A generalised approach to the stability analysis of PWM feedback control systems

A Generalised Approach to the Stability Analysis of PWM Feedback Control Systems” by A. BALESTRINO, A. EISINBERG and L. sc1Av1cc0 Istituto Elettrotecnico University of Jvaples, .Naples, Itab

ABSTRACT:

A unijkd approach to stability analyeie of feedback control systeme with pulse-

width modulators is diecuased. The proposed proceo?ure is developed on the basis of the discrete analog of Lyapunov’a method, with no limiting hypothesis on the structure of the controlled plant and of the modulation law. The analysis is exempli$ed particularly for leud-type PWM control eyetem9. Significant classes are investigated, and the critical values of parameters in a closed form are

plant

determined which oasure

osyrnptotic

stability

of the steady-state

solution,

whatever

the

reference input may be.

1. Zntroduction

This paper deals with the problem of pulse width modulator (PWM) feedback control system stability. From a practical point of view, the interest in such systems is due to the simplicity of the engineering realizations, comparable to that of relay systems, though they are considerably superior to the latter with regard to the dynamic properties. From a theoretical point of view the systems we take into consideration exhibit some nonlinear, very peculiar effects, which have no counterparts in the theory of continuous and discrete systems. Consider a class of modulators in which the width and polarity of pulses of constant amplitude, equally spaced in time, depend on the waveform of the input signal e(t) (see Fig. 1). Let T be the constant interval of sampling, and 7, the width of the pulse u(t) originated from the modulator in the generic interval [nT, nT +T], then the input e(t) and output u(t) can be related as follows : sgn l%(t)] = sgn [WV, 7% = #dIeW’), T,l

I

(1)

with 771E [0, T], and where it has been defined :

sgnGV4 i

+l,

A>O,

0,

h = 0,

- 1,

x < 0.

(2)

If in Eq. (1) the relation between 7n and e(nT) is in implicit form, we speak * This work w&ssupported by the ConsiglioNazionale delle Ricerche.

45

A. Balestrino, A. Eisinberg and L. Sciavicco of natural sampling;

if the relation is in explicit form and, as a result, 7R = Y[e(lzT)]

(3)

then the associated sampling is defined as uniform (16,17). The structure of the control law, introduced from Eq. (l), implies a coilsiderable complication in the study of the dynamics of the feedback system in Fig. 1. The problems of associated analysis are widely dealt with in the literature, with particular regard to matters concerning the existence of limit cycles (3, 5, 15) and of the stability of steady-state solutions. In the case of PWM systems with uniform sampling and null reference input, many authors have investigated their stability making use of a variety of techniques ranging from linearization (1) to the Lyapunov method (2,6,7), from stability criteria of the Popov type (8, 9) to techniques of functional analysis (4, 10, 11). A limited number of papers deal with the problems of stability of systems with a constant reference input (12) and of systems with natural sampling modulators (13, 14).

-I

I

FIG. 1. Pulse-width

modulated

feedback system.

In this paper, unlike the references cited, the problem of asymptotic st’ability of control systems with width modulators is considered in a general context. Once the pulse width modulator is defined in a formal way, the analysis of stability is developed with a constant reference input without any restrictive hypothesis on the eigenvalues of the dynamic matrix of the controlled plant. The analytical procedure is based on the Lyapunov direct method and can provide a parametric check of the definite positiveness of nonhomogeneous quadratic forms. In this manner, it is possible to treat stability matters concerning all the pulse elements commonly used, with both uniform and natural sampling. As an example, the application of this procedure is presented in detail in the case of lead-type PWM; controlled plants are considered as first- and second-order plants with an integrator.

ZZ. Problem

Formulation

Let the controlled plant of order m, single input and single output, be linear and time-invariant, completely observable and controllable with lowpass characteristics.

46

Journal

of The Franklin

Institute

Stability Analysis of PWM

Feedback Control Systems

The equations describing the dynamic behaviour Fig. 1 are

of the control system in

k(t) = Ax(t) + bu(t),

(4)

y(t)= C'W), e(t) = r-Y(t),

1

where x E 9P is the state, y E 93 and u E9 are, respectively, the output and the input of the plant and r E9 is the reference input. As the input of the plant is width modulated and the pulses have a constant repetition period T, it is convenient to consider the discrete model of the Fig. 1 system. As a result of this, we can write X n+1 =

@LX?&

+w?Ll,

(5)

where 4;J4 exp (AT) ;

(6)

w, takes into account the structure of the modulator and depends on the polarity and width of the nth pulse. Introduced the auxiliary variable pn E [0,00) defined as Tn = T sat pn,

(7)

where

sat (X)g

+1,

h>I,

h,

IhI< I,

I - 1,

h< -1.

We define as a generalized width modulator relation is characterized by the mapping: U: x,+x,U

(8)

a device whose input-output

= [~~,sgn(r--c’xn)]

(9)

through the equation k:,x,+d,

= 0,

(IO)

where the range of the mapping is 9P and k, and d, are both vectorial and scalar functions of the image element of (9).Further, it will be recalled that as (10) might be satisfied for more values of pm, the value pn to be considered in (9)is the smallest of those satisfying (10). Let us suppose also that lim Id,] = co, pn+m

(11)

lim 11 k, 11< co. pn+m

(12)

We emphasize that all the width modulators commonly used, both the natural sampling type and the uniform type, can lead to such a formulation. Relations (5) and (10) constitute, therefore, the mathematical discrete model of the control system of Fig. 1.

1-d. 298, No. 1, July 1974

47

A. Bdestrino,

A. Eisinberg and L. Sciavicw

The steady-state solution, of which we plan to check the stability, originate from Eqs. (5) and (lo), thereby obtaining (1-0)x, k:,x,+d,

= aw,,

‘I

= 0.

!

must

(13)

111. Stability Analysis

The asymptotic stability of the steady-state discrete analog of Lyapunov’s direct method. Let K=

solution is studied via the

Q&+1 -x,)-(x,--X,)'Q(x,-xX,) = - x:, Px, - sg:, x, - h,, where

PAQ-CP'Q+P,

(16) (17)

g,4g(x,W = V'--Y&x,-+'Q+w,, h,&h(x,U)

(15)

= (2x:,&+-w;WQO)w,.

(18)

A sufficient condition for the approximate asymptotic stability of the steadystate solution is - Al$ = x:, Px, + Z&(x% U) x, + h(x, 77)> 0, (19) VX,E.G@‘m, x, # x,. 1 In general, x, U is not always obtainable in an explicit form, which hampers the checking of (19). Furthermore, when x,, U is obtainable in an explicit form and, as a consequence, should be replaceable in (19), considerable computational difficulties are generated and the problem has no general solution. With a view to obtaining a unifying procedure, it is more suitable to retain in (19) a quadratic structure ; this is possible only by fixing the value of x, U in (19). This choice implies that the check must be carried out on the subset of S2mcongruent with the mapping Eq. (9). Then it is possible to state the following theorem-omitting henceforth for notational simplicity, the indication of dependent variables and the index n. Theorem I. A sufficient condition for the approximate of the steady-state solution (13) is

asymptotic

stability

x’Px+2g’x+h>O v XE&f,

V p~B?i+,

~%?LL{x: k’x+d

V sgn(.)EYil{+l,O,

where the equality holds only for

48

= 01,

x

=

(20) -I>,

x,.

Journal

of The Franklin

Institute

Stability Analysis of PWM

Feedback Control Systems

In order to exploit fully the power of the method outlined in Theorem I, it is convenient to replace the check of (20), corresponding to the set X, with an unconstrained check. Let us consider for that purpose the subset of 9P defined as 2”~

x: x =

,

Sz-&k

(21)

v ZEsi?,

i where S is an m x m matrix defined as 1

S&I-r;i-j;kk’. The way S is constructed however,

(Sk=O,

(22)

implies that the matrix has rank (m - 1). Actually,

VkE9’m;

Sp=p,

Vp:p’k=O)*rankS=m-1.

(23)

This implies that 2* is a linear variety in 3%. And we can immediately verify by simple substitution that Z* coincides with ~2’. As a consequence of this, Eq. (20) is equivalent to f[z,p,sgn(.)]

=z’SPSz+2

\

g’-&k’P)Sz

(24)

d d 2 + k’k k’Pk-2k,kg’k+haI, ( ) V pE9+,

V zE*,

V sgn(.)E9.

I

The advantage of this formulation is that the variable z is structurally independent of the parameters p and sgn ( . ). Defining J, the (m+ 1) x (m + 1) matrix associated with the quadratic form (24), yields (25)

where J,, = SPS, J12 = JL, =

is an m x m matrix,

S(g-&Pk),

‘k’Pk-2&--‘k+h, Thus the following

(26)

is an m-vector,

is a scalar.

(27)

(28)

theorem can be stated.

Theorem II. Sufficient conditions for the asymptotic stability of the steadystate solution Eq. (13) are that (a) Choose Eq. (14) as a Lyapunov function, the matrix J is semidefinite positive V pE[O,CO), V sgn(.)E9;

Vol.298,No.1,July1974

49

A. Bale&Go,

A. Eisinberg and L. Sciavicco

(b) define the set 9

as

.F&(x,~,sgn(.):

AV(x,P,sgn(.))

= 0, k’x+d

= O>

do not exist (x, p, sgn ( . )) E.F belonging to closed trajectories with (5) and (10) except for the steady-state solution.

congruent

Proof: Since one of Eq. (24) must be verified ‘v’ z E 9F in correspondence with the generic value of p and sgn (. ), it follows the condition of positive semidefiniteness of matrix J. Part (b) of Theorem II represents the discrete equivalent of the Barbashin-Krasowskij condition (18). Remarks. In the hypotheses with (YE 9, we have lim(-ork’k+d) oL+m

(11) and (12), substituting

= O=z-limar =f+li-& a-+*

in (10) x = - ak,

= limd p+mk k=

O”’

As a consequence of Theorem II, since J is positive semidefinite, the principal minor Jll must be positive semidefinite and Jz2> 0 ; furthermore, from Eq. (28), we have k’ Pk

l-29)

and therefore if we want to assure the approximate it must be necessarily verified ; k’Pk>o.

asymptotic

stability (30)

Of course, the above-mentioned conditions are satisfied if the matrix P is positive semidefinite. The conditions expressed in Theorem II are easier to deal with than those given in Theorem I, but imply the positive semideliniteness check of the (m+ 1) x (m+ 1) matrix J. An outstanding simplification occurs assuming that rank SPS = m - 1 as, under this hypothesis, the quadratic form (24) has only one minimal value with respect to z. In fact, as SPS is positive semidefinite for the hypothesis, the quadratic form (24) has the minimum. In addition to this, we verify that

f[z,p,sgn(.)l

=f[z+Bkp,sgn(.)l,

V BEG.

(31)

If we impose that the gradient with respect to z of one of Eq. (24) must become null, it will ensue that the vectors 2, in correspondence with which the minimum occurs, satisfy the equation: d S PS%+g-k’kPk

=

Sl = 0,

that is to say that 1 is a vector parallel to k [see Eq. (23)]. As rank SPS = m - 1, then (32) gives (m - 1) linearly independent equations to determine the components of 2. As, from (31), the component of 2 parallel

50

Journal of The

Franklin Institute

Stability Analysis of P WM Feedback Control Systems to k can be fixed arbitrarily and as rank [SPS, I-S] = m, it is possible to obtain, in an unequivocal manner, for each /3~9’, a vector f, such that the global minimum value of the quadratic form is attained. If rank P = m, substituting in (32) 1 = yk, where y is a scalar, so far unspecified, we obtain

SZ = -P-‘g+&k++lk. The scalar y can be determined

from

k’s2 = -k’P-lg+d+yk’P-lk

= 0.

Making use of Eqs. (33) and (34), from (24) the explicit global minimum is

f[&P,sgn(.)]

(34) expression

of the

= h-g’P-lg+&k(d-g’P-lk).

As a result of this, then should P be invertible, as follows in the form of stability criterion.

Theorem II can be restated

Theorem III. A sufficient condition for approximate asymptotic stability of the steady-state solution of (5), choosing Eq. (14) as a Lyapunov function, on the hypothesis that P be a positive definite matrix, is

(d-gP-lk)>O, h-g’P-lg+&k

V (P,sgn(.))E@+x$.

Condition (36), for the special case r = 0, has been obtained approach in (13). VI. Control Systems

with Lead-type

Pulse-width

(36) by another

Modulators

As an application of the method presented, control systems with a leadtype PWM are considered. Such a modulator is described by M sgn [e(nT)], u(t) =

tE[nT+Tsatp,,(n+l)T).

( 0, Equation

t E [nT, nT + T sat p,) (37)

(5), specified for this case, becomes x,+~ = ax, + M Qlv(p,) sgn

MnT)l,

(33)

where T sat pn vb,)

=

s

exp(-Ao)bdo 0

or, if the dynamic matrix A is invertible,

(39)

we have

v(p,) = A-l[I - exp (-AT

sat P,J] b.

(40)

The laws of uniform and natural sampling can be easily put into the form (10).

Vol. 298, No. 1, July 1974

51

A. Balestrino, A. Eisinberg and L. Sciavicco If the sampling is of uniform type, we have TV = T sat [K 1e(nTI )] and Eq. (10) is specialized k;x,+d,

(41)

in

= c’x,+

ip,sgn[e(nT)]-r

I

= 0.

(42)

If the sampling is of the natural type, there results sgn [e(nT)] = sgn [e(nT + T sat p,)]

7n = T sat [Ke 1(nT + T,) I],

(43)

and Eq. (10) is rewritten as kk x, + d, = c’ exp (AT sat p,) x, + i pn sgn [e(nT)] 1 +Mc’exp(ATsatp,)v(p,)sgn[e(nT)]-r]

= 0.

(44)

Later, an application of the method is developed in the case of a controlled plant of first- and second-order with an integrator. First-order systems The plant is characterized

by a transfer function: G(s) = c/(s+a),

i.e., with the adopted notation, A=-a, Without

b=l,

by

c=c>O,

loss of generality,

0=exp(-UT),

y=cx.

we suppose r > 0; in the steady state, we have

M exp (aT sat pm) - 1 x, = exp (UT) - 1 a ’

sgn(r-cz,)

= 1,

(45)

where pm characterizes the value of p corresponding to the steady state. Suppose that the modulator is of a uniform sampling type. In such a case, from (13), the reference input value congruent with the steady state is r=-

Mc exp (aTsat p,) - 1 +& a exp (UT) - 1

When r = pm = 0, x, = 0, condition notation, becomes

(46)

(36), particularized

with the adopted

- (JzZYK~)~ [exp (aT sat p) - 112exp ( - 2aT) + 2MTKc exp ( - 2aT) paT [exp (aT sat p) - l] +(paT)2[l-exp(-2aT)]>0, Equation

52

(47)

(47), when resolved, gives -aT
a condition

Y pE9?+.

obtained in

l+exp(-aT) l-exp(-UT)’

(43)

(11)and (13) through other methods.

Journal of The Franklin

Institute

In addition, on the other hand, a problem of outstanding practical interest is the determination of the conditions which would warrant asymptotic stability of the steady-state solution, whatever the reference input may be. This problem can be solved by insisting that Eq. (24) will be verified not only V (p, sgn ( . )), but also V pm EL%‘+; actually the check of (24), whatever the reference input may be, is from (46), equivalent to a check V pco~9+. In the case of a first-order controlled plant with a uniform sampling modulator, checking with a piecewise strategy, we consider the three possible cases due to the function sgn separately. If sgn (r-ccx) = - 1, Eq. (36) is positive whatever r may be, provided that Eq. (47) is satisfied; therefore, relation (48) must hold. If sgn (r - cx) = 0, from (36) and (46), we have the condition MTKc < [l + exp ( - uT)]. (49) Finally, if sgn (r - cx) = + 1, then Eq. (36) will become - (MTKc)~ exp (- 2aT) [exp (aT sat pm) - exp (UT sat p)12 + 2MTKc exp ( - 2uT) uT(pm - p) [exp (UT sat pm) - exp (UT sat p)] V p~k%?+, V pm~9i’+

+u2T2(p,-p)2[1-exp(-2uT)]>0 from which the following

constraints

(50)

originate

-[l-exp(-aT)]
(51)

The latter condition, being more restrictive than (48) and (49), gives the boundaries within which the system is stable whatever the reference input may be. This result is partially proved through another method in (11). Let the modulator be of the natural sampling type. In this case, from (13), the reference input congruent with the steady state will be r=-

Mcexp(-aTsatp,)-1 exp(-uT)-1 a

+p,$.

(52)

When r = pm = 0 and x, = 0, Eq. (36) is rewritten as (MTKc)~ [exp (UT sat p) - 112+ 2aTp[l-

exp ( - uT sat p)] MTKc

+ (uT~)~ exp ( - 2aT sat p) [ 1 - exp ( - 2aT)] > 0,

V p E L%+

(53)

from which we have the constraint: -[l--exp(-uT)]
(54)

Proceeding in an analogous manner to the case of uniform sampling, with the unique difference of considering Eq. (36) in respect of (45) and (52), the procedure is carried out taking into account separately the three possible values of the function sgn. If sgn (r - cx) = - 1, then we have that Eq. (36) is satisfied for any value of reference input, provided that (53) is satisfied ; hence, condition (54) must be satisfied. If sgn (r-ccz) = 0, then from (36) and (52) we have the condition

Vol.

298,

No.

1, July

1974

53

A. Balestrino, A. Eisinberg and L. Sciavicco and then the limitation - [ 1 - exp ( - 2aT)] < MTKc

< co.

(56)

Let sgn(r-cz) = + 1. If pm~(O, 1) and p~:(0, l), and setting at zero the derivative of (36) with respect to pm, it can be shown that the more critical condition occurs for p = pm. Equation (36) being positive definite for p = pm, it is sufficient to verify Eq. (36) for pm = 1 and p = 0, i.e. - (MTKc)z

[ 1 - exp ( - UT)] + 2MTKc exp ( - aT) + 1 + exp ( - aT) > 0. (57)

The latter gives the constraints

Equation

(58) combined

1


l+exP(-aT)

(58)

l-exp(-UT)’

with Eq. (54) gives the required relation

-[l-exp(-aT)]
l+exp(-aT) l-exp(-UT)’

It is of some value to stress how Eq. (59) is less restrictive than (51). We wish to point out in addition that, as Eq. (36) is independent of the choice of parameter defining the Lyapunov function, the conditions obtained are necessary and sufficient. In Fig. 2 the stability boundaries are shown.

G(s)=c/s+o

FIG. 2. Stability

54

boundaries for first-order PWM control system with uniform a.nd natural (NS) sampling.

(US)

Journal of The Franklin Institute

Stability Analysis of P WM Feedback Control Systems Second-order systems with integrator Consider the control transfer function :

system in Fig. 1 with a plant characterized

G(s)=

by a

1+ s/a0

s(l +s/a)’

i.e. using the adopted notation

A=;I -“,,I The steady-state

a>O;

1

II

C=

c ’

c=a-I;

b=

;. I

a0

I

(60)

solution is pm = 0,

xl,m

= r,

x2,m

= 0,

(‘31)

where the suffixes 1,2 indicate the components of the state vector x. Furthermore, for the uniform sampling case, Eq. (42) becomes x,+cs,-r+$psgn(r-x,-w,)

= 0

(62)

while, for the natural case, (44) gives x,+cexp(-aTsatp)s2-r+$psgn(r-x1-cx2) l-exp(-aTsatp) aT To check the stability,

1

sgn(r-2,

-cz2)

= 0.

(63)

let

By specializing matrix J and making use of (62), in the case of uniform sampling, stability will be assured if MTK

q exp ( - aT)

((satP)‘+q[

exp (aT sat p) - 1 aT

-csatp

1

‘+q[l-exp(-2aT)]

exp (aT sat p) - 1 2 aT ] ]1<2qP-exp(-2aT)]psatp,

v Let

q = I c I aT/[exp

(64)

(aT) - 11, from (64) the following boundaries are obtained

2aT o
Vol.298,Nal,July 1974

PE%‘+.

(65)

55

A. Balestrino, A. EL&berg

and L. Sciavicco

By using (63) in the case of natural sampling, the stability will be assured if: exp (aT sat p) - 1 2

I

Cexp(-aTsatp)-qexp(-2aT)
aT

I

1+2PL

sat p MTK

(

exp(aTsatp)-1 [

aT

I 1 “1 v Ed+ -qexp( -2aT) . 1’ P I Let q = (c 1exp (aT), from (66) the following boundaries are obtained exp(aTsatp)aT

if c> -&[l-exp(-aT)J,

00,

O
(66)

l-exp(-aT) 21cj-l+exp(-aT)’

ifc<

-&[I-exp(-aT)].

(67)

results are shown in Fig. 3.

FIG 3. Stability boundaries for a uniform (-) tandnatural (- - - - -) sampling PWM control system with a second-ordercritical plant and T$0.

V. Conclusions

In this paper the authors have pointed out a procedure for the analysis of the stability of control systems with pulse-width modulation. On the basis of the Lyapunov direct method, conditions sufficient for asymptotic stability have been determined. The proposed method is based essentially on a check

56 Journal of The Franklin Institute

Stability Analysis of P WM Feedback Control Systems of the definiteness of nonhomogeneous quadratic forms in the state constrained to subsets of p, where such subsets are generally determined in parametric form. With such a procedure, it is possible to overcome the difficulties due to the intrinsic nonlinearity of the system and to face in a unifying manner both the uniform and the natural sampling case-with constant reference input whatever may be the structure of the dynamic matrix associated with the controlled plant. In particular, the cases of first- and critical second-order controlled plants with lead-type modulators have been developed in a detailed manner; it has been shown that control systems behave differently with natural and uniform sampling modulators as far as stability properties are concerned. Some of the results are found in the literature, others are original. Further studies will be toward application of the proposed method with the Lyapunov function presenting a more complex form.

Acknowledgement

The authors wish to express their gratitude to Professor E. I. Jury and Professor A. Ruberti for their valuable discussions during the preparation of this manuscript.

References E. Andeen, “Bnalysis of pulse duration sampled-data systems with linear elements”, IRE Trans. AC, Vol. 5, No. 3, pp. 306-313, 1960. (2) T. T. Kadota and H. C. Bourne, “Stability conditions of PWM systems through the second method of Lyapunov”, IRE Trans. AC, Vol. 6, No. 3, pp. 266-276, 1961. (3) F. R. Delfeld and G. J. Murphy, “Analysis of PWM control systems”, IRE Trans. on AC, Vol. 6, No. 3, pp. 283-292, 1961. (4) E. Polak, “Stability end graphical analysis of first order PWM sampled data regulator systems”, IRE Trans. AC, Vol. 6, No. 3, pp. 276-282, 1961. (5) E. I. Jury and T. Nishimura, “Stability study of PWM feedback systems”, Trans. ASME, pp. 80-86, March 1964. (6) G. J. Murphy and S. H. Wu, “A stability criterion for PWM feedback control systems”, IEEE Trans. AC, Vol. 9, No. 4, pp. 434-441, 1964. (7) G. J. Murphy and S. H. Wu, “The use of Lure forms to establish a sufficient condition for stability of a class of discrete feedback systems with parallel nonlinear elements”, Proc. J ACC, pp. 755-763, 1965. (8) E. I. Jury and B. W. Lee, “On the absolute stability of multinonlinear systems”, Automatica i TeEem,ekhanika, Vol. 26, No. 6, pp. 945965, 1965. (9) E. I. Jury and B. W. Lee, “A stability theory for multinonlinear control systems”, Third IFAC Cong., London., 1966. (10) R. A. Skoog, “On the stability of PWM feedback systems”, IEEE Trans. AC, Vol. 13, No. 5, pp. 532-538, 1968. (11) R. A. Skoog and G. L. Blankenship, “Generalized pulse modulated feedback systems: norms, gains, Lipschitz constants, and stability”, IEEE Trans. AC, Vol. 15, No. 3, pp. 300-315, 1970. (1) R.

Vol. 298, No. 1. July 1974

57

A. Belestrino, A. Eisinberg and L. Sciavicco

(13) (14) (15) (16) (17) (18)

58

V. M. Kuntsevich and Yu. N. Chekhovoi, “Fundamentals of nonlinear control systems with pulse frequency and pulse width modulation”, Automatica, Vol. 7, No. 1, pp. 73-81, 1971. A. Balestrino and L. Sciavicco, “A stability criterion for PWM systems with natural sampling”, Ricer&e di Automatica, Vol. 3, No. 2, pp. 97-112, 1972. A. BaleEitrino and L. Sciavicoo, “On the stability of PWM systems with natural sampling”, Ricerche di Automatica, Vol. 3, No. 3, pp. 1-18, 1972. K. B. Datta, “Periodic modes of PWM control systems”, J. Franklin Inst., Vol. 294, No. 2, pp. 113-122, 1972. P. F. Panter, “Modulation, Noise and Spectral Analysis”, pp. 534-546, McGrawHill, New York, 1965. K. W. Cattermole, “Principles of Pulse Code Modulation”, pp. 114-124, Iliffe Books, London, 1969. W. Hahn, “Stability of Motion”, pp. 10%109, Springer, Berlin, 1967.

Journal

of The Franklin

Institute