SYSTEMS Identification

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

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

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

4. General Aspects Let Zt as input-output data. A mathematical model for the system is a function from these data to the output at time t, y(t), in general 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. Models for Non-Linear Systems Topics to be covered include: • General Aspects • Black-box models • Choice of regressors and nonlinear function • Functions for a scalar regressor • Expansion into multiple regressors • Examples of “named” structures • Grey-box Models • Physical modeling • Semi-physical modeling • Block oriented models • Local linear models

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

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

8. Several Regressors Expansion into multiple regressors In the multi dimensional case (d>1), gk is a function of several variables:

9. Some non-linear model Examples of “named” structures

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. 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. Network of non-linear systems

13. Recurrent networks

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

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. Grey-box Models Physical modeling

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

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. Grey-box Models Linear regression Linear regression means that the prediction is linear in parameters The key is how to choose the function φi(ut,yt-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. 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. 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. Grey-box Models Local linear models To built the model, we need: 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. If the predicted corresponding to is linear in the parameters, the whole model will be a linear regression.