Stability Analysis and Constrained Control of a Class of Fuzzy Positive Systems with Delays Using Linear Copositive Lyapunov Functional

This paper deals with the stability of nonlinear continuous-time positive systems with delays represented by the Takagi–Sugeno (T-S) fuzzy model. A simpler sufficient condition of stability based on linear copositive Lyapunov functional (LCLF) is derived which is not relevant to the magnitude of delays. Based on the result of stability, the problem of controller design via the so-called parallel distributed compensation (PDC) scheme is solved. The control is under a positivity constraint, which means that the resulting closed-loop systems are not only stable, but also positive. Constrained positive control is also considered, further requiring that the trajectory of the closed-loop system is bounded by a prescribed boundary if the initial condition is bounded by the same boundary. The stability results are formulated as linear programs (LPs) and linear matrix inequalities (LMIs), and the control laws can be obtained by solving a set of bilinear matrix inequalities (BMIs). A numerical example and a real plant are studied to demonstrate the efficiency of the proposed method.     

Control for stability and positivity: equivalent conditions and computation

This paper investigates the stabilizability of linear systems with closed-loop positivity. A necessary and sufficient condition for the existence of desired state-feedback controllers guaranteeing the resultant closed-loop system to be asymptotically stable and positive is obtained. Both continuous and discrete-time cases are considered, and all of the conditions are expressed as linear matrix inequalities which can be easily verified by using standard numerical software. Numerical examples are provided to illustrate the proposed conditions.     

Stability and Constrained Control of a Class of Discrete-Time Fuzzy Positive Systems with Time-Varying Delays

This paper deals with the stability of nonlinear discrete-time positive systems with time-varying delays represented by the Takagi–Sugeno (T–S) fuzzy model. The time-varying delays in the systems can be unbounded. Sufficient conditions of stability which are not relevant to the magnitude of delays are derived by a solution trajectory. Based on the stability results, the problems of controller design via the parallel distributed compensation (PDC) scheme are solved. The control is under the positivity constraint, which means that the resulting closed-loop systems are not only stable, but also positive. Constrained control is also considered, further requiring that the state trajectory of the closed-loop system be bounded by a prescribed boundary if the initial condition is bounded by the same boundary. The stability results and control laws are formulated as linear matrix inequalities (LMIs) and linear programs (LPs). A numerical example and a real plant are studied to demonstrate the application of the proposed methods.     

Robust Positive Real Synthesis for 2D Continuous Systems via State and Output Feedback

This paper considers the problem of positive real control for uncertain twodimensional (2D) continuous systems described by the Roesser state-space model. The parameter uncertainties are assumed to be norm bounded in both state and measurement output equations. The purpose is the design of controllers such that the resulting closed-loop system is asymptotically stable and strictly positive real for all admissible uncertainties. A version of the positive realness of 2D continuous systems is established. Then, sufficient conditions for the solvability of the positive real control problem via state feedback and dynamic output feedback controllers, respectively, are proposed. A linear matrix inequality approach is developed to construct the desired controllers. Finally, an illustrative example is provided to demonstrate the applicability and effectiveness of the proposed method.     

Finite-Time Passivity and Passification for Stochastic Time-Delayed Markovian Switching Systems with Partly Known Transition Rates

Finite-time passivity and passification is assessed for stochastic time-delayed Markovian switching systems with partly known transition rates. By employing an appropriate mode-dependent Lyapunov function and some appropriate free-weighting matrices, a state feedback controller is constructed such that the resulting closed-loop system is finite-time bounded and satisfies the given passive constraint condition. Expressed as linear matrix inequalities, some sufficient conditions for solvability of the problem are derived. Finally, an example is given to demonstrate the validity of the main results.     

Passivity-based Control for Markovian Jump Systems via Retarded Output Feedback

This paper is concerned with the problem of passivity-based control for Markovian jump systems via retarded output feedback controllers. A delay-dependent passivity criterion is obtained in terms of linear matrix inequalities. Based on this, a sufficient condition is proposed for the design of a retarded output feedback controller which ensures that the closed-loop system is passive. By using the sequential linear programming matrix method, a desired retarded output feedback controller can be constructed. Numerical examples are provided to demonstrate the advantage and effectiveness of the proposed method.     

Distributed stabilisation of spatially invariant systems: positive polynomial approach

The paper gives a computationally feasible characterisation of spatially distributed controllers stabilising a linear spatially invariant system, that is, a system described by linear partial differential equations with coefficients independent on time and location. With one spatial and one temporal variable such a system can be modelled by a 2-D transfer function. Stabilising distributed feedback controllers are then parametrised as a solution to the Diophantine equation ax + by = c for a given stable bi-variate polynomial c. The paper is built on the relationship between stability of a 2-D polynomial and positiveness of a related polynomial matrix on the unit circle. Such matrices are usually bilinear in the coefficients of the original polynomials. For low-order discrete-time systems it is shown that a linearising factorisation of the polynomial Schur-Cohn matrix exists. For higher order plants and/or controllers such factorisation is not possible as the solution set is non-convex and one has to resort to some relaxation. For continuous-time systems, an analogue factorisation of the polynomial Hermite-Fujiwara matrix is not known. However, for low-order systems and/or controller, positivity conditions on the closed-loop polynomial coefficients can be invoked. Then the computational framework of linear matrix inequalities can be used to describe the stability regions in the parameter space using a convex constraint.     

Delay-Dependent H∞ Control and Filtering for Uncertain Markovian Jump Systems With Time-Varying Delays

This paper deals with the problems of delay-dependent robust H_infin control and filtering for Markovian jump linear systems with norm-bounded parameter uncertainties and time-varying delays. In terms of linear matrix inequalities, improved delay-dependent stochastic stability and bounded real lemma (BRL) for Markovian delay systems are obtained by introducing some slack matrix variables. The conservatism caused by either model transformation or bounding techniques is reduced. Based on the proposed BRL, sufficient conditions for the solvability of the robust H_infin control and H_infin filtering problems are proposed, respectively. Dynamic output feedback controllers and full-order filters, which guarantee the resulting closed-loop system and the error system, respectively, to be stochastically stable and satisfy a prescribed H_infin performance level for all delays no larger than a given upper bound, are constructed. Numerical examples are provided to demonstrate the reduced conservatism of the proposed results in this paper.     

Passivity Analysis and Passification of Markovian Jump Systems

This paper is concerned with the problems of delay-dependent robust passivity analysis and robust passification for uncertain Markovian jump linear systems (MJLSs) with time-varying delay. The parameter uncertainties are time varying but norm bounded. For the robust passivity problem, the objective is to seek conditions such that the closed-loop system under the state-feedback controller with given gains is passive, irrespective of all admissible parameter uncertainties. For the robust passification problem, desired passification controllers will be designed which guarantee that the closed-loop MJLS is passive. By constructing a proper stochastic Lyapunov–Krasovskii function and employing the free-weighting matrix technique, delay-dependent passivity/passification performance conditions are formulated in terms of linear matrix inequalities. Finally, the effectiveness of the proposed approaches is demonstrated by a numerical example.     

部分转移概率离散奇异Markov跳变系统的镇定

针对具有更广泛意义的部分转移概率未知情况下的离散时间奇异 Markov跳变线性系统,给出了使开环系统正则、因果、随机稳定的充分条件,并表示为严格线性矩阵不等式形式,且消除了等式约束条件。部分转移概率未知的条件包括了完全开关系统和随机跳变转移概率完全未知的情况,适用范围更广泛;在此基础上,通过求解严格线性矩阵不等式的可行解,设计了模式依赖状态反馈控制器,使闭环系统正则、因果、随机稳定,实现了系统的镇定;最后,给出一个仿真算例验证了所提方法的有效性。     

Quantized Output Feedback Control of Uncertain Discrete-Time Systems with Input Saturation

This paper deals with the problem of quantized output feedback control for uncertain discrete-time systems with input saturation. Input quantization case and output quantization case are studied, respectively. The purpose of the study was to design of dynamic output feedback controllers such that all the trajectories of the closed-loop system converge to a small ellipsoid for every initial condition starting from large admissible domain. By solving the optimization problem, the corresponding domains can be obtained. Finally, two simulation examples are provided to demonstrate the applicability of the proposed approach.     

Finite-Time Stability Analysis of Impulsive Switched Discrete-Time Linear Systems: The Average Dwell Time Approach

This paper investigates the finite-time stability problem for a class of discrete-time switched linear systems with impulse effects. Based on the average dwell time approach, a sufficient condition is established which ensures that the state trajectory of the system remains in a bounded region of the state space over a pre-specified finite time interval. Different from the traditional condition for asymptotic stability of switched systems, it is shown that the total activation time of unstable subsystems can be greater than that of stable subsystems. Moreover, the finite-time stability degree can also be greater than one. Two examples are given to illustrate the merit of the proposed method.     

Managing quality-of-control in network-based control systems by controller and message scheduling co-design

In network-based control systems (NCSs), plant sensor-controller-actuator nodes in closed-loop operation drive principal network traffic. The quality-of-control (QoC) in an NCS, i.e., the performance delivered by each closed-loop operation, depends not only on the controller design but also on the message scheduling strategy. In this paper, we show that the co-design of adaptive controllers and feedback scheduling policies allows for the optimization of the overall QoC. First, we discuss the limitations of standard discrete-time control models for controllers of control loops that are closed over communication networks. Afterwards, we describe an approach to adaptive controllers for NCS that: 1) overcomes some of the previous restrictions by online adapting the control decisions according to the dynamics of both the application and executing platform and 2) offers capabilities for dynamic management of QoC through message scheduling.     

Analysis and Design of $$H_{\infty }$$ Controllers for 2D Singular Systems with Delays

The \(H_{\infty }\) control design problem is solved for the class of 2D discrete singular systems with delays. More precisely, the problem addressed is the design of state-feedback controllers such that the acceptability, internal stability and causality of the resulting closed-loop system are guaranteed, while a prescribed \(H_\infty \) performance level is simultaneously fulfilled. By establishing a novel version of the bounded real lemma, a linear matrix inequality condition is derived for the existence of these \(H_\infty \) controllers. They can then be designed by solving an iterative algorithm based on LMI optimizations. An illustrative example shows the applicability of the algorithm proposed.     

Delay-Dependent Robust Control for Uncertain Stochastic Time-Delay Systems

This paper is concerned with controller design for uncertain stochastic systems with time-varying delays. The parametric uncertainties which appear in all system matrices are assumed to be norm bounded. Two cases of time-varying delays are investigated. Based on the slack matrix technique, delay-dependent stability criteria for the delayed stochastic systems are derived. By using the analysis results, linear matrix inequality-based controllers are designed. Three numerical examples are provided to demonstrate the effectiveness of the proposed method. This work was supported by the National Natural Science Foundation of China under Grants 60434020 and 60604003.     

Observer-Based H<Subscript>∞</Subscript> Controller for Perturbed System with State and Input Delay

This paper deals with the robust stability problem for continuous-time linear systems with external perturbation and time delay in state and control input. We derive a sufficient condition that guarantees the closed-loop delayed system stability and a prescribed H norm bound of the closed-loop transfer matrix from the disturbance to the controlled output. Based on this derivation, the proposed observer-based robust controller not only reduces the influence of the disturbance, but also guarantees the closed-loop system stability. An example is given to illustrate the availability of the proposed design method.     

Robust Exponential Stabilization for Sampled-Data Systems with Variable Sampling and Packet Dropouts

This paper investigates the problem of robust exponential stabilization for sampled-data systems with variable sampling and packet dropouts. It is assumed that the system parameter uncertainties are norm-bounded and appear in both the state and input matrices. An input delay approach is adopted to model the sample-and-hold behavior with a time-varying delayed control input, and a switched system approach is proposed to model the data-missing phenomenon. On this basis, the sampled-data control system with variable sampling and packet dropouts is modeled as a switched system with time-varying delay. The objective is to design a sampled-data controller to guarantee the robust exponential stability of the resulting closed-loop system. Based on a new piecewise time-dependent Lyapunov functional, novel sufficient conditions are derived for the existence of robustly exponentially stabilizing sampled-data controllers. It is shown that the controller gains can be obtained by solving a set of linear matrix inequalities. Two simulation examples are given to demonstrate the effectiveness of the proposed method.     

Frequency domain computations for nonlinear steady-state solutions

Steady-state analysis and Fourier analysis play a major role in linear signal processing. In response to a bounded input, a steady-state solution exists if all the poles of the discrete-time linear system are inside the unit circle. Despite the fact that there is no principle of superposition for nonlinear systems, under appropriate sufficient conditions (including all poles inside the unit circle for the linear part of the nonlinear system), there is a bounded solution for all time in response to a bounded input for all time for a time-varying nonlinear difference equation. All solutions that start sufficiently close to this unique solution converge to it as time goes to infinity. This steady-state solution can be computed by applying Fourier and inverse Fourier transforms to each step in a Picard process. In this paper, we develop an algorithm to compute (approximate) steady-state solutions for discrete-time, nonlinear difference equations by employing fast Fourier transforms and inverse fast Fourier transforms at each step of the iterative process. Simulations are provided to illustrate our algorithm