Minimal conjunctive normal expression of continuous piecewise affine functions

Reference:
J. Xu, T.J.J. van den Boom, B. De Schutter, and X. Luo, "Minimal conjunctive normal expression of continuous piecewise affine functions," IEEE Transactions on Automatic Control, vol. 61, no. 5, pp. 1340-1345, May 2016.

Abstract:
Continuous piecewise affine (PWA) functions arise in many aspects of control. For this kind of function, we propose the minimal conjunctive normal expression (CNE). The CNE can be expressed as the minimum of a collection of terms, each of which is the maximum of a set of affine functions. The minimal CNE is defined to contain the smallest number of parameters. Analogous to Boolean algebra, we propose implicants and prime implicants for continuous PWA functions. After obtaining all prime implicants, the problem of finding minimal CNEs can then be cast as a binary programming problem. A sharp bound on the number of boolean variables in the binary programming problem is given. In two worked examples, minimal CNEs are derived for given continuous PWA functions.