Low-order Controller Design Via Polynomial Approach

This thesis discusses a unique control method called "polynomial approach" and its application in motion control. Because of its closed-loop-based design procedure, polynomial approach is a general and powerful method for low-order controller design.Polynomial approach is an alternative control design method besides classical control and modern control. Started from the proposal of Routh stability criteria in the late19th century, the coefficients of characteristic polynomial began to be associated with system stability. In the past decades, the advantages of polynomial approach began to be realized by more and more researchers. This method has straightforward design criteria. And its design parameters have clear physical meanings, which makes the method convenient to reach a good compromise among various tradeoffs during controller design.In the past, the theoretical foundation of polynomial approach was gradually established. However, compared with the well-developed classical and modern control, the theory of polyno-mial approach is still at its early stage. In order to further extend its applications, the author feels it is important to further clarify the theoretical aspects of the polynomial approach.Specifically, in the thesis, detailed discussions are first provided for obtaining the nominal characteristic ratio assignment [2.5,2,2,...]. Then, the damping and robustness analysis for all-pole systems shows that characteristic ratios with lower indexes have a stronger influence on the system’s overall performance. In order to deal with pole-zero interaction, a method called "asymptotic Bode curve" is applied in the assignment of time constant. In order to demonstrate the application of polynomial approach in real controller design,"vibration control of two-mass system" is chosen, which is a typical benchmark control problem. Two SISO (Single-Input Single-Output) control configurations (Proportional-Integral-Derivative control, resonant ratio control), and two MIMO (Multi-Input Multi-Output) control configurations (inertia ratio control, state-space-based control) are proposed. All the controllers are designed via polynomial approach and validated by simulation and experimental results.It is shown that, for a two-mass system with high inertia ratio, due to the strong tendency of pole-zero cancelation, the SISO configurations are impossible to provide a balanced tradeoff among damping, stability, robustness and response speed. The IP (Integral-Proportional) con-figuration can only provide sufficient damping when system’s inertia ratio is within5/16. Even though the m-IPD (modified-Integral-Proportional-Derivative) and resonant ratio control can always provide sufficient damping; however, due to the existence of positive derivative feedback loops, both configurations have very poor robustness when the inertia ratio is high. With ad-ditional feedback signals, the MIMO configurations show significant improvements compared with the previous SISO configurations. The inertia ratio control can effectively suppress the resonant vibrations even when the inertia ratio is as high as0.8. Using state-space-based con-trol, the transient response can be precisely designed if all the internal states are measurable. Meanwhile, due to the pole-zero interaction, the response speed needs to be properly specified in order to have a smooth transient response.