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Constrained control of uncertain linear timeinvariant 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. Extensions 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. 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 • Problem formulation • Constrained control MPC Vertex control • Interpolation based control Maximal admissible set Control invariant set Implicit solution Explicit solution • • • • • • • Uncertainty and disturbances Output feedback Interpolation control via LMI Interpolation with cost Example: truck-dolly-trailer Conclusions References 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 Regulate to the origin under the polytopic state and control constraints Extensions - Polytopic uncertainty and polytopic disturbances - Output feedback, by non-minimal state space representation with xT(k) = [y(k) y(k-1) … u(k-1) ….] - Trajectory tracking - Ellipsoidal constraint sets 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 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 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: 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 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 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 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 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 Challenges Advantage • fast on-line computations • computation of vertex control values ui at vertices • slow convergence, essentially P-control 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 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 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 = 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 uo as possible by minimizing c. Let time: k+1 = xv+ Note: Clearly xv+ CN and xo+ 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 xv+ CN and xo+ Since the origin , the vertex control decomposition is feasible: x(k+1) = (k+1)v(k+1), where v CN 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 uo= Kx takes over, with x remaining in . 0 v v v 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: 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. By LP, compute x(k) as the convex combination of the vertices of CN, where vi(j) denotes vertex i in sector j. 3. Compute the vertex control component uv(k)= =1 where ui denotes the pre-computed vertex control value at vertex i. 4. Determine the optimal c* by LP. 5. 6. What for the next sampling instant k:= k+1 Example: uo= [-0.5609 -0.9758]x, x , from unconstrained LQ with Q=I, R=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 Advantage • fast on-line computations Challenges • computation of vertex control values ui at vertices • slow convergence, essentially P-control 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 CN\ • Alternatively, steer the state s.t. maximal contraction w.r.t. CN is achieved, recalling that the Lyaponov function level curves of the vertex control law are shrunken images of CN. Choose u such that the Minkowsky functional 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*, xv*, xo*, s.t. x=c*xv*+(1-c*) xo* 3. Find uv, by LP, as the minimizer of the Minkowski functional 4. 5. What for the next sampling instant k:= k+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 Comp. time [ms]/sampling interval Advantage • fast on-line computations Challenges • pre-computation of vertex control values ui at vertices • slow convergence, essentially P-control 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 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 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 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 MPC: 97 cells Explicit Interpolating Control: 25 cells 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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. 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 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