Lecture 19: Model Predictive Control

Transcription

1 Lecture 19: Model Predictive Control

2 Constrained Optimal Control Problem minimize J = s.t. l(x t, u t ) t=0 x t+1 = f (x t, u t ), t; x 0 is given u t U t, x t X t, t Time-invariant or varying dynamics f (assume f (0, 0) = 0) U t, X t : feasible sets of control & state at time t (assume 0 U t, 0 X t ) - Example: u i [u i, u i ] (box); {x Gx g} (polytope) l(, ) 0: running cost (assume l(0, 0) = 0) - Example: l(x, u) = x T Qx + u T Ru for Q 0 and R 0 Denote J (x 0 ) the optimal cost 2 / 19

3 Convex Constrained Optimal Control Problem minimize J = l(x t, u t ) t=0 s.t. x t+1 = Ax t + Bu t, t; x 0 is given u t U t, x t X t, t Affine dynamics (could be time-varying and with additive perturbations) Both U t and X t are convex sets, t Running cost l(x, u) 0 is a convex function of x and u Convex optimization problem ( -dimensional) Optimal cost J (x 0 ) is a convex function of x 0 Generally, still very difficult to solve 3 / 19

4 Two Solution Methods Dynamic Programming: find value function J ( ) via Bellman eq. J (x) = Assume U t U, X t = X inf [l(x, u) + u U, Ax+Bu X J (Ax + Bu)], x X Optimal control policy u ( ) also obtained from Bellman equation Difficult to solve and represent J ( ); requires infinite number of iterations Greedy Control: minimize immediate cost by choosing control as { u t (x t ) = arg min l(xt, u) Ax t + Bu X } u U Very easy to compute, but often performs poorly Example: x t+1 = 2x t + u t, l(x t, u t ) = x t 2 + u t 2 4 / 19

5 Finite Horizon Approximation minimize J N (x 0 ) = N 1 t=0 l(x t, u t ) s.t. x t+1 = Ax t + Bu t, t = 0,..., N 1; x 0 is given, x N = 0 u t U, x t X, t = 0,..., N 1 Solve over a finite horizon N, with an additional terminal constraint Apply obtained optimal control u 0,..., u N 1, followed by 0, 0,... Resulting in a cost J N (x 0) that is an approximation of J (x 0 ) Online (specific) solution: optimal u t for a specific x 0 Offline (policy) solution: optimal state feedback policy u t ( ) 5 / 19

6 Important Special Case minimize J N (x 0 ) = N 1 t=0 (x T t Qx t + u T t Ru t ) s.t. x t+1 = Ax t + Bu t, t = 0,..., N 1; x 0 is given, x N = 0 Gu t g 0, Hx t h 0, t = 0,..., N 1 Linear dynamics, quadratic cost, polytopic state and control constraints With u = [ u0 T un 1] T T [ ] and x = x T 1 xn T T, the problem is a linearly constrained quadratic programming (QP) problem : min u,x x T Qx + u T Ru s.t. x = Eu + Fx 0, Gu g 0 1, Hx h 0 1, x N = 0 An instance of the more general muliparametric programming Multi-Parametric Programming Matlab Toolbox 6 / 19

7 Important Special Case (cont.) Online solution for any specific x 0 is very efficient e.g. Matlab command quadprog Offline (policy) solution has a nice structure Feasible initial set X 0 : set of all x 0 for which solution is feasible. There exists a polyhedral partition of X 0, X 0 = i X i, s.t. JN (x) is continuous, convex, and piecewise quadratic J N(x 0 ) = x T 0 P i x 0 + 2p T i x 0 + r i, x 0 X i u (x) is continuous, and piecewise affine u (x 0 ) = K i x 0 + k i, x 0 X i If l(x, u) is affine, then both JN ( ) and u N ( ) are piecewise affine The explicit linear quadratic regulator for constrained systems, A Bemporad, M Morari, V Dua, EN Pistikopoulos, Automatica, 38 (1), / 19

8 Model Predictive Control (MPC) Idea: (also called receding horizon control) at any time t, solve the problem over a prediction horizon N 1 minimize J N (x t ) = l(x t+k t, u t+k t ) k=0 s.t. x t+k+1 t = f (x t+k t, u t+k t ), k = 0,..., N 1 u t+k t U, x t+k t X, k = 0,..., N 1 x t t = x t, x t+n t = 0 x t+k+1 t and u t+k t, 0 k N 1, are planned states and controls Implement only the first control u t t at time t u t = u t t (x t), x t+1 = f (x t, u t ) t t + 1 and repeat the above procedure Tutorial overview of model predictive control, J.B. Rawlings, IEEE Control Systems, / 19

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10 Explicit MPC Linear dynamics f (x, u) = Ax + Bu Quadratic (or linear) running cost l(x, u) Polyhedral constraint sets X, U From previous discussions, we know At each t, solve a quadratic/linear program Optimal control u ( ) at any time is piecewise affine Key is to determine which polyhedral partition x t belongs to Warning: Complexity (number of partitions) could still be high Methods exist to reduce number of partitions with sacrifice in optimality Online solution computed in O(N(n + m) 3 ) time using interior point method 10 / 19

11 Example (Boyd) Random 3-state 2-input LTI system (A, B) with l(x, u) = x 2 + u 2 and constraint x 1, u 1; x 0 = [ ] T. costs vs. time horizon N Finite horizon approximation (dashed) becomes infeasible for N 9 MPC performance V N (x 0) (solid) very close to V (x 0 ) for all N / 19

12 Stabilization by MPC Assumption: l(x, u) 0 implies x 0 and u 0 For the given x 0, the N-horizon problem at t = 0 is feasible Fact: the closed-loop solution u t and x t under the MPC satisfy Persistent feasibility: u t U, x t X, t Stability: u t 0 and x t 0 as t Proof: show that V N (x t) is a Lyapunov function Let u 0 0,..., u N 1 0, and x 1 0,..., x N 0 = 0 be the solutions at t = 0 After applying u 0 = u 0 0, x 1 = x 1 0 At time t = 1, u 1 0,..., u N 1 0, 0, and x 2 0,..., x N 0 = 0, 0 are feasible V N (x 1) V N (x 0) l(x 0, u 0) V N (x t) 0 non-increasing implies l(x t, u t) 0 as t 12 / 19

13 Relaxed Terminal Constraint Imposing x t+n t = 0 may make the problem infeasible for short horizon N Control invariant set C X for the dynamics x + = f (x, u) satisfies x C, u U s.t. f (x, u) C Equivalently, C pre(c), the 1-step backward reachable set of C Fact: If we replace the terminal constraint x t+n t = 0 in the MPC problem by x t+n t X N, where X N is a (maximum) control invariant set, then the MPC solution is persistent feasible if x 0 X 0, where Note that X N X 0 X k = pre(x k+1 ) X, k = N 1,..., 0 13 / 19

14 Stability Condition With the relaxed terminal condition x t+n t X N, how to guarantee stability of MPC solution? Idea: Add a terminal cost ρ N (x t+n t ) in J N, where ρ N ( ) satisfies It is a locally positive definite (LPD) funciton on X N For any x X N, there exists u U such that ρ N (x) ρ N (f (x, u)) l(x, u) Fact: with the above terminal cost, the MPC solution is stable, x 0 X 0 Proof: Show that V N (x t+1) V N (x t) l(x t, u t ), t, still holds 14 / 19

15 MPC with Uncertainty Optimal control problem with uncertainty minimize J = l(x t, u t, w t ) t=0 s.t. x t+1 = f (x t, u t, w t ), t; x 0 is given u t U, x t X, t w t models perturbation at time t Minimize E[J] if w t is a stochastic process with known distributions MPC can be adapted to solve the above problem Assume forecast of perturbations over the lookahead horizon is available Most MPC practical applications follow this formulation 15 / 19

16 MPC Solution of Switched LQR Problem ) minimize J = (xt T Q σt x t + ut T R σt u t t=0 s.t. x t+1 = A σt x t + B σt u t, t = 0, 1,... x 0 is given Continuous control u t and discrete control (mode) σ t No input or state constraints Optimal cost J (x 0 ) can be very complicated MPC solution: Solve N-horizon value function JN ( ) and optimal control policy u N ( ) We know JN ( ) is piecewise quadratic and u N ( ) is piecewise linear Adopt un ( ) as the static state-feedback control policy: u t = un (x t) Cost of closed-loop solution approaches J as N 16 / 19

17 MPC in Switching Stabilization of SLSs To stabilize SLS x t+1 = A σt x t + B σt u t, at any time t solve the problem: min (x t+n t ) T P(x t+n t ) u t+k t,σ t+k t s.t. x t+k+1 t = A σt+k t x t+k t + B σt+k t u t+k t, k = 0,..., N 1 x t t = x t Minimize a quadratic function of the state N steps later MPC solution can be applied every N time steps V (x) = x T Px becomes a quadratic periodic Lyapunov function Periodic stabilization of discrete-time switched linear systems, D.-H. Lee and J. Hu, TACl, / 19

18 Constrained Switched LQR Problem minimize J = ) (xt T Q σt x t + ut T R σt u t t=0 s.t. x t+1 = A σt x t + B σt u t, t = 0, 1,... G σt u t g 0,σt, H σt x t h 0,σt, t = 0, 1,... x 0 is given Polyhedral state and control constraints, possibly mode dependent Finite horizon optimal cost JN ( ) continuous, piecewise quadratic, but in general not convex Optimal control u ( ) is piecewise affine, but not continuous Predictive Control for Linear and Hybrid Systems, F. Borrelli, A. Bemporad and M. Morari, / 19

19 References Tutorial overview of model predictive control, J.B. Rawlings, IEEE Control Systems, MPC Matlab Toobox Predictive Control for Linear and Hybrid Systems, F. Borrelli, A. Bemporad and M. Morari, Multi-Parametric Matlab Toolbox Stabilizing model predictive control of hybrid systems, M. Lazar, W. Heemels, S. Weiland, and A. Bemporad, TAC / 19