Download A Root-Locus Technique for Linear Systems with Delay k(0

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1.
FEBRUARY
205
1982
REFERENCES
M Aoki. “Control of large-scaledynamicsystemsb)awegation.”
IEEE Trum.
Aurornor. C o m r . vol. AC-13. pp. 246-153. June 1968
E C. Y.Tse, J. V. Medanlc. and W.R Perkins. ”Generalized Hessenberg transformation, for reduced-order modelling of large-scale systems.” In!. J . Conrr.. vol. 27.
pp 493-512. 4 p r . 1978.
A. Arhel and E Tse. “Reduced-order models. canonical forms and observers.” lnr. J .
Cmrr.. vol. 30, pp. 513-53 I . Sept. 1979.
P. R. Halmoa. Fr~~tre-D~n~e~trronuI
l’error Spocer. Princeton. NJ: Van No5irand.
1958
W.M Wonham. I ~ m o r.3iulrrcariuhle Comrol: A Ceomern~Appromh. Ncw Yorh:
Springer-Verlag. 1973
Fig. 1 Closed-loopsqstem w ~ t hdelay in control.
J. Hlckln and K K. Sinha. “Canonical forms for aggegated models.“ Inr. 1. Comr.,
vol. 27. pp. 473-185. Mar 1978.
“Modelreductionlorlinearmultivariablesystems.”
IEEE Trolls. A l ~ r ~ ~ ? w l .
11. ROOT-LOCUS
FOR SYSTEMS WITH TIME-DELAYS
Conrr., vol AC-25. pp. 1121-1 127. Dec 1980
-. “Figenvalue awgnment by reduced-order modela.” Elerrrotl. 1x11.vol. 1I . pp. .4. Svsrenrs n.ith Delu! irr Control
318-319. July 1975
J. M Siret. G . Mchailezco.and P Bertrand.“Representation of lineardynamical
Consider the linear system with delay in control shown in Fig. I ,
s y t e m s hy aggregated models.” h r . J Conrr.. vol 26, pp. 121-128. Julr 1977.
C F. Chen and L. S . Shieh. ”A novel approach to linear model smplification..’ in
dynamics is given by
Prw- Jowr .4uronmr. C u l m C O I ~Ann
. . Arbor. M I . June 1968.
L.A. Zadehand C. A. D e s c e r . LIneur Swrenl Theon. Neu York: McGraw-Hill.
-.
\vhose
i=AAx+bu(r-h)
1963
y = CX .
In this case. the plant transfer function G J s ) is
Let the open-loop transfer function of the control system be given by
A Root-Locus Technique for Linear Systems with Delay
IL HONG SUH AND ZEUNGNAM BIEN
Ahstruct-A new method of plotting the root-loci is developed for the
linear control system with delay in control or in state. In case of the system
with delay in control, the root-locus plot starts from neighborhoods of the
of theopen-looppoles
and thustheeffect
of
open-loopzerosinstead
time-delay is easily handled. In case of the system nith delay in state, the
open-loop poles are first]! found by appl!ing the root-locus method for the
system with delay in control and then the desired root-loci are found
by
starting the root-loci plot from the open-loop poles.
where K is the open-loop gain. D ( 5 ) and N(s)are polynomial functions of
s of degree 17 and 07. respectively. It is desired to plot the locus of the
poles of the closed-loop transfer function
as K varies from zero to infinity.
Xote that, for each K . the characteristic equation of the control system
I. INTRODUCTIOK
The root-locus method has been used as an invaluable design tool for
linearfeedbacksystems.Althoughtheprocedure
of constructingthe
root-lociforfinitedimensionallinearsystemis
well established,the
root-locus plot for systems with time delay is not easily obtained because
the solution of a trancendental equation is involved. For linear systems
with time-delay in state variable,forexample.
no definitemethod is
known to the authors. In case of feedbacksystemswithtimedelay
in
control variable. several methods exist for computing the root locus plot
such as branch following methods in [I]. [2].and [3] or grid search method
in [4]. One of the common features of these methods is that the root-loci
so thatthosebranches
\vhich arenot
start fromtheopen-looppoles
directly related to the open-loop poles may not be effectively constructed.
In this note. a new method of plotting the root-locus is suggested for
the closed-loop systems with time-delay in control or in state. Based on
the idea of Pan and Chao [j].the method when applied for the system
nith time-delay in control renders the branches starting from the neighborhoodpoints
of the open-loopzeros.Comparedwiththeexisting
methods, this method reflects more clearly the influence of the time-delay
termand
reveals moreabout those branchesnotdirectlyrelated
to
open-loop poles.
The root-locus plot of the system with delay in state variable is also
obtained essentially in the same manner as inthe case with delay in
control.
In the sequal. A. h. and c stand for n X n. 17 X I. and I X I I constant
matrices, respectively. and x. .y. N , and h denote ?]-dimensional state
vector. scaler output. scaler input, and time delay. respectively.
Manuscnpt received August 19. 19x0: revised h.larch 2, 1981
The authors are with the Department of Electrical Snence. Korea Advanced Institute
Science and Technology. Chongyangm. Seoul. Korea.
of
is a transcendental equation in s and thus may include an infinite number
of roots. Therefore. the number of root-locus branches of (5) as K varies
from 0 to co is infinite. If such an infinite number of branches must he
determined for thedesign of controlsystemsinvolvingtimedelay.the
root locus method would be an impractical tool. Fortunately, it is known
that the number of zeros of g ( s . K ) each of whose real partis greater than
any given real number is finite if. Y ( s ) / D ( s )is strictly proper rational [6].
and that all the zeros of g(s, K ) except some finite number around the
origin lie in the left half of s-plane [8]. Thus, mostof the zeros of g(s, K ) ,
being located far from the imaginary axis. do not contribute much in the
system performance, and so only a finite number of root-loci near the
origin may be needed in determining the characteristicsof the closed-loop
system with delay as depicted in Fig. I.
The technique of constructing root-loci developed by Pan and Chao [5],
which was proposed to handle the finite dimensional systems is extended
in the following to solve (5).
First, observe that solving (5) is equivalent to solving the equation
As in [j].introduce a new independent variable “ z ” and show that the
problem of finding the roots of f ( s . k )=O in ( 6 ) for each k is equivalent
to the problem of solving the following simultaneous nonlinear differential equations:
dK --=I,
dt
001 8-9Z86/8Z/O~~-O205$00.7~01982 IEEE
k(0)=ko.
IEEE TRANSACTIONS ON AUTOMATIC COhTROL, VOL. AC-27, NO.
206
Here N ‘ ( s )s d , y ( s ) / d s and D ’ ( s ) g d D ( s ) / d s . Also s,, is a root of (6) for
an initial gain K,,. Once the initial values koand s,, are known. then the
I ) . which are the solutions of (6). are obtained by a
trajectories s( I ) and
numerical integration. Thus. it remains to determine the initial value of s,
for a given k = K O satisfying (6). i.e., it needs to solve
1, FEBRUARY 1982
but
Then the root-locus plot off(s,
k )=O
contains h’branches intersecting at
s = s*.
The proof is similar to the one in [5, p. 8581 and hence omitted.It
of (7) canbe defollows from the above corollary thatthesolutions
termined by (15) and (16) whether they are singular points or not.
Therefore. if the computed zero is found to be a singularpoint.some
modifications must be made as in [5] when plotting the root-locus as k
increases as folloxvs:
For this, observe that
where y is a real constant. If we define T ( s )as
i t is obvious that the poles of T ( s ) in ( I O ) when
k = K , , are the same as
the roots of (8) and can be obtained by plotting the locus of the poles of
T ( s )as d varies from zero to ko.The root-locus of ( I O ) can be obtained
as in [ 5 ] by solving the folloxving nonlinear differential equations:
Here A s is a sufficiently small vector which is tangential to the locus at
the singular point. and soid denotes the singular point.
B. SFsrenrr wirA Delux it? Stute
In this subsection. it is shown that the method developed in Section
11-A can be used to obtain the root-locus plot of a class of system with
delay in state.
Consider a single-input. single-output linear control system with delay
in state whose dynamics is given by
i=A,*+.4,x(r-h)+hu
1’ = cs.
It is assumed that the plant transfer function
expressed in the form of
It is observed that. while the initial conditions s(0) and k(0)in ( 7 ) were
( 1 I ) and (12) are
functionally related by (8). theinitialconditionsfor
independently given so that s( t ) and K ( r ) readily obtained by numerical
integration. Here y is usually chosen to be I or - 1 since the roots of
e - s h - 1 = 0 are easily found.
Aa commented in [ 5 ] ,when the denominator of the right-hand side of
( 7 ) or (11) becomes zero at a point s* for some k. it is noted that (7) or
( 1 1) is not valid. Such a point s* is called a singular point [5]. and may
exist when
~
(18)
G J s ) g ~ ( s ) /u ( s ) can be
where P , ( s ) . P z ( s ) . R,(s\. and R z ( s ) are polynomials of 5 . Such a case
element. First
occurs. for example. when A , has onlyonenonzero
consider the case tvhen P2( s) is not identically equal to zero so that the
open-loop transfer-function of the control system is given as
Here K is theopen-loop gain. D(s) and ,V(s) aresomepolynomial
functions of s. and a(s) is of the form
To handle this singular case, the results on the characterizationof singular
points in [ 5 ] are extended as follows.
Theorem I : Let
(21)
m(s)=~~(s)+,~;~(s)e~’~.
It is desired to plot the poles as K varies for the closed-loop transferfunction
f ( s . k ) = k ~ ( s ) N(s)e-’*
+
lvhere the degrees of polynomials D ( s ) and N ( s) are rf and m . respectively. with n nl. Then the number of roots tbith multiplicity B of the
equationf(s. k)=O for fixed k are at most + nf -2- 3’. w-here
2G A’<
it
+ nf + 1.
(14)
The proof of Theorem 1 is given in the Appendix. It is easily ve-rified
from Theorem 1. that for each k. the transcendental equationj(s. K ) = O
ha> at most a finite set of multiple roots. and if they exist. the multiplicity
is also finite. Based on the above theorem and the results on the singular
points characterized by the properties of higher order derivatives given in
[5]. the following corollary is deduced.
Corollu?: Suppose, for some k qs* is a singular point of the root-locus
of ( 6 ) such thatj(s*. k)=Oand
The desired root-loci of the characteristic equation for system (22) can be
obtained by solving the follouing set of nonlinear differential equations:
ds-- dr
D’(S)+[S’(s)-h,~(s)]e-”h+Km’(s)
1
s(O)=s,
(23)
wherc
and
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1982
XL
Sd
rso4
Y
so7
Fig. 3.
Root-locusofs2-s+0.5e-2’-0.5K=0.0~K<~
Then closed-loop transfer-function T(s) is
0.5
T(s)=
s2
For simplicity. the initial starting point for (24) is chosen to be zero. i.e..
s(O)= so for (23) can be
K(O)= KO =O. Then the initial starting point
found by solving the equation
q(s) g D ( s ) + N ( s ) e - S h = O .
(25)
Here the roots of (25) can be found by applying the root-locus method
developed in Section 11-A.
To handle the case when P2(s)in (19) is equal to zero, the closed-loop
transfer-function in (22) is rearranged as follows: with K = I/K.
+s +0.5F2’
Before solving (23). it is necessary to determine initial starting point. Let
K(O)=O. The initial starting point so at K,=O in (24) is obtained by
solving the equation
q(s)=s2+s+0.5e-*”=0.
For this. (29) is now considered as thecharacteristicequation
system urith delay in control whose transfer-function is given by
111. AN EXAMPLE
To show the use of the root-locus technique developed in
the note. a
simple example is now presented.
Consider the linear control system with delay in state whose open-loop
transfer function is given by
(29)
of the
1
i(s) =
e-2r
I+K-
Then the root-locus of the characteristic equation of the system in (26)
can be obtained as before by replacing D ( s ) , N ( s ) , ~(s),and K i n (23)
with D , ( s ) , YI(s). D ( s ) . and k i n (26). respectively. It is remarked that,
as in the case of the systems with delay in control,onlysomefinite
number of branches of root-locus needs to be considered for the design of
the systems with delay instate if the degree of D ( s ) is greater than that of
N ( 5 ) and ,V,(s), and if l i m ~ - ~ ~ ~ ( ~ ~ A’(j~)e-/’’’’i=O.
) ~ / ~ D ( ~ ~ ~ ) +
+0SK
s(s
+ 1)
with K = O . 5 . But the root-locus of (30) for O< Kc 1 8 0 is easily obtained
11-A:*where y is chosen to be - I , and is
by the methodisSection
sketched in Fig. 2. The roots at K =0.5 are the initial starting points for
the root-loci of (28) as K varies from zero to infinity. The root-locus of
(28) is obtained via (24) and sketched in Fig. 3 with the starting points
denotedass:,i=O,1;..,7.
IV. CONCLUDING
REMARKS
A root-locustechnique was developedfor the linearcontrolsystems
with delay in control or in state by modifying Pan and Chao’s method in
[5]. The technique may be extended to the system with multiple delays in
control and/or in state. and is found to be particularly useful in designing
controllers with intentional time-delay [7].
APPENDIX
P(oofof Theorem I : Let K D i s ) k E ( 3 ) . and denotef(s, k)=j ( s ) for
fixed K. If s is a multiple root off(s). whose order is greater or equal to 2,
then j(s)=O and j ’ ( s ) = O . Eliminating eFrh from these two equations,
208
VOL. AC-27, KO.
IEEE TRANSACTIONS ON CONTROL,
AUTOMATIC
one finds that
[,V’(~)-/l~(s)lE(s)-E’(s)S(s)=O.
(A- 11
Uniform Controllability of a Class of
Linear Time-Varying Systems
G. KERN
Since (A-4) is a polynomial of ( I I + n~)th-order.the number of multiple
roots of order two or greater than two are at most rI + 0 1 .Note that
(A-2)
Now conbider the ( n - I)th differentiation of ](s).
times gives rise to
Abstract-An applicable criteria for uniform con~pletecontrollability to
a class of linear time-varying systems is presented.
INTRODUCTION
I.
Differentiating ](s)
(11 - I )
where/,(s) is a first-order polynomial. I f s is a multiple root of multiplicity ( 1 7 A l ) or greater than ( r l + l ) , then j ‘ ” - ” ( s ) = O and ](“)(.y)=O.
Eliminating e-h’ from these two equations. one obtains
where/:( x ) is a first-order polynomial. Since. for any integer 1. (( d / & is an nl th-order polynomial. (A-4) is an ( n t - I)th-order polynomial. Thus. the number of multiple points of multiplicity ( I 1 + I ) or
greater than ( I1 - 1 ) are at most (nz + 1). Consider ( n 1 )-times differentiation ofj(s). Le..
/ I )‘.Y(s)
-
1. FEBRUARY 1982
The problem of stabilizing “uniformly controllable” finite dimensional
linear time-varying systems has been studied in various papers [I]. [2]. We
notethat in the application of these results the problem of deciding
whether a prescribed pair ( A ( r ) . B( t ) ) is uniformly completely controllableis often difficult. since it may require calculation of the transition
matrix.Inordertoapplythe
results in stability analysis or system
synthesis. it is useful to have criteria for uniform complete controllability
which do not require calculation of the transition matrix. Silverman and
Anderson [3]gave such criteria. which are alsoapplicable in other
problems which involve uniformcomplete controllability. but the constraints on the pair ( A ( r ) . B ( r ) ) are quite restrictive. A weaker condition
under which the results hold are presented only for single-input systems.
In this paper we present a broad class of linear time-varying systems for
which the criteriaforuniformcomplete
Controllability areapplicable.
because the Gramian can be computed without knowledge of the transition matrix of the time-varying part.
11.
SYSTEMDESCRIPTION
AND RESULTS
Consider the linear time-vqing system
Suppose 5 is a root of (A-5). Thenj‘”+l’((s)=O.If s is a multiple roots of
order ( 1 1 +3) greater than ( I I +3).f‘“+’O(s)=O, which gives
i=A(r)r-B(r)u
(1)
\vhere s ( r ) . an 17-vector. is the state of the system at time r . and u ( t ) . an
r-vector. is theinput.The
matrices A ( r ) and B ( r ) are of appropriate
dimensions and their elements are piecewise continuous functions.
It mill be assumed that system ( I ) is a bounded realization. that is, there
exists a constant K such that for all f
IA(r)!<K.
Clearly (A-6) is an ( m - I)th-order polynomial. Thus multiple roots of
multiplicity (17 + 3) greater than ( I I +3) are at most (nz - I). In a similar
way. if s is a multiple root of multiplicity ( n + nt +2). thenj‘”f”’+ll(s)=
0. which gives
~[($-h)”+’N(~)]=nomeroconstant.
(A-7)
Thus, from (A-7). one may conclude that there exists no multiple root of
multiplicity ( n - nl + 2) satisfyingf(s)=O. This completes the proof.
IB(r),<K.
For any fixed sE J . where J is the internal fo < r < r I . we can write ( I ) in
the form
m=A(s)x+[A(r)-A(5)]r+B(r)u.
(1’)
Suppose that the reduced system
.t=A(s)x+B(r)u
sEJfixed.
(2)
is controllable over some interval [ t o .rl]: thus. the system (2) canbe
driven from any initial state s o at time ro to the final state x1 at time rl;
or. equivalently 141. suppose that the symmetric matrix
REFERENCES
M . . ( ~ ~ . ~ ~ ) = J ~ ~ ~ . ~ ~ ~ ~ ~ ~ ~ - ~ ) ~ ( ~
R. H. Ash and G . R.Ash. “Numerical computation of root-loci using the Newton
Raphson technique.“ I E E E Tranx. Auromar Conrr.. vol. AC-13. pp. 576-582, 1968
S E Williamson. ”Accurate root locusplotting includingthe effects of puretime
dcla).” Pruc I E E . vol. 116. no. 7. pp. 1269-1271. 1969.
M. J. Underhill.“Translentandfrequencyresponses
of s?sterns \\ithtime-dela!~.“
E/e<rrow I z r r . . vol. 14. no. 9. pp 284-286. 1978.
F. S . Ftlb?. “I R L. an interactive programforcontrolsystemdesign
using the
root-locus technque.” presented at I E E Int Conf. Comput -Ad D e > . Southampton.
England. Apr. 1972.
C. T Pan and K. S . Chao.“A computer-aidedroot-locusmethod.”
I E E E Trmls.
Auronzar Conrr.. w l AC-23.pp. X56-860.1978.
A . M. Krall. “Stability criteria for feedback systems s i t h a time lag.“ J S I A M . Conrr..
ser. A. vol 2. pp. 160- 170. 1965.
1. H Suh and Z Blen. “Use of time-delay actlons inthe controller-design.’’ I E E E
Tram. Auromor. Cotztr.. vol. AC-25. pp. 600-603. June 1980.
R. Bellman and K. L Cooke. Dlffe~efirra/-D~ffere~e,lrr
Equanom. New York: Academic.
1963
(3)
10
is nonsingular. Then we candeducethat if the matrix A ( r ) satisfies a
Lipschitz condition. the original system (1’) is also controllable.
Theorem 1: Suppose I+,’(
t o . r I ) is nonsingular. and suppose the matrix
A ( r ) satisfies a 1-ipschitz condition
(i)
~A(r)-~(r’)~<L,r~r’~forallr.r’€J.
Then the system (1’) is completely controllable at time to.
Manubcript receiredOctober I . 1980: revised July9. 1981.
Theauthor IS withInatltutfurMathematik
11. TechnlicheUniversitit
Austna
00l8-9186/S2/0200-0208$00.7j
F 1982 IEEE
Ciraz. Graz.