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Sannomiya, T. Ohta, and H. Tanaka, “A method for auto- tuning of PID control parameters,” Automatica, vol. 20, pp. 321–332, May 1984. Discrete-Time State Feedback With Velocity Estimation Using a Dual Observer: Application to an Underwater Direct-Drive Grinding Robot Philippe Hamelin, Pascal Bigras, Julien Beaudry, Pierre-Luc Richard, and Michel Blain Abstract—Hydro-Qu´ ebec’s Research Institute has designed a robot to perform grinding tasks on underwater structures. This unique system is equipped with direct-drive linear motors, which have many useful dynamic characteristics. Since they lack intrinsic stiffness, however, their robustness to external disturbances must be achieved through the controller. Their lack of stiffness is a major disadvantage, because grinding generates very strong disturbance forces. Moreover, controller performance in such a system is limited by velocity feedback, which is usually derived from position encoder data. Though the state observer is recognized as an effective way to estimate velocity from position feedback without delay, it is not robust when applied to a system sensitive to external disturbances. The dual observer, which combines a state observer and a perturbation observer, aims to solve this problem. The simultaneous estimation of the state and disturbance not only improves state observer robustness, but also helps to compensate for disturbances in the controller. This paper presents the design of a discrete- time state-feedback controller with velocity estimation through a discrete- time dual observer. The design is validated by extensive comparative testing for a task that is as intensive as underwater grinding. Index Terms—Discrete-time perturbation observer (POB), grinding, hy- droelectric dams, linear motors, machine tool control. I. INTRODUCTION Hydro-Qu´ ebec manages more than 500 dikes and dams across the province of Qu´ ebec in Canada. Most were commissioned in the mid1900s, and so require periodic repair to ensure their long-term Manuscript received November 17, 2010; revised March 30, 2011; accepted April 23, 2011. Date of publication June 7, 2011; date of current version January 9, 2012. Recommended by Technical Editor C. A. Kitts. P. Hamelin, J. Beaudry, P.-L. Richard, and M. Blain are with the Robotics and Civil Engineering Group of Hydro-Qu´ ebec’s Research Institute, Varennes, QC J3X 1S1, Canada (e-mail: hamelin.philippe@ireq.ca; beaudry.julien@ireq.ca; richard.pierre-luc@ireq.ca; blain.michel@ireq.ca). P. Bigras is with the Department of Automated Production Engineering, ´ Ecole de Technologie Superieure, Universit´ e du Qu´ ebec, 1100 Rue Notre-Dame Ouest, Montr´ eal, QC H3C 1K3, Canada (e-mail: pascal.bigras@etsmtl.ca). Digital Object Identifier 10.1109/TMECH.2011.2154338 operability. One expensive task is grinding the surface of underwa- ter metal structures. This requires the target structure to be dewatered and numerous precautions to be taken to ensure worker safety. The repair cost can amount to more than $1 million. To reduce this cost, the Hydro-Qu´ ebec’s Research Institute (IREQ) has developed a sub- mersible grinding robot prototype to automate the task without dewa- tering the generating units. Robotic grinding is an emerging technology in many industries and is amply demonstrated by applications in the field of hydroelectricity production. Although underwater grinding tests have been performed by temporarily covering an industrial robot [1], sealing industrial robots to perform underwater grinding in deep water is an unsustainable ap- proach. This has led to the development of a unique system (Patent pending 2 694 883), since currently no other robot is specifically de- signed to perform this task. The system is equipped with direct-drive linear motors, which give it several advantages, including high band- width, high accuracy, and low friction [2]. However, since this type of actuator does not have a gearbox, external disturbances, such as those generated by the underwater environment and the grinding process, act directly on the motor’s shaft. Fortunately, the stiffness of direct-drive motors is dynamically configured through the controller. This allows the disturbance sensitivity, as well as the grinding vibrations, to be reduced, which may usually be amplified for the resonance frequencies of the gearbox coupled with a conventional motor. With the exception of the control law described in our recent publica- tions [3], [4], to our knowledge none exists that has been designed and implemented specifically for this class of system. However, the control of machine tools equipped with direct-drive linear motors is a research field upon which to draw. With the growing popularity of these actua- tors in machine tools, many control algorithms have been implemented. Some researchers have proposed H ∞ controllers [5]–[7], which usu- ally use frequency weighting functions to perform the synthesis. These algorithms have good potential, but their performance is closely related to the choice of weighting functions, which is not a straightforward one. Sliding mode controllers (SMCs) [7]–[9] can also achieve high perfor- mance in terms of robustness to external disturbances. Their structure is simple, and their gains are easy to determine. While recent SMC designs lead to a linear controller structure [7], [8], some of them [9] approximate the discontinuous sign function in order to minimize chat- tering. This, however, limits achievement of the best possible position- tracking performance. Adaptive robust controllers [10], [11] have also been proposed as a bridge between SMCs and adaptive controllers. The conventional linear controllers [12], [13] are still popular for con- trolling linear motors. Despite their simplicity, they give good results, particularly, when coupled with other algorithms, such as feedforward, friction compensator [14], and disturbance observer (DOB) [15]. In [9], a cascaded P/PI controller was compared to an SMC on a high- speed milling table driven by direct-drive linear motors. The authors subsequently added a DOB to the P/PI controller [12] to enhance its robustness to external disturbances. The same comparison as in [9] was made in [4] on the underwater grinding robot, but using the enhanced version of the P/PI, i.e., P/PI combined with a DOB [12]. For equivalent design specifications, the P/PI controller was shown to perform better. However, as is often the case for linear motors, velocity was derived from the linear position encoder output with a fixed time step. This produces undesirable noise and even mechanical vibrations, which depend on both the encoder resolution and the sampling rate. This is a major limiting factor for the performance of trajectory tracking. An easy way to reduce this noise is to use a low-pass filter. However, a filter inevitably adds a delay, which limits controller performance. An alternative to this would be to use a state observer to estimate the velocity without any delay. This approach has the disadvantage, 1083-4435/$26.00 © 2011 IEEE