DTIC ADA442139: Adaptive Backstepping With Magnitude, Rate, and Bandwidth Constraints: Aircraft Longitude Control

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by  Defense Technical Information Center

Adaptive backstepping with magnitude, rate, and
bandwidth constraints: Aircraft longitude control
Jay Farrelly, Marios Polycarpouz, and Manu Sharmax
y Department of Electrical Engineering, University of California, Riverside, CA 92521 USA
z Dept. of Elect. and Comp. Engineering and Computer Science, Univ. of Cincinnati, Cincinnati, OH 45221, USA
x Barron Associates, Inc., 1160 Pepsi Place, Suite 300, Charlottesville, Virginia 22901
Abstract the reference input to the control loop. The third set
Thisarticlepresentsanadaptivebacksteppingapproach of of techniques, referred to as pseudo-control hedging
to the longitudinal ((cid:176);fi;Q) control of an aircraft that (PCH) [7, 10], only alters the commanded input to the
directlyaccommodatesmagnitude,rate,andbandwidth loop. The goal in PCH is to decrease the command to
constraints on the aircraft states and the actuator sig- the loop to a point where it is implementable without
nals. Thearticleincludesdesignofthecontrollaw,Lya- saturation. PCH does not directly change the signal
punov analysis of the stability properties of the closed used in the parameter adaptation laws. The work in [7]
loop system, and simulation based analysis of the per- develops and analyzes the PCH method to extend the
formance. control approach of [3, 14] to accommodate magnitude
and rate limits on the commanded surface de(cid:176)ections.
1 Introduction
Although the second and third approaches start with
The introduction of uninhabited combat air vehicles
radically difierent philosophies, for flrst order systems,
(UCAV’s)hasgeneratedinterestinadaptivecontrolfor
itcanbeshownthatthesetwoapproachesareidentical.
aircraft that are capable of maintaining trajectory fol-
For higher order or nonlinear systems, the comparison
lowing control even subsequent to faults or battle dam-
is not as straightforward.
age. In addition, physical limitations impose magni-
tude, rate, and bandwidth constraints on the control Thisarticlepresentsanadaptivebacksteppingapproach
surfacede(cid:176)ectionsandtheaircraftstates. Inthisarticle, forlongitudinalcontrolofanaircraftthathasstateand
we present analysis and simulation results for an adap- actuator constraints. Both theoretical and simulation
tive backstepping approach that is designed to accom- analysesareincluded. Anovelaspectofthisapproachis
modate magnitude, rate, and bandwidth constraints on theabilitytoaccommodatemagnitude, rate, andband-
the actuator signals as well as the intermediate control widthconstraintsontheactuatorsignalsandeachofthe
variables used in the backstepping control approach. intermediate control variables (i.e., aircraft state vari-
ables)usedinthebacksteppingdesign. Thisresultisof
Adaptive control has been applied to aircraft in, for ex-
interest in its own right. This approach has similarities
ample [2, 3, 4, 14, 15, 18]. Piloted hardware in the loop
with both the TSH and PCH methods, since it corrects
simulation and (cid:176)ight testing is described in [2, 4, 18].
both the training signal and the reference input at each
The approach of [3, 4] used a dynamic inversion based
step of the backstepping procedure. Our ultimate goal
nonlinear controller with a neural network approximat-
is to extend this approach to a full vehicle controller
ing the error in the inversion process. Adaptive back-
using on-line function approximation [5, 11, 12] to ac-
steppinghasbeenappliedtoaircraftinforexample[15].
commodate battle damage types of events.
Thatarticledidnotconsidertheefiectsofsaturationon
the closed loop performance.
Control signal rate and amplitude constraints in adap-
2 Problem Formulation
tive control are addressed in, for example, [1, 7, 9, 10,
17]. Theflrsttypeofmethodistocompletelystopadap- Due to space limitations, the discussion of this article
tationundersaturationconditions. Thisad-hocmethod will focus on a simplifled aircraft model with the same
does prevent the tracking errors induced by actuator structure as the linearized, rigid body, aircraft longitu-
constraints from corrupting the parameter adaptation, dinal dynamics [16]. The model is just complex enough
but is undesirable as it does not allow any theoreti- to allow a presentation and analysis of the control de-
cal stability guarantees. The second set of approaches sign approach. Since the model has the same structure
(training signal hedging (TSH)), see e.g. [1, 9], cor- as aircraft longitudinal dynamics, the approach can be
rects the tracking error only where it is used in the pa- extended to a realistic nonlinear aircraft model. This
rameter adaptation laws. It does not directly change extensionisdiscussedfurtherintheconclusionssection.
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1. REPORT DATE 3. DATES COVERED
2006 2. REPORT TYPE 00-00-2006 to 00-00-2006
4. TITLE AND SUBTITLE 5a. CONTRACT NUMBER
Adaptive backstepping with magnitude, rate, and bandwidth constraints:
5b. GRANT NUMBER
Aircraft longitude control
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Standard Form 298 (Rev. 8-98)
Prescribed by ANSI Std Z39-18
The model that we will consider has the form 3 Adaptive Backstepping with Saturation
The backstepping control laws are summarized below.
(cid:176)_ = L +L fi The operating envelope requires fi to remain in the in-
o fi
fi_ = Q¡(Lo+Lfifi) Ttehrveaalc[tfiucLa;tfiorcUd]aesnidgnQrteoqureirmesai–ntionrtehmeaininteirnvatlh[eQiLcnt;eQrvUca].l
Q_ = M +M Q+M –;
o Q – [–L;–U]. In addition, the actuator design will impose
c c
rate and bandwidth limitations on –, which in turn will
wherethe(normalized1)liftforceismodeledasL +L fi
o fi result in rate and bandwidth limitations on the inter-
and the (normalized2) pitch moment is modeled as mediate variables Q and fi . The above operating en-
M +M Q+M –. In this expression, (cid:176) is the (cid:176)ight c c
o Q – velopeandsurfacede(cid:176)ectionconstraintsareassumedto
path angle, fi is the angle of attack, Q is the pitch rate,
be known and flxed.
and – is the control surface de(cid:176)ection. At an operating
point, L , L , M , M andM areunknown, constant, The nominal backstepping commands are
o fi o Q –
scalar parameters. Additional terms can be added to ‡ ·
1
the lift or moment models without changing the theo- fi0 = ¡L^ +(cid:176)_ ¡k (cid:176)~+· ¡´
retical approach. Most importantly, the approach di- c L^ o c (cid:176) (cid:176) fi
‡fi ·
rectly extends to the case where multiple independent Q0 = L^ +L^ fi ¡k fi~+· +fi_ ¡(cid:176)„L^ ¡´
surfaces are available. These generalizations are not in- c o fi fi fi c fi Q
‡ ·
cluded here due to space limitations. –0 = 1 ¡M^ ¡M^ Q¡k Q~+· ¡fi„+Q_
c M^ o Q Q Q c
Thecorrespondingtrackingerrordynamicequationsare –
(cid:176)~_ = L +L (fi +fi~)¡(cid:176)_ withk(cid:176);kfi;kQ >0. Therobustifyingterms(·(cid:176);·fi;·Q)
o fi c c
aremotivatedintheproofofthetheoremtofollow. The
fi~_ = ¡(L +L fi)+Q +Q~¡fi_
o fi c c amplitude, rate, and bandwidth limited control signals
Q~_ = M +M Q+M –¡Q_ can be deflned by
o Q – c
¡ ¡ ¢ ¢
fiw~h=erfie¡thfie t;rQa~ck=inQg¡erQror.s are deflned as (cid:176)~ = (cid:176) ¡ (cid:176)c; • efi‚ = •sat fic0;ficL;ficU‚¡• xfi1 ‚
c c x_ 0 1 x
fi1 = fi1
Our goal is to generate a backstepping controller that x_ 0 ¡2‡ ! x
willcause(cid:176) totrackaninputsignal(cid:176) . Weassumethat fi2 • ‚fi nfi fi2
c 0
(cid:176)c is smooth and that (cid:176)_c is known. In the backstepping + !2 sat(efi;¡Lfi;Lfi)
approach, the (cid:176) controller will generate a commanded • ‚ • nfi‚• ‚
mvaalunedfifoc0rfoQr.fiT.hAenQficcoonnttrroolllelerrwwililllggeenneerraatteetaheQc0cocnotrmo-l fifi_cc = 10 01 xxfifi12
surface command –0 to achieve the Q command.
c
Sinceanaircraftoperatingenvelopeincludesconstraints wheretheparametersofthisfllteraredeflnedbelow. In
on both actuators and states, the controller must work this fllter, efi is the error between the flrst fllter state
in the presence of magnitude, rate, and bandwidth lim- and the magnitude limited fllter input. Since the flrst
its on – as well as operating envelope constraints on state of the fllter is used in the computation of efi, the
Q and fi. In the formulation to follow, we will treat characteristic equation of the fllter is s2 +2‡fi!nfis+
(Lo;Lfi;Mo;MQ;M–) as parameters to be identifled. !n2fi =0, in the linear portion of the two sat functions.
Therefore, we are implementing adaptive backstepping The sat function in the second equation enforces the
with control variable and state constraints. In the ac- rate limit. At the fllter output, fic is magnitude, rate,
tual application, this set of linearization ‘parameters’ is and bandwidth limited. The command fllters for Qc
a function of the operating conditions. Since the oper- and –c are deflned similarly as
atingconditionisslowlytimevarying,theadaptivelaws
¡ ¡ ¢ ¢
wouldberequiredtoadapttheparameterstomatchthe e = sat Q0;QL;QU ¡x
• Q‚ • c c c‚• Q1 ‚
local model. In the flnal full vehicle implementation,
x_ 0 1 x
these‘linearizedparameters’wouldbeapproximatedas Q1 = Q1
x_ 0 ¡2‡ ! x
a function of the (cid:176)ight condition. Q2 • ‚Q nQ Q2
0
+ !2 sat(eQ;¡LQ;LQ)
1Tosimplifythenotation,the 1 factorattheoperatingpoint • ‚ • nQ‚• ‚
mV
isincludedintheliftfunction. Q 1 0 x
2To simplify the notation, the Iy1y factor is included in the Q_cc = 0 1 xQQ12
momentfunction.
¡ ¡ ¢ ¢
e = sat –0;–L;–U ¡x controlrate, bandwidth, andmagnitudeconstraints. In
• ‚– • c c c ‚• –1 ‚
x_ 0 1 x an implementation, deadzones should be used to avoid
x_–1 = 0 ¡2‡ ! x–1 parameter drift due to noise; and, projection methods
–2 • ‚– n– –2 must be used to ensure that the estimation transient
+ !02 sat(e–;¡L–;L–) does not cause L^fi or M^– to change sign.
n– • ‚
£ ⁄ £ ⁄ x 4 Tracking Error Dynamics
–c = 1 0 x–1 : Representingtheactualparametersastheestimatedpa-
–2 rameter minus parameter estimation error: L = L^ ¡
o o
In these expressions, L = 2‡fiRfi, L = 2‡QRQ, L~o, Lfi =L^fi¡L~fi, Mo = M^o¡M~o, MQ =M^Q¡M~Q,
fi !n2fi Q !n2Q M =M^ ¡M~ ; the dynamics of the modifled tracking
L = 2‡–R– and(R ;R ;R )areratelimitson(fi;Q;–), – – –
– !n2– fi Q – errors are
respectively; (! ;! ;! ) are bandwidth limits on
nfi nQ n–
((fific;LQ;Q;–Lc);;–cL(fi)cUa;rQe Uclo;w–ecUr)liamreitsuponpe(rfil;iQm;it–s),oannd(fis;aQt;d–e)-; (cid:176)„_ == (cid:176)(~_L¡+´_(cid:176)L fi¡(cid:176)_ )¡‡¡k ´ +L^ ¡fi ¡fi0¢·
notes the saturation function that is linear with unity o fi c (cid:176) (cid:176) fi c c
¡ ¢
slope between its lower and upper limits. This para- = L^ +L^ fi0+fi~ ¡(cid:176)_ ¡L~ ¡L~ fi+k ´
o fi c c o fi (cid:176) (cid:176)
graph has presented one command fllter approach,
= ¡k (cid:176)„+L^ fi„¡L~ ¡L~ fi+· (4)
many other fllters are possible without afiecting the va- (cid:176) fi o fi (cid:176)
lidity of the stability proofs. If the surface position can fi„_ = fi~_ ¡´_fi
be measured, then the last fllter is not required. Also = ¡k fi„+L~ +L~ fi¡(cid:176)„L^ +Q„+· (5)
fi o fi fi fi
the parameters of the above fllters are not completely Q„_ = Q~_ ¡´_
independent. For example, since the derivative of fi is Q
closely related to Q, the rate limit Rfi should be less = ¡kQQ„¡fi„¡M~o¡M~QQ¡M~––+·Q: (6)
than or equal to the magnitude limit QU.
c
The flnal numbered equations will be used in the sub-
The vector ´=[´ (t);´ (t);´ (t)] is generated by
(cid:176) fi Q sequent stability analysis.
¡ ¢
´_ = ¡k ´ +L^ fi ¡fi0
(cid:176) (cid:176) (cid:176) ¡fi c ¢c 5 Lyapunov Analysis
´_ = ¡k ´ + Q ¡Q0 The objective of this section is to prove the stability
fi fi fi c¡ c ¢
´_ = ¡k ´ +M^ – ¡–0 : oftheclosedloopsystemusingLyapunov-likemethods.
Q Q Q – c c The results are summarized in the following theorem.
These signals are flltered versions of the efiect of state
andcontrolrate,bandwidth,andmagnitudeconstraints Theorem 5.1 The closed loop system with dynamics
on the variable that is being controlled. Finally, we de- describedinSection2andcontrollerdescribedinSection
flne the vector of modifled tracking errors x„ = [(cid:176)„;fi„;Q„] 3 has the following properties:
as
(cid:176)„ = (cid:176)~¡´ 1. L~o;L~fi;M~o;M~Q;M~– 2L1
(cid:176)
fi„ = fi~¡´fi 2. (cid:176)„, fi„, and Q„ 2L1
Q„ = Q~¡´ :
Q
3. (cid:176)„, fi„, and Q„ converge to zero with (cid:176)„, fi„, and Q„ 2
When there is no control variable saturation, x„ con- L2.
verges to x~. In the presence of state and control vari-
able constraints, the signal ´ is designed to remove the
Inaddition,althoughwedonotshowithere,itispossi-
efiects of the saturation from x~ to generated x„, which
ble to show that given certain persistence of excitation
will be used in the parameter adaptation laws. The
conditions,theparametererrorsconvergeexponentially
parameter adaptation laws are:
to zero.
M^_o =¡3Q„ L^_o =¡1((cid:176)„¡fi„) (1) Proof: Deflne the (cid:176)-Lyapunov function as
M^_Q =¡4Q„Q L^_fi =¡2((cid:176)„¡fi„)fi (2) V = 1¡(cid:176)„2+fi„2+Q„2¢+ 1£~T¡¡1£~
M^_– =¡5Q„–; (3) 2 2
where ¡i for i = 1;:::;5 are positive adaptation gains. where £T =[Lo;Lfi;Mo;MQ;M–], £^ is the estimate of
The subsequent sections will show that the above con- £, and £~ = £^ ¡£. Taking the time derivative of V,
trol laws are stable even in the presence of state and using eqns. (4-6) to eliminate ((cid:176)„_;fi„_;Q„_), and using eqns.
(1-3)toeliminate£~_,aftersomealgebraicmanipulations Each graph contains three curves. The dotted curve is
yields the command. The solid curve is the magnitude, rate,
and bandwidth limited command. The dashed curve
V_ = ¡k Q„2¡k fi„2¡k (cid:176)„2 is the actual state or actuator variable. Even though
Q fi (cid:176)
+(cid:176)„· +fi„· +Q„· : (7) the initial parameter error is large, the transient in the
(cid:176) fi Q
controlresponseisreasonable. Figure2includesgraphs
Therefore, the designer selects (· ;· ;· ) as odd, dif- of the three components of x~ in the left column and the
(cid:176) fi Q
ferentiable functions with sign opposite to ((cid:176)„;fi„;Q„), re- three components of x„ in the right column. Note that
spectively. The magnitude of these robustifying terms as the theory predicted, since there is no saturation,
wouldbeselectedtoensureboundednessoftrackinger- x~!x„!0.
rors in the presence of model error, measurement noise, The convergence of x„ to zero is not monotonic. How-
or disturbances. For the results of Section 6, all three ever, the Lyapunov function is non-increasing. The left
of these terms are set to zero. column of Figure 3 plots the norm of the parameter er-
Since V(t) is nonincreasing, we conclude that ror, the norm of x„, and the norm deflned by the square
(Q„;fi„;(cid:176)„;£~T) are each bounded. Barbalat’s Lemma im- root of the Lyapunov function. Notice that although
plies that the origin of the (Q„;fi„;(cid:176)„) variables is asymp- the norm of x„ is both increasing and decreasing, the
totically stable. Finally, because 0•V(t)•V(0), inte- Lyapunov function never increases. The right column
gration of both sides of (7) shows that (Q„;fi„;(cid:176)„)2L2:} of Figure 3 plots each component of £^.
This theorem shows that even when the nominal com- 6.2 Example: Substantial Saturation
mands (fic0;Q0c;–0) exceed the state and control limita- For this set of simulations, (cid:176)r is a 25s square wave with
tions, the quantities x„ = [(cid:176)„;fi„;Q„]T and £~ do not di- peak magnitude of §10 deg. The commands (cid:176) and (cid:176)_
c c
verge. In fact,pthe cumulative norm of these quantities are the states of a second order, relative degree two,
as deflned by V(t) will never increase. Nothing is unity gain prefllter with ‡ =1, and ! =k .
n (cid:176)
proved about the tracking error x~, which may increase
Figure4containsplotsofthestatesandactuatorsignal
duringperiodswhenthestateorcontrollimitationsare
for the flrst and last 25 seconds of a 150 s simulation.
in efiect, because the desired control signal is not being
Even though the initial parameter error is large, the
implemented (i.e., u 6=u0). The vector ´ is deflned to
c c transient in the control response is reasonable. Note
remove from x~ the portion of the tracking error that is
that the large (cid:176) commands result in rate, magnitude,
duetoviolationofthestateorcontrolsignallimitations.
or bandwidth constraining the fi , Q , and – variables.
The compensated error x„ is used to adapt the parame- c c c
Figure 5 includes graphs of the three components of x~
ters, even during periods in which the state or control
intheleftcolumnandthethreecomponents ofx„ inthe
signallimitationspreventimplementationofthedesired
right column. Note that as the theory predicted, since
control signals. When the state or control signal limi-
there is saturation, x„ ! 0, but x~ does not converge to
tations are not in efiect, (i.e., u = u0), ´ approaches
c c zero. In spite of this fact, the parameter estimates con-
zero, and x~ converges towards x„.
verge toward the correct values. In fact, as shown in
6 Simulation Example the right column of Figure 6, the parameters converge
Thissectionanalyzesresultsfromasimulationexample. fasterthantheydidinFigure3. Thereisasigniflcantly
TheparametersforthesimulationweredeflnedinTable higher level of excitation in the present example, but
1. Thecontrolgains areselectedtoenforceatimescale the parameter estimates will only converge if the train-
separation between the three loops of the backstepping ing error is properly compensated to remove the efiects
controller. Thesectioncontainstwosetsofresults. The of the rate, magnitude, and bandwidth limitations on
flrstsetofresultspertaintoasituationwheresaturation thecontrolsignals. TheleftcolumnofFigure6contains
does not occur. The second set of results pertain to a plots of the norm of the parameter error, the norm of
situationwheresigniflcantsaturationoccursinallthree x„, and the norm deflned by the square root of the Lya-
control loops. The parameters of the command fllters punov function. Note that the Lyapunov function is
deflned in Section 3 are ‡fi = ‡Q = 1, !nfi = kfi, and non-increasing even during the periods of saturation.
! =k .
nQ Q
7 Conclusions
6.1 Example: No saturation
This article has presented and analyzed an adaptive
For this set of simulations, (cid:176) is a 25s square wave with
r backstepping approach to the control of longitudinal
peak magnitude of §5 deg. The commands (cid:176) and (cid:176)_
c c aircraft dynamics that directly accommodates magni-
are the states of a second order, relative degree two,
tude, rate, and bandwidth limits on the aircraft states
unity gain prefllter with ‡ =1, and ! =k .
n (cid:176) and actuators. For this preliminary analysis, we have
Figure1containsplotsofthestatesandactuatorsignal only considered a linearized aircraft point design. Due
for the flrst and last 25 seconds of a 150 s simulation. to the positive analysis of the performance, we are cur-
rently extending this research in four directions. First, 6 10
we are extending the adaptive state and actuator con- 4 8
2
strained backstepping control approach to include con- g, deg−02 a, deg 46
trol of lateral-directional dynamics. Second, we are ex- −4 2
tending the linearized adaptive approach to a nonlin- −60 5 10 15 20 25 00 5 10 15 20 25
time, s time, s
ear on-line function approximation based approach [6].
10 60
Third, we are extending the theoretical analysis as a
5 40
gneanlleyr,awleexhtaevnesioexntoenfdthede bthacekrsetseuplptsinpgraespepnrtoeadchh[e1r3e]i.nFtio- Q, deg −05 d, deg 200
−10 −20
the case where the vehicle has multiple, redundant and
independent actuators. −150 5 10time, s15 20 25 −400 5 10time, s15 20 25
8 Acknowledgement (a) Initialstateandcontrol
This research supported by University of California{
6 12
Riverside,andUniversityofCincinnati,andBarronAs- 4 10
sociates Inc. through Air Force PRDA 01-03-VAK. g, deg−022 a, deg 468
References −4 2
[1] A.M.AnnaswamyandJ.E.Wong.\Adaptivecontrol −6125 130 135 140 145 150 0125 130 135 140 145 150
time, s time, s
in the presence of saturation nonlinearity", Int. J. of Adap-
15 30
tive Control and Signal Processing, Vol. 11: 3{19, 1997. 10 20
5 10
fl[2g]urabJl.eB(cid:176)irginhktecronantrdolKla.wWoinset,h\eFXli-g3h6tttaeisltleinssgaoifrcararfte,c"onJ-. Q, deg −05 d, deg−−21000
−10 −30
of Guidance, Control, and Dynamics, Vol. 24(5): 903{909,
−15125 130 135 140 145 150 −40125 130 135 140 145 150
2001. time, s time, s
[3] A. J. Calise, R. T. Rysdyk, \Nonlinear Adaptive (b) Finalstateandcontrol
Flight Control using Neural Networks," IEEE Control Sys-
tems Magazine, pp. 14{25, Dec. 1998. Figure 1: Plots of the aircraft state and control signal
[4] A. J. Calise, S. Lee, and M. Sharma, \Development at the beginning and end of a 150 s simulation
of a reconflgurable (cid:176)ight control law for the X-36 tailless with (cid:176)r 2 §5 deg: command (dotted), flltered
aircraft,"J.ofGuidance,Control,andDynamics,Vol.24(5): command (solid), actual (dashed).
896{902, 2001.
[5] J. A. Farrell,\Stability and Approximator Conver-
Parametrized On-Line Approximators," Int. J. of Control,
genceinNonparametricNonlinearAdaptiveControl,"IEEE
vol. 70(3): 363-384, 1998.
Trans. on Neural Networks, Vol. 9(5): 1008-1020, 1998.
[13] M. Polycarpou, J. A. Farrell, and M. Sharma, \On-
[6] J.A.Farrell,M.Sharma,andM.Polycarpou,\Longi-
Line Approximation Control of Uncertain Nonlinear Sys-
tudinal Flight-path Control using on-line Function Approx-
tems: IssueswithControlInputSaturation"InProc.ofthe
imation,"AIAAJ.ofGuidance,ControlandDynamics,ac-
2003 American Controls Conference.
cepted for publication Jan. 2003.
[14] R. T. Rysdyk, Adaptive Nonlinear Flight Control,
[7] E. Johnson and A. Calise. \Neural network adaptive
Ph.D. Thesis, Georgia Institute of Technology, School of
controlofsystemswithinputsaturation,"InAmericanCon-
Aero. Eng., Atlanta, GA, 1998.
trol Conference, June 2001.
[15] S. N. Singh and M. Steinberg, \Adaptive Control of
[8] M.Krsti¶c,I.Kanellakopoulos,andP.Kokotovi¶c,Non-
Feedback Linearizable Nonlinear Systems with Application
linearandAdaptiveControlDesign,NewYork,JohnWiley,
to Flight Control," J. of Guid, Control, and Dynamics,
1995.
19(4): 871{877, 1996.
[9] S.P.K¶arasonandA.M.Annaswamy.\Adaptivecon-
[16] B. L. Stevens and F. L. Lewis, \Aircraft Control and
trol in the presence of input constraints," IEEE Trans. on
Simulation," John Wiley, 1992.
Automatic Control, Vol. 39 (11): 2325{2330, 1994.
[17] H.WangandJ.Sun,\Modifledmodelreferenceadap-
[10] R.Monopoli.\Adaptivecontrolforsystemswithhard
tivecontrolwithsaturatedinputs,"InConf.onDecisionand
saturation," In IEEE Conf. on Decision and Control, pp.
Control, pp. 3255{3256, 1992.
841{843, 1975.
[18] D. G. Ward, J. F. Monaco and M. Bodson, \Devel-
[11] M. Polycarpou and P. Ioannou, \A Robust Adaptive
opment and (cid:176)ight testing of a parameter identiflcation al-
NonlinearControlDesign,"Automatica,vol.32(3): 423-427,
gorithm for reconflgurable control," AIAA J. of Guidance,
1996.
Control, and Dynamics, Vol. 21(6): 948{956, 1998.
[12] M. Polycarpou and M. Mears, \Stable Adap-
tive Tracking of Uncertain Systems Using Nonlinearly
Table 1: Simulation Parameters: The flrst row of cells deflnes the parameters of the model. The second and third rows of
cells deflne controller parameters. The fourth row of cells deflnes the state constraints and saturation limits.
Parameter: L L M M M
o fi o Q –
Value: -0.1 1.0 0.1 -0.02 1.0
Parameter: ¡1 ¡2 ¡3 ¡4 ¡5
Value: 0.4 16.0 4.0 20.0 30.0
Parameter: k k k
(cid:176) fi Q
Value: 1.3 3.0 30.0
Parameter: fiL fiU Q limits – limits
c c c c
Value: -8.0 deg 15 deg § 15 deg § 45 deg
s
2 1
1
g, degtilde−01 g, degbar−01
−2 −2
0 50 100 150 0 50 100 150
time, s time, s
4 4
, degtilde02 , degbar02
a a
−2 −2
0 50 100 150 0 50 100 150
time, s time, s
1 1
Q, deg/stilde−00..055 Q, deg/sbar−00..055
−1 −1
0 50 100 150 0 50 100 150
time, s time, s
(a) Trackingerrorsx~. (b) Trackingerrorsx„.
Figure 2: Plots of the aircraft tracking errors for (cid:176) 2§5 deg.
r
meter error00..34 Lo−0.01
m of para00..12 −0.220 50 100 150
Nor 00 50 100 150 La1.5
0.08 1
Norm of state error000...000246 Mo00..001200 5500 110000 115500
0
0 50 100 150
Mq−0.02
0.4
Lyapunov norm000...123 −0M.d041230 50 100 150
0 0 50 100 150
0 50 100 150 time, s
(a) Lyapunovquantities. (b) Parameterestimates.
q
Figure 3: For (cid:176)r 2p§5 deg: Left top - Plot of £~(t)T¡¡1£~(t) versus t. Left middle - Plot of kx„k versus t. Left bottom -
Plot of V(t) versus t. Right - Plots of each component of £^ versus t.
15 15 15 20
10 10 10 15
5 5 10
g, deg −05 a, deg 05 g, deg −05 a, deg 05
−10 −5 −10 −5
−15 −10 −15 −10
0 5 10 15 20 25 0 5 10 15 20 25 125 130 135 140 145 150 125 130 135 140 145 150
time, s time, s time, s time, s
10 50 20 50
5
10
Q, deg/s−−1005 d, deg 0 Q, deg/s 0 d, deg 0
−10
−15
−20 −50 −20 −50
0 5 10 15 20 25 0 5 10 15 20 25 125 130 135 140 145 150 125 130 135 140 145 150
time, s time, s time, s time, s
(a) Initialstateandcontrol (b) Finalstateandcontrol
Figure 4: Plotsoftheaircraftstateandcontrolatthebeginningandendofa150ssimulationfor(cid:176)r 2§10deg: magnitude
limited command (dotted), flltered command (solid), actual (dashed)
5 2
1
g, degtilde0 g, degbar−01
−5 −2
0 50 100 150 0 50 100 150
time, s time, s
10 6
a, degtilde05 a, degbar024
−5 −2
0 50 100 150 0 50 100 150
time, s time, s
10 4
Q, deg/stilde05 Q, deg/sbar02
−5 −2
0 50 100 150 0 50 100 150
time, s time, s
(a) Trackingerrorsx~. (b) Trackingerrorsx„.
Figure 5: Plots of the aircraft tracking errors for (cid:176) 2§10 deg.
r
meter error00..34 Lo−−0.00.51
m of para00..12 −0.1520 50 100 150
Nor 00 50 100 150 La1.5
Norm of state error0000....00002468 Mo0.000..5101200 5500 110000 115500
0
0 50 100 150
Mq 0
0.4
Lyapunov norm000...123 −0M.d051230 50 100 150
0 0 50 100 150
0 50 100 150 time, s
(a) Lyapunovquantities. (b) Parameterestimates.
q
Figure 6: For (cid:176)r 2p§5 deg: Left top - Plot of £~(t)T¡¡1£~(t) versus t. Left middle - Plot of kx„k versus t. Left bottom -
Plot of V(t) versus t. Right - Plots of each component of £^ versus t.