Adaptive Neural Event-Triggered Control of MIMO Pure-Feedback Systems With Asymmetric Output Constraints and Unmodeled Dynamics

In this paper, the issue of adaptive neural event-triggered control (ETC) is studied for uncertain block-structure multi-input multi-output (MIMO) constrained non-affine nonlinear systems with unmodeled dynamics. A dynamic signal produced by the auxiliary system based on the property of unmodeled dynamics is employed to solve the dynamical disturbances. The unknown continuous function obtained at each step of recursion is estimated by using radial basis function neural networks (RBFNNs). Utilizing logarithmic function as an invertible mapping, the uncertain constrained MIMO non-affine system is changed into a novel unconstrained block-structure MIMO nonaffine system. Using improved dynamic surface control (DSC) strategy, adaptive event-triggered control scheme is developed for the transformed non-affine system based on relative threshold mechanism. According to the Lyapunov method, all the signals in the closed-loop system are shown to be semi-globally uniformly ultimately bounded (SGUUB). Output constraint requirements are not triggered, and Zeno behavior is avoided. A constrained pure-feedback system and a kind of 2-DOF flexible manipulator system are used to illustrate the theoretical findings.


I. INTRODUCTION
Since backstepping was proposed for a class of feedback linearizable systems in [1], it has widely been applied to construct adaptive controller for nonlinear systems as a popular tool in [2], [3]. Because it needs to differentiate for the designed virtual control at recursive each step, the controller design is quite complicated. This is its disadvantage in [1]- [3]. In order to remove this defect, dynamic surface control was proposed by introducing first-order filter in [4]. The computation complex in conventional backstepping is removed by using algebra operation to replace differential operation in DSC. Using mean value theorem and Nussbaum function, adaptive neural DSC method was developed for non-affine nonlinear systems in [6]. As we know that RBFNNs are a universal approximator. Based on the simple The associate editor coordinating the review of this manuscript and approving it for publication was Ludovico Minati . structure and infinite derivable properties, the RBFNNs has been widely used in adaptive backstepping control or DSC design. In [7], two robust adaptive DSC schemes were investigated by using RBFNNs for a class of pure-feeback systems with input nonlinearity and perturbed uncertainties. In addition, there widely exists unmodeled dynamics in modern industrial process, which usually degrades the system performance, sometimes, and leads to be instable for system. To deal with the impact of dynamical disturbances on the system, dynamic signal method in [8], [9] and Lyapunov function description in [10] as well as and normalization signal in [11] were usually employed to dispose of state and input unmodeled dynamics, respectively. However, the proposed design method was for the unmodeled dynamics in autonomous form in [8]. The dynamic signal method was developed for the unmodeled dynamics in nonautonomous form in [9].
Barrier Lyapunov function (BLF) in [12], [13], integral barrier Lyapunov function (iBLF) in [14] and nonlinear mapping (NM) in [15]- [18] were used to hand output constraints and full state constraints. However, single-input single-output (SISO) systems were discussed in [12]- [18]. In [19], fault-tolerant control was proposed by using BLF for uncertain parametric strict-feedback MIMO nonlinear systems with output restriction. Furthermore, based on BLF, decentralized adaptive finite time control was developed for uncertain pure-feedback system in [20]. Based on a novel BLF, robust adaptive control was investigated for uncertain MIMO nonlinear systems with state and input constraints in [21]. By introducing NM, adaptive neural control was proposed for MIMO nonlinear systems with time-varying output constraints in [22]. However, the dynamical disturbances were not studied in [19]- [22]. In [23], adaptive control scheme was developed for uncertain output constrained MIMO nonlinear systems in strict-feedback form based on both BLF and modified DSC. Furthermore, adaptive neural control was proposed for the MIMO pure-feedback systems with output constraints and dynamic uncertainties using both invertible nonlinear mapping and modified DSC in [24].
It is well known that the designed controller based on event-triggering mechanism can reduce energy consumption and occupation rate of transmission bandwidth. Eventtriggered control (ETC) has become research hot topic in the past decade. By taking a given invariable difference value of the state norm as toggle condition, a simple event-triggered scheduler was proposed for a class of linear systems in [25]. In [26], integrating with model-based networked control theory, a novel ETC strategy with time-varying network delays was developed to a class of linear systems, and two fire-new error discriminants of states, fixed threshold strategy and relative threshold strategy, were designed as the toggle condition. In [27], [28], an event-sampled NN was invented to estimate the unknown terms in a class of strict-feedback systems. In [29]- [32], by structuring state-depended event-triggered condition, some adaptive control schemes were designed for a variety of uncertain systems. However, the above-mentioned achievements were dependent on the assumption that the closed loop system is input-to-state stable (ISS). To solve this problem, a new ETC controller design method was proposed for a class of affine systems with unknown parameters, and a new event-triggered condition called switching threshold strategy was proposed in [33]. Hereafter, some significant achievements based on [33] can be seen in [34]- [36]. However, the controlled plants in [34]- [36] were all SISO systems. Using RBFNNs to estimate the model had been done in many studies such as [37], [38].
In this paper, adaptive control is proposed based on event-triggering mechanism for uncertain constrained blockstructure MIMO non-affine systems. To the best of our knowledge, the relative threshold strategy is first extended and applied to constrained MIMO uncertain pure-feedback systems. RBFNNs are used to appoximate the unknown continuous function vector, which is produced in the process of controller design. The unknown system functions have not been directly estimated by RBFNNs. The main contributions are listed as follows: (1) Adaptive event-triggered DSC is proposed for output constrained block-structure pure-feedback nonlinear systems with dynamic uncertainties based on relative threshold strategy. By constructing first-order auxiliary system to produce a measurable signal, the dynamic uncertain terms are effectively dealt with. Furthermore, to fulfill the constraint requirements, using invertible nonlinear mapping (INM), the constrained block-structure non-affine nonlinear system is changed into an unconstrained one. Furthermore, the controller design is simplified based on the transformed system. (2) The improved DSC approach is applied to the transformed unconstrained block-structure non-affine system, and the first-order auxiliary system is utilized to eliminate the repeated derivation of the intermediate variable in conventional backstepping. (3) The updating laws with the only one tuning parameter for each approximated unknown function at recursive each step are proposed. Furthermore, a simple eventtriggered control vector is developed using modified DSC and the property of hyperbolic tangent function in the final step.

II. PROBLEM STATEMENT AND PRELIMINARIES
Consider the following block-structure MIMO pure-feedback nonlinear systems ∈ R m , (i = 1, . . . , n) are the unknown smooth nonlinear function vectors, ∈ R m are the unknown smooth nonlinear dynamic disturbances, ζ ∈ R n 0 is unmodeled dynamics, Q(ζ,x n , t) ∈ R n 0 is an unknown smooth function vector satisfying the Lipschitz condition, ∈ R m is the output. In this paper, A = tr(A T A) denotes the Frobenius norm of the matrix A, ξ = n i=1 ξ 2 i stands for the Euclidean norm of the vector ξ = [ξ 1 , . . . , ξ n ] T ∈ R n , L ∞ denotes the set of all bounded functions.
Remark 2: , l ij is the number of nodes in the ijth neural networks, is the approximation error.

III. ADAPTIVE EVENT-TRIGGERED DYNAMIC SURFACE CONTROL
For convenience, we introduce the following notations andβ ij are the estimates of β * i and β * ij at time t. The coordinate change and the error signals are defined as follows: . . .
. . ,h n will be given later.

IV. MAIN RESULTS
Define V and the compact set A n as follows: where p > 0, p n = nm + m(2n − 1) + 1.
For V ≤ p, all signals at the right side of (77) are bounded, sov j (t) is bounded. Furthermore, there exists a positive constant C j such that v j (t) ≤ C j . In addition, since E j (t jk ) = 0 and lim t→t − j,k+1 E j (t) = δ j |u j (t jk )| + m j1 , for t j,k+1 ∈ (t jk , t j,k+1 ), using mean value theorem and (76), we have where t * j ∈ (t jk , t j,k+1 ). Let t j,k+1 tend to t − j,k+1 above both sides of the inequality, we have Obviously, (t j,k+1 − t jk ) has a positive lower bound t * j = m j1 C j > 0, and Zeno behavior can be successfully avoided. Remark 4: RBFNNs are used to appoximate the unknown continuous function vector, which is produced in the process of controller design, and the unknown system functions have not been directly estimated by them in this paper, wheras RBFNNs are used to estimate unknown system functions in [37], [38]; Unmodeled dynamic and asymmetric output constraints are considered in our paper, wheras they are not considered in [37], [38]; The considered systems are class of MIMO systems with block-structure, wheras the considered systems are a class of MIMO Systems with similar structure in [37], and they are a class of SISO systems without uncertainties in [38]. Therefore, with the help of event-triggered controller, our proposed method is more practical and the systems in this paper are more general.
Remark 5: Compared with the approach proposed in [24], a new nonlinear mapping based on hyperbolic tangent function is introduced to handle the asymmetric output constraints. By using this novel nonlinear mapping, the origin point between the original and transformed system can be mapped. It means that the controllability of the original system is guaranteed. Different from the time-driven control method proposed in [24], the new event-triggered approach can reduce energy consumption and occupation rate of transmission bandwidth without affecting the tracking accuracy and the system stability.  on the size of the defined compact set A n . In addition, 0 does not include the constants r i1 , . . . , r im , i = 1, . . . , n. Therefore, we can choose large enough constants r i1 , . . . , r im , i = 1, . . . , n for any given design constants σ i1 , . . . , σ im , i = 1, . . . , n and p > 0 such that ≥ 0 p andV ≤ 0 hold.

V. NUMERICAL SIMULATION
To illustrate the theoretical findings of the proposed eventtriggered control, a constrained pure-feedback system and a kind of 2-DOF flexible manipulator system are discussed. Example 1: Consider the following constrained secondorder uncertain MIMO system: where Q(ζ, y, t) = −ζ + y 2 sin t + 0 Case 2: If output constraints and event-triggered controller are not considered, the design constants are chosen as K 1 = diag [3.25, 3.25], K 2 = diag [4,4], σ 1 = σ 2 = 0.5, τ 2 = 0.001, and the other conditions are the same, the simulation results without output constraints and eventtriggered controller are shown in Figures 1 and 2.  From Figures. 1 and 2, it is clearly to know that the output signal vector y can well track the desired trajectory and all the states are within the constraints. The tracking performance of case 1 is better than the tracking performance of case 2. In Figures. 5 and 6, we can find that the event-triggered numbers are 632 and 1411 in 30 seconds, respectively.

VI. CONCLUSION
Combining dynamic surface control technique with relative threshold strategy, adaptive neural event-triggered control has been developed for block-structure MIMO non-affine nonlinear systems including output restriction and dynamical uncertainties. The first-order auxiliary system designed based on property of unmodeled dynamics is employed to dispose of the dynamical uncertain terms. The output constraints can be carried out based on invertible nonlinear mapping. All the signals in the designed control system have been proved to be semi-global uniform ultimate bounded. A constrained purefeedback system and a kind of 2-DOF flexible manipulator system are provided to verify the effectiveness of the designed adaptive event-triggered control algorithm.