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Title: Robust Adaptive Control : The Search for the Holy Grail


1
  • Robust Adaptive Control The Search for the Holy
    Grail
  • By
  • Petros Ioannou
  • University of Southern California
  • Los Angeles, CA 90089
  • Email ioannou_at_usc.edu

2
Outline
  • Adaptive Control Focus on Parametric Uncertainty
  • Why Adaptive Control?
  • Adaptive Control Identifier based
  • Global Results
  • Instabilities
  • Robust Adaptive Control
  • Adaptive Control Non Identifier based, Switching
    Schemes, Multiple Models
  • Adaptive schemes with mixing
  • Library of Tools
  • Example of Applications
  • Adaptive control toolbox
  • Conclusions

3
Adaptive Control Focus on Parametric Uncertainty
  • Adaptive control was motivated as an extension of
    gain scheduling for aircraft autopilot design
  • Off-line design
  • Cannot handle unpredictable changes
  • Complexity Large look up tables

4
Why adaptive control?
  • No fixed gain can stabilize the system if
    or an upper bound for is unknown.
  • The adaptive control input
  • guarantees signal boundedness and

See book Ioannou and Fidan 2006
5
Adaptive Control Identifier based
  • Indirect Schemes
  • Applicable to any controller structure provided
    calculations of controller parameters can be done
    on line.

6
Adaptive Control Identifier based
  • Direct Schemes
  • Applicable to control structures where the
    parametrization
  • is possible.
  • Possible for minimum phase plants with model
    reference control objectives.

7
Certainty Equivalence and Challenges
  • Adaptive control schemes combinations of an on
    line parameter estimator with a control law of
    the same form as the one used if parameters were
    known.
  • Closed loop system becomes nonlinear time varying
  • Most of the practical control design tools
    incorporating robustness and performance
    specifications used for LTI systems are not
    applicable
  • The properties of the on line parameter estimator
    or adaptive law is key to the stability and
    performance of the overall system

8
Early Adaptive Laws Sensitivity Methods
Adaptive Law
Approximate sensitivity functions
If adaptation slow, initial parameter estimates
close to true ones, small reference signals rich
in frequency then local results. Instability
examples
Ref Cruz (1973), Kokotovic (1964), James (1971),
Mareels et.al(1986)
9
Adaptive Laws Parametric Models and Estimation
Error
Linear Parametric Models Bilinear Parametric Models

Ref. Books by Ioannou Fidan (2006), Tao(2003),
Ioannou and Sun(1996), Sastry and Bodson (1989),
Astrom and Wittenmark (1989), Narendra and
Annaswamy (1985)
10
Adaptive Laws Parametric Models and Estimation
Error

Estimation Error Gradient Properties using Lyapunov Estimation Error LyapunovSPR Properties using Lyapunov
Ref. Books by Ioannou and Sun(1996), Ioannou
Fidan (2006), Tao(2003), Sastry and Bodson
(1989), Astrom and Wittenmark (1989), Narendra
and Annaswamy (1985), Egardt (1978), Goodwin and
Sin (1983)
11
Model Reference Adaptive Control (MRAC)
Indirect
Direct
Results All signals are bounded and the plant
output tracks the output of the reference model
asymptotically with time for any input command
If input command is sufficiently rich in
frequencies then convergence is exponential and
have parameter convergence too.
12
Globally stable schemes
  • MRC structure

Minimum phase plant Known relative degree
Ref Monopoli, Morse, Narendra, Astrom, Landau,
Kreisselmeier, Goodwin, Mayne, Egardt and co
workers Unified in Ioannou Sun 1996 and Ioannou
Fidan 2006
13
Adaptive Pole Placement Control (APPC)
Controller has same form as in known parameter
case i.e. LQ, Pole Placement, Observer based
state feedback etc
Direct
Indirect with normalization
In general not possible unless MRAC
Results If the estimated plant polynomials are
strongly coprime each time t then all signals are
bounded and the plant output follows the
reference trajectory asymptotically with time
If the reference trajectory is sufficiently rich
in frequencies then convergence is exponential
and have parameter convergence too.
14
Global Results
Direct
Indirect
Exponentially stable
Swapping Lemmas
Small-Gain or Bellman-Gronwall Lemma
tracking error
Numerous analysis techniques. Simplified unified
proofs include Morses tunability, Ioannou
Tsakalis norm and i/o results from
Desoer and Vidyasagar book
15
Discrete time Adaptive Control
  • A discrete time model of the plant can be used to
    design adaptive controllers. ARMA models,
    discrete time state space models etc OR
  • The continuous time adaptive laws can be
    discretized using Eulers backward derivative
    approximation method

Leading to the same form as discrete time
adaptive laws derived using discrete time models
i.e. see Ioannou and Fidan for details
16
Global Results
  • Results of the late 70s were major achievements
  • Anderson, Astrom, Egardt, Goodwin,
    Kreisselmeier, Landau, Monopoli, Morse, Narendra,
    Wittenmark and co workers.
  • Remarkable
  • Ability to design control schemes for plants with
    completely unknown parameters that guarantee
    boundedness and convergence of tracking error to
    zero even when the estimated parameters do not
    necessarily converge to true values.

17
Fun is Over Instabilities and Non Robust Behavior
Simple estimator
and
as
Ideal Case
But for
it is possible to have
and
Explanation
Bo Egardt 1978 had first parameter drift example.
18
Instability phenomena
Plant
Adaptive Controller
and
as
Properties
For
we have
as
By the early 80s several instability examples
including the famous Rohrss example, Ioannou and
Kokotovic, Astrom, Anderson etc with explanations
of instability mechanisms were published
19
Robust Adaptive Control
  • Modify Adaptive Laws
  • Dominantly rich excitation signals
  • Dead Zone adapt only when signal is large
    relative to modeling errors
  • Leakage Modify the pure integral action of the
    adaptive law by a small feedback gain. Fixed
    , Switching , e-modification
  • Projection Constraint parameter estimates to lie
    within bounded sets in parameter space
  • Dynamic normalization slow down adaptation
    relative to the speed of growth of modeling
    errors

Ref. Egard, Praly, Ioannou, Narendra, Annaswami,
Goodwin, Tsakalis, Kokotovic, Datta, Tao, Sun,
Ystie and many others
20
Robust Adaptive Control
  • Plant

Apply adaptive control with robust adaptive laws
Result If
then all signals are bounded and tracking error
satisfies
If dead zone is used the tracking error can be
guaranteed to be inside a dead zone but this
bound may be conservative. Calculation of
robustness bounds difficult but possible
(Tsakalis )
21
Burst phenomena
Simulations by K. Tsakalis
22
Robust Adaptive Control
  • Extensions and New Directions
  • Control of linear time varying systems with
    unknown parameters. (see book Tsakalis and
    Ioannou 1993)
  • Multivariable Plants ( see book G. Tao 2003)
  • Relaxation of Assumptions ( Nussbaum, Morse,
    Miller, Willems, Byrnes, Martenson, Barmish etc)
  • Control of classes of nonlinear plants with
    unknown parameters. Adaptive Backstepping, tuning
    functions etc (Kokotovic, Morse, Praly,
    Kanellakopoulos, Krstic, Khalil, and co workers
    see book Krstic et. all 1995)
  • Neuro adaptive for classes of non linear systems
  • (Narendra, Polycarpou, Christodoulou,
    Kosmatopoulos, Lewis, Ioannou, Calise and co
    workers)

23
Robust Adaptive Control
  • Open problems
  • General APPC schemes are faced with
    stabilizability issues
  • Closed loop system is time varying and non linear
    making it difficult if at all possible to apply
    practical control design and analysis tools
    established for LTI systems
  • The use of sufficiently rich signals is in
    conflict with control objective

24
Adaptive Control Non Identifier based, Switching
Chemes, Multiple models
Ref. S. Morse. Barmish, Fu, Mayne, Middleton,
Goodwin, Hespanha, Narendra, Anderson, Liberzon,
Athans, Fekri, Safonov Ioannou, Kuipers and
coworkers
25
Adaptive Control Non Identifier based, switching
schemes, multiple models
Switching schemes
Supervisory adaptive control (Morse, Hespanha,
Liberzon, Narendra, Anderson, Mayne, Goodwin
etc) Switches candidate controllers into the loop
by comparing normed estimation errors
Unfalsified adaptive control (Safonov et
al) Non-identifier based scheme that performs
out-of-the-loop evaluations of candidate
controllers
Supervisor
Supervisor
µ1
e1
E1
-

Unfalsification Algorithm Plus Switching Logic


Switching Logic
µN
eN
EN

-
ea
µa
Ea(?)
-

Unknown Plant


d
n
d
n
26
Search methods and switching schemes
Robust multiple model adaptive control (RMMAC)
(Athans, Fekri) Dynamic hypothesis testing scheme
that weights the output of each candidate
controller with the probability its model is the
true model
  • Pi(t) is the probability that model i is the
    true.
  • Soft switching in the sense that it is
    intended that for


27
Comparison
Identifier based adaptive control Switching-based MMAC with fixed candidate controllers RMMAC
Advantages Large parametric uncertainty, unpredictable changes Most control scheme can be made adaptive Stability and robustness results Use powerful robust control techniques Potential for rapid adaptation to large parameter jumps No stabilizability issues Stability and robustness results Use powerful robust control tools Good performance subject to accurate disturbance model and initial conditions of the Kalman filters No stabilizability issues
Disadvantages With the exception of MRAC, stabilizabity issues Some transients during learning Cannot take advantage of practical LTI tools Cannot handle unpredictable changes Large transients due to a prolonged switch to an incorrect controller before reliable data is available Performance may be sensitive to model assumptions (disturbance model, initial conditions, etc.) Cannot handle unpredictable changes Currently, no stability results
28
Adaptive schemes with mixing (Kuipers and
Ioannou)
Plant
1. Parameter subsets
2. Candidate Controllers
3. Multicontroller
LTI tools
State-space realization
C1(s)
Mixing strategy
C2(s)
C3(s)
Mixing strategy must ensure stability on the
overlapping regions.
y
29
Multicontroller Mixing strategy
Interpolate C2(s) and C3(s)
Interpolate C1(s) and C2(s)
  • Numerous controller interpolation approaches have
    been proposed for gain scheduling that are also
    suitable for constructing the multicontroller.
    One intuitive approach, as in RMMAC, is output
    blending
  • Output blending does not always provide stable
    controller interpolation unless controllers are
    design to account for it.

30
Benchmark example by Athans
Increased disturbance power
Plant output
Mixed-mu ? Candidate controllers
Nominal disturbance model
AMC or RMMAC
AMC and RMMAC supervisor output
Correct Controller
Mixed-mu
31
Benchmark example
Adaptive mixing control with fixed and adaptive
candidate controllers
Simulation results for
Plant output
RMMAC
Adaptive pole-placement controller Ca provides
stability for
AMC
32
Brief History Generated Library of Tools
1960s Sensitivity Method, MIT Rule Limited
stability analysis Whitaker, Kalman, Parks,
1970s Lyapunov based Passivity based Morse,
Narendra, Landau,
Late 1970s - 1980s Global stability Morse,
Narendra, Egardt, Landau, Goodwin, Keisselmeier,
Anderson, Astrom.
Early 1980s Robustness issues, Instability Egard,
Rohrs, Ioannou, Athans, Anderson, Astrom,
1980s Robust adaptive control Praly, Ioannou,
Narendra, Tsakalis, Annaswamy, Sun, Tao,
Goodwin, Middleton etc
1990s Nonlinear adaptive control
Adaptive backstepping Krstic, Kanelakopoulos,
Kokotovic, Zhang,
1990s Search methods, Multiple models, Switching
techniques Martenson, Miller, Barmish,Morse,Narend
ra, Morse, Andreson, Safonov, .
Neuro-Adaptive control Fuzzy-Adaptive
control Narendra, Lewis, Polycarpou,
Kosmatopoulos, Xu, Ioannou, Wang, Lavresky,
Hovakimyan
33
Adaptive control of an Airbreathing Hypersonic
Aircraft
Rigid-body Longitudinal dynamics
Unstable MIMO system
34
Simulation Results
Non-adaptive Mixed-mu
K1
Mixing strategy
K8
Non-adaptive Mixed-mu
Kuipers, et. al. Robust adaptive multiple model
controller design for an airbreathing hypersonic
vehicle model. Proc. AIAA GNC Conf., Honolulu,
Hawaii, August 2008.
35
HDD application
  • Hard disk drives (HDD) are a form of data storage
  • Data is written in concentric tracks on a disk
    The actuator controls the position of the head
    over the tracks
  • Track-following is tackled and formulated as a
    disturbance rejection problem
  • Disturbance consists of
  • RRO (repeatable runout) imperfections on the
    tracks, etc.
  • NRRO (non-repeatable runout) vibrations,
    ball-bearing imperfections, etc.

Inject negative of est. disturbance
Disturbance model is made of radial basis
functions
36
HDD experimental results
RRO and NN turned on
Baseline
RRO
RRO NN
  • NN disturbance rejection increases performance
    on various head and track combinations.
  • Experiments conducted at the University of
    California, Los Angeles

37
Adaptive notch filter
  • Wide notch filter
  • Account for variations in modal frequency
  • Adds phase lag which limits bandwidth which
    limits performance
  • Adaptive notch filter
  • Dynamically tracks modal frequency
  • Adds less phase lag increase in bandwidth and
    performance

38
Fast steering mirror application
  • Goal zero tracking error for SISO system with
    uncertain flexible mode
  • Compare ANF to non-adaptive NF
  • ANF has better performance even when non-adaptive
    notch filter has correct center frequency

Loop closed for ANF
ANF vs Non-adaptive with correct center frequency
Non-adaptive with and without correct center
frequency
Image taken from N.O.P. Arancibia, S. Gibson, and
T.-C. Tsao, Saturation and frequency weighting
in adaptive control of laser beam jitter, Proc.
Of SPIE, Vol 6709, 2007.
39
Adaptive control toolbox
A New Toolbox for use with MATLAB and Simulink
P.A. Ioannou B. Fidan
  • Parameter Identification
  • Adaptive Control
  • in both Continuous-Time and
    Discrete-Time

Parameter Identification
Adaptive Law Modifications
  • Gradient Methods
  • Least-Squares Methods
  • SPR-Lyapunov Approaches
  • Normalization
  • Static, Dynamic
  • Parameter Projection
  • Robustness Modifications
  • s, e, Dead-zone

(Adaptive) Control
  • Model Reference (Adaptive) Control
  • Direct, Indirect
  • (Adaptive) Pole Placement Control
  • Polynomial, State-Variable,
  • Linear Quadratic
  • Minimum Prediction Error Control
  • Direct, Indirect, Semi-direct

Other
  • (Adaptive) State Estimation
  • Output Prediction (ARMA)
  • Parametric Model Order Reduction
  • Polynomial Algebra

40
On line Parameter Estimators
Simulink Blocks
Parametric Model Signals
Parametric Model Signals
Plant I/O Signals
Parameter Estimate
Normalizing Signal
Plant I/O Signals
Original Parametric Model Signals
Reduced-Order Parametric Model Signals
Parameter Estimate
Normalizing Signal
MATLAB Commands
Gradient Methods Function
Purpose ucgrad,ucgradbk,ucgradint
Continuous-time gradient algorithms dgradb,
dgradl, udgrad Discrete-time gradient
algorithms dprojmod, dprojorth,
Discrete-time projection algorithms dprojpure,
udproj Least-Squares (LS) Methods Function
Purpose ucrls,
urlsarg Continuous-time
LS algorithms drls, udrls, urlsarg
Discrete-time LS algorithms
Model Conversion/Model Order Reduction Function
Purpose utf2lm, io2lm
Transfer function to linear parametric
model uarma2lm, io2lm ARMA to linear
parametric model lmred
Parametric model order reduction
41
Adaptive control
Simulink Block
Plant Output
Control Signal
Parametric Model Signals
Reference Signal
Parameter Estimate
Dynamic Normalizing Signal
MATLAB Commands
Model Reference Adaptive Control (MRAC) Function
Purpose mrcpoly
MRC/MRAC design Polynomial Approach dmrc
Discrete-time MRC/MRAC
design umrcdrb, umrcdrl Continuous-time
direct MRAC umrcidr
Continuous-time indirect MRAC udmracdr
Discrete-time direct MRAC udmracidr
Discrete-time indirect
MRAC One-Step Ahead Adaptive Control
(OSAAC) (Discrete Time) Function
Purpose dosac
OSAC/OSAAC design udosacdr
Direct OSAAC udosacidr
Indirect OSAAC udosaclcf
OSAAC Linear control form approach
Adaptive Pole Placement Control (APPC) Function
Purpose ppcpoly, ppcssv
Continuous-time PPC/APPC design dppcdo,
dppcimp, Discrete-time PPC/APPC design
dppcdo, dppcimp ppcclq
Continuous-time (adaptive) LQC design ppcdlq
Discrete-time (adaptive) LQC
design uppcpoly, uppcrsf PPC/APPC
(adaptive) LQC implementation Polynomial
Algebra Tools Function
Purpose bezout, diophant Solving
Diophantine equations euclid
Greatest common divisor of two
polynomials polylcm Lowest
common multiple of two polynomials
42
Key features
  • Numerical implementation of an extensive set of
    parameter
  • identification and adaptive control schemes.
  • Simulink blocks with user-friendly GUIs to
    implement the
  • schemes and to select/tune the design
    parameters easily.
  • Ability to implement the same schemes using the
    provided
  • MATLAB commands in order to have flexibility.
  • Applicability to both continuous-time and
    discrete-time plants.
  • Normalization, parameter projection, and robust
    modification
  • capabilities to guarantee stability and
    robustness.
  • Basic polynomial algebra tools to design
    controllers based on
  • model matching and pole placement techniques.

? Process for k 1N_final-1, theta(,k)
ucrls('parameter',x_est,ntheta) thetau,
thetay, thetar, RETYPE mrcpoly(theta(1,k),
1 theta(2,k),Zm,Rm,Lambda_c)
theta_c(,k) thetau() thetay()
thetar() u(k) umrcidr('control',xc,y(k)
r(k),n m,Lambda_c,Lambda_p,theta_c(,k))
dxc umrcidr('state',xc,u(k) y(k),n
m,Lambda_c,Lambda_p) z(k), phi(,k)
umrcidr('output', xc,u(k) y(k),n
m,Lambda_c,Lambda_p) xc xc dxcdt
dxp ufilt('state', xp, u(k), b, a) y(k1)
ufilt('output',xp, u(k), b, a) xp xp
dxpdt dxm ufilt('state', xm, r(k), Zm,
Rm) ym(k1) ufilt('output', xm, r(k), Zm,
Rm) xm xm dxmdt dx_est ucrls('state',
x_est,z(k),phi(,k), 0,ArgLS,,) x_est
x_est dx_estdt end ?
43
Conclusions
  • Major Achievements
  • Plethora of control design and analysis tools,
    practical and theoretical are generated
  • Over the years some will be forgotten, others
    will be rediscovered but many will be used to
    advance the field further.
  • Rich library of tools to choose from. Choosing
    the right control design for a particular
    application is often an art

44
Conclusions
  • Challenges
  • On line learning via data processing may take
    time leading to bad transients till the
    appropriate controller is found
  • I/O data may contain corrupted information due to
    external disturbances, modeling errors i.e. low
    signal/noise leading to the choice of a wrong
    controller again causing bad transients
  • The time varying and nonlinear nature of adaptive
    control makes it difficult if at all possible to
    use the well established practical tools for LTI
    plants to meet robustness and performance
    specifications.

45

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