Let's say that we have a matrix of transfer functions: $$G(s) = C(sI-A)^{-1}B + D$$
And we create the sensitivity matrix transfer function:
$$S(s) = (I+GK)^{-1}$$
Where $K$ is our controller gain matrix.
We also create the complementary sensitivity transfer function matrix:
$$T(s) = (I+GK)^{-1}GK$$
We also create the weighting transfer function matrices: $$W_u(s) \\ W_T(s) \\ W_P(s)$$
You can see them as the tunning matrices. This picture below representing the $H_{ \infty}$ controller. Where $z$ is our performance output. Only for analysis. The $G$ is our transfer function matrix and $K$ is as mention before, our controller gain matrix. $w$ it's a vector of disturbance. Notice that $\omega \neq w$
This whole picture can be described as: $$\begin{bmatrix} z_1 = W_uu\\ z_2 = W_TGu\\ z_3 = W_Pw + W_PGu\\ v = w + Gu \end{bmatrix}$$
And we can create our generalized plant P:
$$P = \begin{bmatrix} 0 & WuI \\ 0& W_TG\\ W_PI & WpG\\ I & G \end{bmatrix}$$
$$z = Pw = \begin{bmatrix} 0 & WuI \\ 0& W_TG\\ W_PI & WpG\\ I & G \end{bmatrix} \begin{bmatrix} w\\ u \end{bmatrix}$$
We ca partioning the generalized plant P by saying that:
$$P = \begin{bmatrix} P_{11} & P_{12}\\ P_{21}& P_{22} \end{bmatrix} = \begin{bmatrix} A & B_1 & B_2\\ C_1 & D_{11} & D_{12} \\ C_2 & D_{21} & D_{22} \end{bmatrix}$$
Then we can say:
$$P_{11} = \begin{bmatrix} A \end{bmatrix} = \begin{bmatrix} 0\\ 0\\ W_PI \end{bmatrix}$$ $$P_{12} = \begin{bmatrix} B_1 & B_2 \end{bmatrix} = \begin{bmatrix} W_uI\\ W_TG\\ W_PG \end{bmatrix} $$ $$P_{21} = \begin{bmatrix} C_1\\ C_2 \end{bmatrix} = \begin{bmatrix} I \end{bmatrix} $$ $$ P_{22} = \begin{bmatrix} D_{11} & D_{12} \\ D_{21} & D_{22} \end{bmatrix} = \begin{bmatrix} G \end{bmatrix}$$
So. Now to the question! In Robust Control, it's something called detectable and stabilizable. I wonder what it are. Accoring to a book I have, this:
$$A, B_2, C_2$$
and
$$A, B_1, C_1$$
needs to be detectable and stabilizable. The definition of stabilizable is:
A system is stabilizable if all unstable states are controllable.
The definition of detectable is:
A system is detectable if all unstable states are observable.
I know how to find out if a system is controllable and observable. That is very easy!
To check controllibility:
$$C_o \equiv \begin{bmatrix} B & AB & A^2B & \dots & A^{n-1}B \end{bmatrix}$$
And then check the rank $$rank(C_o) = n$$
To check observbility:
$$O_o \equiv \begin{bmatrix} C\\ CA\\ CA^2\\ \vdots\\ CA^{n-1} \end{bmatrix}$$
And then check the rank $$rank(O_o) = n$$
Question: Do you know any formula or method to check if all unstable states are controllable and observable? Because i cannot find any states in my system which are unstable. If I find a unstable state, that means my system matrix $A$ has some positive eigenvalues? Right?
I found the answer after hours of resarsch. I was looking in Essentials of Robust Control by Doyle and Zhou.
How to check if the system is stabilizable:
The matrix $$rank\begin{bmatrix} (\lambda_i I-A) & B \end{bmatrix} = n$$
has full-row rank $n$ for all eigenvalues $ \lambda_i \in \Re \geq 0$
How to check if the system is detectable:
The matrix $$rank\begin{bmatrix} (\lambda_i I-A) \\ C \end{bmatrix} = n$$
has full-row rank $n$ for all eigenvalues $ \lambda_i \in \Re \geq 0$