On the design of optimal output feedback control systems†

For a linear control system with quadratic performance index the optimal control takes a feedback form of all state variables. However, if there are some states which are not fed in the control system, it is impossible to obtain the optimal feedback control by using the usual mathematical optimization technique such as dynamic programming or the maximum principle.

This paper presents the optimal control of output feedback systems for a quadratic performance index by using a new parameter optimization technique.

Since the optimal feedback gains depend on the initial states in the output feedback control system, two cases where (1) the initial states are known, and (2) the statistical properties of initial states such as mean and covariance matrices are known, are considered here. Furthermore, the proposed method for optimal output feedback control is also applied to sampled-data systems.     

On the design of optimal and sub-optimal control systems†

Though dynamic programming is an excellent and versatile algorithm, there are many cases in which we cannot apply this algorithm favourably to control system design because of the limited capacity of a digital computer and other difficulties.

Considering these facts, the successively optimizing method and the combined algorithm of dynamic programming and successively optimizing method are presented in this paper.

One of the objects of this paper is to present a practical method for obtaining the optimal solution of control systems with an integral form performance index by the use of dynamic programming.

The other is to present sub-optiinal solutions by the successively optimizing method and the combined algorithm.

These methods are very easy when compared with dynamic programming. Furthermore, these sub-optimal solutions can be made to converge to the optimum by using the recurrent equations repeatedly.     

Application of mathematical programming in the design of optimal control systems†

The problem of applying the various computational methods of mathematical programming in the design of an optimal control system is discussed. A general case of non-linear, non-autonomous, state equations, subject to inequality constraints on both state and control variables, is considered. Both continuous and discrete time systems are investigated. In case of discrete time systems, the sampling intervals are assumed generally unequal and aperiodic, with inequality constraints imposed upon them.

Systems like these impose considerable computational difficulties when treated by the maximum principle or dynamie programming. Using mathematical programming, one may simplify a wide class of those computational problems.

Several examples of applying mathematical programming to particular control problems are presented.     

Linear control of saturating control systems†

The problem is considered of replacing optimum non-linear control of saturating, single-variable systems by sub-optimal linear control. Both the optimal and sub-optimal controllers are designed to minimize the integral-square-error of the systems in response to a step input, and a comparison is made of the values of this integral using the two forms of control. The values of the integral are obtained to high accuracy using numerical minimization techniques.

Consideration is also given to the stability of the systems under the linear controller.     

Sensitivity design of optimal linear systems†

A measure of sensitivity is introduced into the performance index of the optimal linear system. The sensitivity vector is adjoined to the system state vector and the overall system is optimized. A procedure for selecting the weighting matrix elements is given. Finally a design procedure for obtaining approximately optimal feedback controllers with bounds on sensitivity is given. The methods are illustrated with a first-order example.     

State-feedback control of non-linear systems†

A design method for state-feedback controllers for single-input non-linear systems is proposed. The method makes use of the transformations of the non-linear system into ‘controllable-like’ canonical forms. The resulting non-linear state feedback is designed in such a way that the eigenvalues of the linearized closed-loop model are invariant with respect to any constant operating point. The method constitutes an alternative approach to the design methodology recently proposed by Baumann and Rugh. Also a review of different transformation methods for non-linear systems is presented. An example and simulation results of different control strategies are provided to illustrate the design technique.     

Design of dominant-type control systems with large parameter variations†

This paper presents a design procedure for dominant-type systems with large plant parameter variations. The principal contribution is that a fourth-order approximation is used in the dominant region instead of a third-order, which up to now had been the most advanced method. The s-domain specifications of the system are assumed to be in the form of an acceptable dominant closed-loop pole region with bounds on the location of the ‘far-off’ closed-loop poles. The design philosophy is to place compensation zeros within the acceptable dominant closed-loop pole region such that the dominant closed-loop poles remain within their prescribed region despite the large variation in the plant parameters. The design procedure is for plants with simultaneous independent variation in the gain factor and a pair of poles. The design is such as to minimize the sensitivity of the system to internal noise.     

Controllability of linear time-varying systems with delay in control†

This paper is concerned with the study of controllability of linear systems with delay in the control function. It has been illustrated that many of the techniques which proved to be useful in the study of linear systems with no delay (alman et al. 19G2, Kroindlcr and Sarachik 1964) can be generalized when dealing with systems having delay in the control.

An explicit expression is given for transferring a given state to any desired state using minimum control energy.

The corresponding conditions for linear time-invariant systems (ebakhy and Bayoumi 1971) are obtained as a special case. Extensions to multiple- delays systems are also included.

A new degree of controllability is introduced and the corresponding criteria are obtained.     

The bounded energy optimal control for a class of distributed parameter systems†

The bounded energy optimal control for one-dimensional linear stationary distributed parameter system is solved here. The criterion function is a quadratic functional of the output.

Obtaining the optimal control involves the computation of the solution of a certain non-linear integral equation. The method of solving this integral equation is approximating the kernel of the integral operator by a sequence of degenerate kernels. It is shown that the sequence of approximate solutions of the approximate integral equations converges to the optimal solution; and that the sequence of approximate values of the criterion, converges to the optimal value of the criterion.     

Nonsmooth finite-time control of uncertain second-order nonlinear systems

Nonsmooth finite-time stabilizing control laws have been developed for the double integrator system. The objective of this paper is to further explore the finite-time tracking control problem of a general form of uncertain second-order affine nonlinear system with the new forms of terminal sliding mode (TSM). Discontinuous and continuous finite-time controllers are also developed respectively without the singularity problem. Complete robustness can be acquired with the former, and enhanced robustness compared with the conventional boundary layer method can be expressed as explicit bounded function with the latter. Simulation results on the stabilizing and tracking problems are presented to demonstrate the effectiveness of the control algorithms.     

Optimal control of distributed parameter systems with discrete constrained inputs†

The optimal control of linear distributed parameter systems, which are represent able by a linear vector integral equation, is investigated. Restricting the control action to be discrete in time, the problem of minimizing the mean-square error, between specified desired final state functions and the actual state functions at a prescribed final time, subject to an energy constraint on the controlling functions, is treated. A necessary and sufficient condition from functional analysis is used to derive an equation whose solution yields the optimal control vector. Two convergence properties for the discrete problem are established which can be used to determine a good approximate solution to the corresponding measurable optimal control problem. An illustrative example is given.     

Feedback control of distributed parameter systems with spatially concentrated controls†

The problem of designing an optimal feedback controller is discussed for a distributed parameter system governed by a partial differential equation of parabolic type. A quadratic performance index is evaluated with spatially concentrated and time-discrete control, and controllability of this system is investigated. Also an estimator with multi-pointwise observation is constructed. Computational algorithms of these controller and estimator are derived, and to show their validities, a few numerical examples are given     

On modifying the prediction control of a second-order plant†

The switching behaviour of a second-order plant under the combined influence of system non-linearities such as Coulomb friction, stiction, backlash, dead zone, hysteresis and switching delay is analytically examined in this paper.

In the first instance, the mathematical model which characterizes the system dynamics is formulated on a more realistic basis by considering the effect of these non-linearities as integrated parameters rather than isolated phenomena. This is then followed by the derivation of the modified switching equation which, upon careful mathematical manipulation, may be expressed in terms of a power series as a function of the system velocity. Such an approach not only leads to a better understanding of the general switching behaviour during the change over of forward to reversed drive, but also provides a more practical solution for the bang-bang control problem     

Closed-loop control of a class of non-linear systems†

A technique is described for the closed-loop control of a class of non-linear systems iii a potentially large neighbourhood of a nominal trajectory. By the introduction of extra states it is shown how to reduce non-linear differential equations to a related canonical linear form. Since the linear form is not readily amenable to standard design techniques a new synthesis procedure is proposed for the construction of a closed-loop control. The proposed technique relies heavily on computer computation. The computation for the closed-loop control of a lifting body, a system not completely controllable, is given in detail to illustrate the applicability of the method.     

Local uncontrollability for affine control systems with jumps†

This paper investigates affine control systems with jumps for which the ideal I_f(g₁, …, g_m) generated by the drift vector field f in the Lie algebra L(f, g₁, …, g_m) can be imbedded as a kernel of a linear first-order partial differential equation. It will lead us to uncontrollable affine control systems with jumps for which the corresponding reachable sets are included in explicitly described differentiable manifolds.     

On the separation theorem of stochastic optimal control†

This paper presents an application of results proved elsewhere for linear control problems formulated as minimum norm problems in Hilbert space, to extend the classical separation theorem to a larger class of systems. This larger class is not required to have gaussian statistics, and the observables are allowed to depend in an arbitrary linear fashion on present and past values of the state vector. The results obtained are a consequence of the Hilbert space projection theorem, and the separation theorem is proved without appeal to either the explicit form of the estimator or dynamic programming and the principle of optimality.     

Graphical analysis of a class of non-linear feedback control systems with step input†

The class of systems considered in this investigation is a cascade combination of a linear memory system and a non-linear no-memory system in the forward path of a unity feedback control system. The output of the non-linear no-memory system is assumed to be a polynomial function of the input. Regardless of the exact nature of the non-linearity, the objective of this method of analysis is to predict the behaviour of higher-order non-linear systems with different initial conditions for step inputs.

Two different cascade combinations of linear and non-linear blocks in the forward path are considered. For both configurations a similar non-linear differential equation is obtained for some variable in the system. The non-linear differential equation is further reduced to a first-order equation, explicitly independent of the independent variable, time t. Treating all other coefficients as parameters and eliminating each in turn, finally the required phase-plane trajectory is obtained.     

Feedforward—feedback control of distributed parameter systems†

Regulatory control of distributed systems subjected to load disturbances is considered by using feedforward and state measure control configurations. Dynamic compensation of the feedforward signal is accomplished with a lead-lag function, the time constants of which are determined by means of a numerical search technique. Compensation of the state measure signal is provided by the distributed nature of the process itself. Exit temperature regulation of a tubular heat exchanger acted upon by velocity and inlet temperature disturbances is considered as an application for feedforward control. Considerably better performance is obtained with the addition of dynamic compensation to the feedforward signal. State measure control is applied to the exchanger for a feed temperature upset and the effects of sensor location on outlet performance are investigated. An optimal sensor location is determined which minimizes the integral-square error at the outlet.     

A computational method for convex optimal control problems involving linear hereditary systems†

We consider a class of convex optimal control problems involving a linear hereditary system. The main aim of the paper is to devise a computational algorithm for generating a minimizing sequence of controls such that the sequence converges to the optimal control in both the weak? topology of L∞ and the almost-everywhere topology.