This paper is concerned with further studies on control synthesis of discrete-time Takagi-Sugeno (T-S) fuzzy systems. To do this, a novel slack variable technique, which is homogenous polynomially parameter-dependent on both the current-time normalized fuzzy weighting functions and the past-time normalized fuzzy weighting functions with arbitrary degrees, is presented by developing an efficient augmented multi-indexed matrix approach. Under the framework of homogenous matrix polynomials, the algebraic properties of both the current-time normalized fuzzy weighting functions and the past-time normalized fuzzy weighting functions are collected into sets of augmented multi-indexed matrices. Thus, more information about the underlying normalized fuzzy weighting functions is involved into control synthesis. Consequently, the relaxation quality of control synthesis of discrete-time T-S fuzzy systems is improved significantly. Finally, a numerical example is provided to illustrate the effectiveness of the proposed method.
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