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Stability analysis of lateral dynamics of a vehicle model utilizing state feedback with integral control and bifurcation method
利用积分控制和分岔方法的状态反馈对车辆模型的横向动力学进行稳定性分析

Rahul Prakash, Dharmendra Kumar Dheer *
拉胡尔·普拉卡什 （Rahul Prakash），达曼德拉·库马尔·迪尔 （Dharmendra Kumar Dheer） *Department of Electrical Engineering, NIT Patna, Patna, Bihar, India
印度比哈尔邦巴特那 NIT 巴特那电气工程系

ARTICLE INFO  文章信息

Keywords:  关键字：

Unmanned ground vehicle  无人地面车辆
Lateral and yaw dynamics  横向和偏航动力学
Control and stability  控制和稳定性
Bifurcation analysis  分岔分析
State feedback controller
状态反馈控制器

Abstract  抽象

In this work, a novel control strategy is proposed to obtain the stability boundary in addition to reduce the transients around the equilibrium points. To encounter the described problem, a new approach of combining the bifurcation analysis with the state feedback controller is proposed. A bifurcation analysis at different equilibrium points is performed to obtain the stable region of operation. In addition to this, the transients behavior of the system is also obtained simultaneously in the form of eigenvalues plots. The objective of the proposed controller is to generate a control law and state variables to reduce the transients keeping the system within the stability boundary by tuning the reference input matrix. From the obtained simulation results, it is seen that, by combining the bifurcation analysis with state feedback controller, the transients and the steady state error are reduced by selecting the purely negative real eigenvalues and reference input matrix respectively. The obtained closed loop control law and the state variables utilizing Ackerman’s Formula are found within the stability limit. A sensitivity index obtained from local sensitivity analysis verifies the relationship between the stability boundary at different longitudinal velocity on a low-friction road obtained from bifurcation analysis.
在这项工作中，提出了一种新的控制策略来获得稳定性边界，同时减少平衡点周围的瞬态。针对所述问题，提出了一种将分岔分析与状态反馈控制器相结合的新方法。在不同平衡点进行分岔分析以获得稳定的操作区域。除此之外，系统的瞬态行为也以特征值图的形式同时获得。所提出的控制器的目标是生成控制律和状态变量，以通过调整参考输入矩阵来减少瞬态，使系统保持在稳定边界内。从得到的仿真结果可以看出，通过将分岔分析与状态反馈控制器相结合，分别选择纯负实数特征值和参考输入矩阵来减小瞬态和稳态误差。获得的闭环控制律和利用 Ackerman 公式的状态变量位于稳定性极限内。从局部敏感性分析中获得的敏感性指数验证了分岔分析获得的低摩擦道路上不同纵向速度下的稳定性边界之间的关系。

1. Introduction  1. 引言

The development of effective control strategies and simultaneously satisfying the stability boundary for trajectory tracking controllers remains a significant hurdle in autonomous systems. This area has garnered significant attention, leading to numerous findings across various types of autonomous systems [1]. In the context of controlling an Automated Guided Vehicle (AGV) along a designated path, the objective of the vehicle is to precisely track the reference path. This is achieved by applying suitable steering actions that steer the vehicle along the path while also ensuring the vehicle maintains its stability. To accomplish this, a range of control techniques have been utilized, including proportional-integral-derivative (PID) control, adaptive control, sliding mode control (SMC), and model predictive control (MPC) [2].
开发有效的控制策略并同时满足轨迹跟踪控制器的稳定性边界仍然是自主系统的一个重大障碍。这一领域引起了广泛关注，在各种类型的自主系统中取得了大量发现 [1]。在沿指定路径控制自动导引车 （AGV） 的情况下，车辆的目标是精确跟踪参考路径。这是通过应用适当的转向动作来实现的，这些动作使车辆沿着路径转向，同时确保车辆保持其稳定性。为了实现这一目标，人们使用了一系列控制技术，包括比例积分微分 （PID） 控制、自适应控制、滑模控制 （SMC） 和模型预测控制 （MPC） [2]。
The stability boundary of the vehicle is performed with the help of constraints handling in MPC to design a path tracking controller. To prevent the vehicle from drifting as the steering angle increases, the stability boundary is defined with respect to ( β − r ) ( β − r ) (beta-r)(\beta-r) phase plane in [3]. This phase plane is utilized to depict the shift of equilibrium point based on the change in the steering angle. Authors in [4], presented a state feedback controller is designed based on regional pole placement considering the uncertainty in vehicle parameters. A lateral stability analysis is done on a non-linear vehicle model with the help of phase
在 MPC 中的约束处理的帮助下，可以执行车辆的稳定性边界，以设计路径跟踪控制器。为了防止车辆随着转向角的增加而漂移，在 [3] 中定义了相对于 ( β − r ) ( β − r ) (beta-r)(\beta-r) 相平面的稳定性边界。该相位平面用于描述基于转向角变化的平衡点偏移。[4] 中的作者提出了一种基于区域极放置的设计状态反馈控制器，该控制器考虑了车辆参数的不确定性。在相位的帮助下对非线性车辆模型进行横向稳定性分析
plane and bifurcation analysis to find the stable region based on the movements of equilibrium points in [5]. A similar work is presented in [6] to obtain the vehicle stability region utilizing bifurcation analysis in combination with the dynamics of equilibrium points for a 5 DOF model considering the coupling of lateral and longitudinal characteristics. This stability region is obtained by utilizing the change in dynamics of the phase trajectories and the equilibrium points on the phase plane. In [7] authors have designed a SMC for the control of unstable subcritical Hopf bifurcation to a stable supercritical Hopf bifurcation on a chaotic autonomous system. A bifurcation analysis is done on a 2 DOF vehicle model to find the type of bifurcation and limit cycle that exist for the design of active safety systems [8]. Similarly in [9], the analysis is done to demonstrate the stability at handling limits and obtaining stable and unstable steady states. To analyze the effects of different driving modes, the authors in [10], have obtained the bifurcation characteristics for the 7 DOF non-linear vehicle model based on steering and driving torque. A similar analysis is done based on both parameters in [11], to analyze the 5 DOF vehicle model by combining the real coded Genetic Algorithm with Quasi Newton Algorithm.
平面和分岔分析，根据 [5] 中平衡点的运动找到稳定区域。[6] 中提出了类似的工作，利用分岔分析结合平衡点动力学获得车辆稳定性区域，用于考虑横向和纵向特性耦合的 5 DOF 模型。这个稳定区域是通过利用相轨迹的动力学变化和相平面上的平衡点获得的。在 [7] 中，作者设计了一个 SMC，用于控制混沌自治系统上不稳定的亚临界 Hopf 分岔到稳定的超临界 Hopf 分岔。对 2 DOF 车辆模型进行分叉分析，以找出主动安全系统设计中存在的分叉类型和极限循环 [8]。同样，在 [9] 中，进行分析是为了证明在处理极限下的稳定性以及获得稳定和不稳定的稳态。为了分析不同驾驶模式的影响，[10] 中的作者获得了基于转向和驱动扭矩的 7 DOF 非线性车辆模型的分叉特性。根据 [11] 中的两个参数进行了类似的分析，通过将真实编码的遗传算法与准牛顿算法相结合来分析 5 DOF 车辆模型。
For the stability analysis of the vehicle model with respect to the lateral dynamics at higher longitudinal velocity on low friction roads,
对于车辆模型在低摩擦道路上较高纵向速度下的横向动力学稳定性分析，

Nomenclature  命名法

α f α f alpha_(f)\alpha_{f} Front Tire Slip Angle  前轮胎打滑角
α i α i alpha_(i)\alpha_{i} Tire Slip Angle, i = i = i=i= Front ( f f ff ), Rear ( r r rr )
轮胎滑移角， i = i = i=i= 前 （ f f ff ）， 后 （ r r rr ）
α r α r alpha_(r)\alpha_{r} Rear Tire Slip Angle  后轮胎打滑角
β β beta\beta Sideslip Angle at Center of Gravity (CG)
重心侧滑角 （CG）
β r e f β r e f beta_(ref)\beta_{r e f} Reference Sideslip Angle  参考侧滑角
δ δ delta\delta Steering Angle of the Vehicle
车辆的转向角
δ s s δ s s delta_(ss)\delta_{s s} Steady State Steering Angle
稳态转向角
γ ( t ) γ ( t ) gamma(t)\gamma(t) Reference Variable of the vehicle model
车辆模型的参考变量
μ μ mu\mu Tire Road Friction Coefficient
轮胎路面摩擦系数
A Linearized System Matrix  线性化系统矩阵
B B BB Linearized Input Matrix  线性化输入矩阵
C Output Matrix
C α f C α f C_(alpha_(f))C_{\alpha_{f}} Front Tire Cornering Stiffness
前轮胎转弯刚度
C α r C α r C_(alpha_(r))C_{\alpha_{r}} Rear Tire Cornering Stiffness
后轮胎转弯刚度
F y f F y f F_(yf)F_{y f} Front Lateral Tire Force  前侧轮胎力
F y f F y f F_(yf)F_{y f} Rear Lateral Tire Force  后侧向轮胎力
I z I z I_(z)I_{z} Moment of Inertia around z axis
绕 z 轴的惯性矩
K v K v K_(v)K_{v} Understeer Gradient  欠转向梯度
L L LL Total Distance between Front and Rear Axle
前后桥之间的总距离
l f l f l_(f)l_{f} Distance of CG from Front axle
重心与前轴的距离
l r l r l_(r)l_{r} Distance of CG from Rear axle
重心与后桥的距离
m m mm Mass of the vehicle  车辆质量
r ( t ) r ( t ) r(t)r(t) Reference State Variables of the vehicle model
车辆模型的引用 State Variables
r r rr Yaw Rate  偏航率
r r e f r r e f r_(ref)r_{r e f} Reference Yaw Rate  参考偏航率
V x V x V_(x)V_{x} Longitudinal Velocity at Center of Gravity
重心处的纵向速度
x ( t ) x ( t ) x(t)x(t) State variables (r, β β beta\beta ) of the vehicle model
车辆模型的状态变量 （r， β β beta\beta ）
x i ( t ) x i ( t ) x_(i)(t)x_{i}(t) Integrator State Variables ( r , β r , β r,beta\mathrm{r}, \beta ) of the vehicle model
车辆模型的积分器状态变量 （ r , β r , β r,beta\mathrm{r}, \beta ）
y ( t ) y ( t ) y(t)y(t) Output Variable of the vehicle model
车辆模型的 Output Variable
alpha_(f) Front Tire Slip Angle alpha_(i) Tire Slip Angle, i= Front ( f ), Rear ( r ) alpha_(r) Rear Tire Slip Angle beta Sideslip Angle at Center of Gravity (CG) beta_(ref) Reference Sideslip Angle delta Steering Angle of the Vehicle delta_(ss) Steady State Steering Angle gamma(t) Reference Variable of the vehicle model mu Tire Road Friction Coefficient A Linearized System Matrix B Linearized Input Matrix C Output Matrix C_(alpha_(f)) Front Tire Cornering Stiffness C_(alpha_(r)) Rear Tire Cornering Stiffness F_(yf) Front Lateral Tire Force F_(yf) Rear Lateral Tire Force I_(z) Moment of Inertia around z axis K_(v) Understeer Gradient L Total Distance between Front and Rear Axle l_(f) Distance of CG from Front axle l_(r) Distance of CG from Rear axle m Mass of the vehicle r(t) Reference State Variables of the vehicle model r Yaw Rate r_(ref) Reference Yaw Rate V_(x) Longitudinal Velocity at Center of Gravity x(t) State variables (r, beta ) of the vehicle model x_(i)(t) Integrator State Variables ( r,beta ) of the vehicle model y(t) Output Variable of the vehicle model| $\alpha_{f}$ | Front Tire Slip Angle | | :---: | :---: | | $\alpha_{i}$ | Tire Slip Angle, $i=$ Front ( $f$ ), Rear ( $r$ ) | | $\alpha_{r}$ | Rear Tire Slip Angle | | $\beta$ | Sideslip Angle at Center of Gravity (CG) | | $\beta_{r e f}$ | Reference Sideslip Angle | | $\delta$ | Steering Angle of the Vehicle | | $\delta_{s s}$ | Steady State Steering Angle | | $\gamma(t)$ | Reference Variable of the vehicle model | | $\mu$ | Tire Road Friction Coefficient | | A | Linearized System Matrix | | $B$ | Linearized Input Matrix | | C | Output Matrix | | $C_{\alpha_{f}}$ | Front Tire Cornering Stiffness | | $C_{\alpha_{r}}$ | Rear Tire Cornering Stiffness | | $F_{y f}$ | Front Lateral Tire Force | | $F_{y f}$ | Rear Lateral Tire Force | | $I_{z}$ | Moment of Inertia around z axis | | $K_{v}$ | Understeer Gradient | | $L$ | Total Distance between Front and Rear Axle | | $l_{f}$ | Distance of CG from Front axle | | $l_{r}$ | Distance of CG from Rear axle | | $m$ | Mass of the vehicle | | $r(t)$ | Reference State Variables of the vehicle model | | $r$ | Yaw Rate | | $r_{r e f}$ | Reference Yaw Rate | | $V_{x}$ | Longitudinal Velocity at Center of Gravity | | $x(t)$ | State variables (r, $\beta$ ) of the vehicle model | | $x_{i}(t)$ | Integrator State Variables ( $\mathrm{r}, \beta$ ) of the vehicle model | | $y(t)$ | Output Variable of the vehicle model |
it is more informative to utilize the nonlinear tire model. The trajectory of the point on the plane can be used to express the vehicle’s transition state, and the stable state of the vehicle is represented by a point on the plane [12]. In [13], the ( β − β ˙ ) ( β − β ˙ ) (beta-beta^(˙))(\beta-\dot{\beta}) phase plane is utilized for the study of the vehicle lateral stability during cornering. An active braking system is implemented to achieve the yaw moment control system during critical cornering. The ( β − β ˙ ) ( β − β ˙ ) (beta-beta^(˙))(\beta-\dot{\beta}) phase plane is also utilized in [14-16]. In [17], the ( β − r ) ( β − r ) (beta-r)(\beta-r) phase plane is utilized if the stability analysis is performed for different values of constant longitudinal velocity. Since it is easy to sense yaw rate while driving, the ( β − r β − r beta-r\beta-r ) phase plane is arguably more intuitive rather than ( α f − α r ) , ( β − β ˙ ) α f − α r , ( β − β ˙ ) (alpha_(f)-alpha_(r)),(beta-beta^(˙))\left(\alpha_{f}-\alpha_{r}\right),(\beta-\dot{\beta}) available in the literature. Therefore, authors in [18] have implemented ( β − r β − r beta-r\beta-r ) phase plane to analyze the vehicle lateral stability and shown that achieving the vehicle’s lateral and yaw stability is more critical on a low friction road [19]. If the equilibrium point is at origin of ( β − r ) ( β − r ) (beta-r)(\beta-r) plane, then it can be interpreted that the vehicle is following a straight line trajectory. A spinning behavior of the vehicle occurs if only a unstable equilibrium point exits for the set of bifurcation parameter value and can be mathematically analyzed as a saddle node bifurcation [20,21]. In [22], a subcritical Hopf bifurcation in the nonlinear system is interpreted if the perturbations are large then the system is pushed outside the limit cycle and will diverge away from stable equilibrium. In [23], the authors have performed the bifurcation analysis on the understeering and oversteering vehicles for an in-depth understanding of the stability of steady-state motion of cars. It is found that, for an understeering vehicle, a supercritical Hopf bifurcation usually occurs, and an oversteering vehicle shows a subcritical Hopf bifurcation.
使用非线性轮胎模型提供的信息量更大。该点在平面上的轨迹可以用来表示飞行器的过渡状态，飞行器的稳定状态由飞行器上的一个点表示 [12]。在 [13] 中， ( β − β ˙ ) ( β − β ˙ ) (beta-beta^(˙))(\beta-\dot{\beta}) 相位平面用于研究转弯过程中车辆的横向稳定性。实施主动制动系统以实现关键转弯期间的偏航力矩控制系统。相位 ( β − β ˙ ) ( β − β ˙ ) (beta-beta^(˙))(\beta-\dot{\beta}) 平面也用于 [14-16]。在 [17] 中，如果对不同的恒定纵向速度值进行稳定性分析，则使用 ( β − r ) ( β − r ) (beta-r)(\beta-r) 相平面。由于在驾驶时很容易感应到偏航角速率，因此 （ β − r β − r beta-r\beta-r ） 相位平面可以说更直观，而不是 ( α f − α r ) , ( β − β ˙ ) α f − α r , ( β − β ˙ ) (alpha_(f)-alpha_(r)),(beta-beta^(˙))\left(\alpha_{f}-\alpha_{r}\right),(\beta-\dot{\beta}) 在文献中可用。因此，[18] 中的作者实施了 （） β − r β − r beta-r\beta-r ） 相平面来分析车辆的横向稳定性，并表明在低摩擦道路上实现车辆的横向和偏航稳定性更为重要 [19]。如果平衡点位于平面的 ( β − r ) ( β − r ) (beta-r)(\beta-r) 原点，则可以解释为车辆正在遵循直线轨迹。如果只有一个不稳定的平衡点退出分岔参数值集，就会发生车辆的旋转行为，并且可以在数学上分析为鞍座节点分岔[20,21]。在 [22] 中，如果扰动很大，则解释非线性系统中的亚临界 Hopf 分岔，则系统被推到极限循环之外，并偏离稳定平衡。在 [23] 中，作者对转向不足和转向过度车辆进行了分叉分析，以深入了解汽车稳态运动的稳定性。 研究发现，对于转向不足的车辆，通常会发生超临界 Hopf 分岔，而过度转向车辆则表现出亚临界 Hopf 分岔。
A presence saddle point in the phase plane indicates the divergence of the vehicle state and the position is affected by the longitudinal velocity, steering angle and friction coefficient [24]. Further, an adaptive sliding mode control is designed satisfying the stability index obtained from phase plane for vehicle lateral stability. In [25], a linear quadratic regulator and an adaptive proportion integral differential (PID) fuzzy controller is designed to achieve the yaw stability for the unstable region obtained by phase plane analysis. In [26], a slide model control and backstepping controllers are designed for coordination between the active steering, differential braking and active suspension system. A novel control-dependent barrier function (CDBF) is proposed [27] to address the vehicle lateral stability control problem, in which control inputs affect stability regions defined by CDBFs.
相平面中存在的鞍点表示车辆状态的发散，位置受纵向速度、转向角和摩擦系数的影响[24]。此外，设计了一种自适应滑模控制，满足从相位平面获得的车辆横向稳定性指标。在 [25] 中，设计了线性二次调节器和自适应比例积分微分 （PID） 模糊控制器，以实现通过相平面分析获得的不稳定区域的偏航稳定性。在 [26] 中，滑动模型控制和倒退控制器设计用于主动转向、差速制动和主动悬架系统之间的协调。提出了一种新的控制依赖性障碍函数 （CDBF） [27] 来解决车辆横向稳定性控制问题，其中控制输入会影响由 CDBF 定义的稳定性区域。
Due to these physical significance of phase plane and bifurcation analysis, authors have primarily focussed on the achieving the vehicle lateral stability utilizing the phase plane and bifurcation analysis instead of defining the stability region based only on the vehicle parameters [28,29].
由于相平面和分岔分析的这些物理意义，作者主要集中在利用相平面和分叉分析来实现车辆横向稳定性，而不是仅根据车辆参数定义稳定性区域[28,29]。
The vehicle’s lateral dynamics are affected by the changes in the road friction coefficient. There is a high chance of lateral instability on a low friction road [30]. The controlling of yaw rate on the slippery road by applying constraint utilizing sideslip angle is performed in [31]. It is difficult to change the longitudinal velocity of the vehicle while driving on slippery roads, therefore the V x V x V_(x)V_{x} is chosen as 20 m / s 20 m / s 20m//s20 \mathrm{~m} / \mathrm{s} and μ μ mu\mu = 0.3 = 0.3 =0.3=0.3. Extreme driving condition generates a fast transitions between stable and unstable vehicle states, necessitating dynamic requirements for the overall stability of the vehicle including lateral and yaw stability [32]. The longitudinal velocity V x = 22.22 m / s V x = 22.22 m / s V_(x)=22.22m//sV_{x}=22.22 \mathrm{~m} / \mathrm{s} with road friction coefficient of 0.35 are chosen for longitudinal and lateral stability. In [33], authors have chosen different longitudinal velocity (up to 35 m / s 35 m / s 35m//s35 \mathrm{~m} / \mathrm{s} ) and high/low friction coefficient to depict the stability boundary. To track the reference yaw rate at the longitudinal velocity of 22.22 m / s 22.22 m / s 22.22m//s22.22 \mathrm{~m} / \mathrm{s} and a low friction coefficient μ = 0.3 μ = 0.3 mu=0.3\mu=0.3 is utilized [34]. Model predictive controller (MPC) based active front steering control (AFS) is exploited to maintain the yaw stability. The stability boundary of the yaw dynamics is obtained based on the vehicle parameters. Active rearsteering and direct-yaw moment coordinated control were implemented based on the nonlinear fuzzy observation, and the Lyapunov method was utilized to demonstrate the stability of the fuzzy sliding mode controller in order to enhance the control of roll and yaw stability under low friction coefficient of μ = 0.3 μ = 0.3 mu=0.3\mu=0.3 [35]. A model predictive controller is designed to achieve lateral and yaw stability [36]. The double lane change maneuver is utilized for the control and stability on a low friction road μ = 0.5 μ = 0.5 mu=0.5\mu=0.5 and V x = 33.33 m / s V x = 33.33 m / s V_(x)=33.33m//sV_{x}=33.33 \mathrm{~m} / \mathrm{s}. The lateral stability controller is designed based on sliding model control to control the yaw rate of the vehicle [37]. An icy pavement with friction coefficient of 0.1 and 0.2 with longitudinal velocity of up to 27.77 m / s 27.77 m / s 27.77m//s27.77 \mathrm{~m} / \mathrm{s} are chosen for the stability purpose. In [38], for a yaw rate tracking is performed on low friction of 0.4 and longitudinal velocity of 16.66 m / s 16.66 m / s 16.66m//s16.66 \mathrm{~m} / \mathrm{s}. A front steering and yaw moment control of the vehicle are designed based on Linear Quadratic control and the tracking of yaw rate is performed by slide model controller. Different controllers including pure pursuit method, Stanley method, PID control, linear quadratic regulator, sliding mode control and model predictive control are designed for yaw rate path tracking on low friction roads [39]. The longitudinal velocity of 16.66 m / s 16.66 m / s 16.66m//s16.66 \mathrm{~m} / \mathrm{s} and friction coefficient of 0.4 are utilized as the operating conditions for the vehicle stability control.
车辆的横向动力学受道路摩擦系数变化的影响。在低摩擦力的道路上，横向不稳定的可能性很高 [30]。在 [31] 中，通过应用利用侧滑角的约束来控制湿滑路面的偏航率。在湿滑的道路上行驶时很难改变车辆的纵向速度，因此选择 20 m / s 20 m / s 20m//s20 \mathrm{~m} / \mathrm{s} 和 μ μ mu\mu = 0.3 = 0.3 =0.3=0.3 V x V x V_(x)V_{x} 。极端驾驶条件会在车辆稳定和不稳定状态之间产生快速转换，因此需要对车辆的整体稳定性（包括横向和偏航稳定性）提出动态要求[32]。选择道路摩擦系数为 0.35 的纵向速度 V x = 22.22 m / s V x = 22.22 m / s V_(x)=22.22m//sV_{x}=22.22 \mathrm{~m} / \mathrm{s} 以确保纵向和横向稳定性。在 [33] 中，作者选择了不同的纵向速度（高达 35 m / s 35 m / s 35m//s35 \mathrm{~m} / \mathrm{s} ）和高/低摩擦系数来描述稳定性边界。为了跟踪纵向速度和 22.22 m / s 22.22 m / s 22.22m//s22.22 \mathrm{~m} / \mathrm{s} 低摩擦系数 μ = 0.3 μ = 0.3 mu=0.3\mu=0.3 下的参考偏航角速率 [34]。利用基于模型预测控制器 （MPC） 的主动前转向控制 （AFS） 来保持偏航稳定性。偏航动力学的稳定性边界是根据飞行器参数得到的。基于非线性模糊观测实现了主动后转向和直接偏航力矩协调控制，并采用Lyapunov方法证明了模糊滑模控制器的稳定性，以增强在低摩擦系数 μ = 0.3 μ = 0.3 mu=0.3\mu=0.3 [35]下对横滚和偏航稳定性的控制。模型预测控制器旨在实现横向和偏航稳定性 [36]。 双变道机动用于低摩擦路 μ = 0.5 μ = 0.5 mu=0.5\mu=0.5 面的控制和稳定性 V x = 33.33 m / s V x = 33.33 m / s V_(x)=33.33m//sV_{x}=33.33 \mathrm{~m} / \mathrm{s} 。侧向稳定控制器基于滑动模型控制设计，用于控制车辆的偏航率 [37]。为了达到稳定性目的，选择摩擦系数为 0.1 和 0.2、纵向速度 27.77 m / s 27.77 m / s 27.77m//s27.77 \mathrm{~m} / \mathrm{s} 高达 0.2 的结冰路面。在 [38] 中，对于偏航角速率，在 0.4 的低摩擦力和 纵向速度 上进行跟踪 16.66 m / s 16.66 m / s 16.66m//s16.66 \mathrm{~m} / \mathrm{s} 。基于线性二次控制设计了车辆的前转向和偏航力矩控制，偏航率的跟踪由滑模控制器进行。为低摩擦道路上的偏航速率路径跟踪设计了不同的控制器，包括纯追踪法、斯坦利法、PID控制、线性二次调节器、滑模控制和模型预测控制[39]。纵向速度和 16.66 m / s 16.66 m / s 16.66m//s16.66 \mathrm{~m} / \mathrm{s} 摩擦系数 0.4 用作车辆稳定性控制的工作条件。
Based on the literature review, followings are the research gaps identified:
根据文献综述，以下是确定的研究差距：
  • The design of an integral based state feedback controller using pole placement technique for the vehicle model at higher longitudinal velocity is not explored. The stability boundaries on lateral and yaw dynamics during the controller design are formulated by utilizing the vehicle parameters without focussing on the stability through bifurcation analysis.
    没有探讨使用极点放置技术为较高纵向速度的车辆模型设计基于积分的状态反馈控制器。控制器设计过程中横向和偏航动力学的稳定性边界是利用车辆参数制定的，而没有通过分岔分析关注稳定性。
  • The state feedback control via pole placement technique is exploited to place the chosen desired negative real eigenvalues based on the design specifications to generate the control law. However, in the designing of controller for vehicle model the information about the design specifications including rise time, peak time, maximum overshoot are not provided. Instead, the reference values and stability boundary of the control input and the state variables are provided. Therefore, depending on the information about the system given, the controller design needs to improve to implement the stability criteria to enhance the controller performance.
    利用通过极点放置技术进行的状态反馈控制，根据设计规范放置选定的所需负实数特征值，以生成控制律。但是，在车辆模型控制器的设计中，没有提供有关设计规格的信息，包括上升时间、峰值时间、最大过冲。相反，它提供了控制输入和状态变量的参考值和稳定性边界。因此，根据给定的系统信息，需要改进控制器设计以实现稳定性标准以提高控制器性能。
  • Most of the literatures have focussed on analyzing the vehicle lateral and yaw stability for a specific longitudinal velocity. Therefore literature lacks the stability analysis for a wider range of a longitudinal velocity with low friction coefficient. Also, the importance of yaw dynamics over lateral dynamics at low friction roads of the vehicle is not discussed.
    大多数文献都集中在分析特定纵向速度下的飞行器横向和偏航稳定性。因此，文献缺乏对具有低摩擦系数的更宽范围纵向速度的稳定性分析。此外，在车辆的低摩擦道路上，偏航动力学相对于横向动力学的重要性没有得到讨论。
To fulfill the research gap found in the literature, following contributions are made in this manuscript:
为了填补文献中发现的研究空白，本手稿做出了以下贡献：
  • Eigenvalues plays a significant role for the stability analysis of the vehicle model. Therefore, an integral based state feedback controller via pole placement technique is exploited to achieve the stability. The effectiveness of the controller is tested for a wide range of longitudinal velocity on low friction roads.
    特征值在车辆模型的稳定性分析中起着重要作用。因此，利用一种基于积分的状态反馈控制器，通过极点放置技术来实现稳定性。控制器的有效性在低摩擦道路上的宽范围纵向速度下进行了测试。
  • A novel control strategy is proposed for a 2 DOF non-linear vehicle model by combining bifurcation analysis and an integral based state feedback controller via pole placement technique. Thus, the control law and state variables generated by the controller are within the stability boundary by tuning a reference input matrix.
    该文通过极点放置技术将分岔分析和基于积分的状态反馈控制器相结合，提出了一种新的 2 DOF 非线性车辆模型控制策略。因此，通过调整参考输入矩阵，控制器生成的控制律和状态变量位于稳定性边界内。
  • To observe yaw dynamics and the stability boundary of vehicle on low friction roads, a wider range of longitudinal velocity is utilized and the designed controller is tested to bound the vehicle within the stability limit.
    为了观察车辆在低摩擦道路上的偏航动力学和稳定性边界，利用了更宽的纵向速度范围，并测试了设计的控制器以将车辆限制在稳定性极限内。
The rest of this paper is organized as follows: Section 2 describes the mathematical modeling for the analysis and controller design. Phase plane and Bifurcation analysis are defined in Section 2.2. Section 3 deals with the integral state feedback control. Simulation results and discussions are presented in Section 4. Finally, Section 5 ends with the conclusion.
本文的其余部分组织如下：第 2 节描述了分析和控制器设计的数学建模。相平面和分岔分析在 2.2 节中定义。第 3 节涉及积分状态反馈控制。仿真结果和讨论在第 4 节中介绍。最后，第 5 节以结论结束。

2. Model architecture and analysis
2. 模型架构和分析

In this section a detailed description of the vehicle model and the mathematical modeling are provided. Additionally, a phase plane analysis is performed to find the equilibrium points and bifurcation theory is implemented on the obtained parameters values at stable equilibrium point.
本节提供了车辆模型和数学建模的详细说明。此外，还进行了相平面分析以找到平衡点，并在稳定平衡点处对获得的参数值实施了分岔理论。

2.1. System description  2.1. 系统描述

The work is focussed on the analyzing lateral and yaw dynamics of the non-linear vehicle to obtain the stability boundary and to design a controller. Therefore, to develop the mathematical model replicating the dynamics of the vehicle, some assumptions have been made based on the literatures available [40,41]:
这项工作的重点是分析非线性飞行器的横向和偏航动力学，以获得稳定性边界并设计控制器。因此，为了开发复制车辆动力学的数学模型，已经根据现有文献做出了一些假设 [40,41]：
  • A simplified vehicle model with fewer degrees of freedom (lateral and yaw dynamics) by ignoring the longitudinal dynamics and roll dynamics.
    通过忽略纵向动力学和侧倾动力学，具有较少自由度（横向和偏航动力学）的简化车辆模型。
  • In this 2 DOF planar model, the longitudinal velocity is considered as constant to avoid complexities in the controller design.
    在这个 2 DOF 平面模型中，纵向速度被视为常数，以避免控制器设计的复杂性。
  • The left and right tires are assumed to have similar behavior and lumped to form single front and rear tires.
    假设左轮胎和右轮胎具有相似的行为，并集中形成单个前轮胎和后轮胎。
Fig. 1. Vehicle bicycle model.
图 1.车辆自行车模型。
The Steering angle has an important effect on vehicle lateral stability region [6]. Therefore, to obtain the stability region of the vehicle in terms of lateral and yaw dynamics, steering angle is chosen as the control parameter (bifurcation parameter) for different longitudinal velocities in this work.
转向角对车辆横向稳定区域有重要影响 [6]。因此，为了获得车辆在横向和偏航动力学方面的稳定区域，本研究选择转向角作为不同纵向速度的控制参数（分叉参数）。

2.2. Mathematical modeling
2.2. 数学建模

The 2-DOF vehicle bicycle model is utilized to explore the lateral characteristics as shown in Fig. 1. The governing equations of the vehicle’s handling dynamics is described by Newton-Euler law. The vehicle lateral and yaw dynamics [12] are defined by Eq. (1):
利用 2-DOF 车辆自行车模型来探索横向特性，如图 1 所示。车辆操控动力学的控制方程由牛顿-欧拉定律描述。车辆横向和偏航动力学 [12] 由方程 （1） 定义：
m V x β ˙ = F y f + F y r − m V x r m V x β ˙ = F y f + F y r − m V x r mV_(x)beta^(˙)=F_(yf)+F_(yr)-mV_(x)rm V_{x} \dot{\beta}=F_{y f}+F_{y r}-m V_{x} r
I z r ˙ = l f F y f − l r F y r I z r ˙ = l f F y f − l r F y r I_(z)r^(˙)=l_(f)F_(yf)-l_(r)F_(yr)I_{z} \dot{r}=l_{f} F_{y f}-l_{r} F_{y r}
where, β β beta\beta is the sideslip angle, m m mm is the mass of the vehicle, V x V x V_(x)V_{x} is the longitudinal velocity, F y f F y f F_(yf)F_{y f} is the front lateral tire force, F y r F y r F_(yr)F_{y r} is the rear lateral tire force, r r rr is the yaw rate, I z I z I_(z)I_{z} is the moment of inertia along z -axis, l f l f l_(f)l_{f} is the distance of center of gravity (CG)from front axle, l r l r l_(r)l_{r} is the distance of center of gravity from rear axle. The lateral tire forces F y f , F y r F y f , F y r F_(yf),F_(yr)F_{y f}, F_{y r} are the function of front tire slip angle α f α f alpha_(f)\alpha_{f}, rear tire slip angle α r α r alpha_(r)\alpha_{r} considered by the nonlinear Magic formula [42]. The lateral tire forces are expressed by Eq. (2):
其中， β β beta\beta 是侧滑角， m m mm 是车辆的质量， V x V x V_(x)V_{x} 是纵向速度， F y f F y f F_(yf)F_{y f} 是前侧向轮胎力， F y r F y r F_(yr)F_{y r} 是后侧向轮胎力， r r rr 是偏航率， I z I z I_(z)I_{z} 是沿 z 轴的惯性矩， l f l f l_(f)l_{f} 是重心 （CG） 与前轴的距离， l r l r l_(r)l_{r} 是重心与后轴的距离。轮胎侧向力 F y f , F y r F y f , F y r F_(yf),F_(yr)F_{y f}, F_{y r} 是前轮胎滑移角 α f α f alpha_(f)\alpha_{f} 、后轮胎滑移角 α r α r alpha_(r)\alpha_{r} 的函数，由非线性魔术公式 [42] 考虑。轮胎侧向力由方程 （2） 表示：
F y , i = μ D i sin ⁡ { C i tan − 1 ⁡ [ B i α i − E i ( B i α i − tan − 1 ⁡ ( B i α i ) ) ] } F y , i = μ D i sin ⁡ C i tan − 1 ⁡ B i α i − E i B i α i − tan − 1 ⁡ B i α i F_(y,i)=muD_(i)sin{C_(i)tan^(-1)[B_(i)alpha_(i)-E_(i)(B_(i)alpha_(i)-tan^(-1)(B_(i)alpha_(i)))]}F_{y, i}=\mu D_{i} \sin \left\{C_{i} \tan ^{-1}\left[B_{i} \alpha_{i}-E_{i}\left(B_{i} \alpha_{i}-\tan ^{-1}\left(B_{i} \alpha_{i}\right)\right)\right]\right\}
where, i = f , r i = f , r i=f,ri=f, r. While the tire cornering stiffness is varying in nature, but for mathematical complexity it is assumed to be constant for this study. For low adhesion road μ = 0.3 μ = 0.3 mu=0.3\mu=0.3, the tire parameters are chosen as C i = 1.56 , C f = C r , D f = − 2574 , D r = − 1749 , B f = 11 , B r = 18 C i = 1.56 , C f = C r , D f = − 2574 , D r = − 1749 , B f = 11 , B r = 18 C_(i)=1.56,C_(f)=C_(r),D_(f)=-2574,D_(r)=-1749,B_(f)=11,B_(r)=18C_{i}=1.56, C_{f}=C_{r}, D_{f}=-2574, D_{r}=-1749, B_{f}=11, B_{r}=18, E f = − 1.999 , E r = − 1.79 E f = − 1.999 , E r = − 1.79 E_(f)=-1.999,E_(r)=-1.79E_{f}=-1.999, E_{r}=-1.79. For small angle assumption, β ≈ ( V y / V x ) β ≈ V y / V x beta~~(V_(y)//V_(x))\beta \approx\left(V_{y} / V_{x}\right), the tire slip angles are expressed by Eq. (3):
其中， i = f , r i = f , r i=f,ri=f, r .虽然轮胎转弯刚度本质上是不同的，但就数学复杂性而言，它假设在本研究中是恒定的。对于低附着力路 μ = 0.3 μ = 0.3 mu=0.3\mu=0.3 面，轮胎参数选择为 C i = 1.56 , C f = C r , D f = − 2574 , D r = − 1749 , B f = 11 , B r = 18 C i = 1.56 , C f = C r , D f = − 2574 , D r = − 1749 , B f = 11 , B r = 18 C_(i)=1.56,C_(f)=C_(r),D_(f)=-2574,D_(r)=-1749,B_(f)=11,B_(r)=18C_{i}=1.56, C_{f}=C_{r}, D_{f}=-2574, D_{r}=-1749, B_{f}=11, B_{r}=18 ， E f = − 1.999 , E r = − 1.79 E f = − 1.999 , E r = − 1.79 E_(f)=-1.999,E_(r)=-1.79E_{f}=-1.999, E_{r}=-1.79 。对于小角度假设， β ≈ ( V y / V x ) β ≈ V y / V x beta~~(V_(y)//V_(x))\beta \approx\left(V_{y} / V_{x}\right) 轮胎滑移角由方程 （3） 表示：
α f = δ − l f r V x − V y V x = δ − l f r V x − β α f = δ − l f r V x − V y V x = δ − l f r V x − β alpha_(f)=delta-(l_(f)r)/(V_(x))-(V_(y))/(V_(x))=delta-(l_(f)r)/(V_(x))-beta\alpha_{f}=\delta-\frac{l_{f} r}{V_{x}}-\frac{V_{y}}{V_{x}}=\delta-\frac{l_{f} r}{V_{x}}-\beta
α r = l r r V x − V y V x = l r r V x − β α r = l r r V x − V y V x = l r r V x − β alpha_(r)=(l_(r)r)/(V_(x))-(V_(y))/(V_(x))=(l_(r)r)/(V_(x))-beta\alpha_{r}=\frac{l_{r} r}{V_{x}}-\frac{V_{y}}{V_{x}}=\frac{l_{r} r}{V_{x}}-\beta
where, δ δ delta\delta is the steering input to the vehicle, V y V y V_(y)V_{y} is the lateral velocity at CG.
其中， δ δ delta\delta 是车辆的转向输入， V y V y V_(y)V_{y} 是重心处的横向速度。
Eq. (1) is rewritten as below Eq. (4):
方程 （1） 改写为如下方程 （4）：
β ˙ = F y f + F y r m V x − r β ˙ = F y f + F y r m V x − r beta^(˙)=(F_(yf)+F_(yr))/(mV_(x))-r\dot{\beta}=\frac{F_{y f}+F_{y r}}{m V_{x}}-r
r ˙ = l f F y f − l r F y r I z r ˙ = l f F y f − l r F y r I z r^(˙)=(l_(f)F_(yf)-l_(r)F_(yr))/(I_(z))\dot{r}=\frac{l_{f} F_{y f}-l_{r} F_{y r}}{I_{z}}
The reference values of the state variables [43] are obtained from Eq. (5):
状态变量 [43] 的参考值从方程 （5） 中获得：
r r e f = V x L + K v V x 2 δ s s r r e f = V x L + K v V x 2 δ s s r_(ref)=(V_(x))/(L+K_(v)V_(x)^(2))delta_(ss)r_{r e f}=\frac{V_{x}}{L+K_{v} V_{x}^{2}} \delta_{s s}
β r e f = l r − Q V x 2 L + K v V x 2 δ s s β r e f = l r − Q V x 2 L + K v V x 2 δ s s beta_(ref)=(l_(r)-QV_(x)^(2))/(L+K_(v)V_(x)^(2))delta_(ss)\beta_{r e f}=\frac{l_{r}-Q V_{x}^{2}}{L+K_{v} V_{x}^{2}} \delta_{s s}
    • Corresponding author.  通讯作者。
    E-mail address: dkdheer@nitp.ac.in (D.K. Dheer).
    电子邮件地址：dkdheer@nitp.ac.in （D.K. Dheer）。
    https://doi.org/10.1016/j.jocs.2024.102321
    Received 10 November 2023; Received in revised form 7 March 2024; Accepted 9 May 2024
    2023 年 11 月 10 日收到;2024 年 3 月 7 日以修订版形式收到;录用日期 2024 年 5 月 9 日
    Available online 15 May 2024
    2024 年 5 月 15 日在线提供
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