离散变分积分器与最优控制理论

Variational integrator for fractional Pontryagin's systems. Existence of a discrete fractional Noether's theorem

Fractional Pontryagin's systems emerge in the study of a class of fractional optimal control problems but they are not resolvable in most cases. In this paper, we suggest a numerical approach for these fractional systems. Precisely, we construct a variational integrator allowing to preserve at the discrete level their intrinsic variational structure. The variational integrator obtained is then called shifted discrete fractional Pontryagin's system. We provide a solved fractional example in a certain sense. It allows us to test in this paper the convergence of the variational integrator constru…

流形上加速优化的时间自适应拉格朗日变分积分器

Time-adaptive Lagrangian Variational Integrators for Accelerated Optimization on Manifolds

针对流形上加速优化，提出时间自适应拉格朗日变分积分器框架，以保持几何性质并高效求解赋范向量空间和李群上的优化问题。

Accelerated Optimization on Riemannian Manifolds via Projected Variational Integrators

A variational formulation of accelerated optimization on normed spaces was recently introduced by considering a specific family of time-dependent Bregman Lagrangian and Hamiltonian systems whose corresponding trajectories converge to the minimizer of the given convex function at an arbitrary accelerated rate of $\mathcal{O}(1/t^p)$. This framework has been exploited using time-adaptive geometric integrators to design efficient explicit algorithms for symplectic accelerated optimization. It was observed that geometric discretizations were substantially less prone to stability issues, and were t…

Variational integrators for underactuated mechanical control systems with symmetries

Optimal control problems for underactuated mechanical systems can be seen as a higher-order variational problem subject to higher-order constraints (that is, when the Lagrangian function and the constraints depend on higher-order derivatives such as the acceleration, jerk or jounces). In this paper we discuss the variational formalism for the class of underactuated mechanical control systems when the configuration space is a trivial principal bundle and the construction of variational integrators for such mechanical control systems. An interesting family of geometric integrators can be defined…

Solving Parameter Estimation Problems with Discrete Adjoint Exponential Integrators

The solution of inverse problems in a variational setting finds best estimates of the model parameters by minimizing a cost function that penalizes the mismatch between model outputs and observations. The gradients required by the numerical optimization process are computed using adjoint models. Exponential integrators are a promising family of time discretizations for evolutionary partial differential equations. In order to allow the use of these discretizations in the context of inverse problems adjoints of exponential integrators are required. This work derives the discrete adjoint formulae…

On the Benefits of Surrogate Lagrangians in Optimal Control and Planning Algorithms

This paper explores the relationship between numerical integrators and optimal control algorithms. Specifically, the performance of the differential dynamical programming (DDP) algorithm is examined when a variational integrator and a newly proposed surrogate variational integrator are used to propagate and linearize system dynamics. Surrogate variational integrators, derived from backward error analysis, achieve higher levels of accuracy while maintaining the same integration complexity as nominal variational integrators. The increase in the integration accuracy is shown to have a large effec…

Efficient Computation of Higher-Order Variational Integrators in Robotic Simulation and Trajectory Optimization

This paper addresses the problem of efficiently computing higher-order variational integrators in simulation and trajectory optimization of mechanical systems as those often found in robotic applications. We develop $O(n)$ algorithms to evaluate the discrete Euler-Lagrange (DEL) equations and compute the Newton direction for solving the DEL equations, which results in linear-time variational integrators of arbitrarily high order. To our knowledge, no linear-time higher-order variational or even implicit integrators have been developed before. Moreover, an $O(n^2)$ algorithm to linearize the DE…

具有二次拉格朗日项的控制仿射系统新拉格朗日方法的变分积分器

Variational integrators for a new Lagrangian approach to control affine systems with a quadratic Lagrange term

本文针对控制仿射系统且控制代价为二次型的最优控制问题，提出新拉格朗日方法的精确离散与半离散变分积分器，推导离散最优性条件并证明其与连续形式等价，构造低阶积分格式求解低推力轨道转移，并研究了诺特定理。

SO上最优控制问题的离散变分积分器（3）

A Discrete Variational Integrator for Optimal Control Problems on SO(3)

本文研究刚体离散变分最优控制问题。要最小化的成本是施加的外部扭矩移动刚体从一个初始条件到一个预先指定的终端条件。而不是离散化的运动方程，我们使用离散方程从离散拉格朗日-达朗贝尔原理，一个过程，更好地近似的运动方程。在离散时间设置中，这两种方法一般不等价。使用自然的李代数公式离散运动学，该公式保证流保持在李群SO（3）及其代数SO（3）上。我们利用变分法中约束问题的拉格朗日方法导出离散时间的必要条件。我们给出了一个三维刚体机动的数值例子。

负虚系统的数字控制：离散时间混合积分器-增益系统方法

Digital control of negative imaginary systems: a discrete-time hybrid integrator-gain system approach

本文研究负虚系统的数字控制，提出使用离散时间混合积分器-增益系统（HIGS）控制器，证明其本身为步进超前负虚系统，并证明其能渐近稳定最小线性NI系统，通过质量-弹簧系统示例验证了有效性。

Discrete Variational Optimal Control

This paper develops numerical methods for optimal control of mechanical systems in the Lagrangian setting. It extends the theory of discrete mechanics to enable the solutions of optimal control problems through the discretization of variational principles. The key point is to solve the optimal control problem as a variational integrator of a specially constructed higher-dimensional system. The developed framework applies to systems on tangent bundles, Lie groups, underactuated and nonholonomic systems with symmetries, and can approximate either smooth or discontinuous control inputs. The resul…

Structured Linearization of Discrete Mechanical Systems for Analysis and Optimal Control

Variational integrators are well-suited for simulation of mechanical systems because they preserve mechanical quantities about a system such as momentum, or its change if external forcing is involved, and holonomic constraints. While they are not energy-preserving they do exhibit long-time stable energy behavior. However, variational integrators often simulate mechanical system dynamics by solving an implicit difference equation at each time step, one that is moreover expressed purely in terms of configurations at different time steps. This paper formulates the first- and second-order lineariz…

Second-order variational problems on Lie groupoids and optimal control applications

In this paper we study, from a variational and geometrical point of view, second-order variational problems on Lie groupoids and the construction of variational integrators for optimal control problems. First, we develop variational techniques for second-order variational problems on Lie groupoids and their applications to the construction of variational integrators for optimal control problems of mechanical systems. Next, we show how Lagrangian submanifolds of a symplectic groupoid gives intrinsically the discrete dynamics for second-order systems, both unconstrained and constrained, and we s…

Certifying the Intersection of Reach Sets of Integrator Agents with Set-valued Input Uncertainties

We consider the problem of verifying safety for a pair of identical integrator agents in continuous time with compact set-valued input uncertainties. We encode this verification problem as that of certifying or falsifying the intersection of their reach sets. We transcribe the same into a variational problem, namely that of minimizing the support function of the difference of the two reach sets over the unit sphere. We illustrate the computational tractability of the proposed formulation by developing two cases in detail, viz. when the inputs have time-varying norm-bounded and generic hyperrec…