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Singular SystemsDifferential Algebraic Equations Introduction Systems Control Theory Chemical Engineering DAE Index for DAE Optimal Control Directions for the Future Linear Time-Invariant DAEControl System Theory Such that rank E is not maximal $s such that det(sE – A) 1 0 Linear Time-Invariant DAE Control System Theory Linear Time-Invariant DAE Control System Theory Chemical Engineering Example More on indices. Differentiation Index for Nonlinear DAE Differentiation Index for Nonlinear DAE More on indices. e-suboptimal solution to the minimum time optimal control Problem Chemical Engineering Examples Open Questions and Research Topics * * Presented by Henry Potrykus, Qin Group The above is equivalent to the system of equations: Also equivalent to: gives the controllable subspace of the state space, while the unobservable subspace is given by: For the synthesis of controllers and observers for the singular system, one finds: Fixed-volume condenser with negligible liquid hold-up. Differentiating the last two equations gives: This is an index 2 DAE. An index 1 DAE of the form Is, by definition, a system of equations such that Dx2g(x1,x2) is of Full rank: Thus,by the implicit function theorem, it is a system of equations such that, from the get-go, g may be solved (locally) for x2 in terms of x1. The Implicit Function Theorem x1 x2 G(x1,x2) = 0 Graph of x2(x1) The differentiation index may not be well defined. Note that the following example has only a bilinear nonlinearity. This type of nonlinearity occurs frequently in chemical engineering modeling: Differentiating the constraint twice gives: Which shows that the index is 3 unless $ t such that u(t) = 0 in which case the index cannot be 3. Optimal Control of LTI-DAE Stabilizability: rank[sE-A,B] = n, s ? C30 Impulse Controllability: Im N = Im[NB2,N2B2,…,Nn-1,B2] T The system is equivalent to And finding a minimizer for the LQR problem with DAE dynamics is equivalent to an LQR problem for (x1