Edward J. Davison

On "A method for simplifying linear dynamic systems"

Often it is possible to represent physical systems by a number of simultaneous linear differential equations with constant coefficients, \dot{x} = Ax + r but for many processes (e.g., chemical plants, nuclear reactors), the order of the matrix A may be quite large, say 50×50, 100×100, or even 500×500. It is difficult to work with these large matrices and a means of approximating the system matrix by one of lower order is needed. A method is proposed for reducing such matrices by constructing a matrix of lower order which has the same dominant eigenvalues and eigenvectors as the original system.

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.

Robust control of a general servomechanism problem: The servo compensator

The robust control of a general servomechanism problem, which is an extension to the results of [1], is considered in this paper. Necessary and sufficient conditions, together with a characterization of all robust controllers which enables asymptotic tracking to occur, independent of disturbances in the plant and perturbations in the plant parameters and gains of the system, are obtained. A new type of compensator, introduced in [1], called a servo-compensator which is quite distinct from an observer is shown to play an essential role in the robust servomechanism problem. It is shown that this compensator, which corresponds to an integral controller in classical control theory, must be used in any servomechanism problem to assure that the controlled system is stabilizable and achieves robust control; in particular, it is shown that a robust controller of a general servomechanism problem must consist of two devices (i) a servo-compensator and (ii) a stabilizing compensator. A study of the stabilizing compensator is made; in particular, it is shown that a new type of stabilizing compensator called a complementary controller, may be used together with the servo-compensator to form a robust controller for the servo-mechanism problem. A study of the case when perturbations in the robust controller are also allowed is then made; this leads to the Strong robust servo limitation theorem which imposes a fundamental limitation on the ability of practical servomechanisms to regulate a system.

A formula for computation of the real stability radius

This paper presents a readily computable formula for the real stability radius with respect to an arbitrary stability region in the complex plane.

Performance limitations of non-minimum phase systems in the servomechanism problem

Abstract This paper studies the cheap regulator problem and the cheap servomechanism problem for systems which may be non-minimum phase. The study extends some well-known properties of “perfect regulation” and the “perfect tracking and disturbance rejection” of minimum phase systems to non-minimum phase systems. It is shown that perfect rejection to disturbances applied to the plant input can be achieved no matter whether the system is minimum phase or non-minimum phase, whereas a fundamental limitation exists in the achievable transient performance of tracking and rejection to disturbances applied to the plant output for a non-minimum phase system, and that this limitation can be simply and completely characterized by the number and locations of those zeros of the system which lie in the right half of the complex plane. Furthermore, this limitation provides a quantitative measure of the “degree of difficulty” which is inherent in the control of such non-minimum phase systems.

The numerical solution of A'Q+QA =-C

The eigenvalues of the system matrix are used to construct a matrix which is a transformation in state space between two most frequently used canonical forms.

The robust decentralized control of a general servomechanism problem

The decentralized robust control of a completely general servomechanism problem is considered in this paper. Necessary and sufficient conditions, together with a characterization of all decentralized robust controllers which enables asymptotic tracking to occur, independent of disturbances in the plant and perturbations in the plant parameters and gains of the system, is obtained. A new type of compensator called a decentralized servo-compensator which is quite distinct from an observer is introduced. It is shown that this compensator, which corresponds to an integral controller in classical control theory, must be used in any decentralized servomechanism problem to assure that the controlled system is stablizable and achieves robust control; in particular, it is shown that a decentralized robust controller of a general servomechanism problem consists of two devices (i) a decentralized servo-compensator and (ii) a decentralized stabilizing compensator. It is then shown that, under certain mild conditions, there almost always is a solution to the robust decentralized servomechanism problem for any composite system consisting of a number of subsystems interconnected in any arbitrary manner. This last observation has important implications for process control.

On pole assignment in linear multivariable systems using output feedback

The general problem of pole assignment in a linear, time-invariant multivariable system via output feedback is considered. It is shown that given a controllable-observable system ( C,A,B ) in which A \in R^{n \times n} , rank B = m , rank C = r , then for almost all ( B,C ) pairs, \min (n,m + r-1) poles can be assigned arbitrarily close to \min (n, m + r-1 ) specified symmetric values by using output feedback.

An adaptive controller which provides an arbitrarily good transient and steady-state response

A model reference adaptive control problem is posed. In the problem, the objective is not the usual one of forcing the error between the plant output and the reference model output asymptotically to zero, but instead, it is that of forcing this error to be less than a (arbitrarily small) prespecified constant after a (arbitrarily short) prespecified period of time, with a (arbitrarily small) prespecified upper bound on the amount of overshoot. It is shown that to achieve this goal for a stabilizable and detectable, single-input single-output linear time-invariant (LTI) plant, it is necessary and sufficient that the plant be minimum phase. Knowledge of an upper bound on the plant order, of the relative degree, and of the sign of the high-frequency gain is not required. The controller proposed consists of an LTI compensator together with a switching mechanism to adjust the compensator parameters. If an upper bound on the relative degree is available, the compensator has dynamics of order equal to this upper bound less one; otherwise, the order of the compensator is adjusted as well as its parameters. >

The output control of linear time-invariant multivariable systems with unmeasurable arbitrary disturbances

Necessary and sufficient conditions are derived for a minimal order linear time-invariant differential feedback control system to exist for a linear time-invariant multivariable system with unmeasurable arbitrary disturbances of a given class occurring in it, such that the outputs of the system asymptotically become equal to preassigned functions of a given class of outputs, independent of the disturbances occurring in the system, and such that the closed-loop system is controllable. The feedback gains of the control system are obtained so that the dynamic behavior of the closed-loop system is specified by using either an integral quadratic optimal control approach or a pole assignment approach. The result may be interpreted as being a generalization of the single-input, single-output servomechanism problem to multivariable systems or as being a solution to the asymptotic decoupling problem.