**Abstract**

This chapter proposes a nonlinear robust *H*-infinity control approach to enhance the trajectory-following capabilities of autonomous ground electric vehicles (AGEV). Given the inherent influence of driving maneuvers and road conditions on vehicle trajectory dynamics, the primary objective is to address the control challenges associated with trajectory-following, including parametric uncertainties, system nonlinearities, and external disturbance. Firstly, taking into account parameter uncertainties associated with the tire's physical limits, the system dynamics of the AGEV and its uncertain vehicle trajectory-following system are modeled and constructed. Subsequently, an augmented system for control-oriented vehicle trajectory-following is developed. Finally, the design of the nonlinear robust *H*-infinity controller (NRC) for the vehicle trajectory-following system is carried out, which is designed based on the *H*-infinity performance index and incorporates nonlinear compensation to meet the requirements of the AGEV system. The controller design involves solving a set of linear matrix inequalities derived from quadratic *H*-infinity performance and Lyapunov stability. To validate the efficacy of the proposed controller, simulations are conducted using a highfidelity CarSim® full-vehicle model in scenarios involving double lane change and serpentine maneuvers. The simulation results demonstrate that the proposed NRC outperforms both the linear quadratic regulator (LQR) controller and the robust *H*-infinity controller (RHC) in terms of vehicle trajectory-following performance.

**Keywords:** autonomous vehicles, electric vehicles, trajectory-following, robust control, nonlinear control

### **1. Introduction**

In recent years, the emergence of AGEV has attracted significant attention from the experts and scholars [1, 2]. AGEV technology offers notable benefits such as reducing traffic congestion, minimizing air pollution, and enhancing road safety. One key area of research focus is the application of active front steering (AFS) as a chassis active control technology for AGEV steering systems. AFS employs adaptive steering

gear ratio to improve vehicle stability and active safety. The integration of AFS systems in AGEV provides substantial advantages in terms of driver safety, handling flexibility, and trajectory-following performance for AGEV [3]. The inherent features of AFS, including its rapid response and precise execution, contribute to enhanced active safety and superior trajectory-following performance for AGEV [4, 5].

Extensive researches have been conducted in the literatures on the trajectoryfollowing control of AGEV with AFS system [6–12]. For achieving trajectoryfollowing for AGEV with the AFS system, a controller utilizing the Kalman filter with multi-rate is designed to account for the motor control period and the sampling time of the camera [6]. To address the challenges of the control distribution between steering and the control system for AGEV, a model predictive control (MPC) method is proposed in Ref. [7], which reallocates the braking and steering control based on tire force to precisely follow the desired trajectory. Aiming to enhance steering stability for AGEV, a variable steering ratio AFS controller is developed in Ref. [8], it establishes a mapping between vehicle velocity and steering wheel angle. Based on the linearization of the vehicle's model, the vehicle front steering angle is gained by the AFS system to follow the desired trajectory on slippery roads [9]. Moreover, the advanced steering capabilities of the AFS system have proven valuable in other application areas related to trajectory-following control [10–12].

Despite the success achieved in trajectory-following, there remain challenges in handling system nonlinearity, external disturbances, and uncertain model parameters [13, 14]. For example, researchers have employed various control strategies. Nonlinear model predictive control (NMPC) has been utilized to solve the system nonlinearity and ensure feasibility and convergence [15]. A combination of sliding mode and observer technique is applied to estimate model errors and disturbances for enhancing the system's stability [16]. In the context of Markov jump cyber-physical systems, an adaptive sliding mode control (SMC) framework is proposed to handle safety issues arising from actuator failures and external attacks [17]. For uncertain challenges of robotic arm systems, a switchable neural networks-based SMC framework has been developed to accurately track motion trajectories, which can provide real time control to enhance the stability of the trajectory-following control system by adaptive algorithm [18]. An adaptive fuzzy controller (FC) is developed to address the challenge nonlinear trajectory-following system, and the stability of system is guaranteed by Lyapunov method [19]. Some extensions of FC can be obtained from Refs. [20, 21]. Furthermore, active disturbance rejection control is employed to dynamically estimate and offset unmodeled system dynamics and unpredictable external disturbances, it enhances the stability of vehicle trajectory-following system [22]. Speed MPC strategies are proposed to achieve accurate trajectory-following for AGEV [23]. In the milling system, the optimal control and time delay techniques are used to suppress chatter by adaptive extreme value algorithm [24]. To handle the problems of parameter jump in complex nonlinear systems, an adaptive control method with multi-model switching is presented. The least squares technique and some lemmas are also utilized to develop an adaptive control law [25]. For dealing with the system disturbances, a novel optimal control based on iterative techniques is proposed in [26], and it provides the conditions of system asymptotic stability and the *H*-infinity control. An output feedback-based global adaptive control strategy is proposed to handle system nonlinear time-varying parameters [27]. Moreover, robust control strategies have also been implemented in trajectory-following control, providing benefits in addressing the challenges of system nonlinearity, parameter variation, and external disturbance [28].

*Nonlinear Robust Control of Trajectory-Following for Autonomous Ground Electric Vehicles DOI: http://dx.doi.org/10.5772/intechopen.112049*

Therefore, this chapter develops a novel nonlinear robust control framework for AGEV to address the challenges associated with trajectory-following control, including system nonlinearities, uncertain parameter, and disturbances. Firstly, the dynamics of the AGEV and the trajectory-following system are formulated. Subsequently, taking into account the *H*-infinity performance criterion, nonlinear system compensation, and aim of the trajectory-following, a nonlinear robust controller for trajectory-following is designed. Then, the robustness and effectiveness of the nonlinear robust controller is validated through MATLAB/Simulink/Carsim Cosimulation platform under two scenes. The following sections are structured as follows: Section 2 introduces vehicle trajectory-following modeling. Section 3 outlines the design of the nonlinear robust controller. Section 4 analyzes and discusses the simulation outcomes, and Section 5 presents the conclusions.

### **2. Vehicle trajectory-following model**

The primary focus of the chapter revolves the trajectory-following problem for AGEV. It is assumed that the suspension is a rigid structure, and under normal driving conditions, the slip angle tends to be small. For facilitating the analysis of vehicle actual motion, the bicycle model is selected:

$$m\left(\dot{\nu}\_{\rm x} - \nu\_{\rm y}\dot{\rho}\right) = F\_{\hat{f}\hat{\rm y}}\sin\delta\_{\hat{f}} + F\_{\hat{f}\hat{\rm x}}\cos\delta\_{\hat{f}} + F\_{r\infty} \tag{1}$$

$$m(\ddot{\mathbf{y}} + \boldsymbol{\nu}\_x \dot{\boldsymbol{\rho}}) = F\_{f\mathbf{y}} \cos \delta \boldsymbol{\rho} + F\_{f\mathbf{x}} \sin \delta \boldsymbol{\delta} + F\_{r\mathbf{y}} \tag{2}$$

$$I\_x \ddot{\rho} = l\_f \left( F\_{f\circ} \cos \delta\_{\hat{f}} + F\_{f\circ} \sin \delta\_{\hat{f}} \right) - l\_r F\_{r\circ} \tag{3}$$

This model incorporates variables such as mass *m*, yaw angle *φ*, lateral velocity *vx* and longitudinal velocity *vy*, lateral tire forces *Fiy*, longitudinal tire forces *Fix*, moment of inertia *Iz*. Specifically, *Fy* can be expressed:

$$F\_{\mathcal{Y}} = f\_{\mathcal{Y}}(a, F\_x, \mathfrak{s}\_r, \mu) \tag{4}$$

The computation of *α* is:

$$a = \tan^{-1} \left(\frac{v\_{w\eta}}{v\_{w\alpha}}\right) \tag{5}$$

$$\begin{cases} \boldsymbol{\upsilon}\_{w\circ\mathbf{x}} = \boldsymbol{\upsilon}\_{\mathbf{x}} \cos \delta\_{\hat{f}} + \left(\boldsymbol{\upsilon}\_{\mathbf{y}} + l\_{\hat{f}} \dot{\boldsymbol{\rho}}\right) \sin \delta\_{\hat{f}}\\ \boldsymbol{\upsilon}\_{w\circ\mathbf{x}} = \boldsymbol{\upsilon}\_{\mathbf{x}} \cos \delta\_{\hat{f}} + \left(\boldsymbol{\upsilon}\_{\mathbf{y}} - l\_{r} \dot{\boldsymbol{\rho}}\right) \sin \delta\_{\hat{f}} \end{cases} \tag{6}$$

$$\begin{cases} \boldsymbol{\upsilon}\_{w\circ\boldsymbol{\gamma}} = \boldsymbol{\upsilon}\_{\boldsymbol{x}} \cos \delta\_{\boldsymbol{f}} - \left(\boldsymbol{\upsilon}\_{\boldsymbol{\gamma}} + l\_{\boldsymbol{f}} \dot{\boldsymbol{\rho}}\right) \sin \delta\_{\boldsymbol{f}}\\ \boldsymbol{\upsilon}\_{w\circ\boldsymbol{\gamma}} = \boldsymbol{\upsilon}\_{\boldsymbol{x}} \cos \delta\_{\boldsymbol{f}} - \left(\boldsymbol{\upsilon}\_{\boldsymbol{\gamma}} - l\_{\boldsymbol{r}} \dot{\boldsymbol{\rho}}\right) \sin \delta\_{\boldsymbol{f}} \end{cases} \tag{7}$$

*sr* is defined as follows:

$$s\_r = \begin{cases} 1 - \frac{v\_{w\text{x}}}{r w\_w} (r w\_w \neq 0, r w\_w > v\_{w\text{x}})\\ \frac{r w\_w}{v\_{w\text{x}}} - 1 (v\_{w\text{x}} \neq 0, r w\_w < v\_{w\text{x}}) \end{cases} \tag{8}$$

where the tire's radius and angular velocity are represented by *r* and *ww*, respectively. *FZ* is given by the following equation:

$$\begin{cases} F\_{\text{Zfl},\text{Zfr}} = \frac{l\_r mg}{2\left(l\_r + l\_f\right)} \mp \Delta F\_1 + \Delta F\_2 \\\\ F\_{\text{Zrl},\text{Zrr}} = \frac{l\_f mg}{2\left(l\_r + l\_f\right)} \mp \Delta F\_2 + \Delta F\_1 \end{cases} \tag{9}$$

Δ*F*1 and Δ*F*2 are calculated using specific equations:

$$\begin{cases} \Delta F\_1 = \frac{m\left(\dot{\nu}\_x - \nu\_\jmath \omega\right) h\_{\text{cog}}}{2\left(l\_r + l\_f\right)}\\\\ \Delta F\_2 = \frac{m\left(\dot{\nu}\_x - \nu\_\jmath \omega\right) h\_{\text{cog}}}{2L\_h} + \frac{gm\_s h\_\varsigma}{2L\_h} \left(\frac{h\_{\text{cog}} \xi}{\mathcal{g}} - \sin \xi\right) \end{cases} \tag{10}$$

Under the assumption that *α* is small under driving conditions, the tire forces can be simplified:

$$F\_{i\circ} = \mathcal{N}\_{ai}a\_i, (i = f, r) \tag{11}$$

where *Nαi* is the cornering stiffness. Slip angle *αi* can be gained:

$$a\_f = \delta\_f - \frac{v\_\circ + l\_f \dot{\rho}}{v\_\times}, a\_r = -\frac{v\_\circ + l\_r \dot{\rho}}{v\_\times} \tag{12}$$

Taking into account the small front wheel angle, we can approximate cos*δf* ≈ 1 and sin*δf* ≈ 0. Additionally, we assume AGEV only moves in the horizontal plane, and it neglects the influence of other effects. Consequently, based on Eqs. (11) and (12), Eqs. (2) and (3) can be rewritten:

$$\ddot{y} = -\frac{N\_{\rm af} + N\_{\rm af}}{m v\_{\rm x}} \dot{y} - \left(\frac{l\_f N\_{\rm af} - l\_r N\_{\rm af}}{m v\_{\rm x}} + v\_{\rm x}\right) \dot{\rho} + w\_2 \tag{13}$$

$$\ddot{\rho} = -\frac{l\_f \mathcal{N}\_{af} - l\_r \mathcal{N}\_{af}}{I\_x \upsilon\_x} \dot{\rho} - \frac{l\_f^2 \mathcal{N}\_{af} + l\_r^2 \mathcal{N}\_{af}}{I\_x \upsilon\_x} \dot{\rho} + \mathcal{w}\_4 \tag{14}$$

where *w*2 and *w*4 represent the model state error.

During the trajectory-following process of the AGEV, it is crucial to consider state information of the vehicle. **Figure 1** depicts the diagram illustrating the trajectory-following process of the AGEV. The current and expected yaw angles are represented by *φ* and *φr*, respectively. Furthermore, the derivatives of *ye* and *φe* can be expressed:

$$
\dot{\mathcal{Y}}\_{\varepsilon} = \boldsymbol{\upsilon}\_{\mathcal{Y}} - \boldsymbol{\upsilon}\_{\mathcal{X}} \boldsymbol{\varrho}\_{\varepsilon} \tag{15}
$$

$$
\dot{\varphi}\_{\epsilon} = \dot{\varphi} - \dot{\varphi}\_{r} = \dot{\varphi} - \chi \dot{s} \tag{16}
$$

The derivative information of *s* can be expressed as:

$$
\dot{s} = v\_{\mathfrak{x}} + v\_{\mathfrak{z}} \rho\_{\mathfrak{e}} \tag{17}
$$

*Nonlinear Robust Control of Trajectory-Following for Autonomous Ground Electric Vehicles DOI: http://dx.doi.org/10.5772/intechopen.112049*

**Figure 1.** *The diagram illustrating the trajectory-following process of the AGEV.*

$$
\ddot{s} = \dot{\nu}\_{\text{x}} + \dot{\nu}\_{\text{\textdegree}} \rho\_{\text{e}} + \nu\_{\text{\textdegree}} \dot{\rho}\_{\text{e}} \tag{18}
$$

By utilizing Eqs. (15)–(18), the derivatives of *ye* and *φe* are transformed into Eqs. (19) and (20), respectively.

$$
\ddot{\jmath}\_{\varepsilon} = \dot{\upsilon}\_{\mathfrak{y}} - \dot{\upsilon}\_{\mathfrak{x}} \varrho\_{\varepsilon} - \upsilon\_{\mathfrak{x}} \dot{\varrho}\_{\varepsilon} \tag{19}
$$

$$
\ddot{\rho}\_{\epsilon} = \dot{\zeta} - \dot{\chi}\dot{s} - \chi\ddot{s} \tag{20}
$$

The vehicle dynamics mentioned can be reformulated into a state space representation:

$$
\dot{\mathbf{x}} = A\mathbf{x} + Bu + B\_{\text{w}}w \tag{21}
$$

$$A = \begin{bmatrix} \boldsymbol{\nu}\_{\epsilon} \\ \dot{\boldsymbol{\nu}}\_{\epsilon} \\ \dot{\boldsymbol{\nu}}\_{\epsilon} \\ \dot{\boldsymbol{\nu}}\_{\epsilon} \end{bmatrix}, A = \begin{bmatrix} \mathbf{0} & \mathbf{1} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & -\frac{N\_{\rm{gf}} + N\_{\rm{ar}}}{m\nu\_{\rm{x}}} & \frac{N\_{\rm{gf}} + N\_{\rm{ar}}}{m} & -\frac{l\_{f}N\_{\rm{af}} - l\_{r}N\_{\rm{ar}}}{m\nu\_{\rm{x}}} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{1} \\ \mathbf{0} & -\frac{l\_{f}N\_{\rm{af}} - l\_{r}N\_{\rm{ar}}}{I\_{z}\nu\_{\rm{x}}} & \frac{l\_{f}N\_{\rm{af}} - l\_{r}N\_{\rm{ar}}}{I\_{z}} & -\frac{l\_{f}^{2}N\_{\rm{af}} + l\_{r}^{2}N\_{\rm{ar}}}{I\_{z}\nu\_{\rm{x}}} \end{bmatrix}, B\_{\rm{w}} = \begin{bmatrix} \mathbf{0} \\ \mathbf{1} \\ \mathbf{0} \\ \mathbf{1} \end{bmatrix}^{T}$$

$$B = \begin{bmatrix} \mathbf{0} & \frac{N\_{\rm{gf}}}{m} & \mathbf{0} & \frac{l\_{f}N\_{\rm{af}}}{I\_{z}} \end{bmatrix}^{T}$$

The correlation between front wheel angle *δf* and steering wheel angle *δa* can be represented:

$$
\delta\_{\mathfrak{d}} = \mathfrak{r}\_{\mathfrak{f}} \delta\_{\mathfrak{f}} \tag{22}
$$

where *τf* is the gear ratio.

As AGEV navigate through complex and dynamic road conditions, *Ni* (*i = αf, αr*) vary and remain within certain bounds. This variation can be addressed in the following manner:

$$N\_i = \overline{N}\_i + n\_i \tilde{N}\_i, |n\_i| < \mathbf{1}(i = af, ar) \tag{23}$$

$$
\overline{N}\_{\text{ax}} = \frac{N\_{\text{ax\\_min}} + N\_{\text{ax\\_max}}}{2} (z = f, r) \tag{24}
$$

$$
\tilde{N}\_{\text{ax}} = \frac{N\_{\text{ax\\_max}} - N\_{\text{ax\\_min}}}{2} (z = f, r) \tag{25}
$$

The maximum and minimum values of *Ni* (*i = αf, αr*) are denoted as *Nimax* and *Nimin*, respectively. The time-varying parameters *ni* of the system satisfy the condition | *ni* | ≤ 1(*i = αf, αr*).

The system model (21) can be modified as follows:

$$
\dot{\mathbf{x}} = A\_d \mathbf{x} + B\_d \mathbf{u} + B\_w \mathbf{w} \tag{26}
$$

The arguments in the equation have the following significance:

$$\begin{aligned} &A\_{d}=A\_{d}+\Delta A,B\_{d}=\overline{B}\_{d}+\Delta B,[\Delta A \quad \Delta B \quad \begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & -\frac{\overline{M}\_{d}\overline{f}+\overline{N}\_{wr}}{mv\_{x}} & \frac{\overline{N}\_{d}\overline{f}+\overline{N}\_{wr}}{m} & -\frac{l\_{f}\overline{N}\_{d}\overline{f}\_{d}-l\_{r}\overline{N}\_{wr}}{I\_{z}} \\ 0 & 0 & 0 & 1 \\ 0 & -\frac{l\_{f}\overline{N}\_{d}\overline{f}\_{d}-l\_{r}\overline{N}\_{wr}}{I\_{z}v\_{x}} & \frac{l\_{f}\overline{N}\_{d}\overline{f}\_{d}-l\_{r}\overline{N}\_{wr}}{I\_{z}} & -\frac{l\_{f}^{2}\overline{N}\_{d}\overline{f}\_{d}+l\_{r}^{2}\overline{N}\_{wr}}{I\_{z}v\_{x}} \end{bmatrix},\overline{B}\_{d}=\begin{bmatrix} 0\\ \overline{N}\_{d}\\ \overline{m}\\ 0\\ 0\\ l\_{f}\overline{N}\_{d}\\ \end{bmatrix} \\\ &E\_{A}=\begin{bmatrix}0&-\frac{1}{m}&\frac{1}{m}&-\frac{l\_{f}}{m}\\ 0&-\frac{1}{mv\_{x}}&\frac{1}{m}&\frac{l\_{r}}{mv\_{x}}\\ 0&-\frac{l\_{f}}{I\_{x}v\_{x}}&\frac{l\_{f}}{I\_{x}}&-\frac{l\_{f}^{2}}{I\_{x}v\_{x}}\\ 0&-\frac{l\_{f}}{I\_{x}v\_{x}}&\frac{l\_{f}}{I\_{x}}&-\frac{l\_{f}^{2}}{I\_{x}v\_{x}}\\ 0&\frac{l\_{r}}{I\_{x}v\_{x}}&-\frac{l\_{r}}{I\_{x}}&-$$

### **3. The design of nonlinear robust controller**

### **3.1 Robust feedback control design**

To achieve the desired trajectory tracking, an error function is defined and a robust linear feedback gain is designed as follows:

$$J\_s = \int\_0^\infty q\_1 \dot{\mathbf{y}}\_\epsilon^2 + q\_2 \dot{\mathbf{y}}\_\epsilon^2 + q\_3 \rho\_\epsilon^2 + q\_4 \dot{\rho}\_\epsilon^2 + q\_4 \delta^2 dt \tag{27}$$

*Nonlinear Robust Control of Trajectory-Following for Autonomous Ground Electric Vehicles DOI: http://dx.doi.org/10.5772/intechopen.112049*

where the letter symbols in the equation hold the following meanings:

$$\mathcal{Y}\_{\varepsilon} = \mathcal{Y}\_{a} - \mathcal{Y}\_{d}, \dot{\mathcal{Y}}\_{\varepsilon} = \mathcal{v}\_{\mathcal{Y}} - \mathcal{v}\_{\ge} \boldsymbol{\rho}\_{\varepsilon}, \boldsymbol{\rho}\_{\varepsilon} = \boldsymbol{\rho} - \boldsymbol{\rho}\_{r}, \dot{\boldsymbol{\rho}}\_{\varepsilon} = \dot{\boldsymbol{\rho}} - \boldsymbol{\chi}\boldsymbol{\varepsilon}$$

The equation presented above can be expressed as follows:

$$\begin{split} J &= \int\_{0}^{\infty} \left[ \left( \overline{\mathbf{C}}\_{1} \mathbf{x} + \overline{D}\_{11} \mathbf{w} \right)^{T} U \left( \overline{\mathbf{C}}\_{1} \mathbf{x} + \overline{D}\_{11} \mathbf{w} \right) + \mathbf{u}^{T} V \mathbf{u} \right] dt \\ &= \int\_{0}^{\infty} \left[ \left( U^{\frac{1}{2}} \overline{\mathbf{C}}\_{1} \mathbf{x} + U^{\frac{1}{2}} \overline{D}\_{11} \mathbf{w} \right)^{T} \left( U^{\frac{1}{2}} \overline{\mathbf{C}}\_{1} \mathbf{x} + U^{\frac{1}{2}} \overline{D}\_{11} \mathbf{w} \right) + \int\_{0}^{\infty} \left( V^{\frac{1}{2}} u \right)^{T} \left( V^{\frac{1}{2}} u \right) dt \end{split} \tag{28}$$

The arguments in the equation hold the following significance:

$$
\overline{C}\_1 = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}, \overline{D}\_{11} = 1,\\ U = \begin{bmatrix} q\_1 & 0 & 0 & 0 \\ 0 & q\_2 & 0 & 0 \\ 0 & 0 & q\_3 & 0 \\ 0 & 0 & 0 & q\_4 \end{bmatrix} \\ 0, V = q\_5.
$$

The control output z can be obtained:

$$z = C\_1 \mathfrak{x} + D\_{11} \mathfrak{w} + D\_{12} \mathfrak{u} \tag{29}$$

where:

$$\mathbf{C}\_{1} = \begin{bmatrix} U^{\ddagger} \overline{\mathbf{C}}\_{1} \\ \mathbf{0} \end{bmatrix}, \mathbf{D}\_{11} = \begin{bmatrix} U^{\ddagger} \overline{\mathbf{D}}\_{11} \\ \mathbf{0} \end{bmatrix}, \mathbf{D}\_{12} = \begin{bmatrix} \mathbf{0} \\ V^{\ddagger} \end{bmatrix}.$$

The error cost function *J* is related to the control output *z* in the following manner:

$$J = \left\| z \right\|\_{2}^{2} \tag{30}$$

By utilizing the aforementioned system model (26) and the control output of the system (29), the problem of trajectory-following can be reformulated as a standard *H*infinity control problem.

$$\begin{cases} \dot{\mathbf{x}} = A\_d \mathbf{x} + B\_w w + B\_d u \\\\ \mathbf{z} = \mathbf{C}\_1 \mathbf{x} + D\_{11} w + D\_{12} u \end{cases} \tag{31}$$

In accordance with *H*∞ control theory, the aim of this trajectory-following system is to devise a controller *uL = Kx* that satisfy the requirements of trajectory-following for AGEV.

By utilizing Eq. (31) and the state feedback controller *uL = Kx*, it can derive the vehicle trajectory-following system.

$$\begin{cases} \dot{\mathbf{x}} = \mathbf{A}\_s \mathbf{x} + \mathbf{B}\_s \mathbf{w} \\\\ \mathbf{z} = \mathbf{C}\_s \mathbf{x} + \mathbf{D}\_s \mathbf{w} \end{cases} \tag{32}$$

where:

$$A\_s = \overline{A}\_d + \overline{B}\_d K + H\_d F\_d (E\_A + E\_B K), \\ B\_s = B\_w, \\ C\_s = C\_1 + D\_{12} K, \\ D\_s = D\_{11}$$

Within this investigation, *w* is regarded as an external disturbance affecting the system. The representation of the system's transfer function (32) can be formulated as follows:

$$T(\mathbf{s}) = \mathbf{C}\_{\mathbf{s}}(\mathbf{s}I - \mathbf{A}\_{\mathbf{s}})\mathbf{B}\_{\mathbf{s}} + D\_{\mathbf{s}} \tag{33}$$

*w*(*t*) and *z*(*t*) are presented:

$$\left\|\boldsymbol{\Xi}\right\|\_{2}^{2} = \int\_{0}^{+\infty} \boldsymbol{\Xi}^{T}(t)\boldsymbol{\Xi}(t)dt, \boldsymbol{\Xi} = \boldsymbol{w}, \boldsymbol{z} \tag{34}$$

The definition of the *H*-infinity norm is given by:

$$||T(s)||\_{\
\circ} = \sup\_{w \neq 0} \frac{||z||\_2}{||w||\_2} \tag{35}$$

In other words, the *H*-infinity norm represents system maximum singular value. The object of the *H*-infinity is to find *K* that satisfies the desired following performance while constraining the impact of disturbances on the output to a specific level. Hence, the subsequent *H*-infinity performance index is chosen:

$$\frac{\int\_{0}^{\infty} z^{T}(t)z(t)dt}{r^{2}} < \int\_{0}^{\infty} w^{T}(t)w(t)dt\tag{36}$$

In order to demonstrate the stability and *H*∞ performance of the system (33), several lemmas will be presented.

**Lemma 1** [14, 21]: Given matrix *P* ¼ *P*11 *P*12 *P*21 *P*22 � �, where *P* = *PT* , the conditions (37)–(39) are equivalent:

$$P < 0 \tag{37}$$

$$P\_{11} < 0, P\_{22} - P\_{12}^T P\_{11-1} P\_{12} < 0 \tag{38}$$

$$P\_{22} < 0, P\_{11} - P\_{12} P\_{22} - P\_{12}^T < 0 \tag{39}$$

**Lemma 2** [26, 27]: Let *F* be an appropriately dimensioned matrix such that *P* = *PT* . Suppose *M* and *N* are symmetric real matrices, and *FT F* ≤ 1. Under these conditions, the following inequality holds:

$$P + MFN + N^T F^T M^T < 0\tag{40}$$

The existence of a positive value *α* > 0 satisfying the following inequality is both sufficient and necessary conditions:

$$P + a\mathbf{M}\mathbf{M}^T + a^{-1}\mathbf{N}^T\mathbf{N} < \mathbf{0} \tag{41}$$

The *H*-infinity control gain *K* for trajectory-following system (33) can be obtained by applying the developed Theorem 1.

*Nonlinear Robust Control of Trajectory-Following for Autonomous Ground Electric Vehicles DOI: http://dx.doi.org/10.5772/intechopen.112049*

**Theorem 1**: In order to ensure both the stability and *H*∞ performance of the system, certain conditions need to be satisfied. These conditions involve the existence of symmetric matrices *Y* > 0, *X* > 0, and positive values *γ* and *η*, which satisfy the inequality (42). Furthermore, there should exist a *H*∞ control gain *K* for the system that fulfills the following equation.

$$
\begin{bmatrix}
\gamma \mathfrak{ms} \left( \overline{A}\_d Y + \overline{B}\_d X \right) & B\_w & Y\_1 & \varepsilon H\_d & Y\_2 \\
& & & -\gamma I & D\_{11}^T & \mathbf{0} & \mathbf{0} \\
& & & \ast & -\gamma I & \mathbf{0} & \mathbf{0} \\
& & \ast & \ast & -\eta I & \mathbf{0} \\
& & \ast & \ast & \ast & -\eta I
\end{bmatrix} < \mathbf{0} \tag{42}
$$

where:

$$\text{syms}(\ast) = \ast + \ast^T, \Upsilon\_1 = Y\mathbf{C}\_1^T + X^T \mathbf{D}\_{12}^T, \Upsilon\_2 = Y\mathbf{E}\_A^T + X^T \mathbf{E}\_B^T$$

*Proof***:** The condition of stability and *H*∞ performance for system (32) is that there exists a symmetric matrix *U* > 0 and value *γ* that satisfies:

$$
\begin{bmatrix}
UA\_s + A\_s^T U & UB\_s & C\_s^T \\
\* & -\gamma I & D\_s^T \\
\* & \* & -\gamma I
\end{bmatrix} < \mathbf{0} \tag{43}
$$

Inequation (43) can be written:

$$\begin{bmatrix} \gamma \text{ms} \left( U \left( \overline{A}\_d + \overline{B}\_d K \right) + U H\_d F\_d (E\_A + E\_B K) \right) & UB\_w & \left( C\_1 + D\_{12} K \right)^T \\\\ \* & -\gamma I & D\_{11} T \\\\ \* & \* & -\gamma I \end{bmatrix} < 0 \tag{44}$$

Inequation (44) further rewrite:

$$
\begin{bmatrix}
\text{syms}\left(U(\overline{A}\_d + \overline{B}\_d K)\right) & \text{UB}\_w & \mathbb{C}\_s^T \\
\text{\*} & -\gamma I & D\_{11}^T \\
\text{\*} & \text{\*} & -\gamma I
\end{bmatrix} + \begin{bmatrix}
\text{syms}\left(UH\_d F\_d (E\_A + E\_B K)\right) & \mathbf{0} & \mathbf{0} \\
\text{\*} & \mathbf{0} & \mathbf{0} \\
\text{\*} & \text{\*} & \mathbf{0}
\end{bmatrix} < \mathbf{0} \tag{45}
$$

Inequation (45) is equivalent to:

$$\begin{aligned} \begin{bmatrix} \gamma yms \left( U \left( \overline{A}\_d + \overline{B}\_d K \right) \right) & UB\_W & C\_s^T \\ & \ast & -\gamma I & D\_{11}^T \\ & \ast & \ast & -\gamma I \end{bmatrix} + \begin{bmatrix} UH\_d \\ \mathbf{0} \\ \mathbf{0} \end{bmatrix} F\_d \begin{bmatrix} (E\_A + E\_B K) & \mathbf{0} & \mathbf{0} \end{bmatrix} \\ + \begin{bmatrix} \left( E\_A + E\_B K \right)^T \\ \mathbf{0} \\ \mathbf{0} \end{bmatrix} F\_d^T \begin{bmatrix} (UH\_d)^T & \mathbf{0} & \mathbf{0} \end{bmatrix} < \mathbf{0} \end{aligned} \tag{46}$$

Assume that:

$$
\overline{H}\_d = \begin{bmatrix} \mathbf{U}H\_d & \mathbf{0} & \mathbf{0} \end{bmatrix}^T,\\ \overline{F}\_d = F\_d \tag{47}
$$

$$
\overline{E}\_{AB} = \begin{bmatrix}
(E\_A + E\_B K) & \mathbf{0} & \mathbf{0}
\end{bmatrix} \tag{48}
$$

$$
\overline{\boldsymbol{\sigma}} = \begin{bmatrix}
\text{syms}\left(\boldsymbol{U}(\overline{\boldsymbol{A}}\_{d} + \overline{\boldsymbol{B}}\_{d}\boldsymbol{K})\right) & \boldsymbol{U}\boldsymbol{B}\_{w} & \boldsymbol{C}\_{cl}^{T} \\
\text{ $\ast$ } & -\boldsymbol{\gamma}\boldsymbol{I} & \boldsymbol{D}\_{11}\boldsymbol{T} \\
\text{ $\ast$ } & \ast & -\boldsymbol{\gamma}\boldsymbol{I}
\end{bmatrix} \tag{49}
$$

Inequation (46) can be written as the following conditions:

$$-\varpi + \overline{H}\_d \overline{F}\_d \overline{E}\_{AB} + \overline{E}\_{AB}^T \overline{F}\_d^T \overline{H}\_d^T < 0 \tag{50}$$

Based on lemma 2, there exists *η* > 0 satisfy:

$$
\sigma \sigma + \eta \overline{H}\_d \overline{H}\_d^T + \eta^{-1} \overline{E}\_{AB} \overline{E}\_{AB}^T < 0 \tag{51}
$$

Inequation (51) can be gained from Lemma 1.

$$
\begin{bmatrix}
\boldsymbol{\varpi} & \overline{\boldsymbol{H}}\_d & \overline{\boldsymbol{E}}\_{AB}^T \\
\ast & -\eta^{-1}\boldsymbol{I} & \mathbf{0} \\
\ast & \ast & -\eta\boldsymbol{I}
\end{bmatrix} < \mathbf{0} \tag{52}
$$

Expand the inequality (52), according to the property of linear matrix inequality and diag{*U*�1 , *I*, *I*, *η*, *I*}. Let *Y* = *U*�1 , *KY* = *X*, Theorem 1 can be obtained.

### **3.2 Nonlinear robust control design**

Subsequently, in order to enhance system's rapid response and minimize overshoot, the design of the nonlinear compensation feedback control part will be formulated as follows:

$$
\mu\_{\text{nla}} = \rho(r, h) \mathbf{B}^T \mathbf{Q} \mathbf{x} \tag{53}
$$

Here, the nonlinear compensation function *ρ*(*r*, *h*) is introduced, and it depends on the error state *x*, the reference value *r* and actual value *h*. *Q* can be obtained:

$$\mathbf{A}\_s^T \mathbf{Q} + \mathbf{Q} \mathbf{A}\_s + \mathbf{W} = \mathbf{0} \tag{54}$$

$$W = 100^l \cdot I \tag{55}$$

where *l* is adjustable value.

The nonlinear compensation part is as follows:

$$\rho(r,h) = -\beta \frac{1}{1 - e^{-1}} \left( e^{-a \left| 1 - \frac{h - h\_0}{r - h\_0} \right|} - e^{-1} \right) \tag{56}$$

where *α* and *β* are positive adjustable parameters.

*Nonlinear Robust Control of Trajectory-Following for Autonomous Ground Electric Vehicles DOI: http://dx.doi.org/10.5772/intechopen.112049*

By integrating the linear part and the nonlinear part, the actuator's output is ultimately derived in the subsequent expression (57). The utilization of linear feedback part facilitates swifter system response within the trajectory-following, while concurrently, the nonlinear compensation part attains stable output and diminishes system overshoot.

$$\mu\_{final} = \mathbf{K}\mathbf{x} + \rho(r, h)\mathbf{B}^T\mathbf{Q}\mathbf{x} \tag{57}$$

Based on the aforementioned nonlinear compensation part, and taking into account the saturation of the system's actuator output, the nonlinear robust control system model can be reformulated in the subsequent manner.

$$\begin{cases} \dot{\mathbf{x}} = A\_d \mathbf{x} + B\_w \mathbf{w} + B\_d \text{sat}\left(u\_{final}\right) \\\\ \mathbf{z} = \mathbf{C}\_1 \mathbf{x} + D\_{11} \mathbf{w} + D\_{12} \text{sat}\left(u\_{final}\right) \end{cases} \tag{58}$$

$$\text{sat}\left(u\_{final}\right) = \begin{cases} u\_{\text{max}}, & u\_{final} > u\_{\text{max}} \\\\ \mathbf{K} \mathbf{x} + \rho(r, h) \mathbf{B}^T \mathbf{Q} \mathbf{x}, & \left| u\_{final} \right| < u\_{\text{max}} \\\\ -u\_{\text{max}}, & u\_{final} < -u\_{\text{max}} \end{cases} \tag{59}$$

Taking into account the saturation of the front wheel angle, the actual expression for the nonlinear compensation can be represented as follows:

$$
\mu\_{nla} = \mathfrak{sat}(\mathfrak{u}) - \mathrm{Kx} \tag{60}
$$

Based on the aforementioned conditions, the expression for *unla* can be formulated as follows.

$$0 < |u\_{nlm}| < \rho(r, h)\mathbf{B}^T\mathbf{Q}\mathbf{x} \tag{61}$$

Subsequently, the impact of the nonlinear compensation on *H*-infinity performance and stability of the system will be demonstrated.

*Proof*: The Lyapunov functional *V* is defined as follows:

$$\mathbf{V} = \mathbf{x}^T \mathbf{S} \mathbf{x} \tag{62}$$

$$\dot{\mathbf{V}} = \dot{\mathbf{x}}^T \mathbf{S} \mathbf{x} + \mathbf{x}^T \mathbf{S} \dot{\mathbf{x}} = \left(\overline{\mathbf{A}} \mathbf{x} + \overline{\mathbf{R}} \mathbf{x} + \mathbf{B}\_w w\right)^T \mathbf{S} \mathbf{x} + \mathbf{x}^T \mathbf{S} (\overline{\mathbf{A}} \mathbf{x} + \overline{\mathbf{R}} \mathbf{x} + \mathbf{B}\_w w)$$

$$= \mathbf{x}^T \overline{\mathbf{A}}^T \mathbf{S} \mathbf{x} + \mathbf{x}^T \overline{\mathbf{R}}^T \mathbf{S} \mathbf{x} + w^T \mathbf{B}\_w^T \mathbf{S} \mathbf{x} + \mathbf{x}^T \mathbf{S} \overline{\mathbf{A}} \mathbf{x} + \mathbf{x}^T \mathbf{S} \overline{\mathbf{R}} \mathbf{x} + \mathbf{x}^T \mathbf{S} \mathbf{B}\_w w$$

$$= \mathbf{x}^T \left(\overline{\mathbf{A}}^T \mathbf{S} + \mathbf{S} \overline{\mathbf{A}}\right) \mathbf{x} + \mathbf{x}^T \left(\overline{\mathbf{R}}^T \mathbf{S} + \mathbf{S} \overline{\mathbf{R}}\right) \mathbf{x} + w^T \mathbf{B}\_w^T \mathbf{S} \mathbf{x} + \mathbf{x}^T \mathbf{S} \mathbf{B}\_w w$$

where:

$$
\overline{A} = A\_d + B\_d \\
K, \overline{R} = B\_d \\
\mu\_{nla}
$$

When *w* = 0:

$$\dot{V} = \mathbf{x}^T \left( \overline{\mathbf{A}}^T \mathbf{S} + \mathbf{S} \overline{\mathbf{A}} \right) \mathbf{x} + \mathbf{x}^T \left( \overline{\mathbf{R}}^T \mathbf{S} + \mathbf{S} \overline{\mathbf{R}} \right) \mathbf{x} \tag{64}$$

Assuming that:

$$\mathbf{V\_1 = x^T \left(\overline{A}^T \mathbf{S} + \mathbf{S}\overline{A}\right) x} \tag{65}$$

It can be inferred that *V*1 > 0. Assuming that

$$\mathbf{V}\_2 = \mathbf{x}^T \left( \overline{\mathbf{R}}^T \mathbf{S} + \mathbf{S} \overline{\mathbf{R}} \right) \mathbf{x} = 2 \mathbf{x}^T \mathbf{S} \mathbf{B} \left( \mathbf{s} \text{at} \left( \mathbf{K} \mathbf{x} + \rho \mathbf{B}^T \mathbf{S} \mathbf{x} \right) - \mathbf{K} \mathbf{x} \right) = 2m \left( \mathbf{s} \text{at} \left( \mathbf{n} + \rho \mathbf{m} \right) - \mathbf{n} \right) \tag{66}$$

where:

$$\boldsymbol{m} = \mathbf{x}^T \mathbf{S} \mathbf{B}, \boldsymbol{n} = K \mathbf{x}$$

When actuator output is not saturated:

$$|\mathfrak{sat}(n+\rho m)| < u\_{\text{max}}\tag{67}$$

At this time:

$$V\_2 = 2m(\mathfrak{sat}(n + \rho m) - n) = 2\rho m^2 < = 0\tag{68}$$

Therefore:

$$\dot{V} = V\_1 + V\_2 = \mathbf{x}^T \left( \overline{\mathbf{A}}^T \mathbf{S} + \mathbf{S} \overline{\mathbf{A}} \right) \mathbf{x} + 2m(\mathbf{s}\mathbf{x}(n + \rho m) - n)<\tag{69} = \mathbf{x}^T \left( \overline{\mathbf{A}}^T \mathbf{S} + \mathbf{S} \overline{\mathbf{A}} \right) \mathbf{x}<\tag{69}$$

When actuator output is saturated:

$$|\mathfrak{sat}(n+\rho m)| \ge \mathfrak{u}\_{\text{max}}\tag{70}$$

Suppose that *j* is saturated and *ρm* = 0, the system is asymptotically stable. When *j* is not subjected to saturation, the output can take on the following two forms:

$$\begin{cases} \begin{aligned} m < 0, \text{sat}(n + \rho m) - n \ge 0, \qquad &When \ n + \rho m \ge u\_{\text{max}} \text{ and } n > 0 \\ m > 0, \text{sat}(n + \rho m) - n \le 0, \qquad &When \ n + \rho m \le -u\_{\text{max}} \text{ and } n < 0 \end{aligned} \end{cases} \tag{71}$$

It can be observed from the inequality condition (71): *V*2 ¼ 2*m sat n* ð Þ ð Þ� þ *ρm n* < ¼ 0 (72)

Thus:

$$\dot{V} = V\_1 + V\_2 = \mathbf{x}^T \left( \overline{\mathbf{A}}^T \mathbf{S} + \mathbf{S} \overline{\mathbf{A}} \right) \mathbf{x} + 2m(\text{sat}(n + \rho m) - n)<\tag{72} \\ = \mathbf{x}^T \left( \overline{\mathbf{A}}^T \mathbf{S} + \mathbf{S} \overline{\mathbf{A}} \right) \mathbf{x}<\tag{73}$$

Thus, the system with a nonlinear compensation function is asymptotically stable without interference.

*Nonlinear Robust Control of Trajectory-Following for Autonomous Ground Electric Vehicles DOI: http://dx.doi.org/10.5772/intechopen.112049*

Next, the stability and *H*-infinity performance of the system with a nonlinear compensation function under external disturbances will be demonstrated.

Let initially establish a cost function *Jf*:

$$J\_f = \dot{V} + \mathbf{z}^T \mathbf{z} - \chi^2 w^T w \tag{73}$$

Since the system exhibits asymptotic stability, then if the *H*-infinity satisfies the following inequality:

$$\left\|\left\|\mathbf{z}\right\|\right\|^2 < \chi^2 \left\|\left\|w\right\|\right\|^2\tag{74}$$

Then, inequality (76) exists

$$J\_f = \dot{V} + \mathbf{z}^T \mathbf{z} - \chi^2 w^T w < \mathbf{0} \tag{75}$$

The above inequality (76) can be further rewritten:

$$J\_f = \begin{bmatrix} \mathbf{x} \\ \mathbf{w} \end{bmatrix}^T \left( \begin{bmatrix} \overline{\mathbf{A}}^T \mathbf{S} + \mathbf{S} \overline{\mathbf{A}} + 2 \mathbf{S} \overline{\mathbf{R}} & \mathbf{S} \mathbf{B}\_w \\ \mathbf{s} & \mathbf{0} \end{bmatrix} + \begin{bmatrix} \mathbf{C}^T \mathbf{C} & \mathbf{C}^T \mathbf{D} \\ \mathbf{s} & \mathbf{D}^T \mathbf{D} \end{bmatrix} + \begin{bmatrix} \mathbf{0} & \mathbf{0} \\ \mathbf{s} & \mathbf{y}^2 \end{bmatrix} \right) \begin{bmatrix} \mathbf{x} \\ \mathbf{w} \end{bmatrix} \tag{76}$$

Based on inequality (77) and the characteristics of quadratic form, it can establish the following inequality:

$$
\boldsymbol{\Psi} = \begin{bmatrix} \overline{\boldsymbol{A}}^T \boldsymbol{S} + \mathbf{S} \overline{\boldsymbol{A}} + 2 \mathbf{S} \overline{\boldsymbol{R}} & \mathbf{S} \mathbf{B}\_w \\ \boldsymbol{\ast} & \mathbf{0} \end{bmatrix} + \begin{bmatrix} \mathbf{C}^T \mathbf{C} & \mathbf{C}^T \mathbf{D} \\ \boldsymbol{\ast} & \mathbf{D}^T \mathbf{D} \end{bmatrix} + \begin{bmatrix} \mathbf{0} & \mathbf{0} \\ \boldsymbol{\ast} & \mathbf{\mathcal{T}} \end{bmatrix} \tag{77}
$$

Based on Lemma 1:

$$
\Psi = \begin{bmatrix}
\overline{A}^T \mathcal{S} + \mathcal{S}\overline{A} + 2\mathcal{S}\overline{R} & \mathcal{S}B\_w & \mathcal{C}^T \\
& ^\* & \mathcal{Y}^T & D^T \\
& \* & ^\* & -I
\end{bmatrix} \tag{78}
$$

Let *U* = *S,* Eq. (79) is negative from Theorem 1, inequality (66), (68), and (72). Thus, the stability and *H*-infinity performance of system with nonlinear compensation function is proved.

### **4. Simulation and analysis**

This section simulates and validates the proposed nonlinear robust H-infinity state-feedback controller on the MATLAB/Simulink-Carsim®. The simulation framework is implemented using MATLAB/Simulink, while the high-fidelity dynamics model for AGEV trajectory-following is provided by CarSim® software. **Figure 2** illustrates the simulation flowchart, and **Table 1** defines the key parameters of AGEV.

The simulation scenarios include double lane change (DLC) road and serpentine road scenes, with a constant forward speed of 54 km/h. These road scenes are chosen to evaluate the controller's robust following ability and steady-state response

### **Figure 2.** *Flowchart of system simulation framework.*

### **Table 1.**

*The key parameters of the vehicle.*

performance. For comparison purposes, the performance of the proposed controller is also compared with that of the LQR and RHC controllers.

### **4.1 Double lane change scene**

The simulation results for double lane change (DLC) are presented in **Figures 3**–**9**, depicting global trajectories, lateral errors, road curvature, front wheel angle, yaw, yaw error, linear angle, and nonlinear compensation part. **Figures 3** and **4** show the global trajectories and lateral errors obtained from three controllers during DLC scenario. All three controllers exhibit satisfactory tracking performance. The maximum of the lateral error for LQR controller is approximately 0.4 m, while for the RHC controller it is around 0.24 m. Notably, the NRC controller achieves a smaller maximum lateral error compared to the other two controllers, indicating its superior tracking performance. Furthermore, **Figure 3** demonstrates that NRC maintains exceptional system response within the range of 45 to 55 meters, further it highlights NRC has ability to enhance the transient performance of the system.

*Nonlinear Robust Control of Trajectory-Following for Autonomous Ground Electric Vehicles DOI: http://dx.doi.org/10.5772/intechopen.112049*

**Figure 3.** *Global trajectories of three controllers under DLC scene.*

**Figure 4.** *Lateral errors of three controllers under DLC scene.*

The road curvature and front wheel angle for AGEV during DLC scene are illustrated in **Figures 5** and **6**. **Figure 6** indicates that the front wheel angle of the NRC controller consistently falls between that of the LQR and RHC controllers. It is attributed to the fact that a too small front wheel angle would result in a slow system response, while a too large front wheel angle would lead to significant overshoot. The NRC controller incorporates a linear feedback part to enhance the system response and a nonlinear compensation part to mitigate excessive overshoot. As a result, the NRC controller demonstrates excellent trajectory-following capabilities.

**Figure 5.** *Curvature of road under DLC scene.*

**Figure 6.** *Front wheel angle of three controllers under DLC scene.*

**Figure 7.** *Yaw of three controllers under DLC scene.*

**Figure 8.** *Yaw error of three controllers under DLC scene.*

**Figures 7** and **8** depict the yaw and yaw error of the NRC controller, and the NRC controller exhibits smaller yaw error and excellent trajectory-following capabilities compared to the RHC and LQR controllers. The angle of the linear feedback and nonlinear compensation of the NRC controller are illustrated in **Figure 9**. Notably, while the lateral error is smaller, the system nonlinear part of the NRC controller is significant. Conversely, as the vehicle lateral error increases, the system nonlinear part gradually decreases. It aligns with the design intention of the NRC controller, wherein the system exhibits fast response under increasing error scenes and small overshoot when the error is minimal.

*Nonlinear Robust Control of Trajectory-Following for Autonomous Ground Electric Vehicles DOI: http://dx.doi.org/10.5772/intechopen.112049*

**Figure 9.** *Steering angle of NRC under DLC scene.*

### **4.2 Serpentine scene**

**Figures 10** and **11** illustrate the global trajectories and lateral errors during serpentine tracking. It can be observed that the NRC controller exhibits smaller maximum lateral errors compared to the LQR and RHC controllers. Furthermore, the NRC controller demonstrates higher response speed and superior transient performance in comparison with the other two controllers. These findings indicate that the NRC controller outperforms the LQR and RHC controllers in terms of tracking performance on serpentine roads.

**Figure 10.** *Global trajectories of three controllers under serpentine scene.*

**Figure 11.** *Lateral errors of three controllers under serpentine scene.*

**Figures 12** and **13** present the road curvature and front wheel angle of the serpentine scene. The maximum of serpentine road curvature is approximately 0.01(1/*m*). Similar to the DLC scene, the front wheel angle value of the NRC lies within the range of the LQR and RHC controllers. The inclusion of nonlinear compensation enables NRC to demonstrate stable trajectory-following capability.

In **Figures 14** and **15**, it can be observed that NRC responds quickly with minimal yaw error when tracking a trajectory with large curvature. This results in low yaw error and ensures stable tracking performance. **Figure 16** illustrates the angle of linear and nonlinear feedback of NRC under the serpentine scene. The value of the nonlinear

**Figure 12.** *Curvature of road under serpentine scene.*

**Figure 13.** *Front wheel angle of three controllers under serpentine scene.*

**Figure 14.** *Yaw of three controllers under serpentine scene.*

*Nonlinear Robust Control of Trajectory-Following for Autonomous Ground Electric Vehicles DOI: http://dx.doi.org/10.5772/intechopen.112049*

**Figure 15.** *Yaw error of three controllers under serpentine scene.*

**Figure 16.** *Steering angle of NRC under serpentine scene.*

compensation function aligns with the trend of linear feedback, and it contributes to enhanced system response speed and trajectory tracking accuracy.

The mean absolute lateral error (MAE), maximum lateral error (ME), and rootmean-square lateral error (RMSE) are used to quantitatively analyze the trajectoryfollowing performance of NRC, and the RHC and LQR controllers are utilized as comparative test.

**Table 2** presents the values of ME, MAE, RMSE, RI, and RII for the lateral displacement in both DLC and serpentine scenes. The data clearly indicate NRC achieves smaller ME, MAE, and RMSE compared to LQR and RHC in both scenarios.

### **Table 2.**

*Quantitatively analyze of lateral error under two scenes.*

The larger errors observed in the DLC scene can be attributed to the significant lateral displacement in this scenario, which lead to greater trajectory-following errors. In terms of performance improvement, NRC demonstrates an overall enhancement of over 50% compared to LQR in the DLC scene, and over 57% improvement in the serpentine scene. Additionally, under the DLC scene, NRC exhibits a 20.96% higher ME than RHC, which indicates its faster system response in trajectory-following with large model state errors. Furthermore, the MAE of NRC is approximately 14.49% in the DLC scene, which is higher than RHC, it highlights its smaller errors compared to RHC. Overall, the proposed controller outperforms RHC and LQR by offering advantages such as fast response speed and reduced overshoot.


3.4. Number of ancestors and evaluation value of forest property

Table 4 summarizes the land characteristics of all properties in 2014. In 2014, there were 56,329 ancestors, and 93% (52,327) of ancestors had land property. The inheritance tax is necessary in cases when the total evaluation of properties exceeds the value of several exemptions. However, the inheritance tax tends to be applicable when there is land included as property.

Of the 52,327 ancestors with land property, 51,513 cases included housing land, meaning that in most cases, including forest land, the land property included housing land. This is in agreement with the results of the 2013 Housing and Land Statistics conducted by the Statistics Bureau of the Ministry of Internal Affairs and Communications, which estimated that 2,673,000 households owned forest land, of which 2,569,400 households (96.1%) had their own house. Therefore, only a small percentage of cases included inherited land property that was only forest land.

In total, 20.9% of ancestors had land properties that included forest land. However, these properties only accounted for 1.4% of the total value of properties. The average value of forest land per ancestor was 6.8 million yen, much less than the values of housing land, farmland, and rice fields.

Figure 3 shows the number of ancestors with property that included forested land and the value of forest land per ancestor from 1988 to 2014.

About 10,000 inheritances that included forested land were required to pay inheritance tax. The evaluated value of forest land per inheritance has decreased since 1992. For example, when the value in 1992 is fixed to 100, the value in 2014 is 17.4. The reason for this decrease is the decrease in the value of forest land, since the values of both forest land and housing land have decreased. Figure 4 shows the number of ancestors and value of housing land. The pattern of the graph is similar to that of forest land, and when the peak value in 1992 is fixed to 100, the value in 2014 is 36.1.

Figure 5 shows the number of ancestors and value of the inherited property when only standing trees are considered and forest land is excluded, both of which are decreasing.

Figure 4. Number of ancestors and value of housing land. Source: National Tax Agency Annual Statistics Report.

Figure 3. Number of ancestors and value of forest land. Source: National Tax Agency Annual Statistics Report.

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The value of forest land per ancestor has decreased since 1992 (Figure 3). The value in the peak year (1992) was 39 million yen, which decreased to 6.8 million yen in 2014. The reason for this decrease is a decrease in the standard value at the standard final cutting age due to decreases

The stumpage prices of C. japonica and C. obtusa peaked in 1980, and have been decreasing since. Setting the stumpage price in 1980 to 100, the stumpage price in 2016 was 12.3 for C. japonica and 14.4 for C. obtusa. Compared to the stumpage prices in 1960, 7,148 yen and 7,966 yen for C. japonica and C. obtusa, respectively, the stumpage prices in 2016 were 39.2 and 77.2% of those in 1960, respectively. Since this represents the nominal price, a long-term increase in prices should be considered. Using the Corporate Goods Price Index of 2005 by the Bank of Japan, the index was 50.8 and 105.0 in 1960 and 2011, respectively. Using the Consumer Price Index of 2010 by the

3.5. Decrease in the value of forest and stumpage price

in both forest land and stumpage prices (Figure 6).

Source: National Tax Agency Annual Statistics Report.

Note: The total of number of ancestors is the actual number of ancestors. In cases of rice fields and farmland, cultivation rights and perennial tenant rights are included. In the case of housing land, leaseholding is included.

Table 4. Land property in 2014.

Figure 3. Number of ancestors and value of forest land. Source: National Tax Agency Annual Statistics Report.

Figure 4. Number of ancestors and value of housing land. Source: National Tax Agency Annual Statistics Report.

Figure 5 shows the number of ancestors and value of the inherited property when only standing trees are considered and forest land is excluded, both of which are decreasing.

#### 3.5. Decrease in the value of forest and stumpage price

3.4. Number of ancestors and evaluation value of forest property

value of forest land per ancestor from 1988 to 2014.

to 100, the value in 2014 is 36.1.

Source: National Tax Agency Annual Statistics Report.

Table 4. Land property in 2014.

and rice fields.

166 Taxes and Taxation Trends

Table 4 summarizes the land characteristics of all properties in 2014. In 2014, there were 56,329 ancestors, and 93% (52,327) of ancestors had land property. The inheritance tax is necessary in cases when the total evaluation of properties exceeds the value of several exemptions. However, the inheritance tax tends to be applicable when there is land included as property.

Of the 52,327 ancestors with land property, 51,513 cases included housing land, meaning that in most cases, including forest land, the land property included housing land. This is in agreement with the results of the 2013 Housing and Land Statistics conducted by the Statistics Bureau of the Ministry of Internal Affairs and Communications, which estimated that 2,673,000 households owned forest land, of which 2,569,400 households (96.1%) had their own house. Therefore, only

In total, 20.9% of ancestors had land properties that included forest land. However, these properties only accounted for 1.4% of the total value of properties. The average value of forest land per ancestor was 6.8 million yen, much less than the values of housing land, farmland,

Figure 3 shows the number of ancestors with property that included forested land and the

About 10,000 inheritances that included forested land were required to pay inheritance tax. The evaluated value of forest land per inheritance has decreased since 1992. For example, when the value in 1992 is fixed to 100, the value in 2014 is 17.4. The reason for this decrease is the decrease in the value of forest land, since the values of both forest land and housing land have decreased. Figure 4 shows the number of ancestors and value of housing land. The pattern of the graph is similar to that of forest land, and when the peak value in 1992 is fixed

Note: The total of number of ancestors is the actual number of ancestors. In cases of rice fields and farmland, cultivation

rights and perennial tenant rights are included. In the case of housing land, leaseholding is included.

a small percentage of cases included inherited land property that was only forest land.

The value of forest land per ancestor has decreased since 1992 (Figure 3). The value in the peak year (1992) was 39 million yen, which decreased to 6.8 million yen in 2014. The reason for this decrease is a decrease in the standard value at the standard final cutting age due to decreases in both forest land and stumpage prices (Figure 6).

The stumpage prices of C. japonica and C. obtusa peaked in 1980, and have been decreasing since. Setting the stumpage price in 1980 to 100, the stumpage price in 2016 was 12.3 for C. japonica and 14.4 for C. obtusa. Compared to the stumpage prices in 1960, 7,148 yen and 7,966 yen for C. japonica and C. obtusa, respectively, the stumpage prices in 2016 were 39.2 and 77.2% of those in 1960, respectively. Since this represents the nominal price, a long-term increase in prices should be considered. Using the Corporate Goods Price Index of 2005 by the Bank of Japan, the index was 50.8 and 105.0 in 1960 and 2011, respectively. Using the Consumer Price Index of 2010 by the

reduce the standing tree value before their death.17 Since many plantation forests planted after World War II are now reaching the standard cutting age, and thinning has been promoted by the Forestry Agency, the value of standing trees is increasing. To reduce the burden of the inheritor further, forest owners may sell not only standing trees, but also the forest land. It has been noted that some forest owners, for example, older owners without an inheritor, have stopped managing their forest with the intent of selling their forest land along with standing trees to logging companies;18 however, it is difficult to determine the statistics behind this trend. It should be mentioned that the reason for such action is not limited to inheritance, as the decrease in stumpage price (Figure 6) and decrease in log price have led to decreases in income from timber sales. Table 5 shows statistical indexes of forest management. Net income (subtracting column (B) from column (A)) has recently decreased. The percentage of tax in

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Source: Ministry of Agriculture, Forestry and Fisheries, Statistics of Forest Management.19

17Sugano and Tani ([14], p. 32) introduced the following management example: a forest owner conducted clear-cutting of 40- to 50-year-old C. japonica and C. obtusa, and then planted low-value broad-leaved trees. In addition to the value of standing trees, the total value of the inherited properties decreased. As a result, the tax rate (see Table 1) decreased. Nagata [15] mentioned a similar problem at the time of the post-war tax reform, in that there was a large imbalance between cases in which the forest owners conducted clear-cutting and cases in which the forest owners maintained the

19Statistics Bureau, Ministry of Internal Affairs and Communications, e-Stat, http://www.e-stat.go.jp/SG1/estat/List.do?

Note: In this case, tax refers to taxes, public dues, various burden charges, etc.

18Such situations have been mentioned before 2000 (e.g., [16], p. 2).

bid=000001047783&cycode=0 [Accessed: April 18, 2017] (in Japanese)

Table 5. Index of forest management.

forest without clear-cutting.

Figure 5. Number of ancestors and value of standing trees. Source: National Tax Agency Annual Statistics Report.

Figure 6. Stumpage price. Source: Forestry Agency, Annual Statistics of Forestry, Annual Statistics of Forest and Forestry; original source is the Japan Real Estate Institute. Note: As of March 31.

Bank of Japan, the index was 19.1 and 99.7 in 1960 and 2011, respectively. In other words, between 1960 and 2011, the Corporate Goods Price Index increased twofold and the Consumer Price Index increased fivefold. Under such long-term increases in the price index, the nominal prices of 2016 were 39.2 and 77.2% of those in 1960, showing a decrease in price.

The large decrease in stumpage price has affected the value of forests during evaluations for inheritance tax. Since the area of inherited forest is not included in the National Tax Agency Annual Statistics Report, further statistical analysis is difficult, and analyses related to on-site surveys is a topic of future research.

Given the large decreases in stumpage price, it is possible that the stumpage price will decrease beyond the break-even point. When forest owners cannot expect any income from cutting after inheritance, the inheritance tax is only a burden to the inheritor. This would create a scenario where there is a probability that some forest owners would cut standing trees to reduce the standing tree value before their death.17 Since many plantation forests planted after World War II are now reaching the standard cutting age, and thinning has been promoted by the Forestry Agency, the value of standing trees is increasing. To reduce the burden of the inheritor further, forest owners may sell not only standing trees, but also the forest land. It has been noted that some forest owners, for example, older owners without an inheritor, have stopped managing their forest with the intent of selling their forest land along with standing trees to logging companies;18 however, it is difficult to determine the statistics behind this trend. It should be mentioned that the reason for such action is not limited to inheritance, as the decrease in stumpage price (Figure 6) and decrease in log price have led to decreases in income from timber sales. Table 5 shows statistical indexes of forest management. Net income (subtracting column (B) from column (A)) has recently decreased. The percentage of tax in

Source: Ministry of Agriculture, Forestry and Fisheries, Statistics of Forest Management.19 Note: In this case, tax refers to taxes, public dues, various burden charges, etc.

Table 5. Index of forest management.

Bank of Japan, the index was 19.1 and 99.7 in 1960 and 2011, respectively. In other words, between 1960 and 2011, the Corporate Goods Price Index increased twofold and the Consumer Price Index increased fivefold. Under such long-term increases in the price index, the nominal

Figure 6. Stumpage price. Source: Forestry Agency, Annual Statistics of Forestry, Annual Statistics of Forest and Forestry;

Figure 5. Number of ancestors and value of standing trees. Source: National Tax Agency Annual Statistics Report.

The large decrease in stumpage price has affected the value of forests during evaluations for inheritance tax. Since the area of inherited forest is not included in the National Tax Agency Annual Statistics Report, further statistical analysis is difficult, and analyses related to on-site

Given the large decreases in stumpage price, it is possible that the stumpage price will decrease beyond the break-even point. When forest owners cannot expect any income from cutting after inheritance, the inheritance tax is only a burden to the inheritor. This would create a scenario where there is a probability that some forest owners would cut standing trees to

prices of 2016 were 39.2 and 77.2% of those in 1960, showing a decrease in price.

original source is the Japan Real Estate Institute. Note: As of March 31.

surveys is a topic of future research.

168 Taxes and Taxation Trends

17Sugano and Tani ([14], p. 32) introduced the following management example: a forest owner conducted clear-cutting of 40- to 50-year-old C. japonica and C. obtusa, and then planted low-value broad-leaved trees. In addition to the value of standing trees, the total value of the inherited properties decreased. As a result, the tax rate (see Table 1) decreased. Nagata [15] mentioned a similar problem at the time of the post-war tax reform, in that there was a large imbalance between cases in which the forest owners conducted clear-cutting and cases in which the forest owners maintained the forest without clear-cutting.

18Such situations have been mentioned before 2000 (e.g., [16], p. 2).

19Statistics Bureau, Ministry of Internal Affairs and Communications, e-Stat, http://www.e-stat.go.jp/SG1/estat/List.do? bid=000001047783&cycode=0 [Accessed: April 18, 2017] (in Japanese)

which forest inheritance tax is not included in the statistics based on the total cost is almost 6–8%; however, the amount of tax currently yields a net income. This management situation is the foundation of the current abandonment of forest management.


*3.4. Trajectory Optimized*

In the stage of the node expansion, the discretized steering angles would select the steering angle from a set of twenty steering angles from −35 degrees to 35 degrees. This would result in the unnatural swerve steering of the articulated steering vehicle by noncontinuous steering actions. Moreover, the Hybrid A-star algorithm aims to produce the shortest path, causing the planned route would be very close to the obstacles. Therefore, the planned courses need to be improved regarding smoothness and safety.

Polynomial equations are often used to describe the trajectory of mobile transportation using the fifth-order and seventh-order polynomials. The polynomial trajectory is expressed by the *n*-order polynomial as follows:

$$\begin{split} p(t) &= p\_0 + p\_1 \times t + p\_2 \times t^2 + \cdots + p\_n \times t^n \\ &= \sum\_{i=0}^n p^i \times t^i \end{split} \tag{11}$$

where *p*0, *p*1, ... , *pn* are trajectory parameters. As the single polynomial cannot describe the complex trajectory, the entire trajectory could be divided into k-segment polynomials, and each segment is allocated with a certain time step as follows:

$$p(t) = \begin{cases} \left[1, t, t^2, \dots, t^n\right] \cdot p\_1 & t\_0 \le t \le t\_1 \\ \left[1, t, t^2, \dots, t^n\right] \cdot p\_2 & t\_1 \le t \le t\_2 \\ \dots & \\ \left[1, t, t^2, \dots, t^n\right] \cdot p\_k & t\_{k-1} \le t \le t\_k \end{cases} \tag{12}$$

where *k* denotes the number of segments, and *pi* = [*pi*0, *pi*1, ... , *pin*] *T* denotes the polynomial coefficients of the *i th* segment. Then, the trajectory optimization process can be transformed into an optimization problem for obtaining the feasible coefficient *p*1, *p*2, ... , *pk* that minimizes the integration of the square of the fourth derivative of position, namely, *snap*. The optimal problem needs to consider the constraints, such as the continuity at the junction of adjacent segments and the limits on the velocity and acceleration. In all, this optimal problem has been formulated as a constrained quadratic problem with equality constraints and inequality constraints in this work as follows:

$$\begin{cases} \min \quad \mathcal{J}(p) = p^T \mathbf{Q} p\\ \text{s.t.} \quad A\_{c\eta} \boldsymbol{q} = \mathbf{b}\_{c\eta} \\ \quad A\_{i\alpha\eta} \boldsymbol{q} \le \mathbf{b}\_{i\alpha\eta} \end{cases} \tag{13}$$

where the matrices *Q*, *Aeq* and *Aineq* are functions of the time allocation *δt* - [*δt*1 , ... , *δtk* ]. The equality constraint limits the states of movement, including the position, velocity, and acceleration within the segments, and ensures the continuity between the segments. Meanwhile, the inequality constraints form a trajectory corridor, which keeps a distance from the obstacles. To assign the appropriate time for each trajectory segment, the Euclidean distance between way-points is used to allocate time proportionally.

#### **4. Control Design**

In this section, we describe the design of the ATV trajectory tracking control scheme. We have divided the control framework into two parts. The first part deals with the longitudinal speed control of ATVs by adjusting the rotating speeds of the tracks. The second one is the steering motion of the ATVs by adjusting the articulation angular rate.

The control system produces feedback control actions by using an adaptive MPC algorithm. This system regulates the states of the tracking deviation system through feedback control. An error model for the tracking system has been proposed in Equation (6). To predict the system state, the MPC systems need future information about the planned trajectory to analyze the evolution of the tracking error over time. The path planning module could provide the control system with the possible disturbance on the ATVs. The path planner determines the disturbance vector *r* by the longitudinal velocity and yaw rate. *u* is a feedback control action aimed at minimizing the tracking error in the presence of disturbance.

#### *4.1. Reference Trajectory*

A path-planning method has been proposed in Section 3 to derive a smooth trajectory for ATVs. The reference states *x*, *y*, *θ*, *ψ* will be used to determine the ATVs' kinematic control. Based on the planned trajectory, we obtain the reference states *υr* and ˙ *θr*. Planner subsystems formulate time-based reference trajectories by evaluating a given reference trajectory (*xt r*, *yt r*, *θr*, *ψr*) and its derivatives. The reference speeds *υr* and the reference yaw rates ˙ *θr* could be calculated as follows

$$\begin{aligned} \upsilon\_r &= \sqrt{(\dot{x}\_r^t)^2 + (\dot{y}\_r^t)^2} \\ \dot{\theta}\_r &= \frac{\dot{y}\_r^t \dot{x}\_r^t - \dot{x}\_r^t \dot{y}\_r^t}{(\dot{x}\_r^t)^2 + (\dot{y}\_r^t)^2} \end{aligned} \tag{14}$$


**Table 2.**

*Physicochemical characterization of wastewater.*

the wastewater, the parameters of TN and TP were also evaluated. Because the wastewater exceeds the maximum permissible limits of contaminants in water, a treatment must be carried out which reduces the degree of contamination, since it has a large contribution of essential nutrients for the growth of organisms, such as nitrogen and phosphorus. These can stimulate the growth of macro and photosynthetic microorganisms in harmful quantities, which would trigger a eutrophic environment in water bodies.

On the other hand, the parameter ζ was measured because it is directly related to the ionic properties of the dissolved solids present in the wastewater.

Solid/aqueous interfaces suffer from a pH-dependent surface charge, that is, they tend to have a positive charge at acidic pH and a negative charge at basic pH. The sign and magnitude of the surface charge govern the adsorption of ionic species in solution and the physical properties of dispersions (e.g., their stability against coagulation). That is why, in the coagulation process, the suspended solids are destabilized by varying the pH of the water and this tends to zero when it approaches the isoelectric point. A simple variation of pH could be enough to stabilize or destabilize the solids that are dissolved in the water [10]. Therefore, the pH value can control both the charge density of the bio-polyelectrolyte and the surface charge of the suspended particles (**Figure 1**).

The surface charge of both coagulants and colloids in water depend on pH and their behavior having a great influence and performance in the coagulationflocculation processes, which is why ζ measurements are required to characterize the colloidal system and understand repulsion and aggregation between colloidal particles [11].

The dosing strategy was determined by measuring the ζ of both the organic matter colloids and chitosan, and also by the optimum pH value to reach the isoelectric point. The purpose of the chitosan dose is to reduce interparticle repulsion by neutralizing negatively charged particles of organic matter.

In general, the electro-neutrality zone for the chitosan-organic matter system is below pH 6 as shown in **Figure 2**. Working at this pH improves the efficiency of the coagulation-flocculation process, since chitosan achieves a higher charge density in an acid medium, therefore its concentration can be decreased.

The best clarification of the wastewater was at pH 5.9, at this pH the positive charges in the chitosan lead to an increase in the hydrodynamic diameter (Dh) due to the repulsion between the intramolecular protonated amino group and this is beneficial in the effect of dosing in turbidity removal efficiency [8]. Furthermore, at this pH, the system reaches the maximum surface contact between chitosan and organic matter in the wastewater [12].

**Figure 1.** *Ionic behavior of different polyelectrolytes at different pH values.*

*Physicochemical Insights of the Organic Matter Particles Dispersed in Wastewaters Induced… DOI: http://dx.doi.org/10.5772/intechopen.99026*

**Figure 2.**

*Effect of chitosan dosage on turbidity removal efficiency. (A) Sanguaza process; (B) flour process. BP: Bio-polyelectrolyte, TSS: Total Suspended Solids.*

The optimal dose of chitosan was 10 ppm, thus obtaining a turbidity removal efficiency of around 97% at this dose, the surface charges of the organic matter particles were neutralized by chitosan molecules resulting in a ζ value close to zero.

Particle size influences many properties of particulate materials and is a valuable indicator of their quality and performance. Larger, spherical particles will flow more easily than small ones, but smaller particles dissolve more quickly and result in higher suspension viscosities than large particles. Smaller particle sizes and a high surface charge (ζ) typically improve the stability of the particles in solution [13–15].

For the measurement of the particle size, the dynamic light scattering technique (DLS) was used, as can be seen in **Figure 3**, as the chitosan dose is increased, the size of the particles in solution increases due to the adsorption of organic matter by the active functional groups of chitosan and in turn the decrease in the ζ. This indicates that the system is unstable and that this in turn, as the dose increases, will reach the isoelectric point, generating electroneutrality in the system. Therefore, the suspended solids by gravity and their own weight will settle.

The performance of the coagulation-flocculation process was evaluated with different parameters such as COD, COT and TSS. In addition, different doses of polymer through profiles of ζ and pH were evaluated (**Figure 4**).

Electro-neutrality occurs at ζ = − 0.3 at pH = 5.7 and required a dose of 100 ppm of chitosan. At this point, the lowest turbidity was observed in the supernatant (4 FAU) compared to the original turbidity (369 FAU) in the wastewater suspension at room temperature. Furthermore, it can be observed that the floc formation is of higher quality when the isoelectric point is reached. For the COD measurements, the potassium permanganate (KMnO4) procedure was chosen, in which a known amount of KMnO4 is added, which is assessed with a primary calcium oxalate standard. The excess KMnO4 reacts with the excess oxalate and finally, the excess oxalate is titrated

with the permanganate and this reaction was carried out at elevated temperature to accelerate the process, obtaining an efficient coagulation-flocculation process, removing a total of 93% of total COD and 100% of TSS in jar tests (**Figure 5**).

**Figure 3.** *Effect of chitosan dosage on particle size and Z potential.*

#### **Figure 4.**

ζ *and evaluation of water quality parameters as a function of chitosan dose in jar tests. (A) Sanguaza process; (B) Flour process. z: zeta potential, BP: Bio-polyelectrolyte, Chemical Oxygen Demand, TSS: Total Suspended Solids.*

**Figure 5.** *Bio-polyelectrolyte contaminant removal efficiencies.*

*Physicochemical Insights of the Organic Matter Particles Dispersed in Wastewaters Induced… DOI: http://dx.doi.org/10.5772/intechopen.99026*

**Table 3.**

*Evaluation of the quality of the wastewater treated by coagulation-flocculation process.*

The main objective of coagulation-flocculation, as it is a primary treatment in wastewater, is to eliminate the suspended solids present in the water as shown in **Table 3**. When carrying out a good dosage of polyelectrolytes, the removal efficiency can be as high as 100%. In the best dose of chitosan, the coagulationflocculation stage leads to the sedimentation of the suspended pollutants, reaching 92% and 76% removal of COD and TOC, respectively. However, the effluent has a high content of TN and TP represents a risk of eutrophication in the environment, which makes it necessary to couple the coagulation-flocculation process to a secondary treatment.