Impulsive Control of a Mechanical Oscillator with Friction

Transcription

1 9 Aerican Control Conference Hyatt Regency Riverfront, St. Louis, MO, USA June -, 9 ThC8. Ipulsive Control of a Mechanical Oscillator with Friction Yury Orlov, Raul Santiesteban, and Luis T. Aguilar Abstract Undesired dynaics (stic-slip otions, liit cycles, etc.) that appear in friction systes result in steadystate errors, liiting achievable perforance. To enhance the perforance a novel control approach counteracting friction effects is developed. The approach is based on an ipulsive actuation while the underlying syste is getting stuc due to the presence of dry friction. A spring-ass syste serves as a siple test bed. Asyptotic stabilization is achieved not only for the noinal odel but also for its perturbed version provided that external disturbances affecting the syste are of sufficiently sall agnitude. I. INTRODUCTION Friction is a highly nonlinear phenoenon, producing undesired stic-slip otions, thereby resulting in steady state errors. To reduce friction effects without resorting to high gain control loops suitable odeling is required for friction copensation []. Alternatively [5], [6], a secondorder sliding ode control algorith, robust to discrepancies of friction odeling, can be utilized. In this paper, we develop a novel control approach counteracting friction effects. The approach is based on an ipulsive actuation while the underlying syste is getting stuc due to the presence of dry friction. A spring-ass syste serves as a siple test bed. Stabilization of this syste is under study. In contrast to the afore-entioned friction copensation ethods and sliding ode approach, static position feedbac is constructed provided that the inforation is available on whether the syste is in steady state. Robustness against sall paraeter variations and external disturbances is additionally provided. II. PROBLEM STATEMENT The spring-ass syste (Figure ), affected by Coulob friction, is governed by the differential equation ẍ = x α sign(ẋ) + w + u () where x is the displaceent, ẋ is the velocity, > is the ass of the syste, α > is the Coulob friction level, > is the stiffness coefficient, u denotes the control input, and w stands for external disturbances. As equation () appears with discontinuous right-hand side, its eaning is defined in the sense of Filippov []. The wor was supported by CONACYT under grant nuber 459. Y. Orlov and R. Santiesteban are with CICESE Research Center, Electronics and Telecounication Departent, P.O. BOX 44944, San Ysidro, CA, , (e-ail: yorlov{rsanties}@cicese.x) L.T. Aguilar is with Centro de Investigación y Desarrollo de Tecnología Digital, Instituto Politécnico Nacional, Avenida del Parque Mesa de Otay, Tijuana 5 México (e-ail: luis.aguilar@ieee.org). The Coulob friction odel has been chosen for treatent. Although augenting with viscous friction is iportant fro practical standpoint, we preferred not to do so as such an extension would be rather technical. Instead, we preferred to facilitate exposition and to focus on essential features of the general treatent. x = Fig.. u Spring-ass syste. In order to account for odel discrepancies, an unnown external disturbance w(t) has been introduced into odeling. Throughout, the aplitude of the disturbance is assued to be saller than half the Coulob level α, i.e., w(t) α < α () for all t and soe constant α >. Assuption () is ade for a technical reason that becoes clear as far as the controller derivation goes. Being represented in the state space for, the above syste () is odified to ẋ = x ẋ = [ x α sign(x ) + w + u] () where x = x and x = ẋ. Since the unforced syste () is dissipative and its dissipation is lower bounded by the positive constant α α, syste () under u = is getting stuc in finite at the disturbance-dependent zone (see Figure ) S w S = {(x, x ) : x α + α, x = } IR. (4) Assuing that the syste is enforced by an ipulsive x /9/$5. 9 AACC 494

2 x III. CONTROLLER DESIGN Let the ipulsive controller (5) be specified with α x x v(x ) = sign(x ) (8) α+α α+α x and let it be applied to syste () at the instants t i, i =,,... such that x (t i ) α + α, x (t i ) =. (9) Fig.. Phase trajectories of the unforced spring-ass syste (). actuator u = v(x )δ(t t i ) (5) i= utilizing a continuous position feedbac v(x ) and being applied at soe state-dependent instants t i (x, x ), i =,,..., the closed-loop syste appears to exhibit discretecontinuous dynaics ẋ = x, ẋ = ( x α sign(x ) + w), t t i, (6) x (t i +) = x (t i ), x (t i +) = x (t i ) + v(x (t i )), i =,,.... (7) It is clear that dealing with the position feedbac v(x ) yields the above restitution rule (7) because the ipulsive input (5) results in the corresponding instantaneous change of the velocity while the position dynaics and, hence, the position feedbac v(x (t)) reain continuous in. It is worth noticing that applying a state feedbac v(x, x ) would ipose a certain nonlinear restitution rule, caused by an ill-posed product of the Dirac function δ(t τ), localized at a instant τ, and the function v(x (t), x (t)), discontinuous at t = τ (see, e.g., [4] for details on the nonlinear ipulse response). Our objective is to design an ipulsive controller (5) such that the closed-loop syste (7) is asyptotically stable around the origin, regardless of whichever external disturbance () affects the syste. The current position x (t) is assued to be available for easureent whereas the only available inforation on the velocity x (t) is the nowledge of whether it is nullified or it is not. 495 The dynaics of the closed-loop syste (5) (9) is then as follows. Once the underlying syste () hits the stuc zone (4), it is enforced by the ipulsive controller (5) that changes the velocity of the syste instantaneously (see Figure ). The controller aplitude (8) has been pre-specified in such a anner to bring the underlying syste () fro the stuc zone (4) to the phase trajectory that while being disturbancefree arrives at the origin without oscillations. It is shown that the asyptotic stabilization is thus achieved not only for the disturbance-free syste () with w = but also for its perturbed version provided that external disturbances affecting the syste eet the nor upper bound (). The following result is in order. Theore : Let the friction oscillator () be driven by the ipulsive controller (5), specified with (8), (9). Then the closed-loop syste (5) (9) is globally asyptotically stable provided that the nor upper bound () holds for the external disturbance affecting the syste. Proof: It has been entioned that while being unforced, syste () possesses a globally finite stable invariant anifold S w, localized according to (4) within the estiated stuc zone S. Once the closed-loop syste (5) (9) attains S, say at x (t ) = ξ such that ξ α+α, the ipulsive controller (5), (8), (9) is applied at the instant t. The restitution rule (7) is then specified as x (t ) = ξ, α ξ (ξ ) x (t +) = sign(ξ ). () If the closed-loop syste is disturbance-free, the state trajectory, re-initialized with (), would ono-directionally arrive at the origin in finite- (see Figure ). However, if an adissible disturbance () affects the closed-loop syste, the state trajectory would hit the stuc zone at x (t ) = ξ provided that x (t ) =. Our goal is to deonstrate that, even in the worstdisturbance case where w = α or w = α, the following inequality holds: ξ α α ξ. ()

3 x above relation yields x (t ) + (α α )x (t ) + α ξ =. (6) α+α Setting ξ = x (t ) and taing into account that the case where ξ < is under study, it follows that α+α x ξ = α α (α α) + + α ξ. (7) Substituting (7) into () for ξ, we arrive at the inequality Fig.. Phase portrait of the ipulsive closed-loop syste (5) (9): dotted lines are for jups of the velocity, solid lines are for the perturbed trajectories, and dashed line is for the unperturbed trajectory. α α + (α α) + α ξ α α ξ (8) to be verified. To validate (8) it suffices to represent it in the for Then by iteration on i, siilar relations ξ i+ α α ξ i, i =,,.... () could be obtained for the state positions ξ i = x (t i ) at the instants t i, i =,,... of reaching the stuc zone (4). Since q = α α < by assuption, it would follow that li x (t i ) = li q i ξ =, li x (t i +) = () i i i where the relations x (t i +) = α ξ i (ξ i) sign(ξ i ), i =,,..., siilar to (), have been taen into account. For the purpose of validating (), let us copute the value of ξ as a function of ξ, assuing, for certainty, that ξ <, and perforing siilar coputation, otherwise. In the case where w = α the syste dynaics between successive ipacts are governed by = x = ( x α + α ), t (t, t ). (4) Initialized with (), the solution to the perturbed syste (4) is given by ( x αx + α x ) = α ξ + x. (5) Being confined to the instant t when x (t ) =, the (α α) + α [ ξ α α ξ + α α ], (9) and to observe that (9) is equivalent to the inequality α ξ 4(α ) α ξ + 4α (α α ) ξ () α whose validation is reduced to the obvious inequality (α α ), () α resulted fro (). Thus, inequality () is verified in the case of w = α. It reains to verify inequality () in the case of w = with the syste dynaics between successive ipacts governed by = x = ( x α α ), t (t, t ). () The solution of the above syste, initialized with (), is given by ( x αx α x ) = α ξ + x. () Specified at the next ipact instant t when x (t ) =, relation () yields x (t ) + (α + α )x (t ) α ξ =. (4) 496

4 Setting ξ = x (t ), it follows that ξ = α + α + (α + α) α ξ (5) provided that the case where ξ < is under study. Substituting (5) into () for ξ, we arrive at the inequality α + α (α + α) α ξ α α ξ (6) Fig. 4. Throttle echanis (translational linear equivalent). to be verified. To validate (6) it suffices to represent it in the for [ α + α α ] α ξ (α + α) α ξ, (7) and to observe that (7) is equivalent to the inequality α ξ + 4(α ) α ξ 4α (α + α ) ξ. (8) α Since ξ is within the estiated stuc zone (4), i.e., ξ, the validation of (8) is reduced to the inequality α+α + α (α + α ) α (α + α ), (9) α straightforwardly resulted fro (). Thus, inequality () is verified in the case of w = α, too. This copletes the proof of the theore because as pointed out, inequality () leads to the global asyptotic stability of the the closed-loop syste (5) (9). IV. NUMERICAL RESULTS Perforance issues and robustness properties of the proposed ipulsive controller are additionally tested in nuerical experients. In the siulations, perfored with MAT- LAB, the diensionless spring-ass odel () is studied with the paraeters =, =, and α =. The initial position and the initial velocity are set to x () =.5 and x () =, respectively. The ipulsive controller (5), (8), (9) is first applied to the disturbance-free syste. In order to test the controller robustness a haronic external disturbance w =.7 sint is then applied to the closed-loop syste. Good perforance and desired robustness properties of the controller are concluded fro Figures 5 and 6. Alternative application of ipulsive control is in the throttle syste (Figure 4) which is governed by the differential equation (see []) where, if x x =, otherwise. () In the siulations, the paraeters x =, =, =, =, and α = were selected. The initial position and the initial velocity are set to x () = and x () =. Siulation results are provided in Figures 7 and 8 for the unperturbed and perturbed cases, respectively. V. CONCLUSIONS A novel approach, counteracting friction affects, is developed for and tested on a spring-ass syste. The approach is based on an ipulsive actuation while the underlying syste is getting stuc due to the presence of dry friction. Global asyptotic stability of the closed-loop syste and its robustness against external disturbances of sufficiently sall agnitude are carried out in the theoretical study. The effectiveness of the proposed approach is illustrated in nuerical siulations. The approach is expected to enhance control of echanical anipulators with relatively strong Coulob friction forces. REFERENCES [] A.F. Filippov, Differential equations with discontinuous right-hand sides. Dordrecht: Kluwer Acadeic Publisher, 988. [] H. Olsson, K. Astro, C. Canudas de Wit, M. Gafvert, and P. Lischinsy, Friction odels and friction copensation, Eur. J. Contr., vol. 4, pp , 988. [] C. Canudas de Wit, I. Kolanosvy and J. Sun, Adaptive pulse control of electronic throttle, Proc. of the Aerican Control,pp ,. [4] Y. Orlov, Instantaneous ipulse response of nonlinear systes, IEEE Trans. Auto. Ctrl., vol. 45, no. 5, pp. 999,. [5] Y. Orlov, Finite- stability and robust control synthesis of uncertain switched systes, SIAM Journal on Control and Optiization, vol. 4, pp. 5 7, 5. [6] Y. Orlov, Discontinuous Systes: Lyapunov Analysis and Robust Synthesis under Uncertainty Conditions. London: Springer, 8. ẍ = (x x ) α sign(ẋ) + u + w () 497

5 4 4 x x x x x 4 x x 4 x u u Fig. 5. Ipulsive stabilization of the spring-ass syste with Coulob friction: the no disturbance case. Fig. 6. Ipulsive stabilization of the spring-ass syste with Coulob friction: the disturbance case. 498

6 Fig. 7. Ipulsive stabilization of the electronic throttle syste with Coulob friction: the no disturbance case. Fig. 8. Ipulsive stabilization of the electronic throttle syste with Coulob friction: the disturbance case. 499