Norio Fukuma

Simple stability criteria for single and composite linear systems with time delays

Easy ways to test the stability of systems involving time delays have been sought. In this paper, several stability conditions with an extremely simple form are provided. First, some criteria for single linear systems with time delays are presented. Then these results are extended to the composite linear systems with time delays which exist in each subsystem and in the interconnections between subsystems. Among these criteria, those which are expressed by scalar inequalities also permit us to assess the transient behaviour of systems.

On an estimate of the decay rate for stable linear delay systems

In this paper it is shown that for a certain class of stable linear delay systems we can obtain an estimate of the decay rate of the solution by a simple computation.

A stability criterion for linear time-varying systems

In this paper, we derive an algebraic stability criterion of linear time-varying systems employing a specific metric in state-space. Though the used metric includes arbitrary parameters as weights, the criterion itself does not incorporate them. However, if we ask for these parameters, an estimate of the transient state of the systems can be obtained through them. Three examples are illustrated to clarify the character of the obtained criterion.

Upper and lower bounds for the solution to the discrete Lyapunov matrix equation

Both upper and lower bounds for each eigenvalue of the solution to the discrete Lyapunov matrix equation are derived.

Convergence condition for convex combinations of two polynomials

A necessary and sufficient condition is derived for a convex combination of two given polynomials to be a convergent polynomial, i.e. a polynomial having roots all within the unit circle in the complex plane. This can be regarded as a discrete counterpart to the existing result for the continuous case. A simple numerical example is provided to demonstrate the theorem obtained.

Estimate for the root-location of linear systems via the lyapunov matrix equation

Abstract It is well known that the Lyapunov matrix equation can be utilized to obtain the estimate for the transient behavior of linear constant systems. This paper shows that the equation gives information not only on the maximum real part of the characteristic roots of system matrix but also on other extremal values to these roots.

Number of positive eigenvalues of the solution to the discrete Lyapunov matrix equation

It is known that the discrete Lyapunov matrix equation has a positive semidefinite symmetric solution under an appropriate condition. This note proves the fact that the number of positive eigenvalues of the solution is equal to the rank of the controllability matrix formed from the coefficient matrices.

On the Conditions for the Characteristic Roots of a Matrix in a Sector

Abstract Davison and Ramesh expressed equivalently the condition for the characteristic roots of a real matrix to lie within a sector by the condition for a matrixof doubled order to be Hurwitzian (1). A further generalization of their result is given in this note.