**5.2. Feedforward control**

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

**Figure 20.** Vibration damping action. Transfer function from a force applied on the stage to the velocity measured on the stage ( ) *S S q F* . Numerical response. Solid line: closed-loop; Dashed line: Open-loop.

**Figure 21.** Active isolation action. Transfer function from a simulated ground velocity to the velocity measured on the stage ( ) *S G q q* . Numerical response. Solid line: Open loop configuration. Dashed line:

The active isolation action is verified by simulating the excitation coming from the ground. The experimental test in this case has not been performed since in reality it is difficult to excite the machine from the ground in a controlled and effective way. Nevertheless the model is reliable as proved in Figure 14 and the obtained results can be assumed as a good

Closed loop configuration.

validation of the control action.

Although the feedback control explained in Section 5.1 is strongly effective for external disturbances coming from the ground, it could not be sufficient to make the machine completely isolated from the direct disturbance generated by the movement of the payload. It is indeed possible that in the case of high precision requests, feedback control approaches such as PID, Lead-Lag or LQR are not able to satisfy by themselves severe specifications. Hence different schemes, operating selectively on the stage direct disturbances, are required.

In this section an off-line feedforward scheme allowing to isolate the machine from the action of payload direct disturbance in operating condition is proposed. The scheme is not classical, i.e. the command is not generated on-line but it is computed in advance on the basis of the data response to the direct disturbance and the transfer function between the control command and the controlled output. As illustrated in Figure 3, the action of feedforward control is superimposed to the one of the Lead-Lag feedback control and acts exclusively on the disturbance acting from the payload.

The technique is based on the complete knowledge of the fixed pattern followed by the payload of the machine during operations. Since also the operation timing is known, it is possible to compute in advance a feedforward command, so as to be able to suppress the effects of the direct disturbance that are generated by the payload movements, and that cannot be measured. These commands are stored in the electronic control unit and are summed to the feedback control action at the appropriate time.

The model used to design the control law is the four degrees of freedom model exposed in Section 4.1. Being the XZ-plane and YZ-plane symmetric, just the latter is considered in the design phases.

The controlled output is the velocity measured on the stage ( ) *<sup>s</sup> v s* and it can be considered as the sum of two contributions: the effect of the direct disturbance on the output ( ) *Ds v s* and the effect of the feedforward action on the output ( ) *FFs v s* . Then the total response is:

$$\boldsymbol{\upsilon}\_s(\mathbf{s}) = \boldsymbol{\upsilon}\_{Ds}(\mathbf{s}) + \boldsymbol{\upsilon}\_{FFs}(\mathbf{s}) = \boldsymbol{\upsilon}\_{Ds}(\mathbf{s}) + h(\mathbf{s})\boldsymbol{\mu}\_{F\uparrow}(\mathbf{s})\tag{35}$$

where *h s*( ) is the transfer function between the control command ( ) *FF u s* to the controlled output ( ) *FF v s* .

The control signal is:

$$
\mu\_{\rm FF}(\mathbf{s}) = -h(\mathbf{s})^{-1} \upsilon\_{\rm Ds}(\mathbf{s}) \tag{36}
$$

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 when the payload is moving.

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

Bode diagram of *h(s)* is reported in Figure 22 (feedback control is on, vibrations coming from the ground are damped).

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 signal reproduces a standard laser cut periodic profile.

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 and direct disturbance coming from the stage.

**Figure 23.** a) Feedforward control: disturbance profile; b) Control signal. Solid line: feedforward off, dashed line: feedforward on; c) Controlled output: stage velocity. Solid line: feedforward off, dashed line: feedforward on.
