Lyapunov Based Redesign

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Lyapunov Based Redesign

• Motivation

Consider
But the real system is
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Problem

• Approach
• (i) chosen so that nominal closed loop system is
• asymptotically stable.
• (ii) chosen so as to cancel the effect of
  uncertainty.

Find a state feedback controller so that the
closed loop system is stable in a sufficiently
strong sense.
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Assume that results in the
uniformly asymptotically stablenominal closed
loop system,
Solution
Suppose also that is a Lyapunov
function that proves the following.
is strictly increasing.
where , i.e.
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Solution (Continued)
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Solution (Continued)
Let
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Solution (Continued)
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Smooth Control

• Smooth Control ( case)

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Smooth Control (Continued)
Then take large, so that
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- When is chosen small, we can arrive at a
sharper result.
Assume that such that
where
Then, when
where is positive definite if Thus choosing
we have
Also when We
conclude which shows that the
origin is uniformly asymptotically stable.
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Example
Ex
Choose
where
are chosen so that
is Hurwitz.
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Example (Continued)
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Example (Continued)
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Backstepping
Consider a system
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Backstepping (Continued)
(1)
A
(1),(2)
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Backstepping (Continued)
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Backstepping (Continued)
which is similar to the original system but ? has
an asymptotically stable origin when the input is
0.
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Lemma Example
Lemma
(1), (2).
(1)
A
(1), (2)
Ex
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Example (Continued)
Lets consider
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Recursive Backstepping
Consider the following strict feedback system
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Recursive procedure
Recursive procedure
? Consider
Then using the previous result, obtain
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Recursive procedure (Continued)
Next consider
Then we recognize that
Thus, similarly, obtain the state feedback control
and
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Extended Linearization (Gain scheduling method)

• Motivation
• Plant nonlinear
• Controller linear
• Design method classical linearization
• Assumption no single linear controller
  satisfies the performance specification
• Idea design a set of controllers, each good at
  a particular operating point, and switch
  (schedule) the gains of the controllers
  accordingly
• Problem now we have a nonlinear (piecewise
  linear) system with time dependent jump
• Solution no good tool but some theory is being
  developed mostly simulation in the past

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Structure Examples
Structure

• -Examples
• Tank system

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Control Goal
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Nonlinear Actuator

• A different angle nonlinear actuator

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Step Responses
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Approximation
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Results
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Classification
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Issues
Controller
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Example
Ex
Theorem
Proof See Ch 5 in Nonlinear System Analysis
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Formalization

• A version of scheduling on the output

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Block Diagram
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Conditions