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Finite‐time stabilisation of switched linear input‐delay systems via saturating actuators

2018; Institution of Engineering and Technology; Volume: 12; Issue: 15 Linguagem: Inglês

10.1049/iet-cta.2018.5402

1751-8652

Xiangze Lin, Shuaiting Huang, Shihua Li, Yun Zou,

Control and Stability of Dynamical Systems

IET Control Theory & ApplicationsVolume 12, Issue 15 p. 2127-2137 Research ArticleFree Access Finite-time stabilisation of switched linear input-delay systems via saturating actuators Xiangze Lin, Corresponding Author Xiangze Lin xzlin@njau.edu.cn College of Engineering, Nanjing Agricultural University, Nanjing, 210031 People's Republic of ChinaSearch for more papers by this authorShuaiting Huang, Shuaiting Huang College of Engineering, Nanjing Agricultural University, Nanjing, 210031 People's Republic of ChinaSearch for more papers by this authorShihua Li, Shihua Li School of Automation, Southeast University, Nanjing, 210096 People's Republic of ChinaSearch for more papers by this authorYun Zou, Yun Zou School of Automation, Nanjing University of Science and Technology, Nanjing, 210094 People's Republic of ChinaSearch for more papers by this author Xiangze Lin, Corresponding Author Xiangze Lin xzlin@njau.edu.cn College of Engineering, Nanjing Agricultural University, Nanjing, 210031 People's Republic of ChinaSearch for more papers by this authorShuaiting Huang, Shuaiting Huang College of Engineering, Nanjing Agricultural University, Nanjing, 210031 People's Republic of ChinaSearch for more papers by this authorShihua Li, Shihua Li School of Automation, Southeast University, Nanjing, 210096 People's Republic of ChinaSearch for more papers by this authorYun Zou, Yun Zou School of Automation, Nanjing University of Science and Technology, Nanjing, 210094 People's Republic of ChinaSearch for more papers by this author First published: 10 August 2018 https://doi.org/10.1049/iet-cta.2018.5402Citations: 6AboutSectionsPDF 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 In this study, the problem of finite-time feedback stabilisation for switched linear input-delay systems with saturating actuators is addressed by virtue of the comparison-function-like method. Finite-time state feedback stabilisation is discussed and controllers are designed to make the closed-loop systems finite-time stable. Moreover, for the case that the states are unmeasured, observers are constructed to estimate the unavailable states and the observer–controller compensator strategy is proposed. Based on the results proposed in this study, information of transient performance of the controlled switched systems can be obtained and bounds of system trajectory are estimated. An example is employed to illustrate the effectiveness of the proposed approach. 1 Introduction Finite-time stability means that the states of the system do not exceed prescribed bounds during a fixed time interval [1]. It should be noted that finite-time stability and Lyapunov stability are two different concepts [2–4]. The main differences are reflected in the running time, parameter settings (prescribed or not) and the analysis emphasis (qualitative or quantitative). Unlike Lyapunov stability, finite-time stability focuses on the quantitative behaviour of the system over a fixed finite-time interval and has been applied in many practical systems. For example, in missile guidance system [5], once the bound of the launch and target area have been set in advance, quantitative information of the transient behaviours of the dynamical system in a fixed time interval are focused on, and the length of the time interval is the time it takes from missile launch to hit target. In addition, finite-time stability is more suitable than Lyapunov stability in many other practical systems such as chemical process, manufacturing systems and power systems [6]. Research on finite-time stability problems dates back to the 1960s, since the work of Weiss and Infante [7], Angelo [8], Amato et al. [9]. Due to its theoretical and practical importance, finite-time stability has been greeted with great enthusiasm from researchers [10–26]. In recent years, by virtue of linear matrix inequality theory, research of finite-time stability has been revisited and many results have been proposed for different kinds of systems such as continues time systems [1, 10, 11, 15], discrete-time systems [12, 16, 17], stochastic systems [18, 19], chaotic systems [20] and systems with impulsive effects [21, 22]. It is worth mentioning that the results of finite-time stability in [23–26], which imply Lyapunov stability and finite-time convergence, are different from that in this paper. As we all known, switched systems are very common in many engineering fields, included automotive industry, switching power converters, traffic control and others. Stability analysis and feedback stabilisation of switched systems has attracted much attention and is still a hot topic [27–33]. Recently, the problem of finite-time stability of switched system has been discussed by researchers and many results have been reported [10, 11, 34–37]. However, to the best of the authors' knowledge, the research on finite-time stability of input-delay systems is in its infancy. Input delay is a natural phenomenon, which cannot be ignored in various engineering systems such as long transmission lines in pneumatic systems, nuclear reactors, rolling mills, hydraulic systems and manufacturing processes [38]. It is well known that the effects of delay on system dynamics are bilateral. It may cause performance degradation or undesired dynamics, and even may change the stability of system [39, 40]. Moreover, actuator saturation is usually an unavoidable feature in control design because of the physical limitation. And saturation non-linearity always brings adverse effect to the performance and stability of systems [13, 41, 42]. In most of the existing research results of the stabilisation of systems with saturating actuators, the designed feedback controllers are all linear control. However, these linear control methods cannot be directly applied to deal with the switched systems with saturating actuators. In many practical cases, input-delay and actuator saturation exist at the same time in one system. For instance, in the research of agricultural vehicle navigation, input-delay which is the time from agricultural vehicle controllers emit control signals to the mechanical actuators cannot be neglected, and actuator saturation does exists for steering maximum torque is finite. Therefore, it is challenging and interesting to discuss the switched systems with actuator saturation and input time-delay. Moreover, how to make the switched input-delay systems finite-time stable with actuator saturations is not a trivial task. Many results have been reported on the stabilisation of input-delay system with saturating actuator during the recent year [13, 43–49]. Most of these results focus on designing the corresponding controllers to make the system asymptotically stable. Some results are heuristic methods such as that in [45], other methods are obtained in a systemic method such as that in [46–48]. However, to the best of our knowledge, results of finite-time stabilisation for a class of switched linear input-delay systems with saturating actuators in a heuristic way have not been reported. In this paper, a comparing function is constructed to estimate the transient response of switched input-delay system and the time-domain analysis method is applied to investigate the finite-time stabilisation of the switched system with saturating actuators. The main contributions are as follows: (i) by virtue of the time-domain approach, finite-time stabilisation of switched input-delay system with non-linear saturating actuators is discussed and a state feedback controller has been designed to make the closed-loop systems finite-time stable; (ii) an observer is presented to estimate the unavailable states and the observer–controller compensator strategy is proposed. The main results are expressed by scalar inequalities which show the transient performance of the controlled switched systems clearly. Moreover, the bound of the trajectory can also given simultaneously by the proposed methods which leads us easily to judge the finite-time stability of the closed-loop switched systems and deeply to see the insight into the finite-time stabilisation of the switched linear input-delay systems. It should be pointed out that, in [50], finite-time feedback control of input-delay system without switching with saturating actuators has been addressed. For the general continuous time system discussed in [50], only one controller needs to be designed. However, for the switched systems discussed in this note, the design of the controllers and the switching signals should be taken into full consideration. Hence, it is more complicated than that for the general continuous time system. Moreover, when the switched system dwells on one subsystem during the finite-time interval, it degenerates to the system discussed in [50]. The paper is organised as follows. In Section 2, some notations and preliminaries of this problem are provided. The main results of this paper are presented in Section 3. A state feedback controller and the observer–controller compensator strategy are proposed. In Section 4, an example is presented, and simulation result shows the effectiveness of the proposed results. Concluding remarks are presented in Section 5. 2 Preliminaries and problem formulation Consider an input-delay system with saturating actuator described by the following equations: (1)where is the state, and is the control input signal, is the output, are positive real numbers and represent delay time, is the switching signal which is a piecewise constant function depending on time t or state , and are constant real matrices for and . is a continuous vector-valued initial function. Moreover, (respectively, ) denotes the maximum (respectively, minimum) value of the . Saturation function is as follows: (2)Assuming that only a finite part of the non-linearity is considered during the actual system operation, i.e. the operation of the saturation is inside the sector . Corresponding to the switching signal , we have the following switching sequence: which means that subsystem is activated when . In this paper, we assume that the state of switched linear system (1) does not jump at switching instants, i.e. the trajectory is everywhere continuous, and switching signal has finite switching number in any finite interval time. The assumption is common in many articles such as that in [27, 29–31]. Let S denote the set of switching signal which has finite number of switchings on any finite interval time. In this paper, we only consider the switching signal which belongs to S, i.e. . Substituting into system (1), the switched linear systems can be transformed as follows: (3) First, let us review the definition of finite-time stability of switched linear input-delay systems (3) which is extended from [51] and the special case of Definition 1 in [13]. Definition 1.Given three positive scalars with , a positive definite matrix R and a given switching signal , switched linear systems (3) are said to be finite-time stable with respect to , if , where .Without loss of generality, we assume that is controllable, and is observable. State feedback and observer–controller compensators are proposed in the following context. First, we consider the case where the state feedback control law given by (4)such that the input-delay system (1) and (4) are finite-time stable.Then, as the case that the state x is unavailable, a combined observer–controller compensator will have to be used. The problem is to determine and of the following control law: (5)such that the input-delay closed-loop system (1) and (5) are finite-time stable and is a vector-valued initial function. In order to prove the main results in this note, the so-called Bellman–Gronwall's inequality, which can be found in [52], is presented as follows. Lemma 1 (Bellman–Gronwall's inequality [52].Suppose that is real continuous function of t, b,c are constants and , if , then . 3 Main results Now, let us discuss finite-time stabilisation of switched linear input-delay system (1). Sufficient conditions for finite-time stabilisation for the two different cases are presented in the following subsections. State feedback controllers are proposed in the first subsection, and in another subsection the observer–controller compensators are to be designed when the states may be unavailable. 3.1 Finite-time stabilisation with state feedback The closed-loop system (1) and (4) can be transformed as follows: (6)where . Based on the definition of the norm function [53], it is easy to obtain that (7) Let us consider the solution of switched linear input-delay system (6) over the fixed interval . According to the definition of switching signal , for any , there exists k such that . Then, the solution of switched linear input-delay system (6) in the fixed interval is given by (8)where denotes the number of switchings of switched linear input-delay system (6) during . We first choose so that the matrices satisfying the inequality as follows: (9) Remark 1.It should be noted that controllability of subsystems is not necessary. If subsystems are not controllable, the uncontrollable parts are only required to meet the condition (9). Moreover, the whole switched system also does not need to be controllable, but the trajectory of the switched systems with some switching signals under the controllers (4) or (5) can be bounded by a function which is given in the following discussion.We can choose that (10)which means that is the maximum of positive real part of the eigenvalue of all the matrices that lies most far from the imaginary axis. Sufficient conditions for finite-time stability of switched linear input-delay system (6) are presented in the following theorem. Theorem 1.Assume is chosen such that the matrices satisfy the inequality (9). If the unique positive solution of the following equation with respect to : (11)satisfies (12)then the switched linear input-delay system (6) with the controller (4) is finite-time stable with respect to , where denotes the switching number during and , is the upper bound of the switching number of the system during . Proof.Taking norms of the two sides of (8) and based on inequality (7) (13) Combining (9) with (13), we have (14) where Define a scalar function as (15)where satisfies (11) and (12).By direct substitution, it is not difficult to verify that is the solution to the following equation: (16)Let , then based on (12), we can see that , then for (17)Subtracting (16) from (14), we have (18)Now, we consider . It follows that the term in the right-hand side of (18) is always negative or zero. Hence (19)where .Multiplying both sides of (19) by , we have (20)Applying Lemma 1, inequality (20) reduces to (21)From inequality (21), it not difficult to get . In the same way, it is easy to infer that on the intervals , , up to . Therefore, for all , i.e. .On the other hand That is Moreover From (12), we have (22)From the above discussion, it is not difficult to reach the conclusion that input-delay system (6) is finite-time stable with respect to . □ The graphical illustration of solution in (11) is shown in Fig. 1. Remark 2.In the proof procedure of Theorem 1, a comparison function (15), which is used to estimate the upper bound of the trajectory of switched systems in the worst case, is defined. If the upper bound estimate remains within prescribed bounds in the fixed-time interval, switched systems are definitely finite-time stable. The approach presented in this paper is a commonly used estimation method in classical control method, e.g. the estimation of setting time. Remark 3.For Theorem 1, some points should be clarified as follows: (1) The choice of matrix influences the value of parameter . If the line and the curve are too flat, the intersection of them may also be larger than the value In this case, it is not easy for us to reach the conclusion that the closed-loop system is finite-time stable. (2) The switching signals are an important factor for finite-time stability of switched systems. In this paper, the switching number in finite-time interval rather than switching signal is given to guarantee finite-time stability of switched systems. Since the switching number has been known, it is easy to design the switching signal, such as the periodic switching signals with dwell time no longer than or the switching signals with average dwell time . In the following simulations, dwell-time method is used to design the switching signal, see in Section 4. The same switching signal design method can also be used to the following theorems and corollaries. Fig. 1Open in figure viewerPowerPoint Graphical illustration of solution In the above analysis, the saturation is considered to be inside the limited sector . If the function operates without a limited range , we have the following corollary from Theorem 1. Corollary 1.Assume is chosen such that the matrices satisfy the inequality (9). If the unique positive solution of the following equation with respect to : (23)satisfies (24)then the switched linear input-delay system (6) with the controller (4) is finite-time stable with respect to , where , , denotes the switching number during and , is the upper bound of the switching number of the system during . 3.2 Finite-time stabilisation with observer–controller compensators When the states are unavailable, a combined observer–controller compensator will have to be used. In this subsection, finite-time stability of the input delay system (1) with observer–controller compensator is considered. The closed-loop system described by (1) and (5) can be rewritten as follows: (25)where Let us consider the solution of switched linear input-delay system (25) over the fixed interval . According to the definition of switching signal , for any , there exists k such that . Then, the solution of switched linear input-delay system (25) in the fixed interval is given by (26)where denotes the number of switchings of switched linear input-delay system (25) during . We choose and so that the matrices satisfy the inequality as follows: (27) We see that (28)which means that is the maximum of positive real part of the eigenvalue of all the matrices that lies most far to the imaginary axis. The following theorem gives sufficient conditions to guarantee the finite-time stability of input-delay system (25). Theorem 2.Assume and in (5) are chosen such that the matrices satisfy the inequality (27). And is the unique positive solution of the following equation with respect to : (29)which satisfies (30)then the switched linear input-delay system (1) with the controller (5) is finite-time stable with respect to , where , denotes the switching number during and . is the upper bound of the switching number of the system during . Proof.The proof procedure is similar to that of Theorem 1. Based on the properties of norm function and combining (5), (7), (25), one obtains (31) Taking norms of the two sides of (26) and based on (31), we have (32)Applying (27) into (32) yields the following inequality: (33) where Define a scalar function as (34)where satisfies (29)and (30).It is verify that is the solution to the following equation: (35)Let , then based on (30), we can see that , then for (36)Subtracting (35) from (33), we have (37)Now, we consider . It follows that the term in the right-hand side of (37) is always negative or zero. Hence (38)where .Multiplying both sides of (38) by , we have (39)Applying Lemma 1, inequality (39) reduces to (40)From inequality (40), it not difficult to get . In the same way, it is easy to infer that on the intervals , , up to . Therefore, for all , i.e. .On the other hand That is Moreover From (30), we have (41)From the above discussion, it is not difficult to reach the conclusion that input-delay system (25) is finite-time stable with respect to . □ 4 Numerical simulation Now, a numerical example is presented to show the efficiency of the method proposed in this paper. Let us consider the following switched systems with saturating actuator with , Here, the induced norm of matrix A is defined as where is the maxium eigenvalue of the matrix . 4.1 Finite-time stabilisation by state feedback The corresponding parameters are specified as follows: Choosing that it is not difficult to obtain that , from (10) and Theorem 2. Moreover, the intersection point of the line and the curve is which is smaller than Then, by virtue of Theorem 1, it is not difficult to reach the conclusion that the above switched system is finite-time stable with the controllers Without loss of generality, let , the simulation results of switched systems with the initial state under a periodic switching signal with interval time s are presented in Figs. 2, 3–4. Fig. 2Open in figure viewerPowerPoint Time response of states Fig. 3Open in figure viewerPowerPoint Time response of Fig. 4Open in figure viewerPowerPoint Time response of control input 4.2 Finite-time stabilisation by observer–controller compensators The corresponding parameters are specified as follows: Choosing that it is not difficult to obtain that from (28) and Theorem 2. Moreover, the intersection point of the line and the curve is which is smaller than Then, by virtue of Theorem 2, it is not difficult to reach the conclusion that the above switched system is finite-time stable with the controllers Without loss of generality, let , the simulation results of switched systems with the initial state under a periodic switching signal with interval time s are presented in Figs. 5, 6–7. Remark 4.From simulation results, see Figs. 3 and 6, it is easy to get that the maximum values of and for the switched systems of Sections 4.1 and 4.2 are less than the value of . Because the definition of finite-time stability is that all states do not exceed the prescribed bounds in the fixed time interval, it is not difficult to reach the conclusion that the switched systems in Section 4.1 are the finite-time stability under the control of state feedback and output feedback. Remark 5.In Fig. 5, as time goes by, the observed states and do not approach to the actual states and . It should be pointed out that the observer states in this method are only required to guarantee the states of the closed-loop system be bound, and the error between observed state and actual state x is not necessary to converge to zero. It is also one of the differences between finite-time stability and Lyapunov stability. Fig. 5Open in figure viewerPowerPoint Time response of states Fig. 6Open in figure viewerPowerPoint Time response of Fig. 7Open in figure viewerPowerPoint Time response of 4.3 Finite-time stabilisation by designed controllers with disturbances In this subsection, external disturbances are taken into account to show the robustness property of our method. The switching systems with disturbances are as follows: (42)where is the disturbance. First, we present simulation results about finite-time stabilisation of the switched system (42) by state feedback. The corresponding parameters are same as those in Section 4.1 and the parameters about and are as follows: Simulation results are shown in Figs. 8 and 9. Fig. 8Open in figure viewerPowerPoint Time response of states with Fig. 9Open in figure viewerPowerPoint Time response of with Now, simulation results about finite-time stabilisation of the switched system (42) by observer–controller compensators are presented in Figs. 10 and 11. The corresponding parameters are same as those in Section 4.2. Fig. 10Open in figure viewerPowerPoint Time response of states with Fig. 11Open in figure viewerPowerPoint Time response of with From Figs. 9 and 11, it is not difficult to see that for the switched system (42) with disturbances , and are still smaller than . Hence, the following conclusion can be drawn that the switched systems with the proposed controllers are robust to external disturbances to some extent. 5 Conclusion This paper has investigated the problem of finite-time stabilisation of switched linear input-delay systems with saturating actuators. Sufficient conditions for finite-time stability of the switched systems have been derived by constructing a comparing function. Furthermore, by virtue of a time domain approach, state feedback controllers and observer–controller compensators have been designed to finite-time stabilise switched linear input-delay systems for two cases: states available and states unavailable, respectively. The future work is to discuss the applications of the proposed theory and how to finite-time stabilise switched linear input-delay systems with saturating actuators by using sampled-data control techniques. 6 Acknowledgments This work was supported by the Natural Science Foundation of China (grant no. 61773216) and Natural Science Foundation of Jiangsu Province of China (grant no. BK20171386). 7 References 1Amato F. Ariola M., and Dorato P.: ‘Finite-time stabilization via dynamic output feedback’, Automatica, 2006, 42, pp. 337– 342 2Shevitz D., and Paden B.: ‘Lyapunov stability theory of nonsmooth systems’, IEEE Trans. Autom. 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