DNT 354 - Control Principle Frequency Response Techniques DNT 354 - Control Principle

Contents Introduction Concept of Frequency Response Plotting Frequency Response Asymptotic Approximations for Bode Plots

Introduction Frequency response techniques presents the design of feedback control systems through gain adjustment and compensation networks from the perspective of frequency response. The techniques which were developed by Nyquist and Bode in 1930s are older than that of root locus method, which was discovered by Evans in 1948. Advantages of frequency response techniques: When modelling transfer functions from physical data When designing lead compensators to meet a steady-state error requirement and a transient requirement When finding the stability of non-linear systems In settling ambiguities when sketching a root locus

Concept of Frequency Response In the steady state, sinusoidal inputs to a linear system generate sinusoidal responses of the same frequency, but differs in amplitude and phase angle. The differences are in functions of frequency. Sinusoids can be represented as complex numbers called phasors. The magnitude, M of the complex number is the amplitude the sinusoid, and the angle, Φ of the complex number is the phase angle of the sinusoid. Thus,

Concept of Frequency Response Example: Mechanical System

Concept of Frequency Response If the input force, f(t) is sinusoidal, the stead-state output response, x(t), is also sinusoidal and at the same frequency as the input. The input and out are represented by phasors, and the steady-state output sinusoid is shown below. The system function is thus given by, and,

Concept of Frequency Response M(ω) is called the magnitude frequency response and Φ(ω) the phase frequency response. The combination of magnitude and phase frequency response is called the frequency response given by, Magnitude frequency response: The ratio of the output sinusoid’s magnitude to the input sinusoid’s magnitude. Phase frequency response: Difference in phase angle between the output and the input sinusoids.

Plotting Frequency Response G(jω) can be plotted in several methods: As a function of frequency As polar plot As a function of frequency, the magnitude curve can be plotted in decibels (dB) vs. log ω, while the phase curve is plotted as phase angle vs. log ω. In polar plot, phasor length is the magnitude and the phasor angle is phase.

Plotting Frequency Response Example: Find the analytical expression for the magnitude frequency response and phase frequency response for a the following system. Also, plot both the separate magnitude and phase diagram, as well as polar plot.

Plotting Frequency Response Plot: As function of frequency

Plotting Frequency Response Plot: Polar form

Bode Plots Bode plots or Bode diagrams are the log-magnitude and phase frequency response curves as functions of log ω. Consider the following, The magnitude frequency response is the product of the magnitude frequency response for each term, Converting the magnitude response into dB,

Asymptotic Approximations Sketching Bode plots can be simplified because they can be approximated as a sequence of straight lines. There are 4 rules of sketching the Bode plot: G(s) = (s + a) G(s) = 1 / (s + a) G(s) = s G(s) = 1 / s

Asymptotic Approximations Normalized and scaled Bode plots for: G(s) = s G(s) = 1/s G(s) = (s+a) G(s) = 1/(s+a)

Asymptotic Approximations Example: Draw the Bode plots for the system shown below.

Asymptotic Approximations Bode Plot: Magnitude frequency response (components and composite)

Asymptotic Approximations Bode Plot: Phase frequency response (components and composite)

GAIN & PHASE MARGINS IN BODE PLOT GM – gain margin ΦM – phase margin GM- Gain crossover frequency ΦM- Phase crossover frequency Note that, negative gain or phase margin means that the system is not stable

GAIN & PHASE MARGINS IN BODE PLOT Example: If K=200, find the gain margin and phase margin .

GAIN & PHASE MARGINS IN BODE PLOT

GAIN & PHASE MARGINS IN BODE PLOT 6.02dB -6.02dB -165º 15º 7rad/sec 5.5rad/sec

PID CONTROLLER Proportional compensator (P). Use to improve steady state error type 0. Consider P-compensator transfer function as: High Kp gives better steady state but poor transient response. Too high Kp can cause instability. + R(s) E(s) A(s) Y(s) - B(s)

PID CONTROLLER Integral compensator (I). Use to improve steady state error type 0. Consider the I-compensator and actuating signal of : Slow response, can be used with P-compensator to remedy this problem. + R(s) A(s) Y(s) E(s) - B(s)

PID CONTROLLER Derivative compensator (D). Consider the D-compensator as and actuating signal as: Quick response. No effect at steady state because no error signal. Useful for controlling type 2 together with a P-controller. Response only to rate of change and no effect to steady state. + R(s) A(s) Y(s) E(s) - B(s)

PID CONTROLLER Proportional-integral-derivative compensator (PID) . Involves three separate parameters; the proportional, the integral and derivative values.