Nicos Karcanias

Poles and zeros of linear multivariable systems : a survey of the algebraic, geometric and complex-variable theory

This survey is concerned with the definition and main properties of sets of specific values of complex frequency, called poles and zeros, which may be associated with a matrix-valued function of a complex frequency variable. The main emphasis is on the physical interpretation of invariant zeros in terms of the general zero-output behaviour of a linear dynamical system. A discussion is given of the relationship between this definition of a zero and the various other forms adopted in the current literature. The relationship is considered between poles and zeros defined by algebraic means and the standard complex variable theory of algebraic functions. It is shown that the poles and zeros of a square matrix-valued function of a complex variable G(s) are the same as the poles and zeros of an associated algebraic function g(s).

Structure and Smith-MacMillan form of a rational matrix at infinity

The use of special row and column operations in the reduction of a rational matrix to its Smith-MacMillan form at infinity is investigated. The connections between this reduction procedure and the valuation approach are established. A graphical method for finding the Smith-MacMillan form of a rational matrix at infinity from its Bode magnitude array, and some new results on realization theory for polynomial matrices are presented.

Matrix pencil characterization of almost ( A, Z) -invariant subspaces : A classification of geometric concepts

The equivalence between the algebraic, matrix pencil, characterization of the sub-spaces of the ‘ extended ’ geometric theory and their dynamic characterization 13 established. As a result, a complete classification of Almost ( A, B) -invariant, ( A, B) -invariant, Almost controllability and controllability subspaces is derived in terms of matrix pencil invariants. The frequency propagation aspects of infinite spectrum ( A, B)-invariant subspaces are investigated and it is shown that they are limits of closed-loop eigenspaces with arbitrarily largo eigenvalues. Finally, the importance of the infinite frequency subspaces in the study of the asymptotic behaviour of the closed-loop eigenspaces and eigenvalues under scalar gain output feedback is discussed.

Grassmann invariants, almost zeros and the determinantal zero, pole assignment problems of linear multivariable systems

The strucxpects of the rationxr spaces x c, x fwhere x c is the columxof the transfer function matrix G(s) and x f, is the space associated with right matrix fracxcriptions of G(s), are investigated. For x f gcanonixqb;s] Grassmgesentax g(x c) and g(x f) arxd, and are shown xmplete basis free invariants for x c and x f respectively. The almost zeros (AZ) and almost decoupling zeros (ADZ) of G(s)gdefinex local minima of a normgn defix g(x c) and g(x f) respectively. The computation, and certain aspects of the distribution in the complex plane of AZs and ADZs are examined. The role of AZs and ADZs in the determinantal zero and pole assignment problems respectively is examined next. Two important families of systems are defined : the strongly zero non-assignable (SZNA) and the strongly pole non-assignable (SPNA) systems. For SZNA and SPNA systems minimal radius discs Dem e[z, R em e(z)] and Dem f[zb centred at an AZ and ADZ respectively are defined. It is shown that Dem e[R em e(z)] contains at least one zer...

The output zeroing problem and its relationship to the invariant zero structure : a matrix pencil approach

Multivariable zeros have boon defined in a multitude of ways and of these the physical definition of zeros through the problem of zeroing outputs is preferred here. The extension of this definition, from the external to the internal description undertaken, proves the zeros with the corresponding zero directions to be dual concepts to the poles and corresponding modes. The treatment, adopted in this paper leads to the definition of the zero pencil, Z(s) which through the theory of matrix pencils, proves to be an effective means for the analysis of the zero system structure. Use of the Kroneeker canonical form of Z(s) enables the zero properties of the system to be related to the geometric theory of Wonham and Morse. A practical application of the results concerning the placement of zeros brings the paper to a conclusion.

Global asymptotic linearisation of the pole placement map: a closed-form solution for the constant output feedback problem

The problem of pole assignment, by constant output feedback controllers, is studied for minimal systems described by a proper transfer function matrix G(s) ϵ Rm × p(s) with McMillan degree n. A new method is presented based on asymptotic linearisation (around a degenerate point) of the pole placement map related to the problem. The essence of the present approach is to reduce the multilinear nature of the problem to one of solving a linear set of equations, and this is achieved without losing any of the degrees of freedom in the controller. The solution is given in closed form in terms of a one-parameter (ϵ) family of static feedback compensators, for which the closed-loop poles approach the required ones as ϵ → 0. Conditions for the generic, as well as exact, solvability of the arbitrary pole placement problem are given in terms of the numbers m, p, n and the rank of a new system invariant, the D-restricted Plucker matrix. It is shown that the method works generically when mp > n, which (along with the boundary case mp = n) is the best possible condition as far as the number of states of the open-loop system is concerned, for achieving arbitrary pole placement via constant output feedback.

The use of zeros and zero-directions in model reduction

The concept of zeros and zero-directions of a linear time-invariant multivariable system is applied to the problem of shifting eigenvectors into the kernel of the information extraction map C. By making the maximum number of the closed system modes unobservable, a lower-order transfer-function matrix is obtained. The state feedback structure reducing the model to one of lower dimensions does not exploit all of the degrees of freedom. A part of them are used for pole allocation of the remaining observable poles using a dyadic feedback technique. The suggested technique is applied to reduce a broad class of multivariable systems into first-order multivariable systems. Finally, a dynamical structure realizing the same objectives is introduced.

Necessary and sufficient conditions for zero assignment by constant squaring down

The problem of zero assignment by constant squaring down (CZAP) is studied for minimal systems described by a transfer function G(s)ϵRm×l(s), m > l, using tools from exterior algebra and algebraic geometry. Conditions for its solvability are given in terms of the numbers m, l, the Forney dynamical order δ, and the rank ϱδ of the Plucker matrix Pδ. For the cases l = 1 or l = m−1, it is shown that G(s) is completely zero-assignable (CZA) if and only if ϱδ = δ+1. For the cases l≠1, m−1, it is proved that G(s) is CZA, or generically zero-assignable (GZA), under a complex squaring down if and only if l(m−l)⩾δ+1 and ϱδ = δ+1. The latter conditions also provide necessary conditions for the existence of a real solution. For a generic G(s), sufficient conditions for solvability of CZAP by a real squaring down are that l(m−l)⩾δ+1 and that the number g(a0,...,al−1)=(δ+1)!/a0! ⋯. al−1!Πi >j (ai−aj) is odd for some set {a0,...,al−1} satisfying δ+1=∫l−1i=0ai− l(l−1)/2 and 0⩽a0<a1<⋯<al−1⩽m−1. Apart from the existence results, the present approach also allows the computation of the solutions (whenever they exist). Finally, it is proved that the only fixed zeros of CZAP are the zeros of G(s), and a procedure for computing the solutions of CZAP, whenever such solutions exist, based on an optimization problem is given.

A matrix pencil based numerical method for the computation of the GCD of polynomials

The paper presents a new numerical method for the computation of the greatest common divisor (GCD) of an m-set of polynomials of R[s], P/sub m,d/, of maximal degree d. It is based on a previously proposed theoretical procedure (Karcanias, 1989) that characterizes the GCD of P/sub m,d/ as the output decoupling zero polynomial of a linear system S(A/spl circ/,C/spl circ/) that may be associated with P/sub m,d/. The computation of the GCD is thus reduced to finding the finite zeros of the pencil sW-AW, where W is the unobservable subspace of S(A/spl circ/,C/spl circ/). If k=dim W, the GCD is determined as any nonzero entry of the kth compound C/sub k/(sW-A/spl circ/W). The method defines the exact degree of GCD, works satisfactorily with any number of polynomials and evaluates successfully approximate solutions. >

Decentralized determinantal assignment problem: fixed and almost fixed modes and zeros

The decentralized determinantal assignment problem (DDAP) is defined as the unifying description for the study of pole and zero assignment problems under decentralized output, state feedback (DOF, DSF) and decentralized ‘squaring down’ (DSD), respectively. DDAP is reduced to a linear problem of zero assignment of polynomial combinants and a multilinear problem of restricted decomposability of multivectors. The decentralization characteristic (DC) and the decentralized polynomial Grassmann representative )D — ℝ[s] — GR) of DDAP are defined. The fixed zero polynomial of DDAP is then determined as the zero polynomial of D— ℝ[s]—GR. The canonical D—ℝ[s]—GR, (CD—R[s]—GR) and the decentralized Plucker matrix (DPM) of DDAP are introduced and necessary conditions for arbitrary assignment of the non-fixed zeros are given in terms of the DPM. The family of strongly zero non-assignable (SNA( systems is defined, and for such systems the notion of the fixed zero is extended to that of the ‘almost fixed zero’ (AFZ). An...