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The purpose of this monograph is to describe a class of com-putational methods, based on polynomial matrices, for thedesign of dynamic compensators for linear multi-variablecontrol systems. The design of the compensator, which may beeither analogue or digital, is based on pole assignment. Amatrix fraction description, which employs polynomial matri-ces, is used to represent the system. The design comptuta-tion, however, employs matrices of real numbers rather thanpolynomial matrices. This simplifies the computational pro-cedures which can thus be implementedin commercially-avai-lable software packages. Both transient and steady-stateperformace specifications are included in the design proce-dure which is illustrated by four detailed examples.The monograph should be of interest to research workers andengineers in the field fo multi-variable control. For theformer it provides some new computational tools for the ap-plication of algebraic methods, for both groups it introdu-ces some new ideas for a more-direct approach to compensatordesign.