Automatica 44 (2008) 738 – 744 www.elsevier.com/locate/automatica Brief paper Explicit construction of H ∞ control law for a class of nonminimum phase nonlinear systems  Weiyao Lan a , Ben M. Chen b, ∗ a Department of Automation, Xiamen University, Xiamen, Fujian 361005, China b Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117576 Received 27 April 2006; received in revised form 23 May 2007; accepted 25 June 2007 Available online 26 November 2007 Abstract This paper addresses a nonlinear H ∞ control problem for a class of nonminimum phase nonlinear systems. The given system is first transformed into a special coordinate basis, in which the system zero dynamics is divided into a stable part and an unstable part. A sufficient solvability condition is then established for solving the nonlinear H ∞ control problem. Moreover, based on the sufficient solvability condition, an upper bound of the best achievable L 2 gain from the system disturbance to the system controlled output is estimated for the nonlinear H ∞ control problem. The proofs of our results yield explicit algorithms for constructing required control laws for solving the nonlinear H ∞ control problem. In particular, the solution to the nonlinear H ∞ control problem does not require solving any Hamilton–Jacobi equations. Finally, the obtained results are utilized to solve a benchmark problem on a rotational/translational actuator (RTAC) system.  2007 Elsevier Ltd. All rights reserved. Keywords: Disturbance attenuation; Nonlinear systems; L 2 gain; Nonminimum phase 1. Introduction The nonlinear H ∞ control problem is to design a feedback control law for a nonlinear system such that the closed-loop sys- tem is internally stable, and has an L 2 -gain, from its disturbance input to its system output, less than or equal to a prescribed value  > 0. This problem has attracted much research effort since the works of Van der Schaft (1991, 1992), and many in- teresting results are available in the literature (see, for example, Battilotti, 1996; Isidori, Schwartz, & Tarn, 1999; Jiang & Hill, 1998; Van der Schaft, 2000 and references therein). If  > 0 is arbitrary, the problem is also known as the problem of almost disturbance decoupling with internal stability. It was shown that the almost disturbance decoupling problem is solvable if the disturbance input does not affect the unstable part of zero dy- namics of the system, see, for example, Isidori (1996a, 1996b),  This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Murat Arcak under the direction of Editor Hassan Khalil. ∗ Corresponding author. Tel.: +65 6516 2289; fax: +65 6779 1103. E-mail addresses: wylan@xmu.edu.cn (W. Lan), bmchen@nus.edu.sg (B.M. Chen). 0005-1098/$ - see front matter  2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2007.06.019 Marino, Respondek, van der Schaft, and Tomei (1994). When the unstable zero dynamics of the system is affected by distur- bance input, the almost disturbance decoupling problem is also solvable for a special class of nonlinear systems whose zero dy- namics contains a chain of integrators affected by disturbance (Lin, 1998). However, for more general situations, disturbance decoupling is generally not feasible. One has to seek to design a controller that achieves a pre-specified L 2 -gain  >  ∗ , where  ∗ is the best achievable performance for the problem, i.e., the problem is solvable for  >  ∗ and not for  <  ∗ . For linear systems, the optimal value  ∗ can be perfectly calculated by solving two Lyapunov equations if the system is single-input and single-output (SISO) (see, for example, Chen, 2000; Chen, Saberi, & Ly, 1992a, 1992b; Peterson, 1987; Scherer, 1992). It is shown in Chen (2000) and Chen et al. (1992b) that the optimal value is only related to the unsta- ble zero dynamics of the given system even for the singular problem. For nonlinear systems, the problem of estimating the optimal  ∗ is investigated in Isidori et al. (1999) and Ji and Gao (1995). An estimation of optimal H ∞ -gain for nonlinear H ∞ control problem is obtained in Ji and Gao (1995). In Isidori et al. (1999), the authors computed an upper bound of the