Comprehensive Guide to Model Reference Adaptive Control

1. Chapter 4 Model Reference Adaptive Control

2. Table of Contents Introduction Simple MRAC Schemes MRC for SISO Plants Direct MRAC Direct MRAC Indirect MRAC Robust MRAC Case Study

3. Introduction In this chapter, we design and analyze a wide class of adaptive control schemes based on model reference control (MRC) referred to as model reference adaptive control (MRAC). In MRC, the desired plant behavior is described by a reference model and is driven by a reference input. The control law is then developed so that the closed-loop plant has a transfer function equal to .This transfer function matching guarantees that the plant will behave like the reference model for any reference input signal.

4. Introduction Model reference control

5. Introduction Goal: This goal, implies that to satisfy certain assumptions. These assumptions enable the calculation of the controller parameter vector as The above goal guarantees that the tracking error converges to zero for any given reference input signal r. If is known, then can be calculated using and the controller can be implemented.

6. Introduction When is unknown the use of certainty equivalence (CE) approach, where the unknown parameters are replaced with their estimates, leads to the adaptive control scheme referred to as indirect MRAC.

7. Introduction Another way of designing MRAC schemes is to parameterize the plant transfer function in terms of the desired controller parameter vector . The structure of the MRC law is such that we can write In this case, the controller parameter is updated directly without any intermediate calculations, and for this reason the scheme is called direct MRAC.

8. Introduction Direct MRAC

9. Introduction Various classes of MRAC

10. Simple MRAC SchemesScalar Example: Adaptive Regulation Consider the scalar plant The control objective is to determine a bounded function such that the state is bounded and converges to zero for any . A possible procedure to follow in the unknown parameter case is to use the same control law but with k* replaced by its estimate k(t) i.e., we use and search for an adaptive law to update k(t).

11. Simple MRAC SchemesScalar Example: Adaptive Regulation

12. Simple MRAC SchemesScalar Example: Adaptive Regulation Barbalat's lemma

13. Simple MRAC SchemesScalar Example: Adaptive Regulation We have shown that the combination of the control law with the adaptive law meets the control objective in the sense that it guarantees boundedness for all signals and forces the plant state to converge to zero. It is worth mentioning that, as in the parameter identification problems, we cannot establish that k(t)converges to k*. The lack of parameter convergence is less crucial in adaptive control than in PI, because in most cases the control objective can be achieved without requiring the parameters to converge to their true values.

14. Simple MRAC SchemesScalar Example: Direct MRAC without Normalization Consider the following first-order plant: where a, b are unknown parameters but the sign of b is known. The control objective is to choose an appropriate control law u such that all signals in the closed-loop plant are bounded and x tracks the state of the reference model given by or We propose the control law:

15. Simple MRAC SchemesScalar Example: Direct MRAC without Normalization are calculated so that the closed-loop transfer function from r to x is equal to that of the reference model, i.e., plant is controllable parameters a, b are unknown control law

16. Simple MRAC SchemesScalar Example: Direct MRAC without Normalization where are the estimates of , respectively, and search for an adaptive law to generate .

17. Simple MRAC SchemesScalar Example: Direct MRAC without Normalization where are the parameter errors. adaptive laws

18. Simple MRAC SchemesScalar Example: Direct MRAC without Normalization

19. Simple MRAC SchemesScalar Example: Indirect MRAC without Normalization Consider the same problem in last example where are generated by an adaptive law that we design. SSPM

20. Simple MRAC SchemesScalar Example: Indirect MRAC without Normalization

21. Simple MRAC SchemesScalar Example: Indirect MRAC without Normalization The boundedness of depend on and then . The requirement that be bounded away from zero is a controllability condition for the estimated plant. One method for preventing from going through zero is to modify the adaptive law as below. Such a modification is achieved using the following a priori knowledge:

22. Simple MRAC SchemesScalar Example: Indirect MRAC without Normalization Using the same arguments If the reference input signal is sufficiently rich of order 2, then and therefore converge to zero exponentially fast.

23. Simple MRAC SchemesScalar Example: Direct MRAC with Normalization Consider the same first-order plant: Control law: or as before or :Reference model B-SPM

24. Simple MRAC SchemesScalar Example: Direct MRAC with Normalization Using the PI techniques, the adaptive law is given by normalizing signal: Independent of the boundedness of , the above adaptive law guarantees that:

25. Simple MRAC SchemesScalar Example: Direct MRAC with Normalization We can use properties (i)-(ii) of the adaptive law to first establish signal boundedness and then convergence of the tracking error e to zero. It can be follow in ref.

26. Simple MRAC SchemesScalar Example: Indirect MRAC with Normalization Consider the same first-order plant: SPM the gradient algorithm As last, the above adaptive law guarantees that

27. Simple MRAC SchemesScalar Example: Indirect MRAC with Normalization Due to division by , the gradient algorithm has to guarantee that does not become equal to zero. Therefore, instead of above adaptive law we use

28. Simple MRAC SchemesScalar Example: Indirect MRAC with Normalization As shown in last chapter, the above adaptive law guarantees that By applying some lemma and theorem, it can be conclude that:

29. Simple MRAC SchemesVector Case: Full-State Measurement Consider the nth-order plant where are unknown matrices and is controllable. The control objective is to choose the input vector such that all signals in the closed-loop plant are bounded and the plant state follows the state of a reference model: where is a stable matrix, , and is a bounded reference input vector. The reference model and input r are chosen so that represents a desired trajectory that has to follow.

30. Simple MRAC SchemesVector Case: Full-State Measurement Control Law: If the matrices A, B were known, we could apply the control law closed-loop plant Comparison with (*) Matching condition

31. Simple MRAC SchemesVector Case: Full-State Measurement In general, no may exist to satisfy the matching condition (*), indicating that the above control law may not have enough structural flexibility to meet the control objective. In some cases, if the structure of is known, , may be designed so that (*) has a solution for . Let us assume that in (*) exist, and propose the following control law to be generated by an appropriate adaptive law.

32. Simple MRAC SchemesVector Case: Full-State Measurement ±

33. Simple MRAC SchemesVector Case: Full-State Measurement It depends on the unknown matrix B. In the scalar case we manage to get away with the unknown B by assuming that its sign is known. An extension of the scalar assumption to the vector case is as follows: Let us assume that L* is either positive definite or negative definite and where if L* is positive definite and if L* is negative definite. Then and above error dynamic becomes

34. Simple MRAC SchemesVector Case: Full-State Measurement We propose the following Lyapunov function candidate: where satisfies the Lyapunov equation Then are bounded and Note that The assumption may not be realistic.

35. MRC for SISO Plants • In the general case, the design of the control law is not as straightforward as it appears in last examples. • At firs we formulate the MRC problem for a general class of LTI SISO plants and solve it for the case where the plant parameters are known exactly. • The significance of the existence of a control law that solves the MRC problem is twofold: • It demonstrates that given a set of assumptions about the plant and reference model, there is enough structural flexibility to meet the control objective • It provides the form of the control law that is to be combined with an adaptive law to form MRAC schemes in the case of unknown plant parameters.

36. MRC for SISO Plants Problem Statement Consider the SISO LTI plant where are monic polynomials and is a constant referred to as the high-frequency gain. The reference model, selected by the designer to describe the desired characteristics of the plant, is described by: where are monic polynomials and is a constant.

37. MRC for SISO Plants Problem Statement The MRC objective is to determine the plant input so that all signals are bounded and the plant output, , tracks the reference model output as close as possible for any given reference input . We refer to the problem of finding the desired , to meet the control objective as the MRC problem. In order to meet the MRC objective with a control law that is free of differentiators and uses only measurable signals, we assume that the plant and reference models satisfy the following assumptions.

38. MRC for SISO Plants Plant assumptions: Reference model assumptions:

39. MRC for SISO Plants Remark: Assumption P1 requires that the plant be minimum phase and no assumptions about the location of the poles of plant; i.e., the plant is allowed to have unstable poles. Note that we allow the plant to be uncontrollable or unobservable, since, by assumption P1, all the plant zeros are in LHP, any zero-pole cancellation can occur only in LHP, which implies that the plant is both stabilizable and detectable. Assumption P1 is a consequence of the control objective which is met by designing an MRC control law that cancels the zeros of the plant and replaces them with those of the reference model. For stability, such cancellations should occur in LHP, which implies the assumption P1.

40. MRC for SISO Plants MRC Schemes: Known Plant Parameters Let us consider the feedback control law as: Structure of the MRC scheme

41. MRC for SISO Plants MRC Schemes: Known Plant Parameters control law

42. MRC for SISO Plants MRC Schemes: Known Plant Parameters The closed-loop plant should be equal with reference: Choosing and using or matching equation

43. MRC for SISO Plants MRC Schemes: Known Plant Parameters Equating the coefficients of the powers of s on both sides, we can express it in terms of the algebraic equation

44. MRC for SISO Plants MRC Schemes: Known Plant Parameters A state-space realization of the above control law:

45. MRC for SISO Plants MRC Schemes: Known Plant Parameters

46. MRC for SISO Plants MRC Schemes: Known Plant Parameters We obtain the state-space representation of the overall closed-loop plant by augmenting the stateof the plant with the states of the controller, i.e.,

47. MRC for SISO Plants MRC Schemes: Known Plant Parameters reference model realization

48. MRC for SISO Plants MRC Schemes: Known Plant Parameters state error: output tracking error:

49. MRC for SISO Plants MRC Schemes: Known Plant Parameters Let us consider the second-order plant Example: and the reference model choosing and the control input

50. MRC for SISO Plants MRC Schemes: Known Plant Parameters Computing the closed-loop transfer function and the matching equation