Synchronization of Chaotic Gyros Based on Robust Nonlinear Dynamic Inversion

Abstract

A robust nonlinear dynamic inversion (RNDI) technique is proposed in order to synchronize the behavior of chaotic gyros subjected to uncertainties such as model mismatches and disturbances. Gyro is a crucial device that measures and maintains the orientation of a vehicle. By Leipnik and Newton in 1981, chaotic behavior of a gyro under specific conditions was established. Hence, controlling and synchronizing a gyro that shows irregular (chaotic) motion are very important. The proposed synchronization method is based on nonlinear dynamic inversion (NDI) control. NDI is a nonlinear control technique that removes the original system dynamics into the user-defined desired dynamics. Since NDI removes the original dynamics directly, it does not need linearizing and designing gain-scheduled controllers for each equilibrium point. However, achieving perfect cancellation of the original nonlinear dynamics is impossible in real applications due to model uncertainties and disturbances. This paper proposes the robustness assurance method of NDI based on sliding mode control (SMC). Firstly, similarities of the conventional NDI control and SMC are provided. And then the RNDI control technique is proposed. The feasibility and effectiveness of the proposed method are demonstrated by numerical simulations.

1. Introduction

Chaotic behavior is a widely observed phenomenon in nature as well as nonlinear systems. This presents a challenging problem as direct control of chaotic systems is very difficult. Chaos synchronization is a control process in which the motions of coupled chaotic systems under different initial conditions are synchronized [1–10]. Since chaotic systems are very sensitive to the initial conditions, system behaviors initiated distinctly cause irregular different motions. Figure 1 shows the irregular motions of two chaotic systems, called drive and response systems, under different initial conditions. Since the work of Pecora and Carroll in 1990, various chaos synchronization techniques based on nonlinear control theory have been developed. As shown in Figure 2, chaos synchronization of two chaotic systems entails controlling the motion of the response system by introducing an additional control input, such as adaptive control [2], passive control [3], backstepping control [4, 5], and sliding mode control [6, 7]. The feasibility of controlling chaotic systems has been explored in various fields of science and engineering, such as secure communications, chemical reactions, power converters, biological systems, and information processing [2, 5–10].

Figure 1 

Behaviors of unsynchronized chaotic systems.

Figure 2 

Behaviors of synchronized chaotic systems.

In this paper, a robust nonlinear dynamic inversion (RNDI) technique is proposed for the synchronization of chaotic gyros perturbed by bounded uncertainties. Gyro is a device that measures and maintains the orientation of a vehicle. For this reason, gyro is one of the most crucial instruments in safety-critical systems such as aircrafts, spacecrafts, and underwater vehicles. Since the work of Leipnik and Newton [11] who verified chaotic behavior of gyro under specific conditions in 1981, controlling and synchronizing a gyro has received considerable attention [5, 6, 10]. However, synchronization of gyros showing irregular (chaotic) motions is still a challenging problem.

The proposed RNDI control method is an extension of nonlinear dynamic inversion (NDI). NDI is a nonlinear control technique that removes the original system dynamics into the user-defined desired dynamics [12]. Different from many other nonlinear controllers that linearize the original system in order to design gain-scheduled controllers for each equilibrium points, NDI does not need designing gain-scheduled controllers, since it removes the system original dynamics directly. Hence, NDI avoids difficulties of ensuring stability between operating points. For this reason, NDI is widely applied in the aerospace industry [13–16] that operates in various equilibrium points. However, the main drawback of NDI is poor robustness. That is, achieving perfect cancellation of original nonlinear dynamics is impossible in real applications due to the presence of model uncertainties and disturbances acting on the system. Hence, the robustness issue has received a great deal of interest in designing NDI [12–16]. One of the most widely used methods for solving this issue is employing an additional linear controller to form an outer-loop controller while NDI works as an inner-loop controller. However, the main drawback of this method is increased order of the control system. For instance, the controller order increases to 14 when using an 𝐻_∞-based outer-loop controller for X-38 [12]. Recently, Yang et al. proposed the robust dynamic inversion (RDI) control method for linearized systems. The proposed RDI controller guarantees stability against uncertainties without using any outer-loop controller [14]. This paper proposes an extension of the RDI controller for nonlinear systems. Similarities of the NDI and the SMC are provided firstly. And then the RNDI control technique is proposed by following the design method of the SMC provided in [17]. Numerical simulations of the synchronization problem with chaotic gyros show that the proposed method achieves the desired control performance, although the systems are perturbed by uncertainties.

2. The Proposed Robust Nonlinear Dynamic Inversion

2.1. Conceptual Design of Nonlinear Dynamic Inversion

Let us consider the following nonlinear dynamical system: ̇(𝑥)(𝑥)(𝑥),𝑥=𝑓+𝑔𝑢,𝑦=ℎ (1) where 𝑥∈𝐗, an open set of 𝑅^𝑛, is a state vector and 𝑢∶𝑅→𝑅 is an input. 𝑓 and 𝑔 defined on 𝐗 are a smooth nonlinear state dynamic function and a smooth nonlinear control distribution function, respectively. Further, 𝑦∈𝐘, an open set of 𝑅, is an output and ℎ∶𝑅^𝑛→𝑅 is a smooth nonlinear function. If  𝐿_𝑔ℎ(𝑥)≠0 for all 𝑥∈𝐗, then the NDI control input for a single-input and single-output (SISO) system is designed as follows: 𝑢_NDI=𝐿_𝑔ℎ(𝑥)⁻¹𝑦̇_des−𝐿_𝑓ℎ(𝑥), (2) where 𝐿_(⋅) is the Lie derivative with respect to (⋅) and 𝑦_des represents the desired dynamics that determines the system response after canceling the original dynamics. By substituting (2) into (1), the dynamic system controlled by NDI can be represented as follows: 𝑦̇=𝑦̇_des. (3) Hence, the original nonlinear dynamics are replaced with the desired dynamics. However, the robustness issue must be considered in designing an NDI controller as perfect cancelation of the original dynamics cannot be achieved in real applications.

2.2. The Proposed Robust Nonlinear Dynamic Inversion

Let 𝜎∶𝐘→𝑅 be a smooth function such that {𝑦∣𝜎(𝑦)=0} is a smooth manifold. In general, 𝜎(𝑦)=0 is known as a sliding surface or sliding manifold in sliding mode control (SMC) theory [17, 18]. SMC consists of two phases called reaching and sliding phases [18]. One method for designing a sliding surface is using equivalent control. This method, defined as ideal sliding motion, determines an equivalent input, 𝑢_eq, that forces the output to stay on the sliding surface 𝜎(𝑦)=0. In this approach, the equivalent input can be analyzed by means of the manifold invariant conditions [17]: 𝜎̇𝑦=𝜎̇(ℎ(𝑥))=𝜕𝜎𝐿𝜕𝑦^𝑓ℎ(𝑥)+𝐿_𝑔ℎ(𝑥)𝑢_eq=0,⟹𝑢_eq=−𝜕𝜎𝐿𝜕𝑦^𝑔ℎ(𝑥)⁻¹𝜕𝜎𝐿𝜕𝑦^𝑓ℎ(𝑥), (4) where [(𝜕𝜎/𝜕𝑦)𝐿_𝑔ℎ(𝑥)] is assumed to be nonsingular. Then, the dynamics of the system on the sliding surface is governed by 𝑦̇=𝐼−𝐿_𝑔ℎ(𝑥)𝜕𝜎𝐿𝜕𝑦^𝑔ℎ(𝑥)⁻¹𝜕𝜎𝐿𝜕𝑦^𝑓ℎ(𝑥). (5) The dynamics (5) is defined as the ideal sliding dynamics. As shown in (5), the characteristics of the ideal sliding dynamics are determined by the sliding surface. Hence, choosing a sliding surface that makes the system stable is critical in the design of a sliding mode controller.

For the desired dynamics, 𝑦_des∈𝐘, let 𝑦^∗=𝑦_des−𝑦=𝑦_des−ℎ(𝑥)∈𝐘. Then, the equivalent input can be obtained as follows: 𝜎̇𝑦^∗=𝜕𝜎𝜕𝑦^∗𝑦̇^∗=𝜕𝜎𝜕𝑦^∗𝑦̇_des−𝜕𝜎𝜕𝑦^∗𝐿_𝑓ℎ(𝑥)+𝐿_𝑔ℎ(𝑥)𝑢_eq=0,⟹𝑢_eq=𝜕𝜎𝜕𝑦^∗𝐿_𝑔ℎ(𝑥)⁻¹×𝜕𝜎𝜕𝑦^∗𝑦̇_des−𝜕𝜎𝜕𝑦^∗𝐿_𝑓ℎ(𝑥). (6) In (6), if the sliding surface is identity (i.e., 𝜎(𝑦^∗)=𝑦^∗), then the equivalent input yields 𝑢_eq=𝐿_𝑔ℎ(𝑥)⁻¹𝑦̇_des−𝐿_𝑓ℎ(𝑥). (7) Comparing (7) with (2), the equivalent input designed with an identity sliding surface can be considered as the conventional NDI input.

Definition 1. Let 𝜎_NDI∶𝐘→𝑅 be a smooth function. Then 𝜎_NDI=0 is defined as the NDI surface if 𝜎_NDI(𝑦)=𝑦. That is, if the sliding surface is identity, then the sliding surface is defined as the NDI surface.
As the NDI input is represented as the equivalent input of the identity sliding surface, it controls the output on the NDI surface. Moreover, since the error between the desired dynamics and output is forced to be zero by the NDI input, the output is driven by the desired dynamics.

Theorem 2. The NDI law is well defined if and only if 𝐿_𝑔ℎ(𝑥)≠0,  for all 𝑥∈𝐗.

Proof. Suppose that the NDI law is well defined; that is, 𝑢_NDI exists uniquely. Then, it is clear that 𝐿_𝑔ℎ(𝑥)≠0 by (7). Because if 𝐿_𝑔ℎ(𝑥)=0, then 𝑦̇_des−𝐿_𝑓ℎ(𝑥) has to be zero. However, it is impossible to achieve the system 𝑦̇_des−𝐿_𝑓ℎ(𝑥)=0 in real applications. Moreover, if 𝑦̇_des−𝐿_𝑓ℎ(𝑥)=0, then the NDI input exists trivially. Thus, it contradicts the hypothesis of uniqueness. Hence, if the NDI law is well defined, then 𝐿_𝑔ℎ(𝑥)≠0.
Conversely, suppose that 𝐿_𝑔ℎ(𝑥)≠0. Then, from (7), 𝑢_NDI exists. To prove its uniqueness, it is assumed that 𝑢_NDI,1 and 𝑢_NDI,2 are two distinct NDI inputs. From the manifold invariant condition, 𝑦̇_des−𝐿_𝑓ℎ(𝑥)−𝐿_𝑔ℎ(𝑥)𝑢_NDI,1=0=𝑦̇_des−𝐿_𝑓ℎ(𝑥)−𝐿_𝑔ℎ(𝑥)𝑢_NDI,2. Hence, [𝐿_𝑔ℎ(𝑥)](𝑢_NDI,1−𝑢_NDI,2)=0. Since 𝐿_𝑔ℎ(𝑥)≠0, then 𝑢_NDI,1=𝑢_NDI,2. This contradicts the condition of 𝑢_NDI,1≠𝑢_NDI,2. Thus, if 𝐿_𝑔ℎ(𝑥)≠0, then the NDI law is well defined.
Hence, the NDI law is well defined if and only if 𝐿_𝑔ℎ(𝑥)≠0, for all 𝑥∈𝐗.

The proposed robust nonlinear dynamic inversion law, 𝑢, can be obtained by taking the extreme control values [17]: 𝑢𝑢=⁺,if𝑦<𝑦_des,𝑢⁻,if𝑦>𝑦_des, (8) where 𝑢⁺≠𝑢⁻. The following is then satisfied: lim_{ℎ(𝑥)→𝑦+des}𝐿_{𝑓+𝑔𝑢+}ℎ(𝑥)>𝑦̇_des,lim_{ℎ(𝑥)→𝑦−des}𝐿_{𝑓+𝑔𝑢−}ℎ(𝑥)<𝑦̇_des. (9) Then, the point of the controlled vector fields 𝐿_{𝑓+𝑔𝑢+}ℎ and 𝐿_{𝑓+𝑔𝑢−}ℎ moves toward the NDI surface. Figure 3 illustrates the robust nonlinear dynamics inversion regime on the NDI surface.

Figure 3 

The nonlinear dynamic inversion regime on the NDI surface.

Theorem 3. If 𝑢⁺>𝑢⁻, then 𝐿_𝑔ℎ(𝑥)>0 and, conversely, if 𝑢⁺<𝑢⁻, then 𝐿_𝑔ℎ(𝑥)<0.

Proof. It is clear directly from (9). Since 𝐿_𝑓ℎ(𝑥)+𝐿_𝑔ℎ(𝑥)𝑢⁺>𝑦̇_des and 𝐿_𝑓ℎ(𝑥)+𝐿_𝑔ℎ(𝑥)𝑢⁻<𝑦̇_des, then 𝐿_𝑔ℎ(𝑥)(𝑢⁺−𝑢⁻)>0. Hence, if 𝑢⁺>𝑢⁻, then 𝐿_𝑔ℎ(𝑥)>0 and, conversely, if 𝑢⁺<𝑢⁻, then 𝐿_𝑔ℎ(𝑥)<0.

Theorem 4. The RNDI law exists locally if and only if min𝑢⁻,𝑢⁺<𝑢_𝑁𝐷𝐼<max𝑢⁻,𝑢⁺. (10)

Proof. Suppose that the NDI law exists and 𝑢⁺>𝑢⁻, that is, 𝑢⁺=max{𝑢⁻,𝑢⁺} and 𝑢⁻=min{𝑢⁻,𝑢⁺}. From (2) and (9), 𝐿_𝑓ℎ(𝑥) + 𝐿_𝑔ℎ(𝑥)𝑢⁺>𝑦̇_des⇒𝐿_𝑔ℎ(𝑥)𝑢⁺>𝑦̇_des−𝐿_𝑓ℎ(𝑥) = 𝐿_𝑔ℎ(𝑥)𝑢_NDI. Since 𝑢⁺>𝑢⁻, by Theorem 3, 𝐿_𝑔ℎ(𝑥)>0. Hence, 𝑢⁺>𝑢_NDI. Similarly, 𝐿_𝑓ℎ(𝑥) + 𝐿_𝑔ℎ(𝑥)𝑢⁻<𝑦̇_des⇒𝐿_𝑔ℎ(𝑥)𝑢⁻<𝑦̇_des−𝐿_𝑓ℎ(𝑥)=𝐿_𝑔ℎ(𝑥)𝑢_NDI. Since 𝑢⁺<𝑢⁻, 𝐿_𝑔ℎ(𝑥)>0. Hence, min{𝑢⁻,𝑢⁺}=𝑢⁻<𝑢_NDI(𝑥)<𝑢⁺=max{𝑢⁻,𝑢⁺}.
Conversely, if 𝑢⁺<𝑢⁻, then 𝑢⁻=max{𝑢⁻,𝑢⁺} and 𝑢⁺=min{𝑢⁻,𝑢⁺}. From (9), 𝐿_𝑓ℎ(𝑥) + 𝐿_𝑔ℎ(𝑥)𝑢⁻<𝑦̇_des⇒𝐿_𝑔ℎ(𝑥)𝑢⁻<𝑦̇_des−𝐿_𝑓ℎ(𝑥)=𝐿_𝑔ℎ(𝑥)𝑢_NDI. Since 𝑢⁻>𝑢⁺, by Theorem 3, 𝐿_𝑔ℎ(𝑥)<0. Hence, 𝑢⁻>𝑢_NDI⇒max{𝑢⁻,𝑢⁺}>𝑢_NDI. Similarly, 𝐿_𝑓ℎ(𝑥) + 𝐿_𝑔ℎ(𝑥)𝑢⁺>𝑦̇_des⇒𝐿_𝑔ℎ(𝑥)𝑢⁺>𝑦̇_des−𝐿_𝑓ℎ(𝑥) = 𝐿_𝑔ℎ(𝑥)𝑢_NDI. Since 𝐿_𝑔ℎ(𝑥)<0, 𝑢⁺<𝑢_NDI. Thus, min{𝑢⁻,𝑢⁺}<𝑢_NDI. Hence, min{𝑢⁻,𝑢⁺}<𝑢_NDI<max{𝑢⁻,𝑢⁺}.
Conversely, suppose that 𝑢_NDI is a smooth feedback function satisfying (2) and (10). Then, we have 0<𝑢_NDI−min𝑢⁻,𝑢⁺<max𝑢⁻,𝑢⁺−min𝑢⁻,𝑢⁺. (11) Simply, set 𝑢⁻=min{𝑢⁻,𝑢⁺} and 𝑢⁺=max{𝑢⁻,𝑢⁺}. Then (11) is converted as follows: 0<𝑢_NDI−𝑢⁻<𝑢⁺−𝑢⁻⟹0<𝑤_NDI𝑢∶=^NDI−𝑢⁻𝑢⁺−𝑢⁻<1. (12) This leads to 0<1−𝑤_NDI<1. Moreover, the following relationships are satisfied: 𝑦̇_des−𝐿_𝑓ℎ(𝑥)+𝐿_𝑔ℎ(𝑥)𝑢_NDI=𝑤_NDI𝑦̇_des−𝐿_𝑓ℎ(𝑥)+𝐿_𝑔ℎ(𝑥)𝑢⁺+1−𝑤_NDI𝑦̇_des−𝐿_𝑓ℎ(𝑥)+𝐿_𝑔ℎ(𝑥)𝑢⁻=0. (13) From (13), 𝑦̇_des−𝐿_𝑓ℎ(𝑥)−𝐿_𝑔ℎ(𝑥)𝑢⁺ and 𝑦̇_des−𝐿_𝑓ℎ(𝑥)−𝐿_𝑔ℎ(𝑥)𝑢⁻ have opposite signs. Since the orientation of NDI is arbitrary, it is reasonable that 𝐿_𝑓ℎ(𝑥)+𝐿_𝑔ℎ(𝑥)𝑢⁻<𝑦̇_des and 𝐿_𝑓ℎ(𝑥)+𝐿_𝑔ℎ(𝑥)𝑢⁺>𝑦̇_des. Thus, (9) is satisfied. Hence, if the control law 𝑢_NDI satisfies the condition of (10), then the RNDI law exists.

Corollary 5. The RNDI law exists locally if the RNDI law is designed as 𝑢_{𝑅𝑁𝐷𝐼}=𝑢_𝑁𝐷𝐼+𝑘𝐿_𝑔ℎ(𝑥)⁻¹sgn𝑦_𝑑𝑒𝑠−ℎ(𝑥),𝑘>0. (14)

Proof. If 𝑦_des−𝑦>0, then 𝑢_RNDI=𝑢_NDI+𝑘[𝐿_𝑔ℎ(𝑥)]⁻¹≡𝑢⁺. If 𝑦_des−𝑦<0, then 𝑢_RNDI = 𝑢_NDI−𝑘[𝐿_𝑔ℎ(𝑥)]⁻¹≡𝑢⁻. Thus, min{𝑢_NDI−𝑘[𝐿_𝑔ℎ(𝑥)]⁻¹, 𝑢_NDI+𝑘[𝐿_𝑔ℎ(𝑥)]⁻¹}<𝑢_NDI < max{𝑢_NDI−𝑘[𝐿_𝑔ℎ(𝑥)]⁻¹, 𝑢_NDI+𝑘[𝐿_𝑔ℎ(𝑥)]⁻¹}. Hence, by Theorem 4, the RNDI law exists locally.

Stability of the closed-loop system designed by the proposed RNDI control law is proven by a Lyapunov stability criterion.

Theorem 6. Consider the following system: 𝑥̇(𝑡)=𝑓(𝑥(𝑡))+𝑔(𝑥(𝑡))𝑢(𝑡)+𝑑_𝑋(𝑡),𝑦(𝑡)=ℎ(𝑥(𝑡))+𝑑_𝑌(𝑡), (15) where 𝑑_𝑋(𝑡)∈𝑅^𝑛 and 𝑑_𝑌(𝑡)∈𝑅 are bounded disturbances. For 𝜉(𝑡)∈𝑅 such that 𝜉(𝑡)=(𝜕ℎ/𝜕𝑥)𝑑_𝑋(𝑡)+𝑑̇_𝑌(𝑡), if  |𝜉(𝑡)|<𝑘 for the positive switching gain 𝑘; then the system (15) designed by the proposed RNDI control law (14) is globally stable.

Proof. Suppose that 𝑦^∗=𝑦_des−𝑦. To prove the stability of the closed-loop system designed by the proposed RNDI control law, a Lyapunov stability criterion is considered. Choose a Lyapunov candidate as 𝑉𝑦^∗(𝑡=1)2𝑦^∗(𝑡)𝑦^∗(𝑡). (16) Then, the derivative of 𝑉(𝑦^∗(𝑡)) yields 𝑉̇𝑦^∗(𝑡)=𝑦^∗(𝑡)𝑦̇^∗(𝑡)=𝑦^∗(𝑡)𝑦̇_des(𝑡)−𝑦̇(𝑡)=𝑦^∗(𝑡)𝑦̇_des(𝑡)−𝐿_𝑓ℎ(𝑥)+𝐿_𝑔ℎ(𝑥)𝑢_RNDI+𝜕ℎ𝑑𝜕𝑥^𝑋(𝑡)+𝑑̇_𝑌(𝑡)=𝑦^∗(𝑡)𝑦̇_des(𝑡)−𝐿_𝑓ℎ(𝑥)+𝐿_𝑔ℎ(𝑥)×𝑢_NDI+𝑘𝐿_𝑔ℎ(𝑥)⁻¹×sgn𝑦^∗(𝑡)+𝜉(𝑡)=𝑦^∗(𝑡)𝑦̇_des(𝑡)−𝐿_𝑓ℎ(𝑥)+𝐿_𝑔ℎ(𝑥)𝑢_NDI+𝑘sgn𝑦^∗(𝑡)+𝜉(𝑡)=𝑦^∗(𝑡)−𝑘sgn𝑦^∗(𝑡(𝑡=𝑦)−𝜉)_des(𝑡)(𝑡×𝑦−𝑦)−𝑘sgn_des(𝑡)(𝑡)(𝑡≤𝜉(𝑡𝑦−𝑦−𝜉))_des(𝑡)(𝑡𝑦−𝑦)−𝑘_des(𝑡)−𝑦(𝑡)sgn𝑦_des(𝑡)(𝑡)=𝜉(𝑡𝑦−𝑦)_des(𝑡)−𝑦(𝑡)−𝑘𝑦_des(𝑡)(𝑡=𝜉(𝑡𝑦−𝑦))−𝑘_des(𝑡)−𝑦(𝑡)<0. (17) Hence, the system controlled by the proposed RNDI controller is globally stable.

3. Simulation Results

In this section, the performance of the proposed robust nonlinear dynamic inversion technique is evaluated with application to the synchronization of chaotic gyros.

3.1. Dynamics of Chaotic Gyros

The dynamic equation of chaotic gyros with linear-plus-cubic damping mounted on a vibrating platform (Figure 4) can be represented as follows [10]: 𝜃̈+𝛼²(1−cos𝜃)²sin³𝜃−𝛽sin𝜃+𝑐₁𝜃̇+𝑐₂𝜃̇³=𝑓sin𝜔𝑡sin𝜃, (18) where 𝑐₁𝜃̇ and 𝑐₂𝜃̇³ are linear and nonlinear damping terms, respectively. Further, 𝛼²[(1−cos𝜃)²/sin³𝜃]−𝛽sin𝜃 is a nonlinear resilience force and 𝑓sin𝜔𝑡 is a parametric excitation.

Figure 4 

Schematic diagram of a symmetric gyroscope [10].

By letting 𝑥₁₁=𝜃, 𝑥₁₂=𝜃̇, and 𝑔(𝜃)=−𝛼²[(1−cos𝜃)²/sin³𝜃], the state equation of chaotic gyros can be transformed into the following normal form: 𝑥̇₁₁=𝑥₁₂,𝑥̇₁₂=𝑔𝑥₁₁−𝑐₁𝑥₁₂−𝑐₂𝑥³₁₂+𝛽+𝑓sin𝜔𝑡sin𝑥₁₁,𝑦₁₁=𝑥₁₁. (19) It is well known that for particular values of 𝛼²=100, 𝛽=1, 𝑐₁=0.5, 𝑐₂=0.05, 𝜔=2, and 𝑓=35.5, the gyro system exhibits chaotic behavior [11]. Figures 5 and 6 illustrate the irregular motion of the states and the phase portrait under the initial condition of [𝑥₁₁(0),𝑥₁₂(0)]=[1,−1], respectively. Let 𝑥₁=[𝑥₁₁,𝑥₁₂]^𝑇 and 𝑥₂=[𝑥₂₁,𝑥₂₂]^𝑇 denote the states of the drive and response systems, respectively. Then, the dynamic equations of the drive and response systems with model mismatches and disturbances are represented as follows.

(a)

(b)

Figure 5 

Behaviors of the chaotic gyros: (a) trajectories of 𝜃 of the chaotic gyro, (b) trajectories of 𝜃̇ of the chaotic gyro.

Figure 6 

Phase portrait of the chaotic gyro.

Drive system 𝑥̇₁₁=𝑥₁₂,𝑥̇₁₂=𝑔𝑥₁₁−𝑐₁𝑥₁₂−𝑐₂𝑥³₁₂+𝛽+𝑓sin𝜔𝑡sin𝑥₁₁+Δ𝑓𝑥₁₁,𝑥₁₂+𝑑_𝑥1(𝑡),𝑦₁₁=𝑥₁₁, (20) where Δ𝑓(𝑥₁₁,𝑥₁₂) and 𝑑_𝑥1(𝑡) are assumed to be bounded model mismatches and bounded disturbance, respectively.

Response system 𝑥̇₂₁=𝑥₂₂,𝑥̇₂₂=𝑔𝑥₂₁−𝑐₁𝑥₂₂−𝑐₂𝑥³₂₂+𝛽+𝑓sin𝜔𝑡sin𝑥₂₁+Δ𝑓𝑥₂₁,𝑥₂₂+𝑑_𝑥2(𝑡)(𝑡),𝑦+𝑢₂₁=𝑥₂₁, (21) where Δ𝑓(𝑥₂₁,𝑥₂₂) and 𝑑_𝑥2(𝑡) also denote model mismatches and disturbance, respectively, and are also assumed to be bounded. 𝑢(𝑡) is the control input provided by the proposed RNDI controller to synchronize the chaotic gyros.

If the errors between the states of the drive and response systems are defined as 𝑒_𝑥𝑖=𝑥_2𝑖−𝑥_1𝑖 and 𝑒_𝑦𝑖=𝑦_2𝑖−𝑦_1𝑖, then the error dynamics can be represented as follows: 𝑒̇_𝑥1=𝑒_𝑥2,𝑒̇_𝑥2=𝑔𝑥₁₁,𝑥₂₁−𝑐₁𝑒_𝑥2−𝑐₂𝑥³₂₂−𝑥³₁₂,+𝛽+𝑓sin𝜔𝑡sin𝑥₂₁−sin𝑥₁₁+Δ𝑓𝑥₁,𝑥₂(𝑡)(𝑡),𝑒+𝑑+𝑢_𝑦1=𝑦₂₁−𝑦₁₁, (22) where 𝑔(𝑥₁₁,𝑥₂₁)=𝑔(𝑥₂₁)−𝑔(𝑥₁₁), Δ𝑓(𝑥₁,𝑥₂)=Δ𝑓(𝑥₂₁,𝑥₂₂)−Δ𝑓(𝑥₁₁,𝑥₁₂), and 𝑑(𝑡)=𝑑_𝑥2(𝑡)−𝑑_𝑥1(𝑡). Since Δ𝑓(𝑥₁₁,𝑥₁₂), Δ𝑓(𝑥₂₁,𝑥₂₂), 𝑑_𝑥1(𝑡), and 𝑑_𝑥2(𝑡) are bounded, Δ𝑓(𝑥₁,𝑥₂) and 𝑑(𝑡) are also bounded. From (20) to (22), it is clear that the synchronization problem of the two chaotic systems is replaced with a stabilization problem of the error dynamics.

To synchronize the chaotic gyros, the proposed RNDI controller is adopted in the response system in 40 sec. It is assumed that |𝑑_𝑥1(𝑡)+Δ𝑓(𝑥₁₁,𝑥₁₂)|≤0.5 and |𝑑_𝑥2(𝑡)+Δ𝑓(𝑥₂₁,𝑥₂₂)|≤0.7. The initial conditions are 𝑥₁₁(0)=1, 𝑥₁₂(0)=−1, and 𝑥₂₁(0)=1.6, 𝑥₂₂(0)=0.8.

3.2. Simulation Results

In this section, numerical simulation results are provided in order to demonstrate the feasibility and effectiveness of the proposed RNDI controller.

Figures 7 and 8 illustrate the state and error trajectories of 𝜃 and 𝜃̇ of the unsynchronized chaotic gyros, respectively; that is, no control action is performed in this case. The behaviors of the unsynchronized chaotic gyros oscillate independently due to the distinct initial conditions. Since there is no control action for synchronization, the errors of 𝜃 and 𝜃̇ between the drive and response systems do not converge to zero as shown in Figures 7(b) and 8(b). Actually, the behaviors during 𝑡=5--17 sec seem to be synchronized in these figures. However, these are not the result of synchronization but are instead due to dynamic characteristics of the considered systems.

(a)

(b)

Figure 7 

Simulation results of 𝜃 and error trajectories of unsynchronized chaotic gyros: (a) trajectories of 𝜃 of drive and response systems, (b) error behavior of 𝜃.

(a)

(b)

Figure 8 

Simulation results of 𝜃̇ and error trajectories of unsynchronized chaotic gyros: (a) trajectories of 𝜃̇ of drive and response systems, (b) error behavior of 𝜃̇.

Figures 9 and 10 show the synchronized results of chaotic gyros under distinct initial conditions. In this case, the control input generated by the proposed RNDI controller forces the states of the response system to track those of the drive system. It is assumed that the RNDI controller is activated in 𝑡=40 sec to distinguish the dynamic behavior during 𝑡=5--17 sec from the synchronized behavior. Once the controller is activated in 𝑡=40 sec, 𝜃 and 𝜃̇ of the response system can track those of the drive system as shown in Figures 9(a) and 10(a). Moreover, the errors of 𝜃 and 𝜃̇ between the drive and response systems converge to zero within a 1 sec (Figures 9(b) and 10(b)). Hence, the proposed RNDI technique achieves the synchronization of chaotic systems containing uncertainties of model mismatches and disturbances under distinct initial conditions.

(a)

(b)

Figure 9 

Simulation results of 𝜃 and error trajectories of synchronized chaotic gyros: (a) trajectories of 𝜃 of drive and response systems, (b) error behavior of 𝜃.

(a)

(b)

Figure 10 

Simulation results of 𝜃̇ and error trajectories of synchronized chaotic gyros: (a) trajectories of 𝜃̇ of drive and response systems, (b) error behavior of 𝜃̇.

4. Conclusion

In this paper, a robust nonlinear dynamic inversion (RNDI) method has been proposed to solve the synchronization problem of chaotic gyros. The proposed RNDI improves robustness of NDI. Simulation results with synchronization of chaotic gyros using the proposed method show that accurate control performance can be achieved even in the presence of uncertainties without any additional outer-loop controller.

Acknowledgment

This research was supported by the the Ministry of Knowledge Economy (MKE), Republic of Korea, under the Convergence Information Technology Research Center (CITRC) support Program (NIPA-2013-H0401-13-1005) supervized by the National IT Industry Promotion Agency (NIPA).