Dead-beat Control of Third-order Servo-systems†

In this paper a method of obtaining dead-beat response of third-order servo systems is described. It is shown that among the three state variables of the system, e.g. error, output velocity and output acceleration, the first two, following a step signal input, attain zero values simultaneously—for an instant for certain combinations in the system parameters. For such an initial adjustment of the parameters, the system response can be made dead-beat if the parameters are suitably controlled in a discrete manner from an instant of very small value of the error. A switching circuit is described for achieving the discrete control. Results of studies of a proposed third system simulated on the analogue computer are also given.     

Mathematical modeling of elastic inverted pendulum control system

Inverted pendulums are important objects of theoretical investigation and experiment in the area of control theory and engineering. The researches concentrate on the rigid finite dimensional models which are described by ordinary differential equations(ODEs) .Complete rigidity is the approximation of practical models; Elasticity should be introduced into mathematical models in the analysis of system dynamics and integration of highly precise controller. A new kind of inverted pendulum, elastic inverted pendulum was proposed, and elasticity was considered. Mathematical model was derived from Hamiltonian principle and variational methods, which were formulated by the coupling of partial differential equations (PDE) and ODE. Becausse of infinite dimensional, system analysis and control of elastic inverted pendulum is more sophisticated than the rigid one.     

Design of feedback systems with non-minimum-phase unstable plants†

Feedback systems with right half-plane poles and zeros may have inherently very poor sensitivity properties. In the design procedure presented, the closed-loop poles are-restricted to two possible regions in the complex plane. One region is s≤ ?σ. σ>0. A second is the interior and boundary of a circle in the left half-plane. The design is optimum in the sense of maximizing the gain factor uncertainty, for which the restriction is satisfied. The design procedure is very simple to execute and results in loop transmission poles and zeros which are symmetrical with respect to the boundary of the forbidden region. The closed-loop poles lie entirely on the boundary, over the range of gain uncertainty.     

Design of Non-linear Multivariable Control Systems†

Given a piece-wise single valued non-linear multivariable plant a design procedure is suggested to meet the desired specifications for step inputs. The system is first designed as a linear system satisfying the design specifications and its performance as a non-linear system is checked. The method may involve trial and error. An example is worked out Illustrating the procedure. Analogue computer results are also included.     

Parameterized Feedback Control of Non-linear Processes†

A linear optimal controller which is effective in a suitable neighbourhood of a reference optimal trajectoiy is derived. This is accomplished by parameterizing the canonical equations for this field. It is shown that depending on the parameters chosen, different feedback matrices can be calculated and an example comparing the performance of a process subject to a control function based on two possible matrices is presented.     

Optimum Control of a Certain Class of Servomechanism†

Linear control systems governed by the vector matrix differential equation x = A x + B u have been considered. It has been shown how to find the optimum control u so that the system, starting from an initial position x(0), is steered to a state specifying the first p coordinates of the system in time t _o fixed in advance, the values attained by the (n—p) coordinates being immaterial, where n is the dimension of the system. The optimization considered here is with regard to the norm of u supposed to belong to L _m E _r space.     

Control system synthesis and analysis by approximation to third-order systems†

This paper presents a set of design charts relating the time response characteristics and the parameters of the closed-loop transfer function possessing a conjugate pair of poles, a real pole and a real zero. It involves most practical cases and can be applied in both analysis and synthesis of linear control systems for more accurate results. The procedure is direct and quite simple. Three examples are given.     

Fuzzy control of an inverted pendulum with fuzzy compensation of friction forces†

The purpose of this paper is to compare the stabilization, with friction compensation, of an unstable mechanical system, an inverted pendulum, by two methods: the state feedback method, which belongs to the mathematical model-based approach, and fuzzy control. In both cases, the friction forces are compensated either by a simple method, the switching of a constant threshold, or a more sophisticated one: a disturbance observer in the model-based approach and a fuzzy compensator in the fuzzy control situation. The results obtained with the latter are comparable with or even better than those obtained with the former.     

Optimal Time Control of Non-normal Linear Systems†

The optimal time control problem is considered for time-invariant linear systems, [xdot]=Ax+Bu, which do not satisfy the La Salle definition of a normal system (La Salle 1960). It is shown that, for single–input systems, all controllable states must be contained in a certain hyperplane in state space. Furthermore, it is proved that, for completely controllable multiple-input systems, unique time-optimal controls may be determined by a technique developed previously by the author (Thau 1964), Examples are included to illustrate the method.     

Finite Time Optimization of Bang-bang Control Systems†

A Simple technique is demonstrated for synthesizing the optimum control of a bong-bung system, the; control acting for a finite time. The solution is based on the assumption that, in lower-order systems anyway, the control surface converges quickly to the steady-state surface.     

Optimum reduction of large dynamic system†

A performance index connecting the correlation measures of feedback errors of two unity feedback systems is suggested for obtaining the optimum simplified models of large, linear, time-invariant systems. To ensure the suitability of the index a simplification procedure by eliminating the remote poles in sequence is adopted. The examples indicate that the index is a useful criterion for optimum reduction of large dynamic systems.     

Sub-optimal Policies for Two Classes of Control†

Sub-optimal policies for the fuel optimal control and energy optimal control problems for a linear-time invariant system [xdot] = Ax = Bu, ‖u‖ ≤ M are obtained through the use of Liapunov functions. The closed-loop sub-optimal policies are employed for the design of controllers with the help of a special class of solutions of the Hamilton-Jacobi equation. The theory for both classes of optimal-control problems is illustrated by examples.     

Synthesis of a tim-optimal control system†

A second-order system has been taken for the synthesis of time-optimal control. Important elements in the closed-loop system are the relay element, the controlled element and the controlling device performing time-optimal control. An equation for the switching boundary has been obtained by the phase-space technique and verified by a state-variable approach. The realized time-optimal system is found to give better response than any other non-optimal system.     

Optimal Control of Systems with Random Gain of Plant †

A theoretical study is made of optimizing an nth order system which has a random variation of gain of the plant of (a) Gaussian white noise, or (b) generalized Poisson process. The problem is formulated by dynamic programming.     

Concerning Optimum Control of Linear Distributed Parameter Systems†

In this paper distributed parameter systems governed by systems of linear partial differential equations have been treated. The admissibly control functions are assumed to belong to a closed ball in the appropriate Banach space. Formulae have been derived, such that the norm of the control function in the Banach space in which they are assumed to belong is minimized, such that the control becomes simultaneously time optimal.     

The singular-G method for unstable non-minimum-phase plants†

In linear time-invariant feedback systems with plants which have both poles and zeros in the right half-plane, it is always possible to stabilize the system for a fixed plant. But in the previous optimum techniques, the stability margins might be so small as to render the design wholly impractical. This problem was overcome in the X-29 aircraft in a multiple-input-multiple-output (MIMO) setting, by use of a singular-G (compensation) matrix inside the loop. Excellent stability margins were then achievable over a wide plant parameter range, by means of a fixed-G compensation matrix.

This paper extends the singular-G technique to the single-input-single-output (SISO) plant. The latter is converted into an equivalent N × N MIMO plant by means of N parallel independent time-varying modulators acting on the plant output, a technique previously used for non-linear network synthesis. The singular-G method is then applicable to the equivalent N × N MIMO plant. The detailed design procedure is presented by means of an example with N = 2.     

Multi-variable Time-shared Control Systems and the Reduction of Interaction†

A study of feed back control systems with time-shared analogue elements is described. Continuous, ideal impulse commutated and finite pulse commutated systems have been compared for specific examples.

Particular emphasis has been placed on the practical problem of reducing interaction between loops and in particular the effect of varying the commutation duration has been determined.

A design procedure for time-sharing systems is suggested.     

Time optimal control of a linear hyperbolic system†

The time optimal control of transmission lines with amplitude constraints on the control is considered as a typical problem involving systems governed by hyperbolic partial differential equations. Using a Laplace transformation formulation to yield a time ‘optimal’ solution, it is shown how this sub-optimal control which is bang-bang develops into an optimal control which is not always at its limiting values—demonstrating the effect which the nature of the differential equation has on the form of the optimal control. A simple physical interpretation of the results is given.     

Control with stochastic stopping time†

The usual (non-stochastic stopping) control problem is extended to the case of random terminal time. The more general model presented hero should be particularly useful when a system will change while being controlled. Systems which are otherwise deterministic, and systems with additive noise in the dynamic equation are considered.

Results concerning the relevant aspects of reliability and Markov process theories are presented. We show that stochastically stopped control optimality conditions are simply extensions of the usual conditions and that the limits of the criteria and optimal control, as the variance of the stopping probability distribution approaches zero, are the corresponding quantities for the non-stochastic problem.     

Comments on ‘ Properties of the system matrix of a generalized state-space system’†

The purpose of this correspondence is to draw attention to significant extensions of the results presented by Verghese, et al. (1979). These extensions were obtained after the above paper was written, and are described in more detail in Verghese (1978) and Verghese and Kailath (1979).