The mission of automatic steering control system is to ensure that the unmanned electric vehicles accurately track the expected road in a continuous and slippy way, while improving the stability of vehicles [1, 30]. However, unmanned electric vehicles possess the properties of time-varying, external disturbances and parametric uncertainties, the proposed automatic steering control strategy should availably conquer these dynamic characteristics.

As shown in Figure 3, a neural fuzzy based adaptive sliding mode automatic steering control strategy of unmanned electric vehicles is proposed to supervise the lateral dynamics of vehicles. In this architecture, a new sliding surface is designed, then, the adaptive control gain (ACG) of variable structure control law is approximated via the neural network system, in real-time, and the variable boundary layer (ABL) is introduced and adaptively regulating by the fuzzy theory. This novel adaptive sliding mode controller can guarantee the stability of closed-loop automatic steering control system of vehicles.

### 3.1 Traditional SMC Controller

It is well known that the basic idea of SMC is to force its movement in the sliding mode surface, therefore, the construction of sliding surface is a crucial part of SMC to achieve the desired control specifications and performances. A novel proportional-integral-differential sliding surface is designed as

$${\varvec{s}}(t) = {\varvec{k}}_{{\varvec{p}}} {\varvec{e}}(t) + {\varvec{k}}_{{\varvec{i}}} \int {{\varvec{e}}(t){\kern 1pt} } {\text{d}}t + {\varvec{k}}_{{\varvec{d}}} \frac{{\text{d}}}{{{\text{d}}t}}{\varvec{e}}(t),$$

(16)

where *k*_{p} is the positive proportional gain matrix, *k*_{i} is the positive integral gain matrix, and *k*_{d} is the positive derivative gain matrix. \({\varvec{e}}(t) = {\varvec{\upsilon}}(t) - {\varvec{\upsilon}}_{{\varvec{d}}} (t)\) denotes the tracking error, and *υ*_{d}(*t*)=0, *s*=[*s*_{1}, *s*_{2}].

The time derivative of the sliding surface (16) can be obtained as

$$\begin{aligned} {\varvec{\dot{s}}}(t) = & {\varvec{k}}_{d} {\varvec{\ddot{e}}}(t) + {\varvec{k}}_{p} {\varvec{\dot{e}}}(t) + {\varvec{k}}_{i} {\varvec{e}}(t) \\ = & {\varvec{k}}_{d} \left( {({\varvec{A}} + \Delta {\varvec{A}})\dot{\upsilon }(t) + ({\varvec{E}} + \Delta {\varvec{E}})\upsilon + ({\varvec{B}} + \Delta {\varvec{B}}){\varvec{u}} + {\varvec{d}}(t)} \right) & \\ & + {\varvec{k}}_{p} \dot{\upsilon }(t) + {\varvec{k}}_{i} \upsilon (t) \\ = & \left( {{\varvec{k}}_{d} ({\varvec{A}} + \Delta {\varvec{A}}) + {\varvec{k}}_{p} } \right)\dot{\upsilon }(t) + \left( {{\varvec{k}}_{d} ({\varvec{E}} + \Delta {\varvec{E}}) + {\varvec{k}}_{i} } \right)\upsilon (t) \\ & + {\varvec{k}}_{d} ({\varvec{B}} + \Delta {\varvec{B}})u + {\varvec{k}}_{d} {\varvec{d}}(t) \\ \end{aligned}$$

(17)

Setting Eq. (17) to zero as

$$\left. {\dot{\user2{s}}(t)} \right|_{{{\varvec{u}}(t) = {\varvec{u}}_{{{\varvec{eq}}}} (t)}} = 0.$$

(18)

The external disturbance and uncertain terms is ignored, and the equivalent control law is derived as

$${\varvec{u}}_{{{\varvec{eq}}}} (t) = - ({\varvec{k}}_{{\varvec{d}}} {\varvec{B}})^{ - 1} (({\varvec{k}}_{{\varvec{d}}} {\varvec{A}} + {\varvec{k}}_{{\varvec{p}}} )\dot{\user2{\upsilon }} + ({\varvec{k}}_{{\varvec{d}}} {\varvec{E}} + {\varvec{k}}_{{\varvec{i}}} ){\varvec{\upsilon}}).$$

(19)

Equivalent control term is invalid when the tracking errors are away from the sliding surface. Hence, to direct the system errors to the sliding surface, an additional variable structure control term can be designed as

$${\varvec{u}}_{r} \left( t \right) = - {\varvec{K}}\left( t \right)sign\left( {s\left( t \right)} \right),$$

(20)

where *K*(*t*) denotes the control gain, the sign function \(sign( \cdot )\) can be written as

$$sign({\varvec{s}}(t)) = \left\{ {\begin{array}{*{20}l} {1,} \hfill & {\text{if s} \left( t \right) > 0,} \hfill \\ {0,} \hfill & {\text{if s} \left( t \right) = 0,} \hfill \\ { - 1,} \hfill & {\text{if s }\left( t \right) < 0}. \hfill \\ \end{array} } \right.{\kern 1pt}$$

(21)

.

Combining Eqs. (19) and (20), hence, the total automatic steering control law

$${\varvec{u}} = {\varvec{u}}_{{{\varvec{eq}}}} + {\varvec{u}}_{{\varvec{r}}} .$$

(22)

### Remark 1

The control gain *K*(*t*) is involved with the bound of parametric uncertainties, which is hard to be acquired precisely in practical use. Hence, the larger control gain will produce the grievous chattering and excite unstable system dynamics.

### 3.2 NN-based Adaptive Control Gain

The magnitude of external disturbances, parametric uncertainties and signal noises has a vital impact on the control gain *K*(*t*). For purpose of dealing with this problem, an adaptive control gain scheme using the neural network (NN) technique is proposed in this subsection. Specifically, the switching control gain *K*(*t*) of variable structure control term is estimated by a neural network inference system to alleviate the chattering and enhance the dynamic property of the automatic steering controller.

### Definition 1

Let *S* be a compact simply connected set of *R*^{n}. In general, for a continuous function *f*(*x*), there exists an adjustable weight vector *R* and a radial basis Gaussian vector \({\varvec{\phi}}(x)\) such that \({\varvec{f}}(x) - {\varvec{R}}^{{\text{T}}} {\varvec{\phi}}(x)\) has a NN functional reconstruction error vector \(\varepsilon (x)\).

The optimal weight vector \({\varvec{R}}^{*}\) is given as

$${\varvec{R}}^{*} = \arg \min \left\{ {\mathop {\sup }\limits_{{x \in {\varvec{S}}}} \left| {{\varvec{f}}(x) - {\varvec{R}}^{{\text{T}}} {\varvec{\phi}}(x)} \right|} \right\}.$$

(23)

Neural network system has well function approximation capability and fault-tolerance capability at the same time [31, 32]. In this section, to acquire an excellent approximation to the unknown and variable control gain, the control gain *K*(*t*) is directly depended on a RBF neural network algorithm, as follows

$$\hat{\user2{K}}(t) = \sum\limits_{i = 1}^{m} {\hat{\user2{w}}_{{\varvec{i}}} } {\varvec{h}}_{{\varvec{i}}} ({\varvec{s}}) = \hat{\user2{W}}^{{\text{T}}} {\varvec{H}}({\varvec{s}}),$$

(24)

\(\hat{\user2{W}} = \left[ {\hat{w}_{1} , \cdots ,\hat{w}_{m} } \right]^{{\text{T}}}\) is the adjustable weight vector of RBF, \({\varvec{H}}({\varvec{s}}) = \left[ {h_{1} ({\varvec{s}}), \cdots ,h_{m} ({\varvec{s}})} \right]^{{\text{T}}}\) is the Gaussian function, which can be in the form as

$$h_{i} ({\varvec{s}}) = \exp \left( { - \frac{{\left\| {{\varvec{s}} - c_{i} } \right\|}}{{b_{i} }}} \right),\, i = 1,2, \cdots ,m,$$

(25)

where *c*_{i} and *b*_{i} are the parameters of the Gaussian function.

The desired control gain \({\varvec{K}}^{*} (t)\) is given as

$${\varvec{K}}^{*} (t) = {\varvec{W}}^{{*}{\text{T}}} {\varvec{H}}({\varvec{s}}) + \varepsilon^{\prime},$$

(26)

where \(\user2{\varepsilon^{\prime}}\) is the approximation error, \({\varvec{W}}^{*} (t)\) is a constant ideal weight matrix, and there exists a position constant \({\varvec{\eta}}\) which satisfies

$${\varvec{K}}^{*} (t) = {\varvec{\eta}}.$$

(27)

In order to guarantee the asymptotical stability of \(s\left( t \right) \to 0\) as \(t \to \infty\), for the controlled system in Eq. (14), an adaptive control law of parameter \(\hat{\user2{W}}\) can be obtained as

$$\user2{\dot{\hat{W}}} = {\varvec{rk}}_{d} {\varvec{BsH}}({\varvec{s}}),$$

(28)

where *r* is the positive constant, and the weight estimation errors is \(\tilde{\user2{W}} = \hat{\user2{W}} - {\varvec{W}}^{*}\).

### Remark 2

It is interesting to note that the finite nodes of RBF neural network can lead to a small modeling error \(\user2{\varepsilon^{\prime}}\), but, an RBF neural network can be found such that \(\left\| {\user2{\varepsilon^{\prime}}} \right\| \le {\varvec{\varepsilon}}_{{\varvec{N}}}\), where \(\left\| \cdot \right\|\) is the Euclidean norm of a vector,\({\varvec{\varepsilon}}_{{\varvec{N}}}\) denotes a positive constant.

### Remark 3

It is well known that the adaptive law may cause drift of the tunable parameters in the case of the presence of measurement errors. In order to deal with this drawback, the adaptive law could be updated as *σ* modification technique [33].

### 3.3 FL-based Adaptive Boundary Layer

To further relieve the chattering phenomenon and enhance the dynamic performance, a boundary layer is designed in the variable structure control term, as follows

$${\varvec{u}}_{{\varvec{r}}} = \hat{\user2{K}}\left( t \right)sat\left( {\frac{{\varvec{s}}}{{{\varvec{\Delta}}}}} \right).$$

(29)

The function \(sat\left( \cdot \right)\) represents the actuator saturation. It is interesting to note that the chattering phenomenon can be alleviated, however, the unreasonable saturated parameter vector **Δ**=[Δ_{1}, Δ_{2}] can damage the dynamic performance and robustness.

In order to deal with the above problems and achieve the better tracking performance, the adaptive adjusting strategies of thickness of boundary layer are attracting more attention [34, 35]. Here, an adaptive variable boundary layer is designed by the fuzzy logic approach. The input variable is the absolute value of sliding surface \(\left| {s_{i} } \right|\), and the output variable is the thickness Δ_{i}. The membership functions of the linguistic terms very small (VS), small (S), medium (M), large (L), very large (VL) are assigned to the input variable \(\left| {s_{i} } \right|\), and the linguistic terms very narrow (VN), narrow (N), medium (M), wide (W), very wide (VW) are assigned to the output variable Δ_{i}. As illustrated in Figure 4, the input and output variables are fuzzified by the triangular and trapezoidal membership functions.

Fuzzy partitions with triangular membership functions could produce entropy equalization, triangular membership functions are employed to give an error-free reconstruction in this paper.

When the system states are going to be outside the boundary layer, for realizing the fast convergence to the sliding surface, the thickness of boundary layer should be narrowed. When the system states are within the boundary layer, in order to alleviate the chattering and avoid overshoot, the thickness of boundary layer should be widened. Based on these principles, the linguistic fuzzy rules are expressed as the following procedure:

*R*_{1}: if \(\left| {s_{i} } \right|\) is very large then \(\left| {\Delta_{i} } \right|\) is very narrow

*R*_{2}: if \(\left| {s_{i} } \right|\) is large then \(\left| {\Delta_{i} } \right|\) is narrow

*R*_{3}: if \(\left| {s_{i} } \right|\) is medium then \(\left| {\Delta_{i} } \right|\) is medium

*R*_{4}: if \(\left| {s_{i} } \right|\) is small then \(\left| {\Delta_{i} } \right|\) is wide

*R*_{5}: if \(\left| {s_{i} } \right|\) is very small then \(\left| {\Delta_{i} } \right|\) is very wide

The fuzzy rules is designed in accordance with the principle that the thickness of boundary layer is decreased gradually when the vehicle lateral control system tends toward steady state. In the first rule *R*_{1}, if the absolute value of sliding surface \(\left| {s_{i} } \right|\) is very large, then the thickness \(\left| {\Delta_{i} } \right|\) of boundary layer is very narrow. Fuzzy rule *R*_{1} narrows the thickness of boundary layer to accomplish the goal of achieving the sliding surface rapidly. In the second rule *R*_{2}, if the absolute value of sliding surface \(\left| {s_{i} } \right|\) is large, then the thickness \(\left| {\Delta_{i} } \right|\) of boundary layer is narrow. In the third rule *R*_{3}, if the absolute value of sliding surface \(\left| {s_{i} } \right|\) is medium, then the thickness \(\left| {\Delta_{i} } \right|\) of boundary layer is medium. In the fourth rule *R*_{4}, if the absolute value of sliding surface \(\left| {s_{i} } \right|\) is small, then the thickness \(\left| {\Delta_{i} } \right|\) of boundary layer is wide. In the fifth rule *R*_{5}, if the absolute value of sliding surface \(\left| {s_{i} } \right|\) is very small, then the thickness \(\left| {\Delta_{i} } \right|\) of boundary layer is very wide. Fuzzy rule *R*_{5} broadens the thickness to accomplish the goal of alleviating the chattering phenomena.

### Remark 4

The presented adaptive boundary layer approach follows the principles that when the absolute value of sliding surface \(\left| {\varvec{s}} \right|\) tends to zero, the thickness is decreased gradually*.*

### 3.4 Stability Analysis

### Lemma 1 [36]

*If a scalar function *
\({\varvec{V}}(x,t)\)
* satisfies the following conditions*

(i) \({\varvec{V}}(x,t)\) *is lower bounded*.

(ii) \(\dot{\user2{V}}(x,t)\) *is negative semi-definite*.

(iii) \(\dot{\user2{V}}(x,t)\) *is uniformly continuous in time*.

Then, \(\dot{\user2{V}}(x,t) \to 0\) as \(t \to \infty\).

### Lemma 2 [36]

(*Barbalat) If the differentiable function *\(f(t)\)* has a finite limit as *\(t \to \infty\)*, and if f is uniformly continuous, then *\(f(t) \to 0\)* as* \(t \to \infty\).

### Lemma 3 [36]

*If the vector function *\(f(x,t)\)* has continuous and bounded first partial derivatives with respect to x and t, for all x in a ball B*_{r}* and for all *\(t \ge 0\)*, then the equilibrium point at the origin is exponentially stable, if and only if, there exists a function *\({\varvec{V}}(x,t)\)* and strictly positive constants *\(a_{1} ,a_{2} ,a_{3} ,a_{4}\)* such that* \(\forall x \in B_{r}\),\(\forall t \ge 0\):

$$\begin{gathered} a_{1} \left\| x \right\|^{2} \le {\varvec{V}}(x,t) \le a_{2} \left\| x \right\|^{2} , \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \dot{\user2{V}} \le - a_{3} \left\| x \right\|^{2} , \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left\| {\frac{{\partial {\varvec{V}}}}{\partial x}} \right\| \le a_{4} \left\| x \right\|. \hfill \\ \end{gathered}$$

(30)

### Theorem 1

For the steering system Eq. (14) of automated electric vehicles, if the automatic steering controller is established as Eq. (22) with Eqs. (19) and (29), and the weight adaptive control law is given as Eq. (28), then the tracking errors will converge to zero.

### Proof

The Lyapunov function candidate is considered as [34]

$${\varvec{V}}(t) = \frac{1}{2}{\varvec{s}}^{2} + \frac{1}{{2{\varvec{r}}}}\tilde{\user2{W}}^{T} \tilde{\user2{W}}.$$

(31)

Owing to the fact that \(\tilde{\user2{W}} = \hat{\user2{W}} - {\varvec{W}}^{*}\), then, the derivative of Eq. (31) can be obtained as

$$\dot{\user2{V}}(t) = \user2{s\dot{s}} + \frac{1}{{\varvec{r}}}\tilde{\user2{W}}^{T} \user2{\dot{\hat{W}}}.$$

(32)

Rewriting Eq. (14), yields

$$\begin{gathered} \user2{\ddot{\upsilon }} = \user2{A\dot{\upsilon }} + \user2{E\upsilon } + {\varvec{Bu}} +\Delta \user2{A\dot{\upsilon }} +\Delta \user2{E\upsilon } +\Delta {\varvec{Bu}} + {\varvec{d}}(t) \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = \user2{A\dot{\upsilon }} + \user2{E\upsilon } + {\varvec{Bu}} + {\varvec{L}}_{{\varvec{r}}} , \hfill \\ \end{gathered}$$

(33)

where \({\varvec{L}}_{r} = \Delta \user2{A\dot{\upsilon }} +\Delta \user2{E\upsilon } +\Delta {\varvec{Bu}} + {\varvec{d}}(t).\) Based on the Assumption 1, it is interesting to note that the disturbance \({\varvec{L}}_{{\varvec{r}}}\) is bounded.

Substituting Eq. (33) into the function \(s\dot{s}\), then

$$\begin{aligned} \user2{s\dot{s}} & = {\varvec{s}}\left( {{\varvec{k}}_{{\varvec{d}}} \user2{\ddot{e}}(t) + {\varvec{k}}_{{\varvec{p}}} \dot{\user2{e}}\left( t \right) + {\varvec{k}}_{{\varvec{i}}} {\varvec{e}}\left( t \right)} \right) \\ {\kern 1pt} & = {\varvec{s}}\left( {{\varvec{k}}_{{\varvec{d}}} ({\varvec{A}} + \Delta {\varvec{A}})\dot{\user2{\upsilon }}\left( t \right) + {\varvec{k}}_{{\varvec{d}}} ({\varvec{E}} + \Delta {\varvec{E}}){\varvec{\upsilon}}\left( t \right)} \right) \\ {\kern 1pt} & + {\varvec{s}}\left( {{\varvec{k}}_{{\varvec{d}}} ({\varvec{B}} + \Delta {\varvec{B}}){\varvec{u}} + {\varvec{k}}_{{\varvec{d}}} {\varvec{d}}\left( t \right)} \right) + {\varvec{s}}\left( {{\varvec{k}}_{{\varvec{p}}} \dot{\user2{\upsilon }}(t) + {\varvec{k}}_{{\varvec{i}}} {\varvec{\upsilon}}\left( t \right)} \right) \\ & {\kern 1pt} = {\varvec{s}}\left( {{\varvec{k}}_{{\varvec{d}}} \user2{A\dot{\upsilon }}\left( t \right) + {\varvec{k}}_{{\varvec{d}}} \user2{E\upsilon }\left( t \right) + {\varvec{k}}_{{\varvec{d}}} {\varvec{Bu}} + {\varvec{k}}_{{\varvec{p}}} \dot{\user2{\upsilon }}(t) + {\varvec{k}}_{{\varvec{i}}} {\varvec{\upsilon}}\left( t \right)} \right) + {\varvec{sk}}_{{\varvec{d}}} {\varvec{L}}_{{\varvec{r}}} . \\ \end{aligned}$$

(34)

Substituting the equivalent control law Eq. (19) and variable structure control term Eqs. (29) to (34), as follows:

$$\begin{gathered} \dot{\user2{V}}{\kern 1pt} {\kern 1pt} = {\varvec{s}}\left( {{\varvec{k}}_{{\varvec{d}}} \user2{A\dot{\upsilon }}\left( t \right) + {\varvec{k}}_{{\varvec{d}}} \user2{E\upsilon }\left( t \right) + {\varvec{k}}_{{\varvec{d}}} {\varvec{Bu}} + {\varvec{k}}_{{\varvec{p}}} \dot{\user2{\upsilon }}\left( t \right) + {\varvec{k}}_{{\varvec{i}}} {\varvec{\upsilon}}\left( t \right)} \right) \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + {\varvec{sk}}_{{\varvec{d}}} {\varvec{L}}_{{\varvec{r}}} + \frac{1}{{\varvec{r}}}\tilde{\user2{W}}^{{\text{T}}} \user2{\dot{\hat{W}}} \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = {\varvec{s}}\left( {{\varvec{k}}_{{\varvec{d}}} \user2{A\dot{\upsilon }}\left( t \right)} \right) + {\varvec{k}}_{{\varvec{d}}} \user2{E\upsilon }\left( t \right) + {\varvec{k}}_{{\varvec{p}}} \dot{\user2{\upsilon }}\left( t \right) + {\varvec{k}}_{{\varvec{i}}} {\varvec{\upsilon}}\left( t \right)) \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + {\varvec{s}}({\varvec{k}}_{{\varvec{d}}} {\varvec{B}}( - ({\varvec{k}}_{{\varvec{d}}} {\varvec{B}})^{ - 1} (({\varvec{k}}_{{\varvec{d}}} {\varvec{A}} + {\varvec{k}}_{{\varvec{p}}} )\dot{\user2{\upsilon }}(t) \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + ({\varvec{k}}_{{\varvec{d}}} {\varvec{E}} + {\varvec{k}}_{{\varvec{i}}} ){\varvec{\upsilon}}\left( t \right)) - \hat{\user2{K}}_{{\varvec{r}}} \left( t \right){\text{sat}} ({{\varvec{s}} \mathord{\left/ {\vphantom {{\varvec{s}} \Delta }} \right. \kern-\nulldelimiterspace} \Delta }))) \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + {\varvec{sk}}_{{\varvec{d}}} {\varvec{L}}_{{\varvec{r}}} + \frac{1}{{\varvec{r}}}\tilde{\user2{W}}^{{\text{T}}} \user2{\dot{\hat{W}}}{,} \hfill \\ \end{gathered}$$

(35)

then,

$$\begin{aligned} {\kern 1pt} \dot{\user2{V}} & \le {\varvec{sk}}_{{\varvec{d}}} {\varvec{B}}( - \hat{\user2{K}}(t) + {\varvec{K}}^{*} (t) - {\varvec{K}}^{*} (t)) + {\varvec{sk}}_{{\varvec{d}}} {\varvec{L}}_{{\varvec{r}}} + \frac{1}{{\varvec{r}}}\tilde{\user2{W}}^{{\text{T}}} \user2{\dot{\hat{W}}} \\ & {\kern 1pt} \le {\varvec{sk}}_{{\varvec{d}}} {\varvec{B}}( - \tilde{\user2{W}}^{{\text{T}}} {\varvec{H}}({\varvec{s}}) + \user2{\varepsilon^{\prime}} - {\varvec{\eta}}) + {\varvec{sk}}_{{\varvec{d}}} {\varvec{L}}_{{\varvec{r}}} + \frac{1}{r}\tilde{\user2{W}}^{{\text{T}}} ({\varvec{rsk}}_{{\varvec{d}}} {\varvec{BH}}({\varvec{s}})) \\ & {\kern 1pt} \le {\varvec{sk}}_{{\varvec{d}}} {\varvec{B}}(\user2{\varepsilon^{\prime}} - {\varvec{\eta}}) + {\varvec{sk}}_{{\varvec{d}}} {\varvec{L}}_{{\varvec{r}}} \\ {\kern 1pt} & \le - {\varvec{k}}_{{\varvec{d}}} \user2{B\eta }\left| {\varvec{s}} \right| + {\varvec{k}}_{{\varvec{d}}} {\varvec{L}}_{{\varvec{r}}} \left| {\varvec{s}} \right| + \left| {\varvec{s}} \right|{\varvec{k}}_{{\varvec{d}}} \user2{B\varepsilon^{\prime}}{.} \\ \end{aligned}$$

(36)

Selecting \(\user2{B\eta } > {\varvec{L}}_{{\varvec{r}}}\), it can be found that

$$\user2{\dot{V} < }0.$$

(37)

Thus, the proposed control scheme can ensure the asymptotic stability of the closed-loop automatic steering control system.