State Space Analysis & State Transition Matrix (STM) - Topic wise GATE Questions on Control Systems (from 1987) | GATE 2023 - Important Concepts, Short cuts, Complete Solutions, Examples and Practice Problems

2003
1. The zero input response of a system given by the state space equation

Solution : https://www.youtube.com/watch?v=BzTXlRJ5M9k

2004
2. The state variable equations of a system are:
ẋ₁ = -3x₁ -x₂ + u
ẋ₂ = 2x₁
y = x₁ + u
Then the system is
a) controllable but not observable
b) observable but not controllable
c) neither controllable nor observable
d) controllable and observable

Solution : https://www.youtube.com/watch?v=CVH0VsecVuQ

3. The state transition matrix e^At for the given matrix A, is

Solution : https://www.youtube.com/watch?v=w9zVGFsZ3Bw

2005
4. A linear system is equivalently represented by two sets of state equations
Ẋ = AX + BU and Ẇ = CW + DU.
The eigen values of the representations are also computed as |λ| and |μ|. Which one of the following statements is TRUE?

Answer :
Solution :

2006
5. A linear system is described by the following state equation, then the state transition matrix of the system is

Solution : https://www.youtube.com/watch?v=SdKm17kCbFg

2007
6. The state space representation of a separately excited DC servo motor dynamics is given as
where ω is the speed of the motor, i_a is the armature current and u is the armature voltage. The transfer function of the motor is

Solution : https://www.youtube.com/watch?v=jTt-sWQZS0Y
Statement for Linked Answer Questions 7 & 8:

7. The eigen value and eigen vector pairs (λ_i, v_i) for the system are

Solution :
8. The system matrix A is

Solution :

2008
9. A signal flow graph of a system is given below.

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

Solution : https://www.youtube.com/watch?v=cxlX8Rpj9cs

2009
10. consider the system,

where p and q are arbitrary real numbers, which one of the following statements about the controllability of the system is TRUE?
a) the system is completely state controllable for any non zero values of p and q
b) only p=0 and q=0 result in controllability
c) the system is uncontrollable for all values of p and q
d) we cannot conclude about controllability from the given data

Solution : https://www.youtube.com/watch?v=MvBkDRPjIcY

2010
Common Data Questions:(11 and 12)
The signal flow graph of a system is shown below.

11. The state variable representation of the system can be

Solution : https://www.youtube.com/watch?v=-4PqIhkiPD0
12. The transfer function of the system is

Solution : https://www.youtube.com/watch?v=kT5TsnSEl9g

2011
13. 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 system in terms of the state vector ẋ and the vector ẏ = [y₁ y₂]^T is

Solution : https://www.youtube.com/watch?v=CsciojvPQF8

2012
14. The state variable description of an LTI system is by

where y is the output and u is the input. The system is controllable for
a) a₁≠ 0, a₂ = 0, a₃ = 0
b) a₁ = 0, a₂≠ 0, a₃≠ 0
c) a₁ = 0, a₂≠ 0, a₃ = 0
d) a₁≠ 0, a₂≠ 0, a₃ = 0

Solution : https://www.youtube.com/watch?v=dPCv96oD6Ow

2013
15. The signal flow graph for a system is given below.

The transfer function Y(s)/U(s) for this system is

Solution : https://www.youtube.com/watch?v=ndeVYkf7j6w

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

a. State variable equations of the system shown in the above figure is

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

Solution (a & b) : https://www.youtube.com/watch?v=wnfqGfnPFcs

2014
17. A system is described by the following differential equation, where u(t) is the input to the system and y(t) is the output of the system.

When y(0) = 1 and u(t) is a unit step function, y(t) is

Solution : https://www.youtube.com/watch?v=6tsdXBAfClY

18. consider the state space model of a system, as given below.

The given system is
a) controllable and observable
b) uncontrollable and observable
c) uncontrollable and unobservable
d) controllable and unobservable

Solution : https://www.youtube.com/watch?v=MqFMHFSyGgk

19. An unforced linear time invariant (LTI) system is represented by

If the initial conditions are x1(0) = 1 and x2(0) = - 1, then the solution of the state equation is

Solution : https://www.youtube.com/watch?v=PW4_N3VeGPs

20. Consider the state space system expressed by the signal flow diagram shown in the figure.

The corresponding system is
a) always controllable
b) always observable
c) always stable
d) always unstable

Solution : https://www.youtube.com/watch?v=kx74sDKotOs

21. The state equation of a second order linear system is given by

Solution : https://www.youtube.com/watch?v=2Yx8ux4wY4U

22. The state transition matrix Φ(t) of a system,

Solution : https://www.youtube.com/watch?v=tTLuvoMF_Jc

2015
23. The state variable representation of a system is given as

The response y(t) is
a) sin (t)
b) 1 – e^t
c) 1 – cos (t)
d) 0

Solution : https://www.youtube.com/watch?v=Ax6VxSUUJtA

24. A network is described by the state model as

then the transfer function H(s) = (Y(s)/U(s)) is

Solution : https://www.youtube.com/watch?v=z5t9oMUeA-U

State Space Analysis & State Transition Matrix (STM) - Topic wise GATE Questions on Control Systems (from 1987)

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