The state variable description of an autonomous system is, X = AX where X is a twodimensional vector and A is a matrix given by
The eigen values of A are
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
The system equations are given by
y(t) = [1 0]x(t)
The transfer function of the above system is
Given,
Transfer function of the given system is
Now,
∴
= s^{2} + 3s + 2
Now,
Now,
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?
The poles of the system can be obtained from eigen values of the system matrix. Hence, statement1 is false.
State space techniques can be applied to linear or nonlinear, time variant or time invariant systems.
Hence, statement4 is false.
Consider the system shown in figure below:
The system is
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.
Consider the following matrix:
x(t) is given by
Given,
So, x(t) = ϕ(t).x(0)
= state transition equation
The transfer function of the system shown below is
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,
The state equation for the circuit shown below is
Let us select the state variables as V_{c} and i_{L}.
Applying KVL in the mesh2, 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:
The state space representation of the system represented by the SFG shown below is
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
}
The state variable representation of a system is given by:
The system is
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.
The zeros of following system are located at
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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