Multiple Linear Copositive Lyapunov Function Research Articles

Almost output tracking control for switched positive systems: A hysteresis switching strategy

AbstractThis work studies the almost output tracking control problem via the co‐design of controllers and a switching strategy for switched positive systems. For devising a hysteresis switching strategy, the nonnegative state space is divided into some subdomains, where each subdomain corresponds to an activated subsystem. The main advantage of the proposed methodology is that the switching frequency can be reduced in contrast to the min‐switching based designs, and the almost output tracking control problem is allowed to be unsolvable for each subsystem. The almost output tracking criteria, based on multiple linear copositive Lyapunov functions are established in linear vector inequalities form, which ensures asymptotic convergence of the output tracking error to zero, the positivity as well as a prescribed ‐gain index in presence of disturbances. Finally, the theoretical findings are applied to the output tracking control of the boost converter circuit.

Formula omitted] reliable control for switched positive systems with actuator faults and state-dependent switchings

In this paper, the issues of stability and L1-gain performance analysis, by means of the multiple linear copositive Lyapunov function method, for switched positive systems defined on the nonnegative state space, are addressed. A vital feature of the developed switching strategy lies in relaxing the restriction that connecting adjacent Lyapunov functions at switching points is required. This allows us to introduce a gap at switching points to prevent chattering behavior. Then, with the help of the designed switching policy, sufficient conditions of guaranteeing the solvability of the problems are established, and the proposed methodology is also extended to the case where all subsystems consist of interval type parameter uncertainties. Finally, through two examples, the validity and feasibility of the proposed methods are demonstrated.

Output feedback [formula omitted] control of positive Markov jump systems: A dynamic event-triggered method

The output feedback l1 control problems of positive Markov jump systems(PMJSs) are studied using event-triggered method in this paper. Firstly, two novel dynamic output event-triggered control strategies(DOETCS) are proposed for continuous-time PMJSs as well as discrete-time PMJSs, which can reduce the event-triggered frequency and save network resources. Then, by constructing multiple linear copositive Lyapunov functions, two sufficient conditions about positivity and stochastic stability with l1-gain performance are presented for the continuous-time closed-loop PMJSs and the discrete-time closed-loop PMJSs. Simultaneously, two desired output feedback l1 controllers and two DOETCS are also co-designed to stabilize the PMJSs in continuous and discrete cases, respectively. All the conditions obtained in this paper are in the form of linear programming(LP). Further more, the influence of parameters in the novel DOETCS on the event-triggered frequency is investigated in detail in the section of numerical simulation. Finally, two numerical examples are provided to verify the effectiveness of the proposed design schemes.

Actuator saturation control of continuous-time positive switched T–S fuzzy systems

The saturated control problem is investigated for positive switched Takagi–Sugeno (T–S) fuzzy systems with partially controllable subsystems in this paper. Based on the parallel distribution compensation (PDC) algorithm and the convex hull technology, new fuzzy control schemes are proposed for continuous-time positive switched T–S fuzzy systems (PSTSFSs) with actuator saturation. By the multiple linear co-positive Lyapunov function and the slow-fast combined mode-dependent average dwell time (MDADT) approach, sufficient conditions for the stability of continuous-time closed-loop PSTSFSs are developed, which is an extension of the results in previous literature. Furthermore, the least conservative estimation of the attraction domain of PSTSFSs is transformed into an optimization problem. Finally, three simulation examples are given to illustrate the effectiveness of the proposed saturated control schemes.

Robust exponential stabilization of positive uncertain switched neural networks with actuator saturation and sensor faults

This article focuses on the robust exponential stabilization of positive uncertain switched neural networks subject to actuator saturation and sensor faults. Given the existence of interval uncertainty and the constraint concerning positivity of the original system, a new positive state-bounding observer is constructed to guarantee the coinstantaneous estimation of system state and sensor faults. To deal with actuator saturation, the convex hull scheme is employed. By designing the state-feedback controller and utilizing the multiple time-varying linear co-positive Lyapunov function, sufficient conditions for the robust exponential stability on the studied system are established under dwell-time switching. Furthermore, for optimizing the observer matrix, an iterative algorithm is developed. Eventually, a numerical example is exploited to illuminate the feasibility and effectiveness of both the deduced results and the proposed approaches.

Controller synthesis for discrete‐time positive switched T–S fuzzy systems with partially controllable subsystems

AbstractThe controller synthesis problem for discrete‐time positive switched T–S fuzzy systems (PSTSFSs) is investigated in this paper. Unlike previous literature, subsystems are assumed to be partially controllable in this paper. A novel slow–fast combined mode‐dependent average dwell time (SFCMDADT) approach is proposed to obtain sufficient conditions for the stability of general positive switched nonlinear systems with partially stable subsystems. Then based on the obtained results, a parallel distributed compensation (PDC)‐based fuzzy controller is designed, and sufficient conditions for the stability of PSTSFSs with partially controllable subsystems are developed with the aid of a time‐scheduled multiple linear copositive Lyapunov function. Furthermore, sufficient conditions for the stability of the positive switched T–S fuzzy system are given respectively for the two cases that the subsystems are all controllable and all uncontrollable. Finally, two simulation examples are provided to illustrate the effectiveness of the proposed method.

Stochastic stability of positive Markov jump linear systems with fixed dwell time

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.

Stability Analysis of Switched Positive Systems with an Impulse Interval

The stability analysis issue of switched positive systems (SPSs) with an impulse interval is discussed in this paper for the first time. All subsystems are allowed to be unstable. Unlike previous studies, the impulse is restricted to an interval, that is, the impulse interval. By dividing the state space on the nonnegative orthant and establishing multiple linear copositive Lyapunov functions, sufficient linear programming conditions are obtained to ensure that SPSs with an impulse interval are asymptotically stable. A hysteresis state-dependent switching law is designed to prevent the chattering behavior caused by frequent switching. Finally, two numerical examples are given to validate the theoretical findings.

Finite-time interval observer design for discrete-time switched systems: A linear programming approach

This paper deals with the finite-time interval observer design method for discrete-time switched systems subjected to disturbances. The disturbances of the system are unknown but bounded. The framework of the finite-time interval observer is established and the sufficient conditions are derived by the multiple linear copositive Lyapunov function. Furthermore, the conditions which are expressed by the forms of linear programming are numerically tractable by standard computing software. One example is simulated to illustrate the validity of the designed observer.

Dwell-Time-Based Exponential Stabilization of Switched Positive Systems With Actuator Saturation

This article investigates the exponentially stabilization problem for a class of switched positive systems with actuator saturation under dwell time switching signal. The state feedback controllers for each subsystem are designed by proposing a new type of multiple time-varying linear co-positive Lyapunov functions (MTVLCLFs), and sufficient conditions for the exponentially stabilization of the studied systems are proposed under dwell-time switching. In order to cope with actuator saturation, the convex hull technique is utilized. Finally, an example of a turbofan engine model is presented to show the effectiveness of our proposed method.

Stability analysis of totally positive switched linear systems with average dwell time switching

This paper investigates the stability of totally positive switched linear systems (TPSLSs), for the continuous-time case and the discrete-time case. The multiple linear copositive Lyapunov function (MLCLF) is used to study exponential stability of the considered system with average dwell time. Then the case with common linear copositive Lyapunov function (CLCLF) is also established. Sufficient conditions to ensure the stability are given. Finally, numerical examples are presented to validate the efficiency of the derived results.

Stability of switched positive linear delay systems with mixed impulses

ABSTRACTThis paper investigates the global exponential stability of both continuous-time and discrete-time switched positive linear delay systems with the coexistence of destabilising and stabilising impulses. Based on the comparison principle, with the help of discretised multiple linear copositive Lyapunov functions, some delay-independent stability results are obtained. It should be noted that it is the first time that discrete-time switched delay systems with asynchronous switching and impulses are considered, and the switching effect can be stabilising in the obtained results of this paper. Some numerical examples are provided to demonstrate the effectiveness and superiority of the derived results.

Almost Output Regulation for Switched Positive Systems With Different Coordinates Transformations and its Application to a Positive Circuit Model

In this article, the issue of almost output regulation for continuous-time switched positive systems is investigated. We take into consideration the different coordinates transformations in which the subsystems are switched corresponding to the individual coordinate system. A new design scheme is firstly proposed in this manner. Furthermore, each group of the regulator equations has its own solution. Two types of control strategies, including a state feedback law and an error feedback law, are adopted. In contrast to the nominal output regulation problem, we consider the case where an unknown external disturbance affects the systems and the exogenous systems. Therefore, a concept of the almost output regulation for the considered systems is proposed. Then, all tractable conditions guaranteeing the solvability of the addressed problem are presented in terms of linear matrix equalities as well as linear vector inequalities by means of the constructed multiple linear copositive Lyapunov functions. A positive circuit is finally used to illustrate the applicability and availability of the developed theoretical findings.

Nonfragile observer-based guaranteed cost finite-time control of discrete-time positive impulsive switched systems

Abstract This paper considers the nonfragile observer-based guaranteed cost finite-time control of discrete-time positive impulsive switched systems(DPISS). Firstly, the positive observer and nonfragile positive observer are designed to estimate the actual state of the underlying systems, respectively. Secondly, by using the average dwell time(ADT) approach and multiple linear co-positive Lyapunov function (MLCLF), two guaranteed cost finite-time controller are designed and sufficient conditions are obtained to guarantee the corresponding closed-loop systems are guaranteed cost finite-time stability(GCFTS). Such conditions can be solved by linear programming. Finally, a numerical example is provided to show the effectiveness of the proposed method.

Reachable set estimation for switched positive systems

ABSTRACTThis paper focuses on the problem of reachable set estimation for discrete-time switched positive systems under two possible classes of exogenous disturbance. The multiple linear copositive Lyapunov function approach is applied to determine the bounding hyper-pyramids for the reachable set. Based on some Lyapunov-based inequalities and the linear version of the S-procedure technique, the bounding hyper-pyramids for the reachable set can be determined by solving a set of inequalities. Two optimisation methods are adopted to make the bounding hyper-pyramids as small as possible. Genetic Algorithm (GA) is utilised to search for the optimal value of the decision variables in the obtained inequalities. Finally, numerical examples are given to illustrate the effectiveness of the theoretical findings.

Robust stability of switched positive linear systems with interval uncertainties via multiple time-varying linear copositive Lyapunov functions

This paper studies the robust stability analysis of a class of switched positive linear systems with uncertainties in the framework of dwell time switching. The uncertainties refer to interval uncertainties. Two classes of dwell time switching signals are considered in this paper: (i) the first class is confined by a certain pair of upper and lower bounds; (ii) the other is the minimum dwell time. First, a class of multiple time-varying linear copositive Lyapunov functions is constructed to analyze the robust stability of the studied switched system. Then, under the pre-given dwell time switching, the sufficient conditions are obtained by restricting the decay of the Lyapunov functions of the active subsystem and forcing “energy” of the overall switched system to decrease at switching instants by the proposed Lyapunov functions. All present conditions are solvable in terms of linear programming. An example is considered in order to emphasize the effectiveness of the proposed approach.

Stability Analysis and Delay Control for Switched Positive Linear Systems

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 linear copositive Lyapunov function used in the literature can be viewed as a special case of our proposed MPLCLF, and our result can obtain a tighter bound of the 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.

Stabilisation for positive switched T-S fuzzy delayed systems under standard L1 and L∞ performance

ABSTRACTIn this paper, the stabilisation problem for positive switched T-S fuzzy delayed systems under standard L1 and L∞ performance is addressed. First, by proposing a time-scheduled multiple linear copositive Lyapunov function which is time-varying during the mode-dependent minimum dwell-time interval, sufficient conditions of asymptotical stability for positive switched T-S fuzzy delayed system under dwell-time switching are derived for the first time. Then, sufficient conditions for the underlying system to be asymptotically stable with standard L1 and L∞ performance are obtained, respectively. Based on the obtained results, a fuzzy state-feedback controller scheme is designed to stabilise the positive switched T-S fuzzy delayed system, while guaranteeing the positivity and asymptotical stability with standard L1 performance in closed loop. Furthermore, the stabilisation problem under standard L∞ performance is also solved. All the results are presented in the linear programming form. Finally, two numerical examples are presented to demonstrate the effectiveness of the obtained theoretical results.

Input-output Finite-time Control of Uncertain Positive Impulsive Switched Systems with Time-varying and Distributed Delays

This paper is concerned with the input-output finite-time control of uncertain positive impulsive switched systems(UPISS) with time-varying and distributed delays. Firstly, the definition of input-output finite-time stability is extended to UPISS, and the proof of the positivity of UPISS is also given. Then, by constructing multiple linear copositive Lyapunov functions and using the mode-dependent average dwell time(MDADT) approach, a state feedback controller is designed, and sufficient conditions are derived to guarantee that the corresponding closedloop system is input-output finite-time stable(IO-FTS). Such conditions can be easily solved by linear programming. Finally, two examples are given to demonstrate the effectiveness of the proposed method.

Asynchronous L1-gain control of uncertain switched positive linear systems with dwell time

In this paper, dwell time (DT) stability, L1-gain performance analysis and asynchronous L1-gain controller design problems of uncertain switched positive linear systems (SPLSs) are investigated. Via a time-scheduled multiple linear co-positive Lyapunov function (TSMLCLF) approach, convex sufficient conditions of DT stability and L1-gain performance of SPLSs with interval and polytopic uncertainties are presented. Furthermore, by utilizing the feature that the TSMLCLF keeps decreasing even if the controller is running asynchronously with the system, the asynchronous L1-gain controller design problem of SPLSs with interval and polytopic uncertainties is investigated. Convex sufficient conditions of the existence of time-varying asynchronous state-feedback controller which can ensure the closed-loop system's positivity, stability and L1-gain performance are established, and the controller gain matrices can be calculated instantaneously online. The obtained L1-gain in the paper is standard. All the results are presented in terms of linear programming. A practical example is provided to show the effectiveness of the results.