Method and system for adaptive compensation of dry friction

Abstract

A dry friction compensation method for at least one mass or inertia M mobile under the effect of at least one effector element controlled by a force or torque control signal U, the motion of the mass or inertia being characterized by a motion signal Y chosen among one or several of the position X, the speed V and the acceleration, the method includes: defining an ideal model of the closed loop; defining a dry friction compensation control law; the dry friction compensation control law being based on the following friction model: $P=\min \left(\max \left(\frac{\mathrm{VM}}{}+U,-S\right),S\right),$ where V is the speed of the mobile mass or inertial subjected to the friction and a minor time constant, and S is a parameter of dry friction.

Claims

1. A dry friction compensation method for a mechanical system of an apparatus that includes a mass or inertia M that is mobile under effect of an effector element controlled by a force or torque control signal U, the mass or inertia M having to move according to instructions of a setpoint signal C.sub.r chosen among at least one of a position X.sub.r, a speed V.sub.r, and an acceleration, a motion of the mass or inertia being characterized by a motion measurement signal Y chosen among at least one of the position X, the speed V, and the acceleration, the method comprising: defining an ideal model of a closed loop, by receiving as an input the setpoint signal C.sub.r, and producing as an output an ideal motion signal Ym relating to the motion of the mass or inertia according to an ideal model of the mechanical system; and defining a dry friction compensation control law, by receiving as an input the setpoint signal C.sub.r and producing as an output the control signal U for the effector element that puts the mass or inertia in motion and whose motion is measured by a motion measurement sensor, said sensor producing a motion measurement signal Y, the dry friction compensation control law being adaptive as a function of a parameter of a dry friction value S, said dry friction compensation control law being based on the following friction model: $P=\min \left(\max \left(\frac{\mathrm{VM}}{}+U,-S\right),S\right),$ where V is the speed of the mobile mass or inertia subjected to the friction and a minor time constant; comparing the ideal motion signal Ym and the motion measurement signal Y with each other to produce an error signal s(t); calculating the estimation of the dry friction S as a function of the error signal s(t); and using said estimation {umlaut over (S)} as a parameter of dry friction value in the dry friction compensation control law in order to correct the control signal as a function of the estimation of the dry friction value.

2. The method according to claim 1, wherein the estimation of the dry friction value is calculated from:
{dot over ({circumflex over (S)})}=.Math..sub.f(t).Math.(t) with: being a strictly positive scalar corresponding to the adaptation gain of the adaptive compensation control law, and
(t)=sgn(.sub.r(t).Math.+V(t)) where V(t) is a speed motion measurement signal, is a minor time constant, and .sub.r is an acceleration piloting signal.

3. The method according to claim 2, wherein the adaptation gain is variable.

4. The method according to claim 2, wherein a state-feedback control law is implemented and the acceleration piloting signal is calculated by
.sub.r=U.sub.r+K.sub.C1(Y.sub.rY)+K.sub.C2(V.sub.rV) with X.sub.r, V.sub.r being setpoint signals, X, V being motion measurement signals, and K.sub.C1 and K.sub.C2 being state-feedback gains.

5. The method according to claim 2, wherein a PID corrector, of proportional-integral-derivative functions, is implemented, and wherein the proportional-integral-derivative functions of the PID are added to produce the acceleration control signal .sub.r.

6. The method according to claim 2, wherein a PID corrector, of proportional-integral-derivative functions, is implemented, and wherein the PID produces, by combination of the proportional-integral-derivative functions, an intermediate control signal U.sub.PID, said intermediate control signal U.sub.PID used in the adaptive dry friction compensation control law to produce the acceleration control signal .sub.r after passage through a high-pass filter.

7. The method according to claim 2, wherein an RST or LQG, H.sub. corrector is implemented.

8. The method according to claim 1, wherein a speed-feedback control is implemented, and wherein, in the motion measurement signals, the speed is chosen among: a speed of the mass or inertia measured by a sensor, a speed of the mass or inertia calculated from the position of the mass or inertia by derivative calculation, and a speed {circumflex over (V)} of the mass or inertia by estimated calculation by means of a state observer.

9. The method according to claim 1, wherein a position-feedback control is implemented, and wherein, in the motion measurement signals, the position X is chosen among: a position of the mass or inertia measured by a sensor, and a position of the mass or inertia calculated from a measurement of a motion signal other than the position of the mass or inertia.

10. The method according to claim 1, wherein the estimation of the dry friction value is calculated from:
{dot over ({circumflex over (S)})}=.Math.(t).Math.(t), with: being a strictly positive scalar corresponding to the adaptation gain of the adaptive compensation control law, and ${}_f\left(t\right)=\frac{G\left(s\right)}{1+G\left(s\right).Math.F\left(s\right)}.Math.\left(t\right)$ where F(s) is the transfer function of the feedback part of the corrector, G(s) is the input-output transfer function of the system to be piloted, and in the case of a state-feedback position control with force or torque control:
F(s)=M.Math.(K.sub.c2.Math.s+K.sub.c1) $G\left(s\right)=\frac{1}{M.Math.s^3}$
(t)=sgn(.sub.r(t).Math.+V(t)) where V(t) is a speed motion measurement signal, is a minor time constant, and .sub.r is an acceleration piloting signal.

11. The method according to claim 10, wherein the adaptation gain is variable.

12. The method according to claim 10, wherein a state-feedback control law is implemented and the acceleration piloting signal is calculated by
.sub.r=U.sub.r+K.sub.c1(Y.sub.r=Y)+K.sub.c2(V.sub.rV) with X.sub.r, V.sub.r being setpoint signals, X, V being motion measurement signals, and K.sub.C1 and K.sub.C2 being state-feedback gains.

13. The method according to claim 10, wherein a PID corrector, of proportional-integral-derivative functions, is implemented, and wherein the proportional-integral-derivative functions of the PID are added to produce the acceleration control signal .sub.r.

14. The method according to claim 10, wherein a PID corrector, of proportional-integral-derivative functions, is implemented, and wherein the PID produces, by combination of the proportional-integral-derivative functions, an intermediate control signal U.sub.PID, said intermediate control signal U.sub.PID used in the adaptive dry friction compensation control law to produce the acceleration control signal .sub.r after passage through a high-pass filter.

15. The method according to claim 10, wherein an RST or LQG, H.sub. corrector is implemented.

16. A device, comprising: a calculator; a motion measurement sensor configured to produce a mass or inertia motion measurement signal Y corresponding to a motion of a mass or inertia M that is mobile under effect of an effector element and moving according to instructions of a setpoint signal C.sub.r chosen among at least one of a position X.sub.r, a speed V.sub.r, and an acceleration, said motion measurement signal Y corresponding to at least one of the position X, the speed V, and the acceleration of the mass or inertia; and means for real time calculation in the calculator according to an adaptive dry friction compensation control law to produce a control signal U for the effector element, the calculator and means for real time calculation being configured to define the dry friction compensation control law by receiving the setpoint signal C.sub.r as input and producing as an output the control signal U for the effector element that puts the mass or inertia in motion and whose motion is measured by the motion measurement sensor to produce the motion measurement signal Y, the dry friction compensation control law being adaptive as a function of a parameter of dry friction value S, said dry friction compensation control law being based on the following friction model: $P=\min \left(\max \left(\frac{\mathrm{VM}}{}+U,-S\right),S\right),$ where V is the speed of the mobile mass or inertia subjected to the friction and a minor time constant, the calculator and means for real time calculation configured to: compare an ideal motion signal Ym and the motion measurement signal Y with each other to produce an error signal s(t), the ideal motion signal Ym relating to the motion of the mass or inertia according to an ideal model of a closed loop of the mechanical system based on the setpoint signal C.sub.r as input, calculate the estimation of the dry friction S as a function of the error signal s(t), and use said estimation as a parameter of dry friction value in the dry friction compensation control law in order to correct the control signal as a function of the estimation of the dry friction value.

17. An apparatus, comprising: a mechanical system including at least one mass or inertia M that is mobile under the effect of an effector element controlled by a control signal U, the mass or inertia having to move according to the instructions of a setpoint signal C.sub.r chosen among at least one of a position X.sub.r, a speed V.sub.r, and an acceleration; a motion measurement sensor that measures a motion of the mass or inertia and produces a motion measurement signal Y corresponding to at least one of the position X, the speed V and the acceleration of the mass or inertia; and a calculator, including means for adaptive compensation calculation according to an adaptive dry friction compensation control law to produce the control signal U, the calculator and means for adaptive compensation calculation configured to define the dry friction compensation control law by receiving the setpoint signal C.sub.r as input and producing as an output the control signal U for the effector element that puts the mass or inertia in motion and whose motion is measured by the motion measurement sensor to produce the motion measurement signal Y, the dry friction compensation control law being adaptive as a function of a parameter of dry friction value S, said dry friction compensation control law being based on the following friction model: $P=\min \left(\max \left(\frac{\mathrm{VM}}{}+U,-S\right),S\right),$ where V is the speed of the mobile mass or inertia subjected to the friction and a minor time constant, the calculator and means for adaptive compensation calculation configured to: compare an ideal motion signal Ym and the motion measurement signal Y with each other to produce an error signal s(t), the ideal motion signal Ym relating to the motion of the mass or inertia according to an ideal model of a closed loop of the mechanical system based on the setpoint signal C.sub.r as input, calculate the estimation of the dry friction S as a function of the error signal s(t), and use said estimation as a parameter of dry friction value in the dry friction compensation control law in order to correct the control signal as a function of the estimation of the dry friction value.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) The present invention, without being limited thereby, will now be exemplified by the following description of embodiments and implementation modes in relation with:

(2) FIG. 1, which is a block diagram of the dry friction model used within the framework of the present invention,

(3) FIG. 2, which shows the dry friction compensation control law such as proposed by Ph. de Larminat,

(4) FIG. 3, which shows a schematic diagram of the adaptive dry friction compensation control law of the invention,

(5) FIG. 4, which shows as a functional block diagram a first adaptive dry friction compensation control law based on a first calculated estimation {dot over ()} of the Coulombian friction module S, for a position-feedback control,

(6) FIG. 5, which shows as a functional block diagram a second adaptive dry friction compensation control law based on a second calculated estimation {dot over ()} of the Coulombian friction module S, for a position-feedback control,

(7) FIG. 6, which shows as a functional block diagram a third adaptive dry friction compensation control law based on a PID corrector and on the second calculated estimation {dot over ()} of the Coulombian friction module S, for a position-feedback control,

(8) FIG. 7, which shows as a functional block diagram a fourth adaptive dry friction compensation control law based on a PID corrector and on the second calculated estimation {dot over ()} of the Coulombian friction module S, for a position- and speed-feedback control, with an estimation for this latter,

(9) FIG. 8, which shows as a functional block diagram a fifth adaptive dry friction compensation control law for a speed-feedback control and based on the first calculated estimation {dot over ()} of the Coulombian friction module S,

(10) FIG. 9, which shows as a functional block diagram a sixth adaptive dry friction compensation control law for a speed-feedback control and based on the second calculated estimation {dot over ()} of the Coulombian friction module S, and

(11) FIG. 10, which shows as a functional block diagram a seventh adaptive dry friction compensation control law for a speed-feedback control and based on a PID corrector and on the second calculated estimation {dot over ()} of the Coulombian friction module S.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

(12) The detailed description of the invention will first begin by a presentation of the model of Ph. de Larminat before passing to an exemplary embodiment of the invention in which the compensation control law is made adaptive for the model in question.

(13) The model presented hereinafter is that of a mass or inertia subjected to a force by an actuator, for example a mobile element operated by an electrical motor and in contact with a wall, this contact causing dry frictions.

(14) It is a particularly simple model, which can concern both translational and rotational motions.

(15) Let's consider the following types of data:

(16) M: Mass or inertia of the mobile mechanical system,

(17) S: Coulombian friction module

(18) U: Driving force corresponding to a effort/force or torque control, not to be mixed

(19) up with U.sub.r (acceleration setpoint) that will be introduced later,

(20) P: Disturbing force or torque due to the dry friction,

(21) V: Speed of the system measured by a sensor,

(22) X: Position of the system measured by a sensor,

(23) According to the fundamental relation of the dynamics, we have:

(24) $\begin{array}{cc}\stackrel{.}{V}=\frac{1}{M}\left(U-P\right)& \left(1\right)\end{array}$

(25) where {dot over (v)} is an acceleration.

(26) The friction model proposed by Ph. de Larminat is the following:

(27) $\begin{array}{cc}P=\min \left(\max \left(\frac{\mathrm{VM}}{}+U,-S\right),S\right)& \left(2\right)\end{array}$

(28) where V is the speed of the mobile mass or inertia subjected to the friction.

(29) In equation (2), is a minor time constant, for example of a few milliseconds.

(30) By combining (1) and (2), we obtain:

(31) $\begin{array}{cc}\stackrel{.}{V}=\frac{1}{M}\left(U-\min \left(\max \left(\frac{\mathrm{VM}}{}+U,-S\right),S\right)\right)& \left(3\right)\end{array}$

(32) The block diagram corresponding to the model (3) of Ph. de Larminat is shown in FIG. 1.

(33) To this raw model may be added various elements, for example a load disturbance C or also various feedbacks depending on X and/or V (return forces, viscous friction, non linearities . . . ) noted W, so that the equation becomes:

(34) $\begin{array}{cc}\stackrel{.}{v}=\frac{1}{M}\left(U-P-C-W\right)& \left(4\right)\end{array}$

(35) Moreover, a more complete model has been developed by the same author for taking into account the Stribeck effect.

(36) The dry friction compensation law developed in the same book, which is not adaptive, is based on a state-feedback control of the type:
U=M.Math.(U.sub.r+K.sub.c1(X.sub.rX)+K.sub.c2(V.sub.rV))(5)
where
X.sub.r: Position setpoint by a reference coming from a trajectory generator, for this position control law,
V.sub.r: Speed setpoint by a reference coming from a trajectory generator,
U.sub.r: Acceleration setpoint by a reference coming from a trajectory generator,
K.sub.C1 and K.sub.C2 are state-feedback gains,
X: a position signal of the mass or inertia subjected to friction,
V: a speed signal of the mass or inertia subjected to friction.
The speed and position signals of the mass or inertia that are motion measurement signals may come from sensors or be calculated: the speed V that is a motion measurement signal can be estimated from X if no speed sensor provides this information.

(37) To the raw state-feedback control law of equation (5), it is possible to add a compensation for the load disturbances C and other feedbacks W so that we have:
U=M.Math.(U.sub.r+K.sub.c1(X.sub.rX)+K.sub.c2(V.sub.rV))+C+W(Y.sub.r,V)(6)

(38) The friction compensation in the above-mentioned book is made by adding in equation (6) a term:
S.Math.sgn(V+.sub.r.Math.)(7)
with:
.sub.r=U.sub.r+K.sub.c1(X.sub.rX)+K.sub.c2(V.sub.rV)(8)

(39) sgn( ) being the function sign.

(40) Finally, the friction compensation law proposed by Ph. de Larminat is written:
U=M.Math..sub.r+C+W(Y.sub.r,V)+S.Math.sgn(V+.sub.r.Math.)(9)

(41) The block diagram corresponding to this friction compensation law (9) of Ph. de Larminat is shown in FIG. 2.

(42) The friction compensation law in equation (9) has the drawback not to be adaptive and it has therefore a limited industrial interest, because the variations of the friction parameters are significant over the life of a product.

(43) We will now explain the method of the invention that allows, by making the compensation law (9) adaptive, obtaining a better operation of the machines including controlled mobile parts undergoing friction. The invention hence allows obtaining an adaptive friction compensation control law based on a non-adaptive dry friction compensation law (9) such as that of Ph. de Larminat.

(44) This adaptive compensation control law estimates S (Coulombian friction Module) in real time, the estimate being denoted g, so as to inject this parameter into the compensation law.

(45) If the control law (5) is applied to the system described by equation (1), with the hypothesis P=0, a perfect follow-up of the setpoint instruction Y.sub.r by Y is obtained, i.e.:
Y(t)=Y.sub.r(t)t(10)

(46) In the presence of dry friction, i.e. (t)0, equation (10) is no longer verified, if there is not the compensation device described by equations (7) and (9).

(47) Let's define the variation in the setpoint (t) between the output of an ideal model of the closed loop Y.sub.m(t) and the output of the real closed loop Y(t) (including the system subjected to the dry frictions), subjected to the same setpoint Y.sub.r(t):
(t)=Y.sub.r(t)Y(t)(11)

(48) The adaptive control law that we propose to develop has for objective to minimize a criterion relating to (t). For example, the minimization may relate: .sup.2(t) at each instant of time, .sup.2(t)dt.

(49) Other minimization criteria based on (t) may also be used.

(50) More generally, (t) may be defined as a calculated variation between an output Y.sub.m of an ideal model of the desired closed loop and an outputalso called feedback signal/signals , in particular by measurement, Y of the mobile mechanical system, both subjected to the setpoint Y.sub.r, i.e.:
(t)=Y.sub.mY(12)

(51) Equation (11) corresponds to a particular case in which the ideal model of the closed loop corresponds to a unit gain.

(52) It is to be understood that the variation may be calculated between any kind of outputs of the same type of the ideal model and of the mechanical system, and not only of the position X type. Hence, the variation may be calculated by difference between outputs of the speed type.

(53) The adaptive law consists, from (t), in determining , estimate of the Coulomb module S and in injecting this estimate into equation (9).

(54) The schematic representation of the principle of the adaptive dry friction compensation control law is given in FIG. 3.

(55) The non-adaptive control law according to equation (9) proposed by Ph. de Larminat provides a perfect compensation for the friction model of equation (3), provided that the estimate (t) of S(t) is exact. In this case, we have simply:
{dot over (V)}=.sub.r(13)

(56) Generally, the estimation of S is not perfect, and the estimate variation is defined:
{tilde over (S)}=S(14)

(57) From the moment that the estimation (t) of S(t) is imperfect, it is obtained by combining equations (9), (3) and (14):

(58) 0$\begin{array}{cc}M.Math.\stackrel{.}{V}={}_r\left(S+\tilde{S}\right).Math.\mathrm{sgn}\left({}_r.Math.+V\right)-\min \left(\max \left(U+\frac{M}{},-S\right),S\right)& \left(15\right)\end{array}$

(59) But, as S.Math.sgn(.sub.r +V) compensates for

(60) $\min \left(\max \left(U+\frac{M}{},-S\right),S\right),$
it is deduced therefrom:
M.Math.{dot over (V)}=.sub.r+{tilde over (S)}.Math.sgn(.sub.r.Math.+V)(16)

(61) The second term of the right part of equation (16) may be considered as an additive disturbance at the input of the system acting in the closed loop.

(62) Let's call d this additive disturbance:
d(t)={tilde over (S)}.Math.sgn(.sub.r.Math.+V)(17)

(63) Moreover, when only the linear part of the control law, i.e. equation (6), is considered, it can be seen that this law can be decomposed into an anticipating action or feedforward part T(s) and a retroaction or feedback part F(s), a law that can be expressed by means of the Laplace variable s and having for variables setpoint and measurement inputs Y(t) and Y.sub.r (t), respectively. In particular, this law may include an observer of the load disturbance.

(64) Hence, in the most general way, the control U(t) may be expressed as:
U(t)=T(s).Math.Y.sub.r(t)F(s).Math.Y(t)(18)

(65) If considering the control law of equation (9) restricted to its linear components, and also with omitting the term of load disturbance C, we have:
T(s)=M.Math.(s.sup.2+K.sub.c2.Math.s+K.sub.c1)
F(s)=M.Math.(K.sub.c2.Math.s+K.sub.c1)

(66) And the transfer between U(t) and Y(t) may also be modelled by a very simple transfer function, by a double integrator, G(s):

(67) $\begin{array}{cc}Y\left(t\right)=G\left(s\right).Math.U\left(t\right)=\frac{1}{M.Math.s^2}.Math.U\left(t\right)& \left(19\right)\end{array}$

(68) The transfer function between the additive disturbance d(t) and the closed-loop output Y(t) is written:

(69) $\begin{array}{cc}Y\left(t\right)=\frac{G\left(s\right)}{1+G\left(s\right).Math.F\left(s\right)}d\left(t\right)+\frac{G\left(s\right).Math.T\left(s\right)}{1+G\left(s\right).Math.F\left(s\right)}.Math.Y_r\left(t\right)& \left(20\right)\end{array}$

(70) The transfer function of the ideal model of the closed loop (see equation 12), corresponds by definition to the transfer between Y.sub.r(t) and Y(t):

(71) $\begin{array}{cc}{Y}_m\left(t\right)=\frac{G\left(s\right).Math.T\left(s\right)}{1+G\left(s\right).Math.F\left(s\right)}.Math.Y_r\left(t\right)& \left(21\right)\end{array}$

(72) In this case, the error between the output of the ideal model Y.sub.m(t) and the closed-loop output of the system Y(t), by combination of equations 12, 20, 21, is written:

(73) $\begin{array}{cc}\mathrm{.Math.}\left(t\right)=-\frac{G\left(s\right)}{1+G\left(s\right).Math.F\left(s\right)}d\left(t\right)& \left(22\right)\end{array}$

(74) d(t) being expressed according to the expression (17).

(75) In this equation (17), it is noted that d(t) is affine in S.

(76) This leads in proposing as a law of estimation of S, the following law:
{dot over ({circumflex over (S)})}=.Math.(t).Math.(t)(23)

(77) with:

(78) a strictly positive scalar, which is by definition the gain of adaptation,

(79) =sgn(.sub.r(t).Math.+V(t)) if the speed V(t) is measured, otherwise an estimate {circumflex over (V)}(t) of V(t) is used instead of V(t), and which is calculated for example by means of a state observer.

(80) The application of equation (23) of estimation of S to the adaptive dry friction compensation control law of FIG. 3 applied to the dry friction compensation model of Ph. De Larminat of equation (9)/FIG. 2 gives the adaptive compensation law represented as blocks in FIG. 4 and which is of the state-feedback type.

(81) In the left part of FIG. 4, we find setpoint input signals, herein U.sub.r, V.sub.r, Xr. Towards the right of this same Figure, we find other measurement signals obtained from the measurements on the mobile mechanical system or that are calculated with herein the measured speed V or the estimation thereof and the measured position X. At the output, we find the control signal U intended to control at least one motor/effector moving a mass or inertia subjected to friction, wherein the mass or inertia can comprise mobile elements of the motor/effector.

(82) On the top of FIG. 4, we find the block corresponding to the ideal model of the closed loop that corresponds to the transfer function given at equation (21) hereinabove. This ideal model receives as an input the setpoint input signals U.sub.r, V.sub.r, Xr. This ideal model produces as an output an ideal motion signal relating to the motion of the mass or inertia according to the ideal model (supposed without friction) of the system and that is herein of the position type: Y.sub.m.

(83) Towards the bottom of FIG. 4, we find the blocks of calculation and compensation according to the dry friction compensation model of Ph. de Larminat to produce the control signal U. We find blocks corresponding to the calculations of equation (9) with, on the side of the setpoint instructions and measurements, blocks K.sub.C1 and K.sub.C2, which are the state-feedback gains already mentioned. We find blocks corresponding to the mass or inertia M of the mobile element in motion and to the minor time constant . We finally find the specific blocks of calculation of the estimation {dot over ()} calculated according to a first calculation mode presented above and using directly (t) (a second calculation mode {dot over ()} will be seen in relation with FIG. 5).

(84) The study of the convergence of the law (23) may be made using the passivity theory. It is shown in particular that a sufficient condition of convergence is that the transfer function

(85) $\frac{G\left(s\right)}{1+G\left(s\right)F\left(s\right)}$
is positive real, that is to say that the Nyquist locus of said transfer function must be integrally comprised in the right half-plane of the complex plane.

(86) This convergence condition is potentially penalizing, so we have interest in releasing it by substituting .sub.f(t) to (t) in the following equation (24):

(87) $\begin{array}{cc}{}_f\left(t\right)=\frac{G\left(s\right)}{1+G\left(s\right).Math.F\left(s\right)}.Math.\left(t\right)& \left(24\right)\end{array}$

(88) The law of estimation of S is then written:
{dot over ({circumflex over (S)})}=.Math..sub.f(t).Math.(t)(25)

(89) The application of equation (25) of estimation of S to the adaptive dry friction compensation control law of FIG. 3 applied to the dry friction compensation model of Ph. De Larminat of equation (9)/FIG. 2 gives the adaptive compensation law represented as blocks in FIG. 5.

(90) The adaptive dry friction compensation control law represented as blocks in FIG. 5 corresponds to the representation of FIG. 4, except for what concerns the specific blocks of calculation of the estimation {dot over ()} that is herein calculated according to a second mode of calculation, where .sub.f (t) has been substituted to (t).

(91) Calculation algorithms may be implemented, in particular in real time, based on the adaptive dry friction compensation control laws shown in FIGS. 4 and 5.

(92) It can be shown that the algorithms based on these FIGS. 4 and 5, and hence on the respective equations (23) and (25), tend to minimize .sup.2(t) at each instant and is likenable to the gradient algorithm.

(93) It is possible to complexify the preceding adaptive compensation control laws by making the adaptation gain variable, in order to minimize for example the sum of squares of (t), or the sum of squares of (t) weighted by a forgetting factor.

(94) In this latter case, the adaptation gain may, for example, be expressed according to equation (26):
{dot over ()}(t)=(1).Math.F.sup.1(t)+.sup.2(t)(26)
Or equation (27):
{dot over ()}(t)=(1).Math.F.sup.1(t)+.sub.f.sup.2(t)(27)

(95) In equations (26) and (27), the coefficient is the forgetting factor mentioned hereinabove, with: 0<<1.

(96) In the feedback control field, the PID (proportional-integral-derivative) correctors are by far the most commonly used. Therefore, it is desirable to be able to have an adaptive friction compensation control law of this type. A law of the type will now be described, still on the friction model of Ph. de Larminat.

(97) The control signal of a PID corrector may be expressed as follows:

(98) $\begin{array}{cc}{U}_{\mathrm{PID}}\left(t\right)=\left(K_p+\frac{K_i}{s}+\frac{K_d.Math.s}{.Math.s+1}\right).Math.\left(Y_r-Y\right)& \left(28\right)\end{array}$

(99) with:

(100) K.sub.p: proportional coefficient,

(101) K.sub.L: integral coefficient,

(102) K.sub.d: derivative coefficient,

(103) : filtering time constant of the derivative action.

(104) It is to be noted that this representation is not unique, and that there exists a great number of possibilities of implementation of a PID, in particular parallel, series, series-parallel, nevertheless a compensation law based on the structure (28) will be described hereinafter without this description be limitative, because any implementation of a PID corrector may be substituted to (28).

(105) If referring to the scheme of FIG. 2 to transpose a PID corrector thereto, it may be considered that the signal .sub.r associated with the friction corrector is simply the signal U(t) of equation (28), from which has been subtracted the integral action of the PID corrector. Indeed, the integral action is that which has for role to compensate for the static disturbances. Now, the hypothesis has been made of a static load disturbance C(t), which means that the main role of the integral action is to compensate for the load disturbance.

(106) In these conditions, .sub.r(t) is written:

(107) $\begin{array}{cc}{}_r\left(t\right)=\left(K_p+\frac{K_d.Math.s}{.Math.s+1}\right).Math.\left(Y_r-Y\right)& \left(29\right)\end{array}$

(108) Hence, the adaptive dry friction compensation control structure is deduced immediately and the PID adaptive friction compensation control law represented as blocks in FIG. 6 is obtained.

(109) In order to obtain an estimation of the speed {circumflex over (V)} from information of position X obtained by a position sensor, a speed estimator filter is implemented in the obtained control law.

(110) Incidentally, in other implementations, the speed estimator filter may be a high-pass filter or a state observer, and may further have as additional input the control signal U.

(111) It can be noted that the control law of this FIG. 6 uses, similarly to that of FIG. 5, the application of equation (25) of estimation of S to the control low, which introduces a block

(112) 0$\frac{G\left(s\right)}{1+G\left(s\right)F\left(s\right)}$
in the law. It is well understood that, in an alternative embodiment, this block

(113) $\frac{G\left(s\right)}{1+G\left(s\right)F\left(s\right)}$
can be omitted to obtain a control law that is likened to that of FIG. 4, but in the context of a PID corrector and, in this case, the right part of FIG. 6 uses the block structure of the right part of FIG. 4.

(114) In this PID control law structure, it is also possible to add anticipating action or feedforward blocks to the position, speed and/or acceleration setpoint(s), to form the signal .sub.r.

(115) As indicated hereinabove, there exists a great number of PID corrector forms. Whatever said form is, the signal .sub.r corresponds to the control part with no load disturbance compensation.

(116) Incidentally, it may be noted by comparison with equation (18) that, in the case of the PID control law implemented as (28), we also have:

(117) $F\left(s\right)=\left(K_p+\frac{K_i}{s}+\frac{K_d.Math.s}{.Math.s+1}\right),$
where F(s) is the above-mentioned retroaction or feedback part.

(118) In some cases, it is not possible to have access to the internal signals of the PID corrector, in particular when the PID is a product of the market included in a specific electronic casing, so that it may be necessary to calculate the signal .sub.r from the position control signal produced by the PID corrector in its entirety: U.sub.PID(t). By combining (28) and (29), we find the relation:

(119) $\begin{array}{cc}\frac{{}_r}{U_{\mathrm{PID}}}=\frac{\left(K_p+K_d\right).Math.s^2+s}{\left(K_p+K_d\right).Math.s^2+\left(K_1+K_p\right).Math.s+K_I}& \left(30\right)\end{array}$

(120) The transfer function of equation (30) is that of a high-pass filter whose gain is 1 when s.fwdarw..

(121) More generally, the signal .sub.r may be calculated from the position control signal U.sub.PID by a high-pass filtering that does not necessarily obey to that of equation (30). Here also, the PID corrector may be implemented under various forms with or without anticipating actions (feedforward) blocks.

(122) By way of example, it has been shown in FIG. 7, as a block diagram, a PID adaptive friction compensation control law in which the signal .sub.r is calculated by a high-pass filter form the position control signal U.sub.PID produced by the PID corrector. As hereinabove, in an alternative embodiment, the block

(123) $\frac{G\left(s\right)}{1+G\left(s\right)F\left(s\right)}$
may be omitted.

(124) The shown adaptive control laws may also be implemented with a speed feedback.

(125) In a speed-feedback structure, i.e. in which the signal to be processed by the closed loop, is the speed V(t), la structure of the adaptive friction compensation control laws described in relation with FIGS. 4, 5, 6 and 7 do not change fundamentally.

(126) Indeed, the transfer function G(s) of equation (19) is then simply written:

(127) $\begin{array}{cc}G\left(s\right)=\frac{1}{M.Math.s}& \left(31\right)\end{array}$

(128) The transfer function F(s) is then also simplified.

(129) The setpoint signal Y.sub.r disappears, as well as the measurement signal Y. Remain the setpoint signals V.sub.r and possibly U.sub.r. The feedback signal measured is herein V. Moreover, the output of the ideal model of the closed loop is a speed signal V.sub.m and the signal (t) is calculated by making the difference between V V.sub.m. Finally, an estimator of V becomes useless due to the fact that this feedback value is necessarily measured.

(130) Hence, the control law schematized in FIG. 4 applied to a context of speed-feedback control becomes the control law schematized in FIG. 8. Similarly, the control law schematized in FIG. 5 applied to a context of speed-feedback control becomes the control law schematized in FIG. 9. Finally, for the PID laws, the control low schematized in FIG. 6 applied to a context of speed-feedback control becomes the control law schematized in FIG. 10. As indicated hereinabove, this structure of FIG. 10 for a PID-feedback control may be modified, in particular by omission of the derivative action and/or by addition of anticipating action (feedforward) blocks.

(131) The adaptive friction compensation control law that has been presented up to now has been so for a base corrector of the state-feedback type, and, in an alternative embodiment, with a PID corrector. In other embodiments, it is possible to use this compensation law with other correction structures, for example RST, LQG, H.sub.. In any case, it is necessary to calculate the signal .sub.r that corresponds to the control with no load disturbance.

(132) It may also be contemplated to extend this compensation law to an acceleration-feedback control, provided however that information about the speed of the system is known. Indeed, if the system has only one acceleration sensor and no speed sensor, said speed is not observable within the meaning defined by Kalman.