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Journal of the Franklin Institute 356 (2019) 6106–6144 www.elsevier.com/locate/jfranklin

Sufficient bit rate conditions to stabilize an uncertain scalar nonlinear system based on event triggeringR Rundong Dou, Jiayu Chen, Qiang Ling∗ University of Science and Technology of China, Hefei 230027, China Received 19 September 2018; received in revised form 21 January 2019; accepted 20 March 2019 Available online 8 June 2019

Abstract This paper considers to stabilize an uncertain scalar continuous-time nonlinear system with bounded network delay and process noise, which transmits all feedback signals through a digital communication network. In order to save the bandwidth of the feedback network, stability is expected to be maintained at as low as possible feedback bit rate. Based on event triggering, this paper proposes a model-based event-triggered sampling strategy to guarantee the desired input-to-state stability of the concerned system. Due to the bounded network delay, the receiving time instant of a feedback packet cannot be precisely controlled by the sensor, i.e., the receiving time instant is not always equal to its sampling time instant. Their gap, i.e., the network delay, determines how much information can be carried through the receiving time instant and makes great impact on the system’s stability. Sufficient bit rate conditions to stabilize that system are derived. The conditions are determined by the parameter of Lipschitz condition, the upper bound of the network delay and the system uncertainty. Compared with the periodic sampling strategies, a lower bit rate is required by the proposed event-triggered strategy. Simulations are done to verify the effectiveness of the achieved stabilizing bit rate conditions. © 2019 Published by Elsevier Ltd on behalf of The Franklin Institute.

R This work was supported by the National Natural Science Foundation of China, the Intelligent Networked Electric Vehicle Key System Integration Development and Industrialization Project, and the National Key Research and Development Program of China (No. 2016YFC0201003). ∗ Corresponding author. E-mail address: [email protected] (Q. Ling).

https://doi.org/10.1016/j.jfranklin.2019.03.025 0016-0032/© 2019 Published by Elsevier Ltd on behalf of The Franklin Institute.

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1. Introduction During the past decades, Networked Control Systems (NCS), where feedback control loops are closed via digital communication networks, have attracted much attention from various fields. Generally, NCS consist of several sensors, actuators, and controllers, which can share information with each other through a shared communication network. Compared with traditional point-to-point control systems, NCSs provide several advantages, such as low cost, high reliability, increased system flexibility, and decreased volume of wiring. Benefiting from these advantages, NCSs are widely used in various applications, such as the aircraft control, the manufacturing in factories and so on. Some previous results are reported in [1–4] and the references therein. In NCSs, periodic sampling is usually implemented. Different from periodic sampling strategies, Event-Triggered Control (ETC) strategies could be more efficient and appealing by generating sporadic sampling, which can well balance the communication resource utilization and control performance [5]. A recent overview is presented in [6]. In [7], the L p stabilization of nonlinear systems by using output-based dynamic event-triggered controllers is studied. In ETC, the sensor continuously measures the system state, but only sends the measurements to the controller when certain pre-specified conditions are violated. In other words, the feedback loop is only closed when an event indicates that information exchange is necessary in order to maintain a required control performance. Some stability concepts, such as Input-to-State Stability (ISS) and L2 stability, have been studied in the framework of ETC [8]. Generally speaking, ETC makes good use of the data obtained from the last recent sampling to calculate the control input. If the control input is held constant till the next sampling, this control method is called Zero-order-hold. Garcia et al. investigated the model-based eventtriggered control system with the network delay, and studied the asymptotic stability of a linear system by using the static logarithmic quantizer [9]. It is worth mentioning that the model-based control method is also implemented to estimate the system state in order to extend the inter-sampling intervals as long as possible. In [10], this method is applied to the periodic event-triggered control system. Due to the advantages of this control method, we apply the model-based event-triggered control strategy to our system. In [11], noise is added into a linear system without the network delay. In [12], the time-varying network delay and the system uncertainty are considered at the same time. In NCSs, feedback signals are transmitted through digital communication networks. As the feedback network bandwidth is often limited, it is of particular importance to reduce the occupied feedback bit rate, which is denoted as R in the sequel. The recent results regarding the feedback bit rate saving can be classified into three types, which are briefly introduced below. 1) The transmission time instants are fixed and R is finite: In [13], the minimum bit rate conditions for stabilizing a linear discrete-time system are given. For a scalar continuous-time linear system, it can be described as R > R0 =

max(0, A ) , log(2)

(1)

where log(·) represents the base-e logarithm, and A is the eigenvalue of the linear system. For a nonlinear system in [14], the global asymptotic stabilization is guaranteed if the number of quantization levels adopted by the encoder, the sampling period, and a system parameter can

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satisfy certain conditions. Moreover, when the concerned nonlinear control system degenerates into a linear one, the lower bound on the stabilizing bit rate in [14] is the same as the linear results in [13], which will be discussed in the latter section. 2) The transmission time instants are adjustable and R is infinite: In [12], the transmission time instants are equal to the sampling time instants due to the assumption of the zero network delay. When the triggering condition is satisfied, sampling is triggered and the data packet will be encoded and sent to the controller instantaneously and perfectly. For a nonlinear system in [15], the existence of a minimum inter-sampling interval and the ISS of the closed-loop system are guaranteed by the ETC strategy. This ETC method will achieve longer intersampling intervals than periodic sampling control methods. 3) The transmission time instants are adjustable and R is finite: In [16], the concept of “free symbol” is given. It means that the absence of transmission can still convey information. For noiseless linear time-invariant systems, some stabilizing bit rate conditions are provided [16]. In [17], for a linear time-invariant system, sampling is triggered when the output reaches a certain level and one bit is taken to represent the sign of the output change. When the system is noise-free, the initial system state can be precisely known after transmitting a finite number of bits. In other words, the average occupied bit rate R is 0. In [18], in consideration of the bounded processing delay and the bounded process noise, some stabilizing bit rate conditions are provided for a scalar continuous time linear system. These conditions are only up to the system matrix and the bound of processing delay. When the processing delay is very small, the required bit rate is very closed to 0. Moreover, in [19], the network delay is taken into account and the system can also achieve ISS under some bit rate conditions. In [16–19], the lower bit rate bound R0 in Eq. (1) is broken, which can save network bandwidth resources in guaranteeing stability. For the nonlinear system in [20], by considering the effects of the network delay and the quantization error, an upper bound on the stabilizing bit rate related to the norm of the state is obtained. Furthermore, that stabilizing bit rate can achieve the efficient attentiveness [21], i.e., the closer the state is near to the origin, the lower the stabilizing bit rate will be. To the best of our knowledge, there are rare results regarding stabilizing bit rate conditions for nonlinear systems. In this paper, in view of the bounded network delay and process noise, we want to derive stabilizing bit rate conditions for a scalar continuous time nonlinear system with system uncertainty. In [20], it is proved that some bit rate conditions to stabilize the nonlinear system are efficient. But these conditions depend on either the system state or the quantized system state. This paper derives sufficient stabilizing bit rate conditions that are only determined by the parameter of Lipschitz condition, the bounded network delay and the system uncertainty. It will be shown that the stabilizing bit rate of our method is lower than that of the strategy in [14]. For a scalar continuous time linear system, i.e., a special nonlinear system, the stabilizing bit rate of our method is much lower than that of periodic sampling control mechanisms in Eq. (1). The remainder of this paper is organized as follows. In Section 2, some fundamental notation and mathematical models are presented. Some sufficient bit rate conditions for stabilizing a noise-free scalar continuous time nonlinear system with network delay are provided in Section 3. Section 4 presents sufficient bit rate conditions in existence of both network delay and process noise. In Section 5, sufficient bit rate conditions are derived for an uncertain noise-free scalar continuous time nonlinear system with network delay. Section 6 provides sufficient bit rate conditions for an uncertain scalar continuous time system under network delay and process noise. The obtained stabilizing bit rate conditions are fur-

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Fig. 1. An uncertain scalar continuous time nonlinear system.

ther validated through numerical simulations in Section 7. Some final remarks are placed in Section 8.

2. Problem description Before presenting the concerned mathematical models, we briefly introduce the concerned notation in this paper. | · | denotes the absolute value operation. R represents the set of real number. A K function h(x) is continuous, strictly increasing and h(0) = 0. A KL function f(x, y) is a K function with respect to x for any fixed y, and is decreasing with respect to y and limy→∞ f (x, y) = 0 for any fixed x. log2 (·) represents the base-2 logarithm. · stands for the ceil function, i.e., x stands for the smallest integer being no less than x. This paper considers the uncertain scalar continuous time nonlinear system in Fig. 1. In that system, a sensor detects the system state x(t). If the triggering condition is not satisfied, the feedback links (dotted lines) will not be closed. When the triggering condition is satisfied, sampling is triggered instantaneously at time sk , where sk stands for the kth sampling instant. Without considering the influence of processing delay, the encoder encodes and transmits the data at sk . To save the feedback network bandwidth, the encoder only transmits a single bit symbol θ k to the controller. At time rk , the decoder receives θ k , then decodes it to generate a state estimate the controller. The controller updates u(t) for the nonlinear system. As is shown in [22], the control input can also act as an acknowledgment (ACK) of the feedback packets. We assume that the sensor receives the ACK instantaneously. The state estimates of the sensor and the controller will be updated at time rk . The network delay is denoted as dk = rk − sk . Note that rk can be regarded as a quantized value of sk . The system in Fig. 1 is governed by the following dynamics, d x(t ) = f (x(t )) + u(t ) + ω(t ), dt

(2)

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where x(t ) ∈ R, u(t ) ∈ R. The process noise ω(t) is bounded by |ω(t )| ≤ W , ∀t ≥ 0.

(3)

We also assume that the plant model dynamics differ from the actual plant dynamics in an additive fashion: f (x(t )) = fˆ(x(t )) + ς (x(t )),

(4)

where fˆ(x(t )) is the nominal system dynamics, and ς (x(t)) is the system uncertainty. In Eq. (2), in order to ensure the ISS [23], the state x(t) has to satisfy |x(t )| ≤ β(|x(0)|, t ) + γ ( sup |ω(τ )| ), ∀t ≥ 0, t≥τ ≥0

(5)

where β( · , · ) is a KL function, and γ (·) is a K function. As is shown in Fig. 1, the control input u(t) is updated by the state estimate xˆ(t ). Due to the ACK, the sensor and the controller can update their state estimates at the same time rk . Their state estimates will keep the same. The state estimation dynamics can be described as d xˆ(t ) = fˆ(xˆ(t )) + u(t ), xˆ(0) = 0. dt

(6)

For the system in Eq. (2), the following assumptions are taken. 1. f(·) and ς (·) are globally Lipschitz,1 i.e., | f (x) − f (y)| ≤ λ|x − y|,

(7)

|ς (x) − ς (y)| ≤ λ0 |x − y|,

(8)

where λ and λ0 are positive constant. 2. The network delay dk = rk − sk is bounded, i.e., dk ≤ D.

(9)

3. The initial system state is bounded, i.e., |x(0)| ≤ X0 .

(10)

The estimation error between the system state and the model state is defined as x˜e (t ) = x(t ) − xˆ(t ).

(11)

The control input is computed as u(t ) = −Gxˆ(t ) − f (0),

(12)

where G is the control gain and satisfies the following conditions λ1 = λ + λ0 − G < 0.

(13)

1 The globally Lipschitz condition of f(·) can be relaxed into the locally Lipschitz condition as shown by Corollaries 4, 7, 10 and 13.

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3. Sufficient bit rate conditions for D ≥ 0, W = 0 and λ0 = 0 For simplicity, we start with a scalar continuous time nonlinear system without either process noise or system uncertainty, i.e, W = 0 and λ0 = 0. We choose the following eventtriggering function ∗

L(t ) = X0 e−λ t ,

(14)

∗

where λ is a small positive number that measures the speed of exponential convergence and satisfies λ∗ < −λ1 ,

(15)

where λ1 is defined in Eq. (13). (1) Event-triggered strategy: A sensor can measure and access the system state continuously. An event is triggered at time sk when the estimation error x˜e (t ) satisfies the following condition, |x˜e (sk )| = L(sk ).

(16)

Then the encoder generates a symbol θ k according to the following rule and immediately transmits it to the controller, θk = sign(x˜e (sk )),

(17)

where sign(·) represents the sign function ⎧ ⎨1, z > 0, sign(z) = 0, z = 0, ⎩ −1, z < 0.

(18)

(2) State estimation and updating rules: Due to the network delay, the feedback packet θ k , which is transmitted at sk , is received by the controller at rk . Then the event-trigger and the controller immediately update the state estimate xˆ(rk+ ) as xˆ(rk+ ) = xˆ(rk− ) + αr θk L(rk ), ∗

(19) ∗

where αr = 21 (e(λ+λ )D + e(−λ+λ )D ), rk− and rk+ denote the time instants immediately before and after rk , respectively. (3) A lower bound on the inter-sampling interval: The symbol θ k is transmitted to the network at time sk . When the next event is triggered, the encoder will send θk+1 to the network at time sk+1 . So the inter-sampling interval is defined as Tk = sk+1 − sk .

(20)

The following lemma places a lower bound on Tk , whose proof can be found in the Appendix. Lemma 1. ∗

Tk ≥

T1∗

−log( 21 (e(λ+λ )D − e(−λ+λ = (λ + λ∗ )

∗

)D

))

, ∀k ≥ 1,

(21)

In order to avoid re-trigger another event before the last event is responded, we requires T1∗ ≥ D, where T1∗ is the lower bound on the inter-sampling intervals and defined in Eq. (21).

(22)

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Based on the above event-triggering strategy, we can ensure the desired ISS, which is stated below. Its proof is moved into the appendix to improve readability. Theorem 2. The nonlinear system can achieve ISS defined in Eq. (5) under the eventtriggering function in Eq. (14) and the strategy in Eq. (16). Note that the encoder transmits only one bit to the controller in each feedback packet θ k . So the occupied bit rate satisfies R ≤ R1 ,

(23) ∗

λ ) ∗ where R1 = T1∗ = −log( 1 (e(λ+(λ+ approach 0, we can find the folλ∗ )D −e (−λ+λ∗ )D )) . By requiring λ 1 2 lowing limit of R1 ,

RD = ∗lim + R1 λ →0

=

λ . −log( 21 (eλD − e−λD ))

(24)

Based on the above discussion, the following bit rate condition to stabilize a scalar continuous time nonlinear system is given. Proposition 3. The nonlinear system in Eq. (2) can achieve ISS when R > RD .

(25)

By Eq. (24), it is clear that RD is only concerned with λ and D. RD is not related to the system state or quantized system state at all. Provided that there is a very ideal situation that D approaches zero, i.e., the receiving time instant is almost equal to the sampling time instant, we get lim RD = 0.

(26)

D→0+

It implies that the required bit rate for stabilizing a scalar continuous time nonlinear system can be very close to zero if the network delay is very small. The globally Lipschitz condition in Eq. (7) can be relaxed into the locally Lipschitz condition for the noise-free scalar continuous time nonlinear system. One relaxation is presented in the following corollary, whose proof can be found in the Appendix. Corollary 4. Let λ be the locally Lipschitz constant for the function f(·) in the region {x: |x| ≤ B1 }, where B1 is a constant and satisfies B1 ≥ e(λ+λ

∗

)D

X0 +

1 1 − e ( λ1 + λ

∗ )T ∗ 1

αr X0 .

(27)

The nonlinear system in Eq. (2) with W = 0 and λ0 = 0 can achieve asymptotic stability under the bit rate condition in Eq. (25). 4. Sufficient bit rate conditions for D ≥ 0, W ≥ 0 and λ0 = 0 Now we consider the scalar continuous time nonlinear system with bounded process noise. Its dynamics becomes

R. Dou, J. Chen and Q. Ling / Journal of the Franklin Institute 356 (2019) 6106–6144

d x(t ) = f (x(t )) + u(t ) + ω(t ), dt

6113

(28)

where |ω(t)| ≤ W. As Section 3, we need to choose an appropriate event-triggering strategy to reduce the bit rate as low as possible. The triggering condition and function are designed as |x˜e (sk )| = LW (sk ),

(29)

LW (t ) = L(t ) + αW ,

(30)

where L(t) is defined in Eq. (14) and α is a positive constant to be determined later. The sensor and the controller can also update their state estimates at the same time rk due to the ACK. Their state estimates will also remain the same. The updating policy is xˆ(rk+ ) = xˆ(rk− ) + αr θk LW (rk ), 1 (λ+λ∗ )D (e 2

(31)

(−λ+λ∗ )D

where αr = +e ). On the basis of the event-triggering strategy in Eqs. (29) and (30), we can obtain a lower bound on the inter-sampling interval Tk = sk+1 − sk , which is presented in the following lemma and whose proof is given in the Appendix. Lemma 5. Tk ≥

T2∗

−log αr − e−λD = , (λ + λ∗ )

where α satisfies α>

e(λ+λ

∗ )T ∗ 2

(32)

D+T2∗

eλτ dτ . ∗ × (αr − e−λD ) − eλT2 (αr − e−λD ) 0

(33)

Moreover, the ISS in Eq. (5) is ensured and the occupied bit rate R is upper bounded as R ≤ R2 =

1 . T2∗

(34)

As Eq. (22), T2∗ is also required be longer than D as T2∗ ≥ D.

(35)

Eq. (35) can ensure that the current k-th control action has been enforced before the next ((k + 1)-th) event is triggered. Note that when λ∗ → 0, T2∗ can be arbitrarily close to T1∗ . Therefore We can get the following stabilizing bit rate condition, whose proof is straightforward and omitted here. Theorem 6. When the required bit rate R satisfies Eq. (25), the nonlinear system in Eq. (2) with bounded process noise can achieve ISS. The globally Lipschitz condition in Eq. (7) can also be extended to the locally Lipschitz condition for the scalar continuous time nonlinear system with bounded process noise, which is presented below and proven in the Appendix.

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Corollary 7. Let λ be the locally Lipschitz constant for the function f(·) in the region {x: |x| ≤ B2 }, B2 is a positive constant and satisfies

D 1 ∗ (X B2 ≥ e(λ+λ )D + α + αW ) + eλτ W dτ. (36) r 0 ∗ ∗ 1 − e(λ1 +λ )T2 0 Then the nonlinear system in Eq. (2) can achieve ISS. Moreover, we can ensure |x(t )| ≤ B2 , ∀t ≥ 0.

(37)

5. Sufficient bit rate conditions for D ≥ 0, W = 0 and λ0 ≥ 0 In this section, we consider an uncertain scalar nonlinear system without process noise for simplifying our derivation. Due to the system uncertainty, extra bit rate is required compared with bit rate conditions in Section 3. Notice that the event-triggering function and the eventtriggering strategy are still selected as Eqs. (14) and (16). The updating policy is also chosen as Eq. (19). As Lemma 1, we derive an lower bound T3∗ for the inter-sampling time interval Tk = sk+1 − sk . T3∗ needs to satisfy the following equation: 1 = φ1 + φ2 + φ3 , where

(38)

φ1 = e(λ+λ

(λ1 +λ∗ )(D+T3∗ )

∗

)T3∗

∗

∗

1 (λ+λ )D (e − e(−λ+λ )D ), φ2 2 λ0 (λ+λ∗ )T3∗ (λ1 +λ∗ )T3∗ φ3 = G−λ0 αr (e −e ).

=

(λ1 +λ∗ )T3∗

λ0 α e G−λ0 r 1−e(λ1 +λ∗ )T3∗

(e(λ+λ

∗

)(D+T3∗ )

−

), and According to the above expressions of φ 1 , φ 2 and φ 3 , we see that φ 1 , φ 2 and φ 3 are monotonically increasing with respect to T3∗ . When the network delay D is not very large, the solution of Eq. (38), T3∗ , is large, i.e., the event-triggering time instants are sparsely spaced. Of course, when D is small, Eq. (38) does have a positive solution which satisfies e

T3∗ ≥ D.

(39)

Then we obtain the following lemma, whose proof can be found in the Appendix. Lemma 8. According to the Lipschitz conditions in Eqs. (7) and (8), assume that Eq. (38) has a positive solution being no smaller than D. The ISS in Eq. (2) can be guaranteed and Tk ≥ T3∗ .

(40)

By Eq. (17), when every event is triggered, only θ k is transmitted through the network. We have the following bit rate condition without proof. Theorem 9. Suppose that Eq. (38) has a positive solution to satisfy Eq. (39). The ISS in Eq. (2) can be ensured with the following upper bound on the occupied bit rate R, R ≤ R3 =

1 . T3∗

(41)

Similarly, the global Lipschitz condition can be relaxed into the local one. We have the following corollary, whose proof is given in the Appendix. Corollary 10. Let λ be the locally Lipschitz constant for the function f(·) in the region {x: |x| ≤ B3 }, B3 is a positive constant and satisfies B3 ≥ (ϕ + η)X0 .

(42)

R. Dou, J. Chen and Q. Ling / Journal of the Franklin Institute 356 (2019) 6106–6144 ( λ1 + λ∗ ) T ∗

∗

∗

6115

∗

λ0 1 where ϕ = e(λ+λ )D + G−λ αr e (λ1 +λ∗ 3)T ∗ (e(λ+λ )D − e(λ1 +λ )D ) and η = ∗ ∗ αr . Then the 0 3 1−e 1−e(λ1 +λ )T3 nonlinear system in Eq. (2) can achieve ISS. Moreover, we can ensure

|x(t )| ≤ B3 , ∀t ≥ 0.

(43) ∗

Remark 1. In Eq. (38), the right-hand side contains three parts. By requiring λ approach 0 and comparing Eqs. (24) and (41) under the same D, it is easy to see that R3 ≥ RD . So some extra bit rate is required to deal with the system uncertainty. But if we select a very large G, φ 2 and φ 3 will approach zero and R3 is very close to RD , which will be illustrated in Section 7. 6. Sufficient bit rate conditions for D ≥ 0, W ≥ 0 and λ0 ≥ 0 Now we consider an uncertain scalar nonlinear system with process noise. Compared with Section 5, more bit rate is needed to deal with the process noise. Moreover, the event-triggering function and the event-triggering strategy are also selected as Eqs. (29) and (30) here. The state estimate is still updated by Eq. (31). As Lemma 1, we derive an lower bound T4∗ for the inter-sampling time interval Tk = sk+1 − sk . T4∗ needs to satisfy the following equation, 1 = σ1 + σ2 + σ3 , where σ3 =

σ1 = e(λ+λ

λ0 α G−λ0 r (λ+λ∗ )T4∗

(44) ∗

)T4∗

(αr − e−λD ),

∗

σ2 =

(λ1 +λ∗ )T4∗

λ0 α e G−λ0 r 1−e(λ1 +λ∗ )T4∗

(e(λ+λ

∗

)(D+T4∗ )

− e ( λ1 + λ

∗

)(D+T4∗ )

),

∗

×(e − e(λ1 +λ )T4 ). When D is not very large, Eq. (44) does have a positive solution satisfying T4∗ ≥ D. Therefore we can get the following lemma, whose proof is given in the Appendix. Lemma 11. Under the Lipschitz conditions in Eqs. (7) and (8), assume that Eq. (44) has a positive solution, which is no smaller than D. The ISS in Eq. (2) can be guaranteed and Tk ≥ T4∗ .

(45)

where α satisfies that D+T4∗ λτ e dτ 0 α≥ σ1 + σ2 + σ3 − σ4 − σ5 − σ6 ∗

and σ4 = eλT4 (αr − e−λD ), σ5 = e

λ1 T4∗

).

(46) ∗

λ1 T λ0 α e 4 G−λ0 r 1−eλ1 T4∗

∗

∗

(eλ(D+T4 ) − eλ1 (D+T4 ) ), σ6 =

∗ λ0 α (eλT4 G−λ0 r

−

As each feedback packet θ k consists of only 1 bit, we can place the following bit rate conditions without proof. Theorem 12. Provided that Eq.(44) has a positive solution, which satisfies that T4∗ ≥ D, the ISS in Eq. (2) can be ensured with the following upper bound on the occupied bit rate R, 1 R ≤ R4 = ∗ . (47) T4 Similarly, the global Lipschitz condition can be relaxed into the local one. We have the following corollary, whose proof is given in the Appendix.

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Corollary 13. Let λ be the locally Lipschitz constant for the function f(·) in the region {x: |x| ≤ B4 }, B4 is a positive constant and satisfies B4 ≥ (γ + ζ )X0 + (σ + γ + ζ )αW .

(48) ∗

where σ = σ1 + σ2 + σ3 − σ4 − σ5 − σ6 , γ = γ1 + γ2 , γ1 = e(λ+λ )D , γ2 = ∗ ∗ λ0 e(λ1 +λ )T4 1 (λ+λ∗ )D ( λ1 + λ∗ ) D α (e − e ) , and ζ = α . Then the nonlinear system in ∗ ∗ r G−λ0 r 1−e(λ1 +λ∗ )T4∗ 1−e(λ1 +λ )T4 Eq. (2) can achieve ISS. Moreover, we can ensure |x(t )| ≤ B4 , ∀t ≥ 0.

(49)

7. Simulations To verify the effectiveness of the proposed event-triggering methods and evaluate their bit rate advantages against conventional periodic sampling methods, simulations are done here. One example of the system in Eq. (2) is considered, d x(t ) = ax(t ) + bcos(x(t )) + λ0 sin2 (x(t )) + u(t ) + ω(t ), (50) dt where a ∈ R and b ∈ R. So f (t ) = ax(t ) + bcos(x(t )) + λ0 sin2 (x(t )). The state estimation model or the nominal model is d xˆ(t ) = axˆ(t ) + bcos(xˆ(t )) + u(t ). (51) dt According to Eqs. (4), (50) and (51), we have d x˜e (t ) ≤ |a((x(t )) − xˆ(t ))| + |b(cos(x(t )) − cos(xˆ(t )))| + λ0 |sin2 (x(t )) − sin2 (xˆ(t ))| dt x(t ) − xˆ(t ) x(t ) + xˆ(t ) sin +|λ0 sin2 (xˆ(t ))| + |ω(t )| ≤ |a||x˜e (t )| + |b|−2sin 2 2 +λ0 |(sin(x(t )) + sin(xˆ(t )))(sin(x(t )) − sin(xˆ(t )))| + λ0 |xˆ(t )| + |ω(t )| x(t ) − xˆ(t ) + λ0 ∗ 2 ∗ 2cos x(t ) + xˆ(t ) sin x(t ) − xˆ(t ) ≤ |a||x˜e (t )| + |b| ∗ 2 2 2 2 +λ0 |xˆ(t )| + |ω(t )| ≤ |a||x˜e (t )| + |b||x˜e (t )| + 2λ0 |x˜e (t )| + λ0 |xˆ(t )| + |ω(t )| ≤ (|a| + |b| + 2λ0 )|x˜e (t )| + λ0 |xˆ(t )| + W . (52) So the Lipschitz constant λ in Eq. (7) is λ = |a| + |b| + 2λ0 .

(53)

The implemented control law is u(t ) = −Gxˆ(t ) − f (0),

(54)

G = |a| + |b| + 3λ0 + 1.

(55)

In simulations, the following parameters are taken λ∗ = 0.1, D = 0.05 s, X0 = 100.

(56)

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Fig. 2. The simulation results of a noise-free scalar continuous time nonlinear system with a = 3 and b = 2.

We will simulate the following six example systems: 1) 2) 3) 4) 5) 6)

A noise-free nonlinear system. A nonlinear system with bounded process noise. An uncertain noise-free nonlinear system. An uncertain nonlinear system with bounded process noise. A practical nonlinear system with bounded process noise. A linear system with bounded process noise.

Now we introduce the periodic sampling method in [14], which will be compared with our methods in terms of stabilizing bit rates. For the concerned scalar nonlinear system in Eq. (50), the periodic sampling in [14] requires R=

log2 eλτ λτ log2 N λ ≥ ≥ ≥ = Rλ , τ τ τ log(2) log(2)

(57)

where N is the number of quantization levels, τ is the sampling period, and λ is the Lipschitz parameter. 7.1. A nonlinear system with W = 0 and λ0 = 0 We select a = 3, b = 2. Then λ = |a| + |b| = 5, G = |a| + |b| + 1 = 6 and λ1 = λ − G = −1. The dynamics of the concerned system becomes d x(t ) = 3x(t ) + 2cos(x(t )) − 6xˆ(t ) − 2. (58) dt The state estimation error is shown in Fig. 2(a) while the state is shown in Fig. 2(b). It can be seen that the ISS in Eq. (5) is achieved. Fig. 2(c) presents Tk , which shows that Tk ≥ T1∗ = 0.2688 s, and confirms Eq. (21) predicted by Lemma 1 and Theorem 2. Therefore the required the bit rate is upper bounded by R1 = 3.7201 bits/s, which is lower than Rλ = 7.2135 bits/s, the lower bound on stabilizing bit rates under the periodic sampling method. 7.2. A nonlinear system with W ≥ 0 and λ0 = 0 We choose W = 0.2, a = 3 and b = 2. Then λ = |a| + |b| = 5, G = |a| + |b| + 1 = 6 and λ1 = λ − G = −1. According to Eqs. (32) and (33), we choose α = 30.3584. Then the

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Fig. 3. The simulation results of a scalar continuous time nonlinear system with a = 3, b = 2 and W = 0.2.

Fig. 4. The simulation results of an uncertain scalar continuous time nonlinear system with a = 3 and b = 2.

dynamics of the concerned system can be expressed as d x(t ) = 3x(t ) + 2cos(x(t )) − 6xˆ(t ) − 2 + ω(t ). dt

(59)

Fig. 3(a) represents the state estimation error. The state is shown in Fig. 3(b). It can be seen that the ISS in Eq. (5) is achieved. Fig. 3(c) presents Tk , which shows that Tk ≥ T2∗ = 0.2658 s and confirms Eq. (32) predicted by Lemma 5. The upper bound on the occupied bit rates is R2 = 3.7619 bits/s, which is much lower than Rλ = 7.2135 bits/s, the lower bound on stabilizing bit rates under the periodic sampling method. Due to the process noise, R2 > R1 . But R2 is still lower than Rλ . 7.3. An uncertain nonlinear system with W = 0 and λ0 ≥ 0 We pick a = 3, b = 2 and λ0 = 0.15. Then λ = |a| + |b| + 2λ0 = 5.3, G = |a| + |b| + 3λ0 + 1 = 6.45 and λ1 = λ + λ0 − G = −1. The dynamics of the concerned system can be described as d x(t ) = 3x(t ) + 2cos(x(t )) + 0.15sin2 (x(t )) − 6.45xˆ(t ) − 2. dt

(60)

The state estimation error is shown in Fig. 4(a) while the state is shown in Fig. 4(b), which confirms that the ISS in Eq. (5) is achieved. Fig. 4(c) presents Tk and shows that Tk ≥ T3∗ = 0.1497s, i.e., Eq. (38) predicted by Lemma 8 and Theorem 9 is confirmed. The upper bound on the occupied bit rate is R3 = 6.6818 bits/s, which is lower than Rλ = 7.6463/s, the

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Fig. 5. The simulation results an uncertain scalar continuous time nonlinear system with a = 3, b = 2 and W = 0.2.

Fig. 6. A nonlinear circuit.

lower bound on stabilizing bit rates under the periodic sampling method.2 Due to the system uncertainty, R3 is higher than R1 , but still lower than Rλ . So extra bit rate is needed to handle the uncertainty. 7.4. A nonlinear system with W ≥ 0 and λ0 ≥ 0 Here both the process noise and the system uncertainty are considered. We choose a = 3, b = 2, λ0 = 0.15 and W = 0.2. Then λ = |a| + |b| + 2λ0 = 5.3, G = |a| + |b| + 3λ0 + 1 = 6.45 and λ1 = λ − G = −1. According to Eqs. (45) and (46), we choose α = 7.6463. d x(t ) = 3x(t ) + 2cos(x(t )) + 0.15sin2 (x(t )) − 6.45xˆ(t ) − 2 + ω(t ). (61) dt Fig. 5(a) and (b) presents the state estimation error and the state, respectively. It can be seen that the ISS in Eq. (5) is guaranteed. Fig. 5(c) presents Tk , which confirms Tk ≥ T4∗ = 0.1474 s predicted by Lemma 11 and Theorem 12. The upper bound on the occupied bit rate is R4 = 6.7824 bits/s, which is lower than Rλ = 7.6463 bits/s, the lower bound on stabilizing bit rates under the periodic sampling method. 7.5. A practical nonlinear system with W ≥ 0 Here we consider a practical nonlinear system to further verify the effectiveness of the proposed event-triggering strategy. The system is shown in Fig. 6, which contains a nonlinear Josephson junction [24], a resistance R, and a current source is (t) as the input. That Josephson junction is modeled as [24] iL = I0 sin(φL (t )),

(62)

2 Note that R is derived when there is no system uncertainty. It is quite possible that the model uncertainty may λ increase the stabilizing bit rate.

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Fig. 7. The simulation results of a nonlinear circuit with R = 2, I0 = −1A and W = 10−10 A.

where I0 is a constant. The system in Fig. 6 suffers from the current variation iω (t), which can be regarded as a bounded process noise, |iω(t ) | ≤ W .

(63)

For the system in Fig. 6, we want to adjust is (t) to control the magnetic flux φ L (t) at the desired φˆL (t ) based on the proposed event-triggering strategy. The system dynamics can expressed as d φL (t ) = −RI0 sin(φL (t )) − Riω (t ) + Ris (t ). (64) dt When iω (t) ≡ 0, suppose the the desired φˆL (t ) is achieved by is (t ) = iˆs (t ). Then d ˆ φL (t ) = −RI0 sin(φˆL (t )) + Riˆs (t ). (65) dt Under iω (t) = 0, define the state as x(t ) = φL (t ) − φˆL (t ) and the input as u(t ) = is (t ) − iˆs (t ). By Eqs. (64) and (65), we can obtain d x(t ) = f (x, φˆL (t )) − Riω (t ) + Ru(t ), dt

where f (x, φˆL (t )) = RI0 sin (φˆL (t )) − sin φˆL (t ) + x(t ) and ensures f (x, φˆL (t )) ≤ |RI0 ||x(t )|,

(66)

(67)

i.e., the required global Lipschitz condition in Eq. (7) is satisfied and the proposed eventtriggering strategy in Section 4 can be implemented. For the simulations of the practical system in Fig. 6, R = 2, I0 = −1A, W = 10−10 A and φˆL (t ) = 10−7W b while λ∗ = 0.1 and D = 0.05 s as before. By means of Eqs. (32) and (33), the parameter α = 42.4867. The initial magnetic flux φ L (0) is bounded by 10−5W b. The simulation results are shown in Fig. 7. By Fig. 7(b), we see that the ISS in Eq. (5) is achieved so that the magnetic flux φ L (t) stays close to the desired φˆL (t ). Moreover, Fig. 7(c) presents Tk , which satisfies Tk ≥ T2∗ = 1.0723 s and confirms Eq. (32) predicted by Lemma 5. The upper bound on the occupied bit rates is R2 = 0.9326 bits/s, which is much lower than Rλ = 2.8854 bits/s. 7.6. A linear example with W ≥ 0 and λ0 = 0 We choose a = 3 and b = 0, under which the nonlinear system in Eq. (50) is boiled down to a linear system. For that linear system, we choose G = |a| + 1 = 4 and λ1 = λ − G = −1.

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Fig. 8. The simulation results of a scalar continuous time linear system with bounded network delay and process noise.

According to Eqs. (32) and (33), we choose α = 35.5123. Then the concerned linear system is governed by the following dynamics, d x(t ) = 3x(t ) − 4xˆ(t ) + ω(t ). (68) dt The state estimation error x˜e (t ) is shown in Fig. 8(a). The state x(t) is shown in Fig. 8(b). It can be seen the ISS in Eq. (5) is achieved. Fig. 8(c) shows that Tk and confirms Tk ≥ T2∗ = 0.6001 s, which is predicted by Lemma 5 and Theorem 6. Now we compare the stabilizing bit rate of our method with that of the periodic sampling method in [13], which requires R > R0 = 4.3281 bits/s,

(69)

The actually occupied bit rate R by our method is bounded as R ≤ R2 = 1.6664 bits/s,

(70)

which is much lower than R0 in Eq. (69). Note that the linear system in Eq. (68) can also be handled by the event-triggered method in [19]. By implementing the method in [19] with 0 = 0 and D = 0.05 s, we can obtained that the required stabilizing bit rate will be no higher than 1.4420 bits/s, which is lower than R2 in Eq. (70). This stabilizing bit rate inferiority comes from the fact that the method in this paper is initially designed for nonlinear systems. When that nonlinear method is implemented to the linear system in Eq. (68), it does not make full use of the linearity of that system and yields more conservative results than [19], which focuses on linear systems. Of course, the method in [19] is difficult to be implemented to the nonlinear example in Eq. (59). 7.7. Stabilizing bit rates under different parameters According to Theorem 12, the parameters, such as G, D, and λ0 , play a very important role in the stabilizing bit rate. So we want to investigate stabilizing bit rates under different parameters. We still consider the nonlinear system in Eq. (50) with a = 3, b = 2, λ∗ = 0.1. 7.7.1. Stabilizing bit rates under various D We pick λ0 = 0.15, then λ and G become 5.3 and 6.45, respectively. Under the periodic sampling method in [14], the stabilizing bit rate is lower bounded by Rλ = 7.2135 bits/s by Eq. (57). Rλ is constant with respect to the network delay D. Under different network delay

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Fig. 9. Stabilizing bit rates under various D, G and λ0 .

D, we compute Rλ and

1 , T4∗

and show them in Fig. 9(a). We can see that when D increases

from 0 s to 0.054 s, is lower than Rλ , i.e., our ETC method requires a lower stabilizing bit rate than the periodic sampling method in [14]. Such stabilizing bit rate saving comes from the fact that under our ETC method, extra state information is actually extracted from the receiving time instants {rk } by Eqs. (19) and (31). 1 T4∗

7.7.2. Stabilizing bit rates under various G We choose λ0 = 0.15, D = 0.05 s. Rλ = 7.2135 bits/s in [14] is still a constant under different G. As shown by Fig. 9(b), under our ETC method, the bigger G is, the lower bit rate 1 will be. This stabilizing bit rate saving comes from Eq. (44), which shows that a larger T∗ 4

T4∗ can be obtained under a larger G. When G is larger than 6.3, larger than 20,

1 T4∗

1 T4∗

≤ Rλ . When G is further

is almost constant with respect to G.

7.7.3. Stabilizing bit rates under various λ0 Here we choose D = 0.05 s and G = 20, which is guided by the analysis in Section 7.7.2. Fig. 9(c) shows when λ0 is no larger than 1.22, the stabilizing bit rate T1∗ under our method 4 is lower than Rλ under the periodic sampling method. In other words, our strategy is quite robust against the system uncertainty. 8. Conclusion This paper aims to stabilize an uncertain scalar continuous time nonlinear system with bounded process noise based on event-triggering. Such system suffers from bounded network delay, which yields the difference between the receiving time instant and the sampling time instant of each feedback packet, and limits the state information to be extract from the receiving time instants of feedback packets. We propose some event-triggered methods and derive some sufficient stabilizing bit rate conditions for the concerned nonlinear system. The obtained conditions are only up to the Lipschitz constant of the nonlinear system, the upper bound on the network delay and the system uncertainty. Simulations confirm that the proposed eventtriggered methods require lower stabilizing bit rates than some periodic sampling methods when the network delay is not too large. When there are bounded process noise and system uncertainty, a higher stabilizing bit rate is required. Furthermore, we discuss stabilizing bit rate conditions under different parameters.

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Note that the proposed strategies aim to handle scalar nonlinear systems. When general high-order nonlinear systems are considered, the coupling between multiple state variables makes it difficult to find appropriate event-triggering conditions and functions for various state variables. Such event-triggering stabilization problem of high-order nonlinear systems will be investigated in the future. Appendix A. Proof of Lemma 1 Proof. According to the global Lipschitz condition in Eq. (7), the dynamics of the state estimation error x˜e (t ), together with Eqs. (2) and (6), yields d x˜e (t ) = f (x(t )) − f (xˆ(t )) dt ≤λx˜e (t ). (A.1) Now we show sk+1 − rk ≥ T1∗ for 2 cases: 1. When x˜e (sk ) > 0: According to the triggering strategy in Eqs. (16) and (A.1), we know sign(x˜e (sk )) = θk = 1 and e−λ(rk −sk ) x˜e (sk ) ≤ x˜e rk− ≤ eλ(rk −sk ) x˜e (sk ). (A.2) Based on the updating rule in Eq. (19), we get (zk,2 − αr L(rk ))θk ≤ x˜e rk+ ≤ (zk,1 − αr L(rk ))θk . λ(rk −sk )

(A.3)

−λ(rk −sk )

where zk,1 = e L(sk ), zk,2 = e L(sk ). 2. When x˜e (sk ) < 0: We get sign(x˜e (sk )) = θk = −1 and eλ(rk −sk ) x˜e (sk ) ≤ x˜e rk− ≤ e−λ(rk −sk ) x˜e (sk ), (zk,1 − αr L(rk ))θk ≤ x˜e rk+ ≤ (zk,2 − αr L(rk ))θk ,

(A.4) (A.5)

where zk,1 and zk,2 are defined as Eq. (A.3). Due to the bound in Eqs. (9), (A.3) and (A.5) yield 1 (λ+λ∗ )D ∗ (e − e(−λ+λ )D )L(rk ). 2 From Eq. (A.1), we know, for t ∈ [rk , rk + T1∗ ), |x˜e (t )| ≤eλ(t−rk ) x˜e rk+ 1 ∗ ∗ ∗ ≤ (e(λ+λ )D − e(−λ+λ )D )e(λ+λ )(t−rk ) L(t ) 2 1 ∗ ∗ ∗ ∗ < (e(λ+λ )D − e(−λ+λ )D )e(λ+λ )T1 L(t ) 2 ≤L(t ), |x˜e (rk+ )| ≤

(A.6)

(A.7)

where the last inequality comes from the definition of T1∗ in Eq. (21). The proof is completed.

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Appendix B. Proof of Theorem 2 Proof. Following the proof of Lemma 1, we get sk+1 − rk ≥ T1∗ , which guarantees |x˜e (t )| ≤ L(t ), t ∈ [rk , sk+1 ).

(B.1)

According to Eq. (A.1), for t ∈ [sk+1 , rk+1 ), we get |x˜e (t )| ≤ eλ(t−sk+1 ) |x˜e (sk+1 )| ∗ ≤ e(λ+λ )(t−sk+1 ) L(t ) ≤ e(λ+λ

∗

)D

L(t ),

(B.2)

where the last inequality comes from rk+1 − sk+1 ≤ D. According to Eqs. (B.1) and (B.2), it is easy to derive an upper bound on x˜e (t ), |x˜e (t )| ≤ e(λ+λ

∗

)D

L(t ), ∀t ≥ 0.

(B.3)

Then we want to show the exponential convergence of xˆ(t ). The state estimation is governed by the following dynamics d xˆ(t ) = f (xˆ(t )) + u(t ) dt = f (xˆ(t )) − Gxˆ(t ) − f (0).

(B.4)

From Eq. (7), we see that | f (xˆ(t )) − f (0)| ≤ λ|xˆ(t )|.

(B.5)

When xˆ(t ) ≥ 0, −λxˆ(t ) ≤ f (xˆ(t )) − f (0) ≤ λxˆ(t ) and d xˆ(t ) ≤ (λ − G )xˆ(t ), dt When xˆ(t ) < 0, λxˆ(t ) ≤ f (xˆ(t )) − f (0) ≤ −λxˆ(t ) and (−λ − G )xˆ(t ) ≤

d xˆ(t ) ≤ (−λ − G )xˆ(t ). dt According to Eqs. (B.6) and (B.7), we derive that, for t ∈ [rk , rk+1 ), (λ − G )xˆ(t ) ≤

d xˆ(t ))/(xˆ(t )) ∈ [−λ − G, λ − G]. dt Define xˆ(t ) k (t ) = + , t ∈ [rk , rk+1 ). xˆ rk (

(B.6)

(B.7)

(B.8)

(B.9)

k (t) is bounded as e(−λ−G )(t−rk ) ≤ k (t ) ≤ e(λ−G )(t−rk ) , t ∈ [rk , rk+1 ). For t ∈ [0, r1 ), xˆ(t ) = 0. When t = xˆ r1+ = αr θ1 L(r1 ). For t ∈ [r1 , r2 ), we have xˆ(t ) =1 (t )xˆ r1+

r1+ ,

(B.10)

by the updating rule in Eq. (19), we have (B.11)

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=1 (t )αr θ1 L(r1 ).

6125

(B.12)

By repeating the above procedure, we get, for t ∈ [rk , rk+1 ), xˆ(t ) =k (t )k−1 (rk )k−2 (rk−1 ) · · · 1 (r2 )αr θ1 L(r1 ) + k (t )k−1 (rk )k−2 (rk−1 ) · · · 2 (r3 )αr θ2 L(r2 ) .. . + k (t )αr θk L(rk ).

(B.13)

Notice that k (t) ≥ 0, every symbol θ i will not change its sign. From Eqs. (13) and (B.10), we derive that |xˆ(t )| ≤|eλ1 (t−rk ) eλ1 (rk −rk−1 ) eλ1 (rk−1 −rk−2 ) · · · eλ1 (r2 −r1 ) θ1 |αr L(r1 ) + |eλ1 (t−rk ) eλ1 (rk −rk−1 ) eλ1 (rk−1 −rk−2 ) · · · eλ1 (r3 −r2 ) θ2 |αr L(r2 ) .. . + |eλ1 (t−rk ) θk |αr L(rk ).

(B.14)

According to Eq. (B.14), we have, for t ∈ [rk , rk+1 ), |xˆ(t )| ≤

k

eλ1 (t−ri ) αr L(ri )

i=1

≤

k

e ( λ1 + λ

∗

)(t−ri )

e ( λ1 + λ

∗

)(k−i)T1∗

αr L(t )

i=1

≤

k

αr L(t )

i=1

≤

1 1 − e ( λ1 + λ

∗ )T ∗ 1

αr L(t ),

(B.15)

where the third inequality makes use of sk+1 − rk ≥ T1∗ . Notice that |x(t )| ≤ |x˜e (t )| + |xˆ(t )|. Considering the exponential convergence of L(t) in Eq. (14), we combine Eqs. (B.3) and (B.15) to yield |x(t )| ≤ e(λ+λ

∗

)D

L(t ) +

1 1−

∗ ∗ e(λ1 +λ )T1

αr L(t ),

∀t ≥ 0.

The above equation guarantees the desired ISS in Eq. (5).

(B.16)

Appendix C. Proof of Corollary 4 Proof. Its proof is close to that of Theorem 2. Their difference lies in the difference of the globally Lipschitz condition and the locally Lipschitz condition. Under the globally Lipschitz condition in Eqs. (7), (B.15) and (B.16) ensure that, ∀t ≥ 0,

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|x(t )| ≤ e(λ+λ

∗

≤ e(λ+λ

∗

)D

)D

L(t ) + X0 +

1 1−

∗ ∗ e(λ1 +λ )T1

1 1−

∗ ∗ e(λ1 +λ )T1

αr L(t )

αr X0

≤ B1 ,

(C.1)

and |xˆ(t )| ≤

1− ≤ B1 .

1 ∗ ∗ e(λ1 +λ )T1

αr L(t ) (C.2)

The parameters that X0 , α r , λ, λ∗ , λ1 , D, T1∗ are known and can determine B1 , which satisfies ∗ 1 e(λ+λ )D X0 + ∗ ∗ αr X0 ≤ B1 < +∞. 1−e(λ1 +λ )T1 Although Eqs. (A.1) and (B.5) are built upon the globally Lipschitz condition, they can be extended to the locally Lipschitz condition. More specifically, let λ be the locally Lipschitz constant for the function f(·) in the region B1 = {x : |x| ≤ B1 }.

(C.3)

Then Eqs. (A.1) and (B.5) still hold in the region B1 under the locally Lipschitz condition. We see that x(0) ∈ B1 and xˆ(0) = 0 ∈ B1 . By repeating the process in Lemma 1 and Theorem 2, we can still guarantee both Eqs. (B.15) and (B.16), and obtain x(t ) ∈ B1 , ∀t ≥ 0.

(C.4)

Then the ISS in Eq. (5) can be similarly ensured. As t → ∞, x(t) → 0. Then the equilibrium point x = 0 of Eq. (2) is asymptotically stable. Appendix D. Proof of Lemma 5 Proof. Under the global Lipschitz condition in Eq. (7), the state estimation error x˜e (t ) is governed by d x˜e (t ) ≤ f (x(t )) + ω(t ) − f (xˆ(t ))) dt ≤ f (x(t )) − f (xˆ(t )) + ω(t ) ≤λx˜e (t ) + W . (D.1) We need to analyze x˜e (t ) for t ∈ [rk , sk+1 ). From Eq. (D.1), we obtain when x˜e (t ) ≥ 0, −λx˜e (t ) ≤ f (x(t )) − f (xˆ(t )) ≤ λx˜e (t ),

(D.2)

d x˜e (t ) ≤ λx˜e (t ) + W . dt

(D.3)

λx˜e (t ) ≤ f (x(t )) − f (xˆ(t )) ≤ −λx˜e (t ),

(D.4)

−λx˜e (t ) − W ≤ when x˜e (t ) < 0,

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λx˜e (t ) − W ≤

d x˜e (t ) ≤ −λx˜e (t ) + W . dt

6127

(D.5)

From Eqs. (29) and (D.1), together with Eqs. (D.3) and (D.5), we can derive that when x˜e (sk ) > 0, i.e., sign(x˜e (sk )) = θk = 1,

rk

rk pk2 θk − eλ(rk −τ )W dτ ≤ x˜e (rk− ) ≤ pk1 θk + eλ(rk −τ )W dτ, (D.6) sk

sk

where pk1 = eλ(rk −sk ) LW (sk ), pk2 = e−λ(rk −sk ) LW (sk ). By the updating policy in Eq. (31),

rk

rk λ(rk −τ ) + μk2 θk − e W dτ ≤ x˜e (rk ) ≤ μk1 θk + eλ(rk −τ )W dτ,

(D.7)

where μk1 = eλ(rk −sk ) LW (sk ) − αr LW (rk ), μk2 = e−λ(rk −sk ) LW (sk ) − αr LW (rk ). When x˜e (sk )<0, i.e., sign(x˜e (sk )) = θk = −1, we can similarly obtain

rk

rk pk1 θk − eλ(rk −τ )W dτ ≤ x˜e (rk− ) ≤ pk2 θk + eλ(rk −τ )W dτ,

(D.8)

sk

sk

sk

μk1 θk −

sk

rk sk

eλ(rk −τ )W dτ ≤ x˜e (rk+ ) ≤ μk2 θk +

rk

eλ(rk −τ )W dτ.

(D.9)

sk

Then we want to derive an upper bound of |x˜e (rk+ )|. First of all, we need to derive upper bounds on μk1 and μk2 . According to rk − sk ≤ D, we have |μk1 | =(eλ(rk −sk ) LW (sk ) − αr LW (rk )) =eλ(rk −sk ) L(sk ) − αr L(rk ) + eλ(rk −sk ) αW − αr ∗ αW =eλ(rk −sk ) L(sk ) − αr L(rk ) + eλ(rk −sk ) αW − αr ∗ αW 1 ∗ ∗ ≤ (e(λ+λ )D − e(−λ+λ )D )L(rk ) + |eλD − αr |αW 2 =μ¯ k1 , (D.10) |μk2 | =(e−λ(rk −sk ) LW (sk ) − αr LW (rk )) =e−λ(rk −sk ) L(sk ) − αr L(rk ) + e−λ(rk −sk ) αW − αr ∗ αW =e−λ(rk −sk ) L(sk ) − αr L(rk ) + e−λ(rk −sk ) αW − αr ∗ αW 1 ∗ ∗ ≤ (e(λ+λ )D − e(−λ+λ )D )L(rk ) + |e−λD − αr |αW 2 =μ¯ k2 . Due to

(D.11)

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1 ∗ |eλD − αr | =eλD − (eλD + e−λD )eλ D 2 1 λD ≤e − (eλD + e−λD ) 2 1 λD ≤ (e − e−λD ), 2 and 1 ∗ |e−λD − αr | = (eλD + e−λD )eλ D − e−λD 2 1 ≥ (eλD + e−λD ) − e−λD 2 1 ≥ (eλD − e−λD ). 2 Define uk = max(μ¯ k1 , μ¯ k2 ). According to Eqs. (D.10)–(D.13), we have μ¯ k2 ≥ μ¯ k1 , uk = μ¯ k2 .

(D.12)

(D.13)

(D.14)

Then we place an upper bound on

rk + |x˜e (rk )| ≤ uk + eλ(rk −τ )W dτ

|x˜e (rk+ )|

as

sk

1 ∗ ∗ ≤ (e(λ+λ )D − e(−λ+λ )D )L(rk ) + |e−λD − αr |αW + 2

rk

eλ(rk −τ )W dτ.

(D.15)

sk

For t ∈ [rk , rk + T2∗ ], combining Eqs. (32), (33) and (D.1), due to αr − e−λD ≥ 21 (e(λ+λ ∗ ∗ ∗ ∗ e(−λ+λ )D ) and e(λ+λ )T2 ≥ eλT2 , we can obtain

t |x˜e (t )| ≤eλ(t−rk ) |x˜e (rk+ )| + W eλ(t−τ ) dτ rk

rk 1 (λ+λ∗ )D ∗ (e − e(−λ+λ )D )L(rk ) + |e−λD − αr |αW + W eλ(rk −τ ) dτ ≤eλ(t−rk ) 2 sk

t eλ(t−τ ) dτ +W

∗

)D

−

rk

1 (λ+λ∗ )D ∗ (e − e(−λ+λ )D )L(rk ) + eλ(t−rk ) |e−λD − αr |αW 2 t rk

λ(t−rk ) λ(rk −τ ) +W e e dτ + eλ(t−τ ) dτ

≤eλ(t−rk ) ×

sk

rk

t 1 (λ+λ∗ )D (λ+λ )(t−rk ) (−λ+λ∗ )D λ(t−rk ) −λD λ(t−τ ) × (e −e )L(t ) + e |e − αr |αW + W e dτ ≤e 2 sk

D+T2∗ 1 ∗ ∗ ∗ ∗ ∗ ≤e(λ+λ )T2 × (e(λ+λ )D − e(−λ+λ )D )L(t ) + eλT2 (αr − e−λD )αW + W eλτ dτ 2 0 ∗

∗

∗

∗

)T2∗

∗

≤e(λ+λ )T2 × (αr − e−λD )L(t ) + eλT2 (αr − e−λD )αW

∗ ∗ ∗ + e(λ+λ )T2 × (αr − e−λD ) − eλT2 (αr − e−λD ) αW ≤e(λ+λ

≤LW (t ).

× (αr − e−λD )LW (t )

(D.16)

R. Dou, J. Chen and Q. Ling / Journal of the Franklin Institute 356 (2019) 6106–6144

6129

It is obvious that |x˜e (t )| ≤ LW (t ), t ∈ [rk , rk + T2∗ ].

(D.17)

For t ∈ [sk+1 , rk+1 ), according to Eq. (D.1), we can get that |x˜e (t )| ≤eλ(t−sk+1 ) |x˜e (sk+1 )| +

t

eλ(t−τ )W dτ

sk+1

rk+1 −sk+1

LW (sk+1 ) + eλτ W dτ 0

D (λ+λ∗ )D ≤e LW (t ) + eλτ W dτ,

≤e

λ(t−sk+1 )

(D.18)

0 ∗

where the last inequality comes from LW (sk+1 ) ≤ LW (t )eλ (t−sk+1 ) and rk+1 − sk+1 ≤ D. With the similar method in the proof of Theorem 2, we get, for t ∈ [rk , rk+1 ), xˆ(t ) =k (t )k−1 (rk )k−2 (rk−1 ) · · · 1 (r2 )αr θ1 LW (r1 ) + k (t )k−1 (rk )k−2 (rk−1 ) · · · 2 (r3 )αr θ2 LW (r2 ) .. . + k (t )αr θk LW (rk ).

(D.19)

Therefore |xˆ(t )| is bounded as |xˆ(t )| ≤

k

eλ1 (t−ri ) αr LW (ri )

i=1

≤

k

e ( λ1 + λ

∗

)(t−ri )

e ( λ1 + λ

∗

)(k−i)T2∗

αr LW (t )

i=1

≤

k

αr LW (t )

i=1

≤

1 1−

∗ ∗ e(λ1 +λ )T2

αr LW (t ),

(D.20)

where the third inequality makes use of sk+1 − rk ≥ T2∗ . Because of |x(t )| ≤ |xˆ(t )| + |x˜e (t )| and the exponential convergence of LW (t) in Eq. (30), combining Eqs. (D.17) and (D.19) yields

D 1 ∗ |x(t )| ≤e(λ+λ )D LW (t ) + α L (t ) + eλτ W dτ r W ∗ )T ∗ ( λ + λ 1 2 1−e 0

D α ∗ r LW (t ) + ≤ e(λ+λ )D + eλτ W dτ. ∗ ∗ 1 − e(λ1 +λ )T2 0 So the ISS in Eq. (5) is guaranteed.

(D.21)

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Appendix E. Proof of Corollary 7 Proof. From the proof in Lemma 5, it is apparent to see that, ∀t ≥ 0,

D αr ∗ |x(t )| ≤ e(λ+λ )D + (t ) + eλτ W dτ L W ∗ ∗ 1 − e(λ1 +λ )T2 0

D αr ∗ ≤ e(λ+λ )D + + αW ) + eλτ W dτ (X 0 ∗ )T ∗ ( λ + λ 1 2 1−e 0 ≤B2 ,

(E.1)

and |xˆ(t )| ≤

1

1− ≤B2 .

∗ ∗ e(λ1 +λ )T2

αr LW (t )

As X0 , α r , λ, λ∗ , λ1 , D, T2∗ are known, B2 can be computed and satisfies

D 1 ∗ B2 ≥ (e(λ+λ )D + α )( X + αW ) + eλτ W dτ. r 0 ∗ ∗ 1 − e(λ1 +λ )T2 0

(E.2)

(E.3)

Again we defined a region B2 = {x : |x| ≤ B2 }.

(E.4)

We see that x(0) ∈ B2 and xˆ(0) = 0 ∈ B2 . therefore Eqs. (B.5) and (D.1) are hold in the region B2 . We see that x(0) ∈ B2 and xˆ(0) = 0 ∈ B2 . By repeating the processes in Lemma 5, we can still guarantee both Eqs. (E.1) and (E.2), and obtain x(t ) ∈ B2 , ∀t ≥ 0. Then the ISS in Eq. (5) can be similarly ensured.

(E.5)

Appendix F. Proof of Lemma 8 Proof. Considering the global Lipschitz condition in Eq. (7), according to Eqs. (2) and (6),the estimation error x˜e (t ) evolves as d d d x˜e (t ) = x(t ) − xˆ(t ) dt dt dt = | f (x(t )) − fˆ(xˆ(t ))| ≤ | f (x(t )) − f (xˆ(t ))| + |ς (xˆ(t ))| ≤ λ|x˜e (t )| + λ0 |xˆ(t )|.

(F.1)

Similar as the above discussion, we want to analyze x˜e (t ) for t ∈ [rk , sk+1 ). According to Eq. (F.1), we get that when x˜e (t ) ≥ 0, −λx˜e (t ) ≤ f (x(t )) − f (xˆ(t )) ≤ λx˜e (t ),

(F.2)

R. Dou, J. Chen and Q. Ling / Journal of the Franklin Institute 356 (2019) 6106–6144

−λ0 |xˆ(t )| ≤ ς (xˆ(t )) ≤ |ς (xˆ(t ))| ≤ λ0 |xˆ(t )|,

6131

(F.3)

and −λx˜e (t ) − λ0 |xˆ(t )| ≤

d x˜e (t ) ≤ λx˜e (t ) + λ0 |xˆ(t )|, dt

(F.4)

and when x˜e (t ) < 0, λx˜e (t ) ≤ f (x(t )) − f (xˆ(t )) ≤ −λx˜e (t ),

(F.5)

−λ0 |xˆ(t )| ≤ ς (xˆ(t )) ≤ |ς (xˆ(t ))| ≤ λ0 |xˆ(t )|,

(F.6)

and d x˜e (t ) ≤ −λx˜e (t ) + λ0 |xˆ(t )|. (F.7) dt Combining Eqs. (F.4) and (F.7), according to the triggering method in Eq. (16), we know that λx˜e (t ) − λ0 |xˆ(t )| ≤

1. When x˜e (sk ) ≥ 0: e−λ(rk −sk ) x˜e (sk ) − λ0

rk sk

eλ(rk −τ ) |xˆ(τ )|dτ ≤ x˜e (rk− ) ≤ eλ(rk −sk ) x˜e (sk ) + λ0

rk

eλ(rk −τ ) |xˆ(τ )|dτ,

sk

(F.8) On the basis of the updating policy in Eq. (19), we have

rk (zk,2 − αr L(rk ))θk − λ0 eλ(rk −τ ) |xˆ(τ )|dτ ≤ x˜e (rk+ ) ≤ (zk,1 − αr L(rk ))θk sk

rk + λ0 eλ(rk −τ ) |xˆ(τ )|dτ,

(F.9)

sk

2. When x˜e (sk ) < 0: eλ(rk −sk ) x˜e (sk ) − λ0

rk sk

eλ(rk −τ ) |xˆ(τ )|dτ ≤ x˜e (rk− ) ≤ e−λ(rk −sk ) x˜e (sk ) + λ0

rk

eλ(rk −τ ) |xˆ(τ )|dτ,

sk

(F.10)

(zk,1 − αr L(rk ))θk − λ0

rk

sk

+ λ0

eλ(rk −τ ) |xˆ(τ )|dτ ≤ x˜e (rk+ ) ≤ (zk,2 − αr L(rk ))θk rk

eλ(rk −τ ) |xˆ(τ )|dτ,

(F.11)

sk

where zk,1 , and zk,2 are defined as Eq. (A.3). Then we need to derive an upper bound of x˜e (rk+ ). By Eqs. (F.9) and (F.11), we know that

rk 1 ∗ ∗ |x˜e (rk+ )| ≤ (e(λ+λ )D − e(−λ+λ )D )L(rk ) + λ0 eλ(rk −τ ) |xˆ(τ )|dτ. (F.12) 2 sk

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In order to show that sk+1 − rk ≥ T3∗ , first of all, we have to analyze xˆ(t ). The state estimation is governed by the following, d xˆ(t ) = fˆ(xˆ(t )) − Gxˆ(t ) − f (0) dt = f (xˆ(t )) − ς (xˆ(t )) − Gxˆ(t ) − f (0).

(F.13)

According to Eqs. (7) and (8), we see that | f (xˆ(t )) − f (0)| ≤ λ|xˆ(t )|.

(F.14)

when xˆ(t ) ≥ 0, −λxˆ(t ) ≤ f (xˆ(t )) − f (0) ≤ λxˆ(t ),

(F.15)

−λ0 xˆ(t ) ≤ ς (xˆ(t )) ≤ λ0 xˆ(t ),

(F.16)

and −λxˆ(t ) − λ0 xˆ(t ) − Gxˆ(t ) ≤

d xˆ(t ) ≤ λxˆ(t ) + λ0 xˆ(t ) − Gxˆ(t ), dt

(F.17)

when xˆ(t ) < 0, λxˆ(t ) ≤ f (xˆ(t )) ≤ −λxˆ(t ),

(F.18)

λ0 xˆ(t ) ≤ ς (xˆ(t )) ≤ −λ0 xˆ(t ),

(F.19)

and d xˆ(t ) ≤ −λxˆ(t ) − λ0 xˆ(t ) − Gxˆ(t ). dt Combining Eqs. (F.17) and (F.20), we have that for t ∈ [rk , rk+1 ),

λxˆ(t ) + λ0 xˆ(t ) − Gxˆ(t ) ≤

d xˆ(t )/(xˆ(t )) ∈ [−λ − λ0 − G, λ + λ0 − G]. dt Define k (t ) =

xˆ(t ) , t ∈ [rk , rk+1 ), xˆ(rk+ )

(F.20)

(F.21)

(F.22)

where k (t) satisfies e(−λ−λ0 −G )(t−rk ) ≤ k (t ) ≤ e(λ+λ0 −G )(t−rk ) . For t ∈ [0, r1 ), xˆ(t ) = 0. When t = xˆ(r1+ ) = αr θ1 L(r1 ).

r1+ ,

(F.23)

by the updating rule in Eq. (19), we have (F.24)

For t ∈ [r1 , r2 ), we have xˆ(t ) =1 (t )xˆ(r1+ ) =1 (t )αr θ1 L(r1 ).

(F.25)

R. Dou, J. Chen and Q. Ling / Journal of the Franklin Institute 356 (2019) 6106–6144

6133

+ By repeating the above procedure, we get, for t ∈ [rk−1 , rk ),

xˆ(t ) =k−1 (t )k−2 (rk−1 )k−3 (rk−2 ) · · · 1 (r2 )αr θ1 L(r1 ) + k−1 (t )k−2 (rk−1 )k−3 (rk−2 ) · · · 2 (r3 )αr θ2 L(r2 ) .. . + k−1 (t )αr θk−1 L(rk−1 ).

(F.26)

Note that k (t) ≥ 0, every symbol θ i will not change its sign. From Eqs. (13) and (F.23), we derive that |xˆ(t )| ≤ |eλ1 (t−rk−1 ) eλ1 (rk−1 −rk−2 ) eλ1 (rk−2 −rk−3 ) · · · eλ1 (r2 −r1 ) θ1 |αr L(r1 ) + |eλ1 (t−rk−1 ) eλ1 (rk−1 −rk−2 ) eλ1 (rk−2 −rk−3 ) · · · eλ1 (r3 −r2 ) θ2 |αr L(r2 ) .. . + |eλ1 (t−rk−1 ) θk−1 |αr L(rk−1 ) ≤

k−1

eλ1 (t−ri ) αr L(ri ).

(F.27)

i=1

where λ1 = λ + λ0−G . For t ∈[rk , sk+1 ], according to Eq. (F.27), we have that |xˆ(t )| ≤

k−1

eλ1 (t−ri ) αr L(ri ) + eλ1 (t−rk ) |αr θk L(rk )|.

(F.28)

i=1

According to Eqs. (F.12) and (F.28), for t ∈ [rk , sk+1 ), we derive that

t λ(t−rk ) + |x˜e (t )| ≤e |x˜e (rk )| + λ0 e(λ(t−τ )) |xˆ(τ )|dτ rk

rk

t λ(t−rk ) 1 (λ+λ∗ )D (−λ+λ∗ )D λ(rk −τ ) (e −e )L(rk ) + λ0 e |xˆ(τ )|dτ + λ0 eλ(t−τ ) |xˆ(τ )|dτ ≤e 2 sk rk

t 1 ∗ ∗ ≤eλ(t−rk ) × (e(λ+λ )D − e(−λ+λ )D )L(rk ) + λ0 eλ(t−τ ) |xˆ(τ )|dτ 2 sk

t k−1 1 ∗ ∗ ∗ ≤e(λ+λ )(t−rk ) (e(λ+λ )D − e(−λ+λ )D )L(t ) + λ0 eλ(t−τ ) eλ1 (τ −ri ) αr L(ri )dτ 2 sk i=1

t eλ(t−τ ) eλ1 (τ −rk ) |αr L(rk )|dτ + λ0 rk

≤m0 (t ) + m1 (t ) + m2 (t ),

(F.29)

where m0 (t ) = e(λ+λ

m1 (t ) = λ0

∗

)(t−rk ) 1

2 t

sk

eλ(t−τ )

(e(λ+λ k−1 i=1

∗

)D

− e(−λ+λ

∗

)D

)L(t ),

eλ1 (τ −ri ) αr L(ri )dτ,

(F.30)

(F.31)

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R. Dou, J. Chen and Q. Ling / Journal of the Franklin Institute 356 (2019) 6106–6144

m2 (t ) = λ0

t

eλ(t−τ ) eλ1 (τ −rk ) |αr θk L(rk )|dτ.

(F.32)

rk

With some mathematical operations, we obtain that

t k−1 λ1 (−ri ) λt m1 (t ) =λ0 αr e e e(λ0 −G )τ L(ri )dτ sk

i=1

= = = =

λ0 αr λ0 − G

k−1

λ0 αr λ0 − G

k−1

λ0 αr λ0 − G

k−1

λ0 αr G − λ0

k−1

eλ1 (−ri ) eλt (e(λ0 −G )t − e(λ0 −G )sk )L(ri )

i=1

(eλ1 (t−ri ) − eλ(t−sk ) eλ1 (sk −ri ) )L(ri )

i=1

eλ1 (sk −ri ) (eλ1 (t−sk ) − eλ(t−sk ) )L(ri )

i=1

eλ1 (sk −ri ) (eλ(t−sk ) − eλ1 (t−sk ) )L(ri )

i=1

λ0 ∗ ∗ ∗ αr L(t ) e(λ1 +λ )(sk −ri ) (e(λ+λ )(t−sk ) − e(λ1 +λ )(t−sk ) ), G − λ0 i=1 k−1

=

where the last equality comes from L(ri ) = L(t )eλ From Eq. (F.32), we see

t m2 (t ) =λ0 αr L(rk )eλ1 (−rk ) eλt e(λ+λ0 −G−λ)τ dτ =λ0 αr L(rk )eλ1 (−rk ) eλt

rk t

∗

((t−sk )+(sk −ri ))

(F.33)

.

e(λ0 −G )τ dτ

rk

λ0 αr L(rk )eλ1 (−rk ) eλt (e(λ0 −G )t − e(λ0 −G )rk ) λ0 − G λ0 ∗ ∗ = αr L(t )(e(λ+λ )(t−rk ) − e(λ1 +λ )(t−rk ) ), G − λ0 =

(F.34) ∗

where the last equality makes use that L(rk ) = L(t )eλ (t−rk ) . For t ∈ [r1 , r1 + T3∗ ] with k = 1, we have m1 (t ) = 0. Considering Eqs. (38) and (F.29), we derive that

t ∗ |x˜e (t )| ≤eλ(t−r1 ) |(e(λ+λ )(r1 −s1 ) θ1 L(r1 ) − αr θ1 L(r1 ))| + λ0 eλ(t−τ ) |αr θ1 L(r1 )|eλ1 (τ −r1 ) dτ r1

t 1 ∗ ∗ eλ(t−τ ) |αr θ1 L(r1 )|eλ1 (τ −r1 ) dτ ≤eλ(t−r1 ) (e(λ+λ )D − e(−λ+λ )D )L(r1 ) + λ0 2 r1 λ0 1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ≤e(λ+λ )T3 × (e(λ+λ )D − e(−λ+λ )D )L(t ) + αr L(t )(e(λ+λ )T3 − e(λ1 +λ )T3 ) 2 G − λ0 ≤(φ1 + φ3 )L(t ) ≤ L(t ). (F.35)

R. Dou, J. Chen and Q. Ling / Journal of the Franklin Institute 356 (2019) 6106–6144

6135

So s2 − r1 ≥ T3∗ . Then we need to derive that sk+1 − rk ≥ T3∗ by mathematical induction. Suppose that sk − rk−1 ≥ T3∗ , we have sk − ri ≥(k − i)T3∗ ,

(F.36)

and k−1

e

(λ1 +λ∗ )(sk −ri )

≤

k−1

i=1

e

(λ1 +λ∗ )(k−i)T3∗

i=1

≤

e ( λ1 + λ

∗

)T3∗

1 − e ( λ1 + λ

∗ )T ∗ 3

.

(F.37)

where the inequalities make use that λ1 + λ∗ < 0. Therefore we can obtain that ∗

m1 (t ) ≤

∗

e(λ1 +λ )T3 λ0 ∗ ∗ αr (e(λ+λ )(t−sk ) − e(λ1 +λ )(t−sk ) )L(t ). ∗ )T ∗ ( λ + λ G − λ0 1 − e 1 3

(F.38)

According to Eqs. (F.29), (F.34) and (F.38), together with Eq. (38), we have that for t ∈ [rk , rk + T3∗ ], |x˜e (t )| ≤ e

(λ+λ∗ )(t−rk ) 1

2 −e(λ1 +λ

∗

∗

(e

(λ+λ∗ )D

)(t−sk )

−e

)L(t ) +

(−λ+λ∗ )D

∗

λ0 e(λ1 +λ )T3 ∗ )L(t ) + αr (e(λ+λ )(t−sk ) G − λ0 1 − e(λ1 +λ∗ )T3∗

λ0 ∗ ∗ αr (e(λ+λ )(t−rk ) − e(λ1 +λ )(t−rk ) )L(t ) G − λ0 ∗

∗

λ0 e(λ1 +λ )T3 1 ∗ ∗ ∗ ∗ ≤e × (e(λ+λ )D − e(−λ+λ )D )L(t ) + αr (e(λ+λ )(D+T3 ) 2 G − λ0 1 − e(λ1 +λ∗ )T3∗ λ0 ∗ ∗ ∗ ∗ ∗ ∗ −e(λ1 +λ )(D+T3 ) )L(t ) + αr (e(λ+λ )T3 − e(λ1 +λ )T3 )L(t ) G − λ0 ≤ (φ1 + φ2 + φ3 )L(t ) ≤ L(t ), (F.39) (λ+λ∗ )T3∗

which yields that sk+1 − rk ≥ T3∗ , ∀k ≥ 1.

(F.40)

From Eq. (F.40), we know that for t ∈ [rk , sk+1 ), |x˜e (t )| ≤ L(t ).

(F.41)

when t ∈ [sk+1 , rk+1 ), according to Eq. (F.1), we have that |x˜e (t )| ≤ n1 (t ) + n2 (t ), where n1 (t ) = eλ(t−sk+1 ) |x˜e (sk+1 )|,

n2 (t ) = λ0

t

eλ(t−τ ) |xˆ(τ )|dτ.

(F.42)

(F.43)

sk+1

Because rk+1 − sk+1 ≤ D, n1 (t ) ≤ e(λ+λ

∗

)(t−sk+1 )

L(t ) ≤ϕ1 L(t ).

(F.44)

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where ϕ1 = e(λ+λ )D . Considering that λ1 + λ∗ < 0, sk+1 − ri ≥ (k + 1 − i)T3∗ , and rk+1 − sk+1 ≤ D, together with Eq. (F.27), we have, for t ∈ [sk+1 , rk+1 ),

t k n2 (t ) ≤λ0 eλ(t−τ ) eλ1 (τ −ri ) αr L(ri )dτ sk+1

≤

λ0 αr G − λ0

i=1 k

e ( λ1 + λ

∗

)(sk+1 −ri )

(e(λ+λ

∗

)(t−sk+1 )

− e ( λ1 + λ

∗

)(t−sk+1 )

)L(t )

i=1 ∗

∗

e(λ1 +λ )T3 λ0 ∗ ∗ ≤ αr (e(λ+λ )(t−sk+1 ) − e(λ1 +λ )(t−sk+1 ) )L(t ) G − λ0 1 − e(λ1 +λ∗ )T3∗ ≤ϕ2 L(t ). ∗ ∗ e(λ1 +λ )T3 ( λ1 + λ∗ ) T ∗ 3 1−e

λ0 where ϕ2 = G−λ αr 0 Therefor we obtain that

(e(λ+λ

∗

)D

− e ( λ1 + λ

∗

)D

(F.45)

).

|x˜e (t )| ≤ ϕL(t ), ∀t ≥ 0.

(F.46)

where ϕ = ϕ1 + ϕ2 . Now we want place an upper bound on |xˆ(t )| for t ∈ [rk , rk+1 ). Considering t − ri ≥ (k − i)T3∗ and λ1 + λ∗ < 0, from Eq. (F.27), we have |xˆ(t )| ≤

k

eλ1 (t−ri ) αr L(ri )

i=1

≤

k

∗

eλ1 (k−i)T3 αr L(ri )

i=1

≤

k

e ( λ1 + λ

∗

)(k−i)T3∗

αr L(t )

i=1

≤ηL(t ).

(F.47)

1 where η = ∗ ∗ αr . 1−e(λ1 +λ )T3 Note that |x(t )| ≤ |x˜e (t )| + |xˆ(t )|, so we have

|x(t )| ≤ |x˜e (t )| + |xˆ(t )| ≤ (ϕ + η)L(t ), which indicates the ISS in Eq. (5).

(F.48)

Appendix G. Proof of Corollary 10 Proof. According to the Lemma 8, it is obvious that |x˜e (t )| ≤ ϕL(t ) ≤ B3 ,

(G.1)

and |xˆ(t )| ≤ B3 .

(G.2)

If X0 , α r , λ, λ0 , λ∗ , λ1 , D, T3∗ , are known and can determine B3 , which satisfies Eq. (42). We define a region B3 = {x : |x| ≤ B3 }.

(G.3)

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6137

We see that x(0) ∈ B3 and xˆ(0) = 0 ∈ B3 . Moreover, Eqs. (F.1) and (F.14) are still hold in the region B3 . By repeating the processes in Lemma 8, we can still guarantee both Eqs. (G.1) and (G.2), and obtain x(t ) ∈ B3 , ∀t ≥ 0,

(G.4)

which implies the ISS in Eq. (5).

Appendix H. Proof of Lemma 11 Proof. According to the global Lipschitz condition in Eq. (7), together with Eqs. (2) and (6), the estimation error x˜e (t ) evolves as d d d x˜e (t ) = x(t ) − xˆ(t ) dt dt dt = | f (x(t )) − fˆ(xˆ(t ))| ≤ | f (x(t )) − f (xˆ(t ))| + |ς (xˆ(t ))| + |ω(t )| ≤ λ|x˜e (t )| + λ0 |xˆ(t )| + |ω(t )|.

(H.1)

Similar as the above discussion, we want to analyze x˜e (t ) for t ∈ [rk , sk+1 ). According to Eq. (H.1), we get that when x˜e (t ) ≥ 0, −λx˜e (t ) ≤ f (x(t )) − f (xˆ(t )) ≤ λx˜e (t ),

(H.2)

−λ0 |xˆ(t )| ≤ ς (xˆ(t )) ≤ |ς (xˆ(t ))| ≤ λ0 |xˆ(t )|,

(H.3)

−W ≤ ω(t ) ≤ W ,

(H.4)

and −λx˜e (t ) − λ0 |xˆ(t )| − W ≤

d x˜e (t ) ≤ λx˜e (t ) + λ0 |xˆ(t )| + W , dt

(H.5)

and when x˜e (t ) < 0, λx˜e (t ) ≤ f (x(t )) − f (xˆ(t )) ≤ −λx˜e (t ),

(H.6)

−λ0 |xˆ(t )| ≤ ς (xˆ(t )) ≤ |ς (xˆ(t ))| ≤ λ0 |xˆ(t )|,

(H.7)

−W ≤ ω(t ) ≤ W ,

(H.8)

and d x˜e (t ) ≤ −λx˜e (t ) + λ0 |xˆ(t )| + W . (H.9) dt Combining Eqs. (H.5) and (H.9), according to the triggering method in Eq. (29), we know that λx˜e (t ) − λ0 |xˆ(t )| − W ≤

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1. When x˜e (sk ) ≥ 0:

rk −λ(rk −sk ) e x˜e (sk ) − λ0 eλ(rk −τ ) |xˆ(τ )|dτ − qk ≤ x˜e (rk− ) ≤ eλ(rk −sk ) x˜e (sk ) sk

rk + λ0 eλ(rk −τ ) |xˆ(τ )|dτ + qk ,

(H.10)

sk

On the basis of the updating policy in Eq. (31), we have

rk (pk,2 − αr LW (rk ))θk − λ0 eλ(rk −τ ) |xˆ(τ )|dτ − qk ≤ xe (rk+ ) ≤ (pk,1 − αr LW (rk ))θk sk

rk + λ0 eλ(rk −τ ) |xˆ(τ )|dτ + qk ,

(H.11)

sk

2. When x˜e (sk ) < 0:

rk λ(rk −sk ) e x˜e (sk ) − λ0 eλ(rk −τ ) |xˆ(τ )|dτ − qk ≤ x˜e (rk− ) ≤ e−λ(rk −sk ) x˜e (sk ) sk

rk + λ0 eλ(rk −τ ) |xˆ(τ )|dτ + qk ,

(H.12)

sk

rk

(pk,1 − αr LW (rk ))θk − λ0

sk

+ λ0

eλ(rk −τ ) |xˆ(τ )|dτ − qk ≤ x˜e (rk+ ) ≤ (pk,2 − αr LW (rk ))θk rk

eλ(rk −τ ) |xˆ(τ )|dτ + qk ,

(H.13)

sk

r where pk,1 , pk,2 are defined as Eq. (D.6), and qk = skk eλ(rk −τ )W dτ . Then we need to derive an upper bound of x˜e (rk+ ). According to Eqs. (D.10)–(D.13), together with Eqs. (H.11) and (H.13), we know that

rk 1 ∗ ∗ |x˜e (rk+ )| ≤ (e(λ+λ )D − e(−λ+λ )D )L(rk ) + |e−λD − αr |αW + λ0 eλ(rk −τ ) |xˆ(τ )|dτ + qk . 2 sk (H.14) With the similar method in the proof of Lemma 8, we get, for t ∈ [rk−1 , rk ), xˆ(t ) =k−1 (t )k−2 (rk−1 )k−3 (rk−2 ) · · · 1 (r2 )αr θ1 LW (r1 ) + k−1 (t )k−2 (rk−1 )k−3 (rk−2 ) · · · 2 (r3 )αr θ2 LW (r2 ) .. . + k−1 (t )αr θk−1 LW (rk−1 ). Therefore we derive that |xˆ(t )| ≤|eλ1 (t−rk−1 ) eλ1 (rk−1 −rk−2 ) eλ1 (rk−2 −rk−3 ) · · · eλ1 (r2 −r1 ) θ1 |αr LW (r1 ) + |eλ1 (t−rk−1 ) eλ1 (rk−1 −rk−2 ) eλ1 (rk−2 −rk−3 ) · · · eλ1 (r3 −r2 ) θ2 |αr LW (r2 ) .. . + |eλ1 (t−rk−1 ) θk−1 |αr LW (rk−1 )

(H.15)

R. Dou, J. Chen and Q. Ling / Journal of the Franklin Institute 356 (2019) 6106–6144

≤

k−1

eλ1 (t−ri ) αr LW (ri ).

6139

(H.16)

i=1

where λ1 = λ + λ0 − G. For t ∈ [rk , sk+1 ], according to Eq. (H.16), we have that |xˆ(t )| ≤

k−1

eλ1 (t−ri ) αr LW (ri ) + eλ1 (t−rk ) |αr θk LW (rk )|.

(H.17)

i=1

Combining Eqs. (H.1), (H.14), and (H.17), for t ∈ [rk , sk+1 ), we derive that

t

t |x˜e (t )| ≤eλ(t−rk ) |x˜e (rk+ )| + λ0 eλ(t−τ ) |xˆ(τ )|dτ + eλ(t−τ )W dτ rk rk

rk

λ(t−rk ) 1 (λ+λ∗ )D (−λ+λ∗ )D −λD ≤e (e −e )L(rk ) + |e − αr |αW + λ0 eλ(rk −τ ) |xˆ(τ )|dτ + qk 2 sk

t

t + λ0 eλ(t−τ ) |xˆ(τ )|dτ + eλ(t−τ )W dτ rk

rk

1 ∗ ∗ ≤eλ(t−rk ) × (e(λ+λ )D − e(−λ+λ )D )L(rk ) + eλ(t−rk ) |e−λD − αr |αW 2

t

t + λ0 eλ(t−τ ) |xˆ(τ )|dτ + eλ(t−τ )W dτ sk

sk

t 1 ∗ ∗ ∗ ≤e(λ+λ )(t−rk ) (e(λ+λ )D − e(−λ+λ )D )L(t ) + eλ(t−rk ) |e−λD − αr |αW + eλ(t−τ )W dτ 2 sk

t

t k−1 + λ0 eλ(t−τ ) eλ1 (τ −ri ) αr LW (ri )dτ + λ0 eλ(t−τ ) eλ1 (τ −rk ) |αr LW (rk )|dτ sk

≤e

(λ+λ∗ )(t−rk ) 1

2

+ λ0

rk

i=1

t

e

λ(t−τ )

sk

+ λ0 αW

(e

(λ+λ∗ )D k−1

e

−e

(−λ+λ∗ )D

λ1 (τ −ri )

sk

eλ(t−τ )

|e

−λD

− αr |αW +

t

αr L(ri )dτ + λ0

eλ(t−τ ) eλ1 (τ −rk ) |αr L(rk )|dτ

k−1

eλ1 (τ −ri ) αr dτ + λ0 αW

i=1

t

eλ(t−τ ) eλ1 (τ −rk ) αr dτ

rk

where m0 (t), m1 (t), m2 (t) are defined in Eqs. (F.30)–(F.32),

t k−1 m3 (t ) = λ0 αW eλ(t−τ ) eλ1 (τ −ri ) αr dτ,

m4 (t ) = λ0 αW

t

eλ(t−τ )W dτ

rk

≤m0 (t ) + m1 (t ) + m2 (t ) + m3 (t ) + m4 (t ) + m5 (t ) + m6 (t ),

sk

t sk

i=1 t

)L(t ) + e

λ(t−rk )

(H.18)

(H.19)

i=1

eλ(t−τ ) eλ1 (τ −rk ) αr dτ,

(H.20)

rk

m5 (t ) = eλ(t−rk ) |e−λD − αr |αW ,

(H.21)

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R. Dou, J. Chen and Q. Ling / Journal of the Franklin Institute 356 (2019) 6106–6144

t

m6 (t ) =

eλ(t−τ )W dτ.

(H.22)

sk

With some mathematical operations, we obtain that

t k−1 λ1 (−ri ) λt m3 (t ) =λ0 αr e e e(λ0 −G )τ αW dτ sk

i=1

λ0 αr eλ1 (−ri ) eλt (e(λ0 −G )t − e(λ0 −G )sk )αW λ0 − G i=1 k−1

=

λ0 = αr (eλ1 (t−ri ) − eλ(t−sk ) eλ1 (sk −ri ) )αW λ0 − G i=1 k−1

=

λ0 αr eλ1 (sk −ri ) (eλ1 (t−sk ) − eλ(t−sk ) )αW λ0 − G i=1

=

λ0 αr eλ1 (sk −ri ) (eλ(t−sk ) − eλ1 (t−sk ) )αW G − λ0 i=1

k−1

k−1

From Eq. (H.20), we see m4 (t ) =λ0 αr αW eλ1 (−rk ) eλt =λ0 αr αW eλ1 (−rk ) eλt

t

rk

t

(H.23)

e(λ+λ0 −G−λ)τ dτ e(λ0 −G )τ dτ

rk

λ0 αr αW eλ1 (−rk ) eλt (e(λ0 −G )t − e(λ0 −G )rk ) λ0 − G λ0 = αr αW (eλ(t−rk ) − eλ1 (t−rk ) ), G − λ0 =

(H.24)

For t ∈ [r1 , r1 + T4∗ ] with k = 1, we have m1 (t ) = 0 and m3 (t ) = 0. Considering Eqs. ∗ ∗ (44) and (46), together with Eq. (H.18), due to αr − e−λD ≥ 21 (e(λ+λ )D − e(−λ+λ )D ) and ∗ ∗ ∗ e(λ+λ )T4 ≥ eλT4 , we derive that

t 1 ∗ ∗ |x˜e (t )| ≤eλ(t−r1 ) (e(λ+λ )D − e(−λ+λ )D )L(r1 ) + eλ(t−rk ) |e−λD − αr |αW + eλ(t−τ )W dτ 2 s1

t

t + λ0 eλ(t−τ ) |αr θ1 L(r1 )|eλ1 (τ −r1 ) dτ + λ0 αW eλ(t−τ ) αr eλ1 (τ −r1 ) dτ r1

≤e(λ+λ

∗

r1

1 λ0 ∗ ∗ ∗ ∗ × (e(λ+λ )D − e(−λ+λ )D )L(t ) + eλT4 ∗ |e−λD − αr |αW + αr L(t )(e(λ+λ )T4 2 G − λ0

)T4∗

− e ( λ1 + λ +

∗

)T4∗

)

λ0 αr αW (eλ(t−r1 ) − eλ1 (t−r1 ) ) + G − λ0

≤e(λ+λ

∗

)T4∗

D+T4∗

eλτ W dτ

0

|e−λD − αr |L(t ) + eλT4 ∗ |e−λD − αr |αW +

λ0 ∗ ∗ ∗ ∗ αr L(t )(e(λ+λ )T4 − e(λ1 +λ )T4 ) G − λ0

R. Dou, J. Chen and Q. Ling / Journal of the Franklin Institute 356 (2019) 6106–6144

6141

D+T4∗ λ0 ∗ ∗ αr αW (eλT4 − eλ1 T4 ) + eλτ W dτ G − λ0 0 λ0 ∗ ∗ ∗ ∗ (λ+λ∗ )T4∗ −λD (αr − e )L(t ) + αr L(t )(e(λ+λ )T4 − e(λ1 +λ )T4 ) ≤e G − λ0 +

∗

+

∗

λ0 e(λ1 +λ )T4 ∗ ∗ ∗ ∗ αr (e(λ+λ )(D+T4 ) − e(λ1 +λ )(D+T4 ) )L(t ) ∗ )T ∗ ( λ + λ 1 G − λ0 1 − e 4 ∗

+ eλT4 ∗ (αr − e−λD )αW +

λ0 λ0 eλ1 T4 ∗ ∗ ∗ αr αW (eλT4 − eλ1 T4 ) + αr (eλ(D+T4 ) G − λ0 G − λ0 1 − eλ1 T4∗

∗

− eλ1 (D+T4 ) )αW + (σ1 + σ2 + σ3 − σ4 − σ5 − σ6 )αW ≤ (σ1 + σ2 + σ3 )LW (t ) ≤ LW (t ). (H.25)

Therefore s2 − r1 ≥ T4∗ . Then we need to derive that sk+1 − rk ≥ T4∗ by mathematical induction. Suppose that sk − rk−1 ≥ T4∗ , we have sk − ri ≥(k − i)T4∗ ,

(H.26)

and k−1

eλ1 (sk −ri ) ≤

i=1

k−1

∗

eλ1 T4

∗

eλ1 (k−i)T4 ≤

i=1

∗

1 − eλ1 T4

.

(H.27)

where the inequalities make use that λ1 < 0. Therefore we can obtain that ∗

eλ1 T4 λ0 m3 (t ) ≤ αr (eλ(t−sk ) − eλ1 (t−sk ) )αW . G − λ0 1 − eλ1 T4∗

(H.28)

According to Eqs. (F.30), (F.33) (F.34), (H.18), (H.21), (H.22), (H.24) and (H.28), together with Eq. (44), we have that for t ∈ [rk , rk + T4∗ ], |x˜e (t )| ≤e(λ+λ

∗

)T4∗

(αr − e−λD )L(t ) + ∗

+

λ0 ∗ ∗ ∗ ∗ αr L(t )(e(λ+λ )T4 − e(λ1 +λ )T4 ) G − λ0

∗

λ0 e(λ1 +λ )T4 ∗ ∗ ∗ ∗ αr (e(λ+λ )(D+T4 ) − e(λ1 +λ )(D+T4 ) )L(t ) + eλT4 ∗ (αr − e−λD )αW ∗ )T ∗ ( λ + λ G − λ0 1 − e 1 4 ∗

λ0 λ0 eλ1 T4 ∗ ∗ ∗ ∗ αr αW (eλT4 − eλ1 T4 ) + αr (eλ(D+T4 ) − eλ1 (D+T4 ) )αW G − λ0 G − λ0 1 − eλ1 T4∗ + (σ1 + σ2 + σ3 − σ4 − σ5 − σ6 )αW ≤(σ1 + σ2 + σ3 )LW (t ) ≤ LW (t ), (H.29) +

which yields that sk+1 − rk ≥ T4∗ , ∀k ≥ 1.

(H.30)

From Eq. (H.30), we know that for t ∈ [rk , sk+1 ), |x˜e (t )| ≤ LW (t ).

(H.31)

when t ∈ [sk+1 , rk+1 ), according to Eq. (H.1), we have that |x˜e (t )| ≤ ρ1 (t ) + ρ2 (t ) + ρ3 (t ),

(H.32)

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R. Dou, J. Chen and Q. Ling / Journal of the Franklin Institute 356 (2019) 6106–6144

where ρ1 (t ) = eλ(t−sk+1 ) |x˜e (sk+1 )|,

t

ρ2 (t ) = λ0

(H.33)

eλ(t−τ ) |xˆ(τ )|dτ,

(H.34)

eλ(t−τ )W dτ.

(H.35)

sk+1

t

ρ3 (t ) = λ0

sk+1

Because rk+1 − sk+1 ≤ D, ρ1 (t ) ≤ e(λ+λ

∗

)(t−sk+1 )

L(t ) + eλ(t−sk+1 ) αW ≤e(λ+λ

∗

)D

L(t ) + eλD αW ≤ γ1 LW (t ).

(H.36)

∗

where γ1 = e(λ+λ )D . Considering that λ1 + λ∗ < 0, sk+1 − ri ≥ (k + 1 − i)T4∗ , and rk+1 − sk+1 ≤ D, together with Eq. (H.16), we have, for t ∈ [sk+1 , rk+1 ),

ρ2 (t ) ≤λ0

t

e

λ(t−τ )

sk+1

≤

k

eλ1 (τ −ri ) αr LW (ri )dτ

i=1

λ0 αr G − λ0

k

e ( λ1 + λ

∗

)(sk+1 −ri )

(e(λ+λ

∗

)(t−sk+1 )

− e ( λ1 + λ

∗

)(t−sk+1 )

)L(t )

i=1

λ0 αr eλ1 (sk+1 −ri ) (eλ(t−sk+1 ) − eλ1 (t−sk+1 ) )αW G − λ0 i=1 k

+

∗

∗

e(λ1 +λ )T4 λ0 ∗ ∗ ≤ αr (e(λ+λ )(t−sk+1 ) − e(λ1 +λ )(t−sk+1 ) )L(t ) G − λ0 1 − e(λ1 +λ∗ )T4∗ ∗

λ0 eλ1 T4 αr (eλ(t−sk+1 ) − eλ1 (t−sk+1 ) )αW G − λ0 1 − eλ1 T4∗ ≤γ2 LW (t ), +

∗ ∗ e(λ1 +λ )T4 ( λ1 + λ∗ ) T ∗ 4 1−e

λ0 where γ2 = G−λ αr (e(λ+λ 0 ρ 3 (t) can be bounded as

t ρ3 (t ) ≤λ0 eλ(t−τ )W dτ sk+1 D

≤λ0

0

eλτ W dτ ≤ λ0

∗

D+T4∗

)D

− e ( λ1 + λ

∗

)D

(H.37)

).

eλτ W dτ ≤ σ αW .

(H.38)

0

where σ = σ1 + σ2 + σ3 − σ4 − σ5 − σ6 . Therefor we obtain that |x˜e (t )| ≤ γ LW (t ) + σ αW , ∀t ≥ 0.

(H.39)

where γ = γ1 + γ2 . Now we want place an upper bound on |xˆ(t )| for t ∈ [rk , rk+1 ). Con∗ sidering t − ri ≥ (k − i)T4∗ , LW (ri ) ≤ LW (t )eλ (t−ri ) and λ1 + λ∗ < 0, from Eq. (H.16), for t ∈ [rk , rk+1 ), we have

R. Dou, J. Chen and Q. Ling / Journal of the Franklin Institute 356 (2019) 6106–6144

|xˆ(t )| ≤

k

e ( λ1 + λ

∗

)(t−ri )

e ( λ1 + λ

∗

)(k−i)T4∗

6143

αr LW (t )

i=1

≤

k

αr LW (t )

i=1

≤ζ LW (t ).

(H.40)

1 where ζ = ∗ ∗ αr . 1−e(λ1 +λ )T4 Note that |x(t )| ≤ |x˜e (t )| + |xˆ(t )|, so we have

|x(t )| ≤ |x˜e (t )| + |xˆ(t )| ≤ (γ + ζ )LW (t ) + σ αW , which indicates the ISS in Eq. (5).

(H.41)

Appendix I. Proof of Corollary 13 Proof. According to Lemma 11, it is obvious that |x˜e (t )| ≤ γ LW (t ) + σ αW ≤ B4 ,

(I.1)

and |xˆ(t )| ≤ B4 .

(I.2)

Supposing that X0 , α r , λ, λ0 , λ∗ , λ1 , D, T4∗ , are known and can determine B4 , which satisfies Eq. (48). We define a region B4 = {x : |x| ≤ B4 }.

(I.3)

We see that x(0) ∈ B4 and xˆ(0) = 0 ∈ B4 . Moreover, Eqs. (H.1) and (F.14) are still hold in the region B4 . By repeating the processes in Lemma 11, we can still guarantee both Eqs. (I.1) and (I.2), and obtain x(t ) ∈ B4 , ∀t ≥ 0, which implies the ISS in Eq. (5).

(I.4)

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