Tianjin University 2022~2023 First Semester Final Exam
modern control theory
Exam time: November 5, 2022
1. Short answer questions (25 points in total, 5 points for each small question)
1. Try to explain that the linear transformation does not change the transfer function of the linear system.
2. Briefly describe the principle of duality, what is the relationship between the transfer function matrix of the dual system?
3. What is a minimal implementation? What are the necessary and sufficient conditions for an implementation to be a minimal implementation?
4. Try to write down the specific steps of using the Cayley-Hamilton theorem to find the state transition matrix.
5. For a linear system, how to judge that the system state feedback can be stabilized?
2.
The flow chart of a system is shown in the figure above. Try to find the state space expression of the system, and judge the controllability and observability of the system. (12 points)
3. Known system x ˙ = [ 0 0 − 1 1 0 − 3 0 1 0 ] x + [ 1 1 0 ] u , y = [ 0 1 − 2 ] x \dot{x} =\begin{bmatrix} 0 & 0 & -1\\ 1 & 0 & -3\\ 0 & 1 &0\end{bmatrix}x+\begin{ bmatrix}1 \\1 \\0\end{bmatrix}u，y=\begin{bmatrix} 0 & 1 &-2\end{bmatrix}xx˙=⎣⎡010001−1−30⎦⎤x+⎣⎡110⎦⎤u，y=[01−2]x
Try to judge the controllability of the system and decompose the controllability of the system to find the transfer function matrix of the system (12 points)
4. Known nonlinear system { x ˙ 1 = x 2 x ˙ 2 = − ( 1 + x 2 ) 2 x 2 − x 1 \left\{\begin{matrix}\dot{x}_{1}=x_{2} \\\dot{x}_{2}=-\left ( 1+ x_{2}\right ) ^{2}x_{2}-x_{1}\end {matrix}\right.{
x˙1=x2x˙2=−(1+x2)2x2−x1
Try to use Lyapunov's second method to analyze the stability of the system (10 points)
五、 known systemx ˙ = [ 0 1 − 2 − 3 ] x + [ 0 1 ] u , y = [ α 1 ] x \dot{x}=\begin{bmatrix} 0 & 1\\ -2 &-3 \end{bmatrix}x+\begin{bmatrix} 0\\ 1 \end{bmatrix}u，y=\begin{bmatrix} \alpha &1 \end{bmatrix}xx˙=[0−21−3]x+[01]u，y=[a1]x
(1) Try to judge the controllability of the system, if the system is completely observable, findα \alphaα value range (5 points)
(2) Set state feedbacku = kx + vu=kx+vu=kx+v , find the state feedback matrixk = [ k 0 k 1 ] k=\begin{bmatrix} k_{0}&k_{1} \end{bmatrix}k=[k0k1] so that the pole of the system configuration is− 2 ± j 2 -2\pm j2−2±j 2 (6 points)
六、 known systemx ˙ = [ 0 1 1 0 ] x + [ 1 0 ] u , y = [ α 1 ] x \dot{x}=\begin{bmatrix} 0 & 1\\ 1 &0 \end{bmatrix}x+\begin{bmatrix} 1\\ 0 \end{bmatrix}u，y=\begin{bmatrix} \alpha &1 \end{bmatrix}xx˙=[0110]x+[10]u，y=[a1]x
(1) Design a full-dimensional observer so that its poles are located at− 5 -5−5， − 5 -5 − 5 (9 points)
(2) Design the state feedback for the state observer so that the poles of the system are configured at− 2 -2−2， − 3 -3 − 3 (9 points)