Takehiro Mori

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.

Stability of x(t)=Ax(t)+Bx(t- tau )

A stability criterion for linear time-delay systems described by a differential difference equation of the form dx(t)=Ax(t)+Bx(t- tau ) is proposed. The result obtained includes information on the size of the delay and therefore can be a delay-dependent stability condition. Its relation to existing delay-independent stability criteria is also discussed. >

Brief paper: A way to stabilize linear systems with delayed state

A simple method is proposed for stabilizing linear systems with delayed state, using linear feedback. The method requires checking the negativity of a matrix containing two free parameters, and solving a matrix Lyapunov equation with these parameters. A known stabilizability criterion and a simple stability theorem for the open-loop system are obtained as special cases of the main result.

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.

Delay-independent stability criteria for discrete-delay systems

Several sufficient conditions for stability of linear discrete-delay systems are derived. Since these conditions are independent of the delay and possess simple forms, they will provide useful tools to check stability of the systems at the first stage.

Convergence property of interval matrices and interval polynomials

In association with robust control-system design and analysis, the Hurwitz property of interval matrices and interval polynomials has recently been actively investigated. However, its discrete counterpart, the convergence property, has seemingly not been much discussed. In this paper, this property is studied in comparison with the Hurwitz counterpart. Some conditions under which interval matrices or interval polynomials are convergent are derived.

Explicit solution and eigenvalue bounds in the Lyapunov matrix equation

An explicit solution to the algebraic Lyapunov matrix equation is obtained in terms of the controllability matrix of the pair of coefficient matrices. This enables us to determine the number of positive eigenvalues of the positive semidefinite solution through the controllability matrix. Based on this explicit formula, upper and lower bounds for each eigenvalue of the solution are derived, which always give nontrivial estimates.

On the discrete Riccati equation

Several bounds have been reported recently for the trace of the solution to the discrete algebraic matrix Riccati equation. This note adds an alternative one to them.

On the discrete Lyapunov matrix equation

Some bounds for the arithmetic and the geometric means of the characteristic roots of the positive semidefinite solution to the discrete Lyapunov matrix equation are derived.

Bounds in the Lyapunov matrix differential equation

Upper and lower bounds for the trace of the solution of the Lyapunov matrix differential equation are derived. It is shown that they are obtained as solutions to simple scalar differential equations. As a special case, the bounds for the stationary solution give ones for the solution to the Lyapunov algebraic equation.