Adv Syst Sci Appl 2021; 03:1–11 Published online at https://ijassa.ipu.ru. Simultaneously Robust Chaos Synchronization and Identification of Unknown Parameters for Nonlinear Gyro System Abolghasem Daeichian1,2*, Shahram Aghaei3 1Department of Electrical Engineering, Faculty of Engineering, Arak University, Arak, 38156-8-8349, Iran 2Institute of Renewable Energy, Arak University, Arak, 38156-8-8349, Iran 3Department of Electrical Engineering, Yazd University, Yazd, Iran Abstract: This paper concerns robust synchronization and parameter identification for nonlinear gyroscope systems. Gyros are widely utilized in navigational applications where synchronization plays a vital role. A system of nonlinear dynamical equations with some parameters presents a model of gyro systems. The parameters of gyro can vary in time, which can lead to desynchronization of the gyro systems. In this paper, we assume that a gyro system has bounded time-varying unknown parameters and the synchronization problem is considered in two situations. First, the synchronization of two gyroscopes with identical dynamical model and, second, the synchronization of a gyroscope with the Rössler system. The Lyapunov stability theory with control terms is employed to cope with the problem. Also, the identification of time- varying unknown parameters is the side goal of the paper. The proposed scheme synchronizes chaotic nonlinear systems in both situations appropriately. In addition, the slave parameters converge to the nominal values of master parameters despite uncertainty. Simulation results illustrate the superiority of the proposed method. Keywords: robust synchronization, chaos, parameter identification, gyroscope, Rössler system 1. INTRODUCTION Numerous studies confirm that complex and chaotic behaviors are observed in physics, mechanics, engineering, etc. [1–4]. The chaotic behavior of dynamical systems is interested in many researches [5,6]. Also, chaos synchronization in applied systems is studied [7]. The intent of chaos synchronization is to synchronize the states of at least two chaotic systems despite the difference in initial conditions, the difference in model parameters, and even the difference in dynamical models. How to reach synchronization under such distinctions is a challenging crux [8]. The identification of unknown system parameters is also the side goal of many studies. Many schemes have been proposed to synchronize two chaotic systems, such as adaptive control [9–11], sliding mode control [12, 13], robust methods [14, 15], intelligent base methods [16–19] and Lyapunov base methods [20–23]. The gyroscopes are widely employed in navigational, aeronautical, and space engineering. Those have complex dynamical motion including periodic, sub-harmonic, quasi-periodic, and chaotic [24]. Gyro synchronization for a couple of chaotic systems with one-way linking is considered in [25] and it is extended in [26] by applying an active control. Fuzzy sliding mode control is employed to synchronize uncertain chaotic nonlinear gyros in [27]. A variable structure control approach using Neural Network with multi-quadratic radial basis ∗Corresponding author: a-daeichian@araku.ac.ir, a.daeichian@gmail.com 2 A. DAEICHIAN, S. AGHAEI function is introduced in [28] where gyros with known and unknown system parameters are synchronized. The existence of external disturbance is investigated in [29], where fast non- singular terminal sliding mode surface is employed to derive an adaptive finite-time control in order to synchronize the gyro systems. Synchronization and parameter identification for the symmetric gyroscope system with constant parameters have been done in [30]. It is not robust with respect to parameter’s variation and does not converge to the nominal value of parameter. This paper concerns developing robust synchronization based on the Lyapunov theorem with control terms for nonlinear gyroscope systems with unknown time-varying parameters. In addition, unknown parameter identification is expected. The idea is to find the control input u such that the parameters’ uncertainty eliminates in the Lyapunov function. Two situations are considered for the slave system. First, a similar gyro dynamical system, and second, the Rössler system which has another dynamical equation of master gyro. The parameters of slave system or control input converge to the nominal value of master system parameters by a parameter updating law in spite of the uncertainty. It is proved by the Lyapunov stability theorem that the proposed controller can synchronize chaotic systems with time-varying unknown parameters. Simulation results show that method in [30] has a larger margin of error than the proposed robust method here. Section 2 states the problem. Controller and the parameter identifier are designed in section 3. Simulation results verify achieving synchronization in section 4. Finally, Section 5 concludes the paper. 2. PROBLEM STATEMENTS Fig. 2.1 shows a gyro which is established on a vibrating base. The dynamical equations of the system are expressed by the Euler’s angles θ, φ and ψ and a multiple harmonic equation∑n k=1Ak sinωkt which represents base vibration. If we define x1 = θ, x2 = θ̇ and x3 = φ̇, the dynamical equations of the system in the state space framework are [30, 31]: ẋ1 = x2 ẋ2 = −(βφ − βϕ cosx1)(βϕ − βφ cosx1) I2 sin3 x1 − C1(t) I x2 + Mgl I sinx1 − Mg I n∑ k=1 Ak sin (ωkt) sinx1 ẋ3 = −2 cosx1 sinx1 x2x3 + βϕx2 I sinx1 (2.1) where I , Mg, and l denote the polar moments of gyro inertia, the gravity force, and the distance between origin and the center of gravity. βφ and βϕ are constants of the motion. C1(t) is a parameter vector which is unknown, time varying, and uncertain. It worth noting that x1 = θ, which is known as nutation angle, is the angle between theXY Z (fixed) axis and xyz (body) axis. It is shown in Fig.(11-16) of [31] that θ(t) is limited by two extreme values 0 < θ1 < θ(t) < θ2 < π which correspond to the turning points of the central-force. This fact prevents Eq. (2.1) to become singular. For more details on the derivation of this model we refer to chapter 11 of [31]. However, C(t) has the nominal value C with uncertainty or disturbance ∆C(t), that is to say C1(t) = C + ∆C(t). It is assumed that the uncertainty is bounded, namely: Assumption 2.1: (Uncertainty boundedness) ‖∆C‖ ≤ k ∀t ∈ R+ (2.2) Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) SIMULTANEOUSLY ROBUST CHAOS SYNCHRONIZATION AND IDENTIFICATION OF UNKNOWN 3 where k is a known, positive, and real number. Fig. 2.1. Schematic of a gyroscope on a vibrating base Eq. (2.1) is the master (drive) system model. The major goal is that the states of a slave (response) system synchronize with the states of the master system. Here, we consider two different dynamical systems as slave system. First, a gyro with state space representation similar to the master system: ẋs1 = xs2 + u1 ẋs2 = −(βφ − βϕ cosxs1)(βϕ − βφ cosxs1) I2 sin3 xs1 − C2(t) I xs2 + Mgl I sinxs1 − Mg I n∑ k=1 Ak sin (ωkt) sinxs1 + u2 ẋs3 = −2 cosxs1 sinxs1 xs2x s 3 + βϕx s 2 I sinxs1 + u3, (2.3) Second, the Rössler system as slave system: ẋs1 = −xs2 − xs3 + u1 ẋs2 = xs1 + b1x s 2 + u2 ẋs3 = a1 + xs3 (xs1 − c1) + u3 (2.4) where a1, b1, and c1 are the Rössler system parameters. Three control inputs u1, u2 and u3 are augmented to the slave dynamical equations to control the synchronization of the systems. C2(t) is the parameter vector of the slave system. The aim is to not only synchronize the states of two gyros but also C2(t) converges to C (nominal value of the unknown parameter), simultaneously. Considering assumption 2.1, any robust controller has to satisfy the following conditions: 1. limt→∞ ‖e(t)‖ = 0 2. limt→∞ ‖C̃(t)‖ = 0 where e(t) = xs(t)− x(t) and C̃(t) = C2(t)− C. Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) 4 A. DAEICHIAN, S. AGHAEI 3. DESIGN OF CONTROLLER AND IDENTIFIER 3.1. Gyro system as slave If gyro dynamical equations (2.3) are considered as a slave system, then the error dynamic can be written in the form: ė1 = e2 + u1 ė2 = −(βφ − βϕ cosxs1)(βϕ − βφ cosxs1) I2 sin3 xs1 + (βφ − βϕ cosx1)(βϕ − βφ cosx1) I2 sin3 x1 −C2 I xs2 + C1 I x2 + Mgl I (sinxs1 − sinx1) −Mg I n∑ k=1 Ak sin (ωkt)(sinx s 1 − sinx1) + u2 ė3 = −2 cosxs1 sinxs1 xs2x s 3 + 2 cosx1 sinx1 x2x3 + βϕx s 2 I sinxs1 − βϕx2 I sinx1 + u3 (3.5) where e1 = xs1 − x1, e2 = xs2 − x2, and e3 = xs3 − x3. The following theorem introduces control inputs u1, u2, u3, and parameter identification updating law such that limt→∞ ‖e(t)‖ = 0 and limt→∞ ‖C̃(t)‖ = 0. Theorem 3.1: Consider the master system (2.1) with unknown parameter C1(t) which satisfies Assumption 2.1 and the slave system (2.3). Then, there exist the control laws u1 = −e2 − e1 u2 = (βφ − βϕ cosxs1)(βϕ − βφ cosxs1) I2 sin3 xs1 − (βφ − βϕ cosx1)(βϕ − βφ cosx1) I2 sin3 x1 +( C2 I − 1)e2 − Mgl I (sinxs1 − sinx1)− ke2x2 I‖e2x2‖ x2 + Mg I n∑ k=1 Ak sin (ωkt)(sinx s 1 − sinx1) u3 = 2 cosxs1 sinxs1 xs2x s 3 − 2 cosx1 sinx1 x2x3 − βϕx s 2 I sinxs1 + βϕx2 I sinx1 − e3 (3.6) and the parameter updating law Ċ2 = e2x2 I (3.7) which leads to master-slave synchronization. Moreover, the unknown parameter of the slave system converges to the nominal value of the master one. Proof Consider the following Lyapunov function: V (e1, e2, e3, C̃) = 1 2 (e21 + e22 + e23 + C̃2) (3.8) Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) SIMULTANEOUSLY ROBUST CHAOS SYNCHRONIZATION AND IDENTIFICATION OF UNKNOWN 5 where C̃(t) = C2(t)− C. The first derivation of the Lyapunov function along the error dynamic Eq.(3.5) is: V̇ = e1(e2 + u1) + e2(− (βφ − βϕ cosxs1)(βϕ − βφ cosxs1) I2 sin3 xs1 + (βφ − βϕ cosx1)(βϕ − βφ cosx1) I2 sin3 x1 − C2 I xs2 + C1 I x2 + Mgl I (sinxs1 − sinx1)− Mg I n∑ k=1 Ak sinωkt(sinx s 1 − sinx1) + u2) +e3(− 2 cosxs1 sinxs1 xs2x s 3 + 2 cosx1 sinx1 x2x3 + βϕx s 2 I sinxs1 − βϕx2 I sinx1 + u3) + C̃Ċ2 = −e21 − e22 − e23 + ∆C I x2e2 − ke22x2 2 I‖e2x2‖ (3.9) Taking into account the inequality ‖A+B‖ ≤ ‖A‖+ ‖B‖, we obtain the inequality: V̇ ≤ −‖e‖2 + ‖∆C I ‖‖e2x2‖ − ‖ k I ‖‖e2x2‖ (3.10) Finally, assumption 2.1, gives: V̇ ≤ −‖e‖2 (3.11) So, V̇ < 0 for ‖e‖ 6= 0. This Lyapunov function and its derivation demonstrate that the error dynamic Eq.(3.5) is asymptotically stable, that is to say, the master-slave synchronization is achieved. It also guarantees the convergence of the slave system parameter C2(t) to the nominal value of the master system parameter C. 3.2. Rössler system as slave Now, the Rössler system is considered as a slave. The error dynamic in this situation is: ė1 = −xs2 − xs3 − x2 + u1 ė2 = xs1 + b1x s 2 + C1 I x2 − Mgl I sinx1 + (βφ − βϕ cosx1)(βϕ − βφ cosx1) I2 sin3 x1 + Mg I n∑ k=1 Ak sin (ωkt)(sinx1) + u2 ė3 = a1 + xs3(x s 1 − c1) + 2 cosx1 sinx1 x2x3 − βϕx2 I sinx1 + u3 (3.12) The control inputs and parameter identification updating law are given as the following theorem. Theorem 3.2: Consider the master system (2.1) with unknown parameter C1(t) which satisfies Assumption Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) 6 A. DAEICHIAN, S. AGHAEI 2.1 and the slave system (2.4). Then, there exist the control laws u1 = xs2 + xs3 + x2 − e1 u2 = −xs1 − b1xs2 − C2 I x2 − ke2x2 I‖e2x2‖ x2 + Mgl I sinx1 −(βφ − βϕ cosx1)(βϕ − βφ cosx1) I2 sin3 x1 − Mg I n∑ k=1 Ak sin (ωkt)(sinx1)− e2 u3 = −a1 − xs3(xs1 − c1)− 2 cosx1 sinx1 x2x3 + βϕx2 I sinx1 − e3 (3.13) and the parameter updating law, Ċ2 = e2x2 I (3.14) which leads to master-slave synchronization. Moreover, the unknown parameter of the slave system converges to the nominal value of the master one. Proof The proof can be written down in the same way as the proof of Theorem 1 by considering the error dynamic Eq.(3.12). So, the two gyro systems synchronize despite initial state and parameter mismatches. The next section shows the simulation results of the proposed method. Furthermore, the outcomes of the proposed method are compared with the results of [30]. 4. SIMULATION Consider the dynamical model parameters βφ = 2, βϕ = 5, I = 1,Mg = 4, l = 0.25, ω1 = 1, and A1 = 12.1. The Rössler system is chaotic where a1 = 0.2, b1 = 0.2, and c1 = 5.7. Also, we consider different initial conditions for the master gyro as x1(0) = −0.5, x2(0) = −1.2, x3(0) = 10, slave gyro as xs1(0) = xs2(0) = xs3(0) = 0.1, and slave Rössler system as xs1(0) = −0.1, xs2(0) = 0.2, xs3(0) = 0.5. Assuming the unknown, time varying, and uncertain parameter C1(t) = 0.5 + 0.7 sin t where the nominal value is C = 0.5 and ∆C = 0.7 sin t. Thus, the maximum norm of ∆C can be set as k = 0.7. The initial value of slave system parameter is also assumed as C2 = 0. Moreover, the simulation time step is chosen as 0.0001 (sec). The states of two master and slave gyros as well as the error of their synchronization by the method of [30] are depicted in Fig. 4.2. The mean square errors (MSE) of the states synchronization are 6.617× 10−4, 1.151× 10−2, and 1.802× 10−1 for the e1, e2, and e3, respectively. Since the unknown parameter is in the dynamical equation of x2, the steady state synchronization error is observed in e2. This can be quantified by neglecting transient time, i.e. considering MSE for data obtained after 10 seconds from the beginning which are 1.721× 10−12, 7.749× 10−03, and 4.686× 10−10 for e1 to e3, respectively. It can be seen in Fig. 4.2 that e2 does not converge to zero and it has fluctuations between −0.120 and 0.265. This fact is due to the steady state error in the parameter identification algorithm which is shown in Fig.4.5. The mean value of the identified parameter by the method of [30] after initial transient time is 0.447 which suffers approximately 10.6% steady state error. Theorems 1 and 2 assert that control inputs (3.6) and (3.13) with the parameter-updating laws (3.7) and (3.14) not only synchronize the master and slave systems but also guarantee the identification of the unknown parameter of the master system correctly. Figs. 4.3 and 4.4 illustrate the states and synchronization error for gyro-gyro and gyro-Rössler systems by the proposed method, respectively. The simulation results show that synchronization Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) SIMULTANEOUSLY ROBUST CHAOS SYNCHRONIZATION AND IDENTIFICATION OF UNKNOWN 7 0 50 100 150 -3 -2 -1 0 x 1 0 50 100 150 -5 0 5 x 2 0 50 100 150 0 10 20 x 3 0 50 100 150 -3 -2 -1 0 x 1s 0 50 100 150 -5 0 5 x 2s 0 50 100 150 0 10 20 x 3s 0 50 100 150 Time (sec) -0.5 0 0.5 1 e 1 0 50 100 150 Time (sec) -0.5 0 0.5 1 1.5 e 2 0 50 100 150 Time (sec) -10 -5 0 e 3 Fig. 4.2. Method of [30]: row 1: Master system states, row 2: Slave gyro system states, row 3: Synchronization error 0 50 100 150 -3 -2 -1 0 x 1 0 50 100 150 -5 0 5 x 2 0 50 100 150 0 10 20 x 3 0 50 100 150 -3 -2 -1 0 x 1s 0 50 100 150 -5 0 5 x 2s 0 50 100 150 0 10 20 x 3s 0 50 100 150 Time (sec) -0.5 0 0.5 1 e 1 0 50 100 150 Time (sec) -0.5 0 0.5 1 1.5 e 2 0 50 100 150 Time (sec) -10 -5 0 e 3 Fig. 4.3. Proposed method (Gyro-Gyro): row 1: Master system states, row 2: Slave gyro system states, row 3: Synchronization error error asymptotically tends to zero for all states regardless of uncertainty in parameters. In the case of gyro-gyro synchronization, the MSE of the synchronized x2 reduces to 1.019× 10−3 which shows a decrease of more than 91%. Omitting initial transient time give MSE equal to 3.45× 10−5, which is almost 99.5% more accurate than the previous method. Synchronization MSEs for the Gyro-Rössler situation are 2.941× 10−4, 1.202× 10−3, and 1.659× 10−1 related to e1, e2, and e3, respectively. The MSEs decrease to 0.765× 10−12, 0.399× 10−4, and 4.315× 10−10 if neglecting the initial transient time. The gyro-Rössler Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) 8 A. DAEICHIAN, S. AGHAEI 0 50 100 150 -3 -2 -1 0 x 1 0 50 100 150 -5 0 5 x 2 0 50 100 150 0 10 20 x 3 0 50 100 150 -3 -2 -1 0 x 1s 0 50 100 150 -5 0 5 x 2s 0 50 100 150 0 10 20 x 3s 0 50 100 150 Time (sec) -0.5 0 0.5 1 e 1 0 50 100 150 Time (sec) -0.5 0 0.5 1 1.5 e 2 0 50 100 150 Time (sec) -10 -5 0 e 3 Fig. 4.4. Proposed method (Gyro-Rössler): row 1: Master system states, row 2: Slave gyro system states, row 3: Synchronization error 0 100 200 300 Time (sec) -0.2 0 0.2 0.4 Id e n ti fi e d V a lu e 0 100 200 300 Time (sec) 0 0.2 0.4 0.6 Id e n ti fi c a ti o n e rr o r Fig. 4.5. Gyro-gyro synchronization: Left: Parameter identification, Right: Identification error; Dashed-dotted line (black): Nominal value, Dashed line (blue): Proposed method, Solid line (red): Method of [30] 0 100 200 300 Time (sec) -0.2 0 0.2 0.4 Id e n ti fi e d V a lu e 0 100 200 300 Time (sec) 0 0.2 0.4 0.6 Id e n ti fi c a ti o n e rr o r Fig. 4.6. Gyro-Rössler synchronization: Left: Parameter identification, Right: Identification error; Dashed- dotted line (black): Nominal value, Dashed line (blue): Proposed method, Solid line (red): Method of [30] synchronization also ensures convergence to zero of all errors. The MSEs for all situations are written in table 4.1. The unknown parameter is also identified perfectly by the developed method. Figs. 4.5 and 4.6 indicate the convergence of the unknown parameter to the nominal value C as well as the error of estimation for the proposed method and that of [30] in the situation of gyro-gyro and gyro-Rössler systems, respectively. It can be seen that the fluctuations in the e2 decrease as time goes by and the parameter identification error converges to zero in the proposed method while the method of [30] suffers lasting fluctuations as well as steady state error. Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) SIMULTANEOUSLY ROBUST CHAOS SYNCHRONIZATION AND IDENTIFICATION OF UNKNOWN 9 Table 4.1. MSE of synchronization MSE MSE for time>10 (sec) Method e1(×10−4) e2(×10−3) e3(×10−1) e1(×10−12) e2(×10−4) e3(×10−10) Method of [30] 6.617 11.51 1.802 1.721 77.49 4.686 Proposed Gyro-Gyro 6.617 1.019 1.802 1.721 0.345 4.686 method Gyro-Rössler 2.941 1.202 1.659 0.765 0.399 4.315 5. CONCLUSION The proposed synchronization of gyro system has been founded based on the elimination of dynamical equations’ nonlinearity by the controller. Simultaneously, a parameter identification algorithm estimates the unknown, time-varying, and uncertain parameters. The proposed method not only guarantees convergence of synchronization error to zero but also ensures the convergence of unknown parameter estimation to the nominal value. Considering a typical gyroscope, the synchronization error in order of magnitude 10−12, 10−5, and 10−10 for e1, e2, and e3, respectively, are obtained in the situation of gyro-gyro and gyro-Rössler synchronization. Moreover, the unknown parameter is estimated at time 300 second with less than 2.7% of error. However, the proposed algorithm requires the upper bound of the parameter uncertainty to be determined. Another point is that the control inputs are a complex nonlinear function of states that impose high computation load. Nomenclature Parameter Definition X, Y, Z Inertia base vectors (Fixed) x, y, z Body axis θ, φ, ψ The Euler’s angles I The polar moments of gyro inertia l The distance between origin and the center of gravity M The mass of gyro g Gravitational acceleration constant ωk k-th harmony of the base vibration frequency Ak Amplitude of base vibration with frequency ωk βφ, βϕ The constants of the motion a1, b1, c1 The Rössler system parameters xi i-th state of the master system (i=1,2,3) xsi i-th state of the slave system (i=1,2,3) REFERENCES 1. Rakkiyappan, R., Latha, V. P., Zhu, Q., & Yao, Z. (2017) Exponential synchronization of markovian jumping chaotic neural networks with sampled-data and saturating actuators. Nonlinear Analysis: Hybrid Systems, 24, 28–44. 2. Aghaei, S., Daeichian, A., & Puig, V. (2020) Hierarchical decentralized reference governor using dynamic constraint tightening for constrained cascade systems. Journal of the Franklin Institute, 357, 12495–12517. 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(2016) Synchronization and chaos control of heavy symmetric chaotic unknown gyroscope using mqrbvsc. In 2016 International Conference on Intelligent Control Power and Instrumentation (ICICPI), 12–16, IEEE. 29. Yin, L., Deng, Z., Huo, B., & Xia, Y. (2017) Finite-time synchronization for chaotic gyros systems with terminal sliding mode control. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 49, 1131–1140. 30. Ge, Z.-M. & Lee, J.-K. (2005) Chaos synchronization and parameter identification for gyroscope system. Applied Mathematics and Computation, 163, 667–682. 31. Thornton, S. T. & Marion, J. B. (2004) Classical dynamics of particles and systems. Brooks/Cole—Thomson Learning. Copyright © 2021 ASSA. Adv Syst Sci Appl (2021) Introduction Problem statements Design of controller and identifier Gyro system as slave Rössler system as slave Simulation Conclusion