1

Nonlinear Control Via Approximation Input-Output Linearization: The Ball

and Beam ExampleA paper1 by John Hauser, Shankar Sastry, and Petar Kokotovic

Presented by Christopher K. JohnsonMEAM 61325 April 2002

1IEEE Transactions on Automatic Control, Vol. 37, No. 3, March 1992

2

Introduction Presentation of a method for constructing

approximate systems for nonlinear systems that do not have a well-defined relative degree.

The approximate system can be used to control the original nonlinear system.

Outline: Ball and Beam example Approximate I/O linearization of the Ball and Beam Presentation of simulation results Analysis of Approximate Linearization

3

System Dynamics

u

xxGxxB

x

xxxx

1000

0

sin

4

3241

2

4

3

2

1

r

Ball and Beam

uJJMr

MGrrMr

b )(

cos22

Define new input

(invertible)

1)()()(

xxhyuxgxfx

Define states

4

3

2

1

rr

xxxx

cos2

sin0

2

22

MGrrMr

JJMr

MrMGrMRJ

b

b

4

Standard Input-Output Linearization Usual Procedure for Linearization

uxBxxBGxxBxy

xBGxBxy

xyxxhy

4134242

)3(

3241

2

1

2cos

sin

)(

b(x) a(x)u

)()(

1 xbvxa

u

Relative degree is undefined when r = 0 or

d/dt = 0

problem beam and ball thelinearizefully cannot

involutivenot , , , 2 gadgadgspan nff

5

Approximate I/O Linearization Achieve approximate I/O linearization by choosing a

control law to exactly linearize a system that is close to the true N.L. system

Change coordinates =(x) and choose input to make original system look like the approximate system perturbed by higher order terms():

)()(

)()(

1

1)0(

04)3(

3)4(

ddd yyyv

xbvxa

u

1/s 1/s 1/s 1/s4 3 2 1v y

small N.L. terms ()

6

Approximation 1 (modification of f)

Neglect higher-order centrifugal force term Well-defined relative degree Choice of what to neglect determines .

uxBGxBGx

xBGx

xBxxBG

x

x

)cos(sin

cos

sin

3344

343

24132

21

11

vaub :4

43

232

21

(ignore)

7

SIMULINK® Simulation Model

8

Simulation Results: Approx #1 Small

tracking error (increases nonlinearly with A)

0.2 rad = 11.5 deg

Successful tracking control

3 ,2 ,1 , 5 cos)(

AtAtyd

9

Approximation 2 (modification of g)

Only neglect terms that prevent a well-defined relative deg.

Well-defined relative degree of 4

uxBxxBGxxBBxxB

uxBxxBxxBGx

xBxxBG

x

x

)2cos(sin)1(

2cos

sin

423

324

441

24

41242343

24132

21

11

vaub :4

443

32

21

(ignore)

10

Simulation Results: Approx #2

Smaller tracking errors than Approx #1

0.2 rad = 11.5 deg

Very successful tracking control

3 ,2 ,1 , 5 cos)(

AtAtyd

11

Approximation 3 (Jacobian)

Neglect terms higher order than O(x) or O(). (Standard “Jacobian” linearization). Note , not u.

Purpose is for a baseline comparison. Well-defined relative degree of 4

...

)sin(

1

2

4

43

3324132

21

11

TOHJJ

BGxJJ

BMG

BGx

xxBGxBxBGx

x

x

bb

(ignore)

b(x)+a(x)

12

Simulation Results: Jacobian Approx

Large Error for A = 1,2

Unstable for A = 3

Relatively poor control scheme

3 ,2 ,1 , 5 cos)(

AtAtyd

Larg

er s

cale

tha

n pr

evio

us g

raph

s

13

Maximum Error e=(yd-x1)

A Approx #1 Approx #2 Approx #3(Jacobian)

1 3.1 x 10-4 1.3 x 10-4 2.6 x 10-2

2 2.5 x 10-3 1.0 x 10-3 4.6 x 10-1

3 8.4 x 10-3 3.6 x 10-3 Unstable

6 7.0 x 10-2 3.3 x 10-2 Unstable

14

Analysis of Approximate Linearization For systems where the relative degree fails to exist:

)()()( 01 xxxh

uxLxLx gf )()()( 111

),()()( 121 uxxxL guf

),()()( 1 uxxxL iiiguf

),()()()( uxuxaxbxL guf

Choose function to approximate the output.

O(x)2

Differentiate along trajectories

All O(x) and higher from Lg and (if desired) any O(x)2 or higher from Lf

Stop when Lg term is O(1) a(x) is O(1).

ROBUST RELATIVE DEGREE

15

Analysis of Approximate Linearization

Characterize the robust relative degree:

Thm 4.1: The robust relative degree of the nonlinear system is equal to the relative degree of the Jacobian linearized system whenever either is defined.

Corollary 4.2: The robust relative degree of a nonlinear system is invariant under a state-dependent change of control coordinates of the form u(x,v) = a(x) + b(x)v [a,b smooth and a(0)=0 and b(0)<>0].

)()()(

xhyuxgxfx

czybuAzz

Nonlinear System: Linearized System: (x=u=0)

16

Analysis of Approximate Linearization Show that neglecting i produces an approximation to the true

system i(.) can be used as part of a local change of coordinates Frobenius Thm, we can complete the N.L change of coordinates with a

set of functions i(x), i = 1,…,n- s.t. Lgi(x)=0

),(),(

),(),(),(

,

,

)()()()(

01

1

11

121

1111

uxyq

uxuab

ux

ux

xxxx Tn

Tn

New Coordinates Normal Form +

PerturbationsI of O(x,u)2

17

Analysis of Approximate Linearization

This choice of u will linearize the nonlinear system from v to y up to terms O(x,u)2 [Byrnes and Isidori].

If the relative degree = n, then the system is approximately full-state linearizable. Previous work by Krener shows that with the satisfaction of a

controllability condition and with order involutive, there exists an output function h(x) w.r.t which the system has robust relative degree n and such that the remainder functions are O(x,u).

This paper is different in that it constructs an approximate system.

• For the ball and beam, involutivity is satisfied with =3 and the system can be input-output and full-state linearized up to terms of order 3.

vba

u ),(),(

1

18

Analysis of Approximate Linearization Choice of terms included from Lf can be used to improve

the approximation. No choice with terms from Lg.

We cannot guarantee that we can get an approximation to an arbitrary degree.

“In specific applications, [this method] may produce better approximations than the Jacobian approximation. Furthermore, the resulting approximations may be valid on larger domains than the Jacobian approximation.” The following slides make this notion more precise…

19

Analysis of Approximate LinearizationRemark: Since the largest ball that fits in Uε is Bε, the set Uε must get smaller in at least one direction as ε is decreased

UBUU

U n

:sup and whenever : thatprovided Envelopes Operating offamily a is 0, :Def

uUxuxKux

BU n

n

,for )(),(:such that )K( , offunction increasing

monotone a ,0 somefor if,0 ,on Order Higher Uniformly

be tosaid is :function A :Def Remark: If ψ(x,u) is uniformly higher order on Uε x Bε, then it is O(x,u)2

The functions ψi(x,u) that are omitted in the approximation are O(x,u)2 in the neighborhood of the origin. To extend the approx to higher regions, the following definition is used:

20

Analysis of Approximate Linearization If our approximate system is exponentially minimum

phase and the (ignored) ψi terms are uniformly higher order on Uε x Bε, we use the stable tracking control law for the approximate system (with Hurwitz alphas):

)()(

),(),(

1

10)1(

1)(

ddd yyyv

bva

u

)( be error will tracking theand boundedremain willsystem loopclosed theof states thes,derivative and valuessmallly sufficient withies trajectordesiredfor and smallly sufficient for Then,

on order higher uniformly are ),( that suppose andLipschitz q and stablelly exponentia is),0 that suppose

and envelopes operating offamily a be,0 ,Let :Thm

i

O

BUuxq(

U

)( be error will tracking theand boundedremain willsystem loopclosed theof states thes,derivative and valuessmallly sufficient withies trajectordesiredfor and smallly sufficient for Then,

on order higher uniformly are ),( that suppose andLipschitz q and stablelly exponentia is),0 that suppose

and envelopes operating offamily a be,0 ,Let :Thm

i

O

BUuxq(

U

21

Analysis of Approximate Linearization Proof of Theorem:

(Outline)

Remarks: The actual restriction on the class of trajectories

that can be tracked is related to how large the functions ψi are when the approximate state ξi is close to the desired trajectory/derivative y(i)

In certain cases where the ψi functions depend only on the derivative of the output, the main restriction is on the derivatives of the desired trajectory rather than it value

IPAPAP

VPeeeV

qyxuxAee

T

T

d

,0

)(),(

),()),(,(

2