Hybrid Systems Course Lyapunov stability...STABILITY OF LINEAR CONTINUOUS SYSTEMS Theorem (necessary...
Transcript of Hybrid Systems Course Lyapunov stability...STABILITY OF LINEAR CONTINUOUS SYSTEMS Theorem (necessary...
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Hybrid Systems CourseLyapunov stability
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OUTLINE
Focus: stability of an equilibrium point
• continuous systems decribed by ordinary differentialequations (brief review)
• hybrid automata
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OUTLINE
Focus: stability of an equilibrium point
• continuous systems decribed by ordinary differentialequations (brief review)
• hybrid automata
![Page 4: Hybrid Systems Course Lyapunov stability...STABILITY OF LINEAR CONTINUOUS SYSTEMS Theorem (necessary and sufficient condition): The equilibrium point xe =0 is asymptotically stable](https://reader035.fdocuments.in/reader035/viewer/2022062416/610f34e3ddd6f7027d2b6772/html5/thumbnails/4.jpg)
ORDINARY DIFFERENTIAL EQUATIONS
An ordinary differential equation is a mathematical model of acontinuous state continuous time system:
X = <n ´ state spacef: <n! <n ´ vector field (assigns a “velocity” vector to each x)
Given an initial value x0 2 X,an execution (solution in the sense of Caratheodory) overthe time interval [0,T) is a function x: [0,T) ! <n such that:
• x(0) = x0
• x is continuous and piecewise differentiable
•
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ODE SOLUTION: WELL-POSEDNESS
Theorem [global existence and uniqueness non-blocking,deterministic, non-Zeno]If f: <n! <n is globally Lipschitz continuous, then 8 x0 thereexists a single solution with x(0)=x0 defined on [0,1).
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STABILITY OF CONTINUOUS SYSTEMS
with f: <n! <n globally Lipschitz continuous
Definition (equilibrium):xe 2 <n for which f(xe)=0
Remark: {xe} is an invariant set
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Definition (stable equilibrium):
Graphically:
δxe
equilibrium motion
perturbed motion
small perturbations lead to small changes in behavior
execution startingfrom x(0)=x0
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Definition (asymptotically stable equilibrium):
and can be chosen so that
Graphically:
δxe
equilibrium motion
perturbed motion
small perturbations lead to small changes in behaviorand are re-absorbed, in the long run
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Definition (asymptotically stable equilibrium):
and can be chosen so that
Graphically:
small perturbations lead to small changes in behaviorand are re-absorbed, in the long run
δxe
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EXAMPLE: PENDULUM
m
l
frictioncoefficient (α)
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EXAMPLE: PENDULUM
unstable equilibrium
m
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EXAMPLE: PENDULUM
as. stable equilibriumm
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EXAMPLE: PENDULUM
m
l
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Let xe be asymptotically stable.
Definition (domain of attraction):The domain of attraction of xe is the set of x0 such that
Definition (globally asymptotically stable equilibrium):xe is globally asymptotically stable (GAS) if its domain ofattraction is the whole state space <n
execution startingfrom x(0)=x0
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EXAMPLE: PENDULUM
as. stable equilibrium
small perturbations areabsorbed, not allperturbations not GAS
m
m
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Let xe be asymptotically stable.
Definition (exponential stability):xe is exponentially stable if 9 , , >0 such that
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STABILITY OF CONTINUOUS SYSTEMS
with f: <n! <n globally Lipschitz continuous
Definition (equilibrium):xe 2 <n for which f(xe)=0
Without loss of generality we suppose that
xe = 0if not, then z := x -xe! dz/dt = g(z), g(z) := f(z+xe) (g(0) = 0)
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STABILITY OF CONTINUOUS SYSTEMS
with f: <n! <n globally Lipschitz continuous
How to prove stability?find a function V: <n! < such that
V(0) = 0 and V(x) >0, for all x 0V(x) is decreasing along the executions of the system
V(x) = 3
V(x) = 2
x(t)
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STABILITY OF CONTINUOUS SYSTEMS
execution x(t)
candidate function V(x)
behavior of V along theexecution x(t): V(t): = V(x(t))
Advantage with respect to exhaustive check of all executions?
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with f: <n! <n globally Lipschitz continuous
V: <n! < continuously differentiable (C1) function
Rate of change of V along the execution of the ODE system:
(Lie derivative of V with respect to f)
STABILITY OF CONTINUOUS SYSTEMS
gradient vector
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with f: <n! <n globally Lipschitz continuous
V: <n! < continuously differentiable (C1) function
Rate of change of V along the execution of the ODE system:
(Lie derivative of V with respect to f)
STABILITY OF CONTINUOUS SYSTEMS
gradient vector
No need to solve the ODE for evaluating if V(x) decreasesalong the executions of the system
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LYAPUNOV STABILITY
Theorem (Lyapunov stability Theorem):Let xe = 0 be an equilibrium for the system and D½ <n an open
set containing xe = 0.If V: D! < is a C1 function such that
Then, xe is stable.
V positive definite on D
V non increasing alongsystem executions in D(negative semidefinite)
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EXAMPLE: PENDULUM
m
l
frictioncoefficient (α)
energy function
xe stable
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LYAPUNOV STABILITY
Theorem (Lyapunov stability Theorem):Let xe = 0 be an equilibrium for the system and D½ <n an open
set containing xe = 0.If V: D! < is a C1 function such that
Then, xe is stable.
If it holds also that
Then, xe is asymptotically stable (AS)
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LYAPUNOV GAS THEOREM
Theorem (Barbashin-Krasovski Theorem):Let xe = 0 be an equilibrium for the system.
If V: <n! < is a C1 function such that
Then, xe is globally asymptotically stable (GAS).
V positive definite on <n
V decreasing alongsystem executions in <n
(negative definite)
V radially unbounded
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STABILITY OF LINEAR CONTINUOUS SYSTEMS
• xe = 0 is an equilibrium for the system
• the elements of matrix eAt are linear combinations of tKei(A)t,k=0,…,hi, i=1,2,…,n
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STABILITY OF LINEAR CONTINUOUS SYSTEMS
• xe = 0 is an equilibrium for the system
• xe =0 is asymptotically stable if and only if A is Hurwitz (alleigenvalues with real part <0)
• asymptotic stability GAS
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STABILITY OF LINEAR CONTINUOUS SYSTEMS
• xe = 0 is an equilibrium for the system
• xe =0 is asymptotically stable if and only if A is Hurwitz (alleigenvalues with real part <0)
• asymptotic stability GAS
Alternative characterization…
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STABILITY OF LINEAR CONTINUOUS SYSTEMS
Theorem (necessary and sufficient condition):
The equilibrium point xe =0 is asymptotically stable if and onlyif for all matrices Q = QT positive definite (Q>0) the
ATP+PA = -Q
has a unique solution P=PT >0.
Lyapunov equation
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STABILITY OF LINEAR CONTINUOUS SYSTEMS
Theorem (necessary and sufficient condition):
The equilibrium point xe =0 is asymptotically stable if and onlyif for all matrices Q = QT positive definite (Q>0) the
ATP+PA = -Q
has a unique solution P=PT >0.
Remarks:Q positive definite (Q>0) iff xTQx >0 for all x 0Q positive semidefinite (Q¸ 0) iff xTQx ¸ 0 for all x andxT Q x = 0 for some x 0
Lyapunov equation
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STABILITY OF LINEAR CONTINUOUS SYSTEMS
Theorem (necessary and sufficient condition):
The equilibrium point xe =0 is asymptotically stable if and onlyif for all matrices Q = QT positive definite (Q>0) the
ATP+PA = -Q
has a unique solution P=PT >0.
Proof.
(if) V(x) =xT P x is a Lyapunov function
Lyapunov equation
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STABILITY OF LINEAR CONTINUOUS SYSTEMS
Theorem (necessary and sufficient condition):
The equilibrium point xe =0 is asymptotically stable if and onlyif for all matrices Q = QT positive definite (Q>0) the
ATP+PA = -Q
has a unique solution P=PT >0.
Proof.
(only if) Consider
Lyapunov equation
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STABILITY OF LINEAR CONTINUOUS SYSTEMS
Theorem (necessary and sufficient condition):
The equilibrium point xe =0 is asymptotically stable if and onlyif for all matrices Q = QT positive definite (Q>0) the
ATP+PA = -Q
has a unique solution P=PT >0.
Proof.
(only if) Consider
P = PT and P>0 easy to show
P unique can be proven by contradiction
Lyapunov equation
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STABILITY OF LINEAR CONTINUOUS SYSTEMS
Remarks: for a linear system
• existence of a (quadratic) Lyapunov function V(x) =xT P x is anecessary and sufficient condition for asymptotic stability
• it is easy to compute a Lyapunov function since the Lyapunovequation
ATP+PA = -Q
is a linear algebraic equation
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STABILITY OF LINEAR CONTINUOUS SYSTEMS
Theorem (exponential stability):
Let the equilibrium point xe =0 be asymptotically stable. Then,the rate of convergence to xe =0 is exponential:
for all x(0) = x0 2 <n, where 0 2 (0, mini |Re{i(A)}|) and >0is an appropriate constant.
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STABILITY OF LINEAR CONTINUOUS SYSTEMS
Theorem (exponential stability):
Let the equilibrium point xe =0 be asymptotically stable. Then,the rate of convergence to xe =0 is exponential:
for all x(0) = x0 2 <n, where 0 2 (0, mini |Re{i(A)}|) and >0is an appropriate constant.
Re
Im
o
o
o o
eigenvalues of A
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STABILITY OF LINEAR CONTINUOUS SYSTEMS
Theorem (exponential stability):
Let the equilibrium point xe =0 be asymptotically stable. Then,the rate of convergence to xe =0 is exponential:
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STABILITY OF LINEAR CONTINUOUS SYSTEMS
Proof (exponential stability):
A + 0 I is Hurwitz (eigenvalues are equal to (A) + 0)
Then, there exists P = PT >0 such that
(A + 0I)T P + P (A + 0I) <0
which leads to
x(t)T[AT P + P A]x(t) < - 2 0 x(t)T P x(t)
Define V(x) = xT P x, then
from which
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STABILITY OF LINEAR CONTINUOUS SYSTEMS
(cont’d) Proof (exponential stability):
thus finally leading to
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STABILITY OF LINEAR CONTINUOUS SYSTEMS
Theorem (exponential stability):
Let the equilibrium point xe =0 be asymptotically stable. Then,the rate of convergence to xe =0 is exponential:
for all x(0) = x0 2 <n, where 0 2 (0, mini |Re{i(A)}|) and >0is an appropriate constant.
Remark:
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STABILITY OF LINEAR CONTINUOUS SYSTEMS
• xe = 0 is an equilibrium for the system
• xe =0 is asymptotically stable if and only if A is Hurwitz (alleigenvalues with real part <0)
• asymptotic stability GAS exponential stability GES
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OUTLINE
Focus: stability of an equilibrium point
• continuous systems decribed by ordinary differentialequations (brief review)
• hybrid automata
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HYBRID AUTOMATA: FORMAL DEFINITION
A hybrid automaton H is a collection
H = (Q,X,f,Init,Dom,E,G,R)• Q = {q1,q2, …} is a set of discrete states (modes)
• X = <n is the continuous state space
• f: Q£ X! <n is a set of vector fields on X
• Init µ Q£ X is a set of initial states
• Dom: Q! 2X assigns to each q2 Q a domain Dom(q) of X
• E µ Q£ Q is a set of transitions (edges)
• G: E! 2X is a set of guards (guard condition)
• R: E£ X! 2X is a set of reset maps
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q = q1
q = q2
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HYBRID TIME SET
A hybrid time set is a finite or infinite sequence of intervals
= {Ii, i=0,1,…, M } such that
• Ii = [i, i’] for i < M• IM = [M, M’] or IM = [M, M’) if M<1• i’ = i+1
• i · i’
[ ]
[ ]
[ ]
τ0
I0
τ0’
τ1 τ1’I1
I2 τ2 = τ2’
τ3 τ3’I3
t1
t2
t3
t4
t1 Á t2 Á t3 Á t4
time instants in arelinearly ordered
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HYBRID TIME SET: LENGTH
Two notions of length for a hybrid time set = {Ii, i=0,1,…, M }:
• Discrete extent:<> = M+1
• Continuous extent:|||| = i=0,1,..,M |i’-i|
number of discrete transitions
total durationof intervals in
<> = 4|||| = 3’ - 0
[ ]
[ ]
[ ]
τ0
I0
τ0’
τ1 τ1’I1
I2 τ2 = τ2’
τ3 τ3’I3
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HYBRID TIME SET: CLASSIFICATION
A hybrid time set = {Ii, i=0,1,…, M } is
• Finite: if <> is finite and IM = [M, M’]
• Infinite: if <> is infinite or |||| is infinite
• Zeno: if <> is infinite but |||| is finite
finite infinite
infinite ZenoZeno
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HYBRID TRAJECTORY
A hybrid trajectory (, q, x) consists of:
• A hybrid time set = {Ii, i=0,1,…, M }• Two sequences of functions q = {qi(¢), i=0,1,…, M } and x =
{xi(¢), i=0,1,…, M } such that
qi: Ii! Q
xi: Ii! X
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HYBRID AUTOMATA: EXECUTION
A hybrid trajectory (, q, x) is an execution (solution) of thehybrid automaton H = (Q,X,f,Init,Dom,E,G,R) if it satisfiesthe following conditions:
• Initial condition: (q0(0), x0(0)) 2 Init
• Continuous evolution:for all i such that i < i’
qi: Ii! Q is constantxi:Ii! X is the solution to the ODE associated with qi(i)xi(t) 2 Dom(qi(i)), t2 [i,i’)
• Discrete evolution:(qi(i’),qi+1(i+1)) 2 E transition is feasiblexi(i’) 2 G((qi(i’),qi+1(i+1))) guard condition satisfiedxi+1(i+1) 2 R((qi(i’),qi+1(i+1)),xi(i’)) reset condition satisfied
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HYBRID AUTOMATA: EXECUTION
Well-posedness?
Problems due the hybrid nature:
for some initial state (q,x)• infinite execution of finite duration Zeno• no infinite execution blocking• multiple executions non-deterministic
We denote by
H(q,x) the set of (maximal) executions of H starting from (q,x)
H(q,x)1 the set of infinite executions of H starting from (q,x)
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STABILITY OF HYBRID AUTOMATA
H = (Q,X,f,Init,Dom,E,G,R)
Definition (equilibrium):xe =0 2 X is an equilibrium point of H if:• f(q,0) = 0 for all q 2 Q• if ((q,q’)2 E) Æ (02 G((q,q’)) ) R((q,q’),0) = {0}
Remarks:
• discrete transitions are allowed out of (q,0) but only to (q’,0)• if (q,0) 2 Init and (, q, x) is an execution of H starting from
(q,0), then x(t) = 0 for all t2
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EXAMPLE: SWITCHED LINEAR SYSTEM
H = (Q,X,f,Init,Dom,E,G,R)
• Q = {q1, q2} X = <2
• f(q1,x) = A1x and f(q2,x) = A2x with:
• Init = Q £ {x2 X: ||x|| >0}
• Dom(q1) = {x2 X: x1x2 · 0} Dom(q2) = {x2 X: x1x2 ¸ 0}
• E = {(q1,q2),(q2,q1)}• G((q1,q2)) = {x2 X: x1x2 ¸ 0} G((q2,q1)) = {x2 X: x1x2 · 0}
• R((q1,q2),x) = R((q2,q1),x) = {x}
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x1
x2
EXAMPLE: SWITCHED LINEAR SYSTEM
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EXAMPLE: SWITCHED LINEAR SYSTEM
H = (Q,X,f,Init,Dom,E,G,R)
• Q = {q1, q2} X = <2
• f(q1,x) = A1x and f(q2,x) = A2x with:
• Init = Q £ {x2 X: ||x|| >0}
• Dom(q1) = {x2 X: x1x2 · 0} Dom(q2) = {x2 X: x1x2 ¸ 0}
• E = {(q1,q2),(q2,q1)}• G((q1,q2)) = {x2 X: x1x2 ¸ 0} G((q2,q1)) = {x2 X: x1x2 · 0}
• R((q1,q2),x) = R((q2,q1),x) = {x}
xe = 0 is an equilibrium: f(q,0) = 0 & R((q,q’),0) = {0}
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H = (Q,X,f,Init,Dom,E,G,R)
Definition (stable equilibrium):Let xe = 0 2 X be an equilibrium point of H. xe = 0 is stable if
STABILITY OF HYBRID AUTOMATA
set of (maximal) executionsstarting from (q0, x0) 2 Init
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H = (Q,X,f,Init,Dom,E,G,R)
Definition (stable equilibrium):Let xe = 0 2 X be an equilibrium point of H. xe = 0 is stable if
Remark:
• Stability does not imply convergence
• To analyse convergence, only infinite executions should beconsidered
STABILITY OF HYBRID AUTOMATA
set of (maximal) executionsstarting from (q0, x0) 2 Init
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H = (Q,X,f,Init,Dom,E,G,R)
Definition (stable equilibrium):Let xe = 0 2 X be an equilibrium point of H. xe = 0 is stable if
Definition (asymptotically stable equilibrium):Let xe = 0 2 X be an equilibrium point of H. xe = 0 is asymptotically
stable if it is stable and >0 that can be chosen so that
STABILITY OF HYBRID AUTOMATA
set of (maximal) executionsstarting from (q0, x0) 2 Init
set of infinite executionsstarting from (q0, x0) 2 Init
1 := i(i’-i)continuous extent1 <1 if Zeno
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H = (Q,X,f,Init,Dom,E,G,R)
Definition (stable equilibrium):Let xe = 0 2 X be an equilibrium point of H. xe = 0 is stable if
Question:
xe = 0 stable equilibrium for each continuous systemdx/dt = f(q,x) implies that xe = 0 stable equilibrium for H?
STABILITY OF HYBRID AUTOMATA
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EXAMPLE: SWITCHED LINEAR SYSTEM
H = (Q,X,f,Init,Dom,E,G,R)
• Q = {q1, q2} X = <2
• f(q1,x) = A1x and f(q2,x) = A2x with:
• Init = Q £ {x2 X: ||x|| >0}
• Dom(q1) = {x2 X: x1x2 · 0} Dom(q2) = {x2 X: x1x2 ¸ 0}
• E = {(q1,q2),(q2,q1)}• G((q1,q2)) = {x2 X: x1x2 ¸ 0} G((q2,q1)) = {x2 X: x1x2 · 0}
• R((q1,q2),x) = R((q2,q1),x) = {x}
xe = 0 is an equilibrium: f(q,0) = 0 & R((q,q’),0) = {0}
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x1
x2
EXAMPLE: SWITCHED LINEAR SYSTEM
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EXAMPLE: SWITCHED LINEAR SYSTEM
Swiching between asymptotically stable linear systems.
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q1: quadrants 2 and 4q2: quadrants 1 and 3
Switching between asymptotically stable linear systems, butxe = 0 unstable equilibrium of H
q1
q1
q2
q2
EXAMPLE: SWITCHED LINEAR SYSTEM
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||x(i+1)|| > ||x(i)||
overshootssum up
EXAMPLE: SWITCHED LINEAR SYSTEM
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q1: quadrants 1 and 3q2: quadrants 2 and 4
q1
q1
q2
q2
EXAMPLE: SWITCHED LINEAR SYSTEM
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||x(i+1)|| < ||x(i)|||
EXAMPLE: SWITCHED LINEAR SYSTEM
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LYAPUNOV STABILITY
H = (Q,X,f,Init,Dom,E,G,R)
Theorem (Lyapunov stability):Let xe = 0 be an equilibrium for H with R((q,q’),x) = {x}, 8 (q,q’)2 E, and
D½ X=<n an open set containing xe = 0.Consider V: Q£ D! < is a C1 function in x such that for all q 2 Q:
If for all (, q, x) 2 H(q0,x0) with (q0,x0) 2 Init Å (Q£ D), and all q’2 Q, thesequence {V(q(i),x(i)): q(i) =q’} is non-increasing (or empty),
then, xe = 0 is a stable equilibrium of H.
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LYAPUNOV STABILITY
H = (Q,X,f,Init,Dom,E,G,R)
Theorem (Lyapunov stability):Let xe = 0 be an equilibrium for H with R((q,q’),x) = {x}, 8 (q,q’)2 E, and
D½ X=<n an open set containing xe = 0.Consider V: Q£ D! < is a C1 function in x such that for all q 2 Q:
If for all (, q, x) 2 H(q0,x0) with (q0,x0) 2 Init Å (Q£ D), and all q’2 Q, thesequence {V(q(i),x(i)): q(i) =q’} is non-increasing (or empty),
then, xe = 0 is a stable equilibrium of H.
V(q,x) Lyapunov functionfor continuous system q) xe =0 is stableequilibrium for system q
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LYAPUNOV STABILITY
H = (Q,X,f,Init,Dom,E,G,R)
Theorem (Lyapunov stability):Let xe = 0 be an equilibrium for H with R((q,q’),x) = {x}, 8 (q,q’)2 E, and
D½ X=<n an open set containing xe = 0.Consider V: Q£ D! < is a C1 function in x such that for all q 2 Q:
If for all (, q, x) 2 H(q0,x0) with (q0,x0) 2 Init Å (Q£ D), and all q’2 Q,the sequence {V(q(i),x(i)): q(i) =q’} is non-increasing (or empty),
then, xe = 0 is a stable equilibrium of H.
V(q,x) Lyapunov functionfor continuous system q) xe =0 is stableequilibrium for system q
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LYAPUNOV STABILITY
H = (Q,X,f,Init,Dom,E,G,R)
Sketch of the proof.
V(q(t),x(t))V(q1,x(t))
[ ][ ][ ][q(t)= q1 q(t)= q1V(q2,x(t))
0 0’=1 1’=2 2’=3
Lyapunov function forsystem q1! decreaseswhen q(t) = q1, but canincrease when q(t) q1
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LYAPUNOV STABILITY
H = (Q,X,f,Init,Dom,E,G,R)
Sketch of the proof.
V(q(t),x(t))V(q1,x(t))
[ ]0 0’=1
[ ][ ]1’=2 2’=3
[q(t)= q1 q(t)= q1
{V(q1,x(i))}non-increasing
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LYAPUNOV STABILITY
H = (Q,X,f,Init,Dom,E,G,R)
Sketch of the proof.
V(q(t),x(t))V(q1,x(t))
[ ]0 0’=1
[ ][ ]1’=2 2’=3
[q(t)= q1 q(t)= q1
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LYAPUNOV STABILITY
H = (Q,X,f,Init,Dom,E,G,R)
V(q(t),x(t)) Lyapunov-like function
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H = (Q,X,f,Init,Dom,E,G,R)
• Q = {q1, q2} X = <2
• f(q1,x) = A1x and f(q2,x) = A2x with:
• Init = Q £ {x2 X: ||x|| >0}
• Dom(q1) = {x2 X: Cx ¸ 0} Dom(q2) = {x2 X: Cx · 0}
• E = {(q1,q2),(q2,q1)}• G((q1,q2)) = {x2 X: Cx · 0} G((q2,q1)) = {x2 X: Cx ¸ 0}, CT2 <2
• R((q1,q2),x) = R((q2,q1),x) = {x}
EXAMPLE: SWITCHED LINEAR SYSTEM
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H = (Q,X,f,Init,Dom,E,G,R)
q1q2
EXAMPLE: SWITCHED LINEAR SYSTEM
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x1
x2
EXAMPLE: SWITCHED LINEAR SYSTEM
Cx = 0
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Proof that xe = 0 is a stable equilibrium of H for any CT2 <2 :
• xe = 0 is an equilibrium: f(q1,0) = f(q2,0) = 0
R((q1,q2),0) = R((q2,q1),0) = {0}
EXAMPLE: SWITCHED LINEAR SYSTEM
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Proof that xe = 0 is a stable equilibrium of H for any CT2 <2 :
• xe = 0 is an equilibrium: f(q1,0) = f(q2,0) = 0
R((q1,q2),0) = R((q2,q1),0) = {0}
• xe = 0 is stable:
consider the candidate Lyapunov-like function:
V(qi,x) = xT Pi x,
where Pi =PiT >0 solution to Ai
T Pi + Pi Ai = - I
(V(qi,x) is a Lyapunov function for the asymptotically stablelinear system qi)
In each discrete state, the continuous system is as. stable.
EXAMPLE: SWITCHED LINEAR SYSTEM
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Proof that xe = 0 is a stable equilibrium of H for any CT2 <2:
• xe = 0 is an equilibrium: f(q1,0) = f(q2,0) = 0
R((q1,q2),0) = R((q2,q1),0) = {0}
• xe = 0 is stable:
consider the candidate Lyapunov-like function:
V(qi,x) = xT Pi x,
where Pi =PiT >0 solution to Ai
T Pi + Pi Ai = - I
EXAMPLE: SWITCHED LINEAR SYSTEM
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Test for non-increasing sequence condition
Let q(i)=q1 and x(i)=z.
EXAMPLE: SWITCHED LINEAR SYSTEM
Cx = 0
zi
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Test for non-increasing sequence condition
Since V(q1,x(t)) is not increasing during [i,i’], then, when x(t)intersects the switching line at i’, it does at z with 2 (0,1],hence ||x(i+1)|| = ||x(i’)|| · ||x(i)||. Let q(i+1)=q2
Cx = 0
-z
i
EXAMPLE: SWITCHED LINEAR SYSTEM
i’=i+1
z
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Test for non-increasing sequence condition
Since V(q2,x(t)) is decreasing during [i+1,i+1’], then, when x(t)intersects the switching line at i+1’,||x(i+2)|| = ||x(i+1’)|| · ||x(i+1)|| · ||x(i)||
Cx = 0
-z
i
i’=i+1
i+2
EXAMPLE: SWITCHED LINEAR SYSTEM
z
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Test for non-increasing sequence conditionFrom this, it follows that V(q1,x(i+2)) · V(q1,x(i))
Cx = 0
-z
i
i’=i+1
i+2
EXAMPLE: SWITCHED LINEAR SYSTEM
z
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LYAPUNOV STABILITY
H = (Q,X,f,Init,Dom,E,G,R)
Drawbacks of the approach based on Lyapunov-like functions:
• In general, it is hard to find a Lyapunov-like function
• The sequence {V(q(i),x(i)): q(i) =q’} must be evaluated, whichmay require solving the ODEs
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LYAPUNOV STABILITY
H = (Q,X,f,Init,Dom,E,G,R)
Corollary (common Lyapunov function):Let xe = 0 be an equilibrium for H with R((q,q’),x) = {x}, 8
(q,q’)2 E, and D½ X=<n an open set containing xe = 0.
If V: D! < is a C1 function such that for all q 2 Q:
then, xe = 0 is a stable equilibrium of H.
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LYAPUNOV STABILITY
H = (Q,X,f,Init,Dom,E,G,R)
Corollary (common Lyapunov function):Let xe = 0 be an equilibrium for H with R((q,q’),x) = {x}, 8
(q,q’)2 E, and D½ X=<n an open set containing xe = 0.
If V: D! < is a C1 function such that for all q 2 Q:
then, xe = 0 is a stable equilibrium of H.
independent of q
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LYAPUNOV STABILITY
H = (Q,X,f,Init,Dom,E,G,R)
Corollary (common Lyapunov function):Let xe = 0 be an equilibrium for H with R((q,q’),x) = {x}, 8
(q,q’)2 E, and D½ X=<n an open set containing xe = 0.
If V: D! < is a C1 function such that for all q 2 Q:
then, xe = 0 is a stable equilibrium of H.
V(x) common Lyapunovfunction for all systems q
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LYAPUNOV STABILITY
H = (Q,X,f,Init,Dom,E,G,R)
Corollary (common Lyapunov function):Let xe = 0 be an equilibrium for H with R((q,q’),x) = {x}, 8
(q,q’)2 E, and D½ X=<n an open set containing xe = 0.
If V: D! < is a C1 function such that for all q 2 Q:
then, xe = 0 is a stable equilibrium of H.
Proof: Define W(q,x) = V(x), 8 q 2 Q and apply the previous theorem
V(x) common Lyapunovfunction for all systems q
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t
))(( txVsame V function+ identity reset map
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COMPUTATIONAL LYAPUNOV METHODS
HPL = (Q,X,f,Init,Dom,E,G,R)
non-Zeno and such that for all qk2 Q:• f(qk,x) = Ak x
(linear vector fields)• Init ½ [q2 Q {q } £ Dom(q)
(initialization within the domain)• for all x2 X, the set
Jump(qk,x):= {(q’,x’): (qk,q’)2 E, x2G((qk,q’)), x’2R((qk,q’),x)}has cardinality 1 if x 2 Dom(qk), 0 otherwise(discrete transitions occur only from the boundary of the domains)
• (q’,x’) 2 Jump(qk,x)! x’2 Dom(q’) and x’ = x(trivial reset for x)
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COMPUTATIONAL LYAPUNOV METHODS
HPL = (Q,X,f,Init,Dom,E,G,R)
non-Zeno and such that for all qk2 Q:• f(qk,x) = Ak x
(linear vector fields)• Init ½ [q2 Q {q } £ Dom(q)
(initialization within the domain)• for all x2 X, the set
Jump(qk,x):= {(q’,x’): (qk,q’)2 E, x2G((qk,q’)), x’2R((qk,q’),x)}has cardinality 1 if x 2 Dom(qk), 0 otherwise(discrete transitions occur only from the boundary of the domains)
• (q’,x’) 2 Jump(qk,x)! x’2 Dom(q’) and x’ = x(trivial reset for x)
For this class of (non-blocking, deterministic) Piecewise Linear hybridautomata computationally attractive methods exist to construct theLyapunov-like function
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GLOBALLY QUADRATIC LYAPUNOV FUNCTION
HPL = (Q,X,f,Init,Dom,E,G,R)
Theorem (globally quadratic Lyapunov function):
Let xe = 0 be an equilibrium for HPL.
If there exists P=PT >0 such thatAk
T P+ PAk < 0, 8 k
Then, xe = 0 is asymptotically stable.
Remark:
V(x)=xTPx is a common Lyapunov function xe = 0 is stable
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GLOBALLY QUADRATIC LYAPUNOV FUNCTION
Proof (showing exponential stability):There exists >0 such that Ak
T P+ PAk + I · 0, 8 k
There exists a unique, infinite, non-Zeno execution (,q,x) forevery initial state with x: ! <n satisfying
where k: ! [0,1] is such that k k(t)=1, t2 [i,i’].
Let V(x) = xT Px. Then, for t2 [i,i’).
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GLOBALLY QUADRATIC LYAPUNOV FUNCTION
Proof. (cont’d)Since min ||x||2 · V(x) · max ||x||2, then
and, hence,
which leads to
Then,
Since 1 =1 (non-Zeno), then ||x(t)|| goes to zeroexponentially as t! 1
min and max eigenvalues of P
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GLOBALLY QUADRATIC LYAPUNOV FUNCTION
q1q2
conditions of the theorem satisfied with P = I
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GLOBALLY QUADRATIC LYAPUNOV FUNCTION
q1q2
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GLOBALLY QUADRATIC LYAPUNOV FUNCTION
HPL = (Q,X,f,Init,Dom,E,G,R)
Theorem (globally quadratic Lyapunov function):
Let xe = 0 be an equilibrium for HPL.
If there exists P=PT >0 such thatAk
T P+ PAk < 0, 8 k
Then, xe = 0 is asymptotically stable.
Remark:
A set of LMIs to solve. This problem can be reformulated as aconvex optimization problem. Efficient solvers exist.
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GLOBALLY QUADRATIC LYAPUNOV FUNCTION
Suppose that Ak , k=1,2,…,N, are Hurwitz matrices.
Then, the set of linear matrix inequalities
AkT P+ PAk < 0, k=1,2,…,N,
where P is positive definite symmetric is not feasible if andonly if there exist positive definite symmetric matrices Rk,k=1,2,…,N, such that
Proof. Based on results in convex analysis
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GLOBALLY QUADRATIC LYAPUNOV FUNCTION
q1q2
stable node stable focus
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GLOBALLY QUADRATIC LYAPUNOV FUNCTION
q1q2
no globally quadratic Lyapunov function exists although xe = 0 stableequilibrium
stable node stable focus
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PIECEWISE QUADRATIC LYAPUNOV FUNCTION
• Idea:consider different quadratic Lyapunov functions on eachdomain and glue them together so as to provide a (non-quadratic) Lyapunov function for H that is continuous at theswitching times
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PIECEWISE QUADRATIC LYAPUNOV FUNCTION
• Idea:consider different quadratic Lyapunov functions on eachdomain and glue them together so as to provide a (non-quadratic) Lyapunov function for H that is continuous at theswitching times
• Developed for piecewise linear systems with structureddomain and reset
• LMIs characterization
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COMPUTATIONAL LYAPUNOV METHODS
HPL = (Q,X,f,Init,Dom,E,G,R)
non-Zeno and such that for all qk2 Q:• f(qk,x) = Ak x
(linear vector fields)• Init ½ [q2 Q {q } £ Dom(q)
(initialization within the domain)• for all x2 X, the set
Jump(qk,x):= {(q’,x’): (qk,q’)2 E, x2G((qk,q’)), x’2R((qk,q’),x)}has cardinality 1 if x 2 Dom(q), 0 otherwise(discrete transitions occur only from the boundary of the domains)
• (q’,x’) 2 Jump(qk,x)! x’2 Dom(q’) and x’ = x(trivial reset for x)
![Page 103: Hybrid Systems Course Lyapunov stability...STABILITY OF LINEAR CONTINUOUS SYSTEMS Theorem (necessary and sufficient condition): The equilibrium point xe =0 is asymptotically stable](https://reader035.fdocuments.in/reader035/viewer/2022062416/610f34e3ddd6f7027d2b6772/html5/thumbnails/103.jpg)
PIECEWISE QUADRATIC LYAPUNOV FUNCTION
HPL = (Q,X,f,Init,Dom,E,G,R)
satisfies the following additional assumptions:• Dom(q) = {x 2 X: Eq1x ¸ 0, Eq2 x ¸ 0, … , Eqn x¸ 0}
(each domain is a polygon)Eq = [Eq1
T Eq2T … Eqn
T] T 2 <n£ n defines the domain.
• (q’,x’) 2 Jump(q,x) Fq’x = Fqx, q’q, x’=x(matching condition at the boundaries of the domain)
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PIECEWISE QUADRATIC LYAPUNOV FUNCTION
HPL = (Q,X,f,Init,Dom,E,G,R)
Theorem (piecewise quadratic Lyapunov function):
Let xe = 0 be an equilibrium for HPL.
If there exists Uk=UkT, Wk=Wk
T, with all non-negative elements and
M=MT, such that Pk=FkTMFk satisfies
AkT Pk+ PAk + Ek
TUkEk < 0
Pk – EkTWkEk > 0
Then, xe = 0 is asymptotically stable.
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PIECEWISE QUADRATIC LYAPUNOV FUNCTION
HPL = (Q,X,f,Init,Dom,E,G,R)
Theorem (piecewise quadratic Lyapunov function):
Let xe = 0 be an equilibrium for HPL.
If there exists Uk=UkT, Wk=Wk
T, with all non-negative elements and
M=MT, such that Pk=FkTMFk satisfies
AkT Pk+ PAk + Ek
TUkEk < 0
Pk – EkTWkEk > 0
Then, xe = 0 is asymptotically stable.
Proof based on the fact that V(x)=xTPkx, xDom(qk) is aLyapunov-like function for HPL, strictly decreasing along itsexecutions
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PIECEWISE QUADRATIC LYAPUNOV FUNCTION
HPL = (Q,X,f,Init,Dom,E,G,R)
Theorem (piecewise quadratic Lyapunov function):
Let xe = 0 be an equilibrium for HPL.
If there exists Uk=UkT, Wk=Wk
T, with all non-negative elements and
M=MT, such that Pk=FkTMFk satisfies
AkT Pk+ PAk + Ek
TUkEk < 0
Pk – EkTWkEk > 0 Pk positive definite within Dom(k)
Then, xe = 0 is asymptotically stable.
Proof based on the fact that V(x)=xTPkx, xDom(qk) is aLyapunov-like function for HPL, strictly decreasing along itsexecutions
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PIECEWISE QUADRATIC LYAPUNOV FUNCTION
HPL = (Q,X,f,Init,Dom,E,G,R)
Theorem (piecewise quadratic Lyapunov function):
Let xe = 0 be an equilibrium for HPL.
If there exists Uk=UkT, Wk=Wk
T, with all non-negative elements and
M=MT, such that Pk=FkTMFk satisfies
AkT Pk+ PAk + Ek
TUkEk < 0 AkT Pk+ PAk < 0 within Dom(k)
Pk – EkTWkEk > 0 Pk positive definite within Dom(k)
Then, xe = 0 is asymptotically stable.
Proof based on the fact that V(x)=xTPkx, xDom(qk) is aLyapunov-like function for HPL, strictly decreasing along itsexecutions
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PIECEWISE QUADRATIC LYAPUNOV FUNCTION
HPL = (Q,X,f,Init,Dom,E,G,R)
Theorem (piecewise quadratic Lyapunov function):
Let xe = 0 be an equilibrium for HPL.
If there exists Uk=UkT, Wk=Wk
T, with all non-negative elements and
M=MT, such that Pk=FkTMFk satisfies
AkT Pk+ PAk + Ek
TUkEk < 0
Pk – EkTWkEk > 0
Then, xe = 0 is asymptotically stable.
Proof based on the fact that V(x)=xTPkx, xDom(qk) is aLyapunov-like function for HPL, strictly decreasing along itsexecutions
continuitity of V(x)
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GLOBALLY QUADRATIC LYAPUNOV FUNCTION
q1q2
level curves of thepiecewise quadraticLyapunov function(red lines)
phase plot of somecontinuous statetrajectories(blue lines)
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REFERENCES
• H.K. Khalil.Nonlinear Systems.Prentice Hall, 1996.
• S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan.Linear Matrix Inequalities in System and Control Theory.SIAM, 1994.
• M. Branicky.Multiple Lyapunov functions and other analysis tools for switched andhybrid systems.IEEE Trans. on Automatic Control, 43(4):475-482, 1998.
• H. Ye, A. Michel, and L. Hou.Stability theory for hybrid dynamical systems.IEEE Transactions on Automatic Control, 43(4):461-474, 1998.
• M. Johansson and A. Rantzer.Computation of piecewise quadratic Lyapunov function for hybridsystems.IEEE Transactions on Automatic Control, 43(4):555-559, 1998.
• R.A. Decarlo, M.S. Branicky, S. Petterson, and B. Lennartson.Perspectives and results on the stability and stabilization of hybridsystems.Proceedings of the IEEE, 88(7):1069-1082, 2000.