# Control Synthesis as Machine Learning Control by Symbolic Regression Methods
## 2 Problem Statement of Control Synthesis as MLC  
In general, machine learning aims to establish some functional dependency that determines the relationship between input and output data. MLC is aimed to design a control law for some object to achieve given goals optimally in terms of some formulated functional. The control law in general case is a multi-dimensional vector-function that should be defined. Today machine learning, especially in the field of control, is mostly used for the optimal tuning of parameters, like the parameters of a neural network or some given regulator. As a result of learning, an estimate of the unknown parameter vector is received. With the new possibilities that symbolic regression methods open up in machine learning, we are now able to define more broadly a machine learning problem that is to find an unknown function, including both the optimal structure and its parameters. For this, let us introduce the following definitions. **Definition 1**. : _A set of computational procedures, that transforms a vector \(\mathbf{x}\) from an input space \(\mathbf{X}\) to a vector \(\mathbf{y}\) from an output space \(\mathbf{Y}\), and there isn't any mathematical expression \(\mathbf{y}=\mathbf{f}(\mathbf{x})\) for them, is called an unknown function._  
Denote the unknown function between input vector \(\mathbf{x}\) and output vector \(\mathbf{y}\) as  
\[\mathbf{y}=\alpha(\mathbf{x}). \tag{1}\]  
The unknown function (1) can be presented as some device, or collection of experimental results. It is called a black box, because the exact description of it is not known. Let a set of input vectors be determined  
\[\tilde{\mathbf{X}}=\{\mathbf{x}^{1},\dots,\mathbf{x}^{N}\}\subseteq\mathbf{X}. \tag{2}\]  
For every input vector, an output vector is determined by the unknown function (1)  
\[\tilde{\mathbf{Y}}=\{\mathbf{y}^{1}=\alpha(\mathbf{x}^{1}),\dots,\mathbf{y}^{ N}=\alpha(\mathbf{x}^{N})\}\subseteq\mathbf{Y}. \tag{3}\]  
**Definition 2**. : _A pair of sets of compatible dimensions_  
\[(\tilde{\mathbf{X}},\tilde{\mathbf{Y}}) \tag{4}\]  
_is called a training set if \(\tilde{\mathbf{X}}\subseteq\mathbf{X}\), \(\tilde{\mathbf{Y}}\subseteq\mathbf{Y}\) and it is assumed that there is a one-to-one mapping \(\mathbf{X}\rightarrow\mathbf{Y}\)._  
Therefore, from the search of the unknown control function perspective, machine learning can be divided into two main classes: supervised learning, when there is a training sample, and unsupervised, when there is no training sample but some estimate function is given. So machine learning problems can be formulated as follows. **Definition 3**. : _Unsupervised machine learning consists in finding a function_  
\[\mathbf{y}=\beta(\mathbf{x},\mathbf{q}), \tag{5}\]  
_and parameters \(\mathbf{q}\), \(\mathbf{q}=[q_{1}\dots q_{p}]^{T}\), such that for some given estimate \(\gamma:\mathbf{Y}\rightarrow\mathbb{R}^{1}\), \(\forall\mathbf{x}\in\mathbf{X}\) it is true_  
\[\|\gamma(\mathbf{y})-\gamma(\beta(\mathbf{x},\mathbf{q}))\|\leq\delta, \tag{6}\]  
_where \(\delta\) is a small positive value._  
In machine learning control problems, such an evaluation criterion is a functional. **Definition 4**. : _Supervised machine learning consists in building a training set (4) and finding a function (5) such that if the total error for the training set is less than the given value \(\varepsilon\)_  
\[\sum_{i=1}^{N}\|\mathbf{y}^{i}-\beta(\mathbf{x}^{i},\mathbf{q})\|\leq\varepsilon, \tag{7}\]  
_then for \(\forall\mathbf{x}^{\star}\) not included in the training set \(\mathbf{x}^{\star}\
otin\tilde{\mathbf{X}}\) the following inequation is fulfilled_  
\[\|\mathbf{y}^{\star}-\beta(\mathbf{x}^{\star},\mathbf{q})\|\leq\delta, \tag{8}\]  
_where \(\mathbf{y}^{\star}=\alpha(\mathbf{x}^{\star})\)._  
The function \(\beta(\mathbf{x},\mathbf{q})\) in (5) includes a vector of parameters \(\mathbf{q}\).