The modified repetitive control system is a type of servomechanism for the periodic reference input. That is, the modified repetitive control system follows the periodic reference input with small steady state error, even if a periodic disturbance or an uncertainty exists in the plant. Using previously proposed modified repetitive controllers, even if the plant does not include time-delay, transfer functions from the periodic reference input to the output and from the disturbance to the output have infinite numbers of poles. When transfer functions from the periodic reference input to the output and from the disturbance to the output have infinite numbers of poles, it is difficult to specify the input-output characteristic and the disturbance attenuation characteristic. From the practical point of view, it is desirable that the input-output characteristic and the disturbance attenuation characteristic are easily specified. In order to specify the input-output characteristic and the disturbance attenuation characteristic easily, transfer functions from the periodic reference input to the output and from the disturbance to the output are desirable to have finite numbers of poles. From this viewpoint, Yamada et al. proposed the concept of simple repetitive control systems such that the controller works as a modified repetitive controller and transfer functions from the periodic reference input to the output and from the disturbance to the output have finite numbers of poles. In addition, Yamada et al. clarified the parameterization of all stabilizing simple repetitive controllers. However, the method by Yamada et al. cannot be applied for the plant with uncertainty. The purpose of this paper is to propose the parameterization of all robust stabilizing simple repetitive controllers for the plant with uncertainty.

The simple multi-period repetitive control system proposed by Yamada and Ta kenaga is a type of servomechanism for the periodic reference input. That is , the simple multi-period repetitive control system follows the periodic reference input with small steady state error, even if a periodic disturbance or uncertainty exists in the plant. In addition , simple multi-period repetitive control systems make transfer functions from the periodic reference input to the output and from the disturbance to the output have finite numbers of poles. Yamada et al. clarified the parameterization of all stabilizing simple multi period repetitive controllers. Recently, the parameterization of all robust stabilizing simple multi-period repetitive controllers for the plant with uncertainty was clarified by Yamada et al. How ever, their method cannot be applied to multiple-inputlmultiple output plants. In this paper, we propose the parameterization of all robust stabilizing simple multi-period repetitive controllers for multiple-inputlmultiple-output plants.

Multi-period repetitive controllers improve the disturbance attenuation characteristic of the modified repetitive control system that follows the periodic reference input with a small steady state error. Recently, the parameterization of all stabilizing multi-period repetitive controllers was studied. However, when the parameterization of all stabilizing multi-period repetitive controllers is used, the input-output characteristic and the feedback characteristic cannot be specified separately. From the practical point of view, it is desirable to specify the input-output characteristic and the feedback characteristic separately. In addition, the parameterization is useful to design stabilizing controllers. Therefore, the problem of obtaining the parameterization of all stabilizing two-degree-of-freedom multi-period repetitive controllers that can specify the input-output characteristic and the disturbance attenuation characteristic separately is important to solve. In this paper, we propose the parameterization of all stabilizing two-degree-of-freedom multi-period repetitive controllers.

This paper considers the design of high-order modified repetitive control systems for periodic reference inputs with uncertain period-time. The objective of this work was to develop a new design method so that the closed-loop high-order modified repetitive control system is robustly stable with high control precision for periodic reference inputs with uncertain period-time. The parametrization of all stabilizing controllers containing three free parameters is proposed based on the Youla-Kucera parameterization. Moreover, to obtain the free parameters, the constraint conditions were converted into stability conditions in the form of Bilinear Matrix Inequalities that can be solved using the available software. In addition, the high control precision is guaranteed by designing the free parameters after the control characteristic of this control system. The validity and effectiveness of the proposed design method were verified by numerical examples.

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