1
Hybrid Lyapunov Theory
Meeko Oishi, Ph.D.
Electrical and Computer Engineering
University of British Columbia, BC
http://www.ece.ubc.ca/~elec571m.html
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
EECE 571M / 491M Winter 2007 3
! 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
EECE 571M / 491M Winter 2007 4
! 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
EECE 571M / 491M Winter 2007 5
! For the dynamical system
! consider the quadratic Lyapunov function
! whose time-derivative
! can be written as
Review: Lyapunov equation
EECE 571M / 491M Winter 2007 6
! 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.
EECE 571M / 491M Winter 2007 7
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
EECE 571M / 491M Winter 2007 8
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.
Review: Lyapunov stability
EECE 571M / 491M Winter 2007 9
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
Review: Lyapunov stability
EECE 571M / 491M Winter 2007 10
! For the difference equation
! consider the quadratic Lyapunov function
! whose difference
! can be written as
Review: Discrete-time Lyap.
EECE 571M / 491M Winter 2007 11
! 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
EECE 571M / 491M Winter 2007 12
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
EECE 571M / 491M Winter 2007 13
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
EECE 571M / 491M Winter 2007 14
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# = #
EECE 571M / 491M Winter 2007 15
Hybrid stability
Controller 1dx/dt = A1x
Controller 2dx/dt = A2x
STABLE STABLE
UNSTABLE
Controller 1dx/dt = A1x
Controller 2dx/dt = A2x
STABLE STABLE
STABLE
EECE 571M / 491M Winter 2007 16
Hybrid stability
Controller 1dx/dt = A1x
Controller 2dx/dt = A2x
UNSTABLE UNSTABLE
UNSTABLE
Controller 1dx/dt = A1x
Controller 2dx/dt = A2x
UNSTABLE UNSTABLE
STABLE
EECE 571M / 491M Winter 2007 17
Hybrid stability: Example 1
Controller 1dx/dt = A1x
Controller 2dx/dt = A2x
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)
EECE 571M / 491M Winter 2007 18
Hybrid stability: Example 1
Controller 1dx/dt = A1x
Controller 2dx/dt = A2x
STABLE STABLE
UNSTABLE
EECE 571M / 491M Winter 2007 19
Hybrid stability: Example 1
Controller 1dx/dt = A1x
Controller 2dx/dt = A2x
STABLE STABLE
UNSTABLE
EECE 571M / 491M Winter 2007 20
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)
EECE 571M / 491M Winter 2007 21
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.
EECE 571M / 491M Winter 2007 22
Multiple Lyapunov functions
! 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)
EECE 571M / 491M Winter 2007 23
! Candidate Lyapunov function
! Check whether the function meets the requirements for stability
! Lyapunov function in each mode
! Sequence for each mode
Hybrid stability: Example 1
EECE 571M / 491M Winter 2007 24
Hybrid stability: Example 1
! With Q1=Q2=I
! Lyapunovfunctions ineach mode
! Level sets areellipses
EECE 571M / 491M Winter 2007 25
Hybrid stability: Example 1
! 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
EECE 571M / 491M Winter 2007 26
Hybrid stability: Example 1
Controller 1dx/dt = A1x
Controller 2dx/dt = A2x
STABLE STABLE
STABLE
EECE 571M / 491M Winter 2007 27
Hybrid stability: Example 1
! 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
EECE 571M / 491M Winter 2007 28
Multiple Lyapunov functions
! 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)
EECE 571M / 491M Winter 2007 29
! 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