On “type l” multivariable linear systems

The purpose of this paper is to summarize results for linear multivariable systems that are analogous to those single input single output systems which are “type l”, i.e. that contain l integrators. Both frequency domain and time-domain properties are given. Appropriate conditions that guarantee zero steady state error in tracking vector polynomial inputs are presented.     

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.     

On the invertibility of linear multivariable systems

Some results concerning invertibility of a class of linear, time-invariant systems are presented. It is shown that by an appropriate factorization of the transfer matrix of such a system, the problem of cheeking its invertibility can be reduced to that of checking the invertibility of a lower-order system. A sufficient condition for invertibility is also obtained.     

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.     

On structural invariants and the root-loci of linear multivariable systems

The geometric nature of the infinite zeros of the root-loci of linear multivariate systems is investigated using the canonical form derived by Morse (1973). It is shown that an invertible system S(A, B, C) has integer-order infinite zeros in the generic case equal to the controllability indices of a pair (A + KC, B), that suitable choice of proportional output feedback guarantees the absence of other than integer-order zeros and that the orders and asymptotic directions of the infinite zeros are independent of constant state feedback and output injection.     

On the input and output reducibility of multivariable linear systems

By introducing into a constant linear system (F, G, H) with input vectoruand output vectoryan open-loop controlu = Pvand observerz = Qy, a new constant linear system (F, GP, QH) results which has input vectorupsilonand output vectorz. The problem investigated is one of constructing (F, GP, QH) so thatupsilonandzhave minimal dimension, subject to the condition that the controllability and observability properties of (F, G, H) are preserved. It is shown that when the scalar fieldF(over which the system is defined) is infinite, the minimal dimensions ofupsilonandzare essentially independent of the specific values of the input and output matricesGandH. It is also shown that this is not the case whenFis finite. Furthermore, an algorithm is presented for the construction of the minimal input (minimal output) (F, GP, QH), which is directly represented in a useful canonical form.     

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     

On the robustness of multivariable linear feedback systems in state-space representation

A singular values-based approach to specify the robustness of a multivariable linear feedback system in state-space representation is investigated. The robustness measure which is considered is the largest spectral norm of an additive uncertainty in the closed-loop system matrix, for which stability is guaranteed. It is shown that under the constraint of prescribed pole placement, a lower bound for the robustness measure is maximized, when the Frobenius norm of the closed-loop system matrix is minimized.     

The stability of linear multivariable systems

A general criterion is presented for establishing the stability of a multivariable system from the stability of the simplified system consisting of its diagonal elements. This new criterion includes all known criteria of stability based on the diagonal system elements such as Rosenbrock's (1974), Nwokah's (1980) anil Koussiouris's (1980). For the class of matrices satisfying this criterion an equivalent result in terms of eigenloci can also be stated     

Diagonal stabilization of linear multivariable systems

A new procedure is presented for the synthesis of diagonal compensators for N × N linear multivariable systems that are free of fixed modes with respect to constant diagonal output feedbacks. The synthesis procedure employs simple polynomial algebra and it is in the form of an N-step algorithm. The geometric configurations 2N- and 2^N-cells in N-space are shown to be especially suitable for visualizing diagonal feedback and aiding the application of the synthesis algorithm.     

Decentralized control of linear multivariable systems

This paper studies the effect of decentralized feedback on the closed-loop properties of jointly controllable, jointly observablek-channel linear systems. Channel interactions within such systems are described by means of suitably defined directed graphs. The concept of a complete system is introduced. Complete systems prove to be precisely those systems which can be made both controllable and observable through a single channel by applying nondynamic decentralized feedback to all channels. Explicit conditions are derived for determining when the closed-loop spectrum of ak-channel linear system can be freely assigned or stabilized with decentralized control.     

Model reduction for linear multivariable systems

A method has been proposed for obtaining reduced-order models for multivariable systems. Termed nonminimal partial realization, this method is shown to encompass the methods of aggregation (or eigenvalue preservation) and moments matching (Padé-type approximation about more than one point). In particular, a promising subclass of reduced models, called aggregated partial realizations, is discussed in detail and appears to combine the good points of the aggregation as well as the moments matching methods.     

Canonical forms for linear multivariable systems

In this paper a class of well-known canonical forms for single-input or single-output controllable and observable systems are extended to multivariable systems. It is shown that, unlike the single-variable case, the canonical forms are generally not unique, but that the structure of the canonical form can be controlled to some extent by the designer. A major result of the paper is that a multi-input system can be transformed to a set of coupled single-input subsystems.     

Dynamic controllers in linear multivariable systems

The theory of pole assignment in multivariable stationary linear systems using a controller of given order is reassessed. Novel proofs, both simpler and more general than hitherto are given, enabling efficient design algorithms to be formulated. Examples of such algorithms are worked out.     

Compensator design for multivariable linear systems

The problem of the specification of the order and structure of a linear dynamic compensator in order to obtain arbitrary pole placement in a closed-loop linear system comprised of the compensator in cascade with a linear plant is discussed. A significant application of the theory is to the design of optimal systems in those cases where not all the state variables of the plant can be measured. These results permit a completely algorithmized approach to the design of compensators for linear systems.     

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.