To improve vehicle rollover stability, as shown in Figure 1, the motion planning strategy of the intelligent heavy truck includes the rollover index, the obstacle artificial potential field, the MPC motion planner, and the path tracking with multiple dynamic constraints.
4.1 Rollover Index of the Intelligent Heavy Truck
As shown in Eq. (17), the most commonly used vehicle rollover index LTR can be expressed as
$$LTR = \frac{{F_{{\text{L}}}  F_{{\text{R}}} }}{{F_{{\text{L}}} + F_{{\text{R}}} }},$$
(17)
where, F_{L} and F_{R} are the left wheel vertical load and the right wheel vertical load. It is difficult to measure the wheel vertical loads in real time, especially in some emergency cases [31]. Therefore, a novel rollover index should be derived from the model of the intelligent heavy truck.
As shown in Figure 2, the difference between the front left vertical load F_{L1} and the front right vertical load F_{R1}, and the difference between the rear left vertical load F_{L2} and the rear right vertical load F_{R2} can be expressed as follows.
$$\left\{ \begin{gathered} \left( {F_{{{\text{L1}}}}  F_{{{\text{R1}}}} } \right)T_{wf} /2 = k_{{\text{f}}} \left( {\varphi_{{{\text{sf}}}}  \varphi_{{{\text{uf}}}} } \right) + l_{{\text{f}}} \left( {\dot{\varphi }_{{{\text{sf}}}}  \dot{\varphi }_{{{\text{uf}}}} } \right), \hfill \\ \left( {F_{{{\text{L2}}}}  F_{{{\text{R2}}}} } \right)T_{wr} /2 = k_{{\text{r}}} \left( {\varphi_{{{\text{sr}}}}  \varphi_{{{\text{ur}}}} } \right) + l_{{\text{r}}} \left( {\dot{\varphi }_{{{\text{sr}}}}  \dot{\varphi }_{{{\text{ur}}}} } \right), \hfill \\ \end{gathered} \right.$$
(18)
where, T_{wf} is the front track width, and T_{wr} is the rear track width.
In addition, the sum of the left vertical load and the right vertical load is equal to the load weight. So,
$$\left\{ \begin{gathered} F_{{{\text{L1}}}} + F_{{{\text{R1}}}} = m_{{\text{f}}} g, \hfill \\ F_{{{\text{L2}}}} + F_{{{\text{R2}}}} = m_{{\text{r}}} g, \hfill \\ \end{gathered} \right.$$
(19)
where, m_{f} and m_{r} are the front axle mass and the rear axle mass.
Substituting Eqs. (4) and (5) into Eq. (18), the difference between the left vertical load and the right vertical load can be rewritten as follows.
$$\left\{ \begin{gathered} F_{{{\text{L1}}}}  F_{{{\text{R1}}}} = \frac{  2}{{T_{{{\text{wf}}}} }}\left[ \begin{gathered} I_{{{\text{Xf}}}} \ddot{\varphi }_{{{\text{sf}}}}  m_{{{\text{sf}}}} gh_{{\text{f}}} \varphi_{{{\text{sf}}}} \hfill \\  m_{{{\text{sf}}}} h_{{\text{f}}} a_{{\text{y}}} + k_{{\text{b}}} \left( {\varphi_{{{\text{sf}}}}  \varphi_{{{\text{sr}}}} } \right) \hfill \\ \end{gathered} \right], \hfill \\ F_{{{\text{L2}}}}  F_{{{\text{R2}}}} = \frac{  2}{{T_{{{\text{wr}}}} }}\left[ \begin{gathered} I_{{{\text{Xr}}}} \ddot{\varphi }_{{{\text{sr}}}}  m_{{{\text{sr}}}} gh_{{\text{r}}} \varphi_{{{\text{sr}}}} \hfill \\  m_{{{\text{sr}}}} h_{{\text{r}}} a_{{\text{y}}} + k_{b} \left( {\varphi_{sr}  \varphi_{{{\text{sf}}}} } \right) \hfill \\ \end{gathered} \right]. \hfill \\ \end{gathered} \right.$$
(20)
From Eqs. (17), (19), and (20), the front rollover index and the rear rollover index can be obtained,
$$\left\{ \begin{gathered} RI_{{\text{f}}} = \frac{{  \frac{2}{{T_{{{\text{wf}}}} }}\left[ \begin{gathered} I_{{{\text{Xf}}}} \ddot{\varphi }_{{{\text{sf}}}}  m_{{{\text{sf}}}} gh_{{\text{f}}} \varphi_{{{\text{sf}}}} \hfill \\  m_{{{\text{sf}}}} h_{{\text{f}}} a_{{\text{y}}} + k_{{\text{b}}} \left( {\varphi_{{{\text{sf}}}}  \varphi_{{{\text{sr}}}} } \right) \hfill \\ \end{gathered} \right]}}{{m_{{\text{f}}} g}}, \hfill \\ RI_{{\text{r}}} = \frac{{  \frac{2}{{T_{{{\text{wr}}}} }}\left[ \begin{gathered} I_{{{\text{Xr}}}} \ddot{\varphi }_{{{\text{sr}}}}  m_{{{\text{sr}}}} gh_{{\text{r}}} \varphi_{{{\text{sr}}}} \hfill \\  m_{{{\text{sr}}}} h_{{\text{r}}} a_{{\text{y}}} + k_{b} \left( {\varphi_{sr}  \varphi_{{{\text{sf}}}} } \right) \hfill \\ \end{gathered} \right]}}{{m_{{\text{r}}} g}}. \hfill \\ \end{gathered} \right.$$
(21)
To evaluate rollover risk of the intelligent heavy truck, a novel rollover index (NRI) is defined from Eq. (21).
$$NRI = \left\{ {\begin{array}{*{20}l} {  1,\quad \left( {aRI_{{\text{f}}} + bRI_{{\text{r}}} } \right) \le  L,} \hfill \\ {\frac{{aRI_{{\text{f}}} + bRI_{{\text{r}}} }}{L},  L < \left( {aRI_{{\text{f}}} + bRI_{{\text{r}}} } \right) < L,} \hfill \\ {1,\quad \;L \le \left( {aRI_{{\text{f}}} + bRI_{{\text{r}}} } \right),} \hfill \\ \end{array} } \right.$$
(22)
where, L is the wheelbase.
Compared with the problem that the LTR is difficult to measure in realtime, the state quantity in NRI can be obtained through the sensor to ensure realtime performance.
To verify the accuracy and the feasibility of the NRI in evaluating the rollover of the heavy truck, three cases and a “2A truck” model in TruckSim are used.
Fishhook is one of the worst untripped rollover maneuver. In this case, the amplitude of steering angle is set to five degrees. The results of the vehicle NRI and the vehicle LTR at different running speeds are illustrated in Figure 3.
Figure 3 shows that the curves of LTR and NRI are highly consistent with each other. The peak value of the NRI is − 0.7857, and that of the LTR is − 0.7559, as shown in Figure 3a. Also, the error between the NRI and the LTR is less than 4%. In addition, when the vehicle rolls over, as shown in Figure 3b, NRI and LTR reach − 1 at the same time.
4.2 Obstacle’s Artificial Potential Field
The basic idea of the artificial potential field (APF) is to construct a field including the repulsion and the attractor poles. The exponential and the trigonometric functions are applied in this study, to define an APF that can accurately and comprehensively reflect the integration of vehicle and road in dangerous driving state according to the information of road structure parameters, traffic running conditions and the positions of the obstacles. The APF includes road markers or boundaries (P_{R}), crossable area (P_{C}), and noncrossable area (P_{NC}), as shown in Eq. (23).
$$APF{ = }\sum\limits_{q} {P_{{R_{q} }} } + \sum\limits_{j} {P_{{C_{j} }} } + \sum\limits_{i} {P_{{NC_{i} }} ,}$$
(23)
where q, j, and i denote the indices of the road markers or boundaries, the crossable area, and the noncrossable area, respectively.

(1)
Noncrossable area means the obstacles that the vehicle must avoid, such as vehicles and pedestrians. In this paper, the potential field of the noncrossable area is defined as Ref. [32]:
$$\left\{ \begin{gathered} P_{{NC_{i} }} (X,Y) = \frac{{a_{i} }}{{s_{i} (\frac{{\rm {d}X}}{{X_{{{\text{si}}}} }},\frac{{\rm {d}Y}}{{Y_{{{\text{si}}}} }})^{{b_{i} }} }}, \hfill \\ X_{{{\text{si}}}} = X_{0} + uT_{0} + \frac{{\Delta u_{{a_{i} }}^{2} }}{{2a_{{\text{n}}} }}, \hfill \\ Y_{{{\text{si}}}} = Y_{0} + (u\sin \theta + u_{oi} \sin \theta )T_{0} + \frac{{\Delta v_{{a_{i} }}^{2} }}{{2a_{{\text{n}}} }}, \hfill \\ \end{gathered} \right.$$
(24)
where a_{i} and b_{i} are the parameters of the P_{NC}, a_{n} denote the vehicle comfortable acceleration, X_{0} is the minimum longitudinal distance, and Y_{0} is the minimum lateral distance, respectively, dX denotes the longitudinal distance from the ego vehicle to the obstacle, and dY denotes the lateral distance from the ego vehicle to the obstacle, T_{0} represents the safe time interval, θ is the heading angle of the ego vehicle toward the obstacle, X_{si} denotes the safe longitudinal distance, and Y_{si} denotes the safe lateral distance, u_{oi} represents the movement obstacle speed, Δu_{ai} is the longitudinal relative speed between the ego vehicle and the obstacle, and Δv_{ai} is the lateral relative velocity between the ego vehicle and the obstacle.

(2)
Crossable area is less dangerous aera. The potential field can be achieved by the safe distance s_{j} between the obstacle and the ego vehicle.
$$P_{{C_{j} }} (X,Y) = a_{j} \exp \left( {  b_{j} s_{j} \left( {\frac{{\rm{d}X}}{{X_{{{\text{si}}}} }},\frac{{\rm{d}Y}}{{Y_{{\text{si}}} }}} \right)} \right),$$
(25)
where a_{j} and b_{j} represent the parameters of P_{C}.

(3)
Road markers or boundaries are also not allowed to be crossed by vehicle, except that vehicle actively change lanes. The quadratic form function is used to obtain the potential field of the road markers or boundaries as follows.
$$P_{{R_{q} }} (X,Y) = \left\{ \begin{gathered} a_{q} \left( {s_{Rq} (X,Y)  D_{a} } \right)^{2} \begin{array}{*{20}c} {} & {s_{Rq} (X,Y) < D_{a} ,} \\ \end{array} \hfill \\ \begin{array}{*{20}c} 0 & {} & {} & {\begin{array}{*{20}c} {\begin{array}{*{20}c} {} & {} & {} \\ \end{array} } & {} & {s_{Rq} (X,Y) \ge D_{a} ,} \\ \end{array} } \\ \end{array} \hfill \\ \end{gathered} \right.$$
(26)
where a_{q} denotes the intensity parameter of P_{R}, s_{Rq} and D_{a} represent the safe lateral distance and the permitted lateral distance from the ego vehicle to the road boundary, respectively.
An example of the potential field of the noncrossable area is shown in Figure 4a, and the compound potential field of P_{Cj} and P_{Rq} of the twolane road is shown in Figure 4b.
4.3 MPC Motion Planner of the Intelligent Heavy Truck
MPC algorithm is used to realize the motion planning and trajectory tracking for the intelligent heavy truck. The cost function includes the trajectory tracking, the vehicle NRI, and the obstacle’s APF. This will enable the intelligent heavy truck to follow the road regulations and track the global path, prevent rollover, and avoid the obstacles. In addition, the dynamics of the intelligent heavy truck in the future can be predicted from the vehicle rollover model. Finally, the control outputs of F_{xT} and δ_{f} can be obtained by rolling the optimization the cost function using the nonlinear optimal solver.
In order to simplify the calculation of MPC motion planner, it is necessary to linearize the rollover model of the intelligent heavy truck.
The state space model of the intelligent heavy truck can be achieved by sorting out the motion differential Eqs. (1)–(16).
$$\dot{\user2{x}} = {\varvec{f}}({\varvec{x}},{\varvec{u}}_{{\varvec{c}}} ),$$
(27)
where,
$${\varvec{x}} = \left[ {\begin{array}{*{20}c} u & v & r & X & Y & \psi & {\dot{\varphi }_{sf} } & {\dot{\varphi }_{sr} } & {\varphi_{sf} } & {\varphi_{sr} } & {\varphi_{uf} } & {\varphi_{ur} } \\ \end{array} } \right]^{\rm{T}} ,\;{\varvec{u}}_{{\varvec{c}}} = \left[ {\begin{array}{*{20}c} {F_{xT} } & {\delta_{f} } \\ \end{array} } \right]^{\rm{T}} ,\;{\varvec{y}} = \left[ {\begin{array}{*{20}c} u & v & r & Y & {\dot{\varphi }_{sf} } & {\dot{\varphi }_{sr} } & {\varphi_{sf} } & {\varphi_{sr} } \\ \end{array} } \right]^{\rm{T}} .$$
After linearizing Eq. (27), the linear state space model of the intelligent heavy truck can be obtained as follows.
$$\left\{ \begin{gathered} \dot{\user2{x}} = {\varvec{Ax}} + {\varvec{Bu}}_{{\varvec{c}}} , \hfill \\ {\varvec{y}} = \user2{Cx,} \hfill \\ {\varvec{A}} = \frac{{\partial {\varvec{f}}}}{{\partial {\varvec{x}}}}, \hfill \\ {\varvec{B}} = \frac{{\partial {\varvec{f}}}}{{\partial {\varvec{u}}_{{\varvec{c}}} }}, \hfill \\ \end{gathered} \right.$$
(28)
where, the expressions of matrices A and B are very complex and difficult to obtain, so they are calculated by numerical method.
$${\mathbf{C}} = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ \end{array} } \right].$$
According to the first order differential quotient method, the discrete statespace equation can be described as follows:
$$\left\{ \begin{gathered} {\varvec{x}}(k + 1) = {\varvec{A}}_{k} {\varvec{x}}(k) + {\varvec{B}}_{k} {\varvec{u}}_{{\varvec{c}}} (k), \hfill \\ {\varvec{y}}(k) = {\varvec{C}}_{k} {\varvec{x}}(k), \hfill \\ \end{gathered} \right.$$
(29)
where, \(A_{k} = E + t_{s} A,B_{k} = t_{s} B,C_{k} = t_{s} C\), E is the unit matrix with same matrix order as A, and t_{s} =0.001 is discrete sampling time.
To design model predictive controller, setting \({\varvec{\chi}}(k) = \left[ {\begin{array}{*{20}c} {{\varvec{x}}(k)} & {{\varvec{u}}_{{\varvec{c}}} (k)} \\ \end{array} } \right]^{\rm{T}}\), the statespace equation of prediction model can be transformed into
$$\left\{\begin{gathered} {\varvec{\chi}}(k + 1) = \hat{{A}}_{k} {\varvec{\chi}}(k) + \hat{{B}}_{k} \Delta {\varvec{u}}_{{\varvec{ck},}} \hfill \\ {\varvec{y}}(k) = \hat{\user2{C}}_{k} {\varvec{\chi}}(k), \hfill\\ \end{gathered} \right.$$
(30)
where,
$$\hat{\user2{A}}_{k} = \left[ {\begin{array}{*{20}c} {{\varvec{A}}_{k} } & {{\varvec{B}}_{k} } \\ {0_{1 \times 12} } & {1_{1 \times 2} } \\ \end{array} } \right],\hat{\user2{B}}_{k} = \left[ {\begin{array}{*{20}c} {{\varvec{B}}_{k} } \\ {1_{1 \times 2} } \\ \end{array} } \right],\hat{\user2{C}}_{k} = \left[ {\begin{array}{*{20}c} {{\varvec{C}}_{k} } & 0 \\ \end{array} } \right].$$
Therefore, the output prediction equation is
$${\varvec{Y}}_{{\varvec{c}}} \left( k \right) = {\varvec{\psi}}\left( k \right){\varvec{\chi}}\left( k \right) + {{\varvec{\Theta}}}\left( k \right){{\varvec{\Delta}}}{\varvec{U}}_{{\varvec{c}}} \left( k \right)$$
(31)
where,
$$\begin{gathered} \hfill \\ {{\varvec{\Theta}}}\rm{(}k\rm{) = }\left[ {\begin{array}{*{20}c} {\hat{\user2{C}}_{k} \hat{\user2{B}}_{k} } & 0 & 0 & 0 \\ {\hat{\user2{C}}_{k} \hat{\user2{A}}_{k} \hat{\user2{B}}_{k} } & {\hat{\user2{C}}_{k} \hat{\user2{B}}_{k} } & 0 & 0 \\ \cdots & \cdots & \ddots & \cdots \\ {\hat{\user2{C}}_{k} \hat{\user2{A}}_{k}^{{C_{h}  1}} \hat{\user2{B}}_{k} } & {\hat{\user2{C}}_{k} \hat{\user2{A}}_{k}^{{C_{h}  2}} \hat{\user2{B}}_{k} } & \cdots & {\hat{\user2{C}}_{k} \hat{\user2{B}}_{k} } \\ {\hat{\user2{C}}_{k} \hat{\user2{A}}_{k}^{{C_{h} }} \hat{\user2{B}}_{k} } & {\hat{\user2{C}}_{k} \hat{\user2{A}}_{k}^{{C_{h}  1}} \hat{\user2{B}}_{k} } & \cdots & {\hat{\user2{C}}_{k} \hat{\user2{A}}_{k} \hat{\user2{B}}_{k} } \\ \vdots & \vdots & \ddots & \vdots \\ {\hat{\user2{C}}_{k} \hat{\user2{A}}_{k}^{{P_{h}  1}} \hat{\user2{B}}_{k} } & {\hat{\user2{C}}_{k} \hat{\user2{A}}_{k}^{{P_{h}  2}} \hat{\user2{B}}_{k} } & \cdots & {\hat{\user2{C}}_{k} \hat{\user2{A}}_{k}^{{P_{h}  C_{h}  1}} \hat{\user2{B}}_{k} } \\ \end{array} } \right], \hfill \\ \end{gathered}$$
$${{\varvec{\Delta}}}{\varvec{U}}_{{\varvec{c}}} (k) = \left[ {\begin{array}{*{20}c} {{{\varvec{\Delta}}}{\varvec{u}}_{{\varvec{c}}} (k)} & {{{\varvec{\Delta}}}{\varvec{u}}_{{\varvec{c}}} (k + 1)} & \cdots & {{{\varvec{\Delta}}}{\varvec{u}}_{{\varvec{c}}} (k + C_{h} )} \\ \end{array} } \right]^{\rm{T}} ,$$
$${\varvec{Y}}_{{\varvec{c}}} (k) = \left[ {\begin{array}{*{20}c} {{\varvec{y}}(k + 1)} \\ {{\varvec{y}}(k + 2)} \\ \cdots \\ {{\varvec{y}}(k + C_{h} )} \\ \cdots \\ {{\varvec{y}}(k + P_{h} )} \\ \end{array} } \right],\begin{array}{*{20}c} {} & {} \\ \end{array} {\varvec{\psi}}(k) = \left[ {\begin{array}{*{20}c} {\hat{\user2{C}}_{k} \hat{\user2{A}}_{k} } \\ {\hat{\user2{C}}_{k} \hat{\user2{A}}^{2}_{k} } \\ \cdots \\ {\hat{\user2{C}}_{k} \hat{\user2{A}}^{{C_{h} }}_{k} } \\ \cdots \\ {\hat{\user2{C}}_{k} \hat{\user2{A}}^{{P_{h} }}_{k} } \\ \end{array} } \right].$$
It is very necessary to constrain the vehicle kinematics and dynamics in the MPC motion planning of the intelligent heavy truck. Firstly, the control outputs F_{xT} and δ_{f} need to meet the following vehicle dynamics constraints.
$$\left\{ \begin{gathered} F_{{{\text{xT}}}} \le F_{{{\text{xT\_max}}}} = \frac{{T_{\max } }}{{R_{{\text{w}}} }}, \hfill \\ \left {\delta_{{\text{f}}} } \right \le \delta_{{{\text{fmax}}}} , \hfill \\ \left {\Delta \delta_{{\text{f}}} } \right \le\Delta \delta_{{{\text{fmax}}}} , \hfill \\ \end{gathered} \right.$$
(32)
where, T_{max} is the maximum driving torque, F_{xT_max} is the maximum longitudinal force, R_{w} denotes the wheel radius, δ_{fmax} denotes the maximum front wheel steering angle, and Δδ_{fmax} is the maximum front wheel steering angle increment.
In addition, the tire longitudinal force and the tire lateral force shall meet the friction ellipse constraint [33]. For brevity, the details of the specific derivation of the friction ellipse will not be expanded here.
$$\left\{ \begin{gathered} \left( {\frac{{F_{{{\text{xT}}}} }}{{F_{{{\text{xT\_max}}}} }}} \right)^{2} + \left( {\frac{{F_{{{\text{Y1}}}} }}{{F_{{{\text{Y1\_max}}}} }}} \right)^{2} \le \mu^{2} , \hfill \\ \left( {\frac{{F_{{{\text{xT}}}} }}{{F_{{{\text{xT\_max}}}} }}} \right)^{2} + \left( {\frac{{F_{{{\text{Y2}}}} }}{{F_{{{\text{Y2\_max}}}} }}} \right)^{2} \le \mu^{2} , \hfill \\ \end{gathered} \right.$$
(33)
where, μ represents the road adhesion coefficient, F_{Y1_max} and F_{Y2_max} are the maximum lateral forces of the front tire and the rear tire, respectively.
Considering the vehicle dynamic constraints, the cost function of the MPC motion planner can be set up as follows.
$$\begin{gathered} J{ = }\sum\limits_{k = 1}^{{P_{h} }} {\left( \begin{gathered} \left\ {Y(t + k)  Y_{{{\text{ref}}}} (t + k)} \right\_{{{\varvec{Q}}_{{1}} }}^{2} + \left\ {APF(t + k)} \right\_{{{\varvec{Q}}_{{3}} }}^{2} \hfill \\ { + }\left\ {u(t + k)  u_{{{\text{ref}}}} (t + k)} \right\_{{{\varvec{Q}}_{{2}} }}^{2} \hfill \\ { + }\left\ {NRI(t + k)  NRI_{{{\text{ref}}}} (t + k)} \right\_{{{\varvec{Q}}_{{4}} }}^{2} \hfill \\ \end{gathered} \right)} \hfill \\ \begin{array}{*{20}c} {} & {} \\ \end{array} + \sum\limits_{k = 1}^{{C_{h}  1}} {\left\ {\Delta {\varvec{u}}_{{\varvec{c}}} (t + k)} \right\_{{\varvec{R}}}^{2} + \rho \varepsilon^{2} ,} \hfill \\ \end{gathered}$$
(34)
where, P_{h} and C_{h} are the prediction and the control horizons, respectively, Q_{1}, Q_{2}, Q_{3} and Q_{4} are the output state coefficient matrices, ρ denotes the weight coefficient, ε denotes the relaxation factor, R denotes the control weight coefficient matrix.
The designed lateral displacement Y_{ref} and the designed longitudinal speed u_{ref} can be obtained by global planning as shown in Figure 1. The designed NRI_{ref} is setting to 0.
In combination with the vehicle constraints shown in Eqs. (32) and (33), a series of control increments in the control time domain can be obtained by rolling the optimization the cost function using the nonlinear optimal solver Fmincon.
$${{\varvec{\Delta}}}{\varvec{U}}_{c}^{*} (k) = \left[ {\begin{array}{*{20}c} {{{\varvec{\Delta}}}{\varvec{u}}_{{\varvec{c}}}^{\user2{*}} (k)} & {{{\varvec{\Delta}}}{\varvec{u}}_{{\varvec{c}}}^{\user2{*}} (k + 1)} & \cdots & {{{\varvec{\Delta}}}{\varvec{u}}_{{\varvec{c}}}^{\user2{*}} (k + C_{h} )} \\ \end{array} } \right]^{\rm{T}} .$$
(35)
After obtaining the sequence of the control increments, the actual control input increment of the system can be calculated by the first element of the control sequence.
$${\varvec{u}}_{{\varvec{c}}} (k) = {\varvec{u}}_{{\varvec{c}}} (k  1) + \Delta {\varvec{u}}_{{\varvec{c}}}^{\user2{*}} (k).$$
(36)
As the same way, the remaining input control increment of the system can be generated in turn by predicting the next cycle output according to the state information.