Adaptive Inverse Optimal Control of a Novel Fractional-Order Four-Wing Hyperchaotic System with Uncertain Parameter and Circuitry Implementation

An efficient approach of inverse optimal control and adaptive control is developed for global asymptotic stabilization of a novel fractional-order four-wing hyperchaotic system with uncertain parameter. Based on the inverse optimal control methodology and fractional-order stability theory, a control Lyapunov function (CLF) is constructed and an adaptive state feedback controller is designed to achieve inverse optimal control of a novel fractional-order hyperchaotic system with four-wing attractor. Then, an electronic oscillation circuit is designed to implement the dynamical behaviors of the fractional-order four-wing hyperchaotic system and verify the satisfactory performance of the controller. Comparing with other fractional-order chaos control methods which may have more than one nonlinear state feedback controller, the inverse optimal controller has the advantages of simple structure, high reliability, and less control effort that is required and can be implemented by electronic oscillation circuit.


Introduction
Fractional calculus is a classical mathematical theory with history of more than 300 years.Fractional-order differential equations can describe many systems in the real world more adequately, such as electrical circuits [1][2][3], polymer material [4], finance systems [5], and population models [6].
Comparing with integer-order chaotic systems which exhibit complex nonlinear phenomena, fractional-order and multi-wing chaotic systems exhibit more complex and richer dynamical behaviors.It is expected that those chaotic systems will have a certain theoretical and practical significance for secure communication, control processing, and some other engineering applications.There exist some well-known fractional-order systems and multi-wing systems, such as the fractional-order Chua's circuit [7], the fractional-order Rössler system [8], the fractional-order Chen system [9],the fractional-order Lu system [10], the first true four-wing attractor [11], a family of hyperchaotic systems with four-wing attractor [12], among many others [13][14][15][16][17][18][19][20][21][22].
The applications of fractional-order differential equations in control processing developed rapidly in the last two decades.Fractional-order control methods and researches on the stability of fractional-order systems have become the frontier problem in modern nonlinear dynamics [23][24][25].Podlubny and his colleagues proposed the fractionalorder proportional-integral (PI) and proportional-integralderivative (PID) controllers which are named PI  and PI  D  controllers with the orders  and  [26] and designed analogue circuits to implement fractional-order controllers [27].Hamamci presented a method to stabilize a given fractional dynamic system using fractional-order PI  and PI  D  controllers [28].In [29], the author proposed a solution scheme for a class of fractional optimal control problems.In [30], authors designed a fractional-order sliding mode controller to stabilize a fractional-order hyperchaotic system.
Nonlinear controllers have been adopted in many fields in spite of having the complex structure and being not easy to obtain.Optimal control guarantees several desirable properties for the closed-loop system, including stability margins and robustness.To circumvent the task of solving a Hamilton-Jacobi-Bellman equation whose solution is nonexistent or nonunique in generally, the inverse optimal control technique based on input-to-state stability concept was developed [31].
In this article, based on inverse optimal control methodology and fractional-order stability theory, we construct an adaptive state feedback controller to achieve the global asymptotic stabilization of a novel fractional-order fourwing hyperchaotic system.An electronic oscillation circuit is designed to realize the dynamical behaviors of the fractionalorder system and verify the satisfactory performance of the controller.Comparing with other control methods, for example, feedback control [32], active control [33], impulsive control [34], back-stepping control [35], and sliding mode control [36], the inverse optimal controller has the advantages of simple structure, high reliability, and less control effort that is required.
The rest of this paper is organized as follows.In Section 2, the inverse optimal control methodology and fractionalorder stability theory are introduced.In Section 3, firstly, a novel pseudo four-wing hyperchaotic system is analyzed.Then a derivative fractional-order four-wing hyperchaotic system is implemented by numerical simulations, and the adaptive inverse optimal control is applied to stabilize an unstable equilibrium point of the fractional-order four-wing hyperchaotic system.In Section 4, circuitry implementations are given for verifying the feasibility.Finally, the conclusion part summarizes the whole development process and presents some concluding remarks and comments.

Preliminaries
The inverse optimality approach of fractional-order fourwing hyperchaotic system used in this paper requires the knowledge of control Lyapunov function and fractionalorder stability theory.

Inverse Optimal Control Approach.
Consider the following nonlinear system: where  ∈ R  denotes the state vector and  ∈ R  denotes the control vector, respectively.Moreover,  : R  → R  is vector function with (0) = 0 and  : R  → R × is matrix-valued function.
Definition 1 (see [37]).() is a smooth, positive definite, and radially unbounded function.Taking the time derivative of (), one obtains where For all  ̸ = 0, () is a control Lyapunov function (CLF) for system (1), if it satisfies Lemma 2 (see [38]).Suppose that the static state feedback control law where  : R  → R × is a positive definite matrix-valued function, stabilizes system (1) with respect to a positive definite radially unbounded Lyapunov function ().Then the control law is optimal with respect to the cost where 2.2.The Fractional-Order Stability Theory.In the theory of fractional calculus,     represents an arbitrary order differintegral operator.It is a notation for taking both fractional integrals and derivatives in one single expression and can be defined as where  ∈ R is the order of the operation  and  are the bounds of the operation.There are some different definitions for fractional derivatives [39].The most frequently used definitions are Riemann-Liouville definition and Caputo definition.
Riemann-Liouville definition is given as where  − 1 <  < ,  is an integer number, and Γ(⋅) is the Gamma function.If all the initial conditions are zero, the Laplace transform of Riemann-Liouville functional derivative is given as Another alternative definition of Riemann-Liouville definition of fractional-order derivative was reported by Caputo as follows: Since the Caputo definition is more convenient for initial conditions problems, in this paper, the operator   denotes the -order Caputo fractional derivative.
Then some stability theorems for fractional-order systems are introduced.

Adaptive Inverse Optimal
Control of a Novel Fractional-Order Four-Wing Hyperchaotic System with Uncertain Parameter
Figure 3 is the bifurcation diagram of  3 versus .The image shows complex and rich dynamical behaviors.
In the process of investigating this hyperchaotic system, we found this odd: different initial values would generate different area of attractor.With the aforementioned set of parameters and the initial values (0.1, 0.4, 0.9, 1), (0.1, 0.4, −0.9, 1), one can get the strange attractor as shown in Figure 4, where the upper one is indicated by a solid line and the lower one is indicated by a dotted line.
To explain this weird phenomenon, detailed mathematical deduction and simulations will be given.
To produce a real four-wing attractor, we introduce a simple linear state feedback  4 to the third equation of system (15).Then one gets the following system: With the aforementioned set of parameters and  = 1, the real four-wing attractor is shown in Figure 6. Figure 7: Fractional-order four-wing attractor of system (29) for  = 0.9.

Parameter b perturbation
The fractional-order four-

Adaptive Inverse Optimal Control of the Fractional-Order Four-Wing Hyperchaotic System with Uncertain Parameter.
The fractional-order hyperchaotic system with four-wing attractor can be constructed as the following form: The predictor-corrector method is used to implement the fractional-order four-wing hyperchaotic system with the order  = 0.9.The fractional-order four-wing attractor of system (29) with the aforementioned set of parameters is shown in Figure 7.
Then the inverse optimal control methodology is developed to achieve the global asymptotic stabilization of the fractional-order four-wing hyperchaotic system with uncertain parameter.The closed-loop system with a controller  is described by With the aforementioned set of parameters, () is an time-varying uncertain parameter described by () =  + Δ(),  is positive definite, and Δ() is a bounded function which satisfied |Δ()| ≤ .Theorem 6.The fractional-order hyperchaotic system with four-wing attractor can achieve global asymptotical stability by the following linear state feedback control law: where  1 is the estimate value of the unknown parameter  and  > 0. The parameter estimation update law ḃ 1 is Proof.According to Theorem 5, one considers the integerorder dynamical system as follows: Construct a Lyapunov function for system (33).Consider where  > 0,  > 0, ℎ > 0,  > 0, and b =  −  1 .Obviously, () is positive definite.The derivative of () along the time is Substituting ( 32) and ( 33) into (35) yields Equation ( 36) can be rewritten as According to Definition 1, one can obtain It is easy to verify that    < 0 for    = 0. Then () is indeed a control Lyapunov function (CLF).
Define a state feedback controller according to the results of Lemma 2: where  is a positive constant and () −1 is a positive definite function of .Consider   Substituting (39) into (37) yields According to the definition of Δ(), if  satisfies  ≥ |Δ()| = , it can be proved that V() is negative definite for all  ̸ = 0 easily.That means the close-loop system can achieve global asymptotical stabilization with the controller.

Circuit Experimental Researches
In this part, an electronic oscillation circuit is designed to stabilize the equilibrium of the fractional-order four-wing hyperchaotic system based on inverse optimal control law.As shown in Figure 11, the electronic oscillation circuit has two parts: one is the fractional-order four-wing hyperchaotic circuit and the other is an adaptive inverse optimal controller.The circuit consists of resistors, capacitors, operational amplifiers (AD741), and multipliers (AD633).The circuit parameters are as follows:  To implement the electronic circuit of the fractionalorder system, the frequency domain approximation method is adopted.In [42], an effective algorithm is developed to approximate fractional-order transfer functions.From Table 1 in [14], one can obtain an approximation of 1/ 0.9 with an error of about 2 dB as follows: The fractional calculus unit is shown in Figure 12, which is the chain fractance.
One can get the transfer function between A and B. Consider  By circuit simulation, the real four-wing attractor without the controller is shown in Figure 13.
When  = 200 s, the controller starts to work.As shown in Figure 14, state variables are closer to zero.In other words, the fractional-order four-wing hyperchaotic system is stabilized to its unstable equilibrium point.
Then an electronic oscillation circuit is constructed to implement the fractional-order four-wing attractor and the adaptive inverse optimal controller.
As shown in Figure 15, because the output of the circuit is limited precisely for using low-cost components, the experimental phase portraits of the fractional-order fourwing attractor are approximately in agreement with circuitry simulations.
The waveforms of state variables in time domain are shown in Figure 16.When the controller is added to the circuit, the waveforms are close to zero.It verifies the satisfactory performance of the proposed control law and proves the robustness of the system to some extent.

Conclusions
In this paper, combining the adaptive inverse optimal control with the stability theory of fractional-order system, a linear state feedback controller is designed to make the fractionalorder four-wing hyperchaotic system with uncertain parameter stable in the unstable equilibrium point.Through circuit simulations and circuit implementations, the method turned out to be workable.It is remarkably simple as comparing with other fractional chaos control methods which may have more than one nonlinear state feedback controller.This research has a certain theoretical and practical significance for the application of the adaptive inverse optimal control in nonlinear circuits, security communication, and many other engineering applications.

Figure 9 :
Figure 9: Time waveform of state variables of the controlled fractional-order system with the controller started at  = 200 s.

Figure 10 : 10 MathematicalFour-wing ex 4 Figure 11 :
Figure 10: The time evolutions of each state variable starting at time  = 200 s.

Figure 15 :
Figure 15: Phase portrait of the real four-wing attractor by circuitry implementation.

Figure 16 :
Figure 16: Time waveform of state variables of the controlled fractional-order system by circuitry implementation.
(14)ystem(13)is asymptotically stable, in the range of state variable  (except the origin), all the real part of matrix ()'s eigenvalues   are not more than zero.Theorem 5.If system (13) is asymptotically stable, fractionalorder system(14)is also asymptotically stable.Proof.
are negative roots.For equilibrium points  2 and  3 , eigenvalues of matrix  are  ,  2 , and  3 are all unstable.