Positive Fuzzy Systems Research Articles

Practical consensus of T-S fuzzy positive multi-agent systems using linear programming

This paper presents the practical consensus of T-S fuzzy positive multi-agent systems with interval type-1 and type-2 fuzzy sets using observer-based control protocols. A fuzzy positive observer is first constructed for the systems. An observer-based fuzzy control protocol is designed, where an additional constant term is introduced. Some linear programming conditions are established to achieve the positivity of the original system and its observer. Then, the practical consensus of the original system is transformed into the stability of a dynamic system, where a set of new variables is defined by combining the constant term and the states of the agents. In the first step, the positivity of the new variables is guaranteed. In the second step, the stability of the dynamic system consisting new variables is addressed by using a fuzzy copositive Lyapunov function. The gain matrices of observer and control protocols are formulated based on a matrix decomposition approach. All positive and consensus conditions are described via linear programming. Finally, two examples are provided to verify the validity of the obtained results.

Event-Triggered Consensus of T–S Fuzzy Positive Multi-Agent Systems Based on Compensator and Disturbance Observer

Event-Triggered Consensus of T–S Fuzzy Positive Multi-Agent Systems Based on Compensator and Disturbance Observer

Practical consensus of time-varying fuzzy positive multi-agent systems

<p>This paper considers the practical consensus of time-varying fuzzy positive multi-agent systems. A novel error variable is introduced by adding an additional constant term. Under the framework of time-varying fuzzy copositive Lyapunov functions, a fuzzy control protocol with time-varying gain matrices is designed in terms of matrix decomposition technique. Some consensus conditions are addressed via time-varying linear programming. Moreover, the design is developed for false data injection attacks. Finally, two examples are provided for verifying the validity of the design.</p>

Stability analysis and $$L_{2}$$-gain control for positive fuzzy systems by applying a membership-function-dependent Lyapunov function

Stability analysis and $$L_{2}$$-gain control for positive fuzzy systems by applying a membership-function-dependent Lyapunov function

Stability and L1-gain analysis for switched positive fuzzy systems with time-delay: A state-dependent switching policy

This work exploits a state-dependent switching policy to deal with stability analysis and L1-gain performance of switched positive fuzzy systems with time-delay. By dividing up the nonnegative state space, an improved state-dependent switching policy is devised. The main advantages of the proposed switching policy are that the chattering behavior commonly arisen in traditional state-dependent switching strategies is reduced and allows each subsystem to be unstable. Also, sufficient conditions guaranteeing asymptotic stability with an L1-gain performance are derived by the usage of multiple linear copositive Lyapunov-Krasovskii functionals method. Moreover, the proposed approach is also extended to cope with the concerned system with uncertainties, in particular interval ones. Finally, the Lotka-Volterra model is adopted to verify the validity and feasibility of the conclusions.

Exponential stability and stabilization of positive T‐S fuzzy delayed systems

AbstractThe analysis of positive nonlinear delayed systems is of great importance for many real‐world applications. Such systems' stability and stabilization assessment is still an open topic, and there is limited literature on this field. Moreover, further convergence conditions should be considered for many experimental processes, such as exponential stability analysis, which is highly important. Considering the above, we deal in this study with the problem of exponential stability and stabilization of nonlinear fuzzy positive systems with delay. We establish exponential stability criteria using Lyapunov–Krasovskii functional (LKF) and a delay bi‐decomposition approach for bounded and time‐varying delayed systems. The obtained results are then extended to the exponential stabilization case. The control law is designed using Parallel distributed compensation (PDC). The proposed approach, formulated in terms of linear matrix inequalities (LMIs), allows reducing the conservativeness of the delay‐dependent conditions. A comparative study is presented to illustrate the superiority of our method. Moreover, simulation results for the two tanks process show the advantages of the proposed control design.

Hybrid [formula omitted]-gain analysis of T-S fuzzy positive impulsive systems

The finite hybrid L1×l1-gain problem for T-S fuzzy positive impulsive systems is proposed. The problem is solved in the framework of discretized copositive Lyapunov functions. In order to highlight the effect of interaction between continuous and discrete dynamics on system performance, both the impulse state and the discrete-time input are included in the introduced copositive Lyapunov functions. By selecting appropriate auxiliary functions for adjusting the variation of copositive Lyapunov function at impulse instants, the introduced copositive Lyapunov functions turn out to be continuous along the solutions of the system. Through employing the novel discretized function, new criteria for exponential stability and finite hybrid L1×l1-gain are derived in terms of finite linear programming. Numerical examples show that the new criteria can achieve higher accuracy than those of the existing ones.

Stabilization of Positive Systems With Time Delay via the Takagi-Sugeno Fuzzy Impulsive Control.

In this study, the Takagi-Sugeno (T-S) fuzzy impulsive control problem is investigated for a class of nonlinear positive systems with time delay. The time delay under consideration is both in the continuous-time dynamics and at the impulsive instants, which can model practical systems more accurately. An impulse-time-dependent copositive Lyapunov function (IDCLF) is constructed, and the Razumikhin technique is adopted to develop conditions that ensure the globally exponential stability of T-S fuzzy positive systems with delayed impulses. The size constraint between the impulse delay and the bound of impulsive intervals is removed. A T-S fuzzy impulsive controller is designed in terms of the solutions to certain vector inequalities that are readily solvable. Numerical examples and a practical example of lipoprotein metabolism and potassium ion transfer model are given to demonstrate the effectiveness, advantages, and practicality of the proposed results.

Finite time stability analysis for T-S fuzzy positive systems and its application in pest control

This paper is considered with the finite time stability (FTS) for positive nonlinear systems. First, the fuzzy copositive Lyapunov function (FCLF) is established to reduce the conservatism of FTS conditions. Then, by using FCLF, a new FTS criteria of the studied systems is established. Since, the time-derivatives (TD) of membership functions are considered in the stability analysis, our results are less conservative. Besides, the obtained conditions are expressed via the linear programming technique, that can be verified effectively. Finally, the pest-predator system is used to reflect the effectiveness of the results.

Finite Time Annular Domain Robust Stability Analysis and Controller Design for T-S Fuzzy Interval Positive Systems

This paper is concerned with the finite time annular domain robust stability (FTADRS) analysis and controller design for T-S fuzzy positive systems with interval uncertainties. The concept of finite time annular domain stability is first introduced for positive systems. Based on this and using the copositive Lyapunov function approach, some sufficient conditions for FTADRS are derived. Subsequently, the finite time annular domain robust controller is designed via the linear programming technique. Finally, two numerical examples and an application example are employed to show the effectiveness of our results.

Dissipativity analysis and synthesis for positive Roesser systems under the switched mechanism and Takagi-Sugeno fuzzy rules

This paper addresses the dissipativity analysis and controller design for a class of T-S fuzzy positive systems which are described by the Roesser model. First, sufficient conditions obtained by utilizing the minimum dwell time method and co-positive Lyapunov function approach are given to guarantee the considered system is 2-D (ζ,η)-α-dissipative. Then via the state feedback fuzzy controller and the static output feedback fuzzy controller, criteria are also presented under which the resulting closed-loop systems are positive and 2-D (ζ,η)-α-dissipative. Moreover, the explicit design schemes for the desired controller parameters are also presented in the form of linear programming. Finally, a simple illustrative example is provided to show the validity of the approach outlined in this article.

Stability analysis and synthesis for switched Takagi–Sugeno fuzzy positive systems described by the Roesser model

This article focuses on stability analysis and controller synthesis of a class of Roesser-type switched Takagi–Sugeno fuzzy systems with delays. First, sufficient conditions derived by use of the co-positive Lyapunov function and the average dwell time method are presented to guarantee the proposed system is positive and exponentially stable. Then via a state feedback fuzzy controller, criteria are also given under which not only the resulting closed-loop system is positive and exponentially stable but also the desired controller gain matrices can be designed by solving a set of coupled matrix inequalities. Finally, two illustrative examples are provided to show the effectiveness of the approach outlined in this article.

Controller synthesis for switched T–S fuzzy positive systems described by the Fornasini–Marchesini second model

In this paper, controller synthesis problem is investigated for a class of Fornasini–Marchesini second type switched systems with time-varying delays under the Takagi–Sugeno fuzzy rules. Both the cases of state feedback controller and static output feedback controller are designed. By utilizing the common Lyapunov function approach and the multiple Lyapunov functions method, sufficient criteria are derived in the form of linear matrix inequalities to assure the resulting closed-loop 2-D systems to be positive and asymptotically stable, which can be easily computed using the existing numerical algorithms. Explicit design schemes for the state feedback/static output feedback controllers gain matrices are also presented. Finally, advantage and effectiveness of the proposed criteria are shown by a numerical example.

Switched mechanisms for stability and l1-gain analysis of T-S fuzzy positive systems described by the F-M second model

In this paper, the Takagi–Sugeno fuzzy positive system is analyzed under the switched mechanisms, which is described by the discrete-time Fornasini–Marchesini Second model. By utilizing the co-positive Lyapunov function and the novel average dwell time method, sufficient conditions are derived under which the proposed system is positive, exponentially stable and has l1-gain bound. Furthermore, the criteria obtained are presented in the form of matrix inequalities which can be solved by the standard software. Finally, a numerical example is given to demonstrate the effectiveness of the approach outlined in the paper.

Stability analysis and observer-based controllers design for T–S fuzzy positive systems

Abstract This paper investigates the stability analysis and observer-based controllers design for T–S fuzzy positive systems. A fuzzy copositive Lyapunov function is first proposed to analyze the stability of the T–S fuzzy positive systems via linear programming. In terms of the property of the fuzzy membership functions, the fuzzy copositive Lyapunov function is employed to derive the less conservative stability conditions. Then, the line-integral Lyapunov function is presented for the stability analysis. The time-derivatives of the membership functions do not appear in the stability analysis of the T–S fuzzy positive systems, therefore, the proposed stability conditions are more relaxed than those of the conventional Lyapunov function approaches. Based on the obtained stability conditions, observer-based control schemes are designed such that the resultant closed-loop systems are both stable and positive. Finally, two examples are provided to validate the effectiveness of the results proposed in this paper.

ℓ1-induced state-bounding observer design for positive Takagi–Sugeno fuzzy systems

In this paper, the design problem of the ℓ1-induced positive observer is studied for positive Takagi–Sugeno (T–S) fuzzy systems. First, a ℓ1-induced performance index is presented for positive systems and the performance indicator matches well to the linear Lyapunov functions. Then, we explore a novel description of the ℓ1-induced performance for positive T–S fuzzy systems. Moreover, the design problem of the ℓ1-induced positive observer is solved for positive fuzzy systems under a proposed linear programming method. To estimate the state of positive fuzzy systems, a couple of state-bounding positive observers are constructed. In the end, a numerical example is given to show the effectiveness of the proposed approach.

Exponential stability analysis and [formula omitted] synthesis of positive T-S fuzzy systems with time-varying delays

The α-exponential stability analysis and l1 controller synthesis of positive Takagi–Sugeno fuzzy discrete-time systems with time-varying delays are discussed in this paper. A sufficient condition of α-exponential stability for autonomous positive fuzzy systems is first established by constructing a new Lyapunov–Krasovskii functional, which depends on a given scalar 0<α<1 and the bound of time-varying delays. Then, a sufficient condition for controlled systems with l1 gain constraint is also presented while imposing α-exponential stability, upon which, a constrained l1 controller is designed for both single-input and multi-input cases. Finally, an example is given to show the feasibility of the theoretical results.

Mixed L − / L 1 fault detection filter design for fuzzy positive linear systems with time‐varying delays

This study investigates the problem of fault detection for a class of fuzzy positive systems with time-varying delays. A new fault sensitivity measure, called L− index, is proposed. The L− fault detection filter design and multiobjective L− /L1 fault detection filter design problems are addressed. A co-positive type Lyapunov–Krasovskii functional is constructed for the performance analysis and fault detection filter design. Sufficient conditions for the existence of such two filers are firstly given, and then the design methods are presented. Finally, two examples are provided to show the effectiveness and applicability of the proposed method.

Stabilization of positive constrained T–S fuzzy systems: Application to a Buck converter

This paper deals with the problem of constrained stability and tracking of Takagi–Sugeno (T–S) fuzzy positive systems. Linear programming (LP) is used to insert the constraints in the design phase while imposing positivity in closed-loop. The theoretical results are applied to the buck DC–DC converter which is widely used in the photovoltaic generators. Based on the simulation results success of the method is shown for this application.

Stability analysis of discrete-time fuzzy positive systems with time delays

This paper addresses stability analysis for fuzzy positive systems with time delays represented by Takagi-Sugeno type models. Based on the fuzzy linear Lyapunov functions or functionals, we give stability conditions for fuzzy positive systems with constant time delays. Based on the so-called solution trajectory approach, we give stability conditions for fuzzy positive systems with time-varying delays. All the stability results are delay independent and formulated as linear programmings LPs or linear matrix inequalities LMIs that are numerically feasible. This paper also discusses the conservatism for these different approaches. Moreover, this paper proposes some necessary conditions of stability for fuzzy positive systems with time delays.