# Control Synthesis as Machine Learning Control by Symbolic Regression Methods
## 2 Problem Statement of Control Synthesis as MLC
Often in the control tasks the structure of this function is defined beforehand on the basis of experience or intuitively, and it is necessary to find only values of some parameters, for example, coefficients of a PIDcontroller [20; 21] or parameters of some feedback NN-based controller [22; 23; 24]. Symbolic regression methods allow to search for both the very structure of the control function and its parameters. The problem of finding a control function (or a control law) that depends on the state of an object in control theory is called a control synthesis problem. For example, the design task of a feedback controller is a control synthesis problem since such controllers produce control signal based on the object state. The search of such a control function should be considered as machine learning control problem. Consider the formal statement of control synthesis problem. A mathematical model of control object is given  
\[\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x},\mathbf{u}), \tag{9}\]  
where \(\mathbf{x}\) is a vector of the state space, \(\mathbf{x}\in\mathbb{R}^{n}\), \(\mathbf{u}\) is a vector of control, \(\mathbf{u}\in\mathrm{U}\subseteq\mathbb{R}^{m}\), \(\mathrm{U}\) is a compact set, \(m\leq n\). We limit our analysis to all those control problems where \(\mathrm{U}\) is compact, because this encompasses real examples of control problems where the control is bounded, not infinite, and the boundary values also belong to the admissible control. An area of initial conditions is given  
\[\mathsf{X}_{0}\subseteq\mathbb{R}^{n}. \tag{10}\]  
A terminal conditions is given  
\[\tilde{\mathbf{X}}=\{\mathbf{x}^{1},\dots,\mathbf{x}^{N}\}\subseteq\mathbf{X}. \tag{2}\]0  
where \(t_{f}\) is a time of reaching the terminal condition from the initial area. This time isn't given, but is limited, \(t_{f}=\leq t^{+}\), \(t^{+}\) is a given limited time of reaching the terminal condition. It is necessary to find one control function in the form  
\[\tilde{\mathbf{X}}=\{\mathbf{x}^{1},\dots,\mathbf{x}^{N}\}\subseteq\mathbf{X}. \tag{2}\]1  
where \(\mathbf{h}(\mathbf{x}):\ \mathbb{R}^{n}\rightarrow\mathbb{R}^{m}\), that makes the object described by the dynamic model  
\[\tilde{\mathbf{X}}=\{\mathbf{x}^{1},\dots,\mathbf{x}^{N}\}\subseteq\mathbf{X}. \tag{2}\]2  
achieve the terminal goal \(\mathbf{x}^{f}\in\mathbb{R}^{n}\) from any initial condition \(\mathbf{x}^{0,i}\in\mathsf{X}_{0}\subseteq\mathbb{R}^{n}\) with the optimal value of the given quality criterion. \[\tilde{\mathbf{X}}=\{\mathbf{x}^{1},\dots,\mathbf{x}^{N}\}\subseteq\mathbf{X}. \tag{2}\]3  
The described MLC problem of control synthesis can be solved as either supervised or unsupervised machine learning control. The supervised approach is machine learning with application of a training set. A training set for control synthesis problem can be obtained by constructing optimal trajectories from multiple points of the given initial set. For this it is necessary to solve the optimal control problem for each particular initial condition from \(\mathbf{x}^{0,i}\in\mathsf{X}_{0}\) and to receive sets of optimal controls  
\[\tilde{\mathbf{X}}=\{\mathbf{x}^{1},\dots,\mathbf{x}^{N}\}\subseteq\mathbf{X}. \tag{2}\]4  
and optimal trajectories  
\[\tilde{\mathbf{X}}=\{\mathbf{x}^{1},\dots,\mathbf{x}^{N}\}\subseteq\mathbf{X}. \tag{2}\]5Therefore, to solve the control synthesis problem and to find the control function in the form (12) it is enough to approximate by any symbolic regression method the training set (16) on a criterion  
\[\tilde{\mathbf{X}}=\{\mathbf{x}^{1},\dots,\mathbf{x}^{N}\}\subseteq\mathbf{X}. \tag{2}\]6  
where \(t_{0}=0\), \(\mathbf{x}(t,\mathbf{x}^{0,i})\) is a partial solution of the Equation (13) with the initial conditions \(\mathbf{x}^{0,i}\), \(\tilde{\mathbf{x}}(t,\mathbf{x}^{0,i})\) is a trajectory from (16), \(i\in\{1,\ldots,K\}\). The unsupervised machine learning for control synthesis is a direct search of the control function (12) on the basis of the quality criterion minimization.