ECE7850 Lecture 8. Nonlinear Model Predictive Control: Theoretical Aspects

Transcription

1 ECE7850 Lecture 8 Nonlinear Model Predictive Control: Theoretical Aspects Model Predictive control (MPC) is a powerful control design method for constrained dynamical systems. The basic principles and theoretical results for MPC are almost the same for most nonlinear systems, including discrete-time hybrid systems. The particular underlying model (e.g. linear or switched affine, or piecewise affine, or hybrid systems), mainly affects the computational aspects for MPC. This lecture focuses on principles for general nonlinear MPC; next lecture will cover computational aspects, especially for MPC of hybrid systems. 1

2 Lecture Outline Formulation and Related Definitions for General MPC Persistent Feasibility of MPC Stability Analysis of MPC Analysis Without Terminal Constraint/Cost Other Selected Topics 2

3 General discrete-time nonlinear systems: Formulation and Related Definitions x(t +1)=f(x(t),u(t)) y(t) =h(x(t)),t Z + (1) State and Control constraints: x(t) X u(t) U (2) In general, X R n Q and U R m Σ have both continuous and discrete components. To simplify presentation, we assume X R n (but u can have both continuous and discrete components) Assume full state information available, unless otherwise stated. (e.g. y(t) =x(t) or h( ) is bijection) Formulation and Related Definitions 3

4 Basic ideas for MPC: Receding Horizon Control (RHC) At time t, solves a finite horizon optimal control problem based on the system model Apply the first step of the optimal control sequence At time t +1, horizon is shifted and the optimal control problem is solved again using newly obtained state measurements Formulation and Related Definitions 4

5 Formalizing the idea: At time t: solve the following N-horizon optimal control problem: P N (x(t)) : V N (x(t)) = min u J(x(0), u) J f (x N )+ N 1 k=0 l(x k,u k ) subj. to: x k+1 = f(x k,u k ),k =0,...,N 1 x k X, u k U, k =0,...,N 1 x N X f, x 0 = x(t) (3) X f X: terminal state constraint set Assume nonnegative cost functions: J f : X R + and l : X U R + Given x(0), optimal control sequence u 0,...,u N 1 can be found via numerical optimization Formulation and Related Definitions 5

6 Problem (3) can also be solved using DP, leading to optimal control laws μ j(z) = argmin u U(z) {l(z, u)+v j 1 (f(z, u))},j =1,...,N Finding {u k } is much easier than finding μ j ( ); Why we care about {μ j ( )}? For many cases, μ j (z) may have appealing analytical structures that can simplify the online computation for {u k } (e.g.: explicit MPC) Enable various analysis for feasibility, performance, and stability of MPC At any time ˆt, if the state is x(ˆt), then MPC controller will apply uˆt = μ N ( x (ˆt )) Formulation and Related Definitions 6

7 Two fundamental issues for MPC Persistent Feasibility: Problem P(x(t)) remains feasible for all t Closed-Loop Stability: x(t +1)=f(x(t),μ N (x(t))) is stable We need to introduce some key concepts and definitions: One-step backward reachable set: Pre(S) ={x R n : u U s.t. f(x, u) S} One-step forward reachable set: Reach(S) ={x R n : u U, z Ss.t. x = f(z, u) S} Formulation and Related Definitions 7

8 N-step backward reachable set subject to system constraints: R j+1(s) =Pre ( R j (S) ) X, R 0 (S) =S (4) N-step forward reachable set subject to system contratins: R + j+1 (S) =Reach ( R + j (S)) X, R + 0 (S) =S (5) Positive Invariant Set: Given an constrained autonomous system x(t +1)=f a (x(t)), t Z +,x(t) X (6) a set O X is called a positive invariant set if x(0) O x(t) O, t Z + Formulation and Related Definitions 8

9 Maximal Positive Invariant Set O : positive invariant + contains all invariant sets contained in X C X is called a Control Invariant Set for the constrained system (1) if x(t) C u(t) U such that f(x(t),u(t)) C, t Z + Maximal control invariant set C : control invariant + contains all control invariant sets contained in X Efficient algorithms to compute (control) invariant sets are available for particular classes of constrained systems Formulation and Related Definitions 9

10 Persistent Feasibility of MPC Feasible set X k : the set of feasible state x k at prediction step k for which (3) is feasible X k = {x X : u U, s.t. f(x, u) X k+1 }, with X N = X f (7) can be equivalently defined as: X k = Pre(X k+1 ) X, with X N = X f Persistent Feasibility: start from any x(0) X 0 evolve under MPC control law, i.e., x(t +1)=f(x(t),μ N(x(t))) feasibility is guaranteed at all time, i.e. x(t) X 0, for all t Z +. Persistent Feasibility of MPC 10

11 Lemma 1 If X 1 is control invariant, then receding horizon control problem P N (x(t)) is persistently feasible. Proof: X 1 Pre(X 1 ) X = X 0 x X 0, apply the first-step of MPC control u 0.Wehavex+ = f(x, u 0 ) X 1 X 0 However, the above condition is hard to verify as X 1 is defined recursively from X f. A condition directly imposed on X f is desirable. Persistent Feasibility of MPC 11

12 Theorem 1 If X f is control invariant, then the receding horizon optimization P N (x(t)) is persistently feasible. Persistent Feasibility of MPC 12

13 Stability of MPC Stability analysis of MPC: closed-loop system: x(t +1)=f(x(t),μ N (x(t)) persistently feasible and stable characterize domain of attraction Assumption 1 (i) X and U contains origin in their interior. (ii) x =0,u=0is an equilibrium f(0, 0) = 0; (iii) β l z l(z, u) β + l z, for all z X, u U. Stability of MPC 13

14 Theorem 2 Assume conditions in Assumption 1 hold and 1. X f X is control invariant 2. Terminal cost function J f is a local control Lyapunov function satisfying J f (z) β f z, and μ such that: J f (z) J f (f(z, μ(z))) l(z, μ(z)), z X f (8) Then the origin of the closed-loop system under MPC control is exponentially stable with region of attraction X 0. Sketch of proof Pick an arbitrary z X 0, let u 0,...,u N 1 and x 0,x 1,...,x N be the corresponding optimal control and trajectory. Stability of MPC 14

15 Need to show V N (z) is a control Lyapunov function on X 0 Let ˆμ be the law satisfying (8). Construct a new control sequence û = {u 1,u 2,...,u N 1, ˆμ(x N )}. Starting from x 1, the control sequence û results in state trajectory {x 1,x 2,...,x N,f(x N, ˆμ(x N))} By optimality of V N,wehave V N (x 1 ) J N(x 1, û) =N 1 l(x k,u k )+l(x N, ˆμ(x N )) + J f(f(x N, ˆμ(x N ))) k=1 = V N (x 0 ) l(x 0,u 0 ) J f(x N )+l(x N, ˆμ(x N )) + J f(f(x N, ˆμ(x N ))) Notice that x 0 = z, x 1 = f(z, μ N(z)), and x N X f. By (8), we have V N (z) V N (f(z, μ N(z))) l(z, u 0) β l z The conditions on l and J f, along with the boundedness of state trajectory within the N prediction steps clearly indicates that V N (z) β z, forz X 0 and some β<. The stability the follows from the ECLF theorem. Stability of MPC 15

16 Analysis Without Terminal Constraint/Cost Computation of control invariant set is very challenging (except for some simple cases such as linear systems with polytopic constraints) Even the control invariant set is available, using it as a terminal constraint set can lead to poor numerical performance for both online and offline optimization algorithms Using control invariant set as terminal constraint set also shrinks feasible set X 0. Analysis Without Terminal Constraint/Cost 16

17 Lemma 2 (i) If X f = R n, then C X 0 X 1 X N = X f ; (ii) If X f is control invariant, then X C X 0 X 1 X N = X f Persistent feasibility and closed-loop stability can also be guaranteed without terminal constraint set Analysis Without Terminal Constraint/Cost 17

18 We consider MPC with no terminal constraint: P nt N (z) : V N (z) = min u J N (z, u) J f (x N )+ N 1 k=0 l(x k,u k ) subj. to: x k+1 = f(x k,u k ),k =0,...,N 1 x k X, u k U, k =0,...,N 1 x 0 = z (9) With slight abuse of notation, let μ N be the MPC law, i.e., μ N (z) = argmin u U(z) {l(z, u) + V N 1 (f(z, u))}, where V N is defined above. CL-system under MPC: x(t +1)=f(x(t),μ N (x(t))) Define the infinite-horizon version of the above problem: V (z) = inf u 0,u 1,... k=0 l(x k,u k ): subj. to constraint in (9) Analysis Without Terminal Constraint/Cost 18

19 We want PN nt (x(t)) persistently feasible along cl-traj The cl-system is exponentially stable (10) We shall establish conditions to ensure (10) for two different cases: Case I: J f (z) 0, namely, MPC with neither terminal constraint nor terminal cost. Case II: J f (z) is an ECLF (MPC without terminal constraint but with nontrivial terminal cost) Assumption 2 (i) J f (z) β f + z ; (i) βl z l(z, u) β l + z ; (iii) V (z) β z, z X As discussed in Lecture Note 7, condition (iii) is almost equivalent to exponential stabilizability of the constrained system (1). Of course, Assumption 1 is always assumed as well. Analysis Without Terminal Constraint/Cost 19

20 The key property for both cases is the convergence of the optimal trajectory and value iteration as horizon length N increases Lemma 3 (Convergence of Optimal Trajectory): Under Assumption 2, x(t; z, π N ) c xγ t x z, for all z X and t =0,...,N 1 if additionally J f (z) β l z, then the above inequality also holds for t = N Without condition J f (z) βl z, the final state x(n; z, πn ) may be arbitrarily large. Example 1 x(t +1)=x(t)+u(t), fort = {0, 1,...,N 1}, with X = U = R. Let L(x, u) =x 2, ψ 0. Fix an initial state x(0) = z. It can be easily verified that the N-horizon control sequence is of the form { z, 0,...,0,c} is optimal for all c R. Therefore, the terminal state of the corresponding optimal trajectory is equal to c and can be made arbitrarily large. Analysis Without Terminal Constraint/Cost 20

21 Lemma 4 (Value Iteration Convergence): Under Assumption 2, Value iteration converges exponentially, V N (z) V (z) c V γ N V z ; N 0 such that V N (z) is an ECLF for all N N 0 ; sketch of proof: Part I: show special case with J f 0. General case can be found in zhang09 Analysis Without Terminal Constraint/Cost 21

22 Part II: show V N eventually becomes an ECLF for sufficiently large N Analysis Without Terminal Constraint/Cost 22

23 Theorem 3 (Main Result for Case I): Under Assumption 2 with J f (z) 0, there exists N 0 < such that (10) is guaranteed for all N N 0 with region of attraction L α = {z R n : V N (z) α} for any α that ensures L α X. sketch of proof: Analysis Without Terminal Constraint/Cost 23

24 Remarks for Main Result I: With neither terminal constraint nor terminal cost, persistent feasibility and cl-stability can still be guaranteed as long as the prediction horizon N is sufficiently large N can be determined by checking whether V N is an ECLF, which can be done through LMIs in some special cases Large N may cause issues for both online optimization and offline explicit MPC solutions. As a compromise, we can add the terminal cost back while still omitting the terminal constraint, which leads us to Case II. Analysis Without Terminal Constraint/Cost 24

25 Theorem 4 (Main Result for Case II): Under Assumption 2 with J f (z) being an ECLF satisfying condition (8) over some neighborhood of origin X Jf, there exists N 0 < such that (10) is guaranteed for all N N 0 with region of attraction L α = {z R n : V N (z) α} for any α that ensures L α X sketch of proof: Select a sublevel set: ˆXf = {x R n : J f (x) α} X Jf ˆX f is control invariant. Need to guarantee the optimal prediction trajectory always hits ˆX f at the end, i.e., x N ˆX f. This can be achieved by choosing a sufficiently large N. Analysis Without Terminal Constraint/Cost 25

26 Relationship With Unconstrained Problem State and control constraint sets X and U often cause significant challenge in finding control invariant sets and control Lyapunov functions. Dropping these constraints leads to unconstrained MPC that is easier to solve. Solution to the unconstrained MPC can be used to generate control invariant set and control Lyapunov function for the constrained MPC. Consider unconstrained system: x(t +1)=f(x(t),u(t)),x R n,u R m (11) Relationship With Unconstrained Problem 26

27 Unconstrained N-horizon optimal control: V nc N (z) = min u J N (z, u) = N 1 k=0 l(x k,u k ) subj. to x k+1 = f(x k,u k ),k =0,...,N 1 By Lemma 4, we know V nc N becomes an ECLF of (11) for large N Suppose V nc N 0 is an ECLF of (11), define μ nc N 0 (z) = argmin u U(z) {l(z, u)+v nc N 0 (z)} namely, VN nc 0 (z) VN nc ( ( )) 0 f z, μ nc N 0 l(z, μ nc N 0 (z)) Relationship With Unconstrained Problem 27

28 Assume: μ nc N 0 (z) β μ z,z X Due to exponential stability, unconstrained cl-system trajectory and control satisfy: x nc (t; z, μ nc N 0 ) c x r k z, u nc (t; z, μ nc N 0 ) c u r k z Lemma 5 There exists a neighborhood around the origin X nc X, such that the unconstrained cl-trajectory and control are feasible with respect to state and control constraints X and U, i.e., x nc (t; z, μ nc N 0 ) X, and u nc (t; z, μ nc N 0 ) U, z X nc, k Z + Relationship With Unconstrained Problem 28

29 Definition 1 Given a control law μ, a positive invariant set Ω of the cl-system x(t +1) = f(x(t),μ(x(t)) is called constraint admissible if Ω X, and {μ(z) :z Ω} U Theorem 5 Consider the constrained MPC problem defined in (3). Persistent feasibility and cl-stability can be guaranteed under either of the following two conditions: 1. J f (z) = V nc N 0 (z) and X f is a constraint admissible positive invariant set of the unconstrained system under control law μ nc ˆN 2. J f (z) =V nc N 0 (z), X f = R n, and N is sufficiently large Relationship With Unconstrained Problem 29

30 Remarks about Theorem 5: N 0 is the chosen to make unconstrained value function an ECLF, while N is the horizon size for the constrained MPC problem Under the first condition, X f is control invariant and V nc N is an ECLF on X f w.r.t. the constrained system. The desired result follows directly from Theorem 2. Under the second condition, there is no terminal constraint; the desired result follows from Theorem 4. Relationship With Unconstrained Problem 30

31 Summary: MPC: solve N-horizon constrained optimal control problem P N (z) and apply the first optimal control action cl-system under MPC: x(t +1)=f(x(t),μ N (x(t))) Persistent feasible: x(t) X 0, where X 0 denotes the set of initial state for which P N is feasible. Stability of MPC: cl-system asymptotically (or exponentially) stable Persistent feasibility and cl-stability are guaranteed if either of the following holds: X f is control invariant and J f is a local ECLF satisfying (8) on X f ; J f 0, X f = R n, and N is sufficiently large J f is an ECLF satisfying (8) locally and N is sufficiently large J f = VN nc 0 and X f is a constraint admissible positive invariant set of the unconstrained cl-system under μ nc N 0 J f = VN nc 0, X f = R n, and N is sufficiently large. Relationship With Unconstrained Problem 31