Linear Matrix Inequality Stability Conditions Research Articles

Delay-Dependent Stability Improvement for Networked Control Systems: a Sampled Data Approach

This paper concerns the stability conditions and controller design for sampled-data networked control systems (NCSs) model subject to network communication delays, the main objective is to guarantee the maximum allowable upper bounds of network-induced time-varying delays that keep the NCSs stable. First, Lyapunov-Krasovskii functional with simple and double-integral terms is constructed considering both upper and lower bounds of network delay. Then, less conservative Linear Matrix Inequalities (LMIs) stability conditions are established using null terms to introduce free weighting matrices based on the Leibniz-Newton formula. Furthermore, Finsler's lemma is used for the relaxation of the obtained LMI's stability conditions using slack decision variables. It is also used to decouple Lyapunov-Krasovskii matrices from the system ones. The application of the proposed approach for different NCSs gives higher upper delay bounds compared with other methods.

Simple LMIs for stability of stochastic systems with delay term given by Stieltjes integral or with stabilizing delay

This paper introduces stability conditions in the form of linear matrix inequalities (LMIs) for general linear retarded systems with a delay term described by Stieltjes integral. The derived LMIs provide in the unified form conditions for both discrete and distributed delays. Two Lyapunov-based methods for the asymptotic mean square stability of stochastic linear time-invariant systems are presented. The first one employs neutral type model transformation and augmented Lyapunov functionals. Differently from the existing LMI stability conditions based on neutral type transformation, the proposed conditions do not require the stability of the corresponding integral equations. Moreover, it is shown that in the simplest existing LMIs based on non-augmented Lyapunov functionals, the stability analysis of the integral equation can be omitted. The second method is based on a stochastic extension of simple Lyapunov functionals depending on the state derivative. The same two methods are further applied to delay-induced stability analysis of stochastic vector second-order systems, simplifying the recent results via neutral type transformation and leading to new conditions for stochastic systems via the second method. Numerical examples give comparison of results via different methods.

Design of robust <i>H</i><SUB align="right">∞ fuzzy output feedback controller for affine nonlinear systems: fuzzy Lyapunov function approach

In this paper, we propose a new systematic approach based on non-quadratic Lyapunov function and technique of introducing slack matrices, for a class of affine nonlinear systems with disturbance. To achieve the goal, first, the affine nonlinear system is represented via Takagi-Sugeno (T-S) fuzzy bilinear model. Subsequently, the robust H∞ controller is designed based on parallel distributed compensation (PDC) scheme. Then, the stability conditions are derived in terms of linear matrix inequalities (LMIs) by utilising Lyapunov function. Moreover, some slack matrices are proposed to reduce the conservativeness of the LMI stability conditions. Finally, for illustrating the merits and verifying the effectiveness of the proposed approach, the application of an isothermal continuous stirred tank reactor (CSTR) for Van de Vusse reactor is discussed in details.

Design of robust H_ fuzzy output feedback controller for affine nonlinear systems: Fuzzy Lyapunov function approach

In this paper, we propose a new systematic approach based on non-quadratic Lyapunov function and technique of introducing slack matrices, for a class of affine nonlinear systems with disturbance. To achieve the goal, first, the affine nonlinear system is represented via Takagi-Sugeno (T-S) fuzzy bilinear model. Subsequently, the robust H∞ controller is designed based on parallel distributed compensation (PDC) scheme. Then, the stability conditions are derived in terms of linear matrix inequalities (LMIs) by utilising Lyapunov function. Moreover, some slack matrices are proposed to reduce the conservativeness of the LMI stability conditions. Finally, for illustrating the merits and verifying the effectiveness of the proposed approach, the application of an isothermal continuous stirred tank reactor (CSTR) for Van de Vusse reactor is discussed in details.

Hierarchical Stability Conditions for a Class of Generalized Neural Networks With Multiple Discrete and Distributed Delays.

This brief investigates the analysis issue for global asymptotic stability of a class of generalized neural networks with multiple discrete and distributed delays. To tackle delays arising in different neuron activation functions, we employ a generalized model with multiple discrete and distributed delays which covers various existing neural networks. We then generalize the Bessel-Legendre inequalities to deal with integral terms with any linearly independent functions and nonlinear function of states. Based on these inequalities, we design the Lyapunov-Krasovskii functional and derive hierarchical linear matrix inequality stability conditions. Finally, three numerical examples are provided to demonstrate that the proposed method is less conservative with a reasonable numerical burden than the existing results.

Chain observer for Lipschitz non‐linear systems with long time‐varying delayed measurements

The main contribution of this study is to present a chain observer for Lipschitz non-linear systems when the output measurements are affected by a long known and bounded time-varying delay. The proposed chain observer is composed by a set of observers with similar algorithms. Each observer estimates the state in a time horizon, while the first observer of the chain estimates the current state. The structure of each observer of the chain is based on the presence of a dynamical term which permits the compensation of the time-varying delay. The observer gain is computed by solving parameterised linear matrix inequalities (LMIs) which depend on the delay. The less conservative Lipschitz condition allows to manage less restrictive LMI stability conditions, which leads to a more general class of Lipschitz non-linear systems. A Lyapunov-Krasovskii functional is used to demonstrate that the estimation error converges asymptotically to zero. The performance of the proposed observer is evaluated through numerical examples. The observer exhibits good estimation of the system state, even in the presence of significant delayed measurements.

A Novel Actuator Fault-tolerant Control Strategy of DFIG-based Wind Turbines Using Takagi-Sugeno Multiple Models

In this paper, a new combined fuzzy observer-based fault-tolerant tracking control scheme is proposed for a doubly fed induction generator (DFIG) based wind turbine (WT) subject to actuator faults. The main contribution consists of the proposal of a novel fault-tolerant fuzzy tracking controller combined with a nominal control law. The control objective is to ensure good state references tracking regardless of the actuator faults effects and simultaneous system state and faults estimation. This later requires the knowledge of the occurrence of actuator faults which are estimated from a Takagi-Sugeno Fuzzy Proportional Integral Observer (T-S FPIO). Within this control scheme, a T-S FPIO has been developed to provide stability tracking error dynamics even the system is subjected to different actuator faults. A compensation term is appended to the composite controller and to ensure robustness against actuator faults. Stability and tracking analysis properties are demonstrated through a quadratic Lyapunov function, which are formulated in terms of Linear Matrix Inequalities (LMIs). The observer gains are determined based on the proposed LMIs stability conditions. A numerical simulation is carried out on a typical 1.5 MW DFIG based WT system to access the effectiveness of the proposed control scheme in comparison to the existing results.

Stabilization of rigid body attitude motion with time-delayed feedback

A continuous nonlinear full-state time-delayed feedback control scheme is designed within the framework of geometric mechanics to stabilize the rigid body attitude motion, which is subject to an unknown constant time delay in feedback measurement. The attitude kinematics is globally described on the matrix Lie group SO(3) of rigid body rotations. A Morse–Lyapunov–Krasovskii functional is utilized to guarantee the asymptotic stability of the system, which yields the control gain matrices via linear matrix inequality stability conditions. In light of this result, a delayed feedback control scheme is also designed for the planar rotational motion to examine the almost global asymptotic stability of the system in the presence of an unknown time delay in feedback measurement. Simulations are performed for the proposed control schemes based on the discretized models of the controlled systems and the performance of the proposed controllers are evaluated by employing large maneuvers.

Modified Sliding Mode Control for Mismatched Uncertain Large-Scale Systems

In this paper, a modified sliding mode control is proposed to stabilize a class of large-scale systems with mismatched uncertainties, unknown exogenous disturbances and without the measurements of the states. Based on the Lyapunov method, appropriate linear matrix inequality stability conditions are derived to guarantee the stability of the system. Then, a modified adaptive sliding mode control law is proposed to guarantee the finite time reachability of the system states by using output feedback only. Final, a double-inverted pendulums system connected by a spring is simulated to demonstrate the efficacy of the proposed method.

Cluster synchronization in nonlinear complex networks under sliding mode control

This paper investigates the cluster synchronization for network of nonlinear systems via differential mean value theorem method and sliding mode control strategy. Because of the existence of the nonlinear dynamics, the differential mean value theorem method is used to transform the nonlinear error complex network system into a linear parameter-varying system. Since the robustness to the uncertainties of the sliding mode, we can deal with the linear parameter-varying system which contains the time-varying mismatched uncertainties by applying the sliding mode control strategy through the equivalent transformation. In addition, appropriate linear matrix inequality stability condition by the Lyapunov method is derived such that each subsystem in the new sliding mode is completely invariant to both matched and mismatched uncertainties. Finally, simulation example is shown to illustrate the effectiveness of the proposed method.

A multitask sliding mode control for mismatched uncertain large-scale systems

A new sliding mode control (SMC) approach, output variables only, single phase only and chattering phenomenon free, is presented for a class of mismatched uncertain large-scale systems. For a new multitask SMC, it is not required that the system states are available. Moreover, the sliding function in this study just depends on output variables. Using an exponential type sliding surface, the system states are always in the sliding mode at the beginning time t = 0. Using a newly appropriate linear matrix inequality stability conditions by the Lyapunov method are derived such that each subsystem in the new sliding mode is completely invariant to matched uncertainties. As a result, robustness of the mismatched uncertain large-scale systems can be assured throughout an entire response of the system starting from the initial time t = 0. In every subsystem, a scheme of decentralised control using only output states is proposed. In addition, a continuous controller is finally designed for chattering removal. Finally, a numerical example is used to demonstrate the efficacy of the proposed method.

Robust observer-based H control for nonlinear T-S fuzzy time-delay systems

This paper considers the design problem of observer-based H output feedback control over the nonlinear Takagi-Sugeno (T-S) fuzzy systems with distributed delays．Firstly, delay-dependent sufficient conditions satisfy the H performance, are given for the stability analysis in terms of non-strict linear matrix inequality; then, the equivalent linear matrix inequality stability conditions can be obtained. Based on this, the solvable of the observer-based H output feedback control problem is proposed. Finally, a numerical example is given to illustrate the effectiveness of the design approach here.

Linear Matrix Inequality Stability Conditions from a Parameter-Dependent Lyapunov Functional for Linear Systems with a Time-Varying Delay

This paper proposes new stability conditions for linear systems with a time-varying state delay. The time-varying delay is treated as a polytopic uncertainty parameter, and a Lyapunov functional that is dependent on the delay parameter is constructed. Linear matrix inequality conditions of system stability and robust stability are derived. Numerical and practical examples are given to show the effectiveness and non-conservativeness of the proposed conditions.

Fuzzy Controller Design Subject to Actuator Saturation for Dynamic Ship Positioning Systems with Multiplicative Noises

This paper deals with the fuzzy controller design problem for the dynamic ship positioning systems with multiplicative noises. The Takagi—Sugeno (T—S) fuzzy model is employed to represent the non-linear dynamic ship positioning systems. Subject to actuator saturation, some sufficient conditions are derived to guarantee the stability of the closed-loop systems. The stability conditions developed in this paper can be coordinated to the linear matrix inequality (LMI) constraints. By solving these LMI stability conditions, a parallel distributed compensation (PDC)-based fuzzy controller is obtained to stabilize the T—S fuzzy models with multiplicative noises. In order to illustrate the availability and practicability of proposed fuzzy controller design approach, a numerical simulation is provided to express the controlled responses of dynamic ship positioning systems with multiplicative noises.

Pseudo-state feedback stabilization of commensurate fractional order systems

This paper addresses the problem of pseudo-state feedback stabilization of commensurate fractional order systems (FOS). In the proposed approach, Linear Matrix Inequalities (LMI) formalism is used to check if the pseudo-state matrix eigenvalues belong to the FOS stability region of the complex plane. A review of LMI stability conditions is first proposed for fractional order 0 < ν < 1 and 1 < ν < 2 . The paper then focuses particularly on the case 0 < ν < 1 as the stability region is non-convex and associated LMI condition is not as straightforward to obtain as in the case 1 < ν < 2 . A new LMI stability condition is thus proposed. Based on this condition, a necessary and sufficient LMI method for the design of stabilizing controllers is given. This method paves the way for extension to FOS of various LMI-based results. Among these possible extensions, a first result on robust control of polytopic fractional order systems is given in this paper.

LMI stability conditions for fractional order systems

After an overview of the results dedicated to stability analysis of systems described by differential equations involving fractional derivatives, also denoted fractional order systems, this paper deals with Linear Matrix Inequality (LMI) stability conditions for fractional order systems. Under commensurate order hypothesis, it is shown that a direct extension of the second Lyapunov’s method is a tedious task. If the fractional order ν is such that 0<ν<1, the stability domain is not a convex region of the complex plane. However, through a direct stability domain characterization, three LMI stability analysis conditions are proposed. The first one is based on the stability domain deformation and the second one on a characterization of the instability domain (which is convex). The third one is based on generalized LMI framework. These conditions are applied to the gain margin computation of a CRONE suspension.