control switching

8/10/2019 control switching

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A Switching Controller for Uncertain Nonlinear SystemsBy H.K. Lam, Frank H.F. Leung, and Peter K.S. Tam

Control of nonlinear systems is difficult because no

systematic mathematical tools exist to help find

necessary and sufficient conditions to guarantee

their stability and performance. The problem becomes

more complex if some of the parameters of the plant are un-

known. By using a Takagi-Sugeno-Kang (TSK) fuzzy plant

model [1]-[2], [8], [13], a nonlinearsystem canbe expressed

as a weighted sum of some simple subsystems. This model

gives a fixed structure to some nonlinear systems and thus

facilitates analysis. There are two ways to obtain the fuzzy

plant model: 1) by applying identification methods with in-

put-output data from theplant[1]-[2], [8], [13] or 2) directly

from the mathematical model of the nonlinear plant.In recentinvestigationsinto thestabilityof fuzzysystems

formedby a fuzzyplantmodeland a fuzzycontroller, several

stability conditions have been obtained [4]-[5], [9]-[12],

[16]-[19]. A linear controller [14] was also proposed to con-

trol the plant represented by the fuzzy plant model. Most of

the fuzzycontrollersproposed arefunctions of thegrades of

membership of the fuzzy plant model. Hence, the member-

ship functionsof thefuzzy plant model must be known.This

means that the parameters of the nonlinear plant must be

known or must be constant when the identification method

is used to derive the fuzzy plant model. Practically, the pa-

rametersof manynonlinear plants willchangeduring opera-

tion (e.g.,the load of a dc-dc power converter orthenumberof passengers on board a train). In these cases, the robust-

ness property of the fuzzy controller is an important con-

cern. Moreover, the investigations [4]-[5], [9]-[12] tackled

only a regulation problem such that thecontrollers drive all

thesystem statesto zero. Inpractice,we mayface a nonzero

set-point regulation problem or a tracking problem. To

tackle these problems, some algorithms integrating fuzzy

logic with adaptive control theory [3], [7], [15] or with H

control theory [6] can be found.

In this article, a switching controller is proposed to con-

trol nonlinear plants subject to unknown parameterswithin

known bounds. The nonlinear plant is represented by a

fuzzy plant model. This switching controller is able to drive

the system states to follow those of a reference model. The

switching controller consists of several linear controllers.

One of the linear controllers is employed at each moment

according to a switchingscheme, which is derived based on

Lyapunov stability theory.

Theremainder of thearticle is organized as follows. First,

we describe a reference model, a fuzzy plant model, and a

switching controller. Next, we investigate the system stabil-

ity of the switching control system. The switching scheme

will be derived based on Lyapunov stability theory, and the

gains of theswitching controllerswill be designed.Then, we

provide an application example of an inverted pendulum on

a cart. Finally, we present our conclusions.

Reference Model, Fuzzy Plant Model,and Switching ControllerThe nonlinear plant to be tackled is of the following form:

( ) ( ( )) ( ) ( ( )) ( )x A x x B x ut t t t t = + (1)

where A x( ( ))t n n andB x( ( ))t n m are the system ma-

trix and input matrix, respectively, both of which have

known structure but may be subject to unknown parame-

ters (all matrices considered herein are real matrices);

x( )t n 1 is the system state vector; and u( )t m 1 is the

input vector. The system of (1) is represented by a fuzzy

plant model that expresses the multivariable nonlinear sys-

tem as a weighted sum of linear systems. A switching con-

troller is to be designed to close the feedback loop of the

nonlinearplant basedon thefuzzyplant modelsuch that the

system states follow those of a reference model.

Reference ModelThe reference model is a stable linear system given by

( ) ( ) ( )x H x B rt t tm m= + , (2)

where H mn n is a constant stable system matrix,

Bmn m is a constant input matrix, ( )x t n 1 is the sys-

tem state vector of this reference model, and r( )t m 1 is

the bounded reference input.

Fuzzy Plant ModelLetting p be the number of fuzzy rules describing the

multivariable nonlinear plant of (1), theith rule is of the fol-

lowing format:

Rulei: IFf t1( ( ))x isM 1i and ... andf t( ( ))x isM

i

THEN ( ) ( ) ( )x A x B ut t ti i= + , (3)

whereM i is a fuzzy term of ruleicorresponding to the func-

tionf t ( ( ))x in terms of the system states and the unknown

parameters of the nonlinear plant, = =1 2 1 2, , , , , , , ; i p is

February 2002 IEEE Control Systems Magazine 7

Lam ([email protected]), Leung, and Tam are with the Department of Electronic and Information Engineering, The Hong Kong Poly-

technic University, Hung Hom, Kowloon, Hong Kong.

LECTURE NOTE

Authorized licensed use limited to: University of Michigan Library. Downloaded on May 19,2010 at 19:24:43 UTC from IEEE Xplore. Restrictions apply.


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a positive integer, and Ain n and Bi

n m are known sys-tem and input matrices, respectively, of theith rule subsys-

tem. The system dynamics are described by

( )( ) ( ( )) ( ) ( )x x A x B ut w t t t i

p

i i i= +=1 , (4)

where

[ ]w t w t i ii

p

i

= =

1

1 0 1( )x x( ) , ( ( )) for all ,(5)

and

w t

f t f t f

i

i i i

( ( ))

( ( ( ))) ( ( ( ))) (

x

x x

=

M M M1 21 2 ( ( )))

( ( ( ))) ( ( ( ))) ( (

x

x x x

t

f t f t f k k k

M M M1 21 2 ( )( ))tk

p

=1

(6)

are nonlinearfunctions of thesystemstatesand theunknown

parameters. (For more details, see [1]-[2], [8], and [13].)

In this article, the fuzzy plant model of (4) is assumed to

have the following properties:

A A H B Di i m m i i p= = =, , ,...,1 2 (7)

B x x B x B( ( )) ( ( )) ( ( ))t w t t ii

p

i m= ==1

,(8)

where Dim n i p = , , ,...,1 2 , are constant matrices.( ( ))x t is

an unknown nonzero scalar (because w ti( ( ))x is unknown)

but with known bounds and sign. It should be noted that be-

cause ( ( ))x t 0 is required, B x 0( ( ))t is assumed.

Switching ControllerA switching controller is employed to control the nonlinear

plant of (1). The switching controller consists of some sim-

plesubcontrollersthat will be switched from oneto another

to control the system of (1). The switching controller is de-

scribed by

( )u x G x r( ) ( ( )) ( )t m t t j jj

p

= +=

1

,

(9)

where m t j pj( ( )), , ,... ,x =1 2 , takes thevalueof 1/ min or 1/ minaccording to the switching scheme to be discussed later,

ministheminimum valueof( ( ))x t ,andGjm n j p = , , ,...,1 2 ,

are the feedbackgains to be designed. It can be seenthat(9)

is a linear combination of plinear state-feedback control-

lers. At each moment, one of the linear state-feedback con-

trollerswill be chosento control thenonlinear plant accord-

ing to the switching scheme.

Stability Analysis and Design of theSwitching ControllerIn this section, the switching controller will be designed to

consider the system stability. The analysis results are sum-

marized by the following lemma.

Lemma 1: The system states of the nonlinear plant of (1)

represented by the fuzzy plant model of (4)will followthose

of the reference model of (2) if the fuzzy plant model satis-

fies the conditions of (7) and (8) and the switching control-

ler is designed by choosing

i) P 0 >n n

ii)( )( )

mt t t

i

Tm i

= +sgn

sgn

e PB G x r ( ) ( ) ( )

( ) min i p=1 2, ,...,

iii)

( )Q H P PH 0= + >

m

T

miv) G Di i i p= =, , ,... ,1 2 , where

( ) [ ]sgn zz

z=

>

1 0

1 0, ,min max and max min .> >0

See the Appendix for the proof (it should be noted that Q

is needed to prove stability, not for control design). The de-

sign procedure of the linear controller is summarized in the

following steps:

Step I:Obtain the fuzzy plant model of a nonlinear

plant by means of the methods in [1]-[2], [8], [13] or

other suitable ways.

Step II:Choose a reference model in the form of (2).

Step III:Check whether the fuzzy plant model satisfies

conditions (7) and (8).

Step IV:Design the switching controller according to

conditions i) through iv) of Lemma 1.

Application ExampleAn application example will be given here to show the de-

sign procedure of the switching controller. A cart-pole in-

verted pendulum system [5] is shown in Fig. 1. A switching

controller will be designed for it by followingthe designpro-

cedure given in the previous section.

Step I:The dynamic equation of the cart-pole inverted

pendulum system is given by

( ) sin( ( )) ( ) sin( ( ))/ cos( ( ))

t g t aml t t a t =

2 2 2 u tl aml t

( )

/ cos ( ( )),

4 3 2 (10)

where is the angular displacement of the pendulum,g=9 8.m/s

2is the acceleration due to gravity, [ ]m m m =min max

[ ]0 5 2. kg is the mass of the pendulum, [ ]M M M min max[ ]= 8 80 kgis the massof the cart, a =1/( ),m M+ 2 1l= m is the

length of the pendulum, anduis the force applied to the cart.

8 IEEE Control Systems Magazine February 2002



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an d M M1 1

11 11k f t f t ( ( ( ))) ( ( ( )))x x= fo rk =3 4,;

M2

2

2 2

2 2

k f tf t f

f f( ( ( )))

( ( ))max

max min

xx

= +

fork =13,

and M M2 2

12 21k f t f t ( ( ( ))) ( ( ( )))x x= fork =2 4, .

Step II: The system matrix and the in-

put vector of the reference model are

chosen as follows:

Hm=

0 1

8 8 (14)

B m=

0

1 . (15)

It can be seen that the reference model

is a stable system.

Step III : Conditions i) and ii) of

Lemma 1 are satisfied if we choose

[ ] [ ]D D1 2 1 8 8 17 8= = + =f min (16)

[ ] [ ]D D3 4 1 8 8 28 8= = + =f max . (17)

It can be seen that

( ( )) ( ( ))x xt f t=


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We let [ ]x( ) ( ) ( ) [ ( ) ( )]t x t x t t t T T= =1 2 , [ ] ( ) min maxt =[ ]( / ) ( / )22 45 22 45 , and [ ]( ) [ ]min max t = 5 5 . Thenthestate-space representation of (10) is given by

( ) ( ) ( ) ( ( ))( )

( ( ))x Ax B

x xt t u t

f tx t

f t= + =

+0 10 01 2

u t( ).

(11)

By comparing (11) to (10), we can see that

f t g amlx t x t

l aml x t 1

2

2

1

2

14 3( ( ))

( ) cos( ( ))

/ cos ( ( ))

sx =

in( ( ))

( )

x t

x t1

1

and

f t a x t

l aml x t

21

2

14 3

( ( )) cos( ( ))

/ cos ( ( ))

x =

.

Theinvertedpendulum of (10) canbe modeled by a fuzzy

plant model having four rules.Theith rule can be written as

follows:

Rulei: IFf t1( ( ))x is M1i ANDf t2( ( ))x is M2

i

THEN ( ) ( ) ( )x A x Bt t u t i i= + for i=123 4, , , (12)

so that the system dynamical behavior

is described by

( )( ) ( ) ( )x A x Bt w t u t i i ii= +=14

,(13)

where

A A

A A

1 21

3 41

0 1

0

0 1

0

= =

= =

f

f

min

max

;

and

B B1 3 2

0

= =

f min and B B2 4

2

0

= =

f max ;

f f t1 19min ( ( ))= x andf f t1 220max ( ( ))= x ;

f f t

f f t

2 2

2 2

0 2

0001

min

max

. ( ( ))

. ( ( ));

=

=

x

x

and

and

wf t f t

f ti

i i

l

=

M M

M M

1 2

1 2

1 2

1

( ( ( ))) ( ( ( )))

( ( ( )))

x x

x( )l f tl

( ( ( )))21

4

x

=

.

The membership functions, shown in Fig. 2, are given as fol-lows:

M1

1

1 1

1 1

k f tf t f

f f( ( ( )))

( ( ))max

max min

xx

= +

fork =1 2,


M

m gsin ( )

x1=

u

l

x2= .

Figure 1. A cart-pole-type inverted pendulum system.

GradeofMembe

rship

GradeofMembership

1

0.8

0.6

0.4

0.2

08 10 12 14 16 18 20

1

0.8

0.6

0.4

0.2

00.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0

x ( ) [rad]1 t

x ( ) [rad/s]2 t

(a)

(b)

Figure 2. Membershipfunctionsforthe fuzzyplantmodeloftheinvertedpendulumon acart.



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To show the merits of the proposed

switching controller, we compare the

simulation results of the system under

theswitchingcontroller tothoseundera

linear state-feedback controller. The lin-

earstate-feedback controlleris designed

based on the linearized model of the in-

verted pendulum around the origin. The

dynamics of the linearized model of the

inverted pendulum are given by

( ) ~

( ) ~

( )x Ax Bt t u t = + , (20)

where

~,

~A B= +

=

0 1

0

0

1M m

M l

g

M l

o o

o o

,

m m m

o= +min max

2, and M

M Mo=

+min max2

.

The linear state feedback controller

output is given by

u t t M lr o( ) ( )= Gx . (21)

The feedbackgainis set asG=[ . . ]1300 320such that the dynamics of the closed-

loop system are the same as that of the

reference model. The simulation resultsare obtained by using the actual plant of

(10) and the switching controller of (19)

or the linearstate-feedback controller of

(21). A disturbance is also injected into

thesimulated system to reflect the prac-

tical situationof thereal process.Distur-

bances are injected into the system

states and the control input. The actual

system state used by the switching con-

troller in the simulation is x x( ) ( )t td+ .

Here, x d t( ) is thedisturbance of thesys-

tem states defined as [ ]x dT

t( )= 0 0

when t15 and [ ]x dTt( ) ( / )= 20 0 when

15 20< t . x d t( )reachesabout10%of the

maximum value ofx t1( ). The actual con-

trolsignalfed tothe input ofthe inverted

pendulum is u t u t d( ) ( )+ in the simula-

tion. u td( )is the disturbance defined as

u td( )=0 when t10, u td( )=1000 when

10 15< t , and u td( )=0 when 15 20< t .

u td( )reachesabout20%of themaximum

value of u t( ). Figs. 3 and 4 show the re-


0 2 4 6 8 10 1214 16 182015

10

5

0

5

10

15

2 1

3

0 2 4 6 8 10 12 14 16 182030

20

10

0

10

20

3021

3

x(

)(rad)

1

t

x

()

(rad/s)

2

t

Time [s] Time [s]

(a) (b)

Figure 6. Responses ofx( )t and ( )x t with the linear state-feedback controller under theinitial conditions of x( ) [ . . ]0 0 5 0 5= T (response 1), x ( ) [( / ) . ]0 22 45 0 5= T (response 2), ( ) [ ]x 0 0 0= T (response 3), m = 0.5 kg, M = 80 kg, andr t( )=8.

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

2

1

3

0 2 4 6 8 10 12 14 16 18201.5

1

0.5

0

0.5

1

31

2x

()(rad)

1

t

x

()(rad/s)

2

t

Time [s] Time [s]

(a) (b)

Figure 7. Responses of x( )t and ( )x t with the switching controller under the initialconditions of x( ) [ . . ]0 0 5 0 5= T (response 1), x( ) [( / ) . ]0 22 45 0 5= T (response 2), ( ) [ ]x 0 0 0= T (response 3), m = 2 kg, M = 8 kg, andr t( )=8.

0 2 4 6 8 10 12 14 16 18 202

1.5

10.5

0

0.5

1

1.5

2

2 3

1

0 2 4 6 8 10 12 14 16 182054321

012345

3

12

x

(

)(rad)

1

t

x

()

(rad/s)

2

t

Time [s] Time [s]

(a) (b)

Figure 8. Responses ofx( )t and ( )x t with the linear state-feedback controller under theinitial conditions ofx( ) [ . . ]0 0 5 0 5= T (response 1), x( ) [( / ) . ]0 22 45 0 5= T (response 2), ( ) [ ]x 0 0 0= T (response 3), m = 2 kg, M = 8 kg, andr t( )=8.



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sponses of x( )t and ( )x t under the initial conditions of

[ ]x( ) . .0 0 5 0 5= T

(dotted lines), [ ]x( ) ( / ) .0 22 45 0 5= (dashedlines), and [ ]( )x0 0 0=

T(solid lines), withm=05. kg,M= 8 kg,

and r t( )= 8.Figs. 5 and 6 show the responses for m=05. kgand

M= 80 kg. Figs. 7 and 8 show the responses for m =2 kg and

M= 8 kg. Figs. 9 and 10 show the responses for m =2 kg and

M= 80 kg. The simulation results show that the proposed

switchingcontroller gives a better performance under different

combinations of the values of the uncertain parameters than

the linear state-feedback controller.

ConclusionA switchingcontrollerhasbeendesignedfornonlinear plants

subject tounknown parameters. Under some conditions, this

switchingcontroller hastheability to drive thesystemstates

to follow those of a reference model. An application example

of an inverted pendulum on a cart has been given.

AcknowledgmentsThe work described in this article was substantially sup-ported by a Research Grant of The Hong Kong Polytechnic

University (Project G-S888).

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6, pp. 1029-1039, 1997.

[10] C.M. Cheng and N.W. Rees, Stability analysis

of fuzzy multivariable systems: Vector Lyapunov

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[12] H.J. Kang, C. Kwon, H. Lee, and M. Park, Ro-bust stability analysis and design method for the

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[13] E. Kim, M. Park, S. Ji, and M. Park, A new ap-proach to fuzzy modeling, IEEE Trans. Fuzzy Syst.,

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[14] S.H. ,Zak Stabilizing fuzzy system models us-

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no. 6, pp. 825-840, Nov. 2000.

[18] H.K. Lam, F.H.F. Leung, and P.K.S. Tam, Fuzzy

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ear state feedback controller for nonlinear sys-

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657-661, Aug. 2001.

12 IEEE Control Systems Magazine February 2002

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

2

1

3

0 2 4 6 8 10 12 14 16 18201

0.80.60.40.2

00.20.40.60.8

1

31

2x()(rad)

1

t

x

()(rad/s)

2

t

Time [s] Time [s]

(a) (b)

Figure 9. Responses of x( )t and ( )xt with the switching controller under the initialconditions of x( ) [ . . ]0 0 5 0 5=

T(response 1), x( ) [( / ) . ]0 22 45 0 5=

T(response 2),

( ) [ ]x 0 0 0= T

(response 3), m = 2 kg, M = 80 kg, and r t( ) =8.

0 2 4 6 8 10 12 14 16 18 2015

10

5

0

5

10

15

2 1

3

0 2 4 6 8 10 12 14 16 18 2030

20

10

0

10

20

3021

3

x

()(rad)

1

t

x

()(ra

d/s)

2

t

Time [s] Time [s]

(a) (b)

Figure 10. Responses ofx( )t and ( )x twith the linear state-feedback controller under theinitial conditions ofx( ) [ . . ]0 0 5 0 5= T (response 1), x( ) [( / ) . ]0 22 45 0 5= T (response 2),

( ) [ ]x 0 0 0= T (response 3), m = 2 kg, M = 80 kg, and r t( ) =8.



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