Rectified deep neural networks overcome the curse of dimensionality for nonsmooth value functions in zero-sum games of nonlinear stiff systems

In this paper, we establish that for a wide class of controlled stochastic differential equations (SDEs) with stiff coefficients, the value functions of corresponding zero-sum games can be represented by a deep artificial neural network (DNN), whose complexity grows at most polynomially in both the dimension of the state equation and the reciprocal of the required accuracy. Such nonlinear stiff systems may arise, for example, from Galerkin approximations of controlled stochastic partial differential equations (SPDEs), or controlled PDEs with uncertain initial conditions and source terms. This implies that DNNs can break the curse of dimensionality in numerical approximations and optimal control of PDEs and SPDEs. The main ingredient of our proof is to construct a suitable discrete-time system to effectively approximate the evolution of the underlying stochastic dynamics. Similar ideas can also be applied to obtain expression rates of DNNs for value functions induced by stiff systems with regime switching coefficients and driven by general L\'{e}vy noise.Comment: This revised version has been accepted for publication in Analysis and Application

Boundary control of parabolic PDE using adaptive dynamic programming

In this dissertation, novel adaptive/approximate dynamic programming (ADP) based state and output feedback control methods are presented for distributed parameter systems (DPS) which are expressed as uncertain parabolic partial differential equations (PDEs) in one and two dimensional domains. In the first step, the output feedback control design using an early lumping method is introduced after model reduction. Subsequently controllers were developed in four stages; Unlike current approaches in the literature, state and output feedback approaches were designed without utilizing model reduction for uncertain linear, coupled nonlinear and two-dimensional parabolic PDEs, respectively. In all of these techniques, the infinite horizon cost function was considered and controller design was obtained in a forward-in-time and online manner without solving the algebraic Riccati equation (ARE) or using value and policy iterations techniques. Providing the stability analysis in the original infinite dimensional domain was a major challenge. Using Lyapunov criterion, the ultimate boundedness (UB) result was demonstrated for the regulation of closed-loop system using all the techniques developed herein. Moreover, due to distributed and large scale nature of state space, pure state feedback control design for DPS has proven to be practically obsolete. Therefore, output feedback design using limited point sensors in the domain or at boundaries are introduced. In the final two papers, the developed state feedback ADP control method was extended to regulate multi-dimensional and more complicated nonlinear parabolic PDE dynamics --Abstract, page iv

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

Model based fault diagnosis and prognosis of class of linear and nonlinear distributed parameter systems modeled by partial differential equations

With the rapid development of modern control systems, a significant number of industrial systems may suffer from component failures. An accurate yet faster fault prognosis and resilience can improve system availability and reduce unscheduled downtime. Therefore, in this dissertation, model-based prognosis and resilience control schemes have been developed for online prediction and accommodation of faults for distributed parameter systems (DPS). First, a novel fault detection, estimation and prediction framework is introduced utilizing a novel observer for a class of linear DPS with bounded disturbance by modeling the DPS as a set of partial differential equations. To relax the state measurability in DPS, filters are introduced to redesign the detection observer. Upon detecting a fault, an adaptive term is activated to estimate the multiplicative fault and a tuning law is derived to tune the fault parameter magnitude. Then based on this estimated fault parameter together with its failure limit, time-to-failure (TTF) is derived for prognosis. A novel fault accommodation scheme is developed to handle actuator and sensor faults with boundary measurements. Next, a fault isolation scheme is presented to differentiate actuator, sensor and state faults with a limited number of measurements for a class of linear and nonlinear DPS. Subsequently, actuator and sensor fault detection and prediction for a class of nonlinear DPS are considered with bounded disturbance by using a Luenberger observer. Finally, a novel resilient control scheme is proposed for nonlinear DPS once an actuator fault is detected by using an additional boundary measurement. In all the above methods, Lyapunov analysis is utilized to show the boundedness of the closed-loop signals during fault detection, prediction and resilience under mild assumptions --Abstract, page iv