1. Singular Systems Differential 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 DAE Control & System Theory Such that rank E is not maximal &  s such that det(sE – A)  0 The above is equivalent to the system of equations:

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

4. Linear Time-Invariant DAE Control & System Theory 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:

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

6. More on indices. An index 1 DAE of the form Is, by definition, a system of equations such that D x2 g(x 1,x 2 ) 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 x 2 in terms of x 1.

7. The Implicit Function Theorem x1x1 x2x2 G(x 1,x 2 ) = 0 Graph of x 2 (x 1 )

8. Differentiation Index for Nonlinear DAE

10. More on indices. 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.

11. Optimal Control of LTI-DAE Stabilizability: rank[sE-A,B] = n,  s  C  0 & Impulse Controllability: Im N = Im[NB 2,N 2 B 2,…,N -1,B 2 ]  The system is equivalent to And finding a minimizer for the LQR problem with DAE dynamics is equivalent to an LQR problem for (x 1,v)

12.  -suboptimal solution to the minimum time optimal control Problem See slide #3 Note that the fast subsystem must be held at zero for some interval [0,  ) after is T opt x1 is reached. Within a ‘singular perturbation’ mode of reasoning, this action is required to quash the fast, small dynamics approximated by the algebraic constraint.

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

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

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

16. Open Questions and Research Topics We have a Liapunov-type stability criterion for LTIDAE: The LTI-DAE is asymptotically stable iff  P such that A T PE + E T PA < 0 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