# Stabilization to Exponential Input-to-State Stability of a Class of Neural Networks with Delay by Observer-Based Aperiodic Intermittent Control

Authors: Mengyue Li; Biwen Li; Yuan Wan
Journal: Discrete Dynamics in Nature and Society
 (2021)

---

## Abstract

This study is devoted to investigating the stabilization to exponential input-to-state stability (ISS) of a class of neural networks with time delay and external disturbances under the observer-based aperiodic intermittent control (APIC). Compared with the general neural networks, the state of the neural network investigated is not yet fully available. Correspondingly, an observer-based APIC is constructed, and moreover, neither the observer nor the controller requires the information of time delay. Then, the stabilization to exponential ISS of the neural network is realized severally by the observer-based time-triggered APIC (T-APIC) and the observer-based event-triggered APIC (E-APIC), and corresponding criteria are given. Furthermore, the minimum activation time rate (MATR) of the observer-based T-APIC and the observer-based E-APIC is estimated, respectively. Finally, a numerical example is given, which not only verifies the effectiveness of our results but also shows that the observer-based E-APIC is superior to the observer-based T-APIC and the observer-based periodic intermittent control (PIC) in control times and the minimum activation time rate, and the function of the observer-based T-APIC is also better than the observer-based PIC.


---

## Body

## 1. Introduction

Neural network, a mathematical model for information processing, can better simulate the working mechanism of the brain and plays a crucial role in artificial intelligence. The research on the dynamic behaviors of the neural network is the premise of the successful application of the neural network in many fields, such as associative memory, optimization problems, image recognition, different learning tasks, and so on [1–7]. In recent years, the research on the dynamic behaviors of the neural network mainly includes synchronization, stability, periodicity, and other dynamic behaviors, and some new and excellent results are emerging continuously [2, 8–11].As an important dynamic behavior of the neural network, stability has always been concerned by scholars. At present, the research on stability mainly includes asymptotic stability, exponential stability, Lagrange stability, and other stabilities [12–18]. Exponential stability has attracted much attention due to its advantages of fast convergence speed and fast achieving the expected goal in practical application. By using a new multiple Lyapunov function and linear matrix inequalities, Lian and Zhang [14] discussed the exponential stability of a class of uncertain switched Cohen–Grossberg neural networks with interval time-varying, distributed delay, and average dwell time. In [16], a class of hybrid neural networks with inertia term and mixed time-varying delays has been noticed. Then, by establishing appropriate Lyapunov functionals and inequality technique, some sufficient conditions have been given to implement exponential stability of the system.For one thing, as a result of the limitations of existing technical means, the switching speed of hardware devices is limited, which leads to the time delay phenomenon ubiquitous and inevitable in all kinds of engineering application. The existence of time delay is often one of the important factors affecting the stability of dynamic systems. For another, the system will inevitably be affected by external disturbances in practical application. For example, the communication signals will be interfered by other signals in the process of transmission, the machine will be affected by vibration in the process of operation, etc. Therefore, it is more practical to study the stability of neural networks with time delay and external disturbances. Halanay inequality and the generalization of Halanay-type inequality [8] are commonly used in the study of time delay systems. Input-to-state stability has attracted much attention because it can describe the impact of external disturbances on the dynamic system. In other words, input-to-state stability means that the system is probably asymptotically stable without disturbances, and when the input of the system is bounded, the response of the system is also bounded. In [19], the input-to-state stability, a class of delayed dynamical systems with bounded disturbances, was concerned. The input-to-state stability analysis of a class of impulsive stochastic Cohen–Grossberg neural networks with mixed delays was discussed in [20]. Based on stochastic analysis theory, impulse theory, Itô formula, Dynkin formula, and so on, the mean square exponential input-to-state stability of systems with/without impulsive control was obtained.In order to ensure the stable operation of the system, it is frequently necessary to exert external control on the system. It is a remarkable fact that the control in some practical applications is frequently discontinuous, such as vehicle control, the management of wind power generation, refrigeration system of a refrigerator, and the orbit adjustment of spacecraft. In some practical applications, as mentioned above, discontinuous control is not only less costly but also more practical than continuous control. Intermittent control and impulsive control belong to discontinuous control. Intermittent control and impulsive control are similar but different. Intermittent control is executed in some time intervals, while impulsive control is activated at instants. At present, numerous researchers have studied the impulsive control based on the stability of the impulsive system and achieved good results [21–25]. Yet, in some practical applications, it may be difficult to complete a control task in a flash, for example, vehicle acceleration and deceleration control, where the control is intermittent rather than instantaneous. If not, the movement of the vehicle will bump and shock. In this situation, intermittent control is obviously better than impulsive control. As far as we know, compared with impulsive control, the research results of intermittent control are relatively few.In this study, we will consider studying the ISS of the neural network by intermittent control. In recent years, periodic intermittent control has been gradually used to realize some common dynamic behaviors of the neural network [26]. However, in some practical applications, demand of periodicity is unnecessary and unreasonable, such as the wind power generation mentioned above. The aperiodic intermittent control has the characteristics that the duration of each control and the time interval of two adjacent controls are not fixed, so its flexibility is better and practicability is stronger. Apparently, in order to eliminate the limitation of periodicity, aperiodic intermittent control is a better choice. What is noteworthy is that multitudinous scholars focus on the synchronization problem of the system under aperiodic intermittent control [27–29], while the research on the stability is relatively less. Using the condition of converse average dwell time, Liu et al. [30] designed a time-triggered aperiodic intermittent controller to realize exponential input-to-state stability of the continuous-time system. In addition, to reduce the conservatism existing in the time-triggered aperiodic intermittent controller, Liu et al. proposed the event-triggered aperiodic intermittent control. It is worth noting that the state of the system discussed in [30] is measurable, and the controllers use the real state of the system to control. However, due to some physical limitations or measurement technology limitations, the state of the system is often not completely obtained, which will lead to the failure of the feedback control based on the state of the system. Inspired by [30], it is very interesting to study the exponential ISS of the neural network by using the aperiodic intermittent control when the system state is not measurable. Building a state observer is one of the common methods to solve the problem of unmeasurable system state, and many excellent and interesting results have been obtained in many fields [31–33]. Furthermore, as far as we know, almost no scholars use aperiodic intermittent control to achieve the exponential ISS of neural networks with time delay and external disturbances when the system state is unmeasurable.According to the above elaboration, this study mainly focuses on exponential input-to-state stability of a class of neural networks with time delay and external disturbances under observer-based aperiodic intermittent control. The main work of this study is summarized as follows:(1)
A state observer independent of time delay information is constructed, which is more practical because time delay information is often not and is not easy to obtain completely in some practical applications(2)
Combining the time-triggered mechanism with the aperiodic intermittent control based on the state observer, an observer-based time-triggered controller is put forward. Then, the activation time and control duration of the controller are designed by using the infimum of the single activation time rate of the controller and the converse average dwell time method to guarantee the exponential ISS of the neural network. Furthermore, the minimum activation time rate of the controller is estimated.(3)
In order to reduce the conservatism existing in the observer-based time-triggered aperiodic intermittent control, an observer-based event-triggered control mechanism is proposed. Meanwhile, the criterion to ensure the exponential input-to-state stability of the studied network is given, and the minimum activation time rate of the controller is evaluated.The arrangement of the rest of this study is as follows. The second part is the symbol description and the construction of neural networks. The third and fourth parts are the derivation of the results obtained in this study. In the fifth part, a numerical example is given to verify the effectiveness of the results. The conclusion and prospect are given in the sixth part.


## 2. Preliminaries

Letℝ and ℝn denote the set of real numbers and the n-dimensional real vector space, respectively. I represents the identity matrix, ℝ+=0,∞, and ℕ=0,1,…,n. λmaxQ stands for the maximum eigenvalue of the symmetric matrix Q. For a matrix A=aijn×m, its norm is defined as A=λmaxATA1/2. Denoted by C−τ,0,ℝn, the Banach space composed of all continuous vector-valued functions φ:−τ,0⟶ℝn.The symbolL∞,loc represents the set of Lebesgue measurable functions, which are composed of locally essentially bounded functions ι:ℝ+⟶ℝn. For ι∈L∞,loc, set ιs1,s2=esssups1≤t≤s2ιt and ι∞=esssupt∈ℝ+ιt.In this study, we discuss the following neural network model with time delay and external disturbances.(1)x˙t=−Cxt+Afxt+Bgxt−τ+u1t+ωt,t≥t0,xt0+θ=φθ∈C−τ,0,ℝn,where xt=x1t,x2t,…,xntT∈ℝn is the state vector of the neural network; xt0+θ=φθ, θ∈−τ,0 is the initial value condition of (1); fxt=f1x1t,f2x2t,…,fnxntT and gxt−τ=g1x1t−τ,g2x2t−τ,…,gnxnt−τT are the activation functions at time t and delay time t−τ; C=diagc1,c2,…,cn is the real diagonal matrix, where ci>0,i=1,2,…,n; A,B are the weight matrices with corresponding dimensions; the control input of the neural network is represented by u1t; ωt∈L∞,loc is the external disturbances; τ>0 is a transmission delay.Suppose that (1) with u=0 is not ISS. The purpose of this study is to design the APIC to ensure that (1) is exponential ISS. Furthermore, considering that, in a great deal of engineering applications, due to some physical limitations or other reasons, the system state may be extremely difficult or almost impossible to obtain completely, which leads to the malfunction of feedback control based on the real state of the system. Therefore, we construct a state observer (2):(2)x˜˙t=−Cx˜t+Afx˜t+Bgx˜t+u2t,t≥t0,x˜t0=ψ∈ℝn,where x˜t∈ℝn is the estimation of state xt; x˜t0=ψ∈ℝn is the initial value condition of (2).Remark 1.
Because the state value is unknown, it is very likely that the time delay cannot be known, so it is uncertain whether to adopt time-varying delay, distributed constant delay, or other time-delay in the observer system. Thus, the time delay information is not used to design an observer to estimate the state of (1). According to this, we consider the above state observer. Moreover, when the time delay is measurable, it can be similar to the following discussion about the unmeasurable time delay. Therefore, the study only considers the case where the delay information is unmeasurable.
Letti denote the starting instant of the controller and ϑi represent the control duration of the controller. Then, u2t and u1t are designed as(3)u2t=K2x˜t,ti≤t<ti+ϑi,0,ti+ϑi≤t<ti+1,(4)u1t=K1x˜t,ti≤t<ti+ϑi,0,ti+ϑi≤t<ti+1,where K2 and K1 are the control gain matrices.Remark 2.
Considering the immeasurability of system state, the real state of the system cannot be used as feedback control; hence, we consider to regulate and control observer (2) and real system (1) by the observation value. Furthermore, the control input u1t of (1) is APIC. Hence, in order to construct a proper error system, the control input u2t of observer (2) is also designed to be APIC.Remark 3.
Compared with the periodic intermittent control (PIC) proposed in [26], where(5)ut=Kxt,iT≤t<iT+ϑ,0,iT+ϑ≤t<i+1T,the time span of each control in APIC is more flexible. In the above formula, T>0 stands for the period of control and 0<ϑ<T denotes activation time of ut.
Combining (2) and (3), observer (2) can be rewritten as(6)x˜˙t=−C−K2x˜t+Afx˜t+Bgx˜t,ti≤t<ti+ϑi,x˜˙t=−Cx˜t+Afx˜t+Bgx˜t,ti+ϑi≤t<ti+1,i∈ℕ,x˜t0=ψ∈ℝn.
Similarly, combining (1) and (4), system (1) becomes the following form:(7)x˙t=−Cxt+K1x˜t+Afxt+Bgxt−τ+ωt,ti≤t<ti+ϑi,x˙t=−Cxt+Afxt+Bgxt−τ+ωt,ti+ϑi≤t<ti+1,i∈ℕ,xt0+θ=φθ∈C−τ,0,ℝn.
Letet=xt−x˜t denote the error between neural network (1) and observer (2); then, the dynamic of the error system can be described as(8)e˙t=−Cet+Afxt−fx˜t+K1−K2x˜t+Bgxt−τ−gx˜t+ωt,ti≤t<ti+ϑi,e˙t=−Cet+Afxt−fx˜t+Bgxt−τ−gx˜t+ωt,ti+ϑi≤t<ti+1,et0+θ=ϕ˜θ=φθ−ψ∈C−τ,0,ℝn,where i∈ℕ.
It is noteworthy that the real system (1) can be stable when both the observation system (2) and error system (8) are stable. Hence, we consider realizing the stability of (1) by realizing the stability of the augmented system composed of (2) and (8).
Letzt=x˜Tt,eTtT and zt0+θ=Φθ=ψT,ϕ˜TθT∈C−τ,0,ℝ2n. Then, combining (6) and (8), the following argument system can be obtained:(9)z˙t=C1zt+A1Ft+B1Gt−τ+Wt,ti≤t<ti+ϑi,z˙t=C2zt+A1Ft+B1Gt−τ+Wt,ti+ϑi≤t<ti+1,i∈ℕ,zt0+θ=Φθ∈C−τ,0,ℝ2n,where(10)Wt=0,ωTtT,Ft=fTx˜t,fTxt−fTx˜tT,Gt=gTx˜t,gTxt−τ−gTx˜tT,A1=A00A,B1=B00B,C1=−C+K20K1−K2−C,C2=−C00−C.
Assume thatzt=zt,t0,Φ is the solution of (9) with the initial condition zt0+θ=Φθ∈C−τ,0,ℝ2n.Definition 1.
[30]. System (1) is exponential input-to-state stable via observer-based APIC (4) if system (9) is exponentially ISS with respect to ωt, i.e., there exist M>0,r>0, and function Ψ∈K∞, such that the solution zt of (9) satisfies(11)zt≤Mzt0e−rt−t0+Ψωt0,t,t≥t0.
Now, we give the following assumptions and lemmas which are extremely necessary for our follow-up derivation.Assumption 1.
Functionsf⋅, g⋅∈ℝn with f0=0, and g0=0 are Lipschitz continuous, that is, ∀xi,yi∈ℝ; there exist l1i>0,l˜2i>0 satisfy(12)fixi−fiyi≤l1ixi−yi,i=1,2,…,n,gixi−giyi≤l˜2ixi−yi,i=1,2,…,n.
According to Assumption1, let l1=maxl11,l12,…,l1n and l2=maxl˜21,l˜22,…,l˜2n.Assumption 2.
Suppose there are matricesK1, K2 and positive constants g1, g2 that satisfy the following conditions:(13)C1+C1T+A1A1T+B1B1T+l12+3l22+g1I≤0,(14)2C2+A1A1T+B1B1T+l12+3l22−g2I≤0.Lemma 1.
(see [34]). For any X,Y∈ℝn, we have(15)XTY+YTX≤εXTX+ε−1YTY,where ε is any positive constant.Lemma 2.
LetV:t0−τ,+∞⟶0,+∞ and E:t0,+∞⟶0,+∞. If the derivative of V with respect to t satisfies the following inequality:(16)V˙t≤αVt+βVt−τ+Et,t≥t0,where α∈ℝ and β>0, τ>0, then, for any t≥t0, the following result can be obtained:(17)Vt≤Vt0+∫t0te−rs−t0Esdsert−t0,when −r+α+βmaxe−rτ,1≤0.Proof.
Let(18)V¯t=Vt,t∈t0−τ,t0,Vte−rt−t0,t≥t0.
From (16) and (18), it follows that(19)V¯˙t=V˙te−rt−t0−rVte−rt−t0≤α−rVte−rt−t0+βVt−τe−rt−t0+e−rt−t0Et,t≥t0.
(19) is discussed as follows:(a)
Ift−τ∈t0−τ,t0, then Vt−τ=V¯t−τ, t−t0∈0,τ, and we have(20)V¯˙t≤α−rV¯t+βV¯t−τmaxe−rτ,1+e−rt−t0Et,t≥t0.(b)
Ift−τ∈t0,+∞, then(21)V¯˙t≤α−rV¯t+βVt−τe−rt−τ−t0e−rτ+e−r1t−t0Et≤α−rV¯t+βV¯t−τmaxe−rτ,1+e−rt−t0Et,t≥t0.
Hence, for anyt≥t0, we get(22)V¯˙t≤α−rV¯t+βmaxe−rτ,1V¯t−τ+e−rt−t0Et.
Let(23)Ut=V¯t,t∈t0−τ,t0,V¯t−∫t0te−rs−t0Esds,t≥t0,then, for any t≥t0,(24)U˙t≤α−rV¯t+βmaxe−rτ,1V¯t−τ≤α−r+βmaxe−rτ,1supV¯t,V¯t−τ≤0,since −r+α+βmaxe−rτ,1≤0. Therefore, it follows that(25)Ut≤Ut0=V¯t0,for any t≥t0. Then, combining (23), for any t≥t0, we have(26)V¯t−∫t0te−rs−t0Esds≤V¯t0=Vt0,and so(27)V¯t≤Vt0+∫t0te−rs−t0Esds,which means (17) holds. The deducing procedure is finished.Remark 4.
Lemma2 obtained the variation of Vt in the interval t0,+∞ on the premise that the derivative of Vt satisfies condition (16). For the variation of Vt in the interval t0−τ,t0, it is clear that Vt≤supζ∈−τ,0Vt0+ζ. Moreover, the value of Vt in the interval t0−τ,t0 can be regarded as the initial value of differential equation (16). Because the initial value is arbitrary and is predetermined, it is not necessary to know the variation of Vt exactly when t∈t0−τ,t0. Instead, we should pay attention to the variation of Vt when t≥t0, and the function Vt satisfies a certain differential equation.
For the stability of system (9) under the observer-based APIC K1,K2,ti,ϑi, let K1 and K2 satisfy (13) and (14) first. The next step is how to construct ti and ϑi.
To begin with, ensure that the observer-based APICK1,K2,ti,ϑi is designed with the following characteristics:(1)
No Zeno behavior(2)
The controller works intermittently
ti should satify the following condition:(28)ti<ti+1,∀i∈ℕ;limi⟶∞ti=∞.
The duration of controlϑi should satify(29)0<ϑi<Δi≜ti+1−ti,∀i∈ℕ,and let ϑmin=mini∈ℕϑi>0,ϑmax=maxi∈ℕϑi<∞.
Therefore, the problem of stabilization to ISS by observer-based APIC formulated the issue of minimum activation time rate (MATR):(30)Rmin=minti,ϑilimi⟶∞inf∑j=0iϑj∑j=0iΔj,where ti satisfies (28) and ϑi satisfies (29). Additionally, for a controller, if the duration of each control is ϑi, then the activation time rate of the controller during t,s is defined as Rt=∑i=0N−1ϑi/s−t×100%, where the notation N signifies the number of times the controller is activated during the period.


## 3. Stabilization to Exponential ISS via Observer-Based T-APIC

In this section, we devote ourselves to achieving exponential ISS of system (1) via the observer-based T-APIC and consider the following two cases to design the observer-based T-APIC.Case 1.
ti is given to satisfy (28). Here, we design the observer-based time-triggered aperiodic intermittent control by using infimum of single activation time rate γinf≜infi∈ℕϑi/Δi and the converse average dwell time, respectively, and give the criterion of exponential input-to-state stability of neural network (1) under the control. In addition, we estimate the minimum activation time rate Rmin of the control.Theorem 1.
Suppose thatti is given to satisfy (28) and Assumptions 1 and 2 hold. Moreover, there exist r˜1<0 and r2>0, such that(31)−r˜1−g1−ε1+4l22e−r˜1τ≤0,(32)−r2+g2+ε2+4l22≤0,where ε1 and ε2 are the appropriate constants satisfying 0<ε1<g1 and 0<ε2<g2. Then, system (1) can achieve stabilization to exponential ISS under observer-based T-APIC (4), if ϑi satisfies (29) and the following condition:(33)Ru∗=r2r1+r2<γinf=infi∈ℕϑiΔi<1,where r1=r˜1. In addition, we have(34)Rmin>Ru∗.Proof.
LetVzt=zTtzt.
For anyti≤t<ti+ϑi, computing the derivative of Vzt with respect to the time t, we get that(35)V˙zt=2zTtz˙t=2zTtC1zt+2zTtA1Ft+2zTtB1Gt−τ+2zTtWt.
According to Assumption1 and Lemma 1, it can deduce that(36)2zTtA1Ft=A1TztTFt+FTtA1Tzt≤zTtA1A1Tzt+FTtFt≤zTtA1A1Tzt+l12zTtzt.
Analogously, we have that(37)2ztTB1Gt−τ=B1TztTGt−τ+GTt−τB1Tzt≤zTtB1B1Tzt+GTt−τGt−τ≤zTtB1B1Tzt+l22x˜t2+∑i=1nl2i2xit−τ−x˜it2≤zTtB1B1Tzt+l22x˜t2+∑i=1n2l2i2xit−τ2+x˜it2≤zTtB1B1Tzt+3l22x˜t2+∑i=1n2l2i2xit−τ2≤zTtB1B1Tzt+3l22x˜t2+∑i=1n4l2i2eit−τ2+x˜it−τ2≤zTtB1B1T+3l22Izt+4l22zTt−τzt−τ,(38)2ztTWt=zTtWt+WTtzt≤ε1zTtzt+ε1−1WtTWt.
Applying Assumption2 and combining with (35)–(38), we get that(39)V˙t≤zTtC1+C1T+A1A1T+B1B1T+l12+3l22+g1Izt−g1−ε1zTtzt+4l22zTt−τzt−τ+ε1−1WTtWt≤−g1−ε1Vzt+4l22Vzt−τ+ε1−1ωti,t2.
Letht=ε1−1ωti,t2. From Lemma 2, for any ti≤t<ti+ϑi, the following inequality(40)Vzt≤Vzti+∫tite−r˜1s−tihsdser˜1t−ti=Vzti+∫titer1s−tihsdse−r1t−ti,holds. On account of the continuity of Vzt, one has(41)Vzti+ϑi≤Vztie−r1ϑi+e−r1ϑi∫titi+ϑier1s−tihsds≤Vztie−r1ϑi+ε1−1r1−1ωti,ti+ϑi2.
Similarly, for anyti+ϑi≤t<ti+1, we have(42)V˙zt=2zTtz˙t=2ztTC2zt+2ztTA1Ft+2ztTB1Gt−τ+2ztTWt≤zTt2C2+A1A1T+B1B1T+l12+3l22−g2Izt+g2+ε2zTtzt+4l22zTt−τzt−τ+ε2−1WTtWt≤g2+ε2Vzt+4l22Vzt−τ+ε2−1ωti+ϑi,t2.
Leth˜t=ε2−1ωti+ϑi,t2. By Lemma 2, for any ti+ϑi≤t<ti+1, we get(43)Vzt≤Vzti+ϑi+∫ti+ϑite−r2s−ti−ϑih˜sds×er2t−ti−ϑi.
SinceVzt is continuous, substituting (41) into (43), one has(44)Vzti+1≤Vzti+ϑier2ti+1−ti−ϑi+eti+1−ti−ϑi∫ti+ϑiti+1e−r2s−ti−ϑih˜sds≤Vzti+ϑier2ti+1−ti−ϑi+ε2−1r2−1er2ti+1−ti−ϑiωti+ϑi,ti+12≤Vztie−r1+r2ϑi+r2ti+1−ti+ε1−1r1−1+ε2−1r2−1er1ϑi−r1+r2ϑi+r2ti+1−tiωti,ti+12.
Letη=ε1−1r1−1+ε2−1r2−1,η1=ε1−1r1−1,andρijt≜−r1+r2∑k=ji−1ϑk+r2t−tj.
By using (44), we get(45)Vzti≤Vzti−1e−r1+r2ϑi−1+r2ti−ti−1+ηer1ϑi−1−r1+r2ϑi−1+r2ti−ti−1ωti−1,ti2≤Vzti−2e−r1+r2ϑi−2+ϑi−1+r2ti−ti−2+ηer1ϑi−2−r1+r2ϑi−2+ϑi−1+r2ti−ti−2+er1ϑi−1−r1+r2ϑi−1+r2ti−ti−1ωti−2,ti2≤Vzti−3e−r1+r2ϑi−3+ϑi−2+ϑi−1+r2ti−ti−3+ηer1ϑi−3−r1+r2ϑi−3+ϑi−2+ϑi−1+r2ti−ti−3+er1ϑi−2−r1+r2ϑi−2+ϑi−1+r2ti−ti−2+er1ϑi−1−r1+r2ϑi−1+r2ti−ti−1ωti−3,ti2⋯≤Vzt0eρi,0ti+η∑j=0i−1er1ϑj+ρi,jtiωt0,ti2.
Similar to the reasoning process of (44) and combined with (43) and (45), for any ti+ϑi≤t<ti+1, the following inequality(46)Vzt≤Vzt0eρi+1,0t+η∑j=0ier1ϑj+ρi+1,jtωt0,t2,holds.
Combining (40) and (45), for any ti≤t<ti+ϑi, we get(47)Vzt≤Vztie−r1t−ti+ε1−1r1−1ωti,t2≤Vzti+ε1−1r1−1ωti,t2≤Vzt0eρi,0t+η1+η∑j=0i−1er1ϑj+ρi,jtωt0,t2.
From (46) and (47), one can see that the stabilization relies largely on ρijt. Hence, it is utterly momentous to evaluate ρijt.
By (33), there exists λ∈0,r1, such that(48)γinf=infi∈ℕϑiΔi=λ+r2r1+r2,thus,(49)λ+r2r1+r2Δi≤ϑi,i∈ℕ.
By using (49) and the expression of ρi,jt, for any ti≤t<ti+ϑi and i>j, we get(50)ρijt≤−λ+r2∑k=ji−1Δk+r2t−tj=−λ+r2ti−tj+r2t−tj≤−λt−tj+λ+r2ϑi.
For anyti+ϑi≤t<ti+1, i≥j. Based on the definition of ρijt and the derivation of (50), we have(51)ρi+1,jt≤ρi+1,jti+1≤−λti+1−tj.
Combining (47) and (50), for any ti≤t<ti+ϑi, we obtain that(52)Vzt≤Vzt0e−λt−t0+λ+r2ϑi+η1+η∑j=0i−1er1ϑj+r2+λϑi−λt−tj×ωt0,t2≤Vzt0e−λt−t0+λ+r2ϑmax+η1+η∑j=1ier1+r2+λϑmax−jλϑmin×ωt0,t2≤Vzt0e−λt−t0+λ+r2ϑmax+η1+er1+r2+λϑmax1−e−λϑminωt0,t2.
Similarly, combining (46) and (51), for any ti+ϑi≤t<ti+1, one has(53)Vzt≤Vzt0e−λti+1−t0+η∑j=0ier1ϑj−λti+1−tjωt0,t2≤Vzt0e−λt−t0+η∑j=1i+1er1ϑmax−jλϑminωt0,t2≤Vzt0e−λt−t0+ηer1ϑmax1−e−λϑminωt0,t2.
From (52) and (53), for any t≥t0, we have(54)Vzt≤ξ1Vzt0e−λt−t0+ξ2ωt0,t2,where ξ1=eλ+r2ϑmaxandξ2=η1+er1+r2+λϑmax/1−e−λϑmin.
Hence, for anyt≥t0,(55)zt≤ξ1zt0e−λ/2t−t0+ξ2ωt0,t,which means that neural network (1) is the stabilization to exponential ISS under observer-based T-APIC (4).
In addition, from (49), one has(56)λ+r2r1+r2∑j=0iΔj≤∑j=0iϑj,i∈ℕ.
Therefore, we get(57)Ru∗=r2r1+r2<λ+r2r1+r2≤Rmin.
The deducing procedure is finished.Remark 5.
Assumeϑi satisfies (29), and for any i∈N, the following inequality,(58)λ+r2r1+r2Δi≤ϑi<Δi,holds, where λ is a sufficiently small constant satisfying 0<λ<r1; then, we can deduce (33), and so, Theorem 1 is still true.
Because (9) is a switched system under observer-based APIC K1,K2,ti,ϑi, we will next give a sufficient condition for the exponential ISS of (9) by using the converse average dwell time.
Set notationNs,t represents the control times of the observer-based APIC during s,t.Theorem 2.
Assume thatti is given to satisfy (28) and the converse average dwell time condition for appropriate ϖ>0, d≥0, i˜≥1, and for any ti≤t<ti+1, i≥i˜,(59)Ns,t≥t−sϖ−d,t≥s≥t0.
Under Assumptions1 and 2, (9) can realize stabilization to exponential ISS via observer-based T-APIC K1,K2,ti,ϑi, if ϑi satisfies (29) and(60)Ru∗=r2r1+r2<ϑminϖ<1,where r1 and r2 are the same as those in Theorem 1.
Moreover, we get(61)Rmin>Ru∗.Proof.
Likewise, letVzt=zTtzt. The definition of function ρijt is same as that in Theorem 1. Meanwhile, by Assumptions 1 and 2 and the reasoning process of Theorem 1, we still get (40)–(47).
From (60), there is λ satisfying 0<λ<r1 and the following equation:(62)r1+r2λ+r2ϑmin=ϖ.
Then, converse average dwell time condition (59) can be rewritten as, for any ti≤t<ti+1, i≥i˜,(63)Ns,t≥t−sr1+r2/λ+r2ϑmin−d,t≥s≥t0.
Letd1=ϑminr1+r2d+λ+r2Δmax, where Δmax=maxΔi=maxti+1−ti,i∈ℕ. When i≥i˜ and for any ti≤t<ti+1, tj+ϑj≤s<tj+1 satisfying i>j, one has Ns,t=i−j. Similar to Theorem 1, we now need to estimate ρijt.
For anyti≤t<ti+ϑi, by using (63), we have(64)ρijt=−r1+r2∑k=ji−1ϑk+r2t−tj≤−r1+r2i−jϑmin+r2t−tj≤−t−sλ+r2+ϑminr1+r2d+r2t−tj=−t−tj+tj−sλ+r2+ϑminr1+r2d+r2t−tj≤−λt−tj+λ+r2Δj+ϑminr1+r2d≤−λt−tj+d1.
For anyti+ϑi≤t<ti+1, from the definition of ρi,jt and the reasoning process of (64), we get(65)ρi+1,jt≤ρi+1,jti+1≤−λti+1−tj+d1.
Wheni≥i˜, for any ti≤t<ti+ϑi, based on (47), (64), and the derivation of (47), we deduce(66)Vzt≤ed1Vzt0e−λt−t0+η1+er1ϑmax+d11−e−λϑminωt0,t2.
Analogously, combining (46) and (65), we get(67)Vzt≤ed1Vzt0e−λt−t0+ηer1ϑmax+d11−e−λϑminωt0,t2,for any ti+ϑi≤t<ti+1.
Hence,∀t≥ti˜; from (66) and (67), one has(68)Vzt≤ed1Vzt0e−λt−t0+η1+er1ϑmax+d11−e−λϑminωt0,t2.
Then, we analyse the case oft0≤t<ti˜.
According to (40), (46), (47), and the definition of ρijt, it follows that(1)
Whent0≤t<t0+ϑ0,(69)Vzt≤Vzt0e−r1t−t0+e−r1t−t0∫t0ter1s−t0hsds≤Vzt0e−r1t−t0+η1ωt0,t2≤Vzt0e−λt−t0+η1ωt0,t2.(2)
Whent0+ϑ0≤t<t1,(70)Vzt≤Vzt0eρ1,0t+ηer1ϑ0+ρ1,0tωt0,t2≤Vzt0e−r1+r2ϑmin+r2t−t0+η1+er1ϑmax+r2t1−t0−r1+r2ϑminωt0,t2.(3)
Whent1≤t<t1+ϑ1,(71)Vzt≤Vzt0eρ1,0t+η1+ηer1ϑ0+ρ1,0tωt0,t2≤Vzt0e−r1+r2ϑmin+r2t−t0+η1+er1ϑmax+r2t1+ϑ1−t0−r1+r2ϑminωt0,t2.(4)
Whent1+ϑ1≤t<t2,(72)Vzt≤Vzt0eρ2,0t+η∑j=01er1ϑj+ρ2,jtωt0,t2≤Vzt0e−r1+r22ϑmin+r2t−t0+η1+er1ϑmax+r2t2−t0∑j=12e−r1+r2jϑminωt0,t2.(5)
Whent2≤t<t2+ϑ2,(73)Vzt≤Vzt0eρ2,0t+η1+η∑j=01er1ϑj+ρ2,jtωt0,t2≤Vzt0e−r1+r22ϑmin+r2t−t0+η1+er1ϑmax+r2t2+ϑ2−t0∑j=12e−r1+r2jϑminωt0,t2.By inducing, we get(6)
Whenti˜−1≤t<ti˜−1+ϑi˜−1,(74)Vzt≤Vzt0eρi˜−1,0t+η1+η∑j=0i˜−2er1ϑj+ρi˜−1,jt×ωt0,t2≤Vzt0e−r1+r2i˜−1ϑmin+r2t−t0+η1+er1ϑmax+r2ti˜−1+ϑi˜−1−t0∑j=1i˜−1e−r1+r2jϑmin×ωt0,t2.(7)
Whenti˜−1+ϑi˜−1≤t<ti˜,(75)Vzt≤Vzt0eρi˜,0t+η∑j=0i˜−1er1ϑj+ρi˜,jtωt0,t2≤Vzt0e−r1+r2i˜ϑmin+r2t−t0+η1+er1ϑmax+r2ti˜−t0∑j=1i˜e−r1+r2jϑminωt0,t2.
Letξ3=max1,e−r1+r2ϑmin+r2+λti˜−t0. From the above derivation, for any t∈t0,ti˜, we have(76)Vzt≤ξ3Vzt0e−λt−t0+η1+er1ϑmax+r2ti˜−t0−r1+r2ϑmin1−e−r1+r2ϑminωt0,t2.
Letα=η1+er1ϑmax+r2ti˜−t0−r1+r2ϑmin/1−e−r1+r2ϑminandβ=η1+er1ϑmax+d1/1−e−λϑmin. Hence, for any t≥t0, one has(77)Vzt≤ξ1∗Vzt0e−λt−t0+ξ2∗ωt0,t2,where ξ1∗=maxed1,ξ3 and ξ2∗=maxα,β.
So, for anyt≥t0,(78)zt≤ξ1∗zt0e−λ/2t−t0+ξ2∗ωt0,t,which indicates that (1) achieves stabilization to exponential ISS under observer-based T-APIC (4).
Next, we estimateRmin.
From (59), it follows that(79)ϖi+1≥∑j=0iΔj−dϖ,which means that(80)ϖ≥limsupi⟶∞∑j=0iΔj−dϖi+1=limsupi⟶∞∑j=0iΔji+1.
It follows from (30), (60), and (80) that(81)Rmin≥minti,ϑiϑminlimi⟶∞sup∑j=0iΔj/i+1≥ϑminϖ>Ru∗.
The deducing procedure is finished.Remark 6.
Whenti is given to satisfy (28) or satisfy (28) and converse average dwell time (59), by Theorem 1 and Theorem 2, there exists nontrival ϑi, such that (1) is stabilized to exponential ISS via observer-based T-APIC (4). Also, we get that MATR Rmin>Ru∗. Comparing (59) with (60), it is worth mentioning that if converse average dwell time (59) holds, we can use the average dwell time ϖ of Δi substitute for the single Δi in (58) to deduce observer-based T-APIC K1,K2,ti,ϑi.Case 2.
ϑi is given to satisfy (29).
Here, we will give homologous criteria to ensure that (1) is stabilized to exponential ISS via observer-based T-APIC (4) and estimate Rmin.Theorem 3.
Supposeϑi is given to satisfy (29) and Assumptions 1 and 2 hold. (1) is stabilized to exponential ISS via observer-based T-APIC (4), if ti satisfies (28) and one of the following conditions:(a)
(82)1<γinf−1=mini∈ℕΔiϑi<Ru∗−1.(b)
For appropriateλ satisfying 0<λ<r1 and any i∈ℕ,(83)ϑi<Δi≤r1+r2λ+r2ϑi.(c)
Converse average dwell time (59) holds and ϖ satisfies(84)ϑmin<ϖ<ϑmin⋅Ru∗−1,r1 and r2 in the above conditions are the same as those in Theorem 1. Moreover, we can obtain that Rmin>Ru∗.Proof.
For (a) and (b), it can be similar to the proof in Theorem1, and for (c), by using the similar proof of Theorem 2, we can get the result.Remark 7.
From Theorems1–3, it is not difficult to find that Ru∗=r2/r1+r2 is an extremely significant index for Rmin of observer-based T-APIC (4). In addition, one can see that neural network (1) realizes stabilization to exponential ISS via observer-based T-APIC (4) when Ru∗<Rmin.Corollary 1.
Under Assumptions1 and 2, for any i∈ℕ, if ϑi=ϑ and ti satisfy(85)ti+1<ti+1+r1r2ϑ,then system (1) realizes stabilization to exponential ISS via observer-based T-APIC (4).Proof.
From (85), we can deduce(86)1>mini∈ℕϑΔi>r2r1+r2.
Then, by (a) in Theorem3, it follows that Corollary 2 is true.
PIC as an especial case of APIC, next, a corresponding result obtained by the observer-based PIC will be given.Corollary 2.
Under Assumptions1 and 2, if control period T and control width ϑ of observer-based PIC satisfy(87)ϑ<T<ϑ+r1r2ϑ,then (1) can achieve stabilization to exponential ISS via the observer-based PIC.Proof.
Letϑi=ϑ and T be the maximum value of Δi in (82), namely, T=maxΔi. According to (82) of Theorem 3 and condition (87), Corollary 2 is correct.


## 4. Stabilization to Exponential ISS via Observer-Based E-APIC

Compared with the time-triggered control mechanism, the event-triggered control mechanism can not only save unnecessary adjustment times but also control intensity is more appropriate. Therefore, in Theorem4, we propose the observer-based E-APIC to realize the exponential ISS of the system (1).The following is the construction of the observer-based E-APIC.Suppose Assumptions1 and 2 hold. The observer-based E-APIC depends on two indices: threshold value δmax>1 and check period Δ>0, which is generally a comparatively great real number.LetVTzt=zTtzt. To ensure that the observer-based E-APIC satisfies (28) and (29), let ϑi satisfy(88)0<lnδmaxr1<ϑmin=minϑi≤ϑi≤Δ−lnδmaxr2.The event-triggered mechanism is as follows:(89)ti+1=mint:t∈ψi^ti+ϑi,ti+Δ,ifψi^ti+ϑi,ti+Δ≠∅;ti+Δ,ifψi^ti+ϑi,ti+Δ=∅,where(90)ψi^s,t≜θ:s<θ≤t,Vzθ≥δmaxVzs+ξmωs,θ2,with ξm≥maxε1−1r1−1,ε2−1r2−1, 0<ε1<g1, 0<ε2<g2, and r1 and r2 are the same as those in Theorem 1.Remark 8.
Sinceω∈L∞,loc, replacing ωti+ϑi,t with ω∞ will simplify the calculation in (88) and (89).
The event-triggered control is closely related to the system state, that is, the controller is only activated when the systematic deviation reaches or exceeds the preset value given. The systematic deviation is usually expressed aset=xt−xti, and the trigger condition is et≥m, where m stands for the threshold value satisfying m>0 and ti>t. Moreover, ti+1=mint:t>ti,et≥m. Since the state of the system studied can not be obtained completely, the error system between the observer and the plant is constructed, and the corresponding augmented system is constructed by using the observed value and the error. Significantly, there is a special case that et<m always holds, which indicates that the trigger condition has not appeared, that is, the controller has not been activated, which may lead to system instability. The check period proposed in [30] can effectively keep from this situation. Introducing a comparatively great check period to the observer-based E-APIC means eventhough the trigger condition is not satisfied during a check period, there will be a control input at the end of the check period to ensure the ISS of the system.Theorem 4.
Under Assumptions1 and 2, (1) can realize stabilization to exponential ISS via observer-based E-APIC (88) and (89), and the MATR Rmin of the observer-based E-APIC satisfies(91)ϑminΔ≤Rmin≤r˜2r1ϑminr˜2+r1lnδmax,where r˜2 is a positive constant satisfying r˜2>r2 and −r˜2+g2+ε2−1+ε2−1ξm−1+4l22≤0.Proof.
From (88) and (89), one can see that the interval between two adjacent controls Δi=ti+1−ti>ϑi>lnδmax/r1>0, which means the observer-based E-APIC will not be triggered countless times in a limited time. So, there is no Zeno behavior under the observer-based E-APIC. Furthermore, one can see that ϑi is nontrival.
For anyti+ϑi≤t<ti+1, from (42), we have(92)V˙zt≤g2+ε2Vzt+4l22Vzt−τ+ε2−1ωti+ϑi,t2.
From (92), whenever Vzt≥ξmωti+ϑi,t2, we get(93)V˙zt≤g2+ε2+ε2−1ξm−1Vzt+4l22Vzt−τ.
Additionaly, sincer2≥g2+ε2+4l22, apparently there exists r˜2>0, satisfying r˜2>g2+ε2+ε2−1ξm−1+4l22; then, we can choose a positive constant r˜2 satisfying r˜2>r2 and −r˜2+g2+ε2+ε2−1ξm−1+4l22<0.
By using Lemma2 and (93), for any ti+ϑi≤t<ti+1, one has(94)Vzt≤Vzti+ϑier˜2t−ti−ϑi.
By the continuity ofVzt, we get(95)Vzti+1≤er˜2ti+1−ti−ϑiVzti+ϑi≤er˜2ti+1−ti−ϑiVzti+ϑi+ξmωti+ϑi,ti+12.
On the one hand, by (89), if ψi^ti+ϑi,ti+Δ≠∅, which means ti+1≤ti+Δ, one has(96)Vzti+1=δmaxVzti+ϑi+ξmωti+ϑi,ti+12,since Vzt is continuous.
Then, combining (95) and (96), we have(97)δmax≤er˜2ti+1−ti−ϑi,which implies that(98)0<lnδmaxr˜2≤ti+1−ti−ϑi≤Δ−ϑmin,for any i∈ℕ.
On the other hand, ifψi^ti+ϑi,ti+Δ=∅, then ti+1=ti+Δ. By (88), we get(99)0<lnδmaxr˜2<lnδmaxr2≤Δ−ϑi=ti+1−ti−ϑi≤Δ−ϑmin,for any i∈ℕ.
Hence, for anyi∈ℕ, we get (98) holds. Combining (88) and (98), we can obtain(100)0<1r1+1r˜2lnδmax<ti+1−ti≤Δ.
Next, we show that (1) is stabilized to exponential ISS by the observer-based E-APIC.
For anyti≤t<ti+ϑi, by (40), one has(101)Vzt≤δmaxVzti+η1ωti,t2.
In addition, from (41), we have(102)Vzti+ϑi≤Vztie−r1ϑi+η1ωti,ti+ϑi2.
Letξ=maxη1,ξm and ξ˜=δmaxη1+ξ. For any ti+ϑi≤t<ti+1,i∈ℕ, by (89) and (102), we get(103)Vzt≤δmaxVzti+ϑi+ξmωti+ϑi,t2≤δmaxe−r1ϑiVzti+ξ˜ωti,t2.
Letbi=Vzti and pi=δmaxe−r1ϑi, for any i∈ℕ; by (89) and the continuity of Vzt, one has(104)bi+1≤pibi+ξ˜ωti,t2.
Letλ˜=r1ϑmin−lnδmax and ξ˜∗=ξ˜1−e−λ˜−1. It follows from (88) that λ˜>0; in addition, by (104), for any i∈ℕ, we can obtain(105)bi≤e∑j=0i−1lnδmax−r1ϑjb0+ξ˜1+∑j=1i−1e∑k=ji−1lnδmax−r1ϑiωt0,ti2≤e−λ˜ib0+ξ˜1+∑j=1i−1ejlnδmax−r1ϑminωt0,ti2≤e−λ˜ib0+ξ˜∗ωt0,ti2.
By using (101) and (103), for any ti≤t≤ti+1,i∈ℕ, we get(106)Vzt≤δmaxVzti+ξ˜ωti,t2.
Combining (105) and (106), for any t≥t0, we get(107)Vzt≤δmaxe−λ˜iVzt0+ξ˜+δmaxξ˜∗ωt0,t2≤δmaxeλ˜e−λ˜/Δt−t0Vzt0+ξ˜+δmaxξ˜∗ωt0,t2.
Hence, for anyt≥t0, by (107), we have(108)zt≤δmaxeλ˜e−λ˜/2Δzt0+ξ˜+δmaxξ˜∗ωt0,t,which means that (1) realizes stabilization to exponential ISS under observer-based E-APIC (88) and (89). In addition, let ϑi=ϑmin and combining (30) and (100), (91) holds. The deducing procedure is finished.
The following is the contrast among observer-based PIC, observer-based T-APIC, and observer-based E-APIC. For the convenience of exposition, denote byRminP, RminT, and RminE, respectively, MATR of observer-based PIC, observer-based T-APIC, and observer-based E-APIC.
Set symbolNPt, NTt, and NEt, respectively, the control times of the observer-based PIC, the observer-based T-APIC, and the observer-based E-APIC on interval t0,t. Let ϑminT and ϑminE denote the minimum activation time of observer-based T-APIC and observer-based E-APIC, respectively.Theorem 5.
Letϑi=ϑ,i∈ℕ, in observer-based T-APIC and observer-based E-APIC as in observer-based PIC. Then, if Assumptions 1 and 2 hold, there is an observer-based E-APIC (88) and (89) satisfying(109)RminE≤Ru∗≤minRminP,RminT.Proof.
Follow Theorems1–3 and Corollary 2; there exists a fully small ε satisfying 0<ε≪r1, such that minRminP,RminT≥r2+ε/r1+r2. Let δmax=er1r˜2ϑr1+r2∖ε+r2r˜2+r1 in observer-based E-APIC (88) and (89); then, by (91) in Theorem 4, we have RminE≤ε+r2/r1+r2=Ru∗+ε/r1+r2. Hence, (109) holds when ε⟶0.Theorem 6.
(a)
IfσE≜1/r1+1/r˜2lnδmax≥T, then, for any t≥t0, one has(110)t−t0Δ≤NEt≤t−t0σE≤NPt.(b)
Let observer-based T-APIC satisfyϑminT>0 and converse average dwell time NTt≥t−t0/σT, where σT≜r1+r2/λ+r˜2ϑminT with λ defined by (44). If lnδmax≥σT1/r1+1/r˜2−1, then NEt≤NTt.(c)
If the threshold valueδmax in observer-based E-APIC (88) and (89) satisfies lnδmax≥maxσT,T1/r1+1/r˜2−1, then, for any t≥t0, we get(111)NEt≤minNPt,NEt.Proof.
By (100), we can derive (a). From (a) and σE>σT, we have (b), and (c) follows from (a) and (b).Remark 9.
By Theorem6, the control times of observer-based E-APIC are less than those of observer-based PIC and observer-based T-APIC when we choose an appropriate δmax.


## 5. Example

In this section, we provide a numerical example to simulate the validity of the obtained results.Example 1.
Consider a two-dimensional neural network with time delay and external disturbances, which is described as(112)x1tx2t=−1001x1tx2t+10.10.11f1x1tf2x2t+−10.10.1−1g1x1t−τg2x2t−τ+u11tu12t+ω1tω2t.
Then, the corresponding state observer is designed by(113)x˜1tx˜2t=−1001x˜1tx˜2t+10.10.11f1x˜1tf2x˜2t+−10.10.1−1g1x˜1tg2x˜2t+u21tu22t.
Let(114)fixi=sinxi,i=1,2,gixi=0.1tanhxi,i=1,2,and the delay of τ is set to be 0.1, ω1t,ω2tT=sint,rand1T. Moreover, solving matrix inequalities (13) and (14), we have g1=9.9796,g2=5.0204,K1=−19.5I, and K2=−19.6I. Let ε1=ε2=1. By solving inequalities (31) and (32), we have r˜1≥−8.8 and r2≥6.0604, choose r˜1=−5.28, r2=7.42, and then r1=r˜1=5.28.
Here, letti+1=ti+4+0.55×rand1. In order to satisfy the conditions in Theorem 1, we choose ϑi=2.7+0.1×rand1; then, neural network (112) can realize stabilization to exponential ISS via the observer-based T-APIC. The numerical simulations are shown in Figures 1 and 2 .
According to (28) and (59), let ti+1=ti+3+rand1. If ϑi is designed as ϑi=2+rand1, then the conditions (29) and (60) of Theorem 2 can be satisfied, which means that neural network (112) can realize stabilization to exponential ISS via the observer-based T-APIC. The state trajectories of (112) and the starting instant of each control of the observer-based T-APIC are shown in Figures 3 and 4 .
Let the threshold valueδmax=15, check period Δ=10, and ξm=0.5≥max0.1894,0.1348. According to Theorem 4, we choose ϑi=2.5+0.1×rand1; then, under the event-triggered mechanism (89), system (112) realizes stabilization to exponential ISS. The numerical simulations are shown in Figures 5 and 6 .
To better compare observer-based PIC, observer-based T-APIC, and observer-based E-APIC, let control durationϑi=1.5 and the remaining parameters remain unchanged. (a)
For observer-based PIC, according to Corollary2, let T=2. Then, neural network (112) can realize stabilization to exponential ISS via observer-based PIC. The state trajectories of (112) and the starting instant of each control of the observer-based PIC are shown in Figures 7 and 8, respectively.(b)
For observer-based T-APIC, according to Theorem3, let ti be designed as ti+1=ti+2+0.566×rand1. Then, neural network (112) can realize stabilization to exponential ISS via observer-based T-APIC. The numerical simulations refer to Figures 9 and 10, respectively.(c)
For observer-based E-APIC, let the threshold valueδmax=15, check period Δ=10, and ξm=0.5. From Theorem 4, system (112) is stabilization to exponential ISS under the event-triggered mechanism (89). The state trajectories of (112) and the starting instant of each control of the observer-based E-APIC are shown in Figures 11 and 12, respectively.
As follows, the three kinds of controllers are compared from the control times and activation time rate in the same time. Here, we extract the relevant data of 100 s in the numerical simulations, as given in Table1.
From Table1, one can see that the control times of observer-based E-APIC are least and the activation time rate of the observer-based E-APIC is lowest, which show that the observer-based E-APIC is superior to the observer-based T-APIC and the observer-based PIC in control times and activation time rate; in the meanwhile, we can get the observer-based T-APIC has some better performances than the observer-based PIC.Figure 1
State trajectoriesxii=1,2 of system (112) under the observer-based T-APIC.Figure 2
The starting instant of each control of the observer-based T-APIC.Figure 3
State trajectoriesxii=1,2 of system (112) under the observer-based T-APIC.Figure 4
The starting instant of each control of the observer-based T-APIC.Figure 5
State trajectoriesxii=1,2 of system (112) under the observer-based E-APIC.Figure 6
The starting instant of each control of the observer-based E-APIC.Figure 7
State trajectoriesxii=1,2 of system (112) under the observer-based PIC.Figure 8
The starting instant of each control of the observer-based PIC.Figure 9
State trajectoriesxii=1,2 of system (112) under the observer-based T-APIC.Figure 10
The starting instant of each control of the observer-based T-APIC.Figure 11
State trajectoriesxii=1,2 of system (112) under the observer-based E-APIC.Figure 12
The starting instant of each control of the observer-based E-APIC.Table 1
Contrasts among observer-based PIC, observer-based T-APIC, observer-based E-APIC.
Observer-based PICObserver-based T-APICObserver-based E-APICN100504415R10075%66%22%Ru∗≈58%


## 6. Conclusions

This study focuses on the exponential input-to-state stability of a class of neural networks with time delay and external disturbances under the observer-based APIC. Considering that the state of the system and the delay information may not be accurately obtained in practical application, a state observer independent of historical information is structured. For realizing exponential input-to-state stability of the studied network, the observer-based T-APIC is designed by using the infimum of single activation time rate and the condition of converse average dwell time, respectively. An observer-based E-APIC is advanced to solve the problem that the results obtained under the observer-based T-APIC may be conservative. In addition, the MATR of the observer-based T-APIC and the observer-based E-APIC are estimated. The results show that the observer-based E-APIC is superior to the observer-based PIC and the observer-based T-APIC in terms of control times and activation time rate, and the observer-based T-APIC is also superior to the observer-based PIC. Compared with the disturbances considered, the random disturbance makes the system have more abundant and interesting trajectories. Therefore, it is our next work to study the stability of neural networks with random disturbance under aperiodic intermittent control.



---

*Source: 9923792-2021-08-06.xml*