Kalman Filter for Discrete-Time Stochastic Linear Systems Subject to Intermittent Unknown Inputs

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Systems Subject to Intermittent Unknown Inputs

Jean-Yves Keller, Dominique Sauter

To cite this version:

Jean-Yves Keller, Dominique Sauter. Kalman Filter for Discrete-Time Stochastic Linear Systems

Subject to Intermittent Unknown Inputs. IEEE Transactions on Automatic Control, Institute of

Electrical and Electronics Engineers, 2013, 58 (7), pp.1882-1887. �10.1109/TAC.2013.2264739�. �hal-

00846428�



Abstract— State estimation of stochastic discrete-time linear systems subject to persistent unknown inputs has been widely studied but only few works have been dedicated to the case where unknown inputs may be simultaneously or sequentially active or inactive. In this paper, a Kalman filter approach is proposed for state estimation of systems with unknown intermittent inputs. The design is based on the minimisation of the trace of the state estimation error covariance matrix under the constraint that the state estimation error is decoupled from the unknown inputs corrupting the system at the current time. The necessary and sufficient stability conditions are established considering the upper bound of the prediction error covariance matrix.

Index Terms— Kalman filter, intermittent unknown inputs, linear system, covariance matrices.

I. INTRODUCTION

ALMAN filtering plays an essential role in systems theory and has found a wide range of applications [19]

and [29]. The problem of optimal state filtering in the presence of persistent unknown inputs, representing unknown disturbances or unmodeled dynamics, has received considerable attention in the last three decades. The most common approach consists in representing the unknown inputs as a deterministic or stochastic bias model, augment the state of the system with the bias, and apply the Kalman filter to the augmented state model of the system and finally reduce the computational time requirement [10], [1], [20], [17], [18] and [23]. When no prior information about the unknown input is available, an optimal recursive state filter is presented in [24]

so that the state estimation error is decoupled from unknown inputs. Another approach which consists in transforming a standard system with unknown inputs into a singular system without unknown inputs was introduced in [6]. Other optimal filters closer to the standard Kalman filter have been derived by minimising the estimation error covariance matrix with respect to a reduced state feedback gain representing the

Manuscript received May 12, 2011; revised September 25, 2011 and March 21, 2012; Accepted July 10, 2012 ???.

D. Sauter is with University of Lorraine, CRAN-CNRS UMR 7039, BP 239, 54506 Vandoeuvre les Nancy, France (phone: +33(0)3 83 68 44 67;

fax: +33(0)3 83 68 44 62; e-mail: dominique.sauter@ univ-lorraine.fr).

J. Y. Keller is with Universityof Lorraine, CRAN-CNRS UMR 7039, BP 239, 54506 Vandoeuvre les Nancy, France.

degrees of freedom in the design [5], [7] and [15]. The closely related problem of joint input and state estimation for linear discrete-time systems, which are of great importance for Fault Tolerant Control (FTC) when each component of the unknown inputs vector represents actuator or component faults [2], has been recently studied in [11], [12] and [8]. Note that the unknown inputs decoupling constraint used in [24] can be viewed as a limit case of a more general assignment taken into account in the design of stochastic detection filters [26], [22]

and [21] for Fault Detection and Isolation (FDI).

Recent technological advances are revolutionizing our ability to build massively distributed Networked Control Systems (NCS). They present challenging problems arising from the fact that sensors, actuators and controllers exchange information via a digital communication network. The most common network effect can be categorized into additional network induced delays, packets losses, jitters,… In the framework of this paper, we are mainly concerned by packets losses, leading to intermittent data transmission. The state of the art on design control systems that take into account the effects of packet loss and packet delay in networked control systems have been surveyed in [16]. In particular, Kalman filtering with random lost of observations represented by Markovian or Bernoulli processes has been studied in [9] and [30], extended later to include both packet loss and random delay [27], [28] and more recently in [31] when the arrival of observations is driven by a semi-Markov chain.

This paper proposes a state filtering strategy for discrete- time linear stochastic systems in the case where the arrival of unknown inputs is described by a known binary sequence.

Instead of using the parameterized approach proposed in [7]

which requires to pre-compute off-line the structure of the state feedback gain for each combinatorial situation of the binary sequence, the intermittent unknown inputs decoupling constraint will be parameterized from two constant size matrices, called the free part and the constrained part of the filter gain. The constrained gain, structurally dependent on the binary sequence, will be linked to the intermittent unknown inputs estimator. From a two-stage optimisation strategy very similar to those described in [10], the free gain and the constrained gain will be both used to minimize the trace of the state estimation error covariance matrix and the trace of the unknown inputs estimation error covariance matrix. The obtained filter, called the Intermittent unknown Input Kalman Filter (IIKF), will take the form of a standard Kalman filter

Kalman filter for discrete-time stochastic linear systems subject to intermittent unknown inputs

J.Y. Keller, and D. Sauter, Member, IEEE

K

updated online from the intermittent unknown inputs estimation. The stability of the IIKF will be studied on the upper bound of the prediction error covariance matrix via the stability and convergence conditions established for the Unknown Input Kalman Filter (UIKF) designed under persistent unknown inputs.

The paper is organised as follows: Section 2 explains the state filtering problem in the presence of intermittent unknown inputs. Section 3 solves the state filtering problem and studies the filter’s stability before conclusions in Section 4.

II. PROBLEM STATEMENT

Let us consider the following linear discrete-time stochastic systems

k k k k

k Ax Bu Fd w

x    

1 (1.a)

k k

k Cx v

y   (1.b)

where xkⁿ, uk^d, yk^m and dk^^q are the state, control, measurement and unknown input vectors with ^q^m. Matrices A, B, ^C and F are of appropriate dimensions with

q F rank CF

rank( ) ( ) . The process and sensor noises w_kⁿ and

k m

v  are zero mean uncorrelated Gaussian random sequences with

j k T

j j k k

I W v w v

E w _,

0 0



 































 



 with ^W0 (1.c)

The initial state x0, assumed to be uncorrelated with wk and

vk, is a Gaussian random variable with ^E ^{x0 x0 and}



^{(0 0)(0 0)}



⁰

0E x x x x ^T 

P .

The term Fd^_k in (1.a) is used to describe additive unknown inputs as for example interconnecting external inputs in the context of large scale NCS. It is further assumed that d^_k, due to possible unreliable data transmissions, is an intermittent unknown inputs vector

qT k q k i k i k k

k kd d d

d 



   

 ^{1 1} .. .. (1.d)

depending on the known binary variables



^{k ik kq}



k   

  ¹,.., ,.., (1.e)

where _kⁱ¹ when the ith component of d^_k is active and ⁱ_k⁰

otherwise.

As illustrated by Fig. 1, uk represents the control signals sent by the local controller to the plant and ^d_k the control signals sent by remote controller with _kⁱ¹ when ^di_k is received by the plant or _kⁱ0 when d_kⁱ is lost through the unreliable network. The unreliable network is a multichannel network where each component of d^_k may be lost independently from the others. The acknowledgement signals

k indicating the status of reception/delivery (TCP for example) and the distribution matrix F of d^_k are both assumed known to the local controller.

Fig. 1. Intermittent unknown inputs in NCS.

The arrival sequence

 

0¹

 k

j of unknown inputs is considered as deterministic in the design of the following state filter

ˆ ) ˆ (

ˆk^_/kx^k_/k_1Kk^ ykCxk^_/k_1

x (2.a)

T k k T k k k k k

k I K CP I K C K K

P^_/ (  ^ ) ^_{/ 1}(  ^ )  ^{ } (2.b)

k k k k

k Ax Bu

xˆ^_1/  ˆ^/  (2.c)

W A AP

P_k^_1/k _k^_/k ^T (2.d)

where ^xˆ_k/k1 is the state prediction of covariance matrix



^{k k k k k k T}



k

k E x x x x

P^_{/ 1} ( ˆ^_{/ 1})( ˆ^_{/ 1}) based on measurements available until time k1 and

 

0¹

 jk

 , where ^xˆ_k/k is the state estimate of covariance matrix ^Pk/kE



^{(xkxˆk/k)(xkxˆk/k)T}



^based

on measurements available until time ^k and

 

0¹

 jk

 . From (1) and (2), the state prediction error e^k_1/kxk_1xˆ_k^_1/k and the state estimation error e^k_/kxkxˆ^_k/k propagate as

k k k k k

k Ae Fd w

e^_1/  ^_/  ^ (3.a)

k k k k k k

k I K Ce K v

e^_/ (  ^ )^_{/ 1} ^ (3.b)

Under ^E

 

^{ek/k1Fdk1 with F}_^{f1 .. fi .. fq}_^ , we have

 

^{ek/k (IKkC)E}

 

^{ek/k10}

E (and thus ^E

 

^{ek1/k Fdk} ) if and only if the state feedback gain ^K_k^^n,m satisfies the unknown inputs decoupling constraint (IK_k^C)Fd^_k10 in [24] relaxed under (1.d) as





 _k1 _k1

kCF F

K with F_k^_1_^¹_k1f¹ .. _kⁱ_1fⁱ .. _k^q_1f^q_^ (4) Under rank(CF)rank(F)q the existence condition

k k

k rankF s

CF

rank( ^_1) ( ^_1) with 

 

^q i

i k k s

1 1 for a solution to (4) is satisfied k1. This paper proposes to parameterize the solution to equation (4) from constant size matrices as follows





 _k _{k k}

k K G

K  ⁰ (5)

with ^K_k⁰^n,m, ^_k(IK_k⁰C)F_k^_1 and ^G_k^^q,m. Substituting (5) in (4), we obtain







 

_kG_kCF_k1 _k (6)

With Ik_ diag_^ k1_ ⁱk_1 _k^q_1^ 1

1   

 so that ^_kI^_k1^_k since

_k has a special structure with zero columns when ^d_k1 has Plant

Local controller

IIKF Network

Remote controller con

k d

uk

yk

zero components, we deduce from (6) that the state feedback gain (5) satisfies (4) if and only if ^G_k satisfies





 _k1 _k1

kCF I

G (7)

Suggested by the structure of the state feedback gain (5), let

ˆ ) ˆ (

1 / /

1  

 k k k k k

k G y Cx

d (8.a)







  _{ }



T T k k k k k

k E

Q^_1/ ^_1/ ^_1/ (8.b)

with ^_1/ ˆ^_1/ ^₁





 _k _{k k} _k

k d d where ^E

   

^{dˆk1/k GkCEek/k1dk1 under}

(4) and (7). The state estimator (2) and the intermittent unknown inputs estimator (8) will be designed by minimising the trace of ^P_k^_/k and ^Q_k1/k with respect to ^K_k and ^G_k under (4) and (7). The necessary and sufficient conditions so that



 



 

k k k

P _1/

lim for any sequence

 

j^0 will be established.

III. KALMAN FILTER FOR STOCHASTIC LINEAR SYSTEM SUBJECT TO INTERMITTENT UNKNOWN INPUTS The proposed IIKF is described by theorem 3.1.

Theorem 3.1: The Unbiased Minimum Variance (UMV) state estimate xˆ^_k/kof covariance P_k^_/k is generated by the following modified Kalman filter

k k k k k k k k k

k I K C x F d K y

xˆ^_/ (  ⁰ )(ˆ^_{/ 1} ^_1ˆ^_1/ ) ⁰ (9.a)

T k k T k T k k k k k k k k

k I K C P F Q F I K C K K

P^_/ (  ⁰ )( ^_{/ 1} ^_1 ^_1/ ^_1)(  ⁰ )  ^{0 0} (9.b)

k k k k

k Ax Bu

xˆ^_1/  ˆ^_/ 

^{P AP A W}

T k k k

k^_1/  ^_/  (10)

with K_k⁰P_k^_/k1C⁽CP_k^_/k1C^TI^)1, updated online from the additive quantities ^F_k^_1^dˆ_k^_1/k and ^F_k^_1^Q_k^_1/k^F_k^_^T₁ depending on the unknown inputs estimate ^dˆ_k1/k of covariance ^Q_k1/k given by

 

 

















] ) (

) [(

ˆ ) ˆ (

1 1 1 / 1 /

1

1 / /

1















k T k k T k k k

k k k k k k

CF I C CP CF Q

x C y G d

(11)

with G_k^Q_k^_1/k⁽CF_k^_1^)T(CP_k^_/k1C^TI^)1. The ith component ^dˆ_k^i_1/k of ^dˆ_k1/k represents the estimate of _kⁱ_1d_kⁱ_1 (with d^ˆ_k^_ⁱ_1/k⁰

when _kⁱ_1⁰) and the ith component ^Q_kⁱ_1/kon the diagonal part of ^Q_k^_1/krepresents the variance of ^dˆ_k^i_1/k (with Q_k^_ⁱ_1/k⁰

when ⁱ_k1⁰). The IIKF is initialized by xˆ0^/_1x0, P0^/_1P0⁰

and ^1^0,..,0,..,0.

Proof. Let ^Xk



^{xTk dkT}¹



^T, ^Xˆk/k



^{xˆkT/k dˆkT}^1/k



^T,



































 







T

k k

k k k k

k k k

k

e

E e _





 



 ₁^/_{/ 1}^/_/

/ and ^Lk



^{KkT GkT}



^{T. The state}

estimator (2) and the unknown inputs estimator (8) can then be jointly expressed

ˆ ) ˆ (

0

ˆ/ / 1  / 1



  







 k k k k _{k k}

k

k I x L y Cx

X (12.a)

T k k T k k k k k

k I LC LL

P C

I L_{    }

  



 



 



 



 

 _ )

(0 0 )

( _{/ 1}

/ (12.b)

  k k ^k k

k A X Bu

xˆ^_1/  0ˆ^_/ 

^Pk^_1/k   ^A 0^k_/k^A 0^T^W (12.c)

Under ^tr(P_k^_/k1⁾ minimum, Xˆk^_/k is the UMV estimate of ^X_k (and thus ^tr(P_k^_1/k⁾minimum) if and only if the augmented gain



^{kT kT}



^T

k K G

L^ ^{ } is solution to the constrained optimization problem

)

( _/

min 

 ^{k k}

Lk

tr s.t. _^













 

 



1 1 1

k k k

k I

CF F

L (13)

The solution to (13) is difficult to obtain since K^_k depends on G_k^ from K^_kK⁰_k^_kG_k^. Let X^kTkX_k^, Xˆ^k_/k TkXˆ^_k/k

 ,

kT k k k k

k T ^ T

/   /

 and L^kTkL^_k where Tk is an arbitrary non- singular transformation matrix of appropriate dimension. The filter (12) can then be equivalently rewritten

ˆ ) ˆ (

0 ˆ

1 / 1

/

/   

     







 k k k k k _{k k} k

k I x L y Cx

T

X (14.a)

T k k T k k

k k k k

k

k I LC L L

T P C I L

T ^{    }

  







 



 



 

 _ )

( 0 0 )

( _{/ 1}

/ (14.b)

 k k k ^k k

k A T X Bu

x^_1/  ^1ˆ^_/  ˆ 0

(14.c)

 ^{A T T}  ^{A W}

P_k^_1/k 0 _k^1^_{k/k k}^T 0^T (14.d) Under ^tr(P_k^_/k1⁾ minimum, Xˆ^_k/k is the UMV estimate of ^X_k (and thus ^tr(P_k^_1/k⁾ minimum) if and only if the transformed gain ^L_k is solution to

)

( _/

min 

 ^{k k}

Lk

tr s.t. _^









 



 

 



1 1 1

k k k k

k I

T F CF

L (15)

With











 

 I

T_k I ^k 0

 so that ^LkTk



^{KkT GkT}

 

^{TKk0T GkT}



^{T where}

0

Kk is now decoupled from ^G_k , we can verify that the transformed algebraic constraints in (15) reduce to





 _k1 _k1

kCF I

G . So, after straightforward manipulations, (15) becomes

)

( _/

min 0



 k k

Gk Kk

tr















 s.t. G^_kCF_k^_1I_k^_1 (16)

with















 







 







 

k k T

k k T k k k

T k k k T k k k

k k

k G P C K H Q

G H K C P P

/ 1 0

1 /

0 1 / 0

/

/ ( )

)

( (17)

where ^P_k⁰_/k^(I^K_k^0C)P_k^_/k1^(I^K_k^0C)T^K_k^0K_k^0T, Q^k_1/kG^kHkG^_k^T and

I C CP

H_k _k^_/k1 ^T . With ^tr(_k^_/k⁾^tr(P_k⁰_/k⁾^tr(Q_k^_1/k⁾ obtained from (17), we deduce that the global solution to (16) corresponds to the local solutions of the following decoupled optimisation problems

) ( ⁰_/ min

0 k k

Kk P

tr (18.a)

and

) ( min ^_1/

 k k

Gk Q

tr _ s.t. ^G_k^CF_k^i_1^I_k^_ⁱ₁ for ⁱ^1,..,q (18.b) where ^F_k^_ⁱ₁ and ^I_kⁱ₁ represent the ith column of ^F_k^_1 and ^I_k1.

The unique solution to (18.a) coincides with the Kalman filter’s gain / 1 / 1 ¹

0P _C(CP _C I)^

K_{k k}^_{k k}^_k ^T . The existence of the ith