(644e) Efficient Coordination of Plantwide Distributed MPC: The Lyapunov Envelope Algorithm

In model predictive control (MPC), control decisions are determined by solving the optimal control problems at sampling times associated with the predicted trajectories of the system, which can become challenging for large-scale integrated processes. Distributed MPC [1], which adopts a decomposition of the entire process into subsystems and a coordination algorithm to update subsystem controllers’ decisions through information exchange, offers a structured and flexible paradigm of optimally controlling large-scale systems. Methods of automatically generating suitable decompositions based on network theory [2] and distributed optimization algorithms (primal-dual iterative algorithms that aims at an optimal solution of the monolithic problem based on subproblem solutions) [3] have been extensively investigated in the recent years. However, for iterative distributed optimization algorithms, the computation time for the iterations to converge will usually occupy a considerable proportion of the sampling interval (one or a few minutes) and hence result in a prohibitive delay. In other words, distributed optimization is usually difficult to implement in real time.

To overcome this limitation, in this work, we propose a Lyapunov envelope algorithm that is real-time implementable under tight computational time restrictions without sacrificing the closed-loop stability under distributed MPC. Specifically, in the proposed algorithm:

1. The interconnecting relations among the subsystems in the predicted horizon are not required to be satisfied exactly but only considered as soft constraints, i.e., the violations to these equalities are penalized in the objective function. Under this penalization, a primal iterative algorithm analogous to forward-backward splitting [4] is adopted and allowed to be early-terminated before full convergence. The violations to the interconnecting relations are regarded as perturbations.
2. The effect of perturbations originating from the early-terminated algorithm on the closed-loop stability of the entire system, as characterized by the Lyapunov function value difference between the realized and the predicted trajectories, is analyzed based on incremental dissipativity of the interconnected subsystems. Such a dissipativity analysis establishes that the objective function with penalty, under proper assumptions, acts as a plantwide Lyapunov function, called the Lyapunov envelope.
3. An arbitrary number of iterations result in a monotonically decreasing Lyapunov envelope, hence maintaining the closed-loop stability of the plantwide process. Naturally, larger numbers of iterations allow larger decrease and thus improve the resulting control performance.

We apply the Lyapunov envelope to the distributed MPC of a plantwide benchmark process, namely the vinyl acetate monomer plant [5], whose optimal control problems at each sampling time have numbers of variables and constraints on the order of 10⁴. Our simulation studies demonstrate a significant improvement of computational performance and real-time implementability compared to centralized MPC. To our best knowledge, there has not been any other work to apply distributed MPC on a comparable scale.

References

[1] Christofides, P. D., et al. (2013). Distributed model predictive control: A tutorial review and future research directions. Comput. Chem. Eng., 51, 21-41.

[2] Daoutidis, P., et al. (2019). Optimal decomposition of control and optimization problems by network structure: Concepts, methods and inspirations from biology. AIChE J., 65(10), e16708.

[3] Tang, W., & Daoutidis, P. (2021). Fast and stable nonconvex constrained distributed optimization: the ELLADA algorithm. Optim. Eng., doi:10.1007/s11081-020-09585-w

[4] Chen, G. H., & Rockafellar, R. T. (1997). Convergence rates in forward-backward splitting. SIAM J. Optim., 7(2), 421-444.

[5] Chen, R., Dave, K., McAvoy, T. J., & Luyben, M. (2003). A nonlinear dynamic model of a vinyl acetate process. Ind. Eng. Chem. Res., 42(20), 4478-4487.