Decentralized Dynamic Event-Triggered Communication and Active Suspension Control of In-Wheel Motor Driven Electric Vehicles with Dynamic Damping

I. Introduction

THE automotive industry has witnessed a substantial increase in the popularity of advanced technologies such as replacement of the internal combustion engine vehicles with electricity driven motors. Electric vehicles provide a promising solution to the challenges offered by the internal combustion engine vehicles such as environmental friendliness and energy efficiency, and therefore considerable research has been conducted on the development and optimization of electric vehicles [1]. Based on the propulsion system, electric vehicles can be roughly classified into centralized motor driven electric vehicles and in-wheel-motor (IWM) driven electric vehicles. Specifically, IWM configuration offers several advantages over centralized configuration (similar to the conventional internal combustion vehicles), such as generation of precise and fast torque, while not effecting the driveshaft stiffness. Different control strategies for electric vehicles, including vehicle dynamics control [2], steering control [3], and lateral control [4], [5], have been developed. However, installing a motor in a wheel results in an increase in the unsprung mass which in turn aggravates the road holding and ride comfort performance. Therefore, the influence of an IWM on ride comfort and vehicle safety is required for suitable treatment in suspension system design. For example, to improve road holding performance, a dynamic-damping in-wheel motor driven system is developed in [6] such that the system suspends the shaftless direct drive motor and isolates it from the unsprung mass.

Note that road holding and ride comfort are in conflict with each other. Such a conflicting issue therefore needs to be carefully addressed during suspension system analysis and design. Among its other counterparts, active suspension has been acknowledged as a promising means to provide a compromise between these contradictions. Naturally, much attention has been paid to the development of active suspension systems. For example, several active control techniques, such as T-S fuzzy control [7], [8], robust $ H_{\infty} $ control [9], [10], linear optimal control [11], neural-network-based adaptive control [12], [13], have been developed to improve the suspension control performance. To overcome the disadvantages of an IWM in electric vehicles, active suspension under dynamic damping also has been intensively studied [14]-[16] with an aim to manage the trade-off between ride comfort and road holding performance of IWM-driven electric vehicles. For example, by installing a dynamic damper, a vibration mitigation method was presented in [16] for electric vehicles with an in-wheel switched reluctance motor. On the other hand, to obtain better disturbance attenuation performance, the active suspension control design can be explored in a constrained frequency range rather than the entire frequency range [15], [17], [18]. For example, the finite-frequency $ H_{\infty} $ control problem of active suspension system was studied in [17], where the $ H_{\infty} $ norm of the disturbance to control output is minimized in a finite frequency range to preserve ride comfort improvement. Based on static output feedback, a robust finite-frequency $ H_{\infty} $ active suspension control approach was proposed in [18] to improve the ride comfort performance of the suspension system while preserving the hard constraints.

There has been a considerable growth in intra-vehicular networked control systems with the ever-increasing demand of vehicle safety and performance in modern vehicles. A key component is the network architecture that connects the intra-vehicular control system components, such as sensors, actuators and electronic control units. Automotive industry is continuously striving for substantial progress of wired and wireless data communication infrastructure. A widely adopted technology is the controller area network (CAN) that requires the data to be digitized and sampled before broadcasting and transmission over the CAN buses. Today’s autonomous vehicles have a myriad of applications and tasks enabled by CAN, which makes the essentially finite CAN resources more scarce than ever. However, most of existing results on vehicle active suspension control have assumed continual data transmissions among sensors, controllers and actuators. Due to the limited in-vehicle network resources, the data transmissions to be broadcast and transmitted over CAN should be wisely selected and scheduled. Traditionally, a periodic (or time-triggered) communication mechanism has been widely adopted to perform the data transmissions and/or the executions of control tasks at only discretized instants, leading to the so-called sampled-data control for vehicle active suspension systems [10], [19]-[21]. Nevertheless, the sampling period under the conventional sampled-data control is often pre-set in the worst scenario to guarantee the desired system performance. Furthermore, a small sampling period results in a large number of redundant sampled data transmissions, causing excessive occupancy of the limited network resources and also the frequent changes in the actuator states (or known as the actuator attrition) at every sampling instant.

An alternative communication mechanism called an event-triggered communication mechanism has been put forward in the literature to eliminate the limitation of the traditional periodic communication mechanisms. Under such an event-triggered mechanism, data transmissions are invoked only after the occurrence of some well-defined events. As such, event-triggered communication mechanisms can generally achieve significantly improved communication efficiency while not sacrificing too much of desired system performance. Generally, the threshold parameter employed in an event trigger decides how often current data packets should be transmitted [22], [23]-[25]. However, it is noteworthy that most existing event-triggered communication mechanisms employ a pre-set constant threshold parameter in the event trigger design, and thus are often regarded as static event-triggered communication mechanisms (e.g., [26]-[29]). However, there are several practical scenarios that fixing the threshold parameter permanently is not preferable; see the justifications in [30], [31] and also the recent surveys [25], [32]. By inserting some extra internal dynamic variable into a triggering condition, the so-called dynamic event-triggered communication mechanism was developed for various networked dynamical control systems [25], [33]-[36]. On the other hand, an adaptive event-triggered control problem was addressed in [37] for vehicle active suspension system subject to uncertainties and actuator faults. The threshold parameter selected for the event trigger was adaptively adjusted with an aim to reduce data transmissions as much as possible. In [9], a dynamic event-triggered $ H_{\infty} $ control approach was developed for active suspension systems. In [38], an adaptive hybrid event-triggered mechanism was studied to tackle the reliable active suspension control of a cloud-aided quarter-car suspension system in a finite frequency range. A common feature of the designed event triggers in [9], [38] is that they involve a non-increasing dynamic threshold parameter and thus release more sampled data packets than the traditional static event trigger with a fixed threshold parameter. Furthermore, it should be also pointed out that the event-triggered active suspension control studies [9], [37], [38] are all conducted in a centralized manner, which means that the full state of the concerned vehicle suspension system has to be measured by a single sensor and a global event trigger is required to verify the triggering condition at the same time. Apparently, the complete state information of the vehicle suspension system may not be available for the single sensor at all times due to the limited sensor energy and sensing/computing capabilities. In this sense, a decentralized event-triggered communication mechanism in a multi-sensor setting is highly preferable to allow each sensor’s data transmissions to be selected independently. To the best of the authors’ knowledge, the decentralized dynamic event-triggered communication and active suspension control co-design problem for IWM electric vehicles with dynamic damping has not been explored in the existing literature, which motivates this study.

In this paper, we address the dynamic event-triggered active suspension control problem for an uncertain IWM electric vehicle with dynamic damper and actuator fault. To achieve a better trade-off analysis between suspension performance requirements and communication resource usage, the active suspension control issue of the IWM electric vehicle is accomplished under novel decentralized and centralized dynamic event-triggered communication mechanisms. The main contributions of the paper are summarized as follows.

1) A general active suspension model is established for an in-vehicle networked 3-degrees-of-freedom (DOF) quarter IWM electric vehicle subject to uncertain sprung mass, unsprung mass and dynamic damper mass, uncertain actuator fault, and intermittent data transmissions.

2) A novel decentralized dynamic event-triggered communication mechanism is developed to determine independently whether or not each sensor’s sampled data packets should be broadcast and transmitted over the CAN bus. When all system state components are simultaneously measurable, two centralized dynamic event-triggered communication mechanisms are also proposed to regulate the sensor’s data transmissions.

3) New co-design criteria for the desired dynamic event-triggered mechanisms and the admissible active suspension controllers are derived to guarantee prescribed suspension performance and satisfactory communication efficiency.

Furthermore, comprehensive comparisons between different event-triggered mechanisms and various road disturbance responses of an uncertain quarter vehicle suspension model are presented to substantiate the merits of the proposed results.

The rest of the paper is organized as follows. Section II provides the problem formulation. Section III performs the stability and performance analysis as well as the co-design of dynamic event-triggered communication mechanisms and fuzzy active suspension controller. Section IV presents the comprehensive simulations for verifying the derived main results. Section V concludes the paper.

Notation: $ {\mathbb{R}}^{m} $ represents the m-dimensional Euclidean space and $ {\mathbb{R}}^{m\times n} $ represents the set of $ m\times n $ real matrices. $ \|\cdot\| $ represents the Euclidean norm. $ \|w(t)\|_2 = \sqrt{\int_0^\infty w^T(t)w(t)dt} $ represents the norm of $ w(t) $, where $ w(t)\in L_2[0,\infty) $ is the space of the square-integrable vector over $ [0,\infty) $. $ {\bf{I}} $ represents the identity matrix of an appropriate size. $ {\bf{0}} $ represents the zero vector or matrix of an appropriate size. diag$\{\cdots\}$ represents the block diagonal matrix and col$\{\cdots\}$ represents the column vector. The superscript ‘T’ is used to represent the transpose of a matrix. The matrix used in this paper is assumed to be of suitable dimension unless otherwise specified.

II. Problem Formulation

A In-Wheel ADM-Based Active Suspension System

A 3-DOF quarter vehicle active suspension system with an advanced-dynamic-damper-motor (ADM) is shown in Fig. 1, where $ m_s(t) $ represents the uncertain suspended mass of a quarter vehicle body, $ m_u(t) $ represents the uncertain non-suspended mass, i.e., mass of a single tyre, spring and damper, $ m_r(t) $ represents the uncertain mass of the ADM, $ c_s $ is the sprung mass damping coefficient, $ c_r $ is the ADM damping coefficient, $ c_t $ is the tyre damping coefficient, $ k_s $ is the suspension stiffness, $ k_t $ is the tyre stiffness, $ k_{r} $ is the ADM stiffness, $ u(t) $ (correspondingly, $ u_f(t) $) is the control input in fault-free (correspondingly, faulty) case, $ x_s(t) $ is the sprung mass displacement, $ x_u(t) $ is the unsprung mass displacement and $ x_r(t) $ is the ADM displacement.

Figure  1.  A decentralized event-triggered communication mechanism for in-vehicle networked active suspension with an ADM

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Denote by $ x_1(t) = x_s(t)-x_u(t) $ the suspension deflection, $ x_2(t) = x_r(t)-x_u(t) $ the dynamic vibration absorber deflection, $ x_3(t) = x_u(t)-x_t(t) $ the tyre deflection, $ x_4(t) = \dot{x}_s(t) $ the sprung mass speed, $ x_5(t) = \dot{x}_u(t) $ the unsprung mass speed, and $ x_6(t) = \dot{x}_r(t) $ the dynamic vibration absorber speed, respectively. By the Newton’s second law, the dynamical model of the suspension system is given by

Let $ x(t) = [x_1(t),x_2(t),x_3(t),x_4(t),x_5(t),x_6(t)]^T $ be the augmented state vector of the system and $ w(t) = \dot{x}_t(t) $ denote the road disturbance input. From the dynamics (1), it is easy to derive the following state space model

where

Remark 1: The installation of an electric motor in the wheel increases the unsprung mass, which inevitably deteriorates the road holding performance. This thus discourages the application of in-wheel motors in electric vehicles. On that account, a motor is often isolated from the wheel by installing some ADM which serves as an additional vibration isolation module. Specifically, the ADM is capable to cancel the vibration of the road/tyre, thereby leading to satisfactory road holding performance. For a simple demonstration, we consider a 3-DOF quarter vehicle active suspension system whose dynamic parameters are given in Table I. The frequency responses from the bump road disturbance, $\dot{x}_t(t) = H\pi\frac{V}{L}\sin(2\pi \frac{V}{L}t),\; \forall\; t\in[0, \frac{L}{V}]$, to the sprung mass acceleration, suspension deflection and tyre deflection of the ADM electric vehicles and conventional electric vehicles under active suspension are shown in Fig. 2. Further comparisons of the conventional electric vehicle, conventional in-wheel-motor driven vehicle and ADM driven vehicle were presented in [14], [39], which is omitted for brevity. It can be seen from Fig. 2 that the introduction of ADM into the vehicle suspension system is beneficial for reducing the sprung mass acceleration near the resonance of the unsprung mass as compared with the conventional electric vehicle suspension system.

Table  I.  Suspension System Parameters

$ m_s(t) $	$ m_u(t) $	$ m_r(t) $	$ k_s $	$ k_t $
340 kg	50 kg	30 kg	32 kN/m	200 kN/m
$ k_r $	$ c_s $	$ c_t $	$ c_r $
30 kN/m	1496 Ns/m	10 Ns/m	1000 Ns/m

 | Show Table

DownLoad: CSV

Figure  2.  Frequency responses of (a) Sprung mass acceleration; (b) Suspension deflection; (c) Tyre dynamic force, for a conventional electric vehicle (Con-EV) and an advanced-dynamic-damper-motor electric vehicle (ADM-EV)

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B T-S Fuzzy Modeling

In this section, a T-S fuzzy modeling approach is utilized to deal with the uncertain masses, $ m_s(t) $, $ m_u(t) $ and $ m_r(t) $, that vary from time to time. For simplicity but without loss of generality, the uncertain masses are assumed to be bounded, i.e., $ m_s(t)\in[\underline{m}_s,\overline{m}_{s}] $, $ m_u(t)\in[\underline{m}_{u},\overline{m}_{u}] $ and $ m_r(t)\in[\underline{m}_{r},\overline{m}_{r}] $ with the ‘underline’ denotes the lower bounds of the masses and the ‘overline’ denotes the upper bounds of the masses.

Define the maximum and minimum values of $ \frac{1}{m_s(t)} $, $ \frac{1}{m_u(t)} $ and $ \frac{1}{m_r(t)} $ as

where $ \frac{1}{m_s(t)} $, $ \frac{1}{m_u(t)} $ and $ \frac{1}{m_r(t)} $ can be represented, by using the sector nonlinear approach [40], as follows

with $ \xi_1(t) = \frac{1}{m_s(t)} $, $ \xi_2(t) = \frac{1}{m_u(t)} $ and $ \xi_3(t) = \frac{1}{m_r(t)} $ being the premise variables. Denote by $ M_1(\xi_1(t)) $, $ M_2(\xi_1(t)) $, $ N_1(\xi_2(t)) $, $ N_2(\xi_2(t)) $, $ H_1(\xi_3(t)) $, and $ H_2(\xi_3(t)) $ the membership functions satisfying

Then it is easy to verify that

The membership functions $ M_1, N_1, H_1 $ and $ M_2, N_2, H_2 $ are termed as “Light” and “Heavy”, respectively. The following fuzzy If-Then rules are defined to represent the nonlinear system.

Model Rule 1: If $ \xi_{1}(t) $ is Light, $ \xi_{2}(t) $ is Light and $ \xi_{3}(t) $ is Light, Then $ \dot{x}(t) = A_{1}x(t)+B_{1}u(t)+E_{1}w(t) $;

Model Rule 2: If $ \xi_{1}(t) $ is Light, $ \xi_{2}(t) $ is Light and $ \xi_{3}(t) $ is Heavy, Then $ \dot{x}(t) = A_{2}x(t)+B_{2}u(t)+E_{2}w(t) $;

Model Rule 3: If $ \xi_{1}(t) $ is Light, $ \xi_{2}(t) $ is Heavy and $ \xi_{3}(t) $ is Light, Then $ \dot{x}(t) = A_{3}x(t)+B_{3}u(t)+E_{3}w(t) $;

Model Rule 4: If $ \xi_{1}(t) $ is Light, $ \xi_{2}(t) $ is Heavy and $ \xi_{3}(t) $ is Heavy, Then $ \dot{x}(t) = A_{4}x(t)+B_{4}u(t)+E_{4}w(t) $;

Model Rule 5: If $ \xi_{1}(t) $ is Heavy, $ \xi_{2}(t) $ is Light and $ \xi_{3}(t) $ is Light, Then $ \dot{x}(t) = A_{5}x(t)+B_{5}u(t)+E_{5}w(t) $;

Model Rule 6: If $ \xi_{1}(t) $ is Heavy, $ \xi_{2}(t) $ is Light and $ \xi_{3}(t) $ is Heavy, Then $ \dot{x}(t) = A_{6}x(t)+B_{6}u(t)+E_{6}w(t) $;

Model Rule 7: If $ \xi_{1}(t) $ is Heavy, $ \xi_{2}(t) $ is Heavy and $ \xi_{3}(t) $ is Light, Then $ \dot{x}(t) = A_{7}x(t)+B_{7}u(t)+E_{7}w(t) $;

Model Rule 8: If $ \xi_{1}(t) $ is Heavy, $ \xi_{2}(t) $ is Heavy and $ \xi_{3}(t) $ is Heavy, Then $ \dot{x}(t) = A_{8}x(t)+B_{8}u(t)+E_{8}w(t) $.

Then, the blended T-S fuzzy model of the system can be described as

where $ \lambda_1 = M_1(\xi_1(t))\cdot N_1(\xi_2(t))\cdot H_1(\xi_3(t)) $, $\lambda_2 = M_1(\xi_1(t))\cdot N_1(\xi_2(t)) \cdot H_2(\xi_3(t))$, $ \lambda_3 = M_1(\xi_1(t)) \cdot N_2(\xi_2(t)) \cdot H_1(\xi_3(t)) $, $\lambda_4 = M_1\,(\xi_1\,(t)\,)\, \cdot N_2\,(\xi_2\,(t)\,) \cdot H_2\,(\xi_3\,(t)\,)$, $\lambda_5 = M_2\,(\xi_1\,(t)\,)\, \cdot N_1\,(\xi_2\,(t)\,)\, \cdot H_1(\xi_3(t))$, $ \lambda_6 = M_2(\xi_1(t)) \cdot N_1(\xi_2(t)) \cdot H_2(\xi_3(t)) $, $\lambda_7 = M_2(\xi_1(t)) \cdot N_2(\xi_2(t)) \cdot H_1(\xi_3(t))$, $ \lambda_8 = M_2(\xi_1(t)) \cdot N_2(\xi_2(t)) \cdot H_2(\xi_3(t)) $, and weighting function satisfies $ {\lambda}_{m}\ge 0 $ and $ \sum_{m = 1}^{8}{\lambda}_{m} = 1 $.

C Actuator Fault

Under the actuator fault, the actual control input $ u_{f}(t) $ can be modeled as

where $ \rho(t) $ represents the uncertain fault in the actuator, and the parameters $ \rho_L $ and $ \rho_U $ characterize the lower and upper bounds of the fault, respectively. Note that the above uncertain fault model incorporates the following three particular cases of actuator operating scenarios:

$ 1) $ when $ \rho_{L} = \rho_{U} = 1 $, then $ \rho(t) = 1 $ and the actuator is in the normal mode of operation;

$ 2) $ when $ 0<\rho_{L}<\rho_{U}<1 $, then $ 0<\rho(t)<1 $ and the actuator loses its effectiveness; and

$ 3) $ when $ \rho_{L} = \rho_{U} = 0 $, then $ \rho(t) = 0 $ and the actuator is fully blocked.

Furthermore, it is clear that the uncertain fault $ \rho(t) $ can be rewritten as $ \rho(t) = \overline{\rho}_{0}+\underline{\rho}_{0}\rho_{0}(t) $ by introducing the following new variables

where $ \rho_{0}(t) $ represents the normalized unknown parameter.

D A Novel Decentralized Dynamic Event-Triggered Communication Mechanism

Traditionally, a single sensor is adopted to observe and measure the whole system state $ x(t) $. In this sense, the sensor is required to access all six state components $x_1(t),\ldots,x_6(t)$ at all continuous times $ t\in{\mathbb{R}} $. Apparently, this may be inapplicable or even invalid for CAN-based vehicle suspension systems because of at least the following three reasons: 1) a single sensor is more susceptible to sensor failure or malfunction; 2) a single sensor inevitably exhibits high energy expenditure for measuring all six system state components; and 3) CAN represents a typical digital and broadcasting network medium that permits only sampled data packets. As a result, a decentralized multi-sensor-based sampling configuration is more preferable for the concerned CAN-based vehicle suspension system. Furthermore, to avoid excessive consumption of intra-vehicular limited communication resources, a decentralized dynamic event-triggered mechanism is introduced to transmit only ‘necessary’ sampled data while preserving the desired suspension performance. We introduce the following notations for a better understanding of the proposed decentralized dynamic event-triggered communication paradigm of the CAN-based vehicle suspension system.

• $ \{mh|m\in {\mathbb{N}}\} $: This set defines the sampling instants of the system state components, where $ h>0 $ is the constant sampling period. Specifically, six sensors are deployed to monitor and sense the vehicle state components. For any $i = \{1,\ldots,6\}$, sensor i periodically samples the continuous-time system state component $ x_{i}(t) $ into $ x_{i}(mh) $ at instant $ mh $. The current sampled-data $ x_{i}(mh) $ and its time-stamp m are then encapsulated into a single data packet $ (m,x_{i}(mh)) $ for possible network transmission. Whether or not the sampled data packet $ (m,x_{i}(mh)) $ should be broadcast and transmitted is determined by the triggering mechanism.

• $ \{t^{i}_{k}h|t^{i}_{k}\in {\mathbb{N}}\} $: This set defines the event triggering instants, where $ t^{i}_{k}h $ represents the time instant when the $ i{{\rm{th}}} $ event trigger successfully broadcasts the sampled-data over the CAN. For simplicity, it is assumed that data packets are not delayed, lost or disordered during network transmission. This allows us to focus on the development of the decentralized dynamic event-triggered mechanism.

• $ \{s^{i}_{k}h|s^{i}_{k} = t^{i}_{k}+l,\; l = 0,1,2,\dots,t^{i}_{k+1}-t^{i}_{k}-1; s^{i}_{0} = t^{i}_{0}\} $: This set defines the time sequence from the last triggering instant $ t^{i}_{k}h $ to the next triggering instant $ t^{i}_{k+1}h $.

• $ \{j_r|r = 0,1,2,\dots; j_0 = 0\} $: This set defines the time sequence that stores the instants when new augmented packets are generated by a delicate data packet identifier (DPI). Specifically, the DPI detects any newly arrived data packets for generating a new augmented data packet which is then fed into the ZOH for computing the desired control input. Given that the event triggers $ 1,\ldots,6 $ work in an asynchronous manner, the released data packets on each event trigger essentially arrive at the DPI in a different temporal order. The employed DPI is thus required to have the capability to verify the newly arrived data packets and further feed the controller with most latest yet available data packets. Therefore, the main function of the DPI is to group the available data packets into a new data packet according to

By virtue of the property of the ZOH, we then have $\tilde{x}(t) = \tilde{x}(j_r)$, $ t\in[j_r,j_{r+1}) $. The timing diagram of sensor data sampling, triggering and arriving is depicted in Fig. 3. It can be seen that during the interval $ (0,2h) $, the DPI does not receive any data packet from the event triggers $ 1,\dots,6 $ and does not generate any new data during $ (j_0,j_1) $; at time $ 2h $, the DPI receives the released data packets from all event triggers and generates the new data $\tilde{x}(j_1) = \tilde{x}(2h) = {\rm{col}}\{x_1(t^1_1h), x_2(t^2_1h),\dots, x_6(t^6_1h)\}$ for ZOH; at time $ 6h $, the DPI receives only the released data packet from the event trigger $ 2 $ and thus the newly arrived packet $ x_2(t^2_3h) $ will be used only to form the new data $ \tilde{x}(2h) = {\rm{col}}\{x_1(t^1_2h),x_2(t^2_3h),\dots,x_6(t^6_2h)\} $ while the previously stored $ x_1(t^1_2h),x_3(t^3_2h),\ldots,x_6(t^6_2h) $ will be recalled instead. We further provide Algorithm 1 for detailing the mechanism of the developed DPI.

Figure  3.  A timing diagram of sensor data sampling, triggering and arriving

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Algorithm 1 Mechanism of DPI

1: Given $ h>0 $ and the runtime $ T_s $, set $ r=0 $, $ j_r=0 $, $ t_0^i=0 $, $ new\_{id}=0 $

2: for $ m=1:1:\frac{T_s}{h} $ do

3: 　　for $ i=1:6 $ do

4: 　　　if A new data packet $ (t_k^i,x(t_k^ih)) $ from the $ i^{th} $ event trigger arrives at DPI at time $ mh $ then

5: 　　　　　　Set $ new\_{id}=1 $

6: 　　　　　　Update the DPI with the newly arrived packet $ (t_k^i,x(t_k^ih)) $

7: 　　　end if

8: 　　end for

9: 　　if $ new\_{id}=1 $ then

10: 　　　　Set $ r=r+1 $ and $ j_r\leftarrow mh $

11: 　　　　Generate and output the new data ${\tilde{x}(j_r)\leftarrow {\rm{col}}\{x_1(t^1_kh),}$${x_2(t^2_kh),\dots,x_6(t^6_kh)\}} $

12: 　　　　Reset $ new\_{id}=0 $

13: 　　end if

14: end for

For each sensor i, $i\in\{1,\ldots,6\}$, we are now in a position to develop a decentralized dynamic event trigger i, as shown in Fig. 1, to determine whether or not the current sampled data packet $ (s_k^i,x_{i}(s_k^ih)) $ should be transmitted to the controller for calculating the desired control input. The event releasing instants of the $ i{{\rm{th}}} $ decentralized dynamic event trigger are recursively determined as follows

where $ e_{i}(s_k^ih) = x_{i}(s_k^ih)-x_{i}(t^{i}_{k}h) $ is the state error between the current sampling instant $ s_k^ih $ and last transmitted instant $ t^{i}_{k}h $ of the $ i{{\rm{th}}} $ sensor; $ \Phi_{i}>0 $ is the weighting scalar to be determined; and $ \sigma_{i}(s_k^ih)>0 $ is the dynamic threshold parameter whose value is adaptively adjusted according to

where $ \underline{\sigma}_i,\overline{\sigma}_i\in[0,1) $ are the lower and upper bounds of the threshold parameter, respectively, and $ \varrho_i\geq0 $ is a prescribed constant. It can be seen from (10) that $ \sigma_{i}(s_k^ih)\in[\underline{\sigma}_i,\overline{\sigma}_i] $.

Remark 2: It is clear from the dynamic threshold parameter (10) that if $ \varrho_i = 0 $ and $ \overline{\sigma}_i = \sigma_0 $, then $ \sigma_i(s_k^ih)\equiv\sigma_0 $, which means that the proposed decentralized dynamic event-triggered mechanism covers the widely studied static event-triggered mechanisms [23], [24], [26], [27], [41] as its special case. Furthermore, by forcing $ \sigma_0 = 0 $, the decentralized dynamic triggering mechanism reduces to the traditional time-triggered (periodic) mechanism where all the sampled data packets are transmitted over the CAN. Also note that the dynamic threshold parameter adaptively adjusts its value on the interval $ [\underline{\sigma}_i,\overline{\sigma}_i] $ based on the weighting state error in terms of $ e_{i}(s_k^ih) $. More specifically, when $ e_{i}(s_k^ih) $ suffers from significant changes, a smaller value of $ \sigma_i(s_k^ih) $ will be chosen for verifying the event trigger (9). Generally, a smaller threshold parameter leads to more data packets to be transmitted over the network with an aim to achieve better convergence performance of the resulting closed-loop system. On the other hand, when $ e_{i}(s_k^ih) $ experiences little fluctuations, it may mean that the system is now approaching its steady-state and thus a larger threshold parameter $ \sigma_i(s_k^ih) $ should be selected to reduce unnecessary data packet transmissions. From this perspective, the proposed decentralized dynamic event triggers in (9) offer certain adaptiveness between maintaining desired control performance and communication efficiency.

E A Decentralized Event-Triggered Fuzzy Controller

We are interested in constructing and designing the following event-triggered fuzzy state feedback controller

Control Rule 1: If $ \xi_{1}(t) $ is Light, $ \xi_{2}(t) $ is Light and $ \xi_{3}(t) $ is Light, Then $ u_{f}(t) = \rho(t) K_{1}\tilde{x}(t) $;

Control Rule 2: If $ \xi_{1}(t) $ is Light, $ \xi_{2}(t) $ is Light and $ \xi_{3}(t) $ is Heavy, Then $ u_{f}(t) = \rho(t) K_{2}\tilde{x}(t) $;

Control Rule 3: If $ \xi_{1}(t) $ is Light, $ \xi_{2}(t) $ is Heavy and $ \xi_{3}(t) $ is Light, Then $ u_{f}(t) = \rho(t) K_{3}\tilde{x}(t) $;

Control Rule 4: If $ \xi_{1}(t) $ is Light, $ \xi_{2}(t) $ is Heavy and $ \xi_{3}(t) $ is Heavy, Then $ u_{f}(t) = \rho(t) K_{4}\tilde{x}(t) $;

Control Rule 5: If $ \xi_{1}(t) $ is Heavy, $ \xi_{2}(t) $ is Light and $ \xi_{3}(t) $ is Light, Then $ u_{f}(t) = \rho(t) K_{5}\tilde{x}(t) $;

Control Rule 6: If $ \xi_{1}(t) $ is Heavy, $ \xi_{2}(t) $ is Light and $ \xi_{3}(t) $ is Heavy, Then $ u_{f}(t) = \rho(t) K_{6}\tilde{x}(t) $;

Control Rule 7: If $ \xi_{1}(t) $ is Heavy, $ \xi_{2}(t) $ is Heavy and $ \xi_{3}(t) $ is Light, Then $ u_{f}(t) = \rho(t) K_{7}\tilde{x}(t) $;

Control Rule 8: If $ \xi_{1}(t) $ is Heavy, $ \xi_{2}(t) $ is Heavy and $ \xi_{3}(t) $ is Heavy, Then $ u_{f}(t) = \rho(t) K_{8}\tilde{x}(t) $.

Therefore, the blended T-S fuzzy controller is given as

where $ K_n $ are the controller gains to be designed. Furthermore, the arriving intervals at the ZOH can be divided into the following equidistant subintervals

where $ r = {0,1,\dots} $ and $ \epsilon_{r}:= (j_{r+1}-j_r)/h $. Define a delay function $ d(t) $ as

Then $ 0\le d(t)<h $, $ \forall\; t\in[j_r+(l-1)h,j_{r}+lh) $. Furthermore, $ d(t) $ is discontinuous at $ t = j_r+(l-1)h $ and $ \dot{d}(t) = 1 $ for $ t\neq j_r+(l-1)h $. The state error is then given by

Let $e(t - d(t))\triangleq {\rm{col}}\{e_1(t - d(t)),\dots,e_6(t - d(t))\}$ and $x(t-d(t))\triangleq {\rm{col}}\{x_1(t-d(t)),\dots,x_6(t-d(t))\}$. Then the controller (11) becomes

F Suspension Performance Requirements

The aim of this paper is to preserve the following performance requirements [10], [14], [17], [19] for the concerned uncertain IWM quarter vehicle active suspension system in the subsequent analysis and design procedures.

$ 1) $ Ride comfort: A key feature of the suspension system is to ensure passenger’s ride comfort. Ride comfort can be defined by the vehicle body vertical acceleration $ \ddot{x}_s(t) = \dot{x}_4(t) $, which is considered as a performance output of the suspension system (5). This is given by

where $ \tilde{C} = [-\frac{k_s}{m_s},0, 0, -\frac{c_{s}}{m_s}, \frac{c_{s}}{m_s}, 0] $ and $ D = 1/m_s $.

Ride comfort can thus be pursued by minimizing the vehicle’s body vertical acceleration, namely, $\min\{\ddot{x}_s(t)\} = \min\{\dot{x}_4(t)\}$, which is further interpreted as the following $ H_{\infty} $ performance index

for a prescribed level $ \gamma>0 $ and all nonzero road disturbance input $ w(t)\in L_2[0,\infty) $.

$ 2) $ Road holding: For a safety reason, the contact of vehicle wheels to road should be preserved firmly uninterrupted, which requires the dynamic tyre load to be always less than the minimal static one. This is given by

which can be written as

should be satisfied at all time $t \in {\mathbb{R}}$, where $C_1 = [0, 0, \frac{k_t}{(\underline{m}_s+\underline{m}_u+\underline{m}_r)g} 0 , 0, 0]$.

$ 3) $ Suspension stroke: Due to the space limitation and to avoid structural damages, the suspension deflection $x_s(t)- x_u(t)$ of the vehicle suspension system often needs to be confined to an allowable maximum limit $ x_{max} $, i.e.,

which can be represented as

for any $ t\in{\mathbb{R}} $, where $ C_{2} = [1/x_{max},0,0,0,0,0] $.

$ 4) $ Actuator power limitation: It is acknowledged that the high power consumption is deemed as a key disadvantage of an active suspension system. Hence, the actuator power imposed by active suspension control should be limited to some prescribed bound, which means that

for any $ t\in{\mathbb{R}} $.

G Problem Description

Substituting the controller (15) into the system (5), we have the resulting closed-loop system as follows

where the initial condition is supplemented as $ x(s) = \psi(s) $, $ s\in[-h,0] $ with $ \psi(s) $ being a continuous function.

The main problem is now described as follows.

Problem 1: The control objective is to design an appropriate event-triggered fuzzy controller in the form of (15) under the desired decentralized dynamic event-triggered mechanism (9) (s.t. (10)) such that

$ 1) $ The asymptotic stability of the resulting closed-loop suspension system (22) is ensured when $ w(t)\equiv 0 $; and

$ 2) $ Under zero initial condition, the following $ H_{\infty} $ performance index

is guaranteed under a prescribed disturbance attenuation level $ \gamma>0 $ and nonzero $ w(t)\in L_{2}[0,\infty) $; and

$ 3) $ The inequalities

are satisfied, where ${\textit{z}}_{1}(t)$ and ${\textit{z}}_{2}(t)$ are given in (18) and (20), respectively.

III. Main Results

A Stability and Performance Analysis: The Decentralized Case

In this section, the feasibility of the concerned problem is studied formally under the decentralized dynamic event-triggered communication mechanism (9), followed by a sufficient condition that ensures the asymptotic stability of the closed loop suspension system (22) under the prescribed performance requirements.

Theorem 1: For given positive scalars, h, $ \gamma $, $ \kappa $, $ \rho(t)\in[\rho_L,\rho_U] $, diagonal matrix $ \overline{\sigma} $ and control gain matrices $ K_n,\; n = {1,\dots,8} $, the closed-loop system (22) is asymptotically stable under the dynamic threshold parameter (10) and the prescribed $ H_{\infty} $ disturbance performance level $ \gamma $ if there exist appropriate matrices $ P>0 $, $ Q>0 $, $ R_1>0 $, $ R_2>0 $ and a diagonal matrix $ \Phi>0 $ such that

where

Proof: See Appendix A.■

B Co-Design: The Decentralized Case

The following theorem provides the desired communication and control co-design criterion for simultaneously determining the desired control gain matrices $ K_n $, $ n = 1,\dots,8 $ and weighting matrix $ \Phi $ in the decentralized dynamic event-triggered mechanism (9).

Theorem 2: For given positive scalars h, $ \alpha $, $ \beta $, $ \gamma $, $ \kappa $, $ \overline{\rho}_0 $, $ \underline{\rho}_0 $, $ \rho_U $ and diagonal matrix $\overline{\sigma} = {{{\rm{diag}}}} \{\overline{\sigma}_1,\overline{\sigma}_2,\ldots,\overline{\sigma}_6\}$, the closed-loop system (22) is asymptotically stable under the dynamic threshold parameter (10) with the prescribed $ H_{\infty} $ disturbance performance level $ \gamma $ if there exist appropriate matrices $ \bar{P}>0 $, $ \bar{Q}>0 $, $ \bar{R}_1>0 $, $ \bar{R}_2>0 $, $ \bar{\Phi}>0 $, $ \bar{K}_n,\; n = 1,\dots,8 $ and a scalar $ \varepsilon_1 $, such that

where

Then, the admissible controller gain $ K_n = \bar{K}_n^{-1}\bar{P} $ and the weighting diagonal matrix $ \Phi = \bar{\Phi}^{-1} $ for the desired event-triggered mechanism can be determined.

Proof: See Appendix B. ■

C Co-Design: The Centralized Case

In this section, we consider that all six state components of the vehicle suspension system (5) are monitored and measured by a single sensor. This case leads to the centralized event-triggered state feedback control. We next develop two centralized dynamic event-triggered communication mechanisms that regulate the full state transmissions over the in-vehicle CAN. In particular, the event releasing instants are generated recursively according to the following triggering law

where $ e(s_{k}h) = x(s_{k}h)-x(t_{k}h) $ and $ \Phi $ is a weighting matrix to be determined. In the sequel, we construct two different types of dynamic threshold parameters (DTPs) $ \sigma(s_kh) $, each of which has a dynamically adjustable law. Specifically,

where $ \underline{\sigma}\in[0,1) $ and $ \overline{\sigma}\in[0,1) $ are the lower and upper bounds of the threshold parameter, respectively, and $ \varrho>0 $ is a constant; $ \delta>0 $ is a prescribed constant and $ \sigma(0) = \underline{\sigma} \in [0,1) $ is the given initial condition.

Remark 3: We note the following three main features for the constructed DTP 1 and DTP 2. First, when $ \varrho = 0 $ in (32) and $ \delta = 0 $ in (33), then $ \sigma(s_kh) $ becomes constant and the triggering mechanism (31) degenerates into the static one. Second, by forcing $ \sigma(s_kh) = 0 $, DTP 1 and DTP 2 reduce to the traditional time-triggered mechanism. Third, as stated in Remark 2, the DTP 1, whose value varies on the interval $ [\underline{\sigma},\overline{\sigma}]\subset[0,1) $, allows the desired control performance and communication resource utilization to be scheduled in an adaptive manner. On the other hand, an important property of $ \sigma(s_kh) $ in DTP 2 lies in its non-increasing monotonic property [9], [42], which leads to $ 0\leq\sigma(s_kh)\leq\sigma_0<1 $. This corresponds to the communication resource abundance scenario under which more sampled data packets allow to be transmitted to preserve better control performance. Hence, the proposed dynamic event-triggered mechanisms based on DTP 1 or DTP 2 offer more flexibility than the existing static event- and time-triggered mechanisms.

The following theorems provide two co-design criteria for simultaneously determining the control gain matrices $ K_n $ and the trigger weighting matrix $ \Phi $ under DTP 1 and DTP 2.

Theorem 3: For given positive scalars h, $ \overline{\sigma} $, $ \alpha $, $ \gamma $, $ \kappa $, $ \overline{\rho}_0 $ and $ \underline{\rho}_0 $, the closed-loop system (22) is asymptotically stable under the DTP 1 in (32) with prescribed $ H_{\infty} $ disturbance performance level $ \gamma $ if there exist appropriate matrices $ \bar{P}>0 $, $ \bar{Q}>0 $, $ \bar{R}_1>0 $, $ \bar{R}_2>0 $, $ \bar{\Phi}>0 $, $ \bar{K}_n,\; n = 1,\dots,8 $ and a scalar $ \varepsilon_2 $, such that (28) and (29) and

where

Then, the admissible controller gain $ K_n = \bar{K}_n^{-1}\bar{P} $ and the weighting matrix $ \Phi = \bar{P}^{-1}\bar{\Phi}\bar{P}^{-1} $ in the case of DTP 1 can be determined.

Proof: See Appendix C.■

Theorem 4: For given positive scalars h, $ \sigma_0 $, $ \alpha $, $ \gamma $, $ \kappa $, $ \overline{\rho}_0 $ and $ \underline{\rho}_0 $, the closed-loop system (22) is asymptotically stable under the DTP 2 in (33) with prescribed $ H_{\infty} $ disturbance performance level $ \gamma $ if there exist appropriate matrices $ \bar{P}>0 $, $ \bar{Q}>0 $, $ \bar{R}_1>0 $, $ \bar{R}_2>0 $, $ \bar{\Phi}>0 $ and $ \bar{K}_n,\; n = 1,\dots,8 $, and a scalar $ \varepsilon_2 $, such that (28) and (29) and

where $ \bar{\chi}_1 $ is derived from $ \chi_1 $ in (34) by replacing $ \overline{\sigma} $ with $ \sigma_0 $. Then, the admissible controller gain $ K_n = \bar{K}_n^{-1}\bar{P} $ and the weighting matrix $ \Phi = \bar{P}^{-1}\bar{\Phi}\bar{P}^{-1} $ in the case of DTP 2 can be determined.

Proof: The proof is similar to that of Theorem 3 and thus omitted.■

Remark 4: With Theorem 2, Theorem 3 and Theorem 4, the minimal $ H_{\infty} $ performance level $ \gamma_{min} $ can be determined from the following optimization problem

where $ \tilde{\gamma} = \gamma^2 $, and thus the optimal $ H_{\infty} $ performance level can be obtained as $ \gamma_{min} = \sqrt{\tilde{\gamma}} $.

IV. Simulation Results

In this section, a quarter vehicle model whose parameters are given in Table I is exploited to show the effectiveness and advantages of the proposed dynamic event-triggered control method under various event-triggered mechanisms and different road disturbances. Specifically, for comparison purposes, we examine the following event triggering mechanisms.

• DDEM (decentralized dynamic event-triggered mechanism): which is based on the proposed decentralized dynamic event trigger (9) with (10). Moreover, we let $ \varrho_i = 200 $, $ \underline{\sigma}_i = 0.02 $ and $ \overline{\sigma}_i = 0.2 $ in (10);

• DSEM (decentralized static event-triggered mechanism): which is derived from (9) by setting $ \sigma_i(s_k^ih)\equiv \overline{\sigma}_i = 0.2 $ in (10);

• CDEM1 (centralized dynamic event-triggered mechanism 1): which is based on the proposed centralized dynamic event trigger (31) with DTP 1 in (32). Moreover, we set $ \varrho = 50 $, $ \underline{\sigma} = 0.1 $ and $ \overline{\sigma} = 0.2 $;

• CDEM2 (centralized dynamic event-triggered mechanism 2): which is based on the proposed centralized dynamic event trigger (31) with DTP 2 in (33). Moreover, we choose $ \delta = 0.2 $;

• CSEM (centralized static event-triggered mechanism): which is derived from (31) by setting $ \sigma(s_kh)\equiv \underline{\sigma} = 0.1 $;

• CTTM (centralized time-triggered mechanism): which can be derived from (31) by letting $ \varrho = 0 $, $ \underline{\sigma} = 0 $ and $ \overline{\sigma}\rightarrow0 $.

In the subsequent simulation cases, the simulation time is set as $ T = 5 $ sec and the sampling period is chosen as $ h = 1 $ ms. Then, the total number of the sampled data packets within $ 5 $ sec is $ T_{m} = 5000 $. For better comparative studies between the above triggering mechanisms, we introduce the following transmission rate as a performance metric for the communication efficiency

where $ N_{oE} $ denotes the number of events released by the relevant trigger. In the decentralized case, it should be noted that the above communication performance metric can be refined as the average transmission rate of the form $ \digamma = \frac{1}{6}\sum_{i = 1}^{6}\frac{N_{oE}^i}{T_{m}}\cdot 100\% $ with $ N_{oE}^i $ denoting the number of events on the i-th sensor or event trigger.

The sprung, unsprung and dynamic damper masses are uncertain and assumed to vary in the ranges, $ [238\;{\rm{kg}}, 442\;{\rm{kg}}] $, $ [40\;{\rm{kg}}, 60\;{\rm{kg}}] $ and $ [25\;{\rm{kg}}, 35\;{\rm{kg}}] $, respectively, as shown in Fig. 4, where the weighting functions are also depicted. The maximum suspension deflection is set as $ x_{max} = 0.05 $ m and the maximum control force is limited to $ u_{max} = 1.5 $ kN. The minimal static tyre load can be easily obtained as $ (\underline{m}_s+\underline{m}_u+\underline{m}_r)g = 2969.4 $ N. We further assume a persistent $ 50\% $ loss of efficiency in the actuator, i.e., $ \rho(t) = 0.5 $.

Figure  4.  The variable sprung mass $ m_s(t)\in[238\;{\rm{kg}}, 442\;{\rm{kg}}] $, unsprung mass $ m_u(t)\in[40\;{\rm{kg}}, 60\;{\rm{kg}}] $ and dynamic mass $m_r(t)\in [25\;{\rm{kg}}, 35\;{\rm{kg}}]$, and the weighting functions $ \lambda_m(t), m = 1,\dots,8 $

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A Bump Responses

Road inputs can generally be assumed as the discrete events of severe intensity and short span. In this case, an isolated bump input of the following form is considered

where $ H = 0.06 $ m is the height of the bump, $ L = 5 $ m is the length of the bump, and $ V = 45 $ km/h is the vehicle velocity.

The effectiveness of the proposed DDEM can be verified by solving the optimization problem (36) in terms of Theorem 2. The inequalities in Theorem 2 are found feasible under $ \kappa = 0.001 $, $ \alpha = 13 $, $ \beta = 0.09 $, and the prescribed $ H_{\infty} $ performance index $ \gamma_{min} = 45 $. Fig. 5 illustrates the simulation results on the bump responses of the resulting closed-loop suspension system under Theorem 2. It can be seen that, even in the presence of the actuator fault and limited network data transmissions: 1) the sprung mass acceleration $ \ddot{x}_s(t) $ is reduced significantly under the proposed DDEM as compared to the open-loop (or passive suspension) scenario; 2) the vehicle suspension deflection is kept below the maximum allowable limit $ 0.05 $ m; 3) the dynamic tyre load is well less than the static tyre load $ 2969.4 $ N; and 4) the actuator control force is less than the maximum allowable limit $ 1.5 $ kN.

Figure  5.  Bump responses of vehicle body vertical acceleration, suspension deflection, dynamic tyre load and actuator force of open- and closed-loop systems under DDEM and DSEM

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Furthermore, we perform quantitative analysis and comparison between DDEM and DSEM to demonstrate the effectiveness of the designed decentralized event-triggered controller from Theorem 2. A simple calculation shows that $ \digamma_{\rm{DDEM}} = 6.62\% $ and $ \digamma_{\rm{DSEM}} = 6.46\% $. This indicates that both DDEM and DSEM save a significant amount of communication resources by transmitting a much less number of sampled data packets via CAN. Furthermore, although more network communication resources can be saved under DSEM over DDEM, the ride comfort performance is sacrificed to some extent, as clearly depicted in Fig. 5 where the body vertical acceleration under DSEM suffers from more fluctuations than the DDEM case. For an intuitionistic evaluation of the communication performance of the designed DDEM, the triggering instants and intervals of the individual system state components and the corresponding dynamic threshold parameters are provided in Fig. 6, from which one can see that the data transmissions on sensors are invoked in a dramatically sporadic manner.

Figure  6.  Triggering instants and intervals as well as the corresponding dynamic threshold parameters $ \sigma_i(s_k^ih) $ on individual sensors/triggers under DDEM

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B Comparison Between Different Centralized Event-Triggered Communication Mechanisms

In this case, we evaluate the suspension performance and communication efficiency under different centralized triggering mechanisms. Other parameters are the same as given in the previous section, unless otherwise specified. Solving the optimization problem (36) in terms of Theorem 3 and Theorem 4 under $ \kappa = 0.01 $, $ \alpha = 9.8 $, and $ \gamma_{min} = 42 $, it is found that the problem is feasible in both cases. Applying the designed controller to the quarter vehicle suspension system, we obtain the simulation results of each triggering mechanism, as shown in Fig. 7. It can be seen that 1) ride comfort is improved more under the CDEM2 compared with that of CDEM1 and CSEM. The main reason is the CDEM2 permits more sampled data packets to be broadcast over the CAN to maintain better suspension performance; and 2) the other constraint requirements such as suspension deflection, dynamic tyre load and actuator control force are also kept well below the specified limits.

Figure  7.  Bump responses of vehicle body vertical acceleration, suspension deflection, dynamic tyre load and actuator force of open- and closed-loop systems under CDEM1, CDEM2, CSEM and CTTM

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We next perform a quantitative comparison among the event-triggered mechanisms. A simple calculation based on the average transmission rate indicates that CDEM2 transmits $ 14.40\% $, CSEM transmits $ 5.98\% $ and CDEM1 transmits $ 5.04\% $ of the total sampled data packets. In other words, $ 85.60\% $, $ 94.02\% $ and $ 94.96\% $ of the total sampled data are not transmitted over the CAN under CDEM2, CSEM and CDEM1, respectively. It should be noted that the traditional CTTM transmits all the sampled data ($ \digamma = 100\% $) over the CAN and thus guarantees the best suspension system control performance than the other triggering mechanisms, as clearly shown in Fig. 7.

C Random Responses

While running on a real road, a vehicle may subject to unknown and persistent road disturbances. To further evaluate the suspension system performance under the proposed co-design approach, we study the response of the system under different random road disturbances in this case. Specifically, a random road disturbance is deemed as random vibration which can typically be described by a random process [43] as

where $ n_0 = 0.1 \;{\rm{m}}^{-1} $ is the standard cut-off spatial frequency, $ G_q(n_0) $ is the road roughness coefficient, V is the vehicle forward velocity and $ \omega(t) $ is the zero mean white Gaussian noise in the time domain. To validate the effectiveness of the proposed approach, we have selected four different types of road roughness coefficients among the ISO-2631 classified $ G_q(n_0) $ classes, i.e., $ G_q(n_0) = 16 \times 10^{-6}\;{\rm{m}}^3 $ (Class-B, Good), $ G_q(n_0) = 64 \times 10^{-6}\;{\rm{m}}^3 $ (Class-C, Average), $G_q(n_0) = 256 \times 10^{-6}\;{\rm{m}}^3$ (Class-D, Poor) and $ G_q(n_0) = 1024 \times 10^{-6}\;{\rm{m}}^3 $ (Class-E, Very Poor). Moreover, the ride comfort of the passenger depends on the intensity of the vertical acceleration which can be obtained as the root mean square (RMS) value.

Solving the optimization problem (36) under Theorem 2, Theorem 3 and Theorem 4, and performing the simulation for $ 5 $ sec under different random road profiles, it can be seen from Fig. 8 that the suspension deflections, dynamic tyre loads and control inputs are all regulated below the maximum allowable limits. Furthermore, the RMS values of vehicle body vertical acceleration, suspension deflection and dynamic tyre load in the decentralized and centralized triggering cases are provided in Table II and Table III, respectively. It can be seen that 1) the vehicle body vertical acceleration is reduced under the proposed DDEM for all road roughness classes. A comparison of RMS value of the vehicle body vertical acceleration shows that an average of $ 30.2\% $ improvement is noticed under DDEM related to the passive suspension system; and 2) the improvements in the vertical acceleration are more prominent in the case of CDEM1, CDEM2, CSEM and CTTM from Table III. It also reveals that the CDEM2 reduces the acceleration by $ 27.0\% $ by triggering more data as compared to the CSEM ($ 26.7\% $) and CDEM1 ($ 25.7\% $) but less than the CTTM ($ 27.7\% $) which releases the sampled data packets periodically. This further verifies the effectiveness of the proposed DDEM and CDEMs under ISO-classified random road disturbance. Other performance requirements such as suspension deflection, dynamic tyre load and actuator power limitation can also be easily verified under the concerned random road disturbances, and thus are omitted for brevity.

Figure  8.  Random responses of vehicle body vertical acceleration, suspension deflection, dynamic tyre load and actuator force of open- and closed-loop systems under DDEM, CDEM1, CDEM2 and CSEM

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Table  II.  RMS Values of Vertical Acceleration, Suspension Deflection and Dynamic Tyre Load Under Decentralized Case

Roughness coefficient	$ \ddot{x}_s(t) $ ($ {\rm{m/s}}^2 $)			$ x_s(t)-x_u(t) $ ($ {\rm{m}} $)			$ k_t (x_u(t)-x_t(t)) $ ($ {\rm{N}} $)
	Passive	DDEM	$ \uparrow $ (%)	Passive	DDEM	$ \uparrow $ (%)	Passive	DDEM	$ \uparrow $ (%)
Grade B	1.3422	1.0270	23.4	0.0108	0.0081	25.0	848.7011	828.3152	2.4
Grade C	1.5241	1.0872	28.6	0.0122	0.0092	24.6	879.5630	844.0222	4.0
Grade D	1.7300	1.1636	32.7	0.0128	0.0098	23.4	898.0119	858.5714	4.4
Grade E	1.8775	1.2530	33.2	0.0157	0.0126	19.7	954.2535	876.8398	8.1

 | Show Table

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Table  III.  RMS Values of Body Vertical Acceleration ${\ddot{x}_s(t)}$ (${ {\rm{m/s}}^2 }$) Under Different Random Road Disturbances and Event-triggered Mechanisms

Roughness coefficient	Passive	Active (Centralized)
		CDEM1	$ \uparrow $ (%)	CDEM2	$ \uparrow $ (%)	CSEM	$ \uparrow $ (%)	CTTM	$ \uparrow $ (%)
Grade B	1.2574	0.9853	21.6	0.9721	22.7	0.9728	22.6	0.9606	23.6
Grade C	1.3634	1.0415	23.6	1.0228	25.0	1.0261	24.7	1.0088	26.0
Grade D	1.5655	1.1166	28.6	1.0997	29.7	1.1012	29.6	1.0915	30.2
Grade E	1.6305	1.1779	27.7	1.1520	29.3	1.1626	28.7	1.1411	30.0

 | Show Table

DownLoad: CSV

Summarizing the above simulation results, it can be concluded that the proposed communication and control co-design approach makes a wise use of the precious communication resources over the CAN, while sacrificing a negligible amount of suspension performance. Furthermore, the proposed DDEM and CDEMs provide a better trade-off between communication efficiency and suspension control performance than the traditional static and periodic counterparts.

D Frequency Responses

Finally, to further validate the effectiveness of the co-design criteria, the performance of the suspension system is analyzed in the frequency domain. Fig. 9 (a) shows the frequency response of the sprung mass acceleration of the DDEM in comparison to the open-loop (passive suspension) case. The vertical dashed lines represents the lower and the upper bound of the frequency range, i.e., 4–8 Hz in which the human body is sensitive to external vibration. It can be explicitly observed that the sprung mass acceleration is improved to a large extent under the DDEM as compared to the passive system. On the other hand, Fig. 9 (b) presents a comparison among CDEM1, CDEM2 and the open-loop case. The improvement is more pronounced under CDEM2 as compared to the open-loop case. It is obvious from the above simulation results that the sprung mass acceleration is significantly improved under DDEM, CDEM1 and CDEM2 as compared to the passive suspension system in the prescribed frequency region.

Figure  9.  (a) Frequency response of the sprung mass acceleration of the open- and closed-loop systems under DDEM; (b) Frequency response of the sprung mass acceleration under CDEM1 and CDEM2 ($ m_s = 238 $ kg, $ m_u = 60 $ kg, $ m_r = 35 $ kg)

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V. Conclusions

The problem of joint dynamic event-triggered communication and active suspension control for an in-wheel-motor driven electric vehicle equipped with advanced-dynamic-damper-motor has been investigated. A T-S fuzzy model for the uncertain system has been derived. To reduce unnecessary data transmissions over the in-vehicle network, a decentralized event-triggered communication mechanism for the individual system state components and two centralized event-triggered communication mechanisms for the overall system state have been developed. The threshold parameters of the event-triggered communication mechanisms can both be regulated dynamically according to the system state variations and performance requirements. Furthermore, a communication scheduling and active suspension control co-design approach has been proposed to guarantee the desired system performance while maintaining satisfactory communication efficiency. Finally, comprehensive simulation studies have been exploited to show the effectiveness and advantages of the derived main results. While the proposed co-design approach shows its promising performance and merits in simulations, there remain a number of issues for future research. For example, this study has been carried out in the entire frequency range under the state feedback control analysis. To obtain better system and suspension performance and render practical applicability, it would be interesting to develop an observer-based dynamic event-triggered controller in a finite/constrained frequency range. Furthermore, as shown in the proof of Theorem 1, the extended Wirtinger inequality [44], [45] has been employed to derive the main results on stability analysis and co-design. To further improve the conservatism of the derived criteria, the existing looped functional approach and its variants [46], [47] in the literature of sampled-data systems could be adopted and further explored.

APPENDIX APROOF OF THEOREM 1

Consider the following Lyapunov functional candidate

where

where $ V_a $ represents a continuous Lyapunov function, $ V_b $ denotes a discontinuous Lyapunov function, $\varrho(q) = x(q)- x(j_r+(l-1)h)$ and P, Q, $ R_1 $ and $ R_2 $ are positive definite matrices to be determined. Notice that the discontinuous function $ V_b $ can be further divided into a continuous term $ V_{b1} $ and a discontinuous term $ V_{b2} $

For $ t\in \Omega_l^{j_{r}} $, we have $ \dot{\varrho}(q) = \dot{x}(q) $. Note that $ \varrho(t)|_{t = j_r+(l-1)h} = 0 $. Using the extended Wirtinger inequality [44], one has that

which leads to $ V_{b2}(t,x,\dot{x})\ge 0 $. Also $ V_{b2}(t,x,\dot{x}) $ vanishes at $ t = j_r+(l-1)h $, thus the condition

holds, which further means that the constructed Lyapunov functional $ V(t,x,\dot{x}) $ does not increase at $ t = j_r+(l-1)h $. Calculating the time derivative of $ V(t,x,\dot{x}) $ along the system model (22) and applying the Jensen inequality yield

Furthermore, from the triggering condition (9) and dynamic threshold parameter (10), one has

It follows from (46) and (47), that

where $ \xi(t) = [x^T(t),x^T(t-d(t)),x^T(t-h),e^T(t-d(t)),w^T(t)]^T $ and $ \varkappa_{11} $, $ \varkappa_{12} $, $ \varkappa_{13} $ and $ \varkappa_{14} $ are given by (27) in Theorem 1.

We first show that the system (22) is asymptotically stable. It can be easily seen from (27) that by making $ w(t) = 0 $, we can get $V(t,x,\dot{x})\le\sum_{m = 1}^8\sum_{n = 1}^8\;\lambda_m\;\lambda_n\;\varsigma^T(t)\big( \varkappa_1\;+\;\varkappa_{2}^TR_1^{-1}\varkappa_{2} \;+ \varkappa_{3}^TR_2^{-1}\varkappa_{3}\big)\varsigma(t)$ by virtue of Schur complement, where $ \varsigma = [x^T(t),x^T(t-d(t)),x^T(t-h),e^T(t-d(t))]^T $ and $ \varkappa_1,\varkappa_2,\varkappa_3 $ are derived from $ \varkappa_{11},\varkappa_{12},\varkappa_{13} $ by eliminating the last row and column. This leads to $ V(t,x,\dot{x})<0 $. The closed-loop system (22) is thus asymptotically stable without disturbance.

Next, the $ H_{\infty} $ performance of the closed-loop system (22) is established under zero initial condition and nonzero $ w(t)\in L_2[0,\infty) $. Recalling (27) and applying Schur complement, one has from (48) that

Integrating both sides of the inequality from $ t = 0 $ to $ t\rightarrow \infty $, one has that

which means that

under zero initial condition. It is clear that (51) further implies that the pursued $ H_{\infty} $ performance index $ ||y(t)||_2^2\le \gamma^2||w(t)||_2^2 $ is satisfied.

Finally, we show that the performance requirements in (24) are preserved. It can be seen from (49) that $\dot{V}(t,x,\dot{x})- \gamma^2w^T(t)w(t) < 0$. Integrating both sides from $ 0 $ to $ t>0 $, we get

From the definition of Lyapunov functional candidate in (40), one has that $ V(t,x,\dot{x})\geq x^{T}(t)Px(t) $. Therefore, $x^{T}(t)Px(t) < \kappa$ and also $ \tilde{x}^{T}(t)P\tilde{x}(t)<\kappa $ with $ \kappa = \gamma^2w_{max}+V\big(0,x(0),\dot{x}(0)\big) $. Then we have

where $ \theta_{max}(\cdot) $ is the maximum eigenvalue. We know from the above inequalities that performance requirements are guaranteed if

By multiplying both sides of the inequalities above by $ P^{1/2} $ and using the Schur complement, it is readily observed that (56) and (57) are equivalent to the inequalities (25) and (26). ■

APPENDIX B

The following lemma which is useful for deriving our main result in Theorem 2 is recalled.

Lemma 1 [48]: For a time-varying diagonal matrix $ \phi(t) = diag\{\phi_1(t),\phi_2(t),\dots,\phi_p(t)\} $ and two matrices S and R with appropriate dimensions, if $ \|\phi(t)\|\le V $, where $ V>0 $ is a known diagonal matrix, then for any scalar $ \varepsilon>0 $, it is true that $ S \phi R + R^{T} \phi^{T} S^{T}\le \varepsilon R V R^{T}+ \varepsilon^{-1} S^{T} V S $.

PROOF OF THEOREM 2

Pre- and post-multiplying the inequalities (25) and (26) by diag$ \{P^{-1},{\bf{I}}\} $ and (27) by diag$ \{P^{-1} $, $ P^{-1} $, $ P^{-1} $, $ P^{-1} $, $ {\bf{I}} $, $ {\bf{I}} $, $ {\bf{I}} $, $ {\bf{I}}\} $, respectively and letting $ \bar{P} = P^{-1} $, $ \bar{Q} = \bar{P}Q\bar{P} $, $ \bar{R}_1 = \bar{P}R_1\bar{P} $, $ \bar{R}_2 = \bar{P}R_2\bar{P} $, $ \bar{\Phi} = \Phi^{-1} $ and $ \bar{K}_n = K_n\bar{P} $, we can easily obtain (28), (29) and

where $ \hat{\varkappa}_{11} = \left[ { $$\begin{array}{*{20}{c}} \hat{\varpi}_{11}&\hat{\varpi}_{12}&\hat{\varpi}_{13}\\ *&-\bar{P}\bar{\Phi}^{-1}\bar{P}&0\\ *&*&-\gamma^2 I \end{array}$$ } \right] $,

$ \hat{\varpi}_{11} = \left[ { $$\begin{array}{*{20}{c}} \Upsilon_{1}&B_m\rho(t)\bar{K}_n&\bar{R}_1\\ *&\Upsilon_{2}&\frac{\pi^2}{4}\bar{R}_2\\ *&*&\Upsilon_{3} \end{array}$$ } \right] $, $ \Upsilon_{1} = A_m\bar{P}+\bar{P}A_m^T+\bar{Q}-\bar{R}_1 $,

$ \Upsilon_{2} = \overline{\sigma}\bar{P}\bar{\Phi}^{-1}\bar{P}-\frac{\pi^2}{4}\bar{R}_2 $, $ \Upsilon_{3} = -\bar{Q}-\bar{R}_1-\frac{\pi^2}{4}\bar{R}_2 $,

$ \hat{\varpi}_{12} = [-(B_m\rho(t)\bar{K}_n)^T,0,0]^T $, $ \hat{\varpi}_{13} = [E_m^T,0,0]^T $, $\hat{\varkappa}_{12} = [hA_m\bar{P}, hB_m\rho(t)\bar{K}_n,0,-hB_m\rho(t)\bar{K}_n,hE_m]^T$ and $\hat{\varkappa}_{13} = [\tilde{C}_m\bar{P},D_m\rho(t)\bar{K}_n, 0,-D_m\rho(t)\bar{K}_n,0]^T$. To handle the nonlinear terms $ -\bar{P}\Delta^{-1}\bar{P} $, where $ \Delta $ stands for $ \bar{R}_1,\bar{R}_2,\bar{\Phi} $, respectively, it is known that $ (\Delta-\upsilon\bar{P})\Delta^{-1}(\Delta-\upsilon\bar{P})\ge0 $ always holds for any matrix $ \Delta>0 $ and scalar $ \upsilon>0 $, which implies that $ \Delta-2\upsilon\bar{P}\ge-\upsilon^2\bar{P}\Delta^{-1}\bar{P} $. We further note that the inequality (58) depends on the uncertain fault $ \rho(t) $ and thus cannot be solved out directly. Using Lemma 1 and Schur complement and further applying the above bounding inequalities to handle the nonlinear terms, we arrive at

which can be rewritten as

where $ \bar{\Xi}_{mn} $ can be obtained from (58) by replacing $ \rho(t) $ with $ \rho_0 $, $ \bar{\chi}_{1m} = [\varepsilon_1 B_m^T,0,0,0,0,\varepsilon_1 hB_m^T $, $ \varepsilon_1 hB_m^T $, $ \varepsilon_1 D_m^T]^T $ and $\bar{\chi}_{2n} = [0,\underline{\rho}_0\bar{K}_n,0,-\underline{\rho}_0\bar{K}_n,0,0,0,0]^T$. It is then not difficult to derive (30) from (60). ■

APPENDIX CPROOF OF THEOREM 3

Consider a similar discontinuous Lyapunov functional candidate as given in the proof of Theorem 1. Recalling the triggering instant (32) and denoting $ s_{k}h = t-d(t) $, we obtain the following condition $\sum_{m = 1}^8 \sum_{n = 1}^8 \lambda_m \lambda_n\big\{\overline{\sigma}(x(t-d(t))- e(t- d(t)))^{T}\Phi(x(t-d(t))-e(t-d(t)))-e^{T}(t-d(t))\Phi e(t-d(t))\big\}\ge0$. Following the similar lines as proof of Theorem 1, we obtain that

where $ {\Theta}_{mn} = \left[ { $$\begin{array}{*{20}{c}} \tilde{\varkappa}_{11}&\tilde{\varkappa}_{12}&\tilde{\varkappa}_{13}&\tilde{\varkappa}_{14}\\ *&-R_{1}&0&0\\ *&*&-R_{2}&0\\ *&*&*&-I \end{array}$$ } \right] $,

$ \tilde{\varkappa}_{11} = \left[ { $$\begin{array}{*{20}{c}} \tilde{\varpi}_{11}&\tilde{\varpi}_{12}&\tilde{\varpi}_{13}\\ *&(\overline{\sigma}-1)\Phi&0\\ *&*&-\gamma^2 I \end{array}$$ } \right] $,

$ \tilde{\varpi}_{11} = \left[ { $$\begin{array}{*{20}{c}} \Upsilon_{1}&PB_m\rho(t)K_n&R_1\\ *&\Upsilon_{2}&\frac{\pi^2}{4}R_2\\ *&*&\Upsilon_{3} \end{array}$$ } \right] $, $ \Upsilon_{1} = PA_m+A_m^TP+Q-R_1 $,

$ \Upsilon_{2} = \overline{\sigma}\Phi-\frac{\pi^2}{4}R_2 $, $ \Upsilon_{3} = -Q-R_1-\frac{\pi^2}{4}R_2 $, $\tilde{\varpi}_{12} = [-(PB_m\rho(t)K_n)^T, -\overline{\sigma}\Phi,0]^T$, $ \tilde{\varpi}_{13} = [(PE_m)^T,0,0]^T $, $ \tilde{\varkappa}_{12} = [hR_1A_m $, $ hR_1B_m\rho(t)K_n $, $ 0,-hR_1B_m\rho(t)K_n,hR_1E_m]^T $, $ \tilde{\varkappa}_{13} = [hR_2A_m $, $ hR_2B_m\rho(t)K_n $, $ 0 $, $ -hR_2B_m\rho(t)K_n $, $ hR_2E_m]^T $, and $ \tilde{\varkappa}_{14} = [\tilde{C}_m,D_m\rho(t)K_n,0 $, $ -D_m\rho(t)K_n $, $ 0]^T $. Similar to the proof of Theorem 2, performing some congruence transformation and letting $ \bar{P} = P^{-1} $, $ \bar{Q} = \bar{P}Q\bar{P} $, $ \bar{R}_1 = \bar{P}R_1\bar{P} $, $ \bar{R}_2 = \bar{P}R_2\bar{P} $, $ \bar{\Phi} = \bar{P}\Phi \bar{P} $ and $ \bar{K} = K\bar{P} $. Then applying Lemma 1 and Schur complement, we get the inequality (34). ■