Multi-Input Multi-Output Repetitive Control Theory And Taylor Series Based Repetitive Control Design
de Kevin Xu
Repetitive control (RC) systems aim to achieve zero tracking error when tracking a periodic command, or when tracking a constant command in the presence of a periodic disturbance, or both a periodic command and periodic disturbance. This dissertation presents a new approach using Taylor Series Expansion of the inverse system z-transfer function model to design Finite Impulse Response (FIR) repetitive controllers for single-input single-output (SISO) systems, and compares the designs obtained to those generated by optimization in the frequency domain. This approach is very simple, straightforward, and easy to use. It also supplies considerable insight, and gives understanding of the cause of the patterns for zero locations in the optimization based design. The approach forms a different and effective time domain design method, and it can also be used to guide the choice of parameters in performing in the frequency domain optimization design. Next, this dissertation presents the theoretical foundation for frequency based optimization design of repetitive control design for multi-input multi-output (MIMO) systems. A comprehensive stability theory for MIMO repetitive control is developed. A necessary and sufficient condition for asymptotic stability in MIMO RC is derived, and four sufficient conditions are created. One of these is the MIMO version of the approximate monotonic decay condition in SISO RC, and one is a necessary and sufficient condition for stability for all possible disturbance periods. An appropriate optimization criterion for direct MIMO is presented based on minimizing a Frobenius norm summed over frequencies from zero to Nyquist. This design process is very tractable, requiring only solution of a linear algebraic equation. An alternative approach reduces the problem to a set of SISO design problems, one for each input-output pair. The performances of the resulting designs are studied by extensive examples. Both approaches are seen to be able to create RC designs with fast monotonic decay of the tracking error. Finally, this dissertation presents an analysis of using an experiment design sequence for parameter identification based on the theory of iterative learning control (ILC), a sister field to repetitive control. This is suggested as an alternative to the results in optimal experiment design. Modified ILC laws that are intentionally non-robust to model errors are developed, as a way to fine tune the use of ILC for identification purposes. The non-robustness with respect to its ability to improve identification of system parameters when the model error is correct is studied. It is demonstrated that in many cases the approach makes the learning particularly sensitive to relatively small parameter errors in the model, but sensitivity is sometimes limited to parameter errors of a specific sign.
