1. SYSTEMS Identification Ali Karimpour Assistant Professor Ferdowsi University of Mashhad Reference: “System Identification Theory For The User” Lennart Ljung(1999)

2. lecture 5 Ali Karimpour Nov 2010 2 Lecture 5 Models for Non-Linear Systems Topics to be covered include: v General Aspects v Black-box models Choice of regressors and nonlinear function Functions for a scalar regressor Expansion into multiple regressors Examples of “named” structures v Grey-box Models Physical modeling Semi-physical modeling Block oriented models Local linear models

3. lecture 5 Ali Karimpour Nov 2010 3 Models for Non-Linear Systems Topics to be covered include: v General Aspects v Black-box models Choice of regressors and nonlinear function Functions for a scalar regressor Expansion into multiple regressors Examples of “named” structures v Grey-box Models Physical modeling Semi-physical modeling Block oriented models Local linear models

4. lecture 5 Ali Karimpour Nov 2010 4 General Aspects A mathematical model for the system is a function from these data to the output at time t, y(t), in general Let Z t as input-output data. A parametric model structure is a parameterized family of such models: The difficulty is the enormous richness in possibilities of parameterizations. There are two main cases: Black-box models: General models of great flexibility Grey-box models: Some knowledge of the character of the actual system.

5. lecture 5 Ali Karimpour Nov 2010 5 Models for Non-Linear Systems Topics to be covered include: v General Aspects v Black-box models Choice of regressors and nonlinear function Functions for a scalar regressor Expansion into multiple regressors Examples of “named” structures v Grey-box Models Physical modeling Semi-physical modeling Block oriented models Local linear models

6. lecture 5 Ali Karimpour Nov 2010 6 Black-box models Let the output is scalar so: There are two main problems: 1.Choose the regression vector φ(t) A parametric model structure is a parameterized family of such models: 2. Choose the mapping g(φ,θ) Regression vector φ(t)ARX, ARMAX, OE, … For non-linear model it is common to use only measured (not predicted) ????? Choice of regressors and nonlinear function

7. lecture 5 Ali Karimpour Nov 2010 7 Black-box models There are two main problems: 1.Choose the regression vector φ(t) 2. Choose the mapping g(φ,θ) Basis functions Coordinates Scale or dilation Location parameter Global Basis Functions: Significant variation over the whole real axis. Local Basis Functions: Significant variation take place in local environment. Functions for a scalar regressor

8. lecture 5 Ali Karimpour Nov 2010 8 Several Regressors In the multi dimensional case (d>1), g k is a function of several variables: Expansion into multiple regressors

9. lecture 5 Ali Karimpour Nov 2010 9 Some non-linear model Examples of “named” structures

10. lecture 5 Ali Karimpour Nov 2010 10 Simulation and prediction Let The (one-step-ahead) predicted output is: A tougher test is to check how the model would behave in simulation i.e. only the input sequence u is used. The simulated output is: There are some important notations:

11. lecture 5 Ali Karimpour Nov 2010 11 Choose of regressors There are some important notations: Regressors in NFIR-models use past inputs Regressors in NARX-models use past inputs and outputs Regressors in NOE-models use past inputs and simulated outputs Regressors in NARMAX-models use past inputs and predicted outputs Regressors in NBJ-models use all four types.

12. lecture 5 Ali Karimpour Nov 2010 12 Network of non-linear systems

13. lecture 5 Ali Karimpour Nov 2010 13 Recurrent networks

14. lecture 5 Ali Karimpour Nov 2010 14 Models for Non-Linear Systems Topics to be covered include: v General Aspects v Black-box models Choice of regressors and nonlinear function Functions for a scalar regressor Expansion into multiple regressors Examples of “named” structures v Grey-box Models Physical modeling Semi-physical modeling Block oriented models Local linear models

15. lecture 5 Ali Karimpour Nov 2010 15 Grey-box Models Physical modeling Perform physical modeling and denote unknown physical parameters by θ So simulated (predicted) output is: The approach is conceptually simple, but could be very demanding in practice.

16. lecture 5 Ali Karimpour Nov 2010 Grey-box Models Physical modeling

17. lecture 5 Ali Karimpour Nov 2010 17 Grey-box Models Semi physical modeling First of all consider a linear model for system The model can not fit the system so: So we have: And also So we have Let x(t): Storage temperature Exercise1: Derive (I) Solar heated house

18. lecture 5 Ali Karimpour Nov 2010 18 Grey-box Models Block oriented models It is common situation that while the dynamics itself can be well described by a linear system, there are static nonlinearities at the input and/or output. Hammerstein Model: Wiener Model : Hammerstein Wiener Model : Other combination

19. lecture 5 Ali Karimpour Nov 2010 19 Grey-box Models Linear regression Linear regression means that the prediction is linear in parameters The key is how to choose the function φ i (u t,y t-1 ) GMDH-approach considers the regressors as typical polynomial combination of past inputs and outputs. For Hammerstein model we may choose For Wiener model we may choose Exercise2: Derive a linear regression form for equation (I) in solar heated house.

20. lecture 5 Ali Karimpour Nov 2010 20 Grey-box Models Local linear models Non-linear systems are often handled by linearization around a working point. Local linear models is to deal with the nonlinearities by selecting or averaging over some linearized model. Example: Tank with inflow u and outflow y and level h: Operating point at h * is: Linearized model around h * is:

21. lecture 5 Ali Karimpour Nov 2010 21 Grey-box Models Local linear models Sampled data around level h * leads to: Total model Let the measured working point variable be denoted by. If working point partitioned into d values, the predicted output will be:

22. lecture 5 Ali Karimpour Nov 2010 22 Grey-box Models Local linear models To built the model, we need: If the predicted corresponding to is linear in the parameters, the whole model will be a linear regression. It is also an example of a hybrid model. Sometimes the partition is to be estimated too, so the problem is considerably more difficult. Linear parameter varying (LPV) are also closely related.