Sampled-Data Piecewise Affine Slab Systems: A Time-Delay Approach

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Sampled-Data Piecewise Affine Slab Systems: A Time-Delay Approach

  1. 1. Sampled-Data Piecewise Affine Slab Systems: A Time-Delay Approach Behzad Samadi Luis Rodrigues Department of Mechanical and Industrial Engineering Concordia University ACC 2008, Seattle, WA
  2. 2. Outline of Topics
  3. 3. Practical Motivation c Quanser Memoryless Nonlinearities Saturation Dead Zone Coulomb & Viscous Friction
  4. 4. Motivational example Toycopter, a 2 DOF helicopter model
  5. 5. Motivational 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)
  6. 6. 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
  7. 7. Objective To propose a stability analysis method for sampled-data PWA systems using convex optimization time-delay approach Continuous−time PWA systems PWA controller Hold
  8. 8. Piecewise Affine Systems PWA systems are in general nonsmooth nonlinear systems.
  9. 9. 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
  10. 10. Sampled-Data PWA Systems: A Time-Delay Approach PWA slab system ˙x = Ai x + ai + Bu, for x ∈ Ri with the region Ri defined as Ri = {x | σi < CRx < σi+1}, where CR ∈ R1×n and σi for i = 1, . . . , M + 1 are scalars such that σ1 < σ2 < . . . < σM+1
  11. 11. Sampled-Data PWA Systems: A Time-Delay Approach PWA slab system ˙x = Ai x + ai + Bu, for x ∈ Ri with the region Ri defined as Ri = {x | σi < CRx < σi+1}, where CR ∈ R1×n and σi for i = 1, . . . , M + 1 are scalars such that σ1 < σ2 < . . . < σM+1 Continuous-time PWA controller u(t) = Ki x(t) + ki , x(t) ∈ Ri
  12. 12. 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.
  13. 13. Sampled-Data PWA Systems: A Time-Delay Approach The closed-loop system can be rewritten as ˙x(t) = Ai x(t) + ai + B(Ki x(tk) + ki ) + Bw, 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.
  14. 14. Sampled-Data PWA Systems: A Time-Delay Approach Theorem (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}
  15. 15. Sampled-Data PWA Systems: A Time-Delay Approach for all i ∈ I(0), Ωi + τMM1i + τMM2i < 0   Ωi + τMM1i τM Ni 0 τM NT i 0 −τMR   < 0 for all i /∈ I(0), ¯Λi ≻ 0, Ωi + τMM1i + τMM2i < 0     Ωi + τMM1i τM   Ni 0 0   τM NT i 0 0 −τMR     < 0
  16. 16. 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
  17. 17. 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
  18. 18. 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
  19. 19. Summary of the contributions: Formulating stability analysis of sampled-data PWA slab systems as a convex optimization problem Future work: Formulating controller synthesis for sampled-data PWA slab systems as a convex optimization problem

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