STUPIDSID
1(a)
What are the advantages of modern control theory over conventional control theory?
5 M
1(b)
For the system shown, write the state equations satisfied by them. Bring these equations in vector matrix form, Take R = 1 MΩ and C = 1 μF
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:!mage
7 M
1(c)
A feedback system is characterized by the closed loop transfer function, \[T(s)=\dfrac{s^2+3s+3}{s^3+2s^2+3s+1}\]
Draw the signal flow graph and obtain the state model in second companion form.
Draw the signal flow graph and obtain the state model in second companion form.
8 M
2(a)
Obtain the state space representation in the given system in Jordan canonical form. \[\dfrac{y(s)}{U(s)}=\dfrac{2s^2+6s+7}{(s+1)^2(s+2)}\]
12 M
2(b)
Obtain the transfer function for the state model represented by, x = Ax + Bu, y = Cx + DU, where \( A=\begin{bmatrix}
-1 & 1\\
-1 & -10
\end{bmatrix},B=\begin{bmatrix}
0\\
10
\end{bmatrix},C=\begin{bmatrix}
1\\
0
\end{bmatrix},D=[0]\)
8 M
3(a)
Prove that the model matrix M diagonalizes the system matrix A.
4 M
3(b)
For the matrix, \( A=\begin{bmatrix}
0 & 1 & 0\\
0 & 0 & 1\\
-2 & -4 & -3
\end{bmatrix},\) find i)Eigen values ii) Eigen vectors iii) Modal matrix
8 M
3(c)
Compute the state transition matrix for, \(A=\begin{bmatrix}
0 & -3\\
1 & -4
\end{bmatrix} \) using i) Laplace - transformation method. ii) Cayley-Hamilton method.
8 M
4(a)
Define state transition matrix and list its properties.
4 M
4(b)
A linear time invariant system is characterized by, \[\begin{bmatrix}
\dot{x_1}\\
\dot{x_2}
\end{bmatrix}=\begin{bmatrix}
1 & 0\\
1 & 1
\end{bmatrix}\begin{bmatrix}
x_1\\
x_2
\end{bmatrix}+\begin{bmatrix}
-1\\
2
\end{bmatrix}[u];\ \ y=[1\ \ -1]\begin{bmatrix}
x_1\\
x_2
\end{bmatrix}\]
Compute the response y(t) to a unit step input assuming \( X(0)=\begin{bmatrix} 1\\ 0 \end{bmatrix}.\)
Compute the response y(t) to a unit step input assuming \( X(0)=\begin{bmatrix} 1\\ 0 \end{bmatrix}.\)
12 M
4(c)
Evaluate the controllability of the system with \( \dot{x}=Ax+BU\ \text{where}\ A=\begin{bmatrix}
0 & 0 & -6\\
1 & 0 & -11\\
0 & 1 & -6
\end{bmatrix},B\begin{bmatrix}
1\\
0\\
0
\end{bmatrix}.\)
4 M
5(a)
A system is described by following state model: \[\begin{bmatrix}
\dot{x_1}\\
\dot{x_2}\\
\dot{x_3}
\end{bmatrix}=\begin{bmatrix}
0 & 1 & 0\\
0 & 0 & 1\\
-1 & -5 & -6
\end{bmatrix}\begin{bmatrix}
x_1\\
x_2\\
x_3
\end{bmatrix}+\begin{bmatrix}
0\\
0\\
1
\end{bmatrix}u\]
Compute the state feedback gain matrix "K" so that the control law to u = -Kx places the closed loop poles at -2±j4, -5, using direct substitution method.
Compute the state feedback gain matrix "K" so that the control law to u = -Kx places the closed loop poles at -2±j4, -5, using direct substitution method.
10 M
5(b)
Consider the system \(\dot{X}=AX+Bu\ \text{and}\ y=CX \) \[\text{where}\ A=\begin{bmatrix}
0 & 20.6\\
1 & 0
\end{bmatrix},B=\begin{bmatrix}
0\\
1
\end{bmatrix},C=[0\ \ 1]\]
Design a full order state observer using Ackermann's formula.
Design a full order state observer using Ackermann's formula.
10 M
6(a)
What is PI and pD controller? What are its effect on system performance?
6 M
6(b)
Discuss pole placement by state feedback. What is the necessary condition for design using state feedback?
6 M
6(c)
Explain Backlash and Jump resonance with respect to non-linear systems.
8 M
7(a)
What are singular points? Explain different singular points based on the location of Q point.
8 M
7(b)
A linear second order servo system is described by the state equation, \[\ddot{e}+2\xi \omega _n\dot{e}+\omega ^2_ne=0\]
where \( \xi =0.15\) and ωn = 1 rad/sec, e(0) = 1.5 and e(0) = 0. Construct the phase trajectory using the method of isocline.
where \( \xi =0.15\) and ωn = 1 rad/sec, e(0) = 1.5 and e(0) = 0. Construct the phase trajectory using the method of isocline.
12 M
8(a)
Define : i) positive definiteness ii) Negative definiteness iii) Positive semidefiniteness iv) Negative semidefiniteness v) Indefiniteness.
5 M
8(b)
Explain Kravoski's theorem with example.
7 M
8(c)
Examine the stability of a non-linear system governed by the equations, \[\dot{x_1}=-x_1+2^2_1x_2;\ \ \dot{x_2}=-x_2,\ \ \text{Assume}\ 2x_1x_2<1.\]
8 M
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