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Which of the following properties are associated with the state transition matrix ϕ(t)?
1. ϕ(0) = I
2. ϕ(t_{2}  t_{1}) = ϕ(t_{2}) . ϕ^{1}(t_{1})
3. ϕ(t_{2} + t_{1}) = ϕ(t_{2}) . ϕ(t_{1})
4. ϕ(t_{2}  t_{1}) = ϕ(t_{2}  t_{0}). ϕ(t_{0}  t_{1})
Select the correct answer using the codes given below:
The state transition matrix ϕ(t):
The statetransition matrix is defined as a matrix that satisfies the linear homogeneous state equation.
It represents the free response of the system.
The statetransition matrix ϕ(t) completely defines the transition of the states from the initial time t = 0 to any time t when the inputs are zero.
The state transition matrix is given by
ϕ(t) = L1 [sI  A]1 = eAt
Where A = state matrix
The statetransition matrix is dependent only upon the matrix A and, therefore, is sometimes referred to as the state transition matrix of A.
Properties of ϕ(t):
Application:
For statement 2,
ϕ(t_{2}  t_{1}) = ϕ(t_{2}) . ϕ^{1}(t_{1})
consider
ϕ(t_{2}  t_{1}) = ϕ (t_{2} + (t_{1}))
By property 4
⇒ ϕ(t_{2}  t_{1}) = ϕ (t_{2} + (t_{1})) = ϕ(t_{2}) . ϕ(t_{1})
By property 2
⇒ ϕ(t_{2}  t_{1}) = ϕ(t_{2}) . ϕ(t_{1}) = ϕ(t_{2}) . ϕ^{1}(t_{1})
Read the statements regarding controllability and observability and mark the answer as true and false.
The correct answer is option '2'
Concept:
Controllability:
A system is said to be controllable if it is possible to transfer the system state from any initial state x(t0) to any desired state x(t) in a specified finite time interval by a control vector u(t)
Kalman’s test for controllability:
ẋ = Ax + Bu
Q_{c} = {B AB A^{2}B … An^{1} B]
Q_{c} = controllability matrix
If Q_{c} = 0, system is not controllable
If Q_{c}≠ 0, system is controllable
Observability:
A system is said to be observable if every state x(t_{0}) can be completely identified by measurement of output y(t) over a finite time interval.
Kalman’s test for observability:
Q_{0} = [C^{T} A^{T}C^{T} (A^{T})^{2}C^{T} …. (A^{T})^{n1} C^{T}]
Q_{0} = observability testing matrix
If Q_{0} = 0, system is not observable
If Q_{0} ≠ 0, system is observable.
The transfer function G(S) = C(SI  A)^{1}b of the system
x' = Ax + bu
y = Cx + du
has no polezero cancellation. The system
State space representation:
ẋ(t) = A(t)x(t) + B(t)u(t)
y(t) = C(t)x(t) + D(t)u(t)
y(t) is output
u(t) is input
x(t) is a state vector
A is a system matrix
This representation is continuous timevariant.
Controllability:
A system is said to be controllable if it is possible to transfer the system state from any initial state x(t_{0}) to any desired state x(t) in a specified finite time interval by a control vector u(t)
Kalman’s test for controllability:
ẋ = Ax + Bu
Q_{c} = {B AB A^{2}B … A^{n1} B]
Q_{c} = controllability matrix
If Q_{c} = 0, system is not controllable
If Q_{c} ≠ 0, the system is controllable
Observability:
A system is said to be observable if every state x(t_{0}) can be completely identified by measurement of output y(t) over a finite time interval.
Kalman’s test for observability:
Q_{0} = [C^{T} A^{T}C^{T} (A^{T})^{2}C^{T} …. (A^{T})^{n1} C^{T}]
Q_{0} = observability testing matrix
If Q_{0} = 0, system is not observable
If Q_{0} ≠ 0, system is observable.
Duality property of controllability and observability:
If there are no polezero cancellations in the transfer function then the system is completely controllable and observable.
Which of the following can be extended to timevarying systems?
The statespace analysis is a very useful technique of analyzing the control system, it is based on the concept of space and applies to the LTI system as well in non – linearly timevarying system & MIMO system.
It can be used for both continuoustime as well as discretetime systems.
Where, x_{1}, x_{2}, x_{3} …. x_{n} are state variables
A is state matrix
B is the input matrix
Output equation: y(t) = CX(t)+DU(t)
Consider a system governed by the following equations
The initial conditions are such that and Which one of the following is true?
State variable description of an LTI system is given by
Where Y is the output and u is input. System is controllable for
System is controllable if Q_{C} ≠ 0
a_{1}a_{2} (0 – a_{2}) ≠ 0
Hence condition for controllability is
a_{1} ≠ 0, a_{2 }≠ 0, a_{3} = 0
The state transition matrix of a control system is . The system matrix A is
Concept:
The state transition matrix ϕ(t):
The statetransition matrix is defined as a matrix that satisfies the linear homogeneous state equation.
It represents the free response of the system.
The statetransition matrix ϕ(t) completely defines the transition of the states from the initial time t = 0 to any time t when the inputs are zero.
The state transition matrix is given by
ϕ(t) = L^{1} [sI  A]^{1 }= e^{At}
Where A = state matrix
The statetransition matrix is dependent only upon the matrix A and, therefore, is sometimes referred to as the state transition matrix of A.
Properties of ϕ(t):
Calculation:
Given that
State transition matrix ϕ(t) =
Consider the sixth property of the state transition matrix
⇒ At t = 0, dϕ /dt = A
Where A is system or state matrix,
Consider the following properties attributed to state model of a system:
Which of the above statements are correct?
The linear time invariant system is represented by the state space model as
Consider n=number of state variables, m = number of inputs, p= number of outputs. The state transition matrix Φ( t) Φ( t) is given by:
Concept:
The state transition matrix [ϕ(t)] is given by:
ϕ(t) = e^{At} = L^{1} [(SIA)]^{1}
where, A = System matrix
I = Identity matrix
Properties of state transition matrix:
(1) State transition matrix at t = 0 is always equal to the identity matrix.
ϕ(0) = e^{A0 }= I
(2) The differentiation of the state transition matrix at t = 0 is always equal to its system matrix.
A system is represented by what is the transfer function to the system?
Concept:
A transfer function (TF) is defined as the ratio of Laplace transform of the output to the Laplace transform of the input by assuming initial conditions are zero.
TF = L[output] / L[input]
For unit impulse input i.e. r(t) = δ(t)
⇒ R(s) = δ(s) = 1
Now transfer function = C(s)
Therefore, transfer function is also known as impulse response of the system.
Transfer function = L[IR]
IR = L^{1} [TF]
Calculation:
Given differential equation is,
Laplace transform of the above equation is given by
⇒ 3s Y(s) + 2 Y(s) = U(s)
⇒ Y(s) (3s + 2) = U(s)
∴ Transfer function is given by
22 docs274 tests

22 docs274 tests
