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Three bounded proofs for nonlinear multi-input multi-output approximate dynamic programming based on the Lyapunov stability theory

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OPTIMAL CONTROL APPLICATIONS & METHODS 2018年第1期39卷 35-50页

作者： Huang, Zhijian Xiong, Xuemei Chen, Wentao Zhang, Qin Liu, Yihua Chen, Yuli Shanghai Maritime Univ Merchant Marine Coll Shanghai 201306 Peoples R China Shanghai Jiao Thng Univ Inst Power Plant & Automat Shanghai 200030 Peoples R China

The approximate dynamic programming needs 2 prerequisites to be an effective optimal control method. Firstly, it must be assured to be stable and convergent before application. Secondly, the control system should mainly he a nonlinear multi-input multi-output form. Thus, this paper introduces a nonlinear multi-input multi-output approximate dynamic programming and proves that it is stable in Lyapunov sense, therefore it is convergent. Besides, the Lyapunov function design is also analyzed. These proofs are based on the Lyapunov stability theory in the form of the utility function of quadratic, square-weighted sum, and absolute value. Thereafter, 3 typical control examples of nonlinear multi-input multi-output approximate dynamic programming arc offered to show their applications and verify the proofs. The proof overcomes the complex derivation, and the results contain 3 practical and systematic bounded proofs. It is for the first time that the proof focuses on nonlinear multi-input multi-output approximate dynamic programming from the view of utility function. What is more, the results can also serve as an effective analysis and guide for the utility function design and the stability criterion of nonlinear multi-input multi-output approximate dynamic programming as well.

关键词： approximate dynamic programming bounded proof convergent emission glider idling Lyapunov stability theory multi-input multi-output nonlinear utility function

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The derivation of iterative convergence calculation for a nonlinear MIMO approximate dynamic programming approach

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APPLIED MATHEMATICS AND COMPUTATION 2013年第9期219卷 4495-4502页

作者： Huang, Zhijian Ma, Jie Huang, He Shanghai Jiao Tong Univ Ocean Engn State Key Lab Shanghai 200030 Peoples R China Shanghai Maritime Univ Merchant Marine Coll Shanghai 200135 Peoples R China

The standard approximate dynamic programming has only one action output. It's applied to single control variable system, such as inverted pendulum. For multi-input multi-output system, approximate dynamic programming needs a complex scheme. Few papers have derived its iterative convergence calculation, or the presented algorithm lacks rigorous mathematical basis. This paper fist researches matrix analysis foundation for the derivation of multi-input multi-output approximate dynamic programming. The research finds flaws in mathematics of a typical algorithm of its derivation. Hence, we promote approximate dynamic programming to multi-input multi-output form. The detailed iterative convergence calculation of it is derived. An experiment shows its effect. This algorithm is proved to be rigorous in mathematics and not complicated. It is effective for the iterative convergence calculation of multi-input multi-output approximate dynamic programming. (C) 2012 Elsevier Inc. All rights reserved.

关键词： Adaptive critic designs approximate dynamic programming Iterative convergence calculation Neural network

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Choice of approximator and design of penalty function for an approximate dynamic programming based control approach

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JOURNAL OF PROCESS CONTROL 2006年第2期16卷 135-156页

作者： Lee, JM Kaisare, NS Lee, JH Georgia Inst Technol Sch Chem & Biomol Engn Atlanta GA 30332 USA

This paper investigates the choice of function approximator for an approximate dynamic programming (ADP) based control strategy. The ADP strategy allows the user to derive an improved control policy given a simulation model and some starting control policy (or alternatively, closed-loop identification data), while circumventing the 'curse-of-dimensionality' of the traditional dynamic programming approach. In ADP, one fits a function approximator to state vs. 'cost-to-go' data and solves the Bellman equation with the approximator in an iterative manner. A proper choice and design of function approximator is critical for convergence of the iteration and the quality of final learned control policy, because an approximation error can grow quickly in the loop of optimization and function approximation. Typical classes of approximators used in related approaches are parameterized global approximators (e.g. artificial neural networks) and nonparametric local averagers (e.g. k-nearest neighbor). In this paper, we assert on the basis of some case studies and a theoretical result that a certain type of local averagers should be preferred over global approximators as the former ensures monotonic convergence of the iteration. However, a converged cost-to-go function does not necessarily lead to a stable control policy on-line due to the problem of over-extrapolation. To cope with this difficulty, we propose that a penalty term be included in the objective function in each minimization to discourage the optimizer from finding a solution in the regions of state space where the local data density is inadequately low. A nonparametric density estimator, which can be naturally combined with a local averager, is employed for this purpose. (c) 2005 Elsevier Ltd. All rights reserved.

关键词： approximate dynamic programming k-nearest neighbor neural network

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An Optimal approximate dynamic programming Algorithm for Concave, Scalar Storage Problems With Vector-Valued Controls

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL 2013年第12期58卷 2995-3010页

作者： Nascimento, Juliana Powell, Warren B. Kimberly Clark Inc BR-04551000 Sao Paulo Brazil Princeton Univ Dept Operat Res & Financial Engn Princeton NJ 08544 USA

We prove convergence of an approximate dynamic programming algorithm for a class of high-dimensional stochastic control problems linked by a scalar storage device, given a technical condition. Our problem is motivated by the problem of optimizing energy flows for a power grid supported by grid-level storage. The problem is formulated as a stochastic, dynamic program, where we estimate the value of resources in storage using a piecewise linear value function approximation. Given the technical condition, we provide a rigorous convergence proof for an approximate dynamic programming algorithm, which can capture the presence of both the amount of energy held in storage as well as other exogenous variables. Our algorithm exploits the natural concavity of the problem to avoid any need for explicit exploration policies.

关键词： approximate dynamic programming resource allocation storage

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An approximate dynamic programming approach for improving accuracy of lossy data compression by Bloom filters

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2016年第3期252卷 985-994页

作者： Yang, Xinan Vernitski, Alexei Carrea, Laura Univ Essex Dept Math Sci Wivenhoe Pk Colchester CO4 3SQ Essex England Univ Reading Dept Meteorol Reading RG6 6BB Berks England Univ Essex Sch Comp Sci & Elect Engn Wivenhoe Pk Colchester CO4 3SQ Essex England

Bloom filters are a data structure for storing data in a compressed form. They offer excellent space and time efficiency at the cost of some loss of accuracy (so-called lossy compression). This work presents a yes-no Bloom filter, which as a data structure consisting of two parts: the yes-filter which is a standard Bloom filter and the no-filter which is another Bloom filter whose purpose is to represent those objects that were recognized incorrectly by the yes-filter (that is, to recognize the false positives of the yes-filter). By querying the no-filter after an object has been recognized by the yes-filter, we get a chance of rejecting it, which improves the accuracy of data recognition in comparison with the standard Bloom filter of the same total length. A further increase in accuracy is possible if one chooses objects to include in the no-filter so that the no-filter recognizes as many as possible false positives but no true positives, thus producing the most accurate yes-no Bloom filter among all yes-no Bloom filters. This paper studies how optimization techniques can be used to maximize the number of false positives recognized by the no-filter, with the constraint being that it should recognize no true positives. To achieve this aim, an Integer Linear Program (ILP) is proposed for the optimal selection of false positives. In practice the problem size is normally large leading to intractable optimal solution. Considering the similarity of the ILP with the Multidimensional Knapsack Problem, an approximate dynamic programming (ADP) model is developed making use of a reduced ILP for the value function approximation. Numerical results show the ADP model works best comparing with a number of heuristics as well as the CPLEX built-in solver (B&B), and this is what can be recommended for use in yes-no Bloom filters. In a wider context of the study of lossy compression algorithms, our research is an example showing how the arsenal of optimization methods can be applied

关键词： Lossy compression Bloom filter Integer linear program approximate dynamic programming Heuristics

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Network-Level Infrastructure Management Using approximate dynamic programming

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JOURNAL OF INFRASTRUCTURE SYSTEMS 2010年第2期16卷 103-111页

作者： Kuhn, Kenneth D. Univ Canterbury Dept Civil & Nat Resources Engn Christchurch 8041 New Zealand

This research introduces the use of approximate dynamic programming to overcome a variety of limitations of distinct infrastructure management problem formulations. The form, as well as the parameters, of a model specifying the long-term costs associated with alternate infrastructure maintenance policies are learned via simulation. The introduced methodology makes it possible to manage large heterogeneous networks of facilities related by budgetary restrictions and resource constraints as well as by dependencies in maintenance costs or deterioration. In addition, the methodology is particularly well suited to consideration of multiple types of infrastructure condition data at the same time, including continuous-valued data and relevant historical data. Introduced techniques will prove valuable when high-quality deterioration and cost estimation models are available but are ill suited for use in a Markov decision problem framework. Computational studies show that the introduced approach is able to find an optimal solution to a relatively simple infrastructure management problem, and is able to find increasingly good solutions to a more complex problem.

关键词： Infrastructure management approximate dynamic programming Markov decision problem Infrastructure maintenance

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Markov Decision Process-Based Resilience Enhancement for Distribution Systems: An approximate dynamic programming Approach

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IEEE TRANSACTIONS ON SMART GRID 2020年第3期11卷 2498-2510页

作者： Wang, Chong Ju, Ping Lei, Shunbo Wang, Zhaoyu Wu, Feng Hou, Yunhe Hohai Univ Coll Energy & Elect Engn Nanjing 211100 Peoples R China Univ Michigan Dept Elect Engn & Comp Sci Ann Arbor MI 48109 USA Iowa State Univ Dept Elect & Comp Engn Ames IA 50011 USA Univ Hong Kong Dept Elect & Elect Engn Hong Kong Peoples R China

Because failures in distribution systems caused by extreme weather events directly result in consumers' outages, this paper proposes a state-based decision-making model with the objective of mitigating loss of load to improve the distribution system resilience throughout the unfolding events. The system topologies including on/off states of feeder lines are modeled as Markov states, and the probabilities from one Markov state to another Markov state throughout the unfolding events are determined by the component failure caused by the unfolding events. A recursive optimization model based on Markov decision processes (MDP) is developed to make state-based actions, i.e., system reconfiguration, at each decision time. To overcome the curse of dimensionality caused by enormous states and actions, an approximate dynamic programming (ADP) approach based on post-decision states and iteration is used to solve the proposed MDP-based model. IEEE 33-bus system and IEEE 123-bus system are used to validate the proposed model.

关键词： Markov processes Load modeling Meteorology Topology Power systems Resilience Fault tolerance approximate dynamic programming distribution systems Markov decision processes resilience enhancement

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An approximate dynamic programming based approach to dual adaptive control

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JOURNAL OF PROCESS CONTROL 2009年第5期19卷 859-864页

作者： Lee, Jong Min Lee, Jay H. Georgia Inst Technol Sch Chem & Biomol Engn Atlanta GA 30332 USA Univ Alberta Edmonton AB T6G 2G6 Canada

In this paper, an approximate dynamic programming (ADP) based strategy is applied to the dual adaptive control problem. The ADP strategy provides a computationally amenable way to build a significantly improved policy by solving dynamic programming on only those points of the hyper-state space sampled during closed-loop Monte Carlo simulations performed under known suboptimal control policies. The potentials of the ADP approach for generating a significantly improved policy are illustrated on an ARX process with unknown/varying parameters. (C) 2009 Elsevier Ltd. All rights reserved.

关键词： Dual control Stochastic optimal control Adaptive control Stochastic dynamic programming approximate dynamic programming

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Time-dependent green vehicle routing problem with stochastic vehicle speeds: An approximate dynamic programming algorithm

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TRANSPORTATION RESEARCH PART D-TRANSPORT AND ENVIRONMENT 2017年第Jul.期54卷 82-98页

作者： Cimen, Mustafa Soysal, Mehmet Hacettepe Univ Management Sci Ankara Turkey Hacettepe Univ Operat Management Ankara Turkey

This paper addresses a Time Dependent Capacitated Vehicle Routing Problem with stochastic vehicle speeds and environmental concerns. The problem has been formulated as a Markovian Decision Process. As distinct from the traditional attempts on the problem, while estimating the amount of fuel consumption and emissions, the model takes time dependency and stochasticity of the vehicle speeds into account. The Time Dependent Capacitated Vehicle Routing Problem is known to be NP-Hard for even deterministic settings. Incorporating uncertainty to the problem increases complexity, which renders classical optimization methods infeasible. Therefore, we propose an approximate dynamic programming based heuristic as a decision aid tool for the problem. The proposed Markovian Decision Model and approximate dynamic programming based heuristic are flexible in terms that more environmentally friendly solutions can be obtained by changing the objective function from cost minimization to emissions minimization. The added values of the proposed decision support tools have been shown through computational analyses on several instances. The computational analyses show that incorporating vehicle speed stochasticity into decision support models has potential to improve the performance of resulting routes in terms of travel duration, emissions and travel cost. In addition, the proposed heuristic provides promising results within relatively short computation times. (C) 2017 Elsevier Ltd. All rights reserved.

关键词： Green vehicle routing Stochastic time dependent capacitated vehicle routing problem approximate dynamic programming Markovian Decision Process

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Data Driven State Reconstruction of dynamical System Based on approximate dynamic programming and Reinforcement Learning

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IEEE ACCESS 2021年 9卷 73299-73306页

作者： Da Silva, Fabio Nogueira Da Fonseca Neto, Joao Viana Univ Fed Maranhao Grad Program Elect Engn BR-65080805 Sao Luis Maranhao Brazil

A data-driven approximation formulation for the state reconstruction problem of dynamical systems is presented in this paper. Without the assumption of an explicit mathematical model, the Hamilton-Jacobi-Bellman (HJB) based approach of a data-driven state reconstruction design method for monitoring and output-feedback control of dynamical systems is presented. The proposed state reconstruction design is based on a dynamic programming approach. To evaluate the proposed state reconstruction, computational experiments are conducted using only dynamical system model output data. The sensitivity of the algorithm parameters is also analyzed and discussed. The performance evaluation is analyzed in terms of the error metrics of the discrete linear quadratic regulator with output feedback under the value iteration algorithm through a reinforcement learning strategy.

关键词： Mathematical model Observers Heuristic algorithms Optimal control dynamical systems Approximation algorithms Sensors approximate dynamic programming data driven controller optimal control output feedback reinforcement learning state observer state reconstruction

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