DOI 10.1007/s11071-010-9801-8 O R I G I N A L PA P E R
Variable structure based robust backstepping controller
design for nonlinear systems
Chao-Chung Peng· Albert Wen-Jeng Hsue · Chieh-Li Chen
Received: 13 January 2010 / Accepted: 6 August 2010 / Published online: 5 September 2010 © Springer Science+Business Media B.V. 2010
Abstract This study presents robust control architec-ture in the sense of variable strucarchitec-ture control via a backstepping design. By using systematic backstep-ping design techniques, closed-loop behavior of an n-order nonlinear system can be transformed into a stability and convergence problem of a fast switched 2nd order system. There are two main parts contained within the proposed control algorithm; one is a nom-inal control effort generated according to the Lya-punov stability criterion during recursive backstepping processes, and the other belongs to a smooth robust control law designed to eliminate the effects of un-known lumped perturbations. Finally, a Genesio sys-tem is used as an illustrated example to demonstrate the robustness of the control algorithm. The feasibility and properties of the proposed method are given by numerical simulations.
Keywords Chattering· Backstepping · Variable structure· Nonlinear system · Chaotic
C.-C. Peng· C.-L. Chen ()
Department of Aeronautics and Astronautics,
National Cheng Kung University, Tainan, Taiwan, R.O.C. e-mail:[email protected]
A.W.-J. Hsue
Department of Mechatronic Technology, Dahan Institute of Technology, Hualien County, Taiwan, R.O.C.
1 Introduction
A chaotic dynamic is a highly complex nonlinear phe-nomenon which exhibits a number of interesting char-acteristics, including but not limited to unpredictable behavior and excessive sensitivity to different initial conditions. The behavior of a chaotic system is some-times undesirable, however, owing to its powerful ap-plications in engineering (e.g., chemical reactions, bi-ological systems, and secure communications, etc.); controlling these nonlinear chaotic dynamics for ex-tensive application fields has become an attractive area of study.
Recently, the backstepping design technique has been widely used to stabilize and synchronize a va-riety of chaotic systems [1–5]. However, rejection due to model uncertainties or disturbances has not been ad-dressed in these studies. In realistic, complete knowl-edge of the system parameters is not an easy task es-pecially for control practices. In regard to chaotic sys-tems subjected to model uncertainties, several adap-tation laws have been developed to estimate the un-certain parameters [6–9]. On the other hand, when the system is affected by unknown perturbations, vari-able structure control (VSC) theory [10–12] is a good choice to handle these changes in chaotic systems ow-ing to its inherent advantages, which include fast re-sponse, good performance and insensitivity to para-meters (e.g., deviation and exogenous disturbances). Especially, it is simple to implement. Unfortunately,
VSC also includes an inherent drawback, the chatter-ing phenomenon.
In theory, VSC offers stability and robustness to systems through high gain with infinite fast switching. However, it is sometimes difficult to realize in some physical systems due to its undesirable chattering ac-tion which may excite high order unmodeled dynam-ics, damage actuators, and even cause instability. For these reasons, there are two main approaches to cope with chatter: (1) set a boundary layer (or a sigmoid function) around the switching surface for smooth-ing the control effort inside the boundary, and (2) de-rive a higher order sliding mode under specific con-ditions where the value of augmented sliding state is available [3,13] or needed to be detected online [14]. Therefore, it is highly desirable to develop a new ro-bust chattering free controller for chaotic systems to not only preserve the inherent advantages of VSC, but also avoids chatter and releases the limitation of knowing the derivation of the augmented sliding vari-able.
In this work, a robust nonlinear backstepping con-trol algorithm is proposed. The feature of the proposed method is that a stability problem of an n-order system can be transformed into a stability and convergence problem of a fast switched 2nd system. In addition, the proposed control law involves neither exact parame-ters nor additional information of system states. The developed control algorithm also makes the system in-sensitive to model uncertainties and external perturba-tions when infinity switching control is applied to the auxiliary system. Finally, control signals generated by the proposed algorithm are smooth. Therefore, com-pared with the VSC, the developed controller is more adequate for practical implementations.
2 Problem formulation for uncertain nonlinear systems by backstepping
Consider the following n-order system with strict feedback form: ˙x1= f1(x1)+ g1(x1)x2 ˙x2= f2(x1, x2)+ g2(x1, x2)x3 ˙x3= f3(x1, x2, x3)+ g3(x1, x2, x3)x4 .. . (1) ˙xn−1= fn−1(x1, x2, x3, . . . , xn−1) + gn−1(x1, x2, x3, . . . , xn−1)xn ˙xn= fn(x1, x2, . . . , xn)+ gn(x1, x2, . . . , xn)u+ H (•) where x∈ Rn, u∈ R. With fi(0)= 0 and gi(0)= 0 for i= 1, . . . , n, fi, gi are smooth functions and are differentiable. The unknown but smooth lumped per-turbation is denoted as H (•), which includes model uncertainties and exogenous disturbances.
In the following, a coordinated transformation will be applied to the system (1) by utilizing backstepping design scheme.
First, treat the system state x2as an independent in-put, then there exists a state feedback stabilizing con-trol law φ1(x1)which is of the form
x2= φ1(x1)= 1 g1(x1) −f1(x1)− k1x1 (2) where k1>0. Consider a Lyapunov function of the subsystem x1 of the form V1= x12/2. From (2), the derivative of the Lyapunov candidate is
˙V1= x1˙x1= −k1x21≤ 0 (3) Substitute (2) into (1), by adding and subtracting g1(x1)φ1(x1)(i.e., a virtual control law) to the sub-system x1and define a new error variable z1= x2−
φ1(x1), the subsystem (x1, z1)can then be represented as
˙x1= f1(x1)+ g1(x1)φ1(x1)+ g1(x1)z1 ˙z1= f2(x1, x2)+ g2(x1, x2)x3− ˙φ1(x1)
(4) In a similar manner, taking x3as an independent input such that a stabilizing control law φ2(x1, x2)can be found as x3= φ2(x1, x2) = 1 g2(x1, x2) × −∂V1 ∂x1 g1(x1)+ ˙φ1(x1)− f2(x1, x2)− k2z1 (5) Substitute (5) into (4) and consider a Lyapunov func-tion candidate V2= V1+12z12, then one can derive
where k2>0. Define an error variable as z2= x3−
φ2(x1, x2). In the control point of view, it can be found that if the error variables z2approaches zero, the as-ymptotical stability of the subsystem (x1, z1)can be guaranteed. Consider (4), by adding and subtracting g2(x1, x2)φ2(x1, x2)to the subsystem z1, then the sub-system (z1, z2)can be represented as
˙x1= f1(x1)+ g1(x1)φ1(x1)+ g1(x1)z1 ˙z1= f2(x1, x2)+ g2(x1, x2)φ2(x1, x2)
+ g2(x1, x2)z2− ˙φ1(x1)
˙z2= f3(x1, x2, x3)+ g3(x1, x2, x3)x4− ˙φ2(x1, x2) (7)
By using the recursive manner up to the final subsys-tem, then one can obtain the following coordinated n− 1 order system: ˙x1= f1(x1)+ g1(x1)φ1(x1)+ g1(x1)z1 ˙z1= f2(x1, x2)+ g2(x1, x2)φ2(x1, x2) + g2(x1, x2)z2− ˙φ1(x1) ˙z2= f3(x1, x2, x3)+ g3(x1, x2, x3)φ3(x1, x2, x3) + g3(x1, x2, x3)z3− ˙φ2(x1, x2) ˙z3= f4(x1, x2, x3, x4) + g4(x1, x2, x3, x4)φ4(x1, x2, x3, x4) + g4(x1, x2, x3, x4)z4− ˙φ3(x1, x2, x3) .. . ˙zn−2= fn−1(x1, . . . , xn−1) + gn−1(x1, . . . , xn−1)φn−1(x1, . . . , xn−1) + gn−1(x1,· · · , xn−1)zn−1 − ˙φn−2(x1, . . . , xn−2)
and a final subsystem in which the control input ap-pears is
˙zn−1= fn(x1, . . . , xn)+ gn(x1, . . . , xn)u
− ˙φn−1(x1, . . . , xn−1)+ H(•) (8) Note that with the aid of the recursive manner, the sta-bility of the n-order system has simplified as a regula-tion problem on the last transformed state zn−1, which is the resulting transformed state based on the Lya-punov stability requirements for each subsystem. In
explicit words, the state zn−1 can be considered as a performance index such that once zn−1 subjected to unknown perturbations can be forced to zero, then the asymptotical stability of the system (1) can be guaran-teed.
With respect to the final subsystem (8), design a superior dynamic algorithm in the form of
˙vs= ξ sgn(zn−1) (9)
where ξ denotes a robust gain, which satisfies a spe-cific condition, will be introduced later.
The robust backstepping control algorithm for (8) is designed as follows: u= 1 gn(x1, . . . , xn) −fn(x1, . . . , xn) + ˙φn−1(x1, . . . , xn−1)− knzn−1− vs (10) where gn(x1, . . . , xn) = 0 and fn(x1, . . . , xn) are known functions.
Different from the relevant studies presented in [3] and [13], the control law given in (32) does not involve extra state measurement, so the modified controller is adequate for practical realization.
The following work is going to show that by using the proposed control law (10); the model uncertainties can be eliminated as well as external disturbances such that the controlled system can track arbitrary desired orbit.
3 An auxiliary system
Based on the previous result, by substituting the pro-posed algorithm (10) into (8), it leads the following auxiliary system ˙E ˙zn−1 = 0 0 −1 −kn E zn−1 + ξsgn(zn−1)− Ω(•) 0 (11) where E= vs− H (•) is defined as an estimated error and Ω(•) = ∂H(•)/∂t belongs to an unknown contin-uous function which satisfies that
Ω(•)∞= sup
t∈[0,∞)
for some control regions of interest and γ is a bounded positive constant.
3.1 Stability and convergence of the auxiliary system Differentiating the subsystem˙zn−1one more time, one can derive the a 2nd as follows:
¨zn−1+ kn˙zn−1= −ξsgn(zn−1)+ Ω(•) (13) where ξ satisfies the condition that ξ > γ >Ω(•)∞. For the system with discontinuous right-hand side, one condition is that
⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ¨zn−1+ kn˙zn−1= −ξ + Ω(•)∞= −μ−, for zn−1≥ 0 ¨zn−1+ kn˙zn−1= ξ − Ω(•)∞= μ−, for zn−1<0 (14)
and another possible condition is that ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ¨zn−1+ kn˙zn−1= −ξ − Ω(•)∞= −μ+, for zn−1≥ 0 ¨zn−1+ kn˙zn−1= ξ + Ω(•)∞= μ+, for zn−1<0 (15)
In summary, the switched system structures given in (14)–(15) can be dominated by the following com-pact form:
¨zn−1+ kn˙zn−1= −μ sgn(zn−1) (16) where μ∈ [μ−, μ+] and μ+= ξ +Ω(•)∞> μ−= ξ− Ω(•)∞>0.
For the second order system (16), we propose the following Lyapunov candidate:
Vs= 1 2[zn−1 ˙zn−1] α1 α3 α3 α2 zn−1 ˙zn−1 + α4μ|zn−1| (17) where αj with j= 1, . . . , 4 are undetermined positive parameters satisfying α1α2> α32.
Taking the time derivative of (17) gives ˙Vs= α1zn−1˙zn−1+ α2˙zn−1 −kn˙zn−1− μ sgn(zn−1) + α3˙z2 n−1+ α3zn−1 −kn˙zn−1− μ sgn(zn−1) + μα4˙zn−1sgn(zn−1) = α1zn−1˙zn−1− α2kn˙z2n−1− μα2˙zn−1sgn(zn−1) + α3˙z2 n−1− knα3zn−1˙zn−1 − μα3zn−1sgn(zn−1) + μα4˙zn−1sgn(zn−1) = (α1− knα3)zn−1˙zn−1− (α2kn− α3)˙z2n−1 + μ(α4− α2)˙zn−1sgn(zn−1) − μα3|zn−1| (18)
For a given kn, if there exist proper values of αj such that the following (in)equalities:
α1= knα3 α2= α4, α2kn> α3 α1α2> α32 (19) are satisfied, then (18) reduces to
˙Vs= −μα3|zn−1| − (α2kn− α3)˙z2n−1 (20) As a consequence, the origin of (16) is a stable equi-librium point and thereby the asymptotic stability is achieved.
3.2 Control in the sense of high gain
Equation (16) clearly indicates that the stability of an n-order system has simplified as the stability problem of a switched system. In order to illustrate the aver-age characteristic of (13), a phase portrait is taken into account. Figure1 is made up of two regions, a left-hand side and a right-left-hand side, which are conducted by (16). The phase trajectories illustrate that every dif-ferent initial point will be guided to the origin. To elu-cidate the convergent behavior of such the switched by means of high gain manner, the system (13) is reconsidered. Suppose that the initial values of the auxiliary system are all zero and then it can be re-formulated into a standard 2nd order system with a feedback controller depicted in Fig.2. Regarding this standard control architecture, the controller κ(zn−1), which is a function of zn−1, represents the action of
ξsgn(•). Note that κ(zn−1) can be treated as a self-manipulated logical gain. It possesses a nature that the magnitude varies with respect to the size of zn−1 au-tomatically. In more explicit words, the state depen-dent gain κ(zn−1)has the feature that κ(zn−1)·zn−1=
Fig. 1 Phase trajectories of switched system: (a) full region (b) partial region
Fig. 2 Representation of the auxiliary system
Fig. 3 Characteristic of the logical gain κ(zn−1)with respect to|zn−1|
Fig.3, where the rectangular areas are always equiva-lent, i.e., Aabcd= Aabcd= κ · |zn−1| = ξ. The repre-sentation of the nonlinear component is analogous to the analysis technique adopted to forecast whether a limit cycle occurs when a system is subjected to in-put nonlinearities [15]. In the following, the stability and convergence feature of (13) is addressed in the sense of high gain control. First, according to Fig.2, it is confident that the system is stable by choosing
Fig. 4 Roots loci of the 2nd order auxiliary system
kn>0. Once knhas been determined, the correspond-ing roots loci is drawn Fig.4. In addition refer to (16), owing to μ > 0, the control force always counteracts the direction of system output zn−1. In other words, the state zn−1will be forced toward zn−1= 0. Mean-while, consider Fig.3, owing to the nature of the log-ical control gain κ(zn−1), it reveals that the smaller the |zn−1|, the larger the κ(zn−1) such that the un-desirable effects caused by external disturbance can be attenuated as small as possible by means of high gain control. Another alternative to analyze the type-1 system, i.e., (16), subject to a relay control input in frequency domain is the use of describing function method [16]. It can be shown that the intersection of the nonlinear control component, i.e., the relay con-trol, and the type-1 system is at origin. This reveals that the resulting limit cycle is with infinite large fre-quency but zero magnitude, which is consistent with the result presented in Sect.3.1.
Fig. 5 Phase trajectories of the auxiliary system: (a) full region (b) partial region
Fig. 6 Time responses of zn−1
3.3 Properties of the auxiliary system
In order to illustrate the phenomena of the auxiliary system, numerical simulations were performed and the corresponding parameters in (11) were set to be: kn= 6, ξ = 5 and Ω(•)∞= 2. The initial condi-tion was placed at (0.5, 0.3). Figure5(a) shows the phase portrait of the auxiliary system and its conver-gent behavior. According to the phase portrait analy-sis, one can find that zn−1˙zn−1<0 in both the re-gions I and II as depicted in 5(b), the slope of the dashed line is−1/kn. A larger kn causes the dashed line rotating counterclockwise and thereby results in
Fig. 7 Phase trajectories of the auxiliary 2nd system
a larger convergent rate. By the geometric sense, in-creasing kn expands the convergent areas I and II. Note that in the control point of view, the state E, which includes unknown perturbations, is not avail-able in actual. Figure6illustrates that the closed-loop response of zn−1 behaves a 2nd order liked response and Fig.7 shows the corresponding convergent phe-nomena of (16) by using phase portrait versus time. The applied control effort on the auxiliary 2nd or-der system (not for the real control systems) is de-picted in Fig.8and it indicates that robustness of the auxiliary system is achieved by the fast switching ac-tions.
Fig. 8 Applied switching force on the auxiliary 2nd system
Remark The proposed control strategy acts in a sim-ilar manner as behaves in the VSC that the stabil-ity and convergence is guaranteed by infinite fast switching control. However, in this study, the switch-ing effort is applied on the auxiliary system (or so called an augmented system) instead of the real n-order system. So, the actual control force generated by (9) is smooth without serious chattering phenom-enon. It should be emphasized that the stability proof addressed in Sect.3.1is based on the condition given in (12). For any given nonlinear systems, (12) might only be available for some local regions of control in-terest and thereby the proposed controller provides lo-cal robustness.
4 An illustrated example
In this section, Genesio system is taken as a paradigm for demonstrating the proposed robust smooth back-stepping control algorithm. The mathematical model for the Genesio system is described as follows:
˙x1= x2 ˙x2= x3
˙x3= −cx1− bx2− ax3+ x2 1
(21)
The parameters of used in (21) are: a= 1.2, b = 2.92, c= 6 and the initial position is located at (3, −4, 2). The corresponding dynamic behavior in phase plane (x1, x2, x3)is shown in Fig.9(a). Consider the chaotic
system subjected to external perturbation d(t) and control input as the following form:
˙x1= x2 ˙x2= x3
˙x3= −cx1− bx2− ax3+ x2
1+ d(t) + u(t)
(22)
Let (xd1, xd2, xd3)as the reference signal and then
de-fine tracking error as e1= xd1−x1, e2= xd2−x2, e3= xd3− x3so that the system can be represented in the
form of error dynamics ˙e1= ˙xd1− ˙x1= xd2− x2= e2 ˙e2= ˙xd2− ˙x2= xd3− x3= e3 ˙e3= ˙xd3− ˙x3= ˙xd3+ cx1+ bx2 + ax3− x2 1− d(t) − u(t) = ˙xd3+ (ˆc + ˜c)x1+ ( ˆb + ˜b)x2+ (ˆa + ˜a)x3 − (1 − β)x2 1− d(t) − u(t) = ˙xd3+ ˆcx1+ ˆbx2+ ˆax3+ ˜cx1+ ˜bx2+ ˜ax3 − x2 1+ βx 2 1− d(t) − u(t) = f (x, xd)+ H (x, t) − u(t) (23)
where the nominal term f (x, xd)= ˙xd3+ ˆcx1+ ˆbx2+
ˆax3+ βx2
1 is known and the lumped perturbation is defined as H (x, t)= ˜cx1+ ˜bx2+ ˜ax3−x12−d(t). Note that ˆa, ˆb and ˆc are the nominal values of a, b, and c, respectively. The corresponding estimated deviations ˜a, ˜b, ˜c are defined as ˜a = a − ˆa, ˜b = b − ˆb, ˜c = c − ˆc and 0 < β < 1.
The transformed procedures via backstepping tech-nique are listed as follows.
Step 1 Consider the system state e2 as an indepen-dent input and let
e2= φ1(e1)= −k1e1, k1>0 (24) Select a Lyapunov function V1= e21/2. It can then be obtained that
˙V1= e1˙e1= −k1e21≤ 0 (25) and state e1is asymptotically stable.
Step 2 Actually, there may be different between e2 and φ1(e1). Therefore, defining a new error variable
Fig. 9 The chaotic trajectories: (a) without disturbance (b) with disturbance
z1 = e2− φ1(e1) which presents the difference be-tween the stabilizing control law φ1(e1) and the er-ror state e2. It is obvious that the inequality (25) can be satisfied if z1 tends to zero. By adding and sub-tracting the virtual control law φ1(e1)to the first equa-tion of (23), one can get the dynamic of the subsystem (e1, z1)as follows:
˙e1= z1+ φ1(e1) ˙z1= e3− ˙φ1(e1)
(26) In a similar manner, treat the state e3as an independent input which is of the form
e3= φ2(e1, z1)= −e1− k2z1+ ˙φ1(e1),
k2>0 (27)
Select a Lyapunov candidate to the subsystem (e1, z1) in the form of
V2= e21/2+ z21/2 (28)
and the derivative of (28) is ˙V2= e1˙e1+ z1˙z1 = e1φ1(e1)+ z1 + z1e3− ˙φ1(e1) = e1φ1(e1)+ z1 + z1−e1− k2z1+ ˙φ1(e1)− ˙φ1(e1) = −k1e21− k2z21≤ 0 (29) Thus, the subsystem (e1, z1)is asymptotically stable.
Step 3 By adding and subtracting the virtual control law φ2(e1, z1)to the subsystem z1and defining an er-ror variable as z2= e3− φ2(e1, z1), one can further derive ˙e1= z1+ φ1(e1) ˙z1= z2+ φ2(e1, z1)− ˙φ1(e1) (30) and ˙z2= f (x, xd)+ H (x, t) − u(t) − ˙φ2(e1, z1) (31) Based on the backstepping design procedure, the transformed system is asymptotically stable if z2can be suppressed to zero. Therefore, the resulting control algorithm for (31) is
u= k3z2− ˙φ2(e1, z1)+ f (x, xd)+ v
= (k1+ k2+ k3)e3+ (1 + k1k2+ k2k3+ k1k3)e2 + (k3+ k1k2k3)e1+ f (x, xd)+ ν (32)
where the extra control effort ν is generated by
˙ν = ξ sgn(z2) (33)
Substitute (32) and (33) into (31), one can get the same form as well as (11) ˙E ˙z2 = 0 0 −1 −k3 E z2 + ξsgn(z2)− Ω(x, t) 0 (34)
where E= ν − H (x, t) and Ω(x, t) = ∂H(x, t)/∂t. In accordance with the previous analysis, one can con-clude that the auxiliary system (34) is stable and z2 will approach to zero.
5 Numerical simulations
In this section, simulation results of the Genesio chaotic system are performed to demonstrate the fea-sibility of the proposed method. In this case, n= 3. The disturbance term applied in (22) is selected as d(t )= 1.1 cos(0.86t) and the perturbed trajectory is illustrated in Fig.9(b). Control gains and nominal pa-rameters used in (32) and (33) are: k1= 2, k2= 4,
k3= 6, ˆa = 1.1, ˆb = 2.9, ˆc = 5.8, β = 0.85 and
ξ= 25. Since kn= k3= 6, it can be checked that there exist proper values of aj satisfying (25). Desired tra-jectory is given by xd= sin(1.5πt). With the aid of the proposed control law, from Fig.10, one can find that the system states track the desired trajectory precisely when the control law presented in (32) is in action (after t = 4). The corresponding error responses are illustrated in Fig.11, which clearly illustrates that the tracking errors converge towards zero by the use of the proposed controller. According to the proceeding analysis, the asymptotically stability can be guaran-teed once z2approaches to zero. Figure12shows the convergent response of the transformed state z2 and Fig.13is the actual applied control effort, which acts smoothly without chatter.
Fig. 10 Controlled states responses
Fig. 11 Error responses
Fig. 12 Response of z2
6 Conclusions
In this paper, a robust control design for an uncer-tain chaotic system is proposed. The advantages of this control architecture can be summarized as fol-lows: (a) control problem of an n-order system can be simplified as a stability and convergence problem of a 2nd order system; (b) it provides robustness to a nonlinear system subjected to model uncertainties and exogenous disturbance; (c) no information of exact pa-rameters, external disturbance and augmented state are involved in the proposed control algorithm; and (d) fi-nally, the control law has high robustness and is prac-tical realizable. These advantages have been verified through numerical simulations.
Acknowledgements The work was partly supported by the National Science Council, Taiwan, under the grant No. NSC95-2221-E006-276.
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