**Direct Method & Constructing of Liapunov for the Linear & Non-Linear System**

**1. If the Liapunov’s function cannot be found then the system is:**

a) Stable

b) Unstable

c) Conditionally stable

d) Marginally stable

**Answer:** b

**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.

**2. Liapunov’s stability analysis is for the :**

a) LTI system

b) Time variant system

c) Non-linear system

d) Linear system

**Answer:** c

**Explanation:** Liapunov’s stability criterion is for the non-linear system.

**3. The system is unstable for:**

a) W (x)>0; x not equal to zero

b) W (0) =0

c) W (x) has continuous partial derivative with respect to all components of x

d) All of the mentioned

**Answer:** d

**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.

**4. Liapunov’s stability for non-linear system is same as the Routh Hurwitz criteria for the linear system.**

a) True

b) False

**Answer:** a

**Explanation:** Liapunov stability is similar in function as the Routh Hurwitz criteria and Silvester’s theorem can be used to prove it.

**5. Conditions of ___________ are necessary and sufficient condition for the asymptotic stability of the system.**

a) Linear system

b) Krasovskii’s method

c) positive definiteness

d) Variable gradient method

**Answer:** c

**Explanation:** For the system to be asymptotic stable which is desired for the Liapunov stability positive definiteness of the system must be fulfilled.

**6. It is difficult to form Liapunov’s function for:**

a) Linear system

b) Non-linear

c) Time variant systems

d) Time –invariant systems

**Answer:** b

**Explanation:** For non-linear system it is difficult to form the Liapunov’s function And we use various method to do this.

**7. If the V is positive definite, for the system to be asymptotically stable, Q should be negative definite.**

a) Krasovskii’s method

b) Variable gradient method

c) Constant method

d) Non-variable gradient method

**Answer:** a

**Explanation:** 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.

**8. The method which provides considerable flexibility in finding the Liapunov’s function is:**

a) Krasovskii’s method

b) Variable gradient method

c) Constant method

d) Non-variable gradient method

**Answer:** b

**Explanation:** Variable gradient method provides considerable flexibility in finding the Liapunov’s function as the quadratic form approach is too restrictive.

**9. Liapunov stability analysis is different from the classical theories approach of stability.**

a) True

b) False

**Answer:** a

**Explanation:** Liapunov’s stability theorem is applied for the non-linear systems but other stability theorems are applied for the linear systems.

**10. For non-linear systems the equation for damping factor as in linear system is called _________**

a) Krasovskii’s equation

b) Vander Pol’s equation

c) Constant method

d) Non-variable gradient equation

**Answer:** b

**Explanation:** For non-linear systems the equation for damping factor as in linear system is called Vander pol’s equation which depends upon x and damping factor has large positive values for this.