Asian Journal of Control, Vol. 6, No. 4, pp. 447-453, December 2004 447

ADAPTIVE SLIDING MODE BACKSTEPPING CONTROL OF NONLINEAR SYSTEMS WITH UNMATCHED UNCERTAINTY

Ali J. Koshkouei, Alan S. I. Zinober, and Keith J. Burnham

ABSTRACT

This paper considers an adaptive backstepping algorithm for designing the control for a class of nonlinear continuous uncertain processes with dis-turbances that can be converted to a parametric semi-strict feedback form. Sliding mode control using a combined adaptive backstepping sliding mode control (SMC) algorithm, is also studied. The algorithm follows a systematic procedure for the design of adaptive control laws for the output tracking of nonlinear systems with matched and unmatched uncertainty.

KeyWords: Adaptive systems, backstepping, sliding mode control, nonlinear systems, Lyapunov method.

I. INTRODUCTION

The backstepping procedure is a systematic design technique for the globally stable and asymptotically adaptive tracking control of a class of nonlinear systems. Adaptive backstepping algorithms have been applied to systems which can be transformed into a triangular form, in particular, the parametric pure feedback (PPF) form and the parametric strict feedback (PSF) form [9]. This method has been studied widely in recent years [8,9, 16-19]. When plants include uncertainty with lack of information about the bounds of unknown parameters, adaptive control is more convenient; whilst, if some information about the uncertainty, such as bounds, is available, robust control is usually employed.

If a plant has matched uncertainty, the system may be stabilized via state feedback control [3]. Some techniques have been proposed for the case of plants containing unmatched uncertainty [5]. The plant may contain unmodelled terms and unmeasurable external disturbances, bounded by known functions.

The stabilization of dynamical systems with uncer- tainties has been studied in the recent years [1,3,4,6,14].

Most approaches are based upon Lyapunov and lineari-sation methods to design a control. In the Lyapunov approach, it is very difficult to find a Lyapunov function to find a control to stabilise the system. The linearisation approach yields local stability. The backstepping ap-proach presents a systematic method for designing a control to track a reference signal, by selecting an ap-propriate Lyapunov function by changing the coordinate [8,9]. The robust output tracking of nonlinear systems has been studied by many authors [2,12,16].

Sliding mode control (SMC) is a robust control method and backstepping can be considered to be a method of adaptive control. The combination of these methods yields benefits from both approaches.

A systematic design procedure has been proposed to combine adaptive control and SMC for nonlinear sys-tems with relative degree one [21]. The sliding mode backstepping approach has been extended to some classes of nonlinear systems which need not be in the PPF or PSF forms [16-19]. A symbolic algebra toolbox allows straightforward design of dynamical backstep-ping control [15].

The adaptive sliding backstepping control of semi-strict feedback systems (SSF) [20] has been stud-ied by Koshkouei and Zinober [10]. The method ensures that the error state trajectories move on a sliding hyper-plane.

In this paper we develop the backstepping approach for SSF systems with unmatched uncertainty. We also design a controller based upon sliding backstepping mode techniques so that the state trajectories approach a

Manuscript received January 15, 2003; revised April 1,2003; accepted March 19, 2004.

Ali J. Koshkouei and Keith J. Burnham are with Control Theory and Applications Centre, Coventry University, Coven-try CV1 5DB, UK.

Alan S. I. Zinober is with Department of Applied Mathe-matics, The University of Sheffield, Sheffield S10 2TN, UK.

448 Asian Journal of Control, Vol. 6, No. 4, December 2004

specified hyperplane. These systematic methods do not need any extra conditions on the parameters and also any sufficient conditions for the existence of the sliding mode.

A backstepping method for designing a sliding mode control for a class of nonlinear system without uncertainties, has been presented by Rios-Bolívar and Zinober [15-17]. Their method needs some sufficient conditions for existence of the sliding mode. In this pa-per a method for designing sliding mode control is pro-posed which assures that the sliding mode occurs with-out satisfying these conditions and also guarantees the stability of the system.

We extend the classical backstepping method to the parametric semi-strict feedback form in Section 2 to achieve the output tracking of a dynamical reference signal. The sliding mode control design based upon the backstepping approach is presented in Section 3. We consider an example to illustrate the results in Section 4 with some conclusions in Section 5.

II. ADAPTIVE BACKSTEPPING CONTROL

Consider the uncertain system

),,()()()( twQuGGF ρ+ρ+θρ+ρ=ρ (1)

which is transformable into the semi-strict feedback form (SSF) [10,11,20]

1 1 2( , , , ) ( , , ), 1 1Ti i i i ix x x x x x w t i n+= + ϕ θ + η ≤ ≤ −

( ) ( ) ( ) ( , , )Tn n n n nx f x g x u x x w t= + + ϕ θ + η

1xy = (2)

where x =[x1 x2 … xn]T is the state, y the output, u the scalar control, and ϕi (x1, …, xi) ∈ Rp, i = 1, …, n, are known functions which are assumed to be sufficiently smooth. θ ∈ Rp

is the vector of constant unknown pa-rameters and ηi(x, w, t), i = 1, …, n, are unknown nonlinear scalar functions including all the disturbances. w is an uncertain time-varying parameter.

Assumption 1. The functions ηi(x, w, t), i = 1, …, n are bounded by known positive functions hi(x1, … xi) ∈

R, i.e.

nixxhtwx iii ,,1),,(|),,(| 1 =≤η (3)

The output y should track a specified bounded reference signal yr(t) with bounded derivatives up to n-th order.

The system (1) is transferred into system (2) if there exists a diffeomorphism x = x (ρ). The conditions of the existence of a diffeomorphism x = x (ρ) can be found in [13] and the input-output linearisation results in [7].

First, a classical backstepping method will be ex-tended to this class of systems to achieve the output tracking of a dynamical reference signal. The sliding mode control design based upon backstepping tech-niques is then presented in Section 3.

2.1 Backstepping algorithm

The design method based upon the adaptive back-stepping approach is now presented [10,11]. The func-tions that compensate the system disturbances, are con-tinuous. This method ensures that the output tracks a desired reference signal. Step 1. Define the error variable z1 = x1 − yr then

rT ytwxxxz −η+θϕ+= ),,()( 11121 (4)

From (4)

1 2 1 1 1ˆ ( , , )T T

rz x x w t y= + ω θ + η − + ω θ (5)

with ω1(x1) = ϕ1(x1) and ˆθ = θ − θ where )(~ tθ is an

estimate of the unknown parameter vector θ. Consider the stabilization of the subsystem (4) and

the Lyapunov function

θΓθ+=θ − ~~21

21)ˆ,( 12

111TzzV (6)

where Γ is a positive definite matrix. The derivative of V1 is

)ˆ(~)),,(ˆ()ˆ,( 111

112111 θ−ωΓΓθ+−η+θω+=θ − zytwxxzzV Tr

T

(7) Define τ1 = Γω1z1. Let

　 21 1 1 1 1 1 1

ˆ ˆ( , , )4

T atnx t c z h z eα θ = −ω θ − − (8)

with c1, a and positive numbers. Define the error variable

rytxxz −θα−= ),ˆ,( 1122

22 1 1 1 1 1

ˆ4

T atr

nx c z y h z e= + ω θ + − + (9)

Then

21 1 1 2 1 1 1 1( , , )

4T atnz c z z x w t h z e= − + + ω θ + η − (10)

and 1V is converted to

2 11 1 1 1 1 2 1

ˆ ˆ( , ) ( )at TV z c z z z en

− −θ ≤ − + + + θ Γ τ − θ

Step k. (1 < k ≤ n − 1). The time derivative of the error variable zk is

A. Koshkouei et al.: Adaptive Sliding Mode Backstepping Control of Nonlinear Systems 449

1( )1 1

1 11

ˆ ˆ ( )ˆ

kT kk k

k k k i k ri i

z x x y tx

−− −

+ +=

∂α ∂α= + ω θ − − θ + ξ −

∂ ∂θ∑

1T kk t

−∂α+ω θ −

∂ (11)

where 1

11 1

1

( , , ) ( , , )k

kk k k i i

i i

x x x xx

−−

=

∂αω = ϕ − ϕ

∂∑… …

212 21

14

kat k

k k ii i

n e h hx

−−

=

⎛ ⎞⎛ ⎞∂α⎜ ⎟ζ = + ⎜ ⎟⎜ ⎟∂⎝ ⎠⎝ ⎠∑

i

k

i i

kkk x

η∂α∂

−η=ξ ∑−

=

−1

1

1 (12)

Define zk+1 = xk+1 − αk − yr(k)

where

11

1 2 1 11

ˆ ˆ( , , , , , )k

T kk k k k k k i

i i

x x x t z c z xx

−−

− +=

∂αα θ = − − − ω θ +

∂∑…

21 1

11

ˆ ˆ

kk k i

k k k i ki

z zt

−− −

+=

⎛ ⎞∂α ∂α ∂α+ − ζ + τ + Γω⎜ ⎟∂ ∂θ ∂θ⎝ ⎠

∑ (13)

with ck > 0. Then the time derivative of the error vari-able zk is

1 1T

k k k k k k k k kz z c z z z− += − − + + ω θ + ξ − ζ

21

11

ˆ( )ˆ ˆ

kk i

k i ki

z−

−+

=

⎛ ⎞∂α ∂α− θ − τ + Γω⎜ ⎟

∂θ ∂θ⎝ ⎠∑ (14)

Consider the extended Lyapunov function

2 21

1

1 12 2

kT

k k k ii

V V z z−=

= + = + θ Γθ∑ (15)

The time derivative of Vk is

21

1

( 1)2

kat

k i i k ki

k kV c z z z en

−+

=

+≤ − + +∑

)ˆ(ˆ)ˆ(~ 1

11

1 θ−τ⎟⎠

⎞⎜⎝

⎛θ∂

α∂+θ−τΓθ+ ∑

−

=+

−k

k

ii

ik

T z (16)

since

∑=

− ωΓ=ωΓ+τ=τk

iiikkkk zz

11 . (17)

Step n. Define ( )

1n

n n n rz x y−= − α −

with αn−1 obtained from (13) for k = n. Then the time derivative of the error variable zn is

θθ∂

α∂−

∂α∂

−θω++= −−

=+

−∑ ˆˆ

ˆ),()()( 11

11

1 nn

ii

i

nTnnnn x

xtxuxgxfz

( )1 ( , )T nnn n rx t y

t−∂α

− + ω θ + ξ −∂

(18)

where )ˆ,( θω xn is defined in (12) for k = n. Extend the Lyapunov function to be

2 21

1

1 ( 1) 1 ( 1)2 2 2 2

i nat T at

n n n ii

n nV V z e z ea a a

=− −

−=

+ += + + = + θ Γθ +∑

(19) The time derivative of Vn is

1( 1)

2at

n n n nnV V z z e−

−+

= + −

2

2 11

1 1

ˆ ˆ( ) ( )ˆ

n nTi

i i i n ni i

c z z−

−+

= =

∂α⎛ ⎞≤ − − θ − τ + θ Γ τ − θ⎜ ⎟∂θ⎝ ⎠∑ ∑

(20) where

nTnnn zωΓ+τ=τ −1 (21)

We select the control

⎢⎣

⎡∂α∂

+θω−−−−= ∑−

=+

−−

1

11

11

ˆ)()(

1 n

ii

i

nTnnnnn

nx

xxfzcz

xgu

2

( )1 11

1ˆ ˆ

nnn n i

n i n r n ni

z y zt

−− −

+=

⎤⎛ ⎞∂α ∂α ∂α+ τ + − Γω + − ζ ⎥⎜ ⎟∂∂θ ∂θ ⎥⎝ ⎠ ⎦

∑

(22)

with cn > 0. Taking θτ=θ~,ˆ

n is eliminated from the right-hand side of (20). Then

0|||| 2

1

2 <−≤−≤ ∑=

zczcVn

iiin (23)

where inii

cc≤≤

= min . This implies that limt→∞ zi = 0, i =

1,2, …, n, particularly lim t→∞ (x1 − yr) = 0.

III. SLIDING BACKSTEPPING CONTROL

Sliding mode techniques yielding robust control and adaptive control techniques are both popular when there is uncertainty in the plant. The combination of these methods has been studied in recent years [16-19]. In general, at each step of the backstepping method, the new update tuning function and the defined error vari-ables (and virtual control law) take the system to the equilibrium position. At the final step the system is sta-bilized by suitable selection of the control.

The adaptive sliding backstepping control of SSF systems has been studied by Koshkouei and Zinober [10,11]. The controller is based upon sliding backstep-ping mode techniques so that the state trajectories ap-proach a specified hyperplane. The sufficient condition for the existence of the sliding mode, given by Rios-Bolívar and Zinober [15-17], is no longer needed.

To provide robustness, the adaptive backstepping

450 Asian Journal of Control, Vol. 6, No. 4, December 2004

algorithm can be modified to yield an adaptive sliding output tracking controller. The modification is carried out at the final step of the algorithm by incorporating the following sliding surface defined in terms of the error coordinates

01111 =+++=σ −− nnn zzkzk … (24)

where ki > 0, i = 1, …, n − 1, are real numbers. Addi-tionally, the Lyapunov function is modified as follows

12 2 1

1

1 1 1 ( 1)ˆ ˆ( ) ( )2 2 2 2

nT at

n ii

nV z ea

−− −

=

−= + σ + θ − θ Γ θ − θ +∑

(25) Let

1 1 1

11 1 1

n n n

n n n i i i i n i ii i i

k z k− − −

−= = =

⎛ ⎞⎛ ⎞ ⎛ ⎞τ = τ + Γσ ω + ω = Γ ω + σ ω + ω⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠

∑ ∑ ∑

(26) The time derivative of Vn is

)( 1122111

1

1

2−−−

−

=+++−−≤ ∑ nnn

n

iiin zkzkzkzzcV …

1

1 11 1

1

ˆ ˆ( ) ( ) ˆn

T n nn n n n i

i iz f x g x u x

x

−− −

− +=

⎡ ∂α ∂α+σ + + + ω θ − − θ⎢ ∂ ∂θ⎣

∑

( ) 21 11 2 1 1 1 1 1

ˆ ( )ˆ 4n atn n

n rny k z c z h z e

t− −∂α ∂α

− θ − + ξ − + − − + η∂∂θ

11

1 12

ˆ( )ˆ

ni

i i i i i i i i ii

k z c z z z−

−− +

=

∂α⎛+ − − + + ξ − ζ − θ − τ⎜ ∂θ⎝∑

2 2 1

1 11 1 1

ˆ ˆ

i n nl i

i l i n i il i i

w z z k− − −

+ += = =

⎤⎞ ⎛ ⎞ ⎛ ⎞∂α ∂α+Γ − Γ ω + ω ⎥⎟ ⎜ ⎟ ⎜ ⎟

∂θ ∂θ ⎥⎠ ⎝ ⎠ ⎝ ⎠ ⎦∑ ∑ ∑

21

11

ˆ ˆ( ) ( )ˆ

nTi

i n ni

z−

−+

=

∂α− θ − τ + θ Γ τ − θ

∂θ∑ (27)

since from (24), zn = σ − k1z1 − k2z2 − … − k n −1 zn −1.

Setting θτ=θ~,ˆ

n is eliminated from right-hand side of (27). Consider the adaptive sliding mode output tracking control

11 1

1 11

1 ˆ( ) ˆ( )

nT n n

n n n n iin i

u z f x xg x x

−− −

− +=

∂α ∂α⎡= − − − ω θ + τ +⎣ ∂∂θ∑

( ) 211 1 1 2 1 14

n atnr

ny k c z z h z et

− ⎛ ⎞∂α+ + − − + −⎜ ⎟∂ ⎝ ⎠

11

1 12

( )ˆ

ni

i i i i i i i n ii

k z c z z z−

−− +

=

∂α⎛− − − + − ζ − τ − τ⎜ ∂θ⎝∑

2 2 1

1 11 1 1ˆ ˆ

i n nl i

l i i n i il i i

z z k− − −

+ += = =

⎞∂α ∂α⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ Γω + Γ ω + ω⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟∂θ ∂θ ⎝ ⎠⎝ ⎠ ⎝ ⎠⎠∑ ∑ ∑

⎥⎦

⎤σ⎟

⎠⎞

⎜⎝⎛ +−σ− ∑

=)sgn(

1ii

n

ivkKW (28)

where kn = 1, K > 0 and W ≥ 0 are arbitrary real num-bers and

.1,11

1nih

xhv j

i

ki

jii ≤≤

∂α∂

+= −−

=∑ (29)

Then substituting (28) in (27) yields

1( 1)

2at

n nnV V e−

−−

= + σσ −

2121121 ||][][ σ−σ−−≤ −− WKzzzQzzz T

nn ……

where

1

2

1 2 1 1

0 00 0

n n

cc

Q

k k k c− −

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥⎢ ⎥

+⎢ ⎥⎣ ⎦

(30)

which is a positive definite matrix. 2

1 2 1 1 2 1Let [ ] [ ] | | . thenTn n nW z z z Q z z z K W− −= + σ + σ… …

0n nV W≤ − < (31)

which yields limt→∞ σ = 0 and lim t→∞ zi = 0, i =1, 2, …,

n−1. Particularly, lim t→∞ (x1 − yr) = 0. Since zn = σ − k1z1 − k2z2 − … − kn − 1 zn − 1, lim t→∞

zn = 0. Therefore,

the stability of the system along the sliding surface σ = 0 is guaranteed.

There is a close relationship between W ≥ 0 and K > 0. To reduce the chattering obtained from the discon-tinuous term, they should be tuned so that the desired performances are achieved. If K is very large with re-spect to W, unwanted chattering is produced. If K is sufficiently large, one can select W so that stability with a chattering reduction is established. W also affects the reaching time of the sliding mode. By increasing the value W, the reaching time is decreased. Remark 1. Alternatively, one can apply a different pro-cedure at the n-th step which yields

[ nnT

nnnn

xfzxg

u τθ∂

α∂+θω−−−= −

− ˆˆ)(

)(1 1

1

1( )1 1

11

nnn n

i ri i

x yx t

−− −

+=

∂α ∂α+ + +

∂ ∂∑

2

1 1 1 2 1 14atnk c z z h z e

⎛ ⎞− − + −⎜ ⎟

⎝ ⎠

1

1 1 12

n

i i i i i ii

k z c z z z−

− +=

⎛− − − + − ζ⎜⎝

∑

21

11

( )ˆ ˆ

ii l

n i l il

z−

−+

=

⎞⎛ ⎞∂α ∂α− τ − τ + Γω ⎟⎜ ⎟ ⎟∂θ ∂θ⎝ ⎠ ⎠

∑

A. Koshkouei et al.: Adaptive Sliding Mode Backstepping Control of Nonlinear Systems 451

2 1

11 1ˆ

n ni

i n i ii i

z k− −

+= =

∂α⎛ ⎞ ⎛ ⎞+ Γ ω + ω⎜ ⎟⎜ ⎟∂θ ⎝ ⎠⎝ ⎠∑ ∑

⎥⎦

⎤σ⎟

⎠⎞

⎜⎝⎛ +−σ− ∑

=

n

iii vkWK

1)sgn( (32)

with kn = 1, K > 0 and W ≥ 0 arbitrary real numbers and for all i, 1 ≤ i ≤ n

1

2 21

14

iat i

i k ij j

nv e h hx

−−

=

⎛ ⎞∂α⎜ ⎟= + +⎜ ⎟∂⎝ ⎠

∑ (33)

IV. EXAMPLE

Consider the second-order system in SSF form

1 2 1 1 2( , )x x x x x= + θ + η

ux =2 (34)

where |η| ≤ 2x12.We have

211 2xh = , 11 x=ω

ryxz −= 11 , 42 2 1 1 1 1 1

2ˆ atrz x x c z x z e y= + θ + + −

41 1 1 1 1 1

2ˆ atx c z x z eα = − θ − − , 11

12 x

x∂α∂

−=ω

2 1 1 2 2( )z zτ = Γ ω + ω , 2

12 1

1

2 ate xx

⎛ ⎞∂αζ = ⎜ ⎟∂⎝ ⎠

Then the control law (22) becomes

22)2(1

21

21

12221 ˆ

ˆ zyt

xx

zczu rT ζ−+

∂α∂

+τθ∂

α∂+

∂α∂

+θω−−−=

(35)

Simulation results showing desirable transient responses are presented in Fig. 1 with yr = 0.4, a = 0.1, = 10, Γ = 1, c1 = 12, c2 = 0.1 and η(x1, x2) = 2x1

2 cos (3x1 x2).

Alternatively, we can design a sliding mode controller for the system. Assume that the sliding surface is σ = k1z1 + z2 = 0 with k1 > 0. The adaptive sliding mode con-trol law (28) is

Fig. 1. Regulator responses with nonlinear control (35) for

PSSF system.

)2(12

12

1

1221111 ˆ

ˆ)1( rT y

tx

xzkzkcu +

∂α∂

+τθ∂

α∂+

∂α∂

+θω−−−=

2 11 1 1 1

1

1 sgn( )2

ath z e W K k hx

⎛ ⎞∂α+ − σ − + + σ⎜ ⎟⎜ ⎟∂⎝ ⎠

(36)

where τ2 = Γ (z1ω1 + σ(ω2 + k1ω1)). Simulation results showing desirable transient responses are shown in Fig. 2 with the same values as the case without sliding mode and k1 = 1, K = 10, W = 0. The simulation results with K = 10, W = 5, are shown in Fig. 3. If W > 0 the chattering of the sliding motion is reduced and also the reaching time is shorter than when w = 0.

V. CONCLUSION

Backstepping is a systematic Lyapunov method to design control algorithms which stabilize nonlinear sys-

Fig. 2. Tracking responses with sliding control (36) for

PSSF system with K = 10 and W = 0.

Fig. 3. Tracking responses with sliding control (36) for PSSF

system with K = 10 and W = 5.

452 Asian Journal of Control, Vol. 6, No. 4, December 2004

tems. Sliding mode control is a robust control design method and adaptive backstepping is an adaptive control design method. In this paper the control design has benefited from both design approaches. Backstepping control and sliding backstepping control were developed for a class of nonlinear systems which can be converted to the parametric strict feedback form. The plant may have unmodelled or external disturbances. The discon-tinuous control may contain a gain parameter for the designer to select the velocity of the convergence of the state trajectories to the sliding hyperplane. We have extended the previous work of Rios-Bolívar and Zinober [15-17] and Koshkouei and Zinober [10], and have re-moved the sufficient existence condition for the sliding mode to guarantee that the state trajectories converge to a given sliding surface.

REFERENCES

1. Barmish, B.R. and G. Leitmann, “On Ultimate Boundedness Control of Uncertain Systems in the Ab-sence of Matching Assumption,” IEEE Trans. Automat, Contr., Vol. 27, pp. 153-158 (1982).

2. Behtash, S., “Robust Output Tracking for Nonlinear Systems,” Int. J. Contr., Vol. 5, pp. 1381-1407 (1990).

3. Corless, M. and G. Leitmann, “Continuous State Feedback Guaranteeing Uniform Ultimate Bounded-ness for Uncertain Dynamical Systems,” IEEE Trans. Automat. Contr., Vol. 26, pp. 1139-1144 (1981).

4. Chen, Y.H., “Robust Control Design for a Class of Mismatched Uncertain Nonlinear Systems,” Int. J. Op-tim. Theory Appl., Vol. 90, pp. 605-625 (1996).

5. Freeman, R.A. and P.V. Kokotović, “Tracking Con-trollers for Systems Linear in Unmeasured States,” Automatica, Vol. 32, pp. 735-746 (1996).

6. Gutman, S., “Uncertain Dynamical Systems-A Lyapunov Min-max Approach,” IEEE Trans. Automat. Contr., Vol. 24, pp. 437-443 (1979).

7. Isidori, A., Nonlinear Control Systems, Springer Ver-lag, Berlin, Germany (1995).

8. Krstić, M., I. Kanellakopoulos, and P.V. Kokotović, “Adaptive Nonlinear Control without Overparametri-zation,” Syst. Contr. Lett., Vol. 19, pp. 177-185 (1992).

9. Kanellakopoulos, I., P.V. Kokotović, and A.S. Morse, “Systematic Design of Adaptive Controllers for Feed-back Linearizable Systems,” IEEE Trans. Automat. Contr., Vol. 36, pp. 1241-1253 (1991).

10. Koshkouei, A.J. and A.S.I. Zinober, “Adaptive Sliding Backstepping Control of Nonlinear Semi-Strict Feed-back Form Systems,” Proc. 7th IEEE Mediter. Contr. Conf., Haifa, pp. 2376-2383 (1999).

11. Koshkouei, A.J. and A.S.I. Zinober, “Adaptive Output Tracking Backstepping Sliding Mode Control of Nonlinear Systems,” Proc. 3rd IFAC Symp. Robust Contr. Design, Prague, Vol. 1, pp. 167-172 (2000).

12. Li, Z.H., T.Y. Chai, C. Wen, and C.B. Hoh, “Robust Output Tracking for Nonlinear Uncertain Systems,” Syst. Contr. Lett., Vol. 25, pp. 53-61 (1995).

13. Mario, R. and P. Tomei, “Robust Stabilization of Feedback Linearizable Time-varying Uncertain Nonlinear Systems,” Automatica, Vol. 29, pp. 181-189 (1993).

14. Qu, Z., “Global Stabilization of Nonlinear Systems with a Class of Unmatched Uncertainties,” Syst. Contr. Lett., Vol. 18, pp. 301-307 (1992).

15. Rios-Bolívar, M., and A.S.I. Zinober, “A Symbolic Computation Toolbox for the Design of Dynamical Adaptive Nonlinear Control,” Appl. Math. Comp. Sci., Vol. 8, pp. 73-88 (1998).

16. Rios-Bolívar, M., and A.S.I. Zinober, “Dynamical Adaptive Sliding Mode Output Tracking Control of a Class of Nonlinear Systems,” Int. J. Robust Nonlin. Contr., Vol. 7, pp. 387-405 (1997).

17. Rios-Bolívar, M. and A.S.I. Zinober, “Dynamical Adaptive Backstepping Control Design Via Symbolic Computation,” Proc. 3rd Eur. Contr. Conf., Brussels, paper no. 704 (1997).

18. Rios-Bolívar, M., A.S.I. Zinober, and H. Sira-Ramírez, “Dynamical Sliding Mode Control via Adaptive Input- Output Linearization: A Backstepping Approach,” in Robust Control via Variable Structure and Lyapunov Techniques, F. Garofalo and L. Glielmo, Eds., Springer-Verlag, U.S.A., pp. 15-35 (1996).

19. Rios-Bolívar, M. and A.S.I. Zinober, “Sliding Mode Control for Uncertain Linearizable Nonlinear Systems: A Backstepping Approach,” Proc. IEEE Workshop Robust Contr. Variable Struct. Lyapunov Techniques, Benevento, Italy, pp. 78-85 (1994).

20. Yao, B. and M. Tomizuka, “Adaptive Robust Control of SISO Nonlinear Systems in A Semi-Strict Feedback Form,” Automatica, Vol. 33, pp. 893-900 (1997).

21. Yao, B., and M. Tomizuka, “Smooth Adaptive Sliding Mode Control of Robot Manipulators with Guaranteed Transient Performance,” ASME J. Dyn. Syst. Man Cy-bern., Vol. SMC-8, pp. 101-109 (1994).

A. Koshkouei et al.: Adaptive Sliding Mode Backstepping Control of Nonlinear Systems 453

Ali Koshkouei received the Ph.D. degree in Control Theory from University of Sheffield in 1997. He worked at the Department of Applied mathematics from 1997 until 2003 as a Research Associate. Since 2003, he has worked at Con-

trol Theory and Applications Centre, Coventry University as a Senior Research Assistant. He has published about 50 papers in the international journals, conferences and as chapters of books. His research interests are mainly stabi-lisation of nonlinear systems and sliding mode con-trol/observers.

Alan S. I. Zinober After obtaining his Ph.D. degree from Cambridge University in 1974, Alan Zinober was appointed Lecturer in the De-partment of Applied and Computa-tional Mathematics at the Univer-sity of Sheffield in 1974, promoted to Senior Lecturer in 1990, Reader in Applied Mathematics in 1993

and Professor in 1995. He has been the recipient of a num-ber of EPSRC research grants and has published over 160 journal and conference publications. The central theme of his research is in the field of variable structure sliding mode control systems theory. Discontinuous and continu-ous control algorithms have been developed as part of a CAD control package. Other nonlinear control areas stud-ied include sliding observers, nonlinear H infinity control, frequency shaped sliding control, adaptive backstepping techniques and the realization of nonlinear systems. An-other area of research is in deterministic Operations Re-search, in particular the field of combinatorial optimization. In 1990 he edited a research monograph, Deterministic Control of Uncertain Systems, and Variable Structure and Lyapunov Control, was published in 1994. He has been an invited speaker at a number of international conferences. He organised the IEEE International Workshop on Variable Structure and Lyapunov Control of Uncertain Dynamical Systems in September 1992, and the Fourth NCN Work-shop in 2002, and has organized invited sessions at many IFAC and IEEE Conferences. He is a past Chairman of the IEEE (UK and RI) Control Systems Chapter and the IMA (UK) Control Theory Committee.

Keith J Burnham has been Professor of Industrial Control Systems, School of Mathematical and Information Sci-ences, Coventry University, and Direc-tor of the University’s Control Theory and Applications Centre since 1999. This is a multidisciplinary research centre in which effective collaboration

takes place amongst staff from across the University. There are currently a number of research programmes with UK based industrial organisations, many of which are involved with the design and implementation of adaptive control systems. Keith Burnham obtained his BSc (Mathematics), MSc (Control Engineering) and PhD (Adaptive Control) at Coventry University in 1981, 1984 and 1991, respectively. He is regularly consulted by industrial organisations to provide advice in areas of advanced algorithm development for control and condition monitoring. Currently, he is a Member of the Editorial Board of the Transactions of the Institute of Measurement and Control. He is also a Member of the Institution of Electrical Engineers, a Member of the Institute of Measurement and Control, and a Member of the Institute of Mathematics and its Applications.