Solve two differential equations simultaneously

I need to design a controller for and simulate a pneumatic control system. Specifically I need to simulate these system equations and eventually design my own controller.

Since the position equations (x) are a function of the time derivative of the pressure (Pdot) and vice versa, what is the best way of simulating this in Matlab? I have solved differential equations before using ode45, but nothing like this.

Thanks for any help.

To clarify, P1 and P2 pertain to the pressures on each side of a pneumatic actuator. X is the displacement variable for the actuator. m_dot1 and m_dot2 are the mass flow rates into each side of the pneumatic actuator from the control valve.

Cited paper: folk.ntnu.no/skoge/prost/proceedings/acc04/Papers/0905_FrM16.4.pdf

[Torsten]

If you insert v_p for (x_p)_dot in the pressure equations, what is the problem ?

Best wishes

Torsten.

[J Krause]

That part I understand.

I am unsure of the best way to interface the (P)dot (pressure) equations with the (v_p)dot (position) equation, since they depend on each other. The volumes (V1 and V2) are dependent on x_p. Where x_p = v_p.

[Torsten]

I don't understand your problem.

If you insert v_p for (x_p)_dot in the pressure equations, the 4 ODEs are explicit in y(1)=x_p, y(2)=v_p, y(3)=P1 and y(4)=P2.

So they are easily set up for ODE45 as

fun=@(t,y)[y(2);1/M*(-b*y(2)+A*y(3)-A*y(4)-F1-F2);gamma*R*T/V1(y(1))*m1dot-alpha*gamma*y(3)*A/V1(y(1))*y(2);-gamma*R*T/V2(y(1))*m2dot+alpha*gamma*y(4)*A/V2(y(1))*y(2)];

Best wishes

Torsten.