1 Introduction

Under the dual crisis of serious environmental pollution and oil resource shortage, many auto manufacturers have dedicated to the development of the electric vehicles (EVs) to reduce vehicle exhaust emissions and fuel consumption [1, 2]. For the purposes of stability and safety of the EVs, the electric vehicle control system needs to be further studied. Due to its unique dynamic system and drive system, the FWID-EVs have independent controllable torque of each drive wheel compared with the ordinary EVs, which brings enormous advantages to the vehicle control system [3]. In addition, the path-following performance is a crucial factor of the FWID-EV control systems that concern with vehicle active safety and did draw much attention in automotive technology research in recent years [4,5,6]. Consequently, this research intends to study the FWID-EVs to enhance the vehicle path-following performance.

Regarding the practical application of control system of the EVs, the vehicle parameters are uncertain, such as the vehicle mass, moment of inertia, and tire cornering stiffness [7,8,9]. A T–S fuzzy-based sliding mode control approach was adopted to address the parameter uncertainty problem of tire cornering stiffness and ensure control performance [10]. These uncertain parameters will impact the analysis of the vehicle lateral dynamics and controller performance. The observer-based fuzzy controller for EVs was constructed with the uncertainties of vehicle mass and moment of inertia in consideration [11]. By means of the IT-2 fuzzy method, Ren et al. discussed the lateral stability control issue for EVs with actuator saturation and sensor failures [12]. To address the path-following problem in autonomous ground vehicles, a linear fractional transformation (LFT)-based \(H_{\infty }\) SOF controller was developed [13].

Even though the above research can properly handle the parameters problem, the velocity is regarded as fixed in most studies, which does not correspond to the actual situation. And the gain scheduling control strategy that utilizes polytope to deal with the varying velocity is regarded as an effective method [14, 15]. Therefore, this paper intends to combine the LFT strategy and the gain scheduling control strategy to deal with the EVs’ uncertainty problem with varying speed.

In the existing investigations, vehicle state information is usually used for constructing the feedback loop of the control system. For FWID-EVs, accurate vehicle state measurements are required for the path-following control [16, 17]. And the lateral velocity is closely associated with the vehicle motion state. Because of the complex working environment, it is difficult to guarantee the precise measurement of the lateral velocity [18]. Zhang \(et\ al.\) transformed the output to state feedback problem and proposed an observer-based output feedback control method for a class of interconnected systems [19]. Chang et al. discussed the output feedback stabilization problem for a class of cascade nonlinear ODE-PDE systems [20].

However, the LMI variables constructed by the above methods tend to have more structural constraints that do not provide more degrees of freedom to solve the inequality. Recent research indicates that the Robust H-infinity output feedback method can properly manage uncertain system models and noise characteristics, and effectively reduce the structural constraints on LMI variables and improve the degree of freedom of LMI solutions [21,22,23,24]. Zhao et al. proposed an H-infinity SOF controller based on multi-objective frequency domain to obtain unmeasured active suspension systems status information, thereby improving the stability of the suspension [25]. Since the conceptual simplicity, ease of implementation of the H-infinity output feedback method, the robust H-infinity fuzzy static output feedback (RHF-SOF) control strategy which can online obtain the state information of the lateral velocity is investigated in this work.

With the measured state variables, the control variables of the control strategy can be confirmed. Then, the output variables are generated and transmitted to the actuator. The desired control effect will be attenuated when the actuator is saturated, because of the certain physical limitation of the actuator [26, 27]. In [28], an active vibration control approach was presented to handle the actuator saturation of the seismic excited building. For nonlinear 3-DOF helicopters, the distributed cooperative control with actuator saturation and faults was investigated in [29]. Moreover, the synchronous optimal control problem for the actuator saturation of nonlinear systems was discussed in [30]. So far, the research on the actuator saturation of FWID-EVs is rather limited.

Hence, the actuator saturation is still a challenging issue for the path-following control design of nonlinear FWID-EVs.

Inspired by all the problems mentioned above, this research describes the RHF-SOF path-following control strategy, and the actuator saturation of the nonlinear FWID-EVs are considered. The innovative points of this work can be summarized as follows:

  1. (1)

    An RHF-SOF control approach is developed for nonlinear FWID-EVs with consideration of parameter uncertainties, nonlinear characteristics, and actuator saturations;

  2. (2)

    A fuzzy optimization model is developed by combining the LFT strategy and the gain scheduling control strategy to handle the nonlinearities of the parameters;

  3. (3)

    A norm-bounded strategy is designed to guarantee the stability of the controller in the presence of actuator saturation.

The rest research is organized as follows. The system modeling is covered in Section 2. The main results of the RHF-SOF are explained in Section 3. In Section 4, experimental results are discussed. Finally, Section 5 presents the conclusion.

2 Problem formulation

2.1 Description of Vehicle Lateral Behavioral Model

In this work, lateral dynamics is taken into account and the typical vehicle lateral behavioral model is shown in Fig. 1. The lateral behavioral model can be derived as [31] follows:

Fig. 1
figure 1

Vehicle lateral behavioral model

Full size image
$$\left\{ \begin{gathered} \dot{v}_{y} = F_{{yf}} /m + F_{{yr}} /m - v_{x} r + d_{1} \hfill \\ I_{z} \dot{r} = M_{z} + l_{f} F_{{yf}} - l_{r} F_{{yr}} + d_{2} \hfill \\ \dot{\psi } = \dot{\psi }_{h} - \dot{\psi }_{d} = r - \rho v_{x} \hfill \\ \dot{e} = v_{y} + v_{x} \psi + d_{3} \hfill \\ \end{gathered} \right.$$
(1)

Among them, m, \(v_{x}\), \(v_{y}\), \(I_{z}\), \(F_{y f}\), \(F_{y r}\), r, \(M_{z}\) and e are unknown parameters that change over time. All unknown parameters are explained in Nomenclature. Choose \(x(t)=\left[ \begin{array}{llll}v_{y}&r&\psi&e\end{array}\right] ^{T}\) as the state variable. Then, motivated by the above discussions, the dynamic description of the vehicle model is described as follows:

$$\begin{aligned} \dot{x}(t)=A x(t)+B_{1} \omega (t)+B_{2} u(t) \end{aligned},$$
(2)

where \(\omega (t)=\left[ \begin{array}{llll} d_{1}&d_{2}&\rho&d_{3}\end{array}\right] ^{T}\), \(u(t)=\left[ \begin{array}{ll}\delta&\varDelta M_{z}\end{array}\right] ^{T}\), and

$$\begin{aligned} \begin{aligned}&A_{i}=\left[ \begin{array}{cccc}-\frac{C_{f}+C_{r}}{m v_{x}} &{} -\frac{l_{f} C_{f}-l_{r} C_{r}}{m v_{x}}-v_{x} &{} 0 &{} 0 \\ -\frac{l_{f} C_{f}-l_{r} C_{r}}{I_{z} v_{x}} &{} -\frac{l_{f}^{2} C_{f}+l_{r}^{2} C_{r}}{I_{z} v_{x}} &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 \\ 1&{} 0 &{}v_{x} &{}0\end{array}\right] ,\\&B_{1}=\left[ \begin{array}{cccc}1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} v_{x} &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 \end{array}\right] , B_{2}=\left[ \begin{array}{cc}\frac{C_{f}}{m} &{} 0 \\ \frac{l_{f} C_{f}}{I_{z}} &{} \frac{1}{I_{z}} \\ 0 &{} 0 \\ 0 &{} 0\end{array}\right] \end{aligned} \end{aligned}$$

2.2 Fuzzy Model of FWID-EV

Fig. 2
figure 2

Triangular polytope of the longitudinal velocity parameter

Full size image

As shown in Fig. 2, the parameter uncertainty of the vehicle longitudinal velocity is represented by a triangular polytope [32]. Time-varying parameters \(v_{x}\) and \(1 / v_{x}\) are bounded in \(\left[ v_{x \min } \ v_{x \max }\right]\) and \(\left[ 1 / v_{x \max } \ 1 / v_{x \min }\right]\). In triangular polytope, it covers all the possible choices for the parameter pair \(\left[ v_{x} \ 1 / v_{x }\right]\). The vertex coordinates are expressed as \(\varLambda _{i}=\left( \varphi _{x i}, \varphi _{y i}\right) , (i=1,2,3)\). The intersection of the tangents of the velocity curve at points \(\varLambda _{1}\), \(\varLambda _{2}\) , and \(\varLambda _{3}\).

The coordinates of \(\varLambda _{i}\) can be given as follows:

$$\begin{aligned}&\varphi _{x 1}=v_{x \min }, \varphi _{x 2}=v_{x \max }, \varphi _{x 3}=\frac{2 v_{x \max } v_{x \min }}{v_{x \min }+v_{x \max }} \nonumber \\&\varphi _{y 1}=\frac{1}{v_{x \min }}, \varphi _{y 2}=\frac{1}{v_{x \max }}, \varphi _{y 3}=\frac{2}{v_{x \min }+v_{x \max }} \nonumber \end{aligned}.$$

The sum of the vertex coordinates is utilized to represent the premise vector \(v_{x}\) and \(1 / v_{x}\) by

$$\begin{aligned} v_{x}=\sum _{i=1}^{3} \alpha _{i}(t) \varphi _{x i}, \frac{1}{v_{x}}=\sum _{i=1}^{3} \alpha _{i}(t) \varphi _{y i} \end{aligned},$$
(3)

where the weighting functions \(\alpha _{i}(t)\) are chosen as

$$\begin{aligned} \alpha _{1}=\frac{L_{34}}{L_{35}} \frac{L_{25}}{L_{12}}, \alpha _{2}=\frac{L_{34}}{L_{35}} \frac{L_{15}}{L_{12}}, \alpha _{3}=\frac{L_{45}}{L_{35}} \nonumber \end{aligned}$$

with \(L_{i j}\) representing the distance between the points \(\varLambda _{i}\) and \(\varLambda _{j}\). For \(i=1,2,3\), Eq. (3) meets \(\alpha _{i}(t)>0\), \(\sum _{i=1}^{3} \alpha _{i}(t)=1\), and the vehicle model is characterized as

$$\begin{aligned} \dot{x}(t)=\sum _{i=1}^{3} \alpha _{i}(t)\left[ A_{i} x(t) +B_{1 i} \omega (t)+B_{2} u(t)\right] \end{aligned},$$
(4)

where

$$\begin{aligned} \begin{aligned} A_{i}\!&=\!\left[ \begin{array}{cccc} \!-\frac{\varphi _{y i}\left( C_{J}+C_{r}\right) }{m}&{}\!-\frac{\varphi _{y i}\left( l_{f} C_{J}-l_{r} C_{r}\right) }{m}-\varphi _{x i} &{}\!0 &{}\!0 \\ \!-\frac{\varphi _{v i}\left( l_{f} C_{I}-l_{r} C_{r}\right) }{I_{z}} &{}\!-\frac{\varphi _{y i}\left( l_{f}^{2} C_{J}+l_{r}^{2} C_{r}\right) }{I_{z}} &{}\!0 &{}\!0 \\ \!0 &{} 1 &{} 0 &{} 0 \\ \!1 &{} 0 &{} \varphi _{x i} &{} 0 \end{array}\right] \!, \\ B_{1i}\!&=\left[ \begin{array}{cccc}1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} -\varphi _{x i} &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 \end{array}\right] , \end{aligned} \end{aligned}$$

Considering the variation of the driving condition, the parameters of the tire cornering stiffness are unknown and need to be considered in the concerned system. Therefore, it is defined that

$$\begin{aligned} \left\{ \begin{array}{l}C_{f}=C_{f 0}+\lambda _{f} \varDelta C_{f} \\ C_{r}=C_{r 0}+\lambda _{r} \varDelta C_{r}\end{array}\right. \end{aligned}$$
(5)

where the time-varying parameters \(\lambda _{f}\) and \(\lambda _{r}\) meet \(|\lambda _{i}|\le 1,(i=f, r)\), and

$$\begin{aligned}&C_{f 0}=\frac{\left( C_{f \max }+C_{f \min }\right) }{2}, \varDelta C_{f}=\frac{\left( C_{f \max }-C_{f \min }\right) }{2},\\&C_{r 0}=\frac{\left( C_{r \max }+C_{r \min }\right) }{2}, \varDelta C_{r}=\frac{\left( C_{r \max }-C_{r \min }\right) }{2} \end{aligned}.$$

In addition, vehicle mass m and moment of inertia \(I_{z}\) also change since the payload variation. Then, 1/m and \(1 / I_{z}\) can be represented as follows:

$$\begin{aligned} \frac{1}{m}=\frac{1}{m_{0}}+\lambda _{m} \frac{1}{\varDelta m}, \frac{1}{I_{z}}=\frac{1}{I_{z 0}}+\lambda _{I_{z}} \frac{1}{\varDelta I_{z}} \end{aligned},$$
(6)

where \(\lambda _{m}\) and \(\lambda _{I_{z}}\) are time-varying parameters that meeting \(|\lambda _{i}|\le 1,\left( i=m, I_{z}\right)\), and

$$\begin{aligned}&\frac{1}{m_{0}}=\frac{\left( \frac{1}{m_{\max }}+\frac{1}{m_{\min }}\right) }{2}, \frac{1}{\varDelta m}=\frac{\left( \frac{1}{m_{\min }}-\frac{1}{m_{\max }}\right) }{2},\\&\frac{1}{I_{z 0}}=\frac{\left( \frac{1}{I_{z \max }}+\frac{1}{I_{z \min }}\right) }{2}, \frac{1}{\varDelta I_{z}}=\frac{\left( \frac{1}{I_{z \min }}-\frac{1}{I_{z \max }}\right) }{2} \end{aligned}.$$

Furthermore, the varying \(C_{f} / m\), \(C_{r} / m\), \(C_{f} / I_{z}\) , and \(C_{r} / I_{z}\) can be constructed by

$$\begin{aligned} \frac{\text{C}_{f}}{m}&=\frac{\text{C}_{f 0}}{m_{0}} +\frac{\lambda _{m} \text{C}_{f 0}}{\varDelta m} +\frac{\lambda _{f} \varDelta \text{C}_{f}}{m_{0}} +\frac{\lambda _{f} \lambda _{m} \varDelta \text{C}_{f}}{\varDelta m} \\ {}&=\frac{\text{C}_{f 0}}{m_{0}}+\lambda _{1} \vartheta _{1}\\ \frac{\text{C}_{r}}{m}&=\frac{\text{C}_{r 0}}{m_{0}} +\frac{\lambda _{m} \text{C}_{r 0}}{\varDelta m} +\frac{\lambda _{r} \varDelta \text{C}_{r}}{m_{0}} +\frac{\lambda _{r} \lambda _{m} \varDelta \text{C}_{r}}{\varDelta m}\\&=\frac{\text{C}_{r 0}}{m_{0}}+\lambda _{2} \vartheta _{2}\\ \frac{\text{C}_{f}}{I_{z}}&=\frac{\text{C}_{f 0}}{I_{z 0}} +\frac{\lambda _{I} \text{C}_{f 0}}{\varDelta I_{z}} +\frac{\lambda _{f} \varDelta \text{C}_{f}}{I_{z 0}} +\frac{\lambda _{f} \lambda _{I_{z}} \varDelta \text{C}_{f}}{\varDelta I_{z}} \\ {}&=\frac{\text{C}_{f 0}}{I_{z 0}}+\lambda _{3} \vartheta _{3}\\ \frac{\text{C}_{r}}{I_{z}}&=\frac{\text{C}_{r 0}}{I_{z 0}} +\frac{\lambda _{m} \text{C}_{r 0}}{\varDelta I_{z}} +\frac{\lambda _{r} \varDelta \text{C}_{r}}{I_{z 0}} +\frac{\lambda _{r} \lambda _{I_{z}} \varDelta \text{C}_{r}}{\varDelta I_{z}} \\ {}&=\frac{\text{C}_{r 0}}{I_{z 0}}+\lambda _{4} \vartheta _{4} \end{aligned},$$

where the unknown parameters \(|\lambda _{i}|<1,(i=1,2,3,4)\), and

$$\begin{aligned}&\vartheta _{1} =\frac{\text{C}_{f 0}}{\varDelta m} +\frac{\varDelta \text{C}_{f}}{m_{0}} +\frac{\varDelta \text{C}_{f}}{\varDelta m}, \\&\vartheta _{2} =\frac{\text{C}_{r 0}}{\varDelta m} +\frac{\varDelta \text{C}_{r}}{m_{0}} +\frac{\varDelta \text{C}_{r}}{\varDelta m},\\&\vartheta _{3} =\frac{\text{C}_{f 0}}{\varDelta I_{z}} +\frac{\varDelta \text{C}_{f}}{I_{z 0}} +\frac{\varDelta \text{C}_{f}}{\varDelta I_{z}}, \\&\vartheta _{4} =\frac{\text{C}_{r 0}}{\varDelta I_{z}} +\frac{\varDelta \text{C}_{r}}{I_{z 0}} +\frac{\varDelta \text{C}_{r}}{\varDelta I_{z}} \end{aligned}.$$

Without loss of universality, \(\lambda _{1}=\lambda _{2}=\lambda _{3} =\lambda _{4}\) can be set, and the coefficient matrix can be written as \(A_{i}=A_{0 i}+\varDelta A_{i}\), \(B_{2}=B_{20}+\varDelta B_{2}\), and

$$\begin{aligned} \begin{aligned} A_{0 i}&\!=\!\left[ \begin{array}{cccc}-\frac{\varphi _{y i}\left( C_{f 0}+C_{r 0}\right) }{m_{0}} &{}\quad -\frac{\varphi _{y i}\left( l_{f} C_{f 0}-l_{r} C_{r 0}\right) }{m_{0}} -\varphi _{x i} &{}\quad 0 &{}\quad 0 \\ frac{\varphi _{y i}\left( l_{f} C_{f 0}-l_{f} C_{r 0}\right) }{I_{z 0}} &{}\quad -\frac{\varphi _{y i}\left( l_{f}^{2} C_{f 0}+l_{r}^{2} C_{r 0}\right) }{I_{z 0}} &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 \\ 1 &{}\quad 0 &{}\quad \varphi _{x i} &{}\quad 0\end{array}\right] , \\ \varDelta A_{i}&\!=\!\left[ \begin{array}{cccc}-\varphi _{y i}\left( \lambda _{1} \vartheta _{1} \!+\! \lambda _{2} \vartheta _{2}\right) &{}\quad -\varphi _{y i}\left( l_{f} \lambda _{1} \vartheta _{1}-l_{r} \lambda _{2} \vartheta _{2}\right) &{}\quad 0 &{}\quad 0 \\ varphi_{y i}\left( l_{f} \lambda _{3} \vartheta _{3}\!-\!l_{r} \lambda _{4} \vartheta _{4}\right) &{}\quad -\varphi _{y i} \! \left( l_{f}^{2} \lambda _{3} \vartheta _{3} \!+ \!l_{r}^{2} \lambda _{4} \vartheta _{4}\right) &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\end{array}\right] ,\\ B_{20}&\!=\!\left[ \begin{array}{cc}\frac{C_{f 0}}{m_{0}} &{}\quad 0 \\ \frac{l_{f} C_{f 0}}{I_{z 0}} &{}\quad \frac{1}{I_{z 0}} \\ 0 &{}\quad 0 \\ 0 &{}\quad 0\end{array}\right] , \varDelta B_{2}=\left[ \begin{array}{cc}\lambda _{1} \vartheta _{1} &{}\quad 0 \\ l_{f} \lambda _{3} \vartheta _{3} &{}\quad \frac{\lambda _{I_{z}}}{\varDelta I_{z}} \\ 0 &{}\quad 0 \\ 0 &{}\quad 0\end{array}\right] \end{aligned} \end{aligned}.$$

The vehicle model can therefore be expressed as follows:

$$\begin{aligned} \begin{aligned} \dot{x}(t)&=\sum _{i=1}^{3} \alpha _{i}(t) \cdot \left[ \left( A_{i}+\varDelta A_{i}\right) x(t)+B_{1 i} \omega (t)\right. \\&\quad \left. +\left( B_{20}+\varDelta B_{2}\right) u(t)\right] \end{aligned} \end{aligned},$$
(7)

where \(\varDelta A_{i}\) and \(\varDelta B_{i}\) represent the uncertainty caused by \(\varDelta C_{f}\), \(\varDelta C_{r}\), \(\varDelta m\) , and \(\varDelta I_{z}\), which can be formatted as

$$\begin{aligned} \left[ \begin{array}{ll}\varDelta A_{i}&\varDelta B_{2}\end{array}\right] =\left[ \begin{array}{ll}H \varGamma E_{a i}&H \varGamma E_{b}\end{array}\right] \end{aligned},$$
(8)

where

$$\begin{aligned} \begin{aligned}&E_{a i}=\left[ \begin{array}{cccc}-\varphi _{y i}\left( \vartheta _{1}+\vartheta _{2}\right) &{}\quad -\varphi _{y i}\left( l_{f} \vartheta _{1}-l_{r} \vartheta _{2}\right) &{}\quad 0 &{}\quad 0 \\ -\varphi _{y i}\left( l_{f} \vartheta _{3}-l_{r} \vartheta _{4}\right) &{}\quad -\varphi _{y i}\left( l_{f}{ }^{2} \vartheta _{3}+l_{r}{ }^{2} \vartheta _{4}\right) &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\end{array}\right] ,\\&H=\left[ \begin{array}{lll}1 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 1 &{}\quad 1 \\ 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0\end{array}\right] , \varGamma ={\text {diag}}\left\{ \lambda _{1}, \lambda _{1}, \lambda _{I_{z}}\right\} , E_{b} =\left[ \begin{array}{cc}\vartheta _{f} &{}\quad 0 \\ l_{f} \vartheta _{3} &{}\quad 0 \\ 0 &{}\quad \frac{1}{\varDelta I_{z}}\end{array}\right] \end{aligned} \end{aligned}.$$

The actuator input should be bounded in the controller design process due to the physical limits of the actuation system. Inspired by Du et al. [33], the saturation function \({\text {sat}}(u)\) meets

$$\begin{aligned} {\text {sat}}(u)= {\left\{ \begin{array}{ll}-u_{\lim }, &{} u(t) \in \left( -\infty ,-u_{\lim }\right) \\ u(t), &{} u(t) \in \left( -u_{\text{lim}}, u_{\lim }\right) \\ u_{\lim }, &{} u(t) \in \left( u_{\lim },+\infty \right) \end{array}\right. } \end{aligned},$$
(9)

where \(u_{\lim }>0\) is the actuator input limitation.

Taking into account the nonlinearity of the system and the saturation function, the closed-loop system can be derived as follows:

$$\begin{aligned} \begin{aligned} \dot{x}(t)&=\sum _{i=1}^{3} \alpha _{i}(t)\left[ \left( A_{i}+\varDelta A_{i}\right) x(t)+B_{1 i} \omega (t)\right. \\&\quad \left. +\left( B_{20}+\varDelta B_{2}\right) \bar{u}(t)\right] \\&=\sum _{i=1}^{3} \alpha _{i}(t)\left[ \left( A_{i}+\varDelta A_{i}\right) x(t)+B_{1 i} \omega (t)\right. \\&\quad +\left( B_{20}+\varDelta B_{2}\right) \frac{1+\varepsilon }{2} u(t) \\&\quad \left. +\left( B_{20}+\varDelta B_{2}\right) \left( \bar{u}(t)-\frac{1+\varepsilon }{2} u(t)\right) \right] \end{aligned} \end{aligned},$$
(10)

where \({\text {sat}}(u)=\bar{u}(t), 0<\varepsilon <1\).

2.3 Fuzzy SOF Controller

Regarding the vehicle system model’s state variable, it is a remarkable fact that the lateral velocity \(v_{y}\) is hard to available online. Hence, the measured output is expressed as

$$\begin{aligned} y(t)=\left[ \begin{array}{lll}r&\psi&e\end{array}\right] =C_{y} x(t) \end{aligned},$$
(11)

where

$$\begin{aligned} C_{y}=\left[ \begin{array}{llll}0 &{}\quad 1 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 1\end{array}\right] \end{aligned}.$$

The SOF controller can then be acquired as follows:

$$\begin{aligned} \begin{aligned} u(t)&=\sum _{j=1}^{3} \alpha _{j}(t) K_{j} y(t) \\&=\sum _{j=1}^{3} \alpha _{j}(t) K_{j} C_{y} x(t) \\&=K_{s o f} C_{y} x(t) \end{aligned} \end{aligned},$$
(12)

where \(K_{s o f}=\sum _{j=1}^{3} \alpha _{j}(t) K_{j}\) and \(K_{j}\) are the SOF controller gains to be designed.

The purpose of this work is to offer a RHF-SOF control strategy for the concerned vehicle dynamic system. Meanwhile, in the linear matrix inequalities control frame, the vehicle system in Eq. (10) is asymptotically stable and \(H_{\infty }\) performance is satisfied, namely

$$\begin{aligned} \gamma ^{2} \int \limits _{0}^{\infty } \omega ^{T}(t) \omega (t) \mathrm{{d}}t \ge \int \limits _{0}^{\infty } z^{T}(t) z(t) \mathrm{{d}}t \end{aligned},$$
(13)

where \(\gamma\) represents the attenuation level of the concerned vehicle system.

Remark 1

The main challenge of this paper is devoted to propose a reasonable approach to solve the problem of numerous parameter uncertainties in vehicle systems and to design a controller based on the norm-bounded strategy to deal with the actuator saturation problem.

3 Main Result

In this section, the RHF-SOF control strategy for concerned vehicle dynamic systems with uncertain parameters and input saturation is adopted to enhance the path-following performance, and following lemmas are given to prove the main theorem.

Lemma 1

[34] For the saturation function \({\text {sat}}(u)\), \(\left\| \frac{1-\varepsilon }{2} u(t)\right\| \ge \left\| \bar{u}(t)-\frac{1+\varepsilon }{2} u(t)\right\|\) holds for all input constraints satisfying \(|u(t)|\le \frac{{{u}_{lim}}}{\varepsilon }\), if and only if the following inequalities are true:

$$\begin{aligned} \begin{aligned}&\left( \frac{1-\varepsilon }{2}\right) ^{2} u^\mathrm{{T}}(t) u(t) \\&\quad \ge {\left[ \bar{u}(t)-\frac{1+\varepsilon }{2} u(t)\right] ^\mathrm{{T}}\left[ \bar{u}(t)-\frac{1+\varepsilon }{2} u(t)\right] } \end{aligned} \end{aligned}.$$
(14)

Lemma 2

[33] Given real matrices R and S with suitable dimensions, for all matrices F if \(\Vert F\Vert \le I\) holds, then exists a prescribed scalar \(\xi >0\) such that

$$\begin{aligned} R F S+S^\mathrm{{T}} F^\mathrm{{T}} R^\mathrm{{T}} \le \xi ^{-1} R R^\mathrm{{T}}+\xi S^\mathrm{{T}} S \end{aligned}$$
(15)

3.1 Stability Analysis and Controller Design

Theorem 1

The concerned vehicle system in Eq. (10) is asymptotically stable and meets the desired performance in Eq. (13), for given constants \(\varepsilon\), \(\mu _{l i m}\), \(\varphi >0\), if symmetric positive definite matrix Q, output feedback gains \(\bar{K}_{j}\) and positive scalars \(\gamma\), \(\sigma\), \(\xi _{1}\), \(\xi _{2}\), \(\xi _{3}\) exist and such that \((i, j=1,2,3)\) if and only if the following inequalities are true:

$$\begin{aligned}{} & {} \left[ \begin{array}{ccc}\varGamma _{i j}^{11} &{}\quad \varGamma _{i j}^{12} &{}\quad 0 \\ * &{}\quad \varGamma _{i j}^{22} &{}\quad \varGamma _{i j}^{23} \\ * &{}\quad * &{}\quad -\xi _{3}^{-1} I\end{array}\right] <0 \end{aligned}$$
(16)
$$\begin{aligned}{} & {} \left[ \begin{array}{cc}-\left( \frac{\mu _{\lim }}{\varepsilon }\right) ^{2} I &{}\quad \bar{K}_{j} \\ * &{}\quad -\varphi ^{-1} Q\end{array}\right] <0 \end{aligned},$$
(17)

where

$$\begin{aligned} \varGamma _{i j}^{11}&=\left[ A_{0 i} Q\right] _{s}+ \left[ \frac{1+\varepsilon }{2} B_{20} \bar{K}_{j}\right] _{\varepsilon }+\xi _{1}^{-1} H H^\mathrm{{T}}\\&\quad +\xi _{2}^{-1} H H^\mathrm{{T}}+\xi _{3}^{-1} H H^\mathrm{{T}},\\ \varGamma _{i j}^{12}&=\left[ \begin{array}{llllll}B_{1 i}&B_{20}&\frac{1-\varepsilon }{2} \bar{K}_{j}{ }^\mathrm{{T}}&Q C_{1}{ }^\mathrm{{T}}&Q E_{a i}^\mathrm{{T}}&\frac{1+\varepsilon }{2} \bar{K}_{j}{ }^\mathrm{{T}} E_{b}^\mathrm{{T}}\end{array}\right] ,\\ \varGamma _{i j}^{22}&={\text {diag}}\left\{ \begin{array}{lllllll}-\gamma ^{2} I&-\sigma I&-\sigma ^{-1} I&-I&-\xi _{1}^{-1} I&-\xi _{2}^{-1} I\end{array}\right\} ,\\ \varGamma _{i j}^{23}&=\left[ \begin{array}{llllll}0&E_{b}&\quad 0&\quad 0&\quad 0&\quad 0\end{array}\right] ^\mathrm{{T}} \end{aligned}.$$

The robust H-infinity method is based on the principle that the robust H-infinity performance problem of the system is transformed into the solution problem of a linear matrix inequality system using Lyapunov stability theory. Therefore, Eqs. (16) and (17) are the necessary conditions for the computation of the output feedback gains in the LMIs designed in this paper.

Proof

A Lyapunov function is chosen as follows:

$$\begin{aligned} V(t)=x^\mathrm{{T}}(t) P x(t) \end{aligned},$$
(18)

where \(P>0\) is a symmetric matrix. The derivative of V(t) is obtained as

$$\begin{aligned} \dot{V}(t) & = \dot{x}^{T} (t)Px(t) + x^{T} (t)P\dot{x}(t) \\ & = \sum\limits_{{i = 1}}^{3} {\alpha _{i} } \left[ {x^{T} (t)\left( {\left( {A_{{0i}} + \Delta A_{i} } \right)^{T} P + P\left( {A_{{0i}} + \Delta A_{i} } \right)} \right)x(t)} \right. \\ & \quad + \omega ^{T} (t)B_{{1i}} ^{T} Px(t) + x^{T} (t)PB_{{1i}} \omega (t) \\ & \quad + \frac{{1 + \varepsilon }}{2}x^{T} (t)P\left( {B_{{20}} + \Delta B_{2} } \right)u(t) + \frac{{1 + \varepsilon }}{2}u^{T} (t)\left( {B_{{20}} + \Delta B_{2} } \right)^{T} Px(t) \\ & \quad + \left( {\bar{u}(t) - \frac{{1 + \varepsilon }}{2}u(t)} \right)^{T} \left( {B_{{20}} + \Delta B_{2} } \right)^{T} Px(t) \\ & \quad \left. { + x^{T} (t)P\left( {B_{{20}} + \Delta B_{2} } \right)\left( {\bar{u}(t) - \frac{{1 + \varepsilon }}{2}u(t)} \right)} \right] \\ \end{aligned}$$
(19)

According to Eqs. (12), (19) can be expressed as

$$\begin{aligned}\dot{V}(t)&=\dot{x}^\mathrm{{T}}(t) P x(t)+x^\mathrm{{T}}(t) P \dot{x}(t) \\&=\sum _{i=1}^3 \sum _{j=1}^3 \alpha _j \alpha _i\left[ x ^ \mathrm{{T}} (t) \left( \left( A_{0 i}+\varDelta A_i\right) ^\mathrm{{T}} P\right. \right. \\&\quad \left. +P\left( A_{0 i}+\varDelta A_i\right) \right) x(t)+\omega ^\mathrm{{T}}(t) B_{1 i}^\mathrm{{T}} P x(t) \\&\quad +x^\mathrm{{T}}(t) P B_{1 i} \omega (t) \\&\quad +\frac{1+\varepsilon }{2} x^T(t) P\left( B_{20}+\varDelta B_2\right) K_j C_y x(t) \\&\quad +\frac{1+\varepsilon }{2} K_j^T C_y^\mathrm{{T}}(t) x^\mathrm{{T}}(t)\left( B_{20}+\varDelta B_2\right) ^\mathrm{{T}} P x(t) \\&\quad +x^\mathrm{{T}}(t) P\left( B_{20}+\varDelta B_2\right) \left( \bar{u}(t)-\frac{1+\varepsilon }{2} u(t)\right) \\&\quad +\left( \bar{u}(t)-\frac{1+\varepsilon }{2} u(t)\right) ^\mathrm{{T}}\left( B_{20}+\varDelta B_2\right) ^\mathrm{{T}} P x(t) \end{aligned}$$
(20)

According to Lemma 2, let \(R=\left( \bar{ue}(t)-\frac{1+\varepsilon }{2} u(t)\right) ^T\), \(F=I\) , and \(S=\left( B_{20}+\varDelta B_2\right) ^T P x(t)\), then

$$\begin{aligned} \dot{V}(t)&=\dot{x}^\mathrm{{T}}(t) P x(t)+x^\mathrm{{T}}(t) P \dot{x}(t) \\&=\sum _{i=1}^3 \sum _{j=1}^3 \alpha _j \alpha _i\left[ x ^ \mathrm{{T}}( t ) \left( \left( A_{0 i}+\varDelta A_i\right) ^\mathrm{{T}} P\right. \right. \\&\quad \left. +P\left( A_{0 i}+\varDelta A_i\right) \right) x(t)+\omega ^\mathrm{{T}}(t) B_{1 i}^T P x(t) \\&\quad +x^\mathrm{{T}}(t) P B_{1 i} \omega (t) \\&\quad +\frac{1+\varepsilon }{2} x^\mathrm{{T}}(t) P\left( B_{20}+\varDelta B_2\right) K_j C_y x(t) \\&\quad +\frac{1+\varepsilon }{2} K_j^\mathrm{{T}} C_y^\mathrm{{T}}(t) x^\mathrm{{T}}(t)\left( B_{20}+\varDelta B_2\right) ^\mathrm{{T}} P x(t) \\&\quad +\sigma \left( \bar{u}(t)-\frac{1+\varepsilon }{2} u(t)\right) ^\mathrm{{T}}\left( \bar{u}(t)-\frac{1+\varepsilon }{2} u(t)\right) \\&\quad \left. +\sigma ^{-1} x^\mathrm{{T}}(t) P\left( B_{20}+\varDelta B_2\right) \left( B_{20}+\varDelta B_2\right) ^\mathrm{{T}} P x(t)\right] \end{aligned} .$$
(21)

According to Lemma 1, we have

$$\begin{aligned} \dot{V}(t)&=\dot{x}^\mathrm{{T}}(t) P x(t)+x^\mathrm{{T}}(t) P \dot{x}(t) \\&=\sum _{i=1}^3 \sum _{j=1}^3 \alpha _j \alpha _i\left[ x ^ \mathrm{{T}}( t ) \left( \left( A_{0 i}+\varDelta A_i\right) ^\mathrm{{T}} P\right. \right. \\&\quad \left. +P\left( A_{0 i}+\varDelta A_i\right) \right) x(t)+\omega ^\mathrm{{T}}(t) B_{1 i}^\mathrm{{T}} P x(t) \\&\quad +x^\mathrm{{T}}(t) P B_{1 i} \omega (t) \\&\quad +\frac{1+\varepsilon }{2} x^\mathrm{{T}}(t) P\left( B_{20}+\varDelta B_2\right) K_j C_y x(t) \\&\quad +\frac{1+\varepsilon }{2} K_j^\mathrm{{T}} C_y^\mathrm{{T}}(t) x^\mathrm{{T}}(t)\left( B_{20}+\varDelta B_2\right) ^\mathrm{{T}} P x(t) \\&\quad +\sigma { }^{-1} x^\mathrm{{T}}(t) P\left( B_{20}+\varDelta B_2\right) \left( B_{20}+\varDelta B_2\right) ^\mathrm{{T}} P x(t) \\&\quad \left. +\sigma \left( \frac{1-\varepsilon }{2}\right) ^2 u^\mathrm{{T}}(t) u(t)\right] \end{aligned}.$$
(22)

From Eq. (12), it can be shown that

$$\begin{aligned} \dot{V}(t)&=\dot{x}^\mathrm{{T}}(t) P x(t)+x^\mathrm{{T}}(t) P \dot{x}(t) \\&=\sum _{i=1}^{3} \sum _{j=1}^{3} \alpha _{i} \alpha _{j}\left[ x ^ \mathrm{{T}}(t) \left( \left( A_{0 i}+\varDelta A_{i}\right) ^\mathrm{{T}} P\right. \right. \\&\quad +P\left( A_{0 i}+\varDelta A_{i}\right) +\frac{1+\varepsilon }{2} C_{y}^\mathrm{{T}} K_{j}^{T}\left( B_{20}+\varDelta B_{2}\right) ^\mathrm{{T}} P \\&\quad +\frac{1+\varepsilon }{2} P\left( B_{20}+\varDelta B_{2}\right) K_{j} C_{y} \\&\quad +\sigma ^{-1} P\left( B_{20}+\varDelta B_{2}\right) \left( B_{20}+\varDelta B_{2}\right) ^\mathrm{{T}} P \\&\quad \left. +\sigma \left( \frac{1-\varepsilon }{2}\right) ^{2} C_{y}^\mathrm{{T}} K_{j}^\mathrm{{T}} K_{j} C_{y}\right) x(t) \\&\quad \left. +\omega ^\mathrm{{T}}(t) B_{1 i}{ }^\mathrm{{T}} P x(t)+x^\mathrm{{T}}(t) P B_{1 i} \omega (t)\right] \end{aligned}$$
(23)

Then, the following expression holds

$$\begin{aligned} \begin{aligned} J_{z \omega }&=\dot{V}(t)+z^\mathrm{{T}}(t) z(t)-\gamma ^{2} \omega ^\mathrm{{T}}(t) \omega (t) \\&\le \text{X}^\mathrm{{T}}(t)\left( \sum _{j=1}^{3} \sum _{i=1}^{3} \alpha _{i} \alpha _{j} \varLambda _{i j}\right) \text{X}(t) \end{aligned} \end{aligned},$$
(24)

where

$$\begin{aligned}{} & {} \begin{aligned} \varLambda _{i j}&=\left[ \begin{array}{cc} \varLambda _{i j 1} &{}\quad P B_{1 i} \\ * &{}\quad -\gamma ^{2} I \end{array}\right] , \text{X}(t)=\left[ \begin{array}{c} x(t) \\ \omega (t) \end{array}\right] ,\\ \end{aligned}\\{} & {} \begin{aligned} \varLambda _{i j 1}&=\left[ P\left( A_{0 i}+\varDelta A_{i}\right) \right] _{s}\\&\quad +\frac{1+\varepsilon }{2}\left[ P\left( B_{20}+\varDelta B_{2}\right) K_{j} C_{y}\right] _{s} +C_{1}^\mathrm{{T}} C_{1}\\&\quad +\sigma ^{-1} P\left( B_{20} +\varDelta B_{2}\right) \left( B_{20}+\varDelta B_{2}\right) ^\mathrm{{T}} P \\&\quad +\sigma \left( \frac{1-\varepsilon }{2}\right) ^{2} C_{y}^\mathrm{{T}} K_{j}^\mathrm{{T}} K_{j} C_{y} \end{aligned} \end{aligned}.$$

Using the Schur complement, it can be concluded that

$$\begin{aligned} \varLambda _{i j}\!=\!\left[ \begin{array}{ccccc}\zeta _{1} &{}\quad P B_{1 i} &{}\quad P\left( B_{20}+\varDelta B_{2}\right) &{}\quad \frac{1-\varepsilon }{2} C_{y}{ }^\mathrm{{T}} K_{j}{ }^\mathrm{{T}} &{}\quad C_{1}{ }^\mathrm{{T}} \\ * &{}\quad -\gamma ^{2} I &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ * &{}\quad * &{}\quad -\sigma I &{}\quad 0 &{}\quad 0 \\ * &{}\quad * &{}\quad * &{}\quad -\sigma ^{-1} I &{}\quad 0 \\ * &{}\quad * &{}\quad * &{}\quad * &{}\quad -I\end{array}\right] \end{aligned},$$
(25)

where \(\zeta _{1}\!=\!\left[ P\!\left( A_{0 i}\!+\!\varDelta A_{i}\right) \right] _{s} +\frac{1+\varepsilon }{2}\!\left[ P\left( B_{20}\!+\!\varDelta B_{2}\right) \! K_{j} \!C_{y}\right] _{s}\).

Define \(Q=P^{-1}\), \(\bar{K}_{j}=K_{j} C_{y} Q\) , and \(J_{1}={\text {diag}}\{Q\; I\;\) \(I\; I\; I\}\). Pre- and post-multiplying both sides of Eq. (25) with \(J_{1}\), the matrix \(\varLambda _{i j}\) is equivalent to

$$\begin{aligned} \varPhi _{i j}\!=\!\left[ \begin{array}{ccccc}\zeta _{2} &{}\quad B_{1 i} &{}\quad B_{20}+\varDelta B_{2} &{}\quad \frac{1-\varepsilon }{2} \bar{K}_{j}^\mathrm{{T}} &{}\quad Q C_{1}^\mathrm{{T}} \\ * &{}\quad -\gamma ^{2} I &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ * &{}\quad * &{}\quad -\sigma I &{}\quad 0 &{}\quad 0 \\ * &{}\quad * &{}\quad * &{}\quad -\sigma ^{-1} I &{}\quad 0 \\ * &{}\quad * &{}\quad * &{}\quad * &{}\quad -I\end{array}\right] \end{aligned},$$
(26)

where \(\zeta _{2}=\left[ \left( A_{0 i}+\varDelta A_{i}\right) Q\right] _{s}+ \left[ \frac{1+\varepsilon }{2}\left( B_{20}+\varDelta B_{2}\right) \bar{K}_{j}\right] _{s}\).

In light of Eq. (8), the following expressions hold

$$\begin{aligned} \begin{aligned} \varPhi _{i j}&=\begin{bmatrix}\zeta _{3} &{}\quad B_{1 i} &{}\quad B_{20} &{}\quad \frac{1-\varepsilon }{2} \bar{K}_{j}{ }^\mathrm{{T}} &{}\quad Q C_{1}^{T} \\ * &{}\quad -\gamma ^{2} I &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ * &{}\quad * &{}\quad -\sigma I &{}\quad 0 &{}\quad 0 \\ * &{}\quad * &{}\quad * &{}\quad -\sigma ^{-1} I &{}\quad 0 \\ * &{}\quad * &{}\quad * &{}\quad * &{}\quad -I\end{bmatrix},\\&+\begin{bmatrix}\begin{bmatrix}H^\mathrm{{T}}&\quad 0&\quad 0&\quad 0&\quad 0\end{bmatrix}^\mathrm{{T}} \varGamma \begin{bmatrix}E_{a i} Q&\quad 0&\quad 0&\quad 0&\quad 0\end{bmatrix}\end{bmatrix}_{s}\\&+\begin{bmatrix}\frac{1+\varepsilon }{2} \begin{bmatrix}H^\mathrm{{T}}&\quad 0&\quad 0&\quad 0&\quad 0\end{bmatrix}^\mathrm{{T}} \varGamma \begin{bmatrix}E_{b i} \bar{K}_{j}&\quad 0&\quad 0&\quad 0&\quad 0\end{bmatrix}\end{bmatrix}_{s}\\&+\begin{bmatrix}\begin{bmatrix}H^\mathrm{{T}}&\quad 0&\quad 0&\quad 0&\quad 0\end{bmatrix}^\mathrm{{T}} \varGamma \begin{bmatrix}0&\quad 0&\quad E_{b i}&\quad 0&\quad 0 \end{bmatrix}\end{bmatrix}_{s} \end{aligned} \end{aligned},$$
(27)

where \(\zeta _{3}=\left[ A_{0 i} Q\right] _{s}+ \left[ \frac{1+\varepsilon }{2} B_{20} \bar{K}_{j}\right] _{s}\).

From Eq. (27), it can be shown that \(\phi _{i j}<0\) is equivalent to Eq. (16), then \(\dot{V}(t)<0\). The concerned vehicle system is asymptotically stable and the \(H_{\infty }\) performance criterion is satisfied. Symbol \([.]_{s}\) denotes the sum of its internal matrix and its transpose.

Moreover, under zero initial state, processing Eq. (24) by integrating the two sides from 0 to t, it can be concluded that \(V(t)=x^\mathrm{{T}}(t) P x(t) \le \gamma ^{2} \int _{0}^{t} \omega ^\mathrm{{T}}(t) \omega (t) \mathrm{{d}}t \le \gamma ^{2}\Vert \omega (t)\Vert _{2}^{2} \le \gamma ^{2} \varphi\).

Then, the following expressions hold:

$$\begin{aligned} \max _{t \ge 0}u(t)^{2}&=\max _{t \ge 0}\left\| u^\mathrm{{T}}(t) u(t)\right\| _{2} \\ {}&=\max _{t \ge 0}\left\| x^\mathrm{{T}}(t) C_{y}^\mathrm{{T}} K_{j}^\mathrm{{T}} K_{j} C_{y} x(t)\right\| _{2} \\ {}&=\max _{t \ge 0}\left\| \bar{x}^\mathrm{{T}}(t) P^{-1 / 2} C_{y}^\mathrm{{T}} K_{j}^\mathrm{{T}} K_{j} C_{y} P^{-1 / 2} \bar{x}(t)\right\| _{2} \\ {}&\le \varphi \lambda _{\max }\left( P^{-1 / 2} C_{y}^\mathrm{{T}} K_{j}^\mathrm{{T}} K_{j} C_{y} P^{-1 / 2}\right) \end{aligned},$$
(28)

where \(\lambda _{\max }(\cdot )\) denotes the maximum value of a matrix eigenvalue. Furthermore, the saturation constraint is satisfied when the following inequality is true.

$$\begin{aligned} P^{-1 / 2} C_{y}^\mathrm{{T}} K_{j}^\mathrm{{T}} K_{j} C_{y} P^{-1 / 2} \le \left( \frac{\mu _{\lim }}{\varepsilon }\right) ^{2} I \end{aligned}$$
(29)

Employing the Schur complement, it can be concluded that

$$\begin{aligned} \left[ \begin{array}{cc}-\left( \frac{\mu _{\lim }}{\varepsilon }\right) ^{2} I &{} K_{j} C_{y} P^{-1 / 2} \\ * &{} -\varphi ^{-1}\end{array}\right] <0 \end{aligned}.$$
(30)

Denote \(J_{2}={\text {diag}}\left\{ I, P^{-1 / 2}\right\}\), when Eq. (30) is pre- and post-multiplied by \(J_{2}\), Eq. (30) is equivalent to Eq. (17). The proof is now complete.□

Remark 2

Given that there are two unknown matrices \(K_{j}\) and Q in element \(\bar{K}_{j}\), the conditions in Theorem are not convex anymore. So, calculating the output feedback gain matrix \(K_{j}\) directly by Theorem 1 is infeasible, and sufficient conditions will be presented in Theorem 2 for controller optimization and computation.

3.2 Controller Optimization and Computation

Theorem 2

The concerned vehicle system in Eq. (10) is asymptotically stable and meets the desired performance in Eq. (13), for given constants \(\varepsilon\), \(\mu _{l i m}\), \(\varphi >0\), if symmetric positive definite matrix \(X_{W}\), \(X_{R}\), feedback gains \(Y_{R j}\) and positive scalars \(\gamma\), \(\sigma\), \(\xi _{1}\), \(\xi _{2}\), \(\xi _{3}\) exist and such that \((i, j=1,2,3)\).

$$\begin{aligned}{} & {} \left[ \begin{array}{ccc} \bar{\varGamma }_{i j}^{11} &{} \bar{\varGamma }_{i j}^{12} &{} 0 \\ * &{} \varGamma _{i j}^{22} &{} \varGamma _{i j}^{23} \\ * &{} * &{} -\xi _{3}^{-1} I \end{array}\right] <0 \end{aligned}$$
(31)
$$\begin{aligned}{} & {} \left[ \begin{array}{cc} -\left( \frac{\mu _{\lim }}{\varepsilon }\right) ^{2} I &{} Y_{R j} R^\mathrm{{T}} \\ {\ast} & {} -\varphi ^{-1}\left( W X_{W} W^{T}+R X_{R} R^\mathrm{{T}}\right) \end{array}\right] <0 \end{aligned},$$
(32)

where

$$\begin{aligned}\bar{\Gamma }_{i j}^{11}=&\left[ A_{0 i}\left( W X_{W} W^{T}+R X_{R} R^{T}\right) \right] _{s} +\left[ \frac{1+\varepsilon }{2} B_{20} \bar{K}_{j}\right] _{s}\\ +&\xi _{1}^{-1} H H^{T}+\xi _{2}^{-1} H H^{T}+\xi _{3}^{-1} H H^{T} \text{, } \\ \bar{\varGamma }_{i j}^{12} \! = \!&\left[ \begin{array}{llll} B_{1 i}, \! B_{20}, \! \frac{1-\varepsilon }{2}\! \left( Y_{R j} R^{T}\right) ^{T} \! , \! \left( W X_{W} W^{T} \! + \! R X_{R} R^{T}\right) C_{1}^{T}, \end{array}\right. \\&\left. \left( W X_{W} W^{T}+R X_{R} R^{T}\right) E_{a i}^{T}, \frac{1+\varepsilon }{2}\left( Y_{R j} R^{T}\right) ^{T}E_{b}^{T}\right] \end{aligned}$$

Then, the SOF control gains \(K_{j}\) is easy to calculate by \(K_{j}=Y_{R j} X_{R}^{-1}\).

Proof

From [35], the matrices Q and \(\bar{K}_{j}\) can be written as follows:

$$\left\{ {\begin{array}{*{20}l} {Q = WX_{W} W^{{\text{T}}} + RX_{R} R^{{\text{T}}} } \hfill \\ {\bar{K}_{j} = Y_{{Rj}} R^{{\text{T}}} } \hfill \\ \end{array} } \right.$$
(33)

where W is a matrix satisfying \(C_{y} W=0\). R is defined as \(R=C_{y}^{\dagger }+W M\), where M is a given matrix with suitable dimensions, and \(C_{y}^{\dagger }=C_{y}^\mathrm{{T}}\left( C_{y} C_{y}^\mathrm{{T}}\right) ^{-1}\).

Motivated by the above definition, \(C R\!=\!I\) and \(C Q\!=\!0\) can be derived. Furthermore, the following equations hold:

$$\begin{aligned}{} & {} C_{y} Q=C_{y}\left( W X_{W} W^\mathrm{{T}}+R X_{R} R^\mathrm{{T}}\right) =X_{R} R^{T} \end{aligned}$$
(34)
$$\begin{aligned} \bar{K}_{j}&=Y_{R j} R^\mathrm{{T}}=K_{j} C_{y} Q\\ \Leftrightarrow K_{j}&=Y_{R j} R^\mathrm{{T}}\left( X_{R} R^\mathrm{{T}}\right) ^{-1}=Y_{R j} X_{R}^{-1} \end{aligned}$$
(35)

Hence, the desired output feedback gain \(K_{j}\) can be acquired.□

Remark 3

The H-infinity output feedback control algorithm is employed to design the controller, and the actuator saturation is also considered in the controller design, which not only reduces the measurement cost and maintains the feasibility of the controller, but also enhances the control effect and maximizes the overall performance of the system.

4 Experiment

Fig. 3
figure 3

Physical diagram of the HIL experimental test

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Fig. 4
figure 4

Structure diagram of the HIL experimental test

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Table 1 Model parameter values of FWID-EVs
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In this part, in order to evaluate the feasibility of the proposed controller, two typical maneuvers, comprising a single-lane change maneuver and a J-turn maneuver, are executed on hardware-in-the-loop (HIL) experimental platform bases on the LabVIEW RT system. Figure 3 depicts the HIL experimental platform. The experiment platform consists of a G29 simulator, cRIO, PXI, host computer, etc. The vehicle model and road information in CarSim are embedded into PXI RT system, and the control algorithm is embedded into LabVIEW RT system by compiling of MATLAB. The steering angle, which is calculated by cRIO, acts on the G29 driving simulator. The communication between components is performed through CAN signals.

Since Carsim includes modeling functions for road scenarios and working conditions, different road environments can be established. A variety of scenarios and working conditions can be designed to meet the development and verification of the control system under different environments. Therefore, the experimental tests can be implemented by selecting corresponding vehicle model and parameters, and then the effectiveness of the designed controller can be verified by performing vehicle path tracking performance analysis on the selected road.

Figure 4 shows the structure diagram of the HIL experimental test. Table 1 lists the parameter values of the FWID-EVs used in this experiment. Considering that the parameters m, \(I_{z}\), \(C_{f}\) , and \(C_{r}\) are nonlinear and time-varying in real practice, the nominal values of these parameters are provided first, as listed in Table 1. Specially, \(v_{x}\) is set as 20 m/s initially but varies within the upper and lower limits. The uncertainty of cornering stiffness is regarded as \(20 \%\) of the nominal value for both the front and rear wheels, whereas the moment of inertia and vehicle mass are regarded as \(20 \%\) variances. Among them, the upper and lower speed limits are also given together in the table. Moreover, the saturation bounds of steering angle and external yaw moment are defined as 25deg and 3000 N, respectively (i.e., \(\mu _{l i m}= \left[25 \quad 3000\right] ^\mathrm{{T}}\)). Besides, \(\varphi =0.02\) and \(\varepsilon =0.95\) are chosen. The external disturbances are chosen as \(d_{1}\) = 0.001 m/s\(^{2}\), \(d_{2}\) = 0.002 rad/s\(^{2}\) and \(d_{3}\) = 0.003 m/s. Then, Controller A is the name of the RHF-SOF controller proposed in this research, and the controller gain matrix is determined as

$$K_{\text{sof}} = \left[\begin{array}{*{20}c} - 1.05 & - 23.84 & - 0.74 \\ - 119466 & - 1610969 & - 47660 \\ \end{array} \right] .$$

To emphasize the advantage of Controller A presented in this research, the output feedback controller (i.e., Controller B) with input saturation presented in [36] is adopted as a comparative counterpart. The SOF gains can be acquired as

$$K_{B}=\left[ \begin{array}{ccc} -7.64 &{} -74.60 &{} -4.76 \\ -22354 &{} -60880 &{} -1029 \end{array}\right].$$

4.1 Example: 1

Fig. 5
figure 5

Example 1: a road curvature, b longitudinal velocity

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The single-lane change maneuver is to operate the vehicle through a single-lane change to follow the target lane, and therefore reflects vehicle tracking accuracy and handling stability. As shown in Fig. 5a, the response of FWID-EVs under the single-lane change maneuver is demonstrated. The longitudinal velocity is described in Fig. 5b. It is noteworthy that the longitudinal velocity of FWID-EVs is accelerated at first and decelerated later.

Fig. 6
figure 6

Example 1: a lateral velocity, b yaw rate, c lateral offset, d heading error

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The system performance indexes and performance comparisons of FWID-EVs under the control of different controllers can be acquired, which is shown in Fig. 6a, c, and d exhibit that Controller A can obviously decrease the lateral velocity, lateral offset, and heading error to achieve dynamic stability faster as compared with Controller B. Figure 6b shows that the yaw rate produced in both the control strategies are close. In terms of preserving the vehicle’s handling stability, the path-following performance of Controller A is superior to that of Controller B, as shown in Fig. 6.

Table 2 presents the root mean square (RMS) values of system performance for various controllers for additional quantitative investigation. Although the lateral velocity and yaw rate show minor improvement, as compared to Controller B, the lateral offset and heading error of Controller A are reduced by \(9.81\%\) and \(40.22\%\), respectively. It can be concluded that the proposed controller can maintain more accurate tracking performance and simultaneously improve the handling stability of the vehicle during lane changing maneuver.

Fig. 7
figure 7

Example 1: a front wheel steering input, b external yaw moment

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Fig. 8
figure 8

Example 2: (a) road curvature, (b) longitudinal velocity

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Table 2 RMS values of path-following errors in Example 1
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Figure 7 depicts the input situation of actuators with different control strategies. It can be found that the saturation conditions of actuators in all control strategies are bounded in the given limits.

4.2 Example: 2

The J-turn maneuver is produced from the ramp steering input, and this maneuver is used to examine the cornering stability under extreme steering conditions. Therefore, the dynamic response of the vehicle during a J-turn maneuver is studied. Figure 8 shows the longitudinal velocity and road curvature change.

Fig. 9
figure 9

Example 2: a lateral velocity, b yaw rate, c lateral offset, d heading error

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Fig. 10
figure 10

Example 2: (a) front wheel steering input, (b) external yaw moment

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Table 3 RMS values of path-following errors in Example 2
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Figure 9 illustrates the system performance indexes for the J-turn maneuver. As demonstrated in Fig. 9, despite the fact that the lateral velocity and yaw rate of Controller A have a slight performance improvement Controller B, lateral offset and heading error exhibit notable enhancements. This demonstrates that the path-following performance is greatly improved with the proposed control strategy. The RMS values of system performance indexes in Example 2 are listed in Table 3, which shows that Controller A can reduce the heading error and the lateral offset by \(14.03\%\) and \(82.35\%\), respectively, as compared with Controller B.

In J-turn maneuver, the longitudinal speed is lower than that in the single-lane change maneuver, but the road curvature is higher, and the actuator saturation is more likely to happen. Figure 10 shows the outcomes of the front wheel steering and the external yaw moment. The external yaw moment is far from ± 3000 N and both controllers are able to operate the actuator within a reasonable range. It shows that the designed controller is effective in avoiding actuator saturation even under extreme conditions (i.e., high curvature road conditions).

Remark 4

The objective of this work is to enhance the path-following performance of the vehicle while ensuring actuator saturation. From the point of view of the control signal, a higher execution action amplitude can be obtained from the proposed strategy and thus better performance can be guaranteed with satisfied saturation condition. This benefits from the valid sufficient condition in Theorem 2.

5 Conclusion

In this research, a RHF-SOF control strategy was designed to improve the path-following performance for nonlinear FWID-EVs with actuator saturation.

According to experimental results, the proposed controller shows advantages in achieving path-following performance. The following conclusions can be drawn:

  1. (1)

    The proposed simulation result shows that the vehicle performance with the proposed RHF-SOF control approach is superior to output feedback control approach in [36]. This implies that the proposed controller has better robustness and can solve the problem of parameter uncertainty and nonlinearity;

  2. (2)

    The fuzzy optimization model integrates LFT and gain scheduling, which greatly helps to achieve parameter estimation, including vehicle longitudinal speed, vehicle mass, moment of inertia, longitudinal speed, and tire turning stiffness. This means that a more accurate vehicle model can be deployed to the vehicle controller;

  3. (3)

    The norm-bounded strategy is effective for vehicle nonlinear actuator saturation problems. This prevents unstable, fluctuations and other factors that can seriously affect controller performance.

Considering that the model is built based on active steering in this paper, more effort will be devoted to the investigation of the controller design for human–machine cooperative driving in the future work.