Canonical representation: from piecewise-linear function to piecewise-smooth functions

The canonical representation of piecewise-linear (PWL) functions provides a global compact formulation of continuous PWL functions, which has significant advantages in the research and applications concerning nonlinear systems. This work studies the generalization of the canonical representation from PWL functions to piecewise-smooth (PWS) functions. First a class of PWS functions, called the regular PWS functions, is defined as a generalization of the continuous PWL functions. An important example of the regular PWS functions is the continuous piecewise-polynomial function. The continuous PWL function with a PWL partition is also covered by the regular PWS function. Then the canonical representation of the PWS function is defined and the existence conditions are discussed. The PWS generalization of the canonical representation is significant in applications where a PWS scheme can improve the performance of a PWL scheme in the approximation of a nonlinear function, i.e., in approximating the input/output (I/O) relation of a nonlinear system or a mapping neural network or in nonlinear signal processing