On Saturation, Discontinuities and Time-Delays in iiss and ISS Feedback Control Redesign

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1 2010 American Control Conference Marriott Waterfront, Baltimore, MD, USA June 30-July 02, 2010 WeA05.5 On Saturation, Discontinuities and Time-Delays in iiss and ISS Feedback Control Redesign P. Pepe H. Ito Abstract This paper addresses input-to-state stability ISS) and integral ISS iiss) redesign to enhance robustness of feedback control systems in the face of actuator disturbances. The idea of the redesign is to modify control laws available for disturbance-free systems a priori in order to achieve robustness against the disturbance. In practice, control system design often encounters time-delays, discontinuities and actuator limitations at the same time. This paper proposes a unified treatment of these issues in the redesign for actuator disturbances. For the new class of systems, we pursue LgV-type formulas which are widely used for ordinary differential equations. In addition, we incorporate saturation into the LgV-type control for implementation in the presence of actuator limitations. To derive such a redesign formula, we develop a framework of ISS/iISS Lyapunov-Krasovskii functionals allowing for the Filippov solutions. This paper demonstrates that the notion of invariantly differentiable functionals is useful for deriving LgVtype control laws and dealing with discontinuities directly. The iiss formulation which includes the ISS as a special case allows us to address actuator limitations in the redesign problem in a flexible manner. I. INTRODUCTION In 1989 Sontag showed in the paper [33] that finite dimensional, control-affine, nonlinear systems, whose dynamics is described by smooth functions, which are smoothly feedback globally stabilizable, are also smoothly input-tostate stabilizable with respect to disturbances adding to the control input. As well known, those disturbances are very frequent in practice, because of actuator errors. Analogous results hold see [34] and the survey [22]), by considering continuous feedback control laws, for control-affine, nonlinear systems whose dynamics is described by locally Lipschitz functions. Many contributions concerning the state feedback stabilization and the input-output state feedback linearization of nonlinear time-delay systems can be found in the literature see, for instance, [5][6][10][11][20][23][25][28][43]). In [42] a state feedback control law for systems described by retarded functional differential equations with discontinuous right-hand side is proposed, using the framework of the retarded differential inclusions. The idea of control Lyapunov function based domination redesign has been exploited to stabilize time delay systems in [11], where a L g V -type control see [8], [16], [17], [40], [32] and references therein) The work of P. Pepe was supported by the Center of Excellence for Research DEWS, L Aquila, Italy. The work of H. Ito was supported in part by Grant-in-Aid for Scientific Research of JSPS under grant P. Pepe is with the Department of Electrical and Information Engineering, University of L Aquila, Poggio di Roio, L Aquila, Italy pepe@ing.univaq.it. H. Ito is with the Department of Systems Design and Informatics, Kyushu Institute of Technology, Kawazu, Iizuka, Fukuoka , Japan, hiroshi@ces.kyutech.ac.jp. using Lyapunov Razumikhin functions is derived. Lyapunov- Razumikhin and Lyapunov-Krasovskii methodologies see [7]) for the ISS of time-delay, nonlinear systems have been studied in [39] and in [13], [27], respectively. The Lyapunov- Razumikhin approach is used in the paper [19] in order to deal with quantization and time delays. A new methodology to analyze ISS of nonlinear systems with time-varying delays based on the use of nonlinear matrix inequalities is proposed in [4], following a Lyapunov-Krasovskii based approach. In [24] the ISS with respect to actuator errors of a class of nonlinear systems with suitably bounded delay in the feedback channel are considered. A Lyapunov-Krasovskii functional for proving the ISS of the closed loop delayed) system is provided on the basis of the Lyapunov function available for the unforced, delay-free, closed loop system. In [41] the relationship between the exponential stability in the unforced case and the ISS of time-delay nonlinear systems is investigated. Moreover a contribution to the theory of the input-to-state stabilization with respect to actuator errors, for a class of stabilizable time-delay nonlinear systems, is given. Such class consists of time-delay, control-affine, nonlinear systems for which there exists a state feedback control law such that the closed loop system with no disturbance) is linear, delay-free and asymptotically stable. In [31] the inputto-state stabilization of time-invariant state feedback globally stabilizable, control-affine, nonlinear systems with constant time delays is investigated. It is proved that a state feedback globally stabilizable in the case of disturbance equal to zero) time-delay, control-affine, nonlinear system admits an inputto-state stabilizing with respect to a disturbance adding to the control input) state feedback control law, provided that the following main hypothesis is satisfied: there exists a suitable Frechét differentiable or invariantly differentiable Lyapunov-Krasovskii functional for the unforced, disturbance equal to zero) control system obtained by closing the loop with the stabilizing state feedback control law. The formula for such input-to-state stabilizing state feedback control law is provided. In this paper we implement the idea of the ISS feedback control redesign indicated by [33] for systems described by retarded differential equations with possibly discontinuous right hand side and input magnitude constraints. The formula of the redesign is presented in the form of the LgV-type control by means of invariantly differentiable functionals. Invariantly differentiable functionals introduced in [14][15] are popularly used in the literature for studying the stability of time-delay systems. However, to the best of our knowledge, related definitions have not been very popular for designing /10/$ AACC 190

2 controllers although they are applicable to a reasonably broad class of systems. In order to fulfill input magnitude constraints, we incorporate saturation into the LgV-type control. This paper allows the state time-delays to be time-varying of any size and of both discrete and distributed character as far as they are differentiable. Since actuator disturbances, time-delays, discontinuities for instance introduced with the sliding mode control methodology, see [21][26][38]) and input constraints are very frequent problems in practice, a unified treatment of these problems achieved by the proposed redesign method has the potential to lead to a breakthrough in many practical applications. In the presence of actuator limitations, a control of arbitrarily large magnitude and gain yielding an arbitrarily strong convergent property cannot be always achieved. In order to take these circumstances into account, this paper formulates the redesign problem in the framework of iiss. It is also shown how to achieve ISS with a larger feedback gain within a specified maximum magnitude of the input, provided that its maximum magnitude is allowed to be larger than the one of the disturbances. The ISS feedback control redesign proposed here coincides with the one provided in [33] and in [31] for the systems studied there. An example showing the effectiveness of the developed theory is provided. Input time delays are not considered in this paper. The reader can refer to [19] [24] for this important problem. For lack of space, proofs are not reported here and will be published elsewhere. Notations R denotes the set of real numbers, R denotes the extended real line [, ], R + denotes [0, ). The symbol stands for the Euclidean norm of a real vector, or the induced Euclidean norm of a matrix. The essential supremum norm of an essentially bounded function is indicated with the symbol. A function v : R + R m, m positive integer, is said to be essentially bounded if ess sup t 0 vt) <. For given times 0 T 1 <T 2,we indicate with v [T1,T 2) : R + R m the function given by v [T1,T 2)t)=vt) for all t [T 1,T 2 ) and =0elsewhere. An input v is said to be locally essentially bounded if, for any T>0, v [0,T) is essentially bounded. For a positive integer n, for a positive real maximum involved time-delay), C, W 1, and Q denote the space of the continuous functions mapping [, 0] into R n, the space of the absolutely continuous functions, with essentially bounded derivative, mapping [, 0] into R n and the space of the bounded and continuous functions mapping [, 0) into R n, respectively. For φ C, φ [,0) is the function in Q defined as φ [,0) τ)=φτ), τ [, 0). For a function x :[,c) R n, with 0 <c, for any positive real t, x t is the function in C defined as x t τ) =xt + τ), τ [, 0]. Let us here recall that a function γ : R + R + is: of class P if it is continuous, zero at zero, and positive at any positive real; of class K if it is of class P and strictly increasing; of class K if it is of class K and it is unbounded; of class L if it is continuous and it monotonically decreases to zero as its argument tends to. A function β : R + R + R + is of class KL if β,t) is of class K for each t 0 and βs, ) is of class L for each s 0. The symbol a indicates any semi-norm in C such that, for some positive reals γ a, γ a, the following inequalities hold γ a φ0) φ a γ a φ, φ C 1) For example, the M2 norm, given by φ M2 = φ0) 2 + ) 1 0 φτ) 2 2 dτ, satisfies 1) and thus belongs to the set of the a semi-norms in C. The function sgn : R { 1, 0, 1} is defined as: 1, s>0, sgns)= 0, s=0, 2) 1, s<0 The symbols and denote logical sum and logical product, respectively. II. EQUATIONS WITH TIME-DELAYS AND DISCONTINUOUS RIGHT-HAND SIDE Let us consider the following retarded functional differential equation ẋt)=ft, x t )+gt, x t )vt), t 0, a.e., xτ)=ξ 0 τ), τ [, 0], ξ 0 C, 3) where: xt) R n, n is a positive integer; is the maximum involved time-delay; f is a not necessarily continuous) functional mapping R + C R n, g is a Lipschitz on bounded sets functional mapping R + C R n m, m is a positive integer; vt) R m is a measurable, locally essentially bounded input. The explicit dependence of f,g on the time t is here for, and only for, taking into account of time-varying differentiable) time-delays. We suppose that the functional f maps bounded sets of R + C into bounded sets of R n, and that ft, 0) = 0, t 0. The equation 3) can be studied in the framework of differential inclusions [3], [18], [37]). The retarded inclusion corresponding to 3) is given by ẋt) Ψt, x t,vt)), t 0, a.e., xτ)=ξ 0 τ), τ [, 0], ξ 0 C, 4) where: for t, φ, v) R + C R m, Ψt, φ, v) is the set given by Ψt, φ, v)={ξ + gt, φ)v, ξ F[f]t, φ)}; 5) F[f]t, φ) is the convex hull of all limit values of the functional f at the point t, φ) R + C see [37]). In the following, as usual see [2]), by a Filippov solution of 3) we mean a solution of 4). We assume the following. Assumption 1: For each t, φ) R + C, the set F[f]t, φ) is compact in R n ; for each bounded set U R + C, the set t,φ) U F[f]t, φ) is bounded; the multimap t, φ) F[f]t, φ) satisfies the Carathéodory conditions see Sections 4.2, 4.3, pp , in [18]). Remark 2: By Assumption 1 it follows that see Theorem 4.1, p. 123, in [18]): 1) there exists at least one locally absolutely continuous) solution for 4) on a maximal time interval [0,b), 0 <b ; 2) if the maximal interval [0,b) is bounded, at least one solution is unbounded; 3) the solutions set depends upper semicontinuously on the initial conditions. 191

3 III. ISS, IISS DEFINITIONS AND TECHNICAL LEMMAS The notions of ISS and iiss are defined for the system 3) in a usual way as follows: Definition 3: see [33][9][27]) The system described by 3) is said to be input-to-state stable ISS) if there exist a KL function β and a K function γ such that, for any initial state ξ 0 and any measurable, locally essentially bounded input v, the corresponding Filippov solutions exist for all t 0 and furthermore satisfy xt) β ξ 0,t)+γ ) v [0,t) 6) Definition 4: see [35][1][27]) The system described by 3) is said to be integral input-to-state stable iiss) if there exist a K function χ, akl function β and a K function γ such that, for any initial state ξ 0 and any measurable, locally essentially bounded input v, the corresponding Filippov solutions exist for all t 0 and furthermore satisfy χ xt) ) β ξ 0,t)+ t 0 γ vτ) ) dτ 7) In this paper, we also consider situations where the essential bounds of the input vt) are known in the following sense: v i ess inf t R + v i t), v i ess sup t R + v i t), 8) Here, v i, v i R, i =1, 2,...,m. Byv i, v i R we mean that the bounds of v i t) are given a priori. The system 3) is said to be ISS with respect to the disturbance vt) with the essential bounds 8) if the property 6) in Definition 3 holds for any v satisfying 8). It is worth noting that this property is implied by the ISS property of the system ẋt)=ft, x t )+gt, x t )Bvt)), t 0, a.e., Bv)=[B 1 v 1 ),B 2 v 2 ),...,B m v m )] T,v i R v i, s [v i, ) Bv i )= v i, s v i, v i ) v i, s,v i ], without a priori bounds of vt). The iiss property with given essential bounds of vt) is defined in the same manner. For an invariantly differentiable functional V : R + R n Q R +, let D + V : R + C R m R be defined, for t, φ, d) R + C R m, as follows involved y R n ), D + V t, φ, d)= Vθ,φ0),φ [,0)) θ θ=t V t,y,φ [,0) ) + sup ξ F[f]t,φ) y ξ + gt, φ)d) y=φ0) + V t, φ0), φ h) [,0) ) h, 9) h=0 where, for 0 <h<, φ h C is given by { φs + h) s [,h) φ h s)= φ0) s [ h, 0] 10) The following Lemma is the key to the iiss/iss redesign of the L g V -type to be proposed in this paper. In fact, the Lemma ensures that the derivative D + V tells the behavior of the trajectories xt) as in the classical Lyapunov theory for ordinary differential equations. The first part of the following Lemma, i.e. the one concerning the inequality 11) below, is stated in [36] without the input v see also [12][29][30]). Lemma 5: Let xt) be any of the solutions in a maximal time interval [0,b), 0 <b, of the functional differential inclusion 4). Let V : R + R n Q R + be an invariantly differentiable functional. Then, almost everywhere in [0,b), the following inequality holds lim sup V t + h, xt + h), xt+h ) [,0) ) h h V t, xt), x t ) [,0] ) ) D + V t, x t,vt)) 11) Moreover, if the initial condition ξ 0 W 1,, then the functional t V t, xt), x t ) [,0) ) is locally absolutely continuous in [0,b). The following three lemmas will be used in the next section for characterizing iiss and ISS properties of control systems by means of Lyapunov-Krasovskii functionals V. Lemma 6: The system described by 3) is ISS iiss) with initial conditions ξ 0 C if and only if it is ISS iiss, respectively) with initial conditions ξ 0 W 1,. Lemma 7: If there exist a locally Lipschitz, invariantly differentiable functional V : R + R n Q R +, functions α 1, α 2, α 3 of class K, and a function ρ of class K such that: H 1 ) α 1 φ0) ) V t, φ0),φ [,0) ) α 2 φ a ), t R +,φ C; H 2 ) D + V t, φ, v) α 3 φ a )+ρ v ); then, the system described by 3) is ISS. Lemma 8: If there exist a locally Lipschitz, invariantly differentiable functional V : R + R n Q R +, functions α 1, α 2, of class K, a function α 3 of class P, a function ρ of class K such that: H 1 ) α 1 φ a ) V t, φ0),φ [,0) ) α 2 φ a ), t R +,φ C; H 2 ) D + V t, φ, v) α 3 φ a )+ρ v ); then, the system described by 3) is iiss. The following Lemma provides a technique to derive the LgV-type redesign formula proposed in the next section. Lemma 9: Let L : R c, d), c<0 <d, be a continuous, strictly increasing, surjective function, with L0) = 0. Let L 1 :c, d) R be the inverse function of L. Then, for any s 1 R, s 2 c, d), the inequality holds s 1 s 2 Ls 1 )s 1 + L 1 s 2 )s 2 12) IV. IISS AND ISS REDESIGN In this section we consider the system described by the following functional differential equation, corresponding to 3) when the input v is given as the sum of the control input and of the disturbance, ẋt)=ft, x t )+gt, x t )ut)+dt)), t 0, a.e., xτ)=ξ 0 τ), ξ 0 C, 13) where ut) R m is the control input, dt) R m is the measurable, locally essentially bounded disturbance. In the 192

4 following, d i, d i R, i =1, 2,...,m, denote upper and lower essential bounds for the i-th component d i t) of the disturbance, i.e. d i ess inf t R + d it), and S R m is the set d i ess sup t R + d i t), 14) S =d 1, d 1 ) d 2, d 2 ) d m, d m ) 15) Now, we introduce a functional k which represents a control given a priori. The control law k can be discontinuous as in the case of sliding mode control. Let k : R + C R m be a given not necessarily continuous) functional. Let K[k]t, φ) be the convex hull of all limit values of the functional k at the point t, φ) R + C. We assume the following properties for the functional k. Assumption 10: For each t, φ) R + C, the set K[k]t, φ) is compact in R n ; for each bounded set U R + C, the set t,φ) U K[k]t, φ) is bounded; the multimap t, φ) K[k]t, φ) satisfies the Carathéodory conditions see Sections 4.2, 4.3, pp , in [18]). For an invariantly differentiable functional V : R + R n Q R +, let D + V gas) : R + C R be defined for t R + and φ Cinvolved x R n )as D + V gas) t, φ)= V θ,φ0),φ [,0)) θ θ=t + sup ξ 1 F[f]t,φ), ξ 2 K[k]t,φ) V t, x, φ [,0) ) x + Vt, φ0), φ h) [,0) ) h ξ 1 + gt, φ)ξ 2 ) x=φ0), 16) h=0 where, for 0 <h<, φ h C is given by 10). We assume further that the controller k has the following property. Assumption 11: There exist an invariantly differentiable functional V : R + R n Q R +, functions α 1, α 2 of class K, a function α 3 of class P such that, for any t R +, φ C: i) α 1 φ a ) V t, φ0),φ [,0) ) α 2 φ a ); ii) D + V gas) t, φ) α 3 φ a ). In order to be ready for the redesign to be proposed in Theorem 12, let us introduce the parameters a i, b i and the functions Y i, i =1, 2,...,m, fulfilling the following properties P 1 and P 2. P 1 ) a i,b i R, i =1, 2,...,m, satisfy a i < 0 <b i, a i <d i ) a i = )) b i > d i ) b i = )); 17) P 2 ) Y i : R a i,b i ), i =1, 2,...,m, are strictly increasing, locally Lipschitz functions such that Y i 0) = 0, lim Y is)=a i, s lim s Y is)=b i For notational simplicity, we use Y : R m S defined for v = [ v 1 v 2... v m ] T R m as Y v)= [ Y 1 v 1 ) Y 2 v 2 )... Y m v m ) ] T ; 18) where S R m is S =a 1,b 1 ) a 2,b 2 ) a m,b m ) 19) We also use Y 1 : S R m which denotes the inverse function of Y, which is given, for v = [ v 1 v 2... v m ] T S, by Y 1 v)= [ Y 1 1 v 1 ) Y 1 2 v 2 )... Y 1 m v m ) ] T, 20) where Y 1 i :a i,b i ) R is the inverse function of Y i,i= 1, 2,...,m. Finally, let σ be a function of class K such that Y 1 v) T v σ v ) 21) holds for any v S. We are now ready to present the main result of this paper. The following Theorem provides a formula for the iiss/iss redesign of control laws. Theorem 12: Let h i : R + C R, i =1, 2,...,m,be functionals defined, for t R +, φ C,as h i t, φ)= Vt,y,φ [,0)) y gt, φ)e i, 22) y=φ0) where y R n, V is the invariantly differentiable functional in Assumption 11, and e i, i =1, 2,...,m, are the canonical basis in R m. ForY i, i =1, 2,...,m, fulfilling P 1 and P 2, assume that the functionals t, φ) Y i h i t, φ)), i = 1, 2,...,m, are Lipschitz on bounded sets. Then the closed loop system consisting of the system 13) and the feedback control law ut)=kt, x t )+pt, x t ), 23) where, for t R +, φ C, pt, φ)= [ Y 1 h 1 t, φ)) Y m h m t, φ)) ] T, 24) is iiss with respect to the disturbance dt) with the essential bounds 14), and a pair of the functions χ, γ in 7) is given by χs)=α 1 γ a s), γs)=2σs), s 0 25) Moreover, if the function α 3 in the Assumption 11 is of class K and lim α 3s)= s lim α 3s) > s sup 2σ d ) 26) d i [d i,d i] holds, then the closed loop system consisting of 13) and 23) is ISS with respect to the disturbance dt) with the essential bounds 14) and a gain function γ satisfying 6) is given by γs)= 1 α 1 1 α 2 α 1 3 2σs)), s 0 27) γ a Since we can modify α 3 in Assumption 11 by rescaling the functional V, we can obtain the following result. Proposition 13: Let h i, i = 1, 2,...,m, be defined by 22). Suppose that the functionals t, φ) h i t, φ), i =1, 2,...,m, are Lipschitz on bounded sets. Pick Y i, 193

5 i =1, 2,...,m, satisfying P 1 and P 2. Pick a Lipschitz continuous non-decreasing function λ: R + R +, satisfying 0 <λs), s 0, ), lim s λα 1s))α 3 s)= 28) Then, the closed loop system consisting of the system 13) and the feedback control law 23) with pt, φ)= [Y 1 λv t, φ0),φ [,0) ))h 1 t, φ)),..., Y m λv t, φ0),φ [,0) ))h m t, φ))] T, 29) is ISS with respect to the disturbance dt) with the essential bounds 14). Moreover, a gain function γ satisfying 6) is given by γs)= 1 α 1 1 α 2 ˆα 1 3 2σs)), s 0, 30) γ a where ˆα 3 is any class K function satisfying ˆα 3 s) λα 1 s))α 3 s), s 0 31) It is stressed that the existence of a class K function satisfying 31) is ensured by 28) and the positive definiteness of α 3. The existence of a Lipschitz continuous non-decreasing function λ fulfilling 28) is also guaranteed by the positive definiteness of α 3. Thus, Proposition 13 demonstrates that we can achieve ISS of the closed loop system at the price of larger control gain even if α 3 in Assumption 11 does not satisfy 26). To see this point through the role of λ, let us consider λs) =β for a positive real β. It is observed that the gain of the control law in 29) becomes large if we use a large β. Now, the property 28) can be achieved by β>0 if and only if lim s α 3 s)=. Thus, in order to achieve 28) for α 3 which is not of class K, the function λs) needs to be unbounded as s tends to, which implies the increase of the control gain in 29). Next, suppose that α 3 is of class K and that λs)=β. Using ˆα 3 = βα 3 we obtain γ in 30) as γs)= 1 α 1 1 α 2 α 1 3 γ a ) 2σs) β 32) Hence, the larger β is, the smaller the gain γ is. The same discussion applies to the general choice of λs). It is stressed that a smaller γ does not imply a tolerance to the disturbance d of larger maximum magnitude. A smaller γ implies that the effect of the disturbance on the state x is rendered smaller for each magnitude of the disturbance as far as it is below the fixed maximum magnitude given by 14). Although the magnitude of the redesign term in 23) is bounded by the values determined by the functions Y i, the gain of the redesign control law 23) becomes larger when we use a larger λs) yielding better attenuation of the effect of the disturbance. In the situation where we have a constraint on the control input as an allowable maximum gain, we can use smaller λs) by settling for iiss of the closed loop system instead of ISS. V. EXAMPLE Let us consider a system described by the following uncertain scalar retarded functional differential equation ẋt)=γxt ht))sgnxt)) + xt) ut)+dt)), xτ)=ξ 0 τ),τ [, 0], ξ 0 C, 33) where: is an unknown positive real, ht) is an unknown continuously differentiable function, describing the timevarying time-delay, satisfying dht) 0 ht), 1 dt 2 ; 34) γ is an unknown real in the interval 1, 1). Suppose that dt) which is unknown is a Lebesgue measurable disturbance satisfying ess sup t [0, ) dt) d<1. Thus d 1 = d, d 1 = d are upper and lower bounds for the disturbance. Let the actuator allow values of the control input in the interval [ 3, 3]. We want to build a control law such that the closed loop system is ISS with respect to the disturbance dt). Let, for t R +, φ C, kt, φ) = 2sgnφ0)). Then, in this case, the unforced closed loop system ut)= kt, x t ), dt) 0) is described by the retarded equation with discontinuous right-hand side ẋt)=γxt ht))sgnxt)) 2sgnxt)) xt) 35) Let V : R + C R + be the invariantly differentiable functional given by, for t R +, φ C, V t, φ)=φ 2 0) + ht) φ 2 τ) + τ 2 + τ + ) φ 2 τ)dτ, 36) Then, by considering the retarded inclusion corresponding to 35), by 16) we obtain, for t R +, φ C, D + V gas) t, φ) 4φ 2 0) + 2 γ φ0) φ ht)) +φ 2 0) 1 dht) ) φ 2 ht)) dt +φ 2 0) φ 2 ) 1 2 φ 2 τ)dτ 37) By Young s inequality, taking into account of bound limits for γ and dht) dt, we obtain D + V gas) t, φ) 21 γ )φ 2 0) 1 2 φ 2 τ)dτ 38) Therefore, by using as a semi-norm the M2 norm, we obtain α 1 φ M2 ) V t, φ) α 2 φ M2 ), D + V gas) t, φ) α 3 φ M2 ), with α 1 s) = 1 2 s2, α 2 s)=2s 2, α 3 s)=min { 1 21 γ ), 2 } s 2, s 0. Let us now pick a 1, b 1, and Y 1 in accordance with P 1, P 2 for d 1, d 1 as a 1 = 1,b 1 =1and s s+1, s 0 Y 1 s)= s s 1 s<0 Let λs)=β>0. By 22), 29), we obtain pt, φ)= 39) { 2βφ 2 0) 1+2βφ 2 0), φ0) 0 2βφ 2 0) 1+2βφ 2 0), φ0) < 0 40) 194

6 for φ C. Notice that pt, φ) is Lipschitz on bounded sets thus zero at zero). The feedback control law 23) is ut)= 2sgnxt)) 2βx2 t) 1+2βx 2 sgnxt)) 41) t) Such control law adheres to the actuator limitations. By Proposition 13 we conclude that, for any given positive real, for any given time-delay function ht) satisfying 34) and for any parameter γ satisfying γ < 1, the closed-loop system 33),41) is ISS with respect to the disturbance dt), provided its maximum amplitude is less than d. The ISS gain is given by 32). Increasing β, the disturbance effect on the state x can be arbitrarily attenuated provided that the disturbance satisfies ess sup t [0, ) dt) d<1. REFERENCES [1] D. Angeli, E.D. Sontag, Y. Wang, A characterization of integral input to state stability, IEEE Trans. on Automatic Control, Vol. 45, pp , [2] A. Bacciotti, L. Rosier, Liapunov Functions and Stability in Control Theory, Springer-Verlag, Berlin Heidelberg, [3] A.F. 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