Time-optimal Problem Research Articles

Optimal quantum controls robust against detuning error

Abstract Precise control of quantum systems is one of the most important milestones for achieving practical quantum technologies, such as computation, sensing, and communication. Several factors deteriorate the control precision and thus their suppression is strongly demanded. One of the dominant factors is systematic errors, which are caused by discord between an expected parameter in control and its actual value. Error-robust control sequences, known as composite pulses, have been invented in the field of nuclear magnetic resonance (NMR). These sequences mainly focus on the suppression of errors in one-qubit control. The one-qubit control, which is the most fundamental in a wide range of quantum technologies, often suffers from detuning error. As there are many possible control sequences robust against the detuning error, it will practically be important to find ‘optimal’ robust controls with respect to several cost functions such as time required for operation, and pulse-area during the operation, which corresponds to the energy necessary for control. In this paper, we utilize the Pontryagin’s maximum principle (PMP), a tool for solving optimization problems under inequality constraints, to solve the time and pulse-area optimization problems. We analytically obtain pulse-area optimal controls robust against the detuning error. Moreover, we found that short-CORPSE, which is the shortest known composite pulse so far, is a probable candidate of the time optimal solution according to the PMP. We evaluate the performance of the pulse-area optimal robust control and the short-CORPSE, comparing with that of the direct operation.

Multi-parameter optimization of polarization gradient cooling for 87Rb atoms based on reinforcement learning.

Polarization gradient cooling (PGC) plays an important role in many cold atom applications including the formation of Bose-Einstein condensates (BECs) and cooling of single atoms. Traditional parameter optimization of PGC usually relies on subjective expertise, faces challenges in fine manipulation, and exhibits low optimization efficiency. Here, we propose a segmented control method that differs from the traditional PGC process by expanding the experiment parameters from 3 to 30. Subsequently, the conventional timing optimization problem is reformulated as a Markov decision process (MDP), and the experiment parameters are optimized using a reinforcement learning model. With proper settings of hyperparameters, the learning process exhibits good convergence and powerful parameter exploration capabilities. Finally, we capture ∼4.3 × 108 cold atoms, with a phase space density of ∼7.1 × 10-4 at a temperature of ∼3.7 µK in ∼18.8 min. Our work paves the way for the intelligent preparation of degenerate quantum gas.

Convex Approach to Real-Time Multiphase Trajectory Optimization for Urban Air Mobility

Electric vertical takeoff and landing technology has emerged as a promising solution to alleviate ground transportation congestion. However, the limited capacity of onboard batteries enforces hard time constraints for vehicle operations in a complex urban environment. Therefore, a constrained, real-time trajectory optimization method is necessary to enable safe and energy-efficient vehicle flight. Unfortunately, most existing trajectory optimization methods suffer from low computational efficiency, unpredictable convergence processes, and strong dependence on a good initial trajectory. In this paper, we employ a successive convex programming approach to solve the multiphase minimum-energy-cost electric vertical takeoff and landing vehicle trajectory optimization problem for urban air mobility applications. The main contribution is the development and validation of two novel convexification methods that find approximate optimal solutions to the multiphase nonlinear trajectory optimization problem in real time by solving a sequence of convex subproblems. Specifically, the first method transforms the original nonlinear problem into a sequence of second-order cone programming problems through a convenient change of variables and lossless convexification, while the second approach achieves a similar goal without the need for control convexification. The resulting convex subproblems can be solved reliably in real time by advanced interior point methods. The proposed methods are demonstrated through numerical simulations of two different urban air mobility scenarios and compared with the solution obtained from GPOPS-II, a state-of-the-art general-purpose optimal control solver. Our proposed sequential convex programming methods can obtain near-optimal solutions with faster computational speed than GPOPS-II.

Time-Optimal Motions of a Mechanical System with Viscous Friction

Optimal control is a critical tool for mechanical robotic systems, facilitating the precise manipulation of dynamic processes. These processes are described through differential equations governed by a control function, addressing a time-optimal problem with bilinear characteristics. Our study utilizes the classical approach complemented by Pontryagin’s Maximum Principle (PMP) to explore this inverse optimal problem. The objective is to develop an exact piecewise control function that effectively manages trajectory control while considering the effects of viscous friction. Our simulations demonstrate that the proposed control law markedly diminishes oscillations induced by boundary conditions. This research not only aims to delineate the reachability set but also strives to determine the minimal time required for the process. The findings include an exact analytical solution for the stated control problem.

On smoothness conditions and selection of the edge of a scattering surface in one class of 3D time-optimal problems

A class of time-optimal problems in three-dimensional space with a spherical velocity vectogram is considered. The target set is a parametrically defined smooth curve. Numerical and analytical approaches to constructing the bisector of the target set — the scattering surface — in the time-optimal problem are proposed. The algorithms are based on formulas for the edge points of the scattering surface, written in terms of curve invariants. It is shown that these points form the edge of the bisector and lie at the centers of the touching spheres to the curve. A theorem on sufficient conditions for the smoothness of a scattering surface is proven. The equations of the tangent plane to the bisector are found for those points from which exactly two optimal trajectories emerge. An example of solving a time-optimal problem in the form of a set of level surfaces of the optimal result function is given, highlighting the surface of their non-smoothness.

Kriging-based Model Predictive Control for Lower-limb Rehabilitation Robots

Model predictive control (MPC) has emerged as a predominant method in the realm of control systems; yet, it faces distinct challenges. First, MPC often hinges on the availability of a precise and accurate system model, where even minor deviations can drastically affect the control performance. Second, it entails a high computational load due to the need to solve complex optimization problems in real time. This study introduces an innovative method that harnesses the probabilistic nature of Gaussian processes (GPs), offering a solution that is robust, adaptive, and computationally efficient for optimal control. Our methodology commences with the collection of data to learn optimal control policies. We then proceed with offline training of GPs on these data, which enables these processes to accurately grasp system dynamics, establish input–output relationships, and, crucially, identify uncertainties, thereby informing the MPC framework. Utilizing the mean and uncertainty estimates derived from GPs, we have crafted a controller that is capable of adapting to system deviations and maintaining consistent performance, even in the face of unforeseen disturbances or model inaccuracies. The convergence of the closed-loop system is assured through the application of the Lyapunov stability theorem. In our numerical experiments, the exemplary performance of our approach is demonstrated, notably in its capacity to adeptly handle the complexities of dynamic systems, even with limited training data, underlining a significant leap forward in MPC strategies.

A new early warning system based on metaheuristics in CPS architecture

ABSTRACT Cyber security is the most studied in recent years by the association of metaheuristic techniques thanks to their ability to solve optimization problems in a limited time and NP-hard problems. The Early Warning System (EWS) can identify malicious exercises and anomalies in the CPS and provide robust network systems protection. In addition, grouping attacks in EWS is important for defining defense policies. Information security in the CPS architecture is about securing information during data aggregation, large-scale processing and sharing in the network environment, especially low-linkage open networks. Motivated by these considerations, we propose a new security approach based on the Artificial Bee Colony (ABC) metaheuristic algorithm to strengthen the security framework of the CPS architecture against the system and within CPS attacks. In this work, we propose a new Early Warning System approach based on metaheuristics. Based on two fundamental concepts of clustering, coherence and separation, an integrated index was proposed based on Davies-Bouldin (DB) and Silhouette Index (SI). An improved ABC algorithm was proposed based on these indices. After having identified two different types of an objective function, based on the above concepts, and after having tested the algorithm on DBSI, the results of the experiments found are auspicious.

Conditional Beta Approximation: Two Applications

Suppose that X,Y are two independent positive continuous random variables. Let P=\frac{X}{X+Y} and Z=X+Y. If X, Y have gamma distributions with the same scale parameter, then P distribution will be beta and P,\ Z are independent. In the case that the distributions of these two variables are not gamma, the P distribution is well approximated by the beta distribution. However, P,\ Z are dependent. According to matching moment method, it is necessary to compute the moments of conditional distribution for beta fitting. In this paper, some new methods for computing moments of conditional distribution of P given Z are proposed. First of all, it is suggested to consider the regression method. Then Monte Carlo simulation is advised. The Bayesian posterior distribution of P is suggested. Applications of differential equations are also reviewed. These results are applied in two applications namely variance change point detection and winning percentage of gambling game are proposed. The probability of change in variance in a sequence of variables, as a leading indicator of possible change, is proposed. Similarly, the probability of winning in a sequential gambling framework is proposed. The optimal time to exit of gambling game is proposed. A game theoretic approach to problem of optimal exit time is proposed. In all cases, beta approximations are proposed. Finally, a conclusion section is also given.

Distributed Swarm Trajectory Planning for Autonomous Surface Vehicles in Complex Sea Environments

In this paper, a swarm trajectory-planning method is proposed for multiple autonomous surface vehicles (ASVs) in an unknown and obstacle-rich environment. Specifically, based on the point cloud information of the surrounding environment obtained from local sensors, a kinodynamic path-searching method is used to generate a series of waypoints in the discretized control space at first. Next, after fitting B-spline curves to the obtained waypoints, a nonlinear optimization problem is formulated to optimize the B-spline curves based on gradient-based local planning. Finally, a numerical optimization method is used to solve the optimization problems in real time to obtain collision-free, smooth and dynamically feasible trajectories relying on a shared network. The simulation results demonstrate the effectiveness and efficiency of the proposed swarm trajectory-planning method for a network of ASVs.

Workshop Facility Layout Optimization Based on Deep Reinforcement Learning

With the rapid development of intelligent manufacturing, the application of virtual reality technology to the optimization of workshop facility layout has become one of the development trends in the manufacturing industry. Virtual reality technology has put forward engineering requirements for real-time solutions to the Workshop Facility Layout Optimization Problem (WFLOP). However, few scholars have researched such solutions. Deep reinforcement learning (DRL) is effective in solving combinatorial optimization problems in real time. The WFLOP is also a combinatorial optimization problem, making it possible for DRL to solve the WFLOP in real time. Therefore, this paper proposes the application of DRL to solve the dual-objective WFLOP. First, this paper constructs a dual-objective WFLOP mathematical model and proposes a novel dual-objective DRL framework. Then, the DRL framework decomposes the WFLOP dual-objective problem into multiple sub-problems and then models each sub-problem. In order to reduce computational workload, a neighborhood parameter transfer strategy is adopted. A chain rule is constructed for the appealed sub-problem, and an improved pointer network is used to solve the bi-objective WFLOP of the sub-problem. Finally, the effectiveness of this method is verified by using the facility layout of a chip production workshop as a case study.

Time optimal problems on Lie groups and applications to quantum control

<abstract><p>In this paper we introduce a natural compactification of a left (right) invariant affine control system on a semi-simple Lie group $ G $ in which the control functions belong to the Lie algebra of a compact Lie subgroup $ K $ of $ G $ and we investigate conditions under which the time optimal solutions of this compactified system are "approximately" time optimal for the original system. The basic ideas go back to the papers of R.W. Brockett and his collaborators in their studies of time optimal transfer in quantum control (<sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup>, <sup>[<xref ref-type="bibr" rid="b2">2</xref>]</sup>). We showed that every affine system can be decomposed into two natural systems that we call horizontal and vertical. The horizontal system admits a convex extension whose reachable sets are compact and hence posess time-optimal solutions. We then obtained an explicit formula for the time-optimal solutions of this convexified system defined by the symmetric Riemannian pair $ (G, K) $ under the assumption that the Lie algebra generated by the control vector fields is equal to the Lie algebra of $ K $.</p> <p>In the second part of the paper we applied our results to the quantum systems known as Icing $ n $-chains (introduced in <sup>[<xref ref-type="bibr" rid="b2">2</xref>]</sup>). We showed that the two-spin chains conform to the theory in the first part of the paper but that the three-spin chains show new phenomena that take it outside of the above theory. In particular, we showed that the solutions for the symmetric three-spin chains studied by (<sup>[<xref ref-type="bibr" rid="b3">3</xref>]</sup>, <sup>[<xref ref-type="bibr" rid="b4">4</xref>]</sup>) are solvable in terms of elliptic functions with the solutions completely different from the ones encountered in the two-spin chains.</p></abstract>

Extremals in the Markov – Dubins Problem with Control on a Triangle

We formulate a time-optimal problem for a differential drive robot with bounded positive velocities of the driving wheels. This problem is equivalent to a generalization of the classical Markov – Dubins problem with an extended domain of control. We classify all extremal controls via the Pontryagin maximum principle. Some optimality conditions are obtained; therefore, the optimal synthesis is reduced to the enumeration of a finite number of possible solutions.

Energy-optimal trajectories for skid-steer rovers

This paper presents the energy-optimal trajectories for skid-steer rovers on hard ground, without obstacles. We obtain 29 trajectory structures that are sufficient to describe minimum-energy motion, which are enumerated and described geometrically; 28 of these structures are composed of sequences of circular arcs and straight lines; there is also a special structure called whirls consisting of different circular arcs. Our analysis identifies that the turns in the trajectory structures (aside from whirls) are all circular arcs of a particular turning radius, R′, the turning radius at which the inner wheels of a skid-steer rover are not commanded to turn. This work demonstrates its paramount importance in energy-optimal path planning. There has been a lack of analytical energy-optimal trajectory generation for skid-steer rovers, and we address this problem by a novel approach. The equivalency theorem presented in this work shows that all minimum-energy solutions follow the same path irrespective of velocity constraints that may or may not be imposed. This non-intuitive result stems from the fact that with this model of the system the total energy is fully parameterized by the geometry of the path alone. With this equivalency in mind, one can choose velocity constraints to enforce constant power consumption, thus transforming the energy-optimal problem into an equivalent time-optimal problem. Pontryagin’s Minimum Principle can then be used to solve the problem. Accordingly, the extremal paths are obtained and enumerated to find the minimum-energy path. Furthermore, our experimental results by using Husky UGV provide the experimental support for the equivalency theorem.

Global Trajectory Optimization of Multispacecraft Successive Rendezvous Using Multitree Search

This paper investigates the global optimization of multispacecraft successive rendezvous trajectories, which is divided here into three subproblems: target assignment, sequence optimization, and rendezvous time optimization. A method consisting of two novel algorithms is proposed to solve these subproblems. First, a multitree search framework is developed to assign multiple targets to each spacecraft and simultaneously optimize the rendezvous sequence for every single spacecraft. Specifically, a novel algorithm of local search combined with beam search is proposed. Second, this paper converts the rendezvous time optimization problem into a multistage decision problem. Based on a critical rendezvous-epoch-dependent characteristic found in this subproblem, the number of state variables is thereby reduced. A novel dual dynamic programming algorithm is proposed and combined with dynamic programming to solve for the globally optimal rendezvous epochs efficiently. Global optimality is guaranteed by Bellman’s principle of optimality, which is the first time in such a problem to our knowledge. The proposed method achieves state-of-the-art performance in several typical fuel-optimal scenarios of active debris removal. This open-sourced method is non-database-dependent and contains only one design stage, which is expected to be adopted in other successive rendezvous missions.

RETRACTED: Data-driven model predictive control for real-time planned lead time optimization in a reconfigurable flow line

Mass customization has enabled smart manufacturing systems to develop reconfigurable machines and fixtures to fulfill customized products with high process variety. Coupled with these reconfigurable components, diversified design and operational problems should be dealt with such that the efficiency of reconfigurations can be maximized. This paper investigates a planned lead time (PLT) optimization problem after addressing the pre-determined reconfigurations and real-time disruptions in a flow line. A data-driven model predictive control (d-MPC) is developed for this problem such that the total expected cost for work-in-process inventory, final-product inventory, and tardiness penalty can be minimized. Considering the predictive multi-stage transitions and reconfigurable behaviors, d-MPC utilizes max-plus algebra to generate a time-varying state-space equation for a flow line. In this regard, a systematic method for permanent production loss identification can be formulated for real-time disruption diagnosis. This identification is useful in providing feedback signals for d-MPC and thus a closed-loop feedback control for PLT optimization can be achieved. A case study of an industrial welding line is reported to illustrate the operational benefits of d-MPC. The results demonstrate that d-MPC can efficiently cooperate with PLT optimization with machine reconfiguration such that the total expected cost can be minimized. Compared with continuous re-optimization, d-MPC adopts an event-driven mechanism to respond to real-time disruptions and can decrease the nervousness of flow lines with lower operational costs.

Time-Optimal Problem in the Roto-Translation Group with Admissible Control in a Circular Sector

We study a time-optimal problem in the roto-translation group with admissible control in a circular sector. The problem reveals the trajectories of a car model that can move forward on a plane and turn with a given minimum turning radius. Our work generalizes the sub-Riemannian problem by adding a restriction on the velocity vector to lie in a circular sector. The sub-Riemannian problem is given by a special case when the sector is the full disc. The trajectories of the system are applicable in image processing to detect salient lines. We study the local and global controllability of the system and the existence of a solution for given arbitrary boundary conditions. In a general case of the sector opening angle, the system is globally but not small-time locally controllable. We show that when the angle is obtuse, a solution exists for any boundary conditions, and when the angle is reflex, a solution does not exist for some boundary conditions. We apply the Pontryagin maximum principle and derive a Hamiltonian system for extremals. Analyzing a phase portrait of the Hamiltonian system, we introduce the rectified coordinates and obtain an explicit expression for the extremals in Jacobi elliptic functions. We show that abnormal extremals are of circular type, and they correspond to motions of a car along circular arcs of minimal possible radius. The normal extremals in a general case are given by concatenation of segments of sub-Riemannian geodesics in SE2 and arcs of circular extremals. We show that, in a general case, the vertical (momentum) part of the extremals is periodic. We partially study the optimality of the extremals and provide estimates for the cut time in terms of the period of the vertical part.

Time-optimal path planning and tracking based on nonlinear model predictive control and its application on automatic berthing

Autonomous shipping has been identified as the way forward in maritime transport. There are numerous technologies being developed over the past decades for maritime autonomous surface ships, including advanced automatic manoeuvring control. As one of the essential indices to evaluate ships’ manoeuvrability, the time-optimal problem remains a challenging issue in the process of path planning and path tracking, which has not been properly addressed for the maritime autonomous surface ships. To fill this knowledge gap, we propose a novel time-optimal path planning and tracking control method based on nonlinear model predictive control (MPC) and spatial reformulation. The proposed time-optimal controller is designed as a two-level hierarchical controller to plan the time-optimal path in spatial coordinates and track it in temporal coordinates when considering the low-speed manoeuvring model. At the high level, the nonlinear MPC based on a spatial formulation is used to generate a planned path of minimum manoeuvring time for a manoeuvring ship. At the low level, the optimal planning trajectory is interpolated by a constant time iteration and the vessel tracks the time-optimal planning trajectory by implementing the nonlinear MPC based on a temporal formulation. As an advanced predictive controller, the nonlinear MPC optimises the berthing path to a minimum time mathematically at the planning stage and compensates for the uncertain disturbance during the tracking process. To validate the time-optimal control method, a typical scenario of low-speed berthing operation is analysed. The results of the case study show that the proposed control method can be successfully applied to time-optimal manoeuvring problems.

An Adaptive Cuckoo Algorithm-Based Optimization Method for Tourist Traffic Routes

This paper presents in-depth research and analysis of the optimization method of tourist traffic routes using the adaptive cuckoo algorithm. The traditional cuckoo algorithm has the disadvantages of slow convergence speed and easy falling into local extremes. This paper proposes the iterative adaptive CHSACS algorithm based on iteration. Based on the original CS algorithm, a dynamic adaptive step control amount and a segmented weighted position update formula are introduced to give a class of improved CS algorithms to coordinate the problem of local search and global search of the CS algorithm and speed up the convergence speed at the later stage. To address the problem that the out-of-bounds nests interfere with the convergence of the algorithm, a memory strategy is introduced to relocate the out-of-bounds nests in the search space to improve the stability of the algorithm. Experiments are conducted on the iterative adaptive-based conductive CHSACS algorithm with test function sets. Compared with the original CS algorithm, and ACO algorithm, the CHSACS algorithm has faster convergence, higher search accuracy and a better ability to avoid local optima when dealing with continuous function optimization problems. For the dynamic travel path problem of travel time optimization, the road travel time and sightseeing time of different periods are predicted based on historical data, and the most traveled distance of travel time between two sights is dynamically searched by using the search mechanism based on time windows. The iterative adaptive CHSACS algorithm is used to continuously find the tourist traffic route with the shortest travel time travel path from all combinations of tour sequences. This paper combines the characteristics of bus vehicle scheduling itself, takes into account the interests of both bus companies and tourists, establishes a bus vehicle scheduling model with the departure interval as the independent variable, introduces this hybrid cuckoo algorithm into bus scheduling, verifies the scientific and feasibility of the algorithm through examples, and provides a new idea for solving bus scheduling optimization problems.

Multi-Objective Adaptive Optimization Model Predictive Control: Decreasing Carbon Emissions from a Zinc Oxide Rotary Kiln

The zinc oxide rotary kiln, as an essential piece of equipment in the zinc smelting industrial process, is presenting new challenges in process control. China’s strategy of achieving a carbon peak and carbon neutrality is putting new demands on the industry, including green production and the use of fewer resources; thus, traditional stability control is no longer suitable for multi-objective control tasks. Although researchers have revealed the principle of the rotary kiln and set up computational fluid dynamics (CFD) simulation models to study its dynamics, these models cannot be directly applied to process control due to their high computational complexity. To address these issues, this paper proposes a multi-objective adaptive optimization model predictive control (MAO-MPC) method based on sparse identification. More specifically, with a large amount of data collected from a CFD model, a sparse regression problem is first formulated and solved to obtain a reduction model. Then, a two-layered control framework including real-time optimization (RTO) and model predictive control (MPC) is designed. In the RTO layer, an optimization problem with the goal of achieving optimal operation performance and the lowest possible resource consumption is set up. By solving the optimization problem in real time, a suitable setting value is sent to the MPC layer to ensure that the zinc oxide rotary kiln always functions in an optimal state. Our experiments show the strength and reliability of the proposed method, which reduces the usage of coal while maintaining high profits.

Optimal Standoff Distance of Subsonic Unpowered Gliding Vehicle

The inverted Y-tail joint standoff weapon is adopted as the Subsonic Unpowered Gliding Vehicle (SUGV) in this article to gain maximum standoff distance. To obtain the optimal standoff distance of SUGV, a direct approach i.e., homogeneous control vector parameterization (CVP), is employed. To achieve the maximum standoff distance, the SUGV requires to spend a maximum amount of time in the air to conduct a gliding flight, resulting in an unspecified final time optimal problem. To deal with this problem, a time scaling approach is adopted in conjunction with homogeneous CVP to achieve the maximum standoff distance. In accordance with the homogeneous CVP technique, the standoff distance of the SUGV is comparatively long in comparison to the maximum step input, and the SUGV reaches a standoff distance of 121 km with an increment of 1.77%. The optimal simulation outcomes indicate that homogeneous CVP delivers a smooth gliding flight as compared to the maximum step input, and this smoother flight results in increased standoff distance. Furthermore, to evaluate the efficiency between the homogeneous CVP approach and maximum step input, the burden on the elevators in the form of integral square value and time scaling parameter is chosen as a criterion for quantitative analysis, and this analysis reveals that the homogeneous CVP approach has less burden on the elevators than the maximum step input.