Dynamic mode decomposition with application to optimal control

optimal control;model reduction;system identification;Ginzburg-Landau;persistence of excitation;dynamic mode decomposition;adjoint method

Banks, Michael J ; Bodony ; Daniel J

关键词: optimal control;    model reduction;    system identification;    Ginzburg-Landau;    persistence of excitation;    dynamic mode decomposition;    adjoint method;   

Others  :  https://www.ideals.illinois.edu/bitstream/handle/2142/105720/BANKS-THESIS-2019.pdf?sequence=1&isAllowed=y

美国|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

High-dimensional fluid dynamics systems are central to a variety of modern engineering challenges. Direct numerical simulation of these processes is possible, but the high computational cost of these simulations is always an important consideration. Optimal control of these high-dimensional fluid dynamics problems is especially cost prohibitive. In order to keep computational cost within a reasonable level, we explore the potential of merging data-driven model reduction with optimal control. In particular, we demonstrate the application of Dynamic Mode Decomposition (DMD) to the adjoint-based optimal control of the Ginzburg-Landau equation.The adjoint method uses the governing equations of the system to derive a gradient in the space of admissible inputs. This gradient points toward the choice of input which minimizes a cost functional. A gradient descent algorithm can then be employed to arrive at the optimum. Since the gradient comes from the governing equations, it can be difficult to derive and costly to compute. Dynamic Mode Decomposition computes a linear, low-dimensional approximation of the dynamics which allows the adjoint gradient to be computed more easily. We demonstrate that a ten-fold dimensional reduction of the Ginzburg-Landau system can be used to approximate the full-state gradient to within a margin of error of 0.02%.The error in the reduced order gradient is dependent on the quality of the reduced order model (ROM). In order to compute an accurate gradient, we require that the ROM be robust to the range of possible dynamics in the search path of the gradient descent algorithm. This is difficult because one data set does not always capture the rich variety of possible behaviors of the system. Moreover, a reduced order model constructed from data acquired at a single operating point may not be robust to changes in control inputs or boundary conditions, thereby limiting the model’s utility in control.In order to evaluate the quality of a given data set, we use a condition called Persistence of Excitation (PE). When a data sample satisfies the PE property, it guarantees that the data represents the dynamics well and that the hidden model parameters of the system can be approximated using methods from adaptive control. We prove that the persistence of excitation condition ensures that DMD-based reduced order models derived from PE data optimally approximate the true low-rank dynamics of the system. This method is system agnostic and is based on the idea of Persistence of Excitation. Since PE is often not possible to achieve for many systems, we propose an optimization problem which, when solved, specifies an input designed to drive the dynamical system toward a more excited state. We call this Optimally Persistent Excitation (OPE). The act of applying our OPE-enriched data to DMD is called PE-informed DMD. To demonstrate our method, we apply PE-informed DMD to the simulation and closed loop control of the Ginzburg-Landau equation. Our results show that when we start with a poorly representative baseline data set, we can improve the resulting DMD approximation of the low-rank state transition matrix by 20%.