1. Laplace Transforms of Linear Control Systems Eng R. L. Nkumbwa Copperbelt University 2010

2. 3/1/2016 7:44:37 PMEng. R. L. Nkumbwa @CBU 2010 2 Transforms So, What are Transforms? A transform is a mathematical tool that converts an equation from one variable (or one set of variables) into a new variable (or a new set of variables). To do this, the transform must remove all instances of the first variable, the "Domain Variable", and add a new "Range Variable".

3. 3/1/2016 7:44:37 PMEng. R. L. Nkumbwa @CBU 2010 3 Transforms Integrals are excellent choices for transforms, because the limits of the definite integral will be substituted into the domain variable, and all instances of that variable will be removed from the equation. An integral transform that converts from a domain variable a to a range variable b will typically be formatted as such:

4. 3/1/2016 7:44:37 PMEng. R. L. Nkumbwa @CBU 2010 4 Transforms

5. 3/1/2016 7:44:37 PMEng. R. L. Nkumbwa @CBU 2010 5 Mathematical Transformations

6. 3/1/2016 7:44:37 PMEng. R. L. Nkumbwa @CBU 2010 6 Why use the Laplace Transform? In many cases, the indirect Laplace transform approach is easier than the direct approach. From the transformed algebraic equation, we get a transfer function, which represent the input- output relation of the system. Classical control theory has been built on the concept of transfer function. Frequency response (useful for analysis and/or design) can be obtained easily from the transfer function.

7. 3/1/2016 7:44:37 PMEng. R. L. Nkumbwa @CBU 2010 7 Laplace Concepts The Laplace transform (LT) is a mathematical transformation. Basically, the Laplace transform allows us to represent a signal, f(t), as a continuum of damped sinusoids for t ≥ 0. Calculus (derivatives, integrals) becomes algebra in the Laplace-domain, or s- domain.

8. 3/1/2016 7:44:37 PMEng. R. L. Nkumbwa @CBU 2010 8 Laplace Transforms The Laplace Transform converts an equation from the time-domain into the so-called "s- domain", or the Laplace domain, or even the "Complex domain". Transform can only be applied under the following conditions:

9. 3/1/2016 7:44:37 PMEng. R. L. Nkumbwa @CBU 2010 9 Transforms Conditions Transform can only be applied under the following conditions: – The system or signal in question is analog. – The system or signal in question is Linear. – The system or signal in question is Time-Invariant. – The system or signal in question is causal.

10. 3/1/2016 7:44:37 PMEng. R. L. Nkumbwa @CBU 2010 10 System in time-domain

11. 3/1/2016 7:44:37 PMEng. R. L. Nkumbwa @CBU 2010 11 In the time domain where “ * ” represents a convolution operation, which involves an integral. It is usually difficult to model a system represented by a differential equation as a block diagram.

12. 3/1/2016 7:44:37 PMEng. R. L. Nkumbwa @CBU 2010 12 System in the Laplace (or s) domain

13. 3/1/2016 7:44:37 PMEng. R. L. Nkumbwa @CBU 2010 13 In the Laplace (or s) domain This is a convenient form as the input, output and system are separate entities. This is particularly convenient to represent the interconnection of several subsystems.

14. 3/1/2016 7:44:37 PMEng. R. L. Nkumbwa @CBU 2010 14 Definition of the Laplace Transform We consider a function, f(t) that satisfies:

15. 3/1/2016 7:44:37 PMEng. R. L. Nkumbwa @CBU 2010 15 Definition of the Laplace Transform Laplace transform results have been tabulated extensively. More information on the Laplace transform, including a transform table can be found in Mathematics books. H.K Dass and Stroud are recommended.

16. 3/1/2016 7:44:37 PMEng. R. L. Nkumbwa @CBU 2010 16 Laplace Transformation

17. 3/1/2016 7:44:37 PMEng. R. L. Nkumbwa @CBU 2010 17 Laplace Transformation

18. 3/1/2016 7:44:37 PMEng. R. L. Nkumbwa @CBU 2010 18 Note: The Laplace domain is sometimes called the complex frequency domain, to differentiate it from the “simple” frequency domain obtained when using the Fourier transform.

19. 3/1/2016 7:44:37 PMEng. R. L. Nkumbwa @CBU 2010 19 Inverse Laplace Transform

20. 3/1/2016 7:44:37 PMEng. R. L. Nkumbwa @CBU 2010 20 Useful Laplace transform pairs/tables

21. 3/1/2016 7:44:37 PMEng. R. L. Nkumbwa @CBU 2010 21 Properties of the Laplace Transform Convolution/Product Equivalence Differentiation Theorem (Important) Linear Superposition and Homogeneity Time and Frequency Shift Theorems Initial Value Theorem (Important) Final Value Theorem (Important)

22. 3/1/2016 7:44:37 PMEng. R. L. Nkumbwa @CBU 2010 22 Superposition: – {a f1(t ) + b f2(t )} = a F1(s ) + b F2(s ). Time delay: – { f (t − τ )} = e−sτ F (s ).

23. 3/1/2016 7:44:37 PMEng. R. L. Nkumbwa @CBU 2010 23 Research Activity In groups of three, do a detailed research on Laplace Transforms and Inverse Transforms with full knowledge of their properties mentioned above.

24. 3/1/2016 7:44:37 PMEng. R. L. Nkumbwa @CBU 2010 24 Partial Fraction Expansion Laplace transform pairs are extensively tabulated, but frequently we have transfer functions and other equations that do not have a tabulated inverse transform. If our equation is a fraction, we can often utilize Partial Fraction Expansion (PFE) to create a set of simpler terms that will have readily available inverse transforms.

25. 3/1/2016 7:44:37 PMEng. R. L. Nkumbwa @CBU 2010 25 Note This topic is purely mathematics and your are advised to consult your Mathematics Lecturer for detailed knowledge.