1 DAE Optimization towards real-time trajectory generation for flat nonlinear control systems Sachin Kansal, Fraser Forbes University of Alberta Martin Guay Queen’s University

2 ] DAE optimization problems ] Current solution techniques ] Proposed method - Basic Idea ] Flatness of dynamics? ] Normalized NLP solution ] Current and future work Sachin Kansal, Fraser Forbes - U of A Martin Guay - Queen’s U. Talk Outline

3 DAE Optimization * General DAOP Structure ] Infinite Dimensional Problem ] Integration at every iteration ] Algebraic path constraints Sachin Kansal, Fraser Forbes - U of A Martin Guay - Queen’s U. * Solution Difficulties

4 Polynomial u(t) & numerical integration [Hicks & Ray, 1971] Solution Techniques Discretization and numerical integration [Ray, 1981] Sachin Kansal, Fraser Forbes - U of A Martin Guay - Queen’s U. CVI CVP Discretization of control over a region, region contraction and numerical integration [Luus, 1989] IDP Polynomials for states and control & discretization of DE and constraints [Cuthrell & Biegler, 1987] CBT Polynomial structure for pseudo- outputs and subsequent discretization of path constraints [Kansal, Forbes & Guay, 2000] NNLP

5 NNLP - Basic Idea DE elimination from DAOP DAOP AOP System flatness Parameterization AOP NNLP Discretization Sachin Kansal, Fraser Forbes - U of A Martin Guay - Queen’s U.

6 System Flatness ] System flatness permits transformation of system space to a pseudo (flat)-output space ] Flat output space contains all information on system dynamics! DE Equivalents in flat-output space Sachin Kansal, Fraser Forbes - U of A Martin Guay - Queen’s U.

7 Parameterization A polynomial structure with unknown coefficients is chosen for the flat output which defines: Hence defining: Sachin Kansal, Fraser Forbes - U of A Martin Guay - Queen’s U. ] ] ]

8 Normalized AOP DAOP AOP Sachin Kansal, Fraser Forbes - U of A Martin Guay - Queen’s U.

9 ] Continuous-time inequality constraints are enforced at the grid points ] Time is discretized Discretization ] Affects only path inequality constraints Sachin Kansal, Fraser Forbes - U of A Martin Guay - Queen’s U.

10 NNLP Solution Solved with standard NLP solvers, e.g., SQP methods NNLP t0t0 tftf Optimal Input Trajectory Sachin Kansal, Fraser Forbes - U of A Martin Guay - Queen’s U.

11 Results ] Method tested on a number of benchmark DAOP problems ] Consistent or Better Results ] Significant Improvement in Computation Time/Load ] Particularly suitable for RTO Sachin Kansal, Fraser Forbes - U of A Martin Guay - Queen’s U.

12 ? Solution Issues ] Discretization of continuous algebraic constraints ] Initial conditions and bounds for decision variables ? Real-Time Implementation ] Problems that can be handled ] Major implementation and solution issues ? Extension to approximately flat systems Current Work Sachin Kansal, Fraser Forbes - U of A Martin Guay - Queen’s U.

13 Questions? Sachin Kansal, Fraser Forbes - U of A Martin Guay - Queen’s U.