Copyright © IFAC System Identification, Kitakyushu, Fukuoka, Japan, 1997 ADAYnVE ZERO PHASE ERROR TRACKING CONTROLLER WITH PRECISION TRACKING PERFORMANCE Manabu Yamada, Zaier Riadh and Naoki Mizuno Department of Mechanical Engineering, Nagoya Institute of Technology, Showa, Nagoya 466, JAPAN e-mail: yamam@eine.mech.nitech.acjp, riadh@eine.mech.nitech.acjp, mizuno@eine.mech.nitech.acjp Abstract: This paper deals with a discrete-time tracking control where the desired output to be tracked is partially known a priori. The problem of designing prefilter which provides the overall system with the following frequency characteristics is considered: 1) The phase is zero for all frequencies, 2) The gain at given particular frequencies is set to unity, 3) The maximum error over a given frequency range between the gain and unity is less than an arbitrary given positive number. The prefilter satisfying this problem is given in an explicit form, and is implemented in adaptive control scheme. Keywords: discrete-time systems, non-minimum phase systems, frequency responses, adaptive control, dead-beat control, tracking characteristics. 1. INTRODUCTION In this paper, a tracking control problem for a discrete time non-minimum phase system is considered. When the desired trajectory is entirely known in advance, the perfect tracking can be achieved (Jayasuriya and Tornizuka, 1993). This paper deals with the tracking control problem in which the desired trajectory is not entirely known in advance but a finite steps of future desired output is assumed known. This problem is often fonnulated as a design problem of preview feedforward controllers in the framework of frequency domain. Along this line, the most attractive feedforward controller is the Zero Phase Error Tracking Controller, abbr., ZPETC, which has been first proposed by Tomizuka (1987). The ZPETC can provide the overall system from the desired output to the controlled one with frequency characteristics such that the phase is zero for all frequencies and the gain is unity at only zero frequency. However, there has been no discussion nor description of the gain characteristics except at zero frequency. Therefore, the resulting control system may has undesirable gain characteristics. In order to improve the gain characteristics, many new types of ZPETC's have been proposed until now. Funahashi and Yamada 837 (1993) proposed the optimal ZPETC minimizing the integral of the squared error between the gain and unity over a given frequency range. However, the design procedure requires to solve an optimization problem with a troublesome inequality constraint. Torfs, et al. (1992) proposed a ZPETC based on an expansion of the inverse system in power series. However, if the power series diverges, the gain becomes worse than that of Tomizuka (1987). To overcome this problem Funahashi, et af. (1995) proposed a ZPETC so that the convergence of the power series is always guaranteed. Moreover Yamada et a/. (1997) presented a simple design method of obtaining the ZPETC such that the maximum error between the gain and unity is less than an arbitrary given positive number. On the other hand, it has been widely accepted that the parameter uncertainty or changes in the plant may cause significant deterioration of the tracking performance. The ZPETC's proposed by Tornizuka (1987), Funahashi and Yamada (1993) and Funahashi et al . (1995) were made adaptive by Tsao and Tornizuka (1987), Yamada et al . (1995) and Yamada et af. (1996), respectively. Copyright © IFAC System Identification, Kitakyushu, Fukuoka, Japan, 1997 ADAYnVE ZERO PHASE ERROR TRACKING CONTROLLER WITH PRECISION TRACKING PERFORMANCE Manabu Yamada, Zaier Riadh and Naoki Mizuno Department of Mechanical Engineering, Nagoya Institute of Technology, Showa, Nagoya 466, JAPAN e-mail: yamam@eine.mech.nitech.acjp, riadh@eine.mech.nitech.acjp, mizuno@eine.mech.nitech.acjp Abstract: This paper deals with a discrete-time tracking control where the desired output to be tracked is partially known a priori. The problem of designing prefilter which provides the overall system with the following frequency characteristics is considered: 1) The phase is zero for all frequencies, 2) The gain at given particular frequencies is set to unity, 3) The maximum error over a given frequency range between the gain and unity is less than an arbitrary given positive number. The prefilter satisfying this problem is given in an explicit form, and is implemented in adaptive control scheme. Keywords: discrete-time systems, non-minimum phase systems, frequency responses, adaptive control, dead-beat control, tracking characteristics. 1. INTRODUCTION In this paper, a tracking control problem for a discrete time non-minimum phase system is considered. When the desired trajectory is entirely known in advance, the perfect tracking can be achieved (Jayasuriya and Tornizuka, 1993). This paper deals with the tracking control problem in which the desired trajectory is not entirely known in advance but a finite steps of future desired output is assumed known. This problem is often fonnulated as a design problem of preview feedforward controllers in the framework of frequency domain. Along this line, the most attractive feedforward controller is the Zero Phase Error Tracking Controller, abbr., ZPETC, which has been first proposed by Tomizuka (1987). The ZPETC can provide the overall system from the desired output to the controlled one with frequency characteristics such that the phase is zero for all frequencies and the gain is unity at only zero frequency. However, there has been no discussion nor description of the gain characteristics except at zero frequency. Therefore, the resulting control system may has undesirable gain characteristics. In order to improve the gain characteristics, many new types of ZPETC's have been proposed until now. Funahashi and Yamada 837 (1993) proposed the optimal ZPETC minimizing the integral of the squared error between the gain and unity over a given frequency range. However, the design procedure requires to solve an optimization problem with a troublesome inequality constraint. Torfs, et al. (1992) proposed a ZPETC based on an expansion of the inverse system in power series. However, if the power series diverges, the gain becomes worse than that of Tomizuka (1987). To overcome this problem Funahashi, et af. (1995) proposed a ZPETC so that the convergence of the power series is always guaranteed. Moreover Yamada et a/. (1997) presented a simple design method of obtaining the ZPETC such that the maximum error between the gain and unity is less than an arbitrary given positive number. On the other hand, it has been widely accepted that the parameter uncertainty or changes in the plant may cause significant deterioration of the tracking performance. The ZPETC's proposed by Tornizuka (1987), Funahashi and Yamada (1993) and Funahashi et al . (1995) were made adaptive by Tsao and Tornizuka (1987), Yamada et al . (1995) and Yamada et af. (1996), respectively.