- Email: [email protected]

Model-free control of affine chaotic systems ✩ Guoyuan Qi a,∗ , Zengqiang Chen b , Zhuzhi Yuan b a Department of Automation, Tianjin University of Science & Technology, Tianjin 300222, People’s Republic of China b Department of Automation, Nankai University, Tianjin 300071, People’s Republic of China

Received 28 January 2005; received in revised form 8 June 2005; accepted 22 June 2005 Available online 5 July 2005 Communicated by A.P. Fordy

Abstract In practice, there are many chaotic systems whose models are usually unknown or partially unknown. However, the majority of control schemes focus on model-dependent techniques. The model-free controlling problem for affine chaotic systems is investigated in this Letter. An adaptive higher-order differential feedback controller (HODFC), which does not depend on the model of the controlled chaotic system, is presented. The controller utilizes the information of the measured output and the given objective as well as extracted differentials of those via higher-order differentiator (HOD). Stability, convergence and robustness of the closed-loop system are investigated. The presented adaptive HODFC can successfully control the uncertain Lorenz system, the Chen system, the Duffing–Holmes system, the Rössler system, and a new coined 4-dimensional chaotic system, and can drive their trajectories to desired steady states, unstable periodic orbits, or new chaotic states. Importantly, the controller does not use model functions of these systems above. The control scheme is suitable for both chaotic affine systems and ordinary affine systems without knowing their precise models. 2005 Elsevier B.V. All rights reserved. Keywords: Chaotic system; Higher-order differentiator; Adaptive higher-order differential feedback controller; Model-free control; Stability; Convergence

1. Introduction Recently, control and synchronization of chaotic systems have received a great deal of attention. Re✩ This research is supported by a grant from National Nature Science Foundation of China (60374037) and by a grant from Science and Technology Development Foundation of Tianjin Colleges (20051528). * Corresponding author. E-mail addresses: [email protected] (G. Qi), [email protected] (Z. Chen).

0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.06.073

calling many proposed control schemes, we found that the important characteristic of chaos control is that the designed controllers mainly rely on known models of the chaotic systems. For example, Sanchez et al. [1] proposed inverse optimal controller and Guan et al. [2] presented impulsive controller rely on the known model of the Chen system. Based on the Lorenz system model, Yu [3] developed sliding mode control. Agiza and Yassen [4] achieved synchronization between the Rössler and the Chen system using an active controller which also depends on the known models of

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the plants. Solak et al. [5] presented an observer-based feedback approach to control a class of affine chaotic systems, such as Duffing equation and the van der Pol oscillator. The observer and the controller also rely on the model of the controlled chaotic system. In practice, the model of a plant is usually unknown or only partially unknown. Sometimes, we can only obtain some measured variables of a system in experiments or in nature. There are many chaotic behaviors in commutation, laser, and power systems, in ecological systems, and in transmission through the Internet, etc. [6–10]. Even though some kinds of techniques such as neural networks [11–13], fuzzy neural networks [14], etc., can be utilized to identify system models, these techniques still contain bias, moreover, the system itself is also time-varying in some circumstance. Importantly, chaotic systems are difficult to be identified because they are very sensitive to small perturbations and model bias [7–10]. At present, some control approaches for chaotic systems consider uncertainties, such as Zhang et al. [15] used adaptive control, Lu et al. [16] and Yau et al. [17] using the sliding mode control approach. But their robust controllers are mainly based on known normal models in which unknown uncertainties are relatively smaller as compared to the known normal models. In fact, an important characteristic of modern control theory is model relying, which extremely influences schemes of chaos control. Control of chaos has received a great deal of attention in the last 15 years. The majority of methods focus on model-dependent techniques, to build some kinds of feedback control laws. However, the OGY control method is a kind of model-free approach, which is built from analysis of chaotic properties using Poincaré maps [7–10]. Let us review the method. Shinbrot, Ott, Grebogi and Yorke utilized extreme sensitivity to perturbations and ergodicity of chaos to propose a time-delay embedding technique (the OGY method) [7–10]. Supposes that the original model of an n-dimensional (nD) chaotic system is unknown, but the time series data s k ∈ R n−1 obtained from a Poincaré map is measured in an experiment. Then, the discrete (n − 1)-dimensional time-delay embedding model with parameter pk , can be constructed as sk+1 = F (s k , pk ).

(1)

A desired periodic orbit with parameter p0 and fixed point s ∗ are selected. Then, linearize model (1) on p0 and s ∗ to obtain s k+1 − s ∗ = A s k − s ∗ + B(p − p0 ), (2) where A = ∂F /∂s, B = ∂F /∂p. The chaotic ergodicity guarantees that the system trajectory will eventually fall into a neighborhood of the desired periodic orbit. When it does so, the control law is adjusting the parameter pk (called small perturbation) in the form of feedback control, which is written as pk − p0 = −K T s k − z∗ , (3) where K makes the matrix A − BK T Hurwitz, to direct the chaotic trajectory onto the desired orbit. The OGY method does not rely on the original model of the controlled chaotic system. However, the OGY method has some limitations: the constructed model (1) is (n − 1)-dimensional, which is only a map of the original system with nD. Control law (3) is derived from the constructed model. However, both the original system and the given objective are in the nD space, so it is not effective enough to use such a control law to control a complicated chaotic system and a disturbed system. In fact, OGY method can only control lower-dimensional chaos. The OGY method still relies on the constructed model which is not easy to be obtained. When the target is not in the scope of the attractor, the ergoticity does not guarantee the trajectory eventually to move into the neighborhood of the desired orbit. In this Letter, we consider a new approach of controlling chaos where the controller does not rely on both the original model of chaotic system and the delay embedding model constructed by using data from the Poincaré map. The controller depends only on the measured information of the original controlled system and the extracted differentials of the information up to nth order. We consider single-input and single-output (SISO) and multi-input and multi-output (MIMO) affine systems with unknown models and unknown bounded disturbances. In practice, there exist many SISO affine systems in mechanics such as the Duffing–Holmes system, van der Pol equation, robot systems, and flexible-joint mechanisms [18]. The Rössler system, Chua’s circuit and the Lorenz system family [6,19] can be considered as first-order MIMO

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affine systems. We have found that for many kinds of information such as outputs and targets of the controlled systems, their differentials up to the nth order and control inputs are very valuable. For example, the system output differentials up to the (n − 1)st order not only are varying rates of output but also are the inner states of the system. Furthermore, the nth-order differential is just the function on the right-hand side of the given system, i.e., the inner running rule. In design, we have to deal with two problems: (1) Design a higher-order differentiator (HOD) to extract differentials from the measured signal including the output and objective of the system. (2) Apply sufficient measured signals and the extracted differential information to design a higherorder differential feedback controller (HODFC), which does not rely on the model, to drive the system output to achieve the given objective. Since the output of an affine system, in the socalled Brunovsky form, and its differentials up to the (n − 1)st order, are all the states of the system, many kinds of observers were designed to estimate these differentials [5,20–24]. However, these observers mainly rely on the estimated plants. Furthermore, the nthorder differential of the output and each order differential of the given objective cannot be estimated by these kinds of model-dependent observers. Therefore, it is necessary to design a high-quality differentiator to extract differentials of the measured signals. Recently, we designed a stable tracking-differentiator (TD), which can extract the first-order differential [25]. However, if we obtain the nth-order differentials, then n series TD must be linked, which will result in bad differential impulses. Therefore, lately we proposed a higher-order differentiator (HOD), which can extract differentials up to the nth order with higher accuracy and better filtering properties on noises [26]. Using the extracted differentials obtained via HOD, a model-free controller was proposed [26]. However, its parameters need be tuned experimentally, and no analysis of stability of the closed-loop system was given. In this Letter, we will propose a new control technique and apply it to chaos control. Based on the HOD, adaptive higher-order differential feedback controllers (HODFC) for SISO and MIMO affine systems, respec-

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tively, are presented, which do not rely on the models of the systems. Theoretical analysis shows that the closed-loop systems under the HODFC are asymptotically stable and robust for the disturbances and functional variances of the systems. The outputs of the controlled plant converge to any given smooth objectives. Numeric simulations demonstrate that some chaotic systems such as the Lorenz system family [6,19], Duffing–Holmes system [5] and a new fourdimensional chaotic system proposed by Qi et al. [27], can be controlled to any given target point, any desired unstable periodic orbit, even a chaotic orbit, under the proposed adaptive HODFC. This Letter is organized as follows: Section 2 describes the problem under investigation, introduces and modifies the HOD in [26]. Section 3 presents the adaptive HODFC for SISO and MIMO systems, respectively. In Section 4, numerical results demonstrate the validity of the designed controllers for several chaotic systems. Section 5 concludes the Letter.

2. Problem statement 2.1. SISO affine system with disturbance Differential equation of SISO affine system is depicted as : y (n) = f (x, t) + d(t) + u, (4) s where notation s denotes SISO system (4), u ∈ R is the control input, y ∈ R is the system output, t ∈ Ut ⊂ R, x ∈ Ux ⊂ R n , and T x = [x1 , x2 , . . . , xn ]T = y, y (1) , . . . , y (n−1) denotes output differential vector, and is also system state vector, y (i) denotes the ith differential of y, f (·) is an unknown time-varying bounded smooth nonlinear function, and satisfies Lipchitz increasing condition. d(t) is bounded disturbance. Initial values are x(t0 ) = x 0 . System (4) is converted into the following state space model x˙ = x , 1 i n − 1, 1 i+1 x˙n = f (x, t) + d(t) + u, : (5) s y = x1 .

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The system is also called as Brunovsky canonical form which was widely studied [20–23]. The kind of affine system has widely background in practice. It is interesting that many chaotic systems are the SISO affine system [5,24] or MIMO affine system [6,19,27–29]. Usually, controllers of system (5) are mostly based on known nonlinear function f (·) and state x. Some literatures [5,24] designed a controller based on the observer. However, both the observer and the controller mostly rely on the model. We consider the function f (·) and the states x2 , x3 , . . . , xn can be unknown, and only require that output y (i.e., x1 ) is known. Obviously, the controlled plant s is the function of the control input and the output signals and their differentials up to nth order. The high order differential information y (1) , . . . , y (n−1) in Eq. (4) are very significant. They represent not only the varying rates of the system output, but also the inner states of the system. Moreover, the nth order differential of output, i.e., y (n) , is more significant, since y (n) equals the function on the right-hand side which reflects the dynamical running regular of the system to some extent. In addition, control input u is also important information. We can fully utilize all the information rather than model function. Without loss of generality, suppose the given ob(n) jective yr has differentials up to nth order, and yr is continuous. Otherwise, we can soften the signal yr till satisfying these conditions. The same, the information (1) (n) yr , yr , . . . , yr should also be utilized to design control law. Let T r = yr , yr(1) , . . . , yr(n−1) ∈ Ur ⊂ R n , T T x¯ = x, y (n) r¯ = r T , yr(n) , (6) be the given input differential vector, the given input extended differential vector and the output extended differential vector, respectively. Let e=r −x

T = [e1 , e2 , . . . , en ]T = e, e(1) , . . . , e(n−1) ∈ R n , T e¯ = r¯ − x¯ = e¯ T , e(n) (7)

be the error differential vector, and the error extended differential vector, where e = yr − y. In general, the output y and the given input yr are known, but x¯ and r¯ are unknown. Let

T x¯ˆ = y, ˆ yˆ (1) , . . . , yˆ (n) , T r¯ˆ = yˆr , yˆr(1) , . . . , yˆr(n)

(8)

be the estimations of x¯ and r¯ , respectively. Note that yˆ (i) denotes the estimate of y (i) , rather than the ith differential of y. ˆ In the following, we will firstly introduce how to obtain the estimating vector x¯ˆ or r¯ˆ by y and yr . 2.2. High order differentiator In [26], to extract differentials up to nth order for y and yr , we proposed HOD which is described by combined expression, i.e., connecting n0 order dynamic system (9) with n + 1 order algebraic equation (10): z˙ i = zi+1 + ai (y − z1 ), 1 i n0 − 1, (9) z˙ n0 = an0 (y − z1 ), yˆ = z1 , (10) yˆ (i) = zi+1 + ai (y − z1 ), i = 1, . . . , n,

where n0 is the order of the system , generally, n + 1. z , z , . . . , z are the states of the system n 0 1 2 n 0

, ai (i = 1, . . . , n0 ) are the parameters. Obviously, we can obtain z1 , z2 , . . . , zn0 based on the measured signal y via (9), furthermore, calculate the estimated differentials y, ˆ . . . , yˆ (n) via (10). If the parameters ai (i = 1, . . . , n0 ) are not correctly given, all extracted differentials by the HOD are possibly not satisfactory, and even then the sys

tem becomes unstable. The parameters are given by the following form by analysis of stability and convergence [26] a i−1 , ai = KCni−1 0 −1 K = n0 0 a/(n0 − 1)n0 −1 , i = 1, 2, . . . , n0 , n

(11)

where Cji denotes the combination expression. Note that HOD has been simplified into two adjustable parameter n0 and a. We theoretically had the following remarks for HOD under the parameters formula (11) (see [26]): Remarks. (1) HOD does not rely on the model of the estimated system s , and it is additional system based on signal y or yr .

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(2) HOD is asymptotically stable system. (3) HOD holds higher convergence, and satisfies lim yˆ (i) = y (i) ,

t→∞

i = 0, . . . , n,

(12)

ˆ Usually, taking a ∈ [5, 30] via where yˆ (0) equals y. experience, it has higher precision. HOD is easily realized by computer or circuits. (i) (4) All extracted differentials yˆ (i) , yˆr , i = 1,

. . . , n, are smooth. Since has n0 integrators, which means even y or yr is not smooth signals, both yˆ and yˆr is n0 order differentiable. Furthermore, n0 > n + 1 is satisfied, so Remarks (4) holds. In fact, the HOD itself is an accuracy filter. When HOD is in initial process, and x¯ˆ in HOD will ¯ so produce peaking impulse. In this rapidly track x, Letter, to overcome the phenomenon, we modify HOD by adding a restraint device in (10), which is amended as yˆ = xˆ1 , yˆ (i) = xˆi+1 + ai (y − xˆ1 )σi (t), i = 1, . . . , n, (13) σi (t) = (1 − exp(−βt 2i ))/(1 + exp(−βt 2i )). The modified HOD is combined by (9) and (13). We call σi (t) as restraint, and β is a large positive constant. The effectiveness of the restraint is shown as follows: (a) yˆ (i) is depressed to large degree, since σi (t) is almost zero during the initial period. After this pe-

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riod, σi (t) promptly approximates to 1, which means Eq. (13) approximates to Eq. (10). (b) The amended HOD is stable, because dynamical system (9) is not changed when the impulse restraint device is applied in (13). In the following, we give an example to verify the modified HOD. Example 1. Consider SISO Duffing–Holmes chaotic system [5] x˙ = x , 1

2

x˙2 = f (x1 , x2 ) + u, y = x1 ,

(14)

where x1 , x2 are states and f (x1 , x2 ) = p1 x1 − px2 − x13 + q cos(ωt). Take parameters p1 = 0, p = 0.25, q = 11, ω = 1, and initial values x10 = 0.4, x20 = 0.2. Since we consider the estimate problem, let u = 0. Solak et al. [5] designed an observer to estimate x2 (i.e., y (1) ) based on model f (x1 , x2 ) and y. The problem is that if f (x1 , x2 ) is unknown, only the output y is known, how we obtain state x2 , even x˙2 . Noticing that x2 is differential of y, we use the HOD based on y to obtain xˆ2 , furthermore obtain xˆ˙ 2 (i.e., yˆ (2) ). Therefore, the condition that function f (x1 , x2 ) is known is avoided. Let all initials states be zero in Eq. (9). We use amended HOD with parameters n0 = 5, a = 8, β = 100 to estimate y, ˆ yˆ (1) , yˆ (2) by output y. The estimating results are shown in Fig. 1(a) (where Est. denotes

Fig. 1. (a) Comparing plots between the signal y, y (1) , y (2) and its estimate y, ˆ yˆ (1) , yˆ (2) using HOD for uncontrolled Duffing system. (b) Comparing plot between y − y (1) and yˆ − yˆ (1) on phase portrait.

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estimate), in which the curves y, ˆ yˆ (1) , yˆ (2) almost are (1) identical to the curves y, y , y (2) , respectively, and there are no peaking impulses in initial transient time. If we use the HOD rather than the amended HOD, peaking impulse for yˆ (2) in initial transient time is produced. The comparing curves between y − y (1) and estimate yˆ − yˆ (1) on phase portrait all are shown in Fig. 1(b), where solid line denotes y − y (1) and dotted line denotes yˆ − yˆ (1) . Both Figs. 1(a) and (b) show the estimating accuracy of HOD which does not rely on the model function f (x1 , x2 ).

3. High order differential feedback control 3.1. High order differential feedback control for SISO nonlinear system Assumption 1. The extended output differential vector x¯ and extended given input differential vector r¯ are known, and y (n) is yr(n) continuous. Theorem 1. For the time-varying nonlinear affine system (4) with unknown model, HODFC is described by ¯ e¯ + u, u=K ˆ (15) ¯ where K = [kn , kn−1 , . . . , k1 , 1] make the polynomial sn + k1 sn−1 + · · · + kn be a Hurwitz polynomial, and uˆ denotes the filtering signal of the control u, satisfying uˆ˙ = −λuˆ + λu,

(16)

or other filtering equation, where λ is a large positive constant. Then HODFC has following properties:

Setting K = [kn , kn−1 , . . . , k1 ] ∈ R 1×n , 0 1 0 ··· 0 0 0 1 ··· 0 . .. ∈ R n×n , . A= . . 0 0 0 ··· 1 0 0 0 ··· 0 0 0 b = 0 ∈ Rn, 0 1 0 1 0 ··· 0 0 0 1 ··· 0 . . ∈ R n×n. .. Am = .. 0 0 0 ··· 1 −kn −kn−1 −kn−2 · · · −k1 From (18) and definition of vectors e and e¯ in Eq. (7), we have e˙ = Ae + b(yr(n) − y (n) + y (n) − f (x, t) + d(t) + u) = Am e + b Ke + yr(n) − y (n) + y (n) − f (x, t) + d(t) + u ¯ e¯ + y (n) − f (x, t) + d(t) + u , = Am e + b K (19) ¯ makes + k1 + · · · + kn be a Hurwitz polyK nomial. Identically, K makes Am be a Hurwitz matrix. Let ¯ e¯ + y (n) − f (x, t) + d(t) + u = 0. K (20) 1s n

s n−1

(1) HODFC makes the closed-loop system asymptotically stable, and satisfies the following convergence

We have the stable control law ¯ e¯ + y (n) − f (x, t) + d(t) . u=K

lim lim x = r.

If the sum item f (x, t) + d(t) is unknown, the control law is unable to be realized. In another word, this is a model-depending control law. From system (4), we have y (n) − f (x, t) + d(t) = u. (22)

t→∞ λ→∞

(17)

(2) All system variables are bounded. (3) The controller is strongly robust for the function f (·) and bounded disturbance d(t). Proof. From Eq. (4) and the definition of ei in Eq. (7), we have e˙i = ei+1 , i = 1, . . . , n − 1, (n)

e˙n = yr − y (n) (n)

= yr − y (n) + y (n) − (f (x, t) + d(t) + u).

(18)

(21)

But u is a control law to be solved, so it is still unable to be realized. Considering lag property of the filtering in (16), replacing u by u, ˆ we will obtain uˆ ≈ y (n) − f (x, t) + d(t) . (23)

G. Qi et al. / Physics Letters A 344 (2005) 189–202

195

Substituting (23) into (21), we gain the controller (15). Substituting (15) into (20), and using (22), we have ˆ e˙ = Am e + b(u − u).

(24)

From (16), the filtering uˆ is realized via integrator, so the filtering uˆ is necessarily continuous no matter whether u is continuous or not. Furthermore, from As(n) sumption 1, y (n) and yr are continuous, so y (i) and (i) yr (i = 0, . . . , n − 1) must be continuous, too, which means that e¯ is continuous. Therefore, from (15), the control law u must be continuous. From (16) and the continuity of u, we obtain lim uˆ = u.

λ→∞

t→∞ λ→∞

(4) When the particular form is expressed as a nonminimum phase form y (n) = f (x, t) − u + d(t).

(27)

(25)

From (24), (25) and Am being a Hurwitz matrix, the closed-loop control system is asymptotically stable and satisfies lim lim e = 0.

Fig. 2. The realized diagram of adaptive high order differential feedback control based on HOD and HODFC for SISO system.

(26)

Obviously, r is bounded, and from (17), x is bounded. From Assumption 1 and (15), the control law u is bounded. Furthermore, controller (15) does not rely on the system model, so the controller is strongly robust to the function f (·) and bounded disturbance d(t). 2 Remarks. (1) Here, λ → ∞ is only a rigorous expression for mathematics meanings, in general, λ ∈ [5, 50]. The filter (16) is not unique. The filtering uˆ can be completely replaced by other filtering equation, such as the HOD. (2) From (17), by the HODFC, not only the system output y can track given input yr , but also the differentials up to (n − 1)th order of the output y (i) can track those of given input yr , respectively, which is different from the general control objective that the system output only tracks given input. (3) The HODFC has distinct physical meanings. The control law has two terms: uˆ overcomes or elimi¯ e¯ ensures that the nates sum term f (x, t) + d(t), and K closed-loop system is asymptotically stable. Therefore, HODFC is different from parameters tuning in PID control.

System (27) can be controlled by changing control law of HODFC into ¯ e¯ + u. u = −K ˆ

(28)

From Assumption 1, the HODFC is nonadaptive for e¯ = x¯ − x¯ r . When e¯ is unknown, we estimate it via the HOD to obtain eˆ¯ = xˆ¯ − xˆ¯ r by error e = yr − y. (n) From Remark (4) in Section 2.2, yˆ (n) and yˆr are continuous, so all conditions of Assumption 1 can be converted into realization via HOD. The adaptive controller is written as ¯ eˆ¯ + u. u=K ˆ

(29)

Hence, we yield adaptive HODFC. Fig. 2 shows the adaptive HODFC for nonlinear system.

3.2. High orders differentials feedback control for MIMO nonlinear system Consider MIMO nonlinear time-varying system with unknown model (ni )

yi

= fi (X, t) + ui + di (t),

i = 1, . . . , p,

(30)

where X = [X T1 , X T2 , . . . , XTp ]T is the output differential vector, (1) (n −1) T X i = yi , yi , . . . , y i i (31) is the ith output differential vector, yi and ui are the ith output and input, respectively, di (t) is the unknown bounded disturbance, and the initial condition is X(t0 ) = X0 . Assuming that fi (·) is an unknown

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time-varying bounded smooth nonlinear function, and satisfies Lipchitz increasing condition. Let yri , i = 1, . . . , p, be the ith given input, and let (1) (n −1) Xri = [yri , yri , . . . , yri i ]T be the ith given input ¯ ri = [X T , y (ni ) ]T , X ¯i = differential vector, and let X ri ri (n ) [XiT , yi i ]T be the ith given input extended differential vector, and the output extended differential vector, respectively. Let E i = [Ei1 , Ei2 , . . . , Eini ]T = Xri − Xi (1) (n −1) T = εi , εi , . . . , εi i ∈ R ni , ¯ ri − X ¯ i = E Ti , ε (ni ) T ¯i =X E i

(32)

¯ i and X ¯ ri are completely Assumption 2. Consider X measurable. We have the following theorem. Theorem 2. For the time-varying MIMO nonlinear system (30) with unknown model and unknown disturbance, the multivariable HODFC (MHODFC) is described by i = 1, . . . , p,

(33) s ni −1

where K i = [kini , . . . , ki1 , 1] makes + ki1 + · · · + kini be a Hurwitz polynomial, and uˆ i is the filtering value of the control ui . The MHODFC has following properties: s ni

(1) It is able to realize linearized decoupling control. (2) It makes the closed-loop system asymptotically stable, and satisfies the following convergence lim Xi = Xri .

t→∞

i = 1, . . . , p,

(34)

εi

(ni −1)

= −ki1 εi

˙ i = Ami E i + bi δi . E

(35)

(39)

Where Ami ∈ R ni ×ni is controllable normal matrix with parameters kini , . . . , ki1 , and bi = [0, . . . , 0, 1]T ∈ R ni ×1 . Similar to the proof of Theorem 1, δi → 0. Furthermore, s ni + ki1 s ni −1 + · · · + kini is a Hurwitz polynomial, from (39), we obtain lim E i = 0.

(40)

t→∞

It means the 2nd property holds. Meanwhile, according to Assumption 2 and the 2nd property, control input ui is bounded. Therefore, all variables are bounded. Since MHODFC does not rely on the model of system (30), MHODFC has strong robustness for variance of function fi (·) and disturbance di (t). When ¯ ri − X ¯ i in (33) are unknown, we obtain the es¯i =X E ˆ¯ − X ˆ¯ using HOD based on ˆ¯ = X timating vector E i ri i εi = yri − yi . Therefore we obtain the following adaptive MHODFC

4. Application

i = 1, . . . , p.

(38)

From Eq. (38) and the definition of vector E i in Eq. (32), we easily obtain the following important equation

Proof. Substituting (33) into (30), we have ¯ iE ¯ i + uˆ i ) + di (t), = fi (X, t) + (K

(37)

E˙ ini = −ki1 Eini − · · · − kini Ei1 + δi .

ˆ¯ + uˆ ), ¯ iE ui = (K i i

(ni )

− · · · − kini εi + δi .

It means

(3) All system variables are bounded. (4) It is strongly robust for the function f (·) and bounded disturbance d(t).

yi

(36)

where, δi = uˆ i − ui . From (36), we obtain p linear decoupling differential equations. From Eq. (36) and the definition of scalars εi and Eik in Eq. (32), we have (ni )

be the ith error differential vector, and the ith error extended differential vector, respectively, where εi = yri − yi . Assume yri has bounded differentials up to ni th order.

¯ iE ¯ i + uˆ i ), ui = (K

From Eqs. (35) and (30), we have (ni ) (n −1) + · · · + kini yi yi + ki1 yi i (n ) (n −1) = yri i + ki1 yri i + · · · + kini yri + δi ,

i = 1, . . . , p.

2

(41)

In this section, we study application of the ideas presented in the previous sections.

G. Qi et al. / Physics Letters A 344 (2005) 189–202

Example 2. Consider controlled Duffing system (14) in Example 1. Set 1: p1 = 0, q = 11, ω = 1, p = 0.25: the uncontrolled Duffing system is chaotic, as shown in Fig. 1. Set 2: p1 = 0, q = 11, ω = 1, p = 1.45: the uncontrolled Duffing system has a stable periodic orbit. The objective is to make the response of the controlled system with set 1 at initial values x10 = 0.4, x20 = 0.2 approach periodic orbit of the Duffing system with set 2 at initial states xr10 = 0.4, xr20 = 0.

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Yau et al. [17] and Solak et al. [5] achieved the control based on the known function f (x1 , x2 ). In the following, we examine the adaptive HODFC which only rely on the state x1 . The controller is adaptive HODFC given by (29) and (16). Take λ = 10 (2) (1) and K = [225, 30, 1]. Here, eˆ¯ = [yˆr − y (2) , yˆr − (1) T y , yˆr − y] . We use the amended HOD to estimate eˆ¯ by the signal y and yr , with its parameters and initial values same as in Example 1. The comparing curves between the system time responses y, y (1) and (1) given objectives yr , yr are shown in Fig. 3(a), (b).

(1)

Fig. 3. The comparing plots between the system responses and given objectives under control. (a) y and yr , (b) y (1) and yr , (c) y − y (1) and (1) (1) on yr − yr on phase portrait, (d) y − y (1) and on yr − yr of the controlled Duffing system with disturbance on phase portrait.

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Fig. 4. The uncontrolled Lorenz system, with parameters a = 10, b = 8/3, c = 28 and initial values y10 = 0.3, y20 = 0.3, y30 = 2. (a) 3D view, (b) time response.

Obviously, the control accuracy is high. Although there are initial errors among amended HOD, the controlled system and given objective, initial oscillations are small and convergence is rapid. The comparing (1) curves in phase portraits on y − y (1) and yr − yr are shown in Fig. 3(c). Here, we omit the initial process to clearly observe periodic orbit. The initial processes can clearly be shown in Fig. 3(a), (b). When the disturbance f (x1 , x2 ) = 0.1 sin(4πx1 ) sin(πx2 ) is added at the second equation of system (14). But all parameters of the controller are not changed. The controlled result is shown in Fig. 3(d). There is almost no influence via comparing Fig. 3(c) with Fig. 3(d), since the controller itself does not rely on the model. Example 3. Consider the MIMO controlled Lorenz chaotic system [28] y˙ = a(y − y ) + u , 1 2 1 1 y˙2 = cy1 − y2 − y1 y3 + u2 , (42) y˙3 = y1 y2 − by3 + u3 . The autonomous Lorenz system (without control u1 , u2 , u3 ) is symmetric with respect to the z axis. There are three equilibria S0 = [0, 0, 0]T ,

S1 = [p, p, q]T , S2 = [−p, −p, −q]T , p = b(c − 1), q = c − 1. (43)

If take a = 10, b = 8/3, c = 28, we have p = 8.4853, q = 27. Fig. 4(a), (b) show the Lorenz attractor with initial values y10 = 0.3, y20 = 0.3, y30 = 2, where symbols ‘◦’, ‘∗’ and ‘3 denote S0 , S1 and S2 , respectively. We clearly see that the trajectory of the uncontrolled system move around equilibria S1 and S2 with butterfly form, but it cannot reach any one of the three equilibria. Yu [3] used slide mode control method to stabilize the chaotic system based on the model (42). We use adaptive MHODFC to achieve stabilization. The stabilization objective is that the trajectory of the controlled system (42) switches among the three equilibria, which means that the trajectory firstly approximates S0 , secondly S2 , thirdly S1 , at last comes back to S0 , with staying 15 (s) at each equilibrium. We describe the objective with three variables, yielding [y¯r1 (t), y¯r2 (t), y¯r3 (t)]T = [0, 0, 0]T , when 0 t < 15, (t), y¯r2 (t), y¯r3 (t)]T = [−p, −p, q]T , [ y ¯ r1 when 15 t < 30, (44) [y¯r1 (t), y¯r2 (t), y¯r3 (t)]T = [p, p, q]T , when 30 t < 45, T T [y¯r1 (t), y¯r2 (t), y¯r3 (t)] = [0, 0, 0] , when 45 t < 60. To avoid oscillation in switching transient instant, we soften the given objective via transfer function G(s) = 9/(s 2 + 6s + 9), i.e., the softened variables are written

G. Qi et al. / Physics Letters A 344 (2005) 189–202

199

Fig. 5. The controlled Lorenz system with parameters a = 10, b = 8/3, c = 28 and initial values y10 = 2, y20 = 1, y30 = 2. (a) 3D view, (b) time response.

as [yr1 , yr2 , yr1 ]T = G(s)[y¯r1 , y¯r1 , y¯r1 ]T .

described by

(45)

Therefore, the stabilization objective is that the trajectories y1 , y2 , y3 of the controlled system (42) track the softened objectives yr1 , yr2 , yr3 , respectively. The ˆ¯ = controller is adaptive MHODFC (41), where E i (1) (1) T ¯ i = [20, 1], i = [yri − yi , yˆr − yˆi ] and taking K 1, 2, 3. Take the parameters n0 = 5, a = 5 of the HOD. uˆ i is filtering values of ui , which can be obtained via HOD with the parameters n0 = 5, a = 10 based on past information ui . Control results of 3D view is shown in Fig. 5(a), in which we can see the trajectory of the system starts with initial point, then rapidly approximates S0 , then S2 , S1 in turn, at last comes back to S0 . Fig. 5(b) shows time responses of the three variables, which clearly display that the switching processes among three equilibria are all smooth and rapid, and all variables reach the given location precisely and stay the required time, the transient time is only 0.2 (s), and oscillations are small. Note that the controller (41) does not rely on the model of the Lorenz system. If the parameters a, b, c of the Lorenz are taken as other values, even the model function is changed, so much as the controlled plant is other system, such as the Chen system [29], which is

y˙ = a(y − y ), 1 2 1 y˙2 = (c − a)y1 − y1 y3 + cy2 , y˙3 = y1 y2 − by3 ,

(46)

when a = 35, b = 8/3, c = 28, the system is chaotic. Fig. 6(a) shows the 3D state space diagram with initial values y10 = 2, y20 = −1, y30 = 3. The controller is still adaptive MHODFC (41), and all parameters are not changed. The objective is still the trajectory of the Chen system switching among the three equilibria, and is described by (44) and (45). It is notable that the values of p, q in the Chen system are different from those in the Lorenz system. The control result is same as the Lorenz system, which is shown in Fig. 6(b), and further verifies the controller is strongly robust. If the control law is added the Chen system after 3 (s), the initial process moves along the trajectory of the chaotic attractor, but when it is controlled, the orbit of the Chen system rapidly approximates equilibria S0 , and wander among the three equilibria according objective requirement, as shown in Fig. 6(c), (d). The same, if the controlled system is changed into the Rössler system, but controller is not changed including all parameters, the control result still holds ideally (the diagrams of the controlled result are omitted).

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Fig. 6. The Chen system with a = 35, b = 8/3, c = 28 and y10 = 2, y20 = −1, y30 = 3. (a) 3D view without control. (b) 3D view under control after initial instant. (c) 3D view under control after 3 (s), (d) time responses of trajectory under control after 3 (s).

Recently, Qi et al. constructed a new 4D autonomous chaotic system [27], which has a cubic cross-product smooth nonlinearity in each equation. The system can generate complex dynamics with wide parameters ranges, such as chaos, Hopf bifurcation, period-double bifurcation, periodic orbit, sink and source et al. found by means of Lyapunov exponents and bifurcations. Now, we control the system using adaptive MHODFC. Example 4. The controlled system is a new 4D chaotic system by us [27], is written as

x˙ = a(x2 − x1 ) + x2 x3 x4 + u1 , 1 x˙2 = b(x1 + x2 ) − x1 x3 x4 + u2 , x˙3 = −cx3 + x1 x2 x4 + u3 , x˙4 = −dx4 + x1 x2 x3 + u4 ,

(47)

where xi (i = 1, 2, 3, 4) are the state variables of the system, and a, b, c, d are positive real constant parameters. Setting two parameter sets: Set 1: a = 35,

b = 10,

c = 1,

d = 10,

Set 2: a = 35,

b = 10,

c = 1,

d = 26. (48)

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201

Fig. 7. Comparing curves between responses and objectives of controlled system (47). (a) Case 1: 3D views of driving the periodic orbit mode to chaos mode, (b) Case 2: 3D views of driving the chaotic mode to periodic mode.

When the parameters are taken set 1 and set 2 respectively, system (47) without control is in chaos with positive Lyapunov exponent λ1 = 3.3152 and in periodic orbit mode, respectively, as shown in Fig. 7(a), (b) (red curves).1 Let [x1 , x2 , x3 , x4 ]T and [xr1 , xr2 , xr3 , xr4 ]T be the system response (or called output) and the objective (or called given input), respectively. The control objectives are divided two cases: Case 1: The control objective is to drive the trajectory of the system (47) from periodic-orbit mode with parameter set 2 to chaotic mode with parameter set 1. Case 2: The objective is right contrary to case 1. ˆ¯ = Controller is adaptive MHODFC (41), where E i (1) (1) ¯ i = [20, 1], i = 1, 2, 3, 4. [xri − xi , xˆri − xˆi ]T , K The parameters of the HOD are taken as n0 = 5, a = 15, same as Example 2. Using higher accuracy filter HOD with the parameters n0 = 5, a = 30 to estimate uˆ i , the whole running time is 20 (s). The control results are shown in Fig. 7(a), (b), respectively, where red solid line and blue dotted line describe response [x2 , x3 , x4 ]T and objective

1 For interpretation to colour in this figure, the reader is referred

to the web version of this Letter.

[xr2 , xr3 , xr4 ]T of system (47), respectively. Both cases show that the control results are very ideal. It is notable that the controllers of two cases are completely same. Via calculating four variables, the average steady absolute errors are listed as follows: |e¯1 | = 6.4347 × 10−5 ,

|e¯2 | = 1.2718 × 10−4 ,

|e¯3 | = 3.9252 × 10−5 ,

|e¯4 | = 3.8 × 10−3 .

5. Conclusions We have found that information about the measured system output, given target, their differentials up to the nth order, and control input are very useful for adaptive control of uncertain (chaotic) systems. Based on the information, a model-free controller, i.e., an adaptive higher-differential feedback controller (HODFC), for nonlinear SISO and MIMO affine systems has been presented in detail. Theoretically analysis shows that the HODFC makes the closed-loop system asymptotically stable, and reaches the objective in the sense that the system output and its differentials up to (n − 1)th order asymptotically track those of the given objective, respectively. The controller has clear physical meanings and its parameters are closely related to the performance of the closed-loop system.

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The Duffing–Holmes system can be successfully controlled to a given orbit or other objectives by applying the designed adaptive HODFC. Some MIMO chaotic systems such as the Lorenz system family, Chen system, Rössler system, a new four-dimensional chaotic system, can all be controlled to any given target points or any desired orbits even chaotic orbits under the proposed adaptive MHODFC. The HODFC has strong robustness against disturbances and changes of the system structures. It is notable that the proposed modelfree control scheme was derived for general affine systems, so it is suitable for both nonchaotic systems and chaotic systems.

Acknowledgement This research is supported by a grant from National Nature Science Foundation of China 60374037 and by a grant from Science and Technology Development Foundation of Tianjin Colleges 20051528.

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