Systems with delays frequently appear in engineering. The presence of delays makes system analysis and control design much more complicated. In this thesis, robust control problems for such systems will be studied in the frequency domain. The emphasis is on the systems with a single delay. The delay can be an input or output delay or a delay that is a part of a controller.

For plants with a single input or output delay, the Nehari problem, an extended Nehari problem, the one-block problem and the standard H-infinity control problem are studied. The J-spectral factorization of regular para-Hermitian transfer functions is developed before solving these problems. The chain-scattering approach is applied to reduce the standard H-infinity control problem to a delay-free problem and a one-block problem, which is then further reduced to an extended Nehari problem. After solving the extended Nehari problem, the controllers for the one-block problem and the standard H-infinity problem are recovered. The sub-optimal controllers for these problems have the same structure, incorporating a modified Smith predictor (MSP). 

The implementation of MSP (i.e., a distributed delay) is not trivial because of the inherent hidden unstable poles. Two different approaches are proposed to approximate it and, furthermore, to implement it in the z-domain and in the s-domain. The reported instability problem due to the approximation error does not exist, provided that the number of approximation steps is large enough. The steady-state performance of the system is also guaranteed. Two improved formulae for the rectangular rules are obtained as a by-product. 

Another problem associated with the MSP is identified: the MSP may run into numerical problems. A new predictor, the unified Smith predictor (USP), is proposed to overcome this problem. Some well-studied problems are revisited, to clarify their solutions based on a USP. 
Repetitive control is one of the techniques which introduce a delay element into a controller. It is very effective to track or reject periodic signals. This technique, together with H-infinity control, are applied to DC-AC converters to improve the voltage quality in micro-grids.