Interval Linear Systems Research Articles

Stabilization of linear time-invariant interval systems via constant state feedback control

This paper investigates the stabilization problem of linear uncertain systems via constant state feedback control. The systems under consideration contain time-invariant uncertain parameters whose values are unknown but bounded in given compact sets and are thus called interval systems. The criterion for the asymptotic stability of the closed-loop system, obtained when a state feedback control is applied, is that all the eigenvalues of the resulting system matrix are in the strict left half of the complex plane. First, the author shows that to insure an interval system stabilizable, some entries of the system matrices must be sign invariant. More precisely, the number of the least-required, sign-invariant entries in system matrices is equal to the system order. Then, the author studies the stabilizability of a set of interval systems called standard systems which contain sufficient numbers of sign-invariant entries in proper locations. After dividing all standard systems into some subsets by the uncertainty locations, the author then derives necessary and sufficient conditions under which every system in a subset is stabilizable, regardless of its parameter varying bounds. The conditions show that all uncertain entries in system matrices should form a particular geometrical pattern called a "generalized antisymmetric stepwise configuration". For an interval system satisfying the stabilizability conditions, a computational control design procedure is also provided and illustrated via an example. The result is further generalized for nonstandard systems via linear transformation.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Frequency domain criteria for strict aperiodicity of linear interval systems

Abstract Interval systems arise when systems experience parameter variations. This paper derives frequency domain criteria for interval systems to be strictly aperiodic. The frequency domain criteria only requires one frequency domain plot to check for strict aperiodicity of linear interval systems. Furthermore, the largest allowable perturbation bounds can be graphically estimated from the same frequency domain plot.

On the solution of interval linear systems

In the literature efficient algorithms have been described for calculating guaranteed inclusions for the solution of a number of standard numerical problems [3,4,8,11,12,13]. The inclusions are given by means of a set containing the solution. In [12,13] this set is calculated using an affine iteration which is stopped when a nonempty and compact set is mapped into itself. For exactly given input data (point data) it has been shown that this iteration stops if and only if the iteration matrix is convergent (cf. [13]). In this paper we give a necessary and sufficient stopping criterion for the above mentioned iteration for interval input data and interval operations. Stopping is equivalent to the fact that the algorithm presented in [12] for solving interval linear systems computes an inclusion of the solution. An algorithm given by Neumaier is discussed and an algorithm is proposed combining the advantages of our algorithm and a modification of Neumaier's. The combined algorithm yields tight bounds for input intervals of small and large diameter. Using a paper by Jansson [6,7] we give a quite different geometrical interpretation of inclusion methods. It can be shown that our inclusion methods are optimal in a specified geometrical sense. For another class of sets, for standard simplices, we give some interesting examples.

Interval linear systems with symmetric matrices, skew-symmetric matrices and dependencies in the right hand side

The methods of Interval Arithmetic permit to calculate guaranteed a posteriori bounds for the solution set of problems with interval input data. At present, these methods assume that all input data vary independently between their given lower and upper bounds. This paper shows for special interval linear systems how to handle the case where dependencies of the input data occur.

Preconditioners for the Interval Gauss–Seidel Method

Interval Newton methods in conjunction with generalized bisection can form the basis of algorithms that find all real roots within a specified box ${\bf X} \subset {\bf R}^n $ of a system of nonlinear equations $F(X) = 0$with mathematical certainty, even in finite-precision arithmetic. In such methods, the system $F(X) = 0$ is transformed into a linear interval system $0 = F(M) + {\bf F'}({\bf X})({\bf \bar X} - M)$ ; if interval arithmetic is then used to bound the solutions of this system, the resulting box ${{\bf \bar X}}$ contains all roots of the nonlinear system. The interval Gauss–Seidel method is a reasonable way of finding such solution bounds. For the overall interval Newton/bisection algorithm to be efficient, the image box ${{\bf \bar X}}$ should be as small as possible. To do this, the linear interval system is multiplied by a preconditioner matrix Y before the interval Gauss–Seidel method is applied. In this paper, a technique for computing such preconditioner matrices Y is described. This technique involves optimality conditions expressible as linear programming problems. In many instances, the resulting preconditioners give an ${{\bf \bar X}}$ of minimal width. They can also be applied when ${{\bf F'}}$ approximates a singular matrix, and the optimality conditions can be altered to describe preconditioners with a given structure. This technique is illustrated with some simple examples and with numerical experiments. These experiments indicate that the new preconditioner results in significantly less function and Jacobian evaluations, especially for ill-conditioned problems, but it requires more computation to obtain.

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.

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.

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 ε 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.

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'

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.

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

In a recent paper by Jiang (1988), a necessary and sufficient condition is given for a set of real interval matrices to have all their eigenvalues in the unit circle. This result is shown by counter-example to be false and the flaw in the proof is pointed out.

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

Jiang (1988) gives a sufficient and necessary condition for the asymptotic stability of a discrete linear interval system. It is here demonstrated through counter-examples that the condition is not sufficient. The flaw leading to this erroneous result is also indicated.

Sufficient and necessary condition for the asymptotic stability of discrete linear interval systems

In this paper, a sufficient and necessary condition for the asymptotic stability of the discrete linear interval system X(k + 1) = A1X(k) is presented; i.e. all the eigenvalues of each matrix AϵA1, (an interval matrix) have magnitudes less than 1 if and only if all the eigenvalues of every ‘extreme matrix’ of the interval matrix A1, have magnitudes less than 1.

Interval linear systems with prescribed column sums

Nonnegative solutions of an interval linear system A I x = b I ( A I being an interval matrix and b I an interval vector) with additional column sum restrictions of the type σ i α ij a ij ∈[ c -j , c j ] ∀j are described by a system of linear inequalities with auxiliary variables.