Distributed state estimation with moving horizon observers Marcello Farina, Giancarlo...

Distributed state estimation with moving horizon observers

Marcello Farina, Giancarlo Ferrari-Trecate, Riccardo ScattoliniDipartimento di Elettronica e Informazione,

Politecnico di Milano

23rd June 2009

K.U. Leuven Optimization in Engineering Center

Leuven, 23rd June 2009 Marcello Farina 2

Outline

• Introduction

• Statement of the problem

• State of the art

• The DMHE algorithm

• Some improvements

• Conclusions


Outline

• Introduction





• Conclusions


Introduction

A sensor network is composed by a large number of sensor nodes

Features of the sensor nodes in a sensor network:

1. Cooperative effort

2. They locally carry out computations and transmit only required and partially processed data

3. Sensor network algorithms and protocols must possess self-organization capabilities

Advances in wireless communicationsAdvances in electronics

Development of sensor nodes, with sensing, data processing, and communicating components:low cost

low dimension

low power consumption

low memory

low computational power

+

+

+

-

-


Introduction

Applications:

• Health: sensor nodes can be deployed to monitor patients

• Environmental monitoring: preventing forest fires, forecast pollutant distribution over

regions

• Home: Improve quality and energy efficiency of environmental controls (air conditiong,

ventilation systems, …), while allowing reconfiguration and customization, besides saving

wiring costs.

distributed state estimation in sensor networks is a key problem

Advantages (and challenges) of sensor networks :• Build large-scale networks

• Implement sophisticated protocols

• Reduce amount of communication required to perform tasks by distributed and/or local

computations.

• Implement complex power saving modes of operation


Introduction

1 3

2

4

Base station

Introductory example [3]: temperature measurementDynamic evolution of the temperature:

Measured by 4 sensors (i=1, ..., 4), with sensing models:

To obtain a more reliable temperature estimate: DATA FUSION

CASE I: centralized estimator (base station)

where: l : optimal Kalman gain

have the same variance


Introduction

1 3

2

4

CASE II: distributed estimation• The sensors are arranged in a communication graph configuration• The measurements can not be sent simultaneously and instantaneously to a base station

Each sensor computes a local estimate of Tk, based on the available information:

It is obtained according to the equation:

mean upon “regional” quantities measures estimates

Average between regional values obtained with consensus.

Increasing the number of transmissions (NT) among neighbors (in a sampling time):

and the local filters become “optimal”, with performances of the centralized filter


Introduction

Consensus: provides agreement among local variables

1 3

2

4

measures:

Averaging is carried out by means of matrix K:

{stochastic:

compatible with the graph:

graph:

vertices:

edges:

Note: it is doubly stochastic if also:

Take x0=y. A “static” consensus (on measurements) algorithm is such that, at each consensus step:

xs+1=Kxs, s¸ 0


Outline

• Introduction





• Conclusions


Statement of the problem

•The measured system evolves according to the dynamics:

constrained state

constrained disturbance

random variable with mean and var(x0)=0

wt white noise with covariance

• The system is sensed by M nodes, with sensing models:

white noise with covariance


Statement of the problem•The communication network is described by a directed graph

: set of vertices

: set of edges

Neighbors:

Example:

1 2 3 4

We associate to the graph the matrix

{stochastic:

compatible with the graph:

NT: number of transmissions occurring in a sampling interval


Statement of the problemLocal, regional, collective quantities

Assumption: at a generic time instant t, a sensor i collects its measures and the ones of its neighbors.

Example: NT=2

1 2 3 4

Node 4 local measure:

Node 4 regional measure:

Network’s collective measure:

Definition: The system is

• locally observable by node i if is observable

• regionally observable by node i if is observable

• collectively observable if is observable

Notation:

• local quantity (wrt i): referred to node i solely (indicated with zi)

• regional quantity (wrt i): referred to (indicated with )

• collective quantity: referred to the whole network (indicated with )


Statement of the problem

i-th sensor regional observability matrix:

Further definitions

is the i-th sensor regionally unobservable subspace

is the orthogonal projection matrix on

Example: if the system is observable by all nodes of the graph, then:


Statement of the problemIsolated and strongly connected subgraphs

is not strongly connected

are strongly connected subgraphs

is the only isolated subgraph (no incoming paths)


Outline

• Introduction





• Conclusions


State of the art: Distributed Kalman filtersStarting point: CENTRALIZED Kalman filter (base station)

INFORMATION FILTER

PREDICTOR

CORRECTORto carry out this step one needs collective data:


State of the art: Distributed Kalman filters

estimate of xk1 carried out by sensor i at instant k2

• Local estimates obtained with consensus filters on the basis of regional measurements. • If regional observability does not hold, the algorithm does not give reliable results

• Good approximation for NT>>1

Th 1:

Th 2: if A is stable, then the local state estimates converge in mean to the centralized state estimate.

DECENTRALIZED Kalman filter (DKF), [7,8,10]

PREDICTOR

CORRECTOR

locally by the M sensors



Transmission of measurements and estimates [10]

PREDICTOR

CORRECTOR

+ CONSENSUSON ESTIMATES

CONSENSUS

ON MEASURES

NOTE THAT• The proof of the convergence of the estimate for a similar (continuous-time) algorithm is given, if collective observability is satisfied. This does not guarantee the convergence of the discrete-time algorithm• The optimal gain, in the case of a distributed algorithm, is not the Kalman gain (Carli et al). Therefore, optimality is not guaranteed by this algorithm.



To guarantee optimality of DKF, the Kalman gain and the weights of the sensor’s estimates should be a result of an optimization [1,2]

INFORMATION UPDATE

(consensus on measures)

CONSENSUS ON

ESTIMATES (REGIONAL)

PREDICTION

Optimization of Gi and K={kij} to minimize the steady state error

covariance matrices

NOTE THAT

• The minimization problem on Gi and K={kij} is not convex [3]. A bootstrap (iterative 2-step) algorithm is proposed [1,2], which does not necessarily guarantee optimality• Convergence of the algorithm has not been addressed.


State of the art: Moving Horizon EstimatorsOUR starting point: Moving Horizon Estimators [11,12]

Given , the optimal state estimate is obtained from the

solution of:

subject to

where:

(1)

are the optimizers.

The sequence can be obtained by the dynamic constraints given by the model equations.


State of the art: Moving Horizon Estimators

We rearrange the problem by breaking the interval [1,t] into [1,t-N] and [t-N+1,t], and so:

Note that (1) is equivalent to

where

Arrival cost

In the unconstrained case, t-N/t-N(x) is obtained as:

where t-N/t-N is Kalman filter error covariance matrix.



In the constrained case, only an approximation of t-N/t-N(x) can be given, with t-N(¢) (initial penalty term).

Is sufficient to make the MHE convergent.

The choice:

In this way, the MHE cost function is:



smoothing update

An alternative approach is also proposed (smoothing update [12]) TRANSIT COST

t t+Nt-N

estimate is passedbetween adjacent data windows

tt-N t+1t-N+1

estimate is passedbetween overlapping data windows


Outline

• Introduction





• Conclusions


The DMHE algorithm

• The basic algorithm has been designed for NT=1 [5]

• For N¸ 1, each sensor performs the local MHE-i constrained minimization problem:

REGIONAL MEASUREMENTS

(consensus on measures)

Under the constraints:

Center of mass of the estimates provided by the i-th neighbors at time t-1 (of xt-N)

(smoothing update to use recent information for the consensus)

t-Ni is a initial penalty and consensus on estimates term


The DMHE algorithm

HOW TO COMPUTE IT?I) LOCAL UPDATE (Riccati-like update)

II) GLOBAL UPDATE (Consensus-on-estimates)

The matrix t-N/t-1 is the result of

Subject to the LMI:

IN THIS FORMULATION, IT IS NOT

A DISTRIBUTED PROBLEM

Proposition: the following distributed update satisfies the previous LMI

,

for all i


The DMHE algorithmCan we chose K such that the matrices i

t-N/t-1 remain bounded?

Theorem 1: If is non-empty and, for all , there exists a path to i from an observable sensor, then there exists K, compatible with such that i

t-N/t-1 are bounded.

: set of observable nodes : set of unobservable nodes

How to obtain a conservative choice of K? Algorithm 1

• Terms on the lower triangular part of K can be “re-added”

• Theorem 1 states that one can chose non-zero upper triangular terms, and still guarantee boundedness


The DMHE algorithmConvergence issues

Definition: The system is:

with wt=0 and initial condition x0. We denote x(t,x0) its solution (assumed feasible).

DMHE is convergent if:

we define


Lemma 1: if t-N/t-1i are chosen as previously stated, and are bounded, then:

where

and

Theorem 2: if t-N/t-1i are chosen as previously stated, and are bounded, then DMHE is

convergent if the matrix is Schur.


The DMHE algorithmExample

1=0.9264

2=0.4517

3=0.99+0.4795 i

4=0.99-0.4795 i

unstable system

Regional measurements

x3 and x4 are not regionally observable to node 2 (unstable subsystem)

x1 and x2 are not regionally observable to node 4

The state is regionally observable to node 1

The state is regionally observable to node 3


The DMHE algorithm

With the choice:

EIGENVALUES OF t-1/t+Ni

x3 and x4 are not regionally observable to node 2 (unstable subsystem)

x1 and x2 are not regionally observable to node 4 (stable subsystem)

it-1/t-N are bounded


The DMHE algorithm

Let et be a white noise signal with covariance matrix

UNCONSTRAINED CASE CONSTRAINED CASE

sensor 4’s estimates

sensor 2’s estimates


Outline

• Introduction





• Conclusions


Some improvements

If state constraints are not required [4], Lemma 1 and Theorem 2 hold also if t-N/t-1 is not a bounded sequence, that is:

Lemma 1*: if t-N/t-1i are chosen as previously stated, then:

Theorem 2*: if t-N/t-1i are chosen as previously stated, then DMHE is convergent if the

matrix is Schur.

This result is obtained by defining an alternative initial penalty term, i.e.:

OBSERVABLE PARTUNOBSERVABLE PART


Some improvements How to enforce state estimation [6] by exploiting a better transmission of measurements?

1. If NT¸1, the set of regional measurements becomes larger as NT increases

2. If NT=1 use past measurements (as also suggested in [1])

Two transmission protocols are introduced:P1) NT¸1. Sets of collected regional measurements at time t:

P2) NT=1. Sets of collected regional measurements at time t:

EXAMPLE (P2)

Augmented system approach [1]


Some improvements

basic algorithm

collects the regional measurements of the set

its size pit-k is non-decreasing as (t-k)

increases

In general, it is a time-varying system

1) The dynamic constraints change

2) The consensus-on-estimate matrix changes

1-step consensus matrix:

K

with NT communication steps:

K* compatible with graph , generated by KNT

(K*= KNT is a possible choice)

3) The regional observability properties change

The s-steps observability

matrix is:The system is regionally observable by sensor i on horizon N if


Some improvementsAccordingly, the following are redefined:

is the i-th sensor unobservable subspace on horizon N


The transition matrix of the error dynamics changes:

If the graph is strongly connected and collectively observable:

if N (and/or NT) increases: if NT increases the eigenvalues of K* decreaseFor example K*=KNT.

j=eig(K), (| j |· 1 by definition)

jNT=eig(K*)


Outline

• Introduction





• Conclusions


Conclusions• The problem of state estimation in distributed systems is a challenging task:

– convergence– optimality

• The problem must be rigorously stated– Local, regional, collective variables and observability notions– Graph and subgraphs

• We analyze the available distributed estimation algorithms (DKF)– advantages– limitations

• We rely on moving horizon approach to state estimation

• The DMHE algorithm for constrained systems and convergence results

• Some improvements– If state constraints are not given– If the graph is strongly connected and collectively observable, we can always

design a DMHE converging observer


References

1. Alriksson, P. and Rantzer, A. (2006) Distributed Kalman filtering using weighted averaging. In Proc. of the 17° International symposium on Mathematical Theory of Networks and Systems.

2. Alriksson, P. and Rantzer, A. (2008). In Proc. IFAC World Congress.

3. Carli, R. and Chiuso, A. and Schenato, L. and Zampieri, S. (2008). Distributed Kalman filtering based on consensus strategies. IEEE Journal on selected Areas in Communications, 4.

4. Farina, M. and Ferrari-Trecate, G. and Scattolini, R. (2009) A moving horizon scheme for distributed state estimation. Submitted

5. Farina, M. and Ferrari-Trecate, G. and Scattolini, R. (2009) Distributed moving horizon estimation for linear constrained systems. Submitted.

6. Farina, M. and Ferrari-Trecate, G. and Scattolini, R. (2009) Distributed moving horizon estimation for sensor networks. IFAC Workshop on Estimation and Control of Networked Systems (NecSys'09), 24-26 September, 2009, Venice (Italy). To appear.

7. Kamgarpour, M. and Tomlin, C. (2008) Convergence properties of a decentralized Kalman filter. Proc 47° IEEE CDC Conference

8. Olfati-Saber, R. (2005) Distributed Kalman filter with embedded consensus filters. Proc 44° IEEE CDC Conference.

9. Olfati-Saber, R. and Shamma, J. (2005) Consensus filters for sensor networks and distributed sensor fusion. Proc 44° IEEE CDC Conference.

10. Olfati-Saber, R.(2007) Distributed Kalman filtering for sensor networks. Proc 46° IEEE CDC Conference.

11. Rao, C. V., Rawlings, J. and Lee, J. (1999). Stability of constrained linear moving horizon estimation. Proc. ACC.

12. Rao, C. V., Rawlings, J. and Lee, J. (2001). Constrained linear state estimation - a moving horizon approach. Automatica, 37.

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