indirect adaptive robust control of siso nonlinear systems ...byao/research/arc/iarc.pdf ·...

Indirect Adaptive Robust Control of SISO Nonlinear Systems in Semi Strict Feedback Forms

Bin Yao

Intelligent and Precision Control LaboratorySchool of Mechanical Engineering

Purdue UniversityWest Lafayette, IN47907

presented at

15th IFAC World Congress

July 23, 2002

OUTLINE

Practical Motivations

Theoretical Problem Formulation

Indirect Adaptive Robust Control (IARC)–– Tracking Error Dynamics based ZTracking Error Dynamics based Z--swapping Identifierswapping Identifier

–– Actual System Dynamics based XActual System Dynamics based X--swapping Identifierswapping Identifier

Experimental Results–– Precision Motion Control of Linear Motor Drive SystemsPrecision Motion Control of Linear Motor Drive Systems

–– Precision Motion Control of ElectroPrecision Motion Control of Electro--Hydraulic SystemsHydraulic Systems

Conclusions

Smart Machines and Systems

Performance Oriented–– Precision manufacturing demands machines with positioning Precision manufacturing demands machines with positioning

accuracy down to accuracy down to subsub--micrometersmicrometers (for quality) at (for quality) at higher higher acceleration/speedacceleration/speed (for productivity) (for productivity)

–– ……

Intelligent and User Friendly –– Being able to deal with Being able to deal with various working conditionsvarious working conditions

–– Machine health monitoringMachine health monitoring and selfand self--fault detections fault detections

–– PrognosticsPrognostics capability for oncapability for on--demand services demand services

–– ……

Performance Oriented Control Issues

Inherent Process Nonlinearities–– Stringent performance requirements demand Stringent performance requirements demand explicit explicit

compensationcompensation of process of process nonlinearitiesnonlinearities

(e.g., HDD pivot bearing friction modeling and compensation)(e.g., HDD pivot bearing friction modeling and compensation)–– ……

Modeling Uncertainties–– Simply attenuating modeling uncertainties via linear highSimply attenuating modeling uncertainties via linear high--

gain robust feedback is not sufficientgain robust feedback is not sufficient

–– Needs toNeeds to reduce model uncertainties reduce model uncertainties through onthrough on--line line adaptation or learningadaptation or learning

Accurate Parameter Estimation

Provide Improved Control Performance–– Can be used to achieve a better model compensation to reduce theCan be used to achieve a better model compensation to reduce the

effect of model uncertainties effect of model uncertainties

–– Enable onEnable on--line tuning of robust feedback gains to maximize the line tuning of robust feedback gains to maximize the attenuation level of highattenuation level of high--gain robust feedbackgain robust feedback

Essential for Adding Intelligent Features–– Features like prognostics, machine health monitoring, fault Features like prognostics, machine health monitoring, fault

detection, etc, can be added when one knows the time history of detection, etc, can be added when one knows the time history of certain critical parameterscertain critical parameters

–– NonNon--Conservative Task PlanningConservative Task Planning

–– ……

System

Theoretical Problem Formulations

1 1 2 1 1 1( , ) ( , )

( , ) ( , )

Tn

Tn n n n n n

x b x x t x t

x b u x t x t

ϕ θ

ϕ θ

= + + ∆

= + + ∆where

1[ , , ] : unknown parameters and gainsT Tb nb bθ θ= …

( , ) : uncertain nonlinearities/disturbancesi nx t∆

Nonlinearities

System


1 1 2 1 1 1( , ) ( , )

( , ) ( , )

Tn

Tn n n n n n

x b x x t x t

x b u x t x t

ϕ θ

ϕ θ

= + + ∆

= + + ∆where



Parametric uncertaintiesSystem


1 1 2 1 1 1( , ) ( , )

( , ) ( , )

Tn

Tn n n n n n

x b x x t x t

x b u x t x t

ϕ θ

ϕ θ

= + + ∆

= + + ∆where



Nonlinearity uncertaintiesSystem


1 1 2 1 1 1( , ) ( , )

( , ) ( , )

Tn

Tn n n n n n

x b x x t x t

x b u x t x t

ϕ θ

ϕ θ

= + + ∆

= + + ∆where



Unmatched Uncertainties

System


1 1 2 1 1 1( , ) ( , )

( , ) ( , )

Tn

Tn n n n n n

x b x x t x t

x b u x t x t

ϕ θ

ϕ θ

= + + ∆

= + + ∆where



System


1 1 2 1 1 1( , ) ( , )

( , ) ( , )

Tn

Tn n n n n n

x b x x t x t

x b u x t x t

ϕ θ

ϕ θ

= + + ∆

= + + ∆

,bb θθ ∈Ω

where

Assumptions

–– A1:A1:

–– A2: A2: ( )| ( , ) | ( ), i n i i ix t x d t iδ∆ ≤ ∀ Semi-strict assumption

A known bounded convex set



Primary Objectives– In general, output x1 tracks any feasible trajectory x1d(t)

with a guaranteed transient performance and final tracking accuracy with all signals in the system being bounded

– Asymptotic output tracking in the presence of parametric uncertainties only, i.e.,

Control Design Objectives

0, ,i i∆ = ∀1 1 1 0 asdz x x t= − → →∞ when

ˆ ( ) asb bt tθ θ→ →∞

Secondary Objective–– Accurate onAccurate on--line estimation of unknown parameters, line estimation of unknown parameters,

i.e., i.e.,

Backstepping Adaptive Designs (KKK’s book’95, …)

–– Consider parametric uncertainty only (i.e., )Consider parametric uncertainty only (i.e., )

Robust Adaptive Backstepping Designs(Polycarpou and Ioannou’93, Pan and Basar’96, Freeman, et al’96,

Marino and Tomei’98, …)

–– Robust stability; Achievable performances in terms of Robust stability; Achievable performances in terms of LL∞∞ norm are not so transparent norm are not so transparent

–– Accurate parameter estimation is not emphasizedAccurate parameter estimation is not emphasized

Direct Adaptive Robust Control (DARC)(Yao and Tomizuka’94, 01, Yao’97, …)

–– Excellent Output Tracking PerformanceExcellent Output Tracking Performance–– Verified through Several Applications Verified through Several Applications

Literature Survey

0,i i∆ = ∀

High Speed Linear Motor Driven Precision Positioning Stage

Tracking Errors for Typical Industrial Motion(Point-to-Point with Velocity of 1m/sec and Acceleration of 12m/sec2)

0 1 2 3 4 5 6 7 8 9 10 11−20

−15

−10

−5

0

5

10

15

20

time (s)

Witho

ut load

0 1 2 3 4 5 6 7 8 9 10 11−20

−15

−10

−5

0

5

10

15

20

time (s)

With l

oadTracking error in µm (DARC)

Parameter Estimations of DARC (Loaded)

0 2 4 6 8 102

4

6

8

10

12

14

16

18

20

time (s)

θ 1

0 2 4 6 8 10

1

2

3

4

5

6

7

8

9

10

time (s)

θ 2

0 2 4 6 8 10−10

−8

−6

−4

−2

0

2

4

6

8

10

time (s)

θ 3

0 2 4 6 8 10

10

15

20

25

30

35

40

45

50

time (s)

b

Parameter estimates with load (DARC)

Practical Issues of DARC Design

Individual Parameter Estimates Seldom Converge– The design of control law and parameter estimation law are

synthesized jointly through an “energy” function with reducing output tracking error being the sole objective

– Gradient type estimation algorithm only with certain actual tracking error as driving signal

– Small actual tracking error in implementation prone to be corrupted by neglected factors such as sampling delay and noises

– Explicit monitoring of signal excitation level not possible as otherwise one loses the integral type fast dynamic compensation capability of parameter adaptation process

Indirect Adaptive Robust Control

Total Separation of Parameter Estimation from Robust Control Law Design

– Theoretical boundedness of parameter estimates and their derivatives achieved through the use of a rate-limited projection type adaptation law structure with preset adaptation rate limits, as opposed to the traditional way of using complicated mathematical derivations such as the normalization and/or nonlinear damping in the modular backstepping adaptive designs that may not work for systems with disturbances and uncertain nonlinearities

Use Actual System Dynamics to Construct Reliable Parameter Estimation Model

Projection Type Adaptation Law with Rate Limits

Adaptation Law Structure

( )( )ˆˆ ˆPr ( ) , (0)

bM bb bsat oj t θθ θθ τ θ= Γ ∈ Ω

where

( ) 0 0

1,

,,M

M

MM

sat s sθ

ζ θζ ζ θ ζ θ

ζ

≤= = >

A preset rate limit,A designer choice !

( )ˆb

ˆ ˆ ˆ

ˆbˆ ˆ

ˆif or 0

Prˆ and 0

b b

b b b

b b

b b

T

T

TT

n

oj n nI n

n n

q q

q q qq q

q q

x q x

xx q x

Ï Œ∞W £ÔÔÊ ˆ= ÌÁ ˜- G Œ∂W >ÔÁ ˜GÔË ¯Ó

Benefits of Rate Limited Adaptation Law Structure

P1: Parameter Estimates always within Bounded Set

P2: Parameter Adaptation Rate always within Preset Rate

P3: Ideal Performance of Parameter Adaptation Preserved

ˆ ( ) ,bb t tθθ ∈ Ω ∀

Note:

( )( )-1ˆPr 0,b

Tb ojθθ τ τ τΓ Γ − ≤ ∀

ˆ ( ) ,b Mt tθ θ≤ ∀

»» P1 and P2 enable the use of the same ARC design technique P1 and P2 enable the use of the same ARC design technique as in direct approach to construct an ARC control law that as in direct approach to construct an ARC control law that achieves a guaranteed transient and final tracking accuracy, achieves a guaranteed transient and final tracking accuracy, independent of specific estimation function independent of specific estimation function ττ to be usedto be used

Adaptive Robust Control LawControl Functions

Robust Performance Conditions

( )1

1 11 1 1

1

1 2 1 2 2

ˆˆ, , ,

1 ˆ ˆˆ ˆ

1 , ,

ˆ

i i i ia i s

iTi i

ia l l i i il li

i s i s i s i s i i i s i s i

i

x b t

b x b zx tb

k z k zb

α θ α α

α αα φ θ

α α α α α

−− −

+ − −=

= +

∂ ∂= + − − ∂ ∂

= + = − = −

∑

11

2 1 1 11

121 1

1

ˆ

ˆ ˆ ˆ ˆ

iTi

i i i s l l i i i i ial l

ii i

l i c i d i il l

z b b x b z bx

b db

αα φ θ α

α α θ ε εθ

−−

+ − −=

−− −

=

∂+ − − − ∂∂ ∂− − + ∆ ≤ +

∂∂

∑

∑where

( )1

1 11 1

1 1

ˆˆ, , , ,

i i i

i ii i

i i i i l i i ll ll l

z x

x b tx x

αα αφ θ ϕ ϕ

−

− −− −

= =

= −∂ ∂= − ∆ = ∆ − ∆

∂ ∂∑ ∑

Theoretical Performance of ARC LawTheorem 1

Practical Significance

–– Guaranteed transient and final tracking accuracy, important for Guaranteed transient and final tracking accuracy, important for applicationsapplications

With projection type adaptation law with rate limits, in which could be any identifier function, the ARC law u= guarantees that, in general, all signals are bounded with tracking errors bounded a

n

τα

bove by 2 2 ( )

0( ) (0) 2 ( )V V

tt tn n Vz t e z e dλ λ τ ε τ τ− − −≤ + ∫

21 1

where 2min , and n

V n V ci di iik k dλ ε ε ε

== = +∑…

2( ) , an index for tracking accuracynz t

2(0)nz

2( )nz t 2

( )nz ∞

2 max

The exponential convergening rate and the bounds of final tracking errors, 2

( ) , are related to controller design parameters in a known form

V

Vn

V

z

λελ

∞ ≤

Actual System Dynamics Based Identifier

Original System Model in Linear Parametrization Form

Construct Filters to Generate Implementable Prediction Error Model

Linearly Parametrized Static Prediction Output Model,

0 ( , ) ( , ) , assuming 0,Tn n n b ix f x u F x u iθ= + ∆ = ∀

1 2 0

( , )

0

T

Tn

Tn

x

F x u

u

ϕ

ϕ

=

where

0 0 0

( , ), any stable matrix

( ) ( , )

T T Tn

n n

A F x uA

A x f x u

Ω = Ω +Ω = Ω + −

0

0ˆ , where ˆˆ

n T Tb n bT

b

y xy y x

yε θ ε ε θ

θ= + Ω

⇒ = − = Ω − = + Ω − Ω= Ω

, 0Aε ε ε= →

Estimation Functions

Gradient Type

Least Square Type

2

1, 0

1F

τ ε νν

= − Ω ≥+ Ω

1

1 Ttrτ ε

ν= − Ω

+ Ω Γ Ω

( ) ˆmax, if ( ) and Pr ( )1

0 otherwise

b

T

M MTt oj

tr θα λ ρ τ θν

ΓΩΩ ΓΓ − Γ ≤ Γ ≤ + Ω ΓΩΓ =

Theoretical Performance of IARC Law

Lemma

Proof

–– The proof for the above results with projection type The proof for the above results with projection type adaptation law is well established in the literature.adaptation law is well established in the literature. The key The key difficulty here is to show that, even with any predifficulty here is to show that, even with any pre--set set adaptation rate limit, the above results still hold for the adaptation rate limit, the above results still hold for the proposed rated limited projection type adaptation law.proposed rated limited projection type adaptation law. The The technical details are rather involved and given in the papertechnical details are rather involved and given in the paper

When the rate limited projection type adaptation law with either the gradient estimator or the least square estimator is used, the following results hold:

2

2

[0 , ) [0 , )

ˆ [0 , ) [0 , )b

L L

L L

ε

θ∞

∞

∈ ∞ ∞

∈ ∞ ∞

∩

∩

Theoretical Performance of IARC Law

Theorem 2

Sketch of Proof

–– The proof follows a similar procedure as in the modular adaptiveThe proof follows a similar procedure as in the modular adaptivebacksteppingbackstepping designs, which uses some Nonlinear Swapping Lemmas designs, which uses some Nonlinear Swapping Lemmas and the results of the previous lemma to show that and the results of the previous lemma to show that . . Thus, as , the theorem can be proven by uThus, as , the theorem can be proven by using sing Barbalat’sBarbalat’slemma. The details are rather technical and given in the paper. lemma. The details are rather technical and given in the paper.

When the rated-limited projection type adaptation law with either gradient or least-square type identifier function, in the presence of parametric uncertainties only, in addition to the robust performance results in Theorem 1, an improved tracking performance, asymptotic output tracking (i.e., 0), is also acheived. nz →

2[0, )nz L∈ ∞[0, )nz L∞∈ ∞

Note: »» The above results indicate that the ideal performance results ofThe above results indicate that the ideal performance results of

adaptive designs are preserved. adaptive designs are preserved.

Precision Control of Linear Motors

Point-to-Point Motion Trajectory

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

Time (sec)

Pos

itsio

n (m

)

0 1 2 3 4 5 6 7 8 9 10−1

−0.5

0

0.5

1

Time (sec)

Vel

ocity

(m/s

)

0 1 2 3 4 5 6 7 8 9 10

−10

−5

0

5

10

Time (sec)

Acc

eler

atio

n (m

/s2 )


0 1 2 3 4 5 6 7 8 9 10 11−20

−15

−10

−5

0

5

10

15

20

25

30

time (s)

DARC

Tracking error without load in µm

0 1 2 3 4 5 6 7 8 9 10 11−20

−15

−10

−5

0

5

10

15

20

25

30

time (s)

IARC


0 1 2 3 4 5 6 7 8 9 10 11−25

−20

−15

−10

−5

0

5

10

15

20

time (s)

DARC

0 1 2 3 4 5 6 7 8 9 10 11−25

−20

−15

−10

−5

0

5

10

15

20

time (s)

IARC

Tracking error with load in µm

0 2 4 6 8 10

10

15

20

25

30

35

40

45

50Parameter estimates without load (IARC)

time (s)

b

Red: Actual value

Blue: Z−Grad

Black: Z−Ls

Green: X−Grad

Magenta: X−Ls

Parameter Estimations of IARC (No Load)

0 2 4 6 8 10

10

15

20

25

30

35

40

45

50Parameter estimates with load (IARC)

time (s)

b

Red: Actual value

Blue: Z−Grad

Black: Z−Ls

Green: X−Grad

Magenta: X−Ls

Parameter Estimations of IARC (Loaded)

Desirable Features of IARC Design

Accurate Parameter Estimates – The design of control law and parameter estimation law are

totally separated – no more one stone two birds problem.

– The parameter estimation law allows for estimation algorithms with faster parameter convergence properties.

– Driving signal is based on actual system dynamics, not tracking error dynamics, which is less sensitive to neglected factors.

– Explicit monitoring of signal excitation level can be used to improve parameter convergences.

Conclusions

A Theoretical Framework for IARC is established – Adopts a more practical and cleaner interface in separating

the design of control law and parameter estimation law than traditional indirect adaptive control designs – the use of rate-limited projection type adaptation law structure with the adaptation rate-limit being pre-set by practicing engineers for a more controlled adaptation process

– Theoretically guarantees transient and final tracking accuracy for both parameter variations and disturbances, while preserving the ideal asymptotic output tracking performance of adaptive designs.

Experimental Results verify both the excellent output tracking performance as well as the improved parameter estimations of IARC

AcknowledgementsAcknowledgements

The project is supported in part byThe project is supported in part by

National Science FoundationNational Science Foundation---- CAREER grant CMS 9734345CAREER grant CMS 9734345

Purdue Research Foundation

Mathematical Model

1

22

21

xy

FFFBxuxM

xx

drfn

=

+−−−==

where: position M : mass of load: velocity u : input voltage

y : output B : viscous friction const

: nonlinear friction : force ripple

: lumped disturbance

1x

2x

dFfnF rF

Model of Friction Force

is discontinuous at zero velocity

A continuous friction model is used to approximate

fnF

fnF fnF

2 2( ) ( )f n f fF x A S x=

Ffn

V(m/s)Ffn

Af

ARC Controller Design Model

where

where

1 2

2

1

T

x x

x bu

y x

ϕ θ=

= + + ∆=

2 2 2( )f fM x u B x A S x d⇒ = − − +

( )fn fn rd F F F= − − + ∆

⇒

[ ]1 11 , ,f nnM M

A dBb d d dM M M Mθ = = ∆ = = −

In semi strict feedback from

0 2 4 6 8 102

4

6

8

10

12

14

16

18

20

time (s)

θ 1Parameter estimates without load (IARC)

Red: Actual value

Blue: Z−Grad

Black: Z−Ls

Green: X−Grad

Magenta: X−Ls


0 2 4 6 8 10

1

2

3

4

5

6

7

8

9

10

time (s)

θ 2Parameter estimates without load (IARC)

Red: Actual value

Blue: Z−Grad

Black: Z−Ls

Green: X−Grad

Magenta: X−Ls


0 2 4 6 8 10−10

−8

−6

−4

−2

0

2

4

6

8

10Parameter estimates without load (IARC)

time (s)

θ 3

Red: Actual value

Blue: Z−Grad

Black: Z−Ls

Green: X−Grad

Magenta: X−Ls


0 2 4 6 8 102

4

6

8

10

12

14

16

18


time (s)

θ 1

Red: Actual value

Blue: Z−Grad

Black: Z−Ls

Green: X−Grad

Magenta: X−Ls


0 2 4 6 8 10

1

2

3

4

5

6

7

8

9


time (s)

θ 2

Red: Actual value

Blue: Z−Grad

Black: Z−Ls

Green: X−Grad

Magenta: X−Ls


0 2 4 6 8 10−10

−8

−6

−4

−2

0

2

4

6

8


time (s)

θ 3Red: Actual value

Blue: Z−Grad

Black: Z−Ls

Green: X−Grad

Magenta: X−Ls


Parameter Estimations of DARC (Unloaded)

0 2 4 6 8 102

4

6

8

10

12

14

16

18

20

time (s)

θ 1

0 2 4 6 8 10

1

2

3

4

5

6

7

8

9

10

time (s)

θ 2

0 2 4 6 8 10−10

−8

−6

−4

−2

0

2

4

6

8

10

time (s)

θ 3

0 2 4 6 8 10

10

15

20

25

30

35

40

45

50

time (s)

b

Parameter estimates without load (DARC)

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