Analysis and Control of Linear Systems - Chapter 1 pptx

The use of the convolution relation leads us to conclude that the unit-step response is the integral of the impulse response: This response is generally characterized by: – the rise time

System Analysis

Chapter 1 Transfer Functions and Spectral Models

1.1 System representation

A system is an organized set of components, of concepts whose role is to perform one or more tasks The point of view adopted in the characterization of systems is to deal only with the input-output relations, with their causes and effects, irrespective of the physical nature of the phenomena involved

Hence, a system realizes an application of the input signal space, modeling magnitudes that affect the behavior of the system, into the space of output signals, modeling relevant magnitudes for this behavior

Figure 1.1 System symbolics

In what follows, we will consider mono-variable, analog or continuous systems which will have only one input and one output, modeled by continuous signals

Chapter written by Dominique BEAUVOIS and Yves TANGUY

1.2 Signal models

A continuous-time signal (t ∈ is represented a priori through a function x(t) R)defined on a bounded interval if its observation is necessarily of finite duration When signal mathematical models are built, the intention is to artificially extend this observation to an infinite duration, to introduce discontinuities or to generate Dirac impulses, as a derivative of a step function The most general model of a continuous-time signal is thus a distribution that generalizes to some extent the concept of a digital function

1.2.1 Unit-step function or Heaviside step function U(t)

This signal is constant, equal to 1 for the positive evolution variable and equal to

0 for the negative evolution variable

U(t)

1

t

Figure 1.2 Unit-step function

This signal constitutes a simplified model for the operation of a device with a very low start-up time and very high running time

1.2.2 Impulse

Physicists began considering shorter and more intense phenomena For example,

an electric loading Mµ can be associated with a mass M evenly distributed according to an axis

What density should be associated with a punctual mass concentrated in 0? This density can be considered as the bound (simple convergence) of densities Mµ σn( )

verifying:

( )

2( ) 0 elsewhere

( )

δ σ σδ

δ is called a Dirac impulse and it represents the most popular distribution This

impulse δ is also written δ( )t

For a time lag t o, we will use the notations δ(t−t o) or δ( )t o ( )t ; the impulse is graphically “represented” by an arrow placed in t =t o, with a height proportional to the impulse weight

In general, the Dirac impulse is a very simplified model of any impulse phenomenon centered in t =t o, with a shorter period than the time range of the systems in question and with an area S

Figure 1.3 Modeling of a short phenomenon

We notice that in the model based on Dirac impulse, the “microscopic” look of the real signal disappears and only the information regarding the area is preserved Finally, we can imagine that the impulse models the derivative of a unit-step function To be sure of this, let us consider the step function as the model of the real signal u o (t) represented in Figure 1.4, of derivative u o′(t) Based on what has been previously proposed, it is clear that

f designates the frequency expressed in Hz, ωo = 2π f designates the impulse o

expressed in rad/s and ϕ the phase expressed in rad

A real value sine-wave signal is entirely characterized by f o (0≤ f o ≤+∞), by

A (t = t o), by ϕ (−π ≤ϕ≤+π) On the other hand, a complex value sine-wave signal is characterized by a frequency f o with −∞≤ f o ≤+∞

1.3 Characteristics of continuous systems

The input-output behavior of a system may be characterized by different relations with various degrees of complexity In this work, we will deal only with

linear systems that obey the physical principle of superposition and that we can

define as follows: a system is linear if to any combination of input constant

coefficients ∑a i x i corresponds the same output linear combination,

( )

∑a i y i =∑a i G x i

Obviously, in practice, no system is rigorously linear In order to simplify the

models, we often perform linearization around a point called an operating point of

the system

A system has an instantaneous response if, irrespective of input x, output y depends only on the input value at the instant considered It is called dynamic if its

response at a given instant depends on input values at other instants

A system is called causal system if its response at a given instant depends only

on input values at previous instants (possibly present) This characteristic of causality seems natural for real systems (the effect does not precede the cause), but, however, we have to consider the existence of systems which are not strictly causal

in the case of delayed time processing (playback of a CD) or when the evolution variable is not time (image processing)

The pure delay system τ >0 characterized by y( ) (t =x t−τ) is a dynamic system

1.4 Modeling of linear time-invariant systems

We will call LTI such a system The aim of this section is to show that the output relation in an LTI is modeled by a convolution operation

input-1.4.1 Temporal model, convolution, impulse response and unit-step response

We will note by hτ(t) the response of the real impulse system represented in Figure 1.5

1–

τ

Figure 1.5 Response to a basic impulse

Let us approach any input x (t) by a series of joint impulses of width τ and amplitude x k ( )τ

x(t)

Figure 1.6 Step approximation

By applying the linearity and invariance hypotheses of the system, we can approximate the output at an instant t by the following amount, corresponding to the recombination of responses to different impulses that vary in time:

where h (t), the response of the system to the Dirac impulse, is a characteristic of

the system’s behavior and is called an impulse response

If we suppose that the system preserves the continuity of the input, i.e for any convergent sequence x t we have n( )

response, which considers the past in order to provide the present, is a causal function and the input-output relation has the following form:

1.4.3 Unit-step response

The unit-step response of a system is its response i (t) to a unit-step excitation The use of the convolution relation leads us to conclude that the unit-step response is the integral of the impulse response:

This response is generally characterized by:

– the rise time t , which is the time that separates the passage of the unit-step m

response from 10% to 90% of the final value;

– the response timet r, also called establishment time, is the period at the end of which the response remains in the interval of the final value ±α% A current value

of α is 5% This time also corresponds to the period at the end of which the impulse response remains in the interval ±α%; it characterizes the transient behavior of the system output when we start applying an excitation and it also reminds that a system has several inputs which have been applied before a given instant;

– the possible overflow defined as

)(

)(

A system is labeled as stable around a point of balance if, after having been subjected to a low interference around that point, it does not move too far away from

it We talk of asymptotic stability if the system returns to the point of balance and of stability, in the broad sense of the word, if the system remains some place near that point This concept, intrinsic to the system, which is illustrated in Figure 1.7 by a ball positioned on various surfaces, requires, in order to be used, a representation by equations of state

Asymptotic stability Stability in the broad sense Unstable

Figure 1.7 Concepts of stability

Another point of view can be adopted where the stability of a system can be defined simply in terms of an input-output criterion; a system will be called stable if

its response to any bounded input is limited: we talk of L(imited) I(nput) L(imited)

R(esponse) stability

1.4.4.2 Necessary and sufficient condition of stability

An LTI is BIBO (bounded input, bounded output) if and only if its impulse response is positively integrable, i.e if:

To do this, let us demonstrate the opposite proposition: if the impulse response

of the system is not absolutely integrable:

θ θ

+

−∞

∀ ∃K T, ∫ T h( ) d >K

there is a bounded input that makes the output diverge

It is sufficient to choose input x such that:

X LTI Y

)_(

*)_()

)_()

This formally defined transform ratio is the transform of the impulse response

and is called a transfer function of LTI

The use of transfer functions has a considerable practical interest in the study of system association as shown in the examples below

1.4.5.1 Cascading (or serialization) of systems

Let us consider the association of Figure 1.8

Figure 1.8 Serial association

Hencey3(_)=h3(_)*(h2(_)*(h1(_)*x1(_)) ) This leads, in general, to a rather complicated expression

In terms of transfer function, we obtain:

)_()_()_()

1.4.5.2 Other examples of system associations

LTI1

+

+ LTI2

Figure 1.9 Parallel association

) _ ( H ) _ ( H ) _ ( E

) _ ( Y )

_

(

) _ ( H )

_ ( E

) _ ( Y )

_

(

H

2 1

2

H

1.4.5.3 Calculation of transfer functions of causal LTIs

In this section, we suppose the existence of impulse response transforms while keeping in mind the convergence conditions

Using the Fourier transform, we obtain the frequency response H ( f):

The notations used present an ambiguity (same H) that should not affect the

informed reader: when the impulse response is positively integrable, which corresponds to a stability hypothesis of the system considered, we know that the Laplace transform converges on the imaginary axis and that it is mistaken with Fourier transform through p= 2πj f Hence, the improper notation (same H):

On the other hand, the frequency responses, which are defined by the Fourier transform of the impulse response, even considered in the distribution sense, do not always exist The stability hypothesis ensures the simultaneous existence of two transforms

EXAMPLE 1.1.– it is easily verified whether an integrator has as an impulse response the Heaviside step function h(t)=u(t) and hence:

2

12

An LTI with localized constants is represented through a differential equation with constant coefficients with m< : n

( )t b y( )( )t a x( )t a x( )( )t

y

By supposing that x (t) and y (t) are continuous functions defined from −∞ to +∞ , continuously differentiable of order m and n, by a two-sided Laplace transform

we obtain the transfer function H (p):

n n

m m

m m n

n

p b p b b

p a p a a

p

H

p X p a p a a p Y p b p

b

b

+++

++

=

++

=+

++

1 0

1 0 1

0

)

(

)()(

)()(

Such a transfer function is called rational in p The coefficients of the numerator

and denominator polynomials are real due to their physical importance in the initial

differential equation Hence, the numerator roots, called zeros, and the denominator roots, called transfer function poles, are conjugated real or complex numbers

If x (t) and y (t) are causal functions, the Laplace transform of the differential equation entails terms based on initial input values x(0), x′(0), x(m−1)(0) and output values y(0), y′(0), y(n−1)(0); the concept of state will make it possible to overcome this dependence

1.4.6 Causality, stability and transfer function

We have seen that the necessary and sufficient condition of stability of an SLI is for its impulse response to be absolutely integrable: ∫−+∞∞h( )θ dθ <+∞

The consequence of the hypothesis of causality modifies this condition because

we thus integrate from 0 to+∞

On the other hand, if we seek a necessary and sufficient condition of stability for the expression of transfer functions, the hypothesis of causality is determining Since the impulse response h( )θ is a causal function, the transfer function H (p)

is holomorphic (defined, continuous, derivable with respect to the complex number

p) in a right half-plane defined byRe( )p >σo The absolute integrability of h( )θentails the convergence of H (p) on the imaginary axis

A CNS of EBRB stability of a causal LTI is that its transfer function is

holomorphic in the right half-plane defined by Re(p)≥0

When:

( )p e D N( ) ( )p p

where N (p) and D (p) are polynomials, it is the same as saying that all the transfer

function poles are negative real parts, i.e placed in the left half-plane

We note that in this particular case, the impulse response of the system is a function that tends infinitely toward 0

1.4.7 Frequency response and harmonic analysis

1.4.7.1 Harmonic analysis

Let us consider a stable LTI whose impulse response h( )θ is canceled after a period of time t R For the models of physical systems, this period of time t R is in fact rejected infinitely; however, for reasons of clarity, let us suppose t R as finite, corresponding to the response time to 1% of the system

When this system is subject to a harmonic excitation x t( )= Ae 2 jf tπ 0 from 0

Φ( )f0

A module or gain Φ=argH( )f o phase

We note that H ( f) is nothing else but the Fourier transform of the impulse response, the frequency response of the system considered

1.4.7.2 Existence conditions of a frequency response

The frequency response is the Fourier transform of the impulse response It can

be defined in the distribution sense for the divergent responses in tα but not for exponentially divergent responses (e bt) However, we shall note that this response

is always defined under the hypothesis of stability; in this case and only in this case,

we pass from transfer functions with complex variables to the frequency response by determining that p= 2πj f

EXAMPLE 1.2.– let h(t)=u(t) be the integrator system:

( )p p

j f f

2

12

1 because the system is not EBRB stable

=

( ) ( ( ))

H p TL u t is defined according to the functions in the half-plane

>

Re( ) 0p , whereas H f( )=TF u t( ( )) is defined in the distribution sense

Unstable filter of first order: h( )t =e t t ≥0

Hence, even if the system is unstable, we can always consider the complex

number obtained by formally replacing p by 2πj f in the expression of the transfer

function in p The result obtained is not identified with the frequency response but

may be taken as a harmonic analysis, averaging certain precautions as indicated in the example in Figure 1.11

Let us consider the unstable causal system of transfer function

p – 1

Figure 1.11 Unstable system inserted into a loop

The transfer function of the system is

1

2

+

p The looped system is stable and

hence we can begin its harmonic analysis by placing an input sine-wave signal

2

=

jf A

=Φ

12

2arg

122

0

0+

−

=

jf

jf A

12

122arg

Table 1.1 sums up the features of a system’s transfer function, the existence conditions of its frequency response and the possibility of performing a harmonic analysis based on the behavior of its impulse response

1.4.7.3 Diagrams

Table 1.1 Unit-step responses, transfer functions and

existence conditions of the frequency response

H(p) has its poles on the left of

the imaginary axis

TF exists Possible direct analysis

H(p) has poles on the right of

the imaginary axis

Possible analysis if the system is introduced in a stable looping and

Frequency responses are generally characterized according to impulse

ω= 2 j fπ and data H(jω) and Φ(jω) grouped together as diagrams The following are distinguished:

– Nyquist diagram where the system of coordinates adopts in abscissa the real

part, and in ordinate the imaginary part H p( ) p j=ω;

– Black diagram where the system of coordinates adopts in ordinate the module

expressed in decibels, like:

ω

= 10

20 log ( ( ) )

p j

H p and in abscissa arg ( )H p p j=ω expressed in degree;

– Bode diagram which consists of two graphs, the former representing the

module expressed in decibels based on log ( ) and the latter representing the 10 ωphase according to log ( ) Given the biunivocal nature of the logarithm function 10 ωand in order to facilitate the interpretation of the diagram, the axes of the abscissas are graduated in ω

The unit-step response is a slope ramp K: (t)=KtU(t)

The frequency response, which is the Fourier transform of the impulse response,

is defined only in the distribution sense: