# Control Synthesis as Machine Learning Control by Symbolic Regression Methods
## 1 Introduction  
From the moment of its appearance, when the first regulators for machines with steam engines appeared, and in more than a century and a half of its development, the theory of automatic control has come a long way of transformation from a scattered set of control methods for mechanical, hydrodynamic and other systems to fundamental science. Almost all scientific research in control was carried out within the framework of studying the possibility of creating various innovative technical solutions from machine tools at the inception stage to automatic flying and space vehicles or nuclear power plants. The 20 th century is famous for the creation of automatic control systems for industrial complexes and production processes using computers. However, the 21 st century prepares new challenges associated with the emergence of universal objects, such as autonomous robots and robotic systems, capable of autonomously performing completely different tasks in different conditions and environments. Modern control systems must be able to quickly change, refine, learn. This circumstance requires both the universalization and automation of the very process of developing control systems that are not tied to the physics of the control object, operating with laws and patterns that are valid for objects of any complexity and nature. Machine learning control (MLC) meets these new challenges. In control theory, two main tasks for machine learning are distinguished--the identification of the control object model and the synthesis of the control system. In both cases, the task involves finding an unknown function. The function can be set up to parameters, then machine-learning techniques are used only to adjust the parameters [1]. In general case, both the structure of the function and its parameters should be found. Typically, machine learning implies the use of neural network technologies [2; 3; 4; 5; 6]. At first glance, it seems that a variety of neural network structures can satisfy any control problems. However, in fact, the structure of the neural network is determined by the developer and it is a given structure, in which only parameters are configured, while thestructure itself remains unchanged. It is difficult to even guess whether this structure is optimal for a given task. In addition, for complex tasks, a neural network has a complex structure, and for a development engineer who is used to describing objects and systems with some functions that have physical meaning and geometric representation, working with a neural network seems to be a kind of black box. Inspite of the popularity of the neural networks, tutorials on machine learning [7; 8] indicate that neural networks are only part of machine learning and there are other learning technologies. Today, more and more examples appear on the application of symbolic regression methods to MLC problems. Symbolic regression methods look for a control function in the form of code. The variety of symbolic regression methods is defined by different ways of coding functions. The search for the optimal control function is carried out on the code space using evolutionary algorithms. The genetic programming (GP) [9] was the first symbolic regression method capable to search for mathematical expressions. Now there is a variety of such methods [10; 11; 12; 13; 14; 15; 16]. The present paper is focused on the application of symbolic regression to the solution of the control synthesis. The successful solutions of applied control problems have so far been demonstrated mainly by the method of genetic programming [17; 18; 19]. In this paper, we want to show that there are many different symbolic regression methods that can cope with solving MLC problems. For this purpose, the mathematical formulation of the ML is introduced in Section 2 either for supervised machine learning or unsupervised one (or commonly known reinforcement learning). Then the problem of control synthesis is addressed as MLC.