1654 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 54, NO. 3, JUNE 2007 Trajectory Tracking by TP Model Transformation: Case Study of a Benchmark Problem Zoltán Petres, Student Member, IEEE, Péter Baranyi, Member, IEEE, Péter Korondi, Member, IEEE, and Hideki Hashimoto, Fellow, IEEE Abstract—The main objective of this paper is to study the recently proposed tensor-product-distributed-compensation (TPDC)-based control design framework in the case of tracking control design of a benchmark problem. The TPDC is a combina- tion of the tensor product model transformation and the parallel distributed compensation framework. In this paper, we investigate the effectiveness of the TPDC design. We study how it can be uniformly and readily executed without analytical derivations. We show that the TPDC is straightforward and numerically tractable, and is capable of guarantying various different control performances via linear matrix inequality (LMI) conditions. All these features are studied via the state feedback trajectory control design of the translational oscillations with an eccentric rotational proof mass actuator system. The trajectory tracking capability for various tracking commands is optimized here by decay rate LMI conditions. Constraints on the output and control of the closed-loop system are also considered by LMI conditions. We present numerical simulations of the resulting closed-loop system to validate the control design. Index Terms—Linear matrix inequalities (LMIs), parallel dis- tributed compensation (PDC), tensor product (TP) model trans- formation, trajectory command tracking, translational oscillations with an eccentric rotational proof mass actuator (TORA). NOMENCLATURE M Mass of cart. k Linear spring stiffness. m Mass of the proof mass actuator. I Moment of inertia of the proof mass actuator. e Distance between the rotation point and the center of the proof mass. N Control torque applied to the proof mass. q Translational position of the cart. θ Angular position of the rotational proof mass. Manuscript received October 25, 2005; revised April 12, 2006. This work was supported in part by the Hungarian Scientific Research Fund (OTKA) under Grant F 049838 and Grant K 62836, and in part by the János Bolyai Postdoctoral Scholarship of the Hungarian Academy of Sciences. Z. Petres is with the Computer and Automation Research Institute, Hungarian Academy of Sciences, 1111 Budapest, Hungary (e-mail: petres@ tmit.bme.hu). P. Baranyi is with the Computer and Automation Research Institute, Hungarian Academy of Sciences, 1111 Budapest, Hungary, and also with the Integrated Intelligent Systems Japanese–Hungarian Laboratory, Budapest University of Technology and Economics, 1111 Budapest, Hungary (e-mail: baranyi@sztaki.hu). P. Korondi is with Budapest University of Technology and Economics, 1111 Budapest, Hungary. H. Hashimoto is with the Institute of Industrial Science, The University of Tokyo, Tokyo 153-8505, Japan (e-mail: hashimoto@iis.u-tokyo.ac.jp). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIE.2007.894697 I. I NTRODUCTION T HE tensor product (TP) model form is a dynamic model representation whereupon linear matrix inequality (LMI)- based control design techniques [1]–[5] can immediately be executed. It describes a class of linear-parameter-varying (LPV) models by the convex combination of linear time invariant (LTI) models, where the convex combination is defined by the weighting functions of each parameter separately. An important advantage of the TP model forms is that the convex hull of the given dynamic LPV model can readily be determined and analyzed in the TP model representation. Furthermore, the feasibility of the LMIs can be considerably relaxed in this representation via modifying the convex hull of the LPV model. The TP model transformation is a recently proposed numer- ical method to transform LPV models into TP model form [6]–[8]. It is capable of transforming different LPV model representations (such as physical model given by analytic equa- tions, fuzzy, neural network, genetic algorithm based models) into TP model form in a uniform way. In this sense it replaces the analytical derivations and affine decompositions (that could be a very complex or even an unsolvable task), and auto- matically results in the TP model form. Execution of the TP model transformation takes a few minutes by a regular Personal Computer. The TP model transformation minimizes the number of the LTI components of the resulting TP model. Furthermore, the TP model transformation is capable of resulting different convex hulls of the given LPV model. One can find a number of LMIs under the parallel distrib- uted compensation (PDC) framework that can immediately be applied to the TP model according to various control design specifications. Therefore, it is worth linking the TP model transformation and the PDC design framework [9]. In this paper, we revisit the multicriteria nonlinear control problem of the translational oscillations with an eccentric ro- tational proof mass actuator (TORA) system and derive con- trollers to track the general type of trajectory commands. We derive a controller that is capable of asymptotically converging to the command signal. The design also considers constraints on the control and output values. To optimize the tracking performance under the given constraints, we execute decay rate controller design. The remainder of this paper is organized as follows. Section II introduces the tensor product distributed compen- sation (TPDC)-based controller design framework. Section III first describes the TORA system and discusses the goals and specifications of the controller. Then, the convex polytopic 0278-0046/$25.00 © 2007 IEEE