1572 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. 31, NO. 12, DECEMBER 1990

[6], [7] called SWITCAP2 allows simulation of these types of SC neural Indeed, a 20-neuron Hopfield-me auto- associative memory has already beep simulated in 181 using a Tsividis-like structure, and it has been shown to work.

V I _, “SWITCAP: A switched-capacitor network analysis program- Part 11: Advanced applications,” IEEE Circuits Syst. Mag., vol. 5 , pp. 41-46, Dec. 1983.

[8] K. Suyama, “Analysis, simulation, and application of linear and nonlin- ear switched-capacitor/digital networks,” Ph.D. dissertation, Columbia Univ., New York, 1989.

VII. CONCLUDING REMARKS

A SCBAM has been presented. Two SC structures have been proposed that support easy implementation of the SCBAM. Experimental results with 3- and 4-neuron SCBAM’s and vari- ous input vectors validate all our developments. Like all artifi- cial neural networks currently in existence, the SCBAM is no exception and has several limitations. One limitation of the SCBAM at present is the sizes of the input arrays as the number of neurons and stored patterns grow. As such its storage capac- ity is severely limited by the amount of neurons used. It may make sense splitting up the arrays into individual chips. Its feasibility in this respect for VLSI implementation is currently being investigated. Also, this type of network does not readily allow for learning, a much talked-about topic in the literature to date. On the positive side of things, however, because the stored matrix M changes with different input patterns, it makes sense having a programmable switching matrix for interconnections, along with programmable weights. Programmable weights could be accomplished by using binary capacitive arrays. Weights values could then be stored in a semi-static shift register. So, too, could the interconnection matrix be stored. Simple switches could make up the interconnection matrix. This would then make the SCNN a mixed SC/digital network.

The problems of large op amps having limited bandwidth, offset voltages, and noise could be alleviated by using high gain inverters. Inverters, however, suffer from the disadvantage that they have limited dynamic range and once switching thresholds are set they cannot be changed. In certain applications of BAM’s, it is desirable to have varying thresholds in each pro- cessing element. When this happens, the BAM is referred to as a nonhomogeneous $AM, and it contains a richer set of dynamic properties and larger storage capacity than the normal BAM. Coupled with this, it is more difficult to control the sigmoidal function in an inverter than in an op amp.

The field of SCNN’s is far from over, but it is the view of the authors and others that thrusts made in the direction to prove that such networks can be made fault-tolerant and easily adapt- able to VLSI, and have some degree of learning ability, will greatly assist in implementing analog circuitry for the field of artificial neural networks.

REFERENCES B. Kosko, “Bidirectional associative memories,” IEEE Trans. Sys., Man, Cybem., vol. 18, pp. 49-60, Jan./Feb. 1988. -, “Constructing an associative memory,” BYTE, vol. 12, pp. 137-144, Sept. 1987. Y. P. Tsividis and D. Anastassoiu, “Switched capacitor neural networks,” Electron. Lea., ~ Q I . 23, pp. 958-959, July 1987. J. E. Hansen, J. K. Skelton, and D. J. AJlstot, “A time-multiplexed switched-capacitor circuit for neural network applications,” in Proc. IEEE ISCAS ’89, pp. 2177-2180. G. Mathai and B. R. Upadhyaya, “Performance analysis and application of the bidirectional associative memory to isdustrial spectral signatures,” IcJArN ‘89, vol. 1, pp. 1-33-1-37. S. C. Fang, Y. P. Tsividis, and 0. Wing, “SWITCAP: A switched-capaci- tor network analysis program-Part I: Basic features,” IEEE Circuits Syst. Mag., vol. 5 , pp. 4-10, Sept. 1983.

Strict Aperiodic Property of a Polynomial Set with Coefficients on a Diamond and Generalizations

C. B. SOH

Abstract-Recently, Soh and Berger 111, Soh 121, and Garloff and Bose 151 have shown that a family of interval (characteristic) polynomi- als, i.e., with coefficients in a box, is strictly aperiodic if and only if two specified polynomials are strictly aperiodic. This note derives some similar results for a polynomial set with coemcients in a diamond to be strictly aperiodic. A procedure to obtain the maximal diamond in the coefficient space containing only the Coefficients of strictly aperiodic polynomials is also derived. The results and procedure obtained are then generalized to be applicable to a wider class of regions in parame- ter space.

I. INTRODUCT~ON

Consider the characteristic polynomial of a linear continuous time system

P ( s ) = t , s ” + t , s “ - ’ + . . . + t , , t , > O ( 1 )

or the characteristic polynomial of a linear discrete time system

t o > 0 (2) Q(z) = tOz“ + tl( - l)’t“-’ + . . . + t,( - l ) , ,

where

t T = ( t O . . . t , ) .

The real vector t can be represented by a point in ( n + l ) - dimensional Euclidean space. The polynomial ( 1 ) is said to be strictly aperiodic if the roots of (1) are all real, negative, and distinct [2]. Similarly, a linear discrete time system is strictly aperiodic if the roots of (2) are all real, positive, and distinct in the range (0,l) [2]. The problem of aperiodicity arises in obtain- ing a response that has no oscillations or that has oscillations of a finite number only [2].

Recently, Soh and Berger [l], Soh [2], and Garloff and Bose [5] have shown that a family of interval (characteristic) polyno- mials is strictly aperiodic if and only if two specified polynomials are strictly aperiodic. However, interval polynomials are restric- tive in the sense that the coefficients are constrained to be independent of each other.

Consider the polynomials in (1) and (2) where the coefficients t , , i = 0,. . . , n are varying in the specified (n + 1)-dimensional diamond

Manuscript received December 20, 1989; revised June 14, 1990. This paper

The author is with the School of Electrical and Electronic Engineering,

IEEE Log Number 9039236.

was recommended by Associate Editor P. H. Bauer.

Nanyang Technological Institute, Singapore 2263.

0098-4094/90/1200-1572$01.00 01990 IEEE

IEEE T R A N S A ~ I O N S ON CIRCUITS AND SYSTEMS, VOL. 37, NO. 12, DECEMBER 1990 1573

where f,, i = 0,. . . , n denotes the nominal value in the interval in which t , varies and r is the radius of the diamond D.

This note derives some similar results for a polynomial set with coefficients in a diamond to be strictly aperiodic. A proce- dure to obtain the maximal diamond in the coefficient space centered at t"= (to . . . ;,,I and that contains only the coefficients of strictly aperiodic polynomials is also given. The results and procedure obtained are then shown to be applicable to a wider class of regions in parameter space.

The related results on Hurwitz stability have been obtained by Bose and Kim [ 3 ] .

11. MAIN RESULTS Let

n

P o ( s ) = t , s " - J r = O

and n

Q,(z) = ( - l) ' t ,zn-' . r = O

Define

Pl(S) = PdS) + P2( s) = Po( s) - r

p3( s) = P,( s) + r ( - 1 ) " ~ ~

p4(s) = p o ( s ) - r( - I)~S'

P 5 ( S ) = p o ( s > + r ( - 1 ) " + ' s f l ~ '

P 6 ( s ) = P o ( s ) - r ( - l ) " + ' s n - ' .

We first derive the necessary and sufficient conditions for every polynomial with coefficients in the diamond D to have only real, negative, and distinct roots in the range [ T , 51.

Theorem I : Every polynomial (of the form (1)) with coeffi- cients in the diamond D has only real, negative, and distinct roots in the range [ T , 51 if and only if the four polynomials P,(s) , P2(s) , P3(s), and P J s ) have only real, negative, and distinct roots in the range [ T , [ ] .

Proof: (Necessity) The necessity is obvious because the co- efficients of p , ( ~ ) , p , ( ~ ) , p&), and p,(~) lie in the diamond D.

(Sufficiency) Let Po( s) represent a polynomial with coeffi- cients in the diamond D. First note that

P 2 ( x ) < p D ( x ) d PI(,), ( 3 ) P 4 ( X ) Q P D ( X ) Q P 3 ( X ) , - - m < x < - l . (4)

- 1 d x < 0

Thus the graph of P J x ) lies between the graph of P , ( x ) and P , ( x ) for x E [ - 1,O). Similarly, the graph of P,(x) lies between the graph of P , ( x ) and P4(x) for x E ( - C O , - 11.

Soh [ 2 ] has shown that every polynomial in the family of polynomials

P ( s , h , ) = P , , ( s ) + h , s " ~ ' , A , E [ b , , a , ]

has only real, negative, and distinct roots in the range [ 7 , 6 ] if and only if the polynomials P ( s , a , ) and P ( s , b,) have only real, hegative, and distinct roots in the range [ T , ( ] . Combining the above result for i = 0 and i = n with (3) and (4) ensures that the graph of P J x ) intersects the real segment [ T , ( ] n times if the graphs of P l ( x ) , P , ( x ) , P , ( x ) , and P4(x) also intersect the real

This completes the proof. Remark: Further simplification is possible when T > - 1. From

the proof of Theorem 1, only P,(s) and P,(s) are required to have only real, negative, and distinct roots in the range [ ~ , 5 ] .

Similarly, from the proof of Theorem 1, for 5 Q - 1, only P3(s) and P4(s) are required to have only real, negative, and distinct roots in the range [ T , 51.

We now derive the result for a linear continuous time system with perturbed coefficients in a diamond to be strictly aperiodic.

Corollary I : Every polynomial (of the form (1)) with coeffi- cients in the diamond D has only real, negative, and distinct roots if and only if the four polynomials P,(s), P2(s) , P&), and P,( s) have only real, negative, and distinct roots.

Proof: This is the special case of Theorem 1 where the range [T , ,$ ] is chosen to be (-qO).

This completes the proof.

We now consider the case where the coefficient t , equals Po. For example, to = 1 for monk polynomials.

Theorem 2: Every polynomial (of the form (1) and t , = fo) with coefficients in the diamond D has only real, negative, and distinct roots in the range [T,[] if and only if the four polynomi- als P,(s), P2(s) , PJs ) , and P6(s) have only real, negative, and distinct roots in the range [ T , 51.

Proof: The proof is similar to the proof of Theorem 1 with (4) replaced by

P6( x ) Q PD( x ) < P5( x ) , - -m < x Q - 1. (5)

This completes the proof. Remark: Similar simplifications are possible as discussed in

the remark of Theorem 1 with P J s ) replaced by P&s) and P&S) replaced by PJs).

Corollary 2: Every polynomial (of the form (1) and to = io) with coefficients in the diamond D has only real, negative, and distinct roots if and only if the four polynomials P,(s) , P2(s) , P5(s), and P6(s) have only real, negative, and distinct roots.

Proof: This is the special case of Theorem 2 where the range [ ~ , 5 ] is chosen to be ( - - m , O ) .

This completes the proof.

Define Q,(z) to have the same values of t , as P,(s), i = 1,. . . ,6, respectively. Note the difference between Q(z ) (2) and P ( s > (1).

We now show that for each result for polynomials of the form (1) to have only real, negative, and distinct roots in the range [T,(], there exists an equivalent result for polynomials of the form ( 2 ) to have only real, positive, and distinct roots in the range [ - 5, - TI.

We are only required to show that P ( s ) (1) has distinct, real, and negative roots V,,. . . , K, if and only if Q(z) (2) has distinct, real, and positive roots - V,, . * . , - V,.

We define a new variable z such that

s = - z . (6)

Substituting (6) into (1) and dividing by (- l)", we have

Since ( - 1)' = ( - l)-', the above polynomial is the same as Q(z ) - . -

segment [T, 61 n times. ( 2 ) . By (61, Q(z) (2 ) has distinct, real, and positive roots

1574 IEEE TRANSACTIONS ON C I R C I J I T ~ AND SYSTEMS, VOI . 37, NO. 12, DFCFMBER 1990

-VI;. ., - V, if and only if P ( s ) (1) has distinct, real, and negative roots VI; . ., V,.

Define

Q, (z ) = P ~ ( s = - z)( -1)",

We now state the useful equivalent results.

Theorem 3: Every polynomial (of the form (2)) with coeffi- cients in the diamond D has only real, positive, and distinct roots in the range [ - 5, - T ] if and only if the four polynomials Q,(z), Q,(z ) , Q,(z), and Q&z) have only real, positive, and distinct roots in the range [ - 6, - T I .

Theorem 4: Every polynomial (of the form (2) and t,, = f,,) with coefficients in the diamond D has only real, positive, and distinct roots in the range [ - [, - T I if and only if the four polynomials Q,(z), Q,(z), Qs(z) , and Q,(z) have only real, positive, and distinct roots in the range [ - [, - T I .

We now derive the result for a linear discrete time system with perturbed coefficients in a diamond to be strictly aperiodic.

Corollary 3: Every polynomial (of the form (2)) with coeffi- cients in the diamond D has only real, positive, and distinct roots in the range (0,l) if and only if the two polynomials Q,(z) and Q2(z ) have only real, positive, and distinct roots in the range (0,1).

Proof First note that from the remark of Theorem 1, the counterpart of Theorem 1 in Theorem 3 can also be simplified when - T < 1. For this case, only two polynomials Q,(z) and Q,(z) are required to have only real, positive, and distinct roots in the range [ - [, - T]. Corollary 3 is the special case of this simplification where the range [ - [, - T ] is chosen to be (0,l).

Remark: Note that Corollary 3 is also applicable to handle polynomials of the form (2) and t,, = f,,. This is because, from the remark of Theorem 2, the counterpart of Theorem 2 in Theorem 4 can also be similarly simplified where - T < 1.

k = l;.. ,6.

111. MAXIMAL STRICTLY APERIODIC D I A M O N D

We now develop the procedure to obtain the maximal dia- mond centered at t" and containing only the coefficients of strictly aperiodic polynomials.

Let

P ( S , T I ) = P , ~ ( S ) + T I S n - ' (7) dP ds

P ' ( . s , T ~ ) = - ( s , T ~ ) = P ~ ( s ) + T , ( ~ - ~ ) s " - ' - ' . (8)

We first establish a result for P(s ,T , ) to have only real and distinct roots.

Lemma 1: Suppose P&s) has only real and distinct roots. Then, the largest positive value of T,* for all P ( s , T , ) to have only real and distinct roots for all T , E (- T,*, T,*) is given by

where x I , . . ., x , are the real roots of the equation

1 X (9) -( xP6( x ) - ( n - i ) P o ( x ) ) .

Proof First note that real roots of P ( s ) can only leave the real axis in pairs. This implies that T: is given by the minimum absolute real value of T; where P ( s , T ; ) has a multiple real root at s = x .

From (7),

or

It is known that P'(s ,T,) also has a real root at x [4]. Substitut- ing (10) into (8) for s = x and rearranging gives

1 -( xP,;( x ) - ( n - i ) P " ( X ) ) = 0 X

which is the same as (9). Since xl; . ., x,, are the real roots of (9), the minimum

absolute real value of T , where P ( s , T , ) has a multiple real root is given by (10). That is,

7: = min { 1 - P " ( X , ) 1; . .,I ~ f'n(x,) I } . X 7 - I X:,-(

This completes the proof.

continuous time systems.

Theorem 5: Suppose P,(s ) has only real, negative, and dis- tinct roots. Then, the largest diamond centered at r" and whose interior only contains coefficients of polynomials (of the form (1)) having only real, negative, and distinct roots has a radius R given by

~ = m i n ( t 4 , , f 4 , , ~ ~ , ~ , : )

where T X and T,: are defined in Lemma 1.

Proof First, note that it is necessary to have R < f,, to guarantee all the polynomials in the interior of the diamond D to be nth order. Therefore, we assume R < fl, in the remainder of the proof.

Using Corollary 1, the diamond D only contains the coeffi- cients of polynomials having only real, negative, and distinct roots if and only if P(s , T J and P ( s , T ~ ~ ) have only real, negative, and distinct roots for all T [ ) , T,, E [ - R , RI.

Using Lemma 1 and the fact that P ( S , T , ~ ) and P(s,T, ,) have a zero root when f , = O , the largest value of R such that the interior of the diamond only contains the Coefficients of polyno- mials having only real, negative, and distinct roots is as given in Theorem 5.

We now derive the maximal strictly aperiodic diamond for

This completes the proof.

Theorem 6: Suppose P J s ) has only real, negative, and dis- tinct roots. Then, the largest diamond centered at 1" with t , = fl, and whose interior only contains coefficients of polynomials (of the form (I)) having only real, negative, and distinct roots has a radius R given by

R = min { f,, , T , T ~ : }

where T ? and T,: are defined in Lemma 1.

Proof: Since t , , = t",, all the polynomials with coefficients in the diamond are nth order. The proof then follows similar arguments used in the proof of Theorem 5 using Corollary 2 and Lemma 1.

This completes thc proof. We now derive the maximal strictly aperiodic diamond for

discrete time systems.

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. 37, NO. 12, DECEMBER 1990

Theorem 7: Suppose Q&) has only real, positive, and dis- tinct roots in the range (0,l). Then, the largest diamond cen- tered at t“ and whose interior only contains coefficients of polynomials (of the form (2)) having only real, positive, and distinct roots in the range (0,l) has a radius R given by

R = min { t“,, t“,, T,*, Q,(l))

where T,* is defined in Lemma 1.

Proof: First note that it has been shown that P(s,T,) has only real, negative, and distinct roots in the range ( - 1,O) if and only if (Q, (z )+(- l I n ~ , ) has only real, positive, and distinct roots in the range (0,l). Furthermore, P(s,7,) has a negative root at - 1 if and only if

Qo( 1) + ( - l ) ,~ , = O

or

I7,!=Q,(l).

The proof then follows similar arguments used in the proof of Theorem 5 using Corollary 3 and Lemma 1.

This completes the proof.

Theorem 8: Suppose Q&) has only real, positive, and dis- tinct roots in the range (0,l). Then, the largest diamond cen- tered at t“ with t , = t”, and whose interior only contains coeffi- cients of polynomials (of the form (2)) having only real, positive, and distinct roots in the range (0 , l ) has a radius R given by

R = min { t“,, T,*, e,( I)}

where T,* is defined in Lemma 1.

Proofi First note that to = f,, implies that all the polynomi- als with coefficients in the diamond are nth order.

The proof then follows similar arguments used in the proof of Theorem 7 using Corollary 3, the remark of Corollary 3, and Lemma 1.

IV. GENERALIZATIONS

Consider the polynomials in (1) and (2) where the coefficients t , , i = 0; . . , n are varying in the ( n + 1)-dimensional region

0, = ( t : (It,, - for”+ It, - ? , I P + . . . + It, - < r } .

It is easily shown that DpCDl for all positive p less than 1. Furthermore, the coefficients of P,(s ) and Qi(z) , i = 1,. . . ,6

are also part of Dp for all positive p < 1. This implies that the results in Sections I1 and 111 are also

applicable to ensure 0, is strictly aperiodic for all positive p less than 1.

V. AN EXAMPLE

To provide an illustration of the use of the results obtained, consider the following polynomial that has only real, negative, and distinct roots

P,(s) = s 2 +3s +2.

F e are interested in finding the largest diamond centered at t r = [ l 3 21 and whose interior only contains coefficients of polynomials (of the form (1)) having only real, negative, and distinct roots. In accordance with Theorem 5,

t4, = 1

and

Using Lemma 1,

where

or

i2 = 2.

P d ( x , ) = 2x1 +3 = 0

-3 2 ’

x =-

Substituting x1 = -3/2 into (11) gives 1 4

7T -

Similarly, using Lemma 1,

1575

where 1 - ( - 3 x - 4) = 0 x1

or - 4 3

X I = - .

Substituting x1 = -4/3 into (12) gives 1 8 ‘

= -

Finally, applying Theorem 5, the largest diamond has a radius R given by

VI. CONCLUSION

Simple necessary and sufficient conditions for a polynomial set with coefficients in a diamond to be strictly aperiodic are derived. A procedure to obtain the maximal diamond containing only the coefficients of strictly aperiodic polynomials is also given. The results and procedure obtained are shown to be applicable to a wider class of regions in parameter space.

ACKNOWLEDGMENT

The author would like to thank Professor Bose for a reprint of reference [3].

REFERENCES C. B. Soh and C. S . Berger, “Strict aperiodic property of polynomials with perturbed coefficients,” IEEE Trans. Automat. Contr., vol. 34, pp. 546-548, May 1989. C. B. Soh, “Parameter space approach to control problems,” Ph.D. dissertation, Monash Univ., Clayton, Victoria, Australia, 1986. N. K. Bose and K. D. Kim, “Stability of a complex polynomial set with coefficients in a diamond and generalizations,” IEEE Tram. Circuits Syst., vol. 36, pp. 1168-1174, Sept. 1989. S . Barnard and J. M. Child, Higher Algebra. New York Macmillan, 1959. J . Garloff and N. K. Bose, “Boundary implications for stability proper- ties: Present status,” in Reliability in Computing: The Role of Interval Mefhods in Scientific Computing. San Diego, CA: Academic, Harcourt Brace Jovanovich, pp. 391-402, 1988.