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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.
... 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: ...
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 ...
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. ...
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. ...
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. ...
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). ...
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). ...
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]. ...
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] . ...
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 ...
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. ...
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. ...
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] . ...
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. ...
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.
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.
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.
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.
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.
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.
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.
The last chapter has discussed adaptive control design methods for switched nonlinear systems with uncertainties.
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].
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.
A novel method to H ∞ output feedback control of continuous-time switched linear systems is investigated in this paper. Base on multiple Lyapunov function method combined with the Projection lemma and Finsler's lemma, the resulting closed-loop system is stable and has a desire H ∞ performance level γ under static and dynamic H ∞ output feedback, respectively. Both these controllers are solved from strict linear matrix inequalities, which does not decompose Lyapunov matrix and its inverse matrix while a state-coordinate transformation is not required. The validity and advantage of the proposed method are illustrated using an example.
This paper addresses the stability problem associated with a class of switched positive nonlinear systems in which each vector field is homogeneous, cooperative, and irreducible. Instead of using the Lyapunov function approach, we fully establish the invariant ray analysis method to establish several stability conditions that depend on the states, rays, and/or times. We illustrate the efficiency of our proposed approach using the example of a chemical reaction.
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.
The present work addresses the stability issue of switched linear systems (SLSs) subject to event-triggered control. We presume that the feedback controller acquires the switching signals and systems state information only at event-triggered sampling instants. An event triggering strategy is presented for both control and switching. Meanwhile, sufficient condition is obtained to stabilize the event-triggered switched linear systems. In addition, we prove that the Zeno behavior can be excluded through the proposed event triggering strategy. Finally, the speed regulation of a turbofan engine model shows that the derived technique is practicable.
In this note, we will devote to investigate the stability of discrete-time switched positive linear time-varying systems (PLTVSs). Firstly, a new asymptotic stability criterion of discrete-time PLTVSs is obtained by using time-varying copositive Lyapunov functions (TVCLFs) and this criterion is then extended to the switched case based on the multiple TVCLFs. Furthermore, the sufficient conditions are derived for stability of discrete-time switched PLTVSs with stable subsystems by means of function-dependent average dwell time and function-dependent minimum dwell time. In addition, the stability sufficient conditions are drawn for the switched PLTVSs which contain unstable subsystems. It is worth noting that the difference of TVCLFs and multiple TVCLFs are both relaxed to indefinite in our work. The theoretical results obtained are verified by two numerical examples.
This paper is concerned with the controller synthesis issue for enforced positive switched linear systems via output feedback. First, the stabilization problem is studied with output feedback under average dwell time switching signal, and the controllers we proposed guarantee stability and positivity of the closed-loop systems. Second, the output feedback stabilization issue is investigated by introducing special form of diagonal matrices, and the constraints on states and control inputs are solved based on limited initial conditions. Then, the derived conditions are described via linear programming, also extending the theoretical findings to constrained output issue. Finally, the simulation results demonstrate the feasibility of the control strategy.
In this paper, a complete procedure for the study of the output regulation problem is established for a class of positive switched systems utilizing a multiple linear copositive Lyapunov functions scheme. The feature of the developed approach is that each subsystem is not required to has a solution to the problem. Moreover, two types of controllers and switching laws are devised. The first one depends on the state together with the external input and the other depends only the error. The conditions ensuring the solvability of the problem for positive switched systems are presented in the form of linear matrix equations plus linear inequalities under some mild constraints. Two examples are finally given to show the performance of the proposed control strategy.
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.
This paper investigates the stability, stabilization and L1-gain of linear periodic piecewise positive systems. The monotonicity of linear periodic piecewise positive systems is first studied. Then a time-varying co-positive Lyapunov function for periodic piecewise positive systems is employed and a sufficient condition for the asymptotic stability of the system is established. Based on the provided co-positive Lyapunov function and the sufficient stability condition, a state-feedback periodic piecewise controller to stabilize the system is formed and an upper bound of L1-gain of the system is given. Finally, numerical examples are given to illustrate the theoretical results.
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.
This paper concerns the problem of stability for the discrete-time switched positive linear systems with mode-dependent average dwell time. By proposing a novel multiple discontinuous co-positive Lyapunov function approach, the conditions of stability are established for such system based on linear programming. Compared with the existing work, it turns out that our proposed approach produces less conservative results because of providing smaller bounds on the dwell time. A numerical example is given to show the effectiveness of our approach.
This note considers the problem of asymptotic stability for switched positive linear time-varying (LTV) systems. Some improved stability conditions are given for the positive LTV systems and switched positive LTV systems by using time-varying copositive Lyapunov functions (CLF) and switched time-varying CLF, respectively. Comparing with the existing results, the conditions obtained allow the time derivative of Lyapunov functions of subsystems (or systems) to be indefinite, and all subsystems to be unstable. Furthermore, using function-dependent dwell time methods, two improved stability conditions are also proposed for switched positive LTV systems. The conditions obtained allow the average dwell time of systems to have greater range than that of the existing results. Finally, a numerical example is given to illustrate the theoretical results.
Average dwell time (ADT) of switched systems is dependent on two parameters: the decay coefficient of the subsystems and the increase coefficient of the Lyapunov functions of different subsystems, namely, λ and μ, respectively. In the literature, the two parameters are prescribed because popular computation techniques cannot deal with bilinear conditions easily. A new method for the computation of ADT is presented in this study, where a suggested optimisation algorithm is proposed to obtain a loose ADT condition. The advantages of the method lie in twofold: (i) the parameters λ and μ do not need to be given, and (ii) the presented ADT conditions are tractable to solve. The method is also extended to mode-dependent ADT (MDADT), ADT in asynchronous control, and ADT and MDADT of positive switched systems.
Switched systems provide a unified framework for mathematical modeling of many physical or man-made systems displaying switching features such as power electronics, flight control systems, and network control systems. The systems consists of a collection of indexed differential or difference equations and a switching signal governing the switching among them. The various switching signals differentiate switched systems from the general time-varying systems, because the solutions of the former are dependent on not only the system’s initial conditions but also the switching signals.
This paper studies the problem of stochastic stability and stabilization for positive Markov jump systems with distributed time delay and incomplete known transition rates. By using delay partitioning technique, a new form of Lyapunov functional is constructed and a stochastic stability criterion is formulated for positive Markov jump systems with fully known transition rates. Based on this, the case of incomplete known transition rates is discussed. Then a state feedback controller is designed to guarantee the corresponding closed-loop system positive and stochastically stable. The results in this paper are expressed in the form of linear programming problems, and can be easily solved by standard software.
In this paper we consider the class of discrete-time switched systems switching between p autonomous positive subsystems. First, sufficient conditions for testing stability, based on the existence of special classes of common Lyapunov functions, are investigated, and these conditions are mutually related, thus proving that if a linear copositive common Lyapunov function can be found, then a quadratic positive definite common function can be found, too, and this latter, in turn, ensures the existence of a quadratic copositive common function. Secondly, stabilizability is introduced and characterized. It is shown that if these systems are stabilizable, they can be stabilized by means of a periodic switching sequence, which asymptotically drives to zero every positive initial state. Conditions for the existence of state-dependent stabilizing switching laws, based on the values of a copositive (linear/quadratic) Lyapunov function, are investigated and mutually related, too. Finally, some properties of the patterns of the stabilizing switching sequences are investigated, and the relationship between a sufficient condition for stabilizability (the existence of a Schur convex combination of the subsystem matrices) and an equivalent condition for stabilizability (the existence of a Schur matrix product of the subsystem matrices) is explored.
This paper investigates the exponential stability analysis problem for continuous-time and discrete-time positive systems with delays. Necessary conditions for exponential stability of such systems are presented based on the asymptotical stability results of linear positive time-delay systems, and sufficient conditions are provided by using Lyapunov-Krasovskii functional method. It is shown that, unlike asymptotical stability of positive time-delay systems, which is independent of delays, exponential stability of positive time-delay systems has to do with the magnitude of delays. Some illustrative examples are given to show the correctness of the obtained theoretical results.
In this paper we establish the convergence to an optimal non-interfering channel allocation of a class of distributed stochastic algorithms. We illustrate the application of this result via (i) a communication-free distributed learning strategy for wireless channel allocation and (ii) a distributed learning strategy that can opportunistically exploit communication between nodes to improve convergence speed while retaining guaranteed convergence in the absence of communication
This brief addresses stability of the discrete-time positive systems with bounded time-varying delays and establishes some necessary and sufficient conditions for asymptotic stability of such systems. It turns out that, for any bounded time-varying delays, the magnitude of the delays does not affect the stability of these systems. In other words, system stability is completely determined by the system matrices.
This technical note addresses controller design for continuous-time delayed linear systems. The control is under positivity constraint, which means that the resulting closed-loop systems are not only stable, but also positive. Bounded control is also considered, further requiring that the trajectory of the closed-loop system is bounded by a prescribed boundary, if the initial condition is bounded by the same boundary. The contribution lies in three aspects. First, some necessary and sufficient conditions are presented for the existence of controllers satisfying the positivity constraint. Second, a necessary and sufficient condition is established, determining whether the trajectory of a positive system can be bounded by a prescribed boundary, provided that the initial condition is bounded by the same boundary. Third, a sufficient condition is presented for bounded control, and it is shown to be also necessary if asymptotic stability is not required.
We consider n -dimensional positive linear switched systems. A necessary condition for stability under arbitrary switching is that every matrix in the convex hull of the matrices defining the subsystems is Hurwitz. Several researchers conjectured that for positive linear switched systems this condition is also sufficient. Recently, Gurvits, Shorten, and Mason showed that this conjecture is true for the case n = 2, but is not true in general. Their results imply that there exists some minimal integer np such that the conjecture is true for all n < n<sub>p</sub>, but is not true for n = n<sub>p</sub>. We show that n<sub>p</sub> = 3.
We consider planar bilinear control systems with measurable controls. We show that any point in the reachable set can be reached by a ldquonicerdquo control, specifically, a control that is a concatenation of a bang arc with either 1) a bang-bang control that is periodic after the third switch; or 2) a piecewise constant control with no more than two discontinuities. Under the additional assumption that the bilinear system is positive (or invariant for any proper cone), we show that the reachable set is spanned by a concatenation of a bang arc with either 1) a bang-bang control with no more than two discontinuities; or 2) a piecewise constant control with no more than two discontinuities. In particular, any point in the reachable set can be reached using a piecewise-constant control with no more than three discontinuities. Several known results on the stability of planar linear switched systems under arbitrary switching follow as corollaries of our result. We demonstrate this with an example.
This paper is concerned with the design of observers and dynamic output-feedback controllers for positive linear systems with interval uncertainties. The continuous-time case and the discrete-time case are both treated in a unified linear matrix inequality (LMI) framework. Necessary and sufficient conditions for the existence of positive observers with general structure are established, and the desired observer matrices can be constructed easily through the solutions of LMIs. An optimization algorithm to the error dynamics is also given. Furthermore, the problem of positive stabilization by dynamic output-feedback controllers is investigated. It is revealed that an unstable positive system cannot be positively stabilized by a certain dynamic output-feedback controller without taking the positivity of the error signals into account. When the positivity of the error signals is considered, an LMI-based synthesis approach is provided to design the stabilizing controllers. Unlike other conditions which may require structural decomposition of positive matrices, all proposed conditions in this paper are expressed in terms of the system matrices, and can be verified easily by effective algorithms. Two illustrative examples are provided to show the effectiveness and applicability of the theoretical results.
Continuous-time positive systems, switching among p subsystems, are introduced, and a complete characterization for the existence of a common linear copositive Lyapunov function for all the subsystems is provided. When the subsystems are obtained by applying different feedback control laws to the same continuous-time single-input positive system, the above characterization leads to a very easy checking procedure.
This paper addresses the stabilization problem of positive linear systems which have nonnegative states whenever the initial conditions are nonnegative. The synthesis of static output-feedback controllers that ensure the positivity and asymptotic stability of the closed-loop system is investigated. It is shown that this important problem is completely solved for single-input and single-output positive systems. The proposed approach can be applied to multi-input positive systems with controllers having one rank gains. All the provided conditions are necessary and sufficient and can be solved in terms of Linear Programming.
This paper is a tutorial on the positive realization problem, that is the problem of finding a positive state-space representation of a given transfer function and characterizing existence and minimality of such representation. This problem goes back to the 1950s and was first related to the identifiability problem for hidden Markov models, then to the determination of internal structures for compartmental systems and later embedded in the more general framework of positive systems theory. Within this framework, developing some ideas sprang in the 1960s, during the 1980s, the positive realization problem was reformulated in terms of a geometric condition which was recently exploited as a tool for finding the solution to the existence problem and providing partial answers to the minimality problem. In this paper, the reader is carried through the key ideas which have proved to be useful in order to tackle this problem. In order to illustrate the main results, contributions and open problems, several motivating examples and a comprehensive bibliography on positive systems organized by topics are provided.
This brief concerns the stabilisation problem for a class of slowly switched positive linear systems in discrete-time context. The average dwell-time switching associated with the corresponding state-feedback controllers is designed to stabilise the closed-loop systems and keep the states non-negative. The developed conditions are formulated as linear matrix inequalities, which can be directly used for controller synthesis and switching designing. Finally, a numerical example is presented to show the feasibility of the obtained theoretical results.
This study investigates the stabilisation problem of discrete-time switched positive linear systems by means of piecewise linear copositive Lyapunov functions. Two stabilisation strategies are designed under time- and state-dependent switching cases, respectively. The former case aims at determining an upper bound of the minimum dwell time to guarantee that the underlying system is stable for any switching signal with dwell time greater than this bound. The latter case is focused on deriving a state-dependent switching law stabilising the underlying system from the solution of a family of so-called linear copositive Lyapunov-Metzler inequalities. In each case, a sufficient stabilisation condition is given first, then based on which an associated guaranteed cost is further proposed. A practical system derived from the distributed power control in communication networks is given to illustrate the effectiveness and applicability of the theoretical results.
In this study, the state estimation problem for a class of slowly switched positive linear systems (SPLS) is investigated. A multiple linear copositive Lyapunov function (MLCLF) is preliminarily established, by which the state estimator is designed for the underlying systems with an average dwell-time switching. Then, by reducing MLCLF to the common linear copositive Lyapunov function, the results for the SPLS under arbitrary switching can be easily obtained. Finally, a numerical example is given to show the fesibility of the obtained theoretical findings.
In this paper, the stability analysis problem for a class of switched positive linear systems (SPLSs) with average dwell time switching is investigated. A multiple linear copositive Lyapunov function (MLCLF) is first introduced, by which the sufficient stability criteria in terms of a set of linear matrix inequalities, are given for the underlying systems in both continuous-time and discrete-time contexts. The stability results for the SPLSs under arbitrary switching, which have been previously studied in the literature, can be easily obtained by reducing MLCLF to the common linear copositive Lyapunov function used for the system under arbitrary switching those systems. Finally, a numerical example is given to show the effectiveness and advantages of the proposed techniques.
This paper is concerned with the model reduction of positive systems. For a given stable positive system, our attention is focused on the construction of a reduced-order model in such a way that the positivity of the original system is preserved and the error system is stable with a prescribed H∞ performance. Based upon a system augmentation approach, a novel characterization on the stability with H∞ performance of the error system is first obtained in terms of linear matrix inequality (LMI). Then, a necessary and sufficient condition for the existence of a desired reduced-order model is derived accordingly. Furthermore, iterative LMI approaches with primal and dual forms are developed to solve the positivity-preserving H∞ model reduction problem. Finally, a compartmental network is provided to show the effectiveness of the proposed techniques.
In this paper, the problem of exponential H∞ filter problem for a class of discrete-time polytopic uncertain switched linear systems with average dwell time switching is investigated. The exponential stability result of the general discrete-time switched systems using a discontinuous piecewise Lyapunov function approach is first explored. Then, a new µ-dependent approach is proposed, which means the analysis or synthesis of the underlying systems is dependent on the increase degree µ of the piecewise Lyapunov function at the switching instants. A mode-dependent full-order filter is designed such that the developed filter error system is robustly exponentially stable and achieves an exponential H∞ performance. Sufficient existence conditions for the desired filter are derived and formulated in terms of a set of linear matrix inequalities, and consequently the minimal average dwell time and the corresponding filter are obtained from such conditions for a given system decay degree. A numerical example is presented to demonstrate the potential and effectiveness of the developed theoretical results. Copyright © 2007 John Wiley & Sons, Ltd.
We consider stabilization of equilibrium points of positive linear systems which are in the interior of the ÿrst orthant. The existence of an interior equilibrium point implies that the system matrix does not possess eigenvalues in the open right half plane. This allows to transform the problem to the stabilization problem of compartmental systems, which is known and for which a solution has been proposed already. We provide necessary and suucient conditions to solve the stabilization problem by means of aane state feedback. A class of stabilizing feedbacks is given explicitly. c 2001 Elsevier Science B.V. All rights reserved.
This brief addresses the stability problem of discrete-time switched positive systems. The main contribution lies in two aspects. First, a novel method [the switched linear copositive Lyapunov function (SLCLF)] is proposed to reduce the conservatism of the stability criterion obtained by means of the common linear copositive Lyapunov function. Second, some necessary and sufficient conditions are established for the existence of the switched linear copositive Lyapunov function, assuming that a switched linear positive system is given. A numerical example shows the advantage of the obtained results.
Dwell time results are established for a class of switched linear positive systems. Results are also given for a class of time-delayed switched positive systems. An upper bound to the L<sub>1</sub>-induced norm of switched linear positive systems is computed. Examples are also given to illustrate the efficacy of our results.
This note studies the asymptotic stability of switched positive linear discrete systems whose subsystems are ( sp ) matrices. Such a matrix is the character of a kind of asymptotically stable linear systems and it is very easy to test. A new definition of ( sp ) matrix is given by means of graph theory. Based on an approaching using partially ordered semigroups and Lie algebras, we present several new criteria for asymptotic stability. We also derive an algebraic condition and discuss a kind of higher order difference equation. Our results have a robustness property to some extent.
We study the common linear copositive Lyapunov functions of positive linear systems. Firstly, we present a theorem on pairs of second order positive linear systems, and give another proof of this theorem by means of properties of geometry. Based on the process of the proof, we extended the results to a finite number of second order positive linear systems. Then we extend this result to third order systems. Finally, for higher order systems, we give some results on common linear copositive Lyapunov functions.
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.
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.
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”.
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.
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
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.
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.
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...