synergetic optimal controllers
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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
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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
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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.
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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,
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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
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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
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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.
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_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.
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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.
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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.
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0 1 2 3 4 5 66
4
2
0
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4
6
8
t [sec]
synergetic
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,
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Fig. 13 Comparison oftand _ttrajectories driven by synergeticand sliding mode controllers.
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30
25
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t [sec]
x3
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x3 sliding
Fig. 14 Comparison ofx3trajectories driven by synergetic and slidingmode controllers.
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