Robustness of a model following PID control system

The paper deals with properties of a two-loop structure of control system containing the model of controlled plant and two PID controllers. Special attention is paid to high robustness of considered structure to perturbations of the controlled plant in relation to its nominal model and to good reduction of disturbances. On the basis of presented simulation results one can compare properties of the structure under consideration with properties of the classic control system structure with single feedback loop. The proposed model-following control structure (MFC) may find wide application in new intelligent controllers to robust control parameter-varying process plants. 1 Structure of the considered control system The structure of the proposed control systems shows Fig. 1. uR(s) um(s) ym(s) y(s) z(s) r(s) Rm(s) M(s) R(s) P(s) Figure 1. The structure of considered control system The plant P(s) is controlled by the same signal um(s) which controls model M(s). The um(s) is obtained from model controller Rm(s), like in [1, 2]. Due to existence of second loop containing controller R(s) the plant P(s) is also excited by additional signal uR(s). According to assumed control algorithm the additional control signal uR(s) is created from the difference between model ym(s) and plant y(s) output signals. In [3] the application of such a model following control structure to servomechanism control without any detailed analysis of its properties has been proposed. The most important feature of the presented structures is, in our opinion, they robustness meant as a satisfying quality of the control process ensured by the system independently of perturbations influencing the plant. In order to make analysis of system properties let us assume that linear and stable process is described by transfer function P(s) and its nominal model M(s) is known. The following equations result from Fig. 1 (s)M(s) R (s)M(s) R r(s) (s) y m m m + = 1 (1) ( ) ( )( ) R(s)P(s) z(s) (s)M(s) R R(s)P(s) R(s)M(s) (s)P(s) R r(s) y(s) m m + + + + + = 1 1 1 1 (2) R(s) ≠ Rm(s) As can be seen, for R(s)=Rm(s) the MFC system turns into the classic feedback control loop. The control system sensitivity to reference signal r(s) and disturbance z(s) can be determined from equation (2). It is pertinent to note that, if effects of disturbances may be neglected and if the loop controlling the model includes an integral element, the steady state error of the control system e(’ U ’ \ ’ is equal to 0 [4]. It results from Fig. 1 that signal y(s) follows ym(s). Thus, taking into account (1), the equation (2) can be expressed in a form R(s)P(s) z(s) R(s)P(s) R(s)M(s) M(s) P(s) (s) y y(s) m + + + + = 1 1 1 (3) Assuming that nominal process determined by the model M(s) is subject to perturbation û V , the transfer function of the perturbed process P(s) can be expressed in the following form [5, 6] ( ) ) ( 1 ) ( ) ( s û s M s P + = (4) Thus, the equation (3) can be expressed as ( ) ( ) ) ( 1 ) ( ) ( 1 1 ) ( ) ( 1 ) ( ) ( 1 ) ( 1 ) ( ) ( s s M s R s z s s M s R s s y s y m ∆ + + + +       ∆ + + ∆ + = (5) 2 Properties of the system It means that if the process P(s) is identical to its model M(s), ( û(s)=0 ) the plant output follows exactly the model output. In addition, the system sensitivity to disturbances can be influenced independently by an appropriate choice of the controller transfer function R(s). Furthermore, it can be shown that high controller gains ( |R(s)M(s)|>>1 ) guarantee very good follow up properties which are practically independent of process perturbations. The performance, robustness and stability of the proposed control system structure can be examined if the transfer function of the nominal plant M(s) (model) and that of controller R(s) are specified. Let the transfer functions of process and its nominal model have the following forms