Adaptive neural control of vehicular platoons with unknown functions and full state constraints

1. INTRODUCTION

Adaptive control is a control strategy that dynamically adjusts its parameters in response to changes in the system, with the goal of enhancing controller performance in uncertain and evolving environments. This approach finds extensive application in fields such as industrial automation^[1], cruise system design^[2], and traffic management^[3,4]. Especially when dealing with model uncertainties and external disturbances, this approach can effectively ensure the stability and tracking performance of the closed-loop system. With the development of modern industry and intelligent transportation systems, the demand for control of complex dynamic systems is increasing, and the research on adaptive control is particularly important. By continuously optimizing and improving adaptive control algorithms, the response speed and accuracy of the system can be enhanced, which provides a theoretical basis and technical support for achieving efficient and safe automation operations. Recently, the rapid acceleration of urbanization has inevitably resulted in increased road congestion and environmental pollution. This phenomenon has adversely impacted travel efficiency, economic growth, and environmental sustainability, placing unprecedented strain on traditional transportation systems. To address these challenges, many cities are exploring intelligent transportation systems and sustainable travel solutions. Among numerous studies, connected automatic vehicles (CAVs) can achieve efficient information sharing and intelligent decision-making through vehicle-to-vehicle (V2V) communication, reduce traffic accidents, execute intelligent scheduling and dynamic route planning, and reduce energy consumption^[5–8]. Note that distributed control of vehicular systems can achieve the control objective of the entire system through communication and coordination, and there are significant achievements^[9–14]. Specifically, References^[9,10] constructed the platoon as a dynamically decoupled system and used distributed model predictive control (DMPC) technology to solve flexible platooning scheduling problems. Reference^[11] mainly investigated the distributed adaptive fixed time queue tracking problem of third-order completely non-uniform nonlinear vehicles, and solves the interference problem through robust $$ H_{\infty} $$ control theory. Distributed platoon control relies on communication between vehicles, such as V2V. In order to ensure the security of this highly connected network, references^[12–14] discussed the vehicular system under network attacks, designed the adaptive distributed control framework, and utilized Lyapunov theory to prove the asymptotic stability. Based on these vehicular control theory results, the initial motivation of this paper is to design an adaptive control scheme for a third-order vehicle system and complete stability analysis. Due to model errors and parameter variations, the existence of unknown functions has brought great difficulties to the stability analysis of vehicle platoons. Usually, there are two methods for handling unknown functions: adaptive model predictive control (MPC)^[15,16] and function approximation^[17–23]. For example, the reference^[15] proposed a robust adaptive MPC algorithm to achieve good tracking performance and parameter set estimation accuracy, while the tube-based adaptive MPC scheme designed by^[16] ensured the asymptotic stability of the dynamic uncertainty closed-loop system. Differently^[17,18], utilized the neural network (NN) technique to approximate unknown dynamics. However, references^[15–18] considered linear systems or simple nonlinear systems, which might not be applicable to vehicular systems with relatively high complexity. In reference^[19], the proposed new scheme combined local motion planning and adaptive MPC, guaranteeing the motion behavior of vehicles traveling at the desired speed. In addition, adaptive MPC requires high accuracy in system modeling, and additional disturbances can affect the control effect. Hence, reference^[20] approximated unknown nonlinear functions through fuzzy logic systems (FLS) and proved the individual and string stability of the entire heterogeneous vehicle group using predefined time stability criteria and Lyapunov functions. By simpler single critic NN, reference^[21] achieved the approximation of unknown functions and proposed an improved weight update law to ensure the stability of closed-loop systems without the need for difficult-to-find initial allowable control schemes^[22]. developed a platoon tracking control strategy based on an adaptive NN algorithm, and then^[23] extended this scheme to vehicular systems with limited communication resources. Therefore, this paper will investigate vehicle platoons with unknown functions and utilize the NN approximation technique to solve the problem.

On the other hand, the safety issue also needs to be taken seriously, which was not discussed in the above articles^[24]. proposed an intermittent privacy protection mechanism to ensure the security of communication data between subsystems. References^[25,26] aimed to achieve specified performance queue control and strictly maintain tracking errors within the specified region^[27]. constrains the safe distance between vehicles, and^[28] simultaneously established constraints on the position and speed of the vehicle platoon. However, references^[25–28] ignored the constraint condition on acceleration, which may affect the control performance of the vehicle during rapid braking. Besides, imposing constraints is certainly beneficial for system safety, but at the same time, it can also generate redundant terms, which can cause trouble for control design. Notably, using barrier Lyapunov functions (BLFs) in the process of backstepping design is an effective method to overcome the shortcomings of state constraints. For instance, reference^[29] introduced time-varying integral BLFs (IBLFs) into adaptive control design, which ensured the stability of the closed-loop system without violating the constraints of the states^[30]. obtained the result of asymptotic convergence of tracking error to zero through logarithm BLFs (log-BLFs) theory analysis. In^[31–33], tangent BLFs (tan-BLFs) were used to design an adaptive controller. Subsequently, reference^[34]applied BLFs to vehicle platoons and completed stability analysis.

Motivated by the above discussion, this paper is dedicated to researching adaptive neural control of vehicular platoons with unknown functions and full state constraints. Main contributions are summarized as follows.

$$ (1) $$ Different from^[15,16], we establish an effective adaptive NN control framework for third-order vehicle platoons with external disturbances, which avoids excessive assumptions about unknown functions and simplifies the design process.

$$ (2) $$ Unlike traditional vehicle control^[9,10], we additionally consider safety factors during the driving process. By applying time-varying constraints on the position, velocity, and acceleration of the vehicle, it can ensure that the system states don't fluctuate significantly and remain within predefined boundaries, thereby improving predictability to a certain extent.

$$ (3) $$ Compared to the constant constraints investigated by^[30], time-varying state boundaries are more difficult to handle. We adopt tan-BLFs, which allow the controller to quickly adjust the control input when system state approaches constraints, bringing the system state back within a safe range. This helps to improve the response speed and stability of the system and avoid singularity phenomena.

The composition of the paper is summarized as follows. Section 2 introduces the third-order vehicular platoon and key technologies. Section 3 provides the design process of the adaptive NN controller. Section 4 presents the stability analysis. Section 5 shows an example to validate the effectiveness of the control approach. Section 6 concludes this paper.

2. PROBLEM FORMULATION

As shown in Figure 1, we consider the platoon of CAVs with a bidirectional communication topology of the vehicle ahead and the leading reference signal. $$ \beta_{1, i} $$, $$ \beta_{2, i} $$ and $$ \beta_{3, i} $$ denote the position, velocity, and acceleration of vehicle $$ i, i\in\{1, \cdots, N\} $$, respectively. $$ d_{i} $$ is the distance between two CAVs, and $$ d_{i, des} $$ is a desired spacing between two CAVs. $$ y_{c0} $$ is the reference signal. The considered CAV system is equipped with road maps, GPS, and onboard sensors. In addition, each vehicle is capable of V2V communication and receiving its own controller signals and communication between the two vehicles. During the driving process, the spacing can be kept within a safe range, that is $$ 0<d_{i, \min}<d_{i}<d_{i, \max} $$, where $$ d_{i, \min} $$ and $$ d_{i, \max} $$ are the minimum safety spacing and maximum distance, respectively.

Figure 1. Longitudinal motion of the vehicle platoon.

3. ADAPTIVE NEURAL CONTROL

In this section, we will design an adaptive NN controller to ensure the boundedness of the vehicular system with unknown functions and full state constraints. To proceed, given the following coordinate transformation

where $$ \omega_{1, i}(t) $$ and $$ \omega_{2, i}(t) $$ are virtual controllers, and $$ \epsilon_{j, i}(t), j=1, 2, 3 $$ are error variables.

Step 1. Based on (1) and (11), one has

By the RBFNN technique, unknown functions can be approximated by

where $$ Z_{i, 1}=[\bar\beta_{1, i}(t), y_{c0}(t), \dot y_{c0}(t)]^T $$, $$ W_{1, i} $$ is the NN weight vector, $$ S_{1, i}(Z_{1, i}) $$ is the basis function vector, and $$ \Lambda_{1, i}(Z_{1, i}) $$ satisfies $$ |\Lambda_{1, i}(Z_{1, i})| $$$$ \leq \xi_{1, i} $$ with $$ \xi_{1, i}>0 $$.

Next, we select the following tan-BLF

where $$ \delta_{1, i}>0 $$ represents the design parameter, $$ \tilde{W}_{1, i}=W_{1, i}-\hat W_{1, i} $$ is the estimation error and $$ \hat W_{1, i} $$ is the evaluation of $$ W_{1, i} $$. Besides, according to Assumption 2, we can obtain that $$ |\epsilon_{1, i}(t)|\leq p_{ai}(t) $$, where $$ p_{ai}(t)=f_{s}(t)-B_{1} $$ and $$ B_{1} $$ is a positive constant.

Then we further get

In view of $$ \chi_{1, i}(Z_{i, 1})=W_{1, i}^TS_{1, i}(Z_{1, i})+\Lambda_{1, i}(Z_{1, i}) $$, by Young's inequality, one has

Substituting (16) into (15), we can obtain

The virtual controller and the adaptive law are designed as

where $$ \alpha_{1, i}>0 $$ and $$ \rho_{1, i}>0 $$ are tuning parameters. Besides, $$ \Gamma_{1, i}(t) $$ is defined as

in which $$ \Gamma_{1, i}(t)>0 $$ and $$ \theta_{1, i}>0 $$.

Accordingly, it is obvious to deduce that

Furthermore, we substitute (18), (19) and (21) into (17), then

Finally, by defining $$ \Pi_{1, i}=\min\{\alpha_{1, i}, \rho_{1, i}, \delta_{1, i}\} $$, (22) can be rewritten as

Remark 4.In step 1, the virtual controller (18) is ultimately derived through coordinate transformation (11), tan-BLF (14), and the RBFNN technique. The basic design principle is to construct a suitable Lyapunov function, ensuring that its derivative is negative definite or semi-negative definite. Meanwhile, the design of (18) should also incorporate the adaptive law (19) to ensure the convergence of parameter estimation errors and state errors. Therefore, the role of each virtual controller is to stabilize the current subsystem and provide reference inputs for the next subsystem, ultimately achieving the stability of the entire system.

Remark 5.Compared to log-BLF, tan-BLF grows faster when the state approaches the constraint boundary, making the gradient of the Lyapunov function steeper and actively preventing constraint violations. This plays a significant role in systems with full state constraints. In addition, the structure of tan-BLF is more suitable for time-varying constraints because its trigonometric properties can simplify the adaptation to changing boundaries, which is beneficial in constraining dynamic environments that evolve over time.

Remark 6.During the design process, the singularity of the virtual controller has been avoided. When $$ \epsilon_{1, i}(t)\rightarrow 0 $$, we use the L'H$$ \hat{o} $$pital's rule to acquire

Therefore, the singularity problem will not happen in (18).

Step 2. Through (1) and (11), one has

By the RBFNN technique, unknown functions can be approximated by

where $$ Z_{2, i}=[\beta_{1, i}(t), \beta_{2, i}(t), p_{ai}(t), \dot p_{ai}(t), \ddot p_{ai}(t), y_{c0}(t), \dot y_{c0}(t), \hat{W}_{1, i}]^T $$, $$ W_{2, i} $$ is the NN weight vector, $$ S_{2, i}(Z_{2, i}) $$ is the basis function vector, and $$ \Lambda_{2, i}(Z_{2, i}) $$ satisfies $$ |\Lambda_{2, i}(Z_{2, i})|\leq \xi_{2, i} $$ with $$ \xi_{2, i}>0 $$.

Then we select the following tan-BLF

in which $$ \delta_{2, i}>0 $$ is the design parameter, $$ \tilde{W}_{2, i}=W_{2, i}-\hat{W}_{2, i} $$ is the estimation error and $$ \hat{W}_{2, i} $$ is the evaluation of $$ W_{2, i} $$. From Assumption 2, we have $$ |\epsilon_{2, i}(t)|\leq p_{bi}(t) $$, where $$ p_{bi}(t)=f_{v}(t)-B_{2} $$ and $$ B_{2} $$ is a positive constant.

Correspondingly, we further get

In view of $$ \chi_{2, i}(Z_{2, i}) $$$$ =W_{2, i}^TS_{2, i}(Z_{2, i})+\Lambda_{2, i}(Z_{2, i}) $$, by Young's inequality, one has

By (29) and (28), one can deduce that

Besides, the virtual controller and the adaptive law are constructed as follows

in which $$ \alpha_{2, i}>0 $$ and $$ \rho_{2, i}>0 $$ are tuning parameters. In addition, $$ \Gamma_{2, i}(t) $$ is

where $$ \Gamma_{2, i}(t)>0 $$ and $$ \theta_{2, i}>0 $$.

Naturally, we can obtain

Substituting (23), (31), (32) and (34) into (30) will yield

We define $$ \Pi_{2, i}=\min\{\Pi_{1, i}, \alpha_{2, i}, \rho_{2, i}, \delta_{2, i}\} $$, (35) can be changed to

Step 3. Based on (1) and (11), we have

Unknown functions can be approximated by the following RBFNN technique

where $$ Z_{3, i}=[\beta_{1, i}(t), \beta_{2, i}(t), \beta_{3, i}(t), p_{ai}(t), \dot p_{ai}(t), \ddot p_{ai}(t), \dddot p_{ai}(t), y_{c0}(t), \dot y_{c0}(t), \ddot y_{c0}(t), p_{bi}(t), \dot p_{bi}(t), \ddot p_{bi}(t), \hat{W}_{1, i}, \hat{W}_{2, i}]^T $$, $$ W_{3, i} $$ is the NN weight vector, $$ S_{3, i}(Z_{3, i}) $$ is the basis function vector, and $$ \Lambda_{3, i}(Z_{3, i}) $$ satisfies $$ |\Lambda_{3, i}(Z_{3, i})|\leq \xi_{3, i} $$ with $$ \xi_{3, i}>0 $$.

Consider the following tan-BLF

in which $$ \delta_{3, i}>0 $$ is the design parameter, $$ \tilde{W}_{3, i}=W_{3, i}-\hat{W}_{3, i} $$ is the estimation error and $$ \hat{W}_{3, i} $$ is the evaluation of $$ W_{3, i} $$. By Assumption 2, we can obtain that $$ |\epsilon_{3, i}(t)|\leq p_{ci}(t) $$, where $$ p_{ci}(t)=f_{v}(t)-B_{3} $$ and $$ B_{3} $$ is a positive constant.

Hence, the derivative of $$ Q_{3, i}(t) $$ is

According to $$ \chi_{3, i}(Z_{3, i})=W_{3, i}^TS_{3, i}(Z_{3, i})+\Lambda_{3, i}(Z_{3, i}) $$, Assumption 3 and Young's inequality, then we have

Based on (41), (42) and (40), $$ \dot Q_{3, i}(t) $$ satisfies

Subsequently, we construct the virtual controller and the adaptive law as

in which $$ \alpha_{3, i}>0 $$ and $$ \rho_{3, i}>0 $$ are tuning parameters. In addition, $$ \Gamma_{3, i}(t) $$ is

where $$ \Gamma_{3, i}(t)>0 $$ and $$ \theta_{3, i}>0 $$.

Besides, the following inequality is valid

When we substitute (36), (44), (45) and (47) into (43), then

4. STABILITY ANALYSIS

In this section, we summarize the main results in the following Theorem 1 and prove the stability of the closed-loop system.

Theorem 1.Under Assumptions 1-3, we consider the plant (1), virtual controllers (18), (31), adaptive laws (19), (32), (45) and the controller (44). If the initial conditions satisfy $$ |\beta_{1, i}(0)|<f_{s}(0) $$, $$ |\beta_{2, i}(0)|<f_{v}(0) $$ and $$ |\beta_{3, i}(0)|<f_{a}(0) $$, the designed control scheme can ensure that

(i) All closed-loop signals are bounded.

(ii) Error signals are kept in bounded sets $$ \Omega_{\epsilon} $$.

(iii) System states remain within constrained boundaries.

Proof. (i) Firstly, we select the Lyapunov function as $$ Q(t)=Q_{3, i}(t) $$. So the derivative of $$ Q(t) $$ satisfies

By defining $$ \Pi=\min\{\Pi_{2, i}, \alpha_{3, i}, \rho_{3, i}, \delta_{3, i}, i=1, \cdots, n\} $$ and $$ \Xi=\sum_{j=1}^{3}\dfrac{1}{2}\left(\rho_{j, i}\Vert W_{j, i}\Vert^2+\xi_{j, i}^2\right) $$, then (49) can ultimately be written as

Next we multiply both sides of (50) by $$ e^{\Pi t} $$, and integrate it over $$ [0, t] $$, then

Moreover, the following inequality can also be obtained

Based on the definition of $$ Q(t) $$, we can conclude that all signals are bounded.

(ii) According to (52), one has

Therefore, tracking errors are maintained in the following sets

(iii) According to Assumption 2, one gets that

Recalling the transformation (11), we have

Based on the boundedness of $$ \epsilon_{1, i}(t) $$, $$ \epsilon_{2, i}(t) $$ and $$ \epsilon_{3, i}(t) $$ and Assumption 1, we further get

where $$ \bar\omega_{1, i} $$ and $$ \bar\omega_{2, i} $$ are the upper boundaries of virtual controllers. Hence, system states $$ \beta_{1, i}(t) $$, $$ \beta_{2, i}(t) $$ and $$ \beta_{3, i}(t) $$ are bounded. This ends the proof. □

Remark 7.Reference^[23] studied the CAV platoon with unknown dynamics and communication resource limitations. In contrast, we consider the situation where unknown functions and full state constraints coexist, and design a control scheme based on the RBFNN technology, tan-BLF, and backstepping method. Compared with^[25], we investigate more complex third-order vehicle platoons with unknown functions, which makes the proposed control strategy more applicable. In addition, through Lyapunov analysis, it can be proven that both error signals and system states are constrained within the set range, ensuring the safe driving of vehicles.

Remark 8.This paper investigates the vehicle platoon under full state constraints. Firstly, in (4), we provide positive time-varying constraints on states and propose Assumptions 1-2 to ensure the relationship between the constraint boundaries and tracking signals and error signals. Subsequently, suitable tan-BLFs are defined in the backstepping design. In the proof of Theorem 1, we first demonstrate the boundedness of the closed-loop system, and then prove that error signals can converge to bounded compact sets. Finally, based on Assumptions 1-2, the coordinate transformation (11), and bounded compact sets of error signals, it can be concluded that system states always remain within predefined boundaries.

5. SIMULATION

In this section, we will employ an example to illustrate the effectiveness of our control strategy. Next, consider the following vehicular model referred to in^[28].

where the external disturbance is $$ \tau_{i}(t)=0.8e^{-0.5t} $$, and the reference signal is $$ y_{c0}(t)=0.5\sin(t) $$.

The time-varying boundaries of system states are presented as

Besides, the bounds of error signals are

The parameters of controllers are selected as: $$ \alpha_{11}=16 $$, $$ \alpha_{12}=\alpha_{13}=\alpha_{14}=15 $$, $$ \alpha_{22}=20 $$, $$ \alpha_{21}=\alpha_{23}=\alpha_{24}=30 $$, $$ \alpha_{31}=\alpha_{33}=100 $$, and $$ \alpha_{32}=\alpha_{34}=120 $$. The parameters related to adaptive law are $$ \delta_{11}=\delta_{21}=0.3 $$, $$ \delta_{14}=0.4 $$, $$ \delta_{13}=\delta_{22}=\delta_{31}=\delta_{32}=0.5 $$, $$ \delta_{12}=\delta_{33}=\delta_{34}=0.6 $$, $$ \delta_{23}=0.7 $$, $$ \rho_{11}=0.3 $$, $$ \rho_{12}=0.8 $$, $$ \rho_{14}=\rho_{21}=\rho_{22}=0.5 $$, $$ \rho_{13}=\rho_{23}=\rho_{24}=\rho_{31}=\rho_{32}=\rho_{34}=0.6 $$ and $$ \rho_{33}=0.7 $$. Initial values are $$ \beta_{1i}(0)=0(i=1, 2, 3, 4) $$, $$ \beta_{21}(0)=\beta_{22}(0)=\beta_{24}(0)=\beta_{31}(0)=\beta_{33}(0)=\beta_{34}(0)=0.1 $$, $$ \beta_{23}(0)=0.01 $$, $$ \beta_{32}=0.2 $$, $$ \hat W_{j, 1}=0.2 $$, $$ \hat W_{j, 2}=0.4 $$, $$ \hat W_{j, 3}=0.5 $$, and $$ \hat W_{j, 4}=0.4 $$ with $$ j=1, 2, 3 $$.

Figure 2-Figure 9 demonstrate the simulation results. Specifically, Figure 2-Figure 4 show the trajectories of the position, velocity and acceleration of the vehicle platoon, which also indicate that system signal can track the reference signal well and the states are kept in the designed bounds. Figure 5-Figure 7 imply that the error signals have not exceeded the boundaries. Figure 8 shows the trajectories of adaptive laws, while Figure 9 depicts the control inputs. Notably, it can be observed from these results that all signals are bounded and control inputs have good convergence performance. Therefore, this bidirectional communication topology enables vehicles to wirelessly exchange information about their position, velocity and acceleration, reducing reliance on a single preceding vehicle and avoiding instability of the platoon caused by a single point of failure. In addition, by fusing the information of the front and rear vehicles, disturbance propagation can be suppressed more quickly. This provides a suitable scheme design and safety considerations for practical vehicle research.

Figure 2. Trajectories of the position $$ \beta_{1, i} $$.

Figure 3. Trajectories of the velocity $$ \beta_{2, i} $$.

Figure 4. Trajectories of the acceleration $$ \beta_{3, i} $$.

Figure 5. Trajectories of the error $$ \epsilon_{1, i} $$.

Figure 6. Trajectories of the error $$ \epsilon_{2, i} $$.

Figure 7. Trajectories of the error $$ \epsilon_{3, i} $$.

Figure 8. Adaptive laws of $$ \hat{W}_{1i} $$, $$ \hat{W}_{2i} $$, and $$ \hat{W}_{3i} $$.

Figure 9. Control inputs of $$ u_{1} $$, $$ u_{2} $$, $$ u_{3} $$, and $$ u_{4} $$.

6. CONCLUSIONS

This paper proposes an adaptive neural control strategy for a third-order vehicle platoon with unknown functions and full-state constraints. For safety considerations, we impose time-varying constraints on the position, velocity, and acceleration of the vehicle. Meanwhile, the redundant terms caused by constraints can be eliminated by selecting appropriate tan-BLFs and the singularity issue can be avoided. Besides, we use the NN technique to approximate unknown nonlinearities instead of making too many assumptions, which also simplifies the design process. Finally, an example is utilized to verify the feasibility of the proposed control scheme. Notably, this article does not consider the optimal control scheme and the need to reduce the burden on the platoon communication network. In future research, we will introduce a dynamic programming and event-triggering mechanism to further improve the adaptability of the control algorithm.

DECLARATIONS