(347b) Local Optimization of Dynamic Systems with Guaranteed Feasibility of Path Constraints

Inequality path constraints in optimal control problems often represent safety and product quality limits and their strict satisfaction is crucial. A control vector parameterization is applied to reduce the infinite-dimensional optimal control problem to an approximate problem with finitely many decision variables. The path constraints still have to be enforced along the whole time horizon leading to a semi-infinite program (SIP). We present an algorithm that terminates in finitely many steps under relatively mild assumptions returning a guaranteed feasible solution.

The solution satisfies the first-order Karush-Kuhn-Tucker (KKT) optimality conditions to a user-specified tolerance. The algorithm is based on [1] and is an adaption of the global optimization algorithm for SIP proposed in [2]. The semi-infinite constraints are approximated by simultaneous restriction and relaxation: While the right-hand-sides of the path constraints are restricted by ε, the constraints are relaxed by enforcing them at finitely many time points only.

This approximated optimal control problem can be solved by existing dynamic optimization codes. We use the single shooting approach implemented in DyOS [3]. The output is checked for constraint violations and the time sets of the path constraints are populated iteratively by adding time points where the constraint violation is maximal. The restriction ε is reduced if no violation can be determined.

Numerical case studies with models from chemical engineering are presented and the performance is compared with existing algorithms, among others [4].

References

[1]   Fu J, Faust, JMM, Chachuat B, Mitsos A. Local optimization of dynamic programs with guaranteed satisfaction of path constraints. August 2014;submitted.

[2]   Mitsos A. Global optimization of semi-infinite programs via restriction of the right-hand side. Optimization 2011;60(10-11):1291–308.

[3]   Schlegel M, Stockmann K, Binder T, Marquardt W. Dynamic optimization using adaptive control vector parameterization. Computers & Chemical Engineering 2005;29(8):1731–51.

[4]   Chen TW, Vassiliadis VS. Inequality path constraints in optimal control: a finite iteration ε-convergent scheme based on pointwise discretization. Journal of Process Control 2005;15(3):353–62.