Multivariate root loci: a unified approach to finite and infinite zeros

Abstract

A MacLaurin's series expansion is used to approximate the individual sheets of the characteristic gain function of a linear multivariable system in terms of its state space matrices, at the point at infinity on the frequfency plane for smlall values of characteristic gain. This yields the asymptotic root locus directions for systems with one or more sets of closed loop poles going to infinihy for large feedback gains. A bilinear transformation on the frequency variable is used to allow the approximation at any point on the frequency plane. This gives the angles of approach to finite zeros in terms of the state space matrices, and gives a simple way of calculating zeros for square systems. A bilinear transformation on the gain variable is then introduced allowing the characteristic gain function to be approximated at any values of frequency and gain. This enables the directions of departure from the open-loop poles to be determined in terms of the state space matrices of the system.

Low order solutions to the perfect regulation problem with bounded peaking

“…Much of the literature on multivariable root locus assumes that the system satisfies the SNS assumption [10,14,15]. We show in this section that the proposed controller (4.35) ensures that the augmented plant satisfies SNS.…”

“…We follow the developments of [14] closely and explore the asymptotic behaviour of the multivariable root locus first. It will be shown that the unbounded closed-loop poles approach infinity in several Butterworth patterns whose orders, under certain assumptions, are equal to the orders of infinite zeros of the system.…”

“…Thus high gain systems can naturally induce a singularily perturbed system. From this point of view, the feedback gain matrix is scaled with a high gain parameter, and closely resembles the formulation of the multivariable root locus problem [14] where instead of state feedback, output feedback is used. This connection was made explicit in [29].…”

Applications of Singular Perturbation Techniques to Control Problems

“…They have been extensively studied in the control literature. Typical references include Shaked (1976Shaked ( ,1978, Kouvaritakis (1978), Kouvaritakis and J. M. Edmunds (1979) , and more recently Sastry and Desoer (1983). A geometric approach was developed by Willems (1981Willems ( ,1982.…”

Asymptotic unbounded root loci - Formulae and computation

“…The asymptotic behavior has been extensively studied by Kouvaritakis and Shaked [3], Kouvaritakis [4], Kouvaritakis and Edmunds [5], Owens [11,12], MacFarlane and co-workers [2,19], Kokotovic, et. al [22], and by Brockett and Byrnes [10].…”