9.2 The Input-Output Description of a System

Transcription

1 Lecture Notes on Control Systems/D. Ghose/ The Input-Output Description of a System The input-output description of a system can be obtained by first defining a delta function, the representation of the input signal as a collection of delta functions, the impulse response of the system, and the convolution integral. Finally, we will show how Laplace transform is used as a tool to find the system output of a signal easily Delta Function Consider a pulse of width Δ and of height 1/Δ applied at time t i. 1 Δ + Δ time Figure 9.6: A unit pulse δ Δ (t t i ) 1 Δ [u(t t i) u(t t i Δ), when t<t i 1 Δ, when t i t<t i +Δ, when t t i +Δ The delta function has the following properties: 1. The pulse δ Δ (t t i ) has a unit area for all values of Δ. 2. In the limit, as Δ, the pulse δ Δ (t t i ) becomes the impulse function δ(t t i ), which is also called the Dirac delta function or simply the δ-function. In the limit, as Δ, the Laplace transform of δ Δ (t t i )ise st i, which is the Laplace transform of an impulse δ(.) delayed by the time t i. We will show this below.

2 Lecture Notes on Control Systems/D. Ghose/ L[δ Δ (t t i ) L[ 1 Δ {u(t t i) u(t t i Δ)} 1 [ 1 Δ s e st i 1 i+δ) s e s(t 1 [ ( ) e st i 1 e Δs Δs As Δ, we get the impulse function at t t i. So, the Laplace transform of the impulse function is, lim L[δ Δ(t t i ) 1 Δ s e st i 1 ( lim ) 1 e Δs Δ Δ 1 [ ( s e st i 1 lim 1 1 sδ+ s2 Δ 2 Δ Δ 2! e st i s3 Δ 3 3! + ) So, e st i is the Laplace transform of an impulse δ(.), delayed by time t i.or, L[δ(t t i ) e st i If t i, then e st i 1 and, L[δ(t) 1 3. For any piecewise continuous function f(t), δ(t t i ) satisfies the sifting or the picking property, f(t)δ(t t i )dt f(t i ) Representation of a signal as a sum of delta functions This way we may approximate a (piecewise continuous) function r(t), where r(t) for t<, as a series of pulse functions. Each pulse is given by, r Δ (t i )r(t i )δ Δ (t t i )Δ where, δ Δ (t t i )Δ is a pulse of unit amplitude. Then, r(t) may be approximated as, r(t i )δ Δ (t t i )Δ i Thus, r(t) can be represented as a linear combination of an infinite number of delta functions. And as Δ, r(t) is given by an integral, r(t) lim r(t i )δ Δ (t t i )Δr(t i )δ Δ (t t i )Δ r()δ(t )d Δ i

3 Lecture Notes on Control Systems/D. Ghose/ Figure 9.7: Approximation of a signal with pulses The Convolution Integral Let G be a functional 1 that specifies uniquely the system output in terms of the input. Assume that the system, represented by G, has the following properties: Linear: G(α 1 r 1 + α 2 r 2 )α 1 G(r 1 )+α 2 G(r 2 ) Time Invariant: The input-output relationship can be shifted in time. So, If r(t) produces y(t), then r(t Δ) produces y(t Δ). Relaxed: The system is relaxed, which means that the output y(t) fort issolelyand uniquely excited by the input r(t) applied at t. Now, if we approximate r(t) by the sum of the pulse functions as r(t) r Δ (t i ) r(t i )δ Δ (t t i )Δ then the system output y(t) may be approximated as, [ y(t) G r Δ (t i ) i G i [ i r(t i )δ Δ (t t i )Δ i 1 A functional is a function of a function. For example, G[r(t) is a functional.

4 Lecture Notes on Control Systems/D. Ghose/ Figure 9.8: The time invariance property G [δ Δ (t t i ) r(t i )Δ i Now, if we take Δ, Here, y(t) g(t )r()d, g(t ) G[δ(t ) t> is the impulse response and is the output at time t due to an impulse δ(t ) applied at. Since the system is causal (that is, inputs after time t cannot affect the output at time t), we can write, y(t) g(t )r()d (9.2) So, the output y(t) due to an input r() applied from to t is given by (9.2), which is called the convolution integral, and is often denoted as, y(t) g(t) r(t) g(t )r()d g()r(t )d The last equality also demonstrates an important property of the convolution integral 2. 2 Consider g(t )r()d

5 Lecture Notes on Control Systems/D. Ghose/ Let us find the Laplace transform of y(t). [ L[y(t) L g(t )r()d [ L g()r(t )d [ L g()r(t )d We can write the last part since r(t ) for>t(since the system is relaxed). Now. applying the Laplace transform formula, [ L[y(t) g()r(t )d e st dt Changing the order of integration, L[y(t) g()r(t )e st dtd g()r(t )e st ddt g()r(t )e s e s(t ) dtd { } g()e s r(t )e s(t ) dt d { g()e s r(t )e s(t ) dt + Since the system is relaxed, we have r(t ) fort<, and so, { } L[y(t) g()e s r(t )e s(t ) dt d Performing a change of variables, α t dt dα, { } L[y(t) g()e s r(α)e sα dα d So, Let t β. Then, d dβ. So, g()e s d Y (s) G(s)R(s) r(α)e sα dα } r(t )e s(t ) dt d g(t )r()d t g(β)r(t β)dβ g(β)r(t β)dβ which proves the property.

6 Lecture Notes on Control Systems/D. Ghose/ Thus, Convolution in the time domain multiplication in the s-domain So, if the system response in the time domain is given by the integral y(t) then in the s domain it is simplified to, g(t )u()d Y (s) G(s)U(s) where, G(s) is called the system transfer function and is the Laplace transform of the impulse response function g(t). 9.3 Performance Criteria Systems modelled by linear time-invariant differential equations are generally evaluated against the following criteria: 1. Time response 2. Behaviour due to disturbances 3. Stability Laplace transform is a tool that allows easy evaluation of these criteria. 1. It transforms differential and integral operations to algebraic operations. 2. It transforms linear time-invariant (LTI) differential equations into algebraic equations. 3. It has a provision to include initial conditions systematically. 4. It views system behaviour in the frequency domain rather than in the time domain. 9.4 Laplace Transforms of Derivatives and Integrals L [ df df dt dt e st dt

7 Lecture Notes on Control Systems/D. Ghose/ Integrating by parts, L [ df dt f(t)e st f(t)e st ( s)dt f() + sf (s) sf (s) f() In general, [ d n f L s n F (s) s n 1 f() s n 2 f (1) () f (n 1) () dt n where, f (i) () di f dt i t For integrals, [ L f()d F (s) s Some Examples. dx(t) dt + 1 x(t) br(t), x() x Let L[x(t) X(s) andl[r(t) R(s). Taking Laplace transforms on both sides, [ dx(t) L + 1 dt x(t) L [br(t) [ dx(t) L + 1 L [x(t) bl [r(t) dt Now, Find the inverse Laplace transform, sx(s) x() + 1 X(s) br(s) ( X(s) s + 1 ) x() + br(s) X(s) x() s b s + 1 R(s)

8 Lecture Notes on Control Systems/D. Ghose/ x(t) x e t + b e t α r(α)dα How did we obtain the above? By using the convolution integral 3. 3 We know, Further, L [ e at 1 s + a Y (s) G(s)U(s) y(t) g(t α)u(α)dα