A Stochastic Approach to Optimal Soaring Problem andRobust Adaptive LQG Control

Nazli E. Kahveci∗ and Petros A. Ioannou†

University of Southern California, Los Angeles, CA, 90089

Maj D. Mirmirani‡

California State University, Los Angeles, CA, 90032

The soaring performance of a glider Unmanned Aerial Vehicle (UAV) depends a great deal on the ther-mal characteristics including thermal strength and location. Onboard sensor measurements are typically cor-rupted by noise whereas gusts and turbulence effects can cause significant performance degradation duringinter-thermal glide.

In this paper we provide a stochastic approach to the optimal soaring problem. We quantify the perfor-mance losses due to incorrect thermal data, consider the effect of stochastic gust disturbances, and simulatethe deterioration in the system response in the presence of additional sensor inaccuracies. Although the re-covery of losses due to incorrect thermal data is possible only if data resources being used could be enhanced,an adaptive tracking control implied by a Linear Quadratic Regulator (LQR) design inherently provides theoptimal control when the aircraft is subject to gust represented by a Gaussian white noise. The LQR design,however, needs to be improved if in addition stochastic noise is posed at each sensor measurement channel. Byincluding an adaptive Kalman-Bucy filter and modifying the adaptive law and our optimum trajectory gener-ation algorithm inputs accordingly, we obtain an adaptive Linear Quadratic Gaussian (LQG) control design.As a result, the aircraft response meets the performance requirements in the presence of stochastic process andmeasurement noise.

I. Introduction

FLIGHT velocity enhancements as well as range and endurance improvements can be achieved by UAVs in powerlessflight if dynamic1–11 or static10–14 soaring methods are employed. In static soaring the UAVs utilize rising columns

of warm air called thermals. These applications are modeled in general based on the assumption that the flight fromone thermal location to the next is in calm air. The inter-thermal glide, however, is more realistically characterized byrandom gusts acting on the aircraft dynamics. Gust is by definition a strong, abrupt rush of wind, and it usually refersto a single pulse. When the profile is continuous, the gust structure is generally spoken of as turbulence.15

The atmosphere is in a continuous state of motion, and the wind gusts created by the movement of atmosphericair masses can degrade the performance and flying qualities of an airplane, whereas wind shears created by thun-derstorm systems have been identified as the major contributor to several airline crashes.16 From the aircraft designstandpoint, atmospheric turbulence can contribute to aircraft structural fatigue and even structural damage or failure ifthe turbulence spectrum has larger scales.17 Airplanes having low wing loadings are particularly more responsive tothe influence of vertical gusts. In order to reduce the negative effects of these atmospheric phenomena on the flightperformance, proper modeling is required.

This work was supported by the National Aeronautics and Space Administration under Grant UAS/USC-220843.∗Ph.D. Student, Electrical Engineering, University of Southern California, Student Member, AIAA†Professor, Electrical Engineering, University of Southern California, Member, AIAA‡Professor, Mechanical Engineering, California State University, Senior Member, AIAA

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A sharp-edged gust encountered by an airplane can be modeled by a step function profile. The atmosphericturbulence experienced by an airplane is modeled in [18] as a combination of discrete patches of gusts, which isreplaced later on in a limiting-process sense by a model which has a continuously variable distribution of Root MeanSquare (RMS) gust velocity. Alternatively, idealization of a typical gust profile as a stationary Gaussian randomprocess is in wide use.15 The two reasons discussed are the facts that it is vastly more realistic than the simplediscrete-gust idealizations and that the statistical characteristics of the airplane response can be determined directlyfrom the statistical description of the gust velocity profile. The equations of motion in a nonuniform atmosphereare summarized in [16] where pure vertical or plunging motion is discussed for sinusoidal or sharp-edged gusts, andthe velocity variations in a turbulent flow are described. To study the influence of atmospheric disturbances on thedynamics of the aircraft, we modify the aircraft model accordingly and consider a zero-mean Gaussian white noise torepresent the effect of the vertical gusts on the longitudinal dynamics.

Aircraft sensors are utilized for required measurements during flight. In-flight calibration is not applicable at alltimes, and the inherent inaccuracies in these sensors are thus to be taken into account in the control design phase.Sensors might suffer accuracy losses and very slow response times. If the sensor data is collected at periodic rates, thisalso adds to the uncertainty. The accelerations and the angular rates can be measured and used to obtain the aircraftposition, velocity and altitude. These measurements are likely to be corrupted by noise, and some degree of robustnessagainst the sensor noise is desired. If the data from redundant or complementary sensors are combined, multi-sensordata fusion possibly provides more precise information. Nevertheless, implementing such advanced measurementdevices might be both complicated and expensive. In our work maximizing the use of the available information isencouraged, but no redundant or complementary sensors are included.

We focus on the control design for the UAVs in static soaring applications. If the location of the next thermaland the corresponding climb rate called the strength are known, an optimum trajectory for the UAV can be generatedpossibly considering also the safety restrictions posed by the expected altitude loss in the inter-thermal glide. Formore information on how the thermals can be remotely detected, interested reader might refer to [19–24]. The recentimprovements in these remote detection methods are out of the scope of our work. For our design purposes it sufficesto make a quantification of the performance deterioration due to the erroneous thermal strength data.

The linear aircraft model is described and a brief overview of the optimum soaring trajectory generation algorithmis given in Section II. An adaptive controller design is considered in Section III in order to cope with the uncertaintiesin the UAV model which is subject to further deviations due to changing flight conditions. Possible performancelosses due to incorrect thermal data and the noise effect of vertical gusts acting on the aircraft dynamics are discussedin Section IV and Section V respectively. In order to reduce the performance deterioration due to sensor inaccuraciesindicated in Section VI, the LQR design is shifted to an LQG controller in Section VII where the statistical propertiesof the practical gust disturbances and sensor inaccuracies can be exploited to obtain sharper performance. In SectionVIII comparative simulation results are presented. Conclusions are drawn in Section IX.

II. Aircraft Model and Optimum Soaring Trajectory

The linear aircraft model introduced in [25] is used in this paper to describe the longitudinal dynamics of a smallglider UAV. The deviations in the aircraft velocity (V ), angle of attack (α), pitch rate (q), pitch angle (θ) and altitude(h) form the incremental state vector, xs = [ δV δα δq δθ δh ]T . We define the horizontal velocity componentas Vh = Vh0 + δVh where Vh0 is the horizontal velocity at the trim conditions, and δVh is the deviation. If δVh isapproximated as a linear function of the incremental states, the performance output, zp can be chosen as δVh = Cpxs

where Cp is the performance output matrix. The states are available for measurement and are defined as the output ofthe system through the identity C matrix. The resulting linear incremental state space model can be summarized as

xs = Axs + Bu + Gω + HδT (1)y = Cxs

zp = Cpxs

where A,B,G,H are system matrices, xs is the incremental state vector for the longitudinal aircraft model, and u isthe control input which is the elevator deflection. The influence of the thermal strength function, ω is incorporated

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into the aircraft dynamics through the G matrix. It is effective while the aircraft is soaring at a thermal location, and iszero otherwise. The negative throttle setting, δT =−0.3 is applied to cancel the effect of the thrust input for the trimconditions and thus obtain a zero-thrust (gliding) aircraft model.

For the set of trim conditions determined such that [V0 α0 q0 θ0 h0 ]T = [40.00004 0.06308 0 0.00994 4000]T ,relevant nominal A,B,Cp, G and H matrices are

A =

−0.08029 0.44721 0 −0.56001 0.00005−2.30141 −6.87585 0.84355 0.04017 0.001410.00002 −32.58844 −7.80618 −0.00007 0

0 0 1 0 0−0.04997 −0.69718 0 0.69718 0

B =

[0.01158 −0.69954 −37.47948 0 0

]T

Cp =[1.0000 −0.0006 0 0.0006 0

]G=

[0.63434 −9.45270 −44.06030 0 0

]T

H =[4.0986 0 0 0 0

]T

The performance of a glider can be characterized by its polar curve, a graph that shows its sink rate at differentairspeeds. A quadratic polar curve in the form

Vz = aV 2h + bVh + c (2)

a = −0.000391 , b = 0.026 , c = −6.3

is shown to be a good approximation to the dynamics of the glider model via simulations.We assume that the strength of the thermal ahead and the distance to that thermal location are known as data inputs.

The horizontal velocity for the inter-thermal glide is chosen in [25] to minimize the total flight time which includes thetime spent during the inter-thermal glide and the thermal soaring time required to recover the corresponding altitudeloss. This approach is based on the idea that the faster the aircraft glides, the more it sinks, and the longer it needs toclimb at the arrived thermal location. The optimal horizontal velocity for the inter-thermal glide is given by

Vopt =

√c−W

a,

c−W

a> 0 (3)

where W is the strength of the thermal ahead. The optimum flight trajectory can be generated if Vopt is calculated foreach inter-thermal glide towards the destination. The trajectory generation algorithm should also consider the distanceto the next thermal location, current altitude and the velocity of the aircraft for safety concerns. If the optimal velocityis not allowed, a proper flight velocity should be assigned. Reaching the destination set a priori safely in optimumtime is regarded as glider’s optimum performance which can be achieved if the velocity of the aircraft matches thereference velocity for approaching each thermal location on the flight path. An adaptive tracking controller that can beused to achieve this goal in the presence of uncertainties in the dynamics of the vehicle is described in the next section.

III. Robust Adaptive Linear Quadratic Control Design

If the reference horizontal flight velocity is Vopt = Vh0 + δVopt , a set of compatible states, xr can be obtainedusing the left pseudo inverse of the Cp matrix as a Command State Generator (CSG) such that

δVopt = Cpxr (4)⇒ xr = CT

p (CpCTp )−1δVopt (5)

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As the negative throttle setting is applied in the form of a constant disturbance throughout our applications, atracking control design with constant disturbance rejection is desired. We augment the system as

xaug = Aaugxaug + Baugv (6)zaug = Caugxaug (7)

where

xaug =

[e

zaug

], e = xs − xr , v = u

Aaug =

[A 0Cp 0

], Baug =

[B

0

], Caug =

[0 0 0 0 0 1

]If (Aaug, Caug) is detectable and (Aaug, Baug) is stabilizable, the unique positive definite solution P to the Alge-

braic Riccati Equation (ARE)

ATaugP + PAaug + Qz − PBaugR

−1z BT

augP = 0 (8)

Qz =

[0 00 I

]×Q , Q > 0 , Rz > 0

defines the feedback gain, K = R−1z BT

augP . The control input for the augmented system is then chosen as

v = −K xaug = − [ K1 K2 ] xaug = −K1 e−K2 zaug , K1 ∈ R1x5, K2 ∈ R (9)

which asymptotically stabilizes the augmented closed loop system. The control input

u = −K1e−K2

∫ t

0

Cpe(τ)dτ (10)

thus provides tracking control with disturbance rejection for the original system.The tracking control implied by the regulator design for the augmented dynamics is referred to in this paper as the

LQR design in comparison to an LQG controller where filter dynamics are also involved.The state vector, xs can be measured, the throttle setting, δT is known, and the thermal strength function, ω is

remotely detected. For any fixed flight conditions we thus obtain

z = θ∗T φ , θ∗T =[A B

], φ =

1s + λ

[ xs1 xs2 xs3 xs4 xs5 u ]T (11)

where φ is the regressor vector, xsiis the ith element of xs ∈ R5, and λ > 0 is a design parameter. The estimation

model can be defined as

zi = θTi φ , θi(t) =

[ai1 ai2 ai3 ai4 ai5 bi

]T

(12)

for i = 1, 2, ..., 5 where θi(t) is the estimate of θ∗i at time t , and θ∗i ∈ R6 is the ith column of θ∗ ∈ R6×5 . Note thataij is the estimate of aij where aij is the ijth element of A ∈ R5×5 ; bi is the estimate of bi where bi is the ith elementof B ∈ R5.

We use the discrete approximation of the continuous-time online parameter estimator based on the continuous-timemodel of the system. Let k = 1, 2, ... be the discrete-time index and T be the sampling period where estimation isdone at time t = kT. The discrete-time estimation model is thus zi(kT ) = θT

i (kT )φ(kT ) , i = 1, 2, ..., 5 .

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For the adaptive law we employ the discrete version of the least-squares algorithm modified with robust weightinggiven in [26] as follows

Pi(kT ) = Pi([k − 1]T ) − ci(kT )Pi([k − 1]T )φ(kT )φT (kT )Pi([k − 1]T )m2(kT ) + ci(kT )φT (kT )Pi([k − 1]T )φ(kT )

(13)

εi(kT ) =zi(kT )− θT

i ([k − 1]T )φ(kT )m2(kT )

(14)

θi(kT ) = θi([k − 1]T ) +√

ci(kT )Pi(kT )φ(kT )εi(kT ) (15)

⇒ θi(kT ) = θi([k − 1]T ) +ci(kT )Pi([k − 1]T )φ(kT )[zi(kT )− θT

i ([k − 1]T )φ(kT )]m2(kT ) + ci(kT )φT (kT )Pi([k − 1]T )φ(kT )

(16)

for each i = 1, 2, ..., 5 where

ci(kT ) =

ci1 , φT (kT )Pi([k − 1]T )φ(kT ) ≥ δi

ci2 , otherwise(17)

is the robust weighting for ci1 >> ci2 > 0 , δi > 0 , and Pi(0) , ci1 , ci2 , δi , i = 1, 2, ..., 5 are design parameters.The estimation error, εi(kT ) is normalized where m2(kT ) = 1 + φT (kT )φ(kT ) is a proper normalization signal.

To eliminate possible parameter drift due to modeling errors, estimates of parameters are forced to lie inside acompact set by using parameter projection so that their boundedness is guaranteed. It is assumed that lower and upperlimits of each element θ∗ij of θ∗i , j = 1, 2, ..., 6 , are known so that Lij ≤ θ∗ij ≤ Uij for some Lij , Uij ∈ R , i =1, ..., 5 , j = 1, 2, ..., 6 . As a result the orthogonal term-by-term projection is given by

θij(kT )=Prθij(kT )=

θij(kT ) , Lij ≤ θij ≤ Uij

Lij , θij(k) < Lij

Uij , θij(k) > Uij

(18)

where θij(kT ) is the estimate of θ∗ij at time t = kT.

IV. Performance Loss Due to Incorrect Thermal Data

Until rather large errors are involved in the thermal strength data, the optimal reference velocity can be shownto remain almost unaffected. However, the corresponding errors might severely affect the overall performance of theUAV if the optimum trajectory generation algorithm is conservative or aggressive in general, and it might become oneof the critical issues as the optimum overall soaring performance is desired. On the other hand, the thermal data shouldalso be used to define the maximum allowed horizontal velocity that meets the safety requirements. This implies thefact that the effect of the thermal data is twofold.

A discussion on how to achieve improvements in the available thermal data resources is out of the scope of thispaper. It is, however, possible to quantify the effect of the data error on the optimal velocity calculation, and it mightindeed be helpful to estimate the resulting performance loss or the risk taken with the corresponding speed-to-flydecision.

If the thermal strength, W is incorrectly estimated as W + ∆We, where ∆We is the additive error, the optimalhorizontal velocity would be chosen as

Vh|W+∆We=

√c−W −∆We

a(19)

which allows us to calculate the effect of any additive error in the thermal strength data on the corresponding flighttime. The total flight time is originally defined as

ttotal = tglide + tthermal =∆x

Vh+

hloss(Vh, Vz)W

(20)

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hloss(Vh, Vz)=−∆x× Vz

Vh(21)

Vz∼= aV 2

h + bVh + c (22)

where ∆x is the inter-thermal glide distance. Equations (20,21,22) can be modified to include the incorrect data

ttotale = tglidee+ tthermale =

∆x

Vh|W+∆We

+hloss(Vh|W+∆We

, Vz|W+∆We)

W(23)

hloss(Vh|W+∆We, Vz|W+∆We

)=−∆x×Vz|W+∆We

Vh|W+∆We

(24)

Vz|W+∆We

∼= aVh|2W+∆We+ bVh|W+∆We

+ c (25)

The relative error in the total flight time, which can be confirmed by simulations, is then calculated as

η =∣∣∣∣ ttotal − ttotale

ttotal

∣∣∣∣ (26)

The incorrect data such as smaller or larger strength estimation for the thermals on the flight path can be shown toextend the overall flight time needed to reach the destination. If the distance to the next thermal location is estimatedwith error, this does not directly affect the optimum flight velocity, but the safety requirements might need to bemodified. An estimate of the distance to the next thermal location which is smaller than its real value might increasethe allowed maximum velocity and result in an early landing or an aircraft crash. On the other hand, if an estimationlarger than the real distance is taken for granted, the trajectory generation algorithm might conclude that Vopt inEquation (3) is not allowed due to safety considerations and define a reference slower than the real optimum flightvelocity resulting in significant performance loss.

A simulation scenario is chosen where three gliders start together, but with different strength data for the same ther-mal located at a distance of ∆x = 3, 000ft. The first glider has the correct thermal strength data, W = 10ft/sec. Thesecond glider underestimates the climb rate of the thermal as W = 5 ft/sec , whereas the third glider overestimatesit to be W = 15 ft/sec and flies aggressively. After climbing at the thermal location and recovering their corre-sponding altitude losses, the gliders all fly at the same speed-to-fly for the next thermal ahead which has a strength ofW = 5 ft/sec .

The simulation results in Figure (1) reveal that the first glider displays the best performance as expected. It reachesthe thermal location by losing more altitude than the second glider, but it recovers that altitude loss and still finishesearlier. The second glider flies overcautiously and deviates less from the straight flight path. The third glider arrives atthe thermal location earlier than the other two as it flies faster. Meanwhile, it loses more altitude and thus spends more

0 5 10 15 20 25 30 35 40 45 500

1000

2000

3000

4000

5000

6000

7000

time (sec)

dist

ance

cov

ered

, δx

(ft)

With Correct Thermal DataWith Smaller Thermal Strength EstimationWith Larger Thermal Strength Estimation

0 5 10 15 20 25 30 35 40 45 50−600

−500

−400

−300

−200

−100

0

100

time (sec)

altit

ude

chan

ge, δ

h (f

t)

With Correct Thermal DataWith Smaller Thermal Strength EstimationWith Larger Thermal Strength Estimation

Figure 1. The flight distance covered and the corresponding altitude loss for different thermal data provided. (Section IV)

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0 5 10 15 20 25 30 35 40 45 500

1000

2000

3000

4000

5000

6000

7000

time (sec)

dist

ance

cov

ered

, δx

(ft)

For No Noise CaseIn the Presence of Process Noise

0 5 10 15 20 25 30 35 40 45 50−500

−400

−300

−200

−100

0

100

time (sec)

altit

ude

chan

ge, δ

h (f

t)

For No Noise CaseIn the Presence of Process Noise

Figure 2. The flight distance covered and the corresponding altitude loss using the adaptive LQR in the presence of vertical gust distur-bances. (Section V)

time for climbing at the thermal location in comparison with the other two. The shortest total flight time is observedfor the first glider which flies optimally based on the correct thermal data.

Using the polar curve approximation the relative error, η defined by Equation (26) is calculated to be 0.02 forW = 5 ft/sec , and 0.01 for W = 15 ft/sec . The simulation results demonstrate higher relative errors of 0.03 and0.02 for W = 5 ft/sec and W = 15 ft/sec respectively. This discrepancy is partly due to the quadratic polar curveapproximation used to represent the aircraft dynamics in calculations and partly due to the transients observed in thesimulations while tracking the desired trajectory. The relative error provides valuable information to evaluate the riskbeing taken in terms of reliability assessment. Moreover, the value of the relative error proposed by Equation (26) canbe shown to match the simulation results more closely if longer glide distances are considered where the transientsbecome negligible in comparison to the total flight time.

V. Performance Loss Due to Vertical Gust Disturbances in Inter-thermal Glide

We consider gust fields that create effective pitching gusts rather than rolling. The gust disturbance, ωg in inter-thermal glide can thus be incorporated into the system dynamics in Equation (1) through the G matrix similar to thethermal strength function, ω . As the G matrix is independent of the state vector, xs and the control input, u, thismodification defines an additive noise problem. In this paper we do not discuss multiplicative disturbances and onlythe additive ones as gust effects. It can be shown that the optimal control for this stochastic problem is identical to theoptimal control for the deterministic case where ωg(t) = 0 at all times. The details given in [27] prove that for the LQRproblems in the presence of additive disturbance signals modeled as zero-mean Gaussian white noise processes where

0 5 10 15 20 25 30 35 40 45 500

50

100

150

200

250

time (sec)

horiz

onta

l vel

ocity

, Vx (

ft/se

c)

For No Noise CaseIn the Presence of Process Noise

0 5 10 15 20 25 30 35 40 45 50−800

−600

−400

−200

0

200

400

600

800

time (sec)

cont

rol i

nput

, u

For No Noise CaseIn the Presence of Process Noise

Figure 3. The flight velocity and the control input using the adaptive LQR in the presence of vertical gust disturbances. (Section V)

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0 5 10 15 20 25 30 35 40 45 500

1000

2000

3000

4000

5000

6000

7000

time (sec)

dist

ance

cov

ered

, δx

(ft)

No noise caseProcess Noise and Measurement Noise

0 5 10 15 20 25 30 35 40 45 50−600

−500

−400

−300

−200

−100

0

100

time (sec)

altit

ude

chan

ge, δ

h (f

t)

No noise caseProcess Noise and Measurement Noise

Figure 4. The flight distance covered and the corresponding altitude loss using the adaptive LQR in the presence of vertical gust distur-bances and measurement noise. (Section VI)

the system states are available for feedback uncorrupted by any disturbance, the validity of certainty equivalence canbe established and the optimal control can be shown to be identical to the optimal control for the deterministic case.

The results in Figure (2) and Figure (3) are from simulations run in the presence of a gust disturbance, ωg rep-resented as a white noise signal with unity power spectral density. For reference input shaping, a simple first orderlow-pass prefilter is included in our simulations, and the resulting optimal horizontal flight velocity is used as thereference. A smooth aircraft response is obtained using the LQR design, and the process noise does not cause anyconsiderable effect on the motion of the aircraft. If the power spectral density of the process noise is chosen to belarger, further deviations from the optimal performance would be inevitable. Still, acceptable performance is observedunless very significant gust disturbances are present as process noise.

VI. Performance Loss Due to Sensor Inaccuracies

If perfect measurements are available where the output views all the states with perfect sensors, the LQR designinherently provides the optimal control in the case of a Gaussian white noise disturbance. Under the more practicalassumption of imperfect sensory information, however, the optimal control addressed by the LQR is no more satis-factory. If the states and thus the output of the system are corrupted by disturbance signals, those noise-corruptedmeasurements might severely affect the performance of the aircraft, not only through the state feedback, but alsothrough the adaptive law and the safety issues involved in the optimum trajectory generation algorithm being used.

0 5 10 15 20 25 30 35 40 45 50−50

0

50

100

150

200

250

time (sec)

horiz

onta

l vel

ocity

, Vx (

ft/se

c)

No noise caseProcess Noise and Measurement Noise

0 5 10 15 20 25 30 35 40 45 50−1000

−800

−600

−400

−200

0

200

400

600

800

1000

time (sec)

cont

rol i

nput

, u

No noise caseProcess Noise and Measurement Noise

Figure 5. The flight velocity and the control input using the adaptive LQR in the presence of vertical gust disturbances and measurementnoise. (Section VI)

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An additive zero-mean white Gaussian sensor noise with unity power spectral density is applied separately at eachstate sensor for simulation purposes. The corresponding results using the LQR design are presented in Figure (4)and Figure (5) in comparison to the aircraft response where there is no gust disturbance and perfect measurementsare available. As can be observed from these results, the oscillations in the control signal and the altitude change areunacceptable.

VII. Performance Loss Recovery: Adaptive LQG Solution

As the states are available for measurement, at first sight there is no need to design an observer to estimate the fullstate vector, xs(t) in an attempt to make use of the advantages state-feedback design has to offer. When the sensornoise acts to impair these measurements, however, the states are driven by noise and are thus also random processes.The design of an estimator is required for such stochastic systems.

We use the LQG problem setup based on the stochastic-deterministic dualism and the separation principle to designthe feedback controller and the observer separately. Note that if the estimates provided by a Kalman-Bucy filter areused in the adaptive law, the update mechanism would also become more accurate.

If the measurement noise, γ is posed at the sensors modifying the output relation, the aircraft response can bediscussed using the stochastic dynamical system

xs = Axs + Bu + ξ (27)ξ = Gωg

y = Cxs + γ

where the gust disturbance, ωg ∈ R and the measurement noise, γ ∈ R5 are assumed to be zero-mean Gaussian whitenoise with constant power spectral density matrices. Assuming that ξ and γ are uncorrelated, we define

Eωg(t)ωg(τ) = Ωgδ(t− τ) (28)Eγ(t)γT (τ) = Γδ(t− τ) (29)

Eξ(t)γT (τ) = 0 (30)Eγ(t)ξT (τ) = 0 (31)

where Ωg ∈ R and Γ ∈ R5×5 are spectral density matrices, E. is the expectation operator, and δ(t − τ) is a deltafunction. Since the stochastic process, ωg(t) is incorporated into the dynamics through the G matrix, for the resultingstochastic process noise, ξ we obtain

Eξ(t)ξT (τ) = EGωg(t)ωg(τ)GT = GGT Eωg(t)ωg(τ) = GGT Ωgδ(t− τ) (32)

where Ξ = GGT Ωg ∈ R5×5 is the spectral density matrix of ξ .The levels of process and measurement noise might be environmental dependent. If the process and measurement

noise covariance matrices are not available, they could be estimated, and several such methods are discussed in [28–31].A fuzzy adaptive Kalman filter with weaker reliance on the prior statistical information is presented in [32]. On theother hand as stated in [33], it is not unreasonable to assume that the measurement noise have zero mean since thereshould be no bias on the measuring instruments.

We can calculate the optimal filter-gain matrix, Kf through the relation

Kf = SCT Γ−1 (33)

where S satisfies the Filter Algebraic Riccati Equation (FARE)

0 = AS + SAT + Ξ− SCT Γ−1CS (34)

The Kalman-Bucy filter can then be implemented in the form of a state observer as

˙xs = Axs + Bu + Kf (y − Cxs) (35)

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Figure 6. Adaptive LQG Control Structure

where xs ∈ R5 is the vector of state estimates. Note that although the constant negative throttle setting term is ignoredin the filter design phase, it is included in the simulations and asymptotically rejected by the integrator in the controlleras explained in Section III.

The LQR design is based on the assumption that the system is stabilizable and detectable. To extend the LQRsolution to the LQG controller, we further assume that (AT , CT ) is stabilizable and (AT , E) is detectable whereΞ = ET E and Γ > 0 . These assumptions guarantee the existence of the solution S > 0 that results in a stable(A−KfC) matrix as discussed in [27].

If the gust noise, ωg is not white, the noise-shaping filter proposed in [34] can be employed as

xω = Aωxω + Bωn (36)ωg = Cωxω + Dωn (37)

where n(t) is a white noise input, and ωg(t) is the filter output. The filter matrices, Aω, Bω, Cω, Dω can be determinedbased on factoring the spectral density of the noise, ωg(t) as discussed in [35]. The aircraft model and the noise-shapingfilter can then be combined to obtain the augmented dynamics

xωaug= Aωaug

xωaug+ Bωaug

u + Gaugn (38)y = Cωaug

xωaug+ γ (39)

where

xωaug=

[xs

xω

], Aωaug

==

[A GCω

0 Aω

], Bωaug

=

[B

0

], Gaug =

[CDω

Bω

], Cωaug

=[C 0

](40)

A nonwhite γ(t) signal can still be addressed by a similar method. By solving the FARE for the augmented system,the same filter design method can be applied with the required modifications.

The adaptive LQG control structure is demonstrated in Figure (6). The parameters of the Kalman-Bucy filter alongwith the parameters of the linear quadratic controller are updated using the model parameter estimates generated bythe same robust adaptive law and the corresponding solutions of the FARE and the ARE.

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0 5 10 15 20 25 30 35 40 45 500

1000

2000

3000

4000

5000

6000

7000

time (sec)

dist

ance

cov

ered

, δx

(ft)

LQRLQG

0 5 10 15 20 25 30 35 40 45 50−600

−500

−400

−300

−200

−100

0

100

time (sec)

altit

ude

chan

ge, δ

h (f

t)

LQRLQG

Figure 7. The flight distance covered and the corresponding altitude loss comparison in the presence of vertical gust disturbances andmeasurement noise. (Section VIII)

VIII. Adaptive LQR and Adaptive LQG Performance Comparison

The performances of the adaptive LQR and the adaptive LQG controller are compared via simulations in the pres-ence of vertical gust disturbances and sensor measurement noise where Ξ = GGT and identity Γ are the correspondingpower spectral densities. Figure (7) and Figure (8) display the responses of the two systems. If the adaptive Kalman-Bucy filter is included in the dynamics, there is a short delay observed when the UAV leaves the thermal location toglide towards the next thermal, which is definitely an acceptable compromise. The adaptive LQG controller resultsin reliable aircraft performance as the oscillations are efficiently suppressed. The system response becomes ratherpredictable in the presence of random noise.

IX. Conclusion

In optimal static soaring applications the control objective is to track the optimal velocity along the flight path.The remote thermal data is in general assumed to be accurate, atmospheric disturbances acting on the system duringthe inter-thermal flight are ignored, and perfect sensory information is required for the controller design and imple-mentation. A more realistic flight environment can be described, however, only if these assumptions are relaxed.Furthermore, a quantification of the effect of erroneous thermal data on the performance of the glider UAV can be

0 5 10 15 20 25 30 35 40 45 50−50

0

50

100

150

200

250

time (sec)

horiz

onta

l vel

ocity

, Vx (

ft/se

c)

LQRLQG

0 5 10 15 20 25 30 35 40 45 50−1000

−800

−600

−400

−200

0

200

400

600

800

1000

time (sec)

cont

rol i

nput

, u

LQRLQG

Figure 8. The flight velocity and the control input comparison in the presence of vertical gust disturbances and measurement noise. (SectionVIII)

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supportive in reliability assessment.We model the vertical gust disturbances and the sensor measurement noise as stochastic processes. An adaptive

LQG design including an optimal state estimator in addition to an optimal state feedback controller is implemented.The sensor measurements are reconstructed using the noise rejection capabilities of an adaptive Kalman-Bucy filter.These enhanced signals are incorporated with the feedback control, the adaptive law and the trajectory generationalgorithm. The simulation results demonstrate that the performance achieved is comparable to the case with no pro-cess or measurement noise, and thus the adaptive control scheme is improved by exploiting the practical statisticalproperties of random noise.

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