Example 1. A system is described by the state equation X' = AX + BU. The output is given by Y = CX, where
And C = [1 0]
The transfer function of the system is:
(a) s/s2 + 5s + 7
(b) 2s/s2 + 5s + 7
(c) 1/s2 + 5s + 7
(d) s/s2 + 3s + 5
Correct Answer is option (a)
The equation to find the transfer function of the system is given by:
C [SI - A]-1B + D
Where,
I is always an identity matrix.
Computing the values of A, B, I, and C in the above equation, we get:
Transfer function = C [SI - A]-1B + D
State variable analysis: Examples with explanation= s/s2 + 5s + 7
= s/s2 + 5s + 7
Example 2. Find the sum of the Eigen values of the matrix given below:
(a) 15
(b) 12
(c) -10
(d) 10
Correct Answer is option (d)
The shortcut approach to find the sum of the Eigen values is to add the elements present in the diagonal of the given matrix.
The diagonal elements are 2, 1, 3, and 4. It sum is equal to 2 + 1 + 3 + 4 = 10
Hence, the correct answer is 10.
Example 3, A system represented by the equations is given by:
And Y = [1 0]
Find the equivalent transfer function of the system.
(a) s/s2 + 3s + 3
(b) 1/s2 + 3s + 2
(c) 5/s2 + 5s + 2
(d) 2/s2 + 3s + 5
Correct Answer is option (b)
The equation to find the transfer function of the system is given by:
C [SI - A]-1B + D
Where,
I is always an identity matrix.
Computing the values of A, B, I, and C in the above equation, we get:
Transfer function = C [SI - A]-1B + D
= 1/s2 + 3s + 2
Example 4. The observability condition in the state space representation can be determined with the condition given by:
(a) [CT ATCT]
(b) [BT ATBT]
(c) [AT ATCT]
(d) None of these
Correct Answer is option (a)
The observability condition can be determined from the condition [CT ATCT].
Example 5. The system model given by the equation is:
Y '= [1 1] x
(a) Controllable
(b) Observable
(c) Both (a) and (b)
(d) Neither (a) nor (b)
Correct Answer is option (c)
We know that the equation is represented in the form of AX + BU.
By comparing, we get:
C = [1 1]
The controllability of the given state equation can be checked by forming the matrix [B AB]. The observability of the given state equation can be checked by forming the matrix [CT ATCT]. If the determinant of these matrixes is not equal to zero, it is said to observable or controllable.
The matrix so formed for the controllability is given by:
[B AB]
The determinant of the above matrix is: 0 x -3 - 1 = -1
It is not equal to 0. Hence, the matrix is controllable.
Now, let's check for observability.
The matrix so formed for the observability is given by:
[CT ATCT]
The determinant of the above matrix is: 1 x -2 - 2 = -4
It is not equal to 0. Hence, the matrix is observable.
Thus, the system model described by the given state equation is both controllable and observable.
Example 6. The state of a system equation can be written in the form of:
(a) First order differential equation
(b) Second order differential equation
(c) Third order differential equation
(d) None of the above
Correct Answer is option (a)
The state of a system equation is written in the form of first order differential equation using the state variables x1, x2, x3 ? xn.
Example 7: A LTI system is described by the state model given by:
Find if the system is:
(a) Completely controllable
(b) Completely observable
(c) Not completely controllable but completely observable
(d) Not completely observable but completely controllable
Correct Answer is option (c)
The controllability of the given state equation can be checked by forming the matrix [B AB]. The observability of the given state equation can be checked by forming the matrix [CT ATCT].
We will use the above two conditions to determine controllability and observability.
We can check either the determinant or the rank of the formed matrix. If the determinant of these matrixes is not equal to zero, it is said to observable or controllable. If the rank of the matrix is equal to its order, it is said to be observable or controllable.
Let's start.
By comparing the given state model, the parameters are:
Now, the matrix [B AB] is given by:
The determinant of the above matrix is equal to 0. Hence, it is not completely controllable. We can also say that the rank of the matrix is not equal to its order (2).
Now, the matrix [CT ATCT] is given by:
The determinant of the above matrix is not equal to 0. Hence, it is completely observable. We can also say that the rank of the matrix is equal to its order (2).
Thus, the given state model is completely observable but not completely controllable.
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