New Cascade Approach for Global -Exponential ... - IEEE Xplore
Fig. 13. Nominal velocity (dashed line) and velocities without (top) and with (bottom) feedback friction compensation and velocity control: ! = 20 rad=s (average over three experiments).
VIII. CONCLUSION Single and multistate integral friction models have been developed in this work, based on the integral solution of the Dahl model. The advantage of a closed form integral formulation with respect to a differential formulation is evident in terms of computational efficiency and accuracy, with beneficial effects in friction compensation. Almost the same results as with the LuGre model were obtained in simulation with the single state model, but while the LuGre model may fail with fixed step numerical integration algorithms, the integral formulation does not. The multistate integral model, accounting for the hysteresis behavior with non local memory, has been derived by combining the Dahl and the Maxwell slip model. Experiments have been performed on an industrial servomechanism, endowed with a high resolution (but commercial) encoder. On the basis of the experimental outcomes a main issue for future research has emerged. An adaptive identification of the friction model is needed for practical implementation, in order to face variations with respect to load, wear and temperature. For example, an online, recursive identification of the hysteresis function has been proposed in [16]. In this respect, the simplicity of the proposed model could be suitably exploited.
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[8] L. Lampaert, J. Swevers, and F. Al-Bender, “Modification of the Leuven integrated friction model structure,” IEEE Trans. Automat. Contr., vol. 47, pp. 683–687, Apr. 2002. [9] P. Dupont, V. Hayward, B. Armstrong, and F. Altpeter, “Single state elastoplastic friction models,” IEEE Trans. Automat. Contr., vol. 47, pp. 787–792, May 2002. [10] R. Stribeck, “Die wesentlichen Eigenschaften der Gleit-und Rollenlager,” Zeitschrift des Vereines Deutscher Ingenieure, vol. 46, no. 38, 39, pp. 1342–1348, 1902. [11] B. Armstrong-Hélouvry, Control of Machines With Friction. Boston, MA: Kluwer, 1991. [12] D. Karnopp, “Computer simulation of slip-stick friction in mechanical dynamic systems,” ASME J. Dyna. Syst., Meas. Control, vol. 107, no. 1, pp. 100–103, 1985. [13] C. Walrath, “Adaptive bearing friction compensation based on recent knowledge of dynamic friction,” Automatica, vol. 20, no. 6, pp. 717–727, 1984. [14] N. Barabanov and R. Ortega, “Necessary and sufficient conditions for passivity of the LuGre friction model,” IEEE Trans. Automat. Contr., vol. 45, pp. 830–832, Apr. 2000. [15] I. D. Mayergoyz, Mathematical Models of Hysteresis. New York: Springer-Verlag, 1991. [16] L. Lampaert and J. Swevers, “On-line identification of hysteresis functions with nonlocal memory,” in Proc. 2001 IEEE/ASME Int. Conf. Adv. Mech., AIM’01, Como, Italy, 2001, pp. 833–837.
New Cascade Approach for Global -Exponential Tracking of Underactuated Ships Ti-Chung Lee and Zhong-Ping Jiang Abstract—This note investigates the fast tracking control problem of underactuated ships via persistent excitation (PE) conditions. By combining a novel transformation with the computed torque method, a decoupling controller related to the surge force is given first to decompose the error model into two cascade subsystems. Then, a stabilizing controller involving the yaw moment is designed. With the help of the proposed cascaded structure, a weaker PE condition than those given in past literature can be used to verify an integral detectability and guarantee global -exponential convergence by employing several newly developed stability criteria. A new feature of the obtained results is that only one of these reference signals is needed to satisfy the usual PE condition. Simulation results are provided to validate the effectiveness of the proposed scheme. Index Terms—Cascaded systems, global -exponential stability, persistent excitation (PE) condition, underactuated ships.
REFERENCES [1] B. Armstrong-Hélouvry, P. Dupont, and C. C. D. Wit, “A survey of models, analysis tools and compensation methods for the control of machines with friction,” Automatica, vol. 30, no. 7, pp. 1083–1138, 1994. [2] H. Olsson, K. J. Åström, C. C. de Wit, M. Gäfvert, and P. Lischinsky, “Friction models and friction compensation,” Eur. J. Control, vol. 4, no. 3, pp. 176–195, 1998. [3] P. Dahl, “A solid friction model,” The Aerospace Corporation, El Segundo, CA, Tech. Rep. TOR-0158(3107-18), 1968. [4] P. A. Bliman and M. Sorine, “Easy-to-use realistic dry friction models for automatic control,” in Proc. 3rd Eur. Control Conf., ECC’95, Rome, Italy, 1995, pp. 3788–3794. [5] C. C. de Wit, H. Olsson, K. J. Åström, and P. Lischinsky, “A new model for control of systems with friction,” IEEE Trans. Automat. Contr., vol. 40, pp. 419–425, Mar. 1995. [6] J. Swevers, F. Al-Bender, C. G. Ganseman, and T. Prajogo, “An integrated friction model structure with improved presliding behavior for accurate friction compensation,” IEEE Trans. Automat. Contr., vol. 45, pp. 675–686, Apr. 2000. [7] G. Ferretti, G. Magnani, G. Martucci, P. Rocco, and V. Stampacchia, “Friction model validation in sliding and presliding regimes with high resolution encoders,” in Experimental Robotics VIII. ser. STAR, B. Siciliano and P. Dario, Eds. New York: Springer-Verlag, 2003, pp. 328–337.
I. INTRODUCTION This note investigates the fast tracking control problem for underactuated ships. The goal is to achieve global -exponential convergence via a relaxed persistent excitation (PE) condition involving reference surge and yaw velocities. The main motivation behind the tremendous Manuscript received April 2, 2004; revised June 22, 2004 and September 29, 2004. Recommended by Associate Editor Hua Wang. T.-C. Lee is with the Department of Electrical Engineering, Ming Hsin University of Science and Technology, Hsinchu, Taiwan 304, R.O.C. (e-mail: [email protected]). Z.-P. Jiang is with the Department of Electrical and Computer Engineering, Polytechnic University, Brooklyn, NY 11201 USA (e-mail: zjiang@ control.poly.edu). This work was supported in part by the NSC, Taiwan, R.O.C., under Contract NSC-91-2213-E-159-004, and in part by the National Science Foundation under Grants ECS-0093176 and INT-9987317. Digital Object Identifier 10.1109/TAC.2004.839632
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efforts in studying such nonholonomic systems is the well-known fact that these systems do not satisfy the necessary condition proposed by Brockett [2]. Consequently, there is no continuous time-invariant state-feedback law to guarantee the asymptotic stability in the sense of Lyapunov for the stabilization problem. Recently, several authors have shown that this obstacle can be overcome if we adopt feedback laws of new kinds that may be time-varying and/or discontinuous; see, for instance, the survey paper [11] and the numerous references therein. In parallel, the tracking control problem has also received a great deal of attention because of its practical importance; see [7] and the references therein. In most of the published papers on the tracking control of underactuated ships, Lyapunov direct method is used and some kind of PE conditions involving reference surge and/or yaw velocities are imposed to guarantee the exponential stability of the closed-loop system. A similar observation was previously made for the tracking control of mobile robots [3] (also see [9]). Roughly speaking, the fulfillment of a PE condition implies that the desired reference trajectory is “a moving trajectory,” instead of a fixed set-point. In the case of ship tracking control, the PE condition used in [18], [8] requires that the desired yaw angular velocity rd be positive or negative at any time instant. Thus, the proposed controllers cannot be used to track a straight line path in which rd = 0. In [15], the reference yaw angular velocity rd is allowed to change the sign and a novel cascade controller design is proposed. However, the case of tracking a straight line is excluded. This restriction was removed in [4] by introducing for the first time a universal controller that solves simultaneously the problems of stabilization and tracking control for underactuated ships. Although this is a nice addition to the field of controlling underactuated mechanical systems, the PE conditions introduced there appear to be unnecessarily restrictive (see Remark 3). In fact, excluded are some interesting cases when the reference signals are sinusoidal. The latter may happen for some planned trajectories, when one would like to apply the motion planning technique proposed in [14] to study the parking problem. In [1], instead of studying exponential stability, an exponential practical stability result was proposed. Other related but independent work include the way-point tracking [17] and the tracking control on a linear course [6], in which PE condition naturally holds because the forward velocity is a nonzero constant. The purpose of this note is to study the general tracking control problem with global exponential convergence for underactuated ships by introducing a relaxed PE condition. Our PE condition related to the reference surge velocity ud and reference yaw angular velocity rd is precisely stated here. H0): Suppose there exist four positive constants 1 ; 2 ; T , and " such that t+ T t
jrd ( )j
+
jud ( )j
d
"
8t 0:
(1)
Hypothesis H0) holds when at least one of the reference inputs ud and rd is a linear combination of sinusoidal functions with different frequencies. Moreover, it holds in case of the PE conditions previously proposed in past literature [4], [8], [15], [18]. The new approach presented in this note will decompose the error model into two cascade subsystems by using a novel transformation and a decoupling controller related to the surge force. Then, a stabilizing controller involving the yaw moment will be designed. The proposed cascaded structure is different from [15] so that the PE condition H0), weaker than the one proposed in [15], can be used to guarantee the global -exponential stability; see [19] for a precise definition of -exponential stability. Particularly, the proposed result can achieve global -exponential convergence rather than merely an asymptotic or practical convergence result in case of tracking a straight-line path. This is a new result. The whole stability analysis will employ several newly de-
veloped stability criteria rather than using the direct Lyapunov method. Indeed, we will show that a subsystem of the closed-loop system falls into a class of systems studied in [12]. Then, hypothesis H0) will be used to verify an integral detectability condition. Thus, the global uniform asymptotic stability (GUAS) for the subsystem can be guaranteed by employing a stability criterion from [12]. Noticing an interesting relation between GUAS and global -exponential stability, and the cascaded structure of the closed-loop system, the global -exponential stability can be achieved and, therefore, our control problem is solved. Simulation results will be given to validate the effectiveness of the proposed scheme in Section IV. II. PRELIMINARY: A STABILITY CRITERION FOR INTERCONNECTED SYSTEMS
A
CLASS
OF
In this section, a stability criterion for a class of time-varying interconnected systems will be proposed. It is a simple consequence of the result given in [12]. Indeed, consider a system of the form
z_1 = A(t; z )z1 + B (t; z1 ; z2 )T z2 z_2 = 0B (t; z1 ; z2 )z1
(2a) (2b)
where z1 2
0 80 t 0 t < 1 :
p
It is easy to check that assumption A1) implies rd (t) r ; 8t 0, by the Mean-Value Theorem and the continuity of rd . In this case, hypothesis H0) must hold. On the other hand, a weaker, asymptotic tracking, result than global -exponential stability was achieved using the same controller (and choosing some parameters that may depend on various conditions) under the following condition: t jrd ( )j d 1 A2) jud (t)j u > 0 and t for some; u > 0 some 1 > 0 and all 0 t0 t < 1: It is straightforward to see that hypothesis H0) also holds when condition A2) is true. In contrast with the result given in [4], the tracking control problem with global -exponential convergence is solvable (and indeed solved) according to Theorem 1. Moreover, there exist certain tracking signals that satisfy the proposed hypothesis H0) but both conditions A1) and A2) do not hold. For example, consider the tracking signals rd (t) = 0 and ud (t) = sin(!t) for some ! > 0 and all t 0. It is easy to show that A1) and A2) are not true but H0) does hold. Therefore, H0) provides a weaker PE condition than those proposed in past literature [4], [8], [15].
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Fig. 2. Tracking a straight-line path. (a) Errors of displacements and yaw angle. (b) Errors of surge, sway, and angular velocities. (c) Time history of force and moments. (d) Displacement variations for tracking a straight-line path.
Remark 4: Notice that in this note we do not need the boundedness d that was assumed in [4] and [8]. It reduces the requirement of of u regularity for reference surge force.
ACKNOWLEDGMENT The author would like to thank the anonymous referees for their valuable comments and suggestions.
IV. SIMULATION RESULTS By means of simulations, the performance of the proposed tracking controllers is demonstrated by considering circular and straight line reference trajectories. For simulations, we take the same parameters as in [4], i.e., considering a monohull ship with a length of 32 m, a mass of 3 3 3 118 2 10 kg. In this case, m11 = 120 2 10 kg, m22 = 172:9 2 10 5 2 2 0 1 kg, m33 = 636 2 10 kgm ; d11 = 215 2 10 kgs ; d22 = 97 2 103 kgs01 , and d33 = 802 2 104 kgm2 s01 . The initial conditions are set as follows: [x(0); y (0); (0); u(0); v (0); r (0)] = [1:5
m; 02:5 m; 0:7 rad; 0 m/s; 0:5 m/s; 0 rads]:
p
The constants are chosen as (k2 ; k3 ; k4 ) = (5=cM ; 5= 2; 1) with cM = max(m11 ; m22 )=d22 . It can be seen from the simulation results in Figs. 1 and 2 that fast convergence is attained based on our approach. It is also worth noting that in either case, global -exponential convergence is guaranteed using the same controller and parameters. On the contrary, in past literature, switching controllers are usually necessary to track circular and straight-line trajectories at the same time. V. CONCLUSION A global -exponential tracker has been proposed for underactuated ships by using a decoupling method. A novel stability analysis was given based on several newly developed stability criteria. A general PE condition that extends several different PE conditions proposed in present literature was given to guarantee global -exponential stability. Simulation results have shown that the proposed tracking controller efficiently solves the tracking problem. A topic of future research is to address the robust control problem under the wave, wind, and current disturbances for underactuated ships.
REFERENCES [1] A. Behal, D. M. Dawson, W. E. Dixon, and F. Yang, “Tracking and regulation control of an underactuated surface vessel with nonintegrable dynamics,” IEEE Trans. Automat. Contr., vol. 47, pp. 495–500, Mar. 2002. [2] R. W. Brockett, “Asymptotic stability and feedback stabilization,” in Differential Geometric Control Theory, R. W. Brockett, R. S. Millman, and H. H. Sussmann, Eds. Cambridge, MA: Birkhäuser, 1983, pp. 181–191. [3] W. E. Dixon, D. M. Dawson, F. Zhang, and E. Zergeroglu, “Global exponential tracking control of a mobile robot system via a PE condition,” IEEE Trans. Syst., Man, Cybern. B, vol. 30, pp. 129–142, Jan. 2000. [4] K. D. Do, Z. P. Jiang, and J. Pan, “Universal controllers for stabilization and tracking of underactuated ships,” Syst. Control Lett., vol. 47, pp. 299–317, 2002. [5] T. I. Fossen, Guidance and Control of Ocean Vehicles. New York: Wiley, 1994. [6] G. Indiveri, M. Aicardi, and G. Casalino, “Robust global stabilization of an underactuated marine vehicle on a linear course by smooth timeinvariant feedback,” in Proc. 39th IEEE Conf. Decision Control, Sydney, NSW, Australia, 2000, pp. 2156–2161. [7] Z. P. Jiang, “Lyapunov design of global state and output feedback trackers for nonholonomic control systems,” Int. J. Control, vol. 73, pp. 744–761, 2000. [8] , “Global tracking control of underactuated ships by Lyapunov’s direct method,” Automatica, vol. 38, pp. 301–309, 2002. [9] Z. P. Jiang and H. Nijmeijer, “Tracking control of mobile robots: A case study in backstepping,” Automatica, vol. 33, pp. 1393–1399, 1997. [10] H. K. Khalil, Nonlinear Systems. New York: Macmillian, 1992. [11] I. Kolmanovsky and N. H. McClamroch, “Developments in nonholonomic control problems,” IEEE Contr. Syst. Mag., vol. 15, pp. 20–36, Jan. 1995. [12] T. C. Lee, “On the equivalence relations of detectability and PE conditions with applications to stability analysis of time-varying systems,” in Proc. 2003 Amer. Control Conf., Denver, CO, 2003, pp. 1873–1878. [13] T. C. Lee, D. C. Liaw, and B. S. Chen, “A general invariance principle for nonlinear time-varying systems and its applications,” IEEE Trans. Automat. Contr., vol. 46, pp. 1989–1993, Nov. 2001.
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[14] T. C. Lee, C. Y. Tsai, and K. T. Song, “A motion planning approach to fast parking control of mobile robots,” in Proc. 2003 IEEE Int. Conf. Robotics and Automation, Taipei, Taiwan, 2003, pp. 905–910. [15] E. Lefeber, “Tracking Control of Nonlinear Mechanical Systems,” Ph.D. dissertation, Univ. Twente, Twente, The Netherlands, 2000. [16] E. Panteley and A. Loria, “On global uniform asymptotic stability of nonlinear time-varying systems in cascade,” Syst. Control Lett., vol. 33, pp. 131–138, 1998. [17] K. Y. Pettersen and E. Lefeber, “Way-point tracking control of ships,” in Proc. 40th IEEE Conf. Decision Control, Orlando, FL, 2001, pp. 940–945. [18] K. Y. Pettersen and H. Nijmeijer, “Tracking control of an underactuated surface vessel,” in Proc. 37th IEEE Conf. Decision Control, Tampa, FL, 1998, pp. 4561–4566. [19] O. J. Sordalen and O. Egeland, “Exponential stabilization of nonholonomic chained systems,” IEEE Trans. Automat. Contr., vol. 40, pp. 802–819, July 1995.
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unstable, the tracking error remains bounded and the iterative process may converge. Finally, we would like to highlight the fact that our result, stated in [5, Th. 1], has been introduced in [4]. We would like also to point out that we were not aware of [3] at the time we submitted our paper for publication. Nevertheless, our design approach is different from the one proposed in [3]. In fact, it is assumed in [3] that the feedback controller C is available and satisfies the robust stability condition (see [3, Ass. A3]), and the weighting function W1 is designed in order to satisfy
W1 S
1 + 1W2 T e0j!^ 1 < 1:
In our approach, we set W1 as close as possible to 1 and design the feedback controller C to satisfy the robust performance condition
kjW Sj + jW T jk1 < 1: 1
Authors’ Reply to “Comments on ‘Robust iterative learning control design is straightforward for uncertain LTI systems satisfying the robust performance condition’”
(1)
2
(2)
Moreover, in order to guarantee that the least upper bound of the
L -norm of the final tracking error is less than the least upper bound of the L -norm of the initial tracking error, we propose to design 2
2
the feedback controller C satisfying a modified robust performance condition as stated in [5, Th. 2].
A. Tayebi and M. B. Zaremba Index Terms—Iterative learning control (ILC), robust performance, uncertain linear time-invariant (LTI) systems.
The authors appreciate Dr. Doh’s comment [2], which basically states that the condition in [5, Th. 1] is not just sufficient but is also necessary. First of all, we would like to point out that the proof presented in [2] [with actually an error in (4), where a factor = is missing] is not new; it has been already used verbatim in [3]. We agree with [2] that the condition in [5, Th. 1] is necessary in the case where the trial-time is of infinite length and we disagree with it in the case where the trial-time is finite. Of course, a finite trial-time is a more realistic scenario, since, in practice, iterative learning control (ILC) algorithms are applied for systems performing the same task repeatedly over a finite time interval. In fact, our result stated in [5, Th. 1] holds for the infinite-time case (since the robust performance condition is evaluated for all frequencies), and therefore holds for the finite-time case by the truncation argument. It is worth noting that the finite-time case does not require to satisfy (see the convergence condition for all frequencies, but only for ! [1]). Consequently, the robust performance condition used in [5, Th. 1] is too strong and is not necessary for the finite-time case. On the other hand, over a finite-time interval, even though the closed-loop system is
1 (2 )
!1
Manuscript received November 17, 2003; revised October 7, 2004. Recommended by Associate Editor Hong Wang. For reasons beyond the authors’ control, it was not possible to publish this reply in the same issue as [2]. A. Tayebi is with the Department of Electrical Engineering, Lakehead University, Thunder Bay, ON P7B 5E1, Canada (e-mail: [email protected]). M. B. Zaremba is with the Département d’Informatique et d’Ingénierie, Université du Québec, Hull, QC J8X 3X7, Canada (e-mail: [email protected]). Digital Object Identifier 10.1109/TAC.2004.839233
REFERENCES approach [1] N. Amann, D. H. Owens, E. Rogers, and A. Wahl, “An to linear iterative learning control design,” Int. J. Adapt. Control Signal Processing, vol. 10, pp. 767–781, 1996. [2] T.-Y. Doh, “Comments on ‘Robust iterative learning control design is straightforward for uncertain LTI systems satisfying the robust performance condition’,” IEEE Trans. Automat. Contr., vol. 49, pp. 629–630, Apr. 2004. [3] Q. Hu, J.-X. Xu, and T. H. Lee, “Iterative learning control design for Smith predictor,” Syst. Control Lett., vol. 44, pp. 201–210, 2001. [4] A. Tayebi and M. B. Zaremba, “Internal model-based robust iterative learning control for uncertain LTI systems,” in Proc. 39th IEEE Conf. Decision and Control, Sydney, Australia, 2000, pp. 3439–3444. [5] A. Tayebi and M. B. Zaremba, “Robust iterative learning control design is straightforward for uncertain LTI systems satisfying the robust performance condition,” IEEE Trans. Automat. Contr., vol. 49, pp. 101–106, Jan. 2003.
Correction to “Quadratic Stability of a Class of Switched Nonlinear Systems” Jun Zhao and Georgi M. Dimirovski
The correct affiliation of the first author in [1] should read as follows: J. Zhao is with the School of Information Science and Engineering, Northeastern University, Shenyang 110004, China. REFERENCES [1] J. Zhao and G. M. Dimirovski, “Quadratic stability of a class of switched nonlinear systems,” IEEE Trans. Automat. Contr., vol. 49, pp. 574–578, Apr. 2004.
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