1. Singular SystemsDifferential Algebraic Equations Introduction Systems & Control Theory Chemical Engineering & DAE Index for DAE Optimal Control Directions for the Future Presented by Henry Potrykus, Qin Group

2. Linear Time-Invariant DAEControl & System Theory The above is equivalent to the system of equations: Such that rank E is not maximal & $s such that det(sE – A) ¹ 0

3. Also equivalent to: Linear Time-Invariant DAE Control & System Theory

4. For the synthesis of controllers and observers for the singular system, one finds: Linear Time-Invariant DAE Control & System Theory gives the controllable subspace of the state space, while the unobservable subspace is given by:

5. Fixed-volume condenser with negligible liquid hold-up. Chemical Engineering Example Differentiating the last two equations gives: This is an index 2 DAE.

6. An index 1 DAE of the form More on indices. 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.

7. The Implicit Function Theorem x2 G(x1,x2) = 0 x1 Graph of x2(x1)

8. Differentiation Index for Nonlinear DAE

9. Differentiation Index for Nonlinear DAE

10. 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: More on indices. 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.

11. Optimal Control of LTI-DAE Stabilizability: rank[sE-A,B] = n, "s Î C³0 & Impulse Controllability: Im N = Im[NB2,N2B2,…,Nn-1,B2] Þ The system is equivalent to And finding a minimizer for the LQR problem with DAE dynamics is equivalent to an LQR problem for (x1,v)

12. Note that the fast subsystem must be held at zero for some interval [0, e) after is Toptx1 is reached. Within a ‘singular perturbation’ mode of reasoning, this action is required to quash the fast, small dynamics approximated by the algebraic constraint. e-suboptimal solution to the minimum time optimal control Problem See slide #3

13. Maximum Principle for Index 1 DAE A local minimizer satisfies

14. x(t) p(b) C Supporting hyperplane

15. An isothermal CSTR in which the following reaction occurs Has as a model an index 2 DAE: Chemical Engineering Examples

16. We have a Liapunov-type stability criterion for LTIDAE: The LTI-DAE is asymptotically stable iff $ P such that ATPE + ETPA < 0 Open Questions and Research Topics Other linear system properties expressible as Linear Matrix Inequalities and Differential Inclusions can likewise be generalized to DAE. With this in mind, we would like to work out state constrained (LTI-DAE) MPC as an LMI programming problem