SECTION 7: FREQUENCY‐RESPONSE ANALYSIS

MAE 4421 – Control of Aerospace & Mechanical Systems 2 Introduction

K. Webb MAE 4421 Introduction

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 We have seen how to design feedback control systems using the root locus  In this section of the course, we’ll learn how to do the same using the open‐loop frequency response

 Objectives:  Review frequency‐response fundamentals  Relate a system’s frequency response to its transient response  Determine static error constants from the open‐loop frequency response  Determine closed‐loop stability from the open‐loop frequency response

K. Webb MAE 4421 System Response to a Sinusoidal Input

4  Consider an ‐order system  poles: ,,…  Real or complex  Assume all are distinct  Transfer function is:

(1) ⋯

 Apply a sinusoidal input to the system

sin

 Output is given by

∙ (2) ⋯

K. Webb MAE 4421 System Response to a Sinusoidal Input

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 Partial fraction expansion of (2) gives

⋯ (3)

 Inverse transform of (3) gives the time‐domain output

⋯ cos sin (4)

transient steady state

 Two portions of the response:  Transient  Decaying exponentials or sinusoids –goes to zero in steady state  Natural response to initial conditions  Steady state  Due to the input – sinusoidal in steady state

K. Webb MAE 4421 Steady‐State Sinusoidal Response

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 We are interested in the steady‐state response

cos sin (5)

 A trig. identity provides insight into : cos sin sin where tan

 Steady‐state response to a sinusoidal input sin

is a sinusoid of the same frequency, but, in general different amplitude and phase

sin Where (6)

and tan

K. Webb MAE 4421 Steady‐State Sinusoidal Response

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sin → sin

 Steady‐state sinusoidal response is a scaled and phase‐shifted sinusoid of the same frequency  Equal frequency is a property of linear systems

 Note the term in the numerator of (3)  will affect the residues  Residues determine amplitude and phase of the output  Output amplitude and phase are frequency‐dependent

sin

K. Webb MAE 4421 Steady‐State Sinusoidal Response

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sin Linear System sin

 Gain –the ratio of amplitudes of the output and input of the system

 Phase – phase difference between system input and output

 Systems will, in general, exhibit frequency‐dependent gain and phase

 We’d like to be able to determine these functions of frequency  The system’s frequency response

K. Webb MAE 4421 9 Frequency Response

A system’s frequency response, or sinusoidal transfer function, describes its gain and phase shift for sinusoidal inputs as a function of frequency.

K. Webb MAE 4421 Frequency Response

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 System output in the Laplace domain is ⋅

 Multiplication in the Laplace domain corresponds to convolution in the time domain ∗

 Consider an exponential input of the form

where is the complex Laplace variable:

 Now the output is ∗ ⋅ (1)

K. Webb MAE 4421 Frequency Response

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⋅ (1)

 We’re interested in the steady‐state response, so let the upper limit of integration go to infinity ⋅ ⋅ (2)

 Time‐domain response to an exponential input is the time‐domain input multiplied by the system transfer function  What is this input? (3)

 If we let →0, i.e. let →, then we have ⋅ (4)

K. Webb MAE 4421 Euler’s Formula

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 Recall Euler’s formula:

cos sin (5)

 From which it follows that

cos (6) and

sin (7)

K. Webb MAE 4421 Frequency Response

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 We’re interested in the sinusoidal steady‐state system response, so let the input be cos 2

 A sum of complex exponentials in the form of (3)  We’ve let →in the first term and →in the second (8)

 According to (4) the output in response to (8) will be

⋅ ⋅ (9)

K. Webb MAE 4421 Frequency Response

14 ⋅ ⋅ (9)

 is a complex function of frequency  Evaluates to a complex number at each value of  Has both magnitude and phase  Can be expressed in polar form as

(10) where and ∠

 It follows that

(11)

K. Webb MAE 4421 Frequency Response

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 Using (11), the output given by (9) becomes

(12)

(13) where, again

and ∠ (14)

K. Webb MAE 4421 Frequency response Function –

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 is the system’s frequency response function  Transfer function, where →

|→ (15)

 A complex‐valued function of frequency

 at each is the gain at that frequency  Ratio of output amplitude to input amplitude

 ∠ at each is the phase at that frequency  Phase shift between input and output sinusoids

 Another representation of system behavior  Along with state‐space model, impulse/step responses, transfer function, etc.  Typically represented graphically

K. Webb MAE 4421 Plotting the Frequency Response Function

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 is a complex‐valued function of frequency  Has both magnitude and phase  Plot gain and phase separately  Frequency response plots formatted as Bode plots  Two sets of axes: gain on top, phase below  Identical, logarithmic frequency axes  Gain axis is logarithmic –either explicitly or as units of decibels (dB)  Phase axis is linear with units of degrees

K. Webb MAE 4421 Bode Plots

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Units of magnitude are dB Magnitude plot on top

Logarithmic frequency axes

Units of Phase plot phase are below degrees

K. Webb MAE 4421 Interpreting Bode Plots

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Bode plots tell you the gain and phase shift at all frequencies: choose a frequency, read gain and phase values from the plot

For a 10MHz For a 10KHz sinusoidal sinusoidal input, the input, the gain is ‐32dB gain is 0dB (1) (0.025), and and the phase the phase shift is 0°. shift is ‐176°.

K. Webb MAE 4421 Interpreting Bode Plots

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K. Webb MAE 4421 Decibels ‐ dB

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 Frequency response gain most often expressed and plotted with units of decibels (dB)  A logarithmic scale  Provides detail of very large and very small values on the same plot  Commonly used for ratios of powers or amplitudes

 Conversion from a linear scale to dB:

20⋅log

 Conversion from dB to a linear scale:

10

K. Webb MAE 4421 Decibels –dB

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 Multiplying two gain values corresponds to adding their values in dB  E.g., the overall gain of cascaded systems

⋅

 Negative dB values corresponds to sub‐unity gain  Positive dB values are gains greater than one

dB Linear dB Linear 60 1000 62 40 100 ‐3 1/√20.707 20 10 ‐60.5 01 ‐20 0.1

K. Webb MAE 4421 Value of Logarithmic Axes ‐ dB

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 Gain axis is linear in dB  A logarithmic scale  Allows for displaying detail at very large and very small levels on the same plot

 Gain plotted in dB  Two resonant peaks clearly visible

 Linear gain scale  Smaller peak has disappeared

K. Webb MAE 4421 Value of Logarithmic Axes ‐ dB

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 Frequency axis is logarithmic  Allows for displaying detail at very low and very high frequencies on the same plot

 Log frequency axis  Can resolve frequency of both resonant peaks

 Linear frequency axis  Lower resonant frequency is unclear

K. Webb MAE 4421 Gain Response –Terminology

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 Corner frequency, cut off frequency, ‐3dB frequency:  Frequency at which ~5 of peaking gain is 3dB below its low‐frequency value

2 1.45  This is the bandwidth 0.23 of the system 2

 Peaking  Any increase in gain above the low frequency gain

K. Webb MAE 4421 26 Frequency‐Response Factors

K. Webb MAE 4421 Transfer Function Factors

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 Numerator and denominator of a transfer function can be factored into first‐ and second‐order terms

⋯ 2 2 ⋯ ⋯ 2 2 ⋯

 Can think of the transfer function as a product of the individual factors  For example, consider the following system

2

 Can rewrite as 1 1 ⋅ ⋅ 2

K. Webb MAE 4421 Transfer Function Factors

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 Think of this as three cascaded transfer functions

, ,

or

1 1 2

K. Webb MAE 4421 Transfer Function Factors

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 In the Laplace domain, transfer function of a cascade of systems is the product of the individual transfer functions  In the time domain, overall impulse response is the convolution of the individual impulse responses

 Same holds true in the frequency domain  Frequency response of a cascade is the product of the individual frequency responses  Or, the product of individual factors

K. Webb MAE 4421 Frequency Response Components ‐ Example

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 Consider the following system 20 20 1 100

 The system’s frequency response function is 20 20 1 100

 As we’ve seen we can consider this a product of individual frequency response factors 1 1 20 ⋅ 20 ⋅ ⋅ 1 100

 Overall response is the composite of the individual responses  Product of individual gain responses –sum in dB  Sum of individual phase responses

K. Webb MAE 4421 Frequency Response Components ‐ Example

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 Gain response

K. Webb MAE 4421 Frequency Response Components ‐ Example

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 Phase response

K. Webb MAE 4421 33 Bode Plot Construction

In this section, we’ll look at a method for sketching, by hand, a straight‐line, asymptotic approximation for a Bode plot.

K. Webb MAE 4421 Bode Plot Construction

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 We’ve just seen that a system’s frequency response function can be factored into first‐ and second‐ order terms  Each factor contributes a component to the overall gain and phase responses

 Now, we’ll look at a technique for manually sketching a system’s Bode plot  In practice, you’ll almost always plot with a computer  But, learning to do it by hand provides valuable insight

 We’ll look at how to approximate Bode plots for each of the different factors

K. Webb MAE 4421 Bode Form of the Transfer function

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 Consider the general transfer function form: ⋯ 2 ⋯ ⋯ 2 ⋯

 We first want to put this into Bode form:

2 1 1 ⋯ 1 ⋯ 2 1 1 ⋯ 1 ⋯

 The corresponding frequency response function, in Bode form, is

2 1 1 ⋯ 1 ⋯ 2 1 1 ⋯ 1 ⋯

 Putting into Bode form requires putting each of the first‐ and second‐order factors into Bode form

K. Webb MAE 4421 First‐Order Factors in Bode Form

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 First‐order frequency‐response factors include:

, ,  For the first factor, , is a positive or negative integer  Already in Bode form

 For the second two, divide through by , giving

1 and

 Here, , the corner frequency associated with that zero or pole, so

1 and

K. Webb MAE 4421 Second‐Order Factors in Bode Form

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 Second‐order frequency‐response factors include:

2 and  Again, normalize the coefficient, giving

/ 1 and

 Putting each factor into its Bode form involves factoring out any DC gain component  Lump all of DC gains together into a single gain constant,

⋯ ⋯ ⋯ ⋯

K. Webb MAE 4421 Bode Plot Construction

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 Frequency response function in Bode form

⋯ ⋯ ⋯ ⋯

 Product of a constant DC gain factor, , and first‐ and second‐order factors

 Plot the frequency response of each factor individually, then combine graphically  Overall response is the product of individual factors  Product of gain responses –sum on a dB scale  Sum of phase responses

K. Webb MAE 4421 Bode Plot Construction

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 Bode plot construction procedure: 1. Put the sinusoidal transfer function into Bode form 2. Draw a straight‐line asymptotic approximation for the gain and phase response of each individual factor 3. Graphically add all individual response components and sketch the result

 Next, we’ll look at the straight‐line asymptotic approximations for the Bode plots for each of the transfer function factors

K. Webb MAE 4421 Bode Plot –Constant Gain Factor

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 Constant gain

 Constant Phase

K. Webb MAE 4421 Bode Plot – Poles/Zeros at the Origin

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 0:  zeros at the origin

 0:  poles at the origin

 Gain:  Straight line  Slope ⋅20 ⋅6

 0 at 1

 Phase: ∠ ⋅ 90°

K. Webb MAE 4421 Bode Plot –First‐Order Zero

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 Single real zero at

 Gain: 1  0 for  20 6 for  Straight‐line asymptotes intersect at ,0  Phase:

 0° for 0.1  45° for  90° for 10 °  for 0.1 10

K. Webb MAE 4421 Bode Plot –First‐Order Pole

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 Single real pole at

 Gain:  0 for 1 1  20 6 for  Straight‐line asymptotes intersect at ,0  Phase:

 0° for 0.1  45° for  90° for 10 °  for 0.1 10

K. Webb MAE 4421 Bode Plot –Second‐Order Zero

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 Complex‐conjugate zeros:

,

 Gain:

 0 for ≪ 2 1  40 12 for ≫  Straight‐line asymptotes intersect at ,0

 ‐dependent peaking around

 Phase:

 0° for ≪

 90° for

 180° for ≫

 ‐dependent slope through

 Sketch as step‐change at for low , 90°/ for high , or in between

K. Webb MAE 4421 Bode Plot –Second‐Order Pole

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 Complex‐conjugate poles:

,

 Gain:

 1 0 for ≪ 2  40 12 for ≫ 1  Straight‐line asymptotes intersect at ,0

 ‐dependent peaking around

 Phase:

 0° for ≪

 90° for

 180° for ≫

 ‐dependent slope through

 Sketch as step‐change at for low , 90°/ for high , or in between

K. Webb MAE 4421 Bode Plot Construction –Example

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 Consider a system with the following transfer function

10 20 400

 The sinusoidal transfer function:

10 20 400

 Put it into Bode form

10 ⋅ 20 1 0.5 1 20 20 ⋅ 400 1 ⋅ 1 400 400

 Represent as a product of factors

1 1 0.5 ⋅ 1 ⋅ ⋅ 20 1 400

K. Webb MAE 4421 Bode Plot Construction –Example

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K. Webb MAE 4421 Bode Plot Construction –Example

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K. Webb MAE 4421 49 Polar Frequency Response Plots

K. Webb MAE 4421 Polar Frequency Response Plots

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 is a complex function of frequency  Typically plot as Bode plots  Magnitude and phase plotted separately  Aids visualization of system behavior

 A real and an imaginary part at each value of  A point in the complex plane at each frequency  Defines a curve in the complex plane  A polar plot  Parametrized by frequency –not as easy to distinguish frequency as on a Bode plot

 Polar plots are not terribly useful as a means of displaying a frequency response  However, an important concept later, when we introduce the Nyquist stability criterion

K. Webb MAE 4421 Polar Frequency Response Plots

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 Identical frequency responses plotted two ways:  Bode plot and polar plot  Note uneven frequency spacing along polar plot curve  Dependent on frequency rates of change of gain and phase

K. Webb MAE 4421 Relationship between Frequency 52 Response and Transient Response

K. Webb MAE 4421 Transient/Frequency Response Relationship

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 We have seen relationships –some exact, some approximate – between closed‐loop pole locations and closed‐loop transient response

 Also have relationships between closed‐loop frequency response and closed‐loop transient responses

 Applicable to second‐order systems: 2

 Also applicable to higher‐order systems that are reasonably approximated as second‐order  Systems with a pair of dominant second‐order poles

K. Webb MAE 4421 Transient/Frequency Response Relationship

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 Qualitative 2nd‐order time/freq. response/pole relationships  Damping ratio vs. overshoot vs. peaking  Natural frequency vs. risetime vs. bandwidth

K. Webb MAE 4421 Frequency Response Peaking

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 For systems with 0.707, the gain response will exhibit peaking

 Can relate peak magnitude to the damping ratio 1 2 1

 Relative to low‐frequency gain

 And the peak frequency to the damping ratio and natural frequency

12

K. Webb MAE 4421 Transient/Frequency Response Relationship

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 Can also relate a system’s bandwidth (i.e., ‐3dB frequency, ) to the speed of its step response

 Bandwidth as a function of and

12 4 4 2

 Bandwidth as a function of 1% settling time and 4.6 12 4 4 2

 Bandwidth as a function of peak time and 12 4 4 2 1

K. Webb MAE 4421 57 Steady‐State Error from Bode Plots

K. Webb MAE 4421 Static Error Constants

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 For unity‐feedback systems, open‐loop transfer function gives static error constants  Use static error constants to calculate steady‐state error lim →

lim → lim →

 We can also determine static error constants from a system’s open‐loop Bode plot

K. Webb MAE 4421 Static Error Constant –Type 0

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 For a type 0 system

→

 At low frequency, i.e. below any open‐loop poles or zeros

100 30 3 200  Read directly from the open‐loop Bode plot  Low‐frequency gain

K. Webb MAE 4421 Static Error Constant –Type 1

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 For a type 1 system

lim →  At low frequencies, i.e. below any other open‐loop poles or zeros and

 A straight line with a slope of /  Evaluating this low‐frequency asymptote at yields the velocity constant,  On the Bode plot, extend the low‐frequency asymptote to  Gain of this line at 1is

K. Webb MAE 4421 Static Error Constant –Type 1

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85 0.1 50 10 125

K. Webb MAE 4421 Static Error Constant –Type 2

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 For a type 2 system lim →  At low frequencies, i.e. below any other open‐loop poles or zeros and

 A straight line with a slope of /  Evaluating this low‐frequency asymptote at yields the acceleration constant,  On the Bode plot, extend the low‐frequency asymptote to  Gain of this line at 1is

K. Webb MAE 4421 Static Error Constant –Type 2

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1600 0.1 5 100

K. Webb MAE 4421 64 Nyquist Stability Criterion

K. Webb MAE 4421 Stability

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 Consider the following system

 We already have a couple of tools for assessing stability as a function of loop gain,  Routh Hurwitz  Root locus  Root locus:  Stable for some values of  Unstable for others

K. Webb MAE 4421 Stability

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 In this case gain is stable below some value  Other systems may be stable for gain above some value  Marginal stability point:  Closed‐loop poles on the imaginary axis at  For gain

K. Webb MAE 4421 Open‐Loop Frequency Response & Stability

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 Marginal stability point occurs when closed‐loop poles are on the imaginary axis

 Angle criterion satisfied at

1 and ∠ 180°

 Note that

 is the open‐loop frequency response  Marginal stability occurs when:  Open‐loop gain is:  Open‐loop phase is:

K. Webb MAE 4421 Stability from Bode Plots

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 Varying simply shifts gain response up or down  Here, stable for smaller gain values  0 when ∠ 180°

 Often, stable for larger gain values  0 when ∠ 180°

 Root locus provides this information  Bode plot does not

K. Webb MAE 4421 Open‐Loop Frequency Response & Stability

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 A method does exist for determining stability from the open‐loop frequency response:  Nyquist stability criterion  Graphical technique  Uses open‐loop frequency response  Determine system stability  Determine gain ranges for stability  Before introducing the Nyquist criterion, we must first introduce the concept of complex functional mapping

K. Webb MAE 4421 Complex Functional Mapping

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 Consider a complex function ⋯ ⋯

 Takes one complex value, , and yields a second complex value,  In other words, it maps to

K. Webb MAE 4421 Mapping of Contours

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 provides a mapping of individual points in the s‐plane to corresponding points in the F‐plane

 Can also map all points around a contour in the s‐ plane to another contour in the F‐plane

K. Webb MAE 4421 Mapping of Contours

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 Recall how we approached the application of the angle criterion  Vector approach to the evaluation of a transfer function at a particular point in the s‐plane

∏ ∏

∠ Σ∠ Σ∠

 Can take the same approach to evaluating complex functions around contours in the s‐plane

K. Webb MAE 4421 Mapping Contours –Example 1

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 Map contour by in a clockwise direction  Contour does not enclose the zero  Here, , so and ∠ ∠

 As is evaluated around , ∠ never exceeds 0° or 180°  does the same:  Does not rotate through a full 360°  Contour does not encircle the origin

K. Webb MAE 4421 Mapping Contours –Example 2

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 Map contour by in a clockwise direction  Contour does not enclose the pole  Here, 1/, so 1/ and ∠ ∠

1

 ∠ oscillates over some range well within 0° and 180°  rotates through the negative of the same range  Contour does not encircle the origin

K. Webb MAE 4421 Mapping Contours –Example 3

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 Now, contour encloses a single zero

 , so and ∠ ∠

 rotates through a full 360° in a clockwise direction  does the same:  Contour encircles the origin in a clockwise direction

K. Webb MAE 4421 Mapping Contours –Example 4

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 Now, contour encloses a single pole

 1/, so 1/ and ∠ ∠

1

 rotates through a full 360° in a clockwise direction  rotates in the opposite direction  Contour encircles the origin in a CCW direction

K. Webb MAE 4421 Mapping Contours –Example 5

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 Now, contour encloses two poles  , so and ∠ ∠ ∠

1

 and each rotate through a full 360° in a clockwise direction  rotates in the opposite direction  Contour encircles the origin twice in a CCW direction

K. Webb MAE 4421 Mapping Contours –Example 6

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 Now, contour encloses one pole and one zero

 , so and ∠ ∠ ∠

 ∠ and∠ rotate through 360° in a CW direction  Their contributions rotate in opposite directions  ∠ does not rotate through a full 360°  Contour does not encircle the origin

K. Webb MAE 4421 Complex Functional Mapping of Contours

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 Some observations regarding complex mapping of contour in a CW direction to contour :  If does not enclose any poles or zeros, does not encircle the origin  If encloses a single pole, will encircle the origin once in a CCW direction  If encloses two poles, will make two CCW encirclements of the origin  If encloses a pole and a zero, will not encircle the origin

 Next, we’ll use these observations to help derive the Nyquist stability criterion

K. Webb MAE 4421 Nyquist Stability Criterion

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 Our goal is to assess closed‐loop stability  Determine if there are any closed‐loop poles in the RHP

 Consider a generic feedback system:

 Closed‐loop transfer function 1

 Closed‐loop poles are roots (zeros) of the closed‐loop characteristic polynomial: 1

K. Webb MAE 4421 Nyquist Stability Criterion

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 Can represent the individual transfer functions as and  The closed‐loop polynomial becomes 1 1

 From this, we can see that:  The poles of 1 are the poles of , the open‐loop poles  The zeros of 1 are the poles of , the closed‐loop poles

K. Webb MAE 4421 Nyquist Stability Criterion

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 To determine stability, look for RHP closed‐loop poles

 Evaluate 1 CW around a contour that encircles the entire right half‐plane  Evaluate 1 along entire ‐axis  Encircle the entire RHP with an infinite‐radius arc

 If 1 has one RHP pole, resulting contour will encircle the origin once CCW

 If 1 has one RHP zero, resulting contour will encircle the origin once CW

K. Webb MAE 4421 Nyquist Stability Criterion

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 Total number of CW encirclements of the origin, , by the resulting contour will be  # of RHP poles of 1  # of RHP zeros of 1

 Want to detect RHP poles of , zeros of 1 , so

 # of closed‐loop RHP poles

 # of open‐loop RHP poles

 # of CW encirclements of the origin

K. Webb MAE 4421 Nyquist Stability Criterion

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 Basis for detecting closed‐loop RHP poles  Map contour encircling the entire RHP through closed‐ loop characteristic polynomial  Count number of CW encirclements of the origin by resulting contour  Calculate the number of closed‐loop RHP poles:

 Need to know:  Closed‐loop characteristic polynomial  Number of RHP poles of closed‐loop characteristic polynomial

K. Webb MAE 4421 Nyquist Stability Criterion

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 Instead, map through

 Open‐loop transfer function  Easy to use for mapping –we know poles and zeros  Resulting contour shifts left by 1 –that’s all

 Now, count encirclements of the point

K. Webb MAE 4421 Nyquist Stability Criterion

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 Nyquist stability criterion  If a contour that encloses the entire RHP is mapped through the open‐loop transfer function, , then the number of closed‐loop RHP poles, , is given by

where # of CW encirclements of # of open‐loop RHP poles

K. Webb MAE 4421 Nyquist Stability Criterion

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 Want to detect net clockwise encirclements # CW encirclements ‐ # CCW encirclements

 Draw a line from in any direction 2  Count number of times contour crosses the line in each direction

K. Webb MAE 4421 Nyquist Diagram

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 The contour that results from mapping the perimeter of the entire RHP is a Nyquist diagram  Consider four segments of the contour: 1) Along positive ‐axis, we’re evaluating ) . Open‐loop frequency response ① ④ 2) Here, → ② . Maps to zero for any physical system

3) Here, evaluating ) ③ . Complex conjugate of segment ① . Mirror ① about the real axis 4) The origin . Sometimes a special case – more later

K. Webb MAE 4421 Nyquist Criterion –Example 1

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 Apply the Nyquist criterion to determine stability for the following system

 First evaluate along segment ①, ‐axis  This is the frequency response  Read values off of the Bode plot

K. Webb MAE 4421 Nyquist Criterion –Example 1

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 Segment ① is a polar plot of the frequency response  All of segment ②, arc at , maps to the origin

K. Webb MAE 4421 Nyquist Criterion –Example 1

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 Segment ③ is the complex conjugate of segment ①  Mirror about the real axis

K. Webb MAE 4421 Nyquist Criterion –Example 1

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 Count CW encirclements of  Draw a line from 1in any direction  Here,  Closed‐loop RHP poles given by:

 No open‐loop RHP poles, so

 Two RHP poles, so system is unstable

K. Webb MAE 4421 Nyquist Criterion –Example 2

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 This system is open‐loop stable  Stable for low enough  Nyquist plot will not encircle

 Three poles and no zeros  Unstable for above some value  Nyquist plot will encircle

K. Webb MAE 4421 Nyquist Criterion –Example 2

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 For , , and the system is stable  Modifying simply scales the magnitude of the Nyquist plot

K. Webb MAE 4421 Nyquist Criterion –Example 2

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 Here, the Nyquist plot crosses the negative real axis at

 As gain increases real‐ axis crossing moves to the left

 Increasing by 2x or more results in two encirclements of

 Unstable for 60  More later …

K. Webb MAE 4421 Nyquist Diagram –Poles at the Origin

96

 We evaluate the open‐loop transfer function along a contour including the ‐axis  is undefined at the pole  Must detour around the pole  Consider the common case of a pole at the origin

 This is the special case for segment ④ ④

1 2

K. Webb MAE 4421 Nyquist Diagram –Poles at the Origin

97  Segment ④ contour: for 0° 90°

 Evaluate around segment ④ as

1 2

 Magnitude: 1 1 ④ 2 2

 As →0 lim ∞ →  Maps to an arc at

K. Webb MAE 4421 Nyquist Diagram –Poles at the Origin

98

 Segment ④ traversed in a CCW direction

 varies from

 Phase of the resulting contour: ∠ ④

 Negative because it is angle from a pole  Extra phase from additional pole

 maps segment ④ to:  An arc at  Rotating CW from 0° to 90°

K. Webb MAE 4421 Nyquist Criterion –Example 3

99

 Apply the Nyquist criterion to determine stability for the following system

 Use Bode plot to map segment ①  Infinite DC gain  Starts at at for

K. Webb MAE 4421 Nyquist Criterion –Example 3

100  Segment ① starts at at  Heads to the origin at  All of segment ②, arc at , maps to the origin

K. Webb MAE 4421 Nyquist Criterion –Example 3

101

 Segment ③ is the complex conjugate of segment ①  Mirror about the real axis

K. Webb MAE 4421 Nyquist Criterion –Example 3

102  Segment 4 maps to a CW arc at  CW, so it does not encircle  Can’t draw to scale

 Here,  No open‐loop RHP poles, so

 No RHP poles, so system is stable

K. Webb MAE 4421 103 Stability Margins

K. Webb MAE 4421 Stability Margins

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 Recall a previous example

 According to the Nyquist plot, the system is stable  How stable?

 Two stability metrics  Both are measures of how close the Nyquist plot is to encircling the point 1  Gain margin and phase margin

K. Webb MAE 4421 Crossover Frequencies

105

 Two important frequencies when assessing stability:

 Gain crossover frequency  The frequency at which the open‐loop gain crosses

 Phase crossover frequency  The frequency at which the open‐loop phase crosses

K. Webb MAE 4421 Gain Margin

106

 An open‐loop‐stable system will be closed‐loop stable as long as its gain is less than unity at the phase crossover frequency

 Gain margin, GM  The change in open‐ loop gain at the phase crossover frequency required to make the closed‐loop system unstable

K. Webb MAE 4421 Phase Margin

107

 An open‐loop‐stable system will be closed‐loop stable as long as its phase has not fallen below at the gain crossover frequency

 Phase margin, PM  The change in open‐ loop phase at the gain crossover frequency required to make the closed‐loop system unstable

K. Webb MAE 4421 Gain and Phase Margins from Bode Plots

108

K. Webb MAE 4421 Phase Margin and Damping Ratio,

109

 PM can be expressed as a function of damping ratio, , as tan

 For 65° or so, we can approximate: 100 or

K. Webb MAE 4421 110 Frequency Response Analysis in MATLAB

K. Webb MAE 4421 bode.m

111 [mag,phase] = bode(sys,w)

 sys: system model –state‐space, transfer function, or other  w: optional frequency vector –in rad/sec  mag: system gain response vector  phase: system phase response vector –in degrees

 If no outputs are specified, bode response is automatically plotted –preferable to plot yourself

 Frequency vector input is optional  If not specified, MATLAB will generate automatically

 May need to do: squeeze(mag) and squeeze(phase) to eliminate singleton dimensions of output matrices

K. Webb MAE 4421 nyquist.m

112 nyquist(sys,w)

 sys: system model –state‐space, transfer function, or other  w: optional frequency vector –in rad/sec

 MATLAB generates a Nyquist plot automatically

 Can also specify outputs, if desired: [Re,Im] = nyquist(sys,w)

 Plot is not be generated in this case

K. Webb MAE 4421 margin.m

113 [GM,PM,wgm,wpm] = margin(sys)

 sys: system model –state‐space, transfer function, or other  GM: gain margin  PM: phase margin –in degrees  wgm: frequency at which GM is measured, the phase crossover frequency –in rad/sec  wpm: frequency at which PM is measured, the gain crossover frequency

 If no outputs are specified, a Bode plot with GM and PM indicated is automatically generated

K. Webb MAE 4421