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

3/1/2016 7:44:37 PMEng. R. L 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".

3/1/2016 7:44:37 PMEng. R. L 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:

3/1/2016 7:44:37 PMEng. R. L Transforms

3/1/2016 7:44:37 PMEng. R. L Mathematical Transformations

3/1/2016 7:44:37 PMEng. R. L 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.

3/1/2016 7:44:37 PMEng. R. L 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.

3/1/2016 7:44:37 PMEng. R. L 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:

3/1/2016 7:44:37 PMEng. R. L 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.

3/1/2016 7:44:37 PMEng. R. L System in time-domain

3/1/2016 7:44:37 PMEng. R. L 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.

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

3/1/2016 7:44:37 PMEng. R. L 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.

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

3/1/2016 7:44:37 PMEng. R. L 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.

3/1/2016 7:44:37 PMEng. R. L Laplace Transformation

3/1/2016 7:44:37 PMEng. R. L Laplace Transformation

3/1/2016 7:44:37 PMEng. R. L 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.

3/1/2016 7:44:37 PMEng. R. L Inverse Laplace Transform

3/1/2016 7:44:37 PMEng. R. L Useful Laplace transform pairs/tables

3/1/2016 7:44:37 PMEng. R. L 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)

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

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

3/1/2016 7:44:37 PMEng. R. L 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.

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