The state model of linear control system is $\Sigma:\dot{x}=Ax+Bu$, where $x$ is the state and $u$ is the input. A state feedback $F:u=Fx+v$ can be treated as a transformation which maps $\Sigma$ to $\Sigma_F:\dot{x}=(A+BF)x+Bv$. Prove the following conclusions:
All feedback transformation form a group which is called feedback group;
The controllability is invariant under the transformation.
The first question is obviously because the transformations form the linear transformation in $\mathbb R^n$. However what is the controllability in Math? And how can we prove that it is invariant under the transformation.
Any advice is helpful. Thank you.
I'm not exactly sure what the group operation is. Since the feedback is a linear function of the state, the collection of $F$ form a subspace, perhaps the group operation is addition?
The second part is more straightforward.
By definition, $(A,B)$ is cc. iff for all $x_0,x_1$ there exists some $u$ such that with input $u$ the solution to the differential equation $\dot{x}=Ax+Bu$ satisfies $x(0) = x_0, x(1) = x_1$ (in other words, $u$ 'steers' the system from $x_0$ at time zero to $x_1$ at time one).
Suppose $(A,B)$ is cc. and consider the system $(A+BF,B)$. Choose $x_0,x_1$ and let $u$ be control that steers $x_0$ to $x_1$ in the system $(A,B)$, and let $x_u$ be the solution of the system $(A,B)$ with input $u$, starting at $x_0$. Then the control $u^* = u-Fx_u$ will steer the system $(A+BF,B)$ from $x_0$ at time zero to $x_1$ at time one. To see this, note that $x_u$ is a solution of the system $\dot{x}=(A+BF)x+Bu$ with input $u^*$ starting at $x_0$. Hence $(A+BF,B)$ is cc.
(Note: Since $A = (A+BF)-BF$, we see that the above is an equivalence, that is, $(A,B)$ is cc. iff $(A+BF,B)$ is cc. for any $F$.)