Quantized-Feedback-Based Adaptive Event-Triggered Control of a Class of Uncertain Nonlinear Systems

: A quantized-feedback-based adaptive event-triggered tracking problem is investigated for strict-feedback nonlinear systems with unknown nonlinearities and external disturbances. All state variables are quantized through a uniform quantizer and the quantized states are only measurable for the control design. An approximation-based adaptive event-triggered control strategy using quantized states is presented. Compared with the existing recursive quantized feedback control results, the primary contributions of the proposed strategy are (1) to derive a quantized-states-based function approximation mechanism for compensating for unknown and unmatched nonlinearities and (2) to design a quantized-states-based event triggering law for the intermittent update of the control signal. A Lyapunov-based stability analysis is provided to conclude that closed-loop signals are uniformly ultimately bounded and there exists a minimum inter-event time for excluding Zeno behavior. In simulation results, it is shown that the proposed quantized-feedback-based event-triggered control law can be implemented with less than 10% of the total sample data of the existing quantized-feedback continuous control law.


Introduction
As networked control systems including digital communication channels have been successfully utilized in various industrial applications, significant research efforts have been devoted to control designs using input and state quantization [1][2][3]. Concurrently, nonlinear systems have been widely studied and significant progress has been achieved in the analysis and the control of nonlinear systems. On the other hand, recursive control techniques have attracted much attention as effective ways for dealing with the unmatched nonlinearities of lower-triangular nonlinear systems [4][5][6]. Based on these recursive control techniques, adaptive control strategies were presented for input-quantized lower-triangular nonlinear systems with unknown parameters [7,8] and completely unknown nonlinearities [9][10][11][12]. Fundamentally, the controllers designed in [9][10][11][12] assume that state variables are continuously measurable. As opposed to the previous works [7][8][9][10][11][12] considering input quantization, the control design problem of state-quantized nonlinear systems in a lower-triangular form has received limited attention. In [13], an adaptive quantized feedback backstepping controller using quantized state variables was presented under the assumption that the partial derivatives of recursive virtual controllers were constants. Thus, the result [13] is only usable for systems with nonlinear functions matched to a control input. In order to overcome this restriction, an adaptive quantized feedback control design methodology adopting the command filtered backstepping technique for lower-triangular nonlinear systems was recently presented in [14]. Despite this progress, two aspects still need to be addressed to realize further improvement in the quantized feedback control design [14].
where i = 1, . . . , n − 1,x j = [x 1 , . . . , x j ] ∈ R j , j = 1, . . . , n, are state variable vectors, d j are unknown time-varying disturbances satisfying |d j | ≤ d * j with unknown constants d * j > 0, u ∈ R is the control input, and f j (·) : R j → R are unknown C 1 nonlinear functions. In this paper, the control input u is an intermittently updated signal in time by an event-triggering law to be designed later. In addition, u is designed based on the quantized state variables obtained through the following uniform quantizer where i = 1, . . . , n, l ∈ Z + , δ is the length of the quantization interval, χ 1 = δ, and χ l+1 = χ l + δ. Note that q(x i ) is in a countable set Q = {0, ±χ l } and the quantization error κ x,i x i − x q i has the property |κ x,i | ≤ δ where x q i q(x i ) [17]. Assumption 1. Ref. [13] The quantized states x q i , i = 1, . . . , n, are available for feedback, instead of x i . Assumption 2. Ref. [5] The reference signal r and its time derivativesṙ andr are bounded. Lemma 1. Ref. [29] For any η > 0 and ν ∈ R, it is ensured that 0 ≤ |ν| − ν tanh(ν/η) ≤ 0.2785η.

Lemma 2.
Ref. [30] When a matrix A ∈ R n×n is Hurwitz, it is satisfied that e At ≤ β 1 e −β 2 t with β 1 = λ max (G)/λ min (G) and β 2 = 1/λ max (G). Here, G is a symmetric positive definite matrix such that A G + GA = −2I where I is an identity matrix of order n. In addition, λ max (G) and λ min (G) are the maximum and minimum eigenvalues of G, respectively. Problem 1. Consider the uncertain strict-feedback nonlinear system (1) with unknown nonlinearities and state quantizer (2). Our control problem is to provide a quantized-feedback-based event-triggered tracking law u so that the state x 1 follows the reference signal r while all the closed-loop signals remain bounded.

Remark 1.
Several real-world applications such as robot manipulations, electrical power systems, and aircraft systems can be modeled as system (1) [4,5]. Recent advances in the network technology enable the control of these systems over a network with limited communication resources. Then, the proposed theoretical result can be applied to these network-based practical control problems.

Radial Basis Function Neural Networks
According to the universal approximation property of radial basis function neural networks (RBFNNs) [31], if the number of neural nodes N is sufficiently large and the basis functions s i , i = 1, . . . , N, are appropriately chosen, there exists an ideal bounded weight vector W * ∈ R N , that satisfies W * ≤W with a constantW, such that where f ( ) : Ω → R is an unknown function; Ω ⊂ R M is a compact set, = [ 1 , . . . , M ] is an input vector with M elements, ε represents an approximation reconstruction error satisfying |ε| ≤ ε * with a constant ε * > 0, and S( ) = [s 1 ( ), . . . , s N ( )] ∈ R N is a basis function vector. In this study, s i ( ) are chosen as Gaussian functions s i ( ) = e − −c i 2 /ϕ 2 where i = 1, . . . , N, c i = [c i1 , . . . , c iM ] ∈ R M is the center of the receptive field and ϕ is the width of the Gaussian functions. Note that Gaussian basis function vector is bounded as S( ) ≤ S * where S * is a constant [32,33].
Step 1: Consider the first error surface µ 1 . From (1) and (4), we haveμ 1 Then, employing an RBFNN to estimate the unknown function f 1 , the time derivative of V 1 is given bẏ where W * 1 is an optimal weight, S 1 (x 1 ) denotes a basis function vector, and ε 1 is the reconstruction error for estimating f 1 .
The virtual control law α 1 is designed as where k 1 > 0 is a control gain,Ŵ 1 is the estimate of W * 1 ,b 1 are the estimate of an unknown constant b * 1 to be defined later, and tanh 1 = tanh(µ 1 /η 1 ); η 1 > 0 is a design parameter. Applying (7) into (6), we havė Step j (j = 2, . . . , n − 1): From (5), we haveα j,1 =α j,2 . Thus, the time derivative of V j = (1/2)µ 2 j is obtained asV where W * j is an optimal weighting vector, S j (x j ) is a basis function vector, and ε j is the reconstruction error for approximating f j . Now, we choose the virtual control law α j as follows: where k j > 0 is a control gain andŴ j is the estimate of W * j , andb j is the estimate of b * j to be defined later, and tanh j = tanh(µ j /η j ); η j > 0 is a design parameter.
Step n: Consider a Lyapunov function candidate V n = (1/2)µ 2 n . Similar to the previous steps, the time derivative of V n satisfieṡ where W * n is an optimal weight, S n is a basis function vector, and ε n is the reconstruction error. In order to design an actual control law u based on the quantized states, quantized-states-based error surfaces µ q i , virtual control laws α q j , and adaptation laws forŴ j andb j are defined as follows: where i = 1, . . . , n, j = 1, . . . , n − 1, S q j = S j (x withα q j,1 (0) = α q j (0) andα q j,2 (0) = 0. Then, a quantized-feedback-based adaptive event-triggered actual control law u with a triggering law is presented as where u e (t) = u(t) −α q n,1 (t), l ∈ Z + , t 1 = 0, t l denotes the lth event time, θ 1 , θ 2 > 0 are design parameters for the triggering law (19),Ŵ n is the estimate of W * n , andb n is the estimate of unknown constant b * n to be defined later, tanh q n = tanh(µ q n /η n ); η n > 0 is a design constant, S q n = S n (x q n ); , k n > 0 is a control gain, γ w,n > 0 and γ b,n > 0 are tuning gains, and σ w,n > 0 and σ b,n > 0 are small constants for σ−modification. Here,α q n,1 in u e is a filtered signal of α q n given bẏα where ω n and ζ n are filter design parameters,α q n,1 (0) = α q n (0), andα q n,2 (0) = 0. Note that the actual input u is fixed as a constant valueα q n,1 (t l ) until the next event occurs at t l+1 and each event time is determined by checking the condition in (19). The block diagram of the proposed control scheme consisting of (14)-(23) is shown in Figure 1. Define an ideal control signal α n as and its filtered signalsα n,1 andα n,2 obtained from the following filteṙα whereα n,1 (0) = α n (0) andα n,2 (0) = 0. Note that the following property holds.

Remark 3.
In the proposed triggering law (19), the adaptive terms depend on the time-varying error surface recursively. In addition, α q n obtained from (20) is employed in u e of (19). Therefore, it concludes that the triggering law (19) depends on the information of all adaptation parametersŴ i andb i where i = 1, . . . , n.
Owing to ζ i > 0 and ω i > 0, A i are Hurwitz matrices. Then, for any matrix For the stability analysis of the closed-loop system, three lemmas (i.e., Lemmas 3-5) are presented. Lemmas 3 and 4 give the boundedness of the estimation errorsW i andb i , respectively, where i = 1, . . . , n. In Lemma 5, we show that the errors between the quantized signals µ (15) and (21). Then, there exists a compact set

Proof. Let us consider a Lyapunov function candidate
Here, each term can be represented by −σ w,i |µ Here, since the optimal weights W * i and the basis function vectors S i are bounded, there exist constantsW i and S * i satisfying W * i ≤W i and S q i ≤ S * i , respectively. Based on these facts,V w,i satisfieṡ From this inequality, we have thatV w, ∈ Ω w,i andW i finally remains within Ω w,i . Consequently, ifW i (0) ∈ Ω w,i ,W i (t) ∈ Ω w,i for all t ≥ 0 which completes the proof. (16) and (22). Then, there exists a compact set

Lemma 4. Consider the adaptation laws
Proof. Similar to the proof of Lemma 3, a Lyapunov function candidate Let
(iii) According to the similar recursive derivation procedure, it holds that κ µ,i , κ α,i , and κα ,i , i = 3, . . . , n, are bounded as This completes the proof of Lemma 5.
Choose a Lyapunov function candidate V as Theorem 1. Consider the uncertain strict-feedback nonlinear system (1) with the uniform state quantizer (2). Then, for any initial conditions satisfying V(0) ≤ ς, the quantized-feedback-based adaptive event-triggered tracker consisting of the command filters (17) and (23), the virtual control laws (14), the actual event-triggered control law (18)- (20) with the adaptation laws (15), (16), (21) and (22) ensures that all the closed-loop signals are uniformly ultimately bounded, the tracking error µ 1 converges to an adjustable compact set around zero, and the inter-event times t l+1 − t l are lower bounded by the minimum inter-event time t min > 0 where l ∈ Z + .

Comparision with the Recent Work
In this section, the proposed control scheme is compared with the recent adaptive quantized feedback control scheme in [14]. In the recent work [14], an adaptive quantized feedback recursive controller was designed for nonlinear systems described bẏ where i = 1, . . . , n − 1, g j , j = 1, . . . , n and h are known nonlinearities, and ϑ is an unknown parameter vector. Here, the unmatched nonlinear functions g j should be known and satisfy the Lipschitz conditions with known Lipschitz constants. On the contrary, the proposed controller is designed for system (1) involving completely unknown unmatched nonlinear functions f j and disturbances d j . Therefore, no information of the nonlinearities is necessary for designing the proposed quantied feedback controller. To deal with these nonlinearities and disturbances in the quantized feedback recursive control design, we present an adaptive function approximation technique using quantized-states-based adaptation laws (see (14) and (15)) and prove the boundedness between α j and α q j .
On the other hand, the quantized feedback controller in [14] was designed as follows: where j = 1, . . . , n − 1, k j , k n , γ ϑ , and σ ϑ , are design parameters,θ is the estimate of ϑ, g q i = g i (x It should be emphasized that u in (51) is continuously updated. Therefore, this control scheme increases the load in communication through the controller-to-actuator channel which is unfavorable in practical networked control systems with limited communcation resources. In order to reduce this load, we developed our control scheme in an event-driven manner which means that u in (18) is updated only when the triggering condition (19) is satisfied.
In summary, compared with [14], the proposed controller can handle uncertain nonlinear systems with strict-feedback unknown nonlinearities while saving the communication resources by reducing the update of the control input u.

Remark 4.
In the existing event-triggered controller design, the existence of the minimum inter-event time is necessary to avoid the Zeno behavior. To prove the existence of this minimum inter-event time, the triggering error signals should generally be differentiable [21][22][23][24][25][26]. However, the quantized feedback control laws reported in [13,14] were not differentiable because of the quantized state variables. To overcome this problem, we employ the auxiliary first-order low-pass filter (23) for the quantized-feedback-based control law α q n in (20). Subsequently, the differentiable signalα q n,1 is used in the event-triggered actual control law u in (18). Consequently, the existence of the minimum inter-event time can be ensured by the analysis using (49). This design difficulty comes from the simultaneous handling of the quantized-feedback-based control and event-triggered control.

Remark 5.
In the proposed quantized-feedback-based event-triggered tracking scheme, the selection of the design parameters is sufficient conditions. The guidelines for the selection of these parameters are based on the proof of Theorem 1 as follows: (i) As the level of the quantizer δ in (2) decreases with the performance of digital devices or the communication environment, C can be reduced and thus the convergence bound √ 2C/k can be reduced. (ii) As γ w,i and γ b,i , i = 1, . . . , n, increase while fixing σ w,i and σ b,i as small constants, the tuning speed of the estimated parametersŴ i andb i and k can be increased and thus the bound √ 2C/k can be reduced.
(iii) The eigenvalues of M i can be increased by adjusting the filter parameters ζ i and ω i , i = 1, . . . , n, and the control gains k i can be increased. Then, the bound √ 2C/k can be reduced by increasing the control gains k i .
(iv) Reducing the design parameters η i helps to reduce C, which subsequently reduces √ 2C/k. (v) Adjusting the triggering parameters θ 1 and θ 2 manipulates the number of event times along the limited network communication resources in transient and steady-state responses.

Simulation Results
In this section, a numerical example and a hydraulic servo system were simulated to validate the proposed quantized-feedback-based adaptive event-triggered control result. For the two simulations, the sampling time t s was set to t s = 2 ms. Thus, the quantized feedback triggering law (19) was monitored every 2 ms. Furthermore, the tracking performance of the proposed quantized feedback control scheme was compared with that of the previous adaptive quantized feedback control scheme reported in [14]. We show that although the proposed event-triggered control scheme was designed in the presence of unknown nonlinearities, its tracking performance was similar to the performance of the previous continuous controller [14] designed in the presence of known nonlinear functions.
The tracking results and errors are compared in Figure 2a,b, respectively. In each figure, the upper one is the result of the proposed quantized feedback controller and the lower one is the result of the previous quantized feedback controller [14]. As shown in Figure 2, the quantized feedback tracking performances of both controllers were similar, although the proposed quantized feedback approach considers the unknown nonlinearities and the event-triggered inputs. In Figure 3,b i andŴ i are depicted where the adaptive parameters were bounded even though the quantized states were used to update them. Figure 4a displays the control signalα q 3,1 and its triggered signal u. Figure 4b depicts the inter-event times where the maximum inter-event time was 0.3 s which is sixty times longer than the sampling time. The triggering error u e and the triggering threshold θ 1 |µ q 3 | + θ 2 are shown in Figure 5a and the cumulative number of triggering instants of ours is displayed in Figure 5b. From Figures 4 and 5, it is shown that the control input u is updated when |u e | reaches θ 1 |µ q 3 | + θ 2 and the total number of events is 974 which implies that 6.49% = 974 30 t s × 100 of the total sampled data of α q 3,1 are only transmitted through a communication channel during 30 s. Based on these figures, we can conclude that the tracking of uncertain strict-feedback nonlinear systems can be achieved although the quantized state variables x q i , i = 1, 2, 3, were used, the control input was intermittently updated via the triggering law (19), and the inherent nonlinearities and disturbances were completely unknown.

Example 2
Consider a servo system driven by a hydraulic actuator where an inertia load is held by a spring-damper and a hydraulic actuator is placed in parallel to the spring-damper. The dynamic model of hydraulic servo systems is described by [34] where ϑ is the displacement of the inertia load, m s is the mass of the load, k s and b s are the spring constant and the damping constant, respectively, F = AP L is the driving force produced by the hydraulic actuator; A is the ram area and P L is the pressure difference of the hydraulic actuator, d denotes the friction inside the cylinder, V t is the volume of the cylinder, β e is the effective bulk modulus of oil, C t is the total internal leakage factor, and Q L is the supply input flow. For more information about the dynamics of the hydraulic servo systems, see [34]. Let us define the state variables and the control input u as x 1 = ϑ, x 2 =θ, x 3 = AP L /m s , and u = 4Aβ e Q L /(m s V t ). Then, (53) can be rewritten bẏ For the simulation, f 2 , f 3 , and d 2 are assumed to be unknown and the system parameters are set to m s = 300 kg, b s = 1500 N·s/m, k s = 9000 N/m, A = 1.2656 × 10 −4 m 2 , V t = 6.5312 × 10 −3 m 3 , β e = 6.9861 × 10 8 N/m 2 , and C t = 4 × 10 −13 [34]. The friction term is set to d = sign(x 2 )(20 + 22e −100|x 2 | )N. The reference signal r is given by r = 0.1 cos(t) and the initial conditions of the state variables arē x 3 (0) = [0.12, 0, 0] . The design parameters are chosen as δ = 0.001, k 1 = 5, k 2 = 3, k 3 = 20, γ w,i = 30, σ w,i = 0.00001, γ b,i = 20, σ b,i = 0.001, η i = 0.5, ω 1 = 10, ω 2 = 70, ω 3 = 150, ζ 1 = ζ i = 0.707, θ 1 = 10, and θ 2 = 0.1 where i = 2, 3. Similar to the previous example, the simulation results of the proposed controller are compared with those of the controller in [14] with the same design parameters k i , ω j , and ζ j with i = 1, 2, 3 and j = 1, 2 and the known information of f 2 , f 3 , and d 2 . In Figure 6, the tracking results and errors are compared where the initial error under the proposed controller converges close to zero within a few seconds and the tracking performance of the proposed controller is similar to that of the controller in [14]. These figures reveal that the function approximation using quantized states can effectively compensate for the uncertainties f 2 , f 3 , and d 2 . In Figure 7, the estimation parameters Ŵ i andb i , i = 2, 3 are shown. Figure 8a,b depict the input triggering results and the inter-event times, respectively, under the proposed control scheme. The triggering error and the triggering threshold are demonstrated in Figure 9a and the cumulative number of events is displayed in Figure 9b where the total number of events of ours is 1388. Thus, only 9.25% = 1388 30 t s × 100 of the total sampled data ofα q 3,1 during 30 s are released to the communication channel. As illustrated in these figures, we can achieve a good tracking performance for uncertain hydraulic servo systems with state quantization and unknown uncertainties.

Conclusions
A quantized-feedback-based adaptive event-triggered tracking strategy has been provided for state-quantized nonlinear systems in strict-feedback form with unknown nonlinearities. Different from the existing control methods, an adaptive approximation-based controller has been designed by deriving quantized-states-based adaptive laws and the event triggering issue has firstly been addressed in the quantized feedback control field. The closed-loop stability of the quantized-feedback-based event-triggered recursive control system has been analyzed with three lemmas. Further studies on the quantized-feedback-based adaptive event-triggered tracking problem of robotic systems and nonlinear multi-agent systems are recommended as future works.
Author Contributions: Conceptualization, Formal analysis, methodology, software, writing-original draft preparation, Y.H.C.; Conceptualization, supervision, Validation, writing-review and editing, S.J.Y. All authors have read and agreed to the published version of the manuscript.