1. LINEAR CONTROL SYSTEMS
Ali Karimpour
Assistant Professor
Ferdowsi University of Mashhad
<<<1.1>>>
###Control System Design###
{{{Control, Design}}}

2. Different representations of control systems
Lecture 3
Different representations of control systems
Topics to be covered include:
High Order Differential Equation. (HODE model)
State Space model. (SS model)
Transfer Function. (TF model)
State Diagram. (SD model)

3. High order differential equation (HODE).
معادلات ديفرانسيل مرتبه بالا
HODE model

4. Example 1: A high order differential equation (HODE).
مثال 1: يک معادله ديفرانسيل مرتبه بالا
ورودی
خروجی
HODE model

5. Example 2: Another high order differential equation.
مثال2: مثالی ديگر از معادله ديفرانسيل مرتبه بالا
خروجی
ورودی
HODE model eb

6. State Space Models (SS)
معادلات فضای حالت
For continuous time systems
SS model
For linear time invariant continuous time systems
<<<3.6>>>
###State Space Models###
SS model

7. State Space Models معادلات فضای حالت فرم کلی فضای حالتی سيستمهای LTI
General form of LTI systems in state space form
فرم کلی فضای حالتی سيستمهای LTI
<<<3.6>>>
###State Space Models###
SS model

8. Example 3: A linear time invariant continuous time systems
مثال3: يک سيستم خطی غير متغير با زمان (LTI)
output
SS model

9. Example 3: Continue مثال3: ادامه
The equations can be rearranged as follows:ساده سازی
We have a linear state space model with مدل خطی فضای حالتی

10. A demonstration robot containing several servo motors
مثالی از موتور dc در ربات

11. Example 4: DC motor مثال 4: موتور DC e(t) - the electrical torque
J - be the inertia of the shaft
e(t) - the electrical torque
ia(t) - the armature current
k1; k2 - constants
R - the armature resistance
HODE model SS model

12. Different representations
نمايشهای مختلف
SD model
HODE model
TF model
SS model

13. Linear high order differential equation
معادلات ديفرانسيل مرتبه بالای خطی
The study of differential equations of the type described above is a rich and interesting subject. Of all the methods available for studying linear differential equations, one particularly useful tool is provided by Laplace Transforms.
<<<4.2>>>
###Linear Continuous Time Models###
مطالعه معادلات ديفرانسيل مرتبه بالای خطی فوق موضوع بسيار جالبی
بوده و يک روش خاص حل آن استفاده از تبديل لاپلاس است.

14. Table : Laplace transform table
جدول تبديل لاپلاس
u(t) - u(t-τ)

15. Table : Laplace transform properties.
خواص تبديل لاپلاس
y(t-τ) u(t-τ)

16. Transfer Function model
مدل تابع انتقال
TF model
<<<4.5>>>
###Transfer Functions### Or
Input-output model
HODE model TF model

17. Different representations
نمايشهای مختلف
SD model
HODE model
TF model
SS model

18. But how can we change SS model to TF model?
اما چگونه معادلات فضای حالت را به تابع انتقال تبديل کنيم
Use Laplace transform
Then
Let initial condition zero
SS model TF model

19. Different representations
نمايشهای مختلف
SD model
HODE model
TF model
SS model

20. TF model properties
خواص مدل تابع انتقال
1- It is available just for linear systems.
2- It is derived by zero initial condition.
3- It just shows the relation between input and output so it may lose some information.
4- It can be used to show delay systems but SS can not.

21. Example 5: A system with pure time delay
مثال 5: سيستمی با تاخير ثابت

22. State diagram
دياگرام حالت
s -1
x2(0)
x1(s)
x2(s)
SD model

23. State diagram to state space
دياگرام حالت به معادلات حالت
SD model SS model

24. Different representations
نمايشهای مختلف
SD model
HODE model
TF model
SS model

25. Different representations
نمايشهای مختلف
HODE SS
Realization SD TF
Mason’s rule
Discussed in this lecture
Will be discussed in next lecture

26. Exercises 2-1 Find the SS model and TF function model for example 1.
2-3 Find the TF function model for example 3.
2-4 Find the TF function model for example 4. a) Suppose angular position as output b) Suppose angular velocity as output
2-5 In example 5 find the output a) the input is e-2t b) the input is unit step

27. Exercises (Continue)
2-6 In the different representation slide show the validity of red directions
2-7 Change the equations derived in example 2 to one order 3 HODE.