Reinforcement adaptive fuzzy control of wing rockphenomena

Z.L. Liu

Abstract: Wing rock phenomena is a limit cycle roll oscillation experienced by aircraft withslender delta wings at a high angle of attack. A new reinforcement adaptive fuzzy control scheme isproposed for suppressing or tracking wing rock phenomena. A fuzzy logic system is used toapproximate a unknown nonlinear dynamics in which a variable universe technique is applied tomodify the premise parameters of FLS for improving its interpolation precision, and the derivedadaptive law combining with a reinforcement-learning strategy is applied to tune its consequentparameters. The novelty of the proposed control scheme is mainly in its simplicity, robustness andstability so that it can be applied for online learning and real-time control. Three cases of time-varying wing rock control under disturbances are simulated to confirm the effectiveness androbustness of the proposed scheme.

1 Introduction

The phenomenon of wing rock is manifested by a limit cycleoscillation predominantly in roll about the body axis ofaircraft [1]. The control of many aircraft at a high angle ofattack (AOA) is limited by the wing-rock phenomenon.High-speed civil transport and combat aircraft can fly inconditions where this self-induced oscillatory rollingmotion is observed. Such oscillations lead to a significantloss in lift and can cause a serious safety problem duringmanoeuvres such as landing or takeoff [2].To suppress wing rock phenomena, several nonlinear

control schemes [1–6] have been proposed, based on theanalytical models from the free-to-roll test of 80� flat-platedelta wing. Although these control schemes can suppresswing rock, the tracking control performance may be poor ifthe uncertainty and unknown disturbance are considered.More importantly, this analytical model may not be suitablefor practical aircraft control. For example, recent free-to-rolltest results for four military aircraft models (AV-8B, F/A-18C, pre-production F/A-18E, and F-16C) at transonicconditions have shown that five types of rolling motions areobserved during the tests [7]. The test results indicate that anaircraft’s wing rock phenomenon is a complex, uncertain,and time-varying nonlinear system, and therefore its preciseanalytical modelling is unavailable. To guarantee therobustness and stability of the control system, a newreinforcement adaptive fuzzy control scheme is proposedfor suppressing or tracking the wing rock phenomenon.Fuzzy logic control (FLC) has recently emerged as one of

the most active and fruitful approaches to control complexnonlinear systems. The motivation is often due to the fact

that the knowledge and dynamic behaviour of systems arequalitative and uncertain, and the fuzzy set theory appears toprovide a suitable representation of such knowledge. In theFLC, the selection of appropriate membership functions isan important issue as a change of fuzzy membershipfunction may alter the performance of the fuzzy controllersignificantly.

In this study, the variable universe technique [8], basedon the interpolation mechanism of the fuzzy logic system(FLS), is applied to modify the premise parameters of FLSbut to keep fuzzy rules the same. The remarkable advantageof the variable universe fuzzy control is to use only a fewfuzzy rules but maintain a high approximate precision,especially suitable for online learning and real-time control.

For the consequent parameters of FLC, the adaptivecontrol law is derived to compensate for the uncertaindynamics of the wing rock phenomenon. The learningcapability of the adaptive control is achieved by employingan appropriate adaptation law to tune the parameters of thefunction approximator [9]. However, in most of the adaptivecontrol schemes the control input might become quite large,and they may be limited to constant disturbances [10].

To tune the consequent parameters of the FLC, the adaptivelaw is combined with a reinforcement-learning (RL) strategy[11, 12] to obtain fast andprecise online adaptive ability for thenonlinear systemwith unknowndisturbances. TheRLstrategycan be used to learn the unknown desired outputs by means ofthe controller receiving a reinforcement signal (reward orpunishment) according to the last action it has performed in theprevious state and then the controller adjusting itself withsuitable evaluation of its performance. The proposed controlalgorithm is derived by the Lyapunov theorem to ensure thestability of the closed-loop system and the convergence of thetracking error. In particular, the uniformultimate boundedness(UUB) of the tracking error with this controller is guaranteed.

2 Preliminaries

2.1 FLS

An FLS is mainly concerned with imprecision andapproximate reasoning. As we know, the fuzzy inferenceengine uses the fuzzy IF–THEN rules to perform a mapping

q IEE, 2005

IEE Proceedings online no. 20045072

doi: 10.1049/ip-cta:20045072

The author is with the Department of Mechanical Engineering, ConcordiaUniversity, 1455 de Maisonneuve Blvb. W., Montreal, Quebec, Canada,H3G 1M8

E-mail: liu_22001@yahoo.com

Paper first received 5th June and in revised form 2nd November 2004

IEE Proc.-Control Theory Appl., Vol. 152, No. 6, November 2005 615

from an input linguistic vector x ¼ ðx1; . . . ; xnÞ 2 U ¼U1 � � � � � Un � Rn to an output variable y 2 V � R.

The ith fuzzy rule is written as

Ri : IF xi is Ai1 and . . . and xn is Ai

n THEN y is wi ð1Þwhere Ai

j is a fuzzy variable and wi is a singleton number.If the fuzzifier is a singleton, the inference engine adopts

a minimum operator and the defuzzifier is a centre average,then the fuzzy system can be formulated by

yðxÞ ¼PN

i¼1 wimiPNi¼1 mi

¼ WTFðxÞ ð2Þ

where mi ¼ minðmi1ðx1Þ; . . . ; mi

nðxnÞÞ; mij is the membership

function value of fuzzy variable xj; N is the number of fuzzy

rules, WT ¼ ½w1 w2 � � � wN � is an adjustable conse-quent parameter vector, and F ¼ ½F1 F2 � � � FN �T ;where Fi ¼ mi=

PNi¼1 m

i is a fuzzy basis function. For thiscase, [13] proves the theorem of universal approximators.

2.2 Variable universe technique

Let Uj ¼ ½�Lj Lj� be the universe of input variable xj inwhich Lj is a constant determined by designers according tothe change range of xj: The variable universe technologybased on an interpolation mechanism of FLS is used tomodify the premise parameter Lj of the fuzzy approximator,but the fuzzy rules stay the same. A so-called variableuniverse means that some universe Uj can change alongwith changing of variable xj; denoted by [8]

�UUj ¼ ½�ajðxjÞLj ajðxjÞLj � ð3Þ

where aj>0 is called a contraction–expansion factor of theuniverse �UUj: The original universe Uj is naturally called theinitial universe.

Assume that the input membership functions are simpletriangles. We let P and N be fuzzy subsets of the inputvariable xj and define its membership functions mPðxjÞ andmNðxjÞ as

mPðxjÞ ¼0 xj < � Lj

ðxj þ LjÞ=ð2LjÞ �Lj � xj � Lj

1 xj>Lj

8><>: ð4Þ

mNðxjÞ ¼ 1� mPðxjÞ ð5Þ

The following contraction–expansion factor in this paper issuggested:

aiðxiÞ ¼ ei þ ðjxij=LiÞti ð6Þwhere ei is a very small constant for computing efficiency,for example, taking e1 ¼ 0:001; 0:1< ti < 1 is suggested.Reference [8] also proved several theorems to show theconvergence of this algorithm. Figure 1 shows the member-ship function (4) and illustrates the above variable universeidea.

2.3 Nonlinear systems

Consider second-order nonlinear systems of the form

€xxðtÞ ¼ f ðxÞ þ uðtÞ þ dðtÞyðtÞ ¼ xðtÞ

ð7Þ

where f(x) is an unknown continuous function, xðtÞ ¼½x1ðtÞ x2ðtÞ�T ¼ ½xðtÞ _xxðtÞ�T 2 R2 is the state vector of

the system, d(t) is an unknown disturbance, uðtÞ 2 R andyðtÞ 2 R are the input and output of the system, respectively.It is assumed that f(x) and d(t) have upper bounds �ff ðxÞðtÞand �dd; respectively, i.e. j f ðxÞj � �ff ðxÞ and jdðtÞj � �dd:

3 Controller design

The control objective in this paper is to use an adaptivefuzzy control method combining with RL strategy such thatthe system output x(t) can track the desired trajectory xdðtÞunder the condition of the uncertainties and unknowndisturbances.

Denote the tracking error e(t) and evaluating performancesignal s(t) as

eðtÞ ¼ xdðtÞ � xðtÞ ð8Þ

sðtÞ ¼ _eeðtÞ þ lðeÞeðtÞ ð9Þwhere lðeÞ ¼ kl=ðjej þ elÞ; kl is a given positive constant,and el is a small constant, for example, let el ¼ 0:2. Notethat the idea of varying lðeÞ is similar to tuning the slope ofthe sliding surface [14]. The larger it is, the faster will thesystem response be, but a too large value of lðeÞ can causeovershoot, or even instability. It would, therefore, beadvantageous to vary the slope adaptively in such a waythat the slope is increased as the magnitude of the error getssmaller.

It is typical to define an evaluating performance signals(t) as a performance measurement, and when theperformance signal s(t) is small, the system performanceis better; thus, differentiating s(t), with (8) and (9), thenonlinear system (7) can be expressed as

_ssðtÞ ¼ gð�xxÞ � uðtÞ � dðtÞ ð10Þwhere the unknown function gð�xxÞ is given by

gð�xxÞ ¼ €xxdðtÞ þ lðeÞ_eeðtÞ þ _llðeÞe ðtÞ � f ðxÞ ð11Þwhere �xx ¼ ½€xxd eðtÞ xðtÞ�T :

Assume that the terms lðeÞ_eeðtÞ þ _llðeÞeðtÞ � f ðxÞ in (11)can be represented by an ideal fuzzy approximator termWTFðxÞ as follows:

gð�xxÞ ¼ €xxdðtÞ þ lðeÞeðtÞ þ WTFðxÞ þ eðxÞ ð12Þwhere eðxÞ is an approximation error, WT ¼½w1 w2 � � � wn� is an ideal weight matrix, and FðxÞis a fuzzy base function. If WT can be estimated by WW

T; the

N

N

0

0

1

1

P

P

–Lj –Lj

0.5

0.5

m

m

xj

xj–ajLj ajLj

contractinguniverse

expandinguniverse

Fig. 1 Contracting–expanding universe Li

IEE Proc.-Control Theory Appl., Vol. 152, No. 6, November 2005616

function gð�xxÞ can be identified by the following terms:

ggð�xxÞ ¼ €xxdðtÞ þ WWTFðxÞ ð13Þ

Define the control law

uðtÞ ¼ KsðtÞ þ ggð�xxÞ � ddðtÞ ð14Þ

where the positive K represents the fixed gain in the overallcontrol scheme, the output vector of ggð�xxÞ estimates gð�xxÞ andddðtÞ; to be derived later, is the robustifying term to attenuatedisturbances. We should note that the first term Ks(t) in (14)is similar to a proportional and derivative (PD) controller,which maintains the system stability; thus, weight valuescan be initialised to be zero. This means that there is no needfor offline learning or a trial-and-error phase. Figure 2 showsthe architecture of the closed-loop control system.Substituting (14) into (10), we obtain the closed-loop

dynamics as

_ssðtÞ ¼ �KsðtÞ þ ~ggð�xxÞ þ ddðtÞ � dðtÞ ð15Þ

where the estimation error ~ggð�xxÞ is denoted as

~ggð�xxÞ ¼ gð�xxÞ � ggð�xxÞ ¼ ~WWTFðxÞ þ eðxÞ ð16Þ

with ~WW ¼ W � WW and the robustifying term is given by

ddðtÞ ¼ �ddsign�sðtÞ�

ð17Þ

As we see, (15) implies that the overall system is driven bythe estimation error ~ggð�xxÞ and disturbance compensationerror ½ddðtÞ � dðtÞ�:To achieve fast and precise tracking of an uncertain

nonlinear system, the incomplete experts knowledge maynot be enough. RL strategy is used in the adaptive fuzzycontrol scheme. In RL, an internal evaluator called thecritic, which can predict the future system performance, isused. This prediction is needed to obtain an internalreinforcement, which can be used to adapt the critic andthe controller [12, 15].We assume a simple reinforcement signal in this paper to

be

rðtÞ ¼ sðtÞ ð18Þ

where r(t) is used to update the weight vector WW :Before the stability analysis, we have two assumptions:

(a) kWkF � Wm with a known Wm (the Frobenius norm is

defined by kWkF ¼ ½trðWT WÞ�1=2 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP

i; j w2ij

q), and (b)

the approximate error and disturbance compensation error isbounded, i.e. jeðxÞ þ dd � dj � eb:

Theorem: If the weight tuning law provided for the fuzzyapproximator is

_WWWW ¼ rðtÞKwFðxÞ � �KwjsðtÞjWW ð19Þ

where Kw is a positive and diagonal constant matrix, � is apositive constant, and the r(t) given by (18) is thereinforcement signal, then the control input u(t) given by(14) guarantees that the s(t) and WW are uniformly ultimatelybounded (UUB).

Proof: Define the Lyapunov function candidate

VðtÞ ¼ 12s2 þ 1

2tr ~WWT K�1

w~WW

� �ð20Þ

where ~WW ¼ W � WW : Evaluating the time derivative of V(t)along the trajectories of the weight tuning laws (19) yields

_VVðtÞ ¼ s_ss þ tr ~WWTK�1

w_~WW~WW

n oð21Þ

With (15) and (17)–(19), noting that _~WW~WW ¼ _WW � _WWWW with_WW ¼ 0; (21) can be rewritten as

_VVðtÞ ¼ sð�Ks þ ~WWTFþ dd � dÞ� tr ~WWT K�1

w ðrKwF� �KwjsjWWÞ� �

¼ �Ks2 þ s ~WWTFþ seb � trfr ~WWTF� �jsj ~WWT WWg¼ �Ks2 þ seb þ �jsjtrf ~WWT WWg

ð22Þ

Applying the matrix inequalities ~WWTWW � k ~WWkFWm �

k ~WWk2F; (22) becomes

_VVðtÞ � �Ks2 þ seb þ �jsj k ~WWkFWm � k ~WWk2F� �

� �Ks2 � �jsj ðk ~WWkF � Wm=2Þ2 � eb=� � W2m=4

� �ð23Þ

Therefore, _VVðtÞ is guaranteed to be negative as long as either(24) or (25) holds:

fuzzycontroller

plantKKs

d

–d

e, e S u(t) e + le

g(x)xd

xd

x

x

dsign(s)

Fig. 2 Proposed control scheme

Fig. 3 Coefficients aiðtÞ against AOA (deg) in (27)

IEE Proc.-Control Theory Appl., Vol. 152, No. 6, November 2005 617

jsj � eb þ �W2m=4

� �=K ¼ Bs ð24Þ

k ~WWkF � Wm=2þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffieb=� þ W2

m=4q

¼ Bw ð25Þ

Clearly, from inequalities (24) and (25), we can define acompact set O : O ¼ fðjsj; k ~WWkFÞj0 � jsj � Bs and k ~WWkF

� Bwg: By the Lyapunov theory, for sðt0Þ and ~WWðt0Þ 2 O;there exists numbers T(Bs; Bw; sðt0Þ; ~WWðt0ÞÞ such that 0 �jsj � B and k ~WWkF � Bw for all t � t0 þ T: In other words,_VVðs; ~WWÞ is negative outside the compact set O; and then sand ~WW are UUB.

Finally, for the practical implementation of the tuningrule, the weight updating (19) can be converted into adiscrete form

WWTðn þ 1Þ ¼ WWTðnÞ þ t½KwrðnÞFðnÞ � �KwjsðnÞjWWTðnÞ�¼ WWTðnÞð1� t�KwjsðnÞjÞ þ tKwrðnÞFðnÞ

ð26Þwhere t is the sample time.

4 Simulations of wing rock control

4.1 Wing rock model

For verifying proposed controller, we use the wing rockmodel given by [6]

€ffþ a0fþ a1_ffþ a2j _ffj _ffþ a3f

3 þ a4f2 _ff ¼ uðtÞ ð27Þ

where fðtÞ is the roll angle, u(t) is the control input, andaiðtÞði ¼ 0; 1; 2; 3; 4Þ are the time-varying parameters to berelative to free-to-roll experiment conditions [16]. A typicalset of coefficients aiðtÞ (at Reynolds number ¼ 636 000) isillustrated in Fig. 3. A cubic interpolation function is used tostructure a time-varying wing rock model, which is apiecewise constant model with interpolation per 0:5� from25 to 45�: Figure 4 demonstrates the time history of the time-varying wing rock where we assume the initial conditions tobe fð0Þ ¼ 30� and _ffð0Þ ¼ 0; and the run time to be 2000time steps.

If we consider the model (27) with the disturbance d(t),(27) is then modified as

€ff ¼ �ða0fþ a1_ffþ a2j _ffj _ffþ a3f

3 þ a4f2 _ffÞ

þ uðtÞ þ dðtÞ ð28Þ

0

0

0

–1

–2

–3

–4

–5

0

–1

–2

–3

–4

–5

0

–1

–2

–3

–4

–5

10 20 30 40

–2

–4

–6

–8

6

4

2

0

–2

–4

–6

–8

0 10 20 30 40 50

0

–1

–2

–3

–4

–50 10 20 30 40 50

q,deg

q.

time stepsu

(t)

a

c

b

W1(

t)W

2(t)

W3(

t)W

4(t)

time steps

Fig. 5 Simulation results of wing rock suppression

a Control response on phase plane (_yy; rad/sec against y; deg)b Control input u(t)c Weight updating WWT ðtÞ ¼ ½ww1 ww2 ww3 ww4�Fig. 4 Time-varying wing rock behaviour

IEE Proc.-Control Theory Appl., Vol. 152, No. 6, November 2005618

Fig. 7 Simulation results for wing rock tracking xd ¼ �20�

a Control response on phase plane (_yy; rad/sec versus y; deg)b Control response (y; deg versus time steps)c Weight updating WWT ðtÞ ¼ ½ww1 ww2 ww3 ww4�

40

20

0

0 10 20 30 40 50

–20

–40

Xd

X

q (t

)

time stepsa

c

time stepsb

15

10

5

0

0 10 20 30 40 50

–5

–10

–15

–20

u(t)

10

5

0

–5

–10

W1(

t)W

3(t)

W4(

t)

10

5

0

–5

–1010

5

0

0 10 20 30 40 50

–5

–10

W2(

t)

10

5

0

–5

–10

time steps

Fig. 6 Simulation results for wing rock tracking xd ¼ 40�

sin 0:05pt

a Control response (y (deg) against time steps)b Control input u(t)c Weight updating WWT ðtÞ ¼ ½ww1 ww2 ww3 ww4�

IEE Proc.-Control Theory Appl., Vol. 152, No. 6, November 2005 619

Comparing (28) with (7), we know f ðxÞ ¼�ða0fþ a1

_ffþ a2j _ffj _ffþ a3f3 þ a4f

2 _ffÞ. For numericalsimulation purposes, (28) is regarded as an aircraft’s wingrock model.

4.2 Simulation results

To evaluate the robustness of the proposed control scheme,we assume the disturbance to be dðtÞ ¼ 2 sin 2pt þ sin 5ptþ1:5; which is added in the system at t ¼ 20 time steps forall cases. The initial condition we choose is fð0Þ ¼ 40� and_ffð0Þ ¼ 0: The parameters used in this simulation are L1 ¼0:7 rad; L2 ¼ 1 rad=s; e1 ¼ e2 ¼ 0:001; and t1 ¼ t2 ¼ 0:9in (6), kl ¼ 10 and el ¼ 0:2 for solving lðeÞ; and t ¼ 0:01;Kw ¼ K ¼ 50; and � ¼ 0:08 in (26).

The numerical simulations are implemented in Matlaband Simulink environments.

Case 1: Wing rock suppressionFigure 5a shows the output response of wing rock suppressionon the phase plane. The results show that the proposed controlscheme can suppress wing rock quickly and can maintain thestates at the equilibrium position with an error 0< e< 0:14�;even though the time-varying system undergoes disturbances.Figure 5b shows the corresponding control input, and Fig. 5cshows the four stable adaptive weights over time.

Case 2: Tracking control for xdðtÞ ¼ 40� sin 0:05ptTo verify the tracking performances of the proposedcontrol scheme, we assume the time-varying trajectoryxdðtÞ ¼ 40� sin 0:05pt: Figure 6a shows satisfactory per-formance where the output roll angle follows the desiredtrajectory almost simultaneously. If the external disturbanceis added in the time-varying system at t ¼ 20 time steps, thecontrol output keeps almost the same performance with anerror jej< 0:15�: Figure 6b shows the corresponding controlinput. The weight updating is shown in Fig. 6c, in which weobserve that the four weights with respect to the four fuzzyrules alternately require tuning when the system states enterthe different phases.

Case 3: Tracking a constant trajectory xdðtÞ ¼ �20�

In fact, tracking a constant trajectory for wing rock motioncontrol usually results in a big tracking error. Figure 7ashows the output response of tracking xdðtÞ ¼ �20� on thephase plane; Fig. 7b shows that if the disturbance is added inthe system at t ¼ 20 time steps, the proposed control canreject disturbances scheme and can maintain the trackingerror in the small range jeðtÞj< 0:23�. Figure 7 shows thecorresponding weights updating.

5 Conclusions

A new reinforcement adaptive fuzzy control scheme isproposed for suppressing or tracking the wing rock

phenomenon in aircraft in which the uncertainty andunknown disturbance are considered. The fuzzy approx-imator is used to identify the unknown nonlinear function.The variable universe technique is applied to modify thepremise parameters of the approximator. Its consequentparameters can be tuned by the reinforcement adaptive law,which is the online adaptive algorithm derived fromLyapunov stability theory. Therefore, all the signals in theclosed-loop system are bounded even in the presence ofuncertainties and disturbances. Simulation results confirmthe effectiveness and robustness of the proposed scheme.Even though the fuzzy approximator uses only four fuzzyrules, it can achieve precise tracking. Moreover, if thesystem contains unknown disturbances, it can still maintainhigh tracking performances.

6 References

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