We show that the Glover-Doyle algorithm can be formulated simply by using the (J, J')-lossless factorization method and chain scattering matrix description. This algorithm was first stated by Glover and Doyle in 1988. Because the corresponding diagonal block of the (J,J')lossless matrix in the general 4-block H" control problem of the Glover-Doyle algorithm is not square, a new type of chain scattering matrix description is developed. With this description in hand, we obtain two types of state-space solution, which are similar to each other. Thus a similarity transformation between these solutions in the 4-block H" control problem can also be obtained. The main idea of the solution is illustrated by means of block diagrams.
1. Introducdon
Since Zames (1981) proposed the concept of sensitivity minimization in the H" domain, many researchers have made valuable contributions to the study of the H" domain. Transparent controllers for the standard 4-block H" problem were not obtained until Glover and Doyle (1988,1989) developed their well-known dual GD algorithms.
After Glover and Doyle (1989), Green et al. (1990) and Kimura (1991a) offered alternative developments using a J-spectral factorization, a characteristic of a (J, J')-lossless matrix. These methods are all based on the model-matching problem. Green (1992) combined an analytic system with J-lossless factorization to solve the H" control problem, which gradually yielded a problem in the form of the model-matching problem. Using (J, /')-lossless factorization and a chain-scattering matrix description, Kimura (1991b) and Ball et al. (1991) gave a fictitious signal method for solving the 4-block case of the problem. Furthermore, Kondo and Hara (1990) and Tsai and Tsai (1993) obtained results similar to those of Green (1992).
However, in these papers the (1,l) block or the (2,2) block of the (J, J')-lossless matrix is required to be square or to need additional fictitious signals. Consequently, the results obtained by using the (J, J')-lossless factorization method to solve the H" control problem were not the same as those obtained by the Glover-Doyle algorithms. In this paper we combine a normalized coprime factorization of the plant and (J,J')-factorization of one of the coprime factors, together with an alternative type of chain matrix description to recover precisely the results of Glover and Doyle (1988) (by using a left-coprime factorization) and Glover and Doyle (1989) (by using a right-copime factorization).
Despite the specific features of the two cases, the transfer functions for the resulting compensators turn out to be the *