Robust Adaptive Control with Improved Transient …tgibson/RobustAdaptiveControl_MIT... · •...

BOEING is a trademark of Boeing Management Company.Copyright © 2012 Boeing. All rights reserved.

Robust Adaptive Control with Improved Transient Performance

Eugene Lavretsky

MITMay 03, 2012

2

Engineering, Operations & Technology | Boeing Research & Technology

Presentation Overview

• Introduction

• Adaptive Control Development @ Boeing

• Transient Dynamics in Adaptive Control• Motivating Example

• Transient Analysis with All States Accessible

• Adaptive Output Feedback Design Extension

• Conclusions, Comments, and Future Research Directions

3


Introduction

• Classical Model Reference Adaptive Control (MRAC)• Originally proposed in 1958 by Whitaker et al., at MIT• Main idea: Specify desired command-to-output performance of a servo-

tracking system using reference model– Defines ideal response of the system due to external commands– Later called “explicit model following” MRAC

• First proof of closed-loop stability using Lyapunov theory was given in 1965 – 66 by Butchart, Shackcloth, and Parks

Process

Reference Model

Controller

Adaptive Law

External Command

System Response

ControlCommand

Ref. ModelOutput

SystemResponse

Process

Reference Model

Controller

Adaptive Law

External Command

System Response

ControlCommand

Ref. ModelOutput

SystemResponse

4


Robust and Adaptive Flight Control Technology Transitions: Advanced Aircraft and Weapon Systems

• Technology Maturation & Transitions– Extended to Munitions (00-02)– Boeing IRAD Improvements Focus on System ID, Implementation, and Actuator

Saturation Issues– Design Retrofits onto Existing Flight Control Laws– Flight Proven on X-36, MK-84, MK-82, MK-82L, MK-84 IDP 2000, Boeing Phantom

Ray, NASA AirStar– Transitioned to JDAM production programs

93 94 95 96 97 98 99 00 01 02 03

Intelligent Flight Control System (NASA/Boeing)

F-15 ACTIVE

04

MK-82 L-JDAM

Reconfigurable Control For Tailless

Fighters (AFRL-VA/Boeing)

X-36 MK-84 JDAM

Adaptive Control For Munitions

(AFRL-MN/GST//Boeing)MK-84

05

• Gen I, flown 1999, 2003• Gen II, 2002 – 2006

•flight test 4th Q 2005• Gen III, 2006

RobustAdaptiveControl Technology Transition Timeline

MK-82 JDAM

X-45C

X-45A

J-UCAS & Phantom Ray

06 07 12

Boeing IRAD/CRAD

Theoretically justified, numerically efficient, and flight proven technology

Theoretically justified, numerically efficient, and flight proven technology

MK-84 IDP 2000

X-36 RESTORE

MK-82 Laser Seeker

08

Phantom Ray

5


Motivating Example

• Adaptive Servomechanism for Scalar dynamics• Global asymptotic closed-loop stability• Bounded tracking in the presence of constant unknown parameters

Process :Ref. Model :

ˆ ˆController :ˆ

Adaptive Law :ˆ

Benefits : lim lim 0

ref ref ref ref

x r

x x ref

r r ref

ref reft t

x a x bux a x b r t

u k x k r t

k x x x

k r x x

e t x t x t x x r

External Command

Lyapunov-based

6


Motivating Example (continued)

• Tuning MRAC• Increase adaptation gains to get desired (fast) tracking

performance

• Design Tradeoff• Large adaptation gains lead to oscillations (undesirable transients)

• Cause and effect• Reference and transient (error) dynamics have the same time constant

• Need transient dynamics to be faster than reference model• Similar to state observer design

– separation between controller and observer poles reduces transients

,x r

Bounded Signal

Reference Dynamics :

Transient Dynamics :

ref ref ref ref

ref x r

x a x b r

e a e b k x k r

1e refa

7



• Reference Model in MRAC• Similar to Open-Loop Observer

• Add Observer-like Error feedback Term to Reference Model• Similar to Closed-loop Observer

• Properties• Error feedback regulates transients• Converges to “ideal” reference model• No changes to control input• Retains stability and tracking

• Main Benefit• Control of transients

Error Feedback Term

ref ref ref ref e refx a x b r k x x

ref ref ref refx a x b r

ref e x re a k e b k x k r

Error Feedback Gain

Process

Reference Model

Controller

Adaptive Law

External Command

System Response

ControlCommand

Ref. ModelOutput

SystemResponse

ek

Process

Reference Model

Controller

Adaptive Law

External Command

System Response

ControlCommand

Ref. ModelOutput

SystemResponse

ek

8



• Simulation Data• Tracking step-inputs

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-3

-2

-1

0

1

2

3

Time, sec

e =

x - x

ref

ke = 0

ke = 10

ke = 80

Process : 3

Ref. Model : 10 10

ˆ ˆController :ˆ 10

Adaptive Law :ˆ 10

ref ref e ref

x r

x ref

r ref

x x u

x x r k x x

u k x k r

k x x x

k r x x

Trac

king

Err

or

ke = 0 (MRAC)

ke = 10

ke = 80

Increasing Observer Feedback Gain Reduces Transient

Oscillations

Need: Formal Analysis of Transient Dynamics

9


0 1 2 3 4 5 6 7 8 9 10-30

-20

-10

0

10

20

30

x

Commandke = 0

ke = 80

0 1 2 3 4 5 6 7 8 9 10-300

-200

-100

0

100

200

300

400

500

Time, sec

u

ke = 0

ke = 80


• Simulation Data• Tracking performance and control input

ke = 80

Syst

em S

tate

Con

trol

Inpu

t

ke = 0 (MRAC)

10


Transient Analysis for Scalar Dynamics

• Observer-like ref model Error Dynamics System State Transient Dynamics

0ref

ke a e t

max

ref e x r

t

e a k e b k x k r

0

System State Asymptotic TrackingReference StateTransient Dynamics

O o 1 Otk

refx t x t e

o(1) and uniformly bounded

0 max0

0

tke t e e

k

From Lyapunov analysis

0O o 1 Otk

e t e

Error Feedback Term

ref ref ref ref e refx a x b r k x x

Error Dynamics

Reference Model0

ek

k

11


Transient Analysis for Scalar DynamicsAn Alternative Approach

• Error Feedback Gain in Observer-like Ref Model

• Transient Dynamics Singular Perturbation Model

• Fast (Boundary Layer) Dynamics = Transients

0e

kk

Positive constant

Small Parameter

0

o 1 ,as , fixed 0

ref x r

t

e a k e b k x k r

From Lyapunov Analysis

ref e x re a k e b k x k r

Slow Dynamics : 0 refe x x

o 1ref refx a x b r

t

0

d ek e

d

0

System State Asymptotic TrackingTransient Dynamics

+o 1 Otk

refx t x t e

Stretched Time Boundary Layer Dynamics

Assume to be uniformly continuous and bounded

12


MIMO Generalization: State Feedback

• What• Adaptive state feedback

servomechanism design for MIMO dynamical systems in the presence of matched uncertainties

• Why• Improves and streamlines

adaptive design tuning

• How• Model Reference Adaptive Control• Observer-like reference model Reduced and Quantifiable Transients

K

13


System Dynamics and Control

• Open-Loop Dynamics

• Control Objective• Design control input u such that the regulated output z tracks bounded time-varying

command zcmd with bounded errors, reduced transients, and while operating in the presence of matched uncertainties

Matched Uncertainty

Hurwitz

0 00

0

p

ref ref

z

m mz I m m p z I m m Tcmd

n mp p I p p p p p

x xA B B

m m p

C

Ie C eu x z

x B K A B K x B

z C x

Command

KnownRegressor

Unknown Parameters

Uncertain Control Effectiveness

Regulated Output

Plant State

Integrated Tracking

Error

Tref ref cmd

z

x A x B u x B z

z C x

Hurwitz

Matched Uncertainties

Regulated Output

Command

14


• System Dynamics

• Reference Model• Plant w/o uncertainties

• Tracking Error

• Adaptive Control Input

• Tracking Error Dynamics

• Algebraic Lyapunov Equation Adaptive Laws Closed-Loop Stability

• Adaptive Control Tuning Cycle

MRAC @ a Glance: How is It Currently Done ?

Tref ref cmdx A x B u x B z

ref ref ref ref cmdx A x B z

refe x x

ˆ Tu x

Trefe A e B x

0Tref refP A A P Q ˆ Tx e P B lim 0

te t

Rates of adaptation Q P B

e

15


Few Thoughts …


• Reference Model ~ Luenberger Open-Loop Observer

• Tracking Error Dynamics = Transient Error Dynamics

• Need to be “faster” than system dynamics minimizes unwanted transients

• Main Idea: Use Closed-Loop Luenberger Observer as Reference Model



Trefe A e B x

Innovation Term

ref ref ref ref cmd refx A x B z L x x

Observer Gain

16


• Open-Loop Plant

• Observer-like Reference Model

• Tracking Error

• Adaptive Control Input

• Error Dynamics

• Observer Riccati Equation Adaptive Laws Closed-Loop Stability• With prescribed degree of stability

MRAC with Observer–like Reference Model


ref ref ref ref cmd v refx A x B z L x x

refe x x

ˆ Tu x

Hurwitzv

Tref v

A

e A L e B x

1 0T

v ref n n ref n n v v v v vP A I A I P P R P Q

1ˆ Tvx e P B

lim 0t

e t

e

1, traceT TvV e e P e

Lyapunov function

1Tv v v v v v v vP A A P P R P Q

1 1 1 1 1 0Tv v v v v v v vP A A P R P Q P

Global Asymptotic TrackingAdaptive Laws

Observer-like Gain

17


Observer-Like MRAC State Feedback Design Summary

• System Dynamics• Regulated Output

• No Uncertainties LQR PI Ref Dynamics• Baseline Closed-Loop System

• Solve Observer ARE, Compute Observer Gain, and Form Ref model

• Adaptive Control• State-feedback

1ˆ

ˆ

Tv

T

x e P B

u x

, dim dim

Tp ref cmd

z

x A x B u x B z

z C x z u

ref ref ref ref cmd v refx A x B z L x x

1

ref

Tlqr

A

K

Tref ref ref ref ref cmdx A B R B P x B z

1 0T

v ref n n ref n n v v v v vP A I A I P P R P Q

refx x

System State Asymptotic Reference Model Tracking Bounded Command Tracking

1v v vL P R

~ref cmdz z z

Ratesof adaptation

1,v v v v vQ R L P R Design Cycle

18


Transient Analysis for MIMO Dynamics

• Closed-Loop Transient (Error) Dynamics

• Singular Perturbed System

• Stretched Time Boundary Layer Dynamics = Transients

1

= Uniformly Bounded Function of TimeObserver Gain:

Hurwitz Matrix

v

Tref v v

tL

e A P R e B t x t

1v n nvR I

v

11ref ve A P e tv

0 O , as 0vP P v v

01 Orefv e v A v P v e v t

Positive Definite Symmetric

00 0 0

Asymptotic TrackingTransient Dynamics

, O o 1 expref reft t

x t v x t v P x t x tv

0t d e P ev d

19


Closed-loop Reference Model (CRM) in Adaptive ControlTravis E. Gibson (PhD Student, MIT)

• Uncertain Plant

• Reference Model

• Control Input

• Tracking Error• Lyapunov Equation with prescribed degree of stability

• Adaptive Law

• Tuning Knobs: Observer Gain and Adaptation Rate

• Main Result

• Asymptotic Bounds on Control Rate Transients

Trefx A x B u x

ref ref ref refx A x B r l x x

T

ref n n ref n n n nA l I P P A l I I

refe x x

ˆ ˆProj , Tx e P B

ˆ Tu r x

l

0 4 4( ) O , ( ) = O

e et t

lu t u tl

Observer-like Gain

20


Bounds on Control Rate Transients with CRM*Travis E. Gibson (PhD Student, MIT)

• Ref Model dominating eigenvalue

• Error time constant:

• Time constant associated with Aref :

• Inequality enforced by design

Copyright © 2009 Boeing. All rights reserved.

min Real i refiA

10 e ref

1ref

1e l

0 4 4( ) = O , ( ) = O

e et t

lu t u tl

* T.E. Gibson, E. Lavretsky and A.M. Annaswamy, Closed-Loop Reference Models in Adaptive Control: Stability, Robustness, and Transient Performance, CDC 12 submitted

• Main Result: Bounds on control rate

20

10, 100010, 10

ll

u

t

4 e

21


Can We Extend Observer-like State Feedback MRAC Design To Adaptive Output Feedback ?

* E. Lavretsky, “Adaptive Output Feedback Design Using Asymptotic Properties of LQG / LTR Controllers,”

IEEE Transactions on Automatic. Control, Jun, 2012

22


What, Why, and How

• Problem• Output feedback design for MIMO

systems in the presence of “unknown unknowns”

• Aerospace Applications• Very Flexible Aerial (VFA)

platforms.– System dynamics exhibit no

frequency separation between primary and flex modes

– Flex modes are not available online, have low damping ratios, and must be actively controlled / stabilized

• Control Design Architecture• Robust LQG/LTR + Adaptive

output feedback augmentation • Based on asymptotic properties of

LQG/LTR regulators

POLECAT

HELIOS

23


Problem Formulation

• Plant Dynamics• Restrictions: Observable, Controllable, Minimum-Phase

• Control Problem• Using output measurements y, design control input u such that the regulated output

z tracks its bounded time-varying command zcmd with bounded errors, while operating in the presence of “unknown unknowns”

0 00 0

, 0

p

p p

ref

z

d x

m m p m mz I z I m m Td d p cmd

n m p n mp p p

x x BA B

m m p

C

C Ie eu x zAx x B

y C x z C x

MatchedUncertainty Command

KnownRegressor

Unknown Parameters

Uncertain Control Effectiveness

Measured Output

Controlled Output

Plant State

Integrated Tracking

Error

24


Reference Model Construction

• System Dynamics w/o Uncertainties

• Controller Algebraic Riccati Equation

• Reference / Baseline LQR PI Controller Reference Model

1

ref

Tref ref ref ref ref cmd

A

x A B R B P x B z

1 0T Tref ref ref ref ref refP A A P P B R B P Q

Hurwitz


Satisfies Model Matching Conditions by Design

ref ref ref ref cmdx A x Bu B z

1

Tlqr

T Tref ref ref ref lqr ref

K

u R B P x K x

LQR Gain

25


Open-Loop Dynamics Reformulation

• Using Reference Model Data

,

Tref ref cmd

z

x A x B u x B z

y C x z C x

T Tx d

d pT

x

T Tref x d d p ref cmd

xK

x

x A x B u K x x B z

Hurwitz

Measured Regulated

26


• Sufficient Condition for Closed-Loop Stability

• State Feedback Adaptive Law

• Output Measurements

• Output error

• (State Output) Adaptive Feedback

Design Idea

1ˆ Tvx e P B

0T

v ref n n ref n n vP A I A I P

y C x

ˆ ˆye y y C x x C e

1 TvP B C W

ˆ T T Tyx e C W x e W

~SPR

27


Open-Loop DynamicsAssumptions and Squaring-up Method

• Controllable & Observable

• Number of measured outputs p is no less than number of control inputs m

• Achieving Nonzero High Frequency Gain and Minimum-phase Dynamics

,

Tref ref cmd

z

x A x B u x B z

y C x z C x

Measured Regulated output embedded into system dynamics

dim dim dimy p m u z

1

2

det det

det 0 det 0, Re 00

n n

p pB

s I A C s I A B

s I A Bp m C B B s

C

Squaring-Up ProblemFind B2 such that

Rosenbrock System Matrix is Nonsingular in the RHP

No Transmission Zeros in the RHP

Allows to control non-minimum phase dynamics with relative degree greater than 1

28


Adaptive Output Feedback


• Luenberger-type State Observer

• Control Input Closed-Loop Observer Dynamics

• Closed-Loop Plant Dynamics

,

Tref ref cmd

z

x A x B u x B z

y C x z C x

ˆ ˆˆ ˆ ˆ ˆ

ˆ ˆ

Tref v ref cmdx A x B u x L y y B z

y C x

ˆ ˆTu x ˆ ˆ ˆref ref cmd vx A x B z L y y

Observer Gain

ˆ ˆT Tref ref cmdx A x B z B x x

Estimated Parameters

ˆxe x x

Observer Error

29


Adaptive Output Feedback (continued)

• Closed-Loop Observer Dynamics

• Closed-Loop Plant Dynamics

• Observer Error

• Observer Error Dynamics

• Design Task – Reduce Observer Error• Choose Observer Gain• Adapt Parameters

ˆ ˆ ˆref ref cmd vx A x B z L y y

ˆ ˆT Tref ref cmdx A x B z B x x

ˆxe x x

ˆ ˆT Tx ref v xe A L C e B x x

Estimated ParametersObserver Gain

1xe

30


Adaptive Output Feedback, (Observer Design)

• “Squaring-Up” (if p > m)

• Choose parameter–dependent (v) weights

• Solve Filter Algebraic Riccati Equation

• Calculate Observer Gain, (parameter-dependent)1T

v v vL P C R

1 0T T

v ref n n ref n n v v v v vP A I A I P P C R C P Q

0 01 ,

1T

v vv vQ Q B B R R

v v

“Small” Positive Parameter

Positive constant Enforces prescribed degree of stability

1det 0 det det 0 , Re 0p m C B s I A C s I A B s

2B B B

No Transmission Zeros in the RHP

Nonzero High Frequency Gain Free to Choose

31


Adaptive Output Feedback, (Observer Design)

• Parameter–Dependent Algebraic Riccati Equation, with prescribed degree of stability

• Theorem• Inverse solution exists

– Symmetric, positive-definite

• Asymptotic relations take place, as

10 0

1 0T T T

v ref n n ref n n v v vvP A I A I P Q P C R C P B B

v

1 10 O , as 0vP P v v

12

0 OTvP B C R W v

1v vP P

Computable

Dominating term, for small v

0 OT TvP C B W R v

0v

12

2 0 OTvP B B C R W v

12

0

Computable

O0m mT

vp m m

IP B C R W v

Enable MRAC design with output feedback Tuning “knob”

Defines output measurements

32


Adaptive Output Feedback, (Completed)

• Parameter–Dependent Algebraic Riccati Equation

• Asymptotic Relation for Stability Proofs

• Theorem Stability & Bounded Tracking• Parameter Adaptation with Projection Operator

• Adaptive Output Feedback Control

1 0T T

v ref n n ref n n v v v v vP A I A I P P C R C P Q

12

0 OTvP B C R W S v

1 2 0T Tv v v v v v v v vP A A P P C R C P Q P

1

v

Tv ref v v

L

A A P C R C

HurwitzObserver Gain

Computable

Inverse ARE Solution

12

0ˆ ˆ ˆ ˆProj , Tx y y R W S

ˆ ˆTu x ˆ ˆ ˆref ref cmd vx A x B z L y y

SmallReference Model

Lyapunov-based Stability Proof

0 01 ,

1T

v vv vQ Q B B R R

v v

33


Adaptive Output Feedback Design Summary

• System Dynamics• Measured and Regulated Output

• Set Uncertainties to Zero, Design LQR PI Controller, and Create Reference Dynamics

• Baseline Closed-Loop System

• Compute B2 such that:

• Choose Small Parameter• Solve Filter ARE, Compute Kalman Gain and Form State Observer

• Output Feedback Adaptive Laws

• Output Feedback Control

12


ˆ ˆTu x

, , dim dim

Td d p ref cmd

z

x A x B u x B z

y C x z C x y z


0v

1

Tlqr

ref

Tref ref ref ref ref cmd

K

A

x A B R B P x B z

2det 0

det 0, Re 00

B

n n

p p

C B B

s I A Bs

C

10 0

1 1 0T T T

v ref n n ref n n v v vv vP A I A I P P C R C P Q B B

v v

LQG

/ LT

R D

esig

n Ite

ratio

ns

34


Key Design Features

• Adaptive laws and Control Input Do Not explicitly depend of the tuning parameter

• System Dynamics Reformulated Imbeds Desired Reference Model

• LQG / LTR observer tuning leads to improved reference model tracking

12


ˆ ˆTu x

Td d p ref cmdx A x B u x B z


0v

ˆˆ ref

ref

x xx x

x x

Squared-up LTI Dynamics LQG/LTR Observer Output Feedback Adaptive Controller Reference Model Tracking

Tref ref cmdx A x B z B u x

35


Conclusions• Constructive Methods to Design Adaptive State and Output Feedback Controllers for MIMO

Systems with Matched Uncertainties and Quantifiable Transients• Based on asymptotic properties of LQG / LTR regulators• Observer-like reference model modification

• Ongoing Work• Robust and adaptive control for Very Flexible Aerial Platforms

• Future Work• Output Feedback Adaptive Control with Nonparametric Uncertainties

– State Limiter (keeps system state within bounded approximation set)• Combined / Composite Output Feedback Adaptive Design

– Using tracking and prediction errors in adaptive laws

K

36


• Open-Loop Plant

• Observer-like Reference Model

• Tracking Error

• Adaptive Control Input• …• …• STOP RIGHT HERE !!!

• This is a Cancelation-Based Design May have 0 margins Recovering “ideal” control may lead to loss of robustness – A Controversy ?!

• Need Optimal / Robust Control Solutions• Are NOT cancellation-based• Have nonzero gain and time-delay margins

• Question: Can MRAC solutions be formulated using Optimal Control ?

A Technical Challenge


ref ref ref ref cmd refx A x B z L x x

refe x x

ˆ Tu x

e

37


Boeing in Seattle

38

Phantom Ray

First Flight, 04-27-2011

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