Robust stability of 2-D discrete systems employing generalized overflow nonlinearities: An LMI approach

Digital Signal Processing 21 (2011) 262–269

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Digital Signal Processing www.elsevier.com/locate/dsp

Robust stability of 2-D discrete systems employing generalized overflow nonlinearities: An LMI approach Anurita Dey ∗ , Haranath Kar Department of Electronics and Communication Engineering, Motilal Nehru National Institute of Technology, Allahabad-211004, India

a r t i c l e

i n f o

a b s t r a c t

Article history: Available online 1 July 2010

This paper deals with the problem of global asymptotic stability of a class of uncertain discrete systems described by the Fornasini–Marchesini second local state-space (FMSLSS) model using generalized overflow nonlinearities. The systems under investigation involve parameter uncertainties that are assumed to be deterministic and norm bounded. An LMI-based criterion for the global asymptotic stability of such systems is proposed. The proposed criterion is compared with previously reported criteria. © 2010 Elsevier Inc. All rights reserved.

Keywords: Linear matrix inequality Lyapunov stability Nonlinear systems Robust stability 2-D systems Uncertain systems

1. Introduction Due to the rapid increase of the applicability of two-dimensional (2-D) discrete systems theory in many areas such as image data processing and transmission, seismographic data processing, radar and sonar processing, thermal processes in chemical reactors, heat exchangers and pipe furnaces, 2-D digital control systems [1–4], grid based wireless sensor network modeling [5], river pollution modeling [6], etc., the research on 2-D systems has received considerable attention in recent years. In the implementation of linear discrete systems, signals are usually represented and processed in a finite wordlength format which frequently produces several kinds of nonlinearities, such as overflow and quantization. Such nonlinearities may lead to instability in the designed system [7–9]. The common types of overflow nonlinearities are saturation, zeroing, two’s complement, and triangular. The stability properties of 2-D discrete systems described by the Fornasini–Marchesini second local state-space (FMSLSS) model [10] have been studied in [11–28]. Several publications [11,15,19–24] relating to the issue of the global asymptotic stability of 2-D discrete systems described by the FMSLSS model using overflow nonlinearities have appeared. Parametric uncertainties, which are inherent features of many physical systems, may lead to instability and poor performance of the system. Such uncertainties may arise due to the modeling errors, variations in system parameters and some ignored factors [25–29]. The stability properties of uncertain 2-D discrete systems described by the FMSLSS model employing finite wordlength nonlinearity have been studied in [25–28]. This paper deals with the problem of global asymptotic stability of a class of 2-D discrete uncertain systems described by the FMSLSS model employing generalized overflow nonlinearities. Parametric uncertainties involved in the system under investigation are assumed to be deterministic and norm bounded. The paper is organized as follows. In Section 2, the description of the system under consideration and previously reported results are given. In Section 3, we present the main results. The presented approach enables the formulation of the criterion based on the linear matrix inequality (LMI) [30,31],

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Corresponding author. Fax: +91 532 2545341. E-mail addresses: [email protected] (A. Dey), [email protected] (H. Kar).

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©

2010 Elsevier Inc. All rights reserved.

A. Dey, H. Kar / Digital Signal Processing 21 (2011) 262–269

263

without a need for searching for the scaling parameter as in [26], which is beneficial in terms of numerical complexity. To establish the importance of the present method, in Section 4 we compare our results with previously reported results [26,27]. 2. System description and previous results The following notations are used throughout the paper: R n×n Rn I 0 GT G >0 G <0

.

set of n × n real matrices set of n × 1 real vectors identity matrix of appropriate dimension null matrix or null vector of appropriate dimension transpose of the matrix (or vector) G G is positive definite symmetric matrix G is negative definite symmetric matrix any vector or matrix norm

The 2-D discrete uncertain system to be studied presently is described by the FMSLSS model employing generalized overflow arithmetic. Specifically, the system under consideration is given by





 





x11 (i , j ) = f y (i , j ) = f 1 y 1 (i , j )



f 2 y 2 (i , j )

...



T

f n y n (i , j )



y ( i , j ) = ( A 1 +  A 1 ) x ( i , j + 1) + ( A 2 +  A 2 ) x ( i + 1, j ) = y 1 ( i , j ) i  0,

,

y 2 (i , j )

...

j0

T y n (i , j ) ,

(1a) (1b) (1c)

n×n

n×n

where xξ τ (i , j ) = x(i + ξ, j + τ ), x(i , j ) ∈ R is the state vector; A 1 ∈ R , A2 ∈ R are the known constant matrices;  A 1 ∈ R n×n ,  A 2 ∈ R n×n are the unknown matrices representing parametric uncertainties in the state matrices. The nonlinearities under consideration are generalized overflow nonlinearities given by n

⎫  y k (i , j ) > 1 ⎪ ⎬  −1  yk (i , j )  1 f k yk (i , j ) = yk (i , j ), ⎪   ⎭ − L 2  f k yk (i , j )  − L , yk (i , j ) < −1 

L  f k y k (i , j )  L 1 ,



k = 1, 2, . . . , n

(2a)

where

−1  L  1,

L  L 1  1,

L  L 2  1.

(2b)

With the appropriate choice of L, L 1 , and L 2 , (2a) represents the usual types of overflow arithmetics employed in applications such as saturation (L = L 1 = L 2 = 1), zeroing (L = L 1 = L 2 = 0), triangular (L = −1, L 1 = L 2 = 1), two’s complement (L = −1, L 1 = L 2 = 1), etc. The uncertain matrices  A 1 and  A 2 are assumed to satisfy

[ A 1  A 2 ] = M F (i , j )[ N 1

N 2]

(3a)

where M , N 1 , N 2 are known constant matrices with appropriate dimension and F (i , j ) is an unknown matrix representing parameter uncertainty which satisfies

F T (i , j ) F (i , j )  I .

(3b)

It is assumed [19,21,22,24,27,28] that system has a finite set of initial conditions, i.e., there exist two positive integers K and L such that

x(i , 0) = 0,

i  K;

x(0, j ) = 0,

j  L.

(4)

The zero solution of the 2-D discrete system (1)–(4) is said to be globally asymptotically stable if the following holds [24,32]: (1) it is stable in the sense of Lyapunov i.e., for every ε > 0, there exists a δ = δ(ε ) > 0 such that x(i , j ) < ε for all i  0, j  0, whenever x(i , 0) < δ for 0  i  K and x(0, j ) < δ for 0  j  L, where K and L are specified in (4), and (2) every solution of (1a) tends to the origin as i → ∞ and/or j → ∞, i.e.,

lim

i →∞ and/or j →∞

x( i , j ) =

lim x(i , j ) = 0

i + j →∞

for system (1a) for any set of initial conditions satisfying (4).

264

A. Dey, H. Kar / Digital Signal Processing 21 (2011) 262–269

The problem of global asymptotic stability of the system described by (1)–(4) has been studied in [26,27]. The main result of [26] may be stated as follows. Theorem 1. (See [26].) The zero solution of the perturbed nonlinear 2-D filter characterized by (1)–(4) is globally asymptotically stable if, for some positive scalar α (0 < α < 1), there exist a positive scalar σ and a positive definite symmetric matrix P = [ pkl ] ∈ R n×n such that n

(1 + L ) pkk  2

| pkl |,

k = 1, 2, . . . , n

(5a)

l=1, l=k

and

⎡

−P

⎢ AT P ⎢ 1 ⎢ T ⎣ A2 P

P A1

P A2

PM

−α P + σ N 1T N 1

0

σ N 2T N 1

σ N 1T N 2 −(1 − α ) P + σ N 2T N 2

0

0

−σ I

MT P

⎤

⎥ ⎥ ⎥ < 0. 0 ⎦

(5b)

A sufficient condition for the global asymptotic stability of system (1)–(4) employing saturation nonlinearities has been presented in [28]. Unlike [26], the approaches [27,28] are free from searching the scaling parameter α , 0 < α < 1. However, the conditions reported in [27,28] involve, in addition to the matrix P , some other unknown auxiliary matrix or matrices. 3. Main results In this section, the global asymptotic stability conditions of system (1)–(4) are established. Before presenting our main results, we recall the following lemmas. Lemma 1. (See [19].) An n × n positive definite symmetric matrix P = [ pkl ] satisfies







| pkl |,

k = 1, 2, . . . , n .



y T (i , j ) P y (i , j ) − f T y (i , j ) P f y (i , j )  0

(6a)

if and only if n

(1 + L ) pkk  2

(6b)

l=1,l=k

Lemma 2. (See [27,29,30].) Let Σ , Γ , F and R be real matrices of appropriate dimensions with R satisfying, R = R T , then

R + Σ FΓ + Γ T F T ΣT < 0

(7)

for all F T F  I , if and only if there exists a positive scalar σ such that

R + σ −1 ΣΣ T + σ Γ T Γ < 0.

(8)

Next, we present the main results of the paper. Theorem 2. The system described by (1)–(4) is globally asymptotically stable if there exist a positive scalar σ and n × n positive definite symmetric matrices P 1 , P = [ pkl ] such that n

(1 + L ) pkk  2

| pkl |,

k = 1, 2, . . . , n

(9a)

l=1, l=k

and

⎡

−P

⎢ AT P ⎢ 1 ⎢ T ⎣ A2 P MT P

P A1

P A2

PM

− P 1 + σ N 1T N 1

σ N 1T N 2 −( P − P 1 ) + σ N 2T N 2

0

0

−σ I

σ

N 2T 0

N1

⎤

⎥ ⎥ ⎥ < 0. 0 ⎦

(9b)

A. Dey, H. Kar / Digital Signal Processing 21 (2011) 262–269

265

Proof. Consider a 2-D quadratic Lyapunov function

V ξ τ (i , j ) = xξTτ (i , j ) P xξ τ (i , j ).

(10)

Now, following [17] (see [12,15,27] also), we define  V (i , j ) as

 V (i , j ) = V 11 (i , j ) − V˜ 01 (i , j ) − V˜ 10 (i , j )

(11a)

T V˜ 01 (i , j ) = x01 (i , j ) P 1 x01 (i , j ),

(11b)

T V˜ 10 (i , j ) = x10 (i , j )( P − P 1 )x10 (i , j ).

(11c)

where

Eq. (11a), in view of (1a), takes the form

    T T  V (i , j ) = f T y (i , j ) P f y (i , j ) − x01 (i , j ) P 1 x01 (i , j ) − x10 (i , j )( P − P 1 )x10 (i , j )

(12)

which can be rearranged as T

 V (i , j ) = −ˆx (i , j ) W xˆ (i , j ) − α

(13)

where









α = y T (i , j ) P y (i , j ) − f T y (i , j ) P f y (i , j ) , T

xˆ (i , j ) =



W =



T x01 (i ,

j)

T x10 (i ,

(14)

 j) ,



P 1 − ( A 1 +  A 1 )T P ( A 1 +  A 1 ) −( A 1 +  A 1 )T P ( A 2 +  A 2 ) . ( P − P 1 ) − ( A 2 +  A 2 )T P ( A 2 +  A 2 ) −( A 2 +  A 2 )T P ( A 1 +  A 1 )

(15) (16)

In view of Lemma 1, (9a) implies that the quantity α is nonnegative. Therefore, if W > 0, then  V (i , j )  0, where the equality sign holds only when x01 (i , j ) = 0, x10 (i , j ) = 0. From (11) and condition  V (i , j )  0, it follows that

V (i + 1, j + 1)  x T (i , j + 1) P 1 x(i , j + 1) + x T (i + 1, j )( P − P 1 )x(i + 1, j ).

(17)

For any positive integer d  max{ K , L }, with K and L defined by (4), we have



V (i , j ) = V (0, d + 1) + V (1, d) + V (2, d − 1) + · · · + V (d, 1) + V (d + 1, 0)

i + j =d+1

 xT (1, d − 1)( P − P 1 )x(1, d − 1) + xT (1, d − 1) P 1 x(1, d − 1) + xT (2, d − 2)( P − P 1 )x(2, d − 2) + xT (2, d − 2) P 1 x(2, d − 2) + xT (3, d − 3)( P − P 1 )x(3, d − 3) + · · · + xT (d − 1, 1) P 1 x(d − 1, 1) = xT (1, d − 1) P x(1, d − 1) + xT (2, d − 2) P x(2, d − 2) + · · · + xT (d − 1, 1) P x(d − 1, 1)

= V (i , j )

(18)

i + j =d

where use has been made of (17) and x(0, d) = x(d, 0) = x(0, d + 1) = x(d + 1, 0) = 0. Consequently,

lim

i →∞ and/or j →∞

x(i , j ) =

lim x(i , j ) = 0.

(19)

i + j →∞

Thus condition W > 0 is a sufficient condition for the global asymptotic stability of the system described by (1)–(4). Using the well known Schur’s complement the condition W > 0 can equivalently be expressed as

⎡

P

⎢ ⎣ −( A 1 +  A 1 )T P −( A 2 +  A 2 )T P

− P ( A1 +  A1) − P ( A2 +  A2) P1

0

0

P − P1

⎤ ⎥ ⎦ > 0.

(20)

Using (3a), condition (20) can be rewritten in the following form:

R + M F N + NT F T MT < 0

(21a)

266

A. Dey, H. Kar / Digital Signal Processing 21 (2011) 262–269

where

⎡

⎤

−P

P A1

P A2

R = ⎣ A 1T P

−P1

0

0

−( P − P 1 )

⎢

A 2T P





MT = MT P N = [0

⎥ ⎦,

(21b)

0 0 ,

(21c)

N 2 ].

N1

(21d)

Applying Lemma 2, (21) becomes

R + σ −1 M M T + σ N T N < 0.

(22)

By using Schur’s complement, it is easy to show that (22) is equivalent to (9b). This completes the proof of Theorem 2.

2

Remark 1. To test the global asymptotic stability of system (1)–(4) via Theorem 2, one should first solve the condition (9b) for σ > 0, P 1 = P 1T > 0, and P = P T > 0 using Matlab LMI Toolbox [30,31] and also check if there is a solution P = P T > 0 meeting (9a). Thus, this method essentially involves repeated searching of σ > 0, P 1 = P 1T > 0, and P = P T > 0 satisfying (9b) until a solution P = P T > 0 meeting (9a) is found. If a feasible solution of (9) exists, then the system (1)–(4) is globally asymptotically stable. If there are no solutions of (9), then one cannot draw any conclusion regarding the global asymptotic stability of the system under consideration. In the following, as an extension of the present approach, we establish a criterion which is true LMI-based and computationally simpler than Theorem 2. Suppose P = [ pkl ] ∈ R n×n is a matrix characterized by n

pkk = gk +

(αkl + βkl ),

k = 1, 2, . . . , n ,

(23a)

l=1, l=k

pkl = plk =

(1 + L ) 2

αkl = αlk > 0, g k > 0,

(αkl − βkl ),

βkl = βlk > 0,

k, l = 1, 2, . . . , n (k = l),

(23b)

k, l = 1, 2, . . . , n (k = l),

(23c)

k = 1, 2, . . . , n .

(23d)

For n = 3, the matrix P takes the form

⎡ ⎢

g 1 + α12 + β12 + α13 + β13 (1 + L )

P =⎣

2

(α12 − β12 )

2

(α13 − β13 )

(1 + L )

(1 + L ) 2

(α12 − β12 )

g 2 + α12 + β12 + α23 + β23 (1 + L ) 2

(α23 − β23 )

(1 + L ) 2

(α13 − β13 )

(1 + L ) 2

(α23 − β23 )

⎤ ⎥ ⎦

(24)

g 3 + α13 + β13 + α23 + β23

where

αkl = αlk > 0,

βkl = βlk > 0,

g k > 0,

k, l = 1, 2, 3 (k = l).

Now, we prove the following facts: Fact 1. The matrix P = [ pkl ] ∈ R n×n defined by (23) is positive definite. Proof. Note that

ξT Pξ =

=

n

gk ξk2 +

n n −1



k =1

k=1 l=1, l>k

n

n −1

k =1

gk ξk2

+

n

k=1 l=1, l>k

  ξk2 + ξl2 (αkl + βkl ) + ξk ξl (1 + L )(αkl − βkl )

2     (1 + L )(αkl − βkl ) (1 + L )2 (αkl − βkl )2 2 ξl ξl + 1 − (αkl + βkl ) ξk + 2(αkl + βkl ) 4(αkl + βkl )2

(25)

where ξ T = [ξ1 ξ2 . . . ξn ]. Clearly, the right-hand side of (25) is nonnegative; ξ T P ξ > 0 for all ξ = 0 and is zero only for ξ = 0. Hence, P is positive definite matrix. 2

A. Dey, H. Kar / Digital Signal Processing 21 (2011) 262–269

267

Fact 2. The matrix P = [ pkl ] ∈ R n×n defined by (23) satisfies (9a). Proof. Using (23), we obtain





n

(1 + L ) pkk = (1 + L ) gk +

(αkl + βkl )

l=1, l=k

> (1 + L )

n

n

|αkl − βkl | = 2

l=1, l=k

| pkl |,

k = 1, 2, . . . , n .

(26)

l=1, l=k

2

Therefore, P satisfies (9a).

Fact 3. Any positive definite symmetric matrix P = [ pkl ] ∈ R n×n satisfying (9a) can always be expressed in the form of (23). Proof. For a given positive definite symmetric matrix P = [ pkl ] ∈ R n×n satisfying (9a), there always exist αkl = αlk > 0, βkl = βlk > 0, k, l = 1, 2, . . . , n (k = l) and gk > 0, k = 1, 2, . . . , n satisfying (23). To express a given positive definite symmetric matrix P satisfying (9a) in the form of (23), one may select, for instance,

αkl = αlk =



2

(1 + L )

 | pkl | + εkl ,

2

βkl = βlk = εkl , (1 + L )  gk = where

pkk −

2

(1 + L )

k, l = 1, 2, . . . , n (k = l),

(27)

k, l = 1, 2, . . . , n (k = l), n

 | pkl | −

l=1, l=k

4

(1 + L )

n

(28)

εkl , k = 1, 2, . . . , n

(29)

l=1, l=k

εkl > 0. This completes the proof. 2

In the light of above facts (Facts 1–3), Theorem 2 can equivalently be stated as follows. Theorem 3. The system described by (1)–(4) is globally asymptotically stable if there exist a positive scalar σ and n × n positive definite symmetric matrices P 1 , P satisfying (9b) where P is defined by (23). Remark 2. Theorem 3 may be treated as an LMI version of Theorem 2. By taking P in the form of (23), the matrix inequality (9b) becomes linear in the variables σ , gk (k = 1, 2, . . . , n), αkl , βkl (k, l = 1, 2, . . . , n (k = l)), P 1 and consequently, it can be solved using the Matlab LMI Toolbox [30,31]. By contrast, the approach of [26] does not provide a true LMI-based solution to this end. Remark 3. From Fact 2, it is clear that the matrix P characterized by (23) has a built-in feature of satisfying (9a), therefore, one needs not bother about (9a) while using Theorem 3. In view of this fact and Remark 1, Theorem 3 is computationally simpler than Theorem 2. 4. Comparative evaluation It is worth comparing Theorem 2 (or equivalently, Theorem 3) with Theorem 1. Note that (9a) is same as (5a). Further, (5b) is recovered from (9b) by restricting P 1 = α P subject, of course, to 0 < α < 1. Therefore, Theorem 1 is recovered from Theorem 2 as a special case. The main difference between Theorems 2 and 1 is that Theorem 2 does not involve searching of scaling parameter, namely α , as required in Theorem 1. Moreover, the use of a more general and flexible matrix P 1 instead of a scaled matrix α P as used in Theorem 1 makes the present approach less conservative than that of [26]. As an illustration of this, consider a specific example of 2-D discrete uncertain system (1)–(4) with



A1 =





0.8 1 , 0 0.1

A2 =





0.1 1 , 0 0.8

M=



0 , 0.01

N 1 = N 2 = [0.01 0],

L = −1.

(30)

Using the Matlab LMI Toolbox [30,31] and applying Theorem 3, it can easily be verified that (9b) is feasible for the following values of unknown parameters.

α12 = 1.4278, 

P1 =

β12 = 1.4278, 

3.6737 0.8302 , 0.8302 103.6793

g 1 = 1.4192,

σ = 124.3086.

g 2 = 621.6740, (31)

268

A. Dey, H. Kar / Digital Signal Processing 21 (2011) 262–269

Thus, according to Theorem 3, the system under consideration is globally asymptotically stable. On the other hand, it is verified that Theorem 1 fails to determine the robust stability of the present system. We would also like to point out that the presented approach may provide results which are not covered by the approach of [27]. For example, consider a 2-D discrete uncertain system (1)–(4) with



A1 =



1.1 −0.5 , 0.5 −0.1



A2 =



0.01 −0.01 , 0 0.01



M=



0 , 0.01

N 1 = N 2 = [0.01 0],

L = 0.

(32)

Applying Theorem 3, one can find that (9b) is feasible for the following values of unknown parameters.

α12 = 0.2022, 

P1 =

β12 = 8.3506, 

9.0121 −4.0887 , −4.0887 5.7928

g 1 = 0.7049,

σ = 6.1914.

g 2 = 0.1068, (33)

By contrast, it can be verified that [27, Theorem 5] fails as a global asymptotic stability test for the system described by (1)–(4). 5. Conclusions A criterion for the global asymptotic stability of uncertain 2-D discrete systems described by the FMSLSS model employing generalized overflow arithmetic has been proposed (Theorem 2). A computationally simpler version of the proposed criterion has been brought out (Theorem 3). In the presented approach, there is no need for searching the scaling parameter (i.e., α , 0 < α < 1) as in [26], which is beneficial in terms of numerical complexity. The proposed criteria always lead to a larger stability region in the parameter space, as compared to that obtainable via the approach of [26] and may provide results not covered by [27]. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]

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Anurita Dey was born in Allahabad, India, in 1979. She received the B.Sc., B.Tech., and M.Tech. degrees from the University of Allahabad, Allahabad, in 2000, 2003, and 2006, respectively. In 2000, she received the Meghnad Saha Centenary Gold Medal from the Physics Society, University of Allahabad. She is currently working toward the Ph.D. degree at Motilal Nehru National Institute of Technology, Allahabad. Her interests are in nonlinear systems, robust systems, and multidimensional systems. Haranath Kar was born in Bankura, India, in 1968. He received the B.E. degree from Bengal Engineering College in 1989, the M.Tech. degree from Banaras Hindu University, Varanasi, India, in 1992, and the Ph.D. degree from the University of Allahabad, Allahabad, India, in 2000. After spending a brief period at the Defense Research and Development Organization as a Scientist B, he joined Motilal Nehru National Institute of Technology (formerly known as M.N.R. Engineering College), Allahabad, as Lecturer in 1991, where he became Assistant Professor in 2001, Associate Professor in 2006, and Professor in 2007. He spent two years with the Atilim University, Turkey (2002–2004) as an Assistant Professor. Presently, he is also the Chairman of the Senate Post-Graduate Committee, Motilal Nehru National Institute of Technology. His current research interests are in digital signal processing, nonlinear dynamical systems, delayed systems, robust stability, guaranteed cost control, and multidimensional systems. He is a recipient of the 2002–2003 IEE Heaviside Premium Award. In 2005, he received the D.N. Agrawal Award of excellence for his outstanding services rendered in the field of science and technology.