LINEAR CONTROL SYSTEMS

1. LINEAR CONTROL SYSTEMS Ali Karimpour Assistant Professor Ferdowsi University of Mashhad

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

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

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

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

8. output Example 3: A linear time invariant continuous time systems مثال3: يک سيستم خطی غير متغير با زمان (LTI) 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. HODE model SS model Example 4: DC motor مثال 4: موتورDC J - be the inertia of the shaft e(t) - the electrical torque ia(t) - the armature current k1; k2 - constants R - the armature resistance

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

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. مطالعه معادلات ديفرانسيل مرتبه بالای خطی فوق موضوع بسيار جالبی بوده و يک روش خاص حل آن استفاده از تبديل لاپلاس است.

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

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

16. HODE model TF model Transfer Function model مدل تابع انتقال TF model Or Input-output model

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

18. SS model TF model 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

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

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

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

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

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

26. Exercises 2-1 Find the SS model and TF function model for example 1. 2-2 Find the SS model and TF function model for example 2. 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 outputb) Suppose angular velocity as output 2-5 In example 5 find the output a) the input is e-2tb) 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.