Dwell‐time‐dependent stability results for impulsive systems
2017; Institution of Engineering and Technology; Volume: 11; Issue: 7 Linguagem: Inglês
10.1049/iet-cta.2016.1350
ISSN1751-8652
Autores Tópico(s)Stability and Control of Uncertain Systems
ResumoIET Control Theory & ApplicationsVolume 11, Issue 7 p. 1034-1040 Research ArticleFree Access Dwell-time-dependent stability results for impulsive systems Hanyong Shao, Corresponding Author Hanyong Shao hanyongshao@163.com College of Engineering, Qufu Normal University, Rizhao, 276826 People's Republic of ChinaSearch for more papers by this authorJianrong Zhao, Jianrong Zhao College of Engineering, Qufu Normal University, Rizhao, 276826 People's Republic of ChinaSearch for more papers by this author Hanyong Shao, Corresponding Author Hanyong Shao hanyongshao@163.com College of Engineering, Qufu Normal University, Rizhao, 276826 People's Republic of ChinaSearch for more papers by this authorJianrong Zhao, Jianrong Zhao College of Engineering, Qufu Normal University, Rizhao, 276826 People's Republic of ChinaSearch for more papers by this author First published: 09 March 2017 https://doi.org/10.1049/iet-cta.2016.1350Citations: 7AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract This study is concerned with the dwell-time stability for impulsive systems under periodic or aperiodic impulses. A Lyapunov-like functional approach is established to study the stability for impulsive systems. The Lyapunov-like functional is time-varying and decreasing but is not imposed definite positive nor continuous. A specific Lyapunov-like functional is constructed by introducing the integral of state together with the cross-terms of the integral and impulsive states. A tight bounding is obtained for the derivative of this functional with the help of the improved Jensen inequality reported recently and the integral equation of the impulsive system. On the basis of the Lyapunov-like functional approach, new dwell-time-dependent stability results with ranged dwell-time, maximal dwell-time and minimal dwell-time are derived for periodic or aperiodic impulsive systems. The stability results have less conservatism than some existing ones, which are illustrated by numerical examples. 1 Introduction Impulsive systems [1–7] are a class of hybrid systems whose trajectories are discontinuous at some certain instants. Stability with minimal dwell-time or maximal dwell-time for impulsive systems was defined and discussed based on the characters of the system matrices in [4]. The stability of impulsive systems has been paid attention due to their applications in many fields such as epidemiology [8], forestry [9], power electronics [10], networked control systems [11, 12], sampled-data systems [13–16] and so on [17, 18]. For example, in [16], the sampled-data systems were formulated as impulsive systems, and robust stability of sampled-data systems was investigated. The discrete-time method is a basic approach to the stability of impulsive systems, by which the impulsive system is transformed into discrete-time systems, and then the eigenvalue analysis is conducted. This method, however, encounters difficulties when impulsive systems involve uncertainties or aperiodic impulses. To deal with this problem, a Lyapunov functional method was proposed in [16] to address the stability for impulsive systems, and the stability results were applied to sampled-data systems with uncertainties. To improve the Lyapunov functional method further, the looped-functional approach was proposed to deal with the stability for impulsive systems, and stability results with ranged dwell-time, maximal dwell-time and minimal dwell-time were derived. Convex dwell-time characterisations were given for the linear impulsive systems with uncertainties in [19, 20], where the stability analysis was formulated as an infinite dimensional feasibility problem, which was hard to solve. By contrast, the stability results obtained in [21, 22] were in the form of linear matrix inequalities and could be checked easily. Note that in [21, 22] the integral of the system state was not included in the functional. Moreover, when estimating the derivative of the functional, the Jensen inequality [23] was used. With improved inequalities over Jensen inequality reported recently in [15, 24], there still is some room for the results in [21, 22] to improve. On the other hand, very recently the Lyapunov-like functional method was proposed in [25] to derive desired stability results for sampled-data systems. Considering sampled-data systems are special case of impulsive systems, a natural question is: Is it possible for the Lyapunov-like functional method to be extended to impulsive systems? In this paper, the Lyapunov-like functional method is extended from sampled-data systems into impulsive systems. A framework of the Lyapunov-like functional is established in a lemma to analyse the stability for impulsive systems under periodic impulses or aperiodic impulses. A concrete Lyapunov-like functional is constructed by including the integral of the state as well as the cross-terms of this integral and the impulsive state. Then, the derivative of the functional is bounded by taking advantage of the improved Jensen inequality and the integral equation of the impulsive system. By the lemma stability results with ranged dwell-time, maximal dwell-time and minimal dwell-time are derived for impulsive systems. Numerical examples are given to illustrate the less conservatism of the stability results. Notations: Throughout this paper, and denote the transposition and the inverse of the matrix X, respectively. in the symmetric matrix stands for the symmetric terms. For a symmetric matrix X, denotes X is positive definite (negative definite). I denotes the identity matrix with appropriate dimensions. is the Euclidean norm for a vector and is the induced matrix norm. , , refer to the set of n dimension vectors, matrices and non-negative integers, respectively. , and denote the smallest eigenvalue, largest eigenvalue and the spectral radius of a real symmetric matrix. For a given matrix A, stands for . For the impulsive instant , , . 2 Problem formulations Consider the following linear impulsive system: (1)where is the state vector, are constant matrices, is an increasing sequence of impulse instants. The dwell-time satisfies (2)where and are known constants. In the case of periodic impulses, the dwell-time , where h is the period. In this paper, we will study the stability for the impulsive system (1) and (2). For this end, we need the lemmas. Lemma 1.Consider the following non-linear system : where satisfies (2); satisfies , and for there exists such that . Assume that there exist positive scalars , a piecewise continuous functional and another time-varying one , satisfying: (i) (ii) and (iii) For , , . Then impulsive system is asymptotically stable. We first give the following definition of stability for impulsive systems. Definition 1.For system , given the initial condition , regard the state trajectory as a sequence of functions in a lifted domain, and then system is globally asymptotically stable if for all , there exists such that (i) , and (ii) as . Now, based on the definition we prove Lemma 1 to be true. Proof.From (iii), we have By (ii) it is derived that This together with (i) means for all , there exists such that (3)On the other hand, for integrating the first equation of system one can obtain Then By Grown–Bellman Lemma [26], we have Therefore This and (3) imply for all there exists such that and By Definition 1, system is asymptotically stable. This completes the proof. □ Remark 1.Lemma 1 extends the Lyapunov-like functional method from sampled-data systems [25] to impulsive systems and in that the state of the latter is discontinuous. The lemma features constructing a piecewise functional that is time-varying and decreasing but not imposed positive definite or continuous. The functional is different from a usual Lyapunov functional, so we call it a Lyapunov-like functional in the following. Note that for linear impulsive systems, the stability was investigated in a framework of the looped functional recently in [21, 22], while the Lyapunov-like functional has been extended to non-linear impulsive systems. Lemma 2 [15].(The improved Jensen inequality): For a given matrix , the following inequality holds for all continuously differentiable function in : where and Remark 2.If the second term of the right-hand side is deleted, Lemma 2 reduces to Jensen inequality, thus the name ‘the improved Jensen inequality’. Obviously the improved Jensen inequality can give a tighter bound for the integral, compared with Jensen inequality. 3 Main results For , a Lyapunov-like functional in the form of , as in Lemma 1, is constructed for system (1) (4)with and where the related matrices are defined in the following Theorem 1. Remark 3.Compared with the Lyapunov functional in [21, 22, 25], the integral of the state is introduced, and the cross-terms of this integral and the state at impulsive instants are included in Lyapunov-like functional (4). The Lyapunov-like functional is an extension of those in [21, 22]. It also includes that in [25] as a special case. Now employing the Lyapunov-like functional (4), we are in a position to propose a stability result for systems (1) and (2) as follows. Theorem 1.For given , systems (1) and (2) are asymptotically stable, if there exist symmetric matrices P, , Q, S, Z, , matrices , , with , , such that for (5) (6)where Proof.By the definition of Lyapunov-like functional (4), it is obvious that Lyapunov-like functional (4) satisfies (i) and (ii) in Lemma 1. In the following, we will prove (iii) in Lemma 1 true for the Lyapunov-like functional. At first, define and For , computing the time derivative of Lyapunov-like functional (4) along the trajectory of system (1), and using the notations in Theorem 1, we represent the derivative as (7)with Now, we deal with the integrals in the time derivative of Lyapunov-like functional (4). First, define Then by Lemma 2 we have (8)From this inequality it follows that: (9)where and are defined in Theorem 1. Note that there exist , such that the following inequalities hold [15]: (10)Therefore (11)Second, integrating both sides of system (1) from to , we obtain the integral equation (12)Therefore, there exists such that (13)Finally, based on Jensen inequality [23] it is derived that (14)From (7), (11), (13) and (14) it follows that: (15)where (16)with given in Theorem 1.On the other hand, from (5) and (6), by Schur complement lemma, it is derived that for and Noting and are affine with respect to , we have for and By the definition of in (16), it follows that and . Since is affine with respect to t, for . So, from (15) it is concluded for . By Lemma 1, systems (1) and (2) are asymptotically stable. This completes the proof. □ Remark 4.Since Lyapunov-like functional (4) includes the integral of the state, we can take advantage of an improved Jensen inequality in [15] together with the integral equation (12) to estimate tightly the derivative of the Lyapunov-like functional. To apply the convex combination technique to the estimated derivative of the Lyapunov-like functional, the improved Jensen inequality is transformed into a form that is affine in , where the right limit of the state at the impulsive instant is expressed using the left limit. This technique is expected to lead to a less conservative result. In fact, let , , with sufficiently small and then Theorem 1 reduces to the stability result in [22], which is stated as follows. Corollary 1 [22].For given , systems (1) and (2) are asymptotically stable, if there exist symmetric matrices P, , Q, S, , matrices , , with , , such that for where Remark 5.Recently advanced inequalities have been reported to deal with the integrals appearing in Lyapunov functional approaches [15, 24, 27, 28]. The inequalities in [27, 28] target to the integral on constant intervals, while those in [15, 24] target to the integral on varying intervals. If the inequality in [24], instead of that in [15], is employed to estimate the derivative of the Lyapunov-like functional (4), an even less conservative stability result can be expected but at cost of more complexity. When , by Theorem 1 we can get a stability result for impulsive systems with periodic impulse. Corollary 2.(periodic impulses case): For given , system (1) with the period h is asymptotically stable, if there exist symmetric matrices P, , Q, S, Z, , matrices , , with , , satisfying (5) and (6). If matrix A is Hurwitz or anti-Hurwitz, impulsive systems (1) and (2) will be asymptotically stable with the minimal dwell-time or the maximal dwell-time. The stability result is stated as follows. Theorem 2.(maximal dwell-time): For given , assume that there exist symmetric matrices P, , Q, S, Z, , matrices , , with , , satisfying (5) and (6) and . Then, for any and any impulsive sequence satisfying , impulsive system (1) is asymptotically stable. Proof.The idea is the same as the argument of Lemma 1. On the one hand, similar to the proof of Theorem 1, under the condition of Theorem 2, the Lyapunov functional (4) for impulsive system (1) satisfies (i), (ii) and (iii) of Lemma 1 with replaced with , and based on the proof of Lemma 1 we can obtain That is Define , and then it is not difficult to find that under . So, for any and any impulsive sequence satisfying Then, as .On the other hand, let , and then for Therefore Now, from above it can be drawn that impulsive system (1) is asymptotically stable with the maximal dwell-time. This ends the proof. □ Theorem 3.(minimal dwell-time): For given , assume that there exist symmetric matrices P, , Q, S, Z, , matrices , , with , , satisfying (5) and (6) and . Then, for any impulsive sequence satisfying , impulsive system (1) is asymptotically stable. The proof is similar to that of Theorem 2, and is omitted here. 4 Examples In this section, five examples are given to demonstrate the proposed results have applications in fish population control, and have less conservatism than some existing ones. Example 1.In this example, we present a fish population model [29], which exhibits impulsive behaviour in its state variable and may have applications in fisheries management. The fish population in a lake can be described in the form of the impulsive systems (1) and (2) with where is the population of two kinds of fish at time t, is the natural growth rate of the fish population, , indicates how the natural growth of the fish population is disturbed by making catches and adding fish brood at times . Assume is the expected fish population in the lake. To make the fish population , let , and then it is suffice to stabilise by applying to the system. Now consider the impulsive system By Corollary 2 we can obtain the maximum admissible impulsive period for the impulsive system to be stable. Choose and the initial condition to describe the states for the impulsive system as well as the system . In Figs. 1 and 2, it is concluded that the impulse can stabilise the system without impulses, and the proposed method can be employed in the management of fish population growth. Fig. 1Open in figure viewerPowerPoint Trajectory of the impulse system Fig. 2Open in figure viewerPowerPoint Trajectory of the system Example 2.Consider the following impulsive system with: From A it is seen that the first continuous-time dynamic is stable but the second is not. By contrast, the first discrete-time dynamic is not stable while the second is stable, as seen from J. The impulsive system will be unstable if the dwell-time is too small or too large. A dwell-time range is expected for the system to be stable. By the stability result Theorem 1 and those in [21, 22], we can obtain the dwell-time range for aperiodic impulsive systems and the period range for periodic impulsive systems, respectively, which are listed in Table 1.As is shown in Table 1, the dwell-time range obtained by Theorem 1 and the periodic ranges obtained by Corollary 2 in this paper cover those by the stability results in [21, 22]. In this sense, the stability results Theorem 1 and Corollary 2 in this paper have less conservatism than those in [21, 22]. Example 3.Consider impulsive system (1) with Since A is anti-Hurwitz and J is Schur, the dwell-time should be sufficiently small for the impulsive system to be stable. The impulsive system is expected to be asymptotically stable with the dwell-time , where h is referred to the maximal dwell-time. By Theorem 2 and the stability results in [21, 22], the maximal dwell-time h can be obtained; see Table 2.As is shown in Table 2, the maximal dwell-time h obtained by Theorem 2 in this paper is larger than those by the stability results in [21, 22]. In terms of the maximal dwell-time, Theorem 2 in this paper has less conservatism than the stability results in [21, 22]. Example 4.Consider impulsive system (1) with Contrary to Example 3, in this example A is Hurwitz and J is anti-Schur, the dwell-time should be sufficiently large for the impulsive system to be stable. The impulsive system is expected to be asymptotically stable with the dwell-time , where h is referred to the minimal dwell-time. By Theorem 3 and the stability results in [21, 22], the minimal dwell-time h can be obtained as in Table 3.As seen from Table 3, the minimal dwell-time h given by Theorem 3 in this paper is smaller than those in [21, 22]. Therefore, the stability result Theorem 3 in this paper has less conservatism than those in [21, 22]. Table 1. Dwell-time range and the periodic range Methods [21] [22] This paper dwell-time range [0.0802, 1.2082] [0.0780, 2.3214] [0.0780, 2.6417] periodic range [0.0780, 4.0867] [0.0780, 4.6337] [0.0780, 5.2789] Table 2. Maximal dwell-time for aperiodic impulsive systems Methods [21] [22] Theorem 2 in this paper h 0.4471 0.4483 0.4618 Table 3. Minimal dwell-time for aperiodic impulsive systems Methods [21] [22] Theorem 3 in this paper h 1.2323 1.232 1.1417 To end this section, we apply the main result Theorem 1 to sampled-data systems. Example 5.Consider the following sampled-data system: with Let , the sampled-data system can be transformed into an impulsive system in the form of (1) with For , the sampling interval can be obtained as by Theorem 1 in this paper, which is larger than and in [21, 22], respectively. So, Theorem 1 is less conservative than the stability results in [21, 22]. For , the sampling period obtained by Corollary 2 is larger than and by the stability results in [21, 22], respectively. In this regard, the stability result Corollary 2 has less conservatism, compared with those in [21, 22]. 5 Conclusion The dwell-time-dependent stability has been addressed in a framework of Lyapunov-like functional for impulsive systems under periodic or aperiodic impulses. An augmented Lyapunov-like functional was constructed by adding the integral of the state as well as the cross-terms of the integral and the impulsive state. Less conservative estimation for the derivative of the Lyapunov-like functional was conducted employing the improved Jensen inequality together with the integral equation of the impulsive system. New stability results with ranged dwell-time, maximal dwell-time and minimal dwell-time were obtained for periodic or aperiodic impulsive systems. One practical example and some numerical ones were listed to illustrate the stability results’ applications and less conservatism than some existing stability results. 6 References 1Bainov D., and Simeonov P.: ‘ Systems with impulse effects: stability, theory and application’ ( Halsted Press, New York, 1989) 2Yang T.: ‘ Impulsive control theory’ ( Springer-Verlag, Berlin, 2001) 3Cai C. Teel A.R., and Goebel R.: ‘ Converse Lyapunov theorems and robust asymptotic stability for hybrid systems’. Proc. American Control Conf., Portland, Oregon, USA, June 2005, pp. 12– 17 4Hespanha J.P. 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