This paper provides a detailed survey and extension of certain properties of the stable regulator problem: determine \min\inf{u} \int\min{0}\max{\infty} x'Qx + u'u dt subject to \dot{x} = Fx + Gu; x(0) = x_{0}; \liminf{t \rightarrow \infty} x(t) = 0 where Q is not necessarily sign definite. First, equivalence conditions recently given by Willems for the existence of the minimum are extended to include statements in terms of the Hamiltonian matrix and spectral factorization. This provides a precise relation between the time-domain and frequency-domain solutions to the problems. Second, the inverse problem of whether a given feedback u = -Kx is optimal for some Q is easily resolved, as is the redundancy problem of distinct Q 1 and Q 2 , resulting in the same optimal control.

The stable regulator problem and its inverse

Survey paper: A survey of some recent results in linear multivariable feedback theory

Stabilization of a synchronous machine through the control of its exciotation is discussed in this paper. The control laws are derived from the linearized equation of performance of the synchronous machine by the minimization of a scalar performance index. The manipulated variable is an increment to the exciter reference voltage having an amplitude constraint. Control laws are dependent on the selection of the weighting coefficients of the states and the input. Guidelines are given for the selection of these coefficients so that both the small signal dynamic performance and the large signal transient performance of the machine are improved. It is shown that suboptimal controls corresponding to the feedback of machine slip velocity and its derivatives produce substantial improvement of machine response. Different transient conditions are simulated on the nonlinear machine model and the extent of improvement of machine performance by using the proposed control laws is demonstrated.

Stabilization of a Synchronous Machine Through Ooutput Feedback Control

By using active control technology (ACT) on a large flexible aircraft it is possible to effect some alleviation of dynamic loads to which the wing is subjected when the aircraft performs some commanded manœuvre or when it encounters unwanted atmospheric turbulence. The design of such a load alleviation control scheme by means of linear optimal control theory is described in this paper. Of particular concern are the effects upon the aircraft's structural behaviour of using impaired optimal feedback control caused either by the use in the design method of simpler mathematical models to describe the aircraft dynamics, or by the practical impossibility of obtaining for measurement the complete state vector of the aircraft, or by the failure of particular motion sensors. A simple algebraic method of accounting for the deficit in steady-state load values, which occurs as a result of having to use such reduced- order feedback control, is presented in addition to several results which illustrate the effectiveness...

A structure load alleviation control system for a large aircraft

For a linear Nth-order time-invariant system with scalar control variable and quadratic performance index, a procedure is given for computing the optimal value of the feedback vector k. Instead of determining, as a first, step, the N(N+1)/2 elements of a symmetric matrix, the K values of k are computed directly, leading to substantial simplifications, and making the method well suited for higher-order systems. In contrast to other methods starting from the same premise, no restrictions are imposed on the form of matrix A. In addition, and based on the previous considerations, a solution of the inverse optimal control problem is also presented. The form of the weighting matrix in the performance index can be chosen arbitrarily (e.g. as diagonal). In contrast to other known procedures, the computations are straightforward, and again no restrictions are imposed on the form of the system.

Optimal control of linear multivariable systems with quadratic performance index, and the inverse optimal control problem

The optimal control of continuous-time linear multivariable systems with finite- and infinite-time quadratic performance is investigated using the maximum principle. The time-varying optimal-control law is shown to be related to the matrix Riccati differential equation, and a solution is obtained in terms of the components of the transition matrix associated with the linear system of 2n differential equations in the n state variables and the n adjoint variables. The time-varying transition-matrix formulation is also used to develop a solution of the steady-state matrix Riccati and associated Lyapunov equations. The infinite-time optimal-control solution is derived in terms of partitioned eigenvector components related to the stable modes of the augmented system, and the technique is shown to be similar in principle to the method previously developed for reducing the order of a set of matrix differential equations by neglecting high-order modes. The results are also shown to be related to the solution of the algebraic Riccati equation and are used to develop an eigenvector solution for the Lyapunov equation. The techniques are extended to consider the optimal control of the linear system containing input derivatives. Results are obtained for the optimal control of a 1st- and 2nd-order linear system and for a 6th-order model representing the dynamics of a superheater system. The various techniques available for obtaining a solution of the linear optimal-control problem, namely the recurrence-type control algorithm based on dynamic programming, the eigenvector solution of the steady-state Riccati equation and the solution obtained by numerical integration of the time-varying Riccati equation, are compared.

S.A. Marshall

Papers

Optimal control of linear multivariable systems with quadratic performance criteria