Event-Based Control of Nonlinear Systems with Partial State and Output Feedback

Transcription

1 Event-Based Control of Nonlinear Systems with Partial State and Output Feedback Tengfei Liu a, Zhong-Ping Jiang b a State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang, , China b Polytechnic Institute of Engineering, New York University, 6 MetroTech Center, Brooklyn, NY 11201, USA Abstract This paper studies the event-triggered control problem for nonlinear systems with partial state and output feedback. We first consider the control systems that are transformable into an interconnection of two input-to-state stable (ISS) subsystems with the sampling error as the external input. It is shown that infinitely fast sampling can be avoided and asymptotic stabilization can be achieved by appropriately choosing the decreasing rate of the threshold signal of the event-trigger. Then, we focus on the event-triggered output-feedback control problem for nonlinear uncertain systems in the output-feedback form. The key idea is to introduce a novel nonlinear observer-based control design and to transform the control system into the form of interconnected ISS systems. ISS small-gain methods are used as a fundamental tool in the discussions. Keywords: Event-triggered control, nonlinear systems, partial state feedback, output feedback, input-to-state stability (ISS), small-gain theorem. 1. Introduction Tremendous effort has been made for improved performance of sampled-data control systems. As an alternative to the traditional periodic data-sampling, the aperiodic event-triggered data-sampling depends on the real-time system state, and in this way, takes into account the system behavior between the sampling time instants. Such new data-sampling strategy has been proved to be quite useful in reducing the waste of computation and communication resources in feedback control systems. Early results in this direction include Gupta (1963); Liff and Wolf (1966); Tomovic and Bekey (1966); Ciscato and Martiani (1967); Mitchell and McDaniel Jr. (1969). Due to the increasing popularity of networked control systems, recent years have seen a renewed interest in eventtriggered control of linear and nonlinear systems. Significant contributions have been made to the literature; see, e.g., Arzen (1999); Åström and Bernhardsson (2002); Yook et al. (2002); Tabuada (2007); Heemels et al. (2008); Henningsson et al. (2008); Gawthrop and Wang (2009); Lunze and Lehmann (2010); Donkers and Heemels (2012); Marchand et al. (2013); Tallapragada and Chopra (2013) and the references therein. The designs have been extended to distributed control in Wang and Lemmon (2011); Fan et al. (2013); Seyboth et al. (2013); Guinaldo et al. (2013), decentralized and distributed control in Donkers and Heemels (2012); De Persis et al. (2013); Mazo Jr. and Cao (2013), systems with quantized measurements in Garcia and Antsaklis (2013), periodic event-triggered control in Heemels et al. (2013), and designs based on hybrid systems theory Postoyan et al. (2011a,b), to name a few. See also Heemels et al. (2012) for a literature review and tutorial of event-triggered control, For practical implementation of event-triggered control, infinitely fast sampling should be avoided. In other words, the intervals between all the sampling time instants should be guaranteed to be lower bounded by some positive constant; see e.g., the discussions in Lemmon (2010). One special case of infinitely fast sampling is the Zeno behavior, This research was supported partly by NSF grants ECCS and ECCS , partly by NSFC grant , partly by the Fundamental Research Funds for the Central Universities under Grant N in China, and partly by the Project 111 (No. B08015) at the State Key Laboratory of Synthetical Automation for Process Industries. addresses: neuralliu@gmail.com; tfliu@mail.neu.edu.cn (Tengfei Liu), zjiang@nyu.edu (Zhong-Ping Jiang) Preprint submitted to November 12, 2014

2 i.e., there is an infinite number of sampling time instants which converge to a finite value; see, e.g., Goebel et al. (2009) for the discussions on the Zeno behavior in hybrid systems. In most of the existing results, the events of data-sampling are triggered by comparing the real-time system state and a threshold signal, and the event-triggered control problems are transformed into problems of choosing appropriate threshold signals to avoid infinitely fast sampling. Due to the hybrid nature, the forward completeness of the event-triggered control systems is a complex issue. This paper focuses on the event-triggered control problem for nonlinear systems with partial state and output feedback. Several recent results can be found in Åström (2008); Molin and Hirche (2010); Donkers and Heemels (2012); Heemels and Donkers (2013); Li and Lemmon (2013). Specifically, in Åström (2008); Molin and Hirche (2010); Heemels and Donkers (2013); Li and Lemmon (2013), observers are employed to reconstruct the plant state by using the sampled data of the outputs. The paper of Donkers and Heemels (2012) studies general dynamic outputfeedback controllers and uses the theory of hybrid systems to analyze the stability property of the resulted closed-loop systems. In Lehmann and Lunze (2011), a prediction mechanism is employed in the controller to estimate the system output between the sampling time instants, which can be considered as the output-feedback version of Lunze and Lehmann (2010). Note that the event-triggered output-feedback control results mentioned above mainly focus on linear systems. Also, it seems that most of the existing results on partial state and output feedback event-based control for continuous-time, deterministic systems can only guarantee practical convergence. In Abdelrahim et al. (2013), the event-triggered control of nonlinear singularly perturbed systems is studied based only on the slow dynamics, and global asymptotic stabilization can be achieved if a positive transfer interval exists. The notion of input-to-state stability (ISS), invented by Sontag, is a powerful tool to describe the stability property of nonlinear systems with external inputs; see Sontag (2007). For event-triggered control, ISS has been used to describe the influence of data-sampling to control system performance; see, e.g., Tabuada (2007); Anta and Tabuada (2010); Mazo Jr. et al. (2010). In this framework, it is often assumed that the plant has a known input-to-state stabilizing controller with the sampling error considered as the external input. Then, asymptotic stabilization can be achieved if one can find an event-trigger such that the influence of the sampling error is attenuated and the closed-loop system augmented with the event-trigger is asymptotically stable at the origin. In the very recent paper De Persis et al. (2013), the small-gain argument is applied to guarantee the stability of the overall system composed of interacting ISS subsystems, and a parsimonious event triggering mechanism is developed to avoid the Zeno behavior. In our recent paper Liu and Jiang (2013b), we proposed an ISS gain condition for event-triggered control of nonlinear uncertain systems with full-state feedback without using Lyapunov arguments. This paper takes a step forward toward solving the event-triggered control problem for nonlinear systems with partial state and output feedback. The contribution of the paper is twofold: A novel ISS small-gain approach to event-trigger design is proposed for asymptotic convergence of interconnected nonlinear systems with partial state and output feedback; A constructive event-triggered output-feedback control design is developed for a class of nonlinear uncertain systems. The problems are studied in this paper by using the Lyapunov-based ISS arguments. To the best of our knowledge, both of the contributions are new. At the same time, this paper fully takes into account the forward completeness issue caused by the hybrid dynamics of event-triggered control systems. We first consider the control systems that are transformable into an interconnection of two input-to-state stable (ISS) subsystems with the sampling error as the external input. It is also assumed that only the state of one subsystem is available for event-trigger design. Motivated by Guinaldo et al. (2013); Seyboth et al. (2013), we propose an eventtrigger design method with the threshold signal generated by an asymptotically stable system. By considering the closed-loop event-triggered system as a dynamic network, we use the ISS small-gain methods developed in Jiang et al. (1994, 1996); Jiang and Wang (2008); Liu et al. (2011) to fine-tune the parameters of the event-trigger to avoid infinitely fast sampling, and at the same time, guarantee asymptotic convergence of the system state. Exponentially converging threshold signals are used in Guinaldo et al. (2013); Seyboth et al. (2013). But, with an elementary example in this paper, we show that non-exponentially converging threshold signals may be preferred for nonlinear systems. As a special case, if the subsystems are linear with quadratic ISS-Lyapunuov functions, one can find exponentially decreasing threshold signals. It should be noted that time-varying threshold signals has also been used in Postoyan et al. (2011b); Girard (2013); Mazo Jr. and Cao (2013). However, the designs in these papers seem not directly 2

3 applicable to the case of partial state feedback. The new design has also been extended to a more general case in which the dynamics of the threshold signal subsystem also depends on the available state information of the controlled system. Note that this paper uses Lyapunov-based formulations and focuses on the problems caused by partial state and output feedback, which differs from our recent paper Liu and Jiang (2013b) focusing on full state feedback event-triggered and self-triggered control. The second contribution of the paper lies in a new event-triggered output-feedback control design for nonlinear uncertain systems in the output-feedback form; see Krstić et al. (1995). The proposed controller is composed of a novel ISS-induced nonlinear observer and a nonlinear control law. By appropriately choosing the parameters, the controlled system can be ultimately transformed into an interconnection of two ISS subsystems with the sampling error of the output as the external input. Then, an event-trigger design is proposed for output-feedback stabilization. The rest of the paper is organized as follows. Section 2 presents the notations and reviews ISS and a cyclic-smallgain result that will be used for the designs in this paper. In Section 3, we propose a new event-trigger design for a class of control systems transformable into an interconnection of two ISS subsystems. In Section 4, we focus on the event-triggered output-feedback control problem of nonlinear uncertain systems in the output-feedback form. A simulation example is given in Section 5 to show the need of a non-exponentially decreasing threshold signal for event-triggered partial state feedback control. Section 6 contains some concluding remarks. 2. Preliminaries We recall the ISS cyclic-small-gain theorem proposed in Teel (2003); Jiang and Wang (2008); Liu et al. (2011) that plays a crucial role in stability analysis and control synthesis of large-scale systems. Consider a large-scale nonlinear system taking the following form: ẋ i = f i (x, u i ), i = 1,..., N (1) where x i R n i, x = [ x1 T,..., T N] xt, ui R m i and f i : R n+m i R n i with n = Σ N j=1 n j is a locally Lipschitz map. It is assumed that each x i -subsystem (i = 1,..., N) in (1) admits an ISS-Lyapunov function V i : R n i R which is locally Lipschitz on R n i \{0} and satisfies 1. there exist α i, α i K such that 2. there exist γ i j K {0} ( j i) and γ u i K {0} such that α i ( x i ) V i (x i ) α i ( x i ), x i R n i ; (2) V i (x i ) where α i is continuous and positive definite. { ( ( )) max γi j V j x j, γ u i ( u i ) } j=1,...,n; j i V i (x i ) f i (x, u i ) α i (V i (x i )) a.e. (3) In this paper, a.e. stands for almost everywhere, and V means the gradient of V. The functions γ i j, γ u i are known as the ISS gains of the subsystems. With respect to the smooth Lyapunov characterization introduced by Sontag and Wang (1996), we use only locally Lipschitz ISS-Lyapunov functions because of our constructions in (5), (8), (9), (41) and (133). The following theorem in Jiang and Wang (2008) presents a cyclic-small-gain condition to guarantee the ISS property of the large-scale system (1) with state x and input u = [u T 1,..., ut N ]T. Theorem 1. Consider the large-scale system (1). Assume each x i -subsystem admits an ISS-Lyapunov function V i satisfying (2) and (3). Then, the large-scale nonlinear system (1) is ISS if for r = 2,..., N, where 1 i k N and i k i k if k k for 1 k r. γ i1 i 2 γ i2 i 3... γ ir i 1 < Id (4) 3

4 By considering the subsystems (1) as vertices and the gains as the weights of the directed connections between the subsystems, the interconnection structure of the large-scale nonlinear system can be represented with a system digraph. Condition (4) is called cyclic-small-gain condition and means that the composition of the ISS gains along every simple cycle in the large-scale nonlinear system is less than the identity function Id. For the ISS gains γ i j s (1 i N, j i) satisfying condition (4), according to (Jiang et al., 1996, Lemma A.1), we can find K functions ˆγ i j s (1 i N, j i) which are continuously differentiable on (0, ) and slightly larger than the corresponding γ i j s such that condition (4) still holds by replacing the γ i j s with the ˆγ i j s. Motivated by the ISS-Lyapunov function construction in Jiang et al. (1996), an ISS-Lyapunov function can be constructed for the large-scale system (1) as where σ i s are specific compositions of the ˆγ ( ) s. The influence of the external input u can be represented as V(x) = max i=1,...,n {σ i(v i (x i ))} (5) θ(u) = max i=1,...,n {σ i γ u i ( u i )}. (6) Denote f (x, u) = [ f T 1 (x, u 1),..., f T N (x, u N)] T. With the Lyapunov-based ISS cyclic-small-gain theorem presented in Liu et al. (2011), we have V(x) θ(u) V(x) f (x, u) α(v(x)) a.e. (7) with α being a continuous and positive definite function. Based on the Lyapunov-based ISS cyclic-small-gain theorem, we have the following lemma. Lemma 1. Consider the large-scale system (1). Assume that each x i -subsystem admits an ISS-Lyapunov function V i satisfying (2) and (3), and the cyclic-small-gain condition (4) is satisfied. Then, V defined in (5) is an ISS-Lyapunov function of the large-scale system, and at the same time, for any nonempty A, B {1,..., N} satisfying A B = {1,..., N} and A B =, V A (x A ) = max i(v i (x i ))} i A (8) V B (x B ) = max j(v j (x j ))} j B (9) are ISS-Lyapunov functions of the two systems composed of the x i -subsystems with i A and the x j -subsystems with j B, respectively, where x A and x B represent the states of the two subsystems. Moreover, the interconnection gains can be made less than Id. Lemma 1 is used to transform the closed-loop event-triggered control system considered in Section 4 into the form composed of two subsystems, which can be handled by the method developed in Section 3. The proof of Lemma 1 is omitted due to space limitation. The interested reader may consult (Liu et al., 2012b, Section 4) for related discussions. Also see Lemma 2 for more details on the interconnection gains. 3. Event-Trigger Design with Partial State Information In this section, we study an event-trigger design problem for interconnected nonlinear systems. The objective is to develop an ISS gain condition for event-triggered control to avoid infinitely fast sampling and achieve asymptotic convergence Problem Formulation We consider the case in which a well-designed control system takes the following feedback form: ż(t) = h(z(t), x(t), w(t)) (10) ẋ(t) = f (x(t), z(t), w(t)) (11) 4

5 where [z T, x T ] T with z R m and x R n is the state, w R n represents the sampling error of x, h : R m R n R n R m and f : R n R m R n R n represent the system dynamics with h(0, 0, 0) = 0 and f (0, 0, 0) = 0. Here, z is considered to be unavailable for the event-trigger design. For convenience of notations, define x = [z T, x T ] T. One example of such control system that is transformable into the form of (10) (11) is given in Section 4. The sampling error w(t) is defined as w(t) = x(t k ) x(t), t [t k, t k+1 ), k S (12) where {t k } k S is the sequence of the sampling time instants with S = {0, 1, 2,...} Z + being the set of the indices of all the sampling time instants and t 0 = 0. Suppose that x(t) is right maximally defined for all t [0, T max ) with 0 < T max. With respect to the possible finite-time accumulation of t k and finite-time escape of x(t), we consider three cases: (a) S = Z + and lim k t k <, which means Zeno behavior. (b) S = Z + and lim k t k =. In this case, x(t) is defined on [0, ). (c) S is a finite set {0,..., k } with k Z +, i.e., there is a finite number of sampling time instants. In this case, t k < T max and we set t k +1 = T max for convenience of discussions. It should be noted that, in any case, x(t) is defined for all t k S[t k, t k+1 ) [0, T max ). With an appropriate event trigger design, we can guarantee that inf k S {t k+1 t k } > 0, which means that Case (a) is impossible. Also, as shown in the following discussions, by means of small-gain arguments, we can prove that T max = for Case (c). In event-triggered control, the sampling time instants are triggered by comparing the sampling error x(t k ) x(t) with a continuous, positive threshold signal. Define µ : R + R + as the threshold signal. Then, the sampling time instants is generated by Recall the definition of w(t) in (12). We have t k+1 = inf t>t k { x(t k ) x(t) = µ(t)}, k S. (13) w(t) µ(t) (14) for all t k S[t k, t k+1 ). In this paper, the event-trigger design problem is solved for system (10)-(11) by finding an appropriate threshold signal µ(t) for the event-trigger such that the following two objectives are achieved at the same time: Objective 1: z(t) and x(t) are defined for all t 0 and the closed-loop event-triggered system with [z T, x T, µ] T as the state is globally asymptotically stable at the origin. Objective 2: For specific (z(0), x(0)) and specific µ(0) > 0, the intervals between the sampling time instants are lower bounded by a positive constant. Objective 2 aims to avoid infinitely fast sampling. Here, we do not require the existence of some constant t > 0 such that for any (z(0), x(0)) and any µ(0) > 0, t k+1 t k t for all k S. We assume that both the z-subsystem and the x-subsystem are ISS. More precisely, we make the following assumption on the Lyapunov-based ISS properties of the subsystems. Assumption 1. Both the z-subsystem and the x-subsystem are ISS with ISS-Lyapunov functions V z : R m R + and V x : R n R + which are locally Lipschitz on R m \{0} and R n \{0}, respectively, and satisfy where α z, α z, α x, α x K and γ x z, γ w z, γ z x, γ w x K {0}. α z ( z ) V z (z) α z ( z ), (15) V z (z) max{γ x z (V x (x)), γ w z ( w )} V z (z)h(z, x, w) α z (V z (z)), a.e. (16) α x ( x ) V x (x) α x ( x ), (17) V x (x) max{γ z x(v z (z)), γ w x ( w )} V x (x) f (x, z, w) α x (V x (x)), a.e. (18) 5

6 We employ Example 1 to show how an event-triggered control system can be transformed into the form of (10) (11) satisfying Assumption 1. The system in Example 1 is also used to show the need of a threshold signal which does not decrease exponentially. Example 1. Consider system ż(t) = z 3 (t) (19) ẋ(t) = u(t) + z(t) (20) where z R and x R are the state variables, u R is the control input. We consider the case where only x is available for feedback control design. We employ the feedback control law where {t k } k S represents the sequence of sampling time instants. By using (12) and (21), we have u(t) = x(t k ), t [t k, t k+1 ), k S (21) ẋ(t) = x(t) w(t) + z(t). (22) Thus, the controlled system composed of (19) and (22) is in the form of (10) (11) with h(z, x, w) = z 3 and f (x, z, w) = x w + z. To verify the satisfaction of Assumption 1, we define V z (z) = z and V x (x) = x. Clearly, V z and V x are locally Lipschitz. It can be directly checked that V z and V x satisfy (15) and (17), respectively, with α z, α z, α x, α x = Id. Also, direct calculation yields: Then, property (24) implies V z (z)h(z, x, w) = z 3 = V 3 z (z) a.e. (23) V x (x) f (x, z, w) x + w + z = V x (x) + V z (z) + w V x (x) + 2 max {V z (z), w } a.e. (24) V x (x) 4 max {V z (z), w } V x (x) f (x, z, w) 0.5V x (x) a.e. (25) Thus, properties (16) and (18) are satisfied with γ x z (s) = 0, γ w z (s) = 0, α z (s) = s 3, γ z x(s) = 4s, γ w x (s) = 4s and α x (s) = 0.5s for s R +. In Guinaldo et al. (2013); Seyboth et al. (2013), the event-triggered control problem was studied and exponentially converging threshold signals were used in the context of distributed control. Based on this idea, we first try the threshold signal µ(t) defined by µ(t) = µ(0)e ct (26) for all t 0, with initial state µ(0) > 0 and constant c > 0. Equivalently, µ(t) is the solution of the initial value problem µ(t) = cµ(t) (27) for all t 0. However, Example 2 shows that an exponentially converging µ(t) may lead to infinitely fast sampling. Example 2. Consider the system composed of (19) and (22), which is in the form of (10) (11) with h(z, x, w) = z 3 and f (x, z, w) = x w + z. It is shown in Example 1 that the system satisfies Assumption 1. Moreover, the system 6

7 is a cascade connection of the z-subsystem and the x-subsystem, and thus the nonlinear small-gain condition between γ z x and γ x z, i.e., γ z x γ x z < Id (28) holds trivially. As a consequence, the cascade system is ISS with w as the input. See Jiang et al. (1994, 1996) for the details. However, we show that for some initial states, there does not exist an exponentially converging threshold signal in the form of (26) to avoid infinitely fast sampling. Based on the discussions in Appendix A, for any specific z(0), x(0), µ(0) and constant c, there exist constants m 1, m 2, m 3, c such that for all t k S[t k, t k+1 ). Moreover, there exist z(0), x(0), µ(0) such that for all t k S[t k, t k+1 ). Properties (29) and (30) together imply f (x(t), z(t), w(t)) m 1 e t + m 2 e c t m 3 e ct (29) m 1 0, m 2 > 0, m 2 2m 3, 2c c, (30) x(t) > z(t) > 0, (31) f (x(t), z(t), w(t)) < 0 (32) f (x(t), z(t), w(t)) m 2 2 e c t for all t k S[t k, t k+1 ). Given t k, we estimate the upper bound of δt k = t k+1 t k. With property (32), by using the triggering condition (13), we have Then, by using (26) and (33), we have which implies µ(t k+1 ) = x(t k ) x(t k+1 ) tk+1 = f (x(τ), z(τ), w(τ))dτ = t k tk+1 (33) t k f (x(τ), z(τ), w(τ)) dτ. (34) µ(0)e c(t k+δt k ) m 2 2 e c (t k +δt k ) δt k, (35) δt k e c (t k +δt k ) 2µ(0) m 2 (36) by using 2c c. Now, we show S = Z +. Suppose S = {0, 1,..., k } with k being a positive integer. In this case, by using the standard small-gain theorem in Jiang et al. (1994), we can guarantee the boundedness of z(t) and x(t) for all t [0, T max ). Due to continuation, z(t) and x(t) are defined for all t [0, ), i.e., T max =. Recall that x(t k ) > 0 (see (31)). Since the z-subsystem is globally asymptotically stable at the origin, one can find a finite time t t k such that for all t [t, ), and thus z(t) 1 2 x(t k ) (37) ẋ(t) = x(t k ) + z(t) 1 2 x(t k ) (38) for all t [t, ). This contradicts with the boundedness of x(t). Thus, S = Z +. Suppose that infinitely fast sampling does not occur. Then, lim k t k = and inf k Z+ δt k > 0. However, if lim k t k =, then property (36) implies lim k δt k = 0. Thus, infinitely fast sampling occurs. 7

8 Note that the system dynamics are assumed to be known in Example 2. The problem would be more complicated for nonlinear uncertain systems. In particular, in the setting of event-triggered control, because the sampling error w is updated on discrete time instants (which depend on x and µ), the forward completeness of the system may not be trivially guaranteed. From the discussions in Example 2, it can be observed that the problem is caused by the nonlinearity z 3 of the z-subsystem. Intuitively, the signal z(t) does not converge to the origin exponentially, and the exponential convergence of µ(t) is too fast compared with the convergence rate of f (x(t), z(t), w(t)). To overcome the limitation of the exponentially decreasing threshold signal, we consider threshold signals generated by more general dynamic systems in the form of µ(t) = Ω(µ(t)) (39) with initial condition µ(0) > 0, where Ω : R + R + is Lipschitz on compact sets and positive definite. Clearly, (27) is a special case of (39). In the following discussions, the initial condition µ(0) is only required to be strictly larger than zero, and our event triggering mechanism is proved to be valid as long as this condition is satisfied. Under Assumption 1, we develop a condition on the ISS gains of the subsystems under which event-triggered control can be realized without infinitely fast sampling. We consider the interconnected system composed of the z-subsystem (10), the x-subsystem (11) and the µ- subsystem (39) subject to (14). Under Assumption 1, if w(t) is well defined for all t 0 and the small-gain condition (28) is satisfied, then the interconnected system is asymptotically stable at the origin. Moreover, according to the Lyapunov-based ISS cyclic-small-gain theorem in Liu et al. (2011), we can construct a Lyapunov function for the interconnected system: V 0 (z, x, µ) = max {ˆγ z x(v z (z)), V x (x), ˆγ w x (µ), ˆγ z x ˆγ w z (µ) }. (40) If γ ( ) ( ) is nonzero, then the corresponding ˆγ( ) ( ) in (40) is chosen such that ˆγ( ) on (0, ) and slightly larger than its corresponding γ ( ) ; if γ( ) ( ) ( ) K and it is continuously differentiable = 0, then ˆγ( ) ( ) = 0. Moreover, ˆγz x satisfies ˆγ z x γz x < Id. Define γ w x (s) = max{ˆγ w x (s), ˆγ z x ˆγ w z (s)} for s R +. Clearly, γ w x is a K function being locally Lipschitz on (0, ). It is a standard result that V(z, x, µ) = ( γ w ) 1 x (V 0 (z, x, µ)) = max { w) ( γ 1 x ˆγ z x(v z (z)), ( γ w ) 1 x (V x (x)), µ } =: max {σ z (V z (z)), σ x (V x (x)), µ} (41) is also a Lyapunov function of the interconnected system. Note that σ z and σ x are locally Lipschitz on (0, ) and thus continuously differentiable almost everywhere on (0, ). Moreover, Lemma 2 shows that the interconnection ISS gains are less than Id if V z (z) = σ z (V z (z)) and V x (x) = σ x (V x (x)) are considered as the ISS-Lyapunov functions of the z-subsystem and the x-subsystem, respectively. This result is used in the following discussions on the convergence rate of the closed-loop event-triggered system; see the proof of Lemma 3. Lemma 2. Under Assumption 1, if (28) is satisfied, then there exist K functions γ x z, γ w z, γ z x, γ w x, all less than Id, such that V z (z) max{ γ x z ( V x (x)), γ w z ( w )} V z (z)h(z, x, w) ᾱ z ( V z (z)) a.e. (42) V x (x) max{ γ z x( V z (z)), γ w x ( w )} V x (x) f (x, z, w) ᾱ x ( V x (x)) a.e. (43) where ᾱ z can be any continuous and positive definite function satisfying ᾱ z (s) σ z (σ 1 z (s))α z (σ 1 z (s)) for almost all s > 0 and ᾱ x can be any continuous and positive definite function satisfying ᾱ x (s) σ x (σ 1 x (s))α x (σ 1 x (s)) for almost all s > 0. 8

9 Proof. We only prove property (42). Property (43) can be proved similarly. Since σ z K, V z is positive definite and radially unbounded. We choose γ x z, γ w z K such that ( γ w x ) 1 ˆγ z x γ x z γ w x γ x z < Id, (44) (ˆγ w z ) 1 γ w z γ w z < Id. (45) The existence of such γ z x and γ z w can be guaranteed as ( γ w ) 1 x ˆγ z x γz x γ w x < ( γ w ) 1 x γ w x = Id and (ˆγ ) 1 z w γ w z < Id. Then, V z (z) γ z x ( V x (x)) implies V z (z) ( γ w ) 1 x ˆγ z x γz x γ w x ( V x (x)). By using the definitions of V z and V x, we have ( γ w ) 1 x ˆγ z x(v z (z)) ( γ w ) 1 x ˆγ z x γz x γ w x ( γ w ) 1 x (V x (x)), which implies V z (z) γz x (V x (x)). Also, V z (z) γ z w ( w ) implies ( γ w ) 1 x ˆγ z x(v z (z)) (ˆγ ) 1 z w γ w z ( w ) and thus V z (z) (ˆγ z ) 1 x γ w x (ˆγ ) 1 z w γ w z ( w ) γz w ( w ). It is proved that V z (z) max{ γ x z ( V x (x)), γ w z ( w )} V z (z) max{γ x z (V x (x)), γ w z ( w )}. (46) Property (42) can then be proved by using (16). This ends the proof of Lemma 2. It is shown in the following discussions that, to avoid infinitely fast sampling and at the same time, to guarantee k S[t k, t k+1 ) = [0, ), the decreasing rate of µ(t) should be chosen in accordance with the decreasing rate of V(z(t), x(t), µ(t)). We first study the decreasing rate of V(z(t), x(t), µ(t)) in Subsection 3.2. Then, the event trigger design problem is solved for system (10) (11) in Subsection Decreasing Rate of V(z(t), x(t), µ(t)) According to the definition of V in (41), the decreasing rate of V(z(t), x(t), µ(t)) depends on the decreasing rates of V z (z(t)), V x (x(t)) and µ(t). Lemma 3 gives a condition on Ω under which the lower bound of the decreasing rate of V(z(t), x(t), µ(t)) can be estimated by Ω. Lemma 3. Consider the interconnected system composed of (10), (11), (12) and (39) subject to (14). Under Assumption 1, if (28) is satisfied, and if Ω is Lipschitz on compact sets and positive definite, then there exists a continuous, locally Lipschitz and positive definite α V such that for any V(z(0), x(0), µ(0)), V(z(t), x(t), µ(t)) η(t) (47) holds for all t k S[t k, t k+1 ), where η(t) is the solution of the initial value problem η(t) = α V (η(t)) (48) with initial condition η(0) = V(z(0), x(0), µ(0)). Moreover, if there exists a constant > 0 such that (A) Ω satisfies Ω(s) min { σ z (σ 1 z for almost all s (0, ) with σ z and σ x defined in (41), and (s))α z (σ 1 (s)), σ x (σ 1 x (s))α x (σ 1 x (s)) } (49) (B) there exists a T O k S[t k, t k+1 ) such that V(z(t), x(t), µ(t)) for all T O t < sup k S[t k, t k+1 ), then (47) holds for all T O t < sup k S[t k, t k+1 ) with η(t) being the solution of the initial value problem defined by (48) with α V = Ω for T O t < sup k S[t k, t k+1 ) with initial condition η(t O ) = V(z(T O ), x(t O ), µ(t O )). Proof. The result of (47) can be proved by using a combination of the Lyapunov-based cyclic-small-gain theorem and the comparison principle, and is omitted here. See, e.g., (Khalil, 2001, Lemma 3.4), for the comparison principle. Suppose that Conditions (A) and (B) are satisfied. Then, with Lemma 2, (42) and (43) hold with z ᾱ z = ᾱ x = Ω (50) 9

10 for all [z T, x T, µ] T satisfying V(z, x, µ) <. For convenience of discussions, define v 1 (t) = V z (z(t)), v 2 (t) = V x (x(t)), v 3 (t) = µ(t), and v(t) = V(z(t), x(t), µ(t)). Then v(t) = max{v 1 (t), v 2 (t), v 3 (t)}. Now, we prove D + v(t) Ω(v(t)) (51) for all T O t < sup k S[t k, t k+1 ), where D + represents the upper right-hand derivative and is defined by D + v(t) = lim sup h 0 + v(t + h) v(t). (52) h Consider a specific time instant t satisfying T O t < sup k S[t k, t k+1 ). Since γ x z < Id and γ w z < Id, when v 1 (t) = v(t) > 0, V z (z(t)) = V(z(t), x(t), µ(t)) > 0, there exists a neighborhood Θ of [z T (t), x T (t), µ(t)] T such that V z (p z ) max{ γ x z ( V x (p x )), γ w z (p µ )} and V z (p z ) exists for all [p T z, p T x, p µ ] Θ. Then, due to the continuity of V z (z(t)), V x (x(t)) with respect to t, there exists a t > t such that V z (z(τ)) max{ γ x z ( V x (x(τ))), γ w z ( µ(τ) )} max{ γ x z ( V x (x(τ))), γ w z ( w(τ) )} for all τ (t, t ), which implies for all τ (t, t ), according to (42). Then, we have Following the same reasoning, if v 2 (t) = v(t), then V z (z(τ))h(z(τ), x(τ), w(τ)) Ω( V z (z(τ))) (53) D + v 1 (t) Ω(v 1 (t)). (54) D + v 2 (t) Ω(v 2 (t)). (55) Note that D + v 3 (t) = Ω(v 3 (t)) holds automatically for v 3 (t) = µ(t). For T O t < sup k S[t k, t k+1 ), define I(t) = {i {1, 2, 3} : v i (t) = v(t)}. Then, by using (Giorgi and Komlósi, 1992, Lemma 2.9), we have D + v(t) = max{d + v i (t) : i I(t)} max{ Ω(v i (t)) : i I(t)} = max{ Ω(v i (t)) : v i (t) = v(t), i = 1, 2, 3} = Ω(v(t)). (56) Property (51) is proved. Then, by directly applying the comparison principle, the proof of Lemma 3 follows readily. Remark 1. Condition (A) in Lemma 3 is concerned with Ω. Given specific σ z (σ 1 z (s))α z (σ 1 z (s)) and σ x (σ 1 x (s))α x (σ 1 x (s)), one can always find a Ω satisfying the condition. In the following subsection, we will show how Condition (B) can be satisfied by appropriately choosing Ω for the event-triggered control system Event-Trigger Design The main result of this section is given in Theorem 2. Theorem 2. Consider the interconnected system composed of (10), (11), (14) and (39) with Assumption 1 and (28) satisfied. Then, Objectives 1 and 2 given in Subsection 3.1 are achievable if Ω is chosen to be Lipschitz on compact sets and positive definite, there exists a constant > 0 such that Ω(s)/s is nondecreasing for s (0, ] and Ω satisfies (49) for almost all s (0, ), and 10

11 ( σ z α z ) 1 and ( σx α x ) 1 are Lipschitz on compact sets. Proof. Due to the positive definiteness of Ω, the µ(t) generated by (39) satisfies 0 µ(t) µ(0) (57) for all t 0. Moreover, since Ω is chosen to be Lipschitz on compact sets, there exists a constant c > 0 such that for 0 s µ(0), and thus Ω(s) cs (58) µ(t) = Ω(µ(t)) cµ(t) (59) along the trajectory of µ with initial state µ(0). A direct application of the comparison principle implies for all τ t 0. Also, by using the event-trigger (13), we have µ(τ) µ(t)e c(τ t) (60) µ(tk+1 ) = x(t k+1) x(t k ) tk+1 = f (x(τ), z(τ), w(τ))dτ t k tk+1 t k f (x(τ), z(τ), w(τ)) dτ. (61) If the conditions of Theorem 2 are satisfied, then with Lemma 3, the function V defined in (41) has property (47) for all t k S[t k, t k+1 ) with η(t) being generated by (48). Due to the positive definiteness of α V, for any initial condition V(z(0), x(0), µ(0)), V(z(t), x(t), µ(t)) V(z(0), x(0), µ(0)) (62) for all t k S[t k, t k+1 ). If V(z(0), x(0), µ(0)), then define T = 0; otherwise, define T as the first time instant such that η(t ) =, where η(t) is the solution of the initial value problem (48) with η(0) = V(z(0), x(0), µ(0)). Recall that, according to Lemma 3, property (47) holds for all t k S[t k, t k+1 ). Let us estimate the lower bound of δt k = t k+1 t k by considering the cases of t k T and t k > T separately. Case 1: t k T. In this case, we prove that given specific z(0), x(0), µ(0) and specific T, there exists a δ 0 > 0 such that δt k δ 0. In this way, we can also guarantee that z(t), x(t) and w(t) are defined for all t [0, T ] and thus T k S[t k, t k+1 ). Property (62) means that there exists a finite s > 0 depending on the initial state such that [z T (t), x T (t), µ(t)] T s (63) for all t k S[t k, t k+1 ). Thus, there exists a f such that for all t k S[t k, t k+1 ). Then, by also using properties (60) and (61), we have µ(0)e c(t k+δt k ) f (z(t), x(t), w(t)) f (64) tk+1 t k f (x(τ), z(τ), w(τ)) dτ (t k+1 t k ) f = δt k f, (65) 11

12 i.e., δt k e c(t k+δt k ) µ(0) f. (66) If t k T, it is concluded that δt k e c(t +δt k ) µ(0) f. (67) Then, δ 0 can be chosen to satisfy δ 0 e c(t +δ 0 ) = µ(0)/ f. Case 2: t k > T. In this case, we prove that given specific z(t ), x(t ) and µ(t ), there exists a δ 1 > 0 such that δt k δ 1. By using (47) and the definition of T, we have for all T t < sup k S[t k, t k+1 ). Consider an η 1 (t) defined by V(z(t), x(t), µ(t)) (68) η 1 (t) = Ω(η 1 (t)) (69) for all t > T with η 1 (T ) = V(z(T ), x(t ), µ(t )). Then, by using Lemma 3, we have V(z(t), x(t), µ(t)) η 1 (t) for all T < t < sup k S[t k, t k+1 ). Also, by using the definition of V in (41), we have V(z(t), x(t), µ(t)) µ(t) for all t k S[t k, t k+1 ). Thus, µ(t) V(z(t), x(t), µ(t)) η 1 (t) (70) for all T < t < sup k S[t k, t k+1 ). With a similar reasoning as for (60), it can be proved that the η 1 (t) defined by (69) is strictly positive for all T < t < sup k S[t k, t k+1 ). Define Then, according to (70), k µ 1. We prove that k µ = η 1(T ) µ(t ). (71) η 1 (t) k µ µ(t) (72) for all T < t < sup k S[t k, t k+1 ). Since Ω(s)/s is non-decreasing for all s (0, ], we have Ω(η 1 ) Ω ( ) η1 /k µ, (73) η 1 η 1 /k µ which implies Ω(η 1 )/k µ Ω(η 1 /k µ ) for η 1 (0, ]. Then, by using (69), we have 1 η 1 (t) = 1 ( ) 1 Ω(η 1 (t)) Ω η 1 (t) k µ k µ k µ (74) for all T < t < sup k S[t k, t k+1 ). Property (72) can then be proved by using the comparison principle for system (74) with η 1 /k µ as the state and system (39) with µ as the state. If ( σ z α z ) 1 and ( σx α x ) 1 are Lipschitz on compact sets, then one can find constants kz, k x > 0 such that η 1 (t) V(z(t), x(t), µ(t)) max {k z ( z(t) ), k x ( x(t) )}. (75) 12

13 for all T < t < sup k S[t k, t k+1 ). Then, property (72) implies µ(t) 1 k µ max {k z ( z(t) ), k x ( x(t) )} (76) for all T < t < sup k S[t k, t k+1 ). By using the locally Lipschitz property of f, there exists a constant k f > 0 such that for all (z, x, µ) satisfying V(z, x, µ) V(z(T ), x(t ), µ(t )). Then, properties (61), (76) and (77) together imply Also note that (60) means Thus, we have i.e., µ(t k+1 ) (t k+1 t k )k f f (z, x, w) k f max { z, x, µ} (77) (t k+1 t k )k f max t k τ t k+1 { z(τ), x(τ), µ(τ)} { max kµ µ(τ)/k z, k µ µ(τ)/k x, µ(τ) } t k τ t k+1 δt k k f max { k µ /k z, k µ /k x, 1 } µ(t k ). (78) µ(t k+1 ) e cδt k µ(t k ). (79) e cδt k δt k k f max { k µ /k z, k µ /k x, 1 }, (80) δt k e cδt k 1 k f max { }. (81) k µ /k z, k µ /k x, 1 Then, δ 1 can be chosen such that δ 1 e cδ 1 1/k f max { k µ /k z, k µ /k x, 1 }. We now consider the above-mentioned three cases: (a) S = Z + and lim k t k < ; (b) S = Z + and lim k t k = ; (c) S = {0,..., k } with k Z +. (1) Suppose that Case (a) happens. Then, we have inf k S {t k+1 t k } = 0, which contradicts with (67) and (81). Thus, Case (a) is impossible. (2) In Case (b), we have k S[t k, t k+1 ) = [0, ), which means that (47) holds for all t [0, ). (3) In Case (c), since t k +1 = T max, we have k S[t k, t k+1 ) = [0, T max ), and thus (47) holds for all t [0, T max ). By the continuation of solutions, this implies that x(t) is defined for all t [0, ), i.e., T max =. This ends the proof of Theorem 2. Example 3. The infinitely fast sampling problem arising in Example 2 can be readily solved by Theorem 2. By using the V z and V x defined in Example 1, we choose ˆγ z x(s) = 5s, ˆγ w z (s) = 0 and ˆγ w x (s) = 5s for s R +. According to (40), we define By choosing γ w x (s) = 5s for s R +, we have Thus, σ z (s) = s and σ x (s) = s/5 for s R +. V 0 (z, x, µ) = max{5v z (z), V x (x), 5µ}. (82) V(z, x, µ) = max{v z (z), V x (x)/5, µ}. (83) It can be verified that ( σ z α z ) 1 (s) = s and ( σx α x ) 1 (s) = 2s for s R+ are Lipschitz on compact sets. 13

14 We choose Ω(s) = min { σ z (s)α z (σ 1 z (s)), σ x (s)α x (σ 1 x (s)) } = min{s 3, s/2} (84) for s R +. Then, Ω is positive definite and locally Lipschitz, and satisfies (49). Also, Ω(s)/s is non-decreasing for s (0, ). A simulation example is given in Section 5. Remark 2. A special case is that both the z-subsystem and the x-subsystem are linear and their ISS-Lyapunov functions are in the quadratic form. In this case, Assumption 1 can be modified with α z (s) = a z s 2, α z (s) = a z s 2, α x (s) = a x s 2, α x (s) = a x s 2, γz x (s) = bz x s, γz w (s) = b w z s 2, γ z x(s) = b z xs, γ w x (s) = b w x s 2, α z (s) = a z s and α x (s) = a x s for s R + with a z, a z, a x, a x, bz x, b w z, b z x, b w x, a z, a z being positive constants. For the linear case, the small-gain condition (28) is equivalent to b z xbz x < 1. Assume that the small-gain condition is satisfied. Then, there exists an ɛ > 0 such that (b z x + ɛ)bz x < 1. We choose ˆγ z x(s) = (b z x + ɛ)s =: ˆb z xs, ˆγ w x (s) = (b w x + ɛ)s 2 =: ˆb w x s 2, ˆγ z w (s) = (b w z + ɛ)s =: ˆb w z s 2 for s R +. Then, γ w x can be written in the form γ w x (s) = b w x s 2 with b w x being a positive constant. It can be calculated that σ z (s) = ˆb z xs/ b w x and σ x (s) = s/ b w x. Then, the right-hand side of (49) equals min { a z /ˆb z x, a x } b w x s/2. Thus, one can find a positive constant c such that Ω(s) = cs satisfies (49). Remark 3. The event-trigger design in this section considers the systems which are input-to-state stabilizable with the sampling error w as the input. This is closely related to the topic of control under sensor noise if we consider the sampling error as a special sensor noise. One of the previous results can be found in Jiang et al. (1999), which considers nonlinear systems composed of two subsystems, one is ISS and the other one is input-to-state stabilizable with respect to the sensor noise. In Jiang et al. (1999), the ISS of the closed-loop system is guaranteed by using the ISS small-gain theorem in Jiang et al. (1994, 1996). In Section 4, we show that the proposed condition for event-triggered control can also be satisfied by a class of nonlinear uncertain systems through a new nonlinear output-feedback control design An Extension In the discussions above, we consider µ(t) to be generated by system (39) with the system dynamics depending solely on µ(t). A more general case is that µ(t) is generated by system µ(t) = Ω(µ(t), x(t)) (85) with Ω : R + R n R being an appropriately chosen function and µ(0) > 0. With such modification, the events of sampling are triggered based on both µ and x, such that the inter-sampling times could be further increased. In this case, the structure of the interconnected system composed of (10), (11) and (85) subject to (14) is shown in Figure 1. µ z x Figure 1: The structure of the interconnected system composed of (10), (11) and (85). Under Assumption 1, the basic idea is to choose Ω such that for all µ R + and all x R n, Ω(µ, x) Ω(µ) (86) where Ω is chosen to satisfy the conditions given in Theorem 2; 14

15 system (85) is ISS with µ as the state and x as the input, and moreover, µ χ x µ( x ) Ω(µ, x) α µ (µ) (87) where α µ is a continuous, positive definite function and χ x µ is a K function and satisfies γ w x χµ x α 1 x < Id, (88) γ z x(γz w (χµ(α x 1 x ))) < Id. (89) Then, the system composed of subsystems (10), (11) and (85) subject to (14) can be considered as an interconnection of ISS subsystems, and conditions (28), (88) and (89) form the cyclic-small-gain condition given in Jiang and Wang (2008) for the interconnected system. With µ(t) generated by (85), one can still guarantee the boundedness of µ(t) for all t k S[t k, t k+1 ). And with a similar reasoning as for (59), one can find a c > 0 such that µ(t) cµ(t) (90) for all t k S[t k, t k+1 ). Then, the validity of (85) can be proved in the same way as in the proof of Theorem 2. Note that in this case, Lemma 3 should also be generalized with (48) replaced by η(t) = Ω(η(t), x(t)). (91) Clearly, with the extension, the decreasing rate of µ could be reduced, and thus the inter-sampling times could be increased. One realization of the Ω satisfying (86) and (87) is Ω(µ, x) = Ω(µ) χ 1 ( max { χ2 χ x µ( x ) χ 2 (µ), 0 }) (92) where Ω is chosen to satisfy the conditions given in Theorem 2, χ 1 and χ 2 can be any K function. Clearly, with such design, condition (87) is satisfied with α µ = Ω. 4. Event-Triggered Output-Feedback Control The discussions in Section 3 are based on Assumption 1. In this section, we show how the condition is satisfied for an event-triggered output-feedback control system. We consider the popular class of nonlinear uncertain systems in the output-feedback form Krstić et al. (1995): ẋ i = x i+1 + i (x 1 ), i = 1,..., n 1 (93) ẋ n = u + n (x 1 ) (94) y = x 1 (95) where x := [x 1,..., x n ] T with x i R is the state, y R is the output, u R is the control input, and 1,..., n : R R are uncertain functions. It is assumed that only y is available for feedback. Note that systems in the form of (93) (95) have been widely studied in the literature of nonlinear control. In this section, we propose a nontrivial modification of the conventional design for the implementation of event-based control. The following assumption is made on 1,..., n. Assumption 2. For each i = 1,..., n, there exists a known ψ i K being Lipschitz on compact sets such that for all r R. i (r) ψ i ( r ) (96) Based on the achievements in Section 3, the objective of this section is to design an event-triggered outputfeedback controller for system (93) (95) so that the closed-loop system is in the form of (10) (11) and moreover the conditions for event-triggered control are satisfied. 15

16 4.1. Observer-Based Output-Feedback Controller For convenience of notations, we define y m = y + w (97) where y m represents the sampled value of y and w is the sampling error. In the setting of event-triggered control, only the sampled data of y can be used for feedback control. Specifically, owing to the output-feedback structure, we design a nonlinear observer for system (93) (95) as ξ 1 = ξ 2 + L 2 ξ 1 + ρ 1 (ξ 1 y m ) (98) ξ i = ξ i+1 + L i+1 ξ 1 L i (ξ 2 + L 2 ξ 1 ), 2 i n 1 (99) ξ n = u L n (ξ 2 + L 2 ξ 1 ) (100) where ρ 1 : R R is an odd and strictly decreasing function, and L 2,..., L n are positive constants. In the observer, ξ 1 is an estimate of y, and ξ i is an estimate of x i L i y for 2 i n. For convenience of discussions, we define observation errors: ζ 1 = y ξ 1, (101) ζ i = x i L i y ξ i, i = 2,..., n. (102) Based on the estimation by the observer, we design a nonlinear control law in the following form: e 1 = y (103) e 2 = ξ 2 κ 1 (e 1 ζ 1 ) (104) e i = ξ i κ i 1 (e i 1 ), i = 3,..., n (105) u = κ n (e n ) (106) where κ 1,..., κ n are continuously differentiable, odd, strictly decreasing and radially unbounded functions. Note that e 1 ζ 1 = y ζ 1 = ξ 1. Thus, the control law uses only the state of the observer (ξ 1,..., ξ n ), and is realizable. The desired system structure is shown in Figure 2, which is in accordance with the general structure given by (Åström, 2008, Fig. 2). u plant y trigger ym control law ξ 1,..., ξ n observer observer-based controller Figure 2: The structure of the event-triggered output-feedback control system. Remark 4. Equation (98) of the observer is constructed to estimate y by using the available y m. The function ρ 1 in (98) is used to assign an appropriate, probably nonlinear, gain to the observation error system, more precisely, equation (110), to satisfy the cyclic-small-gain condition. Equations (99) (100) are in the same spirit of the reduced-order observer in Jiang et al. (2001). Slightly differently, we use ξ 1 instead of the unavailable y in (99) (100). 16

17 4.2. ISS Property of the Subsystems In this subsection, we show that there exist ρ 1, L 2,..., L n, κ 1,..., κ n to transform the closed-loop system into a network of ISS subsystems. In particular, we prove that the subsystems with ζ 1, ζ 2, e 1,..., e n as the states are ISS, where ζ 2 = [ζ 2,..., ζ n ] T. For the subsystems, we define the following ISS-Lyapunov function candidates: V ζ1 (ζ 1 ) = ζ 1, (107) V ζ 2 ( ζ 2 ) = ( ζ T 2 P ζ 2 ) 1 2, (108) V ei (e i ) = e i, i = 1,..., n. (109) The matrix P for V ζ 2 is positive definite and is determined later. Based on the result in this subsection, in the following subsection, an ISS cyclic-small-gain method is employed to ultimately transform the closed-loop system into the form of (10) (11) with the conditions for event-triggered control satisfied The ζ 1 -subsystem By taking the derivative of ζ 1, we have where ζ 1 = ρ 1 (ζ 1 + w) + φ 1 (ζ 1, ζ 2, e 1 ) (110) φ 1 (ζ 1, ζ 2, e 1 ) = L 2 ζ 1 + ζ (e 1 ). (111) Under Assumption 2, it can be directly verified that there exists a ψ φ1 K being Lipschitz on compact sets such that φ 1 (ζ 1, ζ 2, e 1 ) ψ φ1 ( [ζ 1, ζ 2, e 1 ] T ). System (110) is in the form of system (B.1), with ζ 1, w and ρ 1 (ζ 1 +w) corresponding to η, ω m+1 and κ, respectively. With Lemma 4 in the Appendix, one can find a continuously differentiable ρ 1 such that for any constant 0 < c < 1, l ζ1 > 0 and any χ ζ 2 ζ 1, χ e 1 ζ 1 K which are Lipschitz on compact sets, the ζ 1 -subsystem is ISS with V ζ1 (ζ 1 ) as an ISS- Lyapunov function, which satisfies V ζ1 (ζ 1 ) max { χ ζ 2 ζ 1 (V ζ 2 ( ζ 2 )), χ e 1 ζ 1 (V e1 (e 1 )), χ w ζ 1 ( w ) } V ζ1 (ζ 1 ) (ρ 1 (ζ 1 + w) + φ 1 (ζ 1, ζ 2, e 1 )) lv ζ1 (ζ 1 ) (112) for almost all ζ 1, where χ w ζ 1 (s) = s c (113) for s R The ζ 2 -subsystem By taking the derivative of ζ 2, we can write the ζ 2 -subsystem as ζ 2 = A ζ 2 + φ 2 (ζ 1, e 1 ) (114) where L 2 φ 2 (ζ 1, e 1 ) = I n 1 A =. L n 1 L n 0 0 φ i2 (ζ 1, e 1 ). φ n (ζ 1, e 1 ) (115) (116) 17

18 with φ i (ζ 1, e 1 ) = (L i+1 L i L 2 )ζ 1 L i 1 (e 1 ) + i (e 1 ), i = 2,..., n 1 (117) φ n (ζ 1, e 1 ) = L n L 2 ζ 1 L n 1 (e 1 ) + n (e 1 ). (118) By choosing L 2,..., L n such that A is Hurwitz, there exists a positive definite matrix P = P T R (n 1) (n 1) satisfying PA + A T P = 2I n 1. Define V 0 ζ ( ζ 2 ) = ζ T 2 2 P ζ 2. Then, there exist α 0 ζ, α 0 ζ 2 2 K such that α 0 ζ ( ζ 2 ) V 0 ζ ( ζ 2 ) 2 2 α 0 ζ 2 ( ζ 2 ). With direct calculation, we have where ψ ζ 1 φ 2 and ψ e 1 φ 2 are K functions such that V 0 ζ 2 ( ζ 2 ) ζ 2 = 2 ζ T 2 ζ ζ T 2 P i φ 2 (ζ 1, e 1 ) ζ 2 T ζ 2 + P 2 φ 2 (ζ 1, e 1 ) 2 1 ( λ max (P) V0 ζ 2 ( ζ 2 ) + P 2 ) φ ( ζ 1 ) + ψ e 1 2 φ ( e 1 ) 2, (119) ψ ζ 1 φ 2 (ζ 1, e 1 ) 2 ψ ζ 1 φ 2 ( ζ 1 ) + ψ e 1 φ 2 ( e 1 ) (120) for all ζ 1, e 1 R. Moreover, under Assumption 2, ψ ζ 1 φ and ψ e 1 2 φ can be written as ψ ζ 1 2 φ 2 = ψ ζ 1 φ (s 2 ) and ψ e 1 2 φ 2 = ψ e 1 φ (s 2 ) for 2 s R +, with ψ ζ 1 φ and ψ e 1 2 φ being Lipschitz on compact sets. 2 This means that the ζ 2 -subsystem is ISS with V 0 ζ as an ISS-Lyapunov function. The ISS gains can be chosen as 2 follows. Define χ ζ 1 ζ = 3λ max (P) P 2 ψ ζ 1 2 φ and χ e 1 2 ζ = 3λ max (P) P 2 ψ e 1 2 φ. Then, 2 V 0 ζ 2 ( ζ 2 ) max { } χ ζ 1 ζ (V ζ1 (ζ 1 )), χ e 1 2 ζ (V e1 (e 1 )) 2 V 0 ζ 2 ( ζ 2 ) ( A ζ 2 + φ 2 (ζ 1, e 1 ) ) l ζ 2 V 0 ζ 2 ( ζ 2 ) (121) where l ζ 2 = 1/3λ max (P i ). Hence, there exist χ ζ 1 ζ, χ e 1 2 ζ K being Lipschitz on compact sets and a continuous, positive definite 2 α ζ 2 such that { } V ζ 2 ( ζ 2 ) max χ ζ 1 ζ (V ζ1 (ζ 1 )), χ e 1 2 ζ (V e1 (e 1 )) 2 V ζ 2 ( ζ 2 ) ( A ζ 2 + φ 2 (ζ 1, e 1 ) ) α ζ 2 (V ζ 2 ( ζ 2 )) a.e. (122) The e i -subsystems (i = 1,..., n) It can also be proved that the (e 1,..., e n )-subsystem can be derived into the form ė 1 = κ 1 (e 1 ζ 1 ) + ϕ 1 (e 1, e 2, ζ 2 ) (123) ė 2 = κ 2 (e 2 ) + ϕ 2 (e 1, e 2, e 3, ζ 1, w) (124) ė i = κ i (e i ) + ϕ i (e 1, e 2,..., e i+1, ζ 1 ), i = 3,..., n. (125) For convenience of notations, we denote ė 1 = h 1 (e 1, e 2, ζ 1, ζ 2 ), ė 2 = h 2 (e 1, e 2, e 3, ζ 1, w) and ė i = h i (e 1,..., e i+1, ζ 1 ) for i = 3,..., n. Moreover, under Assumption 2, there exist ψ ϕ1,..., ψ φn K being Lipschitz on compact sets such that ϕ 1 (e 1, e 2, ζ 2 ) ψ ϕ1 ( [e 1, e 2, ζ 2 ] T ) (126) ϕ 2 (e 1, e 2, e 3, ζ 1, w) ψ ϕ2 ( [e 1, e 2, e 3, ζ 1, w] T ) (127) ϕ i (e 1, e 2,..., e i, ζ 1 ) ψ ϕi ( [e 1, e 2,..., e i, ζ 1 ] T ), i = 3,..., n. (128) 18