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

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
CG
 Wm
1  CG
MODEL REFERENCE CONTROL
The plant model is to be minimum phase, i.e., have stable zeros.
The design of C(  *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
 *c = 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.
*
*
c
One way of dealing with the unknown parameter case is to use the certainty
*
equivalence approach to replace the unknown c in the control law with its
estimate  c (t ) 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 
( A1) Plant Assumptions
The plant is a sin gle  input , sin gle  output ( SISO ), linear time 
in var iant ( LTI ) system, described by a transfer function
The plant is strictly proper and min imum phase.
The sign of the so  called high frequency gain K p is
known and , without loss of generality , we will assume
K p  0.
Assumptions
( A2) Re ference Model Assumptions
The reference mod el has the same deg ree as the
correspond ing plant polynomial . The reference mod el
is stable, min imum phase and k m  0.
( A3) Re ference Input Assumptions
The reference input r (t ) is piecewise continuous
and bounded on R.
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  j (0) |  *j |, c j (0)  c*j , 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
(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
Control Law:   yme
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
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
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
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
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  j (0) |  *j |, c j (0)  c*j , 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
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
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).