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LINEAR CONTROL SYSTEMS

LINEAR CONTROL SYSTEMS. Ali Karimpour Associate Professor Ferdowsi University of Mashhad. Topics to be covered include : State space representation.  State transition matrix. Calculation of state transition matrix for time varying system.

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LINEAR CONTROL SYSTEMS

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

  2. Topics to be covered include: State space representation.  State transition matrix. Calculation of state transition matrix for time varying system. Significance of state transition matrix and its property. State transition equation. State transition equation determined by state diagram. Eigenvalues of the matrix A in state space form and poles of transfer function Similarity transformation. Lecture 6

  3. If initial condition and input are defined, then x(t), y(t) ? Start with homogeneous equation با معادله همگن شروع کنید. اگر شرائط اولیه و ورودی مشخص باشد مقدار چند است؟ x(t)و y(t) State Space Models مدل فضای حالت General form of state space model LTI systems

  4. ماتریس انتقال حالت State transition matrix ماتریس انتقال حالت If t0 is not zero

  5. Determination of state transition matrix تعیین ماتریس انتقال حالت Method I روش اول

  6. Determination of state transition matrix تعیین ماتریس انتقال حالت Method II روش دوم

  7. ith position Determination of state transition matrix تعیین ماتریس انتقال حالت Method III روش سوم So ith column of Φ is the response of system to

  8. مثال 1: ماتریس انتقال حالت و را بیابید. x(t) Example 1 State transition matrix and x(t)

  9. 1- Φ(0) = I Property of state transition matrix for LTI systems خواص ماتریس انتقال حالت در سیستمهای LTI 2- Φ-1(t)= Φ(-t) 3- Φ(t2-t1)Φ(t1-t0) = Φ(t2-t0) 4- Φk(t) = Φ(kt)

  10. State transition matrix for time varying systems ماتریس انتقال حالت برای سیستمهای متغیر با زمان

  11. Convolution integral State transition equation معادله انتقال حالت State transition equation

  12. Example 2 Find state transition equation for unit step مثال 2: معادله انتقال حالت را با فرض اعمال پله واحد بیابید.

  13. Example 2 Find state transition equation for unit step مثال 2: معادله انتقال حالت را با فرض اعمال پله واحد بیابید.

  14. مثال 3: معادله و را بیابید. x2(s) x1(s) Example 3 Find x1(s) and x2(s)

  15. s -1 s -1 s -1 s -1 1 1 1 1 u(s) -2 -3 ِ Example 4: Derive x1(s) by using state diagram.

  16. Poles of G(s) Eigenvalues of A Relation between SS model and TF model. ارتباط بین معادلات فضای حالت و تابع انتقال Poles of G(s) ? Eigenvalues of A (modes) ? ارتباط بین قطبهای تابع انتقال و مقادیر ویژه A ؟ 16

  17. eig(A) Example 5: Find the poles of transfer function and the eigenvalues of A مثال5: مطلوبست قطبهای تابع انتقال و مقادیرویژه A g=tf(num,den) [num,den]=ss2tf(A,b,c,0), 17 , g=minreal(g)

  18. Poles of systemقطبهای سیستم Eigenvalues of Aمقادیر ویژه A Example5: Continue مثال5: ادامه What happened to -1 ??!!?? 18

  19. Change of variables (Similarity transformation) تغییر متغیرها ( تبدیلات همانندی) 19

  20. Eigenvalues of Aمقادیر ویژه A Eigenvalues of مقادیر ویژه Eigenvector corresponding to -2 Example 6: A similarity transformation example مثال 6: یک مثال از تبدیل همانندی 20 New system is very simpler than older

  21. Similarity transformation and eigenvalues Eigenvalues of مقادیر ویژه تغییر متغیرها و مقادیر ویژه Eigenvalues of Aمقادیر ویژه A Similarity transformation does not affect the eigenvalues of the system 21

  22. 2- Find the transfer function of the following system.a) By use of the formula between the two representation.b) By use of state diagram 3- Find y(t) for initial condition [1 3 -1]T andthe unit step as input. Exercise 1- Find the state diagram of the following system

  23. 4- The state transition matrix and state transition equation of following system. 5- Suppose x1(0)=1, x2(0)=3 and x3(0)=2 and r(t)=u(t) . Find x(t) and c(t) for t≥0 Answer is: Exercise (cont.)

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