**2.2. Selection of controller architecture**

This section presents an overview of controller algorithms for active vibration and noise control systems. In general, the controllers are designed for two different tasks: tracking a trajectory or rejecting disturbances. For active vibration suppression systems, the task is the rejection of external disturbances and reduction of vibration levels. These can be achieved via feedback and feedforward controllers. The generic block diagrams [7] of these controller algorithms are presented in Fig. 2a and Fig. 2b. Basically, the feedback controller generates controller output signal based on summation of plant response and external disturbance. On the other hand, feedforward controller generates controller output by measuring external disturbance and predicting the plant response.

**Figure 1.** Control System Design Steps

The feedback controller algorithms can be divided into two categories [13] : active damping systems and model-based controllers. In an active damping system, the vibration of the host structure (acceleration, velocity or displacement) can be suppressed around the resonance frequencies. The closed-loop transfer function of the active damping system can be derived by using the block diagram shown in Fig. 2a as it follows:

$$
\hat{y} = d + \hat{u}\mathcal{G}(\mathbf{s})\tag{1}
$$

Here, the sensor output signal is denoted by y, external disturbance is shown with d and controller output is u. The open-loop system is presented with G(s) in Laplace domain. After algebraic manipulations, one can obtain the following relation for the closed-loop system:

$$\text{cy} = \frac{1}{1 + \mathcal{C}(\text{s})\mathcal{G}(\text{s})}d\tag{2}$$

In Equation 2, the closed-loop transfer function shows that the effect of external disturbance on the sensor output can be minimized by increasing the magnitude of the C(s)G(s) as the phase and gain margin of the open-loop system G(s) allows. Since the collocated sensor and actuator pair provides infinite gain and phase margins, the active damping system works very efficiently when the collocated sensor and actuator pair is used.

**Figure 2.** a) Feedback control block diagram b) Feedforward control block diagram

In the case of model-based controller, the open-loop system dynamics is represented with the state-space model as in Equation 3:

$$\begin{aligned} \dot{\mathbf{x}} &= \mathbf{A}\mathbf{x} + \mathbf{B}\mathbf{u} + \mathbf{B}\_{\mathbf{d}}\mathbf{d} \\ \mathbf{y} &= \mathbf{C}\mathbf{x} + \mathbf{D}\mathbf{y} \end{aligned} \quad \text{(a)} \tag{3}$$

In this state-space form, ܠ is the state variable**,** ܡ is the measured output, ܝ is the input signal and ܌ is the external disturbance signal. In addition to these, ۯ presents the state matrix, ۰ is the controller input matrix**,** ۰܌ is the disturbance matrix, ۱is the output matrix and ۲ is the feed-through matrix. For full-state feedback controller ሺݑ ൌ െ۹ݔሻǡ the closed-loop system can be obtained as

$$\begin{aligned} \dot{\mathbf{x}} &= (\mathbf{A} - \mathbf{B}\mathbf{K})\mathbf{x} + \mathbf{B}\_{\mathbf{d}}\mathbf{d} \\ \mathbf{y} &= (\mathbf{C} + \mathbf{D}\mathbf{K})\mathbf{x} \end{aligned} \quad \text{(a)} \tag{4}$$

The aim of this state-space closed-loop system is to eliminate the effect of disturbance on output signal similar to the active damping. The gain matrix ۹ can be determined by applying pole-placement or optimal control design methodologies.

#### **2.3. Design of controller architecture**

682 Smart Actuation and Sensing Systems – Recent Advances and Future Challenges

disturbance and predicting the plant response.

**Figure 1.** Control System Design Steps

by using the block diagram shown in Fig. 2a as it follows:

phase and gain margin of the open-loop system

trajectory or rejecting disturbances. For active vibration suppression systems, the task is the rejection of external disturbances and reduction of vibration levels. These can be achieved via feedback and feedforward controllers. The generic block diagrams [7] of these controller algorithms are presented in Fig. 2a and Fig. 2b. Basically, the feedback controller generates controller output signal based on summation of plant response and external disturbance. On the other hand, feedforward controller generates controller output by measuring external

The feedback controller algorithms can be divided into two categories [13] : active damping systems and model-based controllers. In an active damping system, the vibration of the host structure (acceleration, velocity or displacement) can be suppressed around the resonance frequencies. The closed-loop transfer function of the active damping system can be derived

Here, the sensor output signal is denoted by y, external disturbance is shown with d and controller output is u. The open-loop system is presented with G(s) in Laplace domain. After algebraic manipulations, one can obtain the following relation for the closed-loop system:

In Equation 2, the closed-loop transfer function shows that the effect of external disturbance

actuator pair provides infinite gain and phase margins, the active damping system works

G(s)

� � �

on the sensor output can be minimized by increasing the magnitude of the

very efficiently when the collocated sensor and actuator pair is used.

� � � � ��(�) (1)

���(�)�(�) � (2)

allows. Since the collocated sensor and

C(s)G(s)

as the

For control applications, it is necessary to understand the system dynamics appropriately since the controller design parameters are determined based on the system dynamics. As stated in section 2.2 there are different types of the control algorithms and for each type of the controller, system identification is essential. This system identification can be carried out via a vibration testing & analysis methods. Aim of such method is to extract Frequency Response Functions (FRF) where these functions presents the system response to a specific input in frequency range of interest by means of magnitude and phase. A representative FRF is shown in Fig. 3 for a structure where piezoelectric materials are used as bonded patch actuators (to excite the structure) and LDV as velocity sensor (to measure the vibration response) .

**Figure 3.** Example FRF of a system

The modal (resonance) frequencies of the systems are indicated in magnitude plot of FRFs as sharp peaks where the active damping systems are most effective. In active damping systems, the higher frequency modes may deteriorate the stability due to phase lag. Such effects of the higher frequency modes can be suppressed by including the low-pass filters. On the other hand, model based controllers use mathematical relations which in turn represents the system response based on FRF. Since the scope of this chapter is devoted only to proportional analog velocity feedback controller, more detailed information on modelbased controllers and mathematical system representations can be found in references [6, 7, 9, and 12].
