# Synergetic Synthesis of Nonlinear Laws of Throttle Control of a Pneumatic Drive
## 2 Methods
### Procedure for Synergistic Synthesis of Nonlinear Laws of Throttle Control of EPS
Proceeding from the fact that a change in the flow area \(U_{1f}\) (\(x_{n}\)) on PR1 affects the dynamics of pressure change in the filling chamber \(x_{3}\), and a change in the flow area \(U_{2f}\)(\(x_{n}\)) on PR2 affects the dynamics of pressure change in the exhaust chamber \(x_{4}\), it is advisable to introduce the following collection of invariant varieties:  
\[\left\{ \begin{array}{l}\psi_{1}=x_{4}-x_{4}^{*}=0\\ \psi_{2}=x_{3}-\varphi_{1}(x_{1},x_{2})=0.\end{array} \right. \tag{4}\]  
The second manifold contains some function \(\varphi_{1}(x_{1},x_{2})\), which determines the desired character of the pressure change \(x_{3}\) at the intersection of the invariant manifolds \(\psi_{1}=0\) and \(\psi_{2}=0\). The function \(\varphi_{1}(x_{1},x_{2})\) is determined during the synthesis of control laws based on conditions for the fulfillment of invariant (2). According to the ADAR method, the macro variables \(\psi_{1}\) and \(\psi_{2}\) must satisfy the solution of the system of basic functional equations:  
\[\left\{ \begin{array}{l}\mathrm{T}_{1}\dot{\psi}_{1}(t)+\psi_{1}=0,\\ \mathrm{T}_{2}\dot{\psi}_{2}(t)+\psi_{2}=0,\end{array} \right. \tag{5}\]  
where \(\mathrm{T}_{1}>0\) and \(\mathrm{T}_{2}>0\) provide the conditions for the asymptotic stability of the motion of the system. When the RP system hits the intersection of the manifolds \(\psi_{1}=\psi_{2}=0\), an exact dynamic decomposition of the system (1) occurs, and the equations describe the dynamics of the closed-loop system:  
\[\left\{ \begin{array}{l}\dot{x}_{1}(t)=x_{2};\\ \dot{x}_{2}(t)=a_{21}\cdot\varphi_{1}(x_{1},x_{2})-a_{22}\cdot x_{4}^{*}-a_{23} \cdot x_{2}-a_{24}.\end{array} \right. \tag{6}\]  
The function \(\varphi_{1}(x_{1},x_{2})\), in the decomposed system (6), can be considered as an \"internal\" control that directly affects the movement and speed of the rod. At the second synthesis stage, to search for control and, therefore, the definition of the function \(\varphi_{1}(x_{1},x_{2})\), an additional invariant manifold is introduced, which should ensure the stability of the closed-loop system and the fulfillment of the technological invariant (2). Based on this, we introduce the manifold:  
\[\{\psi_{3}=x_{2}-k(x_{1}-x_{1}^{*})=0. \tag{7}\]  
The following equation describes the dynamics of the system on this manifold:  
\[\dot{x}_{1}=k(x_{1}-x_{1}^{*}), \tag{8}\]  
the stability condition of which, with respect to \(x_{1}=x_{1}{}^{*}\), is the inequality \(k<0\). For the condition \(\psi_{3}=0\) to be fulfilled, the macro variable \(\psi_{3}\) must satisfy the solution of the functional equation:  
\[\mathrm{T}_{3}\dot{\psi}_{3}(t)+\psi_{3}=0, \tag{9}\]  
where \(\mathrm{T}_{3}>0\) is the condition of asymptotic stability of the motion of the representing point of the system to the invariant manifold. Substitute (7) into (9) to define the internal control \(\varphi_{1}\):  
\[\mathrm{T}_{3}(\dot{x}_{2}-k\dot{x}_{1})+x_{2}-k(x_{1}-x_{1}^{*})=0. \tag{10}\]  
Further, the equations of the decomposed system (6) are substituted into Equation (10), as a result of which the following expression is obtained:  
\[\mathrm{T}_{3}[(a_{21}\varphi_{1}(x_{1},x_{2})-a_{22}x_{4}^{*}-a_{23}x_{2}-a_{ 24})-kx_{2}]+x_{2}-k(x_{1}-x_{1}^{*}). \tag{11}\]Let us express the internal control \(\varphi_{1}\) from the expression (11):  
\[\varphi_{1}=\frac{\text{T}_{3}\cdot\left[a_{22}x_{4}^{*}+a_{23}x_{2}+a_{24}+kx_{ 2}\right]-x_{2}+k\left(x_{1}-x_{1}^{*}\right)}{T_{3}d_{21}}. \tag{12}\]  
The desired control law is found as a joint solution of the system of Equations (5), (6), and (12) and has the following form:  
\[