Handout2

Handout2 - Part IB Paper 6: Information Engineering LINEAR...

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Unformatted text preview: Part IB Paper 6: Information Engineering LINEAR SYSTEMS AND CONTROL Glenn Vinnicombe HANDOUT 2 Impulse responses, step responses and transfer functions. G(s) transfer function g(t) impulse response u(s) u(t) Laplace transform pair y(s) = G(s) u(s) y(t) = integraldisplay t u()g(t )d = u(t) g(t) = g(t) u(t) 1 Summary The impulse response, step response and transfer function of a Linear, Time Invariant and causal (LTI) system each completely characterize the input-output properties of that system. Given the input to an LTI system, the output can be deterermined: In the time domain: as the convolution of the impulse response and the input. In the Laplace domain: as the multiplication of the transfer function and the Laplace transform of the input. They are related as follows: The step response is the integral of the impulse response. The transfer function is the Laplace transform of the impulse response. 2 Contents 2 Impulse responses, step responses and transfer functions. 1 2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.1 Definition of the impulse function . . . . . . . . . . 4 2.1.2 Properties of the impulse function . . . . . . . . . . 5 2.1.3 The Impulse and Step Responses . . . . . . . . . . . 6 2.2 The convolution integral . . . . . . . . . . . . . . . . . . . . 7 2.2.1 Direct derivation of the convolution integral . . . . . 7 2.2.2 Alternative statements of the convolution integral . 8 2.3 The Transfer Function (for ODE systems) . . . . . . . . . . . 9 2.3.1 Laplace transform of the convolution integral . . . . 10 2.4 The transfer function for any linear system . . . . . . . . . 11 2.5 Example: DC motor . . . . . . . . . . . . . . . . . . . . . . . 13 2.5.1 Impulse Response of the DC motor . . . . . . . . . . 15 2.5.2 Step Response of the DC motor . . . . . . . . . . . . 16 2.5.3 Deriving the step response from the impulse response 17 2.6 Transforms of signals vs Transfer functions of systems . . 17 2.7 Interconnections of LTI systems . . . . . . . . . . . . . . . . 18 2.7.1 Simplification of block diagrams . . . . . . . . . . . 19 2.8 More transfer function examples . . . . . . . . . . . . . . . 20 2.9 Key Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3 2.1 Preliminaries 2.1.1 Definition of the impulse function The impulse can be defined in many different ways, for example: a) t 1 T or b) t 1 2 T as . (area = 1 in each case.) t T taking limits, as = (t T) impulse at t = T (impulse occurs when argument = ) Note: The convergence, as , occurs in the sense that both functions a) and b) have the same properties in the limit: 4 2.1.2 Properties of the impulse function Consider a continuous function f(t) , and let h (t) denote the pulse approximation to the impulse ( (a) on previous page): h (t) = 1 / if / 2 < t < / 2 otherwise t 1...
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Handout2 - Part IB Paper 6: Information Engineering LINEAR...

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