1. State Space Analysis Hany Ferdinando Dept. of Electrical Engineering Petra Christian University

2. State Space 2 - Hany Ferdinando2 Overview State Transition Matrix Time Response Discrete-time evaluation

3. State Space 2 - Hany Ferdinando3 State Transition Matrix The solution of is If the initial condition x(0), input u(  ) and the state transition matrix  (t) are known the time response of x(t) can be evaluated

4. State Space 2 - Hany Ferdinando4 State Transition Matrix The  (t) is inverse Laplace Transform of  (s) and When the input u(t) is zero, then

5. State Space 2 - Hany Ferdinando5 State Transition Matrix From the equation above we can expand the matrix into (for example, two elements)

6. State Space 2 - Hany Ferdinando6 State Transition Matrix The  11 (s) can be evaluated from the relation between X 1 (s) and x 1 (0), the  12 (s),  21 (s) and  22 (s) can be evaluated with the same procedure

7. State Space 2 - Hany Ferdinando7 Time Response It is the time response of X(t). First, find  (t) from  (s). It is simply the inverse Laplace Transform of  (s). Do the inverse Laplace Transform for each element of  (s).

8. State Space 2 - Hany Ferdinando8 Example i(t)

9. State Space 2 - Hany Ferdinando9 Example If x 1 = v C and x 2 = i L then

10. State Space 2 - Hany Ferdinando10 Example For R = 3, L = 1 and C = 0.5,

11. State Space 2 - Hany Ferdinando11 Example I(s)V(s) s -1 1/C -1/C 1/L -R/L X 1 (s)X 2 (s) R x 1 (0)/sx 2 (0)/s

12. State Space 2 - Hany Ferdinando12 Example s -1 -1/C 1/L -R/L X 1 (s)X 2 (s) x 1 (0)/sx 2 (0)/s When U(s) = 0

13. State Space 2 - Hany Ferdinando13 Example  11 (s) is transfer function of X 1 (s)/x 1 (0). Here, use the Mason Gain Formula to get  11 (s)

14. State Space 2 - Hany Ferdinando14 Example  1 (s) is path cofactor of ,  is 1 + 3s -1 + 2s -2  1 (s) = 1 + 3s -1

15. State Space 2 - Hany Ferdinando15 Example With the same procedures, find the  12 (s),  21 (s) and  22 (s)!

16. State Space 2 - Hany Ferdinando16 Example

17. State Space 2 - Hany Ferdinando17 Example Then the X(t) can be calculated with

18. State Space 2 - Hany Ferdinando18 Discrete-time Evaluation For discrete-time, use the approximation

19. State Space 2 - Hany Ferdinando19 Discrete-time Evaluation

20. State Space 2 - Hany Ferdinando20 Example With the same example above and T = 0.2s,

21. State Space 2 - Hany Ferdinando21 Matlab Use function expm to calculate the  (t) A = [0 -2; 1 -3]; T = 0.2 psy = expm(A*T)