(629a) Steady-State Optimization with Guaranteed Stability under Parametric Uncertainties

Stability is a primary objective in the design and operation of dynamic process systems. As a result, there have been many works to ensure Lyapunov stability in process design. "Global stability" or "stability in the large" is the most desirable stability, but they are very difficult to check and ensure. Just ensuring "local stability" or "stability in the small" requires serious computations because of the need of finding eigenvalues of a numerically undetermined matrix.

Without any knowledge on the nature of uncertainties, the most conservative approach available is required. However, if parametric uncertainties are the only source of uncertainties (or every other uncertainties can be properly translated into or approximated by parametric uncertainties), the conservatism can be spared. The resulting process design problem is a bilevel optimization program, mostly due to an embedded eigenvalue optimization problem.

In this work, the bilevel program is reformulated as a single-level semi-infinite program (SIP) by extending the recent work of [1] from the case of model uncertainty to the case of parametric uncertainty. The reformulation makes use of the Routh-Hurwitz stability criteria which should be satisfied at all possible (thus infinitely many) equilibrium points around a steady-state solution. A row and column generation solution algorithm is proposed to solve this SIP problem. The solution algorithm relies on the branch-and-reduce global optimization algorithm [2] to solve each relaxation and separation problem to global optimality. The proposed algorithm is illustrated with traditional chemical as well as biochemical process design problems. We present extensive computation results, including new robustly stable solutions for these problems.

References:

[1] Chang, Y. and N. V. Sahinidis, Optimization of metabolic pathways under stability considerations, Computers & Chemical Engineering, 29(3), 467-479, 2005.

[2] Tawarmalani, M. and N. V. Sahinidis, Global optimization of mixed-integer nonlinear programs: A theoretical and computational study, Mathematical Programming, Ser. A, 99(3), 563-591, 2004.