Proposed Stability Conditions Research Articles

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.

Design of Fuzzy Functional Observer-Controller via Higher Order Derivatives of Lyapunov Function for Nonlinear Systems.

In this paper, we investigate the stability of Takagi-Sugeno fuzzy-model-based (FMB) functional observer-control system. When system states are not measurable for state-feedback control, a fuzzy functional observer is designed to directly estimate the control input instead of the system states. Although the fuzzy functional observer can reduce the order of the observer, it leads to a number of observer gains to be determined. Therefore, a new form of fuzzy functional observer is proposed to facilitate the stability analysis such that the observer gains can be numerically obtained and the stability can be guaranteed simultaneously. The proposed form is also in favor of applying separation principle to separately design the fuzzy controller and the fuzzy functional observer. To design the fuzzy controller with the consideration of system stability, higher order derivatives of Lyapunov function (HODLF) are employed to reduce the conservativeness of stability conditions. The HODLF generalizes the commonly used first-order derivative. By exploiting the properties of membership functions and the dynamics of the FMB control system, convex and relaxed stability conditions can be derived. Simulation examples are provided to show the relaxation of the proposed stability conditions and the feasibility of designed fuzzy functional observer-controller.

Stability of Sampled-data Systems with Uncertain Time-varying Delays and Its Application to Consensus Control of Multi-agent Systems

In this paper, we propose a new stability condition for sampled-data systems with uncertain time-varying delays less than a sampling interval. The derivation is based on the robustness of the corresponding discrete-time systems subject to perturbation caused by an uncertain time-varying delay. Then, we apply the proposed stability condition to a consensus control problem of two-wheeled robots. Numerical simulations and experiments show the validity and usefulness of the derived condition.

Fuzzy Static Output Feedback H∞ Control for Nonlinear Systems Subject to Parameter Uncertainties

This study is concerned with the stabilization issue of nonlinear systems subject to parameter uncertainties. An interval type-2 T-S fuzzy model is used to represent the nonlinear systems subject to parameter uncertainties. An interval type-2 fuzzy static output feedback controller is designed to synthesize the interval type-2 T-S fuzzy systems. The membership-function-dependent stability conditions are derived by utilizing the information of upper and lower membership functions. The proposed stability conditions are presented in the form of linear matrix inequalities (LMIs). LMI-based stability conditions for interval type-2 fuzzy static output feedback H∞ control synthesis are also developed. Several simulation examples are given to show the superiority of the proposed approach.

Stability analysis and stabilization of Markovian jump systems with time‐varying delay and uncertain transition information

SummaryThe paper investigates the problems of stability and stabilization of Markovian jump systems with time‐varying delays and uncertain transition rates matrix. First, the stochastic scaled small‐gain theorem is introduced to analyze the stability of the Markovian jump system. Then, a new stability criterion is proposed by using a new Lyapunov‐Krasovskii functional combined with Wirtinger‐based integral inequality. The proposed stability condition is demonstrated to be less conservative than other existing results. The merit of the proposed approach lies in its reduced conservatism, which is made possible by a new precise triangle inequality and a new Lyapunov‐Krasovskii functional. Moreover, a controller design criterion is presented according to the stability criterion. Furthermore, the transition rate matrix is treated as partially known and with uncertainty, and the relevant stability and stabilization criteria are proposed. Finally, 3 numerical examples are provided to illustrate the superior result of the stability criteria and the effectiveness of the proposed controller design method.

Event-triggered [formula omitted] filtering for delayed neural networks via sampled-data

This paper is concerned with event-triggered H∞ filtering for delayed neural networks via sampled data. A novel event-triggered scheme is proposed, which can lead to a significant reduction of the information communication burden in the network; the feature of this scheme is that whether or not the sampled data should be transmitted is determined by the current sampled data and the error between the current sampled data and the latest transmitted data. By constructing a proper Lyapunov–Krasovskii functional, utilizing the reciprocally convex combination technique and Jensen’s inequality sufficient conditions are derived to ensure that the resultant filtering error system is asymptotically stable. Based on the derived H∞ performance analysis results, the H∞ filter design is formulated in terms of Linear Matrix Inequalities (LMIs). Finally, the proposed stability conditions are demonstrated with numerical example.

Dynamic stability of rolling missile with proportional navigation & PI autopilot considering parasitic radome loop

The dynamic stability of coning motion for a radar homing rolling missile is studied in this paper, and the parasitic radome loop is considered and introduced into the guidance system of the radar homing rolling missile. A mathematical model with complex summation considering the parasitic radome loop is proposed, and the sufficient and necessary conditions for the dynamic stability of the coning motion for the rolling missile are analytically derived by using the stability criterion of third-order characteristic equation with complex coefficients. Numerical simulations with different parameters are conducted to demonstrate the effectiveness of the proposed stability condition. The stability condition proposed in this paper can be used in the guidance and control system design of a radar homing rolling missile while considering the parasitic radome loop.

Dwell-time stability and stabilization conditions for linear positive impulsive and switched systems

Several results regarding the stability and the stabilization of linear impulsive positive systems under arbitrary, constant, minimum, maximum and range dwell-time are obtained. The proposed stability conditions characterize the pointwise decrease of a linear copositive Lyapunov function and are formulated in terms of finite-dimensional or semi-infinite linear programs. To be applicable to uncertain systems and to control design, a lifting approach introducing a clock-variable is then considered in order to make the conditions affine in the matrices of the system. The resulting stability and stabilization conditions are stated as infinite-dimensional linear programs for which three asymptotically exact computational methods are proposed and compared with each other on numerical examples. Similar results are then obtained for linear positive switched systems by exploiting the possibility of reformulating a switched system as an impulsive system. Some existing stability conditions are retrieved and extended to stabilization using the proposed lifting approach. Several examples are finally given for illustration.

Stability of nonlinear time-delay systems satisfying a quadratic constraint

The stability problem of nonlinear time-delay systems is addressed. A quadratic constraint is employed to exploit the structure of nonlinearity in dynamical systems via a set of multiplier matrices. This yields less conservative results concerning stability analysis. By employing a Wirtinger-based inequality, a delay-dependent stability criterion is derived in terms of linear matrix inequalities for the nominal and uncertain systems. A numerical example is used to demonstrate the effectiveness of the proposed stability conditions in dealing with some larger class of nonlinearities.

Relaxed Stabilization Conditions for the T–S Fuzzy System with Input Constraints

The stabilization conditions of the discrete Takagi–Sugeno (T–S) fuzzy system are reduced by considering possible switching subregions. In addition, the stabilization conditions for the T–S fuzzy system are relaxed by representing the interactions among the fuzzy subsystems in a single matrix. However, these two concepts have not been applied together to the discrete T–S fuzzy system with constraints on the control input. The aim of this paper is to relax the stabilization conditions for the discrete T–S fuzzy system with constraints on the control input. The possible switching subregions fired by two successive states of the system are analyzed and utilized to reduce the stabilization conditions. The interactions of fuzzy subsystems within two subregions are integrated into a single matrix to relax the stabilization conditions. The relaxation and effectiveness of the proposed stabilization conditions are demonstrated by a numerical example and a mass-spring-damper system.

Robust H∞ stabilization conditions for a class of uncertain T–S fuzzy neutral systems with disturbance

In this paper, the H∞ stabilization problems for a class of uncertain Takagi–Sugeno (T–S) fuzzy neutral systems are investigated. Based on homogeneous polynomials and Pólya׳s theorem, a novel stabilization condition for uncertain T–S fuzzy neutral systems is proposed in terms of a linear matrix inequality (LMI) to guarantee the asymptotic stabilization of fuzzy neutral systems. In addition, by using a novel polynomial slack matrix, a more relaxed stabilization condition is developed. Lastly, a truck–trailer example and a numerical example are given to demonstrate that the proposed stabilization condition gives longer allowable delay times and lower desired level of disturbance attenuation γ than those obtained using some existing methods, showing the effectiveness and applicability of the proposed method.

Exponential stability of memristor-based synchronous switching neural networks with time delays

In this paper, we study the existence, uniqueness and stability of memristor-based synchronous switching neural networks with time delays. Several criteria of exponential stability are given by introducing multiple Lyapunov functions. In comparison with the existing publications on simplice memristive neural networks or switching neural networks, we consider a system with a series of switchings, these switchings are assumed to be synchronous with memristive switching mechanism. Moreover, the proposed stability conditions are straightforward and convenient and can reflect the impact of time delay on the stability. Two examples are also presented to illustrate the effectiveness of the theoretical results.

A novel stabilization condition for a class of T–S fuzzy time-delay systems

In this paper, a relaxed stabilization problem for a class of Takagi and Sugeno (T–S) fuzzy time-delay systems is explored. By utilizing homogeneous polynomials scheme and Pólya׳s theorem, a relaxed delay-dependent stabilization condition is proposed. In addition, a novel slack matrix scheme is presented for stabilization condition of a class of T–S fuzzy time-delay systems in terms of linear matrix inequalities (LMIs). Lastly, a well-know numerical example and a truck-trailer example are given to demonstrate that the proposed stabilization condition can provide a longer allowable delay time than some existing studies.

Exponential Stability and Stabilization of Delayed Memristive Neural Networks Based on Quadratic Convex Combination Method.

This paper is concerned with the exponential stability and stabilization of memristive neural networks (MNNs) with delays. First, we present some generalized double-integral inequalities, which include some existing inequalities as their special cases. Second, combining with quadratic convex combination method, these double-integral inequalities are employed to formulate a delay-dependent stability condition for MNNs with delays. Third, a state-dependent switching control law is obtained for MNNs with delays based on the proposed stability conditions. The desired feedback gain matrices are accomplished by solving a set of linear matrix inequalities. Finally, the feasibility and effectiveness of the proposed results are tested by two numerical examples.

비선형 시스템을 위한 Takagi-Sugeno 퍼지 샘플치필터

This paper presents the stability conditions of the Takagi-Sugeno (T-S) fuzzy sampled-data filter. The error system between the T-S fuzzy system and fuzzy filter is presented. In the sense of the Lyapunov stability analysis, the stability conditions are given, which can be represented in terms of linear matrix inequalities (LMIs). The proposed stability conditions utilize the different approach from the conventional methods, and have better performance than that of the conventional ones. The simulation example is given to show the effectiveness of the proposed method.

New results on stability analysis and stabilization of time-delay continuous Markovian jump systems with partially known rates matrix

Summary In this note, the problems of stability analysis and controller synthesis of Markovian jump systems with time-varying delay and partially known transition rates are investigated via an input–output approach. First, the system under consideration is transformed into an interconnected system, and new results on stochastic scaled small-gain condition for stochastic interconnected systems are established, which are crucial for the problems considered in this paper. Based on the system transformation and the stochastic scaled small-gain theorem, stochastic stability of the original system is examined via the stochastic version of the bounded realness of the transformed forward system. The merit of the proposed approach lies in its reduced conservatism, which is made possible by a precise approximation of the time-varying delay and the new result on the stochastic scaled small-gain theorem. The proposed stability condition is demonstrated to be much less conservative than most existing results. Moreover, the problem of stabilization is further solved with an admissible controller designed via convex optimizations, whose effectiveness is also illustrated via numerical examples. Copyright © 2015 John Wiley & Sons, Ltd.

State estimation incorporating infrequent, delayed and integral measurements

This paper investigates the problem of state estimation incorporating infrequent, delayed and integral measurements. In chemical process, there often exist two types of measurements. On one hand, the measurements for process variables such as flow rates and pressures are sampled frequently and are available nearly instantaneously. On the other hand, the measurements for quality variables such as concentration are sampled infrequently and are available with a delay due to long analysis time involved. Moreover, due to the interval time taken by chemical sample collection, the measurements for some quality variables have another important characteristic: it is a function of the states of the compositions over a period of time. This paper formulates the process with infrequent, delayed and integral measurements as an equivalent variable dimension system, whose measurements are both delay and integration free. Based on the new model, a variable dimension unscented Kalman filter (VD-UKF) is proposed to estimate the states. Furthermore, the stability of the proposed VD-UKF is analyzed. Compared with the existing results, the proposed stability condition is significantly relaxed and the invertibility condition of Jacobian matrices is no longer needed. Finally, a simulation example demonstrates the effectiveness of the proposed method.

Further Results on Exponential Robust Stability Analysis for Recurrent Neural Networks With Time-Varying Delay

In this paper, the problems of determining the robust exponential stability and estimating the exponential convergence rate for recurrent neural networks (RNNs) with parametric uncertainties and time-varying delay are studied. The relationship among the time-varying delay, its upper bound, and their difference is taken into account. The developed stability conditions are in terms of linear matrix inequalities (LMIs) and the integral inequality approach (IIA), which can be checked easily by recently developed algorithms solving LMIs. Furthermore, the proposed stability conditions are less conservative than some recently known ones in the literature, and this has been demonstrated via four examples with simulation.

Finite-step Terminal Ingredients for Stabilizing Model Predictive Control

This paper proposes a novel construction of the terminal cost and terminal set in stabilizing model predictive control that uses finite-step Lyapunov functions and finite-step invariant sets. This construction results in a periodic terminal set constraint associated with a finite sequence of terminal sets, out of which none is required to be invariant. We argue that constructing such sets is easier and more scalable compared to invariant sets, while a comparable region of attraction is obtained for the same prediction horizon. In the one step case the proposed stability conditions reduce naturally to the standard terminal cost and constraint set stability conditions.

Stability investigation of the Generalised-α time integration method for dynamic coupled consolidation analysis

In this paper, the stability of the Generalised-α time integration method (the CH method) for a fully coupled solid-pore fluid formulation is analytically investigated for the first time and the corresponding theoretical stability conditions are proposed based on a rigorous mathematical derivation process. The proposed stability conditions simplify to the existing ones of the CH method for the one-phase formulation when the solid–fluid coupling is ignored. Furthermore, by degrading the CH method to the Newmark method, the stability conditions are in agreement with the ones proposed in previous stability investigations on coupled formulation for the Newmark method. The analytically derived stability conditions are validated with finite element (FE) analyses considering a range of loading conditions and for various soil permeability values, showing that the numerical results are in agreement with the theoretical investigation. Then, the stability characteristics of the CH method are explored beyond the limits of the theoretical investigation, assuming elasto-plastic soil behaviour which is prescribed with a bounding surface plasticity constitutive model. Since the CH method is a generalisation of a number of other time integration methods, the derived stability conditions are relevant for most of the commonly utilised time integration methods for the two-phase coupled formulation.