This paper addresses the mode switching problem in piecewise smooth implicit multilinear initial value problems (IVPs), which are relevant for modeling hybrid dynamical systems like HVAC and power systems. Unlike traditional switched systems with explicit mode descriptions, this work focuses on systems where mode information is implicitly encoded in binary-valued variables and switching conditions are defined by inequality constraints. The paper investigates the transversal motion discontinuities that occur when the system meets the boundary surfaces defined by these constraints. A method is presented to determine the discontinuous motion by analyzing the total derivative of the inequality constraints. The modeling framework utilizes hybrid implicit multilinear time-invariant (iMTI) functions and describes the system using inequality-constrained index-1 differential-algebraic equations (DAEs). The Jacobian matrices and thus the total derivatives can be estimated algebraically, due to the use of multilinear functions. To handle the combinatorial complexity associated with the binary variables during mode switching, the paper proposes using sparsity pattern analysis to identify and solve sub-problems more efficiently. The presented method is applied to a two-point temperature-controlled three-tank system, and simulations are performed using the MTI-Toolbox for MathWorks MATLAB.
Warnecke et al. (Wed,) studied this question.