1. Feedback Control Systems (FCS) Dr. Imtiaz Hussain email: imtiaz.hussain@faculty.muet.edu.pkimtiaz.hussain@faculty.muet.edu.pk URL :http://imtiazhussainkalwar.weebly.com/ Lecture-36-37 Transfer Matrix and solution of state equations

2. Transfer Matrix (State Space to T.F) Now Let us convert a space model to a transfer function model. Taking Laplace transform of equation (1) and (2) considering initial conditions to zero. From equation (3) (1) (2) (3) (4) (5)

3. Transfer Matrix (State Space to T.F) Substituting equation (5) into equation (4) yields

4. Example#1 Convert the following State Space Model to Transfer Function Model if K=3, B=1 and M=10;

5. Example#1 Substitute the given values and obtain A, B, C and D matrices.

6. Example#1

11. Example#2 Obtain the transfer function T(s) from following state space representation. Answer

12. Forced and Unforced Response Forced Response, with u(t) as forcing function Unforced Response (response due to initial conditions)

13. Solution of State Equations Consider the state equation given below Taking Laplace transform of the equation (1) (1)

14. Solution of State Equations Taking inverse Laplace State Transition Matrix

15. Example-3 Consider RLC Circuit obtain the state transition matrix ɸ(t). VcVc + - + - VoVo iLiL

16. Example-3 (cont...) State transition matrix can be obtained as Which is further simplified as

17. Example-3 (cont...) Taking the inverse Laplace transform of each element

18. Example#4 Compute the state transition matrix if Solution

19. State Space Trajectories The unforced response of a system released from any initial point x(t o ) traces a curve or trajectory in state space, with time t as an implicit function along the trajectory. Unforced system’s response depend upon initial conditions. Response due to initial conditions can be obtained as

20. State Transition Any point P in state space represents the state of the system at a specific time t. State transitions provide complete picture of the system P( x 1, x 2 ) t0t0 t1t1 t2t2 t3t3 t4t4 t5t5 t6t6

21. Example-5 For the RLC circuit of example-3 draw the state space trajectory with following initial conditions. Solution

22. Example-5 (cont...) Following trajectory is obtained

23. Example-5 (cont...)

24. Equilibrium Point The equilibrium or stationary state of the system is when

25. Solution of State Equations Consider the state equation with u(t) as forcing function Taking Laplace transform of the equation (1) (1)

26. Solution of State Equations Taking the inverse Laplace transform of above equation. Natural Response Forced Response

27. Example#6 Obtain the time response of the following system: Where u(t) is unit step function occurring at t=0. consider x(0)=0. Solution Calculate the state transition matrix

28. Example#6 Obtain the state transition equation of the system

29. END OF LECTURES-36-37 To download this lecture visit http://imtiazhussainkalwar.weebly.com/