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

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

Comment on ‘Stability of interval matrices’

Liao Xiao Xin (1987) gave necessary and sufficient conditions for the stability of a class of interval matrices. It can easily be shown that these results follow directly from a theorem relating stable Metzler matrices and quasidominant negative diagonal matrices by McKenzie (1966) which states (see also Siljak 1978)

A Metzler matrix A is stable if and only if it is quasidominant negative diagonal.

As the author considers Metzler matrices, Theorems 1,2 and 3 are obvious. Also, a theorem by Fiedler and Ptak (1962) concerning a pair of Minkowski matrices (see also Siljak 1978) leads to the same results.     

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 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.     

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 ‘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.     

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.     

Comments on ‘Stability analysis of interval matrices: improved bounds’

It is shown that the main result of the previous paper is correct only for interval matrices with linearly varying elements, not for the general class of interval matrices.     

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.     

Comments on ‘Stability test and stability conditions for delay differential systems’

It is shown that a first-order linear stationary system with two commensurate constant delays cannot be alternately asymptotically stable as the delay increases.     

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.     

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

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.     

Note on ‘ On the definition of transfer function ’

An algorithm for directly obtaining the canonical state-space model corresponding to the Cauer third form of continued fraction expansion (CFE) from a given general state-space model is presented. The algorithm can be used to determine the transfer function of linear time-invariant system from its state-space model as well as to obtain the reduced order models..Two new similarity matrices, one transforms a state-space equation from a general form to a Cauer third CFE canonical form, and the other transforms a state-space model in phase-variable form to a state-space model in a Cauer third CFE canonical form, are derived. Using these matrices an approximate relationship between the original state vector and the state-vector of reduced model obtained by the method of Cauer third CFE is established     

Comments on ‘On the loop transfer recovery’

Results on loop transfer recovery with eigenstructure assignment obtained by Kazerooni and Houpt (1986) are shown to be incorrect by a counter-example. The reasons for these erroneous results are identified, and improved results are presented.     

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.