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
Trang 1ADAPTIVE 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
Thestabilizationofdynamicalsystemswithuncer-
tainties has been studiedintherecentyears[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.
Trang 2specified 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
) , , ( ) ( ) (
)
=
which is transformable into the semi-strict feedback
form (SSF) [10,11,20]
1 T( , ,1 2 , ) ( , , ), 1 1
x =x+ + ϕ x x x θ + η x w t ≤ ≤ − i n
( ) ( ) T( ) ( , , )
x = f x +g x u+ ϕ x θ + η x w t
1
x
where x =[x1 x2 … x n]T is the state, y the output, u the
scalar control, and ϕi (x1, …, x i) ∈ Rp , i = 1, …, n, are
known functions which are assumed to be sufficiently
smooth θ ∈ R pis 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
bounded by known positive functions h i (x1, … x i) ∈
R, i.e
n i
x x h
t
w
i( , , )| ( , ), 1, ,
The output y should track a specified bounded reference
signal y r (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
r
T x x w t y x
z1= 2+ϕ1( 1) +η1( , , )− (4) From (4)
1 2 1Tˆ 1( , , ) 1T
r
z =x + ω θ + η x w t −y + ω θ (5) with ω1(x1) = ϕ1(x1) and θ = θ − θˆ where ~θ(t) is an estimate of the unknown parameter vector θ
Consider the stabilization of the subsystem (4) and the Lyapunov function
θ Γ θ +
=
2
1 2
1 ) ,
1
where Γ is a positive definite matrix The derivative of
V1 is
) (
~ ) ) , , ( ˆ ( ) ,
1 z θ =z x +ω θ+η x w t −y +θ Γ− Γω z −θ
r T
Define τ1 = Γω1z1 Let
1( , , )1 ˆ 1 ˆ 1 1 1 1
4
with c1, a and positive numbers Define the error
variable
r
y t x x
z2= 2−α1( 1,θˆ, )−
4
r
n
Then
2
4
z = −c z + + ω θ + ηz x w t − h z e (10)
and V1 is converted to
1( , )1 ˆ 1 1 1 2 at T ( 1 ˆ)
n
variable z k is
Trang 3( )
1
ˆ
k
x
−
=
∑
T k1
−
∂α
+ω θ −
where
1 1
1
k k
x
−
−
=
∂α
∂
∑
2 1
1
4
k
n
x
−
−
=
i k
i i
k k
k
x η
∂
α
∂
−
η
=
=
− 1
1
Define z k+1 = x k+1 − αk − y r (k)where
1 1
1
x
−
−
=
∂α
∂
∑
2
1 1
k
i
t
−
+
=
with c k > 0 Then the time derivative of the error
vari-able z k is
T
z = −z− −c z +z+ + ω θ + ξ − ζ z
2 1
1 1
k
i
z
−
−
+
=
Consider the extended Lyapunov function
1
1
k T
i
V V− z z
=
The time derivative of V k is
2
1 1
( 1) 2
k
at
k i i k k
i
k k
V c z z z e
n
− +
=
+
⎠
⎞
⎜
⎝
⎛ θ
∂
α
∂ + θ
− τ
Γ
θ
−
k k
i i
i k
since
∑
=
− +Γω =Γ ω
τ
=
i i i k
k k
1
Step n Define
( )
z =x − α −− y
with αn−1 obtained from (13) for k = n Then the time
derivative of the error variable z n is
θ θ
∂
α
∂
−
∂
α
∂
− θ ω + +
−
ˆ ˆ
) , ( ) (
)
n
i i i
n T
n
n
x t
x u x g
x
f
z
n 1 T( , ) ( )n
n x t n y r
t
−
∂α
where ω x n( ,θ) is defined in (12) for k = n Extend the
Lyapunov function to be
1
1
i n
i
=
−
=
(19)
The time derivative of V n is
1
( 1) 2
at
n
−
+
2
1
ˆ
T i
−
− +
∂α
∂θ
where
n T n
n=τ +Γω z
We select the control
⎢
⎣
⎡
∂
α
∂ + θ ω
−
−
−
−
−
−
1
1
) (
n T
n n n n n n
x x x
f z c z x g u
2
( )
1 1
n
n
i
t
−
+
=
⎤
∂
with c n > 0 Taking θˆ =τn,θ~ is eliminated from the
right-hand side of (20) Then
0
||
|| 2 1
2≤− <
−
=
z c z c
i i i
n
i c
c
≤
= min This implies that limt→∞z i = 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
Trang 4algorithm 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
0 1 1 1
=
where k i > 0, i = 1, …, n − 1, are real numbers
Addi-tionally, the Lyapunov function is modified as follows
1
1
n
i
n
a
−
=
−
= ∑ + σ + θ − θ Γ θ − θ +
Let
1
−
τ = τ + Γσ ω +⎜ ω = Γ⎟ ⎜⎜ ω + σ ω +⎜ ω⎟⎟⎟
(26)
The time derivative of V n is
)
1 1
1
2
−
−
−
−
=
+ + +
−
−
n
i i i
1
x
−
=
n r
n
t
∂
∂θ
1
1
2
ˆ
n
i
i
−
−
=
∂α
⎛
∂θ
⎝
∑
⎤
2
1 1
1
ˆ
n
T i
i
z
−
− +
=
∂α
∂θ
since from (24), z n = σ − k1z1 − k2z2 − … − kn−1 z n− 1
Setting θˆ =τn,~θ is eliminated from right-hand side of
(27) Consider the adaptive sliding mode output tracking
control
1
1
( )
n
i
−
=
⎡
∂
4
r
n
t
∂α
1
1
2
ˆ
n
i
i
−
−
=
∂α
⎛
∂θ
⎝
⎞
⎥
⎦
⎤ σ
⎟
⎠
⎞
⎜
⎝
⎛ +
−
σ
1 i i
n
i
v k K
where k n = 1, K > 0 and W ≥ 0 are arbitrary real num-bers and
1 ,
1 1
1
n i h
x h
i k i j i
∂
α
∂ +
=
Then substituting (28) in (27) yields
1
( 1) 2
at
n
−
= + σσ −
2 1
2 1 1 2
−
n
… where
1 2
c c Q
+
which is a positive definite matrix
2
W = z z …z− Q z z …z− +K σ + σW
which yields limt→∞ σ = 0 and lim t→∞z i = 0, i =1, 2, …,
n−1 Particularly, lim t→∞ (x1 − yr ) = 0 Since z n = σ −
k1z1 − k2z2 − … − k n − 1 z n − 1, lim t→∞z n = 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
n
x f z x g
θ
α
∂ + θ ω
−
−
−
) (
1
1 1
n
n
i r
i i
x y
−
+
=
∑
4
at
n
k ⎛ c z z h z e ⎞
1
2
n
i
−
=
⎛
2 1
1 1
i
l z
−
−
+
=
⎞
Trang 52 1
1
i
+
∂α
⎥
⎦
⎤ σ
⎟
⎠
⎞
⎜
⎝
⎛ +
− σ
=
n
i i i v k W K
1
)
with k n = 1, K > 0 and W ≥ 0 arbitrary real numbers and
for all i, 1 ≤ i ≤ n
1
1
4
i
n
x
−
−
=
∂
IV EXAMPLE
Consider the second-order system in SSF form
1 2 1 ( ,1 2)
x =x + θ + ηx x x
u
where |η| ≤ 2x1.We have
2
1 2x
r
y
x
2
r
z =x + θ +x c z + x z e −y
4
2
1
1
x
∂
α
∂
−
= ω
2 ( 1 1z 2 2z )
τ = Γ ω + ω ,
2 1
1
2 at
x
⎛ ∂α ⎞
∂
Then the control law (22) becomes
2 2 ) 2 ( 1 2 1 2 1
1 2 2
2
z
∂ α
∂ + τ θ
α
∂ +
∂ α
∂ + θ ω
−
−
−
=
(35)
Simulation results showing desirable transient responses
are presentedinFig.1withy r= 0.4, a= 0.1, = 10,
Γ = 1, c1 = 12, c2 = 0.1 and η(x1, x2) = 2x1 cos (3x1x2)
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 ( 1 2 1 2 1
1 2 2 1 1 1
t
x x z
k z k c
∂ α
∂ + τ θ
α
∂ +
∂ α
∂ + θ ω
−
−
−
1
1
sgn( ) 2
at
x
∂
(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 algorithmswhichstabilizenonlinearsys-
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
Trang 6tems 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 7 th 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 3 rd 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 3 rd 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)
Trang 7Ali 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 ZinoberAfter 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
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