Performance oriented anti-windup for a Performance oriented anti-windup for a class of neural network controlled systemsclass of neural network controlled systems
G. HerrmannM. C. Turner and I. Postlethwaite
Control and Instrumentation Research GroupUniversity of Leicester
SWAN 2006SWAN 2006 -
Automation and Robotics Research Institute, UTA
Anti-windup for a class of neural network controlled systems2
1. Motivation
2. The plant: A linear plant with matched unknown non-linearities
3. The nominal control system: Linear Control with augmented NN-controller for disturbance rejection
4. Controller conditioning for anti-windup:
– Preliminaries: Constrained multi-variable systems
– Non-linear Controller Conditioning
– Linear Controller Conditioning
5. An Example
6. Conclusions
Anti-windup for a class of neural network controlled systems3
Unknown Nonlinearity
++
MotivationMotivation
LinearPlant
LinearController +
NNcompen-
sation
-
Adap-tation
NN-Control- Examples :S. S. Ge, T. H. Lee, and C. J. Harris, Adaptive Neural Network Control of Robotic Manipulators. World Scientific, Singapore, 1998.
Y. Kim and F.L. Lewis, High-Level Feedback Control with Neural Networks," World Scientific, Singapore, 1998.
??Cu u
Linear control performance in combination with NN-control – Examples of practical validation:G. Herrmann, S. S. Ge, and G. Guo, “Practical implementation of a neural network controller in a hard disk drive,” IEEE Transactions on Control Systems Technology, 2005. ——, “A neural network controller augmented to a high performance linear controller and its application to a HDD-track following servo system,” IFAC 2005 (under journal review).
Anti-Windup (AW)(AW) Control - a possible approach to overcome controller saturationG. Grimm, J. Hatfield, I. Postlethwaite, A. R. Teel, M. C. Turner, and L. Zaccarian, “Antiwindup for stable linear systems with input saturation: An LMI based synthesis,” IEEE Trans. on Autom. Control, vol. 48, no. 9, pp. 1509–1525, 2003.Alternative for NN:W. Gao; R.R. Selmic, "Neural network control of a class of nonlinear systems with actuator saturation Neural Networks", IEEE Trans. on Neural Networks, Vol. 17, No. 1, 2006.
Anti-windup for a class of neural network controlled systems4
LinearPlant
LinearController
+ -
LinearAW-Compen-
sator
Cu u
Motivation: Principle of anti-windup compensationMotivation: Principle of anti-windup compensation
Anti-windup for a class of neural network controlled systems5
The plantThe plant
,
,
);(
pyu
Pp
ppPpP
nnn
xCy
yfBuBxAx
Stable, minimum-phase, strictly proper with matched nonlinear disturbance f(y)
upp nBCrank )(
Anti-windup for a class of neural network controlled systems6
The plantThe plant
*W - optimal (constant) weight matrix
)(s - neural network basis function vector,
- neural network modelling error
so that it can be arbitrarily closely modelled by a neural network approach:
;)()( * ysWyfT ;lssT fKK
The disturbance is continuous in y and bounded:
fKyf )(
Anti-windup for a class of neural network controlled systems7
The Nominal Controller – Linear Control ComponentThe Nominal Controller – Linear Control Component
,
;
,
,
dDyDxCu
dByBxAx
dCCCCL
dCCCCC
PCPPCP
PCCCL CDBACB
CBAA is assumed to be Hurwitz stable
d - exogenous demand signal
The linear controller component defines the closed loop steady state:
dDB
BACy
dCP
dCCLPd
,
,10 dDB
BA
x
x
dCP
dCCL
dP
dC
,
,1
,
,
dPP
dCC
P
C
xx
xx
x
x
,
,
~
~and the controller error:
Anti-windup for a class of neural network controlled systems8
The Nominal Controller – Non-Linear Control ComponentThe Nominal Controller – Non-Linear Control Component
LNL uuu
Estimation algorithm: )(0000ˆdii yyNs(y)ΓW
lliΓ R is symmetric, positive definite Learning Coefficient Matrix
, ]ˆˆˆ[ˆ21 unWWWW ][ 31
unWWWW
unWWWWW
~~~ˆ21 - Estimation error
estimate -compensatesfor non-linearity
discontinuous sliding mode component - compensates for modeling error
NMatrix is a design parameter
)(
)( )(ˆ
d
dTNL yyN
yyNKysWu
Anti-windup for a class of neural network controlled systems9
The Nominal ControllerThe Nominal Controller
dPP
dCC
P
C
xx
xx
x
x
,
,
~
~
can asymptotically track the signal yd so that the controller error:
becomes zero.
unWWWWW
~~~ˆ21
The estimation error
remains bounded.
Anti-windup for a class of neural network controlled systems10
+
NNcompen-
sation
-
Unknown Nonlinearity
++
Controller conditioningController conditioning
LinearPlant
LinearController
Adap-tation
Cu u
Non-linear
Algorithm
Linear AW-comp.
+ -
Anti-windup for a class of neural network controlled systems11
Controller conditioning - PreliminariesController conditioning - Preliminaries
Multi-variable Saturation Function:
)(
)(
)(
,
1,
,
11
uununnuu
uu
uu
usat
usat
uSat )),,max(min()(
, iiiiuuuuuusat
ii
iu
iu
iu
iuSymmetric Multi-variable Saturation Function:
)()( :,
uSatuSatuuuuu
The Deadzone - Counter-part of a Saturation Function:
)()(
);()(,,
uSatuuDZ
uSatuuDZ
uu
uuuu
Anti-windup for a class of neural network controlled systems12
Controller conditioning - Controller conditioning - AssumptionsAssumptions
+
NNcompen-
sation
-
Unknown Nonlinearity
++
LinearPlant
LinearController
Adap-tation
Saturation Limit:Saturation Limit: Cu u
);( CuuSatu
unu
u
u 1
Disturbance LimitDisturbance Limit
)(min,...,1
ini
f uKu
The controller amplitude is large enough to compensate for the unknown non-linearity.
Permissible Range of Tracking Control SystemPermissible Range of Tracking Control System
dDdDB
BACDCu dC
dCP
dCCLPCCdL ,
,
,1,
We do not assume that
the transient behaviour has to satisfy this constraint.0 small design parameter );(0 ,)1( dLKu
uDZf
Anti-windup for a class of neural network controlled systems13
Controller conditioning – Controller conditioning – Non-linear Control ElementNon-linear Control Element
)(
)(||)~(||1-
))~())(~((~
||)~(||
2
d
df
NLKfNLKf
fNLNL yyN
yyNK
NLuc
uDzKuDzK
Kuu
NLuc
ff
)(
)()(ˆ~
d
dTNL yyN
yyNKysWu
iidLKuidii WΓuDZyyNs(y)ΓNL
ucWf
ˆ)()(0000||)~(||ˆ,)1(
0 is a small design dependent constant
0||)~(|| ~ , ~ NL
ucuKu NLfNL
and replaced by a high gain controller. The NN-estimation algorithm is slowed down.The NN-controller is cautiously disabled
0)~( NLK uDzf
1||)~(|| ;~ NL
ucuu NLNL NN-control is usedNN-control is used
)1( fNL Ku
Anti-windup for a class of neural network controlled systems14
Controller conditioning – Controller conditioning – Linear Control ElementLinear Control Element
Linear controller
;)(
))((42
41
32
31
21
2
1
Luuuu
AA
uDZ
x
v
vx
NLNL
AW-compensator:
in practice 0
Note that ,0))(())((
NLLuuuuuuuSatDZ
NLNLThe control limits are satisfied
to be designed
,
,
dDyDxCu
dByBxAx
dCCCCCL
dCCCCCC
Closed Loop: ,)( ; ))(( NLLuuuuC uwuSatuyy
NLNL
compensation
,
;
2,
1,
wvdDyDxCu
vdByBxAx
dCCCCCL
dCCCCCC
withcompensation signals
Anti-windup for a class of neural network controlled systems15
Controller conditioning – Controller conditioning – AW-Compensator Design TargetAW-Compensator Design Target
+
NNcompen-
sation
-
Unknown Nonlinearity
++
LinearPlant
LinearController
Adap-tation
Linear AW-comp. + -
Cu u
Non-linear
Algorithm
++ -
wz
dy
Linear AW-comp.
where PzCz xCxCz ~ ~21 is a designer chosen performance output
Design target for linear Design target for linear AW-compensator:AW-compensator:
0, ;)()( 2
0
22
0
2
dsswdsszMinimize for
This L2-gain optimization target ensures recovery of the nominal controller performance.
Anti-windup for a class of neural network controlled systems16
Controller conditioning – Controller conditioning – AW-Compensator Design TargetAW-Compensator Design Target
The conditioned linear control uL term operating in connection with the constrained
NN-controller uNL, will track asymptotically any permissible steady state.
The NN-weight estimates will remain bounded.
Design target for overall AW-compensator:Design target for overall AW-compensator:
+
NNcompen-
sation
-
Unknown Nonlinearity
++
LinearPlant
LinearController
Adap-tation
Linear AW-comp. + -
Cu u
Non-linear
Algorithm
++ -d
y
Anti-windup for a class of neural network controlled systems17
A Simulation ExampleA Simulation Example
Simulation for a direct drive DC-torque motor[12] Hsieh & Pan (2000)
Hsieh & Pan (2000) [12]:
6-th order model to include issues of static friction, i.e. the pre-sliding behaviour:
650;=k
0.175;=C
;015.54=m
1
s
-4
The nominal model used for linear controller design
;1010
2
11
2
1 umx
x
m
C
m
kx
xs
Other parameters:
80000;=
50000;=k
2.5;=
454.5;=
4;=n
2
Assume both angle position x1 and angle velocity x2 are measurable
Anti-windup for a class of neural network controlled systems18
A Simulation ExampleA Simulation Example
10000
1041943101 57
CA
00
0134217728CB
32
0,dCB
107 101250.31045.7 CC 0101 9CD 0, dCD
Nominal linear Controller:
Nominal NN-Controller:
2
222
2
22
1
2
1
ˆ)ˆ()(
exp
cxcxx
xsi
Gaussian RadialBasis Function
0005.0
001.0
2422191714129742
25212016151110651
cccccccccc
cccccccccc
;3.0ˆ ;15.0ˆˆ
;0ˆ ;15.0ˆ ;3.0ˆ
25212016
151110651
cccc
cccccc
;1.0ˆ;3/001.0
Anti-windup for a class of neural network controlled systems19
A Simulation ExampleA Simulation Example
Saturation limit:5.1u
Conditioning of NN-Controller:
0005.0 ;1.0 ;3.1 KK f
1~xz
Linear AW-Compensator design:
Anti-windup for a class of neural network controlled systems20
A Simulation ExampleA Simulation ExampleControl signalPosition signal
0 0.02 0.04 0.06
-2
0
2
time
u
Unconstrained Response
0 0.02 0.04 0.06-3
-2
-1
0
1
2
3
u
time
Constrained Response
0 0.01 0.02 0.03 0.04 0.05 0.06-2
-1
0x 10
-4
x 1
time
Anti-windup for a class of neural network controlled systems21
A Simulation ExampleA Simulation Example
0 0.02 0.04 0.06-1
-0.8
-0.6
-0.4
-0.2
0x 10
-3
x 1
time
Control signalPosition signal
0 0.02 0.04 0.06-15
-10
-5
0
5
10
15
time
u
Unconstrained Response
0 0.02 0.04 0.06-15
-10
-5
0
5
10
15
u
time
Constrained Response
Anti-windup for a class of neural network controlled systems22
ConclusionsConclusions
I. Development of a conditioning method for a linear controller & robust NN-controller combination:1. Nominal NN-controller: Add-on to a linear controller for compensation of
matched unknown non-linearities/disturbances
2. Linear controller conditioning: Specially structured AW-controller (considering former results)
3. NN-controller conditioning: The unknown non-linearity is bounded and can be counteracted by a variable structure component; once the NN-controller exceeds the bound.
II. Design target: 1. Retain asymptotic tracking for permissible demands and keep NN-estimates
bounded
2. Optimization of linear AW-controller according to an L2-constraint
III. Simulation Result: Performance similar for un/conditioned controller
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