A differential-algebraic perspective on nonlinear controller design methodologies

Abstract

Considerable activity has occurred independently in the fields of nonlinear geometric controller design and the solution of differential-algebraic systems of equations. Recently, a differential-algebraic approach to nonlinear controller design has been proposed. In this paper, the formal relationship between these approaches is identified. In particular, it is shown that the index of the nonlinear inversion problem is equal to ρ + 1, where ρ is the relative order of the process. The merits of the primary nonlinear control algorithms are assessed from a differential-algebraic perspective. Error trajectory controllers offer the advantage of being index one differential-algebraic problems with no associated initialization difficulties. In contrast, the nonlinear inversion and sliding mode control designs result in higher-order index problems with initialization restrictions. The restrictions identified for the nonlinear inverse design are related to the concept of functional controllability from the nonlinear systems literature. The relationship between the differential-algebraic design approach and the nonlinear geometric approach is extended to a class of processes described by nonlinear differential-algebraic equations. Finally, the nonlinear controller design problem is analyzed graphically, highlighting the differences between the various approaches.