1. 1 Chapter 2 1. Parametric Models

2. 2 Parametric Models The first step in the design of online parameter identification (PI) algorithms is to lump the unknown parameters in a vector and separate them from known signals, transfer functions, and other known parameters in an equation that is convenient for parameter estimation. In the general case, this class of parameterizations is of the form where is the vector with all the unknown parameters and are signals available for measurement. We refer it as the linear "static "parametric model (SPM).

3. 3 Parametric Models The SPM may represent a dynamic, static, linear, or nonlinear system. Example: where x, u are the scalar state and input, respectively, and a, b are the unknown constants we want to identify online using the measurements of x, u.

4. 4 Parametric Models

5. 5 Another parameterization of the above scalar plant is In the general case, the above parametric model is of the form

6. 6 Parametric Models Where are signals available for measurement and is a known stable proper transfer function, where q is either the shift operator in discrete time (i.e., q = z) or the differential operator (q = s) in continuous time. We refer to this model as the linear "dynamic"parametric model (DPM). The importance of the SPM and DPM is that the unknown parameter vector appears linearly. So we refer to SPM and DPM as linear in the parameters parameterizations.

7. 7 Parametric Models We can derive SPM from DPM if we use the fact that is a constant vector and redefine to obtain In a similar manner, we can filter each side of SPM and DPM using a stable proper filter and still maintain the linear in the parameters property and the form of SPM, DPM. This shows that there exist an infinite number of different parametric models in the form of SPM, DPM for the same parameter vector.

8. 8 Parametric Models In some cases, the unknown parameters cannot be expressed in the form of the linear in the parameters models. In such cases the PI algorithms based on such models cannot be shown to converge globally. A special case of nonlinear in the parameters models for which convergence results exist is when the unknown parameters appear in the special bilinear form bilinear static parametric model (B-SPM) or bilinear dynamic parametric model (B-DPM)

9. 9 Parametric Models bilinear static parametric model (B-SPM) bilinear dynamic parametric model (B-DPM) where are signals available for measurement at each time t, and are the unknown parameters. The transfer function is a known stable transfer function.

10. 10 Parametric Models state-space parametric models (SSPM) In some applications of parameter identification or adaptive control of plants of the form whose state x is available for measurement, the following parametric model may be used: where is a stable design matrix; are the unknown matrices; and are signal vectors available for measurement. The model may be also expressed in the form

11. 11 Parametric Models state-space parametric models (SSPM) It is clear that the SSPM can be expressed in the form of the DPM and SPM.

12. 12 Parametric Models bilinear state-space parametric models (B-SSPM). Another class of state-space models that appear in adaptive control is of the form where B is also unknown but is positive definite, is negative definite, or the sign of each of its elements is known. The B-SSPM model can be easily expressed as a set of scalar B-SPM or B-DPM.

13. 13 Parametric Models PI Problem For the SPM and DPM: Given the measurements, generate, the estimate of the unknown vector, at each time t. The PI algorithm updates with time so that approaches or converges to. Since we are dealing with online PI, we would also expect that if changes, then the PI algorithm will react to such changes and update the estimate to match the new value of. PI

14. 14 Parametric Models PI Problem For the B-SPM and B-DPM: Given the measurements generate estimates respectively, at each time t the same way as in the case of SPM and DPM. PI

15. 15 Parametric Models PI Problem For the SSPM:: Given the measurements generate estimates of, (and hence the estimates, respectively)at each time t the same way as in the case of SPM and DPM. PI

16. 16 Parametric Models PI Problem The online PI algorithms generate estimates at each time t, by using the past and current measurements of signals. Convergence is achieved asymptotically as time evolves. For this reason they are referred to as recursive PI algorithms to be distinguished from the non-recursive ones, in which all the measurements are collected a priori over large intervals of time and are processed offline to generate the estimates of the unknown parameters.

17. 17 Parametric Models Example 1: Mass-Spring-Dashpot System Let us assume that M, f, k are the constant unknown parameters that we want to estimate online. express in the form of SPM

18. 18 Parametric Models Example 1: Mass-Spring-Dashpot System Measurements: To avoid of derivatives, we filter both sides with the stable filter

19. 19 Parametric Models Example 1: Mass-Spring-Dashpot System Another possible parametric model is:

20. 20 Parametric Models Example 2: Cart with two inverted pendulums

21. 21 Parametric Models Example 2: Cart with two inverted pendulums

22. 22 Parametric Models Example 2: Cart with two inverted pendulums To avoid of derivatives, we filter both sides with the stable filter,

23. 23 Parametric Models Example 2: Cart with two inverted pendulums If in the above model we know that is nonzero, redefining the constant parameters as we obtain the following B-SPM: where,

24. 24 Parametric Models Example 3: second-order autoregressive moving average (ARMA) model This model can be rewritten in the form of the SPM as: where,

25. 25 Parametric Models Example 3: second-order autoregressive moving average (ARMA) model If one of the constant parameters, i.e.,, is nonzero. Then we can obtain a model of the system in the B-SPM form as follows: where,

26. 26 Parametric Models Example 4: nonlinear system Filtering both sides of the equation with the filter, we can express the system in the form of the SPM where,

27. 27 Parametric Models Example 5: dynamical system in the transfer function form where the parameter b is known and a, c are the unknown parameters. Rewrite it as: where, divide to

28. 28 Parametric Models Example 6: DPM model If we want W(s) to be a design transfer function with a pole, say at we write:

29. 29 Parametric Models Example 6: DPM model

30. 30 Parametric Models Example 7: SSPM model where,

31. 31 Parametric Models Example 8: n-th order-SISO LTI system where,

32. 32 Parametric Models Example 8: n-th order-SISO LTI system Filtering by where,

33. 33 Exercises From reference 1, chapter 2, Choose 5 problems from 8 problems.

34. 34 END