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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. [ABSTRACT FROM AUTHOR]

In this study, we consider the extended optimal control problem and search for a control function in the class of feasible functions for a real control object. Unlike the classical optimal control problem, the control function should depend on the state, not time. Therefore, the control synthesis problem for the initial-state domain should be solved, instead of the optimal control problem with one initial state. Alternatively, an optimal trajectory motion stabilisation system may be constructed. Both approaches—control and trajectory motion stabilisation system syntheses—cannot be applied to real-time control, as the task is too complex. The minimum threshold of quality criteria is searched for in the space of mathematical expression codes. Among other problems, the search space is difficult to define and the gradient is hard to determine. Therefore, the advanced control object model is used to obtain a feasible control function. The advanced model is firstly obtained before solving the optimal control problem and it already includes a trajectory motion stabilisation system; in particular, this stabilisation system is synthesised in advance at the control system design stage. When the optimal control problem appears, it is solved in real time in the classical statement, and a control function is searched for as a function of time. The advanced control object model also uses the reference model to generate the optimal trajectory. The search for the optimal control function is performed in real time and considers the synthesised stabilisation system of motion along a determined trajectory. Machine learning control via symbolic regression, namely, the network operator method, is used to directly solve the control synthesis problem. An example solution of the optimal control problem, with an advanced model moving in the environment with obstacles for a group of two mobile robots, is presented. The obtained solution is a control function for a reference model that generates a trajectory from a class of trajectories stabilised with the object's control system. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The extended optimal control problem is considered. It is necessary to find an optimal control function, that not only provides the achievement of terminal state with optimal value of the given quality criterion, but also is implemented in the control system of a real object. It means, that the control function should depend on the state space vector, and the optimal solution should keep optimality property at small perturbations of the found solution. To solve this problem machine learning control by symbolic regression is used. In the extended optimal control problem, the problem statement of stabilization system synthesis for movement along the optimal trajectory is included. Synthesis problem is solved by the network operator method. In the synthesis problem a domain of initial conditions is considered instead of one point of initial state. It provides less sensitivity of found solution to perturbations of initial states. An example of solving the extended optimal control problem with complex phase constraints in the form of bottleneck for four quadcopters is presented. [ABSTRACT FROM AUTHOR]