Closed-loop Eigenstructure Research Articles

Parametric eigenstructure assignment by output-feedback control: the case of multiple eigenvalues

This paper deals with closed-loop eigenstructure assignment by output-feedback control in linear multivariable systems for the general case of multiple eigenvalues. The approach is algebraic in nature and operates directly on the closed-loop characteristic equation using fundamental properties of determinants and their derivatives. A compact form for the controller gain matrix is derived and expressed in terms of a set of parameter vectors that specify the non-uniqueness of the solution of the eigenvalue-assignment problem. Some of these parameter vectors assume restricted forms while others are free. A computational scheme is given in which the free parameter vectors are determined in such a way as to assign the corresponding eigenvectors and generalized eigenvectors in a weighted least-square-error sense. An illustrative numerical example is included.

Sensitivity Reduction by State Derivative Feedback

This paper shows that the sensitivity of state feedback control systems can be reduced by additional state derivative feedback, for a fixed closed loop eigenstructure. The price of this sensitivity reduction is in general noise response amplification. Two indices which quantify stability robustness and response sensitivity are given for time invariant continuous time and discrete time systems, together with an index of response to disturbances and noise. Closed form expressions for the gradients of these indices are given. A two step design procedure is proposed which consists of first selecting a closed loop eigenstructure, then minimizing one of the sensitivity indices under a magnitude constraint on the noise response. Examples are given to illustrate this original design procedure.

Eigenstructure assignment by output feedback in descriptor systems

Some necessary and sufficient conditions for closed-loop eigenstructure assignment by output feedback inlinear time-invariant descriptor systems are presented. These generalise in a natural way the known corresponding results for state-space systems. However, our arguments extend somewhat existing work on eigenstructure assignment by state feedback in descriptor systems. The main issue throughout the paper is closed-loop regularity, i.e. uniqueness of solutions of the initial value problem, which cannot be taken for granted.

Detection filter design: Spectral theory and algorithms

A new formulation of the detection filter problem is generated by assignment of the closed-loop eigenstructure under certain constraints. Detection filters, which are actually a specific class of observers, fix the output error direction of the system so that it can be associated with a particular failure mode and its known design failure direction. The derivation of detection filters from an eigensystem assignment approach permits a very transparent theory. The detection filter gains and closed-loop eigenvectors are obtained from a set of simultaneous equations. Necessary and sufficient conditions for the solution of these algebraic equations are determined which produce a complete theory for detection filters.

Eigenvalue-generalized eigenvector assignment by output feedback

This note generalizes the previous results of the closed-loop eigenstructure assignment via output feedback in linear multivariable systems. Necessary and sufficient conditions for the closed-loop eigenstructure assignment by output feedback are presented. Some known results on entire eigenstructure assignment are deduced from this result.

Sequential decentralized control with a prescribed degree of stability

In this paper, a new approach to decentralized stabilization and control is proposed. This approach combines a suboptimal high-gain output-feedback controller design with a sequential high-gain decentralized control scheme such that at each stage of the sequential scheme, the retained optimal closed loop eigenstructure has a prescribed degree of stability. This new algorithm is illustrated on a practical problem, the decentralized stabilization and control of a multi-machine power system.

Eigenstructure assignment by output feedback and residue analysis

This note deals with the use of feedback to approximate the closed-loop eigenstructure of a system to a prescribed set of values. A new method is proposed to reduce the controller complexity, based on eigenvalue sensitivity concepts. A flight control example is presented as an application.

Some necessary and sufficient conditions for eigenstructure assignment

Abstract Some necessary and sufficient conditions for closed-loop eigenstructure assignment by output feedback in time-invariant linear multivariable control systems are presented. A simple condition on a square matrix necessary and sufficient for it to be the closed-loop plant matrix of a given system with some output feedback is the basis of the paper. Some known results on entire eigenstructure assignment are deduced from this. The concept of an inner inverse of a matrix is employed to obtain a condition concerning the assignment of an eigenstructure consisting of the eigenvalues and a mixture of left and right eigenvectors.

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.

Design of linear quadratic regulators with assigned eigenstructure

A recursive design procedure is presented which assigns the closed-loop eigenstructure in linear quadratic regulators. At each stage, the required solution for the steady-state Riccati matrix which shifts a pole or pole pair to specified values is obtained. For pole pair placement, a free parameter in the solution permits selection of closed-loop eigenvectors. The interactive design procedure is summarized and illustrated by an example

Eigenstructure assignment in linear multivariable systems--A parametric solution

This paper generalizes a recently reported method [1] of closed-loop eigenstructure assignment via state feedback in a linear multivariable system (with n states and r control inputs). By introducing a lemma on the differentiation of determinants, the class of assignable eigenvectors and generalized eigenvectors associated with the assigned eigenvalues is explicitly described by a complete set of n r -dimensional free parameter vectors. This parametric characterization conveniently organizes the nonuniqueness of the solution of the eigenvalue-assignment problem and thereby provides an efficient means of further modifying the system dynamic response. A numerical example is worked out to demonstrate the feasibility of the method.

On the assignability of infinite root loci in almost disturbance decoupling

In this paper, we investigate in detail the almost disturbance decoupling problem with state feedback. An alternative proof of the sufficiency part of the original result by Willoms (1981) on this problem, requiring the construction of a sequence of feedback matrices, is given. It is shown that one can achieve almost disturbance decoupling by letting some of the closed loop eigenvalues run to infinity according to a spatial arrangement determined by the infinite zero structure. The proofs of our results use the fact that almost controllability subspaces can be approximated by controlled invariant subspaces. Our results on the closed-loop eigenstructure are stated in terms of such approximating sequences. We also prove a generalization of a theorem by Wonham (1979), used in the context of minimal order observer design.

Dead-beat control of linear discrete-time systems

The paper considers the problem of determining the various classes of state-feed back dead-beat controllers for linear multivariable discrete-time systems. Unlike earlier work, attention is not restricted to the special case of minimum-time dead-beat controllers. A general dead-beat control problem is posed and solved. The solution is based principally on an approach of closed-loop eigenstructure assignment. The minimum of free parameters in each class are specified and used in an explicit parametric form for the feedback gain matrix. The concept of the greatest minimum of free parameters is introduced. Finally, the method is illustrated by a numerical example

On eigenstructure assignment in linear multivariable systems

The eigenvalue-assignment approach of Brogan [1], [2] is generalized and extended to the assignment of the entire closed-loop eigenstructure of linear multivariable systems. The set of assignable eigenvectors and generalized eigenvectors emerges naturally in the solution and is given, moreover, in an explicit parametric form. Two numerical examples are worked out to demonstrate the application of the procedure.

Asymptotic properties of linear multivariable discrete-time tracking systems incorporating fast-sampling error-actuated controllers

Abstract In this paper, the asymptotic properties of tracking systems incorporating linear multivariable plants which are amenable to fast-sampling error-actuated control (Bradshaw and Porter 1980) are characterized in terms of the eigenstructure of the closed-loop plant matrices. It is shown that the closed-loop eigenstructure is such that the tracking behaviour of systems incorporating fast-sampling error-actuated controllers designed in accordance with the synthesis technique of Bradshaw and Porter (1980) is increasingly dominated by the modes associated with characteristic roots of the form λ=l+σ as the sampling frequency is increased and that increasingly ‘tight’ control is therefore achieved. These theoretical results are illustrated in the case of a closed-loop system incorporating a third-order plant and a fast-sampling error-actuated controller by the presentation of the computed closed-loop eigenstructure and of the results of computer simulation studies.

Asymptotic properties of linear multivariable continuous-time tracking systems incorporating high-gain error-actuated controllers

In this paper, the asymptotic properties of tracking systems incorporating linear multivariable plants which are amenable to high-gain error-actuated control (Porter and Bradshaw 1979) are characterized in terms of the eigenstructure of the closed-loop plant matrices. It is shown that the closed-loop eigenstructure is such that the tracking behaviour of systems incorporating high-gain error-actuated controllers designed in accordance with the synthesis technique of Porter and Bradshaw (1979) is increasingly dominated by the modes associated with the first-order infinito characteristic roots as the gain is increased and that increasingly ‘ tight ’ control is therefore achieved. These theoretical results are illustrated in the case of a closed-loop system incorporating a third-order plant and a high-gain error-actuated controller by the presentation of the computed closed-loop eigenstructure and of the results of computer simulation studies.

Synthesis of output-feedback control laws for linear multivariable continuous-time systems

It is known (Porter and Bradshaw 1978 a) that, in the case of self-conjugate distinct eigenvalue spectra, the closed-loop eigenstructure assignable by output feedback is constrained by the requirement that the eigenvectors and reciprocal eigenvectors lie in well-defined subspaces. In this paper a technique is presented which can be used to select the eigenvectors and reciprocal eigenvectors from these subspaces in the case of appropriately augmented (Kimura 1975) controllable and observable continuous-time systems. Thin technique is ideally suited to digital-computer implementation and therefore greatly facilitates the synthesis of both static (Porter and Bradshaw 1978 a) and dynamic (Porter and Bradshaw 1978 b) output-feedback controllers.

Closed-loop eigenstructure assignment by state feedback in multivariable linear systems with slow and fast modes

It is shown that the recent results of Moore (1976) and Porter and D'Azzo (1977) can be readily extended so as to provide a computationally attractive method of closed-loop eigenstructure assignment by state feedback in multivariable systems with ‘ slow ’ and ‘ fast ’ modes. The general results are illustrated by designing a state-feedback controller for a third-order system with ‘ slow ’ and ‘ fast ’ modes which assigns asymptotically the entire closed-loop eigenstructure as characterized by both ‘ slow ’ and ‘ fast ’ eigenvalues and eigenvectors.

Algorithm for closed-loop eigenstructure assignment by state feedback in multivariable linear systems

Abstract In this paper an algorithm is presented which greatly facilitates the complete exploitation of state feedback in the assignment of the entire closed-loop eigenstructure of controllable multi-input systems. This algorithm is a generalization of the algorithm of MacLane and Birkhoff (1968) for the computation of a basis for the null space of a matrix and is ideally suited to digital computer implementation. The algorithm readily yields the vectors which are required (Porter and D'Azzo 1978) for the simultaneous assignment of Jordan canonical forms, eigenvectors, and generalized eigenvectors to the plant matrices of closed-loop controllable multivariable linear systems. The effectiveness of the algorithm is illustrated by assigning the entire closed-loop eigenstructure of a third-order two-input discrete-time system in such a way that the resulting closed-loop system exhibits time-optimal behaviour.

Design of linear multivariable continuous-time output-feedback regulators

In this paper, the method of entire eigenstructure assignment (Kimura 1975, Moore 1976, Porter and D'Azzo 1917) is applied to the design of linear multivariable continuous-time output-feedback regulators. It is shown that, in the case of self-conjugate distinct eigenvalue spectra, the closed-loop eigenstructure assignable by output feedback is constrained by the requirement that the eigenvectors and reciprocal eigenvectors lie in well-defined subspaces. The method is illustrated by designing an output-feedback regulator for a third-order continuous-time system.