President UniversityErwin SitompulSMI 10/1 Lecture 10 System Modeling and Identification Dr.-Ing. Erwin Sitompul President University

President UniversityErwin SitompulSMI 10/2 Homework 9 Chapter 6Identification from Step Response Time Percent Value Method Determine the approximation of the model in the last example, if after examining the t/t table, the model order is chosen to be 4 instead of 5.

President UniversityErwin SitompulSMI 10/3 t/τ Table 5 values of t i /τ are to be located for n = 4 Result: Solution to Homework 9 Chapter 6Identification from Step Response

President UniversityErwin SitompulSMI 10/4 Solution to Homework 9 Chapter 6Identification from Step Response : 5 th order approximation : 4 th order approximation

President UniversityErwin SitompulSMI 10/5 Least Squares Methods Chapter 6Least Squares Methods The Least Squares Methods are based on the minimization of squares of errors. The errors are defined as the difference between the measured value and the estimated value of the process output, or between y(k) and y(k). There are two version of the methods: batch version and recursive version. ^

President UniversityErwin SitompulSMI 10/6 Least Squares Methods Chapter 6Least Squares Methods Consider the discrete-time transfer function in the form of: The aim of Least Squares (LS) Methods is to identify the parameters a 1,..., a n, b 1,..., b m from the knowledge of process inputs u(k) and process output y(k). As described by the transfer function above, the relation of process inputs and process outputs is:

President UniversityErwin SitompulSMI 10/7 Least Squares Methods Chapter 6Least Squares Methods This relation can be written in matrix notation as: where: Vector of Parameters Vector of Measured Data Hence, the identification problem in this case is how to find θ based on the actual process output y(k) and the measured data from the past m(k).

President UniversityErwin SitompulSMI 10/8 Least Squares Methods Chapter 6Least Squares Methods Assuming that the measurement was done for k times, with the condition k ≥ n + m, then k equations can be constructed as: or:

President UniversityErwin SitompulSMI 10/9 If M is nonsingular, then the direct solution can be calculated as: In this method, error is minimized as a linear function of the parameter vector. The disadvantage of this solution is, that error can be abruptly larger for t > k. Least Squares Methods Chapter 6Least Squares Methods Least Error (LE) Method, Batch Version

President UniversityErwin SitompulSMI 10/10 Least Squares Methods Chapter 6Least Squares Methods A better way to calculate the parameter estimate θ is to find the parameter set that will minimize the sum of squares of errors between the measured outputs y(k) and the model outputs y(k) = m T (k)θ ^ The extreme of J with respect to θ is found when:

President UniversityErwin SitompulSMI 10/11 The derivation of J(θ) with respect to θ can be calculated as: Least Squares Methods Chapter 6Least Squares Methods if A symmetric Least Squares (LS) Method, Batch Version

President UniversityErwin SitompulSMI 10/12 Performing the “Second Derivative Test”, Least Squares Methods Chapter 6Least Squares Methods Second Derivative Test If f ’ (x) = 0 and f ” (x) > 0 then f has a local minimum at x If f ’ (x) = 0 and f ” (x) < 0 then f has a local maximum at x If f ’ (x) = 0 and f ” (x) = 0 then no conclusion can be drawn Always positive definite is a solution that will minimize the squares of errors

President UniversityErwin SitompulSMI 10/13 In order to guarantee that M T M is invertible, the number of row of M must be at least equal to the number of its column, which is again the number of parameters to be identified. More row of M increase the accuracy of the calculation. In other words, the number of data row does not have to be the same as the sum of the order of numerator and denominator of the model to be identified. If possible, rows with any value assumed to be zero (because no measurement data exist) should not be used. Least Squares Methods Chapter 6Least Squares Methods

President UniversityErwin SitompulSMI 10/14 The parameters of a model with the structure of: Example: Least Squares Methods Chapter 6Least Squares Methods are to be identified out of the following measurement data: Perform the batch version of the Least Squares Methods to find out a 1, a 2, and b 2. Hint: n + m =  At least 3 measurements must be available/ utilized. Hint: If possible, avoid to many zeros due to unavailable data for u(k) = 0 and y(k) = 0, k < 0.

President UniversityErwin SitompulSMI 10/15 Using the least allowable data, from k = 2 to k = 4, the matrices Y and M can be constructed as: Example: Least Squares Methods Chapter 6Least Squares Methods

President UniversityErwin SitompulSMI 10/16 Example: Least Squares Methods Chapter 6Least Squares Methods

President UniversityErwin SitompulSMI 10/17 Homework 10 Chapter 6Least Squares Methods Redo the example, utilizing as many data as possible. Does your result differ from the result given in the slide? What could be the reason for that? Which result is more accurate?

President UniversityErwin SitompulSMI 10/18 Homework 10A Chapter 6Least Squares Methods Redo the example, utilizing least allowable data, if the structure of the model is chosen to be After you found the three parameters a 1, a 2, and b 1, for G 2 (z), use Matlab/Simulink to calculate the response of both G 1 (z) and G 2 (z) if they are given the sequence of input as given before. Compare y(k) from Slide 10/15 with y 1 (k) and y 2 (k) from the outputs of the transfer functions G 1 (z) and G 2 (z). Give analysis and conclusions. Odd-numbered Student-ID Even-numbered Student-ID