CREATAS  IEEE CONTROL SYSTEMS MAGAZINE FEBRUARY  X
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CREATAS IEEE CONTROL SYSTEMS MAGAZINE FEBRUARY X

00575132007IEEE Lead and Lag Compensators with Complex Poles and Zeros DESIGN FORMULAS FOR MODELING AND LOOP SHAPING WILLIAM C MESSNER MARK D BEDILLION LU XIA and DUANE C KARNS n classical loop shaping compensator structures are typically cascaded to

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CREATAS IEEE CONTROL SYSTEMS MAGAZINE FEBRUARY X




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CREATAS 44 IEEE CONTROL SYSTEMS MAGAZINE FEBRUARY 2007 1066-033X/07/$25.002007IEEE Lead and Lag Compensators with Complex Poles and Zeros DESIGN FORMULAS FOR MODELING AND LOOP SHAPING WILLIAM C. MESSNER, MARK D. BEDILLION, LU XIA, and DUANE C. KARNS n classical loop shaping, compensator structures are typically cascaded to modify the gain and phase char- acteristics of the open-loop frequency response. These alterations are used to achieve closed-loop perfor- mance specifications for disturbance rejection, refer- ence following, noise rejection, and gain and phase

margins. Lead and lag compensators are standard tools employed in the loop-shaping process. A lead compensator has the transfer function ,( whose maximum phase occurs at the frequency pz and > . The gain of (1) is unity at . The maximum phase lead of the lead compensator is arctan arctan (3) Since arctan ( )< arctan ( )< 90 , the maxi- mum phase lead is less than 90. Furthermore, , , and are related by (2) and
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sin sin Figure 1 shows the Bode plot for a lead compensator with a maximum phase lead of 45 at rad/s. The pole is at = 41 , and the zero is at =

41 . The transfer function of a lag compensator is the reciprocal of a lead compensator. Lead compensators are used to increase the phase mar- gin for a given 0-dB crossover frequency. Lag compensators are used to increase the low-frequency gain for improved disturbance rejection or to decrease the high-frequency gain for improved noise rejection or augmented gain margin. Lead and lag compensators can also be used in modeling to match gain and phase features of frequency-response data. For phase lead greater than or equal to 90, multiple lead compensators must be cascaded together. In

practice, cascad- ed lead compensators are also used for phase lead less than 90 because the high-frequency gain of a single lead compen- sator with maximum phase lead of more than 60 is signifi- cantly larger than the high-frequency gain of cascaded lead compensators each contributing 1/ of the same total phase lead. Figure 2 shows the high-frequency gain of a single lead compensator as a function of the total phase lead. The simplest cascade is the square of a lead compen- sator, called a double lead compensator . The transfer function of the double lead compensator, which

has maximum phase lead , is double (5) where , , and . The poles and zeros of the double lead compensator are real and have multi- plicity of two. The last expression in (5) suggests a variation of the double lead compensator in which the polynomials of the second-order biproper transfer function have damping ratios less than one. In particular, the complex lead com- pensator introduced in [1] is a variation of the double lead compensator in which the poles and zeros are complex conjugates with the same damping ratio. This transfer function has the form complex ,( where is the undamped natural

frequency of the poles, and is the undamped natural frequency of the zeros. The equal damping ratios provide a symmetric phase peak. The gain of (6) is unity at , the frequency of maxi- mum phase lead. The maximum phase lead is designated as for notational simplicity in the following develop- ment. The transfer function of a complex lag compensator is the reciprocal of a complex lead compensator. Figure 3 shows the Bode plot of a complex lead com- pensator with damping ratio and maximum phase lead 90 at rad/s. For comparison, Figure 3 also shows the Bode plot of a double lead

compensator with maximum phase lead 90 at rad/s. The Bode plots in Figure 3 illustrate several notable dif- ferences between the double lead compensator and the complex lead compensator. First, the phase peak of the complex lead compensator is sharper than the phase peak of the double lead compensator. Next, the magnitude tran- sition region of the complex lead compensator is narrower than the magnitude transition region of the double lead FEBRUARY 2007 IEEE CONTROL SYSTEMS MAGAZINE 45 FIGURE 1 Bode plot of the lead compensator of (1). The maximum phase lead is 45 at rad/s. The

pole is at = 41 , and the zero is at = 41 10 10 Magnitude (dB) 10 10 10 10 10 30 60 Phase ( Frequency (rad/s) FIGURE 2 High-frequency gain of the single lead compensator (1) as a function of its maximum phase lead . The high-frequency gain increases nonlinearly with increasing phase lead. Lead com- pensators are often cascaded in practice to reduce the high-fre- quency gain for a desired maximum phase lead. 30 40 50 60 70 80 10 15 20 25 30 35 40 45 Maximum Phase Lead ( High Frequency Gain (dB)
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compensator. Furthermore, the ratio of the high-frequency gain asymptote to the

low-frequency gain asymptote is smaller for the complex lead compensator than for the double lead compensator. Finally, the slope of the magni- tude at the frequency is steeper for the complex lead compensator than for the double lead compensator. The smaller ratio between the high-frequency and low- frequency gain asymptotes of the complex lead compen- sator is an advantage compared to the double lead compensator when the complex lead compensator is used to increase the phase margin for a given 0-dB crossover frequency. However, the narrower phase peak of the com- plex lead compensator

implies that the phase of the com- pensated open-loop attains 180 crossover at a lower frequency when using a complex lead compensator than when using a double lead compensator. Also, the rate at which the open-loop magnitude drops above the 0-dB crossover frequency is smaller for a complex lead compen- sator than for a double lead compensator. These latter two features might result in a smaller gain margin or lower noise rejection near the 0-dB crossover frequency [2]. For a complex lag compensator, the steeper magnitude slope and narrower phase notch are advantages compared to the double lag

compensator, while the smaller difference between the high-frequency and low-frequency gain is a dis- advantage. The smaller difference in the gain asymptotes of the complex lag compensator tends to reduce the low-fre- quency gain and decrease the low-frequency disturbance rejection compared with the double lag compensator. How- ever, the steeper magnitude slope and the narrower phase notch mean that the complex lag can be applied to increase the rate at which the open-loop magnitude drops above the 0- dB crossover frequency, while changing the 180 crossover frequency very little. When trading

off gain margin for phase margin, a complex lag compensator can provide a larger gain margin compared with a double lag compensator. Another application for which the complex lag has similar advantages over the double lag compensator is to improve the perfor- mance of single-input, single-output (SISO) systems that have multiple 0-dB crossover frequencies [3]. Complex lead and complex lag compensators are also useful for system modeling. The Bode plots of measured frequency response data often exhibit narrow phase peaks and notches with complicated magnitude behavior in the vicinity of the

peaks and notches. The complex lead and complex lag compensators can be used to replicate these Bode plot features in many cases. We provide simple closed-form expressions for deter- mining the transfer function parameters of complex lead and lag compensators for a specified phase lead or lag at a particular frequency of maximum phase lead or lag. For each compensator, we characterize the relationship between the damping ratio and the ratio of the asymptotic gains, the slope of the magnitude plot at the frequency of maximum phase lead or lag, the width of the phase peak, and the width of the

magnitude transition region. FIGURE 4 The pole-zero plot for the complex lead compensator of Figure 3. The maximum phase lead is 90 at rad/s. The points and are the locations of the pole and zero of the lead compensator having maximum phase lead of 45 at . The point = is the center of the circle , which passes through the points , , and = 41 . The conjugate poles and of the complex lead compensator lie at the intersection of and the lines corresponding to the damping ratio . Likewise, the point is the center of the circle , which passes through the points , , and = 41 . The conjugate

zeros and of the complex lead compensator lie at the intersection of and the lines corresponding to the damping ratio. 1.5 0.5 0.5 1.5 Real Imaginary jw jw PZ FIGURE 3 Bode plots of the double lead compensator (5) and the complex lead compensator (6). The phase peak and magnitude transition region of the complex lead compensator are narrower than the phase peak and magnitude transition region of the dou- ble lead compensator. The ratio of the high-frequency gain asymptote to the low-frequency gain asymptote is smaller for the complex lead compensator than for the double lead compen- sator. The

slope of the gain at the frequency of maximum phase is steeper for the complex lead compensator than for the double lead compensator. 20 10 10 20 Magnitude (dB) 10 10 10 10 10 45 90 135 Phase ( Frequency (rad/s) Double Lead Complex Lead 46 IEEE CONTROL SYSTEMS MAGAZINE FEBRUARY 2007
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ANALYSIS OF COMPLEX LEAD AND LAG COMPENSATORS Geometric Relationships Figure 4 shows the pole-zero plot of the complex lead com- pensator of Figure 3. A close geometric relationship exists between the locations of the poles and zeros of a complex lead compensator with maximum phase lead at the

frequency and a lead compensator having maximum lead phase lead at the same frequency. The points and are the locations of the pole and the zero of the lead compensator. The point is the center of the circle which passes through the points , and . The conjugate poles and of the complex lead compensator lie at the intersection of and the lines corresponding to the damping ratio . Likewise, the point is the center of the circle , which passes through the points , and . The conjugate zeros and of the complex lead compensator lie at the intersection of and the lines corre- sponding to the damping

ratio. We begin by deriving properties of the single lead com- pensator. The expression sin sin sin sin sin sin sin cos cos tan (7) is a variation of equation (7.4) in [4]. The point is defined as tan . The distance between the point and the point is therefore cos . It follows from the Pythagorean theorem and the trigonometric identity (SI) (see “Useful Trigonometric Identities”) that the distance between and is , while the distance between the point and the point is also . Thus, is center of the circle defined by the points , and . The radius of is The same procedure for the zeros yields cos

tan .( The point tan is the center of the circle with radius defined by the points , and The poles and zeros of the complex lead compensator lie at the intersections of the two circles and the lines corre- sponding to the damping ratio . The undamped natural frequency of the poles is , which is the distance from the poles to the origin . The law of cosines for the triangle Op leads to the relation cos Op ). ( Solving this quadratic equation for , substituting for and , applying SI, and using the relation cos Op implies tan tan .( 10 Following the same procedure for the zeros yields tan tan .(

11 Multiplying, subtracting, adding, and summing the squares of (10) and (11) leads to the relations (12) tan (13) tan (14) tan (15) Equation (12) shows that the frequency of maximum phase lead is the geometric mean of the natural frequencies of the poles and zeros, which is analogous to (2) for a lead com- pensator with real poles and real zeros. The relations (13)–(15) are useful in the proof below that is the fre- quency of maximum phase lead. Determining the Frequency of Maximum Phase Lead The complex lead compensator phase angle ω) is given by ω) arctan arctan (16) FEBRUARY 2007

IEEE CONTROL SYSTEMS MAGAZINE 47 Useful Trigonometric Identities everal trigonometric identities involving the tangent function are useful in the derivations. The derivations of the expres- sions (10) and (11) for and employ tan sin cos cos cos cos .( S1 The formula for the tangent of the difference of two angles is tan tan tan tan tan .( S2 Closely related to (S2) is the double angle formula for tangents tan tan tan .( S3 The derivation of the frequency of maximum phase lead uses an identity for the difference of two arctangents given by arctan arctan arctan tan arctan arctan ))) arctan uv .(

S4
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where the subscript complex is suppressed for notational convenience. Applying the formula for the difference of arctangents (S4) and the relations (12)–(15) to (16) and col- lecting terms leads to ω) arctan arctan tan tan arctan tan tan 17 Setting in (17) and applying the double angle for- mula for tangents (S3) shows that For convenience, we rewrite (17) as ω) ,( 18 where tan 19 and tan .( 20 Differentiating (17) with respect to leads to ω) arctan dx dy 21 where dx tan ,( 22 dy tan ω. ( 23 Evaluating (19)–(23) at shows that ))/ ω) . Since

ω))/ ω) < for all ω> , and ω) )/( ω) > for all ω< the positive phase at must be the global maximum. BODE PLOT FEATURES AND DAMPING RATIO For modeling and design, it is use- ful to determine the relationship between the parameters of the complex lead compensator and features of the compensator Bode plot. Figure 5 indicates the features examined in this section. The ratio of the high-frequency magnitude asymptote to the low- frequency magnitude asymptote of the complex lead compensator determines the reduction in the low-frequency disturbance rejec- tion,

high-frequency noise rejec- tion, and gain margin when the compensator is used for loop shap- ing. For a complex lag compen- sator, the same ratio determines the increase in low-frequency dis- turbance rejection, high-frequency noise rejection, and gain margin. The expression for this ratio is FIGURE 5 Some features of the Bode plot of a complex lead compensator. The Bode plot fea- tures of the complex lead compensator are functions of both the maximum phase lead and the damping ratio. Features of interest on the magnitude plot are the ratio of asymptotic gains, the slope of the magnitude plot

at the compensation frequency, and the width of the transition region from the intersection of the asymptotes to , the frequency of maximum phase lead. The width of the phase peak is the frequency width from 4 phase lead to the phase-lead peak expressed in decades or, equivalently, as the full width at half maximum phase lead also expressed in decades. 15 10 10 15 Magnitude (dB) 10 10 10 10 45 90 135 Phase ( Frequency (rad/s) Ratio of Asymptotic Gains (dB) Slope at Peak (dB/dec) Intersection of Asymptotes to Peak (dec) 4 to Peak (dec) Width at Half Max (dec) 10 48 IEEE CONTROL

SYSTEMS MAGAZINE FEBRUARY 2007
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tan tan tan tan tan tan tan tan tan tan tan tan tan tan (24) FIGURE 6 Ratios of asymptotic gains. This plot shows the ratio of the asymptotic gains as a function of the damping ratio for four phase- lead values. The ratio in dB is nearly a linear function of the damp- ing ratio, especially for phase-lead values less than 90. 0.2 0.4 0.6 0.8 1 10 15 20 25 30 35 40 45 50 Ratio of Asymptotic Gains (dB) Phase Lead = 30 Phase Lead = 60 Phase Lead = 90 Phase Lead = 120 Damping Ratio FIGURE 7 Full width of the phase peak at half of the maximum

phase lead for damping ratios between 0.1 and 1. The plot shows the full width at half maximum as a function of the damping ratio for five different maximum phase-lead values. 0.2 0.1 0.2 0.3 0.4 0.5 Damping Ratio 0.6 0.7 0.8 0.9 1 0.4 0.6 0.8 1.2 1.4 1.6 1.8 Full Width at Half Max Peak (dec) Phase Lead = 30 Phase Lead = 60 Phase Lead = 90 Phase Lead = 120 Phase Lead = 135 FIGURE 8 Full width of the phase peak at half of the maximum phase lead for damping ratios between 0.001 and 0.1. The plot shows the full width at half maximum as a function of the damping ratio for five maximum phase-lead

values, which is useful for model- ing resonance/antiresonance pairs with low damping. 0.1 Phase Lead = 30 Phase Lead = 60 Phase Lead = 90 Phase Lead = 120 Phase Lead = 135 0.05 0.1 0.15 0.2 0.02 0.04 Damping Ratio 0.06 0.08 Full Width at Half Max Peak (dec) FIGURE 9 Width of the phase peak. This plot shows the ratio of the 4 phase-lead frequency to the frequency of maximum phase lead in decades as a function of the damping ratio for several phase-lead values. 0.5 1.5 0.2 0.4 Damping Ratio 0.6 0.8 Four Degrees to Peak (dec) Phase Lead = 30 Phase Lead = 60 Phase Lead = 90 Phase Lead =

120 The smaller ratio between the high-frequency and low-frequency gain asymptotes of the complex lead compensator is an advantage compared to the double lead compensator. FEBRUARY 2007 IEEE CONTROL SYSTEMS MAGAZINE 49
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Figure 6 shows the ratio of the asymptotic gains in dB as a function of the damping ratio. The width of the phase peak can be quantified in several ways. The full width at half maximum (FWHM) is the fre- quency span in decades (dec) between the frequencies at which the phase is one-half of its maximum. This measure of the peak width is useful for model- ing

since it is easily measured from experimental data. Figure 7 shows the numerically determined FWHM peak width for several lead compensators as a function of damp- ing ratio. Figure 8 shows the peak width for damping ratios between 0.001–0.1, which is useful for modeling resonance and antiresonance pairs with low damping ratio. 50 IEEE CONTROL SYSTEMS MAGAZINE FEBRUARY 2007 FIGURE 10 The magnitude transition region width. The width is mea- sured by the frequency span between the intersection of the low- frequency and mid-frequency asymptotes and . Note the nearly linear relationship with

respect to the damping ratio. 0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.2 0.4 Damping Ratio 0.6 0.8 Transition Width (dec) Phase Lead = 30 Phase Lead = 60 Phase Lead = 90 Phase Lead = 120 FIGURE 12 Magnitude slope at the frequency of maximum phase lead for damping ratios between 0.1 and 0.5. The slope at the fre- quency of maximum phase lead increases dramatically for low damping ratios, even for low phase-lead values. Phase Lead = 30 Phase Lead = 60 Phase Lead = 90 Phase Lead = 120 200 180 160 140 120 100 80 60 Magnitude Slope at Max Phase Lead (dB/dec) 40 20 0.1 0.15 0.2 0.25 Damping Ratio 0.3 0.35

0.4 0.45 0.5 FIGURE 11 Magnitude slope at the frequency of maximum phase lead for damping ratios between 0.5 and 1. The slope at the fre- quency of maximum phase lead decreases by 10–15 dB/dec as the damping ratio increases from 0.5–1.0. Phase Lead = 30 Phase Lead = 60 Phase Lead = 90 Phase Lead = 120 55 50 45 40 35 30 25 20 15 10 0.5 0.6 0.7 0.8 Damping Ratio 0.9 1 Magnitude Slope at Max Phase Lead (dB/dec) FIGURE 13 Frequency response of an actuated tape guide. A second-order model captures the low-frequency response. The two prominent peaks at approximately 1160 rad/s and 1940 rad/s arise

from mechanical resonances in the actuator structure. 40 20 20 40 60 80 Magnitude (dB) 10 10 10 10 10 225 180 135 90 45 Phase ( Frequency (rad/s) Second-Order Model Experimental Data Lead compensators are used to increase the phase margin for a given 0-dB crossover frequency.
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Another way to measure the width of the phase peak is the frequency span from a specified small phase lead to the maximum phase lead. This quantity is often useful for designing lag compensators when the goal is to increase the gain margin with minimal effect on the phase margin. Figure 9 shows the

numerically determined width from 4 to the phase peak in decades as a function of damping ratio. The width of the magnitude transition region is another important property of lead compensators because it determines the fre- quency range over which the compensator affects the slope of the open-loop magnitude response. The width of the magnitude transi- tion region is quantified here by the frequency span from the intersection of the asymptotes to . The intersection of the low-frequency and mid-frequency asymptotes is related to the damping ratio and maximum phase lead by tan tan .( 25

Figure 10 shows the width of the magnitude transition region in decades as a function of damping ratio. The maximum slope of the complex lead compensator magnitude determines how much the slope of the open-loop magnitude response changes when the compensator is applied. For lag compensators, a larger magni- tude slope is desirable for increasing low-fre- quency disturbance rejection or increasing high-frequency noise rejection near the fre- quency of the maximum phase lag. Converse- ly, a steeper magnitude slope for a lead compensator limits the design by reducing the slope of the open-loop

magnitude at the open- loop 0-dB crossover frequency. The expression for the magnitude slope at the compensation frequency in decibels per decade is 20 log ω) log (ω) 20 .( 26 FEBRUARY 2007 IEEE CONTROL SYSTEMS MAGAZINE 51 FIGURE 14 Closeup of the first phase peak of Figure 13. The phase peak is 135 above the baseline phase of 180 . The 50% frequencies are the two frequen- cies at which the phase is above the baseline by half of the difference between the baseline and peak values. 15 10 10 15 Magnitude (dB) 10 225 180 135 90 45 Phase ( requency (rad/s) Fr equency: 1080 rad/s

Phase: 113 ? Fr equency: 1130 rad/s Phase: 45 Freque ncy: 1260 rad/s Phase: 113 FIGURE 15 Measured and modeled frequency responses in the vicinity of the two phase peaks. The frequency response of the second-order model augmented by the two complex lead compensators is a good match to the experimental fre- quency response. 15 10 10 15 Magnitude (dB) 10 225 180 135 90 45 Phase ( requency (rad/s) Experimental Data Complex Lead For a complex lag compensator, the steeper magnitude slope and narrower phase notch are advantages compared to the double lag compensator.
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Figures 11 and

12 show the maximum magnitude slope in decibels per decade obtained numerically for several lead compensators as a function of damping ratio. MODELING EXAMPLE We consider an actuated tape guide used to actively steer moving tape. The tape guide is used in a high-precision tape transport system [5] for a prototype multiterabyte tape sys- tem under construction at Carnegie Mellon. Figure 13 shows the frequency response of the device as well as the frequency response of a second-order transfer function manually fit to the data. The system exhibits two resonance/antiresonance pairs for which

finding good transfer function models is often difficult without the use of automated fitting techniques. Figure 14 shows a closeup of the experimental frequency response of the first resonance/antiresonance pair. The estimated phase lead is 135 . The FWHM is log 10 50% 50% log 10 1260 1080 067 dec ,( 27 where 50% is the higher of the two frequencies at which the phase peak is 50% of the maximum and 50% is defined analogously. The estimated damping ratio obtained from Figure 8 is 0.03. The estimated value of is at the geo- metric mean of the 50% phase-peak frequencies 50% 50% 167 rad

.( 28 Figure 15 shows the frequency response in the vicinity of both resonance/antiresonance pairs with the second- order system model augmented by two complex lead com- pensators obtained by the process described above. The parameters of the second complex lead compensator are 117 , 02 , and 1940 rad . The complete transfer function model is 19 21 14 5625 65 06 176 75 21 555 75 14 527 80 17 017 .( 29 DESIGN EXAMPLE This design example applies the complex lag compen- sator to the focus control of a microscope objective lens. FIGURE 16 Frequency response of the focus control plant. A

model containing two resonances matches the experimental response up to 1100 Hz. 30 20 10 10 20 Magnitude (dB) 10 10 10 540 450 360 270 180 90 Phase ( Fr equency (Hz) Experimental Data Tr ansfer Function Model FIGURE 17 Frequency responses of the compensated open loop. The large phase margin of the nominal controller allows the use of lag compensators for trading off phase margin for gain margin. 20 15 10 Magnitude (dB) 10 10 315 270 225 180 135 90 45 Phase ( requency (Hz) No Lag Complex Lag Lag 52 IEEE CONTROL SYSTEMS MAGAZINE FEBRUARY 2007 For modeling and design, it is useful to determine

the relationship between the parameters of the complex lead compensator and features of the compensator Bode plot.
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The example is taken from a heat-assisted mag- netic recording (HAMR) media tester at the Seagate Research Center in Pittsburgh, Penn- sylvania. A permanent magnet is used to apply a magnetic field to a portion of the disc. Writ- ing is achieved by focusing a laser spot on the magnetized portion of the disc with a micro- scope objective lens. The laser spot size defines the dimensions of the written data bits. Effec- tive writing requires that the position of the

microscope objective lens relative to the disc surface is maintained to within the depth of focus of the beam. A servo system is used to mitigate distur- bances due to spindle wobble and disc vibration. The servo system uses a piezo actuator to move the objective lens and a four-quadrant detector with a knife edge to measure the spot size. Figure 16 shows the mea- sured and modeled plant responses. Below the resonance frequency near 1100 Hz, the measured transfer function closely matches a fourth-order system with 100- s delay. Above the resonance frequency, the measurement of plant dynamics

is uncertain; hence, the controller must provide adequate robustness at high frequencies. A proportional-integral (PI) compensator is used to improve disturbance rejection below 100 Hz, while a 19.5-dB notch filter is used to remove the primary reso- nance. A first-order lowpass filter with a 300-Hz cutoff frequency suppresses the high-frequency response. The blue line in Figure 17 shows the loop shape with the PI compensator, the notch filter, and the low-pass filter applied. Figure 17 shows that the gain margin is 4.8 dB at 957 Hz, while the phase margin is 86 at 492 Hz. Experi-

mental results using this controller indicate that most of the error occurs in the frequency range of the sensitivity peak. Use of the complex lag compensator is intended to decrease sensitivity peaking while improving high-fre- quency robustness. The complex lag compensator enables a tradeoff between gain margin and phase margin that improves high-frequency robustness. Figure 18 shows the fre- quency responses of a 25 standard lag compensator at 1,100 Hz and of a 25 complex lag compensator at 600 Hz with damping ratio 0.3. Both compensators provide 19 of phase lag at

500 Hz. The complex lag compen- sator exhibits a small peak and notch in its magnitude because its damping ratio is less than 0.7. The phase notch of the complex lag compensator is much narrower than that of the standard lag compensator, while its magnitude slope is much steeper than the standard lag compensator’s, between 500 and 700 Hz. However, the standard lag compensator has the larger difference between asymptotic gains. Figure 17 also shows the compensated loop shapes employing the standard lag and complex lag compen- sators. Both compensators result in a phase margin of FEBRUARY 2007

IEEE CONTROL SYSTEMS MAGAZINE 53 FIGURE 18 Bode plots for the lag and complex lag compensators. The complex lag compensator magnitude response shows a peak and notch because the compensator damping ratio is below 0.7. The complex lag compensator has a steeper magnitude slope and narrower phase peak than the lag compensator. Magnitude (dB) 10 10 10 10 10 10 30 20 10 Phase ( requency (Hz) Complex Lag Lag FIGURE 19 Experimental sensitivity function magnitudes for the three loop-shaping designs. The low-frequency disturbance rejection of the complex lag compensator design is similar to that of the

nominal design. However, the sensitivity peak of the complex lag compen- sator design is smaller than that of the nominal design. The standard lag compensator design has better low-frequency disturbance rejec- tion than the nominal design but has more sensitivity peaking. 10 10 10 25 20 15 10 10 Magnitude (dB) Fr equency (Hz) No Lag Complex Lag Lag The complex lag compensator enables a tradeoff between gain margin and phase margin that improves high-frequency robustness.
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54 at 500 Hz. However, use of the complex lag com- pensator results in a gain margin of 7.9 dB at

900 Hz, while the gain margin of the system using the standard lag compensator is only 3.6 dB at 807 Hz. The perfor- mance of the complex lag compensator design is similar to the performance of the nominal controller alone at low frequencies. However, the complex lag compen- sator controller provides about 3.1 dB more gain mar- gin and improved robustness to unmodeled high-frequency dynamics compared to the nominal con- troller because its magnitude drops faster between 500 and 700 Hz. The standard-lag compensator design has a smaller gain margin than the complex lag compensator design and

larger phase loss over a wider frequency range. However, the standard-lag design has larger magnitude at low frequencies. Figure 19 shows the magnitudes of the sensitivity functions from experimental data. The complex lag com- pensator design has a smaller peak (4.8 dB) relative to the nominal design (7.2 dB) and the design employing the standard lag compensator (7.9 dB). However, the stan- dard lag compensator design has more disturbance rejec- tion at low frequencies. The complex lag compensator design is a better choice in this situation because the lag compensator is needed only to reduce

sensitivity peaking and obtain high-frequency robustness, while its low-fre- quency disturbance rejection is not significantly worse than that of the nominal controller. CONCLUSIONS AND FUTURE WORK Complex lead and lag compensators are new additions to the repertoire of compensator structures for loop shaping. This article facilitates the use of these compensators by providing explicit formulas that relate the parameters of the compensators to features of their frequency responses. Two examples illustrate the utility of these compensators for system modeling and controller design. While the

examples involve low-order plants, the principles of employing the complex lead and lag compensators remain the same for higher-order systems. We plan to use these compensators as weighting func- tions with automated robust design tools. A weighting function is a transfer function whose frequency response magnitude is used to bound closed-loop response or mod- eling uncertainty. The complex lead and lag compensators provide new degree of freedom for selecting weighting functions. In particular, the steep magnitude slope in the transition region of these compensators more closely approximates

an ideal step function than weighting func- tions appearing in the literature [6]. ACKNOWLEDGMENT W. Messner and L. Xia thank Prof. J. Wickert and V. Kartik for their assistance in designing the actuated tape guide. AUTHOR INFORMATION William C. Messner received a B.S. in mathematics from M.I.T. in 1985 and M.S. and Ph.D. degrees in mechanical engineering from the University of Califor- nia at Berkeley in 1989 and 1992, respectively. He is a professor of mechanical engineering at Carnegie Mel- lon University. He is the leader of the servo control effort at Carnegie Mellon’s Data Storage

Systems Cen- ter. He is a fellow of the American Society of Mechani- cal Engineers and the American Association for the Advancement of Science. Mark D. Bedillion (mark.bedillion@seagate.com) received the B.S.M.E. degree in 1998, the M.S.M.E. degree in 2001, and the Ph.D. degree in 2005 from Carnegie Mel- lon University. He joined Seagate Research in 2002, where he works on the development of novel storage devices. His research interests include control applica- tions in data storage, distributed manipulation, and hybrid systems. He can be contacted at Seagate Technolo- gy, 1251 Waterfront Pl.,

Pittsburgh, PA 15222 USA. Lu Xia received the B.S. degree in industrial automa- tion from Beijing Institute of Technology, Beijing, China in 1998 and the M.S.E.E. degree from the University of Notre Dame, Indiana in 2002. She is currently pursuing the Ph.D. degree with the department of Electrical and Computer Engineering at Carnegie Mellon University. Her research interests include control applications in data storage systems, robust control, and signal identifi- cation and processing. Duane C. Karns served in the U.S. Army’s 82nd Air- borne Division from 1986 to 1990. He received a B.S. in

engineering science from Pennsylvania State University in 1995. He received the M.S. in 1997 and a Ph.D. in 2000, both in electrical and computer engineering from Carnegie Mellon University. He is currently a research staff member at Seagate Research in Pittsburgh, Pennsyl- vania, specializing in the recording physics of magnetic and optical systems. REFERENCES [1] W. Messner, “The development, properties, and application of the com- plex phase lead compensator,” in Proc. 2000 American Control Conf. , Chicago, IL, July, 2000, pp. 2621–2626. [2] W. Messner, “Some advances in loop shaping with

applications to disk drives, IEEE Trans. Magn. , vol. 37, no. 2, pp. 651-656, Mar 2001. [3] W. Messner and R. Oboe, “Phase stabilized design of a hard disk drive servo using the complex lag compensator, “ in Proc. 2004 American Controls Conf. , Boston, MA, 30 June–2 July, 2004, pp. 1165–1170. [4] G.F. Franklin, J.D. Powell, and A. Emani-Naeini, Feedback Control of Dynamic Systems, 3rd ed Reading, MA: Addison-Wesley, 1994. [5] D. Richards, J. Anderson, and L. Erickson, “Dynamic Tape Path Adjust- ment,” US Patent 6 690 531, Feb. 10, 2004. [6] A. Packard, G. Balas, M. Safonov, R. Chiang, P.

Gahinet, and A. Nemirovs- ki, Robust Control Toolbox}, The Mathworks. [Online]. Available: http://www. mathworks.com/access/helpdesk/help/toolbox/robust/ 54 IEEE CONTROL SYSTEMS MAGAZINE FEBRUARY 2007