P. Ramesh
tomatic controls. The existence of real and distinct roots in the negative real axis determines the aperiodic beha-
vior of a linear system. The presence of any complex roots shows that the system is aperiodicallly stable. To
analyze the aperiodic stability, a generalized method was inv estigated in the literature by szaraniec. A thre e-step
transformation procedure is presented by fuller, which develops a polynomial whose number of right hand poles
equals the number of complex roots present in the original polynomial.
In the Rout h-Hurwitz stabilit y criterion, the co efficient of the p olyno mial i s arra nged i n two ro ws. W hen n i s
even, the Sn row is formed by coefficients of even order terms (i.e., coefficients of even power of S) and Sn−1 row
is formed by coefficients of odd order terms (i.e., coefficients of odd power of S). When n is odd, the Sn ro w is
formed by coefficients of odd terms (i.e., coefficients of odd power of S) and Sn−1 row is formed by coefficients
of even order terms (i.e., coefficients of even powers of S). In our proposed scheme, the analysis of aperiodic
stability of a given stable linear discrete system is presented with a help of fuller s e quation a s follows it accounts
all the coefficients of the equivalent one dimensional characteristics equations in the Sn row when n is either
even or odd and Sn−1 row is formed using the coefficients by differentiating the equivalent one dimensional cha-
racteristics equation. The other rows of routh array up to S0 ro w can be formed by nor mal Ro uth -Hurwitz stabil-
ity criterion. I llust rative exam ples s how the applicability of the proposed scheme.
2. Literature Survey
The stabilit y analysis is the mo st important test that should be considered in LTIDS analysis and desi gn. There-
fore stability determination is very important. Hence many researches had been conducted to ascertain, the sta-
bility in last few years. A new for mulation of the cr itical constrai ns for stabilit y limits matr ix and its bi-alterna-
tive product was discussed by Jury i n [1]. Bistritz had introduced in [2] the efficient stability test that involves
real univariate polynomials and real arithmetic only. It also examines the doubling degree technique in the de-
velopment of two dimensional stability test discussed by Bistritz in [3]. The possible root locations of two di-
mensional polynomials were proposed by Khargonekerin [4] had proved the critical constrains of obtained from
the polyno mial and also ob tained the s tabilit y within the un it c ircle. A new method to c o mpute the stab ilit y mar-
gin of two dimensional continuous system was provided by Mastorakis in [5] and illustrated and discussed. A
discussion of stability test was obtained for two dimensional and sufficient conditions for asymptotic stability
were easy to checked by Anderson et al. in [6] [7]. Results o btained in [8] proved that double bi-line ar tr a nsfor-
mation does not pr eserve the stability in either direc tion. i.e. Continuous to discrete domain and vice versa pro-
posed by Jury et al. The algorithm was proposed by Bose et al. in [9 ] explains the positive d efiniteness of arb i-
trary quadratic forms that is expressed in terms of inner wise number. The necessary and sufficient conditions
are us ed to form t he inne r of square matrix dis cussed by Jury in [10]. A new algebraic procedure was solving the
prob lem of stability in very low co unt of arit hmetic oper ations was given by B istritz and Ahmed in [11] [12] . A
simplify algebraic equations were Presented by Jury in [13] it was analyzed for its consistency and stability.
Goodman had focused in [14] the suitable difference between the one dimensional and two dimensional test
cases were presented. Two dimensional recursive filtering were proposed by Huang in [15]. It derived and sim-
plified version of stability theorem. The Hurwitz character of the system was determined by Bauer in [16] and
Also proved a relationship between the Hurwitz character of the denominator polynomial of the two dimensional
transfer function. Vimalsingh had proposed in [17] a new criterion for the general asymptotic stability of two
dimensional discrete system discussed by the roesser model using saturation arithmetic was proposed. The im-
plementation procedure for stab ility test for two di mensional digital filter was presented by katbab et al. in [18]
it had reduced the computations, to determine a conservative coefficient space within the coefficient of a real
two dimensional d igital filter. Kamat et al. in [19] had presented the root distr ibution polyno mial with real coef-
ficients with respect to the unit circle is equivalent to one proposed with the same degree of complexity. The
condition for aperiodicity was discussed by szaraniec in [20]. Fullers proposed methodology in [21] to check
whet her a c ontr ol syst em wa s ha ving the de ad-beat conditio n (Aperiod ic stability) a nd revealed that the charac-
teristics equation with real coefficients and can transformed into characteristics equation with complex coeffi-
cients. Later, szaraniec methodology was analyzed and modified by jury in [22] and it exp lains the ne w theo-
rem for the aperiodicity. Jury et al. in [23] proposed a simple stability test similar to routh table was being in-
troduced for linear discrete systems. Bose et al. in [24] presented a procedure to determine whether or not a
polynomial in several real varia bles is globally positive. A method for determining stability b y use of a table
form has been presented by jury in [25], the table had also been used in determining the roots distribution within
the unit circle.
