Automatica 44 (2008) 282–285www.elsevier.com/locate/automatica

Technical communique

Stability analysis of a class of digital filters utilizing single saturationnonlinearity�

Vimal Singh∗

Department of Electrical-Electronics Engineering, Atilim University, Ankara 06836, Turkey

Received 30 January 2007; received in revised form 30 March 2007; accepted 24 April 2007Available online 8 August 2007

Abstract

A novel criterion for the global asymptotic stability of a class of digital filters utilizing single saturation nonlinearity is presented. An exampleshowing the effectiveness of the present criterion is given.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Asymptotic stability; Digital filters; Discrete-time dynamical systems; Finite wordlength effects; Limit cycles; Lyapunov method; NonlinearSystems; Saturation nonlinearity

1. Introduction

When designing a digital filter using fixed-point arithmetic,nonlinearities due to finite wordlength, namely, quantizationand overflow are produced. The limit cycle phenomenon, whichis a characteristic of nonlinear systems, may possibly occur inthe designed filter if the filter parameters are not chosen prop-erly (Ebert, Mazo, & Taylor, 1969). The zero-input limit cyclesrepresent an unstable behavior and are undesirable in digitalfilters. An important objective in the design of a digital filter is,therefore, to find the ranges of the values of the filter parame-ters and choose the values so that the designed filter is free oflimit cycles (Ebert et al., 1969). Various models of digital filterhave been considered in the literature. These include a class ofdigital filters involving single nonlinearity (Ebert et al., 1969;Johnson & Sandberg, 1995; Kar & Singh, 2005; Mitra, 1978;Sandberg, 1979; Singh, 1990) and a class of state-space digi-tal filters involving multiple nonlinearities (Kar & Singh, 2004;Liu & Michel, 1992, 1994; Ooba, 2003a, 2003b; Singh, 2006).The quantization and overflow nonlinearities may interact witheach other. However, if the total number of quantization steps is

� This paper was not presented at any IFAC meeting. This paper wasrecommended for publication in revised form by Associate Editor Zhihua Quunder the director of Editor André Tits.

∗ Tel.: +90 312 586 8391; fax: +90 312 586 8091.E-mail addresses: vsingh11@rediffmail.com, vimal_singh@atilim.edu.tr

0005-1098/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.automatica.2007.04.015

large or, in other words, the internal wordlength is sufficientlylong, then the effects of these nonlinearities can be regardedas decoupled or noninteracting and can be investigated sepa-rately. Under this decoupling approximation, quantization ef-fects may be neglected when studying the effects of overflow(Butterweck, Ritzerfeld, & Werter, 1988; Classen, Mecklen-bräuker, & Peek, 1976).

This communique considers a class of digital filters involvingsingle overflow nonlinearity (Ebert et al., 1969; Johnson &Sandberg, 1995; Kar & Singh, 2005; Mitra, 1978; Sandberg,1979; Singh, 1990). It is assumed that the effects of quantizationare negligible. Specifically, the system under consideration isgiven by

G(z) = h1z−n + h2z

−(n−1) + · · · + hnz−1

y(r) = output of G(z)

f (y(r)) = input to G(z)

}. (1)

We assume the saturation overflow arithmetic characterized by

f (y(r)) ={ 1, y(r) > 1,

y(r), −1�y(r)�1,

−1, y(r) < − 1.

(2)

Throughout this communique, it is understood that the lin-earized system (i.e., whenf (y(r)) = y(r)) is stable, i.e.,

(zn − hnzn−1 − hn−1z

n−2 − · · · − h2z − h1) �= 0

for all |z|�1. (3)

V. Singh / Automatica 44 (2008) 282–285 283

It was shown previously (Ebert et al., 1969) that the second-order version (n = 2) of system (1)–(3) is free of overflowoscillations. In subsequent works (Kar & Singh, 2005; Mitra,1978; Sandberg, 1979; Singh, 1990), the frequency-domaincriteria for the elimination of overflow oscillations in system(1)–(3) have been presented. In this communique, we derive astate-space criterion for the elimination of overflow oscillationsin system (1)–(3). Like most existing approaches, the presentapproach yields the global asymptotic stability result, whichmeans that the system will be free of overflow oscillations.

The communique is organized as follows. Section 2 presentsa brief discussion on the problem of overflow oscillation.Section 3 gives the main result of the communique. An exam-ple showing the effectiveness of the present approach is givenin Section 4.

2. Problem of overflow oscillation

Consider a stable linear system given by

x(r + 1) = Mx(r) + bu(r). (4)

System (4) cannot have nontrivial zero-input periodic solutions.With the finite wordlength number representations, however,the digital filter becomes a nonlinear system of the form:

x(r + 1) = F(x(r), u(r)). (5)

The dynamic range of each stored variable is finite. The “statespace” of the digital filter (5) is, therefore, a cube (Mills, Mullis,& Roberts, 1978):

C = {x ∈ Rn: |xi | < 1 for each i}. (6)

The digital simulation will faithfully represent the linear system(4) as long as the trajectories of (4) remain inside this cube.However, the simulation will produce large errors if any internalregisters overflow, which will be the case if a trajectory of (4)leaves the cube. In the event of the input being zero, these errorscan lead to oscillation, involving repeated overflows (Classen,Mecklenbräuker, & Peek, 1975; Ebert et al., 1969; Fettweis &Meerkötter, 1975; Mills et al., 1978; Mitra, 1977; Sandberg,1969; Willson, 1972).

Since the amplitude of the oscillation is typically very largeand dominates the effects due to the input, in practice it is usu-ally essential to avoid the possibility of overflow oscillation. Byresorting to the use of the extreme scaling rules, overflow oscil-lation can be avoided. This, however, results in high roundoffnoise (Jackson, 1970; Mullis & Roberts, 1976). In fact, sucha rule often is so conservative that only a small portion of thecube is ever visited, resulting in a “storage-inefficient” design(Mills et al., 1978).

With a view to reducing roundoff noise, it would seem betterto use a less restrictive scaling rule and accept the possibilityof overflow provided that (1) the overflow probability is verysmall, and (2) overflow oscillations are impossible (Mills et al.,1978). The following question then arises: Are there realizationsin which overflow oscillations are impossible? This questionhas generated a great deal of interest. Over the years, a variety

of digital filter structures which are free of overflow oscillationshave been realized. The examples mentioned in Section 1 arejust a few.

3. Main result

Let

A =

⎡⎢⎢⎢⎢⎣

00... In−10h1 h2 · · · hn

⎤⎥⎥⎥⎥⎦ , (7)

where In−1 denotes the (n − 1) × (n − 1) identity matrix. Themain result is given in the following theorem.

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

P = PT = (pij )n×n > 0 (8)

subject to

pnn �n−1∑i=1

|pni |, (9)

such that

P − ATPA > 0, (10)

where A is defined by (7), the superscript ‘T’ to any vector (ormatrix) denotes the transpose of that vector (or matrix), and> 0 denotes that the matrix is positive definite.

Proof. The state-space representation of the system is

x1(r+1)=x2(r), x2(r+1)=x3(r), . . . , xn−1(r+1)=xn(r),

xn(r + 1) = f (h1x1(r) + h2x2(r) + · · · + hnxn(r)). (11)

In view of (2) and (11), one has

|xi(r)|�1, i = 1, 2, . . . , n. (12)

Note that even if the system trajectory lies out of the region(12) initially, (2) and (11) ensure that the trajectory will enterthe region (12) and will not leave this region once it enters inthis region. In other words, for our present objective of arrivingat the global asymptotic stability result, the initial conditionmay be assumed to be such as to satisfy (12). It is convenientto express (11) as

x(r + 1) = f(y(r)) = [f1(y1(r)) f2(y2(r)) · · · fn(yn(r))]T,

(13)

where y(r) = [y1(r) y2(r) · · · yn(r)]T = Ax(r), fi(yi(r)) =yi(r) = xi+1(r), i = 1, 2, . . . , (n − 1), fn(yn(r)) = f (y(r)) =f (hTx(r)), h = [h1 h2 · · · hn]T, x(r) = [x1(r) x2(r) · · ·

284 V. Singh / Automatica 44 (2008) 282–285

xn(r)]T, and A is defined by (7). Choose a quadratic Lyapunovfunction

v(x(r)) = xT(r)Px(r), (14)

where P = PT = (pij )n×n is positive definite. Along the trajec-tories of (13), one has

�v(x(r)) = v(x(r + 1)) − v(x(r))

= fT(y(r))Pf(y(r)) − xT(r)Px(r). (15)

Eq. (15) can be rearranged as

�v(x(r)) = − [xT(r) fT(y(r))]×

[P −ATP

−PA P

] [x(r)

f(y(r))

]− �, (16)

where

�=2[yn(r)−fn(yn(r))][pnnfn(yn(r))+

n−1∑i=1

pnixi+1(r)

]

=2[yn(r)−fn(yn(r))][pnnfn(yn(r))+

n−1∑i=1

pnifi(yi(r))

].

(17)

Note that � can also be expressed as

� = xT(r)ATPf(y(r)) + fT(y(r))PAx(r) − 2fT(y(r))Pf(y(r)).(18)

From (17) it is easy to observe that, owing to (2) and (12), thequantity � is nonnegative if condition (9) is satisfied (note thatP = PT > 0 automatically implies pnn > 0). Therefore, if weimpose the conditions (9) and[

P −ATP−PA P

]> 0, (19)

then �v(x(r)) is negative definite. Thus, under the conditions(8), (9), and (19), the null solution of the system under con-sideration is globally asymptotically stable. In the light of thewell-known Schur complements, (19) is equivalent to (10). Thiscompletes the Proof of Theorem 1. �

Remark 1. Exploiting the equivalence between (17) and (18)together with the property ��0 is a distinct and novel fea-ture of the present approach. Note that only the last elementfn(yn(r))=f (y(r)) of the vector f(y(r)) is the nonlinearity (sat-uration nonlinearity) and the remaining elements fi(yi(r)) =yi(r), i = 1, 2, . . . , (n − 1), are linear functions.

Remark 2. Consider the condition

pii �n∑

j=1,j �=i

|pij |, i = 1, 2, . . . , n. (20)

Eq. (20) includes (9). Thus, system (1)–(3) will be globallyasymptotically stable if the conditions given by (8), (10), and(20), with A defined by (7), are satisfied. Whereas (20) is aset of n inequalities, (9) is single inequality. Therefore, as far

as the global asymptotic stability of system (1)–(3) is con-cerned, one will use {(8).(10)}, where A is given by (7). How-ever, {(8), (10), (20)} has its own importance and utility. Aswas shown previously (Liu & Michel, 1992), {(8), (10), (20)}is a sufficient condition for the global asymptotic stability of annth order state-space digital filter with n state saturation non-linearities and with A being allowed the general form A=(aij ),i, j = 1, 2, . . . , n.1

Remark 3. Pertaining to n=2, (3) becomes the region |h1| < 1,|h2| < 1 − h1. It was shown previously (Ebert et al., 1969) thatsecond-order (n = 2) digital filter {(1), (2)} is free of overflowoscillations in whole of this region. It is routine to verify thatthis result can be arrived at via Theorem 1.

4. Illustrative example

Consider a third-order system (1) with

h1 = 0.008, h2 = −0.815, h3 = 1.51. (21)

We will first apply some frequency-domain criteria in thisexample. First recall the well-known Tsypkin’s criterion. Per-taining to system (1)–(3), Tsypkin’s criterion takes the form

1 + ReG(z)

1 − G(z)> 0 ⇒ 1 − Re G(z) > 0 for all |z| = 1.

(22)

Next, consider Jury–Lee’s criterion (Jury & Lee, 1964; Singh,1985) which, by way of incorporating the sector as well as theslope information of the nonlinearity, was developed as a gener-alization and improvement over Tsypkin’s criterion. Pertainingto saturation nonlinearity, Jury–Lee’s criterion takes the form(Jury & Lee, 1964; Singh, 1985):

1 − Re[1 + �(z − 1)]G(z) − 0.5�|(z − 1)G(z)|2 > 0

for all |z| = 1, (23)

where ��0. Finally, consider the criterion (Kar & Singh, 2005):

1 + Re(1 − k)G(z)

1 − G(z)> 0 for all |z| = 1, (24)

where k is given by k=1/∑n

i=1|hi |. The criterion (24) is basedon a novel characterization, namely, the “reduced sector” char-acterization of the nonlinearity. In other words, the nonlinearityin system (1)–(3) can be viewed (Kar & Singh, 2005) as con-fined to the sector [k, 1], 0 < k < 1, rather than being confinedto the sector [0, 1].

It can be easily verified that each of the criteria (22)–(24) failsin the example given by (21). On the other hand, by choosing

P =[ 10 −20 10

−20 120 −9010 −90 100

], (25)

it turns out that (8)–(10) are satisfied.

1 Note that the present result pertains to a class of nth order systemsinvolving single saturation nonlinearity. By contrast, Liu–Michel’s (1992)result as well as its modified version (Singh, 2006) pertains to a differentclass of nth order systems, i.e., which involve n saturation nonlinearities.

V. Singh / Automatica 44 (2008) 282–285 285

5. Conclusion

A criterion for the global asymptotic stability of a class ofdigital filters using single saturation nonlinearity has been pre-sented. The criterion is in a marked contrast to the previouscriteria. The usefulness of the criterion is illustrated with thehelp of an example.

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