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
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∆
Parametric uncertaintiesSystem
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∆
Nonlinearity uncertaintiesSystem
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∆
Unmatched Uncertainties
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∆
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
= + + ∆
= + + ∆
,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
1[ , , ] : unknown parameters and gainsT Tb nb bθ θ= …
( , ) : uncertain nonlinearities/disturbancesi nx t∆
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∆ = ∀
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.
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 )
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
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
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 11−25
−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
Parameter Estimations of IARC (No Load)
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
Parameter Estimations of IARC (No Load)
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
Parameter Estimations of IARC (No Load)
0 2 4 6 8 102
4
6
8
10
12
14
16
18
20Parameter estimates with load (IARC)
time (s)
θ 1
Red: Actual value
Blue: Z−Grad
Black: Z−Ls
Green: X−Grad
Magenta: X−Ls
Parameter Estimations of IARC (Loaded)
0 2 4 6 8 10
1
2
3
4
5
6
7
8
9
10Parameter estimates with load (IARC)
time (s)
θ 2
Red: Actual value
Blue: Z−Grad
Black: Z−Ls
Green: X−Grad
Magenta: X−Ls
Parameter Estimations of IARC (Loaded)
0 2 4 6 8 10−10
−8
−6
−4
−2
0
2
4
6
8
10Parameter estimates with load (IARC)
time (s)
θ 3Red: Actual value
Blue: Z−Grad
Black: Z−Ls
Green: X−Grad
Magenta: X−Ls
Parameter Estimations of IARC (Loaded)