12-1

High Gain Observer

Introduction

Cxy

uygAxx

),(

where (A,C) is observable.

Consider the following system

This form is special because g depends only on y and u.

Taking the observer as

)ˆ(),(ˆˆ xCyHuygxAx

we obtain that satisfiesxxx ˆ~

xHCAx ~)(~

Therefore, designing C such that AHC is Hurwitz guarantees asymptoticerror convergence.

12-2

High Gain Observer (Continued)

)ˆ(),(ˆˆ 0 xCyHuygxAx

However, any error in modeling g will be reflected in the estimation errorequation. Thus,

where is a nominal model of g. Hence0g

We give a special design of the observer gain that makes the observer robust

to uncertainties in modeling the nonlinear functions. The technique, called as

high-gain observers, works for a wide class of nonlinear systems and guarantees

that the output feedback controller recovers the performance of the state feedback

controller when the observer gain is sufficiently high.

),(),(~)(~0 uyguygxHCAx

12-3

High Gain Observer (Continued)

The main result is a separation principle that allows us to separate the design into

two tasks. First, design a state feedback controller that stabilizes the system and

meets other design specifications. Then, obtain an output feedback controller by

replacing x by provided by the high-gain observer. A key property that makes

this separation possible is the design of the state feedback controller to be globally

bounded in x.

x̂

12-4

Example

Ex:

1

2

21

),(

xy

uxx

xx

Assume that u = (x) is a local state feedback control law that stabilizes the origin.

To implement this control law using only y, we use the observer

)ˆ(),ˆ(ˆ

)ˆ(ˆˆ

1202

1121

xyhuxx

xyhxx

where is a nominal model of the nonlinear function),(0 ux ).,( uxThen

)~,(~~

~~~

122

2111

xxxhx

xxhx

where

))ˆ(,ˆ())ˆ(,()~,(

ˆ~ ,ˆ~222111

xxxxxx

xxxxxx

12-5

Example (Continued)

We want to design such that ThhH 21 .0)(~lim

txt

In the absence of , asymptotic error convergence is achieved by designingH such that

0

1

2

10 h

hA

is Hurwitz. In the presence of , we need to design H with the goal of rejectingthe effect of on .~x

This is ideally achieved, for any , if the transfer function

121

20

11)(

hshshssG

from to is ideally zero. x~

12-6

Example (Continued)

)(sup 0 jwGRw

While this is impossible, we can make arbitrarily small by choosing .112 hh

Taking ,10 and 0,for , 2122

21

1

hh it can be shown that

121

20 )()(

ssssG

Hence .0)(lim 00

sG

Define the scaled estimation errors

221

1~ ,

~x

x

Then the newly defined variables satisfy the singularly perturbed equation.

)~,(122

2111

xx

This equation shows clearly that reducing diminishes the effect of .

12-7

Example (Continued)

Notice, however, that will be whenever )0(1 )/1( O ).0(ˆ)0( 11 xx

Consequently, the solution contains a term of .0 with )/1(

aeat

In fact, .0 asfunction impulsean approaches )/(

at

ea

This behavior is known as the peaking phenomenon.

12-8

Globally Stabilized by State Feedback Controller

1

322

21

xy

uxx

xx

Let’s consider the system

which can be globally stabilized by the state feedback controller

2132 xxxu

The output controller is taken as

)ˆ)(1(ˆ

)ˆ)(2(ˆˆ

ˆˆˆ

122

121

2132

xyx

xyxx

xxxu

where the observer gain assigns the eigen values of .1 and 1at 0 A

12-9

State Feedback Controller

The above figure shows a counter intuitive behavior as 0. Since decreasing

causes the estimation error to decay faster toward zero, one would expect the

response under output feedback to approach the response under state feedback as

decreases. This is the impact of peaking phenomenon. Fortunately, we can

overcome the peaking phenomenon by saturating the control outside a compact

region of interest in order to create a buffer that protects the plant from peaking.

12-10

State Feedback Controller

Suppose the control is saturated as)ˆˆˆ(sat 21

32 xxxu

The above figure shows the performance of the system under saturated state

and output feedback. The control u is shown on a shorter time interval that

exhibits control saturation during peaking. The peaking period decreases with .

12-11

Stabilization

),(

),,(

),,(

zxq

Cxy

uzxz

uzxBAxx

Consider the MIMO nonlinear system

(1)

bygiven are ,, The s. variablestate the

esconsititut and and outputs, measured are

and input, control theis where

CBA

RzRx

RRyRul

SmP

12-12

Stabilization (Continued)

,diagblock 1 mAAA

ii

iA

00

10

0100

0010

,diagblock 1 mm BBB

11

0

0

i

iB

,diagblock 1 mm CCC iiC 1001

. and ,,1 where 21 mmi

12-13

Output Feedback Controller

The functions , and q are locally Lipschitz and (0,0,0)=0, (0,0,0)=0,

q(0,0)=0. Our goal is to find an output feedback controller to stabilize the

origin.

We use a two-step approach to design the output feedback controller.

(i) A partial state feedback controller using x and is designed

to asymptotically stabilize the origin.

(ii) A high-gain observer is used to estimate x from y.

The state feedback controller can be shown as

),,(

),,(

xru

x

where r, are locally Lipschitz in their arguments over the domain of interestand globally bounded functions of x. Moreover, r(0,0,0)=0 and (0,0,0)=0.

12-14

Output Feedback Controller (Continued)

For convenience, we write the closed-loop system under state feedback as

)(XfX (2)

where X = (x, z, ).

The output controller is taken as

),ˆ,(

),ˆ,(

xru

x

(3)

observergain high by the generated is ˆ where x

)ˆ(),,ˆ(ˆˆ 0 xCyHuxBxAx

12-15

Output Feedback Controller (Continued)

The observer gain H is chosen as

,diagblock 1 mHHH

i

ii

i

i

iH

2

2

1

.,,for OLHP in the are

0

of roots thesuch thatchosen

are and specified be toconstant positive a is where

11

1

mii

sss iii

ij

ii

ii

0. )0,0,0( and in boundedglobally andinterest of

domain over the arguments itsin Lipschitzlocally iswhich

),,( of model nominal a is ),,(function The

0

0

x

uxux

12-16

Theorem

Theorem: Consider the closed-loop system of the plant (1) and the output

feedback controller (3). Suppose the origin of (2) is asymptotically

stable and RR is its region of attraction. Let S be any compact set in the

interior of RR and Q be any compact subset of Then,.R

.0 allfor bounded are ,in starting system, loop-closed theof

))(ˆ),(( solutions the,0every for ,such that 0 exists there *1

*1

tQS

txtX

2

*2

2*2

,)(ˆ and )(

satisfy ,in

starting system, loop-closed theof solutions the,0every for that,

such ,on dependent both ,0 and 0exist there,0any given

TttxtX

QS

T

12-17

Theorem (Continued)

.attraction ofregion its ofsubset a is and stablelly exponentia

is system loop-closed theoforigin the,0every for ,such that

0 exists re then thestable,lly exponentia is oforigin theif*4

*4

QS

).0(at starting ofsolution theis )( where

0 ,)()(

satisfy ,in

starting system, loop-closed theof solutions the,0every for

,such that ,on dependent ,0exist there,0any given *3

*3

XtX

ttXtX

QS

r

r

(2)

Proof: See Ch 14.5

(2),

12-18

Results

The theorem shows that the output feedback controller recovers the

performance of the state feedback controller for sufficiently small .Note that

(i) recovery of exponential stability

(ii) recovery of the region of attraction in the sense

that we can recover any compact set in its interior

(iii) the solution under output feedback approaches the

solution under the state feedback as 0.

12-19

Example

Ex: Consider the following plant:

1

213

232

121

sin

sin

xy

uxx

uxxx

txxx

Note : the given system is in triangular form. Thus, it is stabilizableby

33221121 xkxkxkxu

The output feedback controller is

)ˆˆˆˆ(sat 332211213 xkxkxkxu

12-20

Example (Continued)

with

)ˆ(ˆˆ

)ˆ(ˆˆ

)ˆ(ˆˆ

1332

13

122

32

11

21

xyh

uxx

xyh

xx

xyh

xx

For simulation:

All system eigenvalues at 5.

All observer eigenvalues at 5/ with = 0.05.

)15 ,75 ,125( 321 kkk

)125 ,75 ,15( 321 hhh

12-21

Simulation Results

ix

ix̂