1. EE 401 Control Systems Analysis and Design A Review of The Laplace Transform Wednesday 27 Aug 2014 EE 401: Control Systems Analysis and Design Slide 1 of 18

2. The Laplace Transform Wednesday 27 Aug 2014 Question:  What is the utility of this mathematical tool (the  T)? Answer: 1.It greatly simplifies the process of solving Linear Time- Invariant (LTI), homogeneous, Ordinary Differential Equations (ODEs) 2.Provides the basis for a “qualitative” evaluation of linear systems. The  T converts differential equations into algebraic equations EE 401: Control Systems Analysis and Design Slide 2 of 19

3. The Laplace Transform Wednesday 27 Aug 2014 EE 401: Control Systems Analysis and Design Slide 3 of 19

4. Some L T Examples Wednesday 27 Aug 2014 For now we will ignore this Region of Convergence. Region of Convergence 0 iff Re{s}>a1 EE 401: Control Systems Analysis and Design Slide 4 of 19

5. Some L T Examples Wednesday 27 Aug 2014 EE 401: Control Systems Analysis and Design Slide 5 of 19

6. Some L T Examples Wednesday 27 Aug 2014 Recall Euler’s Formula EE 401: Control Systems Analysis and Design Slide 6 of 19

7. Some L T Examples Wednesday 27 Aug 2014 u dv Integration by Parts EE 401: Control Systems Analysis and Design Slide 7 of 19

8. Using MATLAB Directly Symbolic Toolbox Using MATLAB MuPAD notepad MATLAB Symbolic Editor Sometimes different syntax Some L T Examples The MATLAB Symbolic Math Toolbox Wednesday 27 Aug 2014 EE 401: Control Systems Analysis and Design Slide 8 of 19

9. Some L T Examples The MATLAB Symbolic Math Toolbox Wednesday 27 Aug 2014 Using Mathematica EE 401: Control Systems Analysis and Design Slide 9 of 19

10. Some L T Examples Wednesday 27 Aug 2014 Note: We need to include t = 0 in the integral. See Handout “Laplace_Transform_Table.pdf” for a table of Laplace Transforms.Laplace_Transform_Table.pdf” EE 401: Control Systems Analysis and Design Slide 10 of 19

11. Some Properties of the L T Wednesday 27 Aug 2014 Linearity: Example: EE 401: Control Systems Analysis and Design Slide 11 of 19

12. Some Properties of the L T Wednesday 27 Aug 2014 Differentiation: du v F(s)F(s) EE 401: Control Systems Analysis and Design Slide 12 of 19 InputOutput  + - InputOutput

13. Some Properties of the L T Wednesday 27 Aug 2014 Integration: u dv InputOutput EE 401: Control Systems Analysis and Design Slide 13 of 19 InputOutput

14. Some Properties of the L T Wednesday 27 Aug 2014 Convolution: InputOutput EE 401: Control Systems Analysis and Design Slide 14 of 19 InputOutput

15. Inverting the Laplace Transform Wednesday 27 Aug 2014 Inverting the Laplace Transform: Use the tables instead!! DO NOT USE THIS FORMULA EE 401: Control Systems Analysis and Design Slide 15 of 19

16. Inverting the Laplace Transform Example #1: Wednesday 27 Aug 2014 Solve the first order ODE (ordinary differential equation) Take the L T of the equation. = 1 Unforced Response (due to initial conditions) Forced Response (due to input 3u(t)) EE 401: Control Systems Analysis and Design Slide 16 of 19

17. Inverting the Laplace Transform Example #2: Wednesday 27 Aug 2014 Solve the second order ODE (ordinary differential equation) Take the L T of the equation. EE 401: Control Systems Analysis and Design Slide 17 of 19

18. Inverting the Laplace Transform Example #3: Wednesday 27 Aug 2014 Partial fraction expansion for the case of complex roots EE 401: Control Systems Analysis and Design Slide 18 of 19

19. Inverting the Laplace Transform Example #3: Wednesday 27 Aug 2014 Partial fraction for the case of complex roots This result can be further simplified: EE 401: Control Systems Analysis and Design Slide 19 of 19