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General question on making state space equations for a DC motor. In the link below, why are both the electrical and dynamics equations 3 and 4 used to make a state space eqn.8, instead of combining them together into a single? You could combine them by setting one of the equations to theta_dot, and inserting into the theta_dot the other equation.
http://ctms.engin.umich.edu/CTMS/index.php?example=MotorSpeed§ion=SystemModeling
You have a two state system, so (generically) you cannot replace the coupled two state system (Equation (8) in your link) by a single first order differential equation.
However, since the system is completely controllable, you can transform the system into controller canonical form and write the system as a single second order differential equation. In this case we can read off the equation from the transfer function.
You can see from Equation (7) that we have the system \$ ((J s+b)(Ls + R) + K^2) \hat{\omega} = K \hat{V}\$, where \$\hat{\omega}, \hat{V}\$ are the Laplace transforms of the rotational speed and input voltage respectively. This gives the following differential equation \$ JL \ddot{\omega}(t) + (JK+bL)\dot{\omega}(t) + (K^2+bK) \omega(t) = K v(t) \$.
It is not clear that this is more convenient that the state space form, but perhaps gives more insight into the rotational dynamics.
You could combine them by setting one of the equations to theta_dot, and inserting into the theta_dot the other equation
In differential equations you solve for \$\theta(t)\$, not for \$\dot{\theta}(t)\$, and as their values are interconnected the solution is time dependent. For linear differential equations, like the ones in the link, the solution is an exponential function.
In general you could, however, do as you say, but you would get mathematically stuck with a coupled differential equation that's quite difficult to solve. Start by fixing \$\dot{\theta}(t)\$:
$$\dot{\theta}(t) = \frac{K i(t) - J\ddot{\theta}(t)}{b} $$
and then substitute in the other one, getting:
$$ L \frac{di(t)}{dt} + Ri(t) = V - K\frac{K i - J\ddot{\theta}(t)}{b}$$
You need \$\ddot{\theta}(t)\$ to find \$i(t)\$. You can't proceed.
The most common approach to solve the interconnected system for its states' dynamics is to stack the multiple linear differential equations in a single state space (eqs. 8-9 in the link) and solve it in its matrix form.
Your intuition can however be used to find the steady-state (final values, after dynamics) of the system. This is done by setting all derivative to 0 and solving for the states' final values.
