Test: State Space Analysis

The state transition matrix ϕ(t):
The state-transition matrix is defined as a matrix that satisfies the linear homogeneous state equation.
It represents the free response of the system.
The state-transition 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 = eAt
Where A = state matrix
The state-transition 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):

1. ​ϕ(0) = I
2. ϕ^-1(t) = ϕ(-t)
3. ϕ(t₂ - t₁) = ϕ(t₂ - t₀). ϕ(t₀ - t₁)
4. ϕ(t₂ + t₁) = ϕ(t₂) . ϕ(t₁)
5. [ϕ(t)]^k = ϕ(kt)
6. At t =0, dϕ / dt = A
    

Application:
For statement 2,
ϕ(t₂ - t₁) = ϕ(t₂) . ϕ^-1(t₁)
consider
ϕ(t₂ - t₁) = ϕ (t₂ + (-t₁)) 
By property 4
⇒ ϕ(t₂ - t₁) = ϕ (t₂ + (-t₁)) = ϕ(t₂) . ϕ(-t₁)
By property 2
⇒  ϕ(t₂ - t₁) = ϕ(t₂) . ϕ(-t₁) =  ϕ(t₂) . ϕ^-1(t₁)

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 A²B … 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(t₀) can be completely identified by measurement of output y(t) over a finite time interval.
Kalman’s test for observability:
Q₀ = [C^T A^TC^T (A^T)²C^T …. (A^T)^n-1 C^T]
Q₀ = observability testing matrix
If |Q₀| = 0, system is not observable
If |Q₀| ≠ 0, system is observable.

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 time-variant.
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 A²B … A^n-1 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(t₀) can be completely identified by measurement of output y(t) over a finite time interval.
Kalman’s test for observability:
Q₀ = [C^T A^TC^T (A^T)²C^T …. (A^T)^n-1 C^T]
Q₀ = observability testing matrix
If |Q₀| = 0, system is not observable
If |Q₀| ≠ 0, system is observable.
Duality property of controllability and observability:

• If the pair (A, B) is controllable then the pair (A^T, B^T) is observable.
• If the pair of (A, C) is observable then the pair of (A^T, C^T) is controllable.

If there are no pole-zero cancellations in the transfer function then the system is completely controllable and observable.

The state-space 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 time-varying system & MIMO system.
It can be used for both continuous-time as well as discrete-time systems.


Where, x₁, x₂, x₃ …. x_n are state variables
A is state matrix
B is the input matrix
Output equation: y(t) = CX(t)+DU(t) 

System is controllable if |Q_C| ≠ 0
a₁a₂ (0 – a₂) ≠ 0

Hence condition for controllability is
a₁ ≠ 0, a₂ ≠ 0, a₃ = 0

Concept:
The state transition matrix ϕ(t):
The state-transition matrix is defined as a matrix that satisfies the linear homogeneous state equation.
It represents the free response of the system.
The state-transition 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 state-transition 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,

• In the transfer function model, initial conditions are assumed to be zero. In the state-space model, transfer function is not the starting point for the system analysis, and it considers the initial conditions of the system.
• The transfer function for the system is unique.
• The transfer function is applicable only for linear time-invariant systems. The state-space model is applicable to linear time-variant systems also.
• State-space representation of a system is not unique, it may have more than one representation.
• State model can be derived from transfer function of the system.

Concept:
The state transition matrix [ϕ(t)] is given by:
ϕ(t) = e^At = L^-1 [(SI-A)]^-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.

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