Frequency response optimisation of a movement simulator by adaptive sinusoidal reference tracking
The control device of a movement simulator, between its input formed of a control signal and its output formed of a measured magnitude among its speed, its position and its acceleration, by a transfer function B(z −1 )/A(z −1 ) includes a corrector synthesized from the modelling of the movement simulator and equivalent to a RST controller. The controller includes a reference input, a retro-action input whereon is applied the measured magnitude and an output producing the control signal. The transfer function between the reference input of the controller and the measured magnitude is H(z −1 )/W(z −1 ). The device has a reference filter whose input is a sinusoidal reference signal c(t) at a frequency wc and applying at output a filtered reference signal on the reference input of the controller. The reference filter has a transfer function Rr(z −1 )/Fr(z −1 ), whose denominator and numerator are adapted to ensure, for the frequency wc, effective tracking by the movement simulator of the sinusoidal reference.
1. 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,
wherein Te is a sampling period,
D ( z −1 )=(1− z −1 )·(1−2 cos( wc·Te )· z −1 +z −2 ),
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
wherein a numerator Rr(z −1 ) of the reference filter is such as an 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 );
and in that said 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 −1 ) are 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 the device includes a means to determine Rr(z −1 ) by solving the Bezout equation according to a parameter formed by the frequency wc of said reference signal.
2. The device according to claim 1 , further comprising: a means for estimating a pulse by determining a value of the pulse of the reference applied at input of said reference filter.
3. The device according to claim 2 , wherein said estimating means of the pulse is based on a recursive identification algorithm comprising at least an extended least square algorithm.
4. 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.
5. The device according to claim 4 , wherein said estimating means of the residual attenuation and phase-shift is based on a recursive identification algorithm comprising at least a least square algorithm.
6. 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 +z 2 ),
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.
7. The method according to claim 6 , wherein:
the behaviour of the assembly formed by the controller and by the movement simulator is simulated by means of a transfer function H(z −1 )/W(z −1 ) so as to provide an ideal measured magnitude ym(t);
a residual attenuation At and the residual phase-shift φ between the actual movement y(t) of the movement simulator and the ideal measured magnitude ym(t) is estimated from an actual measured magnitude y(t) and the ideal measured magnitude ym(t);
said numerator Rr(z −1 ) is determined in relation to the frequency wc taken as a parameter, of the residual attenuation and phase-shift determined in the estimating step, by on-line resolution of a system of equations:
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Fr
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the filtering of the reference is adapted with the determined value of Rr(z −1 ).
8. The method according to claim 7 , wherein said estimating means of the residual attenuation and phase-shift is based on a recursive identification algorithm comprising at least a least square algorithm.
9. The control method according to claim 6 , further comprising: an initial step for estimating a pulse of said reference signal.
10. The method according to claim 9 , wherein said estimating means of a pulse is based on a recursive identification algorithm comprising at least an extended least square algorithm.
11. The method according to claim 6 , wherein the method implemented by a computer program comprising a series of instructions loaded into a memory of a computer and of being executed by a calculator of said computer for generating a control signal u(t) of the movement simulator.
12. The method according to claim 7 , further comprising: an initial step for estimating a pulse of said reference signal.
13. The method according to claim 8 , further comprising: an initial step for estimating a pulse of said reference signal.
14. The method according to claim 7 , wherein the method is implemented by a computer program comprising a series of instructions loaded into a memory of a computer and executed by a calculator of said computer generating a control signal u(t) of the movement simulator.
15. The method according to claim 8 , wherein the method is implemented by a computer program comprising a series of instructions loaded into a memory of a computer and executed by a calculator of said computer generating a control signal u(t) of the movement simulator.
16. The method according to claim 9 , wherein the method is implemented by a computer program comprising a series of instructions loaded into a memory of a computer and executed by a calculator of said computer generating a control signal u(t) of the movement simulator.
17. The device according to claim 4 , further comprising: a means for estimating a pulse by determining a value of the pulse of the reference applied at input of said reference filter.
18. The device according to claim 17 , wherein said estimating means of the pulse is based on a recursive identification algorithm comprising at least an extended least square algorithm.