Comments on “direct bilinear routh stability criteria for discrete systems”
2
Sysrcms & Control Lcttcrs 8 (1986) RS-R6 North-Holland Comments on “Direct bilinear Routh stability criteria for discrete systems” * Rcccivcd 20 January 1986 Rcviscd 9 April 1916 Ahscrucr: The above paper ’ dlscusxs useful Roulh-hkc stabll- ity &&a for discrete systems. Unfortunately. however. prevl. ous work on the subject escaped rhc attention of the author. This note co-ordlnolcs previous work wllh results obtained m Ihc above paper. II also points OUI some of the more recent developments on lhc SubJccl. Kqtwords: Routh-Hunvitz. Stabthry. In the above paper ‘, the ideas of mixed bilin- ear continued fractions leading to Theorem 3 have already been reported along with a proof of the Theorem in [ 1 ] and [ 21. Moreover, the computation algorithms dis- cussed in the paper are well-known. The algorithm for Theorem 1. which is very similar to that for Theorem 2, was published by Steffen (31 nine years ago whereas the algorithm for Theorem 3 has recently been reported in [l] and [4]. More importantly, it is worth noting that in [l], [3] and [4], the presentation of these algorithms in a table form leads to z-domain stability tests that are analogous to the well-known r-domain Routh (S] and generalized Routh 161 tables. Theorem 4 in the above paper provides a crite- rion to determine the number of zeros of a poly- nomial that are located outside the unit circle in the z-domain. It is interesting to note that the theorem is analogous to a well-known theorem in the s-domain that determines the number of zeros l This work was supported by the National Science Founda- tion under the Presidential Young Investigator Grant No. ECS-8451103. ’ Y. Bistria. Systems and Control Letters 4 (1984) 265-271. in the right half s-plane ([ll] in the above paper). More recently [8], a generalized test based on bilinear continued fractions has been reported. The test provides information regarding the num- ber of zeros outside the unit circle as well as the number of real and complex roots appearing in- side the unit circle. The z-domain continued fractions discussed in the paper are based on the bilinear s to z trans- form [2,7]. Other z-domain continued fraction ex- pansions have recently been reported [9,10.11,12]. In [9.10,11] the expansions are based on the loss- less-discrete-integrator (LDI) s to z transform whereas the expansions in 1121 are based on the forward difference and backward difference s to z transforms. The development of Routh-like stabil- ity tables for discrete systems based on these expansions could be a useful topic for future research. References [l] M. Ismail. Some new results on the continued fraction method and IIS applications IO continuous and discrete systems. Ph.D. Disscrration. Dcpr. of Electrical Enginccr- ing. Univ of Mamtoba. Winnipc8 (1983). 121 M. Ismcul and H.K. Kim. A slmplificd stability ICSI for discrete systems using a new z-domain continued fraction method. IEEE 7jou.s. Cwcuiu olrd .Qsrcmc 30 (1983) 505-507. [3] P Steffen. An algorithm for fating stability of discrete systems. IEEE Trans. Acow. Speech S:gnol Pracess. 25 (1977) 454-456. 141 M. Ismail and H.K. Kim. A simplified algorithm for testing stability of discrete systems. in: Prac. Allerron Co~tjerencr on Cammuntcorton. Con& and Campwing, Vol. 21 (1983) 774-77s. [S] EJ. Routh. A Trearisr on the Stobiliry o/o Gwen State a/ Marion. Adams Prize Essay (MacMillan, New York. 1977). [6] L.S. Shieh and M.J. Goldman, A mixed Cauer form for linear system reduction. IEEE Trans. Sysrcnu Man C@emtr. 4 (1974) 584-588. [7] J. Szaupah. S.K. Mitrr and EI. Jury. Some new ruull~ on discrete system stability. IEEE Trans. Accausr. Speech Signal Pracuss. 25 (1977) 101-102. 0167-6911/86/f3.50 0 1986. Elscvier Science Publishers B.V. (North-Holland)
-
Upload
mohammed-ismail -
Category
Documents
-
view
213 -
download
1
Transcript of Comments on “direct bilinear routh stability criteria for discrete systems”
Sysrcms & Control Lcttcrs 8 (1986) RS-R6
North-Holland
Comments on “Direct bilinear Routh stability criteria for discrete systems” *
Rcccivcd 20 January 1986 Rcviscd 9 April 1916
Ahscrucr: The above paper ’ dlscusxs useful Roulh-hkc stabll- ity &&a for discrete systems. Unfortunately. however. prevl.
ous work on the subject escaped rhc attention of the author. This note co-ordlnolcs previous work wllh results obtained m Ihc above paper. II also points OUI some of the more recent
developments on lhc SubJccl.
Kqtwords: Routh-Hunvitz. Stabthry.
In the above paper ‘, the ideas of mixed bilin- ear continued fractions leading to Theorem 3 have already been reported along with a proof of the Theorem in [ 1 ] and [ 21.
Moreover, the computation algorithms dis- cussed in the paper are well-known. The algorithm for Theorem 1. which is very similar to that for Theorem 2, was published by Steffen (31 nine years ago whereas the algorithm for Theorem 3 has recently been reported in [l] and [4]. More importantly, it is worth noting that in [l], [3] and [4], the presentation of these algorithms in a table form leads to z-domain stability tests that are analogous to the well-known r-domain Routh (S] and generalized Routh 161 tables.
Theorem 4 in the above paper provides a crite- rion to determine the number of zeros of a poly- nomial that are located outside the unit circle in the z-domain. It is interesting to note that the theorem is analogous to a well-known theorem in the s-domain that determines the number of zeros
l This work was supported by the National Science Founda- tion under the Presidential Young Investigator Grant No.
ECS-8451103. ’ Y. Bistria. Systems and Control Letters 4 (1984) 265-271.
in the right half s-plane ([ll] in the above paper). More recently [8], a generalized test based on bilinear continued fractions has been reported. The test provides information regarding the num- ber of zeros outside the unit circle as well as the number of real and complex roots appearing in- side the unit circle.
The z-domain continued fractions discussed in the paper are based on the bilinear s to z trans- form [2,7]. Other z-domain continued fraction ex- pansions have recently been reported [9,10.11,12]. In [9.10,11] the expansions are based on the loss- less-discrete-integrator (LDI) s to z transform whereas the expansions in 1121 are based on the forward difference and backward difference s to z transforms. The development of Routh-like stabil- ity tables for discrete systems based on these expansions could be a useful topic for future research.
References
[l] M. Ismail. Some new results on the continued fraction method and IIS applications IO continuous and discrete
systems. Ph.D. Disscrration. Dcpr. of Electrical Enginccr- ing. Univ of Mamtoba. Winnipc8 (1983).
121 M. Ismcul and H.K. Kim. A slmplificd stability ICSI for
discrete systems using a new z-domain continued fraction
method. IEEE 7jou.s. Cwcuiu olrd .Qsrcmc 30 (1983) 505-507.
[3] P Steffen. An algorithm for fating stability of discrete systems. IEEE Trans. Acow. Speech S:gnol Pracess. 25 (1977) 454-456.
141 M. Ismail and H.K. Kim. A simplified algorithm for testing stability of discrete systems. in: Prac. Allerron Co~tjerencr on Cammuntcorton. Con& and Campwing, Vol. 21 (1983) 774-77s.
[S] EJ. Routh. A Trearisr on the Stobiliry o/o Gwen State a/ Marion. Adams Prize Essay (MacMillan, New York. 1977).
[6] L.S. Shieh and M.J. Goldman, A mixed Cauer form for
linear system reduction. IEEE Trans. Sysrcnu Man C@emtr. 4 (1974) 584-588.
[7] J. Szaupah. S.K. Mitrr and EI. Jury. Some new ruull~ on discrete system stability. IEEE Trans. Accausr. Speech
Signal Pracuss. 25 (1977) 101-102.
0167-6911/86/f3.50 0 1986. Elscvier Science Publishers B.V. (North-Holland)
[a] Y.S. Lai, A generalized WSI for dixrctc system stabihry. [l l] M. Ismnil. Synthesis of discrete-time rcnctoncc functions. IEEE Trans. Acow. Speech Sigmd Process. 32 (1984) Proc. IEEE IN. Swp. ut, Cvcurts u/d ~WCI)L( (May 1242-1243. 1986)
[9] A.M. Davis, A new r-domain conttnucd fraction cxpan- 1121 M Ismail. New z-domain continued lrxlion expansions. sion. IEEE Trans. Ctrcw/s and Sntcnu 29 (1982) 658-662. IEEE Truns. Cwcwrs ud ~I.SIC~U 32 (19RS) 754-758
IlO] Y. Bistria. Z-domun continued fraction cxpnnsions for stable discrclc systems polynomials. IEEE Twts. C~rcuirs ondSwmcs 32(19RS) 1162-1166.
