Control of nonlinear descriptor systems, a computer algebra based approach

Abstract

Many problems in mathematical modeling of lumped parameter systems lead to sets of mixed ordinary differential and algebraic equations. A natural generalization are so called descriptor systems or sets of implicit ordinary differential equations, which are linear in the derivatives. This contribution deals with the geometric control of descriptor systems. Based on the presented geometric framework using the mathematical language of Pfaffian systems, we derive a canonical form of a descriptor system under some mild rank conditions. This form is equivalent to an explicit system, whenever some integrability conditions are met. This approach allows us to extend the well known concepts of accessibility, observability, equivalence by static feedback, etc., to the class of descriptor systems. The Euler-Lagrange and Hamilton-Jacobi equations for optimal control problems with descriptor systems are also derivable from this canonical form similar to the case of explicit control systems. In addition, this approach offers computer algebra based algorithms, which permit to apply the presented methods efficiently to real world problems.