Master-slave H-Infinity robust controller design for synchronization of chaotic systems

This paper is devoted to robust master-slave controller design for generalized chaotic systems synchronization. The closed-loop system is asymptotically stable when the robust stability conditions hold and while the H∞ norm of the closed-loop transfer function with respect to defined output and input is strictly less than γ > 0. In this paper a modified L2 gain approach is used and an original design procedure is proposed to decrease the conservativeness of the former method. The effectiveness of the proposed method is shown in numerical examples.


Introduction
The chaotic synchronization gets much more attention due to the powerful applications in biological systems, chemical reactions, information processing, power converters, secure communications and others.The mentioned topic has attracted many researchers since the original paper by Pecora and Carroll (1990) dedicated to chaotic synchronization and it still belongs to the interesting problems in nonlinear oscillator control.The main idea of synchronization is to design the controller for a slave chaotic system, that the slave output can follow the master system output while the defined error is asymptotically stable, (Wang et al., 2012).Chaos synchronization opens up huge perspective to optimize nonlinear oscillated systems.Electrical power system belongs to this type of nonlinear dynamic systems.Study of the chaos and its control (Shahverdiev et al., 2008) could avoid undesirable power system dynamic and behaviour, which should lead to the power system blackout.One of the essences of the proposed method mentioned also in the paper Song and Yu (2003) is that the tracking error of master-slave system is pushed and forced to the pre-selected invariant manifold.The necessary chaos synchronization condition is that the conditional Lyapunov exponents are negative.Robust H ∞ controller design for synchronization of masterslave dynamic chaotic systems using LMI and feedback linearization has been previously made in Wang and Balakrishnan (2002) and further results to synchronization of Lure systems with time delay in Zeng et al. (2015).The slave control algorithm has been given for the case of fourth order chaotic system with rather complicated control algorithm.Chaos synchronization of Rossler systems, that has origins in chemical kinetics is described in Farghaly (2013).The controller used to synchronize two identical Rossler dynamic systems has been determined by the Lyapunov function.The sufficient global stability criterion for synchronizing Liu chaotic system has been obtained based on a linear feedback control method and Lyapunov function in Chen (2009).Disturbance observer based controller could be found in Mobayen and Javadi (2015).Chaotic system control with Lipschitz nonlinearities and state feedback is described in Mobayen and Tchier (2017).New chaotic regimes in the Lorenz and Chen systems are described in the paper Sprott (2015).The paper describes new regimes and show that Lorenz and Chen chaotic systems admits chaotic solution.The review of chaotic systems and synchronization control should also be consulted in Boccaletti et al. (2002) and excellent book of L Fradkov (2007).On the basis of the above small observation the following problem is studied in this paper: Develop a new design procedure for the given dynamics of chaotic system (8) to design the robust H ∞ controller based on the LMI, feedback linearization and bounded real lemma with less conservativeness than known methods from references using bilinear matrix inequalities (BMIs).
The contribution of this paper is to provide robust performance conditions by minimization of H ∞ gain with respect to the calculated output and disturbance input for the feedback interconnection of a chaos slave system and designed controller which ensures the synchronization in master-slave systems.
The remainder of the paper is organized as follows.In Section 2 we present preliminaries and recall design procedure for H ∞ gain.In Section 3 we address the state error tracking feedback to obtain the new less conservative robust H ∞ controller design procedure.Finally, in Section 4, the proposed design procedure is verified by the examples.
Our notation used in the paper is standard, P ∈ R m×n denotes the set of real m × n matrices, P 0 (P 0) ∈ R n×n is a real symmetric, positive definite (semidefinite) matrix.Furthermore, "*" in matrices denotes the respective transposed (conjugate) term to make matrix symmetric.Finally, I m is an m × m identity matrix, and 0 m is an m × m zero matrix.

Preliminaries and problem formulation
Consider the following chaotic system (Wang et al., 2012) in the form where q ∈ R n is the state vector, v ∈ R m is the control input vector, and w ∈ R k ∈ L 2 [t 0 , ∞) is an exogenous input vector.The system (1) one can split to slave system qsi = q si+1 , i = 1, 2, . . ., n − 1 qsn = f 1 (q s ) + w + u, and to master closed-loop system qmi = q mi+1 , i = 1, 2, . . ., n − 1 qmn = f 2 (q m ), where f i (.), i = 1, 2 are nonlinear function variables, wherein q s ∈ R n and q m ∈ R n are the state vectors of slave and master chaotic systems, and u ∈ R m is the control input vector of the slave system.The slave (2) and master (3) systems can be transformed to matrix form as qs = A s q s + Bu + B 1 w + A sn (q s ), ( 4) where .
Such models can represent many types of chaotic systems.The above mentioned idea of two chaotic systems synchronization is to design the controller for slave system such that the defined output (states) of the slave system can follow the corresponding output (states) of master systems.That implies that the tracking error for both systems will be asymptotically stable lim The tracking error is defined as Subtracting equations ( 4) and ( 5) one obtains where z ∈ R lz is the performance output vector, y ∈ R l is the measurable output vector, A = A s , and The L 2 norm of h ∈ R n is defined as: Definition 2.2.(L 2 gain, Boyd et al. (1994)) The L 2 gain of system (8) (which also equals to the H ∞ norm of the system's transfer matrix) is defined as the quantity where the supremum is taken over all nonzero trajectories of system (8), starting from e(0) = 0.
Based on the foregoing, the following problem is studied in this paper.
Problem 2.3.For system (8) design an outputfeedback controller, with gain matrix K ∈ R m×n , defined as u = Ky − F (q s , q m ), ( 11) such that the closed-loop system is asymptotically stable, and the H ∞ norm of the system's transfer matrix is strictly less than γ, where γ ≥ 0 is a known constant defined by the designer.
Substituting the control law (11) to the system (8) one can obtain where Remark 2.4.Control law ( 13) is defined in a static output-feedback (SOF) form.Let's remark that many controller structures can be transformed to the form (13) (like PI, PID, PD, even full/reduced order dynamic output-feedback controllers), by augmenting the system with additional state variables.For more info, see Ilka (2018) or Veselý and Rosinová (2013).
Lemma 2.5.Let γ ≥ 0 be a known constant scalar.If there exists a controller gain matrix K and a positive definite matrix X satisfying then the closed-loop system formed by system (12) and control law ( 13) is asymptotically stable and the H ∞ norm of its closed-loop transfer function is strictly less than γ.
The Inequality ( 14) in Lemma 2.5 is known as the bounded real lemma (Isidori, 2011(Isidori, , 2017;;Krokavec and Filasova, 2016).The problem with the application of the above inequality to design the robust H ∞ controller is that it gives rather conservative results.In this paper we propose a new robust H ∞ controller design procedure, using modified bounded real lemma and feedback linearization, to control the chaotic system (1) with less conservativeness.

Robust H ∞ ontroller Design
This section formulates the theoretical approach to robust controller design with less conservative results.
Theorem 3.1.Chaotic system (8) with any initial conditions can be stabilized by control algorithm ( 11) with H ∞ norm of the system's transfer matrix strictly less than γ, if there exists a symmetric positive definite matrix P , matrices N i , i = 1, 2, 3 and controller gain matrix K such that the following inequality holds where 16), (17), and (18).
Proof.Suppose that there exists a quadratic function V (e) = e T P e, P 0, and some γ ≥ 0 such that: Integrating (20) from 0 to T , with initial condition e(0) = 0, we can get: Since P is positive definite then V (e(T )) > 0, which implies Furthermore, the inequality ( 22) and the Definition 2.2 implies that the L 2 gain of the system (12) (which also equals to the H ∞ norm of the system's transfer matrix (Boyd et al., 1994)), is strictly less than γ.By using the auxiliary matrices N 1 , N 2 ∈ R n×n and N 3 ∈ R n×k , and by substituting the control law (13) to the system (12), we can get: where A f and B f are defined in ( 15) and ( 16).Since H = 0, we can write where for d T = [ ėT , e T , w T ], V (e) = ėT P e + e T P ė = d T   0, P, 0 P, 0, 0 0, 0, 0 wherein, C f and D f are defined in ( 17) and ( 18).Finally, from (24) we can get (19), which completes the proof.
Theorem 3.1 is formulated as a feasibility problem, for given known γ.The next Corollary completes it with minimization of γ to obtain minimal H ∞ norm.
Corollary 3.2.If the following optimization problem has a solution, then the closed-loop system formed by system (8) and controller (11) will be stable with minimal H ∞ norm min F,P,N1,N2,N3,γ (γ) (29) 4 Examples Example 4.1.In order to evaluate the conservativeness of the previous proposed method (Corollary 3.2) the COMPl e ib library (Leibfritz, 2004) has been used (Table 1).In order to better highlight the advantages of the proposed method beside approaches from (Hoi et al., 2003, Problem 2) and from (Isidori, 2017, Theorem 3.1), which are also based on bilinear matrix inequalities, we have decided to include results from other approaches like the HIFOO toolbox (Burke et al., 2006) and the BMISolver toolbox (Dinh et al., 2011a) as well.The HIFOO toolbox is based on a hybrid algorithm for nonsmooth, nonconvex optimization, which uses several techniques, namely quasi-Newton updating, bundling and gradient sampling.The BMISolver toolbox is combining convex-concave decompositions and linearization approaches for solving BMIs.Numerical solutions for the proposed method (Corollary 3.2) have been carried out by PENBMI 2.1 solver (Henrion et al., 2005) under Matlab R2018a (The Mathworks, Inc., 2018) using YALMIP (Löfberg, 2004) on NEOS Server Version 5.0 (Czyzyk et al., 1998).Numerical solutions for HIFOO and BMISolver Toolboxes as well as for the BMI formulation based on (Hoi et al., 2003, Problem 2) have been taken over from (Dinh et al., 2011b, Table 2).Finally, numerical solutions for the BMI formulation based on (Isidori, 2017, Theorem 3.1) have been carried out by PENLAB solver (Fiala et al., 2013) under Matlab R2017a using YALMIP on HP EliteBook 820 notebook.Table 1 indicates that the proposed approach (Corollary 3.2) is less conservative compared to other approaches using BMIs ( (Hoi et al., 2003, Problem 2) and (Isidori, 2017, Theorem 3.1)) since it can handle much more examples from the COMPl e ib library.Table 1 also proves that the proposed approach with the used relaxation (slack variable approach) combined with PENBMI global solver outperforms the majority of the algorithms in most cases.In addition, thanks to the reduced conservativeness, it can handle much more examples from the COMPl e ib library compared to other BMI-based approaches, which indicates that the proposed approach could be an efficient and reliable computer-aided control system design tool for small and medium sized problems, with a potential for realistic industrial applications as well.
Example 4.2.Numerical simulation of the well known Van der Pol oscillator model has been used to verify the results of the Theorem 3.1 in the second example of this article.The master Van der Pol chaotic system can be described as qm1 = q m2 (33) and the slave system as wherein m > 0 and s > 0. By performing the operation (7) the model for tracking error can be obtained as which can be transformed to the form (8) with Note that matrix A is unstable.The control law ( 13) is described by the following equation u = Ky − F (q s , q m ) = Ke − s q 2 s1 q s2 + m q 2 m1 q m2 , where the obtained gain matrix K (using Theorem 3.1), for different initial parameters are • for 0 < P < ρI, ρ = 100, and γ = 0.4151 The maximal real part of the eigenvalues of the closed-loop system is λ m = −1.0377.
• for ρ = 500 and γ = 0.326 The maximal real part of the eigenvalues of the closed-loop system is λ m = −0.8281.
Behavior of the chaotic system Note that minus sign of matrix K provides negative feedback.
The behaviour of the chaotic system for non-zero initial conditions is shown in Fig. 1.The time responses of the tracking error are shown in Fig. 2 (for (39)) and in Fig. 4 (for (40)).The measurements of γ according to ω (rad s −1 ) are shown in Fig. 3 (for (39)) and in Fig. 5 (for (40)).In both cases the measured value γ is less than the defined maximum value.
The obtained control algorithm (using Theorem 3.1) is u =Kx + 0.1q m2 − q s1 + 0.8q 3 s1 + 2cos(1.5t)+ 1.1q m1 − q 3 m1 − 2.1cos(1.8t), where the gain matrix for γ = 0.3497 and ρ = 50.The maximal real part of the eigenvalues of the closed-loop system is λ m = −1.3044.Dynamic behaviour of the tracking error is shown in Fig. 8, and the measurement of γ according to ω in Fig. 9.In all cases the tracking error converged to zero (Fig. 2, Fig. 4, Fig. 6, and Fig. 8) which means that the controlled slave system tracked the master system.Also in all cases the measured γ was lower than the numerically computed maximum.

Conclusion
In this paper a new bounded real lemma is proposed for robust master-slave controller design that is used for synchronization of generalized chaotic systems.The modified L 2 gain approach and the new bounded real lemma ensures less conservative BMI-based controller design due to the introduced auxiliary matrices.When the robust stability conditions hold the closed-loop system is asymptotically stable and the H ∞ norm of closed-loop transfer function with respect to the defined output and input is strictly less than γ ≥ 0. Numerical examples show the effectiveness of the proposed method.

Figure 1 :
Figure 1: Behaviour of the first chaotic system.

Figure 3 :
Figure 2: Time response of the controlled output variable.

Figure 5 :
Figure 4: Time response of the controlled output variable.
Figure 6: Time response of the controlled output variable.

Table 1 :
H ∞ benchmarks on COMPl e ib plants