Robust L∞-induced filtering and deconvolution of a wide class of linear discrete-time stochastic systems

Author’s Accepted Manuscript Robust L∞-induced filtering and deconvolution of a wide class of linear discrete-time stochastic systems Mehrdad Tabarraie, Seyedabdolkhalegh Mozaffari Niapour www.elsevier.com/locate/sigpro

PII: DOI: Reference:

S0165-1684(15)00432-6 http://dx.doi.org/10.1016/j.sigpro.2015.12.005 SIGPRO6008

To appear in: Signal Processing Received date: 10 October 2014 Revised date: 29 July 2015 Accepted date: 14 December 2015 Cite this article as: Mehrdad Tabarraie and Seyedabdolkhalegh Mozaffari Niapour, Robust L∞-induced filtering and deconvolution of a wide class of linear discrete-time stochastic systems, Signal Processing, http://dx.doi.org/10.1016/j.sigpro.2015.12.005 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Robust L∞-induced filtering and deconvolution of a wide class of linear discrete-time stochastic systems Mehrdad Tabarraie a,*, Seyedabdolkhalegh Mozaffari Niapour b a

Department of Electrical Engineering, Golpayegan Branch, Technical and Vocational University, Golpayegan 87717, Iran b Department of Electrical and Computer Engineering, Northeastern University, Boston, MA 02115, USA *Corresponding author E-mail address: [email protected] Tel/Fax: (+98) 31 57482370

Abstract The problems of stationary robust L  -induced filtering and deconvolution are addressed for discrete-time linear systems with deterministic and stochastic uncertainties in the state–space model. Stochastic uncertainties are in the form of state- and input-dependent multiplicative white noises which appear in both state and the measurement equations. The deterministic part of the system matrices and covariance matrices of the stochastic parameters is unknown and resides in a given polytopic-type domain. For this system a new lemma is derived which characterizes the induced L  norm disturbance attenuation performance by linear matrix inequalities (LMIs). According to this lemma, the problem of estimator design is solved for stochastic uncertain systems based on the notion of quadratic stability. To further reduce the overdesign in the quadratic framework, this paper also proposes a parameter-dependent design procedure, which is much less conservative than the quadratic approach. The proposed estimators guarantee the mean-square exponential stability of estimation error dynamics and satisfy the prescribed induced L  performance index. Two examples are used to demonstrate the proposed methods and their effectiveness.

: Induced L  norm performance; Deconvolution, Filtering; Discrete-time system; Stochastic uncertainty; Parameter-dependent approach

1. Introduction The deconvolution problem is the estimation of the input signal transmitted through a system (channel) from the noise corrupted measurement of the system output [1]. This problem has widespread applications in environments such as channel equalization, image restoration, speech processing, and fault detection; see [1-4]. In many existing works on the deconvolution problem in the literature, it is assumed that the channel model is precisely known. However, the system model is always uncertain in the real world. The uncertainty is due to the modeling errors that arise from system identification procedure or the time variations of model parameters during the operation [5]. The usual causes of these phenomena are due to the perturbation

1

of transmission medium, linearization and model reduction [6]. Another case which usually occurs is the uncertainty in the covariances of the noises due to the variation of environment [6]. Parametric uncertainties have been considered in the form of deterministic uncertainties or stochastic uncertainties in the robust estimation literature. Two types of deterministic uncertainties including polytopic uncertainty and norm-bounded uncertainty have been extensively used for studying robust filtering problems in the literature ([7-10]. Stochastic uncertainties can be represented by stochastic parameters fluctuating around some deterministic nominal values [11]. These uncertainties can be also viewed as multiplicative white noises to the system. This kind of uncertainties have been observed in several engineering applications, such as communication channel equalization [12], image processing [13], fault detection [4,14], and structural vibration [15]. However, in most practical situations, parametric uncertainties are best modeled as a mix of deterministic and stochastic uncertainties, see [1,10,16]. In a typical wireless communication channel, for example, the channel coefficients can be modeled as Gaussian random variables whose mean and variance are uncertain [10]. This yields a model that has a mixture of deterministic and stochastic uncertainties. Example in Section 5 illustrates this case. Therefore, from the practical viewpoint, the filtering and deconvolution problems deserve to be investigated for linear systems with both deterministic and stochastic uncertainties. On the one hand, the filtering (state estimation) problem for linear discrete-time systems with stochastic uncertainties has drawn a great deal of attention in the last past years under different hypothesis and performance criteria. In the case where the exogenous disturbances have known statistical properties, the H 2 (or Kalman) filtering approach arises as an efficient strategy [10,17]. In many cases, however, the statistical nature of the external disturbances is not easily known. To solve this difficulty, alternative approaches have been developed such as H  and L2  L filters [10,18-20] in which disturbances are only supposed to be energy bounded. On the other hand, the deconvolution (input estimation) problem for linear discrete-time systems with random modeling uncertainties received less attention in the literature. Robust minimax deconvolution problem for uncertain linear discrete time systems with the autoregressive moving average (ARMA) model description has been studied in [1,6]. The ARMA model is transformed to the state-space model including state-dependent noise in [6], and then the design of deconvolution filter is investigated under the minimax mean-square error criterion by the saddle-point theory. Unlike the work in [6], a polynomial approach in [1] is adopted to design a minimax deconvolution filter for the ARMA model of the stochastic system. Based on the discrete- and continuous time version of the stochastic bounded real lemma [8,21], robust H  deconvolution filters by the use of the quadratic (parameter-independent) Lyapunov function (QLF) approach have been designed for linear stochastic

2

systems with deterministic uncertainties in [22,23]. It is well known that the QLF approach is conservative because the fixed Lyapunov matrix must be used for the entire uncertainty domain [4,24]. In order to reduce the overdesign of the QLF approach, the design problem of H  and L2  L deconvolution filters via a parameter-dependent Lyapunov function (PDLF) approach has been developed in [24] for continuous-time stochastic systems with polytopic uncertainties. Very recently, a deconvolution estimator in the minimum mean-square error (MMSE) sense [25] has been derived for discrete-time systems with random parametric uncertainties in the measurement matrices. In H  and L2  L estimation schemes, the exogenous (input or disturbance) signals are considered to be of L 2 type. However, the signals often do not satisfy this condition and act more or less continuously over time [26]. Such signals are known as persistent bounded ( L  type) signals. Hence, the approaches based on the H  and L2  L performance criteria are not well suitable for the persistent bounded signals. A solution to treat this difficulty is to address the estimation problem in the induced L  (peak-to-peak) norm setting, see e.g., [26-28] in the deterministic system case. For the continuous-time stochastic systems, Berman and Shaked [29] have recently designed an induced L  filter which ensures an upper bound on the peak value of mean-square estimation error for persistent bounded disturbances. Results of [29] have been extended in [4] where the robust peak-to-peak deconvolution problem has been developed via two QLF and PDLF approaches. In this paper, we address the filtering and deconvolution problems under induced L  norm disturbance attenuation performance criterion for a general class of discrete-time linear stochastic systems with deterministic polytopic uncertainties and a multiple set of state- and input-dependent multiplicative white noise which appears in both state and measurement equations. To the best of authors’ knowledge, so far there has been no result on the peak-to-peak filtering and deconvolution for such general systems in the existing literature. For the first time, we establish a new lemma which provides the sufficient conditions in terms of LMIs for the guarantee of an upper bound on the induced L  norm of this class of systems. This lemma by itself has theoretical importance in peak-to-peak control and estimation of systems of this kind. By using this new lemma, the design problem of the induced L  estimator is first solved for stochastic systems with polytopic uncertainties based on the notion of quadratic stability. To further reduce the overdesign in the quadratic framework, this paper also proposes a new parameter-dependent Lyapunov function method, which is much less conservative than the quadratic approach. Finally, the effectiveness and advantages of the proposed design methods are shown via two numerical examples.

Notation. The superscript T shows matrix transposition, n determines the n -dimensional Euclidean space, Euclidean vector norm, and n m is the set of all n  m real matrices.





is the

is the set of nonnegative integers, and the notation

3

P  0 for P n n means that P is symmetric and positive definite. The symbol * is used for the symmetric terms in a

symmetric matrix, I n is the n  n identity matrix, and  (k )

(, , ) , where  is the sample space,

Given a probability space

is a  -algebra of subsets of the sample space, and

, we denote the space of bounded n -valued functions by l  (, n ) . By

on

 -algebras

k



l

: supk





we denote an increasing family of

. E {}  stands for the expectation operator with respect to the probability measure

denote the space of all non-anticipative stochastic processes

f

 k k 

is the probability measure

E f (k )

2

f

(k )k 



in n with respect to

. Let l  (

 k k 





; n )

satisfying

. 2. Problem formulation

We consider the following linear discrete-time stochastic system with state- and exogenous input-dependent noises: N N     x (k  1)   A ( )   A 0t ( ) t (k )  x (k )   B 1 ( )   B 0t ( ) t (k )   (k ), t 1 t 1     N N     y (k )  C 2 ( )  C 0t ( ) t (k )  x (k )   D 21 ( )   D 0t ( ) t ( k )   (k ), t 1 t 1     z (k )  C 1x (k )  D11 (k ), k  

where x (k ) n is the state vector, and  (k )  l  (



(1a-c)

; q ) is the exogenous signal that includes both the input signal and

the measurement noise. y (k ) r is the measurement vector. z (k ) m is the signal to be estimated, where C 1 and D11 are predefined matrices. The case D11  0 corresponds to estimating of a combination of the states (filtering problem), whereas C 1  0 corresponds to a estimation of a combination of the inputs (deconvolution problem). The variables  t (k ) , t  N are

scalar zero-mean white-noise sequences that satisfy E  i (k ) j (l )   (i  j ) (k  l ) . The initial condition x (0)  x 0 is a random vector. The stochastic processes

 t (k )

( t  N ), and the random vector x 0 are mutually independent.

  [1 , 1 ,...,s ] s is the vector of constant uncertain parameters that belongs to the unit simplex  , given by   :  s :  i , 

s

 i 1

i

  1,  i  0  

(2)

A ( ) , B1 ( ) , C 2 ( ) , D 21 ( ) , A0t ( ) , B 0t ( ) , C 0t ( ) and D 0t ( ) ( t  N  1, 2,..., N  ) are uncertain matrices of

appropriate dimensions which belong to the following convex-bounded polytopic-type domain  : s    : ( ) ( )   i i ;     i  1  

(3)

4

where i :  Ai , B1i , B 2i ,C 2i , D 21,i , A0,1 i ,..., A0,Ni , B 0,1 i ,..., B 0,Ni ,C 0,1 i ,...,C 0,Ni , D0,1 i ,..., D0,Ni  , i  1,..., s denoting the polytope vertices, are known matrices with compatible dimensions.

Remark 1. It is worth mentioning that regarding deconvolution approaches for uncertain stochastic systems in [4,6,22-24], none of them includes stochastic uncertainties in the measurement equation which may not be the case in practice. Furthermore, the optimal input estimator in [25] is applied to linear stochastic systems in which state- and input-dependent noises have been appeared only in the measurement equation and have no same noise term with each other. In the stochastic system (1a-c), the variables  t (k ) , t  N appear in both the state- and input-dependent noise terms. This setting allows the treatment of general situations when the state-dependent and input-dependent noises are same or different in both state and measurement equations, which is commonly encountered in practice, see e.g., [10,12]. For the first time, this paper investigates the deconvolution problem for this wide class of stochastic systems.

0  1

 (k )  0  1

E

 x (k )    2

k

x0

2

k

x 0 n



 

We consider the ESMS system of (1a-c) and assume a full-order general-type estimator with the proper structure xˆ (k  1)  A f xˆ (k )  B f y (k ), xˆ 0  0, zˆ (k )  C f xˆ (k )  D f y (k )

(4)

where xˆ(k ) n and zˆ(k ) m .When z (k ) is the combination of the states ( D11  0 ), the estimator (4) acts as the filter. While z (k ) is defined in the form of a combination of the inputs ( C 1  0 ), the estimator (4) operates as the deconvolution filter. Denoting z (k )  z (k )  zˆ (k ) and for a given scalar   0 , the following cost function is defined: J S : z

Denoting    x

T

l

 

(5)

l

T

xˆ  , the estimation error system is obtained as follows: T



N





N





t 1





t 1



 (k  1)   A ( )   A 0t ( ) t (k )   (k )   B 1 ( )   B 0t ( ) t (k )   (k )     z (k )  C 1 ( )  C 0t ( ) t (k )   (k )   D11 ( )   D 0t ( ) t (k )   (k ) t 1 t 1     N

N

(6)

5

where 0  A ( ) A ( )    B f C 2 ( ) A f

 A 0t ( )  B 0t ( )  0   B1 ( )  t t , B (  )  , A (  )  , B (  )     , 1 o 0   B D ( )  t t   f 21   B f C 0 ( ) 0   B f D 0 ( ) 

(7)

C 1 ( )  C 1  D f C 2 ( ) C f  , D11 ( )  D11  D f D 21 ( ), C ( )  D f C ( ) 0  , D ( )  D f D ( ) t 0

t 0

t 0

t 0

Given the prescribed scalar   0 , our objective in this paper is to design a estimator in the form of (4) such that the estimation error system (6) is ESMS and J S

(k )  l  (



; q ) and

of (5) is negative

for all nonzero

  .

∞

N N     x (k  1)   A ( )   A 0t ( ) t (k )  x (k )   B 1 ( )   B 0t ( ) t (k )   (k ) t 1 t 1     N N     z (k )  C 1 ( )  C 0t ( ) t (k )  x (k )   D11 ( )   D 0t ( ) t (k )   (k ) t 1 t 1    

(8a,b)

and the following performance index for a prescribed   0 :

J E : z

l

 

l

Q ( )  0 Q ( )  A T ( )Q ( )A ( )    A 0t ( )  QA 0t ( )  0,    N

T

(9)

t 1

Proposition

gives sufficient conditions for an upper bound of the peak-to-

peak norm of the stochastic system (8a,b)

Remark 2. Related results with the following lemma in the deterministic context are given in [27,31] for the discrete-time case and in [32] for the continuous-time case. Recently, a similar result in [29] determines an upper bound on the induced L  norm for the continuous-time stochastic systems. However, the model (8a,b) is wider and more flexible than the one considered for the continuous-time systems in [29]. Consequently, the following lemma yields more general results than the continuous-time counterpart in [29].

6

Lemma 1. Given   0 the system (8a,b) is ESMS, and J E is negative for all nonzero  (k )  l  (



  0,1   max eig (A  ( )  

if there exist Q ( )  0 following two matrix inequalities

2





; q ) and

 

   0,   that satisfy the

  :

N N T T  T  t t A T ( )Q ( )B 1 ( )    A 0t ( )  Q ( )B 0t ( )  A ( )Q ( )A ( )  Q ( )  Q ( )    A 0 ( )  QA 0 ( )  t 1 t 1 0 1 :  N N T T   T t t T t t B1 ( )Q ( )A ( )    B 0 ( )  Q ( )A 0 ( )   I q  B1 ( )Q ( )B1 ( )    B 0 ( )  Q ( )B 0 ( )   t 1 t 1  

(10)

and N N T T   1 T 1 t t  1C 1T ( )D11 ( )   1  C 0t ( )  D 0t ( ) Q ( )   C 1 ( )C 1 ( )    C 0 ( )  C 0 ( )  t 1 t 1 0  2 :  N N T T   1 T 1 t t 1 T 1 t t (   )I q   D11 ( )D11 ( )     D 0 ( )  D 0 ( )    D11 ( )C 1 ( )     D 0 ( )  C 0 ( ) t 1 t 1  

(11)

Proof First, using Proposition 1 we can follow from the first diagonal block in (10) that the system (8a,b) is ESMS. V  x (k )   x T (k )Q ( )x (k )

Q ( )  0 we define

V  x (k )  : x T (k  1)Q ( )x (k  1)  x T (k )Q ( )x (k )

 k 

(12)

(t N )

N T    E V  x (k )   E x T (k )  A T ( )Q ( )A ( )  Q ( )    A 0t ( )  Q ( )  A 0t ( )   x (k ) t 1    N T   +x T (k )  A T ( )Q ( )B 1 ( )    A 0t ( )  Q ( )  B 0t ( )    (k ) t 1   N T     (k )  B 1T ( )Q ( )A ( )    B 0t ( )  Q ( )  A 0t ( )   x (k ) t 1  

(13)

T

N T    T (k )  B 1T ( )Q ( )B 1 ( )    B 0t ( )  Q ( )  B 0t ( )    (k )  t 1   

Defining  :  x T , T  and denoting X k  E V  x (k )  and using the fact that X k 1  X k  E V  x (k )  , we obtain T

from (10) that E T 1  0 which implies that



X k 1  X k   X k   E  (k )

2

  0, k 



, X0 0

(14)

Noting under zero initial condition we have X 0  0 . Using  : 1   and multiplying (14) by   k we have



  k X k 1    k 1X k    k E  (k )

2

, k 



, X0 0

(15)

Taking the sum of both sides of the latter, from 0 to k  1 , we obtain

7

k 1



j

X

j 0

j 1

 X k  

k 1

Xk  

     k 1

   j 1X

j

2



E ( j )

j 0



sup k E  (k )

l

j

2

  k 1

j

j 0

2

  

Xk  

 k 1

k 1 j 0

2 l





X k  X 0      j E ( j )



k 1

1    1 

2



k

(16)

1   1

1  k 1 

If 0    1 , it holds that Xk 

  

2

(17)

l

On the other hand, we obtain from (8b) that N N T T      E z T (k )z (k )  E x T (k ) C 1T ( )C 1 ( )   C 0t ( )  C 0t ( )  x (k )  x T (k ) C 1T ( )D11 ( )   C 0t ( )  D 0t ( )   (k ) t 1 t 1      N N T T   T   T  T (k )  D11 ( )C 1 ( )    D 0t ( )  C 0t ( )  x (k )  T (k )  D11 ( )D11 ( )    D 0t ( )  D 0t ( )   (k )  t 1 t 1     

(18)

Applying the Schur complement formula, we have from (11) Q  0 

T t T N  C ( )  0  1 C 1 ( )  1    T  C 1 ( ) D11 ( )      T t (   )I  t 1  D11 ( )   D 0 ( ) 

0

  C t ( ) D t ( )   0 0   0 

(19)

Multiplying the two sides of (19) by T and  , from the left and the right, respectively, and taking expectation, we find that



 X k       E  (k )

2

   E  z (k )  , 1

2

k 

(20)



Substituting (17) in (20) yields 

Taking the supremum over k 



2 l



      E  (k )

yields z

l

 

l

2

   E  z (k )  , 1

2

k 



(21)

.

0    1 implies 0    1 . The inequality (10) can be rewritten by using Schur complements as follows:  Q ( )  Q ( )  A T ( )Q ( )A ( ) A T ( )Q ( )B 1 ( ) A 0T ( )    T T B1 ( )Q ( )A ( )   I q  B1 ( )Q ( )B1 ( ) B 0T ( )   0   A 0 ( ) B 0 ( ) Q ( )  

(22)

where Q ( )  diag Q ( ),...,Q ( )  , T T T A 0 ( )   A 01 ( )  ,  A 02 ( )  ,...,  A 0N ( )   ,   T

T T T B 0 ( )   B 01 ( )  ,  B 02 ( )  ,...,  B 0N ( )    

T

8

The first diagonal block in (22) implies that (1   )Q ( )  A T ( )Q ( )A ( )  0 . Therefore,  1   max eig (A ( )) 



is bounded by



   . That is,  should be constrained to the interval 0,1   max eig  A ( )   . The proof is

2

2

completed.

Remark 3. Note that the inequalities (10) and (11) are nonlinear duo to products between  and Q . Therefore, (10) and (11) become LMIs only if the variable  is fixed.

∞

In this section, we present two approaches for designing robust L 

estimators for system (1a-c) based on either

quadratic or parameter-dependent stability notion.

4.1. QLF approach To facilitate the presentation of our results, first we consider the stochastic system (1a-c) is without polytopic uncertainties, that is,

A , B ,C 1

2

, D 21 , A01 ,.., A0N , B 01 ,..., B 0N ,C 01 ,...C 0N , D01 ,..., D 0N   , t  N , is arbitrary but fixed. Now, we give the

following theorem based on the Lemma 1.

Theorem 1. We consider the ESMS system (1a-c) without polytopic uncertainties and the estimator of (4). Let   0 and



   0,1  max eig (A ) 

  be given constants. Then, the following holds: 2

(a) The estimation error system (6) is ESMS, and J S

of (5) is negative for all nonzero

k   l  (



; q )

if there exist R  R T n n , W W T n n , Z n r , S n n , T m n , Dˆ f m r , and    0,   such that the following two LMIs are satisfied: 1  R ,W , Z , S ,    0,





 2 R ,W ,T , Dˆ f ,   0

(23a,b)

where

9

    1 R  0   0 1 :  RA  WA  ZC  S 2  1 

*

   1W 0 0 S 0

* *  I q RB 1 WB 1  ZD 21 2

* * * R 0 0

*   *  *   *  *    3 

* * * * W 0

and R   0  0  2 :   C 1  Dˆ f C 2 T  Ш1 

*

*

W 0 T 0

*

* (   )I q D  Dˆ D 11

f

21

Ш2

*

*

*

*

*

Im

*

0

 I m N

       

where  1,1  1,2 1      1, N

   2,1   Ш1,1   Ш 2,1     Ш     ,    2,2  , Ш   1,2  , Ш   Ш 2,2  , 2 1 2                Ш Ш  2, N    1, N   2, N    R 0  0   R  3  diag    ,...,  0 W   , 0  W      RA 0t   RB 0t  1,t   ,   ,  2,t t t  t t  W A 0  ZC 0  W B 0  ZD 0  Ш1,t   Dˆ f C 0t , Ш 2,t  Dˆ f D 0t

(b) If (23a,b) is satisfied, an ESMS estimator in the form of (4) achieving J S  0 is given by Af  W 1S , B f  W 1Z , C f  T , Df  Dˆ f

Proof. This theorem can be proved

(24)

similar to those used in [27,29].

(a) According to Lemma 1 and using Schur complements, the estimation error system (6) is ESMS, and J S of (5) is negative for all nonzero

k   l  (



if there exist Q  0

; q )



   0,1  max eig (A ) 

  2

0     satisfying    1Q  0   QA   QA 0

0  I q QB1 QB 0

AT Q B 1T Q Q 0

A 0T Q   B 0T Q  0 0   Q 

(25)

and Q   0  C1   C 0

0 (   )I q D11 D0

C 1T D11T Im 0

C 0T D 0T 0  I m N

  0   

(26)

where

10

  ,A 

Q  diag Q ,...,Q  , A 0   A 01 

  ,B 

B 0   B 01 

T

2 0

T

  ,D 

D 0   D 01 

T

2 0

T









T

,..., B 0N ,..., D 0N

T

2 0

,..., A 0N

T

T

T

 , 

  , C 

T

 , C   C1 0   0

T

2 0

T

 

,..., C 0N

T

T

 , 

T

 

 X Let Q and Q 1 be partitioned in the form of Q :  T M

Y Define matrices K :  T N

 

T

M Y and Q 1 :  T  U  N

N V

  , where we require that X Y 

1

.

In  , K : diag  K ,..., K  , K : diag  K , I q , K , K  , and Kˆ : diag K , I q , I m , I m N  . Per- and 0 

post-multiplying (25) by K T and K , and (26) by Kˆ T and Kˆ , respectively, and taking into account (7) and carrying out some multiplications and through the substitution of Z : MB f , Z : C f N T , Zˆ : MAf N T

(27)

(28) and (29) are obtained as follows:     1Y    1 I n    0  AY   ˆ  X AY  ZC 2Y  Z  1

Y   *  *   *  * 

* 0 A

* *  I q B1

* * * Y

* * * *

X A  ZC 2 2

X B 1  ZD 21 3

I n 0

X 0

   1 X

I n X

0

Y C 1T Y C 2T D fT  Z T

0

C 1T  C 2T D fT

*

(   )I q

T D11T  D 21 D fT

*

*

Im

*

*

*

    0   *  4  * * * *

(28)

И1   И2  И3   0  0   I m N 

(29)

where  1,1  1,2 1     1, N 1,t

   2,1    3,1       I   Y  ,    2,2  ,     3,2  ,   diag   Y  2 3 4  ,...,  I       I  X          2, N     3, N        A 0tY A 0t B 0t  ,  3,t   ,  ,  2,t   t t t t  t t  X A Y  ZC Y X A  ZC X B  ZD 0 0 0  0   0  0

I   , X  

И1   И1,1 , И1,2 ,..., И1, N  , И 2   И 2,1 , И 2,2 ,..., И 2, N  , И 3   И 3,1 , И 3,2 ,..., И 3, N  И1,t  Y C 0t  D fT , И 2,t   C 0t  D fT , И 3,t    D 0t  D fT T

R Define H :  0

T

T

R  , H : diag  H ,..., H  , H : diag H , I q , H , H  , and Hˆ : diag H , I q , I m , I m N  . Performing I n 

congruence transformations to (28) by H and to (29) by Hˆ , and substituting ˆ , T : ZR , R :Y S : ZR

1

, Dˆ f : Df , W  X  R

(30)

(23a) and (23b) are achieved.

11

(b) If there exists a solution to (23a,b), from (27) we obtain that ˆ T , B  M 1Z , C  ZN Af  M 1ZN f f

T

(31)

Applying (28) in the transfer function matrix of the estimator (4), we find that H zyˆ (z )  C f (zI  A f ) 1 B f  D f

  Z  z (I

 Z zMN T  Zˆ n



1

Z  Df

 X Y )  Zˆ



1

(32)

Z  Df

Now using (30), we obtain 1 H zyˆ (s )  T  z (R  X )  S  Z  Dˆ f

(33)

 T  zI  (R  X ) 1 S  (R  X ) 1 Z  Dˆ f 1

which gives the estimator matrices of (24). This completes the proof.



Remark 4. Similar to the Lemma 1, the tuning scalar  in Theorem 1 is bounded in the open interval  0,1  max eig (A ) 





i.e.,   0,1  max  eig (A ) , eig (Af ) 



  . Hence, during designing of estimator, 2

  , 2

 must necessarily lies in the interval



  0,1   max eig (A )  . 2

Remark 5. Similar to Remark 3, the inequalities (23a,b) are LMIs only if  is fixed, and hence can be readily solved by employing the feasp solver in the Matlab LMI toolbox [33]. Moreover, as the scalar  enters the LMIs (23a,b) affinely for a given  , it can be included as an optimization variable to obtain a tighter upper bound of the peak-to-peak norm of the estimation error system (6). Then, selecting the suitable value of  and using the mincx solver in the Matlab LMI toolbox [33], the minimum  can be easily found. The best achievable upper bound of  can be found by combining the minimization





of  ( ) under the LMI constraints (23a,b) for fixed  with a line search over   0,1  max  eig (A ) 

  , see [32]). 2

Due to the fact that LMIs (23a,b) are affine with respect to all the matrices involved, Theorem 1 can be extended for the case which these matrices are uncertain and lie in the polytope (3), which only requires to investigate the LMIs at the vertices of the uncertainty polytope.

12

Corollary 1. Consider the ESMS system (1a-c) over the polytope (3) and the estimator of (4). Given   0 and

 

   0,1  max eig (A j ) 

k   l  (





2

 , j  1, 2..., s , The estimation error system (6) is ESMS, and J is negative for all nonzero  S 

 

; q )

if two LMIs (23a,b) are satisfied by a single set of

 R ,W , Z , S ,T , Dˆ ,   for all the polytope vertices. In the latter case, the robust estimator matrices are obtained via (24). f

Remark 6.

Q

and consequently matrices R

,W , Z, S,

T

4.2. PDLF approach In order to decrease the conservativeness of the QLF approach in subsection 4.1, we utilize a PDLF approach by partitioning the Lyapunov function motivated by the work [34,35] and an idea of structured parameter dependent matrices inspired from [36,37]. Utilizing these ideas, some new sufficient conditions are obtained in terms of LMIs for the existence of the desired estimator. ’

  T n n

G m n

G n

x T x  0, x n : Gx  0, x  0 L n m :   LG  G T LT  0

the uncertain stochastic system (1a-c).

2. We consider the ESMS uncertain stochastic system (1a-c) and the estimator of (4). Given   0 , the following holds: (a) The estimation error system (6) is ESMS, and J S of (5) is negative for all nonzero k   l  (





 

; q ) and

if, for known positive constants  1 ,  2 , and  so that   1  max eig (A ( ))



2

for all

13

 M 11 ( ) n n

   , there exist N ( ) 2n 2n  0 , M ( ) : 

 M 21 ( ) 

n n

M 12 n n  n n n r m n  , Af  , B f  , C f  , M 12 n n 

D f m r , and    0,   such that the following two LMIs are satisfied:

 Њ  M ( ), N ( ),C



 M ( ), N ( ), A f , B f ,   0, f

(34a,b)



, Df ,   0

where  M ( )  M T ( ) * * * *     N ( )     1    M ( ) 2 N ( )  * * * 1 1     2  ,  :  0 0  I q * *    Џ1 ( ) 0 Џ 3 ( ) N ( ) *     Џ 2 ( ) 0 Џ 4 ( ) 0 Nˆ ( ) 

 M ( )  M T ( ) * * *   N ( )       M ( ) 2 N ( ) * * 2 2   2  Њ :  0 0 (   )I q *   Æ1 ( ) 0 D11  D f D 21 ( )  I m  Æ2 ( ) 0 Æ3 ( ) 0 

* * * *

 I m N

   ,     

 Џ 2,1 ( )   Џ 4,1 ( )   Æ2,1 ( )   Æ3,1 ( )   Џ ( )   Џ ( )   Æ ( )    2,2  , Џ ( )   4,2  , Æ ( )   2,2  , Æ ( )   Æ3,2 ( )  , Џ 2 ( )   4 2 3                 Џ (  ) Џ (  ) Æ (  ) Æ (  )  2, N   4, N   2, N   3, N  ˆ N ( )  diag N ( ),..., N ( ) ,





 M 11 ( )A 0t ( )  B f C 0t ( ) 0   M ( )A ( )  B f C 2 ( ) A f  Џ1 ( )   11 ,  , Џ 2,t ( )   t t  M 21 ( )A ( )  B f C 2 ( ) A f   M 21 ( )A 0 ( )  B f C 0 ( ) 0   M 11 ( )B 0t ( )  B f D 0t ( )   M ( )B 1 ( )  B f D 21 ( )  Џ 3 ( )   11 ,  , Џ 4,t ( )   t t  M 21 ( )B 1 ( )  B f D 21 ( )   M 21 ( )B 0 ( )  B f D 0 ( )  Æ1 ( )  C 1  D f C 2 ( ) C f  , Æ2,t ( )   D f C 0t ( ) 0  , Æ3,t ( )  D f D 0t ( )

(b) If (34a,b) is satisfied, the matrices of an admissible robust L  -induced estimator in the form of (4) can be extracted using the following equations: Af  M 121Af , B f  M 12 1B f , C f  C f , Df  Df

(35)

(a) According to Lemma 1, for given  and  , the mean-square exponential stability of (6) and the negativity of J S for all nonzero k   l  ( satisfied for



; q ) and

 

are provided that the following inequalities are

  :

14

   1Q ( ) 0 A T ( )Q ( ) A 0T ( )Q ( )    0  I q B1T ( )Q ( ) B 0T ( )Q ( )   0  Q ( )A ( ) Q ( )B 1 ( )  Q ( ) 0   0 Q ( )   Q ( )A 0 ( ) Q ( )B 0 ( )

(36)

and Q ( ) 0  0 (    )I q   C 1 ( ) D11 ( )  D 0 ( )  C 0 ( )

C 1T ( ) C 0T ( )   T D11 ( ) D 0T ( )  0 Im 0   0  I m N 

(37)

where



Q ( )  diag Q ( ),...,Q ( )  , A 0 ( )   A 01 ( ) 



 ,B



 ,D

B 0 ( )   B 01 ( ) 

T

D 0 ( )   D 01 ( ) 

T

2 0

( )

2 0



( )

T











,..., B 0N ( )

T

,..., D 0N ( )

 , A T

2 0

( )

T



T

,..., A 0N ( )





 , C

 , C ( )   C 1 ( ) 0   0

T

T

T



T

2 0

T

 , 

( )



T



,..., C 0N ( )



T

T

 , 

T

 

Q ( ) Q ( )  M T ( )N 1 ( )M ( )

N ( ) is a symmetric positive definite matrix and

(38)

M ( )





 N  1 ( )     

 0      1 M ( ) 1 ( )   2  0   M ( )A ( )   M ( )A 0 ( )

1T ( )1 ( )1 ( )  0

(39)

2T ( )2 ( )2 ( )  0

(40)

I 2n 1

( )M ( )

0

0

Iq

0

0

0

0

 1 2

0

M T ( ) 0 0 0 0

    0 0  1 N ( )M ( ) 0  0 N 1 ( )M ( )  0

0

0

0

 A 0T ( ) M T ( )    0 0 0    I q B 1T ( )M T ( ) B 0T ( ) M T ( )   M ( )B 1 ( ) N ( ) 0  M ( )B 0 ( ) 0 N ( )  0

A T ( )M T ( )

15

 N   2 ( )     

 0     M ( )  2 ( )   2  0   C 1 ( )   C 0 ( )

 2

I 2n 1

0

( )M ( )

0  0  0   0  I m N 

0

0

0

0

Iq

0

0

0

Im

0

0

0

M T ( )

0

0

0

0 0

(   )I q D11 ( )

0

D 0 ( )

 C 1T ( ) C 0T ( )   0 0    T T D11 ( ) D 0 ( )  Im 0   0  I m N 

N ( )  diag N ( ),..., N ( ) M ( )  diag M ( ),..., M ( )

It is worth noting that the non-singular matrix M ( ) is considered constant for the entire uncertainty domain in [34,35]. Inspired from [36,37], we only need to set part of M ( ) to be constant, which can yield less conservative results. More specifically, we partition M ( )

the following structure:  M ( ) M 12  M ( ) :  11   M 21 ( ) M 22  M 12

(41)

M 22

Define the following matrices 0 I  J 1 :  , J 2  diag J 1 ,..., J 1  , J : diag J1 , J 1 , I q , J 1 , J 2  , J=diag J1 , J 1 , I q , I m , I m N  -T T  0 M M  22 12 

(42)

performing congruence transformations to 1 ( ) by J and to 2 ( ) by J , respectively, the matrix inequalities

T J T 1 ( )J   0,   J 11 ( ) ,  (2N  4) n q ,   0

(43)

 T J T 2 ( )J   0,   J 12 ( ) ,  ( N 1) m q  2n ,   0

(44)

G1 ( )   M ( ) N ( ) 0 0 0

G2 ( )   M ( ) N ( ) 0 0 0

G1 ( )1 ( )  0

G 2 ( )2 ( )  0 , we obtain

 :   J  :   J

1 ( ) ,   0   : G1 ( )J   0,   0

(45)

2 ( ) ,   0   : G 2 ( )J   0,   0

(46)

1

1

16

T J T 1 ( )J   0,  (2N 6) n q : G1 ( )J   0,   0

(47)

 T J T 2 ( )J   0,  ( N 1) m q  4n : G 2 ( )J   0,   0

(48) L1 and L 2

J T 1 ( )J  L1G1 ( )J  J T G1T ( )L1T  0

(49)

J T 2 ( )J  L2G 2 ( )J  J T G 2T ( )L2T  0

(50)

L1 and L 2 , respectively, as L1  J L1 and L2  J L 2 with T

L1   I

1I

0 0 0 ,

L 2   I

 2I

0 0 0

T

T

T

(49) and (50) can be , respectively, expressed as

J T 1 ( )J  0

(51)

J T 2 ( )J  0

(52)

where  M ( )  M T ( ) * * * *     N ( )     1    M ( ) 2 N ( )  * * * 1 1     2  1 ( )    0 0  I q * *    M ( )A ( ) 0 M ( )B 1 ( ) N ( ) *    M ( )A 0 ( ) 0 M ( )B 0 ( ) 0 N ( )  

 M ( )  M T ( ) * *    N ( )       M ( ) 2 N ( ) * 2 2   2  2 ( )   0 0 (   ) I q   C1 ( ) 0 D11 ( )  C0 ( ) 0 D0 ( ) 

* * *  Im 0

*   *    *  *    I m N 

 M ( ) M 12  T M ( ) :  11   J 1 MJ 1 , M (  ) M  21 12  N ( ) : J 1T N ( )J 1 , Af  C f

B f   M 12  Df   0

0  Af I  C f

(53a-c)

B f  M M  D f   0 T 22

T 12

0  I

and taking into account (7), it can be readily verified that (51) and (52) are equivalent to (34a) and (34b), respectively. (b) If there exists a solution to (34a,b), from (53c) we obtain that

17

 Af C  f

B f   M 121  D f   0

0   Af  I  C f

T B f   M 12T M 22  Df   0

0  I

(54)

Applying (54) in the transfer function matrix of the estimator yields T H zyˆ (s )  C f M 12T M 22  zI  M 121Af M 12T M 22T  M 121B f  Df 1

(55)

After carrying out some transformations, we find that T H zyˆ (s )  C f  zI  M 12T M 22 M 121A f



 C f sI  M 121A f



1



1

T M 12T M 22 M 121B f  D f

(56)

M 121B f  D f

Which gives the estimator parameters of (35) and the proof is completed.

Remark 7. The LMI conditions in Theorem 2 still cannot be implemented due to their infinite-dimensional nature in the parameter  . Inspired by the work in [36,37], the infinite-dimensional conditions in Theorem 2 can be cast into finitedimensional LMI conditions that depend only on the vertex matrices of the polytope  , as given in the following Corollary.

2. We consider the ESMS uncertain stochastic system (1a-c) and the estimator of (4). Given   0 , the following holds: (a) The estimation error system (6) is ESMS, and J S of (5) is negative for all nonzero k   l  ( if,

 

   0,1  max eig (A j ) 



2

for

known

positive

constants

1 ,

2 ,

n n  , j  1, 2..., s , there exist N 2 n 2 n  0 , M :  M 11,i     i i n n   M 21,i  



and

 

; q ) and



so

that

M 12  n n  n n  , Af  , M 12  n n 

B f n r , C f m n , D f m r , and    0,   such that the following two LMIs are satisfied: ij   ji  0, Њi  0,

i

1  i

 j  s ,

(57a,b)

 1, 2,..., s  ,

where  M i  M iT  N     1    M 1 i  i  2  ij :  0   Џ1,ij   Џ 2,ij 

*

*

*

21N i

*

*

0

 I q

*

0

Џ 3,ij

N i

0

Џ 4,ij

0

*   *    *  *   Nˆ i 

(58)

18

 M i  M Ti   N       M 2 i  i  2  Њi :  0   Æ1,i  Æ2,i 

Џ 2,ij

*

*

*

*

2 2 N i

*

*

*

0 0 0

(   )I q D11  D f D 21,i Æ3,i

* Im 0

* *

 I m N

    ,    

(59)

j i  Џ ij2,1   Џ ij4,1   Æ2,1   Æ3,1   ij   ij   j   i  Џ Џ Æ Æ   2,2  , Џ 4,ij   4,2  , Æ2,j   2,2  , Æ3,i   3,2  ,          ij   ij   j   i   Џ 2,N   Џ 4,N   Æ2,N   Æ3,N 





Nˆ i  diag N i ,..., N i ,  M A  B f C 2, j Џ1,ij   11,i j  M 21,i A j  B f C 2, j M B Џ 3,ij   11,i 1, j  M 21,i B 1, j

 M 11,i A 0,t j  B f C 0,t j Af  ij  , Џ 2,t   t t A f   M 21,i A 0, j  B f C 0, j  M 11,i B 0,t j  B f D 0,t j   B f D 21, j  ij ,  , Џ 4,t   t t   B f D 21, j   M 21,i B 0, j  B f D 0, j  C f  , Æi2,t   D f C 0,t i

Æ1,i  C 1  D f C 2,i

0 , 0 

i 0  , Æ3,t  D f D 0,t i

(b) If (57a,b) is satisfied, the matrices of an admissible robust improved L  -induced estimator in the form of (4) are given by (35).

Proof. According Theorem 2, for given  ,  1 and  2 , there exists a robust peak-to-peak estimator in the form of (4) that leads to an ESMS estimation error system such that J S of (5) is negative for all nonzero

k   l  (



; q ) and

 

 M ( ) M 12  if there exist N ( )  0 , M ( ) :  11  , A f , B f , C f , D f , and  satisfying  M 21 ( ) M 12 

(34a,b). We assume that the matrices N ( ) and M ( ) have the following structure: s

N ( )    i N i , i 1

s s M M ( )    i M i    i  11,i i 1 i 1  M 21,i

(60)

M 12   M 12 





By considering (60),   M ( ), N ( ), Af , B f ,   and Њ M ( ), N ( ),C f , D f ,  in (34a,b) are written





s

s

s 1

s

 M ( ), N ( ), Af , B f ,    i  j ij  i2 ij   i 1 j 1

i 1

    s

i 1 j  i 1

i

j

ij

  ji



(61)

and





s

Њ M ( ), N ( ),C f , D f ,   i Њi

(62)

i 1

where ij and Њi are defined in (58) and (59).

19

ii  0,

i

 1,..., s  ,

ij   ji  0,

1  i

(63)

 j s





It is observed that inequalities (63) and (57b) guarantee   M ( ), N ( ), Af , B f ,    0 and Њ M ( ), N ( ),C f , D f ,   0 , respectively, and this completes the proof.

Remark 8. The scalar parameters  1 ,  2 provide extra degrees of freedom in the solution space and, hence, reduction of the conservativeness of the solutions. The question arises how to find the optimal combination of  1 ,  2 and  in order to obtain a tighter upper bound of the peak-to-peak norm. Inspired by [38,39], we first solve the feasibility problem of the LMIs (57a,b) with i  1,..., s using Matlab LMI toolbox [33] and obtain a set of initial scaling parameters. Then, applying a numerical optimization algorithm, such as the function fminsearchbnd of Matlab [40], a locally convergent solution to the problem is obtained.

Remark 9. As can be seen in (60), the matrices M ( ) and N ( ) used in Corollary 2 are linearly dependent on  . Very recently, the polynomial parameter-dependent approach has been proposed, which may lead to better results, see ,e.g., [41]. By using this approach, the parameter-dependent matrices are represented in the form of homogeneous polynomials with respect to the uncertain parameter  of arbitrary degree. This technique can be also applied to Theorem 2.

Remark 10. The proposed methods in this paper can also be applied and extended to design the induced L  filter for Takagi– Sugeno fuzzy stochastic systems (see [19,20]) which is an issue for a further research.

5. Illustrative examples In this section we bring two examples that are used to illustrate the effectiveness and benefits of the proposed approaches. The first example serves the purpose of comparing the proposed induced L  filter with the existing L2  L filter in [20] for the state estimation of an F-404 aircraft engine system. The second example presents an application of the proposed deconvolution technique on designing an equalizer for a wireless communication channel and compares the results with those obtained by the existing H 2 / H  estimators in [10,42].

20

0 0.111  0   0.931  0.051  -0.025 0.004 0.03  x (k  1)  0.008 0.98 0.017  x (k )  0.049 0.03   (k )   -0.0025 0.065 0.035 x (k ) (k ) 0.014  0.048 0.037   0.0030 -0.004 0.2  0 0.895  0  1 0 0  0.011 y (k )   x (k )    (k )  0.021 0 1 0  0

(64)

 (k )

 (k )

1 (k ), 2 (k )

T

1 (k )  sin (0.1k)

 1, 1

2 (k )

z (k )  I 3 x (k )

We will design a linear dynamic filter of the form (4) that estimates the vector z (k )  I 3 x (k ) . is obtained   2.2693 for   0.005 , and the

Theorem 1, the minimum L  -induced peak-to-peak filter matrices are as follows:

-0.0678 -0.0251 0.1434   0.9935 0.0293 A f   0.6341 0.2580 -0.1689 , B f   -0.6194 0.7996  0.0971 -0.2996 0.8782   -0.0694 0.3098 0  0.0004 0.0002  0.9996 -0.0002  C f   0.0651 0.0333 0  , D f  -0.0617 0.9370   0.0638 -0.1738 0.5912  -0.0943 0.2866 

L 2  L

xˆ (k  1)  A f xˆ (k )  B f y (k ), zˆ (k )  Lf xˆ (k )

L2  L disturbance attenuation level

-0.1652 A f   0.3622  -0.7953 -0.9654 L f   -0.1058 -0.0476

(65)

  0.2345

-0.0616 0.1134   -1.1095 -0.0668 0.5046 -0.1005 , B f   0.3399 -0.5096  -0.8739 -0.4640 -0.4587 0.7629  -0.0366 0.0232  -0.7547 0.1610  0.0014 -0.3102

To evaluate performances of different methods, the mean-square error (MSE) criterion is adapted. The averaged MSE value at time step k is computed as

21

MSE (k )  E

 z (k )   N1  z 2

N

i 1

i

(k )

2

where N is the total number of samples, z i (k ) : z i (k )  zˆi (k ) denotes the ith sample of the estimation error vector at time step k . peak-to-peak filter and the L2  L filter in [20] 2

supk E z (k ) ) is bounded. However, the comparison result clearly shows that the peak value of the MSE of the filter designed by the proposed method is much smaller than the other one. The comparison result is consistent with our design strategy, since the disturbance



 (k ) is persistent bounded.

0.1 0.5 1 x (k  1)   x (k )    s (k )   0 0.1 1 y (k )  1   (k ) 1   (k )  x (k )   5   (k )  s (k )  0.3 (k )

(66)

 (k ) is the measurement noise which corrupts the

s (k ) received signal y (k ) .

s (k )

 (k )

12  1

 (k ) characterizes the channel identification error, which is a combination of both deterministic and

 22  1

stochastic parametric uncertainties as  (k )  0.5 d (k )  0.3 s (k ) where  d is deterministic uncertainty and satisfies  d  1 for all k 



and  s is a zero-mean white noise process with variance  32  1 and is independent of  (k ) and s (k ) .

We will design a linear time-invariant equalizer for the channel (66) in order to estimate the transmitted signal s (k ) .

x (k  1)  Ax (k )  B 1 (k ),

y (k )  C 2  C 01 s (k )  x (k )   D 21  D 01 s (k )   (k ),

(67)

z (k )  C 1x (k )  D11 (k )

where

22

 s (k ) 

0.1 0.5 1 0  , B1    , 0.1 1 0  1  0.5 d  , C 01   0.3 0.3

 (k )   , A   0  (k )  

C 2  1  0.5 d

D 21  5  0.5 d

0.3 , D 01   0.3 0 , C 1   0 0 , D11  1 0

Now, we compare the three following estimators to estimate the transmitted signal s (k ) Robust H 2 / H  deconvolution filter in [42]:

does not take the

multiplicative noise  s (k ) into account. The H 2 / H  deconvolution filter minimizes an upper-bound on the

 2  0.1241 while satisfying H  -norm disturbance attenuation level    1 . The H 2 / H  deconvolution filter has the structure similar to that of (4) and its matrices are  -0.0991 0.3026   0.1995  Af   , Bf     -0.1776 -0.1019  0.1887  C f   -0.1875 -0.1819 , D f  0.1871

2) Robust stochastic H 2 / H  filter in [10]: In order to employ this filter, the input estimation problem is first converted to a state estimation problem, resulting in the following modified channel model: 0.1 0.5 1 0  0   0 0.1 1 0     x (k )  0  s (k  1) x (k  1)   0 1  0 0 0     0 1 0 0 0  y (k )  1   (k ) 1   ( k ) 5   ( k ) 0  x ( k )  0.3 ( k )

(68)

z (k )   0 0 0 1 x (k )

A robust H 2 / H  filter with the structure similar to that of (65) is designed to estimate z (k ) . This filter minimizes an upper-

 2  0.1833 while satisfying H  attenuation

bound on the

 1.

The associated filter matrices are  0.3626  -0.3059 Af    0.0716  -0.1907 C f   -0.1121

0.2845   -0.1152  -0.2395 0.0030   , Bf    0.0256  -0.0288 0.3938 0.0277     -0.3150 0.0986 -0.3089  -0.2828 -0.2575 0.0294 -0.3082 0.2399

0.3657

-0.0553 0.1205

3) Robust peak-to-peak deconvolution filter using QLF approach: First, the proposed QLF and PDLF approaches in

 It can be seen that the

designing the robust peak-to-peak deconvolution filter are compared in PDLF approach (Corollary 2) w

Corollary 1

n order to obtain a desirable estimate of the signal s (k ) , the minimum peak-to-peak norm

is obtained

  0.9146 (for the suitable choice   0.1 ) by the PDLF approach. The corresponding filter matrices are as follows:

23

 0.0232 0.1746   -0.1881 Af   , Bf      -0.1178 -0.0091 -0.1749 C f   0.1430 0.0914 , D f  0.1317



MSE (k )  E e (k )

2

  N1  e (k ) N

i 1

ei (k ) : s i (k )  sˆi (k )

2

i

 d  0.9999 peak-to-peak deconvolution filter, robust H 2 / H  deconvolution filter in [42] and robust stochastic H 2 / H  filter in [10] In relation of simulation results, we note that the objective of H 2 / H  estimators in [10,42] is to minimize an upper-bound (  2 ) on the asymptotic mean square value of the estimation error lim E z (k ) k 

2

, hile a prescribed H  attenuation level is

assured. The H  norm constraint is due to its inherent robustness against the possibility that the statistics of the transmitted signal s or the receiver noise  are not actually white [10]. From Fig. 2, we find that the MSE of the deconvolution filter [42] is not bounded by  2  0.1241 , because

does not take account of the multiplicative noise  s (k ) . In

contrast, it is clear from Fig. 2 that the MSE of the stochastic filter [10] is bounded by  2  0.1833 , as guaranteed by the design procedure. In addition, the MSE of the filter [10] is smaller than that by the deconvolution filter [42]. Fig. 2 indicates that the peak value of the MSE is smallest for the proposed peak-to-peak deconvolution approach. It is not surprising since the objective of the proposed approach is to minimize an upper-bound on the peak value of the MSE. However, the performance may be degraded by the uncertainties due to the variations of the statistical properties of the input signal and the additive and multiplicative measurement noises. In this regard, we consider the case which the receiver noise

(k )  1 (k )  2 (k )

1 (k )

 22  2.25

 1.5, 1.5

2 (k )

The multiplicative noise  s is assumed to be a zero-

mean white noise process with variance  32  2 The corresponding simulation results are given in Fig. 3. We find from Fig. 3 that the filter [10] is more insensitive to the variations of statistical properties of noises than the deconvolution filter [42]. It is evident from Fig. 3 that the proposed deconvolution filter has the least sensitivity to such variations. Because no statistical assumption on the exogenous noise and input signals is needed for the proposed peak-to-peak method, while on the contrary, two H 2 / H  estimators in [10,42] were designed for Gaussian white noise models in the measurement noise and the exogenous input. Furthermore, the MSE of the proposed deconvolution filter is still the smallest in comparison with two other approaches.

24

Both simulation results confirm that the proposed peak-to-peak methods are not only more robust for dealing with deterministic and stochastic parametric uncertainties but are also less sensitive to variations of the statistical properties of input and noise signals.

In this paper,

the problem of induced L  filtering and deconvolution for stationary discrete-time linear

stochastic uncertain systems with unknown-but-bounded inputs and external noises. Stochastic and deterministic parametric uncertainties appear in both state and measurement equations in the state–space model of the system. We proved a new lemma that presents sufficient conditions in terms of LMIs for a prescribed upper bound of the induced L  norm of this class of systems. Based on this key lemma, two design methods have been proposed based on quadratic and parameter-dependent stability ideas with different degrees of conservativeness and computational complexity. Finally, two numerical examples have been given to illustrate the obvious advantages of the proposed approaches. Simulation results have confirmed that the proposed peak-to-peak estimators has more robustness compared to L2  L and H 2 / H  design methods against parametric (deterministic or stochastic) uncertainties and variations of the statistical properties of driving signals and noises.

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Highlights



The problems of filtering and deconvolution are addressed for discrete-time linear systems with deterministic and stochastic uncertainties.



A new lemma is derived which provides the sufficient conditions in terms of LMIs for the guarantee of an upper bound on the induced L  norm of this class of systems



Two induced L  estimator design methods have been proposed based on quadratic and parameter-dependent stability ideas.

Table 1 Minimum peak-to-peak performances for different approaches

Corollary 1

  0.1

  0.3

  0.6

  0.8

1.0018

0.7163

0.6092

0.6497

27

Corollary 2 with 1   2  1

0.9146

0.6011

0.4945

0.5208

Figure 1:

0.2

the method in [11f] the proposed method

0.18 0.16

Averaged MSE

0.14 0.12 0.1 0.08 0.06 0.04 0.02 0

0

50

100

150

200

250

300

350

400

450

500

120

140

160

180

200

Steps

Fig. 1

Figure 2:

0.25

Averaged MSE

0.2

0.15

0.1

0.05

0

0

20

40

60

80

100

Steps

H2 / H

Fig. 2

H2 / H

28

Figure 3:

Averaged MSE

0.25

0.2

0.15

0.1

0.05

0

0

20

40

60

80

100

120

140

160

180

200

Steps

Fig. 3 H2 / H

H2 / H

29