## 2 Robustness Analysis Theory for Flexible Cable-Bar Tensile Structures  
### Fundamental Robustness Analysis Method Based on \(H_{\infty}\) Theory  
Presently, the definition of structural robustness is the capacity of a structure to avoid the continuous collapses resulted from unusual causes, and an evaluation indicator is needed to express the ratio of the damage consequences to the causes. Meanwhile, \(H_{\infty}\) robust control theory is a well-developed theory in the control area and the \(H_{\infty}\) norm is an important performance indicator in the robust control system. The state space model is usually adopted when describing the control system. From the perspective of input and output, the system is regarded as a mapping from input space to output space. The dynamic equation of the system is usually expressed in the form of state equation and output equation when using the state space model to describe a structural system. For a linear time-invariant system \(\sum_{0}\) with \(r\) dimensional input and \(m\) dimensional output, it can be expressed as:  
\[\sum_{0}:\left\{ \begin{array}{c}\dot{x}(t)=A_{0}x(t)+\ B_{0}u(t)\\ y(t)=\mathcal{C}_{0}x(t)+D_{0}u(t)\end{array} \right. \tag{1}\]  
In this formula, \(x(t)=\left[x_{1}(t)x_{2}(t)\cdots x_{n}(t)\right]^{\mathsf{T}}\) denotes the \(n\) dimensional state vector of the system and is a group of vectors with the least number that can fully describe the time-domain behavior of the system. \(u(t)=\left[u_{1}(t)u_{2}(t)\cdots u_{r}(t)\right]^{\mathsf{T}}\) denotes the input vector of the system, and \(y(t)=\left[y_{1}(t)y_{2}(t)\cdots y_{m}(t)\right]^{\mathsf{T}}\) denotes the output vector of the system, which is adopted to depict the external state of the system. The characteristic matrices of system \(A_{0}\), \(B_{0}\), \(C_{0}\), \(D_{0}\) are the matrix of the \(n\times n\) dimensional system matrix, \(n\times r\) dimensional input matrix, \(m\times n\) dimensional output matrix and \(m\times r\) dimensional input-output coupling matrix, respectively. Laplace transformation is carried out on both ends of Equation (1) of the system state space, as follows:  
\[\sum_{0}:\left\{ \begin{array}{c}sX(s)-x(t_{0})=A_{0}X(s)+B_{0}U(s)\\ Y(s)=C_{0}X(s)+D_{0}U(s)\end{array} \right. \tag{2}\]  
In this formula, \(U(s)\) and \(Y(s)\) are the Laplace transform of \(u(t)\) and \(y(t)\) respectively, and \(s\) is the complex frequency. In the formula (2), assuming initial conditions \(x(t_{0})=0\) and \(\dot{x}(t_{0})=0\), it can be obtained that:  
\[Y(s)=\left\{C_{0}(sI-A_{0})^{-1}B_{0}+D_{0}\right\}U(s) \tag{3}\]  
To define the transfer function of the \(i\)th output vector \(y_{i}\) and the \(j\)th input vector \(u_{j}\) in the frequency domain:  
\[G_{0 ij}(s)=\frac{Y_{i}(s)}{U_{j}(s)} \tag{4}\]  
For the linear structure system, it can be obtained by superposition principle that:  
\[Y(s)=G_{0}(s)U(s) \tag{5}\]  
\[G_{0}(s)=C_{0}(sI-A_{0})^{-1}B_{0}+D_{0} \tag{6}\]  
In this formula, \(I\) is the diagonal matrix, \(G_{0}(s)\) is the system transfer matrix composed of \(G_{0 ij}(s)\). \(G_{0}(s)\) describes the mapping relation between input variables \(u(t)\) and output variables \(y(t)\), and is an attribute of the structural system per se and is irrelevant to input variables \(u(t)\). The aforementioned fundamental idea of \(H_{\infty}\) theory is to assume that the structural system is seen as a structural group containing uncertainties. If all objects in the structural group can meet the expected performance indicators through design, the actual structure will also meet the performance requirements. Therefore, the design of the structural robustness can actually be regarded as the process of optimizing the structural group, and the norm of the system transfer function \(H_{\infty}\) is used as the optimization indicator.