Control Input Saturation Research Articles

Time-optimal feedback control of certain fourth-order systems

Abstract Time-optimal feedback control laws are derived for seven fourth-order systems with real non-positive eigenvalues and a single saturable control input. The method is based on the property of time-optimality exhibited by a certain predictive control strategy.

Minimum-time isochronal surfaces for certain third-order systems

Explicit algebraic expressions for the minimum-time isochronal surfaces for certain third-order systems with real eigenvalues and a single saturable control input are presented in this paper. For a system with positive real eigenvalues in simple ratio the region of controllability (an open convex subset of states from which the state origin can be reached in finite time) is precisely defined

Time-optimal control of certain plants with positive real eigenvalues

This note considers the time-optimal control problem for plants with positive real eigenvalues and a single saturable control input. It is demonstrated that a previous analysis (Fuller 1973, 1974 a) of plants with negative real eigenvalues in simple ratio is also applicable, with certain restrictions, to plants with positive real eigenvalues in the same simple ratio. For a second-order plant a region of controllability in the state plane is algebraically determined. For third and higher-order plants, a region of controllability is partially defined.

Time-optimal feedback control laws for certain third-order relay control systems

Abstract Time-optimal feedback control laws for six third-order plants having real, non-positive eigenvalues and a single saturable control input are calculated explicitly by direct application of Gulko's predictive control strategy. Errors in a previous treatment of three of the plants are discussed.

Optimal and Sub-optimal Saturating Control of Plants Containing Lags

For plants containing simple lags in certain ratios, and with saturable control input, the time-optimal control problem is reduced to the corresponding problem for a plant containing pure integrators. The method uses transformations of variables, and applies to systems of general order. Sub-optimal controllers for the plants with lags are also investigated by means of these transformations.

Simplified time-optimal switching functions for plants containing lags

For some cases of plants with lags and with saturable control input, time-optimal switching functions are calculated explicitly. The resulting functions are simple and do not involve transcendental expressions such as logarithms or exponentials. Consequently the systems are easily simulated on a digital computer, and are convenient as test cases in sensitivity investigations and in comparisons with sub-optimal systems.

On Optimization of a Second-order Non-linear Control System for Various Performance Criteria

ABSTRACT The paper deals with a control system which has a plant consisting of two integrators with a saturable control input. With a command signal (which may be a step plus a ramp) the optimum switching curve and the optimum controller for two different performance criteria (‘Minimum integral square of error’ and ‘Minimum integral modulus of error’) are found explicitly by purely algebraic and analytic methods which start from Pontryagin's Maximum Principle and involves Taylor's series expansion of the Hamiltonian system of equations between two arbitrary successive switch points on the forward path of an arbitrary optimum trajectory. The equations of the switching curves in the phase plane obtained by this method are verified with the known results already determined by Brennan and Roberts (1962), Eggleston (1963), Fuller (1960 a) and Wonham (1963) by other methods.

Minimization of Integral-square-error for Non-Linear Control Systems of Third and Higher Order

ABSTRACT The paper deals principally with a control system which has a plant consisting of three integrators with a saturable control input. The command signal input is a step plus a ramp plus a parabola. The switching surface which minimizes integral-square-error is found, partly algebraically and partly numerically, by methods which start from Pontryagin's maximum principle. All optimum trajectories have an infinite number of switches before the origin is reached, except for two trajectories which have no switches. Some optimum trajectories have the property that the ratio of any two successive switching intervals is constant. All other optimum trajectories (apart from the two exceptional cases) converge rapidly towards these constant ratio trajectories. Thus, when finding optimum trajectories by backwards numerical computation from near the origin of the Hamiltonian system of equations, it is necessary to adjust the ‘initial’ values to be, not only small, but also close to a constant ratio trajectory. ...

Further Study of an Optimum Non-linear Control System†

ABSTRACT This paper is the second of two in which the system studied has a plant which consists of two integrators with a saturable control input, and the performance criterion which is to be minimized is integral-error-squared. In the first paper the known optimum controller was shown to satisfy Pontryagin's maximum principle. In the present paper further verification of the optimum system is obtained by showing that the best performance function satisfies Bellman's partial differential equation. It is found that Bellman's equation is even satisfied for points on the switching curve, Results are also given for other performance criteria.

The Absolute Optimality of a Non-linear Control System with Integral-square-error Criterion†

ABSTRACT For a plant consisting of two integrators with saturable control input, a controller is known which gives a minimum value of integral-error-squared. In the present paper it is shown that this value is the absolute minimum (not merely a local minimum), and that the control input which gives it is unique.

Study of an Optimum Non-linear Control System†

ABSTRACT The system studied has a plant which consists of two integrators with a saturable control input. The command signal input is a step plus a ramp. The performance criterion which is to be minimized is integral-error-squared. The optimum controller for this problem was found previously by a very specialized method. In the present paper this optimum controller is verified by means of (i) Pontryagin's maximum principle, (ii) a special method involving variation of the switching curve.