## 4 Reconfigurability Judgment Analysis and Autonomous Control Method  
### Alignment Controllability Judgment Analysis  
For the alignment control system, the air pressure variation vector \(\mathbf{\delta}\mathbf{p}\) is regarded as the input vector. The vertical displacement vector \(\mathbf{z}\) is regarded as the linear function of the input vector. The alignment control vector \(\mathbf{y}\) is regarded as the output vector.  
According to the literature [31], input constraints, input association constraints, and output constraints during the alignment control are  
\[\mathbf{\delta}\mathbf{p_{min}}\leq\mathbf{\delta}\mathbf{p} \leq\mathbf{\delta}\mathbf{p_{max}} \tag{19}\] \[\mathbf{z_{min}}\leq\mathbf{z}(k+1)=\mathbf{D}\mathbf{x_{c0}}+ \mathbf{D}\mathbf{G}\mathbf{c}\mathbf{K}^{-1}\mathbf{N}\mathbf{\delta} \mathbf{p}(k)\leq\mathbf{z_{max}}\]  
According to the literature [39], the boundary for each alignment control component and vertical displacement constraint is  
\[\mathbf{y_{max}}=-\mathbf{y_{min}}=[x_{\max},z_{\max},a_{\max}, \beta_{\max},\chi_{\max}]^{T}\geq 0 \tag{20}\] \[\mathbf{z_{max}}=-\mathbf{z_{min}}=[z_{\max},z_{\max},z_{\max},z_ {\max}]^{T}\geq 0\]  
The air pressure adjustment constraint boundary is  
\[\mathbf{\delta}\mathbf{p_{min}}=\mathbf{P_{min}}-\mathbf{p}_{0}= \left[p_{\min},p_{\min},\dots p_{\min}\right]^{T}-\mathbf{p}_{0} \tag{21}\] \[\mathbf{\delta}\mathbf{p_{max}}=\mathbf{P_{max}}-\mathbf{p}_{0}= \left[p_{\max},p_{\max},\dots p_{\max}\right]^{T}-\mathbf{p}_{0}\]where \(\mathbf{p}_{0}\) is the initial air pressure distribution. The input constraint space, input interconnection constraint space, and output constraint space are defined as  
\[P(\delta P)=\left\{\,\mathbf{s}\mathbf{p}=[\delta p_{1},\delta p_{2} \ldots\delta p_{n}]^{T}|\delta\mathbf{p_{min}}\leq\delta\mathbf{p}\leq\delta \mathbf{p_{max}}\,\right\}\] \[PZ(\delta P)=\left\{\,\mathbf{s}\mathbf{p}=[\delta p_{1},\delta p _{2}\ldots\delta p_{n}]^{T}|\mathbf{z_{min}}\leq\mathbf{z}(k+1)=\mathbf{D} \mathbf{c}_{0}+\mathbf{D}\mathbf{c}_{i}\mathbf{K}^{-1}\mathbf{N}\delta \mathbf{p}(k)\leq\mathbf{z_{max}}\,\right\} \tag{22}\] \[P(Y)=\left\{\,\mathbf{y}=[x_{\epsilon},z_{\epsilon},\alpha,\beta,X]^{T}|\mathbf{y_{min}}\leq\mathbf{y}\leq\mathbf{y_{max}}\,\right\}\]  
The input generation space is defined as  
\[PY(\delta P)=\left\{\,\mathbf{s}\mathbf{p}=[\delta p_{1},\delta p_{2}\ldots \delta p_{n}]^{T}\big{|}\mathbf{y_{min}}\leq\mathbf{y}(k+1)=\mathbf{C}\mathbf{ c}_{0}+\mathbf{C}\mathbf{c}_{i}\mathbf{K}^{-1}\mathbf{N}\delta\mathbf{p}(k)\leq \mathbf{y_{max}}\,\right\} \tag{23}\]  
Taking the two-dimensional domain as an example, the controllability analysis of the HPDME-ASVIS is equivalent to the analysis of the intersection between the convex polyhedron \(P(\delta P)\), \(PZ(\delta P)\) and \(PY(\delta P)\), as shown in Figure 2. If the system is controllable, the intersection is a nonempty set.  
It is assumed assume that the air spring pressure variation, the alignment control vector, and the vertical deformation vector of the air spring are \(\mathbf{\wp}_{s}\), \(\mathbf{y}_{s}\), \(\mathbf{z}_{s}\) in steady state condition. Based on linear programming, the alignment controllability can also be defined as: \"if there is a non-empty set of parameters \(\delta\mathbf{p}_{s}\), \(\mathbf{y}_{s}\), and \(\mathbf{z}_{s}\) that makes the Equation (19) valid, the alignment controllability is true, which also means that the alignment system is controllable; otherwise, the alignment controllability is false, which also means that the alignment system is uncontrollable\".