Hidden attractors in aircraft control systems with saturated inputs

B.R. Andrievsky,1, 2, 3, ∗ E.V. Kudryashova,2, † N.V. Kuznetsov,2, 4, ‡ O.A. Kuznetsova,2, § and G.A. Leonov2, 1, ¶

1Institute of Problems in Mechanical Engineering, the Russian Academy of Sciences, Saint Petersburg, Russia2Mathematics and Mechanics Faculty, Saint Petersburg State University, Russia

3ITMO University, St. Petersburg, Russia4Department of Mathematical Information Technology, University of Jyvaskyla, Jyvaskyla, Finland

In the paper, the control problem with limitations on the magnitude and rate of the control action in aircraftcontrol systems, is studied. Existence of hidden limit cycle oscillations in the case of actuator position and ratelimitations is demonstrated by the examples of piloted aircraft pilot involved oscillations (PIO) phenomenonand the airfoil flutter suppression system.

PACS numbers: 05.45.Xt, 05.45.GgKeywords: Nonlinear dynamics, Limit cycle oscillation, Hidden attractor, Pilot-involved oscillations, Saturated input, Aeroe-lasticity, Flutter

I. INTRODUCTION

In the paper, the control problem with limitations on themagnitude and rate of the control action in aircraft controlsystems, is studied. This problem has long attracted the at-tention of scientists and developers of flight control systems,is still a challenging one, without losing its relevance at thepresent time.

It is shown in the literature that in the motion of an aero-dynamically stable aircraft a stable limit cycle with a smallamplitude and an unstable one with a large amplitude can co-exist. If the aircraft is aerodynamically unstable, then one ofthe two stable limit cycles with a small amplitude can be re-alized. In addition, there is also an unstable limit cycle, thepresence of which makes it necessary to study the stability ofan aircraft with an automatic control system “in large”, i.e.when large perturbations (including those from the side of apilot) are applied to the aircraft, they can lead him beyond theamplitude of the unstable limit cycle.

An influence of non-linearities like a “saturation” in a pilot–aircraft loop is commonly treated as a possible origin of theso-called Pilot Involved Oscillation (PIO), which leads to se-rious degrade of the piloted aircraft performance, up to thestability loss and the aircraft destruction, cf. [1, 2]. This phe-nomenon is characterized by rapidly developing oscillationswith increasing amplitude at angular velocities, overloads andangular movements of the manned vehicle. The main non-linear factor leading to this phenomenon is, as a rule, the limi-tation of the speed of deviation of the aircraft control elements(aerodynamic control surfaces of the aircraft), which can leadto a delay in the response of the aircraft to the pilot’s com-mands. The study of transient regimes with such a motionleads to the need to develop a mathematical theory of globalanalysis of control systems of aircraft.

∗ boris.andrievsky@gmail.com† e.kudryashova@spbu.ru‡ nkuznetsov239@gmail.com§ olga.kuznetsova@spbu.ru¶ g.leonov@spbu.ru

As the results obtained, for studying the processes that canarise in nonlinear control systems of aircraft (including non-linear oscillations), simple computer modeling is an unreliabletool that can lead to wrong conclusions.

The remainder part of the paper is organized as follows.The analytical-numerical method for hidden oscillations lo-calization is briefly recalled in Sec. II. The adopted dynamicactuator model with saturations in the magnitude and rate isgiven in Sec. III. In Sec. III hidden oscillations in the pilot-aircraft loop (related to the PIO phenomenon) are studiedand localized by means of the iterative analytical-numericalmethod. Hidden oscillations in the unstable aircraft angle-of-attack control system with saturated actuator are demonstratedin Sec. IV. Section V is devoted to hidden oscillations in thepilot-aircraft loop. Hidden limit cycle oscillations in the air-foil flutter feedback suppression system are studied in Sec. VI.Concluding remarks and the future work intentions are givenin Sec. VII. Some background information is given in Ap-pendix VII.

II. ANALYTICAL AND NUMERICAL ANALYSIS OFOSCILLATIONS IN NONLINEAR CONTROL SYSTEMS

Further we consider a nonlinear control system with onescalar non-linearity in the Lur’e form

x = Px+qψ(rTx), x ∈ Rn, (1)

where P is a constant (n× n)-matrix, q,r are constant n-dimensional vectors, T denotes transpose operation, ψ(σ) is acontinuous piecewise-differentiable scalar function, ψ(0)= 0.

One of the main tasks of the investigation of nonlinear dy-namical models (1) is the study of established (limiting) be-havior of the system after the transient processes are over, i.e.,the problem of localization and analysis of attractors (limitedsets of system’s states, which are reached by the system fromclose initial data after transient processes). For numerical lo-calization of an attractor one needs to choose an initial pointin the basin of attraction and observe how the trajectory, start-ing from this initial point, after a transient process visualizesthe attractor. Thus, from a computational point of view, it

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is natural to consider the following classification of attractorsbeing either self-excited either hidden based on the simplicityof finding the basins of attraction in the phase space [3–6]:self-excited attractor has the basins of attraction touching anunstable stationary point, thus, can be revealed numericallyby the integration of trajectories1, started in small neighbor-hoods of the unstable equilibrium, while hidden attractor hasthe basin of attraction, which does not touch equilibria, andis hidden somewhere in the phase space [3–6]. For example,hidden attractors in systems (1) correspond to the case of mul-tistability when the stationary point is stable and coexist witha stable periodic orbit, or there are several coexisting periodicorbits. The classification of attractors as being hidden or self-excited was introduced by in connection with the discovery ofthe first hidden Chua attractor [7–13] and has captured muchattention (see, e.g. [14–43]).

A. Analysis of oscillations by the harmonic balance anddescribing function method

The describing function method (DFM) is a widely usedengineering method for the search of oscillations which areclose to the harmonic periodic oscillations. This method is notstrictly mathematically justified and is one of the approximatemethods of analysis of oscillations (see, e.g. [44–47]).

Let us recall a classical way of applying the DFM. Intro-duce a transfer function W (s) = rT

(P− sI

)−1q, where s is acomplex variable, I is a unit matrix. In order to find a periodicoscillation, a certain coefficient of harmonic linearization k isintroduced in such a way that the matrix P0 = P+ kqrT has apair of pure-imaginary eigenvalues ±iω0(ω0 > 0). The num-bers ω0 > 0 and k are defined by

ω0 : ImW (iω0) = 0, k =−(ReW (iω0)

)−1. (2)

If such ω0 and k exist, then system (1) has a periodic so-lution x(t) for which σ(t) = rTx(t) ≈ a0 cosω0t. Considerϕ(σ) = ψ(σ)− kσ , then, following the DFM, the amplitudea0 of oscillation can be obtained from the equation

Φ(a) =∫ 2π/ω0

0ϕ(acos(ω0t)

)cos(ω0t)dt, Φ(a0) = 0, (3)

where Φ(a) is called a describing function. Assume that r =(1,r2..,rn), then one can use the following initial data

x(0) = (a0,0, ...,0) (4)

for numerical localization of an attractor. However it is knownthat classical DMF can suggest the existence of non-existingperiodic oscillation (see, e.g. [48, 49]) and initial data (4) doesnot necessarily lead to the localization of an attractor.

1 Remark that in numerical computation of trajectory over a finite-time in-terval it may be difficult to distinguish a sustained oscillation from a tran-sient oscillation (a transient oscillating set in the phase space, which cannevertheless persist for a long time).

To get initial data in the basin of attraction of an attractor,we can use the following modification of the DFM [3]. Letus change ϕ(σ) by εϕ(σ), where ε is a small parameter, andconsider the existence of a periodic solution for system

x = P0x+ εqϕ(rTx). (5)

Consider a linear non-singular transformation2 x = Sy, suchthat system (5) is transformed to the form

y1=−ω0y2+εb1ϕ(y1+cT3y3), y2=ω0y1+εb2ϕ(y1+cT

3y3),

y3 = A3y3 + εb3ϕ(y1 + c∗3y3),(6)

where y1, y2 are scalars, y3, b3 and c3 are (n−2)-dimensionalvectors, b1 and b2 are scalars; A3 is a constant ((n−2)× (n−2)) matrix all eigenvalues of which have negative real parts.

Theorem 1 ([3]) If there exists a number a0 > 0 such that

Φ(a0) = 0, b1Φ′(a0)< 0, (7)

then system (5) has a stable periodic solution with initial data

x(0) = S(a0 +O(ε), 0, On−2(ε)

)T. (8)

Remark that there are known examples where DFM can-not reveal the existing oscillation (see, e.g. [3, 9, 50]). Forexample, well-known Aizerman’s and Kalman’s conjectureson the absolute stability of nonlinear control systems are validfrom the standpoint of DFM, while there are known variouscounterexamples of nonlinear systems where the only equilib-rium, which is stable, coexists with a hidden periodic oscilla-tion (see, e.g. [3, 9, 51–56]; the corresponding discrete exam-ples are considered in [57, 58]). In this case the DFM can bejustified and analog of Theorem 1 can be obtained for a spe-cial class of nonlinearities only [3]. For example [3, 52, 59],system with transfer function

W (s) =s2(

(s+0.03)2 +0.092)((s+0.03)2 +1.12

)has infinite sector of linear stability, but, e.g., for nonlinearityψ(σ) = sign(σ) a hidden attractor can be found numerically.

B. Localization of hidden oscillations by the continuationmethod and describing function methods

Consider an analytical-numerical procedure for the hiddenattractors localization based on the continuation method andDFM. For that we construct a finite sequence of functions

ϕj(σ) = ε

jϕ(σ), ε

j = j/m or εj = m/ j, j = 1, ..,m

such that system (1) with initial function ϕ1(σ) has a non-trivial attractor A 1, which either is self-excited and can be

2 Such transformation exists for non-degenerate transfer functions

3

visualized by the initial data from small vicinity of one of theequilibria either can be visualized by the initial data (8). Oneach next step of the procedure (i.e. for j > 2), the initial pointfor a trajectory to be integrated is chosen as the last point ofthe trajectory integrated on the previous step. Following thisprocedure and sequentially increasing j, two alternatives arepossible: the points of A j are in the basin of attraction of at-tractor A j+1, or while passing from system (1) with the func-tion ϕ j(σ) to system (1) with the function ϕ j+1(σ), a lossof stability bifurcation is observed and attractor A j vanishes.If, while changing j from 1 to m, there is no loss of stabilitybifurcation of the considered attractors, then an attractor forthe original system with ϕ(σ) (at the end of the procedure) islocalized.

III. MODELING DYNAMICAL ACTUATOR WITHSATURATIONS

In the linear settings, an aircraft actuator is usually modeledby the second-order differential equation as

δ (t)+2ξactωactδ (t)+ω2actδ (t) = ω

2actu(t), (9)

or, in the Laplace transform representation, as

Wact(s) ={

δ

u

}=

ω2act

s2 +2ξactωacts+ω2act

, (10)

where u denotes the commanded controlling surface (elevator)deflection, δ stands for the actual deflection, ωact ξact are theactuator natural frequency and damping ratio, respectively, s∈C denotes the Laplace transform variable.

In the present study, more realistic actuator model involv-ing both magnitude and rate limitations is considered insteadof the linear one (9). Following [60, 61], let us replace (9)by the nonlinear model shown in Fig. 1. This nonlinear modelinvolves two limited integrators which are dynamic nonlinear-ities. Integrator 1 is associated to the rate limitation ¯

δ , whileIntegrator 2 corresponds to the magnitude limitation δ .

The limited nonlinear dynamic integrator with input x(t),output y(t) and the saturation level y (i.e. |y|6 y for all t; weassume that the limitations are symmetrical) is described bythe following model:

y =

{0, if (y > y)∩ (x · y > 0),x, otherwise,

(11)

where ∩ denotes the logical “AND” operation.In the Lur’e form, nonlinear integrator (11) may be approx-

imately described by the following model [60, 61]:

σ = x−λ (σ − y), y = saty σ , (12)

where gain λ > 0 is sufficiently large. More precisely, as fol-lows from [Lemma 5.1][61] for any continuous input signalx(t) with bounded derivatives, the approximation error tendsto zero when the tuning parameter λ increases.

Figure 1. Block diagram of nonlinear actuator model with magnitudeand rate limitations, cf. [60, 61].

Based on the saturated integrator description given by (12),the actuator model pictured in Fig. 1 is represented by the fol-lowing equations:

σ1 = ω2act(u−δ )−2ξactν−λ (σ1−ν),

σ2 = ν−λ (σ2−δ ),

ν = sat ¯δ

σ1.

(13)

IV. UNSTABLE AIRCRAFT ANGLE-OF-ATTACKCONTROL SYSTEM WITH SATURATED ACTUATOR

Consider the following linearized model of short-term dy-namics of an unstable aircraft in the vertical plane as in [61]:{

α(t) =−Yα α(t)+q(t)−Yδ δe(t),q(t) = Mα α(t)+Mqq(t)+Mδ δe(t),

(14)

where al, q and δe denote the angle-of-attack, the pitch-rateand the elevator deflection, respectively; Yα , Yδ , Mα , Mq, Mδ

are the aircraft model parameters which are assumed to beconstant during the considered time slot. Let the followingproportional feedback law be realized by means of the aircraftlongitudinal stability augmentor:

u(t) = Kα α(t)−KcwXcw(t), (15)

where Kα , Kcw are stability augmentor gains, Xcw is the con-trol wheel deflection, produced by a pilot. In the sequel,model (14) parameters are taken as follows (see [61]): Yα =0.47, Yδ = 0.16, Mα = 0.82, Mq =−0.43, Mδ =−4.4 (in SIunits). Note that Mα > 0 and, therefore, the considered air-craft model is weathercock unstable with respect to the angleof attack. Parameter Kα of stability augmentor (15) is takenas Kα = 0.35.

In the present example, the actuator dynamics are describedby (13) taking into account limiting the maximum speed ofmoving the aircraft controlling surface. Such a limitation mayappear due to the saturation of the of hydraulic fluid supplyto the cylinder when the supplying channels are fully opened.The actuator position limitation is assumed to be sufficientlylarge, therefore it is neglected. To simplify the exposition,without loss of generality, gain Kcw in (15) is taken equal toone. The actuator model parameters (in SI units) are taken asωact = 20, ξact = 0.6, ˙

δ = 10/57.3.

4

A. Localization of hidden oscillations

Let us apply to the considered system the method of hiddenoscillations localization, described in Sec. II, cf. [8, 10, 62–65]. To this end represent the system model (13), (14),(15) (assuming up(t) ≡ 0) to the state-space form (1), con-sidering ψ(σ) as an input of the linear part of the system,and σ as its output. Introduce the state-space vector x asx = (σ ,δe,α,q)T ∈ R4, n = 4. This leads to the followingmatrices in (1):

P=

−λ −ω2act Kα ω2

act 00 0 0 00 −Yδ −Yα 10 Mδ Mα Mq

, q=

λ−2ξact ωact100

,

r =(1 0 0 0

)T. (16)

The block diagram of the linear part of the system is demon-strated in Fig. 2 (on this diagram, the Simulink State-Spaceblock represents model (14)). For the given above parame-ter values (Yα = 0.47, Y δ = 0.16, Mα = 0.82, Mq = −0.43,Mδ = −4.4, Kα = 0.35, ωact = 20, ξact = 0.6, λ = 100) onegets

P=

−100 −400 −140 00 0 0 00 −0.16 −0.47 10 −4.4 0.82 −0.43

, q=

76100

,

r =(1 0 0 0

)T. (17)

Application the iterative procedure of Sec. II shows ex-istence in the system the limit cycle oscillation withthe initial point x(0) =

(σ(0) δe(0) α(0) q(0)

)T=(

−0.1745201 0.009257612 0.5652268 0.9543608)T. Pro-

jection of the limit cycle to the subspace (α,q,δ ) is depictedin Fig. 3. Time histories of variables α , q, δ for given x(0)are plotted in Figs. 4, 5. Figure 4 represents the case whenno rate limitations are taken into account (i.e. the case of alinear actuator model (9)), while Fig. 5 refers to the case ofnonlinear actuator model (13) and reflects an influence of therate limitations to the overall system behavior.

V. AIRCRAFT-PILOT LOOP

The problem of control with limitations on the magnitude,rate, and energy of the control action due to its relevance forpractice has attracted the attention of scientists and developersof automatic control systems for aircrafts for a long time, see[66–71] and the references therein.

Influence of non-linearities like the “saturation” can alsocause the so-called Pilot Involved Oscillations (PIOs), whichviolate the aircraft piloting, cf. [66–69, 72–78]. This phe-nomenon is characterized by rapidly developing fluctuationswith increasing amplitude of angular velocities, accelerations,and angular movements of manned aircraft. Despite the na-ture of PIO is not completely clear, it is generally recognizedthat the main factor leading to PIO is limitations of the rate of

deviation of the aircraft control inputs (such as the controllingaerodynamic surfaces). This restriction may result in a delayin the response of the aircraft to the pilot commands.

As noted in [69, 79], PIOs usually arise in situations wherethe pilot is trying to maneuver by aircraft with a high preci-sion. The study of transient regimes with this motion leadsto the need of development a mathematical theory of globalanalysis of flight control systems.

Below the hidden PIO-like oscillations in aircraft-pilot con-tour are studied based by the example of piloted research air-craft X-15. The transfer function of X-15 research aircraft lon-gitudinal dynamics from the elevator deflection δe to pitch an-gle θ is taken as follows [66, 80–83]:

Gθ

δ(s)=

{θ

δe

}=

3.48(s+0.883)(s+0.0292)(s+0.3516)(s+0.02845)

(s2+1.68s+5.29

) ,where δe(t) denotes the elevator deflection with respect to thetrimmed value, θ(t) stands for the pitch angle (all variablesare given in the SI units), s ∈ C.

The pilot is usually modeled as a serial element in theclosed-loop system. Having enough flight skills, the pilot de-velops a stable relationship between his control action and aspecific set of flight sensor signals [84]. Based on [84–86] thefollowing pilot model in the form of a lead-lag-delay unit istaken in the present study:

Gp(s) ={ u

∆θ

}= Kp

TLs+1TIs+1

e−τes, (18)

where ∆θ is the displayed error between desired pitch angleθ T and actual one θ ; u(t) denotes the pilot’s control action,applied to the elevator servo; Kp is the pilot static gain; TLis the lead time constant; TI stands for the lag time constant;τe denotes the effective time delay, including transport delaysand high frequency neuromuscular lags. Since the present pa-per is focused to studying autonomous systems behavior, it isassumed in the sequel that θ T ≡ 0 and, therefore, ∆θ =−θ .

To apply the method of [10, 62, 64, 87], the time-delaytransfer function exp−τes in pilot model (18) is approximatedemploying the first-order Pade (1,1) representation [88] as

e−τes ≈ −τs+2τs+2

. This leads to the following second-order

model of the pilot dynamics:

Gp(s) ={ u

∆θ

}= Kp

(TLs+1)(−τs+2)(TIs+1)(τs+2)

. (19)

Finally, the transfer function of the open-loop aircraft-pilotsystem from elevator deflection δe to the pilot’s control actionu has the following form:

Gp(s) ={

uδe

}= Kp

(TLs+1)(−τs+2)(TIs+1)(τs+2)

Gθ

δ(s). (20)

In the present study the pilot model parameters are taken as:Kp = 1.8, TL = 0.6 s, TI = 0.2 s, τ = 0.2 s. Consequently, one

5

Figure 2. Simulink block diagram of the linear part of the system (13), (14).

Figure 3. Projection of the limit cycle in system (13), (14), (15) to thesubspace (α,q,δ ). Initial point x(0) =

(−0.1745201,0.009257612,

0.5652268,0.9543608)T (marked by a circle).

Figure 4. Time histories of α , q, δ in the system (9), (14) with linearactuator model.

Figure 5. Time histories of α , q, δ in the system with nonlinearactuator model.

obtains the following transfer function G(s):

Gp(s) =−10.428(s−10)(s+1.667)(s+10)(s+5)(s+0.3516)

× (s+0.883)(s+0.0292)(s+0.02845)

(s2 +1.68s+5.29

) . (21)

The actuator is modeled as a second-order dynamical unitwith a rate limitation (13). The actuator model parametersare taken as ωact = 50 rad/s, ξact = 0.6, ¯

δact = 15/57.3 rad/s,λ = 100 s−1. Block-diagram of the closed-loop aircraft-pilotsystem (13), (21) with saturated actuator model is pictured inFig. 6.

A. Localization of hidden oscillations

Let us apply the procedure of hidden oscillations localiza-tion to piloted aircraft control system (13), (21). To this endconsider the linear subsystem with input ψ and output σ (see.Fig. 6) and, following Sec. II, represent it in the state-spaceform (1). In the considered case, n = 8 and in a certain basis

6

Figure 6. Block-diagram of the closed-loop aircraft-pilot system(13), (21) with rate saturation.

of state space variables, for given above numerical values ofthe model parameters, one obtains the following matrices in(1):

P=

−100 0 4.69 ·104−3.48 ·105−1.14 ·106−7.24 ·105−2.02 ·104−25000 −17.1 −86.8 −194 −327 −102 −2.65 10 1 0 0 0 0 0 00 0 1 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 0 0

,

q=[40,0,0,0,0,0,0,−1

]T, r=

[1,0,0,0,0,0,0,0

]T. (22)

If the equilibria of system (1) are stable, for the numeri-cal search of hidden oscillations we can use the continuationmethod of Sec. II.

Several consequent steps of hidden oscillations localizationvia the above procedure are illustrated by Fig. 7, where phasetrajectory projections to subspace (θ ,q,δ ), where q denotesthe pitch rate, for various values of ε are depicted. Initial valueof x is taken as x0 = [0,0,0,0,0,0,0,0.1]T.

Finally, the following point xT0 = [0.00264,0.00292,4.96 ·

10−14,−4.35 · 10−4,7.58 · 10−5,6.46 · 10−5,0.0482,6.45]T

which belongs to the limit cycle has been found after ten iter-ations. The simulation results in the form of phase trajectoryprojections to subspace (θ ,q) for various initial conditionsare plotted in Fig. 8. Thick solid line on the plot correspondsto the limit cycle oscillation. It is seen that the limit cycleis asymptotically orbitally stable, attracting the neighboringtrajectories. The zero equilibrium is asymptotically stable(thin solid line in Fig. 8). The simulations demonstrate thatthe unstable limit cycle for “intermediate” initial conditions,separating attractivity to the stable limit cycle oscillations andthe equilibrium state may also exist.

VI. AIRFOIL FLUTTER SUPPRESSION

A. Aeroelastic model of an airfoil section

An aeroelastic system describes wing dynamics in the pres-ence of a flow field. Interacting forces between the struc-ture, the moment of inertia and air flow destabilize aircraftby producing flutter and limit cycle oscillation [89–91]. Inthe presence of a flow field, the wing at a flight speed U os-cillates along the plunge displacement direction and rotates at

Figure 7. Consequent steps of hidden oscillations localization. Phasetrajectory projections to subspace (θ ,q,δ ).

Figure 8. Phase trajectory projections to subspace (θ ,q). x(0) =1.2xT

0 – dashed line; x(0) = xT0 – thick solid line (limit cycle); x(0) =

0.4xT0 – thin solid line.

7

the pitch angle about the elastic axis. Because flutter can even-tually damage a wing structure, flutter must be shed during aflight.

Let us use the following steady-state model of the airfoilsection flutter dynamics [92, 93]:[

Iα mwxα bmwxα b mt

][α

h

]+

[cα 00 ch

][α

h

]+

[kα(α) 0

0 kh(h)

][α

h

]=

[M−L

](23)

kα(α) = k1 + k2α2, (24)

kh(h) = κ1 +κ2h2. (25)

where h denote the plunge displacement; α stands for thepitch angle; γ and β are angles of the leading and trail-ing edges, respectively. The wing structure includes a lin-ear spring oriented along the plunge displacement direction,a rotational spring along the pitch angle, and correspondingdampers. mt denotes is the total weight of the main wing andsupporter, mw is the weight of the main wing, xα is the di-mensionless distance between the center of mass and the elas-tic axis, Iα is the moment of inertia, b is the midchord, cα

and ch are the damping coefficients of the pitch angle and theplunge displacement respectively, kh and kα(α) are the springstiffness coefficients of the plunge displacement and the pitchangle respectively, and kα(α) is a nonlinear term. The non-linearity αkα(α) of the spring, a hard spring in fact, which isactually a hard spring, is defined as

The aerodynamics force L and torque M in the low-frequency area and subsonic flight may be represented as fol-lows [89, 90, 93]:

L = ρU2bclα sp

(α +

(hU

+(1

2−a)

bα

U

))+ρU2bclβ spβ +ρU2bclγ spγ, (26)

M = ρU2b2cmα-effsp

(α +

(hU

+(1

2−a)

bα

U

))+

ρU2b2cmβ -eff spβ +ρU2b2cmγ-effspγ, (27)

where ρ is air density, U is the flight speed, a is the dimen-sionless distance between the elastic axis and the mid-chord;sp is the windspan length, clα , cmα

are the lift coefficient andmoment coefficient per unit angle of attack respectively; clβ ,cmβ

are the lift coefficient and moment coefficient per unitangle respectively against the trailing edge, respectively; clγ ,cmγ

are the lift coefficient and moment coefficient per unitangle, respectively, against the leading edge; cmα-eff , cmβ -eff ,cmγ-eff are the moment derivative coefficient per unit angle ofattack, trailing edge and leading edge, respectively. Accord-ing to [93], they are defined as

cmα-eff =

(12+a)

clα +2cmα,

cmβ -eff =

(12+a)

clβ +2cmβ,

cmγ-eff =

(12+a)

clγ +2cmγ.

(28)

Figure 9. The airfoil section with controlling surfaces (cf. [93]).

Introducing notations c1 = ρU2bsp, c2 = ρU2b2sp, rewrite(26), (27) in the form

L = c1

(α +

(hU

+(1

2−a)

bα

U

))+c1clβ β + c1clγ γ,

M = c2cmα-eff

(α +

(hU

+(1

2−a)

bα

U

))+

c2cmβ -effβ + c2cmγ-effγ.

(29)

Denote the state-space vector as x =[α α h h

]T; kh(h) ≡kh. Then

x1 = x2,

x2 = cα1x1 + cαnonl1x31 + cα1x2 + ch1x3 + ch1

x4

+cβ1β + cγ1 γ,

x3 = x4,

x4 = cα2x1 + cαnonl2x31 + cα2x2 + ch2x3 + ch2

x4

+cβ2β + cγ2γ,

(30)

where cα1 , cαnonl1 , cα1 , ch1 , ch1, cβ1 , cγ1 , cα2 , cαnonl2 , cα2 , ch2 ,

ch2, cβ2 , cγ2 are model parameters which are assumed to be

constant on the considered time interval.Let us linearize (30) in the vicinity of the origin and rep-

resent it in the vector-matrix form as x(t) = Ax(t) + bu(t),where A is (4× 4) system matrix, b is (4× 1) input ma-trix, u(t) ≡ β (t) denotes the control action (the controllingsurface deflection). Let the airfoil model (30) parametersbe taken as in [93], see Appendix VII. Matrix A eigenval-ues are as s = {3.05± 15i,−4.63± 13.5i} which shows thatsystem (30) origin is unstable in the Lyapunov sense. Mean-while, as shown in the series of papers, cf. [92–111], due topresence of the cubic nonlinearity in the system model, thesystem trajectories are bounded and the limit cycle oscilla-tions (the airfoil flutter phenomenon) arise. An illustrationof the limit cycle oscillations birth for small initial condi-tions is given by Fig. 10, where projection of the free mo-

8

tion phase trajectory to subspace (α, α,h) for α(0) = 0.1 deg,α(0) = h(0) = h(0) = 0 is plotted.

−0.2−0.1

00.1

0.2

−4−2

02

4−8

−6

−4

−2

0

2

4

6

x 10−3

αdα/dt

h

Figure 10. Free motion. Limit cycle oscillation birth; projection ofphase trajectory to subspace (α, α,h). Initial point α(0) = 0.1 deg,α(0) = h(0) = h(0) = 0.

B. Airfoil flutter suppression system

Consider the problem of flutter suppression by controllingthe airfoil trailing edge β (i.e. assume hereafter that γ ≡ 0).

There is a plenty of papers devoted to airfoil flutter ac-tive suppression systems, see [93, 101, 104–106, 110–114]for mentioning a few. To simplify the exposition by avoid-ing unnecessary difficulties in control law synthesis, in thepresent study the widespread LQR-control design techniqueis employed.

Consider the static state feedback control law u =−Kfbx∈R4, where state-space vector x, as in (30), is a measuredplant state, Kfb is (1×4) matrix (row vector) of the controllerparameters. Introduce the linear-quadratic performance in-dex J =

∫∞

0(xTQx+uTRu

)dt with a given positively defined

(4× 4) matrix Q = QT > 0 and a non-negative scalar R > 0.Employing the standard MATLAB routine lqr one obtains thestate feedback vector Kfb, minimizing performance index Q.

Let us pick up Q = diag{1,0.01,1,2 · 10−3}, R =0.5. MATLAB linear-quadratic optimization routine lqrleads then to the following state feedback vector Kfb =[−0.93 −0.17 −7.22 0.062

]]. This gives the closed-loop

linearized system matrix Afb = A− bKbf with the eigenval-ues sfb = {−17.6±9.0i,−1.53±13.6i}, ensuring asymptoticconvergent system behavior in the close vicinity of the ori-gin. Time histories of α(t), h(t) for controlled motion for thecase of the “ideal” static state feedback controller at the initialpoint α(0) = 0.1 deg, α(0) = h(0) = h(0) = 0 are depicted inFig. 11. The case of dynamical actuator with the magnitudeand rate limitations is considered below.

Since we focus our attention to existence of hidden oscil-lation rather than to controller design problems, for simplic-ity, in the above control law synthesis, the actuator dynamicshave been omitted and the assumption that β (t) ≡ u(t) wasadopted. Let us check the control system performance taking

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.02

0.04

0.06

0.08

0.1α, deg

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

−4

−2

0

2

x 10−5

h, m

t, s

Figure 11. Time histories of α(t), h(t) for controlled motion. “Ideal”static state feedback controller. Initial point α(0) = 0.1 deg, α(0) =h(0) = h(0) = 0.

into account the actuator model. To this end a second-orderactuator model with the output magnitude and rate saturations(12), described in Sec. III is used where δ , δ , δ

¯δ are substi-

tuted by β , β , β¯β respectively. In the present study is taken

that β = 0.0873 rad = 5 deg, ¯β = 8.73 rad/s = 500 grad/s.

ξact = 0.6, ωact = 50 rad/s. Gain λ in (12) is set to 100.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0

0.5

1α/max|α|

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−1

−0.5

0

0.5

h/max|h|

t, s

Figure 12. Time histories of α(t)/maxt |α| (upper plot),h(t)/maxt |h| (lower plot). α(0) = 0.1 deg – solid line, α(0) = 4deg – dash-dot line, α(0) = 5 deg – dashed line.

C. Localization of hidden oscillation

The numerical evaluation shows that in the linear approx-imation, the actuator dynamics do not effect significantly theclosed-loop system performance, as is seen from the plots ofFig. 12. In this figure, time histories of variables α(t), h(t)for various α(0) (the other initial state variables are set tozero), normalized with respect to their maximal values areplotted. Comparing the curves depicted on Fig. 11 with thecorresponding (solid) curves of Fig. 12, one may notice that

9

01

23

4

−0.2

−0.1

0

0.1

0.2−4

−2

0

2

4

tα

dα/d

t

Figure 13. System motion in the subspace (t,α, α) for α(0) = 7 deg(trajectories converge to zero) and α(0) = 8 deg (limit cycle oscilla-tion arises).

−0.2−0.1

00.1

0.2

−5

0

5−2

−1

0

1

2

x 10−3

αdα/dt

h

x0

Figure 14. Limit cycle oscillation in the subspace (α, α,h). Initialpoint α(0) =−0.109 rad, α(0) =−3.55 rad/s, h(0) =−9.33 ·10−4

m, h(0) = 0.031 m/s, β (0) =−0.0873 rad, β (0) =−8.723 rad/s.

in the linear consideration of actuator dynamics (when the sat-urations are not active), the unmodelled actuator dynamics (9)do not lead to significant degradation of the overall systemperformance. The case of α(0) ≈ 4 deg may be treated as a“boundary” one, when the nonlinearities become active. If theinitial conditions are sufficiently large, the feedback controllerlooses its stabilizing properties and the limit cycle oscillationappears. This oscillation can not be found by linearizationof the system model in the vicinity of the origin and, there-fore, may be referred to as a “hidden” one. This phenomenonis illustrated by Fig. 13, where the system trajectories in thesubspace (t,α, α) for α(0) = 7 deg and α(0) = 8 deg are plot-ted. In the first case they tend to the origin, in the secondone the limit circle oscillation occurs. By applying the hid-den oscillations localization procedure of Sec. II, the initialpoint of limit cycle oscillation is found as α(0) =−0.109 rad,α(0)=−3.55 rad/s, h(0)=−9.33 ·10−4 m, h(0)= 0.031 m/s,β (0) =−0.0873 rad, β (0) =−8.723 rad/s. The projection ofthe limit cycle phase plot to the subspace (α, α,h) is plottedin Fig. 14.

VII. CONCLUSIONS

In the paper, the control problem with limitations on themagnitude and rate of the control action in aircraft control sys-tems, is studied. Existence of hidden limit cycle oscillationsin the case of actuator position and rate limitations is demon-strated by the examples of piloted aircraft PIO phenomenonand the airfoil flutter suppression system. Hidden oscillationsin the pilot-aircraft loop are studied and localized by means ofthe iterative analytical-numerical method.

ACKNOWLEDGMENTS

This work was supported by Russian Science Foundationproject (14-21-00041).

APPENDIX. AEROELASTIC MODEL PARAMETERS

Table I. Values of initial parameters of aeroelastic model (27), (28).

a −0.6719 cmγ−0.1005

b 0.1905 m Iα (mwx2α b2 +0.009039) kg m2

cα 0.036 kg m2/s kα (α) 12.77+1003α2

ch 27.43 kg/s kh 2844.4 N/mclα 6.757 mt 15.57 kgclβ 3.358 mw 4.34 kg

clγ −0.1566 sp 0.5945 m

cmα0 xα −(0.0998+a)

cmβ−0.6719 ρ 1.225 kg/m3

Table II. Simulation model parameters.

U 19.0625 mw 4.3400 a −0.6719xα 0.5721 cmγ

−0.1005 b 0.1905Iα 0.0606 cα 0.0360 k1 12.77k2 1003 ch 27.4300 kh 2.844 ·103

clα 6.7570 mt 15.57 clβ 3.3580

clγ −0.1566 sp 0.5945 cmα0

cmβ−0.6719 ρ 1.2250 cmα-eff −1.1615

cmβ -eff −1.9210 cmγ-eff −0.1741 c1 50.4130

c2 9.6037 cα1 −211.39 cαnonl1−778.5

cα1 −0.7076 ch1 1.3454 ·103 ch112.3153

cβ1−207.1799 cγ1 −29.7643 cα2 −9.3225

cαnonl223.6498 cα2 −0.1629 ch2 −172.3376

ch2−2.4678 cβ2

−1.5305 cγ2 1.2691

Model (30) parameters are defined by the following expres-

10

sions, see [93]:

cα1 = c2mtcmα-eff + c1mwxα bclα −mtk1,

cαnonl1 =−mtk2,cα1 = c2mtcmα-eff

(12−a)

b1U

+c1mwxα bclα

(12−a)

b1U− cα mt ,ch1 = khmwxα b,

ch1= c2mtcmα-eff

1U

+ c1mwxα bclα1U

+ chmwxα b,

cβ1 = c2mtcmβ -eff + c1mwxα bclβ ,

cγ1 = c2mtcmγ-eff + c1mwxα bclγ ,

cα2 =−c2mwxα bcmα-eff − c1Iα clα +mwxα bk1,

cαnonl2 = mwxα bk2,

cα2 =−c2mwxα bcmα-eff

(12−a)

b1U

−c1Iα clα

(12−a)

b1U

+ cα mwxα b,

ch2 =−khIα ,

ch2=−c2mwxα bcmα-eff

1U− c1Iα clα

1U− chIα ,

cβ2 =−c2mwxα bcmβ -eff − c1Iα clβ ,

cγ2 =−c2mwxα bcmγ-eff − c1Iα clγ .

(31)

The aeroelastic model (27), (28) parameters are taken as in[93], see Tab. I. Calculations according to (28), (31) lead tomodel (30) parameter values, given in Tab. II.

[1] M. Dornheim, Report pinpoints factors leading to YF-22crash, Aviation Week and Space Technology 137 (1992) 53–54.

[2] C. Shifrin, Sweden seeks cause of Gripen crash, AviationWeek and Space Technology 139 (1993) 78–79.

[3] G. A. Leonov, G. V. Kuznetsov, Hidden attractors in dynami-cal systems. From hidden oscillations in Hilbert-Kolmogorov,Aizerman, and Kalman problems to hidden chaotic attractorsin Chua circuits, Int J. Bifurcation and Chaos 23 (1) (2013)1–69. doi:10.1142/S0218127413300024.

[4] N. Kuznetsov, G. Leonov, Hidden attractors in dynamical sys-tems: systems with no equilibria, multistability and coexistingattractors, IFAC Proceedings Volumes 47 (2014) 5445–5454.doi:10.3182/20140824-6-ZA-1003.02501.

[5] G. Leonov, N. Kuznetsov, T. Mokaev, Homoclinic orbits,and self-excited and hidden attractors in a Lorenz-like sys-tem describing convective fluid motion, Eur. Phys. J. SpecialTopics 224 (8) (2015) 1421–1458. doi:10.1140/epjst/

e2015-02470-3.[6] N. Kuznetsov, Hidden attractors in fundamental problems and

engineering models. A short survey, Lecture Notes in Electri-cal Engineering 371 (2016) 13–25, (Plenary lecture at Interna-tional Conference on Advanced Engineering Theory and Ap-plications 2015). doi:10.1007/978-3-319-27247-4\_2.

[7] N. Kuznetsov, G. Leonov, V. Vagaitsev, Analytical-numericalmethod for attractor localization of generalized Chua’s sys-tem, IFAC Proceedings Volumes 43 (11) (2010) 29–33. doi:10.3182/20100826-3-TR-4016.00009.

[8] G. A. Leonov, N. V. Kuznetsov, V. I. Vagaitsev, Localizationof hidden Chua’s attractors, Physics Letters A 375 (23) (2011)2230–2233. doi:10.1016/j.physleta.2011.04.037.

[9] V. Bragin, V. Vagaitsev, N. Kuznetsov, G. Leonov, Algorithmsfor finding hidden oscillations in nonlinear systems. The Aiz-erman and Kalman conjectures and Chua’s circuits, Journal ofComputer and Systems Sciences International 50 (4) (2011)511–543. doi:10.1134/S106423071104006X.

[10] G. A. Leonov, N. V. Kuznetsov, V. I. Vagaitsev, Hidden at-tractor in smooth Chua systems, Physica D 241 (18) (2012)

1482–1486. doi:10.1016/j.physd.2012.05.016.[11] N. Kuznetsov, O. Kuznetsova, G. Leonov, V. Vagait-

sev, Analytical-numerical localization of hidden attractorin electrical Chua’s circuit, Lecture Notes in ElectricalEngineering 174 (4) (2013) 149–158. doi:10.1007/

978-3-642-31353-0\_11.[12] M. Kiseleva, E. Kudryashova, N. Kuznetsov, O. Kuznetsova,

G. Leonov, M. Yuldashev, R. Yuldashev, Hidden andself-excited attractors in Chua circuit: synchronizationand SPICE simulation, International Journal of Par-allel, Emergent and Distributed Systems (2017) 1–11doi:10.1080/17445760.2017.1334776.

[13] N. Stankevich, N. Kuznetsov, G. Leonov, L. Chua, Scenario ofthe birth of hidden attractors in the Chua circuit, InternationalJournal of Bifurcation and Chaos 27 (12), accepted.

[14] I. Burkin, N. Khien, Analytical-numerical methods of findinghidden oscillations in multidimensional dynamical systems,Differential Equations 50 (13) (2014) 1695–1717.

[15] C. Li, J. Sprott, Coexisting hidden attractors in a 4-D simpli-fied Lorenz system, International Journal of Bifurcation andChaos 24 (03), art. num. 1450034.

[16] Q. Li, H. Zeng, X.-S. Yang, On hidden twin attractors andbifurcation in the Chua’s circuit, Nonlinear Dynamics 77 (1-2) (2014) 255–266.

[17] V.-T. Pham, F. Rahma, M. Frasca, L. Fortuna, Dynamicsand synchronization of a novel hyperchaotic system withoutequilibrium, International Journal of Bifurcation and Chaos24 (06), art. num. 1450087.

[18] M. Chen, M. Li, Q. Yu, B. Bao, Q. Xu, J. Wang, Dynam-ics of self-excited attractors and hidden attractors in general-ized memristor-based Chua’s circuit, Nonlinear Dynamics 81(2015) 215–226.

[19] A. Kuznetsov, S. Kuznetsov, E. Mosekilde, N. Stankevich, Co-existing hidden attractors in a radio-physical oscillator system,Journal of Physics A: Mathematical and Theoretical 48 (2015)125101.

[20] P. Saha, D. Saha, A. Ray, A. Chowdhury, Memristive non-linear system and hidden attractor, European Physical Journal:

11

Special Topics 224 (8) (2015) 1563–1574.[21] V. Semenov, I. Korneev, P. Arinushkin, G. Strelkova, T. Vadi-

vasova, V. Anishchenko, Numerical and experimental studiesof attractors in memristor-based Chua’s oscillator with a lineof equilibria. Noise-induced effects, European Physical Jour-nal: Special Topics 224 (8) (2015) 1553–1561.

[22] P. Sharma, M. Shrimali, A. Prasad, N. Kuznetsov, G. Leonov,Control of multistability in hidden attractors, Eur. Phys. J. Spe-cial Topics 224 (8) (2015) 1485–1491.

[23] Z. Zhusubaliyev, E. Mosekilde, A. Churilov, A. Medvedev,Multistability and hidden attractors in an impulsive Goodwinoscillator with time delay, European Physical Journal: SpecialTopics 224 (8) (2015) 1519–1539.

[24] Z. Wei, P. Yu, W. Zhang, M. Yao, Study of hidden attractors,multiple limit cycles from Hopf bifurcation and boundednessof motion in the generalized hyperchaotic Rabinovich system,Nonlinear Dynamics 82 (1) (2015) 131–141.

[25] M.-F. Danca, N. Kuznetsov, G. Chen, Unusual dynamicsand hidden attractors of the Rabinovich–Fabrikant system,Nonlinear Dynamics 88 (2017) 791–805. doi:10.1007/

s11071-016-3276-1.[26] S. Jafari, V.-T. Pham, S. Golpayegani, M. Moghtadaei,

S. Kingni, The relationship between chaotic maps and somechaotic systems with hidden attractors, Int. J. Bifurcat. Chaos26 (13), art. num. 1650211.

[27] T. Menacer, R. Lozi, L. Chua, Hidden bifurcations in the mul-tispiral Chua attractor, International Journal of Bifurcation andChaos 26 (14), art. num. 1630039.

[28] O. Ojoniyi, A. Njah, A 5D hyperchaotic Sprott B system withcoexisting hidden attractors, Chaos, Solitons & Fractals 87(2016) 172–181.

[29] V.-T. Pham, C. Volos, S. Jafari, S. Vaidyanathan, T. Kapita-niak, X. Wang, A chaotic system with different families of hid-den attractors, International Journal of Bifurcation and Chaos26 (08) (2016) 1650139.

[30] R. Rocha, R. O. Medrano-T, Finding hidden oscillations in theoperation of nonlinear electronic circuits, Electronics Letters52 (12) (2016) 1010–1011.

[31] Z. Wei, V.-T. Pham, T. Kapitaniak, Z. Wang, Bifurcation anal-ysis and circuit realization for multiple-delayed Wang–Chensystem with hidden chaotic attractors, Nonlinear Dynamics85 (3) (2016) 1635–1650.

[32] I. Zelinka, Evolutionary identification of hidden chaotic at-tractors, Engineering Applications of Artificial Intelligence 50(2016) 159–167.

[33] M. Borah, B. Roy, Hidden attractor dynamics of a novel non-equilibrium fractional-order chaotic system and its synchro-nisation control, in: 2017 Indian Control Conference (ICC),2017, pp. 450–455.

[34] P. Brzeski, J. Wojewoda, T. Kapitaniak, J. Kurths, P. Per-likowski, Sample-based approach can outperform the classicaldynamical analysis - experimental confirmation of the basinstability method, Scientific Reports 7, art. num. 6121.

[35] Y. Feng, W. Pan, Hidden attractors without equilibrium andadaptive reduced-order function projective synchronizationfrom hyperchaotic Rikitake system, Pramana 88 (4) (2017) 62.

[36] H. Jiang, Y. Liu, Z. Wei, L. Zhang, Hidden chaotic attractors ina class of two-dimensional maps, Nonlinear Dynamics 85 (4)(2016) 2719–2727.

[37] N. Kuznetsov, G. Leonov, M. Yuldashev, R. Yuldashev, Hid-den attractors in dynamical models of phase-locked loop cir-cuits: limitations of simulation in MATLAB and SPICE, Com-mun Nonlinear Sci Numer Simulat 51 (2017) 39–49. doi:

10.1016/j.cnsns.2017.03.010.

[38] J. Ma, F. Wu, W. Jin, P. Zhou, T. Hayat, Calculation of Hamil-ton energy and control of dynamical systems with differenttypes of attractors, Chaos: An Interdisciplinary Journal ofNonlinear Science 27 (5) (2017) 053108. doi:10.1063/1.

4983469.[39] M. Messias, A. Reinol, On the formation of hidden chaotic at-

tractors and nested invariant tori in the Sprott A system, Non-linear Dynamics 88 (2) (2017) 807–821.

[40] J. Singh, B. Roy, Multistability and hidden chaotic attractorsin a new simple 4-D chaotic system with chaotic 2-torus be-haviour, International Journal of Dynamics and Control doi:10.1007/s40435-017-0332-8.

[41] C. Volos, V.-T. Pham, E. Zambrano-Serrano, J. M. Munoz-Pacheco, S. Vaidyanathan, E. Tlelo-Cuautle, Advances inMemristors, Memristive Devices and Systems, Springer,2017, Ch. Analysis of a 4-D hyperchaotic fractional-ordermemristive system with hidden attractors, pp. 207–235.

[42] Z. Wei, I. Moroz, J. Sprott, A. Akgul, W. Zhang, Hiddenhyperchaos and electronic circuit application in a 5D self-exciting homopolar disc dynamo, Chaos 27 (3), art. num.033101.

[43] G. Zhang, F. Wu, C. Wang, J. Ma, Synchronization behaviorsof coupled systems composed of hidden attractors, Interna-tional Journal of Modern Physics B 31, art. num. 1750180.

[44] N. Krylov, N. Bogolyubov, Introduction to non-linear mechan-ics, Princeton Univ. Press, Princeton, 1947.

[45] E. Popov, I. Pal’tov, Approximate methods for studying non-linear automatic systems, Translation Services, Ohio, 1963.

[46] H. Khalil, Nonlinear Systems, Prentice Hall, NJ, 2002.[47] L. P. Liu, E. H. Dowell, J. P. Thomas, A high dimensional har-

monic balance approach for an aeroelastic airfoil with cubicrestoring forces, J. Fluids and Structures 23 (3) (2005) 351–363.

[48] V. Shakhgil’dyan, A. Lyakhovkin, Sistemy fazovoi avtopod-stroiki chastoty (in Russian), Svyaz’, Moscow, 1972.

[49] E. Kudryashova, N. Kuznetsov, G. Leonov, M. Yuldashev,R. Yuldashev, Nonlinear analysis of PLL by the harmonicbalance method: limitations of the pull-in range estimation,IFAC-PapersOnLine 50 (1) (2017) 1451–1456. doi:https:

//doi.org/10.1016/j.ifacol.2017.08.289.[50] Y. Z. Tsypkin, Relay Control Systems, Cambridge Univ

Press., Cambridge, 1984.[51] V. A. Pliss, Some Problems in the Theory of the Stability of

Motion (in Russian), Izd LGU, Leningrad, 1958.[52] R. E. Fitts, Two counterexamples to Aizerman’s conjecture,

Trans. IEEE AC-11 (3) (1966) 553–556.[53] N. E. Barabanov, On the Kalman problem, Sib. Math. J. 29 (3)

(1988) 333–341.[54] J. Bernat, J. Llibre, Counterexample to Kalman and Markus-

Yamabe conjectures in dimension larger than 3, Dynamicsof Continuous, Discrete and Impulsive Systems 2 (3) (1996)337–379.

[55] G. Leonov, V. Bragin, N. Kuznetsov, Algorithm for con-structing counterexamples to the Kalman problem, Dok-lady Mathematics 82 (1) (2010) 540–542. doi:10.1134/

S1064562410040101.[56] G. Leonov, N. Kuznetsov, Algorithms for searching for hid-

den oscillations in the Aizerman and Kalman problems, Dok-lady Mathematics 84 (1) (2011) 475–481. doi:10.1134/

S1064562411040120.[57] R. Alli-Oke, J. Carrasco, W. Heath, A. Lanzon, A robust

Kalman conjecture for first-order plants, IFAC ProceedingsVolumes (IFAC-PapersOnline) 7 (2012) 27–32. doi:10.

3182/20120620-3-DK-2025.00161.

12

[58] W. P. Heath, J. Carrasco, M. de la Sen, Second-order coun-terexamples to the discrete-time Kalman conjecture, Automat-ica 60 (2015) 140 – 144.

[59] G. Leonov, R. Mokaev, Negative solution of the Kalman prob-lem and proof of the existence of a hidden strange attractor viaa discontinuous approximation method, Doklady Mathematics96 (1) (2017) 1–4. doi:10.1134/S1064562417040111.

[60] J. Biannic, S. Tarbouriech, D. Farret, A practical approachto performance analysis of saturated systems with applica-tion to fighter aircraft flight controllers, in: Proc. 5th IFACSymposium ROCOND. IFAC Proceedings Volumes (IFAC-PapersOnline), Toulouse, France, 2006, http://www.ifac-papersonline.net/ Detailed/30006.html.

[61] J. Biannic, S. Tarbouriech, Optimization and implementationof dynamic anti-windup compensators with multiple satura-tions in flight control systems, Control Engineering Practice17 (2009) 703–713.

[62] V. Bragin, V. Vagaitsev, N. Kuznetsov, G. Leonov, Algorithmsfor finding hidden oscillations in nonlinear systems. the aizer-man and kalman conjectures and Chua’s circuits, J. Computerand Systems Sciences International 50 (4) (2011) 511–543.doi:10.1134/S106423071104006X.

[63] D. Dudkowski, S. Jafari, T. Kapitaniak, N. V. Kuznetsov, G. A.Leonov, A. Prasad, Hidden attractors in dynamical systems,Physics Reports 637 (2016) 1 – 50. doi:http://dx.doi.

org/10.1016/j.physrep.2016.05.002.[64] G. A. Leonov, N. V. Kuznetsov, Hidden attractors in dynami-

cal systems. From hidden oscillations in Hilbert-Kolmogorov,Aizerman, and Kalman problems to hidden chaotic at-tractors in Chua circuits, International Journal of Bifurca-tion and Chaos 23 (1), art. no. 1330002. doi:10.1142/

S0218127413300024.[65] G. A. Leonov, N. V. Kuznetsov, Analytical-numerical meth-

ods for investigation of hidden oscillations in nonlin-ear control systems, IFAC Proceedings Volumes (IFAC-PapersOnline) 18 (1) (2011) 2494–2505. doi:10.3182/

20110828-6-IT-1002.03315.[66] B. Andrievsky, N. Kuznetsov, G. Leonov, Methods for sup-

pressing nonlinear oscillations in astatic auto-piloted aircraftcontrol systems, Journal of Computer and Systems SciencesInternational 56 (3) (2017) 455–470.

[67] O. Brieger, M. Kerr, D. Leißling, J. Postlethwaite, J. Sofrony,M. C. Turner, Anti-windup compensation of rate saturationin an experimental aircraft, in: Proc. American Control Conf.(ACC 2007), AACC, 2007, pp. 924–929.

[68] O. Brieger, M. Kerr, J. Postlethwaite, J. Sofrony, M. C. Turner,Flight testing of low-order anti-windup compensators for im-proved handling and PIO suppression, in: American ControlConf. (ACC 2008), AACC, 2008, pp. 1776–1781.

[69] D. T. McRuer and J. D. Warner (Eds.), Aviation Safety and Pi-lot Control: Understanding and Preventing Unfavorable Pilot-Vehicle Interactions, Committee on the Effects of Aircraft-Pilot Coupling on Flight Safety Aeronautics and Space En-gineering Board Commission on Engineering and TechnicalSystems National Research Council National Academy Press,Washington, DC, 1997.URL http://www.nap.edu/catalog/5469.html

[70] I. Queinnec, S. Tarbouriech, G. Garcia, Anti-windup designfor aircraft flight control, in: Proc. IEEE Int. Symp. IntelligentControl, 2006, pp. 2541–2546.

[71] S. Tarbouriech, M. Turner, Anti-windup design: anoverview of some recent advances and open prob-lems, IET Control Theory Appl. 3 (1) (2009) 1–19,https://lra.le.ac.uk/handle/2381/4813.

[72] B. Powers, Space shuttle pilot-induced-oscillation researchtesting, NASA, 1984.

[73] M. Pachter, R. Miller, Manual flight control with saturatingactuators, IEEE Control Syst. Mag. 18 (1) (1998) 10–19.

[74] O. Brieger, M. Kerr, I. Postlethwaite, M. C. Turner, J. Sofrony,Pilot-Involved-Oscillation suppression using low-order anti-windup: Flight-test evaluation, J. Guidance, Control, and Dy-namics 35 (2) (2012) 471–483. doi:10.2514/1.54441.

[75] O. Brieger, M. Kerr, I. Postlethwaite, M. Turner, J. Sofrony,Pilot-involved-oscillation suppression using low-order anti-windup: Flight-test evaluation, J. of Guidance, Control, andDynamics 35 (2) (2012) 471–483.

[76] B. Andrievsky, N. Kuznetsov, G. Leonov, A. Pogromsky, Hid-den oscillations in aircraft flight control system with input sat-uration, IFAC Proceedings Volumes 46 (12) (2013) 75–79.doi:10.3182/20130703-3-FR-4039.00026.

[77] D. M. Acosta, Y. Yildiz, D. H. Klyde, Avoiding pilot-inducedoscillations in energy-efficient aircraft designs, in: T. Samad,A. Annaswamy (Eds.), The Impact of Control Technology –2nd Ed., IEEE CSS, 2014.URL http://ieeecss.org/sites/ieeecss.org/

files/CSSIoCT2Update/IoCT2-RC-Acosta-1.pdf

[78] B. Andrievsky, N. Kuznetsov, G. Leonov, Convergence-basedanalysis of robustness to delay in anti-windup loop of aircraftautopilot, IFAC-PapersOnLine 48 (9) (2015) 144–149.

[79] H. Duda, Flight control system design considering rate satura-tion, Aerospace Science and Technology 4 (1998) 265–215.

[80] I. Alcala, F. Gordillo, J. Aracil, Phase compensation designfor prevention of pio due to actuator rate saturation, in: Proc.American Control Conference (ACC 2004), Vol. 5, 2004, pp.4687–4691 vol.5.

[81] F. Amato, R. Iervolino, M. Pandit, Stability analysis of linearsystems with actuator amplitude and rate saturations, in: Proc.2001 European Control Conference (ECC 2001), IEEE, Porto,Portugal, 2001, pp. 366 – 371.

[82] F. Amato, R. Iervolino, M. Pandit, S. Scala, L. Verde, Anal-ysis of pilot-in-the-loop oscillations due to position and ratesaturations, in: Proc. 39th IEEE Conf. Decision and Con-trol (CDC 2000), IEEE, Sydney, Australia, 2000, p. 6. doi:

10.1109/CDC.2000.912258.[83] R. Mehra, R. Prasanth, Application of nonlinear global anal-

ysis and system identification to aircraft-pilot coupled oscil-lations, in: Proc. Int. Conf. Control Applications (CCA’98),Vol. 2, 1998, pp. 1404–1408. doi:10.1109/CCA.1998.

721691.[84] D. T. McRuer, H. R. Jex, A review of quasi-linear pilot mod-

els, IEEE Trans. Hum. Factors Electron. HFE-8 (3) (1967)231–249.

[85] C. Barbu, R. Reginatto, A. R. Teel, L. Zaccarian, Anti-windupdesign for manual flight control, in: Proc. American ControlConf. (ACC’99), Vol. 5, AACC, 1999, pp. 3186–3190.

[86] M. Lone, A. Cooke, Review of pilot models used in aircraftflight dynamics, Aerospace Science and Technology 34 (2014)55–74.

[87] G. Leonov, N. Kuznetsov, M. Yuldahsev, R. Yuldashev, Com-putation of phase detector characteristics in synchronizationsystems, Doklady Mathematics 84 (1) (2011) 586–590. doi:10.1134/S1064562411040223.

[88] G. Golub, C. Loan, Matrix Computations, Johns Hopkins Uni-versity Press, Baltimore, 1989.

[89] T. Theodorsen, I. Garrick, Mechanism of flutter: a theoreticaland experiment investigation of the flutter problem. TechnicalRept. 685, NACA, 1940.

[90] T. Theodorsen, General theory of aerodynamic instability and

13

the mechanism of flutter. Technical Rept. 496, NACA, 1935.[91] R. Bisplinghoff, H. Ashley, R. Halfman, Aeroelasticity, Dover,

New York, 1996.[92] A. Abdelkefi, R. Vasconcellos, A. Nayfeh, M. Hajj, An ana-

lytical and experimental investigation into limit-cycle oscilla-tions of an aeroelastic system, Nonlinear Dynamics 71 (1-2)(2013) 159–173. doi:10.1007/s11071-012-0648-z.

[93] C.-L. Chen, C. C. Peng, H.-T. Yau, High-order slidingmode controller with backstepping design for aeroelasticsystems, Communications in Nonlinear Science and Numer-ical Simulation 17 (4) (2012) 1813 – 1823. doi:https:

//doi.org/10.1016/j.cnsns.2011.09.011.URL http://www.sciencedirect.com/science/

article/pii/S1007570411005028

[94] S. Price, H. Alighanbari, B. Lee, The aeroelastic response of atwo-dimensional airfoil with bilinear and cubic nonlinearities,J. Fluids Struct. 9 (1995) 17593.

[95] T. ONeil, T. W. Strganac, Aeroelastic response of a rigid wingsupported by nonlinear springs, J. of Aircraft 35 (4) (1998)616–622.

[96] Q. Ding, D.-L. Wang, The flutter of an airfoil with cubicstructural and aerodynamic non-linearities, Aerospace Sci-ence and Technology 10 (5) (2006) 427 – 434. doi:http:

//dx.doi.org/10.1016/j.ast.2006.03.005.[97] B. Lee, P. LeBlanc, N. A. E. (Canada), N. R. C. Canada,

Flutter Analysis of a Two-dimensional Airfoil with CubicNon-linear Restoring Force, Aeronautical note, NationalResearch Council Canada, 1986.URL https://books.google.ru/books?id=

Kkc7PwAACAAJ

[98] S.-J. Zhang, G.-L. Wen, F. Peng, Z.-Q. Liu, Analysis of limitcycle oscillations of a typical airfoil section with freeplay,Acta Mechanica Sinica 29 (4) (2013) 583592. doi:10.1007/s10409-013-0050-1.

[99] S. J. Price, B. H. K. Lee, H. Alighanbari, An analysis of thepost instability behaviour of a two dimensional airfoil with astructural nonlinearity, J. of Aircraft 31 (1995) 1395–1401.

[100] H. Alighanbari, S. J. Price, The post-Hopf-bifurcation re-sponse of an airfoil in incompressible two-dimensional flow,Nonlinear Dynamics 10 (1996) 381–400.

[101] J. Ko, T. Strganac, A. Kurdila, Adaptive feedback lineariza-tion for the control of a typical wing section with structuralnonlinearity, Nonlinear Dynamics 18 (3) (1999) 289–301.doi:10.1023/A:1008323629064.

[102] Y. Zhang, Y. Chen, J. Liu, G. Meng, Highly accurate solutionof limit cycle oscillation of an airfoil in subsonic flow, Ad-vances in Acoustics and Vibration 2011 (ID 926271) (2011)1–10. doi:10.1155/2011/926271.

[103] P. C. Chen, D. D. Liu, K. C. Hall, E. H. Dowell, Nonlinear Re-duced Order Modeling of Limit Cycle Oscillations of AircraftWings. Final report, Vol. AFRL-SR-BL-TR-00-, Ft. Belvoir

Defense Technical Information Center, Scottsdale, AZ., USA,2000, performong Organization: ZONA Technology, Inc.; Incollaboration with Duke Univ., Durham, NC., USA.URL InternetResource,handle.dtic.mil

[104] J. Ko, T. Strganac, J. Junkins, M. Akella, A. Kurdila, Struc-tured model reference adaptive control for a wing section withstructural nonlinearity, J. Vibration and Control 8 (5) (2002)553–573.

[105] D. Li, S. Guo, J. Xiang, Aeroelastic dynamic response andcontrol of an airfoil section with control surface nonlinearities,J. Sound and Vibration 329 (22) (2010) 4756–4771.

[106] Y. Bichiou, M. Hajj, A. Nayfeh, Effectiveness of a nonlin-ear energy sink in the control of an aeroelastic system, Non-linear Dynamics 86 (4) (2016) 2161–2177. doi:10.1007/

s11071-016-2922-y.[107] S. Irani, S. Sazesh, V. Molazadeh, Flutter analysis of

a nonlinear airfoil using stochastic approach, NonlinearDynamics 84 (3) (2016) 1735–1746. doi:10.1007/

s11071-016-2601-z.[108] S. He, Z. Yang, Y. Gu, Nonlinear dynamics of an aeroe-

lastic airfoil with free-play in transonic flow, Nonlin-ear Dynamics 87 (4) (2017) 2099–2125. doi:10.1007/

s11071-016-3176-4.[109] W. Tian, Z. Yang, Y. Gu, X. Wang, Analysis of nonlin-

ear aeroelastic characteristics of a trapezoidal wing in hy-personic flow, Nonlinear Dynamics (2017) 1–28(in press).doi:10.1007/s11071-017-3511-4.

[110] X. Wei, J. E. Mottershead, Robust passivity-based continuoussliding-mode control for under-actuated nonlinear wing sec-tions, Aerospace Science and Technology 60 (2017) 9 – 19.doi:https://doi.org/10.1016/j.ast.2016.10.024.

[111] S. Fazelzadeh, M. Azadi, E. Azadi, Suppression of nonlinearaeroelastic vibration of a wing/store under gust effects usingan adaptive-robust controller, J. Vibration and Control 23 (7)(2017) 1206–1217. doi:10.1177/1077546315591336.

[112] J. Edwards, Unsteady Aerodynamic Modeling and ActiveAeroelastic Control, Department of Aeronautics and Astro-nautics, Stanford University., 1977.URL https://books.google.ru/books?id=

dNYDAAAAIAAJ

[113] T. Strganac, J. Ko, D. Thompson, A. Kurdila, Identifica-tion and control of limit cycle oscillations in aeroelastic sys-tems, in: Proc. 40th AIAA/ASME/ASCE/AHS/ASC Struc-tures, Structural Dynamics, and Materials Conference and Ex-hibit, 1999, aIAA Paper No. 99–1463.

[114] F. Piovanelli, P. Paoletti, G. Innocenti, Enhanced nonlinearmodel and control design for a flexible wing, in: Proc. Euro-pean Control Conference (ECC 2016), EUCA, Aalborg, Den-mark, 2016, pp. 80–85. doi:10.1109/ECC.2016.7810267.