Deadbeat algorithms for output controllers in discrete-time multivariable systems

This paper studies the problem of designing output deadbeat controllers to force the state of discrete-time multivariable systems to zero in a finite number of samples. Two algorithms are considered. The first is based on the fact that the closed-loop eigenstructure assignable by output feedback is constrained by the requirement that the left and right eigenvectors must be in certain subspaces. In the second algorithm, the output gain matrix is computed through the optimization of certain parameters of the controller, while maintaining its structural constraints. Computer programs have been developed to realize the two algorithms and examples are given to illustrate the feasibility of the techniques.     

Deadbeat control in multivariable non-linear time-varying systems with constraints of control inputs

A new approach to deadbeat control of a class of multivariable non-linear time-varying systems is presented. The approach enables us to find a non-linear control law, minimal positive integer N and a piecewise-constant control input vector subjected to constraints such that the output of closed-loop system coincides with given reference vector r( l) for ≥ NT, where r(<) represents the class of vector functions which can be generated in linear time-invariant systems, T is a given sampling period.     

Deadbeat control of linear multivariable generalized state-spacesystems

The classic idea of deadbeat control is extended to linear multivariable discrete-time generalized state-space systems using algebraic methods. The asymptotic properties of the linear quadratic regulator theory are used to obtain the classes of deadbeat controllers using stabilizing full semistate feedback. The solution is constructed from a `cheap control' problem. Both semistate and output deadbeat control laws are considered. The main design criteria are to drive the semistate and/or outputs of the system to zero in minimum time and that the closed-loop system be internally stable. Unique properties of these types of control laws are discussed. For semistate deadbeat control, all the (dynamic) poles including the ones at infinity are moved to the origin, whereas for output deadbeat, some of the finite transmission zeros are canceled. Numerically reliable algorithms are developed to solve both problems     

Zero structures of n-D systems

A detailed definition of the zero structure of an n-D polynomial matrix is given. The nature of the invariance of this zero structure is determined in both the polynomial matrix and polynomial system matrix contexts. Some connections with previous work are made.     

Practical internal stability of n-D discrete systems

This paper deals with the problem of asymptotic stability for n-D discrete systems in the practical sense that the system input and output signals are unbounded in, at most, one dimension. A definition of practical internal stability is introduced, and necessary and sufficient conditions are derived. The obtained results show that practical internal stability is less restrictive and more relevant for practical applications than the conventional two-dimensional (2-D) internal stability     

Multipurpose controllers for multivariable systems

The primary purpose of this paper is to outline a new procedure for designing controllers which simultaneously achieve a variety of desired design goals in deterministic, unity feedback, linear multivariable systems. More specifically, we will present a new algorithm for the systematic design of a "three-part" multivariable controller which simultaneously ensures: 1) a noninteractive or decoupled closed-loop design, 2) complete and arbitrary closed-loop pole placement, which implies desired (single-loop) transient performance as well as closed-loop stability, 3) zero steady-state errors between the plant outputs and any nondecreasing deterministic inputs, 4) complete steady-state output rejection of nondecreasing deterministic disturbances, and 5) robustness with respect to stability, disturbance rejection, and zero error tracking for rather substantial plant parameter variations. Our development will employ the more "modern" (Laplace-transformed) differential operator approach for controller synthesis, which involves transfer matrix factorizations and the manipulation of polynomial matrices in the Laplace operators.     

Parameter estimation for multivariable systems

A method is presented for estimating the system matrix triple{F,G,H}from noisy measurements of the input and output signals. The algorithm which is derived considering stationary, discrete time systems possessing cyclic state space, enables the estimation to be carried out without having to first estimate the output structural indices. The proposed consistent estimator can be easily implemented.     

Deadbeat output regulators for decouplable recursive non-linear systems

The deadbeat output regulator problem is formulated and solved in a straight-forward way for a class of decouplable multivarialbe recursive non-linear systems. It is shown that the decoupling control sequences drive each regulated output to zero and remain at zero thereafter in a minimum number of control iterations; this number is equal to the relative degree of the corresponding output. In addition, we show that the internal stability of a deadbeat closed-loop system is guaranteed only if the system is minimum phase.     

Canonical forms for multivariable systems

A new variant of observability is introduced, minimal observability, which has an interesting practical meaning. This concept is then exploited to obtain a simple and completely specified canonical form for multioutput nonautonomous analytic systems.     

Lyapunov stability of n-D strongly autonomous systems

In this article we look into stability properties of strongly autonomous n-D systems, i.e. systems having finite-dimensional behaviour. These systems are known to have a first-order representation akin to 1-D state-space representation; we consider our systems to be already in this form throughout. We first define restriction of an n-D system to a 1-D subspace. Using this we define stability with respect to a given half-line, and then stability with respect to collections of such half-lines: proper cones. Then we show how stability with respect to a half-line, for the strongly autonomous case, reduces to a linear combination of the state representation matrices being Hurwitz. We first relate the eigenvalues of this linear combination with those of the individual matrices. With this we give an equivalent geometric criterion in terms of the real part of the characteristic variety of the system for half-line stability. Then we extend this geometric criterion to the case of stability with respect to a proper cone. Finally, we look into a Lyapunov theory of stability with respect to a proper cone for strongly autonomous systems. Each non-zero vector in the given proper cone gives rise to a linear combination of the system matrices. Each of these linear combinations gives a corresponding Lyapunov inequality. We show that the system is stable with respect to the proper cone if and only if there exists a common solution to all of these Lyapunov inequalities.     

A self-tuning regulator for multivariable systems

The controller described in this paper is designed for multivariable plants with constant, unknown parameters. The algorithm operates on-line with the a priori information about the time delay. The order of the system may be given a priori. In the case where the order of the system is unknown it can be determined by a generalized likelihood-ratio statistical test which is described in this paper. The multivariable self tuning regulator consists of the two tasks of estimation and regulation. Estimation of the input-output system model parameters is based on the least-squares principle. The control is computed to minimize the combined cost of output deviation and control energy. Asymptotic properties of the estimation are discussed. Usefulness and simplicity of this approach are illustrated by examples.     

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.     

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.     

Robust controllers for multivariable model-matching systems

The robustness properties of multivariable perfect model-matching systems are presented. In order to stabilize the nominal system, a general parametrized 2-D controller is introduced. A necessary and sufficient condition of robust stabilization is derived for the perfect model-matching system under external disturbances and non-linear time-varying perturbations. The results presented could be viewed as the extension of the work by Åström (1980).     

Adaptive controller for multivariable delayed systems

A self-tuning controller is presented for the control of multi-input-multi-output discrete systems with multiple time delays. It is established that under suitable offline choice of the performance weights the plant output deviations converge to zero even for non-minimum-phase systems.     

Multipurpose deadbeat controllers for multivariable systems

A simple and direct deadbeat control design algorithm for discrete-time multivariable systems is introduced. Our brief was to synthesize a realizable controller to achieve the following objectives: (1) input-output decoupling, (2) deadbeat tracking of any prespecified class of changeable deterministic reference signals and (3) changeable deterministic disturbances rejection where the changeable reference signals and disturbances can be different at each channel. Since the internal stability requirement is satisfied, our design algorithm can easily handle both unstable and non-minimum phase systems. Some constraints on the transfer function are also derived to make sure that the derived controller is realizable. Since all solutions can be described in parametric form, some of the performance criteria can be combined.     

Observable state space realizations for multivariable systems

This paper derives two canonical state space forms (i.e., the observer canonical form and the observability canonical form) from multiple-input multiple-output systems described by difference equations. The state space model is expressed by the first-order difference equation and is equivalent to the input–output representation. More specifically, by setting the different state variables, the difference equations or the input–output representations can be transformed into two observable canonical forms and the canonical state space model can be also transformed into the difference equations. Finally, two examples are given.     

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.