Second-order Control System Research Articles

Non-linear sampled-data control systems with arbitrary values of transportation lag

The effect of inherent finite delay which is not an integral multiple of the sampling period on the character of limit cycles in a second-order feedback control system with a relay type non-linearity is analysed. It is found that the addition of transportation log alters the character of the modes sustained by the system. All the modes that can be sustained for each fractional value of the delay which lies between zero and one and two and three sampling periods are determined by the discrete describing function method. In this connection the critical regions developed by the authors for periods ranging from 9T to 20T have been found to be very useful. The different sots of initial conditions which give rise to the various modes are also established via the state transition technique. It is observed that an increase in the transportation lag usually results in an increase in the amplitude as well as the period of oscillation. Square-wave dither is found to produce a large steady-state error for a step inpu...

Optimal and linear sub-optimal control of second-order saturating control systems†

Consideration is given to the problem of replacing optimum non-linear control of saturating, single variable, second-order systems by sub-optimal linear control. The performance criteria adopted are integral-square-error and integral-modulus-error, and comparisons are made of the values of these integrals under the alternative forms of control. Known results for the double integrator plant are extended to an oscillatory plant (Bushaw'a plant) and to a plant comprising an integrator and a lag. Asymptotic properties of the optimal switching curves for these plants are studied incidentally This paper is a companion to a previous paper which considers the same problem for higher-order pure integrator plants.

Determination of terminal switch curve for time-optimal control of second-order systems via a suboptimal control method

The exact terminal switch curve for time-optimal control of a certain class of second-order systems is derived from a suboptimal control method for nonlinear autonomous systems. The suboptimal control method approximates the necessary conditions for optimal control of the nonlinear system and those of the approximating linear system. The linear approximant is found via the least-squares approximation.

Almost everywhere global asymptotic stability†

A new type of global stability property, called ‘almost everywhere global asymptotic (a.e.g.a.) stability) is introduced Roughly speaking, a dynamical system is said to be a.e.g.a. stable if only a finite number of its trajectories do not tend to the origin of the state space as time tends to infinity. As an application, second-order non-linear direct control systems with the feedback characteristic Φ(σ) lying partly outside the Hurwitz sector are considered. It is shown that a.e.g.a. stability is the strongest type of global stability that can possibly be achieved by such systems.

On the transient performance of a class of second-order limit-cycling control systems

An approach is described for the study of the transient performance of a class of second-order limit-cycling control systems. The geometrical features of the problem are used to derive expressions relating the limit-cycle dimensions with the excess overshoot. An example of application is studied using this approach.

A generalized stability criterion for a class of second-order non-linear control systems†

A condition for the existence of a limit cycle for a class of second-order non-linear control systems with a type zero linear plant is derived. The limit cycle stability is assured when F(a, b, α, β)/E curve has a positive slope at the point of intersection of F(a, b, α, β) and K loci. A typical second-order non-linear system is analysed for the purpose of illustrating the method of finding the amplitude and frequency of sustained oscillation. The method of studying the effects of variation of K, α and β on the amplitude and frequency of oscillation of the system is indicated. With regard to this, the proposed method has certain advantages over the usual method of analysis.

A predictive time-optimal controller for second-order systems with time delay

This paper presents a method of interactive or real-time predictive control which can be implemented without a stored program computer. By restricting the process dy namic model to a second-order-plus-dead-time format, a configuration which is satisfactory for many chemical or petroleum processing systems, a time-optimal controller can be built entirely on a small analog/hybrid computer. For set-point changes, the controller utilizes an analog model of the process to search out the predicted optimum input switching sequence and continuously maintains the process on the time-optimal trajectory. Because of the predictive nature of the method there is no necessity to include the time-delay term in the high-speed model; the predicted switches in the process input are simply ad vanced in time by the amount of the time-delay.

Suboptimal approach to the time-optimal control of second-order systems with complex poles

The time-optimal control of second-order systems with complex poles is well reported in technical literature. The switching boundary is complex and too difficult to implement. In this correspondence, new suboptimal systems are proposed and compared to the one available in [1], [2]. The suboptimal systems now proposed yield better response times, and the nonlinearities required to generate the switching boundaries are much simpler.

Game theoretic approach to minimum-effort control of second-order system†

In this paper fuel optimal control has been found for a second-order system. The system equation has an antagonist v. Controls u and v are both bounded. The performance criterion is minimized by u and by maximized by v. It is assumed that the system is brought from an ardbitrary initial state to the origin in bounded time 0 ≤ t ≤ T, where T is the upper bound on t. The state piano has been separated into different regions. For each initial state a specific sequence of strategy has been derived which is to be followed to bring the system to origin.

Time optimal control of second-order systems with transport lag†

The time optimal control of particular second–order systems with constant transport lags is determined by application of Pontryagin's Maximum Principle. A system with a delayed control and a system with a delayed state are considered. The optimal control for these systems is compared with the optimal control of a similar system which does not contain transport lag. It is found that the optimal control is essentially bang–bang but that it is non–unique under certain conditions.

A simpler stability criterion for a class of nonlinear systems

A simpler stability criterion is presented here that is useful for predicting limit cycles in a class of second-order non-linear control systems. The criterion is developed simply from a study of the roots of the system characteristic equation. It is shown that the limit cycle predicted by this criterion is stable if the slope of the a/b^{2} -locus at the intersection point is positive, while a negative slope indicates an unstable limit cycle.

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.

Neo-modern control theory: Linear and nonlinear servomechanism design

In designing and analyzing Type 1, second-order feedback control systems, a linear mode technique is the usual method of approach. Any adverse effects on system performance due to non-linearities are often explained away as the limiting condition in which the system can operate with any degree of predictability. It is the purpose of this paper to present a neomodern control theory with emphasis on a highly predictable technique for nonlinear analysis and design. Practical applications of this technique to the analysis of servo loops in the Central Air Data Computers of the F-111, C-141, and other aircraft present conclusive evidence of its usefulness and its importance to the state-of-the-art of nonlinear systems.

パラメータを操作量とした二次系の最小時間最適制御

Second order control systems having two system prameters as control inputs, x+2u3x+u2x=0, are analysed for the optimal control problems to minimize the time from an arbitrary initial point to the region on a circle around the origin of the state space.Applying Pontryagin's maximum principle to this problem, a unique solution is not necessarily obtained.This paper presents the conditions of the unique solution and a method of obtaining the optimal trajectory when the unique solution is unobtainable and in addition, using analogue computer, the importance of a boundary line named “Separatrix” is shown in getting the optimal trajectory, and it is pointed out that the switching lines obtained by Pontryagin's maximum principle correspond to the extremum of the functional.

Dynamic Characteristics of Second-Order Control System with Nonlinear Restoring Force

Dynamic Characteristics of Second-Order Control System with Nonlinear Restoring Force

Minimum-Fuel Control of Second-Order Systems with Real Poles

The controlled system has the transfer function G(s)=K/(s-? 1 )(s-? 2 ). The control input is u(t), |u(t)|?1, and the out put is y 1 (t). the control which forces any initial state y 1 (0), ? 1 (0) to the terminal state y 1 (T), ? 1 ,(T), such that -K/? 1 ? 2 ?y 1 (T)?K/? 1 ? 2 , ? 1 (T)=0, and which minimizes the fuel, F(T)=? 0 T |u(t)|dt is determined. The phase plane is divided into three sets G - , G + , and G 0 ; if the state is in G - then u(t)=?1 is used; if the state is in G + , then u(t)=+1 is used; if the state is in G 0 , then u(t)=0 is used.

Time-Optimal Control of Second Order Systems with Both Forcing Term and Parameter Variations

Time-Optimal Control of Second Order Systems with Both Forcing Term and Parameter Variations

Discussion on “Effect of a back-e.m.f. non-linearity on responses of a second-order position control system” and “Application of the second method of Lyapunov to the stability of certain position control systems containing a back-e.m.f. non-linearity”

Discussion on “Effect of a back-e.m.f. non-linearity on responses of a second-order position control system” and “Application of the second method of Lyapunov to the stability of certain position control systems containing a back-e.m.f. non-linearity”

Time optimal control of second order systems

Time optimal control of second order systems

Effect of a back-e.m.f. non-linearity on responses of a second-order position control system

In the linearized analysis of control systems containing a d.c. field-excited motor the armature current is assumed constant, allowing motor torque to be taken proportional to field current. Since, in practice, this introduces the complication of current forcing to overcome variations caused by motor back-e.m.f., the question arises of how much armature current forcing, or constancy, is necessary to give desirable system responses. The variation introduces a two-variable (field current, motor speed) non-linearity into the system equation. The paper investigates the effect of the non-linearity on the step function and ramp and frequency responses of an otherwise ideal output-velocity-stabilized second-order position control system. Increase of the non-linearity is initially similar to increased system damping until a heavily non-linear and undesirable type of response is reached; unstable ramp responses can occur. There is, however, a well defined boundary marking the onset of the latter responses; below it responses are comparable with linear, particularly for critical or heavy damping. The range of responses is investigated, together with armature-current behaviour during them. This, together with the boundaries, gives a specification of armature supply requirements for a desirable response. The effects of load viscous friction and field-current saturation are also investigated.