Control theory- basic question on stabilizabilty

Studying some control theory but having difficulty learning because my lecturer doesn't provide solutions to any of his exercises AT ALL. Below I've attached a problem I've just done and my answers and would like to know if I'm on the correct track.

For those who can't view the images:

Questions

1. a) Consider the system

$$\dot{x}=Ax+Bu, x(0)=x_0 \in \mathbb{R}^n ,$$

with unrestricted control u: [0,T] $\rightarrow \mathbb{R}^m, A \in \mathbb{R}^{n \times n}$ and $B \in \mathbb{R}^{n\times m}$.

(i) Define what it means for this system to be stabilizable.

(ii) State at theorem giving a sufficient condition on the matrices A,B for this system to be stabilizable.

(iii) Show that the system $$\dot{x_1}=x_2 +u, \dot{x_2}=x_1$$ where $x_1, x_2,$ and u are real valued functions, is stabilizable.

(i) Consider a closed loop feedback control of the form u=kx. Then the system is stabilizable if there exists a matrix $K \in \mathbb{R}^{m \times n}$ such that the system

$$\dot{x} =(A+BK)x, x(0)=x_0 \in \mathbb{R}^n$$

is asymptotically stable.

(ii) If (A,B) is controllable, then it is stabilizable.

(iii) (A,B) is controllable if and only if rank(G)=n where G is the controllability matrix $G=(B,AB,...,A^{n-1}B) \in \mathbb{R}^{n \times nm}$.

For this particular system one has $A=\begin{bmatrix} 0 & 1 \\1 & 0 \end{bmatrix} B= \begin{bmatrix} 1 & 0 \end{bmatrix}$ transposed

Giving G (not writing out calculation because I'm slow at latex) $=\begin{bmatrix} 1 & 0 \\0 & 1 \end{bmatrix}$.

Clearly (1,0) and (0,1) are linearly independant so rank(G)=2=n and hence (A,B) is controllable, and so is also stabilizable.

^Is this correct? Thanks.

• Your image is unviewable on my device. Please take a moment to write it up. – Emily May 18 '15 at 14:03
• Which image, the questions, my answers or both? – Greg May 18 '15 at 14:10
• It looks good to me. – obareey May 25 '15 at 20:15
• Looks good for me too. – Cybernetician Feb 16 '16 at 3:54