Chapter 3. Dynamics

The modes thus correspond to the terms of the solution to the homogeneous equation D3.6E and the terms of the impulse response D3.15E and the step response.

If s β is a zero of BDsE and uDtE Ceβ t, then it follows that

A zero of BDsE at s β blocks the transmission of the signal uDtE Ceβ t.

3.4 Laplace Transforms

The Laplace transform is very convenient for dealing with linear timeinvariant system. The reason is that it simplifies manipulations of linear systems to pure algebra. It also a natural way to introduce transfer functions and it also opens the road for using the powerful tools of the theory of complex variables. The Laplace transform is an essential element of the language of control.

The Laplace Transform

Consider a function f defined on 0 t [ and a real number σ 0. Assume that f grows slower than eσ t for large t. The Laplace transform F L f of f is defined as

Z [

L f FDsE e−st f DtEdt

0

We will illustrate computation of Laplace transforms with a few examples

Differentiating the above equation we find that the transform of the function f2DtE te −at is

1

F2DsE Ds aE2

Repeated differentiation shows that the transform of the function f3DtE

tn e−at is

Dn − 1E! F3DsE Ds aEn

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3.4 Laplace Transforms

Setting a 0 in f1 we find that the transform of the unit step function f4DtE 1 is

1

F4DsE s

Similarly we find by setting a 0 in f3 that the transform of f5 tn is

n! F5DsE sn 1

Setting a ib in f1 we find that the transform of f DtE e−ibt cos bt − i sin bt is

Separating real and imaginary parts we find that the transform of f6DtE sin bt and f7DtE cos bt are

Proceeding in this way it is possible to build up tables of transforms that are useful for hand calculations.

Properties of Laplace Transforms The Laplace transform also has many useful properties. First we observe that the transform is linear because

Z [ Z [

LDa f bnE aFDsE bFDsE a e−st f DtEdt b e−stnDtEdt

Z [ 0 0

e−stDa f DtE bnDtEEdt aL f bLn

0

Next we will calculate the transform of the derivative of a function, i.e. f PDtE d dtf DtE . We have

where the second equality is obtained by integration by parts. This formula is very useful because it implies that differentiation of a time function corresponds to multiplication of the transform by s provided that the

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Chapter 3. Dynamics

initial value f D0E is zero. We will consider the transform of an integral

Z [

yDtE nDt −τ EuDτ Edτ

0

see D3.18E. We will now consider the Laplace transform of such an expression. We have

Z [ Z [ Z [

YDsE e−st yDtEdt e−st nDt −τ EuDτ Edτ dt

Z0[ Z t 0 0

e−sDt−τ E e−sτ nDt −τ EuDτ Edτ dt

[Z [

e−sτ uDτ Edτ e−stnDtEdt GDsEU DsEZ0 0

The description of a linear time-invariant systems thus becomes very simple when working with Laplace transforms.

Next we will consider the effect of a time shift. Let the number a be positive and let the function fa be a time shift of the function f , i.e.

The Laplace transform of fa is given by

Z [ Z [

FaDsE e−st f Dt − aEdt e−st f Dt − aEdt

Z0[ a Z [

e−as e−sDt−aE f Dt − aEdt e−as e−st f DtEdt e−as FDsE

a 0

D3.20E Delaying a signal by a time units thus correspond to multiplication of its Laplace transform by e−as.

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3.4 Laplace Transforms

The behavior of time functions for small arguments is governed by the behavior of the Laplace transform for large arguments. This is expressed by the so called initial value theorem.

This holds provided that the limit exists.

The converse is also true which means that the behavior of time functions for large arguments is governed by the behavior of the Laplace transform for small arguments. Final value theorem. Hence

These properties are very useful for qualitative assessment of a time functions and Laplace transforms.

Linear Differential Equations

The differentiation property L ddtf sL f − f D0E makes the Laplace transform very convenient for dealing with linear differential equations. Consider for example the system

ddty ay bu

Taking Laplace transforms of both sides give

sYDsE − yD0E aYDsE bU DsE

Solving this linear equation for YDsE gives

YDsE sy−D0Ea s −b a U DsE

Transforming back to time function we find

Z t

yDtE eat yD0E b eaDt−τ EuDτ Gdτ

0

To convert the transforms to time functions we have used the fact that

the transform

1

s − a

corresponds to the time function eat and we have also used the rule for transforms of convolution.

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EXAMPLE 3.2—A TIME DELAY

Consider a system which simply delays the input T time units. It follows from 3.20 that the input output relation is

YDsE e−sT U DsE

The transfer function of a time delay is thus

GDsE UYDDssEE e−sT

It is also possible to calculate the transfer functions for systems described by partial differential equations.

EXAMPLE 3.3—THE HEAT EQUATION

T

GDsE e− sT

GDsE 1T cosh sT

Transfer functions and Laplace transforms are ideal to deal with block diagrams for linear time-invariant systems. We have already shown that a block is simply characterized by

YDsE GDsEU DsE

The transform of the output of a block is simply the product of the transfer function of the block and the transform of the input system. Algebraically this is equivalent to multiplication with a constant. This makes it easy to find relations between the signals that appear in a block diagram. The combination of block diagrams and transfer functions is a very nice combination because they make it possible both to obtain an overview of a system and to guide the derivation of equations for the system. This is one of the reasons why block diagrams are so widely used in control.

Notice that it also follows from the above equation that signals and systems have the same representations. In the formula we can thus consider n as the input and u as the transfer function.

To illustrate the idea we will consider an example.

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Chapter 3. Dynamics

−1

Figure 3.5 Block diagram of a feedback system.

EXAMPLE 3.4—RELATIONS BETWEEN SIGNALS IN A BLOCK DIAGRAM

Consider the system in Figure 3.5. The system has two blocks representing the process P and the controller C. There are three external signals, the reference r, the load disturbance d and the measurement noise n. A typical problem is to find out how the error e related to the signals r d and n? Introduce Laplace transforms and transfer functions. To obtain the desired relation we simply trace the signals around the loop. Starting with the signal e and tracing backwards in the loop we find that e is the difference between r and y, hence E R − Y. The signal y in turn is the sum of n and the output of the block P, hence Y N PDD V E. Finally the signal v is the output of the controller which is given by V PE. Combining the results we get

E R − N PDD C EE

With a little practice this equation can be written directly. Solving for E gives

Notice the form of the equations and the use of superposition.

Simulating LTI Systems

Linear time-invariant systems can be conveniently simulated using Matlab. For example a system with the transfer function

5s 2

GDsE s2 3s 2

is introduced in matlab as

G=tf([5 2],[1 3 2])

The command step(G) gives the step response of the system.

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3.4 Laplace Transforms

Transfer Functions

The transfer function of a linear system is defined as

where U DsE Lu is the Laplace transform of the input u and YDsE L y is the Laplace transform of the output y. The Laplace transforms are computed under the assumption that all initial conditions are zero.

Circuit Analysis

Laplace transforms are very useful for circuit analysis. A resistor is described by the algebraic equation

V RI

but inductors and capacitors are describe by the linear differential equations

Z t

CV IDτ Gdτ

0

L dIdt V

Taking Laplace transforms we get

LV RI

LV sC1 L I

LV sLL I

The transformed equations for all components thus look identical, the transformed voltage LV is a generalized impedance Z multiplied by the transformed current L I. The impedance is

ZDsE R for a resistor

ZDsE sC1 for a capacitor

ZDsE sL for an inductor

Operating with the transforms we can thus pretend that all elements of a circuit is a resistor which means that circuit analysis is reduced to pure algebra. This is just another illustration of the fact that differential equations are transformed to algebraic equations. We illustrate the procedure by an example.

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