On event-triggered control for integral input-to-state stable systems

Systems & Control Letters 123 (2019) 24–32

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Systems & Control Letters journal homepage: www.elsevier.com/locate/sysconle

On event-triggered control for integral input-to-state stable systems✩ ∗

Hao Yu a , Fei Hao a , , Xia Chen b a b

The Seventh Research Division, School of Automation Science and Electrical Engineering, Beihang University, Beijing, 100191, China School of Automation Engineering, Qingdao University of Technology, Qingdao, 266520, China

article

info

Article history: Received 9 January 2018 Received in revised form 12 October 2018 Accepted 31 October 2018 Available online xxxx Keywords: Event-triggered control Nonlinear systems Integral input-to-state stability Model-based control

a b s t r a c t In this paper, the stabilization of nonlinear systems by means of event-triggered control is studied. Two kinds of triggering conditions are proposed for a class of systems that only satisfy the integral inputto-state stability with respect to measurement errors. A constant threshold of measurement errors is involved in the first kind of triggering conditions to ensure the ultimately bounded stability, while the second type of triggering conditions is constructed by using a decreasing extra signal as the threshold. The conditions on the extra signal are proposed to guarantee asymptotic stability and exclude Zeno behavior. Moreover, the relationship, between the proposed results and those based on the input-to-state stability, is discussed. Finally, numerical examples are provided to illustrate the efficiency and feasibility of the obtained results. © 2018 Elsevier B.V. All rights reserved.

1. Introduction The traditional schemes of sampled-data control systems are based on periodic sampling paradigms, i.e., the sampling and updating of signals are executed according to an isometric elapse of time. This may cause a waste of communication resources if the measured signals experience no significant change. As an alternative, event-triggered control has gained more and more attention due to the advantages of reducing transmission rates. In eventtriggered control, the signals are updated only when an event, depending on states or outputs of the systems, occurs (see [1–3] and the references therein). In this way, event-triggered control could be seen as a closed-loop sampling strategy [4] and has the potential to execute the control tasks only when they are needed. Most existing studies on event-triggered control are based on the concept of input-to-state stability (ISS stability) [5]. In detail, they required that the closed-loop system is ISS stable with respect to some measurement errors. Generally, the measurement errors are defined as the deviation between the current state/output and the lastest sampled one. From this concept, several kinds of triggering conditions were proposed to guarantee the stability of systems with state feedbacks [1,6–8], output feedbacks [9–13], disturbances [2,14–17], and so on. Especially, a number of studies on event-triggered linear (multi-agent) systems (see, e.g., [18,19]) ✩ This work was supported by National Nature Science Foundation of China under Grant 61573036, 61174057, and 61703225, and the Academic Excellence Foundation of BUAA for Ph.D. Students. ∗ Corresponding author. E-mail addresses: [email protected] (H. Yu), [email protected] (F. Hao), [email protected] (X. Chen). https://doi.org/10.1016/j.sysconle.2018.10.013 0167-6911/© 2018 Elsevier B.V. All rights reserved.

are under the framework of ISS stability, since the internal stability of linear systems always implies the ISS stability with respect to external inputs [20]. However, the assumption of the closedloop systems being ISS stable with respect to measurement errors is somewhat restrictive for nonlinear systems (see [21] for an example of systems even affine in control). In [22], the authors developed an event-triggered control scheme for nonlinear systems without the limit of ISS stability while the required assumption, that the nominal system is exponentially stable, was also restrictive sometimes. In [23,24], the concept of integral input-to-state stability (iISS stability) was proposed. Intuitively, iISS stability is a kind of ‘‘L2 to L∞ stability’’, that is, the state of an iISS stable system is small if its inputs have finite energy defined by a proper integral [25]. Hence, iISS stability is a natural generalization of ISS stability which requires bounded inputs yielding bounded states. In [25], the authors studied the iISS stabilization of nonlinear systems by introducing iISS-control Lyapunov functions. More importantly, they provided an example of systems that are iISS stabilizable but not ISS stabilizable. Thus, iISS stability is a weaker concept [26] than that of ISS stability from the considerations both of analysis and design. Based on the above observations, iISS stability is available to a wider range of nonlinear systems. Naturally, by investigating the event-triggered control for iISS stable systems, one can expect to improve the applicability and practicability of event-triggered control strategies, which motivates this study. The main contributions of this paper are summarized as follows. First, two kinds of triggering conditions are proposed for the nonlinear systems that only satisfy the iISS stability with respect to measurement errors. A constant threshold of measurement errors

H. Yu, F. Hao and X. Chen / Systems & Control Letters 123 (2019) 24–32

25

is involved in the first kind of triggering conditions to ensure the ultimately bounded stability, while the second type of triggering conditions is constructed by using a decreasing extra signal as thresholds. The conditions on the extra signal are proposed to guarantee the asymptotic stability of systems and exclude Zeno behavior.1 Then, the existence of the designed triggering conditions is discussed in some special situations. Note that the considered triggering conditions are not new. In fact, [3] provided a uniform framework for the analysis on some popular event-triggered control mechanisms, including the ones introduced above. However, [3] mainly focused on the ISS stable systems, and thus, the results cannot be directly extended to those in our paper. Second, the relationships between the proposed results and those based on ISS stability are discussed. On one hand, different from the results based on ISS stability, both of the triggering conditions in general exist only in a semi-global sense. In other words, it may fail to construct a common triggering condition for all initial states such that the corresponding systems are stable. On the other hand, by some further assumptions, the obtained results can reduce into special cases of those for the ISS stable systems in [11]. Thus, the proposed results could be seen as an extension of some existing ones while capturing the particular characteristics of iISS stable systems. In [28], a relative event-triggered control scheme with minimally attentive property was considered, which can be properly extended to the case of iISS stable systems. However, the stability analysis in [28] was still based on the framework of ISS stability, see, [28, Proposition 4.2], and hence, the particular feature on semi-global existence of the triggering conditions for iISS stable systems was not revealed in the existing work. Third, one example with a convergent measurement error (sampling period) yielding divergent states is constructed for the iISS stable systems. The example shows that for some iISS stable systems, even though the utilization of communication resources tends to infinity, an improper schedule of sampling instants would still completely destroy the stability of systems. This is mainly because of the property that only part of convergent signals could drive the iISS stable systems keeping bounded [29]. This phenomenon may rarely occur for the ISS stable systems. Hence, the example illustrates that when only assuming iISS stability, eventtriggered control is necessary to some degree since the system is easier to lose the stability due to an improper schedule of samplings. The remainder of this paper is organized as follows. After the necessary notations are introduced, the event-triggered control systems under the assumption of the iISS stability with respect to measurement errors are formulated in Section 2. The main results of this paper are included in Section 3. In Section 4, simulations are provided to illustrate the efficiency and feasibility of the proposed results. Finally, the conclusions of this paper are drawn in Section 5. All the proofs are provided in Appendix.

definite function γ : R≥0 → R≥0 is one such that γ (0) = 0 and γ (s) > 0 for all s > 0. The inverse function of an invertible function α is denoted by α −1 . A function f : Rn → Rm is Lipschitz continuous on compacts if for every compact set S ⊂ Rn there exists a constant L > 0 such that ∥f (x) − f (y)∥ ≤ L ∥x − y∥ for every x, y ∈ S and L is called (local) Lipschitz constant. Given a piecewise continuous function x : R → Rn and a constant t ∈ R, define x(t − ) := lims↑t x(t). sign(·) denotes the sign function.

Notation. Let R (R≥0 ) be the set of real (nonnegative) numbers and Z≥0 denote the set of all the nonnegative integers. ∥·∥ denotes Euclidian norm of a vector while |·| denotes the absolute value of a scalar. A shorthand notation for (xT1 , xT2 )T is denoted by (x1 , x2 ). SN is the class of continuous derivative, nondecreasing functions from R≥0 to R≥0 . A function γ from R≥0 to R≥0 is said to be of class K (denoted by γ ∈ K) if it is continuous, strictly increasing and satisfy γ (0) = 0. If γ is of class K and γ (s) → ∞ as s → ∞, then γ belongs to class K∞ (denoted by γ ∈ K∞ ). A continuous function β : R≥0 × R≥0 → R≥0 is of class KL (denoted by β ∈ KL) if it satisfies: (i) for each t ≥ 0, β (·, t) is of class K, and (ii) for each s ≥ 0, β (s, ·) is nonincreasing and limt →∞ β (s, t) = 0. A positive

DV (x)f¯ (x, e) ≤ −α3 (∥x∥) + σ (∥e∥),

1 Zeno behavior denotes the phenomenon that an infinite number of samplings occur in a finite time interval, and see, e.g., [27] for more details on Zeno behavior in event-triggered control.

2. Problem formulation Consider the following nonlinear plant, x˙ (t) = f (x(t), u(t)), x(0) = x0 , n

(1)

n

where x ∈ R and x0 ∈ R are the plant’s state vector and the initial state, respectively. u ∈ Rm is the control input. f : Rn × Rm → Rn is Lipschitz continuous on compacts and satisfies f (0, 0) = 0. Now consider the systems with limited communication resources, and then, the controller is designed in a model-based manner [30], i.e., u(t) = k(xm (t)),

(2)

where the gain function k : Rn → Rm will be clarified latter. The model state xm ∈ Rn is generated by the following model: x˙ m (t) = fˆ (xm (t)), t ∈ [tk , tk+1 ); xm (tk ) = x(tk ),

(3)

where the function fˆ : Rn → Rn is Lipschitz continuous on compacts and the monotonically increasing sequence {tk }∞ k=0 denotes the triggering times decided by triggering conditions. Define e(t) := xm (t) − x(t), t ∈ [tk , tk+1 ) as the measurement error. Then the closed-loop system for the plant (1) under the controller (2) has the following form: x˙ (t) = f (x(t), k(x(t) + e(t))) =: f¯ (x(t), e(t)).

(4)

The controller gain function k is Lipschitz continuous on compacts and it is assumed to be designed rendering the closedloop system (4) iISS stable with respect to the measurement error e ∈ Rn . Due to the Lipschitz continuity of functions f and k, f¯ is also Lipschitz continuous on compacts. Thus, according to [24], if the closed-loop system (4) is iISS stable, then it admits an iISSLyapunov function in the following sense.2 Definition 1 ([24]). A continuously differentiable function V : Rn → R≥0 is called an iISS-Lyapunov function for the system (4) if there exist functions α1 , α2 , σ ∈ K∞ , and a positive definite function α3 such that

α1 (∥x∥) ≤ V (x) ≤ α2 (∥x∥),

(5)

n

for all x ∈ R and (6)

for all x, e ∈ Rn . Note that if α3 is also of class K∞ , then V is an ISS-Lyapunov function as well. And in this case, the closed loop system (4) is ISS stable with respect to e (see, e.g., [5,32] for details). Remark 1. The model-based holder manner during two successive triggering instants is not a necessary condition for our results, and thus, there is almost no restriction on the selection of fˆ except the Lipschitz continuity. Especially, in the case of fˆ = 0, the controller (2) would reduce to the common zero-order-holder controller 2 The definition of iISS stability is not explicitly needed in this paper and one can see, e.g., [24,31] for an introduction of related notions.

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H. Yu, F. Hao and X. Chen / Systems & Control Letters 123 (2019) 24–32

u(t) = k(x(tk )), t ∈ [tk , tk+1 ). The reason for introducing (2) and (3) is to increase the designing freedom in Simulation section. This helps us to observe some particular and valuable phenomena for event-triggered iISS stable systems. Furthermore, by utilizing proper fˆ , one could expect to obtain some more less conservative results (see, e.g., [33]). Note that the results in [33] may only apply for linear systems, which are ISS stable as introduced before. Hence, this issue for iISS stable systems needs some further study in the future. Then for clarity, we will introduce the following definitions of ultimately bounded stabilization and asymptotic stabilization [20]. Definition 2. The system (4) is said to be semi-globally ultimately boundedly stabilized if for any compact set S0 ⊂ Rn , there exists a triggering condition such that, for any initial state x0 ∈ S0 , the corresponding solution x(t) satisfies

satisfying V (x) ≤ α1 (r0 ). For such x, we have ∥x∥ ≤ r0 and from Definition 1, it holds that DV (x)f¯ (x, e) ≤ − αr0 (∥x∥) + σ (∥e∥) ≤ −α¯ r0 (V (x)) + σ (∥e∥), (8) where α¯ r0 := αr0 ◦ α2

−1

∈ K. Let

cr0 ∈ [0, k0 (α¯ r0 ◦ α1 (r0 ))]

(9)

with k0 ∈ (0, 1). Then based on the above definitions, we have the following lemma. Lemma 2. For a given r0 > 0, suppose that the initial state of the system (4) satisfies V (x0 ) ≤ α1 (r0 ) and the corresponding solution exists in the interval [0, T¯ ) with T¯ > 0. If σ (∥e(t)∥) ≤ cr0 , t ∈ [0, T¯ ) with cr0 defined in (9), then it leads to cr0

∥x(t)∥ ≤ max{β (∥x0 ∥ , t), b}, t ≥ 0,

V (x(t)) ≤ max{βr0 (V (x0 ), t), α¯ r−0 1 (

with some function β ∈ KL and constant b > 0.

where βr0 ∈ KL satisfies βr0 (s, 0) = s, s ≥ 0, and is generated by applying Lemma 4.4 in [34] with some k0 ∈ (0, 1) to the following auxiliary differential equation,

Definition 3. The system (4) is said to be semi-globally asymptotically stabilized if for any compact set S0 ⊂ Rn , there exists a triggering condition such that, for any initial state x0 ∈ S0 , the corresponding solution x(t) satisfies

∥x(t)∥ ≤ β (∥x0 ∥ , t), t ≥ 0, with some function β ∈ KL. Moreover, if the inequality above holds for any x0 ∈ Rn , the system is globally asymptotically stabilized. Therefore, the main purpose of our paper is designing triggering conditions to schedule the triggering time sequence {tk }∞ k=0 such that the plant (1) with the controller (2) is ultimately boundedly stabilized or asymptotically stabilized, and the triggering time sequence {tk }∞ k=0 does not exhibit Zeno behavior. Moreover, we would like to show the relationship between the proposed results and those obtained under the assumption that the system (4) is ISS stable with respect to e (i.e., α3 ∈ K∞ in Definition 1). Before ending this section, we introduce the following lemma used in the analysis of the main results. Lemma 1 ([20]). For any positive definite function α : R≥0 → R≥0 and scalar r0 > 0, there always exists α¯ ∈ K such that α¯ (s) ≤ α (s) for all s ∈ [0, r0 ]. 3. Main results In this section, we provide the main results of this paper. First, an absolute triggering condition is designed to guarantee the ultimately bounded stability of the system (4). Then, to obtain the asymptotic stability, a triggering condition with decreasing threshold signal is designed. Finally, the existence of the proposed triggering conditions is discussed with respect to different cases of functions in Definition 1. 3.1. Ultimately bounded stabilization To guarantee the ultimately bounded stability, we introduce the following absolute triggering condition, tk+1 = inf{t > tk |σ (∥e(t)∥) > c0 },

(7)

where σ is defined in Definition 1 and c0 > 0 is to be designed. For a given r0 > 0, from Lemma 1, there exists αr0 ∈ K such that αr0 (s) ≤ α3 (s), s ∈ [0, r0 ]. Then, consider an arbitrary state x ∈ Rn

k0

)}, t ∈ [0, T¯ )

w ˙ (t) = −α¯ r0 (w(t)) + cr0 ,

(10)

(11)

i.e., it holds that w (t) ≤ max{βr0 (w (0), t), α¯

−1 cr0 r0 ( k0

)}, t ∈ [0, ∞).

Lemma 2 shows that for any initial state of the system (4), if the threshold of the measurement error e is sufficiently small, one can ensure the boundedness of the state. Thus, by relating c0 with x0 , we provide the main result of this subsection. Theorem 3. Consider the event-triggered control system (4) with the triggering condition (7). If the parameter c0 in (7) is selected as c0 ≤ k0 (α¯ r0 ◦ α1 (r0 )) with k0 ∈ (0, 1) and r0 ≥ α1−1 ◦ α2 (∥x0 ∥), then there is no Zeno behavior and the system (4) is semi-globally ultimately boundedly stabilized, i.e., its state satisfies c0 ∥x(t)∥ ≤ max{β¯ r0 (∥x0 ∥ , t), α1−1 (α¯ r−0 1 ( ))}, t ≥ 0, k0 with β¯ r0 (s, t) := α1−1 (βr0 (α2 (s), t)) and βr0 defined in (10). Moreover, a positive lower bound τa of inter-event times can be given by

τa =

1 L2

ln(1 +

L2 σ −1 (c0 ) L1 r0

),

where L1 , L2 > 0 are, respectively, the local Lipschitz constants of fa (x, e) := fˆ (x + e) − f¯ (x, e) with respect to x and e in the compact set {z ∈ Rn |∥z ∥ ≤ M(x0 , c0 )} with √ c0 M(x0 , c0 ) := 5 max{β¯ r0 (∥x0 ∥ , 0), α1−1 (α¯ r−0 1 ( ))}. k0 Note that the selections of α¯ r0 depend on r0 and α3 . If α3 ∈ / K, although α¯ r0 is of class K, the value of α¯ r0 ◦ α1 (r0 ) may be not monotonically increasing with respect to r0 . Especially, in the case of lims→∞ α3 (s) = 0, the value would approach to zero as r0 goes to infinity. Hence, for the iISS stable system (4), the thresholds in absolute triggering conditions in general cannot be selected arbitrarily. However, if α3 ∈ K∞ , i.e., the system (4) is ISS stable, then for any c0 > 0, there always exists an r0 such that the conditions of Theorem 3 hold. The analysis above shows that, only with the assumption of iISS stability, the existence of the absolute triggering conditions is in a semi-global sense. On one hand, for any initial state x0 ∈ Rn , one can select a c0 > 0 in (7) such that the corresponding system is ultimately bounded. On the other hand, it may fail to provide

H. Yu, F. Hao and X. Chen / Systems & Control Letters 123 (2019) 24–32

a c0 that could guarantee the stability of the system (4) with any x0 ∈ Rn . The latter, however, could be trivially achieved with any c0 > 0 if α3 ∈ K∞ . This agrees with some existing results in, e.g., [2,9] that focused on the ISS stable systems.

27

Define V1 (x) := ρ (V (x)) with ρ (s) := σ −1 ( 21 α∆ (s)) ∈ K. Then for any V1 (x) ∈ [0, ρ (α1 (∆1 ))] and η ≤ ∆2 , the above inequality leads to V1 (x) ≥ η ⇒ DV1 (x)f¯ (x, e) ≤ −αs (V1 (x)),

3.2. Asymptotic stabilization

where the function αs (r) := 21 ρ˙ (ρ −1 (r)) · α∆ (ρ −1 (r)) is positive definite. Then we further propose the following conditions.

Consider the following triggering condition with a decreasing threshold:

Condition 2. There exists a positive scalar ∆3 ≤ ρ (α1 (∆1 )) such that Ω (r) ≤ αs (r), r ∈ [0, ∆3 ].

tk+1 = inf{t > tk |∥e(t)∥ > η(t)},

(12)

with the threshold variable η generated by

η˙ (t) = −Ω (η(t)), η(0) = η0 ,

(13)

where the positive scalar η0 and the positive definite function Ω are to be designed. Moreover, we require that Ω is Lipschitz continuous on compacts. Thus, it holds that the η-system is asymptotically stable and η(t) > 0, t ∈ [0, ∞). Since the triggering condition (12) introduces an auxiliary dynamical system (13), in the sequel of this subsection, we consider the stabilization of the overall ξ -system with ξ := (x, η). And we would like to show that the triggering condition (12) has the potential to stabilize asymptotically the ξ -system. Different from the ISS stable systems, the boundedness of states may not be guaranteed when the iISS stable system is cascaded by an arbitrary asymptotically stable system. Thus, referring to [35], we introduce the following conditions. Condition 1. σ (s) = O(Ω (s)) in a neighborhood of zero, i.e., there exists a nonnegative constant l such that lim sups→0 σ (s)/Ω (s) ≤ l. If Condition 1 holds, according to Proposition 11 in [35], we have that there exists a function p ∈ K∞ such that the positive variable ηˆ (t) := p(η(t)) satisfies

η˙ˆ (t) ≤ −2σ (η(t)), η(t) is defined in (13) and p has the form of p(s) = ∫where s q(τ )dτ , s ≥ 0 with some function q ∈ SN . Then we propose 0 the following lemma for the considered event-triggered control systems.

Lemma 4. Suppose that Condition 1 holds for the event-triggered control system (4) under the triggering condition (12). Then the corresponding solution of the system (4) is forward completed in t direction.

Condition 3. There exists a positive scalar ∆4 ≤ ∆3 such that is nondecreasing for s ∈ (0, ∆4 ].

Ω (s) s

Condition 4. The function (ρ ◦ α1 )−1 is Lipschitz continuous on compacts. Based on the above conditions, we provide the main result of this subsection. Theorem 5. Suppose that Conditions 1–4 hold for the event-triggered control system (4) with the triggering condition (12). Then for any given η0 > 0, the ξ -system is globally asymptotically stabilized and there is a positive lower bound of inter-event times. It can be observed that Conditions 1 and 2 have opposite effects on the selection of Ω in (12), that is, Condition 1 expects a large Ω while a small one is preferred for Condition 2. Actually, these two conditions hold only if σ (s) = O(αs (s)) in a neighborhood of zero. This requirement highly depends on the property of the iISS-Lyapunov function in Definition 1, which may fail for some systems. Hence, in some cases, it would be hard to find a proper Ω such that the conditions in Theorem 5 hold. To deal with this issue, an improved version of Theorem 5 is provided, based on which the existence of Ω is discussed in some special situations. Since Condition 2 only needs to be satisfied in a small range containing the origin, we propose a local assumption on the dynamic of the system (4).

˜ 1 > 0, suppose that if Assumption 1. For a positive constant ∆ ˜ 1 }, then there exist a continuously x ∈ C := {z ∈ Rn |∥z ∥ ≤ ∆ differentiable function V˜ : Rn → R≥0 , functions α˜ 1 , α˜ 2 , σ˜ ∈ K∞ and α˜ 3 ∈ K satisfying α˜ 1 (∥x∥) ≤ V˜ (x) ≤ α˜ 2 (∥x∥), for all x ∈ C and DV˜ (x)f¯ (x, e) ≤ −α˜ 3 (∥x∥) + σ˜ (∥e∥),

Since the solution of (4) exists for any t ∈ [0, ∞), Lemma 4 excludes Zeno behavior from the event-triggered control systems described by (4) and (12). However, this lemma does not give a positive lower bound of all inter-event times. In fact, the lower bound δTˆ in (27) would approach zero when Tˆ goes to infinity. Thus, to provide the positive lower bound of inter-event times, some more conditions are required. For a given ∆1 > 0, according to Lemma 1, there exists αˆ ∆ ∈ K such that α3 (s) ≥ αˆ ∆ (s) for s ∈ [0, ∆1 ]. As a result, when ∥x∥ ∈ [0, ∆1 ] and ∥e∥ ≤ η, it holds that

Assumption 1 is not strong since it always hold with the functions in Definition 1. However, when the state in (4) is small, its dynamic may exhibit some good behaviors that are not captured by Definition 1. Thus, different functions from those in Definition 1 could be selected such that Assumption 1 holds. Based on this fact, one can expect to improve the design freedom of Ω . Parallel to the analysis and proof of Theorem 5, one can obtain the following corollary from Assumption 1.

DV (x)f¯ (x, e) ≤ − αˆ ∆ (∥x∥) + σ (η) ≤ −α∆ (V (x)) + σ (η),

Corollary 6. Suppose that Assumption 1 holds for the event-triggered control system (4) with the triggering condition (12). If the following statements hold:

(14)

where α∆ := αˆ ∆ ◦ α2−1 ∈ K. Thus from Definition 1, for any x ∈ Rn satisfying V (x) ≤ α1 (∆1 ), we have that (14) holds and α∆ (V (x)) ≤ α∆ ◦ α1 (∆1 ) =: ∆s . Then let ∆2 ∈ (0, σ −1 ( 21 ∆s )). According to (14), it holds that when V (x) ≤ α1 (∆1 ) and η ≤ ∆2 , 1 −1 V (x) ≥ α∆ (2σ (η)) ⇒ DV (x)f¯ (x, e) ≤ − α∆ (V (x)). 2

for all x ∈ C and e ∈ Rn .

(1) Condition 1 holds; ˜ 3 ≤ ρ˜ (α˜ 1 (∆ ˜ 1 )) such that Ω (r) ≤ (2) there exists a positive scalar ∆ ˜ 3 ]; α˜ s (r), r ∈ [0, ∆ ˜4 ≤ ∆ ˜ 3 such that Ω (s) is (3) there exists a positive scalar ∆ s ˜ 4 ]; nondecreasing for s ∈ (0, ∆ (4) the function (ρ˜ ◦ α˜ 1 )−1 is Lipschitz continuous on compacts,

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H. Yu, F. Hao and X. Chen / Systems & Control Letters 123 (2019) 24–32

where

α˜ ∆ := α˜ 3 ◦ α˜ 2−1 , 1

ρ˜ := σ˜ −1 ◦ ( α˜ ∆ ),

2 1 ˙ α˜ s := [ρ˜ ◦ ρ˜ −1 ] · [α˜ ∆ ◦ ρ˜ −1 ]. 2 Then for any given η0 > 0, the ξ -system is globally asymptotically stabilized and there is a positive lower bound of inter-event times. Next, based on Corollary 6, we provide the results on the existence of Ω in the following two cases: (i) the function α3 is of class K in Definition 1; (ii) the value of η0 is selected in some small regions around zero. Proposition 1. Consider the event-triggered control system (4) with the triggering condition (12). If α3 ∈ K and Assumption 1 holds, then Condition 1 could be removed from Corollary 6. Note that one can always find Ω (s) such that the item (2) in Corollary 6 holds. Consequently, Proposition 1 implies that if α3 ∈ K, the conditions in Corollary 6 are the same as those in Theorem 2 of [11] (in the state-feedback case), which is derived for the ISS stable systems. Hence, in this case, there is no essential difference between the results for the ISS stable and the iISS stable systems. This is mainly because both kinds of systems have the same cascade properties, that is, when they are driven by any asymptomatically stable systems, the corresponding states would converge to the origin, as showed in [20,36]. Proposition 2. Suppose that Assumption 1 holds for the eventtriggered control system (4) with the triggering condition (12). If η0 in (13) satisfies σ (η0 ) ≤ c0 where c0 is defined in Theorem 3, then the conclusion in Proposition 1 holds and the ξ -system is semi-globally asymptotically stabilized. On one hand, Proposition 2 shows that by selecting a sufficiently small η0 , one can improve the existence of Ω . If we always assume the last two requirements in Corollary 6, then the analysis on Theorem 3 implies that the existence of the triggering condition (12) is generally in a semi-global sense as well. On the other hand, to achieve the global existence of triggering conditions, as what was done for ISS stable systems, more conditions are required in Theorem 5 and Corollary 6. For better property, it is natural to require some more restrictive conditions, although they may fail to meet for some applications. Therefore, compared to the results for ISS stable systems, the assumption of iISS stability would degrade the existence of triggering conditions from a global to a semi-global sense. In other words, the proposed results can apply for a wider range of systems at the cost of requiring some information of initial state.

Fig. 1. Simulation results of the system with x0 = 2 and c0 = 0.3. Table 1 Boundedness of systems with different cases of x0 and c0 . x0 c0 Boundedness

2 0.3

2 1.2

√

√

2 2

2 4

×

×

1 1.2

√

4 1.2

6 1.2

×

×

Thus, by choosing V (x) = 12 ln(1 + x2 ), one can show that (16) is iISS stable with respect to e with α1 (s) = α2 (s) = 12 ln(1 + s2 ), α3 (s) = s , and σ (s) = 2s in Definition 1. Note that lims→∞ α3 (s) = 0 1+s2 and the system (16) cannot be ISS stable with respect to e as shown in [23]. 4.1. Ultimately bounded stability

Consider the following scalar nonlinear plant P1 which is transformed from Example 1 in [29],

First, we provide simulation results about the absolute triggering condition (7). Consider the initial state x0 = 2. Then by calculations in Theorem 3, one can select c0 = 0.3 in (7). Fig. 1 shows the state trajectories and the evolution of inter-event times for the system with x0 = 2 and c0 = 0.3. It can be observed that the state is ultimately bounded and no Zeno behavior occurs since the inter-event times are not less than 0.0304 s. Also, Fig. 1 shows that the inter-event times would increase during the stage of convergency. This is reasonable since with the state decreasing, the measurement error e would change slowly while the threshold keeps constant. Moreover, in Table 1, we provide the simulation results for different cases of x0 and c0 . From this table, one can observe the semi-global existence of triggering conditions for the considered example. On one hand, for a given initial state x0 , a sufficiently small c0 is admitted to ensure the ultimate boundedness, while the stability would be destroyed if c0 is larger than some constant. On the other hand, even if the considered c0 could stabilize the system with some x0 , it may fail to guarantee the boundedness as the initial state increases.

P1 : x˙ (t) = −sat(x(t)) + (u(t) + x(t))(1 + x(t)),

4.2. Asymptotic stability

4. Simulations

(15)

where sat(x) := sgn(x) min{1, |x|}. Then controller gain function in (2) can be selected as k(s) = −s, which renders the closed-loop system (4) as

where e(t) := xm (t) − x(t), t ∈ [tk , tk+1 ) with the model state xm (t) generated by

In this subsection we construct the triggering condition (12) to stabilize asymptotically the plant in P1 . We mainly utilize Corollary 6 since the functions in Definition 1 would lead to quite complex αs and ρ in Theorem 5. Due to σ (s) = 2s, any Ω of linear form could satisfy Condition 1. When |x| ≤ 1, the dynamic of the closed-loop system (16) would be covered by the following uncertain linear system:

x˙ m (t) = −3xm (t), t ∈ [tk , tk+1 ); xm (tk ) = x(tk ).

x˙ (t) = −x(t) − l(t)e(t),

x˙ (t) = −sat(x(t)) − e(t)(1 + x(t)),

(16)

(17)

H. Yu, F. Hao and X. Chen / Systems & Control Letters 123 (2019) 24–32

Fig. 2. Simulation results of the system with Ω (s) =

s , 4

x0 = 6, and η0 = 0.6.

where l(t) ∈ [−2, 2] denotes the uncertainty. Clearly, the system in (17) is ISS stable with respect to e, and hence, one can select V˜ (x) =√ 21 x2 in Assumption 1. By simple calculations, we have √

ρ˜ (r) = 2 , α˜ s (r) = 4r and ρ˜ ◦ α˜ 1 (r) = 42r . Therefore, according to Corollary 6, Ω (s) = 4s in (12) could stabilize asymptotically the closed-loop system with any η0 and x0 . Clearly, one has η(t) = η0 e−t /4 in this case. Fig. 2 provides the simulation results for the system (16) under the triggering condition (12) with η0 = 0.6 and x0 = 6. It is showed that the state converges to zero and the inter-event times are lower bounded by 0.004 s. Recall that the absolute triggering condition (7) with c0 = σ (η0 ) = 1.2 fails to stabilize the system when x0 = 6. Moreover, from Fig. 2, one has that the inter-event times would increase with the state approaching the origin. This phenomenon shall be due to the decrease of the local Lipschitz constants in Theorem 3. To illustrate the effects of Condition 1 on stability, in Fig. 3 we further provide the simulation results for√the triggering conr

3

dition (12) with Ω (s) = s4 , i.e., η(t) = ( 2t + η0−2 )−1 . Note that such an Ω could meet all the requirements in Corollary 6 except for Condition 1. From Fig. 3, the corresponding system is not stable obviously. More importantly, it can be observed that although the inter-event time (as well as the measurement error) goes to zero with the system operating, the state of the closed-loop system would still diverge to infinity. Empirically, the behavior of sampled-data systems with sufficiently small sampling periods (or measure errors) should be close to the one of systems with continuous feedback controllers. Hence, the results in Fig. 3 are quite interesting considering that the plant (15) could be stabilized by the continuous controller u = −x. Comparing Fig. 2 (with 2347 events) and Fig. 3 (with 13 761 events), one has that for some systems, even though the utilization of communication resources tends to infinity, an improper schedule of sampling instants would still completely destroy the stability. This phenomenon may rarely occur for an ISS stable system, since the state of the system driven by bounded measurement errors is always bounded. In fact, one reason of the results in Fig. 3 is that, the following system in cascade x˙ (t) = −sat(x(t)) − z(t)x(t), z˙ (t) = −z 3 (t),

is not globally stable as showed in [29]. According to the analysis above, the iISS stable systems have a stronger demand to relate sampling behaviors with plant states than that of the ISS stable systems. Clearly, this demand is hard to

29

Fig. 3. Simulation results of the system with Ω (s) =

s3 4

, x0 = 6, and η0 = 0.6.

be fulfilled by time-triggered control which uses almost no state information for determining the sampling periods. Therefore, for the sampled-data systems only with iISS stability, event-triggered control seems to be a necessary choice to some degree. 4.3. Existence of Ω In this subsection, we will illustrate the applicability of the conditions on Ω . First, an example is provided to showed that, to achieve the asymptotic stability, the conditions in Section 3.2 are more restrictive than those in Section 3.1. Then, it will be showed that, by using Propositions 1–2, one can improve the range of the admitted Ω in (12) compared to those in Theorem 5. Initially, consider the following plant P2 P2 : x˙ (t) = −sat(x3 (t)) + (u(t) + x(t))(1 + x(t)),

(18) 3

where we replace the term sat(x(t)) in (15) by sat(x (t)). The other implementation is the same as that for (15). The iISS stability for P2 can be checked by the iISS-Lyapunov function V (x) = 14 ln(1 + x4 ) with σ (r) = 2r. Then consider the case of |x| ≤ 1 in Assumption 1. Correspondingly, the dynamic of P2 can be covered by x˙ (t) = −x3 (t) − l(t)e(t), where l(t) ∈ [−2, 2] denotes the uncertainty. Let V (x) = 14 x4 in Assumption 1, and then, by definitions the functions in Corollary 6 can be selected as follows: 3

3

5

α˜ ∆ (r) = c1 r 2 ; ρ˜ (r) = c2 r 4 ; α˜ s (r) = c3 r 3 , where ci , i ∈ {1, 2, 3} are some positive constants. To meet the conditions in Corollary 6, one needs to find an Ω that satisfies si5 multaneously lim supr →0 r /Ω (r) ≤ l1 and lim supr →0 Ω (r)/r 3 ≤ l2 for some nonnegative l1 and l2 . Clearly, such an Ω does not exist. However, since the system in (18) is iISS stable and Lipschitz continuous on compacts, it satisfies the hypotheses in Theorem 3. Therefore, the calculations above provide an example where the results in Section 3.1 can be used while those in Section 3.2 cannot. Next, we will illustrate Propositions 1–2 and consider the following plant P3 , P3 : x˙ (t) = −x(t) + (u(t) + x(t))(1 + x(t)),

(19)

where we replace the term sat(x(t)) in (15) by x(t). The other implementation is the same as that for (15). With the iISS-Lyapunov s2 function V (x) = 12 ln(1 + x2 ), we have α3 (s) = 1+ ∈ K in s2

30

H. Yu, F. Hao and X. Chen / Systems & Control Letters 123 (2019) 24–32

with k0 ∈ (0, 1) for all t ∈ [0, ∞). By definitions, it holds that cr V (x0 ) ≤ α1 (r0 ) and α¯ r−0 1 ( k 0 ) ≤ α1 (r0 ). Hence 0

w(t) ≤ α1 (r0 ), t ≥ 0. Then, from (8) and σ (∥e(t)∥) ≤ cr0 , t ∈ [0, T¯ ), the derivative of V (x(t)) along the solution of the system (4) satisfies V˙ (t) ≤ −α¯ r0 (V (x(t))) + cr0 , t ∈ [0, T¯ ), for any t such that V (x(t)) ≤ w (t) ≤ α1 (r0 ). Therefore, by Comparison Principle [20], we have V (t) ≤ w (t), t ∈ [0, T¯ ) since V (x0 ) = w (0).

Fig. 4. Simulation results of the systems with Ω (s) =

s3 4

Proof of Theorem 3. Since r0 ≥ α1−1 ◦ α2 (∥x0 ∥), it holds that V (x0 ) ≤ α1 (r0 ). Suppose that the solution of the system (4) exists in the interval [0, T¯ ). Then according to the triggering condition (7), it holds that σ (∥e(t)∥) ≤ cr0 , t ∈ [0, T¯ ) with cr0 = c0 . Then from Lemma 2, we have V (t) ≤ w (t), t ∈ [0, T¯ ) with w defined in (11). Next, we prove that there is a positive lower bound of interevent times in the interval [0, T¯ ). From the definition of w (t), it follows that

and η0 = 0.6.

3

Definition 1 for (19). Thus, Ω (s) = s4 satisfies Proposition 1 for (19). Moreover, when x0 = 2, Proposition 2 holds for η0 = 0.6 and 3 Ω (s) = s4 . Fig. 4 depicts the state trajectories of systems under the 3

triggering condition (12) with Ω (s) = s4 and η0 = 0.6. It can be observed that both of the states tend to zero, which illustrates the 3 feasibility of Propositions 1–2. Note that Ω (s) = s4 yields a quite slow convergency rate of the threshold variable η(t) as well as the states in Fig. 4. 5. Conclusions This paper has studied the event-triggered control problem for a class of nonlinear systems that only satisfy the iISS stability with respect to measurement errors. Two kinds of triggering conditions were proposed to ensure ultimately bounded stability and asymptotic stability, respectively. The first one was an absolute triggering condition while the second one involved a decreasing extra signal. The requirements of these two kinds of triggering conditions were provided to ensure the stability and exclude Zeno behavior. Different from the results based on the assumption of ISS stability, both the triggering conditions in general exist only in a semi-global sense. In other words, it may fail to construct the considered triggering conditions for all initial states such that the corresponding systems are stable, although for any fixed initial state, the requirements on the existence of triggering conditions are moderate. Moreover, it was showed that, when assuming α3 is of class K in Definition 1, the obtained results could reduce into special cases of those for ISS stable systems. Finally, simulations were provided to illustrate the efficiency and feasibility of the obtained results. Especially, an example with a convergent measurement error (sampling period) yielding divergent states was constructed, which may rarely occur for the ISS stable systems. Therefore the simulations showed that, for a sampled-data system only satisfying iISS stability, event-triggered control was necessary to some degree since its stability was easier to be destroyed by an improper schedule of samplings.

α1 (∥x(t)∥) ≤ V (t) ≤ α1 (r0 ), t ∈ [0, T¯ ),

(21)

which implies that the state (x(t), e(t)) would be bounded in the compact set {(x, e) ∈ R2n |∥(x, e)∥ ≤ M(x0 , c0 )} for all t ∈ [0, T¯ ). Due to the local Lipschitz continuity of f¯ and fˆ , one has that there exist Lipschitz constants L1 , L2 > 0 corresponding to {z ∈ Rn |∥z ∥ ≤ M(x0 , c0 )} such that

    ∥fa (x(t), e(t))∥ = fˆ (x(t) + e(t)) − f¯ (x(t), e(t)) ≤ L1 ∥x(t)∥ + L2 ∥e(t)∥ for all t ∈ [0, T¯ ). Moreover, by the definition of e(t), it follows that   d   ∥e(t)∥ ≤ ∥˙e(t)∥ = fˆ (x + e) − f¯ (x, e) dt

≤ L1 ∥x(t)∥ + L2 ∥e(t)∥ ,

(22)

for all t ∈ (tk , tk+1 ). By integrating the above differential inequality with the initial state e(tk ) = 0, it holds that

∫

t

∥e(t)∥ ≤

eL2 (t −s) L1 ∥x(s)∥ ds, t ∈ (tk , tk+1 ).

tk

Using the facts of ∥x(t)∥ ≤ r0 , t ∈ [0, T¯ ) and e(tk− ) = σ −1 (c0 ), one can obtain that the inter-event times are lower bounded by τa defined in the theorem. Based on the analysis above, the state is bounded uniformly with respect to T¯ from (21) and there is no Zeno behavior. As a result, the solution of the system (4) is forward completed in t direction. Therefore, V (t) ≤ w (t) for all t ∈ [0, ∞), which completes the proof by using (5). Proof of Lemma 4. Suppose the solution of the system (4) exists in the interval [0, Tˆ ) with Tˆ > 0. Note that despite the system (4), the solution η(t) of (13) is always well-defined for any t ∈ [0, ∞). The triggering condition (12) shows that ∥e(t)∥ ≤ η(t) for all t ∈ [0, Tˆ ). Then let V0 (ξ ) := V (x) + ηˆ . Then by definitions, V0 admits two functions αˆ 1 , αˆ 2 ∈ K∞ such that

αˆ 1 (∥ξ ∥) ≤ V0 (ξ ) ≤ αˆ 2 (∥ξ ∥). The derivative of V0 along the solution of the systems in (4) and (13) becomes V˙ 0 (t) ≤ − α3 (∥x(t)∥) + σ (∥e(t)∥) − 2σ (η(t)),

≤ − α3 (∥x(t)∥) − σ (η(t)),

Appendix Proof of Lemma 2. For the auxiliary system (11), if we set w (0) = V (x0 ), it follows that

w(t) ≤ max{βr0 (V (x0 ), t), α¯ r−0 1 (

cr0 k0

)},

(20)

(23)

which implies that the state of the system in (4) is bounded. And specifically, there exists a function α0 ∈ K∞ satisfying

∥x(t)∥ + ∥η(t)∥ ≤ α0 (∥x(0)∥ + ∥η(0)∥), for all t ∈ [0, Tˆ ).

H. Yu, F. Hao and X. Chen / Systems & Control Letters 123 (2019) 24–32 V1 (T∗ ) η(T∗ )

≥ 1. Then with the same procedure in the proof of Theorem

Hence, to prove the lemma, we only need to show that there is no Zeno behavior. Due to the asymptotic stability of (13) and the Lipschitz continuity of Ω , for any fixed η0 > 0, there exists c¯ > 0 such that

2 in [11], one can show that Condition 3 implies

−Ω (η(t)) ≥ −¯c η(t), t ∈ [0, ∞),

for all t ∈ [T∗ , ∞). As a result, it follows that

which leads to

kw η(t) ≥ V1 (t),

η(τ ) ≥ η(t)e

−¯c (τ −t)

,

(24)

for any t ∈ [0, ∞) and τ ∈ [t , ∞). Consider two successive triggering instants tk , tk+1 ∈ [0, Tˆ ). Then from (24), it follows that

η(tk+1 ) ≥ η(tk )e−¯c (tk+1 −tk ) ≥ η(Tˆ )e−¯c (tk+1 −tk ) ,

(25)

where the last inequality uses the fact of η(t) being monotonically decreasing. Moreover, from the boundedness of x(t) and η(t), the triggering condition (12), and the Lipschitz continuity of fa defined in Theorem 3, there exists ∆f > 0 satisfying

∥fa (x(t), e(t))∥ ≤ ∆f , t ∈ [0, Tˆ ).

kˆ w η(t) ≥ V1 (t),

for all t ∈ [T∗ , ∞) with kw := max{1, kˆ w }. By definitions, we have V1 (x) ≥ ρ ◦ α1 (∥x∥). Thus, from Condition 4, there exists a positive constant kˆ v > 0 satisfies

∥x(t)∥ ≤ (ρ ◦ α1 )−1 (V1 (t)) ≤ kˆ v V1 (t) ≤ kv η(t), for all t ∈ [T∗ , ∞) with kv := kˆ v kw . Moreover due to (12) and the Lipschitz continuity of f¯ , there exists kf > 0 such that

  f¯ (x(t), e(t)) ≤ kf (∥x(t)∥ + η(t)) ≤ kf (1 + kv )η(t),

(29)

for t ∈ [T∗ , ∞). Recall that η(t) is monotonically decreasing. Similar to (26), (29) implies

 −  e(t ) ≤ δ tk kf (1 + kv )η(tk ). k+1

Thus, according to (22),

 −  e(t ) ≤ (tk+1 − tk )∆f . k+1

(26)

The triggering condition (12) implies e(tk−+1 ) = η(tk+1 ). And let δ tk := tk+1 − tk denote the kth inter-event time. Then from (24)–



(26) we have that δ tk should satisfy δ tk ec¯ δ tk ≥



η(Tˆ ) . Namely, for any ∆f

tk , tk+1 ∈ [0, Tˆ ), it holds that δ tk ≥ δTˆ > 0 with δTˆ satisfying

δTˆ ec¯ δTˆ =

31

η(Tˆ ) . ∆f

(27)

This implies that for any Tˆ < ∞ there at most exist Tˆ /δTˆ + 1 triggering instants. And thus no Zeno behavior occurs, which completes the proof. Proof of Theorem 5. Since Condition 1 holds, the asymptotic stability can be directly proved from (23) and Lemma 4. In the rest of this proof, we will provide a positive estimation of the minimum inter-event times. First consider the situation that only a finite number of triggering instants occur. Let Tˆ denote the last one in the triggering time sequence. Then similar to the proof of Lemma 4, the lower bound of inter-event times can be given by δTˆ defined in (27). Otherwise, due to the asymptotic stability, there exists a triggering instant T∗ ∈ {tk } such that V (x(t)) ≤ α1 (∆1 ), η(t) ≤ min{∆2 , ∆4 }, V1 (t) ≤ min{∆3 , ∆4 },

(30)

Thus, combining (25) and (30), we have that, for any tk ≥ T∗ , the inter-event time δ tk is lower bounded by δ0 > which satisfies δ0 ec¯ δ0 = k (11+kv ) . Note that all of the lower bounds δTˆ , δT∗ and δ0 are f positive and the above analysis holds for any given η0 > 0, which completes the proof. Proof of Proposition 1. To prove this proposition, we introduce an improved result on ‘‘changing dissipation rate’’ in the spirit of [37,35]. For any d ≥ 0, let Dd := {z ∈ Rn |∥z ∥ ≤ 1d } and especially, D0 = Rn if d = 0. Definition 4. Consider the system (4). Suppose that there exist a constant d ≥ 0, a continuously differentiable function V : Rn → R≥0 , functions α1 , α2 , σ ∈ K∞ and α3 ∈ K satisfying (5) for any x ∈ Rn and satisfying (6) for any x ∈ Dd and e ∈ Rn . Then the pair of functions (σ , α3 ) is defined as a supply pair for the system (4) on Dd . To deal with iISS-Lyapunov function, we extend the results of [37] to the case that α3 ∈ K and that the inequalities hold only on a compact set of x. For any given d ≥ 0, and functions γ ∈ K∞ and α ∈ K, define the constant Dγ α := γ −1 (lims→ 1 α (s)). d

Lemma 7. Assume that there exists a nonnegative scalar d ≥ 0 such that (γ , α ) with α ∈ K is a supply pair for the system (4) on the set γ (s) Dd . Then for any given γ˜0 ∈ K∞ satisfying lim sups→Dγ α γ˜ (s) ≤ l0 0

(28) for all t ∈ [T∗ , ∞). Next let us study the inter-event times for two different cases of tk < T∗ and tk ≥ T∗ . Case I: tk < T∗ . By definition, it holds that tk , tk+1 ≤ T∗ . Then also utilizing the calculations in Lemma 4, we have the lower bound δT∗ of inter-event times in this case, where δT∗ is generated in (27) by replacing Tˆ as T∗ Case II: tk ≥ T∗ . Hence (28) holds for t ∈ [T∗ , ∞). Note that the triggering condition ensures ∥e(t)∥ ≤ η(t) for all t ∈ [0, ∞). According to the analysis before this theorem, we have that the derivative of V1 (x(t)) along the solution of the system in (4) satisfies V1 (x(t)) ≥ η(t) ⇒ V˙ 1 (t) ≤ −αs (V1 (t)) ≤ −Ω (V1 (t)), for t ∈ [T∗ , ∞), where the last inequality utilizes (28) and Condition 2. If η(T∗ ) ≥ V1 (x(T∗ )), then from the Comparison Principle, it holds that V1 (t) ≤ η(t) for all t ∈ [T∗ , ∞). Otherwise, let kˆ w :=

with some l0 ≥ 0 and any scalar cd ∈ (0, Dγ α ), there exists a function α˜ ∈ K such that (γ˜ , α˜ ) is also a supply pair for (4) on Dd with γ˜ ∈ K∞ satisfying γ˜ (r) = γ˜0 (r), r ∈ [0, cd ].

The proof of the lemma can be found in our previous work [38]. Note that when α ∈ K∞ and d = 0, Lemma 7 would deduce the result that if γ (r) = O[γ˜0 (r)] as r → ∞ then γ˜ is admitted to equal γ˜0 on any range because of Dγ α = ∞. This is the same as Theorem 1 in [37]. Now return to the proof of Proposition 1. According to Lemma 1, for any given positive definite Ω , one can find a function σ1 ∈ K σ (1) such that σ1 (s) ≤ Ω (s) for all s ∈ [0, 1]. Let ks := σ1(1) and define

σ˜ 0 (s) =

{

σ1 (s), s ∈ [0, 1); ks σ (s) s ∈ [1, ∞),

which leads to σ˜ 0 ∈ K∞ . Note that Definition 1 holds for all x ∈ Rn (i.e., d = 0) and α3 ∈ K. Thus, Dσ α3 = σ −1 (lims→∞ α3 (s)) > 0 and let

32

H. Yu, F. Hao and X. Chen / Systems & Control Letters 123 (2019) 24–32

lα := min{Dσ α3 , 1}. By definitions, it holds that lim sups→Dσ α σ (l ) 1 max{ σ˜ (lα ) , k− s } 0 α

3

σ (s) σ˜ 0 (s)

< ∞. Thus, from Lemma 7, for σ˜ 0 there exist functions α˜ ∈ K, σ˜ ∈ K∞ , and a positive scalar cd ≥ 0 such that (σ˜ , α˜ ) is also a supply pair for (4) on Rn with σ˜ (s) = σ˜ 0 (s), s ∈ [0, cd ]. Clearly, one has σ˜ (s) = O(Ω (s)) in a neighborhood of zero, since σ1 (s) ≤ Ω (s) for all s ∈ [0, 1]. Therefore, Condition 1 always ≤

holds and the proof is completed. Proof of Proposition 2. Note that the only use of Condition 1 in Corollary 6 is to ensure the boundedness and convergence of the state in (4). Thus, we will show that just under the hypotheses of this proposition, the boundedness and convergence hold as well. For any given x0 , from Theorem 3 and η(t) being monotonically decreasing, we have that if σ (η0 ) ≤ c0 then the state of the system (4) is bounded. Define the bound as xmax , i.e., ∥x(t)∥ ≤ xmax , t ∈ [0, ∞). Then according to Lemma 1 and Definition 1, there exist functions γ ∈ K∞ and α ∈ K such that (γ , α ) is a supply pair for (4) on Xx := {z ∈ Rn |∥z ∥ ≤ xmax }. γ (s) Moreover, for any γ˜0 ∈ K, it holds that lim sups→r1 γ˜ (s) ≤ γ (r1 ) γ˜0 (r1 )

0

< ∞ with r1 := γ −1 (α (xmax )). Then referring to the proof of Proposition 1, for any given positive definite Ω , one can find functions γ˜ ∈ K∞ and α˜ ∈ K such that (γ˜ , α˜ ) is a supply pair for (4) on Xx , with γ˜ (s) = O(Ω (s)) in a neighborhood of zero. Hence, similar to the proof of Lemma 4, one can prove that the state in (4) would converge to zero by using the supply pair (γ˜ , α˜ ) and the boundedness of x. Therefore in this case, Condition 1 is not required in Corollary 6, which completes the proof. References [1] P. Tabuada, Event-triggered real-time scheduling of stabilizing control tasks, IEEE Trans. Automat. Control 52 (9) (2007) 1680–1685. [2] J. Lunze, D. Lehmann, A state-feedback approach to event-based control, Automatica 46 (1) (2010) 211–215. [3] R. Postoyan, P. Tabuada, D. Nešić, A. Anta, A framework for the event-triggered stabilization of nonlinear systems, IEEE Trans. Automat. Control 60 (4) (2015) 982–996. [4] F. Hao, H. Yu, Stability of model-based event-triggered control systems: a separation property, Int. J. Syst. Sci. 48 (5) (2017) 1035–1047. [5] E.D. Sontag, Smooth stabilization implies coprime factorization, IEEE Trans. Automat. Control 34 (1989) 435–443. [6] T.F. Liu, Z.P. Jiang, A small-gain approach to robust event-triggered control of nonlinear systems, IEEE Trans. Automat. Control 60 (8) (2015) 2072–2085. [7] A. Girard, Dynamic triggering mechanisms for event-triggered control, IEEE Trans. Automat. Control 60 (7) (2015) 1992–1997. [8] H. Yu, F. Hao, A Lyapunov-based small-gain approach on design of triggering conditions in event-triggered control systems, Internat. J. Robust Nonlinear Control 26 (13) (2016) 2938–2960. [9] M.C.F. Donkers, W.P.M.H. Heemels, Output-based event-triggered control with guaranteed L∞ -gain and improved and decentralized event-triggering, IEEE Trans. Automat. Control 57 (6) (2012) 1362–1376. [10] P. Tallapragada, N. Chopra, Event-triggered dynamic output feedback control for LTI systems, in: IEEE Conference on Decision and Control (CDC), Maui, 2012, pp. 6597–6602. [11] T. Liu, Z. Jiang, Event-based control of nonlinear systems with partial state and output feedback, Automatica 53 (7) (2015) 10–22. [12] M. Abdelrahim, R. Postoyan, J. J. Daafouz, D. Nešić, Stabilization of nonlinear systems using event-triggered output feedback controllers, IEEE Trans. Automat. Control 61 (9) (2016) 2682–2687.

[13] V.S. Dolk, D.P. Borgers, W. Heemels, Output-based and decentralized dynamic event-triggered control with guaranteed Lp -gain performance and Zenofreeness, IEEE Trans. Automat. Control 62 (1) (2017) 34–49. [14] W. Wu, R. Sven, G. Daniel, S. Liu, Event-triggered control for discrete-time linear systems subject to bounded disturbance, Internat. J. Robust Nonlinear Control 29 (9) (2016) 1902–1918. [15] M. Abdelrahim, R. Postoyan, J. J. Daafouz, D. Nešić, Robust event-triggered output feedback controllers for nonlinear systems, Automatica 75 (2017) 96–108. [16] P. Zhang, T. Liu, Z.P. Jiang, Input-to-state stabilization of nonlinear discretetime systems with event-triggered controllers, Systems Control Lett. 103 (2017) 16–22. [17] H. Yu, F. Hao, Input-to-state stability of integral-based event-triggered control for linear plants, Automatica 85 (2017) 248–255. [18] X. Wang, M.D. Lemmon, Self-triggered feedback control systems with finitegain L2 Stability, IEEE Trans. Automat. Control 54 (3) (2009) 452–467. [19] J. Almeida, C. Silvestre, A.M. Pascoal, Synchronization of multi-agent systems using event-triggered and self-triggered broadcasts, IEEE Trans. Automat. Control 69 (9) (2017) 4741–4746. [20] H. Khalil, Nonlinear Systems, Prentice Hall Upper Saddle River, NJ, 2002. [21] R. Freeman, Global internal stabilizability does not imply global external stabilizability for small sensor disturbances, IEEE Trans. Automat. Control 40 (12) (1995) 2119–2122. [22] Y.F. Gao, R. Wang, C. Wen, W. Wang, Digital event-based control for nonlinear systems without the limit of ISS, IEEE Trans. Circuits Syst. Express Briefs 64 (7) (2017) 807–811. [23] E.D. Sontag, Comments on integral variants of ISS, Systems Control Lett. 34 (1) (1998) 93–100. [24] D. Angeli, E.D. Sontag, Y. Wang, A characterization of integral input-to-state stability, IEEE Trans. Automat. Control 45 (6) (2000) 1082–1097. [25] D. Liberzon, E.D. Sontag, Y. Wangm, On integral-input-to-state stabilization, in: Proceedings of the American Control Conference, Vol. 3, San Diego, 1999, pp. 1598–1602. [26] C.M. Kellett, P.M. Dower, Input-to-state stability, integral input-to-state stability, and L2 -gain properties: Qualitative equivalences and interconnected Systems, IEEE Trans. Automat. Control 61 (1) (2016) 3–17. [27] D.P. Borgers, W. Heemels, Event-separation properties of event-triggered control systems, IEEE Trans. Automat. Control 59 (10) (2014) 2644–2656. [28] X. Wang, M. Lemmon, Attentively efficient controllers for event-triggered feedback systems, in: 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC), Orlando, 2011, pp. 4698–4703. [29] M. Arcak, D. Angeli, E.D. Sontag, A unifying integral ISS framework for stability of nonlinear cascades, SIAM J. Control Optim. 40 (6) (2002) 1888–1904. [30] E. Garcia, P.J. Antsaklis, Model-based event-triggered control for systems with quantization and time-varying network delays, IEEE Trans. Automat. Control 58 (2) (2013) 422–434. [31] D. Angeli, B. Ingalls, E.D. Sontag, Y. Wang, Separation principles for input– output and integral-input-to-state stability, SIAM J. Control Optim. 43 (1) (2004) 256–276. [32] E.D. Sontag, Y. Wang, On characterizations of the input-to-state stability property, Systems Control Lett. 24 (5) (1995) 351–359. [33] H. Yu, F. Hao, Model-based event-triggered control for linear plant with threshold variable and model states, Internat. J. Robust Nonlinear Control 27 (1) (2017) 135–155. [34] Y. Lin, E.D. Sontag, Y. Wang, A smooth converse Lyapunov theorem for robust stability, SIAM J. Control Optim. 34 (1) (1996) 124–160. [35] A. Chaillet, D. Angeli, Integral input to state stable systems in cascade, Systems Control Lett. 57 (7) (2008) 519–527. [36] H. Ito, A Lyapunov approach to cascade interconnection of integral input-tostate stable systems, IEEE Trans. Automat. Control 55 (3) (2010) 702–708. [37] E. Sontag, A. Teel, Changing supply functions in input/state stable systems, IEEE Trans. Automat. Control 40 (8) (1995) 1476–1478. [38] H. Yu, F. Hao, Improved results of changing supply functions and the applications to nonlinear systems in cascade, in: Chinese Control and Decision Conference (CCDC), Shenyang, 2018, pp. 958–963.