I have a control theory problem much akin to controlling the angle of an electric motor to a reference angle $\gamma_{ref}$. (The "electric motor" exists in software only, and so has very little noise)
Both the angle $x_1=\gamma$ and the speed $x_2=\dot{\gamma}$ are measureable, but only the angle acceleration $\ddot{\gamma}$ is controllable. The motor experiences a friction force $-b\dot{\gamma}$ and has inertia $1/c$. Because I want to eliminate steady-state errors, I've introduced a dummy variable $x_3$ such that $\dot{x_3}=\gamma_{ref}-\gamma$. My open loop system is:
$$ \dot{x}= \begin{bmatrix} 0 & 1 & 0 \\ 0 & -b & 0 \\ -1 & 0 & 0 \end{bmatrix} x + \begin{bmatrix} 0 \\ c \\ 0 \end{bmatrix} u + \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \gamma_{ref} $$
To put the control in canonical form, I've set:
$$u=-l_1\gamma-l_2\dot{\gamma}-l_3x_3$$
and so
$$ \begin{bmatrix} 0 \\ c \\ 0 \end{bmatrix} u = \begin{bmatrix} 0 & 0 & 0 \\ -cl_1 & -cl_2 & -cl_3 \\ 0 & 0 & 0 \end{bmatrix} x $$
which makes my closed loop system
$$ \dot{x}= \begin{bmatrix} 0 & 1 & 0 \\ -cl_1 & -b-cl_2 & -cl_3 \\ -1 & 0 & 0 \end{bmatrix} x + \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \gamma_{ref} = Ax+R\gamma_{ref} $$
To get the characteristic equation, I have to take
$$\det{(A-\lambda I)}=0$$
which I've worked out to be
$$\boxed{\lambda^3+(b+cl_2)\lambda^2+cl_1\lambda-cl_3=0}$$
PROBLEM. How do I actually choose the poles? Using my test environment in Java, I've concluded that for $b=0.1,c=1.0$ the poles
$$ \lambda=-0.63\quad \lambda=-0.085-0.074i\quad \lambda=-0.085+0.074i $$
work fairly nicely... but this is a completely arbitrary trial-and-error method. I've seen some textbooks give answers like "choose one pole -42b" or whatever. But this is still not ideal. What general methods are available to me, where I can pick the poles based on some properties I want, like rise time and overshoot, and so on?
I've tried reading about Bode plots but they sidetrack into theoretical stuff.
Thanks a lot!!
Choice of poles comes from the knowledge of the domain of application, electric motor engineering in the case. The compromises between time constants, overshoot, settling times, noise amplification, and other factors cannot be resolved by theory alone. Trial and error is reasonable, and techniques such as Bode plots and optimal control may help you find the answer.
But it always must come down to the meaning of the variable and the physics of the process.