Automatica 46 (2010) 775–778

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Automatica

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Technical communique

On new estimates for Lyapunov exponents of discrete time varyinglinear systemsI

Adam Czornik ∗, Aleksander NawratDepartment of Automatic Control, Silesian Technical University, ul. Akademicka 16, 44-101 Gliwice, Poland

a r t i c l e i n f o

Article history:Received 30 April 2009Received in revised form23 November 2009Accepted 16 December 2009Available online 18 February 2010

Keywords:Time varying discrete linear systemsLyapunov exponentsCharacteristic exponentsFrozen time method

a b s t r a c t

In this paper, we propose certain new bounds for the Lyapunov exponents of discrete time varying linearsystems. The bounds are expressed in terms of spectral radii of matrix coefficients and therefore may beused to establish the exponential stability of time varying system on the basis of eigenvalues of individualcoefficient. This approach is known in the literature as frozen time method.

© 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Consider the linear discrete time varying system

x(n+ 1) = A(n)x(n), n ≥ 0 (1)

where (A(n))n∈N is a bounded sequence of invertible k-by-k realmatrices. By ‖‖ we denote the Euclidean norm in Rk and theinduced operator norm. The transition matrix is defined as

A(m, k) = A(m− 1) . . . A(k)

for m > k andA(m,m) = I , where I is the identity matrix. For aninitial condition x0 the solution of (1) is denoted by x(n, x0) so

x(n, x0) = A(n, 0)x0.

Let a = (a(n))n∈N be a sequence of real numbers. The number (orthe symbol±∞) defined as

λ(a) = lim sup1n

n→∞

ln |a(n)|

I The research presented here was done as a part of research and developmentproject no. O R00 0021 06 and has been supported by Ministry of Science andHigher Education funds in the years 2008–2010. The material in this paper wasnot presented at any conference. This paper was recommended for publication inrevised form by Associate Editor Mikael Johansson under the direction of EditorAndré L. Tits.∗ Corresponding author. Tel.: +48 32 2371093; fax: +48 32 2372127.E-mail address: Adam.Czornik@polsl.pl (A. Czornik).

0005-1098/$ – see front matter© 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.automatica.2010.01.014

is called the characteristic exponent of sequence (a(n))n∈N . Forx0 ∈ Rk, x0 6= 0 the Lyapunov exponent λ(x0) of (1) is definedas characteristic exponent of (‖x(n, x0)‖)n∈N that is

λ(x0) = lim sup1n

n→∞

ln ‖x(n, x0)‖ .

It is well known (Barreira & Pesin, 2002) that the set of allLyapunov exponents of system (1) contains at most k elements,say −∞ ≤ λ1 (A) < λ2 (A) < · · · < λr (A) < ∞

and the set λ1, λ2, . . . , λr is called the spectrum of (1). TheLyapunov exponents have been studied formany years (seeArnold,Crauel, and Eckmann (1991) and the references therein) and it isknown that it is generally impossible to obtain these quantitiesanalytically. Therefore numerical and estimation methods arerequired for their approximation. A review of numerical methodsfor continuous time systems is included in Diecia and Eliab (2006).The problem of estimating Lyapunov exponents, which is ofhighest importance in a quite variety of subfields of the controltheory, has been investigated, among others, in the followingpapers: Adrianova (1995), Li and Xia (2004) and Smith (1965) forcontinuous time systems, Li and Chen (2004) for discrete systemsand Key (1990) for discrete stochastic systems.Denote byM(n) andm(n) the largest and the smallest absolute

value of eigenvalues of A(n). It is well known (see Agarwal (2000))that, when (A(n))n∈N is a constant sequence equal to A, then λ(x0)is equal to the logarithmof absolute value of one of the eigenvaluesofA, so that lnm(n) ≤ λ(x0) ≤ lnM(n) Themain object of this note

776 A. Czornik, A. Nawrat / Automatica 46 (2010) 775–778

is to discuss sufficient conditions for the following inequality

lim sup1n

n→∞

n−1∑l=0

lnm(l) ≤ λ(x0) ≤ lim sup1n

n→∞

n−1∑l=0

lnM(l), (2)

to hold. As we stated above it is true when (A(n))n∈N is constant,however the next example shows that it may fail for periodic(A(n))n∈N .

Example 1. Consider system (1) with periodic (A(n))n∈N . Theperiod T = 2 and

A(0) =

14

0

14

14

, A(1) =

12 4

012

.Then λ([ 1 1 ]) =

12 ln(

14

√6 + 5

8 ) > 0, whereaslim supn→∞

1n

∑n−1l=0 lnM(l) = −

3 ln 22 < 0 and inequality (1) is

not satisfied.

The inequality (2) is connected to the frozen time approachto the stability analysis of system (1). In this method the timevariable in the system coefficients is viewed as a parameter. Anexamples of this approach are papers Desoer (1970), Fuchs (1982),Gil’ and Medina (2001) and Chapter 10 of Gil’ (2007). The generalresult of this approach is that if the frozen time eigenvalues of thesystem are in the unit circle and if the rate of time variation issufficiently small, then the time varying system is exponentiallystable. Similar results for stabilizability are in Kamen, Khargonekar,and Tannenbaum (1989). If thematrices A(n) are such thatM(n) <δ < 1 for all n = 0, 1, . . . and (2) holds then λ(x0) ≤ ln δ < 0for all x0 ∈ Rk, x0 6= 0 and consequently (1) is exponentiallystable. Even more, on the basis of (2) we will be able to concludethe stability of (2) even if for some A(n) we have M(n) > 1. Thecondition under which (2) holds will be formulated in therms ofthe limit of norm variation of the sequence (A(n))n∈N whereasthe condition for stability formulated in Desoer (1970), Fuchs(1980) and Fuchs (1982) are formulated for individual variation‖A (n+ 1)− A(n)‖ for all natural n. In that sense our result ismore general. Moreover, in contrast with these papers we arenot limited to the situation where M(n) < δ < 1 for all n =0, 1, . . .. Yet another approach to estimation of Lyapunov exponentis presented in Czornik and Jurgas (2008). In this paper a lineardiscrete time varying system with coefficients in given boundedset

∑of invertible matrices it is considered and it is shown

that the interval (ln ρ∗(Σ), ln ρ(Σ)), where ρ∗(Σ), ρ(Σ) are thegeneralized subradius and generalized radius of

∑, is the set of all

possible values of themaximal Lyapunov exponent of systemswithcoefficients in the set

∑.

2. Main result

Let us introduce the following definition.

Definition 2. For positive numbers ε, d a matrix sequence(A(n))n∈N is said to be of type Ω (ε, d) if there exists an increas-ing sequence of natural numbers (nk)k∈N such that

lim supk→∞

knk≤ d (3)

and ‖A(p)− A(q)‖ ≤ ε for all p and q belonging to the sameinterval [nk, nk+1] .

In the proof of the next theorem we will use the followingLemma.

Lemma 3. For all δ > 0 and all square matrices B there exist ζ > 0and a positive definite symmetric matrix P such that the matrix

X = ATPA− ((δ + 1)M(A))2 P

is negative definite for all matrices Awith ‖A− B‖ < 2ζ , whereM(A)is the largest absolute value of eigenvalues of A.

Proof. Matrix

C =1

(δ + 1)M(B)B

has the spectral radius smaller than 1 and therefore there existssymmetric positive definite matrix P such that the followingdiscrete Lyapunov equation

CTPC − P = −1

((δ + 1)M(B))2I.

is satisfied. It implies that

BTPB− ((δ + 1)M(B))2 P = −I.

The eigenvalues of X vary continuouslywith A and as A tends B, thismatrix tends to BTPB − ((δ + 1)M(B))2 P , whose eigenvalues areall −1. The eigenvalues of X are therefore negative through someneighborhood ‖A− B‖ < 2ζ of B.

The next theorem contains the main result of this paper.

Theorem 4. Fix the positive constants K , δ. If there are positiveconstants ε, d such that if (A(n))n∈N is of typeΩ (ε, d) and ‖A(n)‖ ≤K for all n = 0, 1, . . . , then the inequalities

−δ + lim sup1n

n→∞

n−1∑l=0

lnm(l)

≤ λ(x0) ≤ δ + lim sup1n

n→∞

n−1∑l=0

lnM(l) (4)

hold.

Proof. Let us fix K , δ > 0 and a k-by-k real matrix B such that‖B‖ ≤ K . For δ2 and B let ζ > 0 be a constant fromLemma3. DenotebyΓ (B) the set of all k-by-k realmatrices A such that ‖A− B‖ < ζ,then as B varies over the compact set ‖B‖ ≤ K , the sets Γ (B) forman open covering of it and by Heine–Borel Lemma there is a finitecovering Γ (Bi), i = 1, . . . , ν. If ζi, Pi, i = 1, . . . , ν are selected forBi, i = 1, . . . , ν according to Lemma 3, then there exists H > 1such that

H−1 ≤‖Pix‖‖x‖≤ H (5)

for all x ∈ Rk, x 6= 0 and all i = 1, . . . , ν. Let us define ε, d asfollows

ε = min ζi : i = 1, . . . , ν , d =δ

2 lnH. (6)

Suppose that (A(n))n∈N is of type Ω (ε, d) and ‖A(n)‖ ≤ K forall n = 0, 1, . . .. Let (nk)k∈N be an increasing sequence of naturalnumbers such that (3) holds and consider q ∈ [nk, nk+1]. Since‖A(q)‖ ≤ K , we have A(q) ∈ Γ (Bi) for some i and

‖A(p)− Bi‖ ≤ ‖A(p)− A(q)‖ + ‖A(q)− Bi‖ < ε + ζi ≤ 2ζi,

for all p ∈ [nk, nk+1]. Moreover, for p ∈ [nk, nk+1] we have

xTAT (p)PiA(p)x ≤((

δ

2+ 1

)M(p)

)2Vi(x),

A. Czornik, A. Nawrat / Automatica 46 (2010) 775–778 777

where M(p) is largest absolute value of eigenvalues of A(p) andVi(x) = xTPix. Therefore if p ∈ [nk, nk+1] , then

Vi(x(p+ 1))Vi(x(p))

≤

((δ

2+ 1

)M(p)

)2and

Vi(x(p))Vi(x(nk))

≤

p−1∏r=nk

((δ

2+ 1

)M(r)

)2and by (5)

‖x(p)‖‖x(nk)‖

≤ Hp−1∏r=nk

((δ

2+ 1

)M(r)

).

For natural p denote by N(p) the number of elements of (nk)k∈Nsuch that nk ≤ p. By the last inequality we have

‖x(m)‖ ≤ ‖x(n0)‖‖x(m)‖∥∥x(nN(m))∥∥

N(m)−1∏r=0

‖x(nr+1)‖‖x(nr)‖

≤ ‖x(n0)‖HN(m)m−1∏r=0

((δ

2+ 1

)M(r)

). (7)

Since k ≥ nN(k), (3) gives

lim supk→∞

N(k)k≤ lim sup

k→∞

knk≤ d. (8)

Taking lim sup 1n n→∞ ln on both sides of (7) and using (6), (8), weget

λ(x0) ≤ d lnH + ln(δ

2+ 1

)+ lim sup

1n

n→∞

n−1∑r=0

ln (M(r))

≤ δ + lim sup1n

n→∞

n−1∑r=0

ln (M(r)) , (9)

since ln(δ2 + 1

)≤

δ2 . To prove the second part of (4) consider the

adjoint equation

y(n+ 1) = B(n)y(n),

where B(n) =(AT (n)

)−1 and AT denotes the transpose of A. Thearguments leading to (9) remains valid. We can therefore replacex(n), A(n) in (7) by y(n), B(n). SinceM(B(n)) = 1/m(n), it gives

‖y(m)‖ ≤ ‖y(n0)‖HN(m)m−1∏r=0

(δ2 + 1m(r)

). (10)

Moreover, yT (n + 1)x(n + 1) = yT (n)x(n) and thereforeyT (n)x(n) = yT (n0)x(n0). When y(n0) = x(n0), this gives‖y(n)‖ ‖x(n)‖ ≥ ‖x(n0)‖2. From this and (10) it follows that

‖x(m)‖ ≥ ‖x(n0)‖H−N(m)m−1∏r=0

(m(r)δ2 + 1

).

Taking lim sup 1mm→∞ ln on both sides and using (6), (8), we get

λ(x0) ≥ −d lnH − ln(δ

2+ 1

)+ lim sup

1n

n→∞

n−1∑r=0

ln (m(r))

≥ −δ + lim sup1n

n→∞

n−1∑r=0

ln (m(r))

since− ln(δ2 + 1

)≥ −

δ2 .

Let us consider the belongingness of sequence (A(n))n∈N to setΩ (ε, d) as a measure of variation of the sequence, in the sensethat smaller values of ε, d correspond to sequences varying moreslowly, and the constant δ from inequality (4) as a measure ofaccuracy of estimation of Lyapunov exponents by spectral radii.Then, the idea of the Theorem is as follows: on each accuracy levelit is possible to estimate the Lyapunov exponents by spectral radiifor sufficiently slowly varying systems.Denote byΩ0 the set of all bounded sequences (A(n))n∈N which

are of type Ω (ε, d) for all ε, d > 0. Observe that the class Ω0contains all convergent sequences and in particular all sequencesthat are constant starting from certain n0. Class Ω0 contains,however also much more complicated systems not only those thateventually become time invariant (see Example 6). Moreover, theclass Ω0 is closed with respect to addition and multiplication ofmatrices i.e. if (A(n))n∈N , (B(n))n∈N ∈ Ω0 then (A(n)+ B(n))n∈N ∈Ω0 and (A(n)B(n))n∈N ∈ Ω0.If (A(n))n∈N ∈ Ω0 then (4) holds forall δ > 0 and in particular if all matrices A(n) stable (in the discretesense i.e. all eigenvalues have module less than 1) the following istrue:

Corollary 5. If (A(n))n∈N ∈ Ω0 then

lim sup1n

n→∞

n−1∑l=0

lnm(l) ≤ λ(x0) ≤ lim sup1n

n→∞

n−1∑l=0

lnM(l), (11)

and if in addition M(n) < ε < 1 for all n ≥ n0 and certain ε > 0,then (1) is exponentially stable.

Notice that the corollary is a generalization of the result fromDesoer (1970) and Fuchs (1980). In the first paper it has beenshown that if M(n) < ε < 1 for all n ≥ n0 and certain ε > 0 and‖A (n+ 1)− A(n)‖ is sufficiently small, then (1) is exponentiallystable. Whereas in the second paper it has been shown that if(A(n))n∈N tends to a stable matrix then (1) is exponentially stable.Both results are in fact special cases of our corollary.Finally we present a numerical example.

Example 6. Consider system (1) with (A(n))n∈N given by

A(n) =[a bc sinHn

],

where Hn =∑n+1i=1

1i and a, b, c ∈ R, a > 1, 4bc ≥ (a− 1)2.

Observe that for p > q

‖A(p)− A(q)‖ =∣∣sinHp − sinHq∣∣ ≤ (Hp − Hq)

and therefore considering sequence nk = lk, for fixed natural l, weconclude that

(A(n))n∈N ∈ Ω(ε,1l

)for all ε > 0, since Hl(k+1) − Hlk −→

k→∞0. It means that

(A(n))n∈N ∈ Ω0.

Wewill show that for bc−12 > a > 1 all Lyapunov exponents of (1)are positive. It is easy to calculate that

m(l) =

√4bc + (a− sinHn)2 − a− sinHn

2.

Since the function f : [−1, 1] → R, f (x) = (√4bc + (a− x)2 −

a− x)/2 is decreasing we have the following bound form(l)

m(l) ≥

√4bc + (a− 1)2 − a− 1

2> 1.

Now the inequality λ(x0) > 0 follows from (11).

778 A. Czornik, A. Nawrat / Automatica 46 (2010) 775–778

3. Conclusions

In this paper we presented a new bound for Lyapunovexponents of linear discrete time varying systems. The boundis expressed in terms of spectral radii of matrix coefficients. Itenables to use the bound in the frozen time approach to stabilityof the system. In particular from our bound we obtained certaingeneralization of results from Desoer (1970) and Fuchs (1980).

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