## 3 MPC Strategy
### MPCSSystem  
#### 3.1.1 Predictive Model  
Combining Equations (2)-(7), a model can be obtained, which is:  
\[\left\{ \begin{array}{rl}\ddot{z}_{1}=&1/m_{wf}\left(-K_{tf}z_{1}-K_{sf}(z_{1 }-z_{2})-C_{sf}\big{(}\dot{z}_{1}-\dot{z}_{2}\big{)}-F_{fc}\right)\\ \ddot{z}_{2}=&\left(1/m_{b}+a2/I_{p}\right)(K_{sf}(z_{1}-z_{2})+C_{sf}\big{(} \dot{z}_{1}-\dot{z}_{2}\big{)}+F_{fc}\big{)}+\\ &\left(1/m_{b}-ab/I_{p}\right)(K_{sf}(z_{3}-z_{4})+C_{sf}(z_{3}-\dot{z}_{4})+ F_{rc})+a/I_{p}m_{b}gzh_{g}\\ \ddot{z}_{3}=&1/m_{wr}\left(-K_{tr}z_{3}-K_{sr}(z_{3}-z_{4})-C_{sr}(\dot{z}_{3 }-\dot{z}_{4})-F_{rc}\right)\\ \ddot{z}_{4}=&\left(1/m_{b}-ab/I_{p}\right)(K_{sf}(z_{1}-z_{2})+C_{sf}\big{(} \dot{z}_{1}-\dot{z}_{2}\big{)}+F_{fc}\big{)}+\\ &\left(1/m_{b}+b2/I_{p}\right)(K_{sr}(z_{3}-z_{4})+C_{sr}(\dot{z}_{3}-\dot{z}_ {4})+F_{rc})-b/I_{p}m_{b}gzh_{g}\end{array} \right. \tag{8}\]  
Equation (8) can be written in the form of a differential equation as:  
\[\dot{\mathbf{x}}=\mathbf{A}\mathbf{x}+\mathbf{B}\mathbf{u} \tag{9}\]  
where \(\mathbf{x}=\left[\dot{z}_{1},\dot{z}_{2},\dot{z}_{3},\dot{z}_{4},z_{1},z_{2},z_{3 },z_{4}\right]^{\rm T}\) are the state quantities; \(\mathbf{u}=\left[z,f_{fc},F_{rc}\right]^{\rm T}\) are control outputs; \(\mathbf{A}\) and \(\mathbf{B}\) are state matrix and control matrix, respectively. See Appendix A for specific elements. The system is a linear time-invariant system. Equation (9) expresses the continuous system and. using the first-order Euler method for discretization, becomes:  
\[\frac{d\mathbf{x}}{dt}\approx\frac{\mathbf{x}(t+1|t)-\mathbf{x}(t|t)}{T_{s}} \tag{10}\]  
where \(T_{s}\) is the sampling time; \(\mathbf{x}(t\,|\,t)\) and \(\mathbf{x}(t+1\,|\,t)\) are the states at time \(t\) and \(t\)+1, respectively. The form of \(t+i\) in the text refers to the increase in the \(i\)-\(th\) time domain from time \(t\). The actual corresponding time is \(t+iT_{s}\), where \(i=1,2,3\), (... ). Figure 2: Basic block diagram of the model predictive control (MPC) strategy. Combining Equations (9) and (10), a discrete system can be obtained, that is:  
\[\left\{ \begin{array}{l}\mathbf{x}(t+1|t)=\mathbf{A}_{a}\mathbf{x}(t|t)+\mathbf{B}_{b}\mathbf{u}(t|t) \\ \mathbf{A}_{a}=\mathbf{I}_{8\ast 8}+\mathbf{A}T_{s}\\ \mathbf{B}_{b}=\mathbf{B}T_{s}\end{array} \right. \tag{11}\]  
Converting Equation (11) into an equation for controlling the increment and adding the output equation, the state equation of the discrete system is:  
\[\left\{ \begin{array}{l}\mathbf{\xi}(t+1\big{|}t)=\widetilde{\mathbf{A}}_{a}\mathbf{\xi}(t \big{|}t)+\widetilde{\mathbf{B}}_{b}\Delta\mathbf{u}(t\big{|}t)\\ \mathbf{\eta}(t\big{|}t)=\widetilde{\mathbf{C}}_{c}\mathbf{\xi}(t\big{|}t)\end{array} \right. \tag{12}\]  
where \(\mathbf{\xi}(t\!\!\upharpoonright\!t)=[\mathbf{x}(t\!\!\upharpoonright\!t);\mathbf{u}(t-1 \!\!\upharpoonright\!t)]\) are the augmented state quantities of the system; \(\widetilde{A}_{a}=[\mathbf{A}_{a},\mathbf{B}_{b};\mathbf{0}_{3\ast 8},\mathbf{I}_{3\ast 3}]\); \(\widetilde{\mathbf{B}}_{b}=[\mathbf{B}_{b};\mathbf{I}_{3\ast 3}]\); Output matrix \(\widetilde{\mathbf{C}}_{c}=\mathbf{I}_{11\ast 11}\), \(\mathbf{I}\) is the identity matrix; and \(\Delta\mathbf{u}(t|t)=\mathbf{u}(t|t)-\mathbf{u}(t-1\!\upharpoonright\!t)\). Here, all augmented state quantities are output. Let \(Np\) and \(Nc\) be the forecast time domain and control time domain of the system, respectively. The augmented state quantity and system output quantity in the forecast time domain are:  
\[\left\{