# Decoupling Control of Multi-DOF Supporting System of MLDSB
## 4 Decoupling Control of the Multi-DOF System  
### Design of the Decoupling Device  
The electromagnetic force and hydrostatic force of each supporting system are controlled by the dual input differential method, and the inputs can be combined as shown in Equation (11). \[\mathbf{u}_{\mathrm{z}}=\left[ \begin{array}{cccccc}I_{1}+U_{1}&I_{2}+U_{2}&I_{3}+U _{3}&I_{4}+U_{4}\end{array} \right]^{\mathrm{T}} \tag{11}\]  
where \(I_{i}\) is the input current, \(U_{i}\) is the input voltage. The state variable can be selected as follows. \[\xi_{\mathrm{z}}=\left[ \begin{array}{cccccc}x_{c}&y_{c}&\dot{x}_{c}&\dot{y} _{c}&\theta_{x}&\theta_{y}&\dot{\theta}_{x}&\dot{\theta}_{y}\end{array} \right]^ {\mathrm{T}}=\left[ \begin{array}{cccccc}\xi_{1}&\xi_{2}&\xi_{3}&\xi_{4}&\xi_ {5}&\xi_{6}&\zeta_{7}&\xi_{8}\end{array} \right]^{\mathrm{T}} \tag{12}\]  
The output variable can be selected as follows. \[\mathbf{Y}_{\mathrm{z}}=\left[ \begin{array}{cccccc}Y_{1}&Y_{2}&Y_{3}&Y_{4}\end{array}  \right]^{\mathrm{T}}=\left[ \begin{array}{cccccc}x_{A}&x_{B}&y_{A}&y_{B} \end{array} \right]^{\mathrm{T}} \tag{13}\]  
According to Equations (1)-(10), the state space equation can be obtained as follows. \[\left\{ \begin{array}{l}\dot{\xi}_{z}=\mathbf{A}_{z}\xi_{z}+\mathbf{B}_{z}\mathbf{u}_{z} \\ Y_{z}=\mathbf{C}_{z}\xi_{z}\end{array} \right. \tag{14}\]  
The state feedback decoupling strategy is a decoupling method that combines the state feedback control law with the \(\alpha\) inverse decoupling theory. Namely, the control law is shown as follows. \[\mathbf{u}=\mathbf{H}\mathbf{v}-\mathbf{K}\mathbf{x} \tag{15}\]  
The specific structure is shown in Figure 8.  
The state-space expression after construction is shown as follows. \[\left\{ \begin{array}{l}\dot{\mathbf{x}}=(\mathbf{A}+\mathbf{B}\mathbf{K})\mathbf{x}+\mathbf{B}\mathbf{H }\mathbf{v}\\ \mathbf{Y}=\mathbf{C}\mathbf{x}\end{array} \right. \tag{16}\]  
where \(\mathbf{x}\) is \(n\times 1\) state vector, \(\mathbf{u}\) is \(l\times 1\) input vector, \(\mathbf{y}\) is \(m\times 1\) output vector, \(\mathbf{A}\) is \(n\times n\) state matrix, \(\mathbf{B}\) is \(n\times l\) input matrix, \(\mathbf{C}\) is \(m\times n\) output matrix. In the \(\alpha\)-order inverse decoupling theory, the desired closed-loop matrix \(\mathbf{G}_{\mathrm{L}}\)(s) is set to the diagonal form, so that the output matrix \(\mathbf{Y}\) is the \(\alpha\)-order integral of the input matrix \(\mathbf{X}\) with respect to time as shown in Equations (17) and (18). \[\mathbf{G}_{\mathrm{L}}(s)=\left[ \begin{array}{cccccc}\frac{1}{s^{(\mathbf{a}_{1})} }&\frac{1}{s^{(\mathbf{a}_{2})}}&\dots&\frac{1}{s^{(\mathbf{a}_{m})}}\end{array} \right] ^{\mathrm{T}} \tag{17}\]  
\[\mathbf{X}=\mathbf{Y}^{(\mathbf{a})}=\left[ \begin{array}{cccccc}y_{1}^{(\mathbf{a}_{1})}&y_ {2}^{(\mathbf{a}_{2})}&\dots&y_{m}^{(\mathbf{a}_{m})}\end{array} \right]^{\mathrm{T}} \tag{18}\]  
According to Equation (14), the target of multi-DOF decoupling is to transform \(\mathbf{G}\)(s) into the decoupled diagonal matrix form by selecting the feedback matrix \(\mathbf{K}\) and feedforward matrix \(\mathbf{H}\) based on the \(\alpha\)-order inverse decoupling theory. Figure 8: Block diagram of state feedback decoupling method. In the \(\alpha\)-order inverse decoupling theory, non-singular matrices \(\mathbf{E}\) and \(\mathbf{F}\) are defined to solve \(\mathbf{K}\) and \(\mathbf{H}\), and its expressions are defined as follows. \[\mathbf{E}=\mathbf{CA}^{(\alpha_{i})}\mathbf{B} \tag{19}\]  
\[\mathbf{F}=\mathbf{CA}^{(\alpha_{i}+1)} \tag{20}\]  
where, \(\alpha=(\alpha_{i})\) is the decoupling order constant. The solution idea of \(\alpha\) is as follows. 1. For \(\mathbf{G}\)(s), the output equation can be seen from the state-space equation as follows. \[\mathbf{Y}=\mathbf{CX}\] (21) where \(\mathbf{Y}\) contains \(i\) output components and \(\mathbf{C}\) contains the \(i\) row vectors. 2. Take the derivative of both ends of the above equation with respect to time \(t\), and substitute the state equation into Equation (21).