C. D. Johnson

Accomodation of external disturbances in linear regulator and servomechanism problems

Modern linear regulator and servomechanism theories (of the deterministic type) either ignore system disturbances altogether or assume they can be represented by initial conditions on the plant state vector. Controllers designed by such theories may fail to meet performance specifications when the system is subjected to persistently acting disturbances. In this paper, we show how one can modify existing regulator and servomechanism theories to take into account the presence of persistent fluctuating disturbances. By this means one can design a deterministic controller which consistently maintains set-point regulation, or servotracking, in the face of a bread class of realistic external disturbances. In addition, we show how one can systematically exploit any useful effects that may be present in the action of external disturbances.

Optimal control of the linear regulator with constant disturbances

An optimal control problem for a linear regulator with constant external disturbance is formulated. It is shown that, for a suitably selected quadratic-type performance index, the optimal control is not an explicit function of the external disturbance. Moreover, the optimal control can be synthesized as a time-invariant linear function of the state plus the first time integral of a certain other time-invariant linear function of the state.

Further study of the linear regulator with disturbances--The case of vector disturbances satisfying a linear differential equation

In a previous paper [1], the conventional optimal linear regulator theory was extended to accommodate the case of external input disturbances \omega(t) which are not directly measurable but which can be assumed to satisfy d^{m+1}\omega(t)/dt^{m+1} = 0 , i.e., represented as m th-degree polynomials in time t with unknown coefficients. In this way, the optimal controller u^{0}(t) was obtained as the sum of: 1) a linear combination of the state variables x_{i}, i = 1,2,...,n , plus 2) a linear combination of the first (m + 1) time integrals of certain other linear combinations of the state variables. In the present paper, the results obtained in [1] are generalized to accommodate the case of unmeasurable disturbances \omega(t) which are known only to satisfy a given \rho th-degree linear differential equation D: d^{\rho}\omega(t)/dt^{\rho} + \beta_{\rho}d^{\rho-1}\omega(t)/dt^{\rho-1}+...+\beta_{2}d\omega/dt + \beta_{1}\omega=0 where the coefficients \beta_{i}, i = 1,...,\rho , are known. By this means, a dynamical feedback controller is derived which will consistently maintain state regulation x(t) \approx 0 in the face of any and every external disturbance function \omega(t) which satisfies the given differential equation D -even steady-state periodic or unstable functions \omega(t) . An essentially different method of deriving this result, based on stabilization theory, is also described, In each cases the results are extended to the case of vector control and vector disturbance.

Unified canonical forms for matrices over a differential ring

Abstract Linear differential equations with variable coefficients of the vector form (i) x = A (t) x and the scalar form (ii) y (n) +α n (t)y (n−1) +⋯+α 2 (t) y +α 1 (t)y=0 can be studied as operators on a differential module over a differential ring. Using this differential algebraic structure and a classical result on differential operator factorizations developed by Cauchy and Floquet, a new (variable) eigenvalue theory and an associated set of matrix canonical forms are developed in this paper for matrices over a differential ring. In particular, eight new spectral and modal matrix canonical forms are developed that include as special cases the classical companion, Jordan (diagonal), and (generalized) Vandermonde canonical forms, as traditionally used for matrices over a number ring (field). Therefore, these new matrix canonical forms can be viewed as unifications of these classical ones. Two additional canonical matrices that perform order reductions of a Cauchy-Floquet factorization of a linear differential operator are also introduced. General and explicit formulas for obtaining the new canonical forms are derived in terms of the new (variable) eigenvalues. The results obtained here have immediate applications in diverse engineering and scientific fields, such as physics, control systems, communications, and electrical networks, in which linear differential equations (i) and (ii) are used as mathematical models.

On observers for systems with unknown and inaccessible inputs

Many practical applications of modern control theory lead to the problem of observing a plant state x(t) in the presence of unknown, inaccessible inputs which are acting on the plant. This paper traces some early works on that problem and compares those earlier results with results recently published in this Journal by Meditch and Hostetter (1974).

Accommodation of disturbances in optimal control problems

Abstract Modern theories of automatic control have virtually ignored one of the oldest and most fundamental problems of control—the problem of controlling systems subjected to persistently acting, unknown external disturbances. In this paper the problem of control in the face of unknown disturbances is studied from the optimization point of view. It is first shown that a straightforward approach to the problem leads to control laws that are hopelessly unrealizable. A novel alternative approach is then proposed which leads to physically realizable control laws which perform optimally in the face of a broad class of unknown external disturbances. The proposed method is illustrated by posing and solving a modified version of the classical linear-quadratic regulator problem, in which persistently acting disturbances have been added.

Another note on the transformation to canonical (phase-variable) form

Linear transformation of arbitrary, completely controllable, single-input, time-variant linear dynamical system to canonical form

Adaptive controller design using disturbance-accommodation techniques

In this paper, the theory of disturbance-accommodating control (DAC) is extended to include the case of internal ‘disturbances’ arising from uncertain plant-parameter variations. This extension of DAC theory, when combined with existing DAC theory for external disturbances, leads to what is believed to be the first general theory of adaptive control for finite-dimensional linearized dynamical systems.

Stabilization of linear dynamical systems with respect to arbitrary linear subspaces

Abstract The classical notion of stabilizing a controlled dynamical system to some specified equilibrium point is extended to include stabilization to a specified linear subspace. Necessary and sufficient conditions for existence of a solution are derived and an explicit solution recipe is given for one special case.

The unreachable poles defect in LQR theory: analysis and remedy

In virtually every application of optimum linear-quadratic regulator (LQR) theory there exists a hidden region of ‘unreachable poles’ (in the left half-plane) which cannot be realized as optimum closed-loop poles. These regions of unreachable closed-loop poles are not visible using the solution procedures ordinarily employed in LQR applications and their lurking presence has (apparently) been overlooked by many professors, textbook writers and industrial users of LQR control theory for the past 25 years. The existence of these regions of unreachable poles represents a serious defect in the LQR method because those regions may (and often do!) contain closed-loop pole patterns which are considered highly desirable by classical control engineering standards, i.e. by ITAE and other classical standards of ‘ideal’ transient response. We first show how one can identify the regions of unreachable poles in an LQR problem. Then, it is shown how one can modify conventional LQR theory to overcome this defect and make...