# How do you linearize a nonlinear system that does not largely depend on state variables?

I have the dynamics of an aircraft described in terms of three nonlinear differential equations that are largely dependent on external variables. That is, variables that are NOT one of the three state variables. I can describe this thusly:

$V' = f(a, Cd, F, V, \theta)$;

$\psi$' = $f(a, Cl, F, V, b)$;

$\theta ' = f(a, Cl, F, V, \theta, b)$;

The state variables are $V, Psi$, and $\theta$. The relationships between the other variables $a, F, Cd, Cl, b$ are very complicated: $Cl$ is a function of a and $b$,$F$ is a function of Theta, $V, Cl, a,$ etc. Suffice it to say the block diagram of this system is very complex.

I need to linearize this system so that I can tune a two PID controllers. In order to do that I need to first linearize the system (right?). My question is, how do you linearize a system like this when putting it into state space form doesn't seem to capture much of the actual dynamics? Do you just put in constant values for the other external variables that represent a certain mode of flight?

• If you don't model their dynamics, the other "variables" ($a, F, Cd, Cl, b$) are not variables but parameters, and generally yes, you keep them as parameters (meaning you leave them as constants when linearizing) and plug in numbers after. You can then do more complicated stuff, like calculate different sets of PID gains for different conditions, or actually model the dynamics of these parameters, or do robust control, or even not use PID : D – Steve Heim May 17 '17 at 7:49
• About linearization specifically, since some of your "external" variables are functions of your state variables, note that their derivatives will appear in the linearization through the Jacobian (e.g. see here). The other variables will indeed be treated as constants. – user3658307 May 17 '17 at 18:13