SYSTEMS Identification Ali Karimpour Assistant Professor Ferdowsi University of Mashhad Reference: “System Identification Theory For The User” Lennart Ljung

lecture 3 Ali Karimpour Nov Lecture 3 Simulation and prediction Topics to be covered include: v Simulation. v Prediction. v Observers.

lecture 3 Ali Karimpour Nov Simulation Suppose the system description is given by Let Undisturbed output is Let (by a computer) Disturbance is Output is Models used in, say, flight simulators or nuclear power station training simulators are of course more complex, but still follow the same general idea.

lecture 3 Ali Karimpour Nov Prediction (Invertibility of noise model) Disturbance is H must be stable so: Invertibility of noise model With

lecture 3 Ali Karimpour Nov Invertibility of noise model Lemma 3_1 Consider v(t) defined by Assume that filter H is stable and let: Assume that 1/H(z) is stable and: Define H -1 (q) by Then H -1 (q) is inverse of H(q) and Prediction (Invertibility of noise model)

lecture 3 Ali Karimpour Nov Proof Prediction (Invertibility of noise model)

lecture 3 Ali Karimpour Nov Prediction (Invertibility of noise model) Example 3.1 A moving average process Let That is This is a moving average of order 1, MA(1). Then If |c| < 1 then the inverse filter is determined as So e(t) is: Exercise(3E.1): Let Compute H -1 (q).

lecture 3 Ali Karimpour Nov Prediction (One-step-ahead prediction of v) Now we want to predict v(t) based on the pervious observation Now the knowledge of v(s), s ≤t-1 implies the knowledge of e(s), s ≤t-1 according to inevitability. Also we have Suppose that the PDF of e(t) be denoted by f e (x) so: Now we want to know f v (x) so:

lecture 3 Ali Karimpour Nov Prediction (One-step-ahead prediction of v) So the (posterior) probability density function of v(t), given observation up to time t-1, is 1- Maximum a posteriori prediction (MAP): Use the value for which PDF has its maximum. 2- Conditional expectation: Use the mean value of the distribution in question. We use mostly the blocked one. Exercise: Show that conditional expectation minimizes the mean square error of the prediction error. Exercise (3E.4)

lecture 3 Ali Karimpour Nov Prediction (One-step-ahead prediction of v) Conditional expectation Alternative formula Exercise(3T.1): Suppose that A(q) is inversely stable and monic. Show that A -1 (q) is monic.

lecture 3 Ali Karimpour Nov Prediction (One-step-ahead prediction of v) Example 3.2 A moving average process Let That is Conditional expectation Alternative formula

lecture 3 Ali Karimpour Nov Prediction (One-step-ahead prediction of v) Example 3.3 Let That is Conditional expectation Alternative formula

lecture 3 Ali Karimpour Nov Prediction (One-step-ahead prediction of y) Let Suppose v(s) is known for s ≤ t-1 and u(s) are known for s ≤ t. Since

lecture 3 Ali Karimpour Nov Prediction (One-step-ahead prediction of y) Unknown initial condition Since only data over the interval [0, t-1] exist so The exact prediction involves time-varying filter coefficients and can be computed using the Kalman filter. The prediction error So the variable e(t) is the part of y(t) that can not be predicted from past data. It is also called innovation at time t.

lecture 3 Ali Karimpour Nov Prediction (k-step-ahead prediction of y) k-step-ahead predictor of v First of all we need k-step-ahead prediction of v Known at t Unknown at t

lecture 3 Ali Karimpour Nov Prediction (k-step-ahead prediction of y) k-step-ahead prediction of v is: Suppose we have measured y(s) for s≤ t and u(s) is known for s≤ t+k-1. So let

lecture 3 Ali Karimpour Nov Prediction (k-step-ahead prediction of y) k-step-ahead prediction of y is: Define Exercise: Show that the k-step-ahead prediction of Can also viewed as a one-step-ahead predictor associated with the model:

lecture 3 Ali Karimpour Nov Prediction (k-step-ahead prediction of y) Define prediction error of k-step-ahead prediction as: Exercise: Show that prediction error of k-step-ahead prediction is a moving average of e(t+k), …,e(t+1) Exercise(3E.2): Determine the 3 -step-ahead prediction for and respectively. What are the variance of the associated prediction error.

lecture 3 Ali Karimpour Nov Observer In many cases we ignore noises, so deterministic model is used This description used for “computing,” “guessing,” or “predicting”. So we need the concept of observer. As an example let: This means that So we have

lecture 3 Ali Karimpour Nov Observer So we have If input output data are lacking prior to time t = 0, first one suffers from an error, but second one still is correct for t > 0. In the other hand first one is un affected by measurement errors in the output, but second one affected. So the choice of predictor could be seen as a trade-off between sensitivity with respect to output measurement errors and rapidly decaying effects of erroneous initial conditions. Exercise (3E.3): Show that if Then for the noise model H(q)=1, (I) is the natural predictor, whereas the noise model Leads to the predictor (II)

lecture 3 Ali Karimpour Nov Observer A family of predictor for So the choice of predictor could be seen as a trade-off between sensitivity with respect to output measurement errors and rapidly decaying effects of erroneous initial conditions. To introduce design variables for this trade-off, choose a filter W(q) such that Applying it to both sides we have Which means that The right hand side of this expression depends only on y(s), s≤t-k, and u(s). s ≤t-1. So

lecture 3 Ali Karimpour Nov Observer A family of predictor for The trade-off considerations for the choice of W could then be 1. Select W(q) so that both W and WG have rapidly decaying filter coefficients in order to minimize the influence of erroneous initial conditions. 2. Select W(q) so that measurement imperfections in y(t) are maximally attenuated. The later issue can be shown in frequency domain. Suppose that The prediction error is:

lecture 3 Ali Karimpour Nov Observer A family of predictor for The prediction error is: The spectrum of this error is, according to Theorem 2.2: The problem is thus to select W, such that the error spectrum has an acceptable size and suitable shape.

lecture 3 Ali Karimpour Nov Observer Fundamental role of the predictor filter Let Then y predicted as: They are linear filters since: Linear Filter