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Stabilization of Quasi Integrable Hamiltonian SystemsWith Fractional Derivative Damping by Using

Fractional Optimal Control

Fang Hu and Wei Qiu Zhu

Abstract—An innovative procedure for designing fractional optimalcontrol to asymptotically stabilize, with probability one, quasi integrableHamiltonian systems with fractional derivative dampings is proposed. Itis proved that the systems indeed can be stabilized by using the proposedprocedure. An example is given to illustrate the procedure and its effec-tiveness.

Index Terms—Dynamical programming, fractional control, fractionalderivative damping, largest Lyapunov exponent, nonlinear stochasticsystem, stochastic averaging, stochastic stabilization.

I. INTRODUCTION

The dynamics and control of fractional order systems have beenintensively studied in the last decade since various kinds of phys-ical and engineering systems, such as those physical systems withfrequency-dependent damping exhibiting non-local and history-de-pendent properties, are found to be best described by fractionaldifferential equations. Bagley and Torvik [1], [2] have shown thatusing fractional derivative models to describe the viscoelastic dampingof structures have several attractive features. Techniques based onfractional derivative modeling for damping behavior of materials andsystems have also been used by Koeller [3], Makris and Constantinou[4], Shen and Soong [5]. Overview of researches in this area is givenby Rossikhin and Shitikova [6]. Specifically, nonlinear Hamiltonianequations with fractional damping and Duffing system with fractionaldamping have been studied in [7], [8].For the fractional order systems, it has been shown that the perfor-

mance of fractional order control may be better than that of the integerorder control [9]. The stabilization of fractional order systems is oneof the most significant control problems. The problem of pseudo-statefeedback stabilization of commensurate fractional systems wasaddressed by Farges, Moze and Sabatier [10]. Stabilizing a givenfractional-order time delay system by using fractional-order PID con-trollers was proposed by Hamamci [11]. The necessary and sufficientconditions for the robust stability and stabilization of fractional-orderinterval systems was given in [12]. The robust stabilization problemof fractional order systems with interval uncertainties was solvedby Delshad, Asheghan and Beheshti [13] via fractional controllers.However, all the studies in the field of stabilization of fractional orderdynamical systems are limited to deterministic systems. So far, tothe authors’ knowledge, no work on the feedback stabilization offractional order stochastic dynamical system is available.

Manuscript received April 25, 2011; accepted April 16, 2013. Date of publi-cation April 18, 2013; date of current version October 21, 2013. This work wassupported by the National Natural Science Foundation of China under a Keygrant 10932009 and grants 11072212 and 11272279. Recommended by Asso-ciate Editor P. Pepe.F. Hu is with the Department of Mechanics, Zhejiang University, Hangzhou

310027, China (e-mail: sdhzhf9360@yahoo.com.cn).W. Q. Zhu is with the Department of Mechanics, State Key Laboratory of

Fluid Power Transmission and Control, Zhejiang University, Hangzhou 310027,China (e-mail: wqzhu@yahoo.com).Color versions of one or more of the figures in this technical note are available

online at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TAC.2013.2258787

0018-9286 © 2013 IEEE

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 11, NOVEMBER 2013 2969

In the present technical note, a procedure is proposed to asymptoti-cally stabilize, with probability one, multi-degree-of-freedom (MDOF)quasi integrable Hamiltonian system with fractional derivative damp-ings via fractional control. The equations of such a system are firstreduced to partially averaged stochastic differential equations inSection II. Then, the main results of the present technical note, i.e., theformulation and solution of stochastic stabilization are stated in Sec-tion III. In Section IV, an example is worked out in detail to illustratethe application and effectiveness of the proposed control design proce-dure.

II. PARTIAL AVERAGING OF CONTROLLED SYSTEM

Consider an DOF controlled system with fractional derivativedampings under parametric excitations of Gaussian white noises. Theequations of the system are of the form

(1)

where and are generalized displacements and momenta, re-spectively; is a suitably small positive parameter;is twice differentiable Hamiltonian; are differen-tiable functions representing the coefficients of fractional dampings;

are fractional derivative dampings in the sense of Rie-mann-Liouville definition [8]; denote feedbackcontrol forces which are unknown and to be determined by usingthe proposed method; are twice differ-entiable functions representing amplitudes of stochastic excitations;

, and are bounded; are Gaussianwhite noises in the sense of Stratonovich with correlation functions

. System (1) may be unstablewith probability one due to pure parametric excitation of Gaussianwhite noises. The objective of the present study is to stabilize withprobability one the system using fractional optimal feedback control.The Hamiltonian system associated with system (1) is

(2)

Note that lower letters such as denote deterministic quantitieswhile capital letters such as denote random processes. System(2) is assumed to be integrable, i.e., there exist independent integralsof the motion which are in involution. In this case,system (1) is called weakly controlled quasi integrable Hamiltoniansystem. For simplicity, further assume that the Hamiltonian is sep-arable and of the form

(3)

where is the total energy of the Hamiltonian system; ,and represent the total energy, kinetic energy and poten-

tial energy of the th oscillator, respectively; denotes linear ornonlinear restoring force of the th oscillator.Then (2) can be rewritten into the following form:

(4)

Under certain conditions [14], (4) has the following family of periodicsolutons surrounding the origin of the phase plane :

(5)

where

(6)

and is defined by

(7)

and are the so-called generalized harmonic func-tions. and are the amplitude and instantaneous frequencyof system (4), respectively. is phase angle consisting of time-varyingpart and constant part . Expand into the followingFourier series:

(8)

Integrating (8) with respect to from 0 to leads to the followingapproximate average frequency

(9)

of the th oscillators. When is linear, is constant and equal to. For nonlinear , is a periodically time-varying function ofwhile is a simple function of . In the following,

(10)

will be used to replace in (5) in the averaging.When is small, the solution to system (1) is periodically random

processes

(11)

in which are of the form of (6) with replaced byrespectively, are related to in a similar way as (7).

Treating (11) as a generalized van der Pol transformation fromto yields the following stochastic differential equations forand :

(12)

where , ,, and

(13)

2970 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 11, NOVEMBER 2013

It is seen from (10) and (12) that the time rates of and are of theorder of while the time rates of are finite. Thus, and areslowly varying processes while are rapidly varying process.According to the stochastic averaging principle [15]–[18], in non-

resonant case, i.e., , are integers, converges in prob-ability to an -dimensional diffusion Markov process as in atime interval , where . This time intervalcan be extended to semi-infinite, i.e., , since Gaussian whitenoises are mixing [19]. This limiting diffusion process is governed bythe following averaged equations:

(14)

where are standard Wiener processes and

(15)

in which denotes the averaging with respect to .Since and vary slowly with time, the following approx-

imate relation can be obtained by using (10)

(16)

By using the approximate relation in (16) and the following asymp-totic integrals

(17)

in (15) can be obtained as follows:

(18)

Substituting (18) into (15) and completing the averaging with respectto yield the explicit expressions for and . In the fol-lowing, system (14) will be used to replace system (1) for designingthe feedback control to stabilize the system.

III. MAIN RESULT

A. Fractional Ergodic Control With Undetermined Cost Function

Here, the stochastic stabilization problem is formulated as a frac-tional ergodic control with undetermined cost function, since both sta-bilization and ergodic control are conducted in a semi-infinite time-in-

terval. For ergodic control of system (14), the partially averaged per-formance index is of the form

(19)

where is convex function of , is arbitrarypositive real number, and is a positive-definite diag-onal matrix with positive elements ,

. Applyingthe stochastic dynamical programming principle [20] to the controlproblem described by (14) and (19) yields the following dynamicalprogramming equation:

(20)

where is value function and is a constant representingoptimal average cost.Minimizing the left-hand side of (20) with respect to yields op-

timal control law

(21)

It is seen from (21) that are fractional functions of generalized mo-menta and amplitudes . To distinguish this control law from ourprevious control law [21], i.e., , where are non-fractionalfunctions of and , here it is called fractional control. Note thatthe meaning of fractional control here is something different from thatin PID control [9], [11].Substituting (21) into (20) and completing the averaging of the terms

involving lead to the following final dynamical programming equa-tion:

(22)

For ergodic control problem, , , and are given and the optimalcontrol law is obtained by solving (22) and then substituting the resul-tant into (21). For stabilization problem, , , and areunknown and will be determined later by the requirement of stabiliza-tion.

B. The Largest Lyapunov Exponent and Stochastic Stabilization

Substituting the optimal control forces in (21) into partially av-eraged (14) to replace and averaging the termlead to the following completely averaged equations:

(23)

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Since the stochastic excitations in (1) are pure parametric, isthe trivial solution of the controlled system (23) and the coefficients of(23) satisfy the following conditions

(24)

Assumption 1: Assume that the drift and diffusion coefficients of(23) satisfy the following conditions:

(25)

(26)

where is an arbitrary vector and is a scalar, means the normof a vector. Eq. (25) implies that the drift and diffusion coefficients arehomogeneous in of degree one and (26) implies that the diffusionmatrix of (23) is nonsingular. If (25) is not satisfied, then (23) may belinearized at , and so (25) will be satisfied.The expression for the largest Lyapunov exponent of averaged

system (23) can be obtained following a procedure similar to that in[22]. Introduce the following new variables:

(27)

The equations for and are obtained from (23) or its linearizedequation by using differential rule as follows:

(28)

(29)

where and

(30)

Note that , so, only equations for in (29) areindependent. Let and be replaced by

.Integrating (28) from 0 to and dividing by , one obtains

(31)

As , the first term and last term on the right-hand side of (31)vanish, since is bounded while grows as .Define the Lyapunov exponent of (23) or its linearized equation as theasymptotic rate of the exponential growth of , i.e.,

(32)

Assumption 2: -dimensional vector diffusion process isergodic over the interval , . Under this

assumption, the time averaging in (32) can be replaced by assemblyaveraging, i.e.,

(33)

where is the stationary probability density of obtained fromsolving the reduced Fokker-Planck-Kolmogorov (FPK) equation asso-ciated with (29). Based on the multiplicative ergodic theorem dueto Oseledec [23], in (33) approaches to the largest Lyapunov expo-nent .According to Khasminskii’s theorem [24], the necessary and suffi-

cient condition for the asymptotic stability with probability one of thetrivial solution of a stochastic system is that the largest Lyapunov ex-ponent of its linearized system is negative. Therefore, we arrive at thefollowing theorem:Theorem 1: Under assumptions 1 and 2, the necessary and sufficient

condition for the asymptotic stabilization with probability one of thetrivial solution of averaged controlled system (23) is .The averaging principle [15]–[17] states that the difference between

the solution of original system and that of averaged system is of theorder of . Thus, we have the following corollary:Corollary 1: The approximate necessary and sufficient condition for

the asymptotic stabilization with probability one of the trivial solutionof controlled quasi Hamiltonian system (1) is .The procedure to stabilize the quasi integrable Hamiltonian systems

with fractional derivative damping is thus:1) derive the partially averaged stochastic differential equations(14) for given controlled system;

2) obtain the fractional optimal controller, (21);3) derive the completely averaged equations (23);4) derive the largest Lyapunov exponent (33) of the com-pletely averaged equations;

5) choose proper , , and to make negative and min-imized.

IV. EXAMPLE

Consider the feedback stabilization of 2 Duffing oscillators withfractional derivative dampings under parametric excitations ofGaussian white noises. The equations of motion of the system are ofthe form

(34)

where represent the natural frequencies of the degenerated linearsystem; are intensities of nonlinearity; are small damping co-efficients; are constant coefficients of stochastic excitations;are independent Gaussian white noises in the senses of Stratonovichwith intensities . , , and are of the order of , thevalue of which is taken as 0.1 in this example.The Hamiltonian is , with

, ( , 2). System (34) can be rewritten in the form of (1)by letting . Following the pro-cedures of stochastic averaging given in Section II yields the followingpartially linearized averaged equations for

(35)

2972 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 11, NOVEMBER 2013

Fig. 1. Largest Lyapunov exponents and of uncontrolled and op-timally controlled system (34) versus excitation intensity fordifferent . , , , ,

, , , ,.

where

(36)

The dynamical programming equation is of the form of (22) withand defined by (36). The optimal control forces are of the form of(21) with , i.e.,

(37)Thus, as ,

(38)

and is the solution of (22). It is seen that in order to satisfy(22), and as should be of the following asymptoticform:

(39)

where , and are constants. Substituting (36)–(39) into (22)yields the following equations:

(40)

Fig. 2. Largest Lyapunov exponents and of uncontrolled and op-timally controlled system (34) versus fractional derivative orderfor different . , , . Theother parameter are the same as Fig. 1.

and can be obtained by solving (40) numerically for given ,and . The optimal control are then obtained from (37) as follows:

(41)

Then, the fully averaged drift coefficients for can be obtained asfollows:

(42)

Note that the drift coefficients and diffusion coefficientssatisfy the conditions in (24), (25) and (26). By introducing

the transformations (27), the following equations for and areobtained:

(43)

(44)

where

(45)

The approximate largest Lyapunov exponent of controlled system (34)is then

(46)

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 11, NOVEMBER 2013 2973

Fig. 3. Samples of the displacements of system (34): (a) uncontrolled; (b) con-trolled, and (c) sample of control force. , ,, . The other parameter are the same as Fig. 2.

Fig. 4. Largest Lyapunov exponents of optimally controlled system (34)versus for different . , .The other parameter are the same as Fig. 2.

where is obtained from solving the reduced FPK equation asso-ciated with (44). The largest Lyapunov exponent of uncon-trolled system (34) can be obtained using (46) by letting .Numerical results have been obtained for system (34) and shown in

Figs. 1–4. Figs. 1 and 2 show and versus the excitationintensity and the order of fractional derivative

, respectively, for different values of . It is seen that ispositive while is negative. Thus, the system is indeed stabilizedby the feedback control. Samples of the displacements of uncontrolledand controlled systems as well as the optimal control force are shownin Fig. 3, from which the effectiveness of the proposed control strategycan be visualized intuitively. It is seen from Fig. 3(c) that the controlforce is indeed of the order of .The absolute value of the negative largest Lyapunov expo-

nent can be taken as a measure of stability margin. Fig. 1 shows thatwill increases as increases. increases quickly as

increases when is small, while it is almost a constant for differentvalues when gets larger. Thus, is robust to the change of

excitation intensity when is large (e.g., larger than 2.0).It can be seen from Fig. 2 that the stability of uncontrolled system

will increase as the order of fractional damping increases. Fig. 2 alsoshows that for small (about less than1.25), willfirst increase andthen decrease as increases,while for large , willmonotonouslydecrease as increases. Thus, if other parameters are kept unchanged,the systemcouldbemore stableby taking larger valueof .

as functions of for different values of areshown in Fig. 4. It is seen that the feedback stabilization is more ef-fective by taking smaller . It can also be seen that the changing rateof as increasing will decrease as increases. It can be con-cluded from Fig. 4 that the system will be more robust to the changesof when taking a large value of .In summary, system (34) indeed can be stabilized by using the pro-

posed control design procedure, and it will be more stable by takinglarger and (or) smaller . For larger , the stability of the controlledsystem is more robust to the changes of excitation intensity, as well asparameter in the feedback controller.

V. CONCLUSION

In the present technical note, an innovative procedure for designingfractional optimal controller to asymptotically stabilize, with prob-

ability one, quasi integrable Hamiltonian systems with fractionalderivative dampings has been proposed. The procedure consists ofderiving the partially averaged equations, obtaining the fractionaloptimal controller by establishing and solving the dynamical program-ming equation, and determining the stability by evaluating the largestLyapunov exponent. The results of example have further showed thatthe controlled system can be stabilized by using fractional optimalcontrol through proper choice of cost function.

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