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Backstepping for Piecewise Affine Systems: A SOS Approach

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Backstepping for Piecewise Affine Systems: A SOS Approach

  1. 1. Outline Introduction PWA Controller Synthesis Numerical Example Conclusion Backstepping for Piecewise Affine Systems An SOS Approach Behzad Samadi Luis Rodrigues Department of Mechanical and Industrial Engineering Concordia University SMC 2007, Montreal Samadi, Rodrigues Backstepping for Piecewise Affine Systems
  2. 2. Outline Introduction PWA Controller Synthesis Numerical Example Conclusion Outline of Topics Introduction PWA Controller Synthesis Numerical Example Conclusion Samadi, Rodrigues Backstepping for Piecewise Affine Systems
  3. 3. Outline Introduction PWA Controller Synthesis Numerical Example Conclusion Motivational example Tunnel diode circuit Samadi, Rodrigues Backstepping for Piecewise Affine Systems
  4. 4. Outline Introduction PWA Controller Synthesis Numerical Example Conclusion Motivational example PWA characteristic Samadi, Rodrigues Backstepping for Piecewise Affine Systems
  5. 5. Outline Introduction PWA Controller Synthesis Numerical Example Conclusion Motivational example Piecewise affine (PWA) model: ˙x1 = −30x1 − 20x2 + 24 + 20u ˙x2 =    0.05x1 − 0.25x2, x2 < 0.2 0.05x1 + 0.1x2 − 0.07, 0.2 < x2 < 0.6 0.05x1 − 0.2x2 + 0.11, x2 > 0.6 Desired equilibrium point: xcl = 0.3714 0.6429 T Samadi, Rodrigues Backstepping for Piecewise Affine Systems
  6. 6. Outline Introduction PWA Controller Synthesis Numerical Example Conclusion Piecewise Affine Systems A continuous-time PWA system is described as ˙x(t) = Ai x(t) + ai + Bi u(t), if x(t) ∈ Ri Samadi, Rodrigues Backstepping for Piecewise Affine Systems
  7. 7. Outline Introduction PWA Controller Synthesis Numerical Example Conclusion Piecewise Affine Systems A continuous-time PWA system is described as ˙x(t) = Ai x(t) + ai + Bi u(t), if x(t) ∈ Ri The polytopic cells, Ri , i ∈ I = {1, . . . , M}, partition a subset of the state space X ⊂ Rn such that ∪M i=1Ri = X, Ri ∩ Rj = ∅, i = j, where Ri denotes the closure of Ri . Samadi, Rodrigues Backstepping for Piecewise Affine Systems
  8. 8. Outline Introduction PWA Controller Synthesis Numerical Example Conclusion Piecewise Affine Systems A continuous-time PWA system is described as ˙x(t) = Ai x(t) + ai + Bi u(t), if x(t) ∈ Ri The polytopic cells, Ri , i ∈ I = {1, . . . , M}, partition a subset of the state space X ⊂ Rn such that ∪M i=1Ri = X, Ri ∩ Rj = ∅, i = j, where Ri denotes the closure of Ri . Each cell is constructed as the intersection of a finite number of half spaces Ri = {x|Ei x + ei 0} Samadi, Rodrigues Backstepping for Piecewise Affine Systems
  9. 9. Outline Introduction PWA Controller Synthesis Numerical Example Conclusion Piecewise Affine Systems Practical examples: Mechanical systems with hard nonlinearities such as saturation, deadzone, Columb friction Contact dynamics Electrical circuits with diodes Samadi, Rodrigues Backstepping for Piecewise Affine Systems
  10. 10. Outline Introduction PWA Controller Synthesis Numerical Example Conclusion Piecewise Affine Systems PWA systems are in general nonsmooth nonlinear systems. Samadi, Rodrigues Backstepping for Piecewise Affine Systems
  11. 11. Outline Introduction PWA Controller Synthesis Numerical Example Conclusion Piecewise Affine Systems PWA systems are in general nonsmooth nonlinear systems. Controller synthesis methods for PWA systems Hassibi and Boyd (1998) - Quadratic stabilization and control of piecewise linear systems - Limited to piecewise linear controllers for PWA systems with one variable in the domain of nonlinearity 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 Samadi, Rodrigues Backstepping for Piecewise Affine Systems
  12. 12. Outline Introduction PWA Controller Synthesis Numerical Example Conclusion Objective To propose a method for PWA controller synthesis using convex optimization Samadi, Rodrigues Backstepping for Piecewise Affine Systems
  13. 13. Outline Introduction PWA Controller Synthesis Numerical Example Conclusion Objective To propose a method for PWA controller synthesis using convex optimization Convex optimization problems are numerically tractable. Samadi, Rodrigues Backstepping for Piecewise Affine Systems
  14. 14. Outline Introduction PWA Controller Synthesis Numerical Example Conclusion Sum of Squares Decomposition SOS decomposition for polynomials of degree d in n variables: p(x) = m i=1 f 2 i (x) for some polynomials fi Samadi, Rodrigues Backstepping for Piecewise Affine Systems
  15. 15. Outline Introduction PWA Controller Synthesis Numerical Example Conclusion Sum of Squares Decomposition SOS decomposition for polynomials of degree d in n variables: p(x) = m i=1 f 2 i (x) for some polynomials fi SOS polynomials are non-negative. Samadi, Rodrigues Backstepping for Piecewise Affine Systems
  16. 16. Outline Introduction PWA Controller Synthesis Numerical Example Conclusion 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. Samadi, Rodrigues Backstepping for Piecewise Affine Systems
  17. 17. Outline Introduction PWA Controller Synthesis Numerical Example Conclusion Backstepping for PWA systems Consider the following PWA system ˙x1 = A (1) i1 x1 + a (1) i1 + B (1) i1 x2, for E (1) i1 x1 + e (1) i1 > 0 ˙x2 = A (2) i2 X2 + a (2) i2 + B (2) i2 u, for E (2) i2 X2 + e (2) i2 > 0 where ij = 1, . . . , Mj for j = 1, 2 and X2 = x1 x2 Samadi, Rodrigues Backstepping for Piecewise Affine Systems
  18. 18. Outline Introduction PWA Controller Synthesis Numerical Example Conclusion Backstepping for PWA systems Piecewise polynomial Lyapunov functions for PWA systems with continuous vector fields SOS Lyapunov functions for PWA systems with discontinuous vector fields Samadi, Rodrigues Backstepping for Piecewise Affine Systems
  19. 19. Outline Introduction PWA Controller Synthesis Numerical Example Conclusion Continuous PWA systems It is assumed that for the following subsystem ˙x1 = A (1) i1 x1 + a (1) i1 + B (1) i1 x2, for E (1) i1 x1 + e (1) i1 > 0, with i1 = 1, . . . , M1 there exist a continuous piecewise polynomial Lyapunov function V (1)(x1) and a continuous PWA controller x2 = γ(1)(x1) with    V (1)(x1) = V (1) i1 (x1) γ(1)(x1) = K (1) i1 (x1) + k (1) i1 , for E (1) i1 x1 + e (1) i1 > 0, Samadi, Rodrigues Backstepping for Piecewise Affine Systems
  20. 20. Outline Introduction PWA Controller Synthesis Numerical Example Conclusion Continuous PWA systems In addition, the continuous piecewise polynomial V (1) (x1) = V (1) i1 (x1), x1 ∈ Ri1 is a Lyapunov function for the closed loop system satisfying − V (1) i1 .(A (1) i1 x1 + a (1) i1 + B (1) i1 γ (1) i1 (x1)) −Γ (1) i1 (x1).(E (1) i1 x1 + e (1) i1 ) − αV (1) i is SOS where α > 0, Γ (1) i1 (x1) is an SOS function. Samadi, Rodrigues Backstepping for Piecewise Affine Systems
  21. 21. Outline Introduction PWA Controller Synthesis Numerical Example Conclusion Continuous PWA systems Consider now the following candidate Lyapunov function V (2) (X2) = V (1) (x1) + 1 2 (x2 − γ(1) (x1)).(x2 − γ(1) (x1)) Note that V (2)(X2) is a continuous piecewise polynomial function. Samadi, Rodrigues Backstepping for Piecewise Affine Systems
  22. 22. Outline Introduction PWA Controller Synthesis Numerical Example Conclusion Continuous PWA systems The synthesis problem can be formulated as the following SOS program. Find u = γ (2) i2 (X2), Γ (1) i1 (x1), Γ (2) i2 (X2), ci2j2 (X2) s.t. − x1 V (2) i2 .(A (1) i1 x1 + a (1) i1 + B (1) i1 x2) − x2 V (2) i2 .(A (2) i2 X2 + a (2) i2 + B (2) i2 u) −Γ (1) i1 (x1).(E (1) i1 x1 + e (1) i1 ) −Γ (2) i2 (X2).(E (2) i2 X2 + e (2) i2 ) − αV (2) i2 is SOS, Samadi, Rodrigues Backstepping for Piecewise Affine Systems
  23. 23. Outline Introduction PWA Controller Synthesis Numerical Example Conclusion Continuous PWA systems Γ (1) i1 (x1) and Γ (2) i2 (X2) are SOS γ (2) i2 (X2) − γ (2) j2 (X2) = ci2j2 (X2)(E (2) i2j2 X2 + e (2) i2j2 ) where i1 = 1, . . . , M1, i2 = 1, . . . , M2, R (2) i2 and R (2) j2 are level-1 neighboring cells, E (2) i2j2 X2 + e (2) i2j2 = 0 contains their boundary, ci2j2 is an arbitrary polynomial and γ (2) i2 (X2) = K (2) i2 X2 + k (2) i2 Samadi, Rodrigues Backstepping for Piecewise Affine Systems
  24. 24. Outline Introduction PWA Controller Synthesis Numerical Example Conclusion Backstepping for PWA systems If the SOS program is feasible, a controller u = γ (2) i2 (X2) can be found for the original PWA system (sufficient condition). The same procedure can be repeated for PWA systems in strict feedback form ˙x1 = A (1) i1 x1 + a (1) i1 + B (1) i1 x2, for E (1) i1 x1 + e (1) i1 > 0 ˙x2 = A (2) i2 X2 + a (2) i2 + B (2) i2 x3, for E (2) i2 X2 + e (2) i2 > 0 ... ˙xn = A (n) in Xn + a (n) in + B (n) in u, for E (n) in Xn + e (n) in > 0 where ij = 1, . . . , Mj and Xj = [xT 1 . . . xT j ]T for j = 2, . . . , n. Samadi, Rodrigues Backstepping for Piecewise Affine Systems
  25. 25. Outline Introduction PWA Controller Synthesis Numerical Example Conclusion Discontinuous PWA systems For discontinuous PWA systems, an SOS Lyapunov function is constructed using affine controllers in each step. Since the controller in the last step will not be used in the construction of the Lyapunov function, the last controller can be a PWA controller. Samadi, Rodrigues Backstepping for Piecewise Affine Systems
  26. 26. Outline Introduction PWA Controller Synthesis Numerical Example Conclusion Tunnel Diode Consider the tunnel diode PWA model: ˙x1 = −30x1 − 20x2 + 24 + 20u ˙x2 =    0.05x1 − 0.25x2, x2 < 0.2 0.05x1 + 0.1x2 − 0.07, 0.2 < x2 < 0.6 0.05x1 − 0.2x2 + 0.11, x2 > 0.6 Desired equilibrium point: xcl = 0.3714 0.6429 T Samadi, Rodrigues Backstepping for Piecewise Affine Systems
  27. 27. Outline Introduction PWA Controller Synthesis Numerical Example Conclusion Tunnel Diode First step: Consider the following system ˙x2 =    −0.25x2 + 0.05x1, x2 < 0.2 0.1x2 − 0.07 + 0.05x1, 0.2 < x2 < 0.6 −0.2x2 + 0.11 + 0.05x1, x2 > 0.6 Samadi, Rodrigues Backstepping for Piecewise Affine Systems
  28. 28. Outline Introduction PWA Controller Synthesis Numerical Example Conclusion Tunnel Diode First step: Consider the following system ˙x2 =    −0.25x2 + 0.05x1, x2 < 0.2 0.1x2 − 0.07 + 0.05x1, 0.2 < x2 < 0.6 −0.2x2 + 0.11 + 0.05x1, x2 > 0.6 Lyapunov function: V (x2) = 1 2(x2 − 0.6429)2 Samadi, Rodrigues Backstepping for Piecewise Affine Systems
  29. 29. Outline Introduction PWA Controller Synthesis Numerical Example Conclusion Tunnel Diode First step: Consider the following system ˙x2 =    −0.25x2 + 0.05x1, x2 < 0.2 0.1x2 − 0.07 + 0.05x1, 0.2 < x2 < 0.6 −0.2x2 + 0.11 + 0.05x1, x2 > 0.6 Lyapunov function: V (x2) = 1 2(x2 − 0.6429)2 The following control input can stabilize this system to x2 = 0.6429 x1 = γ(x2) = 0.3714 − 4.8344(x2 − 0.6429) Samadi, Rodrigues Backstepping for Piecewise Affine Systems
  30. 30. Outline Introduction PWA Controller Synthesis Numerical Example Conclusion Tunnel Diode Second step: Construct the Lyapunov function Vγ(x) = 1 2 (x2−0.6429)2 + 1 2 (x1−0.3714+4.8344(x2−0.6429))2 Samadi, Rodrigues Backstepping for Piecewise Affine Systems
  31. 31. Outline Introduction PWA Controller Synthesis Numerical Example Conclusion Tunnel Diode Second step: Construct the Lyapunov function Vγ(x) = 1 2 (x2−0.6429)2 + 1 2 (x1−0.3714+4.8344(x2−0.6429))2 Solve the SOS program to compute the control input(α = 0.25): u =    −0.35009 + 1.2572x1 − 0.1216x2, x2 < 0.2 −0.34175 + 1.2603x1 − 0.20165x2, 0.2 < x2 < 0.6 −0.3784 + 1.2567x1 − 0.13739x2, x2 > 0.6 Samadi, Rodrigues Backstepping for Piecewise Affine Systems
  32. 32. Outline Introduction PWA Controller Synthesis Numerical Example Conclusion Tunnel Diode Simulation: x(0) = [0.5 0.1]T Samadi, Rodrigues Backstepping for Piecewise Affine Systems
  33. 33. Outline Introduction PWA Controller Synthesis Numerical Example Conclusion Conclusion: A backstepping technique was developed for continuous and discontinuous PWA systems. Samadi, Rodrigues Backstepping for Piecewise Affine Systems
  34. 34. Outline Introduction PWA Controller Synthesis Numerical Example Conclusion Conclusion: A backstepping technique was developed for continuous and discontinuous PWA systems. The proposed technique consists of a series of convex problems. Therefore, it is computationally efficient. Samadi, Rodrigues Backstepping for Piecewise Affine Systems
  35. 35. Outline Introduction PWA Controller Synthesis Numerical Example Conclusion Conclusion: A backstepping technique was developed for continuous and discontinuous PWA systems. The proposed technique consists of a series of convex problems. Therefore, it is computationally efficient. A stabilizing controller was designed for the tunnel diode example by the proposed method. Samadi, Rodrigues Backstepping for Piecewise Affine Systems

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