However, because the relative velocity is a very complex term and may be spread in many terms to represent the kinetic energy of the connecting-rod, a simplification of the model is necessary and represents the movement of the system as much as using the complex term. Thus, the technique of concentrated masses of an equivalent dynamic model [20] can be applied when three requirements are evaluated to the dynamical equivalence, which are as follows:the final mass of the model must be equal to the total mass of the original link; the CG of connecting-rod must stay in the same original position of the link; the final momentum of inertia of the masses must be equal to the initial link, as shown in Figure 2.

Through this equivalent dynamic model of concentrated masses, the complex rotational and translation motions of the connecting-rod are converted into a rotational motion of the crank and a translational motion of the cylinder.This conversion generates equivalence when the place of the percussion centers of the equivalent masses related to the original link mass is determined. In the typical geometry of a typical connecting-rod, because its masses are higher to the forces of connection with the crank (width and height)its percussion center is closer to the connection extremity of the crank than the CG. Therefore, this geometry allows simplifying and concentrating the masses in the connections with the links and present an error relatively small in the precision of this dynamic model [20]. These connections with links are denoted by The connecting-rod of this work has typical geometry with CGco placed with 𝑙CGcocr to 1/3 of the length 𝑙 of the connection with the crank and 𝑙CGcocy to 2/3 of length 𝑙 of the connection to the cylinder. Introducing these lengths in (19),it is possible to obtain the partial value of the total of masses of the connecting-rod coupled to the links by the connections,as given by This geometric distribution for the connecting-rod results in the equivalent masses for application on the crank with the equivalent mass 𝑚cocr equal to 1/3 of 𝑚co and the equivalent mass to application in the cylinder 𝑚cocy equal to 2/3 of 𝑚co.In this way of distribution of applied masses to the links of crank (20) and cylinder (21), with their pure motions of rotation and translation, possibilities of the obtainment of the kinetic energy equivalent of the connecting-rod link are given by The kinetic energy of the cylinder is also considered the translational movement. The angular position 𝜃 used (6),which determines the linear position of the cylinder CG in function of the crank’s rotational angle. Thus, the kinetic energy of the mono cylinder is given by The potential gravitational energy is given by the connecting-rod link, which possesses the CG motion related to the vertical displacement that generates a variation of energy in relation to the position variation. By geometry,the potential gravitation energy is considered as a vertical projection of 𝑙CGco to the variation of the CG position of the connection-rod link in function of the gravitational force,given by Therefore, (24) can be rewritten in function of the gravitational constant force, given by Substituting the energy terms into (14) in terms of their derivatives and applying Euler-Lagrange equation (8), the governing equation of motion of the crank-connecting-rod system in function of the conservatives and nonconservative forces are obtained and are written in state-space form as (26),

The application of the controller consists of substituting the excitation torque generated by the air force 𝐹a induced by the pressure 𝑃a (12), for the control signal U, given by Introducing the control signal in (26), it has the equation of motion with the control system which can be denoted by The control vector U consists of two parts; U fe and U fo =−(𝑥1), where Ufo is the feed forward control and Ufe is the feedback control obtained through the SDRE control signal,as show in

Therefore, changing the notation of the control signal U =Ufe+Ufo in (29), the final equation of motion of the controlled system can be rewritten by the form of From (31), the movement of the system can be controlled by the angular velocity 𝜃̇ = 𝑥̇1 = 𝑥2, being the main control variable, similar to an automatic vehicular driver. The