composite sliding mode control of chaotic systems

International Journal of Bifurcation and Chaos, Vol. 13, No. 4 (2003) 863–878 c World Scientific Publishing Company

COMPOSITE SLIDING MODE CONTROL OF CHAOTIC SYSTEMS WITH UNCERTAINTIES CHUN-CHIEH WANG∗ and JUHNG-PERNG SU† Department of Electrical Engineering, ∗Graduate School of Engineering Science & Technology (Doctoral Program), National Yunlin University of Science & Technology, 123, Section 3, University Road, Douliu, Yunlin 640, Taiwan, R.O.C. † [email protected] Received August 21, 2001; Revised March 22, 2002 This paper presents a new approach to the design of a composite sliding mode control for a class of chaotic systems with uncertainties. A significant feature of this control scheme is the incorporation of a new complementary sliding variable to the conventional sliding variable in order that a high-performance controller can be obtained. It has been shown that the guaranteed steady-state error bounds are reduced by half, as compared with the conventional sliding control. Moreover, the dynamic responses during the reaching phase are also significantly improved. We used a controlled uncertain Lorenz system and a controlled uncertain Chua’s circuit as illustrative examples to demonstrate the effectiveness of the design. Keywords: Composite sliding mode control; complementary sliding variable; chaotic systems with uncertainties; Lorenz system; Chua’s circuit.

1. Introduction

able system parameter. Through properly selecting Hamilton parameter, Vincent and Yu [1991] matched up a variational principle to design a bangbang controller to tame chaotic systems, showing that it is possible to drive a chaotic system to one of the unstable equilibrium points. Hartley and Mossayebi [1992] presented a classical analysis of the control of Lorenz chaos, based on perturbation linearization. Recently, Fuh and Tung [1995] proposed a differential geometric approach to controlling Lorenz chaos, which is capable of controlling a chaotic system either to an equilibrium state or to any desired periodic orbit. Liaw and Tun [1996] improved the differential geometric method to control a noisy chaotic system, and showed that the overall system was able to reject noise. In publications regarding chaotic system control, the design of controller is often based upon the

Chaotic phenomena are found and studied worldwide in many fields. Due to irregularity and complexity in nonlinear dynamic systems, these should be removed in order to improve system performance to avoid fatigue of mechanical systems in many practical applications, and enable the system to be predictable in accordance with the desired objectives. The issue of controlling or ordering of chaos therefore draws increasing attention within the nonlinear dynamics research area. In 1990, Ott, Grebogi and Yorke [Ott et al., 1990] developed OGY scheme to control the chaotic system, which makes it possible to convert chaotic attractors to any one of a large number of possible attracting time-periodic motions by creating only small time-dependent perturbations of an avail†

Author for correspondence. 863

864 C.-C. Wang & J.-P. Su

assumption that the chaotic model is well known. However, in real physical systems or experimental situations, unmodeled dynamics and parameter variations of a mechanical system do exist. These uncertainties may cause chaotic perturbations to originally regular behavior, or induce additional chaos in originally chaotic but known behavior, generating unknown chaotic motion. Much work concerning chaotic control of well-known models has been done at present, and, much attention has been given to controlling nonlinear chaotic systems, which contain uncertainties. Chen and Dong [1997] included parameter uncertainties in their discussion of an adaptive controller for handling a chaotic Duffing oscillator. In 1998, Qin et al. [1998] used a neural-network based adaptive control for the uncertain Duffing–Holmes systems. Zhang et al. [1998] employed adaptive control to adjust a chaotic Duffing oscillator with additional uncertainties. Ge and Wang [1999, 2000] used adaptive backstepping with tuning functions method for the control of uncertain Chua’s circuits with all the key parameters unknown. Chen et al. [1999, 2000] improved fuzzy modeling, prediction and adaptive control to control the chaotic Duffing oscillator with uncertainties. In 1996, Yu [1996] introduced a sliding mode control strategy that utilized the switching-in of a two-value Lorenz system parameter. Such control strategy generates a sliding region such that, once the system state enters the region, the state will be attracted to an equilibrium point. But the effect of system uncertainties was not dealt with, neither was addressed the chattering caused by the control. Yau et al. [2000] proposed a sliding mode hyperplane design method to track target orbits for a class of uncertain chaotic systems. Tian and Yu [2000] developed an adaptive chaos control method based on the invariant manifold theory for stabilizing chaotic systems with unknown parameters. In this paper, we will present a new robust control scheme using sliding mode approach, for chaotic systems with uncertainties. Sliding mode control is known to be very robust against parameter variations and external disturbances and has been widely accepted as an efficient method for tracking control of uncertain nonlinear systems. It has been shown to be able to achieve “perfect” performance in principle in the presence of parameter uncertainties, bounded external disturbances, etc [Slotine & Li, 1991]. However, in order to account for the presence of parameter

uncertainties and bounded disturbances, a discontinuous switching function is inevitably incorporated into the control law to achieve the so-called sliding condition [Utkin, 1997]. Due to imperfect switching in practice it will raise the issue of chattering, which is usually undesirable. To suppress chattering, a continuous approximation of the discontinuous sliding control is usually employed in the literature. Though, the chattering can be made negligible if the width of the boundary layer is chosen large enough, the guaranteed tracking precision will deteriorate if the available control bandwidth is limited. To reach a better compromise between small chattering and good tracking precision in the presence of parameter uncertainties, various compensation strategies have been proposed. For example, integral sliding control [Lee et al., 1992; Chern & Wu, 1993; Baik et al., 1996], sliding control with time-varying boundary layers [Slotine & Li, 1991] etc., were presented. Alternatively, applying the so-called reaching law approach, Gao et al. [1993] proposed sliding controllers such that the trajectories are forced to approach the sliding surface faster when they are far away from the sliding surface. This approach seems to be an efficient method capable of increasing the approaching speed to the sliding surface; however, the behavior of the system dynamics, governed by the transformed first-order equation, can only be predicted through the measurement of the generalized error; hence the transient response during the reaching phase may not be remarkably improved. In this paper, we will define a new sliding variable as a complement to the conventional sliding variable to form a useful error transformation by which an nth order problem can be transformed into an equivalent first order problem such that an efficient continuous sliding control can be devised to achieve a better performance of the system. We will show that this control law can significantly improve error transient responses during the reaching phase as well as steady-state tracking precision while in the boundary layer. The effectiveness of the newly developed control scheme will be demonstrated through the control of an uncertain Lorenz chaotic system and a type of Chua’s circuit [Chua et al., 1986; Bartissol & Chua, 1988].

2. Problem Statement In this section, we will present the sliding control law for the following cascade of nonlinear systems

Composite Sliding Mode Control of Chaotic Systems with Uncertainties 865

with bounded inputs. Evidently, many electricmechanical plants are found to belong to this class: (n)

(n−1)

y = fm (t, y, y, ˙ ..., y ) (n−1)

+ gm (t, y, y, ˙ . . . , y )Fm (ζ) + dm (t) (p)

(n−1)

(1)

The object is to design a control law such that the output y will approximately track a given signal yd , i.e. limt→∞ e(t) = 0, where

(p−1)

˙ ..., ζ ) ζ = fa (t, y, y, ˙ . . . , y , ζ, ζ, (n−1)

where y is the output and u is the input; d m and da are disturbances; fm , gm : R × Rn → R, Fm : R → R and fa , ga : R × Rn × Rp → R are smooth functions. We assume that the driving force (torque), Fm (ζ), exerted on the mechanical part (1) is not completely known but subjected to a sectorbound condition: α1 ζ 2 ≤ ζFm (ζ) ≤ α2 ζ 2 ,

(3)

we will express Fm (ζ) as Fm (ζ) = ρζ + ∆Fm (ζ)

(4)

|∆Fm (ζ)| ≤ σ|ζ|

(5)

where σ = (α2 − α1 )/2, and 2



δ

if if

(7)

For any auxiliary variable φ, define the auxiliary error z as z(t) = ζ(t) − φ(t) .

(8)

Then, the error equations can be obtained as follows, (n) d

(n)

e = fm (t, x) + gm (t, x)Fm (ζ) + dm − y

(9)

(p)

(p)

z = fa (t, x, z) + ga (t, x, z)u + da − φ .

(10)

3. Composite Sliding Control

with

ρ=

e(t) = y(t) − yd (t) .

(p−1)

˙ . . . , ζ )u + da (t) + ga (t, y, y, ˙ . . . , y , ζ, ζ, (2)

α +α 2  1

min ≤ g (t, x) ≤ β max , 0 < β min ≤ 2. 0 < βm m m a min , β max , and ga (t, x, ζ) ≤ βamax , ∀ t, where βm m βamin , βamax are given constants, time-varying or state-dependent functions. 3. |dm (t)| ≤ Dm , |da (t)| ≤ Da , ∀ t, where Dm , Da are given constants or time functions.

For any λm , λa > 0, define the following transformations n d + λm ξe (t) (11) sm (t) = dt

d + λm dt

sa (t) =

d + λa dt

p

sz (t) =

d + λa dt

p−1

0 < α1 < α2 or α1 < α2 < 0 α1 < 0 < α2 (6)

where δ is any positive number such that δ > σ. Let the functions fm and fa in (1), (2) be expressible in the following forms: fm (t, x) = fˆm (t, x) + ∆fm (t, x), fa (t, x, ζ) = fâ (t, x, ζ) + ∆fa (t, x, ζ), respectively, where x = [x1 , x2 , . . . , xn ]T = [y, y, ˙ ..., ˙ . . . , ζ (p−1) ]T ; y (n−1) ]T , ζ = [ζ1 , ζ2 , . . . , ζp ]T = [ζ, ζ, fˆm , fâ are nominal parts and ∆fm , ∆fa are uncertain parts. The following assumptions about the uncertainties of the system (1) and (2) were made. Assumption 1

1. |∆fm (t, x)| ≤ M (t, x), |∆fa (t, x, ζ)| ≤ N (t, x, ζ), where M , N are known functions.

n−1

se (t) =

d − λm ξe (t) dt

ξz (t)

(12)

(13)

d − λa ξz (t) dt

(14)

where ξe (t) = t e(τ )dτ , ξz (t) = t z(τ )dτ . The following important relationships can be easily established R

R

s˙ e + λm (se + sm ) = s˙ m ,

(15)

s˙ z + λa (sz + sa ) = s˙ a .

(16)

Differentiation of sm and of sa with respect to t and substituting the expressions (4) for F m (ζ) yield s˙ m + λm sm = fˆm + w ˆm + ∆fm + gm ρζ + gm ∆Fm + gm dm

(17)


s˙ a + λa sa = fâ + w â + ∆fa + ga u + ga da ,

(18)

where w ˆm =

n−1 X

n+1

p−1 X

p+1

k=0

w â =

k=0

k+1

k+1

!

(n−k) (k) e λm

!

(k) z λ(p−k) a

(n) yd

+λn+1 m ξe −

By Assumption 1 and Lemma 1, we have (19)

(p)

+λp+1 a ξz − φ . (20)

To proceed with the design of sliding control, we first restrict our attention to the mechanical subsystem (1), where the variable ζ can be regarded as its input. Let Gm := q

1 min β max βm m

.

(21)

−1 ≤ g G −1 ≤ One has 0 < βm βm and 0 < βm m m ≤ q max /β min . Given (gm Gm )−1 ≤ βm , where βm = βm m µm > 0 and k1 := σ/(|ρ|)(0 < k1 < 1), the statefeedback control ζ = φ(x) was designed to consist of the following two terms

φ(x) =

Gm [ˆ u m + vm ] , ρ

where γ is a constant that satisfies γ = e − (γ + 1); i.e. γ = 0.2785.

(22)

se s˙ e + sm s˙ m ≤ −λm (se + sm )2 + βm γµm + |se + sm |[|∆fm | + |dm | + |1 − gm Gm ||fˆm + w ˆm |

+ k1 gm Gm |fˆm + w ˆm |−gm Gm Km ] . For ηm > 0, if Km verifies gm Gm Km ≥ ηm + M + Dm + |1 − gm Gm ||fˆm +w ˆm | + k1 gm Gm |fˆm + w ˆm |

then se s˙ e + sm s˙ m ≤ −λm (se + sm )2 − ηm |se + sm | + −1 G−1 − 1| ≤ β − 1, a sufficient βm γµm . Since |gm m m condition for Km to satisfy (27) is Km ≥ βm (ηm + M + Dm )

+ (βm − (1 − k1 ))|fˆm + w ˆm | .

Φm =

vm = −

(23)

Km (sm + se ) Km tanh . 1 − k1 (1 − k1 )µm

(24)

Substituting ζ = φ(x) (22) into the right-hand side of (17) yields s˙ m + λm sm = (1 − gm Gm )(fˆm + w ˆm ) + ∆fm + gm ∆Fm + gm dm

− ηˆm |se + sm |

For any µm > 0 and any s ∈ R, s tanh(s/µm ) ≥ 0, and the following inequality holds s µm

≤ γµm

(30)

is satisfied for |se + sm | ≥ Φm . We note that the relationship (15) gives se s˙ m + sm s˙ e = (se s˙ e + sm s˙ m ) − λm (s2m − s2e ). We have, for |se + sm | ≥ Φ, (se + sm )(s˙ e + s˙ m ) = 2(se s˙ e + sm s˙ m )

Lemma 1.

(29)

se s˙ e + sm s˙ m ≤ −λm (se + sm )2

We need the following lemma from [Polycarpou, 1996] for subsequent analysis.

0 < |s| − s tanh

βm γµm ϑηm

Then the following condition

Km gm Gm Km (sm + se ) tanh − 1 − k1 (1 − k1 )µm (25)

(28)

For any 0 < θ < 1, set ηˆm := (1 − θ)ηm and

where u ˆm = −fˆm − w ˆm

(27)

(26)

− λm (s2m − s2e )

≤ −2λm (se + sm )2 − 2ˆ ηm |se + sm |

(31)

if |sm | ≥ |se |; and (se − sm )(s˙ e − s˙ m ) = −λm (s2e − s2m ) ≤ −λm Φm |se − sm |

(32)

if |sm | ≤ |se |. From (30)–(32), one can conclude that error-state trajectories start off the boundary


. e

Φ

ε

e

−ε

−Φ

s =Φ s =0 s = −Φ Fig. 1.

Guaranteed error bounds.

layer, |se + sm | ≤ Φm , will move toward the hyperplane, se = 0, sm = 0, and will reach the boundary layer in finite time. When s2e + s2m is considered as an error measure, Eq. (30) suggests the approaching speed of error-state trajectories to the origin will be faster than that of the conventional sliding mode control. From the definitions (11) and (12) of s m and se , respectively, one has sm + s e = 2

d + λm dt

n−1

e.

(33)

By a similar argument made in [Slotine & Li, 1991], whenever in the boundary: |sm + se | ≤ Φm , the guaranteed tracking precision is (i) e ≤ 1 (2λm )i ε , 2

for

i = 0, . . . , (n − 1)

(34)

n−1 . When compared with the where ε = Φm /λm conventional continuous sliding control [Slotine & Li, 1991], Eq. (34) indicates the guaranteed steadystate error bounds are reduced by half. For n = 2

and λm = 1, the guaranteed error bounds are depicted in Fig. 1, where the heavy-shaded region represents the error bound of the proposed control, while the light-shaded region represents that of conventional sliding control without incorporation of a complementary sliding variable. The effectiveness of this design will be illustrated by a simulated example of an uncertain Lorenz system in Sec. 4. Next, let us consider the original overall system (1) and (2), where u is the control input. Rewrite Eq. (17) as follows: s˙ m + λm sm = fˆm + w ˆm + ∆fm + gm ρφ + gm ρz + gm ∆Fm (z + φ) + gm dm , where the auxiliary error z is defined in (8). Substituting the control φ given in (22) and in view of the bound (5) of ∆Fm , one has, whenever |se + sm | ≥ Φm , se s˙ e + sm s˙ m ≤ −λm (se + sm )2 − ηˆm |se + sm | max + (|ρ| + σ)βm |z||se + sm | .

(35)


We summarize the overall controller design in the following theorem. Consider the system (1), (2) which satisfies Assumption 1. Suppose the function F m (ζ) satisfies the sector condition (3) globally. Let the function φ(x) be specified in (22) with u ˆ m , vm defined in (23), (24), respectively. Let the gain K m be designed to satisfy (28). With the definition of w â given in (20), the controller is designed as follows: Theorem 1.

u = Ga [ˆ u a + va ]

(36)

where Ga := q

and

1 βamin βamax

.

(37)

u â = −fâ − w â va = −Ka sat

(38) sa + s z Φa

.

(39)

For ηa > 0, if Ka ≥ βa (ηa + N + Da ) + (βa − 1)|fâ + w â | ,

(40)

q

where βa := (βamax )/(βamin ), then, for any ηˆm > max Φ )/2λp−1 , the tracking error (7) will ((|ρ| + σ)βm a a asymptotically tend to a neighborhood of zero in finite time. Since 0 < βa−1 ≤ ga−1 G−1 ≤ βa and 0 < a ≤ ga Ga ≤ βa , Inequality (40) implies

Proof.

βa−1

Ka ≥

ga−1 G−1 a (ηa +

|ga−1 G−1 a

+ N + Da )

se s˙ e + sm s˙ m ≤ −λm (se + sm )2 − ηˆm |se + sm | +

max Φ (|ρ| + σ)βm a

2λp−1 a

Thus, ga Ga Ka ≥ ηa + N + Da + |(1 − ga Ga )(fâ + w â )| By a similar process as has been done above, one can show that the control law (36) fulfills the following condition (41)

whenever |sz + sa | ≥ Φa . A similar argument as was made above [Eqs. (30)–(32)] for the mechanical subsystem can be extended to the electric subsystem to reach the conclusion that any trajectory

|se + sm |

≤ −λm (se + sm )2 − η˜m |se + sm | , max Φ )/2λp−1 + η provided ηˆm ≥ ((|ρ| + σ)βm ˜m > 0 a a for any η˜m > 0. Consequently, the inequality assures that there exits tc < ∞ such that whenever t > tc ,

(i) e (t) < 1 (2λm )i ε , 0 ≤ i ≤ n − 1 , 2

n−1 . where ε = Φm /λm

4. Controlling the Uncertain Lorenz System and the Uncertain Chua’s Circuit In this section, an uncertain Lorenz system and an uncertain Chua’s circuit are considered.

4.1. The Lorenz system The Lorenz system can be described by x˙ 1 = −σf x1 + σf x2 + d1 , x˙ 2 = rx1 − x2 − x1 x3 + d2 + u , x˙ 3 = x1 x2 − bx3 + d3 ,

− 1||fâ + w â | .

sz s˙ z + sa s˙ a ≤ −λa (sz + sa )2 − ηa |sz + sa |

z(t) will reach the boundary layer, |s z + sa | ≤ Φa , in finite time when it starts off the boundary layer at time t = 0. In fact, in view of (34), we have |z(t)| ≤ 1/2(Φa /λp−1 a ) when in the boundary layer. As a result, it follows, from (35),

(42)

where u is the control input; d1 , d2 and d3 are the external disturbances, bounded by three known constants, δ1 , δ2 , δ3 , respectively, i.e. |d1 | ≤ δ1 , |d2 | ≤ δ, |d3 | ≤ δ3 . The chaotic motion of the Lorenz system, called the butterfly effect, is shown in Fig. 2. with parameters σf = 10, b = 8/3, r = 28 and u = 0. The time responses of the butterfly effect are depicted in Fig. 3. Assume x1r , x2r and x3r are the desired signals, then (x1r , x2r , x3r ) = (x1r , x1r , x21r /b). We note that if d3 = 0, and if x1 x2 is regarded as the input to the third equation of (42), then it is easy to verify that the subsystem, x˙ 3 = x1 x2 − bx3 , is input-to-state stable [Khalil, 1996]. Therefore,


40 20

x2

x3

50

0 20

0 -20

20

0 x2

-20

-20

0 x1

-40 -50

40

40

50

0 x2

50

x3

60

x3

60

0 x1

20

20

0 -50

0 x1 Fig. 2.

0 -50

50

Chaotic trajectories of the Lorenz system without a control signal.

60 50

x3

40 30

x 1,x 2 ,x 3

20 10 0 x1

-10 -20 -30

x2

-40 0

5

10

15

20

25

30

Tim e ( s e c )

Fig. 3. line).

Time responses of the Lorenz system without a control signal (x1 -blue solid line, x2 -red dotted line, x3 -black solid

870 C.-C. Wang & J.-P. Su 60

50

40

x3

30

x1,x2,x3

20

10

0 x2 -10

x1

-20

-30

-40 0

Fig. 4. line).

2

4

6 Time (sec)

8

10

12

Time responses under control for sudden changes of parameters (x1 -blue solid line, x2 -red dotted line, x3 -black solid

we will restrict ourselves to the design of the sliding control of the subsystem x˙ 1 = −σf x1 + σf x2 + d1 , x˙ 2 = rx1 − x2 − x1 x3 + d2 + u ,

(43)

by treating x3 (t) as time-varying parameter. To examine the robustness of the controller against parameter variations, we assume that 10 ≤ σf ≤ 15, and 28 ≤ r ≤ 35. According to the notations given in Sec. 3, we note that fˆm = −ˆ σf x1 = −12.5x1 , gm = σf , Fm (x2 ) = x2 , fâ = rˆx1 − x2 − x1 x3 = 31.5x1 − x2 − x1 x2 and ga = 1. Take M (t, x) = 2.5|x1 |, N (t, x, ξ) = 3.5|x1 | and min ≤ g max = 15; 1 = β min ≤ g ≤ 10 = βm m ≤ βm a qa βamax = 1.5.

min β max = We have Gm = 1/ βm m

q

0.0816, Ga = 1/ βamin βamax = 0.8165 and βm = q q √ √ max /β min = βm 1.5, β βamax /βamin = 1.5. = a m Taking α1 = 1, α2 = 1.01 such that Fm (x2 ) is subjected to a sector-bound condition (3) gives σ = (α2 − α1 )/2 = 0.005. Choose ρ = δ = 1.005 > σ, then k1 = σ/|ρ| = 1/201. By setting λa = 1.0, λm = 0.1, ηa = 1, ηm = 7.695, µ = 15, θ = 0.1, and Φa =6 1, we have Φm = µγβm /θηm = 6.6, and max )/2λ = ηˆm = (1 − θ)ηm = 7.6 > ((|ρ| + σ)Φa βm a 7.575. The functions, d1 = 0.5 cos(5πt), d2 =

0.5 cos(5πt), and d3 = 0.5 cos(5πt) are considered as external disturbances in the simulations. Let Dm = 0.5 = Da . The function φ(x) can be computed from (22)–(24), where Km = βm (ηm + M + Dm ) + (βm − (1 − k1 ))|fˆm + w ˆm |. The overall controller can, therefore, be obtained by Theorem 1 with Ka = βa (ηa + N + Da ) + (βa − 1)|fâ + w â |. We simulated the Lorenz system, using SIMULINK with Runge–Kutta algorithm at a fixed-step integration time of 0.001 s, by considering the following cases, respectively.

4.1.1. Case A Consider the first case that the external disturbances are absent, i.e. d1 = d2 = d3 = 0. The control input is activated at t = 5 s. As can be observed from Fig. 4, the chaotic behaviors of the uncontrolled system disappear after a transient period of about 2 sec, and the steady-state of x 1 , x2 , x3 , are steered to 8.5, 8.5, 27.1, respectively. Figure 4 also depicts the time responses of x 1 , x2 , and x3 from 8.5, 8.5, 27.1 to 8.5, 8.5, 18.1, respectively, when the parameters, σf , b, r, of the system are changed from 10, 8/3, 28 to 15, 4, 35, respectively, at t = 9 s. Figure 5 indicates the control effort u at the corresponding time instants.

Composite Sliding Mode Control of Chaotic Systems with Uncertainties 871 80

0.5 cos(5πt), d2 (t) = 0.5 cos(5πt) and d3 (t) = 0. If the controller is activated at time t = 5 s, Fig. 8 shows the time response of each state, and Fig. 9 displays the control effort. If, instead, d 1 = 0 = d2 , and d3 (t) = 0.5 cos(5πt), the effects on x1 and x2 seem not to be visible, as indicated in Fig. 10. Even for the case: d1 (t) = d2 (t) = d3 (t) = 0.5 cos(5πt), the impact on the responses of x1 , x2 appears not so obvious, as can be observed from Fig. 12. The corresponding controls of the above cases are depicted, respectively, in Figs. 11 and 13. These results demonstrate the good disturbance rejection property of the proposed control law.

60 40 20 0

u

-20 -40 -60 -80 -100 -120 -140 0

2

4

6

8

10

12

Time (sec)

Fig. 5.

Control input u for Case A.

4.2. The Chua’s circuit The dynamic equation of Chua’s circuit is described by

4.1.2. Case B Without external disturbances, we consider the second case that the desired values x1r , x2r , x3r are changed from 8.5, 8.5, 27.1 to 12, 12, 54 at t = 9 s. Then the responses of x1 , x2 , x3 , and the control input u, activated at t = 5 s, are shown in Figs. 6 and 7, respectively.

C1

1 dVc1 = (Vc2 − Vc1 ) − g(Vc1 ) dt R

C2

dVc2 1 = (Vc1 − Vc2 ) + IL dt R

L

4.1.3. Case C

dIL = −Vc2 dt

(44)

where C1 , C2 , L and R are circuit parameters, IL is the current through the inductor L; V c1 and Vc2 are

Third, we examine the disturbance rejection property of the proposed controller. Let d 1 (t) =

60 50 40 30

x3

x 1,x 2,x 3

20 10 0 x1 -10 x2

-20 -30 -40 0

2

4

6

8

10

12

Tim e ( s e c )

Fig. 6. Time responses under control for sudden changes of reference signals (x1 -blue solid line, x2 -red dotted line, x3 -black solid line).


400 350 300 250 200

u

150 100 50 0 -50 -100 -150 0

2

4

6

8

10

12

10

12

Time (sec)

Fig. 7.

Control input u for Case B.

60

50

40 x3

30

x1,x2,x3

20

10

0

x1

-10

-20

x2

-30

-40 0

2

4

6 Time (sec)

8

Fig. 8. Time responses under control for d1 = d2 = 0.5 cos(5πt) and d3 = 0 (x1 -blue solid line, x2 -red dotted line, x3 -black solid line).


60 40 20 0

u

-20 -40 -60 -80 -100 -120 0

2

4

6

8

10

12

Time (sec)

Fig. 9.

Control input u for d1 = d2 = 0.5 cos(5πt) and d3 = 0.

60

50

40 x3

30

x1,x2,x3

20

10

0 x1 -10

-20

x2

-30

-40 0

2

4

6 Time (sec)

8

10

12

Fig. 10. Time responses under control for d1 = d2 = 0 and d3 = 0.5 cos(5πt) (x1 -blue solid line, x2 -red dotted line, x3 -black solid line).


50 40 30 20 10

u

0 -10 -20 -30 -40 -50 0

2

4

6

8

10

12

Time (sec)

Fig. 11.

Control input u for d1 = d2 = 0 and d3 = 0.5 cos(5πt).

60

50

40

x3

30

x1,x2,x3

20

10

0 x2 -10 x1 -20

-30

-40 0

Fig. 12. line).

2

4

6 Time (sec)

8

10

12

Time responses under control for d1 = d2 = d3 = 0.5 cos(5πt) (x1 -blue solid line, x2 -red dotted line, x3 -black solid

Composite Sliding Mode Control of Chaotic Systems with Uncertainties 875 80

5

z3

60 40 20

0

-5 1

0

6 0 .5

4

u

-20

2

0

0

-40 -2

-0.5

-60

-4 z2

-1

-80

z1

-6

Fig. 14. Trajectories of the Chua’s circuit: attractor.

-100 -120

period-1

-140 0

2

4

6

8

10

12

Time (sec)

Fig. 13.

3

Control input u for d1 = d2 = d3 = 0.5 cos(5πt).

2 1 0

z3r

the voltages across C1 and C2 , respectively, and the piecewise linear function g(Vc1 ) describes the V − I characteristics of the Chua’s diode g, as follows

-1 -2 -3 0 .6

1 g(Vc1 ) = Gb Vc1 + (GA − GB )(|Vc1 + E| 2 − |Vc1 − E|) (45) with GA < 0 and GB < 0 being some appropriately chosen constants. Set z1 := IL , z2 := −Vc2 , z3 := Vc1 , and E = 1 V. Let d1 , d2 , and d3 be external disturbances, bounded for all time by three known constants, δ1 , δ2 , δ3 , respectively, i.e. |d1 | ≤ δ1 , |d2 | ≤ δ, |d3 | ≤ δ3 . Then Eq. (44) can be reformulated in the following form z˙1 = b1 z2 + d1 z˙2 = −b2 (z3 + z2 ) − b3 z1 + d2 + u z˙3 = −b4 (z2 + z3 ) − b5 z3

− b6 (|z3 + 1| − |z3 − 1|) + d3

(46)

where b1 = 1/L, b2 = 1/RC2 , b3 = 1/C2 , b4 = 1/RC1 , b5 = GB /C1 , b6 = (GA − GB )/2C1 , and u is the controller. We note that if d3 = 0, and if z2 is regarded as the input to the third equation of (46), then it is easy to verify that the subsystem, z˙ 3 = −b4 (z2 + z3 ) − b5 z3 − b6 (|z3 + 1| − |z3 − 1|), is input-to-state stable [Khalil, 1996]. Therefore, we will restrict ourselves to the design of the sliding control of the subsystem z˙1 = b1 z2 + d1 z˙2 = −b2 (z3 + z2 ) − b3 z1 + d2 + u .

(47)

0 .4 0 .2 0 -0.2 z2r

-0.4 -4

-3

-2

-1

0

1

2

3

4

z1r

Fig. 15. Trajectories of the Chua’s circuit: double-scroll attractor.

We assume that the Chua’s circuit (47), shown in Fig. 14, is originally (u = 0) in the periodic state with parameters C1 = 0.11364, C2 = 1, L = 0.0625, R = 1, GA = −1.143, and GB = −0.714, i.e. (b1 , b2 , b3 , b4 , b5 , b6 ) = (16, 1, 1, 8.7997, −6.2830, −1.8875) [Ge & Wang, 2000; Chua et al., 1986; Bartissol & Chua, 1988]. The objective is to force the output y = z 1 (t) of the controlled Chua’s circuit (46) to asymtotically track the chaotic reference signal y r = z1r (t) generated from another uncontrolled Chua’s circuit in chaotic state, double-scroll attractor, shown in Fig. 15, with parameters C1 = 0.10204, C2 = 1, L = 0.0625, R = 1, GA = −1.143, and GB = −0.714, i.e. (b1 , b2 , b3 , b4 , b5 , b6 ) = (16, 1, 1, 9.8001, −6.9973, −2.1021) [Ge & Wang, 2000]. We take the initial conditions (z1 (0), z2 (0), z3 (0)) = (2, −0.3, 0.4) for simulations with the controlled Chua’s circuit (46) and (z r1 (0), zr2 (0), zr3 (0)) = (0.2, −0.5, 0.3) with the reference Chua’s circuit, respectively. To examine the robustness of the controller against parameter variations, we


assume that 10 ≤ b1 ≤ 20, 0 ≤ b2 ≤ 9 and b3 = 1. Using the notations given in Sec. 3, we take fˆm = 0, Fm = z2 , fâ = −z1 − z2 − z3 , and M = 0, N = 10|z2 + z3 |. We note that gm = b1 , min = 10 ≤ g max and thus βm m ≤ 20 = βm . Consider the constraint for gaq: βamin = 1 ≤ ga ≤ 1.5 = βamax . min β max = 0.070711, G = We have Gm = 1/ βm a m

q

1/ βamin βamax = 0.8165 and βm =

q

max /β min = βm m

q √ √ 2, βa = βamax /βamin = 1.5. Taking α1 = 1.2, α2 = 0.9 such that Fm is subject to a sector-bound condition (3) gives σ = (α2 − α1 )/2 = 0.15. Choose ρ = δ = 1.05 > σ, then k1 = σ/|ρ| = 0.1429. By setting λa = 1.0, λm = 0.1, ηa = 1, ηm = 30, µ = 15, θ = 0.1, and Φa = 2, we have Φm = µγβm /θηm = 2.2072, and ηˆm = (1 − θ)ηm = 27 max /2λ > (|ρ| + σ)Φa βm The signals, a = 26.667. d1 = 0.5 cos(5πt), d2 = 0.5 cos(5πt), and d3 =

4

3

2

z1 and z1r

1

0

-1

-2

-3

-4 0

10

20

30

40

50

Time ( sec )

Fig. 16.

Time responses of y = z1 (t) and the reference signal yr = z1r (t) (z1r -blue solid line, z1 -green dotted line).

1.8 0.02

1.6 0.015

1.4 0.01

1.2 0.005

e= y-yr

e= y-yr

1

0.8

0

0.6 -0.005

0.4 -0.01

0.2 -0.015

0

-0.2

-0.02

0

5

10

15

20

25 Time ( sec )

(a) Fig. 17. scale.

30

35

40

45

50

0

0.5

1

1.5

2

2.5 Time ( sec )

3

3.5

4

4.5

5

(b)

(a) Tracking error of the controlled Chua’s circuit. (b) Tracking error of the controlled Chua’s circuit in another

Composite Sliding Mode Control of Chaotic Systems with Uncertainties 877 3 z3 2

z2

z2 and z3

1

0

-1

-2

-3 0

Fig. 18.

5

10

15

20

25 Time ( sec )

30

35

40

45

50

z2 , z3 of the controlled Chua’s circuit (z2 -blue solid line, z3 -black dotted line).

10 8 6 4

u

2 0 -2 -4 -6 -8 -10 0

10

20

30

40

50

Time ( sec )

Fig. 19.

Control input u of the controlled Chua’s circuit.

0.5 cos(5πt) are taken as external disturbances in the simulations. Take Dm = 0.5 = Da . The function φ(x) can be computed from (22)–(24), where Km = βm (ηm +M +Dm )+(βm −(1−k1 ))|fˆm + w ˆm |. The overall controller can, therefore, be obtained by Theorem 1 with Ka = βa (ηa +N +Da )+(βa −1)|fâ + w â |.

We simulated the Chua’s circuit using SIMULINK with Runge–Kutta algorithm at a fixed-step integration time of 0.001 s. Simulation results are shown in Figs. 16–19. Figure 16 indicates that the output of the controlled Chua’s circuit (46) asymptotically track the reference signal y r = z1r (t) quite well. This good performance can be further


justified from the observation of the tracking error e = y − yr , shown in Fig. 17, where the magnitude of the tracking error seems to fall into the bound: |e(t)| ≤ 0.005 very rapidly (within 0.5 s). As compared with some recent results, e.g. those of the control presented in [Ge & Wang, 2000], where the tracking error seems not to settle down to a comparative small bound until 100 s, we note that the proposed control is promising in this application. Figure 18 shows that the states z2 and z3 remain bounded for all time. The input of the controlled Chua’s circuit is depicted in Fig. 19.

5. Conclusion We have presented a new composite sliding control scheme in this paper for a class of chaotic systems with uncertainties. A significant feature of the control scheme is the incorporation of a new sliding variable as a complement to the conventional sliding variable to form a more meaningful measure of errors such that an efficient control law can be derived. The new proposed control law has been shown to reduce the ultimate bound of tracking error by half, when compared with the conventional method. Moreover, the reaching dynamics during the reaching phase is also significantly improved. To illustrate the effectiveness of the design, the uncertain Lorenz chaotic system and the Chua’s circuit with uncertainties were used as simulated examples. Both theoretical and simulation results reveal that the proposed composite sliding control is promising for controlling uncertain chaotic dynamics, even for higher-dimensional and more complex systems.

References Baik, I. C., Kim, K. H., Kim, H. S., Moon, G. W. & Youn, M. J. [1996] “Robust nonlinear speed control of PM synchronous motor using boundary layer integral sliding control with sliding load torque observer,” IEEE PESC’96 Record, pp. 1242–1247. Bartissol, P. & Chua, L. O. [1988] “The double hook,” IEEE Trans. Circuits Syst. 35, 1512–1522. Chen, C. L. & Lin, W. Y. [1997] “Sliding mode control for nonlinear systems with global invariance,” Proc. Inst. Mech. Engrs. 211, 75–82. Chen, L., Chen, G. & Lee, Y. W. [1999] “Fuzzy modeling and adaptive control of uncertain chaotic systems,” Inform. Sci. 121, 27–37. Chen, L. & Chen, G. [2000] “Fuzzy modeling, predictiion, and control of uncertain chaotic systems based on time series,” IEEE Trans. Circuits Syst. 47, 1527–1531.

Chern, T. L. & Wu, Y. C. [1993] “Design of brushless DC position servo systems using integral variable structure approach,” IEE Proc. Electr. Power Appl. 140, 27–34. Chua, L. O., Komuro, M. & Matsumoto, T. [1986] “The double scroll family. Parts I and II,” IEEE Trans. Circuits Syst. 33, 1073–1118. Fuh, C. C. & Tung, P. C. [1995] “Controlling chaos using differential geometric method,” Phys. Rev. Lett. 75, 2952–2955. Gao, W. & Hung, J. C. [1993] “Variable structure control of nonlinear systems: A new approach,” IEEE Trans. Indust. Elect. 40, 45–55. Ge, S. S. & Wang, C. [2000] “Adaptive control of uncertain Chua’s circuits,” IEEE Trans. Circuits Syst. 47, 1397–1402. Hartley, T. T. & Mossayebi, F. [1992] “Classical control of a chaotic system,” 1st IEEE Conf. Control Applications, Dayton, Ohio, USA, pp. 522–526. Khalil, H. K. [1996] Nonlinear Systems, 2nd edition (Prentice-Hall, Englewood Cliffs, NJ). Lee, J. H., Ko, J. S., Chun, S. K., Lee, J. J. & Youn, M. J. [1992] “Design of continuous sliding mode controller for BLDDM with prescribed tracking performance,” Conf. Record, IEEE PESC’92, pp. 770–775. Liaw, Y. M. & Tung, P. C. [1996] “Extended differential geometric method to control a noisy chaotic system,” Phys. Lett. A222, 163–170. Ott, E., Grebogi, C. & Yorke, J. A. [1990] “Controlling chaos,” Phys. Rev. Lett. 64(11), 1196–1199. Polycarpou, M. M. [1996] “Stable adaptive neural control scheme for nonlinear systems,” IEEE Trans. Automat. Contr. 41, 447–451. Qin, H., Zhang, H. & Chen, G. [1998] “Neural-network based adaptive control of uncertain chaotic systems,” Circuits Syst. 1998. ISCAS’98. Proc. 1998 IEEE Int. Symp., Vol. 3, pp. 318–321. Slotine, J-J. E. & Li, W. [1991] Applied Nonlinear Control (Prentice-Hall, Englewood Cliffs, NJ). Tian, Y. P. & Yu, X. [2000] “Adaptive control of chaotic dynamical systems using invariant manifold approach,” IEEE Trans. Circuits Syst. 47, 1537–1542. Utkin, V. I. [1997] “Survey paper-variable structure systems with sliding modes,” IEEE Trans. Automat. Contr. 22(2), 212–222. Vincent, T. L. & Yu, J. [1991] “Control of a chaotic system,” Dyn. Contr. 1, 35–52. Yau, H. T., Chen, C. K. & Chen, C. L. [2000] “Sliding mode control of chaotic systems with uncertainties,” Int. J. Bifurcation and Chaos 10(5), 1139–1147. Yu, X. [1996] “Controlling Lorenz chaos,” Int. J. Syst. Sci. 27, 355–359. Zhang, H., Qin, H. & Chen, G. [1998] “Adaptive control of chaotic systems with uncertainties,”Int. J. Bifurcation and Chaos 8(10), 2041–2046.