MODEL REFERENCE ADAPTIVE CONTROL

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

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Control System design steps
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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
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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
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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.
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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.
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Direct MRAC
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Indirect MRAC
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Assumptions
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Assumptions
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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
gt0 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.
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General MRAC

• Some of the basic methods used to design
  adjustment mechanism are
• MIT Rule
• Lyapunov rule

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MRAC using MIT Rule
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Sensitivity Derivative
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Alternate cost function
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Adaptation of a feed forward gain
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Adaptation of a feed forward gain using MIT Rule
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Block Diagram Implementation
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MRAC using MIT Rule
Control Law
gamma (g) 1 Actual Kp 2 Initial guessed Kp
1
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Error between Estimated and Actual value of Kp
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Error between Model and Plant
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MRAC for first order system- using MIT Rule
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Adaptive Law- MIT Rule
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Block Diagram
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Simulation
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Error and Parameter Convergence
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Error and Parameter Convergence
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MIT Rule - Remarks

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

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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

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Lyapunov Stability
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Definitions
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Design MRAC using Lyapunov theorem
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Adaptation to feed forward gain
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Design MRAC using Lyapunov theorem
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Adaptation of Feed forward gain
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Simulation
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First order system using Lyapunov
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First order system using Lyapunov, contd.
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First order system using Lyapunov, contd.
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Comparison of MIT and Lyapunov rule
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Simulation
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State Feedback
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Error Function
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Lyapunov Function
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Adaptation of Feed forward gain
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Adaptation of Feed forward gain
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Output Feedback
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Stability Analysis - MRAC - Plant
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MRAC - Model
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MRAC - Simple control Law
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MRAC - Feedback control law
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MRAC - Block diagram
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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
gt0 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.
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Summary of Lyapunov rule for MRAC
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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

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Homework Problem
Design of MRAC using MIT Rule
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Homework Problem
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Homework Problem- contd.
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Homework Problem- contd.
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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).