سیستمهای کنترل خطی پاییز 1389 بسم ا... الرحمن الرحيم دکتر حسين بلندي - دکتر سید مجید اسما عیل زاده

Design Controller Via Root Locus

3 Example: Phase Lead compensator

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5- PID Compensator, adding two zeros and a pole at s = 0

Example: PID controller design

6- Lead-Lag Compensator, adding two zeros and two poles

Example: Lag-Lead compensator design

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Controller design in the frequency domain

Bode Magnitude Plot: plots the magnitude of G(j  ) in decibels w.r.t. logarithmic frequency, i.e., Bode Phase Plot: plots the phase angle of G(j  ) w.r.t. logarithmic frequency, i.e., A better way to graphically display the frequency response! Bode Plots Benefits: –Display the dependence of magnitude of the frequency response on the input frequency better, especially for magnitude approaching zero –Log axis converts the multiplications and divisions into additions and subtractions, which are easier to handle graphically –Allow straight-line approximations for quick sketch

Bode Plots Ex:

Bode Plots of LTI Systems Transfer Function Frequency Response Bode Magnitude Plot Bode Phase Plot

Example Lead Compensator Transfer Function: Plot the straight line approximation of G(s)’s Bode diagram: Phase (deg); Magnitude (dB) Frequency (rad/sec)

1st Order Bode Plots Summary 1st Order Poles –Break Frequency –Mag. Plot Approximation 0 dB from DC to  b and a straight line with  20 dB/decade slope after  b –Phase Plot Approximation 0 deg from DC to. Between and 10  b, a straight line from 0 deg to  90 deg (passing  45 deg at  b ). For frequency higher than 10  b, straight line on  90 deg. 1st Order Zeros –Break Frequency –Mag. Plot Approximation 0 dB from DC to  b and a straight line with 20 dB/decade slope after  b –Phase Plot Approximation 0 deg from DC to. Between and 10  b, a straight line from 0 deg to 90 deg (passing 45 deg at  b ). For frequency higher than 10  b, straight line on 90 deg. Note:By looking at Bode plots you should be able to determine the relative order of the system, its break frequency, and DC (steady-state) gain. This process should also be reversible, i.e., given a transfer function, be able to plot a straight line approximation of Bode plots.

Bode Plots of Complex Poles Standard Form of Transfer Function Frequency Response  n  n nn  n  n Peak (Resonant) Frequency and Magnitude for

2nd Order System Frequency Response Phase (deg); Magnitude (dB) Frequency (rad/sec)  n nn 10  n

2nd Order System Frequency Response A Few Observations: Three different characteristic frequencies: –Natural Frequency (  n ) –Damped Natural Frequency (  d ): –Resonant (Peak) Frequency (  r ): When the damping ratio , there is no peak in the Bode magnitude plot. DO NOT confuse this with the condition for over-damped and under- damped systems: when  the system is under-damped (has overshoot) and when  the system is over-damped (no overshoot). As  ® 0,  r ®  n and  G(j  )    increases; also the phase transition from 0 deg to  180 deg becomes sharper.

Example Second-Order System Transfer Function: Plot the straight line approximation of G(s)’s Bode diagram: Phase (deg); Magnitude (dB) Frequency (rad/sec)

Bode Plots of Complex Zeros Standard Form of Transfer Function Frequency Response Frequency (rad/sec)

Bode Plots of Poles and Zeros Bode plots of zeros are the mirror images of the Bode plots of the identical poles w.r.t. the 0 dB line and the 0 deg line, respectively: LetGs Gs Gj Gj GjGj GjGj GjGj p z p z pz pz pz () () () () ()() log() () ()()     |    | |      Frequency (rad/sec) Phase (deg) Magnitude (dB)  n  n nn

2nd Order Bode Diagram Summary 2nd Order Complex Poles –Break Frequency –Mag. Plot Approximation 0 dB from DC to  n and a straight line with  40 dB/decade slope after  n. Peak value occurs at: –Phase Plot Approximation 0 deg from DC to. Between and    n, a straight line from 0 deg to  180 deg (passing  90 deg at  n ). For frequency higher than    n, straight line on  180 deg. 2nd Order Complex Zeros –Break Frequency –Mag. Plot Approximation 0 dB from DC to  n and a straight line with 40 dB/decade slope after  n. –Phase Plot Approximation 0 deg from DC to. Between and    n, a straight line from 0 deg to 180 deg (passing 90 deg at  n ). For frequency higher than    n, straight line on 180 deg.

Design fundamental of Lead & lag controller (just as a remember)

log a Design fundamental of lag the controller: PM=25° How can the controller help us? PM=40°

1- Phase-lag controller decreases the BW. It makes the system slower. 2- It reduces the noise effect. Decreases the noise affect on the output. Design fundamental of lag the controller:

47 Phase (deg) Magnitude (db) Design fundamental of lag the controller:

φmφm 20 log a PM=25° How can the controller help us? PM=40° Design fundamental of Lead the controller:

49 1- lead controller increases the BW. It makes the system faster. 2- It exaggerates the noise effect. Increases the noise affect on the output. Design fundamental of Lead the controller:

50 Phase (deg) Magnitude (db) Design fundamental of Lead the controller:

Lag controller design

Design procedure of a phase-lag controller in the frequency domain Step 1: Consider with as a phase-lag controller. Step 2: Try to fix k according to the performance request, otherwise let k=1 Step 3: Sketch the Bode plot of the system (with the fixed k ) without controller. Note: If the plant has another gain k, let log a

Step 5: Find the gain of the system at the new gain crossover frequency and let: Step 6: Put the right corner of the controller sufficiently far from Step 7: Check the designed controller. Step 4: Find the system PM and if it is not sufficient choose the new gain crossover frequency such that the PM is ok (reduce it a little). ?

Design a lag controller for the following system such that the phase margin be 45° and the ramp error constant be 100. Find the M p of overall system. + - Step 1: Consider with as a phase-lag controller. Note: If the plant has another gain k, let Step 2: Try to fix k according to the performance request, otherwise let k=1 Example 1:

55 Step 3: Sketch the Bode plot of the system (with the fixed k ) without controller. Step 4: Find the system PM and if it is not sufficient choose the new gain crossover frequency such that the PM is ok (reduce it a little). PM=25°

Step 5: Find the gain of the system at the new gain crossover frequency and let: Step 6: Put the right corner of the controller sufficiently far from

Step 7: Check the designed controller.