Stability is crucial for practical implementation of computed optimal controls onboard, as it compensates for discrepancies that arises, as a result, of objectively existing differences in the models used during control design and the real objects. The synthesized control approach solves optimal control problems by strategically placing stable equilibrium points in state space, activated at equal time intervals. However, stability alone proves insufficient for maintaining optimal performance; equilibrium points must also exhibit specified quality characteristics to minimize sensitivity to external disturbances and model uncertainties. Stable equilibria can be classified as aperiodic or oscillatory. Near oscillatory points, the system’s differential equations fail to ensure strict contractivity in state space mappings, potentially causing unpredictable error growth between the actual object state and its model-based estimate. Thus, oscillatory equilibria are undesirable for control synthesis. This paper imposes additional quality requirements on the stabilization dynamics approaching equilibrium points namely to the object’s movement to the stability point. A numerical approximation method is proposed for synthesizing stabilization systems meeting these criteria. Optimal trajectories are first generated around the target equilibrium for various initial conditions via optimal control solutions. Symbolic machine learning then serves as a universal function approximator, trained on these supervised trajectories to minimize approximation error across the trajectory set. A computational experiment demonstrates stabilization synthesis for a differential-drive mobile robot using symbolic regression, a form of evolutionary machine learning. Results confirm that high quality stabilization system is little sensitive to disturbances.

The problem of building a more accurate mathematical model of the control object in order to solve the problem of navigation over a certain time interval in the absence of signals from global positioning systems is considered. To solve the problem, it is assumed that there are additional results of an experiment in which control actions on an object were measured in order to ensure its movement along a complex trajectory and the position of the control object as its reaction to these control actions. To clarify the model of the control object, the identification problem is solved, in which it is necessary to find the right parts of the differential equations describing the dynamics of the movement of the object, i.e. its acceleration along all three geometric axes. The peculiarity of the work lies in the fact that to approximate the right parts of differential equations, a symbolic regression method is used, which by machine learning finds mathematical expressions in encoded form. A description of the method of symbolic regression of universal code is presented. The description describes in detail the small variations of the universal code that are used in finding the optimal solution. An example of identification of a quadcopter model is given from experimental data obtained when an object moves along complex closed trajectory.

The traffic flow control problem in urban road networks in the case of sudden change of its quantitative characteristics is considered. Traffic flow is usually characterized by density, velocity, etc. These characteristics may change due to road accidents or other temporary events that increase or decrease demand on transportation. Dealing with such situations is not stationary and requires high quality online solutions made by human experts or traffic engineers. It is proposed to consider such a problem as a control synthesis problem and to find a control law that will assist humans in decision making. The searched control is a function of the deviation of the current traffic flow characteristics from that calculated before. The problem is formulated as a control synthesis. To solve the control synthesis problem of traffic flows, a symbolic regression approach is used. It allows to find a mathematical expression of control function, that returns changes of the traffic light phases duration. The traffic light timing obtained is optimal even for states that were not included in the training set. A statement of the control synthesis problem for traffic light timing is presented. In the computational experiment, a test example for an urban network of two intersections is presented.

В статье представлены новые результаты применения методов символьного машинного обучения для автоматизации разработки системы управления применительно к коллаборативной многоагентной системе. Предложена оригинальная математическая постановка задачи оптимального управления коллаборативной многоагентной системой, где коллаборативность подразумевает совместную деятельность роботов и людей в едином сложном фазовом пространстве для достижения цели управления с оптимальным значением заданного критерия качества. В представленной формулировке предлагается рассматривать роль человека в виде фазовых ограничений с неопределенностью и дополнительного вектора управления, с помощью которого уменьшается влияние этих фазовых ограничений в терминах минимизации функционала. Решением задачи является многомерная функция управления, которая переводит технический объект, группу мобильных роботов, в целевое состояние с учетом возможных неопределенностей в состоянии объекта, при этом минимизируя заданный критерий качества процесса. Задача состоит в автоматическом нахождении как оптимальной структуры, так и параметров функции управления. Такая общая постановка задачи управления не может быть решена напрямую существующими аналитическими методами из-за сложности и требует большого ручного труда по выбору соответствующих регуляторов и оптимальной настройке их параметров. В статье предлагается подход, основанный на символьном машинном обучении с помощью символьной регрессии и эволюционной оптимизации для решения такой обобщенно поставленной задачи. Эффективность подхода продемонстрирована численными экспериментами. Приведен пример задачи оптимального управления для трех мобильных роботов с двумя управляемыми фазовыми ограничениями. Прикладная задача рассматривается в общей математической постановке, что позволяет использовать универсальные методы символьного машинного обучения для автоматического поиска решения.

The work is devoted to automatic solving the control object stabilization system synthesis problem at the point of state space. To solve this problem machine learning by symbolic regression is used. Main goal of stabilization system consists of direction the control object to given point in state space independent on its initial state in some given area. Usually at application of symbolic regression one terminal state, a set of initial states and a quality criterion are given, that includes sum values of the criterion estimations of achievement by control object of the terminal state from each initial state. The sum of criterion values hides the individual properties of each motion path from some initial state to the terminal state. Some trajectories don't reach the terminal state or reaches the terminal state on complex path. Unlike the previous approach in this work, the shape of the trajectory of the control object to the terminal state is determined in advance. Initially, the optimal control problem is repeatedly solved for each given initial state according to speed criteria and/or minimum path length. Then, at the second stage, the symbolic regression method searches for one control function as a function of the object state vector that provides an approximation of all previously obtained optimal trajectories. This approach was named supervised machine learning, because the set of optimal trajectories is a training sample.

In the absence of any observation system or low veracity of the data, it is possible to provide control over a limited time interval basing on a high-precision control object model used. The paper proposes to use a multilayer artificial neural network (ANN) to obtain such a model. In this case, instead of the ordinary differential equation (ODE) system in the Cauchy form, we get a mixed ODE-ANN mathematical model, the parameters of which are tuned to the dynamics of a particular control object. The paper describes the process of identification of the control object model including obtaining a sufficient volume of the training sample. General data properties are formulated to obtain a good model from the point of view of use in control problems. The problem of optimal control for an object described with ANN-based model is formulated and numerical approach for its solution is proposed. An example of solving the optimal control problem for a mobile robot based on the identified neural network model is given.

The solution of the optimal control synthesis problem based on the Bellman equation is considered. The Bellman optimal control problem allows us to obtain the control function as a function of the state space coordinates. The essential difficulty of this problem is that the function is unknown. To solve the problem, one of the symbolic regression methods, the network operator method, is used to search for the Bellman function. This approach enables an automatic search for the structure and parameters of the mathematical expression using a special genetic algorithm. The result of the solution is a mathematical expression for the Bellman function. The optimal control found includes the gradient of the Bellman function. This function is described by code in the form of a network operator matrix. Therefore, the gradient of Bellman function is calculated numerically. An example of solving the control synthesis problem via Bellman equation for a mobile robot subjected to disturbance of initial conditions is presented.

It is known that solving the classical problem of optimal control leads to obtaining a control function, as a function of time, which cannot be implemented directly in the control system of a real object, since the resulting control system is open-loop one. It is proposed to use the method of the extended model of the control object. Initially, a universal system for stabilizing the movement of an object along any trajectory in the space of states from a certain class is synthesized for the object model. This stabilization system is built into the control object. The original reference model of the control object is then added to the object with a free control vector on the right side. Thus, the extended object model includes an object model with a motion stabilization system and a reference model. The optimal control problem is solved for the extended model. In the synthesis of a universal stabilization system, machine learning by symbolic regression is used. An example of solving the problem of optimal control of a wheel robot with a differential drive is given.

The widespread and growing use of flying unmanned aerial vehicles (UAVs) is attributed to their high spatial mobility, autonomous control, and lower cost compared to usual manned flying vehicles. Applications, such as surveying, searching, or scanning the environment with application-specific sensors, have made extensive use of UAVs in fields like agriculture, geography, forestry, and biology. However, due to the large number of applications and types of UAVs, limited power has to be taken into account when designing task-specific software for a target UAV. In particular, the power constraints of smaller UAVs will generally necessitate reducing power consumption by limiting functionality, decreasing their movement radius, or increasing their level of autonomy. Reducing the overhead of control and decision-making software onboard is one approach to increasing the autonomy of UAVs. Specifically, we can make the onboard control software more efficient and focused on specific tasks, which means it will need less computing power than a general-purpose algorithm. In this work, we focus on reducing the size of the computer vision object classification algorithm. We define different tasks by specifying which objects the UAV must recognize, and we construct a convolutional neural network (CNN) for each specific classification. However, rather than creating a custom CNN that requires its dataset, we begin with a pre-trained general-purpose classifier. We then choose specific groups of objects to recognize, and by using response-based pruning (RBP), we simplify the general-purpose CNN to fit our specific needs. We evaluate the pruned models in various scenarios. The results indicate that the evaluated task-specific pruning can reduce the size of the neural model and increase the accuracy of the classification tasks. For small UAVs intended for tasks with reduced visual content, the proposed method solves both the size reduction and individual model training problems.

This paper is devoted to the solution of the optimal control problem. The obtained control should be optimal in terms of quality criteria and, at the same time, feasible when implemented in the control object. To solve the optimal control problem in the class of feasible control functions, an advanced mathematical model of the control object is used. Firstly, the universal stabilisation system of the motion along any trajectory from some class is developed via symbolic regression. Then, the obtained stabilisation system is inserted into the right part of the control object model instead of the control vector. A reference model with a free control vector in the right part is added to the model; thus, the advanced mathematical model of the control object is obtained. After this, the optimal control problem is solved with the advanced mathematical model of the control object. The optimal control problem is stated in the classical form when the control is a time function. Here, the control function is searched for the reference model. The preliminary design of the universal stabilisation system for some class of trajectories allows the solution of the optimal control problem via the control object in a reasonable time frame. The proposed methodology is computationally tested for a model of the spatial motion of a quadcopter and a group of two-wheeled mobile robots with a differential drive. The results of the experiments show that the universal stabilisation system ensures the stabilisation of the motion of the objects along optimal trajectories, which are not known beforehand but obtained as a result of solving the problem with an advanced model.

The paper proposes design of an automatic motion stabilization system for a given control object along a desired trajectory of its motion in space. The stabilization system is constructed by means of symbolic regression. In order to obtain control parameters, a reference model is built. In the reference model, the speed of motion is determined by geometrical parameters of the desired trajectory. The considered problem is formulated in this paper as an extended optimal control problem where the cost function is the integrated accuracy of the motion along the desired trajectory. In order to implement the control function for a real control object, the control synthesis problem is solved by applying the variational genetic algorithm. An example of stabilization system designed by proposed method is given for a quadcopter moving along a given trajectory.

The control object mathematical identification problem is considered. It’s supposed, that a mathematical model is an ordinary differential equations system in Cauchi form. Therefore, the problem consists of finding right parts of differential equations system. Machine learning by symbolic regression is used for solving this problem. An example of quad-rotor spatial motion mathematical model identification is presented. Synthetic data obtained from known mathematical model are used for identification of model in the example. To obtain the training data some optimal control programs are set, and then a right parts of differential equation system a found by symbolic regression. To check a quality of identification a new optimal control problem was solved for found mathematical model. After that the found optimal solution in the form of optimal control program is used for control of the known control object model. Results of simulation are compared with a solution of the same optimal control problem for the known mathematical model.

To apply a solution of the optimal control problem directly to the control object for which model this problem was solved, it is necessary to build a system of motion stabilization along the obtained optimal trajectory. However, the construction of such a stabilization system is not provided in the classical formulation of the optimal control problem. Therefore, additional requirements for the properties of the optimal solution have been introduced into the optimal control problem. A formulation of the extended optimal control problem is introduced and the approaches to its numerical solution are discussed. One way to meet the introduced requirements is to build a stabilization motion system of an object along a program trajectory. As far as stabilization to the whole program trajectory, not from point to point on this trajectory, is rather challenging task, so for this, numerical methods of symbolic regression are proposed to be applied. The approach consists in application of machine learning by symbolic regression to the control synthesis problem. Symbolic regression allows to find a mathematical expression for control function as a function of the state space vector. Computational example of solving the extended optimal control problem for a robot group is presented. It is shown experimentally, that the solution of the extended optimal control problem is much less sensitive to disturbances, than a direct solution of the classical optimal control problem.

The paper considers the machine mathematical expression search problem for a control function. To solve this problem machine learning of control by symbolic regression numerical method is used. In difference from other works, where searching mathematical expression is performed by symbolic regression, here not one and some symbolic regression methods are applied sequence for search of one a mathematical expression. The paper contains a description of new symbolic regression method, that is named universal code. The method is constructed on the base the genetic programming and Cartesian genetic programming and it uses the principle small variation of basic solution. The paper presents an example of application of the universal code together with the network operator for stabilization of wheeled robot in the point of the state space with given quality of stabilization, which is needed for small sensitivity of a control function to external disturbances.

The problem of a stabilisation system synthesis for the motion of a control object along a given spatial trajectory is considered. The complexity of the problem is that the preset trajectory is defined in the state subspace and not in time. This paper describes a stabilisation system synthesis for motion along a trajectory specified in time and along a trajectory specified in the form of a manifold in a state space. In order to construct a stabilisation system, it is necessary to determine a distance between an object and the given trajectory at each moment in time. For trajectories that are not given in time, the determination of this distance can be ambiguous. An object may be exactly on a trajectory but at a different time. This paper proposes some approaches to solve the problem. One of the approaches is to transform a given trajectory in a state subspace into a trajectory given in time. A description of a universal method to perform this transformation is presented. In order to solve the synthesis problem automatically, without having to analyse the mathematical model of the control object, it is suggested that machine learning control by symbolic regression is used. In computational experiments, examples of stabilisation system syntheses for quadcopter motion along a given spatial trajectory are presented.

This paper considers the control synthesis problem and its solution using symbolic regression. Symbolic regression methods, which were previously called genetic programming methods, allow one to use a computer to find not only the parameters of a given regression function but also its structure. Unlike other works on solving the control synthesis problem using symbolic regression, the novelty of this paper is that for the first time this work employs a training dataset to address the problem of general control synthesis. Initially, the optimal control problem is solved from each point in a given set of initial states, resulting in a collection of control functions expressed as functions of time. A reference model is then integrated into the control object model, which generates optimal motion trajectories using the derived optimal control functions. The control synthesis problem is framed as an approximation task for all optimal trajectories, where the control function is sought as a function of the deviation of the object from the specified terminal state. The optimization criterion for solving the synthesis problem is the accuracy of the object’s movement along the optimal trajectory. The paper includes an example of solving the control synthesis problem for a mobile robot using a supervised machine learning method. A relatively new method of symbolic regression, the method of variational complete binary genetic programming, is studied and proposed for the solution of the control synthesis problem.

Рассмотрена задача оптимального управления транспортным потоком в сети городских дорог. Управление осуществляется изменением длительностей рабочих фаз светофоров на регулируемых перекрестках. Приведено описание разработанной системы управления. В системе управления предусмотрено использование трех видов управлений: программного, с обратной связью и ручного. При управлении с обратной связью для определения количественных характеристик транспортного потока используются детекторы дорожной инфраструктуры, видеокамеры, индуктивные петлевые и радиолокационные датчики. Обработка сигналов с детекторов позволяет определить состояние транспортного потока в каждый текущий момент времени. Для определения моментов переключения рабочих фаз светофоров количественные характеристики транспортных потоков поступают в математическую модель транспортного потока, реализованную в вычислительной среде системы автоматического управления транспортными потоками. Модель представляет собой систему конечно-разностных рекуррентных уравнений и описывает изменение транспортного потока на каждом участке дороги в каждый такт времени на основе рассчитанных данных по характеристикам транспортного потока в сети, пропускным способностям маневров и распределению потока на перекрестках с альтернативными направлениями движения. Модель обладает свойствами масштабирования и агрегирования. Структура модели зависит от структуры графа управляемой сети дорог, а количество узлов в графе равно количеству рассматриваемых участков дорог сети. Моделирование изменений транспортного потока в режиме реального времени позволяет оптимально определять длительности рабочих фаз светофоров и обеспечивать управление транспортным потоком с обратной связью по его текущему состоянию. В работе рассмотрена система автоматического сбора и обработки данных, поступающих в модель. Для моделирования состояний транспортного потока в сети и решения задачи оптимального управления транспортным потоком разработан программный комплекс CTraf, краткое описание которого представлено в работе. Приведен пример решения задачи оптимального управления транспортным потокам в сети дорог города Москва на основе реальных данных.

The work presents original approach for solving the most popular computational traveling salesman problem. This is a modified genetic algorithm built on the basis of the principle of small variations of the basic solution. Its advantage is that the time of its operation does not depend on the complexity of the problem in this case on the number of cities, but only depends on the parameters of the algorithm. In this algorithm, a possible solution is not an ordered set of visited cities but a set of small variations of some one possible solution. This solution is called basic. It is determined by the researcher as the closest to the optimal solution. In this case, the basic solution is made by a greedy algorithm. During the search process, the basic solution changes to the best current solution found. Together with the genetic algorithm, an accurate overlap algorithm is used, which has polynomial complexity and improves solutions by eliminating the self-intersections of the path.

This paper presents a novel numerical method for solving the control system synthesis problem through the application of machine learning techniques, with a particular focus on symbolic regression. Symbolic regression is used to automate the development of control systems by constructing mathematical expressions that describe control functions based on system data. Unlike traditional methods, which often require manual programming and tuning, this approach leverages machine learning to discover optimal control solutions. The paper introduces a general framework for machine learning in control system design, with an emphasis on the use of evolutionary algorithms to optimize the generated control functions. The key contribution of this research lies in the development of an algorithm based on the principle of small variations in the baseline solution. This approach significantly enhances the efficiency of discovering optimal control functions by systematically exploring the solution space with minimal adjustments. The method allows for the automatic generation of control laws, reducing the need for manual coding, which is especially beneficial in the context of complex control systems, such as robotics. To demonstrate the applicability of the method, the research applies symbolic regression to the control synthesis of a mobile robot. The results of this case study show that symbolic regression can effectively automate the process of generating control functions, significantly reducing development time while improving accuracy. However, the paper also acknowledges certain limitations, including the computational demands required for symbolic regression and the challenges associated with real-time implementation in highly dynamic environments. These issues represent important areas for future research, where further optimization and hybrid approaches may enhance the method's practicality and scalability in real-world applications.

This work describes automatization of creation process of automatic control systems. This needs not only for faster construction of control systems, but also to attract the machine to the solving the control problems, in order to computer found some solutions itself and offered them to developers. Here the optimal control problem in the extended statement is considered, when its solution can be realized directly in a real object. Here, for it purpose it is proposed to synthesize the universal stabilization system of a control object motion along any trajectory from enough wide class. The model of control object with stabilization system and with reference model for generation of a trajectory is called extended model of control object. The optimal control problem with the extended model of control object for realization of its solution it is enough to solve in classical form and to find a control function as a function of time. Universal stabilization system will provide quality motion a control object along obtained trajectory. For synthesis of a universal stabilization system machine learning control by symbolic regression is used. A computational example of application considered approach to a control of spatial motion of a quadcopter group is presented.