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Test: State Space Analysis - Electrical Engineering (EE) MCQ


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10 Questions MCQ Test GATE Electrical Engineering (EE) Mock Test Series 2025 - Test: State Space Analysis

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Test: State Space Analysis - Question 1

Which of the following properties are associated with the state transition matrix ϕ(t)?
1. ϕ(0) = I
2. ϕ(t2 - t1) = ϕ(t2) . ϕ-1(t1)
3. ϕ(t2 + t1) = ϕ(t2) . ϕ(t1)
4. ϕ(t2 - t1) = ϕ(t2 - t0). ϕ(t0 - t1)
Select the correct answer using  the codes given below:

Detailed Solution for Test: State Space Analysis - Question 1

The state transition matrix ϕ(t):
The state-transition matrix is defined as a matrix that satisfies the linear homogeneous state equation.
It represents the free response of the system.
The state-transition matrix ϕ(t) completely defines the transition of the states from the initial time t = 0 to any time t when the inputs are zero.
The state transition matrix is given by 
ϕ(t) = L-1 [sI - A]-1 = eAt
Where A = state matrix
The state-transition matrix is dependent only upon the matrix A and, therefore, is sometimes referred to as the state transition matrix of A.
Properties of ϕ(t):

  1. ​ϕ(0) = I
  2. ϕ-1(t) = ϕ(-t)
  3. ϕ(t2 - t1) = ϕ(t2 - t0). ϕ(t0 - t1)
  4. ϕ(t2 + t1) = ϕ(t2) . ϕ(t1)
  5. [ϕ(t)]k = ϕ(kt)
  6. At t =0, dϕ / dt = A
     

Application:
For statement 2,
ϕ(t2 - t1) = ϕ(t2) . ϕ-1(t1)
consider
ϕ(t2 - t1) = ϕ (t2 + (-t1)) 
By property 4
⇒ ϕ(t2 - t1) = ϕ (t2 + (-t1)) = ϕ(t2) . ϕ(-t1)
By property 2
⇒  ϕ(t2 - t1) = ϕ(t2) . ϕ(-t1) =  ϕ(t2) . ϕ-1(t1)

Test: State Space Analysis - Question 2

Read the statements regarding controllability and observability and mark the answer as true and false.

  • For a system to be controllable, |Qc| = 0.
  • For the system to be observable, |Q0| = 0.
  • For the system to be controllable and observable, |Qc| ≠ 0 and |Qo| ≠ 0.

Detailed Solution for Test: State Space Analysis - Question 2

The correct answer is option '2'
Concept:
Controllability:
A system is said to be controllable if it is possible to transfer the system state from any initial state x(t0) to any desired state x(t) in a specified finite time interval by a control vector u(t)
Kalman’s test for controllability:
ẋ = Ax + Bu
Qc = {B AB A2B … An-1 B]
Qc = controllability matrix
If |Qc| = 0, system is not controllable
If |Qc|≠ 0, system is controllable
Observability:
A system is said to be observable if every state x(t0) can be completely identified by measurement of output y(t) over a finite time interval.
Kalman’s test for observability:
Q0 = [CT ATCT (AT)2CT …. (AT)n-1 CT]
Q0 = observability testing matrix
If |Q0| = 0, system is not observable
If |Q0| ≠ 0, system is observable.

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Test: State Space Analysis - Question 3

The transfer function G(S) = C(SI - A)-1b of the system
x' = Ax + bu
y = Cx + du
has no pole-zero cancellation. The system

Detailed Solution for Test: State Space Analysis - Question 3

State space representation:
ẋ(t) = A(t)x(t) + B(t)u(t)
y(t) = C(t)x(t) + D(t)u(t)
y(t) is output
u(t) is input
x(t) is a state vector
A is a system matrix
This representation is continuous time-variant.
Controllability:
A system is said to be controllable if it is possible to transfer the system state from any initial state x(t0) to any desired state x(t) in a specified finite time interval by a control vector u(t)
Kalman’s test for controllability:
ẋ = Ax + Bu
Qc = {B AB A2B … An-1 B]
Qc = controllability matrix
If |Qc| = 0, system is not controllable
If |Qc| ≠ 0, the system is controllable
Observability:
A system is said to be observable if every state x(t0) can be completely identified by measurement of output y(t) over a finite time interval.
Kalman’s test for observability:
Q0 = [CT ATCT (AT)2CT …. (AT)n-1 CT]
Q0 = observability testing matrix
If |Q0| = 0, system is not observable
If |Q0| ≠ 0, system is observable.
Duality property of controllability and observability:

  • If the pair (A, B) is controllable then the pair (AT, BT) is observable.
  • If the pair of (A, C) is observable then the pair of (AT, CT) is controllable.

If there are no pole-zero cancellations in the transfer function then the system is completely controllable and observable.

Test: State Space Analysis - Question 4

Which of the following can be extended to time-varying systems?

Detailed Solution for Test: State Space Analysis - Question 4

The state-space analysis is a very useful technique of analyzing the control system, it is based on the concept of space and applies to the LTI system as well in non – linearly time-varying system & MIMO system.
It can be used for both continuous-time as well as discrete-time systems.


Where, x1, x2, x3 …. xn are state variables
A is state matrix
B is the input matrix
Output equation: y(t) = CX(t)+DU(t) 

Test: State Space Analysis - Question 5

Consider a system governed by the following equations

The initial conditions are such that  and  Which one of the following is true?

Detailed Solution for Test: State Space Analysis - Question 5





Test: State Space Analysis - Question 6

State variable description of an LTI system is given by


Where Y is the output and u is input. System is controllable for

Detailed Solution for Test: State Space Analysis - Question 6


System is controllable if |QC| ≠ 0
a1a2 (0 – a2) ≠ 0

Hence condition for controllability is
a1 ≠ 0, a2 ≠ 0, a3 = 0

Test: State Space Analysis - Question 7

The state transition matrix of a control system is . The system matrix A is

Detailed Solution for Test: State Space Analysis - Question 7

Concept:
The state transition matrix ϕ(t):
The state-transition matrix is defined as a matrix that satisfies the linear homogeneous state equation.
It represents the free response of the system.
The state-transition matrix ϕ(t) completely defines the transition of the states from the initial time t = 0 to any time t when the inputs are zero.
The state transition matrix is given by 
ϕ(t) = L-1 [sI - A]-1 = eAt
Where A = state matrix
The state-transition matrix is dependent only upon the matrix A and, therefore, is sometimes referred to as the state transition matrix of A.
Properties of ϕ(t):

Calculation:
Given that
State transition matrix ϕ(t) = 
Consider the sixth property of the state transition matrix
⇒ At t = 0, dϕ /dt = A
Where A is system or state matrix,

Test: State Space Analysis - Question 8

Consider the following properties attributed to state model of a system:

  1. State model is unique.
  2. Transfer function for the system is unique.
  3. State model can be derived from transfer function of the system.

Which of the above statements are correct?

Detailed Solution for Test: State Space Analysis - Question 8
  • In the transfer function model, initial conditions are assumed to be zero. In the state-space model, transfer function is not the starting point for the system analysis, and it considers the initial conditions of the system.
  • The transfer function for the system is unique.
  • The transfer function is applicable only for linear time-invariant systems. The state-space model is applicable to linear time-variant systems also.
  • State-space representation of a system is not unique, it may have more than one representation.
  • State model can be derived from transfer function of the system.
Test: State Space Analysis - Question 9

The linear time invariant system is represented by the state space model as

Consider n=number of state variables, m = number of inputs, p= number of outputs. The state transition matrix Φ( t) Φ( t) is given by:

Detailed Solution for Test: State Space Analysis - Question 9

Concept:
The state transition matrix [ϕ(t)] is given by:
ϕ(t) = eAt = L-1 [(SI-A)]-1
where, A = System matrix
I = Identity matrix
Properties of state transition matrix:
(1) State transition matrix at t = 0 is always equal to the identity matrix. 
ϕ(0) = eA0 = I
(2) The differentiation of the state transition matrix at t = 0 is always equal to its system matrix.

Test: State Space Analysis - Question 10

A system is represented by  what is the transfer function to the system?

Detailed Solution for Test: State Space Analysis - Question 10

Concept:
A transfer function (TF) is defined as the ratio of Laplace transform of the output to the Laplace transform of the input by assuming initial conditions are zero.
TF = L[output] / L[input]

For unit impulse input i.e. r(t) = δ(t)
⇒ R(s) = δ(s) = 1
Now transfer function = C(s)
Therefore, transfer function is also known as impulse response of the system.
Transfer function = L[IR]
IR = L-1 [TF]
Calculation:
Given differential equation is,

Laplace transform of the above equation is given by

⇒ 3s Y(s) + 2 Y(s) = U(s)
⇒ Y(s) (3s + 2) = U(s)
∴ Transfer function is given by

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