Jos e Manuel D az Mart nez, Sebasti an Dormido Bencomo ...Speci cacition in Nyquist diagrama 1 1! =...
Transcript of Jos e Manuel D az Mart nez, Sebasti an Dormido Bencomo ...Speci cacition in Nyquist diagrama 1 1! =...
Loopshaping de lazo cerrado interactivo
Jose Manuel Dıaz Martınez, Sebastian Dormido BencomoRamon Costa i Castello
IV Seminario de Innovacion Docente en Automatica
Leon, 10-12 de Enero de 2018
J.M Dıaz , S. Dormido, R. Costa Loopshaping de lazo cerrado interactivo Leon, 10-12 de Enero de 2018 1 / 25
Working Team
Jose Manuel Dıaz Sebastian Dormido Ramon [email protected] [email protected] [email protected]
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Problem FormulationDefinition
Y (s)U (s)
D (s)
N (s)
Di (s)
E (s)R (s)C (s) G (s)
−
+
L(s) = C (s)G (s), S(s) =1
1 + L(s), T (s) =
L(s)
1 + L(s)
Y (s)E (s)U(s)
=
T (s) G (s)S(s) S(s) −T (s)S(s) G (s)S(s) −S(s) −S(s)
C (s)S(s) T (s) −C (s)S(s) −C (s)S(s)
R(s)Di (s)D(s)N(s)
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Frequency Domain Design
Frequency Domain offers a self-contained methods for analysis anddesign
Nyquist criteria :I Closed-loop stability analysisI Robustness analysis
Simple steady-state behavior analysis
Time-response (indirectly)
Design idea:I Shape L(s) according to the specifications
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Open loop Bode : L(s)Specificacition
0dB
|L(jω)|dB
ess
Bandwith
Typical design
Noise attenuation
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Open loop Nyquist : L(s)Specificacition
ρ
−1 β
γ
Typical design
Robustness Noise attenuation
ess
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LCSD : Interactive design
An Interactive and Comprehensive Software Tool to Promote Active Learningin the Loop Shaping Control System Design. Jose M. Diaz; RamonCosta-Castello ; Rocio Munoz ; Sebastian Dormido (2017).
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Closed-loop shaping
Open-loop shaping tries to shape the closed-loop function indirectly.
I Robustness (gain margin, phase margin, distance to -1) is analyzedover L(s).
I Performance is difficult to be analyzed looking at L(s).
Motivation : Closed-loop shaping may be a more natural approach.
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Closed loop : Complementary sensitivity functionSpecificacition in Bode diagram
0dB
|T (jω)|dB
ess
Bandwith
Typical design
Noise amplification
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Closed loop : Complementary sensitivity functionSpecificacition in Nyquist diagram
−1 1
ω = 0
Typical T (s) design
Noise
ess
Resonance
ωnoiseωband
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Closed loop : Complementary sensitivity functionSpecificacition in Nichols diagram
dB
T (jω)
ω = 0
ωnoiseωband
Degree
Typical T (s) design
Resonance
Noise
ess
What about phase ?
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Closed loop : Sensitivity functionSpecificacition
Y (s)U (s)
D (s)
N (s)
Di (s)
E (s)R (s)C (s) G (s)
−
+
E (s) =1
1 + C (s)G (s)R(s) = S(s)R(s) =
DC (s)DG (s)
NC (s)NG (s) + DC (s)DG (s)R(s)
E (s) =−1
1 + C (s)G (s)N(s) = −S(s)N(s) = − DC (s)DG (s)
NC (s)NG (s) + DC (s)DG (s)N(s)
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Closed loop : Sensitivity functionSpecificacition
lims→j∞ S(s) = 1
Robustness
d (−1, L (jω)) = infω| − 1− L (jω) | = inf
ω|1 + L (jω) |
=
[supω
1
|1 + L (jω) |
]−1
= ‖S(s)‖−1∞ .
Waterbed : ∫ ∞0
ln |S (jω) |dω = −κπ2
+ π
nnmp∑k
pk
where pk ∈ C+ are the unstable poles of L(s), nnmp are the numberof unstable poles of L(s) and κ = lims→∞ sL(s).
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Closed loop : Sensitivity functionSpecificacition in Bode diagrama
0dB
|S(jω)|dB
ess Bandwith
Typical design
RobustnessNoise amplification
What about phase ?J.M Dıaz , S. Dormido, R. Costa Loopshaping de lazo cerrado interactivo Leon, 10-12 de Enero de 2018 14 / 25
Closed loop : Sensitivity functionSpecificacition in Nyquist diagrama
−1 1
ω = 0
Typical S(s) design
Robustness
ess
ωnoiseωband
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Closed loop : Sensitivity functionSpecificacition in Nichols diagrama
dB
S(jω)
ω = 0
ωnoise
ωband
Degree
Typical S(s) design
Robustness
ess
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Controller ParametrizationDefinition
For G (s) stable:
C (s)
G (s)Q(s)
G (s)
R(s) Y (s)
−
+
+
+
C (s) =Q(s)
1− G (s)Q(s)
T (s) = Q(s)G (s)
S(s) = 1− Q(s)G (s)
C (s)S(s) = Q(s)
J.M Dıaz , S. Dormido, R. Costa Loopshaping de lazo cerrado interactivo Leon, 10-12 de Enero de 2018 17 / 25
Controller ParametrizationInteractive Tool : Sensitivity function
→ zeros of S(s) are not directly connected to Q(s)
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Controller ParametrizationInteractive Tool : Complementary Sensitivity function
→ Interactivity is straightforward.→ Robustness analysis.
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Controller ParametrizationInteractive Tool : Sensitivity function
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Controller ParametrizationRobust stability (additive uncertainty)
C(s)
Gn(s)Q(s)
Gn(s)
R(s)
W au (s) ∆(s)
Y (s)
−
+
+
+
Robust stability condition:
‖C (s)S(s)W au (s)‖∞ = ‖Q(s)W a
u (s)‖∞ < 1→ |Q(jω)| < 1
|W au (jω)| ∀ω
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Controller ParametrizationRobust stability (multiplicative uncertainty)
Robust stability condition:
‖T (s)Wmu (s)‖∞ = ‖Gn(s)Q(s)Wm
u (s)‖∞ < 1
→ |Gn(jω)Q(jω)| < 1
|W ,u(jω)| ∀ω
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Controller ParametrizationInteractive Tool : Uncertainty modeling
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ConclusionsSome comments
Direct Design.I PerformanceI RobustnessI Limitations are automatically visualized
Sometimes Controller is too high order.I order reduction
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ConclusionsFurther work
On going project.I A set of examples is being developedI Tool functionalitiesI Reviewing concepts
Optimal H∞ solutions
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