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EE4107 -‐ Cybernetics Advanced
Faculty of Technology, Postboks 203, Kjølnes ring 56, N-3901 Porsgrunn, Norway. Tel: +47 35 57 50 00 Fax: +47 35 57 54 01
Exercise 4: Stability Analysis A dynamic system has one of the following stability properties:
• Asymptotically stable system • Marginally stable system • Unstable system
Below we see the behavior of these 3 different systems after an impulse:
Asymptotically stable system:
lim!→!
ℎ 𝑡 = 0
Marginally stable system:
0< lim!→!
ℎ 𝑡 < ∞
Unstable system:
lim!→!
ℎ 𝑡 = ∞
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EE4107 -‐ Cybernetics Advanced
Poles
The poles is important when analysis the stability of a system. The figure below gives an overview of the poles impact on the stability of a system:
Thus, we have the following:
Asymptotically stable system:
Each of the poles of the transfer function lies strictly in the left half plane (has strictly negative real part).
Marginally stable system:
One or more poles lies on the imaginary axis (have real part equal to zero), and all these poles are distinct. Besides, no poles lie in the right half plane.
Unstable system:
At least one pole lies in the right half plane (has real part greater than zero).
Or: There are multiple and coincident poles on the imaginary axis. Example: double integrator 𝐻(𝑠) = !
!!
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EE4107 -‐ Cybernetics Advanced
Feedback Systems
Below we see a typical feedback system:
Where we have the following transfer functions:
Where
𝐻!(𝑠) is the Controller transfer function
𝐻!(𝑠) is the Process transfer function
𝐻!(𝑠) is the Measurement (sensor) transfer function
Here are some important transfer functions to determine the stability of a feedback system:
Loop Transfer function
The Loop transfer function 𝐿(𝑠) (Norwegian: “Sløyfetransferfunksjonen”) is defined as follows:
𝐿 𝑠 = 𝐻! 𝑠 𝐻!(𝑠)𝐻!(𝑠)
Tracking transfer function
The Tracking transfer function 𝑇(𝑠) (Norwegian: “Følgeforholdet”) is defined as follows:
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EE4107 -‐ Cybernetics Advanced
𝑇 𝑠 =𝑦(𝑠)𝑟(𝑠)
=𝐻!𝐻!𝐻!
1 + 𝐻!𝐻!𝐻!=
𝐿(𝑠)1 + 𝐿(𝑠)
= 1 − 𝑆(𝑠)
Sensitivity transfer function
The Sensitivity transfer function 𝑆(𝑠) (Norwegian: “Sensitivitetsfunksjonen/avviksforholdet”) is defined as follows:
𝑆 𝑠 =𝑒(𝑠)𝑟(𝑠)
=1
1 + 𝐿(𝑠)= 1 − 𝑇(𝑠)
Characteristic Polynomial
We have that:
𝐿 𝑠 =𝑛!(𝑠)𝑑!(𝑠)
And:
𝑇 𝑠 =𝑦(𝑠)𝑟(𝑠)
=𝐿(𝑠)
1 + 𝐿(𝑠)=
𝑛!(𝑠)𝑑!(𝑠)
1 + 𝑛!(𝑠)𝑑!(𝑠)
=𝑛!(𝑠)
𝑑! 𝑠 + 𝑛! 𝑠
Where 𝑛!(𝑠) and 𝑑! 𝑠 numerator and the denominator of the Loop transfer function 𝐿(𝑠).
The characteristic polynomial for the control system (tracking function) is defined as:
𝑐 𝑠 = 𝑑! 𝑠 + 𝑛! 𝑠
The stability of the control system is determined by the placement of the roots of the characteristic polynomial in the complex plane.
Task 1: Stability Analysis
Given the following transfer functions:
𝐻 𝑠 =1
𝑠 + 1
𝐻 𝑠 =1𝑠
𝐻 𝑠 =1𝑠!
𝐻 𝑠 =1
𝑠 − 1
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EE4107 -‐ Cybernetics Advanced
Task 1.1
Pen and paper: What are the poles for the different transfer functions above? Plot the poles in the imaginary plane. What are the stability properties of these systems (“asymptotically stable system”, “marginally stable system” or “unstable system”)?
Discuss the results.
Task 1.2
Do the same using MathScript.
Discuss the results.
Tip! Use the built-‐in functions poles and pzgraph.
Task 1.3
Plot the impulse responses of these systems using MathScript. Are they as expected?.
Tip! Use the built-‐in function impulse, which is similar to the step function we have used before.
Task 2: Stability Analysis of Feedback systems
Given the following feedback system:
The transfer function for the process is:
𝐻!(𝑠) =1
𝑠 + 1 !𝑠
The transfer function for the measurement/sensor is:
𝐻! 𝑠 = 𝐾! = 1
The transfer function for the controller is:
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EE4107 -‐ Cybernetics Advanced
𝐻!(𝑠) = 𝐾!
We shall use 3 different values for 𝐾!:
𝐾! = 1
𝐾! = 2
𝐾! = 4
Task 2.1
Find 𝐿(𝑠), 𝑇(𝑠) and 𝑆(𝑠) for the system (both “pen and paper” and in MathScript).
Tip! In MathScript we can use the series and feedback functions in order to find 𝐿(𝑠) and 𝑇(𝑠).
Task 2.2
Plot the step response for the feedback system (𝑇(𝑠)).
Task 2.3
Find the poles and plot the poles in the imaginary plane for the feedback system (𝑇(𝑠)).
Is the system asymptotically stable, marginally stable system or unstable (for the 3 different values of 𝐾!)?
Discuss the results.
Task 3: Control System
Given the following control system:
The transfer functions are as follows:
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EE4107 -‐ Cybernetics Advanced
𝐻! 𝑠 =𝐾!
𝑇!𝑠 + 1𝑒!!"
𝐻! 𝑠 =𝐾!
𝑇!𝑠 + 1𝑒!!"
𝐻! 𝑠 = 𝐾!
𝐻! 𝑠 = 𝐾!𝑇!𝑠 + 1𝑇!𝑠
Task 3.1
Find the loop transfer function 𝐿(𝑠), the tracking transfer function 𝑇(𝑠) and the sensitivity transfer function 𝑆(𝑠) for the system.
Task 4: Stability of Feedback systems
Given the following control system:
The transfer function for the process (including measurement) is:
𝐻!" =2𝑠
Task 4.1
Define the stability properties of this process (is the process stable or not?).
Task 4.2
The transfer function for the controller is:
𝐻! = 𝐾!
1. What is the loop transfer function 𝐿(𝑠)? 2. What is the tracking transfer function 𝑇(𝑠)? 3. What is the characteristic polynomial? 4. What is the systems pole(s)?
Task 4.3
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EE4107 -‐ Cybernetics Advanced
For which values of 𝐾! is the system
• Asymptotically Stable • Unstable • Marginally stable?
Task 4.4
Define the system in MathScript and find the step response for the system for different values of 𝐾!.
Additional Resources
• http://home.hit.no/~hansha/?lab=mathscript
Here you will find tutorials, additional exercises, etc.
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