M. M. Newmann

Optimal and sub-optimal control using an observer when some of the state variables are not measurable†

The effect upon the cost is considered when the unavailable states of a linear plant are estimated with a linear observer and the estimates are used in the feedback control.

Design algorithms for a sensitivity constrained suboptimal regulator

The problem of finding the linear feedback control comprised of state and trajectory sensitivity terms which minimizes an infinite-time quadratic cost functional containing sensitivity variables is discussed. Since trajectory sensitivity cannot be accurately modelled in a closed-loop formulation the control feeds back approximate sensitivity terms. As the minimizing control is dependent on initial conditions, necessary conditions for a minimum are derived which satisfy various alternative sets of initial condition criteria. An efficient computer algorithm, based on a gradient search technique, is proposed for finding the feedback gains. By means of a numerical example it is shown that it is possible to reduce trajectory sensitivity while retaining certain desirable features of the linear optimal regulator design. Finally an algorithm is suggested for finding the linear feedback control comprised of state terms alone which minimizes the augmented cost functional.

Minimal-order observer-estimators for continuous-time linear systems

Abstract This paper considers the problem of designing minimal-order observer-estimators for both deterministic and stochastic time-varying linear continuous-time systems. Use of an equivalent ‘ canonical ’ class of observers does much to unify and clarify previous approaches to the problem. The main result is an optimal unbiased observer-estimator which may be implemented off-line with reduced computational effort Asymptotic behaviour of the deterministic (stochastic) observer-estimator is analysed and related via observability (and controllability) concepts to the structure of the original system.

Specific optimal control of the linear regulator using a dynamical controller based on the minimal-order Luenberger observer†

Abstract The problem is considered of designing the optimal, fixed configuration, dynamical feedback controller of order (n—m) required to control an nth-order linear system with output of order m(m<n) the cost functional for the regulator being of the usual integral quadratic form. It is supposed that only the means and covariances of the system state initial conditions are known. The form of the dynamical controller is assumed to be that of the minimal-order Luenberger observer. Two problems are considered. First, a design algorithm is derived for obtaining the optimal dynamical controller (observer) on the assumption that its output will be fed into the regulator system through the feedback gain matrix normally used for the optimal state regulator. Secondly, the algorithm is extended to the case when the feedback gain matrix is treated as one of the design parameters of the dynamical controller. Although the problem is basically one of parameter optimization it can be considered from the point of view ...

On attempts to reduce the sensitivity of the optimal linear regulator to a parameter change

One suggested method is to adjoin a quadratic form in the sensitivity vector to the integrand of the cost functional and to find the feedback, linear in the state and sensitivity vectors, which minimizes the augmented cost functional. Several authors have incorrectly assumed that the augmented system can be treated as another optimal regulator problem. In this paper it is shown that the problem, when correctly formulated, has no solution in the sense that there can be no finite state-independent feedback matrices which will satisfy the necessary conditions for an optimum.

On the design of discrete-time optimal dynamical controllers using a minimal-order observer

This paper considers the problem of designing optimal dynamical observer-based controllers for discrete-time linear time-invariant systems with inaccessible state. It is assumed that the initial state is an unknown random vector with known mean and covariance. The performance index is taken to be the expectation, with respect to the initial state, of the standard quadratic one for the discrete-time regulator.

A continuous-time reduced-order filter for estimating the state vector of a linear stochastic system†

A design procedure for a, reduced-order filter is suggested and the differential equation obeyed by the estimate error covariance matrix is obtained. The problem of optimizing the filter within the design framework is partially solved.

Polynomial optimization of stochastic discrete-time control for unstable plants

The matrix polynomial method of optimization of linear multivariable discrete-time control for unstable plants with stationary stochastic inputs is derived in a constructive manner. New sufficient conditions are obtained and Kuceras second diophantine equation is found directly from the requirement that the system be stable. Although the second equation must be satisfied, it is proved that it need not be used in the optimization when a certain coprimeness exists.

Analogue Computation of the Switching Surfaces in Three-dimensional Phase Space for the Optimization of Non-Linear Control Systems†

ABSTRACT The paper deals with the method of using an analogue computer to determine the optimum control for systems in which there is a bounded constraint on the control signal. The difficulties arising out of this method are also discussed. Two problems are considered. In each case the criterion of optimality is minimization of integral square error. The first problem concerns the control of a plant consisting of two integrators ; the control to take place in a given finite time, the command signal input being a step or a step plus a ramp. Ail analytic/numerical solution of this problem is given and compared with the analogue solution. The switching boundary is calculated. The second problem concerns the control of a plant consisting of three integrators with a command signal input of a step plns a ramp plus a parabola. The switching surface obtained is only optimal if the initial conditions are specially chosen, otherwise a result is obtained which is similar to Feldbnums (1955) switching enrve for the...

Polynomial optimization of stochastic continuous-time control for unstable plants

The matrix polynomial method of optimization of linear multivariable continuous-time control for unstable plants with stationary stochastic inputs is derived in a constructive manner. New sufficient conditions are included but certain extra conditions imposed previously for continuous-time optimization are shown to be unnecessary. Kuceras second diophantine equation is derived directly from the requirement that the system be stable. As in discrete time, the second equation need not be used in the optimization when a certain coprimeness exists.