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    Frequency Response Techniques

•   The definition of frequency response

•   How to plot frequency response

•   How to use frequency response to analyze stability

•   How to use frequency response to analyze a system's
    transient and steady state error performance

•   How to use frequency response to design the gain to meet
    stability specifications




                                                           Dr Branislav Hredzak
                                                   Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                     Copyright © 2004 by John Wiley & Sons. All rights reserved.
2

                    Introduction

Frequency response technique has distinct advantages:

1. when modeling transfer functions from physical data

2. when designing compensators to meet a steady-state error
   requirement and a transient response requirement

3. when finding the stability of nonlinear systems

4. in settling ambiguities when sketching a root locus




                                                             Dr Branislav Hredzak
                                                     Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                       Copyright © 2004 by John Wiley & Sons. All rights reserved.
3

The Concept of Frequency Response
         t
                The steady-state output sinusoid is:




                                         … magnitude
                                         frequency response


                                               … phase
                                               frequency response




                                             Dr Branislav Hredzak
                                     Control Systems Engineering, Fourth Edition by Norman S. Nise
                                       Copyright © 2004 by John Wiley & Sons. All rights reserved.
4



Analytical expression for the frequency response can be obtained from the
transfer function G(s) of the system as

                       G ( jω ) = G ( s ) | s → jω


 Plotting of the frequency response G ( jω ) = M G ∠φG (ω )

          •   As a function of frequency, with separate magnitude and phase
              plots




                                                                 Dr Branislav Hredzak
                                                         Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                           Copyright © 2004 by John Wiley & Sons. All rights reserved.
5


Problem Find the analytical expression for the magnitude frequency response
and the phase frequency response for a system G(s) = 1 / ( s + 2). Also, plot the
separate magnitude and phase diagrams.


Solution
                                 1
 Substitute s = jω into G ( s) =
                                  s+2

                1    2 − jω
 G ( jω ) =         = 2
              jω + 2 ω + 4

                        1
 G ( jω ) = M (ω ) =
                       ω2 + 4

                ⎛ω ⎞
  φ (ω ) = − tan⎜ ⎟
               ⎝2⎠




                                                                    Dr Branislav Hredzak
                                                            Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                              Copyright © 2004 by John Wiley & Sons. All rights reserved.
6

                          1
20 log M (ω ) = 20 log
                         ω2 + 4




                                          Dr Branislav Hredzak
                                  Control Systems Engineering, Fourth Edition by Norman S. Nise
                                    Copyright © 2004 by John Wiley & Sons. All rights reserved.
7


        Asymptotic Approximations: Bode Plots
•    Bode plots or Bode diagrams: curves of the log-magnitude and
     phase frequency response as functions of log ω


    Consider:


    Then, the magnitude response is given by:




    Converting the magnitude response into dB, we obtain:




                                                                    Dr Branislav Hredzak
                                                            Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                              Copyright © 2004 by John Wiley & Sons. All rights reserved.
8




Phase frequency response
- is the sum of the phase frequency response curves of the zero terms minus
the sum of the phase frequency response curves of the pole terms.




                                                                    Dr Branislav Hredzak
                                                            Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                              Copyright © 2004 by John Wiley & Sons. All rights reserved.
9

           Bode Plots for G(s) = (s + a)



At low frequencies when ω approaches zero:

The magnitude response in dB is:




                                   - low-frequency asymptote
                                                                Dr Branislav Hredzak
                                                        Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                          Copyright © 2004 by John Wiley & Sons. All rights reserved.
10

At high frequencies, where ω>>a :




                                    - high-frequency asymptote
                                                    Dr Branislav Hredzak
                                            Control Systems Engineering, Fourth Edition by Norman S. Nise
                                              Copyright © 2004 by John Wiley & Sons. All rights reserved.
11


Phase response




 From the above equation:
 •   at the break frequency, a, phase is 450
 •   at low frequencies the phase is 00
 •   at high frequencies the phase is 900




                                                       Dr Branislav Hredzak
                                               Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                 Copyright © 2004 by John Wiley & Sons. All rights reserved.
12


•   It is often convenient to normalize the magnitude and scale the
    frequency so that the log-magnitude plot will be 0 dB at a break
    frequency of unity.

•   Normalization and scaling makes it easier to add components to obtain
    the Bode plot of a function such as




•   To normalize (s + a), factor out the quantity a and form a[(s/a)+ 1].

•   The frequency is scaled by defining a new frequency variable, s1 = s/a.

•   The magnitude is divided by the quantity a to yield 0 dB at the break
    frequency.

•   Hence, the normalized and scaled function is (sl + 1).

                                                                  Dr Branislav Hredzak
                                                          Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                            Copyright © 2004 by John Wiley & Sons. All rights reserved.
13




    … original
    magnitude response




… normalized and scaled
  magnitude response




           Dr Branislav Hredzak
   Control Systems Engineering, Fourth Edition by Norman S. Nise
     Copyright © 2004 by John Wiley & Sons. All rights reserved.
14




  … original
  phase response




… normalized and scaled
    phase response




          Dr Branislav Hredzak
  Control Systems Engineering, Fourth Edition by Norman S. Nise
    Copyright © 2004 by John Wiley & Sons. All rights reserved.
15


Similarly, we can find the normalized and scaled Bode plots for:


                G(s) = s                                 G(s) = 1/s




                                                                   Dr Branislav Hredzak
                                                           Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                             Copyright © 2004 by John Wiley & Sons. All rights reserved.
16




G(s) = (s + a)   G(s) = 1/(s + a)




                          Dr Branislav Hredzak
                  Control Systems Engineering, Fourth Edition by Norman S. Nise
                    Copyright © 2004 by John Wiley & Sons. All rights reserved.
17

Problem Draw the Bode plots for the system G(s) = K(s + 3)/[s(s + l)(s + 2)].


Solution

 First we normalize G(s),



                                          •   the break frequencies are at 1, 2, and 3

 To find value of G(s) at ω=0.1 we substitute s=0 for all (s/a+1) terms and
                                3
                                   K
 the actual value of s=j0.1 for 2             term.
                                  s
                           3
   Hence    G ( j 0.1) ≈     K / 0.1 = 15 K
                           2

    For K=1: 20 log(G ( j 0.1)) = 20 log 15 = 23.52dB

                                                                          Dr Branislav Hredzak
                                                                  Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                                    Copyright © 2004 by John Wiley & Sons. All rights reserved.
18




        Dr Branislav Hredzak
Control Systems Engineering, Fourth Edition by Norman S. Nise
  Copyright © 2004 by John Wiley & Sons. All rights reserved.
19


1




            Dr Branislav Hredzak
    Control Systems Engineering, Fourth Edition by Norman S. Nise
      Copyright © 2004 by John Wiley & Sons. All rights reserved.
20

  Bode Plots for G(s) = s2 + 2ζωns + ω2n



At low frequencies:




At high frequencies:




 Break frequency:      ωn

After scaling and normalization


                                           Dr Branislav Hredzak
                                   Control Systems Engineering, Fourth Edition by Norman S. Nise
                                     Copyright © 2004 by John Wiley & Sons. All rights reserved.
21




At low frequencies: Phase = 00
At high frequencies: Phase = 1800
Phase at ωn :
                                    2
                 G ( jω n ) = j 2ζω n

                - hence phase at ωn is 900




                                                     Dr Branislav Hredzak
                                             Control Systems Engineering, Fourth Edition by Norman S. Nise
                                               Copyright © 2004 by John Wiley & Sons. All rights reserved.
22

Corrections to Second-Order Bode Plots G(s) = s2 + 2ζωns + ω2n

•   there is an error between the actual response and the asymptotic
    approximation of the second-order polynomial

    Actual response:




                   a magnitude correction of +20log2ζ can be
                   made at the natural, or break, frequency            Dr Branislav Hredzak
                                                               Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                                 Copyright © 2004 by John Wiley & Sons. All rights reserved.
23

Bode Plots for G(s) = 1/ (s2 + 2ζωns + ω2n)

                              a magnitude correction of -20log2ζ can be
                              made at the natural, or break, frequency




                                                Dr Branislav Hredzak
                                        Control Systems Engineering, Fourth Edition by Norman S. Nise
                                          Copyright © 2004 by John Wiley & Sons. All rights reserved.
24
Problem Draw the Bode log-magnitude and phase plots of

                             ( s + 3)
            G(s) =
                     ( s + 2)( s 2 + 2 s + 25)

Solution




                                                                 Dr Branislav Hredzak
                                                         Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                           Copyright © 2004 by John Wiley & Sons. All rights reserved.
25




        20 log(3 / 50) = −24.4dB




( s 2 + 2 s + 25) ↔ s 2 + 2ζω n + ωn
                                   2


ωn = 5 → ζ = 0.2
 -the correction can be found
 by plotting a point
 -20log(2x0.2)=7.96 dB above
 the asymptotes at ωn=5
                 Dr Branislav Hredzak
         Control Systems Engineering, Fourth Edition by Norman S. Nise
           Copyright © 2004 by John Wiley & Sons. All rights reserved.
26




        Dr Branislav Hredzak
Control Systems Engineering, Fourth Edition by Norman S. Nise
  Copyright © 2004 by John Wiley & Sons. All rights reserved.
27

           lntroduction to the Nyquist Criterion
•       relates the stability of a closed-loop system to open-loop poles location




    •    the Nyquist criterion can tell us how many closed-loop poles are in the
         right-half of the complex plane
    •    it is based on mapping of a contour through the function G(s)H(s)




                                                                        Dr Branislav Hredzak
                                                                Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                                  Copyright © 2004 by John Wiley & Sons. All rights reserved.
28


Mapping

                 Contour A

                      Q

                                                 -1




•   Contour A can be mapped through F(s)=G(s)H(s) into contour B (Nyquist
    diagram) by substituting each point of contour A into the function F(s)
    and plotting the resulting complex numbers.




                                                                Dr Branislav Hredzak
                                                        Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                          Copyright © 2004 by John Wiley & Sons. All rights reserved.
Nyquist Criterion




                               Contour A




                                                       -1




 •   Stability can be determined using equation Z = P – N

     where
        P… the number of open-loop poles of G(s)H(s) enclosed by the contour A
        N…the number of encirclements the Nyquist diagram (contour B) makes
        about (– 1)
        Z…the number of right-half-plane poles of the closed-loop system

Note: Counterclockwise encirclements are positive, and clockwise encirclements are
negative

                                                                           Dr Branislav Hredzak
                                                                   Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                                     Copyright © 2004 by John Wiley & Sons. All rights reserved.
30
               Applying the Nyquist Criterion to Determine Stability

Note: Counterclockwise encirclements are positive, and clockwise encirclements are
negative




                                                            P=0
                                                            N=0
                                                            Z=P-N=0 … system is stable




                                                            P=0
                                                            N= - 2
                                                            Z=P-N=2 … system is unstable


                                                                           Dr Branislav Hredzak
                                                                   Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                                     Copyright © 2004 by John Wiley & Sons. All rights reserved.
31

               Sketching the Nyquist Diagram

•   The contour that encloses the right half-plane can be mapped through the
    function G(s)H(s) by substituting points along the contour into G(s)H(s)


•   Approximations can be made to G(s)H(s) for points around the infinite
    semicircle by assuming that the vectors originate at the origin. Thus, their
    length is infinite, and their angles are easily evaluated.




                                                                       Dr Branislav Hredzak
                                                               Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                                 Copyright © 2004 by John Wiley & Sons. All rights reserved.
32
Problem Sketch the Nyquist diagram for the system

           G(s) = 500/[(s + 1)(s + 3)(s + 10)].

Solution




                                            ω = 43




                                                     ω = 30 / 14 SystemsBranislav Hredzak S. Nise
                                                               Control
                                                                       Dr
                                                                          Engineering, Fourth Edition by Norman
                                                                   Copyright © 2004 by John Wiley & Sons. All rights reserved.
33


•   If there are open-loop poles situated along the contour enclosing the right
    half-plane, then a detour around the poles on the contour is required;
    otherwise, a complete sketch of the Nyquist diagram cannot be made.




                                           •    each pole's vector rotates through
                                                + 1800 as we move around the
                                                contour near that pole




    •    detour must be only an infinitesimal distance into the right half-
         plane, or else some closed-loop, right-half plane poles will be
         excluded in the count




                                                                     Dr Branislav Hredzak
                                                             Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                               Copyright © 2004 by John Wiley & Sons. All rights reserved.
34




                          •   each pole's vector rotates through - 1800
                              as we move around the contour near that
                              pole




•   detour must be only an infinitesimal distance into the left half-
    plane, or else some closed-loop, left-half plane poles will be
    excluded in the count




                                                                Dr Branislav Hredzak
                                                        Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                          Copyright © 2004 by John Wiley & Sons. All rights reserved.
35

Problem Sketch the Nyquist diagram of the unity feedback system
G(s) = (s + 2)/s2.


Solution




                                                                Dr Branislav Hredzak
                                                        Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                          Copyright © 2004 by John Wiley & Sons. All rights reserved.
36

             Stability via the Nyquist Diagram

•   Stability can be determined using equation Z = P – N

    where
       P… the number of open-loop poles of G(s)H(s) enclosed by the contour
       N…the number of encirclements the Nyquist diagram makes about (– 1)
       Z…the number of right-half-plane poles of the closed-loop system


•   If the closed-loop system has a variable gain K in the loop we can visualize
    the Nyquist diagram as expanding (increased gain) or shrinking (decreased
    gain)
•                                     OR

•   we can visualize the Nyquist diagram as remains stationary and the (- 1) point
    is moved along the real axis. First, we set the gain to unity and position the
    critical point at (- 1/K) rather than (- 1).


                                                                       Dr Branislav Hredzak
                                                               Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                                 Copyright © 2004 by John Wiley & Sons. All rights reserved.
37

            Stability via the Nyquist Diagram




P = 2, hence in order to obtain a stable system Z = P – N = 0, therefore N = 2,
which means there must be 2 counterclockwise crossings

                                                                     Dr Branislav Hredzak
                                                             Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                               Copyright © 2004 by John Wiley & Sons. All rights reserved.
38

     Stability via Mapping Only the Positive jω-Axis
    - this is a simplified approach to determine stability

                                                                   - there must be no
                                                                   encirclements for this
                                                                   system to be stable




•    the encirclements of the critical point can be determined from the mapping
     of the positive jω-axis alone

•    this system is stable for the range of loop gain, K, that ensures that the
     open-loop magnitude is less than unity at that frequency where the phase
     angle is 1800 (or equivalently, - 1800)

                                                                       Dr Branislav Hredzak
                                                               Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                                 Copyright © 2004 by John Wiley & Sons. All rights reserved.
39

    Stability via Mapping Only the Positive jω-Axis


                                                                   - there must be two
                                                                   counterclockwise
                                                                   encirclements for this
                                                                   system to be stable




•     this system is stable if the open-loop magnitude is greater than unity at
      that frequency where the phase angle is 1800 (or equivalently, -1800)




                                                                      Dr Branislav Hredzak
                                                              Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                                Copyright © 2004 by John Wiley & Sons. All rights reserved.
40

Problem Find the range of gain for stability and instability, and the gain for
marginal stability, for the unity feedback system G(s) = K / [ ( s2 + 2s + 2)(s+ 2)].
For marginal stability find the radian frequency of oscillation. Use the Nyquist
criterion and the mapping of only the positive imaginary axis.
Solution     First we draw the Nyquist diagram for K = 1




                                         Im[G ( jω ) H ( jω )] = 0 when ω = 6
                                                                          1   1
                                         G ( jω ) H ( jω ) ω =   6
                                                                     =−     =   ∠1800
                                                                          20 20
                                          Hence,
                                          stable for K < 20
                                          unstable K > 20
                                          marginally stable for K = 20, and oscillation is √6
                                                                      Dr Branislav Hredzak
                                                                            Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                                              Copyright © 2004 by John Wiley & Sons. All rights reserved.
41


        Gain Margin and Phase Margin
           via the Nyquist Diagram

•   Gain margin and phase margin can be used to determine how
    stable a system is.

•   Systems with greater gain and phase margins can withstand
    greater changes in system parameters before becoming
    unstable.




                                                          Dr Branislav Hredzak
                                                  Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                    Copyright © 2004 by John Wiley & Sons. All rights reserved.
42

•   Gain margin, GM , is the change in open-loop gain, expressed in
    decibels (dB), required at 1800 of phase shift to make the closed-loop
    system unstable.

•   Phase margin, ΦM ,is the change in open-loop phase shift required at
    unity gain to make the closed-loop system unstable.




                                                                   Dr Branislav Hredzak
                                                           Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                             Copyright © 2004 by John Wiley & Sons. All rights reserved.
43

Problem Find the gain and phase margin for the system
G(s) = K / [ ( s2 + 2s + 2)(s+ 2)] if K = 6 .

Solution:




                                      Im[G ( jω ) H ( jω )] = 0 when ω = 6

                                          G ( jω ) H ( jω ) ω =   6
                                                                      = −0.3

                                        Hence GM = 20 log(1 / 0.3) = 10.45dB

                                     Using computer it can be found that
                                     G ( jω ) H ( jω ) has a unity gain at ω =1.253 rad/s

                                     The phase angle at ω = 1.253 rad/s is -112.30

                                       Hence Φ M = −112.30 − (−1800 ) = 67.7 0
                                                                                Dr Branislav Hredzak
                                                                        Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                                          Copyright © 2004 by John Wiley & Sons. All rights reserved.
44

    Stability, Gain Margin, and Phase Margin via
                     Bode Plots
•   Bode plots are subsets of the complete Nyquist diagram but in another form

•   They can be easily drawn

Evaluating Gain and Phase Margins using Bode Plots




                                                                  Dr Branislav Hredzak
                                                          Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                            Copyright © 2004 by John Wiley & Sons. All rights reserved.
45

        Relation between Closed - Loop Transient
        and Closed - Loop Frequency Responses
    Damping Ratio and Closed-Loop Frequency Response




•    There will be no peak at frequencies above zero if ζ>0.707


                                                                          Dr Branislav Hredzak
                                                                  Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                                    Copyright © 2004 by John Wiley & Sons. All rights reserved.
46


Since ζ is related to percent overshoot, we can plot Mp vs. percent overshoot

                     − ln(%OS / 100)
              ζ =
                    π 2 + ln 2 (%OS / 100)




                      Closed-loop frequency response peak vs
                      percent overshoot for a two-pole system

                                                                        Dr Branislav Hredzak
                                                                Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                                  Copyright © 2004 by John Wiley & Sons. All rights reserved.
47

Response Speed and Closed-Loop Frequency Response

 Bandwidth of the closed-loop frequency response is defined as the frequency,
 ωBW , at which the magnitude response curve is 3 dB down from its value at
 zero frequency

                                          M = 1 / 2 corresponds to - 3 dB




                                                                    Dr Branislav Hredzak
                                                            Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                              Copyright © 2004 by John Wiley & Sons. All rights reserved.
48




        Dr Branislav Hredzak
Control Systems Engineering, Fourth Edition by Norman S. Nise
  Copyright © 2004 by John Wiley & Sons. All rights reserved.
49




        Dr Branislav Hredzak
Control Systems Engineering, Fourth Edition by Norman S. Nise
  Copyright © 2004 by John Wiley & Sons. All rights reserved.
50
         Relation between Closed- and Open-Loop
                   Frequency Responses
Damping Ratio from Phase Margin


                    2                 2
                  ωn                ωn
G ( jω ) =                  =
           jω ( jω + 2ζω n ) − ω 2 + j 2ζω nω




                                                • a phase margin of 65.520 (ζ = 0.707) or larger
                                                is required from the open-loop frequency
                                                response to ensure there is no peaking in the
                                                closed-loop frequency response
                                                                            Dr Branislav Hredzak
                                                                    Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                                      Copyright © 2004 by John Wiley & Sons. All rights reserved.
51
        Response Speed from Open-Loop Frequency Response

•        the closed-loop bandwidth can be estimated from the open-loop frequency
         response
    •     It can be shown that, approximately, the closed-loop bandwidth,
          ωBW,equals the frequency at which the open-loop magnitude response is
          between -6 and -7.5 dB if the open-loop phase response is between -1350
          and -2250




                                               ωBW=3.5rad/s                Dr Branislav Hredzak
                                                                   Control Systems Engineering, Fourth Edition by Norman S. Nise
                                                                     Copyright © 2004 by John Wiley & Sons. All rights reserved.
52

   Steady-State Error Characteristics from
           Frequency Response
Type 0 system
                      Initial value:




                       For Type 0 system:



                                       - which is the same as
                                       the initial value




                                                    Dr Branislav Hredzak
                                            Control Systems Engineering, Fourth Edition by Norman S. Nise
                                              Copyright © 2004 by John Wiley & Sons. All rights reserved.
53


Type 1 system
                The initial -20 dB/decade slope can be
                thought of as originating from a function,




                which crosses the frequency axis (G’=0dB) at




                For Type 1 system:


                               - which is the same as the
                               frequency intercept
                                                Dr Branislav Hredzak
                                        Control Systems Engineering, Fourth Edition by Norman S. Nise
                                          Copyright © 2004 by John Wiley & Sons. All rights reserved.
54


Type 2 system   The initial -40 dB/decade slope can be
                thought of as originating from a function,




                which crosses the frequency axis (G’=0dB) at




                   For type 2 system:




                                                Dr Branislav Hredzak
                                        Control Systems Engineering, Fourth Edition by Norman S. Nise
                                          Copyright © 2004 by John Wiley & Sons. All rights reserved.

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10 elec3114

  • 1. 1 Frequency Response Techniques • The definition of frequency response • How to plot frequency response • How to use frequency response to analyze stability • How to use frequency response to analyze a system's transient and steady state error performance • How to use frequency response to design the gain to meet stability specifications Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 2. 2 Introduction Frequency response technique has distinct advantages: 1. when modeling transfer functions from physical data 2. when designing compensators to meet a steady-state error requirement and a transient response requirement 3. when finding the stability of nonlinear systems 4. in settling ambiguities when sketching a root locus Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 3. 3 The Concept of Frequency Response t The steady-state output sinusoid is: … magnitude frequency response … phase frequency response Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 4. 4 Analytical expression for the frequency response can be obtained from the transfer function G(s) of the system as G ( jω ) = G ( s ) | s → jω Plotting of the frequency response G ( jω ) = M G ∠φG (ω ) • As a function of frequency, with separate magnitude and phase plots Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 5. 5 Problem Find the analytical expression for the magnitude frequency response and the phase frequency response for a system G(s) = 1 / ( s + 2). Also, plot the separate magnitude and phase diagrams. Solution 1 Substitute s = jω into G ( s) = s+2 1 2 − jω G ( jω ) = = 2 jω + 2 ω + 4 1 G ( jω ) = M (ω ) = ω2 + 4 ⎛ω ⎞ φ (ω ) = − tan⎜ ⎟ ⎝2⎠ Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 6. 6 1 20 log M (ω ) = 20 log ω2 + 4 Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 7. 7 Asymptotic Approximations: Bode Plots • Bode plots or Bode diagrams: curves of the log-magnitude and phase frequency response as functions of log ω Consider: Then, the magnitude response is given by: Converting the magnitude response into dB, we obtain: Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 8. 8 Phase frequency response - is the sum of the phase frequency response curves of the zero terms minus the sum of the phase frequency response curves of the pole terms. Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 9. 9 Bode Plots for G(s) = (s + a) At low frequencies when ω approaches zero: The magnitude response in dB is: - low-frequency asymptote Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 10. 10 At high frequencies, where ω>>a : - high-frequency asymptote Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 11. 11 Phase response From the above equation: • at the break frequency, a, phase is 450 • at low frequencies the phase is 00 • at high frequencies the phase is 900 Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 12. 12 • It is often convenient to normalize the magnitude and scale the frequency so that the log-magnitude plot will be 0 dB at a break frequency of unity. • Normalization and scaling makes it easier to add components to obtain the Bode plot of a function such as • To normalize (s + a), factor out the quantity a and form a[(s/a)+ 1]. • The frequency is scaled by defining a new frequency variable, s1 = s/a. • The magnitude is divided by the quantity a to yield 0 dB at the break frequency. • Hence, the normalized and scaled function is (sl + 1). Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 13. 13 … original magnitude response … normalized and scaled magnitude response Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 14. 14 … original phase response … normalized and scaled phase response Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 15. 15 Similarly, we can find the normalized and scaled Bode plots for: G(s) = s G(s) = 1/s Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 16. 16 G(s) = (s + a) G(s) = 1/(s + a) Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 17. 17 Problem Draw the Bode plots for the system G(s) = K(s + 3)/[s(s + l)(s + 2)]. Solution First we normalize G(s), • the break frequencies are at 1, 2, and 3 To find value of G(s) at ω=0.1 we substitute s=0 for all (s/a+1) terms and 3 K the actual value of s=j0.1 for 2 term. s 3 Hence G ( j 0.1) ≈ K / 0.1 = 15 K 2 For K=1: 20 log(G ( j 0.1)) = 20 log 15 = 23.52dB Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 18. 18 Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 19. 19 1 Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 20. 20 Bode Plots for G(s) = s2 + 2ζωns + ω2n At low frequencies: At high frequencies: Break frequency: ωn After scaling and normalization Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 21. 21 At low frequencies: Phase = 00 At high frequencies: Phase = 1800 Phase at ωn : 2 G ( jω n ) = j 2ζω n - hence phase at ωn is 900 Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 22. 22 Corrections to Second-Order Bode Plots G(s) = s2 + 2ζωns + ω2n • there is an error between the actual response and the asymptotic approximation of the second-order polynomial Actual response: a magnitude correction of +20log2ζ can be made at the natural, or break, frequency Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 23. 23 Bode Plots for G(s) = 1/ (s2 + 2ζωns + ω2n) a magnitude correction of -20log2ζ can be made at the natural, or break, frequency Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 24. 24 Problem Draw the Bode log-magnitude and phase plots of ( s + 3) G(s) = ( s + 2)( s 2 + 2 s + 25) Solution Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 25. 25 20 log(3 / 50) = −24.4dB ( s 2 + 2 s + 25) ↔ s 2 + 2ζω n + ωn 2 ωn = 5 → ζ = 0.2 -the correction can be found by plotting a point -20log(2x0.2)=7.96 dB above the asymptotes at ωn=5 Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 26. 26 Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 27. 27 lntroduction to the Nyquist Criterion • relates the stability of a closed-loop system to open-loop poles location • the Nyquist criterion can tell us how many closed-loop poles are in the right-half of the complex plane • it is based on mapping of a contour through the function G(s)H(s) Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 28. 28 Mapping Contour A Q -1 • Contour A can be mapped through F(s)=G(s)H(s) into contour B (Nyquist diagram) by substituting each point of contour A into the function F(s) and plotting the resulting complex numbers. Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 29. Nyquist Criterion Contour A -1 • Stability can be determined using equation Z = P – N where P… the number of open-loop poles of G(s)H(s) enclosed by the contour A N…the number of encirclements the Nyquist diagram (contour B) makes about (– 1) Z…the number of right-half-plane poles of the closed-loop system Note: Counterclockwise encirclements are positive, and clockwise encirclements are negative Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 30. 30 Applying the Nyquist Criterion to Determine Stability Note: Counterclockwise encirclements are positive, and clockwise encirclements are negative P=0 N=0 Z=P-N=0 … system is stable P=0 N= - 2 Z=P-N=2 … system is unstable Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 31. 31 Sketching the Nyquist Diagram • The contour that encloses the right half-plane can be mapped through the function G(s)H(s) by substituting points along the contour into G(s)H(s) • Approximations can be made to G(s)H(s) for points around the infinite semicircle by assuming that the vectors originate at the origin. Thus, their length is infinite, and their angles are easily evaluated. Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 32. 32 Problem Sketch the Nyquist diagram for the system G(s) = 500/[(s + 1)(s + 3)(s + 10)]. Solution ω = 43 ω = 30 / 14 SystemsBranislav Hredzak S. Nise Control Dr Engineering, Fourth Edition by Norman Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 33. 33 • If there are open-loop poles situated along the contour enclosing the right half-plane, then a detour around the poles on the contour is required; otherwise, a complete sketch of the Nyquist diagram cannot be made. • each pole's vector rotates through + 1800 as we move around the contour near that pole • detour must be only an infinitesimal distance into the right half- plane, or else some closed-loop, right-half plane poles will be excluded in the count Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 34. 34 • each pole's vector rotates through - 1800 as we move around the contour near that pole • detour must be only an infinitesimal distance into the left half- plane, or else some closed-loop, left-half plane poles will be excluded in the count Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 35. 35 Problem Sketch the Nyquist diagram of the unity feedback system G(s) = (s + 2)/s2. Solution Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 36. 36 Stability via the Nyquist Diagram • Stability can be determined using equation Z = P – N where P… the number of open-loop poles of G(s)H(s) enclosed by the contour N…the number of encirclements the Nyquist diagram makes about (– 1) Z…the number of right-half-plane poles of the closed-loop system • If the closed-loop system has a variable gain K in the loop we can visualize the Nyquist diagram as expanding (increased gain) or shrinking (decreased gain) • OR • we can visualize the Nyquist diagram as remains stationary and the (- 1) point is moved along the real axis. First, we set the gain to unity and position the critical point at (- 1/K) rather than (- 1). Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 37. 37 Stability via the Nyquist Diagram P = 2, hence in order to obtain a stable system Z = P – N = 0, therefore N = 2, which means there must be 2 counterclockwise crossings Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 38. 38 Stability via Mapping Only the Positive jω-Axis - this is a simplified approach to determine stability - there must be no encirclements for this system to be stable • the encirclements of the critical point can be determined from the mapping of the positive jω-axis alone • this system is stable for the range of loop gain, K, that ensures that the open-loop magnitude is less than unity at that frequency where the phase angle is 1800 (or equivalently, - 1800) Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 39. 39 Stability via Mapping Only the Positive jω-Axis - there must be two counterclockwise encirclements for this system to be stable • this system is stable if the open-loop magnitude is greater than unity at that frequency where the phase angle is 1800 (or equivalently, -1800) Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 40. 40 Problem Find the range of gain for stability and instability, and the gain for marginal stability, for the unity feedback system G(s) = K / [ ( s2 + 2s + 2)(s+ 2)]. For marginal stability find the radian frequency of oscillation. Use the Nyquist criterion and the mapping of only the positive imaginary axis. Solution First we draw the Nyquist diagram for K = 1 Im[G ( jω ) H ( jω )] = 0 when ω = 6 1 1 G ( jω ) H ( jω ) ω = 6 =− = ∠1800 20 20 Hence, stable for K < 20 unstable K > 20 marginally stable for K = 20, and oscillation is √6 Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 41. 41 Gain Margin and Phase Margin via the Nyquist Diagram • Gain margin and phase margin can be used to determine how stable a system is. • Systems with greater gain and phase margins can withstand greater changes in system parameters before becoming unstable. Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 42. 42 • Gain margin, GM , is the change in open-loop gain, expressed in decibels (dB), required at 1800 of phase shift to make the closed-loop system unstable. • Phase margin, ΦM ,is the change in open-loop phase shift required at unity gain to make the closed-loop system unstable. Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 43. 43 Problem Find the gain and phase margin for the system G(s) = K / [ ( s2 + 2s + 2)(s+ 2)] if K = 6 . Solution: Im[G ( jω ) H ( jω )] = 0 when ω = 6 G ( jω ) H ( jω ) ω = 6 = −0.3 Hence GM = 20 log(1 / 0.3) = 10.45dB Using computer it can be found that G ( jω ) H ( jω ) has a unity gain at ω =1.253 rad/s The phase angle at ω = 1.253 rad/s is -112.30 Hence Φ M = −112.30 − (−1800 ) = 67.7 0 Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 44. 44 Stability, Gain Margin, and Phase Margin via Bode Plots • Bode plots are subsets of the complete Nyquist diagram but in another form • They can be easily drawn Evaluating Gain and Phase Margins using Bode Plots Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 45. 45 Relation between Closed - Loop Transient and Closed - Loop Frequency Responses Damping Ratio and Closed-Loop Frequency Response • There will be no peak at frequencies above zero if ζ>0.707 Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 46. 46 Since ζ is related to percent overshoot, we can plot Mp vs. percent overshoot − ln(%OS / 100) ζ = π 2 + ln 2 (%OS / 100) Closed-loop frequency response peak vs percent overshoot for a two-pole system Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 47. 47 Response Speed and Closed-Loop Frequency Response Bandwidth of the closed-loop frequency response is defined as the frequency, ωBW , at which the magnitude response curve is 3 dB down from its value at zero frequency M = 1 / 2 corresponds to - 3 dB Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 48. 48 Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 49. 49 Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 50. 50 Relation between Closed- and Open-Loop Frequency Responses Damping Ratio from Phase Margin 2 2 ωn ωn G ( jω ) = = jω ( jω + 2ζω n ) − ω 2 + j 2ζω nω • a phase margin of 65.520 (ζ = 0.707) or larger is required from the open-loop frequency response to ensure there is no peaking in the closed-loop frequency response Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 51. 51 Response Speed from Open-Loop Frequency Response • the closed-loop bandwidth can be estimated from the open-loop frequency response • It can be shown that, approximately, the closed-loop bandwidth, ωBW,equals the frequency at which the open-loop magnitude response is between -6 and -7.5 dB if the open-loop phase response is between -1350 and -2250 ωBW=3.5rad/s Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 52. 52 Steady-State Error Characteristics from Frequency Response Type 0 system Initial value: For Type 0 system: - which is the same as the initial value Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 53. 53 Type 1 system The initial -20 dB/decade slope can be thought of as originating from a function, which crosses the frequency axis (G’=0dB) at For Type 1 system: - which is the same as the frequency intercept Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  • 54. 54 Type 2 system The initial -40 dB/decade slope can be thought of as originating from a function, which crosses the frequency axis (G’=0dB) at For type 2 system: Dr Branislav Hredzak Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.