synergetic optimal controllers

8/13/2019 Synergetic Optimal Controllers

1/15

Nonlinear Synergetic Optimal Controllers

Nusawardhana, S. H.ak, and W. A. Crossley

Purdue University, West Lafayette, Indiana 47907

DOI:10.2514/1.27829

Optimality properties of synergetic controllers are analyzed using the EulerLagrange conditions and the

HamiltonJacobiBellman equation. First, a synergetic control strategy is compared with the variable structure

sliding mode control. The connections of synergetic control design methodology and the methods of variable

structure sliding mode control are established. In fact, the methods of sliding surface design for the sliding mode

control are essential for designing invariant manifolds in the synergetic control approach. It is shown that the

synergetic control strategy can be derived using tools from the calculus of variations. The synergetic control laws

have a simplestructure because they arederived from theassociatedrst-order differential equation. It is alsoshown

that the synergetic controller for a certain class of linear quadratic optimal control problems has the same structure

as the one generated using the linear quadratic regulator approach by solving the associated Riccati equation. The

synergetic optimal control and sliding mode control methodologies are applied to the nonlinear control of the wing-

rock suppression problem. Twodifferent wing-rock dynamic modelsare used to test the design of thesynergetic and

sliding mode controllers. The performance of the closed-loop systems driven by these controllers is analyzed and

compared.

Nomenclature

A = state matrix inRnn

Ax = state-dependent vector function of dimensionnAr = state matrix of reduced-order design modelB = input matrix inRnm

Bx = state-dependent input matrix function of sizen m

Br = input matrix of reduced-order design modelIm = identity matrix inR

mm

J = numerical value of the cost functional in optimalcontrol problems

K = gain of discontinuous control in sliding modecontroller

L = cost functional integrandM = manifoldfx: x 0gof dimensionn mN = weight matrix for cross term between state and

control variables in LQR integrandO = zero matrixP, ~P = matrices, solutions to Riccati equation, or

algebraic HJB equationQ = weight matrix for state variables in LQR

integrandR = weight matrix for control variables in LQR

integrandS = surface/hyperplane matrixSr = surface/hyperplane matrix in reduced-order

control design

s = Laplace variable, or differentiation operatorsx = state-dependent surface functionT = positive design parameter

T T> >0 = symmetric positive denite constant matrix,design parameter matrix

u = input vector of dimensionmud = discontinuous control componentus = continuous control componentueq = equivalent controlV = value function in optimal control problemWg = matrix orthogonal toB, generalized inverse to

W, so thatWWg Inmx = state variable vector of dimensionnxr = state variable vector of reduced-order dynamic

modeli = ith parameter of wing-rock dynamic model

= parameter related to actuator input in wing-rockdynamic model

a = actuator state variable in wing-rock dynamicmodel

= adjustable parameter used in the design ofS,x = aggregated variable ofx dening the manifold

M

= roll angle_ = roll rate, = variables appearing in algebraic matrix Riccati

equation

I. Introduction

T HE objective of the optimal control problem (see, for example,[1], Chap. 5) is to nd an admissible control strategy thatensures the completion of control objective andleads to optimizationof a given performance index. In most cases, solving the nonlineardynamical systems optimal control problem is not straightforward.Based upon sufcient optimality conditions, solving the associatednonlinear partial differential equation, known as the HamiltonJacobiBellman (HJB) equation [26], provides the solution to suchproblems. A closed-form solution is often difcult to obtainespecially for nonlinear dynamical systems. However for a class ofnonlinear dynamical systems with a special cost functional, solving arst-order differential equation produces a closed-form analyticalsolution to a class of optimal control problems. This rst-orderdifferential equation plays an essential role in the synergetic controlmethodology.

Kolesnikovand coworkers [7] developedthe methodof synergeticcontrol, and they, and others, later applied synergetic control to anumber of industrial processes, including problems in power

Received 15 September 2006; revision received 4 December 2006;accepted for publication 14 January 2007. Copyright 2007 by A.Nusawardhana, S. Zak,and W. Crossley.Publishedby the American Instituteof Aeronautics and Astronautics, Inc., with permission. Copies of this papermaybe made forpersonal or internal use, on condition that the copier pay the$10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 RosewoodDrive, Danvers, MA 01923; include the code 0731-5090/07 $10.00 incorrespondence with the CCC.

Graduate Student, School of Aeronautics and Astronautics, 315 NorthGrant Street. Student Member AIAA.

Professor, School of Electrical and Computer Engineering, ElectricalEngineering Building, 465 Northwestern Avenue.Associate Professor, School of Aeronautics and Astronautics, 315 North

Grant Street. Associate Fellow AIAA.

JOURNAL OFGUIDANCE, CONTROL, AND DYNAMICSVol. 30, No. 4, JulyAugust 2007

1134
http://dx.doi.org/10.2514/1.27829http://dx.doi.org/10.2514/1.27829


2/15

generation [812]. The synergetic control approach was also appliedto adaptive control of nonlinear systems using neural networks [13].The resulting synergetic control structure is attractive, because it isderived from arst-order differential equation related to optimalitycriteria. The authors of the above publications, however, do notelaborate on the derivation of synergetic control. Some state thatsynergetic control can be derived from functional optimization, butthey do not state whether necessary or sufciency conditions lead tothe control structure.

The

rst objective of this paper is to present our derivation of thesynergetic controller and to establish the connections with thevariable structure sliding mode control approach. We show that thesynergetic control for linear systems is a linear approximation of thenonlinear variable structure sliding mode controller. The linearity ofthe synergetic controller for linear systems is used to prove thestability of the closed-loop system driven by the synergeticcontroller. Thesecond objective is to show that synergetic control notonly satises the necessary condition for optimality but also satisesthe sufcient condition for optimality. The third objective is to showthat, for linear time-invariant (LTI) systems and for a special form ofthe performance index, the synergetic control approach yields thesame controller structure as the linear quadratic regulator approach.This is an especially attractive feature of synergetic controllers for aclass of LTI optimal control problems, because the synergetic

controllers result from solving a rst-order differential equationwhile the linear quadratic regulator (LQR) design relies on solvingthe quadratic Riccati equation. The fourth objective of the paper is topresent a synergetic control design algorithm. The nal objective isto apply the proposed method to the regulation of wing-rockproblems and to compare the results using the proposed method withthose using the variable structure control. Some results in this paperwere reported by us in [14].

II. Controlling Nonlinear Systems

Consider the following model of a nonlinear dynamical system:

_x Ax Bxu (1)

wherex2 Rn

and u2 Rm

. Thus, Ax 2 Rn

and Bx 2 Rnm

. Wediscuss now a general method for stabilization and control of suchsystems. We refer to this approach as the stabilization and control viasystem restriction and manifold invariance. It consists of two steps:

1) Construction of a manifold

M fx: sx 0; sx 2 Rmg

so that

det

@

@xBx

0

andthe system restricted toM behaves in a prespecied manner.Forexample, the trajectories of the system restricted to M asymptoti-

cally converge to a predetermined pointx

2 M. In the above, weused the following convention:

@

@x

@

@xsx

>

2 Rmn

2) Construction of a controller that steers the system trajectories tothe manifold M and then forces the trajectories to stay on themanifold thereafter.

A simple condition guaranteeing that the system trajectories aredirected toward the manifold M is

d

dtkk2 2> _< 0 for 0 (2)

The variable structure sliding mode control design is an example ofthis two-step approach to stabilizing nonlinear systems. We brieyelaborateon this approach. We observe that a trajectory is conned tothe manifold M if and only if x 0 and _x 0. Solving

_x 0foru yields a controller that forces a trajectory to lie on alevel surfaceofx. This controller is referredto, in the literature, asthe equivalent controller and is denotedueq. To proceed, let

@

@x

@

@xsx

>

sxx 2 Rmn

Then, we have

_x sx _x sxxAx sxxBxueq 0

Hence,

u eq sxxBx1sxxAx (3)

Thus, the system dynamics restricted to the manifold M fx: sx 0g are described by the equations

0 (4)

_x Ax Bxueq In BxsxxBx1sxxAx

(5)

Combining Eqs. (4) an d (5) gives a reduced-order, n m-

dimensional dynamical system. This order reduction can beeffectively exploited in the construction of the manifoldfx: 0gresulting in a prespecied behavior of the system restricted to themanifold. Once the manifold is constructed, the next step is thecontroller design. Among many possible controller architectures thatsatisfy condition (2), the following variable structure sliding modecontroller was analyzed in depth in [15],

u ueq ud (6)

where

u d sxxBx1 x

k x k (7)

is the discontinuous component ofu. The controller gain > 0is adesign parameter.

It is easy to show that the control law given by Eq. ( 6) satisescondition (2). Indeed,

d

dtkk2 2> _ 2>sxx _x 2

>sxxAx sxxBxu

2>sxxAx sxxBx

sxxBx

1sxxAx

sxxBx1

kk

2

kk2

kk 2kk < 0

for 0

Suppose now that we wish to construct a controller that would

force the trajectories to approach the manifold M f 0gexponentially as described by

T _ 0 (8)

where T T> > 0is a symmetric positive denite matrix, a designparameter matrix. It follows from Eq. (8) that

_ T1 (9)

Note that if we construct a controller that satises Eq. (8), then usingEq. (9), we obtain

d

dtkk2 2> _ 2>T_< 0 for 0

Thus, the design condition(2) is satised for such a control law.We now derive a control that satises the condition (8). Because sx, _ sxx _x, hence Eq. (8) can be represented as

NUSAWARDHANA,AK, AND CROSSLEY 1135


3/15

T _ Tsxx _x sx TsxxAx Bxu sx

TsxxBxu TsxxAx sx 0

The resulting control law obtained from the last equality takes theform,

u TsxxBx1TsxxAx sx

sxxBx1sxxAx sxxBx

1T1sx

ueq us (10)

where

u s sxxBx1T1sx (11)

In Eq. (11), usdenotes a smooth component of the resulting control

law. The control law given by Eq. (10) is referred to as the nonlinearsynergetic controller. This controller forces the system trajectories toapproach the manifold asymptotically. When the trajectory starts onthe manifold, the synergetic controller will maintain it therethereafter. Theadvantage of thesynergetic controller over thecontrollaw given by Eq. (6) is that the synergetic controller is smooth whilethe sliding mode controller is discontinuous. However, the controllaw(6) forces the system trajectory onto the manifold in anite time,and it maintains the trajectory on the manifold thereafter.

When we compare ud, appearing in Eq. (7), and usfrom Eq. (11),we can conclude thatusis a linear approximation to ud. We considera linear case when both udand usarescalar.In this case, sxxBx isa scalar. Plots in Fig. 1 indicate that as Tdecreases, the gain 1=Tincreases and usclosely approximates udin its discontinuous region.

We also show in Fig. 1 a plot of a smooth continuous tangential-sigmoid function approximating ud. Using the tangential-sigmoidfunction to approximate ud, or any other boundary-layer typeapproximation [16], yields a nonlinear controller, even though theapproximation results in a smooth control law. The use of the linearapproximation of ud, on the other hand, allows us to use linearanalysis techniques of closed-loop system stability properties.

In thefollowing section, we consider a linear synergeticcontroller.For the linear case, we present a general algorithm for an invariantmanifold construction and analyze the closed-loop system driven bythe linear synergetic controller.

III. Linear Synergetic Control

We consider a linear time-invariant dynamical system model ofthe form,

_x Ax Bu (12)

wherex xt 2 Rn, u ut 2 Rm, and a linear manifold has theform

x Sx 0

where S2 Rmn. Our goal is to construct a control strategy,u ux, such that for any initial state x0, the closed-loop systemtrajectory approaches the manifold x 0 according to the policyT_ 0. The relation T_ 0 is central to the synergeticcontrol approach. The rate with which the solutions of the abovesystem of therst-order linearordinary differential equation decay tozero can be controlled by appropriately selecting the matrix T. Toproceed, note that

_

@

@xx

>

_x S _x

Substituting the above into Eq. (8) yields

T _ TS _x Sx TSAx Bu Sx 0

We assume thatdetSB 0. Hence,

u TSB1TSA Sx (13)

The advantage of the above control strategy is that it is a linearcontroller that drives the system trajectory toward the manifoldfSx 0g. Once on the manifold, or within a small neighborhoodabout the manifold, the system dynamics are of a reduced order. Thisorder reduction can be exploited by the designer to shape the systemdynamics when the system is moving along the manifold. A numberof methods from the variable structure sliding mode control can beemployed here to construct the matrix S of the linear invariantmanifold; see, for example, Chap. 4 of [17], Secs. 34 in [18], Part IIin [19], and pp. 333336 in [20].

IV. Stability Analysis of a Linear Closed-Loop SystemDriven by Synergetic Controller

The synergetic control strategy (13) forces the system trajectory toasymptotically approach the manifold f 0g. Thus, the closed-loop system will be asymptotically stable provided that the system(12) restricted to the manifold f 0g is asymptotically stable. Inour proof of the above result, weuse the following theorem from[20]on p. 333.

Theorem 1: Consider the system model, _x Ax Bu, whererankB m, the pair A;B is controllable, and a given set ofn mcomplex numbersf1; . . . ; nmgis such that any i whoseimaginary part is nonzeroappears in the above setin a conjugate pair.Then, there exist full-rank matrices W2 Rnnm and Wg 2Rnmn such that

WgW Inm; WgB O

and the eigenvalues ofWg

AWareeig WgAW f1; . . . ; nmg

Furthermore, there exists a matrix S2 Rmn such thatSB Imandthe system _x Ax Bu restricted to fSx 0g has its poles at1; . . . ; nm.

We can now state and prove the following theorem.Theorem 2: If the manifold fSx 0g is constructed so that the

system _x Ax Bu restricted to it is asymptotically stable anddetSB 0, then the closed-loop system

_x Ax Bu; u TSB1TSA Sx

is asymptotically stable. Proof: Suppose that for a given set of complex numbers

1; . . . ; nmwhose real parts are negative, we constructed matricesS2 Rmn, W2 Rnnm, and Wg 2 Rnmn that satisfy theconditions of Theorem 1. Note that we also have SW O. We use

1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 12

1

0

2

ud

andits

approxs.

Decreasing T

Decreasing T

sign()

tansig(T1)

T1

(0.5T)1

Fig. 1 The synergetic controller component us a s a linearapproximation to udfor the scalar case when sxxBx 1 and 1.

1136 NUSAWARDHANA,AK, AND CROSSLEY


4/15


5/15

Proposition 2: For a nonlinear dynamical system model,

_x Ax Bxu

sx

(25)

wheresxxBx is invertible, the control law

u TsxxBx1TsxxAx sx (26)

derived from the rst-order differential equation (24) is the

extremizing controller for the optimal control problem with theperformance index given by Eq. (23).

It follows from the above that the extremizing control law for alinear time-invariant dynamical systemmodeled by Eq. (12)withtheassociated performance index (23) is given by Eq. (13).

VI. Background Results From Optimal Control

The cost functional used to derive the synergetic optimal control isrelated to the formulation of a linear quadratic optimal controlproblem with cross-product terms in the quadratic performanceindex. We briey discuss this as an introduction for the followingsections derivation of synergetic control based upon the sufcientoptimality condition. Comprehensive discussion of optimal controlwith cross-product terms in the quadratic performance index can befound in [23] on pp. 5657.

Consider a linear time-invariant dynamical system modeled byEq. (12) and the associated performance index

J

Z 1

t0

x>Qx 2x>N>u u>Ru dt (27)

whereQ Q> 0,N 2 Rmn, andR R> > 0. The problem ofdetermining an input u ut, t t0, for the dynamicalsystem (12) that minimizes the performance index (27) is calledthe deterministic linear quadratic optimal regulator problem. For theLQR problem, the control law

u RB> ~P Nx (28)

where the matrix ~Pis the solution to

A > ~P ~PA Q B> ~P N>R1B>~P N O (29)

minimizes the performance index (27). The control law u given by

Eq. (28), together with the solution ~P to Eq. (29), solves theassociated HJB equation

minu

dV

dt x>Qx 2x>N>u u>Ru

0 (30)

whereV minuJx>~Px.

VII. Sufciency Condition for Optimality

of Synergetic Controllers

Consider now a continuous-time nonlinear dynamical systemmodeled by

_xt fxt;ut; t0 t 1; xt0 x0 (31)

We wish to nd a control strategy fut: t 2 t0; 1g that togetherwith its corresponding state trajectory fxut: t 2 t0; 1gminimizes the performance index,

J

Z 1

t0

Lxut;ut dt (32)

This is an innite-horizon optimal control problem. One way tosynthesize an optimal feedback control u is to employthe followingtheorem that gives us a sufciency condition for optimality of theresulting controller. This well-known theorem can be found, forexample, in Vol. I, on p. 93 of [24].

Theorem 4: SupposeVx is a solution to the HJB equation,

minu

dVx

dt Lx;u

0 (33)

Vx !0 as t! 1 (34)

Suppose thatu is a minimizer of Eq. (33). Letxt, t2 t0; 1 bethe state trajectory emanating from the given initial condition x0

when the control ut is applied, that is, xt satises _xt fxt;ut subject to the initial conditionxt0 x0. Then Visthe unique solution to the HJB equation, that is,

Vx minu

Jx Jx

Using the above condition, we now provide an optimal controlstructure for a class of nonlinear systems.

Proposition 3: Consider a nonlinear system model given byEq. (25), where sxxBx is invertible, and the associatedperformance index,

J Z 1

t0

Lt; _t dt (35)

where

L; _ _>T>T_ > (36)

The feedback control law,

u sxxBx>T>TsxxBx

1sxxBx>

T>TsxxAx Tsx

minimizes the performance index (35). The above control law canbefurther simplied and represented in the form,

u TsxxBx1TsxxAx sx (37)

The value of the performance index for this control problem is

Vt Jtjutut >tPt (38)

whereP T T>. Proof: Our goal is to nd a feedback controller that minimizes the

performance index (35). Let

Vt minut

Jt >tPt (39)

The HJB equation for the optimal control of dynamical system (25)with the associated performance index (35) is

minut

d

dtVt Lt; _t 0 (40)

To nd the minimizing u, we apply the rst-order necessarycondition for the unconstrained minimization to the expressionwithin the curly bracket. Note thatis not a function ofu. I t i s _thatdepends uponu. We have

@

@u

dVt

dt Lt; _t

@

@u

d>P

dt L; _

@

@uf2 _>P _>T>T_ >g

@

@uf2sxx _x

>P

sxx _x>T>Tsxx _x

>g @

@uf2sxxBxu

>P

2sxxBxu>T>TsxxAx

sxxBxu>T>TsxxBxug 0

Observing that the last term in the above equation is quadratic inu,



6/15

we obtain

sxxBx>P sxxBx

>T>TsxxAx

sxxBx>T>TsxxBxu 0 (41)

Solving the above foru gives

u sxxBx>T>TsxxBx

1sxxBx>Psx

T>Tsx

xAx

Because sx is constructed in such a way that sxBx isinvertible,the controlleru above canfurther be simplied.Note alsothatTis symmetric and invertible. Hence,

u TsxxBx1TTsxxBx

TsxxBx>Psx

T>TsxxAx TsxxBx1TTPsx

TsxxAx (42)

We substitute the control function u from Eq. (42) into the HJBequation (40) and solve the equation forP. In our manipulations weuse the expression, _ T1, to obtain

d

dt>

P L; _ _>

P >

P_ _>

T>

T_ >

T1>P >PT1 > > >TTP

>PT1 2> >TTP PT1 2Im 0

from which we obtain

P T T>; P2 Rmm (43)

When we substitutePfromEq.(43) backinto Eq. (42), we obtainthecontrol law,

u TsxxBx1sx TsxxAx

Note that if we premultiply Eq. (41) by fsxxBx>Pg1 and use1) P T T> and 2) _ sxxAx Bxu, we obtain thedifferential equation that plays a fundamental role in synergeticcontrol,

T _ 0

We use the above equation to construct the optimizing control laws(42) and ultimately Eq. (37).

The following theorem gives a sufcient condition for the LQRcontrol strategy to take the form of the synergetic control law.

Theorem 5: Consider the LTI dynamic system modeled by

_x Ax Bu

Sx (44)whereS is constructed such thatSB is invertible. Associated with theabove LTI system (46) is the cost function

J

Z 1

t0

_>T>T_ > dt

Z 1

t0

_x>S>T>TS _x x>S>Sx dt

(45)

The optimal controller has the form,

u B>S>T>TSB1B>S>PS B>S>T>TSAx (46)

whereP T T> 2 Rmm. The control law (46)hasthesameformas the control law derived using the LQR approach tothe system (44)together with the cost (45) reformulated as

J

Z 1

t0

x>Qx 2x>N>u u>Ru dt (47)

where

Q A>S>T>TSA S>S; N B>S>T>TSA

R B>S>T>TSB

Furthermore, since

~P S>PS

the control law (46) canbe reformulated to the form of the synergeticcontrol strategy,

u TSB1TSA Sx

Proof: We rst construct the optimal feedback controller forEq. (44) that minimizes Eq. (45). That is, we constructu so that

Vxt0 x>S>PSx min

uJxt0

The HJB equation for this LTI case has the form

minu

d

dtx>S>PSx _x>S>T>TS _x x>S>Sxg 0 (48)

from which we obtain the control law (46) using the same argumentsas in the nonlinear case. This controller has a similar structure as thecontrol law (42) in the nonlinear case analyzed above. Next, wesubstitute the control law (46) back into the HJB equation (48) toobtain

_x>S>PSx x>S>PS _x _x>S>T>TS _x x>S>Sx

Ax Bu>S>PSx x>S>PSAx Bu

Ax Bu>S>T>TSAx Bu x>S>Sx x>x 0

where

A

>

S>

PS S>

PSA A>

S>

T>

TSA S>

S TB>S>T>TSB1 (49)

B>S>PS B>S>T>TSA (50)

For the HJB equation to be satised, regardless of the value ofx, theexpression for must be zero, which yieldsP T T> 2 Rmm.To see this, substituteP T T> into the expression for givenby Eq. (50) to obtain

B>S>T>S B>S>T>TSA B>S>T>TSA S (51)

Substituting given by Eq. (51) into the last term of Eq. (49) yields

>B>S>T>TSB1 >TSB1B>S>T>1

S TSA>TSA S S>S A>S>T>S S>TSA

A>S>T>TSA

which cancels out the rst four terms in Eq. (49). These steps lead tothe conclusion thatP T.

Note that because SB is invertible by construction, and P TT> by Eq. (43), the control law (46) simplies to

u TSB1B>S>T>1B>S>T>S TSAx

TSB1TSA Sx

which has the same form as the synergetic control law (13).

The solution to the LQR problem of the system ( 44) with the cost(47) is characterized by the matrix ~Pwhich is the solution to theRiccati equation



7/15

A > ~P ~PA Q B>~P N>R1B> ~P N A> ~P ~PA

A>S>T>TSA S>S B> ~P

B>S>T>TSA>TSB1B>S>T>1B> ~P

B>S>T>TSA O

(52)

Using the matrix ~P, the optimal feedback control of the associated

LQR problem is

u R1B> ~P Nx B>S>T>TSB1B> ~P

B>S>T>TSAx (53)

Comparing Eqs. (49) and (52) together with Eq. (50), we see that~P S>PS. If we set ~P S>PS and take into account thatSB is

invertible, then the above controller takes the form of the synergeticcontrol law.

It follows from the above that the control law derived from therst-order differential equation T_ 0 is not only extremizing,that is, this control law can be obtained from the EulerLagrangeequation, but it is also optimizing in the sense that it can be obtainedfrom the HJB equation.

It is well known that one can obtain the solution to thedeterministic LQR problem from the HJB equation. Therefore, forthe linear plant model and Sx, we can obtain the optimizingcontrol law,

u TSB1TSA Sx

directly from the formula for the LQR control given by Eq. (28) inSec.VIafter making appropriate associations between the matricesof the performance indices,

J

Z 1

t0

x>Qx 2x>N>u u>Ru dt

and

J

Z 1

t0

_>T>T_ >dt

Z 1

t0

S _x>T>TS _x Sx>Sx dt

VIII. Synergetic Optimal Stabilizing Control

The problem of constructing an invariant manifold M fx: 0g such that the system conned to it is asymptotically stable hasbeen addressed in the area of sliding mode control. The manifoldconstruction in sliding mode control, for the LTI case, can be viewedas constructing the matrix S so that the system zeros (or zerodynamics fora generalclassof nonlinear systems) are asymptoticallystable; see, for example, Sec. 6.5 in [20]. Therefore, various methodsfor constructing stable manifolds in sliding mode control are directly

applicable for construction in the synergetic control; see, forexample, Chap. 4 in [17], Secs. 34 in [18], Part II in [19], andpp. 333336 in [20].

In the synergetic control approach, the controlleru is designed bysolving the rst-order differential equation T_ 0, whichinduces the attractivity and invariance of the manifold 0.

We summarize this section with the following two-step designprocedure:

1) Construct a stable hyperplane, or a manifold,M fx: 0g.2) Construct the optimizing control u.We illustrate the above considerations with a simple numerical

example.Example 1: Consider the double integrator state-space model,

_x 0 1

0 0 x 01 u (54)

The control design objective for the system (54) is to regulate the

state to the origin ofR2 with a performance index to be determined.When is notproperlyconstructed, theoverall closed-loop systemisunstable. The construction of, in this example, is concerned withselecting the row vectors. Consider two candidates:

s 1 1 and s 1 1 (55)

that yield sx and sx, respectively. Associated withthese matrices are the following performance indices:

J

Z 1

t0

_2

2

dt

Z 1

t0

x>Qx 2uNx Ru

2

dt

and

JZ 1

t0

_2

2

dt

Z 1t0

x>Qx 2uNx Ru2 dt

where

Q 1 1

1 2

; N 0 1 ; R 1

Q 1 1

1 2

; N 0 1 ; R 1

(56)

We selectedT 1. We used MATLAB, Version 6.5, to constructthe LQR controllers. The results of the optimal control design usingboth the synergetic control and the LQR [Eq. (28)] for both cases s

(denoted P) and s(denoted P) are summarized in Table1. Fromthis table, we can see that the LQR controllers for both cases yieldstabilizing solutions. All closed-loop eigenvalues for both cases arenegative. However, the synergetic controllers are criticallydependent upon the selection of the manifold sx. Whens s, the solution generated using the synergetic controller is thesame as that obtained using the LQR method and results in negativeclosed-loop eigenvalues. However, when s s, the closed-loopsystem driven by the synergetic controller has one positive closed-loop pole. The phase portraits of two closed-loop systems are shownin Fig. 2. Theleft plot corresponds to P, therightone corresponds toP. On both plots, solid lines indicate 0 and 0, whiledashed lines indicate vectors perpendicular to both 0 and 0. Dashed arrows indicate the direction of trajectoriesapproaching planes 0and 0, while solid arrows indicate

the direction of trajectories along manifolds 0and 0.In both cases, trajectories approach the respective manifolds, as

expected. However, in the P case, state trajectories along themanifold 0converge to the origin, while in thePcase, statetrajectories along the manifold 0diverge from the origin. Thisis because motion along the manifold 0is not stable, while themotion along the manifold 0is stable.

IX. Synergetic Control Designfor Wing-Rock Suppression

The wing-rock phenomenon manifests itself in the form of asustained limit-cycle oscillation that limits the potential maneuverperformance of an aircraft. There are diverse nonlinear modelsrepresenting the wing-rock phenomenon [2528]. For the purpose ofcontrol design, we employ two dynamical models: 1) second-ordermodel [29], and 2) third-order model [30,31]. The main distinctionbetween the two models lies in the modeling of the actuator

Table 1 Comparison of the designs using the synergetic andLQR approaches

Problem Controller, closed-loop poles Synergetic LQR

P u 1 2 x 1 2 x

c:l: 1, 1 1,1P u

1 0 x 1 2 xc:l: 1,1 1,1



8/15

dynamics. The second-order model uses a zero-order model for theactuator dynamics, while the third-order model incorporates arst-

order model of theactuator dynamics. We demonstrate thefeasibilityof our proposed synergetic control design based on either of thewing-rock models. Additionally, sliding mode controllers aredeveloped based on the second- and third-order dynamic models.Thesliding surfaceconstruction methodfor sliding mode control canalso be used to constructa stableinvariantmanifoldfor thesynergeticoptimal control. Thus, for the synergetic and the sliding modecontroller designs, we can use the same manifold for each controldesign method. We then compare the performance of eachcontrollerin the closed-loop conguration. Finally, we explore the problem ofcontrol design using a reduced-order model where we use the third-order model as a truth model, while a simplied second-ordermodel as a design model. The controllers designed using the designmodel are tested on the truth third-order model.

A. Control Design Using Second-Order Dynamic Model

The second-order dynamic model [29] of the wing rock usingzero-order dynamic actuator model has the form:

1 2_ 3jj _ 4j _j _ 53 u (57)

where the parameter values ofi are as follows: 1 36:1063,2 0:665537, 3 2:75214, 4 0:00954708, and5 41:6621. We represent the second-order nonlinear equa-tion (57) in the following state-space format:

_x1

_x2

" #

x2

1x1 5x31 2x2 3jx1jx2 4jx2jx2

" #

0

1

" #u

Ax Bu (58)

where x1 and x2 _ are the state variables. We use theprocedure of Slotine and Li [32] to construct the manifold, where

x1; x2

d

dt

x1 _x1 x1 x2 x1 1 |{z}

S

x

(59)

We select 6, which isthe samevalue asin Fernand and Downing[29].

1. Sliding Mode Controller Design

As in [15] on p. 228, we design a sliding mode controller thatconsists of two terms: 1) the equivalent control termueqand 2) thediscontinuous control term ud. The equivalent control term isdetermined by solving _x1; x2 0, from which we obtain

ueq 1x1 5x31 2x2 3jx1jx2 4jx2jx2 x2(60)

Note that if one applies the equivalent control alone, then thetrajectories of the system driven by this equivalent controller willmove parallel to the sliding surface x1; x2 0. This isbecause theequivalent control forces _x1; x2 0. The role of thediscontinuous control termud is to bend the trajectories towardthe sliding surface and to compensate for any matched uncertainties.We use the following form ofud:

ud Ksignx1; x2 (61)

Therefore, the two-term sliding mode control law, consisting of the

terms given by Eqs. (60) and (61) takes the form,

u ueq ud

1x1 5x31 2x2 3jx1jx2 4jx2jx2

x2

Ksignx1 x2 (62)

4 3 2 1 0 1 2 3 4

4

3

2

1

0

1

2

3

4

x1

x2

Closed-Loop Phase Plot with s+=[1 1]

4 3 2 1 0 1 2 3 4

4

3

2

1

0

1

2

3

4

x1

x2

Closed-Loop Phase Plot with s=[1 1]

Fig. 2 Phase portraits of the double integrator closed-loop system.

12 8 4 0 4 8 122

1

0

1

2

x1

x2

Fig. 3 Trajectories of the closed-loop second-order dynamic model ofthe wing rock. Solid line trajectories are produced by the synergetic

controller. Dotted line trajectories are produced by the sliding modecontroller.



9/15


10/15

_a a u (66)

We represent the above model in the following state-space format:

_x1

_x2

_x3

264

375

x2

1x1 2x2 3x32 4x

21x2 5x1x

22 x3

1x3

2664

3775

0

0

1

2664 3775u Ax Bu (67)

where x1 and x2 _. The state x3 a represents therst-order actuator. The values of the model parameters are asfollows: 1 0:0148927, 2 0:0415424, 3 0:01668756,4 0:06578382, 5 0:08578836, 1:5, and 1=15.These values correspond to an aircraftight at an angle of attack of21.5 deg and can be found in [25,28,30,31].

In our construction of the two-dimensional invariant manifold, weagain use the method of Slotine and Li [32] to obtain

x1

; x2

; x3

ddt 2

x1

_x2

2x2

2x1

1x1 2x2 3x32 4x

21x2 5x1x

22 x3

2x2 2x1 (68)

where after some numerical experiments we selected 10 toreduce excessive overshoot. We mention that becausex1; x2; x3

ddt

2x1, the closed-loop system restricted to themanifoldx1; x2; x3 0will have a double pole at 10.

In the following sections, we design sliding mode and synergeticcontrollers around the above invariant manifold.

1. Sliding Mode Controller Construction

Using the manifold dened by Eq. (68), we rst compute the

equivalent control ueq for the third-order model by solving theequation _x1; x2; x3 0, that is,

_x rxx> _x rxx

>Ax Bu 0

where rxx is the gradient of x, with respect tox x1 x2 x3

>. After some manipulations, we obtain

ueq

1x2 2 _x2 33x2 _x2 5x

32 2x1x2 _x2

x3 2 _x2

2x2

(69)

Note that theexpression for_x2isgiven bythe state equation (67).We

construct the same type of sliding mode controller as for the second-order wing-rock dynamical model,

u ueq ud

where

ud Ksignx1; x2; x3; K 10 (70)

Hence the sliding controller for the third-order wing-rock dynamicalmodel takes the form,

u ueq ud

1x2 2 _x2 33x2 _x2

5x32 2x1x2 _x2 x3 2 _x2 2x2

Ksign

_x2 2x2 2x1

(71)

2. Synergetic Controller Construction

The synergetic control law is obtained by solving the rst-orderdifferential equation,

T_x1; x2; x3 x1; x2; x3 0

Solving this differential equation foru, we obtain

u

1x2 2 _x2 33x2 _x2 5x32 2x1x2 _x2

x3 2 _x2

2x2

T1

_x2 2x2

2x1

(72)

where T 0:01. Note again that the expression for_x2 is provided bythe state model (67). We mention again that the second term inEq. (72) can be interpreted as a smooth approximation to thediscontinuous term of the sliding mode controller, that is, the secondterm in Eq. (71).

3. Numerical Results

As predicted, and expected, both controllers drive closed-looptrajectories toward the origin. Figure 6 consists of two subguresdepicting trajectories, in the x1; x2 plane, of the third-order closed-

loop systems driven by the sliding mode and synergetic controllers,respectively.The value of the design parameterT in the synergetic controller

was selected so that the closed-loop system trajectories, driven bythis controller, converge to the origin at the same rate as those drivenby the sliding mode controller. We conclude from plots in Fig. 6 thatto achieve comparable behavior ofx1 to that of the sliding modecontroller, the state x2 of the synergetic controller driven systemundergoes large excursions induced by the actuator state x3. Suchovershoots can be reduced by increasing the value of the parameterT. As a consequence of increasingT, trajectory convergence of thesynergetic controller toward the origin occurs at a lower rate. Notethat increasing Tessentiallycorrelates with decreasing the synergeticcontrol law gains. We can see in Fig. 7 that closed-loop state variabletrajectories of closed-loop wing-rock dynamics driven by either

sliding mode or synergetic controller converge to the origin.However, rapid convergence to theoriginin the synergetic controllerdrivensystem comes with a price in thatx2andx3initially experienceexcessively large overshoots that may not be desirable. A largeexcursion of the actuator statex3results in large deviations of the roll

rate _, that is,x2.When we increase the value of T from T 0:01 to T 2:5,

trajectory overshoots of the closed-loop dynamics driven by thesynergetic controller decrease. Transient discrepancies between

10 5 0 5 1030

20

10

0

10

20

30

x1

x2

Sliding Mode

10 5 0 5 1030

20

10

0

10

20

30

x1

x2

Synergetic

Fig. 6 State variable trajectories of the third-order closed-loop wing-

rock dynamic system in the _coordinates.



11/15


12/15

order wing-rock dynamical model (67) is driven by the control laws(71) and (72) where both controllers were developed using Eq. (67).

Here, we investigate the case of designing synergetic and slidingmode control laws for wing-rock suppression using a lower-ordermodel. Then, these controllers are employed to control a higher-order representation of thedynamics. To serve ourpurpose,the third-order wing-rock model represents the truth model. We obtain ourdesign model from the third-order model by neglecting the actuatordynamics. Thus, the design model has the form,

_xr _x1

_x2

x21x1 2x2 3x

32 4x

21x2 5x1x

22

0

u (73)

0 1

1 5x22 2 3x

22 4x

21

" # x1

x2

" #

0

" #u

Arxr Bru (74)

where the subscript r denotes reduced. The parameters i inEq. (73) have been given previously and are the same as those inEq. (67).

1. Controller Construction Using the Design Model

We nowproceed with thedesign procedure using thesecond-orderdesign model derived above. The invariant manifold is constructedsimilarly to Eq. (59):

x Srxr 1 xr (75)

Using the matricesArand Brdened in Eq. (74) along with SrfromEq. (75), we obtain the synergetic controller

ursynergetic TSrBr1TSrAr Srxr

TSrBr1TSrAr TSrBr

1Srxr (76)

The sliding mode controller has the form,

ursliding mode TSrBr1TSrArxr KsignSrxr (77)

As in the preceding design cases, the synergetic and sliding modecontroller structures are related. Both controllers use the equivalentcontrol component. However, the sliding mode controller uses adiscontinuous component, while the synergetic control employs asmooth approximation to the discontinuous component of the slidingmode controller.

We now connect the control laws (76) and (77) to drive the third-order model (67), so that theclosed-loop third-order systembecomes

_x1

_x2

_x3264 375

x2

1x1 2x2 3x32 4x

21x2 5x1x

22 x3

1x3

2664

3775

0

0

1

2664

3775ur (78)

where urcan be eitherursynergeticorursliding mode. Thereare two designparameters,Tand , whose inuence on the closed-loop dynamics(78) will be investigated next.

2. Numerical Results

Adjustable parameters and Tmust be carefully selected becausehigh values of either orTleadto unstable solutions, while very lowvalues of either orTinduce oscillations with frequency inverselyproportional to thevalue of either orT.InFig. 11,weshowafamilyof stable closed-loop solutions for various values of . Our

experiments indicate that for axed value ofT, where T 0:1, astabilizing controller requires the value ofto be less than 3. Thus,all s such that 0< 3 are feasible values for xed T 0:1.However, as can be seen in Fig. 11, lower values ofinduce higher

frequency oscillations around the manifold M. Hence, slowermotion along the manifold M will reduce oscillations around M.

The parameter T also affects stability and performance of theclosed-loop wing-rock suppression. As Tgets higher, the gain of thesecond term of Eq. (76) becomes smaller. Such low-gain controlaction prevents excessive oscillations. In Fig.12, we show a familyof stable solutions with various values ofTs using 1.Wecanseethat smaller values ofTinduce oscillations around the manifold Mwith higher frequency. Although lower values ofT, yielding highergains for the second term of Eq. (76), lead to rapid motion toward themanifoldM, their effect on higher frequency oscillations aroundMmay not be desirable. As a comparison, the closed-loop solutiondrivenby thesliding mode is also shown in Fig. 12 asthe plotwith thedash-dotted line. We conclude that the selection ofT 0:1 may leadto the synergetic design with comparable performance to that of thesliding mode design.

Plots of state variablesversus time are presentedin Figs. 13 and 14,where we used Sr 1 1 , the sliding mode control gainK 10,

8 6 4 2 0 2 4 6 830

20

10

0

10

20

30

x1

x2

=3

=0.1

=1

Decreasing

Family of manifolds Ms

Fig. 11 The effect of . Higher values of correspond to rapidmovement toward the origin. However, higher values of induceoscillatory dynamics around the manifoldM.

8 7 6 5 4 3 2 1 0 1

0

2

4

6

8

10

12

x1

x2

T=1

T=0.1

T=0.01

slidingmode

T=0.005

Manifold M

Fig. 12 The effect ofTon the performance of the synergetic controller.

A lower value ofTcorresponds to rapid movement toward the manifoldM. However,a lower value ofTinduces oscillation aroundthe manifoldM.



13/15

and the synergetic control design parameterT 0:1. These values

were selected so that the states and _ driven by both synergetic andsliding mode controllers would have approximately similar timeresponses. From Eq.(78),weseethatthecontrollerurdirectlyaffectsthe state variablex3. We show in Fig.14time responses ofx3drivenby both synergetic and sliding mode controllers. Whenx3 is driven

by the sliding mode control, it experiences chattering. However, wesee thatx3is smooth when it is driven by the synergetic control. Thetransient peakofx3when it is driven bythe synergetic control, as it isalso seen in the phase portraits in Figs. 11and12, is larger than thepeak ofx3driven by the sliding mode control.

X. Concluding Remarks

A variety of sliding mode controllers has been proposed andapplied to a large numberof engineering problems [3537] includingnonlinear problems [38,39]. The behavior of nonlinear dynamicsystems driven by the synergetic control to some extent resemblesthat of nonlinear dynamic systems driven by sliding mode control.Hence, one can benet by analyzing synergetic controllers from thepoint of view of sliding mode control combined with an optimalcontrol. Explorations in sliding mode control may, indeed, bebenecial to the synergetic control. Such explorations may revealfurther optimality characteristics. The synergetic control approach

offers a closed-form analytical solution to a class of nonlinearoptimal control problems. The proposed control design methodologyis attractive because it only requires the associated rst-orderdifferential equation. Indeed, as the problem of optimal controldesign is further restricted to linear time-invariant dynamicalsystems, optimal control solution generated by the synergetic controlis the same as the solution generated by the LQR with the specialform of the performance criterion. In summary, the papercontributions are as follows:

1) Synergetic control approach was shown to follow fromboth thenecessary (EulerLagrange) andthe sufciency(HJB) conditionsforoptimality, resulting in therst-order differentialequation to developa synergetic controller.

2) Connections between the synergetic control approach and thevariable structure sliding mode control method were established.

3) Synergetic control was shown to provide the same controller asthe LQR with a special performance index for the case of linear time-invariant dynamic systems.

4) Synergetic control was used to successfully design a nonlinearcontroller for wing-rock stabilization using existing nonlineardynamic models.

5) Synergetic controllers were shown to be effective in thenonlinear wing-rock stabilization application. This includedreducing overshoot in the actuator state variable and by providing

a smoother transition of state variables to the stable condition.

References

[1] Owens, D., Multivariable and Optimal Systems, Academic Press,London, U.K., 1981.

[2] Lee, E., and Markus, L.,Foundations of Optimal Control Theory, TheSIAM Series in Applied Mathematics, Wiley, New York, 1968.

[3] Pinch,E., Optimal Control and Calculus of Variations, OxfordSciencePublications, Oxford University Press Inc., New York, 1993.

[4] Bryson, A., andHo, Y.-C.,Applied Optimal Control, Hemisphere, NewYork, 1975.

[5] Athans, M., and Falb, P., Optimal Control: An Introduction to theTheory and Its Applications, Lincoln Laboratory Publications,McGrawHill, New York, 1966.

[6] Bellman, R., and Kalaba, R., Dynamic Programming and Modern

Control Theory, Academic Press, New York, 1965.[7] Kolesnikov, A. A. (ed.),Modern Applied Control Theory: Synergetic

Approach in Control Theory, TRTU, Moscow-Taganrog, 2000.[8] Kolesnikov, A., Synergetic Control of the Unstable Two-Mass

System,Fifteenth International Symposium on Mathematical Theoryof Networks and Systems, Univ. of Notre Dame Press, South Bend, IN,Aug. 2002.

[9] Kolesnikov, A.,Synergetic Control for Electromechanical Systems,Fifteenth International Symposium on Mathematical Theory ofNetworks and Systems, Univ. of Notre Dame Press, South Bend, IN,Aug. 2002.

[10] Kondratiev, I., Dougal, R., Santi, E., and Veselov, G., SynergeticControl for m-Parallel Connected DC-DC Buck Converters, 35thAnnual IEEE Power Electronics Specialists Conference, IEEE,Piscataway, NJ, 2004, pp. 182188.

[11] Santi, E., Monti, A., Li, D., Proddutur, K., and Dougal, R.,Synergetic

Control for Power Electronics Applications: A Comparison with theSliding Mode Approach, Journal of Circuits, Systems, andComputers, Vol. 13, No. 4, 2004, pp. 737760.

[12] Jiang, Z., and Dougal, R.,Synergetic Control of Power Converters forPulse Current Charging of Advanced Batteries from a Fuel Cell PowerSource, IEEE Transactions on Power Electronics, Vol. 19, No. 4,July 2004, pp. 11401150.

[13] Terekhov, V., Tyukin, I., and Prokhorov, D., Adaptive Control onManifolds with RBF Neural Networks,Proceedings of the 39th IEEEConference on Decision and Control, IEEE, Piscataway, NJ,Dec. 2000, pp. 38313836.

[14] Nusawardhana, and ak, S., Optimality of Synergetic Controllers,Proceedings of the 2006 ASME International Mechanical EngineeringCongress and Exposition, American Society of Mechanical Engineers,Chicago, IL, 510 Nov. 2006.

[15] DeCarlo, R.,ak,S. H., andMatthews, G.,Variable Structure Control

of Nonlinear Multivariable Systems: A Tutorial, Proceedings of theIEEE, Vol. 76, No. 3, March 1988, pp. 212232.[16] Khalil, H.,Nonlinear Systems, 2nd ed., PrenticeHall, Upper Saddle

River, NJ, 1996.

0 1 2 3 4 5 66

4

2

0

2

4

6

8

t [sec]

synergetic

d/dt synergetic

sliding

d/dt sliding

,

d/dt

Fig. 13 Comparison oftand _ttrajectories driven by synergeticand sliding mode controllers.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

30

25

20

15

10

5

0

t [sec]

x3

x3 synergetic

x3 sliding

Fig. 14 Comparison ofx3trajectories driven by synergetic and slidingmode controllers.



14/15

[17] Edwards, C., and Spurgeon, S., Sliding Mode Control: Theory andApplications, Taylor and Francis, London, 1998.

[18] Utkin, V., and Young, K.,Methods for Constructing DiscontinuityPlanes in Multidimensional Variable Structure Systems, Automationand Remote Control (English Translation), Vol. 39, No. 10, 1978,pp. 14661470.

[19] Utkin, V., Sliding Modes in Control and Optimization, SpringerVerlag, Berlin, 1992.

[20] ak, S.H., Systems and Control, OxfordUniv.Press, NewYork,2003.[21] Gelfand, I., and Fomin, S., Calculus of Variations, PrenticeHall,

Englewood Cliffs, NJ, 1963.[22] Leitmann, G., The Calculus of Variation and Optimal Control,Mathematical Concepts and Methods in Science and Engineering,Plenum Press, New York, 1981.

[23] Anderson, B., and Moore, J., Optimal Control: Linear QuadraticMethods, PrenticeHall, Englewood Cliffs, NJ, 1989.

[24] Bertsekas, D.,Dynamic Programming and Optimal Control,Vols.12,Athena Scientic, Belmont, MA, 1995.

[25] Elzebda, J., Nayfeh, A., and Mook, D.,Development of an AnalyticalModel of Wing Rock for Slender Delta Wings, Journal of Aircraft,Vol. 26, No. 8, 1989, pp. 737743.

[26] Guglieri,G., and Quagliotti, F., Analyticaland Experimental Analysisof Wing Rock,Nonlinear Dynamics, Vol. 24, No. 2, 2001, pp. 129146, Kluwer Academic Publishers.

[27] Hsu, C.-H., and Lan, C.,Theory of Wing Rock ,Journal of Aircraft,Vol. 22, No. 10, 1985, pp. 920924.

[28] Konstadinopoulos, P., Mook, D., and Nayfeh, A., Subsonic WingRock of Slender Delta Wings, Journal of Aircraft, Vol. 22, No. 3,1985, pp. 223228.

[29] Fernand, J.,and Downing, D.,Discrete Sliding Mode Control of WingRock, AIAA Guidance, Navigation and Control Conference, AIAA,Washington, D.C., 13 Aug. 1994, pp. 4654; also AIAA Paper 1994-

3543.[30] Gazi, V., and Passino, K.,Direct Adaptive Control Using Dynamic

Structure Fuzzy Systems, Proceedings of the American ControlConference, IEEE, Piscataway, NJ, June 2000, pp. 19541958.

[31] Passino, K., Biomimicry for Optimization, Control, and Automation,SpringerVerlag, London, 2005.

[32] Slotine, J., and Li, W., Applied Nonlinear Control, PrenticeHall,Englewood Cliffs, NJ, 1991.

[33] Corless, M.,Leitmann, G., and Ryan, E., Control of Uncertain Systemswith Neglected Dynamics, edited by A. S. I. Zinober, Chap. 12,

Deterministic Control of Uncertain Systems, Peter Peregrinus Ltd.,Michael Faraday House, Six Hills Way, Stevenage, Herts. SG1 2AY,U.K., 1990, pp. 252268.

[34] ak, S., and Walcott, B., State Observation of Nonlinear ControlSystems via the Method of Lyapunov, edited by A. S. I. Zinober,Chap.16, Deterministic Control of UncertainSystems,Peter PeregrinusLtd., Michael Faraday House, Six Hills Way, Stevenage, Herts. SG12AY, U.K., 1990, pp. 333350.

[35] Hess, R., and Wells, S., Sliding Mode Control Applied toRecongurable Flight Control Design,Journal of Guidance, Control,and Dynamics, Vol. 26, No. 3, MayJune 2003, pp. 452462.

[36] Shima, T., Idan, M., and Golan, O., Sliding-Mode Control forIntegrated Missile AutopilotGuidance,Journal of Guidance, Control,and Dynamics, Vol. 29, No. 2, MarchApril 2006, pp. 250260.

[37] Utkin, V., Guldner, J., and Shi, J., Sliding Mode Control in Electro-Mechanical Systems, Taylor and Francis, Philadelphia, PA, 1999.

[38] Chin, C.L. W.,Adaptive Decoupled Fuzzy Sliding-Mode Control of aNonlinear Aeroelastic System, Journal of Guidance, Control, andDynamics, Vol. 29, No. 1, Jan.Feb. 2006, pp. 206209.

[39] Zhou, D., Mu, C., and Xu, W.,Adaptive Sliding-Mode Guidance of aHoming Missile, Journal of Guidance, Control, and Dynamics,Vol. 22, No. 4, JulyAug. 1999, pp. 589594.



15/15

synergetic optimal controllers

Documents