Input-to-state stability for discrete-time nonlinear systems

Transcription

1 0 0 Abstract Automatica (00) } Input-to-state stability for discrete-time nonlinear systems A short version of this paper was presented at the th Triennial IFAC World Congress, Beijing, July },. This paper was recommended for publication in revised form by Associate Editor H. Nijmeijer under the direction of Editor Hassan K. Khalil. * Corresponding author. Tel.: #--0-; fax: # addresses: zjiang@control.poly.edu (Z.-P. Jiang), ywang@ math.fau.edu (Y. Wang). Zhong-Ping Jiang*, Yuan Wang Department of Electrical and Computer Engineering, Polytechnic University, Six Metrotech Center, Brooklyn, NY 0, USA Department of Mathematics, Florida Atlantic University, Boca Raton, FL, USA Received April ; revised 0 November 000; received in "nal form December 000 The input-to-state stability property and ISS small-gain theorems are introduced as the cornerstone of new stability criteria for discrete-time nonlinear systems. In this work we study the input-to-state stability (ISS) property for discrete-time nonlinear systems. It is shown that most ISS results for continuous-time nonlinear systems in the current literature can be extended to the discrete-time case. Several equivalent characterizations of ISS are introduced and two ISS small-gain theorems are proved for nonlinear and interconnected discrete-time systems. ISS stabilizability is discussed and comparisons with the continuous-time case are made. As in the continuous time framework, where the notion ISS found wide applications, we expect that this notion will provide a useful tool in areas related to stability and stabilization for nonlinear discrete time systems as well. 00 Elsevier Science Ltd. All rights reserved. Keywords: Nonlinear stability; Discrete time systems; Lyapunov functions; Input-to-state stability; Nonlinear small-gain. Introduction The purpose of this paper is to study the input-to-state stability (ISS) property for discrete-time nonlinear systems of the general form x(k#)"f (x(k), u(k)), () where states x(k) are in, and control values u(k) in, for some n and m, and for each time instant k.we assume that f : P is continuous. We were motivated by the corresponding ISS notion which was originally proposed by Sontag (, ) for continuous-time nonlinear systems. The ISS property concerns with the continuity of state trajectories on the initial states and the inputs. Roughly speaking, a system is ISS if every state trajectory corresponding to a bounded 000-/0/$ - see front matter 00 Elsevier Science Ltd. All rights reserved. PII: S ( 0 ) control remains bounded, and the trajectory eventually becomes small if the input signal is small no matter what the initial state is. As shown in Sontag (, ) and many other references (Coron, Praly, & Teel, ; Isidori, ; Jiang & Mareels, ; Jiang, Teel, & Praly, ; Kazakos & Tsinias, ; KrsticH & Li, ; KrsticH, Kanellakopoulos, & Kokotovic, ; Praly & Jiang, ; Praly & Wang, ; Sontag & Wang, ; Sontag & Wang, ; Teel, ), ISS turns out to be a very natural stability property and, indeed, has been successfully employed in the stability analysis and control synthesis of nonlinear systems with complex structure. Our interest in discrete-time nonlinear systems () is due to the fact that discrete-time (or, di!erence) systems have their own interest and have found applications in various "elds (Agarwal, ; LaSalle, ; Lakshmikantham & Trigiante, ). For instance, the stability theory of di!erence systems was recently used to design stabilizing control laws for discrete-time nonlinear systems*see, for instance, Byrnes and Lin (), Chen and Khalil (), Guo (), Kotsios and Kalouptsisis (), Nijmeijer and van der Schaft (), Tsinias, Kotsios, and Kalouptsidis (), Tsinias () and references therein

2 Z.-P. Jiang, Y. Wang / Automatica (00) } 0 0 One of the main results in this work provide an equivalence relation between ISS and ISS-Lyapunov function. The latter concept was introduced in Sontag () and Sontag and Wang () for continuous-time systems. In addition to this interesting result, we prove that various equivalent characterizations of the ISS condition proposed in Sontag and Wang () also hold for discretetime nonlinear systems. As in the continuous-time context, we show that a discrete-time system () can be rendered ISS (or, input-to-state stabilizable) if and only if it is globally stabilizable via state-feedback. This theorem is stated and illustrated in Section. However, new phenomena arise in the extension from continuous-time to discrete-time. For continuous-time a$ne systems, continuous stabilization implies ISS stabilization by means of state-feedback change u"k(x)#v. This is not the case any more for discrete systems as shown by an example. For a discrete-time system, whether it is an a$ne or non-a$ne system, a more complex feedback transformation of the form u"k (x)#k (x)v is in general required. We also show in this work that ISS small gain theorems in Coron et al. () and Jiang et al. () are extendible to the discrete-time setting. The work on small gain theorems for nonlinear systems can be traced back to Mareels and Hill (), where a nonlinear generalization of the classical small-gain theorem was proposed for both continuous- and discrete-time feedback systems within the input}output framework. As a consequence of not specifying the role of initial conditions, no conclusion was drawn in Mareels and Hill () about the internal stability of the interconnected system. Here, with the ISS concept, the role of initial conditions can be made explicit and both external and internal stability properties are established for the whole feedback system. The results in this paper are not surprising, considering the fact that most of them are available for continuous time systems. Most of our results can be considered as discrete analogues to the earlier results for ISS and its related characterizations and ISS nonlinear small gain theorems obtained in Coron et al. (), Jiang, Mareels, and Wang (), Jiang et al. (), Lin, Sontag, and Wang (), Sontag (, ) and Sontag and Wang (, ). Though the conceptual notions are all originated from the continuous case, and many arguments are, in some cases very straightforward or routine, generalizations of their continuous counterparts, many technical results cannot be obtained by the obvious `discretizinga generalization of the arguments used in the continuous framework. In some cases, one has to adopt completely di!erent arguments. Considering the signi"cant role played by ISS in the continuous case and the fact that many technical results need to be carefully examined for the discrete time case, we consider it necessary and appropriate to present the results with thorough proofs. The remainder of the paper is comprised of "ve sections. Primary notations and de"nitions are given in Section. In Section, we introduce the ISS concept for discrete-time systems. Several popular characterizations of ISS in continuous-time are extended to the discretetime case. Section proposes two nonlinear small-gain theorems for discrete-time interconnected ISS systems. Section discusses when a discrete-time system can be made ISS via state-feedback. We close this paper with some brief concluding remarks in Section. The technical results are discussed in Appendices A and B.. Notation and denitions The notations used in this paper are quite standard. We use to denote the set of all nonnegative integers. For any positive real number r, r denotes the largest integer that is less than or equal to r. For any x in, x is its transpose and x its Euclidean norm. For a nm matrix A, A stands for its induced matrix norm. For any function : P, we denote (with slight abuse of notation) "sup(k): k )R. In the case when is bounded, this is the standard l norm. For any k and any function : P, denotes the truncation of at k; i.e., ( j)"( j) if j)k, and ( j)"0 ifj'k. We denote :"!. We let Id denote the identity function from onto, and we use γ γ to denote the composition of two functions γ and γ which are from to. In this paper, controls or inputs are functions u : P. For a given system, we often consider the same system but with controls restricted to take values in some subset ΩL; we use M Ω to denote the set of controls taking values in Ω. For each ξ and each input u, we denote by x( ), ξ, u) the trajectory of system () with initial state x(0)"ξ and the input u. Clearly such a trajectory is de"ned uniquely on. Recall that a function γ : P is a K-function if it is continuous, strictly increasing and γ(0)"0; it is a K -function if it is a K-function and also γ(s)pr as spr; and it is a positivedexnite function if γ(s)'0 for all s'0, and γ(0)"0. A function β : P is a KL-function if, for each "xed t*0, the function β( ), t) isak-function, and for each "xed s*0, the function β(s, ) ) is decreasing and β(s, t)p0 astpr.. Input-to-state stability properties The main concern of this section is to understand the dependence of state trajectories on the magnitude of inputs for systems of the following type: x(k#)"f (x(k), u(k)), ()

3 Z.-P. Jiang, Y. Wang / Automatica (00) } 0 0 where inputs u( ) ) are functions from to. We also assume that f (0, 0)"0, i.e., ξ"0 is an equilibrium of the 0-input system... ISS and ISS}Lyapunov functions We "rst introduce the concepts of ISS and ISS}Lyapunov functions. Other equivalent notions of ISS will be given and demonstrated in the subsequent subsections. Denition.. System () is (globally) input-to-state stable (ISS) if there exist a KL-function β : P and a K-function γ such that, for each input ul and each ξ, it holds that x(k, ξ, u))β(ξ, k)#γ(u) () for each k. Note that, by causality, the same de"nition would result if one would replace () by x(k, ξ, u))β(ξ, k)#γ(u ) () for every k 0. Recall that u denotes the truncation of u at k!. It can be seen from () that the ISS property implies that the 0-input system x(k#)"f (x(k), 0) is globally asymptotically stable (GAS) and that () is `converging-input converging-statea, i.e., every trajectory x(k, ξ, u) goes to 0ifu(k) goes to 0 as kpr. However, the converse is not true. A simple example to consider is x(k#)" (#sin u(k))x(k). It is not hard to see that the system is 0-input GAS, and satis"es the converging-input converging-state property. But the system is not ISS, because for the constant input function u(k)"π/, the trajectory x(k, ξ, u) is identically ξ, which violates any estimation of type (). Denition.. A continuous function < : P is called an ISS}Lyapunov function for system () if the following holds:. There exist K -functions α, α such that α (ξ))<(ξ))α (ξ), ξ. (). There exist a K -function α and a K-function σ, such that <( f(ξ,μ))!<(ξ))!α (ξ)#σ(μ) () for all ξ, for all μ. A smooth ISS}Lyapunov function is one which is smooth. Remark.. As in the case of continuous time, Property in the above de"nition is equivalent to the following property: There exist some K -function α and some K-function χ such that ξ*χ(μ) N <(f(ξ,μ))!<(ξ))!α (ξ). () It should be mentioned that it results in an equivalent property if the function α in () is merely required to be continuous and positive de"nite. See Jiang and Wang (00) for a detailed proof. Example.. As a simple illustration of these notions, we specialize () to linear discrete-time systems: x(k#)"ax(k)#bu(k) () where A is a Schur matrix, i.e., the eigenvalues of A are located strictly inside the unit disk. For such a matrix, there are constants c'0 and 0)σ( such that A)cσ (cf. LaSalle,, Chapter ). From (), we have x(k#)"ax(0)# ABu(j), () from which the ISS property () follows with β(r, k)"cσr, γ(r)" cσbr" cbr!σ. () Next, we show that system () has a quadratic ISS}Lyapunov function. Given a symmetric and positivede"nite matrix Q, let P'0 be the unique solution to the matrix equation APA!A"!Q. Consider the positive-de"nite and radially unbounded function <(x)"xpx, () which satis"es property () with α (r)"λ (P)r and α (r)"λ (P)r. Direct computation shows <(x(k#))!<(x(k)) "!x(k)qx(k)#x(k)apbu(k) #u(k)bpbu(k). () Then, by completing squares, property () holds with α (r)" λ (Q)r, σ(r)" APB λ (Q) #BPB r. Therefore, < de"ned in () is an ISS}Lyapunov function for ()

4 0 Z.-P. Jiang, Y. Wang / Automatica (00) } 0 0 Clearly, if < is an ISS}Lyapunov function for (), then < is a Lyapunov function for the 0-input system x(k#)"f (x(k), 0). As in the classic Lyapunov stability theory, we can prove that a system is ISS if and only if it admits an ISS}Lyapunov function. See the main Theorem below. We "rst demonstrate the su$ciency. Lemma.. If system () admits a continuous ISS}Lyapunov function, then it is ISS. Proof. Assume that system () admits an ISS}Lyapunov function <. Let α (i",, ) and σ be as in () and (). First observe that () can be rewritten as <( f (ξ, μ))!<(ξ))!α (<(ξ))#σ(μ) () for all ξ and μ, where α "α α. Without loss of generality, we assume that Id!α K (cf. Lemma B.). Fix a point ξ and pick an input u. Let x(k) denote the corresponding trajectory x(k, ξ, u) of (). Let ρ be any K -function such that Id!ρK. Consider the set de"ned by D"ξ : <(ξ))b, () where b"αρ σ(u). Claim. If there is some k such that x(k )D, then x(k)d for all k*k. Proof. Assume that x(k )D. Then <(x(k )))b, that is, ρ α (<(x(k ))))σ(u). By (), <(x(k #)))(Id!α )(<(x ))#σ(u) and since Id!α K, we have <(x(k #)))(Id!α )(b)#σ(u) "!(Id!ρ) α (b)#b!ρ α (b)#σ(u) )!(Id!ρ) α (b)#b)b. () Using induction, one can show that <(x(k #j)))b for all j, that is, <(x(k))d for all k*k. We now let j "mink : x(k)d)r. Then it follows from the above conclusion that <(x(k)))γ( (u) for all k*j, where γ( (r)"α( ρ σ(r). For k(j,it holds that ρ α( (<(x ))'σ(u), and hence, <(x(k#))!<(x(k)))!α (<(x(k)))#σ(u) "!(Id!ρ) α (<(x(k)))!ρ α (<(x(k)))#σ(u) )!(Id!ρ) α (<(x(k))). () By a standard comparison lemma (see e.g., Jiang & Wang, 00), there exists some KL-function βk such that <(x(k)))βk(<(x(0)), k) for all 0)k)j #. Thus, <(x(k)))maxβk(<(ξ), k), γ( (u), k. () From this one gets () with β(s, t)"α (βk(α (ξ), t)) and γ(s)"α γ( (s). Remark.. Tracking the functions used in the proof, it can be seen that if () holds with some α K such that Id!α K, then, for any ρk so that Id!ρ is of class K, there is some βkkl such that () holds with γ( "α ρσ. This can be taken as an initial step in computing gain functions γ as in () from a given ISS-Lyapunov function. Remark.. Suppose () holds with some α K such that Id!α K. If one lets ρ"id, then, with b"α σ(u) and x(k )D, estimate () becomes <(x(k #)))b. This shows that the set D"ξ: <(ξ))b is invariant. Still let j "mink : x(k)d)r. Then, for k*j, <(x(k)))α σ(u). For k)j!, () yields <(x(k#)))<(x(k)). Consequently, <(x(k)))<(x(0)) for all 0)k)j. Thus, one gets the following estimate: <(x(k, ξ, u)))max<(ξ), α σ(u) for all k, all ξ and all u. Also note that in the proof of Lemma., it is enough for Id!α( to be nondecreasing instead of being of class K... Asymptotic gains Consider system (). We say that the system has a K- asymptotic gain if there exists some γ K such that lim x(k, ξ, u))γ lim u(k) () for all ξ. We say that system () is uniformly bounded input bounded state (UBIBS) if bounded initial states and controls produce uniformly bounded trajectories, i.e., there exist two K-functions σ and σ such that sup x(k, ξ, u))maxσ (ξ), σ (u, ξ, ul. ()

5 Z.-P. Jiang, Y. Wang / Automatica (00) } 0 0 Again, by the causality property of the system, the above is equivalent to σ (s)*s and x(k, ξ, u)) max σ (ξ),σ (u(j)), ξ, ul, k*. (0) Suppose that the system () is ISS. Without loss of generality, one may assume that the trajectories of the system satisfy the following: x(k, ξ, u))maxβ(ξ, k), γ(u), k () for some βkl, some γk. Clearly, such an estimation implies UBIBS. Also note that, for any k*l, x(k, ξ, u))maxβ(x(l, ξ, u), k!l), γ(u) )maxβ(β(ξ, l),k!l),β(γ(u ),k!l),γ(u). Letting l"k/, we get x(k, ξ, u))maxβi(ξ, k/), β(γ(u ), k/), γ(u ), where βi(s, r)"β(β(s, r), r). From this one can see that lim x(k,ξ, u))γ lim u(k) (). Hence we proved the following result: Lemma.. Suppose system () is ISS. Then it is UBIBS and it admits a K-asymptotic gain. Furthermore, if () holds, then the function γ can be taken as a K-asymptotic gain of the system. The converse of the "rst statement in the above result is also true, see Theorem. Remark.. Suppose < is an ISS}Lyapunov function for the system satisfying an dissipation inequality () for some α K and σk. Suppose also that Id!α K. By Remark., for any ρk such that Id!ρK, there is some βkl such that () holds with γ( "αρσ. Applying the same argument in deriving () from (), one can show that γ( is also a K-asymptotic gain for <(x(k,ξ, u)), that is, lim <(x(ξ, k, u)))α ρσ lim u(k) () for all ξ and all u... Robust stability margins As in the case of continuous time systems, there is also an interesting connection between the ISS and the robust stability. Throughout this section, we let BM denote the closed unit ball of. Let ρ be any K function. Consider the system x(k#)"f (x(k), d(k)ρ(x(k))) :"g(x(k), d(k)), () where dm M. We view the signals d( ) ) as disturbances. For such systems, there are natural de"nitions of global asymptotical stability (GAS) and uniform global asymptotical stability (UGAS), see De"nition A.. We will say that system () is robustly stable if there exists a K function ρ (called a stability margin) such that system () is UGAS. Note that for a nonlinear GAS system, in general only small perturbations can be tolerated while preserving stability. The requirement ρk is thus nontrivial. Assume now system () is robustly stable. Let ρk be the stability margin. This means that the corresponding system () is UGAS. By the Converse Lyapunov Theorem presented in the appendix (also see Jiang & Wang, 00), it follows that there exists some smooth function < such that α (ξ))<(ξ))α (ξ) ξ, () holds for some α,α K ; and furthermore, <( f (ξ, μρ(ξ)))!<(ξ))!α (ξ), ξ, μ) () for some K -function α. Observe that this is equivalent to the following: ν)ρ(ξ) N <( f (ξ, ν))!<(ξ))!α (ξ). () It then follows from Remark. that < is a smooth ISS}Lyapunov function for system (). Thus, one obtains the following result: Lemma.. Suppose that system () is robustly stable. Then it admits a smooth ISS}Lyapunov function. The following is a key lemma in obtaining our main result given in Section.. Its proof is analogous to the proof of the corresponding continuous time result in Sontag and Wang (, Section V), but we provide the details nonetheless, since the lemma is essential in obtaining our main results concerning the ISS property. Lemma.. If system () is UBIBS and if it admits a K- asymptotic gain, then it is robustly stable. Proof. Assume that the system is UBIBS with σ, σ as in (). Let γ be a K-asymptotic gain for the system. Without loss of generality, we assume that σ "γ and σ (s)*s for all s*0. Pick any K -function ρ such that γ(ρ(s)))s/, s*

6 Z.-P. Jiang, Y. Wang / Automatica (00) } 0 0 Below we will show that with such a choice of ρ, the system de"ned by x(k#)"f (x(k), d(k)ρ(x(k)) () is GAS. Recall that dm M. Pick any ξ, dm M. Let x (k) denote the corresponding trajectory of (). Claim. σ (ρ(x (k))))σ (ξ)/ for all k*0. Proof. First notice that the claim is true if ξ"0 since both sides vanish. Assume now that ξo0. Again observe that the claim is true for k"0, because σ (ρ(x (0)))"γ(ρ(x (0))))ξ/)σ (x (0))/. Let k "min k : σ (ρ(x (k))*σ (ξ). Then k '0. Suppose that the claim is false, and hence, k (R. Then for 0)k)k!, it holds that σ (ρ(x (k))))σ (ξ)/, and hence, γ(d(k)ρ(x (k)))) σ (ξ)/ for all 0)k)k!. It then follows from (0) that x (k )) max σ (ξ), σ (d( j)ρ(x ( j))) )σ (ξ), () which, in turn, implies that σ (ρ(x (k ))))x (k )/) σ (ξ)/(σ (ξ)/. This contradicts the de"nition of k. This shows that k "R, i.e., the claim is true. An immediate consequence of the claim is that () holds for all k, and that lim x (k) is "nite for each trajectory. Taking the limits on both sides of (), one sees that lim x (k)) lim γ(d(k)ρ(x (k)))) lim x (k)/. It then follows that lim x (k)"0 for each trajectory. This shows that system () is GAS, which, together with Theorem in the appendix, implies that the system is UGAS... Equivalent characterizations of ISS In this section, we present some equivalence relations among various notions. Theorem. Consider system (). The following are equivalent:. It is ISS.. It is UBIBS and it admits a K-asymptotic gain.. It is robustly stable.. It admits a smooth ISS}Lyapunov function. Proof. We have: [N] (clear by De"nitions), [N] (see Lemma.), [N] (see Lemma.), [N] (see Lemma.). Remark.. A subset A of is 0-input (forward) invariant for system () if it is (forward) invariant for the corresponding 0-input system x(k#)"f (x(k),0), that is, if x(0)a then x(k)a for all k. As in the continuous time case, one may also de"ne ISS and UGAS with respect to such a closed invariant set A. It is routine to generalize Theorem to the case when A is compact, where for statement, robust stability means the existence of a smooth ρk such that the system x(k#)"f (x(k), d(k)ρ(x(k) A )), where d( ) )M M,isUGAS with respect to A. The case when A is not compact is slightly trickier. Still, it can be shown that in that case statements,, and are equivalent; and it is not hard to see that statement is indeed strictly weaker than the other three statements. Even in the simple case when u"0, the UBIBS property in combination with a K-asymptotic gain reduces to the GAS property with respect to A, which is in general strictly weaker than the UGAS property with respect to A... Remarks about gain functions As it is well recognized in the literature, the notion of ISS is a natural nonlinear extension of classical xnite gain stability (Desoer & Vidyasagar, ) in that only linear incremental gains are considered. That is why we refer γ in the ISS property () to as an ISS-gain. Naturally, we raise the question of how to compute such a gain function for a given ISS system (). As in the continuous-time case (Sontag, ), a similar ISS algorithm can be developed on the basis of an ISS}Lyapunov function < satisfying () for some α, α K and the di!erence dissipation inequality <(x(k#, ξ, u))!<(x(k, ξ, u)) )!α(<(x(k, ξ, u)))#σ(u) (0) for some αk, some σk. According to Lemma B., we assume, without loss of generality, that Id!αK. By Remark., one sees that, for any K -function ρ such that Id!ρK, the function α α (Id#ρ) σ(s) can be taken as an ISS-gain function for the system. That is, there is a KL-function β such that x(k, ξ, u))β(ξ, k)#α α(id#ρ) σ(u) for all k, ξ and all u. Lemma.. Suppose that system () admits an ISS}Lyapunov function satisfying () with some α,α K and (0) for some αk and σk. Assume further that

7 Z.-P. Jiang, Y. Wang / Automatica (00) } 0 0 Id!αK. Then it holds that lim <(x(k, ξ, u)))α σ lim u(k) () for all ξ and all u. Consequently, γ (s):"α α σ(s) is a K-asymptotic gain such that () holds. Proof. Indeed, by Remark., for any 0(c(, it holds that lim <(x(ξ, k, u)))α c σ lim u(k) for all ξ and all u. For any given ξ and any u, letting cp, we get ().. Nonlinear small-gain theorems In this section, we will discuss the ISS property for interconnected and nonlinear discrete-time systems x (k#)"f (x (k), v (k), u (k)), x (k#)"f (x (k), v (k), u (k)), () subject to the interconnection constraints v (k)"x (k), v (k)"x (k), () where for i", and for each k, x (k), u (k), and f is continuous in its arguments. In the continuous-time case, the ISS small gain theorems are very powerful in treating stability and stabilization problems for such interconnected systems. The "rst such small gain theorem for nonlinear systems was obtained in Jiang et al. (). In Coron et al. (), the small gain theorem was derived by a much simpler proof in conjunction with the results in Sontag and Wang (). In Jiang et al. (), a small gain theorem was presented in terms of Lyapunov functions. In this section, we will follow the continuous-time approach used in Coron et al. (, Section ) to derive an ISS small-gain theorem for discrete-time nonlinear systems. In particular, we show that Theorems and of Coron et al. () hold in the discrete-time setting as well. In contrast to the nonlinear small-gain theorem of Mareels and Hill () our ISS small-gain theorem takes into account the role of initial conditions and o!ers a result on the internal asymptotic stability for the interconnected discrete-time system (). We "rst introduce a useful technical result that is instrumental in the development of our discrete ISS nonlinear small-gain theorems. The continuous analogue of this result was "rst established in Jiang et al. () under an assumption of observability. Since the completeness of solutions is not an issue in the discrete time case, we do not need the observability assumption in our context. For notational simplicity, we use x (k), i",, to denote the trajectories of each individual x -system in () with inputs (v, u ). If no ambiguity is possible, we also use x(k)"(x (k), x (k)) to mean the solutions of the interconnected system () subject to () and with inputs u"(u, u ). Lemma.. Let h : P and h : P be two continuous mappings. Assume that the following ISS-like properties hold for the solutions of each subsystem in () h (x (k)))maxβ (ξ, k), γ (h (v )), γ (u ), h (x (k)))maxβ (ξ, k), γ (h (v )), γ (u ). If γ γ (s)(s (equivalently, γ γ (s)(s) for all s'0, then, for the composite system () subject to the interconnection constraint (), for all initial condition ξ"(ξ, ξ ) and all u, the trajectories (x (k), x (k)) satisfy the following properties: there exist some σ, γk such that (h (x (k, ξ, u)), h (x (k, ξ, u))))maxσ(ξ), γ(u) k*0 () there exists some λk such that lim (h (x (k)), h (x (k)))) lim λ(u(k)). () Proof. We "rst establish the boundedness property (). Pick any initial condition ξ"(ξ,ξ ) and any input u"(u, u ), by causality, the corresponding solution x(k)"(x (k), x (k)) of () and () satis"es h (x (k)) )maxβ (ξ, k), γ ((h (x )) ), γ (u ), () h (x (k)) )maxβ (ξ, k), γ ((h (x )) ), γ (u ), () which implies that (h (x )) )maxβ (ξ, 0), γ (β (ξ, 0)), γ γ ((h(x )) ), γ γ (u ), γ (u ). () Then, the small-gain condition, i.e., γ γ (s)(s, implies (h (x )) )maxβ (ξ, 0), γ (β (ξ, 0)), γ γ (u ), γ (u ) )maxσ (ξ), γ (u) () for all k*0, where σ (s)"maxβ (s, 0), γ (β (s, 0)), and γ(s)"maxγ γ (s), γ (s). Similarly, one can show that,

8 Z.-P. Jiang, Y. Wang / Automatica (00) } 0 0 for each k*0, (h (x ) ))maxσ (ξ), γ (u) () for some σ, γ K. The combination of () and () then yields () with σ"σ #σ and γ"γ #γ. Observe that σ and γ are both independent of the choices of ξ and u. Next, we establish (). By () and (), one sees that there exists some c'0 (depending on ξ and u) such that h (x (k)))c and h (x (k)))c, () for all k, and hence, both lim h (x (k)) and lim h (x (k)) are "nite. Applying () with x (k/) as the initial state of x,weg h (x (k)))maxβ(x (k/), k/), This together with () yields γ ((h (x )) ), γ (u ) h (x (k)))maxβ(c, k/), γ ((h (x )) ), γ (u ). Taking limits in the above, we conclude that lim h (x (k)) )max γ lim h (x (k)), γ lim u (k). Similarly, we have lim h (x (k)) )max γ lim h (x (k)), γ lim u (k). Combining the fact that both lim h (x (k)) and lim h (x (k)) are well de"ned and the fact that γ γ (s)(s for each s'0, one concludes that lim h (x (k)) )max lim lim h (x (k)) )max lim γ γ (u (k)), lim γ (u (k)), γ γ (u (k)), lim γ (u (k)). The asymptotic gain condition () then follows for some γk. Now, we are ready to state and prove our "rst discrete ISS small-gain theorem. Theorem. Suppose both the subsystems in () are ISS in the sense that x (k, ξ, v, u ))maxβ (ξ, k), γ (v ), γ (u ), x (k, ξ, v, u ))maxβ (ξ, k),γ (v ), γ (u ). If γ γ (s)(s (equivalently, γ γ (s)(s) for all s'0, then the interconnected system () and () is ISS with (u, u ) as input. Proof. It follows immediately from Lemma. by letting h and h be the identity functions. Indeed, in this case, () yields the UBIBS condition; and () yields the "nite K-asymptotic gain condition. The above small-gain theorem can also be stated in terms of ISS}Lyapunov functions. Assume that both the subsystems in () are ISS. Let < and < be ISS}Lyapunov functions for the x and x -subsystem of (), respectively. That is, there exist class K -functions α, σ, ρ and ρ ()i, j)) such that α (ξ))< (ξ))α (ξ) () and < ( f (ξ, ν, μ ))!< (ξ ) )!σ (< (ξ ))#ρ (< (ν ))#ρ (μ ), () < ( f (ξ, ν, μ ))!< (ξ ) )!σ (< (ξ ))#ρ (< (ν ))#ρ (μ ). () In view of Lemma B., we may assume that Id!σ K for i",. Theorem. Assume that x - and x -subsystems of () admit ISS}Lyapunov functions < and < respectively that satisfy () and (), with Id!σ K for i",. If there exists a K -function ρ such that σ (Id#ρ) ρ σ (Id#ρ) ρ (Id, () then the interconnected system () and () is ISS with (u, u ) as the input. Proof. By Remark., for any K -function η such that Id!ηK, there exist β, β KL such that, for )i, j) and joi, it holds that < (x (k, ξ, v, u )) )maxβ (< (ξ ), k),γ (ρ (< (v ))#ρ (u )),

9 Z.-P. Jiang, Y. Wang / Automatica (00) } 0 0 where γ (s)"σ (Id#η)(s), i",. Observe that, for any K-function γ and for any K -function χ, γ(r#s))maxγ (Id#χ)(r),γ (Id#χ)(s). Then, for )i, j) and joi, it follows that < (x (k, ξ, v, u )) )maxβ (< (ξ ), k), σ(id#η) (Id#χ) ρ(< (v )), σ (Id#η) (Id#χ) ρ(u ). () Pick η, χk so that (Id#η) (Id#χ)(s))(Id#ρ)(s) for all s*0. Noticing the relation (), and applying Lemma. to () with h (x )"< (x ), and γ"σ (Id#ρ)ρ for i",, one concludes that there exist some σ, γ, λk such that along the trajectories of the interconnected system () and (), it holds that, for i",, < (x (k, ξ, u)))maxσ(ξ), γ(u), k, ξ, u and lim < (x (k,ξ, u)))λ(u), k, ξ, u. With property (), it follows that system () and () is UBIBS and admits a K-asymptotic gain, and hence, it is ISS. When (u, u )"(0,0) in (), the small-gain condition () can be weakened as follows. Corollary.. Consider the interconnected discrete-time system x (k#)"f (x (k), x (k)), () x (k#)"f (x (k), x (k)). () Assume that both x -subsystem () and x -subsystem () possess ISS}Lyapunov functions in the sense that < ( f (ξ, ξ ))!< (ξ ))!σ (< (ξ ))#ρ (< (ξ )), < ( f (ξ, ξ ))!< (ξ ))!σ (< (ξ ))#ρ (< (ξ )), () () with σ satisfying that Id!σ K. If the following small gain condition holds σ ρ σ ρ (Id, () then the interconnected system () and () is GAS at the origin. Proof. Let χ "σ ρ, χ "σ ρ. By Remark., one sees that < (x (k)))max< (x (0)), χ (< (x ) ), < (x (k)))max< (x (0)), χ (< (x ) ). By the same argument used as in deriving () and () from (), one shows that if (χ χ )(s)(s, then (< (x (k)), < (x (k))))(< (x (0)), < (x (0))) for all k. By Lemma., an asymptotic gain from < (x )to< (x ) for () is χ, and an asymptotic gain from < (x )to< (x ) for () is χ. That is, lim < (x (k)))χ lim < (x (k)) )χ χ lim < (x (k)). Again, the assumption that χ χ (Id implies that lim < (x (k))" lim < (x (k))"0. Thus, the system is GAS. Example.. As an elementary application of Theorem, let us consider a cascade nonlinear discretetime system of the form z(k#)"q(z(k), x(k), u(k)), () x(k#)"g(x(k), v(k)). () Assume that z-system () is ISS with (x, u) as input, and let γ be an ISS-gain with respect to x. Let x-system () be ISS with v as input. Clearly, γ,0isaniss-gain function with respect to z for the x-system since z does not a!ect the x-trajectories. Thus, the small-gain condition holds between γ and γ. As a direct consequence of Theorem, the cascade system () and () is ISS with (u, v) as input.. Input-to-state stabilizability Consider system (). We say that the system is continuously stabilizable if there is a continuous function w : P with w(0)"0 such that under the feedback u"w(x), the closed-loop system x(k#)"f (x(k), w(x(k))) is GAS. We say that system () is continuously ISS stabilizable if there exist a continuous map w : P with w(0)"

10 Z.-P. Jiang, Y. Wang / Automatica (00) } 0 0 and an nn matrix Γ of continuous functions, invertible for each x, such that under the control law u"w(x)#γ(x)v, the closed-loop system x(k#)"f (x(k), w(x(k))#γ(x(k))v(k)) () is ISS (with v as the new input). Clearly, if a system is ISS-stabilizable, then it is stabilizable. The following result shows that the two types of stabilizability are equivalent. Theorem. System () is continuously stabilizable if and only if it is ISS-stabilizable. Proof. It is enough to show that stabilizability implies ISS-stabilizability. Assume that the system is stabilized under the continuous feedback law u"w(x) with w(0)"0, i.e., the system x(k#)"f (x(k), w(x(k))) () is GAS. Applying Theorem to () one knows that there is a Lyapunov function < satisfying <( f (ξ, w(ξ)))!<(ξ)(!α(ξ), ξ, for some positive de"nite function α. De"ne the continuous function δ : P by δ(s, r):" max <(f(ξ, w(ξ)#μ))!<(ξ)#α(ξ). Note then that for every s'0, δ(s,0)(0. By Lemma. in Sontag (), there exist a K -function χ and a smooth function g : P with g(s)" for all s[0,] such that δ(s, g(s)r)(0 whenever s*χ(r). Let now Γ(ξ)"g(ξ)I. We would like to show that with such a choice of Γ, the corresponding closed-loop system () is ISS. For this purpose, we note that whenever ξ*χ(μ), <( f (ξ, w(ξ)#γ(ξ)μ))!<(ξ)#α(ξ))δ(ξ, μ)(0, that is, ξ*χ(μ)n<( f (ξ, w(ξ)#γ(ξ)μ))!<(ξ)(!α(ξ). By Remark., < is an ISS}Lyapunov function for (). Therefore, system () is ISS. Upon specialization of () to a linear system x(k#)"ax(k)#bu(k), () we obtain a checkable necessary and su$cient condition for ISS-stabilizability. Corollary.. System () is ISS-stabilizable if and only if (A, B) is stabilizable, i.e. there is a matrix K so that the eigenvalues of A#BK are inside the unit disk. For the continuous time case, Corollary. can be extended to nonlinear systems a$ne in controls. It was shown in Sontag () that if an a$ne system x "f (x)#g(x)u is stabilizable, then there is some feedback u"k(x) such that the closed-loop system x "f (x)#g(x)(k(x)#w) is ISS with respect to w. In contrast to the continuous time situation, this result fails in the discrete time case. As an illustration, we consider a one-dimensional discrete-time system x(k#)"x(k)#(x(k)#)u. () With the feedback u"!x/(x#), the closed-loop system is GAS. In fact, the closed-loop system is even dead beat, that is, all trajectories reaches the origin in one step. On the other hand, there is no continuous feedback u"k(x)#w for which the closed-loop system is ISS with respect to w. This can be seen as follows. The closed-loop system of () with u"k(x)#w is x(k#)"g(x(k))#(x(k)#)w(k), () where G(x(k))"x(k)#(x(k)#)K(x(k)). Now consider the signal w(k) given by: if G(x(k))*0, w(k)"! if G(x(k))(0. It can be seen that with such a choice of w, x(k#)* x(k)#, and hence, x(k#)pr. This shows that the system fails to be ISS no matter what G is.. Conclusions In this paper, we proved that a system is ISS if and only if it has a smooth ISS-Lyapunov function. It was shown that many recent characterizations and results related to ISS for continuous-time nonlinear systems in the literature can "nd their analogues in discrete-time. As in the continuous-time case, two nonlinear ISS small gain theorems have been obtained for a general interconnection of nonlinear discrete-time systems, in terms of either ISS-gains or ISS}Lyapunov functions; also see Jiang et al. (000). Along the process of extending the corresponding

11 Z.-P. Jiang, Y. Wang / Automatica (00) } 0 0 ISS theory for continuous-time systems to the discretetime setting, new phenomena arise. For instance, every a$ne continuous-time "nite-dimensional system that is stabilizable can be rendered ISS with respect to additive noise v as in u"k(x)#v (see Sontag, ). This is not the case any more for discrete-time systems. In general, a far more complex feedback transformation of the type u"k (x)#k (x)v is required to make a discrete-time nonlinear system ISS with respect to the new input v. While ISS has found wide applications in several control problems for continuous-time systems, we expect that the discrete ISS-related properties developed in this paper will serve as a promising toolkit for discrete-time nonlinear feedback design. Results in this direction will be explored further and reported separately. Acknowledgements The work of the "rst author was supported partly by a start-up grant from Polytechnic University and partly by NSF Grant INT-. The work of the second author was supported in part by NSF Grant DMS-. The authors would like to thank Dr. E.D. Sontag for many helpful discussions. The authors would also like to thank the anonymous reviewers for their constructive comments and criticism. Appendix A. Global asymptotic stability for systems with disturbances This work relies heavily on the work in Jiang and Wang (00). In this appendix, we provide some key results that were used in the present work. For details, we refer the reader to Jiang and Wang (00). Consider the system x(k#)"f (x(k), d(k)), k, (A.) where dm Ω for some compact subset Ω of, and f is assumed to be continuous. We will use x( ),ξ, d) to denote the solution of () with the initial state ξ and the timevarying parameter dm Ω. Denition A.. System (A.) is uniformly globally asymptotically stable if the following two properties hold.. Uniform stability. There exists a K -function δ( ) ) such that for any ε'0, it holds that x(k, ξ, d))ε for all k, all dm Ω, and all ξ)δ(ε).. Uniform global attraction. For any r,ε'0, there exists some ¹ such that for every dm Ω, x(k, ξ, d)(ε for all k*¹ whenever ξ)r. It was shown in Jiang and Wang (00) that system (A.) is UGAS if and only if for some βkl, x(k, ξ, d))β(ξ, k) k, ξ. Denition A.. System () is globally asymptotically stable (GAS) if. for every ε'0, there exists some δ'0 such that x(k, ξ, d)(ε for all k*0, all dm Ω, and all ξ(δ; and. the attraction property lim x(k, ξ, d)"0 holds for all ξ, all dm Ω. The following is one of the main results of Jiang and Wang (00). Theorem. For system (), the following are equivalent:. The system is GAS.. The system is UGAS.. The system admits a smooth Lyapunov function. That is, there exists a smooth function < : P for which the following holds: there exist two K -functions α and α such that for any ξ, α (ξ))<(ξ))α (ξ); there exists some K -function α such that, for any ξ and any μω, <( f (ξ, μ))!<(ξ))!α (ξ). Appendix B. A technical lemma In the proof of Lemma., we used the following seemingly obvious result. To make the work more self contained, we provide the result with a detailed proof. Lemma B.. For any K -function α, there is a K - function α( such that the following holds: α( (s))α(r) for all r*0; and id!α( K. Observe that if a locally Lipschitz function α( K satis"es the condition that α( (r))/ for almost all r, then (r!α( (r))*/, and hence, id!α( K. To prove Lemma B., we "rst prove the following result. Lemma B.. Let λ be a locally Lipschitz K -function. Then, for any 0)a(b(R, there exists a locally Lipschitz function λ K such that. λ (r)"λ (r) on [0, a] and λ (r))λ (r) on [a, b];. λ (r))/ for almost all r*a

12 Z.-P. Jiang, Y. Wang / Automatica (00) } 0 0 Proof. Let λ be a smooth K -function, and let 0)a(b be given. Consider the function κ(r)" λ (r) λ (a)# minλ(s), ds if 0)r)a, if r*a. Then κ is locally Lipschitz on (0,R), κ(r))/ for all r*a, κ(r))λ (r) for all r*0, and κk. IfκK, then we are done with λ "κ. Suppose κ K. Then there exists some b *b! such that κ(b ))λ (b )!. Let λ (r)" if 0)r)b κ(r), κ(b )#(r!b ) if r*b. Then λ is a locally Lipschitz K -function, 0)λ (r))/ for almost every r. It is also not hard to see that λ (r))λ (r) for all r)b. Proof of Lemma B.. Let α be a K function. Without loss of generality, we assume that α is smooth (otherwise, replace α by a smooth K -function majorized by α). Applying Lemma B. to α with a"0, b", we know that there exists some locally Lipschitz function λ K such that Properties and in Lemma B. hold. If λ (r))α(r) for all r*0, then the proof is done with α( "λ. Suppose for some k, we have found locally Lipschitz K -functions λ,,λ and 0(r ((r! such that for each )j)k!, the following holds: r *r #, where r "0; λ "λ on [0, r ], and λ (r))α(r) for all r[0, r ] with λ (r )"α(r ), and λ (r))α(r) for all 0)r)r #; λ (s))/ almost everywhere. If λ )α(r) r*0, (B.) then the proof is done with α( "λ. Suppose (B.) fails. Then there exists some r *r # such that λ (r )"α(r ). Let α be de- "ned by α (r)" λ (r) if 0)r)r, α(r) if r*r. Applying Lemma B. to the locally Lipschitz K -function α with a"r and b"r #, we get some locally Lipschitz K -function λ such that λ "λ on [0, r ] and λ )α for all 0)r)r #, and λ (r))/ almost everywhere. Thus, by induction, we have shown either. for some k, there exists a locally Lipschitz K - function λ such that λ (r))α(r) for all r*0 and λ (r))/ almost everywhere; or. there exist 0(r (r with r *r # so that, for each k, there exists some locally Lipschitz K - function λ such that λ (r))/ almost everywhere, λ (r)"λ (r) on [0, r ], and λ (r))α(r) on [0, r ] with λ (r )"α(r ). In the "rst case, we complete the proof by letting α( "λ. Suppose the latter is the case. De"ne α( by α( (r)"λ (r) on [r, r ). Since r *r #, we see that α( is de- "ned on [0,R). Since λ "λ on [0, r ], α( is locally Lipschitz, and α( (r))/ for almost all r*0. Finally, since α( (r )"α(r ), we see that α( is of class K. References Agarwal, R. P. (). Diwerence equations and inequalities: theory, methods and applications. New York: Marcel Dekker. Byrnes, C. I., & Lin, W. (). Losslessness, feedback equivalence and the global stabilization of discrete-time nonlinear systems. IEEE Transactions on Automatic Control,, }. Chen, F. -C., & Khalil, H. K. (). Adaptive control of a class of nonlinear discrete-time systems using neural networks. IEEE Transactions on Automatic Control,, }0. Coron, J. -M., Praly, L., & Teel, A. (). Feedback stabilization of nonlinear systems: Su$cient conditions and Lyapunov and input}output techniques. In A. Isidori (Ed.), Trends in control. London: Springer. Desoer, C., & Vidyasagar, M. (). Feedback systems: Input}output properties. New York: Academic Press. Guo, L. (). On critical stability of discrete-time adaptive nonlinear control. IEEE Transactions on Automatic Control, (), }. Isidori, A. (). Nonlinear control systems II.. London: Springer. Jiang, Z. P., Lin, Y., & Wang, Y. (000). A local nonlinear small-gain theorem for discrete-time feedback systems and its applications. Proceedings of the third Asian control conference (ASCC +000), Shanghai (pp. }). Jiang, Z. P., & Mareels, I. M. Y. (). A small-gain control method for cascaded nonlinear systems with dynamic uncertainties. IEEE Transactions on Automatic Control,, }0. Jiang, Z. P., Mareels, I. M. Y., & Wang, Y. (). A Lyapunov formulation of the nonlinear small gain theorem for interconnected ISS systems. Automatica,, }. Jiang, Z. P., Teel, A., & Praly, L. (). Small-gain theorem for ISS systems and applications. Mathematics of Control, Signals and Systems,, }0. Jiang, Z. P., & Wang, Y. (00). A converse Lyapunov theorem for discrete time systems with disturbances. Submitted for publication. Kazakos, D., & Tsinias, J. (). The input to state stability conditions and global stabilization of discrete-time systems. IEEE Transactions on Automatic Control,, }. Kotsios, St., & Kalouptsisis, N. (). Adaptive control for a certain class of nonlinear systems. preprint. KrsticH, M., & Li, Z. (). Inverse optimal design of input-to-state stabilizing nonlinear controllers. IEEE Transactions on Automatic Control,, }0. KrsticH, M., Kanellakopoulos, I., & KokotovicH, P. V. (). Nonlinear and adaptive control design. New York: Wiley. Lakshmikantham, V., & Trigiante, D. (). Theory of diwerence equations: Numerical methods and applications. New York: Academic Press

13 Z.-P. Jiang, Y. Wang / Automatica (00) } 0 0 LaSalle, J. P. (). The stability and control of discrete process. New York: Springer. Lin, Y., Sontag, E. D., & Wang, Y. (). A smooth converse Lyapunov theorem for robust stability. SIAM Journal on Control and Optimization,, }. Mareels, I. M. Y., & Hill, D. J. (). Monotone stability of nonlinear feedback systems. Journal of Mathematical Systems and Estimation Control,, }. Nijmeijer, H., & van der Schaft, A. (). Nonlinear dynamical control systems. New York: Springer. Praly, L., & Jiang, Z. P. (). Stabilization by output feedback for systems with ISS inverse dynamics. Systems & Control Letters,, }. Praly, L., & Wang, Y. (). Stabilization in spite of matched unmodeled dynamics and an equivalent de"nition of input-tostate stability. Mathematics of Control, Signals and Systems,, }. Sontag, E. D. (). Smooth stabilization implies coprime factorization. IEEE Transactions on Automatic Control,, }. Sontag, E. D. (). Further facts about input to state stabilization. IEEE Transactions on Automatic Control,, }. Sontag, E. D., & Wang, Y. (). On characterizations of the input to state stability property. Systems & Control Letters,, }. Sontag, E. D., & Wang, Y. (). New characterizations of the input to state stability property. IEEE Transactions on Automatic Control,, }. Teel, A. R. (). A nonlinear small-gain theorem for the analysis of control systems with saturation. IEEE Transactions on Automatic Control,, }0. Tsinias, J. T. (). Versions of Sontag's `input to state stability conditiona and the global stabilization problem. SIAM Journal on Control and Optimization,, }. Tsinias, J., Kotsios, S., & Kalouptsidis, N. (). Topological dynamics of discrete-time systems. Robust control of linear systems and nonlinear control, Proceedings of international symposium MTNS-, vol. II (pp. }). Boston: Birkhauser. Zhong-Ping Jiang received the B.Sc. degree in mathematics from the University of Wuhan, China, in, the M.Sc. degree in statistics from the UniversiteH de Parissud, Paris, France, in, and the Ph.D. degree in automatic control and mathematics from the Ecole des Mines de Paris, Paris, France, in. From to, he held visiting researcher positions in several institutions including INRIA (Sophia-Antipolis), France, the Department of Systems Engineering in the Australian National University, Canberra and the Department of Electrical Engineering in the University of Sydney. In, he also visited several US universities. In January, he joined the Polytechnic University at Brooklyn as an Assistant Professor of Electrical Engineering. His main research interests include stabilization, robust/adaptive nonlinear control and decentralized control with applications to nonholonomic mechanical systems. He has authored or coauthored over 0 refereed technical papers in these areas. Currently, Dr. Jiang is an Associate Editor for Systems & Control Letters, and the International Journal of Robust and Nonlinear Control. He is also on the IEEE Conference Editorial Board. Dr. Jiang is the recipient of a prestigious Queen Elizabeth II Fellowship Award () from the Australian Research Council and a CAREER Award (000) from the U.S. National Science Foundation. Yuan Wang received her Ph.D. Degree in Mathematics from Rutgers University in. Since Dr. Wang has been with the Department of Mathematical Sciences at Florida Atlantic University, where she is currently a Professor. Her research interests lie in several areas of control theory, including realization and stabilization of nonlinear systems. Dr. Wang is an Associate Editor of Systems & Control Letters. She also served on the IEEE Conference Editorial Board from to. She received an NSF Young Investigator Award in.