1. IAAC International Symposium in Systems & Control, 7-8 October 2013, Technion, Haifa, Israel P-O Gutman: Constrained control of uncertain linear time-invariant systems: an interpolation based approach Constrained control of uncertain linear time- invariant systems: an interpolation based approach Per-Olof Gutman Abstract: In this paper, a novel approach to control uncertain discrete-time linear time-invariant systems with polytopic state and control constraints is proposed. The main idea is to use interpolation. The control law has an implicit and explicit form. In the implicit form, at each time instant, at most two linear programming problems are solved on-line. In the explicit form, the control law is given as a piecewise a-ne and continuous function of the state. The design method can be seen as a computationally favorable alternative to optimization-based control schemes such as Model Predictive Control. Proofs of recursive feasibility and asymptotic stability are given. Several simulations demonstrate the performance, also in comparison with MPC. Ext- ensions include output feedback, LPV and time-varying systems, and ellipsoidal constraint sets. Main reference: Hoai-Nam Nguyen, Constrained control of uncertain, time-varying systems: an interpolation based approach, accepted for publication as a Springer book, Lecture Notes in Control and Information Sciences, 2014.

2. IAAC International Symposium in Systems & Control, 7-8 October 2013, Technion, Haifa, Israel P-O Gutman: Constrained control of uncertain linear time-invariant systems: an interpolation based approach Outline Uncertainty and disturbances Output feedback Interpolation control via LMI Interpolation with cost Example: truck-dolly-trailer Conclusions References Problem formulation Constrained control MPC Vertex control Interpolation based control Maximal admissible set Control invariant set Implicit solution Explicit solution

3. IAAC International Symposium in Systems & Control, 7-8 October 2013, Technion, Haifa, Israel P-O Gutman: Constrained control of uncertain linear time-invariant systems: an interpolation based approach Problem formulation under the polytopic state and control constraints  Regulate to the origin  Extensions - Polytopic uncertainty and polytopic disturbances - Output feedback, by non-minimal state space representation with x T (k) = [y(k) y(k-1) … u(k-1) ….] - Trajectory tracking - Ellipsoidal constraint sets

4. IAAC International Symposium in Systems & Control, 7-8 October 2013, Technion, Haifa, Israel P-O Gutman: Constrained control of uncertain linear time-invariant systems: an interpolation based approach Constrained control – an overview Many solutions, among them Anti-reset windup, and over-ride control - Ad-hoc Optimal control - Almost always open loop solution Model Predictive Control - Implicit: optimal control problem over a finite receding horizon solved at each sampling instant - Explicit: piecewise affine state feedback control law computed off-line - Extends with complexity to the uncertain plant case Vertex control (Gutman and Cwikel, 1986) - Computationally cheap with one LP-problem per sampling instant - Covers the uncertain plant case with no additional complexity - No optimization criterion

5. IAAC International Symposium in Systems & Control, 7-8 October 2013, Technion, Haifa, Israel P-O Gutman: Constrained control of uncertain linear time-invariant systems: an interpolation based approach

6. IAAC International Symposium in Systems & Control, 7-8 October 2013, Technion, Haifa, Israel P-O Gutman: Constrained control of uncertain linear time-invariant systems: an interpolation based approach Unconstrained LQ in central orange cell:

7. IAAC International Symposium in Systems & Control, 7-8 October 2013, Technion, Haifa, Israel P-O Gutman: Constrained control of uncertain linear time-invariant systems: an interpolation based approach on

8. IAAC International Symposium in Systems & Control, 7-8 October 2013, Technion, Haifa, Israel P-O Gutman: Constrained control of uncertain linear time-invariant systems: an interpolation based approach

9. IAAC International Symposium in Systems & Control, 7-8 October 2013, Technion, Haifa, Israel P-O Gutman: Constrained control of uncertain linear time-invariant systems: an interpolation based approach

10. IAAC International Symposium in Systems & Control, 7-8 October 2013, Technion, Haifa, Israel P-O Gutman: Constrained control of uncertain linear time-invariant systems: an interpolation based approach

11. IAAC International Symposium in Systems & Control, 7-8 October 2013, Technion, Haifa, Israel P-O Gutman: Constrained control of uncertain linear time-invariant systems: an interpolation based approach Advantage fast on-line computations Challenges computation of vertex control values u i at vertices slow convergence, essentially P-control

12. IAAC International Symposium in Systems & Control, 7-8 October 2013, Technion, Haifa, Israel P-O Gutman: Constrained control of uncertain linear time-invariant systems: an interpolation based approach or, in a similar way, for any other feedback control

13. IAAC International Symposium in Systems & Control, 7-8 October 2013, Technion, Haifa, Israel P-O Gutman: Constrained control of uncertain linear time-invariant systems: an interpolation based approach

14. IAAC International Symposium in Systems & Control, 7-8 October 2013, Technion, Haifa, Israel P-O Gutman: Constrained control of uncertain linear time-invariant systems: an interpolation based approach 

15. IAAC International Symposium in Systems & Control, 7-8 October 2013, Technion, Haifa, Israel P-O Gutman: Constrained control of uncertain linear time-invariant systems: an interpolation based approach It might be desirable to make u as near u o as possible by minimizing c. Let  Note: Clearly x v +   C N and x o +   xv+xv+ time: k+1

16. IAAC International Symposium in Systems & Control, 7-8 October 2013, Technion, Haifa, Israel P-O Gutman: Constrained control of uncertain linear time-invariant systems: an interpolation based approach, cont’d Recall: with x v +   C N and x o +   Since the origin  , the vertex control decomposition is feasible: x(k+1) =  (k+1)v(k+1), where v   C N Then, clearly, c * (k+1)≤  (k+1)  0, as k  , since the vertex control law is asymptotically stabilizing, and hence x(k) reaches  in final time where the stabilizing local control law u o = Kx takes over, with x remaining in . v v  v

17. IAAC International Symposium in Systems & Control, 7-8 October 2013, Technion, Haifa, Israel P-O Gutman: Constrained control of uncertain linear time-invariant systems: an interpolation based approach Calculation of c * Non-linear optimization Linear Programming:

18. IAAC International Symposium in Systems & Control, 7-8 October 2013, Technion, Haifa, Israel P-O Gutman: Constrained control of uncertain linear time-invariant systems: an interpolation based approach

19. IAAC International Symposium in Systems & Control, 7-8 October 2013, Technion, Haifa, Israel P-O Gutman: Constrained control of uncertain linear time-invariant systems: an interpolation based approach Advantage fast on-line computations Challenges computation of vertex control values u i at vertices slow convergence, essentially P-control

20. IAAC International Symposium in Systems & Control, 7-8 October 2013, Technion, Haifa, Israel P-O Gutman: Constrained control of uncertain linear time-invariant systems: an interpolation based approach The vertex control law is but one of several possible in C N \  Alternatively, steer the state s.t. maximal contraction w.r.t.  C N is achieved, recalling that the Lyaponov function level curves of the vertex control law are shrunken images of  C N. Choose u such that the Minkowsky functional

21. IAAC International Symposium in Systems & Control, 7-8 October 2013, Technion, Haifa, Israel P-O Gutman: Constrained control of uncertain linear time-invariant systems: an interpolation based approach 1.Measure the state x(k) 2.Determine, by LP, the optimal c *, x v *, x o *, s.t. x=c * x v * +(1-c * ) x o * 3.Find u v, by LP, as the minimizer of the Minkowski functional 4. 5. What for the next sampling instant k:= k+1

22. IAAC International Symposium in Systems & Control, 7-8 October 2013, Technion, Haifa, Israel P-O Gutman: Constrained control of uncertain linear time-invariant systems: an interpolation based approach Advantage fast on-line computations Challenges pre-computation of vertex control values u i at vertices slow convergence, essentially P-control Comp. time [ms]/sampling interval

23. IAAC International Symposium in Systems & Control, 7-8 October 2013, Technion, Haifa, Israel P-O Gutman: Constrained control of uncertain linear time-invariant systems: an interpolation based approach

24. IAAC International Symposium in Systems & Control, 7-8 October 2013, Technion, Haifa, Israel P-O Gutman: Constrained control of uncertain linear time-invariant systems: an interpolation based approach

25. IAAC International Symposium in Systems & Control, 7-8 October 2013, Technion, Haifa, Israel P-O Gutman: Constrained control of uncertain linear time-invariant systems: an interpolation based approach

26. IAAC International Symposium in Systems & Control, 7-8 October 2013, Technion, Haifa, Israel P-O Gutman: Constrained control of uncertain linear time-invariant systems: an interpolation based approach Comparison with MPC Explicit Interpolating Control: 25 cells Explicit MPC: 97 cells

27. IAAC International Symposium in Systems & Control, 7-8 October 2013, Technion, Haifa, Israel P-O Gutman: Constrained control of uncertain linear time-invariant systems: an interpolation based approach

28. IAAC International Symposium in Systems & Control, 7-8 October 2013, Technion, Haifa, Israel P-O Gutman: Constrained control of uncertain linear time-invariant systems: an interpolation based approach

29. IAAC International Symposium in Systems & Control, 7-8 October 2013, Technion, Haifa, Israel P-O Gutman: Constrained control of uncertain linear time-invariant systems: an interpolation based approach

30. IAAC International Symposium in Systems & Control, 7-8 October 2013, Technion, Haifa, Israel P-O Gutman: Constrained control of uncertain linear time-invariant systems: an interpolation based approach

31. IAAC International Symposium in Systems & Control, 7-8 October 2013, Technion, Haifa, Israel P-O Gutman: Constrained control of uncertain linear time-invariant systems: an interpolation based approach

32. IAAC International Symposium in Systems & Control, 7-8 October 2013, Technion, Haifa, Israel P-O Gutman: Constrained control of uncertain linear time-invariant systems: an interpolation based approach

33. IAAC International Symposium in Systems & Control, 7-8 October 2013, Technion, Haifa, Israel P-O Gutman: Constrained control of uncertain linear time-invariant systems: an interpolation based approach

34. IAAC International Symposium in Systems & Control, 7-8 October 2013, Technion, Haifa, Israel P-O Gutman: Constrained control of uncertain linear time-invariant systems: an interpolation based approach

35. IAAC International Symposium in Systems & Control, 7-8 October 2013, Technion, Haifa, Israel P-O Gutman: Constrained control of uncertain linear time-invariant systems: an interpolation based approach

36. IAAC International Symposium in Systems & Control, 7-8 October 2013, Technion, Haifa, Israel P-O Gutman: Constrained control of uncertain linear time-invariant systems: an interpolation based approach

37. IAAC International Symposium in Systems & Control, 7-8 October 2013, Technion, Haifa, Israel P-O Gutman: Constrained control of uncertain linear time-invariant systems: an interpolation based approach

38. IAAC International Symposium in Systems & Control, 7-8 October 2013, Technion, Haifa, Israel P-O Gutman: Constrained control of uncertain linear time-invariant systems: an interpolation based approach

39. IAAC International Symposium in Systems & Control, 7-8 October 2013, Technion, Haifa, Israel P-O Gutman: Constrained control of uncertain linear time-invariant systems: an interpolation based approach

40. IAAC International Symposium in Systems & Control, 7-8 October 2013, Technion, Haifa, Israel P-O Gutman: Constrained control of uncertain linear time-invariant systems: an interpolation based approach

41. IAAC International Symposium in Systems & Control, 7-8 October 2013, Technion, Haifa, Israel P-O Gutman: Constrained control of uncertain linear time-invariant systems: an interpolation based approach Interpolation with cost

42. IAAC International Symposium in Systems & Control, 7-8 October 2013, Technion, Haifa, Israel P-O Gutman: Constrained control of uncertain linear time-invariant systems: an interpolation based approach

43. IAAC International Symposium in Systems & Control, 7-8 October 2013, Technion, Haifa, Israel P-O Gutman: Constrained control of uncertain linear time-invariant systems: an interpolation based approach

44. IAAC International Symposium in Systems & Control, 7-8 October 2013, Technion, Haifa, Israel P-O Gutman: Constrained control of uncertain linear time-invariant systems: an interpolation based approach

45. IAAC International Symposium in Systems & Control, 7-8 October 2013, Technion, Haifa, Israel P-O Gutman: Constrained control of uncertain linear time-invariant systems: an interpolation based approach A novel interpolation between a global vertex control law and a local control law, that may be locally optimal. A method to avoid the explicit computation of the vertex control values. Like MPC, the new controller tends to get the state away from the constraints when near them, and satisfy performance specifications when near the set point. Proofs of constrained stability for uncertain plants and bounded disturbances, and output feedback Like MPC, the new control law is affine over a polyhedral partition of the feasible control invariant set. The interpolating control law is considerably simpler than MPC with fewer polyhedral cells in the explicit case; and, in the implicit case, with extremely simple and fast LP-computations whose computational requirements are orders of magnitude less than MPC. Extension to LMI based interpolating control with ellipsoidal state constraint sets. Extension to interpolating control with quadratic cost. Extensions to time-varying and LPV systems.

46. IAAC International Symposium in Systems & Control, 7-8 October 2013, Technion, Haifa, Israel P-O Gutman: Constrained control of uncertain linear time-invariant systems: an interpolation based approach