Adaptive Control of Mass-Spring Systems with Unknown Hysteretic Contact Friction

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1 26 American Control Conference (ACC) Boston arriott Copley Place July 6-8, 26. Boston, A, USA Adaptive Control of ass-spring Systems with Unknown Hysteretic Contact Friction ohammad Al Janaideh and Dennis Bernstein Abstract We apply retrospective cost adaptive control (RCAC) to a command-following problem for mass-spring systems with unknown contact friction. Dahl, Lure, and axwell-slip models are used to generate the friction force. We consider a single-degree-of-freedom oscillator with control force applied directly to the mass, as well as a noncolocated two-degree-of-freedom oscillator with control force applied to the secondary mass and performance given by the position of the primary mass. For harmonic command following, we show that RCAC achieves internal model control without knowledge of either the friction force or the friction model. I. INTRODUCTION Virtually all mechanical systems are affected by friction [, 2]. In motion control applications, friction degrades performance and, for multiple reasons, may present a significant challenge. In particular, friction can vary over time due to changes in surface properties and lubrication; it may depend on loads and geometry; it may be difficult to model and thus is usually uncertain; it is difficult to measure or estimate; and it is highly nonlinear and hysteretic [3 8]. In view of these challenges, it is not surprising that a wide variety of feedback control techniques have been used to compensate for friction, including both fixed-gain [9, ] and adaptive methods [ 6]. The present paper takes the latter approach by applying retrospective cost adaptive control (RCAC) to illustrative systems [7 9]. In the present paper, we consider a single-degree-offreedom oscillator with control force applied to the mass, as well as a two-degree-of-freedom oscillator with the control force applied to the secondary mass and the performance given by the position of the primary mass. The challenging aspect of these examples is the fact that all of the masses are subjected to friction, the friction model is unknown and no direct measurement of the friction force available. To generate the friction force, we consider three friction models, namely, the Dahl [2], Lure [8], and axwellslip models [2, 22]. The contribution of the present paper is thus a numerical investigation of the performance of RCAC without knowledge of either the friction force or the underlying friction model. As a performance metric, we use phase shift calculations to determine whether RCAC achieves internal model control with harmonic command following. This method was used in [23] for Hammerstein systems. However, the applications in the present paper are not Hammerstein systems. We stress that the phase-shift. Al Janaideh is with the Department of echanical and Industrial Engineering, University of Toronto, Toronto, ON 5S 38, Canada. D. S. Bernstein is with the Department of Aerospace Engineering, The University of ichigan, Ann Arbor, I 489, USA. calculations are for analysis only; all of the closed-loop simulations are based on nonlinear friction models. II. FRICTION ODELS We consider the mass-spring system shown in Figure (a), q(t) denotes the mass position. With x q and x 2 q, it follows that ẋ (t) = x 2 (t), () ẋ 2 (t) = C x 2(t) K x (t) + F (t) F f(t), (2) y(t) = x (t), (3) y(t) is the position measurement available to the controller, F (t) is the control force, and F f (t) is the friction force. Equations ()-(3) can be expressed as ẋ(t) = A c x(t) + B c [F (t) F f (t)], (4) y(t) = C c x(t), (5) [ ] [ ] A c, B c, and C c [ ]. K C We now review the Dahl, Lure, and axwell-slip friction models used to generate the friction force F f (t). A. Dahl friction model The Dahl friction model is given by F f (t) = σ F f(t) q(t) ε ( sgn F f(t) f C f C ) q(t) q(t), (6) σ > represents the slope of force-deflection curve for F f =, f C is the Coulomb friction force, and ε > influences the shape of the hysteresis loop. For all ε, the Dahl model is a Lipschitz-continuous, rate-independent hysteresis model [8]. We consider the Dahl model with ε =, σ =.75, and f C = N. B. Lure friction model The Lure model [8], which models the asperities of two surfaces as elastic bristles, is given by ṁ(t) = q(t) q(t) m(t), ρ( q(t)) (7) F f (t) = σ m(t) + σ 2 ṁ(t) + σ 3 q(t), (8) ρ( q(t)) f C + f s f C e ( q(t) vs )2, (9) σ σ m(t) is the average deflection of the bristles, σ >, σ 2 >, and σ 3 > represent stiffness, damping, and /$3. 26 AACC 667

2 r z u y c d dt F f F (a) q s Fig. 2. Command-following problem for the force-actuated mass-spring system with retrospective cost adaptive control update of the controller c(s). (b) Fig.. (a) Force-actuated mass-spring system, F f is the unknown friction force and F is the external input force. (b) Force-actuated massspring system, (s) = s 2, q(t) is the output displacement, F is the external force, and F f is the friction +Cs+K force. mass, respectively, f s is the stiction force, and v s is the Stribeck velocity, which is the velocity at which the steadystate friction force starts decreasing. The Lure model is a Lipschitz-continuous, rate-dependent hysteresis model [8]. We consider the Lure model of σ = 5 N/m, σ 2 = 5 N-s/m, σ 3 =.4 N-s/m, v s =. m/s, f s =.5 N, and f c = N. C. axwell-slip model The axwell-slip model with deadband width i R for i =,..., N [2, 22] is given by F f (t) = F f N ( λ i q(t) qi (t) ), () ] [ q+ (t) q i = [(q i (t), q(t), i ) N (q i (t), q(t), i )], q (t) () (q i, q, ) U( q i + q i ), N (q i, q, ) U( q i + q + i ), and {, x, U(x) =, x <. The axwell-slip model is a discontinuous rate-independent hysteresis model [8]. We consider the axwell-slip model with N = 4, λ =, λ 2 =.3, λ 3 =., λ 4 =.4, =.5, 2 =, 3 =.5, and 4 =.5. q III. ADAPTIVE CONTROL OF THE FORCE-ACTUATED ASS-SPRIN SYSTE WITH FRICTION NONLINEARITY A. Problem formulation For sampled-data control, let h be the sampling time and k =,, 2,.... To discretize (5) and (6), consider x(hk + h) x(hk) = ha c x(hk) + hb c [u(hk) F f (hk)], (2) y(hk) = C c x(hk), (3) u = F represents the control force. Then x(k + ) = Ax(k) + B[u(k) F f (k)], (4) A I 2 + ha c, B hb c, (5) and, for convenience x(k) denotes x(kh). We use (5) to discretize ()-(3) as x(k + ) = Ax(k) + B[u(k) F f (k)], (6) y(k) = Cx(k), (7) [ ] [ ] h A Kh Ch, B, C [ ], and z(k) r(k) y(k), (8) z(k) is the command-following error and r(k) is the position command. The goal is to determine u that makes z small. B. RCAC A block diagram for (7)-(2) with RCAC is shown in Figure 2. We assume that the discrete-time transfer function that relates the control force to the displacement of the mass is unknown except for an estimate of a single nonzero arkov parameter as needed for RCAC. The friction model and friction force are also unknown. The adaptive controller has the form u(k) = i (k)u(k i) + N i (k)z(k i) n c + Q i (k)r(k i), (9) 668

3 , for all i =,..., n c, i (k) R, N i (k) R, and Q i (k) R. The control (9) can be expressed as r u y z ur + c+ F u(k) = θ(k)φ(k ), and θ(k) = [ (k) nc (k) N (k) N nc (k) Q (k) Q nc (k) ] R 3nc φ(k ) = [ u(k ) u(k n c) z(k ) z(k n c) r(k ) r(k n c) ] T R 3nc. Next, define the cumulative cost function k J R (θ, k) = φ T (i 2)θ T (k) Û T (i ) 2 i=2 + (θ(k) θ )P (θ(k) θ ) T, (2) is the Euclidean norm. inimizing (2) yields θ T (k) = θ T (k ) + P (k )φ(k 2) [φ T (k )P (k )φ(k 2) + ] [φ T (k 2)θ T (k ) Û T (k )], (2) The error covariance is updated by P (k) = P (k ) P (k )φ(k 2) [φ T (k 2)P (k )φ(k ) + ] φ T (k 2)P (k ). (22) We initialize the error covariance matrix as P = αi 3nc, α >. IV. PERFORANCE ANALYSIS Consider the force-actuated mass-spring system shown in Fig. 2 with the harmonic command r(k) = Re { A r e (jωk)}, A r is a complex number and Ω is the command frequency. Since the input is harmonic, the friction force is harmonic. For analysis only, we consider the main harmonic component in the friction force for the phase-shift calculations. Then a transfer function F that represents the main harmonic component in the friction force can be used to represent the friction force F f. Then the system shown in Fig. 2 with the controller can be represented as in Fig. 3, c ur, (23) + c uy and uy If u is also harmonic, then + c + F. (24) u(k) = Re { A r ur (e jω ) e j(ωk+ ur(ejω )) }, (25) ur (e jω ) and ur (e jω ) are, respectively, the magnitude and phase of ur, at the frequency Ω. Then the Fig. 3. Linearized approximation for the force-actuated mass-spring system with RCAC adaptive control. The friction force F f is approximated by the transfer function F. harmonic steady-state response is given by Thus y(k) = Re { A r ur (e jω ) uy (e jω ) (26) e j(ωk+ ur(ejω )+ uy(e jω )) }. (27) z(k) = Re { A r e (jωk)} Re { A r ur (e jω ) uy (e jω ) e j(ωk+ ur(ejω )+ uy(e jω )) }. (28) Therefore, z(k) = if and only if the magnitude and phase of ur (e jω ) satisfy ur (e jω ) = uy (e jω ), (29) ur (e jω ) = uy (e jω ). (3) In the following examples, we use the Fourier transform to determine the most significant frequency component in the output y. Then, we calculate uy (e jω ), which is the phase between the control force u and output y. Then, we compare uy (e jω ) with ur (e jω ), which represents the phase between the command r and the control force u. In order to verify (28) and (29), we approximate F f by the transfer function F to obtain the phase of uy (e jω ). This provides a technique for determining whether RCAC develops an internal model of the harmonic command. V. SDOF OSCILLATOR We now consider (6)-(8) with h =. sec. Consider (6)-(8) with the magnitude and phase shift of the most significant harmonic component in the command r(k), control signal u(k), and mass position q(k) to determine whether RCAC achieves internal model control. Example 5.: Consider the command r(k) = 5 sin( π hk) with the Dahl model, and let = kg, C = 2 N-s/m, and K = N/m. We use RCAC with n c = 2 and α =.. Figure 4 shows that RCAC drives the command-following error z to zero. Figure 4 shows that the phase shift between the command r(k) and the control u(k) is 7.7 deg, and the phase shift between the control signal u(k) and the mass position q(k) is 7.87 deg. Thus RCAC achieves the correct gain and phase shift that stabilize the mass-spring system with unknown friction force at the command frequency. Example 5.2: Consider the command r(k) = 5 sin( π hk) with the Lure model (8)-(), and let 669

4 (a) (b) (c) (d) (e) (f) (g) Fig. 4. Example 5.: (a) shows the command-following error z with the Dahl model whose input and output are shown in (b) for the closed-loop system with RCAC, (c) shows the friction force, (d) shows the control u, (e) shows the evolution of the controller coefficients θ, (f) shows the relationship between the command r and the control u, and (g) shows the relationship between the control u and the mass position q. = kg, C = 2 N-s/m, and K = N/m. Then ω n = rad/sec, ζ =, and (z) =.2642z+.353 z z We use RCAC with n c = 2 and α =.. Figure 5 shows that RCAC drives the command-following error z to zero. Figure 5 shows that the phase shift between the command input r(k) and the control signal u(k) is 2.38 deg, and the phase shift between the control signal u(k) and the mass position q(k) is 2.47 deg. Thus RCAC achieves the correct gain and phase shift that stabilize the mass-spring system with unknown friction force at the command frequency. (a) (b) (c) (d) (e) (f) (g) Fig. 5. Example 5.2: (a) shows the command-following error z with the Lure model whose input and output are shown in (b) for the closed-loop system with RCAC, (c) shows the friction force, (d) shows the control u, (e) shows the evolution of the controller coefficients θ, (f) shows the relationship between the command signal r and the control signal u, and (g) shows the relationship between the control u and the mass position q. Example 5.3: Consider the command r(k) = 5 sin( π hk) with the axwell-slip model () and Fig. 6. Example 5.3: (a) shows the command-following error z with the axwell-slip model whose input and output are shown in (b) for the closedloop system with RCAC, (c) shows the relationship between the command signal r and the control u, and (d) shows the relationship between the control u and the mass position q. (2), and let = kg, C = N-s/m, and K = N/m. We use RCAC with n c = 8 and α =.. Figure 6 shows that RCAC drives the command-following error z to zero. Figure 4 shows that the phase shift between the command input r(k) and the control signal u(k) is deg, and the phase shift between the control signal u(k) and the mass position q(k) is deg. Thus RCAC achieves the correct gain and phase shift that stabilize the mass-spring system with unknown friction force at the command frequency. Example 5.4: Consider Examples 5. and 5.2 with = kg, C = N-s/m, and K = 2 N/m. Then ω n =.44 rad/sec, ζ =.3536, and (z) =.345z+.226 z z We use RCAC with n c = 2 and α =.. Figure 7 shows that RCAC drives the command-following error z to zero. VI. TWO-DEREE-OF-FREEDO OSCILLATOR As shown in Figure 8, we consider a two-degree-offreedom oscillator with force applied to the mass 2 with performance given by the position of the mass. The friction forces F f and F f2 applied to and 2 are unknown, and the external force F is applied to mass. Let q denote the position of mass, and q 2 the position of mass 2. We consider x(k + ) = Ax(k) + B [ F (k) F f(k) F f2(k) 2 ], (3) y(k) = Cx(k), (32) x T = [ x x 2 x 3 x 4 ], x (k) = q (k), x 3 (k) = q 2 (k), x 2 (k) and x 4 (k) are the velocities of 67

5 z (a) (b) (c) A. Numerical examples: two-degree-of-freedom oscillator All of the examples in this section consider (3)-(33) with h =. sec. Example 6.: We consider the command r(k) = 3 sin(4 π kh) + 2 sin(6 π kh) with Lure model with σ 3 =.4 N-s/m for the friction force F f. We consider F f2 with the Lure model with σ 3 = N-s/m. Let = 2 = kg, C = N-m/s, and K = N/m. We use RCAC with n c = 5 and α =. Figure 9 shows the command-following error z, the friction forces F f and F f2, and the control signal u(k). (d) (e) (f) Fig. 7. Example 5.4: (a) shows the command-following error z with the Lure model whose input and output are shown in (b) for the closed-loop system with RCAC, (b) shows the relationship between the command r and the control u, (c) shows the relationship between the control u the mass position q, (d) shows the command-following error z with the Dahl model whose input and output are shown in (b) for the closed-loop system with RCAC, (e) shows the relationship between the command signal r and the control u, and (f) shows the relationship between the control u and the mass position q. z (a) u (b) Ff Ff q q 2 (c) (d) Fig. 9. Example 6.: (a) shows the error z. The friction forces F f and F f2 shown in (c) and (d), respectively. (b) shows the control u. Fig. 8. Force-actuated mass-spring system, F is the external force, F f and F f2 are unknown friction forces, F is the external force, q is the position of the secondary mass, and q 2 is the position of the primary mass 2. The performance variable is the command-following error for 2. and 2, respectively. Then h A = hk hc hk hc h, B =, hk hc 2 2 hk 2 [ ] C =, The control force u is given by z(k) = r(k) q 2 (k). (33) u(k) = i (k)u(k i) + N i (k)z(k i) n c + Q i (k)q (k i). (34) Example 6.2: We consider the command r(k) = 3 sin(4 π kh) + 2 sin(6 π kh) with the Lure model of Example 5.2 and axwell model of Example 5.3. Let = 2 = kg, C = N-m/s, and K = N/m. We use RCAC with n c = 5 and α =. Figure shows the commandfollowing error z, the friction forces F f and F f2, and the control signal u(k). These examples show that RCAC can stabilize the closedloop system consisting of a two-degree-of-freedom oscillator with unknown friction forces produced by an unknown friction model. VII. CONCLUSIONS Retrospective cost adaptive control (RCAC) was applied to a command-following problem involving a single-degree-offreedom oscillator with control force applied to the mass, as well as a two-degree-of-freedom oscillator with force applied to the secondary mass, measurements of the positions of both masses, and performance given by the position error of the primary mass. The friction forces characterized by the Dahl, Lure, and axwell-slip hysteresis models are unknown. RCAC drives the command-following error of the closedloop system to zero. The numerical results in the paper show that RCAC achieves internal model control. Future work will 67

6 z F f (a) q (c) u F f (b) q 2 Fig.. Example 6.2: (a) shows the error z. The friction forces F f and F f2 are shown in (c) and (d), respectively. (b) shows the control u. consider the more challenging case for the two-degree-offreedom oscillator only the position of the primary mass is measured. (d) [4] C. Canudas de Wit and P. Lischinsky, Adaptive friction compensation with partially known dynamic friction model, Int. J. Adapt. Contr. Sig. Proc., vol., no., pp. 65-8, 997. [5] Y. Tan, C. Jie, and T. Hualin, Adaptive backstepping control and friction compensation for AC servo with inertia and load uncertainties, IEEE Trans. Indus. Electro., vol. 5, no. 5, pp , 23. [6] C. Canudas de Wit, K. Astrom, and K. Braun, Adaptive friction compensation in DC-motor drives, IEEE J. Robot. Auto., vol. 3, no. 6, pp , 987. [7] R. Venugopal and D. S. Bernstein, Adaptive disturbance rejection using ARARKOV system representations, IEEE Trans. Contr. Sys. Tech., vol. 8, no. 2, pp , 2. [8]. A. Santillo and D. S. Bernstein, Adaptive control based on retrospective cost optimization, AIAA J. uid. Contr. Dyn., vol. 33, no. 2, pp , 2. [9] J. B. Hoagg,. A. Santillo and D. S. Bernstein, Discrete-time adaptive command following and disturbance rejection for minimumphase systems with unknown exogenous dynamics, IEEE Trans. Autom. Contr., vol. 53, no. 4, pp , 28. [2] P. Dahl, Solid friction damping of mechanical vibrations, AIAA J., vol. 4, no. 2, pp , 976. [2] D. D. Rizos and S. D. Fassois, Presliding friction identification based upon the axwell slip model structure, Chaos, vol. 4, no. 2, pp , 24. [22] F. Al-Bender, V. Lampaert, and J. Swevers, odeling of dry sliding friction dynamics: From heuristic models to physically motivated models and back, Chaos, vol. 4, no. 2, pp , 24. [23]. Al Janaideh and D. S. Bernstein, Adaptive Control of Hammerstein Systems with Unknown Prandtl-Ishlinskii Hysteresis, Proc. Inst. ech. Eng. I J. Syst. Contr. Eng., vol. 229, no. 2, pp , 25. REFERENCES [] B. Armstrong-Helouvry, Control of achines with Friction, Boston, A, Kluwer, 99. [2] B. Armstrong-Helouvry, P. Dupont, and C. Canudas de Wit, A survey of model, analysis tools and compensation methods for the control of machines with friction, Automatica, vol. 3, no. 7, pp , 994. [3] C. Canudas de Wit, H. Olsson, K. J. Astrom, and P. Lischinsky, A new model for control of systems with friction, IEEE Trans. Autom. Contr., vol. 4, no. 3, pp , 995. [4] J. Swevers, F. Al-Bender, C. anseman, and T. Projogo, An integrated friction model structure with improved presliding behavior for accurate friction compensation, IEEE Trans. Autom. Contr., vol. 45, no. 4, pp , 2. [5] V. Lampaert, J. Swevers, and F. Al-Bender, odification of the Leuven integrated friction model structure, IEEE Trans. Autom. Contr., vol. 47, no. 4, pp , 22. [6] D. S. Bernstein, Ivory host, IEEE Contr. Sys. ag., Vol. 27, pp. 6-7, 27. [7] B. S. R. Armstrong and Q. Chen, The Z-Properties Chart, IEEE Contr. Sys. ag., vol. 28, pp , 28. [8] A. K. Padthe, B. Drincic, J. Oh, D. D. Rizos, S. D. Fassois, and D. S. Bernstein, Duhem odeling of Friction-Induced Hysteresis: Experimental Determination of earbox Stiction, IEEE Contr. Sys. ag., vol. 28, pp. 9-7, 28. [9] S. Southward, C. Radcliffe, and C. accluer, Robust nonlinear stickslip friction compensation, J. Dyn. Sys. eas. Contr., vol. 3, no. 4, pp , 99. [] J. Amin, B. Friedland, and A. Harnoy, Implementation of a friction estimation and compensation technique, IEEE Contr. Sys. ag., vol. 7, no. 4, pp. 7 76, 997. [] S. W. Lee and J.-H. Kim, Robust adaptive stick-slip friction compensation, IEEE Trans. Ind. Electron., vol. 42, no. 5, pp , 995. [2] L. Freidovich, A. Robertsson, A. Shiriaev, and R. Johansson, Luremodel-based friction compensation, IEEE Trans. Contr. Sys. Tech., vol. 8, no., pp. 94 2, 2. [3] L. arton and B. Lantos, Control of mechanical systems with Stribeck friction and backlash, Sys. Contr. Lett., vol. 58, no. 2, pp. 447,