Kalman’s Test for Controllability and Observability
Controllability
The state equations of the LTI system are:
[Tex]\dot{x} = Ax+Bu [/Tex]—- (state equation)
[Tex]y=Cx+Du [/Tex]—- (output equation)
Note
- The order of matrix A is NxN
- The state vector ‘x’ is of order Nx1
- ‘u’ is of order Mx1 (M is the number of inputs)
For the system to be controllable, the rank of the composite matrix QC must be equal to ‘N’
Composite matrix is represented as:
Qc = [B AB A2B ….. An-1B]
Let us consider one example to check the controllability of the control system using Kalman Test.
Example: Find whether the given system is controllable or not using Kalman Test.
[Tex]\dot{x}= \begin{bmatrix} 0 & 1\\ -6 & -5 \end{bmatrix}x + \begin{bmatrix} 0\\1 \end{bmatrix}u [/Tex]
Solution:
[Tex]A= \begin{bmatrix} 0 & 1\\ -6 & -5 \end{bmatrix} [/Tex]
B = [Tex]\begin{bmatrix} 0\\1 \end{bmatrix} [/Tex]
Using the composite matrix equation: Qc = [B AB A2B ….. An-1B]
[Tex]AB= \begin{bmatrix} 1\\-5 \end{bmatrix} [/Tex]
Qc = [B AB ]
[Tex]Q_{c}= \begin{bmatrix} 0 & 1\\ 1 & -5 \end{bmatrix} [/Tex] (rank of Qc =2 i.e., equal to N)
[Tex]|Q_{c}| = (-5*0)-(1*1) [/Tex]
[Tex]|Q_{c}|=-1 [/Tex]
[Tex]|Q_{c}| \neq 0 [/Tex]
Hence the system is controllable.
Observability
The state equations of the LTI system are:
[Tex]\dot{x} = Ax+Bu [/Tex]—- (state equation)
[Tex]y=Cx+Du [/Tex]—- (output equation)
Note
- The order of output vector ‘y’ is Px1
- The matrix ‘C’ is of the order 1xN
For the observable system, the rank of the composite matrix Qo must be equal to ‘N’
Composite matrix is represented as:
Q0 = [CT ATCT ….. (AT)n-1CT]
Let us consider one example to check the observability of the control system using Kalman Test.
Example: Find whether the given system is observable or not using Kalman Test.
[Tex]\dot{x}= \begin{bmatrix} 0 & -6\\ 1 & -5 \end{bmatrix}x + \begin{bmatrix} 6\\1 \end{bmatrix}u [/Tex]
[Tex]y= \begin{bmatrix} 0 &1 \end{bmatrix}x + [0]u [/Tex]
Solution:
[Tex]A= \begin{bmatrix} 0 & -6\\ 1 & -5 \end{bmatrix} [/Tex]
[Tex]C = \begin{bmatrix} 0 &1 \end{bmatrix} [/Tex]
Using the composite matrix equation: Q0 = [CT ATCT ….. (AT)n-1CT]
CT = [Tex]\begin{bmatrix} 0\\1 \end{bmatrix} [/Tex]
ATCT = [Tex]\begin{bmatrix} 1\\-5 \end{bmatrix} [/Tex]
Q0 = [CT ATCT]
Q0 = [Tex]\begin{bmatrix} 0 & 1\\ 1 & -5 \end{bmatrix} [/Tex] (rank of Qo =2 i.e., equal to N)
|Q0| = -1
[Tex]|Q_{0}| \neq 0 [/Tex]
Hence the system is observable.
Controllability and Observability in Control System
The control system is the system that directs the input to another system and regulates its output. It helps in determining the system’s behavior. The controllability and observability help in designing the control system more effectively. Controllability is the ability to control the state of the system by applying specific input whereas observability is the ability to measure or observe the system’s state. In this article, we will study controllability and observability in detail.
Table of Content
- What is Controllability?
- What is Observability?
- Kalman’s Test for Controllability and Observability
- Condition of Controllability and Observability in S-Plane
- Advantages and Disadvantages of Controllability and Observability
- Applications of Controllability and Observability