Friction Induced Hunting Limit Cycles: An Event Mapping Approach

Transcription

1 Friction Induced Hunting Liit Cycles: An Event Mapping Approach Ron H.A. Hensen, Marinus (René) J.G. van de Molengraft Control Systes Technology Group, Departent of Mechanical Engineering, Eindhoven University of Technology (EUT), Eindhoven, P.O.Box 513, 56 MB, The Netherlands. Eail: Abstract This paper studies the occurence of liit cycles for a siple PIDcontrolled otion syste subjected to different shapes of static frictional daping functions. The friction characteristics are copared with respect to the so-called hunting phenoenon. In particular, a classification with respect to the ability to predict both stable and unstable liit cycles is obtained. Furtherore, the local stability of the equilibriu points is discussed and attractor basins are given. To perfor this analysis, the closed-loop dynaics are represented with an equivalent one-diensional ap representing a so-called event ap. 1 Introduction In general, echanical systes exhibit friction. A negative anifest of friction is a degeneration of perforance in ters of increasing tracking errors and friction induced liit cycling, i.e., periodic solutions of a nonlinear autonoous syste, for instance for a servo syste. Due to its oscillatory and persistent behavior, liit cycling is an undesirable phenoenon in controlled servo systes. Here, a friction induced phenoenon called hunting [1] is considered, which is ainly caused by the cobination of a difference in the static friction level and the Coulob friction and an integral action in the control loop for a siple regulator task. The research on friction induced stick-slip oscillations has been extensive over the past decades [1,, 5, 6, 8, 1, 11, 1]. Friction induced liit cycles have been analyzed in literature with different techniques and various results, e.g., the Describing Function Analysis (DFA) [1,, 5] was widely used but resulted only in qualitative prediction and even these were often found to be incorrect [, 13]. Other iportant analysis tools are (i) Phase Plane Analysis (PPA) [6, 1], (ii) exact algebraic analysis [, 11] and (iii) analysis tools originating fro nonlinear dynaics [7, 8, 1]. The latter sees to be the ost powerful due to the ability to deal with nonlinearities in the friction odel, however are ainly based on expensive nuerical investigation of the closed-loop syste. For the hunting phenoenon, [8] shows for two nonlinear friction odels, i.e., the dynaic LuGre friction odel [3] and the static Switch friction odel [1], coparable results with respect to periodic solutions for the nonlinear closed-loop dynaics. A nuerically constructed bifurcation diagra predicts for both odels a disappearance of the hunting liit cycles for certain controller settings which is confor the practically observed behaviour, where industrial regulator probles are tuned by hand and indeed prevent stick-slip oscillations. In contrast to these results are the exact algebraic analysis by [] for a PID-controlled siple ass syste in the presence of static friction and a lower Coulob friction level. For this friction odel with a discontinuous jup in the friction level the authors state that the controlled syste will lead to a frictional liit cycle for all stabilizing controllers. The ajor difference between this idealized friction odel and the static Switch friction odel used in [8] is the shape of the friction curve for velocities unequal to zero. Therefore the focus in this paper will be on the influence of the shape of the friction curve on the dynaic properties of the closed-loop syste, i.e., stability of equilibriu points, types of periodic solutions and the possibility to predict the disappearance of the hunting phenoenon. The various friction curves with a continuous drop in the friction level and a variable shape will be integrated in the Switch friction odel [8, 1] and the resulting nonlinear closed-loop dynaics will be analyzed with an event apping technique [6] which is a Phase Plane Analysis siilar to the technique used in [1]. Here, the onediensional apping technique is used to construct the so-called event ap of the nonlinear closed-loop dynaics. Construction of the event ap will be perfored through nuerical siulation of the closed-loop syste and the ap will give insight into the closedloop dynaics together with stability properties and attractor basins of the periodic solutions and equilibriu set. Besides the nuerical stability analysis of the equilibriu points, the exact algebraic results obtained by [] will be used to perfor an analytic analysis of the equilibriu points for different frictional daping characteristics. The outline of this paper is as follows. A description of the closedloop syste will be given in Section together with the various friction characteristics used. For the different frictional daping curves, a classification will be derived with respect to the local stability of the equilibriu set in Section 3. In Section, the construction of the equivalent one-diensional ap of the nonlinear discontinuous three-diensional autonoous dynaic syste will be discussed. The nuerical results on the location of the liit cycles, both stable and unstable, together with their regies of attraction will be given in Section 5 for the various friction curves. The paper will be concluded in Section 6 and future research topics will be addressed. The Closed-loop Syste For echanical positioning systes, often odeled as a siple otor-driven ass syste subjected to friction as shown in Figure 1, an iportant control proble is the regulator task, where due to a static friction level an integral action in the control loop is necessary to eliinate steady state errors. Hence, the siplest odel of the closed-loop syste reads as follows Z t q + P (q + D _q + I q(fi )dfi )= F where q is the position of the syste with respect to the desired position, F the friction force, the ass of the syste to be controlled and P, I, D the controller paraeters. For certain frictional 67

2 Control effort u Mass q Friction Force F Figure 1: Siple ass syste subjected to friction. daping functions and PID-controller settings a liit cycle will exist around q =, which is called hunting [1]. Sufficient conditions to generate or siulate friction induced hunting liit cycles are: Λ integral control, Λ good stiction properties of the friction odel, i.e., in the stick phase the su of the external forces on the ass syste are copletely copensated with an equivalent friction force and Λ a frictional daping function with a larger static friction level than the Coulob friction level [, 8, 1], i.e., either a continuous or discontinuous drop in the friction force for velocities unequal to zero. An applicable friction odel to siulate stick-slip oscillations, like the hunting phenoenon, is the discontinuous Switch friction odel as deonstrated in [8, 1]. The Switch odel [1] is a static friction odel that can be considered as a odified version of the Karnopp odel [9]. The idea is to odel the friction force as a set-valued function at _q =. The friction force is defined as F (_q; u) =ρ g(_q)sgn(_q) +b _q; 8 _q 6= Slip in(juj;f s)sgn(u); 8 _q = Stick R t where u = P (q + D _q + I q(fi )dfi ) is the applied control effort, F s is the axiu static friction at zero velocity, g(_q) is the frictional daping curve for velocities unequal to zero and b is the viscous daping. The equation of otion, describing the closedloop syste, reads as _x = f Switch (x) = x P (x1 + Dx + Ix3) 1 F (x;u) x (1) where x = [x 1 x x 3] T = [q _q R t q(fi )dfi ]T and the control effort u = P (x 1 + Dx + Ix 3). The existence of solution of this Filippov-type of systes Eq. 1 is not trivial and therefore the uniqueness of solution is also questionable. In [] it is shown that for this syste a solution of the differential inclusion, corresponding to the tie derivative of the second state equation, i.e., _x F Switch (x) = 8< : u g(x ) bx ; x > u +[ F s;f s]; x = u + g(x ) bx ; x < exists if the set-valued function F Switch (x) is upper seicontinuous, closed, convex and bounded for all x R 3. To analyze the existence and the uniqueness of the solution of the differential inclusion Eq., a choice for the frictional daping function g(x ) has to be ade. Here, the frictional daping function is odeled with the widely used Stribeck curve, i.e., g(x ) = F c +(F s F c)exp( j x j fi ),wheref s is the static friction level, F c the Coulob friction level, is the so-called Stribeck velocity and fi deterines the shape of the friction curve () Friction Force [N] Different shaped Stribeck curves F s - - F c Velocity [/s] vs Control Effort u F s F c -F c -F s - li F x switch li F x switch + Velocity [/s] Figure : Different Frictional daping characteristics and the attraction sliding ode. for velocities unequal to zero. Exaples of these friction odels with different values for the shaping paraeter fi are given in the left plot of Figure. For the different daping curves the solution of the differential inclusion Eq. exists since F Switch (x) is upper sei-continuous, non-epty, closed, convex and bounded. The set [ F s;f s] at x =is the sallest closed convex set that contains the left and right liits of F Switch (x) to x =, as depicted in the right plot of Figure. As shown in the figure, an attraction sliding ode at x =exists, since if F s» u» F s,then _x < for x > and _x > for x <. Moreover, at x =a transversal intersection exists if juj >F s. Therefore, uniqueness of the solution of the differential inclusion Eq. and the equation of otion Eq. 1 is ensured. As for the Karnopp friction odel, a narrow band fi 1 around zero velocity is introduced to nuerically integrate the syste of equations of the Switch odel. Due to this narrow stick band the odel distinguishes three situations: (i) the slip phase, (ii) the stick phase and (iii) the transition phase, which all will be described by ordinary differential equations (ODEs). The transition phase describes the transition fro stick to slip or velocity reversals without stiction. In the stick phase, the acceleration of the ass is set to ffx and forces the velocity x to zero for ff>. This acceleration ter ensures that the ODE belonging to the stick phase does not suffer fro nuerical instabilities as in the Karnopp friction odel. 3 Local stability of equilibriu points The equilibriu points of the autonoous syste, as described in Eq. 1, satisfy the condition _x = f Switch (x) =. Hence, x 1 and x should be zero, which results for the closed-loop syste to be in the stick phase and therefore jx 3j» Fs PI. Since it involves a regulator task, it is iportant to realize that the equilibriu points of interest are indeed those for which the position error x 1 and velocity error x are zero. The integral ter x 3 shoulot necessarily be zero. Hence, the set of equilibriu points are located on the line x Λ = fl [1] T ; 8jflj» Fs PI : For the equilibriu points, the applied control effort u = PIx Λ 3 is copensated by an equivalent portion of the static friction F s. The local stability of such a set of equilibriu points on a line is not trivial. Here, the results presented by [], which are obtained with an exact algebraic analysis of the sae closed-loop syste, are used to analyse the equilibriu set. With respect to the ability to proote hunting liit cycling they report: By analyzing the tie-doain response of a servo with 68

3 PID position control and Coulob or Coulob + static friction, it has been shown that no syste with Coulob friction alone will show a liit cycle in the neighbourhood of the origin, and that every syste with inial Coulob friction anonzero static friction has such a liit cycle as a possible otion... when the Coulob + static friction odel is valid and the relative agnitudes of Coulob and static friction lie in their noral range, there is no cobination of PID control paraeters that will eliinate stick-slip., where the ters nonzero static friction and noral range express the range of the paraeters F s >F c >. The basic idea to analyze the local stability of the equilibriu points is to linearize the different frictional daping characteristics fro above and fro below to x =, i.e., linearize the state equations for the slip phase of Eq. 1. If the resulting closed-loop syste, with the linearized slip phase ebedded in Eq. 1, can locally be represented as a PID-controlled syste with Coulob friction alone the equilibriu points are locally stable. The exact algebraic results show for such a closed-loop syste that no hunting liit cycle exist on an open interval near the origin in x 1 direction, i.e., solutions which start in the stick phase. Starting on this open interval the equilibriu set will be reached either with asyptotic convergence or with a contracting sequence of stick and slip transitions. A sufficient condition for these algebraic results is that the linear part of the closed-loop dynaics, i.e., the hoogeneous part of Eq. 1, is asyptotically stable. To perfor the analysis on the local stability of the set of equilibriu points for the above discussed frictional daping functions, i.e., g(x )=F c +(F s F c) exp ( j x j fi ), the following Taylor expansion at x =for fi > 1 can be constructed: g(x fi x + R(x ) ß F s; y= where R(x ) is the rest ter of this expansion with R() = =and vs is supposed to be positive. For fi =1this daping function g(x ) is not differentiable at x =, but let us analyse this function for positive velocity x going to zero: g(x ) = li y# fi x + R(x ) y# ß F s (Fc Fs) x ; where R(x ) is again the rest ter with li y# R(y) j y# =. For negative velocity x going to zero the following approxiation holds g(x ) ß F s (Fc Fs) v x s. Cobining both results for fi =1the linearization can be written as: g(x ) ß F s + (Fc Fs) jx j: For the range < fi < 1 of the shaping paraeter, the function g(x ) is also not differentiable at x = and oreover its first derivative with respect to x is undefined for both positive and negative velocity x approaching zero. Hence, the nonlinear frictional daping function in the slip phase can not be linearized for <fi<1. Using the different linearizations of the nonlinear frictional daping characteristics in the slip of the friction force, the following linearized friction forces are found about x =: F (x ) ß 8< : Fssgn(x) +bx; 8 fi>1 F ssgn(x )+ (Fc Fs) v x + bx s ; if fi =1 undefined; 8 <fi<1 Substitution of either the first or second linearization in the slip phase of the state space representation of Eq. 1, the closed-loop syste is locally reduced to a PID controlled siple ass syste subjected to Coulob friction only, where the Coulob friction level is equal to the axiu static friction level of the original syste. Moreover for fi =1, the linear dynaics in the slip phase of the closed-loop syste is odified by the linearization. Hence, a sufficient condition for the local stability of the equilibriu set can be described as follows: if the linear part of the dynaics in the slip phase are asyptotically stable the equilibriu set is locally stable. For the first and second region of fi, the characteristic polynoial of the hoogeneous part of the linearized closed-loop dynaics about x =are: s 3 +( PD+ b s 3 + PD+ b s + P s + PI 8 fi>1 Fc Fs + )s + P s + PI 8 fi =1: For <fi< 1, the linearization is undefined, which disables us to construct a characteristic polynoial for this region. Using the Routh Hurwitz stability criterion for the linear part of the linearized closed-loop dynaics, the following classification on the stability of the set of equilibria can be obtained (under the assuption that the linear part of dynaics of the original closed-loop syste, i.e., represented by the characteristic polynoial corresponding to fi> 1, are asyptotically stable): Λ Stable for daping functions with continuous stick-slip transitions and fi > 1, since the linear part of the dynaics of the linearized closed-loop syste are the sae as for the original syste. Λ Stable for daping functions with continuous stick-slip transition with fi =1if ( PD+b (Fs Fc) I) >, which is a sufficient condition. Λ For daping functions with continuous stick-slip transitions where < fi < 1, no sufficient condition on the local stability of the set of equilibriu points can be obtained, since the linearization is undefined. Event Mapping: a Phase Plane Analysis To study the global three-diensional non-linear dynaics of the closed-loop syste for the various daping functions, a technique siilar to a Phase Plane Analysis [1] is used. With respect to the friction induced hunting liit cycles the syste ight be described by a one-diensional iterated apping as shown by [1]. Such a reduction of diension can only take place for a syste with sufficiently strong dissipation in the phase space [6]. For the hunting stick-slip oscillations, an infinite dissipation occurs when the closed-loop syste enters the stick phase, i.e., a finite volue in the slip phase is reduced to zero volue in finite tie in the stick phase. To illustrate the event apping technique, as presented by [6], the closed-loop syste of Eq. 1 is considered for a daping function with fi =. The syste is initialized in the stick phase with x()=[:1 ] T and the closed-loop paraeters used in this 69

4 Table 1: Closed-loop syste paraeters. Mass =1 [kg] Static friction level F s = [N] Coulob friction level F c = [N] Stribeck velocity =:1 [/s] Shape paraeter fi = [-] Viscous daping b =1 [N.s/] Controller gain P =1 [N/] Derivative action D =1 [s] Integral action I =1 [1/s] Narrow stick width =1e 8 [/s] Acceleration coefficient ff =1 [1/s] siulation study are given in Table 1. In the upper plot of Figure 3, the tie response of the syste is shown for all three states. The equilibriu points define an continuous line E with an upper and a lower bound for x 3 as depicted in the phase portrait in the lower plot of Figure 3. x3 x x1 3 x Tie [sec] Event1 Event Event3 Event d d -.5 d 1 1 Phase Portrait of Response 3 E d 3 x x Figure 3: Closed-loop state responses. Stick Plane The stick plane defines the phase where (i) the ass has zero velocity and (ii) the control effort u is saller or equal to the axiu static friction level R F s. Depending on the actual state x 1 the integral ter x 3 = e t x t 1dfi will increase or decrease in the stick plane. The integral ter builds up and eventually the control effort will overcoe the axiu static friction level. Hence, the syste enters the slip phase and this transition is defined as the beginning of an event and depicted with a circle in Figure 3. The return transition into the stick phase, depicted with a square, defines the end of an event. Two successive events are separated by one stick phase that can be characterized by the state x 1 = d which is constant during the stick phase. The consecutive values of the state x 1 in stick phase can be represented by a one-diensional ap in the ( ;+1 ) plane. Such a ap is the so-called event ap and its graphical interpretation is in general uch sipler than that of the global otion and oreover the ap is a single valued function P( ). For the otion presented in Figure 3 the equivalent one-diensional ap is given in Figure, where the cobination (d 1 ;d ) is iterated to (d ;d 3 ) and successively (d ;d 3 ) to (d 3 ;d )..1. These three points, depicted with big dots, are part of the event ap and describe only a very sall part of the global dynaics of the syste. P( ) = (d,d ) 3 (d,d ) 1 (d,d ) Figure : Nuerically constructed event ap of the closed-loop syste for a frictional daping function with fi =. To investigate the global dynaics of the closed-loop syste, the function P( ) has to be coputed, which will be constructeuerically since it is ipossible to derive an analytic expression for the event ap of the closed-loop systes with non-linear daping functions. To construct the event ap, the syste is initialized in the stick phase for different initial positions x 1 = d 1 and integrated nuerically until the syste enters the stick phase again. Hence, on a grid d 1 = [; d;:::;d d; d] the next iterated points d of the ap are coputed. These points are eleents of the single valued function P( ) defining the global dynaics of the closed-loop syste. Due to the fact that the closed-loop syste is odd with respect to the origin the function P( ) will also be odd, i.e., P( )= P( ) and therefore the nuerical integration is liited to positive initial positions. The eleents of the function P([d;d + d; : : : ; d d; d]) with d = :1 and d = d = :5, as depicted with sall dots in Figure, give an ipression of the global dynaics. To locate a liit cycle, i.e., a closed trajectory, the second iterated event ap is of interest, since the first iterated ap P( ), as shown in the figure, aps an initial state to a state +1 which has an opposite sign after one event and will not close the trajectory iediately, as can be seen Figure 3. However, the ap describing an initial state to a state + which has possibly the sae sign after two events ight close the trajectory at once when = +. This second iterated ap is defined by + = P(+1 )=P(P( )) = P ( ) and the fixed points of the ap P locate both (i) the equilibriu points of the syste and (ii) the liit cycles. The fixed points of the ap are located at d Λ = f j+ = P ( )= g and the local stability of the fixed points of the ap can be deterined by the contraction theore [1]. A fixed point of the ap are locally stable if the response is locally contracting to this point, i.e., the slope of the ap P in the fixed point of interest has an absolute value saller than 1. The fixed point are locally unstable if the response is locally diverging fro this point, i.e., the slope of the ap P in the fixed point of interest has an absolute value larger than 1. In Figure 5, the second iterative event ap P ( ) is shown together with its fixed points d Λ fd Λ 1;d Λ ;d Λ 3g. The exact location of the liit cycles are coputed by approxiating the single 7

5 value function P ( ) in their vicinity with a third order polynoial through four neighbouring data points. Through the origin, defining the equilibriu set, the function P ( ) is approxiated with a line through three data points including the origin. These approxiation are used to locate the fixed points of the ap and their local stability can be deterined. In Table, the occurring types P (d ) = + n d d Magnified d 1 P (d ) = + n d 3 -d x 1-3 Magnified d d 1 - -d x x 1-3 a) Second iterated event ap for the syste with Magnified P (d ) = + n. d. -d -... d 1 -. Figure 5: Second iterated event ap locating the equilibriu set and liit cycles. Table : Fixed points of the second iterated event ap P ( ). Fixed point Type of solution Local stability d Λ 1 = Equilibriu set Stable d Λ =:3 Liit cycle Unstable d Λ 3 =:53 Liit cycle Stable of solution together with their location and local stability are given. The equilibria of the closed-loop syste, for a daping function with shaping factor fi =, are indeed locally stable as predicted analytically in the previous section. Moreover, there exist for this syste two liit cycles, where one is stable and one is unstable. Another interesting feature of the one-diensional event ap is the ability to describe the attractor basins for the different solutions, i.e., ( d Λ ;d Λ ) will eventually end up in the origin and therefore in the equilibriu set and starting [ :5; d Λ ); (d Λ ; :5] will end up in the stable liit cycle with aplitude d Λ 3. 5 Nuerical Results In this section, the location of the equilibriu points and liit cycles together with their local stability, are deterined for two other daping functions, i.e., with shaping paraeter fi =1and fi = :5. Again the corresponding event ap P ( ) is used to analyse the various systes where the closed-loop syste paraeters are chosen equal to the paraeters in Table 1 except for the shaping paraeter fi. In Figure 6 the various second iterated event aps are given and for the various daping functions the closedloop syste shows only one liit cycle, which are depicted with d Λ, which is clearly different with the two liit cycles found for the daping function with fi =in Section. For fi =1the equilibriu set d Λ 1 is unstable as shown in the agnified part of Figure 6 a), since the slope through the origin is larger than 1. Using the sufficient condition for the stability of the equilibriu set as derived in Section 3, this result can be tested analytically, i.e., ( PD+b I) =1 [1/s] should be larger than (Fs Fc) =[1/s], which is not sat- isfied. The nuerical results are in line with the analytic result, however this condition can not be used in this sense since the condition is only a sufficient one. The details about the fixed points for the event ap for fi = 1 are given in Table 3. For the daping function where fi =:5 an interesting phenoenon appears for the x 1 b) Second iterated event ap for the syste with Figure 6: Second iterated event ap locating the equilibriu set and liit cycles for various daping functions. Table 3: Fixed points of P ( ) for the various daping functions. Shape Fixed point Type of solution Local stability fi =1 d Λ 1 = Equilibriu set Unstable fi =1 d Λ =:31 Liit cycle Stable fi =:5 d Λ 1 = Equilibriu set Unstable fi =:5 d Λ =:1786 Liit cycle Stable equilibriu set, i.e., in the neighbourhood of the origin. The event aps, as depicted in Figure 6 b), see to be discontinuous around the origin predicting unstable sets of equilibriu points. This nuerical result is also in line with the analytic results on the stability of the equilibriu set for daping functions where < fi < 1, since the hoogeneous part for this linearized closed-loop syste are always unstable. Moreover, the attractor basin for the stable liit cycle is the entire doain excluded the origin. The details on these two daping functions are also given in Table 3. An interesting question is what will happen when for instance the controller gain P is altered. Moreover, it isof interest to know how the syste characteristics change for the different daping functions. Hence, the fixed points of the second iterated event aps together with their local stability for the various daping functions, on a grid P = f1;:::;31g [N/] with step-size 1, are coputed and shown in Figure 7. The so-called bifurcation diagra of Figure 7 a), predicts for the frictional daping function with fi =, a disappearance of the hunting liit cycles for controller gains larger than approxiately P =1[N/]. The equilibriu set is for all coputed controller gains locally stable as expected by the analytic results presented in Section 3. Figure 7 b) presents the changing dynaics for the syste with fi =1, where the unstable equilibriu set becoes stable and intersects with the stable liit cycles for controller gain P ß 7 [N/]. Larger controller gains predict only stable equilibriu points ano liit cycles. The sufficient condition derived for this closed-loop syste predicts locally stable 71

6 Fixed Points d [] i i i Stable periodic solutions Stable set of equilibriu points Unstable periodic solutions Controller Gain P [N/] a) Syste with Fixed Points d [] c) Syste with Stable periodic solutions Unstable set of equilibriu points Controller Gain P [N/] Fixed Points d [] Stable periodic solutions Stable set of equilibriu points Unstable set of equilibriu points Controller Gain P [N/] b) Syste with Figure 7: Bifurcation diagra with varying controller gain P. equilibriu points for P > Fs Fc Dv + I s D b D =9[N/]. This bound can be considered as an analytic upper bound due to the fact that this condition is only sufficient anot necessary. The equilibria of the closed-loop systes with fi =:5, as depicted in Figure 7 c), are locally unstable for all coputed controller gains. This is in line with the expected unstable set of equilibriu points as predicted by the analytic results. Fro the bifurcation diagras can be seen that a sall change in the shaping paraeter fi the closed-loop dynaics copletely changes with respect to the existing periodic solutions and stability property of the equilibriu points. 6 Conclusions and Future Topics In this paper, the closed-loop dynaics of a PID-controlled ass syste subjected to friction is represented with an equivalent onediensional ap. This one-diensional ap reveals all the nonlinear dynaical properties such as (i) stability of the equilibriu set and (ii) the existence of liit cycles with their stability properties. The so-called event ap is used to exaine the differences in the non-linear closed-loop dynaics for various frictional daping functions. As presented in the previous section it is essential to consider the different possible solutions for the different frictional daping functions. A classification on the various friction curves with a continuous transition of the friction force fro the stick phase to the slip phase is given in Table with respect to the following dynaical properties: (i) possible existence of stable and unstable liit cycles, (ii) possible disappearance of the hunting phenoenon and (iii) stability property of the equilibriu set. This classification is based on analytic observations presented in Section 3 together with the nuerical results in Section and 5. Fro this table can be concluded that practically identified friction curves where fi lies between zero and one are not able to describe the experientally observed phenoenon that hunting liit cycles can be eliinated by properly chosen PID settings. Overall, it can be concluded that inor changes in the friction odel ight cause severe changes to the closed-loop non-linear dynaics and therefore to the possible solutions. References [1] Arstrong-Hélouvry B., Dupont P., and Canudas de Wit C., A survey of odels, analysis tools and copensation ethods for Table : Classification for frictional daping functions. Frictional daping function with fi>1 Equilibriu set Stable Liit cycles Possible - both stable and unstable Eliination of hunting Possible Frictional daping function with fi =1 Equilibriu set Stable or unstable Liit cycles Possible - only stable Eliination of hunting Possible Frictional daping function with <fi<1 Equilibriu set Unstable Liit cycles Always - stable Eliination of hunting Never the control of achines with friction, Autoatica, 3, , (199). [] Arstrong-Hélouvry B., and Ain B., PID Control in the Presence of Static Friction: A Coparison of Algebraic and Describing Function Analysis, Autoatica, 3, , (1996). [3] Canudas de Wit, C., Olsson, H., Åströ, K.J., & Lischinsky, P. (1995). A New Model for Control of Systes with Friction. IEEE Transactions on Autoatic Control, [] Filippov A.F., Differential Equations with Discontinuous Right-hand Sides, Matheatics and Its Applications, Kluwer Acadeic, Dordrecht, (1988). [5] Gäfvert M., Coparisons of Two Dynaic Friction Models, Proc. of the 6th IEEE Conference on Control Applications, Hartford, (1997). [6] Galvanetto U., and Knudsen C., Event Maps in a Stick-Slip Syste, Nonlinear Dynaics, 13, pages , (1997). [7] Guckenheier J., and Holes P., Nonlinear Oscillations, Dynaic Systes, and Bifuractions of Vector Fields, Applied Matheatical Sciences, New York, (1983). [8] Hensen R.H.A., van de Molengraft H.J.G., Steinbuch M., Friction Induced Hunting Liit Cycles: A Coparison between the LuGre and Switch Friction Model, Subitted for publication in Autoatica. [9] Karnopp D., Coputer Siulation of Stick-Slip Friction in Mechanical Dynaic Systes, ASME Journal of Dynaic Systes, Measureent and Control, 17, pages 1-13, (1985). 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