1. Digital Control Systems
State Space Analysis(2)

2. STATE SPACE REPRESENTATIONS OF DISCRETE-TIME SYS
Nonuniqueness of State Space Representations

3. STATE SPACE REPRESENTATIONS OF DISCRETE-TIME SYS
Nonuniqueness of State Space Representations ≡

4. SOLVING DISCRETE TIE STATE-SPACE EQUATIONS
Solution of LTI Discrete-Tim State Equations x(k) or any positive integer k may be obtined directly by recursion, as follows:

5. SOLVING DISCRETE TIE STATE-SPACE EQUATIONS
State Transition Matrix It is possible to write the solution of the homogeneous state equation as state transition matrix(fundamental matrix) :

6. SOLVING DISCRETE TIE STATE-SPACE EQUATIONS
State Transition Matrix

7. SOLVING DISCRETE TIE STATE-SPACE EQUATIONS
z Transform Approach to the Solution of Discrete-Time State Equations

8. SOLVING DISCRETE TIE STATE-SPACE EQUATIONS
z Transform Approach to the Solution of Discrete-Time State Equations Example: a) b)

9. SOLVING DISCRETE TIE STATE-SPACE EQUATIONS
z Transform Approach to the Solution of Discrete-Time State Equations Example: a)

10. SOLVING DISCRETE TIE STATE-SPACE EQUATIONS
z Transform Approach to the Solution of Discrete-Time State Equations Example: a)

11. SOLVING DISCRETE TIE STATE-SPACE EQUATIONS
z Transform Approach to the Solution of Discrete-Time State Equations Example: a)

12. SOLVING DISCRETE TIE STATE-SPACE EQUATIONS
z Transform Approach to the Solution of Discrete-Time State Equations Example: a)

13. SOLVING DISCRETE TIE STATE-SPACE EQUATIONS
Solution of LTV Discrete-Time State Equations solution of x(k) may be found easily by recusion State transition matrix

14. SOLVING DISCRETE TIE STATE-SPACE EQUATIONS
Solution of LTV Discrete-Time State Equations

15. SOLVING DISCRETE TIE STATE-SPACE EQUATIONS
Solution of LTV Discrete-Time State Equations

16. SOLVING DISCRETE TIE STATE-SPACE EQUATIONS
Solution of LTV Discrete-Time State Equations Properties of

17. PULSE TRANSFER FUNCTION MATRIX

18. PULSE TRANSFER FUNCTION MATRIX
Similarity Transformation:
The pulse transfer function matrix is invariant under simiarity transformation.
The pulse transfer function does not depend on the particular state vector.

19. DISCRETIZATION OF CONT. TIME STATE SPACE EQUATIONS

20. DISCRETIZATION OF CONT. TIME STATE SPACE EQUATIONS
Solution of Continuous Time State Equations
Properties of matrix exponential

21. DISCRETIZATION OF CONT. TIME STATE SPACE EQUATIONS
Solution of Continuous Time State Equations

22. DISCRETIZATION OF CONT. TIME STATE SPACE EQUATIONS
Discrete-time representation of
Discretization of Continuous Time State Equations

23. DISCRETIZATION OF CONT. TIME STATE SPACE EQUATIONS
Discretization of Continuous Time State Equations

24. DISCRETIZATION OF CONT. TIME STATE SPACE EQUATIONS
Multiplying (2) by eAT and subtracting it from (1) gives:
Discretization of Continuous Time State Equations
Remember:
(1)
(2)

25. DISCRETIZATION OF CONT. TIME STATE SPACE EQUATIONS
Discretization of Continuous Time State Equations
G(T),H(T) depend on the sampling period C and D are constant matrices and do not depend on the sampling period T.

26. DISCRETIZATION OF CONT. TIME STATE SPACE EQUATIONS
Discretization of Continuous Time State Equations
Example:
This result agrees with the z transform of G(s), where it is preceded by a sampler and zero order hold

27. DISCRETIZATION OF CONT. TIME STATE SPACE EQUATIONS
Discretization of Continuous Time State Equations
Example:

28. DISCRETIZATION OF CONT. TIME STATE SPACE EQUATIONS
Discretization of Continuous Time State Equations
Example:

29. DISCRETIZATION OF CONT. TIME STATE SPACE EQUATIONS
Discretization of Continuous Time State Equations
Example:
When T=1

30. DISCRETIZATION OF CONT. TIME STATE SPACE EQUATIONS
MATLAB Approach to the Discretization of Continuous Time State Equations
Note:
Default format is format short
For more accuracy use format long
Example:
G and H differs for a different sampling period