Control Requirements

Let’s consider the simple mechanical system in Figure 1.

The signals are described in table 1.

For the nano-hexapod alone, we have the following equations: \[ \begin{align*} x &= \frac{1}{ms^2 + k} F + \frac{1}{ms^2 + k} F_d + \frac{k}{ms^2 + k} x_\mu \\ F_m &= \frac{ms^2}{ms^2 + k} F - \frac{k}{ms^2 + k} F_d + \frac{k m s^2}{ms^2 + k} x_\mu \\ d &= \frac{1}{ms^2 + k} F + \frac{1}{ms^2 + k} F_d - \frac{ms^2}{ms^2 + k} x_\mu \end{align*} \]

We can write the equations function of \(\omega_\nu = \sqrt{\frac{k}{m}}\): \[ \begin{align*} x &= \frac{1/k}{1 + \frac{s^2}{\omega_\nu^2}} F + \frac{1/k}{1 + \frac{s^2}{\omega_\nu^2}} F_d + \frac{1}{1 + \frac{s^2}{\omega_\nu^2}} x_\mu \\ F_m &= \frac{\frac{s^2}{\omega_\nu^2}}{1 + \frac{s^2}{\omega_\nu^2}} F - \frac{1}{1 + \frac{s^2}{\omega_\nu^2}} F_d + \frac{k \frac{s^2}{\omega_\nu^2}}{1 + \frac{s^2}{\omega_\nu^2}} x_\mu \\ d &= \frac{1/k}{1 + \frac{s^2}{\omega_\nu^2}} F + \frac{1/k}{1 + \frac{s^2}{\omega_\nu^2}} F_d - \frac{\frac{s^2}{\omega_\nu^2}}{1 + \frac{s^2}{\omega_\nu^2}} x_\mu \end{align*} \]

Assumptions:

• the forces applied by the nano-hexapod have no influence on the micro-station, specifically on the displacement of the top platform of the micro-hexapod.

This means that the nano-hexapod can be considered separately from the micro-station and that the motion \(x_\mu\) is imposed and considered as an external input.

The nano-hexapod can thus be represented as in Figure 2.

What we measure is not the absolute motion \(x\), but the relative motion \(x - w\) where \(w\) is the motion of the granite.

Also, \(w\) induces some motion \(x_\mu\) through the transmissibility of the micro-station.

Let’s define the mass and stiffness of the nano-hexapod.

m = 50; % [kg]
k = 1e7; % [N/m]

Let’s define the Plant as shown in Figure 2:

Gn = 1/(m*s^2 + k)*[-k, k*m*s^2, m*s^2; 1, -m*s^2, 1; 1, k, 1];
Gn.InputName  = {'Fd', 'xmu', 'F'};
Gn.OutputName = {'Fm', 'd', 'x'};

Now, define the transmissibility transfer function \(T_\mu\) corresponding to the micro-station motion.

wmu = 2*pi*50; % [rad/s]

Tmu = 1/(1 + s/wmu);
Tmu.InputName = {'r1'};
Tmu.OutputName = {'ymu'};

w0 = 2*pi*40;
xi = 0.5;
Tw = 1/(1 + 2*xi*s/w0 + s^2/w0^2);
Tw.InputName = {'w1'};
Tw.OutputName = {'dw'};

We add the fact that \(x_\mu = d_\mu + T_\mu r + T_w w\):

Wsplit = [tf(1); tf(1)];
Wsplit.InputName = {'w'};
Wsplit.OutputName = {'w1', 'w2'};

S = sumblk('xmu = ymu + dmu + dw');

Sw = sumblk('y = x - w2');

Gpz = connect(Gn, S, Wsplit, Tw, Tmu, Sw, {'Fd', 'dmu', 'r1', 'F', 'w'}, {'Fm', 'd', 'y'});

Let’s consider a feedback loop using \(d\) as shown in Figure 5.

Let’s apply a direct velocity feedback by deriving \(d\): \[ F = F^\prime - g s d \]

Thus: \[ d = \frac{1}{ms^2 + gs + k} F^\prime + \frac{1}{ms^2 + gs + k} F_d - \frac{ms^2}{ms^2 + gs + k} x_\mu \]

\[ F = \frac{ms^2 + k}{ms^2 + gs + k} F^\prime - \frac{gs}{ms^2 + gs + k} F_d + \frac{mgs^3}{ms^2 + gs + k} x_\mu \]

and \[ x = \frac{1}{ms^2 + k} (\frac{ms^2 + k}{ms^2 + gs + k} F^\prime - \frac{gs}{ms^2 + gs + k} F_d + \frac{mgs^3}{ms^2 + gs + k} x_\mu) + \frac{1}{ms^2 + k} F_d + \frac{k}{ms^2 + k} x_\mu \]

\[ x = \frac{ms^2 + k}{(ms^2 + k) (ms^2 + gs + k)} F^\prime + \frac{ms^2 + k}{(ms^2 + k) (ms^2 + gs + k)} F_d + \frac{mgs^3 + k(ms^2 + gs + k)}{(ms^2 + k) (ms^2 + gs + k)} x_\mu \]

And we finally obtain: \[ x = \frac{1}{ms^2 + gs + k} F^\prime + \frac{1}{ms^2 + gs + k} F_d + \frac{gs + k}{ms^2 + gs + k} x_\mu \]

K_dvf = 2*sqrt(k*m)*s;
K_dvf.InputName = {'d'};
K_dvf.OutputName = {'F'};

Gpz_dvf = feedback(Gpz, K_dvf, 'name');

Now let’s consider that \(x_\mu = d_\mu + T_\mu r\)

\[ x = \frac{1}{ms^2 + gs + k} F^\prime + \frac{1}{ms^2 + gs + k} F_d + \frac{gs + k}{ms^2 + gs + k} d_\mu + T_\mu \frac{gs + k}{ms^2 + gs + k} r \]

And \(\epsilon = r - x\): \[ \epsilon = \frac{1}{ms^2 + gs + k} F^\prime + \frac{1}{ms^2 + gs + k} F_d + \frac{gs + k}{ms^2 + gs + k} d_\mu + \frac{ms^2 + gs + k - T_\mu (gs + k)}{ms^2 + gs + k} r \]

Let’s consider a feedback loop using \(d\) as shown in Figure 5.

Let’s apply integral force feedback by integration \(F_m\): \[ F = F^\prime - \frac{g}{s} F_m \]

And we finally obtain: \[ x = \frac{1}{ms^2 + mgs + k} F^\prime + \frac{1 + \frac{g}{s}}{ms^2 + mgs + k} F_d + \frac{k}{ms^2 + mgs + k} x_\mu \]

K_iff = 2*sqrt(k/m)/s;
K_iff.InputName = {'Fm'};
K_iff.OutputName = {'F'};

Gpz_iff = feedback(Gpz, K_iff, 'name');

Now let’s consider that \(x_\mu = d_\mu + T_\mu r\)

\[ x = \frac{1}{ms^2 + mgs + k} F^\prime + \frac{1 + \frac{g}{s}}{ms^2 + mgs + k} F_d + \frac{k}{ms^2 + mgs + k} d_\mu + \frac{T_\mu k}{ms^2 + mgs + k} r \]

And \(\epsilon = r - x\): \[ \epsilon = \frac{1}{ms^2 + mgs + k} F^\prime + \frac{1 + \frac{g}{s}}{ms^2 + mgs + k} F_d + \frac{k}{ms^2 + mgs + k} d_\mu + \frac{ms^2 + mgs + k - T_\mu k}{ms^2 + mgs + k} r \]

Instead of a pure integrator, let’s use a low pass filter with a cut-off frequency above the bandwidth of the micro-station \(\omega_mu\)

% K_iff = (2*sqrt(k/m)/(2*wmu))*(1/(1 + s/(2*wmu)));
% K_iff.InputName = {'Fm'};
% K_iff.OutputName = {'F'};

% Gpz_iff = feedback(Gpz, K_iff, 'name');

Let’s first consider that only the output \(x\) is used for feedback (Figure 8)

We then have: \[ \epsilon &= r - G_{\frac{x}{F}} K \epsilon - G_{\frac{x}{F_d}} F_d - G_{\frac{x}{x_\mu}} d_\mu - G_{\frac{x}{x_\mu}} T_\mu r \]

And then:

With \(S = \frac{1}{1 + G_{\frac{x}{F}} K}\), we have: \[ \epsilon = - S G_{\frac{x}{F_d}} F_d - S G_{\frac{x}{x_\mu}} d_\mu + S (1 - G_{\frac{x}{x_\mu}} T_\mu) r \]

We have 3 terms that we would like to have small by design:

• \(G_{\frac{x}{F_d}} = \frac{1}{ms^2 + k}\): thus \(k\) and \(m\) should be high to lower the effect of direct forces \(F_d\)
• \(G_{\frac{x}{x_\mu}} = \frac{k}{ms^2 + k} = \frac{1}{1 + \frac{s^2}{\omega_\nu^2}}\): \(\omega_\nu\) should be small enough such that it filters out the vibrations of the micro-station
• \(1 - G_{\frac{x}{x_\mu}} T_\mu\)

\[ 1 - G_{\frac{x}{x_\mu}} T_\mu = 1 - \frac{1}{1 + \frac{s^2}{\omega_\nu^2}} T_\mu \]

We can approximate \(T_\mu \approx \frac{1}{1 + \frac{s}{\omega_\mu}}\) to have:

In our case, we have \(\omega_\nu > \omega_\mu\) and thus we cannot lower this term.

Some implications on the design are summarized on table 2.

Controller for the damped plant using DVF.

wb = 2*pi*50; % control bandwidth [rad/s]

% Lead
h = 2.0;
wz = wb/h; % [rad/s]
wp = wb*h; % [rad/s]

H = 1/h*(1 + s/wz)/(1 + s/wp);

% Integrator until 10Hz
Hi = (1 + s/2/pi/10)/(s/2/pi/10);

K = Hi*H*(1/s);

Kpz_dvf = K/abs(freqresp(K*Gpz_dvf('y', 'F'), wb));
Kpz_dvf.InputName = {'e'};
Kpz_dvf.OutputName = {'F'};

Controller for the damped plant using IFF.

wb = 2*pi*50; % control bandwidth [rad/s]

% Lead
h = 2.0;
wz = wb/h; % [rad/s]
wp = wb*h; % [rad/s]

H = 1/h*(1 + s/wz)/(1 + s/wp);

% Integrator until 10Hz
Hi = (1 + s/2/pi/10)/(s/2/pi/10);

K = Hi*H*(1/s);

Kpz_iff = K/abs(freqresp(K*Gpz_iff('y', 'F'), wb));
Kpz_iff.InputName = {'e'};
Kpz_iff.OutputName = {'F'};

Loop gain

Let’s connect all the systems as shown in Figure 8.

Sfb = sumblk('e = r2 - y');

R = [tf(1); tf(1)];
R.InputName = {'r'};
R.OutputName = {'r1', 'r2'};

Gpz_fb_dvf = connect(Gpz_dvf, Kpz_dvf, R, Sfb, {'r', 'dmu', 'Fd', 'w'}, {'y', 'e', 'Fm', 'd'});
Gpz_fb_iff = connect(Gpz_iff, Kpz_iff, R, Sfb, {'r', 'dmu', 'Fd', 'w'}, {'y', 'e', 'Fm', 'd'});