Singular perturbation analysis of cheap control problem for sampled data systems

2209 Fig. 1. Bode plot of the true system and mean errors. The reference signal r and the noise e 0 were chosen as independent, zero mean, Gaussian white noise signals, with variances 1 (=8r(!)) and 0.01 (= 0), respectively. A Monte Carlo simulation consisting of 1024 different runs was performed. In each run we generated N = 1024 data points and identified the system directly using second-order ARX and output error models and indirectly using a second-order model of the kind (35) where we fixed H(q;) = 1. For the direct method with an output error method we can expect biased results since for this model H(q;) = 1 6 = H 0 (q) [cf. (23)]. For the ARX model, on the other hand, there should ideally be no bias error since the chosen model structure coincides with the one used to generate the data. In Fig. 1 we plotted the true system together with the mean errors in the estimated models. The results for the ARX and the indirect cases are similar. The differences are likely to be due to numerical problems (poor initial conditions, problems with local minima, etc.) in the estimation routine used for the indirect method. These problems are not present in the ARX case since there the prediction error estimate is found without iterations by solving a standard least squares problem. In Fig. 1 we also included a plot of the theoretical bias error according to (22) for the output error model. As can be seen from the figure, the obtained bias error is close to the theoretical value. VII. CONCLUSION By studying the bias error due to feedback in the estimated transfer functions when using the direct method we have obtained a nonstandard motivation for the indirect approach to closed-loop identification. This method gives consistency regardless of the noise color but requires perfect knowledge of the regulator and gives suboptimal accuracy. REFERENCES [1] M. Gevers, " Toward a joint design of identification and control, " in Iterative weighted least-squares identification and weighted LQG control design, " Automatica, Abstract—This paper studies the discrete-time cheap control problem for sampled data systems using the theory of singular perturbations. It is shown, by using the two timescale property of singularly perturbed systems, that the problem can be solved in terms of two reduced-order subproblems for which computations can be done in parallel, thus increasing the computational speed. …