I haven't really encountered control theory before. Could you possibly recommend some papers/textbooks that start with the very basics of the field? I have a couple of months to get to grips with the fundamentals so it doesn't matter if it's quite a dense introduction.
I've got a starting problem (the nonlinear system described below), and if you could give me some pointers I'd be really grateful. I'd like to design a state feedback controller which can reposition the eigenvalues of the linearized system so that oscillations decay to zero in less than 1s, with a period of $\pi$s.
$\dot{x}_1=x_{2}$
$\dot{x}_{2}=-x_{1}+x_{2}(1-x_{1}^{2}-2x_{2}^{2})+2u$
Many thanks
Let me rewrite the system as follows at first.
$$\dot{x} = f(x) + Bu \Leftrightarrow \begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \end{bmatrix} = \begin{bmatrix} x_{2} \\ -x_{1}+x_{2}(1-x_{1}^{2}-2x_{2}^{2}) \end{bmatrix} + \begin{bmatrix} 0 \\ 2 \end{bmatrix} u$$
First you need to determine around which point you want to linearize the system. In this case you proberbly want to linearize the system around the one of the equilibrium points (also called the fixed points). You find these by $f(x) = 0$ i.e. $\dot{x}_1 = 0$ and $\dot{x}_2 = 0$. In this case we have that the fixed point $(x_1^*,x_2^*) = (0,0)$.
You want to linearize the system now around the fixed point. Let $x = x^* + z$. Then the linearization is given by $\dot{z} = J_F(x)|_{x=x^*}z$. herein $J_F(x)|_{x=x^*}$ means the jacobian of $f(x)$ evaluated at $x^*$.
You will then obtain your $A$ and $B$ matrices for the state space system. Since you didn't tell in which states you are interested i.e. what the output is, I will assume that you are interested in both states hence i.e. $y = z$ thus $C = \begin{bmatrix} 1 & 1 \end{bmatrix}$.
To check if you even can apply state feedback your system needs to be controllable. To check if your system is controllable you need to compute $$\mathcal{C} = \begin{bmatrix}B & AB & A^{2}B & \cdots & A^{n-1}B\end{bmatrix}$$ In your case the order of the system $n = 2$ e.g. $\mathcal{C} = \begin{bmatrix}B & AB \end{bmatrix}$. For this matrix you will need to check if it has full rank. If it does not has full rank it is not controllable. But this does not yet mean that you can't design a state feedback. It may be the case namely that only 1 state, $z_1$ or $z_2$, is not controllable. Now if that uncontrollable state is however stable by itself then you can still design a state feedback for the other (unstable) state.
First you check the eigenvalues $\lambda_i$ of your $A$ matrix. Then you check using the Hautus test if the rank of the matrix $\begin{bmatrix} A - \lambda_i I & B \end{bmatrix}$ is $n$ for all $\lambda_i$, the order of the system. If this is the case you can still create a state feedback.
Then let $u = -Kz$, with $K = \begin{bmatrix} k_1 & k_2 \end{bmatrix}$. The system then becomes $\dot{z} = (A - BK)z$, and $y = (C - DK)z$ if $D \neq 0$, else just $y = Cz$. You can now compute your state feedback by computing the roots of $\det(\lambda I - (A - BK))$ which are a function of $k_1$ and $k_2$, which you can then adjust such that the poles/eigenvalues of the system are at your desired location.