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IEEE TRANSACTIONS ON AUTOMATIC CONTROL. VOL. AC-27. NO. 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 IEEE TRANSACTION ON AUTOMATIC CONTROL. VOL. AC-27.NO. I , FEBRUARY 207 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.