lecture 11 lti frequency models 2

Lecture 11 – Linear Systems. Concepts and Representations. - Part 2: Frequency representation models -

ContentFrequency response of continuous–time systems

Principle of the frequency representationFrequency response of systemGraphical representation of frequency response

Frequency response of discrete–time systemsMinimum phase systemsNon-parametric system modelsNon-parametric model conversionsDiscretization of continuous-time models

Summary

1. Frequency response of continuous–time systems

1.1. Principle of the frequency representation

A mathematical model must describe the transfer from input u(t) to output y(t). In time domain, the reference form of the model is the differential equation of order n , which has a rational transfer function. There is a class of systems where the physical variables depend also of spatial coordinates, the so-called systems with distributed parameters. For such systems, the reference model is the differential equation with partial derivatives, and the representations in “s” domain generate irrational transfer function.

Frequency models have another point of view in description of the transfer from input to output. The variables u(t) and y(t) are described by spectral functions U (ω) and, respectively, Y (ω) (The Fourier transforms of input and output variables). These complex functions describe:

(1) The characteristics of the amplitude spectral densities: and

(2) The phase characteristics: .

The frequencies model is defined by two characteristics or diagrams:

- Amplification (amplitude) characteristic (diagram), A(), which allows the

computation of the function , knowing (v. fig. 1):

(1)

- Phase characteristic (more exactly, phase difference), φ (ω ) , which allow the

computation of function , knowing :

2 Signals and Systems. Complements

(2)

Figure 1: The frequency representation of system

So, as basics, the frequency model defines the relation between signals u(t) and y(t), with spectral models, by using frequency characteristics: of amplitude and phase. In using such a representation of the model, does not care what kind of system we have, with lamped (concentrated) or distributed parameters, so these models are more general then those in time.

1.2.Frequency response of system 1

Let be H(s) the transfer function of the considered system. The input signal is

(3)

For linear systems and stationary regime, the output signal is also sinusoidal:

(4)

In order to find the parameter Y and of the output signal, without integration of the differential equation of the system, we rewrite the variables u(t) and y(t) as:

;

Based on superposition principle, we may consider that term output is the effect

of the input signal . By using the convolution to express the input-output relation we obtain

(5)

where we can replace u(t) by and y(t) by . It results1 Raspunsul la frecventa al sistemului

Lecture 11: Linear systems. Part 2: Frequency domain models 3

(6)where

(7)

Because the amplification of the system at frequency is the amplitude ratio of Y and U, and the difference between two sinusoidal signals is the phase of the output signal y(t), from Eq. (6) it results

(8)

(9)

The function H(j) – called frequency response – generates the two characteristics (diagrams) which describes the input-output transfer: amplification characteristic, A(), and phase characteristic, (). It can be rewritten as

(10)Also, the following relation is valid

(11)because A() is an even function and () is odd.

The frequency response could also written as

(12)where P() is even and Q() –odd.

1.3. Graphical representation of frequency response

A. Transfer place (locus) ( Nyquist characteristic ) is the geometric locus in complex plane of the peak of vector H(j), when ω is changing from - ∞ to ∞. Generally, only the positive frequency is used [0 +∞). Fig. 2 presents an example of such transfer locus.

For a point M of transfer locus, with a corresponding frequency ω* , the amplitude and the

phase is directly computed from diagram: amplification A(*) is done by

and phase by

.

Example: Plot the Nyquist characteristics for a system described by the transfer function

Figure 2: Transfer place (locus)


and H (s )= 2 s+1

0.5 s2+1.3 s+1 .

□Solution: Function nyquist is used to calculate the Nyquist frequency response and draw the Nyquist characteristic. It is possible to impose a specific frequency domain, w. Clik right the mouse on the figure to use some specialized functions.

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num = [2 1]; den = [1 2.5 3 1];

sys = tf(num,den); % for continuous time-systems; subplot(221), nyquist(sys); w = 0:0.1:100; grid[Re, Im] = nyquist(sys, w);re = squeeze(Re); % convert 1x1xN to 1xNim = squeeze(Im); % convert 1x1xN to 1xNsubplot(222), plot(re, im); grid; den = [0.5 1.3 1]; sys = tf(num,den); subplot(223), nyquist(sys); w = 0:0.1:100; grid[Re, Im] = nyquist(sys, w);re = squeeze(Re); % convert 1x1xN to 1xNim = squeeze(Im); % convert 1x1xN to 1xNsubplot(224), plot(re, im); grid;


B. Bode diagrams

The use of Bode diagrams to represent the frequency response supposes two distinct characteristics:

- Magnitude response in [dB} as function of frequency [rad/s];

(1)- Phase response

(2)In both diagrams, logarithmically spaced frequency is used.

Let be H(j) the frequency response of the system, written as the product of two terms

and . The Bode diagrams for the initial system are computed on the Bode

diagrams of the sub-systems described by and , by summing point by point on frequency axis:

(3)

(4)or

, (5)

Example: Plot the Nyquist characteristics for a system described by the transfer function

H 1 (s )= 2 s+1

0.5 s2+1.3 s+1 and H 2 (s )= 1

s+1.

□Solution: Function bode is used to calculate the frequency response, as magnitude and phase. Run the code below. It is possible to impose a specific frequency domain, w. Clik right the mouse on figure to use some specialized functions.


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2. Frequency response of the discrete–time systems

Let H(z) the transfer function of discrete-time system, with z=esT e

and Te as

sampling period. By substitution we impose the current working point from complex

s plane to be placed on the imaginary axis. Because the input-output signals are sampled, it is

necessary as [−ωe

2

ωe

2 ], which corresponds to the base band (Fig. 3.a).

num_1 = [2 1]; den_1 = [0.5 1.3 1]; % S1

sys_1 = tf(num_1,den_1); num_2 = [1]; den_2 = [1 1]; % S2sys_2 = tf(num_2,den_2); w = 1e-2:0.1:100; figure(1); bode(sys_1, 'b', sys_2, 'r', w); grid;legend('S1', 'S2'); [A1, P1] = bode(sys_1, w); A1 = squeeze(A1);[A2, P2] = bode(sys_2, w); A2 = squeeze(A2);G1= 20*log10(A1); % dBG2= 20*log10(A2); % dB figure(2); semilogx(w, G1, 'b', w, G2, 'r'); grid;xlabel('Frequency[rad/s])'); ylabel('Magnitude [dB]'); legend('S1', 'S2');


Frequency response of time-discrete systems is obtained by replacing the variable z

with in the transfer function

(1)Magnitude and phase diagrams are defined by relations:

(2)

; (3)

Figure 3: Trajectories of current point in s and z planes, associated with frequency response

Example: Below is described the Bode diagram for a time-discrete system described by a

transfer function H (s )= 0.5 s+0.2

s2−1.4 s+0.7 with Te = 0.1 The representation of the frequency

response is up to ωe/2 = π/Te = 31.4 rad/s.

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num = [0.5 0.2]; den = [1 -1.4 0.7]; Te = 0.1; sys = tf(num,den, Te); bode(sys); grid;


3. Minimum phase systems

Let consider the next transfer functions:

(1)

All of them have the same value of the magnitude : , because

(2)

But he phase diagrams are different, as

(3)

Frequency representations have physical sense only for stable systems, for which the sinusoidal permanent regime exists. As consequence, the condition of poles to be in the left side plan is considered as satisfied.

The minimum phase system is a system which has the minimum phase response, in absolute value, from the set of all systems with the same magnitude diagram. The main conditions for a minimum phase system are:

- No delay;- No- zeros in the right semi plane;- No – poles in the right semi plane (stable system);

For a minimum phase system, only one diagram (magnitude or phase) could describe the system. In fact, there are relations which allow the conversion from one to another diagram.

Let the magnitude diagram and the phase diagram. For minimum phase systems:

(4)

where H {. } is the operator of the Hilbert transform 2. More, if P(ω) and Q(ω) are the real and imaginary part of the frequency response, P(ω) and Q(ω) is a Hilbert pair.

(5)

2 By definition,


3.1. Parametric frequency representation of minimum-phase systems

Frequencies models described by graphical representations (Bode) have arbitrary shapes and are considered non-parametric models. There it is possible to have a parametric representation of such a (frequency) model: asymptotic Bode diagram. (see Fig. 4). This diagram has a standard shape (case-linear), i.e. a set of segment with a slope of

integer). The graphical representation has (2n) parameters, the same as the

transfer function. The parameters are: the slopes , the cut-off

3

frequencies

, plus a parameter which defines the vertical position of the diagram and is

assimilated as DC gain.

Figure 4: Parametric Bode diagram

4. Non-parametric system models

Non-parametric models are graphical representations with arbitrary shapes. These could be in ''t'' domain or in '''' domain.

From the building mechanism, we distinguish:

1). Graphical representations from experiments, as example: impulse response or step response (for all “t” domain) or experimental frequency diagrams (for '''' domain).

2). Graphical representations of impulse response or step response, computed on wide set of points (as example: the results generated by Matlab function: impulse and step) or graphical representations of computed frequency diagrams (as example: the results generated by Matlab function: Nyquist or Bode);

5. Non-parametric model conversions

3 Frecvente de frangere


Non-parametric models are difficult to use in analysis and synthesis because it is necessary to a lot numerical values. It is useful then to make conversions from non-parametric to parametric models. We distinguish:

A. Non-parametric to non- parametric model conversions;B. Non-parametric to non-parametric with parameters (few number) model conversions;C. Non-parametric to parametric model conversion.

A. Non-parametric to non-parametric model conversion ( h(t) < - > H() )

Let consider a stable system, which satisfies a finite length of the impulse response, so

. This function accepts both Fourier and Laplace transform. It results:

(1)

(2)

; (3)

Let be time when the response is practical zero, so when . Numerical

computation of the frequency diagrams starting from impulse response, uses the basic

relations:

(4)

(5)

For the invers conversion, from frequency diagrams of minimum phase systems to impulse response, we start from the inverse Fourier transform:

After few processing steps, we have:


; (6)

Numeric calculus use finite limits for integrals:

; (7)

where is the frequency for which:

; . (8)

B. Non-parametric to non-parametric with parameters model conversion

The most used non–parametric model is the impulse response, h(t). The simplest

parameterization is the sampled form of the impulse response: h(iTe) h(i), .

h( iTe )|i≥n=0Example: FIR filter. Let consider an ideal low pass filter, as in Fig. 1. The impulse response of the filter is computed by the inverse Fourier transform of the frequency response of this filter:

(1)

Figure 1: Frequency response of the ideal low pass filter

The impulse response, presented in Fig. 2, shows that the filter in physical unrealizable, because it is non-causal: it responds before applying the input, at t=0. In order to obtain a physical implementation we consider a finite observation rectangular window, wR(t), starting from t = 0, as it is draw in Fig. 2. The observed answer is possible to FIR systems and under digital implementation only. As effect, the response impulse is sampled and the set of samples is

(2)and represented in Fig. 3. These samples represent the non-parametric model of FIR filter.

1


The transfer function of FIR filter is:

(3)By considering an odd value for M, we have symmetry:

(4)

-5 0 5 10

0

0.2

0.4

0.6

0.8

1

h(t)

wR(t)

Figure 2: Impulse response and rectangular window

-1 0 1 2 3 4 5 6-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

h(0) h(1)

h(2)

h(3) h(4)

h(M)

h(2M)

Te M.Te

h(k)

t

k

Figure 3: Impulse response of FIR filter■

C. Non-parametric to parametric model conversion

We consider the problem of parameterization of the impulse response with a few parameters. A solution to reduce the number of parameters is the approximation of the impulse response


by a sum of finite number of terms, following a finite number of linear independent functions

. Let this approximation be

(1)

where the parameters which define the representation of the system are: : a0. a1, a2, …aN-1. The number of parameters is less than the number of numerical values from the initial representation (the sampled impulse response). The choice of orthogonal functions, as in (1), makes the simplification in the computing process of the N parameters ai. Taking Laplace transform in (1) we obtain:

(2)

and the structure of the system is presented in Fig. 1

By choosing of the function system we obtain various parameterization solutions,

each with specific error of approximation of the initial nonparametric model. A simple

approximation solution of the impulse response is presented in Fig. 2.

Figure 1: The parameterization principle

Figure 2: Approximation by constant impulses of the impulse response model


Let consider the impulse response under a set of numerical values obtained by sampling period Te. Because this data are accepted as initial parameters, then by choosing the width of

the rectangular functions equal with r.Te, it results a decreasing of number of

parameters of r times. The transfer functions are

; (3)

and the structure of the parameterized system is presented in Fig. 3.

Figure 3: The structure of the parameterized system by using the set function (3)Other variants of parameterizations of non-parametric model use the approximation of the impulse response by orthogonal poliynom, e.g. Laquerre polynoms. The functions

are obtained by the filter bank of Fig.4.

Figure 4: Laguèrre filter bank

For discrete time, the transfer functions of the filter bank are:

... (4)

The structure of the non-parametric model, under parameterizations of the (3), presented in Fig. 5.


Figure 5: The structure of the parameterized system, with Laquerre polynoms

D. Non-parametric to parametric (transfer function) model conversion

The non-parametric frequency model to transfer function model conversion is considered. The conversion procedure requires the following epochs:

1). Approximation of the logarithmic frequency diagram, , by an asymptotic diagram, linear on intervals. Linear segment are used with slopes of ±k.20 dB/dec, where k=0,1,2,...

2). Based on obtained asymptotic diagram write the transfer function of the system, under the minimum phase system.

3). Draw the phase diagram ϕ1 (ω ) of the considered minimum phase system. By comparing

the initial phase diagram of the system, ϕ (ω ) , with ϕ1 (ω ), two situations could appear:

3.a). Appropriate diagrams, including high frequencies; the transfer function is that of

stage (2);

3.b). The real diagram ϕ (ω )

is below ϕ1 (ω )

, with higher difference at high

frequencies; the system is of non-minimum phase, may having elements with dead time

and/or dynamic subsystems with rational transfer functions with zeros on right semi plane (all

pas filters, of order 1 or 2). Identification of the subsystems which generates only phases is

made based on diagram Δ ϕ (ω)=ϕ1 (ω)−ϕ ( ω)

:

3.b.1). If then the system has dead time;


3.b.2). then some additional terms must be considered as

or

. Details could be found in [1].

6. Discretization of continuous-time systems

Two approaches are possible: (1) the use of pair sampler-holder; (2) analytic relation which link variables s and z, which are independent variables of the transfer functions..

6.1. The sampler-holder approach

Let H(s) be the transfer function of time-continue system. By discretization, a discrete time system is build, having sampled input-output signals u(k) and y(k). The time-discrete signal u(k) is obtained by sampling. Because the system H(s) is time-continuous, a holder element is used immediately after sampler. Thus, the input signal in the continue-time system is u(t). At the output of the system, a virtual sampler is used, which generate the discrete time variable y(k).

Figure 1: Time continue system (a) discretization by sampler-holder approach (b)

The reconstruction of the signal u(t) from sampled signal u(k) can be analyzed in two ways:1). Ideal sampling (as theoretical reference), with u(t) output signal of the estimator;

2). Real case sampling (real case). The output of the estimator is , and discrete –

time system approximates the initial system, the approximation error being in strong link with

the reconstruction error of time-continue signal starting from discrete time one.Case 1: The ideal reconstruction is made by an ideal low pass filter, with frequency diagram of figure 2.

Figure 2: Ideal low pass filter (LPF) frequency diagrams

The frequency response and the impulse response of the low pass filter are


and (1)

This response reflects the non-causal character of the low pass ideal filter (i.e.

), and – thus – the existence of finite error in discretization process.

The estimation elements used for time continue signal , are composed of extrapolators,

with transfer function , having input as and output by , is:

(2)

For zero-order holder, the response impulse is presented in Fig. 3 and has a transfer function as

(3)

The transfer function of the discrete system is

(4)

In order to evaluate the error of estimation, the frequency diagrams of zero-order extrapolator and ideal low pass filter will be compared. The magnitude and phase diagrams, computed

from the frequency response, ,, are

, (5)

As presented in Fig. 3 the approximation could be accepted only on the low frequency range.


Figure 3: Impulse response and frequency diagrams of the zero-order extrapolator

For first-order extrapolator, represented in Fig. 4, the impulse reasons could be approximate by

(6)

where is a triangular function with slope 1/T e . The transfer function of extrapolator is then:

(7)

Figure 4: Impulse response and frequency diagrams for the first -order extrapolator

The general expressions for magnitude and phase diagrams are:

(8)

Presented also in Fig. 4. The result shows the superiority of the first order extrapolator with reference to the zero order.


6.2. Tustin method

The relation is processed as

(9)Taking the first two terms from the series we obtain:

(10)Selecting the variable s:

(11)

A bilinear transformation is obtained, which project the imaginary axis on the unit

ration circle . The points of the right side of “s” semi plane are translated into the

interior of this circle. So, the method called Tustin consist n replacing the variable s from transfer function H(s) by (11) :

(13)Example : Let be a first order system, with gain 1 and unit time constant. Using sampling period as Te=0.2 s, the frequency diagrams are considered for frequencies up to ωe/2 for both systems: the continuous and discretized one. As presented in figure, there are errors around the limit ωe/2. num = [1]; den = [1 1]; sys = tf(num,den); Ts = 0.2; i = 1; method = 'zoh'; sys_d1 = c2d(sys, Ts, method);i = i+1; method = 'foh'; sys_d2 = c2d(sys, Ts, method);i = i+1; method = 'tustin'; sys_d3 = c2d(sys, Ts, method); w = 1e-2:0.01: pi /Ts;bode(sys, w); hold on;bode(sys_d1,w); hold on;bode(sys_d2,w); hold on;bode(sys_d3,w); hold off; grid;legend('(1)-cont', '(2) d-zoh', '(3) d-foh', '(4)d-tustin');


Figure: Magnitude and phase diagrams for continuous-time (1) and discretized by (2) zero-order (3) first order extrapolator; (4) Tustin

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References

[1]. Emil Ceanga, Analiza si sinteza circuitelor si sistemelor, 2010.

lecture 11 lti frequency models 2

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