STUPIDSID
Attempt any four:
1 (a)
Define singular point of the system. How do you identify them in the phase portrait?
5 M
1 (b)
Draw the sinusoidal response of saturation with dead zone nonlinearity. Write the response equations.
5 M
1 (c)
Define positive definite matrix. What are the properties of the positive definite matrix if it is symmetric?
5 M
1 (d)
Compute the 2-norm for the matrices \[ i) \ A=\begin{bmatrix}
0 &1 \\3 &5
\end{bmatrix} \\ ii) \ F=\begin{bmatrix}
1 &0 \\0 &5
\end{bmatrix} \]
5 M
1 (e)
What are the limitations of plant inverse controller?
5 M
1 (f)
Obtain the linear system matrix at the operating point \( x^T_0 = [1 \ \ 0.5 \ \ 0.5] \) for the system of equations. \[ \begin{bmatrix}
x_1 \\ x_2 \\ x_3
\end{bmatrix} = \begin{bmatrix}
x^2_2\\x^2_3
\\ -3x^2_1 - 11x^2_2 - 12x_3
\end{bmatrix} \] Comment whether the operating point is stable?
5 M
2 (a)
Draw the phase trajectory for the following system using delta method. Assume initial condition. \( x=1, \ \dot{x}=0 \ \ \ddot{x}+2\dot{x}+4x=0 \)
10 M
2 (b)
Derive the describing function for relay with dead zone nonlinearity.
10 M
3 (a)
Obtain via analytical method the solution of the following system and write the equation of trajectory. Assume initial condition \( x_{10}=1, \ x_{20}=0. \\ \dot{x_1}= x_2 \\ \dot{x}_2 = -2x_1 - 3x_2 \)
10 M
3 (b)
Explain Lyapunov stability analysis with neat phase trajectories.
10 M
4 (a)
Design IMC controller for plant model. \[ \tilde{G}(s) = \dfrac {(-s+1)}{2s+1} \] in order to achieve the response with time constant of 1.5 sec.
10 M
4 (b) (i)
Explain choice of filters in IMC for step and ramp reference inputs.
5 M
4 (b) (ii)
What is NMP system? Explain inverse response.
5 M
5 (a)
Design the optimal controller via Riccati equation for the system \[ \dot {x} = \begin{bmatrix}0 &1 \\2 &-1 \end{bmatrix} x+ \begin{bmatrix}0\\1 \end{bmatrix}u \] to minimize the performance index \[ J=\int^\infty_0 (x^2_1 + x^2_2 + u^2)dt. \]
10 M
5 (b)
Write the steps for constructing the Lyapunov function via Krasovskii method.
10 M
6 (a)
Investigation stability of the given system using describing function method.
10 M
6 (b)
Write steps for linearizing the nonlinear system using feedback linearization: \( \dot{x}=f(x)+g(x)u \\ y=h(x) \)
10 M
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