An H∞ Loop-Shaping Design Procedure forAttitude Control of an AUV

Scott B. Gibson and Daniel J. StilwellBradley Department of Electrical and Computer Engineering

Virginia Tech{sgibson2, stilwell}@vt.edu

Abstract—This work provides a process to design stabilizingrobust H∞ attitude controllers for autonomous underwater vehi-cles (AUVs) using classical loop-shaping techniques. We adopt theloop-shaping design ideas introduced by McFarlane and Glover[1]. In this paper, we present two control topologies and pro-cedures for designing robust attitude controllers for streamlinedAUVs. We describe how to achieve a desired controller bandwidthduring the design process using open-loop shaping guidelines.Our results are verified by evaluating performance in the field ofseveral attitude controllers that are designed for a streamlinedtail-controlled AUV. The experimental results also show the trade-off between reference tracking performance and control effort.

I. INTRODUCTION

An attitude controller for an autonomous underwater vehicle(AUV) uses actuators to stabilize the vehicle’s heading andpitch while in motion. The principal contribution of our workis a process for designing a robust attitude controller basedon loop-shaping ideas for a streamlined tail-controlled AUV.Our approach uses a loop-shaping procedure where the open-loop frequency response of the system is adjusted to achievedesired closed-loop performance and robustness goals. Wespecifically address the challenge of adjusting the desiredactuator bandwidth in a loop-shaping design framework. Wepresent two control topologies that enable the designer toreduce or increase actuator bandwidth.

As the actuator bandwidth of an attitude control systemincreases, a system will generally exhibit improved trackingperformance at the expense of actuators using more energyand wearing out sooner. For applications where tracking per-

Fig. 1. Virginia Tech 690s AUV

formance can be relaxed, actuator bandwidth can be smaller,leading to slower actuator movement and lower energy con-sumption. The trade-off between tracking performance andenergy efficiency introduces a design requirement for actuatorbandwidth guided by mission objectives.

Robust H∞ control design has been addressed previouslyin [2]–[7], but loop-shaping designs have not previously beenconsidered for streamlined AUV applications. We use a loop-shaping design procedure proposed by McFarlane and Glover[1] that has been successfully implemented in a variety ofapplications such as [8] and [9]. The design procedure syn-thesizes an H∞ controller to robustly stabilize the vehiclewith respect to coprime factor uncertainty using classical loop-shaping techniques [10]. Weighting filters are selected to shapethe open-loop response of the plant. We provide a method ofchoosing the weighting filters to shape the plant and adjust theactuator bandwidth.

A second topology, referred to as a two degree of freedomcontroller, is proposed by Hoyle [11] and Limebeer [12]. Forthis controller, the commands and feedback enter the controllerseparately and are independently processed. A reference modelis introduced and represents the desired closed-loop transferfunction. The two degree of freedom controller requires moredesign elements, but can be used if input tracking requirementsare not satisfied by the loop-shaping design proposed byMcFarlane and Glover.

In both control topologies, the prefilter normally includesan integrator. In some conditions, the integrator can wind updue to actuator saturation and compromise the stability ofthe AUV. For practical applications, we also present an anti-windup scheme that prevents integrator windup in the eventthat the vehicle’s actuators saturate.

The Virginia Tech 690s AUV is shown in Figure 1 and isa smaller variant of the Virginia Tech 690 AUV presented in[13]. A series of attitude controllers were designed for the Vir-ginia Tech 690s, using both of the approaches proposed herein,and evaluated experimentally. We present experimental datathat confirms the efficacy of our proposed design proceduresand our ability to achieve the required actuator bandwidth. Ourresults confirm that a controller with lower actuator bandwidthreduces actuator movement compared to a controller withhigher actuator bandwidth. The results also demonstrate thetrade-off between reference tracking performance and energyefficiency.

K G

d

n

r u + y

+

−

Fig. 2. Control system topology

II. LOOP-SHAPING METHODOLOGY

A simple control loop topology is shown in Figure 2 whereG is the system to be controlled, K is the controller, r isthe reference input, u is the control input to the system, dis the output disturbance, y is the output, n is the outputnoise, and GK is the open-loop transfer function. Fromthe block diagram, we can calculate the sensitivity functionS = (I + GK)−1 and the closed-loop transfer functionT = GK(I + GK)−1. With the sensitivity function and theclosed-loop transfer function, we form the output and controlinput equations,

Y (s) = T (s)R(s) + S(s)D(s)− T (s)N(s) (1)U(s) = K(s)S(s)[R(s)−N(s)−D(s)] (2)

The maximum and minimum singular values of a matrixare denoted σ(·) and σ(·) respectively. From (1) and (2),we determine that for disturbance rejection σ(S) must besmall for the frequency range of the disturbance, for noiseattenuation σ(T ) must be small in the frequency range ofthe noise, for reference tracking σ(T ) ≈ σ(T ) ≈ 1, andfor lower bandwidth or control effort σ(KS) must be small.The closed-loop requirements for σ(S), σ(T ), and σ(KS)cannot be satisfied simultaneously. However, the closed-looprequirements can be achieved over certain frequency ranges.For example, noise is usually relevant at high frequencies,while disturbance rejection is important at low frequencies.

In classical loop-shaping, the magnitude of the open-looptransfer function GK is modified by appending weightingfilters to the open-loop system GK. The largest singular valuesof S and T are related to the singular values of the open-loopsystem by approximating

σ(S) ≈ 1/σ(GK) (3)

at frequencies where σ(GK)� 1 and

σ(T ) ≈ σ(GK) (4)

when σ(GK) � 1 [14]. The closed-loop requirements canbe converted to open-loop objectives using (3) and (4). Whenshaping the open-loop transfer function GK at frequencieswhere GK � 1, σ(GK) must be large for good disturbancerejection and good reference tracking. When shaping the open-loop transfer function GK at frequencies where GK � 1,σ(GK) must be small for good noise attenuation and σ(K)must be small for energy efficiency. Finally for single-inputsingle-output systems, closed-loop stability is related to the

W1 GKsW2r

−

K

(a) Shaped plant and controller.

W1

Ks W2

GKs(0)W2(0)r − u y

+

(b) Alternate loop-shaping controller topology

Fig. 3. Two loop-shaping design topologies

roll-off rate, open-loop gain and open-loop phase near thecrossover frequency [14].

III. LOOP-SHAPING DESIGN PROCEDURE

We adopt the loop-shaping design procedure proposed byMcFarlane and Glover [1]. The procedure shapes the plantG with a prefilter W1 and a postfilter W2 shown in Figure3(a) where Gs = W2GW1. The open-loop singular valuesof the shaped plant Gs are shaped according to the classicalshaping guidelines mentioned in Section II to achieve thedesired performance.

After shaping Gs by selection of W1 and W2, an H∞controller Ks is synthesized that robustly stabilizes the systemwith respect to the normalized coprime factorization for theshaped plant Gs = M−1s Ns. The H∞ controller Ks canbe synthesized by using the expressions in [10]. The finalcontroller for the plant G is K = W1KsW2 as shown inFigure 3(a). After generating K, the coprime uncertainty γ iscomputed as shown in [10]. For robust stability, γ ≥ 1 shouldbe as small as possible, and usually γ < 4 is required whichcorresponds to 25% allowed coprime uncertainty.

A systematic approach to selecting the weighting filters W1

and W2 is presented in [8] and [14]. To satisfy the open-loop objectives, the singular values of Gs = W2GW1 usuallyrequire high gain at low frequencies, roll-off rate of lessthan 40 dB/decade at the crossover frequency, and low gainat high frequencies. The postfilter W2 is normally chosento be a constant matrix. The weights in W2 indicate therelative importance of signals to be controlled. The prefilterW1 = WbwWpWi contains the dynamic shaping and ischosen to include integral action, phase-advance, phase-lag,and bandwidth gain. Integral action Wi increases trackingperformance at low frequencies. Phase-advance Wp reducesthe roll-off rate at the desired crossover frequency. Phase-lag Wp increases the roll-off rate at higher frequencies. Thebandwidth gain Wbw is applied last to change the crossoverfrequency of the singular values. Some iteration is required toshape the singular values of Gs.

ρI K1 Ns M−1s ρI

K2

Tref

∆Ns ∆Ms

r β + u

+

y+ z

+ −

φ−

−+

(a) Two degree of freedom design configuration

W1

K2

GK1Wkr + u y

+

K

(b) Two degree of freedom controller topology

Fig. 4. Two degree of freedom controller design configuration and topology

An alternate topology shown in Figure 3(b) was proposedby [8]. The alternate topology moves the synthesized H∞controller Ks to the feedback path so that changes in thereference commands do not excite the dynamics of the con-troller. Moving the controller to the feedback path can reducelarge overshoot caused by changes in reference commands.Assuming integral action in W1 or G, a gain Ks(0)W2(0) isadded to achieve a zero steady-state error.

IV. TWO DEGREE OF FREEDOM CONTROLLER

If strict time-domain requirements exist, the loop-shapingdesign procedure may not be sufficient. In Section III, theloop-shaping design procedure produces a one degree offreedom controller that only utilizes the error signal between areference command and a measurement. A two degree of free-dom controller inputs both the feedback and the commands.An extension to the H∞ loop-shaping design procedure ofMcFarlane and Glover is proposed by [11] and [12]. Theproblem is to find a stabilizing controller K for the shapedplant Gs = M−1s Ns which minimizes the H∞ norm between[rT φT]T and [uT yT zT]T where φ represents the uncertaintyin the nominal plant model G and z is the output of thedesign configuration shown in Figure 4(a). The two degreeof freedom controller design procedure employs a referencemodel Tref chosen by the designer for the closed-loop systemto follow. Another parameter ρ is also introduced to weighrelative importance of robust stabilization as compared withmodel reference matching. A thorough treatment of the theorypresented by [11] and [12] is omitted from this paper. How-ever, a systematic design procedure is presented and followsthe proposed procedure in [14].

Wp Wbw1s G

ksat

+ u

− ++

W1

Fig. 5. W1 with anti-windup topology

First, the open-loop singular values of Gs are shaped bydesigning a pre-compensating filter W1 as explained in SectionIII. Then a desired closed-loop reference model Tref is chosenbetween the commands and the outputs. The speed of theresponse of the closed-loop reference model Tref must berealistic, or the closed-loop system will have poor robuststability properties and the controller will produce excessivecontrol signals [11]. The design parameter ρ is selected be-tween 1 and 3 to emphasize robust stability or model matching.With these design choices, one can setup the standard Ricattiequations associated with H∞ controller synthesis whosesolutions yield K1 and K2 in the Figure 4(b). A pre-filter Wk

is required to achieve good model matching at steady-statewhere Wk = [(I − Gs(0)K2(0))−1Gs(0)K1(0)]−1Tref (0).In our iterative approach, the performance of the synthesizedcontroller is evaluated and the values of ρ, W1, and Tref areadjusted if another control design iteration is required.

V. ANTI-WINDUP SCHEME

Windup occurs by the interaction of actuator saturation andintegral action of a controller. When an actuator saturates, acontroller with an integrator will continue to integrate the errorand may cause the integral term to become large, which isknown as wind-up.

In this paper, we implement a back calculation anti-windupscheme. The weighting filter W1 from Section III usually hasan integrator to improve low frequency tracking performance.The integrator from W1 can be separated as shown in theblock diagram in Figure 5. When an actuator saturates, thedifference between the control input after saturation u and thecontrol input before saturation is multiplied by a saturationgain ksat. The output of the block ksat is added before theintegrator. With a sufficiently large ksat, the back calculationscheme prevents integrator wind-up.

VI. 690S CONTROLLER DESIGN

The Virginia Tech 690s AUV shown in Figure 1 was usedto evaluate the controller design procedure proposed herein.The AUV diameter is 17.53 centimeters (6.90 inches), lengthis 1.55 meters, and weight is 27.67 kilograms. It is ballastedto be slightly positively buoyant. The Virginia Tech 690s AUVhas a single propeller at the stern of the vehicle, and it has fourcontrol surfaces at the tail. The vehicle also has an attitude and

ω (rad/sec)10-2 10-1 100 101 102

SinglularValues

(dB)

-50

-40

-30

-20

-10

0

10

20

30

Pitch

Yaw

Fig. 6. Nominal largest singular values for the pitch and yaw channels

TABLE ITHE COPRIME UNCERTAINTY γ FOR EACH CONTROLLER DESIGN

Procedure 1.0 rad/sec 1.5 rad/sec 2.0 rad/secOne degree of freedom 2.1363 2.0115 1.9559Two degree of freedom 2.6118 2.3877 2.2896

heading reference sensor (AHRS) that measures attitude, bodyrelative rotation rates, and translation accelerations.

The system identification technique presented in [15] wasused to generate the 690s AUV system model. For the 690sdynamic model, there are two actuator commands u = [δe δr]

T

and four measurements y = [q θ r ψ]T, where δe is the

elevator deflection, δr is the rudder deflection, q is the AUVbody pitch rate, θ is the AUV pitch angle, r is the AUV bodyyaw rate, and ψ is the AUV heading or yaw angle. The largestsingular values of the nominal system plant G are plotted inFigure 6.

In the following sections, the designs of one and two degreeof freedom controllers with actuator bandwidths of 1.0 rad/sec,1.5 rad/sec, and 2.0 rad/sec are presented for the VirginiaTech 690s AUV. All 690s attitude controllers are designedto control the pitch angle θ and yaw angle ψ of the vehicle.The weighting filter W1 is the same for all controllers.

A. Loop-Shaping Design Procedure

Recall that W1 = WbwWpWi. For good tracking perfor-mance and disturbance rejection, the smallest singular valuesat low frequencies must be large. An integrator increases thesingular values at low frequencies and thus improves lowfrequency tracking performance. Both pitch and yaw channelsinclude an integrator Wi = 1/s. The roll-off rate increasesdue to the addition of an integrator. The weight

Wp =(s+ 0.5)(s+ 0.6)

s+ 2(5)

is used to add phase-lead that reduces the roll-off rate atthe desired crossover frequency and to add phase-lag that in-creases the roll-off rate at high frequencies. Desired controller

ω (rad/sec)10-2 10-1 100 101 102

SinglularValues

(dB)

-40

-30

-20

-10

0

10

20

30

40

50

Pitch 1.0 rad/sPitch 1.5 rad/sPitch 2.0 rad/sYaw 1.0 rad/sYaw 1.5 rad/sYaw 2.0 rad/s

Fig. 7. Largest singular values of the shaped plants for the pitch and yawchannels at bandwidths of 1.0 rad/sec (solid line), 1.5 rad/sec (dashed line),and 2.0 rad/sec (dashed-dot line)

bandwidths are selected to be 1.0 rad/sec, 1.5 rad/sec, and2.0 rad/sec to confirm that a controller with lower actuatorbandwidth reduces actuator movement compared to a con-troller with a higher bandwidth. By plotting the largest singularvalues of GWpWi, we chose the bandwidth gain Wbw to movethe crossover frequency to the desired controller bandwidth.Bandwidth gains of Wbw = 3.073 dB, Wbw = 6.637 dB,and Wbw = 8.843 dB change the open-loop bandwidth of oursystem to 1.0 rad/sec, 1.5 rad/sec, and 2.0 rad/sec respectivelyfor the pitch channel. Similarly, bandwidth gains Wbw = 4.164dB, Wbw = 7.544 dB, and Wbw = 9.534 dB change the open-loop bandwidth of our system to 1.0 rad/sec, 1.5 rad/sec, and2.0 rad/sec respectively for the yaw channel.

Finally, we select W2 = diag {1, 2, 1, 2} because pitch andyaw angle tracking are more important than nonzero angularrates. The largest open-loop singular values of the shaped plantGs = W2GW1 are shown in Figure 7. With the shaped plantsGs designed with different crossover frequencies, an H∞controller is synthesized for each shaped plant. The coprimeuncertainty γ is γ = 2.1363, γ = 2.0115, and γ = 1.9559for controller bandwidths of 1.0 rad/sec, 1.5 rad/sec, and2.0 rad/sec respectively. The step responses for the alternatecontrol topology using the synthesized H∞ controllers areshown in Figure 8(a).

B. Two Degree of Freedom Controller

If the controller developed by using the loop-shaping designprocedure proposed by McFarlane and Glover does not meetperformance requirements, a two degree of freedom controllercan be designed. For our purposes, we seek to design con-trollers with the same bandwidths as the one degree of freedomcontroller and compare actuator movement. As such, we selecta reference model

Tref =1

s+ 1(6)

for a good closed-loop response that is achievable and a ρ = 1for more emphasis on robust stability. A higher value of ρ

0 5 10 15

Amplitude

0

0.5

1

1.5

Pitch 1.0 rad/s

Pitch 1.5 rad/s

Pitch 2.0 rad/s

Time (s)0 5 10 15

Amplitude

0

0.5

1

1.5

Yaw 1.0 rad/s

Yaw 1.5 rad/s

Yaw 2.0 rad/s

(a) Step responses using the loop-shaping design procedure for the pitchand yaw channels at bandwidths of 1.0 rad/sec, 1.5 rad/sec, and 2.0 rad/sec

0 5 10 15

Amplitude

0

0.5

1

1.5

Pitch 1.0 rad/sPitch 1.5 rad/sPitch 2.0 rad/sTref

Time (s)0 5 10 15

Amplitude

0

0.5

1

1.5

Yaw 1.0 rad/sYaw 1.5 rad/sYaw 2.0 rad/sTref

(b) Step responses using a two degree of freedom design for the pitch andyaw channels at bandwidths of 1.0 rad/sec, 1.5 rad/sec, and 2.0 rad/sec

Fig. 8. Closed-loop step responses for one and two degree of freedomcontrollers

would put more emphasis on reference model matching. UsingTref , ρ, and the same prefilter W1 from Section VI-A, an H∞controller is synthesized for each shaped plant. The coprimeuncertainty γ is γ = 2.6118, γ = 2.3877, and γ = 2.2896for controller bandwidths of 1.0 rad/sec, 1.5 rad/sec, and 2.0rad/sec respectively. After synthesizing the H∞ controller, thestep responses using the synthesized H∞ controllers are shownin Figure 8(b).

VII. FIELD TRIALS

After generating the controllers from Section VI-A andSection VI-B, the controllers were implemented on the Vir-ginia Tech 690s AUV and subsequently evaluated in the field.Data was acquired during experiments performed at ClaytorLake in Pulaski County, Virginia. The data collected fromthe experiments consisted of level-flight at a 2.0 meter depthfor 3 minutes. For all controllers, the 690s operated at apropeller speed of 1500 rpm, which corresponds to a surge

70 80 90 100 110-5

0

5

Pitch

(deg)

One Degree of Freedom Controller

70 80 90 100 110Time (s)

35

40

45

Yaw

(deg)

CommandMeasured 2.0 rad/sMeasured 1.0 rad/s

(a) Attitude tracking at a steady-state depth for the one degree of freedomcontrollers.

70 80 90 100 110-5

0

5

Fins(deg)

2.0 rad/s

70 80 90 100 110Time (s)

-5

0

5

Fins(deg)

1.0 rad/s

(b) Fin angles at a steady-state depth for the one degree of freedomcontrollers.

Fig. 9. Attitude and fin angle at a steady-state depth using a one degree offreedom controller design.

velocity of approximately 1.65 meters/sec. Only the resultsfor the 1.0 rad/sec and 2.0 rad/sec controllers are shownto illustrate differences between tracking performance andactuator movement.

For a performance metric to compare controllers, we employthe average of the squared values defined

P =

n∑j=1

[e(j)]2

n(7)

where n is the number of samples, e(j) is the performancesignal, and P is the average of the squared values of theperformance signal. We use pitch error θe, yaw error ψe,elevator rate δ̇e, and rudder rate δ̇r as performance signalse(j) to compare the attitude controllers.

Figure 9(a) and 10(a) show the one and two degree offreedom attitude tracking results after achieving a steady-statedepth. Figure 9(b) and 10(b) show the value of the four fincommand angles after achieving a steady-state depth. For all

TABLE IIVIRGINIA TECH 690S AUV CONTROLLER PERFORMANCE P

Controller Average of squared θe Average of squared ψe Average of squared δ̇e Average of squared δ̇r1D 1.0 rad/sec 0.384 9.036 28.978 3.4761D 2.0 rad/sec 0.065 0.041 145.285 38.6882D 1.0 rad/sec 1.322 0.333 36.905 1.9542D 2.0 rad/sec 0.185 0.056 178.912 58.451

70 80 90 100 110-5

0

5

Pitch

(deg)

Two Degree of Freedom Controller

70 80 90 100 110Time (s)

5

10

15

Yaw

(deg)

CommandMeasured 2.0 rad/sMeasured 1.0 rad/s

(a) Attitude tracking at a steady-state depth for the two degree of freedomcontrollers.

70 80 90 100 110-5

0

5

Fins(deg)

2.0 rad/s

70 80 90 100 110Time (s)

-5

0

5

Fins(deg)

1.0 rad/s

(b) Fin angles at a steady-state depth for the two degree of freedomcontrollers.

Fig. 10. Attitude and fin angle at a steady-state depth using a two degree offreedom controller design.

the controllers evaluated, the 690s AUV was able to achieveand maintain stable flight in response to reference commands.Our results show that both design procedures generate robuststabilizing attitude controllers.

For both the one and two degree of freedom controllers,the fin movement in Figure 9(b) and 10(b) is reduced asthe controller bandwidth decreases. Comparing the value ofP for the 2.0 rad/sec to the 1.0 rad/sec controllers in TableII, P decreases for elevator and rudder rate, meaning theactuator rate or movement is higher for 2.0 rad/sec controllers.

0 10 20 30 40-40

-20

0

20

Pitch

(deg)

Two Degree of Freedom Controller

Command

Measured 2.0 rad/s

Measured 1.0 rad/s

0 10 20 30 40-40

-20

0

20

Pitch

(deg)

One Degree of Freedom Controller

Command

Measured 2.0 rad/s

Measured 1.0 rad/s

Fig. 11. Transient pitch tracking for one and two degree of freedomcontrollers.

However, there is a trade-off between tracking performanceand bandwidth. The controllers with the lowest bandwidthhave less actuator movement, but the yaw and pitch referencetracking is worse than the controllers with the highest band-width. In Table II, the value of P for 2.0 rad/sec controllersis smaller than for the 1.0 rad/sec controllers, meaning thereis less tracking error.

From Figure 9(a), 10(a) and Table II, the steady-state resultsfor 1.0 rad/sec one and two degree of freedom controllerscan be compared. The 1.0 rad/sec one degree of freedompitch steady-state error is smaller than the two degree offreedom pitch steady-state error. Recall that the two degree offreedom ρ chosen for this design emphasized robust stabilityinstead of reference tracking. For more reference control, ρcan be increased and Tref can be changed, but this reducesthe importance of robust stability.

The 690s AUV is given a depth command that generates thepitch command input for the H∞ attitude controller. Step pitchcommands were not attempted. A 2.0 meter depth commandgenerated the transient pitch commands and measured pitchangles in Figure 11. The pitch commands are not step inputs,but the tracking performance for the controllers is shown. Asexpected, the 2.0 rad/sec one degree of freedom controllerhas a faster rise time and smaller steady-state error comparedto the 1.0 rad/sec controller. For the two degree of freedomcontrollers, both of the controllers appear to have similar risetimes, but the 2.0 rad/sec controller has smaller steady-stateerror.

VIII. CONCLUSION

The loop-shaping design procedure provides a way toproduce stabilizing robust H∞ attitude controllers for AUVsusing classical loop-shaping techniques. We confirmed throughour results that a controller with a lower actuator bandwidthreduces actuator movement compared to a controller witha higher actuator bandwidth. The experimental results alsoshowed the trade-off between reference tracking performanceand control effort.

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