Fixed-Order Robust Controller Design by Convex Optimization Using Spectral Models

This thesis proposes a new method to design fixed-order controllers in frequency domain using convex optimization. The method is based on the shaping of open-loop transfer function in the Nyquist diagram with infinity norm constraints on weighted closed-loop transfer functions. A parametric model is not required in this method as it directly uses frequency-domain data. Furthermore, systems with multi-model uncertainty as well as systems with frequency-domain uncertainties can be considered. Fixed-order linearly parameterized controllers are designed with the proposed method for single-input single-output (SISO) linear time-invariant plants. The shaping of the open-loop transfer function is performed based on the minimization of the difference with a desired open-loop transfer function under H∞ constraints on the closed-loop sensitivity functions. Since these constraints represent a nonconvex set in the space of the controller parameters, an inner convex approximation of this set is proposed using the desired open-loop transfer function. This approximation makes the problem of robust fixed-order controller design a convex optimization problem. An extension of the method is proposed to design two-degree-of-freedom (2DOF) controllers for SISO plants. The method is also extended to tune fixed-order linearly parameterized multivariable controllers for multiple-input multiple-output (MIMO) linear time-invariant plants where the stability of the closed-loop system is guaranteed using Gershgorin bands. The control problem is solved only using a finite number of frequency-domain samples. However, the stability and performance conditions between frequency samples are also verified if a frequency-domain uncertainty is considered. It is shown that this adds some conservatism to the solution. The proposed frequency-domain method has been tested on many simulation examples. The method has been applied to a flexible transmission benchmark for robust controller design giving extremely good results. Additionally, the method has also been implemented on an experimental high-precision double-axis positioning system. These results show the effectiveness of the proposed methods.