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# On Linear Copositive Lyapunov Functions and the Stability of Switched Positive Linear Systems

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## Abstract

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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... Such a class of systems has been widely applied in chemical engineering [4], ecology [5], network employing [6], and so on. Some important results on positive switched systems [7][8][9][10][11] have been reported. A human immunodeficiency viral mutation model was constructed via positive switched systems [7]. ...
... Gurvits et al. [8] stated that the stability of the convex hull of system matrices cannot guarantee the stability of positive switched systems. Using a common copositive Lyapunov function [9], the exponential stability of positive switched systems was presented. The state-feedback stabilization of positive switched systems was explored using a piecewise linear Lyapunov function [10]. ...
... This together with (8a) gives The Lyapunov function (12) is a switched copositive Lyapunov function [12], which is different from common copositive Lyapunov function [9,11] and multiple copositive Lyapunov functions [14,16,18]. Under the common copositive Lyapunov function, a switched system is stable for arbitrary switching. ...
Article
This paper proposes a reliable control of positive switched systems with random nonlinearities which may induce the security problem of the systems. The random nonlinearities are governed by stochastic variables obeying the Bernoulli distribution. A switched linear copositive Lyapunov function is employed for the systems. Using a matrix decomposition approach, the gain matrix of controller is formulated by the sum of nonnegative and non-positive components. A reliable controller is designed for positive switched systems with actuator faults by virtue of linear programming. Under the designed reliable controller, the systems can resist some possible security risks triggered by random nonlinearities and actuator faults. The obtained approach is developed for systems subject to exogenous disturbances. Finally, two examples are provided to verify the validity of the obtained results.
... This class of systems comprises finite time delay PLSs with a switching signal to switch among them. This system has been used in various applications like compartment systems, epidemic dynamics, congestion control, and drug delivery control of mammillary systems [10][11][12][13]. ...
... Proof. Consider the L-K function, which is defined in Equation (13). By differentiation from the L-K candidate along the trajectory of system (1), the following relations can be evaluated for V 1 (the time differentiation ofV 2 ,V 3 ,V 4 is presented in Equation (14)): ...
... The function (t ) is presented in (13). The time derivative of the L-K function candidate is the same as its derivative in Theorem 1 for V 2 , V 3 , V 4 (Equation (13)). ...
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Abstract This paper investigates a bumpless transfer control problem for uncertain switched positive linear time‐delay systems (SPLTDSs) with discrete and distributed delays. The innovation of this study is to develop a linear feedback controller and design a switching law that ensures lower controller signal bumps due to the switches of the SPLTDSs and also guarantee the L1–gain performance of the system. To this end, a bumpless transfer performance is proposed, and stabilisation constraints for SPLTDSs with synchronous switching are developed. Dwell time criteria are utilised in the stability analysis. Also, the results are improved to cover the uncertain SPLTDSs with interval or polytopic uncertainties. Then, asynchronous switching is considered, and the proposed method is extended to deals with asynchronous switching mode. Asynchronous switching means switches of the controller have lagged behind the subsystem switches. All the stability conditions are derived by multiple co‐positive Lyapunov–Krasovskii functional for synchronous and asynchronous switching conditions. At last, two numerical examples are presented to show the effectiveness of the proposed method compared with the leading existing solutions.
... The family of the Metzler matrices play a crucial role in many circumstances, as explained and illustrated for instance in the contributions [15], [7], [9], [10], [16], [1]. In particular, they are crucial in the theory of the positive systems and they can be used to establish stability results for both continuous and discrete-time systems [20], [23], [4], [9], possibly with delay [12], [25]. ...
... with R defined in (44) is satisfied and H defined in (16). ...
... where H is the matrix defined in (16) and R is the matrix defined in (44), is an interval observer for the system (48), i.e. if the inequalities ...
... In the linear case, the stability analysis of this kind of systems is characterized by their joint spectral radius (JSR) whose approximation has been studied for years (see [13] for a survey). In this paper, we consider the templates of primal and dual linear copositive functions (already considered in [17]), studying their analytical properties and the corresponding lifts. As final by-product of our techniques, we propose a new hierarchy of linear programs (based on pathcomplete Lyapunov certificates using primal and dual norms templates) in order to approximate the JSR of a set of nonnegative matrices up to an arbitrary accuracy. ...
... See Figure 6 for a graphical interpretation of this class of functions. In the context of positive switching systems, copositive norms as in (12) were considered, among many other examples, in [17,9]. In order to highlight the closure properties of these templates, we adopt the following notation; ...
... (1) In this table, in the line denoted by ρ G we reported the optimal values of the LPs described by (16), (17) for the corresponding (primal and dual) De Bruijn graphs. We have thus proven that ρ(A) ∈ [1.065, 1.070], having an instability certificate for the positive switching system (11) defined by A. It is interesting to note how, in this particular case, the conditions arising from the primal De Bruijn graphs and the template of dual copositive norms provide better upper bounds for the JSR. ...
Preprint
This paper investigates, in the context of discrete-time switching systems, the problem of comparison for path-complete stability certificates. We introduce and study abstract operations on path-complete graphs, called lifts, which allow us to recover previous results in a general framework. Moreover, this approach highlights the existing relations between the analytical properties of the chosen set of candidate Lyapunov functions (the template) and the admissibility of certain lifts. This provides a new methodology for the characterization of the order relation of path-complete Lyapunov functions criteria, when a particular template is chosen. We apply our results to specific templates, notably the sets of primal and dual copositive norms, providing new stability certificates for positive switching systems. These tools are finally illustrated with the aim of numerical examples.
... Here, two positive L 1 fuzzy filters are constructed to estimate the states of system (6). It follows from Lemma 1 that the filters (20) and (21) are positive ifĀ θα and A θα are Metzler matrices,B θα ≥≥ 0, B θα ≥≥ 0,C θα ≥≥ 0, and C θα ≥≥ 0. ...
... Substituting (22) into Corollary 1, it will result in the solution for positive L 1 fuzzy upper-bounding filter (20) in Theorem 3. ...
... Proof: From (33), we obtain thatĀ θα is the Metzler matrix. From (34), it implies thatB θα ≥≥ 0 andC θα ≥≥ 0, which means positivity of the filter (20). ...
Article
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In this article, the issue of positive L₁ filter design is investigated for positive nonlinear stochastic switching systems subject to the phase-type semi-Markov jump process. Many complicated factors, such as semi-Markov jump parameters, positivity, T-S fuzzy strategy, and external disturbance, are taken into consideration. Practical systems under positivity constraint conditions and unpredictable structural changes are characterized by positive semi-Markov jump systems (S-MJSs). First, by the key properties of the supplementary variable and the plant transformation technique, phase-type S-MJSs are transformed into Markov jump systems (MJSs), which means that, to an extent, these two kinds of stochastic switching systems are mutually represented. Second, with the help of the normalized membership function, the associated nonlinear MJSs are transformed into the local linear MJSs with specific T-S fuzzy rules. Third, by choosing the linear copositive Lyapunov function (LCLF), stochastic stability (SSY) criteria are given for the corresponding system with L₁ performance. Some solvability conditions for positive L₁ filter are constructed under a linear programming framework. Finally, an epidemiological model illustrates the effectiveness of the theoretical findings.
... The literature devoted to the behaviour of type (1) offers analysis tools for both the general case of switching systems (with n n i   A ℝ arbitrary stable matrices) such as [1]- [5], and the particular case of switching positive systems (with n n i   A ℝ nonnegative / essentially nonnegative stable matrices), such as [6]- [15]. Most of the analysis tools provide tests for Lyapunov function candidates of different forms, which rely on various types of algebraic conditions. ...
... Moreover, this case can be particularized for system (1) with nonnegative dynamics. The particularization provides the same algebraic conditions as presented in [11] for discrete-time ( 0 1 r   ), and in [6], [8], [9] for continuous-time ( 0 r  ), for characterizing the existence of linear copositive Lyapunov functions of form 1 1 1 ...
... SPSs are usually made up of a switching signal and a bunch of positive subsystems. The SPSs can be discovered in plenty of engineering areas such as networks utilising TCP, image processing and formation control of unmanned aerial systems [14][15][16][17]. Note that owning to the required positive property of SPSs, the analysis and stabilisation strategies pertained to switched systems are not suitable for the former systems any more. ...
... and (14)- (16) hold true for nonnegative matrices B u , C u , D u , F u , (I + Q), then system (1) is GES and meanwhile the desired system performance can also be ensured if the switching signal (t ) satisfies (17). Moreover, the controller can be calculated as ...
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Abstract The current study is dedicated to addressing the stability and weighted L1‐gain performance analysis of switched impulsive positive systems (SIPSs). Firstly, by designing a novel multiple piecewise‐continuous copositive linear Lyapunov function and using the mode‐dependent average dwell time (MDADT) switching method, improved stability conditions that are able to achieve a tighter dwell time bound are developed. It is shown that the system under study is stable and possesses an attenuation property under the designed switching signals. Secondly, with the above stability conditions, a more effective controller design strategy has been proposed. The solved controllers are both quasi‐time dependent and mode‐dependent. Moreover, the constraint on the rank of the controller and the computation burden caused by iteration algorithm used in literature are relaxed. Finally, the feasibility and superiority of the proposed strategies are verified by some simulations.
... Because of the existence of positive constraints in the switched systems, numerous results from the study on normal switched systems may not be applicable to SPSs. 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]. ...
Article
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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.
... This approach is used in [24] to study data describing daily precipitation measurements. Further applications of the nonnegative factorization of completely positive matrices can be found in data mining and clustering ( [28]), and in automatic control ( [13,44]). ...
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We aim to factorize a completely positive matrix by using an optimization approach which consists in the minimization of a nonconvex smooth function over a convex and compact set. To solve this problem we propose a projected gradient algorithm with parameters that take into account the effects of relaxation and inertia. Both projection and gradient steps are simple in the sense that they have explicit formulas and do not require inner loops. Furthermore, no expensive procedure to find an appropriate starting point is needed. The convergence analysis shows that the whole sequence of generated iterates converges to a critical point of the objective function and it makes use of the Łojasiewicz inequality. Its rate of convergence expressed in terms of the Łojasiewicz exponent of a regularization of the objective function is also provided. Numerical experiments demonstrate the efficiency of the proposed method, in particular in comparison to other factorization algorithms, and emphasize the role of the relaxation and inertial parameters.
... Different from general systems [3,4], positive switched systems have their own research approaches. For example, linear Lyapunov functions are usually employed [5,6], linear programming is chosen as the computation method [7,8], and linear L 1 -gain performance is used for the disturbance attenuation analysis [9,10]. Some research issues have been explored for positive systems such as stability [11,12], performance analysis [13,14], control synthesis [15,16], and so on. ...
Article
This paper proposes a novel double switchings reliable control for positive switched systems. Double switchings refer to the switchings of subsystems and faults. Considering the actuator faults during the system operation, a class of periodic switched actuator faults is first considered. A reliable controller is designed using linear copositive Lyapunov functions and an average dwell time switching rule is presented for the systems. Then, two average dwell time switching rules are given for the subsystems and the faults, respectively. A corresponding reliable controller is proposed in terms of linear programming. Compared with existing results, the novel approach contains three advantages: (i) the actuator faults of subsystems are allowed to be varying, (ii) two classes of detailed double switchings rules are designed, and (iii) the presented reliable control framework provides a new methodology to deal with the reliable control problem of general switched systems with faults. Finally, two examples are given to illustrate the validity of the presented results.
... Loop this sequence and the initial state be [1,4] T , the simulation result is given in Figure 1, which shows the convergence of the system. The corresponding Lyapunov function is illustrated in Figure 1B. Figure 1B shows that switching behavior between subsystem 1 and subsystem 2 is destabilizing, which causes an increase in the energy function V(t). ...
Article
This article studies the exponential stability of continuous‐time switched positive systems consisting of unstable subsystems. Different from the existing results, both stabilizing and destabilizing switching behaviors act in the switching sequences. By employing multiple composite copositive Lyapunov functions, sufficient condition is derived to ensure the exponential stability of the system, which evaluates the ratio of stabilizing switching behaviors to compensate the state divergence caused by either unstable subsystems or destabilizing switching behaviors. Simulations demonstrate the effectiveness of the result.
... In a linear switched positive system with time-varying delay, every subsystem itself is positive. Some theoretical and practical results for discretetime linear switched positive systems with delay have been investigated in literature [20,21]. An important point about switched positive systems with delay is that the control design method should cause the closed-loop switched delay system to be positive. ...
... Notably, the switched systems in which all subsystems are positive are called switched positive systems. Some applications of switched positive systems can be found in congestion control (Bolajraf et al., 2010), wireless power control (Mason & Shorten, 2007), the epidemiological models and the power regulation of transmitters (Blanchini et al., 2015), Takagi-Sugeno (T-S) fuzzy model (S. Li & Xiang, 2018). ...
Article
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In this paper, the problem of finite-time stability and finite-time boundedness for a class of linear switched singular positive time-delay systems with finite-time unstable subsystems is investigated. Necessary and sufficient positivity conditions for the systems are firstly presented by using the state-space singular value decomposition and monomial coordinate transformation methods. A class of quasi-alternative switching signals is then designed to analyze the switching behaviours of the systems. Mode-dependent average dwell time (MDADT) switching, consisting of a slow mode-dependent average dwell time (SMDADT) switching and a fast mode-dependent average dwell time (FMDADT) switching, is applied to the systems whose subsystems are stable and unstable. By establishing a suitable copositive Lyapunov-Krasovskii functional and adopting the MDADT switching strategy, all sufficient conditions guaranteeing the finite-time stability and finite-time boundedness of the considered systems are formulated in terms of linear vector inequalities as well as linear matrix inequalities. Several numerical examples are provided to illustrate the effectiveness of the proposed methods. Notably, a positive electrical circuit model is utilized to demonstrate the applicability of the theoretical results.
... Switched systems, which are composed of a family of subsystems and a switching rule orchestrating the switching among subsystems, are an important class of hybrid systems [5][6][7]. 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]. ...
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 internally positive linear timeinvariant systems vastly appear in various fields such as engineering, economics, pharmacy, and chemistry. An intensive amount of research work has been done for the analysis and synthesis of linear time-invariant systems by making use of convex optimization techniques [3][4][5][6][7][8]. If the output to the linear time-invariant system is non-negative for the non-negative input with zero initial rates, then such a system is externally positive [1,2]. ...
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In this article, we present a stability analysis of linear time-invariant systems in control theory. The linear time-invariant systems under consideration involve the diagonal norm bounded linear differential inclusions. We propose a methodology based on low-rank ordinary differential equations. We construct an equivalent time-invariant system (linear) and use it to acquire an optimization problem whose solutions are given in terms of a system of differential equations. An iterative method is then used to solve the system of differential equations. The stability of linear time-invariant systems with diagonal norm bounded differential inclusion is studied by analyzing the Spectrum of equivalent systems.
... Therefore, consider the following three-phase iterative procedure (Algorithm 1): Phase 1 (Stability): Define a maximum number of iterations it max , the accuracy of the guaranteed cost tol and a sufficiently large initial value for ρ = ρ 0 . Initialize m = 0 (iteration counter), B g (ᾱ) and B t (α) as in (15) satisfying (18), apply a relaxation in the stabilizability constraint by solving (7) (replacing A cl (α) byĀ(α) = (1 − r)A cl (α)), (8) and (9), and consider (19) instead of (10). ...
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This paper investigates the problem of designing H2 robust (or gain-scheduled) static output-feedback (or state-feedback) controllers for discrete-time positive linear systems affected by time-invariant (or time-varying) parameters belonging to a polytope. For this purpose, an iterative procedure based on robust (parameter-dependent) linear matrix inequalities is proposed. Unlike most approaches, where the controller is obtained by means of a change of variables, the synthesis conditions deal with the control gain directly as an optimization variable, which is specially appealing to cope with closed-loop positivity or structural constraints. The existence of feasible initial conditions for the iterative procedure and some relaxation strategies adopted to reduce the conservativeness of the method are also discussed. Numerical examples borrowed from the literature and statistical comparisons show that the proposed technique is in general less conservative than other approaches, providing solutions and handling cases where traditional design techniques for positive systems cannot be applied.
... In practice, many control systems contain quantities taking nonnegative values such as communication networks, biological systems, industrial processes, etc. Positive systems can describe the systems consisting of nonnegative quantities [5]- [7]. Consequently, a new class of switched systems, named positive switched systems, was introduced [8]- [10]. In [11] and [12], a common linear copositive Lyapunov function was used for the stability of positive switched systems under arbitrary switching. ...
Article
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This paper investigates the double switchings reliable control of positive switched systems with actuator faults. Different from existing fixed actuator faults, a class of unfixed actuator faults is introduced, where each subsystem can have several faults rather than only a fault. First, a periodic occurring fault is considered for the subsystems. Using linear Lyapunov functions with linear programming, a set of reliable controllers is designed for positive switched systems. Then, an average dwell time based switching law between the normal and faulty modes of subsystems is established. In detail, the considered systems consist of several switched subsystems, where the subsystems have normal and faulty modes. In such a case, two classes of multiple linear Lyapunov functions are constructed for the subsystems and the whole switched systems, respectively. Accordingly, two classes of reliable controllers are designed for the normal and faulty modes, respectively. Meanwhile, two classes of average dwell time switching laws, named double switchings laws, are designed for the subsystems and the whole switched systems, respectively. Finally, two examples are provided to illustrate the effectiveness of the proposed design.
... As customary, the stability of an equilibrium point of a given dynamic system can be studied through the Lyapunov function. Lyapunov functions can be seen as stability certificates for ordinary differential equations (ODEs) [12], [13]. The problem of finding the Lyapunov function is generally complex, and it has been the scope of many research papers in the control community [14]. ...
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In this note, a novel methodology that can extract a number of analysis results for linear time-invariant systems (LTI) given only a single trajectory of the considered system is proposed. The superiority of the proposed technique relies on the fact that it provides an automatic and formal way to obtain valuable information about the controlled system by only having access to a single trajectory over a finite period of time (i.e., the system dynamics is assumed to be unknown). At first, we characterize the stability region of LTI systems given only a single trajectory dataset by constructing the associated Lyapunov function of the system. The Lyapunov function is found by formulating and solving a linear programming (LP) problem. Then, we extend the same methodology to a variety of essential analysis results for LTI systems such as deriving bounds on the output energy, deriving bounds on output peak, deriving $\mathbf{L}_2$ and RMS gains. To illustrate the efficacy of the proposed data-driven paradigm, a comparison analysis between the learned LTI system metrics and the true ones is provided.
... Considerations of switching behaviors and their applications are generally investigated in the community of mathematicians, scientists, and engineers. Especially, switched positive systems (SPSs) comprising all individual positive subsystems have been successfully applied and solved in many real-world problems, for instance, formation flying [4], network communication [5], wireless power control [6], congestion control [7], compartmental model [8], water-quality model [9], and positive circuit model [10]. ...
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Many practical systems can be modeled in terms of uncertainties, which refer to the differences or errors between actual data and mathematical simulations. However, systems including slight uncertainties and exogenous disturbances may lead to the instability of those systems. Besides, the behavior of systems is preferable to investigate within a prescribed bound over a fixed time interval. Therefore, in this paper, we study a robust finite-time control of discrete-time linear switched positive time-varying delay systems with interval uncertainties and exogenous disturbance. A distinctive feature of this research is that the considered systems consist of finite-time bounded subsystems and finite-time unbounded subsystems. A class of quasi-alternative switching signals is validly designed to analyze the mechanism and switching behaviors of the systems among their subsystems. By utilizing a copositive Lyapunov–Krasovskii functional method combined with the slow mode-dependent average dwell time and the fast mode-dependent average dwell time switching techniques, new sufficient conditions containing several symmetric negative-definite matrices are derived to guarantee robust finite-time control of the systems. These results are applied to a water-quality controllability model in streams to the standard level. Finally, the consistent results between the theoretical analysis and the corresponding numerical simulations are shown.
... When all subsystems of switched systems are positive under the specific switching rules, the systems are well known as switched positive systems (SPSs). Their applications can encounter in various areas, such as compartmental model [5], water-quality model [6], formation flying [7], congestion control [8], wireless power control [9], and network communication using transmission control protocol [10]. Furthermore, the system's behavior that relies not only on the present state but also on the past state is discovered in many situations, for example, fluid and mechanical transmissions, metallurgical processes, and networked communications. ...
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The problem of finite-time boundedness for a class of linear switched positive time-varying delay systems with interval uncertainties and exogenous disturbance is addressed. This characteristic research is that the studied systems include the finite-time bounded subsystems and finite-time unbounded subsystems. Both a slow mode-dependent average dwell time and a fast mode-dependent average dwell time switching techniques are utilized reasonably. And by applying a copositive Lyapunov-Krasovskii functional, novel delay-dependent sufficient criteria are derived to guarantee such systems to be finite-time bounded concerning the given parameters and designed switching signal. Furthermore, new finite-time boundedness criteria of the systems without interval uncertainties are also obtained. Finally, the efficiency of the theoretical results is presented in two illustrative examples.
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The stability problem for positive switched systems with two subsystems is investigated with an improved mode-dependent average dwell time (MDADT) scheme. By utilizing system positivity and the new scheme, the stability conditions of the considered system are addressed. The proposed switching law is more applicable in practice than the average dwell time (ADT) and MDADT schemes with certain conditions, and those existing methods can be regarded as some special cases of the theoretical findings in the paper. Then, the new stability criteria are extended to non-autonomous systems. Finally, two simulation examples with some comparisons to the classical results are given to illustrate the advantages of our obtained results.
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This paper investigates the robustness of exponential stability of a class of switched systems described by linear functional differential equations under arbitrary switching. We will measure the stability robustness of such a system, subject to parameter affine perturbations of its constituent subsystems matrices, by introducing the notion of structured stability radius. The lower bounds and the upper bounds for this radius are established under the assumption that certain associated positive linear systems upper bounding the given subsystems have a common linear copositive Lyapunov function. In the case of switched positive linear systems with discrete multiple delays or distributed delay the obtained results yield some tractably computable bounds for the stability radius. Examples are given to illustrate the proposed method.
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This paper investigates the event-triggered control of positive switched systems with randomly occurring actuator saturation and time-delay, where the actuator saturation and time-delay obey different Bernoulli distributions. First, an event-triggering condition is constructed based on a 1-norm inequality. Under the presented event-triggering scheme, an interval estimation method is utilized to deal with the error term of the systems. Using a co-positive Lyapunov functional, the event-triggered controller and the cone attraction domain gain matrices are designed via matrix decomposition techniques. The positivity and stability of the resulting closed-loop systems are reached by guaranteeing the positivity of the lower bound of the systems and the stability of the upper bound of the systems, respectively. The proposed approach is developed for interval and polytopic uncertain systems, respectively. Finally, two examples are provided to illustrate the effectiveness of the theoretical findings.
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In this paper, the global asymptotic stability (GAS) of continuous-time and discrete-time nonlinear impulsive switched positive systems (NISPS) are studied. For continuous-time and discrete-time NISPS, switching signals and impulse signals coexist. For both of these systems, using the multiple max-separable Lyapunov function method and average dwell-time (ADT) method, some sufficient conditions on GAS are given. Based on these, the GAS criteria are also given for continuous-time and discrete-time linear impulsive switched positive systems (LISPS). From our criteria, the stability of the systems can be judged directly from the characteristics of the system functions, switching signals and impulse signals of the systems. Finally, simulation examples verify the validity of the results.
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In the framework of discrete-time switching systems, we analyze and compare various stability certificates relying on graph constructions. To this aim, we define several abstract expansions of graphs (so-called lifts), which depend on the chosen family of candidate Lyapunov functions (the template). We show that the validity of a given lift is linked with the analytical properties of the template. This allows us to generate new lifts, and as a byproduct, to obtain comparison criteria that go beyond the concept of simulation recently introduced in the literature. We apply our constructions to the case of copositive linear norms for positive switching systems, leading to novel stability criteria that outperform the state of the art. We provide further results relying on convex duality and we demonstrate via numerical examples how the comparison among different stability criteria is affected by the properties of the copositive norms template.
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This paper is concerned with the event-triggered L1-gain control of a class of nonlinear positive switched systems. First, an event-triggering condition in the form of 1-norm is presented for the systems. By virtue of the event-triggering strategy, the original system is transformed into an interval uncertain system. An event-triggered L1-gain controller is designed by decomposing the controller gain matrix into the sum of nonnegative and non-positive components. Under the design controller, the resulting closed-loop systems are positive and L1-gain stable. The obtained approach is developed for the systems subject to input saturation. All presented conditions are solvable in terms of linear programming. Finally, two examples are provided to verify the effectiveness of the design.
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In this paper, the asynchronous ℓ1 filtering issue is investigated for a class of discrete-time switched positive systems. An improved semi-time-dependent framework is established, which is compatible with the switching feature of asynchronously switched systems. First of all, the stability and ℓ1-gain analysis for switched positive linear systems are studied by using a semi-time-dependent multiple copositive Lyapunov function. On this foundation, both upper-bound and lower-bound positive filters are designed, which can ensure the positivity, stability and non-weighted ℓ1-gain of the corresponding filtering error systems. Moreover, the developed bounded filtering scheme is semi-time-dependent, which can achieve less conservative performance compared with the conventional time-independent methodology. Finally, an example is provided to demonstrate the potentials and merits of the proposed filtering scheme.
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This paper deals with the stabilization problem for non-smooth variable-order Riemann-Liouville fractional switched systems with all modes unstable in the presence of unknown nonlinearity. A controller containing the discontinuous switching item and Riemann-Liouville fractional-order derivative term is firstly designed. By applying fractional order calculation, non-smooth analysis theory and Lyapunov stability theory, some criteria are established under the joint design of controller and state-dependent switching law. An application to variable-order fractional switched permanent magnet synchronous motors is demonstrated and relevant numerical simulations for considered system are given to verify the validity of our designed scheme.
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This paper studies the stochastic stability of positive Markov jump linear systems with a fixed dwell time. By constructing an auxiliary system that originated from the initial system with state jumps, sufficient and necessary conditions of stochastic stability for positive Markov jump linear systems are obtained with both exactly known and partially known transition rates. The main idea in the latter case is applying a convex combination to convert bilinear programming into linear programming problems. On this basis, multiple piecewise linear co-positive Lyapunov functions are provided to achieve less conservative results. Then state feedback controller is designed to stabilize the positive Markov jump linear systems by solving linear programming problems. Numerical examples are presented to illustrate the viability of our conclusions.
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This paper studies the exponential stability of switched positive system consisting of unstable subsystems with distributed time-varying delay. Unlike the existing results concerning delays, switching behaviors dominating the system can be either stabilizing or destabilizing. The distributed delay is supposed to be slowly varying and upper-bounded. To tackle the difficulties brought by both the switching behaviors with mixed effects and the distributed delay, a multiple discretized Lyapunov-Krasovskii functional is employed to derive sufficient conditions for the exponential stability of the system. Specifically, by adjusting the ratio of the stabilizing switching behaviors, the state divergence caused by unstable subsystems and destabilizing switching behaviors can be compensated. Simulation examples demonstrate the effectiveness of the results.
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This paper is concerned with the stability analysis for positive switched systems, based on a so-called Φ-dependent average dwell time scheme that covers the classical average dwell time and mode-dependent average dwell time ones as two special cases. By a modified multiple linear copositive Lyapunov functions approach, stability criteria with the new scheme are established for positive switched systems in both autonomous and non-autonomous cases. Two numerical examples with different Φ are finally given to illustrate the effectiveness of our results. Some comparisons among the existed correlative results show the good points of the new approach.
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This paper deals with designing a unilateral tracking controller for the Switching Positive Linear Systems (SPLSs). Positive systems are widespread, and most biological, economic systems, and so on, with nonnegative variables, belong to this class of systems. System states are assumed partially unmeasurable; thus, a reduced‐order positive switching observer is utilized to improve the closed‐loop system response. Also, interval uncertainty is considered to provide a more practical result. In addition, a unilateral tracking controller is developed in this study. The unilateral tracking controller keeps the system and observer states in the same direction as the reference system and system output. Also, the unilateral tracking method avoids overshoot. System stability is presented in synchronous and asynchronous switching conditions. In asynchronous switching mode, controller switches have a time lag behind the system's switching. The switching time lag in asynchronous mode is assumed unknown with a known upper bound for each subsystem. The closed‐loop system's stability is guaranteed by utilizing multiple Lyapunov functions, and the Mode Dependent Average Dwell Time (MDADT) strategy. The designed unilateral tracking controller is developed to track a reference system. Also, sufficient conditions are provided to guarantee the L1‐gain performance of the closed‐loop system. Finally, an illustrative example is provided to demonstrate the proposed scheme's effectiveness compared with the foremost existing solutions.
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This paper focuses on mean exponential stability and L1-gain analysis for nonlinear positive Markov jump systems (NPMJSs) based on a switching transition probability (STP), where sector nonlinear functions and delays are adopted. By developing a nonlinear stochastic copositive Lyapunov-Krasovskii functional (NSCLKF) approach, a sufficient condition for mean exponential stability of nonlinear positive time-delay Markov jump system is first presented under mode-dependent average dwell time (MDADT) switching by using linear programming (LP) approach. Further, L1-gain performance analysis is obtained. Moreover, the corresponding results for NPMJSs are given under average dwell time (ADT) switching. Illustrative results are provided to verify the validity of the theoretical results.
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The Hankel-type Lq/Lp induced norms across a single switching over two linear time-invariant (LTI) positive systems are discussed. The norms are defined as the induced norms from vector-valued Lp-past inputs to vector valued Lq-future outputs across a switching at the time instant zero. The Hankel-type L2/L2 induced norm across a single switching for general LTI systems is studied in details to evaluate the performance deterioration caused by switching. Thanks to the strong positivity property, we successfully characterize the Hankel-type Lq/Lp induced norms for the positive system switching even for p, q being 1, 2, ∞. In particular, we will show that some of them are given in the form of linear program and semidefinite program (SDP). The SDP-based characterizations are useful for the analysis of the Hankel-type Lq/Lp induced norms where the systems of interest are affected by parametric uncertainties.
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This paper investigates the exponential stability of positive switched linear systems with impulse. By using the switched linear time‐varying copositive Lyapunov function method and the analytical method developed in positive systems, we establish new explicit criteria for exponential stability of the positive switched linear systems. Then, it is applied to the consensus of multi‐agent systems under the mode‐dependent interval dwell time case. The proposed conditions mean that all switched subsystems could be unstable. Finally, three examples illustrate the effectiveness and superiority of the proposed method.
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This paper deals with stabilising uncertain Switched Positive Linear Systems (SPLSs) with a bumpless transfer controller. The contribution of this study is to improve switching controller robustness at the switching instances and also guarantee the L1–gain performance of the positive systems. Positive systems are widespread; most of the biological, economic systems, disease, etc., have nonnegative variables belonging to this class of systems. Positive Systems modelling encounters different types of uncertainties; therefore, stability analysis and stabilisation of uncertain SPLSs are crucial in analysing positive systems. The stability of the SPLSs can be guarantee by using multiple copositive Lyapunov functions with the Dwell Time (DT) strategy. The stability of the SPLS has been investigated under synchronous and asynchronous switching modes. The so-called asynchronous switching means that the switches between the candidate controllers and system modes are asynchronous. Also, the L1–gain performance of the system is derived. Finally, two illustrative examples are provided to demonstrate the proposed scheme’s effectiveness compared with the foremost existing solutions.
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This study focuses on the controller synthesis issues for constrained switched linear systems with uncertainties under mode-dependent average dwell time (MDADT) switching strategy. First, output feedback controllers ensure that the closed-loop systems are positive and asymptotically stable. Second, the bounded controllers are acquired based on system states with interval and polytopic uncertainties. Also, the proposed approach can be applied to the systems with the constrained output. Then, the presented conditions can be formulated in terms of linear programming. Finally, illustrative example is provided to show the effectiveness of the theoretical results.
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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.
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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
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In this paper we define strong and weak common quadratic Lyapunov functions (CQLFs) for sets of linear time-invariant (LTI) systems. We show that the simultaneous existence of a weak CQLF of a special form, and the non-existence of a strong CQLF, for a pair of LTI systems, is characterised by easily verifiable algebraic conditions. These conditions are found to play an important role in proving the existence of strong CQLFs for general LTI systems.
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It was recently conjectured that the Hurwitz stability of the convex hull of a set of Metzler matrices is a necessary and sufficient condition for the asymptotic stability of the associated switched linear system under arbitrary switching. In this note, we show that (1) this conjecture is true for systems constructed from a pair of second-order Metzler matrices; (2) the conjecture is true for systems constructed from an arbitrary finite number of second-order Metzler matrices; and (3) the conjecture is in general false for higher order systems. The implications of our results, both for the design of switched positive linear systems, and for research directions that arise as a result of our work, are discussed toward the end of the note.
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In this note, the problem of determining necessary and sufficient conditions for the existence of a common quadratic Lyapunov function for a pair of stable linear time-invariant systems whose system matrices A<sub>1</sub> and A<sub>2</sub> are in companion form is considered. It is shown that a necessary and sufficient condition for the existence of such a function is that the matrix product A<sub>1</sub>A<sub>2</sub> does not have an eigenvalue that is real and negative. Examples are presented to illustrate the result.
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It was recently shown that a family of exponentially stable linear systems whose matrices generate a solvable Lie algebra possesses a quadratic common Lyapunov function, which implies that the corresponding switched linear system is exponentially stable for arbitrary switching. In this paper we prove that the same properties hold under the weaker condition that the Lie algebra generated by given matrices can be decomposed into a sum of a solvable ideal and a subalgebra with a compact Lie group. The corresponding local stability result for nonlinear switched systems is also established. Moreover, we demonstrate that if a Lie algebra fails to satisfy the above condition, then it can be generated by a family of stable matrices such that the corresponding switched linear system is not stable. Relevant facts from the theory of Lie algebras are collected at the end of the paper for easy reference. Key words. switched system, asymptotic stability, Lie algebra AMS subject classifications. 93D20, 93B25, 93B12, 17B30 PII. S0363012999365704 1.
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1. The field of values 2. Stable matrices and inertia 3. Singular value inequalities 4. Matrix equations and Kronecker products 5. Hadamard products 6. Matrices and functions 7. Totally positive matrices.
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In this paper we derive a necessary and sufficient condition for the existence of a common diagonal quadratic Lyapunov function for a pair of positive linear time-invariant (LTI) systems.
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We present a result on the existence of a common quadratic Lyapunov function for a pair of linear time-invariant systems. We show that this result charac-terises, generalises, and provides new perspectives on several well-known stability results. In particular, new time-domain formulations of the Circle Criterion and Meyer's extension of the KYP lemma are presented.
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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. This article surveys recent 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 study communication networks that employ drop-tail queueing and Additive-Increase Multiplicative-Decrease (AIMD) congestion control algorithms. It is shown that the theory of nonnegative matrices may be employed to model such networks. In particular, important network properties such as: (i) fairness; (ii) rate of convergence; and (iii) throughput; can be characterised by certain non-negative matrices. We demonstrate that these results can be used to develop tools for analysing the behaviour of AIMD communication networks. The accuracy of the models is demonstrated by several NS-studies.
Conference Paper
In this paper, we present a number of results concerned with the stability of positive switched linear systems. In particular, we show that a recent conjecture concerning the existence of common quadratic Lyapunov functions (CQLFs) for positive LTI systems is true for second order systems, and establish a class of switched linear systems for which CQLF existence is equivalent to exponential stability under arbitrary switching. However, this conjecture is false for higher dimensional systems and we illustrate this fact with a counterexample. A number of stability criteria for positive switched linear systems based on common diagonal Lyapunov functions (CDLFs) are also presented, as well as a necessary and sufficient condition for a general pair of positive LTI systems to have a CDLF To the best of the authors' knowledge, this is the first time that a necessary and sufficient condition for CDLF existence for n-dimensional systems has appeared in the literature.
Conference Paper
We examine the well known distributed power control (DPC) algorithm proposed by Foschini and Miljanic and show via simulations that it may fail to converge in the presence of time-varying channels and handoff, even when the feasibility of the power control problem is maintained at all times. Simulation results also demonstrate that the percentage of instability is a function of the variance of shadow fading, interference and the target signal to interference plus noise ratio. In order to better explain these observations and provide a systematic framework to study the stability of distributed power control algorithms in general, we present the problem in the context of switched systems, which can capture the time variations of the channel and handoffs. This formulation leads to interesting stability problems, which we address using common quadratic Lyapunov functions and M-matrices.
Conference Paper
Vicsek et al. proposed (1995) a simple but compelling discrete-time model of n autonomous agents {i.e., points or particles} all moving in the plane with the same speed but with different headings. Each agent's heading is updated using a local rule based on the average of its own heading plus the headings of its "neighbors". In their paper, Vicsek et al. provide simulation results which demonstrate that the nearest neighbor rule they are studying can cause all agents to eventually move in the same direction despite the absence of centralized coordination and despite the fact that each agent's set of nearest neighbors change with time as the system evolves. This paper provides a theoretical explanation for this observed behavior. In addition, convergence results are derived for several other similarly inspired models. The Vicsek model proves to be a graphic example of a switched linear system which is stable, but for which there does not exist a common quadratic Lyapunov function.
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In a recent Physical Review Letters article, Vicsek et al. propose a simple but compelling discrete-time model of n autonomous agents (i.e., points or particles) all moving in the plane with the same speed but with different headings. Each agent's heading is updated using a local rule based on the average of its own heading plus the headings of its "neighbors." In their paper, Vicsek et al. provide simulation results which demonstrate that the nearest neighbor rule they are studying can cause all agents to eventually move in the same direction despite the absence of centralized coordination and despite the fact that each agent's set of nearest neighbors change with time as the system evolves. This paper provides a theoretical explanation for this observed behavior. In addition, convergence results are derived for several other similarly inspired models. The Vicsek model proves to be a graphic example of a switched linear system which is stable, but for which there does not exist a common quadratic Lyapunov function.
Topics in Matrix Analysis Co-ordination of groups of mobile autonomous agents using nearest neighbour rules
• R Horn
• C A Johnson