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THE APPLICATION OF OPTIMAL-CONTROL METHODOLOGY TO NON-LINEAR programming-PROBLEMS

MATHEMATICAL programming 1981年第3期21卷 331-347页

作者： MURRAY, DM YAKOWITZ, SJ Systems and Industrial Engineering Department University of Arizona 85721 Tucson AZ USA

dynamic programming techniques have proven to be more successful than alternative nonlinear programming algorithms for solving many discrete-time optimal control problems. The reason for this is that, because of the stagewise decomposition which characterizes dynamic programming, the computational burden grows approximately linearly with the numbern of decision times, whereas the burden for other methods tends to grow faster (e.g.,n3 for Newton's method). The idea motivating the present study is that the advantages of dynamic programming can be brought to bear on classical nonlinear programming problems if only they can somehow be rephrased as optimal control *** shown herein, it is indeed the case that many prominent problems in the nonlinear programming literature can be viewed as optimal control problems, and for these problems, modern dynamic programming methodology is competitive with respect to processing time. The mechanism behind this success is that such methodology achieves quadratic convergence without requiring solution of large systems of linear equations.

关键词： Optimization Nonlinear programming Unconstrained Optimization Discrete Optimal Control differential dynamic programming

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A MULTIOBJECTIVE APPROACH TO THE SHORT-TERM SCHEDULING OF A HYDROELECTRIC POWER-SYSTEM

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IEEE TRANSACTIONS ON POWER SYSTEMS 1995年第4期10卷 1750-1755页

作者： LYRA, C FERREIRA, LRM COPEL SOSDPPOBR-80430 CURITIBAPARANABRAZIL

This paper is concerned with the short-term scheduling of the Iguacu river hydroelectric power system, in Southern Brazil. The system comprises four plants with an unusual high coupling, held by two different utilities. Even though government regulations establishes operation guide lines, utilities have their own interests. A multiobjective framework is adopt to study the operation of the system. dynamic programming is used to find optimal solutions, with an implementation based on the concept of differential dynamic programming. A case study shows the possibility of obtaining energetic gains with the adoption of non-conventional operation rules and the need of negotiation to avoid wasting energy.

关键词： OPTIMAL HYDROELECTRIC SCHEDULING SHORT-TERM SCHEDULING MULTIOBJECTIVE OPTIMIZATION dynamic programming differential dynamic programming

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A feasibility-driven approach to control-limited DDP

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AUTONOMOUS ROBOTS 2022年第8期46卷 985-1005页

作者： Mastalli, Carlos Merkt, Wolfgang Marti-Saumell, Josep Ferrolho, Henrique Sola, Joan Mansard, Nicolas Vijayakumar, Sethu Heriot Watt Univ Sch Engn & Phys Sci Inst Sensors Signals & Syst Edinburgh Midlothian Scotland Univ Edinburgh Sch Informat Edinburgh Midlothian Scotland Univ Oxford Oxford Robot Inst Oxford England Univ Politecn Cataluna Inst Robot & Informat Ind Barcelona Spain LAAS CNRS Toulouse France

differential dynamic programming (DDP) is a direct single shooting method for trajectory optimization. Its efficiency derives from the exploitation of temporal structure (inherent to optimal control problems) and explicit roll-out/integration of the system dynamics. However, it suffers from numerical instability and, when compared to direct multiple shooting methods, it has limited initialization options (allows initialization of controls, but not of states) and lacks proper handling of control constraints. In this work, we tackle these issues with a feasibility-driven approach that regulates the dynamic feasibility during the numerical optimization and ensures control limits. Our feasibility search emulates the numerical resolution of a direct multiple shooting problem with only dynamics constraints. We show that our approach (named Box-FDDP) has better numerical convergence than Box-DDP+ (a single shooting method), and that its convergence rate and runtime performance are competitive with state-of-the-art direct transcription formulations solved using the interior point and active set algorithms available in KNITRO. We further show that Box-FDDP decreases the dynamic feasibility error monotonically-as in state-of-the-art nonlinear programming algorithms. We demonstrate the benefits of our approach by generating complex and athletic motions for quadruped and humanoid robots. Finally, we highlight that Box-FDDP is suitable for model predictive control in legged robots.

关键词： Optimal control differential dynamic programming Feasibility Direct multiple shooting Control limits

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Cheap Newton steps for optimal control problems: Automatic differentiation and Pantoja's algorithm

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OPTIMIZATION METHODS & SOFTWARE 1999年第5期10卷 729-743页

作者： Christianson, B Univ Hertfordshire Numer Optimisat Ctr Hatfield AL10 9AB Herts England

In this paper we discuss Pantoja's construction of the Newton direction for discrete time optimal control problems. We show that automatic differentiation (AD) techniques can be used to calculate the Newton direction accurately, without requiring extensive re-writing of user code,and at a surprisingly low computational cost: for an N-step problem withp control variables and q state variables at each step, the worst case cost is 6(p + q + 1) times the computational cost of a single target function evaluation, independent of N, together with at most p(3)/3 + p(2)(q + 1) + 2p(q + 1)(2) + (q + 1)(3), i.e. less than (p + q + 1)(3), floating point multiply-and-add operations per time step. These costs may be considerably reduced if there is significant structural sparsity in the problem dynamics. The systematic use of checkpointing roughly doubles the operation counts, but reduces the total space cost to the order of 4pN floating point stores. A naive approach to finding the Newton step would require the solution of an Np x Np system of equations together with a number of function evaluations proportional to Np, so this approach to Pantoja's construction is extremely attractive, especially if q is very small relative to N. Straightforward modifications of the AD algorithms proposed here can be used to implement other discrete time optimal control solution techniques, such as differential dynamic programming (DDP), which use state-control feedback. The same techniques also can be used to determine with certainty, at the cost of a single Newton direction calculation, whether or not the Hessian of the target function is sufficiently positive definite at a point of interest. This allows computationally cheap post-hoc verification that a second-order minimum has been reached to a given accuracy, regardless of what method has been used to obtain it.

关键词： automatic differentiation checkpoints differential dynamic programming Newton's method optimal control Pantoja's algorithm verification

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Fourier-Hermite dynamic programming for Optimal Control

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL 2023年第10期68卷 6377-6384页

作者： Hassan, Syeda Sakira Sarkka, Simo Aalto Univ Dept Elect Engn & Automat Espoo 02150 Finland

In this article, we propose a novel computational method for solving nonlinear optimal control problems. The method is based on the use of Fourier-Hermite series for approximating the action-value function arising in dynamic programming instead of the conventional Taylor-series expansion used in differential dynamic programming. The coefficients of the Fourier-Hermite series can be numerically computed by using sigma-point methods, which leads to a novel class of sigma-point-based dynamic programming methods. We also prove the quadratic convergence of the method and experimentally test its performance against other methods.

关键词： Approximate dynamic programming differential dynamic programming Fourier-Hermite series sigma-point dynamic programming trajectory optimization

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DOC: Differentiable Optimal Control for Retargeting Motions onto Legged Robots

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ACM TRANSACTIONS ON GRAPHICS 2023年第4期42卷 1-14页

作者： Grandia, Ruben Farshidian, Farbod Knoop, Espen Schumacher, Christian Hutter, Marco Bacher, Moritz Disney Res Zurich Switzerland Swiss Fed Inst Technol Zurich Switzerland

Legged robots are designed to perform highly dynamic motions. However, it remains challenging for users to retarget expressive motions onto these complex systems. In this paper, we present a Differentiable Optimal Control (DOC) framework that facilitates the transfer of rich motions from either animals or animations onto these robots. Interfacing with either motion capture or animation data, we formulate retargeting objectives whose parameters make them agnostic to differences in proportions and numbers of degrees of freedom between input and robot. Optimizing these parameters over the manifold spanned by optimal state and control trajectories, we minimize the retargeting error. We demonstrate the utility and efficacy of our modeling by applying DOC to a Model-Predictive Control (MPC) formulation, showing retargeting results for a family of robots of varying proportions and mass distribution. With a hardware deployment, we further show that the retargeted motions are physically feasible, while MPC ensures that the robots retain their capability to react to unexpected disturbances.

关键词： differentiable optimal control differential dynamic programming motion retargeting model-predictive control

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NUMERICAL TRAJECTORY OPTIMIZATION FOR STOCHASTIC MECHANICAL SYSTEMS

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2019年第4期41卷 A2065-A2087页

作者： Boutselis, George, I Pan, Yunpeng Theodorou, Evangelos A. Georgia Inst Technol Dept Aerosp Engn Atlanta GA 30332 USA Magna Autonomous Syst LLC Palo Alto CA 94304 USA

In this paper we develop a novel optimal control framework for uncertain mechanical systems. Our work extends differential dynamic programming and handles uncertainty through generalized polynomial chaos (gPC) theory. The obtained scheme is able to influence the probabilistic evolution of nonlinear systems with stochastic model parameters. Its scalable, fast-converging nature plays a key role when dealing with gPC expansions of high-dimensional problems. Based on Lagrangian principles, we also prove that variational integrators can be designed to properly propagate and linearize gPC representations in discrete time. This observation allows us to further improve the efficiency of our trajectory-optimization methodology. Numerical simulations support the aforementioned arguments, while demonstrating the benefits of our approach over standard optimization methods. Last but not least, a complete analysis is provided that studies the properties of the proposed algorithm.

关键词： trajectory optimization polynomial chaos differential dynamic programming discrete mechanics

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SECURITY-LEVEL STRATEGIES FOR NON-LINEAR STOCHASTIC PURSUIT-EVASION differential-GAMES

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OPTIMAL CONTROL APPLICATIONS & METHODS 1981年第2期2卷 107-116页

作者： CHOMPAISAL, A LEONDES, CT University of California Los Angeles California 90024 U.S.A

A first-order algorithm is developed to solve for the security-level strategies of differential games without an assumption that a saddle point exists. The inner optimization uses differential dynamic programming together with a convergence control method. The outer optimization uses a gradient method. The application of the algorithm is demonstrated by solving for numerical strategies for a realistic missile-antimissile pursuit-evasion problem. Two major advantages for this algorithm are that the initialization requires only a rough estimate of the nominal control and that the Hessian matrix of the Hamiltonian with respect to the control vector is positive (negative) definite for the minimization (maximization) problem only on the optimal path.

关键词： differential games Pursuit-evasion problems differential dynamic programming

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A globally convergent and efficient method for unconstrained discrete-time optimal control

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JOURNAL OF GLOBAL OPTIMIZATION 2002年第3-4期23卷 401-421页

作者： Ng, CK Liao, LZ Li, D Hong Kong Baptist Univ Dept Math Kowloon Hong Kong Peoples R China Chinese Univ Hong Kong Dept Syst Engn & Engn Management Shatin Hong Kong Peoples R China

Shift schemes are commonly used in non-convex situations when solving unconstrained discrete-time optimal control problems by the differential dynamic programming (DDP) method. However, the existing shift schemes are inefficient when the shift becomes too large. In this paper, a new method of combining the DDP method with a shift scheme and the steepest descent method is proposed to cope with non-convex situations. Under certain assumptions, the proposed method is globally convergent and has q-quadratic local conve rgence. Extensive numerical experiments on many test problems in the literature are reported. These numerical results illustrate the robustness and efficiency of the proposed method.

关键词： unconstrained discrete-time optimal control non-convex optimization differential dynamic programming steepest descent global convergence quadratic convergence

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A learning-based model predictive control scheme for injection speed tracking in injection molding process

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COMPLEX & INTELLIGENT SYSTEMS 2024年第6期10卷 7845-7861页

作者： Ren, Zhigang Cai, Jianpu Zhang, Bo Wu, Zongze Guangdong Univ Technol Sch Automat Guangdong Key Lab IoT Informat Technol Guangzhou 510006 Guangdong Peoples R China Guangdong Univ Technol Guangdong HongKong Macao Joint Lab Smart Discrete Guangzhou 510006 Guangdong Peoples R China Minist Educ Key Lab Intelligent Detect & Internet Things Mfg G Guangzhou 510006 Peoples R China Shenzhen Univ Coll Mechatron & Control Engn Guangdong Lab Artificial Intelligence & Digital Ec Shenzhen 518060 Peoples R China

Injection molding is a pivotal industrial process renowned for its high production speed, efficiency, and automation. Controlling the motion speed of injection molding machines is a crucial factor that influences production processes, directly affecting product quality and efficiency. This paper aims to tackle the challenge of achieving optimal tracking control of injection speed in a standard class of injection molding machines (IMMs) characterized by nonlinear dynamics. To achieve this goal, we propose a learning-based model predictive control (LMPC) scheme that incorporates Gaussian process regression (GPR) to predict and model uncertainty in the injection molding process (IMP). Specifically, the scheme formulates a nonlinear tracking control problem for injection speed, utilizing a GPR-based learning residual model to capture uncertainty and provide accurate predictions. It learns the dynamics model and historical data of the IMM, automatically adjusting the injection speed according to target requirements for optimal production control. Additionally, the optimization problem is efficiently solved using a control-constrained differential dynamic programming approach. Finally, we conduct comprehensive numerical experiments to demonstrate the effectiveness and efficiency of the proposed LMPC scheme for controlling injection speed in IMP.

关键词： Injection molding process Learning-based model predictive control Optimal control Gaussian process regression differential dynamic programming

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