1. MODEL REFERENCE ADAPTIVE CONTROL
(ECES-817)
Presented by : Shubham Bhat

2. Outline Introduction MRAC using MIT Rule
Feed forward example (open loop )
Closed loop First order example
MRAC using Lyapunov Rule
Feed forward example (open loop)
Closed loop first order example
Comparison of MIT and Lyapunov Rule
Homework Problem

3. Control System design steps

4. INTRODUCTION Design of Autopilots – A type of Adaptive Control
MRAC is derived from the model following problem or model reference control (MRC) problem.
Structure of an MRC scheme

5. MRC Objective
The MRC objective is met if up is chosen so that the closed-loop transfer
function from r to yp has stable poles and is equal to Wm(s), the transfer
function of the reference model.
When the transfer function is matched, for any reference input signal r(t), the plant output yp converges to ym exponentially fast.
If G is known, design C such that

6. MODEL REFERENCE CONTROL
The plant model is to be minimum phase, i.e., have stable zeros.
The design of C( ) requires the knowledge of the coefficients of the plant
transfer function G(s).
If is a vector containing all the coefficients of G(s) = G(s; ), then the parameter vector may be computed by solving an algebraic equation of the form
= F( )
The MRC objective to be achieved if the plant model has to be minimum phase and its parameter vector has to be known exactly.

7. MODEL REFERENCE CONTROL
When is unknown, the MRC scheme cannot be implemented because cannot be calculated and is, therefore, unknown.
One way of dealing with the unknown parameter case is to use the certainty equivalence approach to replace the unknown in the control law with its estimate obtained using the direct or the indirect approach.
The resulting control schemes are known as MRAC and can be classified as
indirect MRAC and direct MRAC.

8. Direct MRAC

9. Indirect MRAC

10. Assumptions

11. Assumptions

12. Proofs for the theorems can be found in the reference.
MRAC - Key Stability Theorems
Theorem 1: Global stability, robustness and asymptotic zero tracking performance
Consider the previous system, satisfying assumptions with relative degree being one. If the control input and the adaptation law are chosen as per Lyapunov theorem, then there exists >0 such that for belongs [0, ] all signals inside the closed loop system are bounded and the tracking error will converge to zero asymptotically
Theorem 2: Finite time zero tracking performance with high gain design
Consider the previous system, satisfying assumptions with relative degree being one. If then the output tracking error will converge to zero in finite time with all signals inside the closed loop system remaining bounded.
Proofs for the theorems can be found in the reference.

13. General MRAC
Some of the basic methods used to design adjustment mechanism are
MIT Rule
Lyapunov rule

14. MRAC using MIT Rule

15. Sensitivity Derivative

16. Alternate cost function

17. Adaptation of a feed forward gain

18. Adaptation of a feed forward gain using MIT Rule

19. Block Diagram Implementation

20. MRAC using MIT Rule Control Law: gamma (g) = 1 Actual Kp = 2
Initial guessed Kp = 1

21. Error between Estimated and Actual value of Kp

22. Error between Model and Plant

23. MRAC for first order system- using MIT Rule

24. Adaptive Law- MIT Rule

25. Block Diagram

26. Simulation

27. Error and Parameter Convergence

28. Error and Parameter Convergence

29. NOTE: MIT rule does not guarantee error convergence or stability
MIT Rule - Remarks
NOTE: MIT rule does not guarantee error convergence or stability
usually kept small
Tuning crucial to adaptation rate and stability.

30. MIT Rule to Lyapunov transition
Several Problems were encountered in the usage of the MIT rule.
Also, it was not possible in general to prove closed loop stability, or convergence of the output error to zero.
A new way of redesigning adaptive systems using Lyapunov theory was proposed by Parks.
This was based on Lyapunov stability theorems, so that stable and provably convergent model reference schemes were obtained.
The update laws are similar to that of the MIT Rule, with the sensitivity functions replaced by other functions.
The theme was to generate parameter adjustment rule which guarantee stability

31. Lyapunov Stability

32. Definitions

33. Design MRAC using Lyapunov theorem

34. Adaptation to feed forward gain

35. Design MRAC using Lyapunov theorem

36. Adaptation of Feed forward gain

37. Simulation

38. First order system using Lyapunov

39. First order system using Lyapunov, contd.

40. First order system using Lyapunov, contd.

41. Comparison of MIT and Lyapunov rule

42. Simulation

43. State Feedback

44. Error Function

45. Lyapunov Function

46. Adaptation of Feed forward gain

47. Adaptation of Feed forward gain

48. Output Feedback

49. Stability Analysis - MRAC - Plant

50. MRAC - Model

51. MRAC - Simple control Law

52. MRAC - Feedback control law

53. MRAC - Block diagram

54. Proofs for the theorems can be found in the reference.
MRAC - Stability Theorems
Theorem 1: Global stability, robustness and asymptotic zero tracking performance
Consider the above system, satisfying assumptions with relative degree being one. If the control input is designed as above, and the adaptation law is chosen as shown above, then there exists >0 such that for belongs [0, ] all signals inside the closed loop system are bounded and the tracking error will converge to zero asymptotically
Theorem 2: Finite time zero tracking performance with high gain design
Consider the above system, satisfying assumptions with relative degree being one. If then the output tracking error will converge to zero in finite time with all signals inside the closed loop system remaining bounded.
Proofs for the theorems can be found in the reference.

55. Summary of Lyapunov rule for MRAC

56. References
Adaptive Control (2nd Edition) by Karl Johan Astrom, Bjorn Wittenmark
Robust Adaptive Control by Petros A. Ioannou,Jing Sun
Stability, Convergence, and Robustness by Shankar Sastry and Marc Bodson

57. Design of MRAC using MIT Rule
Homework Problem
Design of MRAC using MIT Rule

58. Homework Problem

59. Homework Problem- contd.

60. Homework Problem- contd.

61. Deliverables Deliverables: Simulate the system in MATLAB/ Simulink.
Design an MRAC controller for the plant using MIT Rule.
Plot the error between estimated and actual parameter values.
Try different reference inputs (ramps, sinusoids, step).