PART B

Explain
(a) Chetaev's Instability theorem. (3)
(0) Aizermann's and Kalman's conjecture. (3)

(௦) Kalman Yakubovich Popov Lemma. . (3)

Definc stability in the sense of Lyapunov. State and prove Lyapunov's theorem on (9)
stability.

(a) Find the sector [a, (31 for which the system with feedback nonlinearity is abso- (6)

lutely stable using Popov criterion. The forward transfer function of the system is
5
G(s) = =

(82-8-1)

(0) State the conditions to be satisfied by a transfer function matrix to be strictly (3)
positive real.

PART C

(a) Explain in detail the concept gain scheduling and the steps involved in the (4)
development of a gain scheduled tracking controller for nonlinear systems.

(b) Consider the system (8)
മുട 2122

22 = ‏إنه‎ + ४
Design a feedback control and a change of variable that linearize the system and
place the poles at —2 + jl
(a) Explain diffeomorphism. (3)
(b) Consider the system (9)

= p*2 _1

22 = ते| + ७
Is this system feedback linearizable? If yes, find a feedback control lawt.hat linearize
the state equation.

9. (a) Explain the design procedure of a backstepping controller for a nonlinear system. (5)

(b) Given the system (7)
21 = 22 + 6x?

£2 = 23
£3 == ४
where 9 € |--| ||, Design a backstepping controller.

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