Handout2

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

This preview shows pages 1–6. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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(τ)g(t − τ)dτ = 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 Δ Δ T or b) t 1 Δ 2 Δ T as Δ → . (area = 1 in each case.) t T taking “limits”, as Δ → = δ(t − T) impulse at t = T (impulse occurs when argument = ) Note: The “convergence”, as Δ → , 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 Δ (t) denote the pulse approximation to the impulse ( (a) on previous page): h Δ (t) = 1 / Δ if − Δ / 2 < t < Δ / 2 otherwise t 1 Δ...
View Full Document

{[ snackBarMessage ]}

Page1 / 22

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

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online