TRACKING CONTROL OF NONLINEAR SYSTEMS: A SLIDING MODE DESIGN VIA CHAOTIC OPTIMIZATION

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

TRACKING CONTROL OF NONLINEAR SYSTEMS: A SLIDING MODE DESIGN VIA CHAOTIC OPTIMIZATION* ZHAO LU and LEANG-SAN SHIEH Department of Electrical and Computer Engineering, University of Houston, Houston, TX 77204-4005, USA JAGDISH CHANDRA School 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 challenges. In this paper, we present a novel scheme for tracking control of a class of affine nonlinear systems with multi-inputs. This effective procedure is based on a new sliding mode design for tracking control of such nonlinear systems. The construction of an optimal sliding mode is a difficult problem and no systematic and efficient method is currently available. Here, we develop an innovative approach that utilizes a chaotic optimizing algorithm, which is then successfully applied to obtain the optimal sliding manifold. The existing efficient reaching law approach is then utilized to synthesize the sliding mode control law. The sliding mode control scheme proposed 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 difficult to construct mathematical models that describe them accurately. Invariably, one encounters significant deviations between the constructed models and the real systems. These uncertainties may be due to unknown or partially known parametric values of the controlled system. Further, the changing environments and unspecified disturbances such as measurement noises may also contribute to the disparities with the real system. Therefore, the robust controller design that effectively deals with such uncertainties of a system becomes an important research subject. ∗

The customary approach for designing control law for nonlinear systems is linearization, such as local linearization and feedback linearization, via which the well-developed linear system theory can be utilized for controlling the nonlinear system [Guo et al., 2000; Jiang, 2002]. However, there are many well-known limitations in applying linearization techniques. For local linearization methods, they are strictly local, potentially unstable and usually not powerful enough to handle broad classes of complex and/or uncertain dynamics. In the past two decades, major progress has been made in the development of a geometric theory of nonlinear feedback. Feedback linearization techniques are based on Lie algebras, and coordinate and control

Supported by the US Army Research Office under Grant DAAD 19-02-1-0321. 1343

1344

Z. Lu et al.

transformations. Systematic design procedures are built on the solid foundation these geometric results form. However, this method has two drawbacks: (1) perfect linearization cannot be obtained because the nonlinear feedback depends on the nonlinear model of the system that necessarily contains uncertainty; (2) the resulting control law is somewhat complex and requires too much control effort because it pursues linearization of the system rather than stabilization or tracking. To deal with the uncertainties in nonlinear systems, three main approaches have been employed commonly: (1) adaptive control; (2) Lyapunovbased control; and (3) sliding mode control (also called variable structure control). The adaptive control utilizes the linear-in-parameter assumption to formulate error equations which relate measurable signals to parameter errors. Then, a parameter update law is formed from the error equations. The objective of either stabilization or tracking is thus achieved during the adaptation process. On the other hand, the Lyapunov-based approach relies on an explicit construction of a energy or Lyapunov function, by which a state-feedback control is synthesized using the bounds on the uncertainties. Finally, the sliding mode control (SMC), which is closely related to the Lyapunov-based approach, exploits the variable structure concept to achieve the goal. It first defines the sliding surface in the error state space and devises a switching statefeedback control law. The high-speed switching control law forces the error state to slide along the surface until it converges and then the tracking is attained. In this paper, we address the issues of designing the sliding mode tracking controller for general affine nonlinear systems with multi-inputs. Most of the existing schemes about sliding mode control for multi-input nonlinear systems concentrate on the equations that can be transformed to a canonical form or are feedback-linearizable. For general affine nonlinear systems, there is lack of effective and systematic approaches for tracking specific trajectories of nonlinear systems, especially chaotic ones. As it is well known, the crux of nonlinear sliding mode control is the construction of a good sliding mode. Hence, in this paper, we utilize the emerging chaotic optimization algorithm (COA) to successfully search and construct the optimal sliding mode for tracking control of the nonlinear systems. In this procedure, the existing effective reaching law

approach is adopted to synthesize the sliding mode control law.

2. Sliding Mode Control System Design Sliding mode control (SMC) evolved from the pioneering work in Russia in the early 1960s [Utkin, 1977]. Yet, SMC did not receive wide acceptance among engineering professionals prior to 1970’s, which is probably due to the lack of an effective design procedure and the presence of considerable chattering in the SMC systems. The situation, however, changed with resultant rise of interest in SMC in the 1970’s as people started recognizing the robustness and invariance of SMC. It is well known that the most distinguishing feature of SMC is its ability to result in very robust control systems. In many cases invariant control system results. Loosely speaking, the term “invariant” means that the system is completely insensitive to parametric uncertainty and external disturbances. The primary characteristic of a variable structure system (VSS) is that the feedback signal is discontinuous, switching on one or more manifolds in state space. When the state crosses each discontinuity surface, the structure of the feedback system is altered. Under certain circumstances, all motions in the neighborhood of the manifold are directed towards the manifold, and thus a sliding motion on a predefined subspace of the state-space is established in which the system state repeatedly crosses the switching surface. This mode has useful invariance properties in the face of uncertainties in the plant model, and therefore is a candidate for robust tracking control of uncertain nonlinear systems.

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 ∈ R m the input vector, n the order of the system, m the number of inputs, m ≤ n, f (x) and g(x) are nonlinear functions of appropriate dimensions. It is known that, in general, the transient dynamics of a SMC system consists of two modes, i.e. a “reaching mode” followed by a “sliding mode”. Therefore the design of SMC usually involves two fundamental steps. The first step is to design an appropriate m-dimensional

Tracking Control of Nonlinear Systems

switching function s(x) such that the system exhibits the desired behavior in the sliding mode. The second step is to determine a control law that guarantees the reaching and sliding conditions. The desired sliding mode dynamics is usually a fast and stable error-free response void of overshoot. For the reaching mode, the desired response is to reach the switching manifold, described by s(x) = Hx = 0, rank(H) = m H ∈ Rm×n , s(x) ∈ Rm ,

(2)

i.e. h11  h21  s(x) =  .  .. hm1 

h12 h22 .. . hm2

··· ··· ··· ···

  h1n x1  x2  h2n     .  = 0. ..  .   ..  hmn xn

(3)

For an m-input system, there are m switch functions and 2m − 1 sliding manifolds of different dimensions. The first m of them are designed as Si = {x|si = hi x = 0} ,

i = 1 to m ,

(4)

where hi = [hi1 hi2 · · · hin ] is a row vector. Si may be called basic sliding manifolds since each of them 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 sliding modes. SE may be called the eventual sliding manifold since it is the manifold that all state trajectories must reach eventually. Obviously, the basic sliding manifold is a (n − 1) dimension dynamics, and the eventual sliding manifold is (n − m) dimension dynamics. There are several possible switching schemes for steering the state to enter various sliding manifolds. Three of them are described briefly here. (1) Fixed-Order Sliding Mode Switching: In this scheme, sliding modes take place in a preassigned order while the state is traversing the state space. For example x(0) → S1 → (S1 ∩ S2 ) → (S1 ∩ S2 ∩ S3 ) → · · · → SE .

(6)

(2) Eventual Sliding Mode Switching: Here the state is driven to the eventual sliding manifold SE , only on which the sliding mode control

1345

takes place. This scheme is simpler in SMC implementation and gives a faster transient than the fixed order schemes. (3) Free-Order Sliding Mode Switching: In this scheme, a sliding mode is switched in whenever the state reaches any switching manifold S i . It is a “first-reach-first-switch” scheme, thus the order of sliding modes is not fixed. For this case, the system state moves into the sliding mode sooner than in previous cases. Therefore, the overall response is more robust and the transient response can be faster. Also, the required control effort is usually smaller in magnitude when compared to the fixed-order scheme, so saturation is less likely to occur. Finally, the associated SMC is easy to be obtained, especially by using the reaching law method.

2.2. Design of sliding manifold The characteristic feature of a variable structure system (VSS) is that the sliding mode occurs on a prescribed switching surface. The surface is the intersection of a set of discontinuity surfaces in the state space of a multiple-state system, where each control input switches between two functions. These discontinuity surfaces are selected so that, while in the sliding mode, the system performance satisfies the design objectives such as stability, performance index minimization, chattering reduction, etc. Mathematically, the sliding manifold is not merely a hypersurface in the original state-space of the plant, but a linear operator that can be represented as a linear, time-invariant dynamic system, acting on the states. During the past decade, the problem of constructing the sliding manifolds in SMC has been studied mainly under the framework established by Utkin and Young [1979]. A coordinate transformation is found to partition the states into a special form referred to as the canonical form. In the new coordinates, design of the switching surface is carried out in a reduced-order system separated from the control variables. This idea has been extensively used in the literature of variable structure systems theory, and has been exploited from different perspectives [Utkin & Young, 1979; Dorling & Zinober, 1986; Kwatny & Kim, 1989; Sira-Ramirez, 1989]. For linear systems, the sliding surface can be designed by pole placement, quadratic optimization, or a geometric approach and eigenstructure assignment technique.

1346

Z. Lu et al.

However, for general affine nonlinear multiinput systems, the design of sliding surface is much more complicated. In general, the nonlinear system (1) needs to be converted into the following regular canonical form   via a nonsingular coordinate transx formation x1 = φ(x): 2

x˙ 1 = f1 (x1 , x2 ) ,

(7)

x˙ 2 = 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 a nonlinear feedback control law x2 = s(x1 ) for the subsystem (7) when x2 is viewed as virtual control. However, the feedback control law x 2 = s(x1 ) for the nonlinear system (7) is usually hard to obtain, and not every system can be easily transformed to a regular canonical form [Gao & Hung, 1993]. The other approach is to use the feedback linearization technique to secure a linear equivalence. Suppose that the nonlinear system (1) is feedback-linearizable. We apply a diffeomorphic transformation z = ϕ(x) so that the new state variable z and control variable ν satisfy the linear differential equation z˙ = Az + Bν , u = ψ(x, ν) ,

(9)

2.3. Design of sliding mode control The condition under which the state will move towards and reach a sliding surface is called a reaching condition. The sliding mode control law is synthesized from the reaching condition. For specifying the reaching condition, three approaches have been proposed (see [Hung et al., 1993] and the references therein). (a) The Direct Switching Function Approach: The earliest reaching condition proposed was si s˙ i < 0,

s(e) ∈ R m , (11)

i = 1, 2, . . . , m .

(12)

This reaching condition is global but does not guarantee a finite reaching time. Moreover, this reaching condition is very difficult to use for the multi-input SMC. (b) The Lyapunov Function Approach: By choosing an appropriate Lyapunov function V (x, t) = sT s ,

(13)

a global reaching condition is given by V˙ (x, t) < 0

(10)

where z ∈ Rn is the transformed coordinates, ν ∈ Rm , A and B are constant matrices in Brunovsky canonical form. With the linearized dynamic equation (9), a sliding surface can easily be found. This idea has been exploited by many researchers [Fu & Liao, 1990; Li & Slotine, 1987], under the assumption that the nonlinear system (1) is feedbacklinearizable, which is unfortunately satisfied only under certain structural conditions. A Lyapunov approach for constructing the sliding manifold was proposed [Su et al., 1996], but Lyapunov functions usually are difficult to construct for general affine nonlinear systems. In this paper, we utilize a chaotic optimization algorithm to search for the optimal sliding manifold, without requiring any transformation of the system. As a result, this approach is more general and is simpler than the other methods. For tracking control of the n-dimensional nonlinear system (1), the sliding surface S(e) is usually defined as S(e) = {e|s(e, t) = He(t) = 0},

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

when s 6= 0 .

(14)

The finite reaching time is guaranteed by modifying the formula above to V˙ (x, t) < −ε when s 6= 0, where ε is positive . (15) Clearly, this approach would lead to the eventual sliding mode switching scheme. (c) The Reaching Law Approach: The reaching law is a set of differential equations which specifies the dynamics of sliding variable s(x). The differential equation of an asymptotically stable s(x) is itself a reaching condition. In addition, by the choice of the parameters in the differential equation, the dynamic quality of the SMC system in the reaching mode can be controlled. A practical general form of the reaching law is s˙ = −Q sign(s) − Kd(s) , where Q = diag[q1 , q2 , . . . , qm ], qi > 0 , sign(s) = [sign(s1 ), . . . , sign(sm )]T , K = diag[k1 , k2 , . . . , km ], ki > 0 ,

(16)

Tracking Control of Nonlinear Systems

d(s) = [d1 (s1 ), . . . , dm (sm )]T , si di (si ) > 0, when si 6= 0, di (0) = 0 . Usually one of the following three forms of the reaching law is adopted: (1) Constant rate reaching: s˙ = −Q sign(s) .

(17)

This law forces the sliding variable s(x) to reach the switching manifold S at a constant rate |s˙ i | = −qi . The merit of this reaching law is its simplicity. But, if qi is too small, the reaching time will be too long. On the other hand, a too 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 switching manifolds faster when s is large. (3) Power rate reaching: s˙ i = −ki |si |α sign(si ), i = 1, 2, . . . , m .

(19)

This reaching law increases the reaching speed when the state is far away from the switching manifold, but reduces the rate when the state is near the manifold. The result is a faster reaching and a lower chattering reaching mode. In this paper, the constant plus proportional rate reaching law is adopted to synthesize the sliding mode control law for tracking control of the nonlinear system (1). The time derivative of s(x) along the reaching mode trajectory needs to be computed to design the control law. From (1) and (18) we have ∂s ∂s f (x) + g(x)u = −Q sign(s) − Ks . (20) ∂x ∂x Noting that the matrix (∂s/∂x)g(x) is nonsingular, and solving Eq. (20) for the control law yields −1    ∂s ∂s g(x) f (x) + Q sign(s) + Ks . u=− ∂x ∂x (21) The reaching law approach not only establishes the reaching condition but also specifies the dynamic characteristics of the system during the reaching phase. Additional merits of this approach include simplification of the solution for SMC and providing a means for the reduction of chattering, i.e. the chattering can be reduced by adjusting the matrices Q and K. Also, the control law obtained

1347

via a reaching law automatically leads to the freeorder switching scheme. From the practical point of view, 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 globally optimizing algorithms during the past three decades. An important and effective approach to global optimum searching problem is the genetic algorithm (GA) [Holland, 1975], in which the solution space is encoded as gene-like strings, and then the procedure of searching the global optimum is carried out by reproduction, mutation, crossover and computing the fitness value. In comparison to other local search algorithms, GA guarantees a higher chance of reaching a global optimum by starting with multiple random search points, and by considering several candidate solutions simultaneously. The unique crossover operator in GA offers the possibility of exchanging attributes among potential solutions. The deficiency of classical GA is mainly attributed to the fact that the diversity of a population relies on mutation only once the population has been initialized. Since mutation must be kept at a low rate (otherwise the offspring do not inherit the characteristics of their parents, leading to a random search), it does not diversify the population effectively once the population has been converged. Moreover, in classical GA, the population size must be large; otherwise GA does not provide a sufficient sample size, causing premature convergence. However, large population size requires more time to converge the population. The rate of convergence is, generally, unacceptably slow. Another means to global optimum searching problem is the simulated annealing algorithm (SA) [Kirkpatrick et al., 1983], in which a stochastic mechanism is introduced to avoid being trapped in a local optimum. Boltzmann machine is the neural network implementation of SA algorithm. The disadvantage of SA is that relatively longer computation time is needed and it can be painfully slow as the problem size increases, which implies that it cannot be used efficiently in large-scale optimizing problem. Recently, a new and quite promising approach for global optimizing has come along, that is, applying chaotic dynamics. This is introduced in order to overcome the local minimum problem and

1348

Z. Lu et al.

to perform efficient search based on the ergodicity of chaos. Some successful applications of chaotic neural network for complicated global optimizing problems have been reported, to solve such difficult problems as the large-scale traveling salesman problem (TSP) [Hasegawa et al., 2002; Kwok & Smith, 2000]. A key characteristic of chaotic dynamics is its self-similarity, which indicates that attractors of chaotic dynamical systems usually have fractal structures. The chaotic dynamics can be used to search solutions only along such a fractal attractor typically with zero Lebesgue measure in a bounded region of the state space. Therefore, the chaotic search is expected to be more efficient than stochastic search schemes, if good solutions are embedded in a searching region. However, the chaotic neurodynamics approach is based, usually, on the architecture of the Hopfield neural network, which leads to the same defect as the other optimizing schemes using Hopfield neural network. That is, the objective functions to be optimized are restricted only to continuously differentiable quadratic functions. Fortunately, an emerging strategy called chaotic optimization algorithm (COA) has been proposed [Li & Jiang, 1998] and successfully applied to some typical nonlinear optimizing problems. A number of simulations have demonstrated that COA is more efficient than GA and SA, and there are no limitations on the type of objective functions to be optimized. The philosophy of chaotic optimizing algorithm is simple, which is based on two main steps: first, a transform from chaotic space to solution space is carried out, and then the searching for global optimum is executed according to the chaotic dynamics itself rather than guiding it at random. The details about the COA will be shown in the next section, by designing the optimal sliding manifold.

 x˙ (t) = a(x2 (t) − x1 (t))   1 x˙ 2 (t) = (c − a)x1 (t) − x1 (t)x3 (t) + cx2 (t)   x˙ 3 (t) = x1 (t)x2 (t) − bx3 (t)

where a = 35, b = 3 and c = 28. If we represent the system by a simple state-space equation: x(t) ˙ = f (x(t)),

x(t) ∈ R 3 .

To illustrate the chaotic optimization algorithm and the proposed sliding manifold design procedure, it is especially convenient to use examples. The wellknown 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 as follows:

(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 x d (t) to be defined 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 x d (t), a sliding surface should be defined ahead. To utilize the COA to search the optimal sliding surface, we need to define the performance index as J(H) =

N X

{kei k + wk∆ei k} ,

(25)

i=1

where N is the duration of the simulation for evaluating the design, i is the time index in simulation, ei is the error vector at simulation step i and ∆e i is the change in error vector, w (being fixed at 1 in this paper) is a bias weighting between e i and ∆ei . The ∆ei term can be distinctively weighted to further suppress oscillations. The optimal sliding surface coefficients H ∈ R 3×3 which minimize the performance index will be searched via COA. Note that there are nine elements in the sliding surface coefficients matrix H ∈ R 3×3 , therefore nine variables are involved in this optimizing problem. The chaotic equation of COA can be selected as Logistic mapping, i.e. νn+1 = µνn (1 − νn ),

3.2. Design of optimal sliding manifold using COA

(22)

µ = 4.

(26)

First, we select nine initial chaotic variables ν1,0 , 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 variables. Then, we set up the ranges for searching the elements in the sliding surface coefficients matrix, namely, the upper-bound values upbound i , i = 1, 2, . . . , 9 and the lower-bound values lowbound i , i = 1, 2, . . . , 9. Assume H ∗ and J ∗ to be the optima. The search procedure by COA starts from n = 0:

Tracking Control of Nonlinear Systems

Step 1. Chaotify the variables: Substitute ν1,n , ν2,n , . . . , ν9,n into (26), then we can get 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 space to 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 of the sliding surface, and then define the sliding mode as: S(e) = {e|s(e) = Hn+1 · e(t) = 0}. Step 4. Synthesis of the sliding mode control law via the reaching law approach: −1 −1 u = (−B −1 )[Hn+1 Q sign(s)+Hn+1 Ks+f (x)− x˙ d ]. (28)

Apply this control law for tracking control of Chen’s chaotic system for the duration of simulation. Step 5. Compute the performance index: According to the (25), we can compute the performance 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, until we get the optima H ∗ and J ∗ . Then the optimal sliding 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 a desired trajectory xd (t).

4. An Illustrative Example Although some nonlinear control strategies for chaotic systems have been put forward [Mascolo & Grass, 1999; Yau et al., 2000], they are only feasible for single-input systems. In our simulations, a complex multi-input chaotic system, Chen’s system, is used to confirm the validity of the approach proposed in this paper.

1349

The Chen’s chaotic system described by (22) has the attractor shown in Fig. 1(a). Unlike the familiar Lorenz system where if any one of the three components x1 , x2 , x3 is controlled then the other states will follow, it has been verified that this chaotic system is not topologically equivalent to the Lorenz attractor, and is known to be more complex dynamically. The goal here is to find a sliding mode control law u(t) ∈ Rm , to guide the system state x(t) to match a prespecified reference signal x d (t). The trajectory shown in Figs. 1(a) and 1(b) can be viewed as some kind of deterministic chaotic motion trajectory before control is applied. In our study, we specify the target reference xd (t) to be a closed orbit corresponding to a periodic 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 condition xd (0) = (−1.7570, −1.9648, 7.9743)T , the threedimensional phase portrait and the time-domain response of reference xd are given in Figs. 2(a) and 2(b), respectively. The objective for control is to guide the chaotic trajectory to settle on this deterministic orbit. An optimal sliding surface is defined in the error state space as (29), i.e. the desired switching function is finally designed as s(e) = H ∗ e = H ∗ (x − xd ) .

(30)

The parameters Q and K in the reaching law are fixed at Q = diag[0.7, 0.7, 0.7] and K = diag[18, 18, 18], which can be determined by bisection search method. In our simulation, no a priori knowledge on the design is assumed and the search ranges for all elements of H are set to be the same, that is, lowbound i = −10, upbound i = 10, i = 1, 2, . . . , 9. By the COA-based search procedure, the coefficients matrix of the optimal sliding surface H ∗ can be obtained. The result is:   −8.9851 −7.3033 7.2837 H ∗ =  −7.2973 −7.2944 −6.7039  . (31) 5.6752 1.5060 −9.4379 Then the sliding mode control law is synthesized as: u = (−B −1 )[H ∗−1 Q sign(s)+H ∗−1 Ks+f (x)− x˙ d ]. (32) The initial condition used is x(0) = (−15, 5, 20) T . The control performance under the sliding mode

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 deterministic chaotic time series of Chen’s system.

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 desired reference orbit xd (t).

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.

Tracking Control of Nonlinear Systems

Fig. 5.

Fig. 6.

The control input signal u versus time.

The time responses of error states.

1353

1354

Z. Lu et al.

controller (32) is visualized by Figs. 3 and 4, which show that the phase plane trajectory is steered to the reference periodic orbit, while Fig. 5 is the control input signals. The time responses of error states are illustrated in Fig. 6, and this demonstrates that error states converge to zero very quickly. All the simulation results have shown that the proposed sliding mode control law design approach is effective in guiding the chaotic trajectories of the chaotic Chen’s system to the intended orbit with desired performance.

5. Conclusion The main contribution of this paper is an efficient approach to design the optimal sliding manifold for robust tracking control of general affine nonlinear systems with multi-inputs. Whereas the sliding mode control theory has been well developed for single-input systems that are in controller canonical form, there does not exist an effective and systematic sliding mode control strategy for general affine nonlinear multi-input systems. The key contribution here is the construction of a good sliding manifold. Towards this goal, we utilize successfully the chaotic dynamics in searching and constructing the optimal sliding manifold for nonlinear systems under study. Compared to other methods, the proposed scheme has better generality and lesser constraints than most existing procedures, which require that the nonlinear systems of interest are canonical-transformable or feedbacklinearizable. Having selected the sliding surface, the sliding mode control law is synthesized via the existing reaching law method, which has the capabilities of simultaneously ensuring the reaching condition, arranging the logic for the free-order switching, influencing the dynamic quality of the system during the reaching phase, and providing the means for controlling the chattering level. The newly proposed control scheme has been successfully applied to Chen’s chaotic attractor, which has recently been shown to be topologically more complicated than the celebrated Lorenz attractor, and therefore presents a more difficult challenge for control.

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

Dorling, C. M. & Zinober, A. S. I. [1986] “Two approaches to hyperplane design in multivariable structure control systems,” Int. J. Contr. 44, 65–82. Fu, L. C. & Liao, T. L. [1990] “Globally stabel robust tracking of nonlinear systems using variable structure control and with an application to a robotic manipulator,” IEEE Trans. Automat. Contr. 35, 1345–1350. Gao, W. B. & Hung, J. C. [1993] “Variable structure control of nonlinear systems: A new approach,” IEEE Trans. Indust. Electron. 40, 45–55 Guo, S. M., Shieh, L. S., Chen, G. & Lin, C. F. [2000] “Effective chaotic orbit tracker: A predictionbased digital redesign approach,” IEEE Trans. Circuits Syst. 47, 1557–1570. Hasegawa, M., Ikeguchi, T. & Aihara, K. [2002] “Solving large scale traveling salesman problems by chaotic neurodynamics,” Neural Networks 15, 271–283. Holland, J. H. [1975] Adaptation in Natural and Artificial Systems (The University of Michigan Press, Ann Arbor, MI). Hung, J. Y., Gao, W. B. & Hung, J. C. [1993] “Variable 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 intelligent methodologies and sliding-mode control: A survey,” 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 control of partially linearizable dynamics,” Proc. American Control Conf., Pittsburgh, PA, pp. 1148–1153. Kwok, T. & Smith, K. A. [2000] “Experimental analysis of chaotic neural network models for combinatorial optimization under a unifying framework,” Neural Networks 13, 731–744. Li, B. & Jiang, W. S. [1998] “Optimizing complex functions by chaos search,” Cybern. Syst. 29, 409–419. Li, D. & Slotine, J. E. [1987] “On sliding control for multi-input multi-output nonlinear systems,” Proc. American Control Conf., Minneapolis, MN, pp. 874–879. Mascolo, S. & Grass, G. [1999] “Controlling chaotic dynamics using backstepping design with application to 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-

Tracking Control of Nonlinear Systems

ture systems: A Lyapunov approach,” Automatica 32, 925–928. Ueta, T & Chen, G. [2000] “Bifurcation analysis of Chen’s equation,” Int. J. Bifurcation and Chaos 10, 1917–1931. Utkin, V. I. [1977] “Variable structure systems with sliding modes,” IEEE Trans. Automat. Contr. 22, 212–222.

1355

Utkin, V. I. & Young, K. D. [1979] “Methods for constructing discontinuity planes in multidimensional variable 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.