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

Off-equilibrium measurements (vehicle longitudinal dynamic)

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