Model-free subspace-based LQG-design

Model-free subspace-based LQG-design - Proceedings of the...

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Proceedings of the American Control Conference San Diego, California June 1999 Model-free subspace-based LQG-design W. Favoreell, B. De Moor2, P. Van Overschee3 Katholieke Universiteit Leuven, Department of Electrical Engineering - ESAT/SISTA, Kardinaal Mercierlaan 94, B-3001 Leuven, BELGIUM M. Gevers4 Universit6 Catholique de Louvain, Centre for Systems Engineering and Applied Mechanics - CESAME Biitiment Euler 4, avenue Georges Lemaitre, B-1348 Louvain-la-Neuve, BELGIUM. Abstract When only input/output data of an unknown sys- tem are available, the classical way to design a linear quadratic Gaussian controller for that system mainly consists of three separate parts. First a system identifi- cation step is performed to find the system parameters. With these parameters a Kalman filter is designed to find an estimate of the state of the system. Finally, this state is then used in an LQ-controller. In the literature these three steps are hardly ever considered as one joint problem. Based on techniques from the field of sub- space system identification the present paper gives a new, much more direct method to calculate a finite- horizon LQG-controller. The three steps of the LQG- controller design, i.e. system identification, Kalman fil- ter and LQ-control design are replaced by a QR- and a SV-decomposition. The equivalence between the new subspacebased approach and the classical approach is proven. 1 Introduction A lot of the modern control methods, such LQG control, use state space models to design a controller. When such a model is not available, the LQG-design first requires a system identification from the available input/output data. In addition, once a state space model is available, the state of the system is estimated 'W. Favoreel is a Research Assistant supported by the I.W.T. e-mail:, http://www.esat.-, Tel.: 32/16/321808, Fax: 32/16/321970 'B. De Moor is a Research Associate of the Fund for Scientific Research (F. W .O.-Vlaanderen) . 3P. Van Overschee is a senior Research Assistant of the Na- tional Fund for Scientific Research (F.W.0.-Vlaanderen) 4M. Gevers is Professor and Chairman of CESAME (Cen- tre for Systems Engineering and Applied Mechanics) Universit6 Catholique de Louvain , with a Kalman filter. Finally, an LQ-controller has to be designed to feed back the estimated state. In this paper we show that, based on techniques from the field of subspace identification, one can bypass these three steps and find an LQG-controller directly from input/output data of the unknown system. We will be interested in the LQG-controller design for linear time-invariant systems that can be described by the following state space equations: Xk+l = AXk -k BUk Kek, (1) Yk = CXk DUk ek (2) where the input Uk E Rm, the output Yk E R', the state xk E Rn and the stationary, ergodic, white noise E R' with the following covariance matrix: E[e,eT] = Sa,. The available input/output data set is supposed to be measured in open-loop. The main problem we are in- terested in is the following: Model-free LQG-control Droblem:
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Model-free subspace-based LQG-design - Proceedings of the...

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