On the optimal control of a linear neutral differential equation arising in economics

In this paper, we apply two optimization methods to solve an optimal control problem of a linear neutral differential equation (NDE) arising in economics. The first one is a variational method, the second follows a dynamic programming approach. Due to the infinite dimensionality of the NDE, the second method requires the reformulation of the latter as an ordinary differential equation in an appropriate abstract space. It is shown that the resulting HJB equation admits a closed-form solution, allowing for a much finer characterization of the optimal dynamics compared to the alternative variational method. The latter is clearly limited by the nontrivial nature of asymptotic analysis of NDEs.Neutral differential equations; economic dynamics; optimal control; calculus of variations; dynamic programming; infinite dimension

모델기반강화학습을이용한공정제어및최적화

학위논문(박사)--서울대학교 대학원 :공과대학 화학생물공학부,2020. 2. 이종민.순차적 의사결정 문제는 공정 최적화의 핵심 분야 중 하나이다. 이 문제의 수치적 해법 중 가장 많이 사용되는 것은 순방향으로 작동하는 직접법 (direct optimization) 방법이지만, 몇가지 한계점을 지니고 있다. 최적해는 open-loop의 형태를 지니고 있으며, 불확정성이 존재할때 방법론의 수치적 복잡도가 증가한다는 것이다. 동적 계획법 (dynamic programming) 은 이러한 한계점을 근원적으로 해결할 수 있지만, 그동안 공정 최적화에 적극적으로 고려되지 않았던 이유는 동적 계획법의 결과로 얻어진 편미분 방정식 문제가 유한차원 벡터공간이 아닌 무한차원의 함수공간에서 다루어지기 때문이다. 소위 차원의 저주라고 불리는 이 문제를 해결하기 위한 한가지 방법으로서, 샘플을 이용한 근사적 해법에 초점을 둔 강화학습 방법론이 연구되어 왔다. 본 학위논문에서는 강화학습 방법론 중, 공정 최적화에 적합한 모델 기반 강화학습에 대해 연구하고, 이를 공정 최적화의 대표적인 세가지 순차적 의사결정 문제인 스케줄링, 상위단계 최적화, 하위단계 제어에 적용하는 것을 목표로 한다. 이 문제들은 각각 부분관측 마르코프 결정 과정 (partially observable Markov decision process), 제어-아핀 상태공간 모델 (control-affine state space model), 일반적 상태공간 모델 (general state space model)로 모델링된다. 또한 각 수치적 모델들을 해결하기 위해 point based value iteration (PBVI), globalized dual heuristic programming (GDHP), and differential dynamic programming (DDP)로 불리는 방법들을 도입하였다. 이 세가지 문제와 방법론에서 제시된 특징들을 다음과 같이 요약할 수 있다: 첫번째로, 스케줄링 문제에서 closed-loop 피드백 형태의 해를 제시할 수 있었다. 이는 기존 직접법에서 얻을 수 없었던 형태로서, 강화학습의 강점을 부각할 수 있는 측면이라 생각할 수 있다. 두번째로 고려한 하위단계 제어 문제에서, 동적 계획법의 무한차원 함수공간 최적화 문제를 함수 근사 방법을 통해 유한차원 벡터공간 최적화 문제로 완화할 수 있는 방법을 도입하였다. 특히, 심층 신경망을 이용하여 함수 근사를 하였고, 이때 발생하는 여러가지 장점과 수렴 해석 결과를 본 학위논문에 실었다. 마지막 문제는 상위 단계 동적 최적화 문제이다. 동적 최적화 문제에서 발생하는 제약 조건하에서 강화학습을 수행하기 위해, 원-쌍대 미분동적 계획법 (primal-dual DDP) 방법론을 새로 제안하였다. 앞서 설명한 세가지 문제에 적용된 방법론을 검증하고, 동적 계획법이 직접법에 비견될 수 있는 방법론이라는 주장을 실증하기 위해 여러가지 공정 예제를 실었다.Sequential decision making problem is a crucial technology for plant-wide process optimization. While the dominant numerical method is the forward-in-time direct optimization, it is limited to the open-loop solution and has difficulty in considering the uncertainty. Dynamic programming method complements the limitations, nonetheless associated functional optimization suffers from the curse-of-dimensionality. The sample-based approach for approximating the dynamic programming, referred to as reinforcement learning (RL) can resolve the issue and investigated throughout this thesis. The method that accounts for the system model explicitly is in particular interest. The model-based RL is exploited to solve the three representative sequential decision making problems; scheduling, supervisory optimization, and regulatory control. The problems are formulated with partially observable Markov decision process, control-affine state space model, and general state space model, and associated model-based RL algorithms are point based value iteration (PBVI), globalized dual heuristic programming (GDHP), and differential dynamic programming (DDP), respectively. The contribution for each problem can be written as follows: First, for the scheduling problem, we developed the closed-loop feedback scheme which highlights the strength compared to the direct optimization method. In the second case, the regulatory control problem is tackled by the function approximation method which relaxes the functional optimization to the finite dimensional vector space optimization. Deep neural networks (DNNs) is utilized as the approximator, and the advantages as well as the convergence analysis is performed in the thesis. Finally, for the supervisory optimization problem, we developed the novel constraint RL framework that uses the primal-dual DDP method. Various illustrative examples are demonstrated to validate the developed model-based RL algorithms and to support the thesis statement on which the dynamic programming method can be considered as a complementary method for direct optimization method.1. Introduction 1 1.1 Motivation and previous work 1 1.2 Statement of contributions 9 1.3 Outline of the thesis 11 2. Background and preliminaries 13 2.1 Optimization problem formulation and the principle of optimality 13 2.1.1 Markov decision process 15 2.1.2 State space model 19 2.2 Overview of the developed RL algorithms 28 2.2.1 Point based value iteration 28 2.2.2 Globalized dual heuristic programming 29 2.2.3 Differential dynamic programming 32 3. A POMDP framework for integrated scheduling of infrastructure maintenance and inspection 35 3.1 Introduction 35 3.2 POMDP solution algorithm 38 3.2.1 General point based value iteration 38 3.2.2 GapMin algorithm 46 3.2.3 Receding horizon POMDP 49 3.3 Problem formulation for infrastructure scheduling 54 3.3.1 State 56 3.3.2 Maintenance and inspection actions 57 3.3.3 State transition function 61 3.3.4 Cost function 67 3.3.5 Observation set and observation function 68 3.3.6 State augmentation 69 3.4 Illustrative example and simulation result 69 3.4.1 Structural point for the analysis of a high dimensional belief space 72 3.4.2 Infinite horizon policy under the natural deterioration process 72 3.4.3 Receding horizon POMDP 79 3.4.4 Validation of POMDP policy via Monte Carlo simulation 83 4. A model-based deep reinforcement learning method applied to finite-horizon optimal control of nonlinear control-affine system 88 4.1 Introduction 88 4.2 Function approximation and learning with deep neural networks 91 4.2.1 GDHP with a function approximator 91 4.2.2 Stable learning of DNNs 96 4.2.3 Overall algorithm 103 4.3 Results and discussions 107 4.3.1 Example 1: Semi-batch reactor 107 4.3.2 Example 2: Diffusion-Convection-Reaction (DCR) process 120 5. Convergence analysis of the model-based deep reinforcement learning for optimal control of nonlinear control-affine system 126 5.1 Introduction 126 5.2 Convergence proof of globalized dual heuristic programming (GDHP) 128 5.3 Function approximation with deep neural networks 137 5.3.1 Function approximation and gradient descent learning 137 5.3.2 Forward and backward propagations of DNNs 139 5.4 Convergence analysis in the deep neural networks space 141 5.4.1 Lyapunov analysis of the neural network parameter errors 141 5.4.2 Lyapunov analysis of the closed-loop stability 150 5.4.3 Overall Lyapunov function 152 5.5 Simulation results and discussions 157 5.5.1 System description 158 5.5.2 Algorithmic settings 160 5.5.3 Control result 161 6. Primal-dual differential dynamic programming for constrained dynamic optimization of continuous system 170 6.1 Introduction 170 6.2 Primal-dual differential dynamic programming for constrained dynamic optimization 172 6.2.1 Augmented Lagrangian method 172 6.2.2 Primal-dual differential dynamic programming algorithm 175 6.2.3 Overall algorithm 179 6.3 Results and discussions 179 7. Concluding remarks 186 7.1 Summary of the contributions 187 7.2 Future works 189 Bibliography 192Docto

Differential Dynamic Programming for time-delayed systems

Trajectory optimization considers the problem of deciding how to control a dynamical system to move along a trajectory which minimizes some cost function. Differential Dynamic Programming (DDP) is an optimal control method which utilizes a second-order approximation of the problem to find the control. It is fast enough to allow real-time control and has been shown to work well for trajectory optimization in robotic systems. Here we extend classic DDP to systems with multiple time-delays in the state. Being able to find optimal trajectories for time-delayed systems with DDP opens up the possibility to use richer models for system identification and control, including recurrent neural networks with multiple timesteps in the state. We demonstrate the algorithm on a two-tank continuous stirred tank reactor. We also demonstrate the algorithm on a recurrent neural network trained to model an inverted pendulum with position information only.Comment: 7 pages, 6 figures, conference, Decision and Control (CDC), 2016 IEEE 55th Conference o