Andrei Dmitruk, Ivan Samylovskiy
We consider the time-optimal problem for a classical system of "double integrator" under the presence of a linear state constraint. By using the Maximum Principle of Dubovitskii and Milyutin, we determine a complete synthesis of optimal trajectories and qualitatively analyze the corresponding Lagrange multipliers.
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In this paper, first of all, we introduce the basic concepts of generating function in combinatorics and some combinatorial identities. In order to facilitate the understanding of m-order time optimal control synthesis function of discrete system (referred as m-order synthesis function), secondly, we introduce the derivation process and control ideas of 2nd-order synthesis function, and then deduce in detail the m-order synthesis function by means of generating function. By use of the m-order tracking-form synthesis function with filter factor, the methods of signal extraction and its predicti…
Cut time and optimal synthesis in sub-Riemannian problem on the group of motions of a plane
本文给出了平面SE（2）的运动群（旋转平移）上的左不变次黎曼问题的一个解。局部和全局极值轨迹的特点。证明了第一共轭时间的上下界上界。的切割时间被证明是等于第一麦克斯韦时间对应的一组离散对称的指数映射。得到了割迹的全局描述。描述了最佳合成。
Global optimality analysis in sub-Riemannian problem on the Lie group SH(2) is considered. We cutout open dense domains in the preimage and in the image of the exponential mapping based on the description of Maxwell strata. We then prove that the exponential mapping restricted to these domains is a diffeomorphism. Based on the proof of diffeomorphism, the cut time, i.e., time of loss of global optimality is computed on SH(2). We also consider the global structure of the exponential mapping and obtain an explicit description of cut locus and optimal synthesis.
This work is devoted to demonstration of the analysis on optimizing the output power harvested from vibration energy harvester.
Nonlinear optimal control synthesis via occupation measures
本文研究了一类非线性最优控制问题，其中所有的问题数据都是多项式。在本文的第一部分中，我们回顾了如何占领措施可以用来逐点逼近的最优值函数的一个给定的OCP，使用线性矩阵不等式（LMI）松弛的层次。在第二部分中，我们扩展的方法来近似的最优值函数在一个给定的集合，我们使用这样一个功能，建设性和计算推导出几乎最优控制法律。数值算例表明了该方法的有效性。
This paper reviews several uses of harmonic information in the synthesis of mechanisms and shows that such information can be put to even greater use in this field. Results are presented for both single and multi-degree of freedom systems which support this claim. In both cases, the inclusion of harmonic information into the objective function aids the search to locate high quality solutions.
For a particular class of planar dynamics that are linear with respect to the control variable, we show that the feedback strategy ''null-singular null'' is minimizing the maximum of a coordinate over infinite horizon, under a L 1 budget constraint on the control. Moreover, we characterize the optimal cost as a function of the budget. The proof is based on an unusual use of the clock form. This result generalizes the one obtained formerly for the SIR epidemiological model to more general Kolmogorov dynamics, that we illustrate on other biological models.
We study the problem of the minimum-time damping of a closed string under a bounded load, applied at a single fixed point. A constructive feedback control law is designed, which allows bringing the system to a bounded neighbourhood of the terminal manifold. The law has the form of the dry friction at the point, where the load is applied. The motion under the control is governed by a nonlinear wave equation. The existence and uniqueness of solution of the Cauchy problem for this equation are proved. The main result is the asymptotic optimality of the suggested control law.
In this paper, first of all, we introduce the basic concepts of generating function in combinatorics and some combinatorial identities. In order to facilitate the understanding of m(greater than 1)-order time optimal control synthesis function of discrete system (referred as m-order synthesis function), secondly, we introduce the derivation process and control ideas of 2nd-order synthesis function fsun(), and then deduce in detail the m-order synthesis function by means of generating function. By use of the m-order tracking-form synthesis function with filter factor, the methods of signal extr…
This paper addresses the problem of control synthesis for nonlinear optimal control problems in the presence of state and input constraints. The presented approach relies upon transforming the given problem into an infinite-dimensional linear program over the space of measures. To generate approximations to this infinite-dimensional program, a sequence of Semi-Definite Programs (SDP)s is formulated in the instance of polynomial cost and dynamics with semi-algebraic state and bounded input constraints. A method to extract a polynomial control function from each SDP is also given. This paper pro…
We consider optimal distributed controller synthesis for an interconnected system subject to communication constraints, in linear quadratic settings. Motivated by the problem of finite heavy duty vehicle platooning, we study systems composed of interconnected subsystems over a chain graph. By decomposing the system into orthogonal modes, the cost function can be separated into individual components. Thereby, derivation of the optimal controllers in state-space follows immediately. The optimal controllers are evaluated under the practical setting of heavy duty vehicle platooning with communicat…
Optimization-based Constrained Funnel Synthesis for Systems with Lipschitz Nonlinearities via Numerical Optimal Control
本文针对具有Lipschitz非线性和有界扰动的系统，提出基于数值最优控制的漏斗综合算法，通过多重打靶法将DLMI约束问题转化为半定规划，兼顾最大化漏斗入口与最小化吸引漏斗尺寸以抑制扰动。
The first part of this paper proposed a family of penalized convex relaxations for solving optimization problems with bilinear matrix inequality (BMI) constraints. In this part, we generalize our approach to a sequential scheme which starts from an arbitrary initial point (feasible or infeasible) and solves a sequence of penalized convex relaxations in order to find feasible and near-optimal solutions for BMI optimization problems. We evaluate the performance of the proposed method on the H2 and Hinfinity optimal controller design problems with both centralized and decentralized structures. Th…
Time Optimal Synthesis for Left--Invariant Control Systems on SO(3)
考虑由$\dot x=x(f+ug)$给出的控制系统，其中$x\in SO(3)$，$|u|\leq 1$和$f,g\in so(3)$定义了两个正交的左不变向量场，它们被归一化，使得$\|f\|=\cos(\al)$和$\|g\|=\sin(\al)$，$\al\in ]0,π/4[$。在本文中，我们提供了一个上界和一个下界$N(α)$，时间最优轨迹切换的最大数量。更准确地说，我们表明，$N_S(\al)\leq N(\al)\leq N_S(\al)+4$，其中$N_S(\al)$是一个合适的整数函数$\al$的$\al\to 0$是为了$π/(4α).$的结果是通过研究的时间最优合成的$R P^2$，其中的投影是由一个适当的Hopf纤维化定义的投影控制问题。最后，我们研究了单位球面$S^2$上的投影控制问题。它表现出有趣的功能，这将部分严格推导和部分描述的数值模拟。
A dynamical system entrains to a periodic input if its state converges globally to an attractor with the same period. In particular, for a constant input the state converges to a unique equilibrium point for any initial condition. We consider the problem of maximizing a weighted average of the system's output along the periodic attractor. The gain of entrainment is the benefit achieved by using a non-constant periodic input relative to a constant input with the same time average. Such a problem amounts to optimal allocation of resources in a periodic manner. We formulate this problem as a peri…
The use of deceptive strategies is important for an agent that attempts not to reveal his intentions in an adversarial environment. We consider a setting in which a supervisor provides a reference policy and expects an agent to follow the reference policy and perform a task. The agent may instead follow a different, deceptive policy to achieve a different task. We model the environment and the behavior of the agent with a Markov decision process, represent the tasks of the agent and the supervisor with reachability specifications, and study the synthesis of optimal deceptive policies for such …
This paper studies the bioremediation, in minimal time, of a water resource or reservoir using a single continuous bioreactor. The bioreactor is connected to two pumps, at different locations in the reservoir, that pump polluted water and inject back sufficiently clean water with the same flow rate. This leads to a minimal-time optimal control problem where the control variables are related to the inflow rates of both pumps. We obtain a non-convex problem for which it is not possible to directly prove the existence of its solutions. We overcome this difficulty and fully solve the studied probl…
Optimal control remains as one of the most versatile frameworks in systems theory, enabling applications ranging from classical robust control to real-time safe operation of fleets of vehicles. While some optimal control problems can be efficiently solved using algebraic or convex methods, most general forms of optimal control must be solved using memory- expensive numerical methods. In this paper we present a theoretical formulation and a corresponding numerical algorithm that can find Pontryagin- optimal inputs for general dynamical systems by using a direct method. Pontryagin-optimal inputs…
In this paper we advocate for Isaacs' method for the solution of differential games to be applied to the solution of optimal control problems. To make the argument, the vehicle employed is Pontryagin's canonical optimal control example, which entails a double integrator plant. However, rather than controlling the state to the origin, we require the end state to reach a terminal set that contains the origin in its interior. Indeed, in practice, it is required to control to a prescribed tolerance rather than reach a desired end state; constraining the end state to a terminal manifold of co-dimen…
Limit Time Optimal Synthesis for a Control-Affine System on $S^2$
对于$α\in(0,π/2)$，令$(Σ)_α$是控制系统$\dot{x}=(F+uG)x$，其中$x$属于二维单位球$S^2$，$u\in [-1,1]$和$F,G$是生成具有相应长度$\cos(α)$和$\sin(α)$的垂直轴的旋转的$3\times3$反对称矩阵。在本文中，我们研究的时间最优综合（TOS）从北极$(0,0,1)^T$相关的$(Σ)_α$，参数$α$趋于零。我们首先证明了TOS的特点是一个“两蛇”的配置在整个$S^2$，除了附近的$U_α$的南极$(0,0,-1)^T$的直径最多$Ø(α)$。接下来我们将说明，在$U_α$内部，TOS取决于$r(α):=π/2α-[π/2α]$和$α$之间的关系。更确切地说，我们的特点是三个主要的关系，考虑序列$(α_k)_{k\geq 0}$满足$(a)$ $r(α_k)=\bar{r}$ ; $(b)$ $r(α_k)=Cα_k$和$(c)$ $r(α_k)=0$，其中$\bar{r}\in (0,1)$和$C>0$。在每种情况下，我们描述的TOS和提供，经过适当的重新缩放，限制行为，$α$趋于零，相应的TOS内$U_α$。
In this paper we consider the minimum time population transfer problem for a two level quantum system driven by {\em two} external fields with bounded amplitude. The controls are modeled as real functions and we do not use the Rotating Wave Approximation. After projection on the Bloch sphere, we tackle the time-optimal control problem with techniques of optimal synthesis on 2-D manifolds. Based on the Pontryagin Maximum Principle, we characterize a restricted set of candidate optimal trajectories. Properties on this set, crucial for complete optimal synthesis, are illustrated by numerical simu…
Optimal synthesis control for evolution equations subject to nonlocal inputs
针对无限维演化方程在分布与非局部输入下的LQ最优控制问题，将记忆项视为状态分量，给出闭环Riccati型完整解。
We study the synthesis of a policy in a Markov decision process (MDP) following which an agent reaches a target state in the MDP while minimizing its total discounted cost. The problem combines a reachability criterion with a discounted cost criterion and naturally expresses the completion of a task with probabilistic guarantees and optimal transient performance. We first establish that an optimal policy for the considered formulation may not exist but that there always exists a near-optimal stationary policy. We additionally provide a necessary and sufficient condition for the existence of an…
Generalised Regret Optimal Controller Synthesis for Constrained Systems
本文提出约束系统的广义后悔最优控制器综合方法，涵盖动态后悔与竞争比框架，支持扰动逐点有界及状态输入约束，数值验证其性能优于经典$\u210b_2$、$\u210b_\infty$控制器。
A deterministic method for synthesizing the interconnect topologies optimized for the required properties is proposed. The method is based on the original description of graphs by projections, on establishing the bijective correspondence of the required properties and the projection properties of the initial graph, on postulating the corresponding restrictions of modified projections and on iteratively applying these restrictions to them either until the projection system is solved and the projections of the desired graph are obtained, or until its incompatibility with the given initial condit…
Constrained Optimal Synthesis and Robustness Analysis by Randomized Algorithms
在本文中，我们考虑使用随机算法的鲁棒控制。本文将现有的顺序统计量分布理论推广到一般情况，即总体分布不连续，且顺序统计量具有一定的约束条件。特别地，我们得到了一个有关顺序统计量的分布不等式。此外，我们还提出了两种不同的方法来寻找可靠的解决方案的鲁棒分析和优化综合问题的约束。此外，最小的计算工作量的调查和界的样本量。
This paper considers the synthesis of optimal safe controllers based on density functions. We present an algorithm for robust constrained optimal control synthesis using the duality relationship between the density function and the value function. The density function follows the Liouville equation and is the dual of the value function, which satisfies Bellman's optimality principle. Thanks to density functions, constraints over the distribution of states, such as safety constraints, can be posed straightforwardly in an optimal control problem. The constrained optimal control problem is then s…
Although quantum computers have the potential to efficiently solve certain problems considered difficult by known classical approaches, the design of a quantum circuit remains computationally difficult. It is known that the optimal gate design problem is equivalent to the solution of an associated optimal control problem, the solution to which is also computationally intensive. Hence, in this article, we introduce the application of a class of numerical methods (termed the max-plus curse of dimensionality free techniques) that determine the optimal control thereby synthesizing the desired unit…
Optimal Switching Synthesis for Jump Linear Systems with Gaussian initial state uncertainty
本文提出了一种方法，用于设计具有高斯初始状态不确定性的跳跃线性系统的最优切换序列，利用Wasserstein度量和滚动时域框架，通过最小化基于Wasserstein距离的目标函数来保证系统均方稳定性。
We propose a framework for synthesizing a feedback control policy that maximizes the region of attraction (ROA) of a closed-loop nonlinear dynamical system. Our synthesis technique relies on stochastic optimization, which involves computation of an objective function capturing the ROA for a feedback control law. We employ a machine learning technique based on deep neural network to estimate the ROA for a given feedback controller. Overall, our technique is capable of synthesizing a controller co-optimizing traditional control objectives like LQR cost together with ROA. We demonstrate the effic…
Stochastic optimal control in Hilbert spaces: $C^{1,1}$ regularity of the value function and optimal synthesis via viscosity solutions
本文用动态规划方法研究无穷维随机最优控制，证明值函数的$C^{1,1}$正则性，并基于$B$-连续粘性解构造最优反馈控制，应用于随机反应-扩散和延迟方程。
Benign Geometry and Distributional Robustness of $\mathcal{H}_2$ Synthesis
本文研究离散时间线性时不变系统的标准与分布鲁棒 $\mathcal{H}_2$ 综合，证明平稳性等价于全局最优性，并推广至分布鲁棒情形，最后刻画三类模糊集的最坏情况协方差以提供性能保证。
Optimal Synthesis on the Special Linear Group
本研究针对带左不变黎曼度量的二阶实特殊线性群，利用欧拉-庞加莱方程及其线性化确定共轭与麦克斯韦时间，证明第一麦克斯韦时间与割时间重合，给出割迹的显式矩阵描述，分析第一麦克斯韦时间函数并确定其全局最小值，刻画割迹与共轭迹的交集，证明所有测地线均为等最优的。
System level synthesis is a promising approach that formulates structured optimal controller synthesis problems as convex problems. This work solves the distributed linear-quadratic regulator problem under communication constraints directly in infinite-dimensional space, without the finite-impulse response relaxation common in related work. Our method can also be used to construct optimal distributed Kalman filters with limited information exchange. We combine the distributed Kalman filter with state-feedback control to perform localized LQG control with communication constraints. We provide a…
In the context of the linear programming (LP) approach to data-driven control, one assumes that the dynamical system is unknown but can be observed indirectly through data on its evolution. Both theoretical and empirical evidence suggest that a desired suboptimality gap is often only achieved with massive exploration of the state-space. In case of linear systems, we discuss how a relatively small but sufficiently rich dataset can be exploited to generate new constraints offline and without observing the corresponding transitions. Moreover, we show how to reconstruct the associated unknown stag…
Robust Nonlinear Optimal Control via System Level Synthesis
针对受范数有界扰动的非线性系统有限时域约束鲁棒最优控制问题，提出基于一阶泰勒展开分解与系统级综合的方法，联合优化多要素以降低规划保守性，并通过刚体旋转控制实验验证了优势。
A major challenge faced in the design of large-scale cyber-physical systems, such as power systems, the Internet of Things or intelligent transportation systems, is that traditional distributed optimal control methods do not scale gracefully, neither in controller synthesis nor in controller implementation, to systems composed of millions, billions or even trillions of interacting subsystems. This paper shows that this challenge can now be addressed by leveraging the recently introduced System Level Approach (SLA) to controller synthesis. In particular, in the context of the SLA, we define sui…
Optimal Kinematic Synthesis and Prototype Development of Knee Exoskeleton
针对膝关节外骨骼旋转范围影响康复效果的问题，构建优化问题求解单自由度外骨骼连杆最优尺寸，提升坐站运动活动范围24%，开展不对准研究并研制原型验证仿真结果。
Spatiotemporal Response Decay for Near-Optimal Distributed LQR via System Level Synthesis
该研究针对网络化LQR问题，用系统级综合方法证明集中式响应的时空指数衰减界，据此设计近最优分布式控制器，数值实验验证了架构选择的指导因素。
Robust Optimal Control for Nonlinear Systems with Parametric Uncertainties via System Level Synthesis
针对带范数有界干扰和参数不确定性的非线性系统，提出一种基于系统层级合成的方法，通过分解系统并凸化不确定性，联合优化轨迹与误差反馈，实现低保守性的鲁棒最优控制。
Optimal Control Synthesis of Closed-Loop Recommendation Systems over Social Networks
本文从控制论角度研究社交网络和电商平台的推荐系统设计问题，将推荐系统设计视为具有性能指标的无限时间状态反馈最优控制问题，通过奖励一致性与参与度、惩罚极化和偏离基线、正则化邻近用户曝光来实现稳定推荐系统。
System Level Synthesis
本文探讨了系统级综合框架，提出了一个全新的视角，用于线性系统中受约束的鲁棒和最优控制器综合。文章强调了SLS方法将控制器综合任务从控制器设计转变为整个闭环系统设计的优势，并突出了其在可扩展性和透明性方面的益处。特别介绍了SLS在大规模分布式最优控制和鲁棒控制中的应用，展示了如何通过SLS实现局部控制器的计算，并将鲁棒和最优控制方法扩展到大规模系统。同时，SLS还允许新的设计方法，首次量化了由于模型不确定性导致的鲁棒控制器性能退化，这种透明性对于使鲁棒控制方法与现代机器学习和统计推断技术相结合至关重要。文中还强调了实际和高效的计算解决方案，并通过易于理解的案例研究展示了方法。
Direct policy search has been widely applied in modern reinforcement learning and continuous control. However, the theoretical properties of direct policy search on nonsmooth robust control synthesis have not been fully understood. The optimal $\mathcal{H}_\infty$ control framework aims at designing a policy to minimize the closed-loop $\mathcal{H}_\infty$ norm, and is arguably the most fundamental robust control paradigm. In this work, we show that direct policy search is guaranteed to find the global solution of the robust $\mathcal{H}_\infty$ state-feedback control design problem. Notice th…
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