controller design for a class of cascade nonlinear systems using

8/12/2019 Controller Design for a Class of Cascade Nonlinear Systems Using

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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 ,

http://www.paper.edu.cn


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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

[1] R. Sepulchre, M. Jankovic, and P.V. KoKotovic, Constructive Nonlinear Control. New York: Spinger-Verlag,1997

[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.


controller design for a class of cascade nonlinear systems using

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