Energy function approximations for differential algebraic polynomial systems of Stokes-type

Abstract Energy functions are generalizations of controllability and observability Gramians to nonlinear systems and as such find applications in both nonlinear balanced truncation and feedback control. These energy functions are solutions to the Hamilton-Jacobi-Bellman (HJB) equations—partial differential equations defined over spatial dimensions determined by the number of state variables in the nonlinear system. Thus, they cannot be resolved with local basis functions, even for problems of modest dimension. In this paper, we extend recent results that utilize Kronecker products to generate polynomial approximations to HJB equations. Specifically, we consider the addition of linear constraints that exhibit a Stokes-type differential-algebraic equation (DAE) structure. This extension leverages the so-called strangeness framework for DAEs to create separate sets of algebraic and differential variables with differential equations that only involve the differential variables and algebraic equations that relate them both. At this point, the existing polynomial approximations using Kronecker products can be applied to find approximations to the energy functions. This approach is demonstrated on two polynomial feedback control problems. While the standard transformation destroys the sparsity in the original system, we present a formulation that preserves the original sparsity structure.