1. THE LAPLACE TRANSFORM AND TRANSFER FUNCTIONS1 / 11 THE LAPLACE TRANSFORM AND TRANSFER FUNCTIONS Prof Kang Li Email: K.Li@qub.ac.uk

2. THE LAPLACE TRANSFORM AND TRANSFER FUNCTIONS2 / 11 Lecture objectives:- Introduce transfer functions Introduce the Laplace Transform

3. THE LAPLACE TRANSFORM AND TRANSFER FUNCTIONS3 / 11 6.1Introduction We have seen how:- Input Actual system Output for example:- VaVa  can be represented by:- Input Physical model Output

4. THE LAPLACE TRANSFORM AND TRANSFER FUNCTIONS4 / 11 which can be analysed to give:- Input Mathematical model Output This form of mathematical model is not very convenient for studying control systems when one wishes to cascade elements to make up a complete system. It would be more convenient to have an algebraic expression which when multiplied by the input gave the output. This algebraic expression is called the transfer function. So the desired mathematical model would be:-

5. THE LAPLACE TRANSFORM AND TRANSFER FUNCTIONS5 / 11 Input Mathematical model Output Input Output ? and output = input X transfer function One method of transforming a differential equation into an algebraic one is the Laplace Transform. We will now introduce this transform.

6. THE LAPLACE TRANSFORM AND TRANSFER FUNCTIONS6 / 11 Definition where s is a complex operator  +j . In control systems it is usual to consider signals which are zero for t<0. So the lower limit in the integral can be changed to zero. This gives the single-sided form of the Laplace Transform. It has the added advantage that initial conditions are easily introduced. 6.2The Laplace Transform A Laplace transform always exists for a linear function of time which is physically realisable.

7. THE LAPLACE TRANSFORM AND TRANSFER FUNCTIONS7 / 11 Physical significance of the Laplace Transform. Just as a repetitive function of time can be represented by a series of sinusoids using Fourier analysis and a function which only repeats with an infinite period can be represented by a spectrum of sinusoids using the Fourier Transform so the Laplace Transform represents functions of time by an infinite series of sinusoids whose amplitudes are exponential functions of time.

8. THE LAPLACE TRANSFORM AND TRANSFER FUNCTIONS8 / 11 Find the Laplace Transform of a unit step. f(t)=0 for t<0 and f(t)=1 for t  0. From the definition: Example 6.1

9. THE LAPLACE TRANSFORM AND TRANSFER FUNCTIONS9 / 11 Find the Laplace Transform of an exponential function of time. f(t)=0 for t<0 and f(t) = for t  0. From the definition: Example 6.2 In this way the Laplace Transform for all common signal can be found and a table of useful transform pairs drawn up. (Homework - acquire a table for your own use.)

10. THE LAPLACE TRANSFORM AND TRANSFER FUNCTIONS10 / 11 Although an inverse Laplace Transform exists to transform a function of s back to a function of time, the usual method is by partial fractions and use of the standard transform pairs from a table. 6.3The Inverse Laplace Transform

11. THE LAPLACE TRANSFORM AND TRANSFER FUNCTIONS11 / 11 Find the function of time corresponding to Example 6.3 This can be expressed:- T t f(t)f(t) A 0.632A T t This is a function which rises exponentially with time.