April 28, 2004 8:54 00990

International Journal of Bifurcation and Chaos, Vol. 14, No. 4 (2004) 1343–1355c© World Scientific Publishing Company

TRACKING CONTROL OF NONLINEAR SYSTEMS:

A SLIDING MODE DESIGN VIA

CHAOTIC OPTIMIZATION*

ZHAO LU and LEANG-SAN SHIEHDepartment of Electrical and Computer Engineering,

University of Houston, Houston, TX 77204-4005, USA

JAGDISH CHANDRASchool of Engineering and Applied Sciences,

George Washington University, Washington DC 20052, USA

Received August 27, 2002; Revised January 27, 2003

The output tracking for a general family of nonlinear systems presents formidable technical chal-lenges. In this paper, we present a novel scheme for tracking control of a class of affine nonlinearsystems with multi-inputs. This effective procedure is based on a new sliding mode design fortracking control of such nonlinear systems. The construction of an optimal sliding mode is adifficult problem and no systematic and efficient method is currently available. Here, we developan innovative approach that utilizes a chaotic optimizing algorithm, which is then successfullyapplied to obtain the optimal sliding manifold. The existing efficient reaching law approachis then utilized to synthesize the sliding mode control law. The sliding mode control schemeproposed here is particularly appropriate for robust tracking of the chaotic motion trajectory.

Keywords : Sliding manifold; reaching law; chaotic optimization; ergodicity.

1. Introduction

For many physical systems, it is usually difficultto construct mathematical models that describethem accurately. Invariably, one encounters signif-icant deviations between the constructed modelsand the real systems. These uncertainties may bedue to unknown or partially known parametric val-ues of the controlled system. Further, the changingenvironments and unspecified disturbances such asmeasurement noises may also contribute to the dis-parities with the real system. Therefore, the robustcontroller design that effectively deals with suchuncertainties of a system becomes an importantresearch subject.

The customary approach for designing con-trol law for nonlinear systems is linearization, suchas local linearization and feedback linearization,via which the well-developed linear system theorycan be utilized for controlling the nonlinear sys-tem [Guo et al., 2000; Jiang, 2002]. However, thereare many well-known limitations in applying lin-earization techniques. For local linearization meth-ods, they are strictly local, potentially unstableand usually not powerful enough to handle broadclasses of complex and/or uncertain dynamics. Inthe past two decades, major progress has been madein the development of a geometric theory of nonlin-ear feedback. Feedback linearization techniques arebased on Lie algebras, and coordinate and control

∗Supported by the US Army Research Office under Grant DAAD 19-02-1-0321.

1343

April 28, 2004 8:54 00990

1344 Z. Lu et al.

transformations. Systematic design procedures arebuilt on the solid foundation these geometric re-sults form. However, this method has two draw-backs: (1) perfect linearization cannot be obtainedbecause the nonlinear feedback depends on the non-linear model of the system that necessarily containsuncertainty; (2) the resulting control law is some-what complex and requires too much control effortbecause it pursues linearization of the system ratherthan stabilization or tracking.

To deal with the uncertainties in nonlinear sys-tems, three main approaches have been employedcommonly: (1) adaptive control; (2) Lyapunov-based control; and (3) sliding mode control (alsocalled variable structure control). The adaptive con-trol utilizes the linear-in-parameter assumption toformulate error equations which relate measurablesignals to parameter errors. Then, a parameter up-date law is formed from the error equations. Theobjective of either stabilization or tracking is thusachieved during the adaptation process. On theother hand, the Lyapunov-based approach relies onan explicit construction of a energy or Lyapunovfunction, by which a state-feedback control is syn-thesized using the bounds on the uncertainties.Finally, the sliding mode control (SMC), which isclosely related to the Lyapunov-based approach,exploits the variable structure concept to achievethe goal. It first defines the sliding surface in theerror state space and devises a switching state-feedback control law. The high-speed switching con-trol law forces the error state to slide along thesurface until it converges and then the tracking isattained.

In this paper, we address the issues of design-ing the sliding mode tracking controller for generalaffine nonlinear systems with multi-inputs. Most ofthe existing schemes about sliding mode control formulti-input nonlinear systems concentrate on theequations that can be transformed to a canonicalform or are feedback-linearizable. For general affinenonlinear systems, there is lack of effective and sys-tematic approaches for tracking specific trajecto-ries of nonlinear systems, especially chaotic ones.As it is well known, the crux of nonlinear slidingmode control is the construction of a good slidingmode. Hence, in this paper, we utilize the emerg-ing chaotic optimization algorithm (COA) to suc-cessfully search and construct the optimal slidingmode for tracking control of the nonlinear systems.In this procedure, the existing effective reaching law

approach is adopted to synthesize the sliding modecontrol law.

2. Sliding Mode Control System

Design

Sliding mode control (SMC) evolved from the pio-neering work in Russia in the early 1960s [Utkin,1977]. Yet, SMC did not receive wide acceptanceamong engineering professionals prior to 1970’s,which is probably due to the lack of an effectivedesign procedure and the presence of considerablechattering in the SMC systems. The situation, how-ever, changed with resultant rise of interest in SMCin the 1970’s as people started recognizing the ro-bustness and invariance of SMC. It is well knownthat the most distinguishing feature of SMC is itsability to result in very robust control systems. Inmany cases invariant control system results. Looselyspeaking, the term “invariant” means that the sys-tem is completely insensitive to parametric uncer-tainty and external disturbances.

The primary characteristic of a variable struc-ture system (VSS) is that the feedback signal isdiscontinuous, switching on one or more manifoldsin state space. When the state crosses each dis-continuity surface, the structure of the feedbacksystem is altered. Under certain circumstances, allmotions in the neighborhood of the manifold aredirected towards the manifold, and thus a slidingmotion on a predefined subspace of the state-spaceis established in which the system state repeatedlycrosses the switching surface. This mode has use-ful invariance properties in the face of uncertain-ties in the plant model, and therefore is a candidatefor robust tracking control of uncertain nonlinearsystems.

2.1. Preliminaries

Considering a class of affine nonlinear system:

x = f(x) + g(x)u , (1)

where x ∈ Rn is the state vector, u ∈ Rm the inputvector, n the order of the system, m the number ofinputs, m ≤ n, f(x) and g(x) are nonlinear func-tions of appropriate dimensions. It is known that,in general, the transient dynamics of a SMC systemconsists of two modes, i.e. a “reaching mode” fol-lowed by a “sliding mode”. Therefore the design ofSMC usually involves two fundamental steps. Thefirst step is to design an appropriate m-dimensional

April 28, 2004 8:54 00990

Tracking Control of Nonlinear Systems 1345

switching function s(x) such that the system ex-hibits the desired behavior in the sliding mode. Thesecond step is to determine a control law that guar-antees the reaching and sliding conditions. The de-sired sliding mode dynamics is usually a fast andstable error-free response void of overshoot. For thereaching mode, the desired response is to reach theswitching manifold, described by

s(x) = Hx = 0, rank(H) = m

H ∈ Rm×n, s(x) ∈ Rm ,(2)

i.e.

s(x) =

h11 h12 · · · h1n

h21 h22 · · · h2n

...... · · ·

...

hm1 hm2 · · · hmn

x1

x2

...

xn

= 0 . (3)

For an m-input system, there are m switchfunctions and 2m − 1 sliding manifolds of differentdimensions. The first m of them are designed as

Si = {x|si = hix = 0} , i = 1 to m, (4)

where hi = [hi1 hi2 · · · hin] is a row vector. Si

may be called basic sliding manifolds since each ofthem is associated with a single switching function.The last one is designated as

SE = {x|s = Hx = 0} = S1 ∩ S2 ∩ · · · ∩ Sm ,

H ∈ Rm×n ,(5)

which is the intersection of all m basic slidingmodes. SE may be called the eventual sliding man-

ifold since it is the manifold that all state trajec-tories must reach eventually. Obviously, the basicsliding manifold is a (n − 1) dimension dynamics,and the eventual sliding manifold is (n−m) dimen-sion dynamics.

There are several possible switching schemes forsteering the state to enter various sliding manifolds.Three of them are described briefly here.

(1) Fixed-Order Sliding Mode Switching: In thisscheme, sliding modes take place in a preas-signed order while the state is traversing thestate space. For example

x(0) → S1 → (S1 ∩ S2) → (S1 ∩ S2 ∩ S3)

→ · · · → SE . (6)

(2) Eventual Sliding Mode Switching: Here thestate is driven to the eventual sliding mani-fold SE, only on which the sliding mode control

takes place. This scheme is simpler in SMC im-plementation and gives a faster transient thanthe fixed order schemes.

(3) Free-Order Sliding Mode Switching: In thisscheme, a sliding mode is switched in wheneverthe state reaches any switching manifold Si. Itis a “first-reach-first-switch” scheme, thus theorder of sliding modes is not fixed. For this case,the system state moves into the sliding modesooner than in previous cases. Therefore, theoverall response is more robust and the tran-sient response can be faster. Also, the requiredcontrol effort is usually smaller in magnitudewhen compared to the fixed-order scheme, sosaturation is less likely to occur. Finally, the as-sociated SMC is easy to be obtained, especiallyby using the reaching law method.

2.2. Design of sliding manifold

The characteristic feature of a variable structuresystem (VSS) is that the sliding mode occurs ona prescribed switching surface. The surface is theintersection of a set of discontinuity surfaces inthe state space of a multiple-state system, whereeach control input switches between two functions.These discontinuity surfaces are selected so that,while in the sliding mode, the system performancesatisfies the design objectives such as stability, per-formance index minimization, chattering reduction,etc. Mathematically, the sliding manifold is notmerely a hypersurface in the original state-space ofthe plant, but a linear operator that can be repre-sented as a linear, time-invariant dynamic system,acting on the states.

During the past decade, the problem of con-structing the sliding manifolds in SMC has beenstudied mainly under the framework established byUtkin and Young [1979]. A coordinate transforma-tion is found to partition the states into a specialform referred to as the canonical form. In the newcoordinates, design of the switching surface is car-ried out in a reduced-order system separated fromthe control variables. This idea has been extensivelyused in the literature of variable structure systemstheory, and has been exploited from different per-spectives [Utkin & Young, 1979; Dorling & Zinober,1986; Kwatny & Kim, 1989; Sira-Ramirez, 1989].For linear systems, the sliding surface can be de-signed by pole placement, quadratic optimization,or a geometric approach and eigenstructure assign-ment technique.

April 28, 2004 8:54 00990

1346 Z. Lu et al.

However, for general affine nonlinear multi-input systems, the design of sliding surface is muchmore complicated. In general, the nonlinear system(1) needs to be converted into the following regularcanonical form via a nonsingular coordinate trans-

formation[

x1

x2

]

= φ(x):

x1 = f1(x1, x2) , (7)

x2 = f2(x1, x2) + g2(x1, x2)u , (8)

where x1 ∈ Rn−m, x2 ∈ Rm and det[g2(x)] 6= 0.A switching surface can be found by designing anonlinear feedback control law x2 = s(x1) for thesubsystem (7) when x2 is viewed as virtual control.However, the feedback control law x2 = s(x1) forthe nonlinear system (7) is usually hard to obtain,and not every system can be easily transformed toa regular canonical form [Gao & Hung, 1993].

The other approach is to use the feedbacklinearization technique to secure a linear equiva-lence. Suppose that the nonlinear system (1) isfeedback-linearizable. We apply a diffeomorphictransformation z = ϕ(x) so that the new statevariable z and control variable ν satisfy the lineardifferential equation

z = Az +Bν , (9)

u = ψ(x, ν) , (10)

where z ∈ Rn is the transformed coordinates, ν ∈Rm, A and B are constant matrices in Brunovskycanonical form. With the linearized dynamic equa-tion (9), a sliding surface can easily be found. Thisidea has been exploited by many researchers [Fu& Liao, 1990; Li & Slotine, 1987], under the as-sumption that the nonlinear system (1) is feedback-linearizable, which is unfortunately satisfied onlyunder certain structural conditions.

A Lyapunov approach for constructing thesliding manifold was proposed [Su et al., 1996],but Lyapunov functions usually are difficult to con-struct for general affine nonlinear systems. In thispaper, we utilize a chaotic optimization algorithmto search for the optimal sliding manifold, withoutrequiring any transformation of the system. As aresult, this approach is more general and is simplerthan the other methods.

For tracking control of the n-dimensional non-linear system (1), the sliding surface S(e) is usuallydefined as

S(e) = {e|s(e, t) = He(t) = 0}, s(e) ∈ Rm ,

(11)

where e = x − xd is the tracking error vector for adesired trajectory xd(t) and H ∈ Rm×n representsthe coefficients of the sliding surface.

2.3. Design of sliding mode control

The condition under which the state will move to-wards and reach a sliding surface is called a reaching

condition. The sliding mode control law is synthe-sized from the reaching condition. For specifyingthe reaching condition, three approaches have beenproposed (see [Hung et al., 1993] and the referencestherein).

(a) The Direct Switching Function Approach: Theearliest reaching condition proposed was

sisi < 0, i = 1, 2, . . . , m . (12)

This reaching condition is global but does notguarantee a finite reaching time. Moreover, thisreaching condition is very difficult to use for themulti-input SMC.

(b) The Lyapunov Function Approach: By choosingan appropriate Lyapunov function

V (x, t) = sT s , (13)

a global reaching condition is given by

V (x, t) < 0 when s 6= 0 . (14)

The finite reaching time is guaranteed by mod-ifying the formula above to

V (x, t) < −ε when s 6= 0, where ε is positive .

(15)

Clearly, this approach would lead to the even-tual sliding mode switching scheme.

(c) The Reaching Law Approach: The reaching law

is a set of differential equations which specifiesthe dynamics of sliding variable s(x). The dif-ferential equation of an asymptotically stables(x) is itself a reaching condition. In addition,by the choice of the parameters in the differen-tial equation, the dynamic quality of the SMCsystem in the reaching mode can be controlled.A practical general form of the reaching law is

s = −Q sign(s) −Kd(s) , (16)

where

Q = diag[q1, q2, . . . , qm], qi > 0 ,

sign(s) = [sign(s1), . . . , sign(sm)]T ,

K = diag[k1, k2, . . . , km], ki > 0 ,

April 28, 2004 8:54 00990

Tracking Control of Nonlinear Systems 1347

d(s) = [d1(s1), . . . , dm(sm)]T ,

sidi(si) > 0, when si 6= 0, di(0) = 0 .

Usually one of the following three forms of thereaching law is adopted:

(1) Constant rate reaching:

s = −Q sign(s) . (17)

This law forces the sliding variable s(x) to reachthe switching manifold S at a constant rate|si| = −qi. The merit of this reaching law isits simplicity. But, if qi is too small, the reach-ing time will be too long. On the other hand, atoo large qi will cause severe chattering.

(2) Constant plus proportional rate reaching:

s = −Q sign(s) −Ks . (18)

Clearly, by adding the proportional rate term−Ks, the state is forced to approach the switch-ing manifolds faster when s is large.

(3) Power rate reaching:

si = −ki|si|αsign(si), i = 1, 2, . . . , m . (19)

This reaching law increases the reaching speed whenthe state is far away from the switching manifold,but reduces the rate when the state is near the man-ifold. The result is a faster reaching and a lowerchattering reaching mode.

In this paper, the constant plus proportionalrate reaching law is adopted to synthesize the slid-ing mode control law for tracking control of the non-linear system (1). The time derivative of s(x) alongthe reaching mode trajectory needs to be computedto design the control law. From (1) and (18) we have

∂s

∂xf(x) +

∂s

∂xg(x)u = −Q sign(s) −Ks . (20)

Noting that the matrix (∂s/∂x)g(x) is nonsin-gular, and solving Eq. (20) for the control law yields

u = −

[

∂s

∂xg(x)

]

−1 [

∂s

∂xf(x) +Q sign(s) +Ks

]

.

(21)

The reaching law approach not only estab-lishes the reaching condition but also specifies thedynamic characteristics of the system during thereaching phase. Additional merits of this approachinclude simplification of the solution for SMC andproviding a means for the reduction of chattering,i.e. the chattering can be reduced by adjusting thematrices Q and K. Also, the control law obtained

via a reaching law automatically leads to the free-order switching scheme. From the practical point ofview, this scheme appears to be the most efficient.

3. Design of Optimal Sliding

Surface Using COA

3.1. Chaotic optimizing algorithm

There has been a rise of interest in developing glob-ally optimizing algorithms during the past threedecades. An important and effective approach toglobal optimum searching problem is the genetic al-gorithm (GA) [Holland, 1975], in which the solutionspace is encoded as gene-like strings, and then theprocedure of searching the global optimum is car-ried out by reproduction, mutation, crossover andcomputing the fitness value. In comparison to otherlocal search algorithms, GA guarantees a higherchance of reaching a global optimum by startingwith multiple random search points, and by con-sidering several candidate solutions simultaneously.The unique crossover operator in GA offers thepossibility of exchanging attributes among potentialsolutions. The deficiency of classical GA is mainlyattributed to the fact that the diversity of a popu-lation relies on mutation only once the populationhas been initialized. Since mutation must be keptat a low rate (otherwise the offspring do not inheritthe characteristics of their parents, leading to a ran-dom search), it does not diversify the populationeffectively once the population has been converged.Moreover, in classical GA, the population size mustbe large; otherwise GA does not provide a suffi-cient sample size, causing premature convergence.However, large population size requires more timeto converge the population. The rate of convergenceis, generally, unacceptably slow.

Another means to global optimum searchingproblem is the simulated annealing algorithm (SA)[Kirkpatrick et al., 1983], in which a stochasticmechanism is introduced to avoid being trapped ina local optimum. Boltzmann machine is the neu-ral network implementation of SA algorithm. Thedisadvantage of SA is that relatively longer compu-tation time is needed and it can be painfully slowas the problem size increases, which implies that itcannot be used efficiently in large-scale optimizingproblem.

Recently, a new and quite promising approachfor global optimizing has come along, that is,applying chaotic dynamics. This is introduced inorder to overcome the local minimum problem and

April 28, 2004 8:54 00990

1348 Z. Lu et al.

to perform efficient search based on the ergodic-ity of chaos. Some successful applications of chaoticneural network for complicated global optimizingproblems have been reported, to solve such diffi-cult problems as the large-scale traveling salesmanproblem (TSP) [Hasegawa et al., 2002; Kwok &Smith, 2000]. A key characteristic of chaotic dy-namics is its self-similarity, which indicates that at-tractors of chaotic dynamical systems usually havefractal structures. The chaotic dynamics can beused to search solutions only along such a frac-tal attractor typically with zero Lebesgue measurein a bounded region of the state space. Therefore,the chaotic search is expected to be more efficientthan stochastic search schemes, if good solutionsare embedded in a searching region. However, thechaotic neurodynamics approach is based, usually,on the architecture of the Hopfield neural network,which leads to the same defect as the other op-timizing schemes using Hopfield neural network.That is, the objective functions to be optimizedare restricted only to continuously differentiablequadratic functions.

Fortunately, an emerging strategy calledchaotic optimization algorithm (COA) has beenproposed [Li & Jiang, 1998] and successfully ap-plied to some typical nonlinear optimizing prob-lems. A number of simulations have demonstratedthat COA is more efficient than GA and SA, andthere are no limitations on the type of objec-tive functions to be optimized. The philosophy ofchaotic optimizing algorithm is simple, which isbased on two main steps: first, a transform fromchaotic space to solution space is carried out, andthen the searching for global optimum is executedaccording to the chaotic dynamics itself rather thanguiding it at random. The details about the COAwill be shown in the next section, by designing theoptimal sliding manifold.

3.2. Design of optimal sliding

manifold using COA

To illustrate the chaotic optimization algorithm andthe proposed sliding manifold design procedure, itis especially convenient to use examples. The well-known Chen’s chaotic system [Chen & Ueta, 1999]is taken as an example for this purpose.

The recently discovered chaotic attractor,Chen’s chaotic system can be described asfollows:

x1(t) = a(x2(t) − x1(t))

x2(t) = (c− a)x1(t) − x1(t)x3(t) + cx2(t)

x3(t) = x1(t)x2(t) − bx3(t)

(22)

where a = 35, b = 3 and c = 28. If we repre-sent the system by a simple state-space equation:

x(t) = f(x(t)), x(t) ∈ R3 . (23)

The nonlinear system to be designed is

x(t) = f(x(t)) +Bu(t) , (24)

where u(t) ∈ R3 and B = I3. Obviously, the system(24) is an affine nonlinear system with multi-input.Assume that the tracking reference is xd(t) to bedefined later.

In order to find a sliding mode control law,u(t) ∈ R3, which can make the system state x(t)to track the prespecified reference signal xd(t), asliding surface should be defined ahead. To utilizethe COA to search the optimal sliding surface, weneed to define the performance index as

J(H) =N

∑

i=1

{‖ei‖ + w‖∆ei‖} , (25)

where N is the duration of the simulation for eval-uating the design, i is the time index in simulation,ei is the error vector at simulation step i and ∆ei

is the change in error vector, w (being fixed at 1in this paper) is a bias weighting between ei and∆ei. The ∆ei term can be distinctively weightedto further suppress oscillations. The optimal slidingsurface coefficients H ∈ R3×3 which minimize theperformance index will be searched via COA. Notethat there are nine elements in the sliding surfacecoefficients matrix H ∈ R3×3, therefore nine vari-ables are involved in this optimizing problem.

The chaotic equation of COA can be selectedas Logistic mapping, i.e.

νn+1 = µνn(1 − νn), µ = 4 . (26)

First, we select nine initial chaotic variablesν1,0, ν2,0, . . . , ν9,0, 0 ≤ vi,0 ≤ 1, i = 1, 2, . . . , 9.Notice that fixed points of Logistic mapping 0.25,0.5, 0.75 cannot be adopted as initial chaotic vari-ables. Then, we set up the ranges for searchingthe elements in the sliding surface coefficients ma-trix, namely, the upper-bound values upboundi, i =1, 2, . . . , 9 and the lower-bound values lowboundi,i = 1, 2, . . . , 9. Assume H∗ and J∗ to be theoptima. The search procedure by COA starts fromn = 0:

April 28, 2004 8:54 00990

Tracking Control of Nonlinear Systems 1349

Step 1. Chaotify the variables:

Substitute ν1,n, ν2,n, . . . , ν9,n into (26), then we canget nine chaotic variables v1,n+1, v2,n+1, . . . , v9,n+1

via the Logistic chaotic equation.

Step 2. Transform from the chaotic space to the

solution space:

Perform the transformation from the chaotic spaceto the solution space by the following formula:

hi,n+1 = lowbound i + (upbound i − lowbound i) ,

×νi,n+1, i = 1, 2, . . . , 9 . (27)

Step 3. Assign the sliding surface coefficients

and define the sliding mode:

Set Hn+1 = {hi,n+1} as the coefficients matrix ofthe sliding surface, and then define the sliding modeas: S(e) = {e|s(e) = Hn+1 · e(t) = 0}.

Step 4. Synthesis of the sliding mode control law

via the reaching law approach:

u=(−B−1)[H−1

n+1Q sign(s)+H−1

n+1Ks+f(x)−xd].

(28)

Apply this control law for tracking control of Chen’schaotic system for the duration of simulation.

Step 5. Compute the performance index:

According to the (25), we can compute the perfor-mance index Jn+1;

If n = 0 then J∗ = Jn+1, H∗ = Hn+1, else

If Jn+1 ≤ J∗ then J∗ = Jn+1, H∗ = Hn+1 else

do nothing

Set n = n+ 1, then return to Step 1.Repeat the procedure above in finite times, un-

til we get the optima H∗ and J∗. Then the optimalsliding manifold can be acquired:

S(e) = {e|s(e, t) = H∗e(t) = 0}, s(e) ∈ Rm ,

(29)

where e = x − xd is the tracking error vector for adesired trajectory xd(t).

4. An Illustrative Example

Although some nonlinear control strategies forchaotic systems have been put forward [Mascolo &Grass, 1999; Yau et al., 2000], they are only fea-sible for single-input systems. In our simulations,a complex multi-input chaotic system, Chen’s sys-tem, is used to confirm the validity of the approachproposed in this paper.

The Chen’s chaotic system described by (22)has the attractor shown in Fig. 1(a). Unlike thefamiliar Lorenz system where if any one of thethree components x1, x2, x3 is controlled then theother states will follow, it has been verified that thischaotic system is not topologically equivalent to theLorenz attractor, and is known to be more complexdynamically. The goal here is to find a sliding modecontrol law u(t) ∈ Rm, to guide the system statex(t) to match a prespecified reference signal xd(t).The trajectory shown in Figs. 1(a) and 1(b) can beviewed as some kind of deterministic chaotic motiontrajectory before control is applied.

In our study, we specify the target referencexd(t) to be a closed orbit corresponding to a pe-riodic solution of the unforced Chen’s equation.Let the parameters of system (22) be a = 45,b = 1.5 and c = 28. Under these parameters,the system (22) generates a periodic solution [Ueta& Chen, 2000]. Starting from the initial conditionxd(0) = (−1.7570, −1.9648, 7.9743)T , the three-dimensional phase portrait and the time-domain re-sponse of reference xd are given in Figs. 2(a) and2(b), respectively. The objective for control is toguide the chaotic trajectory to settle on this deter-ministic orbit.

An optimal sliding surface is defined in the er-ror state space as (29), i.e. the desired switchingfunction is finally designed as

s(e) = H∗e = H∗(x− xd) . (30)

The parameters Q and K in the reaching laware fixed at Q = diag[0.7, 0.7, 0.7] and K =diag[18, 18, 18], which can be determined by bisec-tion search method. In our simulation, no a pri-

ori knowledge on the design is assumed and thesearch ranges for all elements of H are set to be thesame, that is, lowbound i = −10, upbound i = 10,i = 1, 2, . . . , 9. By the COA-based search proce-dure, the coefficients matrix of the optimal slidingsurface H∗ can be obtained. The result is:

H∗ =

−8.9851 −7.3033 7.2837

−7.2973 −7.2944 −6.7039

5.6752 1.5060 −9.4379

. (31)

Then the sliding mode control law is synthesizedas:

u=(−B−1)[H∗−1Q sign(s)+H∗−1Ks+f(x)−xd].

(32)

The initial condition used is x(0) = (−15, 5, 20)T .The control performance under the sliding mode

April 28, 2004 8:54 00990

1350 Z. Lu et al.

(a)

(b)

Fig. 1. (a) The deterministic chaotic attractor of Chen’s system, plotted in the x3 − x1 − x2 space. (b) The deterministicchaotic time series of Chen’s system.

April 28, 2004 8:54 00990

Tracking Control of Nonlinear Systems 1351

(a)

(b)

Fig. 2. (a) The desired reference orbit xd(t), plotted in the x3−x1−x2 space. (b) The deterministic time series of the desiredreference orbit xd(t).

April 28, 2004 8:54 00990

1352 Z. Lu et al.

Fig. 3. The controlled trajectory x(t) of Chen’s chaotic system to the reference orbit xd(t), plotted on x3 − x1 − x2 space.

Fig. 4. The time responses of the controlled system for tracking the reference orbit.

April 28, 2004 8:54 00990

Tracking Control of Nonlinear Systems 1353

Fig. 5. The control input signal u versus time.

Fig. 6. The time responses of error states.

April 28, 2004 8:54 00990

1354 Z. Lu et al.

controller (32) is visualized by Figs. 3 and 4, whichshow that the phase plane trajectory is steered tothe reference periodic orbit, while Fig. 5 is the con-trol input signals. The time responses of error statesare illustrated in Fig. 6, and this demonstrates thaterror states converge to zero very quickly.

All the simulation results have shown that theproposed sliding mode control law design approachis effective in guiding the chaotic trajectories of thechaotic Chen’s system to the intended orbit withdesired performance.

5. Conclusion

The main contribution of this paper is an efficientapproach to design the optimal sliding manifoldfor robust tracking control of general affine nonlin-ear systems with multi-inputs. Whereas the slidingmode control theory has been well developed forsingle-input systems that are in controller canoni-cal form, there does not exist an effective and sys-tematic sliding mode control strategy for generalaffine nonlinear multi-input systems. The key con-tribution here is the construction of a good slidingmanifold.

Towards this goal, we utilize successfully thechaotic dynamics in searching and constructingthe optimal sliding manifold for nonlinear sys-tems under study. Compared to other methods,the proposed scheme has better generality andlesser constraints than most existing procedures,which require that the nonlinear systems of in-terest are canonical-transformable or feedback-linearizable. Having selected the sliding surface, thesliding mode control law is synthesized via the exist-ing reaching law method, which has the capabilitiesof simultaneously ensuring the reaching condition,arranging the logic for the free-order switching, in-fluencing the dynamic quality of the system duringthe reaching phase, and providing the means forcontrolling the chattering level.

The newly proposed control scheme has beensuccessfully applied to Chen’s chaotic attractor,which has recently been shown to be topologi-cally more complicated than the celebrated Lorenzattractor, and therefore presents a more difficultchallenge for control.

References

Chen, G. & Ueta, T. [1999] “Yet another chaotic attrac-tor,” Int. J. Bifurcation and Chaos 9, 1465–1466.

Dorling, C. M. & Zinober, A. S. I. [1986] “Twoapproaches to hyperplane design in multivariablestructure control systems,” Int. J. Contr. 44,65–82.

Fu, L. C. & Liao, T. L. [1990] “Globally stabel robusttracking of nonlinear systems using variable struc-ture control and with an application to a roboticmanipulator,” IEEE Trans. Automat. Contr. 35,1345–1350.

Gao, W. B. & Hung, J. C. [1993] “Variable structurecontrol of nonlinear systems: A new approach,” IEEE

Trans. Indust. Electron. 40, 45–55Guo, S. M., Shieh, L. S., Chen, G. & Lin, C. F.

[2000] “Effective chaotic orbit tracker: A prediction-based digital redesign approach,” IEEE Trans. Cir-

cuits Syst. 47, 1557–1570.Hasegawa, M., Ikeguchi, T. & Aihara, K. [2002] “Solv-

ing large scale traveling salesman problems by chaoticneurodynamics,” Neural Networks 15, 271–283.

Holland, J. H. [1975] Adaptation in Natural and Artifi-

cial Systems (The University of Michigan Press, AnnArbor, MI).

Hung, J. Y., Gao, W. B. & Hung, J. C. [1993] “Vari-able structure control: A survey,” IEEE Trans. Indust.

Electron. 40, 2–22.Jiang, Z. P. [2002] “Advanced feedback control of the

chaotic Duffing equation,” IEEE Trans. Circuits Syst.

49, 244–249.Kaynak, O. [2001] “The fusion of computationally in-

telligent methodologies and sliding-mode control: Asurvey,” IEEE Trans. Indust. Electron. 48, 4–17.

Kirkpatrick, S., Gellatt, J. C. D. & Vecchi, M. P. [1983]“Optimization by simulated annealing,” Science 220,671–680.

Kwatny, H. G. & Kim, H. [1989] “Variable structure con-trol of partially linearizable dynamics,” Proc. Ameri-

can Control Conf., Pittsburgh, PA, pp. 1148–1153.Kwok, T. & Smith, K. A. [2000] “Experimental anal-

ysis of chaotic neural network models for combinato-rial optimization under a unifying framework,” Neural

Networks 13, 731–744.Li, B. & Jiang, W. S. [1998] “Optimizing complex func-

tions by chaos search,” Cybern. Syst. 29, 409–419.Li, D. & Slotine, J. E. [1987] “On sliding con-

trol for multi-input multi-output nonlinear systems,”Proc. American Control Conf., Minneapolis, MN,pp. 874–879.

Mascolo, S. & Grass, G. [1999] “Controlling chaoticdynamics using backstepping design with applicationto the Lorenz system and Chua’s circuit,” Int. J.

Bifurcation and Chaos 9, 1425–1434.Sira-Ramirez, H. [1989] “Nonlinear variable structure

systems in sliding mode: The general case,” IEEE

Trans. Automat. Contr. 34, 1186–1188.Su, W. C., Drakunov, S. V. & Ozguner, U. [1996]

“Constructing discontinuity surface for variable struc-

April 28, 2004 8:54 00990

Tracking Control of Nonlinear Systems 1355

ture systems: A Lyapunov approach,” Automatica 32,925–928.

Ueta, T & Chen, G. [2000] “Bifurcation analysis ofChen’s equation,” Int. J. Bifurcation and Chaos 10,1917–1931.

Utkin, V. I. [1977] “Variable structure systems withsliding modes,” IEEE Trans. Automat. Contr. 22,212–222.

Utkin, V. I. & Young, K. D. [1979] “Methods forconstructing discontinuity planes in multidimensionalvariable structure systems,” Automat. Remote Contr.

39, 1466–1470.Yau, H. T., Chen, C. K. & Chen, C. L. [2000] “Sliding

mode control of chaotic systems with uncertainties,”Int. J. Bifurcation and Chaos 10, 1139–1147.