A control system is a system that directs the input to another system and regulates its output. It helps in determining the system’s behavior. The controllability and observability are two important concepts that help in designing the control system more effectively.
Controllability is the ability to control the state of the system by applying specific input. For example, if we can steer a car to any desired direction by using the steering wheel, then the car is controllable.
Observability is the ability to measure or observe the system’s state. For example, if we can know the speed of a car by looking at the speedometer, then the car is observable.
In this article, we will learn how to check the controllability and observability of a state model in control system using Kalman’s test and some examples.
State Model of a Control System
A state model of a control system is a mathematical representation of the system using state variables, input variables and output variables. The state variables are the minimum set of variables that describe the complete state of the system at any instant of time. The input variables are the external signals that affect the system’s state. The output variables are the signals that reflect the system’s state.
A state model of a control system can be written as:
x'(t) = A x(t) + B u(t)
y(t) = C x(t) + D u(t)
where
- x(t) is the state vector of order n x 1
- u(t) is the input vector of order m x 1
- y(t) is the output vector of order p x 1
- A is the state matrix of order n x n
- B is the input matrix of order n x m
- C is the output matrix of order p x n
- D is the feedforward matrix of order p x m
Kalman’s Test for Controllability and Observability
Kalman’s test is a simple and effective method to check whether a state model of a control system is controllable or observable.
Controllability Test
To check whether a state model is controllable, we form a controllability matrix Qc as:
Qc = [B AB A^2 B … A^(n-1) B]
where n is the order of A matrix.
The rank of Qc matrix must be equal to n for the system to be controllable. In other words, the determinant of Qc matrix must be non-zero.
|Qc| ≠ 0 => System is controllable
|Qc| = 0 => System is uncontrollable
Observability Test
To check whether a state model is observable, we form an observability matrix Qo as:
Qo = [C^T A^T C^T (A^T)^2 C^T … (A^T)^(n-1) C^T]
where n is the order of A matrix and T denotes transpose.
The rank of Qo matrix must be equal to n for the system to be observable. In other words, the determinant of Qo matrix must be non-zero.
|Qo| ≠ 0 => System is observable
|Qo| = 0 => System is unobservable
Examples
Let us see some examples to apply Kalman’s test for controllability and observability.
Example 1:
Consider the following state model:
x'(t) = [0 -1; -2 -3] x(t) + [0;1] u(t)
y(t) = [1 -1] x(t)
Find whether the system is controllable and observable.
Solution:
Here,
A = [0 -1; -2 -3]
B = [0;1]
C = [1 -1]
n = 2
To check controllability, we form Qc as:
Qc = [B AB] = [[0;1] [2;5]]
|Qc| = (0)(5) – (1)(2) = -2 ≠ 0
Hence, rank(Qc) = n = 2 and the system is controllable.
To check observability, we form Qo as:
Qo = [C^T A^T C^T] = [[1; -1] [-3; -2]]
|Qo| = (1)(-2) – (-1)(-3) = -5 ≠ 0
Hence, rank(Qo) = n = 2 and the system is observable.
Example 2:
Consider the following state model:
x'(t) = [0 1; 0 0] x(t) + [0;1] u(t)
y(t) = [1 0] x(t)
Find whether the system is controllable and observable.
Solution:
Here,
A = [0 1; 0 0]
B = [0;1]
C = [1 0]
n = 2
To check controllability, we form Qc as:
Qc = [B AB] = [[0;1] [1;0]]
|Qc| = (0)(0) – (1)(1) = -1 ≠ 0
Hence, rank(Qc) = n = 2 and the system is controllable.
To check observability, we form Qo as:
Qo = [C^T A^T C^T] = [[1; 0] [0; 0]]
|Qo| = (1)(0) – (0)(0) = 0
Hence, rank(Qo) < n and the system is unobservable.
Conclusion
In this blog post, we learned about the controllability and observability of a state model in control system. We learned how to use Kalman’s test to check whether a system is controllable or observable. We also saw some examples to illustrate the application of Kalman’s test.
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