1. Feedback Control Systems (FCS)
Lecture-30-31
Transfer Matrix and solution of state equations
Dr. Imtiaz Hussain
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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

7. Example#1

8. Example#1

9. Example#1

10. 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). Vc + - Vo iL

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(to) 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(x1, x2) t0 t1 t2 t3 t4 t5 t6

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

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End of Lectures-30-31