Discrete Interval Systems Research Articles

Controllability and quadratic stability quadratic stabilization of discrete-time interval systems—an LMI approach

This paper presents necessary and sufficient conditions for discrete-time interval systems to be quadratically stable or quadratically stabilizable, and sufficient conditions for discrete-time interval systems to be controllable. All the results are obtained in terms of linear matrix inequality (LMI). With the LMI toolbox, the problems can be conveniently solved. Illustrative numerical examples show that the results are effective and less conservative to check the robust controllability and stability and to design the stabilizing controller for discrete-time interval systems.

LMI conditions for quadratic and robust -stability of interval systems

Sufficient conditions for the quadratic Open image in new window -stability and further robust Open image in new window -stability of interval systems are presented in this paper. This robust Open image in new window -stability condition is based on a parameter-dependent Lyapunov function obtained from the feasibility of a set of linear matrix inequalities (LMIs) defined at a series of partial-vertex-based interval matrices other than the total vertex matrices as in previous results. The results contain the usual quadratic and robust stability of continuous-time and discrete-time interval systems as particular cases. The illustrative example shows that this method is effective and less conservative for checking the quadratic and robust Open image in new window -stability of interval systems.

Robust static output-feedback stabilization for nonlinear discrete-time systems with time delay via fuzzy control approach

This paper addresses the problem of designing robust static output-feedback controllers for nonlinear discrete-time interval systems with time delays both in states and in control input. In the approach, we do not directly employ the Lyapunov approach, as do in most of traditional fuzzy control design approaches. Instead, sufficient conditions for guaranteeing the robust stability for the considered systems are derived in terms of the matrix spectral norm of the closed-loop fuzzy system. The stability conditions are further formulated into linear matrix inequalities so that the desired controller can be easily obtained by using the Matlab linear matrix inequality (LMI) toolbox. Finally, an example is provided to illustrate the effectiveness of the proposed approach.

Robust [formula omitted]-stability of discrete and continuous time interval systems

Necessary and sufficient conditions are presented for the robust Schur and Hurwitz stability of the dynamic interval systems. An interval matrix can be expressed as a linear fractional transformation of an interconnection matrix with structured real parametric uncertainties. Consequently, the robust stability problem is converted to a robust nonsingular problem. An explicit condition in terms of the structured singular value is provided to ensure the stability robustness of dynamic interval systems. This systematic approach is feasible for both discrete-time and continuous-time cases. The proposed technique can be applied directly to D -stability for performance requirements.

Simultaneous static output-feedback stabilisation for discrete-time interval systems with time delay

The simultaneous static output-feedback stabilisation problem for a collection of discrete-time interval systems with time delays both in states and in control input is considered. A sufficient condition for the existence of static output-feedback simultaneously stabilising controllers is obtained in terms of matrix spectral norms. It is shown that this problem is solvable if a corresponding matrix spectral norm assignment problem is solvable. It is also shown that the matrix spectral norm assignment problem is equivalent to a bilinear matrix inequality (BMI) problem. A sufficient condition for the BMI problem is then derived and the condition is a linear matrix inequality feasibility problem, which can be solved easily. An example is provided to demonstrate the effectiveness of the proposed methodology.

A New Method for Modelling of Large Scale Interval Systems

In recent years, simulation, design and control of systems with parametric uncertainties for the entire range of operation gained much importance. This paper presents a new method for reducing order of Large Scale Interval systems. The proposed method involves simple computations, and is efficient as it needs the formulation of only one Routh type tabulation, avoiding the necessity of formulating two tables viz., γ and δ tables, and avoids the computation of time moments of the original high order Interval system beforehand, unlike other available methods [1,2]. The proposed Simplified Routh Approximation method (SRAM) always generates stable reduced order Interval systems, retaining the initial time moments of the original high order system. The proposed simplified method is extended for the reduction of high order Multivariable and Discrete Interval systems to overcome the limitations and drawbacks of some of the existing methods [1,2] in literature. Typical numerical examples are considered to illustrate the flexibility and effectiveness of the proposed method.

Discrete modelling of uncertain continuous systems having an interval structure using higher-order integrators

In this paper, a higher-order integrator approach is proposed to obtain an approximate discrete-time transfer function for uncertain continuous systems having interval uncertainties. Because of the simple algebraic operations of this approach, the resulting discrete model is a rational function of the uncertain parameters. The problem of non-linearly coupled coefficients with exponential nature occurring in the exact discretetime transfer function is therefore circumvented. Furthermore, the interval structure of the uncertain continuous-time system is preserved in the resulting discrete model by using this approach. Formulae to obtain the lower and upper bounds for the coefficients of the discrete interval system are derived, so that digital simulation and design for the uncertain continuous systems can be performed by using the available robustness results in the discrete-time domain.

New sufficient conditions for the stability of continuous and discrete time-delay interval systems

Based on the Gersgorin theorem, this paper discusses the stability testing problem for continuous and discrete interval systems including a time delay. Several new delay-independent sufficient conditions for preserving the stability of the above systems are developed. By these conditions, criteria that can guarantee the mentioned systems be stable with a decaying rate are also proposed. Finally, numerical examples are given to demonstrate the applicability of the obtained results.

Discrete interval system reduction using Pade approximation to allow retention of dominant poles

This brief presents a method of reduction for discrete interval systems using Pade approximation to allow the retention of dominant poles. For a given system, the denominator of the reduced model is formed by retaining the dominant poles of the original system, while the numerator is obtained by matching interval time moments. Two numerical examples illustrate the procedure.

Comments on ‘on the stability of discrete-time linear interval systems’

Several stability conditions derived for linear discrete-time interval systems by Myszkorowski (Automatica, 30, 913–914, 1994) can be much more compactly rephrased with the aid of M-matrix theory.

Necessary and sufficient conditions for stability of time-varying discrete interval matrices

Sufficient conditions for the stability of time-varying discrete interval systems are presented. The spectral radius of the maximum absolute matrix of all interval matrices is employed as opposed to utilizing the matrix norm or Lyapunov theorem. For special cases, our result becomes a necessary and sufficient condition for stability. Furthermore, the proposed conditions are inclusive of the previous results of stability of time-invariant interval systems.

Comments on ‘Convergence property of interval matrices and interval polynomials’

This note shows that the discrete linear interval system result of Mori and Kokame (1987) is also applicable to linear discrete time systems with linearly dependent perturbations.

Counter-examples to Jiang's results

Abstract This note points out an important mistake in a recent paper by Jiang (1988) on the asymptotic stability of discrete linear interval systems.

Sufficient condition for asymptotic stability of discrete interval systems

Abstract A counter-example is given to a recent result on the stability analysis of interval matrices by Jiang (1988). A sufficient condition is then presented for the asymptotic stability of the discrete-time interval system X(k + 1) = A I X(k), by testing the norms of extreme matrices of the interval matrix A I such that they are all less than one.

Sufficient condition for asymptotic stability of discrete interval systems

Abstract A counter-example is given to a recent result on the stability analysis of interval matrices by Jiang (1988). A sufficient condition is then presented for the asymptotic stability of the discrete-time interval system X(k + 1) = A I X(k), by testing the norms of extreme matrices of the interval matrix A I such that they are all less than one.

Comments on ‘Sufficient and necessary condition for the asymptotic stability of discrete linear interval systems’

Abstract In a recent paper by Jiang (1988), a necessary and sufficient condition for asymptotic stability of the interval matrix of discrete-time systems is presented. In this contribution it is shown that this condition is not true in general. A counter-example is given and additional remarks are made.

Counter-example to ‘Sufficient and necessary condition for the asymptotic stability of discrete linear interval systems’

Abstract In this note, the interval matrix result for discrete linear interval system by Jiang (1988) is shown to be incorrect, i.e. all eigenvalues of each matrix A e A 1 (an interval matrix) have magnitudes less than 1 if and only if all the eigenvalues of every ‘extreme matrix’ of the interval matrix A 1 have magnitudes less than 1.

Counter-example to ‘Sufficient and necessary condition for the asymptotic stability of discrete linear interval systems’

Abstract In Jiang (1988), a condition is given which is supposedly necessary and sufficient for the stability of interval matrices in discrete-time systems. We demonstrate, via a counter-example, that the condition given is, in fact, not sufficient.

Comments on ‘Sufficient and necessary condition for the asymptotic stability of discrete linear interval systems’

Comments on ‘Sufficient and necessary condition for the asymptotic stability of discrete linear interval systems’

Counter-example to 'Sufficient and necessary condition for the asymptotic stability of discrete linear interval systems'

Abstract In Jiang (1988), a condition is given which is supposedly necessary and sufficient for the stability of interval matrices in discrete-time systems. We demonstrate, via a counter-example, that the condition given is, in fact, not sufficient.