In this thesis a novel method for the realisation of Max-Min-Plus-Scaling (MMPS) functions is presented. It has previously been shown that continuous piecewise affine (PWA) functions, conjunctive MMPS functions (also lattices- or min-max functions) and kripfganz MMPS functions (difference between two convex functions) can each describe the same function. Each form has its own specific use cases and benefits and thus it may be desired to rewrite a specific function described in one form, into another. Current techniques are input dependent and not available for each combination, while some techniques blow the number of parameters. The technique presented in this thesis however is input and output independent and rigid in its construction. It fills up the gaps where some realisation were not possible yet, it generates a rigid and predictable output. The necessary and sufficient conditions for the existence of each form are proposed. Additionally, it is explained how redundant terms may be removed in order to ensure a minimal representation. An algorithm is given for the decomposition and for the construction of each canonical form and supported with some worked examples. The decomposition provides the tools to efficiently map between different descriptions.