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.

What is Controllability?

The system is controllable when the desired output is obtained by applying the specific controlled input. It is the ability to control the state of the system. The controllability of the system can be checked using the Kalman Test.

Control System

The given below is the condition for the controllability:

Q₀ = [B AB A²B ….. A^n-1B]

If the determinant of Q₀ is not equal to 0 then the system is controllable.

[Tex]|Q_{0}| \neq 0 [/Tex]  —- (system is controllable)

|Q₀| = 0 —– (system is un-controllable)

What is Observability?

It is the system’s ability to measure or observe the system state. If the internal state of the system is determined using the input and output signals during a finite interval of time then the system is said to be observable. The observability of the system can be checked using the Kalman Test. The given below is the condition for the observability:

Q₀ = [C^T A^TC^T ….. (A^T)^n-1C^T]

Note: A^T,C^T means transpose of the respective matrix

If the determinant of Q₀ is not equal to 0 then the system is controllable.

[Tex]|Q_{0}| \neq 0 [/Tex]—- (system is observable)

|Q₀| = 0 —– (system is not observable)

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 Q_C must be equal to ‘N’

Composite matrix is represented as:

Q_c = [B AB A²B ….. A^n-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: Q_c = [B AB A²B ….. A^n-1B]

[Tex]AB= \begin{bmatrix} 1\\-5 \end{bmatrix} [/Tex]

Q_c = [B AB ]

[Tex]Q_{c}= \begin{bmatrix} 0 & 1\\ 1 & -5 \end{bmatrix}  [/Tex] (rank of Q_c =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 Q_o must be equal to ‘N’

Composite matrix is represented as:

Q₀ = [C^T A^TC^T ….. (A^T)^n-1C^T]

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 = [C^T A^TC^T ….. (A^T)^n-1C^T]

C^T = [Tex]\begin{bmatrix} 0\\1 \end{bmatrix} [/Tex]

A^TC^T = [Tex]\begin{bmatrix} 1\\-5 \end{bmatrix} [/Tex]

Q₀ = [C^T  A^TC^T]

Q₀ = [Tex]\begin{bmatrix} 0 & 1\\ 1 & -5 \end{bmatrix} [/Tex] (rank of Q_o =2 i.e., equal to N)

|Q₀| = -1

[Tex]|Q_{0}| \neq 0 [/Tex]

Hence the system is observable.

Condition of Controllability and Observability in S-Plane

The controllability and observability of the control system can be calculated using the transfer function. To find the same, the following points should be kept in mind:

• The transfer function represents the system in the s-plane. If the numerator and denominator polynomial do not have any common factor except constant terms, then the system is controllable.

• The same condition is applicable for the observability. There must be no pole-zero cancellation. The system cannot be completely observable in the case of incomplete input and output.

Advantages and Disadvantages of Controllability and Observability

There are some list of Advantages and Disadvantages of Controllability and Observability given below :

Advantages

• Controllability provides how the system responds to different inputs. Hence, it helps in providing a flexible system response.

• The controllable systems can adapt to changes when the operating conditions are changed.

• Observability helps in the precise control and monitoring of the system. Hence it helps in the accurate state estimation.

• Observability checks the internal behavior of the system which helps in detecting the faults inside the system.

Disadvantages

• It is very difficult to achieve controllability in a complex system. It leads to increasing the complexity of the design.

• Sometimes, to achieve controllability in the system, additional control inputs are required. These additional inputs lead to an increase in the energy consumption of the system.

• The observability of the system is measured using the sensors of the system. Failures and limitations of the performance of the sensor can reduce the accuracy of the state estimation.

• Sometimes, the observability methods are computationally intensive which increases the cost of the system due to the addition requirement of the computational resources.

Applications of Controllability and Observability

1. Controllability and observability find their application in aerospace engineering. Controllability ensures that the aircraft can be moved to different locations when the specific input is given. On the other hand, observability helps in the accurate estimation of state for navigation.
   
2. It is used in the field of robotics where controllability helps in providing the precise control of the robotic arm motion while observability measures the state estimation for mapping. 
   
3. It is also used in the power system. Controllability helps in regulating the power flow while on the other hand, observability provides accurate monitoring and control of the system.

Conclusion

In this article, we have studied controllability and observability, their application, advantages, and disadvantages. We have also studied how to check whether the system is controllable and observable. This property helps in designing the control system properly.

FAQs on Controllability and Observability in Control System

1. Can a system is observable but not controllable and vice versa? Comment on this.

Yes, it is possible but for the system to work efficiently, the system must be observable and controllable.

2. Has the stability been impacted due to controllability and observability?

Controllability means that the system can be controlled with different given inputs. Controlling the system on the basis of various inputs leads to stability. On the other hand, observability helps in determining the accurate information of the state which helps in providing stability to the system.

3. How is controllability determined in a control system?

Controllability is determined using a controllability matrix.