraction constraints defined in (3.13). However, as

this constraint is strictly convex, the control optimization problem is also convex, and

consequently, one can ensure that any local minimum is a global minimum.

It is important to emphasize here that the slack variables included in Problem 3–1 should be

small enough to minimize the deviation from the original terminal constraints (without slack

variables). They are penalized in the control costs with large weights (Sy, Su andSi) to assure

that the slacks will be different from zero only when it is notpossible to obtain a feasible

solution with null slacks. In practice, if one adopts valuesfor weighting matricesSy, Su and

Si which are several orders of magnitude larger thanR, Qy andQu, it can be verified that the

slack vectors are zeroed in a finite number of time steps. Particularly, if the model of the true

system is one of the models of setΩ, then, one can prove that, after zeroing the slack variables

of the integrating states, the solution of Problem 3–1 will lead the true plant to a steady-state

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where the inputs reach the targets and the outputs lie in their control zones. This statement is

corroborated by the following theorem:

Theorem 3.1For a time delay process with stable and integrating outputs that can be stabi-

lized at a desired steady-state, if at time stepk the optimal solution to Problem 3–1 results in

a slack vector corresponding to the true plant integrating states equal to zero, then, for the

undisturbed system, the solution of Problem 3–1 at any subsequent time stepk + j is feasible

with δi,k+j(ΘT ) = 0 and so it is possible to find a solution to Problem 3–1 at the subsequent

time steps that leads the system in closed loop to the desiredsteady-state.

Proof. The proof of this theorem can be obtained by following the same steps as in Chapter 2.

That is, one proves that the cost function of the true plant model is decreasing and converges to

zero when the desired equilibrium point is reachable, otherwise this cost function is non-zero

and the system will converge to a steady-state that lies at the minimum distance from the desired

equilibrium point.

3.3 Application of the controller to the partial combustion FCC converter

The aim of this section is to describe the main steps for applying the proposed controller to a

challenge case study: the converter of an industrial FCC unit. For the sake of clarity, these steps

are briefly described in the subsections that follow.

3.3.1 The control problem of the FCC systems

The fluidized-bed catalytic cracking unit is one of the most profitable process units and perhaps

the most complex and challenging operating process of any oil refinery. The major goal behind

the cracking process is to transform catalytically low-value raw materials, such as gasoil and/or

atmospheric residue, into lighter and more valuable products, of which the cracked naphtha

(gasoline) and liquefied petroleum gas are the commercial products of highest aggregate value

for the refinery considered here.

Mostly, this kind of industrial process exhibits complex dynamic behavior, strongly interact-

ing variables as well as economic and operating constraintson the process system inputs and

outputs. Because of these characteristics, it is rather unavoidable and justifiable to prefer the

employment of advanced control strategies rather than the conventional PID-based decentral-

ized control techniques, in order to extend the operating campaign, to preserve the mechanical

integrity and mainly to reach economic objectives. Among these strategies, model predictive

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control (MPC) appears to be the standard approach for controlling industrial FCC units.

It is well known that MPC explicitly incorporates a linear ornonlinear process model to

forecast the future behavior of the controlled outputs and to compute appropriate control ac-

tions through an optimization based formulation. In spite of the real model being nonlinear,

MPC based on linear step response models has been thoroughlystudied and implemented in

FCC process systems (see e.g. Grosdidieret al., 1993; Moro and Odloak, 1995; Yanget al.,

1996; Abou-Jeyab and Gupta, 2001; Jiaet al., 2003; De Souzaet al., 2010). FCC units may

be subject to frequent transitions in the operating conditions, caused by changes in the feed

quality or due to the pursuit of economic goals, or both, so that the use of conventional lin-

ear MPC may result in poor control performance. A better performance seems to be achieved

when one considers MPC strategies derived from first-principles nonlinear models (NMPC)

(Khandalekar and Riggs, 1995; Ali and Elnashaie, 1997; Harnischmacher and Marquardt, 2007;

Ansari and Tade, 2000; Romanet al., 2009).

The simulated applications of NMPC to FCC process systems show that the approach can

provide better results than the corresponding ones based onlinear models, but these available

solutions seem still far from the practical application stage, mainly when the following issues

are concerned: systematic methods to tune the controller parameters, and guarantee of stability

and convergence of NMPC coupled to the state estimator. Also, it is worth emphasizing that

some model formulations may lead to unacceptable computational burdens. In order to circum-

vent this drawback, it has been considered the application of empirical nonlinear models such

as artificial neural network (ANN) to represent the process system in the NMPC algorithm.

Several research works that focused on the application of this modeling approach to FCC con-

trol strategies have been reported in the literature, e.g. (Santoset al., 2000; Vieiraet al., 2005).

The major disadvantage of the NMPC based on ANN is that it requires large training data sets,

which is not usually available in the industrial environment. An excellent and detailed review

on the MPC approaches mentioned earlier concerning the FCC system is available in the paper

by Pinheiroet al. (2012).

Another class of advanced control algorithms that can deal with nonlinearities of the process

model is the robust model predictive control where a linear model with varying parameters is

considered to approximate the nonlinear process, and this control strategy has been the focus

of this thesis. In this sense, the controller proposed in thelast section will be applied to the

converter of an industrial FCC unit, which is described in next subsection.

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3.3.2 The control structure of the FCC unit

One of the purposes of this chapter is to study the application of the proposed robust MPC

at the FCC converter unit at the Cubatão (Brazil) refinery. A simplified schematic representa-

tion of this industrial unit is given in Figure 3.1, as well asthe main regulatory control loops

(PID controllers). Summarily, this process unit comprisesthe feed pre-heating furnace, the

reactor-regenerator system, the air blower and the main fractionating column. The feedstock

for this process unit is mainly gasoil, which is pre-heated in the furnace to a temperature of

about 230oC-300oC. A small amount of raw naphtha is also fed to the system. Then,both

gasoil and raw naphtha streams are fed to the riser where the cracking reactions take place in

the presence of hot catalyst that comes from the regenerator-vessel (RG), and a regulatory PID

controller of the riser outlet temperature adjusts the regenerated catalyst plug valve. The prod-

ucts of the cracking reaction are separated from the spent catalyst in the reactor-vessel (RX),

and afterwards they are sent to the main fractionating column with the purpose of separating

the products of the cracking process, such as fuel gas, liquefied petroleum gas (LPG), gasoline

(main product) from the heavier products such as light cycle-oil (LCO) and clarified oil (CLO).

At the bottom of the reactor RX the spent catalyst is held in a fluidized-bed with steam, which

aims to strip the hydrocarbons adsorbed on the catalyst particles. The level of catalyst in the

reactor is controlled through the manipulation of the spentcatalyst plug valve.

Simultaneously, one has the production of coke, which deposits on the catalyst surface,

resulting in its deactivation. For this reason, the spent catalyst must be continuously regenerated

in the regenerator vessel (RG) by burning the coke with air. The regenerator system considered

here operates in a partial-combustion mode (the conversionof coke to CO2 is not complete),

which implies that a large amount of CO is produced. The energyreleased from the coke

combustion is used to supply the required heat to the endothermic catalytic cracking reactions,

while the remaining energy is recovered in a heat recovery boiler to generate high-pressure

steam, where the CO-rich regenerator flue gas is burned.

In the present study, the proposed MPC algorithm focuses on the control of the main pro-

cessing section of this FCC unit, that is, the reactor-regenerator (converter) process system. For

this purpose, the control structure considered here is the same as the FCC unit of Petrobras at

Cubatão refinery, where the main controlled outputs are the following:

• y1 – boiler flue gas velocity;

• y2 – delta pressure on the spent catalyst valve;

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Figure 3.1: Schematic representation of the FCC converter.

• y3 – average temperature of the regenerator dense phase;

• y4 – average temperature of the regenerator dilute phase.

As highlighted in Figure 3.1, the set of manipulated inputs for the proposed MPC controller are

set-points to the following PID controllers:

• u1 – gasoil feed flow rate;

• u2 – temperature of the gasoil feed;

• u3 – temperature at the outlet of the cracking riser;

• u4 – raw naphtha feed flow rate;

• u5 – total air flow rate to the regenerator.

With the proposed4 × 5 control scheme, real plant test were performed and the system model

was obtained at the most probable operating condition, which is denoted here as the nominal

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model (NM). The transfer functions corresponding to NM are given in Table 3.1. Such a model

was found to best describe the dynamic behavior of the FCC at the nominal design conditions.

As far as the author knowledge, all the MPC controllers implemented in industrial FCC systems

are based on the nominal linear model. This means that the nonlinearities of the FCC system

are only moderate, at least at operating points near the design operating conditions, where the

nominal model is identified and fits the nonlinear system quite well. However, when the RTO

layer is present in the plant control structure, the operating point tends to be moved by RTO

towards the optimum economic points that depend on the feedstock composition and market

demands. In this case, although MPC based on the nominal model can still keep the system

under control, the performance of the closed-loop system may deteriorate when compared to

the design performance.

Table 3.1: Process model corresponding to the nominal operating condition.u1 u2 u3 u4 u5

y13× 10−4

s

−4.43× 10−3

s

2.23× 10−2

s

1.05× 10−4

s

−4.89× 10−4

s

y2(3.9s− 0.6)× 10−3

300s2 + 40s+ 1

5× 10−3(−7s+ 1)

300s2 + 40s+ 1

−3.3× 10−2

22s+ 1

−3.8× 10−4

14s+ 1

5.7× 10−4

48s+ 1

y3−2.43× 10−3

s

9× 10−3e−10s

s

−4.92× 10−2

s

−1.9× 10−3

s

1.2× 10−2e−10s

s

y4−2.29× 10−3

s

3.32× 10−2

s

−0.23

s

−2.23× 10−3

s

3.78× 10−3

s

Since the purpose here is to evaluate the control algorithm at real operating scenarios, two

additional models were considered in the design of the robustly stable multi-model predictive

controller. These additional models will represent the lower and upper bounds associated with

the nominal model uncertainty, and they are denoted here as M1 and M2, whose transfer func-

tion matrices are shown in Table 3.2 and Table 3.3, respectively.

From the models presented above, it is clear that the partialcombustion FCC converter shows

an integrating character with respect to all manipulated inputs. Only outputy2, has stable poles

with fast and slow dynamics that appear together. Besides theintegrating nature, this cracking

process also presents some time delays between the process inputs and outputs, which can affect

the performance of the controller. The integrating behavior of the FCC converter when the set-

point to the riser temperature is manipulated will happen only when the carbon present in the

coke deposited on the catalyst is partially converted to carbon dioxide (CO2), and the remaining

is converted to carbon monoxide (CO). A distinct behavior is observed when the FCC converter

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Table 3.2: Process model corresponding to the operating condition M1.u1 u2 u3 u4 u5

y13.6× 10−4

s

−5.20× 10−3

s

2.80× 10−2

s

1.26× 10−4

s

−5.87× 10−4

s

y2(4.9s− 0.76)× 10−3

363s2 + 44s+ 1

(−44s+ 6.3)× 10−3

363s2 + 44s+ 1

−3.96× 10−2

24.2s+ 1

−4.6× 10−4

15.4s+ 1

6.8× 10−4

48s+ 1

y3−2.98× 10−3

s

1.1× 10−2e−12s

s

−5.90× 10−2

s

−2.3× 10−3

s

1.4× 10−2e−8s

s

y4−2.75× 10−3

s

3.98× 10−2

s

−0.27

s

−2.8× 10−3

s

4.54× 10−3

s

Table 3.3: Process model corresponding to the operating condition M2.u1 u2 u3 u4 u5

y12.4× 10−4

s

−3.46× 10−3

s

1.86× 10−2

s

8.4× 10−5

s

−3.91× 10−4

s

y2(3.0s− 0.46)× 10−3

243s2 + 36s+ 1

(−27s+ 3.8)× 10−3

243s2 + 36s+ 1

−2.64× 10−2

19.8s+ 1

−3.04× 10−4

12.6s+ 1

4.56× 10−4

43.2s+ 1

y3−1.94× 10−3

s

7.2× 10−3e−9s

s

−3.94× 10−2

s

−1.5× 10−3

s

9.6× 10−3e−11s

s

y4−1.83× 10−3

s

2.66× 10−2

s

−0.18

s

−1.8× 10−3

s

3.02× 10−3

s

is operated in a total combustion mode (Grosdidieret al., 1993; Yanget al., 1996) or instead

of the riser temperature, the manipulated input is the opening of the catalyst valve. In these

cases the FCC system is stable in open-loop. However, to the author knowledge, the control of

partial combustion FCC has not yet been addressed in literature. As the partial combustion FCC

converter studied in the chapter is an open-loop unstable (integrating) system, while the total

combustion FCC is open-loop stable, the control of the partial combustion converter is more

difficult than that corresponding to the total combustion converter. This difficulty first appears

in the identification of the linear model on which the controller is based. The identification pro-

cedure based on the step tests, which is successfully applied in most open-loop stable systems,

cannot be applied in this case, because the integrating outputs of the partial combustion FCC

would tend to surpass their operating ranges and would jeopardize the integrity of the system.

Furthermore, the poor excitation of the system usually reflects in uncertainty of the linear model

parameters and may also result in a poor performance of the controller. For these reasons, there

is interest in studying the application of a robust MPC strategy in such a system, particularly

when there is severe uncertainties in the process model.

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3.3.3 Simulation results

Here, it is presented the application results of the controller proposed in this chapter to the par-

tial combustion FCC converter described in the previous subsection. The three models identified

from plant tests (listed in tables 3.1, 3.2 and 3.3 constitute the multi-plant setΩ = Θ1,Θ2,Θ3

on which the robust MPC controller is based. In addition, forthe simulated operation scenario

considered here, one takes into account the operation practices of the experienced plant oper-

ators with respect to the bounds of the process variables, which they deem adequate for the

smooth and safe operation of the system. The suggested bounds for the manipulated and con-

trolled variables, which were considered in all the scenarios presented in this study, are shown

in Table 3.4 and Table 3.5, respectively. Moreover, a sampling time (∆t) equal to 1 min was

adopted.

Table 3.4: Operational bounds for the manipulated inputs ofthe FCC process system.Inputs umin umax ∆umax Unitu1 5000 9000 50 m3/du2 170 350 3 Cu3 520 600 2 Cu4 400 2500 50 m3/du5 3500 5000 30 kNm3/d

Table 3.5: Operational bounds for the controlled outputs ofthe FCC process system.Outputs ymin ymax Unity1 0 30 m/sy2 0.4 1.1 kgf/cm2

y3 630 725 Cy4 640 730 C

The performance of the robust MPC proposed here is compared to a simplified version of

the conventional MPC controller that is already implemented in the real FCC process. This

conventional MPC is based on a finite prediction horizon (Hp) and it relies upon only one

model (nominal) to predict the process system behavior. In the existing MPC the controlled

variables are also controlled within zones, but the zone control strategy is implemented in a

different way (Zaninet al., 2002) in which penalization of the future behavior of the outputs in

the cost function of the conventional MPC will only occur when their predictions lie outside

the zones. The deviations of the inputs from their targets are also penalized along the control

horizon as in the proposed approach. The conventional controller also includes the typical input

constraints, represented here by equation (3.10). Then, the control law of the conventional MPC

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results from the solution to the following QP:

Problem3–2

min∆uk

V2,k,

V2,k =

Hp∑

j=0

∥∥∥y(k + j|k)− ysp

∥∥∥

2

Qy

+m−1∑

j=0

∥∥∥∆u(k + j|k)

∥∥∥

2

R+

m−1∑

j=0

∥∥∥u(k + j|k)− udes,k

∥∥∥

2

Qu

,

subject to (3.10).

In the simulations presented here, modelΘ1 is considered to be the nominal model, which

is used in the control cost that is minimized in both controllers. The model that represents the

true process system is unknown, but one assumes that it is represented as

ΘT =FT ,B

sl,T ,B

dl,T ,B

il,T , γi,j,T

, l = 1, . . . , p,

where the pair of gain matrices(Bs

l,T ,Bil,T

)lies in the polytope which is the convex hull defined

byL vertices (total number of elements on the setΩ), i.e.

(Bs

l,T ,Bil,T

)=

L∑

n=1

λn

(Bs

l,T ,Bil,T

), and

L∑

n=1

λn = 1, λn ≥ 0;

whereas the matrices related to the dynamic part of the process system may assume any possible

dynamic of the plants that belong to setΩ, or

(FT ,B

dl,T , γi,j,T

)=

(Fn,B

dl,n, γi,j,n

), for any n = 1, . . . , L.

Also, with the purpose of improving the numerical conditioning of the control optimization

problem, the manipulated inputs, the controlled outputs, as well as the prediction models, are

scaled according to the following expressions:

yscaledi =yi − ymin

i

ymaxi − ymin

i

, i = 1, . . . , ny,

uscaledj =

uj − uminj

umaxj − umin

j

, j = 1, . . . , nu.

The scaling parameters are those related with the operational bounds of the process variables,

whose values are those listed in tables 3.4 and 3.5.

The tuning parameters of the existing MPC through Problem 3–2 are the following:m = 5,

Qy = diag(40, 10, 10, 10), Qu = diag(200, 400, 500, 400, 0), R = diag(20, 30, 8, 5, 10), which

are also used for the application of the proposed robust MPC, and the output prediction horizon

is Hp = 180. These parameters were obtained through a trial and error procedure in the real

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system. To provide a fair comparison between the existing conventional MPC and the proposed

robust MPC, the values ofm, Qy, Qu andR for the robust MPC were kept the same as in

the conventional MPC. The output prediction horizonHp is infinite for the robust controller.

The remaining tuning parameters of the proposed MPC, are:Sy = diag(105, 105, 105, 105),

Si = diag(108, 108, 108, 108) andSu = diag(106, 106, 106, 106, 0).

In general, the existing practical tuning rules for selecting parametersm, Qy, Qu andR

of the conventional finite horizon MPC can also be adopted forthe tuning of the robust MPC

considered here. The remaining tuning parameters of the robust MPC are the slack weights

Sy, Si andSu that, as discussed in the previous section, should be several orders of magnitude

larger than the output weightQy and input weightsQu andR. If this condition is satisfied and

the desired optimum steady-state is reachable, then, it canbe proved that the robust controller

will drive the closed-loop system to the desired steady-state without offset.

Even though there are several disturbances that can affect this kind of industrial process

system, it is considered here only two scenarios that are typical of the operation of the FCC

unit. Namely:

Case 1: Regulator operation.Initially, the process starts from the steady-state definedby

u(0) =[8240 m3/d 267 C 537 C, 1404 m3/d 4592 kNm3/d

]⊤and

y(0) =[

24.9 m/s 0.69 kgf/cm2 713 C 714 C]⊤

,

where the outputs are outside the control zones that are specified in Table 3.6, on which it can

be observed that the velocity of flue gas in the boiler is controlled at a fixed set-point (24 m/s),

turning out to be a target in the control scheme. In this setting, the controllers should drive

the FCC process to a new equilibrium point where the other controlled outputs are inside the

current output zones and the following manipulated inputs should converge to the optimization

targetsu1,des = 7930 m3/d, u2,des = 225.4 C, u3,des = 532 C andu4,des = 1573 m3/d, which

are defined by the RTO layer of the FCC unit; notice that the targeted variables start from initial

values that are not at their desired goals. It is also noteworthy that the number of economic

targets to be pursued is exactly equal to the total number of available degrees of freedom in

this process system (number of manipulated inputs), as would be expected when there is a RTO

layer in the control structure. In this simulated case, the gains of the true process model result

from a convex combination defined by the weighting elementsλ =[0.2 0.5 0.3

], and its

assumed dynamics is the one corresponding to the process model M1.

The simulated responses of the controlled outputs and manipulated inputs are plotted in

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Table 3.6: Output zones of the FCC process system.Outputs ymin ymax Unity1 24 24 m/sy2 0.75 1.0 kgf/cm2

y3 690 710 Cy4 690 710 C

figures 3.2 and 3.3, respectively. It can be seen from these figures that the responses of all

process variables are clearly different for the two controllers, and the robust MPC is more

aggressive and has a better performance than the conventional MPC. All the controlled outputs

are brought to the respective control zones and/or target bythe robust MPC more rapidly when

compared with the conventional MPC. Moreover, despite both controllers behave nicely to bring

the targeted inputs to their desired economic values, the proposed robust MPC can accomplish

this task more efficiently because the conventional MPC takes a longer time to achieve the

economic goals. This good performance conducted by the robust MPC will certainly reflect

in a more efficient operation of the FCC system if the controller is integrated with real time

optimization algorithms of the plant.

Case 2: Zone and target tracking operation.Another typical disturbance to the FCC system

is the re-definition either of the output zones or the economic targets, or both, imposed mainly by

market demand that enforces the plant optimizer to produce different optimum points that must

be tracked by the FCC controller. To simulate this scenario, after 200 time steps, a change in the

control zones and optimum targets is set to the closed-loop system. In addition, the gains of the

true process model are switched to a new model combination whose weights are now specified

by λ =[0.3 0.2 0.5

], and the dynamic part of the plant is assumed to correspond tothe

process model M2. The new upper and lower limits for the controlled outputs are presented in

Table 3.7, including the new target fory1 (22 m/s). New economic targets associated with inputs

u1,des, u2,des, u3,des andu4,des correspond to the following values 8486 m3/d, 295.4C, 540C

and 1308 m3/d, respectively. The definition of these new values of targets and control zones

matches to the usual practice adopted by the process operators. In this sense, the manipulated

inputs u5 plays the role of compensating the effect of the disturbances from this simulated

scenario.

Figures 3.4 and 3.5 illustrate the performance of the robustand conventional MPC for this

tracking operation. In this case, the robust MPC also responds more aggressively than the con-

ventional MPC and is able the reach the desired goals much faster than the existing MPC. As

shown in Figure 3.4, the controllers behave differently from each other to bring the controlled

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85

Figure 3.2: Controlled outputs of the FCC process system in Case 1. Nominal MPC (−−), robust MPC (—),upper lower limits (− · −).

Table 3.7: Modified Output zones of the FCC process system.Outputs ymin ymax Unity1 22 22 m/sy2 0.4 0.7 kgf/cm2

y3 660 690 Cy4 670 680 C

outputs to the new control zones. More specifically, the robust MPC has a superior perfor-

mance since it can stabilize the outputs within the control zones in a much faster pace than in

the conventional MPC. In accordance with Figure 3.5, the robust controller is also capable of

stabilizing the targeted variables in their desired steady-states no later than two and half hours,

while the conventional controller would take practically all the time period of the simulation to

accommodate them at this new point equilibrium. With regardto the non-targeted input (u5),

the conventional controller shows a more sluggish responsewith large overshoots before con-

verging to the required steady-state, and contrarily to this poor behavior, the response of the

robust MPC shows a much smoother and faster trajectory.

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Figure 3.3: Manipulated inputs of the FCC process system in Case 1. Nominal MPC (−−), robust MPC (—),optimum targets (− · −).

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Figure 3.4: Controlled outputs of the FCC process system in Case 2. Nominal MPC (−−), robust MPC (—),upper lower limits (− · −).

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Figure 3.5: Manipulated inputs of the FCC process system in Case 2. Nominal MPC (−−), robust MPC (—),optimum targets (− · −).

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When analyzing the behavior of the robust MPC cost function associated with the true plant

model, shown in Figure 3.6, one can note that for both the simulated scenarios it is not strictly

decreasing, as would be expected. Indeed, such a decrease inthis cost function can be observed

only after twenty-four time steps in the regulatory operation and after three time steps in the

tracking operation. This is justified as the conditions required by Theorem 3.1 are not satisfied,

that is, since at the beginning of each simulation the components of the slack vector related with

the integrating states of the system are still not zeroed, thus there is no guarantee that this cost

function interprets the role of a Lyapunov-like function, as established in Theorem 3.1.

Figure 3.6: Control cost of the robust MPC for both simulatedcases of the FCC process system.

From the preceding simulation results, it is possible to conclude that, although the exist-

ing conventional MPC holds an acceptable control performance for the simulated scenarios,

the proposed robust MPC has a better effectiveness for handling very disturbed and uncertain

scenarios without jeopardizing the converter stable operation. In this way, the proposed robust

control strategy becomes a more reliable tool to reach the economic objectives of the industrial

FCC unit than the existing conventional MPC.

3.4 Conclusion

In this chapter, a robustly stable model predictive controller was described. It extends the con-

troller previously presented in Chapter 2 to the more generalcase of integrating systems. To

this end, the general model formulation proposed in Santoro(2011), which is also based on the

analytical step response model, was used in development of the controller. Mostly, this model

formulation allows to synthesize a one-step formulation based proposed robust MPC. Further-

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90

more, following the approach of the industrial application, the controller proposed here has been

formulated in such a way that inputs and/or some outputs of this process system are guided to

their corresponding optimum targets while the remaining controlled outputs are maintained in-

side specified zones.

In order to evaluate the performance of the proposed controller, it was applied to a key section

of the FCC unit of a Brazilian oil refinery: the regenerator–reactor system (converter), which

operates in the partial combustion mode, thus resulting in integrating behavior of the majority

of the controlled outputs with respect to all manipulated inputs. The robust MPC considered

here was compared to the advanced control system existing inreal process system (conven-

tional MPC) and so, from the simulation results, one concludes that the robust control strategy

provides a reasonably better alternative than the conventional controller for the case in which

there is plant/model mismatch and the system is very disturbed. Since the conventional MPC

can provide a poor control performance in some realistic scenarios of the FCC plant operation,

the robust controller seems be a good and competitive candidate to future implementations in

the real process system.

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CHAPTER 4

One-step formulation of robust MPC forunstable time delay processes

4.1 Introduction

Open-loop unstable time delay processes, such as heat and mass exchange networks, recycle

and exothermic reaction, are commonly found in the process industry. In particular, when an

unstable reactor system, as a CSTR system, has to be controlled, a greatly desired property of

the controller is the guarantee of stability of the closed-loop system. Furthermore, within an

economic standpoint, it is well known that CSTR schemes must often operate at open-loop un-

stable operating regions in order to maximize their profitability (Biagiola and Figueroa, 2004).

On this matter, more sophisticated control strategies should then be designed to drive in a stable

pathway such reactor systems from one unstable to another more profitable steady-state. In

this sense, strategies as robust model predictive control seem fairly appealing and justifiable to

successfully pursue both operational and economic goals ofunstable reactor systems.

On the other hand, as discussed in Chapter 1, some problems in the synthesis of robustly

stabilizing MPC for unstable time delay processes remain open. In this context, this chapter

addresses a solution to the problem of robust MPC of time delay processes with stable and un-

stable modes. The major features of the proposed method are those that aim to overcome many

disadvantages of the approaches available in the literature, namely: (i) the model uncertainty is

incorporated into the controller formulation and is one based upon the multi-plant description

instead of a norm-bounded exogenous disturbance. Then, it is appealing as it can accommodate

91

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92

the model nonlinearities of the process system; (ii) the controller design achieves offset-free

control law through a one-step formulation by considering astate-space model in incremental

form, which means elimination of the target calculation layer (other optimization problem) as-

sociated with the dynamic layer of the controller; (iii) theclosed-loop Lyapunov stability within

the uncertainty description is easily ensured by includingan appropriate set of slacked terminal

constraints into the optimization problem. This avoids non-trivial and off-line calculations of

parameters from the dual-mode MPC formulations.

This chapter has been divided into five parts. The first part details the analytical step response

based state-space model to represent stable and unstable time delay processes. The second

part deals with the development of a stabilizing infinite horizon MPC algorithm in terms of

the proposed model formulation. Then, this nominally stable MPC controller is extended to

the robust case considering the multi-plant model uncertainty, and its closed-loop stability is

analyzed in the third part. In the fourth part, the proposed controllers are applied to simulation

studies of an unstable reactor system, whereas the fifth partpresents the concluding remarks of

the chapter.

4.2 Model description

In a similar approach to the state-space models proposed in chapters 2 and 3, which have their

origin in an analytical form of the step response associatedwith the transfer functions of the

process system, it is presented herein a novel state-space model formulation of stable and un-

stable time delay processes. For this end, let there be a MIMOlinear system withny outputs

andnu inputs, and a transfer function exists for each pair (yi, uj), or

y1(s)...

yi(s)...

yny(s)

=

G1,1(s) · · · G1,j(s) · · · G1,nu(s)...

.. ....

. .....

Gi,1(s) · · · Gi,j(s) · · · Gi,nu(s)...

.. ....

. .....

Gny,1(s) · · · Gny,j(s) · · · Gny,nu(s)

u1(s)...

uj(s)...

unu(s)

. (4.1)

Also, assume that this system has non-repeated stable and unstable poles, as well as time delays

between the outputs and inputs. Then, the transfer functionof an outputyi with respect to an

inputuj can be expressed generically as:

Gi,j(s) =bi,j,0 + bi,j,1s+ . . .+ bi,j,nbs

nb

(s− rsti,j,1) . . . (s− rsti,j,na)(s− runi,j,1) . . . (s− runi,j,nc)e−γi,j ·s, (4.2)

wherena, nb, nc ∈ N|nb < (na+ nc) andrsti,j,1, . . . , rsti,j,na are the distinct stable poles of the

system, whileruni,j,1, . . . , runi,j,nc are the distinct unstable poles of the system. The step response

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93

of Gi,j(s), denoted here asSi,j(s), can be obtained in partial fraction as follows:

Si,j(s) =d0i,js

+na∑

l=1

dsti,j,ls− rsti,j,l

e−γi,js +nc∑

l=1

duni,j,ls− runi,j,l

e−γi,js. (4.3)

Let ∆t be the sampling time, hence (4.3) shall be written in discrete time by the following

expression:

Si,j(k∆t) = d0i,j +na∑

l=1

dsti,j,lersti,j,l

(k∆t−γi,j) +nc∑

l=1

duni,j,leruni,j,l

(k∆t−γi,j). (4.4)

From (4.4) it is then possible to obtain an equivalent discrete time state-space model in incre-

mental form of the inputs given by

xs(k + 1)xst(k + 1)xun(k + 1)z1(k + 1)z2(k + 1)

...zp(k + 1)

︸ ︷︷ ︸

x(k+1)

=

Iny 0 0 Bs1 Bs

2 · · · Bsp−1 Bs

p

0 Fst 0 Bst1 Bst

2 · · · Bstp−1 Bst

p

0 0 Fun Bun1 Bun

2 · · · Bunp−1 Bun

p

0 0 0 0 0 · · · 0 0

0 0 0 Inu 0 · · · 0 0...

......

......

. .....

...0 0 0 0 0 · · · Inu 0

︸ ︷︷ ︸

A

xs(k)xst(k)xun(k)z1(k)z2(k)

...zp(k)

︸ ︷︷ ︸

x(k)

+

Bs0

Bst0

Bun0

Inu0...0

︸ ︷︷ ︸

B

∆u(k),

y(k) =[Iny Ψst Ψun 0 0 · · · 0

]

︸ ︷︷ ︸

C

x(k), (4.5)

where:

x(k) =[xs(k)⊤ xst(k)⊤ xun(k)⊤ z1(k)

⊤ · · · zp(k)⊤]⊤

,

x ∈ Cnx, xs ∈ R

ny, xst ∈ Cnst, xnun ∈ C

nun, z1 · · · zp ∈ Rnu, p = max

i,jγi,j,

nst = ny.nu.max(na), nun = ny.nu.max(nc), nx = ny + nst+ nun+ p.nu,

Fst = diag(e∆t·rst

1,1,1 · · · e∆t·rst1,1,na · · · e∆t·rstny,nu,1 · · · e∆t·rstny,nu,na

)∈ C

nst×nst,

Fun = diag(e∆t·run

1,1,1 · · · e∆t·run1,1,na · · · e∆t·runny,nu,1 · · · e∆t·runny,nu,na

)∈ C

nun×nun,

Ψst =

nu.na︷ ︸︸ ︷

1 1 · · · 10 0 · · · 0...

.... . .

...0 0 · · · 0

· · ·

nu.na︷ ︸︸ ︷

0 0 · · · 00 0 · · · 0...

.... . .

...1 1 · · · 1

∈ Rny×nst,

Ψun =

nu.nc︷ ︸︸ ︷

1 1 · · · 10 0 · · · 0...

.... . .

...0 0 · · · 0

· · ·

nu.nc︷ ︸︸ ︷

0 0 · · · 00 0 · · · 0...

.... . .

...1 1 · · · 1

∈ Rny×nun.

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94

In the model formulation proposed here,xs(k) is related with the artificial integrating modes

produced by incremental form of the inputs,xst(k) andxun(k) correspond to the stable and

unstable states of the system, respectively. Each component zl(k) represents the past input

move delayed by the time periodl, i.e. zl(k) = ∆u(k − l), l = 0, . . . , p. Matrix Bstl , with

l = 0, . . . , p is filled according to the following rule:

• If l = γi,j, then[Bsl ]i,j = d0i,j, else,[Bs

l ]i,j = 0.

Conversely, the construction of matricesBstl andBun

l require a more careful attention. For

instance, in the absence of time delays (l = 0), these matrices would be obtained byBst0 =

DstFstNst andBun0 = DunFunNun, respectively, in which:

Dst = diag(dst1,1,1 · · · d

st1,1,na · · · d

stny,nu,1 · · · d

stny,nu,na

)∈ C

nst×nst,

Nst =

Jst

Jst

...Jst

ny

∈ R

nst×nu, Jst =

na

1 0 · · · 0...

.... ..

...1 0 · · · 0

na

0 1 · · · 0...

.... ..

...0 1 · · · 0

...

na

0 0 · · · 1...

.... ..

...0 0 · · · 1

∈ Rnu.na×nu,

Dun = diag(dun1,1,1 · · · d

un1,1,nc · · · d

unny,nu,1 · · · d

unny,nu,nc

)∈ C

nun×nun,

Nun =

Jun

Jun

...Jun

ny

∈ R

nun×nu, Jun =

nc

1 0 · · · 0...

.... . .

...1 0 · · · 0

nc

0 1 · · · 0...

.... . .

...0 1 · · · 0

...

nc

0 0 · · · 1...

.... . .

...0 0 · · · 1

∈ Rnu.nc×nu.

However, in case ofl 6= 0, Bstl andBun

l would have respectively the same dimension asBst0 =

DstFstNst andBun0 = DunFunNun, whose elements corresponding to the transfer functions

with time delay different froml are then replaced with zeros.

Remark 4.1To synthesize the nominal and robust IHMPC controllers in the framework of the

state-space model proposed earlier, one needs to assure that this model formulation is stabiliz-

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95

able and detectable. On applying both the Popov-Belevitch-Hautus tests for stabilizability and

detectability (Hespanha, 2009), it can be easily verified that the ranks of the matrices

[A− λI B

]and

[A− λI

C

]

, λ ∈ R, |λ| ≥ 0,

are equal tonx, which confirm the required properties of the model.

4.3 The formulation of the nominal MPC

The proposed infinite horizon MPC of stable and unstable timedelay processes considering the

model defined in last section is one based on the following open-loop quadratic cost function:

V1,k =∞∑

j=0

∥∥∥y(k + j|k)− ysp,k − δy,k −Ψun(Fun)(j−m−p)

δun,k

∥∥∥

2

Qy

+∞∑

j=0

∥∥∥u(k + j|k)− udes,k − δu,k

∥∥∥

2

Qu

+m−1∑

j=0

∥∥∥∆u(k + j|k)

∥∥∥

2

R(4.6)

+∥∥∥δy,k

∥∥∥

2

Sy

+∥∥∥δu,k

∥∥∥

2

Su

+∥∥∥δun,k

∥∥∥

2

Sun

,

where:Qy ∈ Rny×ny is a positive definite weighting matrix of the controlled outputs;Qu ∈

Rnu×nu andR ∈ R

nu×nu are positive semi-definite weighting matrices of the inputsand input

moves, respectively;y(k + j|k) is the output prediction vector at time stepk + j computed

at time stepk, taking into account the effects of the future control actions;ysp,k is the output

reference vector (desired set-point) that needs to be computed as additional decision variables of

the control problem;∆u(k + j|k) is the vector of the input moves, in which∆u(k + j|k) = 0

for j ≥ m; udes,k is the vector of input targets;δy,k ∈ Rny, δu,k ∈ R

nu andδun,k ∈ Rnun

are slack variables introduced in the control problem so that it has always a feasible solution;

Sy ∈ Rny×ny, Su ∈ R

nu×nu andSun ∈ Rnun×nun are positive definite weighting matrices

corresponding to those slacks, and should be selected such that the controller tends to reduce

the slacks to zero or at least minimize them in the least square sense.

Considering the control cost defined in (4.6), each term that contains an infinite sum can be

divided into two parts for ease of interpretation. Then, forthe time-delayed system, the first

and third terms on the right hand side of (4.6) are respectively split into a finite sum that ranges

from time stepj = 0 to j = m+ p (j = 0 to j = m− 1) and another sum that ranges from time

stepj = m + p + 1 to ∞ (j = m to ∞). This is so done because beyond the time stepm the

control moves are assumed to be equal to zero and the infinite sums can be simplified. Then,

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96

the proposed control cost can be written as follows:

V1,k =

m+p∑

j=0

∥∥∥y(k + j|k)− ysp,k − δy,k −Ψun(Fun)(j−m−p)

δun,k

∥∥∥

2

Qy

+∞∑

j=1

∥∥∥y(k +m+ p+ j|k)− ysp,k − δy,k −Ψun(Fun)jδun,k

∥∥∥

2

Qy

+m−1∑

j=0

∥∥∥u(k + j|k)− udes,k − δu,k

∥∥∥

2

Qu

+m−1∑

j=0

∥∥∥∆u(k + j|k)

∥∥∥

2

R(4.7)

+∞∑

j=0

∥∥∥u(k +m+ j|k)− udes,k − δu,k

∥∥∥

2

Qu

+∥∥∥δy,k

∥∥∥

2

Sy

+∥∥∥δu,k

∥∥∥

2

Su

+∥∥∥δun,k

∥∥∥

2

Sun

.

From the state-space model defined in (4.5), one has

y(k +m+ p+ j|k) = Cx(k +m+ p+ j|k)

= xs(k +m+ p|k) +Ψst(Fst)jxst(k +m+ p|k) + (4.8)

Ψun(Fun)jxun(k +m+ p|k).

Then, due to the presence of the artificial integrating stateand the unstable state components in

the first infinite sum included in (4.6), as well as the assumption that the input values computed

beyond the control horizon are constant in the other infinitesum of (4.6), the cost function

will be unbounded unless the following terminal equality constraints are included in the control

optimization problem:

xs(k +m+ p|k)− ysp,k − δy,k = 0, (4.9)

xun(k +m+ p|k)− δun,k = 0, (4.10)

u(k +m− 1|k)− udes,k − δu,k = 0. (4.11)

The above equations can be written in terms of the input movesas follows:

Ns

(

Am+px(k)−ApW∆uk

)

− ysp,k − δy,k = 0, (4.12)

Nun

(

Am+px(k)−ApW∆uk

)

− δun,k = 0, (4.13)

I⊤nu∆uk + u(k − 1)− udes,k − δu,k = 0, (4.14)

where:

Ns =

[

Iny 0ny×nst 0ny×nun

p︷ ︸︸ ︷

0ny×nu · · · 0ny×nu

]

,

Nun =

[

0nun×ny 0nun×nst Inun

p︷ ︸︸ ︷

0nun×nu · · · 0nun×nu

]

,

W =[Am−1B Am−2B · · · AB B

],

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97

∆uk =[∆u(k|k)⊤ · · · ∆u(k +m− 1|k)⊤

]⊤, I⊤nu =

[Inu · · · Inu

].

Once the restrictive equations (4.12) to (4.14) are imposedto the control problem, the cost

function can now be expressed as follows:

V1,k =

m+p∑

j=0

∥∥∥y(k + j|k)− ysp,k − δy,k −Ψun(Fun)(j−m−p)

δun,k

∥∥∥

2

Qy

+m−1∑

j=0

∥∥∥u(k + j|k)− udes,k − δu,k

∥∥∥

2

Qu

+m−1∑

j=0

∥∥∥∆u(k + j|k)

∥∥∥

2

R(4.15)

+∥∥∥x

st(k +m+ p|k)∥∥∥

2

Q+∥∥∥δy,k

∥∥∥

2

Sy

+∥∥∥δu,k

∥∥∥

2

Su

+∥∥∥δun,k

∥∥∥

2

Sun

,

in which the weighting terminal matrixQ is determined from the solution to the Lyapunov

equation of the system, or

Q = (Fst)⊤(Ψst)⊤QyFstΨst + (Fst)⊤QFst.

Finally, the control law from the IHMPC proposed here can be formulated as the solution to

the following optimization problem:

Problem4–1

min∆uk,ysp,k,δy,k,δu,k,δun,k

V1,k,

subject to (4.12), (4.13), (4.14) and

∆u(k + j|k) ∈ U, j = 0, . . . ,m− 1. (4.16)

U =

−∆umax ≤ ∆u(k + j|k) ≤ ∆umax

∆u(k + j|k) = 0, j ≥ m

umin ≤ u(k − 1) +∑j

i=0 ∆u(k + i|k) ≤ umax

, (4.17)

ymin ≤ ysp,k ≤ ymax. (4.18)

Remark 4.2The MPC control law based on Problem 4–1 may yield offset if the values associ-

ated with weighting matricesSy, Su andSun are orders of magnitude close to ones ofQy, Qu

andR. As already discussed in Chapter 3, the values adopted in practice for these matrices are

sufficiently large so as to guarantee that slack vectors are non-zero only if the corresponding

terminal constraints are violated. More specifically, if the original, hard-constrained solution

is feasible, then, one would like that the slacked optimization problem provides the same control

action.

Remark 4.3 Problem 4–1 will produce a control law that forces the cost function to decrease

asymptotically only if the slack vector of the unstable states is zeroed. Then, the convergence

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98

and nominal stability of the proposed IHMPC is only obtainedafter the zeroing the prediction of

the unstable states at the end of the control horizon extended to the maximum time delay. From

this observation, one can propose an alternative formulation to Problem 4–1 by appending

to the control problem a convex contracting constraint of the slack vector associated with the

unstable states, i.e.∥∥∥δun,k

∥∥∥

2

Sun

≤∥∥∥δun,k

∥∥∥

2

Sun

, (4.19)

in which the slack vectorδun,k is such that

Nun

(

Am+px(k)−ApW∆uk

)

− δun,k = 0,

and the control sequence is computed from the following solution:

∆uk =[∆u∗(k|k − 1)⊤ · · · ∆u∗(k +m− 2|k − 1)⊤ 0⊤

]⊤.

It is clear that once the slack vector of the unstable states converges to zero, Problem 4–1

reduces to the optimization problem defined below, which provides a stabilizing IHMPC control

law:

Problem4–2

min∆uk,ysp,k,δy,k,δu,k

V2,k,

V2,k =

m+p∑

j=0

∥∥∥y(k + j|k)− ysp,k − δy,k

∥∥∥

2

Qy

+∥∥∥x

st(k +m+ p|k)∥∥∥

2

Q

+m−1∑

j=0

∥∥∥u(k + j|k)− udes,k − δu,k

∥∥∥

2

Qu

+m−1∑

j=0

∥∥∥∆u(k + j|k)

∥∥∥

2

R

+∥∥∥δy,k

∥∥∥

2

Sy

+∥∥∥δu,k

∥∥∥

2

Su

,

subject to (4.12), (4.14), (4.16), (4.17), (4.18) and

Nun

(

Am+px(k)−ApW∆uk

)

= 0. (4.20)

The convergence of the closed-loop system with the infinite horizon MPC represented through

Problem 4–2 can be summarized in the following theorem:

Theorem 4.1Consider an undisturbed time delay process system with stableand unstable poles

and with a stabilizable pair(A,B) of state matrices. Assume that the control horizon is such

that m ≥ nun + 1, and at time stepk Problem 4–2 is feasible. Then, the control actions

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99

obtained from the solution to Problem 4–2 at successive timesteps drives the process system

asymptotically to its desired steady-state, if the input optimizing targets and output zones are

reachable, otherwise, the process system will converge to an equilibrium point (steady-state)

lying at a minimum distance from the desired input targets and output zones.

Proof. The proof follows similar steps as in Muske and Rawlings (1993). Consider that, at

time stepk, the optimal solution to Problem 4–2 is designated as(

∆u∗k,y

∗sp,k, δ

∗y,k, δ

∗u,k

)

, and

at time stepk + 1 a solution inherited from the previous time step is used to show that, for

the undisturbed system, the control cost function is non-increasing. To follow this approach,

consider that this inherited solution is defined as(

∆uk+1, ysp,k+1, δy,k+1, δu,k+1

)

, which is

computed as follows:

∆uk+1 =[∆u∗(k + 1|k)⊤ · · · ∆u∗(k +m− 1|k)⊤ 0⊤

]⊤,

ysp,k+1 = y∗sp,k, δy,k+1 = δ

∗y,k, δu,k+1 = δ

∗u,k.

The next step is then to verify if this inherited solution satisfies all constraints of Problem 4–2.

For instance, for constraint (4.12), which is equivalent to(4.9), one has

xs(k +m+ p+ 1|k + 1)− ysp,k+1 − δy,k+1 = xs(k +m+ p+ 1|k + 1)− y∗sp,k − δ

y,k.

Since the first optimal control move is applied in process at time stepk, the initial state at time

k+ 1, for an undisturbed system, is given byx(k+ 1|k+ 1) = x(k+ 1|k), and so, this implies

that

xs(k +m+ p+ 1|k + 1)− ysp,k+1 − δy,k+1 = xs(k +m+ p+ 1|k + 1)− y∗sp,k − δ

y,k

= xs(k +m+ p|k)− y∗sp,k − δ

y,k.

Analogously, for constraint (4.13), one concludes that

Nun

(

Am+px(k + 1|k + 1)−ApW∆uk+1

)

= Nun

(

Am+p[Ax(k) +B∆u∗(k|k)

]−ApW∆uk+1

)

= NunA(

Am+px(k)−ApW∆u∗

k

)

= 0.

Also, one can verify that constraint (4.14) is satisfied by the proposed solution as follows:

I⊤nu∆uk+1 + u∗(k|k)− udes,k − δu,k+1 = I⊤nu∆uk+1 +∆u∗(k|k) + u(k − 1)− udes,k − δ∗u,k

= I⊤nu∆u∗k + u(k − 1)− udes,k − δ

∗u,k = 0.

With respect to the inequality constraints, it is trivial tosee that the proposed solution also

satisfies these constraints. Therefore, it has been proved that at time stepk + 1, the solution

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100

inherited at time stepk satisfies all the constraints of Problem 4–2. The control cost value

associated with this solution and the optimal control cost value from the time stepk can be

compared through the following relation:

V ∗2,k − V2,k+1 =

∥∥∥y(k)− y∗

sp,k − δ∗y,k

∥∥∥

2

Qy

+∥∥∥u

∗(k|k)− udes,k − δ∗u,k

∥∥∥

2

Qu

+∥∥∥∆u∗(k|k)

∥∥∥

2

R.

Since the weighting matricesQy, Qu andR are assumed positive semi-definite, one can con-

clude thatV2,k+1 ≤ V ∗2,k, and consequentlyV ∗

2,k+1 ≤ V ∗2,k. Therefore, the Lyapunov stability of

the closed-loop system is guaranteed. It is important to bear in mind that the cost function will

converge to a minimum equal to

V ∗2,∞ =

∥∥∥δ

∗y,∞

∥∥∥

2

Sy

+∥∥∥δ

∗u,∞

∥∥∥

2

Su

,

where it is noted that it will be equal to zero if, and only if, the input targets and output zones

are reachable, otherwise it will represent the minimum weighted distance between the desired

point and the reachable space.

4.4 The formulation of the robust MPC

Here, the nominally stable IHMPC proposed in the previous section is extended to the robust

case where the multi-plant uncertainty description is adopted (Badgwell, 1997). In order to

characterize the multi-plant model uncertainty, one defines the setΩ composed of possible lin-

ear models(Θ1, . . . ,ΘL) which may represent the process system in different operating condi-

tions. With the model formulation presented in (4.5) each individual modelΘn is then denoted

by Θn = Fstn ,F

unn ,Bs

l,n,Bstl,n,B

unl,n, γi,j,n, with n = 1, . . . , L andl = 0, . . . , p. Following the

same notation as in chapters 2 and 3, the nominal plant refersto ΘN , while the true plant is

represented byΘT .

At the same time that the stabilizing IHMPC algorithm developed in the previous section

is extended to the robust case, the approach here generalizes the method of Badgwell (1997)

to unstable time delay processes and the case of both optimizing target tracking and control

interval tracking. In this way, the robust IHMPC proposed here seeks to solve the following

optimization problem:

Problem4–3

min∆uk,ysp,k(Θn=1,...,L),δy,k(Θn=1,...,L),δu,k,δun,k(Θn=1,...,L)

V3,k(ΘN),

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101

V3,k(ΘN) =

m+p∑

j=0

∥∥∥yN(k + j|k)− ysp,k(ΘN)− δy,k(ΘN)−Ψun(Fun)(j−m−p)

δun,k(ΘN)∥∥∥

2

Qy

+m−1∑

j=0

∥∥∥u(k + j|k)− udes,k − δu,k

∥∥∥

2

Qu

+m−1∑

j=0

∥∥∥∆u(k + j|k)

∥∥∥

2

R

+∥∥∥x

stN(k +m+ p|k)

∥∥∥

2

Q(ΘN )+∥∥∥δy,k(ΘN)

∥∥∥

2

Sy

+∥∥∥δu,k

∥∥∥

2

Su

+∥∥∥δun,k(ΘN)

∥∥∥

2

Su

,

subject to (4.14), (4.16), (4.17) and, forn = 1, . . . , L,

Ns

(

Am+p(Θn)x(k)−Ap(Θn)W(Θn)∆uk

)

− ysp,k(Θn)− δy,k(Θn) = 0, (4.21)

Nun

(

Am+p(Θn)x(k)−Ap(Θn)W(Θn)∆uk

)

− δun,k(Θn) = 0, (4.22)

ymin ≤ ysp,k(Θn) ≤ ymax, (4.23)∥∥∥δun,k(Θn)

∥∥∥

2

Sun

≤∥∥∥δun,k(Θn)

∥∥∥

2

Sun

, (4.24)

V3,k(Θn) ≤ V3,k(Θn), (4.25)

where each control costV3,k(Θn) is obtained with a solution inherited from Problem 4–3 at

time stepk − 1 and translated to time stepk, that is

∆uk =[∆u∗(k|k − 1)⊤ · · · ∆u∗(k +m− 2|k − 1)⊤ 0⊤

]⊤,

ysp,k(Θn=1,...,L) = y∗sp,k−1(Θn=1,...,L).

The slacksδy,k(Θn=1,...,L), δun,k(Θn=1,...,L) andδu,k are sought to fulfill the respective terminal

equality constraints with the purpose of adapting the current state of the true plant (x(k)) to

other models lying withinΩ, or

Ns

(

Am+p(Θn)x(k)−Ap(Θn)W(Θn)∆uk

)

− ysp,k(Θn)− δy,k(Θn) = 0, n = 1, . . . , L,

Nun

(

Am+p(Θn)x(k)−Ap(Θn)W(Θn)∆uk

)

− δun,k(Θn=1,...,L) = 0, n = 1, . . . , L,

I⊤nu∆uk + u(k − 1)− udes,k − δu,k = 0.

Remark 4.4The introduction of constraint (4.24) aims to force that theslack vector associated

with each model ofΩ will be made equal to zero in a finite number of time steps, such that the

unstable states can be zeroed at the end of the delayed control horizon. Particularly, when this

happens to the true plant model, such a constraint can be satisfied for the subsequent time steps,

and so it is proved that the control cost for the true plant acts as a Lyapunov function and, if the

desired steady-state is reachable, it will converge to zero,as stated in the theorem below.

Theorem 4.2Consider a stable and unstable time delay process, whose true plant model can

assume any of the models lying in the setΩ. Then, if at a given time stepk the slack vector of

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102

the unstable states of the true plant model is a null vector, it will remain null at any subsequent

time stepk + j (j = 1, 2, . . .). Also, for an undisturbed system, and assuming that the process

system is stabilizable at the desired steady-state, the control law produced by the solution to

Problem 4–3 drives the true plant model to this equilibrium point, and as a consequence, the

control cost of the true plant will converge to zero.

Proof. If at time stepk Problem 4–3 is solved and the following vector of variables corresponds

to its optimal solution:

(

∆u∗k,y

∗sp,k(Θn=1,...,L), δ

∗y,k(Θn=1,...,L), δu,k, δ

∗un,k(Θn=1,...,L)

)

.

Also, if the slack vector concerningδ∗un,k(ΘT ) is null, then the cost function of the true plant

corresponding to this solution is given by

V3,k(ΘT ) =

m+p∑

j=0

∥∥∥y

∗T (k + j|k)− y∗

sp,k(ΘT )− δ∗y,k(ΘT )

∥∥∥

2

Qy

(4.26)

+m−1∑

j=0

∥∥∥u

∗(k + j|k)− udes,k − δ∗u,k

∥∥∥

2

Qu

+m−1∑

j=0

∥∥∥∆u∗(k + j|k)

∥∥∥

2

R

+∥∥∥x

stT (k +m+ p|k)

∥∥∥

2

Q(ΘT )+∥∥∥δ

∗y,k(ΘT )

∥∥∥

2

Sy

+∥∥∥δ

∗u,k

∥∥∥

2

Su

.

Now, considering the following solution at time stepk + 1:

∆uk+1 =[∆u∗(k + 1|k)⊤ · · · ∆u∗(k +m− 1|k)⊤ 0⊤

]⊤,

ysp,k+1(Θn=1,...,L) = y∗sp,k(Θn=1,...,L),

whose slacksδy,k+1(Θn=1,...,L), δun,k+1(Θn=1,...,L) andδu,k+1 are such that the following con-

ditions are satisfied:

Ns

(

Am+p(Θn)x(k)−Ap(Θn)W(Θn)∆uk+1

)

−ysp,k+1(Θn)−δy,k+1(Θn) = 0, n = 1, . . . , L,

Nun

(

Am+p(Θn)x(k)−Ap(Θn)W(Θn)∆uk+1

)

− δun,k+1(Θn=1,...,L) = 0, n = 1, . . . , L,

I⊤nu∆uk+1 + u∗(k|k)− udes,k − δu,k+1 = 0.

Then, it is clear that for the true plant model such a solutioninherited from time stepk is a

feasible solution to Problem 4–3 at timek + 1. In addition, once the same input sequence is

used for all models of setΩ and the current statex(k) corresponds to the actual model of the

true plant, it is concluded thatxT (k + j|k + 1) = xT (k + j|k), for anyj ≥ 1, which finally

implies δy,k+1(ΘT ) = δ∗y,k(ΘT ), δun,k+1(ΘT ) = δ

∗un,k(ΘT ) = 0 and δu,k+1 = δ

∗u,k. In this

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103

case, the cost function associated with the true plant for which the proposed solution is feasible

at time stepk + 1 can be represented as follows:

V3,k+1(ΘT ) =

m+p∑

j=0

∥∥∥y

∗T (k + j + 1|k)− y∗

sp,k(ΘT )− δ∗y,k(ΘT )

∥∥∥

2

Qy

(4.27)

+m−1∑

j=0

∥∥∥u

∗(k + j + 1|k)− udes,k − δ∗u,k

∥∥∥

2

Qu

+m−1∑

j=0

∥∥∥∆u∗(k + j + 1|k)

∥∥∥

2

R

+∥∥∥x

stT (k +m+ p|k)

∥∥∥

2

Q(ΘT )+∥∥∥δ

∗y,k(ΘT )

∥∥∥

2

Sy

+∥∥∥δ

∗u,k

∥∥∥

2

Su

.

Considering (4.26) and (4.27), it can be seen that the following relationship holds:

V ∗3,k(ΘT )− V3,k+1(ΘT ) =

∥∥∥y(k)− y∗

sp,k(ΘT )− δ∗y,k(ΘT )

∥∥∥

2

Qy

+

∥∥∥u

∗(k|k)− udes,k − δ∗u,k

∥∥∥

2

Qu

+∥∥∥∆u∗(k|k)

∥∥∥

2

R.

Now, from constraint (4.25), one concludes that the value ofthe cost function for the true plant

with the optimal solution at time stepk + 1 is not greater than the corresponding value with

the optimal solution inherited from time stepk, i.e. V ∗3,k+1(ΘT ) ≤ V3,k+1(ΘT ), which means,

therefore, that

V ∗3,k(ΘT )− V ∗

3,k+1(ΘT ) ≥∥∥∥y(k)− y∗

sp,k(ΘT )− δ∗y,k(ΘT )

∥∥∥

2

Qy

+

∥∥∥u

∗(k|k)− udes,k − δ∗u,k

∥∥∥

2

Qu

+∥∥∥∆u∗(k|k)

∥∥∥

2

R.

Since the right hand side of the equation above is non-negative, as the weighting matrices are

Qy ≥ 0, R ≥ 0 andQu ≥ 0, the control cost will be strictly non-increasing and bounded

below by zero, and so, for a large enough timek, it converges to zero, namelyV ∗3,k(ΘT ) −

V ∗3,k+1

(ΘT ) = 0. This proves the closed-loop convergence of the robust MPC controller.

4.5 Application to an unstable reactor system

The purpose of this section is to apply the proposed control schemes and to evaluate their

performances through simulation of an unstable reactor system. Such a system is a CSTR

in which an exothermic, irreversible, liquid phase, and first order chemical reaction (A→ B)

takes place. By considering the assumptions of fixed volume (under control), perfect mixing

and constant physical parameters, the set of nonlinear differential equations resulting from the

application of the mass and energy balances are as follows (Nagrathet al., 2002):

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dx1

dτ= q(x1f − x1)− φx1κ

dx2

dτ= q(x2f − x2)− δ(x2 − x3)− βφx1κ

dx3

dτ=

qc(x3f − x3)

δ1+

δ(x2 − x3)

δ1δ2

κ = exp x2

1 + x2/γ

(4.28)

whereτ is the dimensionless time,x1 is the dimensionless concentration of reactant A,x2

is the dimensionless reactor temperature,x3 is the dimensionless jacket temperature,q is the

dimensionless feed flow rate to the reactor,qc is the dimensionless jacket flow rate. The meaning

of these dimensionless variables can be found in Embiruçu (1993) as well as the dimensionless

parameters of the CSTR model, whose nominal values are those specified in Table 4.1.

Table 4.1: CSTR model parameters values.β γ δ δ1 δ2 φ x1f x2f x3f

8.0 20 0.3 0.1 0.5 0.072 1.0 0.0 -1.0

The objective of the controller is to guide one of the manipulated inputs to a desired target

and to maintain the controlled outputs, which are more numerous than the inputs, within the cor-

responding desired control zones, thus effectively accommodating the proposed design strategy

within the approaches of the nominal and robust MPC controllers. Before proceeding with the

analysis of the properties of the proposed robust MPC, it willbe justified its use for this specific

system through a performance comparison with the corresponding nominally stabilizing MPC

when there is model uncertainty. To this end, three linear models, which comprise the setΩ,

are obtained from the linearization of the CSTR model defined in (4.28) around the distinct

open-loop unstable equilibrium points. The open-loop unstable behaviors are related with those

steady-states shown in Table 4.2.

Table 4.2: Different equilibrium points for the unstable reactor system.Variables Steady-state 1 Steady-state 2 Steady-state 3x1/(dimensionless) 0.5528 0.5595 0.6364x2/(dimensionless) 2.7517 2.6489 1.9146x3/(dimensionless) 0.0 -0.1181 -0.4823q /(dimensionless) 1.0 0.9485 0.7232qc /(dimensionless) 1.65 1.8824 2.7779

In terms of transfer functions, the models corresponding tothe three operating points, already

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105

including uncertainties on the time delays that are supposed to represent operating conditions

that may happen in practical applications, are representedbelow:

Θ1 =

(0.45s2 + 10.36s+ 5.86)e−0.05s

s3 + 22.86s2 + 5.55s− 12.54

1.04

s3 + 22.86s2 + 5.55s− 12.54

(−2.75s2 − 64.03s− 46.9)e−0.05s

s3 + 22.86s2 + 5.55s− 12.54

(−3s− 5.43)

s3 + 22.86s2 + 5.55s− 12.54

(−16.51s− 12.5)

s3 + 22.86s2 + 5.55s− 12.54

(−10s2 − 34.5s+ 4.13)e−0.2s

s3 + 22.86s2 + 5.55s− 12.54

,

Θ2 =

(0.44s2 + 11.2s+ 5.78)e−0.2s

s3 + 25.16s2 + 6.22s− 11.88

0.86

s3 + 25.16s2 + 6.22s− 11.88

(−2.65s2 − 67.62s− 46.15)e−0.05s

s3 + 25.16s2 + 6.22s− 11.88

(−2.65s− 4.49)

s3 + 25.16s2 + 6.22s− 11.88

(−15.89s− 11.15)

s3 + 25.16s2 + 6.22s− 11.88

(−8.82s2 − 2.98s+ 3.14)e−0.2s

s3 + 25.16s2 + 6.22s− 11.88

,

Θ3 =

(0.48s2 + 21.35s+ 7.45)e−0.1s

s3 + 44.73s2 + 9.20s− 14.26

0.46

s3 + 44.73s2 + 9.20s− 14.26

(−2.67s2 − 119.9s− 60.34)e−0.05s

s3 + 44.73s2 + 9.20s− 14.26

(−1.48s− 2.36)

s3 + 44.73s2 + 9.20s− 14.26

(−15.99s− 8.14)

s3 + 44.73s2 + 9.20s− 14.26

(−4.95s2 − 1.25s+ 1.27)e−0.2s

s3 + 44.73s2 + 9.20s− 14.26

.

The comparison scenario to be simulated is that in which the system starts from the equi-

librium point u(0) =[1.0 1.65

]⊤andy(0) =

[0.8933 0.5193 −0.5950

]⊤, and the ma-

nipulated inputu1 should be driven to targetu1,des = 1.32, while the controlled outputs must

lie inside the control zones specified in Table 4.3. Here, thenominal model used in both MPC

controllers is represented by modelΘ2 and modelΘ1 is assumed to represent the true process.

The controllers have the following tuning parameters:∆t = 0.05, m = 7, Qy = diag(1, 1, 1),

Qu = diag(0.5, 0), R = diag(200, 100), Sy = diag(104, 104, 104), Su = diag(104, 0), and

Sun = diag(103, 103, 103, 103, 103, 103). It is interesting to note that the control horizon should

be at least 7 since this reactor system has six unstable poles. In addition, the input and input

move constraints presented in Table 4.4 must be strictly fulfilled.

Table 4.3: Control zones for the unstable reactor system.Outputs ymin ymax Unity1 0.5 0.6 dimensionlessy2 2.5 3.0 dimensionlessy3 -1.0 -0.8 dimensionless

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Table 4.4: Controller bounds for the manipulated inputs of the unstable reactor system.Inputs umin umax ∆umax Unitu1 0.0 5.0 1.0 dimensionlessu2 0.0 10.0 2.0 dimensionless

The resulting output and input responses are shown in figures4.1 and 4.2, respectively. In

this case, there is a clear superiority of the robust MPC overthe nominal MPC, showing that

this latter cannot stabilize the plant corresponding to model Θ1 when the controller considers

modelΘ2 to forecast the dynamic behavior of the controlled outputs.Then, one can conclude

that, although the nominal IHMPC provides nominal stability, there is no guarantee that stability

will be preserved in the presence of a realistic model uncertainty. On the other hand, the robust

MPC performs the task quite well and drives the three outputsto their control zones as well as

the targeted input to its economic goal.

Figure 4.1: Controlled outputs of the unstable reactor system. Nominal MPC (−−), robust MPC (—), upperand lower limits (− · −).

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107

Figure 4.2: Manipulated inputs of the unstable reactor system. Nominal MPC (−−), robust MPC (—), optimumtarget (− · −).

From now on, an analysis of the behavior of the proposed robust MPC for case of zone con-

trol and target tracking is described. The closed-loop simulation begins atu(0) =[1.32 3.73

]⊤

andy(0) =[0.5717 2.6174 −0.9613

]⊤, and the control zones and input target are the same

as in the previous simulation. At time stepk = 20 the controller is required to stabilize the

reactor system at targetu1,des = 1.22 and the controlled outputs should be driven to inside the

new control zones defined in Table 4.5. Next, at time stepk = 220, after the system has reached

a steady-state, disturbances corresponding to a decrease of 10% in jacket flow rate (u2) and an

increase of 1% in feed flow rate (u1) are introduced in the system.

Table 4.5: New control zones for the unstable reactor system.Outputs ymin ymax Unity1 0.6 0.7 dimensionlessy2 1.8 2.3 dimensionlessy3 -1.2 -0.9 dimensionless

Figures 4.3 and 4.4 display the respective dynamic behaviors of the controlled outputs and

manipulated inputs. From these figures, one verifies that thecontroller effectively attains the

required task. In first simulation scenario, the controllerleads the controlled outputs quickly

to their new zones and the targeted input to its desired value. Note that, initially the controller

increases the feed flow rate to the reactor and the cooling jacket flow rate such that the heat

removed from the reactor also increases, which makes the reactor and jacket temperatures to

decrease and, consequently, the reactant concentration toincrease. Therefore, after this fast

transient period, the controller is able to guide the feed flow rate to its target and to stabilize the

outputs inside their control zones.

With regard to the second simulated scenario, it emphasizesthat the proposed controller

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108

is capable of maintaining the outputs inside the bounds of their current zones immediately

after the introduction of the disturbances. Moreover, the controller also was able to bring back

the targeted input to its desired value since it is moved awayfrom its current position when

the system is under pertubation, and so the reactor converges to a new steady-state because

the non-targeted input accommodates the effect of the inserted disturbances. Indeed, in view

of the fact that the output requirements are more heavily penalized in the control objective

function than the input ones (the components of the weighting matrixQy are larger than theQu

ones), the controller prioratizes to maintain the outputs inside their control zones, and allows

the targeted input is momentarily to move its target value, which implies in the effective usage

of the available degrees of freedom.

Figure 4.3: Controlled outputs of the unstable reactor system for the tracking case. Robust MPC (—), upper andlower limits (− · −).

Finally, Figure 4.5 shows the robust MPC control costs associated with the nominal and true

plant models for the two simulated scenarios described previously. In this regard, it is possible

to observe that the cost function corresponding to the plantmodel is strictly decreasing (so-

called Lyapunov function), whereas the cost function related with the nominal model is not, as

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109

Figure 4.4: Manipulated inputs of the unstable reactor system for the tracking case. Robust MPC (—), optimumtarget (− · −).

would be expected and proved in Theorem 4.2, thus assuring the theoretical properties of the

proposed MPC controller.

Figure 4.5: Control costs for the unstable reactor system. Plant model (—), nominal model (−−).

4.6 Concluding remarks

This chapter presents a robustly stable MPC strategy for thestable and unstable time delay

processes. Within a practical standpoint, as it has been adopted in chapters 2 and 3, the pursuit

of economic goals for some process variables, to be defined bya RTO layer, as well as the zone

control strategy, are explicitly incorporated in the formulation of the proposed controller.

With the intention of circumventing the need to calculate the system steady states that are

sought to eliminate offset in the output tracking case, it has been proposed an analytical state-

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110

space model in incremental form based on step response of theprocess system. Allied to this

matter, the robust stability of the closed-loop system is guaranteed by means of additional

slacked terminal constraints included into the control problem optimization, which is solved

within a one-step formulation at each time step.

The simulation results undoubtedly demonstrate the performance superiority of the proposed

robust MPC as compared with the corresponding nominal MPC for the unstable reactor system

considered herein. In this way, by considering a multi-plant uncertainty description, the robust

controller seems to be an attractive and effective tool for controlling industrial reactor systems

that show several unstable operating points.

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CHAPTER 5

Conclusions and future research directions

5.1 Summary of contributions

This thesis deals with the development of robust model predictive control algorithms with guar-

antee of stability applicable to the stable, unstable and integrating time delay processes. In this

case, model uncertainty is represented by a discrete set of linear models (multi-plant descrip-

tion), which can stand for different operating conditions of the process system. Robustness is

achieved by adding constraints that prevent the controllercost function from increasing for any

plant model in the uncertainty definition set. From an industrial viewpoint, some remarks are

worth emphasizing about the robust controllers proposed herein, and they are the following:

(i) The minimum order state-space model used in the controllers derives from step responses

associated with the transfer function models of the process;

(ii) The inclusion of slack variables enlarges the feasibleregion of the control optimization

problem, by assuring that the controller has always a feasible solution;

(iii) The output zone control and the tracking of optimizingtargets for some process variables

(usually related to economic targets) are easily handled inthe control optimization prob-

lems;

(iv) The resulting optimization problems are convex and highly structured, greatly simplifying

their numerical solution.

In Chapter 2, it is presented a controller obtained from a two-step formulation that extends

successfully an existing controller to process systems with multiple time delays between the

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inputs and outputs. Specifically, the additional states related to the time delays of the pro-

cess system are independent of the model parameters, and consequently, they do not affect the

model uncertainty, which ensures stability and convergence of the controller. Simulation results

demonstrate that this robust MPC algorithm indeed outperforms its equivalent nominally stable

MPC for the examples considered here, including a nonlinearreactor system.

Another version of RMPC algorithm, proposed in Chapter 3, is based on a one-step formu-

lation (only one optimization problem). The model description that takes into account a more

general case of integrating systems, i.e. any output could be an integrating process variable with

respect to all system inputs, thus encompassing the vast majority of the process systems found

in the process industry. Due to its general feature, such a control strategy was tested through

simulation in the key part of the FCC unit, the converter, which operates in a partial combustion

mode and exhibits an integrating character for some controlled outputs with respect to all ma-

nipulated inputs. Performance comparisons consistently show that the proposed RMPC has a

superior performance in comparison to the advanced controlsystem existing in the real process

system (conventional MPC) for all simulated scenarios (regulatory case and optimum target

tracking).

The problem of synthesizing a robust MPC algorithm of unstable time delay processes is

addressed in Chapter 4. This controller assures Lyapunov stability of the closed-loop system

by means of a one-step control formulation. The main idea consists in using slacked terminal

constraints into the optimization problem as a way to cancelthe effect of the unstable states

on the output prediction at any time step beyond the control horizon for any model within the

domain of the model uncertainty. The control simulations ofan unstable reactor system are

exploited to illustrate the effectiveness of this robust controller.

Summarizing, the robustly stabilizing MPC strategies proposed in this work provide useful

solutions for stable, unstable and integrating time delay processes. In addition the simulation

results show that the resulting control performance seems quite acceptable for the industrial

implementation.

5.2 Suggestions for future work

Some lines of research deriving from this thesis can be further looked into. Namely:

1. In the state-space models proposed in this work, the time delays between the system out-

puts and inputs are incorporated to them through an augmented model representation. Such

an approach may affect computational aspects of the controllers, mostly because the num-

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ber of additional states of this system representation increases linearly with the dead-time

length. In this regard, explicit time-delay compensation schemes (e.g. Santoset al., 2012)

concerning the proposed state-space models can be used in order to avoid the augmented

representation, thus numerically improving the stabilizing robust MPC algorithms.

2. In spite of the fact that the proposed MPC strategies guarantee robust Lyapunov stability,

i.e. if the true system is reachable, then it will converge tothe desired equilibrium point,

regardless the tuning parameters of the controller. Nonetheless, the control performance

of these strategies can be improved by selecting systematically the main MPC tuning pa-

rameters with appropriate methods. From this standpoint, one issue that remains open and

needs to be further addressed is to develop a method to optimally tune the parameters of

the robust MPC controllers, and while it is also capable of dealing with the model uncer-

tainty explicitly. An attempt to address this matter in a systematic way could follow the

ideas proposed in the recent paper by Neryet al. (2014).

3. Computational issues involved in the implementation of the proposed robust MPC con-

trollers have not been considered in this work. Due to the inclusion of the non-linear

cost contracting constraints in the robust control formulation, the control action is then

obtained at each sampling time as the solution to a nonlinearprogramming (NLP) prob-

lem, which when applicable to high-order systems can be computationally expensive, even

if the proposed optimization problems are convex and numerically well-conditioned. In-

deed, efficient large scale implementation would require better optimization techniques

than those currently employed in commercial packages of industry. On this subject, a

possible way would be to reformulate the robust MPC problemsas LMI optimization

problems (non-linear constraints strictly convex), whichcan be solved with a significant

reduction of computational burden when compared with the NLP-based control problems

(Capronet al., 2012). Another alternative to be investigated would be to evaluate meta-

heuristic optimization algorithms, such as Particle SwarmOptimization (PSO) and Genetic

Algorithm (GA), which have been preferred over gradient-based ones for more efficiently

solving NLP problems (Datta and Figueira, 2011).

4. A challenging issue to be exploited considering the controllers proposed here is to design

robustly stabilizing MPC algorithms that integrate the economic objectives, usually de-

fined by a RTO layer, into the dynamic control layer (Alamoet al., 2014). In other words,

it will seek single-layer economic MPC controllers that explicitly take into account the

model uncertainty in their formulation, whilst assuring recursive feasibility and stability.

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114

5. So far the developed control methods have only been testedin simulation environments. In

this way, testing the application of the proposed controllers to real process systems, includ-

ing feeforward control actions through a suitable extension of the measured disturbances

on the state-space models, would be a natural consequence ofthis thesis.

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