(b) Consider the system represented by

1 1 0 0
ॐ | 0 1 1 2 + | 1 | ७
0 0 -1 0

Check the controllability of the system and comment on the stabilizability of
the system using controllable decomposition procedure.

(a) What is the significance of a observability gramian matrix. Derive the expression
for the observability gramian matrix of a linear system.

(b) Comment on the controllability of the system, & = Ax + Bu where
1 0 ட்‌
ய
with 20 = ( ‏ب‎ ) and obtain the general solution.
20

PART C(Modules V and VI)

(a) Explain the different companion forms for MIMO systems.

(b) Consider the system 2: = 42 + Bu, y = Cx where,

0 0 -6 1
4 = | 1 0 -11 |, 851 0 |, C=(001)
0 1 -6 0
Design a reduced order observer so that the observer poles are at 8 = —2+ 6

(a) Derive the transfer function of a combined observer controller configuration.

(b) Consider the system & = Ax + Bu, y = Cx where,

(10), (൮).

Using transfer function approach design a full order observer-controller that
makes the estimation error to decay at least as fast as e~! and the closed loop
poles ats = - 1 + ¡1

(a) Explain in detail the optimality criteria for choosing observer poles.

(b) Given the system - Ar + Bu, ‏سے‎ Cx where

1 0 0 1 0
4 == | 0 -1 0 Jand 0ಎ | 0 1 (011)
0 0 2 0 0

Obtain the observable canonical form realization.

Page 2

(6)

(4)
(5)

(4)
(8)

(4)
(8)

(4)
(8)