Exponential Convergence Estimates for a Single ... - Semantic Scholar
IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 25, NO. 7, JULY 2014
1401
Exponential Convergence Estimates for a Single Neuron System of Neutral-Type Xiaofeng Liao, Senior Member, IEEE , Chuandong Li, and Tingwen Huang
Abstract— The future behavior of a dynamical system is determined by its initial state or initial function. Nontrivial neuron system involving adaptive learning corresponds to the memorization of initial information. In this paper, exponential estimates and sufficient conditions for the exponential stability of a single neuron system of neutral-type are studied. Of particular importance is the fact that exponential convergence guarantees that this system is capable of memorizing initial functions. Furthermore, this system is also capable of conveying much more information with respect to the initial functions memorized by neuron system with time delay. The proofs follow some new results on nonhomogeneous difference equations evolving in continuous-time combined with the Lyapunov–Krasovskii functional and the descriptor system approach. The exponential stability conditions are expressed in terms of a linear matrix inequality, which lead to less restrictive and less conservative exponential estimates.
Index Terms— Exponential estimates, linear matrix inequality (LMI), Lyapunov–Krasovskii functional, neuron system, neutral differential equations, nonhomogeneous difference equations. I. I NTRODUCTION Systems with neutral differential equations are discussed for various real world dynamical systems including genomic regulation [1], neurophysiology [2], [3], linguistics [4], and so on. They are generally treated as undesired cases. It is well known that a very important qualitative feature of dynamical systems is stability. However, as we will discuss in this brief, some particular neutral differential systems might be highly useful when the problem is capability to memorize dynamic information. Existences of delays between the interactions of variables are reported for various classes of dynamical systems [14], for example, delays due to translation and transcription in genetics, travel time of impulses from the postsynaptic connections to the presynaptic connections in neurophysiology [2], [3]. It should be pointed out that time delays not only occur in the system states (called as retarded type delays), but also appear in the derivatives of system states (called as neutral type delays). Those delays generally result in neutral differential equation systems that can find a variety of applications in practice such as chemical reactor, distributed networks containing lossless transmission lines, partial element equivalent circuits in VLSI systems, and Lotka–Volterra systems [14]–[16]. Those systems might be capable of memorizing initial functions rather than initial states. Manuscript received May 2, 2013; revised August 31, 2013; accepted November 7, 2013. Date of publication January 2, 2014; date of current version June 10, 2014. This work was supported in part by the National Natural Science Foundation of China under Grant 60973114 and Grant 61170249, in part by the Natural Science Foundation Project of CQCSTC under Grant 2009BA2024, in part by the State Key Laboratory of Power Transmission Equipment and System Security and New Technology, Chongqing University, under Grant 2007DA10512711206, in part by the Program for Changjiang Scholars, and in part by the Specialized Research Fund for priority areas for the Doctoral Program of Higher Education. X. Liao and C. Li are with the College of Electronic and Information Engineering, Southwest University, Chongqing 400715, China (e-mail: [email protected]; [email protected]). T. Huang is with Texas A & M University at Qatar, Doha 23874, Qatar (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TNNLS.2013.2290698
In this paper, we consider a single neuron system of neutral-type with the following form: d [x(t)+ px(t −τ )] = −ax(t)+b tanh[x(t −σ )], t ≥ 0 dt
(1)
where a, τ and σ are the positive constants, b and p are the real numbers with | p | < 1. Note that the inequality is indeed necessary for the stability of the discrete operator D(xt ) = x(t) + px(t − τ ). Furthermore, the stability property can be guaranteed for all values of τ by taking τ as a parameter. For each solution x(t) of system (1), the following initial condition is assumed: x(t) = ϕ(t), t ∈ [−δ, 0]
(2)
where δ = max{τ, σ }, ϕ ∈ ([−δ, 0], R). If p = 0, system (1) becomes the following form: d x(t) = −ax(t) + b tanh[x(t − σ )], dt
t ≥0
(3)
where the initial condition of system (3) is x(t) = ϕ(t), t ∈ [−σ, 0].
(4)
Delay differential equations of various types that contain (3) as a special case have been proposed by many authors for the study of the dynamical characteristics of Hopfield type neural networks [9], [11]. In [5], the asymptotic stability of system (1) with time-varying coefficients was investigated. Then, the dynamic behavior under the condition τ = σ was studied in [6]. Based on a linearization technique, the oscillatory and nonoscillatory behaviors of system (1) were investigated in detail. The sufficient conditions for asymptotic stability are derived using a Lyapunov-like approach [6]. Moreover, Park [7] studied system (1) with the requirement b > 0. Sufficient criteria for delay-dependent asymptotic stability were obtained by using a linear matrix inequality. His results are a generalization of existing ones [5], [6]. Recently, a new delay-dependent criterion for the asymptotic stability of system (1) was established using the neutral transformation and the linear matrix inequality (LMI) approach [8]. Compared with the results in [5]–[7], the new criterion is less conservative. Recently, there are some papers [20], [21] which derived delay-independent sufficient conditions for neutraltype neural networks. The above-mentioned results guarantee the asymptotic stability of neutral-type neuron system, but they do not provide an estimate of the convergence rate, neither the bound of the norm of the solution. For linear time-invariant systems, asymptotic stability is equivalent to exponential stability. Then, one may ask if there exists the possibility of using the LMI approach to derive the exponential estimates for the solutions of the neutral differential systems. However, to the best of our knowledge, these estimates as well as the exponential bounds for the case of neutral-type delay systems have not been addressed in the literature. In terms of rigorous mathematics, our concern is the following. Definition 1 [9], [11], [14]: System (1) is said to be exponentially stable if there exist β > 0 and γ ≥ 1 such that for every solution
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x(t, ϕ), where ϕ ∈ C([−δ, 0], R), the following exponential estimate holds: |x(t, ϕ)| ≤ γ e−βt |ϕ|δ , t ≥ 0.
(5)
The aim of this paper is to determine a lower bound for the decay rate β, and an upper bound for the γ -factor in terms of LMIs derived using the Lyapunov–Krasovskii approach. The rest of this paper is organized as follows. As the difference equation of the neutral system plays a key role in our work, some results concerning the exponential estimates for difference equations evolving in continuous-time domain are given in Section II. In Section III, a delay-dependent solution is obtained in terms of LMIs using a descriptor model transformation of the system. These results lead to less restrictive and less conservative exponential estimates on the solution of a single neutral-type neuron system with two delays. Some comparisons with the known results and two numerical examples are also presented in Section IV.
To use Lemma 1 for determining the exponential estimates of the solution of (9), we search for a solution z(t) in the form Le−ρt with ρ > 0 and L > 0, for inequality (7), i.e., z(t) must satisfy Le−ρt ≥ | p|L
sup
|x(t − θ)| + μe−βt
(6)
where λ is a positive real number, μ ≥ 0 and β > 0. Let z(t), t ≥ −δ be a continuous function such that z(t) ≥ λ
sup
min{τ,σ }≤θ≤δ
z(t − θ) + μe−βt , for t ≥ 0.
(7)
must hold, or equivalently 1 ≥ | p| eρδ +
−δ ≤θ ≤0
μ (ρ−β)t . e L
(13)
Let us define a positive constant α such that | p| = e−αδ . Inequality (13) takes the form μ (14) 1 ≥ e(ρ−α)δ + e(ρ−β)t L and we obtain that for any choice of ρ and L such that (15)
and correspondingly μ . (16) 1 − e(ρ−α)δ Then, condition (7) of Lemma 1 is satisfied. Now, we search for a z(t) such that conditions (15) and (16) hold and that condition (8) of Lemma 1 is also satisfied. We observe that as | p| < 1, μ ≥ 0 and ρ < α, the choice μ }e− min{− ln(| p|)/δ,β}t (17) z(t) = max{|ϕ|δ , 1 − e(ρ−α)δ is such that for −δ ≤ θ ≤ 0 L>
z(θ) ≥ |ϕ(θ)|δ ≥ |ϕ| ≥ |x(θ)|.
Then, if |x(θ)| ≤ z(θ), for
(12)
Le−ρt ≥ | p| Le−ρt eρδ + μe−βt
0 < ρ < min{α, β}
First, we will give a technical result of independent interest, which will be used later for deriving the exponential estimates for difference equations evolving in continuous-time domain. Lemma 1: Given a function x(t), t ≥ −δ, that satisfies the condition min{τ,σ }≤θ≤δ
e−ρ(t −θ) + μe−βt .
Obviously
II. E XPONENTIAL E STIMATES FOR D IFFERENCE E QUATION
|x(t)| ≤ λ
sup
min{τ,σ }≤θ≤δ
(8)
it follows that z(t) defines an upper bound for |x(t)| where t ≥ 0, i.e.,
Hence conditions (7) and (8) of Lemma 1 hold if z(t) satisfies (17). We are now able to work out the exponential estimates for the solution of the nonhomogeneous difference equation. Lemma 2: Consider system (9) with initial condition (11) and | p | < 1. Then, the solution x(t, ϕ) of (9) is such that the inequality
|x(t)| ≤ z(t), for t ≥ 0. Proof: For convenience, we assume τ ∈ [0, δ]. Then, for t ∈ [0, τ ], we have z(t) ≥ λ sup [z(t − θ)] + μe−βt .
|x(t)| ≤ K (ϕ)e−ρt
(18)
ρ < min{α, β}
(19)
holds, with
where α = − ln(| p|)/δ, and
τ ≤θ≤σ
K (ϕ) > max |ϕ|δ ,
As the time argument of z is negative, it follows from (8) that: z(t) ≥ λ sup [|x(t − θ)|] + μe−βt τ ≤θ≤σ
μ 1 − e(ρ−α)δ
.
(20)
Proof: Observe that the solution x(t) of system (9) is such that |x(t)| ≤ | p| |x(t − τ )| + | f (t)|
and (6) implies
≤ | p|
z(t) ≥ |x(t)|. Hence, we obtain the conclusion. We consider the nonhomogeneous difference equation of the form x(t) + px(t − τ ) = f (t)
(9)
| f (t)| ≤ μe−βt , μ ≥ 0, and β > 0.
(10)
where f (t) satisfies
For any continuous function ϕ ∈ C([−δ, 0], R), there exists a unique solution x(t, ϕ) of (9) and (10) satisfying the initial condition x(t, ϕ) = ϕ(θ), θ ∈ [−δ, 0].
(11)
sup
min{τ,σ }≤θ≤δ
|x(t − θ)| + μe−βt
(21)
hence, condition (6) of Lemma 1 holds. Also, observe that for z(t) = K (ϕ)e−ρt , t ≥ −δ with ρ and K (φ) defined in (19) and (20), respectively, it follows from the above argument [see (17)] that z(t) satisfies conditions (7) and (8) of Lemma 1. The result follows directly. In the homogenous case, the result reduces to the following: Corollary 1: Consider system (9) with initial condition (11) and | p| < 1. In the homogeneous case, i.e., μ = 0 and β is arbitrarily large, then the solution x(t, ϕ) is such that |x(t)| ≤ |ϕ|δ eξ t , for t ≥ 0 with ξ = ln | p| /δ.
IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 25, NO. 7, JULY 2014
Proof: The result is obtained by setting μ = 0 and β arbitrarily large in Lemma 2. From the above analysis, we note that | p | < 1 guarantees exponential stability for the nonhomogeneous difference equation (9). Delay-independence with respect to τ also guarantees that small changes in τ do not destabilize the system [9]–[11]. III. E XPONENTIAL E STIMATES FOR A S INGLE N EURON S YSTEM OF N EUTRAL -T YPE In this section, we determine the exponential estimates for the solution of system (1), when the parameters are known. We would like to derive an exponential estimate in the form given by (5) for the solution of system (1). As we will observe below in the stability analysis of our model using Lyapunov–Krasovskii functional, the (Krasovskii) theorem cannot be always applied directly since the natural construction of a Lyapunov candidate leads to a degenerate function, which is not positive-definite in the corresponding stability theorem. In these cases, we need an appropriate stability theory and this has been given in [15]. Rewrite system (1) in the following equivalent descriptor system: y˙ (t) = −ax(t) + b tanh[x(t − σ )] (22) 0 = −y(t) + x(t) + px(t − τ ). Lemma 3: Suppose that the nonlinear time delay system (1) is given. If there exist positive constants α, β, γ , q1 , and real numbers q2 , q3 such that the inequality (α, γ , q1 , q2 , q3 ) < 0
From (28) and (29), we obtain the following inequality q1 |x(t) + px(t − τ )|2 ≤ V (t) ≤ α2 |xt |2δ
= −2q2 y 2 (t) − 2(aq1 + q3 − q2 )y(t)x(t) +2bq1 y(t) tanh[x(t − σ )] + 2q3 x 2 (t) +2 pq2 y(t)x(t − τ ) + 2 pq3 x(t)x(t − τ ). Similarly, one has V˙2 (t) = αx 2 (t)−αx 2 (t −τ )e−2βτ −2αβ
0 x 2 (t +θ)e2βθ dθ
−τ 2 2 +γ tanh [x(t)] − γ tanh [x(t − σ )]e−2βσ 0 −2βγ tanh2 [x(t + θ)]e2βθ dθ. −σ
Moreover, using the relation tanh2 [x(t)] ≤ x 2 (t), we obtain V˙ (t) ≤ −2(q2 − βq1 )y 2 (t) − 2(aq1 + q3 − q2 )y(t)x(t) + 2 pq2 y(t)x(t − τ ) + 2bq1 y(t) tanh[x(t − σ )] + (2q3 + α + γ )x 2 (t) + 2 pq3 x(t)x(t − τ )
(23)
− αx 2 (t − τ )e−2βτ − γ tanh2 [x(t − σ )]
× e−2βσ − 2βq1 y 2 (t) 0 0 − 2βγ tanh2 [x(t +θ)]e2βθ dθ −2αβ x 2 (t +θ)e2βθ dθ
(α, γ , q1 , q2 , q3 ) ⎡ ⎤ −2(q2 −βq1 ) −(aq1 +q3 −q2 ) pq2 bq1 ⎢−(aq1 +q3 − q2 ) 2q3 +α+γ pq3 0 ⎥ ⎥ =⎢ ⎣ pq3 −αe−2βτ 0 ⎦ pq2 0 0 −γ e−2βσ bq1
−σ
then, for any initial condition (2) the solution x(t, ϕ) of system (1) satisfies the inequality α2 −βt e |ϕ|δ (25) |x(t) + px(t − τ )| ≤ q1 where the positive constant α2 is defined as
Consider
the
(26)
following
Lyapunov–Krasovskii
V (t) = V1 (t) + V2 (t)
(27)
−τ
¯ = x¯ T (t)(α, γ , q1 , q2 , q3 )x(t) 0 tanh2 [x(t + θ)]e2βθ dθ −2βγ
(24)
α2 = q1 + τ α + σ γ .
−σ 0
x 2 (t + θ)e2βθ dθ − 2βq1 y 2 (t).
−2βα −τ
Clearly, we have d ¯ V (xt ) + 2βV (xt ) = x¯ T (t)(α1 , β, q1 , q2 )x(t) dt where x(t) ¯ = [y(t), x(t), x(t − τ ), tanh[x(t − τ )], tanh[x(t − σ )]]T and
where
V1 (t) = [y(t), x(t)] V2 (t) = α
0 −τ 0
1 0 0 0
q1 0 q2 q3
(28)
x 2 (t + θ)e2βθ dθ tanh2 [x(t + θ)]e2βθ dθ
+γ
y(t) x(t)
(30)
where α2 is given by (26). The time derivative of functional (28) along the trajectories of system (22) is
q q −ax(t) + b tanh[x(t − σ )] V˙1 = 2 [y(t), x(t)] 1 2 0 q3 −y(t) + x(t) + px(t − τ )
−ax(t)+b tanh[x(t −σ )] = [2q1 y(t), 2q2 y(t)+2q3 x(t)] −y(t)+x(t)+ px(t −τ )
holds, where
Proof: functional
1403
(29)
−σ
with α, β, γ , q1 are the positive constants and q2 , q3 are real numbers of Lemma 3. This functional is degenerated as it is usual for descriptor systems [10], [12].
(α, γ , q1 , q2 , q3 ) ⎡ ⎤ −2(q2 −βq1 ) −(aq1 +q3 −q2 ) pq2 bq1 ⎢−(aq1 + q3 − q2 ) 2q3 + α+γ pq3 0 ⎥ ⎥< 0. =⎢ ⎣ pq3 −αe−2βτ 0 ⎦ pq2 0 0 −γ e−2βσ bq1 Condition (23) implies that d V (xt ) + 2βV (xt ) ≤ 0. dt This inequality leads to the following: V (xt (ϕ)) ≤ e−2βt V (ϕ), t ≥ 0.
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Combining (30) and the previous inequality, we obtain, for t ≥ 0 the estimation
where the positive constant α2 is defined as α2 = q1 + σ γ .
q1 [x(t) + px(t − τ )]2 ≤ V (xt (ϕ))
IV. C OMPARISONS W ITH THE K NOWN R ESULTS
≤ e−2βt V (ϕ) ≤ α2 e−2βt |ϕ|2δ
and we arrive at the conclusion α2 −βt |x(t) + px(t − τ )| ≤ e |ϕ|δ , t ≥ 0 q1 where α2 is given by (26). Theorem 1: Consider system (1) and | p| < 1. If there exist positive constants α, β, γ , q1 and real numbers q2 , q3 such that the inequality (23) is satisfied, where (α, γ , q1 , q2 , q3 ) is given by (24), then for any initial condition (2) the solution x(t, ϕ) of system (1) satisfies inequality (5) with 0 < ρ < min{α, β} and γ >
1 1 − e(ρ−α)δ
(31)
α2 q1
(32)
where α = − ln | p|/δ and the constant α2 is given by (26). Proof: The result follows from Lemmas 2 and 3 by substituting √ μ = α2 /q1 |ϕ|δ in (20). Remark 1: There is a tradeoff in the estimates for the decay rate and the bound on the norm of the solutions. The decay rate can be chosen in the range (0, min{− ln | p|/δ, β}); if ρ is close to zero we obtain a lower γ factor at the cost of slow convergence rate. However, if ρ is close to its upper bound the γ factor increases. Remark 2: As we have observe above, exponential stability problems in system (1) can be formulated using LMI (23) with (24). Clearly, it only makes sense to cast this problem (23) with (24) if this inequality can be solved efficiently and in a reliable way, i.e., a key problem related to the study of LMI (23) with (24) is feasibility, which means that the test whether or not there exist solutions α, γ , q1 , q2 , q3 of (α, γ , q1 , q2 , q3 ) < 0 for a given β is called a feasibility problem. The LMI (23) with (24) is called infeasible if no solutions exist. Problem (23) with (24) is to determine whether the problem is feasible or not. It is called the feasibility problem. The solutions of the problem can be found by solving eigenvalue problem with respect to α, γ , q1 , q2 , q3 , which is a convex optimization [12]. Various efficient convex optimization algorithms can be used to check whether the inequality (23) with (24). We use MATLABs LMI control toolbox [13], which is significantly faster than classical convex optimization algorithms [12]. Therefore, all solutions α, γ , q1 , q2 , q3 can be obtained simultaneously. Similar to Lemma 3, for system (3) the following Theorem 2 is immediate. Theorem 2: Suppose that the nonlinear time delay system (3) is given. If there exist positive constants α, β, γ , q1 , and real numbers q2 , q3 such that the inequality ⎤ ⎡ −2(q2 − βq1 ) −(aq1 + q3 − q2 ) 0 bq1 ⎢−(aq1 + q3 − q2 ) 2q3 + α + γ 0 0 ⎥ ⎥< 0 ⎢ ⎣ 0 ⎦ 0 0 −αe−2βτ bq1
0
0
(35)
−γ e−2βσ
(33) holds, then for any initial condition (4) the solution x(t, ϕ) of system (3) satisfies the inequality α2 −βt |x(t)| ≤ e |ϕ|σ (34) q1
To compare our results with the known results, for convenience, we first give some existing results. In [6], the authors studied the convergence of system (1) when τ = σ and their result is Theorem 3 [6]: Assume that a > 0, b ∈ R and a 2 (1 − p2 ) > b2 . Then, for any solution of system (1) under τ = σ , we have lim x(t) = 0. t →∞
However, in [5], the authors derived the following result. Theorem 4 [5]: Let p < 1, p, a, and b are the positive constants. If pa 2 + b2 (36) 2(1 − pb) then every solution x(t) of system (1) satisfies lim x(t) = 0. t →∞ Park [7] defined the operator D˜ : C → R as t ˜ t ) = x(t) + px(t − τ ) + b tanh x(s)ds. (37) D(x pb < 1 and a > 1 +
t −σ
Hence, system (1) becomes d ˜ D(xt ) = −ax(t) + b tanh x(t), t > 0 (38) dt and the following theorem is derived. Theorem 5 [7]: For given σ > 0, every solution x(t) of system (1) satisfies lim x(t) = 0, if the operator D˜ is stable and there exist t →∞ the positive scalars ε1 , ε2 , ε3 , α and β such that the LMI holds √ √ ⎡ ⎤ b b b −ap −σ ab ⎢ ∗ −ε1 ⎥ 0 0 0 0 ⎢ ⎥ ⎢∗ ⎥ 0 0 0 ∗ −ε 2 ⎢ ⎥ 0 dt (41) then they derived the following result. Theorem 6 [8]: For given a > 0, τ > 0 and σ > 0, system (1) is asymptotically stable if the operator D˜˜ is stable and there exist a constant α with 0 < |α| < 1, and positive constants β, γ , θ and η such that the following linear matrix inequality holds ⎤ ⎡ 11 12 b α − a b(α − a) ⎢ ∗ −β − 2 pα pb −α −bα ⎥ ⎥ ⎢ 2−η ⎢ ∗ b b2 ⎥ ∗ θσ ⎥
1401
Exponential Convergence Estimates for a Single Neuron System of Neutral-Type Xiaofeng Liao, Senior Member, IEEE , Chuandong Li, and Tingwen Huang
Abstract— The future behavior of a dynamical system is determined by its initial state or initial function. Nontrivial neuron system involving adaptive learning corresponds to the memorization of initial information. In this paper, exponential estimates and sufficient conditions for the exponential stability of a single neuron system of neutral-type are studied. Of particular importance is the fact that exponential convergence guarantees that this system is capable of memorizing initial functions. Furthermore, this system is also capable of conveying much more information with respect to the initial functions memorized by neuron system with time delay. The proofs follow some new results on nonhomogeneous difference equations evolving in continuous-time combined with the Lyapunov–Krasovskii functional and the descriptor system approach. The exponential stability conditions are expressed in terms of a linear matrix inequality, which lead to less restrictive and less conservative exponential estimates.
Index Terms— Exponential estimates, linear matrix inequality (LMI), Lyapunov–Krasovskii functional, neuron system, neutral differential equations, nonhomogeneous difference equations. I. I NTRODUCTION Systems with neutral differential equations are discussed for various real world dynamical systems including genomic regulation [1], neurophysiology [2], [3], linguistics [4], and so on. They are generally treated as undesired cases. It is well known that a very important qualitative feature of dynamical systems is stability. However, as we will discuss in this brief, some particular neutral differential systems might be highly useful when the problem is capability to memorize dynamic information. Existences of delays between the interactions of variables are reported for various classes of dynamical systems [14], for example, delays due to translation and transcription in genetics, travel time of impulses from the postsynaptic connections to the presynaptic connections in neurophysiology [2], [3]. It should be pointed out that time delays not only occur in the system states (called as retarded type delays), but also appear in the derivatives of system states (called as neutral type delays). Those delays generally result in neutral differential equation systems that can find a variety of applications in practice such as chemical reactor, distributed networks containing lossless transmission lines, partial element equivalent circuits in VLSI systems, and Lotka–Volterra systems [14]–[16]. Those systems might be capable of memorizing initial functions rather than initial states. Manuscript received May 2, 2013; revised August 31, 2013; accepted November 7, 2013. Date of publication January 2, 2014; date of current version June 10, 2014. This work was supported in part by the National Natural Science Foundation of China under Grant 60973114 and Grant 61170249, in part by the Natural Science Foundation Project of CQCSTC under Grant 2009BA2024, in part by the State Key Laboratory of Power Transmission Equipment and System Security and New Technology, Chongqing University, under Grant 2007DA10512711206, in part by the Program for Changjiang Scholars, and in part by the Specialized Research Fund for priority areas for the Doctoral Program of Higher Education. X. Liao and C. Li are with the College of Electronic and Information Engineering, Southwest University, Chongqing 400715, China (e-mail: [email protected]; [email protected]). T. Huang is with Texas A & M University at Qatar, Doha 23874, Qatar (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TNNLS.2013.2290698
In this paper, we consider a single neuron system of neutral-type with the following form: d [x(t)+ px(t −τ )] = −ax(t)+b tanh[x(t −σ )], t ≥ 0 dt
(1)
where a, τ and σ are the positive constants, b and p are the real numbers with | p | < 1. Note that the inequality is indeed necessary for the stability of the discrete operator D(xt ) = x(t) + px(t − τ ). Furthermore, the stability property can be guaranteed for all values of τ by taking τ as a parameter. For each solution x(t) of system (1), the following initial condition is assumed: x(t) = ϕ(t), t ∈ [−δ, 0]
(2)
where δ = max{τ, σ }, ϕ ∈ ([−δ, 0], R). If p = 0, system (1) becomes the following form: d x(t) = −ax(t) + b tanh[x(t − σ )], dt
t ≥0
(3)
where the initial condition of system (3) is x(t) = ϕ(t), t ∈ [−σ, 0].
(4)
Delay differential equations of various types that contain (3) as a special case have been proposed by many authors for the study of the dynamical characteristics of Hopfield type neural networks [9], [11]. In [5], the asymptotic stability of system (1) with time-varying coefficients was investigated. Then, the dynamic behavior under the condition τ = σ was studied in [6]. Based on a linearization technique, the oscillatory and nonoscillatory behaviors of system (1) were investigated in detail. The sufficient conditions for asymptotic stability are derived using a Lyapunov-like approach [6]. Moreover, Park [7] studied system (1) with the requirement b > 0. Sufficient criteria for delay-dependent asymptotic stability were obtained by using a linear matrix inequality. His results are a generalization of existing ones [5], [6]. Recently, a new delay-dependent criterion for the asymptotic stability of system (1) was established using the neutral transformation and the linear matrix inequality (LMI) approach [8]. Compared with the results in [5]–[7], the new criterion is less conservative. Recently, there are some papers [20], [21] which derived delay-independent sufficient conditions for neutraltype neural networks. The above-mentioned results guarantee the asymptotic stability of neutral-type neuron system, but they do not provide an estimate of the convergence rate, neither the bound of the norm of the solution. For linear time-invariant systems, asymptotic stability is equivalent to exponential stability. Then, one may ask if there exists the possibility of using the LMI approach to derive the exponential estimates for the solutions of the neutral differential systems. However, to the best of our knowledge, these estimates as well as the exponential bounds for the case of neutral-type delay systems have not been addressed in the literature. In terms of rigorous mathematics, our concern is the following. Definition 1 [9], [11], [14]: System (1) is said to be exponentially stable if there exist β > 0 and γ ≥ 1 such that for every solution
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x(t, ϕ), where ϕ ∈ C([−δ, 0], R), the following exponential estimate holds: |x(t, ϕ)| ≤ γ e−βt |ϕ|δ , t ≥ 0.
(5)
The aim of this paper is to determine a lower bound for the decay rate β, and an upper bound for the γ -factor in terms of LMIs derived using the Lyapunov–Krasovskii approach. The rest of this paper is organized as follows. As the difference equation of the neutral system plays a key role in our work, some results concerning the exponential estimates for difference equations evolving in continuous-time domain are given in Section II. In Section III, a delay-dependent solution is obtained in terms of LMIs using a descriptor model transformation of the system. These results lead to less restrictive and less conservative exponential estimates on the solution of a single neutral-type neuron system with two delays. Some comparisons with the known results and two numerical examples are also presented in Section IV.
To use Lemma 1 for determining the exponential estimates of the solution of (9), we search for a solution z(t) in the form Le−ρt with ρ > 0 and L > 0, for inequality (7), i.e., z(t) must satisfy Le−ρt ≥ | p|L
sup
|x(t − θ)| + μe−βt
(6)
where λ is a positive real number, μ ≥ 0 and β > 0. Let z(t), t ≥ −δ be a continuous function such that z(t) ≥ λ
sup
min{τ,σ }≤θ≤δ
z(t − θ) + μe−βt , for t ≥ 0.
(7)
must hold, or equivalently 1 ≥ | p| eρδ +
−δ ≤θ ≤0
μ (ρ−β)t . e L
(13)
Let us define a positive constant α such that | p| = e−αδ . Inequality (13) takes the form μ (14) 1 ≥ e(ρ−α)δ + e(ρ−β)t L and we obtain that for any choice of ρ and L such that (15)
and correspondingly μ . (16) 1 − e(ρ−α)δ Then, condition (7) of Lemma 1 is satisfied. Now, we search for a z(t) such that conditions (15) and (16) hold and that condition (8) of Lemma 1 is also satisfied. We observe that as | p| < 1, μ ≥ 0 and ρ < α, the choice μ }e− min{− ln(| p|)/δ,β}t (17) z(t) = max{|ϕ|δ , 1 − e(ρ−α)δ is such that for −δ ≤ θ ≤ 0 L>
z(θ) ≥ |ϕ(θ)|δ ≥ |ϕ| ≥ |x(θ)|.
Then, if |x(θ)| ≤ z(θ), for
(12)
Le−ρt ≥ | p| Le−ρt eρδ + μe−βt
0 < ρ < min{α, β}
First, we will give a technical result of independent interest, which will be used later for deriving the exponential estimates for difference equations evolving in continuous-time domain. Lemma 1: Given a function x(t), t ≥ −δ, that satisfies the condition min{τ,σ }≤θ≤δ
e−ρ(t −θ) + μe−βt .
Obviously
II. E XPONENTIAL E STIMATES FOR D IFFERENCE E QUATION
|x(t)| ≤ λ
sup
min{τ,σ }≤θ≤δ
(8)
it follows that z(t) defines an upper bound for |x(t)| where t ≥ 0, i.e.,
Hence conditions (7) and (8) of Lemma 1 hold if z(t) satisfies (17). We are now able to work out the exponential estimates for the solution of the nonhomogeneous difference equation. Lemma 2: Consider system (9) with initial condition (11) and | p | < 1. Then, the solution x(t, ϕ) of (9) is such that the inequality
|x(t)| ≤ z(t), for t ≥ 0. Proof: For convenience, we assume τ ∈ [0, δ]. Then, for t ∈ [0, τ ], we have z(t) ≥ λ sup [z(t − θ)] + μe−βt .
|x(t)| ≤ K (ϕ)e−ρt
(18)
ρ < min{α, β}
(19)
holds, with
where α = − ln(| p|)/δ, and
τ ≤θ≤σ
K (ϕ) > max |ϕ|δ ,
As the time argument of z is negative, it follows from (8) that: z(t) ≥ λ sup [|x(t − θ)|] + μe−βt τ ≤θ≤σ
μ 1 − e(ρ−α)δ
.
(20)
Proof: Observe that the solution x(t) of system (9) is such that |x(t)| ≤ | p| |x(t − τ )| + | f (t)|
and (6) implies
≤ | p|
z(t) ≥ |x(t)|. Hence, we obtain the conclusion. We consider the nonhomogeneous difference equation of the form x(t) + px(t − τ ) = f (t)
(9)
| f (t)| ≤ μe−βt , μ ≥ 0, and β > 0.
(10)
where f (t) satisfies
For any continuous function ϕ ∈ C([−δ, 0], R), there exists a unique solution x(t, ϕ) of (9) and (10) satisfying the initial condition x(t, ϕ) = ϕ(θ), θ ∈ [−δ, 0].
(11)
sup
min{τ,σ }≤θ≤δ
|x(t − θ)| + μe−βt
(21)
hence, condition (6) of Lemma 1 holds. Also, observe that for z(t) = K (ϕ)e−ρt , t ≥ −δ with ρ and K (φ) defined in (19) and (20), respectively, it follows from the above argument [see (17)] that z(t) satisfies conditions (7) and (8) of Lemma 1. The result follows directly. In the homogenous case, the result reduces to the following: Corollary 1: Consider system (9) with initial condition (11) and | p| < 1. In the homogeneous case, i.e., μ = 0 and β is arbitrarily large, then the solution x(t, ϕ) is such that |x(t)| ≤ |ϕ|δ eξ t , for t ≥ 0 with ξ = ln | p| /δ.
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Proof: The result is obtained by setting μ = 0 and β arbitrarily large in Lemma 2. From the above analysis, we note that | p | < 1 guarantees exponential stability for the nonhomogeneous difference equation (9). Delay-independence with respect to τ also guarantees that small changes in τ do not destabilize the system [9]–[11]. III. E XPONENTIAL E STIMATES FOR A S INGLE N EURON S YSTEM OF N EUTRAL -T YPE In this section, we determine the exponential estimates for the solution of system (1), when the parameters are known. We would like to derive an exponential estimate in the form given by (5) for the solution of system (1). As we will observe below in the stability analysis of our model using Lyapunov–Krasovskii functional, the (Krasovskii) theorem cannot be always applied directly since the natural construction of a Lyapunov candidate leads to a degenerate function, which is not positive-definite in the corresponding stability theorem. In these cases, we need an appropriate stability theory and this has been given in [15]. Rewrite system (1) in the following equivalent descriptor system: y˙ (t) = −ax(t) + b tanh[x(t − σ )] (22) 0 = −y(t) + x(t) + px(t − τ ). Lemma 3: Suppose that the nonlinear time delay system (1) is given. If there exist positive constants α, β, γ , q1 , and real numbers q2 , q3 such that the inequality (α, γ , q1 , q2 , q3 ) < 0
From (28) and (29), we obtain the following inequality q1 |x(t) + px(t − τ )|2 ≤ V (t) ≤ α2 |xt |2δ
= −2q2 y 2 (t) − 2(aq1 + q3 − q2 )y(t)x(t) +2bq1 y(t) tanh[x(t − σ )] + 2q3 x 2 (t) +2 pq2 y(t)x(t − τ ) + 2 pq3 x(t)x(t − τ ). Similarly, one has V˙2 (t) = αx 2 (t)−αx 2 (t −τ )e−2βτ −2αβ
0 x 2 (t +θ)e2βθ dθ
−τ 2 2 +γ tanh [x(t)] − γ tanh [x(t − σ )]e−2βσ 0 −2βγ tanh2 [x(t + θ)]e2βθ dθ. −σ
Moreover, using the relation tanh2 [x(t)] ≤ x 2 (t), we obtain V˙ (t) ≤ −2(q2 − βq1 )y 2 (t) − 2(aq1 + q3 − q2 )y(t)x(t) + 2 pq2 y(t)x(t − τ ) + 2bq1 y(t) tanh[x(t − σ )] + (2q3 + α + γ )x 2 (t) + 2 pq3 x(t)x(t − τ )
(23)
− αx 2 (t − τ )e−2βτ − γ tanh2 [x(t − σ )]
× e−2βσ − 2βq1 y 2 (t) 0 0 − 2βγ tanh2 [x(t +θ)]e2βθ dθ −2αβ x 2 (t +θ)e2βθ dθ
(α, γ , q1 , q2 , q3 ) ⎡ ⎤ −2(q2 −βq1 ) −(aq1 +q3 −q2 ) pq2 bq1 ⎢−(aq1 +q3 − q2 ) 2q3 +α+γ pq3 0 ⎥ ⎥ =⎢ ⎣ pq3 −αe−2βτ 0 ⎦ pq2 0 0 −γ e−2βσ bq1
−σ
then, for any initial condition (2) the solution x(t, ϕ) of system (1) satisfies the inequality α2 −βt e |ϕ|δ (25) |x(t) + px(t − τ )| ≤ q1 where the positive constant α2 is defined as
Consider
the
(26)
following
Lyapunov–Krasovskii
V (t) = V1 (t) + V2 (t)
(27)
−τ
¯ = x¯ T (t)(α, γ , q1 , q2 , q3 )x(t) 0 tanh2 [x(t + θ)]e2βθ dθ −2βγ
(24)
α2 = q1 + τ α + σ γ .
−σ 0
x 2 (t + θ)e2βθ dθ − 2βq1 y 2 (t).
−2βα −τ
Clearly, we have d ¯ V (xt ) + 2βV (xt ) = x¯ T (t)(α1 , β, q1 , q2 )x(t) dt where x(t) ¯ = [y(t), x(t), x(t − τ ), tanh[x(t − τ )], tanh[x(t − σ )]]T and
where
V1 (t) = [y(t), x(t)] V2 (t) = α
0 −τ 0
1 0 0 0
q1 0 q2 q3
(28)
x 2 (t + θ)e2βθ dθ tanh2 [x(t + θ)]e2βθ dθ
+γ
y(t) x(t)
(30)
where α2 is given by (26). The time derivative of functional (28) along the trajectories of system (22) is
q q −ax(t) + b tanh[x(t − σ )] V˙1 = 2 [y(t), x(t)] 1 2 0 q3 −y(t) + x(t) + px(t − τ )
−ax(t)+b tanh[x(t −σ )] = [2q1 y(t), 2q2 y(t)+2q3 x(t)] −y(t)+x(t)+ px(t −τ )
holds, where
Proof: functional
1403
(29)
−σ
with α, β, γ , q1 are the positive constants and q2 , q3 are real numbers of Lemma 3. This functional is degenerated as it is usual for descriptor systems [10], [12].
(α, γ , q1 , q2 , q3 ) ⎡ ⎤ −2(q2 −βq1 ) −(aq1 +q3 −q2 ) pq2 bq1 ⎢−(aq1 + q3 − q2 ) 2q3 + α+γ pq3 0 ⎥ ⎥< 0. =⎢ ⎣ pq3 −αe−2βτ 0 ⎦ pq2 0 0 −γ e−2βσ bq1 Condition (23) implies that d V (xt ) + 2βV (xt ) ≤ 0. dt This inequality leads to the following: V (xt (ϕ)) ≤ e−2βt V (ϕ), t ≥ 0.
1404
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Combining (30) and the previous inequality, we obtain, for t ≥ 0 the estimation
where the positive constant α2 is defined as α2 = q1 + σ γ .
q1 [x(t) + px(t − τ )]2 ≤ V (xt (ϕ))
IV. C OMPARISONS W ITH THE K NOWN R ESULTS
≤ e−2βt V (ϕ) ≤ α2 e−2βt |ϕ|2δ
and we arrive at the conclusion α2 −βt |x(t) + px(t − τ )| ≤ e |ϕ|δ , t ≥ 0 q1 where α2 is given by (26). Theorem 1: Consider system (1) and | p| < 1. If there exist positive constants α, β, γ , q1 and real numbers q2 , q3 such that the inequality (23) is satisfied, where (α, γ , q1 , q2 , q3 ) is given by (24), then for any initial condition (2) the solution x(t, ϕ) of system (1) satisfies inequality (5) with 0 < ρ < min{α, β} and γ >
1 1 − e(ρ−α)δ
(31)
α2 q1
(32)
where α = − ln | p|/δ and the constant α2 is given by (26). Proof: The result follows from Lemmas 2 and 3 by substituting √ μ = α2 /q1 |ϕ|δ in (20). Remark 1: There is a tradeoff in the estimates for the decay rate and the bound on the norm of the solutions. The decay rate can be chosen in the range (0, min{− ln | p|/δ, β}); if ρ is close to zero we obtain a lower γ factor at the cost of slow convergence rate. However, if ρ is close to its upper bound the γ factor increases. Remark 2: As we have observe above, exponential stability problems in system (1) can be formulated using LMI (23) with (24). Clearly, it only makes sense to cast this problem (23) with (24) if this inequality can be solved efficiently and in a reliable way, i.e., a key problem related to the study of LMI (23) with (24) is feasibility, which means that the test whether or not there exist solutions α, γ , q1 , q2 , q3 of (α, γ , q1 , q2 , q3 ) < 0 for a given β is called a feasibility problem. The LMI (23) with (24) is called infeasible if no solutions exist. Problem (23) with (24) is to determine whether the problem is feasible or not. It is called the feasibility problem. The solutions of the problem can be found by solving eigenvalue problem with respect to α, γ , q1 , q2 , q3 , which is a convex optimization [12]. Various efficient convex optimization algorithms can be used to check whether the inequality (23) with (24). We use MATLABs LMI control toolbox [13], which is significantly faster than classical convex optimization algorithms [12]. Therefore, all solutions α, γ , q1 , q2 , q3 can be obtained simultaneously. Similar to Lemma 3, for system (3) the following Theorem 2 is immediate. Theorem 2: Suppose that the nonlinear time delay system (3) is given. If there exist positive constants α, β, γ , q1 , and real numbers q2 , q3 such that the inequality ⎤ ⎡ −2(q2 − βq1 ) −(aq1 + q3 − q2 ) 0 bq1 ⎢−(aq1 + q3 − q2 ) 2q3 + α + γ 0 0 ⎥ ⎥< 0 ⎢ ⎣ 0 ⎦ 0 0 −αe−2βτ bq1
0
0
(35)
−γ e−2βσ
(33) holds, then for any initial condition (4) the solution x(t, ϕ) of system (3) satisfies the inequality α2 −βt |x(t)| ≤ e |ϕ|σ (34) q1
To compare our results with the known results, for convenience, we first give some existing results. In [6], the authors studied the convergence of system (1) when τ = σ and their result is Theorem 3 [6]: Assume that a > 0, b ∈ R and a 2 (1 − p2 ) > b2 . Then, for any solution of system (1) under τ = σ , we have lim x(t) = 0. t →∞
However, in [5], the authors derived the following result. Theorem 4 [5]: Let p < 1, p, a, and b are the positive constants. If pa 2 + b2 (36) 2(1 − pb) then every solution x(t) of system (1) satisfies lim x(t) = 0. t →∞ Park [7] defined the operator D˜ : C → R as t ˜ t ) = x(t) + px(t − τ ) + b tanh x(s)ds. (37) D(x pb < 1 and a > 1 +
t −σ
Hence, system (1) becomes d ˜ D(xt ) = −ax(t) + b tanh x(t), t > 0 (38) dt and the following theorem is derived. Theorem 5 [7]: For given σ > 0, every solution x(t) of system (1) satisfies lim x(t) = 0, if the operator D˜ is stable and there exist t →∞ the positive scalars ε1 , ε2 , ε3 , α and β such that the LMI holds √ √ ⎡ ⎤ b b b −ap −σ ab ⎢ ∗ −ε1 ⎥ 0 0 0 0 ⎢ ⎥ ⎢∗ ⎥ 0 0 0 ∗ −ε 2 ⎢ ⎥ 0 dt (41) then they derived the following result. Theorem 6 [8]: For given a > 0, τ > 0 and σ > 0, system (1) is asymptotically stable if the operator D˜˜ is stable and there exist a constant α with 0 < |α| < 1, and positive constants β, γ , θ and η such that the following linear matrix inequality holds ⎤ ⎡ 11 12 b α − a b(α − a) ⎢ ∗ −β − 2 pα pb −α −bα ⎥ ⎥ ⎢ 2−η ⎢ ∗ b b2 ⎥ ∗ θσ ⎥
