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

2. lecture 1 Ali Karimpour Nov 2009 2 Lecture 1 Introduction Topics to be covered include: v Dynamic systems. v Models. v An Archetypical problem-ARX models and LSM. v The system identification procedure.

3. lecture 1 Ali Karimpour Nov 2009 3 Dynamic systems System: An object in which variables of different kinds interact and produce observable signals. Stimuli: External signals that affects system. Dynamic System: A system that the current output value depends not only on the current external stimuli but also on their earlier value. Time series: A dynamic system whose external stimuli are not observed.

4. lecture 1 Ali Karimpour Nov 2009 4 Dynamic systems Stimuli It can be manipulated by the observer. Input Disturbance It can not be manipulated by the observer. Measured Unmeasured Dynamic system Input u Measured disturbance w Unmeasured disturbance v Output y

5. lecture 1 Ali Karimpour Nov 2009 5 A solar heated house Dynamic system Pump velocity u Solar radiation w Wind, outdoor temperature v Storage temperature y

6. lecture 1 Ali Karimpour Nov 2009 6 Speech generation Dynamic system chord, vibaration airflow v Sound y Time series: A dynamic system whose external stimuli are not observed.

7. lecture 1 Ali Karimpour Nov 2009 7 Models Model: Relationship among observed signals. Model types 1- Mental models 2- Graphical models 3- Mathematical (analytical) models 4- Software models Split up system into subsystems, Joined subsystems mathematically, 1- Modeling 2- System identification Does not necessarily involve any experimentation on the actual system. Building models It is directly based on experimentation. Input and output signals from the system are recorded. 3- Combined

8. lecture 1 Ali Karimpour Nov 2009 8 The fiction of a true model

9. lecture 1 Ali Karimpour Nov 2009 9 A Basic problem and the linear least squares method Perhaps the most basic relationship between the input and output is the linear difference equation Where

10. lecture 1 Ali Karimpour Nov 2009 10 Least Square Method Suppose that we don’t know the value of parameters θ, but the recorded input and output over a time interval 1 ≤ t ≤ N is: Now define:

11. lecture 1 Ali Karimpour Nov 2009 11 First order difference equation Consider the simple model Now we have Then

12. lecture 1 Ali Karimpour Nov 2009 12 First order difference equation Consider the simple model Exercise: Suppose for t=1 to 6 the value of u and y are: Derive

13. lecture 1 Ali Karimpour Nov 2009 13 Where An Archetypical problem of system identification

14. lecture 1 Ali Karimpour Nov 2009 14 Linear Regression Model structures such as That are linear in θ are known in statistics as linear regressions. The vector φ(t) is called the regression vector and its components are the regressors. In this models y(t) explained with the regression vector φ(t) –which contains older value of y(t)- so it is called Auto-Regression with eXtera inputs. ARX-models Auto Extra

15. lecture 1 Ali Karimpour Nov 2009 15 Model quality and experimental design Consider a finite impulse response (FIR) Observed data generate by e(t) is a white noise sequence with variance λ

16. lecture 1 Ali Karimpour Nov 2009 16 Model quality and experimental design deterministic stochastic The estimate is consequently unbiased. The covariance matrix of the parameter error is: Exercise1 : Derive P N. Exercise2 : Let show that

17. lecture 1 Ali Karimpour Nov 2009 17 Model quality and experimental design

18. lecture 1 Ali Karimpour Nov 2009 18 The system identification procedure The construction of a model from data involves three basic identities 2- The set of candidate models; a model structure. 3- A rule by which candidate models can be assessed using the data, like the least square selection rule. 1- A data set, like Z N. The data must be maximally informative, subject to constraints. The most important and at the same time, the most difficult choice. Model sets with adjustable parameters with physical interpretation may be called gray boxes. Models whose parameters do not reflect physical consideration may be called black boxes.

19. lecture 1 Ali Karimpour Nov 2009 19 2- A model structure. 3- Set the parameters. 1- A data set, like Z N. Model validation We arrived at a particular model. Whether the model is good enough? Whether it is valid for its purpose? Model validation Note: A model can never be accepted as a final and true description of the system. Rather, it can at best be regarded as a good enough description of certain aspects that are particular interest to us.

20. lecture 1 Ali Karimpour Nov 2009 20 Experimental Design Data Choose Model set Choose Criteria to fit Calculate Model Validate Model Prior Knowledge Ok : use it Not Ok : Revise The system identification loop The model may be deficient for variety of reasons: The numerical procedure failed to find the best model according to our criterion. The criterion was not well chosen. The model set was not appropriate. The data set was not informative enough.