DOB Presentation by Shim

7/29/2019 DOB Presentation by Shim

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ANALYSIS AND SYNTHESIS OF DISTURBANCE OBSERVERAS AN ADD-ON ROBUST CONTROLLER

Hyungbo Shim

(School of Electrical Engineering, Seoul National University, Korea)

in collaboration with Juhoon Back, Nam Hoon Jo, and Youngjun Joo

under the support from Hyundai Heavy Industries, Co., LTD.

Oct. 15, 2010, at UCSB


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Overview2

disturbance observer (DOB)

dates back to the Japanese article (K. Ohnishi, 1987)

has been of much interest in control application community, butdid not draw much attention in control theory community

41 pub. record in IEEE Trans. Indus. Elec. (2003-2007)

8 in IEEE Trans. Contr. Sys. Tech. (2003-2007)

3 in IEEE Trans. Automat. Contr. (2003-2007) original idea is intuitively simple, but less analyzed rigorously

This talk is about

developing theoretical framework for DOB approach, and

appreciation of it as a robust control tool presentation of robust stability condition and robust nominal

performance recovery

discussion about various aspects of DOB, and extension to

nonlinear systems


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Overviewproblem statement

Inner-loop dynamic controller:

The same input-output behavior!

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Overviewproblem statement

not only just the same steady-state behavior,but also the same transient behavior(with the same IC between the nominal and the real)

important feature required by industrywhere settling time, overshoot, etc., should be uniformamong many product units

a sharp contrast to other robust or adaptive redesign approach

which possibly changes the transient behavior

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The same input-output behavior!


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Linear DOB: Intuition5


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Linear DOB: Intuition6


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Linear DOB: Intuitive justification7


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More Issues to be Solved8

The argument so far does not present real power and limitations.

robust stability is still of question

what about transient performance recovery?

unanswered observations from practice:

It does not work for non-minimum phase plant.

High bandwidth of Q(s) destabilizes the closed-loop.

It cannot handle large parameter variation. Higher order Q(s) leads to instability.

can we extend the DOB-based controller for uncertain

nonlinear plants?


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Our Starting Point9


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Normal form realization of Q(s)10

Byrnes-Isidori

Normal Form

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Representation of P (nonlinear plant)11

Byrnes-Isidori

Normal Form

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Representation of nominal plant Pn (with Q)12

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Summary: Problem Statement13

State-space realization of DOB structure

?

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* (too brief) Review of Tikhonovs Theorem14

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Robust stability of DOB structure with small

= Robust stability of Boundary-Layer subsystem

+ Robust stability of Quasi-Steady-State subsystem

Closed-loop system in the Standard Singular Perturbation Form15

With coordinate change given by

the overall closed-loop system is expressed by

outer-loopcontroller C

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Quasi-Steady-State Subsystem16

It is the same as the disturbance-free nominal closed-loop! z-dynamics becomes unobservable.

z-dynamics should be ISS (minimum phase in linear case).

implies (so Q-filter with the plant inverse acts

as a state observer!)

QSS subsystem:

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Boundary-Layer Subsystem17

Robust stability condition:

Q-filter should be stable.

No matter how large the uncertainty is, once bounded,there are s so that two polynomials are Hurwitz! proof

Semi-global result for boundedness of

S 18


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Summary:

Robust Stability

Robust stability of the closed-loop with DOB isguaranteed if, on a compact set of the state-space,

1. zero dynamics of uncertain plant is ISS,2. outer-loop controller stabilizes the nominal plant,

3. coefficients of Q-filter are chosen for the RS condition,

4. is sufficiently small.

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Robust Steady-State Performance19

Nominal closed-loop

(slow) transient steady-state

Real closed-loop

(fast) transient (slow) transient steady-state

nominal plant

outer-loop controller C

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Robust (slow) Transient Performance20

I. Tikhonovs theorem guarantees that

if

II. However, (0) and (0) become unbounded as ! 0.

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Peaking phenomenon reflected in coordinate change21

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Robust (slow) Transient Performance

I. Tikhonovs theorem guarantees that

if

II. However, (0) and (0) become unbounded as ! 0.

Lesson: Due to peaking phenomenon, it is not true that

conventional DOB recovers (slow) transient performance.

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Saturating the Peaking Components

Peaking components that affect slow dynamics:

Replace with

and

[Esfandiari & Khalil, 1992]

S bili L i BL S b24


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Stability Loss in BL Subsystemand its recovery by a dead-zone function

BL -dynamics becomes

and loses GES.

Add a dead-zone function:

to recover GES.

S bili A l i f BL S b25


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Stability Analysis of BL Subsystemwith saturation & deadzone

A l f T kh Th26


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Application of Tikhonovs Theoremfor the second interval

Summary 27


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Summary:

Robust Transient Performance Recovery

thanks to saturation & dead-zone functions.

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Proposed Nonlinear DOB Structure [Back/S, AUT, 08]

M S29


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Example: Point Mass Satellite [Back/S, TAC, 09]

l M S ll30


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Example: Point Mass Satellite

Nominal: Real: with DOB (solid=nominal)

Real: w/o DOB:

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Robust Stability Condition for Linear DOB

Shim & Jo, AUT, 2009

back to linear

presents simple linear robust stability conditionworking on frequency domain

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Problem Formulation

Standing Assumption:where is a collection of

Ass: C(s) is designed

for Pn(s).

Problem: Design of

Q(s) for robust stability

Ch h Q( )33


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Characteristic equation with Q(s)

Q(s) =( s)l + bl1( s)

l1 + + b1( s) + a0

( s)l+r + al+r1( s)l+r

1 +

+ a1( s) + a0=:

NQ(s; )

DQ(s; ); > 0

R b S b l f h O ll S34


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Robust Stability of the Overall System

Observation 1:

Observation 2:

[S/Jo, AUT, 09]

D i f Q( ) h35

a0


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Design of Q(s): when

How to choose Q(s):

Q(s) =a0

( s)r + ar1( s)r1 + + a0

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Discussion: Answers to the raised questions

1. It does not work for non-minimum phase plant. Correct, for small .

2. Small destabilizes the closed-loop.

No, if Q(s) satisfies RS condition. It may be the effect of initialpeakings but not a destabilization.

3. It cannot handle large parameter variation.

No in general. By following the design guidelines, any bounded

variation can be handled.

4. Higher order Q(s) leads to instability.

No in general. But, for higher order Q(s), the selection of

coefficients becomes more complicated.

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robust stability condition in action

Experiments

Taken from Yi, Chang, & Shen, IEEE/ASME Trans. Mechatronics, 2009

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St39


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Step responses

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Robust tracking control

Hyundai HS-165 (handling 165kg)

[Bang/S/Park/Seo, IEEE Trans. Industrial Elect., 2010]

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Robust tracking control

Experiment results

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Summary

theoretical study on DOB

robust stability condition

(new even for classical linear DOB) transient performance recovery

(thanks to saturation/dead-zone functions)

DOB = Inner-loop controller = Your first AID kit!


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THANK YOU

Any feedback or comments are welcome

Documents

DOB Presentation by Shim