## 4 Controller Design  
The poles placement method (PPM) is a well-known approach employed in the feedback control. It offers the flexibility to relocate the dominant poles of the closed loop system to achieve desired performances. PPM is very desirable for feedback control, as the location of the poles corresponds directly to the eigenvalues of the system, which controls the characteristics of the closed loop response [29]. It is known that the poles of symmetric structural systems are complex with real and complex parts. The poles of the proposed smart structure are related to modal damping ratios and natural frequencies by Equation (10)  
\[\left\{\begin{array}{l}\lambda_{1,2}=-\xi_{1}\omega_{1}\pm j\omega_{1}\sqrt{1 -\xi_{1}^{2}}=-\xi_{1}\omega_{1}\pm j\omega_{d1}\\ \lambda_{3,4}=-\xi_{2}\omega_{2}\pm j\omega_{2}\sqrt{1-\xi_{2}^{2}}=-\xi_{2} \omega_{2}\pm j\omega_{d2}\end{array}\right. \tag{10}\]  
Except the pole corresponding to the inertia element of the identified model \(G\)(s), the locations of the poles corresponding to the first two modes are plotted in Figure 10.  
Mathematically, if the dominant complex-conjugate poles lie close to the imaginary axis, the system transient response exhibits excessive oscillations and delays slowly. Therefore, to guarantee  
Figure 10: Locations of poles corresponding to the first two modes.  
Figure 9: Comparison of system responses with the validation signal excitation: (**a**) time responses; and (**b**) frequency responses.  
fast delay response characteristics, it is necessary to place the closed-loop poles in a particular region far away from the imaginary axis. It can be observed from Figure 10 that both the two pairs of poles locate fairly close to the imaginary axis. Thus, both the two pairs of poles need to be relocated to get a better performance of the system [30].  
Compared with the polynomial form, the state space approach is more convenient in designing the controller. As it offers the designer to select \(n\) independent gains for a system, the arbitrarily desired poles is achieved. The state space representation of the system is written as Equation (11)  
\[\dot{\mathbf{x}}(t) =\mathbf{A}\mathbf{x}(t)+\mathbf{B}u(t) \tag{11}\] \[y(t) =\mathbf{C}\mathbf{x}(t)+\mathbf{D}u(t)\]  
where \(y(t)\) is the output vector, in our case is the voltage output of strain sensors, and \(u(t)\) is the input vector, which is the pre-amplified voltage to the piezoelectric actuators. \(\mathbf{x}(t)\), \(\mathbf{A}\), \(\mathbf{B}\), \(\mathbf{C}\) and \(\mathbf{D}\) are the state variable vector, state matrix, control matrix, output matrix and direct transfer matrix, respectively. Those parameters are defined as  
\[\mathbf{x}(t)=\left[\begin{array}{c}x_{1}\\ \vdots\\ x_{i}\\ \vdots\\ x  
To achieve the desired damping characteristics of the target modes, without affecting the natural frequencies of the structure, the best solution is to adjust only the real part of the dominant poles. As it is known, the dynamic response of a second order system has no vibration and achieves a stable state in a short time when the damping ratio is 0.707. Thus, the optimal locations of the desired poles can be attained when the closed loop damping ratio is set to be 0.707. The basic idea of this method is illustrated in Figure 11. The comparisons between the relocated closed loop and open loop poles are listed in Table 7.