Design of Adaptive Backstepping controller for Systems with
Mismatched Perturbations to Achieve Asymptotical Stability
Chien-Wen Chung
Department of Electrical Engineering,
Far East University, Tainan, Taiwan
e-mail: cwen.chung@msa.hinet.net
Abstract – Based on the Lyapunov stability theorem, a methodology of designing
the block backstepping controllers for a class
of multi-input multi-output (MIMO) systems
with matched and mismatched perturbations is
proposed in this paper. Some adaptive mechanisms are embedded in both the virtual input and the robust controllers so that not only
are the mismatched perturbations suppressed,
but also the knowledge of upper bound of perturbation is not required except those of the
input uncertainties. Finally, the feasibility of
the proposed methodology can be verified by
controlling of a separated excited DC motor
(SEDCM).
Key Words – Lyapunov stability, backstepping,
SEDCM.
1. Introduction
Backstepping is a recursive procedure that interlaces
the choice of a Lyapunov function with the design of
feedback control [1]. One advantage of this technique
is that it can circumvent the problem of mismatched
perturbations and achieve the property of asymptotical stability. The other advantage is that it can exploit
some extra flexibility that exists with lower order systems.
In recent years, some researchers used MIMO backstepping technique to design robust controllers for systems with perturbations. Wu & Zhou [2] designed an
output feedback controller for a class of MIMO nonlinear systems to achieve uniform ultimate boundedness stability. Lin and Qian [3] proposed the partial
state and output feedback controllers, Liu et al. [4]
designed a H ∞ controller and applied it in a class of
chemical plants. Yao and Tomizuka [5] and Liu et al.
[6] designed the backstepping controllers for a class of
MIMO nonlinear systems.
Although the works proposed in [3,5,6] can be employed for MIMO systems to achieve asymptotical stability; however, some controlled systems have to be in
the strict feedback form [3,4], some control schemes
[3,4] need the existence of the positive definite and
proper Lyapunov function. In addition, only the au-
Yaote Chang
Department of Electrical Engineering,
Kao Yuan University, Kaoshuing, Taiwan
e-mail: yat.chang@msa.hinet.net
thors of [5,6] considered the uncertainties of input gain;
however, the knowledge of the upper bound of perturbations is required.
Due to the limitations of each of the methodologies
mentioned above, in this paper we proposed a design methodology of adaptive block backstepping controllers for a class of MIMO systems with mismatched
perturbations as well as matched input uncertainty to
solve regulation problems. The advantages of this proposed methodology are that not only are asymptotical
stability can still be achieved, but also the knowledge of
the upper bound of perturbations is not required except those of input uncertainties when designing the
controllers. Finally, a SEDCM is demonstrated for
showing the applicability of the proposed technique.
2. System Descriptions and Problem Formulations
Consider the following MIMO nonlinear system:
(1a)
ẋ1 = f1 (t, x1 ) + G(t, x1 )x02 + Θf2 (t, x1 )
h
i
ẋ2 = f3 (t, x) + ∆f4 (t, x) + B(t, x) + ∆B(t, x) u,
(1b)
T
T
x1 xT2
where x =
∈ Rn , n = n1 + n2 represents measurable state vector, x1 ∈ Rn1 , x2 =
h
iT
T
T
∈ Rn2 , x02 ∈ Rn1 , x002 ∈ Rn2 −n1 , n1 ≤
x02
x002
n2 . The vector fi (t, x1 ) ∈ Rn1 , i = 1, 2 and f3 (t, x) ∈
Rn2 are known nonlinearities, where vector fi , i = 1, 2
is once differentiable. The known matrix G(t, x1 ) ∈
Rn1 ×n1 and B(t, x) ∈ Rn2 ×n2 are nonsingular, where
G is once differentiable. u ∈ Rn2 is the control input.
The unknown constant matrix Θ ∈ Rn1 ×n1 is mismatched external disturbance. The matrix or vector
∆(·) denote the unknown model uncertainties and/or
parameter variations in (·).
The objective of this paper is to design the virtual
input x02 and the control effort u so that the system’s state x1 and x002 are able to track the desired
piecewise continuous, bounded, signal y0 ∈ Rn1 and
y00 ∈ Rn2 −n1 asymptotically respectively, where y0 is
twice differentiable, y00 is once differentiable.
3. Design of the Virtual Input and Robust Con-
troller
In order to show that the state x1 can track the signal
y0 asymptotically, one can design the virtual input as
φ(t, x) = x02 − ψ(t, x1 )
ψ(t, x1 ) = −G−1 [Ke1 + f1 + Θ̂(t)f2 − ẏ0 ],
(2)
Theorem 1 Consider the system (1). The lumped
T
, ζ1 ∈
perturbations are defined as ζ , ζ 1 T ζ 2 T
Rn1 , ζ 2 ∈ Rn2 −n1 ,
ζ 1 , ∆f40 + ∆B0 u + Ġζ−1 (Ke1 + f1 + Θ̂f2 − ẏ0 )
+G−1 (KΘf2 + ḟ1ζ + Θ̂ḟ2ζ )
0
n1
where e1 = x1 − y , [e1 e2 · · · en1 ] ∈ R is the
tracking error and K ∈ Rn1 ×n1 is a constant matrix
designed in a way such that Re[λ(K)] > 0. The adaptive gain Θ̂(t) , [θ̂ij (t)] ∈ Rn1 ×n1 , 1 ≤ i, j ≤ n1 in (2)
is defined as
˙
θ̂ij (t)
= f2j
n1
P
pid e1d ,
ζ 2 , ∆f400 + ∆B00 u,
iT
iT
h
h
T
T
∆f4 , ∆f40 T ∆f400 T
, ∆B , ∆B0 ∆B00
,
and ζ can be satisfied by the constraints
(3)
d=1
kζ(t, x, u)k ≤ r0 + r1 kxk + r2 kuk,
n1 ×n1
where f2 , [f21 f22 · · · f2n1 ]. P , [pjd ] ∈ R
,1 ≤
j, d ≤ n1 is a symmetric and positive definite matrix
and can satisfies eT1 Qe1 ≥ 0, Q , PK + KT P.
Before designed the backstepping controllers, we can
also divide dfi /dt, i = 1, 2 into dfi /dt , dfin /dt +
dfiζ /dt, where dfin /dt is the nominal part of dfi /dt,
and dfiζ /dt is the part of dfi /dt which contains pertur∂fi
0
i
bations, i.e., dfin /dt = ∂f
∂t + ∂x1 (f1 + Gx2 ), dfiζ /dt =
∂fi
Similarly, one can also have dG−1 /dt ,
∂x1 Θf2 .
−1
−1
dG−1
n /dt+dGζ /dt and dGn /dt =
Gx02 ), dG−1
ζ /dt =
−1
∂G
∂x1
Θf2 .
00 T T
f3 ] ,
T
[f30
∂G−1
∂t
−1
+ ∂G
∂x1 (f1 +
0
−1
u11 = −f30 − Ġ−1
n (Ke1 + f1 + Θ̂f2 − ẏ ) − G
h
i
˙ + Θ̂ḟ − ÿ0 ,
K(f1 + Gx02 − ẏ0 ) + ḟ1n + Θ̂f
2
2n
x002
00
1 − r2 kB−1 k
`
4. Stability Analysis
= ẋ1 − ẏ0 = −Ke1 + Θ̃f2 + Gφ.
V1 =
(6)
(7)
0T 0
1 T
e1 Pe1 + θ̃ θ̃ ,
2
(8)
,
if kzk =
6 0
, r̂` (t0 ) = 0, (5)
if kzk = 0
where α is a designed positive constant.
= f1 + Gψ + Θf2 + G(x02 − ψ)
= −Ke1 + Θ̃f2 + ẏ0 + Gφ,
where Θ̃ = Θ − Θ̂ , [θ̃ij ] ∈ Rn1 ×n1 , θ̃ij = θij − θ̂ij ,
1 ≤ i, j ≤ n1 , are the adaption errors of unknown
constant Θ. Using (6), the dynamics of the tracking
error e1 is given by
kun k + kB−1 k + η
1
kB−1 k
α−1 kzkkxk
0
ẋ1
Now one can choose a Lyapunov function as
e2 =
− y , r2 <
and η > 0 are the designed
positive constants. The adaptive rules of r̂` (t), 0 ≤ ` ≤
1 in (4) are given by
r̂˙` (t) =
According to (2), (1a) can be rewritten as
ė1
u12 = −GT e1 , u2 = −f300 + ẏ00 ,
( −1
6 0
− (t) + ρ(t) Bkzkz , if kzk =
,
us =
0,
if kzk = 0
h
iT
r2
z = φT e2 T , ρ(t) =
Proof:
Decomposes the vec-
tor f3 as f3 =
where f30 ∈ Rn1 and
00
n2 −n1
f3 ∈ R
. The backstepping controllers are designed as
u = un + us ,
(4)
T
where un = B−1 uT1 uT2
, u1 = u11 + u12 ,
(t) = r̂0 (t) + r̂1 (t)kxk,
where r` , ` = 0, 1, are unknown positive constants and
r2 is given in (4). If the virtual input and the backstepping controllers are designed as (2) and (4) respec
T
tively, the state trajectory of e = e1 T e2 T
will
approach zero asymptotically; and the adaptive gains
θ̂ij , 1 ≤ i, j ≤ n1 , r̂` (t), 0 ≤ ` ≤ 1 are all bounded.
0
where
θ̃ , [θ̃1 θ̃ 2 · · · θ̃n1 ]T ∈ Rn1 n1 ×1 , θ̃ i ,
θ̃i1 θ̃i2 · · · θ̃in1 ∈ R1×n1 , 1 ≤ i ≤ n1 . Differentiating (8) along the trajectory of (7) yields
T
1
V̇1 = − eT1 PK + KT P e1 + f2T Θ̃ Pe1 + eT1 Gφ
2
0T
(9)
+θ̃ θ̃˙ 0 .
˙
˙
According to (3) and θ̂ij = −θ̃ij , 1 ≤ i, j ≤ n1 , one can
r̃˙` = r̂˙` , ` = 0, 1, one can obtain the time derivative of
(14) as
verify that
T
f T Θ̃ Pe1
2n1
P
f2j θ̃1j
=
=
n1
P
j=1
n1
P
d=1
n1
X
f2j θ̃2j
···
j=1
n1
P
p1d e1d
f2j θ̃1j
n1
X
p2d e1d
n1
P
···
n1
X
p1d f1d +
n1
X
f2j θ̃n1 j
˙
θ̃1j θ̃1j −
j=1
n1
X
f2j θ̃2j
j=1
j=1
n1
X
pn1 d e1d
T
1
V̇2 = − eT1 PK + KT P e1 + f2T Θ̃ Pe1 + eT1 Gφ
2
1
X
0T
+θ̃ θ̃˙ 0 + zT ż + α
r̃` r̃˙`
T
`=0
d=1
d=1
+···+
f2j θ̃n1 j
j=1
d=1
j=1
=−
n1
P
n1
X
1
= − eT1 Qe1 + eT1 Gφ + φT (ζ 1 + u12 + B0 us )
2
1
X
+eT2 (ζ 2 + B00 us ) + α
r̃` r̃˙`
p2d e1d
d=1
`=0
pn1 d e1d
d=1
n1
X
n1
X
˙
˙
θ̃2j θ̃2j − · · · −
θ̃n1 j θ̃n1 j
j=1
j=1
0T ˙
= −θ̃ θ̃ 0 .
1
X
1
= − eT1 Qe1 + zT ζ + Bus + α
r̃` r̃˙`
2
`=0
1
≤
(10)
X
1
− eT1 Qe1 + kzkkζk + zT Bus + α
r̃` r̂˙`
2
`=0
1
≤ − eT1 Qe1 + kzk(
2
Using (10), (9) can be rewritten as
V̇1 = −eT1 Qe1 /2 + eT1 Gφ.
(11)
−kzk(
0
According to (4), one can obtain u11 + u12 = B un
T
and u2 = B00 un , where B , B0T B00T
, B0 ∈
Rn1 ×n2 , B00 ∈ R(n2 −n1 )×n2 . From (1) and (4), the
time derivative of φ is
φ̇ = ẋ02 + Ġ−1 Ke1 + f1 + Θ̂f2 − ẏ0
˙ + Θ̂ḟ − ÿ0
+G−1 Kė1 + ḟ1 + Θ̂f
2
2
= f30 + ∆f40 + B0 (un + us ) + ∆B0 u
−1
0
+(Ġ−1
n + Ġζ ) Ke1 + f1 + Θ̂f2 − ẏ
h
+G−1 K(f1 +Gx02 +Θf2 − ẏ0 )+ ḟ1n + ḟ1ζ
i
˙ + Θ̂(ḟ + ḟ ) − ÿ0
+Θ̂f
2
2n
2ζ
= ζ 1 + u12 + B0 us
(12)
= ẋ002 − ẏ00
= f300 + ∆f400 + ∆B00 u + B00 (un + us ) − ẏ00
= ζ 2 + B00 us .
(13)
In order to prove the state trajectory of e will approach
zero asymptotically, one can use the Lyapunov function
V1 given by (8) and define another Lyapunov function
candidate as
1
V2 = V1 + (zT z + α
2
1
X
r̃`2 ),
`
r` kxk + r2 kuk)
`=0
r̂` kxk` + ρ) +
`=0
1
X
(r̂` − r` )kzkkxk`
`=0
1
= − eT1 Qe1 + kzk(−ρ + r2 kuk)
2
n
o
1
≤ − eT1 Qe1 + kzk − ρ + r2 [kun k+kB−1k (ρ + )]
2
1
= − eT1 Qe1
2 h
i
+kzk − ρ(1 − r2 kB−1 k) + r2 (kun k + kB−1 k)
1
= − eT1 Qe1 − ηkzk ≤ 0.
2
and the time derivative of e2 is
ė2
1
X
1
X
(14)
`=0
where z is defined in (4). r̃` , r̂` − r` , 0 ≤ ` ≤ 1 are
the adaptive errors of the unknown constant r` . Using
(4), (5), (12), (13), the result of (11) and noting that
The preceding equation indicates that z and state e1
will approach zero asymptotically. On the other hand,
z reaches zero means that state e2 and φ will reach
zero asymptotically. Therefore one can conclude that
T
the state variable e = e1 T e2 T
will approach
zero as t → ∞. Since V2 > 0 and V̇2 ≤ 0, this indicates
θ̂ij , 1 ≤ i, j ≤ n1 , ĝ` (t), 0 ≤ ` ≤ 1, are all bounded. 5. Application of Controlling SEDCM
In order to demonstrate the applicability of the proposed control scheme in this section, we apply the
backstepping controller to control a SEDCM dynamic
system [7] are described as
dw
1
= (Te − Bw − TLN − ∆TL )
dt
J
1
dia
ua
=
(−RaN ia − E − ∆Ra ia ) +
dt
La
La
1
dif
uf
=
(−Rf N if − ∆Rf if ) +
,
dt
Lf
Lf
(15)
where E = kif w and Te = kia if = E
w ia . The nomenclature of each variable in (15) is listed in Table 1
[7]. ∆Ra = Ra − RaN , ∆Rf = Rf − Rf N and
∆TL = TL − TLN are the unknown deviations from
the nominal values RaN , Rf N and TLN of Ra , Rf and
TL , respectively.
According to the notation of (1), one can rewrite (15)
in the form as (1), i.e., x1 , w, x02 , ia , x002 , if ,
E
L
f1 (t, x1 ) , −(Bx1J+TLN ) , G(t, x1 ) , Jx
, Θ , −∆T
,
J
1
−(RaN x02 +kx1 x00
2)
, f300 (t, x) ,
La
0
−∆Ra x2
−∆Rf x00
2
, ∆f400 (t, x) ,
,
La
Lf
T
, [ua uf ] ,
f2 (t, x1 ) , 1, f30 (t, x) ,
−Rf N x00
2
,
Lf
0
y = wm ,
∆f40 (t, x) ,
y00 = if m , u
B,
"
1
La
0
0
#
1
Lf
,
∆B ,
0
0
0
0
.
Rated
power
Rated
speed
Field
voltage
Armature
resistance
Field resistance
Inertia
3.7kW
1800rpm
240V
1.2Ω
60Ω
0.208kgm2
Rated
voltage
Rated
torque
Armature
inductance
Field inductance
Motor
constant
Damping
coefficient
240V
18Nm
0.01H
60H
0.3Nm/A2
0.011kgm2s−1
The motor parameter data are listed in Table 1 [7].
The design parameters are chosen to be (K, Q, η) =
(3.5, 33, 0.01). Reference speed is initially set up
in wm (y 0 ) =1800r/min and is later modified to
1840r/min at 2 sec. A sudden change of mismatched
disturbance θ (load torque uncertainty ∆TL = 14N ·m)
is applied at t = 6 sec. Fig. 1 shows that the actual
speed x1 achieved the robust performance in a short
transient time even if mismatched load torque existed.
Fig. 2 shows that the trajectory of x002 , which can track
the y 00 robustly. Fig. 3 shows that the adaptive gain θ̂
almost converges to the true θ. The adaptive gain ĝ0 ,
which is bounded, is shown in Fig. 4.
[1] H. K. Khalil, Nonlinear Control, Prentice-Hall, New
Jersey, 1996.
6. Conclusion
An adaptive backstepping controllers for a class of
MIMO nonlinear systems with mismatched perturbations is successfully proposed in this paper. The advantages of the proposed control scheme are not only can
achieve asymptotical stability, but also it can be applied to a more general perturbed systems than those
in [7]. The simulation results of a SEDCM show the
performance of tracking is robust even the mismatched
perturbation exists.
[4] X. Liu, A. Jutan, and S. Rohani, “Almost disturbance decoupling of MIMO nonlinear systems and
application to chemical processes,” Automatica, Vol.
40, pp. 465-471, 2004.
Table 1: Nomenclature of the SEDCM [7]
TL : load torque
B: damp coefficient
if : motor field current
w: motor speed
k: back emf constant
Lf : field inductance
uf : field control input
Te = kia if = E
w ia : develop torque
E = kif w: back emf
ia : motor armature current
Ra : armature resistance
Rf : field resistance
La : armature inductance
ua : armature control input
J: damp inertia
Table 2: Parameters of the SEDCM [7]
References
[2] Y. Wu and Y. Zhou, “Output feedback control for
MIMO non-linear systems with unknown sign of the
high frequency gain matrix,” International Journal
of Control, Vol. 77, No. 1, pp. 9-18, 2004.
[3] W. Lin and C. Qian, “Semi-global robust stabilization of MIMO nonlinear systems by partial state and
dynamic output feedback,” Automatica, Vol. 37, pp.
1093-1101, 2001.
[5] B. Yao and M. Tomizuka, “Adaptive robust control
of MIMO nonlinear systems in semi-strict feedback
forms,” Automatica, Vol. 37, pp. 1305-1321, 2001.
[6] X. Liu, G. Gu, and K. Zhou, “Robust stabilization
of MIMO nonlinear systems by backstepping,” Automatica, Vol. 35, pp. 987-992, 1999.
[7] Z. Z. Liu, F. L. Luo, and M. H. Rashid “Speed
nonlinear control of dc motor drive with field weakening,” IEEE Transaction on Industry Applications,
Vol. 39. No. 2, pp. 417-423, 2003.
1860
1840
1820
x1
y'
1800
1780
0
2
4
6
8
10
Figure 1: Trajectories of angular speeds x1 and y 0 .
0.5
x 2"
y"
0.4
0
2
4
6
Figure 2: Trajectories of field current
8
x”2
10
and y”.
10
0
-10
-20
-30
-40
-50
-60
-70
-80
-90
0
2
4
6
8
10
Figure 3: Adaptive gain θ̂ and θ.
400
g0
350
300
250
200
150
100
50
0
0
2
4
6
Figure 4: Adaptive gain ĝ0 .
8
10