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# Controller synthesis for piecewise affine slab differential inclusions: A duality-based convex optimization approach

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### Controller synthesis for piecewise affine slab differential inclusions: A duality-based convex optimization approach

1. 1. Outline Introduction Stability Analysis L2 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
2. 2. 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
3. 3. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions Motivation Question: What is the dual of a piecewise aﬃne (PWA) system? Samadi, Rodrigues Controller synthesis for Piecewise Aﬃne Systems 3/ 25
4. 4. 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
5. 5. 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 described as ˙x ∈ Conv{Aiκx + aiκ + Buiκ u + Bwiκ w, κ = 1, 2} y ∈ Conv{Ciκx + ciκ + Duiκ u + Dwiκ w, κ = 1, 2} for (x, w) ∈ RX×W i where Conv stands for the convex hull of a set. Samadi, Rodrigues Controller synthesis for Piecewise Aﬃne Systems 4/ 25
6. 6. 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 described as ˙x ∈ Conv{Aiκx + aiκ + Buiκ u + Bwiκ w, κ = 1, 2} y ∈ Conv{Ciκx + ciκ + Duiκ u + Dwiκ w, κ = 1, 2} for (x, w) ∈ RX×W i where Conv stands for the convex hull of a set. RX×W i 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 + 1 are scalars such that σ1 < σ2 < . . . < σM+1 Samadi, Rodrigues Controller synthesis for Piecewise Aﬃne Systems 4/ 25
7. 7. 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 as saturation, deadzone, Columb friction Contact dynamics Electrical circuits with diodes Samadi, Rodrigues Controller synthesis for Piecewise Aﬃne Systems 5/ 25
8. 8. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions Piecewise Aﬃne Slab Diﬀerential Inclusions Hassibi and Boyd (1998) - Quadratic stabilization and control of piecewise linear systems - Limited to piecewise linear controllers for PWA slab systems Johansson and Rantzer (2000) - Piecewise linear quadratic optimal control - No guarantee for stability Feng (2002) - Controller design and analysis of uncertain piecewise linear systems - All local subsystems should be stable Rodrigues and Boyd (2005) - Piecewise aﬃne state feedback for piecewise aﬃne slab systems using convex optimization - Stability analysis and synthesis using parametrized linear matrix inequalities Samadi, Rodrigues Controller synthesis for Piecewise Aﬃne Systems 6/ 25
9. 9. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions Objective To introduce a concept of duality for PWA slab diﬀerential inclusions To propose a method for PWA controller synthesis for stability and L2-gain performance of PWA slab diﬀerential inclusions using convex optimization Samadi, Rodrigues Controller synthesis for Piecewise Aﬃne Systems 7/ 25
10. 10. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions Objective To introduce a concept of duality for PWA slab diﬀerential inclusions To propose a method for PWA controller synthesis for stability and L2-gain performance of PWA slab diﬀerential inclusions using convex optimization Convex optimization problems are numerically tractable. Samadi, Rodrigues Controller synthesis for Piecewise Aﬃne Systems 7/ 25
11. 11. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions Dual parameter set PWA slab diﬀerential inclusion: ˙x ∈ Conv{Aiκx + aiκ, κ = 1, 2}, x ∈ Ri Ri = {x| Li x + li < 1} Samadi, Rodrigues Controller synthesis for Piecewise Aﬃne Systems 8/ 25
12. 12. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions Dual parameter set PWA slab diﬀerential inclusion: ˙x ∈ Conv{Aiκx + aiκ, κ = 1, 2}, x ∈ Ri Ri = {x| Li x + 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
13. 13. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions Dual parameter set Suﬃcient conditions for stability P > 0, AT iκP + PAiκ + αP < 0, ∀i ∈ I(0),    λiκ < 0, AT iκP + PAiκ + αP + λiκLT i Li Paiκ + λiκli LT i aT iκP + λiκli Li λiκ(l2 i − 1) < 0, for i /∈ I(0). Samadi, Rodrigues Controller synthesis for Piecewise Aﬃne Systems 9/ 25
14. 14. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions Dual parameter set Dual parameter set ΩT = AT iκ LT i aT iκ li i = 1, . . . , M, κ = 1, 2 Samadi, Rodrigues Controller synthesis for Piecewise Aﬃne Systems 10/ 25
15. 15. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions Dual parameter set Suﬃcient conditions for stability Q > 0, AiκQ + QAT iκ + αQ < 0, ∀i ∈ I(0), κ = 1, 2    µiκ < 0 AiκQ + QAT iκ + αQ + µiκaiκaT iκ QLT i + µiκli aiκ Li Q + µiκli aT iκ 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
16. 16. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions L2 gain PWA slab diﬀerential inclusion: ˙x ∈ Conv{Aiκx + aiκ + Bwiκ w, κ = 1, 2}, (x, w) ∈ RX×W i y ∈ Conv{Ciκx + ciκ + Dwiκ w, κ = 1, 2} RX×W i = {(x, w)| Li x + li + Mi w < 1} Samadi, Rodrigues Controller synthesis for Piecewise Aﬃne Systems 12/ 25
17. 17. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions L2 gain PWA slab diﬀerential inclusion: ˙x ∈ Conv{Aiκx + aiκ + Bwiκ w, κ = 1, 2}, (x, w) ∈ RX×W i y ∈ Conv{Ciκx + ciκ + Dwiκ w, κ = 1, 2} RX×W i = {(x, w)| Li x + li + Mi w < 1} Parameter set: Φ =      Aiκ aiκ Bwiκ Li li Mi Ciκ ciκ Dwiκ   i = 1, . . . , M, κ = 1, 2    Samadi, Rodrigues Controller synthesis for Piecewise Aﬃne Systems 12/ 25
18. 18. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions L2 gain Suﬃcient conditions for L2 gain performance P > 0, AT iκP + PAiκ + CT iκCiκ ∗ BT wiκ P + DT wiκ Ciκ −γ2 I + DT wiκ Dwiκ < 0, ∀i ∈ I(0, 0),         AT iκP + PAiκ +CT iκCiκ + λiκLT i Li ∗ ∗ aT iκP + cT iκCiκ + λiκli Li λiκ(l2 i − 1) + cT iκciκ ∗ BT wiκ P + DT wiκ Ciκ + λiκMT i Li DT wiκ ciκ + λiκli MT i −γ2 I + DT wiκ Dwiκ +λiκMT i Mi         < 0 and λiκ < 0 for i /∈ I(0, 0). Samadi, Rodrigues Controller synthesis for Piecewise Aﬃne Systems 13/ 25
19. 19. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions Dual parameter set Dual parameter set ΦT =      AT iκ LT i CT iκ aT iκ li cT iκ BT wiκ MT i DT wiκ   i = 1, . . . , M, κ = 1, 2    Samadi, Rodrigues Controller synthesis for Piecewise Aﬃne Systems 14/ 25
20. 20. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions Dual parameter set Suﬃcient conditions for stability Q > 0, AiκQ + QAT iκ + Bwiκ BT wiκ ∗ CiκQ + Dwiκ BT wiκ −γ2 I + Dwiκ DT wiκ < 0, ∀i ∈ I(0, 0)         AiκQ + QAT iκ +Bwiκ BT wiκ + µiκaiκaT iκ ∗ ∗ LiκQ + Mi BT wiκ + µiκli aT iκ µiκ(l2 i − 1) + Mi MT i ∗ CiκQ + Dwiκ BT wiκ + µiκciκaT iκ Dwiκ MT i + µiκli ciκ −γ2 I + Dwiκ DT wiκ +µiκciκcT iκ         < 0 and µiκ < 0 for i /∈ I(0, 0). A new result that extends the result in Hassibi and Boyd (1998) for ci = 0 Samadi, Rodrigues Controller synthesis for Piecewise Aﬃne Systems 15/ 25
21. 21. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions PWA controller synthesis Consider the following system: ˙x ∈ Conv{Aiκx + aiκ + Buiκ u, κ = 1, 2}, x ∈ Ri Ri = {x| Li x + li < 1} The stability conditions corresponding to the dual parameter set is used to formulate the synthesis problem. Samadi, Rodrigues Controller synthesis for Piecewise Aﬃne Systems 16/ 25
22. 22. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions PWA controller synthesis Controller synthesis problem: Q > 0, AiκQ + QAT iκ + Buiκ Yi + Y T i BT uiκ + αQ < 0, for i ∈ I(0), κ = 1, 2 , and µi < 0                 AiκQ + QAT iκ +Buiκ Yi + Y T i BT uiκ +αQ + µi aiκaT iκ +aiκZT i BT uiκ + Buiκ Zi aT iκ +Buiκ Wi BT uiκ       ∗ Li Q + µi li aT iκ +li ZT i BT uiκ µiκ(l2 i − 1)           ≤ 0, for i /∈ I(0) and κ = 1, 2 Samadi, Rodrigues Controller synthesis for Piecewise Aﬃne Systems 17/ 25
23. 23. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions PWA controller synthesis New variables: Yi = Ki Q Zi = µi ki Wi = µi ki kT i There is a problem: Wi is not a linear function of the unknown parameters µi , Yi and Zi . Samadi, Rodrigues Controller synthesis for Piecewise Aﬃne Systems 18/ 25
24. 24. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions PWA controller synthesis Two solutions: Convex relaxation: Since Wi = µi ki kT i ≤ 0, if the synthesis inequalities are satisﬁed with Wi = 0, they are satisﬁed with any Wi ≤ 0. Therefore, the synthesis problem can be made convex by omitting Wi . Rank minimization: Note that Wi = µi ki kT i ≤ 0 is the solution of the following rank minimization problem: min Rank Xi s.t. Xi = Wi Zi ZT i µi ≤ 0 Rank minimization is also not a convex problem. However, trace minimization works practically well as a heuristic solution min Trace Xi , s.t. Xi = Wi Zi ZT i µi ≤ 0 Samadi, Rodrigues Controller synthesis for Piecewise Aﬃne Systems 19/ 25
25. 25. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions L2 gain PWA controller synthesis Consider the following system: ˙x ∈ Conv{Aiκx + aiκ + Buiκ u + Bwiκ w, κ = 1, 2}, y ∈ Conv{Ciκx + ciκ + Duiκ u + Dwiκ w}, for (x, w) ∈ RX×W i = {(x, w)| Li x + li + Mi w < 1} The L2 conditions corresponding to the dual parameter set is used to formulate the synthesis problem. Samadi, Rodrigues Controller synthesis for Piecewise Aﬃne Systems 20/ 25
26. 26. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions L2 gain PWA controller synthesis L2 gain controller synthesis problem: Q > 0,              AiκQ + Buiκ Yi +QAT iκ + Y T i BT uiκ +Bwiκ BT wiκ    ∗ CiκQ + Duiκ Yi +Dwiκ BT wiκ −γ2 I + Dwiκ DT wiκ           < 0 for i ∈ I(0), κ = 1, 2 , and µi < 0                              AiκQ + Buiκ Yi +QAT iκ + Y T i BT uiκ +Bwiκ BT wiκ + µi aiκaT iκ aiκZT i BT uiκ + Buiκ Zi aT iκ       ∗ ∗ LiκQ + Mi BT wiκ +µiκli aT iκ + li ZT i BT uiκ µiκ(l2 i − 1) + Mi MT i ∗    CiκQ + Duiκ Yi +Dwiκ BT wiκ + µiκciκaT iκ ciκZT i BT uiκ + Duiκ Zi aT iκ      Dwiκ MT i +µiκli ciκ +li Duiκ Zi       −γ2 I + Dwiκ DT wiκ +µiκciκcT iκ + ciκZT i DT uiκ +Duiκ Zi cT iκ                            < 0, for i /∈ I(0) and κ = 1, 2 Samadi, Rodrigues Controller synthesis for Piecewise Aﬃne Systems 21/ 25
27. 27. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions Surge model of a jet engine Consider the following model (Kristic et al 1995): ˙x1 = −x2 − 3 2x2 1 − 1 2x3 1 ˙x2 = u A bounding envelope is computed for the nonlinear function f (x1) = −3 2x2 1 − 1 2x3 1 Samadi, Rodrigues Controller synthesis for Piecewise Aﬃne Systems 22/ 25
28. 28. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions Modeling By substituting the PWA bounds in the equations of the nonlinear system, we get a diﬀerential inclusion ˙x ∈ Conv{Aiκx + aiκ + Buu + Bw w}, x ∈ Ri y = Cx + Dw w + Duu (1) where i = 1, . . . , 4, κ = 1, 2 The approximation error of the nonlinear function is considered as the disturbance input (w) and the objective is to limit the L2-gain from w to x1. Samadi, Rodrigues Controller synthesis for Piecewise Aﬃne Systems 23/ 25
29. 29. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions Simulation Samadi, Rodrigues Controller synthesis for Piecewise Aﬃne Systems 24/ 25
30. 30. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions Conclusions: A new concept, dual parameter set, was introduced for PWA diﬀerential inclusions. Samadi, Rodrigues Controller synthesis for Piecewise Aﬃne Systems 25/ 25
31. 31. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions Conclusions: A new concept, dual parameter set, was introduced for PWA diﬀerential inclusions. Using the dual parameter set, suﬃcient conditions for stability and L2 gain performance were obtained. Samadi, Rodrigues Controller synthesis for Piecewise Aﬃne Systems 25/ 25
32. 32. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions Conclusions: A new concept, dual parameter set, was introduced for PWA diﬀerential inclusions. Using the dual parameter set, suﬃcient conditions for stability and L2 gain performance were obtained. Convex methods were proposed for PWA controller synthesis for stability and performance. Samadi, Rodrigues Controller synthesis for Piecewise Aﬃne Systems 25/ 25
33. 33. Outline Introduction Stability Analysis L2 Gain Analysis Controller Synthesis Numerical Example Conclusions Conclusions: A new concept, dual parameter set, was introduced for PWA diﬀerential inclusions. Using the dual parameter set, suﬃcient conditions for stability and L2 gain performance were obtained. Convex methods were proposed for PWA controller synthesis for stability and performance. Note that the dual parameter set does not necessarily deﬁne a PWA system. The questions still is: Does a dual system exist for a PWA system in general? Samadi, Rodrigues Controller synthesis for Piecewise Aﬃne Systems 25/ 25