HybridwLyapunov Theory Revie Review: LQ Lyapunov Theory

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Hybrid Lyapunov Theory

Meeko Oishi, Ph.D.

Electrical and Computer Engineering

University of British Columbia, BC

http://www.ece.ubc.ca/~elec571m.html

[email protected]

EECE 571M/491M, Spring 2007

Lecture 7

Tomlin LN 6

EECE 571M / 491M Winter 2007 2

Announcements

! “Lecture review” examples! Demonstrate main concepts from previous lecture

! 2 each

! Sign-up sheet

! 10-15 minutes, informal

! Homework #1 due Thursday


! Tools to aid in stability analysis

! Phase-plane plots

! Stability in the sense of Lyapunov: Continuous systems

! Direct method: If you can find a Lyapunov function, then you knowthe system is stable.

! Lyapunov functions are “energy-like functions”

! Lypaunov functions are a sufficient condition for stability

! Special case: Lyapunov theory for linear systems

! Necessary and sufficient conditions

! Quadratic Lyapunov functions

Review


! Quadratic Lyapunov functions for linear systems (differential ordifference equations)

! Positive definite matrix properties

! Linear Quadratic Lyapunov theorem

! Linear continuous dynamics

! Quadratic Lyapunov function

! Converse theorems

! Asymptotic stability vs. stability in the sense of Lyapunov

! LQ Lyapunov theorem for difference equations

Review: LQ Lyapunov Theory


! For the dynamical system

! consider the quadratic Lyapunov function

! whose time-derivative

! can be written as

Review: Lyapunov equation


! A matrix P is positive definite if

! A matrix P is positive semi-definite if

! A matrix P is negative definite if

! A matrix P is negative semi-definite if

Review: Positive definite matr.


Theorem: Lyapunov stability for linear systems

! The equilibrium point x*=0 of dx/dt = Ax is asymptoticallystable if and only if for all matrices Q = QT > 0 there existsa matrix P = PT > 0 such that

! For a given Q, P will be unique

! The solution P is given by

! To numerically solve for P, formulate the linear matrix inequality

! And invoke the Matlab LMI toolbox

Review: Lyapunov stability


Summary of linear quadratic Lyapunov theorems (from S. Boyd)

! If P > 0, Q > 0, then system is globally asymptotically stable

! If P > 0, Q ! 0, then system is stable in the sense of Lyapunov

! If P > 0, Q ! 0, and (Q, A) observable, then system is globallyasymptotically stable

! If P > 0, Q ! 0, the sublevel sets of { x | xTPx " a } areinvariant and are ellipsoids

! If P ! 0, Q ! 0, then the system is not stable.



Converse linear quadratic Lyapunov theorems (from S. Boyd)

! If A is stable, then there exists P > 0, Q > 0 that satisfy theLyapunov equation

! If A is stable and Q ! 0, then P ! 0

! If A is stable, Q ! 0, and (Q, A) is observable, then P > 0


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! For the difference equation

! consider the quadratic Lyapunov function

! whose difference

! can be written as

Review: Discrete-time Lyap.

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! Review

! Linear quadratic lyapunov theory

! Lyapunov equation

! LMIs

! Introduction to hybrid stability

! Hybrid equilibrium

! Hybrid stability

! Multiple Lyapunov functions

! Hybrid systems

! Switched systems

Today’s lecture

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Hybrid equilibrium

! Definition:

The continuous state is an equilibriumpoint of the autonomous hybrid automaton

H = (Q, X, f, R, Dom, Init) if!

!

! Assume without loss of generality that x* = 0

! Jumps out of (q,0) are allowed as long as in thenew mode, the continuous state is x = 0

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Hybrid stability

! Definition:

The equilibrium point x* = 0 of the autonomoushybrid automaton H = (Q, X, f, R, Dom, Init) isstable if for all ! > 0 there exists a " > 0 such thatfor all executions (#, q, x) starting from the

state (q0, x0),

! As before, the continuous state must merely staywithin some arbitrary bound of the equilibrium point -- convergence is not required

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Hybrid stability

! Definition:

The equilibrium point x* = 0 of the autonomoushybrid automaton H = (Q, X, f, R, Dom, Init) isasymptotically stable if it is stable and " can bechosen such that for all executions (#, q, x) starting

from the state (q0, x0),

! Note that (#, q, x) is assumed to be an infinite

execution, with

! Recall that for non-Zeno executions, t# = #

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Hybrid stability

Controller 1dx/dt = A1x


STABLE STABLE

UNSTABLE



STABLE STABLE

STABLE

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Hybrid stability



UNSTABLE UNSTABLE

UNSTABLE



UNSTABLE UNSTABLE

STABLE

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Hybrid stability: Example 1



STABLE STABLE

UNSTABLE!

Q = q1,q2{ }

f = {A1x,A2x}, A1 ="1 10

"100 "1

#

$ %

&

' ( , A2 =

"1 100

"10 "1

#

$ %

&

' (

Init =Q){X \ 0}

Dom = {q1,x1x2 * 0}+{q2,x1x2 , 0}

R(q1,{x | x1x2 , 0}) = (q2,x), R(q2,{x | x1x2 * 0}) = (q1,x)

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STABLE STABLE

UNSTABLE

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STABLE STABLE

UNSTABLE

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Multiple Lyapunov functions

Consider a Lyapunov-like function V(q,x):

! When the system is evolving in mode q, V(q,x) must decreaseor maintain the same value

! Every time mode q is re-visited, the value V(q,x) must be lowerthan it was last time the system entered mode q.

! When the system switches into a new mode q’, V may jump invalue

! For inactive modes p, V(p,x) may increase

! Requires solving for V directly

V(x)

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Hybrid Lyapunov stability

Theorem: Lyapunov stability for hybrid systems

! Consider a hybrid automaton H with equilibrium point x*=0.Assume that there exists an open set such that

for some . Let be acontinuously differentiable function in x such that for all :

! If for all (#,q,x) starting from , and all ,the sequence is non-increasing (orempty), then x*=0 is a stable equilibrium point of H.

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! Lyapunov-like function in each mode must

! Decrease when that mode is active

! Enter that mode with a value lower than the last time the modewas entered

! Valid for any continuous dynamics (including nonlinear) and anyreset map (including ones with discontinuities in the state)

! Requires construction of sequence of ‘initial’ values of V(q,x)each time a mode is re-visited

(Defeats goal of bypassing direct integration or solution of x(t) )

V(x)

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! Candidate Lyapunov function

! Check whether the function meets the requirements for stability

! Lyapunov function in each mode

! Sequence for each mode


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! With Q1=Q2=I

! Lyapunovfunctions ineach mode

! Level sets areellipses

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! These Lyapunov-like functionsare not acceptable candidatesto show hybrid stability

! V(q,x) decreases while mode q is active

! But V is higher when mode is re-visited

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STABLE STABLE

STABLE

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! These Lyapunov-like functionsare acceptable candidates toshow hybrid stability

! V(q,x) decreases while mode q is active

! V is lower when mode is re-visited

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! Often difficult to use in practice!

! To yield more practical theorems, focus on specific classes of

! switching schemes (arbitrary, state-based, timed)

! dwell times within each mode

! continuous dynamics (linear, affine)

! Lyapunov functions (common Lyapunov functions, piecewisequadratic Lyapunov functions, etc.)

V(x)

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! Hybrid equilibrium

! Allows switching

! Continuous state must be 0

! Hybrid stability

! Switching allowed so long as the continuous state remainsbounded

! Multiple Lyapunov functions

! Most general form of stability

! Difficult to use in practice

! Narrow according to structure within the hybrid system

Summary

Documents

HybridwLyapunov Theory Revie Review: LQ Lyapunov Theory