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Proceedings

of

the 1998

IEEE

International Conference on Control Applications

Trieste, Italy

1-4

September 199 8

TA05

A n Application of Robust Feedback L inearization

to

a Ball and Beam Control Problem

B. C. Chang

Department

of

Mechanical Engineering

Drexel University, Philadelphia, PA 19104

bchang@coe.drexel.edu

Harry Kwtany Shr-Shiung Hu

Department

of

Mechanical Engineering

Drexel University, Philadelphia, PA

19104

Department of Mechanical Engineering

Drexel University, Philadelphia, PA

19104

hkwatnv~,coe.drexil .edu

sr947cxk@~ost.drexel.edu

Abstract

In

this paper, we present how ,U-s ynthe sis can greatly

improve the robustness against the inexact cancellation arising in

feedback input-output linearization of nonlinear systems. A

simulation of the nonlinear ball and beam tracking problem

illustrates that ,L -synthesis controllers can offer much better

robust stability and robust p erformance than H , controllers.

I. Introduction

Despite

its

limitations, feedback input-output linearization

[1,2] is one of the most important tools in nonlinear control

systems design. The technique is mainly based on the

cancellation of nonlinear terms in the plant dynamics by the

controller. Exact cancellation

is

impossible in practice because

of inaccurate measurements, plant uncertainties, and

disturbances. Although

not

much discussion related

to

the

robustness issue is available in the literature [3-5j,

it

is well

known that the inexact cancellation can greatly hamper the

application of the technique.

The ap proach can be practical if the robustness issues caused

by inexact dynamics cancellation and imperfect state estimation

can be properly addressed. The effect of inexact dynamics

cancellation can be expressed

in

terms of plant uncertainty

or

norm bounded uncertain disturbance by which

a

,U -synthesis

[6-

141

or linear

H ,

control problem [15-221 can be formulated to

address the robustness issue.

In [23],

Hauser et. al. considered a nonlinear ball and beam

control problem in which they pointed out that the relative

degree [ 1,2] of the ball and be am system is not well defined and

thus

not

feedback input-output linearizable.

To

resolve this

difficulty,

a

feedback linearizable nonlinear model was used to

approximate the original ball and beam mode l. Although some

closed-loop tracking simulations with ideal controllers were

given to justify the approximation,

no

practical outer loop

controller design was employed to address the imperfect

dynamics cancellations caused by nonlinear plant uncertainties

and the inaccurate meas urement of the state variables.

In this paper, we use Hauser el.

al.'s

approximate input-

output linearization approach to design an inner-loop nonlinear

controller which approximately linearizes the input-output

relationship of the inne r closed-loop system. Then an outer-loop

linear controller is designed based on ,U -synthesis approach

to

achieve robust stability and robust performance. By computer

simulations, we find that p-synthesis controllers are able

to

provide robust stability and robust performance for reasonably

l a r g e p l a n t u n c e r t a i n t i e s a n d s t a t e v e c t o r m e a s u r e m e n t e r r o rs .

The simulations also show that

,U

-synthesis controllers offer

much better closed-loop robust stability and performance than

H, controllers. Although a

,U

-synthesis controller usually is of

'high order,

it

can be reduced tremendously without degrading

much of the performance.

* This research was supported in part by NASA Langley Research Center

under Contract NCC-1-224 and in part by the Boeing Company under

Contract N A S 1-20220.

The paper is organized

as

follows.-Section'II briefly reviews

feedback input-output linearization and

p

-synthesis. Th e design

of the inner-loop feedb ack linearization controller and the outer-

loop ,U -synthesis controller will be presented

in

Sections

111

and

IV respectively. Section V includes the computer simulations of

the ball and beam tracking for p -synthesis and H , controllers.

Section VI is

a

conclusion.

11. Preliminaries

input-output linearization and ,U -synthesis.

Feedback Input-Output Linearization

The basic concept of feedback input-output linearization

[1,2] is briefly reviewed as follows. Consider a SISO nonlinear

dynamic system with the following form

In this section, we will give a quick review of feedback

x =

f ( x )+ g(x)L

y = h ( x )

2-1)

where

x

is the nxl system state, U is the control input, y

is

the

controlled output, and g, lz are smooth functions of x

Let

L;( )

denote the kth order Lie derivative of the scalar function

@ (r)

with respect to the vector field

f ( x ) .

The

1st

order Lie

derivative is defined as:

(2-22)

and then higher order Lie derivatives are successively defined a s

The relative degree

of

the system is defined as

m

ax

LJ

($1

= <4% >

=

- f ( x )

L; 4) L, L;-y@))= < dL;-l(@),f

>

.

(2-2b)

2-3)

For

the ball and beam example, the relative degree equals

the number of system states if an approximate model

is

used. In

this case, we can define the vector

z

that consists of

and

a

coordinate transfohation x + z hat transforms (2-1)

into

r = inf(k L, (L;?(/Z))

1

z k =L;?(h),

k = 1 , 2

._., (2-4)

i =.Az + E[a(x ) p x ) u ]

y

=

cz

(2-5)

where

A ,

C,

E are constant matrices, and

Q X)

, p x ) are given

as follows,

The nonlinear system (2-1) can be input-output linearized by

letting

.

L

= p - ( x ) [ L t + v - c r (x ) j

which yields

i =

A

+

E L) z +

Ev

y = cz

(2-7)

where

L

is a constant matrix to be chosen

to

place the

eigenvalues

of

A+EL.

p-Syn

thesis

For the purpose of robust stability analysis, all the plant

uncertainties, structured

or

unstructured, unmodelled dynamics

0-7803-4104-X/98/ 10.0001998 IEEE

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or parametric perturbations, can be described by the following

block diagram [7],

L__oo---l

Fig.

2.1

M- A structure for robust stability analysis.

where A(s) =block diug [Al (s ) , A*($), ..,A,,,(s)} and M(s ) is the

nominal linear clos ed-loop system which includes the nominal

plant and the stabilizing controller.

S i nc e M A

(s)

I S stable, due to the fact that both

M

and A are

stable, the closed-loop stability can be ensured if and only if

I

+

M A (j u) remains nonsingular at all frequencies and for

all A

under consideration. With this M - A structure, the structured

singular value (SSV) of

M ,

or

p

( M ) [23] is defined as

p ( M )= [ m i n ( S ( A ) :det ( I +MA) = O}-

VU R , (2-9)

The structured singular value p is a measure of the system

robust stability.

A

smal ler

p

mean s better robust stability. Th e

value of

,U

depends not only on M but also on the structure of

A. Ignoring the structure of uncertainties can result in an

unnecessarily conservative control system design.

I2

 

W

Fig. 2.2 M A structure with input w and output z .

Besides being a good measure of robust stability, the SSV

can be also used for robust performance. Consider the system

shown in Fig. 2.2, where M is the nominal closed loop system

which includes the nominal plant and a controller, and the A

with IIAll_ < 1 represents the system uncertainties. Robust

stability mea ns that the closed-loo p system remains stable for all

uncertainties with IIA II_ < 1 and robust performance means that

the

H ,

norm of the closed loop system from w to

z

remains less

than one for

all

uncertainties with II A II_ < 1. However, by the

following main-loop theorem, robust stability and robust

performance can be put together and measured by a single

structured singular value.

Theorem

2.1:

(Main

Loop

Theorem) [6,9]:

Consider the block diagram in Fig. 2.3. The robust stability

and the robust performance can be guaranteed if and only if

p

( M ) < for all

w

and for al l d iug{ Al ,A 2}, where A,

represe nts perform ance block, and A, is a matrix with block

diagonal stru cture which represents th e system uncertainties.

Fig.

@w

.3 Main

Loop

Theorem

Th e process of ,U-ana lysis is to rearrange a given closed-

loop

system with uncertainties into an appropriate MA structure

and then compute the upper bound of the structured singular

value

p

for the

M A

structure. The process of

p

-synthesis, on

the other hand,

is

to design a controller K(s ) such that the closed-

loop

system M has

a

small upper bound of

y

with respect to the

given structure of A w hich includes the performance and the

plant uncertainty blocks. An existing algorithm for

p

-synthesis

is the D-K iteration algorithm

1141,

which consists of the p -

analysis

(D-Step) and the

H ,

optimization (K-Step).

Although

the D-K i teration algorithm usually does not give an optimal

solution, it has been satisfactory in many applications

[6,13].

111. The Ball and Beam Problem

I

..............................................

I

Fig. 3.1 The ball and beam system.

I n

Fig. 3.1, 

Y

is the position of the ball,

6

the angular

position of the beam, and

z

the torque applied to the beam. The

ball is assumed to roll without slipping on the beam. Let the

mass and moment of inertia of the ball be

M

and

J,>,

respectively, the moment of inertia of the beam be J , the radius

of the ball be R , and the acceleration of gravity be G. Define the

state vector

Then the ball and beam system can be represented by the

following model [23],

x = [ x ,

x, x, x

= [ u

i e 61

(3-

1)

x

=

j x)

g(X)u

3 =

12

 x)

with

f ( x ) = [ x z

B ( x , x ~

Gsinx , ) x4 0IT

g(x)

= i o

0 0

11

h(x)

=

x,

(3-2a)

(3-2b)

( 3 - 2 ~ )

(3-2d)

the torque

where

~=2MxIx,x,+MG~,cosx,+(Mx~J+J , , )u (3-2e)

(3-20

:=M / ( J ,>

R 2+ M )

The objective

of

the ball and beam control problem is to

design a controller so that the position of the ball will follow a

tracking signal that represents the desired trajectory of the ball.

Th e system is nonlinear and the set of ball equilibrium locations

is the straight line defined by the beam. First, in the rest of the

section, we will employ feedback linearization to design an

inner-loop nonlinear controller which render the input-output

relationship of the inner-loop approximately linear. Then, in the

next section, a

p

-synthesis outer-loop controller will be

designed to assure robust stability and robust performance.

As

pointed out by Hauser et.

al.

[23], the relative degree of

the ball and beam system in (3-2) is not well defined. To be able

to employ feedback input-output linearization approach for the

nonlinear ball and beam control problem, the

xp

term in (3-2b)

is ignored as suggested by [23]. With the coordinate

transformation x +

z

defined by the following

zi

= X I , z2

=x

z1 = -BG sin x3, z4 = -BGx4 cosx3

the approximate model of the ball and beam system, i.e., the

mode l of (3-2) with x,x,' term removed , can be rewritten as

(3-3)

z , = z 2 , 2 2

=z3,

z 3

= z 4

i = BGX: sin x, + (-BG

cos

x3 u :=a x )+ p x ) u

( 3 - 4 4

and

Let

Y

=

z ,

(3-4b)

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be tlie inner-loop cont roller, then the input-output relationship of

the inner closed-loop system becomes linear as shown in the

following,

z,

= z 2 ,

z2

= z 3 ,

z3 = z q

(3-6a)

i q =

-0 .0024~1

-0 .0 5 ~ 2

0.3523

4 v

and

y = 2,

(3-6b)

Recall that the success of feedback linearization approach

depends heavily on the dynamics cancellations. The functions

a(x)

and p x) computed based

on

tlie model may not be the

same as those i n the real world; furthermore, the measured state

variables are not the same as the actual state variables. In the

next section, an outer-loop ,U -synthesis controller will designed

to address these robustness issues.

IV. Robustness Considerations

In this section, we will formulate a ,U -synth esis control

problem

so

that an outer-loop linear controller can be

constructed to provide robust stability/performance against the

inexact dynamics cancellation arising in the inner-loop feedback

linearization design.

..................

.

CL .......f i

..........

...............,

. ..........................

Y

: .

* ;

: .

..............................................................

1 I

~ K ;

: - - ;

.

:

........................

i

..............

i Y ?

Fig. 4.1 Formulation of an outer-loop control problem.

In Fig. 4.1, 

PL

stands for the linearized system (3-6). The

objective is to find a controller

K

so

that the closed-lop system is

robustly stable and the displacement of the ball follows

w2,

he

reference signal, as closely as possible.

We

is a weighting

function for the tracking error, usually a low-pass filter;

W,,

is a

weighting function for the measurement noise, usually a high-

pass

filter. Th e combination of W , and A , represents the plant

uncertainty, and usually W, is a high-pass filter.

W e ,

W,,, and W , are weighting functions chosen by the

designers such that the design specifications can be met. We

choose them as follows.

(4- 1)

100(s 100)

We=- 0.3 y , 1 0 w,=

s +0.03

s+10000

We

is a low-pass filter to emp hasiz e the tracking accuracy at low

frequencies.

W,,

is for measurem ent noises, and W s for plant

uncertainties including the inexact cancellation caused by

modeling error or state vector measurement error in the inner-

Combining the plant, fL, nd the weighting functions

W e ,

W,,

and

W,,

we have the generalized plant C in Fig. 4.2.  A i is

a 1x1 plant uncertainty block, and A2 a 2x1 fictitious

performance block.

With the

D-K

iteration algorithm

i n

Section

11, we first

obtained the

H ,

controller

K , ( s )

with tlie optimal

H

norm

equal to 6.714. Since K , ( s ) ignores the structure information of

A and treats A as

a

full matrix, it gives a conservative solution

to the problem. Fig.4-3a show s the plot and

p

plot

of

the

closed loop system.

loop.

.............................

.....,

5,

[A, 0 ] ............

: :

: oA( - . . . -w . .

j

;w2

j :

I .....4

._.,:

; ; ................................. j j

:

i .

......,......

: i -

.

-........

I

-............. I

t 7

w3

............

:

. .

.

..

..... ................

Fig. 4.2 Generalized plant for p -synthesis.

With the

D-K

iteration algorithm in Section

11

we first

obtained the

H,

controller K , ( s ) with the optimal

H,

norm

equal to 6.714. Since

K , ( s )

ignores the structure inform ation of

A

and treats A as a full matrix, it gives

a

conservative solution

to the problem. Fig.4-3a shows the plot and

,U

plot of the

closed loop system.

.

~ ' ~ ' . . . . . .

'

. .

' ~ ~ ' '

1

I

10

I00

I000

As

expected,

,U

plot is lower than

B

plot, i.e.,

l / B

<

l / p ,

at each frequency. This implies that the allowable set of

structured uncertainties is larger than that of unstructured

uncertainties. Next we will continue the

D - K

iteration design.

After five iterations, the process converges to a controller K , ( s )

which gives the 0 plot and

p

plot

of 5,[ (s ) , , ( s ) ]

in Fig.4-

3b, where ~ G, K ] is lower linear fractional transformation.

Note that

K ,

s ) gives much better ,U than K ,(s)

Fig. 4.3a

b

and

p

plots for the

H _

control law.

1 0 8 l l

0 1 10Frequency (radk)

00 1000

Fig. 4.3b and

p

plots for the K,

(s)

p -synthesis control.

In tlie

D-K

iteration, we choose the order of the scaling

function

D ( s )

to be

2

which implies that

6(s),

and hence

K , ( s ) , are

of

order

10. K , ( s )

has ten Hankel singular values as

follows: 6

58e 4

6

27e 4 5 75e 4 1. 89e 4 8 40e 3 4. 34e 3

3

47e 3 7 47et 2

5

45e-3 4 29e- 3

It is easy to see that

K , ( s )

can be reduced to an gLhorder controller K , ( s ) by

truncating its balanced realization. Th e ,U plots for tlie closed-

loop

systems

3 [G s),

, s)] and

&Tr[G(s),

K ,

(s)]

are shown in

Fig. 4.4. The fact that the two plots coincide together reveals

that K , ( s ) is an excellent approximation of

K ,

3).

I

I 5

1

0

' '

'

' '

'

0. 1 10

100

1000

Fig. 4.4 

p

plots for the closed-loop systems

3,[G(s),

,

(s)]

and Zr[C(s), K r

(s)]

.

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Time response simulation for the closed-loop system with

the reduced p -synthesis controller

K ,

(s) will be given in the

next section.

V. Time

Response Simulations

The simulation diagram is shown in Fig. 5.1  i n which the

plant:

Note that the term Bx,~, s not ignored in simulation;

furthermore, we assume there is a perturbation tenn A,, sin t in

the plant. Th e measured state vector contains measurement

errors as follows,

A I =

x,

A = x, A,,? in 1Ot,

= 2,3,4

(5-2)

.

. ...

I

Fig. 5.1 Simulation diagram.

The coordinate transformation

A

-+2 is defined by the

fol owi ng

21

=A,, 22

= A 2

(5-3)

i3

= -BG

sin

i3 14 = -BGi4

cos

A3

The dem ul t iplexer D e m i x extracts

2,

and 2 components out of

the vector 2 in which 2 stands for r , the position of the ball,

and 2 = -BGsin 2, where A is

6 ,

the angular position of the

beam, contaminated with measureme nt error. The inner-loop

feedback linearization nonlinear controller is described by the

foll

owing,

=-[-a(;)-& -0.352, -0.052, -0.0024z1, V I

(5-4)

P ( 3

where p(A) = - B G c o s i 3 , a n d a i ) BGi: sin

A 3 .

The outer-

loop

l inear controller

K , ( s ) ,

of

gLh

rder, was designed in the

previous section by p -synthesis and model reduction.

n the simulation, the system parameters are chosen the sam e

as those in [23]:

M = 0.05

kg, R

= 0.01

m, J

=

0.02 kg i n 2 ,

J,, =

0.000002 kg m 2 , G = 9.81 d s 2 , nd thus,

B

= 0.7143. First of

all, we assume'that A,,, and A,, i n (5-2) and (5-1) are zero; that

means no measurement errors or sinusoidal plant perturbations.

Th e only perturbation considered in the first simulation is the

term Bx,xi hat we ignored in the design model. The tracking

signal is assumed 0.5(1-e- ) which can be regardcd as a

combination of low fr equenc y signals or as the output of the

low-pass filter

I /(s+l)

driven by a step function. Fig.5.2a

shows the tracking response of the ball position for the closed-

loop system with reduced-order p -synthesis controller

K, ( s )

.

We can se e that the tracking error is very small.

0.6

I

I

esponse

10 20 30 40

seconds

Fig. 5.2a Trackin g response for the closed-loop system with

reduced-order p -synthesis controller

K , ( s )

when

A,,,

=O

and A,

=O.

Next, besides the plant perturbation term Bx ,~ :, we assume

that the measurement errors i n (5-2) are 0.2sinlOt and the

sinusoidal perturbation i n (5-1) is 0.3sin t , i.e.,

A

=0.2 and

A,,=0.3. In this case, the dynam ics cancellation in the inner-

loop

feedback linearization is Tar from perfect. Fig.5.2b shows

that the reduced-order p -synthesis outer-loop controller K , ( s )

provides excellent robust tracking performance. The

perturbations and measurement errors ha ve only slight effect

on

the tracking response.

0 6

0 4

0 2

I

0 I O 20 30 40 seconds

0.2

Fig. 5.2b Tracking response for the closed-loop system with

reduced- order p -synthesis controller K , ( s ) when

A,,,=0.2 and A,,=0.3.

For the purpose of comparison, we will design an H _

controller and compare its tracking performance and robustness

handling ability with th e p -synthesis controller. Th e controller

K , ( s ) ,obtained in the first iteration of D-K algorithm, indeed is

an H , controller based

on

the weighting functions (4-1). The

tracking error fo r this controller K , ( s ) is unacceptable. In order

to achieve a decent tracking performance for H , controller, we

modify the weighting functions as follows,

( 5 - 5 )

lO(s 100)

w, - O 3 w,,O .OO l w -

s 0.03

A

- s 10000000

A

61h-order H , controller

K _

(s) is designed based on the block

diagram in Fig. 4.1 with the weighting functions in

(5-1).

Now,

we will replace the K,(s ) in the simulation diagram of Fig. 5.1 

by

K _ ( s )

and repeat what we just did for the p -synthesis

controller.

Again, firstly we assume that A,8,and A,, in (5-2) and

(5-1)

are zero; that means no measurement errors and no sinusoidal

plant perturbations. The only perturbation considered in the first

simulation is the term Bx,x: that we ignored in the design

model. The tracking signal is assumed 0 . 3 -e- ) which can be

regarded as th e output of the low-pass filter 1/ s

1)

driven by

a

step function. Fig.5.2a shows the tracking response

of

the ball

position for the closed-loop system with the H , controller

K _ ( s ) . Although the tracking ability of the H , controller is

not as good as the ,U -synthesis controller, its tracking response

is close to the reference signal.

69

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0.6 I I

reference

esponse

I

10

20 30

40 seconds

Fig. 5.321 Tr acking re spons e for the closed-loop system w ith

U ,

controller

K _

s )

when

A,,t

=0

and

A ,

O.

40 seconds

Fig. 5.3b Track ing response for the closed-loop system with

H _

controller K _ ( s ) when A,,,=0.02 and A,,=0.3.

Next, besides the plant perturbation term

B x , x i ,

we assume

that the measurement error in (5-2) is 0.02sinlOt and the

sinusoidal perturbation in (5-1) is 0.3sin

t ,

i.e.,

A,,,

=0.02 and

A,=0.3. Fig.5.3b shows that the tracking error grows without

bound as time increases. Note that the measurement error in the

simulation of

p

-synthesis controller is 10 times larger; still, the

system remains stable and has excellent robust performance as

shown in Fig. 5.2b. From the comparison, we see that the

p -

synthesis controller provides much better robust stability and

robust performance than the

U ,

controller.

VI. Conclusions

In this paper, we applied approximate feedback linearization

to take care of the nonlinearity of the ball and beam control

system. Then p -synthesis was employed to address the

robustness issues arising in linearization process due to inexact

dynam ics cancellation. Com puter simulations revealed that p

-

synthesis controllers are able to provide robust stability and

robust performance for reasonably large plant uncertainties and

state vector measurement errors. The simulations also show that

p

-synthesis controllers offer much better closed-loop robust

stability and performance than

H ,

controllers

References

A. Isidori, Nonlinear Control SJwterizs, 3rd ed. Berlin:

Springer-Verlag, 1995.

H.

Nijmeijer and A.J. van der Schaft,

Nonlinear D.ynanzical

Control Systems,

Springer-Verlag, 1990

Shankar Sastry, John Hauser, and Petar Kokotovic, Zero

Dynamics of Regularly Perturbed Systems May Be

Singularly Perturbed ,

System & Control Letters

13 (1989)

Sastry, S.S., and Kokotovic, P.V., Feedback Linearization

in the Presence of Uncertainties,

Int. J. Adapt. Contr. &

Signal Processing, Vol. 2, p p 327-346, 1988.

F. Esfandiari and H. Khalil, Output Feedback

Stabilization of F u l l y Linearizable Systems, Zrzt.

J .

Corztr.,

J. Doyle,

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