Nonlinear model-predictive control and dynamic real-time optimization on infinite horizons

这个项目的总体目标是开发有效的算法来解决广泛的模型预测控制问题，特别是经济非线性模型预测控制问题(NMPC)，并基于无限水平策略保证闭环系统的稳定性。经济的NMPC是基于一个由过程运行的收入和成本组成的非线性目标函数，因此它不一定是正定函数。因此，由于目标函数不能作为Lyapunov函数，因此不能应用调节性NMPC的标准稳定性证明。在这个项目的头三年里，我们探索了一种依赖于无限水平方法的替代公式。通过应用贝尔曼最优原理，这种方法自然意味着监管和经济NMPC的稳定性--前提是可以计算出足够准确的数值解。首先，研究了时间轴的变换，得到了一个具有有界费用的有限水平问题。为了应用非线性规划的求解技术，对变换后的有限域问题进行了离散化。随后，研究了在保持足够的解精度的同时减少计算量的几种可能性，例如一种新的控制网格自适应策略和邻域极值更新。最后，比较了无限水平公式和有限水平公式的闭环系统性能。结果表明，对于不存在预先指定的最终时间的连续操作过程，无限视界公式是一种很有前途的替代方法。尽管如此，关于算法和方法细节的问题仍然悬而未决，应在第四年进行调查。首先，对于具有离散时间控制运动的连续时间过程，将得到闭环系统的稳定性。其次，将新的控制网格自适应策略扩展到路径约束的多阶段问题，以保证在地平线的过渡部分也有足够好的分辨率。第三，我们将研究具有有限回报的无限视界方程的数值解。最后，在邻域极值控制器的帮助下，进一步减少了计算时间。

The overall objective of this project is to develop efficient algorithms for a wide range of model-predictive control problems, in particular economic nonlinear model-predictive control problems (NMPC), with guaranteed closed-loop stability based on an infinite horizon strategy. Economic NMPC is based on a nonlinear objective function consisting of revenues and costs for process operation, which is thus not necessarily a positive-definite function. Consequently, standard stability proofs for regulatory NMPC cannot be applied as the objective function cannot serve as a Lyapunov function. Furthermore, finite moving horizon concepts are still computationally involved and not completely satisfactory.In the first three years of this project, we explored an alternative formulation relying on an infinite horizon approach. By applying Bellman's principle of optimality, this method naturally implies stability for regulatory as well as economic NMPC - provided a sufficiently accurate numerical solution can be computed. First, a transformation of the time axis was investigated leading to a finite horizon problem with bounded costs. In order to apply solution techniques from nonlinear programming, the transformed finite horizon problem was discretized. Subsequently, several possibilities to reduce computational load while maintaining sufficient solution accuracy were investigated such as a novel control grid adaptation strategy and neighboring-extremal updates. Finally, the closed-loop performance of the infinite horizon formulation was compared to a finite horizon formulation. It could be shown that the infinite-horizon formulation is a promising alternative for continuously operated processes for which no prespecified final time exists. Nonetheless, there remain open issues regarding algorithmic as well as methodological details which shall be investigated in the fourth year. First, closed-loop stability will be derived for continuous-time processes with discrete-time control moves. Second, the novel control grid adaptation strategy will be extended for path-constrained multi-stage problems in order to guarantee a sufficiently good resolution also in the transient parts of the horizon. Third, we will investigate the numerical solution of the infinite-horizon formulation with finite rewards. Finally, computational time will be further reduced with the help of a neighboring-extremal controller.