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

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

2004
2. The state variable equations of a system are:
1 = -3x1 -x2 + u
2 = 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

3. The state transition matrix eAt for the given matrix A, is

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?
Solution :

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

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, ia is the armature current and u is the armature voltage. The transfer function of the motor is

7. The eigen value and eigen vector pairs (λi, vi) 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

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

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

12. The transfer function of the system is

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

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

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

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

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 eAt 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

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

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

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

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

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

2015
23. The state variable representation of a system is given as
The response y(t) is
a) sin (t)
b) 1 – et
c) 1 – cos (t)
d) 0

24. A network is described by the state model as
then the transfer function H(s) = (Y(s)/U(s)) is

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