On the computation of transmission zeros of linear multivariable systems

Corrected and recomputed transmission zeros of a previously published numerical example of a realistic physical system are published for possible use as a test case for new computational algorithms.     

Computation of zeros of linear multivariable systems

Several algorithms have been proposed in the literature for the computation of the zeros of a linear system described by a state-space model {A, B, C, D}. In this paper we discuss the numerical properties of a new algorithm and compare it with some earlier techniques of computing zeros. The method is a modified version of Silverman's structure algorithm and is shown to be backward stable in a rigorous sense. The approach is shown to handle both nonsquare and/or degenerate systems. Several numerical examples are also provided.     

Computation of the zeros of linear multivariable systems

The various subsets of the entire set of system zeros are unambiguously characterized in a manner which permits the definitive computation of the sets of centralized transmission and decoupling zeros by solving appropriate generalized eigenvalue problems using widely available ‘ expertly written ’ software embodying powerful QZ techniques, Furthermore, it is indicated that the present method is immediately applicable to the computation of the sets of decentralized fixed modes and decentralized transmission zeros with respect to any set of decentralized controllers.     

The role of transmission zeros in linear multivariable regulators

The problem is considered of designing compensators to regulate linear multivariable systems. It is shown that the essential mechanism employed by such a compensator is the multivariable analogue of pole-zero cancellation: the compensator supplies transmission zeros to cancel the unstable poles of the disturbance and reference signals. If the compensator is additionally required to function in the presence of small variations in system parameters then it must supply transmission zeros in greater multiplicity. Finally it is shown that the transmission zeros are generated by an ‘ internal model ’, incorporated in the compensator, of the dynamics of the disturbance and reference signals.     

On transmission zeros and zero directions of multivariable time-invariant linear systems with input-derivative control

The purpose of this paper is to investigate the problem of finding inputs which generate zero outputs in linear systems whose state equation contains the derivative of the input. The method developed makes use of the simplest matrix generalized inverse, the {1}-inverse.     

Computation of invariant zeros of linear,time-invariant,multivariable systems

In this paper a numerical method is presented for computing the invariant zeros of a controllable linear, time-invariant, multivariable system described by the 4-tuplo (A, B, C, D) or the triple (A, B, C). The method is based on the fact. that a controllable system can be made maximally unobservable by means of state variable feedback, thereby causing the cancellation of the invariant zeros by an equal number of the system poles. The invariant zeros are obtained as the eigenvalues of a matrix of the same dimension as the number of invariant zeros. The method is applicable to both multivariable as well as single-input, single-output systems. Examples are given to illustrate the use of the method.     

Comments on "On blocking zeros in linear multivariable systems"

It is suggested that the name "blocking zeros" in the above technical note is not appropriate and should be changed in order to avoid confusion in the literature.     

Finite transmission zeros of linear time-invariant generalized state space systems

In this paper, the problem of finding inputs which generate zero outputs in linear time-invariant control systems of the form E(dx/dt) = Ax + Bu is considered where E and A are not necessarily square matrices. This problem is solved using the theory of matrix generalized inverses.     

Computation of finite and infinite zeros of linear multivariable systems: a state space approach

An iterative algorithm, developed by the authors to compute the finite zeros of a linear multivariable system (Sebakhy et al. 1983), is extended to compute the infinite zeros of the system as well. The method is simple, efficient and suitable for programming on a digital computer     

On multivariable linear systems

A general result in linear system theory concerning the controllability and observability of multi-input multi-output systems is presented. Specifically, it is shown that any controllable observable linear system can be made controllable from any input and observable from any output using output feedback.     

Correspondence: A note on the orders of the infinite zeros of linear multivariable systems

A physical mechanism is suggested for the appearance of non-integer order infinite zeros. It is used to motivate a conjecture that attempts to characterize all possible orders of infinite zeros in terms of system structural invariants.     

Achievable performance of multivariable systems with unstable zeros and poles

This paper examines the fundamental limitations imposed by unstable (right half plane; RHP) zeros and poles in multivariable feedback systems. We generalize previously known controller-independent lower bounds on the H     

Near-decoupling of linear multivariable systems

Available exact-decoupling conditions by constant compensation do not address the important class of ‘nearly decouplable’ systems. For practical systems, parameter perturbations, measurement errors and system-modelling inaccuracies may conceal the system decouplability by constant- or low-order compensation, thus calling for unnecessarily high-order dynamics. In this paper, state-space decoupling conditions formulated as rank-degeneracy conditions are used to detect how close a system is to one that is exactly decouplable. The procedure simultaneously provides the compensators necessary to achieve the indicated level of decoupling. The conditions are extended to cases where it is required to use a fixed constant compensator for decoupling under different sets of perturbations     

A stability condition of zeros of sampled multivariable systems

This note derives a new condition for zeros of sampled multivariable systems to be stable for sufficiently small sampling periods. It is a natural extension of Hagiwara's result for single-input-single-output systems (1993) to multivariable systems     

Further discussion on the calculation of transmission zeros

The transmission zeros of a previous reported boiler system example [1,2] are computed, using double and extended precision arithmetic on an IBM 370/165 digital computer, by the algorithm of [2].