Jury: A Simplified Stability Criterion

in a com-pact and shock-proof form. It requires no heat-up time and the oscillationi can be started instanta-neously under given initial conditions.

In principle any electronic device based upon Halleffect can be replaced by a similar device based uponmagnetoresistance effect. Theoretically, magnetoresist-ance effect devices should have much higher efficiencyaind sensitivity than those of the Hall effect devices,when the operating magnetic field is well over 3000gauss. In another method of making use of the mag-netoresistance effect, oscillators as well as other kindsof devices can be constructed with two magnetoresist-

ance elements connected in a push-pull circuit. This willgreatly improve the performance of the device. Someof these experiments have been carried out and theresults are to be published elsewhere.

ACKNOWLEDGMENTThe author acknowledges with thankis the support of

this work by Prof. J. Dosse, director of the Institute ofHigh-Frequency Engineering, Technische HochschuleStuttgart, Stuttgart, Germany, He also wishes to thankthe German Research Foundation (Deutsche For-schungsgemeinschaft) for its financial support.

A Simplified Stability Criterion for LinearDiscrete Systems*

E. I. JURYt, SENIOR MEMBER, IRE

Summary-In this study a simplified analytic test of stability oflinear discrete systems is obtained. This test also yields the neces-sary and sufficient conditions for a real polynomial in the variable zto have all its roots inside the unit circle in the z plane. The newstability constraints require the evaluation of only half the numberof Schur-Cohn determinants [11, [2]. It is shown that for the test ofa fourth-order system only a third-order determinant is requiredand for the fifth-order, one second-order and one fourth-orderdeterminant are required. The test is applied directly in the zplane and yields the minimum number of constraint terms. Stabilityconstraints up to the sixth-order case are obtained and for thenth-order case are formulated. The simplicity of this criterion issimilar to that of the Lienard-Chipard criterion [31 for the continuouscase which has a decisive advantage over the Routh-Hurwitz cri-terion [41, [51. Finally, general conditions on the number of rootsinside the unit circle for n even and odd are also presented in thispaper.

I. INTRODUCTION

T IS KNOWN THAT linear time-invariant discretesystems can be described by constant coefficientlinear difference equations. These equations can be

easily transformed into the function of the complex vari-able z by the z-transform method. One of the problemsin the analysis of such systems is the test for stability,i.e., to determine the necessary and sufficient conditionsfor the roots of the system characteristic equation to lie

* Received AuguLst 31, 1961; revised manuscript received, Febru-ary 23, 1962. This research was supported by the USAF Office ofScientific Research of the Air Research and Development Commandunder Contract No. AF 18(600)-1521.

t Department of Electrical Engineering, University of California,Berkeley, Calif.

inside the unit circle in the z plane. These stability testsinvolve both graphical procedures such as Nyquistlocus, Bode diagrams, and the root-locus, and analyticalmethods such as Schur-Cohn or Routh-Hurwitz cri-teria. Because of the higher order determinants to beevaluated using the presented form of the Schur-Cohncriterion, many authors in the past have used either aunit shifting transformation [6]1 or bilinear transforma-tion [7]. The latter transformation maps the inside ofthe unit circle in the z = eTs plane into the left half ofthe w plane2 and then applies the Routh-Hurwitz cri-terion. This transformation involves two difficulties:1) algebraic manipulation for higher-order systems be-comes complicated, and 2) the final constraints on thecoefficients in the z plane become unwieldy and requirealgebraic reductions to yield the minimum number ofterms. Because of these limitations this criterion is notusually used for systems higher than second-order.A recent investigation by this author has shown that

the evaluation of the Schur-Cohn determinants can besimplified considerably by making use of the real coeffi-cients of the polynomial in z, so that the manipulationinvolved in testing for the zeros of a polynomial arecomparable to those using the "transformed (or modi-fied) Routh-Hurwitz criterion," thus avoiding the bi-linear transformation [8], [9].

1 This transformation uses p = z -1.2 This transformation uses

w+1w

=1w-1

1962 1493

PROCEEDINGS OF THE IRE

The study in this paper represents a major simplifi-cation of the earlier work [8], [9] where it is shown thatonly half the number of determinants are required forobtaining the stability constraints. This simplificationhas a decisive advantage over the m-odified Routh-Hurwitz criterion and, indeed, higher-order systemscan easily be tackled using the proposed stability test.

II. THEORETICAL BACKGROUND

In this sectioni we review the simplificationis whichhad been obtained in an earlier publication [8], [9] andexplain in detail the manipulations involved.Schur-Cohn criterion [1], [2]:If for the polynomials

F(z) = ao + alz + a2z2 + * * * + a,z"

the conjugate sign in (2) is superfluous. It is the uti-lization of this fact that leads to the simplificationof (2).As noticed from (2), the highest-order determinant

|A,,,l is of order 2n, while the characteristic equation isof order n. Hitherto this contituted one of the dis-couraging facts in widely using the criterion for higher-order sampled-data systems. A recourse to transforma-tion to other planes was therefore attempted to yieldeasier stability tests.

III. SIMPLIFICATION OF THE STABILITYCONSTRAINT EQUATION [8], [10]

Since all ak in (2) are real, the mnatrix can be written as

(1)[Pk QP I

Ak

T pTJall the determiniants of the matrices(4)

Ak =

ao 0

a, ao

ak-1 ak-2

_~ 0

an-l a

an-k+l an-k+2

o 0 ...O

o 0*. .

ak3 .

0 a,, ... an-k+l

0 0

O ... 0 do dl . . . -k-I

0 . .. . .

a,,_k+3 * a,

k = 1, 2, 3, * * n, dk = complex conjugate of akare different from zero, then F(z) has no zeros on thecircle z = 1 and ,u zeros in this circle, ,u being the num-ber of variations in sign in the sequence 1, AlA2 * * A,,.XThe proof of the above theorem isquite involved and is available in the literature [1],[2]. This criterion was first introduced by Cohn in 1922,and since that time neither engineers nlor mnathemuati-cians have simplified it to a usable form.

For a system of order n to be stable, all the n zeros ofits characteristic nth order equation must lie within theunit circle, i.e., the sequences 1, |All, IA21, - * * , |tst|must have n variations in sign. The stability criterioncan, therefore, be expressed by the constraints [4]:

AkI < 0, kodd

Ak > 0, k evenf (3)

For a discrete or a sampled-data system, all the co-efficients of the characteristic equation are real. Hence,

0 0 . . do

(2)

where the superscript T denotes transpose anid

-ao a0 0 01

P a, ao 0 ..Pl.. = al a ° * *° v

Lak- ak-2 ak-3 . ao-

[a,, a,- a,,-k

o a,, . a,-k+2Qk== L * kj

Lo0 O * * a,

(5)

(6)

It is noticed that all the diagonal termiis of each of Pk andQk are equal; each is symmetric with its cross diagonal.It is this characteristic of the Schur-Cohn matrix thatleads to the following simplification [1], using a unitarytransformation.

Ak = Xk + Yk Xk - Yk , (7)

1494 June

an 47n 1. . an-k+ 1]

0

)...1) 2 Ill.

Jury: A Simplified Stability Criterion

where Xk and Yk. are given as follows:

[ao a, a2 * * * ak-Ilo aO a, *.* ak-2

Xk =I0 0 ** a1 J

_O O 0.** .ao _

(8)

3) After collecting the terms of Ak and Bk., replace allthe bi's by the ai's. Hence,

AkI = (At + Bk^)(Ak -Bk) = A,.2 - B,A2 (12)3and for stability from (3) this reduces to

and

an-k+1 *.** an-, a1

[a,,-k,±2 a,-, 0 1(9Yk = n7+ n °(9)

an-l* *0 01

La, *. *0 0°Thus the Schur-Cohn determinant Ak I is reduced to

the product of two k-order determinants [1], [2] whichis considerably easier to evaluate than the 2k-order de-terminant IAk| . If ak is complex, then this simplifica-tion is no longer possible [1], [2].

IV. THE SYMMETRICAL PROPERTIES OF

XXk+ Ykl| Xk-ykl [8]Now IXk+ Yk I is a homogeneous polynomial of di-

mension k in the variables ao, * , a,,. The polynomialXk- Yk is identical to the polynomial XXk+YkIj ex-

cept for a change of sign of those monomial terms whichhave an odd number of elements from Yk,, i.e.,

I Xk + YkI = Ak+ Bk- (10)3

I Ak|, IBkI, k even

Ak < I Bke|, koddJk = 1, 2, *, n. (13)

Therefore, the application of the Schur-Cohn criterionnow reduces to the evaluation of determinants up toorder n only for the nth order polynomial.

It should be noted that the last simplification is use-ful only for design procedures where the coefficients ofF(z) are given in other thani numerical values. How-ever, if ak's are given in numerical values, the use of(10) and (11) is preferred for computer calculations.

VI. EQUIVALENCE OF THE LAST CONSTRAINTA,, B,, TO A SIMPLE AUXILIARY

CONSTRAINT [9 ]This equivalence constitutes the second major simpli-

fication of the original Schur-Cohn criterion. It reducesthe stability test considerably. In this equivalence it hasbeen shown that when the first (n - 1) constraints of theA's and B's are satisfied, then for a,,>0, the followingauxiliary constraints

F(1) > 0, F(-1)> 0, n even

<0, n odd(14)

Xk - Yk I = Ak - Bk (11)

where A,(Bk) is the sumn of all monomial terms whichdo not chanige (do chanige) sign wheii YkI is replaced by-Y,,in IXk+Yk|.

V. IDENTIFICATION OF Ak AND Bk(which we designate as the stability constants) [8].

1) Let all the as's in the matrix Yk, in (9) be denotedby b,'s; then expand the determinant Xk+ YkI interms of ai and bi.

2) After expanisioni, examine every term which is aproduct of ai's and bi's; if it containis an even num-ber (including zero) of bi's, then it is assigned toAk,; otherwise assigni the term to Bk.

From (8) and (9), by replacing all the a's of Y,C by b's, we obtaiinA k anid Bk by first expanding the following determinant:

IXk+ YkI.ao+b.-k+± ai+b,,-k+2 a2+b.,k+3 -

b1,-+2 ao+bn-k+3 ai+bii,k+4 -

bn-kA+3 b??-b+4 ao+bn-k-[ -

bn_l N7 0 O - *b,b, , 0 0 00 0 0 0

0 0= 102,--0

*ak-2+b.-i ak-l+b.a,ak3+ bn ak-2

*ak-4 ak-3

* a, a2

ao a,

0 aO

n - 1, n

are equivalent to A, t B, I, depending on n. For anya, the auxiliary constraint can be written as

F(1) * F(- 1)> O, n even

< O, n odd

In the Section VII the exact mathematical relatioii-ship betweeni A,, - Bn2 and the auxiliary constraint willbe introduced.

VII. REDUC'TION IN THE NUMBER OF DETERMINANTSFOR OBTAINING THE STABILITY CONSTANTS

Ak'S AND Bk'SIn this section we will show that for the stability test

only about half the number of determinants for obtaini-ing the Ak's and B,k's are required. This importanit re-duction is based on certain properties that exist betweenthe Ak's and Bk's. We will indicate these properties firstand then show how they can be used for this majorsimplification.

1) Ak2 Z Bk24* Ak_,Ak+l± BkBk+lk = 2, 3, 4, * *, n-.(5)4

(13a)

4 The above first property is also equivalent to

A12 _ B&2=A,, A,,l-BkBk+l, k = 2, 3, 4, * *, n - 1. (15a)

1962 1495

bn_2

PROCEEDINGS OF THE IRE

The above equivalence is established by expansionfor the first few values of k = 2, 3, 4, 5 and can be gen-eralized similarly for any other values of k.f XVhenk = n -1, the above equivalence is reduced to

(16)

This limiting case has been rigorously proven by deter-minant manipulations using properties 2) and 3). Eq.(15) is the key identity for reducing the determinant ofAk and Bk, for it is noticed that by forcing certain re-strictions on the A's and B's before and after a certaink we can dispense with A e2 < Bk,2. This will be best illus-trated by the few examples to be discussed.

2) An= (ao+ a2+ a4 + * )(An_l- B_1). (17)

Bn = (a, + a3 + as + )(An-, - B,-0),n > 2. (18)

The above identity can be easily verified for the firstfew values of n and can be generalized for any n.6 Theimportance of this property lies in the mathematicalproof of the previous section as follows:

An' >< B.2,~ (ao + a2+ a4++ * * )2

A,,l + Bni1 = (aO + a2 + a4 +± - )A.-2- (a,l+ a.3+ a,5 + -)B,,-2

ABt = (ao+ a2 + a4+ - - - )

-la - (al+ a3B+ a5+ ).An-1 Bn-1

(23)

(24)

(25)

If we multiply (23) by A,,,-Bn_ and use (24) and(25), we readily obtain the limiting case of (1Sa):

A',l- B 21 = A,,B-2-B-BB 2. (26)

The above three properties in combination withthe preceding discussion will nlow be used in obtainingthe new stability constraints for low order systemsand then, by generalization, to obtain the constraintfor an n-order system.

Examples of Low Order Systems: We will apply thereduction properties wherever they are applicable ton =3, 4, 5, 6, and then obtain the stability constraintfor any n. We will assume an >0. This can be easily doneby multiplying F(z) by minus one, if necessary.

a) n = 3, F(z) = ao+ alz + a2Z2 + a3Z3I a3 > 0. (27)

(19) The stability constraints from the modified stabilityequations (11) and (12) are given by the following in-equalities:

An2 - Bn2 0 =F(1) F(-1) < 0 (20)

which verifies the equivalence An I| BJ to theauxiliary constraint.

3) (An-I + Bn-1) = A,-2(ao + a2 + a4 + * - * )

-Bn-2(al+a3+as+ * * ), n>3. (21)

The above relationship can also be written as

A n-I + B5_1 = 2 [F(1) (An-2 -B2)

+ F(-1) (An_2 + Bn_2)J .

Eq. (22) has been verified for anyn > 3 by using cer-

tain determinant manipulations [IIJ. The use of thethird property lies in the fact that one can obtainAn_l+Bn_1 from An2 and Bn_2, and most important,to verify the first property for the limiting case, i.e.,k = n-1, as follows.The third and second properties can be rewritten

as follows:

5 For values of k up to 15, and for certain n, this equivalence hasalso been checked numerically. Although a rigorous proof (other thandeterminant expansion) for (15) has been obtained, by using a heur-istic argument however, it can easily be shown [111 that this identityis not violated for all values of k between 5 and n -1.

6 To establish the identity for the general case, [11] it is easier toshow the following equivalent conditions for (17) and (18):

A. + B. = F(l)(A.-1 - B,,-,); A. - B. = F(-1)(An-I -B_A.

(22)

aoI < a3, | All < I Bil (28)l ao2 -a32| > aOa2- aa3I ,I A21 > |B2j (29)7

F(z) Iz=l > 0, F(z) Iz=-l < 0, or A31 < B31 . (30)

Reduction of the Constraint Conditions: From the firstproperty of the Ak and Bk, we may write in this case

Since B, is positive (i.e., a3>0), we may write for theright side

Al- A3> B3.B,

7To show how to obtain A2 and B2 for this case:1) Expand the determinant of (13a) for k = 2 and n = 3, as follows:

II ao+ b2 a, + b3 |a02+aob2-a,b2-b2121b3 a

o

2) To identify A2 and B2, follow procedure 2 in Section V to obtain

A2 = a2 -b32B2 = aob2- ab3.

3) Replace all the b's of A2 and B2 by the a's to get

A2 = a02- a32

B2 = aoa2- a,a3.

A similar procedure is used for obtaining the Ak's and Bk's for any kand n.

or

A,22> B22 AiA3> BlB3. (31)

(29a)

1496 Juane

A,' 1 > B,I-l .--* An-2An> B,-2Bn-

. (a,+ a3 + a5 . . .)l

Jury: A Simplified Stability Criterion

Using conditions (28) and (30) in combination with(29a) the new stability constraints are:

1) ao < a3, or ao- a3 < 0, ao+ a3 > 0, (32)

2) B3 < 0, (33)3) 1 A3| < B3) , or F(1) > 0, F(-1) <0. (34)

From the second property in (18)

B3 = (a, + a3)(A2- B2) . (35)Since a,+a3 is positive (from (34)) because it is equal toF(1) - F( - 1) >0, therefore B3 is negative only whenA2- B2 <0. It is of interest to note that from the thirdproperty of the stability constants it is readily seenthat A2+B2<0 is identically satisfied from the first andthird conditions. Thus the simplified form of the sta-bility constraint for n =3 reduces to

I aol < a3 (36)

ao2 -a32 < aoa2 - aia3 (37)

It is noticed that for the fourth-order9 case only one de-terminant of third-order for obtaining A3 and B3 is re-quired. All other conditions are very simple.

c) n = 5, F(z) = ao + alz + a2z2 + a3z3 + a4z4 + a5z5,as > 0. (49)

The stability constraints in symbolic form

A1) < B1, B1= a > 0 (50)

|A2| > |B2|

lA3l < |B3|

|A4| > |B4l

F(t) > 0, F(-1) <0, or A5| < B6I.

(51)

(52)

(53)(54)

Reduction of the Constraint Relationship: We maykeep in this case condition (51) but we eliminate (52)by using the first property, i.e.,

A32 < B32' A2A4 < B2B4. (52a)

ao + a, + a2 + a3 > 0, ao - a, + a2- a3 < 0. (38) With A2 negative because of (53),10 the stability con-straint for (50), (52), (52a) and (53) becomes (50),

b) n = 4, F(z) = aO + a1Z + a2Z2 + a3z3 + £4Z4, (51), A4>0, and A42>B42. Using A4>0 in (56), i.e.,

a4 > 0. (39) (A4-B4)(A4+B4)>0, we obtain A4-B4>0 andA4+B4>0. Furthermore, since A2=A12-B12 is to be

The stability constraint negative from (50), it is readily satisfied if the secondI aol < a4, I Al < Bil (40) constraint 0+(51) is equivalent to A2-B2<0, A2+B2

<0. Finally, we obtain for the reduced form the follow-jao2 - a42l > aoa3 - aia4, A21 > B21 (41) ing:

ao3 + aoa2a4 + aia3a4- aoa42- a2a42- aoa32 I< |aO2a4 + ao2a2 + a12a4 -aoa2a4-a43 -aoala3 | (42)

A2- B2 < 0,

A4- B4 > 0,

A2+ B2 < 0

A4 + B4 > 0

or F(1) > 0, F(-1) < 0.|A3) < |B3I (43)

F(z) jZ= > 0, F(z),=.1 > or A4j > B4 . (44)

Reduction of the Constraint Equations: Using property1), we may write, as in the previous case,

A1

A22 > B22 X A1A3 > B1B3, or A3> B3. (45)B1

Eqs. (40)-(42) are now equivalent to (40), B3<0 and(42). Using B3<0 with (43), we finally obtain the re-

duced constraints

aol <a4, or Al| <B1 (46)

A3-B3 > 0, A3+ B3 < 0 (47)8

F(1) > 0, F(-1) > 0. (48)

8 Constraints in (47) can also be written as B3<0, A31 < JB31.The latter has an advantage in numerical testing if B3 <0 is violated.Then the test could be discontinued without having to calculate A3.

It is noticed that in this case only the second- andfourth-order determinants are required. One may alsouse a different form of reduction by eliminating (56).However, this will not yield much simplification over

the previous form because a fourth-order determinantwith a third-order determinant is then required.

g An alternate form which is more advantageous for design canbe obtained for the fourth-order case. This form can be easily ob-tained by using properties 1) and 2). It is given as follows:

1) A2 <0

2) A2<- |B2|

3) A3 - B3 > 0

4) F(1) > 0, F(-1) > 0.

In this case only one third-order equation in 3) and two second-order equations in 2) are to be solved, while in the former case twothird-order equations are to be solved. It should be noted that when2) is satisfied, relationship 1), i.e., A2 <0, becomes redundant. There-fore, the final constraints for n=4 can be summarized as [121:

1) A2 < -I B21 , or A2-B2 < 0, A2 + B2 < 0

2) A3 - B3 > 0

3) F(1) > 0, F(-1) > 0.

10 Note A2=A12-B12.

(55)

(56)

(57)

1962 1497

PROCEELDINGS OF [HEJRE

d) n = 6, F(s) = aO + alz + a2z2 + az3 + a4z4

+ a5z5 + a6z6, a6 > 0.

The stability constrainits in symbolic form are

All < BI, B =a6 > 0 (58)1 A2 > 1B21 (59)|A,, < B31 (60)

A4I > B41 (61)A51 < B51 (62)F(1) > 0, F(-1) > 0, or A6, > B361. (63)

Reduction of the Constraint Relationship: In this casewe mav conveniently eliminate the determninanits of (59)and (61) as follows: usinig first 4 22>1322XA 1 il >B1iB:,'we obtain the equivalent constrainit for (58)-(60), not-inig (45), as follows:

All < Bi, | A3 < |B31 and B <0. (64)To eliiniricate the determuinant of (61), we use the fol-

lowing proveni identity:142 > B424'> A3A5 > B3B5. (65)

Notinig that B3<0, and (60) and (62), the above issatisfied if B5 is positive, and (60) anid (62) are also satis-fied. Therefore the stability constrain-ts for n =6 tnowreduce to

I All < B,

|-31 < B3 I, B1 < 0, or (A3-+ B3) < (,

(A3 -B3) > 0

| A,I < I Bsl , Bs > 0, or (A5+ B5) > 0,

(A5 - B5) <0F(1) > 0, F(-1) > 0. (66)

Following the above reduction we can similarly ob-taini the stability constraints for higher-order systemsfor the general case, i.e., for anv n, by using the firstproperty.The stability constraints cani be presenited in either

one of the following two formiis.The New Stability Criterion: A necessary and sufficieint

conditioni for the polynomiiial F(z) =ao+alz+a2z2±+ -

+akzk+ * +a,,z"l with a,, >0 to have all its rootsiniside the uniit circle is represenited by the followinig coni-strainits for n even anid n odd, respectively:

n even'

F(1) > 07 F(-1) > 0

A1I < B,, B1= a,, > 0, or

A1-B, < 0, A1+ Bs>0

A:, - Ba > 0, A1,; + B3; < (0A,,-B5,< 0 A+B.-,>(),' 7 >0, A-+ 7K< ()

> 0 for n = 4k

A,,, - B,_1 < 0 for any other n

< 0 for n - 4k

An,,1 + Bz,,l > 0 for any other nI,k= 1, 2, 3

n. odd1IF(1) > 0, F(-t) < 0

A,2-B2 < 0, A2+-B < 0

A"'I4-B4 > O, A44+ 4 > ()

A16-36<0, .16+B4 <0

> 0 for n -1 = 4kA,,_- 1, "1

< 0 for any other(n - 1)

> 0 for ,i-1 -4kA,1-1 + B- 1 < () foranv other (n - 1)

k = 1, 2, 3

Alternate Form: Atn alternate equLivalenit miiethodwhich is of advantage if the stability conlstatits evalua-tion is carried out by imiethods othier thani a computer ishereby presented.

it even

F(1) > 0, F(-1) > (0

!Al| < Bi, B I = a,, > (

|1A3 <| B3, B:;<1A51 < BsI, B> 01A71 < B71, 13 < (

< 0, for it 4k,-l,_l < B,_1, Br,-

> B0 for any other n1

k- 1,2,3

ni odd

F(1) > 0, F(-1) < 0

A,2 > B2 , A., < 01 A41 > 1B41, A4 > (0|A6I > 1|B6!, A6< 0

1' Note that Ak+B, and A. -Bk cani be obtained directly from(10) and (11). > 0, for n - 1 = 4k

Jitne1498}

Jury: A Simplified Stability Criterion

|An_l | > | B_ |, A-< 0, for any other n - 1

k= 1,2,3 ...

In the above case the identification of AkI and BkIfrom the determinant Xk+ Yk could be used. Thiscriterion can be applied for design purposes when theak's of F(z) are given other than numerically.Summary of Stability Constraints for Low-Order Sys-

tems: In this section the constraints in terms of the co-efficients of the low-order polynomials are presented.

n = 2

F(z) = ao + alz + a2z2 a2 > 0

1) aol < a2, or ao-a2 <O02) ao+ al+ a2 > 0, ao-al+ a2 >O

n = 3

F(z) = ao + alz + a2z2 + a3z3, a3 > O

1) aol < a3

2) ao2 - a32 < aoa2- a1a3

3) ao+ a,+ a2+ a3 >O, ao-a,+ a2-a3 < 0n = 4

F(z) = ao + alz + a2z2 + a3z3 + a4z4, a4 > 0

1) ao2 - a42 - aoa3 + a1a4 < 0

2) ao2 - a42 + aoa3- aja4 < 0

3) ao3 + 2aoa2a4 + a1a3a4- aoa42- a2a42 - aoa32- a02a4- a02a2- a12a4 + a43 + aoala3 > 0

4) F(1) > 0, F(-1) > 0

Evaluation of the Stability Determinants [17]: To eval-uate the Ak's and Bk's from (13), we need to expand thisdeterminant only for k = n -1. All other values of k canbe obtained from this expansion by a limiting process asfollows:To obtain Xn.3+ Y.-31 from XX-,. + Y.-, I, we let

(in the latter) ao = 0, a1 = 0 and ai->ai2, and then divideby -1/b2. Similarly, we can obtain all the otherXk+ Yk by a similar limiting process.For the case when the coefficients of F(z) are given in

numbers, one can use (13a) only for k =n-1, and onlyby letting the bi's as +aj's obtain for A,-1+Bn_1, and as-ai's obtain for An1-Bn_-. This is evident if the deter-minant is written in detail; one can notice that it in-cludes inside all other determinants for other valuesof k.

Relationship Between the New Criterion and HurwitzDeterminants: In this paper some relationships betweenthe new stability constraints and the Hurwitz con-straints are established. To show this relationship, weuse the bilinear transformation z = (w+ 1)/(w-1) inF(z), (1), with an>O, to obtain the following Hurwitzpolynomial:

Fi(w) = bo + biw + b2w2 + * * + b4wn7

where bk is a linear combination of the ak's in F(z). Thefollowing relationships exist:

a) F(z) z=lbn,

b) n odd

n-= 2(n12)(-')(A.-, - B.,), for n = 4k, k = 1, 2,3

--2(n'2)(n-1)(An-1 -Bn1), for any other n,n even

In-1 = 2( B2)(n--)(An_1-B._1), for n- 1 = 4k,k = 1, 2, 3

= - 2(n/2)(n-)(An1 -Bn.1), for any y other n,

where I\AH1 is the next to last Hurwitz determinant.The above relationships have been verified by expan-sion for low-order systems and can be generalized forany order.

It should be noted that if the constraints of the newstability criterion for n even are written in even form(i.e., in terms of Ak and Bk where k is even), then thelast constraint could be terminated by

>An- -Bn-1 0

This can be easily obtained by using properties 1) and 2).A similar form can be obtained of n odd if its stabilityconstraints are written in terms of odd forms of Ak andBk. Thus, one can show that next to the last Hurwitzconstraint for the transformed polynomial is identical tothe last constraint of the new stability criterion. Thissimplification could be used to advantage in design inorder not to solve two higher-order equations. It is ap-plied to n =4 as shown above. Furthermore, if the sys-tem is initially stable and some of its parameters changeso as to approach instability, the constraint A,- -Bn_is the most critical one. Hence, to obtain the maximumparameter variations for stability limit only the equa-tion of (An-l-Bn-1) need be solved.Number of Roots of a Real Polynomial Inside the Unit

Circle: The preceding criterion yields the necessary andsufficient condition for all the roots of F(z) to lie insidethe unit circle. To obtain information on the number ofroots that lie inside the unit circle if the system is un-stable, we can usefully modify the above discussions tocover this case. Following the original work of Schur-Cohn [1], [2] and its modified form [9], and using thefirst property of the stability constants Ak's and Bk's,one can state the following.The number of roots of F(z) that lie inside the unit

circle is equal to the number of variations of sign of thefollowing constraints:

1962 1499

F (z) z==- I "-'Z (- J)nbo

PROCEEDINGS OF THE IRE

>0A4,, BJ1 zF(1)F(-1)<0

ni odd

1 > 0

A2| |B2|IB3I t-A2A4 B2B4A41 e B4|

B5 |A4A6 Z B4B6

A6| B61

n-1| Bo-_l

A,t tB, F(1)F(-1 < 0

It is noticed from the above that for n eveni one nieednot have to evaluate the even determinants but canl usethe relationship Ak2 t Bk2'1A1_,.+1 tBk_ Bk+1 whiclare obtained from the calculated odd determinianits.Similarly, the odd determiniants are to be avoided from-ln odd. It is assumed that in this counting nieither onie ofthe constants Ak is equivalent to Bk. For such a case

special procedure should be used and this will be (lis-cussed in a somewhat differently related criterioni [161.

VI II. CONCLIUSI()N

From the preceding discussioni, it is shown first thatin the original Schur-Cohn or the modified Routh-Hurwitz criteria, the number of determinants requiredfor the stability is almost halved. The use of the criteriafor both design of discrete systemls as well as for testingroots of a polynomial inside the tllit circle is evidenit.This criterion will nlow be useful in maniy applicationssuch as the stability test of difference equations withconstant and periodically varyinig coefficients, in nloni-

linear discrete systems for the stability studs of lilmlit,cycles, in somiie applicationis of dligital coml1puLters, in thestability test of liniear svstemiis with randonmi\-vfary-1-ingparameters, andcl in niany other- applications, such laseconomiiic andl inivenitory systems. Trhuls it is lhoped thlatthis criterioni will find minany applicationis ini ariousfields in additioni to the above, anid its use by eniginieers,physicists anid mathematicians will be greatl y enlia nce(ld.The evaluationi of the higher order determinants couldalso be reduced to evaluationi of only seconid orcler (leter-tiiinants by usinlg altertnate but equivalenit formiis of thecriterioni. Ihe discussioni of these alterniate forinis hasbeen recently presenited [13 15] and in future publica-tioIns the conniiectioni betweeni the various form-ls of thesestabilityT criteria will be discussed [16|.

AC KNOWLEDGNMENT

The author gratefull'y acknowledges the helpful dis-cussions and aid of M. A. Pai in the preparation of thismaniuscript.

REFERENCESill A. Cohni, "Uberdie Anzahl der WuLrzelnl einier algebraicheni

Gleichtung in einein Kreise," Aath. Zeit., vol. 14, pp. 110-148:August, 1922.

121 M. Mardeni, "Geometry of the Zeros," American MathemiaticalSociety, New York, N. Y.; 1949.

[31 F. R. Gantnmacher, "Theory of Matrices," Chelsea PublishinigCo., New York, N. Y., vol. 2, pp. 221-227; 1959.

[41 E. I. Jury, "Sampled-Data Control Systems," John Wiley anidSonis, Inc., New York, N. Y.; 1958.

[51 Y. Tsypkin, "Theory of Pulse Systems,'" State Press for Physicsand Mathematical Literature, Moscow, U.S.S.R.; 1959.

[6] Van J. Tschaunier, "Die Stabilitat von ImpulssysteleniRegelungtechnik," vol. 8, pp. 42-46; February, 1960.

[7] R. C. Oldenbourg and H. Sartoriotis, "The Dynamics of Auito-matic Conitrols,' Americani Society of Mechanical Enigineers,New York, N. Y., ch. 5; 1948.

[81 E. 1. Jury and B. H. Bharuchai, "Notes on the stability criteriollfor linlear discrete systems," IRE ThIRANS. ON AUTOMAtric CON-TROL, vol. AC-6, pp. 88-90; February, 1961.

[91 E. I. Jury, "Additionis to stability criterioni for liniear discretlesystems," IRE TItANS. ON AUTOMATIC CoNx rlIo -,ol. AC-6, pp.342-343; September, 1961.

[101 B. H. Bhartucha, "Anialysis of Initegral-Siquare Error in Saipled-Data Control Systems," tUniiversity of Californa 13Berkeley.ERL Ser. 60, Issue No. 206; Jutne 30, 1958.

[111 E. I. Jury, "A Simplified Stability Criterioni for linear l)iscreteSystems," University of Californ;ia, Berkeley, FRI. Rept. No.373; J unle, 1961.

[121 E. I. Jurv, "DiscLussioll of a paper by l TIschalunier eiititled "Tlhestability of sampled-data systems," Regelungstlchnnik, vol. 9, pp.340-342; August, 1961.

1131 E. I. Jury anid J. Blanchard, "A stability test for liniear (liscretesystems in table formii," IRoc. IRE, x-ol. 49, pp. 1947--1948;December, 1961.

[141 E. 1. Jury, "A stability test for linaear discrete systenms usilngsimple divisioni," PROC. IRE, v-ol. 49. pp. 1948-1949; December,1961.

[15] E. I. Jury anid S. C. Gupta, "Oni the Stability of linear dliscretesvstems," to be published in the Regelungstechnik.

[16] E. I. Jury and J. Blanchard, "OTT the Roots of a Real PolvnomiiialIniside the Uiiit Circle anid a Stability Criterioni for Liniear Dis-crete Systems," Universit' of California, Berkeley, FRI. Rept.No. 60, Issue No. 425; Decemiiber 26, 1961.

[17] E. 1. Jury', "Oni the evaluation of the stability determiiinaubts inlinear discrete systelms," to be published by IRE 'TRIANS. ONA U"TOMATIC CoNTROl.; JUIV, 1962.

it even

1 >0A 11 ><IBB2| A 121 > B B:A3| .,1B4I AzA5 Q BjB5A5 <1B53s'

I|A2i Z

A4|Z

IA _ | |Bn ,_I>0

An- B,t2 |< 0

A31 t

1A51

A,> 0

An2- Br2 |< 0

1500 J1lnlie