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 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 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.

The area monitoring problem is considered as optimization task. The problem belongs to the class of global optimization. First, the problem of finding optimal trajectories is solved so that the control objects, when moving along these trajectories, observe the entire area, do not collide with each other and do not violate phase constraints. One of the main quality criteria when searching for trajectories is their minimum total length, and the closeness of the trajectory lengths of each control object. Subsequently, the optimal control problem is solved in order to ensure the movement of the object along the obtained optimal trajectory. To solve the problem in class of implemented in real object control functions the stabilisation system of control object motion along a given trajectory is synthesized. When solving the optimal control problem, the same criteria are considered as when finding optimal trajectories, with the addition of the accuracy of passing points on trajectories. An illustrative example of solving the optimal area monitoring problem by two quadcopters with one circular phase constraint is provided. Symbolic regression is employed to synthesise a system for stabilising the movement of an object along the optimal trajectory.

Coverage path planning (CPP) and path motion stabilization (PMS) play crucial roles in a variety of robotics applications, with a particular focus on their significance in precision agriculture. CPP determines optimal paths for covering specified target space, while PMS ensures precise movement along these paths. The paper proposes the advancement of CPP and PMS methodologies through the application of modern metaheuristic algorithms and symbolic regression methods. This study addresses a novel problem where the utilization of these methodologies is pursued to develop a universal stabilization system for moving an object along a trajectory, ensuring complete coverage of target space. Metaheuristic algorithms, such as the Grey Wolf Optimizer used in this study, can effectively solve CPP problems by decomposing the target space into subregions and searching for the optimal trajectory to efficiently explore these subregions. PMS utilizes symbolic regression methods and machine learning control techniques to develop stabilization systems. The control function of the stabilization system is considered as a function of the deviation from the equilibrium state. Consequently, it becomes feasible to explicitly derive a universal stabilization system for subsequent integration into the control object. The proposed CPP and PMS methodologies are evaluated through a computational experiment centered on crop monitoring using a quadcopter, thereby validating their effectiveness in practical applications.

The paper argues that modern machine learning methods make it possible to find mathematical expressions for various multidimensional functions. The article formulates a list of problems in which it is necessary to find a mathematical expression of a multidimensional function as a solution. Searching for control functions also belongs to the class of such problems. The presented list of problems is a proposal to researchers in the field of computational mathematics to develop new effective general methods of machine learning in order to obtain mathematical expressions of the desired functions. To solve the stated problems, the paper presents a structural-parametric approach based on evolutionary machine learning methods using symbolic regression. The paper presents solutions for three example problems of the function search.

The paper deals with the problem of traffic flow control in the urban road networks. The control is performed by traffic lights at intersections. The statement of the optimal control problem is given. A model-based approach is applied. The mathematical model of traffic flow based on the theory of controlled networks is used. A quality criterion is set. It is pointed out that a feature of the control of this object is its cyclical nature. Control is repeated at a certain time interval, which is called a control cycle. The duration of traffic light phases is considered as control. The phases switch sequentially in a certain order. Additional constraints are given that must be taken into account when choosing phase duration. As a result of solution of the optimal control problem the coordination plan for traffic lights is obtained. Coordination plan is cyclically repeated. The control synthesis problem is formulated that accounts for change in initial state. It is proposed to solve the control synthesis problem as a problem of selecting an optimal coordination plan from a set of plans calculated in advance for various initial flow states and model parameters.

Traffic lights at nearby intersections have a notable impact on the traffic flow. To ensure effective performance of the entire traffic system signal timing should be coordinated and synchronized. In this paper, the signal timing problem is considered as an optimal control problem. A universal recurrent traffic flow model is applied. The assumption is that information on the topology of road network, manoeuvre parameters, initial conditions, input flows, capacity constraints on road sections, and traffic light phase duration constraints is available. The mentioned parameters may be obtained from detectors of road infrastructure such as video cameras, loop and radio detectors and documentation. The objective is to develop an optimal traffic coordination plan for all monitored intersections within a specific time frame and minimizing a specified quality criterion. Coordination plan consists of ordered set of phases and their duration. The search space is big enough, so the problem is solved by evolutionary approach. The proposed optimization method is a variational genetic algorithm. The algorithm employs a principle of small variations of a basic solution to generate the population of possible solutions. Using this principle, we select a basic solution that is a current coordination plan. The set of codes that represent small variations of the basic solution are then used to determine all other feasible solutions. The proposed method is implemented to resolve the optimal control problem in an X-type intersection which experiences heavy traffic.

The work is devoted to the numerical complete solution of the optimal control problem. The complete solution means the solution of the optimal control problem together with the solution of the control synthesis problem to stabilize the movement of the control object along the found optimal trajectory. To solve this problem, evolutionary computations and symbolic regression are used. First, the optimal control problem by an evolutionary algorithm in the classical formulation is solved, after that the control synthesis problem by a method of symbolic regression is solved. The statement of the complete optimal control problem is presented. The computational experiment considers the solution of the complete optimal control problem for a quadcopter.

A new numerical method for solving the optimal control problem in class practically implemented solutions is presented. The method uses an approach of the synthesized control and takes account uncertainties of initial states. Like as synthesized control the method move a control object changing location of stable equilibrium point. As a result it chooses from all possible optimal synthesized controls such, that less sensitive to changes of initial states. As an example, the optimal control problem of quadcopter with complex phase constraints in the form of obstacle areas and narrow bottle neck is considered. To solve this problem firstly the synthesis control problem is solved for obtaining stable equilibrium point in the state space by symbolic regression. After that positions of stable equilibrium points are searched according to source functional from the optimal control problem. As an example, the optimal control problem of quadcopter with complex phase constraints in the form of obstacle areas and narrow bottle neck is considered. To solve this problem firstly the synthesis control problem is solved for obtaining stable equilibrium point in the state space by symbolic regression. After that positions of stable equilibrium points are searched according to source functional from the optimal control problem. Additionally at the search of equilibrium point positions the goal functional is calculated as a sum of fuctional values for all given points of initial states.

The problem of numerical solution of the optimal control problem for a group of robots is considered. The main complexity of the problem is the presence of dynamic phase constraints that describe the condition to prevent collisions between robots. Since there are several possibilities to avoid collision, and each of these possibilities corresponds to a local minimum, therefore the objective function has many local minima, thus the problem belongs to the class of global optimization. Precise methods for solving global optimization problems in this case cannot be applied due to the large dimension of the search space. It is proposed to use evolutionary algorithms to solve the optimal control problem of a group of robots. In computational experiment a complex optimal control problem for four quadcopters moving in space with four obstacles was solved by genetic and hybrid algorithms.

The problem of control synthesis for a mobile robot with phase constraints in the form of an arbitrarily located obstacle is formulated. To solve the problem, a numerical method of machine learning based on symbolic regression is used. According to the approach, in addition to its state, the object receives information about the distance to the obstacle and the direction of its location. As a result of solving the synthesis problem, we obtain a nonlinear feedback control function that provides a better estimate of the obstacle bypass according to a given criterion. An example for a mobile robot with a differential drive is considered.

In order to obtain optimal control of a real object, it is necessary to know the precise mathematical model of this control object. In the present study an artificial neural network is used for building a mathematical model of the control object. First, some forms of control are defined, and with the help of these controls, the control object is modeled. The obtained values of the controls and the space state vector are stored to create a training sample. The artificial neural network is then trained on this training set. For a trained neural network, a set of optimal control problems is solved. The optimal control obtained by the trained artificial neural network is applied to a real control object. The accuracy of the approximation of the mathematical model by an artificial neural network can be estimated based on the proximity of the functional values of the control object and the trained neural network.

The optimal control problem is considered in the context of the influence of small perturbations on its solution. The definition of stability for a solution of the optimal control problem is introduced. It is shown that stable solutions are feasible. Assertions are given that in order to obtain feasible solutions to the optimal control problem, it is necessary to ensure the stability property when searching for the optimal solution. A method of obtaining a stable solution of the optimal control problem is proposed. An example of comparing stable and unstable solutions is presented.

The optimal control problem of object motion along the program trajectory is considered. The problem is stated as the optimal control problem with phase constraints in the form of equalities. Initial and terminal states are constrained. The numerical solution to the problem is proposed. Some points are defined on the program trajectory. The quality criterion includes time and hitting accuracy to the terminal state, integral error of motion along the program trajectory and sum of minimal distances to defined points on the program trajectory. To solve the problem direct approach on the base of control approximation by piecewise linear function and synthesized control are used. For synthesized control a method of machine learning control by symbolic regression is applied. In computational experiment the optimal control problem is solved for mobile robot moving along a program trajectory. Solutions obtained by both methods are compared in the presence of perturbations of the mathematical model of control object.

The problem of optimal control of Mecanum-wheeled robot is considered. Mecanum wheels allow to move along the plane in any direction without turning the body of the robot. A feature of this robot model as a control object is that the control vector dimension is greater than the state vector. As a result, the same optimal robot movement can be realized by different control means. To solve the problem of optimal control, two numerical methods are used: direct control and synthesized. To check the efficiency of the obtained control functions, the mathematical model of the control object with both control functions was simulated under disturbances. The evaluation criterion was the change in the value of the quality criterion of the optimal solution depending on the level of perturbation. Experiments have shown that the synthesized optimal control for the Mecanum-wheeled robot is less sensitive to disturbances than the direct optimal control.

A numerical solution of the problem of the general synthesis of a stabilization system by a symbolic regression method is considered. The goal is to automatically find a feedback control function using a computer so that the control object can reach a given terminal position from anywhere in a given region of the initial conditions with an optimal value of the quality criterion. Usually, the control synthesis problem is solved analytically or technically taking into account the specific properties of the mathematical model. We suppose that modern numerical approaches of symbolic regression can be applied to find a solution without reference to specific model equations. It is proposed to use the numerical method of Cartesian genetic programming (CGP). It was developed for automatic writing of programs but has never been used to solve the synthesis problem. In the present work, the method was modified with the principle of small variations in order to reduce the search area and increase the rate of convergence. To apply the general principle of small variations to CGP, we developed special types of variations and coding. The modified CGP searches for the mathematical expression of the feedback control function in the form of a code and, at the same time, the optimal value of the parametric vector which is also a new feature-simultaneous tuning of the parameters inside the search process. This approach enables working with objects and functions of any type, which is not always possible with analytical methods. The need to use the received solution on the onboard processor of the control object imposes certain restrictions on the used basic set of elementary functions. This article proposes the theoretical foundations of the study of these functions, and the concept of the space of machine-made functions is introduced. The capabilities of the approach are demonstrated on the numerical solution of the control system synthesis problems for a mobile robot and a Duffing model.

The paper is devoted to application of Cartesian Genetic Programming (CGP) for generating optimal trajectories of a mobile robots group. The problem of a control system synthesis for a mobile robots group is solved. The proposed algorithm uses numerical approach from the class of symbolic regression methods to which Cartesian Genetic Programming belonging. It allows to receive a control function in the form of a mathematical expression. We consider several stages to get optimal trajectories for mobile robots group moving along which the robots wouldn’t collide with each other and obstacles. Initially, we solve the problem of synthesis for each robot in order to get the stabilized robot control system relative some point in the state space. At the second stage, spatial trajectories are found along which robots move from the current state to the obtained equilibrium points without collisions. It was proposed to improve an initial algorithm by using the principal of small variation of basic solution. There is considered a group of three robots and the control system for them with phase constraints in the paper.

The problem of optimal control for interaction of three robots is considered. To solve the problem, the synthesized optimal control method is used. According to this method, firstly the synthesis problem of control system for each robot is solved. The synthesized control system allows to stabilize robot relatively some point in the state space. On the second stage positions of some stabilization points are found in the state space such that at switching these points from one to the next through a set time interval, robots move a cargo from initial position to terminal one with optimal value of quality criterion. For the solution of the synthesis problem the symbolic regression method is used. All phase constraints describing group interaction and obstacles, have included in quality criterion as a penalty functions. For searching positions of the points the evolutionary algorithm particle swarm optimization is used.

This paper concerns the quasi time-optimal motion in a simplified model of a mobile robot with constraints imposed on the state variables. As it is known, in this kind of problems involving the so-called unicycle type of dynamics, the classical assumptions of regularity with respect to the state constraints are not fulfilled. This fact greatly complicates the analysis of such control problems through the use of Pontryagin’s maximum principle. Moreover, the difficulties are also due to the presence of a singular control mode which is related to the angular velocity. The paper proposes a certain regularization approach in order to overcome the obstacles above and to develop a tool for the subsequent numerical implementation.

This paper presents a new two-step numerical approach to a solution of the optimal control problem with phase constraints named a synthesized optimal control. The stated problem combines two well-known tasks: the optimal control and the control system synthesis. Initially, the synthesis problem is considered and a feedback control is received that provides a steady state for the control object relative to some point in a state space. Then a sequence of points is searched in the state space, so that each of the points is stable in the state space of the object, so the object moves from the initial condition to terminal one by switching from one stabilization point to another in some time interval. It is shown in the paper that such approach allows to receive a solution that does not differ much from those of the optimal control problem considering the value of the quality criterion, but it works more stable in the presence of disturbances. At the same time such approach is much more applicable in real engineering tasks since some distinctions between the mathematical model of the control object and the real control object are smoothed over due to the first stabilization procedure of the approach. The paper includes a mathematical problem statement, the description of modern computational methods for its solution and computational example.

The paper considers the solution of the problem of optimal control system synthesis. It is proposed to solve this problem based on the approximation of the set of optimal trajectories using symbolic regression methods. At the first step the optimal control problem is solved for various initial states; at the second step symbolic regression method is used to approximate the obtained set of optimal trajectories. In the suggested approach the proximity of the solution to the optimal one is determined by the accuracy of the approximation. A computational experiment of solving the applied problem of optimal control system synthesis for a group of car-like mobile robots in space with dynamic and static phase constraints is presented. The experiment showed that the found synthesized control function allows to move robots by the trajectory close to the optimal one for any initial state from a given domain.

This paper is addressed to application of SOMA to the optimal control problem of a group of objects with static and dynamic phase constraints. The optimization problem is not convex and unimodal. In the practical part a synthesized optimal control problem is considered. Firstly, it is necessary to solve a control synthesis problem to stabilize each object relatively a point on the state space. To solve the synthesis problem a network operator method was used. For each control object the coordinates of some stabilization points were found. While moving from point to point in given time interval all objects reach their terminal states without violation of constraints and with optimal quality criterion. A modified SOMA is proposed for the search of stabilization points.

A study is focused on formulation of traffic flow control problems in the urban roads network. The problems of parametric identification, optimal control, and control synthesis are considered. Initially, the problem of obtaining a mathematical model of traffic flows as a control object is described. To obtain a model, it is proposed to use an approach based on the controlled networks theory. Evolutionary algorithms are used to solve the considered control problems.

This paper considers the optimal control problem for a group of interacting robots. To solve the problem, a hybrid evolutionary algorithm is applied that consists of two popular evolutionary algorithms: particle swarm optimization and gray wolf optimizer. When solving the problem, we use the approach of synthesized optimal control. Initially, we make controlled objects stable in the state space with respect to a certain point. Then we look for coordinates of the stabilization points using the hybrid algorithm. Points should be located so that when switching stabilization points after each fixed time interval, the objects reach the control goal without violating the phase constraints with the optimal value of the quality criterion.

The use of various optimization methods and the efficiency of their work strongly depends on the type of functional under investigation. It turned out that in the case of solving optimization problems with a nonlinear compound functional, it is not at all easy to estimate its convexity. And the absence of the unimodality property of the objective function means the instability and low efficiency of the application of gradient extremum search methods. This paper is devoted to the study of the properties of unimodality and convexity of functionals. The nontriviality of the problem of estimating the unimodality of a functional is shown, the concept of a fundamental sequence of functions that are the arguments of the objective functional is introduced, theorems on sufficient conditions for the absence of unimodality of the objective function are formulated and proved.

An experimental comparison of evolutionary algorithms and gradient-based methods for the optimal control problem is carried out. The problem is solved separately by Particle swarm optimization, Grey wolf optimizer, Fast gradient descent method, Marquardt method and Adam method. The simulation is performed on a jet aircraft model. The results of each algorithm performance are compared according to the best found value of the fitness function, the mean value and the standard deviation.

A reliability index of various electronics is determined by the experimental data of tests for different values of parameters of the equipment. The received data are collected in bulky tables and references. This paper presents modern numerical approach, allowing to compile the experimental data on changes of reliability index not in the form of tables but as a function of the operating parameters of the devices. The methodology is based on the method of network operator for the design of the optimal structure of function and selection of its parameters. The network operator method belongs to a class of methods of symbolic regression and provides an evolutionary search for the best compositions of mathematical expressions on the space of elementary structures. The method allows you to automatically receive the required description of the functional dependencies. The effectiveness of the method is demonstrated by the example of searching the law, which describes the change in the failure rate depending on three parameters that characterize its constructive and technological performance and operating conditions.

A synthesis of optimal control for a group of robots is considered. Suppose every robot of group has full information about other robots. A two-stage numerical approach for optimal control synthesis is applied. Firstly, the task of stability in a state space for one robot is decided. For this purpose multilayer network operator is used. Secondly, a problem of designing optimal trajectories is considered. Trajectories in a form of set of points in a state space are set. It is necessary to find these points the way that robots could not encounter in the process of movement. Computational experiment is presented for group of three mobile robots are parking at spatial restricts.

A problem of optimal urban traffic flows control is considered. A mathematical model of control by the traffic lights at intersections using the controlled networks theory is given. It is a system of nonlinear finite-differential equations. To present a large scale road networks the model contains the connection matrices that describe interactions between input and output roads in subnetworks. The traffic flow control is performed by the coordination of active phases of traffic lights. A control goal is to minimize the difference between the total input flow and total output flow for all subnetworks. In this paper, a neural network approach for traffic road network parameters adjustment is presented. A simulation is conducted under a microscopic traffic simulation software CTraf. Results demonstrate that neural network reinforcement training obtained good parameters of the network model.

The paper proposes a new method for synthesis of optimal control systems. The network operator method is used. The intellectual evolution algorithm is applied for searching. This algorithm combines advantages of several evolutionary algorithms. A numerical example of control system synthesis for a wheeled mobile robot is presented.