Nonlinear Adaptive Robust Control
– Theory and Applications
Bin Yao
School of Mechanical Engineering
Purdue University
West Lafayette, IN47907
OUTLINE
„
Motivation
„
General ARC Design Philosophy and Structure
„
Specific Design Issues
„
Application Examples
‹
Motion and Pressure Control of Electro-Hydraulic Systems
‹
Precision Motion Control of Linear Motors
Dynamic Friction Compensation
‹
„
Conclusions
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
2
Obstacles for Control Algorithms Design
„
Inherent Nonlinearities
‹
„
Friction forces; Nonlinear process dynamics; …
Modeling Uncertainties
‹
Unknown but reproducible or slowly changing terms
(e.g., unknown parameters, repeatable run out, …)
‹
Non-reproducible terms
(e.g., random external shock disturbances, non-repeatable run out)
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
3
Strategies For Performance Improvement
ƒ Nonlinear Physical Model Based Analysis and Synthesis
‹
Deal with the inherent physical nonlinearities directly
ƒ Fast Robust Feedback for Maximum Attenuation of Various
Uncertainty Effect
‹
Effective to both repeatable and non-repeatable uncertainties
ƒ Controlled Learning for Uncertainty Reduction
‹
Effective handling of repeatable uncertainties
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
4
Different Types of Race Car Drivers
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
5
Mr. Robust Control (DRC)
A Boxer
Fast
Instantaneous
Reaction !
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
6
Mr. Adaptive Control (AC)
A Thinker
Good Learning
Ability
but
Not so Fast
Instantaneous
Reaction !
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
7
Mr. Adaptive Robust Control
A Thinker with a
good body
Good Learning
Ability
and
Fast
Instantaneous
Reaction !
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
8
What is the order of arrivals of three drivers ?
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
9
Random Road Profile
Order of arrivals of three drivers:
or
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
10
What is the winning order of three drivers in
individual practices ?
Repeatable Road Profile
Winning order of drivers:
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
12
What is the winning order of three drivers in
actual competition ?
Semi-Repeatable Driving Condition
Winning order of drivers:
or
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
14
DETERMINISTIC ROBUST CONTROL (DRC)
ƒ Sliding Mode Control (SMC)
(Utkin’77, Young, Slotine, Sastry, Hedrick, Zinober, Zak, …)
‹
‹
‹
Matching Condition
Control Chattering Due to Discontinuous Control Law
Smoothing Techniques (Slotine’85)
)
Asymptotic Tracking is lost and a trade off exists between the
actuator requirements and the achievable tracking accuracy.
ƒ Lyapunov Function Based Min-Max Methods
(Leitmann, Corless and Leitmann’81, Barmish, Chen, ...)
‹
Unmatched Uncertainties (Qu’93, Freeman and Kokotovic’93,…)
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
15
ADAPTIVE CONTROL OF LINEAR SYSTEMS
„
„
Stable Adaptive Control (late 1950-1980) (Astrom, Narendra,...)
Robust Adaptive Control (1980s)
To achieve robustness to small disturbances, time-varying parameters and
unmodeled dynamics (Rohrs, et al, 1985, Egardt, 1979)
‹
Use Appropriate Reference Input
Persistent Excitation (PE) Conditions (Boyd and Sastry, Annaswamy,…)
‹ Robustify Adaptation Law
(i) Dead zone;
(ii)σ -modification (Ioannou and Kokotovic, 83);
(iii) ε -modification; (iv) Discontinuous Projection (Sastry, Goodwin,…)
„
Recent Trends
‹ Improve Transient Performance (Zhang and Bitmead, 90)
Using VSC (Fu,92, Narendra and Boskovic,87), (Datta,93)
‹ Relax Assumptions
Minimum Phase; Relative Degree; Sign of high-frequency gain; Order.
ADAPTIVE CONTROL OF NONLINEAR SYSTEMS
ƒ Adaptive Control of Robot Manipulators (1985-present)
(Slotine and Li’88, Sadegh and Horowitz’90, Spong’89…)
ƒ Adaptive Control of Feedback Linearizable Systems
--Certainty-Equivalence Based (1987-early 90s)
‹
‹
Nonlinearity-Constrained Schemes (Sastry and Isidori’89, …)
Uncertainty-Constrained Schemes --extended matching condition
ƒ Systematic Design Method--Backstepping (1990-present)
Parametric-strict Feedback Form (Kanellakopoulos, Kokotovic, and Morse’91)
‹
‹
Without Overparametrization (Krstic, et al’92)
Improved Transient Performance (Kanellakopoulos, et al’93)
ƒ Robust Adaptive Control of Nonlinear Systems
Robot (Reed and Ioannou’89, …); Backstepping (Polycarpou and Ioannou’93, Pan and
Basar’96, Freeman, et al’96, Marino and Tomei’98, …)
‹
Robust stability; Achievable performances in terms of L∞ norm are not
so transparent
NONLINEAR ADAPTIVE CONTROL (AC)
•
Reduce/Eliminate uncertainties through parameter
adaptation to achieve asymptotic tracking.
Limitations:
• Need certain invariant properties (e.g., parameters
being unknown but constant).
• Possible instability in the presence of even small
measurement noises and disturbances
DETERMINISTIC ROBUST CONTROL (DRC)
•
Attenuate the effect of model uncertainties through
robust feedback.
Limitations:
• limited final tracking accuracy (tracking errors can
only be reduced by increasing feedback gains)
OUTLINE
ƒ Motivation
ƒ General ARC Design Philosophy and Structure
ƒ Specific Design Issues
ƒ Application Examples
‹
‹
‹
Motion and Pressure Control of Electro-Hydraulic Systems
Precision Motion Control of Linear Motors
Dynamic friction compensation
ƒ Conclusions
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
19
ADAPTIVE ROBUST CONTROL (ARC)
ƒ Problem Formulation
‹
Practical situation that the system is subjected to both
repeatable and non-repeatable uncertainties
ƒ Means Used To Achieve High Performance
‹
‹
Fast Robust Filter Structures to attenuate the effect of
model uncertainties as much as possible
Learning Mechanisms (e.g., parameter adaptation) to
reduce repeatable model uncertainties
ƒ General Design Philosophy
‹
Robust performance provided by robust feedback should not
be lost when introducing learning mechanisms; Learning
mechanisms are introduced only when their destabilizing
effects can be controlled
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
20
θ
Learning Mechanisms
(Parameter Adaptation)
Coordination
Mechanisms
us
r (t )
uf
Adjustable
Model Compensation
Robust
Feedback
u
Plant
x y
General Structure of ARC Controllers
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
21
FIRST-ORDER UNCERTAIN SYSTEM
x = f ( x, t ) + u,
θ:
∆:
f = ϕ T ( x , t )θ + ∆ ( x , t )
unknown parameters
uncertain nonlinearities
Assumptions
θ
∈ Ω θ = (θ min , θ max )
| ∆ | ≤ δ ( x, t )
where Ω θ
and δ ( x, t ) are known
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
22
OBJECTIVE
For any desired output trajectory x d (t ), design a
bounded control input u (t ) such that the tracking error,
e=x−xd, is as small as possible
e (t )
x d (t )
x
t
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
23
θ
θ = Γϕe
Parameter Adaptation Law
−k
Linear Stabilizing Feedback
xd
xd − ϕ T θ
uf
Model Compensation
+
u ∆=0
x = ϕ Tθ + u
x
Plant
+
e
-
xd
Adaptive Control of a First- order Uncertain System
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
24
ADAPTIVE CONTROL METHOD
„
„
„
Assume that there is no uncertain nonlinearity
∆ = 0
On
- line parameter adaptation
Error dynamics
e + k e = -ϕ
„
θ
Lyapunov function
V
V
„
~
T
1
e
2
a
=
a
= e − ϕ
[
2
+
T
1 ~
θ
2γ
~
2
]
θ − ke +
RESULTS
1 ~
γ
~
θ θ = − ke
2
e → 0
and ϕ θ → 0 as t → ∞
i.e., Zero final tracking error for any feedback gain
since the parametric uncertainty is eliminated.
T
LIMITATIONS OF ADAPTIVE CONTROL
•
Transient performance is unknown
e
t
•
Uncertain nonlinearities are not considered
When ∆
•
≠ 0,
what is the performance ?
May be unstable when
∆ ≠ 0
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
26
us2
u s 2 (e, t )
Nonlinear Robust Feedback
us
+
u s1
−k
Linear Stabilizing Feedback
xd − ϕ θ0
T
xd
Fixed Model
Compensation
uf +
u
∆
x = ϕ T θ + ∆ + u x -+
Plant
e
xd
Deterministic Robust Control of a First- order System
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
27
DETERMINISTIC ROBUST CONTROL
•
Error dynamics
~
e + ke = −ϕ T θ 0 + ∆ + u s2
•
Choose the robust control us2 such that
(
)
~
e us2 − ϕ Tθ0 + ∆ ≤ ε
•
where ε is a design parameter.
Example
u s2 = −
•
eu s2 ≤ 0
and
1
4ε
[
θ m a x − θ m in
2
ϕ
2
+δ
2
]e
1 2
Stability analysis by a Lyapunov function Vs = e
2
V s ≤ − 2 kV s + ε
V s ≤ exp ( − 2 kt )V s (0 ) +
ε
1 − exp ( − 2 kt )]
[
2k
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
28
DETERMINISTIC ROBUST CONTROL
•
•
e(0)
Guaranteed transient: exponential convergence
Guaranteed final tracking accuracy:
2
Upper Bound of Tracking Error
[
exp( − 2kt ) e(0) − e(∞ )
2
2
] + e( ∞ )
2
e( ∞ ) ≤
2
e(t )
ε
k
2
t
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
29
ARC DESIGN via SMOOTH PROJECTION
θπ
[ () ]
θ
π (⋅)
θ = Γ − lθ θ + ϕ e
Smooth Projection Modified Adaptation Law
us2
u s 2 (e, t )
Nonlinear Robust Feedback
us
+
u s1
−k
Linear Stabilizing Feedback
x d − ϕ θπ
T
xd
uf
Adjustable
Model Compensation
+
u
∆
x = ϕ T θ + ∆ + u x -+
Plant
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
e
xd
30
SMOOTH PROJECTION
()
πθ
θ max
εθ
θ max
θ min
θ
θ min
PROPERTY
~
Vθ (θ ) is a positive definite function where (Teel’93)
1 θ
~
V θ (θ ) =
π (ν + θ ) − θ )d ν ,
γ ≥ 0
(
∫
γ
~
0
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
31
MODIFICATION OF ADAPTATION LAW
lθ acts as a nonlinear damping and satisfies
( )
( )
lθ θ = 0
~
i i . θ πT lθ θ ≥ 0
i.
if
if
θ ∈Ωθ
θ ∉Ωθ
lθ
θ min
θmax
θ
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
32
ARC DESIGN via SMOOTH PROJECTION
•
Error dynamics
~
e + ke = −ϕ θ π + ∆ + u s 2
T
•
Choose the robust control u s 2 such that
(
~
)
e us 2 − ϕ θπ + ∆ ≤ ε
i.
T
ii. eus 2 ≤ 0
•
Example
[
]
1
2
2
us2 = −
θmax −θmin + εθ ϕ + δ 2 e
4ε
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
33
PERFORMANCE OF ARC
•
In general, achieves the same performance as DRC
‹ Guaranteed transient: exponential convergence
‹ Guaranteed final tracking accuracy
e (0 )
2
U pper B ound of T racking E rror
[
exp ( − 2 kt ) e ( 0 )
2
− e (∞ )
2
]
+ e (∞ )
2
e (∞ )
e (t )
2
≤
ε
k
2
t
•
In addition, achieves asymptotic tracking in the
presence of parametric uncertainties as AC
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
34
PROOF of ARC PERFORMANCE
•
In general, use the same Lyapunov function V s
DRC to recover the results of DRC
(
~
V s = − ke + e − ϕ θ π + ∆ + u s 2
2
T
≤ − 2 kV s + ε
•
as in
)
When ∆ = 0, use a new p.d. function Vt = Vs +Vθ
to obtain asymptotic tracking
~
V t = − ke + eu s 2 − e ϕ θ π +
2
T
()
1 ~
γ
θπθ
~
= − ke 2 + eu s 2 − θ π lθ θ
≤ − ke 2
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
35
ARC DESIGN via DISCONTINUOUS PROJECTION
θ
θ = Pr ojθ (Γ ϕ e )
Modified Adaptation Law
us2
u s 2 (e, t )
Nonlinear Robust Feedback
us
+
u s1
−k
Linear Stabilizing Feedback
x d − ϕ θπ
T
xd
uf
Adjustable
Model Compensation
+
u
∆
x =ϕ θ +∆+u
T
Plant
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
x
+
e
-
xd
36
ARC DESIGN via DISCONTINUOUS PROJECTION
•
Parameter adaptation with projection
qˆ = Pr ojqˆ (Gj e )
Ï 0 if
Ô
Pr ojqˆ (∑i ) = Ì 0 if
i
Ô
Ó∑i
•
Properties
{
qˆi = qˆimax
and ∑i >0
qˆi = qˆimin
and ∑i <0
otherwise
P1.
θ ∈ Ωθ = θ :
P2.
θ ( Γ -1 Pr ojθ (Γ •) − •) ≤ 0,
~
θmin ≤ θ ≤ θmax
}
∀•
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
37
DESIRED COMPENSATION ARC DESIGN
θ
θˆ = Pr ojθˆ (Γ ϕ ( x d ) z )
Modified Adaptation Law
us2
us 2 ( z , t )
Nonlinear Robust Feedback
us
+
u s1
− k s1
Linear Stabilizing Feedback
x d − ϕ ( x d ) θˆ
T
xd
uf
Adjustable
Model Compensation
+
u
∆
x = ϕ ( x )T θ + ∆ + u x - +
Plant
xd
z
SIMULATION OF A FIRST-ORDER SYSTEM
•
The Plant
•
The Controller Parameters
δ =1
θ0 = 2
k = 10
ε 0 = 0 .3
γ = 2000
•
x = θ sin (π x ) + ∆ + u
round ( t )
∆ = ( - 1)
ϕ = sin (π x )
Ω θ = ( 0 , 20 )
θ = 18
D t = 1m s
ε 1 = 0 .001
The Desired Trajectory
x d = 0 .5 (1 − c o s ( 1 . 4 π t ) )
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
39
Adaptive Control (no disturbances) -- Solid
1
Adaptive Control (with disturbances) -- Dashed
Feedback Linearization (no disturbances)
Tracking Error
0.8
0.6
0.4
0.2
0
-0.2
0
0.5
1
1.5
2
2.5
3
3.5
4
Time (s)
Adaptive Control of a First-order Uncertain System
35
Estimated Parameter
Adaptive Control (no disturbances) -- Solid
30
Adaptive Control (with disturbances) -- Dashed
25
Feedback Linearization
20
15
10
5
0
0
0.5
1
1.5
2
2.5
3
3.5
4
Time (s)
Adaptive Control of a First-order Uncertain System
0.04
Tracking Error
0.03
0.02
0.01
0
ε 0 =1
-0.01
-0.02
0
0.5
1
and
ε1
=0.001 (with disturbances) -- Solid
=0.3 and
=0.3 and
=0.001 (with disturbances) -- Dashed
=0.001 (no disturbances) -- Dotted
=0.08 and
=0.001 (with disturbances) -- Dashdot
1.5
2
2.5
Time (s)
3
3.5
4
Deterministic Robust Control of a First-Order System
0.25
Solid: ARC(d)
Dashed: ARC(s)
0.2
Dotted: AC
Tracking Error
0.15
Dashdot: DRC
0.1
0.05
0
−0.05
−0.1
−0.15
0
0.5
1
1.5
2
2.5
3
3.5
4
Time (sec)
Tracking Errors In the Presence of Parametric Uncertainty Only
30
Solid: ARC(d)
Dashed: ARC(s)
Estimated Paramter
25
Dotted: AC
Dashdot: DRC
20
15
10
5
0
0
0.5
1
1.5
2
2.5
3
3.5
4
Time (sec)
Estimates In the Presence of Parametric Uncertainty Only
0.25
Solid: ARC(d)
Dashed: ARC(s)
0.2
Tracking Error
0.15
Dotted: AC
Dashdot: DRC
0.1
0.05
0
−0.05
−0.1
−0.15
0
0.5
1
1.5
2
2.5
3
3.5
4
Time (sec)
In the Presence of Parametric Uncertainty and Small Disturbances
35
Solid: ARC(d)
Estimated Paramter
30
Dotted: AC
25
Dashed: ARC(s)
Dashdot: DRC
20
15
10
5
0
0
0.5
1
1.5
2
2.5
3
3.5
4
Time (sec)
In the Presence of Parametric Uncertainty and Small Disturbances
0.04
0.035
Tracking Error
0.03
Solid: ARC(d) Dashed: ARC(s) Dashdot: DRC
0.025
0.02
0.015
0.01
0.005
0
−0.005
0
0.5
1
1.5
2
2.5
3
3.5
4
Time (sec)
In the Presence of Parametric Uncertainty and Large Disturbances
45
40
Paramter Estimate
35
30
25
20
15
10
5
0
0
Solid: ARC(d) Dashed: ARC(s) Dashdot: DRC
0.5
1
1.5
2
2.5
3
3.5
4
Time (sec)
In the Presence of Parametric Uncertainty and Large Disturbances
ADAPTIVE ROBUST CONTROL (ARC)
„
Departs From Robust Adaptive Control (RAC)
‹
‹
‹
„
Departs From Deterministic Robust Control (DRC)
‹
„
In terms of fundamental view point, puts more emphasis on the
underline robust control law design for a guaranteed robust output
tracking performance in general.
In terms of achievable performance, guarantees a prescribed
transient performance and tracking accuracy.
In terms of design approaches and proof, uses two Lyapunov
functions; one the same as DRC and the other as RAC.
Achieves asymptotic tracking in the presence of parametric
uncertainties without using discontinuous control law or infinite gain
feedback—overcome the design conservativeness.
Departs From Other Existing Combined Schemes
(Narendra and Boskovic’92, Slotine and Li’88, Chen’92, …)
‹ Achieves a guaranteed transient performance and tracking
accuracy
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
49
OUTLINE
ƒ Motivation
ƒ General ARC Design Philosophy and Structure
ƒ Specific Design Issues
ƒ Application Examples
‹
‹
‹
Motion and Pressure Control of Electro-Hydraulic Systems
Precision Motion Control of Linear Motors
Dynamic friction compensation
ƒ Conclusions
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
50
Specific Design Issues
ƒ Means to achieve fast robust feedback
ƒ Learning techniques (e.g., parameter adaptation)
to reduce model uncertainties for an improved
performance
ƒ Desired compensation to alleviate the effect of
measurement noises
ƒ Direct/Indirect and Integrated ARC designs
ƒ Extensions
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
51
Maximizing Disturbances Attenuation
ƒ General Means :
High
- gain feedback to raise the closed
- loop
bandwidth
ƒ Problems:
May run into Control Saturation Problem during
large transients caused by sudden changes of
large command inputs
ƒ Solutions:
Separating the achievable closed- loop bandwidths
to command inputs and disturbances
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
52
Respect System Dynamics
ƒ Design Principle
Only ask system to track feasible reference trajectories
ƒ Solutions
On-line reference trajectory generation
and
Physical model based nonlinear compensation !
ωd
eω = ω − ω d
ω
ω in p u t , Command Input
ω d , Generated reference Input
eω , Tracking error
t
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
53
Nonlinear Feedback Gains for Improved Transient
Performance and Better Trade-off in Meeting
Various Needs
Robust feedback
us
High gain feedback for a better
disturbance rejection to improve
transient performance
tracking error e
Moderate feedback gain for a better
trade-off between noise attenuation
and disturbance rejection
Small feedback gain to avoid control
saturation and instability
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
54
Specific Design Issues
ƒ Means to Achieve Fast Robust Feedback
ƒ Learning techniques (e.g., parameter adaptation)
to reduce model uncertainties for an improved
performance
ƒ Desired compensation to alleviate the effect of
measurement noises
ƒ Direct/Indirect and Integrated ARC designs
ƒ Extensions
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
55
Controlled Learning Process
Use prior knowledge such as physical bounds of
parameter variations to achieve a controlled learning
process; this helps get rid off the destabilizing effect
of on
- line learning and enable a fast adaptation loop
to be used for a better performance !
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
56
Specific Design Issues
ƒ Means to Achieve a Fast Robust Feedback
ƒ Learning techniques (e.g., parameter adaptation)
to reduce model uncertainties for an improved
performance
ƒ Desired compensation to alleviate the effect of
measurement noises
ƒ Direct/Indirect and Integrated ARC designs
ƒ Extensions
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
57
DISCONTINUOUS PROJECTION BASED ARC DESIGN
θ
θˆ = Pr ojθˆ (Γ ϕ ( x ) z )
Modified Adaptation Law
us2
us 2 ( z , t )
Nonlinear Robust Feedback
us
+
u s1
−k
Linear Stabilizing Feedback
x d − ϕ ( x ) θˆ
T
xd
uf
Adjustable
Model Compensation
+
u
∆
x = ϕ ( x )T θ + ∆ + u x -+
Plant
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
z
xd
58
DESIRED COMPENSATION ARC DESIGN
θ
θˆ = Pr ojθˆ (Γ ϕ ( x d ) z )
Modified Adaptation Law
us2
us 2 ( z , t )
Nonlinear Robust Feedback
us
+
u s1
− k s1
Linear Stabilizing Feedback
x d − ϕ ( x d ) θˆ
T
xd
uf
Adjustable
Model Compensation
+
u
∆
x = ϕ ( x )T θ + ∆ + u x -+
Plant
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
z
xd
59
Desired Compensation ARC Structure
‹ Reducing
the effect of measurement noise
) Regressor does not depend on measurements
‹ Fast
adaptation rate in implementation
‹ An
almost total separation of robust control law
design and parameter adaptation design; this
facilitates controller gain tuning process
considerably
‹ Off-
line calculation of regressors
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
60
Specific Design Issues
ƒ Means to Achieve a Fast Robust Feedback
ƒ Learning techniques (e.g., parameter adaptation)
to reduce model uncertainties for an improved
performance
ƒ Desired compensation to alleviate the effect of
measurement noises
ƒ Direct/Indirect and Integrated ARC designs
ƒ Extensions
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
61
Specific Design Issues
ƒ Means to Achieve a Fast Robust Feedback
ƒ Learning techniques (e.g., parameter adaptation)
to reduce model uncertainties for an improved
performance
ƒ Desired compensation to alleviate the effect of
measurement noises
ƒ Direct/Indirect and Integrated ARC designs
ƒ Extensions
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
62
MIMO Semi-Strict FEEDBACK FORM I
i- th Subsystem
xi = fi 0 ( χi , t ) + Fi ( χi , t )θ + Bi ( χi ,θ , t ) xi+1,mi + Di ( χi , t ) ∆i
ηi = Φi0 ( χi , t ) + Φ1i ( χi , t )θ
1 ≤ i ≤ r -1
r- th Subsystem
xr = M−1 ( χi , β, t )  fr0 ( χr , t ) + Fr ( χr , t )θ + Fβ β + Bi ( χr ,θ, β, t ) u + Dr ∆r 
ηr =Φr ( χr ,θ,t )
Output
y = yr =
[
]
T
T T
y1b , … , y rb
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
63
Inputs
χ1
χ2
Outputs
∆1
1-th
SUBSYSTEM
∆2
.
.
u
...
∆r
r-th
SUBSYSTEM
xr
.
.
.
x1 y1b
U2
2-th
SUBSYSTEM
x2
T
N2
y2b
Ur
T
Nr
yrb
MIMO Semi-Strict FEEDBACK FORM I
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
64
ASSUMPTIONS
•
Bi is nonsingular and can be linearly parametrized
•
M is an s.p.d. matrix and can be linearly parametrized
•
The η i − su b system is BIBS stable w.r.t. input
•
There exist known functions δ i ( χ i , t )
∆ i ( χ ,θ , u , t ) ≤ δ i ( χ i , t ) ,
( χi −1 , xi )
such that
i = 1, … , r
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
65
DIFFICULTIES
•
High Relative Degrees
•
Mismatched Uncertainties
Uncertainties do not enter the system in the same channel
as control inputs
•
Coupling and Appearance of Parametric Uncertainties
in the Input Channels of Each Layer
•
Last Layer’s State Equations Cannot be Linearly
Parametrized
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
66
Inputs
χ1
χ2
∆1
u1b
∆2
u2b
...
∆r
urb
Outputs
.
.
r-th
SUBSYSTEM
xr
2-th
SUBSYSTEM
x2
1-th
SUBSYSTEM
x1 y
.
...
MIMO Semi-Strict FEEDBACK FORM II
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
67
Extensions
ƒ Output Feedback ARC of Uncertain Linear Systems with Disturbances
- Need observers for state estimates
ƒ Observer Based ARC of a Class of Nonlinear Systems with Dynamic
Uncertainties
- Partial state feedback
ƒ Adaptive Robust Control without Knowing Bounds of Parameter
Variations
- Use fictitious bounds for a controlled learning process
ƒ Neural Network Adaptive Robust Control
- Integrate the universal approximation capability of neural
networks into ARC design for general nonlinearities
ƒ Adaptive Robust Repetitive Control
ƒ For periodic unknown disturbances and repetitive tasks
APPLICATIONS
ƒ Precision Motion Control of High Speed Machine Tools and
Linear Motors
Combined the design technique with digital control
ƒ Control of Electro-Hydraulic Systems
Hydraulic Servo-systems; Hydraulic Excavators
ƒ Trajectory Tracking Control of Robot Manipulators
ƒ
Motion and Force Control of Robot Manipulators
(a) in contact with a stiff surface with unknown stiffness
(b) in contact with a rigid surface
ƒ Coordinated Control of Multiple Robot Manipulators
ƒ Ultra-Precision Control of Piezo-electrical Actuators; Harddisk Drives; …
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
69
Electro-Hydraulic Experimental Setup
Electro-Hydraulic Experimental Setup
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
71
Non-Circular
Track Profile
Eccentricity
External Vibrations
Flow-induced
Aero-elastic
Forces
Pivot
Friction
Hard-Disk Drive Experimental Setup
OUTLINE
ƒ Motivation
ƒ General Design Philosophy and Structure
ƒ Specific Design Issues
ƒ Application Examples
‹
Motion and Pressure Control of Electro-Hydraulic Systems
‹
Precision Motion Control of Linear Motors
‹
Dynamic Friction Compensation
ƒ Conclusions
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
73
Linear Motion
Table
Gear
Leadscrew
Rotary Servo Motor
FLUX DENSITY (B)
COILCOIL
ASSEMBLY
ASSEMBLY
CURRENT
(I)(I)
CURRENT
FORCE (F=IxBf)
FORCE (F)
MAGNET ASSEMBLY
MAGNETIC
ATTRACTION (Fa)
Linear Motor vs. Rotary Motor
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
74
Advantages of Linear Motor Drive Systems
ƒ Mechanical simplicity (no mechanical transmission
mechanisms), higher reliability, and longer lifetime
ƒ No backlash and less friction, resulting in the
potential of having high load positioning accuracy
ƒ No mechanical limitations on achievable acceleration
and velocity
ƒ Bandwidth is only limited by encoder resolution,
measurement noise, calculation time, and frame
stiffness
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
75
Difficulties in Control of Linear Motors
ƒ Model Uncertainties
Parametric uncertainties (e.g., load inertia)
Discontinuous disturbances (e.g., Coulomb friction);
external disturbances (e.g., cutting force)
ƒ Drawback of without mechanical transmissions
Gear reduction reduces the effect of model uncertainties
and external disturbance
ƒ Significant uncertain nonlinearities due to position
dependent electro
- magnetic force ripples (e.g. iron
core linear motors)
ƒ Implementation issues (e.g., measurement noise)
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
76
Mathematical Model
x1 = x 2
M x 2 = u − B x 2 − F fn ( q ) − F r ( q ) + F d
y = x1
where
x 1 : position
x 2 : velocity
y : output
Ffn : nonlinear friction
Fd : lumped disturbance
M : mass of load
u : input voltage
B : viscous friction const
F r : force ripple
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
77
Model of Friction Force
ƒ F fn is discontinuous at zero velocity
Ffn
Af
V(m/s)
Ffn
ƒ
A continuous friction model
F fn
is used to approximate
F fn
F fn ( x 2 ) = A f S f ( x 2 )
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
78
Force ripple
„ Cogging force is a magnetic force developed due to the
attraction between the permanent magnets and the cores of
the coil assembly. It depends only on the relative position
of the motor coils with respect to the magnets, and is
independent of the motor current
„ Reluctance force is developed due to the variation of self
inductance of the windings, which causes a position
dependent force in the direction of motion when current
flows through the coils
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
79
P
0.06
Fo rce ripple (v)
0.04
0.02
0
-0.02
-0.04
-0.06
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
P o sitio n (m )
Measurement of Force Ripple (X-axis)
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
80
Model of Ripple Forces
„ The permanent magnets of the linear motor are identical and
are equally spaced at a pitch of P :
F r ( x1 + P ) = F r ( x1 )
„ It can be approximated quite accurately by the first several
harmonics, which is denoted as Fr ( x1 ) and represented by
F r ( x 1 ) = A rT S r ( x 1 )
where
A r = [ A r1s , A r1c ,
S r ( x1 ) = [sin(
, A rqs , A rqc ] T
2π
2π
x1 ), cos(
x1 ),
P
P
, sin(
2π q
2π q
x1 )] T
x1 ), sin(
P
P
and q is the numbers of harmonics used to approximate Fr ( x1 )
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
81
ARC Controller Design Model
M x 2 = u − Bx 2 − A f S
⇒
where
f
− A rT S r + d
d = ( F fn − F fn ) + ( F r − F r ) + ∆
x1 = x 2
⇒
θ 1x2 = u − θ 2 x2 − θ 3S f − θ
T
4b
~
S r ( x1 ) + θ 5 + d
y = x1
where
θ1 = M ,
θ 2 = B,
θ3 = Af ,
θ5 = dn,
~
d = d − dn
θ 4b = Ar
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
82
Assumption and Control Objective
„
Assumption:
θ
d
•
∈ Ω θ = {θ :
∈ Ω d = {d :
θ min < θ < θ max }
| d | ≤ δ d ( x,t ) }
Objective:
Synthesize a control input u such that the output y track the
reference motion trajectory yr as closely as possible
yr(t)
e(t)
y
t
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
83
ARC Controller Design
„
Define a switching
- function
- like quantity as
p = e + k1e = x 2 − x 2 eq ,
x 2 eq = y d − k1e
where
e = y − y d (t )
„
Error dynamics
Mp = u + ϕ T θ + d
where
ϕ T = [ − x 2 eq , − x 2 , − S f ( x 2 ), − S r ( x1 ),1]
x 2 eq = y d − k1e
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
84
ARC Controller Design (cont’d)
„
ARC control law
ua = −ϕ Tθˆ
u = ua + u s ,
u s = u s1 + u s 2 ,
„
u s1 = − k 2 p
Error dynamics
Mp + k2 p = us 2 − ϕ Tθ + d
„
Choose robust control us 2 such that
where
„
ε
(
)
i
~
p us 2 − ϕ θ + d ≤ ε
ii
pu s 2 ≤ 0
T
~
is a design parameter
Example
us 2 = −
1
( θ max − θ min ⋅ ϕ + δ d ) 2 p
4ε
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
85
Performance of ARC
If the adaptation function τ is chosen as τ = ϕp , then the
ARC control law guarantees that
„
In general, all signal are bounded. Furthermore, the positive
definite function Vs defined by
Vs =
is bounded by
where
2
Vs ≤ exp( −λt )Vs (0) +
λ = 2 k 2 θ 1 max
p2(t)
1
Mp
2
ε
[1 − exp( −λt )]
λ
upper bound of p2(t)
p 2 (∞ ) ≤
p2(t)
t
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
2ε
λM
86
Performance of ARC (cont’d)
•
In addition, zero final tracking error is achieved in the
presence of parametric uncertainties only
~
)
(i.e., d = 0 , ∀ t ≥ t 0
p2(t)
upper bound of p2(t)
p 2 (∞ ) = 0
t
p2(t)
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
87
Implementation Issues
„
„
Regressor ϕ has to be calculated on-line based on the actual
measurement of the velocity x2
ƒ
The effect of velocity measurement noise may be severe
ƒ
Slow adaptation rate has to be used
Model compensation ua depends on the actual feedback of the state
ƒ
Creates certain interactions between the model compensation
ua and the robust control u s
ƒ
Complicates the controller gain tuning process
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
88
Desired Compensation ARC (DCARC)
„
DCARC control law
u = ua + u s ,
ua = −ϕ d Tθˆ
τ = ϕd p
where
ϕ d T = [ − y d , − y d , − S f ( y d ), − S r ( y d ),1]
„
Error dynamics
M p = u s − ϕ d T θ + (θ 1 k1 − θ 2 ) e + θ 3 [ S f ( y d ) − S f ( x 2 )] + θ 4Tb [ S r ( y d ) − S r ( x1 )] + d
addition
„
terms
Notice that (applying Mean Value Theorem)
S f ( y d ) − S f ( x2 ) = g f ( x2 , t )e,
where
g f ( x2 , t )
S r ( x1 ) − S r ( y d ) = g r ( x 2 , t ) e
and g r ( x 2 , t ) are certain nonlinear functions
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
89
DCARC Controller Design (cont’d)
„
Robust control function
u s = u s1 + u s 2 ,
u s1 = − k s1 p
where k s 1 is a nonlinear gain such that the matrix A defined
below is p.d.

 k s1 − k 2 − θ 1 k 1 + θ 2 + θ 3 g f
A = 
 − 1 (k θ + k θ g − θ T g )
f
r
1 2
1 3
4b
 2
−
1
( k 1θ 2 + k 1θ 3 g f − θ 4Tb g r
2
1
M k 13
2

)



For example
k s1 ≥ k 2 + θ 1 k1 − θ 2 − θ 3 g f +
„
Choose u s 2 such that
i
ii
(
1
T
2
(
θ
k
+
θ
k
g
+
θ
g
)
2 1
3 1 f
4b r
2θ 1 k13
)
~
T ~
p us2 − ϕ d θ + d ≤ ε
pu s 2 ≤ 0
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
90
Performance of DCARC
If the DCARC law is applied, then
„
In general, all signal are bounded. Furthermore, the positive
definite function Vs defined by
Vs =
is bounded by
1
1
2
Mp 2 + Mk 1 e 2
2
2
Vs ≤ exp( −λt )Vs (0) +
where λ = min{ 2k 2
„
θ 1max , k1}
ε
[1 − exp( −λt )]
λ
In addition, zero final tracking error (i.e., asymptotic tracking) is
achieved in the presence of parametric uncertainties only
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
91
Implementation of DCARC Adaptation Law
„
Digital implementation of the adaptation law
θ i min

θˆi [( j + 1) ∆ T ] = θ
i max
 Θ
i

if
if
θˆi = θˆimax
θˆi = θˆimin
and
•>0
and
•<0
otherwise
where
( j +1 ) ∆ T
ˆ
Θ i = θ i ( j∆ T ) + γ i ∫
ϕ d ,i ( e + k1e ) dt
j∆ T
„
Velocity
- free implementation of adaptation law
( j+1)∆T
( j+1)∆T
( j+1)∆T

ˆ
Θi =θi ( j∆T) +γ i k1∫
ϕd,iedt+ϕd,ie j∆T − ∫
ϕd,i (t)edt
j∆T
 j∆T

Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
92
X-Y Linear Motor Driven Positioning Stage
Pentium II 400
MATLAB/Simulink
Real-Time Workshlp
Y-axis
A/D,I/O
MTRACE
Real-Time Interface
Complier
A/D, D/A,
Digital I/O
PWM
Amplifiers
MLIB
Main Processor
Encoder Signals
X-axis
Encoder
Interface
Encoder Singals
5X multiplication
DS1103
ControlDesk
Hall Sensors
Host PC
Servo Controller
X-Y Stage (Driven by two linear motors)
Experimental Setup
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
94
Performance Indexes
e M = max
„
Transient Performance:
„
Final Tracking Accuracy: e F = max
{e (t ) }
t
[
t∈ T f − 2 , T f
„
Average Tracking Performance: L 2 [ e ] =
„
Average Control Input:
„
Degree of Control Chattering:
cu =
∑
L 2 [∆ u ]
L 2 [u ]
,
L 2 [u ] =
L 2 [∆ u ] =
1
N
1
Tf
∫
T
0
f
]{e ( t ) }
1
Tf
∫
T
0
f
e
2
dt
u (t ) dt
2
N
∑ u ( j ∆ T ) − u ( ( j − 1) ∆ T )
2
j =1
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
95
Comparative Experimental Results (Y-axis)
„
PID with model compensation:
u = θˆ1 (0) yd (t ) + θˆ2 (0) yd (t ) + θˆ3 (0) S f ( y ) − K p e − K i ∫ edt − K d e
S f ( y) =
„
„
2
π
arctan(900 y ), K p = 5.4 × 10 3 , K i = 5.4 × 10 5 , K d = 18
ARC:
DRC:
k1 = 400, k 2 = 32, θˆ (0) = [0.05,0.25,0.1,0]T
Γ = diag [5,0,2,1000 ]
• Same controller law with ARC but without parameter adaptation
„
DCARC:
k1 = 400, k2 = 32, θˆ(0) = [0.05,0.25,0.1,0]T
Γ = diag[25,0,5,1000]
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
96
Tracking Sinusoidal Trajectory
„
Set 1: To test the nominal tracking performance, the
motors are run without payload
„
Set 2: To test the performance robustness of the
algorithms to parameter variations, a 20lb payload is
mounted on the motor
„
Set 3: A large step disturbance (a simulated 0.5V
electrical signal) is added at about t=2.5s and removed
at t=7.5s to test the performance robustness of each
controller to disturbance
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
97
60
PID
40
20
0
-20
0
1
2
3
4
5
6
7
8
9
10
6
7
8
9
10
6
7
8
9
10
6
7
8
9
10
60
DRC
(µ m)
40
20
0
-20
Tracking Erro r
0
1
2
3
4
5
60
ARC
40
20
0
-20
0
1
2
3
4
5
60
40
DCARC
20
0
-20
0
1
2
3
4
5
Time (sec)
Tracking errors for yr=0.05sin(4t) without load (Y-axis)
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
98
50
0
PID
-50
0
1
2
3
4
5
6
7
8
9
10
6
7
8
9
10
6
7
8
9
10
6
7
8
9
10
(µ m)
50
0
DRC
-50
T racking Erro r
0
1
2
3
4
5
50
0
ARC
-50
0
1
2
3
4
5
50
0
DCARC
-50
0
1
2
3
4
5
Time (sec)
Tracking errors for yr=0.05sin(4t) with 20lb load (Y-axis)
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
99
50
0
PID
-50
0
1
2
3
4
5
6
7
8
9
10
6
7
8
9
10
6
7
8
9
10
6
7
8
9
10
(µ m)
50
0
DRC
-50
T racking Erro r
0
1
2
3
4
5
50
0
ARC
-50
0
1
2
3
4
5
50
0
DCARC
-50
0
1
2
3
4
5
Time (sec)
Tracking errors for yr=0.05sin(4t) with disturbance (Y-axis)
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
100
180
160
140
120
Set1
Set2
Set3
100
80
60
40
20
0
PID
DRC
ARC
DCARC
Transient Performance eM (Y-axis)
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
101
25
20
Set1
Set2
Set3
15
10
5
0
PID
DRC
ARC
DCARC
Final Tracking Accuracy eF (Y-axis)
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
102
0.45
0.4
0.35
0.3
Set1
Set2
Set3
0.25
0.2
0.15
0.1
0.05
0
PID
DRC
ARC
DCARC
Control Effort L2[u] (Y-axis)
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
103
0.9
0.8
0.7
0.6
Set1
Set2
Set3
0.5
0.4
0.3
0.2
0.1
0
PID
DRC
ARC
DCARC
Degree of Control Chattering Cu (Y-axis)
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
104
Positsion (m)
0.4
0.3
0.2
0.1
0
0
1
2
3
4
5
6
7
8
9
10
6
7
8
9
10
6
7
8
9
10
T ime (sec)
Velocity (m/s)
1
0.5
0
-0.5
-1
0
1
2
3
4
5
Acceleration (m/s 2)
T ime (sec)
10
5
0
-5
-10
0
1
2
3
4
5
T ime (sec)
Point-to-Point Motion Trajectory
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
105
50
0
PID
T racking E rro r (µ m)
-50
0
1
2
3
4
5
6
7
8
9
10
6
7
8
9
10
6
7
8
9
10
50
0
DRC
-50
0
1
2
3
4
5
50
0
DCARC
-50
0
1
2
3
4
5
T ime (sec)
Tracking errors for point-to-point trajectory (Y-axis)
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
106
Conclusion
„
Unique Features of Nonlinear Adaptive Robust
Control Strategy:
• Nonlinear physical model based; easy to incorporate physical
intuition into the controller design stage.
• Address nonlinearities associated with linear motor dynamics
directly.
• Address parameter variations due to change of load, nominal
value of disturbances, …
• Effectively handle the effect of hard-to-model terms (e.g.,
uncompensated friction)
• Achieve high performance through the use of learning techniques
such as on-line parameter estimation.
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
107
Conclusion (cont’d)
ƒ Comparative Experimental Results
• Illustrate the above claims
• The proposed schemes outperform a PID controller with
model compensation significantly in terms of output tracking
accuracy
• Tracking errors for high-speed/high-acceleration movements
are very small, even during transient period. Final tracking
errors are mostly within measurement resolution level of 1µ m
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
108
Acknowledgements
The project is supported in part by
National Science Foundation
-- CAREER grant CMS
- 9734345
- - Program Manager: Dr. Alison Flatau
Purdue Research Foundation
Caterpillar Inc.
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
109
More Information can be downloaded from:
http://widget.ecn.purdue.edu/~byao
Copyright by Bin Yao, School of Mechanical Engineering, Purdue University
110