1. President UniversityErwin SitompulSMI 9/1 Dr.-Ing. Erwin Sitompul President University Lecture 9 System Modeling and Identification http://zitompul.wordpress.com

2. President UniversityErwin SitompulSMI 9/2 Chapter 5Discrete-Time Process Models Homework 8 (a)Find the discrete-time transfer functions of the following continuous-time transfer function, for T s = 0.25 s and T s = 1 s. Use the Forward Difference Approximation (b)Calculate the step response of both transfer functions for 0 ≤ t ≤ 5 s. (c)Compare the step response of both transfer functions with the step response of the continuous-time transfer function G(s) in one plot.

3. President UniversityErwin SitompulSMI 9/3 Solution of Homework 8 (a) Chapter 5Discrete-Time Process Models

4. President UniversityErwin SitompulSMI 9/4 Chapter 5Discrete-Time Process Models Solution of Homework 8

5. President UniversityErwin SitompulSMI 9/5 (b)The step response of both transfer functions for 0 ≤ t ≤ 5 s. Chapter 5Discrete-Time Process Models Solution of Homework 8 Using the following command in Matlab workspace: Y1 = dlsim([0.625],[1 –1.5 1.125],ones(1,21)) Y1 = [0 0 0.6250 1.5625 2.2656 2.2656 1.4746 0.2881 –0.6018 –0.6018 0.3993 1.9010 3.0273 3.0273 1.7602 –0.1403 –1.5658 –1.5658 0.0378 2.4433 4.2473] Using the following command in Matlab workspace: Y2 = dlsim([10],[1 0 9],ones(1,6)) Y2 = [ 0 0 10 10 –80 –80 ]

6. President UniversityErwin SitompulSMI 9/6 Chapter 5Discrete-Time Process Models Solution of Homework 8 (c)Comparing the step responses FDA delivers bad results Possible solutions can be the use of smaller sampling time T s or the use of ZOH or TA T s = 0.25 sT s = 1 s

7. President UniversityErwin SitompulSMI 9/7 Solution of Homework 8 Chapter 5Discrete-Time Process Models FDA with smaller sampling time T s

8. President UniversityErwin SitompulSMI 9/8 Chapter 5Discrete-Time Process Models Solution of Homework 8 Using TA or ZOH, with reasonably large sampling time T s

9. President UniversityErwin SitompulSMI 9/9 Industry processes can be modeled in various ways, such as in state-space description or in transfer functions. The models mostly used for control purposes are in form of linear differential or difference equations, with parameters assumed as known and constant. In real conditions, it is often necessary to measure or estimate these parameters from input and output signals of the process. This case is referred to as parameter estimation or process identification. Chapter 6Process Identification

10. President UniversityErwin SitompulSMI 9/10 Chapter 6Process Identification The objective of process identification is to find a model that can describe the process. The information provided to do that is the inputs and the outputs of the process. independent, arbitrary, measurable, known dependent, measurable, known The ideal result of a process identification will be:

11. President UniversityErwin SitompulSMI 9/11 Identification Procedure Chapter 6Process Identification A general procedure in process identification includes: Determination of model structure → Based on mathematical origin or artificial intelligence Estimation of model parameter → Based on the chosen model structure Model verification → A model must be able to produce accurate output if “unseen” input data is given to it

12. President UniversityErwin SitompulSMI 9/12 Classification of Identification Methods Based on input signals Natural, generated during the process and measured Artificial, generated especially for the identification purpose Based on mathematics point of view Deterministic, assuming exact knowledge about process outputs, inputs, disturbance, etc, and do not consider random sources and influences Stochastic, assuming some properties and some knowledge of random disturbances, statistical approach Based on data processing Batch method, one calculation using the whole data at once, off-line Recursive method, gradual use of data, estimated parameters are improved from each experiment, can be on-line or off-line Chapter 6Process Identification

13. President UniversityErwin SitompulSMI 9/13 Identification from Step Response Chapter 6Process Identification The methods in this category aim to provide first estimate of the process and provide approximate information about the process gain, dominant time constant, and time delay. The input signal used to excite the process is a step change of the process input. It is necessary that the process is in a steady-state before the step change occurs. The measured step response needs to be normalized for unit step change and zero initial conditions.

14. President UniversityErwin SitompulSMI 9/14 “First Order + Time Delay” Approximation The approximation model for the identified process is given in s-Domain as: Chapter 6Identification from Step Response where K is the process gain, τ denotes time constant, and T d is the time delay. The step response of the transfer function G(s) given above in time domain is:

15. President UniversityErwin SitompulSMI 9/15 Chapter 6 Unit step response Approximation of unit step response First order + time delay If the step response is a normalized one, the process gain K is equal to the new steady-state output, K = y(∞). The actual unit step response and its approximation will always have two crossing points. Time constant τ and time delay T d can be calculated if the two crossing points are already chosen. The two crossing points should be chosen thoughtfully, to avoid large difference between the two step responses. “First Order + Time Delay” Approximation Identification from Step Response

16. President UniversityErwin SitompulSMI 9/16 Chapter 6 Unit step response Approximation of unit step response First order + time delay From two freely-chosen points (t 1,y 1 ) and (t 2,y 2 ), after some manipulations, we can also obtain τ and T d through calculations as follows: “First Order + Time Delay” Approximation Identification from Step Response

17. President UniversityErwin SitompulSMI 9/17 Chapter 6 “First Order + Time Delay” Approximation Advantage: Easy calculation, straightforward after two points are chosen Disadvantage: Low accuracy, the higher the process order, the lower the accuracy of the model Time delay will always present in the model Identification from Step Response

18. President UniversityErwin SitompulSMI 9/18 Time-Percent Value Method Chapter 6 The approximation model for the identified process is given in s-Domain as: From the unit step response, empirical values h ∞, t 10, t 30, t 50, t 70, and t 90 are obtained. Step response Identification from Step Response

19. President UniversityErwin SitompulSMI 9/19 Chapter 6 The values of parameters K, τ, and n are determined as follows: K is obtained from the steady-state value of the step response of the process divided by the magnitude of the input step. Using the “t/t Table”, up to 6 points of t i /t j can be located → the model order n can be determined. Using the “t/τ Table”, up to 5 points of t i /τ for the previously determined model order n can be located → the time constant τ can be determined. Time-Percent Value Method Identification from Step Response

20. President UniversityErwin SitompulSMI 9/20 Chapter 6 Time-Percent Value Method t/t Tablet/τ Table Identification from Step Response

21. President UniversityErwin SitompulSMI 9/21 Chapter 6 Example: Time-Percent Value Method A step function u(t) = 3(t) is fed in a process. As the step response, the following graph is obtained. Determine the approximate transfer function of the process by using the Time-Percent Value Method. Identification from Step Response

22. President UniversityErwin SitompulSMI 9/22 Example: Time-Percent Value Method Chapter 6Identification from Step Response

23. President UniversityErwin SitompulSMI 9/23 Example: Time-Percent Value Method Chapter 6 t/t Table From 6 t i /t j points, the most representative order for the model is 5 Identification from Step Response

24. President UniversityErwin SitompulSMI 9/24 t/τ Table 5 values of ti/τ can be located for n = 5 Result: Example: Time-Percent Value Method Chapter 6Identification from Step Response

25. President UniversityErwin SitompulSMI 9/25 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.

26. President UniversityErwin SitompulSMI 9/26 Homework 9 Chapter 6Identification from Step Response “First Order + Time Delay” Approximation Determine the approximation of the model in the last example, using the data from t 1 = 15 s and t 2 = 40 s. Print the graph and draw the response of your model on it. NEW