State Space Model
The single-input and single output continuous time LTI system is described by the differential equation which is given below:
[Tex]\frac{d^Ny(t)}{dt^N}+\frac{d^{N-1}y(t)}{dt^{N-1}}+….+a_{n}y(t)=x(t) [/Tex] —-(1)
The N state variables q1(t), q2(t), ….,qn(t) is defined as:
q1(t)=y(t) , …., qn(t)= y(n-1)(t)
where [Tex]y^k(t)=\frac{d^ky(t)}{dt^k} [/Tex] —–(2)
From equation (1) and equation (2)
[Tex]\dot{q_{n}(t)}= -a_{n}q_{1}(t)-a_{n-1}q_{2}(t)-…….-a_{1}q_{n}(t)+x(t) [/Tex] ——-(3)
and,
y(t)=q1(t) —— (4)
The equation 3 and equation 4 can be expressed in the matrix form as given below where [Tex]\dot{q_{k}(t)}= \frac{dq_{k}(t)}{dt} [/Tex]:
Matrix Form
Hence the final state equation will be:
[Tex]\dot{q(t)} = Aq(t)+Bx(t) [/Tex] —- (state equation)
[Tex]y(t)=Cq(t)+Dx(t) [/Tex] —- (output equation)
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