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QUESTION: 1

The state-space representation for a system is

The transfer function Y(s) /U(s) is

Solution:

Substituting the values,

QUESTION: 2

Determine the state-space representation for the transfer function given in question. Choose the state variable as follows

Q.

Solution:

(s^{3} + a_{2}s^{2} + a_{1}s + a_{0})C(s) = b_{0}R(s)

Taking the inverse Laplace transform assuming zero initial conditions

QUESTION: 3

Determine the state-space representation for the transfer function given in question. Choose the state variable as follows

Q.

Solution:

Fourth order hence four state variable

QUESTION: 4

A state-space representation of a system is given by

The time response of this system will be

Solution:

QUESTION: 5

Solution:

QUESTION: 6

Solution:

QUESTION: 7

Consider the system shown in fig.

The controllability matrix is

Solution:

QUESTION: 8

Consider the system shown in fig.

The observability matrix is

Solution:

QUESTION: 9

Consider the system shown in fig.

The system is

Solution:

det C_{M} = 0. Hence system is not controllable. det O_{M }= 1. Hence system is observable.

QUESTION: 10

Consider the system shown in fig.

Q. The controllability matrix for this system is

Solution:

QUESTION: 11

Consider the system shown in fig.

The observability matrix is

Solution:

QUESTION: 12

Consider the system shown in fig.

Q. The system is

Solution:

Since the determinant is not zero, the 3 x 3 matrix is nonsingular and system is controllable

The rank of O_{M} is 3. Hence system is observable.

QUESTION: 13

A state flow graph is shown in fig.

Q.The state and output equation for this system is

Solution:

QUESTION: 14

A state flow graph is shown in fig.

The system is

Solution:

det O_{M} = 0. Thus system is not observable

det C_{M} = -1. Thus system is controllable.

QUESTION: 15

Consider the network shown in fig. The state-space representation for this network is

Solution:

v_{c} and i_{L} are state variable.

Hence equation are

QUESTION: 16

For the network shown in fig. The output is

i_{R}(t). The state space representation is

Solution:

Hence v_{1} and i_{3} are state variable.

QUESTION: 17

Consider the network shown in fig. This system may be represented in state space representation

Q. The state variable may be

Solution:

Energy storage elements are capacitor and inductor. v_{C }and i_{L} are available in differential form and linearly independent. Hence v_{C }and i_{L} are suitable for state-variable.

QUESTION: 18

Consider the network shown in fig. This system may be represented in state space representation

Q. If state variable are chosen as in previous question, then the matrix A is

Solution:

.....(i)

.....(ii)

Solving equation (i) and (ii)

QUESTION: 19

Consider the network shown in fig. This system may be represented in state space representation

Q. The matrix B is

Solution:

QUESTION: 20

Consider the network shown in fig.

Q. The state variable may be

Solution:

There are three energy storage elements, hence 3 variable. i_{2} ,i_{4} and v_{o} are available in differentiated form hence these are state variable.

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