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| − | <math>\begin{ | + | p1 a) <math>A=\begin{bmatrix} |
| − | x & | + | -1 & 1 \\ |
| − | + | 0 & -2 | |
| − | \end{ | + | \end{bmatrix}</math> |
| + | <math>X(t)=\begin{bmatrix} | ||
| + | X_1(t) \\ | ||
| + | X_2(t) | ||
| + | \end{bmatrix} </math> | ||
| + | |||
| + | <math> \dot{x}_1(t)=-X_1(t)+X_2(t) | ||
| + | \dot{x}_2(t)=-2X_2(t) | ||
| + | |||
| + | <math>\Phi(t)=\begin{bmatrix} | ||
| + | \Phi_1(t) & \Phi_2(t) \\ | ||
| + | \end{bmatrix} </math> | ||
| + | For <math> \Phi_1(t) assume <math>X_(0) =\begin{bmatrix} | ||
| + | 1 \\ | ||
| + | 0 | ||
| + | \end{bmatrix} </math> | ||
| + | |||
| + | |||
| + | |||
| + | \therefore<math>\Phi_1(t)=\begin{bmatrix} | ||
| + | e^-t \\ | ||
| + | 0 | ||
| + | \end{bmatrix} </math> | ||
| + | |||
| + | For<math>\Phi_2(t) assume X_(0)=\begin{bmatrix} | ||
| + | 0 \\ | ||
| + | 1 | ||
| + | \end{bmatrix} </math> | ||
| + | |||
| + | <math>X_2(t)= | ||
| + | |||
| + | <math>\Phi_(t)= | ||
| + | <math>\Phi_(t_1 t)=\Phi_( t)\Phi_(t)^-1 | ||
| + | |||
| + | b) <math>A=\begin{bmatrix} | ||
| + | -cost & cost \\ | ||
| + | 0 & -2cost | ||
| + | \end{bmatrix}</math> | ||
| + | |||
| + | <math>\Phi_(t)=e^\begin{matrix} \int_{0}^{t} A\, \mathrm{d}t \end{matrix}</math> | ||
| + | =\begin{bmatrix} | ||
| + | e^\sin t & 0 \\ | ||
| + | 0 & e^2\sin t | ||
| + | \end{bmatrix}</math> | ||
| + | \begin{bmatrix} | ||
| + | 1 & - \sin t \\ | ||
| + | 0 & 1 | ||
| + | \end{bmatrix}</math>=\begin{bmatrix} e^\sin t & - \sin te^ \sin t \\ | ||
| + | 0 & e^2\sin t | ||
| + | \end{bmatrix}</math> | ||
| + | |||
| + | <math>\Phi_(t_1 t)=\Phi_( t)\cdot \Phi_(t)^-1</math>=\begin{bmatrix} | ||
| + | & \\ | ||
| + | 0 & | ||
| + | \end{bmatrix}</math> | ||
| + | |||
| + | p2 a) <math>\left| {\lambda\Iota-A} \right|=\begin{bmatrix} | ||
| + | \lambda+2 & -2 \\ | ||
| + | 1 & \lambda-1 | ||
| + | \end{bmatrix}</math> | ||
| + | |||
| + | <math>\lambda_1 =0 \lambda_2=-1 | ||
| + | Marginally stable ,not asy ,stable | ||
| + | |||
| + | |||
| + | b) <math>c=\begin{bmatrix} | ||
| + | B & AB | ||
| + | \end{bmatrix}</math>=<math>\begin{bmatrix} | ||
| + | 1 & 0 \\ | ||
| + | 1 & 0 | ||
| + | \end{bmatrix}</math> | ||
| + | rank=1 | ||
| + | not observable, unobservable subspace<math> \left\{ {\begin{bmatrix} | ||
| + | 1 \\ | ||
| + | 1 | ||
| + | \end{bmatrix} } \right\} | ||
| + | c) <math>0=\begin{bmatrix} | ||
| + | C \\ | ||
| + | CA | ||
| + | \end{bmatrix} </math>=<math>\begin{bmatrix} | ||
| + | 1 & -1 \\ | ||
| + | -1 & 1 | ||
| + | \end{bmatrix}</math> | ||
| + | |||
| + | rank=1 | ||
| + | not observable, unobservable subspace<math> \left\{ {\begin{bmatrix} | ||
| + | 1 \\ | ||
| + | -1 | ||
| + | \end{bmatrix} } \right\} | ||
| + | |||
| + | d) | ||
| + | |||
| + | e) \therefore talse | ||
| + | |||
| + | \therefore talse | ||
| + | |||
| + | |||
| + | |||
| + | |||
| + | f) <math>A-BK=\begin{bmatrix} | ||
| + | -2-k_1 & 2-k_2 \\ | ||
| + | -1-k_1 & 1-k_2 | ||
| + | \end{bmatrix}</math> | ||
| + | <math>\left| {\lambda-A+BK} \right|=\lambda^2+\left( {a+b+1} \right)\lambda+3a+3-ab=0 | ||
| + | |||
| + | \lambda_1 =-3 ang \lambda_2=-1 | ||
| + | <math>\begin{cases} | ||
| + | -3a + 9 -ab=0 \\ | ||
| + | 2a - b+3-ab=0 | ||
| + | \end{cases}</math> | ||
| + | |||
| + | a=0 ,b=3 <math>k=\begin{bmatrix} | ||
| + | 0 & 3 \\ | ||
| + | \end{bmatrix} </math> | ||
| + | |||
| + | |||
| + | g)<math>\begin{bmatrix} | ||
| + | \lambda\Iota-A \\ | ||
| + | C | ||
| + | \end{bmatrix}</math>=<math>\begin{bmatrix} | ||
| + | \lambda+2 & -2 \\ | ||
| + | 1 & \lambda-1 \\ | ||
| + | 1 & -1 | ||
| + | \end{bmatrix}</math> | ||
| + | must contain <math>\lambda=0 , no | ||
Revision as of 08:02, 20 May 2017
p1 a) $ A=\begin{bmatrix} -1 & 1 \\ 0 & -2 \end{bmatrix} $ $ X(t)=\begin{bmatrix} X_1(t) \\ X_2(t) \end{bmatrix} $
$ \dot{x}_1(t)=-X_1(t)+X_2(t) \dot{x}_2(t)=-2X_2(t) <math>\Phi(t)=\begin{bmatrix} \Phi_1(t) & \Phi_2(t) \\ \end{bmatrix} $ For $ \Phi_1(t) assume <math>X_(0) =\begin{bmatrix} 1 \\ 0 \end{bmatrix} $
\therefore$ \Phi_1(t)=\begin{bmatrix} e^-t \\ 0 \end{bmatrix} $
For$ \Phi_2(t) assume X_(0)=\begin{bmatrix} 0 \\ 1 \end{bmatrix} $
$ X_2(t)= <math>\Phi_(t)= <math>\Phi_(t_1 t)=\Phi_( t)\Phi_(t)^-1 b) <math>A=\begin{bmatrix} -cost & cost \\ 0 & -2cost \end{bmatrix} $
$ \Phi_(t)=e^\begin{matrix} \int_{0}^{t} A\, \mathrm{d}t \end{matrix} $
=\begin{bmatrix}
e^\sin t & 0 \\
0 & e^2\sin t \end{bmatrix}</math> \begin{bmatrix}
1 & - \sin t \\ 0 & 1
\end{bmatrix}</math>=\begin{bmatrix} e^\sin t & - \sin te^ \sin t \\ 0 & e^2\sin t \end{bmatrix}</math>
$ \Phi_(t_1 t)=\Phi_( t)\cdot \Phi_(t)^-1 $=\begin{bmatrix}
& \\
0 & \end{bmatrix}</math>
p2 a) $ \left| {\lambda\Iota-A} \right|=\begin{bmatrix} \lambda+2 & -2 \\ 1 & \lambda-1 \end{bmatrix} $
$ \lambda_1 =0 \lambda_2=-1 Marginally stable ,not asy ,stable b) <math>c=\begin{bmatrix} B & AB \end{bmatrix} $=$ \begin{bmatrix} 1 & 0 \\ 1 & 0 \end{bmatrix} $ rank=1 not observable, unobservable subspace$ \left\{ {\begin{bmatrix} 1 \\ 1 \end{bmatrix} } \right\} c) <math>0=\begin{bmatrix} C \\ CA \end{bmatrix} $=$ \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix} $
rank=1 not observable, unobservable subspace$ \left\{ {\begin{bmatrix} 1 \\ -1 \end{bmatrix} } \right\} d) e) \therefore talse \therefore talse f) <math>A-BK=\begin{bmatrix} -2-k_1 & 2-k_2 \\ -1-k_1 & 1-k_2 \end{bmatrix} $ $ \left| {\lambda-A+BK} \right|=\lambda^2+\left( {a+b+1} \right)\lambda+3a+3-ab=0 \lambda_1 =-3 ang \lambda_2=-1 <math>\begin{cases} -3a + 9 -ab=0 \\ 2a - b+3-ab=0 \end{cases} $
a=0 ,b=3 $ k=\begin{bmatrix} 0 & 3 \\ \end{bmatrix} $
g)$ \begin{bmatrix} \lambda\Iota-A \\ C \end{bmatrix} $=$ \begin{bmatrix} \lambda+2 & -2 \\ 1 & \lambda-1 \\ 1 & -1 \end{bmatrix} $
must contain $ \lambda=0 , no $
