Adaptive PID Control for Asymptotic Tracking Problem of MIMO Systems

PID control, which is usually known as a classical output feedback control for SISO systems, has been widely used in the industrial world(Astrom & Hagglund, 1995; Suda, 1992). The tuning methods of PID control are adjusting the proportional, the integral and the derivative gains to make an output of a controlled system track a target value properly. There exist much more researches on tuning methods of PID control for SISO systems than MIMO systems although more MIMO systems actually exist than SISO systems. The tuning methods for SISO systems are difficult to apply to PID control for MIMO systems since the gains usually become matrices in such case. MIMO systems usually tend to have more complexities and uncertainties than SISO systems. Several tuning methods of PID control for such MIMO system are investigated as follows. From off-line approach, there are progressed classical loop shaping based methods (Ho et al., 2000; Hara et al., 2006) and H∞ control theory based methods (Mattei, 2001; Saeki, 2006; Zheng et al., 2002). From on-line approach, there are methods from self-tuning control such as the generalized predictive control based method (Gomma, 2004), the generalized minimum variance control based method (Yusof et al., 1994), the model matching based method (Yamamoto et al., 1992) and the method using neural network (Chang et al., 2003). These conventional methods often require that the MIMO system is stable and are usually used for a regulator problem for a constant target value but a tracking problem for a time-varying target value, which restrictions narrow their application. So trying these problems is significant from a scientific standpoint how there is possibility of PID control and from a practical standpoint of expanding applications. In MIMO case, there is possibility to solve these problems because PID control has more freedoms in tuning of PID gain matrices. On the other hand, adaptive servo control is known for a problem of the asymptotic output tracking and/or disturbances rejection to unknown systems under guaranteeing stability. There are researches for SISO systems (Hu & Tomizuka, 1993; Miyasato, 1998; Ortega & Kelly, 1985) and forMIMO systems (Chang D Dang O Johansson, 1987). Their controllers generally depend on structures of the controlled system and the reference system, which features are undesirable from standpoint of utility (Saeki, 2006; Miyamoto, 1999). So it is important to develop the fixed controller like PID controller to solve the servo problem and to show that conditions. But they are difficult to apply to the tuning of PID controller because of differences of their construction. In this paper, we consider adaptive PID control for the asymptotic output tracking problem of MIMO systems with unknown system parameters under existence of unknown disturbances. 9