E. B. Lee

On stability independent of delay for linear systems

A new criterion for asymptotic stability of solutions of certain linear differential-difference equations (independent of the delay duration) is given in terms of solutions of a complex Lyapunov matrix equation.

Approximation of infinite-dimensional systems

A Fourier series-based method for approximation of stable infinite-dimensional linear time-invariant system models is discussed. The basic idea is to compute the Fourier series coefficients of the associated transfer function T/sub d/(Z) and then take a high-order partial sum. Two results on H/sup infinity / convergence and associated error bounds of the partial sum approximation are established. It is shown that the Fourier coefficients can be replaced by the discrete Fourier transform coefficients while maintaining H/sup infinity / convergence. Thus, a fast Fourier transform algorithm can be used to compute the high-order approximation. This high-order finite-dimensional approximation can then be reduced using balanced truncation or optimal Hankel approximation leading to the final finite-dimensional approximation to the original infinite-dimensional model. This model has been tested on several transfer functions of the time-delay type with promising results. >

Observability and related structural results for linear hereditary systems

New results on the theory of observability and deterministic observers for linear time.invariant systems with delays are presented. The results have been established only for the case of point delays which are commensurable ; however, most of the results can be generalized to the situation with distributed delays and to the ease with non-commensurate delays.

Controller synthesis for linear systems with retarded state and control variables and quadratic cost

The problem of optimal control of a system whose mathematical model involves dependence on both a history of state type variables and the control variables is considered. The canonical equations which one obtains by application of a maximum principle for the optimal control problem are transformed into equivalent integral equations which can be solved by use of the Fredholm resolvent theory. A feedback solution is obtained.

A simple method to determine the stability and margin of stability of 2-D recursive filters

The authors propose some simple numerical algorithms for stability testing of 2-D recursive digital filters and estimation of their margin of stability. The basic idea is to take one variable as a parameter and to make the system a continuous function of this parameter. The critical values of the parameter under which the system becomes unstable are then found. The critical values can be computed via checking singularity of the Sylvester resultant matrix or Kronecker sum in some specific domains. This is equivalent to solving a generalized eigenvalue problem. If a filter is stable, the estimation of the margin of stability can be obtained simply by scaling these two variables. If a filter is unstable, this algorithm will show all the poles in the unit bidisk and will also give the maximal analytic bidisk. Three examples are given to show the effectiveness of the algorithm. >

Balanced approximation of two-dimensional and delay-differential systems

A generalized balanced approximation method for reducing two-dimensional (2-D) and delay differential models is given. This is possible because of the form the Gramians of such systems take. In particular it is shown that the reachability and observability Gramians can be expressed as integrals of a certain system related function along the imaginary axis or unit circle of the complex plane. The Gramians provide two sets of system invariants which lead in a natural way to the concept of a generalized balanced realization, which in turn leads to a way to do model reduction. This paper is based on the report [0].

A simplified derivation of the Zeheb-Walach 2-D stability test with applications to time-delay systems

Zeheb and Walach gave a stability test for n-D systems. For 2-D systems, a simpler derivation is presented here using the results of DeCarlo et al. It is then shown how the method may also be used to test for stability (independent of delay) of retarded time-delay systems.

Realizations of 2-D filters and time delay systems

Possible techniques for achieving an absolutely minimal realization of an input/output system described by a matrix of rational functions in two indeterminates are described.

Canonical forms for time delay systems

Polynomial transformations are considered for linear systems with time delays. A technique for state reduction is given as well as schemes for obtaining other useful canonical forms. The results are achieved using a combination of module theoretic ideas (polynomial ring) and functional analysis.

Stability testing of time delay systems

Some new results on stability theory for time delay systems of retarded type are described. Based on Schurs theorem and the bounded real Lemma, a new stability test is then given.