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An Extension of the Nyquist Criterion to a Class of Distributed Parameter Systems
CopHight © IFAC 3rd Svmposium Control of Distributed Parameter Svs[ems Toulouse , France , 1982 '
AN EXTENSION OF THE NYQUIST CRITERION TO A CLASS OF DISTRIBUTED PARAMETER SYSTEMS N. Oztiirk and A. Uraz Department of Electrical and Electronic Engineering , Hacettepe Ullit'a sil)" Ankara , Turkey
Abstract. In this paper, the Nyquist stability criterion has been applied to a class of linear time-invariant distributed parameter feedback systems. l'he open-loop transfer function descl 'iption of these systems is taken to be by an irrational function KG(jS)H(jS) instead of KG(s)H(s). l'he sector \arg/S\= rt/4 in the Fs-plane is map ped onto G();)H(j;)-plane and the Nyquist criterion is applied for t he stability test. ~oreover, the inverse ~yquist stabili:y criterion is also considered. In the direct plot, the sector in the IS-plane i s mapped onto G(y's)H(jS)-plane, whereas in the inverse plot, this sector is mapped onto (G(/S)H(/B»-l_plane. 'l'hus, the applicability of the Nyquist criterion has been extended. An illustrative example is inc l uded. Keywords. Nyquist criterion; stability; distributed parameter systems.
IiHHODUC1'ION
distributed parameter systems (Willems, 1970). Root-locus plot on the IS and
Distributed parameter systems arise in various ap plications, such as chemical process systems, aerospace systems, magneto-hydrodynamic systems, communication systems as well as environmental, physiological and sociological systems. rhey are usually described by irrational or transcendental functions of compl~ variable s in the transfer-function method.
various planes are carried ou t by Uraz (19 81). In this work, it is shown that the classical Nyquist criterion can be extended to the class of distributed parameter systems by mapping the Brin's sector, which is I arg I = 1l/4 in the jS-plane,
.fS
onto open-loop transfer function plane in the direct plot and the inverse open-loop transfer function plane in the case of the inverse plot.
Stability analysis of distributed parameter systems entails the investigations of zeros of a characteristic equation containing double-valued functions of the form;S. As jS is two-valued function
THE NYQ UISl' CRITERION
of s, the s-plane is not sufficient and not suitable to represent these functions. therefore, Fa-plane instead of the s-plane
We consider a linear time-invariant distributed parameter feedback system which has an open-loop transfer function in the the form of KG(/S)H(jS) where K is an
appears to be rather useful for these functions. 'r hus, Brin (1962) defined the transform z=/S, and developed a stability
adjustable para meter and G(jS)l1(jS) is rational function in
criterion, similiar to that of the lIiikhailoT criterion, for characteristic equations which are rational functions in .[S. the liiikhailov criterion, however, is
IS.
B
l'he correspond-
ing cha racteri s tic eq u atior. is 1+ KG(jS)l1(jS)= 0
inconvenient to apply to systems having one adjustable parameter since the kikhailov curve must be plotted each time whenever a system paramet e r is changed. Although tne Nyquist plot shows the influence of only one system parameter, it can not directly be generalized to
As
JS
(1)
is a double-valued fu n ction of s,
so is tr ,e open-loop trans f er function KG(jS)H(jS). In the Nyquist criterion, it is necessary to know that whether the open-loop transfer function has one or more poles on the ima gin ary axis and in 4 51
452
N. Ozturk and A. Uraz
the rie:;ht naIf of tne s-plane. I,hen the open-loop transfer function has some poles on tne imaginary axis of tne s-plane the contour whicn is mapped onto open loop transfer function ~lane mus t be modified by plaring the poles outsiae the contour with semicircles of infinitesimal radii. ~hen tne open-loop transfer function nas some poles in tne right half s-plane, for tile input - output stability it is necessary and sufficient tnat tne plot of the open-loop transfer function encircles (-IlK, jO) point as many times in counterclockwise oirection as tilere are unstab le open -l oop poles. In t Ile case of distri buted parameter systems , zeros and poles of the open-lo op transfer function can not be expressed in the s-plane. thus, the classical nyquist criterion can not directly be ap plied to the distributed parameter systems . fnere is a method of repres~nting a Qouble-valued function as a one - valued function by constructing what is known as niemann surface. riowever , instead of using two- sneeted rliemann surface, it is possible to consider t rl e jS-plane where functions 01' .;;; becor::e single -valued functions. thus, poles and zeros of the open-loop transfer functions can now be easi l y described in tne /S-plane. l'he instability re r ion is no longer tne right half plane in tne IS-plane. rlrin (1962) criterion states that t he input-output staoility of a system possessing an irrational characteristic po l ynomial of t he f orm in Eq . (1) is determined by having all the roots outside the sector larg fSl>1l/4 in the fS-plane as shown in 1"ig. 1.
that the linear time-invariant distribut ed para~et feedback system ~s inpu t-ou t put stable if and only if the correspond ing plot of G(ue ±jIl/4) n ( ue ±jIl/4) for ( -l/ K, j O) point p times
O~u~ ellc lo s es
in c cu nterc lo c kwis e jirec t io n where u = 1.fU lend s = G + j w. On account of
syr.~ etry,
it is sufficien t
to plot the locus G(ue+jTl/4)H(ue+jll/4) which corresponds to t he variation of u=O to u=oo, and then to count the semicircleme n ts of the plot about the (-I/K,jO) point. It should ~lso be noted thet if the function KG(fS)H(/S) has singulariti -
"r : - plane, es a long ue + j III 4 contour 1n tIle:Vs they are handled in the us ual fashion by encircling the singulnr~ties in the seme direction with the circu:Ar arcs of infinitesi~ RI radii. ~or example, if Pj is a pole on the sector larg/SI= Tl/4, then the contour in the vicinity of Pj should be replaced by Pj + re jg where r is a vanishingly smell radius and g varies from 51l/4 to n/4 for Pj 1 0 and froll! -Tl/4 to rr/4 for Pj = 0 in counterclockwise direction. these circular arcs are mApped in the opposite direction onto the G(/8)H(/8) - p la ne as circular arcs of infinite radii, as it will be shown in the following example. The rational parts in the open-loop transfer function can be transformed to the jS- plane by taking s= (/8)2 and the above developments can t hen directly be applied . rhus , the jS- plane c a n be con-
Im
stability
c
Slaered to be more general than t he s-plane.
--~~------*-~r-----~Re
l'he problems in the direct ~yquis t criterion also arise in the inverse Nyquist criterion. Dividing Eq . (1) by KG(jS)n(/S), we obtain 1 + ____....;l:c..-_____
=0
( 2)
AG(!S)H(.jS)
~ig.
1. the stability region in the Js-plane.
Consiaer now a closed contour C as snown in lig. 1. If the open-loop transfer function KG(/S)H(y's) of the linear timeinvarian t distributed parameter feedback system has p poles inside C, then the map of C to the G(jS)H(jS)-plane provides
'r he zeros and poles of Eq . (2) are the saffie as the zero s of Eq. (1) and poles of (KG(jS)H(/S))-\lespect!vely. Supcose that (KG(/s)H(jS) has p ' poles in the contour C as shown in lig. 1. l'hen the inverse nyquist criterion is s t ated as follows ; tne map of C to the (G(fS)H(jS))-l_plar:e provides that the
An Extension of the Nyquist Criterion lif-ea~ time-inva r iant distrib ~ ted feedback system is i~put-output stable if and only if t he correspon ding plot of
453
point is w = 1. 18 2 .
Im
(G(ue ij Tl./4)H(ue±jTl/4»-1 for O~u:::;-oo encloses (-K,jO) point pt time s in countereclockwise dirflction.
fa illustrate the above developments co n sider a clused-Ioop distributed-lumped paramet", r system with an open-loop transfer function as KG(jS ) tiC
(.fi,2
Is) = K Is
0)
2 (ra-I) (js + I )
Is - plane
c
------------~~~~--~~~Re
-J -jo(3
\\
I
,, \\
,'
,,
"
\
'-"
iig. 3. Nyquist plot of Eq. (3). loop transfer function plane. It is shown that when the open-loop transfer function of these systems has some poles on and inside the sector I arg jSl~n/ 4 the developed procedure is more convenient than the classical Nyquist test. REFERENCES
X
.. ole
o
ze :· _·
foIe-zero configuration of iq. (3) in tne IS-plane.
encirclements about the critical point (-l/K,jO) is -1
if
K<0.419
+1
if
K > 0 . 419
wnere minus and plus signs show clockwise and counte rc lockwise encircleme n ts, respectively. Since the open-loop system has one pole inside the sectorlarg jS 1< Tl/4 in the jS- p la&e, tne system is input-output stable if K>0.419. ·l 'he corresponding critica l frequency about the critical
CONCLU S ION The graphical extended to a meter systems larg )SI= Tl/4
( __I_. jO ) / 0.t.19 / ~
Im
iig. 2 .
".,
\ \ \
One pole is at tn e origin and tne contour C avoids it by means of a s mal l circular arc, as shown in Big. 2 . As sign ing different values to u from 0 to 00 , the plot i& ti g . 3 is obtai ned. fh e plot of tig. 3 shows that t ne nUll.ber of
j
-+--------------~~~~-------.Re --",' I
+ 3)
HI
G(~}H(E}-plane
procedure of Nyquist is class of distrib u ted pareby mapping the sector in the jS-plane onto open-
Brin,I.A. (1962). Stability of certain systems with distributed and lumped paremeters. Autom. and Remote Control, 23, 798-807. Uraz,A. (1981). Analysis of linear distrib~ted paramet e r systems. ~ International Conference on Control and its Applications, Coventry, England, 23-25 March 1981, pp.323-32~ Willems,j.L. (1970). Linear time-invariant systems. In R. W. Brockett and H.H. Rosenbrock (Ed.), Stability Theory of Dynamical Systems, Nelson, London, Chap. 3, p.79.
AN EXTENSION OF THE NYQUIST CRITERION TO A CLASS OF DISTRIBUTED PARAMETER SYSTEMS N. Oztiirk and A. Uraz Department of Electrical and Electronic Engineering , Hacettepe Ullit'a sil)" Ankara , Turkey
Abstract. In this paper, the Nyquist stability criterion has been applied to a class of linear time-invariant distributed parameter feedback systems. l'he open-loop transfer function descl 'iption of these systems is taken to be by an irrational function KG(jS)H(jS) instead of KG(s)H(s). l'he sector \arg/S\= rt/4 in the Fs-plane is map ped onto G();)H(j;)-plane and the Nyquist criterion is applied for t he stability test. ~oreover, the inverse ~yquist stabili:y criterion is also considered. In the direct plot, the sector in the IS-plane i s mapped onto G(y's)H(jS)-plane, whereas in the inverse plot, this sector is mapped onto (G(/S)H(/B»-l_plane. 'l'hus, the applicability of the Nyquist criterion has been extended. An illustrative example is inc l uded. Keywords. Nyquist criterion; stability; distributed parameter systems.
IiHHODUC1'ION
distributed parameter systems (Willems, 1970). Root-locus plot on the IS and
Distributed parameter systems arise in various ap plications, such as chemical process systems, aerospace systems, magneto-hydrodynamic systems, communication systems as well as environmental, physiological and sociological systems. rhey are usually described by irrational or transcendental functions of compl~ variable s in the transfer-function method.
various planes are carried ou t by Uraz (19 81). In this work, it is shown that the classical Nyquist criterion can be extended to the class of distributed parameter systems by mapping the Brin's sector, which is I arg I = 1l/4 in the jS-plane,
.fS
onto open-loop transfer function plane in the direct plot and the inverse open-loop transfer function plane in the case of the inverse plot.
Stability analysis of distributed parameter systems entails the investigations of zeros of a characteristic equation containing double-valued functions of the form;S. As jS is two-valued function
THE NYQ UISl' CRITERION
of s, the s-plane is not sufficient and not suitable to represent these functions. therefore, Fa-plane instead of the s-plane
We consider a linear time-invariant distributed parameter feedback system which has an open-loop transfer function in the the form of KG(/S)H(jS) where K is an
appears to be rather useful for these functions. 'r hus, Brin (1962) defined the transform z=/S, and developed a stability
adjustable para meter and G(jS)l1(jS) is rational function in
criterion, similiar to that of the lIiikhailoT criterion, for characteristic equations which are rational functions in .[S. the liiikhailov criterion, however, is
IS.
B
l'he correspond-
ing cha racteri s tic eq u atior. is 1+ KG(jS)l1(jS)= 0
inconvenient to apply to systems having one adjustable parameter since the kikhailov curve must be plotted each time whenever a system paramet e r is changed. Although tne Nyquist plot shows the influence of only one system parameter, it can not directly be generalized to
As
JS
(1)
is a double-valued fu n ction of s,
so is tr ,e open-loop trans f er function KG(jS)H(jS). In the Nyquist criterion, it is necessary to know that whether the open-loop transfer function has one or more poles on the ima gin ary axis and in 4 51
452
N. Ozturk and A. Uraz
the rie:;ht naIf of tne s-plane. I,hen the open-loop transfer function has some poles on tne imaginary axis of tne s-plane the contour whicn is mapped onto open loop transfer function ~lane mus t be modified by plaring the poles outsiae the contour with semicircles of infinitesimal radii. ~hen tne open-loop transfer function nas some poles in tne right half s-plane, for tile input - output stability it is necessary and sufficient tnat tne plot of the open-loop transfer function encircles (-IlK, jO) point as many times in counterclockwise oirection as tilere are unstab le open -l oop poles. In t Ile case of distri buted parameter systems , zeros and poles of the open-lo op transfer function can not be expressed in the s-plane. thus, the classical nyquist criterion can not directly be ap plied to the distributed parameter systems . fnere is a method of repres~nting a Qouble-valued function as a one - valued function by constructing what is known as niemann surface. riowever , instead of using two- sneeted rliemann surface, it is possible to consider t rl e jS-plane where functions 01' .;;; becor::e single -valued functions. thus, poles and zeros of the open-loop transfer functions can now be easi l y described in tne /S-plane. l'he instability re r ion is no longer tne right half plane in tne IS-plane. rlrin (1962) criterion states that t he input-output staoility of a system possessing an irrational characteristic po l ynomial of t he f orm in Eq . (1) is determined by having all the roots outside the sector larg fSl>1l/4 in the fS-plane as shown in 1"ig. 1.
that the linear time-invariant distribut ed para~et feedback system ~s inpu t-ou t put stable if and only if the correspond ing plot of G(ue ±jIl/4) n ( ue ±jIl/4) for ( -l/ K, j O) point p times
O~u~ ellc lo s es
in c cu nterc lo c kwis e jirec t io n where u = 1.fU lend s = G + j w. On account of
syr.~ etry,
it is sufficien t
to plot the locus G(ue+jTl/4)H(ue+jll/4) which corresponds to t he variation of u=O to u=oo, and then to count the semicircleme n ts of the plot about the (-I/K,jO) point. It should ~lso be noted thet if the function KG(fS)H(/S) has singulariti -
"r : - plane, es a long ue + j III 4 contour 1n tIle:Vs they are handled in the us ual fashion by encircling the singulnr~ties in the seme direction with the circu:Ar arcs of infinitesi~ RI radii. ~or example, if Pj is a pole on the sector larg/SI= Tl/4, then the contour in the vicinity of Pj should be replaced by Pj + re jg where r is a vanishingly smell radius and g varies from 51l/4 to n/4 for Pj 1 0 and froll! -Tl/4 to rr/4 for Pj = 0 in counterclockwise direction. these circular arcs are mApped in the opposite direction onto the G(/8)H(/8) - p la ne as circular arcs of infinite radii, as it will be shown in the following example. The rational parts in the open-loop transfer function can be transformed to the jS- plane by taking s= (/8)2 and the above developments can t hen directly be applied . rhus , the jS- plane c a n be con-
Im
stability
c
Slaered to be more general than t he s-plane.
--~~------*-~r-----~Re
l'he problems in the direct ~yquis t criterion also arise in the inverse Nyquist criterion. Dividing Eq . (1) by KG(jS)n(/S), we obtain 1 + ____....;l:c..-_____
=0
( 2)
AG(!S)H(.jS)
~ig.
1. the stability region in the Js-plane.
Consiaer now a closed contour C as snown in lig. 1. If the open-loop transfer function KG(/S)H(y's) of the linear timeinvarian t distributed parameter feedback system has p poles inside C, then the map of C to the G(jS)H(jS)-plane provides
'r he zeros and poles of Eq . (2) are the saffie as the zero s of Eq. (1) and poles of (KG(jS)H(/S))-\lespect!vely. Supcose that (KG(/s)H(jS) has p ' poles in the contour C as shown in lig. 1. l'hen the inverse nyquist criterion is s t ated as follows ; tne map of C to the (G(fS)H(jS))-l_plar:e provides that the
An Extension of the Nyquist Criterion lif-ea~ time-inva r iant distrib ~ ted feedback system is i~put-output stable if and only if t he correspon ding plot of
453
point is w = 1. 18 2 .
Im
(G(ue ij Tl./4)H(ue±jTl/4»-1 for O~u:::;-oo encloses (-K,jO) point pt time s in countereclockwise dirflction.
fa illustrate the above developments co n sider a clused-Ioop distributed-lumped paramet", r system with an open-loop transfer function as KG(jS ) tiC
(.fi,2
Is) = K Is
0)
2 (ra-I) (js + I )
Is - plane
c
------------~~~~--~~~Re
-J -jo(3
\\
I
,, \\
,'
,,
"
\
'-"
iig. 3. Nyquist plot of Eq. (3). loop transfer function plane. It is shown that when the open-loop transfer function of these systems has some poles on and inside the sector I arg jSl~n/ 4 the developed procedure is more convenient than the classical Nyquist test. REFERENCES
X
.. ole
o
ze :· _·
foIe-zero configuration of iq. (3) in tne IS-plane.
encirclements about the critical point (-l/K,jO) is -1
if
K<0.419
+1
if
K > 0 . 419
wnere minus and plus signs show clockwise and counte rc lockwise encircleme n ts, respectively. Since the open-loop system has one pole inside the sectorlarg jS 1< Tl/4 in the jS- p la&e, tne system is input-output stable if K>0.419. ·l 'he corresponding critica l frequency about the critical
CONCLU S ION The graphical extended to a meter systems larg )SI= Tl/4
( __I_. jO ) / 0.t.19 / ~
Im
iig. 2 .
".,
\ \ \
One pole is at tn e origin and tne contour C avoids it by means of a s mal l circular arc, as shown in Big. 2 . As sign ing different values to u from 0 to 00 , the plot i& ti g . 3 is obtai ned. fh e plot of tig. 3 shows that t ne nUll.ber of
j
-+--------------~~~~-------.Re --",' I
+ 3)
HI
G(~}H(E}-plane
procedure of Nyquist is class of distrib u ted pareby mapping the sector in the jS-plane onto open-
Brin,I.A. (1962). Stability of certain systems with distributed and lumped paremeters. Autom. and Remote Control, 23, 798-807. Uraz,A. (1981). Analysis of linear distrib~ted paramet e r systems. ~ International Conference on Control and its Applications, Coventry, England, 23-25 March 1981, pp.323-32~ Willems,j.L. (1970). Linear time-invariant systems. In R. W. Brockett and H.H. Rosenbrock (Ed.), Stability Theory of Dynamical Systems, Nelson, London, Chap. 3, p.79.