recently I faced to a problem in extracting an advantageous mathematical model of a pneumatic servo positioning system which uses a normal cylinder and piston actuated with a proportional pneumatic valve, to be applicable for designing an intelligent neural network-based adaptive controller with the use of sliding mode observer for the aim of trajectory tracking with the presence of matched and unmatched uncertainties and nonlinearities and disturbances. the approach will be considered as a disturbance rejection or fault-tolerant method. But the problem is the emerging complicated high order relations specifically in modeling the proportional valve. we have the mass flow rate ''m dot'' equation which is a function of valve-generated compressed air pressure ''P1 and P2'' flows into both sides of the cylinder and the generated control signal ''u''. but the main equation for describing the whole process is based on the second law of newton which is written as "m(y double dot)=(P1.A1 - P2.A2) - Ff - FL- Fd". the challenge is on the term (P1.A1 - P2.A2) and more specifically on how to define P1 and P2 in order to be as simple as possible and not to increase the order of the system equations. in some researches (like my recent IEEE 2018), P1 and P2 define by using thermodynamic laws and physical description. in these approaches, you may face with the derivative of P1 and P2 and so you forced to differentiate the whole equation and turns it into "m(y third dot)=(P1dot.A1 - P2dot.A2) - Ff - FL- Fd". in other hands in a few types of research in order to avoid the complexity and decrease the order of system equations, the term (P1.A1 - P2.A2) turns into a simple form of b.f(u) in which b is a positive constant and f(u) is a function of control signal. this form of description is a bit conservative but helps a lot in extracting a state-space and control form representation which is of necessities for designing the nonlinear controller.

now the question is if I need a sliding mode observer thorough designing the disturbance rejection intelligent adaptive controller, what can I do and which way is more effective? any other so far so good ideas and comments which are not mentioned in the explanation or any kind of corrections if I am wrong in any parts are all welcomed.