Sensitivity approaches to optimization of linear systems with time delay

Automatica, Vol. 7, pp. 671-679. Pergamon Press, 1971. Printed in Great Britain.

Sensitivity Approaches to Optimization of Linear Systems with Time Delay* Les approches de sensibilit6 l'optimalisation des systems lin6aires avec retard Approximation der Empfindlichkeit bei der Optimierung von linearen Systemen mit Zeitverz/Sgerung 1--[o]Ixo~I qyBCTBHTe.rlbHOCTH K OnTrtMrI3aRrtI, I JIHHeHHbIX CHCTeM C 3 a r t a 3 ~ i B a I - i H e M K. I N O U E t , H. A K A S H I t , K. O G I N O ++ and Y. SAWARAGI++

Sensitivity approaches considerably decrease the computational efforts, while preserving a good performance, in the optimal and sub-optimal control synthesis for linear systems with small time delays. Summary--This paper presents two different sensitivity approaches to synthesis problems of optimal control for linear systems with small time delays. In the first part, the optimal control considering sensitivity is discussed for a minimum energy problem to make up for the undesirable effect caused by a small time delay. In this sensitivity synthesis of optimal control, the central role is played by a combined system which consists of the model neglecting the delay time and the sensitivity model with respect to it. In order to ensure the optimal control with zero-sensitive terminal constraints, the controllability property of the combined system is investigated, and the necessary and sufficient condition for it is derived for a linear system. The second part presents a synthesis method of a suboptimal control for a regulator problem, in which the optimal control is expanded into MacLaurin series in terms of the delay time and the first two or more terms are used to yield a sub-optimal control. Another slightly different approach, in which the singular perturbation method is applied, is also discussed and compared with the other one. As shown by the numerical examples, the present sensitivity approaches decrease considerably the computational efforts, while preserving a good performance in the synthesis of time delay systems. 1. INTRODUCTION SINCE the optimality condition for systems with time delays was derived in the form of the Maximum Principle by KHARATICHVILI [1], optimal control problems of time delay systems have been extensively studied theoretically [2, 3]. Since, however, * Received 4 February 1971; revised 7 June 1971. The original version of this paper was presented at the IFAC Symposium on Systems Engineering Approach to Computer Control which was held in Kyoto, Japan during August 1970. It was recommended for publication in revised form by Associate Editor P. Kokotovi6. t Department of Precision Mechanics, Faculty of Engineering, Kyoto University, Kyoto, Japan. Department of Applied Mathematics and Physics, Faculty of Engineering, Kyoto University, Kyoto, Japan. 671

systems involving time delays are expressed by differential-difference equations or by ordinarypartial differential equations, the practical computation [4-6] of an optimal control from the optimality condition needs enormous efforts, and it becomes almost impossible when the system order is high. Therefore, it is a practical necessity to find feasible approximate methods. As for approximate methods, CHANG JEN-WEI [7] developed a method based on the Dynamic Programming where the initial function was approximated by segments of lines, while SHIMEMURA [8] proposed a method in which the differential-difference equation was approximated by a higher order differential equation and, then, the ordinary optimization technique was adopted. However, these methods, including the methods in Refs. [4-6], do not make use of the advantages which may be obtained when the delay time is small. Taking account of the smallness of the delay time, in this paper, two different sensitivity approaches are developed. In the first part of the paper, a direct extension of the authors' works [9-11] to the system with a time delay is discussed for a minimum energy problem on the assumption that the delay time may be unknown. The combined system which consists of the model without the delay time and the sensitivity model with respect to the delay is an important part, and the controllability property of the combined system plays a basic role. Assuming that the delay time is exactly known, in the second part, approximate methods are developed to synthesize sub-optimal controls under a quadratic performance index. An

672

K. INOUE, H. AKASHI,K. OGINO and Y. SAWARAGI

mportant role is played by the optimal control sensitivity with respect to the delay time. The concept of the optimal control sensitivity is originally treated in the papers by WERNER and CRUZ [12] and by SArqNtJTI and KOKOTOVI6 [13]. The distinctive feature of the approximate methods proposed in this paper is that, by introducing sensitivity functions and sensitivity equations with respect to the delay time, one has only to deal with a higher order differential equation or a few sets of differential equations instead of a differential-difference equation. In what follows, discussions are focused on only a linear system with a time delay for the brevity of presentation. 2. SYNTHESIS O F M I N I M U M E N E R G Y CONTROL WITH ZERO-SENSITIVE T E R M I N A L CONSTRAINTS

Let a time delay system be expressed by a differential-difference equation 5c(t) = A x ( t ) + B x ( t - e) + Cu(t)

(1)

with an initial function x(t) = qb(t),

- e <~t <~O,

" ~(t) = A x ( t ) + B x ( t - e,) - C R - 1C'p(t),

t t

[J(t) = - A'p(t) - B ' p ( t + e),

'x(t)=qb(t),

~p(t)=0,

-e
(5)~ (5)2

x ( T ) = x J,

T
(6)1

(6)2

and the optimal control u*(t) is given by u*(t) = -- R - l C ' p ( t ) ,

(7)

where p ( t ) is the adjoint vector. Even if the delay time e is exactly known, however, this two point boundary value problem (5) and (6) is prohibitively difficult to solve in practice, since (5)~ is of retarded type with the initial function (6)~ while (5)2 is of advanced type with the terminal function (6)2. Besides it is almost impossible to repeat the optimization when the delay time varies from one control period to another. Further, in the case when the delay time is unknown, it is impossible to obtain the optimal control. Thus, it is necessary to find a feasible approximate method.

(2) 2.1. State sensitivity equation with respect to time

where x=col.(xx . . . . . x.), u=col.(ul . . . . . u,), A, B, C and e are respectively the state vector, the control vector, n × n - , nxn-, nxr-eonstant matrices and a scalar time delay parameter, positive and small. It is assumed in this section that the delay time is unknown or that it may vary from the a priori given value from one control period to another. The ordinary synthesis problem of minimum energy control with terminal constraints is written as follows: For the system expressed by (1), synthesize an optimal control u*(t) such that it [1] satisfies the terminal constraint x(T)= x f,

(3)

and [2] minimizes the energy consumption J(u) =

~

T

u'(t)Ru(t)dt,

0

(4)

detay If the delay time e is small, the model system

~=(A+B)x+Cu, x(0)= 4,(0),

(8)

which is obtained by neglecting the delay time e in (1), is often used instead of the actual system (1) to avoid the difficulty of solving the two point boundary value problem of differential-difference equations. The synthesis that adopts the differential equation model (8) as the controlled system will henceforth be called the conventional synthesis. Even if the two point boundary value problem is reducible to that of differential equations, the main object [1] of this synthesis problem is not satisfied because of the neglected delay time. The difference in the state between the actual system (1) and the model system (8) corresponding to a certain control Ax(t, e) = x(t, ~ ) - x(t)

(9)

can be approximated by where T, X f and R are respectively a given final time, a given desired final state and a positive definite matrix. The symbol ..... denotes the transpose of a matrix or a vector. When the delay time e is exactly known, this synthesis problem is reduced to solve the following two point boundary value problem by applying the Maximum Principle derived by KHARATICrIVILI[1].

Ax(t, ~ ) = y ( t ) . e

(10)

by the use of a MacLaurin series expansion, where y(t) is the state sensitivity function with respect to

the delay time e and is defined by y(t) = 8x(t, e)/Oel~=o-

(11)

Sensitivity approaches to optimization of linear s~stems with time delay The sensitivity function y(t) is dominated by the sensitivity equation = - B ( A + B ) x + (A + B ) y - B C u ,

y(0)=0,

(12)

if the continuity [14, 15] and the differentiability of the solution of (1) with respect to e at e = 0 are guaranteed. If the difference Ax at the final time T is estimated to be large, then the main object [1] of this synthesis problem cannot be attained. It is, therefore, desired to develop another synthesis method which is not only able to compensate the variation of the terminal constraint but also able to decrease the computational difficulty of the original synthesis problem. 2.2. Combined system and sensitivity synthesis Since the variation of the terminal constraint due to the existence of a small time delay e is approximated to be Ax(T, e ) = y ( T ) . e , it is desirable to choose the control in such a way that it satisfies y ( T ) = 0 for achieving the main object [1]. In order to do this, consider the sensitivity function y(t) as an additional state and consider the simplified model (8) and the sensitivity model (12) simultaneously,

f

x(0) = ~b(0),

(13)1

~= -B( A + B)x +(A + B)y-BCu,

y(0)=0,

~=(A+B)x+Cu,

S(u) =

q

673

T

(15)

u'(t)Ru(t)dt. o

The distinctive feature of the sensitivity synthesis is that by constructing a combined system of higher order differential equations, the problem is reduced to that of differential equations with enough satisfaction of the main object [I]. 2.3. Controllability o f the combined system It should be noticed here that a new basically important problem comes out in the case of sensitivity synthesis--the problem whether we can find an optimal control which satisfies the terminal constraints (14). It can be said that an optimal control exists if the combined system (13) is completely controllable. Considera simple time delay system

~

=x~+l,

(i=1 . . . . . n - l ) ,

(16)1

Jc.=alx 1+ . .. +a.x.+bkXk(t--e)+CU,

(16)2

where c # 0 and b k # O. Then, the combined system is written by "~=x~+ 1,

(i= 1. . . . . n - 1),

(17)1

~ n = a l x l + . . . +anxn+bkXk+CU,

(17)2

)~=Y~+ 1, (i=1 . . . . . n - l ) ,

(17)3

(13)2 ~n = --bkSCk + a l y l + . . . + a n y . + bky k. (17)4

which will be called the combined system. Then, it may be possible now to control not only the model state x(t) but also its sensitivity function y(t) by a proper control synthesis in the extended space----(x, y). Thus, one can take the effect of the small time delay e, neglected in the model (8), into consideration at the initial stage of the synthesis. This is the basis of the sensitivity synthesis of optimal control [9, 10]. The minimum energy problem with the terminal constraints may thus be restated as follows: For the combined system (13), synthesize the sensitivity optimal control that [1]' satisfies the new terminal constraint y(T) =0

(14)1

x ( T ) = x ~,

(14),~

together with

and [2] minimizes the energy consumption

Simple calculation shows that if and only if the controllability checking determinant ]GI calculated as

[ G ] = ( - 1 ) " 2 + " - l ( a l + b l ) b " l c 2",

( k = l ) , (18)1

IG] = ( - 1)"' +("- 1)k#k~"' 1 Uk~" 2",,

( k ¢ 1), (18)2

)

does not vanish, the combined system (17) is completely controllable. Thus, the necessary and sufficient condition for the combined system to be completely controllable is that a~ + b 1 # 0 for k = 1 and a l # 0 for k # l . 2.4. Example To demonstrate the effectiveness of the sensitivity synthesis, we consider the first order time delay system 5¢=ax + b x ( t - e ) + u ;

x(t) = dp(t), -- e <~t <~O.

(19)

674

K. TNOUE, H. AKASHI, K. OGINO a n d Y. SAWARAG1

(e=0-1) and a model system (~=0) corresponding to the conventional synthesis and the sensitivity

The combined system in this case becomes '~=(a+b)x+u,

t

x(O)=ff(O),

(20)1

~=-b(a+b)x+(a+b)y-bu,

y(O)=O,

(20)2 which is completely controllable. Numerical results are shown in Figs. 1, 2 and Table 1 for the case of ~b(t)= 1, x s = 0, T = 1.0 and R = 1.0. The state trajectories of an actual system

synthesis are shown in Figs. 1 and 2 respectively for the cases o f a = b = 1 a n d a = b = - 1. In Table 1, the values o f the t e r m i n a l c o n s t r a i n t error corr e s p o n d i n g to f o u r actual systems are listed for d e m o n s t r a t i o n o f h o w the m a i n object [1] is satisfied. The results clearly show the superiority o f the sensitivity synthesis to the c o n v e n t i o n a l one in achieving the m a i n object [1], although m o r e energy is necessary for the sensitivity synthesis.

1.0

1"51

Sensitivity synthesis

~ ~

Conventionalsynthesis /M°del (4=0) / Actual (~=0.1)

\\ 1.0

X

/

\\"

~=o.,~~

0'5

A /cu to,

o

.¢.. 03

0-0

0"5

Conventional \

",\\ .

synthesis

~

Model (4=0) '~/~/ Actual (4=0'1)

0

0 '3

Model

I'0

-0.%

/ ///

~ ~

os

t

l

///

o.7 Timer

/"

Fio. 2. State trajectories corresponding to the two methods (a = b = - 1, ~b(t)= 1).

FzG. 1. State trajectories corresponding to the two methods ( a = b = l , ~ ( t ) = l ) .

TABLE 1.

I

(e=O)~" i / J X \ x xx

Actual(¢=Ol)f

0"7

Time,

V,

::=,z:::'

ERROR COMPARISON OF TERMINAL CONSTRAINT BETWEEN THE TWO METHODS

Terminal constraint error Actual system

e=0.05

e=0.10

E=0.15

e=0.20

Energy consumption

a = 1-0

c.s.

0"1686

0"2933

0"3900

0"4674

2-0373

b = 1 "0

s.s.

0-0030

0"0078

0.0123

0-0158

5.4181

a = -- 1 "0

c.s.

--0.0175

--0"0356

--0.0547

--0'0752

0"0373

b=--1.0

s.s.

--0.0030

--0.0122

--0.0277

--0.0502

1.4181

c.s., conventional synthesis; s.s., sensitivity synthesis.

1.0

675

Sensitivity approaches to optimization of linear systems with time delay 3. SYNTHESIS OF SUB-OPTIMAL CONTROL WITH A QUADRATIC PERFORMANCE INDEX

As in the previous section, we also consider here the linear time delay system given by

[l] Continuity assumption Assume that the solutions x(t, E) and p(t, E) of (24) have a limiting property limx(t, 8)=Z(t),

O
(27) 1

O
(27h

C-+0

n(t)=Ax(t)+Bx(t-E)+&(t); x(t)=+(t),

-&< t
but here the delay time E is assumed to be exactly known. The optimization problem considered here is a regulator problem, that is, the problem of finding the optimal control that minimizes the following quadratic performance index J(u)=+

’ [x'(t)Qx(t)+u'(t)Ru(t)]dt s

0

,

(22)

where Q and R are positive definite matrices and the terminal state x(T) is assumed to be free. By applying the Maximum Principle developed by KHARATICHVILI [l], the optimal control is obtained in the form u*(t)= -R-‘C’p(t),

limp(t, s)=p((t),

(21)

1

e+O

where 3(t) and j(t) are the solution of k=(A+B)X-CR-‘C’j!,

1 ;=

-(A’+B’)jb

X(0)=&0), QZ,

F(T) = 0,

(2% (2%

which is obtained by letting s=O in (24) and (25). [2] Differentiability assumption Assume that x(t, E) and p(t, e) are differentiable with respect to E and denote ax& &)/a&= u(t, &), 8p(t, &)/a&= w(t, E),

(2%

then, u(t, E) and w(t, E) should satisfy ~(t)=Au(t)+Bu(t-E)-BB~~(~-E)-CR-~C’W(~),

(3O)l

(23)

G(t)= -A’w(t)-Fw(t+e)--II’fi(t+a)--e,(t), where p(t) is the solution of the canonical equation 5(t)=Ax(t)+Bx(t-E)-CR-%‘p(t),

J Id(t)=

-A’p(t)-B’p(t+~)--x(t)

u(t)=O, --E
(24), (2%

(30),

(

{ W)=&),

w(t)=O, T
1 h(t)=09

(301 (31),

By assuming u(t, E) and w(t, E) are continuous with respect to E at s=O, we have

with x(t)=#J(t), 1 pCt)=O,

--E
(25),

limu(t, &)=0(t),

O
(32)1

limw(t, &)=W((t), O
(32),

E-+0

(25),

Unfortunately again, we have no formula for a general solution of such a system of differentialdifference equation, because (24), is of retarded type with initial function (25), while (24)2 is of advanced type with final function (25),. Hence, to find the solution p(t) is a very difficult task. 3.1. Expansion of optimal control in delay time The approximation method proposed here is to regard the optimal control u*(t) given by (23) as a function of the delay time E and to approximate it by a truncated MacLaurin series in E. Then, the suboptimal control E is expressed by

{

a-0

where u(t) and G(t) are sensitivity functions and satisfy the sensitivity equation i=(A+B)U-CR-‘C’W+&),

{ ti= -(A’+B’)W-

QG+&),

U(O)=O, (33), F(T)=O.

(33),

These are obtained by letting E=0 in (30) and (31). The nonhomogeneous parts, c(t) and c(t), are t(t)= -B:(t)=

-B[(A+B)X(t)-CR-lC’jY(t)],

(34)l ti(t, E)= u*(t, 0) +&*(t, &)/&I, =0X,

c(t)= -B’j(t)=B’[(A’+B’)F(t)+QX(t)],

(26)

under the assumption that the optimal control u*(t, E) is continuous and differentiable with respect to E at s=O.

1

(34),

which are known functions if the solution of (28) is found.

K. INOUE, H. AKASHI,K. OGINO and Y. SAWARAG!

676

3.2. Sub-optimal control synthesis Thus, on the assumption that the optimal control u*(t,e) is continuous and differentiable with respect to e at e=0, the sub-optimal control if(t, e) can be expressed by ~(t, ~)=-R-1C'[ff(t)+~(t).~].

(35)

As is shown in (28) and (33), the functions F(t) and ~(t) are the solution of the two point boundary value problems of the linear differential equations. Therefore, the problems can be reduced to the final value problems [16] by introducing the matrix M defined by /3 = - M 2 ,

(36)

and the vector c defined by = - M~ + c,

(37)

respectively for (28) and (33). Substitution of (36) and (37) into (28) and (33) gives

~I= - M ( A + B ) - ( A ' + B')M-MCR-~C'M+Q, M(T) =0,

(38)

the time delay system (21) may be approximated by a higher order differential equation

fc=Ax+Bz~+Cu,

x(O)= q~(O),

(41) 1

/~1 =X__Z1 ,

zl(0) =(~(__ ~),

(41) 2

~i~zi-

z'(0) = ~( - i;0,

I --Z i,

(i= 2 . . . . . m). (41)3 Thus, the optimal control u+(t,2) for the approximate higher order system (41) will preserve a good performance. Although, in this way, the problem is reduced to one involving differential equations, at the same time, the higher order system means a larger computational task. Taking account of the fact that the parameter 2 is small, because of the smallness of the delay time e, we can overcome this difficulty by applying the singular perturbation method, which was developed by SANNUTI and KOKOTOVI6 [13], to the approximate system (41). Thus, we have the sub-optimal control ti(t, 2)=u+(t, O)+8u+(t, 2)/~2]~= o.2,

(42)

and

c= -[MCR-1C' +(A' + B')]c + M~ +~, c(T)=0.

(39)

(38) is a well-known Riccati type equation and (39) is a linear equation. Once M and c are found by a backward time computation, p and ~ are easily obtained from (36) and (28) and from (37) and (33) respectively. By the property that the homogeneous parts of (28) and (33) are identical, which is a consequence of the linearity of (21), the matrix M is always valid if the higher order terms than the first in e are taken into consideration in (26). Thus, the method proposed here also converts the two point boundary value problem of a set of differential-difference equations (24) into the simple two point boundary value problems of two sets of differential equations (28) and (33) and finally into the two final value problems of differential equations (38) and (39). 3.3. Another

approach*--singular perturbation

method In this section, another slightly different method is developed. By dividing the delay time ~ into m equal sub-intervals of time and denoting

zt(t) = x ( t - i2), 2 = elm, (i= 1. . . . . m), (40) * Independently, SA~m'rl [18] developed a quite similar approach, and discussed th© continuity and differontiability property of the optimal control with respect to the delay time.

for the approximate system (41) and also for the original time delay system (21). The continuity and differentiability of u+(t, 2) with respect to 2 at 2 = 0 can be easily checked by the Tup6iev theorem [17], and it is easily found that the zero-th order optimal control u+(t, 0) coincides with u*(t, 0) in (26). The coefficient of the second term, Ou+(t,2)/32 at 2 = 0 is given by

Ou+(t, 2)/t321~=0 = -R-IC'zr(t),

(43)

where n(t) is the solution of the two point boundary value problem

{

(o=(A+B)to,CR-'C'~z-mBx,

(44)1

~t= - ( A +B ) z - Qo~-mB'fi,

(44)2

09(0) = B [ ,~o ~b(-i2)-mq~(0)], "~n(T)=0. "=

(45), (45)2

In order to compare ~(t, 2) with i(t, e), denoting

~/m= ~, ~/m=~,

(46)

divide (44) and (45) by m. We have

f ~=(A+B)v-CR-1C'#+~(t), ]. ~(0) =

~o(O)/m,

(47)1

~ ~= - ( A' + B')#-Q~+~(t), ~(T)=0,

(47)2

Sensitivity approaches to optimization of linear systems with time delay and the sub-optimal control (42) can be expressed by if(t, 2) =

- R- xC'[ff(t) + n(t).

2]

~(t)-e].

= -R-1C'[ff(t)+

(48)

Comparison of (47) with (33) indicates that the two equations are identical except the initial conditions ~(0) and ~(0). It should be noticed that ~(0) approaches

677

to the original system through the mediation of the higher order approximate system. 3.4. Example As an example, consider the problem of finding the optimal control which minimizes

s(u)=½yl(5x2+u~)dt, for the linear first order system ~(t) = x(t) + 0 . 5 x ( t - e) + u(t).

,.-~olim~0)---Bf°. [tk(t)-~b(0)]dt,

(50) (51)

(49)

as m tends to infinity. So, if ~b(t) is constant, ~(0) coincides with ~(0), that is, if(t, 2) becomes equal to if(t, e). The difference of the initial conditions ~(0) and ~(0) for a general initial function ~b(t) is caused by the fact that in the preceding section the regular perturbation method is applied directly to the original time delay system, while in this section the singular perturbation method is applied indirectly

The values of the performance index corresponding to the three approximation methods versus the delay time e are shown in Figs. 3, 4 and 5 respectively for the three types of initial functions. The effectiveness of the sensitivity synthesis is dearly understood from these results. As is seen in Figs. 3 and 5, for the initial function {1}, if(t, e) yields the better performance than if(t, 2), and the other way around for the initial function {3}. This is due to the difference of the initial conditions ~(0) and ~(0).

5.0

3.5

Initial f u n c t i o n

{I) N

Initial

x

function ,v

{2)

l

1.0

f

I

I

I'O

I I I I

4,0 - -

I I I I I 1 I

3-0

I _

-c

0

• t_

0

o o

n

3.0

/

2.5

J ~'(f,e) 2..0

0.1

0'2

0'3

Delay t i m e ,

FIG. 3. Comparison of performance index values under initial function {1 }.

~0

O

a n d ~'(t, X)

o-I Delay

0.2 time,

•

FIO. 4, Comparison of performance index values under initial function {2}.

0-3

678

K. [NOUE, H. AKASHI, K. OGINO a n d Y. SAWARAGI 30

rnitial function

2e

I

! I

0

- t

25

t/*

b~(r,x)

20

0

0"1 Delay

0'2 time,

0"3

e

FIG. 5. Comparison of performance index values under initial function ~3). 4. CONCLUSIONS The two different m e t h o d s to synthesize approximate optimal controls for the system with a time delay are developed by i n t r o d u c i n g the sensitivity f u n c t i o n a n d the sensitivity e q u a t i o n with respect to the delay time o n the a s s u m p t i o n that the delay time is small. The a d v a n t a g e of the sensitivity methods is that the two p o i n t b o u n d a r y value p r o b l e m of a set of differential-difference equations which is o b t a i n e d by applying the M a x i m u m Principle is reduced to the two p o i n t b o u n d a r y value p r o b l e m of a set of higher order ordinary differential equations or a few sets of o r d i n a r y linear differential equations. This diminishes considerably the c o m p u t a t i o n a l efforts, while preserving a good performance as shown in the examples. The advantage of the sensitivity m e t h o d s will be evet~ more a p p a r e n t when the system to be controlled is nonlinear. All numerical c o m p u t a t i o n s were performed by F A C O M 230-60 at D a t a Processing Center in K y o t o University. REFERENCES [1] G. L. KHARATICHVILI: Maximum principle in the theory of optimal time delay processes. Dokl. Acad. ScL USSR 136, 39--42 (1961). [2] D. H. CI~XrvNGand E. B. LEE: Linear optimal systems with time delay. J. S I A M Control 4, 548-575 (1966).

[3] A. MANITIUS: Optimum control of linear time-lag processes with quadratic performance index. Pre-print of the IVth IFAC Congress, Warszawa, Paper No. 13.2 (1969). [4] D. MACKINNON: Optimal control of systems with ptlrc time delays using a variational programming approach. IEEE Trans. Aut. Control AC-12, 255-262 (1967). [5] D. H. ELLER, J. K. AGGARWALand H. T. BANKS: Optimal control of linear time-delay systems. IEEE Trans. Aut. Control AC-14, 678-687 (1969). [6] Z. V. REKASXUSand G. A. LAWRENCE: Minimum energy control of systems with time delay. IEEE Trans. Aut. Control AC-15, 365-368 (1970). [7] CHAr~G J~N-WH: The problem of synthesizing an optimal controller in systems with time delay. Avtomatika i Telemekhanika 23, 133-137 (1962). [8] E. SHIMEMURA: Approximate method for the optimization of systems with time delay (in Japanese). Pre-print of the 9th JACC, Paper No. 162 (1966). [9] Y. SAWARAGI,K. INOUE and K. ASAI: Synthesis of open-loop optimal control with zero sensitive terminal constraints. Automatica 5, 389-394 (1969). [10] Y. SAWARAOI,K. INOUEand T. OHKr: Sensitivity synthesis of optimal control under the changes of system order. Pre-print of the IVth IFAC Congress, Warszawa, Paper No. 68.1 (1969). [I1] K. INOUIE,K. OG1NO and Y. SAWARAGI: Sensitivity synthesis of optimal input for parameter identification. Pre-print of the 2nd IFAC Symposium on Identification and Process Parameter Estimation, Praha, Paper No. 9.7 (1970). [12] R. A. WERNERand J. B. CRuz: Feedback control which preserves optimality for systems with unknown parameters. 1EEE Trans. Aut. Control AC-13, 621-629 (1968). [13] P. SANNUTIand P. KOKOTOVI6: Singular perturbation method for near optimum design of high-order nonlinear systems. Pre-print of the IVth IFAC Congress, Warszawa, Paper No. 68.5 (1969). [14] R. BELLMANand K. L. COOKE: On the limit of solutions of differential-difference equations as the retardation approaches zero. Proc. natn Acad. Sci. 45, 1026-1028 (1959). [15] S. SUGIVAMA: Continuity properties on the retardation in the theory of difference-differential equations. Proc. Japan Acad. 37, 179-182 (1961). [16] R. E. KALMAN: Contributions to the theory of optimal control. Bol. Soc. Math., Mexico pp. 102-119 (1960). [17] V. A, TuPOEV: Asymptotic behavior of the solution of a boundary value problem for systems of differential equations of first order with a small parameter in the derivative. DokL Akad. Nauk S S S R 143, 1296-1299 (1962). [18] P. SANNU~: Near optimum design of time-lag systems by singular perturbation method. Pre-print of the 1lth JACC Paper No. 20-A (1970). R6sum~-Le pr6sent article pr6sente deux approches differentes de sensibilit6 aux probl6mes de synth~se de commande optimale des syst~mes lin6aires avec de faibles retards. L'article discute d'abord la commande optimale a la lumi6re de la sensibilit6 pour un probl6me d'6nergie minimale afin de compenser l'effet ind6sirable provenant d'un faible retard. Dans cette synth~se de commande optimale du point de rue de la sensibilit6, le r61e central est jou6 par un syst6me combin6 qui comprend le mod61e n6gligeant le retard et le module de sensibilit6 par rapport 5_ ce dernier. Afin d'assurer la commande optimale avec des contraintes terminales b. sensibilit6 nulle, l'aptitude 5_la commande du syst6me combin6 est etudi6e et la condition n6cessaire et suffisante pour sa r6alisation est 6tablie pour un syst~me lin6aire. La deuxi~me partie de l'article pr6sente une m6thode de synth6se d'une commande sous-optimale pour un probl6me de rggulateur, dans lequel la commande optimale est developp6e en s6rie de Mac Laurin selon le retard et les

Sensitivity approaches to optimization of linear systems with time delay premiers deux termes--ou davantage---sont utilis6s pour donner une c o n m a n d e sous-optimale. Une approche 16#rement differente, dans laquelle la m6thode de la perturbation singuli~re est appliqu6e, est 6galement discut6e et compar6e ~ l'autre. Ainsi qu'il est montr6 par les exemples num6riques, les pr6sentes approches de sensibilit6 diminuent considerablement les efforts de calcul tout en conservant de bonnes performances dans la synth~se des syst~mes avec retard. Zusammenfassung--Angegeben werden zwei verschiedene Approximationen der Empfindlichkeit bei Problemen der optimalen Synthese yon Regelungen mit ldeiner Totzeit. Im ersten Teil wird die optimale Regelung in Bezug auf die Empfindlichkeit ftir ein Energieminimumproblem betrachtet, um auf den unerwiinschten, durch die kleine Totzeit hervorgerufenen Effekt vorzubereiten. In dieser Empfindlichkeitssynthese bei der optimalen Regelung spielt ein kombiniertes System die Hauptrolle, alas aus dem Modell unter Vernachl/issigung der Totzeit und dem entsprechenden Empfindlichkeitsmodell im Hinblick auf sie besteht. U m die optimale Regelung zu sichern, wird die Eigenschaft der Steuerbarkeit des kombinierten Systems untersucht und die notwendige und hinreichende Bedingung daftir ftir ein lineares System abgeleitet. Der zweite Teii betrifft eine Synthesemethode fiir eine Regelung, bei tier die optimale Regelung in Mac-LaurinReihen in Termen der Totzeit entwickelt ist. Die ersten zwei oder mehr Terme werden benutzt, um eine suboptimale Regelung zu erzielen. Eine andere wenig verschiedene Approximation, bei der eine StSrungsmethode angewandt wird, wird diskutiert und mit der anderen verglichen. Wie durch numerische Beispiele belegt wird, vermindert die dargelegte Empfindlichkeitsapproximation den Rechen-

679

aufwand betr~ichtlich, und dabeit wird eine gute Durchfiihrung der Synthese yon Systemen mit Totzeit bewahrt. Pe3mMe--HacTomHas CTaTb~I npe~JmraeT ~Ba pa3YmqHbIX HOHXO~a tIyBCTBHTe~HOCTH K Hpo6HeMaM cI4HTe3a OllTHMa.rlbHOrO yrIpaBneHH~lJIHHe~[HbIMHCHCTeMaMH C He6oflbHJHMH 3ana3~bIBaHI~MH. CTaTbH 06cy~aaeT cHa~ana OIITHMaYtbHOeynpaBYleHHe B CI~Te tI~BHTeJ~HOCTH ~JIfl npo6neMH MHHHMaHbHO~ 3HepFHH C TeM qTO6bI KOMneHCHpOBaTb He~KeJIaTeYlbHbI~ D{1D(~eKT BblTeKalOIHHI~ H3 He6onbmoro 3aIIa3~MBaHHg. B 3TOM CHHTe3e OHTHMaYI~HOFOyHpaBneHH~ C TO~IKH3peHH~ qyBCTBHTeflbHOCTH, HeHTpaYlbHyIO poJIb HrpaeT KOM6HHHpoBaHHaa CHCTeMa COCTOm]/aa H3 Mo~(e.rIHnpeHe6peryIoxHeR 3aIIa3~bIBaHHeM H H3 Mo~(eHH ~(BCTBHTe.rlbHOCTH IIO OTHOmeHHIO K nocne~HeMy. ~ n ~ o6ecne~eHH~[ onT~rMa~HOrO yrIpaB0-1eHR~C KOHe~IbIMHorpaHH~eHHgMH HMeIOIIIHMH HyHeBylO ~IyI~TBHTeHBHOCTb, H3y'IaCTC~I CHOGO6HOGTb K yrlpaBYleHHIO KOM6~HHpOBaHHO~ GHCTeMbl

H BbIBO~HTGfl

Heo6xo~Moe H ~ocTaTO~moe yC~oBHe ero ocyIHeCT~IeHH~ HHHeI~IHO~ CHCTeM~. ~IaCTb CTaTI~H iIpe~YlaraeT MeTO~ CHHTe3a c y ~ o o

BTopas

HTIIMaHI,HOFO y n p a a n e i ~ ~ IIpO6JIeMbl perynaTopa, B KOTOpOM OHTHMaJ~HOe ynpaBnenMe pa3naraeTca no p~J1aM MaK-JIopena n 3anI~C~MOCr~ OT 3ana311~mawaa H :ma nepBbtX '~YleHa~H/IH 60~.~, HCIIOYI~3yIOTC~[ H ~ I IIOHytIenH~I CT~OHTHMaJIbHOFO ynpa~eH~n. O6cy~aeTcfl TaK~recncrKa pa3HH'qHl,I~ n o ~ x o ~ B KOTOpOM IlpHMeH~teTC~I MeTO~ KpHTHqeCKOFO BO3MylHeHH~I H OH cpaBHHBaeTcfl C n p e K b l ~ y -

IHHM. KaK 3TO noKa3bIBaIOT tITHCJIOBJ~IenpmvtepsI, HaCTom]me no~xoJ~bI tIyBCTBHTeJIbHOCTH 3HatIHTeJIbHO CoKpau~a~oT BbIq~CJIeHH~I coxpaHas B MeCTe C TeM x o p o I H y I O pa6oTy B CHHTe3e CHCTeM C 3aIIa3~bIBaHHeM.