Hidden Periodic and Chaotic Oscillations in Nonlinear Dynamical

Preprints of the 19th World Congress The International Federation of Automatic Control Cape Town, South Africa. August 24-29, 2014

Hidden attractors in dynamical systems: systems with no equilibria, multistability and coexisting attractors

N.V. Kuznetsov, G.A. Leonov

Department of Applied Cybernetics, Saint-Petersburg State University, Russia Department of Mathematical Information Technology, University of Jyv¨askyl¨a,Finland (e-mail: [email protected])

Abstract: From a computational point of view it is natural to suggest the classiﬁcation of attractors, based on the simplicity of ﬁnding basin of attraction in the phase space: an attractor is called a hidden attractor if its basin of attraction does not intersect with small neighborhoods of equilibria, otherwise it is called a self-excited attractor. Self-excited attractors can be localized numerically by the standard computational procedure, in which after a transient process a trajectory, started from a point of unstable manifold in a neighborhood of unstable equilibrium, is attracted to the state of oscillation and traces it. Thus one can determine unstable equilibria and check the existence of self-excited attractors. In contrast, for the numerical localization of hidden attractors it is necessary to develop special analytical-numerical procedures in which an initial point can be chosen from the basin of attraction analytically. For example, hidden attractors are attractors in the systems with no-equilibria or with the only stable equilibrium (a special case of multistability and coexistence of attractors); hidden attractors arise in the study of well-known fundamental problems such as 16th Hilbert problem, Aizerman and Kalman conjectures, and in applied research of Chua circuits, phase-locked loop based circuits, aircraft control systems, and others.

Keywords: hidden oscillation, hidden attractor, systems with no equilibria, coexisting attractors, coexistence of attractors, multistable systems, multistability, 16th Hilbert problem, nested limit cycles, Aizerman conjecture, Kalman conjecture, absolute stability, nonlinear control system, describing function method, harmonic balance, phase-locked loop (PLL), drilling system, aircraft control systems, Chua circuits

1. INTRODUCTION attracted to the state of oscillation and traces it. Thus self-excited attractors can be easily visualized. An oscillation in dynamical system can be easily local- In contrast, for a hidden attractor its basin of attraction ized numerically if initial conditions from its open neigh- is not connected with unstable equilibria. For example, borhood lead to long-time behavior that approaches the hidden attractors are attractors in the systems with no- oscillation. Such an oscillation (or a set of oscillations) equilibria or with the only stable equilibrium (a special is called an attractor and its attracting set is called the 1 case of multistable systems and coexistence of attractors). basin of attraction . Thus, from a computational point Multistability is often undesired situation in many ap- of view it is natural to suggest the following classification plications, however coexisting self-excited attractors can of attractors, based on the simplicity of finding basin of be found by the standard computational procedure. In attraction in the phase space: contrast, there is no regular way to predict the existence or Definition. An attractor is called a hidden attractor if coexistence of hidden attractors in a system. Note that one its basin of attraction does not intersect with small neigh- cannot guarantee the localization of an attractor by the borhoods of equilibria, otherwise it is called a self-excited integration of trajectories with random initial conditions attractor. (especially for multidimensional systems) since its basin of attraction can be very small. For a self-excited attractor its basin of attraction is connected with an unstable equilibrium and, therefore, self- For numerical localization of hidden attractors it is nec- excited attractors can be localized numerically by the essary to develop special analytical-numerical procedures standard computational procedure, in which after a tran- in which an initial point can be chosen from the basin of sient process a trajectory, started from a point of unstable attraction analytically since there are no similar transient manifold in a neighborhood of unstable equilibrium, is processes leading to such attractors from the neighborhoods of equilibria. 1 Rigorous definitions of attractors can be found, e.g., in (Broer et al., 1991; Boichenko et al., 2005; Milnor, 2006; Leonov, 2008)

van der Pol oscillator Belousov-Zhabotinsky reaction 2-prey 1-predator model 5 0.45 4 3 0.35 15 2 1 0.25 13 y 0 y z 11 −1 0.15 −2 9 4 7 −3 0.05 x 10 5 2 −4 1.5 4 y 3 1 x 10 −5 −0.05 1 0.5 x −3 −2 −1 x0 1 2 3 −0.1 0.1 0.3 x0.5 0.7 0.9 1 0

Fig. 1. Standard computation of classical self-excited periodic oscillations.

Lorenz system Rossler: system 3 Chua’s circuit

50 25 2 20 1 40 15 z 0 30 z z 10 −1 20 5 −2 −3 10 0 8 30 −4 0 4 15 0.5 −20 10 0 10 −10 y −4 5 0 0 −10 y 0 y 10 −8 −5 x −0.5 x 20 −30 −12 −10 −2.5 −1.5 −0.5 0.5 1.5 2.5 x

Fig. 2. Standard computation of classical self-excited chaotic attractors.

2. SELF-EXCITED ATTRACTOR LOCALIZATION ered. Nowadays there is enormous number of publications devoted to the computation and analysis of various self- In the first half of the last century during the initial period exited chaotic oscillations (see, e.g., recent publications of the development of the theory of nonlinear oscillations (Awrejcewicz et al., 2012; Tuwankotta et al., 2013; Zelinka (Timoshenko, 1928; Krylov, 1936; Andronov et al., 1966; et al., 2013; Zhang et al., 2014) and many others). Stoker, 1950) much attention was given to analysis and In Fig. 1 is shown the computation of classical self-exited synthesis of oscillating systems, for which the problem of oscillations: van der Pol oscillator (van der Pol, 1926), the existence of oscillations can be studied with relative one of the modifications of Belousov-Zhabotinsky reaction ease. (Belousov, 1959), two preys and one predator model (Fujii, These investigations were encouraged by the applied re- 1977). search of periodic oscillations in mechanics, electronics, In Fig. 2 is shown the computation of classical self-excited chemistry, biology and so on (see, e.g., (Strogatz, 1994)) chaotic attractors: Lorenz system (Lorenz, 1963), Rössler The structure of many applied systems (see, e.g., Rayleigh system (Rossler, 1976), “double-scroll” attractor in Chua’s (Rayleigh, 1877), Duffing (Duffing, 1918), van der Pol circuit (Bilotta and Pantano, 2008). (van der Pol, 1926), Tricomi (Tricomi, 1933), Beluosov- Zhabotinsky (Belousov, 1959) systems) was such that the 3. HIDDEN ATTRACTOR LOCALIZATION existence of oscillations was almost obvious since the oscillations were excited from unstable equilibria (self-excited oscillations). This allowed one, following Poincare’s advice While the classical attractors are self-excited attractors “to construct the curves defined by differential equations” and, therefore, they can be obtained numerically by the (Poincaré,1881), to visualize periodic oscillations by the standard computational procedure, for the localization of standard computational procedure. hidden attractors it is necessary to develop special procedures since there are no similar transient processes leading Then in the middle of 20th century it was found numer- to such attractors. At first the problem of investigating ically the existence of chaotic oscillations (Ueda et al., hidden oscillations arose in the second part of Hilbert’s 1973; Lorenz, 1963), which were also excited from unstable 16th problem (1900) on the number and possible disposi- equilibria and could be computed by the standard compu- tions of limit cycles in two-dimensional polynomial systems tational procedure. Later many other famous self-excited (see, e.g., (Reyn, 1994; Anosov et al., 1997; Chavarriga attractors (Rossler, 1976; Chua et al., 1986; Sprott, 1994; and Grau, 2003; Li, 2003; Dumortier et al., 2006; Lynch, Chen and Ueta, 1999; Lu and Chen, 2002) were discov- 2010) and authors’ works (Leonov and Kuznetsov, 2007a;

5446 19th IFAC World Congress Cape Town, South Africa. August 24-29, 2014

Kuznetsov and Leonov, 2008; Leonov et al., 2008; Leonov 2004; Leigh, 2004; Gil, 2005; Margaliota and Yfoulis, 2006; and Kuznetsov, 2010; Leonov et al., 2011a; Kuznetsov Brogliato et al., 2006; Michel et al., 2008; Liao and Yu, et al., 2013b; Leonov and Kuznetsov, 2013b)). The prob- 2008; Wang et al., 2009; Llibre et al., 2011; Haddad lem is still far from being resolved even for a simple class and Chellaboina, 2011; Grabowski, 2011; Alli-Oke et al., of quadratic systems. Here one of the main difficulties in 2012; Nikravesh, 2013). The generalization of Kalman’s computation is nested limit cycles — hidden oscillations. conjecture to multidimensional nonlinearity is known as Markus-Yamabe conjecture (Markus and Yamabe, 1960) Later, the problem of analyzing hidden oscillations arose (also proved to be false (Bernat and Llibre, 1996; Cima in engineering problems. In 1956s in M. Kapranov’s work et al., 1997; Leonov and Kuznetsov, 2013b)). (Kapranov, 1956) on the stability of PLL model, the qualitative behavior of this model was studied and the At the end of the last century the difficulties of numerical estimation of stability domain was obtained. In these analysis of hidden oscillations arose in simulations of air- investigations Kapranov assumed that in PLL systems craft’s control systems (anti-windup) (Lauvdal et al., 1997; there were self-excited oscillations only. However, in 1961, Leonov et al., 2012a; Andrievsky et al., 2012, 2013a,b) N. Gubar’ revealed a gap in Kapranov’s work and the and drilling systems (de Bruin et al., 2009; Kiseleva et al., possibility of the existence of hidden oscillations in two- 2012; Leonov and Kuznetsov, 2013b; Kiseleva et al., 2014; dimensional PLL models was showed analytically (Gubar’, Leonov et al., 2014), which caused crashes. 1961; Leonov and Kuznetsov, 2013b). Further investigations on hidden oscillations were greatly In 1957 R.E. Kalman stated the following (Kalman, 1957): encouraged by the present authors’ discovery (Kuznetsov “If f(e) in Fig. 1 [see Fig. 3] is replaced by constants K et al., 2010; Leonov et al., 2010c; Kuznetsov et al., 2011c,b; corresponding to all possible values of f 0(e), and it is found Leonov et al., 2011c,b; Bragin et al., 2011; Leonov and that the closed-loop system is stable for all such K, then it Kuznetsov, 2012; Kuznetsov et al., 2013a; Leonov and intuitively clear that the system must be monostable; i.e., Kuznetsov, 2013a), in 2009-2010 (for the first time), of all transient solutions will converge to a unique, stable chaotic hidden attractor in Chua’s circuits — a simple critical point.” Kalman’s conjecture is the strengthening electronic circuit with nonlinear feedback, which can be described by the following equations f(e) c r Σ e f G(s x˙ = α(y − x − ψ(x)) + y˙ = x − y + z − (1) z˙ = −(βy + γz) ψ(x) = m1x + (m0 − m1)sat(x)

Fig. 3. Nonlinear control system. G(s) is a linear transfer 8 15 function, f(e) is a single-valued, continuous, and 6 10 diﬀerentiable function (Kalman, 1957) 4

2 5 of Aizerman’s conjecture (Aizerman, 1949), in which in z 0 z 0

−2 place of condition on the derivative of nonlinearity it −5 −4 is required that the nonlinearity itself belongs to linear 5 −10 −6 sector. Here the application of widely-known describing 2 0 0 y 2 −8 y −15 −2 −15 −10 −5 −5 function method (DFM) leads to the conclusion on the −6 −4 −2 0 2 4 6 0 5 10 15 absence of oscillations and the global stability of the only x x stationary point, what explains why these conjectures were Fig. 4. a. Self-excited and b. Hidden Chua attractor with put forward. similar shapes In the last century the investigations of Aizerman’s and Kalman’s conjectures on absolute stability have led to Until recently only self-excited attractors have been found the finding of hidden oscillations in automatic control in Chua circuits (see, e.g., Fig. 4 a.). Note that L. Chua systems with a unique stable stationary point and with a himself, in analyzing various cases of attractors existence nonlinearity, which belongs to the sector of linear stability in Chua’s circuit (Chua, 1992), does not admit the ex- (see, e.g., (Leonov et al., 2010a; Bragin et al., 2010; Leonov istence of hidden attractor (being discovered later) in his and Kuznetsov, 2011a; Kuznetsov et al., 2011b; Leonov circuits. Now it is shown that Chua’s circuit and its various and Kuznetsov, 2011b, 2013b) and surveys (Leonov et al., modifications (Kuznetsov et al., 2010, 2011a; Leonov et al., 2010b; Bragin et al., 2011; Leonov and Kuznetsov, 2013b)). 2011c, 2012b; Kuznetsov et al., 2013a) can exhibit hidden chaotic attractors (see, Fig. 4 b., Fig. 5 3 ) with posi- Discussions and recent developments on existence of tive largest Lyapunov exponent (Leonov and Kuznetsov, periodic solution and absolute stability theory, related 2007b) 4 . to Aizerman and Kalman conjectures, are presented, e.g., in (Kaszkurewicz and Bhaya, 2000; Lozano, 2000; 3 Applying more accurate analytical-numerical methods (Lozi and Vidyasagar, 2002; Ackermann and Blue, 2002; Rasvan, Ushiki, 1993), one might here distinguish a few close coexisting attractors. 2 In engineering practice for the analysis of the existence of periodic 4 While there are known the effects of the largest Lyapunov ex- solutions it is widely used classical harmonic linearization and ponent (LE) sign reversal for nonregular time-varying linearizations, describing function method (DFM). However this classical method is the computation of Lyapunov exponents for linearization of nonlinear not strictly mathematically justified and can lead to untrue results autonomous system along non stationary trajectories is widely used

5447 19th IFAC World Congress Cape Town, South Africa. August 24-29, 2014

K = 1.8

x(0) = 2.0848, y(0) = 0.0868, z(0) = −2.819 5

~ ~ ~ ~ x(0) = 0.01, y(0) = 0, z(0) = 0 x(t)

− 0

−5 Circuit 1 x(t) 0 50 100 150 200 250 300 unst A 4 Circuit 2 t M hidden 0.5

2 2 ~ y(t) ~

− 0 0

y(t) y(t) −0.5 0 50 100 150 200 250 300 −2 t

z(t) and z(t) z(t) 5 −4 ~

0.4 z(t) 0 0.2 5 − 0 −5 S 0 z(t) 0 50 100 150 200 250 300 1 ~ −0.2 ~ st y(t) and y(t) −0.4 −5 x(t) and x(t) t M1

st Fig. 6. Synchronization on self-excited attractor. M2 K = 1.8 x(0) = -3.7727, y(0) = -1.3511, z(0) = 4.6657 10 ~ z S ~ ~ ~ x(t) 2 x(0) = 0.01, y(0) = 0, z(0) = 0 − 0 −10 Circuit 1 x(t) 0 50 100 150 200 250 300 10 Circuit 2 t 2 ~

F 5 y(t) ~ 0 − 0 0 y y(t) −2 unst 0 50 100 150 200 250 300 M −5 t

1 and z(t) z(t) 10 −10 ~ 10 z(t) 0 5 10 − 0 −10

0 z(t) x ~ −5 ~ 0 50 100 150 200 250 300 y(t) and y(t) −10 −10 x(t) and x(t) t

Fig. 7. No synchronization on hidden attractor. et al., 2014; Sprott, 2014), see also (Graham and Tl, 1986; Fig. 5. Hidden chaotic attractor (Ahidden — green domain) Banerjee, 1997; Pisarchik et al., 2006; Feudel, 2008; Yang. in classical Chua circuit: locally stable zero equi- et al., 2010; Wei, 2011; Wang and Chen, 2012b,a; Wei, librium F0 attracts trajectories (black) from stable st 2012; Wei and Pehlivan, 2012; Liu et al., 2013; Abooee manifolds M1,2 of two saddle points S1,2; trajectories et al., 2013; Sprott et al., 2013; Galias and Tucker, 2013; unst (red) from unstable manifolds M1,2 tend to infin- Pisarchik, 2014). ity; α = 8.4562, β = 12.0732, γ = 0.0052, m0 = −0.1768, m1 = −1.1468. 4. ANALYTICAL-NUMERICAL PROCEDURE FOR HIDDEN OSCILLATIONS LOCALIZATION Nowadays Chua circuits and other electronic generators of chaotic oscillations are widely used in various chaotic An effective method for the numerical localization of hid- secure communication systems. The operation of such den attractors in multidimensional dynamical systems is systems is based on the synchronization chaotic signals of the method based on a homotopy and numerical con- two chaotic identical generators (transmitter and receiver) tinuation: it is necessary to constructed a sequence of for different initial data (Kapitaniak, 1992; Ogorzalek, similar systems such that for the first (starting) system 1997; Yang, 2004; Eisencraft et al., 2013). The control the initial data for numerical computation of oscillating signal, depending on the difference of signals of transmitter solution (starting oscillation) can be obtained analytically, and receiver, changes the state of receiver. The existence e.g, it is often possible to consider the starting system with of hidden oscillations and the improper choice of the self-excited starting oscillation. Then the transformation form of control signal may lead to inoperability of such of this starting oscillation is tracked numerically in passing systems. For example, consider the synchronization of two from one system to another. Chua system (1), linearly coupled through the second Further, it is demonstrated an example of effective equation by K(y − y˜), with initial data x(0), y(0), z(0) analytical-numerical approach for hidden oscillations lo- in a small neighborhood of the zero equilibrium point calization in the systems with scalar nonlinearity, which is for the first system andx ˜(0), y˜(0), z˜(0) on the attractor based on the continuation principle, the method of small for the second system. Fig. 6 and Fig. 7 show that the parameter and, a modification of the describing function synchronization is acquired for the classical double-scroll method (DFM) 5 . attractor (α = 9.3516, β = 14.7903, γ = 0.0161, m = 0 Consider a system with one scalar nonlinearity −1.1384 m1 = −0.7225) and may not be acquired for the dx hidden attractor. = Px + qψ(r∗x), x ∈ n. (2) dt R Recent examples of hidden attractors can be found in Here P is a constant (n × n)-matrix, q, r are constant n- (Jafari and Sprott, 2013; Molaie et al., 2013; Li and Sprott, dimensional vectors, ∗ is a transposition operation, ψ(σ) is 2014b,a; Lao et al., 2014; Li and Sprott, 2014a; Chaudhuri a continuous piecewise-differentiable scalar function, and and Prasad, 2014; Wei et al., 2014; Li et al., 2014; Wei ψ(0) = 0. for investigation of chaos (Leonov and Kuznetsov, 2007b; Kuznetsov 5 DFM is used here for determining an initial periodic solution for and Leonov, 2005, 2001; Kuznetsov et al., 2014), where the pos- continuation method. However, in many cases the initial periodic itiveness of the largest Lyapunov exponent is often considered as solution is a self-excited oscillation and may be obtained by the indication of chaotic behavior in the considered nonlinear system. standard computational procedure.

5448 19th IFAC World Congress Cape Town, South Africa. August 24-29, 2014

Define a coefficient of harmonic linearization k (suppose real numbers; A3 is an ((n − 2) × (n − 2))-matrix with that such k exists) in such a way that the matrix P0 = P+ all eigenvalues with negative real parts. Without loss of ∗ kqr has a pair of purely imaginary eigenvalues ±iω0 generality, it can be assumed that for the matrix A3 ∗ (ω0 > 0) and the rest eigenvalues have negative real parts there exists a positive number d > 0 such that y3(A3 + ∗ 2 n−2 . Rewrite system (2) as A3)y3 ≤ −2d|y3| , ∀ y3 ∈ R . dx = P x + qϕ(r∗x), (3) Introduce the describing function dt 0 Z 2π/ω0 where ϕ(σ) = ψ(σ) − kσ. Φ(a) = ϕ cos(ω0t)a cos(ω0t)dt, (7) Introduce a finite sequence of the functions 0 and assume the existence of its derivative. ϕ0(σ), ϕ1(σ), . . . , ϕm(σ) such that the graphs of neighboring functions ϕj(σ) and Theorem 4.1. (Leonov, 2010; Bragin et al., 2011) If there ϕj+1(σ) slightly differ from one another, the initial func- is a positive a0 such that tion ϕ0(σ) is small, and the final function ϕm(σ) = ϕ(σ) dΦ(a) Φ(a ) = 0, b < 0 (8) (e.g., below it is considered ϕj(σ) = εjϕ(σ), εj = j/m). 0 1 da a=a Since the initial function ϕ0(σ) is small over its domain of 0 definition, the method of describing functions give mathe- then there is a periodic solution 0 ∗ matically correct conditions of the existence of a periodic x (0) = S(y1(0), y2(0), y3(0)) solution for system with the initial data y1(0) = a0 +O(ε), y2(0) = 0, y3(0) = dx = P x + qϕ0(r∗x) (4) On−2(ε). dt 0 see, e.g. (Leonov, 2010; Leonov et al., 2010c,a). This theorem describes the procedure of the search for stable periodic solutions by the standard describing function Its application allows one to define a stable nontrivial method. periodic solution x0(t) (a starting oscillating attractor is denoted further by A0). Note that condition (8) cannot be satisfied in the case when the conditions of Aizerman’s and Kalman’s conjec- Two alternatives are possible. The first case: all the points tures are fulfilled (i.e. a nonlinearity belongs to the sector of A0 are in an attraction domain of the attractor A1, of linear stability). But it is possible to modify and justify which is an oscillating attractor of the system the describing function method for the special nonlineari- dx = P x + qϕj(r∗x) (5) ties satisfying the conditions of Aizerman’s and Kalman’s dt 0 conjectures Leonov and Kuznetsov (2011a); Bragin et al. with j = 1. The second case: in the passing from system (2011). (4) to system (5) with j = 1 it is observed a loss of stability (bifurcation) and A0 vanishes. In the first case the solution x1(t) can be found numerically by starting a trajectory 5. HIDDEN OSCILLATIONS IN AIRCRAFT of system (5) with j = 1 from the initial point x0(0). If CONTROL SYSTEMS numerical integration over a sufficiently large time interval [0,T ] shows that the solution x1(t) remains bounded and is The presence of control input saturation can dramatically not attracted by an equilibrium, then this solution reaches degrade the closed-loop system performance. Since the an attractor A1. In this case it is possible to proceed to feedback loop is broken when the actuator saturates, the system (5) with j = 2 and to perform a similar procedure unstable modes of the regulator may then drift to undesir- of computation of A2 by starting a trajectory of system able values. A terrifying illustration of this detrimental ef- (5) with j = 2 from the initial point x1(T ) and computing fect is given by the pilot-induced oscillations that entailed a trajectory x2(t). the YF-22 (Boeing) crash in 1992 and Gripen (SAAB) crash in 1993. Following this procedure, sequentially increasing j, and computing xj(t) (a trajectory of system (5) with the initial Consider the following equations (Andrievsky et al., 2013) j−1 data x (T )), one can either find a solution around Am of the aircraft short period angular motion along the (an attractor of system (5) with j = m, i.e. original system longitudinal axis, linearized about trim operating points (3)), or observe at a certain step j a bifurcation, where the αtrim, δe,trim, cf. (Reichert, 1992; Ferreres and Biannic, attractor vanishes. 2007; Biannic and Tarbouriech, 2009): Let ϕ0(σ) = εϕ(σ) with ε being a small parameter. To α˙ = Zαα + q + Zδδe, 0 (9) define the initial data x (0) of the initial periodic solution, q˙ = Mαα + Mqq + Mδδe,. system (4) with the nonlinearity ϕ0(σ) is transformed by 6 Here α, δe are the deviations of the angle-of-attack and a linear nonsingular transformation x = Sy to the form the elevator deflection angle from a trim flying condition, 0 ∗ y˙1 = −ω0y2 + b1ϕ (y1 + c3y3), the variable q represents the body axis pitch rate, Mα, 0 ∗ M , M , Z , Z are the linearized model parameters for y˙2 = ω0y1 + b2ϕ (y1 + c3y3), (6) q δ α δ 0 ∗ the given flight conditions. Let the elevator deflection y˙ 3 = A3y3 + b3ϕ (y1 + c3y3). angle be symmetrically bounded about trim value δe,trim: Here y , y are scalars, y is (n − 2)-dimensional vector; ¯ 1 2 3 |δe| ≤ δe. Suppose that the linear dynamics of the elevator b3 and c3 are (n − 2)-dimensional vectors, b1 and b2 are servosystem between PID-controller output signal u(t) 6 ˜ Such a transformation exists for nondegenerate transfer functions. and unsaturated elevator deflection δe(t) along with the

5449 19th IFAC World Congress Cape Town, South Africa. August 24-29, 2014

ε=0.25 ε=0.5

1 1

=q 0 =q 0 3 3 x x

−1 −1 5 5 −3 0.05 −3 0.05 x 10 0 0 x 10 0 0 x =σ −5 −0.05x =α x =σ −5 −0.05x =α 2 1 2 1 ε=0.75 ε=1

1 1

=q 0 =q 0 3 3 x x

−1 −1 5 5 −3 0.05 −3 0.05 x 10 0 0 x 10 0 0 x =σ −5 −0.05x =α x =σ −5 −0.05x =α 2 1 2 1

Fig. 8. Multistep localization of hidden oscillation: εj = j/4. At the last step (i.e. j = m in (5)) there is stable zero equilibrium point coexist with stable oscillation (hidden oscillation). compensating device model is described by the following poles inside a truncated sector with a minimal degree of transfer function stability η = 5.6 (to ensure fulfillment of the settling time ˜ 2 2 δe k(T s + 2ξ2T2s + 1) specification) and a minimal damping ratio ξ = 0.3 as 2 −1 W (s) = = 2 2 , (10) follows: kP = 5.5, kq = 0.55 s, kI = 19 s . The H∞ gain u T1 s + 2ξ1T1s + 1 of the closed-loop system is found as H∞ = 1.13 and the where k denotes the servo static gain, T1, T1 are the frequency stability margin is 75 deg. time constants and ξ1, ξ2 stand for the damping ratios. Finally, the saturated actuator output signal (the elevator Consider the closed-loop aircraft control system behavior deflection) is as follows imposed by saturation of the elevator servo. The focus of our attention is the possibility of hidden oscillations in δ˜ (t) ¯ e the closed-loop system. Consider behavior analysis of the δe(t) = δe sat ¯ , (11) δe particular control system (9) – (12). where sat(·) denotes the saturation function. At the first step let us find matrices P, q, r in (2) for In the sequel the following parameter values are taken the considered case. After simple calculations one can find −1 −2 (Barbu et al., 1999): Zα = −1.0 s , Zδ ≈ 0, Mα = 15 s , from (9), (10), (12) the following matrices: −1 −2 ¯ Mq = 3.0 s , Mδ = −18 s , δe = 20 deg (≈ 0.35 rad),  Z 0 1 0 0    k = 10, T1 = 0.083 s, T2 = 0.057 s, ξ1 = 0.1, ξ2 = 0.4. α −Zδ  −1 0 0 0 0   0  It may be easily checked that for the given parameters  M 0 M 0 0    P= α q  , q= −Mδ  the aircraft is weathercock unstable; the eigenvalues s1,2 −1 −2  −kP kI −kq −2ξ1T1 −T1   0  of system (9) are taken as s1 = −6, s2 = 2. Let the 0 0 0 1 0 0 control goal be the tracking for the commanded angle-of- ∗ ∗ 2 −2 2 −2 2 −2 attack α (t). The following classical PID controller may r = − kkP T2 T1 , kkI T2 T1 , −kkqT2 T1 , be applied for eliminating the static tracking error and 2 −1 −2 2 −2 −2 the achievement of the desired transient specification for k(2ξ2T2 − 2ξ1T2 T1 )T1 , k(1 − T2 T1 )T1 . the “nominal” (non-saturated) system: The nonlinearity ψ(·) in (2) has a form (11). u(t) = kI σI (t) + kP e(t) + kDq(t), t Z Numerically, for the given above parameter values, matrices P, q, r take the from: σI (t) = e(τ) dτ, σ(0) = 0, (12) 0  −1.0 0 1 0 0   0  ∗ where e(t) = α(t) − α (t) is a tracking error, kP , kq, kI  −1 0 0 0 0   0      are proportional, derivative and integral controller gains P= 15.0 0 −3.0 0 0  , q= 18.0  (respectively).  −5.56 19.1 −0.556 −2.41 −145   0  0 0 0 1 0 0 The gains kI , kP , kD are computed for the nominal ∗ (unsaturated) mode so as to place the nominal closed loop r = (−26.1 89.6 −2.61 54.5, 763) .

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Application of the described above multistep localization Barbu, C., Reginatto, R., Teel, A.R., and Zaccarian, L. procedure allows one to find hidden oscillation (see Fig. 8) (1999). Anti-windup design for manual flight control. in the considered system. In Proc. American Control Conf. (ACC’99), volume 5, 3186–3190. AACC. 6. CONCLUSION Belousov, B.P. (1959). Collection of short papers on radiation medicine for 1958, chapter A periodic reaction Since one cannot guarantee revealing complex oscillations and its mechanism. Moscow: Med. Publ. [in Russian]. regime by linear analysis and standard simulation, rigorous Bernat, J. and Llibre, J. (1996). Counterexample to nonlinear analysis and special numerical methods should Kalman and Markus-Yamabe conjectures in dimension be used for investigation of nonlinear control systems. larger than 3. Dynamics of Continuous, Discrete and Impulsive Systems, 2(3), 337–379. ACKNOWLEDGMENT Biannic, J. and Tarbouriech, S. (2009). 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