# Synchronization of Chaotic Gyros Based on Robust Nonlinear Dynamic Inversion

Authors: Inseok Yang; Dongik Lee
Journal: Journal of Applied Mathematics
 (2013)

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## 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.


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## Body

## 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:(1)x˙=f(x)+g(x)u,y=h(x),
where x∈X, an open set of Rn, 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:Rn→R is a smooth nonlinear function. If  Lgh(x)≠0 for all x∈X, then the NDI control input for a single-input and single-output (SISO) system is designed as follows:
(2)uNDI=[Lgh(x)]-1[y˙des-Lfh(x)],
where L(·) is the Lie derivative with respect to (·) and ydes 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:
(3)y˙=y˙des.
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

Letσ: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, ueq, 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]:
(4)σ˙(y)=σ˙(h(x))=(∂σ∂y)[Lfh(x)+Lgh(x)ueq]=0,⟹ueq=-[(∂σ∂y)Lgh(x)]-1[(∂σ∂y)Lfh(x)],
where [(∂σ/∂y)Lgh(x)] is assumed to be nonsingular. Then, the dynamics of the system on the sliding surface is governed by
(5)y˙=[I-Lgh(x)[(∂σ∂y)Lgh(x)]-1(∂σ∂y)]Lfh(x).
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,ydes∈Y, let y*=ydes-y=ydes-h(x)∈Y. Then, the equivalent input can be obtained as follows:
(6)σ˙(y*)=(∂σ∂y*)y˙*=(∂σ∂y*)y˙des-(∂σ∂y*)[Lfh(x)+Lgh(x)ueq]=0,⟹ueq=[(∂σ∂y*)Lgh(x)]-1×[(∂σ∂y*)y˙des-(∂σ∂y*)Lfh(x)].
In (6), if the sliding surface is identity (i.e., σ(y*)=y*), then the equivalent input yields
(7)ueq=[Lgh(x)]-1[y˙des-Lfh(x)].
Comparing (7) with (2), the equivalent input designed with an identity sliding surface can be considered as the conventional NDI input.Definition 1.
LetσNDI:Y→R be a smooth function. Then σNDI=0 is defined as the NDI surface if σNDI(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 ifLgh(x)≠0,  for all x∈X.Proof.
Suppose that the NDI law is well defined; that is,uNDI exists uniquely. Then, it is clear that Lgh(x)≠0 by (7). Because if Lgh(x)=0, then y˙des-Lfh(x) has to be zero. However, it is impossible to achieve the system y˙des-Lfh(x)=0 in real applications. Moreover, if y˙des-Lfh(x)=0, then the NDI input exists trivially. Thus, it contradicts the hypothesis of uniqueness. Hence, if the NDI law is well defined, then Lgh(x)≠0.
Conversely, suppose thatLgh(x)≠0. Then, from (7), uNDI exists. To prove its uniqueness, it is assumed that uNDI,1 and uNDI,2 are two distinct NDI inputs. From the manifold invariant condition, y˙des-Lfh(x)-Lgh(x)uNDI,1=0=y˙des-Lfh(x)-Lgh(x)uNDI,2. Hence, [Lgh(x)](uNDI,1-uNDI,2)=0. Since Lgh(x)≠0, then uNDI,1=uNDI,2. This contradicts the condition of uNDI,1≠uNDI,2. Thus, if Lgh(x)≠0, then the NDI law is well defined.
Hence, the NDI law is well defined if and only ifLgh(x)≠0, for all x∈X.The proposed robust nonlinear dynamic inversion law,u, can be obtained by taking the extreme control values [17]:
(8)u={u+,ify<ydes,u-,ify>ydes,
where u+≠u-. The following is then satisfied:
(9)limh(x)→ydes+Lf+gu+h(x)>y˙des,limh(x)→ydes-Lf+gu-h(x)<y˙des.
Then, the point of the controlled vector fields Lf+gu+h and Lf+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.
Ifu+>u-, then Lgh(x)>0 and, conversely, if u+<u-, then Lgh(x)<0.Proof.
It is clear directly from (9). Since Lfh(x)+Lgh(x)u+>y˙des and Lfh(x)+Lgh(x)u-<y˙des, then Lgh(x)(u+-u-)>0. Hence, if u+>u-, then Lgh(x)>0 and, conversely, if u+<u-, then Lgh(x)<0.Theorem 4.
The RNDI law exists locally if and only if(10)min{u-,u+}<uNDI<max{u-,u+}.Proof.
Suppose that the NDI law exists andu+>u-, that is, u+=max{u-,u+} and u-=min{u-,u+}. From (2) and (9), Lfh(x) + Lgh(x)u+>y˙des⇒Lgh(x)u+>y˙des-Lfh(x) = Lgh(x)uNDI. Since u+>u-, by Theorem 3, Lgh(x)>0. Hence, u+>uNDI. Similarly, Lfh(x) + Lgh(x)u-<y˙des⇒Lgh(x)u-<y˙des-Lfh(x)=Lgh(x)uNDI. Since u+<u-, Lgh(x)>0. Hence, min{u-,u+}=u-<uNDI(x)<u+=max{u-,u+}.
Conversely, ifu+<u-, then u-=max{u-,u+} and u+=min{u-,u+}. From (9), Lfh(x) + Lgh(x)u-<y˙des⇒Lgh(x)u-<y˙des-Lfh(x)=Lgh(x)uNDI. Since u->u+, by Theorem 3, Lgh(x)<0. Hence, u->uNDI⇒max{u-,u+}>uNDI. Similarly, Lfh(x) + Lgh(x)u+>y˙des⇒Lgh(x)u+>y˙des-Lfh(x) = Lgh(x)uNDI. Since Lgh(x)<0, u+<uNDI. Thus, min{u-,u+}<uNDI. Hence, min{u-,u+}<uNDI<max{u-,u+}.
Conversely, suppose thatuNDI is a smooth feedback function satisfying (2) and (10). Then, we have
(11)0<uNDI-min{u-,u+}<max{u-,u+}-min{u-,u+}.
Simply, set u-=min{u-,u+} and u+=max{u-,u+}. Then (11) is converted as follows:
(12)0<uNDI-u-<u+-u-⟹0<wNDI:=uNDI-u-u+-u-<1.
This leads to 0<1-wNDI<1. Moreover, the following relationships are satisfied:
(13)y˙des-(Lfh(x)+Lgh(x)uNDI)=wNDI{y˙des-(Lfh(x)+Lgh(x)u+)}+(1-wNDI){y˙des-(Lfh(x)+Lgh(x)u-)}=0.
From (13), y˙des-Lfh(x)-Lgh(x)u+ and y˙des-Lfh(x)-Lgh(x)u- have opposite signs. Since the orientation of NDI is arbitrary, it is reasonable that Lfh(x)+Lgh(x)u-<y˙des and Lfh(x)+Lgh(x)u+>y˙des. Thus, (9) is satisfied. Hence, if the control law uNDI satisfies the condition of (10), then the RNDI law exists.Corollary 5.
The RNDI law exists locally if the RNDI law is designed as(14)uRNDI=uNDI+k[Lgh(x)]-1sgn(ydes-h(x)),k>0.Proof.
Ifydes-y>0, then uRNDI=uNDI+k[Lgh(x)]-1≡u+. If ydes-y<0, then uRNDI = uNDI-k[Lgh(x)]-1≡u-. Thus, min{uNDI-k[Lgh(x)]-1, uNDI+k[Lgh(x)]-1}<uNDI < max{uNDI-k[Lgh(x)]-1, uNDI+k[Lgh(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:(15)x˙(t)=f(x(t))+g(x(t))u(t)+dX(t),y(t)=h(x(t))+dY(t),
where dX(t)∈Rn and dY(t)∈R are bounded disturbances. For ξ(t)∈R such that ξ(t)=(∂h/∂x)dX(t)+d˙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 thaty*=ydes-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
(16)V(y*(t))=12y*(t)y*(t).
Then, the derivative of V(y*(t)) yields
(17)V˙(y*(t))=y*(t)y˙*(t)=y*(t)[y˙des(t)-y˙(t)]=y*(t)[(∂h∂x)y˙des(t)-{(∂h∂x)Lfh(x)+Lgh(x)uRNDI+(∂h∂x)dX(t)+d˙Y(t)}]=y*(t)[[Lgh(x)]-1y˙des(t)-{[Lgh(x)]-1Lfh(x)+Lgh(x)×{uNDI+k[Lgh(x)]-1×sgn(y*(t))[Lgh(x)]-1}+ξ(t)}]=y*(t)[(y*(t))(y*(t))+ksgn(y*(t))+ξ(t)Lfh}y˙des(t)-{Lfh(x)+Lgh(x)uNDI+ksgn(y*(t))+ξ(t)Lfh}]=y*(t)[-ksgn(y*(t))-ξ(t)]=[ydes(t)-y(t)]×[-ksgn(ydes(t)-y(t))-ξ(t)]≤|ξ(t)||ydes(t)-y(t)|-k(ydes(t)-y(t))sgn(ydes(t)-y(t))=|ξ(t)||ydes(t)-y(t)|-k|ydes(t)-y(t)|=(|ξ(t)|-k)|ydes(t)-y(t)|<0.
Hence, the system controlled by the proposed RNDI controller is globally stable.


## 2.1. Conceptual Design of Nonlinear Dynamic Inversion

Let us consider the following nonlinear dynamical system:(1)x˙=f(x)+g(x)u,y=h(x),
where x∈X, an open set of Rn, 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:Rn→R is a smooth nonlinear function. If  Lgh(x)≠0 for all x∈X, then the NDI control input for a single-input and single-output (SISO) system is designed as follows:
(2)uNDI=[Lgh(x)]-1[y˙des-Lfh(x)],
where L(·) is the Lie derivative with respect to (·) and ydes 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:
(3)y˙=y˙des.
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

Letσ: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, ueq, 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]:
(4)σ˙(y)=σ˙(h(x))=(∂σ∂y)[Lfh(x)+Lgh(x)ueq]=0,⟹ueq=-[(∂σ∂y)Lgh(x)]-1[(∂σ∂y)Lfh(x)],
where [(∂σ/∂y)Lgh(x)] is assumed to be nonsingular. Then, the dynamics of the system on the sliding surface is governed by
(5)y˙=[I-Lgh(x)[(∂σ∂y)Lgh(x)]-1(∂σ∂y)]Lfh(x).
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,ydes∈Y, let y*=ydes-y=ydes-h(x)∈Y. Then, the equivalent input can be obtained as follows:
(6)σ˙(y*)=(∂σ∂y*)y˙*=(∂σ∂y*)y˙des-(∂σ∂y*)[Lfh(x)+Lgh(x)ueq]=0,⟹ueq=[(∂σ∂y*)Lgh(x)]-1×[(∂σ∂y*)y˙des-(∂σ∂y*)Lfh(x)].
In (6), if the sliding surface is identity (i.e., σ(y*)=y*), then the equivalent input yields
(7)ueq=[Lgh(x)]-1[y˙des-Lfh(x)].
Comparing (7) with (2), the equivalent input designed with an identity sliding surface can be considered as the conventional NDI input.Definition 1.
LetσNDI:Y→R be a smooth function. Then σNDI=0 is defined as the NDI surface if σNDI(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 ifLgh(x)≠0,  for all x∈X.Proof.
Suppose that the NDI law is well defined; that is,uNDI exists uniquely. Then, it is clear that Lgh(x)≠0 by (7). Because if Lgh(x)=0, then y˙des-Lfh(x) has to be zero. However, it is impossible to achieve the system y˙des-Lfh(x)=0 in real applications. Moreover, if y˙des-Lfh(x)=0, then the NDI input exists trivially. Thus, it contradicts the hypothesis of uniqueness. Hence, if the NDI law is well defined, then Lgh(x)≠0.
Conversely, suppose thatLgh(x)≠0. Then, from (7), uNDI exists. To prove its uniqueness, it is assumed that uNDI,1 and uNDI,2 are two distinct NDI inputs. From the manifold invariant condition, y˙des-Lfh(x)-Lgh(x)uNDI,1=0=y˙des-Lfh(x)-Lgh(x)uNDI,2. Hence, [Lgh(x)](uNDI,1-uNDI,2)=0. Since Lgh(x)≠0, then uNDI,1=uNDI,2. This contradicts the condition of uNDI,1≠uNDI,2. Thus, if Lgh(x)≠0, then the NDI law is well defined.
Hence, the NDI law is well defined if and only ifLgh(x)≠0, for all x∈X.The proposed robust nonlinear dynamic inversion law,u, can be obtained by taking the extreme control values [17]:
(8)u={u+,ify<ydes,u-,ify>ydes,
where u+≠u-. The following is then satisfied:
(9)limh(x)→ydes+Lf+gu+h(x)>y˙des,limh(x)→ydes-Lf+gu-h(x)<y˙des.
Then, the point of the controlled vector fields Lf+gu+h and Lf+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.
Ifu+>u-, then Lgh(x)>0 and, conversely, if u+<u-, then Lgh(x)<0.Proof.
It is clear directly from (9). Since Lfh(x)+Lgh(x)u+>y˙des and Lfh(x)+Lgh(x)u-<y˙des, then Lgh(x)(u+-u-)>0. Hence, if u+>u-, then Lgh(x)>0 and, conversely, if u+<u-, then Lgh(x)<0.Theorem 4.
The RNDI law exists locally if and only if(10)min{u-,u+}<uNDI<max{u-,u+}.Proof.
Suppose that the NDI law exists andu+>u-, that is, u+=max{u-,u+} and u-=min{u-,u+}. From (2) and (9), Lfh(x) + Lgh(x)u+>y˙des⇒Lgh(x)u+>y˙des-Lfh(x) = Lgh(x)uNDI. Since u+>u-, by Theorem 3, Lgh(x)>0. Hence, u+>uNDI. Similarly, Lfh(x) + Lgh(x)u-<y˙des⇒Lgh(x)u-<y˙des-Lfh(x)=Lgh(x)uNDI. Since u+<u-, Lgh(x)>0. Hence, min{u-,u+}=u-<uNDI(x)<u+=max{u-,u+}.
Conversely, ifu+<u-, then u-=max{u-,u+} and u+=min{u-,u+}. From (9), Lfh(x) + Lgh(x)u-<y˙des⇒Lgh(x)u-<y˙des-Lfh(x)=Lgh(x)uNDI. Since u->u+, by Theorem 3, Lgh(x)<0. Hence, u->uNDI⇒max{u-,u+}>uNDI. Similarly, Lfh(x) + Lgh(x)u+>y˙des⇒Lgh(x)u+>y˙des-Lfh(x) = Lgh(x)uNDI. Since Lgh(x)<0, u+<uNDI. Thus, min{u-,u+}<uNDI. Hence, min{u-,u+}<uNDI<max{u-,u+}.
Conversely, suppose thatuNDI is a smooth feedback function satisfying (2) and (10). Then, we have
(11)0<uNDI-min{u-,u+}<max{u-,u+}-min{u-,u+}.
Simply, set u-=min{u-,u+} and u+=max{u-,u+}. Then (11) is converted as follows:
(12)0<uNDI-u-<u+-u-⟹0<wNDI:=uNDI-u-u+-u-<1.
This leads to 0<1-wNDI<1. Moreover, the following relationships are satisfied:
(13)y˙des-(Lfh(x)+Lgh(x)uNDI)=wNDI{y˙des-(Lfh(x)+Lgh(x)u+)}+(1-wNDI){y˙des-(Lfh(x)+Lgh(x)u-)}=0.
From (13), y˙des-Lfh(x)-Lgh(x)u+ and y˙des-Lfh(x)-Lgh(x)u- have opposite signs. Since the orientation of NDI is arbitrary, it is reasonable that Lfh(x)+Lgh(x)u-<y˙des and Lfh(x)+Lgh(x)u+>y˙des. Thus, (9) is satisfied. Hence, if the control law uNDI satisfies the condition of (10), then the RNDI law exists.Corollary 5.
The RNDI law exists locally if the RNDI law is designed as(14)uRNDI=uNDI+k[Lgh(x)]-1sgn(ydes-h(x)),k>0.Proof.
Ifydes-y>0, then uRNDI=uNDI+k[Lgh(x)]-1≡u+. If ydes-y<0, then uRNDI = uNDI-k[Lgh(x)]-1≡u-. Thus, min{uNDI-k[Lgh(x)]-1, uNDI+k[Lgh(x)]-1}<uNDI < max{uNDI-k[Lgh(x)]-1, uNDI+k[Lgh(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:(15)x˙(t)=f(x(t))+g(x(t))u(t)+dX(t),y(t)=h(x(t))+dY(t),
where dX(t)∈Rn and dY(t)∈R are bounded disturbances. For ξ(t)∈R such that ξ(t)=(∂h/∂x)dX(t)+d˙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 thaty*=ydes-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
(16)V(y*(t))=12y*(t)y*(t).
Then, the derivative of V(y*(t)) yields
(17)V˙(y*(t))=y*(t)y˙*(t)=y*(t)[y˙des(t)-y˙(t)]=y*(t)[(∂h∂x)y˙des(t)-{(∂h∂x)Lfh(x)+Lgh(x)uRNDI+(∂h∂x)dX(t)+d˙Y(t)}]=y*(t)[[Lgh(x)]-1y˙des(t)-{[Lgh(x)]-1Lfh(x)+Lgh(x)×{uNDI+k[Lgh(x)]-1×sgn(y*(t))[Lgh(x)]-1}+ξ(t)}]=y*(t)[(y*(t))(y*(t))+ksgn(y*(t))+ξ(t)Lfh}y˙des(t)-{Lfh(x)+Lgh(x)uNDI+ksgn(y*(t))+ξ(t)Lfh}]=y*(t)[-ksgn(y*(t))-ξ(t)]=[ydes(t)-y(t)]×[-ksgn(ydes(t)-y(t))-ξ(t)]≤|ξ(t)||ydes(t)-y(t)|-k(ydes(t)-y(t))sgn(ydes(t)-y(t))=|ξ(t)||ydes(t)-y(t)|-k|ydes(t)-y(t)|=(|ξ(t)|-k)|ydes(t)-y(t)|<0.
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 (Figure4) can be represented as follows [10]:
(18)θ¨+α2(1-cosθ)2sin3θ-βsinθ+c1θ˙+c2θ˙3=fsinωtsinθ,
where c1θ˙ and c2θ˙3 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 lettingx11=θ, x12=θ˙, and g(θ)=-α2[(1-cosθ)2/sin3θ], the state equation of chaotic gyros can be transformed into the following normal form:
(19)x˙11=x12,x˙12=g(x11)-c1x12-c2x123+(β+fsinωt)sinx11,y11=x11.
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.Behaviors of the chaotic gyros: (a) trajectories ofθ of the chaotic gyro, (b) trajectories of θ˙ of the chaotic gyro.
(a)
(b)Figure 6
Phase portrait of the chaotic gyro.Drive system(20)x˙11=x12,x˙12=g(x11)-c1x12-c2x123+(β+fsinωt)sinx11+Δf(x11,x12)+dx1(t),y11=x11,
where Δf(x11,x12) and dx1(t) are assumed to be bounded model mismatches and bounded disturbance, respectively.Response system(21)x˙21=x22,x˙22=g(x21)-c1x22-c2x223+(β+fsinωt)sinx21+Δf(x21,x22)+dx2(t)+u(t),y21=x21,
where Δf(x21,x22) and dx2(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 asexi=x2i-x1i and eyi=y2i-y1i, then the error dynamics can be represented as follows:
(22)e˙x1=ex2,e˙x2=g(x11,x21)-c1ex2-c2(x223-x123),+(β+fsinωt)(sinx21-sinx11)+Δf(x1,x2)+d(t)+u(t),ey1=y21-y11,
where g(x11,x21)=g(x21)-g(x11), Δf(x1,x2)=Δf(x21,x22)-Δf(x11,x12), and d(t)=dx2(t)-dx1(t). Since Δf(x11,x12), Δf(x21,x22), dx1(t), and dx2(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|dx1(t)+Δf(x11,x12)|≤0.5 and |dx2(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.Figures7 and 8 illustrate the state and error trajectories of θ and θ˙ 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 θ˙ 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.Simulation results ofθ and error trajectories of unsynchronized chaotic gyros: (a) trajectories of θ of drive and response systems, (b) error behavior of θ.
(a)
(b)Simulation results ofθ˙ and error trajectories of unsynchronized chaotic gyros: (a) trajectories of θ˙ of drive and response systems, (b) error behavior of θ˙.
(a)
(b)Figures9 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 θ˙ 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 θ˙ 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.Simulation results ofθ and error trajectories of synchronized chaotic gyros: (a) trajectories of θ of drive and response systems, (b) error behavior of θ.
(a)
(b)Simulation results ofθ˙ and error trajectories of synchronized chaotic gyros: (a) trajectories of θ˙ of drive and response systems, (b) error behavior of θ˙.
(a)
(b)


## 3.1. Dynamics of Chaotic Gyros

The dynamic equation of chaotic gyros with linear-plus-cubic damping mounted on a vibrating platform (Figure4) can be represented as follows [10]:
(18)θ¨+α2(1-cosθ)2sin3θ-βsinθ+c1θ˙+c2θ˙3=fsinωtsinθ,
where c1θ˙ and c2θ˙3 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 lettingx11=θ, x12=θ˙, and g(θ)=-α2[(1-cosθ)2/sin3θ], the state equation of chaotic gyros can be transformed into the following normal form:
(19)x˙11=x12,x˙12=g(x11)-c1x12-c2x123+(β+fsinωt)sinx11,y11=x11.
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.Behaviors of the chaotic gyros: (a) trajectories ofθ of the chaotic gyro, (b) trajectories of θ˙ of the chaotic gyro.
(a)
(b)Figure 6
Phase portrait of the chaotic gyro.Drive system(20)x˙11=x12,x˙12=g(x11)-c1x12-c2x123+(β+fsinωt)sinx11+Δf(x11,x12)+dx1(t),y11=x11,
where Δf(x11,x12) and dx1(t) are assumed to be bounded model mismatches and bounded disturbance, respectively.Response system(21)x˙21=x22,x˙22=g(x21)-c1x22-c2x223+(β+fsinωt)sinx21+Δf(x21,x22)+dx2(t)+u(t),y21=x21,
where Δf(x21,x22) and dx2(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 asexi=x2i-x1i and eyi=y2i-y1i, then the error dynamics can be represented as follows:
(22)e˙x1=ex2,e˙x2=g(x11,x21)-c1ex2-c2(x223-x123),+(β+fsinωt)(sinx21-sinx11)+Δf(x1,x2)+d(t)+u(t),ey1=y21-y11,
where g(x11,x21)=g(x21)-g(x11), Δf(x1,x2)=Δf(x21,x22)-Δf(x11,x12), and d(t)=dx2(t)-dx1(t). Since Δf(x11,x12), Δf(x21,x22), dx1(t), and dx2(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|dx1(t)+Δf(x11,x12)|≤0.5 and |dx2(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.Figures7 and 8 illustrate the state and error trajectories of θ and θ˙ 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 θ˙ 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.Simulation results ofθ and error trajectories of unsynchronized chaotic gyros: (a) trajectories of θ of drive and response systems, (b) error behavior of θ.
(a)
(b)Simulation results ofθ˙ and error trajectories of unsynchronized chaotic gyros: (a) trajectories of θ˙ of drive and response systems, (b) error behavior of θ˙.
(a)
(b)Figures9 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 θ˙ 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 θ˙ 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.Simulation results ofθ and error trajectories of synchronized chaotic gyros: (a) trajectories of θ of drive and response systems, (b) error behavior of θ.
(a)
(b)Simulation results ofθ˙ and error trajectories of synchronized chaotic gyros: (a) trajectories of θ˙ of drive and response systems, (b) error behavior of θ˙.
(a)
(b)


## 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.



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*Source: 519796-2013-08-28.xml*