J. M. Lemos, J. Gir~ao, A. S. Silva and J. S. Marques · Continuous/discrete lter Prediction step:...

The Fokker-Planck equation in estimation and control

J. M. Lemos, J. Girao, A. S. Silva and J. S. Marques

INESC-ID, IST/Univ. Lisboa, Portugal

Work in the framework of project SPARSIS

3rd IFAC Workshop onThermodynamic Foundation of Mathematical Systems Theory

Louvain-la-Neuve, 3-5 July 2019

J. M. Lemos (INESC-ID, Portugal ) Fokker-Planck equation TFMST 2019 1 / 15

Motivation and objective

Systems composed of many dynamic agents (population of robots,internet users, ...) require probabilistic tools to describe their behaviour inorder to design control and estimation algorithms.These problems may be addressed using the Fokker-Planck equation(FPE).A technique to solve the FPE based on semigroup decomposition ispresented and used to solve problems:

Joint state-parameter estimation

Adaptive control

Tracking groups of targets


Presentation plan

Stochastic agents and the FPE;

FPE integration with semigroup decomposition;

Continuous/discrete filter


Joint state-parameter estimation

Adaptive control

Conclusions.


Stochastic agents and the FPE

Stochastic agents described by a stochastic differential equation

dxt = f (xt)dt + σdwt

The pdf p(x , t) of the state x at time t satisfies the Fokker-Planckequation (scalar case for simplicity)

∂p

∂t= −fx(x)p − f (x)

∂p

∂x+σ2

2

∂2p

∂x2

with initial and boundary condirions

p(x , 0) = px0(x), p(±∞, t) = 0, ∀t > 0


FPE integration with semigroup decomposition (1)

∂p

∂t= Lip

Differential operators

L1p = −fx(x)p L2p = −f (x)∂

∂xp L3p =

σ2

2

∂2p

∂x2

Solution of these 3 PDEs given by the integral operators

p(x , t + ∆) = T 1∆p(x , t) ≈ 1

1 + fx(x)∆p(x , t)

p(x , t + ∆) = T 2∆p(x , t) ≈ p(x − f (x)∆, t)

p(x , t + ∆) = T 3∆p(x , t) = p(x , t) ∗ G (x , ∆)



Approximate solution via Trotter’s formula

p(x , t + ∆) ≈ T 3∆T 2

∆T 1∆p(x , t)

Converges linearly with ∆



Probabilistic interpretation

Continuousstate equation

Discretestate equation

Fokker-Planck equation

Operators fordiscrete timepdf propagation

Discretize intime (1st order)

Discretize intime (operatorcomposition)

Propagate thea priori pdf ofthe state(discrete time)

Propagate the a priori pdf of the state(continuous time)


FPE integration with semigroup decomposition (4)Example: PLL error dynamics

state-20 -15 -10 -5 0 5 10 15 20

pdf

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

monte carlo simfokker planck solution

f (x) = ax − KPLLsin(x)


Continuous/discrete filter

Prediction step: Compute p(x(tk)|Y tk−1) (predicted pdf) bypropagating from time tk−1 until tk the pdf p(x(tk−1)|Y tk−1), usingthe FPE in the time interval [tk−1, tk ], taking as initial conditionp(x(tk−1)|Y tk−1).

Filtering step: Compute the filtered pdf at time tk using Bayes law

p(x(tk)|Y tk ) = K(tk)p(y(tk)|x(tk))p(x(tk)|Y tk−1), (1)

where K is a normalizing constant that depends on time.

[Jazwinsly, 1966]



Ste

ps

16

12

10

14

2

8

6

4

0

87654X axis3

0

5

Tar

get d

istr

ibut

ion


Joint state-parameter estimation (1)

dx

dt= f (x , θ)

For a given parameter, the state has a well defined evolution. If theparameter is a r.v. with a known distribution, how can we compute thestate pdf?

x

timet

x(t,θ)

x

timet

p(x,t)

Each solution is generated for adifferent value of the parameter

Augment the state:

z(t) =

[x(t)θ

]dz =

[f (x , θ)θ

]dt +

[0σ

]dw



Tracking 2 parameters with the FPE filter and KF

t0 2 4 6 8 10 12 14 16 18 20

x

-60

-40

-20

0

20

40

60

80

100


Adaptive control

time0 5 10 15 20 25 30 35 40 45 50

x 1=

a

-1.5

-1

-0.5

0

0.5

1

time0 5 10 15 20 25 30 35 40 45 50

x 2

0

50

100

referencereal stateobservationsfiltered maximum value estimate

time0 5 10 15 20 25 30 35 40 45 50

x 1=

a

-1

0

1

2

time0 5 10 15 20 25 30 35 40 45 50

x 2

0

50

100

referencereal stateobservationsfiltered maximum value estimate


Conclusions

The FPE provides a mean to describe ensembles of stochastic agentsthat can be applied to a variety of fields

The state estimation based on FPE is tightly related to particle filter,without the need to perform Monte Carlo

In adaptive control, advantage can be taken of parameter uncertainty


J. M. Lemos, J. Gir~ao, A. S. Silva and J. S. Marques · Continuous/discrete lter Prediction step:...

Documents

Transcript of J. M. Lemos, J. Gir~ao, A. S. Silva and J. S. Marques · Continuous/discrete lter Prediction step:...