Applied Mathematics and Computation 217 (2010) 2877–2882

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate/amc

LMI approach to stability of a class of discrete-time systemsusing two’s complement arithmetic

Vimal SinghDepartment of Electrical-Electronics Engineering, Atilim University, Ankara 06836, Turkey

a r t i c l e i n f o

Keywords:Asymptotic stabilityDiscrete-time dynamical systemDigital filterFinite word length effectLimit cycleLyapunov method

0096-3003/$ - see front matter � 2010 Elsevier Incdoi:10.1016/j.amc.2010.08.023

E-mail addresses: vsingh11@rediffmail.com, vim

a b s t r a c t

systematic and computationally tractable (i.e., linear matrix inequality based) approach tothe global asymptotic stability of a class of discrete-time systems, implemented with fixed-point arithmetic, utilizing a very strong nonlinearity (which is produced by employingtwo’s complement overflow arithmetic) is presented. Application of the approach to sec-ond-order digital filter is discussed. Two third-order examples illustrating the effectivenessof the approach are given.

� 2010 Elsevier Inc. All rights reserved.

1. Introduction

When a linear time-invariant discrete-time system [1] is implemented on a digital computer or on special-purpose digitalhardware, errors due to finite register length are unavoidable. The signals in discrete-time systems are represented with fi-nite precision arithmetic. Arithmetic operations (e.g., multiplications and additions) performed generally lead to an increasein the required word length. Therefore, precautionary measures have often to be taken for signal word length reduction,namely quantization and overflow corrections. These word length reductions have the effect of inserting nonlinearities(quantization and overflow nonlinearities) [2–4]. The nonlinearities may lead to instability. The limit cycle phenomenon,which is a characteristic of nonlinear systems, may possibly occur in the system if the system parameters are not chosenproperly [5]. The zero-input limit cycles represent an unstable behavior and are undesirable in digital filters [5,6]; for a re-view of the problems arising due to the zero-input limit cycles, the reader is referred to [6]. An important step in the imple-mentation of a discrete-time system using fixed-point arithmetic is, therefore, to find the ranges of the values of the systemparameters and choose the values so that the system is free from limit cycles. The quantization and overflow nonlinearitiesmay interact with each other. However, if the total number of quantization steps is large or, in other words, the internal wordlength is sufficiently long, then quantization effects can be neglected when studying the effects of overflow [2]. Various mod-els of digital filters have been considered in the literature. For example, [5,7–11] have considered direct-form digital filtersutilizing single overflow nonlinearity. On the other hand, state-space digital filters involving multiple nonlinearities are stud-ied in [3,4,6,12–14].

This paper deals with direct-form digital filters, implemented in fixed-point arithmetic, employing single overflow non-linearity. The stability of such filters when the overflow arithmetic is of the saturation type (saturation nonlinearity) hasbeen investigated extensively [7–11]. This paper considers two’s complement overflow arithmetic (which yields a verystrong nonlinearity [2–4]) and presents a new criterion for the global asymptotic stability of the filter with such arithmetic.To the best of author’s knowledge, this paper is the first report on presenting a systematic approach to the stability of direct-form digital filters using two’s complement overflow arithmetic. The criterion takes the form of linear matrix inequality(LMI) and, hence, is computationally tractable. Examples showing the effectiveness of the criterion are given.

. All rights reserved.

al_singh@atilim.edu.tr

2878 V. Singh / Applied Mathematics and Computation 217 (2010) 2877–2882

2. System description

The system under consideration is a direct-form digital filter, involving single overflow nonlinearity, implemented withfixed-point arithmetic [5,7–11]. To be specific, the system is given by

GðzÞ ¼ h1z�n þ h2z�ðn�1Þ þ � � � þ hnz�1;

yðrÞ ¼ output of GðzÞ;f ðyðrÞÞ ¼ input of GðzÞ:

9>=>; ð1Þ

The nonlinearity characterized by

f ðyðrÞÞ ¼ yðrÞ; if jyðrÞj 6 1jf ðyðrÞj 6 1; if jyðrÞj > 1

�ð2Þ

is under consideration. It is assumed that the effects of quantization are negligible. Eq. (2) includes, among others, two’scomplement overflow arithmetic [2–4]. The following condition is assumed to hold:

zn � hnzn�1 � hn�1zn�2 � � � � � h2z� h1–0; for alljzjP 1: ð3Þ

Condition (3) implies that the underlying linear system (i.e., when f(y(r) = y(r)) is stable), i.e., the infinite-precision counter-part of the system is stable.

3. Difference equation representation of the system

Consider

znUðzÞ ¼ FðzÞ ð4Þ

and

YðzÞ ¼ ðh1 þ h2zþ � � � þ hnzn�1ÞUðzÞ: ð5Þ

From (4) and (5) one obtains

GðzÞ ¼ YðzÞFðzÞ ¼

UðzÞFðzÞ �

YðzÞUðzÞ ¼ h1z�n þ h2z�ðn�1Þ þ � � � þ hnz�1: ð6Þ

Taking the inverse z-transform of (4) and (5) yields

uðr þ nÞ ¼ f ðyðrÞÞ ð7Þ

and

yðrÞ ¼ h1uðrÞ þ h2uðr þ 1Þ þ � � � þ hnuðr þ n� 1Þ; ð8Þ

respectively. Thus the difference Eqs. (7) and (8) are an alternative representation of the discrete-time nonlinear feedbacksystem given by (1)–(3), which has a linear block (with the transfer function G(z) = Y(z)/F(z) in the forward path and a non-linear block (with the input y(r) and the output f(y(r)) in the feedback path.

4. Main result

Define the matrix G = GT as

G ¼ GT ¼Cn�1 0

0 cnn

� �; ð9Þ

where Cn�1 ¼ CTn�1 is a (n � 1) � (n � 1) matrix, 0 denotes the null vector or null matrix of appropriate dimension, and the

superscript T to any vector (or matrix) denotes the transpose of that vector (or matrix). The main result of this paper is givenin the following theorem.

Theorem 1. The null solution of the system described by (1)–(3) is globally asymptotically stable if there is a

Cn�1 ¼ CTn�1 > 0 ð10Þ

and a positive constant cnn > 0 such that

G � ATGA > 0; ð11Þ

where G = GT is defined by (9),

V. Singh / Applied Mathematics and Computation 217 (2010) 2877–2882 2879

A ¼

00 In�1

..

.

0h1 h2 � � � hn

266666664

377777775; ð12Þ

In�1 denotes the (n � 1) � (n � 1) identity matrix, and > 0 denotes that the matrix is positive definite.

Proof. The state-space representation of the system is

x1ðr þ 1Þ ¼ x2ðrÞ; x2ðr þ 1Þ ¼ x3ðrÞ; . . . ; xn�1ðr þ 1Þ ¼ xnðrÞ; xnðr þ 1Þ ¼ f ðh1x1ðrÞ þ h2x2ðrÞ þ � � � þ hnxnðrÞÞ: ð13Þ

Eq. (13) can be put in the form

xðr þ 1Þ ¼ f ðyðrÞÞ ¼ f1ðy1ðrÞÞ f2ðy2ðrÞÞ � � � fnðynðrÞÞ½ �T; ð14Þ

where yðrÞ ¼ y1ðrÞ y2ðrÞ � � � ynðrÞ½ �T ¼ AxðrÞ; fiðyiðrÞÞ ¼ yiðrÞ ¼ xiþ1ðrÞ; i ¼ 1;2; . . . ; ðn� 1Þ; fnðynðrÞÞ ¼ f ðyðrÞÞ ¼ f ðhTxðrÞÞ;h ¼ h1 h2� � �hn½ �T; xðrÞ ¼ ðx1ðrÞ x2ðrÞ � � � xnðrÞ½ �T, and A is defined by (12).

Choose a quadratic Lyapunov function

vðxðrÞÞ ¼ xTðrÞGxðrÞ: ð15Þ

Along the trajectories of (14), one has

DvðxðrÞÞ ¼ vðxðr þ 1ÞÞ � vðxðrÞÞ ¼ f TðyðrÞÞGf ðyðrÞÞ � xTðrÞGxðrÞ: ð16Þ

Eq. (16) can be rearranged as

DvðxðrÞ ¼ �xTðrÞðG � ATGAÞxðrÞ � d; ð17Þ

where

d ¼ cnn y2nðrÞ � f 2

n ðynðrÞÞ� �

: ð18Þ

Observe that d can also be expressed as

d ¼ xTðrÞATGAxðrÞ � f TðyðrÞÞGf ðyðrÞÞ; ð19Þ

where G = GT is defined by (9). In fact, (19) is a key and novel step in the present approach.Owing to the conditions Cn�1 ¼ CT

n�1 > 0 and cnn > 0,G = GT (see (9)) is positive definite, i.e., the v(x(r)) function in (15) ispositive definite. In view of the restriction (2), the quantity d is nonnegative. Therefore, if the condition (11) is imposed, thenthe Dv(x(r)) in (17) is negative definite. Thus, under the conditions stated in Theorem 1, the null solution of the system isglobally asymptotically stable. This completes the proof of Theorem 1. h

Remark 1. The matrix inequality (11) is linear in the elements of the matrices P and G, i.e., (11) is an LMI. The elements ofthe matrices P and G are the LMI variables (decision variables). The LMI (11) can be solved using the LMI toolbox [15,16],which has the built-in feature of guaranteeing the existence or nonexistence, whichever is the case, of the values of the pre-vailing unknown constants and yielding the values (if they exist).

Remark 2. The stability of a class of continuous-time as well as discrete-time nonlinear systems incorporating the sector aswell as slope information of the nonlinearity has received considerable attention [17–27]. However, the two’s complementnonlinearity [2–4], which is represented by (2), belongs to a family of very strong nonlinearities and can best be character-ized by the quantity d given by (18) being nonnegative.

Remark 3. The stability of the system described by (1)–(3) with saturation overflow arithmetic has received considerableattention [7–11]. To the contrary, the stability of this system with two’s complement arithmetic has not received as muchattention. The result stated in Theorem 1 which pertains to two’s complement arithmetic is, therefore, the first of its kind.

Remark 4. The contrast between the stability result presented in this paper and a recently reported stability result [28] maybe noted. Whereas [28] deals with state-space digital filters (nth-order system) involving multiple two’s comlement nonlin-earities (n nonlinearities), the present paper deals with direct-form digital filters (nth-order system) involving single two’scomplement nonlinearity.

2880 V. Singh / Applied Mathematics and Computation 217 (2010) 2877–2882

5. Special case

The following corollary is a special case of Theorem 1.

Corollary 1. The null solution of the system described by (1)–(3) is globally asymptotically stable if there is a positive definitediagonal matrix

D ¼ diagðd1;d2; . . . ; dnÞ > 0; ð20Þ

such that the following LMI holds:

D� ATDA > 0; ð21Þ

where A is given by (12).

Proof. By choosing G (see (9)) as a diagonal matrix D, Corollary 1 directly follows from Theorem 1. h

Remark 5. Observe that, pertaining to second-order digital filter, the matrix G is a 2 � 2 diagonal matrix. In other words,pertaining to second-order digital filter, conditions (11) and (21) are one and the same. However, pertaining to third andhigher-order digital filters, the matrix G is more general than being a diagonal matrix, i.e., condition (11) is less restrictivethan condition (21).

6. Application to second-order digital filter

In this section, application of the proposed criterion to second-order digital filter is discussed. Pertaining to second-orderdigital filter (n = 2), condition (3) becomes the region

jh1j < 1; jh2j < 1� h1: ð22Þ

The region given by (22) is the region of stability of second-order digital filter if the filter is implemented with the infiniteprecision. Consider the region

jh1j < 1; jh2j < 1� jh1j; ð23Þ

which is a subset of (22). It is known [5] that, for certain points within the region (22) but out of the region (23), the second-order digital filter with two’s complement overflow arithmetic exhibits zero-input limit cycles (the so-called overflow oscil-lations). This same digital filter with saturation overflow arithmetic is found [5] to be free from overflow oscillations in theentire region (22). Global asymptotic stability criteria have been developed (for example, [7]), whereby the fact that thisfilter with saturation overflow arithmetic is globally asymptotically stable in the entire region (22) is established [7].

Note that since two’s complement overflow arithmetic yields a much stronger nonlinearity than the nonlinearity gener-ated by saturation overflow arithmetic [2–4], the global asymptotic stability region (or the so-called overflow stability re-gion) for two’s complement nonlinearity is expected to be smaller than the respective region for saturation nonlinearity.However, it is worth mentioning that the hardware implementation of two’s complement overflow arithmetic is knownto be simpler and less expensive than that of saturation overflow arithmetic [7].

No point within the region (23) can be found [5] for which this filter with two’s complement overflow arithmetic hasoverflow oscillations. In other words, (23) is the overflow stability region of second-order digital filter with two’s comple-ment arithmetic. In the following, it is shown that the global asymptotic stability criterion presented above successfullyyields this result.

It was shown in [6] that, pertaining to the 2 � 2 coefficient matrix A = (aij)2�2, a necessary and sufficient condition to sat-isfy (21) is the following:

ja11 � a22j < 1� detðAÞ; ð24Þ

where ‘det’ stands for the determinant and it is assumed that A is a Schur stable matrix. For A given in (12) corresponding ton = 2, the condition of A being Schur stable is (22) and (24), in conjunction with (22), reduces to (23). This shows, in light ofCorollary 1, that the second-order digital filter with two’s complement overflow arithmetic is globally asymptotically stablein the region (23). This is also a confirmation of the fact [5] that the second-order digital filter with two’s complement arith-metic is free from overflow oscillations in the region (23).

As pointed out above, for certain points within the region (22) but out of the region (23), the second-order digital filterwith two’s complement overflow arithmetic exhibits zero-input limit cycles (overflow oscillations) [5]. Some of these pointsare found [5] to be near the region (23). This means that the possibility of obtaining any better or substantially better two’scomplement overflow stability region than the region (23) for second-order digital filter is ruled out. In other words, for allpractical purposes, (23) may be considered as a necessary and sufficient condition or a condition being close to necessary andsufficient condition for the overflow stability of second-order digital filter with two’s complement arithmetic. As shown

V. Singh / Applied Mathematics and Computation 217 (2010) 2877–2882 2881

above, Corollary 1 establishes (23) as the global asymptotic stability region for this filter with two’s complement arithmetic.This, therefore, is an ample demonstration of the novelty and effectiveness of the proposed criterion.

7. Third-order examples

Example 1. Consider an example of a third-order digital filter with

h1 ¼ 0:5; h2 ¼ 0:4; h3 ¼ 0; ð25Þ

i.e.,

A ¼0 1 00 0 1

0:5 0:4 0

264

375: ð26Þ

Let

G ¼0:4 0:1 00:1 0:78 00 0 1

264

375: ð27Þ

Then one obtains

G � ATGA ¼0:15 �0:1 0�0:1 0:22 �0:1

0 �0:1 0:22

264

375: ð28Þ

The matrices in (27) and (28) are positive definite. Thus, in light of Theorem 1, the digital filter in this example is globallyasymptotically stable.

Example 2. Consider a third-order digital filter with

h1 ¼ 0; h2 ¼ a; h3 ¼ 0: ð29Þ

In this example, condition (3) becomes

z3 � az ¼ zðz2 � aÞ– 0; for all jzjP 1; ð30Þ

which is satisfied if and only if

�1 < a < 1: ð31Þ

In other words, (31) is a necessary and sufficient condition for the implementation of the digital filter in this example withthe infinite precision. The problem is to determine the condition for two’s complement overflow stability of the digital filterin this example if the filter is implemented with finite precision.

The matrix A (see (12)) in this example is

A ¼0 1 00 0 10 a 0

264

375: ð32Þ

One obtains

D� ATDA ¼d1 0 00 d2 � d1 � d3a2 00 0 d3 � d2

264

375: ð33Þ

For the matrix (33) to be positive definite, it is required to satisfy the following conditions:

d1 > 0; d2 > d1 þ d3a2; d3 > d2: ð34Þ

There exist d1 > 0, d2 > 0, d3 > 0 satisfying (34) for all values of a belonging to (31). For example, let d1 = 1, d2 = 0.5(3 + a2)/(1 � a2), d3 = 2/(1 � a2), where a is given by (31). Note that the quantity (1 � a2) is positive for all values of a belonging to(31). In other words, condition (21) is fulfilled in this example for all values of a belonging to (31). Thus, in light of Corollary1, (31) turns out to be a necessary and sufficient condition for two’s complement overflow stability of the finite-precisiondigital filter in this example. As noted above, (31) is also a necessary and sufficient condition for the stability of the

2882 V. Singh / Applied Mathematics and Computation 217 (2010) 2877–2882

infinite-precision digital filter in this example. This, therefore, is a further illustration of the effectiveness of the proposedcriterion.

8. Conclusion

This paper has addressed the problem of global asymptotic stability of fixed-point digital filters in direct-form realizationemploying two’s complement overflow arithmetic. A novel LMI-based criterion (sufficient condition) for the global asymp-totic stability of such filters is presented. Application of the criterion to derive the two’s complement overflow stability re-gion of second-order digital filter is highlighted. The effectiveness of the criterion is further substantiated via two third-orderexamples.

Acknowledgments

The author is grateful for the constructive comments and suggestions of the reviewer.

References

[1] A.V. Opeinhem, R.W. Schafer, Discrete-Time Signal Processing, second ed., Prentice-Hall, Upper Saddle River, NJ, 1999.[2] H.J. Butterweck, J.H.F. Ritzerfeld, M.J. Werter, Finite Wordlength Effects in Digital Filters – A Review, EUT Report 88-E-205, Eindhoven University of

Technology, Eindhoven, The Netherlands, 1988.[3] D. Liu, A.N. Michel, Asymptotic stability of discrete-time systems with saturation nonlinearities with application to digital filters, IEEE Trans. Circuits

Syst. I 39 (1992) 798–807.[4] D. Liu, A.N. Michel, Dynamical Systems with Saturation Nonlinearities: Analysis and Design, Springer-Verlag, New York, 1994.[5] P.M. Ebert, J.E. Mazo, M.G. Taylor, Overflow oscillations in digital filters, Bell Syst. Tech. J. 48 (1969) 2999–3020.[6] W.L. Mills, C.T. Mullis, R.A. Roberts, Digital filter realizations without overflow oscillations, IEEE Trans. Acoust. Speech Signal Process. 26 (1978) 334–

338.[7] I.W. Sandberg, The zero-input response of digital filters using saturation arithmetic, IEEE Trans. Circuits Syst. 26 (1979) 911–915.[8] V. Singh, A generalized approach for the absolute stability of discrete-time systems utilizing the saturation nonlinearity, based on passivity properties,

IEEE Trans. Circuits Syst. 37 (1990) 444–447.[9] K.K. Johnson, I.W. Sandberg, A separation theorem for finite precision digital filters, IEEE Trans. Circuits Syst. I 42 (1995) 541–545.

[10] H. Kar, V. Singh, Elimination of overflow oscillations in digital filters employing saturation arithmetic, Digital Sign. Process. 15 (2005) 536–544.[11] V. Singh, LMI approach to stability of direct form digital filters utilizing single saturation nonlinearity, IEEE Trans. Circuits Syst. II 54 (2007) 878–881.[12] V. Singh, Elimination of overflow oscillations in fixed-point state-space digital filters using saturation arithmetic, IEEE Trans. Circuits Syst. 37 (1990)

814–818.[13] T. Ooba, Stability of linear discrete dynamics employing state saturation arithmetic, IEEE Trans. Automat. Contr. 48 (2003) 626–630.[14] V. Singh, Modified form of Liu–Michel’s criterion for global asymptotic stability of fixed-point state-space digital filters using saturation arithmetic,

IEEE Trans. Circuits Syst. II 53 (2006) 1423–1425.[15] S. Boyd, L.El. Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, PA, 1994.[16] P. Gahinet, A. Nemirovski, A.J. Laub, M. Chilali, in: LMI Control Toolbox – for Use with Matlab, The Math- Works Inc., Natick, MA, 1995.[17] S.M. Lee, J.H. Park, Delay-dependent criteria for absolute stability of uncertain time-delayed Lur’e dynamical systems, J. Frankl. Inst.-Eng. Appl. Math.

347 (2010) 146–153.[18] S.J. Choi, S.M. Lee, S.C. Won, J.H. Park, Improved delay-dependent stability criteria for uncertain Lur’e systems with sector and slope restricted

nonlinearities and time-varying delays, Appl. Math. Comput. 202 (2009) 520–530.[19] S.M. Lee, O.M. Kwon, J.H. Park, Delay-independent absolute stability for time-delay Lur’e systems with sector and slope restricted nonlinearities, Phys.

Lett. A 372 (2008) 4010–4015.[20] S.M. Lee, J.H. Park, Robust stabilization of discrete-time nonlinear Lur’e systems with sector and slope restricted nonlinearities, Appl. Math. Comput.

200 (2008) 450–457.[21] S.M. Lee, J.H. Park, O.M. Kwon, Improved asymptotic stability analysis for Lur’e systems with sector and slope restricted nonlinearities, Phys. Lett. A 362

(2007) 348–351.[22] P. Park, Stability criteria for sector- and slope-restricted Lur’e systems, IEEE Trans. Automat. Contr. 47 (2002) 308–313.[23] W.M. Haddad, V. Kapila, Absolute stability criteria for multiple slope-restricted monotonic nonlinearities, IEEE Trans. Automat. Contr. 40 (1995) 361–

365.[24] J.A.K. Suykens, J. Vandewalle, B.D. Moor, An absolute stability criterion for the Lur’e problem with sector and slope restricted nonlinearities, IEEE Trans.

Circuits Syst. 45 (1988) 1007–1009.[25] V. Singh, Lyapunov-based proof of Jury–Lee’s criterion: some appraisals, IEEE Trans. Circuits Syst. 32 (1985) 396–398.[26] V. Singh, A stability inequality for nonlinear feedback systems with slope-restricted nonlinearity, IEEE Trans. Automat. Control 29 (1984) 743–744.[27] E.I. Jury, B.W. Lee, On the stability of a certain class of nonlinear sampled-data systems, IEEE Trans. Automat. Contr. 9 (1964) 51–61.[28] V. Singh, Modified criterion for global asymptotic stability of fixed-point state-space digital filters using two’s complement arithmetic, Automatica 46

(2010) 475–478.