Stability and Invariance Analysis of Uncertain ... - Semantic Scholar
Stability and Invariance Analysis of Uncertain Discrete-Time Piecewise Affine Systems Matteo Rubagotti, Sergio Trimboli, Alberto Bemporad∗
Abstract This note proposes a method to analyze uniform asymptotic stability and uniform ultimate boundedness of uncertain piecewise affine systems whose dynamics are only defined in a bounded and possibly non-invariant set X of states. The approach relies on introducing fake dynamics outside X and on synthesizing a piecewise affine and possibly discontinuous Lyapunov function via linear programming. The existence of such a function proves stability properties of the original system and allows the determination of a region of attraction contained in X . The procedure is particularly useful in practical applications for analyzing the stability of piecewise affine control systems that are only defined over a bounded subset X of the state space, and to determine whether for a given set of initial conditions the trajectories of the state vector remain within the domain X .
1
Introduction
In the last decade the interest in studying the dynamical properties of piecewise affine (PWA) systems has increased considerably, due to their powerful modeling capabilities. Discrete-time PWA models are a special class of hybrid systems that can represent combinations of finite automata and linear dynamics, are a good approximation of nonlinear systems [1], and are equivalent to hybrid systems in mixed logical dynamical form [2, 3]. Stability analysis tools for PWA systems are useful to describe the properties of autonomous hybrid systems, or to check a posteriori the stability of a given closedloop system [4, 5]. In particular, stability analysis becomes fundamental when a PWA control law is synthesized without a-priori guarantees of closed-loop stability; an example are explicit model predictive control (MPC) laws designed in applications with short prediction horizons and without terminal constraints to limit the number of partitions [6–8], or approximated in order to reduce their complexity [9, 10]. ∗ This paper is published on the IEEE Transactions on Automatic Control, vol. 58, no. 9, pp. 23592365, 2013. M. Rubagotti is with Nazarbayev University, 53 Kabanbay Batyr Ave, 010000 Astana, Kazakhstan. Email: [email protected]. S. Trimboli is with the Department of Mechanical and Structural Engineering, University of Trento, Via Mesiano 77, 38100 Trento, Italy. Email: [email protected]. A. Bemporad is with IMT Institute for Advanced Studies, P.zza S. Ponziano 6, 55100 Lucca, Italy. Email: [email protected]. This work was partially supported by the European Commission under project FP7-INFSO-ICT-248858 “MOBY-DIC – Model-based synthesis of digital electronic circuits for embedded control”.
The most widely used methods for stability analysis of discrete-time PWA systems are based on piecewise quadratic (PWQ) Lyapunov functions [11] and require the solution of semi-definite programming problems. As highlighted in [12], the search for a PWQ Lyapunov function can be overly conservative, even with the use of the so-called S-procedure (see, e.g., [13]). A valid alternative are PWA Lyapunov functions, computed by linear programming (LP) [12, 14]. Other types of Lyapunov functions can be used for the same purpose, such as piecewise polynomial Lyapunov functions [15]. The approaches for PWA stability analysis recalled above deal with deterministic systems. However, in applications it is often important to determine if the state converges to the origin (or to a terminal set) despite parametric uncertainties and/or external disturbances. Some classical results on stability analysis and control of uncertain linear parameter varying and switched linear systems appeared in [16–19] and [20, Chap. 7]. In analyzing the stability of PWA systems, it is always assumed that the set X of states in which the PWA dynamics are defined is positively invariant, as the notion of stability has no practical relevance if the state trajectory exits the domain of definition of the dynamics [21]. However, often the PWA system to be analyzed is defined in a set X that may not be invariant, meaning that, for certain initial conditions, the corresponding state trajectories are not defined for all time steps. A possible approach to tackle the possible non-invariance of X is to perform a reachability analysis to find the maximum positively invariant subset of X , using a recursive procedure (see [20, Chap. 4-5], [22, 23], and the references therein). Unfortunately this can lead to very involved solutions, due to the exponential complexity of reachability analysis of PWA systems, and in some cases searching the maximum invariant set is even an undecidable problem. This paper proposes a stability analysis framework for (possibly discontinuous) discrete-time PWA systems that are affected by additive disturbances bounded within a polytopic set (a similar framework was proposed in [24] to handle the case of parametric uncertainties). A discontinuous PWA Lyapunov function is synthesized via linear programming (LP) either to determine if the state converges to the origin or to find a terminal set where the state is ultimately bounded. As we assume that the PWA dynamics are defined in a closed polytope X of the state space which may not be invariant, by artificially extending the systems dynamics outside X the proposed method can also determine an invariant subset of X for the original PWA system. In this case the attractiveness of the origin (or that of the terminal set) is determined with respect to such a region of attraction. A preliminary version of this paper focusing only on asymptotic stability of deterministic PWA systems has appeared in [25]. A numerical implementation of the overall stability analysis proposed in this work can be found in the MOBY-DIC Toolbox for MATLAB, described in [26].
2
Basic notation and definitions
Let R, Z, Z≥0 , and Z>0 denote the sets of reals, integers, non-negative integers, and positive integers, respectively. Given a ∈ R, the ceil function dae is defined as the smallest b ∈ Z such that a ≤ b. Given a vector v ∈ Rn , let kvk denote any vector norm, 2
and let kvkW , kW vk∞ , where W ∈ Rn×n is a full rank diagonal matrix. Given two matrices A1 ∈ Rm1 ×n , A2 ∈ Rm2 ×n , [A1 ; A2 ] denotes the matrix [A01 A02 ]0 ∈ R(m1 +m2 )×n . Given a discrete-time signal d : Z≥0 → Rp , the sequence of the values of d from time zero to the k-th instant is denoted by d[k] . Given a set A ⊆ Rn , its ¯ and its convex hull by conv(A). Given interior is denoted by int(A), its closure by A, a finite number of sets Ai , i ∈ Ia = {1, ..., na }, we say that {Ai } is aSpartition of A na Ai = A. if int(Ai ) 6= ∅, int(Ai ) ∩ int(Aj ) = ∅, ∀i, j ∈ Ia with i 6= j, and i=1 If {Ai } is a partition with Ai ∩ Aj = ∅ ∀i, j ∈ Ia with i 6= j, it is denoted as strict partition. A polyhedron is a set given by the intersection of a finite number of (closed or open) half-spaces. A polytope A is a bounded polyhedron, and the set of the vertices ¯ of its closure A¯ is denoted by vert(A). Consider a discrete-time nonlinear system x(k + 1) = ϕ(x(k), d(k)),
(1)
where x ∈ X ⊆ Rn is the state vector, the input d ∈ D ⊂ Rp collects both model uncertainties and exogenous disturbances, and both X and D are compact sets with 0 ∈ int(X ), 0 ∈ D. Definition 1 For system (1) the one-step reachable set from a set A ⊆ X is R(A) , {y ∈ Rn : y = ϕ(x, d), d ∈ D, x ∈ A}. Definition 2 A set B ⊆ X is called robustly positively invariant (RPI) with respect to dynamics (1) if, for all d ∈ D, R(B) ⊆ B. Note that the set X is not assumed to be RPI with respect to dynamics (1), so that some trajectories may possibly leave X and be therefore defined only on a finite time interval of time [0, kmax ]. Definition 3 Consider dynamics (1) and a RPI set B ⊆ X with 0 ∈ int(B) and ϕ(0, d) = 0 for all d ∈ D. System (1) is uniformly asymptotically stable in P (UAS(P)) if there exists a KL-function1 φ such that, for all the initial conditions x(0) ∈ P and for all the sequences d[k] with d(i) ∈ D, i = 0, ..., k, kx(k)k ≤ φ(kx(0)k, k), for all k ∈ Z≥0 . Definition 4 Given two RPI sets P ⊆ X and F ⊆ P, with 0 ∈ int(F), system (1) is uniformly ultimately bounded from P to F (UUB(P, F)) if for all a > 0 there exists T (a) > 0 such that, for every x(0) ∈ P with kx(0)k ≤ a, x(T ) ∈ F for all the sequences d[T ] with d(k) ∈ D, k = 0, ..., T .
3
Stability analysis problem
Given a closed polytope X ⊂ Rn that includes the origin in its interior, consider a partition {Xi } of X that consists of a finite number s of polytopes Xi , {x : Hi1 x ≤ h1i , Hi2 x < h2i }, i ∈ I , {1, . . . , s}, 1 See,
e.g., Section II.A in [27] for a formal definition of KL-functions.
3
(2)
where Hi1 ∈ Rqi,1 ×n , Hi2 ∈ Rqi,2 ×n are constant matrices with qi,1 , qi,2 ∈ Z>0 , i ∈ I, and h1i ∈ Rqi,1 and h2i ∈ Rqi,2 are constant vectors, and assume that {Xi } is a strict partition. The closure X¯i of Xi is denoted by X¯i = {x : Hi x ≤ hi }, with Hi = [Hi1 ; Hi2 ] ∈ Rqi ×n and hi = [h1i ; h2i ] ∈ Rqi . The subset of indices I0 is defined as I0 , {i ∈ I : 0 ∈ X¯i }. Without loss of generality, we assume that 0 ∈ vert(X¯i ) for all i ∈ I0 , which is necessary when PWA Lyapunov functions are considered. If this property does not hold for the given partition {Xi }, it is always possible to obtain it by further partitioning the regions whose indices belongoto I0 . n ˜ , with H ˜ ∈ ˜ ≤h ˜ ∈ Rη×p , and h Consider a closed polytope D = d ∈ Rp : Hd η p R , p, η ∈ Z>0 , and 0 ∈ D. The vertices of D are denoted by d1 , . . . , dη ∈ R , and, since D is compact by definition, D = conv(d1 , . . . , dη ). Consider the autonomous discrete-time uncertain PWA system x(k + 1) = Ai x(k) + ai + Ei d(k) if x(k) ∈ Xi
(3)
with x(k) ∈ X , d(k) ∈ D, Ai ∈ Rn×n , ai ∈ Rn , Ei ∈ Rn×p , and k ∈ Z≥0 . Note that the dynamics (3) are not required to be continuous on the boundaries of the sets Xi . Problem statement: Given the uncertain PWA system (3), for which X is not necessarily a RPI set, prove the properties of stability and convergence to the origin (uniform asymptotic stability or uniform ultimate boundedness) with respect to a RPI set P ⊆ X . In case of ultimate boundedness, find another RPI set F including the origin, where the state is driven in finite time. Nominal stability analysis is considered a special case of the problem when D = {0} or Ei = 0 for all i ∈ I.
4
Extended system and reachability analysis
Consider the one-step reachable set R(X ) from X , R(X ) = {Ai x + ai + Ei d : d ∈ D, x ∈ Xi , i ∈ I}, and define the closed set R∪ (X ) , R(X ) ∪ X .
(4)
The closure R(X ) of R(X ) can be computed as the union of the one-step reachable sets from all the sets X¯i , namely R(X¯i ) , {Ai x + ai + Ei d, d ∈ D, x ∈ X¯i }. By relying on the results in [20, Chap. 6], we can compute the convex sets R(X¯i ) as R(X¯i ) = conv (Ai vi,h + ai + Ei dµ , µ = 1, . . . , η, h = 1, . . . , mi )
(5)
where vi,h represents each of the mi vertices of X¯i , to get R∪ (X ) =
s [ i=1
R(X¯i ) ∪ X .
(6)
As the dynamics (3) are not defined outside X , the proposed strategy consists of defining fake dynamics on R∪ (X ) \ X . Let XH , conv(R∪ (X )), and consider the dynamics x(k + 1) = ρx(k), if x(k) ∈ XE , XH \ X (7) 4
where ρ ∈ [0, 1) is an adjustable parameter of the proposed approach. The region XE , when XE 6= ∅, can be divided into convex polyhedral regions as in [28, Th. 3]. As a result, new regions Xi , i = s + 1, . . . , s˜, are created, that together with the original Xi with i ∈ I define a strict partition {Xi } of XH , with i ∈ I˜ , {1, . . . , s˜}. The dynamics of the extended system on XH are Ai x(k) + ai + Ei d(k) if x(k) ∈ Xi , i ∈ I x(k + 1) = (8) ρx(k) if x(k) ∈ Xi , i ∈ I˜ \ I. For convenience, let Ai , ρI, ai , 0, and Ei , 0, for i ∈ I˜ \ I. Also, notice that, if XE = ∅, then I˜ \ I = ∅, and no additional dynamics are introduced. Lemma 1 The set XH is RPI for the extended dynamics (8). Proof:See Appendix. The definitions of XH and of the dynamics in XE as in (7) are simplistic, yet we will prove their effectiveness. Other choices of XH (see, e.g., [25]) and of the dynamics (7) are possible, provided that Lemma 1 still holds. If X is a RPI set, one has R∪ (X ) = X , and therefore XH = X , because X is a convex set by definition. In this case, the definition of fake dynamics is not necessary, and the same approach hereafter described can be applied with s˜ = s, and XE = ∅. For any pair (i, j) ∈ I˜ × I˜ define the closed transition sets Xij , x ∈ X¯i : ∃d ∈ D : Ai x + ai + Ei d ∈ X¯j (9) of states that can possibly end up in X¯j in one step under dynamics i. Note that XH = Ss˜ Ss˜ i=1 j=1 Xij , but {Xij } in general is not a partition of XH , unless D = {0}, or ˜ To compute the sets Xij we exploit here controllability analysis [20, Ei = 0, ∀i ∈ I. Chap. 5] and consider the disturbance vector d as the external input with respect to which the controllability analysis is carried out. First, we compute the set hj Hj Ai Hj Ei x x Hj ai n+p ¯ Mi (Xj ) = ∈R : + ≤ ˜ ˜ d d 0 0 H h which is defined on the extended space (x, d), and we project it onto the first n coordinates (the state space), obtaining the so-called pre-image set of X¯j with respect to dynamics i Pre(X¯j , i) , x ∈ Rn : ∃d ∈ D : Ai x + ai + Ei d ∈ X¯j . If we impose x ∈ X¯i , we obtain Xij = X¯i ∩ Pre(X¯j , i) .
(10)
˜ In the particular case of no Let mij be the number of vertices of Xij , (i, j) ∈ (I˜ × I). ˜ additive disturbances (i.e., D = {0}, or Ei = 0, ∀i ∈ I), the transition sets Xij can be computed without using of the projection operation, as Hi 0 hi Xij = x ∈ Rn : x+ ≤ . H j Ai Hj ai hj 5
5
PWA Lyapunov analysis of the extended system
Let V : XH → R be a function such that V (x) ≥ α1 kxkW
(11a)
V (0) = 0
(11b)
V (f (x, d)) − λV (x) ≤ δ
(11c)
˜ and ∀d ∈ D, where f : Rn × Rη → Rn is the PWA state update ∀x ∈ Xi (i ∈ I) function defined by (8), α1 > 0, λ ∈ (0, 1), δ ≥ 0. W ∈ Rn×n can be chosen as any full rank diagonal matrix, but its value is fixed a priori as a design parameter. We call a function V : Rn → R satisfying (11) a Lyapunov-like function. Remark 1 Note that often in the literature an upperbound α2 kxkW ≥ V (x) is imposed, with α2 > 0. Here this is not necessary as V will be defined as a PWA function over the bounded set XH . As a consequence, it is always possible to find a posteriori α2 > 0 such that V (x) ≤ α2 kxkW , ∀x ∈ XH , once a Lyapunov-like function V has been determined. Note also that condition (11c) could be replaced by V (f (x, d)) − V (x) ≤ −α3 kxkW + δ
(12)
where α3 = (1−λ)α1 > 0. In fact, by (11a)-(11c), it follows that V (f (x, d))−V (x) ≤ −(1 − λ)V (x) + δ ≤ −(1 − λ)α1 kxkW + δ. To the end of synthesizing a PWA Lyapunov-like function for system (8), define a linear function Vi : X¯i → R for each region Xi , Vi (x) , Fi x + gi
(13a)
˜ where in (13a) Fi ∈ R1×n and gi ∈ R are coefficients to be determined. The for i ∈ I, overall Lyapunov-like function candidate is V (x) = max Vi (x)
(13b)
N (x) , {i ∈ I˜ : x ∈ X¯i }.
(13c)
i∈N (x)
where Note that simply V (x) = Fi x + gi for x ∈ int(Xi ). As for numerical reasons we want to consider closed sets X¯i and Vi (x), Vj (x) may not coincide on common boundaries X¯i ∩ X¯j (unless very conservative continuity conditions are imposed), for the states x on the common boundaries we impose (11a) and (11c) on Vi for all i ∈ N (x), although only one value (the max) is taken in (13b), as V (x) must be single-valued. In the specific case of (11b), we have Vi (0) = 0 for all i ∈ I0 , because the continuity condition is imposed at the origin. The constraints Fi vi,h + gi ≥ α1 kvi,h kW
6
(14a)
˜ h = 1, . . . , mi , while are imposed for all mi vertices vi,h ∈ vert(X¯i ), i ∈ I, Fj (Ai vij,h + ai + Ei dµ ) + gj − λ(Fi vij,h + gi ) ≤ δ
(14b)
˜ and for all vij,h ∈ vert(Xij ), with h = 1, . . . , mij , for all Ai , ai , Ei with i ∈ I, all dµ with µ = 1, . . . , η. We define kmax ∈ R>0 as the maximum value such that H , {x : kxkW ≤ kmax } ⊆ XH , which can be easily determined by bisection. The further constraints δ≥0
(14c)
rδ ≤ (1 − λ)Lα1
(14d)
α1 > 0
(14f)
gi = 0, i ∈ I0
(14e)
are also imposed, where L is the minimum distance between the boundary of the hyperrectangle H and the origin, while r = 1 + ε, where ε > 0 is a fixed parameter, chosen as small as possible (e.g., equal to the machine epsilon). Condition (14d) is imposed to ensure that the set where the state is ultimately bounded is included in XH (the reader is referred to the proof of Lemma 2, after equation (24), for details on this aspect), while (14e) ensures the fulfillment of (11b). The vector of variables to be determined ˜ We define now a procedure is composed by δ, α1 , and the terms Fi and gi , with i ∈ I. to determine a choice for such variables by means of a convex optimization problem. Lemma 2 Consider the linear-fractional program δ α1 subj. to (14)
mininize
(15a) (15b)
associated with the autonomous uncertain PWA dynamics (8) and the candidate Lyapunovlike function (13), and assume it admits a feasible solution. Then: 1. if the optimal δ is equal to zero, the origin is an equilibrium point for system (8), which is UAS(XH ); 2. otherwise (the optimal δ is strictly positive), system (8) is UUB(XH , F), with F = {x ∈ XH : V (x) ≤ VF+ }
(16)
where VF+ , max V (x) x∈R(Q) Q , x ∈ Rn : kxkW ≤
(17) rδ (1 − λ)α1
.
(18)
Proof: See Appendix. 7
Remark 2 Note that (15) is, strictly speaking, a quasi-convex optimization problem. Even though quasi-convex optimization problems are solved in general by bisection (solving a convex optimization problem at each step), linear-fractional programs like (15) can be transformed into an equivalent LP using standard procedures (see [29, ˜ plus an auxiliary Chap. 4]). The variables of the LP are δ, α1 , Fi and gi , with i ∈ I, variable introduced by the formulation of the equivalent problem. The total number of variables is nv = 3 + s˜(n + 1), the total number of constraints is nc = 4 + card(I0 ) + Ps˜ Ps˜ i=1 mi + η j=1 mij . Remark 3 The aim of (15) is to minimize the volume of the hyper-rectangle Q (in fact, the cost function of (15a) is directly proportional to the volume of Q) for the given choice of the sets Xi , in order to obtain a set F that is as small as possible (even if not necessarily the smallest one, since there might be another set F 0 ⊂ F such that system (8) is also UUB(XH , F 0 )). Note that, if (15) is solvable by fixing δ = 0 a priori, (15) becomes a test of asymptotic stability of the origin. Note that in this case the constraints in (14) parameterize all the PWA Lyapunov functions that are defined on the same partition {Xi } of XH . Clearly, a necessary condition for system (8) to be UAS is that Ei = 0, ∀i ∈ I0 , or D = {0}, otherwise the origin would not be an equilibrium. Remark 4 In case (14) is infeasible, besides increasing the value of λ or changing that of ρ, a possibility is to increase the number of regions Xi of XH , therefore providing more flexibility in synthesizing the PWA Lyapunov-like function. A possible way is to consider the sets Xij as the new sets X¯i and restart the one-step reachability analysis. Due to space limitations, we refer the reader to [24] for further details. A practical procedure to calculate F consists of partitioning Q into a number of ˜ and then obtain all the sets R(Qi ) as in (5). subsets Qi , Qi = X¯i ∩ Q with i ∈ I, Then F can be easily computed as the union of the subsets of the sets Xi for which V (x) ≤ Vˆ , i.e., F=
6
s˜ [ i=1
XiF , XiF , {x ∈ Xi : V (x) ≤ Vˆ }.
(19)
Invariance analysis
So far we have analyzed the properties of the extended system (8). We want now to derive conditions on the original system (3). Consider again system (8) in XH , assume that a feasible solution to (14) exists, define (recalling that XE = XH \ X ) inf x∈XE V (x) if XE 6= ∅ − VE , (20) +∞ if XE = ∅ and consider the set P , {x ∈ XH : V (x) < VE− } ⊆ X . 8
(21)
In case X is a RPI set (and hence XE = ∅), one has P ≡ X , since V (x) < ∞ for all x ∈ X . Otherwise, the set P is given by the union of a finite number of polytopes, each of them composed by the points x ∈ Xi , i ∈ I, for which V (x) < VE− . Being the sublevel set of a discontinuous Lyapunov function, the set P in this case may not be convex, and not even connected. The following result related to RPI sets can be easily proven. Lemma 3 Consider system (3), whose dynamics are defined on X , and assume that either the dynamics (7) are defined in XE , or XE = ∅. Then, if a set X˜ ⊆ X is RPI for (8), it is also RPI for (3). Proof: See Appendix. The results for the extended system (8) proved in Lemma 2, and the definition of P in (21) and Lemma 3 are exploited next to state the main result of this paper. Theorem 1 Consider system (3), whose dynamics are defined on X , and assume that either the dynamics (7) are defined in XE , or XE = ∅. Then: 1. if the optimal δ of (15) is zero, then P in (21) is a RPI set for (3), and (3) is UAS(P); 2. otherwise, if VE− ≥ VF+ , the sets P and F are RPI sets with respect to dynamics (3), and system (3) is UUB(P, F).
Proof: See Appendix. Note that, by construction of F, if XE = ∅, then F ⊆ P is always satisfied. Also, note that the result of Theorem 1 refers to the original system (3), and the fake dynamics were only employed to determine a suitable Lyapunov-like function. A graphical representation of the sets XH , X , P, Q and F is shown in Fig. 1, where we show that XH ⊇ X ⊇ P. Also, we recall that Q, F ⊆ XH by definition.
F
Q
P
X
XH
Figure 1: Graphical representation of the sets XH , X , P, Q and F for an example of second-order system
7
Numerical complexity
The procedure described in Section 4 requires four main numerical computations. First, we need the vertex representation of the polytopes in (2). This can be done by solving 9
a vertex enumeration problem (see the approaches in [30] and [31], for example). The second operation is the computation of convex hulls, which is the dual of the vertex enumeration problem [32]. The complexity of these two operations is exponential in the number of facets of the considered polytopes. The third operation is the projection in (10), for which there exist different methods [33] and whose complexity increases quickly with the number of vertices or facets. The fourth computation is to obtain minimal hyperplane representations of polyhedra to eliminate redundant inequalities [34], which can be done by solving one LP for each inequality defining the given polyhedron. In the stability and invariance analysis procedure described in Sections 56, the solution of a single LP is needed to compute V (x), while a number of operations similar to those needed in Section 4 is required to compute the sets P and F.
8
Conclusions
This note has addressed the problem of determining the properties of uniform asymptotic stability and uniform ultimate boundedness of (possibly discontinuous) discretetime PWA systems subject to additive disturbances. Since the dynamics are defined in the set X that is a priori not invariant, partially fictitious dynamics are exploited to define a discontinuous PWA Lyapunov-like function. As an outcome of the optimization problem, the RPI set P ⊆ X is obtained, together with the terminal set F, or the certificate of asymptotic stability.
Appendix Proof of Lemma 1. If x ∈ XH , then either x ∈ X or x ∈ XE . If x ∈ X then the successor state Ai x + ai + Ei d ∈ R∪ (X ) ⊆ XH by definition of XH . If x ∈ XE , the successor state is ρx ∈ XH , because XH is a convex set including the origin. Proof of Lemma 2. To prove the positive definiteness of the Lyapunov-like function, Pmi define βi,h ≥ 0, h=1 βi,h = 1, as a set of coefficients which allow one to express x ∈ X¯i as a convex combination of the vertices of X¯i . Since functions Vi are affine functions defined on convex sets X¯i , the satisfaction of (14a) for all vi,h ∈ vert(X¯i ), ˜ h = 1, . . . , mi , for x = Pmi βi,h vi,h , leads to with i ∈ I, h=1 α1 kxkW
m
i
X
= α1 βi,h vi,h
h=1
≤
W
= Fi
mi X h=1
βi,h α1 kvi,h kW ≤
mi X
! βi,h vi,h
h=1
+ gi
mi X
βi,h (Fi vi,h + gi )
h=1
mi X
βi,h = Fi x + gi . (22)
h=1
For this reason, for x ∈ int(Xi ), since Vi (x) = Fi x+gi , (11a) holds. Moreover, on the boundaries of X¯i , according to (13b), one has α1 kxkW ≤ Fi x + gi for all i ∈ N (x), and therefore α1 kxkW ≤ max Si∈N (x) {Fi x+gi } = V (x). This implies that (11a) holds for all x ∈ XH , since XH = i∈I˜ Xi .
10
for the decay of the Lyapunov-like function, for x ∈ Xij , define βij,h ≥ 0, PmAs ij β h=1 ij,h = 1, as a set of coefficients such that x ∈ Xij can be expressed as a convex combination of the vertices of Xij . Therefore, " ! !# mij η X X V (f (x, d)) = Fj Ai βij,h vij,h +ai + Ei cµ dµ + gj (23) µ=1
h=1
Pη
where cµ ≥ 0, µ=1 cµ = 1, are coefficients used to express the fact that any point d ∈ D can be written as a convex combination of the vertices Pm Pηof D. Recalling that ij (14b) holds for all the vertices of Xij , and that h=1 βij,h = µ=1 ch = 1, from (23) we get " mij !# η X X V (f (x, d)) = Fj βij,h Ai vij,h + ai + Ei cµ dµ + gj µ=1
h=1 mij
= Fj
X
βij,h
Ai vij,h + ai +
=
βij,h
≤
cµ Ei dµ
+ gj
η X
cµ (Fj (Ai vij,h + ai + Ei dµ )) + gj
µ=1
h=1 mij
X
!
µ=1
h=1 mij X
η X
βij,h
h=1
=δ+λ
η X µ=1
mij X h=1
cµ (δ + λ (Fi vij,h + gi ) − gj ) + gj
βij,h (Fi vij,h + gi ) −
= δ + λ Fi
mij X
mij X
βij,h gj + gj
h=1 mij
βij,h vij,h +
X
!
βij,h gi
h=1
h=1
= δ + λ (Fi x + gi ) = δ + λV (x), which proves that (11c) holds for all x ∈ int (Xij ). Also, on the boundaries of Xij , the decreasing condition (11c) is imposed for all (i, j) ∈ N (x) × N (f (x, d)), and therefore max Fj (f (x, d)) + gj ≤ δ + λ max (Fi x + gi ). i∈N (x)
j∈N (f (x,d))
Ss˜
Ss˜ Since by definition of the sets Xij we have that XH = i=1 j=1 Xij , (11c) holds for all x ∈ XH , and then (11) holds for all x ∈ XH . Up to this point, the proven results hold for any δ ≥ 0. Considering the case δ = 0, the decay condition (11c) becomes V (f (x, d)) − λV (x) ≤ 0, which also implies f (0, d) = 0, ∀d ∈ D. From V (f (x, d)) − λV (x) ≤ 0 we can easily obtain V (x(k)) ≤ λk V (x(0)), which leads to α1 kx(k)kW ≤ V (x(k)) ≤ λk V (x(0)) ≤ λk α2 kx(0)kW . Therefore, α2 k λ kx(0)kW , kx(k)kW ≤ α1 11
which implies that system (8) is UAS(XH ), according to Definition 3, with φ(kx(0)k, k) = α2 λk /α1 . Consider now the case δ > 0. By (11a) and (11c) it follows that V (f (x, d)) − V (x) ≤ −(1 − λ)V (x) + δ ≤ −(1 − λ)α1 kxkW + δ = −α3 kxkW + δ, where α3 = (1−λ)α1 . By definition of Q in (18), for all x ∈ XH \Q we obtain −α3 kxkW +rδ < 0, leading to V (f (x, d)) − V (x) < −εδ, x ∈ XH \ Q. (24) By definition of F, the fulfillment of the constraint (14d) implies Q ⊆ H ⊆ XH and, by the invariance of XH (Lemma 1), R(Q) ⊆ XH . This implies that the max in (17) exists (since Q, and therefore R(Q), are closed sets) and therefore F in (16) is well defined. To prove invariance of F, we distinguish two cases: • if x ∈ F ∩ Q, then f (x, d) ∈ R(Q) ⊆ F; • if x ∈ F \ Q, then (24) holds, and therefore V (f (x, d)) < V (x) ≤ VF+ (this latter being defined in (17)), meaning that f (x, d) ∈ F by definition of F. Therefore, F is a RPI set for (8). To prove that system (8) is UUB(XH , F) according to Definition 4, we must show that for all a > 0 there exists T = T (a) such that, for all x(0) ∈ XH with kx(0)kW ≤ a, x(T ) ∈ F for any admissible sequence of the disturbance term d. We define Qa , {x ∈ Rn : kxkW ≤ a}, and distinguish two cases: • If 0 < a ≤ rδ/α3 , condition x(0) ∈ Qa implies x(0) ∈ Q. Therefore, simply taking T (a) = 1, we have that for every x(0) ∈ Qa , x(T ) ∈ F (by definition of F). • If a > rδ/α3 , let VH− , inf x∈XH \Q V (x) > 0. Then, from (24) it follows that ¯ ¯ < V −, ∀x(0) ∈ XH \ Q, ∃k¯ = k(a) ∈ Z>0 : V (x(k)) H
(25)
¯ ∈ Q, and therefore x(k¯ + 1) ∈ F. In order to explicitely find meaning that x(k) ¯ k as a function of a, let Va+ , supx∈Qa \Q V (x). From (24), one has + Va − VH− ¯ . k(a) , rδ ¯ In conclusion, if a > rδ/α3 , there exists T (a) = k(a) + 1 such that, for every x(0) ∈ Qa , x(T ) ∈ F.
Proof of Lemma 3. Since X˜ ⊆ X , from (8) we can easily see that, given x(k) ∈ X˜ , we have that x(k + 1) = Ai x(k) + ai + Ei d(k), with x(k) ∈ Xi and i ∈ I. Notice that dynamics (3) and (8) coincide for x(k) ∈ X˜ . This means that we applied dynamics (3), and the same will hold for all subsequent time instants, since x(k + 1) ∈ X˜ by assumption (X˜ is a RPI set). We conclude that X˜ is a RPI set for dynamics (3). Proof of Theorem 1. The proof consists of showing that the PWA Lyapunov-like function VP (x) : P → R, VP (x) ≡ V (x), ∀x ∈ P 12
(26)
where V (x) is found as in Lemma 2 for the extended system (8) in XH , is a Lyapunovlike function for (3) over the set P. First of all, we prove that P is a RPI set for (8) in XH . In case δ = 0, for all x ∈ P, V (f (x, d)) ≤ λV (x), and then f (x, d) ∈ P by definition of P. In case δ > 0, first of all notice that, by definition of P and F as sublevel sets, it follows that the assumption VE− ≥ VF+ implies F ⊆ P. Then, we distinguish two cases: • if x ∈ F, then f (x, d) ∈ F (which is a RPI set for (8)), and then f (x, d) ∈ P since F ⊆ P; • if x ∈ P \ F, we distinguish again two sub-cases: – if x ∈ Q, then f (x, d) ∈ F ⊆ P, since R(Q) ⊆ F by construction (see (16)-(18)); – if x ∈ / Q, then, from (24), V (f (x, d)) < V (x), leading to f (x, d) ∈ P. Since P ⊆ X , by Lemma 3 we obtain that P is a RPI set for dynamics (3). This fact leads to the possibility of defining a Lyapunov-like function for system (3) in P. Considering that P ⊆ XH , if (11) are satisfied with δ = 0 for all x ∈ XH (and then for all x ∈ P), we conclude that VP (x) is a Lyapunov function for system (3) in P, and that system (3) is UAS(P). In case δ > 0, if (11) hold for all x ∈ XH (and then for all x ∈ P), the invariance of F with respect to dynamics (3) is guaranteed by Lemmas 2 and 3. We have already proved that P is RPI for (3), and we know that F ⊆ P by construction. We also know from Lemma 2 that the extended system (8) is UUB(XH , F), which means that, for all a > 0, there exists T (a) > 0 such that, for every x(0) ∈ XH with kx(0)k ≤ a, x(T ) ∈ F for all the sequences d[T ] with d(k) ∈ D, k = 0, . . . , T . However, since dynamics (3) and (8) coincide in P, one can use Definition 4 again considering dynamics (3). We know that for all x(0) ∈ P, for all a > 0 there exists T (a) > 0 such that, for every x(0) ∈ P with kx(0)k ≤ a, x(T ) ∈ F for all the sequences d[T ] with d(k) ∈ D, k = 0, . . . , T . In conclusion, system (3) is UUB(P, F).
Acknowledgement The authors would like to thank Prof. Stephen Boyd (Stanford University) for the useful discussion on the formulation of Lemma 2.
References [1] E. D. Sontag, “Nonlinear regulation: The piecewise linear approach,” IEEE Transactions on Automatic Control, vol. 26, no. 2, pp. 346–358, 1981. [2] A. Bemporad and M. Morari, “Control of systems integrating logic, dynamics, and constraints,” Automatica, vol. 35, pp. 407–428, 1999.
13
[3] A. Bemporad, “Efficient conversion of mixed logical dynamical systems into an equivalent piecewise affine form,” IEEE Transactions on Automatic Control, vol. 49, no. 5, pp. 832–838, 2004. [4] F. J. Christophersen, M. Baotic, and M. Morari, “Stability analysis of hybrid systems with a linear performance index,” in Proc. Conference on Decision and Control, Atlantis, Paradise Island, Bahamas, December 2004, pp. 4589–4594. [5] A. Bemporad, W. Heemels, and B. De Schutter, “On hybrid systems and closedloop MPC systems,” IEEE Transactions on Automatic Control, vol. 47, no. 5, pp. 863–869, 2002. [6] S. Di Cairano, A. Bemporad, I. Kolmanovsky, and D. Hrovat, “Model predictive control of magnetically actuated mass spring dampers for automotive applications,” Int. Journal of Control, vol. 80, no. 11, pp. 1701–1716, 2007. [7] F. Borrelli, A. Bemporad, M. Fodor, and D. Hrovat, “An MPC/hybrid system approach to traction control,” IEEE Trans. Contr. Systems Technology, vol. 14, no. 3, pp. 541–552, May 2006. [8] N. Giorgetti, A. Bemporad, H. Tseng, and D. Hrovat, “Hybrid model predictive control application towards optimal semi-active suspension,” Int. Journal of Control, vol. 79, no. 5, pp. 521–533, 2006. [9] A. Alessio and A. Bemporad, “A survey on explicit model predictive control,” in Nonlinear Model Predictive Control: Towards New Challenging Applications, ser. Lecture Notes in Control and Information Sciences, L. Magni, F. Allgower, and D. M. Raimondo, Eds. Berlin Heidelberg: Springer-Verlag, 2009, vol. 384, pp. 345–369. [10] A. Bemporad, A. Oliveri, T. Poggi, and M. Storace, “Ultra-fast stabilizing model predictive control via canonical piecewise affine approximations,” IEEE Transactions on Automatic Control, vol. 56, no. 12, pp. 2883–2897, 2011. [11] G. Ferrari Trecate, F. A. Cuzzola, D. Mignone, and M. Morari, “Analysis of discrete-time piecewise affine and hybrid systems,” Automatica, vol. 38, no. 12, pp. 2139–2146, 2002. [12] P. Grieder, M. Kvasnica, M. Baoti´c, and M. Morari, “Stabilizing low complexity feedback control of constrained piecewise affine systems,” Automatica, vol. 41, no. 10, pp. 1683–1694, 2005. [13] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear matrix inequalities in system and control theory. SIAM, 1994. [14] M. Lazar and A. Joki´c, “On infinity norms as Lyapunov functions for piecewise affine systems,” in Proc. Hybrid systems: computation and control, Stockholm, Sweden, April 2010, pp. 417–432.
14
[15] S. Prajna and A. Papachristodoulou, “Analysis of switched and hybrid systems – beyond piecewise quadratic methods,” in Proc. American Control Conference, Denver, CO, June 2003, pp. 2779–2784. [16] A. Polanski, “Lyapunov function construction by linear programming,” IEEE Transactions on Automatic Control, vol. 42, no. 7, pp. 1013–1016, 1997. [17] E. C. Kerrigan and D. Q. Mayne, “Optimal control of constrained, piecewise affine systems with bounded disturbances,” in Proc. Conference on Decision and Control, Las Vegas, NV, December 2002, pp. 1552–1557. [18] G. Zhai, H. Lin, and P. J. Antsaklis, “Quadratic stabilizability of switched linear systems with polytopic uncertainties,” International Journal of Control, vol. 76, no. 7, pp. 747–753, 2003. [19] H. Lin and P. Antsaklis, “Synthesis of uniformly ultimate boundedness switching laws for discrete-time uncertain switched linear systems,” in Proc. Conference on Decision and Control, Maui, HI, December 2003, pp. 4806–4811. [20] F. Blanchini and S. Miani, Set-theoretic methods in control. 2008.
Springer-Verlag,
[21] P. Biswas, P. Grieder, J. L¨ofberg, and M. Morari, “A survey on stability analysis of discrete-time piecewise affine systems,” in IFAC World Congress, Prague, Czech Republic, June 2005. [22] S. V. Rakovi´c, P. Grieder, M. Kvasnica, D. Q. Mayne, and M. Morari, “Computation of invariant sets for piecewise affine discrete time systems subject to bounded disturbances,” in Proc. Conference on Decision and Control, Atlantis, Paradise Island, Bahamas, December 2004, pp. 1418–1423. [23] S. Rakovi´c, E. Kerrigan, D. Mayne, and J. Lygeros, “Reachability analysis of discrete-time systems with disturbances,” IEEE Transactions on Automatic Control, vol. 51, no. 4, pp. 546–561, 2006. [24] S. Trimboli, M. Rubagotti, and A. Bemporad, “Stability and invariance analysis of uncertain PWA systems based on linear programming,” in Proc. Conference on Decision and Control, Orlando, FL, December 2011, pp. 7398–7403. [25] M. Rubagotti, S. Trimboli, D. Bernardini, and A. Bemporad, “Stability and invariance analysis of approximate explicit MPC based on PWA Lyapunov functions,” in Proc. IFAC World Congress, Milan, Italy, August 2011, pp. 5712–5717. [26] A. Oliveri, D. Barcelli, A. Bemporad, B. A. G. Genuit, W. P. M. H. Heemels, T. Poggi, M. Rubagotti, and M. Storace, “MOBY-DIC: A Matlab toolbox for the circuit design of explicit MPC,” in IFAC Conference on Nonlinear Model Predictive Control, Noordwijkerhout, the Netherlands, August 2012. [27] M. Lazar, W. P. M. H. Heemels, and A. R. Teel, “Lyapunov functions, stability and input-to-state stability subtleties for discrete-time discontinuous systems,” IEEE Transactions on Automatic Control, vol. 54, no. 10, pp. 2421–2425, 2009. 15
[28] A. Bemporad, M. Morari, V. Dua, and E. N. Pistikopoulos, “The explicit linear quadratic regulator for constrained systems,” Automatica, vol. 38, no. 1, pp. 3–20, 2002. [29] S. P. Boyd and L. Vandenberghe, Convex Optimization. Press, 2004.
Cambridge University
[30] D. Avis, “LRS: A revised implementation of the reverse search vertex enumeration algorithm,” in Polytopes, Combinatorics and Computation, ser. DMVSeminar 29, G. Kalai and G. Ziegler, Eds. Birkhauser Verlag, 2000, pp. 177–198. [31] K. Fukuda and A. Prodon, “Double description method revisited,” in Combinatorics and Computer Science, ser. Lecture Notes in Computer Science. Springer, 1996, pp. 91–111. [32] K. Fukuda, cdd/cdd+ Reference Manual, Institute for operations Research ETHZentrum, 1997. [33] C. N. Jones, E. C. Kerrigan, and J. M. Maciejowski, “Equality set projection: A new algorithm for the projection of polytopes in halfspace representation,” University of Cambridge, Tech. Rep., June 2004. [34] R. Suard, J. Lofberg, P. Grieder, M. Kvasnica, and M. Morari, “Efficient computation of controller partitions in multi-parametric programming,” in Proc. Conference on Decision and Control, Atlantis, Bahamas, December 2004, pp. 3643– 3648.
16
Abstract This note proposes a method to analyze uniform asymptotic stability and uniform ultimate boundedness of uncertain piecewise affine systems whose dynamics are only defined in a bounded and possibly non-invariant set X of states. The approach relies on introducing fake dynamics outside X and on synthesizing a piecewise affine and possibly discontinuous Lyapunov function via linear programming. The existence of such a function proves stability properties of the original system and allows the determination of a region of attraction contained in X . The procedure is particularly useful in practical applications for analyzing the stability of piecewise affine control systems that are only defined over a bounded subset X of the state space, and to determine whether for a given set of initial conditions the trajectories of the state vector remain within the domain X .
1
Introduction
In the last decade the interest in studying the dynamical properties of piecewise affine (PWA) systems has increased considerably, due to their powerful modeling capabilities. Discrete-time PWA models are a special class of hybrid systems that can represent combinations of finite automata and linear dynamics, are a good approximation of nonlinear systems [1], and are equivalent to hybrid systems in mixed logical dynamical form [2, 3]. Stability analysis tools for PWA systems are useful to describe the properties of autonomous hybrid systems, or to check a posteriori the stability of a given closedloop system [4, 5]. In particular, stability analysis becomes fundamental when a PWA control law is synthesized without a-priori guarantees of closed-loop stability; an example are explicit model predictive control (MPC) laws designed in applications with short prediction horizons and without terminal constraints to limit the number of partitions [6–8], or approximated in order to reduce their complexity [9, 10]. ∗ This paper is published on the IEEE Transactions on Automatic Control, vol. 58, no. 9, pp. 23592365, 2013. M. Rubagotti is with Nazarbayev University, 53 Kabanbay Batyr Ave, 010000 Astana, Kazakhstan. Email: [email protected]. S. Trimboli is with the Department of Mechanical and Structural Engineering, University of Trento, Via Mesiano 77, 38100 Trento, Italy. Email: [email protected]. A. Bemporad is with IMT Institute for Advanced Studies, P.zza S. Ponziano 6, 55100 Lucca, Italy. Email: [email protected]. This work was partially supported by the European Commission under project FP7-INFSO-ICT-248858 “MOBY-DIC – Model-based synthesis of digital electronic circuits for embedded control”.
The most widely used methods for stability analysis of discrete-time PWA systems are based on piecewise quadratic (PWQ) Lyapunov functions [11] and require the solution of semi-definite programming problems. As highlighted in [12], the search for a PWQ Lyapunov function can be overly conservative, even with the use of the so-called S-procedure (see, e.g., [13]). A valid alternative are PWA Lyapunov functions, computed by linear programming (LP) [12, 14]. Other types of Lyapunov functions can be used for the same purpose, such as piecewise polynomial Lyapunov functions [15]. The approaches for PWA stability analysis recalled above deal with deterministic systems. However, in applications it is often important to determine if the state converges to the origin (or to a terminal set) despite parametric uncertainties and/or external disturbances. Some classical results on stability analysis and control of uncertain linear parameter varying and switched linear systems appeared in [16–19] and [20, Chap. 7]. In analyzing the stability of PWA systems, it is always assumed that the set X of states in which the PWA dynamics are defined is positively invariant, as the notion of stability has no practical relevance if the state trajectory exits the domain of definition of the dynamics [21]. However, often the PWA system to be analyzed is defined in a set X that may not be invariant, meaning that, for certain initial conditions, the corresponding state trajectories are not defined for all time steps. A possible approach to tackle the possible non-invariance of X is to perform a reachability analysis to find the maximum positively invariant subset of X , using a recursive procedure (see [20, Chap. 4-5], [22, 23], and the references therein). Unfortunately this can lead to very involved solutions, due to the exponential complexity of reachability analysis of PWA systems, and in some cases searching the maximum invariant set is even an undecidable problem. This paper proposes a stability analysis framework for (possibly discontinuous) discrete-time PWA systems that are affected by additive disturbances bounded within a polytopic set (a similar framework was proposed in [24] to handle the case of parametric uncertainties). A discontinuous PWA Lyapunov function is synthesized via linear programming (LP) either to determine if the state converges to the origin or to find a terminal set where the state is ultimately bounded. As we assume that the PWA dynamics are defined in a closed polytope X of the state space which may not be invariant, by artificially extending the systems dynamics outside X the proposed method can also determine an invariant subset of X for the original PWA system. In this case the attractiveness of the origin (or that of the terminal set) is determined with respect to such a region of attraction. A preliminary version of this paper focusing only on asymptotic stability of deterministic PWA systems has appeared in [25]. A numerical implementation of the overall stability analysis proposed in this work can be found in the MOBY-DIC Toolbox for MATLAB, described in [26].
2
Basic notation and definitions
Let R, Z, Z≥0 , and Z>0 denote the sets of reals, integers, non-negative integers, and positive integers, respectively. Given a ∈ R, the ceil function dae is defined as the smallest b ∈ Z such that a ≤ b. Given a vector v ∈ Rn , let kvk denote any vector norm, 2
and let kvkW , kW vk∞ , where W ∈ Rn×n is a full rank diagonal matrix. Given two matrices A1 ∈ Rm1 ×n , A2 ∈ Rm2 ×n , [A1 ; A2 ] denotes the matrix [A01 A02 ]0 ∈ R(m1 +m2 )×n . Given a discrete-time signal d : Z≥0 → Rp , the sequence of the values of d from time zero to the k-th instant is denoted by d[k] . Given a set A ⊆ Rn , its ¯ and its convex hull by conv(A). Given interior is denoted by int(A), its closure by A, a finite number of sets Ai , i ∈ Ia = {1, ..., na }, we say that {Ai } is aSpartition of A na Ai = A. if int(Ai ) 6= ∅, int(Ai ) ∩ int(Aj ) = ∅, ∀i, j ∈ Ia with i 6= j, and i=1 If {Ai } is a partition with Ai ∩ Aj = ∅ ∀i, j ∈ Ia with i 6= j, it is denoted as strict partition. A polyhedron is a set given by the intersection of a finite number of (closed or open) half-spaces. A polytope A is a bounded polyhedron, and the set of the vertices ¯ of its closure A¯ is denoted by vert(A). Consider a discrete-time nonlinear system x(k + 1) = ϕ(x(k), d(k)),
(1)
where x ∈ X ⊆ Rn is the state vector, the input d ∈ D ⊂ Rp collects both model uncertainties and exogenous disturbances, and both X and D are compact sets with 0 ∈ int(X ), 0 ∈ D. Definition 1 For system (1) the one-step reachable set from a set A ⊆ X is R(A) , {y ∈ Rn : y = ϕ(x, d), d ∈ D, x ∈ A}. Definition 2 A set B ⊆ X is called robustly positively invariant (RPI) with respect to dynamics (1) if, for all d ∈ D, R(B) ⊆ B. Note that the set X is not assumed to be RPI with respect to dynamics (1), so that some trajectories may possibly leave X and be therefore defined only on a finite time interval of time [0, kmax ]. Definition 3 Consider dynamics (1) and a RPI set B ⊆ X with 0 ∈ int(B) and ϕ(0, d) = 0 for all d ∈ D. System (1) is uniformly asymptotically stable in P (UAS(P)) if there exists a KL-function1 φ such that, for all the initial conditions x(0) ∈ P and for all the sequences d[k] with d(i) ∈ D, i = 0, ..., k, kx(k)k ≤ φ(kx(0)k, k), for all k ∈ Z≥0 . Definition 4 Given two RPI sets P ⊆ X and F ⊆ P, with 0 ∈ int(F), system (1) is uniformly ultimately bounded from P to F (UUB(P, F)) if for all a > 0 there exists T (a) > 0 such that, for every x(0) ∈ P with kx(0)k ≤ a, x(T ) ∈ F for all the sequences d[T ] with d(k) ∈ D, k = 0, ..., T .
3
Stability analysis problem
Given a closed polytope X ⊂ Rn that includes the origin in its interior, consider a partition {Xi } of X that consists of a finite number s of polytopes Xi , {x : Hi1 x ≤ h1i , Hi2 x < h2i }, i ∈ I , {1, . . . , s}, 1 See,
e.g., Section II.A in [27] for a formal definition of KL-functions.
3
(2)
where Hi1 ∈ Rqi,1 ×n , Hi2 ∈ Rqi,2 ×n are constant matrices with qi,1 , qi,2 ∈ Z>0 , i ∈ I, and h1i ∈ Rqi,1 and h2i ∈ Rqi,2 are constant vectors, and assume that {Xi } is a strict partition. The closure X¯i of Xi is denoted by X¯i = {x : Hi x ≤ hi }, with Hi = [Hi1 ; Hi2 ] ∈ Rqi ×n and hi = [h1i ; h2i ] ∈ Rqi . The subset of indices I0 is defined as I0 , {i ∈ I : 0 ∈ X¯i }. Without loss of generality, we assume that 0 ∈ vert(X¯i ) for all i ∈ I0 , which is necessary when PWA Lyapunov functions are considered. If this property does not hold for the given partition {Xi }, it is always possible to obtain it by further partitioning the regions whose indices belongoto I0 . n ˜ , with H ˜ ∈ ˜ ≤h ˜ ∈ Rη×p , and h Consider a closed polytope D = d ∈ Rp : Hd η p R , p, η ∈ Z>0 , and 0 ∈ D. The vertices of D are denoted by d1 , . . . , dη ∈ R , and, since D is compact by definition, D = conv(d1 , . . . , dη ). Consider the autonomous discrete-time uncertain PWA system x(k + 1) = Ai x(k) + ai + Ei d(k) if x(k) ∈ Xi
(3)
with x(k) ∈ X , d(k) ∈ D, Ai ∈ Rn×n , ai ∈ Rn , Ei ∈ Rn×p , and k ∈ Z≥0 . Note that the dynamics (3) are not required to be continuous on the boundaries of the sets Xi . Problem statement: Given the uncertain PWA system (3), for which X is not necessarily a RPI set, prove the properties of stability and convergence to the origin (uniform asymptotic stability or uniform ultimate boundedness) with respect to a RPI set P ⊆ X . In case of ultimate boundedness, find another RPI set F including the origin, where the state is driven in finite time. Nominal stability analysis is considered a special case of the problem when D = {0} or Ei = 0 for all i ∈ I.
4
Extended system and reachability analysis
Consider the one-step reachable set R(X ) from X , R(X ) = {Ai x + ai + Ei d : d ∈ D, x ∈ Xi , i ∈ I}, and define the closed set R∪ (X ) , R(X ) ∪ X .
(4)
The closure R(X ) of R(X ) can be computed as the union of the one-step reachable sets from all the sets X¯i , namely R(X¯i ) , {Ai x + ai + Ei d, d ∈ D, x ∈ X¯i }. By relying on the results in [20, Chap. 6], we can compute the convex sets R(X¯i ) as R(X¯i ) = conv (Ai vi,h + ai + Ei dµ , µ = 1, . . . , η, h = 1, . . . , mi )
(5)
where vi,h represents each of the mi vertices of X¯i , to get R∪ (X ) =
s [ i=1
R(X¯i ) ∪ X .
(6)
As the dynamics (3) are not defined outside X , the proposed strategy consists of defining fake dynamics on R∪ (X ) \ X . Let XH , conv(R∪ (X )), and consider the dynamics x(k + 1) = ρx(k), if x(k) ∈ XE , XH \ X (7) 4
where ρ ∈ [0, 1) is an adjustable parameter of the proposed approach. The region XE , when XE 6= ∅, can be divided into convex polyhedral regions as in [28, Th. 3]. As a result, new regions Xi , i = s + 1, . . . , s˜, are created, that together with the original Xi with i ∈ I define a strict partition {Xi } of XH , with i ∈ I˜ , {1, . . . , s˜}. The dynamics of the extended system on XH are Ai x(k) + ai + Ei d(k) if x(k) ∈ Xi , i ∈ I x(k + 1) = (8) ρx(k) if x(k) ∈ Xi , i ∈ I˜ \ I. For convenience, let Ai , ρI, ai , 0, and Ei , 0, for i ∈ I˜ \ I. Also, notice that, if XE = ∅, then I˜ \ I = ∅, and no additional dynamics are introduced. Lemma 1 The set XH is RPI for the extended dynamics (8). Proof:See Appendix. The definitions of XH and of the dynamics in XE as in (7) are simplistic, yet we will prove their effectiveness. Other choices of XH (see, e.g., [25]) and of the dynamics (7) are possible, provided that Lemma 1 still holds. If X is a RPI set, one has R∪ (X ) = X , and therefore XH = X , because X is a convex set by definition. In this case, the definition of fake dynamics is not necessary, and the same approach hereafter described can be applied with s˜ = s, and XE = ∅. For any pair (i, j) ∈ I˜ × I˜ define the closed transition sets Xij , x ∈ X¯i : ∃d ∈ D : Ai x + ai + Ei d ∈ X¯j (9) of states that can possibly end up in X¯j in one step under dynamics i. Note that XH = Ss˜ Ss˜ i=1 j=1 Xij , but {Xij } in general is not a partition of XH , unless D = {0}, or ˜ To compute the sets Xij we exploit here controllability analysis [20, Ei = 0, ∀i ∈ I. Chap. 5] and consider the disturbance vector d as the external input with respect to which the controllability analysis is carried out. First, we compute the set hj Hj Ai Hj Ei x x Hj ai n+p ¯ Mi (Xj ) = ∈R : + ≤ ˜ ˜ d d 0 0 H h which is defined on the extended space (x, d), and we project it onto the first n coordinates (the state space), obtaining the so-called pre-image set of X¯j with respect to dynamics i Pre(X¯j , i) , x ∈ Rn : ∃d ∈ D : Ai x + ai + Ei d ∈ X¯j . If we impose x ∈ X¯i , we obtain Xij = X¯i ∩ Pre(X¯j , i) .
(10)
˜ In the particular case of no Let mij be the number of vertices of Xij , (i, j) ∈ (I˜ × I). ˜ additive disturbances (i.e., D = {0}, or Ei = 0, ∀i ∈ I), the transition sets Xij can be computed without using of the projection operation, as Hi 0 hi Xij = x ∈ Rn : x+ ≤ . H j Ai Hj ai hj 5
5
PWA Lyapunov analysis of the extended system
Let V : XH → R be a function such that V (x) ≥ α1 kxkW
(11a)
V (0) = 0
(11b)
V (f (x, d)) − λV (x) ≤ δ
(11c)
˜ and ∀d ∈ D, where f : Rn × Rη → Rn is the PWA state update ∀x ∈ Xi (i ∈ I) function defined by (8), α1 > 0, λ ∈ (0, 1), δ ≥ 0. W ∈ Rn×n can be chosen as any full rank diagonal matrix, but its value is fixed a priori as a design parameter. We call a function V : Rn → R satisfying (11) a Lyapunov-like function. Remark 1 Note that often in the literature an upperbound α2 kxkW ≥ V (x) is imposed, with α2 > 0. Here this is not necessary as V will be defined as a PWA function over the bounded set XH . As a consequence, it is always possible to find a posteriori α2 > 0 such that V (x) ≤ α2 kxkW , ∀x ∈ XH , once a Lyapunov-like function V has been determined. Note also that condition (11c) could be replaced by V (f (x, d)) − V (x) ≤ −α3 kxkW + δ
(12)
where α3 = (1−λ)α1 > 0. In fact, by (11a)-(11c), it follows that V (f (x, d))−V (x) ≤ −(1 − λ)V (x) + δ ≤ −(1 − λ)α1 kxkW + δ. To the end of synthesizing a PWA Lyapunov-like function for system (8), define a linear function Vi : X¯i → R for each region Xi , Vi (x) , Fi x + gi
(13a)
˜ where in (13a) Fi ∈ R1×n and gi ∈ R are coefficients to be determined. The for i ∈ I, overall Lyapunov-like function candidate is V (x) = max Vi (x)
(13b)
N (x) , {i ∈ I˜ : x ∈ X¯i }.
(13c)
i∈N (x)
where Note that simply V (x) = Fi x + gi for x ∈ int(Xi ). As for numerical reasons we want to consider closed sets X¯i and Vi (x), Vj (x) may not coincide on common boundaries X¯i ∩ X¯j (unless very conservative continuity conditions are imposed), for the states x on the common boundaries we impose (11a) and (11c) on Vi for all i ∈ N (x), although only one value (the max) is taken in (13b), as V (x) must be single-valued. In the specific case of (11b), we have Vi (0) = 0 for all i ∈ I0 , because the continuity condition is imposed at the origin. The constraints Fi vi,h + gi ≥ α1 kvi,h kW
6
(14a)
˜ h = 1, . . . , mi , while are imposed for all mi vertices vi,h ∈ vert(X¯i ), i ∈ I, Fj (Ai vij,h + ai + Ei dµ ) + gj − λ(Fi vij,h + gi ) ≤ δ
(14b)
˜ and for all vij,h ∈ vert(Xij ), with h = 1, . . . , mij , for all Ai , ai , Ei with i ∈ I, all dµ with µ = 1, . . . , η. We define kmax ∈ R>0 as the maximum value such that H , {x : kxkW ≤ kmax } ⊆ XH , which can be easily determined by bisection. The further constraints δ≥0
(14c)
rδ ≤ (1 − λ)Lα1
(14d)
α1 > 0
(14f)
gi = 0, i ∈ I0
(14e)
are also imposed, where L is the minimum distance between the boundary of the hyperrectangle H and the origin, while r = 1 + ε, where ε > 0 is a fixed parameter, chosen as small as possible (e.g., equal to the machine epsilon). Condition (14d) is imposed to ensure that the set where the state is ultimately bounded is included in XH (the reader is referred to the proof of Lemma 2, after equation (24), for details on this aspect), while (14e) ensures the fulfillment of (11b). The vector of variables to be determined ˜ We define now a procedure is composed by δ, α1 , and the terms Fi and gi , with i ∈ I. to determine a choice for such variables by means of a convex optimization problem. Lemma 2 Consider the linear-fractional program δ α1 subj. to (14)
mininize
(15a) (15b)
associated with the autonomous uncertain PWA dynamics (8) and the candidate Lyapunovlike function (13), and assume it admits a feasible solution. Then: 1. if the optimal δ is equal to zero, the origin is an equilibrium point for system (8), which is UAS(XH ); 2. otherwise (the optimal δ is strictly positive), system (8) is UUB(XH , F), with F = {x ∈ XH : V (x) ≤ VF+ }
(16)
where VF+ , max V (x) x∈R(Q) Q , x ∈ Rn : kxkW ≤
(17) rδ (1 − λ)α1
.
(18)
Proof: See Appendix. 7
Remark 2 Note that (15) is, strictly speaking, a quasi-convex optimization problem. Even though quasi-convex optimization problems are solved in general by bisection (solving a convex optimization problem at each step), linear-fractional programs like (15) can be transformed into an equivalent LP using standard procedures (see [29, ˜ plus an auxiliary Chap. 4]). The variables of the LP are δ, α1 , Fi and gi , with i ∈ I, variable introduced by the formulation of the equivalent problem. The total number of variables is nv = 3 + s˜(n + 1), the total number of constraints is nc = 4 + card(I0 ) + Ps˜ Ps˜ i=1 mi + η j=1 mij . Remark 3 The aim of (15) is to minimize the volume of the hyper-rectangle Q (in fact, the cost function of (15a) is directly proportional to the volume of Q) for the given choice of the sets Xi , in order to obtain a set F that is as small as possible (even if not necessarily the smallest one, since there might be another set F 0 ⊂ F such that system (8) is also UUB(XH , F 0 )). Note that, if (15) is solvable by fixing δ = 0 a priori, (15) becomes a test of asymptotic stability of the origin. Note that in this case the constraints in (14) parameterize all the PWA Lyapunov functions that are defined on the same partition {Xi } of XH . Clearly, a necessary condition for system (8) to be UAS is that Ei = 0, ∀i ∈ I0 , or D = {0}, otherwise the origin would not be an equilibrium. Remark 4 In case (14) is infeasible, besides increasing the value of λ or changing that of ρ, a possibility is to increase the number of regions Xi of XH , therefore providing more flexibility in synthesizing the PWA Lyapunov-like function. A possible way is to consider the sets Xij as the new sets X¯i and restart the one-step reachability analysis. Due to space limitations, we refer the reader to [24] for further details. A practical procedure to calculate F consists of partitioning Q into a number of ˜ and then obtain all the sets R(Qi ) as in (5). subsets Qi , Qi = X¯i ∩ Q with i ∈ I, Then F can be easily computed as the union of the subsets of the sets Xi for which V (x) ≤ Vˆ , i.e., F=
6
s˜ [ i=1
XiF , XiF , {x ∈ Xi : V (x) ≤ Vˆ }.
(19)
Invariance analysis
So far we have analyzed the properties of the extended system (8). We want now to derive conditions on the original system (3). Consider again system (8) in XH , assume that a feasible solution to (14) exists, define (recalling that XE = XH \ X ) inf x∈XE V (x) if XE 6= ∅ − VE , (20) +∞ if XE = ∅ and consider the set P , {x ∈ XH : V (x) < VE− } ⊆ X . 8
(21)
In case X is a RPI set (and hence XE = ∅), one has P ≡ X , since V (x) < ∞ for all x ∈ X . Otherwise, the set P is given by the union of a finite number of polytopes, each of them composed by the points x ∈ Xi , i ∈ I, for which V (x) < VE− . Being the sublevel set of a discontinuous Lyapunov function, the set P in this case may not be convex, and not even connected. The following result related to RPI sets can be easily proven. Lemma 3 Consider system (3), whose dynamics are defined on X , and assume that either the dynamics (7) are defined in XE , or XE = ∅. Then, if a set X˜ ⊆ X is RPI for (8), it is also RPI for (3). Proof: See Appendix. The results for the extended system (8) proved in Lemma 2, and the definition of P in (21) and Lemma 3 are exploited next to state the main result of this paper. Theorem 1 Consider system (3), whose dynamics are defined on X , and assume that either the dynamics (7) are defined in XE , or XE = ∅. Then: 1. if the optimal δ of (15) is zero, then P in (21) is a RPI set for (3), and (3) is UAS(P); 2. otherwise, if VE− ≥ VF+ , the sets P and F are RPI sets with respect to dynamics (3), and system (3) is UUB(P, F).
Proof: See Appendix. Note that, by construction of F, if XE = ∅, then F ⊆ P is always satisfied. Also, note that the result of Theorem 1 refers to the original system (3), and the fake dynamics were only employed to determine a suitable Lyapunov-like function. A graphical representation of the sets XH , X , P, Q and F is shown in Fig. 1, where we show that XH ⊇ X ⊇ P. Also, we recall that Q, F ⊆ XH by definition.
F
Q
P
X
XH
Figure 1: Graphical representation of the sets XH , X , P, Q and F for an example of second-order system
7
Numerical complexity
The procedure described in Section 4 requires four main numerical computations. First, we need the vertex representation of the polytopes in (2). This can be done by solving 9
a vertex enumeration problem (see the approaches in [30] and [31], for example). The second operation is the computation of convex hulls, which is the dual of the vertex enumeration problem [32]. The complexity of these two operations is exponential in the number of facets of the considered polytopes. The third operation is the projection in (10), for which there exist different methods [33] and whose complexity increases quickly with the number of vertices or facets. The fourth computation is to obtain minimal hyperplane representations of polyhedra to eliminate redundant inequalities [34], which can be done by solving one LP for each inequality defining the given polyhedron. In the stability and invariance analysis procedure described in Sections 56, the solution of a single LP is needed to compute V (x), while a number of operations similar to those needed in Section 4 is required to compute the sets P and F.
8
Conclusions
This note has addressed the problem of determining the properties of uniform asymptotic stability and uniform ultimate boundedness of (possibly discontinuous) discretetime PWA systems subject to additive disturbances. Since the dynamics are defined in the set X that is a priori not invariant, partially fictitious dynamics are exploited to define a discontinuous PWA Lyapunov-like function. As an outcome of the optimization problem, the RPI set P ⊆ X is obtained, together with the terminal set F, or the certificate of asymptotic stability.
Appendix Proof of Lemma 1. If x ∈ XH , then either x ∈ X or x ∈ XE . If x ∈ X then the successor state Ai x + ai + Ei d ∈ R∪ (X ) ⊆ XH by definition of XH . If x ∈ XE , the successor state is ρx ∈ XH , because XH is a convex set including the origin. Proof of Lemma 2. To prove the positive definiteness of the Lyapunov-like function, Pmi define βi,h ≥ 0, h=1 βi,h = 1, as a set of coefficients which allow one to express x ∈ X¯i as a convex combination of the vertices of X¯i . Since functions Vi are affine functions defined on convex sets X¯i , the satisfaction of (14a) for all vi,h ∈ vert(X¯i ), ˜ h = 1, . . . , mi , for x = Pmi βi,h vi,h , leads to with i ∈ I, h=1 α1 kxkW
m
i
X
= α1 βi,h vi,h
h=1
≤
W
= Fi
mi X h=1
βi,h α1 kvi,h kW ≤
mi X
! βi,h vi,h
h=1
+ gi
mi X
βi,h (Fi vi,h + gi )
h=1
mi X
βi,h = Fi x + gi . (22)
h=1
For this reason, for x ∈ int(Xi ), since Vi (x) = Fi x+gi , (11a) holds. Moreover, on the boundaries of X¯i , according to (13b), one has α1 kxkW ≤ Fi x + gi for all i ∈ N (x), and therefore α1 kxkW ≤ max Si∈N (x) {Fi x+gi } = V (x). This implies that (11a) holds for all x ∈ XH , since XH = i∈I˜ Xi .
10
for the decay of the Lyapunov-like function, for x ∈ Xij , define βij,h ≥ 0, PmAs ij β h=1 ij,h = 1, as a set of coefficients such that x ∈ Xij can be expressed as a convex combination of the vertices of Xij . Therefore, " ! !# mij η X X V (f (x, d)) = Fj Ai βij,h vij,h +ai + Ei cµ dµ + gj (23) µ=1
h=1
Pη
where cµ ≥ 0, µ=1 cµ = 1, are coefficients used to express the fact that any point d ∈ D can be written as a convex combination of the vertices Pm Pηof D. Recalling that ij (14b) holds for all the vertices of Xij , and that h=1 βij,h = µ=1 ch = 1, from (23) we get " mij !# η X X V (f (x, d)) = Fj βij,h Ai vij,h + ai + Ei cµ dµ + gj µ=1
h=1 mij
= Fj
X
βij,h
Ai vij,h + ai +
=
βij,h
≤
cµ Ei dµ
+ gj
η X
cµ (Fj (Ai vij,h + ai + Ei dµ )) + gj
µ=1
h=1 mij
X
!
µ=1
h=1 mij X
η X
βij,h
h=1
=δ+λ
η X µ=1
mij X h=1
cµ (δ + λ (Fi vij,h + gi ) − gj ) + gj
βij,h (Fi vij,h + gi ) −
= δ + λ Fi
mij X
mij X
βij,h gj + gj
h=1 mij
βij,h vij,h +
X
!
βij,h gi
h=1
h=1
= δ + λ (Fi x + gi ) = δ + λV (x), which proves that (11c) holds for all x ∈ int (Xij ). Also, on the boundaries of Xij , the decreasing condition (11c) is imposed for all (i, j) ∈ N (x) × N (f (x, d)), and therefore max Fj (f (x, d)) + gj ≤ δ + λ max (Fi x + gi ). i∈N (x)
j∈N (f (x,d))
Ss˜
Ss˜ Since by definition of the sets Xij we have that XH = i=1 j=1 Xij , (11c) holds for all x ∈ XH , and then (11) holds for all x ∈ XH . Up to this point, the proven results hold for any δ ≥ 0. Considering the case δ = 0, the decay condition (11c) becomes V (f (x, d)) − λV (x) ≤ 0, which also implies f (0, d) = 0, ∀d ∈ D. From V (f (x, d)) − λV (x) ≤ 0 we can easily obtain V (x(k)) ≤ λk V (x(0)), which leads to α1 kx(k)kW ≤ V (x(k)) ≤ λk V (x(0)) ≤ λk α2 kx(0)kW . Therefore, α2 k λ kx(0)kW , kx(k)kW ≤ α1 11
which implies that system (8) is UAS(XH ), according to Definition 3, with φ(kx(0)k, k) = α2 λk /α1 . Consider now the case δ > 0. By (11a) and (11c) it follows that V (f (x, d)) − V (x) ≤ −(1 − λ)V (x) + δ ≤ −(1 − λ)α1 kxkW + δ = −α3 kxkW + δ, where α3 = (1−λ)α1 . By definition of Q in (18), for all x ∈ XH \Q we obtain −α3 kxkW +rδ < 0, leading to V (f (x, d)) − V (x) < −εδ, x ∈ XH \ Q. (24) By definition of F, the fulfillment of the constraint (14d) implies Q ⊆ H ⊆ XH and, by the invariance of XH (Lemma 1), R(Q) ⊆ XH . This implies that the max in (17) exists (since Q, and therefore R(Q), are closed sets) and therefore F in (16) is well defined. To prove invariance of F, we distinguish two cases: • if x ∈ F ∩ Q, then f (x, d) ∈ R(Q) ⊆ F; • if x ∈ F \ Q, then (24) holds, and therefore V (f (x, d)) < V (x) ≤ VF+ (this latter being defined in (17)), meaning that f (x, d) ∈ F by definition of F. Therefore, F is a RPI set for (8). To prove that system (8) is UUB(XH , F) according to Definition 4, we must show that for all a > 0 there exists T = T (a) such that, for all x(0) ∈ XH with kx(0)kW ≤ a, x(T ) ∈ F for any admissible sequence of the disturbance term d. We define Qa , {x ∈ Rn : kxkW ≤ a}, and distinguish two cases: • If 0 < a ≤ rδ/α3 , condition x(0) ∈ Qa implies x(0) ∈ Q. Therefore, simply taking T (a) = 1, we have that for every x(0) ∈ Qa , x(T ) ∈ F (by definition of F). • If a > rδ/α3 , let VH− , inf x∈XH \Q V (x) > 0. Then, from (24) it follows that ¯ ¯ < V −, ∀x(0) ∈ XH \ Q, ∃k¯ = k(a) ∈ Z>0 : V (x(k)) H
(25)
¯ ∈ Q, and therefore x(k¯ + 1) ∈ F. In order to explicitely find meaning that x(k) ¯ k as a function of a, let Va+ , supx∈Qa \Q V (x). From (24), one has + Va − VH− ¯ . k(a) , rδ ¯ In conclusion, if a > rδ/α3 , there exists T (a) = k(a) + 1 such that, for every x(0) ∈ Qa , x(T ) ∈ F.
Proof of Lemma 3. Since X˜ ⊆ X , from (8) we can easily see that, given x(k) ∈ X˜ , we have that x(k + 1) = Ai x(k) + ai + Ei d(k), with x(k) ∈ Xi and i ∈ I. Notice that dynamics (3) and (8) coincide for x(k) ∈ X˜ . This means that we applied dynamics (3), and the same will hold for all subsequent time instants, since x(k + 1) ∈ X˜ by assumption (X˜ is a RPI set). We conclude that X˜ is a RPI set for dynamics (3). Proof of Theorem 1. The proof consists of showing that the PWA Lyapunov-like function VP (x) : P → R, VP (x) ≡ V (x), ∀x ∈ P 12
(26)
where V (x) is found as in Lemma 2 for the extended system (8) in XH , is a Lyapunovlike function for (3) over the set P. First of all, we prove that P is a RPI set for (8) in XH . In case δ = 0, for all x ∈ P, V (f (x, d)) ≤ λV (x), and then f (x, d) ∈ P by definition of P. In case δ > 0, first of all notice that, by definition of P and F as sublevel sets, it follows that the assumption VE− ≥ VF+ implies F ⊆ P. Then, we distinguish two cases: • if x ∈ F, then f (x, d) ∈ F (which is a RPI set for (8)), and then f (x, d) ∈ P since F ⊆ P; • if x ∈ P \ F, we distinguish again two sub-cases: – if x ∈ Q, then f (x, d) ∈ F ⊆ P, since R(Q) ⊆ F by construction (see (16)-(18)); – if x ∈ / Q, then, from (24), V (f (x, d)) < V (x), leading to f (x, d) ∈ P. Since P ⊆ X , by Lemma 3 we obtain that P is a RPI set for dynamics (3). This fact leads to the possibility of defining a Lyapunov-like function for system (3) in P. Considering that P ⊆ XH , if (11) are satisfied with δ = 0 for all x ∈ XH (and then for all x ∈ P), we conclude that VP (x) is a Lyapunov function for system (3) in P, and that system (3) is UAS(P). In case δ > 0, if (11) hold for all x ∈ XH (and then for all x ∈ P), the invariance of F with respect to dynamics (3) is guaranteed by Lemmas 2 and 3. We have already proved that P is RPI for (3), and we know that F ⊆ P by construction. We also know from Lemma 2 that the extended system (8) is UUB(XH , F), which means that, for all a > 0, there exists T (a) > 0 such that, for every x(0) ∈ XH with kx(0)k ≤ a, x(T ) ∈ F for all the sequences d[T ] with d(k) ∈ D, k = 0, . . . , T . However, since dynamics (3) and (8) coincide in P, one can use Definition 4 again considering dynamics (3). We know that for all x(0) ∈ P, for all a > 0 there exists T (a) > 0 such that, for every x(0) ∈ P with kx(0)k ≤ a, x(T ) ∈ F for all the sequences d[T ] with d(k) ∈ D, k = 0, . . . , T . In conclusion, system (3) is UUB(P, F).
Acknowledgement The authors would like to thank Prof. Stephen Boyd (Stanford University) for the useful discussion on the formulation of Lemma 2.
References [1] E. D. Sontag, “Nonlinear regulation: The piecewise linear approach,” IEEE Transactions on Automatic Control, vol. 26, no. 2, pp. 346–358, 1981. [2] A. Bemporad and M. Morari, “Control of systems integrating logic, dynamics, and constraints,” Automatica, vol. 35, pp. 407–428, 1999.
13
[3] A. Bemporad, “Efficient conversion of mixed logical dynamical systems into an equivalent piecewise affine form,” IEEE Transactions on Automatic Control, vol. 49, no. 5, pp. 832–838, 2004. [4] F. J. Christophersen, M. Baotic, and M. Morari, “Stability analysis of hybrid systems with a linear performance index,” in Proc. Conference on Decision and Control, Atlantis, Paradise Island, Bahamas, December 2004, pp. 4589–4594. [5] A. Bemporad, W. Heemels, and B. De Schutter, “On hybrid systems and closedloop MPC systems,” IEEE Transactions on Automatic Control, vol. 47, no. 5, pp. 863–869, 2002. [6] S. Di Cairano, A. Bemporad, I. Kolmanovsky, and D. Hrovat, “Model predictive control of magnetically actuated mass spring dampers for automotive applications,” Int. Journal of Control, vol. 80, no. 11, pp. 1701–1716, 2007. [7] F. Borrelli, A. Bemporad, M. Fodor, and D. Hrovat, “An MPC/hybrid system approach to traction control,” IEEE Trans. Contr. Systems Technology, vol. 14, no. 3, pp. 541–552, May 2006. [8] N. Giorgetti, A. Bemporad, H. Tseng, and D. Hrovat, “Hybrid model predictive control application towards optimal semi-active suspension,” Int. Journal of Control, vol. 79, no. 5, pp. 521–533, 2006. [9] A. Alessio and A. Bemporad, “A survey on explicit model predictive control,” in Nonlinear Model Predictive Control: Towards New Challenging Applications, ser. Lecture Notes in Control and Information Sciences, L. Magni, F. Allgower, and D. M. Raimondo, Eds. Berlin Heidelberg: Springer-Verlag, 2009, vol. 384, pp. 345–369. [10] A. Bemporad, A. Oliveri, T. Poggi, and M. Storace, “Ultra-fast stabilizing model predictive control via canonical piecewise affine approximations,” IEEE Transactions on Automatic Control, vol. 56, no. 12, pp. 2883–2897, 2011. [11] G. Ferrari Trecate, F. A. Cuzzola, D. Mignone, and M. Morari, “Analysis of discrete-time piecewise affine and hybrid systems,” Automatica, vol. 38, no. 12, pp. 2139–2146, 2002. [12] P. Grieder, M. Kvasnica, M. Baoti´c, and M. Morari, “Stabilizing low complexity feedback control of constrained piecewise affine systems,” Automatica, vol. 41, no. 10, pp. 1683–1694, 2005. [13] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear matrix inequalities in system and control theory. SIAM, 1994. [14] M. Lazar and A. Joki´c, “On infinity norms as Lyapunov functions for piecewise affine systems,” in Proc. Hybrid systems: computation and control, Stockholm, Sweden, April 2010, pp. 417–432.
14
[15] S. Prajna and A. Papachristodoulou, “Analysis of switched and hybrid systems – beyond piecewise quadratic methods,” in Proc. American Control Conference, Denver, CO, June 2003, pp. 2779–2784. [16] A. Polanski, “Lyapunov function construction by linear programming,” IEEE Transactions on Automatic Control, vol. 42, no. 7, pp. 1013–1016, 1997. [17] E. C. Kerrigan and D. Q. Mayne, “Optimal control of constrained, piecewise affine systems with bounded disturbances,” in Proc. Conference on Decision and Control, Las Vegas, NV, December 2002, pp. 1552–1557. [18] G. Zhai, H. Lin, and P. J. Antsaklis, “Quadratic stabilizability of switched linear systems with polytopic uncertainties,” International Journal of Control, vol. 76, no. 7, pp. 747–753, 2003. [19] H. Lin and P. Antsaklis, “Synthesis of uniformly ultimate boundedness switching laws for discrete-time uncertain switched linear systems,” in Proc. Conference on Decision and Control, Maui, HI, December 2003, pp. 4806–4811. [20] F. Blanchini and S. Miani, Set-theoretic methods in control. 2008.
Springer-Verlag,
[21] P. Biswas, P. Grieder, J. L¨ofberg, and M. Morari, “A survey on stability analysis of discrete-time piecewise affine systems,” in IFAC World Congress, Prague, Czech Republic, June 2005. [22] S. V. Rakovi´c, P. Grieder, M. Kvasnica, D. Q. Mayne, and M. Morari, “Computation of invariant sets for piecewise affine discrete time systems subject to bounded disturbances,” in Proc. Conference on Decision and Control, Atlantis, Paradise Island, Bahamas, December 2004, pp. 1418–1423. [23] S. Rakovi´c, E. Kerrigan, D. Mayne, and J. Lygeros, “Reachability analysis of discrete-time systems with disturbances,” IEEE Transactions on Automatic Control, vol. 51, no. 4, pp. 546–561, 2006. [24] S. Trimboli, M. Rubagotti, and A. Bemporad, “Stability and invariance analysis of uncertain PWA systems based on linear programming,” in Proc. Conference on Decision and Control, Orlando, FL, December 2011, pp. 7398–7403. [25] M. Rubagotti, S. Trimboli, D. Bernardini, and A. Bemporad, “Stability and invariance analysis of approximate explicit MPC based on PWA Lyapunov functions,” in Proc. IFAC World Congress, Milan, Italy, August 2011, pp. 5712–5717. [26] A. Oliveri, D. Barcelli, A. Bemporad, B. A. G. Genuit, W. P. M. H. Heemels, T. Poggi, M. Rubagotti, and M. Storace, “MOBY-DIC: A Matlab toolbox for the circuit design of explicit MPC,” in IFAC Conference on Nonlinear Model Predictive Control, Noordwijkerhout, the Netherlands, August 2012. [27] M. Lazar, W. P. M. H. Heemels, and A. R. Teel, “Lyapunov functions, stability and input-to-state stability subtleties for discrete-time discontinuous systems,” IEEE Transactions on Automatic Control, vol. 54, no. 10, pp. 2421–2425, 2009. 15
[28] A. Bemporad, M. Morari, V. Dua, and E. N. Pistikopoulos, “The explicit linear quadratic regulator for constrained systems,” Automatica, vol. 38, no. 1, pp. 3–20, 2002. [29] S. P. Boyd and L. Vandenberghe, Convex Optimization. Press, 2004.
Cambridge University
[30] D. Avis, “LRS: A revised implementation of the reverse search vertex enumeration algorithm,” in Polytopes, Combinatorics and Computation, ser. DMVSeminar 29, G. Kalai and G. Ziegler, Eds. Birkhauser Verlag, 2000, pp. 177–198. [31] K. Fukuda and A. Prodon, “Double description method revisited,” in Combinatorics and Computer Science, ser. Lecture Notes in Computer Science. Springer, 1996, pp. 91–111. [32] K. Fukuda, cdd/cdd+ Reference Manual, Institute for operations Research ETHZentrum, 1997. [33] C. N. Jones, E. C. Kerrigan, and J. M. Maciejowski, “Equality set projection: A new algorithm for the projection of polytopes in halfspace representation,” University of Cambridge, Tech. Rep., June 2004. [34] R. Suard, J. Lofberg, P. Grieder, M. Kvasnica, and M. Morari, “Efficient computation of controller partitions in multi-parametric programming,” in Proc. Conference on Decision and Control, Atlantis, Bahamas, December 2004, pp. 3643– 3648.
16
