4- frequency response analysis

Frequency Response Analysis

Outlines

1. Bode Frequency Analysis

2. Nichols Frequency Analysis

3. Nyquist Stability Analysis

4. Obtaining Frequency Response Data


• The frequency response of a dynamic system is the output of a system given unit-amplitude, zero-phase, sinusoidal inputs at varying frequencies.

• You can use the frequency response of a system to locate poles and zeros of a system. Using this information, you then can design a controller to improve unwanted parts of the frequency response.


• When applied to the system, a sinusoidal input with unit amplitude, zero phase, and frequency ω produces the following sinusoidal output.

• A is the magnitude of the response as a function of ω, and φ is the phase. The magnitude and phase of the system output vary depending on the values of the system poles, zeros, and gain.

Bode Frequency Analysis• Use Bode plots of system frequency responses to

assess the relative stability of a closed-loop system given the frequency response of the open-loop system.

• By analyzing the frequency response, you can determine what the open- and closed-loop frequency responses of a system imply about the system behavior.

• Use the CD Bode VI to create a Bode plot.• Use the CD Evaluate at Frequency VI to determine

the frequency at specified values.

Bode Frequency Analysis• For example, consider the following transfer function

that represents a linear time-invariant system.

• Applying the sinusoidal input x(t)=sin(ωt) to this previous system produces the following equation:

• Using this equation, the following equation represents the complex frequency response.

Bode Frequency Analysis• You can separate the complex frequency response

equation into two parts—the magnitude A(ω) and the phase φ(ω).

• These two equations represent the magnitude and the phase of the frequency response, respectively.

• The Bode magnitude plot shows the gain plotted against the frequency.

• The Bode phase plot shows the phase, in degrees, as a function of the frequency.

Bode Frequency Analysis

• Bode plots also illustrate the system bandwidth as the frequency at which the output magnitude is reduced by three decibels.

• You also can use the CD Bandwidth VI to determine the system bandwidth.

Example 1 Bode Magnitude & Phase

Example 2 Bode Magnitude & Phase Margins

LabVIEW MathScript Approach

• Alternatively, you can use the following m-file code in the MathScript Window (Tools »MathScript Window):

num = 50; den = [1 9 30 40]; sys = tf(num,den); bode(sys)

Example3 • This example demonstrates how to create a

Bode plot and a gain and phase margin plot of a system model.

• Using

Nichols Frequency Analysis• Use Nichols frequency analysis to obtain the closed-loop

frequency response of a system from the open-loop response.

• Open-loop response curves, or loci, of constant magnitude and phase often provide reference points that help you analyze a Nichols plot.

• Each point on the open-loop response curve corresponds to the response of the system at a given frequency.

• You then can read the closed-loop magnitude response at that frequency from the Nichols plot by identifying the value of the magnitude locus at which the point on the curve intersects.

Nichols Frequency Analysis• Similarly, you can determine the closed-loop phase by identifying the phase

locus at which the open-loop curve crosses.

• Use the CD Nichols VI to create a Nichols plot and examine system performance in dynamic systems.

• The CD Nichols VI calculates and plots the open-loop frequency response against the gain and phase on the Nichols plot.

• Different points on the plot correspond to different values of the frequency ω.

• Examine the Nichols plot to determine the gain and phase margins, bandwidth, and the effect of gain variations on the closed-loop system behavior.

Nyquist Stability Analysis• Use Nyquist stability analysis to examine the system

performance of dynamic systems.

• Nyquist plots consist of the real part of the frequency response plotted against the imaginary part of the response.

• Nyquist plots indicate the stability of a closed-loop system, given an open-loop system, which includes a gain of K.

Nyquist Stability Analysis

• Use the CD Nyquist VI to create a Nyquist plot.

• The Nyquist stability criterion relates the number of closed-loop poles of the system to the open-loop frequency response.

• On the Nyquist plot, the number of encirclements around (–1, 0) is equal to the number of unstable closed-loop poles minus the number of unstable open-loop poles.

Nyquist Stability Analysis• You can use this criterion to determine how many

encirclements the plant requires for closed-loop stability. • For example, if the plant has all open-loop stable poles, there

are no encirclements. I• f the plant has one open-loop unstable pole, there is one

negative, counter-clockwise encirclement. Figure below shows a system with one unstable pole.

Nyquist Stability Analysis• Often you want to determine a range of gain values

for which the system is stable, rather than testing the stability of the system at a specific value of K. To determine the stability of a closed-loop system, you must determine how a range of gain values affects the stability of the system.

• Consider the following closed-loop transfer function equation with output Y(s) and input U(s), where K is the gain.

Nyquist Stability Analysis

• The closed-loop poles are the roots of the equation 1 + KH(s)=0. The complex frequency response of KH(s), evaluated for s = iω in continuous systems andeiωT

• For discrete systems, encircles (–1, 0) in the complex plane if 1 + KH(s) encircles (0, 0).

• If you examine the Nyquist plot of H(s), you can see that an encirclement of (–1/K,0) by H(s) is the same as an encirclement of (–1, 0) by KH(s).

• Thus, you can use one Nyquist plot to determine the stability of a system for any and all values of K.

Obtaining Frequency Response Data• The Frequency Response VIs discussed return frequency

response data that contains information about the frequency response of all input-output pairs in the model.

• The frequency response information for the CD Bode VIreturns information about the Bode magnitude and Bode phase.

• The frequency response information for the CD Nichols VIreturns information about the real and imaginary parts of the frequency response.

• The frequency response information for the CD Nyquist VIreturns information about the open-loop gain and open-loop phase.

• Use the CD Get Frequency Response Data VI to access this information for a specified input-output pair, a list of input-output pairs, or all input-output pairs of the system.

Obtaining Frequency Response Data

• The CD Get Frequency Response Data VI uses the Frequency Response Data input, which contains the frequency response information for all the input-output pairs of a system model.

• For state-space models, the CD Get Frequency Response Data VI returns the frequency response of the input-state pair(s).

• Because transfer function and zero-pole-gain models do not have states, the frequency response data for an input-state pair of these forms is an empty array.

4- frequency response analysis

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