Further results on Robust MPC using Linear Matrix Inequalities

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1 Further results on Robust MPC using Linear Matrix Inequalities M. Lazar, W.P.M.H. Heemels, D. Muñoz de la Peña, T. Alamo Eindhoven Univ. of Technology, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands, Dept. de Inginería de Sistemas y Automática, Univ. of Seville, Seville, Spain corresponding author: m.lazar@tue.nl Keywords : robust model predictive control (MPC), linear matrix inequalities (LMIs), H control, input-to-state stability (ISS) Abstract : This paper presents a novel method for designing the terminal cost and the auxiliary control law (ACL) for robust MPC of uncertain linear systems, such that ISS is a priori guaranteed for the closed-loop system. The method is based on the solution of a set of LMIs. An explicit relation is established between the proposed method and H control design. This relation shows that the LMI-based optimal solution of the H synthesis problem solves the terminal cost and ACL problem in inf-sup MPC, for a particular choice of the stage cost. This result, which was somehow missing in the MPC literature, is of general interest as it connects well known linear control problems to robust MPC design. 1 Introduction Perhaps the most utilized method for designing stabilizing and robustly stabilizing model predictive controllers (MPC) is the terminal cost and constraint set approach [1]. This technique, which applies to both nominally stabilizing and inf-sup robust MPC schemes, relies on the off-line computation of a suitable terminal cost along with an auxiliary control law (ACL). For nominally stabilizing MPC with quadratic costs, the terminal cost can be calculated for linear dynamics by solving a discrete-time Riccati equation, with the optimal linear quadratic regulator (LQR) as the ACL [2]. In [3] it was shown that an alternative solution to the same problem, which also works for parametric uncertainties, can be obtained by solving a set of LMIs. The design of infsup MPC schemes that are robust to additive disturbances was treated in [4], where it was proven that the terminal cost can be obtained as a solution of a discrete-time H Riccati equation, for an ACL that solves the corresponding H control problem. In this article we present an LMI-based solution for obtaining a terminal cost and an ACL, such that inf-sup MPC schemes [7, 8] achieve input-to-state stability (ISS) [5] for linear systems affected by both parametric and additive disturbances. The proposed LMIs generalize the conditions in [3] to allow for additive uncertainties as well. Moreover, we establish an explicit relation between the developed solution and the LMIbased 1 optimal solution of the discrete-time H synthesis problem corresponding to a specific performance output, related to the MPC cost. This result, which was somehow 1 A similar connection is established in [4], with the difference that the Riccati-based solution to the optimal H synthesis problem is exploited, rather than the LMI-based solution; also, parametric uncertainties are not considered.

2 missing in the MPC literature, adds to the results of [4] and to the well-known connection between design of nominally stabilizing MPC schemes and the optimal solution of the LQR problem. Such results are of general interest as they connect well known linear control problems to MPC design. 2 Preliminary definitions and results 2.1 Basic notions and definitions Let R, R +, Z and Z + denote the field of real numbers, the set of non-negative reals, the set of integer numbers and the set of non-negative integers, respectively. We use the notation Z c1 and Z (c1,c 2] to denote the sets {k Z + k c 1 } and {k Z + c 1 < k c 2 }, respectively, for some c 1, c 2 Z +. For i Z +, let i = 1, N denote i = 1,..., N. For a set S R n, we denote by int(s) the interior of S. A polyhedron (or a polyhedral set) in R n is a set obtained as the intersection of a finite number of open and/or closed half-spaces. The Hölder p-norm of a vector x R n is defined as x p := ( [x] 1 p [x] n p ) 1 p for p Z[1, ) and x := max i=1,...,n [x] i, where [x] i, i = 1,..., n, is the i-th component of x and is the absolute value. For a positive definite and symmetric matrix M, denoted by M 0, M 1 2 denotes its Cholesky factor, which satisfies (M 1 2 ) M 1 2 = M 1 2 (M 1 2 ) = M and, λ min (M) and λ max (M) denote the smallest and the largest eigenvalue of M, respectively. We will use 0 and I to denote a matrix with all elements zero and the identity matrix, respectively, of appropriate dimensions. Let z := {z(l)} l Z+ with z(l) R o for all l Z + denote an arbitrary sequence. Define z := sup{ z(l) l Z + }, where denotes an arbitrary p-norm, and z [k] := {z(l)} l Z[0,k]. A function ϕ : R + R + belongs to class K if it is continuous, strictly increasing and ϕ(0) = 0. A function ϕ : R + R + belongs to class K if ϕ K and lim s ϕ(s) =. A function β : R + R + R + belongs to class KL if for each fixed k R +, β(, k) K and for each fixed s R +, β(s, ) is decreasing and lim k β(s, k) = Input-to-state stability Consider the discrete-time nonlinear system x(k + 1) = Φ(x(k), w(k), v(k)), k Z +, (1) where x(k) R n is the state and w(k) R dw, v(k) R dv are unknown disturbance inputs at the discrete-time instant k. The mapping Φ : R n R o R l R n is an arbitrary nonlinear function. We assume that Φ(0, w, 0) = 0 for all w. Let W and V be subsets of R dw and R dv, respectively. Definition 2.1 We call a set P R n robustly positively invariant (RPI) for system (1) with respect to (W, V) if for all x P it holds that Φ(x, w, v) P for all (w, v) W V. Definition 2.2 Let X with 0 int(x) be a subset of R n. We call system (1) ISS(X, W, V) if there exist a KL-function β(, ) and a K-function γ( ) such that, for each x(0) X, all w = {w(l)} l Z+ with w(l) W, l Z + and all v = {v(l)} l Z+ with v(l) V, l Z + it holds that the corresponding state trajectory of (1) satisfies x(k) β( x(0), k) + γ( v [k 1] ), k Z 1. We call the function γ( ) an ISS gain of system (1). 2

3 2.3 Input-to-state stability conditions for inf-sup robust MPC Consider the discrete-time constrained nonlinear system x(k + 1) = φ(x(k), u(k), w(k), v(k)), k Z +, (2) where x(k) X R n is the state, u(k) U R m is the control action and w(k) W R dw, v(k) V R dv are unknown disturbance inputs at the discrete-time instant k. φ : R n R m R dw R dv R n is an arbitrary nonlinear function with φ(0, 0, w, 0) = 0 for all w W. We assume that 0 int(x), 0 int(u) and W, V are bounded. Next, let F : R n R + and L : R n R m R + with F (0) = L(0, 0) = 0 be arbitrary nonlinear functions. For N Z 1 let ū [N 1] (k) := (ū(k), ū(k + 1),..., ū(k + N 1)) U N = U... U denote a sequence of future inputs and, similarly, let w [N 1] (k) W N, v [N 1] (k) V N denote some sequences of future disturbances. Consider the MPC cost J(x(k), ū [N 1] (k), w [N 1] (k), v [N 1] (k)) N 1 := F ( x(k + N)) + L( x(k + i), ū(k + i)), where x(k + i + 1) := φ( x(k + i), ū(k + i), w(k + i), v(k + i)) for i = 0, N 1 and x(k) := x(k). Let X T X with 0 int(x T ) denote a target set and define the following set of feasible input sequences: U N (x(k)) := {u [N 1] (k) U N x(k + i) X, i = 1, N 1, x(k + N) X T, i=0 x(k) := x(k), w [N 1] (k) W N, v [N 1] (k) V N }. Problem 2.3 Let X T X and N Z 1 be given. At time k Z + let x(k) X be given and infimize sup J(x(k), ū [N 1] (k), w [N 1] (k), v [N 1] (k)) w [N 1] (k) W N, v [N 1] (k) V N over all input sequences ū [N 1] (k) U N (x(k)). Assuming the infimum in Problem 2.3 exists and can be attained, the MPC control law is obtained as u MPC (x(k)) := ū (k), where denotes the optimum 2. Next, we summarize recently developed a priori sufficient conditions for guaranteeing robust stability of system (2) in closed-loop with u(k) = u MPC (x(k)), k Z +. Let h : R n R m denote an auxiliary control law (ACL) with h(0) = 0 and let X U := {x X h(x) U}. Assumption 2.4 There exist functions α 1, α 2, α 3 K and σ K such that: (i) X T X U ; (ii) X T is a RPI set for system (2) in closed-loop with u(k) = h(x(k)), k Z + ; (iii) L(x, u) α 1 ( x ) for all x X and all u U; (iv) α 2 ( x ) F (x) α 3 ( x ) for all x X T ; (v) F (φ(x, h(x), w, v)) F (x) L(x, h(x))+σ( v ), x X T, w W, v V. 2 If the infimum does not exist, one has to resort to ISS results for sub-optimal solutions, see, e.g., [6]. 3

4 In [7, 8] it was shown that Assumption 2.4 is sufficient for guaranteeing ISS of the MPC closed-loop system corresponding to Problem 2.3. Notice that although in Problem 2.3 we have presented the open-loop formulation of inf-sup MPC for simplicity of exposition, Assumption 2.4 is also sufficient for guaranteeing ISS for feedback inf-sup variants of Problem 2.3, see [7, 8] for the details. Remark 2.5 The sufficient ISS conditions of Assumption 2.4 are an extension for robust MPC of the well known terminal cost and constraint set stabilization conditions for nominal MPC, see A1-A4 in [1]. While the stabilization conditions for MPC [1] require that the terminal cost is a local Lyapunov function for the system in closed-loop with an ACL, Assumption 2.4 requires in a similar manner that the terminal cost is a local ISS Lyapunov function [5] for the system in closed-loop with an ACL. 3 Problem formulation For a given stage cost L(, ), to employ Assumption 2.4 for setting-up robust MPC schemes with an a priori ISS guarantee (or to compute state feedback controllers that achieve local ISS), one needs systematic methods for computing a terminal cost F ( ), a terminal set X T and an ACL h( ) that satisfy Assumption 2.4. Once F ( ) and h( ) are known, several methods are available for calculating the maximal RPI set contained in X U for certain relevant subclasses of system (2), in closed-loop with u(k) = h(x(k)), k Z +, see, for example, [9, 10] and the references therein. As a consequence, therefore, we focus on solving the following problem. Problem 3.1 Calculate F ( ) and h( ) such that Assumption 2.4-(v) holds. This problem comes down to computing an input-to-state stabilizing state-feedback given by h( ) along with an ISS Lyapunov function (i.e. F ( )) for system (2) in closedloop with the ACL. This is a non-trivial problem, which depends on the type of MPC cost, system class and on the type of candidate ISS Lyapunov function F ( ). Furthermore, it would be desirable that the MPC cost function is continuous and convex. 3.1 Existing solutions Several solutions have been presented for the considered problem for particular subclasses of system (2). Most methods consider quadratic cost functions, F (x) := x P x, P 0, L(x, u) = x Qx + u Ru, Q, R 0, and linear state feedback ACLs given by h(x) := Kx. (i) The nominal linear case: φ(x, u, 0, 0) := Ax + Bu, A R n n, B R n m. In [2] it was proven that the solutions of the unconstrained infinite horizon linear quadratic regulation problem with weights Q, R satisfy Assumption 2.4-(v), i.e. and K = (R + B P B) 1 B P A P = (A + BK) P (A + BK) + K RK + Q. (3) Numerically, this method amounts to solving the discrete-time Riccati equation (3). (ii) The linear case with parametric disturbances: φ(x, u, w, 0) := A(w)x + B(w)u, A(w) R n n, B(w) R n m are affine functions of w W with W a 4

5 compact polyhedron. In [3] it was proven that P = Z 1 and K = Y Z 1 satisfy Assumption 2.4-(v), where Z R n n and Y R m n are solutions of the linear matrix inequality Z (A(w i)z + B(w i)y ) (R 1 2 Y ) (Q 1 2 Z) (A(w i)z + B(w i)y ) Z 0 0 R 2 1 0, i = 1, E, Y 0 I 0 Q 2 1 Z 0 0 I with w 1,..., w E the vertices of the polytope W. Numerically, this method amounts to solving a semidefinite programming problem. This solution trivially applies also to the case (i) and, moreover, it was extended to piecewise affine discrete-time hybrid systems in [11]. (iii) The nonlinear case with additive disturbances: φ(x, u, 0, v) = f(x)+g 1 (x)u+ g 2 (x)v with suitably defined functions f( ), g 1 ( ) and g 2 ( ). A nonlinear ACL given by h(x) was constructed in [4] using linearization of the system, so that Assumption 2.4- (v) holds for all states in a sufficiently small sublevel set of V (x) = x P x, P 0. Numerically this method amounts to solving a discrete-time H Riccati equation. For the linear case with additive disturbances (i.e. f(x) = A, g 1 (x) = B and g 1 (x) = B 1 ), it is worth to point out that an LMI-based design method to obtain the terminal cost, for a given ACL, was presented in [12]. 4 Main results In this section we derive a novel LMI-based solution to the problem of finding a suitable terminal cost and ACL that applies to linear systems affected by both parametric and additive disturbances, i.e. x(k + 1) = φ(x(k), u(k), w(k), v(k)) := A(w(k))x(k) + B(w(k))u(k) + B 1(w(k))v(k), (4) where A(w) R n n, B(w) R n m, B 1 (w) R n dv are affine functions of w. We will also consider quadratic cost functions, F (x) := x P x, P 0, L(x, u) = x Qx + u Ru, Q, R 0, and linear state feedback ACLs given by h(x) := Kx. 4.1 LMI-based-solution Consider the linear matrix inequalities, Z 0 (A(w i)z + B(w i)y ) (R 2 1 Y ) (Q 2 1 Z) 0 τi B 1(w i) T 0 0 (A(w i)z + B(w i)y ) B 1(w i) Z 0 0 0, R 1 2 Y 0 0 I 0 Q 1 2 Z I i = 1, E, (5) where w 1,..., w E are the vertices of the polytope W, Q R n n and R R m m are known positive definite and symmetric matrices, and Z R n n, Y R m n and τ R >0 are the unknowns. Theorem 4.1 Suppose that the LMIs (5) are feasible and let Z, Y and τ be a solution with Z 0, τ R >0. Then, the terminal cost F (x) = x P x, the stage cost L(x, u) = x Qx + u Ru and the ACL h(x) = Kx with P := Z 1 and K := Y Z 1 satisfy Assumption 2.4-(v) with σ( v ) := τ v 2 2 = τv v. 5

6 Proof: For brevity let (w i ) denote the matrix in the left-hand side of (5). Using W = Co{w 1,..., w E } (where Co{ } denotes the convex hull) and the fact that A(w), B(w) and B 1 (w) are affine functions of w, it is trivial to observe that if (5) holds for all vertices w 1,..., w E of W, then (w) 0 holds for all w W. Applying the Schur complement to (w) 0 (pivoting after diag(z, I, I)) and letting M(w) := A(w)Z + B(w)Y yields the equivalent matrix inequalities: ( ) Z M(w) Z 1 M(w) Z QZ Y RY M(w) Z 1 B 1 (w) B 1 (w) Z 1 M(w) τi B 1 (w) Z 1 0 B 1 (w) and Z 0. Letting A cl (w) := A(w) + B(w)K, substituting Z = P 1 and Y = KP 1, and performing a congruence transformation on the above matrix inequality with diag(p, I) yields the equivalent matrix inequalities: ( ) P Acl (w) P A cl (w) Q K RK A cl (w) P B 1 (w) B 1 (w) P A cl (w)) τi B 1 (w) 0 P B 1 (w) and P 0. Pre multiplying with ( x v ) and post multiplying with ( x v ) the above matrix inequality yield the equivalent inequality: (A cl (w)x+b 1 (w)v) P (A cl (w)x+b 1 (w)v) x P x x (Q+K RK)x+τv v, for all x R n and all v R dv. Hence, Assumption 2.4-(v) holds with σ( v ) = τ v 2 2. Remark 4.2 In [8] the authors established an explicit relation between the gain τ R >0 of the function σ( ) and the ISS gain of the corresponding closed-loop MPC system. Thus, since τ enters (5) linearly, one can minimize over τ subject to the LMIs (5), leading to a smaller ISS gain from v to x. 4.2 Relation to LMI-based H control design In this section we formalize the relation between the considered robust MPC design problem and H design for linear systems. But first, we briefly recall the H design procedure for the discrete-time linear system (4). For simplicity, we remove the parametric disturbance w and consider only additive disturbances v V. However, the results derived below that relate to the optimal H gain also hold if parametric disturbances are considered, in the sense of an optimal H gain for linear parameter varying systems. Consider the system corresponding to (4) without parametric uncertainties, i.e. x(k + 1) = Ax(k) + Bu(k) + B 1 v(k), z(k) = Cx(k) + Du(k) + D 1 v(k), (6) where we added the performance output z R dz. Using the results of [13], [14] it can be demonstrated that system (6) in closed-loop with u(k) = h(x(k)) = Kx(k), k Z +, has an H gain less than γ if and only if there exists a symmetric matrix P such that: P 0 (A + BK) P (C + DK) 0 γi B1 P D1 P (A + BK) P B 1 P 0 C + DK D 1 0 I 0. (7) 6

7 Letting Z = P 1, Y = KP 1 and performing a congruence transformation using diag(z, I, Z, I) one obtains the equivalent LMI: Z 0 (AZ + BY ) (CZ + DY ) 0 γi B1 D1 AZ + BY B 1 Z 0 0. (8) CZ + DY D 1 0 I Indeed, from the above inequalities, where V (x) := x P x, one obtains the dissipation inequality: V (x(k + 1)) V (x(k)) z(k) γ v 2 2. (9) Hence, we can infer that i=0 z(i) 2 2 γ i=0 v(i) 2 2 and conclude that the H norm of the system is not greater than γ. Minimizing γ subject to the above LMI yields the optimal H gain as the square root of the optimum. Remark 4.3 In [13], [14] an equivalent formulation of the matrix inequality (7) is used, i.e. with γi in the south east corner of (7)-(8) instead of I, which leads to the adapted dissipation inequality V (x(k+1)) V (x(k)) γ 1 z(k) 2 2+γ v 2 2. Then, by minimizing over γ subject to the LMIs (8), one obtains the optimal H gain directly as the optimal solution, without having to take the square root. However, regardless of which LMI set-up is employed, the resulting optimal H gain and corresponding controller (defined by the gain K) are the same, with a difference in the storage function V (x) = x P x with a factor γ. Theorem ( 4.4 ) Suppose ( that ) the LMIs (5) without parametric uncertainties and (8) with C = Q 2 1 0, D = 0 R 1 and D 2 1 = 0 are feasible for system (6). Then the following statements are equivalent: 1. Z, Y and τ are a solution of (5); 2. Z, Y and γ are a solution of (8) with C = ( ) ( ) Q 2 1 0, D = 0 R 1 and D 2 1 = 0; 3. System (6) in closed-loop with u(k) = Kx(k) and K = Y Z 1 satisfies the dissipation inequality (9) with storage function V (x) = x P x and P = Z 1, and it has an H norm less than γ = τ; 4. Assumption 2.4-(v) holds for F (x) = x P x, L(x, u) = x Qx + u Ru and h(x) = Kx, with P = Z 1, K = Y Z 1 and σ( v ) = τ v 2 2 = γ v 2 2. The proof of Theorem 4.4 is trivially obtained by replacing C, D and D 1 in (8) and (9), respectively, and using Theorem 4.1 and the results of [13], [14]. Theorem 4.4 establishes that the LMI-based solution for solving Problem 3.1 proposed in this paper guarantees an H gain equal to the square root of the gain τ = γ of the σ( ) function for the system in closed-loop with the ACL. It also shows that the optimal H control law obtained by minimizing γ = τ subject to (8) (for a particular performance output related to the MPC cost) solves the terminal cost and ACL problem in inf-sup robust MPC. These results establish an intimate connection between H design and inf-sup MPC, in a similar way as LQR design is connected to nominally stabilizing MPC. This connection is instrumental in improving the closed-loop ISS gain of inf-sup MPC closed-loop systems as follows: an optimal gain τ = γ of the σ( ) function results in a smaller gain of the function γ( ) of Definition 2.2 for the MPC closed-loop system, as demonstrated in [8]. 7

8 5 Conclusions In this article we proposed a novel LMI-based solution to the terminal cost and auxiliary control law problem in inf-sup robust MPC. The developed conditions apply to a more general class of systems than previously considered, i.e. linear systems affected by both parametric and additive disturbances. Since LMIs can be solved efficiently, the proposed method is computationally attractive. Furthermore, we have established an intimate connection between the proposed LMIs and the optimal H control law. This result, which was somehow missing in the MPC literature, adds to the well-known connection between design of nominally stabilizing MPC schemes and the optimal solution of the LQR problem. Such results are of general interest as they connect well known linear control problems to MPC design. References [1] D.Q. Mayne, J.B. Rawlings, C.V. Rao and P.O.M. Scokaert. Constrained model predictive control: Stability and optimality, Automatica, Vol. 36, No. 6, pp , [2] P.O.M. Scokaert and J.B. Rawlings. Constrained linear quadratic regulation, IEEE Transactions on Automatic Control, Vol. 43, No. 8, pp , [3] M.V. Kothare, V. Balakrishnan and M. Morari. Robust constrained model predictive control using linear matrix inequalities, Automatica, Vol. 32, No. 10, pp , [4] L. Magni, G. De Nicolao, R. Scattolini and F. Allgöwer. Robust MPC for nonlinear discrete-time systems, International Journal of Robust and Nonlinear Control, Vol. 13, pp , [5] Z.-P. Jiang and Y. Wang. Input-to-state stability for discrete-time nonlinear systems, Automatica, Vol. 37, pp , [6] M. Lazar and W.P.M.H Heemels. Predictive control of hybrid systems: Input-to-state stability results for sub-optimal solutions, Automatica, in press. [7] L. Magni, D.M. Raimondo and R. Scattolini. Regional input-to-state stability for nonlinear model predictive control, IEEE Transactions on Automatic Control, Vol. 51, No. 9, pp , [8] M. Lazar, D. Muñoz de la Peña, W.P.M.H. Heemels, T. Alamo. On input-to-state stability of min-max MPC, Systems & Control Letters, Vol. 57, pp , [9] I. Kolmanovsky and E.G. Gilbert. Theory and computation of disturbance invariant sets for discrete-time linear systems, Mathematical Problems in Engineering, Vol. 4, pp , [10] A. Alessio, M. Lazar, W.P.M.H. Heemels and A. Bemporad. Squaring the circle: An algorithm for generating polyhedral invariant sets from ellipsoidal ones, Automatica, Vol. 43, No. 12, pp , [11] M. Lazar, W.P.M.H. Heemels, S. Weiland and A. Bemporad. Stabilizing model predictive control of hybrid systems, IEEE Transactions on Automatic Control, Vol. 51, No. 11, pp , [12] T. Alamo, D. Muñoz de la Peña, D. Limon and E.F. Camacho. Constrained minmax predictive control: Modifications of the objective function leading to polynomial complexity, IEEE Transactions on Automatic Control, Vol. 50, No. 5, pp , [13] I. Kaminer, P.P. Khargonekar and M.A. Rotea. Mixed H 2/H control for discrete time systems via convex optimization, Automatica, Vol. 29, pp , [14] H. Chen and C.W. Scherer, Moving horizon H with performance adaptation for constrained linear systems, Automatica, Vol. 42, pp ,