# GATE Questions & Answers of State variable model and solution of state equation of LTI systems

## What is the Weightage of State variable model and solution of state equation of LTI systems in GATE Exam?

Total 11 Questions have been asked from State variable model and solution of state equation of LTI systems topic of Control Systems subject in previous GATE papers. Average marks 1.91.

The state diagram of a system is shown below. A system is described by the state-variable equations



The State-variable equations of the system shown in the figure above are

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The state diagram of a system is shown below. A system is described by the state-variable equations



The state transition matrix ${e}^{{A}_{t}}$ of the system shown in the figure above is

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The state variable description of an LTI system is given by

$y=\left(\begin{array}{ccc}1& 0& 0\end{array}\right)\left(\begin{array}{c}{x}_{1}\\ {x}_{2}\\ {x}_{3}\end{array}\right)$

where y is the output and u is the input. The system is controllable for

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The block diagram of a system with one input u and two outputs y1 and y2 is given below.

A state space model of the above system in terms of the state vector $\overline{)x}$ and the output vector $\overline{)y}={\left[\begin{array}{cc}{y}_{1}& {y}_{2}\end{array}\right]}^{T}$ is

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The signal flow graph of a system is shown below.

The state variable representation of the system can be

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The signal flow graph of a system is shown below.

The transfer function of the system is

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Consider the system $\frac{dx}{dt}=Ax+Bu$ with $A=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ and $B=\left[\begin{array}{c}p\\ q\end{array}\right]$ where p and q are arbitrary real numbers. Which of the following statements about the controllability of the system is true?

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A single flow graph of a system is given below

The set of equations that correspond to this signal flow graph is

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The state space representation of a separately excited DC servo motor dynamics is given as

$\left[\begin{array}{c}\frac{d\omega }{dt}\\ \frac{d{i}_{a}}{dt}\end{array}\right]=\left[\begin{array}{cc}-1& 1\\ -1& -10\end{array}\right]\left[\begin{array}{c}\omega \\ {i}_{a}\end{array}\right]+\left[\begin{array}{c}0\\ 10\end{array}\right]u$

where $\omega$ is the speed of the motor, ${i}_{a}$ is the armature current and u is the arnature voltage. The transfer function $\frac{\omega \left(s\right)}{U\left(s\right)}$ of the motor is

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Consider a linear system whose state space representation is $\stackrel{·}{\mathbf{x}}\left(t\right)=\mathbf{Ax}\left(t\right)$. If the initial state vector of the system is $\mathbf{x}\left(0\right)\mathbf{=}\left[\begin{array}{c}1\\ -2\end{array}\right]$, then system response is $\mathbf{x}\left(t\right)\mathbf{=}\left[\begin{array}{c}{e}^{\mathit{-}\mathit{2}\mathit{t}}\\ -2{\mathrm{e}}^{\mathit{-}\mathit{2}\mathrm{t}}\end{array}\right]$. If the initial vector of the system changes to $\mathbf{x}\left(0\right)\mathbf{=}\left[\begin{array}{c}1\\ -1\end{array}\right]$, then the system responce becomes $\mathbf{x}\left(t\right)\mathbf{=}\left[\begin{array}{c}{e}^{\mathit{-}\mathit{t}}\\ -{\mathrm{e}}^{\mathit{-}\mathrm{t}}\end{array}\right]$.

The eigenvalue and eigenvector pairs $\left({\lambda }_{i},{v}_{i}\right)$ for the system are

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Consider a linear system whose state space representation is $\stackrel{·}{\mathbf{x}}\left(t\right)=\mathbf{Ax}\left(t\right)$. If the initial state vector of the system is $\mathbf{x}\left(0\right)\mathbf{=}\left[\begin{array}{c}1\\ -2\end{array}\right]$, then system response is $\mathbf{x}\left(t\right)\mathbf{=}\left[\begin{array}{c}{e}^{\mathit{-}\mathit{2}\mathit{t}}\\ -2{\mathrm{e}}^{\mathit{-}\mathit{2}\mathrm{t}}\end{array}\right]$. If the initial vector of the system changes to $\mathbf{x}\left(0\right)\mathbf{=}\left[\begin{array}{c}1\\ -1\end{array}\right]$, then the system responce becomes $\mathbf{x}\left(t\right)\mathbf{=}\left[\begin{array}{c}{e}^{\mathit{-}\mathit{t}}\\ -{\mathrm{e}}^{\mathit{-}\mathrm{t}}\end{array}\right]$.

The system matrix A is