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.     

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 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.     

Comment on ‘Sufficient condition for the asymptotic stability of interval matrices’

A simplified proof is given for the main conclusion of the previous paper.     

Comments on ‘Sufficient condition for the asymptotic stability of interval matrices’

It is asserted that two proofs in the previous paper are unnecessary and that one of the proof procedures is erroneous.     

Comments on ‘Sufficient condition for the asymptotic stability of interval matrices’

Recently, Jiang (1987) has shown that a set of interval matrices is stable if the symmetric part of the vertex matrices are stable. This result of Jiang (1987) is shown to be only applicable to matrices with all the diagonal elements negative.     

Comments on ‘A simple sufficient condition for the stability of linear discrete time systems’

Berger (1982) has found the maximum hyperspheres in the stable domain of parameter space. We show that the maximum stable hyperspheres for odd-order systems contain some unstable polynomials, and propose a correction to Berger's results.     

Further comments on ‘A simple criterion for stability of linear discrete systems’

Some additional comments are offered on the stability condition for linear discrete systems in Dabke (1983). It is pointed out that the condition can be naturally extended to cover time-varying systems.     

Comments on ‘A necessary and sufficient condition for the positive-detiniteness of interval symmetric matrices’

The main results obtained recently by Shi and Gao are already known and can be strengthened.     

Comments on ‘A necessary and sufficient condition for the positive-definiteness of interval symmetric matrices’

Shi and Gao (1986) gave a necessary and sufficient condition for the positive-definiteness of interval symmetric matrices. We point out that their result is not new and the assumptions can be less restrictive.     

Kharitonov’s theorem and routh criterion for stability margin of interval systems

In this paper, it is shown that the gain margin and phase margin of interval system can be determined analytically using Kharitonov’s theorem and V. Krishnamurthi’s corollary on Routh criterion without using graphical and iterative techniques. Further, it is proved that the existing results of Anderson et al. [2] on the stability of low-order interval systems using Kharitonov’s theorem are only applicable for absolute stability of the interval system and it is not applicable for relative stability of the interval systems, i.e., for phase margin. The proposed technique and stability analysis for low-order interval systems are verified with examples.     

Comment on ‘On sufficient conditions for the stability of interval matrices’

Some new sufficient conditions for the stability of interval matrices were given by Argoun (1986). These conditions, however, are incorrect.     

Approximation of discrete linear systems†

Given a multi–input, multi–output linear time–invariant discrete–time system or its weighting matrices, equations are derived to characterize a system of lower dimension which optimally approximates the given system with respect to the sum of squared output errors of the impulse response. These equations are shown to imply that in principle a modified B. L. Ho algorithm can be used to find the approximation. The connection with projection methods of approximation is demonstrated, Some comments on computational difficulties are included.     

Note on 'Time-optimal control strategy for saturating linear discrete systems’

The control policy taking a linear system from a given state to the origin in a minimum number of time steps is not unique. Such flexibility also offers an opportunity to minimize the control effort in a straightforward manner.     

Comment on ‘Stability analysis and controller synthesis for discrete linear time-delay systems with state saturation nonlinearities’

A recently reported paper (Ji, X., Liu, T., Sun, Y., and Su, H. (2011), ‘Stability analysis and controller synthesis for discrete linear time-delay systems with state saturation nonlinearities’, International Journal of Systems Science, 42, 397–406) for the global asymptotic stability analysis and controller synthesis for a class of discrete linear time delay systems employing state saturation nonlinearities is reviewed. It is claimed in Ji, Liu, Sun and Su (2011) that a previous approach by Kandanvli and Kar (Kandanvli, V.K.R and Kar, H. (2009), ‘Robust stability of discrete-time state-delayed systems with saturation nonlinearities: Linear matrix inequality approach’, Signal Processing, 89, 161–173) is recovered from their approach as a special case. It is shown that this claim is not justified.