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

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

Control System design steps

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

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

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.

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.

Direct MRAC

Indirect MRAC

Assumptions

Assumptions

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.

General MRAC Some of the basic methods used to design adjustment mechanism are MIT Rule 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 Control Law: gamma (g) = 1 Actual Kp = 2 Initial guessed Kp = 1

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

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

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

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.

Summary of Lyapunov rule for MRAC

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

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

Homework Problem

Homework Problem- contd.

Homework Problem- contd.

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).