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Controller Design for a class of Cascade Nonlinear Systems using
Backstepping and Control Lyapunov Function Techniques
Pan Feng Gu Rui
(Control Science and Engineering Research Center, Southern Yangtze University,214122, P.R. China)
Abstract:This paper studies controller design approaches for a class of cascade nonlinear systems achievinginput-to-state stability with respect to the bounded disturbance, which can be measured and used for control. A
general form of using backstepping and control Lyapunov Function (CLF) techniques to design state feedback
controllers for cascades is proposed. In this paper, two types of controllers (C1, C2) are discussed. Their
properties are illustrated with simulation. Analysis shows that controller C1has a faster and better state
response with disturbance input increasing than controller C2. On the other hand, controller C1is
dramatically affected by the gain k, while controller C2hardly varies with the gain b varying.
Keywords: Input-to-state stability, Cascade, State feedback control, Backstepping, Control Lyapunov
Function .
1. Introduction.Among studies on the stabilization of cascade systems, a particularly useful result is the
input-to-state stability (ISS) condition of Sontag [10], which states that if subsystems of degree
two are ISS, global asymptotical stability (GAS) respectively, then the general cascade is GAS.
This result has been widely used for nonlinear design. In the literature several successful,
constructive control methods were presented such as backstepping and forwarding (see e.g.
Sepulchre et al[1], Praly et al[5], Arack et al[4], Mazenc.et al[3] ).
In this paper, uniting ISS-CLF and backstepping techniques, we design two different state
feedback controllers (C1, C2) for a class of cascade nonlinear systems achieving input-to-state
stability with control and disturbance input. Analysis characters show that controller C1 has a
faster and better state response with disturbance input increasing than controllerC2, but the state
response of controller C1varies with the gain k. Adding disturbance restriction to the cascade
systems, which vanishes at the origin, we finally render the cascade global asymptotically stable
to the origin.
This paper is organized as follows: in the next section we recall necessary definitions and
review background results. In section 3 we design the stabilizing control laws for a class of
cascade systems. An illustrative example is given in section 4. Some properties of two different
controllers are compared by simulations. Finally, we get some conclusions in section 5.
2. Preliminaries.
A function RRr : is said to be of class K if it is continuous, strictly increasing and
is zero at zero. It is of class K if, in addition, it is unbounded.
A function
RRR : is said to be of classKL if, for each fixed s , the function
),( t decreases to zero at infinity.
stands for the Euclidean norm.
o stands for composition. )( 2121 rrrr =o
Function RRh : and RRf : , the Lie derivative of h alongf , written as hLf ,
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is defined by )()( xfx
hxhLf
= .
A general nonlinear system of the form
dxgxfx )()( +=& (1)
where f is a local Lipschitz function and d is a locally essentially bounded disturbance
input.
Definition1: System (1) is said to be input-to-state stable if for any bounded disturbance d
and initial state )0(x , the solution )(tx satisfies the following estimate:
)(),)0(()( drtxtx + (2)
where is of classKL , r is of classK.
Theorem 1: System (1) is input-to-state stable if and only if there exists an ISS-CLF
RRxV n :)( , a smooth positive definite radially unbounded function, holds for all 0x :
dxdxdVLVLgf
,)()()( ++ (3)
3. Controller design of state feedback control
Consider a cascade nonlinear system of the form
+=++=
+=++=
dxxguxxfuxxdxxgxxfx
dxgxxfxxdxgxfx
),(),,(),(),(),(
)(),()()()(
2122122122122122
1121121111111
&
&
)5(
)4(
This system is represented by the block-diagram, which shows a feedback loop. We need to find
a control law ),( 21 xxuu= via backstepping that ensures the input-to-state stability for the
closed-loop cascade systems.Proposition 1: Consider the system of form:
uxdxgxfx )()()( ++=& (6)
with the following assumptions.
1) System (6) is input-to-state stable.
2) There is a smooth control law ),( dxRu= satisfying 0)0,0( =R .
Under these conditions, the augmented system (4),(5) is input-to-state stable with a smooth
control law ),,( 21 dxxRu=
Proof: We define the following control law ),( 111 dxRu = paralleling the correspondingSontag-type control law )(xRu= [4]
0
0
0
)(
),(1
1
1
4
1
2
11
1
1
1
1
=
++=
VL
VL
VL
VLww
dxR
(7)
Where dVLVLw gf )( 11 11 += , the disturbance can be measured and used for control.
The above control law is smooth on { }0\nR . By theorem 1, we know that systems (4) is ISS,
then it admits a smooth ISS-CLF )(11
xV so that we have
dxdxdxRVLdVLVL gf ,).()(),()()( 1111111 111 +++ .
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First, we design controller C1: ),,( 212 dxxRu= .
From above, we know the system (5) can be stabilized by a smooth state feedback control
law ),( 112 dxRx = with 0)0,0( =R . Rewrite the cascade as
[ ] [ ]
++=
+++=
uxxdxxgxxfx
xRxxdxRxdxgxfx x
),(),(),(
)()(),()()()(
2122122122
1121111111111
&
&
)9(
)8(
The change of variable ),( 112 dxRxz =
Results in the system
[ ]
=
+++=
),(
)(),()()()(
112
1111111111
dxRxz
zxdxRxdxgxfx x&&&
&
The derivative ),( 11 dxR& can be expressed by
211111 )()(),( 111 xRLdRLRLdxR gf ++=& (10)
Taking ),( 112 dxRxv = &
reduce the system to the cascade[ ]
=
+++=
vz
zxdxRxdxgxfx x
&
& )(),()()()( 1111111111
Now construct the positive composite function
2
112122
1)(),( zxVxxV += (11)
The derivative2V along the solution of the system satisfies
[ ]
zvzVLdx
zvzVLRVLdVLVLxxV gf
+++
++++=
)()()(
)()()(),(
11
11111212
1
1111
&
choosing 0,11 >= kkzVLv
yields
)()(),((
)()),(()()()(
2
1
112
1
2
1121
2
12
dx
xd
dxRxk
x
ddxRxkxkzdxV
+
+
++=+&
(12)
where K .
From above, we know ),( 212 xxV is an ISS-CLF for cascade system . Substituting zv,
and 1R& , we obtain the state feedback control law of system
2111121
1
2212 )()()[,(),,( 1111 xRLdRLRLVLxxdxxRu gf +++==
]),(),())(( 212212112 dxxgxxfxRxk (13)
Thus, the control law ),,( 2122 dxxRu = achieves ISS of .
Next, we design the second controller C2: ),,( 2122 dxxRu = .
We define the positive composite function
211211212 )),((
21)(),( dxRxxVxxV += (14)
The derivative2V along the solution of the system satisfies:
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))(()()( 121221112 111 RxRxxVLdVLVLV gf&&& +++=
{ )()()()( 222121111 111 udgfRxRVLdVLVL gf +++++=
}12111 1111 ])()([ VLxRLdRLRL gf +++ (15)
We now choose the control u such that the terms except containing disturbance terms are
controlled inside the bracket (15). In order for a controller to avoid less nonlinear cancellation
than controller C1, it takes the form:
])([),,( 212111
2212 111vfVLxRLRLdxxRu f ++==
(16)
yields
vRxdRLRxdgRxdxV g )())(()()()( 1211221212 1 +++ & (17)
Taking 0),()()( 122
112
2
212 1>= bRxbRLRxgRxv g
yields
])()[()()( 2122
221212 dgRxgRxdxV + &
2
12121
2
1
2
12 )(]))(()[( 11 RxbdRLxRRLRx gg (18)
By using Youngs inequality [9], we have
dgRxdgRx 212222
12 )(4
1)( +
dRLxRdRLRx gg ))((4
1)( 121
22
1
2
12 11+
Applying the inequalities above to (18) we get
[ ]
)(~~)(~
)(
~
)(~)()(2
1)()()(
2
1
12
1
2
121
22
1212
dx
xd
Rxb
x
dRxbxdRxbdxV
+
+
++=++&
(19)
Where K~,~ . Therefore ),( 212 xxV is an ISS-CLF for the system . This implies that the
control law ),,( 212 dxxRu= input-to-state stabilizes cascade system . Thus, the controller C2
should be of the form
2
1221211
1
2212 )()([),,( 111 gRxfVLxRLRLdxxRu f +==
)]()( 122
112 1RxbRLRx g (20)
Remark 3.1: Only if the external disturbance is bounded, the controllers C1, C2can achieve ISS.
Remark 3.2: From the solution )(tx estimation (2) , 0 as t . Since for 0tt we
have )()(
drtx . If there exists a class K function such that for all 0x , the inequality
(3) holds:
0)()(
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Remark 3.3: Although 2C is significantly more complex than 1C , the backstepping approach
has its merit such that repeated application of above proposition, we can design ISS control laws
for extended cascade systems of the form.
++=
++=
++=
uxxdxxgxxfx
xxxdxxgxxfx
xxdxgxfx
nnnnnnn ),,(),,(),,(
),(),(),(
)()()(
111
32122122122
21111111
LLL&
M
M
&
&
4. Example
For a clearer illustration of some properties of two different controllers, we consider the
following example of the cascade nonlinear system of the form:
)23(
)22(:1
2
21212
2
12
211
2
11
++=
++=
uxxdxxxxx
xxdxxx
&
&
Where d is external disturbance input.
A Lyapunov function for (22) is 21112
1)( xxV = .
Using the equality (7), we obtain a control law
=
++++
==
0,0
0,)()()(
),(
1
1
1
4
1
2
1111
111
1
1
1
11111
VL
VLVL
VLdVLVLdVLVL
dxRugfgf
=
++++=
0,0
0,)()][(
1
1
2
1
2
11
x
xxdxdx
The derivative ),( 11 dxR& can be expressed:
])(
21[
2
1
2
1
1
1
1
xdx
dx
x
u
++
++=
(C1) From control law of (13), we obtain controller:
2
1211
2
2
1
2
1
1
2
21
212 ))()(
21(1),,(11
xxxdxxxdx
dxxx
dxxRuC
++++
++==
+++++ dxxxxxdxdxxk 2122
1
2
1
2
112 ])()([
(C2) From control law of (20), we obtain the controller
2
2
1
2
121
2
12
1
2
1
1
2
21
212 ))()(
21(
1),,(
2xxxxxx
xdx
dx
xxdxxRuC +
++
++==
++
++++
+++++
2
2
1
2
1
111
2
21
2
1
2
112
)(
)2(])()([
xdx
dxxxxxbxdxdxx
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The dynamic controllers21
, CC uu provide stabilization for the cascade nonlinear system 1 .
Performances of the two controllers are compared by simulations. The transient responses of the
designed closed-loop system are shown in figure 1
(a) (b)
Figure1. State response of controlled system without disturbance
Remark 4.1: The response is computed with d=0 and Tx ]1,1[)0( = . Figure 1(a) represents the C1
controller with )20,10,5(),,( 210 =kkk and Figure 1(b) represents C2 controller
with )20,10,5(),,( 210 =bbb . The state response of the system driven by C1control law 1Cu has the
same performance as the state response of the system driven by C2control law2C
u achieving
ISS. Note that the state response of controller C1results in the same performance for any positive
values of the gain ),,( 210 kkk . So does controller C2.
(a) (b)
Figure2. State response of controlled system with the same disturbance
Remark 4.2: All the initial conditions are the same as in figure 1. Figure 2 (a) represents the
state response of controller C1that shows a faster and better damped response of the state with
the gain k increasing which the equilibrium point of the perturbed system 1 is not the
origin 0=x because of nonvanishing perturbation. Figure 2 (b) represents the state response of
controller C2 that has similar trajectories to figure 1 (b) besides the equilibrium 02 x of the
state 2x because of nonvanishing perturbation.
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(a) d=2 (b) d=5
(c) d=10
Figure3. State responses of controlled system with different disturbances
Remark 4.3:The response is computed with initial conditions Tx ]1,1[)0( = , 10=k and 1=b .
Solid curves are the state response of the system driven by C1control law1C
u and dotted curves
are the state response of the system driven by C2control law2C
u . Figure 3 shows that the state
1x s response of the two controllers have the same performance, but the state 2x s response of
controller C1have the faster and better damped performance than controller C2 with the external
disturbance increasing.
Figure4. State response of controlled system with the same restricted
)1(2
2
1
xxd += disturbance
Remark 4.4:Figure 4 shows that if the disturbance is restricted such as )1( 22
1 xxd += , the two
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controllers have the same state response converging to the origin which the
disturbance )1( 22
1 xxd += vanishes at the origin.
5. Conclusion
In this paper we design two types of controllers of the cascade nonlinear system.
Performances of these controllers are compared by simulations. We find that each of thecontrollers has its merits respectively. Controller C1shows a faster and better damped response
of the state with the gain k and disturbance increasing, while controller C2 shows a slower
response of the state than controller C1with the disturbance increasing and it hardly vary with
the gain k varying. For some choices of the parameters such as limited disturbance which
vanishes at the origin, the two controllers result in the same performance of the state response
converging to the origin.
REFERENCES
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[2] M. Jankovic and P.V. Kokotovic, Global stabilization of an enlarged class of cascade nonlinear systems
[3] F. Mazeac, R. Secpulchre, and M. Jankovic, Lyapunov Functions for stable cascades and applications to Global stabilization.
IEEE Trans. Automat. Contr.,Vol.44, pp1795-1800, sep. 1999.
[4] M. Arack, A.R. Teel, P.V. Kokotovic, Robust Nonlinear control of feedforward systems with unmodeled dynamics.
Automatica, Vol 37,pp 256-272,2001
[5] L. Praly, R. Ortega, and G. Kaliora, Stabilization of Nonlinear system via forwarding mod{ VLg }. IEEE Trans. Automat.
Contr, Vol.46. pp1461-1466. 2001.
[6] E.D.Sontag. A universal construction of Artsteins theorem on nonlinear stabilization. Systems Control Lett. 13
pp117-123.1989
[7] A.Isidori, Nonlinear Control System II, London, 1999.[8] H.K.Khalil. Nonlinear Systems. Prentice Hall, Englewood Cliffs, NJ, Second edition, 1996.
[9] G.Hardy, J.E. Littlewood, and G. Ploga Inequalities, 2nd.ed. Cambridge, UK, Cambridge Univ Press, 1989.
[10] E.D. Sontag. Smooth stabilization implies coprime factorization. IEE. Trans. Automat. Contr. Vol 34. pp435-443. 1989.
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