Event-Triggered Fixed-Time Control for Steer-by-Wire Systems With Prespeci(cid:12)ed Tracking Performance

This paper addresses the event-triggered output feedback control problem for (steer-by-wire) SbW systems with uncertain nonlinearity and time-varying disturbance. First, a new framework of event-triggered control systems is proposed to eliminate the jumping phenomenon of event-based control input, and the trade-oﬀ between saving communication resources and attenuating jumping phenomenon can be removed. Then, the adaptive disturbance observer and fuzzy-based state observer are developed to estimate the external disturbance and unavailable state of augmented SbW systems, respectively. Third, an event-triggered (cid:12)xed-time control is developed for SbW systems to achieve pre-speci(cid:12)ed tracking accuracy while saving communication resources of the controller area network (CAN). Furthermore, theoretical analysis based on Lyapunov stability theory is provided to verify the tracking error of SbW systems can converge to the prespeci(cid:12)ed neighborhood of the origin in (cid:12)xed time regardless of the initial tracking error. Finally, simulations and experiments are given to evaluate the eﬀectiveness and superiority of the proposed methods.


Introduction
The steer-by-wire (SbW) system is one of the main subsystems of autonomous vehicles, realizing the steering control of front-wheels. Compared with the conventional steering system, SbW systems have two distinct characteristics: 1) the mechanical linkage between the steering wheel and front-wheels is no longer required; 2) the additional sensors, steering motor actuator, control unit, and controller area network (CAN) are necessary for SbW systems. Although the steering control of autonomous vehicles can be achieved through the SbW system, it is still a challenge to achieve accurate modeling and control of the SbW system with measurement and communication limitations [1][2][3][4][5].
Significant contributions have been made to the literature of SbW control systems; see, e.g., [6][7][8][9][10][11][12][13][14][15] and the references therein. Representatively, the model-based control method, such as proportional-integral-derivative control [6], proportional-derivative control with feedforward compensation [7], and model predictive control [8] are nvestigated for SbW systems. Considering the SbW system's parametric uncertainties and disturbances, the fairly accurate state model and robust sliding mode control are first reported in [9]. Then, the adaptive sliding mode control method of SbW systems is investigated in [10][11][12][13] without a priori bounds of uncertainties and disturbances. References [14,15] address the active fault-tolerant control problem of SbW systems subject to modeling uncertainties and disturbances with a priori bounds. A detailed literature review of the SbW control system can be found in Tab.1. Although the methods mentioned above have obtained fruitful achievements in SbW control systems, there are still the following limitations: 1) multi-sensor measurement technology is required in the control application,

Rreference
Technique Main limitations Rreferences [6][7][8] Model-based control method 1) Multi-sensor measurement technology is required in the control application, which increases the hardware complexity and cost of SbW systems, and 2) the uninterrupted transmission of the control input to the steering motor is required, which occupies unnecessary CAN communication resources.
Rreferences [9][10][11][12][13] Sliding mode control for uncertain SbW systems with disturbance Rreferences [14,15] Model predictive control for SbW systems with actuator fault  [16,17] Sliding mode differentiator k-th derivative of signal with the known Lipschitz constant or bounded function. Rreference [18] Sliding mode differentiator for nonlinear mechanical systems The input and output of controlled systems should be bounded.
Rreferences [19,20] Sliding-mode observer of uncertain Lagrangian systems The bounds of uncertainties must be a priori.
Rreferences [35,36] Model-based event-triggered control for nonlinear systems Rreferences [27,28,31,37,38] Adaptive Event-triggered control for uncertan nonlinear systems Rreferences [39,40] Jumping-attenuation eventtriggered control which increases the hardware complexity and cost of SbW systems, and 2) the uninterrupted transmission of the control input to the steering motor is required, which will occupy unnecessary CAN communication resources. In practical applications, the vehicle-mounted CAN with limited communication channel bandwidth is usually shared by different nodes, so addressing the communication constraints is of great significance.
To reduce the use of sensors in control systems, significant contributions have been made to the literature of the state estimator. Representatively, the sliding mode differentiator is proposed in [16] for the signal with the k-th derivative having a known positive Lipschitz constant. Reference [17] develops a robust kth-order differentiation for signals with a given func-tional bound of the (k + 1)-th derivative. Reference [18] proposes the sliding mode differentiator for nonlinear mechanical systems with bounded-input-bounded-state (BIBO) property. Reference [19,20] proposes a global sliding mode observer for nonlinear mechanical systems subject to the Coriolis term and uncertainty with a priori bound. The first-order low-pass is considered the differentiator for signals with bounded derivatives [21][22][23], and the high-gain observer is developed for nonlinear systems with accurate model [24,25]. Although the above state estimators are useful in some particular applications, they are challenging to achieve satisfactory estimation performance for SbW systems without BIBO property, bounded derivatives of state, accurate model, and a priori bounds of uncertainty/disturbance. To this end, the approximator-based observer is proposed in [26][27][28][29][30][31], but the convergence of observation errors is asymptotic rather than lemma.fixed.time -time. For this reason, this paper proposes an adaptive state observer for an uncertain SbW system without the above assumptions and limitations shown in Tab.2.
In network control systems, the communication networks are usually shared by different system nodes, while the network resources including communication channel bandwidth and computation, abilities are limited. To reduce the unnecessary waste of communication resources, great efforts have been made to develop event-triggered control, in which information transmission occurs only when necessary, rather than continuously; see, e.g., [27,28,[31][32][33][34][35][36][37][38][39][40] and the references therein. Specifically, with the assumption of the input-to-state stability (ISS), the event-triggered stabilization and tracking problem of nonlinear systems are addressed in [32][33][34], respectively. In [35,36], the model-based eventtriggered control method is proposed for network-based plants. Considering the model uncertainty of nonlinear systems, the event-triggered state and output feedback control are developed in [37,38] and [27,28,31], respectively. It is worth mentioning that the control input is updated and transmitted only at event-triggered instants, so the jumping phenomenon of control input caused by communication is inevitable. Consequently, the large impulse will be applied to the system, especially in event-triggered control systems with relative threshold, which certainly affects the smoothness of actuator output and degrade the system performance [39,40].
To this end, the switching triggered strategy, including fixed threshold strategy and relative threshold strategy, is proposed in [39,40] to attenuate the jumping phenomenon of event-based control input. As shown in Fig.1(a), the core idea of the switching triggered strategy is that the fixed threshold strategy will be applied when the amplitude of control input is large, and the relative scheme will be applied when the amplitude of control input is small. Thereby, the event-triggered error has been constrained within the bounds of a constant, and the jumping phenomenon can be attenuated while saving communication resources as much as possible in this context. Unfortunately, it is difficult to solve the trade-off between saving communication resources and attenuating the jumping phenomenon, and the jumping phenomenon still exists in [39,40]. This makes it difficult for the existing event-triggered control system to be applied to electromechanical systems with higher requirements for the smoothness of their actuators. Much imoportantly, the practical stability can be achieved asymptotically or within finite-time rather that fixed-time, which cannot guarantee the transient performance of closed-loop systems.
It is essential to save communication resources and reduce hardware costs from applying and developing network-controlled SbW systems. Simultaneously, to ensure vehicles' safety and comfort, it is necessary to achieve fixed-time prespecified tracking performance while guaranteeing the smoothness of the SbW system steering motor's output. As mentioned in Tab. 1 and Tab. 3, the existing SbW control systems and event-triggered control systems cannot meet these requirements and should be improved. For this reason, this paper proposes an even-triggered output feedback control method for SbW systems. The contributions of this paper are summarized in the following aspects: (1) From the perspective of even-triggered control system design: A new framework of the event-triggered control system is proposed in this paper. 1) Combined with the event-triggered control systems in [27,28,[31][32][33][34][35][36][37][38], as shown in Fig.1(b), the jumping phenomenon of control input can be eliminated, and 2) compared with the event-triggered control systems in [39,40], the trade-off between saving communication resources and attenuating jumping phenomenon can be removed.

Relative threshold
Fixed threshold Fixed threshold Fig. 1 Event-triggered control input. (a) Control input in [39,40], (b) Control input in the designed event-triggered control system.
(2) From the perspective of state observe design: An adaptive state observer for an uncertain SbW system without the above assumptions and limitations shown in Tab. 2 is proposed in this paper. 1) Compared with recent researches on state observer [16-19, 41, 42], there is no need for additional assumptions, such that bounded-state property, bounded higher-order derivatives of output, and a priori known nonlinearities, and 2) compared with recent researches on FLS/NN-based state observer [27,28,30,43,44], the convergence speed of the observation error can be improved in this paper, i.e., the observation error can converge to the adjustable neighborhood of the origin in finite-time instead of asymptotically converging to the small neighborhood of the origin.
(3) From the perspective of control method design: An observer-based event-triggered fixed-time control is proposed for uncertain SbW systems regardless of the initial tracking error. 1) Compared with SbW control systems [9][10][11][12][13][14][15], the unnecessary sensor can be removed and CAN communication resources can be saved, 2) compared with event-triggered output feedback control [28,31,40], the prespecified tracking performance can be achieved within fixed time, so both the transient and steady performance of closed-loop systems can be guaranteed, and 3) compared with [27], the bounds of the initial state of the controlled system is no longer required.
The rest of the paper is organized as follows. In Section II, problem formulation and preliminaries are given. The fuzzy-based state-observer and event-triggered fixed time control is presented in Section III. The simulation and experiment are given in Section IV. Section V is the concluding remarks.

Problem formulation
According to the researches [7,9,10,45], the dynamics model of the steering motor can be established as The rotation of the front-wheels around their vertical axes can be modeled as where H f (θ f ,θ f ) = τ e + τ f denotes the uncertain nonlinearity. The transmission ratio between the steering motor and front-wheels is which together with (1)-(2) gets Steering motor with J e = J f + µ 2 J m . For brevity, the dynamics model (4) can be rewritten as     ẋ )/J e denotes the lumped uncertain nonlinearity of the SbW system, g = µ/J e , and d o (t) = µτ d /J e denotes the lumped motor pulsation disturbance [9,10].
Control Objective: This paper addresses the eventtriggered output feedback control problem for SbW systems, such that (1) the tracking error between the front-wheels steering angle and its reference signal can converge to the prespecified neighborhood of the origin in fixed time, and (2) the jumping phenomenon of the control input caused by event-triggered communication can be eliminated.
The following assumptions are made on the reference signal and external disturbance, respectively.

Assumption 1
The reference signal y d (t) is known, and there exist unknown positive constantsȳ d andȳ d such that |y d (t)| ≤ȳ d and |ẏ d (t)| ≤ȳ d .  (5), as- Remark 1 : The above assumptions are relatively general, even compared with the existing researches on time-triggered tracking control of SbW systems [9][10][11][12][13]. Specifically, 1) for the assumption of the external disturbance d(t), the same assumption as Assumption 2 can be found in [9][10][11][12]. Moreover, the assumptions that both the external disturbance and its time derivative are bounded is made in [15]. 2) for the assumption of the reference signal y d (t), both the time derivative of the reference signalẏ d (t) and its second-order time derivativeÿ d (t) are required in the control design [9][10][11][12][13]. It is worth noting that to ensure driving safety and vehicle comfort, the reference path is usually smooth and its change rate is usually bounded in the practice application. Thereby, we can find that the time derivative of the reference signal, i.e.,ẏ d (t), may not be known but can be considered bounded in this context. So compared with the related works, e.g., [9][10][11][12][13], Assumptions 1-2 are rather mild. In [7,10,46], the self-aligning torque τ e can be obtained where β and γ can be obtained from the following the two-degree-of-freedom model of the vehicle where the parameters of the model are defined in Table  4.
Besides, the following simplified model of self-aligning torque can be obtained [9,11,47,48] where ρ τ denotes a time-varying coefficient for various road conditions, and θ f means the steering angle of the front-wheels. From (6)- (7), one can find that τ e is firstorder differentiable. Furthermore, the SbW system is a typical electromechanical system that can be expressed by the general EulerLagrange formulation. According to the existing researches [49,50], the friction torque τ f can be presented as the following parameterized form with α i and β i , i = 1, 2, 3 being the positive constants to be defined, which means that τ f is differentiable. From the above analysis, one can find that the selfaligning torque τ e and friction torque τ f of SbW systems are first-order differentiable, i.e., the nonlinearity f 0 (x 0 ) of (5) is differentiable. From a practical point of view, the self-aligning torque τ e and friction torque τ f of SbW systems have physical meaning. So their rate of change exists and is continuous. Therefore, the assumption that the nonlinearity of the SbW system is first-order differentiable is true in practice.

Fuzzy logic system and Useful Lemmas
A typical FLS consists of four parts: knowledge base, fuzzifier, fuzzy inference engine and defuzzifier. The knowledge base consists of a series of fuzzy IF-THEN inference rules: where χ l , l = 1, · · · , n, and Y denote the inputs and output of FLS, respectively, F j l and G j are fuzzy sets and their membership functions are µ F j l (χ l ) and µ G j (Y), respectively, m is the number of fuzzy rules. Then, through the singleton fuzzifier, center average defuzzification, and product inference, the output of FLS can be expressed as is the parameter vector, and the fuzzy basis function vector is ξ( Lemma 1 (see [51]) Suppose that the input universe of discourse Ω is a compact set in R n . Then, for the continuous function f (χ) on Ω and arbitraryω > 0, there exists an FLS (9) with an optimal parameter vector Θ * such that Lemma 2 (see [52]) Consider a class of systemsẋ = f (x), there exist a smooth positive-definite function V (x) and some positive scalars α > 0, β > 0, p > 0, p > 0 and k > 0 such thaṫ Then the fixed-time stability can be guranteed with the

Lemma 3 Consider a class of systemsẋ
there exist a smooth positive-definite function V (x) and some positive scalars with T 0 being the initial time.
The proof of Lemma 3 is given in Appendix A.

Remark 2
To improve the closed-loop system's transient performance, finite-time control methods have been developed for various nonlinear systems during the past few years [53][54][55]. Generally, the practical finite-time stability of closed-loop systems can be guaranteed for uncertain nonlinear systems. As shown in Tab.5, the conclusion of practical finite-time stability theory can be obtained from the existing research. From Tab. 5, one can find that [53] provides a more general Lyapunov condition of the practical finite-time stability. However, consider the systemẋ = f (x), if there exist continuous positive-definite function V (x) and some scalars α i > 0, i = 1, 2, 3, 1 > β 1 > 0, and 1 > β 2 ≥ 0 such thaṫ when β 2 ̸ = 0, the trajectory of systemẋ = f (x) is difficult to judge based on the conclusions of [53] and [54]. This Lyapunov condition (i.e., β 2 ̸ = 0) is corresponds to the stability analysis of the state observer in this paper. Thus, the following Lemma is given in this paper.

Fuzzy-based state observer design
To eliminate the jumping phenomenon of control input, a new framework of the event-triggered control system, as shown in Fig.3, is proposed. In this context, based on [16,56], the following augmented system (15) can be established from the system (5) by introducing the auxiliary variable ,u is considered as the "new" control signal to be designed.
As shown in Fig. 4, the angle position x 1 of the augmented system (15) can be measured by a linear sensor. In practical applications, the angular velocity of the front wheel's steering angle can be obtained through a sensor such as an encoder or gyroscope. However, the introduction of non-essential sensors will increase the hardware complexity and cost, thereby reducing the reliability of SbW systems. For this reason, the state observer is proposed for the augmented system (15), i.e., where χ = [x 1 ,x 2 ,x 3 ] T and Y = Θ T ξ(χ) denotes the input and output of FLS,x 1 =x 1 − x 1 ,d(t) is the estimation of the disturbance d(t), the parameters k 1 , Besides, the adaptive laws of Θ andd(t) are designed aṡ where ψ(x 1 ) = −♭(t)ϕ 1 (x 1 )/(|ϕ 1 (x 1 )|+ε ob ), σ 1 , σ 2 , γ 1 , γ 2 and ε ob are positive constants, and ♭(t) with ♭ ob > 0 is For the adaptive state observer (16), the following main results can be obtained in this section. (18)

Lemma 4 Consider the adaptive laws
with C ob = max( µ1 λmin(Q) , C o1 )/λ 1/2 min (P ), and with Q being the designed positive definite matrix,Θ has been defined in Lemma 4,Φ =Θ + ∥Θ * ∥ +d +ω,ω has been defined in Lemma 1, andΘ andd have been defined in Lemma 4. Proof : Combined with (5) and (16), the following dynamics of observation errors can be obtained Consider the following Lyapunov function candidate where ζ = [ϕ 1 (x 1 ),x 2 ,x 3 ] T and P = P T > 0. Note that x 1 ≡ 0 meansẋ 1 ≡ 0 is also hold, and from (22), one can find that V ob ≡ 0 will be always hold in this context. Thus the following analysis is given for the condition thatx 1 ̸ = 0. From (22), the following equation can be obtaineḋ with B 1 = [0, 1, 0] T and B 2 = [0, 0, 1] T . Combined with (23) and (24), the derivative of V ob can be obtaineḋ with . This together with Lemma 3 yields This ends the proof of Theorem 1.
Remark 3 : As can be seen from Lemma 1, the approximation property of FLS can only be established in a convex region of interest, which implies that the initial states are within the bounded set. The same is true for all FLS/NN-based methods.

Controller area network (CAN)
Angle sensor
Theorem 2 Consider the augmented system (15) under the developed state observer (16) and event-triggered control (39), the tracking error can converge to the prespecified neighborhood of the origin in fixed-time, i.e., where ℓ 1 has been defined in (29), and T tr is defined as with c 2 = min(η 12 , η 22 , η 32 ) andC ≥ ∥x∥ for x ∈ Ω.
Remark 4 : In the practical application of the digital control system, the following dead-zone technique can be used to prevent the parameter drift problem of the adaptive law (35) where κ(0) = 0, ε c is the small positive constant to be designed. Combined with (48) and the analysis of Theorem 2, it is not difficult to find that V ≤ ε c can be achieved within fixed time. This together with (26) and (45) yields that can be achieved in fixed time.
Remark 5 : In this paper, the contradiction method to prove that Zeno-behavior [57] is avoided. Suppose that ∆t k = t k+1 − t k = 0. Due to the function υ(t) is always continuous, so one has However, it can be seen from (40) that The above analysis indicates that (50) contradicts the event-triggering condition (40), which means that the Zeno-behavior can be strictly avoided under the triggering mechanism (40).

Numerical simulation
(1). Simulation model of the SbW system According to existing researches [10,11], the friction torque τ f in (4) are regarded as τ f = 0.25(tanh(100x 2 )− tanh(x 2 )) + 30 tanh(100x 2 ) + 10x 2 , and self-aligning torque τ e and the parameters of the system (4) can be easily obtained in [10]. The different stiffness coefficients and vehicle velocity with respect to τ e are respectively used in the two simulations The , i = 1, 2, j = 1, 2, 3 and µ F j , j = 1, 2, 3 with the fuzzy set F j i of the inputs. The fuzzy IF-THEN rules are selected as shown in Tab.6, where Table 6 The rule base of the fuzzy logic system G j , j = 1 · · · 27 are the fuzzy sets of the fuzzy output, and the center points of the sets G j are defined as Θ j with Θ j being the j-th element of the vector Θ. The parameters of (26)- (41) and (48)
The adaptive event-triggered control [39] and observerbased fuzzy event-triggered control [27] are chosen for comparison in this paper. For the SbW system (5), the adaptive state observer designed in [27] can be expressed as where ♭ 1 and ♭ 2 satisfy that A = [−k 1 , 1; −k 2 , 0] is Hurwitz, and u f = H L (s)u with H L (s) being the Butterworth low-pass filters (as described in [27]). The eventtriggered control designed in [27] can be expressed as   with ϱ 2 > 0, and r(t) being designed as r(t) = α 2 (t) − ϱ 1 tanh (ς 2 ϱ 1 /ϵ), where ϱ 1 > ϱ 2 > 0, ϵ > 0, ς 2 =x 2 −α 1 , and α 1 and α 2 can be found in [27]. For the eventtriggered control [27], the parameters of equations (7), control with swithcing threshold strategy [39] can be expressed as where e(t) = u(t) − ω(t) δ, m, m 1 and D are positive constants to be designed, and ω(t) being designed as where α 2 can be obtained from [39], and the related parameters in [39] are chosen as c 1 = 180, φ 1 = 0, c 2 = 160, . Simulation results and analysis Fig. 6 and Tab. 7 give the control performance of the different ETCs in the simulation I. From Fig. 6 and Tab. 7, one can find that in the designed event-triggered control system, the tracking error can converge to the smaller residual set of the origin, and the jumping phenomenon of control input can be avoided while saving more communication resources. Fig. 7 and Tab. 8 give the control performance of the different ETCs in simulation II. One can find that the ETC designed in this paper also has better adaptability to model uncertainty. Fig. (8) and Fig. 9 give the observation results in simulation I and simulation II. From Fig. 8, one can find that both the designed adaptive state observer and modelbased HGO can achieve satisfactory observation performance without model uncertainty. As shown in Fig.  9, when the model uncertainty is considered in simulation II, i.e., when the model parameters are changed, the observation performance of the adaptive observer designed in this paper is better than that of the modelbased HGO.

Experiment
The experiment platform of SbW systems is shown in Fig. 10. In this platform, the single board computer (dSPACE-ds1202) is used as the control unit of SbW systems, and the servo motor driver (XiNJE DS2-20P7) is used for driving the steering motor (XiNJE MS80ST-M02430B-20P7) equipped with a reducer. The linear sensor (KTR11-10) fixed on the steering arm measures the steering angle of the front-wheels. A computer is applied to display the experimental results of the experiment on-line and store the experimental data. The sampling period is chosen as 0.001s. (

(2). Parameter selection of the compared methods
To verify the adaptability and superiority of the designed observer, HGO [25] is used to estimate the system (15) for comparison in experiment. According to Theorem 3.2 and the equation (19) of [25], L h = [14.16, 339.4, 7965.86] T is chosen in experiment by solving matrix inequality (24) of [25]. Besides, the following low-pass filter [21] is used for comparison whereẋ is the estimation ofẋ and τ is the small positive constant to be designed. τ = 0.005 is chosen in experiment.
To verify the jumping phenomenon of control input caused by the event-triggered communication can be eliminated under the designed event-triggered control system, the adaptive event-triggered control [39] and observer-based fuzzy event-triggered control [27] are chosen for comparison. For the event-triggerid control [39], the parameters and functions of the equations (5), (7), (21), (22) and (28)  (3).Experiment results and analysis Fig.11 and Tab.9 give control results in experiment I under the different ETCs. It is easy to find that in the designed event-triggered control system, the tracking error can converge to the smaller neighborhood of the origin, and the jumping phenomenon of control input caused by event-triggered communication can be avoided while saving more communication resources can be saved. Fig.12 and Tab.10 give control results in experiment II. It also can be found that the designed ETC can achieve better tracking accuracy, and the control input is without jumping phenomenon. Fig. 13 and Fig.14 give the observation result of the state x 2 and x 3 under the different experiments. From Fig. 13 and Fig.14, one can find that the estimation results under  the state observer are more smooth and suitable for the design of the output feedback controller. Besides, one can also find that the trend of observation results under the designed state observer is more consistent with the estimation results of the low-pass filter than the model-based HGO.

Conclusion
This paper proposes an event-triggered fixed-time control for uncertain SbW systems by considering the bandwidth limitation of CAN. A new framework is proposed to eliminate the jumping phenomenon of the eventbased control input. Then, an adaptive fuzzy-based state observer and disturbance observer are proposed to estimate the unavailable state and disturbance of the augmented SbW system . Furthermore, an adaptive event-triggered control is proposed for SbW systems by considering the effect of observation error and eventtriggered error, such that the prespecified tracking performance can be guaranteed within fixed time. Finally, simulations and experiments are presented to evaluate the effectiveness and superiority of the proposed methods. Future research on the SbW control system will include the following aspects: (1) Input saturation and output constraint of the SbW system should be addressed. In the practical appli-cation, the control input saturation of SbW systems, i.e., output torque saturation of the steering motor, often occurs due to hardware limitations, limiting the SbW system performance severely even leads to the SbW system instability. Besides, the output constraint of SbW systems, i.e., the steering angle constraint of the front-wheels, is inevitable. So, the input saturation and output constraint of SbW systems will be considered in the control design. (2) The time delay phenomenon should be considered.
In the practical application of the SbW system, the time delay phenomenon caused by measurement and communication is inevitable, which will be considered in future research on SbW control systems.

ςα2
. This completes the proof of Case 1 in Lemma 3.
Case 2 β 1 < β 2 . In this case, the inequality (13) can be expressed as (61). Thus, the conclusion as in (64) can be obtained, which completes the proof of Case 2 in Lemma 3. This completes the proof of Lemma 3.