Frequency-Domain of Control Systems

Eng R. L. Nkumbwa Copperbelt University 2010

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Its all Stability of Control Systems

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Introduction

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 In practice, the performance of a control system is more realistically measured by its time domain characteristics.  The reason is that the performance of most control systems is judged based on the time response due top certain test signals.

 In the previous chapters, we have learnt that the time response of a control system is

usually more difficult

to determine analytically, especially for

higher order systems

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Introduction

 In design problems, there are

no unified methods

of arriving at a designed system

that meets the time-domain performance specifications

, such as maximum overshoot, rise time, delay time, settling time and so on.

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Introduction

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 On the other hand, in

frequency domain

, there is a

wealth of graphical methods

available that are not limited to low order systems.  It is important to realize that there are correlating relations between frequency domain performance in a linear system,  So the

time domain properties

of the system can be predicted based on the

frequency domain characteristics

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Example: Gun Positional Control

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Why use Frequency-Domain?

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 With the previous concepts in mind, we can consider the

primary motivation

for conducting control systems

analysis and design

in the frequency domain to be

convenience

and the

availability of the existing analytical tools

.

 Another reason, is that, it presents an alternative point of view to control system problems, which often provides valuable or crucial information in the complex analysis and design of control systems. 4/25/2020 Eng. R. L. Nkumbwa

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Frequency-Domain Analysis

 The starting point for frequency-domain analysis of a linear system is its

transfer system.

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Time & Frequency-Domain Specs.

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 So, what are time-domain specifications by now?

 Ok, what of frequency domain specifications?

 What are they?

 Lets look at the pictorials views… 4/25/2020 Eng. R. L. Nkumbwa

Time-Domain Specifications

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Frequency-Domain Specifications

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Wrap Up…

 The frequency response of a system directly tells us the relative

magnitude and phase

of a system’s output sinusoid if the system input is a sinusoid.   What about output frequency?

If the plant’s transfer function is

G (s),

the open-loop frequency response is

G (jw)

.

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Further Frequency Response

 In previous sections of this course we have considered the use of

standard test inputs

, such as step functions and ramps.  However, we will now consider the steady state response of a system to a

sinusoidal input test signal

.

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Further Frequency Response

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 The

response

of a linear constant-coefficient linear system to a sinusoidal test input is an output sinusoidal signal at the

same frequency

as the input.

 However, the

magnitude

and

phase

of the output signal differ from those of the input sinusoidal signal, and the amount of difference is a function of the input frequency.

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Further Frequency Response

 We will now examine the transfer function G(s) where s = jw and graphically display the complex number G(jw) as w varies.

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 The Bode plot is one of the most powerful graphical tools for analyzing and designing control systems, and we will also consider polar plots and log magnitude and phase diagrams.

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Further Frequency Response

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 How is this different from Root Locus?

 The information we get from frequency response methods is different than what we get from the root locus analysis.

 In fact, the two approaches complement each other.  One advantage of the

frequency response

approach is that we can use data derived from measurements on the physical system without deriving its mathematical model.

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Further Frequency Response

 What is the

Importance of Frequency

methods?

 They are a powerful technique to design a single loop feedback control system.

 They provide us with a viewpoint in the frequency domain.

 It is possible to extend the frequency analysis idea to nonlinear systems (approximate analysis).

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Who Developed Frequency Methods?

 Bode, Nyquist, Nichols and others, in the 1930s and 1940s.  Existed before root locus methods.

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What are the advantages?

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    

We can study a system from physical data and determine the transfer function experimentally.

We can design compensators to meet both steady state and transient response requirements.

We can determine the stability of nonlinear systems using frequency analysis (out of the scope of this lecture).

Frequency response methods allow us to settle ambiguities while drawing a root locus plot.

A system can be designed so that the effects of undesirable noise are negligible.

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What are the disadvantages?

 Frequency response techniques are not as intuitive as root locus.

 Find more cons 4/25/2020 Eng. R. L. Nkumbwa

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Concept of Frequency Response

 The frequency response of a system is the steady state response of a system to a sinusoidal input.

 Consider the stable, LTI system shown below.

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Concept of Frequency Response

 The input-output relation is given by:

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Concept of Frequency Response

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Concept of Frequency Response

 Obtaining Magnitude

M

and Phase

Ø

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Concept of Frequency Response

 For linear systems,

M

the input frequency, w.

and

Ø

depend only on  So, what are some of the frequency response plots and diagrams?

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Frequency Response Plots and Diagrams

 There are

three frequently

used representations of the

frequency response

: 

Nyquist diagram

: a plot on the complex plane (G(jw)-plane) where

M

and

Ø

are plotted on a single curve, and

w

becomes a hidden parameter.

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Frequency Response Plots and Diagrams



Bode plots

: separate plots for

M

and

Ø

, with the horizontal axis being

w

is log scale.

 The vertical axis for the M-plot is given by M is decibels (db), that is 20log 10 (M), and the vertical axis for the

Ø

-plot is

Ø

in degrees.

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Frequency Response Plots and Diagrams



Log-magnitude versus phase plot which is called the Nichols plot.

 Now, let us consider each of the techniques in more detail in the following chapters.

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Frequency-Domain Systems

 We can plot

G(jw)

as a function of

w

in three ways:

–

Bode Plot.

–

Nyquist Plot.

–

Nichols Plot (we may not cover this).

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Nyquist Diagram or Analysis

 The polar plot, or Nyquist diagram, of a sinusoidal transfer function G(jw) is a plot of the magnitude of G(jw) versus the phase angle of G(jw) on polar coordinates as w is varied from zero to infinity.

 Thus, the polar plot is the locus of vectors |G(jw)| LG(jw) as w is varied from zero to infinity. 4/25/2020 Eng. R. L. Nkumbwa

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Nyquist Diagram or Analysis

 The projections of G(jw) on the real and imaginary axis are its real and imaginary components.

 The

Nyquist Stability Criteria

is a test for system stability, just like the Routh-Hurwitz test, or the Root-Locus Methodology.

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Nyquist Diagram or Analysis

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 Note that in polar plots, a positive (negative) phase angle is measured counterclockwise (clockwise) from the positive real axis. In the polar plot, it is important to show the frequency graduation of the locus.

 Routh-Hurwitz and Root-Locus can tell us where the poles of the system are for particular

values of gain

.

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Nyquist Diagram or Analysis

 By altering the gain of the system, we can determine if any of the poles move into the RHsP, and therefore become unstable.

 However, the Nyquist Criteria can also give us additional information about a system.

 The Nyquist Criteria, can tell us things about the

frequency characteristics

of the system.

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Nyquist Diagram or Analysis

 For instance, some systems with constant gain might be stable for

low-frequency

inputs, but become unstable for

high-frequency

inputs.

 Also, the Nyquist Criteria can tell us things about the phase of the input signals, the time-shift of the system, and other important information.

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Nyquist Kuo’s View

 Kuo et al (2003) suggests that, the Nyquist criterion is a semi-graphical method that determines the stability of a closed loop system by investigating the properties of the frequency domain plot, the Nygmst plot of L(s) is a plot of L (jw) in the polar coordinates of M [L(jw)] versus Re[L(jw)] as

w

varies from 0 to ∞.

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Nyquist Xavier’s View

 While, Xavier et al (2004) narrates that, the Nyquist criterion is based on “Cauchy’s Residue Theorem” of complex variables which is referred to as “

Principle of Argument

”.

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The Argument Principle

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  If we have a contour, Γ, drawn in one plane (say the complex laplace plane, for instance), we can map that contour into another plane, the F(s) plane, by transforming the contour with the function F(s). The resultant contour, Γ F(s) will circle the origin point of the F(s) plane N times, where N is equal to the difference between Z and P (the number of zeros and poles of the function F(s), respectively).

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Nyquist Criterion

 Let us first introduce the most

important equation

when dealing with the Nyquist criterion:

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 Where: –

N

is the number of encirclements of the (-1, 0) point.

–

Z

is the number of zeros of the characteristic equation.

–

P

is the number of poles of the open-loop characteristic equation.

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Nyquist Stability Criterion Defined

 A feedback control system is stable, if and only if the contour ΓF(s) in the F(s) plane does not encircle the (-1, 0) point when P is 0.

 A feedback control system is stable, if and only if the contour ΓF(s) in the F(s) plane encircles the (-1, 0) point a number of times equal to the number of poles of F(s) enclosed by Γ.

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Nyquist Stability Criterion Defined

 In other words, if P is zero then N must equal zero. Otherwise, N must equal P. Essentially, we are saying that Z must always equal zero, because Z is the number of zeros of the characteristic equation (and therefore the number of poles of the closed-loop transfer function) that are in the right-half of the s plane.

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Nyquist Manke’s View

 While Manke (1997) outlines that, the Nyquist criterion is used to identify the presence of roots of a characteristic equation of a control system in a specified region of s-plane.  He further adds that although the purpose of using Nyquist criterion is similar to RHC, the approach differs in the following respect: 4/25/2020 Eng. R. L. Nkumbwa

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– The open loop transfer G(s) H(s) is considered instead of the closed loop characteristic equation 1 + G(s) H(s) = 0 – Inspection of graphical plots G(s) H(s) enables to get more than YES or NO answer of RHC pertaining to the stability of control systems. 4/25/2020 Eng. R. L. Nkumbwa

Kuo’s Features of Nyquist Criterion

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 Kuo also outlines the following as the features that make the Nyquist criterion an attractive alternative for the analysis and design of control systems: – In addition to providing the absolute stability, like the RHC, the NC also gives information on the relative of a stable system and the degree of instability.

– The Nyquist plot of G(s) H(s) or of L (s) is very easy to obtain.

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Kuo’s Features of Nyquist Criterion

– The Nyquist plot of G(s) H(s) gives information on the frequency domain characteristics such as Mr, Wr, BW and others with ease.

– The Nyquist plot is useful for systems with pure time delay that cannot be treated with the RHC and are difficult to analyze with root locus method.

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Any more worries about freqtool …

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