Post on 05-Dec-2014
description
AUTHORS: MARCIO RIBEIRO DA SILVA GARCIA ADHEMAR DE BARROS FONTES
CARLOS EDUARDO TRABUCO DÓREA
An Iterative Algorithm For Constrained MPC With Stability Of Bilinear Systems
UNIVERSIDADE FEDERAL DA BAHIADEPARTAMENTO DE ENGENHARIA ELÉTRICA DA ESCOLA POLITÉCNICA
Table Of Contents
2
Purposes;
Predictive Control Concepts Model Prediction Constraints The Iterative Compensation Method Control Law Stability
Example
General Conclusions and Future Works;
References
Purposes3
1. Stablish a new control strategy with focus on reducing the prediction error in nonlinear systems and constraints satisfaction;
2. Formulation of an extension of the linear theory of invariant sets for the bilinear case;
3. Numerical examples regarding performance improvement and stability analysis
Predictive Control4
Features Application
Process model-based prediction;
Airplane system control;
Robustness regarding model –plant mismatch and process noise;
Robots control;
Easy tunning; Solar powerplant process control;
Easy threating of process constraints;
Several process of the chemical and petrochemical industry – FCC, Distillation columns, chemical reactors, furnace, etc;
Incorporation of the nonlinearities in the system model
Concepts5
r(k) - Reference signal;u(k) – Control signal;y(k) – System output; NY – Prediction horizon; NU – Control horizon;
Receiding Horizon: After the optimal sequence calculation, only the first element of the control vector is applied. In the next step, all calculation process is repeated.
FuturePast
Model6
Researches show that linear model-based controllers may present performance issues when applied to strongly nonlinear process;
Bilinear models:
Less complex than most nonlinear models;
More representative than linear models;
Linear in the parameters;
Bilinearity is presented in several process.
Model7
• Bilinear Model:• Linear Model:
Bilinear systems are Linear Parameter Varying (LPV) systems
Prediction8
a) Linear Prediction:
b) Bilinear Prediction:
Constraints9
All process are subjected to constraints;
The constraints are applied to the control signal and the process input / output variables;
Constraints violation may result in physical damage in the actuators or take the process output out of the specifications.
Constraints in predictive controllers are intrinsically related to prediction.
The Iterative Compensation Method10
Constraints satisfaction is easilly obtained in linear systems;
In nonlinear systems, future inputs sequences are unknown in the moment of prediction;
Prediction errors due to model linearization or uncertainties may take the state to violate the constraints.
The iterative compensation method consists in overcoming the prediction issue in bilinear systems so that the calculated control action do not violate the process constraints
11
Flowchart - The Iterative Compensation method
DataData
Calculation Of u(j=0)
u(k+i) = u(k-1), i=1,...,Hu
uj(k+i) = u(k-1)+u(j -1) (k+n),
i=1,...,Hu, n=1,...,i
CalculationOf u(j)
<orj > 20?
uoptimalu(j
)
uoptimalu(j
)
Yes
NoIter
ativ
e co
mpe
nsat
ion
j=j+1
Initi
al
Seq
uenc
e
Stop Criterion
The Iterative Compensation Method12
1-step ahead prediction for an unconstrined system:
Convergence is achieved if, and only if:
The convergence of the prediction depend on the system model, present state and tunning.
Control Law13
The minimum of J is calculated with Quadratic programming
Stability14
Bilinear systems are linear in the parameters;
An invariant set can be calculated in finite time considering that the resulting matrix A belongs to a convex set given by all possible combinations of the bounds on u(k);
The real maximum invariant set is unknow. The calculated set is an inner aproximation of the real maximal set;
The theory of stability, terminal constraints and invariant sets is well defined for linear systems. For bilinear systems, some assumptions may be done:
Stability15
Initial state lie inside the invariant set;
Contraction rate 0 < λ < 1, i.e, x(k) ∈ Ω ⇒x(k+1) ∈ λΩ;
The stronger constraint x(k+i) ∈ λiΩ must be imposed;
Zero prediction error and no model-plant mismatch. In practice this is not possible therefore, one must guarantee minimum erros.
Stability in bilinear systems can be guaranteed if the following conditions are fully satisfied:
Example16
System model:
Constraints:
Initial state = [3 2.8]T
Hp=10, =25:
-- blue line: Quasilinear controller- Black line: Iterative Compensation
State Trajectory
-4< x1(k) <4-4< x2(k) <4- red line: Invariant set for -1< u(k) <1-.- line: Invariant set for -0.5< u(k)
<0.5
Examples17
3 – Convergence of the prediction for a given iteration and effective cost improvements.
Convergence
-- Initial prediction -- Before convergence- After convergence
Cost Improvement
General Conclusion and Future Works
18
The iterative compensation method enables control without constraints violation;
The algorithm presents performance improvements over linear controllers and other technics for nonlinear control;
The calculation of invariant sets considering bilinear systems as LPV systems results in larger sets in comparison to existing strategies;
Future works should focus on the improvement of the developed strategy and a deeper analysis of the algorithm convergence and conservativness of the calculated invariant set.
References19
Bloemen, H. H. J., T. J. J. van den Boom e H. B. Verbruggen (2001). An optimization algorithm dedicated to a mpc problem for discrete time bilinear models, Proceedings of the american control conference. pp. 2376–2381.
Dórea, C.E.T., Hennet, J.C., (a,b)-invariant polyhedra of discrete-time systems. J. Optimiz. Theory Appl. 103(3), 521–542, 1999.
References20
Fontes, Desenvolvimento e Avaliação de Controladores Preditivos Baseados em Modelos Bilineares, PhD Thesis, Universidade Federal do Rio Grande do Norte, Natal-RN – Brasil, 2002.
Goodhart, Burnham, James, Bilinear, Self-tuning control of a high temperature heat treatment plant. IEEE Procedure – Control Theory Applications, vol. 141. N° 1, Jan, 1994.
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