Invariant Ellipsoids Method for Chaos Synchronization in a Class of Chaotic Systems

This paper presents an invariant sets approach for chaos synchronization in a class of master-slave chaotic systems affected by bounded perturbations. The method provides the optimal state-feedback gain in terms of the minimal ellipsoid that guarantees minimum synchronization error bound. The problem of finding the optimal invariant ellipsoid is formulated in terms of a semi-definite programming problem that can be easily solved using various simulation and calculus tools. The effectiveness of the proposed criterion is illustrated by numerical simulations on the synchronization of Chua’s systems.


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International Journal of Robotics and Control Systems Vol. 2, No. 1, 2022, pp. 57-66 ISSN: 2775-2658 case, the optimization on both energy and error can be easily realized. The synchronization of an uncertain slave system following the dynamic of an equivalent master system is developed in [17] by using a Lyapunov-based approach.
Linear state feedback by linear matrix inequalities (LMI) techniques have been proposed in [18], [19] where the synchronization criterion is transformed into LMI form which can be efficiently verified and solved. Most of the schemes to achieve chaos synchronization were originally designed for identical chaotic systems in which it is assumed that systems parameters are completely known in advance. Do to this unnatural assumption several approaches have been recently proposed to deal with this issue [20], [21], [22]. This paper proposes a static state-feedback control for master-slave synchronization designed by invariant ellipsoids techniques. The idea initially established in [23,24,25] and then further investigated over the last decades (see [26], [27] and the references therein) is based on the analysis of reachable and feasible sets for uncertain dynamic models or on the search for their approximations by simple convex domains like boxes, polyhedra or ellipsoids [28].
A set in the state space is said to be positively invariant for a given dynamical system if every trajectory initiated in this set remains inside it at all future time instants. In the presence of disturbances, if the invariance is preserved, the term of robust invariance is used.
The basic idea of the ellipsoidal control is to attract the state trajectory, for any initial state vector outside the ellipsoid, to a small ellipsoid region including the origin. When the trajectory of the system reaches the ellipsoid, it will remain within it for every future time instant. In this framework, the aim is therefore to synthesize a feedback gain that guarantees the synchronisation error belonging to the minimum invariant ellipsoid.
The main advantage of invariant ellipsoids lies in its simple characterization as a solution of LMI in terms of all the varying parameters, which is very convenient to be used in practice [29]. In fact, the control problem is reduced to semidefinite program, i.e., to the optimization of a linear function under LMI constraints.
Although the concept of invariant ellipsoid combined with LMI technique is widely used for analysis and design of control systems, however it found fewer applications in synchronization problems. For this reason, this paper exploits the theoretical framework of invariant ellipsoid control to address the synchronization between two chaotic systems.
The paper is organized as follows. Preliminaries definitions and results together with problem definition are presented in Sec. 2. Sec. 3 contains the main results consisting in finding a minimal invariant ellipsoid and the corresponding state-feedback control. A numerical simulation is reported in Sec. 4 to highlight the characteristics of the method. Finally some concluding remarks are included in the last section.

Preliminaries
Consider the master chaotic system described bẏ where x ∈ R n is the state vector, A ∈ R n×n is a constant matrix, w ∈ R n is the exogenous disturbance which doesn't depend on the state vector x and is bounded at each time instant as The quantity Dw takes into account both uncertainties on linear and nonlinear part of the chaotic system. The continuous nonlinear function g(x) satisfies Lipschitz condition, namely where ρ is the Lipschitz constant. Similar assumptions have been largely adopted in the devoted control literature. In general, as remarked in [20], [30], [31], the last assumption is not restrictive since the trajectories of chaotic systems are always bounded.
The slave chaotic system has the same structure of the master one with different initial conditions and with no information about the disturbance w(t): where the control input u(t) ∈ R n is required to synchronizex(t) with the state of the master system x(t). Based on the linear-state-feedback approach, the control input u(t) is chosen as with K ∈ R n×n . By defining the error e(t) = x(t) −x(t) which characterizes the synchronization accuracy, thenė The goal is to choose a suitable K, such that chaos synchronization is achieved, i.e. the error signal e becomes as small as possible. With this aim we use the framework of invariant ellipsoids including the large deviation case as described in [32].
Remark 1 Note that the term Dw can be regarded both as uncertainty on the master and/or slave system. It is then the difference between the actual state of the system and the state modelled from the nominal model (4). Therefore what the method tries to do is to bring the synchronization closer to the actual state.

Definition 1 The ellipsoid
is invariant for system (6), if two conditions hold: An ellipsoid E is minimal if it has the minimal sum of squares of its halfaxes, i.e. if tr(P ) is minimal.

Linear matrix inequalities
A linear matrix inequality or LMI is a matrix inequality of the form Lemma 1 Schur complements. For a given symmetric matrix where Q(x) = Q(x) T , R(x) = R(x) T and S(x) depends affinely on x, Schur complements states that the LMI M > 0 is equivalent to the matrix inequalities Lemma 2 S-Lemma. Let A, B be symmetric n×n matrices, and assume that the quadratic inequality x T Ax ≥ 0 is strictly feasible (there existsx such thatx T Ax > 0). Then the quadratic inequality if and only if there exists a non-negative τ such that Lemma 3 Matrix square root. Let F be a positive definite Hermitian matrix. Then F has a diagonalization F = P * diag{λ 1 , ..., λ n }P where P is a unitary matrix and λ 1 , ..., λ n are the eigenvalues of F , which are all positive. The square root of F is defined as the matrix

Invariant ellipsoid design
In order to reduce the synchronization error, the dynamic of which is defined in (6), the optimal state-feedback gain K is computed in this section. This controller gain K, obtained in the following theorem, gives the minimum invariant ellipsoid, expressed as the minimum of the trace of P , for the error between master and slave trajectories. The error is reduced both in the case when initial error started within the smallest invariant ellipsoid and in the large deviation case, i.e. when the initial error is outside the optimal invariant set.
subject to on matrix variables Q = Q T , H = H T , Y ∈ R n×n defines the matrixP =Q −1 of the minimal invariant ellipsoid and corresponding filter gain Proof. To satisfy both properties of invariant ellipsoid (7) it is sufficient to suppose that there exists a quadratic Lyapunov function Differentiating V along the error dynamical trajectory (6) and using (3) yieldṡ where Since 2ρ||Qe||||e|| ≤ ||Qe|| 2 + ρ 2 ||e|| 2 then it follows that Due to Schur complements the first inequality in (16) can be rewritten as Also e T Qe ≥ 1 and w T w ≤ 1 can be formulated as From S-Lemma it follows that (18) implies (17) if exists τ ≥ 0 such that Hence By Schur complements Eq. (20) is equivalent to from which (10) follows when Y = QK.
To reduce minimization of tr(Q −1 ) to a linear problem introduce a matrix H = H T such that Q −1 ≤ H. Due to Schur complements this inequality is equivalent to LMI (11). Therefore the problem is reduce to the minimization of tr(H) subject to (10) and (11) and the proof follows.

Remark 2
The introduction of the new variable τ , due to the S-procedure, makes (10) a bilinear matrix inequality (BMI). Because τ is a scalar, its optimal value can be found by executing a simple loop or by using a BMI solver. For τ fixed, the problem defined by (9)-(11) is a semidefinite programming problem (SDP).

Numerical example
In this section the Chua circuit is simulated to illustrate the above control method for chaos synchronization. The simulation is performed within the Matlab 2021a environment while the underlying optimisations are carried out using the Yalmip toolbox [33]. Chua's circuit in Fig. 1 is the simplest and most widely studied real nonlinear dynamical systems [34]. It consists of three energy-store elements (an inductor and two capacitors), a linear resistor and a single nonlinear resistor, called Chua's diode. Using the Kirchhoff's circuit laws the circuit is described by equations: where R is a linear resistance, v C1 and v C2 are the voltages across capacitors C 1 and C 2 , respectively, i L is the current through the inductor L and φ(v C1 ) is the current through the nonlinear resistor as a function of the voltage across capacitor C 1 . This nonlinear function is described by the oddsymmetric piecewise-linear function where G a , G b and B p are three fixed constants of the diode. In particular G a is the slope of the nonlinear function in the inner region and G b the slope in the two outer regions.
The system (22) can be rewritten in dimensionless form as where and Throughout the rest of the section, we will use the dimensionless form with k = 1 rewritten in matrix form asẋ where If α = 15.6, λ = 28, m 0 = −1.143 and m 1 = −0.714 then the Chua system exhibits a double scroll attractor (Fig. 2).
To Trajectories of master and slave systems are depicted in Fig. 3.

Conclusions
A robust state-feedback controller has been proposed for chaos synchronization in a class of chaotic systems. It makes use of the method of invariant ellipsoids which yields the optimal solution in terms of minimal ellipsoid containing the synchronization error. Robustness refers to the ability to maintain the invariance even in the presence of disturbances on the master system. The effectiveness of the proposed approach has been tested in the case of Chua master-slave systems synchronization. Two interesting perspectives are the extension of this method to the case of systems with interval uncertainties and its application to the synchronisation of systems of different types and dimensions.
Funding: This research received no external funding. Conflicts of Interest: The authors declare no conflict of interest.