**5.3. Modal control**

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

signal reproduces a standard laser cut periodic profile.

and direct disturbance coming from the stage.

when the payload is moving.

from the ground are damped).

Since the operation pattern and timing are known (Figure 23 (a)), the transfer function *h s*( ) can be obtained by using an FFT analyzer, the command signal ( ) *FF u s* (Figure 23 (b)) can be computed offline, stored in the control unit and applied to the system at the proper time

It is worthy to notice that the inversion of *h s*( ) leads to a non-causal function with a numbers of zeros equal or higher than the number of poles. This issue is overcome by adding the required number of poles at a frequency sufficiently high (more than 100 Hz), in

Bode diagram of *h(s)* is reported in Figure 22 (feedback control is on, vibrations coming

Figure 23 (c) shows that the proposed technique is effective and allows to isolate the machine from the direct disturbance generated by the payload operations. The excitation

The coupling of this action with the feedback control system permits to obtain a full vibration damping and active isolation from external disturbance coming from the ground

**Figure 22.** Control command to controlled output stage velocity transfer function (h(s)) Bode diagram.

order to make the feedforward filter proper and fit to be used in the control scheme.

The third and last control technique proposed in this chapter is a modal approach to perform a feedback control scheme. This strategy is similar in performance to the Lead–Lag strategy illustrated in Section 5.1, but it simplifies the control design procedure once it gives a direct feeling on actuators action on machine modes.

The method is based on the scheme reported in Figure 24. The goal of the technique is to decouple the rotational and translational motion modes of the machine to direct the action of the controller selectively on the dynamic of interest.

**Figure 24.** Modal control overall scheme.

The eight geophones measurements on stage and frame are elaborated to obtain four velocity differences:

$$\begin{aligned} V\_{DX+} &= V\_{SensSX+} - V\_{SensFX+} \\ V\_{DX-} &= V\_{SensSX-} - V\_{SensFX-} \\ V\_{DY+} &= V\_{SensSY+} - V\_{SensFY+} \\ V\_{DY-} &= V\_{SensY-} - V\_{SensFY-} \end{aligned} \tag{37}$$

These values are then summed and subtracted in order to obtain the motion mode uncoupling.

Rotational mode:

$$\begin{aligned} V\_{RX} &= V\_{DX+} + V\_{DX-} \\ V\_{RY} &= V\_{DY+} + V\_{DY-} \end{aligned} \tag{38}$$

Translational mode

$$\begin{aligned} V\_{TX} &= V\_{DX+} - V\_{DX-} \\ V\_{TY} &= V\_{DY+} - V\_{DY-} \end{aligned} \tag{39}$$

The control dynamic is the same of Lead-Lag approach, the difference consisting in the error fed to the controller. The poles of the system in open and closed loop are reported in Table 4.

**Figure 25.** Modal control. a) Control command to stage-frame velocities difference transfer function. b) Control command to translational dynamics transfer function. c) Control command to rotational dynamics transfer function. Solid line: open loop. Dashed line: closed loop.

Figure 25 shows the motion modes uncoupling and system behaviour in open and closed loop. Figure 25.a illustrates control command to stage-frame velocities difference transfer function where translational and rotational modes are coupled. Figure 25.b and Figure 25.c report the translational ( *VTX TY*, ) and rotational ( *VRX RY* , ) dynamics respectively. It is worthy to notice that the influence of rotational dynamics is dominant, being its response amplitude higher than translational one. Due to this consideration it can be easily explained the low action of the feedback control on the translational dynamics (b)) is compared to the rotational one (c)).
