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 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.     

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 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.     

Properties and calculation of transmission zeros of linear multivariable systems

A new definition of transmission zeros for a linear, multivariable, time-invariant system is made which is shown to be equivalent to previous definitions. Based on this new definition of transmission zeros, new properties of transmission zeros of a system are then obtained; in particular, it is shown that a system with an unequal number of inputs and outputs almost always has no transmission zeros and that a system with an equal number of inputs and outputs almost always has either n−1 or n transmission zeros, where n is the order of the system; transmission zeros of cascade systems are then studied, and it is shown how the transmission zeros of a system relate to the poles of a closed loop system subject to high gain output feedback. An application of transmission zeros to the servomechanism problem is also included. A fast, efficient, numerically stable algorithm is then obtained which enables the transmission zeros of high order multivariable systems to be readily obtained. Some numerical examples for a 9th order system are given to illustrate the algorithm.     

Computation of transfer function matrices of linear multivariable systems

This paper is concerned with the computation of transfer function matrices of linear multivariable systems described by their state-space equations. The algorithm proposed here performs orthogonal similarity transformations to find the minimal order subsystem corresponding to each input-output pair and uses some determinant identities to determine the elements of the transfer function matrices. Numerical examples are included to illustrate the performance of the proposed algorithm.     

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.     

Computation of optimal output feedback gains for linear multivariable systems

For linear systems with quadratic performance indices, it is shown that the optimal output feedback gains can be computed using gradient techniques. Unlike previous algorithms, this approach avoids the solution of nonlinear matrix equations while appearing to ensure convergence. Computational results for a fourth-order system are presented.     

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.     

Computation of the impulsive behavior of multivariable linear systems using a division algorithm

A new method is proposed which allows the evaluation of the impulsive behavior of a linear multivariable system of arbitrary order. The proposed method is in closed matrix form, that is only constant (real) matrix arithmetic (i.e. multiplication and addition) is involved among the polynomial matrices and therefore the algorithm can be easily programmed in a digital computer.     

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.     

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 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.     

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     

Computation of the structural invariants of linear multivariable systems with an extended version of the program

The structural invariants like finite and infinite zeros play an important role in many problems in the analysis of linear systems. In [1] Emami-Naeini and Van Dooren presented the program for computing the finite zeros of a linear system, which in the author's opinion is at present the best commonly available program for this problem. Such a program does not exist for computing the other structural invariants, like the infinite zeros and the Kronecker indices. This paper presents an extended version of the program , which computes the finite zeros as well as the other structural invariants.