System Of Linear Matrix Inequalities Research Articles

On Safe Tractable Approximations of Chance-Constrained Linear Matrix Inequalities

In the paper we consider the chance-constrained version of an affinely perturbed linear matrix inequality (LMI) constraint, assuming the primitive perturbations to be independent with light-tail distributions (e.g., bounded or Gaussian). Constraints of this type, playing a central role in chance-constrained linear/conic quadratic/semidefinite programming, are typically computationally intractable. The goal of this paper is to develop a tractable approximation to these chance constraints. Our approximation is based on measure concentration results and is given by an explicit system of LMIs. Thus, the approximation is computationally tractable; moreover, it is also safe, meaning that a feasible solution of the approximation is feasible for the chance constraint.

Robust synchronization of different coupled oscillators: Application to antenna arrays

This paper deals with the synchronization of a chain of nonlinear and uncertain models of nonidentical oscillators. Using Lyapunov's theory of stability, a dynamical controller guaranteeing the synchronization of the oscillators is determined. The problem of synchronization is transformed into a problem of asymptotic stabilization for a nonlinear system and then is formulated as a system of linear matrix inequalities where the parameter variations of the two oscillators and their differences are modeled by polytopic matrices. The theoretical result is successfully applied to an array of transistor-based oscillators used in smart antenna systems.

On the Low Rank Solutions for Linear Matrix Inequalities

In this paper we present a polynomial-time procedure to find a low-rank solution for a system of linear matrix inequalities (LMI). The existence of such a low-rank solution was shown in the work of Au-Yeung and Poon and the work of Barvinok. In the approach of Au-Yeung and Poon an earlier unpublished manuscript of Bohnenblust played an essential role. Both proofs in the work of Au-Yeung and Poon and that of Barvinok are nonconstructive in nature. The aim of this paper is to provide a polynomial-time constructive procedure to find such a low-rank solution approximatively. Extensions of our new results and their relations to some of the known results in the literature are discussed.

On Lyapunov stability of impulsive Takagi–Sugeno fuzzy systems

We analyze the Lyapunov stability of impulsive Takagi–Sugeno fuzzy systems. Using the direct Lyapunov method, we establish sufficient conditions for the stability of these systems. We show that these conditions can be expressed in terms of a system of linear matrix inequalities. As an example, we consider an impulsive fuzzy control in a two-species “predator–prey” model.

Antenna Arrays Principle and Solutions: Robust Control Approach

This paper treats solutions on the ability of a chain of non identical oscillators to drive antenna arrays. Frequency approaches were studied in order to solve the problem of synchronization of the oscillators. However, in this article, a new structure of chain of oscillators is introduced. Secondly, Lyapunov theory of stability is used to design a dynamical controller guarantying the oscillators synchronization. The problem of synchronization is transformed into a problem of asymptotic stabilization for a nonlinear system. It is formulated as a system of linear matrix inequalities where the parameter variations of the two oscillators and their differences are modeled by polytopic matrices. The theoretical result is successfully applied to an array of transistor-based oscillators used in "smart antenna" systems.

Synthesis of controllers for damping oscillations in dynamical systems under uncertainty

We review the modern approaches to the synthesis of robust H∞ controllers that ensure optimal damping of oscillations in dynamical systems under uncertainty. In the synthesis method based on Riccati equations, these many-parameter equations can be solved only when the parameters are contained in a bounded parallelepiped with given boundaries. The synthesis of a robust H∞ output control for systems with unknown bounded parameters is reducible to the solution of an optimization problem constrained by a system of linear matrix inequalities. The proposed controller synthesis algorithms are implemented using standard MATLAB procedures. The efficiency of the proposed methods and algorithms is demonstrated in application to optimal damping of oscillations in a parametrically excited pendulum.

Numerical methods for estimation of the attraction domain in the problem of control of the wheeled robot

The problem of controlling the wheeled robot was considered. In the robot model used, the current curvature of the trajectory of the objective point which is related by simple algebraic expressions with the angle of rotation of the front wheels was taken as the control parameter. Boundedness of the angle of rotation of the robot front wheels imposes bilateral constraints on the control. The control constraints influence strongly the transients of the robot entering the desired trajectory. Additionally, the nature of the transients depends on the initial conditions. The aim of the paper was to construct the attraction domain guaranteeing the given rate of the transients. The quadratic Lyapunov function was used to approximate this domain. The problem of determining the quadratic Lyapunov functions was reduced to the standard mathematical verification for solvability of the system of linear matrix inequalities. For the case where the aim of control is to drive the robot objective point to the straight segment of the trajectory, the results of calculations of the stability domains were presented for various formulations of the problem.

Robust variance-constrained control for a class of continuous time-delay systems with parameter uncertainties

In this paper, we consider the robust variance-constrained control problem for uncertain linear continuous time-delay systems subjected to parameter uncertainties. The purpose of this multi-objective control problem is to design a static state feedback controller that does not depend on the parameter uncertainties such that the resulting closed-loop system is asymptotically stable and the steady-state variance of each state is not more than the individual pre-specified value simultaneously. Using the linear matrix inequality approach, the existence conditions of such controllers are derived. A parameterized representation of the desired controllers is presented in terms of the feasible solutions to a certain linear matrix inequality system. An illustrative numerical example is provided to demonstrate the effectiveness of the proposed results.

Solvability conditions for a second-order system of linear matrix inequalities

The necessary and sufficient conditions for solvability of a second-order system of linear matrix inequalities that can be reduced to two or three scalar inequalities of the second order are presented.

Uniform stabilization of discrete-time switched and Markovian jump linear systems

The uniform stability of discrete-time switched linear systems, possibly with a strongly connected switching path constraint, and the existence of finite-path-dependent dynamic output feedback controllers uniformly stabilizing such a system are both shown to be characterized by the existence of a finite-dimensional feasible system of linear matrix inequalities. This characterization is based on the observation that a linear time-varying system is uniformly stable only if there exists a finite-path-dependent quadratic Lyapunov function. The synthesis of a uniformly stabilizing controller is done without conservatism by solving any feasible system of linear matrix inequalities among an increasing family of systems of linear matrix inequalities. The result carries over to the almost sure uniform stabilization of Markovian jump linear systems.

PASSIVITY ANALYSIS AND PASSIFICATION OF DISCRETE-TIME HYBRID SYSTEMS

This paper proposes several (sufficient) criteria based on the numerical solution of systems of linear matrix inequalities (LMIs) for proving the passivity of discrete-time hybrid systems in piecewise affine form, and for the synthesis of switched linear control laws that enforce passivity.

Products of positive forms, linear matrix inequalities, and Hilbert 17th problem for ternary forms

A form p on R n (homogeneous n-variate polynomial) is called positive semidefinite (p.s.d.) if it is nonnegative on R n . In other words, the zero vector is a global minimizer of p in this case. The famous 17th conjecture of Hilbert [Bull. Amer. Math. Soc. (N.S.), 37 (4) (2000) 407] (later proven by Artin [The Collected Papers of Emil Artin, Addison-Wesley Publishing Co., Inc., Reading, MA, London, 1965]) is that a form p is p.s.d. if and only if it can be decomposed into a sum of squares of rational functions. In this paper we give an algorithm to compute such a decomposition for ternary forms ( n=3). This algorithm involves the solution of a series of systems of linear matrix inequalities (LMI's). In particular, for a given p.s.d. ternary form p of degree 2 m, we show that the abovementioned decomposition can be computed by solving at most m/4 systems of LMI's of dimensions polynomial in m. The underlying methodology is largely inspired by the original proof of Hilbert, who had been able to prove his conjecture for the case of ternary forms.

Synthesis of Optimal Robust H -Control by Convex Optimization Methods

It is shown that the synthesis of robust H∞-control with respect to the output for systems with unknown limited parameters reduces to the solution of an optimization problem under constraints prescribed by a system of linear matrix inequalities. For this kind of problems, an optimization algorithm implemented with the use of standard procedures of the MATLAB package is suggested.The algorithm effectiveness is illustrated by the example of an optimal dampening of vibrations of a parametrically perturbed pendulum.

An existence result for polynomial solutions of parameter-dependent LMIs

We show in this paper that any system of linear matrix inequalities depending continuously upon scalar parameters and solvable for any value of the latter in a fixed compact set, admits a branch of solutions polynomial with respect to the parameters. This result is useful for studying, e.g. parametric robustness or gain-scheduling issues.

Extended Matrix Cube Theorems with Applications to μ-Theory in Control

In this paper, we study semi-infinite systems of Linear Matrix Inequalities which are generically NP-hard. For these systems, we introduce computationally tractable approximations and derive quantitative guarantees of their quality. As applications, we discuss the problem of maximizing a Hermitian quadratic form over the complex unit cube and the problem of bounding the complex structured singular value. With the help of our complex Matrix Cube Theorem we demonstrate that the standard scaling upper bound on μ(M) is a tight upper bound on the largest level of structured perturbations of the matrix M for which all perturbed matrices share a common Lyapunov certificate for the (discrete time) stability.

On Tractable Approximations of Uncertain Linear Matrix Inequalities Affected by Interval Uncertainty

We present efficiently verifiable sufficient conditions for the validity of specific NP-hard semi-infinite systems of linear matrix inequalities (LMIs) arising from LMIs with uncertain data and demonstrate that these conditions are tight up to an absolute constant factor. In particular, we prove that given an $n\times n$ interval matrix ${\cal U}_\rho=\{A\mid\, |A_{ij}-A^*_{ij}|\leq\rho C_{ij}\}$, one can build a computable lower bound, accurate within the factor ${\pi\over 2}$, on the supremum of those $\rho$ for which all instances of ${\cal U}_\rho$ share a common quadratic Lyapunov function. We then obtain a similar result for the problem of quadratic Lyapunov stability synthesis. Finally, we apply our techniques to the problem of maximizing a homogeneous polynomial of degree 3 over the unit cube.

Static output feedback stabilisation with H∞ performance for a class of plants

In this paper, the problem of static output feedback control of a linear system is considered. The existence of a static output feedback control law is given in terms of the solvability of two coupled Lyapunov inequalities which result in a non-linear optimisation problem. However, using state-coordinate and congruence transformations and by imposing a block-diagonal structure on the Lyapunov matrix, we will see that the determination of a static output feedback gain reduces, for a specific class of plants, to finding the solution of a system of linear matrix inequalities. The class of plants considered is those which are minimum phase with a full row rank Markov parameter. The method is extended to incorporate H ∞ performance objectives. This results in a sub-optimal static H ∞ control law found by non-iterative means. The simplicity of the method is demonstrated by a numerical example.

Robust control of linear uncertain systems with regional pole and variance constraints

This paper concerns the design problem of state feedback controllers which guarantee the closed-loop poles within a specified disc and steady-state variances to be less than a set of given upper bounds for linear systems with norm-bounded parameter uncertainties. Using the linear matrix inequality approach, the existence conditions of such controllers are derived. A parametrized representation of the desired controllers is presented in terms of the feasible solutions to a certain linear matrix inequality system. Based on this, a solution to the minimum-effect guaranteed-performance design problem is presented in the sense that the required control effort is minimized subject to performance constraints.

Optimal guaranteed cost control of linear uncertain systems: LMI approach

Abstract In this paper, we study the guaranteed cost control problem for a class of linear systems with norm-bounded time-varying parameter uncertainty and a quadratic cost function in a linear matrix inequality (LMI) framework. The solvability condition is expressed as a system of LMIs. Efficient convex optimization techniques are available to solve this system. Moreover, its solutions parametrize the set of guaranteed cost controllers. The LMI-based characterization of guaranteed cost controllers enables us to easily solve the guaranteed cost control problem with some additional requirements. As a result we directly handle the optimal guaranteed cost control problem.

LMI approach to an H/sub ∞/-control problem with time-domain constraints over a finite horizon

The discrete time H/sub /spl infin//-control problem is considered with time-domain constraints over a finite horizon. It is shown that a solution of this problem can be obtained from a system of linear matrix inequalities.