ON STABILITY OF LINEAR AND WEAKLY NONLINEAR SWITCHED ...
Accepted Manuscript On stability of linear and weakly nonlinear switched systems with time delay Mohamad S. Alwan, Xinzhi Liu PII: DOI: Reference:
S0895-7177(08)00010-1 10.1016/j.mcm.2007.12.024 MCM 3290
To appear in:
Mathematical and Computer Modelling
Received date: 4 September 2007 Accepted date: 7 December 2007 Please cite this article as: M.S. Alwan, X. Liu, On stability of linear and weakly nonlinear switched systems with time delay, Mathematical and Computer Modelling (2008), doi:10.1016/j.mcm.2007.12.024 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT ON STABILITY OF LINEAR AND WEAKLY NONLINEAR SWITCHED SYSTEMS WITH TIME DELAY
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Mohamad S. Alwan and Xinzhi Liu
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Department of Applied Mathematics, University of Waterloo Waterloo, Ontario, Canada N2L 3G1, Email:[email protected].
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Abstract– This paper studies time-delayed switched systems that include both stable and unstable modes. By using multiple Lyapunov functions technique and a dwell time approach, several criteria on exponential stability for both linear and nonlinear systems are established. It is shown that by suitably controlling the switching between the stable and unstable modes exponential stabilization of the switched system can be achieved. Some examples and numerical simulations are provided to illustrate our results. Keywords: Switched systems, Time delay, Exponential stability
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1. Introduction
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By a switched system, we mean a hybrid system that consists of a finite number of continuous-time dynamical systems and a switching law that organizes the switching between these systems. Here, the switching from one mode to another one is not known a priori, but it is determined by some environmental factors which usually are not a part of the mathematical model under consideration. For some examples of such systems, interested readers may refer to [17],[21]. The importance of studying such systems is two fold. Firstly, a large class of real life or engineering systems are inherently multi-model in the sense that their behavior is represented by several dynamical models; secondly, some systems may be stabilized by several feedback controllers (known as switching control), rather than one controller. In fact, there has been increasing interest in designing an appropriate switching rule in order to stabilize the system [3], [9],[10], [16],[19],[23]. Ordinary differential equations (ODEs) have long played important roles in modeling many physical processes and they will continue to serve as a fundamental tool in future investigations. But sometimes ODEs may be inadequate in modeling systems whose future state depend not only on the present state but also on the past. In those cases, ODEs give only an approximation of some real systems. More realistic models should include some of the historical values of the state that leads us to delay differential equations (DDEs). The early motivations for studying DDEs came from their applications in population dynamics when Volterra investigated the predatorprey model, and in Minorsky’s study of ship stabilization and automatic steering. These studies indicate the importance of considering delay in the feedback mechanism [15]. Another motivation for studying time-delayed systems is that the presence of delay may cause undesirable performance. As we know, in ordinary systems, oscillation phenomena might be seen in some systems with at least two state variables and chaotic behavior is noticed in some systems with at least three state variables, but in some cases these unsuitable behaviors occur in one order delay systems; moreover, small delay may destroy stability of some systems, but large delay may stabilize 1
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others. As a result, there are many studies of systems with time delay. A number of monographes have been devoted to the subject of DDEs. These works include books by Bellman and Cooke [2] and Krasovkii and Brenner [12], texts by Halanay [7], Hale and Lunel [8], Driver [4], El’sgoll’ts and Norkin [5] and Bellen and Zennaro [1]. Some other books dedicated to the applications of DDEs are MacDonald [14], Gopalsamy [6], and Kuang [13]. The study of delay differential equations is currently very heavy due to the growing progress achieved in the understanding of the dynamics of many important time delayed systems. Switched systems and time delay are important issues encountered in many fields. The area of switched delay systems is still in its infancy. Nevertheless, there has been reasonable progress in this field. Stability of switched delay systems has been receiving an increasing attention (see [11],[18],[20],[22]). In this paper, we investigate stability property of switched delay systems consisting of unstable and stable modes. Delay-version comparison lemmas for a single equation and system are developed to help us find the growth rate for the unstable modes. The first lemma is in fact an analog to a Lemma in [7], while the second one is analogous to Lemma 2.3 in [14]. By using multiple Lyapunov functions, a general criteria on exponential stability of these systems is established. Here, our stability results for linear and a special class of nonlinear systems is characterized by the locations of the eigenvalues. The organization of this paper is as follows; in Section 2, we describe switched delay systems and introduce some notation and definitions that will be used in the rest of this paper. In Section 3, we shall state our main results. Numerical examples are also presented to verify our theoretical results. We finally conclude this work in Section 4. 2. Problem Formulation Switched delay system can be described as follows x˙ = fi (t, x, xt ),
t ∈ [tk−1 , tk )
(1)
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where x ∈ Rn with Rn being n-dimensional Euclidean space, i ∈ S = {1, 2, · · · , N}, k = 1, 2, · · · , tk−1 < tk , and limk→∞ tk = ∞. We assume that fi is smooth enough to guarantee the existence of solutions, and fi (t, 0, 0) ≡ 0 to ensure that system (1) has a trivial solution. Let xt = x(t + θ), θ ∈ [−τ, 0], with τ being a positive constant that represents the time delay, x(t) be the solution of (1) with the initial condition xt0 , kxt kτ = supt−τ ≤θ≤t kx(θ)k with kxk being the Euclidean norm of x, P kAk = max1≤j≤n ( ni=1 a2ij )1/2 be the norm of the n × n matrix A = (aij ), AT be the transpose of a matrix A, λ(A) be the eigenvalue of an n×n matrix A, and Re[λ(A)] be the real part of λ(A). A symmetric matrix (i.e. P = P T ) is said to be positive definite if all its eigenvalue are positive. Denote by λmin(P ) and λmax (P ) the minimum and maximum eigenvalue of P respectively. Let Ai be a Hurwitz matrix (i.e. Re[λ(Ai )] < 0). Then, there exist positive definite matrices Pi and Qi satisfying the Lyapunov equation ATi Pi + Pi Ai = −Qi .
Let Vi (x) = xT Pi x. Then, we have for all i ∈ S
λmin (Pi )kxk2 ≤ Vi (x) ≤ λmax (Pi )kxk2 . 2
ACCEPTED MANUSCRIPT It follows that Vj (x) ≤ µVi (x),
≥ 1 with λM = max{λmax (Pi ), ∀i ∈ S}, and λm = min{λmin(Pi ), ∀i ∈
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λM λm
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where µ = S}.
∀i, j
Definition: The trivial solution of system (1) is said to be exponentially stable if there exist positive constants k ≥ 1 and λ such that kx(t)k ≤ kkxt0 kτ e−λ(t−t0 ) ,
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for any solution x(t) of (1).
t ≥ t0
Before stating our main results, we present the following lemmas which are frequently used in proving our theorems.
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Lemma 1: For a ∈ R, with a > 0, and t0 ∈ R+ = [0, ∞), let u : [t0 , t0 + a) → R+ satisfy the following delay differential inequality u(t) ˙ ≤ αu(t) + β sup u(θ), θ∈[t−τ,t]
t ∈ [t0 , t0 + a).
Assume that α + β > 0. Then, there exist positive constants ξ and k such that u(t) ≤ keξ(t−t0 ) ,
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t ∈ [t0 , t0 + a)
(2)
where ξ = α + β and k = supθ∈[t0 −τ,t0 ] u(θ).
Proof: Claim y(t) = keξ(t−t0 ) is a solution of the delay differential equation y(t) ˙ = αy(t) + β sup y(θ), θ∈[t−τ,t]
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with the initial condition
y(t) = u(t),
t ∈ [t0 , t0 + a)
t ∈ [t0 − τ, t0 ].
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To prove the claim, we check that y(t) ˙ = ξkeξ(t−t0 ) sup y(θ) = keξ(t−t0 ) = y(t)
θ∈[t−τ,t]
and (3) becomes
ξkeξ(t−t0 ) = αkeξ(t−t0 ) + βkeξ(t−t0 )
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(3)
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u(θ)eξ(t−t0 ) ,
θ∈[t0 −τ,t0 ]
t ∈ [t0 , t0 + a).
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u(t) ≤ keξ(t−t0 ) =
y(t) ˙ ≤ A(t)y(t) + B(t) sup y(θ)
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T where y(t) = y1 (t), y2 (t), · · · yn (t) ≥ 0 and supθ∈[t−τ,t] y(θ) = supθ∈[t−τ,t] y1 (θ), supθ∈[t−τ,t] y2 (θ), T · · · , supθ∈[t−τ,t] yn (θ) . Then, there exists a ξ > 0 such that
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θ∈[t−τ,t]
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Lemma 2: For a ∈ R with a > 0 and t ∈ [t0 , t0 + a), t0 ∈ R+, let A(t) and B(t) be n × n matrices of continuous functions, α(t) = λ A(t) + AT (t) , kB(t)k ≤ β1 and α(t) + kB(t)k ≤ β2 , (β2 > 0). Assume that the following inequality is satisfied.
ky(t)k ≤ kyt0 kτ eξ(t−t0 ) , where ξ = (β1 + β2 )/2.
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Proof: Let v(t) = y T (t)y(t) = ky(t)k2. Then,
t ∈ [t0 , t0 + a)
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v(t) ˙ = y˙ T (t)y(t) + y T (t)y(t) ˙ T ≤ A(t)y(t) + B(t) sup y(θ) y(t) θ∈[t−τ,t] + y T (t) A(t)y(t) + B(t) sup y(θ) θ∈[t−τ,t] ≤ y T (t) AT (t) + A(t) y(t) + 2kB(t)kky(t)kkytkτ ≤ α(t) + kB(t)k v(t) + kB(t)kkvt kτ ≤ β2 v(t) + β1 kvt kτ .
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By Lemma 1, there exists a ξ > 0 such that v(t) ≤ kvt0 kτ e2ξ(t−t0 ) ,
where 2ξ = β1 + β2 , and hence ky(t)k ≤ kyt0 kτ eξ(t−t0 ) ,
t ∈ [t0 , t0 + a).
In fact, Lemma 2 is an analog to Lemma 2.3 in [14] that shall be used in this paper 4
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Lemma 3:[14] For t ∈ [t0 , ∞), let A(t) and B(t) be n × n matrices of continuous ˙ functions, A(t), B(t), A(t) be bounded and A(t) be Hurwitz. Furthermore, assume that A1) λ(AT (t) + A(t)) ≤ −α(t) < 0. A2) − α(t) + 2kB(t)k ≤ −β < 0, with β being a positive constant. A3) y(t) ˙ ≤ A(t)y(t) + B(t) supt−τ ≤θ≤t y(θ), where y(t) = (y1 (t), y2(t), · · · , yn (t))T ≥ 0, and supt−τ ≤θ≤t y(θ) = (supt−τ ≤θ≤t y1 (θ), supt−τ ≤θ≤t y2 (θ), · · · , supt−τ ≤θ≤t yn (θ))T . Then, there exists a positive constant ξ such that ky(t)k ≤ kyt0 kτ e−ξ(t−t0 ) ,
t ≥ t0 .
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3. Main results In this section, we establish exponential stability of switched delay systems incorporating unstable and stable modes. Let Su = {1, 2, · · · , r} and Ss = {r + 1, r + 2, · · · , N} be respectively the set of indices of the unstable and stable modes. I. Linear systems Consider the following linear switched delay system
t ∈ [tk−1 , tk )
(4)
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x(t) ˙ = Ai x(t) + Bi x(t − τ ),
where for each i ∈ S = Su ∪ Ss , Ai and Bi are n × n constant matrices. Theorem 1: The origin of system (4) is globally exponentially stable if the following assumptions hold. A1-i. For i ∈ Su ,
Re[λ(Ai )] > 0,
and Re[λ(Ai + Bi )] > 0.
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A1-ii. For i ∈ Ss , Ai is Hurwitz and λ (Q ) − β ∗ β ∗ min i i − + i 0 such that Re([Ai − γI]) < 0. With the aid of Lemma 2, where β1 = 2γ + βi∗ /λm and β2 = βi∗ /λm , there exists a ξi > 0 such that Vi (x) ≤ kVitk−1 kτ eξi (t−tk−1 )
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Similarly, for i ∈ Ss , we have
V˙ i (x) ≤ −λmin (Qi )kxk2 + βi∗ kxk2 + βi∗ kxt k2 λ (Q ) − β ∗ β∗ min i i ≤ − Vi (x) + i kVit kτ λM λm λmin (Qi )−βi∗ λM
By Lemma 3, where αi =
and βi =
βi∗ , λm
there exists a ζi > 0 such that
Vi (x) ≤ kVitk−1 kτ e−ζi (t−tk−1 )
Then,
r Y
µeξi (ti −ti−1 ) ×
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VN (x) ≤
i=1
NY −r−1
µeζj τ e−ζj (tj −tj−1 )
j=r+1
×kV1t0 kτ e−ζN (t−tN−1 )
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By condition (5), we have VN (x) ≤
r Y i=1
µ×
NY −r−1 j=r+1
µeζj τ × kV1t0 kτ e−λ
∗ (t−t ) 0
and, by (6) and (7) we get ∗ −ν)(t−t ) 0
VN (x) ≤ kV1t0 kτ e−(λ
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,
t ≥ t0 .
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√
,
t ≥ t0 ,
µ; this shows that the origin of (4) is exponentially stable.
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where K =
kx(t)k ≤ Kkxt0 kτ e−(λ
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Remark 1: Assumption A1 is made to ensure instability and stability of the modes, while Assumption A2 means that the total activation time of stable modes is larger than that of the unstable modes and that the solutions are kept down whenever the modes are switched. The following example illustrates these results.
0.5 0 0 0.5
A2 =
−1 0 0 −1
1 0 0 1
P1 =
,
B1 =
2 0 0 1
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Take γ = 1, Q1 =
A1 =
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Example 1: Consider system (4) where the continuous initial condition x(t) = t + 1, τ = 1,
0.1 0 , B2 = 0 0.1 3 0 and Q2 = . 0 3
Then,
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1 0 0 1
and P2 =
1.5 0 0 1.5
.
A simple check shows that µ = 1.5, the growth rate ξ = 2.5, and the decay rate ζ = 1.3318. Figure (1) shows the solution vanishing exponentially. Obviously, since the growth rate is larger than the decay rate, running stable modes longer than unstable ones is strongly required. II. Nonlinear Systems Consider the following nonlinear switched system with time delay
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x(t) ˙ = Ai x(t) + gi (t, x, xt )
(8)
where i ∈ Su ∪ Ss , and gi (t, 0, 0) ≡ 0. Sufficient conditions to guarantee exponential stability of the origin are stated in the following theorem.
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Theorem 2: The origin of system (8) is globally exponentially stable if the following assumptions hold. A1-i. For i ∈ Su , Re[λ(Ai )] > 0. A1-ii. For i ∈ Ss , Ai is Hurwitz. A2. For each i ∈ S, there exist positive constants ai and bi such that 2xT Pi gi (t, x, xt ) ≤ ai kxk2 + bi kxt k2τ . 7
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0.4
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Time t
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x(t)
0.8
Figure (1): Linear switched delay system with unstable and stable modes
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A3. For i ∈ Ss ,
λ (Q ) − a bi min i i − + < 0. λM λm A4. Assumption A2 of Theorem 1 holds.
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Proof For each i ∈ S, define Vi (x) = xT Pi x. Then, the time derivative along the trajectories of (8) is V˙i (x) = x˙ T Pi x + xT Pi x˙ = [Ai x + gi(t, x, xt )]T Pi x + xT Pi [Ai x + gi (t, x, xt )] = xT [ATi Pi + Pi Ai ]x + 2xT Pi gi (t, x, xt ) For i ∈ Su , there exists γ > 0 such that Re[λ(Ai − γI)] < 0. Then,
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V˙i (x) ≤ 2γxT Pi x + ai kxk2 + bi kxt k2τ ai bi ≤ 2γ + Vi (x) + kVi kτ λmin(Pi ) λmin (Pi ) t Therefore, there is a ξi > 0 such that Vi (x) ≤ kVitk−1 kτ eξi (t−tk−1 )
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For i ∈ Ss ,
λ (Q ) − a bi min i i ˙ Vi (x) ≤ − Vi (x) + kVi kτ λmax (Pi ) λmin (Pi ) t
So that there exists ζi > 0 such that Vi (x) ≤ kVitk−1 kτ e−ζi (t−tk−1 ) 8
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√ where K = µ. The following example illustrates these results.
t ≥ t0
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∗ −ν)(t−t )/2 0
kxk ≤ Kkxt0 kτ e−(λ
Example 2: Consider system (8) where the initial condition x(t) = t + 1, τ = 1 and Mode1
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x˙ 1 = 0.5x1 + 2ln(1 + x1 (t − 1)) x˙ 2 = 0.5x2 + 2sinx2 (t − 1) Mode2
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x˙ 1 = −x1 + 0.1sinx1 (t − 1) x˙ 2 = −x2 + 0.1ln(1 + x2 (t − 1))
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For i = 1 and by taking Q1 = I2 , with I2 being the identity matrix of size 2 × 2, we have P1 = I2 , so that V1 (x) = x21 + x22 . By little effort, one gets a1 = 2, b1 = 2 and hence the growth rate ξ = 2.5. For i = 2, take Q2 = 3I2 to get P2 = 1.5I2 , V2 (x) = 1.5x21 + 1.5x22 , a2 = 0.15, b2 = 0.15 and the decay rate ζ = 1.2997. Figure (2) shows the simulation result of this system. 1.4
1.2
1
x(t)
0.8
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0.6
x1(t) x2(t)
0.4
0.2
0
2
4
6
8
10
12
14
Time t
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Figure (2): Nonlinear switched delay system with unstable and stable modes
4. Conclusion Having developed delay-version comparison lemmas that help us calculate the growth rates of unstable delay systems, we successfully established exponential stability of 9
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switched time delay systems that consist of unstable and stable modes. In fact, we have shown that under a class of switching signal where the total activation time of the stable modes is larger than that of the unstable modes, exponential stability of the entire switched system is ensured by using dwell time approach. The stability result was achieved by applying multiple Lyapunov function technique to the linear and a special class of nonlinear subsystems. Acknowledgment This research was financially supported by Natural Sciences and Engineering Research Council of Canada (NSERC).
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References
A. Bellen and M. Zennaro. Numerical Methods for Delay Differential Equations. Oxford University Press, New York, 2003.
[2]
R. Bellman and K.L. Cooke. Differential-Difference Equations. Academic Press, New York, 1963.
[3]
W.P. Dayawansa and C.F. Martin. A Convers Lyapunov Theorem for a Class of Dynamical Systems which Undergo Switching. IEEE Trans.Automat. Contr., VOL. 44, NO.4, 751-759, 1999.
[4]
R.D. Driver. Ordinary and Delay Differential Equations. Springer-Verlag, New York Inc., 1977.
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L.E. El’sgol’ts and S.B. Norkin. Introduction to the Theory and Application of Differential Equations with Deviating Arguments. Academic Press, New York, 1973.
[6]
K. Gopalsamy. Stability and Oscillations in Delay Differential Equations of Population Dynamics. Kluwer Academic Puplishers, Netherlands, 1992.
[7]
A. Halanay. Differential Equations: Stability, Oscillations, Time Lags. Academic Press, Inc., New York, 1966.
[8]
J.K. Hale and S.M.V. Lunel. Introduction to Functional Differential Equations. Springer-Verlag, New York, 1993.
[9]
J.P. Hespanha and A.S. Morse. Stability of Switched Systems with Average DwellTime. Proc. of the 38th IEEE Conf. on Decision and Control, 2655-2660, Phoenix, AR, 1999.
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[1]
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[10] B. Hu, X. Xu, A.N. Michel and P.J. Antsaklis. Stability Analysis For a Class of Nonlinear Switched Systems. Proc. of the 38th IEEE Conf. on Decision and Control, 43744379, Phoenix, AR, 1999. [11] S. Kim,S.A. Campbell, and X.Z.Liu. Stability of a Class of Linear Switching Systems with Time Delay. IEEE Trans. on Circuits and Systems- I. Regular papers, VOL.53, NO.2, pp.384-393, 1999.
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ACCEPTED MANUSCRIPT [12] N.N. Krasovskii and J.L. Brenner. Stability of Motion: Applications of Lyapunov’s Second Method to Differential Systems and Equations with Delay. Stanford University Press, Stanford, CA, 1963.
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[13] Y. Kuang. Delay Differential Equations with Applications in Population Dynamics. Academic Press Inc., CA, 1993.
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[14] X. Liu, X. Shen and Y. Zhang. Exponential Stability of Singularly Perturbed Systems with Time Delay. Applicable Analysis. VOL.82, NO.2, 117-130, 2003. [14] N. MacDonald. Biological Delay Systems: Linear Stability Theory. Cambridge University Press, Cambridge, 1989.
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[15] N. Minorsky. Self-Excited Oscillations in Dynamical Systems. J. of Applied Mechanics, 9, 65-71, 1942. [16] A.S. Morse. Supervisory Control of Families of Linear Set-Point Controllers-Part I: Exact Matching. IEEE Trans.Automat. Contr., VOL. 41, NO.10, 1413-1431, 1996.
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[17] A.S. Morse, editor. Control Using Logic-Based Switching. Lecture Notes in Control and Information Sciences 222. Springer, New York, 1997. [18] X-M. Sun, G.M. Mimirovski, J. Zhao and W. Wang. Exponential Stability for Switched Delay Systems Based on Average Dwell Time Technique and Lyapunov Function Method. Proc. of the 2006 American Control Conference,pp. 1539-1543, Minneapolis, Minnesota, USA, June 14-16,2006.
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[19] R. Wang, X. Liu and Z. Guan. Robustness and Stability Analysis for a Class of Nonlinear Switched Systems with Impulse Effects. Dynamic Systems and Applications. 14, 233-248, 2004. [20] S. Yang, X. Zhengrong, C. Qingwei and H. Weili. Dynamical Output Feedback Control of Discrete Switched System with Time Delay. 2004 IEEE, Proc. of the 5th World Congress on Intelligent Control and Automation.1088-1091, Hangzhou, China, 2004. ˇ [21] M. Zefran and J.W. Burdick. Design of Switching Controllers for Systems with Changing Dynamics. Proc. of the 37th IEEE Conf. on Decision and Control, 2113-2118, Tampa, FL, 1998.
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[22] Y. Zhang. Stability of Hybrid Systems with Time Delay. Ph.D. Thesis, University of Waterloo, Ontario, Canada, 2004.
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[23] G. Zhai, B. Hu, K. Yasuda and A.N. Michel. Stability Analysis of Switched Systems with Stable and Unstable Subsystems: An Average Dwell Time Approach. International J. of Systems Science, VOl. 32, NO. 8, 1055-1061, 2001.
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S0895-7177(08)00010-1 10.1016/j.mcm.2007.12.024 MCM 3290
To appear in:
Mathematical and Computer Modelling
Received date: 4 September 2007 Accepted date: 7 December 2007 Please cite this article as: M.S. Alwan, X. Liu, On stability of linear and weakly nonlinear switched systems with time delay, Mathematical and Computer Modelling (2008), doi:10.1016/j.mcm.2007.12.024 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT ON STABILITY OF LINEAR AND WEAKLY NONLINEAR SWITCHED SYSTEMS WITH TIME DELAY
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Mohamad S. Alwan and Xinzhi Liu
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Department of Applied Mathematics, University of Waterloo Waterloo, Ontario, Canada N2L 3G1, Email:[email protected].
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Abstract– This paper studies time-delayed switched systems that include both stable and unstable modes. By using multiple Lyapunov functions technique and a dwell time approach, several criteria on exponential stability for both linear and nonlinear systems are established. It is shown that by suitably controlling the switching between the stable and unstable modes exponential stabilization of the switched system can be achieved. Some examples and numerical simulations are provided to illustrate our results. Keywords: Switched systems, Time delay, Exponential stability
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1. Introduction
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By a switched system, we mean a hybrid system that consists of a finite number of continuous-time dynamical systems and a switching law that organizes the switching between these systems. Here, the switching from one mode to another one is not known a priori, but it is determined by some environmental factors which usually are not a part of the mathematical model under consideration. For some examples of such systems, interested readers may refer to [17],[21]. The importance of studying such systems is two fold. Firstly, a large class of real life or engineering systems are inherently multi-model in the sense that their behavior is represented by several dynamical models; secondly, some systems may be stabilized by several feedback controllers (known as switching control), rather than one controller. In fact, there has been increasing interest in designing an appropriate switching rule in order to stabilize the system [3], [9],[10], [16],[19],[23]. Ordinary differential equations (ODEs) have long played important roles in modeling many physical processes and they will continue to serve as a fundamental tool in future investigations. But sometimes ODEs may be inadequate in modeling systems whose future state depend not only on the present state but also on the past. In those cases, ODEs give only an approximation of some real systems. More realistic models should include some of the historical values of the state that leads us to delay differential equations (DDEs). The early motivations for studying DDEs came from their applications in population dynamics when Volterra investigated the predatorprey model, and in Minorsky’s study of ship stabilization and automatic steering. These studies indicate the importance of considering delay in the feedback mechanism [15]. Another motivation for studying time-delayed systems is that the presence of delay may cause undesirable performance. As we know, in ordinary systems, oscillation phenomena might be seen in some systems with at least two state variables and chaotic behavior is noticed in some systems with at least three state variables, but in some cases these unsuitable behaviors occur in one order delay systems; moreover, small delay may destroy stability of some systems, but large delay may stabilize 1
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others. As a result, there are many studies of systems with time delay. A number of monographes have been devoted to the subject of DDEs. These works include books by Bellman and Cooke [2] and Krasovkii and Brenner [12], texts by Halanay [7], Hale and Lunel [8], Driver [4], El’sgoll’ts and Norkin [5] and Bellen and Zennaro [1]. Some other books dedicated to the applications of DDEs are MacDonald [14], Gopalsamy [6], and Kuang [13]. The study of delay differential equations is currently very heavy due to the growing progress achieved in the understanding of the dynamics of many important time delayed systems. Switched systems and time delay are important issues encountered in many fields. The area of switched delay systems is still in its infancy. Nevertheless, there has been reasonable progress in this field. Stability of switched delay systems has been receiving an increasing attention (see [11],[18],[20],[22]). In this paper, we investigate stability property of switched delay systems consisting of unstable and stable modes. Delay-version comparison lemmas for a single equation and system are developed to help us find the growth rate for the unstable modes. The first lemma is in fact an analog to a Lemma in [7], while the second one is analogous to Lemma 2.3 in [14]. By using multiple Lyapunov functions, a general criteria on exponential stability of these systems is established. Here, our stability results for linear and a special class of nonlinear systems is characterized by the locations of the eigenvalues. The organization of this paper is as follows; in Section 2, we describe switched delay systems and introduce some notation and definitions that will be used in the rest of this paper. In Section 3, we shall state our main results. Numerical examples are also presented to verify our theoretical results. We finally conclude this work in Section 4. 2. Problem Formulation Switched delay system can be described as follows x˙ = fi (t, x, xt ),
t ∈ [tk−1 , tk )
(1)
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where x ∈ Rn with Rn being n-dimensional Euclidean space, i ∈ S = {1, 2, · · · , N}, k = 1, 2, · · · , tk−1 < tk , and limk→∞ tk = ∞. We assume that fi is smooth enough to guarantee the existence of solutions, and fi (t, 0, 0) ≡ 0 to ensure that system (1) has a trivial solution. Let xt = x(t + θ), θ ∈ [−τ, 0], with τ being a positive constant that represents the time delay, x(t) be the solution of (1) with the initial condition xt0 , kxt kτ = supt−τ ≤θ≤t kx(θ)k with kxk being the Euclidean norm of x, P kAk = max1≤j≤n ( ni=1 a2ij )1/2 be the norm of the n × n matrix A = (aij ), AT be the transpose of a matrix A, λ(A) be the eigenvalue of an n×n matrix A, and Re[λ(A)] be the real part of λ(A). A symmetric matrix (i.e. P = P T ) is said to be positive definite if all its eigenvalue are positive. Denote by λmin(P ) and λmax (P ) the minimum and maximum eigenvalue of P respectively. Let Ai be a Hurwitz matrix (i.e. Re[λ(Ai )] < 0). Then, there exist positive definite matrices Pi and Qi satisfying the Lyapunov equation ATi Pi + Pi Ai = −Qi .
Let Vi (x) = xT Pi x. Then, we have for all i ∈ S
λmin (Pi )kxk2 ≤ Vi (x) ≤ λmax (Pi )kxk2 . 2
ACCEPTED MANUSCRIPT It follows that Vj (x) ≤ µVi (x),
≥ 1 with λM = max{λmax (Pi ), ∀i ∈ S}, and λm = min{λmin(Pi ), ∀i ∈
T
λM λm
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where µ = S}.
∀i, j
Definition: The trivial solution of system (1) is said to be exponentially stable if there exist positive constants k ≥ 1 and λ such that kx(t)k ≤ kkxt0 kτ e−λ(t−t0 ) ,
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for any solution x(t) of (1).
t ≥ t0
Before stating our main results, we present the following lemmas which are frequently used in proving our theorems.
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Lemma 1: For a ∈ R, with a > 0, and t0 ∈ R+ = [0, ∞), let u : [t0 , t0 + a) → R+ satisfy the following delay differential inequality u(t) ˙ ≤ αu(t) + β sup u(θ), θ∈[t−τ,t]
t ∈ [t0 , t0 + a).
Assume that α + β > 0. Then, there exist positive constants ξ and k such that u(t) ≤ keξ(t−t0 ) ,
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t ∈ [t0 , t0 + a)
(2)
where ξ = α + β and k = supθ∈[t0 −τ,t0 ] u(θ).
Proof: Claim y(t) = keξ(t−t0 ) is a solution of the delay differential equation y(t) ˙ = αy(t) + β sup y(θ), θ∈[t−τ,t]
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with the initial condition
y(t) = u(t),
t ∈ [t0 , t0 + a)
t ∈ [t0 − τ, t0 ].
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To prove the claim, we check that y(t) ˙ = ξkeξ(t−t0 ) sup y(θ) = keξ(t−t0 ) = y(t)
θ∈[t−τ,t]
and (3) becomes
ξkeξ(t−t0 ) = αkeξ(t−t0 ) + βkeξ(t−t0 )
3
(3)
ACCEPTED MANUSCRIPT Hence, y(t) = keξ(t−t0 ) is indeed a solution of (3) where ξ and k are defined above. By Proposition 2 in [7], we have sup
u(θ)eξ(t−t0 ) ,
θ∈[t0 −τ,t0 ]
t ∈ [t0 , t0 + a).
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u(t) ≤ keξ(t−t0 ) =
y(t) ˙ ≤ A(t)y(t) + B(t) sup y(θ)
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T where y(t) = y1 (t), y2 (t), · · · yn (t) ≥ 0 and supθ∈[t−τ,t] y(θ) = supθ∈[t−τ,t] y1 (θ), supθ∈[t−τ,t] y2 (θ), T · · · , supθ∈[t−τ,t] yn (θ) . Then, there exists a ξ > 0 such that
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θ∈[t−τ,t]
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Lemma 2: For a ∈ R with a > 0 and t ∈ [t0 , t0 + a), t0 ∈ R+, let A(t) and B(t) be n × n matrices of continuous functions, α(t) = λ A(t) + AT (t) , kB(t)k ≤ β1 and α(t) + kB(t)k ≤ β2 , (β2 > 0). Assume that the following inequality is satisfied.
ky(t)k ≤ kyt0 kτ eξ(t−t0 ) , where ξ = (β1 + β2 )/2.
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Proof: Let v(t) = y T (t)y(t) = ky(t)k2. Then,
t ∈ [t0 , t0 + a)
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v(t) ˙ = y˙ T (t)y(t) + y T (t)y(t) ˙ T ≤ A(t)y(t) + B(t) sup y(θ) y(t) θ∈[t−τ,t] + y T (t) A(t)y(t) + B(t) sup y(θ) θ∈[t−τ,t] ≤ y T (t) AT (t) + A(t) y(t) + 2kB(t)kky(t)kkytkτ ≤ α(t) + kB(t)k v(t) + kB(t)kkvt kτ ≤ β2 v(t) + β1 kvt kτ .
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By Lemma 1, there exists a ξ > 0 such that v(t) ≤ kvt0 kτ e2ξ(t−t0 ) ,
where 2ξ = β1 + β2 , and hence ky(t)k ≤ kyt0 kτ eξ(t−t0 ) ,
t ∈ [t0 , t0 + a).
In fact, Lemma 2 is an analog to Lemma 2.3 in [14] that shall be used in this paper 4
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Lemma 3:[14] For t ∈ [t0 , ∞), let A(t) and B(t) be n × n matrices of continuous ˙ functions, A(t), B(t), A(t) be bounded and A(t) be Hurwitz. Furthermore, assume that A1) λ(AT (t) + A(t)) ≤ −α(t) < 0. A2) − α(t) + 2kB(t)k ≤ −β < 0, with β being a positive constant. A3) y(t) ˙ ≤ A(t)y(t) + B(t) supt−τ ≤θ≤t y(θ), where y(t) = (y1 (t), y2(t), · · · , yn (t))T ≥ 0, and supt−τ ≤θ≤t y(θ) = (supt−τ ≤θ≤t y1 (θ), supt−τ ≤θ≤t y2 (θ), · · · , supt−τ ≤θ≤t yn (θ))T . Then, there exists a positive constant ξ such that ky(t)k ≤ kyt0 kτ e−ξ(t−t0 ) ,
t ≥ t0 .
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3. Main results In this section, we establish exponential stability of switched delay systems incorporating unstable and stable modes. Let Su = {1, 2, · · · , r} and Ss = {r + 1, r + 2, · · · , N} be respectively the set of indices of the unstable and stable modes. I. Linear systems Consider the following linear switched delay system
t ∈ [tk−1 , tk )
(4)
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x(t) ˙ = Ai x(t) + Bi x(t − τ ),
where for each i ∈ S = Su ∪ Ss , Ai and Bi are n × n constant matrices. Theorem 1: The origin of system (4) is globally exponentially stable if the following assumptions hold. A1-i. For i ∈ Su ,
Re[λ(Ai )] > 0,
and Re[λ(Ai + Bi )] > 0.
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A1-ii. For i ∈ Ss , Ai is Hurwitz and λ (Q ) − β ∗ β ∗ min i i − + i 0 such that Re([Ai − γI]) < 0. With the aid of Lemma 2, where β1 = 2γ + βi∗ /λm and β2 = βi∗ /λm , there exists a ξi > 0 such that Vi (x) ≤ kVitk−1 kτ eξi (t−tk−1 )
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Similarly, for i ∈ Ss , we have
V˙ i (x) ≤ −λmin (Qi )kxk2 + βi∗ kxk2 + βi∗ kxt k2 λ (Q ) − β ∗ β∗ min i i ≤ − Vi (x) + i kVit kτ λM λm λmin (Qi )−βi∗ λM
By Lemma 3, where αi =
and βi =
βi∗ , λm
there exists a ζi > 0 such that
Vi (x) ≤ kVitk−1 kτ e−ζi (t−tk−1 )
Then,
r Y
µeξi (ti −ti−1 ) ×
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VN (x) ≤
i=1
NY −r−1
µeζj τ e−ζj (tj −tj−1 )
j=r+1
×kV1t0 kτ e−ζN (t−tN−1 )
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By condition (5), we have VN (x) ≤
r Y i=1
µ×
NY −r−1 j=r+1
µeζj τ × kV1t0 kτ e−λ
∗ (t−t ) 0
and, by (6) and (7) we get ∗ −ν)(t−t ) 0
VN (x) ≤ kV1t0 kτ e−(λ
6
,
t ≥ t0 .
ACCEPTED MANUSCRIPT Thus, ∗ −ν)(t−t )/2 0
√
,
t ≥ t0 ,
µ; this shows that the origin of (4) is exponentially stable.
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where K =
kx(t)k ≤ Kkxt0 kτ e−(λ
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Remark 1: Assumption A1 is made to ensure instability and stability of the modes, while Assumption A2 means that the total activation time of stable modes is larger than that of the unstable modes and that the solutions are kept down whenever the modes are switched. The following example illustrates these results.
0.5 0 0 0.5
A2 =
−1 0 0 −1
1 0 0 1
P1 =
,
B1 =
2 0 0 1
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Take γ = 1, Q1 =
A1 =
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Example 1: Consider system (4) where the continuous initial condition x(t) = t + 1, τ = 1,
0.1 0 , B2 = 0 0.1 3 0 and Q2 = . 0 3
Then,
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1 0 0 1
and P2 =
1.5 0 0 1.5
.
A simple check shows that µ = 1.5, the growth rate ξ = 2.5, and the decay rate ζ = 1.3318. Figure (1) shows the solution vanishing exponentially. Obviously, since the growth rate is larger than the decay rate, running stable modes longer than unstable ones is strongly required. II. Nonlinear Systems Consider the following nonlinear switched system with time delay
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x(t) ˙ = Ai x(t) + gi (t, x, xt )
(8)
where i ∈ Su ∪ Ss , and gi (t, 0, 0) ≡ 0. Sufficient conditions to guarantee exponential stability of the origin are stated in the following theorem.
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Theorem 2: The origin of system (8) is globally exponentially stable if the following assumptions hold. A1-i. For i ∈ Su , Re[λ(Ai )] > 0. A1-ii. For i ∈ Ss , Ai is Hurwitz. A2. For each i ∈ S, there exist positive constants ai and bi such that 2xT Pi gi (t, x, xt ) ≤ ai kxk2 + bi kxt k2τ . 7
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1
0.6
0.4
0.2
0
0
2
4
6
8
10
12
14
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Time t
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x(t)
0.8
Figure (1): Linear switched delay system with unstable and stable modes
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A3. For i ∈ Ss ,
λ (Q ) − a bi min i i − + < 0. λM λm A4. Assumption A2 of Theorem 1 holds.
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Proof For each i ∈ S, define Vi (x) = xT Pi x. Then, the time derivative along the trajectories of (8) is V˙i (x) = x˙ T Pi x + xT Pi x˙ = [Ai x + gi(t, x, xt )]T Pi x + xT Pi [Ai x + gi (t, x, xt )] = xT [ATi Pi + Pi Ai ]x + 2xT Pi gi (t, x, xt ) For i ∈ Su , there exists γ > 0 such that Re[λ(Ai − γI)] < 0. Then,
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V˙i (x) ≤ 2γxT Pi x + ai kxk2 + bi kxt k2τ ai bi ≤ 2γ + Vi (x) + kVi kτ λmin(Pi ) λmin (Pi ) t Therefore, there is a ξi > 0 such that Vi (x) ≤ kVitk−1 kτ eξi (t−tk−1 )
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For i ∈ Ss ,
λ (Q ) − a bi min i i ˙ Vi (x) ≤ − Vi (x) + kVi kτ λmax (Pi ) λmin (Pi ) t
So that there exists ζi > 0 such that Vi (x) ≤ kVitk−1 kτ e−ζi (t−tk−1 ) 8
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√ where K = µ. The following example illustrates these results.
t ≥ t0
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∗ −ν)(t−t )/2 0
kxk ≤ Kkxt0 kτ e−(λ
Example 2: Consider system (8) where the initial condition x(t) = t + 1, τ = 1 and Mode1
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x˙ 1 = 0.5x1 + 2ln(1 + x1 (t − 1)) x˙ 2 = 0.5x2 + 2sinx2 (t − 1) Mode2
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x˙ 1 = −x1 + 0.1sinx1 (t − 1) x˙ 2 = −x2 + 0.1ln(1 + x2 (t − 1))
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For i = 1 and by taking Q1 = I2 , with I2 being the identity matrix of size 2 × 2, we have P1 = I2 , so that V1 (x) = x21 + x22 . By little effort, one gets a1 = 2, b1 = 2 and hence the growth rate ξ = 2.5. For i = 2, take Q2 = 3I2 to get P2 = 1.5I2 , V2 (x) = 1.5x21 + 1.5x22 , a2 = 0.15, b2 = 0.15 and the decay rate ζ = 1.2997. Figure (2) shows the simulation result of this system. 1.4
1.2
1
x(t)
0.8
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0.6
x1(t) x2(t)
0.4
0.2
0
2
4
6
8
10
12
14
Time t
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Figure (2): Nonlinear switched delay system with unstable and stable modes
4. Conclusion Having developed delay-version comparison lemmas that help us calculate the growth rates of unstable delay systems, we successfully established exponential stability of 9
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switched time delay systems that consist of unstable and stable modes. In fact, we have shown that under a class of switching signal where the total activation time of the stable modes is larger than that of the unstable modes, exponential stability of the entire switched system is ensured by using dwell time approach. The stability result was achieved by applying multiple Lyapunov function technique to the linear and a special class of nonlinear subsystems. Acknowledgment This research was financially supported by Natural Sciences and Engineering Research Council of Canada (NSERC).
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