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IEEE 43rd Conference on Decision and Control December 14-17, 2004 Atlantis, Paradise Island, Bahamas WeA01.5 Computation of Invariant Sets for Piecewise Af ne Discrete Time Systems subject to Bounded Disturbances S. V. Rakovi , P. Grieder, M. Kvasnica, D. Q. Mayne, M. Morari c Abstract Piecewise af ne (PWA) systems are useful models for describing non-linear and hybrid systems. One of the key problems in designing controllers for these systems is the inherent computational complexity of controller synthesis and analysis. These problems are ampli ed in the presence of state and input constraints and additive but bounded disturbances. In this paper we exploit set invariance and parametric programming to devise an ef cient robust time optimal control scheme. Speci cally, the state is driven into the maximal robust invariant set in minimum time. We show how to compute and derive conditions for nite time computation. Keywords: Piecewise Af ne Dynamics, Set Invariance, Constrained Control, Robust Control. I. INTRODUCTION Piecewise af ne (PWA) systems have attracted much interest in the research community since they can approximate non-linear systems [27] and because of their equivalence to many classes of hybrid systems [12]. Optimal control of constrained PWA systems is analyzed in [6], [18] while an explicit characterization of the optimal solution is obtained in [6], [15], [19]. The explicit solution takes the form of a PWA state feedback control law, i.e. the state space is partitioned into polyhedral sets (possibly overlapping) and in each of these sets the optimal control law is an af ne function of the state. On-line implementation of the explicit control law reduces to a simple set-membership test. It is general practice to provide guarantees on constraint satisfaction by adding a (robustly) controlled invariant terminal set constraint to the optimization problem [20]. Although computation of invariant sets has garnered great interest in the control community [5], [8], [14], [16], only few results for obtaining robustly positively invariant sets for PWA systems are reported; the most relevant work is an excellent thesis [14]. Our results are based on the results for linear systems in [16], [21] as well as recent extensions to PWA systems in [10], [15]. An algorithm for computing the maximal robustly pos itively invariant set is described and suf cient conditions for nite termination of this algorithm are given. This research was partially supported by the Engineering and Physical Sciences Research Council, United Kingdom. S. V. Rakovi and D. Q. Mayne are with Department of Electrical and c Electronic Engineering, Imperial College London, United Kingdom. Email: sasa.rakovic@imperial.ac.uk and d.mayne@imperial.ac.uk P. Grieder, M. Kvasnica and M. Morari are with Automatic Control Laboratory, ETH Zurich, Switzerland. grieder@control.ee.ethz.ch, The set is subsequently used to initialize an iterative computation scheme which converges to the maximal ro bustly attractive set K ( ). A similar scheme is applied to obtain the maximal robustly control invariant set C . We also introduce a set of ef cient algorithms to obtain PWA feedback controllers that guarantee robust constraint satisfaction of the closed-loop system. This paper is organised as follows. The preliminaries are given in Section II. Section III contains ef cient algorithms for computational geometry needed in subsequent sections. Section IV presents a general framework for computing the maximal robustly positively invariant set for a subset of the autonomous piecewise af ne system. Section V gives an algorithm for computing moderate complexity controllers for perturbed PWA systems. Section VI provides two interesting examples demonstrating that our algorithms provide low/moderate complexity state feedback control. Section VII contains general conclusions. All proofs as well as fuller exposition of the results of this paper can be found in [24]. Notation and Basic De nitions: Let N {0, 1, . . . , }, {1, 2, . . . , }, Nq {0, 1, 2, . . . , q} and N+ N+ q {1, 2, . . . , q}. We use convh to denote the convex hull. A polyhedron is the (convex) intersection of a nite number of open and/or closed half-spaces, a polytope is a (convex) closed and bounded polyhedron and a P collection is a (possibly non-convex) union of a nite number of polyhedra. Given A and B two subsets of Rn , the following are basic {x Rn | x A}; / set operations: set complement Ac set difference A \ B {x | x A and x B}; symmetric / set difference A B (A \ B) (B \ A) = {x | (x A and x B) or (x B and x A)}; the Pontryagin / / (Minkowski set) difference A B {x Rn | x + b A, b B}; Minkowski set addition A B = {a + b | a A, b B}. II. PRELIMINARIES We consider the discrete time system de ned by: x+ = f (x, u, w), (1) kvasnica@control.ee.ethz.ch and morari@control.ee.ethz.ch where x, u and w denote, respectively, the current state, control and disturbance and x+ denotes the successor state. The function f ( ) is piecewise af ne in each of a nite number of polytopes {Qi }, i N+ , with disjoint interiors q 0-7803-8682-5/04/$20.00 2004 IEEE 1418 that cover the region of state space of interest, i.e. X i N+ Qi . The system satis es: q f (x, u, w) Ai x + Bi u + ci + w, x Qi . (2) System (2) is subject to the following set of constraints: (x, u, w) X U W Rn Rm Rn (3) The sets X, U and W are compact and polytopic and each set contains the origin in its interior. The following de nitions are needed in the sequel: De nition 1: A set X is said to be a robustly controlled invariant set for the PWA system in (2) subject to the constraints in (3) if for every x there exists a u U such that f (x, u, w) , w W. De nition 2: The set C is said to be maximal robustly control invariant for the PWA system in (2) subject to the constraints in (3) if it contains all robustly control invariant sets. De nition 3: For a robustly control invariant set , the k-step robustly attractive set k ( ) for the PWA system (2) subject to the constraints in (3) is de ned by: k ( ) ( k 1 ), k N+ , 0 = and {x X | u U s.t. f (x, u, w) , w W} The maximal robustly attractive set K ( ) is de ned by ( ) k N k . the union of all k , k N, i.e. K III. GEOMETRIC COMPUTATIONS WITH P COLLECTIONS We now present some tools that are required for set computations with P collections. For computation of the set difference of two polyhedra the reader is referred to [3]. The rst two results show how the set difference of a P collection and a polyhedron (or a P collection) may be computed: p Proposition 1: Let C j=1 Cj be a P collection, where all the Cj , j N+ , are non-empty polyhedra. If q p A is a non-empty polyhedron, then C \ A = j=1 (Cj \ A) is a P collection. p Proposition 2: Let the sets C j=1 Cj and D q Dk be P collections, where all the Cj , j N+ , and p k=1 C and Dk , k N+ , are non-empty polyhedra. If E0 q Ek Ek 1 \Dk , k N+ then C \D = Eq is a P collection. q The reader is referred to [25] for proofs and comments on computational ef ciency. If C and D are P collections it follows that C D can be easily veri ed since C D C \ D = , similarly C = D is also easily veri ed since C = D (C \ D = and D \ C = ) C D= [Ac ( B)] . The following algorithm implements the computation of the Pontryagin difference of a P collection C j N+ Cj , where Cj , j N+ are polytopes in Rn , and p p a polytope B Rn . Algorithm 3.1 (C B): 1) Input: P collection C, polytope B 2) H = convh(C) 3) D = H B 4) E = H \ C 5) F = E ( B) 6) G = D \ F 7) Output: P collection G Proposition 3: [24] Consider Algorithm 3.1, then G = C B. Algorithm 3.1 is illustrated on sample P collections in Figures 1(a) to 1(f). It is important to note that in general C B = j N+ (Cj B), but only j N+ (Cj B) p p C B (set equality holds only in a very limited number of cases). Algorithm 3.1 for computation of the Pontryagin difference is conceptually similar to that proposed in [14], [15], [26]. However, computing the convex hull in the rst step signi cantly reduces (in general) the number of sets obtained at step 3, which in turn results in fewer Minkowski set additions. Since computation of Minkowski set addition is expensive, a reasonable runtime improvement is expected. In principle, computation of the convex hull can be replaced by computation of any convex set containing the P collection C an easily computable alternative to the convex hull is H = env(C), where env denotes the envelope [2]. Necessary computations can be ef ciently implemented by using standard computational geometry software such as [9], [17], [28]. IV. INVARIANT SET COMPUTATION FOR PWA SYSTEMS We rst address the computation of a robustly positively invariant set for PWA systems around the origin. We assume that the origin is an equilibrium of system x+ = f (x, u, 0) where f ( ) is de ned in (2), so that ci = 0 for all i N0 , q where N0 N+ , is de ned by: q q N0 q {i N+ | 0 Qi } q c where 0 is the origin of the state space. Following [7], [10], [22], we assume that a stabilizing piecewise linear control law (x) for the nominal system x+ = Ai x + Bi u, x Qi , i N0 along with the associated common quadratic q Lyapunov function can be computed, so that (x) = Ki x if x Q , i N0 i q where Q i {x | x Qi X, Ki x U}, i N0 q X0 i N0 q An ef cient algorithm for computing the Pontryagin (Minkowski Set) difference of a P collection and a polytope is discussed next. If A and B are two subsets of Rn it is known that (see for instance [14], [26]), A B= and we de ne Q i (4) 1419 5 4 3 2 1 0 C2 C1 5 4 3 2 1 5 4 3 2 1 5 5 5 4 3 2 H 4 3 2 1 E x2 4 3 2 1 0 D B 0 0 0 F 1 0 F x2 x2 x2 x2 1 2 3 4 5 5 4 3 2 1 0 1 2 3 4 5 1 2 3 4 5 5 4 3 2 1 0 1 2 3 4 5 1 2 3 4 5 5 1 2 3 1 2 3 4 4 3 2 1 0 1 2 3 4 5 5 5 4 3 2 1 0 1 2 3 4 5 D 4 3 2 1 0 1 2 3 4 5 4 5 5 x2 1 2 3 4 5 5 4 3 2 1 0 1 2 3 4 5 (a) j Cj and B. x1 (b) H = convh(C). x 1 (c) D = H Fig. 1. x1 B. (d) E = H \ C. x 1 (e) F = E ( B). x 1 (f) G = D \ F . x 1 Graphical Illustration of Algorithm 3.1. Thus we consider the following autonomous system: x+ = fa (x, w) (Ai +Bi Ki )x+w, x Q , i N0 . (5) i q and we will aim to compute a robustly positively invariant subset contained in X0 de ned in (4). For any integer k, wk denotes the sequence {w(0), w(1), . . . w(k 1)}, and (k; x0 , wk ) denotes the solution of x+ = fa (x, w) at time k if the initial state is x0 and the disturbance sequence is wk . Given the non-empty set A and a function fa ( ) de ned in (5) let k (A) { (k; x, wk ) | x A, wk Wk } { (k; x, 0) | x A} k is the maximal robustly positively invariant set , = otherwise k 0 k , however if k = for some integer k then the simple conclusion is that = . Lemma 1: Let the set sequence { k } be generated by Algorithm 4.1. Then for any integer k the set k is a P collection. Some of the properties of Algorithm 4.1 may be established, see [1], [4], [8], [14] and also [13] where similar results are reported for unperturbed linear switched systems. Lemma 2: Let the set j for some nite j be a xed point (i.e., j = j+1 ) of Algorithm 4.1, then j is the maximal robustly positively invariant set. B. Finite Termination of the Computation of the Maximal Robustly Positively Invariant Subset We isolate a set of conditions that are suf cient to guarantee nite time termination of Algorithm 4.1. Our rst step is to introduce the augmented system aug x+ =fa (x, w) denote the k step reachable set and let k (A) denote the k step reachable set for the nominal system x+ = fa (x, 0), where for any k N, Wk W W . . . W. De nition 4: The maximal positively invariant set, , for the discrete time system x+ = fa (x, 0), where fa ( ) is de ned in (5) subject to the constraints in (4) is de ned by {x X0 | (k; x, 0) X0 , k N}. De nition 5: The maximal robustly positively invariant set, , for the discrete time system x+ = fa (x, w), where fa ( ) is de ned in (5), subject to the constraints in (4) is de ned by: {x X0 | (k; x, wk ) X0 , wk Wk , k N}. A. The Maximal Robustly Positively Invariant Set Consider now the perturbed discrete-time system, de ned in (5). The set of the states a ( ) that robustly evolve to X0 for all w W in one step is: a ( ) {x X0 | fa (x, w) , w W}. (6) If X0 is a P collection, then the set a ( ) is also a P collection by de nition (6). Algorithm 4.1 provides a procedure for computing the maximal robustly positively invariant subset [1], [4], [8], [14]: Algorithm 4.1 (Computation of ): 1) 0 = X0 2) k+1 = a ( k ) 3) If k+1 = k , return; Else, set k = k + 1 and goto 2. The algorithm generates the set sequence { k } satisfying k+1 k , k N and it terminates if k+1 = k so that Aaug x + w {(Ai + Bi Ki ), i N0 }. q (7a) (7b) A aug A aug This augmented system fa (x, w) corresponds to system n (5) with Qi = R , i.e., any dynamic i N0 may be active at q any time step. Let aug (k, x0 , wk ) denote the set of states which is reachable from the initial state x0 in k steps for aug x+ = fa (x, w) and a disturbance sequence wk . The kstep nominal and robust reachable sets for the augmented system are then given, respectively by: aug (A) k and aug (A) k { aug (k, x0 , 0)|x0 A} { aug (k, x0 , wk )|x0 A, wk Wk } aug Let the set Fk (Fk ) be the k-step disturbance response for system de ned in (5) ((7)) so that Fk+1 aug Fk+1 1 (Fk ), F1 {0}, aug aug aug (Fk ), F1 1 {0}. It follows trivially, from de nitions of the corresponding aug sets, that aug (A) k (A), aug (A) k (A), Fk k k aug Fk for all k N. As shown in [23], the set F aug limk Fk exists and is bounded by a compact robustly aug aug positively invariant set F satisfying Fk Fk F aug F, k > 0, if the nominal system fa (x, 0) in (7) is absolutely asymptotically stable [11]. 1420 Theorem 1: [24] Suppose that there exists a compact aug robustly positively invariant set F satisfying F F interior(X0 ) and that the nominal augmented system (7) with W = {0} is absolutely asymptotically stable [11]; Then, Algorithm terminates 4.1 in nite time. Remark 1: A detailed overview of the properties of aug aug the set sequences {Fk }, {Fk } and the set F aug limk Fk as well as algorithms to compute a robustly aug positively invariant set F F for the augmented system (7) are discussed in more detail in [23]. Corollary 1: Suppose that the nominal system de ned in (5) is asymptotically stable with W = {0} and X is a compact set that contains the origin in its interior. Then algorithm 4.1 computes the maximal positively invariant set (see de nition 4) in nite time. If the original system, de ned in (1) is piecewise linear, i.e. N0 = N+ , then Algorithm 4.1 computes the maximal q q robustly positively invariant set. In general, N0 = N+ so q q that non-existence of for system (5) does not imply non-existence of this set for the original system in (1). However, considering the discrete time system de ned in (1) the issue of nite time termination of Algorithm 4.1 becomes signi cantly more complicated. V. MAXIMAL ROBUSTLY CONTROLLED INVARIANT SET This section shows how computation of permits ( ). A similar procedure can be the computation of K employed for the computation of C . If the computation scheme is applied using multi-parametric programming [3], a robust minimum-time controller is automatically computed by the proposed scheme. The proposed procedure for computation of the state feedback controller is based on the results in [10], [15]. The Algorithm below describes the computation scheme for K ( ), C or a minimum time state-feedback controller for generic PWA systems, depending on the speci c implementation: Algorithm 5.1 (Computation K ( ) or C ): 1) De ne a target set S0 and set S0 = S0 W, k = 0. 2) Compute Sk+1 = (Sk ), where {x X | u U s.t. f (x, u, 0) Sk }, (8) If Sk+1 is computed by solving a multi-parametric program [10], this will automatically yield a PWA control law which drives all states x Sk+1 into the set Sk in one time step. 3) If Sk+1 = Sk , return; Else, set k = k + 1, Sk = Sk W and goto step 2. The sets Sk are, in general, P collections, making Algorithm 5.1 computationally demanding. The following two results are standard and well known results in set invariance or viability theory: Theorem 2: Suppose that S0 = X and that there exists a k N such that Sk = Sk +1 . Then, Algorithm 5.1 terminates and C = Sk . (Sk ) Theorem 3: Suppose that S0 = and that there exists a k N such that Sk = Sk +1 . Then, Algorithm 5.1 terminates and K ( ) = Sk . It is clear that at each time k, the effective target set is l a P collection, i.e. Sk = l Lk Sk where Lk has a nite cardinality, that changes with time k, so that (Sk ) = l Lk l where Sk is a polytopic set. For each time k and for each (i, l) N+ Lk let q l (Sk ). (9) Zk and (i,l) {(x, u) | (x, u) (Qi X) U), l Ai x + Bi u + ci Sk } ProjX Zk (i,l) (10) (11) Xk (i,l) It follows from the de nition of ( ) that (i,l) (i,l) i N+ Xk and (Sk ) = (i,l) (N+ Lj ) Xk . Let q q V (x, u, i) u Ru + (x+ ) Q(x+ ) i i l (Sk ) = 0, R = R 0. where x+ Ai x+Bi u+ci and Q = Q i For each time k and for each (i, l) N+ Lj let the q (i,l) problem Pk (x) be de ned as: o Vk (x, i, l) = min{V (x, u, i) | (x, u) Zk u (i,l) } (12) and let o (x, i, l) = arg min{V (x, u, i) | (x, u) Zk k u (i,l) } (i,l) (13) o denote the argument of Vk (x, i, l). Since the set Zk is a polytopic set and since V (x, u, i) is quadratic in (x, u) it (i,l) follows that the problem Pk (x) is a quadratic program for each k and each (i, l) N+ Lj and therefore can q be solved as multi parametric quadratic program. In order to obtain a controller which drives every feasible state x into a target set in minimum time, it is therefore necessary to compute K ( ) by solving a sequence of multi-parametric programs in Step 2. of Algorithm 5.1. As already remarked, each resulting controller covers a convex set and their union is equal to (Sk ). In on-line application, several controller regions may be associated to any given state, since the different controllers will overlap. In order to obtain closed loop trajectories which enter the initial target set in minimum time, the controller region associated to the lowest iteration must be selected, i.e. k o = min{k N | x Xk k, l (i,l) (i,l) }. where Xk is de ned in (11). The controller partition (i,l) associated to Xko is then activated and o o (x, i, l) applied k in a standard multi-parametric manner [3]. Note that a state may be contained in several partitions with equal iteration number k o . If this is the case, any of the controllers associated to k o may be applied. 1421 Remark 2: If a controller partition computed at iteration k1 is covered by controller regions of equal or lower iteration number k2 k1 , the respective partition may be discarded since it will never be applied on-line. This may lead to drastic decreases in controller complexity. Ef cient methods to check whether a polytope is covered by a P collection are given in [10]. The multi-parametric min-max optimal control scheme in [15] requires each region partition to be maintained from iteration to iteration. Algorithm 5.1 only requires the representation of the feasible set Sk to be stored which is expected to mitigate the combinatorial explosion. It is therefore reasonable to assume that the minimum-time algorithm proposed here will yield controllers of relatively low complexity versus min-max controllers. This is also indicated by results for nominal systems in [10]. Remark 3: Note that K ( ) may not be nitely determined. In order to obtain a controller, it may therefore be necessary to abort Algorithm 5.1 after a prede ned maximum number of iterations or after the state space of interest has been covered. The resulting minimum time controller will then only cover a subset of K ( ). VI. NUMERICAL EXAMPLES In order to illustrate the proposed procedure we consider two second order PWA systems. Our rst example is the following 2-dimensional problem adopted from [18]: x+ = Ai x + Bi u + ci + w where i = 1 if x1 1 and i = 2 if x1 1 and A1 = A2 = 0 1 0.2 0 , , B1 = , c1 = 0 01 1 0.5 0.2 0 0.5 , B2 = , c2 = 0 1 1 0 w {w R | w 2 10 8 6 4 2 2 0 2 4 6 8 x 10 6 4 2 0 2 4 6 8 x 1 Fig. 2. Final robust controller for Example 1. where (14) 1, 2, i= 3, 4, 1 A1 = 0 1 A2 = 0 1 A3 = 0 A4 = if if if if x1 0 & x2 0 x1 0 & x2 0 x1 0 & x2 0 x1 0 & x2 0 1 1 , , B1 = 0.5 1 1 1 , B2 = , 1 0.5 1 1 , B3 = , 1 0.5 , B4 = 1 0.5 . . 1 1 01 and the additive disturbance w is bounded: with the following bounds on the disturbance: (15) w {w R2 | w 0.1}. 0.2}. (17) The system is subject to constraints x1 + x2 15, 3x1 x2 25, 0.2x1 + x2 9, x1 6, x1 8, and 1 u 1, whereas weight matrices for the optimization problem are Q = I and R = 1. We applied Algorithm 5.1 to the PWA system (14) and obtained a controller that guarantees robust constraints satisfaction for all admissible disturbances (15). The terminal set was obtained by computing the maximal robustly positively invariant set [16] for x+ = (A1 + B1 K1 )x + w, where K1 is the Riccati LQR feedback controller. Using the MPT toolbox [17] the algorithm converged after 75 seconds at iteration 15. The resulting PWA control law is de ned over a polyhedral partition consisting of 417 regions and is depicted in Figure 2. Our second example is the following 2-dimensional PWA system with 4 dynamics [10]: x+ = Ai x + Bi u + w (16) One can observe, that the system is a perturbed double integrator in the discrete time domain, with different orientation of the vector eld. The output and input constraints, respectively, are: 5 x1 5, 5 x2 5, 1 u 1, whereas weight matrices for the optimization problem are Q = I and R = 1. By solving the SDP in [10], the following stabilizing feedback controllers are obtained: K1 = [ 0.5897 0.9347], K2 = [0.5897 0.9347], K3 = [0.5897 0.9347] and K4 = [ 0.5897 0.9347]. Subsequently is computed according to Algorithm 4.1 and used as a target set for the robust minimum-time Algorithm 5.1. This technique yields a controller that guarantees robust constraint satisfaction for all possible realizations of the disturbance bounded by (17). Using the MPT toolboxa controller partition with 508 regions is obtained at iteration 17 after 107 seconds of computation time (see Figure 3). 1422 2.5 2 1.5 1 0.5 0 0.5 1 1.5 2 2.5 5 4 3 2 1 0 1 2 3 4 5 x1 Fig. 3. Final robust controller for Example 2. VII. CONCLUSIONS We have introduced ef cient techniques for computations with P collections, which can be combined with set invariance theory to compute non-convex robustly positively invariant sets for piecewise-af ne (PWA) systems. Speci cally we have presented methods to compute the maximal robustly positively invariant subset , the maxi and the maximal robustly mal robustly controllable set C attractive set K ( ) for PWA systems. In addition, suf cient conditions for nite time determination of are given. We have furthermore shown how these methods may be used to obtain robust state feedback controllers for PWA systems if combined with multi-parametric programming techniques. The proposed controller robustly drives the state into the target set in minimum time. ACKNOWLEDGMENT The authors greatfully acknowledge the fruitful discussion with Dr. K. I. Kouramas. All of the proposed algorithms and computational schemes have been implemented by using the Multi-Parametric Toolbox and are available for download [17]. REFERENCES [1] J. P. Aubin, Viability theory, ser. Systems & Control: Foundations & Applications. Birkha ser, 1991. u [2] A. Bemporad, K. Fukuda, and F. Torrisi, Convexity recognition of the union of polyhedra, Computational Geometry, vol. 18, pp. 141 154, Apr. 2001. [3] A. Bemporad, M. Morari, V. Dua, and E. Pistikopoulos, The explicit linear quadratic regulator for constrained systems, Automatica, vol. 38, no. 1, pp. 3 20, Jan. 2002. [4] D. P. Bertsekas, Control of uncertain systems with a set membership description of the uncertainty, Ph.D. dissertation, MIT, 1971. [5] F. 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