Closed Loop Subspace System Identification

Closed Loop Subspace System Identification - Proceedings of...

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Proceedings of the 36th TA05 1o:zo Closed Loop Subspace System Identification1 Conference on Decision & Control San Diego, Califomla USA December 1997 Peter Van Overschee2 & Bart De Moor3 Katholieke Universiteit Leuven, ESAT-SISTA Kardinaal Mercierlaan 94, 3001 Leuven (Heverlee) , Belgium, Tel+32/16/32 10 95 Fax +32/16/32 19 70, .. http://www.esat Abstract peter.vanoverschee@esat wk I \lc In this paper we present a general framework for closed loop subspace system identification. This framework consists of two new projection theo- rems which allow the extraction of non-steady state Kalman filter states and of system related matrices directly from input output data. Three algorithms for the identification of the state space matrices can be derived from these theorems. The similari- ties between the theorems and algorithms, and the corresponding open loop theorems and algorithms in the literature are remarking. 1 Introduction Subspace identification methods have been proven to be a valuable alternative for classical prediction error or instrumental variables methods [4]. Differ- ent breads of subspace identification methods have been published [2,6,10] of which a good overview is presented in [7, 111. All of these subspace methods have a number of nice properties in common: un- like classical methods, they do not suffer from the problems caused by a-priori parametrizations, ini- tial estimates and non-linear optimizations. They also identify multivariable systems in a very simple and elegant way. The trade-off is that the obtained models are sub-optimal, in the sense that they are not the optimal solution of an error-minimization 'This work is supported by the Interuniversity Poles of Attraction Programme (IUAP P4-02 (1997-2001): Model- ing, Identification, Simulation and Control of Complex Sys- tems; and IUAP P4-24 (1997-2001): Intelligent Mechatronic Systems (IMechS) ), the Human Capital and Mobility Net- work SIMONET (System Identification and Modeling Net- work), SCIENCE-ERNS1 (European Research Network for System Identification): SC1-CT92-0779. The scientific re- sponsibility is assumed by its authors. 'Senior research assistant of the FWO 3Senior research associate of the FWO P(z) I-+ t- -1 Figure 1: criterion. Standard feedback setup. We assume that uk and vk are measured and that a limited number of impulse response samples of the controller is given. In practice it is often necessary to perform identifi- cation experiments on systems operating in closed loop. This is especially true when open loop ex- periments are not allowed due to safety (unstable processes) or production (undesirable open loop be- havior) reasons. System identification from closed loop data is thus a relevant topic. All of the published subspace identification meth- ods however have one major drawback, they do not work when the data is gathered under closed loop experimental conditions as in Figure 1. Indeed, when there is feedback, the results of any of the algorithms [2, 6, 101 are asymptotically biased [5].
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Closed Loop Subspace System Identification - Proceedings of...

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