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

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• 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. Outline of Topics
• 3. Practical Motivation c Quanser Memoryless Nonlinearities Saturation Dead Zone Coulomb & Viscous Friction
• 4. Motivational example Toycopter, a 2 DOF helicopter model
• 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. 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. 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. Piecewise Aﬃne Systems PWA systems are in general nonsmooth nonlinear systems.
• 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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