Controller Identification in Closed Loop Based on Closed Loop Output Error Method
This method has been developed in order to identify the model of a controller operating in closed loop with a known plant model based on closed loop output error algorithm (see: controller identification scheme). It will be essentially used in order to reduce the controller order. The main advantage of this method with respect to the open loop type identification methods is that the modeling error (the difference between the nominal model and the reduced order model of the controller) is frequency weighted by the magnitude of two closed loop sensitivity functions. Therefore the modeling error is minimized in the critical frequency zones where the sensitivity functions are large. Hence, we expect that the robustness properties of the nominal controller will be saved also in the reduced order controller.
For this method a recursive parameter adaptation algorithm is considered as follows:
e°(t+1) = u(t+1)-qT(t)f(t)
F-1(t+1) = F-1(t)+lf(t)fT(t)
q(t+1) =q(t)+F(t+1)f(t)e°(t+1)
where e°(t+1) is the a priori closed loop prediction error, q(t) is the vector of the parameters, f(t) is the regressor vector containing the previous values of the input of the adjustable controller model and of the predictor output û(t). F(t) is the adaptation gain and l is the forgetting factor.
In order to have in the reduced order controller the fixed terms of the nominal controller (like an integrator) the filtered output of the controller by Hs/Hr is used in the algorithme instead of u(t), where Hr and Hs are respectively the fixed terms of the numerator and denominator of the controller. Hr and Hs are the polynomials in z-1, so a function named ctod may be used to convert a second order continuous time polynomial with a damping ratio and a natural frequency into a polynomial in z-1.
See: Controller order reduction module .
(file conid.htm)