Previous Year Questions- State Variable Analysis - 2 - Control Systems - Electrical

Q16: A system is described by the following state and output equations
 when u(t) is the input and y(t) is the output
The system transfer function is (2009)
(a)
(b)
(c)
(d)
Ans: (c)
Sol: Selecting x₁(t) and x₂(t) as state variable,

Q17: The state space equation of a system is described by
where x is state vector, u is input, y is output and C = [1 0]
A unity feedback is provided to the above system G(s) to make it a closed loop system as shown in figure.
For a unit step input r(t), the steady state error in the input will be (2008)
(a) 0
(b) 1
(c) 2
(d) ∞
Ans: (a)
Sol: Steady state error, using final value theorem

Q18: The state space equation of a system is described by
where x is state vector, u is input, y is output and C = [1 0]
The transfer function G(s) of this system will be (2008)
(a)
(b)
(c)
(d)
 Ans: (d)
Sol: 
Q19: The state equation for the current I₁ in the network shown below in terms of the voltage V_x and the independent source V, is given by  (2007)
(a)
(b)
(c)
(d)
Ans: (a)
Sol: Any state equation represents the dynamical behaviour of the given network. State equations usually follow a specific 'format' while beong represented. On the left side of each state equation, the derivative of only one variable is used. On the right hand side a mathematical function is represnted involving any or all the state varibles and the sources.
Using KVL in Loop-I
Using KVL in Loop-II,
Eliminating I₂ from equation (i) and (ii), we get,

Q20: For a system with the transfer function  the matrix A in the state space form X = AX + Bu is equal to (2006)
(a) (b) (c) (d) Ans: (b)
Sol: Replacing s by d/dt,

Q21: A state variable system
 with the initial condition  and the unit step input u(t) has the state transition equation  (2005)
(a)
(b)
(c)
(d)
Ans: (c)
Sol: ZIR (zero Input Response) = ϕ(t) × X(0)
ZSR (Zero State Response)
∴ State transition equation

Q22: A state variable system
with the initial condition  and the unit step input u(t) has
The state transition matrix  (2005)
(a)
(b)
(c)
(d)
Ans: (a)
Sol: Comparing equation (i) and (ii), we get

Q23: The state variable description of a linear autonomous system is X = AX,
where X is the two dimensional state vector and A is the system matrix given by The roots of the characteristic equation are  (2004)
(a) -2 and +2
(b) -j2 and +j2
(c) -2 and -2
(d) +2 and -2
Ans: (a)
Sol: 
System matrix  Characteristic equation ⇒ ∣sI −A∣ = 0
Roots of the characteristic equation are -2 and +2.

Q24: The following equation defines a separately excited dc motor in the form of a differential equation
 The above equation may be organized in the state-space form as follows
 where the P matrix is given by  (2003)
(a)
(b)
(c)
(d)
Ans: (a)
Sol: 
Q25: A second order system starts with an initial condition of  without any external input. The state transition matrix for the system is given by The state of the system at the end of 1 second is given by  (2003)
(a)
(b)
(c)
(d)
Ans: (a)
Sol: State transition matrix,
Initial conditions,
Zero input response is given by
State of the system at t = 1s

Q26: For the system  with u as unit impulse and with zero initial state, the output, y, becomes   (2002)
(a) 2e^2t
(b) 4e^2t
(c) 2e^4t
(d) 4e^4t
Ans: (b)
Sol: 
Q27: For the system  which of the following statements is true?  (2002)
(a) The system is controllable but unstable
(b) The system is uncontrollable but unstable
(c) The system is controllable but stable
(d) The system is uncontrollable but stable
Ans: (b)
Sol: ⇒ Uncontrollable.
Characteristic equation:
⇒ eigen values, s = 3.5 ± j0.866
i.e. roots lies on right side of s-plane .
⇒ unstable.

Q28: The state transition matrix for the system  with initial state X(0) is  (2002)
(a) $(sI-A)-1$(sI − A)⁻¹
(b) e^At X(0)
(c) Laplace inverse of [(sI − A)⁻¹]
(d) Laplace inverse of [(sI − A)⁻¹ X(0)]
Ans: (c)
Sol: e^At = L⁻¹[sI−A]⁻¹ 

Q29: Given the homogeneous state-space equaion   the steady state value x_ss = lim⁡_t→∞x(t), given the initial state value of x(t) = [10 − 10]^T, is  (2001)
(a)
(b)
(c)
(d)
Ans: (a)
Sol: