Edward W. Kamen

Linear systems with commensurate time delays: stability and stabilization independent of delay

Notions of exponential stability independent of delay and stabilizability independent of delay are developed for the class of delay differential systems of the retarded type with commensurate time delays. Various criteria for exponential stability independent of delay with a given order are specified in terms of matrices whose entries are functions of a single real parameter and polynomials in one variable whose coefficients are functions of a single real parameter. Sufficient conditions and a necessary condition based on local stabilizability are given for stabilizability independent of delay using state feedback with commensurate time delays. Constructive methods for determining a stabilizing feedback are also presented. The last part of the paper deals with a standard type of observer and regulator with the requirement that the closed-loop system be stable independent of delay.

The poles and zeros of a linear time-varying system

Abstract For linear time-varying discrete-time and continuous-time systems, a notion of poles and zeros is developed in terms of factorizations of operator polynomials with time-varying coefficients. In the discrete-time case, it is shown that the poles can be computed by solving a nonlinear recursion with time-varying coefficients. In the continuous-time case, the poles can be calculated by solving a nonlinear differential equation with time-varying coefficients. The theory is applied to the study of the zero-input response and asymptotic stability. It is shown that if a time-varying analogue of the Vandermonde matrix is invertible, the zero-input response can be decomposed into a sum of modes associated with the poles. Stability is then studied in terms of the components of the modal decomposition.

Stabilization of time-delay systems using finite-dimensional compensators

For linear time-invariant systems with one or more noncommensurate time delays, necessary and sufficient conditions are given for the existence of a finite-dimensional stabilizing feedback compensator. In particular, it is shown that a stabilizable time-delay system can always be stabilized using a finite-dimensional compensator. The problem of explicitly constructing finite-dimensional stabilizing compensators is also considered.

On the control of linear systems whose coefficients are functions of parameters

The stabilization and regulation of linear discrete-time systems whose coefficients depend on one or more parameters is studied. For linear systems whose coefficients are continuous functions of real or complex parameters (respectively, analytic or rational functions of real parameters), it is shown that reachability of the system for all values of the parameters implies that the system can be stabilized using gains that are also continuous (analytic, rational) functions of the parameters. Closed-form expressions for a collection of stabilizing gains are given in terms of the reachability Gramian. For systems which are stabilizable for all values of the parameters, it is shown that continuous (analytic, rational) stabilizing gains can be computed from a finite-time solution to a Riccati difference equation whose coefficients are functions of the parameters. These results are then applied to the problem of tracking and disturbance rejection in the case when both the plant and the exogenous signals contain parameters.

Proper stable Bezout factorizations and feedback control of linear time-delay systems†

This paper deals with the existence and construction of proper stable Bezout factorizations of transfer function matrices of linear time-invariant systems with commensurate time delays. Existence of factorizations is characterized in terms of spectral controllability (or spectral observability)of the co-canonical (or canonical) realization of the transfer function matrix. An explicit procedure for computing proper stable Bezout factorizations is given in terms of a specialized ring of pure and distributed time delays. This procedure is utilized to construct finite-dimensional stabilizing compensators and to construct feedback systems which assign the characteristic polynomial of the closed-loop system.

Multiple target angle tracking using sensor array outputs

The use of the output of an array of sensors to track multiple independently moving targets is reported. The output of each sensor in the array is the sum of signals received from each of the targets. The results of direction-of-arrival estimation by eigenvalue analysis are extended to derive a recursive procedure based on a matrix quadratic equation. The solution of this matrix quadratic equation is used to provide updated target positions. A linear approximation method for estimating the solution of the matrix equation is presented. The algorithm is demonstrated by the simulated tracking of two targets. The main advantage of the algorithm is that a closed-form solution for updating the target angle estimates has been obtained. Also, its application is straightforward, and the data association problem due to uncertainty in the origin of the measurements is avoided. However, it requires the inversion of an N*N as well as other linear operations, so that the computational burden becomes substantial as N becomes very large. >

Control of slowly-varying linear systems

State feedback control of slowly varying linear continuous-time and discrete-time systems with bounded coefficient matrices is studied in terms of the frozen-time approach. This study centers on pointwise stabilizable systems. These are systems for which there exists a state feedback gain matrix placing the frozen-time closed-loop eigenvalues to the left of a line Re s=- gamma >

An efficient algorithm for tracking the angles of arrival of moving targets

A novel technique for tracking the angles of arrival of moving targets is presented. The targets are modeled as signal sources that continuously emit narrowband signals which impinge on an array of sensors. Estimates of target angles are obtained by minimizing the norm of an error matrix function involving the covariance of the sensor outputs. The algorithm yields estimates that are automatically correctly associated with previous estimates. Consequently, the data association problem does not arise, and this results in a much more efficient scheme in comparison to existing methods involving search over N factorial possible measurement/target associations (where N is the number of targets). Performance of the algorithm is illustrated by a simulation example. >

Pointwise stability and feedback control of linear systems with noncommensurate time delays

Feedback control of linear neutral (and retarded) time-delay systems with one or more non-commensurate time delays is studied. A new (algebraic) notion of stability, called pointwise stability, is defined and is shown to be generically equivalent to uniform asymptotic stability independent of delay. Necessary and sufficient conditions are then given for regulability, that is, for the existence of a dynamic output feedback compensator with pure delays such that the closed-loop system is internally pointwise stable (and thus stable independent of delay). Necessary and sufficient conditions involving matrix-fraction descriptions are also given for the existence of a state realization which is regulable. Finally, the problem of stabilization using nondynamic state feedback is briefly considered in the case when the systems input matrix has constant rank.

On the inner and outer poles and zeros of a linear time-varying system

The notion of inner and outer poles and zeros of a linear time-varying discrete-time system is developed in terms of noncommutative factorizations of operator polynomials with time-varying coefficients. Emphasis is placed on left and right poles and zeros, which are referred to as outer poles and zeros. The outer poles and zeros arise in the study of system reducibility, stability, and transmission blocking zeros. A frequency-domain characterization of right poles and zeros is given in terms of a generalized frequency function.<<ETX>>