# UPTU Old Year Question Papers

# B Tech 8th Semester

# Modern Control System 2006-07

**Note : Attempt all questions.**

**1. Attempt any two parts of the following :**

(a) Derive the state space representation of the speed control system shown in Fig. 1(a) Assume torque constant as Kj, and back emf constant as K_{b}.

(b) Consider the state space model of a system with

A =

0 | 1 | 0 | 0 | |

0 | -1 | 1 | , B = | 0 |

0 | -1 | -10 | 10 |

C = [l 0 0] Obtain characteristic polynomial and transfer function of the system.

(c) Consider the system

0 | 0 | -2 | 0 | |||

X = | 0 | 1 | 0 | X | x(0) = | 1 |

1 | 0 | 3 | 5 | 0 |

Obtain the free response of the system.

**2. Attempt any two of the following**

(a) Consider the difference equation y{k + 2) + a_{x}y{k + 1) + a_{2}y(k) =

b_{Q}r(k + 2) + b-f{k + 1) + b_{2}r(k)

(b) Assuming that the system is initially at rest and r(k) = 0 for k < 0 obtain the transfer function G(z) = Y(z)/K(z)

(c) The state variable model of a plant is given by x = Ax + Bu y = Cx

where A =

fo 1 1 | i©
i |
||

1©
1 cn i |
IIcq | ||

1h-1
1 |

C = [l 0]

Obtain its discrete time state model for T = 0.1 sec. Obtain the pulse transfer function of the system shown in Fig. 2(a) for a sampling period T = 1 sec.

**3. Attempt any four parts of the following :**

(a) Consider the system

0 | 0 | -1 | 1 | ||

X = | 1 | 0 | -2 | x + | 2 |

0 | 1 | -2 | 0 |

(b)

u

y = [0 0 l]x

(b) Design an observer that has eigen values at s = -4 – 5 and -6. Show that the following system is uncontrollable :

u

Xj | -0.5 | 0 | Xj | + | 0 | |

x_{2} |
0 | -2 | x_{2} |
1 |

y = [o i]

X-,

Xn

(c) Is the following system observable ?

Xj | 11
h-1 o 1 |
Xj | 10
1 |
||

X_{2} |
0 -2 | X_{2} |
1h-1
1 |

u X-, Xa

(d) Explain the procedure for designing state observer. Use Liapunov’s method to find conditions for the stability of linear system described by the state matrix

-a P-P -a

(e) Distinguish between state and output controllability and give method to test it.

**4. Attempt any four parts of the following :**

(a) The system x = —x + u is to be transferred from lj. Ox(0) = 5 to x(l) = 0 such that J = — I (u) dt is minimized. Find the optimal control. ^{
}

(b) Find the optimal control for the system

0 | 1 | 0 | |||

X = | -10 | 0 | x + | 10 | u |

which minimizes the performance index

1 | 0 | ||

given x(0) = | 1 | , X£ — | 0 |

(c) State the two point boundary value problem and give its solution in terms of Euler-Lagrange equation.

(d) How will you formulate an optimal control problem ? What are standard optimal control problems ?

(e) State and explain Hamilton Jacobi equation.

(f) Explain :

(1) Linear quadratic problem

(2) Constrained optimization

** **

**5. Attempt any four parts of the following :**

(a) What is fuzzy logic based control system ? Explain with example.

(b) Define the term membership function and fuzzy PI controller.

(c) What is the need for an adaptive control system ?

(d) Compare MRAC with self tuning regulator.

(e) List some of the advantages and disadvantages of sampled data control system ?

(f) Give the various controller structures used in adaptive control system.