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.     

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.     

Steady-state decoupling of linear multivariable systems

A constructive criterion for decoupling the steady states of a linear, time-invariant multivariable system is developed. This criterion consists of a set of inequalities which, when satisfied, will cause the steady states to be decoupled. It turns out that pure integrators in the loops play an important role.     

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.     

The design of linear multivariable systems

This paper describes a computer-aided procedure whereby a succession of single-loop designs, using Nyquist loci, yields a multivariable design which is stable and attenuates disturbances. The procedure permits more freedom than previously available for the selection of the compensating matrix. It is shown how a simple modification of the procedure enables security against component failure to be obtained.     

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.     

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.     

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 linear multivariable systems

A configuration and an orderly procedure for the design of linear multivariable control systems are presented. This procedure employs transfer function matrices to specify the system response to set point changes and disturbances. Neither a state-variable formulation nor solution of the matrix Riccati equation are required. The design procedure is illustrated by a simple example involving an unstable system.     

Subspace identification of multivariable linear parameter-varying systems

A subspace identification method is discussed that deals with multivariable linear parameter-varying state-space systems with affine parameter dependence. It is shown that a major problem with subspace methods for this kind of system is the enormous dimension of the data matrices involved. To overcome the curse of dimensionality, we suggest using only the most dominant rows of the data matrices in estimating the model. An efficient selection algorithm is discussed that does not require the formation of the complete data matrices, but processes them row by row.     

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.     

Algebraic theory of linear multivariable feedback systems

This paper presents an algebraic theory for analysis and design of linear multivariable feedback systems. The theory is developed in an algebraic setting sufficiently general to include, as special cases, continuous and discrete time systems, both lumped and distributed. Designs are implemented by construction of a controller with two vector inputs and one vector output. Use of controllers of this type is shown to generate convenient stability results, and convenient global parametrizations of all I/O maps and all disturbance-to-output maps achievable, for a given plant, by a stabilizing compensator. These parametrizations are then used to show that any such I/O map and any such disturbance-to-output map may be simultaneously realized by choice of an appropriate controller. In the special case of lumped systems, it is shown that the design theory. can be reduced to manipulations involving polynomial matrices only. The resulting design procedure is thus shown to be more efficient computationally. Finally, the problem of asymptotically tracking a class of input signals is considered in the general algebraic setting. It is shown that the classical results on asymptotic tracking can be generalized to this setting. Additionally, sufficient conditions for robustness of asymptotic tracking, and robustness of stability are developed.     

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.     

Design of linear multivariable continuous-time tracking systems

In this paper, simple matricial methods are used to develop a basis for the design of multivariable continuous-time tracking systems which are generalized versions of classical single-input single-output servomechanisms.     

Feedback stabilization of multivariable two-dimensional linear systems

A constructive algorithm is presented for solving a two-dimensional (2-D) polynomial matrix equation that plays an important role in the output feedback stabilization of multivariable 2-D linear systems. It is shown that this algorithm can have some computational advantages over the previous known ones based on the Gröbner-basis approach. As an application of the algorithm, the class of all stabilizing strictly causal 2-D compensators is parametrized for a given stabilizable causal (not necessarily strictly causal) 2-D plant.