State Space Representation

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

2. Overview • Introduction • State variable • Signal Flow Graph (revisited) • State Space (SS) matrix • Transfer function from SS • SS in Matlab State Space 1 - Hany Ferdinando

3. Introduction • State Space (SS) represents a dynamic system in matrices • It is MIMO (multi input multi output) system (transfer function is SISO – single input single output) • With SS, one can get future condition of a dynamic system State Space 1 - Hany Ferdinando

4. Examples • On-off switch: • There are only two positions • The state of the switch can assume one of two possible states • if the present state is ‘on’, then one can know the future state State Space 1 - Hany Ferdinando

5. State Variable Definition: is a set of variables such that the knowledge of these variables and the input functions will, with the equations describing the dynamics, provide the future state and output of the system State Space 1 - Hany Ferdinando

6. Model of A Dynamic System • Model of a dynamic system is in nth-order differential equation • The order depends on the number of storage element in that system • SS will change one nth-order differential equation into n first-order differential equations State Space 1 - Hany Ferdinando

7. Example i(t) State Space 1 - Hany Ferdinando

8. Example If x1 = vC and x2 = iL then State Space 1 - Hany Ferdinando

9. Example Matrix A Matrix B State Space 1 - Hany Ferdinando

10. General Form of SS State Space 1 - Hany Ferdinando

11. Signal Flow Graph (revisited) Sum of the forward-path factors Sum of the feedback loop factors State Space 1 - Hany Ferdinando

12. Signal Flow Graph State • From G(s) get the order of the equation • Make the denominator of G(s) in the form of 1-(sum of the feedback loop) by dividing G(s) with sn (n is the order of G(s)) • Draw the new SFG State Space 1 - Hany Ferdinando

13. Example State Space 1 - Hany Ferdinando

14. b3 b2 b1 x4 1/s 1/s 1/s 1/s b0 x3 x2 x1 -a3 U(s) Y(s) -a2 -a1 -a0 Example Phase Variable Format State Space 1 - Hany Ferdinando

15. Example State Space 1 - Hany Ferdinando

16. b3 b2 b1 x4 1/s 1/s 1/s 1/s b0 x3 x2 x1 U(s) -a3 Y(s) -a2 -a1 -a0 Example Input Feedforward Format State Space 1 - Hany Ferdinando

17. Example State Space 1 - Hany Ferdinando

18. TF from SS State Space 1 - Hany Ferdinando

19. SS in Matlab (basic) • Use two functions: • tf2ss to convert transfer function model into state space model • ss2tf to convert state space model into transfer function model State Space 1 - Hany Ferdinando