New zero-input overflow stability proofs based on Lyapunov theory

New zero-input overflow stability proofs based on Lyapunov

theory

Citation for published version (APA):

Werter, M. J., & Ritzerfeld, J. H. F. (1989). New zero-input overflow stability proofs based on Lyapunov theory. In

Proc. ICASSP 89, Int. Conf. Acoustics, Speech, and Signal Processing, Glasgow, Scotland, 23-26 May 1989

(pp. 876-879)

Document status and date:

Published: 01/01/1989

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D2.7

Mew zero-input overflow stability proof8 based cm Lyapunov theory

nichiel J . Werter and John H.F. Ritzerfeld

Department of Electrical Engineering Eindhoven University of Technology

Eindhoven, The Netherlands

ABSTRACT

In this paper we demonstrate some new proofs of sup-

pressing zero-input overflow oscillations in re- sive digital filters. These proofs are based on the second method of Lyapunov

For second-order digital filters with complex conju- gated poles the state describes a trajectory in the

phase plane, spiralling towards the origin, as long

as no overflow correction is applied. Following this

state signal an energy function can be defined,which is a natural candidate for a Lyapunov function. For the second-order direct form digital filter with a saturation characteristic this energy function is a Lyapunov function.

However, this function is not the only possible Lya- punov function of this filter. All energy functions with anenergy matrix that is diagonally dominant, guarantee zero-input stability, if a saturation characteristic is used for overflow correction. In this paper we determine the condition a general

second-order digital filters has to fulfil so that

there exists at least one energy function with a matrix, which is diagonally dominant.

1. Introduction

In this paper we demonstrate sope new proofs of sup-

pressing zero-input overflow oscillations in recur- sive digital filters, which have also been published

in [I]. These proofs are based on the second rethal

of L y e p m v , which starts with a properly chosen

energy function B(n), preferably in quadratic form:

B(n) = ,rT(n) .P.x(n). (1.1)

Without loss of generality, w e choose P syametrical:

PT = P, (1.2)

The Lyapunov theory d-ds that

E = aT.p.X

>

o

r

0,

so that B ( n ) = 0 implies ,r(n) =

0.

follows that P has to be positive definite.

Further, the system dynamics muat be such that if no overflow correction is applied, according to

~ ' ( n ) = A * ~ ( p l ) , (1.4)

the energy strictly decreases with increasing time:

S'(n)

<

where B'(n) is the energy pertaining to the state

~ ' ( n ) . This inequality is satisfied if

P

-

Every matrix P satisfying condition (1.21, (1.3) and

(1.6) defines an energy function E(n), which is a

candidate for a Lyapunov function. If moreover for one such a matrix a subsequent overflow correction

x ( n ) = F{Z'(JJ)} (1.7)

lcuers the energy for all possible states,

8 ( n ) 5 B'(n) for all n (1.8)

the function E(n) is called a L y e p m v function of

the nonlinear system under consideration.

The existence of a Lyapunov function in a digital filter guarantees freedom from zero-input overflow oscillations. The idealized linear filter is assumed

to be stable, so the state in this filter amymptoti-

cally approaches the zero-state, for which also the energy is zero. In the actual digital filter o v e r flow correction lowers the energy B(n), implying that also here the energy asymptotically reaches zero, implying zero-state, which completes the prove of zero-input stability.

Care must be taken if

"<"

(1.5) so that the energy can remain constant. Such a

situation occurs for a marginal choice of the matrix

P, for which the energy function is called a semi-

L y a p m v function. If, moreover, the equality sign

in (1.8) applies it can occur that the energy re-

mains constant, associated with the risk of zero- input oscillations.

2. Overfla, stability in secolld-order filtera

In this section we investigate the overflow stabi-

lity of second-on3er d l g l t a l fllters. For sake of

conciseness, we restrict the following discussion to sections with complex poles. Compared with real poles, they generally favour all forms of parasitic oscillations (particularly for high Q-values) and thus deserve special consideration.

The 2 x 1 state vector x(n) = (x,,~,)~ in an auto-

nomous second-order system satisfies the fundmental difference equation

x ( n ) = FL4.x(wl)l, (2.1)

In this paper it is understood that

[F~A.III~ = F([A.all}, (2.3)

i.e. the individual components of A - x undergo the

same memoryless and local overflow correction.

876

The quastion to be analyzed is: under which circum-

stances (choice of A, F and ~ ( 0 ) ) does (or does not)

(2.1) admit periodic solutions?

Due to the overflow bound, which is henceforth n o r malized to unity, the state variables conform to

Ixt(n)( I 1 , (2.4)

resulting in a state vector confined to the interior

of the unit square (cf. Fig.2.1). Without overflow

(as long as (2.4) holds) the solution of (2.1) is

where q l , q l = eriJO denotes the ccmplex eigenvalues

of A and trij.ti denotes the pertinent eigenvectors.

It is tacitly assumed that

r <

stability. Further, the constants of integration

( X , P) are determined by the initial state ~ ( 0 ) .

If, for the time being, time n is viewed as a conti-

nuous variable, ~ ( n ) describes a trajectory in the

phase plane. For the case

r

lipse with main axes in the direction of tr and ti.

For

r <

circle), we obtain a non-closed, ellipselike curve

spiralling towards the origin, cf. Fig. 2.1.

Fig. 2.1. Trajectory of the state vector r(n)

in the state plane.

Of course, these results only apply to the digital

filter as long as overflow does not occur. If this

is not the case, the linearly determined x(n) leaves

the unit square, and overflow correction has to be applied. This correction introduces one of two basic

atate modifications: (a) x ( n ) is moved towards the

origin, (b) ~ ( n ) is roved away from the origin.

Case (a) is wanted because it supports the natural linear motion; no oscillation occurs if all over- flows are corrected this way.

Case (b) is dangerous, because it colpensates or even overcompensates the linear behaviour and, hence can (but need not) lead to oscillations. Of course. these statements ask for an unambiguous definition of "distance from the origin". Instead of the widely used Euclidean norm our definition is guided by the linear state motion, according to (2.5). Following

~ ( n ) = x ( n ) . [~.c~s(r(n))-~~.sin(r(n))] (2.6)

two variables I(n),f(n) can be associated with each

state ~ ( n ) . Particularly, the variable X(n) is de-

termined from ~ ( n ) as

Comparing (2.6) with (2.5) one recognizes

w(n) = X.ern, (2.8)

i.e. a monotonically decreasing function. Therefore, the function

R(n) = X2(n) (2.9)

is a natural candidate for a Lyapunov function. We

choose x ( n ) as the "distance from the origin".

Overflow correction is now visualized in Fig. 2.2.

An uncorrected state point B is mapped into B', B",

or B"' after applying saturation. zeroing and two's

complement, respectively. For this example all types

lead to an increase of E ( n ) and, hence, to a move-

ment away from the origin. On the other hand, for

point C this is only true for zeroing and two's com- plement overflow correction.

For some ellipse gecmetries it is possible to use appropriate overflow characteristics such that the

state always moves t-ds the origin and wcilla-

tions are suppressed.

Obviously this is not the case for the arbitrarily oriented ellipse of Fig. 2.2. However, it is easily

recognized that for an ellipse whose ax- coincide

with the xI-x.-axes, each of the three overflow cot-

rections satisfies the stability condition, while

for an ellipse with a 450 inclination stabilization

can be obtained at least with a saturation charar teristic.

*2

't

-'

I

Fig. 2.2. Ellipse X = constant in the state plane.

3. Ovema &ability in direct forr filters

In this section we investigate the zero-input

stability of the second-order dfrect form dlgltel

fllter with overflow correction. This filter is described by the syster matrix

It turns out to be free from zero-input overflow oscillations, if saturation is used. This is true for all pairs of filter coefficients a and b in the “stability triangle”, described by

l - l a ( - b > O

l + b

>

The analytic proof of this statement has been re-

ported by Ebert e.a. [Z]. In this section we present

two novel proofs using Lyapunov theory, one proof only for complex conjugated poles, the other for all pairs a and b in the stability triangle.

For complex Conjugated pole pairs q , , q a = er*J’ we

define an energy function E(n), according to (2.10): (3.3) 2

E ( n ) = x,(n)-a.x,(n) .x,(n)-b-x;(n),

(3.4)

This energy function is characterized by the matrix (3.5) which is symmetrical. It is positive definite, due to

Det[P] = -b

-

>

and Tr[P] = 1

-

+

>

The system dynamics is such that if no overflow cor- rection is applied, the energy decreases:

E’(n) = ez.E(n-l)

<

In the nonlinear system, saturation causes an addit- ional energy reduction. If before correction we have xl’(n)>;, then after saturat’ion x,(n)=l. The compo-

nent xt ( n ) can never overflow since x,’(n)=x,(n-l).

So E ( n ) = 1

-

-

E(n)-E’ ( n ) = -[xI’ (01-11 * [l+x,’ (n)-a.x,’(n)]

<

The latter inequality is due to the stability re-

quirement la

<

E(n) is a Lyapunov function, which guarantees zero- input stability in the second-order direct form fil- ter with saturation and complex conjugated poles.

for values o

1

<

The zero-input stability of this filter can also be

proved with another Lyapunov function E(n), which is

characterized by the symnetrical matrix 1-b -a

= [-a 1-b]. (3.11)

Matrix P is positive definite for all pairs of coef-

ficients a and b in the stability triangle, due to

Oet[P] = (l+a-b).(l-a-b)

>

and

Tr[Pl = 2.(1-b)

>

(3.12) (3.13) Without overflow correction the energy cannot in- crease, since

E’(n)-E(n-l) = -(l+b). [a*x,(n-l)+(b-l) .~,(n-l)]~

5 0 for all n. (3.14)

In the nonlinear system saturation causes a reduc-

tion of the energy. If xI’(n)

>

tion a,(n) = 1 and with x,(n) = xa’(n):we have

E(n)-E’ (n)= -[xI’(n)-ll- [(l-b).(l+x,’(n))-2a.x,’(n)]

<

The last inequality is a consequence of the

stability condition 1- a1 b

>

can be drawn for x,’(n] <--1.

In a strict sense function E(n) is not a Lyapunov function since the energy can remain constant. This

situation can only appear if no overflow correction

is applied. But then the filter responds linearly and the state will asymptotically reach zero; no zero-input overflow oscillation is possible in the second-order direct form filter with saturation. This result is now proved for all pairs of filter coefficients a and b in the stability triangle.

4. Overflas stability in filtera with saturation It is a c o m o n property of normal, wave digital and lattice filters of second and higher order that the

”energy, matrix” P is diagonal and that E(n)=constant

are ellipses oriented parallel to the coordinate

axes. Only this ellipse georetry all- for all

overflow characteristics applied to the individual state variables, without risk of zero-input overflow oscillations.

The question arises: Which A matrices a d d t a diago-

nal ”energy matrix”? This question has been solved in [3] for the second-order filter, where it is shown that this is only possible if the system

matrix A fulfils the condition:

laII

-

<

-

If a saturation characteristic is used in a sewnd-

order digital filter the condition (4.1) for zero- input stablility (valid for all overflow character- istics) can be relaxed. We shall prove the following theorem (see also [41):

If E(n) = xT(n) .P.g(n), (4.2)

satisfies all conditions for an energy function (see Section 1) and if matrix P is diagonally dominant:

IPlZl I PI1 and lP1rl I Pra. (4.3)

then the function E ( n ) is a Lyapunov function of the

filter with saturation. Proof:

The difference in energy between the uncorrected

signal ~ ’ ( n ) and the actual state ~ ( n ) is

E’(n)-E(n) =

= pII * (XI *2-x7:)+pa,. (xz* 2-x,)+2p1,. (X1’Xt’-xIxr) 2

,2 2 2 2

1 P I , (XI -XI )+P*r(X,’ -Xr)-21P1,

I

1)

+ l P . t l ~ ~ l ~ I ’ ~ - ~ x t ’ l ~ - ~ I ~ 1 ~ - ~ x z ~ ~

1 .

1 ) .

’

1).

878

The inequality is valid due to s@(~,')=sgn(X,) and

sgn(r,')=sgn(x,). If only one of the c y n e n t s

overflows, f.e. x,'

>

E' (n)-E(n) 2

2

(PII-(PI.

1)

(4.5)

If both components overflow then l x , l =

I X , ~

E' (n)-B(n) 1 (PI ,-1plt

1)

1)

+ IPI.I(IxI'l-lx*'

I)

So in a system with an energy E(n) satisfying (4.3)

saturation causes an additional energy reduction.

Now we solve the question: Which second-order A

matrices posse88 at least one energy function E ( f l )

whose energy matrix P is diagonally dominant? Therefore the energy in the idealized system must strictly decrease, which will be satisfied if

P

-

For a second-order system this condition is equiva-

lent to the inequalities

I Det[P

-

>

-

>

Since e1,-a,,

<

R t e r p , , in such a way that the centre point of the

ellipse ( a - p , ,

,

of the p,,-p,,-plane.

For

lal.l

<

p?,!,

within the e lipse (4.11) satisfying condition (4.3)

if the ellipse curves the line p . , = (pit(. Such a

point of intersection is found if

epl ,2+[b.sgn(p,,)+dlpI ,p.,+[c+e.san(~,,)+flp,,~

<

has real solutions. So

(4.13)

[b.sgn(p,,)+d12 2 4a[c+e.sgn(pl,)+f1, (4.14)

or

[ ( 1+Det [AI 1 2- (a ,+a

,

,

'1

[ (l-Det [AI )2-(a, I - a , , - ~ , , s B n ( p , ,) 1

1

The first factor of (4.15) is positive due to stabi- lity conditions. The second factor shows the reein- ing condition which can be written in the form

2

1al1-aZ11

<

-

The case

4

?.I

case resu ting in the condition

~ a , , - a a z ~

<

The conditions (4.16) and (4.17) form together the

result of this section: All second-order digital

filters with a system matrix A satisfying:

Ia,,-a..I i? 2~min(~eI,~,~a,,J)+1-Det[A], (4.18)

possess at least one energy function E(n), with a matrix P satisfying (4.3), and thus are free from zero-input overflow oscillations for saturation. As it should be, this condition is less restrictive than (4.1). Contrary to the former condition, all stable direct form filter satisfy (4.18).

REFERENCES

[l] M.J. Werter, "Suppression of parasitic oscflla-

tions due t o overflow and quantization In recur-

sfve dfgital filters," Ph.D.-Thesis, Eindhoven

University of Technology, The Netherlands, 1989. [2] P.M. Ebert, J.B. Mazo and M.G. Taylor, "Overflow oscillations in recursive digital filters," Bell

Syst. Tech. J . , vol. 48, pp. 2999-3020, 1969. [31 W.L. Mills, C.T. Mullis and R.A. Roberts, "Di-

gital filter realizations without overflow os-

cillations," IEEE Trans. Acoust., Speech, Signal

Processing, vol. ASSP-26, pp. 334-338, 1978. [ 4 ] J.H.F. Ritzerfeld, "A condition for the overflow

stability of second-order digital filters that is satisfied by all scaled state-space structu- res using saturation," submitted to IEEE Trans.

C f r c u i t s S y s t .

For p , . = 0 the energy matrix P is diagonal which

has a solution for a system matrix A satisfying con-

dition (4.1). If condition (4.1) is & satisfied,

which can only be true for filters with a l , . a l l

<

we have according to (4.12) a discriminant which is

negative, with as result that (4.11) describes the

inner part of an elliptical curve in the p,,-pIz-

plane, which is non-empty, since R

>