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New results on stability of switched positive systems: An average dwell-time approach

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Abstract

This study is concerned with the problem of exponential stability for a class of switched positive linear systems consisting of both stable and unstable subsystems. The sufficient conditions of exponential stability are established in continuous-time and discrete-time domains. Based on the average dwell-time approach, new stability results for such kind of systems are first derived, which allows the ascent of the multiple linear copositive Lyapunov functions caused by unstable subsystems. Furthermore, when all subsystems are stable, the exponential stability condition for switched positive systems is presented. Finally, numerical examples are given to illustrate the effectiveness of the results.

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... For specific switched systems with all individual positive subsystems, these systems are called switched positive systems and widely arise in various applications such as wireless power control [8], congestion control [9], compartmental model [2], water-quality model [6,10], and so on. Stabilities of switched positive systems have been extensively investigated under an appropriate switching rule; for example, the stability analysis of switched positive systems [11][12][13][14][15][16][17], robust stability analysis of switched positive systems [18,19], and L 1 -gain analysis of switched positive systems [19,20]. ...
... (3) Apart from the studies in [12,15,25,36] that proposed stability conditions for switched positive systems with at least one stable subsystem, our research mainly concentrates on the (robust) stability of switched positive time-varying delay systems with all unstable subsystems. Note that conditions in [18] also guarantee globally asymptotically stable of switched positive systems under all unstable subsystems, but the existence of the time-delay was not taken into account. ...
... Since A i is the Metzler matrices and D i 0, ∀i ∈ N, system (1) without interval uncertainties (2) and (3) is positive by employing Lemma 1. Under the same notations and vector function (12) in Theorem 1, we can prove this corollary by using the time-scheduled MCLKF: ...
Article
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In this paper, the problem of robust stability for a class of linear switched positive time-varying delay systems with all unstable subsystems and interval uncertainties is investigated. By establishing suitable time-scheduled multiple copositive Lyapunov-Krasovskii functionals (MCLKF) and adopting a mode-dependent dwell time (MDDT) switching strategy, new delay-dependent sufficient conditions guaranteeing global uniform asymptotic stability of the considered systems are formulated. Apart from past studies that studied switched systems with at least one stable subsystem, in the present study, the MDDT switching technique has been applied to ensure robust stability of the considered systems with all unstable subsystems. Compared with the existing results, our results are more general and less conservative than some of the previous studies. Two numerical examples are provided to illustrate the effectiveness of the proposed methods.
... The positivity requirement of the state variables means that positive systems are defined on cones rather than on linear spaces, thereby resulting in many issues and challenges. In recent years, many researchers have explored the stability analysis and controller design of positive systems [6][7][8][9][10][11][12][13] . ...
... From (11), (12) and (17) , one can obtain that (14) holds, then ...
... by (11), (12), (18) and (28) , it yields that ...
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This study addresses the exponential stability and positive stabilization problems of impulsive positive systems (IPSs) with time delay. Specially, three types of impulses, namely, disturbance, “neutral”, and stabilizing impulses, are considered. For each type of impulsive effect, the exponential stability criterion is established utilizing the Lyapunov-Razumikhin techniques. Moreover, on the basis of the obtained stability results, the state-feedback controller design problem is investigated to positively stabilize the IPSs with time delay under different types of impulsive effects. Finally, numerical examples are provided to illustrate the effectiveness of the theoretical results.
... In the late 1970s, Luenberger [4] proposed a systematic theoretical approach for studying positive systems, afterwards, the theory of positive systems has been extended to many allied areas such as positive delayed systems, positive switched systems, positive Markovian jump systems and positive nonlinear systems. During the past few years, many valuable results have been reported especially on stability analysis and synthesis of positive systems with or without delay in a number of publications such as [5][6][7][8][9][10][11][12][13][14][15][16][17] . ...
... Zhao et al. [9] investigated the stability problem for a class of pos-itive switched linear systems with ADT switching, and obtained some sufficient criteria by means of multiple linear co-positive Lyapunov functions. Furthermore, by using ADT approach, [11] derived new stability conditions for positive switched linear systems consisting of both stable and unstable subsystems (modes), and generalized the results in [9,13] analyzed the L ∞ -gain performance for positive systems with distributed delays. Finite-time control problem for positive Markovian jump systems with constant delay and partly known transition rates was investigated in [14] . ...
... Due to the hybrid characteristic generated by impulse effect, mode-switching as well as positive constraint on system state, the corresponding theoretical analysis will be more challenging. In [11] , Lian et al. discussed a class of positive switched systems with both stable and unstable modes by multiple linear co-positive Lyapunov functions. They concluded that, if the total running time ratio between stable and unstable modes is no less than a specified constant, then the entire system is exponentially stable under certain ADT condition, and the values of Lyapunov functions are allowed to increase in the running time intervals of unstable modes. ...
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This paper studies the stability and L1-gain problems for impulsive positive switched systems (IPSS) consisting of both unstable and stable subsystems, and proposes a mode-dependent impulsive control (MDIC) method for the first time. By means of a piecewise linear co-positive Lyapunov function (PLCLF) and using average dwell time (ADT) approach, some sufficient criteria for global exponential stability (GES) and standard L1-gain are established in terms of linear programming (LP), which can be verified conveniently by Matlab. A special impulsive controller is designed to guarantee the positivity and stability of the addressed system. Several numerical examples with simulations are provided to demonstrate the effectiveness of the obtained results.
... L 1 -and L ∞ -gain based robust stability and stabilization of positive systems were proposed in [9], where all conditions are formulated into linear programming. The linear copositive Lyapunov function and linear programming approaches are applied to analysis and synthesis of positive systems and hybrid positive systems [10][11][12][13][14][15][16]. Some researchers are also interested in the control issues of positive time-delay systems. ...
... By (21) and (22), the condition (10) holds. From (9) and (13), it can follow that the condition (11) holds. (b) For polytopic system (1), we obtain the corresponding closed-loop system: ...
... , that is, the condition (10) holds. The condition (11) is a direct result form the condition (13). □ Remark 2: The conditions (12) and (13) are solvable in terms of linear programming. ...
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This paper proposes the robust model predictive control of positive time-delay systems with interval and polytopic uncertainties, respectively. The model predictive control framework consists of linear constraint, linear performance index, linear Lyapunov function, linear programming algorithm, and cone invariant set. By virtue of matrix decomposition technique, robust model predictive controllers of interval and polytopic positive systems with multiple state delays are designed, respectively. A multi step control strategy is utilized and a cone invariant set is constructed. Linear programming is used for the corresponding MPC conditions. Finally, a numerical example is given to verify the effectiveness of the proposed design.
... Furthermore, in practice, there are switched systems with both Hurwitz stable and unstable subsystems, such as disturbance, unmodeled dynamics and possible faults [10]. There have been some literatures considering the stability problem of SPLSs with both stable and unstable subsystems [11][12]. Based on the ADT approach, the sufficient condition of exponential stability for SPLSs with unstable subsystems is given in [11], which allows the ascent of the multiple linear copositive Lyapunov functions caused by unstable subsystems. ...
... There have been some literatures considering the stability problem of SPLSs with both stable and unstable subsystems [11][12]. Based on the ADT approach, the sufficient condition of exponential stability for SPLSs with unstable subsystems is given in [11], which allows the ascent of the multiple linear copositive Lyapunov functions caused by unstable subsystems. In [12], the problem of stability analysis has been addressed for the discrete-time switched delay positive systems in the presence of unstable subsystems. ...
Article
This paper is concerned with the stability and robust stability of switched positive linear systems whose subsystems are all unstable. By means of the mode-dependent dwell time approach and a class of discretized co-positive Lyapunov functions, some stability conditions of switched positive linear systems with all modes unstable are derived in both the continuous-time and the discrete-time cases, respectively. The co-positive Lyapunov functions constructed in this paper are time-varying during the dwell time and time-invariant afterwards. In addition, the above approach is extended to the switched interval positive systems. A numerical example is proposed to illustrate our approach.
... The unifying approach for the theoretical research of positive systems is first proposed by the famous scholar Luenberguer in his book [6]. Since then, many papers have been concentrated on stability analysis, controller design, and synthesis of positive systems [7][8][9][10][11][12][13][14][15][16][17][18][19][20]. ...
... x t 1 0 and x t 1 0, then, (ii) The exponential stability of the error system (8). ...
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This paper is concerned with the design and the synthesis of the impulsive positive observer (IPO) for positive linear continuous systems. The IPO can estimate the states for positive systems even when the measured output is only available at discrete-time instants. In this paper, a time-varying weighted copositive Lyapunov function is constructed, and the upper and lower bounds of impulsive intervals method combined with the convex combination technique are used to establish sufficient conditions for the existence of the IPO. Furthermore, the design of the dynamic output feedback controller based on the IPO is addressed to positively stabilize positive linear systems. Algorithms are given to design the IPO and the controller, respectively. Finally, two numerical examples are provided to show the effectiveness of the theoretical results. Copyright
... Definition 1 [15] For certain switching signal σ , if there exists θ > 0, ξ > 0 satisfying ...
... with c = ξ λ e N 0 (lnκ+(β+α))ς max . Applying (15) and Definition2, we have exponential stability of the system (6). ...
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In practical applications, it takes time for a switched system to switch from the current subsystem to the candidate subsystem and run the candidate controller, which will cause the controller to be asynchronous with the subsystem mode, namely asynchronous switching. This paper investigates the robust stabilization for switched uncertain systems with input quantization when controller and subsystem mode are subject to asynchronous switching. First of all, the criteria are presented to achieve exponential stability of the closed-loop systems under asynchronous switching. Then, the design method of the controllers is given in which each controller for its corresponding subsystem comprises two parts. One is used to handle model uncertainty and asynchronous switching, and the influence of quantization is eliminated by the other. Finally, the feasibility and effectiveness of the derived technique are verified through a numerical example.
... In [13], the absolute exponential stabilisation of the switched system is obtained by the state feedback and ADT switching strategy. Moreover, some extended results [14,15] are achieved by relaxing the decreasing requirement of a collection of Lyapunv functions. In the meantime, several conditions to establish the ADT stability properties of the switched positive linear system are considered in [15]. ...
... Moreover, some extended results [14,15] are achieved by relaxing the decreasing requirement of a collection of Lyapunv functions. In the meantime, several conditions to establish the ADT stability properties of the switched positive linear system are considered in [15]. ...
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This study presents a result on constructing separable Lyapunov-like functions for the switched positive non-linear system. Firstly, the contraction theory has been introduced to the stability analysis of the switched positive non-linear system. Secondly, the specific forms of a set of separable Lyapunov-like functions have been demonstrated for stable subsystems and unstable subsystems to achieve the asymptotic stability for any switching signal satisfying an average dwell time constraint. Additionally, an algorithm is provided to compute the set of separable Lyapunov-like functions based on the sum of squares programming. Finally, three simulation examples are provided to illustrate the effectiveness of their result.
... In fact, due to positive switched systems owning the double features of switching and non-negativity, they can express a type of systems more accurately, such as traffic congestion model [25] and HIV viral mutation model [26] . In the recent years, positive switched systems become more and more popular, so the research results of them are constantly increasing [27][28][29][30][31][32][33][34][35][36] . ...
... Later on, the average dwell time method are widely used for the analysis and control of positive switched systems. Stability and stabilization [29][30] , robust finite-time stabilization [31] , estimator design [32] and actuator saturation problem [33] of positive switched systems asynchronous 1 L control [34] and output tracking control [35] of delayed switched positive systems, were investigated. As the research going, the multiple linear co-positive or quadratic diagonal Lyapunov functions method is adopted [36][37][38] . ...
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This paper focuses on the design of state-dependent switching law with the dwell time restriction for interval positive switching systems. First, we present the design method of the hybrid-type state- and time-driven switching law, which inherits the two aspect advantages of the state- and time-dependent ones. Second, by multiple diagonal quadratic Lyapunov functions, we conduct the double designs of controllers and the switching signal, and sufficient conditions for stabilization of systems are proposed by the LMIs form. Finally, an example demonstrates the validity of the presented results.
... As a type of complex positive systems, switched positive systems have attracted considerable attention [19][20][21]. The analysis and synthesis for such systems are challenging due to the fact that they possess both characteristics of switched systems and positive ones. ...
... Let M = max {1, M}. By (19) and (20), it follows that ...
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Addressed in this study are the minimal period and stability issues of periodic switched positive systems. With the assumption that all the system matrices, delays and switching signal are periodic, it is shown that the considered switched system is also periodic with the minimal period being a divisor of the least common multiple of the periods of subsystems and switching signal, and an algorithm is presented to determine the minimal period. Then some necessary and sufficient exponential stability conditions are proposed for periodic switched positive systems. These conditions are also extended to a more general class of periodic switched systems. Finally, a numerical example with three cases is provided to demonstrate the effectiveness of the theoretical results and reveals two interesting facts: The system matrices of periodic positive systems are unnecessarily non‐negative and system delays influence system stability. These facts imply that there exist some remarkable differences between general switched positive systems and periodic switched positive systems.
... Besides, several phenomena can be modeled by SPSs, such as compartmental model [30], water-quality model [31], formation flying [32], congestion control [33], wireless power control [34], and network communication using transmission control protocol [35]. Due to the complex dynamics of SPSs and their numerous applications, stability analysis on SPSs has been a significant investigation, and some relevant researches have been reported in [36][37][38][39][40][41][42][43][44]. In addition, most practical systems often contain the term uncertainties, which refer to the differences or errors between models and reality. ...
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The global stability problem for a class of linear switched positive time-varying delay systems (LSPTDSs) with interval uncertainties by means of a fast average dwell time (FADT) switching is analyzed in this paper. A distinctive feature of this research is that all subsystems are considered to be unstable. Both the continuous-time and the discrete-time cases of LSPTDSs with interval uncertainties and all unstable subsystems (AUSs) are investigated. By constructing a time-scheduled multiple copositive Lyapunov-Krasovskii functional (MCLKF), novel sufficient conditions are derived within the framework of the FADT switching to guarantee such systems in the case of continuous-time to be globally uniformly exponentially stable. Based on the above approach, the corresponding result is extended to the discrete-time LSPTDSs including both interval uncertainties and AUSs. In addition, new stability criteria in an exponential sense are formulated for the studied systems without interval uncertainties. The efficiency and validity of the theoretical results are shown through simulation examples.
... By adopting ML-CLF approaches, the stability analysis problem for SPLSs with ADT switching has been investigated in [29] in both continuous-time and discrete-time cases. Then, the authors of [11] extended the results of [29] to SPLSs consisting of both stable and unstable subsystems. It is worth noting in the existing works related to stability analysis of SPLSs under ADT switching that: 1) the used MLCLF for every positive linear subsystem is a continuous function in each activated interval; and 2) how to get a lower bound on the ADT is not studied in those works while obtaining a tighter bound on the ADT guaranteeing the stability of SPLSs is meaningful. ...
Article
In this paper, the problems of stability analysis and delay control for switched positive linear systems (SPLSs) are studied by using average dwell time (ADT) switching. By developing a new class of multiple piecewise-continuous linear copositive Lyapunov function (MPLCLF) approach, an improved stability condition for SPLSs is obtained, which can obtain a tighter bound on the ADT guaranteeing the stability. The obtained stability condition covers the existing results on stability analysis for SPLSs, since the multiple liner copositive Lyapunov function (MLCLF) used in the literature can be viewed as a special case of our proposed MPLCLF, and our result can obtain a tighter bound of ADT. Then, a more efficient controller design method over the existing ones is also established for SPLSs with switching delay to guarantee the underlying system to be exponentially stable. To illustrate the advantages of our proposed results, two numerical examples are finally given.
... Even with this restriction, the result presented here is of significance. Meanwhile, as the procedures of the proof involves in deploying Lyapunov-like functions, it is expected that the results are applicable to other systems such as linear and nonlinear switched descriptor systems [12][13][14][15], with possible development into fractional order switched systems [16,17] etc. ...
... The authors in [8] constructed a linear copositive Lyapunov function for positive systems, which was used to discuss the stability of positive systems [9][10][11][12]. Ait Rami et al. for the first time, addressed a linear programming technique to the stabilisation of positive systems [13,14], which was further used to investigate the output-feedback control [15], ℓ 1 -induced control [16], robust stabilisation [17] and so on [18][19][20][21]. At present, the linear copositive Lyapunov function associated with the linear programming technique is often applied to positive systems though there are some other techniques [22,23]. ...
... Typical time-dependent switchings include dwell time switching, average dwell time switching and mode-dependent averaged dwell time (MDADT) switching. Generally speaking, ADT and MDADT have the advantages of relatively less conservativeness and extra flexibility in system analysis and synthesis [22,[37][38][39][40][41] . Both switching signals have been recently introduced to SPLSs [35,[42][43][44][45] . ...
Article
In this paper, stability analysis for switched positive linear systems (SPLSs) with average dwell time switching is revisited and discussed in both continuous-time and discrete-time contexts. By utilizing system positivity, two simple stability conditions with fewer constraints are first proposed for SPLSs on the basis of multiple linear copositive Lyapunov functions (MLCLFs). Based on the developed results, three novel concepts called closed-chain, l-open-chain and quasi-cyclic switching signals are introduced to derive some results that are not only simple but also less conservative. All the obtained results are formulated in terms of linear matrix inequalities and can guarantee the global exponential stability of SPLSs under ADT switching. To illustrate the advantages of our established results, a numerical example is finally given.
... Thanks to their great research values, SPLSs have drawn many researchers' attention. As a basic characteristic, the stability of SPLSs deserves attention and has been studied in many papers such as [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. The linear co-positive Lyapunov functions (LCLFs) approach utilised in aforementioned papers has shown its effectiveness in analysing the stability of SPLSs. ...
Article
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This study is concerned with dwell time stability and stabilisation problems of switched positive linear systems (SPLSs). The dwell time refers to minimum dwell time and constant dwell time. Several stability conditions for primal and transpose SPLSs with dwell time are presented, and the relation between these conditions is illustrated. Some of these conditions are given in terms of infinite-dimensional linear programming (LP), which cannot be solved directly. Then, by utilising the piecewise linear approach, new alternative convex conditions are formulated in terms of finite-dimensional LP. Compared to the existing literature, results with lower or at least the same conservatism can be obtained under the new conditions for the same discretised order. An algorithm is given to reduce the computational cost. Meanwhile, it is proved that there exists a relation between these convex and non-convex conditions if the discretised order is sufficiently large. By utilising the transpose conditions, alternative convex conditions on stabilisation of SPLSs with dwell time are also presented. The controller gain matrices can be computed by solving a set of LP directly. Finally, the correctness and superiority of the results are verified by numerical examples.
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This paper concentrates on the output tracking control problem with L1-gain performance of positive switched systems. We adopt the multiple co-positive Lyapunov functions technique and conduct the dual design of the controller and the switching signal. Through introducing a new state variable, which is not the output error, the output tracking control problem of the original system is transformed into the stabilization problem of the dynamics system of this new state. The proposed approach is still effective even the output tracking control problem of any subsystem is unsolvable. According to the state being available or not, we establish the solvability conditions of the output tracking control problem for positive switched systems, respectively. In the end, a number example demonstrates the validity of the presented results.
Book
This book offers its readers a detailed overview of the synthesis of switched systems, with a focus on switching stabilization and intelligent control. The problems investigated are not only previously unsolved theoretically but also of practical importance in many applications: voltage conversion, naval piloting and navigation and robotics, for example. The book considers general switched-system models and provides more efficient design methods to bring together theory and application more closely than was possible using classical methods. It also discusses several different classes of switched systems. For general switched linear systems and switched nonlinear systems comprising unstable subsystems, it introduces novel ideas such as invariant subspace theory and the time-scheduled Lyapunov function method of designing switching signals to stabilize the underlying systems. For some typical switched nonlinear systems affected by various complex dynamics, the book proposes novel design approaches based on intelligent control concepts. It is a useful source of up-to-date design methods and algorithms for researchers studying switched systems and graduate students of control theory and engineering. In addition, it is a valuable reference resource for practising engineers working in switched-system control design. Readers should have a basic knowledge of linear, nonlinear and switched systems.
Chapter
Control systems often suffer from various limits or constraints in the operation space [1, 2], which may arise out of performance requirements or physical constraints imposed on the system by its environmental regulations.
Chapter
As mentioned in Chap. 2, for a switched system, even all its subsystems are stable, it may fail to preserve stability under arbitrary switching, but may be stable under restricted switching signals. Therefore, it is of significance to study the controlled-switching stabilization problems of switched systems. The controlled switching may result from the physical constraints of a system or the designers’ intervention [1] which is actually related to the controlled-switching stabilization problem [2]. Generally, the controlled switching in systems could be classified into state-dependent and time-constrained ones.
Chapter
It has been shown in Ye, Neurocomputing 71:3373–3378 (2008), Yu, et al., Neurocomputing 156:245–251 (2015), Peng, et al., Neurocomputing 149:132–141 (2015), Miao, Neurocomputing 111:184–189 (2013), Xu, Neurocomputing 154:337–346 (2015), [1–5] that the adaptive backstepping technique is a powerful tool which has been widely used to solve some complex optimization problems and applied in the fields of industry and engineering.
Chapter
Full-text available
Control synthesis of switched systems is always a hot study topic in the control field for its significance of both theory aspect and practical application. In the past few years, some control problems of switched systems have been successfully solved, but there still are quite many interesting topics deserving further investigation; some of them have been considered in the book.
Chapter
The last chapter has discussed adaptive control design methods for switched nonlinear systems with uncertainties.
Chapter
In a certain sense, switching signals in systems can be classified into autonomous (uncontrolled) or controlled ones [1, 2], which respectively, result from the system itself and the designers’ intervention [3].
Conference Paper
This paper mainly studied the exponential stability of singular switched systems with impulse effect. Without lose of generality, supposed that there are stable and unstable subsystems in the switched systems. Combining the multi-Lyapunov function with mode-dependent average dwell time method, some sufficient conditions for the exponential stability of singular switched systems with impulsive effect are obtained. Then two kinds of controllers are designed to ensure the stable of each subsystem and eliminate the impulse when switch occurs according to the given conditions.
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This paper is concerned with the problem of stability and L 1 -gain characterization for a class of switched positive systems consisting of both stable and unstable subsystems. Such systems can be modeled ingeniously as switched positive systems satisfying persistent dwell time switching. Compared to the widely used dwell time and average dwell time switching in the previous literature, persistent dwell time switching is more general due to its covering such two switchings as special cases. A new sufficient criterion ensuring the stability of switched positive systems is derived by using a persistent dwell time approach. And then an unweighted L 1 -gain is computed by solving a linear programming problem. The presented method in this paper may decrease the conservatism. Finally, the effectiveness and advantage of the provided method are illustrated with an example.
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This study addresses the control synthesis of positive switched systems in both continuous- and discrete-time contexts. First, a novel design approach to the state-feedback controller of continuous-time positive switched systems without time delay is presented by virtue of multiple linear copositive Lyapunov functions associated with linear programming. Then, the design approach is extended to discrete-time positive switched systems. It is shown that the presented approach is less conservative than existing ones through comparisons. Furthermore, the approach is developed to the stabilisation of positive switched systems with time delay. Some discussions on the presented approach are provided to show potential applications in corresponding control issues of positive switched systems. Finally, two examples are given to illustrate the theoretical findings.
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Chapter
This paper investigates the globally uniformly exponential stability of positive switched linear system (PSLS) in both continuous-time and discrete-time contexts. By using the multiple piecewise-continuous linear copositive Lyapunov function (MPLCLF) and exploring mode-dependent average dwell time (MDADT) switching, several stability criteria are developed with a switching strategy where slow switching and fast switching are applied to stable and unstable subsystems respectively. The proposed methods are also used to stabilize PSLS with controllable and uncontrolled subsystems. The obtained results provide lower bounds on MDADT of stable subsystems and higher bounds on MDADT of unstable subsystems and reduce the conservatism compared with the existing results. Finally, two numerical examples are provided to validate the advantages of the obtained results.
Chapter
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This work presents a hybrid nonlinear control methodology for a broad class of switched nonlinear systems with input constraints. The key feature of the proposed methodology is the integrated synthesis, via multiple Lyapunov functions, of “lower-level” bounded nonlinear feedback controllers together with “upper-level” switching laws that orchestrate the transitions between the constituent modes and their respective controllers. Both the state and output feedback control problems are addressed. Under the assumption of availability of full state measurements, a family of bounded nonlinear state feedback controllers are initially designed to enforce asymptotic stability for the individual closed-loop modes and provide an explicit characterization of the corresponding stability region for each mode. A set of switching laws are then designed to track the evolution of the state and orchestrate switching between the stability regions of the constituent modes in a way that guarantees asymptotic stability of the overall switched closed-loop system. When complete state measurements are unavailable, a family of output feedback controllers are synthesized, using a combination of bounded state feedback controllers, high-gain observers and appropriate saturation filters to enforce asymptotic stability for the individual closed-loop modes and provide an explicit characterization of the corresponding output feedback stability regions in terms of the input constraints and the observer gain. A different set of switching rules, based on the evolution of the state estimates generated by the observers, is designed to orchestrate stabilizing transitions between the output feedback stability regions of the constituent modes. The differences between the state and output feedback switching strategies, and their implications for the switching logic, are discussed and a chemical process example is used to demonstrate the proposed approach.
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This paper is concerned with the sliding mode control (SMC) of a continuous-time switched stochastic system. A sufficient condition for the existence of reduced-order sliding mode dynamics is derived and an explicit parametrization of the desired sliding surface is also given. Then, a sliding mode controller is then synthesized for reaching motion. Moreover, the observer-based SMC problem is also investigated. Some sufficient conditions are established for the existence and the solvability of the desired observer and the observer-based sliding mode controller is synthesized. Finally, numerical examples are provided to illustrate the effectiveness of the proposed theory.
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This paper addresses stabilization issue of switched nonlinear systems where some modes are stable and others may be unstable. A new stabilizing switching law that determines the initial states and the switching instants for any given switching sequence is proposed. The developed technique relies on the tradeoff among the functions’ gains of continuous modes, and does not depend on the constant ratio condition required in “dwell-time scheme”.
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This brief addresses the control problem of linear time-invariant discrete-time systems with delays. The control is under positivity constraint, which means that the resulting closed-loop systems are not only stable, but also positive. The contribution lies in three aspects. First, a necessary and sufficient condition is established for the existence of such controllers for discrete-time delayed systems. Second, a sufficient condition is provided under the additional constraint of bounded control, which means that the control inputs and the states of the closed-loop systems are bounded. Third, sufficient conditions are proposed for discrete-time delayed systems with uncertainties, whether or not bounded control is considered. All the results are formulated as linear programming problems, hence easy to be verified. And the controllers are explicitly constructed if existent.
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By a switched system, we mean a hybrid dynamical system consisting of a family of continuous-time subsystems and a rule that orchestrates the switching between them. The article surveys developments in three basic problems regarding stability and design of switched systems. These problems are: stability for arbitrary switching sequences, stability for certain useful classes of switching sequences, and construction of stabilizing switching sequences. We also provide motivation for studying these problems by discussing how they arise in connection with various questions of interest in control theory and applications
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We consider the problem of common linear copositive Lyapunov function existence for positive switched linear systems. In particular, we present a necessary and sufficient condition for the existence of such a function for switched systems with two constituent linear time-invariant systems. Several applications of this result are also given.
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This paper addresses the problem of stability analysis and control synthesis of switched systems in the discrete-time domain. The approach followed in this paper looks at the existence of a switched quadratic Lyapunov function to check asymptotic stability of the switched system under consideration. Two different linear matrix inequality-based conditions allow to check the existence of such a Lyapunov function. The first one is classical while the second is new and uses a slack variable, which makes it useful for design problems. These two conditions are proved to be equivalent for stability analysis. Investigating the static output feedback control problem, we show that the second condition is, in this case, less conservative. The reduction of the conservatism is illustrated by a numerical evaluation.
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The stability properties of linear switched systems consisting of both Hurwitz stable and unstable subsystems are investigated by using piecewise Lyapunov functions incorporated with an average dwell time approach. It is shown that if the average dwell time is chosen su#ciently large and the total activation time ratio between Hurwitz stable and unstable subsystems is not smaller than a specified constant, then exponential stability of a desired degree is guaranteed. The above result is also extended to the case where nonlinear norm-bounded perturbations exist. 1. Introduction By a switched system, we mean a hybrid dynamical system that is composed of a family of continuous-time subsystems and a rule orchestrating the switching between the subsystems. Recently, there has been increasing interest in the stability analysis and switching control design of such systems (see, e.g., [1]-[14] and the references cited therein). The motivation for studying such switched systems stems from the...