State-sensitive event-triggered path following control of autonomous ground vehicles

1. INTRODUCTION

In the past few decades, the autonomous ground vehicles (AGVs) are rapidly developing because of their wide applications in military and civilian fields^[1–4]. Vehicle motion control, as the main function for fulfilling tasks, has become increasingly important in the design of AGVs. Since the sensors, controllers, and actuators are interconnected through communication networks, AGVs have evolved into typical networked control systems (NCSs)^[5,6]. Due to the limited bandwidth in-vehicle network, the network-induced delays and packet dropouts impose additional constraints on the control loop in AGVs. Therefore, the control issues that consider information exchange schemes have gained popularity in designing of AGVs^[6–9].

Path following control, which involves ensuring that a vehicle tracks the desired path with zero-steady state error, is a fundamental ability for AGVs^[10,11]. It is worth noting that the GPS, which is used for positioning and navigation, is important to the path following control of AGVs. Up to now, it should be mentioned that the advent of shared communication brings some new challenges due to the networked integration of path following control. Therefore, the path following control needs to consider not only the in-vehicle communication network but also the out-communication network. In a traditional way, although many efforts have been dedicated to developing path following control strategies, only a few of them can cope with time delays or packet dropouts caused by networks due to the low networked integrations for AGVs^[12,13]. Various control strategies have been developed to deal with such network-introduced difficulties, such as fuzzy control^[14], MPC algorithm^[8], robust control method^[9,15,16], and adaptive control strategy^[17]. However, the results of those methods show some conservations in communication scheduling. Based on the observation of existing works, there is still much room for further research on the co-design of control and communication for networked AGVs. This motivated us to study event-triggered path following control of AGVs.

To address the co-design issues of path following control and communication, the event-triggered communication (ETC) scheme, which transmits its measurement data only according to control requirements, has been proposed and received much attention from the control community. Compared to traditional time-triggered communication schemes, the ETC scheme is efficient in reducing communication frequency, thereby helping to avoid network congestion in in-vehicle networks. In addition, a larger communication interval can provide an extra time slot for scheduling vehicles among AGVs while preserving control performance^[18]. However, the event-triggered control scheme of path following control in AGVs is considered just in a few of works^[19–22], making it meaningful to develop ETC schemes specifically in path following control for AGVs. Furthermore, since functional safety is very important to the motion control of AGVs, a more intelligent ETC scheme should be well developed for path following control. In fact, ETC schemes have been extensively investigated in NCSs due to their resource-saving nature^[23,24] and the references therein. Up to now, several novel ETC schemes have been developed to solve the problems arising from networks. For example, the adaptive ETC scheme that depended on state gradient is proposed to adjust the event threshold in^[25], while a dynamic ETC scheme that introduces an additional positive term in the traditional ETC scheme is presented in^[26]. Resilient ETC schemes are also proposed in^[27,28] to deal with denial of service attacks, and learning-based ETC schemes are developed to cope with deception attacks in^[19]. Furthermore, a memory-based ETC scheme is developed to improve control performance in^[29]. Although the existing works on event-trigged control strategies show a flexible capacity to adjust transmission frequency, such previous works on the design of advanced ETC need extra terms or dynamics to adjust the event-triggered parameters rather than the parameters themselves. Therefore, these adjustment strategies of event-triggered threshold are blind to state perception, which splits the co-design of communication and control to some extent. In addition, due to the limited bandwidth for the in-vehicle communication network and limited capacity of calculation for the electronic control unit (ECU), using ETC to reduce communication and control burden is highly advantageous for path following control of AGVs.

Due to the limitation of the previous works, the state-measurement-based event-triggered control scheme is first developed to make a trade-off between communication efficiency and control performance. The proposed scheme, known as the state-sensitive (SS)-ETC scheme, incorporates the state measurement directly into the adjustment of event-triggered threshold dynamically. The convergence of the dynamic event-triggered thresholds is also removed while the stability of the path following control is not destroyed. The main contributions of this paper can be summarized as follows:

● The SS-ETC scheme is developed to adaptively adjust the event threshold. A barrier-like function is first introduced to indicate the SS-ETC scheme for path following control of AGVs. Compared to the static ETC scheme^[30] with a fixed event-triggered parameter, the proposed SS-ETC can adjust the event-triggered threshold dynamically by introducing such a barrier-like function. Importantly, there is no need to introduce extra terms or dynamics when adjusting the event-triggered threshold. This is very different from the previous works, such as adaptive ETC scheme^[25] and dynamic ETC scheme^[26].

● The state measurement will directly contribute to the adjustment of an event-triggered threshold. Unlike the separate descriptions for physical dynamics and communication intervals in previous works^[27,28], the dynamics of plant and communication scheme are intergraded in a uniform way. This approach gives a clearer presentation on stability properties and communication behavior. In addition, the proposed SS-ETC scheme is based on the sampled-data framework, which is different from the continuous one proposed by^[31].

● The stability of the path following control of AGV is naturally guaranteed by the proposed SS-ETC scheme. With the proposed SS-ETC scheme, all the event-triggered thresholds can be naturally suppressed within a stability region. This approach provides a more flexible way to the co-design of communication and control in path following control of AGV.

Compared to the previous works, the proposed SS-ETC scheme, based on state perception, shows a higher level of adaptiveness on threshold adjustment in order to improve the functional safety of AGVs.

The remainder of this paper is organized as follows: Section Ⅱ gives a more detailed description on the proposed SS-ETC and NCS modeling for path following control of AGVs. Section Ⅲ presents the main results of this work, including the stability analysis and controller design under the proposed SS-ETC. In section Ⅳ, simulation results, which were conducted on a networked path following control of AGVs, are shown to verify the proposed theoretical results. Finally, section Ⅴ concludes this paper.

To the end of this section, some physical meanings of related self-steering control of autonomous vehicles (AVs) are given below Table 1.

2. PROBLEM FORMULATION

In this section, we first present the networked path following control modeling. Then, the SS-ETC scheme is proposed to dynamically adjust the event-triggered parameter. At last, the path following control modeling of AGVs under the proposed SS-ETC scheme is established.

2.1. Networked path following control framework

A networked path following control of AVs can be classified as a typical NCS, where all the measurements are transmitted over a CAN bus and GPS network, among others. The control actions are calculated by an ECU. To optimize the utilization of communication and computation resources, this paper introduces the SS-ETC unit. The introduction of the SS-ETC unit aims to address the following two problems:

● Under the conditions of resource constraints of in-and-out vehicle networks, the SS-ETC unit should regulate its communication behavior to realize a more efficient utilization of resources.

● Considering the control requirements of the path following control, the SS-ETC unit must ensure the stability of the path following control of AVs under resource-constrained environments.

The framework of the networked path following control is shown in Figure 1. Then, the mathematical modeling of path following control of AVs is given as follows

(1) $$ \begin{equation} \left\{ \begin{aligned} \dot{e}&=v_{x}\beta+v_{x}\psi+d_{1}\\ \dot{\psi}&=r-\rho(\sigma)v_{x}\\ \dot{\beta}&=a_{11}\beta+a_{12}r+b_{1}\delta_{f}+d_{2}\\ \dot{r}&=a_{21}\beta+a_{22}r+b_{2}\delta_{f}+d_{3}. \end{aligned} \right. \end{equation} $$

Figure 1. Path following control diagram with SS-ETC scheme.

with

$$ a_{11}=-\frac{(C_{f}+C_{r})}{mv_{x}}, a_{12}=-(1+\frac{(l_{f}C_{f}-l_{r}C_{r})}{mv_{x}^{2}}), \\ a_{21}=\frac{(l_{r}C_{r}-l_{f}C_{f})}{I_{z}}, a_{22}=-(\frac{(l_{f}^{2}C_{f}+l_{r}^{2}C_{r})}{v_{x}I_{z}}), \\ b_{1}=\frac{C_{f}}{mv_{x}}, b_{2}=\frac{l_{f}C_{f}}{I_{z}} .$$

Further, define the state vector $$ x(t)=[e, \psi, \beta, r]^{T} $$, the control input $$ u(t)=\delta_{f} $$, the state-space form of the path following model^[32] of AVs can be given as follows

(2) $$ \begin{equation} \dot{x}(t)=Ax(t)+Bu(t)+w(t) \end{equation} $$

where

$$ \begin{gather*} A=\begin{bmatrix} 0 & v_{x} & v_{x} & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & a_{11}&a_{12}\\ 0 & 0 & a_{21}&a_{22} \end{bmatrix}, B=\begin{bmatrix} 0\\ 0\\ b_{1}\\ b_{2} \end{bmatrix}, w(t)=\begin{bmatrix} d_{1}\\ -\rho(\sigma)v_{x}\\ d_{2}\\ d_{3} \end{bmatrix} \end{gather*} $$

Here, the disturbance $$ w(t)=[d_{1}, -\rho(\sigma)v_{x}, d_{2}, d_{3}]^{T} $$ with modeling errors $$ d_{1} $$, $$ d_{2} $$, and $$ d_{3} $$.

For the purpose of achieving a better understanding of the proposed SS-ETC framework, the following general assumptions are given first.

Assumption 1The sensors are time triggered with a fixed sampling period $$ h $$. Then, the sampling set $$ \mathbb{S}_{1}=\{0, h, 2h, \cdots, kh\} $$ is assumed for all $$ k\in \mathbb{N} $$ under a sampled-data framework.

Assumption 2Under the ETC scheme, not all the sampled data would be transmitted. Therefore, one can assume that the set of successfully transmitted samples is $$ \mathbb{S}_{2}=\{0, t_{1}h, t_{2}h, \cdots, t_{k}h\} $$. Obviously, $$ \mathbb{S}_{2}\subseteq\mathbb{S}_{1} $$.

Assumption 3The controllers and actuators are event-triggered. Then the control input at the actuator is generated by a zero-order holder (ZOH) with its holding interval $$ t\in [t_{k}h+\tau_{t_{k}}, t_{k+1}h+\tau_{t_{k+1}}) $$. Here, $$ \tau_{t_{k}} $$ is the transmission delay.

Remark 1The main difference between $$ \mathbb{S}_{1} $$ and $$ \mathbb{S}_{2} $$ lies in the fact that not all the sampled data would be transmitted for the ETC scheme. In fact, if $$ \mathbb{S}_{1}=\mathbb{S}_{2} $$, it means that all sampled data are transmitted, and the event-triggered transmission becomes a time-triggered transmission for Assumption. 2.

2.2. SS-ETC scheme

The basic idea of the proposed SS-ETC scheme, which adjusts its event-triggered threshold according to the received state measurement, is captured by Figure 2.

Figure 2. The basic idea of the SS-ETC.

The properties of the SS-ETC can be attributed to the following two cases:

● When a larger state measurement $$ ||x(t_{k}h)|| $$ is received, the SS-ETC scheme deems that the physical system becomes unstable, and it would encourage transmissions to stabilize the physical system with a higher control frequency. Then a lower event-triggered is adopted.

● When a smaller state measurement $$ ||x(t_{k}h)|| $$ is received, the SS-ETC scheme deems that the physical system is stable, and some unnecessary transmissions can be reduced. Then an upper event-triggered threshold is maintained.

In order to highlight the SS-ETC scheme, we begin by introducing the following barrier-like function:

(3) $$ \begin{equation} f(x)=\frac{a}{x+b} \end{equation} $$

where $$ a $$ and $$ b $$ are given positive constants, and $$ x\geq0 $$ is an independent variable.

From (3), the following facts can be easily obtained:

● The upper bound of $$ f(x) $$ is $$ \frac{a}{b} $$, while $$ x=0 $$.

● A larger $$ x $$ leads to a smaller $$ f(x) $$ and vice versa. In addition, when $$ x\rightarrow \infty $$, $$ f(x)\rightarrow 0 $$.

● $$ b $$ is used to avoid zero in the denominator.

Since the ETC is an error-oriented communication scheme, then $$ \varsigma(i_{k}h) $$ is generally defined as the state error between the current sampling instant and the latest transmitted sampling instant, i.e.,

(4) $$ \begin{equation} \varsigma(i_{k}h)=x(i_{k}h)-x(t_{k}h) \end{equation} $$

with $$ i_{k}h=t_{k}h+\ell h $$, $$ \ell\in \mathbb{N} $$.

Motivated by (3), the SS-ETC scheme described in Figure 1 is designed as

(5) $$ \begin{equation} t_{k+1}h=t_{k}h+ \min\limits_{\ell}\{\ell h|\varsigma^{T}(i_{k}h)\Phi \varsigma(i_{k}h)\geq \frac{\sigma_{\epsilon}}{||x(t_{k}h)||+\epsilon}\mathcal{X}(t_{k})\} \end{equation} $$

where $$ \mathcal{X}(t_{k})=x^{T}(t_{k}h)\Phi x(t_{k}h) $$. $$ \epsilon>0 $$ is a given constant, $$ \sigma_{\epsilon} $$ is a positive event-triggered parameter related to $$ \epsilon $$, and $$ \Phi $$ is a positive definite matrix to be designed.

The event defined by $$ \frac{\sigma_{\epsilon}}{||x(t_{k}h)||+\epsilon}\mathcal{X}(t_{k}) $$ is important to the transmission under the ETC scheme. It is worth noting that the existing parameter $$ \epsilon $$ is used to avoid zero in the denominator when one wants to design a meaningful event-triggered scheme (5). In addition, once the parameter $$ \epsilon $$ is selected, the corresponding event-triggered parameter $$ \sigma_{\epsilon} $$ can be designed.

Remark 2In fact, the threshold value is important to make a trade-off between control performance and communication efficiency. The basics of the event design are to guarantee the stability of the controlled systems. But it is not easy to seek the best threshold because of time-varying requirements on control performance and communication efficiency. Thus, one can design dynamic ETC schemes or adaptive ETC schemes in order to achieve a better trade-off between control performance and communication efficiency.

Remark 3From the SS-ETC (5), a dynamic event threshold is well characterized by $$ \frac{\sigma_{\epsilon}}{||x(t_{k}h)||+\epsilon} $$, which is related to the parameters $$ \epsilon $$, $$ \sigma_{\epsilon} $$, and $$ ||x(t_{k}h)|| $$. Based on the established SS-ETC scheme, we can arrive at

● The pair ($$ \epsilon $$, $$ \sigma_{\epsilon} $$) gives the basis of the event-triggered parameter design while preserving the input-to-state stability (ISS). For a different $$ \epsilon $$, one can derive a corresponding $$ \sigma_{\epsilon} $$ to guarantee the existing of the controller $$ K $$;

● The $$ \frac{\sigma_{\epsilon}}{\epsilon} $$ supplies a upper bound of event-triggered parameter when $$ ||x(t_{k}h)||=0 $$. This indicates that a lower transmission frequency is expected while the stability of the studied NCS is guaranteed;

● The event-triggered threshold is dynamically adjusted according to the latest available measurement $$ ||x(t_{k}h)|| $$, directly.

Thus, the main advantages of the proposed SS-ETC lie in that: 1) by comparing adaptive ETC proposed in^[25], the lower bound of the event-triggered $$ \sigma_{m} $$ is not needed for the proposed SS-ETC (5); 2) by comparing the dynamic ETC scheme^[26], the extra term $$ \eta(t) $$, which is used to adjust the event threshold, is not necessary for the proposed SS-ETC (5); 3) It is clear that the thresholds of the proposed SS-ETC scheme (5) are always arrested in a stability region while threshold adjusting.

Therefore, the proposed SS-ETC (5) will make a trade-off between communication efficiency and control performance while the desired stability is guaranteed.

3. MAIN RESULTS

In this section, the stability analysis and input to state stabilization issues are conducted with Lyapunov theory. In addition, the co-design algorithm is also presented at the end of this section.

For facilitating the proofs in the following sections, the following useful lemmas, which play important roles in our derivations, are presented first.

Lemma 1^[33] For any positive definite matrix $$ R\in \mathbb{R}^{n\times n} $$ with $$ R=R^{T} $$, a scalar $$ \mu>0 $$ and a vector function $$ \dot{x}:[-\mu, 0]\rightarrow \mathbb{R}^{n} $$ such that the following integration inequality is well defined, i.e.,

where $$ \xi^{T}(t)=\left[ \begin{array}{cc} x(t) & x(t-\eta) \\ \end{array} \right]. $$

Lemma 2^[34] For any matrix $$ R>0 $$, $$ \left[ \begin{array}{cc} R & S \\ S^{T} & R \\ \end{array} \right]> 0 $$, scalars $$ -\eta_{2}\leq\eta(t)\leq -\eta_{1} $$, and a vector function $$ \dot{x}:[-\eta_{2}, -\eta_{1}]\rightarrow \mathbb{R}^{n} $$ such that the following integration inequality is well defined, i.e.,

where $$ \zeta^{T}(t)=\left[ \begin{array}{ccc} x(t-\eta_{1})& x(t-\eta(t)) & x(t-\eta_{2}) \\ \end{array} \right] $$.

4. VERIFICATION EXAMPLES

In this section, simulation results are provided to verify the designed stabilization method under the SS-ETC scheme for path following control of AGVs. The kinematics model of AGVs is depicted in Figure 3, and the detailed parameters can be browed from^[32] with $$ m=1500kg $$, $$ I_{z}=2500kg\cdot m^{2} $$, $$ l_{s}=0.8m $$, $$ l_{f}=1.3m $$, $$ l_{r}=1.4m $$, $$ C_{f}=40000N/rad $$, $$ C_{r}=40000N/rad $$, $$ v_{x}=25km/h $$.

Figure 3. Kinematics model of path following control.

The initial state is given by $$ x(0)=\left[ \begin{array}{cccc} -0.1 & 0 & -0.01 & 0.2\\ \end{array} \right] $$, the external disturbance is $$ w(t)=0.01\sin(t) $$ during $$ t\in[30s, 45s] $$.

Let $$ \tau_{m}=0.1 $$, $$ \tau_{M}=0.2 $$, $$ \alpha=0.32 $$, $$ \epsilon=1 $$. For comparisons, the controller's gain $$ K_{0} $$ based on time delay method is given by

Then according to Algorithm 1 and by solving LMIs (13) in Theorem 2, the controller gain for SS-ETC is obtained that $$ \sigma_{\epsilon}=0.23 $$ and

with

By setting sampling period $$ h=0.1s $$ and simulation length $$ T=150s $$, the following simulation results for comparisons are obtained.

Firstly, the state responses of the path following control are compared under a time-triggered scheme, static ETC scheme, and SS-ETC scheme, respectively. The simulations are given as follows. From the above simulation results from Figure 4$$ \sim $$Figure 7, the control performances, which include convergence and stability prosperities, are indeed improved under the proposed SS-ETC scheme by comparing with static ETC scheme.

Figure 4. Response of lateral offset.

Figure 5. Comparisons of heading error response.

Figure 6. Comparisons of sideslip angle response.

Figure 7. Comparisons of yaw rate response.

Then, we will observe the control performance with the given index $$ J=||x(t)||^{2} $$. The detailed performances are given in Table 2.

Table 2 shows that the control performance under SS-ETC is still less than a time-triggered control scheme. This implies that the functional safety is well trade-off between the time-triggered control scheme and static ETC scheme when the SS-ETC scheme is adopted.

Following this, we will analyze communication behaviors for the time-triggered transmission scheme, static ETC scheme, and SS-ETC scheme.

Based on Figure 8, the transmission times and average periods are given below. From Figure 8 and Table 3, one can see that the transmission times of SS-ETC are more than the ones of static ETC, and a less average transmission period is achieved. However, it is still far less than the time-triggered scheme. This is the reason why the SS-ETC can improve the control performance while saving communication resources. Nevertheless, such extra transmissions are based on the perception of the physical systems.

Figure 8. Release interval comparisons.

In order to present the evolutions of the SS-ETC scheme, the following simulation is used to record the change of event-triggered thresholds of the SS-ETC.

The adaptiveness of the proposed SS-ETC scheme is shown in Figure 9, where the event-triggered threshold would be dynamically adjusted according to the state measurement in real time. Recalling (5), when $$ ||x(t_{k}h)|| $$ becomes a larger one; thus a smaller event threshold is adopted. This is helpful in improving control performance when an unstable measurement is detected. However, the threshold is equal to the static event-triggered parameter when the state measurement $$ ||x(t_{k}h)||\rightarrow 0 $$. This is helpful in reducing transmissions when the system is stable.

Figure 9. Evolutions of threshold comparisons.

From the above simulation results, we can arrive at the following facts.

● If $$ ||x(t_{k}h)||\rightarrow 0 $$, the event-triggered parameter in the SS-ETC scheme will degenerate to the static event-triggered parameter, which is beneficial to reduce transmissions when a stable measurement is observed.

● An adaptive adjustment scheme of the proposed SS-ETC is sensitive to fluctuations in measurements. Once an unstable state measurement is detected, the SS-ETC scheme will adopt a smaller event-triggered threshold to encourage a transmission, where better control performance is expected.

● Under the proposed SS-ETC scheme, the control performance can be well improved, even at the cost of increased communication consumption. This supplies a dynamic way to trade off between control and communication.

Under the above discussions, the proposed SS-ETC scheme provides a more flexible transmission way to accommodate control requirements.

5. CONCLUSIONS

So far, a novel SS-ETC scheme has been well designed for reducing communication load while ISS is achieved. The barrier-like function is first introduced to design such a SS-based ETC scheme. The proposed SS-ETC scheme shows some intelligent features when adjusting the event-triggered threshold by a nonlinear function. The intelligence of the proposed SS-ETC scheme can be summarized as follows:

● The event thresholds of such sampled-data error will be suppressed in a given upper bound, which the stability of the NCS is guaranteed;

● The event thresholds realize the two-way adjustment (decrement and increment) according to the state measurement automatically.

Under the proposed SS-ETC, the networked path following model, which included both communication and control parts, has been accurately described. Then, the stability analysis and controller design have been derived in detail for the purpose of ISS. At last, some simulation experiments have been conducted to confirm the effectiveness of the proposed co-design method for SS-ETC. Different from the previous works on variable-threshold ETC, the designed SS-ETC adjusted event threshold solely based on the physical state perception. In fact, the proposed SS-ETC is a class of state-oriented communication schemes in which the physical security of path following control can be well guaranteed.

However, the proposed SS-ETC scheme failed to discuss the following aspects.

● It is obvious that the parameter $$ \epsilon $$ will make any difference on the event-triggered condition. How to design such a parameter is not included in this paper;

● So far, the proposed SS-ETC scheme only depended on state measurement. Some other more reasonable indexes should be further considered;

● In fact, if the vehicle maneuvers very fast, the nonlinear dynamics of the vehicle is not negligible, and a linear controller is not effective. Therefore, the controller design by considering nonlinear dynamics of the vehicles should be included;

● In addition, due to vulnerability of communication networks, a security SS-ETC scheme, which includes both physical state and communication security perception, is still a challenging problem^[35,36].

The above issues may be left for our future work.

DECLARATIONS

APPENDIX

Proof of Theorem 1: The following Lyapunov-Krasovskii functions are well constructed

where

By differentiating the above designed functionals along with $$ x(t) $$, we can obtain

In view of Lemma 1 and Lemma 2, the following relation holds

and

where $$ \zeta^{T}=\left[ \begin{array}{ccc} x(t-\tau_{m})& x(t-\tau(t))& x(t-\tau_{M}) \\ \end{array} \right] $$.

By summing up (18) $$ \sim $$ (24) and applying $$ \varsigma^{T}(i_{k}h)\Phi \varsigma(i_{k}h)\leq \frac{\sigma_{\epsilon}}{\epsilon}\mathcal{X}(t_{k}) $$, the following relation is achieved

where $$ \chi(t)=col\{ x(t), x(t-\tau_{m}), x(t-\tau(t)), x(t-\tau_{M}), w(t)\} $$ and $$ \Pi_{11} $$ is given in (13).

Using Schur complement for (25), (13) hold when $$ \Pi<0 $$ for all $$ t $$.

Let $$ a=\lambda_{min} (P) $$, $$ b=\lambda_{max}(P)+\eta_{2}\lambda_{max}(Q_{i})+\frac{\eta^{2}_{2}}{2}\lambda_{max}(R_{i}) $$ and $$ ||\overline{w}||=\max(||w(t)||) $$ for $$ i=1, 2 $$, it gives

from (25).

Obviously, (26) shows that the NCS (9) is ISS under the SS-ETC scheme (5).