Inequality Constraints Research Articles

Adaptive robust constraint-following control method for improving carbody hunting stability of high-speed trains

The low-frequency carbody hunting motion of high-speed trains (HSTs) frequently occurs due to the complex operating environment and deterioration of wheel–rail contact conditions. It not only affects ride comfort but also poses a risk of derailment safety accidents for HSTs. In such scenarios, traditional passive suspension systems struggle to ensure satisfactory dynamic performance of HSTs. However, active suspension systems can offer an effective solution. To enhance the low-frequency carbody hunting stability and environmental adaptability of HSTs, this study proposes a novel adaptive robust displacement inequality constraint following control (AICFC) method for active suspension utilising the beneficial nonlinearity inspired by bio-inspired structures (BIS). The effectiveness of the proposed AICFC-BIS method is validated through numerical simulations comparing it with other control strategies. Simulation results show that the AICFC-BIS method can effectively suppress low-frequency carbody hunting motion and improve the dynamic performance of HSTs. Compared with conventional passive suspensions and other active suspensions, HSTs equipped with AICFC-BIS active suspension exhibit significantly reduced operational safety risk and improved ride comfort indices while being energy efficient.

New method for stability of probabilistic Boolean networks with inequality constraints

This paper studies the stability of probabilistic Boolean networks (PBNs) with state inequality constraints by establishing a new algebraic representation. On one hand, combining with the semi-tensor product (STP) of matrices and the law of total expectation, the algebraic representation of PBNs is established. On the other hand, based on the STP of matrices, an algorithm is proposed to determine the solution set of the state inequality constraints. Then, based on the algebraic representation and the inequality constrained solution set, a new form of transition probability matrix is constructed for PBNs. Finally, by the proposed transition probability matrix, new criteria are proposed for the finite-time stability with positive probability of PBNs. Several simulation examples are used to illustrate the effectiveness of the obtained results.

Modified preview repetitive control with equivalent-input-disturbance based on two-dimensional model

This paper deals with the problem of two-dimensional (2D) system-based modified preview repetitive control (MPRC) with equivalent-input-disturbance (EID) for continuous-time systems. First, an EID-based MPRC law is used to improve the tracking performance and disturbance attenuation. Next, by using a new equality constraint, a continuous-discrete 2D system is established and the design problem of the EID-based MPRC law is transformed into the stability problem of the 2D system. Then, we employ the 2D system theory together with the linear matrix inequality (LMI) approach, a sufficient condition is derived to ensure the stability of the closed-loop system. By solving an LMI, the gains of the controller and state observer can be obtained. Finally, two numerical examples are provided to illustrate the effectiveness and superiority of the developed controller design technique.

Dynamic Analysis of a Vehicle–Bridge System Under Excitation of Random Road Irregularities

This paper presents a comprehensive study of the dynamic response analysis of vehicle–bridge coupled systems, with detailed simulation methods for the vehicles, bridges, and wheel–road coupling relationships. The simulation of the entire vehicle–bridge coupling system is carried out using the open-source finite element analysis platform OpenSees. A novel three-dimensional wheel–road coupling element is introduced to model the interactions between the wheel and road nodes. This element facilitates precise computation of the dynamic responses within the vehicle–bridge coupled system, including both vehicle and bridge behaviors, along with the interaction forces between the wheels and the bridge surface. The coupling element consists of a wheel node and all potential road nodes on the bridge surface that the wheel may traverse. This configuration preserves the finite element model of the entire vehicle–bridge coupled system throughout the vehicle’s movement, thereby improving the efficiency of numerical simulations of vehicle–road interactions. The study accounts for the impact of random road irregularities on the dynamic responses of both the vehicle and the bridge. These irregularities are treated as input parameters for the wheel–road coupling element rather than being accounted for through the wheel–road interaction constraint equations, thereby improving the convenience of simulating random road irregularities.

Point-based Models: A Unified Approach for Geometric Constraint Analysis of Planar N-Bar Mechanisms with Rotary, Sliding, and Rolling Joints

Abstract Kinematic simulation of planar n-bar mechanisms has been an intense topic of study for several decades now. However, a large majority of efforts have focused on position analysis of such mechanisms with limited links and joint types. This paper presents a novel, unified approach to analysis of geometric constraints of planar n-bar mechanisms with revolute joint (R-joint), prismatic joint (P-joint), and rolling joint. This work is motivated by a need to create and program a system of constraint equations, which deal with different types of joints in a unified way. A key feature of this work is that the rolling joint constraints are represented by four-point models, which enables us to use the well-established undirected graph rigidity analysis algorithms. As a result, mechanisms with an arbitrary combination of revolute-, prismatic-joints, and wheel/gear/wheel-belt chains without any limitations on their actuation, can be analyzed and simulated efficiently for potential implementation in an interactive computer software and large-scale data generation.

Multiobjective linear fractional programming model with equality and inequality constraints under pentagonal intuitionistic fuzzy environment

Multiobjective linear fractional programming model with equality and inequality constraints under pentagonal intuitionistic fuzzy environment

Research on power metering method for nonlinear loads in power system based on improved S-transform

Abstract Traditional power metering algorithms are no longer suitable for new power systems to address the issue of distorted voltage and current waveforms caused by a growing number of nonlinear loads in modern power systems. The paper proposes a power-metering algorithm based on an Improved Stockwell Transform (IST). Based on the Stockwell transform, several parameters that can control the Gaussian window are introduced to improve the flexibility of the Gaussian window. Additionally, the inequality constraint based on the energy concentration measure is used to optimally select the best window parameters, enhancing the time–frequency domain analysis capability of the IST. The study uses IST to decompose and reconstruct the voltage and current fundamental characteristic signals and distortion characteristic signals in the power system. This provides good time resolution at low frequencies and good frequency-domain resolution at high frequencies. Power metering based on unsteady voltage and current is achieved by calculating the resulting fundamental and harmonic powers, respectively. The simulation results demonstrate that the IST-based power metering method has a higher level of accuracy than the existing metering methods.

A Lagrange barrier approach for the minimum concave cost supply problem via a logarithmic descent direction algorithm

The minimisation of concave costs in the supply chain presents a challenging non-deterministic polynomial (NP) optimisation problem, widely applicable in industrial and management engineering. To approximate solutions to this problem, we propose a logarithmic descent direction algorithm (LDDA) that utilises the Lagrange logarithmic barrier function. As the barrier variable decreases from a high positive value to zero, the algorithm is capable of tracking the minimal track of the logarithmic barrier function, thereby obtaining top-quality solutions. The Lagrange function is utilised to handle linear equality constraints, whilst the logarithmic barrier function compels the solution towards the global or near-global optimum. Within this concave cost supply model, a logarithmic descent direction is constructed, and an iterative optimisation process for the algorithm is proposed. A corresponding Lyapunov function naturally emerges from this descent direction, thus ensuring convergence of the proposed algorithm. Numerical results demonstrate the effectiveness of the algorithm.

Sample Average Approximation for Stochastic Programming with Equality Constraints

Sample Average Approximation for Stochastic Programming with Equality Constraints

Autonomous longitudinal separation control of aircraft in random wind fields based on MPC

Aiming at the problem of autonomous longitudinal separation under random disturbance of flight path, the random factor of high-altitude wind often leads to poor robustness of longitudinal separation between two aircraft. This paper proposes an autonomous longitudinal separation control method for aircraft based on model predictive control. Firstly, the nonlinear kinematic differential equation of the wind field difference between the two aircraft and the longitudinal separation is established, and the linear time-varying prediction model is derived. The longitudinal separation and path deviation distance of the two aircraft are selected as the optimization objectives, the high-altitude wind is the random disturbance, and the true airspeed and yaw angle of the leading aircraft are measurable. Terminal equality constraints are added to the air safety and aircraft performance constraints to maintain the stability of the system. To verify the effectiveness of the algorithm, within the prescribed simulation time of 120 seconds, this paper sets three groups of different expected intervals 12, 13, 14km. Through the designed MPC controller, by controlling the true airspeed and yaw angle of the trailing aircraft within the rolling time domain period, the interval curve between the two aircraft is relatively smooth and is always not less than the minimum safety interval of 10 km. It stabilizes at the expected target interval at the 74th, 90th and 118th seconds respectively, and begins to return to the route at the 58th, 74th and 95th seconds, and finally returns to the center line of the route; two groups of wind field control groups are set up, one group is the forecast wind intensity increased by 2 times, and the other group is the turbulent wind disturbance intensity increased by 8 times. Both can smoothly and stably establish the expected interval of 12 km at the 61st and 72nd seconds respectively.

Constraint Qualifications and Optimality Conditions for Nonsmooth Semidefinite Multiobjective Programming Problems with Mixed Constraints Using Convexificators

In this article, we investigate a class of non-smooth semidefinite multiobjective programming problems with inequality and equality constraints (in short, NSMPP). We establish the convex separation theorem for the space of symmetric matrices. Employing the properties of the convexificators, we establish Fritz John (in short, FJ)-type necessary optimality conditions for NSMPP. Subsequently, we introduce a generalized version of Abadie constraint qualification (in short, NSMPP-ACQ) for the considered problem, NSMPP. Employing NSMPP-ACQ, we establish strong Karush-Kuhn-Tucker (in short, KKT)-type necessary optimality conditions for NSMPP. Moreover, we establish sufficient optimality conditions for NSMPP under generalized convexity assumptions. In addition to this, we introduce the generalized versions of various other constraint qualifications, namely Kuhn-Tucker constraint qualification (in short, NSMPP-KTCQ), Zangwill constraint qualification (in short, NSMPP-ZCQ), basic constraint qualification (in short, NSMPP-BCQ), and Mangasarian-Fromovitz constraint qualification (in short, NSMPP-MFCQ), for the considered problem NSMPP and derive the interrelationships among them. Several illustrative examples are furnished to demonstrate the significance of the established results.

A flexible and efficient algorithm for high dimensional support vector regression

In high dimensional statistical learning, variable selection and handling highly correlated phenomena are two crucial topics. Elastic-net regularization can automatically perform variable selection and tends to either simultaneously select or remove highly correlated variables. Consequently, it has been widely applied in machine learning. In this paper, we incorporate elastic-net regularization into the support vector regression model, introducing the Elastic-net Support Vector Regression (En-SVR) model. Due to the inclusion of elastic-net regularization, the En-SVR model possesses the capability of variable selection, addressing high dimensional and highly correlated statistical learning problems. However, the optimization problem for the En-SVR model is rather complex, and common methods for solving the En-SVR model are challenging. Nevertheless, we observe that the optimization problem for the En-SVR model can be reformulated as a convex optimization problem where the objective function is separable into multiple blocks and connected by an inequality constraint. Therefore, we employ a novel and efficient Alternating Direction Method of Multipliers (ADMM) algorithm to solve the En-SVR model, and provide a complexity analysis as well as convergence analysis for the algorithm. Furthermore, extensive numerical experiments validate the outstanding performance of the En-SVR model in high dimensional statistical learning and the efficiency of this novel ADMM algorithm.

List coloring based algorithm for the Futoshiki puzzle

Given a graph G=(V, E) and a list of available colors L(v) for each vertex v\in V, where L(v) \subseteq {1, 2, ..., k}, List k-Coloring refers to the problem of assigning colors to the vertices of $G$ so that each vertex receives a color from its own list and no two neighboring vertices receive the same color. The decision version of the problem, List k-Coloring, is NP-complete even for bipartite graphs. As an application of list coloring problem we are interested in the Futoshiki Problem. Futoshiki is an NP-complete Latin Square Completion Type Puzzle. Considering Futoshiki puzzle as a constraint satisfaction problem, we first give a list coloring based algorithm for it which is efficient for small boards of fixed size. To thoroughly investigate the efficiency of our algorithm in comparison with a proposed backtracking-based algorithm, we conducted a substantial number of computational experiments at different difficulty levels, considering varying numbers of inequality constraints and given values. Our results from the extensive range of experiments indicate that the list coloring-based algorithm is much more efficient.

Time-optimal ergodic search: Multiscale coverage in minimum time

Search and exploration capabilities are essential for robots to inspect hazardous areas, support scientific expeditions in extreme environments, and potentially save human lives in natural disasters. The variability of scale in these problems requires robots to reason about time alongside their dynamics and sensor capabilities to effectively assess and explore for information. Recent advances in ergodic search methods have shown promise in supporting trajectory planning for exploration in continuous, multiscale environments with dynamics consideration. However, these methods are still limited by their inability to effectively reason about and adapt the time to explore in response to their environment. This ability is crucial for adapting exploration to variable-resolution information-gathering tasks. To address this limitation, this paper poses the time-optimal ergodic search problem and investigates solutions for fast, multiscale, and adaptive robotic exploration trajectories. The problem is formulated as a minimum-time problem with an ergodic inequality constraint whose upper bound specifies the amount of coverage needed. We show the existence of optimal solutions using Pontryagin’s conditions of optimality, and we demonstrate effective, minimum-time coverage numerically through a direct transcription optimization approach. The efficacy of the approach in generating time-optimal search trajectories is demonstrated in simulation under several nonlinear dynamic constraints, and in a physical experiment using a drone in a cluttered environment. We find that constraints such as obstacle avoidance are readily integrated into our formulation, and we show through an ablation study the flexibility of search capabilities at various scales. Last, we contribute a receding-horizon formulation of time-optimal ergodic search for sensor-driven information-gathering and demonstrate improved adaptive sampling capabilities in localization tasks.

Energy optimal path planning for wheeled mobile robots with circular obstacle

AbstractThe energy optimal motion planning algorithm for a differential drive robot with a circular obstacle in the maneuver space is introduced in this article. Rather than the traditional unicycle model, a non‐canonical state space model of the kinematics of a wheeled mobile robot is used to formulate the optimal control problem. The necessary conditions for optimality are formally derived using calculus of variations, resulting in constraints for the costates when the inequality constraint become active, revealing that the controls are required to be continuous. The optimality conditions permit decomposing the motion planning problem in two independent segments: prior to and after activation of the inequality constraints. Analytical expressions for the evolving states and control are found to be a function of Jacobi elliptic functions, permitting reducing the optimal control problem to a root‐finding problem. A comprehensive parametric study is included to demonstrate the impact on the optimal trajectories by varying the radius of the obstacle. The study shows that there are three distinct maneuver strategies with respect to the size of the obstacle. Furthermore, by including entering and exit time as optimization variables, a methodology for generalizing the development to any given motion scenario is developed and numerically illustrated.

Model-predictive optimal control of ferrofluidic microrobots in three-dimensional space

Ferrofluid microrobots have emerged as promising tools for minimally invasive medical procedures. Their unique properties to navigate complex fluids and reach otherwise inaccessible regions of the human body have enabled new applications in targeted drug delivery, tissue engineering, and diagnostics. This paper proposes a model-predictive controller for the external magnetic manipulation of ferrofluid microrobots in three dimensions (3D). The internal optimization routine of the controller determines appropriate changes in the applied electromagnetic field to minimize the deviation between the actual and desired trajectories of the microrobot. A linear system governing locomotion is derived and used as the equality constraints of the optimization problems associated with the feedback index. In addition to ferrofluid droplets, the controller presented in this work may be applied to other magnetically-pulled microrobots. Several experiments are performed to validate the controller and showcase its ability to adapt to changes in system parameters such as the desired tracking trajectory and the size, orientation, deformation, and velocity of the microrobot. The accuracy of the controller is analyzed for each experiment, and the average error is found to be within 0.25 mm for small velocities. An additional experiment is performed to demonstrate significant improvement over a PID controller that is optimally tuned using Bayesian optimization. The results presented in this paper suggest that the proposed control algorithm could enable new microrobotic capabilities in minimally invasive medical procedures, lab-on-a-chip applications, and microfluidics.

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.

Hybrid Orientation and Position Collaborative Motion Generation Scheme for a Multiple Mobile Redundant Manipulator System Synthesized by a Recurrent Neural Network.

To enable distributed multiple mobile manipulator systems to complete collaborative tasks safely and stably, this article investigates and presents a motion generation scheme that considers both orientation and position coordination based on a distributed recurrent neural network. Moreover, physical limits are also considered. Specifically, the orientation and position coordination constraints and physical limits are modeled separately as equality and inequality constraints with coupled variables. Subsequently, a motion generation scheme for multiple mobile manipulators based on quadratic programming is established. Finally, a distributed linear variational inequality-based primal-dual neural network is constructed to solve the motion generation scheme and obtain the motion trajectories of all the mobile manipulators. The simulation results demonstrate that the hybrid orientation and position collaboration motion generation scheme effectively addresses the position and orientation coordination problem for multiple mobile manipulator systems. Compared to other schemes, the proposed scheme based on a distributed computing structure greatly enhances the stability of the system. Additionally, the proposed approach introduces orientation coordination and physical limits, which increases the practicality of the system.

A neurodynamic optimization approach to distributed nonconvex optimization based on an HP augmented Lagrangian function

This paper develops a neurodynamic model for distributed nonconvex-constrained optimization. In the distributed constrained optimization model, the objective function and inequality constraints do not need to be convex, and equality constraints do not need to be affine. A Hestenes–Powell augmented Lagrangian function for handling the nonconvexity is established, and a neurodynamic system is developed based on this. It is proved that it is stable at a local optimal solution of the optimization model. Two illustrative examples are provided to evaluate the enhanced stability and optimality of the developed neurodynamic systems.

Hybrid TBETI domain decomposition for huge 2D scalar variational inequalities

AbstractThe unpreconditioned H‐TFETI‐DP (hybrid total finite element tearing and interconnecting dual‐primal) domain decomposition method introduced by Klawonn and Rheinbach turned out to be an effective solver for variational inequalities discretized by huge structured grids. The basic idea is to decompose the domain into non‐overlapping subdomains, interconnect some adjacent subdomains into clusters on a primal level, and enforce the continuity of the solution across both the subdomain and cluster interfaces by Lagrange multipliers. After eliminating the primal variables, we get a reasonably conditioned quadratic programming (QP) problem with bound and equality constraints. Here, we first reduce the continuous problem to the subdomains' boundaries, then discretize it using the boundary element method, and finally interconnect the subdomains by the averages of adjacent edges. The resulting QP problem in multipliers with a small coarse grid is solved by specialized QP algorithms with optimal complexity. The method can be considered as a three‐level multigrid with the coarse grids split between primal and dual variables. Numerical experiments illustrate the efficiency of the presented H‐TBETI‐DP (hybrid total boundary element tearing and interconnecting dual‐primal) method and nice spectral properties of the discretized Steklov–Poincaré operators as compared with their finite element counterparts.