MODEL REFERENCE ADAPTIVE CONTROL

Presented by : Shubham Bhat

(ECES-817)

•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

Outline

Control System design steps

Design of Autopilots – A type of Adaptive Control

MRAC is derived from the model following problem or model reference control (MRC) problem.

INTRODUCTION

Structure of an MRC scheme

MRC Objective

The MRC objective is met if up is chosen so that the closed-loop transferfunction from r to yp has stable poles and is equal to Wm(s), the transferfunction 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

mWCG

CG

1

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 planttransfer 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.

c*

*

c* *

*

*

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 asindirect MRAC and direct MRAC.

* c*

)(tc

*c

MODEL REFERENCE CONTROL

Direct MRAC

Indirect MRAC

Assumptions

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Assumptions

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 1: Global stability, robustness and asymptotic zero tracking performance

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.

MRAC - Key Stability Theorems

* *

* *(0) | |, (0) ,j j j jc c

Proofs for the theorems can be found in the reference.

General MRAC

Some of the basic methods used to design adjustment mechanism are(i) MIT Rule(ii) Lyapunov rule

MRAC using MIT Rule

Sensitivity Derivative

Alternate cost function

Adaptation of a feed forward gain

Adaptation of a feed forward gain using MIT Rule

Block Diagram Implementation

MRAC using MIT Rule

gamma (g) = 1Actual Kp = 2Initial guessed Kp = 1

Control Law: eym

Error between Estimated and Actual value of Kp

Error between Model and Plant

MRAC for first order system- using MIT Rule

Adaptive Law- MIT Rule

Block Diagram

Simulation

Error and Parameter Convergence

Error and Parameter Convergence

• NOTE: MIT rule does not guarantee error convergence or stability

• usually kept small

• Tuning crucial to adaptation rate and stability.

MIT Rule - Remarks

1. Several Problems were encountered in the usage of the MIT rule.

2. Also, it was not possible in general to prove closed loop stability, or convergence of the output error to zero.

3. A new way of redesigning adaptive systems using Lyapunov theory was proposed by Parks.

4. This was based on Lyapunov stability theorems, so that stable and provably convergent model reference schemes were obtained.

5. The update laws are similar to that of the MIT Rule, with the sensitivity functions replaced by other functions.

6. The theme was to generate parameter adjustment rule which guarantee stability

MIT Rule to Lyapunov transition

Lyapunov Stability

Definitions

Design MRAC using Lyapunov theorem

Adaptation to feed forward gain

Design MRAC using Lyapunov theorem

Adaptation of Feed forward gain

Simulation

First order system using Lyapunov

First order system using Lyapunov, contd.

First order system using Lyapunov, contd.

Comparison of MIT and Lyapunov rule

Simulation

State Feedback

Error Function

Lyapunov Function

Adaptation of Feed forward gain

Adaptation of Feed forward gain

Output Feedback

Stability Analysis - MRAC - Plant

MRAC - Model

MRAC - Simple control Law

MRAC - Feedback control law

MRAC - Block diagram

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 1: Global stability, robustness and asymptotic zero tracking performance

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.

MRAC - Stability Theorems

* *

* *(0) | |, (0) ,j j j jc c

Proofs for the theorems can be found in the reference.

Summary of Lyapunov rule for MRAC

References

1. Adaptive Control (2nd Edition) by Karl Johan Astrom, Bjorn Wittenmark

2. Robust Adaptive Control by Petros A. Ioannou,Jing Sun

3. Stability, Convergence, and Robustness by Shankar Sastry and Marc Bodson

Homework Problem

Design of MRAC using MIT Rule

Homework Problem

Homework Problem- contd.

Homework Problem- contd.

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).

Deliverables