Parameter-dependent Matrix Inequalities Research Articles

Asynchronous secure control for singular nonhomogeneous Markov jump cyber‐physical systems against dual cyber‐attacks

AbstractIn this article, based on the dynamic event‐triggered transmission protocol, the issue of asynchronous secure control for singular nonhomogeneous Markov jump cyber‐physical systems (CPSs) is studied. The matrix that is governed by the time‐varying transition rates lies in a convex polyhedron. To further degrade the transmission percentage over the network, the dynamic event‐triggered transmission protocol is developed. Besides, owing to the vulnerability of the network, the replay attack and deception attack are both considered, the occurrences of which are modelled by mutual independent random variables satisfying Bernoulli distribution. By constructing a parameter dependent and an auxiliary dynamic variable dependent Lyapunov functional, sufficient conditions of stochastic admissible for singular nonhomogeneous Markov jump CPSs are derived by parameter dependent matrix inequalities, which can be converted into the finite linear matrix inequalities. A numerical example is provided to illustrate the feasibility and effectiveness of the theoretical results.

Variable-Parameter-Dependent Saturated Robust Control for Vehicle Lateral Stability

This article proposes a novel gain-scheduling control method to improve vehicle lateral stability based on a variable-parameter-dependent approach. A time-varying velocity-dependent model to describe the vehicle lateral dynamic characteristics is constructed under cornering stiffness uncertainties and controller saturation. A novel Lyapunov function with variable-parameter-dependent Lyapunov matrix is established to develop the conditions. Using the proposed Lyapunov function, a general condition of the variable-parameter-dependent saturated robust yaw moment controller is provided in terms of time-varying parameter-dependent matrix inequalities. In order to obtain a tractable solution, a condition is further developed with finite linear matrix inequalities. Moreover, an optimal distribution method is adopted to generate the desired yaw moment based on torque allocation. These torques are computed by optimizing the distribution errors and tire workloads. Simulations results under both J-turn and double-lane changing scenarios are used to illustrate the merits of the proposed method. The control synthesis approach is also applicable to other applications involving time-varying parameters.

Global Exponential Stability of Delayed Neural Networks Based on a New Integral Inequality

This paper focuses on the problem of exponential stability for a class of neural networks with time-varying delays. A more general inequality is established which extends the auxiliary function-based integral inequality. Based on the inequality and parameter-dependent matrix inequality, an improved delay-dependent stability criterion is obtained by constructing an augmented Lyapunov functional. Three numerical examples are given to illustrate the efficiency of the method.

Advanced stability criteria for static neural networks with interval time-varying delays via the improved Jensen inequality

This paper investigates the asymptotic stability of static neural network (SNN) with interval time-varying delay. A proper Lyapunov-Krasovskii functional (LKF) is presented to coordinate with the features of the improved Jensen inequality. Based on the new proposed LKF together with the improved Jensen inequality and a parameter-dependent matrix inequality, new sufficient conditions ensuring the delay-dependent stability are derived in terms of linear matrix inequality (LMI). By considering a marginal augmented vector with two complementary integral terms to yield more crossing information, a further enhanced stability criterion is obtained. The validity of the theorectical results is demonstrated by two numerical examples.

Output-feedback stabilization of singular LPV systems subject to inexact scheduling parameters

This paper is concerned with the output-feedback stabilization problem for a class of singular linear parameter-varying (LPV) systems. The objective is to design a dynamic output-feedback controller such that the resulting closed-loop system is robustly admissible. Different from the classical gain-scheduling techniques, it is assumed in this paper that the scheduling parameters involved in the plant model are not exactly available for the controller. First, a sufficient and necessary condition for the solvability of the underlying stabilization problem is obtained in the form of general parameter-dependent matrix inequalities. Second, an equivalent transformation is developed to convert the general parameter-dependent conditions into parameter-dependent linear matrix inequalities (LMIs). Third, the singular LPV system whose coefficient matrices take a polytopic form is considered, and the finite LMI-based conditions are derived. Finally, an illustrative example is provided to show the effectiveness of the proposed design methods.

H2 and H∞ digital redesign of analog controllers for continuous-time polytopic systems

The aim of this paper is to investigate the problems of designing H∞ and H2 discrete-time stabilizing controllers for polytopic continuous-time systems through the digital redesign methodology. This technique consists of determining a stabilizing digital gain starting from a predetermined continuous-time controller, such that a performance criterion, associated to the difference between the continuous-time and discrete-time state trajectories, is minimized. In this paper, synthesis conditions are proposed in terms of parameter-dependent matrix inequalities with polynomial variables and a given scalar parameter. Differently from other approaches that minimize the Euclidean norm and can only handle precisely known systems, the proposed method minimizes a bound to the H2 or H∞ norm of a system that represents the dynamics of the error between the polytopic continuous-time plant and the discrete-time representation. Numerical examples are presented to show the advantages of the proposed methodology when compared with other conditions from the literature in the precisely known case, and to illustrate the applicability of the technique for uncertain systems.

H2 and H∞ dynamic output feedback control of continuous-time Markov systems with uncertain rates using LMIs

The aim of this paper is to investigate the problems of H∞ and H2 dynamic output feedback control for continuous-time Markov jump linear systems (MJLS) subject to uncertain transition rates. Synthesis conditions are proposed in terms of parameter-dependent matrix inequalities with a scalar parameter and polynomial variables. The conditions become linear for fixed values of the scalar parameter, which provides an extra degree of freedom to search for feasible solutions and less conservative H∞ and H2 bounds. Differently from the existing approaches, the proposed method provides full order mode-dependent controllers constructed in terms of the state space matrices of the operation modes and partitions of the slack variables introduced in the design conditions. Therefore, the resulting controller does not depend explicitly on the Lyapunov matrix, being robust with respect to the uncertain transition rates. Moreover, the closed-loop MJLS stability and the bounds to the H∞ or H2 norm are certified by means of polynomially parameter-dependent Lyapunov matrices. Numerical examples illustrate the applicability and the advantages of the proposed technique when compared with other approaches from the literature.

New results on H∞ filtering for Markov jump systems with uncertain transition rates

In this paper, we study the H∞ filtering problem for a class of continuous-time Markov jump systems with time-varying uncertainties in transition rates, in which the uncertain transition rates are assumed to be affine parameter-dependent uncertainty models. By converting the affine parameter-dependent uncertainty models for transition rates into time-varying polytopic ones and using the Lyapunov function approach, a sufficient condition on the existence of an H∞ filter is obtained in terms of a parameter-dependent matrix inequality. Also, the parameter-dependent matrix inequality is converted into a set of parameter-free linear matrix inequalities which can be solved numerically. Illustrative examples are given to demonstrate the effectiveness and advantages of the approach.

Stability analysis of continuous-time Markovian jump time-delay systems with time-varying transition rates

This paper investigates the stability problem of Markovian jump systems. The systems under consideration include time-varying delay and time-varying transition rates. The time-varying transition rate matrix in continuous-time domain is considered to lie in a convex bounded domain. By constructing a parameter-dependent Lyapunov functional and fully considering the information about the rate of change of time-varying parameters, a parameter-dependent stochastic stability condition is derived. Furthermore, based on the structure characteristics of Lyapunov matrix and transition rate matrix, the parameter-dependent matrix inequality is converted into a finite set of linear matrix inequalities (LMIs). A numerical example is presented to demonstrate the effectiveness of the theoretical results.

Scheduled control for robust attenuation of non-stationary sinusoidal disturbances with measurable frequencies

Attenuation of sinusoidal disturbances with uncertain yet online measurable frequencies is considered. The disturbances are modeled as the outputs of an undisturbed parameter-dependent exogenous system with a skew-symmetric system matrix, obtained in response to nonzero initial conditions. The problem is formulated for a parameter-dependent plant as the synthesis of a parameter-dependent controller in a way to ensure internal stability as well as a desired level of steady-state disturbance attenuation in the face of all admissible parameter variations. The solvability of this problem is first related to the existence of bounded solutions to a matrix differential regulator equation subject to an asymptotic norm constraint. Reformulating this as a parameter-dependent state-feedback like synthesis, based on which suitable solutions to the differential regulator equation can be obtained online, tractable solvability conditions are then provided in the form of parameter-dependent matrix inequalities. Controllers that solve the generalized asymptotic regulation problem are also parameterized in terms of the suitable solutions of the differential regulator equation and some free parameter-dependent matrices that are to be designed off-line to ensure stability. A procedure is then developed to design the free parameters in a way to achieve desirable transient behavior. The use of the developed synthesis procedure is illustrated on a simplified version of the course control problem in ship steering.

LPV Control for Robust Attenuation of Non-stationary Sinusoidal Disturbances with Measurable Frequencies

Attenuation of sinusoidal disturbances with uncertain and arbitrarily time-varying, yet online measurable frequencies is considered. The disturbances are modeled as the outputs of an autonomous exogenous system, whose system matrix depends on some uncertain parameters and is skew-symmetric for all admissible parameter values. A procedure is then developed for the synthesis of an observer-based controller that uses the online measurements of the uncertain parameters to guarantee a desired level of attenuation at steady-state in the face of all admissible parameter variations. The controller is scheduled by the measurements of the uncertain parameters as well as the matrix-valued outputs of a unit that is also scheduled on the uncertain parameters. The synthesis procedure is based on solving a convex optimization problem in which the variables are subject to a set of parameter-dependent matrix inequality constraints, which can be relaxed into finitely many linear matrix inequalities. The procedure also provides options for improving the transient response.