Frequency response optimisation of a movement simulator by adaptive sinusoidal reference tracking

A retro-action control device of a movement simulator (10) for tracking by a simulator of a reference, said movement simulator being modelled between input formed of a control signal u(t) and output formed of a measured magnitude y(t) comprising one of speed, position and acceleration, by a transfer function B(z−1)/A(z−1), said control device comprising a controller (20) synthesised from said modelling of the movement simulator, said controller comprising a reference input, a retro-action input whereon is applied said measured magnitude y(t) and an output producing a control signal u(t), the transfer function between the reference input of the controller and the measured magnitude then being H(z−1)/W(z−1);

• further comprising:
• a reference filter (30) whose input is a sinusoidal reference signal c(t) at a frequency wc and applying a filtered reference signal c′(t) on the reference input of said controller (20);
• said reference filter having a transfer function Rr(z−1)/Fr(z−1) and being adaptive;
• and
• a means for simulating the assembly formed by the controller and by the physical system, producing an ideal measured magnitude ym(t) and having as a transfer function H(z−1)/W(z−1);
• a means for estimating a residual attenuation and phase-shift capable, from an actual measured magnitude y(t) and the ideal measured magnitude ym(t), to determine the values of the residual attenuation At and of the residual phase-shift φ, between the actual movement y(t) of the movement simulator and the ideal measured magnitude ym(t);
• a means to determine Rr(z−1) by calculating, at the frequency wc of the reference signal, a numerator Rr(z−1) of the transfer function of the reference filter, in relation to the residual attenuation and phase-shift determined by said estimating means, by on-line resolution of a system of equations:
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• the means to determine Rr(z−1) taking also as an input parameter the value of the frequency of said reference signal,
• wherein Te is a sampling period,
• Fr being the denominator of the reference filter,
• W being the denominator of the transfer function between the reference input of the controller and the measured magnitude,
• H being the numerator of the transfer function between the reference input of the controller and the measured magnitude.

A retro-action control method comprising:

• tracking via a movement simulator by a simulator of a reference, said movement simulator being modelled, between input formed of a control signal u(t) and output formed of a measured magnitude y(t) comprises a position, a speed and an acceleration, by a transfer function B(z−1)/A(z−1),
• controlling via a control device including a controller (20) synthesised from said modelling of the movement simulator, said controller comprising a reference input, a retro-action input whereon is applied said measured magnitude y(t) and an output producing a control signal u(t), the transfer function between the reference input of the controller and the measured magnitude then being H(z−1)/W(z−1);
• comprising:
• a sinusoidal reference signal (30) having a frequency wc is filtered for applying a filtered reference signal c′(t) on the reference input of said controller (20);
• the filtering of the reference signal being adaptive,
• wherein the filtering is provided according to a transfer function Rr(z−1)/Fr(z−1),
• wherein Te is a sampling period,
• D(z−1)=(1−z−1)·(1−2 cos(wc·Te)·z−1+z2),
• Lo(Z−1) being a polynomial to be resolved,
• Fr(Z−1) being the denominator of the reference filter,
• W(Z−1) being the denominator of the transfer function between the reference input of the controller and the measured magnitude,
• H(Z−1) being the numerator of the transfer function between the reference input of the controller and the measured magnitude, and
• a numerator Rr(z−1) of the reference filter is determined so that the expression Fr(z−1)W(z−1)−Rr(z−1)H(z−1) is factorised in the form:
• D(z−1)Lo(z−1)=[1−2 cos(wc·Te)·z−1+z−2]Lo(z−1);
• wherein the numerator Rr(z−1) of the transfer function between the reference input of the controller and the measured magnitude and in that said polynomial Lo(Z−) are determined as a solution of a so-called Bezout equation:
• D(z−1)Lo(z−1)+Rr(z−1)H(z−1)=Fr(z−1)W(z−1)
• for the frequency wc of the reference signal and in that Rr(z−1) is determined while solving said Bezout equation according to a parameter formed by the frequency wc of said reference signal.