This chapter discusses H₂ optimal controllers for multivariable plants. The basic regulator problem has been studied using a time domain approach in the state-space and a frequency domain approach using transfer function matrices and Wiener-Hopf theory.

In this chapter two special problems of mixed H₂/H_∞ control have been presented. The corresponding two-degrees-of-freedom controller is made up by a feedforward part solving the underlying H_∞ mixed sensitivity problem and by a feedforward part that can be carried out independently by solving an H₂-norm optimization problem.

The following chapter discusses H_∞ filtering. The solution of the H_∞ filtering problem was obtained in polynomial system form. The H_∞ filter was first proposed by Grimble for use in a range of signal processing problems. The filter enables the power spectrum of the estimation error to be reduced to lower values in chosen frequency ranges than is possible with any other filter.

This chapter discusses eigenstructure assignment in linear systems by state feedback. Assignment of invariant factors in linear systems has been intensively studied in control theory for more than two decades since it is of great importance in many areas of this theory. For instance, such classical tasks as linear quadratic control and deadbeat control lead to specific requirements for poles placement of closed-loop systems.

This chapter surveys several existing methods for the J-spectral factorisation of a polynomial para-Hermitian matrix Z with real coefficients. A polynomial matrix Z is said to be para-Hermitian if Z* = Z, where the polynomial matrix Z* is the adjoint of Z, defined by Z(s)-Zτ(-s).