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QUESTION: 1

The state variable description of an autonomous system is, X = AX where X is a two-dimensional vector and A is a matrix given by

The eigen values of A are

Solution:

Eigen values of A are given by

or, (s - σ)^{2} + ω^{2} = 0

or, s = σ ± jω

or, s = σ + jω and s = σ - jω

QUESTION: 2

The system equations are given by

y(t) = [1 0]x(t)

The transfer function of the above system is

Solution:

Given,

Transfer function of the given system is

Now,

∴

= s^{2} + 3s + 2

Now,

Now,

QUESTION: 3

Consider the following statements related to state space analysis of control systems:

1. The zeros of the system can be obtained from eigen value of the system matrix.

2. A system is said to be observable if every state x_{0} can be exactly determined from the measurement of the output ‘y’ over a finite interval of time 0 ≤ t ≤ t_{f}.

3. The process by which transfer function changes to state diagram or state equations is called decomposition of the transfer function.

4. The state space techniques can be applied to linear and time invariant systems only.

Which of the above statements are correct?

Solution:

The poles of the system can be obtained from eigen values of the system matrix. Hence, statement-1 is false.

State space techniques can be applied to linear or non-linear, time variant or time invariant systems.

Hence, statement-4 is false.

QUESTION: 4

Consider the system shown in figure below:

The system is

Solution:

From given block diagram, the state equations can be written as:

x_{1} = -x_{1} + u and x_{2} = -2x_{2} + 2u

in matrix form,

Also, y = x_{1} + x_{2}

In matrix form,

Thus,

Since, |Q_{c}| ≠ 0 and |Q_{0}| ≠ 0, therefore given system is both controllable and observable.

QUESTION: 5

Consider the following matrix:

x(t) is given by

Solution:

Given,

So, x(t) = ϕ(t).x(0)

= state transition equation

QUESTION: 6

The transfer function of the system shown below is

Solution:

From given block diagram, the state equations can be written as:

Also, output equation is

y = 2x_{1} + x_{2}

In matrix form, we have:

and y = [2 1]x(t)

Now,

∴ Transfer function,

QUESTION: 7

The state equation for the circuit shown below is

Solution:

Let us select the state variables as V_{c} and i_{L}.

Applying KVL in the mesh-2, we have:

or,

.............(1)

Also, by applying KCL at the given node, we get:

or,

..........(2)

From equations (1) and (2), state equations in matrix form can be written as:

QUESTION: 8

The state space representation of the system represented by the SFG shown below is

Solution:

The state equations from the given signal flow graph can be written as:

In matrix form,

Also, output is

y(t) = 6x_{1} + x_{2
In matrix form
}

QUESTION: 9

The state variable representation of a system is given by:

The system is

Solution:

Here,

So, IQ_{C}I = 0 - 0 + 2(0 - 4)

= - 8 ≠ 0

Hence, the system is controllable.

Also, Q_{0} = [C^{T} A^{T}C^{T} (A^{T})^{2} C^{T}]

Here,

and

So,

= 1 ≠ 0

Since |Q_{0}| ≠ 0, therefore given system is observable.

QUESTION: 10

The zeros of following system are located at

Solution:

T.F.,

Hence, zeros are at:

So, zeros are at:

or, -17 -5s + s^{2} + 9s + 20 = 0

or, s^{2} + 4s + 3 = 0

or, (s+1) (s+3) = 0

or, s = -1,- 3

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