Nonlinear Dyn (2015) 81:789–799DOI 10.1007/s11071-015-2029-x

ORIGINAL PAPER

UDE-based linear robust control for a class of nonlinearsystems with application to wing rock motion stabilization

Alon Kuperman · Qing-Chang Zhong

Received: 2 September 2014 / Accepted: 13 March 2015 / Published online: 25 March 2015© Springer Science+Business Media Dordrecht 2015

Abstract In this paper, robust UDE-based linear con-trol strategy is developed for a class of second-ordernonlinear uncertain systems with disturbances. Afterrevealing the control structure and performing stabil-ity analysis, the robust control strategy is used to sup-press wing rock motion. The approach is based on theuncertainty and disturbance estimator,which calculatesand robustly cancels system uncertainties and inputdisturbances via appropriate filtering. The informationregarding the known part of the linear plant dynamicsis used by the controller; the terms containing the non-linear functions are treated as additional uncertaintiesto the system. The algorithm provides excellent per-formance in suppressing the wing rock dynamics andrejecting disturbances. Simulations are given to showthe effectiveness of the strategy via an application toan experimentally derived delta wing rock model.

Keywords Wing rock · Uncertainty and disturbanceestimator (UDE) · Robust control · Nonlinear systems

A. Kuperman (B)Department of Electrical Engineering and Electronics,Ariel University, Ariel, Israele-mail: alonku@ariel.ac.il

Q.-C. ZhongDepartment of Electrical and Computer Engineering,Illinois Institute of Technology, Chicago, USAe-mail: zhongqc@iit.edu

1 Introduction

The uncertainty and disturbance estimator (UDE) strat-egy introduced in [26] is able to quickly estimate uncer-tainties and disturbances and thus provides excellentrobust performance. It is based on the assumption thata continuous signal can be approximated as it is appro-priately filtered. The UDE-based control strategy hasbeen successfully extended to uncertain systems withdelays in [11,22]. Recently, the two-degree-of-freedomnature of UDE controllers has been revealed in [25],which considerably facilitates the design of the con-troller. In [23], the UDE strategywas adopted to formu-late a robust input–output linearized controller, whichwas then applied to control the wing rock motion.

Lightly damped or undamped rolling oscillationsaround the longitudinal axis at moderate to high angleof attack (AOA) exhibited by several modern high-performance aircraft are commonly referred to as wingrock. Such dynamics typically possesses limit cycles,which become stable after a transient phase. In addi-tion to being highly annoying to the pilot, wing rockcan possess a deteriorating effect on an aircraft perfor-mance. For somecases,wing rock is an earlywarningofimminent departure or spin entry. For other cases, theseverity of wing rock could create inertial and kine-matic coupling to cause AOA excursions and lead toloss of control. Handling qualities are clearly com-promised in addition to degradation of maneuveringcapabilities in terms of the maximum achievable AOA[3,4].

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790 A. Kuperman, Q.-C. Zhong

Two different types of wing rock have been men-tioned in the literature. The first type of wing rock isusually associated with low-airspeed, high-AOA flightin gusty conditions and characterized by unsteady lat-eral motions at moderate to high AOA. These motionsshow small-amplitude intermittent non-periodic rolloscillations, assumed to be a function of pilot–vehicleinteraction. Flight procedures can be changed to avoidthis type of wing rock without significantly affectingmission completion. The second type of wing rock ischaracterized by very large changes in roll angle andnormally associated with high-AOAmaneuvering suchas in close-in air combat. For example, if a combataircraft is incapable of tracking a target due to wingrock, significant mission accomplishment degradationis obvious. Moreover, the presence of wing rock duringthe approach or landing phase can have severe influenceon the operational safety of the aircraft. From the sta-bility point of view, wing rock phenomenon arises froma nonlinear aerodynamic mechanism and is associatedwith the nonlinear trend of roll damping derivatives,leading to hysteresis and sign changes of the stabilityparameters when increasing the AOA during aircraftmaneuvers [10,13]. Thewing rock is usually controlledby appropriate ailerons deflection, as shown in Fig. 1.Controlling wing rock is a challenge to both aircraftdesigners and control engineers.

Many researchers have tackled the problem of con-trolling wing rock motion. Adaptive feedback lin-earization approaches were proposed in [8,18], whilesuboptimal and optimal feedback algorithms were dis-cussed in [1,24]. Kalman filter-based control [19] wasshown to be a good candidate to solve the wing rock

Fig. 1 Aileron deflection influence on the roll motion, modifiedfrom http://en.wikipedia.org/wiki/Aileron

motion problem. Fuzzy [20,21], fuzzy adaptive [15]and fuzzy neural [14] approaches gained popularityduring the last decade. Neural network-based control[6,9] and wavelet adaptive backstepping approach [7]were also considered. Recently, several nonlinear con-trol algorithms were shown to be suitable for deal-ing with wing rock [2,16,17]. In this paper, linearUDE-based control strategy is developed for a classof second-order nonlinear uncertain systems with dis-turbances and is directly applied to solving the wingrock motion problem by stabilization, without goingthrough the input–output linearization.

The rest of the paper is organized as follows. InSect. 2, general UDE-based linear control law foruncertain nonlinear systems with disturbances is revis-ited, followed by a particular case of second-order non-linear systems. Wing rock modeling is described inSect. 3, followed by the application of linear UDE-based control to the wing rock problem in Sect. 4. Con-clusions are drawn in Sect. 5.

2 Linear UDE-based control for nonlinear systems

2.1 General case

Consider a nonlinear systemwith uncertainties and dis-turbances described as

x(t) = Ax(t) + g(x(t), t) + Bu(t) + w(x(t),u(t), t)

+ f(x(t),u(t),d(t), t). (1)

Here, x = (x1, . . ., xn)T is the state vector, u =(u1, . . ., ur )T the control input vector, d(t) the unpre-dictable disturbances vector, A the known constantmatrix, B the known constant matrix, g(x, t) theknown smooth nonlinear function of the state vector,w[x(t),u(t), t] the known nonlinear function of thestate vector and control input and f[x(t),u(t),d(t), t]the unknown smooth nonlinear function of state vector,control input and unpredictable disturbances.

Linear reference model is chosen to indicate thedesired closed-loop specifications, as

xm(t) = Amxm(t) + Bmc(t) (2)

while the control objective forces the error e betweenthe states of the reference model and the states of thesystem

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Linear robust control for a class of nonlinear systems 791

e(t) = (e1(t), . . . , en(t))T = xm(t) − x(t)

= (xm1 − x1, . . . , xmn − xn)T (3)

to be stable and satisfy error dynamic equation givenby

e(t) = (Am + K) e(t), (4)

where K is an error feedback gain matrix with appro-priate dimensions, xm = (xm1, . . ., xmn)

T is the refer-ence state vector and c = (c1, . . ., cr )T is a piecewisecontinuous and uniformly bounded reference vector (inorder to simplify the exposition, the arguments of func-tions in the time domain are omitted hereafter). If thereferencemodel is chosen to be stable,Kmaybe chosenas 0; otherwise, if different error dynamics is desired orrequired to guarantee stability, common control strate-gies, e.g., pole placement, can be used to choose K. Itis worth noting that the dimension of c does not have tobe the same as that of u. This provides more freedomfor the choice of Bm.

Combining Eqs. (1)–(4), the following holds,

Amx + Bmc − Ax − g − Bu − w − f = Ke. (5)

Hence, the control signal u needs to satisfy

Bu = Amx + Bmc − Ke − Ax − g − w − f . (6)

In case B is nonzero, it is possible to construct a lin-ear control law for the nonlinear system (1) to achievedesired performance defined by the referencemodel (2)as follows. Denote the terms including uncertainties,external disturbances and known nonlinear dynamicsin (6) as

h �= −g − w − f = −x + Ax + Bu (7a)

with latter equality derived from (1). Hence, theunknown dynamics and disturbances can be obtainedfrom known system dynamics and control signal. How-ever, it cannot be directly used to formulate a controllaw. The UDE control strategy proposed in [26] adoptsan estimation of this signal to construct control laws asfollows. Let g f (t) be the impulse response of a filtermatrixG f (s), whose pass band contains the frequencycontent of h [this requirement will be clarified follow-ing (11)]. Then, h can be accurately estimated from theoutput of the UDE as

hude = h ∗ g f , (7b)

where ‘*’ is convolution operator. Combining (6) with(7), replacing h with hude and rearranging, there is

Bu = Amx + Bmc − Ke − Ax + hude= Amx + Bmc − Ke − Ax

+ (−x + Ax + Bu) ∗ g f . (8)

This results in the following linear UDE control law,after using the Laplace transform:

U(s) = B+(− AX(s) + [AmX(s) + BmC(s)

−KE(s) − sX(s)G f (s)] (

I − G f (s))−1

),

(9)

where B+ = (BTB)−1BT is the pseudo-inverse ofB. Note that if sG f (s) is implementable, there isno need of measuring or estimating the derivative ofstates. Moreover, the control signal is independent onunknown dynamics and disturbances. Since u is anapproximate solution of (5), Eqs. (4) and (5) are notalways met, and when choosing the control parame-ters, the following structure constraint needs to be met:(I − BB+)

(Amx + Bmc − g − Ax − Bu − w

− f − Ke) = 0. (10)

Obviously, if B is square and invertible matrix, theabove structural constraint is always met. If not, thechoice of the reference model and the error feedbackgain matrix is restricted. The unknown dynamics anddisturbances also play a role in the above constraint. Asshown in [26], a system in the canonical form alwayssatisfies this constraint.

The Laplace transform of actual error dynamics isobtained by combining (2), (3) and (9) as

E(s) = − (sI − [Am + K])−1 (I − G f (s)

)H(s)

= Z(s)H(s), (11)

indicating that the signal h is attenuated twice: first bya low-pass filter (sI − [Am +K])−1 and then by a fre-quency selective filter I−G f (s), possessing high-passfilter properties if G f (s) is strictly proper (clarifyingthe above requirement of G f (s) pass band to containthe frequency content of h). Hence, the functions ofthese twofilters are decoupled in the frequency domain.High-frequency components of h are attenuated by thelow-pass filter, while its low-frequency components,after passing through the low-pass filter, are attenuatedby a high-pass filter. In order to assure that the actualsystem (1) follows the reference model (2), the right-hand side of (11) must be close to zero. This is satisfiedif Z(s) is close to zero at the frequency range whereH(s) is nonzero [25].

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792 A. Kuperman, Q.-C. Zhong

Assume that the frequency contents of signal hreside between ω1 and ω2 with 0 ≤ ω1 ≤ ω2 andin order to meet the requirement on the steady statetracking error, it is necessary that |E( jω)| ≤ δ|H( jω)|for |ω| ∈ [ω1, ω2] and δ > 0. Then, the following musthold when designing the controller,

|Z( jω)| =∣∣∣( jωI − [Am + K])−1

∣∣∣∣∣I − G f ( jω)

∣∣≤ δ, |ω| ∈ [ω1, ω2] . (12)

Once the desired error dynamics (4) is chosen, the fil-ter matrixG f (s) can be determined to satisfy this con-straint.

2.2 Second-order SISO nonlinear systems

Consider a particular case of (1), given by a second-order SISO nonlinear system with uncertainties anddisturbances described as

[x1x2

]=

[0 1

−a1 −a2

] [x1x2

]+

[0

g(x1, x2)

]+

[0b

]u

+[

0w(u)

]+

[0

f (x1, x2, u, d)

], (13)

wherea1 anda2 are knownpositive constants, g(x1, x2)is known smooth nonlinear function of the states,w(u)

is known nonlinear function of the control input andf (x1, x2, u, d) is the unknown smooth nonlinear func-tion of states, control input and unpredictable distur-bance d. Selecting reference model and error feedbackgain matrix as

[xm1

xm2

]=

[0 1

−am1 −am2

] [xm1

xm2

]+

[0bm

]c (14)

and

K =[k11 k12k21 k22

], (15)

respectively, choosingG f (s) as the following diagonalmatrix of general first-order filters [5,12],

Gf (s) =[

(1−ε1)T1s+1T1s+1 0

0 (1−ε2)T2s+1T2s+1

], (16)

with 0 < ε1, ε2 < 1, and taking into account theLaplace transform of (14), the control law is derivedfrom (9) as

U (s) = b−1

⎛⎜⎜⎜⎜⎜⎝

(a1ε2−am1+k21

ε2+ −am1+k21

ε2T2· 1s

)X1(s)+(

a2ε2−am2+k22−T−12

ε2+ −am2+k22

ε2T2· 1s + −(1−ε2)

ε2· s

)X2(s)+

s2+(a2−k22)s+a1−k21s2+a2s+a1

(bmε2

+ bmε2T2

· 1s

)C(s)

⎞⎟⎟⎟⎟⎟⎠

= b−1((

KP1 + KI1 · 1s

)X1(s)

+(KP2 + KI2 · 1

s+ KD2 · s

)X2(s)

+HC (s)

(KPC + KIC · 1

s

)C(s)

). (17)

According to (17), state feedbacks x1 and x2 undergoPI and PID filtering, respectively, while the referencecommand c passes through a dual-stage network (pre-filter), consisting of a frequency selective filter HC anda PI filter. The overall control structure is depicted inFig. 2. Note that in case the error gainmatrix is selectedas zero, HC (s) = 0 and c undergoes PI filtering only.Moreover, if ε2 = 1, then PID filter of x2 turns into aPI filter. Note that (1,1) entry ofG f (s) and the first rowof K are irrelevant and may be left as zeros.

3 Modeling of the wing rock motion

The wing rock motion can be described by the follow-ing one-degree-of-freedom nonlinear differential equa-tion [3,16]

φ = u − I−1xx Tr , (18)

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Linear robust control for a class of nonlinear systems 793

c1x2x1b−

( )CH s ( )CPI s

1( )PI s

2 ( )PID s

Eq. (13)

d

Fig. 2 Equivalent closed-loop system structure

where φ [rad] is the roll angle, Ixx [N · m · s2 · rad−1]is the moment of inertia about the roll axis, u [N · m]is the control action created by the ailerons deflectionand Tr [N · m] is the rolling torque evaluated as

Tr = q · S · w · Cr , (19)

where q is the dynamic pressure, S is the wing surfacearea, w is the wing span and Cr is the rolling momentcoefficient. The dynamic pressure is given by

q = 1

2ρv2, (20)

where ρ is the air density and v is the airspeed. Thefollowing expression of the rolling moment coefficientwas experimentally identified in [3,4],

Cr = b0φ + b1φ + b2φ∣∣φ∣∣ + b3φ3 + b4φφ2, (21)

where b0 − b4 are nonlinear functions of aircraft AOAand Reynolds number (Re), which is related to the air-speed as

Re = ρvL

μ(22)

with L and μ being the wing root chord and air viscos-ity, respectively. Substituting (19)–(22) into (18), thewing rock motion can be described as

φ + a0φ + a1φ + a2φ∣∣φ∣∣ + a3φ

3 + a4φφ2 = u (23)

with ai = I−1xx ·q ·S ·w·bi , i = 0, . . . , 4,which are also

nonlinearly dependent on AOA and Re as well as onthe aircraft dimensions and environmental conditions.They are normally determined experimentally.

When there is a disturbance d (e.g., wind gust)acting additively to the input u and there are alsomodel parameters uncertainties in the form ai =

ain + �ai , i = 0, . . . , 4, the wing rock dynamics(23) can be reformulated into the form of system

(1) with x = (φ φ

)T,A =

[0 1

−a0n −a1n

], g =

(0 −a2nφ

∣∣φ∣∣ − a3nφ3 − a4nφφ2)T = (

0 g2)T

,b =(0 1

)T, f = (

0 − �a0φ − �a1φ − �a2φ∣∣φ∣∣

− �a3φ3 − �a4φφ2 + d)T = (

0 f2)T

.

Consider an aircraft with an 80◦ deltawingwith L =479mm and w = 169mm. According to the free-to-roll experiments described in [4] on the delta wing forAOA = 25◦–45◦ and v = 15–40m s−1, correspondingto Re = 486000–1290000, the coefficients a0–a4 weredetermined and are shown in Fig 3. Obviously, theseparameters vary a lot.

It is well known that an uncontrolled wing rockmotion results either in limit cycle oscillations or inunstable divergence depending on the initial condi-tion [14]. This is shown in Fig. 4 for two differ-ent initial conditions. In the case of initial conditionsof x0 = (

35 · 10−3rad 0rad · s−1)T

, the wing rockmotion results in limit cycle oscillations as shown inFig. 3a. Step changes of the Reynold number and theAOA affect the oscillation characteristics but do notcause divergence. In the case of initial conditions of

x0 = (35 · 10−3rad 52 · 10−3rad · s−1

)T, the wing

rock motion results in roll angle divergence as shownin Fig. 4b. Hence, appropriate control strategies shouldbe developed to avoid these effects.

4 UDE-based stabilization of wing rock motion

Assume an external input disturbance d(t) =∑∞k=0 [1(t − 100k) − 1(t − 50 − 100k)] added to the

control input u(1(t) denotes unit step function) andthe steady state error constraint matrix is set to δ =(0.01 0.010.01 0.01

)for |ω| ∈ [

0, 20rad · s−1]. The ref-

erence model is chosen as the second-order system(φm

φm

)=

(0 1

−ω2m −2ξmωm

)(φm

φm

). Here, c =

(0 0

)Tbecause the control objective is to suppress the

wing rock dynamics and the desired steady state stable

operating point for the system states is x = (0 0

)T.

In order to suppress the wing rock within 1 sec with-out overshoot, the referencemodel is chosenwith ξ = 1andωm = 2π rad·s−1. The error feedbackgain is set to

123

794 A. Kuperman, Q.-C. Zhong

Fig. 3 Coefficients a0–a4 corresponding to different AOA and Re

K = 0, leading to the following low-pass filter matrix,

(sI − [Am + K])−1 =⎛⎝

s+4πs2+4πs+4π2

1s2+4πs+4π2

4π2

s2+4πs+4π2s

s2+4πs+4π2

⎞⎠

=(L11(s) L12(s)L21(s) L22(s)

). (24)

In general, all the entries of the high-pass filtermatrix I–Gf(s)maybeof different types and/or orders,and the matrix should be optimized in order to satisfythe given constraintwhile using theminimumresources(memory in case of digital implementation and passivecomponents in case of analog implementation). How-

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Linear robust control for a class of nonlinear systems 795

Fig. 4 Uncontrolled wing rock dynamics

Fig. 5 Bode diagrams ofappropriate entries of Z(jω)

ever, the discussion on the high-pass filter matrix topol-ogy is out of the present paper’s scope. Following (16),selecting the high-pass filter matrix of the form

I − G f (s) =(

ε1T1sT1s+1 0

0 ε2T2sT2s+1

)=

(Y11(s) 0

0 Y22(s)

),

(25)

the error dynamics (11) may be presented as(E1(s)E2(s)

)= −

(Z11(s) Z12(s)Z21(s) Z22(s)

) (0

H(s)

)

= −(Z12(s)Z22(s)

)H(s) (26)

where h(t)�= − f2(t)−g2(t). Hence, combining (24)–

(26), the following relationmust be simultaneously sat-isfied according to (12),( |Z12( jω)|

|Z22( jω)|)

=( |L12( jω)Y22( jω)|

|L22( jω)Y22( jω)|)

<

(0.010.01

), |ω| ∈ [0, 20] . (27)

Combining (24), (25) and (27) further reduces to

( |Z12( jω)||Z22( jω)|

)=

⎛⎝

1s2+4πs+4π2

ε2T2sT2s+1

ss2+4πs+4π2

ε2T2sT2s+1

⎞⎠

<

(0.010.01

), |ω| ∈ [0, 20] (28)

123

796 A. Kuperman, Q.-C. Zhong

Fig. 6 Operatingconditions adopted in thesimulations

Fig. 7 Simulation results

and is satisfied by choosing T2 < 3.1 × 10−3 s andε2 = 1. According to (17), the control law is

U (s) = −a0n X1(s) − a1n X2(s)

+(1 + 1

T2s

)(ω2

mX1(s) + 2ξmωmX2(s))

− 1

T2X2(s), (29)

where x1(t) = φ(t) and x2(t) = φ(t).

The control approach was verified by simulations.During the simulations, the controller was periodi-cally turned on and off to demonstrate the differencesbetween controlled and uncontrolledmotions; theUDEtime constant was set to T2 = 0.003. The appropriateentries of Z( jω) are shown in Fig. 5. It is clear thatmaximum values of both are below 0.01 (−40dB) andthe constraint (27) is satisfied.

123

Linear robust control for a class of nonlinear systems 797

Fig. 8 Zoomed simulationresults

123

798 A. Kuperman, Q.-C. Zhong

The operating conditions were changed during thesimulations, as shown in Fig. 6, in order to enhance thedemonstration of controller performance.

The system states were set to x0 =(0.44rad 0.26rad · s−1

)Tat t = 0 to create a diver-

gence. Then, each time the controller was turned off,and the system states were reset to x0 = (

0.44rad

0rad · s−1)T , bringing an oscillatory behavior. Note

that the stated initial conditions were chosen to allowgoodvisibility of the results. The systempossesses sim-ilar satisfactory performance for different initial con-ditions as well.

The simulation results are shown in Fig. 7, with thezoomed versions of the results presented in Fig. 8. Theproposed controller can stop the divergence as shownin Fig. 8a and oscillations as shown in Fig. 8b.

When the controller is turned on, the control actionconsists of two phases: The first is driving system state

vector to the desired stable operating point of(0 0

)Tand the second is counteracting external disturbance.Note that after the divergence/oscillation is suppressed,the controller keeps canceling the disturbance by feed-ing a signal equal in magnitude but opposite in sign.When the disturbance is removed, the control actionreduces to zero after the state is driven to

(0 0

)T. The

steady state performance of the system when the con-troller is ON is shown in Fig. 8c. Evidently, steady stateerrors are forced to zero after a transient.

In order to demonstrate filtering performance, thefrequency responses of steady state errors are shown inFig. 9 along with the Fourier transform of f. Obviously,the filtering performance is satisfactory as expected.The key component is definitely the UDE filter matrix,which quickly estimates the total uncertainty (seeFig. 10) and cancels it appropriately.

5 Conclusions

In this paper, the UDE-based control has been appliedto the problem of wing rock motion stabilization. Fol-lowing the modeling procedure of wing rock phenom-ena, an appropriate linear UDE-based control strat-egy has been developed. The proposed algorithm hasshown the ability of suppressing the wing rock oscilla-tions. Simulation results have demonstrated the effec-tiveness of the strategy via an application to an exper-imentally derived delta wing rock model in uncertain

Fig. 9 Frequency domain results

Fig. 10 Estimation of total uncertainty by UDE filter

environment with and without external disturbances.In addition, a novel UDE filter matrix was introduced,allowing choosing several filter types instead of onein order to optimize the system. Appropriate designguidelines were presented for the wing rock case. Itmay be concluded that the proposed method may berelated to the class of disturbance-observer-based con-trol algorithms, benefiting from robust performanceand minimum required information regarding the sys-tem under control. On the other hand, disadvantagessuch as increased control bandwidth requirement andpossible instability in case of wide-range parameterdrifts are present here as well. Detailed comparison toother techniques used to solve similar problem is leftas a future work.

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Linear robust control for a class of nonlinear systems 799

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