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- 1. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Stability Analysis and Controller Synthesis for a Class of Piecewise Smooth Systems The Oral Examination for the Degree of Doctor of Philosophy Behzad Samadi Department of Mechanical and Industrial Engineering Concordia University 18 April 2008 Montreal, Quebec Canada
- 2. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Outline
- 3. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Introduction
- 4. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Practical Motivation c Quanser Memoryless Nonlinearities Saturation Dead Zone Coulomb & Viscous Friction
- 5. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Theoretical Motivation Richard Murray, California Institute of Technology: “ Rank Top Ten Research Problems in Nonlinear Control 10 Building representative experiments for evaluating controllers 9 Convincing industry to invest in new nonlinear methodologies 8 Recognizing the diﬀerence between regulation and tracking 7 Exploiting special structure to analyze and design controllers 6 Integrating good linear techniques into nonlinear methodologies 5 Recognizing the diﬀerence between performance and operability 4 Finding nonlinear normal systems for control 3 Global robust stabilization and local robust performance 2 Magnitude and rate saturation 1 Writing numerical software for implementing nonlinear theory ...This is more or less a way for me to think online, so I wouldn’t take any of this too seriously.”
- 6. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Special Structure Piecewise smooth system: ˙x = f (x) + g(x)u where f (x) and g(x) are piecewise continuous and bounded functions of x.
- 7. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Objective To develop a computational tool to design controllers for piecewise smooth systems using convex optimization techniques.
- 8. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Objective To develop a computational tool to design controllers for piecewise smooth systems using convex optimization techniques. Why convex optimization? There are numerically eﬃcient tools to solve convex optimization problems. Linear Matrix Inequalities (LMI) Sum of Squares (SOS) programming
- 9. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Literature 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 How (2003) - Observer-based control of piecewise aﬃne systems - Bilinear matrix inequality 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
- 10. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Major Contributions 1 To propose a two-step controller synthesis method for a class of uncertain nonlinear systems described by PWA diﬀerential inclusions. 2 To introduce for the ﬁrst time a duality-based interpretation of PWA systems. This enables controller synthesis for PWA slab systems to be formulated as a convex optimization problem. 3 To propose a nonsmooth backstepping controller synthesis for PWP systems. 4 To propose a time-delay approach to stability analysis of sampled-data PWA systems.
- 11. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Lyapunov Stability for Piecewise Smooth Systems
- 12. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Lyapunov Stability for Piecewise Smooth Systems Theorem (2.1) For nonlinear system ˙x(t) = f (x(t)), if there exists a continuous function V (x) such that V (x⋆ ) = 0 V (x) > 0 for all x = x⋆ in X t1 ≤ t2 ⇒ V (x(t1)) ≥ V (x(t2)) then x = x⋆ is a stable equilibrium point. Moreover if there exists a continuous function W (x) such that W (x⋆ ) = 0 W (x) > 0 for all x = x⋆ in X t1 ≤ t2 ⇒ V (x(t1)) ≥ V (x(t2)) + t2 t1 W (x(τ))dτ and x → ∞ ⇒ V (x) → ∞ then all trajectories in X asymptotically converge to x = x⋆.
- 13. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Lyapunov Stability for Piecewise Smooth Systems Why nonsmooth analysis? Discontinuous vector ﬁelds Piecewise smooth Lyapunov functions Lyapunov Function Vector Field Continuous Discontinuous Smooth Piecewise Smooth
- 14. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Extension of local linear controllers to global PWA controllers for uncertain nonlinear systems
- 15. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Extension of linear controllers to PWA controllers Objectives: Global robust stabilization and local robust performance Integrating good linear techniques into nonlinear methodologies
- 16. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions PWA Diﬀerential Inclusions Consider the following uncertain nonlinear system ˙x = f (x) + g(x)u Let ˙x ∈ Conv{σ1(x, u), . . . , σK(x, u)} where σκ(x, u) = Aiκx + aiκ + Biκu, x ∈ Ri , with Ri = {x|Ei x + ei ≻ 0}, for i = 1, . . . , M ≻ represents an elementwise inequality.
- 17. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions PWA Diﬀerential Inclusions PWA bounding envelope for a nonlinear function f (x) δ1(x) δ2(x) -4 -3 -2 -1 0 1 2 3 4 -60 -50 -40 -30 -20 -10 0 10
- 18. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Extension of linear controllers to PWA controllers
- 19. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Extension of linear controllers to PWA controllers Theorem (3.2) Let there exist matrices ¯Pi = ¯PT i , ¯Ki , Zi , ¯Zi , Λiκ and ¯Λiκ that verify the conditions for all i = 1, . . . , M, κ = 1, . . . , K and for a given decay rate α > 0, desired equilibrium point x⋆, linear controller gain ¯Ki⋆ and ǫ > 0, then all trajectories of the nonlinear system in X asymptotically converge to x = x⋆.
- 20. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Extension of linear controllers to PWA controllers Theorem (3.2) Let there exist matrices ¯Pi = ¯PT i , ¯Ki , Zi , ¯Zi , Λiκ and ¯Λiκ that verify the conditions for all i = 1, . . . , M, κ = 1, . . . , K and for a given decay rate α > 0, desired equilibrium point x⋆, linear controller gain ¯Ki⋆ and ǫ > 0, then all trajectories of the nonlinear system in X asymptotically converge to x = x⋆. If the conditions are feasible, the resulting PWA controller provides global robust stability and local robust performance.
- 21. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Extension of linear controllers to PWA controllers Theorem (3.2) Let there exist matrices ¯Pi = ¯PT i , ¯Ki , Zi , ¯Zi , Λiκ and ¯Λiκ that verify the conditions for all i = 1, . . . , M, κ = 1, . . . , K and for a given decay rate α > 0, desired equilibrium point x⋆, linear controller gain ¯Ki⋆ and ǫ > 0, then all trajectories of the nonlinear system in X asymptotically converge to x = x⋆. If the conditions are feasible, the resulting PWA controller provides global robust stability and local robust performance. The synthesis problem includes a set of Bilinear Matrix Inequalities (BMI). In general, it is not convex.
- 22. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Controller synthesis for PWA slab diﬀerential inclusions: a duality-based convex optimization approach
- 23. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Controller synthesis for PWA slab diﬀerential inclusions: a duality-based convex optimization approach Objective: To formulate controller synthesis for stability and L2 gain performance of piecewise aﬃne slab diﬀerential inclusions as a set of LMIs.
- 24. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions L2 Gain Analysis for PWA Slab Diﬀerential Inclusions 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 = Aiκ1 aiκ1 Bwiκ1 Li li Mi Ciκ2 ciκ2 Dwiκ2 i = 1, . . . , M, κ1 = 1, 2, κ2 = 1, 2
- 25. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions A duality-based convex optimization approach Dual Parameter Set ΦT = AT iκ1 LT i CT iκ2 aT iκ1 li cT iκ2 BT wiκ1 MT i DT wiκ2 i = 1, . . . , M, κ1 = 1, 2, κ2 = 1, 2
- 26. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions LMI Conditions for L2 Gain Analysis Parameter Set P > 0, AT iκ1 P + PAiκ1 + CT iκ2 Ciκ2 ∗ BT wiκ1 P + DT wiκ2 Ciκ2 −γ2I + DT wiκ2 Dwiκ2 < 0 for i ∈ I(0, 0), κ1 = 1, 2 and κ2 = 1, 2 and λiκ1κ2 < 0 AT iκ1 P + PAiκ1 +CT iκ2 Ciκ2 +λiκ1κ2 LT i Li ∗ ∗ aT iκ1 P + cT iκ2 Ciκ2 + λiκ1κ2 li Li λiκ1κ2 (l2 i − 1) + cT iκ2 ciκ2 ∗ BT wiκ1 P +DT wiκ2 Ciκ2 +λiκ1κ2 MT i Li DT wiκ2 ciκ2 + λiκ1κ2 li MT i −γ2I +DT wiκ2 Dwiκ2 +λiκ1κ2 MT i Mi < 0 for i /∈ I(0, 0), κ1 = 1, 2 and κ2 = 1, 2
- 27. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions LMI Conditions for L2 Gain Analysis Dual Parameter Set Q > 0, Aiκ1Q + QAT iκ1 + Bwiκ1 BT wiκ1 ∗ Ciκ2 Q + Dwiκ2 BT wiκ1 −γ2I + Dwiκ2 DT wiκ2 < 0 for i ∈ I(0, 0), κ1 = 1, 2 and κ2 = 1, 2 and µiκ1κ2 < 0 Aiκ1 Q + QAT iκ1 +Bwiκ1 BT wiκ1 +µiκ1κ2 aiκ1 aT iκ1 ∗ ∗ Li Q + Mi BT wiκ1 + µiκ1κ2 li aT iκ1 µiκ1κ2 (l2 i − 1) + Mi MT i ∗ Ciκ2 Q + Dwiκ2 BT wiκ1 +µiκ1κ2 ciκ2 aT iκ1 Dwiκ2 MT i + µiκ1κ2 li ciκ2 −γ2I +Dwiκ2 DT wiκ2 +µiκ1κ2 ciκ2 cT iκ2 < 0 for i /∈ I(0, 0), κ1 = 1, 2 and κ2 = 1, 2
- 28. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions PWA L2 gain controller synthesis The goal is to limit the L2 gain from w to y using the following PWA controller: u = Ki x + ki , x ∈ Ri New variables: Yi = Ki Q Zi = µi ki Wi = µi ki kT i Problem: Wi is not a linear function of the unknown parameters µi , Yi and Zi .
- 29. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions A duality-based convex optimization approach Proposed 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 .
- 30. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions A duality-based convex optimization approach Proposed 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 (maximization for negative semideﬁnite matrices) works practically well as a heuristic solution max Trace Xi , s.t. Xi = Wi Zi ZT i µi ≤ 0
- 31. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions A duality-based convex optimization approach The following problems for PWA slab diﬀerential inclusions with PWA outputs were formulated as a set of LMIs: Stability analysis (Propositions 4.1 and 4.2) L2 gain analysis (Propositions 4.3 and 4.4) Stabilization using PWA controllers (Propositions 4.5 and 4.6) PWA L2 gain controller synthesis (Propositions 4.7 and 4.8)
- 32. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Backstepping Controller Synthesis for PWP Systems: A Sum of Squares Approach
- 33. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Backstepping Controller Synthesis for PWP Systems: A Sum of Squares Approach Objective: To formulate controller synthesis for a class of piecewise polynomial systems as a Sum of Squares (SOS) programming.
- 34. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Backstepping Controller Synthesis for PWP Systems: A Sum of Squares Approach Objective: To formulate controller synthesis for a class of piecewise polynomial systems as a Sum of Squares (SOS) programming. SOSTOOLS, a MATLAB toolbox that handles the general SOS programming, was developed by S. Prajna, A. Papachristodoulou and P. Parrilo. p(x) = m i=1 f 2 i (x) ≥ 0
- 35. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Backstepping Controller Synthesis for PWP Systems: A Sum of Squares Approach PWP system in strict feedback from ˙x1 = f1i1 (x1) + g1i1 (x1)x2, for x1 ∈ P1i1 ˙x2 = f2i2 (x1, x2) + g2i2 (x1, x2)x3, for x1 x2 ∈ P2i2 ... ˙xk = fkik (x1, x2, . . . , xk ) + gkik (x1, x2, . . . , xk )u, for x1 x2 ... xk ∈ Pkik where Pjij = x1 ... xj Ejij (x1, . . . , xj ) ≻ 0
- 36. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Sampled-Data PWA Systems: A Time-Delay Approach Motivation: Toycopter, a 2 DOF helicopter model
- 37. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Sampled-Data PWA Systems: A Time-Delay Approach Example: Pitch model of the experimental helicopter: ˙x1 =x2 ˙x2 = 1 Iyy (−mheli lcgx g cos(x1) − mheli lcgz g sin(x1) − FkM sgn(x2) − FvMx2 + u) where x1 is the pitch angle and x2 is the pitch rate. Nonlinear part: f (x1) = −mheli lcgx g cos(x1) − mheli lcgz g sin(x1) PWA part: f (x2) = −FkM sgn(x2)
- 38. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Sampled-Data PWA Systems: A Time-Delay Approach x1 f(x1) f (x1) ˆf (x1) -3.1416 -1.885 -0.6283 0.6283 1.885 3.1416 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 PWA approximation - Helicopter model
- 39. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Sampled-Data PWA Systems: A Time-Delay Approach x1 x2 -3 -2 -1 0 1 2 3 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Continuous time PWA controller
- 40. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Sampled-Data PWA Systems: A Time-Delay Approach
- 41. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Sampled-Data PWA Systems: A Time-Delay Approach Sampled-data PWA controller u(t) = Ki x(tk ) + ki , x(tk ) ∈ Ri Continuous−time PWA systems PWA controller Hold
- 42. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Sampled-Data PWA Systems: A Time-Delay Approach The closed-loop system can be rewritten as ˙x(t) = Ai x(t) + ai + Bi (Ki x(tk ) + ki ) + Bi w, for x(t) ∈ Ri and x(tk ) ∈ Rj where w(t) = (Kj − Ki )x(tk ) + (kj − ki ), x(t) ∈ Ri , x(tk ) ∈ Rj The input w(t) is a result of the fact that x(t) and x(tk ) are not necessarily in the same region.
- 43. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Sampled-Data PWA Systems: A Time-Delay Approach Theorem (6.1) For the sampled-data PWA system, assume there exist symmetric positive matrices P, R, X and matrices Ni for i = 1, . . . , M such that the conditions are satisﬁed and let there be constants ∆K and ∆k such that w ≤ ∆K x(tk ) + ∆k Then, all the trajectories of the sampled-data PWA system in X converge to the following invariant set Ω = {xs | V (xs, ρ) ≤ σaµ2 θ + σb}
- 44. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Sampled-Data PWA Systems: A Time-Delay Approach Solving an optimization problem to maximize τM subject to the constraints of the main theorem and η > γ > 1 leads to τ⋆ M = 0.2193
- 45. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Sampled-Data PWA Systems: A Time-Delay Approach x1 x2 -3 -2 -1 0 1 2 3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 Sampled data PWA controller for Ts = 0.2193
- 46. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Sampled-Data PWA Systems: A Time-Delay Approach x1 x2 -3 -2 -1 0 1 2 3 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Continuous time PWA controller
- 47. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Conclusions
- 48. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Summary of Major Contributions 1 To propose a two-step controller synthesis method for a class of uncertain nonlinear systems described by PWA diﬀerential inclusions. 2 To introduce for the ﬁrst time a duality-based interpretation of PWA systems. This enables controller synthesis for PWA slab systems to be formulated as a convex optimization problem. 3 To propose a nonsmooth backstepping controller synthesis for PWP systems. 4 To propose a time-delay approach to stability analysis of sampled-data PWA systems.
- 49. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Publications 1 B. Samadi and L. Rodrigues, “Extension of local linear controllers to global piecewise aﬃne controllers for uncertain nonlinear systems,” accepted for publication in the International Journal of Systems Science. 2 B. Samadi and L. Rodrigues, “Controller synthesis for piecewise aﬃne slab diﬀerential inclusions: a duality-based convex optimization approach,” under second revision for publication in Automatica. 3 B. Samadi and L. Rodrigues, “Backstepping Controller Synthesis for Piecewise Polynomial Systems: A Sum of Squares Approach,” in preparation 4 B. Samadi and L. Rodrigues, “Sampled-Data Piecewise Aﬃne Systems: A Time-Delay Approach,” to be submitted.
- 50. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Publications 1 B. Samadi and L. Rodrigues, “Backstepping Controller Synthesis for Piecewise Polynomial Systems: A Sum of Squares Approach,” submitted to the 46th Conference on Decision and Control, cancun, Mexico, Dec. 2008. 2 B. Samadi and L. Rodrigues, “Sampled-Data Piecewise Aﬃne Slab Systems: A Time-Delay Approach,” in Proc. of the American Control Conference, Seattle, WA, Jun. 2008. 3 B. Samadi and L. Rodrigues, “Controller synthesis for piecewise aﬃne slab diﬀerential inclusions: a duality-based convex optimization approach,” in Proc. of the 46th Conference on Decision and Control, New Orleans, LA, Dec. 2007. 4 B. Samadi and L. Rodrigues, “Backstepping Controller Synthesis for Piecewise Aﬃne Systems: A Sum of Squares Approach,” in Proc. of the IEEE International Conference on Systems, Man, and Cybernetics (SMC 2007), Montreal, Oct. 2007. 5 B. Samadi and L. Rodrigues, “Extension of a local linear controller to a stabilizing semi-global piecewise-aﬃne controller,” 7th Portuguese Conference on Automatic Control, Lisbon, Portugal, Sep. 2006.
- 51. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Questions
- 52. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Open Problems Can general PWP/PWA controller synthesis be converted to a convex problem? What is the dual of a PWA system?
- 53. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Special Structure Type f (x) g(x) Piecewise Smooth Piecewise Continuous Piecewise Continuous Piecewise Polynomial Piecewise Polynomial Piecewise Polynomial Piecewise Aﬃne Piecewise Aﬃne Piecewise Constant Piecewise Linear Piecewise Linear Piecewise Constant Linear Linear Constant
- 54. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Convex Optimization “In fact the great watershed in optimization is not between linearity and nonlinearity, but convexity and nonconvexity.” (Rockafellar, SIAM review, 1993) The hard part is to ﬁnd out if a problem can be formulated as a convex optimization problem.
- 55. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Extension of linear controllers to PWA controllers Piecewise Quadratic Lyapunov function V (x) = xT Pi x + 2qT i x + ri , for x ∈ Ri
- 56. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Extension of linear controllers to PWA controllers Conditions on the PWA controller: ¯Ki = ¯Ki⋆ , if x⋆ ∈ Ri (¯Aiκ + ¯Biκ ¯Ki )¯x⋆ = 0, if x⋆ ∈ Ri (¯Aiκ + ¯Biκ ¯Ki )¯Fij = (¯Ajκ + ¯Bjκ ¯Kj )¯Fij , if Ri Rj = ∅
- 57. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Extension of linear controllers to PWA controllers Conditions on the PWA controller: ¯Ki = ¯Ki⋆ , if x⋆ ∈ Ri (¯Aiκ + ¯Biκ ¯Ki )¯x⋆ = 0, if x⋆ ∈ Ri (¯Aiκ + ¯Biκ ¯Ki )¯Fij = (¯Ajκ + ¯Bjκ ¯Kj )¯Fij , if Ri Rj = ∅ Continuity of the Lyapunov function: ¯FT ij (¯Pi − ¯Pj )¯Fij = 0, if Ri Rj = ∅
- 58. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Extension of linear controllers to PWA controllers Conditions on the PWA controller: ¯Ki = ¯Ki⋆ , if x⋆ ∈ Ri (¯Aiκ + ¯Biκ ¯Ki )¯x⋆ = 0, if x⋆ ∈ Ri (¯Aiκ + ¯Biκ ¯Ki )¯Fij = (¯Ajκ + ¯Bjκ ¯Kj )¯Fij , if Ri Rj = ∅ Continuity of the Lyapunov function: ¯FT ij (¯Pi − ¯Pj )¯Fij = 0, if Ri Rj = ∅ Positive deﬁniteness of the Lyapunov function: ¯Pi ¯x⋆ = 0, if x⋆ ∈ Ri Pi > ǫI, if x⋆ ∈ Ri , Ei x⋆ + ei = 0 Zi ∈ Rn×n, Zi 0 Pi − ET i Zi Ei > ǫI , if x⋆ ∈ Ri , Ei x⋆ + ei = 0 ¯Zi ∈ R(n+1)×(n+1), ¯Zi 0 ¯Pi − ¯ET i ¯Zi ¯Ei > ǫ¯I , if x⋆ /∈ Ri
- 59. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Extension of linear controllers to PWA controllers Monotonicity of the Lyapunov function: for i such that x⋆ ∈ Ri , Ei x⋆ + ei = 0, Pi (Aiκ + BiκKi ) + (Aiκ + BiκKi )T Pi < −αPi for i such that x⋆ ∈ Ri , Ei x⋆ + ei = 0, Λiκ ∈ Rn×n, Λiκ 0 Pi (Aiκ + BiκKi ) + (Aiκ + BiκKi )TPi + ET i ΛiκEi < −αPi for i such that x⋆ /∈ Ri , ¯Λiκ ∈ R(n+1)×(n+1), ¯Λiκ 0 ¯Pi (¯Aiκ + ¯Biκ ¯Ki ) + (¯Aiκ + ¯Biκ ¯Ki )T ¯Pi + ¯ET i ¯Λiκ ¯Ei < −α¯Pi
- 60. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Extension of linear controllers to PWA controllers Consider the following second order system ˙x1 = x2 ˙x2 = −x1 + 0.5x2 − 0.5x2 1 x2 + u with the following domain: X = x1 x2 − 30 < x1 < 30, −60 < x2 < 60
- 61. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Extension of linear controllers to PWA controllers x1 x2 -6 -4 -2 0 2 4 6 -10 -5 0 5 10 Trajectories of the open loop system
- 62. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Extension of linear controllers to PWA controllers x1 x2 -30 -20 -10 0 10 20 30 -60 -40 -20 0 20 40 60 Trajectories of the closed-loop system for the linear controller
- 63. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Extension of linear controllers to PWA controllers x1 x2 -30 -20 -10 0 10 20 30 -60 -40 -20 0 20 40 60 Trajectories of the closed-loop system for the PWA controller
- 64. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Extension of linear controllers to PWA controllers The main limitation: The controller synthesis is formulated as a set of Bilinear Matrix Inequalities (BMI). Linear systems analogy: Consider the linear system ˙x = Ax + Bu with a candidate Lyapunov function V (x) = xTPx and a state feedback controller u = Kx. Suﬃcient conditions for the stability of this system are: Positivity: V (x) = xT Px > 0 for x = 0 Monotonicity: ˙V (x) = ˙xT Px + xT P ˙x < 0 P > 0 (AT + KT BT )P + P(A + BK) < 0 BMIs are nonconvex problems in general.
- 65. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Extension of linear controllers to PWA controllers Contributions: To propose a two-step controller synthesis method for a class of uncertain nonlinear systems described by PWA diﬀerential inclusions. The proposed method has two objectives: global stability and local performance. It thus enables to use well known techniques in linear control design for local stability and performance while delivering a global PWA controller that is guaranteed to stabilize the nonlinear system. Diﬀerential inclusions are considered. Therefore the controller is robust in the sense that it can stabilize any piecewise smooth nonlinear system bounded by the diﬀerential inclusion. Stability is guaranteed even if sliding modes exist.
- 66. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions A duality-based convex optimization approach Linear systems analogy: Consider the dual system ˙z = (A + BK)T z with a candidate Lyapunov function V (x) = xT Qx Suﬃcient conditions for the stability of this system are: Positive deﬁniteness: Q > 0 Monotonicity: (A + BK)Q + Q(AT + KT BT ) < 0 Deﬁne Y = KQ. Positive deﬁniteness: Q > 0 Monotonicity: AQ + BY + QAT + Y T BT < 0 Then K = YQ−1.
- 67. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions A duality-based convex optimization approach PWA slab system: ˙x =Ai x + ai , x ∈ Ri Ri ={x| Li x + li < 1} Literature Hassibi and Boyd (1998) Rodrigues and Boyd (2005)
- 68. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions A duality-based convex optimization approach Suﬃcient conditions for stability P > 0, AT i P + PAi + αP < 0, for 0 ∈ Ri , λi < 0, AT i P + PAi + αP + λi LT i Li Pai + λi li LT i aT i P + λi li Li λi (l2 i − 1) < 0, for 0 /∈ Ri Hassibi and Boyd (1998)
- 69. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions A duality-based convex optimization approach Suﬃcient conditions for stability Q > 0, Ai Q + QAT i + αQ < 0, for 0 ∈ Ri , µi < 0, Ai Q + QAT i + αQ + µi ai aT i QLT i + µi li ai Li Q + µi li aT i µi (l2 i − 1) < 0, for 0 /∈ Ri Hassibi and Boyd (1998)
- 70. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions A duality-based convex optimization approach Parameter set: Ω = Ai ai Li li i = 1, . . . , M Dual parameter set ΩT = AT i LT i aT i li i = 1, . . . , M
- 71. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions A duality-based convex optimization approach PWA slab system: ˙x =Ai x + ai + Bwi w, x ∈Ri = {x| Li x + li < 1}, y =Ci x + Dwi w, Parameter set: Φ = Ai ai Bwi Li li 0 Ci 0 Dwi i = 1, . . . , M Hassibi and Boyd (1998)
- 72. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions A duality-based convex optimization approach Summary: Introducing PWA slab diﬀerential inclusions Introducing the dual parameter set Extending the L2 gain analysis and synthesis to PWA slab diﬀerential inclusions with PWA outputs Extending the deﬁnition of the regions of a PWA slab diﬀerential inclusion Proposing two methods to formulate the PWA controller synthesis for PWA slab diﬀerential inclusions as a convex problem
- 73. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Sum of Squares Programming A sum of squares program is a convex optimization program of the following form: Minimize J j=1 wj αj subject to fi,0 + J j=1 αj fi,j(x) is SOS, for i = 1, . . . , I where the αj ’s are the scalar real decision variables, the wj ’s are some given real numbers, and the fi,j are some given multivariate polynomials.
- 74. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Sum of Squares Programming A sum of squares program is a convex optimization program of the following form: Minimize J j=1 wj αj subject to fi,0 + J j=1 αj fi,j(x) is SOS, for i = 1, . . . , I where the αj ’s are the scalar real decision variables, the wj ’s are some given real numbers, and the fi,j are some given multivariate polynomials. SOSTOOLS, a MATLAB toolbox that handles the general SOS programming, was developed by S. Prajna, A. Papachristodoulou and P. Parrilo.
- 75. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Backstepping Controller Synthesis for PWP Systems: A Sum of Squares Approach Consider the following PWP system: ˙x =fi (x) + gi (x)z, x ∈ Pi ˙z =u where Pi = {x|Ei (x) ≻ 0} where Ei (x) ∈ Rpi is a vector polynomial function of x and ≻ represents an elementwise inequality.
- 76. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Backstepping Controller Synthesis for PWP Systems: A Sum of Squares Approach Backstepping as a Lyapunov function construction method: Consider ˙x = fi (x) + gi (x)z, x ∈ Pi
- 77. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Backstepping Controller Synthesis for PWP Systems: A Sum of Squares Approach Backstepping as a Lyapunov function construction method: Consider ˙x = fi (x) + gi (x)z, x ∈ Pi Assume that there exists a polynomial control z = γ(x) and V (x) is an SOS Lyapunov function for the closed loop system verifying V (x) − λ(x) is SOS −∇V (x)T(fi (x) + gi (x)γ(x)) − Γi (x)TEi (x) − αV (x) is SOS for i = 1, . . . , M and any α > 0, where λ(x) is a positive deﬁnite SOS polynomial, Γi (x) is an SOS vector function
- 78. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Backstepping Controller Synthesis for PWP Systems: A Sum of Squares Approach Backstepping as a Lyapunov function construction method: Consider ˙x = fi (x) + gi (x)z, x ∈ Pi Assume that there exists a polynomial control z = γ(x) and V (x) is an SOS Lyapunov function for the closed loop system verifying V (x) − λ(x) is SOS −∇V (x)T(fi (x) + gi (x)γ(x)) − Γi (x)TEi (x) − αV (x) is SOS for i = 1, . . . , M and any α > 0, where λ(x) is a positive deﬁnite SOS polynomial, Γi (x) is an SOS vector function Consider now the following candidate Lyapunov function Vγ(x, z) = V (x) + 1 2 (z − γ(x))T (z − γ(x)) Note that Vγ(x, z) is a positive deﬁnite function.
- 79. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Backstepping Controller Synthesis for PWP Systems: A Sum of Squares Approach Controller synthesis: The synthesis problem can be formulated as the following SOS program. Find u = γ2(x, z) and Γ(x, z) s.t. −∇xVγ(x, z)T (fi (x) + gi (x)z) −∇zVγ(x, z)T u −Γ(x, z)T Ei (x, z) − αVγ(x, z) is SOS, Γ(x, z) is SOS γ2(0, 0) = 0 where i2 = 1, . . . , M2 and γ2(x1, x2) is a polynomial function of x1 and x2.
- 80. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Backstepping Controller Synthesis for PWP Systems: A Sum of Squares Approach Example: Single link ﬂexible joint robot:
- 81. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Backstepping Controller Synthesis for PWP Systems: A Sum of Squares Approach Example: Single link ﬂexible joint robot: ˙x1 = x2 ˙x2 = − MgL I sin(x1) − K I (x1 − x3) ˙x3 = x4 ˙x4 = − f2(x4) J + K J (x1 − x3) + 1 J u where x1 = θ1, x2 = ˙θ1, x3 = θ2 and x4 = ˙θ2. u is the motor torque and f2(x4) denotes the motor friction which is described by f2(x4) = bmx4 + sgn(x4) Fcm + (Fsm − Fcm) exp(− x2 4 c2 m )
- 82. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Backstepping Controller Synthesis for PWP Systems: A Sum of Squares Approach PWP approximation: x1 f1(x1) f1(x1) ˆf1(x1) -3.1416 -0.8976 0.8976 3.1416 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 x4 f2(x4) f2(x4) ˆf2(x4) -8 -6 -4 -2 0 2 4 6 8 -3 -2 -1 0 1 2 3
- 83. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Backstepping Controller Synthesis for PWP Systems: A Sum of Squares Approach Typical structure of the regions of a PWP system in strict feedback form x1 P11 vvmmmmmmmmmmmmmmmm ((QQQQQQQQQQQQQQQQ (x1, x2) P21 P22 P23 (x1, x2, x3) P31 }}{{{{{{{{ P32 }}{{{{{{{{ !!CCCCCCCC P33 !!CCCCCCCC (x1, x2, x3, x4) P41 P44 P42 P45 P43 P46
- 84. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Backstepping Controller Synthesis for PWP Systems: A Sum of Squares Approach State variables of the nonlinear model - PWA controller: Time x1 Time x2 Time x3 Time x4 0 2 4 60 2 4 6 0 2 4 60 2 4 6 -400 -200 0 200 -1 0 1 2 3 4 -8 -6 -4 -2 0 0 1 2 3 4
- 85. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Backstepping Controller Synthesis for PWP Systems: A Sum of Squares Approach Control input - PWA controller: Time u 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -7 -6 -5 -4 -3 -2 -1 0 1 ×104
- 86. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Backstepping Controller Synthesis for PWP Systems: A Sum of Squares Approach State variables of the nonlinear model - PWP controller: Time x1 Time x2 Time x3 Time x4 0 2 4 60 2 4 6 0 2 4 60 2 4 6 -10 -5 0 5 10 0 1 2 3 -5 -4 -3 -2 -1 0 0 1 2 3 4
- 87. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Backstepping Controller Synthesis for PWP Systems: A Sum of Squares Approach Control input - PWP controller: Time u 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -300 -250 -200 -150 -100 -50 0 50 100 150 200
- 88. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Backstepping Controller Synthesis for PWP Systems: A Sum of Squares Approach Summary of the contributions: Introducing PWP systems in strict feedback form Formulating backstepping controller synthesis for PWP systems as a convex optimization problem Polynomial Lyapunov functions for PWP systems with discontinuous vector ﬁelds PWP Lyapunov functions for PWP systems with continuous vector ﬁelds Numerical tools such as SOSTOOLS and Yalmip/SeDuMi
- 89. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Sampled-Data PWA Systems: A Time-Delay Approach PWA system ˙x = Ai x + ai + Bi u, for x ∈ Ri with the region Ri deﬁned as Ri = {x|Ei x + ei ≻ 0},
- 90. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Sampled-Data PWA Systems: A Time-Delay Approach PWA system ˙x = Ai x + ai + Bi u, for x ∈ Ri with the region Ri deﬁned as Ri = {x|Ei x + ei ≻ 0}, Continuous-time PWA controller u(t) = Ki x(t) + ki , x(t) ∈ Ri
- 91. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Sampled-Data PWA Systems: A Time-Delay Approach Lyapunov-Krasovskii functional: V (xs, ρ) := V1(x) + V2(xs , ρ) + V3(xs , ρ) where xs(t) := x(t) x(tk ) , tk ≤ t < tk+1 V1(x) := xT Px V2(xs , ρ) := 0 −τM t t+r ˙xT (s)R ˙x(s)dsdr V3(xs , ρ) := (τM − ρ)(x(t) − x(tk))T X(x(t) − x(tk )) and P, R and X are positive deﬁnite matrices.
- 92. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Sampled-Data PWA Systems: A Time-Delay Approach for all i such that 0 ∈ Ri , Ψi + τM M1i P 0 Bi + τM I −I XBi BT i P 0 + τM BT i X I −I −γI < 0 Ψi + τM M2i τM AT i KT i BT i RBi + P 0 Bi τM Ni τM BT i R Ai Bi Ki + BT i P 0 τM BT i RBi − γI 0 τM NT i 0 − τM 2 R
- 93. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Sampled-Data PWA Systems: A Time-Delay Approach for all i such that 0 /∈ Ri , ¯Λi ≻ 0, Ψi + τM M1i P 0 0 Bi + τM I −I 0 XBi BT i P 0 0 + τM BT i X I −I 0 −γI < 0 Ψi + τM M2i τM AT i KT i BT i kT i BT i + aT i RBi + P 0 0 Bi τM Ni 0 τM BT i R Ai Bi Ki Bi ki + ai +BT i P 0 0 τM BT i RBi − γI 0 τM NT i 0 0 − τM 2 R < 0
- 94. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Sampled-Data PWA Systems: A Time-Delay Approach Summary of the contributions Formulating stability analysis of sampled-data PWA systems as a convex optimization problem
- 95. Outline Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions Active Suspension System Example: Active suspension with a nonlinear damper −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 −30 −20 −10 0 10 20 30 40 Suspension deflection speed (m/s) DampingForce(N) Nonlinear PWA Approximation

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