1 Hybrid Lyapunov Theory Meeko Oishi, Ph.D. Electrical and Computer Engineering University of British Columbia, BC http://www. ece . ubc .ca/~elec571m.html moishi @ ece . ubc .ca 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 know the 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 or difference 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
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! “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