**State Variable Analysis – Part II**

**1. State variable analysis has several advantages overall transfer function as:**

a) It is applicable for linear and non-linear and variant and time-invariant system

b) Analysis of MIMO system

c) It takes initial conditions of the system into account

d) All of the mentioned

**Answer:** d

**Explanation:** State variable analysis has several advantages overall transfer function as it is applicable for linear and non-linear and variant and time-invariant system, analysis of MIMO system, it takes initial conditions of the system into account.

**2. The minimum number of states require to describe the two degree differential equation:**

a) 1

b) 2

c) 3

d) 4

**Answer:** b

**Explanation:** The number of states required to describe a network is equal to the number of energy storing elements in the electrical network.

**3. For a system with the transfer function H(s) = 3(s-2)/s ^{3}+4s^{2}-2s+1 , the matrix A in the state space form is equal to:**

**Answer:** b

**Explanation:** Taking the inverse Laplace of the equation and converting it into differential equation and solving the state space by considering various matrices.

**4. The transfer function Y(s)/U(s) of a system described by the state equations dx/dt=-2x+2u and y(t) = 0.5x is:**

a) 0.5/(s-2)

b) 1/(s-2)

c) 0.5/(s+2)

d) 1/(s+2)

**Answer:** d

**Explanation:** Taking Laplace transform of the equation X(s) = 2U(s)/(s+2)

Then Y(s)/U(s) =1/(s+2).

**5. A linear time invariant single input single output system has the state space model given by dx/dt=Fx+Gu, y=Hz, where**

** **

**Here, x is the state vector, u is the input, and y is the output. The damping ratio of the system is:**

a) 0.25

b) 0.5

c) 1

d) 2

**Answer:** b

**Explanation:** T(s) = 1/(s^{2}+2s+4)

G = 0.5.

**6. Given the matrix**

**the Eigen value are___________**

a) 1,2,3

b) 1

c) -1,-2,-3

d) 0

**Answer:** c

**Explanation:** Solving the characteristic equation|SI-A| =0.

**7. A transfer function of control system does not have pole-zero cancellation. Which one of the following statements is true?**

a) System is neither controllable nor observable

b) System is completely controllable and observable

c) System is observable but uncontrollable

d) System is controllable but unobservable

**Answer:** b

**Explanation:** If the input-output transfer function of a linear time-invariant system has pole-zero cancellation, the system will be neither controllable nor observable.

**8. The analysis of multiple input multiple output is conveniently studied by;**

a) State space analysis

b) Root locus approach

c) Characteristic equation approach

d) Nicholas chart

**Answer:** a

**Explanation:** State space analysis is different from the transfer function approach as this can be used for the analysis of multiple input and multiple output system.

**9. The state equation in the phase canonical form can be obtained from the transfer function by:**

a) Cascaded decomposition

b) Direct decomposition

c) Inverse decomposition

d) Parallel decomposition

**Answer:** d

**Explanation:** The state equation from transfer function by parallel decomposition in the phase canonical form.

**10. A logarithmic spiral extending out of the singular point is__________**

a) Stable

b) Unstable focus

c) Conditionally stable

d) Marginally stable

**Answer:** b

**Explanation:** Unstable focus is a logarithmic spiral extending out of the singular point.

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