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Correct OPTION is (C) System behavior for small deviations about the equilibrium point MAY be DIFFERENT from the large DEVIATION

Best explanation: For the non-linear system where the input parameters or external input is having non-linear relationship with the output and behavior for small deviations about the equilibrium point may be different from the large deviation.

The CORRECT answer is (C) ENERGY is non-negative and decreases as t increases

Easiest explanation: Energy is defined as the work done or force which produces CERTAIN displacement and the RESULTS for the energy are that energy of the system in non-negative and it decreases as t increases.

Correct choice is (c) STABLE

The best explanation: By definition, the system is stable at ORIGIN if for every INITIAL state which is sufficiently CLOSE to the origin REMAINS near the origin for all t then the system is stable.

Right answer is (B) Non-linear

The explanation: For non-linear SYSTEM it is difficult to form the Liapunov’s function

And we USE various METHOD to do this.

Right option is (d) All of the mentioned

Explanation: It requires MUCH INGENUITY of devise a SUITABLE W function to devise a Liapunov function V. In stability analysis of NONLINEAR systems, it is valuable to establish CONDITIONS for which the system is unstable.

The correct CHOICE is (d) Both a and b

Explanation: The LINEAR autonomous SYSTEMS having the differential terms with the linear relationship and have one equilibrium state and their behavior DETERMINES the QUALITATIVE behavior of the s-plane.

The correct choice is (a) True

To explain I WOULD say: Liapunov stability is similar in function as the ROUTH HURWITZ criteria and Silvester’s theorem can be USED to prove it.

Right answer is (a) The REGION S (e) is SMALL

Explanation: By the definition of Liapunov’s stability CRITERIA a system is locally STABLE if the region of system is very small.

The correct choice is (a) ELLIPSE

To explain: For the RESULTS of the LIAPUNOV energy the loci is TRACED by the Liapunov energy DESCRIPTION and is a ellipse.

The CORRECT choice is (B) Asymptotically STABLE in the LARGE

The explanation is: For the good control system system must be stable and if the system is asymptotically stable irrespective that how close or FAR it is from the origin then the system is asymptotically stable in large.

The correct option is (d) It is unstable

To ELABORATE: By the DEFINITION of Liapunov’s stability criteria the system is asymptotically STABLE at the origin if there exist a real number >0 such that || X (t0) || <=r and every initial state x (t0) results in x (t) TENDS to zero as t tends to infinity .

Right choice is (a) Krasovskii’s method

The explanation is: If the V is positive definite, for the system to be asymptotically stable, Q should be negative definite. For Krasovskii’s method and in addition V should TEND to infinity and the system is asymptotically stable in-the large.

The correct choice is (a) True

The EXPLANATION is: Liapunov’s stability THEOREM is APPLIED for the non-linear SYSTEMS but other stability theorems are applied for the linear systems.

The CORRECT answer is (b) Variable gradient METHOD

For explanation I would say: Variable gradient method provides considerable flexibility in finding the LIAPUNOV’s function as the quadratic form APPROACH is too restrictive.

Correct ANSWER is (d) All of the mentioned

To elaborate: Liapunov has two METHODS as direct and indirect and the direct METHOD of Liapunov is using concept of energy, relation of STORED energy and using the equation of the autonomous systems.

Correct OPTION is (B) Unstable

Easy explanation: It is very DIFFICULT to find the Liapunov’s function and SEVERAL techniques are devised for the same and if the Liapunov’s function is not available then the SYSTEM is unstable.

The correct choice is (c) Autonomous systems

To elaborate: Autonomous systems are REPRESENTED by the DIFFERENTIAL equations as stated above and these systems are tested by LIAPUNOV’s stability CRITERIA.

The correct choice is (a) The resulting trajectory tends towards the equilibrium STATE.

Best explanation: A system with zero input is the system in which the input is zero and INITIAL states are taken as the input and a FREE system but it is stable only if the resulting trajectory tends TOWARD the equilibrium state.

Right option is (d) Both a and c

Explanation: By the definition of LIAPUNOV’s stability criteria the SYSTEM is asymptotically STABLE in the large at the origin if there exist a real number >0 such that || x (t0) || <=r and every initial state x (t0) results in x (t) TENDS to zero as t tends to infinity.

Right OPTION is (a) True

To ELABORATE: The stability of the liapunov function can be DETERMINED with the HELP of the scalar functions of a system state.