Hybrid Systems - Lecture n. 3 Lyapunov stabilityhome.deib.polimi.it/prandini/file/hybrid systems 9...

Hybrid Systems - Lecture n. 3Lyapunov stability

Maria Prandini

DEI - Politecnico di MilanoE-mail: [email protected]

OUTLINE

Focus: stability of equilibrium point

• continuous systems decribed by ordinary differentialequations (brief review)

• hybrid automata

OUTLINE



• hybrid automata

ORDINARY DIFFERENTIAL EQUATIONS

An ordinary differential equation is a mathematical model of a continuous state continuous time system:

X = <n ≡ state spacef: <n → <n ≡ vector field (assigns a “velocity” vector to each x)

ORDINARY DIFFERENTIAL EQUATIONS

An ordinary differential equation is a mathematical model of a continuous state continuous time system:

X = <n ≡ state spacef: <n → <n ≡ vector field (assigns a “velocity” vector to each x)

Given an initial value x0 ∈ X,an execution (solution in the sense of Caratheodory) over the time interval [0,T) is a function x: [0,T) → <n such that:

• x(0) = x0

• x is continuous and piecewise differentiable

•

ODE SOLUTION: WELL-POSEDNESS

Theorem [local existence non-blocking]If f: <n → <n is continuous, then ∀x0 there exists at least a solution with x(0)=x0 defined on some [0,δ).

Theorem [local existence and uniqueness non-blocking, deterministic]If f: <n → <n is Lipschitz continuous, then ∀ x0 there exists a single solution with x(0)=x0 defined on some [0,δ).

Theorem [global existence and uniqueness non-blocking, deterministic, non-Zeno]If f: <n → <n is globally Lipschitz continuous, then ∀ x0 thereexists a single solution with x(0)=x0 defined on [0,∞).

STABILITY OF CONTINUOUS SYSTEMS

with f: <n → <n globally Lipschitz continuous

Definition (equilibrium): xe ∈ <n for which f(xe)=0

Remark: xe is an invariant set

Definition (stable equilibrium):The equilibrium point xe ∈ <n is stable (in the sense ofLyapunov) if

execution startingfrom x(0)=x0

Definition (stable equilibrium):

Graphically:

δxe

equilibrium motion

perturbed motion

ε

small perturbations lead to small changes in behavior

Definition (stable equilibrium):

Graphically:

small perturbations lead to small changes in behavior

δxe

ε phase plot

Definition (asymptotically stable equilibrium):

and δ can be chosen so that

Graphically:

δxe

equilibrium motion

perturbed motion

ε

small perturbations lead to small changes in behaviorand are re-absorbed, in the long run

Definition (asymptotically stable equilibrium):

and δ can be chosen so that

Graphically:

small perturbations lead to small changes in behaviorand are re-absorbed, in the long run

δxe

ε

EXAMPLE: PENDULUM

m

l

frictioncoefficient (α)

EXAMPLE: PENDULUM

unstable equilibrium

m EXAMPLE: PENDULUM

as. stable equilibriumm

EXAMPLE: PENDULUM

m

l

EXAMPLE: PENDULUM

as. stable equilibrium

small perturbations are absorbed, not allperturbations

m

m

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

More definitions: exponentially stable, globally exponentially stable, ...

execution startingfrom x(0)=x0



Definition (equilibrium): xe ∈ <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)



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)


execution x(t)

candidate function V(x)

behavior of V along the execution x(t): V(t): = V(x(t))

Advantage with respect to exhaustive check of all executions?


V: <n → < continuously differentiable (C1) function

Rate of change of V along the execution of the ODE system:


gradient vector

No need to solve the ODE for evaluating if V(x) decreasesalong the executions of the system

LYAPUNOV STABILITY

Theorem (Lyapunov stability Theorem):Let xe = 0 be an equilibrium for the system and D⊂ <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)

LYAPUNOV STABILITY



Then, xe is stable.

Lyapunov functionfor the system and the equilibrium xe

Finding Lyapunov functions is HARD in generalSufficient condition

Proof:

Given ε >0, choose r∈ (0,ε) such that Br = x∈ <n: ||x|| · r ⊂ DSet α : = minV(x): ||x|| = r > 0 and choose c ∈ (0,α). Then,

Ωc := x: V(x) · c ⊂ Br

Since then V(x(t))· V(x(0)), ∀ t≥ 0. Hence, all executions starting in Ωc stays in Ωc.

V(x) is continuous and V(0) = 0. Then, there is δ >0 such that Bδ = x∈ <n: ||x|| · δ ⊂ Ωc .

Therefore, ∀ ||x(0)|| < δ ⇒ ||x(t)|| < ε, ∀ t≥ 0

EXAMPLE: PENDULUM

m

l

frictioncoefficient (α)

energy function

xe stable

LYAPUNOV STABILITY



Then, xe is stable.

If it holds also that

Then, xe is asymptotically stable (AS)

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

STABILITY OF LINEAR CONTINUOUS SYSTEMS

• xe = 0 is an equilibrium for the system

• the elements of matrix eAt are linear combinations of eλi(A)t, i=1,2,…,n



• xe =0 is asymptotically stable if and only if A is Hurwitz (all eigenvalues with real part <0)

• asymptotic stability ≡ GAS



• xe =0 is asymptotically stable if and only if A is Hurwitz (alleigenvalues with real part <0)

• asymptotic stability ≡ GAS

Alternative characterization…


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




ATP+PA = -Q


Remarks: Q positive definite (Q>0) iff xTQx >0 for all x ≠ 0Q positive semidefinite (Q≥ 0) iff xTQx ≥ 0 for all x and xT Q x = 0 for some x ≠ 0

Lyapunov equation




ATP+PA = -Q


Proof.

(if) V(x) =xT P x is a Lyapunov function

Lyapunov equation




ATP+PA = -Q


Proof.

(only if) Consider

Lyapunov equation




ATP+PA = -Q


Proof.

(only if) Consider

P = PT and P>0 easy to show

P unique by contradiction

Lyapunov equation




ATP+PA = -Q


Remarks: for a linear system

• existence of a (quadratic) Lyapunov function V(x) =xT P x is a necessary and sufficient condition

• it is easy to compute a Lyapunov function since the Lyapunovequation is a linear algebraic equation

Lyapunov equation


Theorem (exponential convergence):

Let the equilibrium point xe =0 be asymptotically stable. Then, the rate of convergence to xe =0 is exponential:

for all x(0) = x0 ∈ <n, where -λ0 ∈ (maxi Reλi(A), 0) and µ>0 is an appropriate constant.




for all x(0) = x0 ∈ <n, where -λ0 ∈ (maxi Reλi(A), 0) and µ >0 is an appropriate constant.

Re

Im

o

o

o o

eigenvalues of A





Proof (exponential convergence):

A + λ0 I is Hurwitz (eigenvalues are equal to λ(A) + λ0)

Then, there exists P = PT >0 such that

(A + λ0I)T P + P (A + λ0I) <0which leads to

x(t)T[AT P + P A]x(t) < - 2 λ0 x(t)T P x(t) Define V(x) = xT P x, then

from which


(cont’d) Proof (exponential convergence):

thus finally leading to


(cont’d) Proof (exponential convergence):

thus finally leading to

OUTLINE



• hybrid automata

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 ⊆ Q× X is a set of initial states

• Dom: Q → 2X assigns to each q∈ Q a domain Dom(q) of X

• E ⊆ 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

q = q1

q = q2

HYBRID TIME SET

A hybrid time set is a finite or infinite sequence of intervals

τ = Ii, i=0,1,…, M such that

• Ii = [τi, τi’] for i < M • IM = [τM, τM’] or IM = [τM, τM’) if M<∞

• τi’ = τi+1

• τi · τi’

[ ][ ]

[ ]

τ0

I0

τ0’

τ1 τ1’I1

I2 τ2 = τ2’

τ3 τ3’I3

HYBRID TIME SET

A hybrid time set is a finite or infinite sequence of intervals

τ = Ii, i=0,1,…, M such that

• Ii = [τi, τi’] for i < M • IM = [τM, τM’] or IM = [τM, τM’) if M<∞

• τi’ = τi+1

• τi · τi’

[ ][ ]

[ ]

τ0

I0

τ0’

τ1 τ1’I1

I2 τ2 = τ2’

τ3 τ3’I3

t1

t2

t3

t4

t1 ≺ t2 ≺ t3 ≺ t4

the elements of τ arelinearly ordered

τ∞ := ∑i(τi’-τi)(continuous extent)

HYBRID TRAJECTORY

A hybrid trajectory (τ, q, x) consists of:

• A hybrid time set τ = 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

HYBRID AUTOMATA: EXECUTION

A hybrid trajectory (τ, q, x) is an execution (solution) of the hybrid automaton H = (Q,X,f,Init,Dom,E,G,R) if it satisfies the following conditions:

• Initial condition: (q0(τ0), x0(τ0)) ∈ Init

• Continuous evolution:for all i such that τi < τi’

qi: Ii → Q is constantxi:Ii→ X is the solution to the ODE associated with qi(τi)xi(t) ∈ Dom(qi(τi)), t∈ [τi,τi’)

• Discrete evolution:(qi(τi’),qi+1(τi+1)) ∈ E transition is feasiblexi(τi’) ∈ G((qi(τi’),qi+1(τi+1))) guard condition satisfiedxi(τi+1) ∈ R((qi(τi’),qi+1(τi+1)),xi(τi’)) reset condition satisfied

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)∞ the set of infinite executions of H starting from (q,x)

STABILITY OF HYBRID AUTOMATA

H = (Q,X,f,Init,Dom,E,G,R)

Definition (equilibrium): xe =0 ∈ X is an equilibrium point of H if:• f(q,0) = 0 for all q ∈ Q• ((q,q’)∈ E) ∧ (0∈ G((q,q’)) ⇒ R((q,q’),0) = 0

Remarks:

• discrete transitions are allowed out of (q,0) but only to (q’,0)• for all (q,0) ∈ Init and (τ, q, x) is an execution of H starting

from (q,0), then x(t) = 0 for all t∈ τ

EXAMPLE: SWITCHING LINEAR SYSTEM


• Q = q1, q2 X = <2

• f(q1,x) = A1x and f(q2,x) = A2x with:

• Init = Q × x∈ X: ||x|| >0

• Dom(q1) = x∈ X: x1x2 · 0 Dom(q2) = x∈ X: x1x2 ≥ 0

• E = (q1,q2),(q2,q1)• G((q1,q2)) = x∈ X: x1x2 ≥ 0 G((q2,q1)) = x∈ 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


Definition (stable equilibrium): Let xe = 0 ∈ 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


set of (maximal) executionsstarting from (q0, x0) ∈ Init



Definition (asymptotically stable equilibrium): Let xe = 0 ∈ X be an equilibrium point of H. xe = 0 is asymptotically

stable if δ>0 that can be chosen so that


set of (maximal) executionsstarting from (q0, x0) ∈ Init

set of infinite executionsstarting from (q0, x0) ∈ Initτ∞ < ∞ if Zeno



Question:

xe = 0 stable equilibrium for each continuous system dx/dt = f(q,x) implies that xe = 0 stable equilibrium for H?

STABILITY OF HYBRID AUTOMATA EXAMPLE: SWITCHING LINEAR SYSTEM


• Q = q1, q2 X = <2


• Init = Q × x∈ X: ||x|| >0

• Dom(q1) = x∈ X: x1x2 · 0 Dom(q2) = x∈ X: x1x2 ≥ 0

• E = (q1,q2),(q2,q1)• G((q1,q2)) = x∈ X: x1x2 ≥ 0 G((q2,q1)) = x∈ 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


Asymptotically stable linear systems.


q1: quadrants 2 and 4q2: quadrants 1 and 3

Asymptotically stable linear systems, but xe = 0 unstableequilibrium of H


q1: quadrants 1 and 3q2: quadrants 2 and 4


||x(τi)|| < ||x(τi+1)|| ||x(τi)|| > ||x(τi+1)||overshoots sum up

LYAPUNOV STABILITY


Theorem (Lyapunov stability):Let xe = 0 be an equilibrium for H with R((q,q’),x) = x, ∀ (q,q’)∈ E, and

D⊂ X=<n an open set containing xe = 0. Consider V: Q× D → < is a C1 function in x such that for all q ∈ Q:

If for all (τ, q, x) ∈ H(q0,x0) with (q0,x0) ∈ Init ∩ (Q× D), and all q’∈ Q, the sequence V(q(τi),x(τi)): q(τi) =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

LYAPUNOV STABILITY


Theorem (Lyapunov stability):Let xe = 0 be an equilibrium for H with R((q,q’),x) = x, ∀ (q,q’)∈ E, and

D⊂ X=<n an open set containing xe = 0. Consider V: Q× D → < is a C1 function in x such that for all q ∈ Q:

If for all (τ, q, x) ∈ H(q0,x0) with (q0,x0) ∈ Init ∩ (Q× D), and all q’∈ Q, the sequence V(q(τi),x(τi)): q(τi) =q’ is non-increasing (or empty),


V(q,x) Lyapunov functionfor continuous system q⇒ xe =0 is stableequilibrium for system q

LYAPUNOV STABILITY


Sketch of the proof.

V(q(t),x(t))V(q1,x(t))

[ ][ ][ ][ q(t)= q1 q(t)= q1V(q2,x(t))

τ0 τ0’=τ1 τ1’=τ2 τ2’=τ3

Lyapunov function forsystem q1 → decreaseswhen q(t) = q1, but can increase when q(t) ≠ q1

LYAPUNOV STABILITY




[ ]τ0 τ0’=τ1

[ ][ ]τ1’=τ2 τ2’=τ3

[ q(t)= q1 q(t)= q1

V(q1,x(τi))non-increasing

LYAPUNOV STABILITY




[ ]τ0 τ0’=τ1

[ ][ ]τ1’=τ2 τ2’=τ3

[ q(t)= q1 q(t)= q1

LYAPUNOV STABILITY


V(q(t),x(t)) Lyapunov-like function



• Q = q1, q2 X = <2


• Init = Q × x∈ X: ||x|| >0

• Dom(q1) = x∈ X: Cx ≥ 0 Dom(q2) = x∈ X: Cx · 0

• E = (q1,q2),(q2,q1)• G((q1,q2)) = x∈ X: Cx · 0 G((q2,q1)) = x∈ X: Cx ≥ 0, CT∈ <2

• R((q1,q2),x) = R((q2,q1),x) = x



q1q2


Proof that xe = 0 is a stable equilibrium of H for any CT∈ <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...


Proof that xe = 0 is a stable equilibrium of H for any CT∈ <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


Test for non-increasing sequence condition

The level sets of V(qi,x) = xTPi x are ellipses centered at the origin. A level set intersects the switching line CTx =0 at z and -z.

CTx = 0

z

-z

τi

τi’=τi+1

τi+2


Test for non-increasing sequence condition

The level sets of V(qi,x) = xTPi x are ellipses centered at the origin. A level set intersects the switching line CTx =0 at z and -z.

Let q(τi)=q1 and x(τi)=z.

Since V(q1,x(t)) is not increasing during [τi,τi’], then, when x(t) intersects the switching line at τi’, it does at α z with α ∈ (0,1], hence ||x(τi+1)|| = ||x(τi’)|| · ||x(τi)||. Let q(τi+1)=q2

Since V(q2,x(t)) is decreasing during [τi+1,τi+1’], then, when x(t) intersects the switching line at τi+1’, ||x(τi+2)|| = ||x(τi+1’)|| · ||x(τi+1)|| · ||x(τi)||

From this, it follows that V(q1,x(τi+2)) · V(q1,x(τi))

LYAPUNOV STABILITY


Drawbacks:

• The sequence V(q(τi),x(τi)): q(τi) =q’ must be evaluated, whichmay require solving the ODEs

• In general, it is hard to find a Lyapunov-like function

LYAPUNOV STABILITY


Corollary (Lyapunov stability Theorem):Let xe = 0 be an equilibrium for H with R((q,q’),x) = x, ∀

(q,q’)∈ E, and D⊂ X=<n an open set containing xe = 0.

If V: D → < is a C1 function such that for all q ∈ Q:


Proof: Define W(q,x) = V(x), ∀ q ∈ Q and apply the theorem

V(x) common Lyapunovfunction for all systems q

t

))(( txVsame V function+ identity reset map

COMPUTATIONAL LYAPUNOV METHODS

HPL = (Q,X,f,Init,Dom,E,G,R)

non-Zeno and such that for all qk∈ Q:• f(qk,x) = Ak x

(linear vector fields)• Init ⊂ ∪q∈ Q q × Dom(q)

(admissible initialization)• for all x∈ X, the set

Jump(qk,x):= (q’,x’): (qk,q’)∈ E, x∈G((qk,q’)), x’∈R((q,q’),x)has cardinality 1 if x ∈ ∂Dom(q), 0 otherwise(discrete transitions occur only from the boundary of the domains)

• (q’,x’) ∈ Jump(qk,x) → x’∈ Dom(q’) and x’ = x(trivial reset for x)

For this class of Piecewise Linear hybrid automata computationallyattractive methods exist to construct the Lyapunov-like function

GLOBALLY QUADRATIC LYAPUNOV FUNCTION


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, ∀ k

Then, xe = 0 is asymptotically stable.

Proof.

V(x) = xT Px is a common Lyapunov function


Alternative proof (showing exponential convergence):There exists γ >0 such that Ak

T P+ PAk +γ I · 0, ∀ k

There exists a unique, infinite, non-Zeno execution (τ,q,x) forevery initial state with x: τ → <n satisfying

where µk: τ → [0,1] is such that ∑k µk(t)=1, t∈ [τi,τi’].

Let V(x) = xT Px. Then, for t∈ [τi,τi’).


Sketch of the proof. (cont’d)Since λmin(P) ||x||2 · V(x) · λmax(P) ||x||2, then

and, hence,

which leads to

Then,

Since τ∞ =∞ (non-Zeno), then ||x(t)|| goes to zero exponentially as t→ τ∞


q1q2

conditions of the theoremsatisfied with P = I



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, ∀ k

Then, xe = 0 is asymptotically stable.

Remark:

A set of LMIs to solve. This problem can be reformulated as a convex optimization problem. Efficient solvers exist.


• A globally quadratic Lyapunov function does not exist if and only if there exist positive definite symmetric matrices Rksuch that


q1q2

conditions of the theoremNOT satisfied for any P but xe = 0 stable equilibrium

stable node stable focus

piecewise quadraticLyapunov function

PIECWISE QUADRATIC LYAPUNOV FUNCTION

• Developed for piecewise linear systems with structureddomain (set of convex polyhedra) and reset

• Resulting Lyapunov function is continuous at the switchingtimes

• LMIs characterization

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 and hybrid 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.

Hybrid Systems - Lecture n. 3 Lyapunov stabilityhome.deib.polimi.it/prandini/file/hybrid systems 9...

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