Robust Repetitive Control Design Based On The Continuous-discrete Two-dimensional Model For Linear Plants With Periodic Coefficients

Repetitive control is a method of tracking control for systems that exhibit repetitive behavior, such as servo systems. The theory and its application have been the subject of a great deal of research over the past two decades. A focus of particular concern has been the time delay involved in repetitive control because it is the main factor influencing robust stability; it makes controller design difficult. Currently, the main methods of designing repetitive-control systems only consider the overall effect of the control and learning actions and focus primarily on system stability. This prevents them from exploiting the special features of repetitive control to make fundamental improvements in convergence and tracking performance.This dissertation presents a new repetitive control method based on a continuous-discrete two-dimensional (2D) model. It enables individual adjustment of the control and learning actions, thereby improving the transient performance. This makes the method less conservative than those that ignore the difference between the control and learning actions. The main results and innovations are listed as follows:(1) The design of a robust repetitive-control system based on static output-feedback is presented.For a linear plant with time-varying periodic uncertainties, a repetitive-control system based on static output feedback is described. A continuous lifting technique is used to construct a continuous-discrete 2D model that precisely describes the control and learning actions of the repetitive-control process. A parameter transformation and a singular-value decomposition of the output matrix yield a sufficient condition for the robust stability of the system in terms of an LMI. The stability condition, which is derived using 2D Lyapunov functional, not only guarantees the robust stability of the system, but also ensures that the tracking error decreases monotonically from period to period. A comparison of simulation results for this system and for a system designed using one-dimensional optimal state-feedback repetitive control demonstrates the validity and superiority of the method. (2) The design of a robust repetitive-control system based on reconstruction-state feedback of a state observer is concerted.A combination of state observer and reconstruction-state feedback is used to design a repetitive-control system for a linear plant with time-varying periodic uncertainties. This eliminates the problem of state feedback being difficult to implement in actual systems. A continuous-discrete 2D model is established. A singular-value decomposition of the output matrix and stability theory for 2D systems yield a sufficient condition for the robust stability of the control system and provide an LMI-based design algorithm for the state-observer gain and the feedback control gains. Simulation results show that a repetitive-control system based on a state observer is robustly stable and that the tracking error converges rapidly to zero.(3) The design problem of a robust repetitive-control system that provides a given H?disturbance attenuation performance level is dealt with.Combining the transfer function of the repetitive controller and the weighting matrix yields a disturbance attenuation performance index. A continuous-discrete 2D model based on the reconstruction-state feedback of a state observer and the input of a disturbance signal is built that accurately describes the features of repetitive control, thereby enabling individual adjustment of the control and learning actions. A sufficient condition for the repetitive-control system to have a disturbance-attenuation bound in the H?setting is given in terms of an LMI. It produces the parameters of the repetitive controller and the state observer. A numerical example demonstrates the effectiveness of the method, the main advantage of which is the easy, individual adjustment of control and learning through the tuning of two parameters in the LMI-based condition.(4) The adjustment effectiveness of control and learning is analyzed and an LMI-based method of designing a robust modified repetitive-control system is developed.A modified repetitive controller is constructed by inserting a low-pass filter into the time-delay positive feedback line, which yields a modified repetitive-control system based on state feedback. A continuous-discrete 2D model is established. The continuity of repetitive control and stability theory for time-delay systems are used to derive an LMI-based robust-stability condition. Two tuning parameters in the stability condition enable individual adjustment of the control and learning actions by means of the 2D feedback control gains. To illustrate how the control and learning actions of the modified repetitive-control system can be adjusted, a machine-tool workpiece system is designed. The system exhibits parametric vibrations during the machining of hard metal that periodically forms chips. Two parameters are used to evaluate the control and learning actions. A qualitative analysis of the simulation results shows what effect the adjustment has on the transient response and convergence rate of the tracking error. A performance index is employed to assess the overall effect of those two actions, and is used as a criterion for the selection of the tuning parameters. A design algorithm for a general modified repetitive-control system is also presented.(5) A design algorithm that simultaneously optimizes the maximum cutoff angular frequency of the low-pass filter and the feedback gains is provided.In a modified repetitive-control system, the cutoff angular frequency of the low-pass filter is mainly related to the tracking range and tracking precision, and the feedback gains are mainly related to the stability. These two kinds of parameters influence each other. A continuous-discrete 2D model of a modified repetitive-control system and stability theory on time-delay systems are used to establish two LMI-based sufficient stability conditions. The conditions are used to separately design the cutoff angular frequency of the low-pass filter and the feedback control gains. The features of the conditions are exploited to develop an iterative algorithm that searches for the best combination of the maximum cutoff angular frequency of the low-pass filter and the feedback gains. The control and learning actions are adjusted by changing two tuning parameters in the LMI condition. This method is used to control the speed of a rotational system. Simulation results show that it effectively eliminates the trade-off between stability and steady-state tracking performance.