Some Classical and Some New Ideas for Identification of Linear Systems
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Some Classical and Some New Ideas for Identification of Linear Systems Lennart Ljung
Received: 6 December 2012 / Revised: 31 January 2013 / Accepted: 4 February 2013 / Published online: 2 March 2013 © Brazilian Society for Automatics–SBA 2013
Abstract This paper gives an overview of the identification of linear systems. It covers the classical approach of parametric methods by means of maximum likelihood and predicion error methods, as well all classical non-parametric methods through spectral analysis. It also covers very recent techniques dealing with convex formulations by regularization of FIR and ARX models, as well as new alternatives to spectral analysis, through local linear models. An example of identification of aircraft dynamics illustrates the approaches.
at the same time by u k (t), k = 1, 2, 3. Then assume that we can write
Keywords System identification · Maximimum likelihood · Prediction error methods · Spectral analysis · Regularization · Local polynomial methods
In this simple relationship, we can adjust the parameters to fit the observed data as well as possible by a common least squares fit. We use only the 90 first data points of the observed data. That gives certain numerical values of the nine parameters above:
1 Introduction 1.1 An Introductory Example: Aircraft Dynamics Consider a physical system, with observed input and output signals (see Fig. 1). Let us take a modern military aircraft, like the Swedish fighter Gripen, as an example. From one of the earlier test flights, some data were recorded as depicted in Fig. 2. In order to be able to simulate the aircaft and to design an effective autopilot, it is necessary to understand how, in this case, the pich rate is affected by the three inputs. We need mathematical expressions for this. A fair amount of knowledge exists about aircraft dynamics, but let us just try a simple difference equation relation. Denote the output, the pitch rate, at sample number t by y(t) and three control inputs L. Ljung (B) Division of Automatic Control, Department of Electrical Engineering, Linköping University, Linköping, SE581 83, Sweden e-mail: [email protected]
y(t) = a1 y(t − 1) − a2 y(t − 2) − a3 y(t − 3) + b1,1 u 1 (t − 1) + b1,2 u 1 (t − 2) + b2,1 u 2 (t − 1) + b2,2 u 2 (t − 2) + b3,1 u 3 (t − 1) + b3,2 u 3 (t − 2)
(1)
y(t) − 1.15y(t − 1) + 0.50y(t − 2) − 0.35y(t − 3) = −0.54u 1 (t − 1) + 0.04u 1 (t − 2) + 0.15u 2 (t − 1) + 0.16u 2 (t − 2) + 0.16u 3 (t − 1) + 0.07u 3 (t − 2)
(2)
We may note that this model is unstable—it has a pole in 1.0026, but that is in order, because the pitch channel is unstable at the velocity and altitude in question. How can we test if this model is OK? Since we used only half of the observed data for the estimation, we can test the model on the whole data record. Since the model is unstable and thus simulation is difficult, it is natural to let the model predict future outputs, say five samples ahead, and compare with the measured outputs. That is done in Fig. 3. We see that the simple model (2) provides quite reasonable
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