Difference between revisions of "ECE-QE AC2-2015" - Rhea

Revision as of 04:28, 17 May 2017

AC-2 P1.

$ \mathbf{a)} \quad e^{At}=\begin{bmatrix} -1 & -1 & 1\\ -1 & 1 & 2\\ 1 & 1 & -2\\ \end{bmatrix}\begin{bmatrix} e^{t} & 0 & 0\\ 0 & e^{-t} & 0\\ 0 & 0 & e^{0}\\ \end{bmatrix}\begin{bmatrix} -2 & -\frac{1}{2} & -\frac{3}{2}\\ 0 & \frac{1}{2} & \frac{1}{2}\\ -1 & 0 & -1\\ \end{bmatrix} $

=$ \begin{bmatrix} -e^{t} & -e^{-t} & 1\\ -e^{t} & e^{-t} & 2\\ e^{t} & e^{-t} & -2\\ \end{bmatrix}\begin{bmatrix} -2 & -\frac{1}{2} & -\frac{3}{2}\\ 0 & \frac{1}{2} & \frac{1}{2}\\ -1 & 0 & -1\\ \end{bmatrix} $

=$ \begin{bmatrix} 2e^{t}-1 & \frac{1}{2}e^{t} -\frac{1}{2}e^{-t} & \frac{3}{2}e^{t}-\frac{1}{2}e^{-t}-1\\ 2e^{t}-2 & \frac{1}{2}e^{t}+\frac{1}{2}e^{-t} & \frac{1}{2}e^{t}+\frac{1}{2}e^{-t}-2\\ -2e^{t}+2 & -\frac{1}{2}e^{t}+\frac{1}{2}e^{-t} & -\frac{3}{2}e^{t}+\frac{1}{2}e^{-t}+2\\ \end{bmatrix} $

$ \mathbf{b)}\quad\lambda_1=1,\quad\lambda_2=-1,\quad\lambda_3=0 $, $ \because\quad\lambda_1>0 \quad\therefore\quad unstable. $

$ \mathbf{c)}\quad If \quad we \quad want \quad t \to \infty, X(t) \to0 $

$ \quad Model 1 \quad and \quad Model 3 \quad need \quad to \quad be \quad zero. $

 $ \omega_1^T X_[0]=0 $

 $ \omega_3^T X_[0]=0 $

$ \begin{bmatrix} -2 & -\frac{1}{2} &-\frac{3}{2}\\ -1 & 0 & -1 \\\end{bmatrix}\quad X_[0]=0 $

$ \quad\therefore\quad X_1=-X_3 , X_2=X_3 $

$ \quad \therefore \quad X_[0]=\quad X_3\begin{bmatrix} -1 \\ 1 \\ 1 \\ \end{bmatrix} $

$ \quad The \quad set\quad is\begin{Bmatrix} \begin{bmatrix} -1 \\ 1 \\ 1 \end{bmatrix} \end{Bmatrix} $

$ \quad If \quad we \quad want \quad to\quad remain\quad bounded $

$ \quad Model 2 \quad and \quad Model 3 \quad are \quad already \quad bounded \quad when \quad t>0 $

$ \quad \therefore\omega_1^T X_[0]=0 $

$ \begin{bmatrix} -2 & -\frac{1}{2} &-\frac{3}{2}\\ \end{bmatrix} \quad X_[0]=0 $

$ \quad X_1=-\frac{1}{4} X_2-\frac{3}{4}X_3 $

$ \quad X=\begin{bmatrix} -\frac{1}{4} X_2-\frac{3}{4}X_3\\ X_2\\ X_3\\ \end{bmatrix}=X_2\begin{bmatrix} -\frac{1}{4}\\ 1\\ 0\\ \end{bmatrix} +X_3\begin{bmatrix} -\frac{3}{4}\\ 0\\ 1\\ \end{bmatrix} $

$ \quad \therefore \quad The \quad set\quad is\begin{Bmatrix} \begin{bmatrix} -\frac{1}{4} \\ 1 \\ 0\\ \end{bmatrix}，\begin{bmatrix} -\frac{3}{4}\\ 0\\ 1\\ \end{bmatrix}\end{Bmatrix} $

$ \mathbf{d)} \quad C=\begin{bmatrix} B & AB & A^2B \end{bmatrix}=\begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ -1 & -1 & -1 \end{bmatrix} $

$ \quad rank=1\ne \mbox 3,\quad not\quad controllable $

$ \quad The\quad reachable\quad subspace\quad is \begin{Bmatrix} \begin{bmatrix} 1 \\ 1 \\ -1 \end{bmatrix} \end{Bmatrix} $

$ \mathbf{e)} \quad O=\begin{bmatrix} \quad C\\ \quad CA\\ \quad CA^2 \\ \end{bmatrix} =\begin{bmatrix} 1 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{bmatrix} $

$ \quad rank=1\ne \mbox 3,\quad not \quad observable $

$ \quad unobservable \quad subspace\quad is \begin{Bmatrix} \begin{bmatrix} 0 \\ 1 \\ 0 \\ \end{bmatrix},\begin{bmatrix} -1\\ 0 \\ 1 \\ \end{bmatrix}\end{Bmatrix} $

$ (\quad f-\quad j)\quad Can't \quad resolve. $

P2.

P3.$ \quad Let \quad X[3]= \begin{bmatrix} 1 & 1\\ \end{bmatrix}^{t} ,\quad X[0]=\begin{bmatrix} 0 & 0\\ \end{bmatrix}^{t} $

$ \quad C_3 =\begin{bmatrix} B & AB & A^2B\\ \end{bmatrix} =\begin{bmatrix} 1 & 1 & 0\\ 0 & 1 & 1\\ \end{bmatrix}^{t} $

$ \quad U =\begin{bmatrix} u(2)\\ u(1)\\ u(0)\\ \end{bmatrix} =\quad C_3^{t} ( C_3C_3^{t})^{-1} X[3]=\begin{bmatrix} \frac{2}{3} & -\frac{1}{3}\\ \frac{1}{3} & \frac{1}{3}\\ - \frac{1}{3} & \frac{2}{3}\\ \end{bmatrix}\begin{bmatrix} 1\\ 1\\ \end{bmatrix}=\begin{bmatrix} \frac{1}{3}\\ \frac{2}{3}\\ \frac{1}{3}\\ \end{bmatrix} $

$ \quad \therefore \quad U[0]=\frac{1}{3}, \quad U[1]=\frac{2}{3}, \quad U[2]=\frac{1}{3} $

$ \quad Energy \quad is \quad U^{2}[0] +U^{2}[1] +U^{2} [2]= \frac{6}{9}= \frac{2}{3} $

P4.$ ( X_1^{2}-1)( X_2-2)=0 $

 $ \quad -X_2( X_1^{2}+1)=0 $
 $ \quad X_1=    ,  \quad X_2=0 $

$ \quad X_e1=\begin{bmatrix} 1\\ 0\\ \end{bmatrix},\quad X_e2=\begin{bmatrix} -1\\ 0\\ \end{bmatrix} $

$ D f(X)=\begin{bmatrix} 2 X_1 X_2-4 X_1 & X_1^{2}-1\\ -2 X_1X_2 & -X_1^{2}-1\\ \end{bmatrix} $

$ D f( X_e1)=\begin{bmatrix} -4 & 0 \\ 0 & -2\\ \end{bmatrix} \quad all \quad\lambda_i\quad negative \quad asy \quad stable $

$ \quad D f( X_e2)=\begin{bmatrix} 4  &   0  \\ 0  &   -2\\ \end{bmatrix} \quad has \quad a\quad positive \quad \lambda,\quad unstable $

$ \mathbf{d)} \quad X_1=r,X_2=-s,X_3=s \quad X=r\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}+s\begin{bmatrix} 0 \\ -1 \\ 1 \end{bmatrix}\\ $ $ Subspace\quad is \begin{Bmatrix} \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} ,& \begin{bmatrix} 0 \\ -1 \\ 1 \end{bmatrix} \end{Bmatrix}.\\ $ $ \mathbf{e)} \quad \lambda I-A=\begin{bmatrix} \lambda-1 & -1 & -1 \\ 0 & \lambda & -1 \\ 0 & 0 & \lambda+1 \end{bmatrix}\quad \lambda_1=1,\lambda_2=0,\lambda_3=-1\\ $ $ \qquad for \;\lambda_1=1 \qquad\begin{bmatrix} \lambda I-A & B \end{bmatrix}=\begin{bmatrix} 0 & -1 & -1 & 1 \\ 0 & 1 & -1 & 1 \\ 0 & 0 & 2 & 0 \end{bmatrix}\\ $ $ \qquad for \;\lambda_2=0 \qquad\begin{bmatrix} \lambda I-A & B \end{bmatrix}=\begin{bmatrix} -1 & -1 & -1 & 1 \\ 0 & 0 & -1 & 1 \\ 0 & 0 & 1 & 0 \end{bmatrix}\\ $ $ \qquad for \;\lambda_3=-1 \qquad\begin{bmatrix} \lambda I-A & B \end{bmatrix}=\begin{bmatrix} -2 & -1 & -1 & 1 \\ 0 & -1 & -1 & 1 \\ 0 & 0 & 0 & 0 \end{bmatrix} \qquad rank<3 \qquad must\;contain\;\lambda=-1 \qquad \therefore\; No. $

$ \mathbf{f)} \quad \begin{bmatrix} \lambda I-A \\ C \end{bmatrix}=\begin{bmatrix} \lambda-1 & -1 & -1 \\ 0 & \lambda & -1 \\ 0 & 0 & \lambda+1 \\ 0 & 1 & 1 \end{bmatrix} $

$ for\; \lambda_1=1 \quad \begin{bmatrix} \lambda I-A \\ C \end{bmatrix}=\begin{bmatrix} 0 & -1 & -1 \\ 0 & 1 & -1 \\ 0 & 0 & 2 \\ 0 & 1 & 1 \end{bmatrix} \quad rank<3 \quad must\;have\;\lambda_1=1 $

$ for\; \lambda_2=0 \quad \begin{bmatrix} \lambda I-A \\ C \end{bmatrix}=\begin{bmatrix} -1 & -1 & -1 \\ 0 & 0 & -1 \\ 0 & 0 & 1 \\ 0 & 1 & 1 \end{bmatrix}\\ $

$ for \lambda-3=-1\quad \begin{bmatrix} \lambda I-A \\ C \end{bmatrix}=\begin{bmatrix} -2 & -1 & -1 \\ 0 & -1 & -1 \\ 0 & 0 & 0 \\ 0 & 1 & 1 \end{bmatrix} \quad rank<3 \quad must\;have\;\lambda_3=-1 $

$ \because \;eigenvalues \left\{1,-1,2 \right\}\quad \therefore\;Yes. $

$ A-LC=<math>\begin{bmatrix} 1 & 1-L_1 & 1-L_1 \\ 0 & -L_2 & 1-L_2 \\ 0 & -L_3 & -1-L_3 \end{bmatrix}\quad LC=\begin{bmatrix} 0 & L_1 & L_1 \\ 0 & L_2 & L_2 \\ 0 & L_3 & L_3 \end{bmatrix} $

$ \lambda I-\left(A-LC \right)=\begin{bmatrix} \lambda-1 & L_1-1 & L_1-1 \\ 0 & \lambda+L_2 & L_2-1 \\ 0 & L_3 & \lambda+1+L_3 \end{bmatrix} $

For conditions $ \quad\lambda_1=1,\quad\lambda_2=-1,\quad\lambda_3=-2 $

$ \begin{cases} 3L_2 + 6L_3 = 9 \\ L_2 + 2L_3 = 3 \\ L_2=1 \\ L_3=1 \end{cases} \quad L=\begin{bmatrix} 2 \\ 1 \\ 1 \end{bmatrix} $

$ \mathbf{g)} \quad \because\;\lambda_1=1,\quad Not\;stable. $

$ \mathbf{h)} \quad AU_1=\lambda_1 U_1 \quad AU_2=\lambda_2 U_2 \quad AU_3= \lambda_3 U_3\\ $

$ \begin{alignat}{2} y & = C X_(t) =C[U_1 e^t (\omega_1^T X_{(0)})+U_2 e^0 (\omega_2^T X_{(0)})+U_3 e^{-t} (\omega_3^T X_{(0)})]\\ & = -\omega_2^T \omega_{(0)}\\ \end{alignat} $

$ \therefore\;bounded $

$ :)\;\because \; \frac{1}{s}\; has\; pole=0 \quad\therefore\;Not \; BIBO \;Stable. $

P2.

$ \mathbf{(a)} \quad A=\frac{1}{2}\begin{bmatrix} 1 & -1 \\ -1 & -1 \end{bmatrix},\quad \lambda=0 $

$ \begin{alignat}{1} (0)^k & =\beta_0 ,\; k \to \infty \quad \beta_0=0\;\\ \end{alignat} $