Finite-Time Performance Guaranteed Event-Triggered Adaptive Control for Nonlinear Systems with Unknown Control Direction

This paper studies the issue of (cid:12)nite-time performance guaranteed event-triggered (ET) adaptive neural tracking control for strict-feedback nonlinear systems with unknown control direction. A novel (cid:12)nite-time performance function is (cid:12)rst constructed to de-scribe the prescribed tracking performance, and then a new lemma is given to show the di(cid:11)erentiability and boundedness for the performance function, which is important for the veri(cid:12)cation of the closed-loop stability. Furthermore, with the help of the error transformation technique, the origin constrained tracking error is transformed into an equivalent unconstrained one. By utilizing the (cid:12)rst-order sliding mode di(cid:11)erentiator, the issue of \explosion of complexity" caused by the backstepping design is adequately addressed. Subsequently, an ingenious adaptive updated law is given to co-design the controller and the ET mechanism by the combination of the Nussbaum-type function, thus e(cid:11)ectively handling the in(cid:13)uences of the measurement error resulted from the ET mechanism and the challenge of the controller design caused by the unknown control direction. The presented event-triggered control scheme can not only guarantee the prescribed tracking performance, but also alleviate the communication burden simultaneously. Finally, numerical and practical examples are provid-ed to demonstrate the validity of the proposed control strategy. prescribed performancel · First-order sliding mode di(cid:11)erentiator · Strict-feedback systems · Unknown control direction


Introduction
In the past decades, strict-feedback nonlinear systems (SFNSs), as a special kind of nonlinear systems, have evoked widespread attention because of their powerful capability to model various kinds of practical systems, such as chemical stirred tank reactor [1], flexible joint robotic system [2], hypersonic flight vehicles [3] and so on. As a consequence, for the control problems of SFNSs, many scholars have carried out in-depth research owing to the huge needs in engineering applications. The adaptive backstepping method [4], as a breakthrough method in the field of nonlinear control, has become an effective way to deal with the control issues of SFNSs. With the aid of neural network (NN) [5] or fuzzy logic approximator, the adaptive backstepping method has been widely used to control the SFNSs with unknown nonlinear dynamics [6][7][8][9]. However, the results mentioned above exist the issue of "explosion of complexity" resulted from repeated derivations of the virtual controller in backstepping design. To handle such a problem, many useful approaches have been developed. For example, the dynamic surface control (DSC) method was employed to estimate the derivative of the virtual controller in [10,11]. The authors in [12] use a more efficient technique namely first-order sliding mode differentiator (FOSMD) to avert tedious calculations.
Noting that in some practical systems, the tracking error often needs to meet some specified transient and steady state performance indicators, such as small overshoot, fast convergence speed and small steadystate error, so as to ensure the control performance of the system. In view of such situation, the authors in [13] firstly presented a performance function transformation method, whose basic idea was to convert the constrained tracking error into the unconstrained one. Subsequently, such a method was not only applied to more general systems with miscellaneous engineeringoriented phenomena, such as unmodeled dynamics [14], input saturation [15], actuator failures [16] and input quantization [17], but also widely employed to a great deal of practical systems [18,19]. Unfortunately, the performance functions in the aforementioned works are not concerned about finite time convergence, which limits their engineering applications with high-accuracy control. The prescribed finite time convergence issue is extremely difficult for controller design of nonlinear systems. More recently, this issue were gained some concern, and a few preliminary and valuable results were developed in [20][21][22][23][24]. Specifically, the authors in [20] constructed a finite-time performance function, whose convergence time can be set arbitrarily. This result [20] has been extended to a finite-time fuzzy tracking control scheme for SFNSs with dynamic disturbances [22]. And then, the finite-time consensus tracking control was handled in [24] for multi agent systems with prescribed performance and mismatched uncertainties.
It is worth pointing out that the results mentioned above can only be obtained when the control direction of the system is known in advance. Once the control direction is unknown, these control schemes are no longer feasible. In view of such a case, the authors in [25] firstly developed a Nussbaum-type function (NTF), which effectively handled the difficulty in controller design caused by the unknown control direction. Subsequently, on the basis of NTF, a great number of interesting works have been reported. To mention a few, in combination with NTF and Barrier Lyapunov function, an adaptive tracking control strategy has been constructed for a class of state-constrained SFNSs with unknown control direction [26]. Based on the NTF and fuzzy logic approximator, the DSC fuzzy controller presented in [27] has successfully guaranteed the stability of uncertain non-strict-feedback systems with unknown virtual control coefficients. These NTF-based results does not concern two hot directions: the prescribed finite time convergence and the limited network resources.
Recently, networked control has been developed rapidly due to its peculiarities of low cost, good flexibility, reliable operation, convenient installation, etc. However, the network bandwidth is limited, which inevitably brings some problems including transmission delay [28], packet disorder [29], and so on. To deal with such problems, the authors in [30] presented an event-triggered control (ETC) approach, which efficaciously economizes the network resource. On the basis of the ETC, a series of valuable results have been reported [31][32][33][34][35][36][37]. For instance, a relative threshold ETC method has been developed for SFNSs with unknown parameters in [33], which achieves a good balance between the system performance and the limited network resources. In combination with the relative threshold strategy, some fruitful ETC schemes have been presented for more general nonlinear systems with prescribed performance [34], input saturation [36] and unmeasured state [37]. Unfortunately, these aforementioned ETC methods can not be directly extended to prescribed performanceguaranteed SFNSs with unknown control direction. The main reason lies in the unknown control direction is coupled in the compensation process of the measurement error between the controller and the actuator. In this case, it becomes significantly challenging to construct a suitable ETC scheme to weaken the effect of the measurement error on system stability. As a consequence, it is significant and necessary to investigate the ETC for SFNSs subject to both prescribed performance and unknown control direction.
According to the aforementioned discussion so far, the main objective of this paper is to construct a finitetime performance guaranteed ETC scheme for SFNSs with unknown control direction. Firstly, the finite-time performance function is constructed to guarantee the tracking performance constraint by the aid of the error transformation approach. Secondly, the FOSMD is embedded in the backstepping procedure to cope with the issue of "explosion of complexity", and then an ingenious adaptive law is given to facilitate the co-design of controller and ET mechanism. Finally, a finite-time performance guaranteed ETC scheme is developed based on the novel adaptive law and the NTF, which guarantees the prescribed tracking performance, alleviates the communication burden and compensates the measurement error at the same time. The main contributions are listed as follows: 1) A novel finite-time performance function is developed such that the prescribed tracking performance is achieved in a predetermined finite time instead of the infinite time. Moreover, a new lemma is derived to show the differentiability and boundedness of the constructed performance function, which plays an important role for the system stability; 2) A novel adaptive law is given to estimate the upper bound of the actual control gain, and then the controller and the ET mechanism are co-designed to compensate successfully the measurement error resulted from the ET mechanism; 3) In combination with such an adaptive law and the hyperbolic tangent function, a novel ET actuator under the relative threshold strategy is proposed, thereby achieving the prescribed finite-time tracking performance and saving the communication resource.

System description
This paper considers a class of SFNSs as follows: . . , n), u ∈ R and y ∈ R denote, respectively, the state vector, the system control input and output. f i (x i ) and g i (x i ) stand for the unknown smooth nonlinear functions.

Assumption 2
The reference trajectory y d and its time derivativesẏ d andÿ d are bounded.

Radial basis function NNs
It has been shown in [39] that the radial basis function (RBF) NN has a powerful ability to approximate any unknown smooth nonlinear function T (Z) over a compact set Ω Z as , . . . , Θ io ] T and △ i are the center and width of NN, respectively. W * ∈ R l represents the optimal neural weight vector with l > 1 being the number of neural node, δ(Z) denotes the approximation error satisfying |δ(Z)| ≤δ, whereδ > 0 is an arbitrarily small constant.

Novel finite-time performance function
A novel finite-time performance function is constructed as follows: where ϕ 1 , ϕ 2 and T c are positive constants, tanh(·) represents the hyperbolic tangent function.

Remark 1
Notice that the finite-time performance control problem has been considered in the existing results in [20][21][22], [24]. These results mentioned above require some restrictions. For example, the initial condition ϕ(0) is required to satisfy ϕ(0) ≤ 1 in [20][21][22]; and ϕ(t) depends on the order n of the controlled system [24], which makes the computational complexity of ϕ(t) greatly increase for high-order nonlinear systems. Compared with the existing works [20][21][22], [24], it is obvious from (2) that the finite-time performance function developed in this paper is easy-to-implement due to the mild initial condition ϕ(0) = − tanh(ϕ 1 ) + ϕ 2 + 1 > 0 and the independence of system order n.
In this paper, the tracking error e 1 = y − y d should remain within the predefined performance constraint as follows where ς 1 and ς 2 are both positive design parameters. It can be concluded from (2) and (3) that −ς 1 ϕ(0) and ς 2 ϕ(0) with ϕ(0) = − tanh(ϕ 1 ) + ϕ 2 + 1 denote, respectively, the minimum value of the transient undershoot and the maximum value of the transient overshoot of e 1 . −ς 1 ϕ 2 and ς 2 ϕ 2 represent the low bound and upper bound of the steady-state tracking error e 1 , respectively. Besides, T c stands for the time when tracking error e 1 decays to the steady-state value ϕ 2 .

Useful Definition and Lemmas
Definition 1 [25] Any smooth even function N (φ) can be called as a function of Nussbaum-type when it satisfies lim s→∞ sup 1 where r 1 and r 2 stand for, respectively, a suitable constant and a positive constant. g(x(t)) denotes a unknown smooth function which takes values in the un- The FOSMD [40][41][42] is described as where ϑ 1 , ϑ 2 and ϱ 1 represent the system states, ω 1 and ω 2 stand for the positive design constants, υ denotes the input signal of the FOSMD. Then, we can obtain the following lemmas.
Lemma 3 [41,42] By selecting suitable design constants, the following equalities can be obtained after a finite time in the absence of input noises: Notice that Lemma 3 is derived under the case of no input noises, i.e. υ = υ 0 , when the input noise exists, the following Lemma holds. Lemma 4 [41,42] If the input noise meets |υ − υ 0 | ≤ ι, the following inequalities can be obtained in a finite time: wherel andε are both positive constants exclusively depended on the design constants of the FOSMD.
Step 1: For the purpose of transforming the constrained tracking error e 1 (3) into the equivalent unconstrained variable z 1 , we introduce a smooth and strictly increasing transformation function Υ (z 1 ), which satis- With the help of Υ (z 1 ), (3) can be expressed as where Υ (z 1 ) is constructed as follows Then, it follows from (6) that ) .
Define the Lyapunov function candidate as follows where 1 stands for the weight estimation error withŴ 1 being the estimation of W * 1 . Then, the dynamic of V 1 along (8) iṡ Let unknown function We utilize the RBF NN to approximate T 1 (Z 1 ) as where |δ 1 (Z 1 )| ≤δ 1 .
By substituting (11) into (10) yieldṡ According to Young's inequality, we have where λ 1 > 0 is a design constant. Substituting (13) into (12), one haṡ The virtual law α 1 and neural weight updated lawẆ 1 are constructed as follows where k 1 > 0 and σ 1 > 0 are both design parameters. Based on (15)-(17) and x 2 = z 2 + α 1 , we obtain Substituting (18) and (19) into (14), one can obtaiṅ With the help of Young's inequality, we have Substituting (21) into (20), one haṡ where q 1 = σ 1 ∥W * 1 ∥ 2 /2 +δ 2 1 /2λ 1 and p 1 = min{2k 1 , In order to effectively estimateα i−1 and overcome the issue of explosion of complexity, the FOSMD is constructed as where ϑ i1 , ϑ i2 and ϱ i1 represent the system states, ω i1 and ω i2 are both positive design constants. Combining with (24) and Lemma 3-4, it follows thaṫ where ε i satisfies |ε i | ≤ε i withε i being a positive constant. If the input signal α i−1 of the FOSMD (24) is not influenced by the noise, it can be concluded from Lemma 3 thatε i = 0. Moreover, if the input signal α i−1 is influenced by the bounded noise, we can conclude from Lemma 4 that |ε i | ≤ε i . Consider the Lyapunov function candidate where stands for the weight estimation error withŴ i being the estimation of W * i . Based on (23), we havė where i denotes the ideal weights vector and |δ i (Z i )| ≤δ i .
According to Young's inequality, one can obtain where λ i is a positive design constant. Based on (25) and (28), we geṫ Designing the virtual control law α i and the relative neural weight updated lawẆ i as where k i , µ i and σ i are all positive design parameters. According to (30)-(32), we can conclude that Substituting (33) and (34) into (29), it can be obtained thaṫ With the help of Young's inequality, we have By substituting (36) into (35), one getṡ where Step n: For z n = x n − α n−1 , its dynamics along (1) can be expressed aṡ

Similar to
Step i, the FOSMD is employed to acquirė α n−1 , which can avoid tedious computation: where ϑ n1 , ϑ n2 and ϱ n1 denote the states of FOSMD (39), ω n1 and ω n2 are both positive design parameters. Then, it can be concluded from (39) and Lemma 3-4 thaṫ where ε n satisfies |ε n | ≤ε n withε n being a positive constant.
Let the Lyapunov function candidate as where Γ n = Γ T n > 0 is a constant matrix, G is a positive design constant,W n =Ŵ n − W * n stands for the weight estimation error withŴ n being the estimation of W * n , g n2 =ĝ n2 − g n2 withĝ n2 being the estimation of g n2 .
Furthermore, based on (38), one can deducė where W * T n S n (Z n ) is adopted to approximate f n (x n ) with Z n = [x 1 , x 2 , ..., x n ] T ∈ R n and |δ n (Z n )| ≤δ n .
Based on Young's inequality, we get where λ n > 0 is a design parameter.
Since 0 < g n1 ≤ |g n (x n )| ≤ g n2 , |ζ 1 (t)| ≤ 1 and 0 < ρ < 1, we can conclude that b(x n ) is bounded and there exists unknown positive constants b and b such that b ≤ |b(x n )| ≤ b. Moreover, the sign of b(x n ) is as same as g n (x n ).
Multiplying both sides of (56) by e pnt and integrating it over the interval [0, t], it follows that Then, it can be concluded from (57) and Lemma 2 that V n (t) and φ n (t) are bounded on [0, t f ). Consequently, z n ,Ŵ n andĝ n2 are also bounded on [0, t f ).
Remark 2 From the ET controller (45)-(50), it is obvious that the control signal is sent to the actuator in an aperiodic way, which significantly economizes the communication resource between the controller and actuator. However, due to the unknown direction of control gain g n (x n ), it is quite difficult to construct a suitable controller to compensate the measurement error caused by the ET condition (50). To deal with such a difficulty, two effects have been taken in this paper: 1) the nonlinear term −g n2 θ tanh( θzn π ) is introduced into (53), which effectively compensates the measurement error based on Lemma 5, see (54) for details; 2) an ingenious adaptive lawġ n2 is designed in (48) to estimate the unknown upper bound g n2 of g n (x n ), which makes it possible to co-design the controller and the ET mechanism. As a consequence, the effect of the measurement error on the system stability is handled effectively.

Simulation studies
For demonstrating the effectiveness of our presented method, the following two simulation examples are considered in this section.

Numerical example
Consider the SFNSs [45] as follows   The main goal of this simulation study is to make the system output y track the reference signal y d = 0.8 sin(t) while reduce the communication burden. Meanwhile, the tracking error e 1 satisfies the predefined performance constraint −ς 1 ϕ(t) < e 1 (t) < ς 2 ϕ(t), where ς 1 = 3, ς 2 = 4, and ϕ(t) is defined in (2)   Simulation results are shown in Figs. 1-5. From Figs. 1-2, we can see that system output y can efficaciously track y d and the tracking error e 1 satisfies the predefined performance constraint. Fig. 3 illustrates the curve of the ETC signal u(t). It can be concluded from Fig. 3 that the communication burden between the controller and actuator are significantly reduced. Fig. 4 displays the inter-event times, which indicates the Zeno   phenomenon doesn't happen. The cumulative number of ET instants is shown in Fig. 5. In comparison with the time-triggered control scheme, the ETC scheme p-resented in this paper only requires 905 times signal transmission, which economizes nearly 55% communication resource.

The Spring-mass-damper mechanical vibration system
To further show the applicability of our presented ETC approach, we consider the spring-mass-damper mechanical vibration system [26] as follows where x 1 and x 2 , denote, respectively, the position and velocity. The physical parameters are selected as M = 1, C d = 2 and J s = 8, which are as same as these in [26]. The desired reference output is y d = 1.6 sin(t) + 1.6 sin(0.5t).

Conclusions
This paper considered the finite-time performance guaranteed ETC problem for SFNSs with unknown con-trol direction. An easy-to-implement finite-time performance function has been constructed to depict the predefined performance constraint, and then a related lemma has been developed to guarantee the stability of the considered closed-loop error system. Based on the error transformation technique, the original constrained tracking error has been transformed into an equivalent unconstrained one. Moreover, the introduction of the FOSMD avoided the issue of "explosion of complexity", thereby making the controller design and stability analysis easier. Meanwhile, an ingenious adaptive law has been developed to estimate the upper bound of the actual control coefficient. Subsequently, a novel performance guaranteed ETC scheme has been proposed by the combination of the adaptive law, which has achieved the predefined tracking performance, compensated the measurement error and economized the communication resource.

Data Availability Statement
Data sharing is not applicable to this paper because no datasets are generated or analyzed during the current study. Furthermore, it follows from (66) that lim t→T + cφ (t) = 0. Then, it can be seen that lim t→T + cφ (t) = lim t→T − cφ (t), which means thatφ(t) is continuous at time T c . As such, ϕ(t) is continuous for all t ≥ 0. Subsequently, it can be concluded thatφ(t) is bounded on [0, T c ]. Besides, noting thatφ(t) = 0 holds on (T c , ∞), it is obvious thaẗ ϕ(t) is bounded for all t ≥ 0.