Robust Stabilizability of Switched Linear Time-Delay Systems with Polytopic Uncertainties

This paper concerns the problems of robust stability analysis and stabilization for uncertain linear system with time-varying delay. In order to obtain a robust stability criterion with less conservative for uncertain linear delay system, a quadratic integral inequality is improved by matrix decomposition. Further, two different kinds of controller, i.e., linear memoryless state feedback controller and linear state feedback controller with state delay, are derived in terms of the robust stability criteria. Two numeral examples are provided to demonstrate the validity of the proposed methods.

Studies the optimal guaranteed control problem for a class of linear uncertain time-delay systems. The systems under consideration are described by state differential equations with time-varying unknown but norm-bounded uncertain parameters and time-varying unknown but bounded delay in states. An upper bound of the performance measured by standard quadratic function is constructed and based on the delay-dependent stabilizing approach, a method of designing a state feedback robust quadratic stabilizing control law guaranteeing this quadratic performance index be less than the constructed upper bound is derived. A procedure is given to select an optimal state feedback controller in the sense of minimizing this upper bound of the quadratic performance index. The results depend on the size of delay and are given in terms of several linear matrix inequalities.

An Integro-differential-delay Operator Approach to Transformations of Linear Differential Time-delay Systems

This article is concerned with robust stability of uncertain discrete-time linear systems. The matrix defining the linear system (system matrix) is assumed to depend affinely on a set of time-invariant unknown parameters lying on a known polytope. Robust stability is investigated by checking whether a certain integer power κ of the uncertain system matrix has spectral norm less than one. This peculiar stability test is shown to be equivalent to the positivity analysis of a homogeneous symmetric matrix polynomial with precisely known coefficients and degree indexed by κ. A unique feature is that no extra variables need to be added to the problems being solved. Numerical experiments reveal that the value of κ needed to test robust stability is mostly independent of the system dimension but grows sharply as the eigenvalues of the uncertain system approach the unit circle. By identifying the proposed stability test with a particular choice of a parameter-dependent Lyapunov function, extra variables can be introduced, yielding linear matrix inequalities optimisation problems of improved convergence.

This study aims at the simultaneous solution of several quaternion linear systems with the same Hermitian and positive definite coefficient matrix by employing the conjugate gradient method. We consider the setting when the quaternion Hermitian positive definite coefficient matrix at hand is very large so that direct methods are not applicable. In the study, we first transform linear quaternion systems into real linear systems. The transformed real linear systems have special structure due to the fact that they are real representations of quaternion systems. Benefitting from the special structure, we further reduce the size of these linear systems. Then a block conjugate gradient method is applied to the resulting reduced real linear systems. The solution obtained after applying the conjugate gradient method is a real representation of the solution of the original quaternion problem. Thus, a conversion of this real solution to the quaternion setting is performed in the end.

This research examines new methods for stabilizing linear time-delay systems that are subject to denial-of-service (DoS) attacks. The study takes into account the different effects that a DoS attack can have on the system, specifically delay-independent and -dependent behaviour. The traditional proportional-integral-derivative (PID) acts on the error signal, which is the difference between the reference input and the measured output. The approach in this paper uses what we call the PID state feedback strategy, where the controller acts on the state signal. Our proposed strategy uses the Lyapunov-Krasovskii functional (LKF) to develop new linear matrix inequalities (LMIs). The study considers two scenarios where the time delay is either a continuous bounded function or a differentiable and time-varying function that falls within certain bounds. In both cases, new LMIs are derived to find the PID-like state feedback gains that will ensure robust stabilization. The findings are illustrated with numerical examples.

Stability conditions of T-S fuzzy system based on a common positive definite matrix or a set of positive definite matrices have been proposed in lots of literature. This paper proposes an approach to relax the conservatism of those stability conditions. By using the concept of the efficient maximal overlapped-rules group (EMORG), a relaxed sufficient condition is derived to guarantee quadratic stability of the fuzzy model globally. This condition only requires finding a local common positive definite matrix in each EMORG. Moreover, the proposed stability condition includes previous results as special cases. Several examples are illustrated to show the effectiveness of the proposed stability condition.

A linear control system defined by a stationary differential equation of nth order with several commensurate lumped and distributed delays in state is considered. In the system, the input is a linear combination of m variables and their derivatives of order not more than n−p and the output is a k-dimensional vector of linear combinations of the state and its derivatives of order not more than p−1. For this system, a spectrum assignment problem by linear static output feedback with commensurate lumped and distributed delays is studied. Necessary and sufficient conditions are obtained for solvability of the arbitrary spectrum assignment problem by static output feedback controller. Corollaries on stabilization of the system are obtained.

This paper mainly discusses the robust quadratic stability and stabilization of linear discrete time stochastic systems with state delay and uncertain parameters. By means of the linear matrix inequality (LMI) method, a sufficient condition is respectively obtained for the stability and stabilizability of the considered system. Moreover, an example with simulations is given to verify the effectiveness of our theoretical results.

The design of robust controllers is considered here for a class of two-dimensional (2-D) discrete switched systems described by the Roesser model with polytopic uncertainties. Attention focuses on the design of a switched state feedback controller, which guarantees the robust asymptotic stability and a prescribed performance for the closed-loop system. By using multiple parameter-dependent Lyapunov functionals, and introducing some switched free-weighting matrices, a new sufficient condition for the robust performance analysis of uncertain 2-D discrete switched systems is developed. Furthermore, the design of switched state feedback controller is proposed in terms of linear matrix inequalities (LMIs). Illustrative examples are given to illustrate the effectiveness of the developed theoretical results.

In this paper, the problem of robust H∞ filtering for switched linear discrete-time systems with polytopic uncertainties is investigated. Based on the mode-switching idea and parameter-dependent stability result, a robust switched linear filter is designed such that the corresponding filtering error system achieves robust asymptotic stability and guarantees a prescribed H∞ performance index for all admissible uncertainties. The existence condition of such filter is derived and formulated in terms of a set of linear matrix inequalities (LMIs) by the introduction of slack variables to eliminate the cross coupling of system matrices and Lyapunov matrices among different subsystems. The desired filter can be constructed by solving the corresponding convex optimization problem, which also provides an optimal H∞ noise-attenuation level bound for the resultant filtering error system. A numerical example is given to show the effectiveness and the potential of the proposed techniques. Copyright © 2006 John Wiley & Sons, Ltd.

In this paper, we consider generalized delay-dependent stability conditions of Takagi-Sugeno fuzzy systems with discrete and distributed time-delays. For fuzzy systems with discrete time-delays, both delay-independent stability conditions and delay-dependent stability conditions have already been obtained. However, no stability condition for fuzzy systems with distributed time-delays has appeared in the literature. This is also true in case of the robust stability for uncertain fuzzy systems with distributed time-delays. With the aid of the stability theorems of linear systems with distributed time-delays, we employ a similar Lyapunov functional for the stability of fuzzy time-delay systems. Then we introduce many free matrices to such a Lyapunov functional, in order to obtain delay-dependent stability conditions. These techniques lead to generalized and less conservative stability conditions. In fact, delay-dependent stability conditions thus obtained are given in linear matrix inequalities(LMIs) and guarantee a wide stability region. We also consider the robust stability of fuzzy time-delay systems with uncertain parameters. Applying the same techniques made on the stability conditions, we obtain delay-dependent sufficient conditions for the robust stability of uncertain fuzzy systems with discrete and distributed time-delays. Moreover, we consider the state feedback stabilization. Based on stability and robust stability conditions, we obtain conditions for the state feedback controller to stabilize the fuzzy time-delay systems. Finally, we take some examples to illustrate our results.

The problem of solving linear systems is one of the most fundamental problems in computer science, where given a satisfiable linear system (A,b), for A ∈ ℝ^{n×n} and b ∈ ℝⁿ, we wish to find a vector x ∈ ℝⁿ such that Ax = b. The current best algorithms for solving dense linear systems reduce the problem to matrix multiplication, and run in time O(n^ω). We consider the problem of finding ε-approximate solutions to linear systems with respect to the L₂-norm, that is, given a satisfiable linear system (A ∈ ℝ^{n×n}, b ∈ ℝⁿ), find an x ∈ ℝⁿ such that ||Ax - b||₂ ≤ ε||b||₂. Our main result is a fine-grained reduction from computing the rank of a matrix to finding ε-approximate solutions to linear systems. In particular, if the best known Õ(n^ω) time algorithm for computing the rank of n × O(n) matrices is optimal (which we conjecture is true), then finding an ε-approximate solution to a dense linear system also requires Ω̃(n^ω) time, even for ε as large as (1 - 1/poly(n)). We also prove (under some modified conjectures for the rank-finding problem) optimal hardness of approximation for sparse linear systems, linear systems over positive semidefinite matrices and well-conditioned linear systems. At the heart of our results is a novel reduction from the rank problem to a decision version of the approximate linear systems problem. This reduction preserves properties such as matrix sparsity and bit complexity.

We derive scalable decentralized conditions that can guarantee robust stability for networks of linearly interconnected, stable, linear time invariant dynamical systems. Unlike previous results of this kind, we allow for heterogeneous dynamics on arbitrary interconnection topologies (i.e., linear systems on arbitrary underlying graphs). Robust stability of the entire network is guaranteed by satisfying local rules that involve only an agent and its neighboring dynamics; each new agent will introduce only one additional condition of this kind and hence the stability certificates scale with the network size. An application of this theory is given, where robustness analysis of Internet congestion control protocols is carried out in the general case of dynamics at both users and resources without any global bounds on the dynamics

This paper develops a multiple fixed model blending based adaptive parameter identification scheme for multi-input multi-output (MIMO) systems with polytopic parameter uncertainty. The developed identification scheme is proven to be asymptotically stable for uncertain linear time-invariant MIMO systems, and is shown to provide fast adaptation for even uncertain linear time-varying (LTV) systems. Furthermore, utilizing the proposed parameter identification scheme, a linear-quadratic (LQ) optimal multiple model adaptive control (MMAC) scheme is developed for linear MIMO systems with polytopic uncertainties. The proposed MMAC scheme is applied to tracking control of uncertain lateral vehicle dynamics. A set of simulation test results are presented to verify the stability and effectiveness of the proposed MMAC scheme.