Control of nonlinear systems with friction
(Manuscript Number 97-055R)
Control of nonlinear systems with friction R.M. Hirschorny G. Millerz October 15, 1998
Abstract
In this paper we introduce a new continuous dynamic controller for a class of nonlinear systems which includes mechanical system models with a bristle model for nonlinear friction e ects. We obtain sucient conditions for global stabilization using an estimate for bristle defection and present experimental results illustrating the bene ts of our dynamic controller in the regulation of a high speed linear positioning table. Keywords: nonlinear systems, friction observers, Lyapunov stability.
1 Introduction Using model-based friction compensation in low velocity high precision tracking control can e ectively reduce steady state error [3] without the need for excessive integral gains. Coulomb friction, viscous and static friction, and the Stribeck e ect have been modelled successfully (c.f. [3, 4, 10]). In [1] friction in a lubricated journal bearing is both measured and estimated using a friction observer. System performance substantially improved when the friction estimate was used and the estimated nonlinear friction was found to be consistent with the measurements of the actual friction forces. Generally the overall performance of a model-based friction compensation technique improves with a more complete friction model [3]. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada. y Department of Mathematics and Statistics, Queen's University, Kingston, Ontario K7L 3N6, Canada. E-mail: [email protected] z Industrial Engineering Department, Celestica Inc., North York, Ontario M3C 1V7, Canada. E-mail: [email protected]
1
In [5] Canudas de Wit, Olsson, Astrom, and Lischinsky present a new dynamic friction model which captures dynamics friction e ects (the Dahl e ect, frictional memory, stick-slip motion) as well as steady state friction e ects, including the Stribeck e ect. They use a state variable bristle model, the Lund-Grenoble (LuGre) model, to describe the friction between two sliding surfaces. The bristle de ection cannot be measured directly so they design a nonlinear friction observer. Theorem 1 of [5] shows that, when a PD controller is chosen so that the transfer function relating the position error to the friction estimation error is strictly positive real (SPR), the tracking and observation errors tend to zero. This results in high accuracy position control, but the rate of convergence and robustness of the controller are both adversely a ected by the limitations on the pole locations of the compensated system which are imposed by the requirement that the error transfer function be strictly positive real (SPR) [5]. In [11] Vedagarbha, Dawson, and Feemster design Controller/Observers for mechanical systems which utilize the LuGre nonlinear friction model but need not be (SPR). For a class of mechanical systems they develop several observers, including an exponentially stable observer and, in addition, present adaptive controllers which yield asymptotic position tracking while compensating for certain parameter uncertainties. In this paper we introduce a new Lyapunov-based continuous dynamic controller for a more general class of nonlinear systems than is considered in [5] or [11]. This class does include servo motors with friction e ects modelled via the nonlinear LuGre friction model considered in [5], [11]. For this system the restrictions on the pole locations of the compensated system found in [5] are not needed. Our reduced order observer is, in some sense, a generalization of the exponentially stable observer developed in [11], and for the mechanical systems studied in [11] also yield convergence of state and bristle defection estimation errors to zero at an exponential rate. Our controller is used in the control of a high speed precision linear tracking table and the experimental comparisons between our controller, that of [5], and a tuned PID controller without direct nonlinear friction compensation are presented.
2 Problem Statement In [5] the interface between two surfaces is modeled by contact between sets of bristles. In particular, if z represents the average bristle de ection, v the 2
velocity between the two surfaces, and F the force due to friction, then
z_ = v ? gj(vvj) z; F = 0 z + 1z_ + 2 v:
(1) (2)
This Lund-Grenoble (LuGre) model of friction captures viscous friction, Coulomb friction, as well as the Dahl and Stribeck e ects [5]. A key feature of this model is that the unobserved bristle de ection z enters the equations linearly, and 0 < g(v) < a for some positive constant a. One consequence of this last condition is that z is bounded. Here g(v) models the changeover from resting static to Coulomb friction (Stribeck e ect) (c.f. [5]). We will consider the more general nonlinear system model with input u 2 IRm which admits a state-space representation in IRn (n k) of the form x_ = a1(x; u) + A1 (x; u)z (3) z_ = a2(x; u) + A2 (x; u)z with x 2 IRk ; z 2 IR` ; (x; z )T 2 IRn and where a1 ; a2 ; A1 and A2 are, respectively, a k-vector, a `-vector, a k `-matrix and a ` `-matrix Lipschitz continuous with respect to x and u. Note that continuously di erentiable implies Lipschitz continuous. The assumption contained in (3) is linearity of the equations with respect to the unmeasured states. In [5] it is shown that, in the scalar case, the average bristle de ection is bounded. In particular, there exists a constant d > 0 such that, for jz(t0 )j d, it follows that jz(t)j d 8 t t0 and for all controls u. Our rst assumption is that our generalized model shares this property. Assumption A1 : There exist d > 0 such that, for kz(t0 )k d, every solution (x(t); z (t)) to (3) satis es kz (t)k d 8t t0 : We also suppose that it is possible to design a full state feedback controller unom(x; z ) which regulates the output x such that any solution of the closed-loop system x(t) = (x(t); z (t)) is bounded, and x(t) goes to zero. This is expressed by assumption A2 below via the existence of some Lyapunov function. Assumption A2 lets us conclude that x is bounded and goes to zero via the standard Lyapunov \second" theorem. It is closely related to assumption A2 of [9] but is weaker in the sense that here V (x) need only be a proper function of x, while in [9] V (x; z ) must be a proper function of x and z . We also note that A2 of [9] does not hold for our servo system model incorporating nonlinear friction. 3
Assumption A2 : There exists a continuously di erentiable positive semidefinite function V from IRk to IR, a positive de nite k k matrix W , and a Lipschitz continuous map unom from IRn to IRm such that 1. The function V is proper (i.e. the preimage of a compact set is compact), 2. For all (x; z ) 2 IRn we have
@V (x) [a (x; u (x; z)) + A (x; u (x; z))z] ?xT Wx: 1 nom 1 nom @x We de ne the n n matrix E (x; z ) by "
(4)
#
1 (x; unom (x; z )) E (x; z) = 00 A A2 (x; unom (x; z)) ;
and our third assumption is a kind weak observability condition if we set H = [Ik 0] and consider y = H (x; z)T to be an output (c.f. [9]). Assumption A3 : There exist a constant positive de nite n n matrix Q and an n k matrix K (x; z ) smoothly depending on (x; z ) such that, for any (x; z ),
Q(E (x; z) ? K (x; z)H ) + (E (x; z) ? K (x; z)H )T Q 0
(5)
in the sense of symmetric matrices. We note that Assumption A3 holds if, for some positive de nite ` ` matrix Q2 ,
Q2 A2 (x; unom(x; z)) + A2 (x; unom (x; z))T Q2 0:
(6)
In this case a lower order observer results, but a weaker version of (6) suces for the construction of a reduced order dynamic controller. Assumption A30 : There exists a smooth map h : IRk 7! IR`, a positive de nite ` ` matrix Q2 > 0, and a positive semi-de nite ` ` matrix W2 0 such that, for any (x; z ) 2 IRn,
Q2 (A2 (x; unom (x; z)) ? dhxA1 (x; unom(x; z))) +(A2 (x; unom (x; z )) ? dhx A1 (x; unom (x; z )))T Q2 ?W2 0: We note that (6) results when we pick h(x) 0; W = 0. 4
(7)
3 The controller and main results
3.1 The controller
The state of our dynamic controller will be denoted by (xb; zb), and xe; ze stand for x ? xb; z ? zb respectively. A hat on functions depending on u or z indicates it is evaluated with zb substituted for z and u = unom (x; zb) e.g. Ab1 stands for A1 (x; unom(x; zb)). Note that xe is used as a state in our dynamic controller even though we assume that x is measured directly. Thus xe does not represent our best estimate for x but is an extra state used by the controller. In our reduced order controller xb is not used. Using this notation we propose , as in [9], the dynamic controller u = unom(x; z^) (8) c1 x~ + P1 (x; z^); _x^ = ab1 + Ab1 z^ + K (9) c2 x~ + P2 (x; z^); _z^ = ab2 + Ab2 z^ + K (10) with # " # " 0 P ( x; z ^ ) 1 ? 1 (11) P (x; z^) = P (x; z^) = Q AbT( @V )T : 2 1 @x
3.2 Regulation results
Theorem 3.1 Under assumptions A1, A2 and A3, and for kz(t0 )k d, the continuous dynamic controller (8)-(10) achieves the following property for the closed-loop system: for any initial conditions, the state (x(t); z (t); xb(t); zb(t)) is bounded and x(t) ! 0. To prove the above Theorem the reader is referred to the proofs in [9] and the observation that in our case z and z~ are bounded as a consequence of assumption A1 and hence zb is bounded a priori.
3.3 Reduced order controller
The following theorem is, in some sense, a generalization of the exponentially stable observer introduced in [11] for servo-motor control. Theorem 3.2 Under assumptions A1, A2 and A3 0, and for kz(t0 )k a, the dynamic controller u = unom(x; zb) zb = p + h(x) (12) T + (Ab2 ? dhx Ab1 )h(x) ) p_ = (Ab2 ? dhx Ab1 )p + ab2 ? dhx ab1 + Q?2 1 AbT1( @V @x 5
achieves the following property for the closed-loop system: for any initial conditions, the state (x(t); z (t); zb(t)) is bounded, and x(t) converges to 0. If A3 0 holds with W2 > 0 then both x and z ? zb converge to zero exponentially fast.
Proof Set U (x; z~) = V (x) + 21 z~T Q2z~. A straightforward calculation yields T b _ U_ = ?xTWx + z~T (AbT1 ( @V @x ) + Q2 (ab2 + A2 z ? zb))
and zb_ = p_ + h_ (x) T b = (Ab2 ? dhx Ab1 )zb + ab2 ? dhx ab1 + Q?2 1 AbT1 ( @V @x ) + dhx (ab1 + A1 z ): In particular, U_ = ?xTWx + z~T Q2 (Ab2 ? dhxAb1 )z ? (Ab2 ? dhx Ab1 )zb) = ?xTWx + z~T Q2 (Ab2 ? dhx Ab1 )~z = ?xTWx + 12 z~T (Q2 (Ab2 ? dhx Ab1 ) + (Ab2 ? dhx Ab1 )T Q2 )~z hence U_ ?xTWx ? 21 z~T W2 z~ 0: (13) Since W is positive de nite we see that x 7! 0 as a consequence of Lyapunov's direct method. If W2 is positive de nite then standard arguments show that both x and z ? zb converge to zero exponentially.
3.4 Servo motor control
As in [5] we consider the problem of controlling a servo system with friction. Our model is mx = Ku ? F (x;_ z) (14) z_ = x_ ? gj(x_x_j) z; where the average bristle de ection z is related to the friction F by F (x;_ z) = 0 z + 1 (x_ ? gj(x_x_j) z) + 2 x:_ (15) The positive bounded function g describes the Stribeck e ect and is modelled by [5] x_ 2 g(x_ ) = 1 (F + (F ? F )e?( vs ) ); (16)
0
c
s
6
c
where 0 > 0: Setting x1 = x; x2 = x_ this model takes the form of (3) with 1 jx2j ]. a2 = [x2 ]; A2 = [? gj(xx22j) ]; a1 = [x2 Km u ? 1m+2 x2 ]T ; A1 = [0 ? m0 + mg (x2 ) If we could measure z directly then the feedback controller m ( x + x + 1 F (x;_ z)) (17) unom (x; z) = K 1 2 m achieves x_1 = x2 ; x_2 = x1 + x2 . Thus for = ?2a2 ; and = ?2a (where a > 0) we have system dynamics described by x_ = Ax where the eigenvalues of A are ?a + ai; ?a ? ai. This gives a reasonable tradeo between speed of response and accuracy. If W is a diagonal matrix with diagonal entries 2h > 0 and 2l > 0 then
Q1 =
"
h 3h + al 2a h 2ha2 1 2a ( 2a2 + l) 2a2
#
is the positive de nite solution to Lyapunov's equation AT Q1 + Q1 A = ?W . Thus for u = unom , V = xT Q1 x we have dtd V (x(t)) = ?xT (t)Wx(t) 0: To show that assumptions A1, A2, A3, and A30 hold for this system we can proceed as follows: suppose that d is chosen so that 0 < g(x) d and jz(0)j d. Then dtd z2 (t) < 0 when jz(t)j > g(x2 (t)) and hence jz(t)j d 8t 0 (Section 3 of [5]). Thus assumption A1 holds. For V (x); W , and unom (x; z) as above, A2 holds. Assumption A3 is satis ed for Q = I33 h iT and K (x; z ) = I22 ?A1 (x; unom (x; z )) ; and assumption A30 holds if Q2 = [ ] for any real > 0 and h(x) = 0. The dynamic controller (12) takes the form (here zb = p)
u = Km ( x1 + x2 + m1 (0 zb + 1 (x2 ? gj(xx22j) zb) + 2x2 )); zb_ = x2 ? gj(xx22j) zb + m 2 (?0 + 1 gj(xx22j) )( 2ha2 x1 + 21a ( 2ha2 + l)x2 ):
(18)
Note that the observer proposed in [5] has the form of (18) but uses kx1 instead of 2 jx2j h 1 h T Q?2 1 AbT1( @V @x ) = m (?0 + 1 g(x2 ) )( 2a2 x1 + 2a ( 2a2 + l)x2 ): This results in a somewhat simpler observer, but one which severely restricts the choice of pole locations in the compensated system because in [5] the transfer function relating the estimation error to the tracking error must be strictly positive real. Note that ; h; and l are design parameters. When 02h x1 , hence our dynamic controller x2 = 0 equation (18) becomes zb_ = ? ma
7
will provide, at least momentarily, an integral control action. Thus reducing
or increasing h will e ectively increase the integral gain. If we want x(t) to track a function xd (t) we use states e1 = x1 ? xd and e2 = x2 ? e_d and adjust (17) accordingly. It is clear that Theorems 3.1 and 3.2 can also be applied to tracking problems.
Remark 3.3 We note that A30 also holds for h(x1 ; x2 ) = ? m1 x2 and W2 =
[2 10 ]. For this choice of h the observer (12) takes the form
zb = p + h(x) 0 m0 m p_ = (1 + m 1 )x2 + 1 x1 ? 1 p + 12 x2 2 (?0 + 1 jx2j )( h2 x1 + 1 ( h2 + l)x2 ? m p + m2 x2 ) + m 2a 2a 2 21 g(x2 ) 2a and from (13) we see that
(19)
U_ ?2hx21 ? 2lx22 ? 0 z~2 0: 1
Thus Theorem 3.2 implies that x1 ; x2 ; and zb converge to zero at an exponential rate. The observer-controller (19) resembles the exponentially stable observer described in [11].
4 Experimental Results
4.1 Model Identi cation and Validation
Experiments were conducted using the linear positioning table shown in Figure 1. The apparatus consists of a 2 meter long Techno/Isel Double Rail Linear track and with a Roller Carriage utilizing recirculating ball bearings. For this experiment two carts where locked together to form a positioning table. The carts are joined to a 24 volt DC servo motor by no-slip Synchromesh Drive Cable system. This permits fast and accurate movement of the table. The position of the carts is measured by an optical encoder attached to the DC motor which results in a linear resolution of 19; 000 q-counts per metre. From (2) the steady state bristle de ection, zss; is found to be zss = g(v)sgn(v):From (16), (15), and (4.1) the the steady state friction force is modelled by
Fss = (FC + (FS ? FC )e?(v=vs )2 )sgn(v) + 2 v: 8
(20)
Figure 1: Photo of cart and track assembly This means that there are 6 parameters, 0 ; 1 ; 2 ; FS ; FC ; and vs, to identify. Because equation (20) is linear in all parameters except vs , it lends itself to least squared error parameter estimation. The characteristic Stribeck velocity depends on lubrication and material properties and is usually found empirically. It ranges from 0:00001 to 0:1 m=s (c.f. [3]). In our case vs = 0:005 m=s was found to work well. It follows from (20) that, in the case where v 0,
Fss = FC + (FS ? FC )e?(vss =vs )2 + 2 v;
(21)
where vss represents steady state velocity. In vector form equation (21) can T be expressed as Fss = aT x;where a = 1 ? e?(vss =vs )2 ; e?(vss =vs )2 ; vss ; x = (FC ; FS ; 2 )T : Applying various constant voltages between 4 and 14 volts, the steady state velocity was measured. Twenty experiments were performed and the least squares estimate for x yielded the parameter estimates FC = 2:0 N , FS = 2:15 N and 2 = 4:6 Ns=m: 9
Steady state friction data yields no information about 0 , which represents bristle sti ness, or 1 , which is related to damping during friction transients. On the other hand presliding motion is governed by
mx = ?(0 z + 1z_ + 2x_ )
(22)
within the static friction regime, with u 0. For our motor-cart system x represents the displacement, in meters, of an equivalent 1.62 kg mass. Within a small operating range displacement can be attributed entirely to bristle de ection. Linearizing (22) about z = 0 and x_ = 0 yields mx+(1 +2 )x_ +0 x = 0;a damped second order system with spring constant 0 and viscous damping coecient 1 + 2 . Using 0 = 105 N=m, our model provided a good prediction for the actual system response. Simulations indicate that small variations in 0 do not a ect the open-loop response. Hence 0 = 105 N=m was adopted. The viscous friction 2x_ has little e ect at low velocities, where the damping force depends mainly on 1 . For moderate damping (a damping factor of :5) we have 1 495 Ns=m, 2 = 4:6 Ns=m. Simulations and our experiments revealed that minor variations in 0 and 1 have little a ect on the open and closed loop response but the closed loop response is adversely a ected by overestimation of Fc and FS . The result of underestimating these 2 parameters is a small reduction in performance, while even small overestimation results in undesirable limit cycles. 1.2
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Figure 2: Response to ramp input of u = 2t
Figure 3: Response to decreasing input, u = 10=(1 + 2t)
S The graphs in Figures 2 and 3 indicate our system's actual and predicted open loop response to two di erent inputs. The solid line, labeled 10
\Actual System Response", shows the position of the platform as measured by the encoder. The line labeled \Linear Model Prediction" is the response predicted by a system model with (linear) friction model F = 2 x_ . The line labeled \Nonlinear Model Prediction" indicates the response predicted by the system model with the friction model (15). The response to a ramp input is displayed in Figure 2. The nonlinear model does a good job in modelling the sticking of the platform when the armature voltage is small. The system's response to a decreasing input are illustrated in Figure 3. Here the \sticking" of the platform at lower voltages is successfully captured by our nonlinear model, whose prediction for position is accurate to within 10%. The error in the linear model prediction is almost 100% at t = 1s and increases steadily thereafter. These experiments clearly show that, for our apparatus, the more complicated LuGre model for friction proposed in [5] is far superior to the linear friction model.
4.2 Lund-Grenoble Dynamic Feedback Controller
In this section the dynamic feedback controller proposed by Canudas de Wit, Olssen, Astrom and Lischinsky [5] (abbreviated here as the LuGre controller) is implemented on our apparatus. Th friction observer developed in [5] is as follows:
z^_ = v ? gj(vvj) z^ ? ke; k > 0
(23)
where the state z^ estimates the average bristle de ection z . Theorem 1 of [5] severely restricts the choice of control law by requiring that the transfer function, G(s), between the position error and the observer error be strictly positive real (SPR). Since this sucient condition is not necessary there will be cases where this their controller is e ective even when G(s) is not strictly positive real (SPR).Using a PD controller with friction compensation yields 1 s + 0 : G(s) = ms2+ k2 s + k 1 The requirement that G(s) be SPR implies that k1 > 0; k2 > 01m : In our case this implies that k1 > 0 and k2 > 404. The derivative gain k2 is large and leads one to expect over-damping and noise propagation problems. A value of k = 0:001 was used in the observer for bristle de ection, equation (23). This resulted in a good tradeo between rapid observer convergence and a fast rate of convergence rate of x to zero. Figure 4 shows the response of the LuGre controller to tracking a step function of xd = 10 cm. Here 11
the proportional gain used was k1 = 350 and k2 = 450. The response is reasonably accurate but very slow due to excess damping in the system. 0.2
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Figure 4: Step response Lund-Grenoble Figure 5: Tracking a cosine using a PID controller and new dynamic controller controller
Increasing the proportional gain decreases the response time but results in high frequency vibration which is probably a consequence of the high gain feedback stimulating unmodelled dynamics. Of course there will be cases where the error transfer function is not strictly positive real but nonetheless this controller functions well. In our experiments we found that signi cantly reducing the damping K2 destabilized the control algorithm. In this context our new dynamic feedback controller (17) allows the use of any feedback gains which place poles in the left-half complex plane.
4.3 New Dynamic Feedback Controller
We rst use our controller (17) to track a 10 cm step and compare its performance with that of the Lund-Grenoble controller. We then try to track a cosine curve using our new controller and also a PID controller tuned to perform this task. To track the step function we used a proportional gain of k1 = 600 and a derivative gain of k2 = 50. Since we can use any feedback gains which place poles in the left-half complex plane we are able to choose a small derivative gain and place the second order poles at ?10 10i for a damping ratio of approximately 0:7. Of course we don't known z exactly and thus we have a nonlinear system which, strictly speaking, does not have poles, but, in 12
the case where z^ = z , we have linear dynamics for x and can talk about the system's poles. The performance of our new controller was compared to that of the Lund-Grenoble controller. For these experiments the constants h; l, and of equation (18) were assigned values of h = 5, l = 0:05, and
= 0 : These seemed to result in a reasonable compromise between the rates of convergence of the tracking and bristle estimation errors. Figure 4 shows the response of our system tracking a 10 cm step using the dynamic controller (18) . From Figure 4 we see that both controllers performed well in this experiment, but our new controller was able to respond more quickly because we can employ a more modest level of damping. Next we tuned a PID controller to make our platform track a cosine curve. As above we choose proportional and derivative gains to place the systems poles at ?10 10i and then experiment with various integral gains k. With k = 0 we found signi cant steady state error as well as saturation of the control initially. With k = 500 the steady state error attenuated too slowly. With k = 1000 we achieved reasonable performance (see Figure 5). With k set to 4000 we experiences excessive overshoot and saturation of the control at both +24 and ?24 volts (see Figure 5). On the other hand our dynamic controller (18) did an excellent job in regulating the path of the platform as can be seen in Figure 6. Note that after the target trajectory is reached almost no tracking error occurs at the times of peak platform acceleration. In Figure 7 we compare the performance of the tuned PID controller with our controller and note that both controllers use similar levels of control voltage. 0.15 0.1
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Figure 6: Tracking a cosine using our dynamic controller
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Figure 7: Our dynamic controller vs a PID controller
5 Conclusions We have developed a dynamic controller for a class of nonlinear systems ane in the unmeasured states. This class includes servomechanisms which incorporate the nonlinear Lund-Grenoble friction model introduced in [5]. The parameters in the Lund-Grenoble bristle model of [5] were identi ed for our high speed positioning table and both the open loop performance of the resulting model and the closed loop performance using the LuGre nonlinear controller were examined. The model accurately predicted the actual state evolution, and the controller of [5] achieved high accuracy but modest speed of response due to high damping needed for stability in our application. Our new controller avoids this restriction and was superior to a tuned PID controller in tracking a time varying curve. Both the PID and our dynamic controller required similar levels of control voltage. It was found that the performance of our new dynamic controller was adversely a ected by the overestimation of two of the nonlinear friction parameters Fc and Fs related to the Stribeck e ect. Slightly underestimating Fc and Fs caused only a small degradation in performance. While the design of adaptive controllers is beyond the scope of this work, we did nd that the performance of our controller-observer was relatively una ected by moderately large (30%) deviations in our friction parameter estimates, except for overestimation of the parameters Fc and Fs . Adaptive controller design was beyond the scope of this investigation, but it may well be possible to generalize the approach used in [11] to construct adaptive controllers which yield asymptotic position tracking while compensating for certain parameter uncertainties. We note that in [11] uncertainties in Fc and Fs are not compensated for directly.
References [1] Amin, J., Friedland, B., Harnoy, A., \Implementation of a Friction Estimation and Compensation Technique", IEEE Control Systems Magazine , August, 1997. [2] B. Armstrong-Helouvry, Control of Machines with Friction, Boston, MA: Kluwer, 1991. [3] Armstrong-Helouvry, B., Dupont, P., Canudas de Wit, C., \A Survey of Models, Analysis Tools and Compensation Methods for the Control of Machines with Friction", Automatica, Vol. 7, No. 9, 1994. 14
[4] Canudas de Wit, C., Noel, P., Aubin, A., Brogliato, B., \Adaptive Friction Compensation in Robot Manipulators: Low Velocities", International Journal of Robotics Research, Vol. 10, No. 3, June 1991. [5] Canudas de Wit, C., Olsson, H., Astrom, K.J., and Lischinsky, P., \A New Model for Control of Systems with Friction", IEEE Trans. Control Systems Technology, Vol. 2, Sept. 1995. [6] Isidori, A. and Byrnes, C. I., \Output regulation of nonlinear systems", IEEE Trans. Auto. Control, Vol. AC-35, No. 2, 1990. [7] LaSalle, J.-P., \Stability theory for ordinary di erential equations," Journal of Di erential Equations, Vol. 4, 1968. [8] Praly, L., Bastin, G., Pomet, J.-B. and Jiang, Z. P., \Adaptive Stabilization of Nonlinear Systems", to be published in the series: Lecture Notes in Information and Control, Springer-Verlag, 1991. [9] Pomet, J.-B., Hirschorn, R.M., and Cebuhar, W., \Dynamic Output Feedback Regulation for a Class of Nonlinear Systems",Math. Control Signals Systems, 6:106-125, 1993. [10] Tataryn, P.D., Sepehri, N. Strong, D., \Experimental Comparison of Some Compensation Techniques for the Control of Manipulators With Stick-Slip Friction", Control Engineering Practice, Vol. 4, No. 9, Sept. 1996 [11] Vedagarbha, P., Dawson, D.M., Feemster, M., \Tracking Control of Mechanical Systems in the Presence of Nonlinear Dynamic Friction Effects", Proceedings of the American Control Conference, Albuquerque, NM, June, 1997. [12] Yang, S., Tomizuka, M., \Adaptive Pulse Width Control for Precise Positioning Under the In uence of Stiction and Coulomb Friction", Journal of Dynamic Systems, Measurement, and Control, Vol. 110, Sept. 1988.
15
Control of nonlinear systems with friction R.M. Hirschorny G. Millerz October 15, 1998
Abstract
In this paper we introduce a new continuous dynamic controller for a class of nonlinear systems which includes mechanical system models with a bristle model for nonlinear friction e ects. We obtain sucient conditions for global stabilization using an estimate for bristle defection and present experimental results illustrating the bene ts of our dynamic controller in the regulation of a high speed linear positioning table. Keywords: nonlinear systems, friction observers, Lyapunov stability.
1 Introduction Using model-based friction compensation in low velocity high precision tracking control can e ectively reduce steady state error [3] without the need for excessive integral gains. Coulomb friction, viscous and static friction, and the Stribeck e ect have been modelled successfully (c.f. [3, 4, 10]). In [1] friction in a lubricated journal bearing is both measured and estimated using a friction observer. System performance substantially improved when the friction estimate was used and the estimated nonlinear friction was found to be consistent with the measurements of the actual friction forces. Generally the overall performance of a model-based friction compensation technique improves with a more complete friction model [3]. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada. y Department of Mathematics and Statistics, Queen's University, Kingston, Ontario K7L 3N6, Canada. E-mail: [email protected] z Industrial Engineering Department, Celestica Inc., North York, Ontario M3C 1V7, Canada. E-mail: [email protected]
1
In [5] Canudas de Wit, Olsson, Astrom, and Lischinsky present a new dynamic friction model which captures dynamics friction e ects (the Dahl e ect, frictional memory, stick-slip motion) as well as steady state friction e ects, including the Stribeck e ect. They use a state variable bristle model, the Lund-Grenoble (LuGre) model, to describe the friction between two sliding surfaces. The bristle de ection cannot be measured directly so they design a nonlinear friction observer. Theorem 1 of [5] shows that, when a PD controller is chosen so that the transfer function relating the position error to the friction estimation error is strictly positive real (SPR), the tracking and observation errors tend to zero. This results in high accuracy position control, but the rate of convergence and robustness of the controller are both adversely a ected by the limitations on the pole locations of the compensated system which are imposed by the requirement that the error transfer function be strictly positive real (SPR) [5]. In [11] Vedagarbha, Dawson, and Feemster design Controller/Observers for mechanical systems which utilize the LuGre nonlinear friction model but need not be (SPR). For a class of mechanical systems they develop several observers, including an exponentially stable observer and, in addition, present adaptive controllers which yield asymptotic position tracking while compensating for certain parameter uncertainties. In this paper we introduce a new Lyapunov-based continuous dynamic controller for a more general class of nonlinear systems than is considered in [5] or [11]. This class does include servo motors with friction e ects modelled via the nonlinear LuGre friction model considered in [5], [11]. For this system the restrictions on the pole locations of the compensated system found in [5] are not needed. Our reduced order observer is, in some sense, a generalization of the exponentially stable observer developed in [11], and for the mechanical systems studied in [11] also yield convergence of state and bristle defection estimation errors to zero at an exponential rate. Our controller is used in the control of a high speed precision linear tracking table and the experimental comparisons between our controller, that of [5], and a tuned PID controller without direct nonlinear friction compensation are presented.
2 Problem Statement In [5] the interface between two surfaces is modeled by contact between sets of bristles. In particular, if z represents the average bristle de ection, v the 2
velocity between the two surfaces, and F the force due to friction, then
z_ = v ? gj(vvj) z; F = 0 z + 1z_ + 2 v:
(1) (2)
This Lund-Grenoble (LuGre) model of friction captures viscous friction, Coulomb friction, as well as the Dahl and Stribeck e ects [5]. A key feature of this model is that the unobserved bristle de ection z enters the equations linearly, and 0 < g(v) < a for some positive constant a. One consequence of this last condition is that z is bounded. Here g(v) models the changeover from resting static to Coulomb friction (Stribeck e ect) (c.f. [5]). We will consider the more general nonlinear system model with input u 2 IRm which admits a state-space representation in IRn (n k) of the form x_ = a1(x; u) + A1 (x; u)z (3) z_ = a2(x; u) + A2 (x; u)z with x 2 IRk ; z 2 IR` ; (x; z )T 2 IRn and where a1 ; a2 ; A1 and A2 are, respectively, a k-vector, a `-vector, a k `-matrix and a ` `-matrix Lipschitz continuous with respect to x and u. Note that continuously di erentiable implies Lipschitz continuous. The assumption contained in (3) is linearity of the equations with respect to the unmeasured states. In [5] it is shown that, in the scalar case, the average bristle de ection is bounded. In particular, there exists a constant d > 0 such that, for jz(t0 )j d, it follows that jz(t)j d 8 t t0 and for all controls u. Our rst assumption is that our generalized model shares this property. Assumption A1 : There exist d > 0 such that, for kz(t0 )k d, every solution (x(t); z (t)) to (3) satis es kz (t)k d 8t t0 : We also suppose that it is possible to design a full state feedback controller unom(x; z ) which regulates the output x such that any solution of the closed-loop system x(t) = (x(t); z (t)) is bounded, and x(t) goes to zero. This is expressed by assumption A2 below via the existence of some Lyapunov function. Assumption A2 lets us conclude that x is bounded and goes to zero via the standard Lyapunov \second" theorem. It is closely related to assumption A2 of [9] but is weaker in the sense that here V (x) need only be a proper function of x, while in [9] V (x; z ) must be a proper function of x and z . We also note that A2 of [9] does not hold for our servo system model incorporating nonlinear friction. 3
Assumption A2 : There exists a continuously di erentiable positive semidefinite function V from IRk to IR, a positive de nite k k matrix W , and a Lipschitz continuous map unom from IRn to IRm such that 1. The function V is proper (i.e. the preimage of a compact set is compact), 2. For all (x; z ) 2 IRn we have
@V (x) [a (x; u (x; z)) + A (x; u (x; z))z] ?xT Wx: 1 nom 1 nom @x We de ne the n n matrix E (x; z ) by "
(4)
#
1 (x; unom (x; z )) E (x; z) = 00 A A2 (x; unom (x; z)) ;
and our third assumption is a kind weak observability condition if we set H = [Ik 0] and consider y = H (x; z)T to be an output (c.f. [9]). Assumption A3 : There exist a constant positive de nite n n matrix Q and an n k matrix K (x; z ) smoothly depending on (x; z ) such that, for any (x; z ),
Q(E (x; z) ? K (x; z)H ) + (E (x; z) ? K (x; z)H )T Q 0
(5)
in the sense of symmetric matrices. We note that Assumption A3 holds if, for some positive de nite ` ` matrix Q2 ,
Q2 A2 (x; unom(x; z)) + A2 (x; unom (x; z))T Q2 0:
(6)
In this case a lower order observer results, but a weaker version of (6) suces for the construction of a reduced order dynamic controller. Assumption A30 : There exists a smooth map h : IRk 7! IR`, a positive de nite ` ` matrix Q2 > 0, and a positive semi-de nite ` ` matrix W2 0 such that, for any (x; z ) 2 IRn,
Q2 (A2 (x; unom (x; z)) ? dhxA1 (x; unom(x; z))) +(A2 (x; unom (x; z )) ? dhx A1 (x; unom (x; z )))T Q2 ?W2 0: We note that (6) results when we pick h(x) 0; W = 0. 4
(7)
3 The controller and main results
3.1 The controller
The state of our dynamic controller will be denoted by (xb; zb), and xe; ze stand for x ? xb; z ? zb respectively. A hat on functions depending on u or z indicates it is evaluated with zb substituted for z and u = unom (x; zb) e.g. Ab1 stands for A1 (x; unom(x; zb)). Note that xe is used as a state in our dynamic controller even though we assume that x is measured directly. Thus xe does not represent our best estimate for x but is an extra state used by the controller. In our reduced order controller xb is not used. Using this notation we propose , as in [9], the dynamic controller u = unom(x; z^) (8) c1 x~ + P1 (x; z^); _x^ = ab1 + Ab1 z^ + K (9) c2 x~ + P2 (x; z^); _z^ = ab2 + Ab2 z^ + K (10) with # " # " 0 P ( x; z ^ ) 1 ? 1 (11) P (x; z^) = P (x; z^) = Q AbT( @V )T : 2 1 @x
3.2 Regulation results
Theorem 3.1 Under assumptions A1, A2 and A3, and for kz(t0 )k d, the continuous dynamic controller (8)-(10) achieves the following property for the closed-loop system: for any initial conditions, the state (x(t); z (t); xb(t); zb(t)) is bounded and x(t) ! 0. To prove the above Theorem the reader is referred to the proofs in [9] and the observation that in our case z and z~ are bounded as a consequence of assumption A1 and hence zb is bounded a priori.
3.3 Reduced order controller
The following theorem is, in some sense, a generalization of the exponentially stable observer introduced in [11] for servo-motor control. Theorem 3.2 Under assumptions A1, A2 and A3 0, and for kz(t0 )k a, the dynamic controller u = unom(x; zb) zb = p + h(x) (12) T + (Ab2 ? dhx Ab1 )h(x) ) p_ = (Ab2 ? dhx Ab1 )p + ab2 ? dhx ab1 + Q?2 1 AbT1( @V @x 5
achieves the following property for the closed-loop system: for any initial conditions, the state (x(t); z (t); zb(t)) is bounded, and x(t) converges to 0. If A3 0 holds with W2 > 0 then both x and z ? zb converge to zero exponentially fast.
Proof Set U (x; z~) = V (x) + 21 z~T Q2z~. A straightforward calculation yields T b _ U_ = ?xTWx + z~T (AbT1 ( @V @x ) + Q2 (ab2 + A2 z ? zb))
and zb_ = p_ + h_ (x) T b = (Ab2 ? dhx Ab1 )zb + ab2 ? dhx ab1 + Q?2 1 AbT1 ( @V @x ) + dhx (ab1 + A1 z ): In particular, U_ = ?xTWx + z~T Q2 (Ab2 ? dhxAb1 )z ? (Ab2 ? dhx Ab1 )zb) = ?xTWx + z~T Q2 (Ab2 ? dhx Ab1 )~z = ?xTWx + 12 z~T (Q2 (Ab2 ? dhx Ab1 ) + (Ab2 ? dhx Ab1 )T Q2 )~z hence U_ ?xTWx ? 21 z~T W2 z~ 0: (13) Since W is positive de nite we see that x 7! 0 as a consequence of Lyapunov's direct method. If W2 is positive de nite then standard arguments show that both x and z ? zb converge to zero exponentially.
3.4 Servo motor control
As in [5] we consider the problem of controlling a servo system with friction. Our model is mx = Ku ? F (x;_ z) (14) z_ = x_ ? gj(x_x_j) z; where the average bristle de ection z is related to the friction F by F (x;_ z) = 0 z + 1 (x_ ? gj(x_x_j) z) + 2 x:_ (15) The positive bounded function g describes the Stribeck e ect and is modelled by [5] x_ 2 g(x_ ) = 1 (F + (F ? F )e?( vs ) ); (16)
0
c
s
6
c
where 0 > 0: Setting x1 = x; x2 = x_ this model takes the form of (3) with 1 jx2j ]. a2 = [x2 ]; A2 = [? gj(xx22j) ]; a1 = [x2 Km u ? 1m+2 x2 ]T ; A1 = [0 ? m0 + mg (x2 ) If we could measure z directly then the feedback controller m ( x + x + 1 F (x;_ z)) (17) unom (x; z) = K 1 2 m achieves x_1 = x2 ; x_2 = x1 + x2 . Thus for = ?2a2 ; and = ?2a (where a > 0) we have system dynamics described by x_ = Ax where the eigenvalues of A are ?a + ai; ?a ? ai. This gives a reasonable tradeo between speed of response and accuracy. If W is a diagonal matrix with diagonal entries 2h > 0 and 2l > 0 then
Q1 =
"
h 3h + al 2a h 2ha2 1 2a ( 2a2 + l) 2a2
#
is the positive de nite solution to Lyapunov's equation AT Q1 + Q1 A = ?W . Thus for u = unom , V = xT Q1 x we have dtd V (x(t)) = ?xT (t)Wx(t) 0: To show that assumptions A1, A2, A3, and A30 hold for this system we can proceed as follows: suppose that d is chosen so that 0 < g(x) d and jz(0)j d. Then dtd z2 (t) < 0 when jz(t)j > g(x2 (t)) and hence jz(t)j d 8t 0 (Section 3 of [5]). Thus assumption A1 holds. For V (x); W , and unom (x; z) as above, A2 holds. Assumption A3 is satis ed for Q = I33 h iT and K (x; z ) = I22 ?A1 (x; unom (x; z )) ; and assumption A30 holds if Q2 = [ ] for any real > 0 and h(x) = 0. The dynamic controller (12) takes the form (here zb = p)
u = Km ( x1 + x2 + m1 (0 zb + 1 (x2 ? gj(xx22j) zb) + 2x2 )); zb_ = x2 ? gj(xx22j) zb + m 2 (?0 + 1 gj(xx22j) )( 2ha2 x1 + 21a ( 2ha2 + l)x2 ):
(18)
Note that the observer proposed in [5] has the form of (18) but uses kx1 instead of 2 jx2j h 1 h T Q?2 1 AbT1( @V @x ) = m (?0 + 1 g(x2 ) )( 2a2 x1 + 2a ( 2a2 + l)x2 ): This results in a somewhat simpler observer, but one which severely restricts the choice of pole locations in the compensated system because in [5] the transfer function relating the estimation error to the tracking error must be strictly positive real. Note that ; h; and l are design parameters. When 02h x1 , hence our dynamic controller x2 = 0 equation (18) becomes zb_ = ? ma
7
will provide, at least momentarily, an integral control action. Thus reducing
or increasing h will e ectively increase the integral gain. If we want x(t) to track a function xd (t) we use states e1 = x1 ? xd and e2 = x2 ? e_d and adjust (17) accordingly. It is clear that Theorems 3.1 and 3.2 can also be applied to tracking problems.
Remark 3.3 We note that A30 also holds for h(x1 ; x2 ) = ? m1 x2 and W2 =
[2 10 ]. For this choice of h the observer (12) takes the form
zb = p + h(x) 0 m0 m p_ = (1 + m 1 )x2 + 1 x1 ? 1 p + 12 x2 2 (?0 + 1 jx2j )( h2 x1 + 1 ( h2 + l)x2 ? m p + m2 x2 ) + m 2a 2a 2 21 g(x2 ) 2a and from (13) we see that
(19)
U_ ?2hx21 ? 2lx22 ? 0 z~2 0: 1
Thus Theorem 3.2 implies that x1 ; x2 ; and zb converge to zero at an exponential rate. The observer-controller (19) resembles the exponentially stable observer described in [11].
4 Experimental Results
4.1 Model Identi cation and Validation
Experiments were conducted using the linear positioning table shown in Figure 1. The apparatus consists of a 2 meter long Techno/Isel Double Rail Linear track and with a Roller Carriage utilizing recirculating ball bearings. For this experiment two carts where locked together to form a positioning table. The carts are joined to a 24 volt DC servo motor by no-slip Synchromesh Drive Cable system. This permits fast and accurate movement of the table. The position of the carts is measured by an optical encoder attached to the DC motor which results in a linear resolution of 19; 000 q-counts per metre. From (2) the steady state bristle de ection, zss; is found to be zss = g(v)sgn(v):From (16), (15), and (4.1) the the steady state friction force is modelled by
Fss = (FC + (FS ? FC )e?(v=vs )2 )sgn(v) + 2 v: 8
(20)
Figure 1: Photo of cart and track assembly This means that there are 6 parameters, 0 ; 1 ; 2 ; FS ; FC ; and vs, to identify. Because equation (20) is linear in all parameters except vs , it lends itself to least squared error parameter estimation. The characteristic Stribeck velocity depends on lubrication and material properties and is usually found empirically. It ranges from 0:00001 to 0:1 m=s (c.f. [3]). In our case vs = 0:005 m=s was found to work well. It follows from (20) that, in the case where v 0,
Fss = FC + (FS ? FC )e?(vss =vs )2 + 2 v;
(21)
where vss represents steady state velocity. In vector form equation (21) can T be expressed as Fss = aT x;where a = 1 ? e?(vss =vs )2 ; e?(vss =vs )2 ; vss ; x = (FC ; FS ; 2 )T : Applying various constant voltages between 4 and 14 volts, the steady state velocity was measured. Twenty experiments were performed and the least squares estimate for x yielded the parameter estimates FC = 2:0 N , FS = 2:15 N and 2 = 4:6 Ns=m: 9
Steady state friction data yields no information about 0 , which represents bristle sti ness, or 1 , which is related to damping during friction transients. On the other hand presliding motion is governed by
mx = ?(0 z + 1z_ + 2x_ )
(22)
within the static friction regime, with u 0. For our motor-cart system x represents the displacement, in meters, of an equivalent 1.62 kg mass. Within a small operating range displacement can be attributed entirely to bristle de ection. Linearizing (22) about z = 0 and x_ = 0 yields mx+(1 +2 )x_ +0 x = 0;a damped second order system with spring constant 0 and viscous damping coecient 1 + 2 . Using 0 = 105 N=m, our model provided a good prediction for the actual system response. Simulations indicate that small variations in 0 do not a ect the open-loop response. Hence 0 = 105 N=m was adopted. The viscous friction 2x_ has little e ect at low velocities, where the damping force depends mainly on 1 . For moderate damping (a damping factor of :5) we have 1 495 Ns=m, 2 = 4:6 Ns=m. Simulations and our experiments revealed that minor variations in 0 and 1 have little a ect on the open and closed loop response but the closed loop response is adversely a ected by overestimation of Fc and FS . The result of underestimating these 2 parameters is a small reduction in performance, while even small overestimation results in undesirable limit cycles. 1.2
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Figure 2: Response to ramp input of u = 2t
Figure 3: Response to decreasing input, u = 10=(1 + 2t)
S The graphs in Figures 2 and 3 indicate our system's actual and predicted open loop response to two di erent inputs. The solid line, labeled 10
\Actual System Response", shows the position of the platform as measured by the encoder. The line labeled \Linear Model Prediction" is the response predicted by a system model with (linear) friction model F = 2 x_ . The line labeled \Nonlinear Model Prediction" indicates the response predicted by the system model with the friction model (15). The response to a ramp input is displayed in Figure 2. The nonlinear model does a good job in modelling the sticking of the platform when the armature voltage is small. The system's response to a decreasing input are illustrated in Figure 3. Here the \sticking" of the platform at lower voltages is successfully captured by our nonlinear model, whose prediction for position is accurate to within 10%. The error in the linear model prediction is almost 100% at t = 1s and increases steadily thereafter. These experiments clearly show that, for our apparatus, the more complicated LuGre model for friction proposed in [5] is far superior to the linear friction model.
4.2 Lund-Grenoble Dynamic Feedback Controller
In this section the dynamic feedback controller proposed by Canudas de Wit, Olssen, Astrom and Lischinsky [5] (abbreviated here as the LuGre controller) is implemented on our apparatus. Th friction observer developed in [5] is as follows:
z^_ = v ? gj(vvj) z^ ? ke; k > 0
(23)
where the state z^ estimates the average bristle de ection z . Theorem 1 of [5] severely restricts the choice of control law by requiring that the transfer function, G(s), between the position error and the observer error be strictly positive real (SPR). Since this sucient condition is not necessary there will be cases where this their controller is e ective even when G(s) is not strictly positive real (SPR).Using a PD controller with friction compensation yields 1 s + 0 : G(s) = ms2+ k2 s + k 1 The requirement that G(s) be SPR implies that k1 > 0; k2 > 01m : In our case this implies that k1 > 0 and k2 > 404. The derivative gain k2 is large and leads one to expect over-damping and noise propagation problems. A value of k = 0:001 was used in the observer for bristle de ection, equation (23). This resulted in a good tradeo between rapid observer convergence and a fast rate of convergence rate of x to zero. Figure 4 shows the response of the LuGre controller to tracking a step function of xd = 10 cm. Here 11
the proportional gain used was k1 = 350 and k2 = 450. The response is reasonably accurate but very slow due to excess damping in the system. 0.2
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Figure 4: Step response Lund-Grenoble Figure 5: Tracking a cosine using a PID controller and new dynamic controller controller
Increasing the proportional gain decreases the response time but results in high frequency vibration which is probably a consequence of the high gain feedback stimulating unmodelled dynamics. Of course there will be cases where the error transfer function is not strictly positive real but nonetheless this controller functions well. In our experiments we found that signi cantly reducing the damping K2 destabilized the control algorithm. In this context our new dynamic feedback controller (17) allows the use of any feedback gains which place poles in the left-half complex plane.
4.3 New Dynamic Feedback Controller
We rst use our controller (17) to track a 10 cm step and compare its performance with that of the Lund-Grenoble controller. We then try to track a cosine curve using our new controller and also a PID controller tuned to perform this task. To track the step function we used a proportional gain of k1 = 600 and a derivative gain of k2 = 50. Since we can use any feedback gains which place poles in the left-half complex plane we are able to choose a small derivative gain and place the second order poles at ?10 10i for a damping ratio of approximately 0:7. Of course we don't known z exactly and thus we have a nonlinear system which, strictly speaking, does not have poles, but, in 12
the case where z^ = z , we have linear dynamics for x and can talk about the system's poles. The performance of our new controller was compared to that of the Lund-Grenoble controller. For these experiments the constants h; l, and of equation (18) were assigned values of h = 5, l = 0:05, and
= 0 : These seemed to result in a reasonable compromise between the rates of convergence of the tracking and bristle estimation errors. Figure 4 shows the response of our system tracking a 10 cm step using the dynamic controller (18) . From Figure 4 we see that both controllers performed well in this experiment, but our new controller was able to respond more quickly because we can employ a more modest level of damping. Next we tuned a PID controller to make our platform track a cosine curve. As above we choose proportional and derivative gains to place the systems poles at ?10 10i and then experiment with various integral gains k. With k = 0 we found signi cant steady state error as well as saturation of the control initially. With k = 500 the steady state error attenuated too slowly. With k = 1000 we achieved reasonable performance (see Figure 5). With k set to 4000 we experiences excessive overshoot and saturation of the control at both +24 and ?24 volts (see Figure 5). On the other hand our dynamic controller (18) did an excellent job in regulating the path of the platform as can be seen in Figure 6. Note that after the target trajectory is reached almost no tracking error occurs at the times of peak platform acceleration. In Figure 7 we compare the performance of the tuned PID controller with our controller and note that both controllers use similar levels of control voltage. 0.15 0.1
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Figure 6: Tracking a cosine using our dynamic controller
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Figure 7: Our dynamic controller vs a PID controller
5 Conclusions We have developed a dynamic controller for a class of nonlinear systems ane in the unmeasured states. This class includes servomechanisms which incorporate the nonlinear Lund-Grenoble friction model introduced in [5]. The parameters in the Lund-Grenoble bristle model of [5] were identi ed for our high speed positioning table and both the open loop performance of the resulting model and the closed loop performance using the LuGre nonlinear controller were examined. The model accurately predicted the actual state evolution, and the controller of [5] achieved high accuracy but modest speed of response due to high damping needed for stability in our application. Our new controller avoids this restriction and was superior to a tuned PID controller in tracking a time varying curve. Both the PID and our dynamic controller required similar levels of control voltage. It was found that the performance of our new dynamic controller was adversely a ected by the overestimation of two of the nonlinear friction parameters Fc and Fs related to the Stribeck e ect. Slightly underestimating Fc and Fs caused only a small degradation in performance. While the design of adaptive controllers is beyond the scope of this work, we did nd that the performance of our controller-observer was relatively una ected by moderately large (30%) deviations in our friction parameter estimates, except for overestimation of the parameters Fc and Fs . Adaptive controller design was beyond the scope of this investigation, but it may well be possible to generalize the approach used in [11] to construct adaptive controllers which yield asymptotic position tracking while compensating for certain parameter uncertainties. We note that in [11] uncertainties in Fc and Fs are not compensated for directly.
References [1] Amin, J., Friedland, B., Harnoy, A., \Implementation of a Friction Estimation and Compensation Technique", IEEE Control Systems Magazine , August, 1997. [2] B. Armstrong-Helouvry, Control of Machines with Friction, Boston, MA: Kluwer, 1991. [3] Armstrong-Helouvry, B., Dupont, P., Canudas de Wit, C., \A Survey of Models, Analysis Tools and Compensation Methods for the Control of Machines with Friction", Automatica, Vol. 7, No. 9, 1994. 14
[4] Canudas de Wit, C., Noel, P., Aubin, A., Brogliato, B., \Adaptive Friction Compensation in Robot Manipulators: Low Velocities", International Journal of Robotics Research, Vol. 10, No. 3, June 1991. [5] Canudas de Wit, C., Olsson, H., Astrom, K.J., and Lischinsky, P., \A New Model for Control of Systems with Friction", IEEE Trans. Control Systems Technology, Vol. 2, Sept. 1995. [6] Isidori, A. and Byrnes, C. I., \Output regulation of nonlinear systems", IEEE Trans. Auto. Control, Vol. AC-35, No. 2, 1990. [7] LaSalle, J.-P., \Stability theory for ordinary di erential equations," Journal of Di erential Equations, Vol. 4, 1968. [8] Praly, L., Bastin, G., Pomet, J.-B. and Jiang, Z. P., \Adaptive Stabilization of Nonlinear Systems", to be published in the series: Lecture Notes in Information and Control, Springer-Verlag, 1991. [9] Pomet, J.-B., Hirschorn, R.M., and Cebuhar, W., \Dynamic Output Feedback Regulation for a Class of Nonlinear Systems",Math. Control Signals Systems, 6:106-125, 1993. [10] Tataryn, P.D., Sepehri, N. Strong, D., \Experimental Comparison of Some Compensation Techniques for the Control of Manipulators With Stick-Slip Friction", Control Engineering Practice, Vol. 4, No. 9, Sept. 1996 [11] Vedagarbha, P., Dawson, D.M., Feemster, M., \Tracking Control of Mechanical Systems in the Presence of Nonlinear Dynamic Friction Effects", Proceedings of the American Control Conference, Albuquerque, NM, June, 1997. [12] Yang, S., Tomizuka, M., \Adaptive Pulse Width Control for Precise Positioning Under the In uence of Stiction and Coulomb Friction", Journal of Dynamic Systems, Measurement, and Control, Vol. 110, Sept. 1988.
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