Outline Introduction Stability Analysis L 2 Gain Analysis Controller Synthesis Numerical Example Conclusions Controller synthesis for piecewise affine slab differential inclusions A duality-based convex optimization approach Behzad Samadi Luis Rodrigues Department of Mechanical and Industrial Engineering Concordia University CDC 2007, New Orleans Samadi, Rodrigues Controller synthesis for Piecewise Affine Systems 1/ 25
33
Embed
Controller synthesis for piecewise affine slab differential inclusions: A duality-based convex optimization approach
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions
Controller synthesis for piecewise affine slabdifferential inclusions
A duality-based convex optimization approach
Behzad Samadi Luis Rodrigues
Department of Mechanical and Industrial EngineeringConcordia University
CDC 2007, New Orleans
Samadi, Rodrigues Controller synthesis for Piecewise Affine Systems 1/ 25
Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions
Outline of Topics
1 Introduction
2 Stability Analysis
3 L2 Gain Analysis
4 Controller Synthesis
5 Numerical Example
6 Conclusions
Samadi, Rodrigues Controller synthesis for Piecewise Affine Systems 2/ 25
Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions
Motivation
Question: What is the dual of a piecewise affine (PWA)system?
It is still an open problem.
Samadi, Rodrigues Controller synthesis for Piecewise Affine Systems 3/ 25
Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions
Motivation
Question: What is the dual of a piecewise affine (PWA)system?
It is still an open problem.
Samadi, Rodrigues Controller synthesis for Piecewise Affine Systems 3/ 25
Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions
Piecewise Affine Slab Differential Inclusions
A continuous-time PWA slab differential inclusion is describedas
for (x ,w) ∈ RX×Wi where Conv stands for the convex hull ofa set.
RX×Wi for i = 1, . . . ,M are M slab regions defined as
Ri = (x ,w) | σi < CRx + DRw < σi+1,
where CR ∈ R1×n, DR ∈ R1×nw and σi for i = 1, . . . ,M + 1are scalars such that
σ1 < σ2 < . . . < σM+1
Samadi, Rodrigues Controller synthesis for Piecewise Affine Systems 4/ 25
Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions
Piecewise Affine Slab Differential Inclusions
Practical examples:
Mechanical systems with hard nonlinearities such assaturation, deadzone, Columb friction
Contact dynamics
Electrical circuits with diodes
Samadi, Rodrigues Controller synthesis for Piecewise Affine Systems 5/ 25
Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions
Piecewise Affine Slab Differential Inclusions
Hassibi and Boyd (1998) - Quadratic stabilization and controlof piecewise linear systems - Limited to piecewise linearcontrollers for PWA slab systems
Johansson and Rantzer (2000) - Piecewise linear quadraticoptimal control - No guarantee for stability
Feng (2002) - Controller design and analysis of uncertainpiecewise linear systems - All local subsystems should be stable
Rodrigues and Boyd (2005) - Piecewise affine state feedbackfor piecewise affine slab systems using convex optimization -Stability analysis and synthesis using parametrized linearmatrix inequalities
Samadi, Rodrigues Controller synthesis for Piecewise Affine Systems 6/ 25
Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions
Objective
To introduce a concept of duality for PWA slab differentialinclusions
To propose a method for PWA controller synthesis for stabilityand L2-gain performance of PWA slab differential inclusionsusing convex optimization
Convex optimization problems are numerically tractable.
Samadi, Rodrigues Controller synthesis for Piecewise Affine Systems 7/ 25
Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions
Objective
To introduce a concept of duality for PWA slab differentialinclusions
To propose a method for PWA controller synthesis for stabilityand L2-gain performance of PWA slab differential inclusionsusing convex optimization
Convex optimization problems are numerically tractable.
Samadi, Rodrigues Controller synthesis for Piecewise Affine Systems 7/ 25
Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions
Dual parameter set
PWA slab differential inclusion:
x ∈ ConvAiκx + aiκ, κ = 1, 2, x ∈ Ri
Ri = x | ‖Lix + li‖ < 1
Parameter set:
Ω =
[Aiκ aiκ
Li li
] ∣∣∣∣ i = 1, . . . ,M, κ = 1, 2
Samadi, Rodrigues Controller synthesis for Piecewise Affine Systems 8/ 25
Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions
Dual parameter set
PWA slab differential inclusion:
x ∈ ConvAiκx + aiκ, κ = 1, 2, x ∈ Ri
Ri = x | ‖Lix + li‖ < 1
Parameter set:
Ω =
[Aiκ aiκ
Li li
] ∣∣∣∣ i = 1, . . . ,M, κ = 1, 2
Samadi, Rodrigues Controller synthesis for Piecewise Affine Systems 8/ 25
Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions
Dual parameter set
Sufficient conditions for stability
P > 0,
ATiκP + PAiκ + αP < 0, ∀i ∈ I(0),
λiκ < 0,[AT
iκP + PAiκ + αP + λiκLTi Li Paiκ + λiκliL
Ti
aTiκP + λiκliLi λiκ(l2
i − 1)
]< 0,
for i /∈ I(0).
Samadi, Rodrigues Controller synthesis for Piecewise Affine Systems 9/ 25
Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions
Dual parameter set
Dual parameter set
ΩT =
[AT
iκ LTi
aTiκ li
] ∣∣∣∣ i = 1, . . . ,M, κ = 1, 2
Samadi, Rodrigues Controller synthesis for Piecewise Affine Systems 10/ 25
Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions
Dual parameter set
Sufficient conditions for stability
Q > 0,
AiκQ + QATiκ + αQ < 0, ∀i ∈ I(0), κ = 1, 2
µiκ < 0[AiκQ + QAT
iκ + αQ + µiκaiκaTiκ QLT
i + µiκliaiκ
LiQ + µiκliaTiκ µiκ(l2
i − 1)
]< 0,
for i /∈ I(0).
A new interpretation for the result in Hassibi and Boyd (1998)
Samadi, Rodrigues Controller synthesis for Piecewise Affine Systems 11/ 25
Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions