Lyapunov functionals and matrices€¦ · Here Pand Qare given symmetric matrices of the appropriate dimensions. Vladimir L. Kharitonov (Faculty of Applied Mathematics and Processes

Lyapunov functionals and matrices

Vladimir L. Kharitonov

Faculty of Applied Mathematics and Processes of ControlSaint-Petersburg State University

10-th Elgersburg School, March 4-10, 2018

Vladimir L. Kharitonov (Faculty of Applied Mathematics and Processes of Control Saint-Petersburg State University)Lyapunov functionals and matrices10-th Elgersburg School, March 4-10, 2018 1

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Outline

1 Quadratic performance index

2 Exponential estimates

3 Critical values of delay

4 Robustness bounds

5 Notes and references


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Evaluation of quadratic performance indices

We consider a control system of the form

dx(t)

dt= Ax(t) +Bu(t),

y(t) = Cx(t).

Assume that a control law of the form

u(t) = Mx(t− h), t ≥ 0, (1)

stabilizes the the system.


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The closed-loop system

dx(t)

dt= A0x(t) +A1x(t− h), t ≥ 0, (2)

whereA0 = A, and A1 = BM,

is exponentially stable. Given a quadratic performance index of theform

J(u) =

∞∫0

[yT (t)Py(t) + uT (t)Qu(t)

]dt. (3)

Here P and Q are given symmetric matrices of the appropriatedimensions.


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The value of the index can be presented in the form

J(u) =

∞∫0

[xT (t, ϕ)W0x(t, ϕ) + xT (t− h, ϕ)W1x(t− h, ϕ)

]dt,

where ϕ ∈ PC([−h, 0], Rn) is an initial function of the solution x(t, ϕ)of the closed loop system (2), and matrices W0 = CTPC,W1 = MTQM .


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Theorem 1.4

The value of the performance index (3) for the stabilizing control law(1) is equal to

J(u) = v0(ϕ) +

0∫−h

ϕT (θ)W1ϕ(θ)dθ,

where the delay Lyapunov matrix U(τ) used in the functional v0(ϕ) isassociated with matrix W = W0 +W1 = CTPC +MTQM .


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Exponential estimates

We show here how an exponential estimate for the solutions of system

dx(t)

dt= A0x (t) +A1x (t− h) , t ≥ 0. (4)

can be derived with the use of complete type functionals. Let system(4) be exponentially stable, select positive definite matrices Wj ,j = 0, 1, 2, and define the complete type functional v(ϕ). Thefunctional satisfies the equality

d

dtv(xt) = −w(xt), t ≥ 0,

where

w(xt) = xT (t)W0x(t) +xT (t−h)W1x(t−h) +

0∫−h

xT (t+ θ)W2x(t+ θ)dθ.


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First we show that there exists σ > 0 for which the inequality

dv(xt)

dt+ 2σv(xt) ≤ 0, t ≥ 0, (5)

holds. On the one hand, by Lemma 2.3 there exist δ(0)1 > 0 and

δ(0)2 > 0, such that

v(ϕ) ≤ δ(0)1 ‖ϕ(0)‖2 + δ(0)2

0∫−h

‖ϕ(θ)‖2 dθ, ϕ ∈ PC([−h, 0], Rn).


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On the other hand, it is evident that

w(ϕ) ≥ λmin(W0) ‖ϕ(0)‖2 + λmin(W2)

0∫−h

‖ϕ(θ)‖2 dθ,

If σ > 0 satisfies the inequalities

2σδ1 ≤ λmin(W0), 2σδ2 ≤ λmin(W2).

then (5) holds.


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Now, inequality (5) implies that

v(xt(ϕ)) ≤ v(ϕ)e−2σt, t ≥ 0.

It follows from Corollary 1.3 and Corollary 2.3 that

α1 ‖x(t, ϕ)‖2 ≤ v(xt(ϕ)) ≤ v(ϕ)e−2σt ≤ α2 ‖ϕ‖2h e−2σt, t ≥ 0.

And we arrive at the desired exponential estimate

‖x(t, ϕ)‖ ≤ γ ‖ϕ‖h e−σt, t ≥ 0,

where

γ =

√α2

α1.


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Remark 1.4

It is worth of mentioning that the obtained exponential estimatedepends on the choice of positive definite matrices Wj, j = 0, 1, 2.These matrices may serve as free parameters for optimization of theestimate. Special choice of matrices Wj, j = 0, 1, 2, may result in atighter exponential estimate for the solutions of system (4).


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Example

x(t) = A0x(t) +A1x(t− 1)

A0 =

(0 1−10 0

), A1 =

(0 00 2

)W0 = W1 = W2 =

1

3I

W = I.


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Norm of the Lyapunov matrix


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Example

‖U(0)‖ = 9.522

α1 = 0.071, α2 = 86.364, σ = 0.0029

‖x(t, ϕ)‖ 6√α2

α1e−σt‖ϕ‖H

‖x(t, ϕ)‖ 6 34.96 e−0.0029 t‖ϕ‖H


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Critical values of delay

In this section an interesting connection between the spectrum of theoriginal system (4), and that of the auxiliary delay free system

d

dτY (τ) = Y (τ)A0 + Z(τ)A1,

d

dτZ(τ) = −AT1 Y (τ)−AT0 Z(τ). (6)

will be established. But first we prove the following theorem.

Theorem 2.4

Let λ0 be an eigenvalue of the auxiliary delay free system (6), then −λ0is also an eigenvalue of the system.


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The following theorem provides a connection between the spectrum oftime-delay system (4), and that of delay free system (6). Thisconnection may be effectively used for the computation of critical delayvalues of system (4), i.e., the values for which system (4) admits aneigenvalue on the imaginary axis of the complex plane.

Theorem 3.4

If s0 is an eigenvalue of time-delay system (4), such that −s0 is also aneigenvalue of the system, then s0 and − s0 belong to the spectrum ofdelay free system (6).


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Remark 2.4

The spectrum Λ of the delay system (4) depends on the delay value h,while the spectrum of the delay free system (6) does not depend of h. Inparticular, this means that system (4) remains exponentially stable(unstable) for all values of delay h ≥ 0, if the spectrum of system (6)has no common points with the imaginary axis of the complex plane.On the other hand, the common points represent possible crossingpoints of the imaginary axis through which eigenvalues of system (4)may migrate from one half-plane to the other one, as the system delayh varies.


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Example

x(t) = A0x(t) +A1x(t− h)

A0 =

(0 1−10 0

), A1 =

(0 00 2

)

L =

(I ⊗A0 I ⊗A1

−AT1 ⊗ I −AT0 ⊗ I

)

I ⊗A0 =

0 0 −10 00 0 0 −101 0 0 00 1 0 0

, I ⊗A1 =

0 0 0 00 0 0 00 0 2 00 0 0 2

AT0 ⊗ I =

0 −10 0 01 0 0 00 0 0 −100 0 1 0

, AT1 ⊗ I =

0 0 0 00 2 0 00 0 0 00 0 0 2

Vladimir L. Kharitonov (Faculty of Applied Mathematics and Processes of Control Saint-Petersburg State University)Lyapunov functionals and matrices

10-th Elgersburg School, March 4-10, 2018 18/ 42

Example

Λ(L) ={±i√

10, ±i(√

11− 1), ±i(√

11 + 1)}

f(s) = s2esh + 10esh − 2s

f(iω) = 0 ⇐⇒

(ω2 − 10) cos(ωh) = 0,

(ω2 − 10) sin(ωh) = −2ω


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Example

1 ω =√

10 : f(iω) 6= 0

2 ω =√

11− 1 :

f(iω) = 0 ⇔ h =1√

11− 1

(π2

+ 2πk), k = 0, 1, . . .

3 ω =√

11 + 1 :

f(iω) = 0 ⇔ h =1√

11 + 1

(3π

2+ 2πk

), k = 0, 1, . . .

Critical delay values:

h 0.678 1.092 2.548 3.390 4.004 5.460 . . .

Stability region: h ∈ (0.678, 1.092)


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Robustness bounds

It is well-known that Lyapunov functions for delay free systems areeffectively used for estimation of the robustness bounds for perturbedsystems. The main contribution of the section consists in thedemonstration that the complete type functionals may also providereasonable robustness bounds for time-delay systems.


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Robustness bounds

Consider a perturbed system of the form

dy(t)

dt= (A0 + ∆0) y(t) + (A1 + ∆1) y(t− h), t ≥ 0. (7)

Here matrices, ∆0 and ∆1, are unknown, but such that

‖∆k‖ ≤ ρk, k = 0, 1. (8)

Let system (4) be exponentially stable. We would like to findconditions on ρ0 and ρ1 under which system (7) remains stable for all∆0 and ∆1 satisfying (8).


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Robustness bounds

To this end we will use a complete type functional v(ϕ) defined forsystem (4). We compute the time derivative of the functional along thesolutions of the perturbed system (7).

Lemma 1.4

The time derivative of functional v(ϕ) along the solutions of perturbedsystem (7) is of the form

d

dtv(yt) = −w(yt) + 2 [∆0y(t) + ∆1y(t− h)]T l(yt), t ≥ 0,

where

l(yt) = U(0)y(t) +

0∫−h

U(−h− θ)A1y(t+ θ)dθ.


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Robustness bounds

We remind that v(yt) looks as

v(yt) = yT (t)U(0)y(t) + 2yT (t)

0∫−h

U(−h− θ)A1y(t+ θ)dθ

+

0∫−h

yT (t+ θ1)AT1

0∫−h

U(θ1 − θ2)A1y(t+ θ2)dθ2

dθ1+

0∫−h

yT (t+ θ) [W1 + (h+ θ)W2] y(t+ θ)dθ.


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Robustness bounds

Letν = max

θ∈[0,h]‖U(θ)‖ , a = ‖A1‖ .

Then the following statement holds.

Theorem 4.4

Let system (4) be exponentially stable. Given positive definite matricesW0,W1,W2, system (7) remains exponentially stable for all ∆0 and ∆1

satisfying (8) if the following inequalities hold

1. λmin(W0) > ν [2ρ0 + hρ0a+ ρ1].

2. λmin(W1) ≥ νρ1 [1 + ha].

3. λmin (W2) ≥ ν [ρ0 + ρ1] a.


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Robustness bounds

J1(t) = 2yT (t)∆T0 U(0)y(t)

≤ 2νρ0 ‖y(t)‖2 ,

J2(t) = 2yT (t− h)∆T1 U(0)y(t)

≤ νρ1

[‖y(t)‖2 + ‖y(t− h)‖2

],


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Robustness bounds

J3(t) = 2yT (t)∆T0

0∫−h


≤ hνρ0a ‖y(t)‖2 + νρ0a

0∫−h

‖y(t+ θ)‖2 dθ,

J4(t) = 2yT (t− h)∆T1

0∫−h


≤ hνρ1a ‖y(t− h)‖2 + νρ1a

0∫−h

‖y(t+ θ)‖2 dθ.


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Robustness bounds

From the above inequalities we obtain that

d

dtv(yt) ≤ −w(yt) + ν [2ρ0 + hρ0a+ ρ1] ‖y(t)‖2

+νρ1 [1 + ha] ‖y(t− h)‖2

+ν [ρ0 + ρ1] a

0∫−h

‖y(t+ θ)‖2 dθ,


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Robustness bounds

Remark 3.4

Theorem 4.4 remains true if we assume that the uncertain matrices ∆0

and ∆1 are continuous functions of t and xt.


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Example

x(t) = A0x(t) +A1x(t− 1)

A0 =

(0 1−10 0

), A1 =

(0 00 2

)W = I, ‖U(0)‖ = 9.522

W0 = W1 = W2 =1

3I

0.035 > 4ρ0 + ρ1,

0.035 > 3ρ1,

0.035 > 2ρ0 + 2ρ1

⇔

0.035 > 4ρ0 + ρ1,

0.012 > ρ1


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Notes and references

Yu.M. Repin

The first contribution dedicated to construction of the quadraticLyapunov functionals with a given time derivative was that by Yu.M.Repin (1965). In this seminal contribution a quadratic functional of ageneral form was suggested. The time derivative of the functional hasbeen computed, and then, equating the derivative to the prescribedone, a system of equations for the matrices that define the functionalhas been derived. The system includes a linear matrix partialdifferential equation, ordinary matrix differential equations, as well asalgebraic relations between the matrices.


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Yu.M. Repin

Under some simplifying assumptions the system has been reduced to asystem of two matrix differential equations similar to that given by (6).We should admit that many essential elements needed for computationof Lyapunov matrices, some in an explicit form, and some in an implicitform, can be already found there. Without any doubts this three-pagescontribution had a profound impact on the research in the area.


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R. Datko

In the paper by R. Datko, (1972), the main target was a presentationof an infinite dimensional version of the Lyapunov-Krasovskii approachto the stability analysis of linear time-delay systems. In particular, onecan find there an interpretation from the operator point of view of theresults given by Repin, (1965).


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W.B. Castelan and E. F. Infante

The paper by W.B. Castelan and E. F. Infante (1977) is dedicated tothe following initial value problem

dQ(τ)

dτ= AQ(τ) +BQT (h− τ), Q(

h

2) = K, (9)

where K is a given n× n matrix. It is worth to be mentioned that thedynamic property in [28] was written in this form. It has been shownthere that for any given K the initial value problem admits a uniquesolution. An exhausted analysis of the solution space of the equation(9) is presented in the paper, as well. A reader can find thereinteresting observations on the spectrum of an auxiliary delay freesystem.


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A more interesting for us is the second paper by these Authors, (1978).Here for the first time the three basic properties of Lyapunov matriceshave been explicitly indicated. Once again, follow the traditionestablished by Repin (1965), the dynamic property was written in theform of the equation (9). The symmetry property has not received thedue attention, it was just mentioned as a property of a matrix Q(τ)similar to that of the improper integral

U(τ) =

∞∫0

KT (t)WK(t+ τ)dt.

The algebraic property has been introduced as a bridge connectingmatrices Q(τ), and Q(τ).


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What is very important for us is that we can find there functionalssimilar to that of the complete type. For the functionals quadraticlower and upper bounds have been provided. The main goal of thepaper was to demonstrate that the functionals may be effectivelyapplied for the computation of upper exponential estimates of thesolutions of time-delay systems.


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Unfortunately, at one of the intermediate steps, namely in thecomputation of an upper bound for the functional the desiredexponential estimate has been explicitly used. There we also find aremark without proof that for the case of exponentially stable systemsthe functional v0(ϕ) does not admit quadratic lower bounds.


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W. Huang

Next serious break-through in this direction has been done in the paperby W. Huang (1989), where the existence of lower bounds for thefunctionals of the form v0(ϕ) has been studied. It has beendemonstrated there for the case of exponentially stable systems thatthe functionals admit local cubic lower bounds of the following form

α ‖ϕ(0)‖3 ≤ v0(ϕ), ϕ ∈ C([−h, 0], Rn), and ‖ϕ‖h ≤ H,

where α and H are two positive constants.


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W. Huang

It is important to say here that this result, as well as the other onespresented in this contribution, have been proven for a very generalclass of linear time-delay systems. Probably this is the first paperwhere it has been explicitly stated that Lyapunov matrices arecompletely defined by the three basic properties: the dynamic, thesymmetry, and the algebraic ones. The most important resultpresented there is the existence theorem. The theorem states that if atime-delay system satisfies the Lyapunov condition, then for anysymmetric matrix W there exists a corresponding Lyapunov matrix.And what is more, an explicit frequency domain expression for theLyapunov matrix has been given, as well.


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J.E. Marshall, H. Gorecki, A. Korytowski, and K. Walton

A detail account of the application of the functionals of the form v0(ϕ)to the computation of various quadratic performance indices can befind in an interesting book by J.E. Marshall, H. Gorecki, A.Korytowski, and K. Walton (1992).


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J. Louisell

In the paper by J. Louisell (2001) a relation between the spectrum ofthe time-delay system and that of the auxiliary delay free system ofmatrix equations has been established. Namely, it has been shown thatany pure imaginary eigenvalue of the time-delay system is also aneigenvalue of the auxiliary system. The statement has been obtainedfor the case of neutral type systems with one delay. In some sense thestatement of Theorem 3.4 is a generalization of this important result.


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S. Mondie

The following open problem is one of the most important related withLyapunov matrices, and deserves to be mentioned here: Findconditions of the exponential stability of system (??) expressed in theterms of a Lyapunov matrix U(τ), associated with a positive definitematrix W . The first result in this direction has been obtained by S.Mondie (2012), where some necessary and sufficient conditions havebeen derived for the case of scalar equation.


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Bellman, R. and Cooke, K.L., Differential-Difference Equations,Academic Press, New York, 1963.

Castelan, W. B. and Infante, E. F., On a functional equationarising in the stability theory of difference-differential equations,Quarterly of Applied Mathematics, 35: 311–319, 1977.

Castelan, W. B. and Infante, E. F., A Liapunov functional for amatrix neutral difference-differential equation with one delay,Journal of Mathematical Analysis and Applications, 71: 105–130,1979.

Datko, R., An algorithm for computing Liapunov functionals forsome differential difference equations, In L. Weiss (Ed.), OrdinaryDifferential Equations, Academic Press, New York, 387–398, 1972.

Datko, R., A procedure for determination of the exponentialstability of certain diferential-difference equations, Quarterly ofApplied Mathematics, 36: 279–292, 1978.


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Datko, R., Lyapunov functionals for certain linear delay-differentialequations in a Hilbert space, Journal of Mathematical Analysis andApplications, 76: 37–57, 1980.

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Gu, K., Kharitonov, V.L. and Chen, J., Stability of Time DelaySystems, Birkhauser, Boston, MA, 2003.

Huang, W., Generalization of Liapunov’s theorem in a linear delaysystem, Journal of Mathematical Analysis and Applications, 142:83–94, 1989.

Infante, E.F., Some results on the Lyapunov stability of functionalequations, In K.B. Hannsgen, T.L. Herdmn. H.W. Stech and R.L.Wheeler (Eds), Volterra and Functional Differential Equations,


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Infante, E.F. and Castelan, W.V., A Lyapunov functional for amatrix difference-differential equation, Journal of DifferentialEquations, 29: 439–451, 1978.

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Louisel, J., Growth estimates and asymptotic stability for a class ofdifferential-delay equation having time-varying delay, Journal ofMathematical Analysis and Applications, 164: 453–479, 1992.

Louisell, J., Numerics of the stability exponent and eigenvalueabscissas of a matrix delay system, In L. Dugard and E.I. Verriest(Eds), Stability and Control of Time - delay Systems, LectureNotes in Control and Information Sciences, 228, Springer-Verlag,140–157, 1997.

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Lyapunov functionals and matrices€¦ · Here Pand Qare given symmetric matrices of the appropriate dimensions. Vladimir L. Kharitonov (Faculty of Applied Mathematics and Processes