**4. Active vibration suppression control based on piezoelectric actuator**

### **4.1 Piezoelectric active vibration suppression system**

Passive vibration suppression refers to the introduction of one or more massspring damping systems in the propagation path of the vibration source. Although this technical solution is simple and reliable, it can only effectively attenuate highfrequency vibrations in a wide frequency band. With the rapid development of smart sensors and smart actuators and high-speed microprocessors, active vibration suppression is becoming more and more attractive in vibration suppression. In piezoelectric active control, it is classified according to the control structure, which can be divided into feedforward control and feedback control. In precision vibration isolation, different vibration active control structures need to be adopted for different vibration isolation objects in actual work.

In the piezoelectric active vibration suppression system, the overall control block diagram of feedforward and feedback is shown in **Figure 7**. In **Figure 7**, The vibration suppression closed-loop consists of a table feedback loop with a signal filter function and a ground-based feedforward loop with a signal filter function. The feedback loop is implemented as: The feedback acquisition sensor collects the table vibration, and then filters the excess noise signal, and then transmits it to the feedback controller for algorithm calculation, and adjusts the gain of the entire feedback loop to change the feedback characteristics. The implementation of the feedforward loop is as follows:

*Active Vibration Suppression Based on Piezoelectric Actuator DOI: http://dx.doi.org/10.5772/intechopen.103725*

The feedforward acquisition sensor collects the ground vibration signal, then filters the excess noise signal, and then transmits it to the feed-forward controller for algorithm calculation, and adjusts the feedforward control parameters to change the feedforward characteristics. The piezoelectric feedback active control algorithm can be realized by sky-hook damping, integral force feedback (IFF), and other algorithms, and the feedforward control algorithm can be realized by adaptive control algorithm or phase compensation algorithm. Finally, the hybrid operation of the two active controllers is output to the piezoelectric actuator, and the active vibration suppression system is controlled to complete the active control. Compared with using one of the control structures or algorithms alone, the active hybrid control (AHC) can achieve better vibration suppression performance. The following will introduce a piezoelectric active hybrid control design method, including an IFF control and a recursive least square (RLS) adaptive feedforward control, the feedback control realizes the skyhook damping effect, and the adaptive feedforward control realizes the ground-based advance response.

## **4.2 Piezoelectric IFF control in vibration suppression**

Active feedback control can effectively solve the problem that the signal at the natural frequency is amplified, that is, the problem of formant attenuation. At present, the sky-hook technology is widely used in piezoelectric active feedback control. Generally, the sky-hook effect is established by absolute speed feedback control to reduce the formant peak value while maintaining high-frequency attenuation. This section introduces sky-hook technology based on piezoelectric actuators, which uses a combination of dynamic force sensors and piezoelectric actuators to design an integral force active control algorithm to achieve the control effect of sky-hook.

According to the structure above, the schematic diagram of the piezoelectric IFF control is shown in **Figure 8**. In **Figure 8**, the amount of elongation of the piezoelectric actuator is represented by δ, F represents the dynamic force signal. The active vibration control method of piezoelectric IFF control is: The dynamic force sensor collects the dynamic force signal, and after noise removal and filtering, the integral

**Figure 8.** *Schematic diagram of piezoelectric IFF principle.*

calculation compensation is completed in the active control unit, and the control signal is output to the piezoelectric actuator to complete the piezoelectric active feedback control.

The IFF control law based on sky-hook damping technology is:

$$
\sigma\_{IFF} = \frac{1}{\mathbf{k\_d s}} \cdot \Psi \cdot \mathbf{F} = \frac{1}{\mathbf{k\_d s}} \cdot \mathbf{M\_P} \cdot \Psi \cdot \mathbf{s^2} = \frac{1}{\mathbf{k\_d s}} \cdot \mathbf{k\_i} \cdot \mathbf{s^2} \tag{14}
$$

where ki ¼ MP � Ψ is the integral gain coefficient of the IFF control.

According to **Figure 8**, the motion control equation of the piezoelectric IFF control system can be expressed as:

$$\mathbf{M} \mathbf{s}^2 \mathbf{x}\_0 = -\mathbf{m} \mathbf{s}^2 \mathbf{x}\_i = \mathbf{k}\_d (\mathbf{x}\_i - \mathbf{x}\_m) = \mathbf{F} \tag{15}$$

$$
\delta = \mathbf{x}\_{\bullet} - \mathbf{x}\_{\bullet} \tag{16}
$$

The open-loop transfer function between the elongation *δ* of the piezoelectric actuator and the output *F* of the force sensor can be expressed as:

$$\frac{\text{F}}{\delta} = \text{k}\_{\text{d}} \frac{\text{Mms}^2}{\text{Mms}^2 + \text{k}\_{\text{d}}(\text{M} + \text{m})} \tag{17}$$

According to the integral gain coefficient of the IFF algorithm in Eq. (14), after sorting and calculation, the displacement xm of the middle section can be obtained as:

$$\mathbf{x}\_{\mathbf{m}} = \frac{\mathbf{s}\mathbf{x}\_{\mathbf{o}} + \mathbf{k}\_{\mathbf{i}}\mathbf{x}\_{\mathbf{i}}}{\mathbf{s} + \mathbf{k}\_{\mathbf{i}}} = \frac{\mathbf{s}\mathbf{x}\_{\mathbf{o}} + \mathbf{M}\_{\mathbf{P}} \cdot \Psi \cdot \mathbf{x}\_{\mathbf{i}}}{\mathbf{s} + \mathbf{M}\_{\mathbf{P}} \cdot \Psi} \tag{18}$$

According to the above derivation, the transmissibility curve of the piezoelectric vibration isolation system under the IFF control algorithm is:

$$\mathbf{T\_{C-IFF}(s)} = \frac{\mathbf{cs} + \mathbf{k\_d}}{\mathbf{M\_pS^2} + (\mathbf{c} + \mathbf{k\_i})\mathbf{s} + \mathbf{k\_d}} = \frac{\mathbf{1}}{\mathbf{s^2(1/\alpha\_n^2) + s\left(\Psi/\alpha\_n^2\right) + 1}} \tag{19}$$

$$
\rho\_{\mathbf{n}} = \sqrt{\frac{\mathbf{k\_d}}{\mathbf{M\_p}}} \tag{20}
$$

where *ω*<sup>n</sup> is the natural frequency of the passive vibration isolation system. When the gain factor Ψ ≫ c, the equivalent damping *c* of the system can be ignored.

The natural frequency and damping ratio of the piezoelectric vibration isolation system under the IFF control algorithm are expressed as:

$$
\rho\_{\rm P} = \sqrt{\frac{\mathbf{k}}{\mathbf{M}\_{\rm P}}} = \rho\_{\rm n} \tag{21}
$$

$$\zeta\_{\rm P} = \frac{\mathbf{c} + \mathbf{k\_i}}{2\sqrt{\mathbf{km}}} = \frac{\mathbf{M\_P} \cdot \Psi}{2\sqrt{\mathbf{km}}} = \frac{\Psi}{2} \sqrt{\frac{\mathbf{M\_P}}{\mathbf{k}}} = \frac{\Psi}{2\rho\_{\rm n}}\tag{22}$$

The natural frequency of the vibration isolation system under IFF control is consistent with the natural frequency of the passive system and does not change. The damping ratio of the vibration isolation system under the IFF control is proportional to the integral gain coefficient. By increasing the integral gain coefficient, the formant peak value at the natural frequency can be effectively reduced to achieve the sky-hook effect. It is worth noting that an excessively large gain coefficient will lead to system stability errors, making the vibration isolation system unstable.

According to the theoretical analysis of IFF, the simulation analysis is carried out in Matlab. The passive system parameters are shown in **Table 1**, and the parameters of the subsequent simulation are also consistent with **Table 1**. The simulation results are shown in **Figure 9**. It can be found that with the increase of the integral gain coefficient, the value of the resonance peak of the vibration isolation system decreases continuously, which plays a good role in suppressing vibration. This shows that the piezoelectric IFF control can achieve the effect of sky-hook damping control and can effectively suppress the formant.

#### **4.3 Piezoelectric RLS adaptive feedforward control in vibration suppression**

The most direct way to improve the performance of feedback control is to increase its feedback gain. However, with the increase of the feedback gain, a large steady-state error will be introduced into the system. Therefore, a ground-based feedforward control strategy emerges as the times require. The ground-based feedforward control can effectively improve the local frequency-domain vibration suppression performance of the system by predicting the vibration signal in advance and implementing active control in the active algorithm. In piezoelectric feedforward control, the use of adaptive feedforward control is an extremely effective method. This section introduces an RLS adaptive feedforward control method.

The adaptive controller Fff ¼ F yi ð Þ <sup>k</sup> ,d kð Þ, yoð Þ <sup>k</sup> � � is a finite impulse response filter (FIR), also known as a transversal filter. For the observation signal that changes with time *i*, the tap weight vector w nð Þ of the transversal filter must be time-varying.


**Table 1.**

*Simulation parameters of single-degree-of-freedom passive vibration isolation system*.

**Figure 9.** *Transmittance curve of piezoelectric vibration isolation system under passive control and IFF control.*

**Figure 10.** *Transverse filter with time-varying tap weights.*

To keep the adaptive controller in an optimal state, that is, to keep the gradient function of the cost function close to zero, the variable parameters must converge to the optimal value in real-time. Therefore, the filter used in the RLS algorithm is a transversal filter with time-varying tap weights. **Figure 10** shows the structure diagram of the transversal filter with time-varying tap weights.

The RLS adaptive control algorithm is a transversal filter based on the least-squares criterion. The algorithm recursively deduces the weight vector of the current time according to the filter tap weight vector of the previous time. Assuming that N data *y* (1), *y*(2), … , *y*(*i*), … , *y*(*N*) are known, the data is filtered with an M-order transversal filter with time-varying tap weights to get estimate the desired signals *d*(1), *d*(2), … , *d*(*i*), … , *d*(*N*). Then the estimate of the expected response can be expressed as:

$$\hat{\mathbf{d}}(\mathbf{i}) = \sum\_{\mathbf{j}=\mathbf{0}}^{\mathbf{M}-1} \mathbf{w}\_{\mathbf{j}}(\mathbf{n}) \mathbf{y}(\mathbf{i} - \mathbf{j}) = \mathbf{w}^{\mathrm{T}}(\mathbf{n}) \mathbf{y}(\mathbf{i}) \tag{23}$$

where wjð Þ n is the tap weight of the M-order transversal filter with time-varying tap weights, *w*ð Þ n is the tap weight vector of the filter with time-varying tap weights, and *y*ð Þi is the tap input vector of the filter with time-varying tap weights at the *i-*th time, and are respectively:

$$\mathbf{w(n)} = \begin{bmatrix} \mathbf{w\_0(n)}, \mathbf{w\_1(n)}, \dots \mathbf{w\_{M-1}(n)} \end{bmatrix}^T \tag{24}$$

*Active Vibration Suppression Based on Piezoelectric Actuator DOI: http://dx.doi.org/10.5772/intechopen.103725*

$$\mathbf{y}(\mathbf{i}) = \begin{bmatrix} \mathbf{y}(\mathbf{i}), \mathbf{y}(\mathbf{i-1}), \dots, \mathbf{y}(\mathbf{i-M} + \mathbf{1}) \end{bmatrix}^T \tag{25}$$

Then the estimated error of the filter with time-varying tap weights can be written as:

$$\mathbf{e}(\mathbf{i}) = \mathbf{d}(\mathbf{i}) - \hat{\mathbf{d}}(\mathbf{i}) = \mathbf{d}(\mathbf{i}) - \sum\_{j=0}^{M-1} \mathbf{w}\_{\mathbf{j}}(\mathbf{n}) \mathbf{y}(\mathbf{i} - \mathbf{j}) = \mathbf{d}(\mathbf{i}) - \mathbf{w}^{T}(\mathbf{n}) \mathbf{y}(\mathbf{i}) \tag{26}$$

Then the cost function under the least-squares criterion using the pre-windowing method can be expressed as:

$$\xi(n) = \sum\_{i=1}^{n} \lambda^{n-i} \mathbf{e}^2(i) \tag{27}$$

where *λ* is the forgetting factor, the value range is 0 ≤*λ*≤1.

The RLS adaptive feedforward controller is built-in Matlab for simulation, and the simulation results are shown in **Figure 11**. Under the action of piezoelectric RLS adaptive feedforward control, the effective suppression rate of active control to amplitude can reach 80%, which is obviously better than passive control.

#### **4.4 Piezoelectric active hybrid controller**

After the above description of IFF control and RLS adaptive feedforward control, a design scheme of piezoelectric AHC can be given. The block diagram of the AHC is shown in **Figure 12**. The principle is as follows: Given an additional external excitation signal, the signal *y*(*n*) is measured with a feedforward sensor. Design a transversal filter *C z*�<sup>1</sup> ð Þ with time-varying tap weights, and continuously estimate the expected response *d*(*i*) by fitting the tap weight vector *w*ð Þ n constantly changing. By iteratively deriving the least squares estimated tap weight vector *w*ð Þ n , the square weighted sum of the estimated error *e*(*i*) (that is, the platform vibration error of the load platform) under this system is obtained to be the smallest, thereby, ensuring the platform vibration error of the load platform. The feedforward control loop finally generates

**Figure 11.** *Time domain comparison of RLS adaptive feedforward control and passive control.*

**Figure 12.** *Piezoelectric AHC block diagram.*

the feedforward control signal *uFF*ð Þ n , which is converted by the piezoelectric brake into an actual force acting on the system, which is opposite to the direct interference generated by the feedforward, thereby, eliminating the system vibration caused by the feedforward source. The feedback control loop performs compensation control on the error signal according to the feedback controller.

As shown in **Figure 12**, *G*ð Þs is the structural open-loop transfer function, *P*ð Þs is the structural closed-loop transfer function, *Gc*ð Þ<sup>s</sup> is the feedback controller, *C z*�<sup>1</sup> ð Þ is a transversal filter, *F z*�<sup>1</sup> ð Þ is the transfer function from the feedforward control force to the vibration isolation system, *F z* ^ �<sup>1</sup> ð Þ is the model of *F z*�<sup>1</sup> ð Þ function with M-order identification, *<sup>y</sup>*(*n*) is an additional external stimulus, *<sup>q</sup>* <sup>∗</sup> ð Þ *<sup>n</sup>* is the vibration response of the external excitation signal to the open-loop transfer function *G*ð Þs of the structure, *uFF*ð Þ *n* is the feedforward control output force, *uFB*ð Þ *n* is the feedback control output force, ^*q*ð Þ n is the first response error, and is the residual vibration error of the table after feedforward control, *q*ð Þ n is the second error response, and is the residual vibration error of the table after entering the hybrid control.

According to Eq. (26), combined with the AHC block diagram, the system error *e*0 ð Þ *n* can be equivalent to the first residual vibration error response ^*q n*ð Þ, that is, the table residual vibration error after feedforward control:

$$\mathbf{e}'(\mathbf{n}) = \hat{\mathbf{q}}(\mathbf{n}) = \mathbf{q}^\*(\mathbf{n}) - \mathbf{u}\_{\text{FF}}(\mathbf{n}) = \mathbf{q}^\*(\mathbf{n}) - \mathbf{w}^T(\mathbf{n})\mathbf{y}(\mathbf{n})\tag{28}$$

In the real-time control of the active vibration system, due to the real-time control operation of the system, the sensor can only measure the vibration signal of the ground foundation and the vibration signal after the active hybrid control of the table, and cannot directly measure the vibration response *<sup>q</sup>* <sup>∗</sup> ð Þ *<sup>n</sup>* from the ground vibration signal to the open-loop transfer function *G s*ð Þ. Therefore, the signal transfer can be estimated by the model reference function *F z* ^ �<sup>1</sup> ð Þ:

$$\mathbf{q}^\*\left(\mathbf{n}\right) \dot{=} \mathbf{y}'\left(\mathbf{n}\right) = \mathbf{y}(\mathbf{n})\hat{\mathbf{F}}\left(\mathbf{z}^{-1}\right) \tag{29}$$

Then the system expansion error *e*(*n*) can be equivalent to the second error response *q*(*n*), that is, the residual vibration error of the table after AHC:

$$\mathbf{e}(\mathbf{n}) = \mathbf{q}(\mathbf{n}) = \mathbf{e}'(\mathbf{n}) \frac{\mathbf{P}(\mathbf{s})}{\mathbf{1} + \mathbf{P}(\mathbf{s})\mathbf{G}\_{\mathbf{c}}(\mathbf{s})} = \hat{\mathbf{q}}(\mathbf{n}) \frac{\mathbf{P}(\mathbf{s})}{\mathbf{1} + \mathbf{P}(\mathbf{s})\mathbf{G}\_{\mathbf{c}}(\mathbf{s})} \tag{30}$$

According to Eq. (27), the cost function after using AHC is:

*Active Vibration Suppression Based on Piezoelectric Actuator DOI: http://dx.doi.org/10.5772/intechopen.103725*

$$\xi(\mathbf{n}) = \sum\_{i=1}^{n} \lambda^{\mathbf{n}-i} \mathbf{e}^2(\mathbf{i}) = \sum\_{i=1}^{n} \lambda^{\mathbf{n}-i} \left[ \frac{\mathbf{P}(\mathbf{s})}{\mathbf{1} + \mathbf{P}(\mathbf{s}) \mathbf{G}\_c(\mathbf{s})} \right]^2 \mathbf{e}'^2(\mathbf{i}) \tag{31}$$

The goal of the adaptive feedforward control algorithm is to find an optimal discrete filter and optimal weights so that the objective function of the cost function can be minimized, that is, the gradient of the cost function is zero:

$$\nabla\_{\mathbf{k}}\xi = \frac{\partial\xi}{\partial\mathbf{w}\_{\mathbf{k}}} = -2\left[\frac{\mathbf{P}(\mathbf{s})}{\mathbf{1} + \mathbf{P}(\mathbf{s})\mathbf{G}\_{\mathbf{c}}(\mathbf{s})}\right]^2 \sum\_{i=1}^{N} \left\{\lambda^{\text{N}-i}\mathbf{y}(\mathbf{i}) \left[\mathbf{q}^\*(\mathbf{i}) - \mathbf{w}^T(\mathbf{i})\mathbf{y}(\mathbf{i})\right] \right\} = \mathbf{R}(\mathbf{n})\mathbf{w}(\mathbf{n}) - \mathbf{r}(\mathbf{n})$$

$$\mathbf{\color{red}{32}}$$

$$\mathbf{R(n)} = \sum\_{i=1}^{n} \boldsymbol{\lambda}^{\text{n}-i} \mathbf{y(i)} \mathbf{y}^{\text{T}}(\mathbf{i}) = \lambda \mathbf{R(n-1)} + \mathbf{y(n)} \mathbf{y}^{\text{T}}(\mathbf{n}) \tag{33}$$

$$\mathbf{r(n)} = \sum\_{i=1}^{n} \lambda^{n-i} \mathbf{y(i)} \mathbf{q^\*(i)} = \lambda r(n-1) + \mathbf{y(n)} \mathbf{q^\*(n)}\tag{34}$$

where **R**ð Þ *n* is the autocorrelation matrix of the interference signal, and **r**ð Þ n is the cross-correlation matrix of the interference signal and the feedforward source signal. Then, the gradient of Eq. (32) is zero, and the arrangement can be obtained:

$$
\hat{w}(\mathbf{n}) = \mathbf{R}^{-1}(\mathbf{n})r(\mathbf{n})\tag{35}
$$

For the convenience of description, define an inverse matrix **<sup>B</sup>**ð Þ¼ *<sup>n</sup>* **<sup>R</sup>**�<sup>1</sup> ð Þ *n* , and the following expression can be obtained:

$$\begin{split} \mathbf{B}(\mathbf{n}) &= \mathbf{R}^{-1}(\mathbf{n}) = \lambda^{-1} \left[ \mathbf{B}(\mathbf{n} - \mathbf{1}) - \frac{\mathbf{B}(\mathbf{n} - \mathbf{1}) \mathbf{y}(\mathbf{n}) \mathbf{y}^{\mathrm{T}}(\mathbf{n}) \mathbf{B}(\mathbf{n} - \mathbf{1})}{\lambda + \mathbf{y}^{\mathrm{T}}(\mathbf{n}) \mathbf{B}(\mathbf{n} - \mathbf{1}) \mathbf{y}(\mathbf{n})} \right] \\ &= \lambda^{-1} \left[ \mathbf{B}(\mathbf{n} - \mathbf{1}) - \mathbf{k}(\mathbf{n}) \mathbf{y}^{\mathrm{T}}(\mathbf{n}) \mathbf{B}(\mathbf{n} - \mathbf{1}) \right] \end{split} \tag{36}$$

where **k**ð Þ *n* is called the gain vector, and its expression is:

$$\mathbf{k(n)} = \frac{\mathbf{B(n-1)y(n)}}{\lambda + \mathbf{y^T(n)B(n-1)y(n)}} \tag{37}$$

According to Eqs. (35)–(37), when the optimal solution of the tap weight vector has been obtained, the update formula of the weight vector can be further derived:

$$\hat{\boldsymbol{\mathfrak{o}}}\left(\mathbf{n}\right) = \mathbf{R}^{-1}(\mathbf{n})\boldsymbol{r}(\mathbf{n}) = \mathbf{B}(\mathbf{n})\boldsymbol{r}(\mathbf{n}) = \hat{\boldsymbol{\mathfrak{o}}}\left(\mathbf{n}-\mathbf{1}\right) + \mathbf{k}(\mathbf{n})\left[\mathbf{q}^{\*}\left(\mathbf{n}\right) - \mathbf{y}^{T}(\mathbf{n})\hat{\boldsymbol{\mathfrak{o}}}\left(\mathbf{n}-\mathbf{1}\right)\right] \tag{38}$$

Then, the initialization of the AHC algorithm is set to **<sup>w</sup>**^ ð Þ¼ <sup>0</sup> **<sup>0</sup>**, **<sup>B</sup>**ð Þ¼ <sup>0</sup> *<sup>δ</sup>*�<sup>1</sup> and *<sup>δ</sup>* be very small positive numbers. The iterative formula of the AHC algorithm can be sorted out:

$$\begin{cases} \mathbf{k(n)} = \frac{\mathbf{B(n-1)y(n)}}{\lambda + \mathbf{y^T(n)B(n-1)y(n)}} \\ \mathbf{B(n)} = \lambda^1 \left[ \mathbf{B(n-1)} - \mathbf{k(n)} \mathbf{y^T(n)B(n-1)} \right] \\ \hat{w}(\mathbf{n}) = \hat{w}(\mathbf{n-1}) + \mathbf{k(n)} \left[ \mathbf{q^\*(n)} - \mathbf{y^T(n)} \hat{w}(\mathbf{n-1}) \right] \end{cases} \tag{39}$$

**Figure 13.** *Simulation comparison curves of transmissibility under different control modes.*

The simulation analysis of the AHC is carried out in Matlab, and the transmissibility curve in **Figure 13** can be obtained. In passive control, the vibration of the load platform is not effectively suppressed at all, and it is significantly attenuated at high frequencies. In the IFF control, the resonance peak at the natural frequency of the system is obviously suppressed, but the high-frequency attenuation is not improved. When the AHC is adopted, the formant of the system is further reduced, and the high frequency also shows a higher attenuation. The piezoelectric AHC has a better vibration isolation effect.
