control of nonlinearsystems based on gaussianprocess models · control of nonlinearsystems based on...
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Control of Nonlinear Systemsbased on Gaussian Process
Models
Juš KocijanJožef Stefan Institute, Ljubljana, Slovenia
&University of Nova Gorica, Nova Gorica, Slovenia
Workshop Modellbasierte Kalibrierverfahren fuer Automotive-Systeme, Vienna University of Technology, Vienna, Austria
GP model
•Probabilistic (Bayes) nonparametric model.
•Prediction of the output based on similarity test input – training inputs
•Output: normal distribution �Predicted mean
�Prediction variance
µµµµ-2σσσσ µµµµ+2σσσσµµµµ
GP model attributes (vs. e.g. ANN)
• Smaller number of parameters
• Measure of confidence in prediction, depending on data
• Easy to use (engineering practice)
• Incorporation of prior knowledge *
• Data smoothing
• Computational cost increases with amount of data ⇑
• Recent method, still in development
• Nonparametrical model
* (also possible in some other models)
Applications and domains of use
� dynamic systems modelling
� time-series prediction
� dynamic systems control
� fault detection
� smoothing
� chemical engineering and process control
� biomedical engineering
� biological systems
� environmental system
� power systems and engineering
� motion recognition
� traffic
Dynamic systems control with GP
• Model based predictive control� Internal model control
� General model based predictive control
� Explicit model based predictive control
• Gain-scheduling control
Basics of predictive control
•At every k: calculation ofprediction
•We set
• forminimize
•For control it is used only
Moving horizont strategy
$ ( )y k j+ j N N= 1 2,...,
r k j( )+∆u k j( )+
uNj ...,0=
∆u k( )
k k+N1 k+Nu k+N2preteklost sedaj prihodnost
w
r
u
yJ r k j y k j u k jj N
N
j
Nu
= + −∑ + + +∑= =
−( ( ) $ ( )) ( ( ))
1
2 2 2
0
1β ∆
General model based predictive control
• General model based predictive control principle
• Cost function (PFC)
• constraints on input signal, input signal rate, statesignals, state signals rate and
• constrained optimisation – SAFE CONTROL
2
)()](ˆ)([min PkyPkrJ
kU+−+=
vkPky ≤+ )(ˆvar
Optimisationalgorithm
Referencegenerator
Process
Model Model
w r
yy u
u
n
y
+ +_ _+
+
+
_
J. Kocijan and R. Murray-Smith. Nonlinear predictive control with Gaussianprocess model.In Switching and Learning in FeedbackSystems, volume 3355 of Lecture Notes in Computer Science, Pages 185-200. Springer, Heidelberg, 2005.
B. Likar and J. Kocijan. Predictive control of a gas-liquid separation plant based on a Gaussian process model. Computers and Chemical Engineering, Volume 31, Issue 3, Pages 142-152, 2007.
Dynamic system identification andmodel simulation
•Why does identification of dynamicsystems seem more complex thanmodelling of static functions?
•Simulation
�“naive” ... m(k)
�with propagation
m(k),v(k)
»Analitic app.
>Taylor app.
>“exact”
»MC Monte Carlo
with mixtures
Z
Z
Z
Z
Z
- L
- L
- 1
- 2
- 1
.
.
.
.
.
.
.
.
.
u ( k - 2 )
u ( k - L )
M o d e l n a o s n o v i
G a u s s o v i h p r o c e s o v( m ( k ) , v ( k ) )
u ( k - 1 )
N
( m ( k - 1 ) , v ( k - 1 ) )
( m ( k - 2 ) , v ( k - 2 ) )
( m ( k - L ) , v ( k - L ) )
N
N
N
GP model
pH process: control results –unconstrained case
0 500 1000 1500 2000 2500 3000 3500 4000 4500 50003
4
5
6
7
8Plant output (full line), set-point (dashed line), 95% confidence interval (grey)
Time [sec]
pH
0 500 1000 1500 2000 2500 3000 3500 4000 4500 50008
10
12
14
16Input
Time [sec]
Q3
pH process: control results –unconstrained case
0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000
0.05
0.1
0.15
0.2
0.25Standard deviation
Time [sec]
pH process: control results –constrained case (constraint on variance only)
0 500 1000 1500 2000 2500 3000 3500 4000 4500 50003
4
5
6
7
8Plant output (full line), set-point (dashed line), 95% confidence interval (grey)
Time
0 500 1000 1500 2000 2500 3000 3500 4000 4500 50008
10
12
14
16Input
Time
pH process: control results – constrainedcase (constraint on variance only)
0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000
0.05
0.1
0.15
0.2Standard deviation
Time
Next step: Explicit Nonlinear Predictive Control Based on
Gaussian Process Models
Parametric programming approachesfor explicit constrained nonlinear MPCParametric programming approaches
for explicit constrained nonlinear MPC
Convex problemsConvex problems Non-convex problemsNon-convex problems
Local approximation
with mp-QP
Local approximation
with mp-QP
Approximatemp-NLP 1
(computations at the vertices of X0)
Approximatemp-NLP 1
(computations at the vertices of X0)
Approximate mp-NLP 2(computations at the vertices and several interior points of X0)
Approximate mp-NLP 2(computations at the vertices and several interior points of X0)
A. Grancharova, J. Kocijan and T. A. Johansen.Explicit stochastic predictive control of combustion plants based on Gaussian process models. Automatica, Volume 44, Issue 6, Pages 1621-1631, 2008.
0 20 40 60 80 100 120 140 160 180 200-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.895% confidence interval of x(t) and set point
time instants
-1-0.5
00.5
1
-0.5
0
0.5
-1
-0.5
0
0.5
1
x1
x - space
x2
x 3
State space partition of the explicit reference tracking GP-NMPC controller
The 95% confidence interval of the state variable predicted with the Gaussian process model
Incorporation of local linear models(LMGP model)
• Derivative of function observed besidethe values of function
• Derivatives are coefficients of linearlocal model in an equilibrium point (prior knowledge)
• Covariance function to be replaced; theprocedure equals as with usual GP
• Very suited to data distribution that canbe found in practice
K. Ažman, J. Kocijan.Non-linear model predictive control for models with local information and uncertainties. Trans. Inst. Meas. Control, 2008, vol. 30, no. 5, 371-396.
Gain-scheduling control
• Controller parameters varying based on changes of process model parameters
• Parameter varying model
• Fixed-structure Gaussian process model
Fixed-Structure GP (FSGP) model
• GP models of varying parameters
• Works well with relatively small number of linear local models
• The mechanisms for blending and scheduling of local models are joined together.
• The selection of scheduling variables -the inputs of GP models - the relevance detection capability of GP models.
• Control design methods based on parametric process models.
Example
)()(1
)()( 3
2tu
ty
tyTty +
+=+
00.5
11.5
22.5
0
5
10
15
200
5
10
15
u(t)
Nonlinear system and equilibrium curve
y(t)
y(t+
T)
System:
Two Regions:well modelled: 1.25<u<2.35Not well modelled: 0<u<1.25
Trained GP models
-1.5
-1
-0.5
0
0.5
1
a
Prediction of parameter a=a(y)
-2 0 2 4 6 8 10 12 14 160
0.2
0.4
y
2 σa
pred ± 2σtraining pts
true
prediction
-10
0
10
20
30
40
50
b
Prediction of parameter b=b(u)
-2 -1 0 1 2 3 40
2
4
u
2 σb
pred ± 2σtraining pts
true
prediction
Closed-loop response of gain-scheduling control based on FSGP model
0 20 40 60 80 100 1200
2
4
6
8
10
Time [sec]
Closed-loop response
0 20 40 60 80 100 1200
0.05
0.1
σ a
Time [sec]
0 20 40 60 80 100 1200
0.2
0.4
σ b
Time [sec]
set-point
response
K. Ažman, J. KocijanFixed-structure Gaussian process model.International Journal of Systems Science. Volume 40, Issue 12, Pages 1253–1262.
Application of GP models for fault diagnosis and detection
• Is the fault diagnosed because of the fault occurance or because model is not OK?
Dj. Juričić and J. Kocijan. Fault detection based on Gaussian process model. In I. Troch and F. Breitenecker, editors, Proceedings of the 5th Vienna Symposium on Mathematical Modeling (MathMod), Vienna, 2006.
0 1000 2000 3000 4000 5000 6000 70002
4
6
8
10
12Plant response and model output
Time
0 1000 2000 3000 4000 5000 6000 70000
0.2
0.4
0.6
0.8
1
Standard deviation of the prediction error
Time
process
model mean
mean+2*std
mean-2*std
0 1000 2000 3000 4000 5000 6000 70000
0.5
1
1.5
2
2.5
Test SN
Time
0 1000 2000 3000 4000 5000 6000 70000
0.2
0.4
0.6
0.8
1Index I
Time
Conclusions
• The Gaussian process model is an example of a flexible, probabilistic, nonparametric model with inherent uncertainty prediction
• It is suitable for dynamic systems modelling and control design� Model based predictive control
� Gain-scheduling control
• When to use GP model? � systems: nonlinearity, corrupted data (noise,
uneven distribution), uncertainty
� easy-to-use, measure of prediction confidence
� prior knowledge that can be used
�Engineering applications