State Transition Matrix and its Properties
State Transition Matrix
If the matrix is satisfying the linear homogenous state equation then the matrix is called state transition matrix. Linear homogenous state equation is described below:
[Tex]\dot{x(t)}=\frac{dx(t)}{dt} = Ax(t) [/Tex] —(1)
Let [Tex]\phi(t) [/Tex] be the state transition matrix then
[Tex]\frac{d\phi(t)}{dt} =A\phi(t) [/Tex] —(2)
If [Tex]\phi(t) [/Tex] is the state transition matrix, it will satisfy the given condition:
[Tex]x(t)=\phi(t)x(0) [/Tex] —– (3)
Taking the laplace transform of the equation (1):
sx(s)-x(0) = Ax(s)
x(s)(sI-A) = x(0)
x(s) = (sI-A)-1x(0)
Taking the inverse laplace transform of the above equation:
x(t) = L-1[(sI-A)-1x(0)]
The state transition matrix is:
Φ(t) = L-1[(sI-A)-1] = eAt
State Transition Matrix Properties
There are five important properties of state transition matrix:
- ϕ(t) = I (I is the identity matrix)
- ϕ(t) = eAt = (e-At)-1 = [ϕ(-t)]-1
- ϕ(t2-t0) = ϕ(t2-t1)ϕ(t1-t0)
- ϕ(t-t0) = ϕ(t) ϕ-1(t0)
- ϕ(t+t0) = ϕ(t) ϕ(t0)
What is State Space Analysis ?
The State Space analysis applies to the non-linear and time-variant system. It helps in the analysis and design of linear, non-linear, multi-input, and multi-output systems. Earlier the transfer function applied to the linear time-invariant system but with the help of State Space analysis, it is possible to find the transfer function of the non-linear and time-variant systems. In this article, we will study the State Space Model in control system engineering.
Table of Content
- What is the State Space Analysis?
- State Space Model
- Transfer Function from State Space Model
- State Transition Matrix and its Properties
- Controllability and Observability
- Solved Example on State Space Analysis
- Advantages and Disadvantages of State Space Analysis
- Applications of State Space Analysis
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