Chaos Synchronization Based on Unknown Input Proportional Multiple-Integral Fuzzy Observer

Abstract

This paper presents an unknown input Proportional Multiple-Integral Observer (PIO) for synchronization of chaotic systems based on Takagi-Sugeno (TS) fuzzy chaotic models subject to unmeasurable decision variables and unknown input. In a secure communication configuration, this unknown input is regarded as a message encoded in the chaotic system and recovered by the proposed PIO. Both states and outputs of the fuzzy chaotic models are subject to polynomial unknown input with kth derivative zero. Using Lyapunov stability theory, sufficient design conditions for synchronization are proposed. The PIO gains matrices are obtained by resolving linear matrix inequalities (LMIs) constraints. Simulation results show through two TS fuzzy chaotic models the validity of the proposed method.

1. Introduction

It is well known that the chaotic systems have a complex dynamical behavior and their fundamental characteristic is the chaos. The chaotic systems are highly sensitive to parameters variation and to initial conditions because the chaos is a source of oscillation and instability. Moreover, in the long term the behavior of the chaotic systems becomes difficult to predict which can lead systems to instability and undesirable performance [1]. On the other hand, the chaotic systems constitute a good platform to investigate the nonstandard control problems including synchronization and stabilization.

Since Pecora and Carroll [2, 3] have introduced in 1990 the concept of chaotic synchronization between two chaotic dynamical systems based on the Lorenz’s chaotic system [4], the synchronization and control of chaotic systems attract more and more attention from various disciplines. A great deal of chaos applications have been developed in engineering fields such as secure communication, physical systems, system identification, and biological systems; see, for example, [5–10].

Recently, a particular attention has been paid to the synchronization and control problems for dynamical networks due to their extensive application in fields of science and engineering (see, e.g., [11–13]). In [11] the authors propose a novel concept of bounded 𝐻_∞ synchronization and state estimation to handle the time-varying nature of an array stochastic complex network in discrete-time domain over a finite horizon. The synchronization problem in [12] is considered for a new class of continuous-time neural networks of neutral type with parameters, discrete-time delays, and unbounded distributed time delays being all dependent on the Markovian jumping mode. The sampled-data synchronization control scheme in [13] is studied for a class of dynamical networks with stochastic sampling. In the formwork of state estimation, fault detection, and filtering for a class of nonlinear systems with sensor networks, we can mention the works of [14–17]. Indeed, in [14] the finite-horizon distributed 𝐻_∞ state estimator design scheme is proposed for a class of discrete time-varying nonlinear systems with both stochastic parameters and stochastic nonlinearities. The problem of designing the distributed 𝐻_∞ filters in [15] is considered for a class of polynomial nonlinear stochastic systems which are represented in a state-dependent linear-like form. The distributed finite-horizon filter is proposed in [16] for a class of time-varying systems subject to randomly varying nonlinearities over lossy sensor networks that involve both the quantization errors and successive packet dropouts. The robust fuzzy fault detection filter in [17] is designed for a class of uncertain discrete-time Takagi-Sugeno fuzzy systems with successive packet dropouts which involve both the stochastic multiple time-varying discrete delays and the infinite distributed delays.

Different methods and techniques for chaos synchronization and control have been investigated including impulsive control [18], feedback control [19], adaptive control [20], lag synchronization [21], sliding mode control [22], and fuzzy control [23]. Particularly, in the chaotic secure communication problems, some messages can be masked efficiently and securely [24] by chaotic signals since the chaos has the characteristic of broadband like a noise and is consequently difficult to predict. The message in the secure communications which is recovered by the response system should synchronize with the drive system [25]. In framework of a secure communication, many approaches have been addressed such as chaos modulation [26], chaos shift key [27], and chaos masking [28]. Among different methods of synchronization and control for chaotic systems, the TS fuzzy systems have received much attention from various researches fields since the pioneering work of Takagi and Sugeno (TS) [29]. Indeed, TS fuzzy model can approximate a highly nonlinear analytical relation of chaotic system by fuzzy IF-THEN rules where the implications describe local dynamics as linear models. Then, the nonlinear behavior of the chaotic system can exactly be obtained as an aggregation of local linear models with nonlinear activation functions. TS fuzzy models are widely used as a tool of analysis and design of synchronization and control schemes because of the mathematical analysis simplicity of their simple structure with local dynamics, for example, [30–32].

Since because of, many practical control problems the states are partially or fully unavailable, the state observer methods can be used to estimate the measurements of unavailable or failed sensors. For this reason, it is important to design the observers for state estimation. In relation to that state estimation observers there are many works to that deal with stability analysis and stabilization of TS fuzzy models by applying Lyapunov theory and derive stability conditions in terms linear matrix inequalities constraints [33]; of among this works we can mention the results developed in [34]; When the decision variable is chosen as unmeasurable or unavailable state in activation functions, for example, in [35], the robust observer is designed for unknown inputs TS fuzzy models. Recently, in secure communication field the design problem of unknown input observer has been investigated in [36–39]. The authors propose in [36] a new secured transmission scheme based on smooth adaptive unknown input observers for chaotic synchronization and robust to channel noise. The unknown input observer in [37] is presented with unknown constant disturbance of parameters and unknown input to be recovered as messages in the master-slave configuration. The robust adaptive high-gain fuzzy observer is designed in [32] for chaotic systems where their parameters are assumed unknown and their states unavailable. The author deals with, in [38], the unknown input observer design for fuzzy systems with application to chaotic system reconstruction within both domains, continuous and discrete time, where the sufficient conditions have been derived in terms of linear matrix inequality constraints by using Lyapunov stability theory. Then, in [38, 39] the pole assignment in a LMI region is considered in order to improve the observer performance.

In the context of the unmeasurable decision variable, the synchronization problem for chaotic systems characterized by TS fuzzy models was not addressed by the above works. Our main contribution in this paper is to develop a synchronization procedure which takes into account the unmeasurable decision variables. The effects due to unmeasurable decision variable and the unknown input on the overall synchronization system (chaotic system and observer) are compensated with additional parameter. In addition, in the present study the estimation of the unknown input as a message to be reconstructed within a secure communication concept is considered.

In the framework of a secure communication, the design of unknown input PIO is addressed in this work for two chaotic systems, Lorenz’ system and Rossler’s system, characterized by chaotic TS fuzzy models. These models are subject to unmeasurable decision variable and polynomial unknown input where its 𝑘th derivative is zero. The proposed PIO estimates both the states of the considered chaotic systems and the polynomial unknown input. This latter is considered as a message to encode by the chaotic system and then to reconstruct it by the PIO. Furthermore, the integral action included in the observer structure contributes to reduce the results conservatism due to quadratic function. Indeed, this parameter allows introducing an additional degree of freedom to be determined. By utilizing Lyapunov stability theory, sufficient conditions are derived to design the polynomial unknown input PIO. Then, the PIO gains parameters are resolved in terms of linear matrix inequalities constraints. Moreover, when the decision variables are measurable we also discuss this particular case in our work. Finally, we present simulation results to illustrate the effectiveness of the proposed approach of synchronization and reconstruction.

The rest of this paper is organized as follows. In Section 2, the considered unknown input TS fuzzy model structure is described. This unknown input affects both the dynamics of the TS fuzzy model and the output signal. The unknown input PIO is designed in Section 3. In Section 4, simulation results for two chaotic systems, Lorenz’ system and Rossler’s system, are given. An unknown input, assumed as a message to be received in a secure communication configuration, is perfectly reconstructed. Finally, a conclusion and further works end this paper.

2. Unknown Input TS Fuzzy Model Structure

The TS fuzzy model subject to unknown input is considered as follow: 𝑥̇(𝑡)=^𝑟_𝑖=1𝜇_𝑖(𝑥)𝐴_𝑖𝑥(𝑡)+𝐵_𝑖𝑢(𝑡)+𝑑_𝑖+𝐹_𝑖𝑣(𝑡,𝑦(𝑡)(𝑡)(𝑡),)=𝐶𝑥+𝐹𝑣 (1) where 𝑥(𝑡)∈𝑅^𝑛 represents the state vector, 𝑢(𝑡)∈𝑅^𝑛𝑢 corresponds to known input vector, 𝑣(𝑡)∈𝑅^𝑛𝑣 shows the unknown input, and 𝑦(𝑡)∈𝑅^𝑛𝑦 is the output vector. 𝐴_𝑖∈𝑅^𝑛×𝑛 are the state matrices, 𝐵_𝑖∈𝑅^{𝑛×𝑛𝑢} are the input matrices, 𝑑_𝑖∈𝑅^𝑛 is a vector system dependent, 𝐹_𝑖∈𝑅^{𝑛×𝑛𝑣} and 𝐹∈𝑅^{𝑛𝑦×𝑛𝑣} are the unknown input matrices, and 𝐶∈𝑅^{𝑛𝑦×𝑛} is the output matrix. The activation functions 𝜇_𝑖(𝑥) depend on the state 𝑥(𝑡) of the system and satisfy the following conditions: ^𝑟_𝑖=1𝜇_𝑖(𝑥)=1,∀𝑡≥0,0≤𝜇_𝑖(𝑥)≤1,∀𝑖∈{1,…,𝑟}, (2) where 𝑟 is the number of local models.

Hypothesis 1. Unknown input 𝑣(𝑡) has a polynomial form of 𝑘−1 degree in time whose 𝑘th derivative is equal to zero.

Let the following notations be introduced: 𝑣̇(𝑡)=𝑣₁(𝑡),𝑣̇₁(𝑡)=𝑣₂(𝑡),⋮𝑣̇_𝑘−1(𝑡)=𝑣_𝑘(𝑡),𝑣_𝑘(𝑡)=0. (3) Note that the polynomial form allows considering a wide variety of unknown inputs.

3. Unknown Input Proportional Multiple-Integral Observer Design

The unknown input PIO is considered as follow: 𝑥̇(𝑡)=^𝑟_𝑖=1𝜇_𝑖(𝑥)𝐴_𝑖𝑥(𝑡)+𝐵_𝑖𝑢(𝑡)+𝑑_𝑖+𝐹_𝑖𝑣̂(𝑡)+𝐾_𝑃𝑖𝑦(𝑡)−𝑦(𝑡)+𝑧_𝑥(𝑡),𝑦(𝑡)𝑥(𝑡)𝑣̂(𝑡),𝑣̇̂(𝑡)==𝐶+𝐹_𝑟𝑖=1𝜇_𝑖(𝑥)𝐾_𝐼𝑖𝑦(𝑡)−𝑦(𝑡)+𝑣₁̂(𝑡)+𝑧_𝑣(𝑡),𝑣̇_𝑗̂(𝑡)=^𝑟_𝑖=1𝜇_𝑖(𝑥)𝐾^𝑗_𝐼𝑖𝑦(𝑡)−𝑦(𝑡)+𝑣̂_𝑗+1(𝑡)+𝑧_𝑣𝑗(𝑡)𝑣̂,for𝑗:1,…,𝑘−1if𝑗=𝑘−1,_𝑗+1(𝑡)=0, (4) where 𝐾_𝑃𝑖∈𝑅^{𝑛×𝑛𝑦}, 𝐾_𝐼𝑖∈𝑅^{𝑛𝑣×𝑛𝑦}, and 𝐾^𝑗_𝐼𝑖∈𝑅^{𝑛𝑣×𝑛𝑦} correspond to proportional and integral gains, respectively. Due the effect of the unmeasurable decision variables, the variables 𝑧_𝑥(𝑡), 𝑧_𝑣(𝑡), and 𝑧_𝑣𝑗(𝑡) are introduced in the PIO.

According to Hypothesis 1, TS fuzzy model (1) and unknown input PIO (4) can be rewritten, respectively, as follow: 𝑥̇_𝑎(𝑡)=^𝑟_𝑖=1𝜇_𝑖(𝑥)𝐴_𝑖𝑥_𝑎(𝑡)+𝐵_𝑖𝑢(𝑡)+𝑑_𝑖,𝑦(𝑡)=𝐶𝑥_𝑎(𝑡),𝑥̇_𝑎(𝑡)=^𝑟_𝑖=1𝜇_𝑖(𝑥)𝐴_𝑖𝑥_𝑎(𝑡)+𝐵_𝑖𝑢(𝑡)+𝑑_𝑖+𝐾_𝑖𝑦(𝑡)−𝑦(𝑡)(𝑡),𝑦(𝑡)=+𝑧𝐶𝑥_𝑎(𝑡), (5)(6) where𝑥_𝑎(𝑡)=𝑥(𝑡)𝑣(𝑡)𝑣₁(𝑡)⋯𝑣_𝑘−1(𝑡),𝑥_𝑎(𝑡)=𝑥(𝑡)𝑣̂(𝑡)𝑣₁̂(𝑡)⋯𝑣̂_𝑘−1(𝑡),𝑧(𝑡)=𝑧_𝑥(𝑡)𝑧_𝑣(𝑡)𝑧_𝑣1(𝑡)⋯𝑧_𝑣𝑘−1(𝑡) (7a) with 𝑒_𝑎(𝑡)=𝑥_𝑎(𝑡)−𝑥_𝑎(𝑡),𝑒_𝑎𝑦=𝑦(𝑡)−𝑦(𝑡),𝐴_𝑖=𝐴_𝑖𝐹_𝑖00⋯000𝐼_𝑛𝑣0⋯0000𝐼_𝑛𝑣⋯0⋯⋯⋯⋯⋯𝐼_𝑛𝑣,000000𝐵_𝑖=𝐵_𝑖00⋯0,𝑑_𝑖=𝑑_𝑖00⋯0,𝐾_𝑖=𝐾_𝑃𝑖𝐾_𝐼𝑖𝐾¹_𝐼𝑖⋯𝐾^𝑘−1_𝐼𝑖,𝐶=𝐶𝐹0⋯0, (7b)(7c)

where 𝐼_𝑛𝑣 is an identity matrix.

3.1. Unmeasurable Decision Varaiables

The dynamic error 𝑒_𝑎(𝑡) of state estimation is given by 𝑒_𝑎̇(𝑡)=^𝑟_𝑖=1𝜇_𝑖(𝑥)𝒜_𝑖𝑒_𝑎(𝑡)+Δ𝐴𝑥_𝑎(𝑡)+Δ𝐵𝑢(𝑡)+Δ𝑑−𝑧(𝑡), (8) where 𝒜_𝑖=𝐴_𝑖−𝐾_𝑖𝐶,Δ𝐴=^𝑟_𝑖=1𝜇_𝑖𝐴_𝑖,Δ𝐵=^𝑟_𝑖=1𝜇_𝑖𝐵_𝑖,Δ𝑑=^𝑟_𝑖=1𝜇_𝑖𝑑_𝑖,𝜇_𝑖=𝜇_𝑖(𝑥)−𝜇_𝑖(𝑥). (9)

Remark 1. Since the activation functions satisfy the convex sum property, we can write −1<𝜇_𝑖<1, and the variables matrices Δ𝐴, Δ𝐵, Δ𝑑 are bounded and the following conditions hold: Δ𝐴≤𝛿₁,𝛿₁=^𝑟_𝑖=1𝛿_1𝑖,Δ𝐵≤𝛿₂,𝛿₂=^𝑟_𝑖=1𝛿_2𝑖,Δ𝑑≤𝛿₃,𝛿₃=^𝑟_𝑖=1𝛿_3𝑖 (10) with 𝛿_1𝑖>0, 𝛿_2𝑖>0 and 𝛿_3𝑖>0, are the Euclidian norms of 𝐴_𝑖, 𝐵_𝑖, and 𝑑_𝑖, respectively.

Theorem 2. The dynamic error (8) is asymptotically stable if there exist a common positive definite matrix 𝑃=𝑃^𝑇, matrices 𝑀_𝑖 and the positive scalars 𝛼 and 𝛼₀, for all 𝑖∈{1,…,𝑟}, such that:𝑃𝐴_𝑖+𝐴^𝑇_𝑖𝑃−𝑀_𝑖𝐶−𝐶^𝑇𝑀_𝑖^𝑇+𝛼₀𝛿²₁𝐼𝑃𝑃−𝛼𝐼<0, (11a) where the matrices and parameters 𝐴_𝑖, 𝐶, 𝛿₁ are defined in (7c), and (10), respectively.
The parameters of unknown input PIO (4) are obtained by 𝐾_𝑖=𝑃⁻¹𝑀_𝑖,𝑧=0,𝑖𝑓𝑒_𝑎𝑦<𝜀,𝑧=𝜎₁𝛿²₁𝑥^𝑇_𝑎𝑥_𝑎2𝑒^𝑇_𝑎𝑦𝑒_𝑎𝑦𝑃⁻¹𝐶^𝑇𝑒_𝑎𝑦+𝜎₂𝛿²₂𝑢^𝑇𝑢2𝑒^𝑇_𝑎𝑦𝑒_𝑎𝑦𝑃⁻¹𝐶^𝑇𝑒_𝑎𝑦+𝜎₃𝛿²₃12𝑒^𝑇_𝑎𝑦𝑒_𝑎𝑦𝑃⁻¹𝐶^𝑇𝑒_𝑎𝑦,𝑖𝑓𝑒_𝑎𝑦≥𝜀 (11b)(11c)
with variables 𝑥_𝑎, 𝑧, 𝑒_𝑎𝑦 and the parameters𝛿₂, 𝛿₃ are described in (7a), (7b), (10), respectively, and 𝜎₁=(𝛼₀/𝜆), 𝜎₂=((𝛼𝛼₀𝜆₃)/(𝛼(𝛼₀+𝜆₃(1+𝜆))−𝛼₀𝜆₃)), 𝜎₃=𝜆₃ where 𝜆, 𝜆₃ are positive scalars arbitrarily fixed and 𝜀 is a very small positive threshold.

Proof. The proposed quadratic function of Lyapunov is 𝑉(𝑡)=𝑒^𝑇_𝑎(𝑡)𝑃𝑒_𝑎(𝑡) with 𝑃=𝑃^𝑇>0. The conditions ((11a), (11b), and (11c)) guarantee the asymptotic stability of the dynamic error of state estimation (8). The proof is partially given in the appendix and for more details see [40].

3.2. Measurable Decision Variables Case

The design of unknown input PIO with measurable decision variables represents the particular case of our study developed in this section. In this condition the unknown input PIO is as follow: 𝑥̇(𝑡)=^𝑟_𝑖=1𝜇_𝑖(𝑥)𝐴_𝑖𝑥(𝑡)+𝐵_𝑖𝑢(𝑡)+𝑑_𝑖+𝐹_𝑖𝑣̂(𝑡)+𝐾_𝑃𝑖𝑦(𝑡)−𝑦(𝑡),𝑦(𝑡)𝑥(𝑡)𝑣̂(𝑡),𝑣̇̂(𝑡)==𝐶+𝐹_𝑟𝑖=1𝜇_𝑖(𝑥)𝐾_𝐼𝑖𝑦(𝑡)−𝑦(𝑡)+𝑣₁̂(𝑡),𝑣̇_𝑗̂(𝑡)=^𝑟_𝑖=1𝜇_𝑖(𝑥)𝐾^𝑗_𝐼𝑖𝑦(𝑡)−𝑦(𝑡)+𝑣̂_𝑗+1(𝑡),𝑣̂for𝑗∶1,…,𝑘−1if𝑗=𝑘−1,_𝑗+1(𝑡)=0, (12) where all variables and matrices are defined in relations ((7a), (7b), and (7c)), and the activation functions 𝜇_𝑖(𝑥) depend on the measurable states.

The dynamics of the augmented state estimation error 𝑒_𝑎(𝑡) between the TS fuzzy model (1) and PIO (12) becomes 𝑒_𝑎̇(𝑡)=^𝑟_𝑖=1𝜇_𝑖(𝑥)𝐴_𝑖−𝐾_𝑖𝐶𝑒_𝑎(𝑡). (13)

Theorem 3. The dynamic error (13) is asymptotically stable if there exist a symmetric matrix 𝑄>0 and matrices 𝑁_𝑖 such that the following conditions hold, for all 𝑖∈{1,…,𝑟}: 𝑄𝐴_𝑖+𝐴^𝑇_𝑖𝑄−𝑁_𝑖𝐶−𝐶^𝑇𝑁_𝑖^𝑇<0. (14a) The parameters of the unknown input PIO (12) are given by: 𝐾_𝑖=𝑄⁻¹𝑁_𝑖. (14b)

Proof. Consider the Lyapunov quadratic function 𝑉(𝑡)=𝑒^𝑇_𝑎(𝑡)𝑄𝑒_𝑎(𝑡), where 𝑄=𝑄^𝑇>0. The time derivative of 𝑉(𝑡) allows writing 𝑉̇=^𝑟_𝑖=1𝜇_𝑖(𝑥)𝑒^𝑇_𝑎(𝐴_𝑖−𝐾_𝑖𝐶)^𝑇𝑄+𝑄(𝐴_𝑖−𝐾_𝑖𝐶)_𝑖𝑒_𝑎. (15)
The stability condition 𝑉̇(𝑡)<0 is satisfied if (𝐴_𝑖−𝐾_𝑖𝐶)^𝑇𝑄+𝑄(𝐴_𝑖−𝐾_𝑖𝐶)_𝑖<0; (16) with variables change 𝑁_𝑖=𝑄𝐾_𝑖; we obtain the linear matrix inequalities ((14a) and (14b)). The proof is completed.

The unknown input PIO gains 𝐾_𝑖=𝑄⁻¹𝑁_𝑖 are determined by resolving these constraints. In the following section, a simulation example is given through two chaotic systems in order to validate this proposed approach.

4. Simulation Examples

Two chaotic systems are characterized by TS fuzzy models with unmeasurable decision variables and subjected to unknown input. In a secure communication concept, the unknown input is assumed as a message to be recovered in the PIO after being encoded in the chaotic system by means of public transmission canal. These chaotic systems are used to show the good simultaneous reconstruction of states and message by the proposed unknown input PIO. The first nonlinear model is the Lorenz’s system [41], and the second is the fourth Rossler’s system [21].

4.1. Lorenz Chaotic System

The Lorenz chaotic system [38] is represented by following the dynamic equations: 𝑥̇₁=−10𝑥₁+10𝑥₂,𝑥̇₂=28𝑥₁−𝑥₂−𝑥₁𝑥₃,𝑥̇₃=𝑥₁𝑥₂−83𝑥₃. (17)

4.1.1. TS Fuzzy Model

The Lorenz’s system can be exactly represented by TS fuzzy model with the decision variable 𝑥₁(𝑡)∈[−30,30] as follow [42]: Rule 1: 𝑥₁(𝑡) is 𝜇₁(𝑥₁(𝑡)), THEN 𝑥̇(𝑡)=𝐴₁𝑥(𝑡), Rule 2: 𝑥₂(𝑡) is 𝜇₂(𝑥₁(𝑡)), THEN 𝑥̇(𝑡)=𝐴₂𝑥(𝑡),where 𝑥(𝑡)=[𝑥₁(𝑡),𝑥₂(𝑡),𝑥₃(𝑡)], 𝜇₁(𝑥₁(𝑡))=(30+𝑥₁(𝑡))/60, 𝜇₂(𝑥₁(𝑡))=(30−𝑥₁(𝑡))/60, and 𝐴₁=−1010028−1−30030−83,𝐴₂=−1010028−1300−30−83. (18)

The Lorenz chaotic attractor is given in Figure 1.

Figure 1 

Chaotic behavior of Lorenz fuzzy system.

The TS fuzzy model of the Lorenz chaotic system (17) is 𝑥̇(𝑡)=²_𝑖=1𝜇_𝑖(𝑥)𝐴_𝑖𝑥(𝑡)+𝐸_𝑖𝑣(𝑡,𝑦(𝑡)(𝑡)(𝑡),)=𝐶𝑥+𝐸𝑣 (19) with 𝐵₁=𝐵₂=⁰⁰₀, 𝐸₁=¹¹₁, 𝐸₂=¹₁^0.5, 𝐸=¹₁, 𝐶=⁰⁰¹₀₁₁.

The unknown input 𝑣(𝑡) is assumed as a message to be recovered by the unknown input PIO (4). This observer plays the role of decoder and the chaotic system the encoder within a secure communication configuration.

4.1.2. Unknown Input PIO

The unknown input PIO gains are determined by resolving the LMIs constraints ((11a), (11b), and (11c)) of Theorem 2: 𝐾_𝑖=𝐾^𝑇_𝑃𝑖𝐾^𝑇_𝐼𝑖𝐾^1𝑇_𝐼𝑖^𝑇,𝑍=𝑍^𝑇_𝑥𝑍^𝑇_𝑣𝑍^𝑇_𝑣1^𝑇. (20)

The unknown input PIO gains are given by Table 1.

Table 1 

The unknown input is assumed as a message to be encoded by chaotic system with the second derivative being zero, shown in Figure 2. The simulation results are obtained with the initial conditions 𝑥₀=111𝑥₀=000 and with 𝜀=10⁻³.

Figure 2 

The unknown input and its estimated.

The unknown input and the estimated one are given in Figure 2. Excepted around the time origin, we obtained a good reconstruction of the unknown input. Figures 3(a), 3(b), and 3(c) show the states 𝑥₁(𝑡), 𝑥₂(𝑡), and 𝑥₃(𝑡) and their estimated 𝑥₁(𝑡), 𝑥₂(𝑡), and 𝑥₃(𝑡), respectively.

(a)

(b)

(c)

Figure 3 

(a) The state 𝑥1(𝑡) and its estimated 𝑥1(𝑡). (b) The state 𝑥2(𝑡) and its estimated 𝑥2(𝑡). (c) The state 𝑥3(𝑡) and its estimated 𝑥3(𝑡).

The dynamic errors of the states estimation are given in Figure 4. The obtained simulation results show the good reconstruction of the states and the unknown input.

Figure 4 

The errors between states and their estimated.

Remark 4. Note that applying Theorem 3 instead of Theorem 2 for this example leads to bad estimation. The simulation results for this example, carried out with the same initial conditions: 𝑥₀=111 and 𝑥₀=000, are shown in Figure 5. Indeed, we see clearly that the best estimation (Figure 6) is given by Theorem 2 (dashed line) which takes into account the estimation of decision variables.

Figure 5 

The errors between unknown input and its estimated.

Figure 6 

The zoom on the errors between unknown input and its estimated.

4.2. Fourth Rossler Chaotic System

The fourth Rossler chaotic system [21] is represented by the following dynamic equations: 𝑥̇₁=−𝑥₂−𝑥₃,𝑥̇₂=𝑥₁+0,254𝑥₂+𝑥₄,𝑥̇₃=𝑥₁𝑥₃𝑥̇+3,₄=−0,5𝑥₃+0,05𝑥₄. (21)

4.2.1. TS Fuzzy Model

The fourth Rossler’s system can be exactly described by TS fuzzy model with the decision variable 𝑥₁(𝑡)∈[−80,20] as follow [42]: Rule 1: 𝑥₁(𝑡) is 𝜇₁(𝑥₁(𝑡)), THEN 𝑥̇(𝑡)=𝐴₁𝑥(𝑡)+𝑑₁, Rule 2: 𝑥₂(𝑡) is 𝜇₂(𝑥₁(𝑡)), THEN 𝑥̇(𝑡)=𝐴₂𝑥(𝑡)+𝑑₂,where 𝑥(𝑡)=[𝑥₁(𝑡),𝑥₂(𝑡),𝑥₃(𝑡),𝑥₄(𝑡)], 𝜇₁(𝑥₁(𝑡))=(80+𝑥₁(𝑡))/100,  𝜇₂(𝑥₁(𝑡))=(20−𝑥₁(𝑡))/100, and 𝐴₁=,𝐴0−1−1010.25010020000−0.50.05₂=,𝑑0−1−1010.250100−80000−0.50.05₁=𝑑₂=0030. (22) The fourth Rossler chaotic attractor is given in Figure 7.

(a)

(b)

(c)

(d)

Figure 7 

(a) Chaotic behavior 𝑥4(𝑥1(𝑡),𝑥2(𝑡)) of fourth Rossler fuzzy system. (b) Chaotic behavior 𝑥4(𝑥1(𝑡),𝑥3(𝑡)) of fourth Rossler fuzzy system. (c) Chaotic behavior 𝑥4(𝑥2(𝑡),𝑥3(𝑡)) of fourth Rossler fuzzy system. (d) Chaotic behavior 𝑥3(𝑥1(𝑡),𝑥2(𝑡)) of fourth Rossler fuzzy system.

The TS fuzzy model of the fourth Rossler system (21) is 𝑥̇(𝑡)=²_𝑖=1𝜇_𝑖(𝑥)𝐴_𝑖𝑥(𝑡)+𝐸_𝑖𝑣(𝑡)+𝑑_𝑖,𝑦(𝑡)=𝐶𝑥(𝑡)+𝐸𝑣(𝑡) (23) with 𝐵₁=𝐵₂=⁰⁰₀₀, 𝐸₁=¹¹₁₁, 𝐸₂=¹₁₁^0.5, 𝐸=¹₁, 𝐶=⁰¹⁰¹₀₀₁₀.

The unknown input 𝑣(𝑡) is assumed as a message to be encoded by the TS fuzzy model (23).

4.2.2. Unknown Input PIO

The unknown PIO gains are obtained by resolving the LMIs constraints ((11a), (11b), and (11c)) of Theorem 3: 𝐾_𝑖=𝐾^𝑇_𝑃𝑖𝐾^𝑇_𝐼𝑖𝐾^1𝑇_𝐼𝑖𝐾^2𝑇_𝐼𝑖𝐾^3𝑇_𝐼𝑖^𝑇,𝑍=𝑍^𝑇_𝑥𝑍^𝑇_𝑣𝑍^𝑇_𝑣1𝑍^𝑇_𝑣2𝑍^𝑇_𝑣3^𝑇. (24)​​The unknown PIO gains are given by Table 2.

Table 2 

The unknown input is assumed as a message to be encoded by chaotic system. The best results are obtained with the fourth derivative being zero, shown in Figure 8. The simulation results are obtained with the initial conditions 𝑥₀=00030 and 𝑥₀=11129 and with 𝜀=10⁻³.

Figure 8 

The unknown input and its estimated.

The unknown input and the estimated one are given in Figure 8. Excepted around the time origin, we got a good reconstruction of the unknown input. Figures 9(a), 9(b), 9(c) and 9(d) represent the states 𝑥₁(𝑡), 𝑥₂(𝑡), 𝑥₃(𝑡), and 𝑥₄(𝑡) and their estimated 𝑥₁(𝑡), 𝑥₂(𝑡), 𝑥₃(𝑡), and 𝑥₄(𝑡), respectively.

(a)

(b)

(c)

(d)

Figure 9 

(a) The state 𝑥1(𝑡) and its estimated 𝑥1(𝑡). (b) The state 𝑥2(𝑡) and its estimated 𝑥2(𝑡). (c) The state 𝑥3(𝑡) and its estimated 𝑥3(𝑡). (d) The state 𝑥4(𝑡) and its estimated 𝑥4(𝑡).

The dynamic errors of the states estimation are represented in Figure 10. The obtained simulation results show the good estimation of the states and the unknown input.

Figure 10 

The errors between states and their estimated.

5. Conclusion

In this paper, we have addressed the synchronization and reconstruction problem for chaotic systems. The TS fuzzy models subjected to unmeasurable decision variables and unknown inputs are employed to exactly describe the behavior of two chaotic systems, Lorenz’s system and Rossler’s system. Based on Lyapunov theory and LMI formulation, an unknown input proportional integral observer to achieve the synchronization and the unknown input reconstruction is designed. To take into account a wide variety of unknown inputs, a polynomial form with 𝑘th derivative zero is considered. Moreover, both the measurable and unmeasurable decision variables cases are studied. Simulation results are given to verify the effectiveness of the proposed method by reconstructing both states and unknown inputs. In the secure communication field, the proposed polynomial unknown input PIO with unmeasurable decision variables presents a good synchronization technique and messages recovering.

Motivated by the given results, the problem of diagnosis and fault tolerant control for more complex systems will be considered. Moreover, to reduce the conservatism due to the quadratic approach, nonquadratic Lyapunov functions will be introduced in our further research.

Appendix

By using the quadratic Lyapunov function 𝑉(𝑡)=𝑒^𝑇_𝑎(𝑡)𝑃𝑒_𝑎(𝑡) where 𝑃=𝑃^𝑇>0 and the following lemma.

Lemma A.1. For any matrices X and Y of appropriate dimensions, the following property is satisfied: 𝑋^𝑇𝑌+𝑌^𝑇𝑋≤𝜆𝑋^𝑇𝑋+𝜆⁻¹𝑌^𝑇𝑌with𝜆>0, (A.1)

the time-derivative of 𝑉(𝑡) leads 𝑉̇≤^𝑟_𝑖=1𝜇_𝑖(𝑥)𝑒^𝑇_𝑎𝒜^𝑇_𝑖𝑃+𝑃𝒜_𝑖+𝛼₀𝛿²₁𝐼+∝⁻¹𝑃²𝑒_𝑎+𝜎₁𝛿²₁𝑥^𝑇_𝑎𝑥_𝑎+𝜎₂𝛿²₂𝑢^𝑇𝑢+𝜎₃𝛿²₃𝐼−2𝑒^𝑇_𝑎𝑃𝑧 (A.2) with 𝛼₀=𝜆₁(1+𝜆),∝⁻¹=𝜆₁⁻¹+𝜆₂⁻¹+𝜆₃⁻¹,𝜎₁=𝜆₁1+𝜆⁻¹=𝛼₀𝜆,𝜎₂=𝜆₂=𝛼𝛼₀𝜆₃𝛼𝛼₀+𝜆₃(1+𝜆)−𝛼₀𝜆₃,𝜎₃=𝜆₃. (A.3) And taking into account (11c) we obtain 2𝑒^𝑇_𝑎𝑃𝑧=2𝑒^𝑇_𝑎𝑃𝜎₁𝛿²₁𝑥^𝑇_𝑎𝑥_𝑎2𝑒^𝑇_𝑎𝑦𝑒_𝑎𝑦𝑃⁻¹𝐶^𝑇𝑒_𝑎𝑦+2𝑒^𝑇_𝑎𝑃𝜎₂𝛿²₂𝑢^𝑇𝑢2𝑒^𝑇_𝑎𝑦𝑒_𝑎𝑦𝑃⁻¹𝐶^𝑇𝑒_𝑎𝑦+2𝑒^𝑇_𝑎𝑃𝜎₃𝛿²₃12𝑒^𝑇_𝑎𝑦𝑒_𝑎𝑦𝑃⁻¹𝐶^𝑇𝑒_𝑎𝑦=𝜎₁𝛿²₁𝑥^𝑇_𝑎𝑥_𝑎+𝜎₂𝛿²₂𝑢^𝑇𝑢+𝜎₃𝛿²₃𝐼. (A.4) Then, the relation (A.2) becomes 𝑉̇≤^𝑟_𝑖=1𝜇_𝑖(𝑥)𝑒^𝑇_𝑎𝒜^𝑇_𝑖𝑃+𝑃𝒜_𝑖+𝛼₀𝛿²₁𝐼+∝⁻¹𝑃²𝑒_𝑎. (A.5) The condition of stability 𝑉̇(𝑡)<0(forall𝑖=1,…,𝑟) is satisfied if 𝒜^𝑇_𝑖𝑃+𝑃𝒜_𝑖+𝛼₀𝛿²₁𝐼+∝⁻¹𝑃²+𝛼⁻¹𝑃²<0. (A.6) The Schur complement of condition (A.6) with variables given in (9) allows writing the LMI (11a).