GATE EC|| CONTROL SYSTEM || STATE SPACE ANALYSIS|| PYQS(2000-2025)

The state transition matrix e^At of the system shown in figure above is:

(GATE 2013 || EC || MCQ||2 MARK)

Consider a system S represented in state space as

[Tex]\frac{dx}{dt} = \begin{bmatrix} 0 & -2 \\ 1 & -3 \end{bmatrix} x + \begin{bmatrix} 1 \\ 0 \end{bmatrix} r[/Tex]

Which of the state space representations given below has/have the same transfer function as that of S ?

( GATE 2024 || EC || MSQ||2 MARK)

Consider a system where x1(t),x2(t)x1​(t),x2​(t), and x3(t)x3​(t) are three internal state signals and u(t)u(t) is the input signal. The differential equations governing the system are given by
[Tex]\frac{d}{dt} \begin{bmatrix} x_1(t) \\ x_2(t) \\ x_3(t) \end{bmatrix} = \begin{bmatrix} 2 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} x_1(t) \\ x_2(t) \\ x_3(t) \end{bmatrix} + \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} u(t)[/Tex]

Which of the following statements is/are TRUE?

(GATE 2025 || EC || MCQ||2 MARK)

A network is described by the state model as

X₁ = 2x₁ – x₂ + 3u,

X₂ = – 4x₂ – u,

Y = 3x₁ – 2x₂

The transfer function H(s) = [Tex]\left( \frac{Y(s)}{U(s)} \right)[/Tex]

is:

(GATE 2015 || EC || MCQ||2 MARK)

A second-order linear time-invariant system is described by the following state equations

 [Tex]\frac{d}{dt}[/Tex]

x₁ (t)+ 2x₁ (t) = 3u(t), 

[Tex]\frac{d}{dt}[/Tex]

x₁ (t)+ x₂(t) = u(t); 

where x₁(t) and x₂(t) are the two state variables and u(t) denotes the input if the output c(t) = x₁(t), then the system is 

( GATE 2016 || EC || MCQ||2 MARK)

Consider the state space realization:

[Tex]\begin{bmatrix} x'_1(t) \\ x'_2(t) \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & -9 \end{bmatrix} \begin{bmatrix} x_1(t) \\ x_2(t) \end{bmatrix} + \begin{bmatrix} 0 \\ 45 \end{bmatrix} u(t),[/Tex]

with the initial condition

[Tex]\begin{bmatrix} x_1(0) \\ x_2(0) \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix};[/Tex]

where u(t) denotes the unit step function the value of

[Tex]\lim_{x \to \infty} |\sqrt{x_1^2(t) + x_2^2(t)} [/Tex] is

____________

( GATE 2017 || EC || NAT||2 MARK)

A second-order LTI system is described by the following state equations:

[Tex]\frac{d}{dt} x_1(t) - x_2(t) = 0[/Tex]

[Tex]\frac{d}{dt} x_2(t) + 2x_1(t) + 3x_2(t) = r(t);[/Tex]

Where x₁(t) and x₂(t) are two state variables and r(t) denotes the input. The input c(t) = x₁(t).

The system is

( GATE 2017 || EC || MCQ||2 MARK)

The state equation and the output equation of a control system are given below:

[Tex]\mathbf{x} = \begin{bmatrix} -4 & -1.5 \\ 4 & 0 \end{bmatrix} \mathbf{x} + \begin{bmatrix} 2 \\ 0 \end{bmatrix} u[/Tex]

[Tex]Y = \begin{bmatrix} 1.5 & 0.625 \end{bmatrix} \mathbf{x}[/Tex]

The transfer function representation of the system is:

( GATE 2018 || EC || MCQ||2 MARK)

Let the state-space representation of an LT1 system be x’(t) = Ax (t) + Bu(t), 

y(t) = Cx(t) + du (t) where A, B, C are matrices, d is a scalar, u(t) is the input to the system, and y (t) is its output. Let B = [0 0 1]^T and d = 0. Which one of the following options for A and C will ensure that the transfer function of this LTI system is?

[Tex]H(s) = \frac{1}{s^3 + 3s^2 + 2s + 1}?[/Tex]

(GATE 2019 || EC || MCQ||2 MARK)

The electrical system shown in the figure converts input source current i_S(t) to output voltage v_o(t)

Current i_L(t) in the inductor and voltage v_c(t) across the capacitor is taken as the state variables, both assumed to be initially equal to zero, i.e., i_L(0)  = 0 and v₀(0) = 0 The system is

( GATE 2021 || EC || MCQ||2 MARK)