The behaviour of control systems with definite time delay and random disturbance

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Methods are now well established for analysing the behaviour of closed-sequence control systems in which the relations between the quantities may be represented by differential equations, the inputs being known functions of time.The paper is concerned with the methods of analysis when these conditions are not met, the system having definite time delays or random disturbances or a combination of both. This occurs, in particular, in all manually operated controls, because there is an unavoidable time delay of relatively large duration in all human muscular response, and the behaviour of even a skilled operator is never completely consistent. The variability of response can usually be considered to be equivalent to an additional input of a random nature, corresponding to the operator's inability correctly to assess the situation and to apply the signals corresponding to his assessment. Analogous difficulties arise in process controls and in many servo systems.It is shown how the behaviour of a system with a definite time delay and a random disturbance may be calculated and represented in non-dimensional form for cases in which the remaining part of the transfer function has certain simple but rather general forms. Results are given and compared for two alternative methods, namely the solution of the differential equations and the solution of finite-difference equations which represent approximately the same relations. It is shown that a finite-difference representation using a time interval equal to the finite time delay gives tolerably good results and requires much less labour.Charts are given showing the extent of amplification of the random disturbance.More specifically, the paper presents, in convenient non-dimensional form, a complete picture of the performance of any control system having an open-loop transfer function of the form (1+T1p)(1+T2p)ε−Tp/P2 with a random disturbance. This includes, as special cases, manual controls either of the rate, position or aided-lay varieties, certain types of process control, and, as a first approximation, a considerable range of servo-mechanism systems, since the definite delay term ε−Tp may be considered as an approximation to the effect of a number of small time delays in cascade.A further form of loop transfer function to which some consideration is given is [(1+T1p)(1+T2p)ε−Tp]/[(1+T3p)(1+T4p)P2], which applies to an even wider range of servo mechanisms.