Control theory- basic question on stabilizability

[Nemesis10192]

Please see the below post for details!

 

[Nemesis10192]

Hi all. 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.



Here is an image of the question.

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answers


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


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


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 $\displaystyle G=(B,AB,...,A^{n-1}B) \in \mathbb{R}^{n x nm}$.


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


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


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.

 

[stapel]

Here is an image of the question.
View attachment 5322

I'm sorry, but I can't read this at all. I've even downloaded the image and attempted image-correction, and I still can't read it. My best guess follows:

Consider the system:

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

with unrestricted [??] $\displaystyle n\, :\, [0,\, 7]\, \rightarrow\, \mathbb{R}^{[??]},\, A\, \in\, \mathbb{R}^{n+m},\,$ and $\displaystyle \, B\, \in\, \mathbb{R}^{n+m}$

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

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

iii) Show that the system:

. . . . .$\displaystyle \dot{x_1}\, =\, [??]\, +\, u$

. . . . .$\displaystyle \dot{x_2}\, =\, x_1$

where $\displaystyle \, x_1,\, x_2,\, u\,$ are real-valued functions, is stabilizable.

Please reply with the rest of the information. Thank you.