Nonconvex Control Research Articles

Graph-based semi-supervised learning with non-convex graph total variation regularization

Graph total variation (GTV) is a widely employed regularization in graph-based semi-supervised learning (GSSL), which enforce the piece-wise smoothness of the label values concerning the underlying graph structure. However, due to its inherently biased estimation, GTV regularization tends to cause an underestimation of label values on boundary nodes . In this paper, we propose a novel GSSL model with non-convex GTV regularization. Specifically, we construct the non-convex GTV regularization by subtracting the generalized Moreau envelope from the original GTV regularization term, which can reduce the estimation bias and thus effectively estimate the label values of the boundary nodes. We demonstrate that under certain conditions of the non-convex control parameter, our proposed GSSL model maintains global convexity, thereby providing a theoretical guarantee for algorithm convergence. Additionally, we present an efficient alternating direction multiplier method (ADMM) to solve the proposed model. Finally, we validate the effectiveness on both synthetic data and background subtraction. The proposed GSSL model outperforms state-of-the-art model with a 17.97% improvement in SNR value on synthetic data, and increases average recall, average precision, and average F-measure by 4.92%, 1.19%, and 1.57% respectively on the pan–tilt-zoom(PTZ) challenge of background subtraction.

Convex Hull Relaxation of Optimal Control Problems With General Nonconvex Control Constraints

Convex Hull Relaxation of Optimal Control Problems With General Nonconvex Control Constraints

Stochastic Maximum Principle for Optimal Advertising Models with Delay and Non-Convex Control Spaces

Stochastic Maximum Principle for Optimal Advertising Models with Delay and Non-Convex Control Spaces

СТРАТЕГІЇ КЕРУВАННЯ У ПРОБЛЕМІ ЗБЛИЖЕННЯ КОНФЛІКТНО-КЕРОВАНИХ ОБ’ЄКТІВ

The authors propose a new approach to forming control strategies in the problem of approach of conflict-controlled objects. Modifications of the first direct method is developed for the stroboscopic strategy in the class of counter-controls where the classical Pontryagin’s condition is not satisfied. The lower resolving function is considered, which plays a key role in the formulation of the results and in the general case can be determined using the Minkowski functional of a multivalued mapping. An upper resolving function is introduced and a modified scheme of the method of resolving functions is proposed, which guarantees the termination of the conflict-controlled process in the class of quasi-strategies and counter-controls where the classical Pontryagin’s condition is not satisfied. The guaranteed times for different schemes of the considered methods are compared. The theoretical results are illustrated on a second-order model example with a special non-convex control region of the pursuer. Keywords: control strategy, resolving function, dynamic game problem, problem of approach of controlled objects.

СТРАТЕГІЇ КЕРУВАННЯ У ПРОБЛЕМІ ЗБЛИЖЕННЯ КОНФЛІКТНО-КЕРОВАНИХ ОБ’ЄКТІВ

The authors propose a new approach to forming control strategies in the problem of approach of conflict-controlled objects. Modifications of the first direct method is developed for the stroboscopic strategy in the class of counter-controls where the classical Pontryagin’s condition is not satisfied. The lower resolving function is considered, which plays a key role in the formulation of the results and in the general case can be determined using the Minkowski functional of a multivalued mapping. An upper resolving function is introduced and a modified scheme of the method of resolving functions is proposed, which guarantees the termination of the conflict-controlled process in the class of quasi-strategies and counter-controls where the classical Pontryagin’s condition is not satisfied. The guaranteed times for different schemes of the considered methods are compared. The theoretical results are illustrated on a second-order model example with a special non-convex control region of the pursuer. Keywords: control strategy, resolving function, dynamic game problem, problem of approach of controlled objects.

Abort orbit determination of launch vehicles based on difference of convex programming

Existing exoatmospheric guidance laws become ineffective when a thrust loss fault (like one or more engines unexpectedly shut down) occurs in a launch vehicle, which makes the nominal orbit unreachable. This paper presents a new abort orbit determination process to determine an abort orbit which the launch vehicle is able to reach. It involves solving at most two optimization problems and the abort orbit may be circular or elliptical. We will present how to transform the nonconvex optimization problems into difference of convex (DC) programming problems which can be solved by existing methods with guaranteed convergence. The main contribution of this paper lies in proposing a relaxation technique to convexify a nonconvex control constraint and the combination of the introduction of new constraints and a relaxation technique to deal with the strict terminal constraints. Validity of the relaxation technique is ensured by theoretical analysis. A unique feature of the transformation process is that the feasible sets defined by both the control constraint and the strict terminal constraints are all enlarged. This brings a critical benefit that the resulting DC programming problems can be very efficiently solved. Thus, an abort orbit determination algorithm, which is based on DC programming, can be designed to very reliably and efficiently determine an abort orbit, which will be demonstrated by numerical examples.

Optimal Melanoma Treatment Protocols for a Bilinear Control Model

In this research, for a given time interval, which is the general period of melanoma treatment, a bilinear control model is considered, given by a system of differential equations, which describes the interaction between drug-sensitive and drug-resistant cancer cells both during drug therapy and in the absence of it. This model also contains a control function responsible for the transition from the stage of such therapy to the stage of its absence and vice versa. To find the optimal moments of switching between these stages, the problem of minimizing the cancer cells load both during the entire period of melanoma treatment and at its final moment is stated. Such a minimization problem has a nonconvex control set, which can lead to the absence of an optimal solution to the stated minimization problem in the classes of admissible modes traditional for applications. To avoid this problem, the control set is imposed to be convex. As a result, a relaxed minimization problem arises, in which the optimal solution exists. An analytical study of this minimization problem is carried out using the Pontryagin maximum principle. The corresponding optimal solution is found in the form of synthesis and may contain a singular arc. It shows that there are values of the parameters of the bilinear control model, its initial conditions, and the time interval for which the original minimization problem does not have an optimal solution, because it has a sliding mode. Then for such values it is possible to find an approximate optimal solution to the original minimization problem in the class of piecewise constant controls with a predetermined number of switchings. This research presents the results of the analysis of the connection between such an approximate solution of the original minimization problem and the optimal solution of the relaxed minimization problem based on numerical calculations performed in the Maple environment for the specific values of the parameters of the bilinear control model, its initial conditions, and the time interval.

Formation Trajectory Tracking of Discrete-Time Distributed Multi-AUVs with Nonconvex Control Inputs and Weak Communication

This paper considers the formation trajectory tracking problem of discrete-time distributed multi-AUVs (multiple autonomous underwater vehicles) with control input in a nonconvex set and weak communication. Firstly, a linear model of a single AUV is obtained by the feedback linearization of a single AUV model using the Lie derivative theory. Then, a linear equation of multi-AUV formation is obtained, and the formation coordinated control system of multi-AUVs based on the feedback linearization model is given. Secondly, the formation trajectory tracking problem of multi-AUVs is transformed into the coordination control problem of leader–followers formation. The coordination controller of the leader–followers formation under a weak communication environment is designed, and the controller satisfies the nonconvex constraints. Next, the coordination control problem of leader–followers formation is transformed into the stability of the error of leader and followers at zero by coordinate transformation. By establishing an appropriate Lyapunov–Krasovskii function, the corresponding linear matrix inequalities are obtained, and the condition of zero stability is obtained by solving the linear matrix inequalities of the leader–followers formation. Under this condition, stable trajectory tracking can be achieved in the multi-AUV formation. Finally, the stability of the designed coordination controller is verified by simulation experiments.

Distributed L2 − L∞ containment control with nonconvex constraints for multi-agent systems

This paper studies a distributed L2−L∞ containment control problem with or without considering nonconvex control input constraints. The main challenge is how to address nonconvex constraints while satisfying the L2−L∞ performance index function. A distributed algorithm is proposed using only the local information, and the containment control problem with nonconvex constraints and L2−L∞ performance can be solved. Based on the properties of the projection operator and convexity, the constrained L2−L∞ containment control problem can be solved under the undirected connected graphs. Finally, the effectiveness of the algorithm is verified by a numerical experiment.

Direct Search Algorithm for Load Frequency Control of a Time-Delayed Electric Vehicle Aggregator

This article addresses the topic of frequency regulation of a single-area power system connected to an electric vehicle (EV) aggregator over a non-ideal communication network. It is considered that the command control action is received by the EV aggregator with constant delay and the power system includes uncertain parameters. Due to the presence of uncertainties and the delay term, the frequency regulation problem is non-convex and hard to solve. The present approaches in the literature convert the non-convex control design problem into a convex problem with a set of Linear Matrix Inequalities (LMIs), which is conservative and in many cases results infeasibility. In this paper, an innovative iterative algorithm, called direct search, is employed for the time-delayed system to design the unknown parameters of a pre-assumed controller. The controller choice is not limited and various controllers’ structures can be assumed. Without loss of generality, a proportional-integral (PI) controller is designed. The novel direct search algorithm can determine a feasible solution whenever at least one solution lays in the design space. Hence, by selecting a wide design space, we can anticipate that the PI controller guarantees closed-loop stability. Numerical simulations are carried out to demonstrate the performance of the developed controller compared to the state-of-the-art approach.

Formation Coordination Control of Leaderless Multi-AUV System with Double Independent Communication Topology and Nonconvex Control Input Constraints

In this paper, the formation coordination control of discrete-time distributed leaderless multiple autonomous underwater vehicle (AUV) system with double independent position–velocity communication topology and control inputs on a nonconvex set is studied. Firstly, the problem of formation coordination control of multi-AUV system is transformed into the problem of formation consensus of multi-AUV system, and the consistent state of leaderless multi-AUV system formation was defined. Secondly, considering the existence of bounded communication delay and nonconvex control input constraints for multi-AUV system formation under weak communication conditions, a formation consistent constraint controller algorithm for discrete-time leaderless multi-AUV system with double independent communication topology is proposed by introducing constraint operators. By using the properties of graph theory, random matrix and SIA matrix, and selecting appropriate controller parameters, the multi-AUV system formation can reach the defined consensus state. Furthermore, the unbounded communication delay of multi-AUV system formation is studied. Finally, the simulation results show that the proposed controller constraint algorithm is effective.

Reachable set computation of linear systems with nonconvex constraints via convex optimization

This paper addresses the reachable set computation of a linear system with a nonconvex control constraint and other convex control and state constraints. We propose to convexify the nonconvex constraint by a relaxation technique. We prove that the reachable set of the relaxed constrained system at any given final time is equal to that of the original constrained system at the same final time under certain assumptions, and interestingly it holds even though the admissible control set is expanded. The theoretical result can enable us to use convex optimization to efficiently and accurately compute the reachable set of the original constrained system. We will verify and apply the theoretical result to reachable set computation via two planetary precision landing examples.

Diffusion optical tomography reconstruction based on convex–nonconvex graph total variation regularization

Graph total variation (GTV) is a powerful regularization tool for diffuse optical tomography (DOT) reconstruction since it combines the powerful representation ability of graph and the edge‐preserving ability of total variation (TV) regularization. However, as everyone knows, the classical TV regularization trend underestimates the large edge values. In this paper, we propose a convex–nonconvex graph total variation (CNC‐GTV) regularization for DOT reconstruction. In particular, we construct a nonconvex regularization by subtracting the generalized Huber function from the GTV regularization. We show that the global convexity of the objective function can be guaranteed by adjusting the nonconvex control parameters. Moreover, we present an alternating direction multiplier method (ADMM) to solve the proposed DOT reconstruction model. Numerical experiments show that the proposed model outperforms existing models in visual and numerical results.

Convex Trajectory Planning [About This Issue

This <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">IEEE Control Systems</i> issue includes one feature and one application for control. The feature “Convex Optimization for Trajectory Generation,” by Danylo Malyuta, Taylor P. Reynolds, Michael Szmuk, Thomas Lew, Riccardo Bonalli, Marco Pavone, and Behçet Açikmes¸e, provides a tutorial on the optimization methods that have proven themselves in both theory and practice to be fast and reliable at producing dynamically feasible trajectories for nonlinear systems subject to nonconvex constraints. The authors and their colleagues are original developers of the methods discussed: lossless convexification and two sequential convex programming algorithms called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">SCvx</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">GuSTO</i> . At their core, all three methods leverage a reliable, low-level convex optimizer that solves nonconvex control problems through problem reformulation, linearization, or both. Over the past decade, these methods have generated growing interest across academia and industry, including from organizations such as NASA, Masten Space Systems, SpaceX, and Blue Origin. The article provides the reader with an intimate understanding of what it takes to use—or even extend—each of the methods to generate nontrivial trajectories. Open source code written in the Julia language accompanies the article and can serve as a ready-to-use toolbox for the reader’s further work on sequential convex programming.

Containment Control for Discrete-Time Multi-Agent Systems with Nonconvex Control Input and Position Constraints

With increasing attention on containment control problems in several areas, we investigate this specific problem which can be more practical. Systems with nonconvex input and position constraints are common but can be strongly nonlinear. A distribute algorithm using a projection operator is proposed to ensure that the control input of every follower remains in a nonconvex set and that all followers stay in the closed set given by leaders. In analysis, a model transformation is proposed, and then we introduce a method utilizing two similar triangles to prove the acceptability of the algorithm. The findings of the research could be pragmatic in robotics, astronautics, and so on. At last, numerical simulations are provided to show the contrast and results.

Containment control with input and velocity constraints

In this paper, containment control for double-integrator discrete-time multi-agent systems is investigated with switching graphs, nonconvex input constraints and nonconvex velocity constraints. In order to eliminate the nonlinearities caused by the nonconvex input and velocity constraints, we first introduce two time-varying scaling factors to transform the original constrained system into an unconstrained time-varying linear one. By proposing an inductive method based on the lower bounds of the scaling factors, we decouple the relevance of the two kinds of constraints and transform the obtained time-varying system into a new equivalent system. It is shown that all agents converge into the convex hull of the leaders even when switching graphs, nonconvex control input constraints and nonconvex velocity constraints coexist, as long as there exists at least one directed path from one of the leaders to each agent in the union of the communication graphs. Finally, a simulation example is given to illustrate the obtained theoretical results.

To the dynamic reconstruction of sliding controls

The paper is devoted to the problem of dynamic control reconstruction for controlled deterministic affine systems. The reconstruction has to be carried out in real time using known discrete inaccurate measurements of an observed trajectory of the system. This trajectory is generated by an unknown measurable control with values in a given compact set. A correct formulation of the reconstruction problem for the case of non-convex control restriction set is given. An approach to solving this problem is suggested. This approach is based on auxiliary variational problems with non-classical convex-concave Tikhonov-regularized integral cost. A numerical method for solving dynamic control reconstruction problem is suggested. It reduces the reconstruction problem to solving systems of linear ordinary differential equations. Matching conditions for the approximation parameters (accuracy and step of the known measurements and a Tikhonov regularizing parameter) such that the constructed approximations converge to the solution are obtained. Results of numerical simulation are exposed to illustrate the theory.

Flocking of Multi-Agent Systems with Nonuniform and Nonconvex Input Constraints

This paper investigates the nonuniform and nonconvex input constrained flocking control problem of continuous-time multi-agent systems. A distributed flocking control algorithm is proposed for each agent using the local information from its neighbor agents subject to nonuniform and nonconvex control constraints. Based on a constraint scaling factor and a specific coordinate transformation, a Lyapunov function is constructed to address the nonlinearities caused by input constraints. It is proved that all agents from an explicitly specified region of initial states eventually converge to a common point with the same velocity, while each agent's inputs remain within its own constrained nonconvex set. A simulation example is given to demonstrate the effectiveness of the theoretical results.

Distributed Continuous‐Time Containment Control of Heterogeneous Multiagent Systems with Nonconvex Control Input Constraints

This paper focuses on studying containment control problem with switching communication graphs of continuous‐time heterogeneous multiagent systems where the control inputs are constrained in a nonconvex set. A nonlinear projection algorithm is proposed to address the problem. We discuss the stability and containment control of the system with switching topologies and nonconvex control input constraints under three different conditions. It is shown that all agents converge to the convex hull of the given leaders ultimately while staying in the nonconvex set under the premise that at least one directed path from leaders to the agents exists in each bounded time interval. Finally, the validity of the results obtained in this paper is verified by simulation.

On a discrete game problem with non-convex control vectograms

In a normed space of finite dimension, a discrete game problem with fixed duration is considered. The terminal set is determined by the condition that the norm of the phase vector belongs to a segment with positive ends. In this paper, a set defined by this condition is called a ring. At each moment, the vectogram of the first player's controls is a certain ring. The controls of the second player at each moment are taken from balls with given radii. The goal of the first player is to lead a phase vector to the terminal set at a fixed time. The goal of the second player is the opposite. In this paper, necessary and sufficient termination conditions are found, and optimal controls of the players are constructed.