Available online at www.sciencedirect.com Automatica 39 (2003) 1407–1415 www.elsevier.com/locate/automatica Brief Paper A phase-plane approach to time-optimal control of single-DOF mechanical systems with friction  Dong-Soo Choi, Seung-Jean Kim, In-Joong Ha * ASRI/ACRC, School of Electrical Engineering, Seoul National University, San 56-1 Shinlim-Dong, Kwanak-Ku, Seoul 151-742, South Korea Received 19 July 2001; received in revised form 6 November 2002; accepted 12 March 2003 Abstract In this paper, we attempt to solve the time-optimal control problem for single-degree-of-freedom (DOF) mechanical systems with friction, while taking into account not only the velocity-dependent control input constraint but also the state constraint. Direct application of the Pontryagin’s maximum principle (PMP) leads to a sixth-order nonlinear two-point boundary-value problem (TPBVP) which is very dicult to solve numerically. In this context, we take a phase-plane analysis without resorting to the PMP. Thereby, the exact time-optimal solution can be obtained simply by solving a set of rst-order dierential equations with continuous right-hand sides. We also present some simulation results to demonstrate the practical use of the time-optimal solution. ? 2003 Elsevier Ltd. All rights reserved. Keywords: Time-optimal control; Maximum principle; Friction; Mechanical systems 1. Introduction Most mechanical systems are subject to the eect of fric- tion to some extent. Nonetheless, the eect of Coulomb fric- tion was ignored or precompensated in most prior works on the time-optimal control of mechanical systems (Van Willigenburg & Loop, 1991). This is mainly because the conventional Pontryagin’s maximum principle (PMP) can- not be applied directly to time-optimal control of mechan- ical systems with friction, since their dynamic behavior is governed by second-order dierential equations with discon- tinuous right-hand sides. In this context, a version of PMP for optimal multiprocess (Clarke & Vinter, 1989) was em- ployed in Kim and Ha (2001) to time-optimal control of single-degree-of-freedom (DOF) mechanical systems with friction. In general, the direct application of the PMP leads to a nonlinear two-point boundary-value problem (TPBVP), which cannot be solved analytically. Therefore, it should be solved via sophisticated iterative numerical techniques such as the quasi-linearization and the gradient projection  This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Kenko Uchida under the direction of Editor Tamer Basar. * Corresponding author. Tel.: +82-2-880-7312; fax: +82-2-878-8198. E-mail addresses: coolbart@justek.com (D.-S. Choi), sjkim@asri.snu.ac.kr (S.-J. Kim), ijha@asri.snu.ac.kr (I.-J. Ha). techniques (Kirk, 1970). Usually, these methods require good estimates of the initial trajectory, or may show very poor convergence property (Kirk, 1970). On the other hand, it is well known that the so-called phase-space technique is useful to solve the time-optimal control problem for a spe- cic class of second-order systems (Shin & Mckay, 1985; (Bobrow, Dubosky, & Gibson, 1985). In this paper, we attempt to solve the time-optimal control problem for single-DOF mechanical systems with friction, while taking into account not only the control input constraint but also the state constraint. Specically, we take a phase-plane analysis instead of resorting to the PMP. Thereby, The exact time-optimal solution can be obtained simply by solving a set of rst-order dierential equations with continuous right-hand sides. Hence, it can be constructed numerically via the direct application of the well-known Euler or Runge–Kutta methods. Finally, we present some simulation results to demonstrate the practical use of the time-optimal solution. 2. Problem statement The dynamic behavior of a single-DOF mechanical sys- tem with friction is governed by ˙ x = v; ˙ v = - 1 m F (v;u)+ 1 m u; (1) 0005-1098/03/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0005-1098(03)00112-2