Input-to-state stability of distributed parameter systems

Input-to-state stabilityof

distributed parameter systems

Andrii Mironchenko

Institute of MathematicsUniversity of Würzburg

University of Illinois at Urbana-Champaign25 September 2013

Outline

1 Basic definitions

2 ISS of linear systems

3 ISS of nonlinear systems

4 Impulsive systems

5 Summary and Outlook

Andrii Mironchenko ISS of distributed parameter systems UI at Urbana-Champaign 2 / 30

Class of systems

{x(t) = Ax(t) + f (x(t),u(t)), x(t) ∈ D(A) ⊂ X ,u(t) ∈ U,x(0) = φ0.

X = State spaceUc = PC(R+,U)

Ax = limt→+01t (T (t)x − x).

f (0,0) = 0.

x ∈ C([0,T ],X ) is a mild solution iff

x(t) = T (t)φ0 +

∫ t

0T (t − s)f (x(s),u(s))ds.


Examples of generators


ODEs: A is a matrix, T (t) = etA.Parabolic equations: A = ∆

Hyperbolic equations: A =

(0 I∆ 0

)Schrödinger equation: A = i∆Delay equations: A is a delay operator.


Comparison functions

K∞ := {γ : R+ → R+ | γ(0) = 0, γ is continuous, growing and unbounded}L :=

{γ : R+ → R+ | γ is continuous, strictly decreasing and lim

t→∞γ(t) = 0

}KL := {β : R+ × R+ → R+ | β(·, t) ∈ K, ∀t ≥ 0, β(r , ·) ∈ L, ∀r > 0}


Input-to-state stabilityDefinition (GAS uniform w.r.t. state (0-UGASx))0-UGASx :⇔ ∃β ∈ KL: ∀φ0 ∈ X , ∀t ≥ 0

‖φ(t , φ0,0)‖X ≤ β(‖φ0‖X , t).

Definition (ISS)ISS :⇔ ∃β ∈ KL, γ ∈ K∞: ∀t ≥ 0, ∀φ0 ∈ X , ∀u ∈ Uc

‖φ(t , φ0,u)‖X ≤ max{β(‖φ0‖X , t), γ︸︷︷︸

Gain

(‖u‖Uc )}.

‖x(t)‖X

β(‖φ0‖X , t)

γ(‖u‖Uc )

t


Input-to-state stabilityDefinition (GAS uniform w.r.t. state (0-UGASx))0-UGASx :⇔ ∃β ∈ KL: ∀φ0 ∈ X , ∀t ≥ 0

‖φ(t , φ0,0)‖X ≤ β(‖φ0‖X , t).

Definition (ISS)ISS :⇔ ∃β ∈ KL, γ ∈ K∞: ∀t ≥ 0, ∀φ0 ∈ X , ∀u ∈ Uc

‖φ(t , φ0,u)‖X ≤ max{β(‖φ0‖X , t), γ︸︷︷︸

Gain

(‖u‖Uc )}.

‖x(t)‖X

β(‖φ0‖X , t)

γ(‖u‖Uc )

tAndrii Mironchenko ISS of distributed parameter systems UI at Urbana-Champaign 6 / 30

Fundamentals of ISS-Theory for ODEs

ISSExistence of

ISS-Lyapunovfunction

Characterizationsof ISS

Small-gaintheorems

SoW96

SoW95

JTP94 DRW07 JTP96 DRW06


Stability concepts for∞-dim systems

Σ : x = Ax + Bu, x(0) = x0,B ∈ L(U,X ).

φ(t , x0,u) = T (t)x0 +

∫ t

0T (t − r)Bu(r)dr ,

Fact

0-UGASx ⇔ ∃M, λ > 0 : ‖T (t)‖ ≤ Me−λt ⇔ T exp. stable .

Fact0-GAS ⇔ limt→∞ ‖T (t)x‖ = 0 ∀x ∈ X ⇔ T strongly stable.

For∞-dim systems: GAS 6= 0-UGASx .


ISS of linear systems for Uc := PC(R+,U)

Σ : x = Ax + Bu, x(0) = x0.

φ(t , x0,u) = T (t)x0 +

∫ t

0T (t − r)Bu(r)dr ,

Fact

Σ is 0-UGASx ⇔ ∃M, λ > 0 : ‖T (t)‖ ≤ Me−λt .

‖φ(t , x0,u)‖X ≤ ‖T (t)‖‖x0‖X +

∫ t

0‖T (t − r)‖‖B‖‖u(r)‖Udr ,

≤ Me−λt‖x0‖X︸︷︷︸β(‖x0‖X ,t)

+K supr∈[0,t]

‖u(r)‖U , K > 0.

Lemma

Σ is 0-UGASx ⇔ Σ is ISS w.r.t. L∞⇒6⇐ Σ is 0-GAS


ISS theory for linear systems with bounded inputoperators

0-UGASISS

w.r.t. L∞

wISS

0-GAS

ISSw.r.t. Lp

wISSw.r.t. L1

T is expstable

T isstronglystable

6⇑6⇑


Lyapunov functions


Definition

V : X → R+ is ISS-Lyapunov function iff ∃ψ1, ψ2, χ, α ∈ K∞:

ψ1(‖x‖X ) ≤ V (x) ≤ ψ2(‖x‖X )

V (x) ≥ χ(‖ξ‖U) ⇒ Vu(x) ≤ −α(V (x)),

∀x ∈ X , ∀ξ ∈ U, ∀u ∈ Uc with u(0) = ξ.Here Vu(x) = lim

t→+01t (V (φ(t , x ,u))− V (x)).

Theorem∃ ISS-Lyapunov function⇒ ISS.


Example

{∂s∂t = ∂2s

∂x2 − f (s) + u(x , t), x ∈ (0, π), t > 0,s(0, t) = s(π, t) = 0.

Here f is locally Lipschitz, uneven and monotonically increasing up toinfinity.Let u(·, t) ∈ L2(0, π). DefineAs = d2s

dx2 with D(A) = H10 (0, π) ∩ H2(0, π).

dsdt

= As − f (s) + u, t > 0.

This equation defines a control system with X = H10 (0, π) and

U = L2(0, π).


V (s) =

∫ π

0

(12

s2x (x) +

∫ s(x)

0f (y)dy

)dx .

First property of LF:∫ s(x)

0f (y)dy ≥ 0 ⇒ V (s) ≥

∫ π

0

12

s2x (x)dx =

12‖s‖2H1

0 (0,π).

The derivative of V along the trajectories is equal

V (s) = −∫ π

0(sxx (x)− f (s(x)))2dx︸︷︷︸

I(s)

+

∫ π

0(sxx (x)− f (s(x)))(−u)dx .

Using Cauchy-Schwarz inequality for the second term, we have:

V (s) ≤ −I(s) +√

I(s) ‖u‖L2(0,π).


For s ∈ H10 (0, π) ∩ H2(0, π) using Friedrich’s inequality one can prove:

I(s) ≥∫ π

0s2

xx (x)dx ≥∫ π

0s2

x (x)dx = ‖s‖2H10 (0,π)

.

Define the gain as χ(r) = ar , a > 1.Assuming

‖s‖H10 (0,π)

≥ χ(‖u‖L2(0,π))

we obtain

V (s) ≤ −I(s) +1a

√I(s)‖s‖H1

0 (0,π)≤ −(1− 1

a)I(s) ≤ −(1− 1

a)‖s‖2H1

0 (0,π).

This proves, that V is an ISS-Lyapunov function.


Interconnected systems

Σ :

{Σi : xi = Aixi + fi(x1, . . . , xn,u), xi ∈ Xii = 1, . . . ,n

Xi state space of Σi

Ai infinitesimal generator of C0-semigroup on Xi .

X = X1 × . . .× Xn state space of the whole system.

Xi := X1 × . . .× Xi−1 × Xi+1 × . . .× Xn × Uspace of inputs into i-th subsystem.


ISS-Lyapunov functions for subsystems

Σ :

{Σi : xi = Aixi + fi(x1, . . . , xn,u), xi ∈ Xii = 1, . . . ,n

ISS-LF for Σi

Vi : Xi → R+ is ISS-Lyapunov function for Σi iff∃ψi1, ψi2, αi , χi , χij ∈ K∞, j = 1, . . . ,n:

ψi1(‖xi‖Xi ) ≤ Vi(xi) ≤ ψi2(‖xi‖Xi )

Vi(xi) ≥ max{ n

maxj=1

χij(Vj(xj)), χi(‖ξ‖U)}⇒ Vi(xi) ≤ −αi(Vi(xi)),

∀xi ∈ Xi , ∀xi ∈ Xi , ∀v ∈ PC(R+, Xi) with v(0) = xi .


Small-gain theorem

Gain matrix: ΓM = (χij)i,j=1,...,n, χij ∈ K∞ ∪ {0}.Gain operator: Γ : Rn

+ → Rn+

Γ(s) :=

(n

maxj=1

χ1j(sj), . . . ,n

maxj=1

χnj(sj)

), s ∈ Rn

+.

Theorem (Dashkovskiy, M., MCSS, 2013)

Let Vi be ISS-Lyapunov function for Σi with gains χij .

Γ(s) 6≥ s, ∀s ∈ Rn+\ {0} (SGB)

⇒Σ is ISS.V (x) := maxi{σ−1

i (Vi(xi))} is a Lyapunov function for Σ.

For n = 2 (SGB) ⇔ χ12 ◦ χ21 < id .


Example: interconnected diffusion equations

∂s1∂t = c1

∂2s1∂x2 + a12s2, x ∈ (0,d), t > 0,

s1(0, t) = s1(d , t) = 0;∂s2∂t = c2

∂2s2∂x2 + a21s1, x ∈ (0,d), t > 0,

s2(0, t) = s2(d , t) = 0.

State spaces:X1 = X2 = L2(0,d)

Generators:

Ai = cid2

dx2 with D(Ai) = H10 (0,d) ∩ H2(0,d).

ISS-Lyapunov functions for subsystems:

Vi(si) =1

2ci

(dπ

)2

‖si‖2L2(0,d).


V1(s1) =1c1

(dπ

)2 ∫ d

0s1(x)

(c1∂2s1

∂x2 + a12s2(x)

)dx

= −(

dπ

)2 ∫ d

0

(∂s∂x

)2

dx +1c1

(dπ

)2 ∫ d

0a12s1(x)s2(x)dx

≤ −‖s1‖2L2(0,d) +1c1

(dπ

)2

|a12|‖s1‖L2(0,d)‖s2‖L2(0,d).

Analogously we obtain:

V2(s2) ≤ −‖s2‖2L2(0,d) +1c2

(dπ

)2

|a21|‖s1‖L2(0,d)‖s2‖L2(0,d).

Gains:

χ12(r) =c2

c31

(dπ

)4 ∣∣∣∣ a12

1− ε

∣∣∣∣2 · r , χ21(r) =c1

c32

(dπ

)4 ∣∣∣∣ a21

1− ε

∣∣∣∣2 · r .Andrii Mironchenko ISS of distributed parameter systems UI at Urbana-Champaign 19 / 30

Thus:

V1(s1) ≥ χ12 ◦ V2(s2) ⇒ V1(s1) ≤ −ε‖s1‖2L2(0,d),

V2(s2) ≥ χ21 ◦ V1(s1) ⇒ V2(s2) ≤ −ε‖s2‖2L2(0,d).

Small-gain condition:

χ12 ◦ χ21 < Id ⇔ 1c2

1c22

(dπ

)8 |a12a21|2

(1− ε)4 < 1,

Thus, if

|a12a21| < c1c2

(πd

)4

⇒ 0-UGASx .


Impulsive systems

t0 t

x

x(t0)

t1 t2 t3 t4 t5

x(t) = Ax(t) + f (x(t),u(t)) , t 6∈ {t1, t2, . . .},x(t) = g(x−(t),u−(t)) , t ∈ {t1, t2, . . .}.

u ∈ PC([0,∞),U), x(t) ∈ X , f : X × U → X . x−(t) := lims↗t x(s),

u−(t) := lims↗t u(s).


ISS of impulsive systems

Definition (Input-to-state stability (ISS))ISS (for given T ) :⇔ ∃β ∈ KL, γ ∈ K∞: ∀x ∈ X , ∀u ∈ Uc , ∀t ≥ t0

‖φ(t , t0, x ,u)‖X ≤ max {β(‖x‖X , t − t0), γ(‖u‖Uc )} .

ISS uniform w.r.t. S :⇔ ISS ∀T ∈ S, and β, γ do not depend onT ∈ S.

β(‖x(t0)‖X , t)

γ(‖u‖Uc)

tt0

‖x(t0)‖X

‖x‖X


ISS-Theory for∞-dim systems

ISSExistence of


Construction ofISS-Lyapunov

functions

Dwell-timeconditions


ISS-Lyapunov functions (ISS-LF)

Σ :x(t) = Ax(t) + f (x(t),u(t)), t 6= tk ,x(t) = g(x−(t),u−(t)), t = tk , k ∈ N.

Definition (ISS-Lyapunov function for impulsive systems)

V : X → R+ is ISS-Lyapunov function for Σ if ∃ ψ1, ψ2 ∈ K∞:

ψ1(‖x‖X ) ≤ V (x) ≤ ψ2(‖x‖X ), x ∈ X

and ∃χ, α, ϕ ∈ K∞: ∀x ∈ X , ∀ξ ∈ U and ∀u ∈ Uc with u(0) = ξ

V (x) ≥ χ(‖ξ‖U) ⇒{

Vu(x) ≤ −ϕ(V (x))V (g(x , ξ)) ≤ α(V (x)),

V is exponential ISS-LF for Σ with coefficients c,d ∈ R, if

V (x) ≥ χ(‖ξ‖U) ⇒{

Vu(x) ≤ −cV (x)V (g(x , ξ)) ≤ e−dV (x).


Average dwell time (ADT) condition

N(t , s)

s t

Let N(t , s) be number of impulse times tk in (s, t ].

Theorem (Hespanha, Liberzon, Teel, Automatica 2008)Let V be exponential ISS-LF for Σ with coefficients c,d ∈ R, d 6= 0.∀ µ, λ > 0 S[µ, λ]: ⇔ class of impulse time sequences {tk}:

−dN(t , s)− (c − λ)(t − s) ≤ µ ∀s, t : 0 ≤ s ≤ t . (ADT)

Then Σ is uniformly ISS w.r.t. S[µ, λ].


Generalized ADT condition

Theorem (Dashkovskiy, M., SICON, 2013)

Let V be exponential ISS-LF for Σ with coefficients c,d ∈ R, d 6= 0.∀ h : R+ → (0,∞): ∃g ∈ L: h(x) ≤ g(x) ∀x ∈ R+

S[h]:⇔ class of impulse time sequences:

−dN(t , s)− c(t − s) ≤ ln h(t − s) ∀t ≥ s ≥ t0. (gADT)

Then Σ is uniformly ISS w.r.t. S[h].

Corollary

gADT with h(x) = eµ−λx , x ∈ R+ ⇒ ADT.


ISS and non-exponential ISS-Lyapunov functions

V (x) ≥ γ(‖u‖U)⇒{

Vu(x) ≤ −ϕ(V (x))V (g(x ,u)) ≤ α(V (x)).

Let Sθ :={{ti}∞1 ⊂ [t0,∞) : ti+1 − ti ≥ θ, ∀i ∈ N

}.

Theorem (Dashkovskiy, M., SICON, 2013)

Let V be an ISS-Lyapunov function for Σ.

∃θ, δ > 0 :

∫ α(r)

r

dsϕ(s)

≤ θ − δ, ∀r > 0 (FDT)

⇒ ISS ∀ sequences of impulse times T ∈ Sθ.


Overview of dwell-time conditions

generalized ADT

−dN(t , s) − c(t − s) ≤ ln h(t − s)

Average DT

−dN(t , s) − (c − λ)(t − s) ≤ µ

Fixed DT∫ α(a)a

dsϕ(s) ≤ θ − δ

1θ ≤ c−λ

−d

h(x) := eµ−λx

µ := −d

ϕ := c · idα := e−d · id

for exponential LFs for nonexponential LFs


Summary and Outlook

ISSExistence of


Construction ofISS-Lyapunov

functions

Dwell-timeconditions

Other results in ISS theoryLinearization method for study of LISSSmall gain theorems for impulsive systemsSmall gain theorems for time-delay systems


Plans for the near future

ISS of linear systems x(t) = Ax(t) + Cu(t), with unbounded C.Converse ISS Lyapunov theoremCharacterisation of ISS for∞-dim systems.Applications of ISS TheoryRobust Stabilisation of PDEs"Integral ISS" theory for∞-dim systems.


Input-to-state stability of distributed parameter systems

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