Adaptive Event-Triggered Control for Nonlinear Systems With Asymmetric State Constraints: A Prescribed-Time Approach

Finite/fixed-time control yields a promising tool to optimize a system's settling time, but lacks the ability to separately define the settling time and the convergence domain (known as practically prescribed-time stability, PPTS). We provide a sufficient condition for PPTS based on a new piecewise exponential function, which decouples the settling time and convergence domain into separately user-defined parameters. We propose an adaptive event-triggered prescribed-time control scheme for nonlinear systems with asymmetric output constraints, using an exponential-type barrier Lyapunov function. We show that this PPTS control scheme can guarantee tracking error convergence performance, while restricting the output state according to the prescribed asymmetric constraints. Compared with traditional finite/fixed-time control, the proposed methodology yields separately user-defined settling time and convergence domain without the prior information on disturbance. Moreover, asymmetric state constraints can be handled in the control structure through bias state transformation, which offers an intuitive analysis technique for general constraint issues. Simulation and experiment results on a heterogeneous teleoperation system demonstrate the merits of the proposed control scheme.

parameterized settling time for controller design. In [1] and [2], a classic Lyapunov-based finite-time stability criterion revealed that the upper bound of the settling time can be regulated through initial values and control parameters. Subsequently, various finite-time concepts [3]- [6] were developed to obtain a more accurate settling time estimate [7]- [10]. However, the above estimate results depend on the initial value, leading to estimation conservatism in the presence of sensor noise and partial observations. To address this problem, fixed-time stability [11] was developed to relax the need for prior knowledge on initial conditions, and to give a uniform settling time constrained only by control parameters, which thus facilitates state-feedback control [12]- [14] and observer design [15]. Although the aforementioned works realized fixed-time stability to some extent, one would like to select a settling time arbitrarily according to task requirements rather than constrained by the control parameters. Furthermore, the aforementioned "finite/fixed time" refers to the upper bound of the setting time instead of the actual one, which we would like to attain. Therefore, partial performance metrics cannot be user defined when applying the traditional finite/fixed-time controller. Settling time was used in [16] and [17] as an explicit parameter rather than the corresponding control metric, thereby facilitating a new paradigm switch from finite-time stability to prescribed-time stability (PTS). The time transformation technique was proposed for PTS analysis and control synthesis, which effectively mapped the time domain to the user-defined range [18], [19]. However, few works investigated the user-defined properties for settling time and convergence domain simultaneously, i.e., practically prescribed-time stability (PPTS). In [20] and [21], a time-varying fractional function based control method was proposed to achieve a user-defined settling time for high-order nonlinear systems. However, the convergent accuracy was determined by the upper bound of disturbances. PTS was realized for strict-feedback-like systems in the framework of state-feedback and output-feedback control [22], [23], but lacked the ability for a user to define the convergence behavior of the system, independent of the settling time. Moreover, actuators commanded with the PTS principle behave on the basis of control period (sample time) at the cost of increased communication burden and excessive energy consumption. In contrast to time-based approaches, event-triggered mechanisms create a new paradigm for control systems subject to limited data transmission, which determines the state update or controller implementation according to specialized event conditions [24]. It is shown in [25] that introducing a dynamic event-triggered mechanism offers higher design flexibility to exclude Zeno phenomenon. For instance, the adaptive event-triggered methods in [26]- [28] leverage the communication resources in the actuation channel, reducing the computational burden, though they only achieved asymmetric stability. Compared to continuous triggering, interval actuator triggering will inevitably weaken control performance, leading to a conflict between the number of triggers and stable behavior. Although the PTS concept was successfully introduced in multiagent systems [29], the PPTS property has not been explored in existing event-triggered works. Ensuring the PPTS property under an eventtriggered mechanism therefore remains a challenge.
In addition, the system states are inevitably subject to specific constraints according to physical limitations and motion requirements. As an example, if the end-effector of a space robot moves beyond the field of view of the hand-eye camera, the target will be lost, thereby leading to docking task failure. Therefore, the constraints imposed on the control system facilitate safe and smooth task execution. Since predetermined constraints are incorporated in stability analysis and control synthesis, barrier Lyapunov functions (BLF) become an effective approach to dealing with constraint control problems. As the associated state approaches a predetermined constraint, the BLF will tend to infinity, thereby resulting in a large control input that regulates the system state within the constraint range. Log-type [30] and tan-type [31] BLF were introduced to tackle output state constraints and applied to stochastic nonlinear systems [32]. Recently, we proposed a novel exponential-type BLF (EBLF) applied to state feedback [33], [34], which was beneficial to address nonlinear systems with symmetric state constraints. Several works [35]- [37] attempted to consider asymmetric state constraints using piecewise Lyapunov functions, but the time-derivative terms of a Lyapunov function may increase the control input, leading to increased energy use. In [12], upper and lower bound functions were integrated to a Lyapunov function, whereas the prescribed-time property was not considered.
Motivated by the above observations, we investigate the prescribedtime stabilization of nonlinear MIMO systems, subject to asymmetric constraints. Parametric uncertainty, additive disturbance, and asymmetric output constraints can be addressed effectively in the proposed adaptive event-triggered control scheme, simultaneously guaranteeing the user-defined settling time and convergence domain.
The contributions of this article are threefold 1) A new sufficient condition for PPTS is provided to ensure userdefined characteristics of settling time and convergence domain. These two metrics can be preset simultaneously, which addresses the limit caused by control parameter constraints and thus enables us to decouple response speed and accuracy.
2) The EBLF technique is combined with back-stepping control to impose time-varying asymmetric output constraints, which transforms an asymmetric constraint into a symmetric one by bias state transformation. Different from approaches that deal with asymmetric constraints [30], [35]- [37], the proposed framework eliminates the need to determine the sign of the constraint state or its state transformation variables in real time. 3) In contrast to adaptive event-triggered methods employing the universal approximation theorem [26]- [28], the proposed control scheme can ensure the robustness against parametric uncertainty and additive disturbances while overcoming the contradiction between control accuracy and event-triggered mechanism. This allows convergence accuracy to be quantitatively calculated and artificially predetermined, without depending on disturbance upper bounds.

II. PRELIMINARIES
We use the following notations. Z + denotes the set of positive natural numbers. R ≥0 stands for the set of nonnegative real numbers. I n denotes the identity matrix of dimension n. ∀A ∈ R n×n , A is the Frobenius norm. The superscript (·) (i) represents the ith time derivative. diag{a j } and col{a j } stand for the diagonal matrix and column vector with a j as the jth entry, respectively.
Consider the following nonlinear system: where x(t) ∈ R n is the system state and f : R n × R ≥0 → R n a continuous-differential function. The equilibrium of (1) is PPTS if x(t) ≤ ε for t ≥ t 0 + T , where t 0 is the initial time, ε the convergence domain, and T the settling time, which are defined by the user within an achievable range. T s ≤ T < +∞, with T s denoting the time consumed in the necessary data transmission and processing. Notice that the arbitrary setting of ε and T is emphasized in the concept of PPTS, i.e., ε and T are decoupled. In order to achieve PPTS, we introduce a time-varying piecewise function where t T = t 0 + T and α ∈ (0, ln(2)/T ] is a tuning parameter. Theorem 1: If there exists a positive-definite continuous-differential function V (x(t), t) : R n × R ≥0 → R ≥0 and positive scalars a, b, and c such that a > α, b > c, anḋ then the equilibrium of system (1) is PPTS and the system trajectory will enter the convergence domain The uniformly ultimately bounded (UUB) stability can be guaranteed by applying the Lyapunov theorem [38]. Moreover, with the comparison theorem, integrating both sides of (4) yields α . Therefore, the system trajectory approaches a certain region Ω when t = t T . It is worth pointing out that the boundary of Ω only depends on the user-designed parameters rather than system dynamics. Thus, the convergence accuracy can be user-defined according to practical task requirements.
For t ≥ t T , substituting (2) into condition (3) leads tȯ Furthermore, integrating both sides of (6), we have Therefore, the system trajectory will never leave Ω for t ∈ (t T , +∞), which completes the proof. Remark 1: Different from the UUB stability, the settling time in PPTS is not constrained by the initial range or ultimate bound. In addition, the ultimate bound does not depend on disturbances or system uncertainties. Different from traditional concepts of finite/fixed/prescribed-time stability, the settling time and accuracy are treated as independent parameters in PPTS.
Remark 2: Compared with the existing prescribed-time controllers [33], [34], we have carried out a more rigorous stability analysis of the transient-state process and quantitatively given the selection basis of corresponding parameters in this article. α is adjusted according to a predetermined convergence time rather than a fixed empirical parameter. The scalar function (2) at t > t T is replaced by an exponential form so that the system trajectory will converge in an asymptotic form after the scheduled settling time. In this regard, we reveal the relationship among PPTS, asymptotic stability, and UUB stability in the case t > t T .

Remark 3:
The following properties of the piecewise function (2) are instrumental for the implementation of PPTS in the period of t ∈ [t 0 , t T ): 1) ς(t) is monotonically decreasing and constantly positive; Therefore, C ∞ functions that fulfill these three properties can be also used as (2). In terms of implementation, the right derivative of ς(t) is defined asς(t) at t = t T . Three parameters (α, a, and b) also play an essential role in the realization of PPTS. Given the settling time and accuracy requirements, larger α and a (or smaller b) within the admissible range facilitate accuracy improvement, but may lead to excessive control gain. Hence, there is a tradeoff for parameter regulation according to the practical demand on physically available resources (communication bandwidth, maximum actuation torque, etc.).

III. MAIN RESULT
Consider a class of nonlinear MIMO systems with parametric uncertainty and additive disturbance , t) the unknown nonparametric disturbance, and u(t) ∈ R nn the control input. The following assumptions are needed for i = 1, 2, . . ., n.
Assumption 1: There exist two constants a i andā i such that 0 < Remark 4: Since the associated functions above are composed of system states subject to measurement range, Assumptions 1 and 2 are rational in practical applications [39]. In addition, although gain functions A i (x i (t), t) and appended disturbances h i (x i (t), t) are state dependent with constant upper bounds, they are only used in stability analysis rather than controller implementation or accuracy calculation.
In order to deal with the computational burden induced by frequent controller triggers, a dynamic mechanism is required to determine whether to send updated control input to the plant. Here, we implement an event-triggered control scheme with a time-varying relative threshold [40] as follows: where β j and γ j are positive design parameters, t k is the update time, and u j (t) and τ j (t k ) are the jth element of u(t) and τ (t k ), respectively. Once the mechanism (10) is triggered, the control input u j (t) will be updated by the intermediate virtual control law τ j (t k+1 ). Thus, for t ∈ [t k , t k+1 ), u j (t) remains at τ j (t) updated at the last moment such that which further indicates u j (t) = τ j (t) where 1j (t) and 2j (t) ∈ [−1, 1] are the time-varying threshold parameters. Given ϑ 1 = diag{1/1 + 1j (t)β j } and ϑ 2 = −col{ 2j (t)γ j / 1 + 1j (t)β j }, a more compact form can be deduced as which also implies the boundedness of ϑ 1 and ϑ 2 , namely there exist positive scalars ϑ 1 ,θ 1 , ϑ 2 , andθ 2 such that ϑ 1 ∈ [ϑ 1 ,θ 1 ] and ϑ 2 ∈ [ϑ 2 ,θ 2 ].

A. Controller Design
With the desired state denoted by x id (t) ∈ R n i , the tracking error is represented by e i (t) = x i (t) − x id (t). The control objective can be stated as follows. For the nonlinear MIMO system (8), design the event-triggered control scheme to guarantee the PPTS as well as asymmetric constraint performance of the tracking error. To this end, we utilize the back-stepping approach to conduct the overall stability design and control synthesis. To what follows, the arguments will be omitted without specific notes to avoid the ambiguity, e.g., e i ≡ e i (t).
Step 1: Consider the state transformation z 1 = e 1 + Δk c and z 2 = x 2 −ῡ 1 , where Δk c is the bias constraint function andῡ 1 is the auxiliary stabilizing function to be designed later. Then, consider the EBLF where In general, the constraint function is set to limit state errors. Thus, a monotonically decreasing function that eventually tends to zero is chosen as constraint function. Then, where Recalling z 1 = e 1 + Δk c , the dynamics of where μ c = (k c /k c ) 2 + 1 and 1 > 0 is constant. Note that the following inequalities hold in (15): Setting we conclude from (15) thaṫ Designing the auxiliary stabilizing functionῡ 1 as Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
where a 1 = 1/a 1 and ρ = (a + |ς| ς ).â 1 andθ 1 are the estimation of a 1 and θ 1 , which are updated bẏ with σ a1 and σ θ1 being positive scalars. Hence, combining (18) and (19) yields Consider the Lyapunov function candidate where tr(·) is the trace of the corresponding matrix,ã 1 = a 1 − a 1 , andθ 1 =θ 1 − θ 1 . Using (19)- (22), the time-derivative of V 1 can be calculated aṡ The following inequalities hold in the last two terms of (23): Therefore, we have a more simplified form ofV 1 by substituting (20) and the above inequalities into (23) where . It can be concluded from (26) that the PPTS convergence of z 1 can be guaranteed if z 2 is stabilized.
Step k (2 ≤ k ≤ n − 1 ): Define the state transition z k = x k − υ k−1 , in whichῡ k−1 is the auxiliary stabilizing function. Introducing the Lyapunov function candidate: where σ ak and σ θk are positive scalars,ã k =â k − a k , θ k =θ k − θ k , and a k = 1/a k , whileâ k andθ k will be defined later. Taking the time derivative of V k and using inequalities similar to (16) and (17) yielḋ

Based on Young's inequality, we have
where j > 0 for j = 1, 2, . . ., n. Hence, substituting (28) into (27), one can obtaiṅ where unnumbered eq., shown at the bottom of this page, and κ k is defined as The auxiliary stabilizing function,ῡ k , is then designed as whereâ k andâ k are regulated byȧ k = σ ak z T k υ k − ρâ k andθ k = σ θk z k ψ T k − ρθ k , respectively. Therefore, we can obtain a compact form ofV k through further consolidation and simplificatioṅ j=1 jāj and Young's inequality is utilized.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
It follows from (36) that τ j (t) is bounded and continuously differentiable such that |τ j (t)| ≤ χ j . In view of the fact that τ j (t k ) − u j (t k ) = 0 and lim t→t k+1 τ j (t) − u j (t) = κ i , there exists a positive scalar t such that {t k+1 − t k } ≥ t ≥ κ j /χ j . Hence, Zeno behavior can be effectively eliminated in the proposed control scheme.
Remark 7: Compared to existing works that focused on addressing symmetric constraints [41]- [43], we propose a novel unified control method based on bias state transformation to satisfy asymmetric constraint requirements. To achieve the prescribed asymmetric constraint k cj < e 1j <k cj for j = 1, 2, . . ., n, in whichk cj ∈ R + and k cj ∈ R − are user-defined constraint functions, then k c and Δk c can be calculated as k c = max{k cj } and Δk c = col{Δk cj }, where k cj =k cj − k cj /2 and Δk cj = −k cj − k cj /2, respectively. In particular, the bias constraint function becomes zero in the case of symmetric constraints, namely Δk cj = 0. This approach transforms an asymmetric constraint into a symmetric one by means of bias state transformation, eliminating the need for a real-time determination of the sign of the constraint state or its associated state transformed variables, which thus avoids the singularity problem that may arise from the derivation of the sign function. As a consequence, the proposed control scheme is unified to address nonlinear MIMO systems with symmetric or asymmetric time-varying constraints.

IV. SIMULATION AND EXPERIMENT
We carried out simulations and an experiment in two scenarios, namely planar robot tracking and bimanual-local-single-remote teleoperation, respectively, to test the proposed controller (36).

A. Simulation on 2-DoF Planar Robot
The plant considered in this simulation is a 2-DoF planar serial robot with dynamics   [44].
Simulation is performed with T = 0.6, 1, and 1.5 s. According to Theorem 1, the position synchronization errors need to reach e 1 ≤ 2b/α = 0.028 when arriving at the preassigned settling time. Table I summarizes the transient and steady-state accuracy under the three parameters. We see that the output state can meet the specified accuracy at the specified time and rapidly reach a high convergence accuracy. Since the predetermined accuracy is greater than the value of the constraint function in the case of T = 1.5 s, e 1 follows the constraint function in preference. In the other two cases, e 1 t=T is basically equal, which indicates that the convergence time and accuracy are decoupled. Fig. 1 demonstrates the convergent profiles of the tracking errors, where e 1 = x 1 − x 1 d , e 2 = x 2 −ẋ 1 d , e j1 and e j2 are denoted as the first and second elements of e j (j = 1, 2), respectively. We see that the PPTS characteristic is guaranteed while the tracking error never violates the asymmetric constraints. Fig. 2 depicts the trigger moment and the released intervals of the controller under the action of the event trigger mechanism (9)-(10) in the case of T = 0.6 s. During the transient-state process, the proposed controller takes only 0.166 s to update and act on the driving joints, effectively reducing the computational burden.
We perform comparative simulations of finite-time control method with prescribed performance [6]. The control parameters are set to T = 1 s with the initial value x 1 (0) = [2, −1] T rad. The asymmetry of the constraint function is further enhanced by k c = 3.6 exp(−4.7t) + 10 −4 and Δk c = 0.5 exp(−4t) + 10 −4 . The rest of the parameters are the same as in the simulation above. To ensure a fair comparison, we guarantee a settling time (T = 1 s) driven by finite-time control [6] through the following parameters: α 2 = 5, γ 1 = 4, γ 2 = 5/7, μ = 0.001, and ρ = 9/11. Fig. 3(a) and (b) shows that the proposed method overcomes the overshoot caused by the asymmetric constraints while achieving PPTS convergence. The singularity problem that often occurs in finite/fixed time control has also been effectively circumvented through employing damped reciprocal approach, as shown in Fig. 3(b). Compared with Fig. 1, it can be observed that the settling time in the proposed method is independent of the initial value and state constraints, but only determined by the time parameter T , thus validating the superiority of the proposed controller.

B. Experiment on Heterogeneous Teleoperation System
In the experiment, we asked the human operator to draw a 2-D circle trajectory with 9-cm radius at the local side using two hands simultaneously. The teleoperation setup is composed of local and remote sides via a communication channel, where the operator manipulates the end handles of two 7-DoF Omega 7 haptic devices (Force Dimension Inc., Switzerland) based on visual feedback, as shown in Fig. 4(a). The command signals given by their two hands are integrated utilizing a Kalman filter to attenuate noise in the hands motion. Data transfer between human and the remote robot is realized via TCP/IP protocol, where the control frequency and additional latency are 100 Hz and 0.5 s. Inspired by [45], latency is introduced to test the savings in communication resources and robustness against delay-induced perturbations. The control objective for the remote robot is to track the state signal sent from the human operator. A 2-DoF simulated SCARA Robot is used as the remote site. Considering the workspace limitation, the exponential-decay constraint functions are set as k c = 4 exp(−4.6t) + 10 −3 and Δk c = 0.1 exp(−4.6t) + 10 −3 . The desired joint positions sent from the local site are mapped from Cartesian space to joint one using inverse kinematics. It can be observed in Fig. 4(b) that the position tracking errors converge within the prescribed settling time and never exceed the user-defined asymmetric time-varying constraints. In addition, the velocity tracking performance is guaranteed, as shown in Fig. 4(c). The PPTS characteristic is therefore ensured [i.e., e 1 ≤ 9.38 × 10 −4 (rad), e 2 ≤ 5.5 × 10 −3 (rad/s), t > 1 s].

V. CONCLUSION
This article introduced a new sufficient condition for PPTS of nonlinear continuous-time systems. Based on piecewise scalar function, we propose a new adaptive event-triggered control scheme that ensures the tracking error converges to a prescribed convergence domain within a user-defined settling time. In addition, the EBLFbased control framework can effectively handle asymmetric output constraints with the help of bias state transformation, ensuring that the output state never violates the predefined constraints. This approach can deal with symmetrical and asymmetrical constraints in a unified PPTS control structure with a low computational burden, as a result of setting the bias constraint function. Therefore, critical performance metrics can be arbitrarily prescribed based on task requirements, including the settling time, convergence accuracy, and motion constraint, which provides a quantitative basis for parameter regulation.