## 4 Actuators/Sensors Position Configuration
### Controllability/Observability Criterion for Multimodal Composite Control Improvement
Assume that the first \(m\)-order modal matrix \(z\) of the system is as follows, and at the same time, its modal expression in the form of state space is  
\[z=\left[\dot{y}_{1},\omega_{1}y_{1},\dot{y}_{2},\omega_{2}y_{2},\dots,\dot{y}_ {m},\omega_{m}y_{m}\right]^{T} \tag{17}\]  
\[\dot{z}_{2m\times 1}=A_{2m\times 2m}z_{2m\times 1}+B_{2m\times r}\mu_{r\times 1} \tag{18}\]  
where  
\[z_{2m\times 1}= \begin{cases}y\\ y\end{cases}  \tag{19}\]  
Based on the conversion relationship between the electrical signal and the force signal of the piezoelectric effect of the material, combined with Equation (18), the state space expression of the flexible plate can be obtained. \[ \begin{cases}\dot{z}=Az+B\mu\\ v=Cz\end{cases}  \tag{20}\]In the formula, \(A\) corresponds to the inherent characteristics of the system itself such as modal frequency and damping ratio, \(B\) corresponds to the form and position of the actuator, and \(C\) corresponds to the form and position of the sensor. As shown in the following Equation (21), A is represented as a DIAG diagonal matrix \(diag(A_{i})\), \(b_{i}\) represents the matrix \(\varphi^{T}Q\) in the \(i\)-th row vector, and its length \(n\) corresponds to the number of actuators. \(c_{ij}\) refers to the characteristic constant of the \(j\)-th sensor of the \(i\)-th order modal displacement of the flexible plate. \[ \begin{split} A_{i}&=\begin{bmatrix}-2\xi_{i}^{x} \omega_{i}&-\omega_{i}\\ \omega_{i}&0\end{bmatrix},B=(b_{1},0,b_{2},0,\ldots,b_{m},0)^{T},\\ C&=\begin{bmatrix}0&c_{11}&0&c_{12}&\ldots&0&c_{1n}\\ 0&c_{21}&0&c_{22}&\ldots&0&c_{2n}\\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ 0&c_{m1}&0&c_{m2}&\ldots&0&c_{mn}\end{bmatrix}\end{split}  \tag{21}\]  
In the instantaneous excitation condition, the vibration response of each sensor position at time \(T_{0}\) can be expressed as  
\[J=\int_{0}^{T_{0}}v^{T}(t)v(t)dt \tag{22}\]  
Referring to the [31], the maximum energy in the time domain can be written as  
\[J=\left(e^{At_{f}}z_{0+}-z_{t_{f}}\right)^{T}W_{o}\left(t_{f}\right)^{-1}\left( e^{At_{f}}z_{0+}-z_{t_{f}}\right) \tag{23}\]  
Term \(W_{o}\) is the observability Gramian matrix, and when the time series is positive infinity, \(W_{o}\) results  
\[W_{o}=\int_{0}^{\infty}e^{A^{T}\tau}C^{T}Ce^{A\tau}d\tau \tag{24}\]  
Minimizing the \(W_{o}\) norm yields the sensor position that maximizes the vibration response \(J\) and \(W_{o}\) satisfies the Lyapunov equation. \[AW_{o}+W_{o}A^{T}=-M^{T}M \tag{25}\]  
Solve then can get  
\[W_{o}=diag\left(\frac{c_{n}}{4\xi_{i}\omega_{i}},\frac{c_{n}}{4\xi_{i}\omega_{ i}}\right) \tag{26}\]  
The reasoning analysis process based on the observability Gramian matrix shows that the matrix describes the vibration response of each vibration modal frequency point at the sensor configuration position. For systems or structures with multiorder dispersed modal frequencies, the sensor position is different, and the matrix data change. When the diagonal elements of the matrix are the largest, the sensor combination can obtain the largest system response. The position selection of the actuator is similarly to that of the sensor. It tries to maximize the input energy of the system, and derives the controllability of the system in turn, the Gramian matrix and its solution in the form of a diagonal matrix. \[W_{c}=\int_{0}^{\infty}e^{A^{T}\tau}BB^{T}e^{A\tau}d\tau \tag{27}\]  
\[W_{c}=diag\left(\frac{\beta_{n}}{4\xi_{i}\omega_{i}},\frac{\beta_{n}}{4\xi_{i }\omega_{i}}\right) \tag{28}\]  
\(\xi_{i}\) represented as the damping rate of the \(i\)-th mode, \(\omega_{i}\) represents the \(i\)-th mode frequency, \(c_{n}\) is the vibration response of the \(n\)-th sensor, and \(\beta_{n}\) is the vibration response of the \(n\)-th actuator. In the same way, for systems or structures oriented to multiorder dispersed modal frequencies, the response of the system represented by the matrix is different depending on the position of the actuator.