SENSITIVITY SYNTHESIS OF OPTIMAL CONTROL FOR DISTRIBUTED PARAMETER SYSTEMS
Department of Control Engineering
Toshihiro KOBAYASHI (Received October 4, 1975)
SYNOPSIS
This paper deals with the sensitivity synthesis of optimal control for distributed parameter systems.
The sensitivity synthesis of optimal control is presented to make up for undesirable effects due to system parameter uncertainties and variations, or due to simplified modeling. Sensitivity functions and sensitivity equations are defined and derived for various types of system parameter variations. Controllability is investi- gated for the combined system which consists of a model equation and its parameter sensitivity equation. The sensitivity synthesis is then applied to the problem of terminal control. Numerical results show the superiority of this synthesis to the conventional synthesis.
1. INTRODUCTION
Optimal control problems for distributed parameter systems have been studied mainly from theoretical points of view. Since, however, the systems have the parameters which can not be precisely determined and also vary, we should consider the optimal control problems with consideration of sensitivity to make up for the undesirable effects caused by parameter uncertainties and variations from practical points of view.
Relatively little attention has been given to the sensitivity problem for the distributed parameter system. Porter [ 1,2] formulated sufficient conditions to insure that feedback compensators decrease the effects of parameter variations upon distributed parameter systems. Davis [3] extended Porter's results to the more general distributed systems. Morisue [4] investigated the parameter sensitivity of the time optimal control of a continuous slab heating furnace of plate or strip mill in the iron and steel works. Pritchard and Mayhew [5] considered the sensitivity of the discrete feedback tosystem changes in the parameter of system and the initial state of the system. These works consist in the field of sensitivity analysis.
As for the sensitivity synthesis, Chu and Shin [6] presented a design method for
the low sensitivity optimal model feedback control of a class of linear distributed
parameter systems with distributed controls. Pederson and Nardizzi [7] gave the
synthesis method of a feedback control structure that adapts the nominal open-loop
control in such a manner as to compensate for the errors introduced by model
uncertalntles.
In this paper sensitivity functions and sensitivity equations are, at first, defined and derived for the four types of system parameter variations, that is, the ordinary system parameter variation (a-variation), the variation of intial conditions (B-variation), the variation of time delay parameters(S-variation)and the variation of spatial domains (a-variation).
Next it is pointed out that the concept of the combined system, which consists of a model system equation and its parameter sensitivity one, plays an important role in the sensitivity synthesis of optimal control. For the optimal control with consideration of sensitivity to exist, the controllability of the combined systems are
analyzed by using the results obtained in Kobayashi [8]. The feedforward
sensitivity synthesis of optimal control is introduced into the problem of terminal control in order that terminal constraints are satisfied in spite of the parameter variations.
By comparing with the conventional synthesis, the effectiveness of the sensitivity synthesis is demonstrated by giving several examples.
Moreover the optimal feedback (the-semi-closed) synthesis is discussed, which consists of output feedbacks and prefilters.
2. SENSITIVITY FUNCTIONS AND SENSITIVITY EQUATIONS
In this chapter we shall consider the systems satisfying the following require- ments ( i ) and ( ii) :
(i) Their solutions must exist and should be uniquely determined.
(ii) Their solutions should be differentiable with respect to the parameters and the initial data.
The second requirement is stronger than the well-known Hadamard one. It is, however, a fundamental condition in the following development of our theory. We can show that some systems are satisfied with the above requirements.
Moreover we consider the feedforward synthesis of optimal control in this section.
2.1 a-sensitivity Function and a-sensitivity Equation.
We shall deal with sensitivity for the ordinary system parameter variation without the variation of initial conditions andthe order of the system.
Let us consider the distributed parameter system described by the following vector formed partial differential equation :
-xO-: :-Ejit}-::-{-"( ttX) =H( t, x, u( t, x), f( t, x), to(t, x)) tÅr to, xe2CEM (2. 1)
with the boundary condition
G(t, e, u(t, e), g(t, s), co(t, e))=o tÅr t,, eEr (2.2)
and with the initial condition
u(to, x)=uo(x), (23)
where 9 is a fixed spatial domain and its boundary is r.u(t,x)=col(ui(t,x),•••, un(t, x)) is a state function vector, f(t,x)=col(fi(t,x), •••, f.(t,x)) is a distributed control function vector, g(t,e)=col(gi(t,e),•••, gr,(t,e)) is a boundary control function vector and w(t,x)==col(tui(t,x),•••,toh(t,x)) is a variable parameter function vector. H and G are time-varying spatial differential operators which are continuously differentiable in all arguments.
So far the optimal control problems for distributed parameter systems have been discussed under the assumption that the system parameter tu(t,x) is accurately known.
In practice, however, it is very difficult to decide the accurate value of this parameter. Therefore it is necessary to consider a new synthesis of optimal control to make up for the undesirable effects caused by a parameter uncertainty and variation. From this point of view, it is important to discuss the sensitivity
synthesis of optimal control.
Now let us regard the system described by(2.1),(2.2)and(2.3)with a parameter value to as the model system for a physical one. On the other hand, let us describe the actual system by (2.1), (2.2) and (2.3) with a parameter value to+Atu, where Atu(t, x)= ept(t, x) is a weak variation.
The difference in the state between the actual system and the model one corresponding to a certain control is described by
k
Au(t, x, tu)=u(t, x, tu+Aw)-u(t, x, w)= 2,v(t, x)"i(t, x)+o[e2] , (2.4) i=1
where
iv(t, x)==//.m, I[u(t, x,co +eist)-u(t, x,co)] (2.s)
eiLt(t, x)=col(O, •••, O, s/u,(t, x), O, •••, O).
iv(t,x) is the derivative of u(t,x, to) with respect to toi(t,x) and is called the a-sensitivity function to tui(t, x). The a-sensitivity function iv(t, x) satisfies the following linear partial differential equation :
pmatv(t x) =H.1,.=,,v(t, x)+H.1, =, tÅrt,, xest (2.6)
at
with the boundary condition
G.IA.-oiv(t, e)+G.IA.=o =O tÅr t,, eep (2.7)
and with the initial condition
where H., Gu and Hw,Gw are derivatives with respect to u and w, respectively. The equation(2. 6) is called the a-sensitivity equation to wi(t, x).
2.2B-sensitivity Function and B-sensitivity Equation
Consider the case there is a difference Auo(x) with respect to only the initial
condition between the model system and the actual one, where Auo(x)==en(x) is a
weak variation. Namely the model system is described by (2.1), (2.2) and (2.3), and the actual one is described by (2.1), (2.2) and the initial condition
u(to, x)= uo (x)+A ue (x). (2.9)
In this case, the difference in the state between the actual system and the model one corresponding to certain controls is described by
n
Au(t, x, uo)=u(t, x, uo+Auo) -u(t, x, uo)= 2 iw(t, x)ni(x)+o[e2] , (2.10) i=1
where
iW (t, X) = //-Mo l:- [U(t, X, Uo + E , 77 )- U(t, X, Uo )]
eirp(x)==col(O, ''', O, eni(x), O, ''', O)•
iw(t,x) is the derivative of u(t, x, uo)with respect to uoi(x) and is called the B-sensitivity function to uoi. The B-sensitivity function iw(t,x) satisfies the foilowing linear partial differential equation:
-gO-t!!{-3 liiFl!-LtW(ttX) == H.IAu,=oiw(t, x) tÅr to, xe9 (2.11)
with the boundary condition
G.IA.,=o iw(t, e)==O tÅr t,, eep (2.12)
and with the initial condition iWJ•(to, X)= ! ni(X) (i == j)
(O (itj) (2.13)
.Equation (2.11) is the B-sensitivity equation to uoi.
2.3 S-sensitivity Function 6-sensitivity Equation
Suppose that the actual system is described by the following differential-diffe- rence equatlonl
-=O-::-,s;2-::-L-U(ttX) =H(t, x, u(t, x), u(t-E, x), f(t, x)) tÅr O, xes) (2.14) with the boundary condition
G(t, e, u(t, e), g(t, e)) =o tÅr o, eEr (2.ls)
and with the initial condition
u(t, x)=u(t, x) tE [-e, O] (2.16)
, where E is a time-delay parameter which is positive and in some sense small.
Let us consider the following simplified model system : au(t,x) ..H(t, x, u(t, x), u(t, x), f(t, x))
at -
=:H(t, x, u(t, x), f(t, x)) tÅr O, xe9 (2.17)
with the boundary condition
G(t, e, u(t, e), g(t, e))-O tÅr O, eer (2.18)
and with the initial condition
u(O, x)==a(O, x) (2.19)
which is obtained by letting e=O in (2.14) and (2.16).
In this case, the difference in the state between the actual system and the model system is denoted by
Au(t, x, E)=u(t, x,e)-u(t, x)
=z(t, x)e +o[e2], (2.20)
where z(t,x) is the derivative of u(t,x,E) with respect to e and is called the 6-sensitivity function. The 0-sensitivity function z(t,x) is satisfied with the following 0sensltlvlty equatlon:
-:a=:-'5il-=--`--U(ttX) = [Hu+Hu(t-e)]1..,oz(t,x)+Hu(t-e)le--o fi (t, x, u, f) tÅrO (2.21)
with the boundary condition
Gul.=oz(t, e) =O tÅrO, (2.22)
and with the initial condition
z(O, x) ==O. (2.23) The sensitivity equation (2.21) does not contain the time delay parameter. This is because in the above case the nominal value of the time delay parameter equals zero.
2.4a-sensitivity Function and a-sensitivity Equation
Let us consider the actual system with the spatial domain varying with time :
.:a-::.}siil-::-Lu(ttx) = H(t, ., .(t, x), f(t, x)) in Q == l (t, x)l tÅrt,, oÅq.Åq,(t) I (2.24)
with the boundary condition
Gi(t,u(t,o), gi(t))= O, G,(t, s(t), u(t,s(t)), g2(t)) == O (2.25) and with the initial condition
u(to, x) =uo (x) OSx5s(to) =:so. (2.26)
We shall discuss the special case s(t)=so+et(e:in some sense small). Suppose that the simplified model system is described by (2.24) with Q==l(t, x);tÅrto, OÅqxÅqsOL the boundary condition
Gi(t, u(t, O), g'(t)) =O, G2(t, so, u(t, so), g2(t)) =O (2.27)
and the initial condition (2.26) .
In this cace, the difference in the state between the actual system and the model system is denoted by.
AU(t, X, e) =u(t, x, e) -u(t, x)
=:y(t, x) e+o [E2], (2.28)
where y(t,x) is the derivative of u(t, x,e) with respect to e and is called the a-sensitivity function. The a-sensitivity function y(t, x) satisfies the following a- sensitivity equation :
iY'(EStti LLttX) =H.le=o y(t, x) in Q== l(t, X); tÅr to, OÅqXÅq Sel (229)
with the boundary condition
Giule=o y(t, O) = O, G2.1.=o[y(t, so) + -=a-::-,si}FÅ}y-L-U(txSO) ] +G2.I...o t=O (2.30)
and with the initial condition
y(to, x) =O OSx5 so. (2.31)
3. COMBINED SYSTEMS AND THEIR CONTROLLABILITY
3. 1 Combined Systems
Consider the model equation and the sensitivity equation simultaneously, which is simply called the combined system, and then it may be recognized that there is a possibility to control not only the model state but also its sensitivity function by choosing proper controls f(t,x) and g(t,x). That is, we can take the effect of the small parameter variations into consideration at the initial stage of the synthesis.
This is the basis of the sensitivity synthesis of optimal control, Inoue [9].
Combining the model equation and the a-sensitivity equation, we can get the a-combined system :
:a:- =År3;i-:=-U(ttX) =H(t, x, u(t, x), f(t, x), w(t, x))
G(t, e, u(t, e), g(t, e) , to(t, e)) =O, u(to, x) =uo(x) (3.1)
.gO!.tulLe-LdLLtV(t X) ==HulA.=oiv(t, x) +HtoIAw-o
at
Gulde-oiv(t, e) +G.IA.=o=:O, iv(to,x) =O which governs an extended state (u(t,x),iv(t,x)).
Similarly,if we pay attention to the variation of the initial condition, we can get the B-combined system :
-au(tx)-
H(t, x, u(t, x) ,f(t, x))
at -
G(t, g, u(t, e),g(t, e)) =O, u(to, x) ==uo (x)
-ga-tg!{-)sLl-tt Lw(ttx)=H.I,.,.,,,w(t,.) (3•2)
GulAu,-o ,w(t, e) ==O, iw,•(to,x) =Irpi(x) (i=J) tO (i4 j)
We should consider the following S-combined system, if we pay attention to the variation of the time delay parameter.
-=a==si2-::-L-u(ttx) =H- (t, x, u(t, .) , f(t, x))
G (t, e, u(t, g),g(t, e)) -O, u(O, x) -=-u(o,x)
-=a:-L3iz::"`-Z(ttX) == [Hu+Hu(t-e)]l.=oz(t, x) +H.(,-.)i,=, fi(t, x, u, f) (3'3) Gule-o z(t, e) =:O, z(O, x) =o.
Moreover the a-combined system becomes to
-=a-:=}si2-::-LU(tt X) =H (t, x, u(t, x) ,f(t, x)) in Q= l (t, x) l tÅr t,, oÅqxÅq s,l
Gi(t, u(t, O) ,g'(t)) =O, G2(t, s, u(t, so),g2(t)) ==O u(to, x) ==uo(x)
(3.4)
90:}IS'3:FKL( tt ) =Hule...o y(t, x) in Q= l (t, x); tÅr to, OÅq xÅq sol
Giule-oy(t, O) =O, G2ule-o[y(t, so) + aU(atx' SO) t] +G2xle..o t==O y( to, x) =O.
It should be noticed that a new basic important problem comes out in the case of sensitivity synthesis of optimal control ••• the problem what performance indicies and constraints are to be given for the combined systems. It depends on the control object of the original problem.
It is very important whether the combined system is controllable for the control inputs or not, in the case of the sensitivity synthesis. Therefore we shall investigate controllability of linear combined systems in the following section.
3.2Controllability of Linear Combined Systems
We investigate controllability of linear combined systems by means of the results obtained in (8)
[A] Controllability of the a-combined system
Consider the model system described by an infinite dimensional linear differential equation in the state space X:
d"itt)=A(p) u(t) +B(q) f(t), u(to) ==o, (3.s)
where p is a K-dimensional parameter vector and q is an s-dimensional parameter
vector.
Suppose the i-th component pi is a variable parameter. Then the a-combined system is written by
i:i,;(Y,(,1',)-(i/ll(Pl) .?,))(Y,(,1',)+(,il[z',)f(t),(,,",`j,o,')-(go), (36)
where iv (t) is the a-sensitivity function to pi,
,A(p)=.2h ..,ea {iXe-2-Ap(.)Sdz,,l,,=, andiB(q)=,;iSll;=,sOZSIiiSLLBi.)Caliqq:IAp-o.
If the system(3.5)consists of the infinite set of Lj-dimensional equations which are seperate for each j, the combined system (3.6) becomes to
Slr(,,";)=(,lll %,)(IZI)+(,IIil)fJ•(,,",'((tt,O)))-("8') (3 7)
The combined system (3.6) is controllable (at time ti) if and only if each finite
dimensional subsystem (3.7) is controllable(at time ti)for each j from Theorem 2.1
in (8). Therefore the sufficient condition for the combined system (3.6) to be
controllable at time ti is
rank Q.(t) =rank [Po(t) i Pi(t) I •••1P2LJ•-i] =2L,• for some te(to,ti) (j=-1,2,-) (3.8)
where
Ph+i(t) ==-GJ'(t) Ph(t) +Ph(t) , Po(t) =EJ'(t) , G,(t)=(,AAI. %,), E,(t)=(oBJ' ,OB,).
In the case of A(p) and B(q) being time-invariant, the above condition is equivalent to
rank (Ejl GjEjl •••I Gg•`'-'Ej]=2Lj (7'=1,2, "'). (3•9)
Next suppose the i-th component qi i's a variable parameter. Then in the
controllability condition (3.8)
iAj=.S-I].i-!2A(Ell;i2;ll-LZIZi!i.)ddp,M.IAq=o anbiBj=.2S.i!alfii;l:2LtiB Lq(.)fdlSIiqmiIAq=o.
(Example 3.1)
Consider a simple heating system
=a-::.zii2-=-LU -(ttX)=aAu(t,x)+bf(t,x) tÅro,xEÅí
o2 a2 a2 A :axi +ax; +"'+axh
u(t, e) =O tÅr O, ger, u(O, x) =uo(x), where a is a variable scalar parameter.
The sensitivity equation becomes .a=.xx3t::-L-v(ttx)=aA.(t:x)+Au(t,.)
v(t, e) =o, v(o, x) =o.
The combined system is described by gi(,u,(;j;,))-(a2 2.)(,u,(;;l,))+(bf(6•x)) (3.10) (,",(;;g,')-o, (,",(8;i,))-("o(j))
Using the eigenfunction (Åë,(x)I and the eigenvalue IAil of aA, we get (,"((,`j.X)))-i.;=,(.",i(,`)))Åë,(X), (f('6X))-S.=,(f`(,`))Åëi(x),
(Uo6X?)=;.i:=,(U&i)Åë,(x), aAÅë,(x)=A,dii(x) (i=1,2,-)
The above combined system is written by
Z,T(S,t((,')')-(Ai/.9,)(,",i(`,`)))+b(f6(`)), (Y,i(,O,))-(u&i)(i-i.2,•••)
From (3.9) , the combined system (3.10) is controllable at any time t (tÅrO) if A,# O (i=1,2"••)
(Example 3.2)
Consider the system with a uniformly distributed control
=0-:x-:-:r-L-"(ttX)=agtlZ!{i}YZ-]!-"(i,X)+f(t), tÅro,xe(O,1)
u(t, O) ==u(t, l) =O, u(O, x) =uo(x)
where a is a variable scalar parameter. In this caseÅëi(x) =VITsin(irrx), Ai=-i2rr2a and the combined system becomes
Z,T(,",t(`,`)')-(A/;/.9,)(,",k`,`,')+(g')f(t), (,",`(,O,')-("do), i
ki-sf2-f, sin(irrx)dx {l8 [l•i•ZM).
Therefore the combined system is not controllable.
(Example 3.3)
Let us consider the system with a scanning control
-0=-::-}5i;=-L-U(ttX) =aAu(t,x) +f(t)@S(x-s(t)) tÅrO, xe9, s(t)e9
u(t, g)eP, u(O, x) =uo(x) ,
where a is a variable scalar parameter. Using the eigenfunctions and the eigenvalues of aA, we obtain the following combined system
ST,(,",`((,`)')-(A/;,.R,)+(,",i,(,`,')+(Åët(Z(`")f(t), (,",i,(,O,')-(z`o).
(i=1, 2, •-)
In this case the matrix Q, of the controllability condition (3.8) becomes dÅë, ds(t)
-AiÅëi(s(t)) O Åëi(s(t)) o
dx dt
O O -!i!L:' Åë,(s(t)) O
a
(i=1, 2, •-)
If the rank of this matrix is 2 at some te(O, ti)for each i, the combined system
is controllable at time ti. For this, it is sufficient that Åëi(s(t))tO for somete(O,ti)
t
(a4 1) . Then Åëi(x) = vi-27sin(inx), Ai=-i2rr2a.
Suppose 9= (O, 1) and s(t) = ati
In this case
Åëi(s(t))tO a. e. in (O, ti) for each i.
Hence the combined system is controllable at time ti.
On the other hand, in the cases of s(t)= X (kÅr 1),the nk-th subsystem k
is not controllable (n: positive integer) , because Åëi(s(t)) = Msints' =O i=k, 2k, •••.
From this, the scanning controls are very important and interesting.
(Example 3.4)
Consider a vibrating system with a variable parameter a:
ett gllSs-N-)LU(tt,X) =aAu(t,x) +f(t,x) tÅro, xeg
u(t, e) =O tÅrO, eer, u(o,x) =u,,(x) , -OU(O X) =u,,(x) . This system can be written in a vector form
gi(z;[i;:))==(2.6)(zwtl)+(9)f(t•x)•(za[s;:l)-(ta:[:),)•
The sensitivity equation becomes
gi(:[l;x.))-(2. 6)(:[l;x.I)+(2 8)(z;[i;B)•(:[8;:l)-(8)•
By the eigenfunction expansion with respect to an operator aA, the combined system becomes
41--
dt u,,(t) - A, OOO u,,(t) 1 u,,(O) - u,,,
v, ,( t) - O OO 1 v, ,( t) O i , v,,(O) - o
v2i(t) Ai/aO AiO v2i(t) O v2i(O) O
In this case the controllability condition (3.9)
ra"k(Z S i:,. 11,tia)=` `z=i'2' '
holds. Therefore the combined system is controllable at any time t(tÅr O).
(Example 3.5)
Suppose b is a variable parameter in Example 3.1, then the combined system becomes
ZE(ZI.k,3) :(Ci R,)(%I.[,tl)+(2)f,(t),(zl.[81)-(udi) (i-i,2,•••)
which is not controllable.
(Example 3.6)
Let us consider a heating system
iO:-::-Eisl-:=-U(ttX)==a!t21-!111iiljiU !Z-)(i,X)tÅro,oÅqxÅqi
u(t, O) --O, au(t, 1) + (1-a)-0:-::;;E2L=-U(t 1) =cf(t) ,OÅqaÅq 1, u(O, x) =uo(x) , where a is a variable parameter. The eigenvalues and the eigenfunctions of this system are
Ai=-aB?', Åëi(x)= 2,e,-4,Pik2,e, sinPix;tanJei=a-.iJei (i---i,2,•••).
Then the combined system is written by
ill,T(Zl[ll)=(/iA.tR,)(Åél[`,l)'(S-z[:i,i,',T,tt`j,,',],,,)f(t)•(Åé;[g',)-("st)
(i=1, 2, •••) dAi ,# o.
which is controllable at any time t (tÅrO) , because Åëi(1)-Åë;•(1)tO, da
[B] Controllability of the B-combined system Consider a model system
d
iii u(t) =Au(t) +Bf(t)• u(to) =uo (3.11)
which is an infinite dimensional differential equation. Here uo is a variableinitial value. The B-sensitivity equation is described by
ZiTt w(t) =Aw(t), w(t,)=rp. (3.12)
Therefore the B-combined system becomes
{;,7(k;[t,l)-(g 2)(%[`,l)+(Bf6"), (s[`,:l)=(yo) (3.i3)
which is clearly not controllable, because the model system state and the sensitivity one have no explicit relation each other.
[C] Controllability of the S-combined system
Suppose that a time delay system is expressed by an infinite dimensional
differential-difference equation :
ZiTtu(t) =Au(t) +u(t-e) +Bf(t), (3.14)
u(t) =u(t) , -eStÅqO, u(O) =uO.
A model system is obtained by letting e=O. Then the S-combined system is given
by
Si,(Z(,ll):(A-+.L,O..,)(Z[ll)+(B-flli;,),,),(z(,81)-(u,o) (3.is)
where I is an infinite dimensional identity matrix.
If the system (3.14) is separate, the combined system (3.15) is also written by
Z,T(.";((;)')==(A-tl,4t.L'i, A,O+i,)(.",f'i,`)')+(B-i.fil.I,)),(.",i,',i,(`')=(,"oi),
where Ii i's an LixLi i'dentity matrix.
Then the sufficient condition for the combined system to be controllable at time ti is that for each i (i == 1, 2, •••)
rank Q. (t) ==rank [Po(t) Pi(t) •••P2L,.i(t) ] =2Li te(O, ti) (3.17) where Ph+i(t) =-Gi(t) Ph(t) +Pk(t),Po(t)=Ei(t)
G,(t)-(A-t.+,tt,, .,O.,,), E,(t)-(B-i.,).
(Example 3.7)
Consider a time delay system
-a:-::-Esi2-:!-LU(ttX) ==Au(t,x) +u(t-e,x) +f(t,x) tÅro, xeg
u(t, x) =-u(t, x) -eStÅq O, u(O, x) ==uo(x)
a(e) u(t, e) + =a-=-Årsif-LU(t,e) =o tÅro, eer.
By using the eigenfunction (Åëi(x)i and the eigenvalue (Ail of an operator A, the 6-combined system is written by
St(.";(`;)')-(ti,+,i-, ,,.O,)(.U,I',(;,)) +(-l)f,(t),(.U;,(8,))-(u6o).
If li=-1 (i =1,2,•••), the system is controllable at any time tÅrO.
[D] Controllability of the a-combined system
Since the boundary condition of the o-sensitivity equation is the different form from that in other cases, we can not here investigate the controllability of the a-combined system.
4. SENSITIVITY SYNTHESIS OF OPTIMAL CONTROL
In this section we shall apply the sensitivity synthesis to the problem of terminal control (the minimum energy control problem with terminal constraints).
4.1 Sensitivity Synthesis of Optimal Control for a-parameter Variation
Let us consider the same model system(2.1), (2.2) and(2.3) as in Chapter 2.
The conventional synthesis of the problem of terminal control is stated as follows :
For the model system expressed by(2.1),(2.2)and(2.3),synthesize such controls f'(t, x) and g'(t, e) that
(i) satisfy the terminal constraint
u(ti,x) =ud(x) xe9, (4•1)
and
(ii) minimize
J(f, g) == f,,t'dt f.f '( t, x)R(t, x)f(t, x)dx + f,,t'dt f.gT( t, 6)Q(t, e)g(t, 6)de (4.2)
where ti is a final time, ud(x) a desired final state, R(t,x) an rxr positive definite matrix function and Q(t,g) an r'xr' positive definite matrix function.
The difference Au(t, x, to) in the state between the actual system with the
parameter tu+Aw and the model system with the parameter tu, can be approximated Au(t, x, tu) = 2 iv(t, x) Atui•
If the difference Au(ti,i i,'tu)at the fmal time ti is estimated to be large, then the main object (i) of this synthesis problem cannot be satisfied. It is, therefore,
necessary to develop another synthesis method which is able to compensate the violence of the terminal constraint.
Since the variation of the terminal state due to the small parameter variation Atu is approximated to be
k
Au(t,,x, to) == .Z iV(t,,x) Aw,, (4.3)
t=1
it is, thus, desirable to choose the controls such that they should satisfy
,v(t.x) =O (i=1, 2, •-, k) (4.4)
for the main object (i) being satisfied. From this point of view, the sensitivity synthesis of the problem of terminal control is, then, restated as follows:
For the a-combined system(3.1),synthesize sensitivity optimai controls f."(t,x) and g:(t, e) such that
(i) satisfy the new terminal constraint for the a-sensitivity function
iv(ti, x) =O xest (i=1, 2, •••, k) (4.5)
and
(ii) minimize the energy consumption J(f,g) given by (4.2).
The sensitivity synthesis method proposed can solve the problem of terminal control with enough satisfaction of the main object(4.1). It should be noticed that the restated problem has a solution if the combined system is controllable at time t,. In the case we cannot make sure of the controllability of the combined systems (for example, nonliear systems), we can discuss the sensitivity synthesis by taking another performanceindex Jv in place of the sensitivity terminal constraint(4.5),
Jv=J(f, g) + f.v' (ti, x) P(x) v(ti, x) dx, (4.6)
where P(x)is `an nxn positive matrix function. An optimal weighting function P"(x) may be sought according to synthesis objects.
4.2Sensitivity Synthesis of Optimal Control for the Systems with Small Time Delays
In the following two sections, we shall apply the sensitivity synthesis of optimal control to the linear systems with time delays or moving boundaries from a computational point of view. Since the system with time delays is expressed by a differential-difference, equation, the direct computation of an optimal control from the optimality condition is in practice prohibitive. Therecore, feasible approximate numerical methods are of practical necessity.
Let us consider the following linear time delay system
:0:=:-}5ii--::-`-U(ttX) =Au(t, x) +u(t-e, x) +f t, x) OÅqtÅqti, xest (4.7)
u(t, x) =u(t, x) -eStÅq O, u(O, x) =uo(x) xest (4.8) au(t, {f) +-0:=;ixiu-U(t,.e) =o oÅqtÅqt,, eer, (4.g)
where E may be unknown which is in some sense small, and
A= ,.SII;,Åí;, (a iJ•(x ) ae, )+ao(x ), oe. = ,.$i ,a tj (e) al, cos( v, xJ• )•
For the system expressed by(4.7),(4.8)and(4.9),synthesize such a control f"(t, x) that
(i) satisfies the terminal constraint
U(t,,X) =Ud(X) (4.10) and (ii) minimizes the energy consumption
1(f)= 7f,t' f.f2( t, x)dtdx, (4.11)
where ti, ud(x)and 7 are respectively a given final time, a given desired final state and a positive constant.
In this case, the optimality condition is
f,ti f. f"(t, x)[f(t, x) -f"(t, x)]dtdx=O Vf. (4.12)
Introduce the adjoint state defined by
ga?Ii Rl}i!i-L(tt ) ==-A*p(t, x) +p(t+e, x) oÅqtÅq t,, xeg (4.i3)
p(t, x) =O tiÅqtSti+E, xest (4.14)
ap(t•e) t =0s2sttL-P(t,.{) =o oÅqtÅqt,, eEr. (4.ls)
Then an optimal control is given by
f"(t, x) =--IL p(t, x). (4.i6)
7
The two point boundary value problem (4.7) -- (4.9) and (4.13) •-- (4.15) is prohivitively difficult to solve, since(4.7)is of retarded type with the initial function u(t,x), while (4.13) is of advanced type with the terminal function (4.14).
If the delay time e is smail, the model system
-:a-::-}st-:=U(ttX) ==Au(t, x) +u(t, x) +f(t, x) oÅqtÅqt,, xsg (4.17)
u(O, x) =uo(x) xest (4.18)
au(t,e)+=a-:x:-u-"(,t.6) ==o oÅqtÅqt,, eer (4.ig)
which is obtained by letting e=O in(4.7)--(4.9),is conventionally used instead of the actual system (4.7) -- (4.9) to avoid the difficulty of treating the differential- difference equation. Then, such a synthesis that adopts the differential equation model system (4.17), (4.18) and (4.19) as a controlled system is also called the conventional syethesis. Even if, by this conventional synthesis, the computational effort is greatly decreased, the main object (i) of this synthesis problem cannot be E;atisfied Therefore the sensitivity synthesis is applied to the optimal control problem tbr the systems with small time delays, which is able not only to compensate the violence of the terminal constraint but also to decrease the computational difficulty of the original synthesis problem.
The sensitivity synthesis problem is stated as follows : For the combined system
.a-.,-L:.2-:=U(t X) =Au(t, x) +u(t, x) +f(t, x)
et
(4.20) =a=-L:-L::-LZ(t X) =Az(t, x) +z(t, x) -Au(t, x) -u(t, x) -f(t, x)
at
(4.21) u(O, x) =uo(x), z(O, x) =O
.u(t,e)+=a-:=3tt!-L-U(t,.e)=o, .z(t,e)+=aL::i}tt;uu(t,.e)=o, (4.22) synthesize the sensitivity optimal control such that
(i) satisfies the new terminal constraint Z(ti, X) =O
together with U(ti, X) ==Ud(X)
(ii)minimizes the energy consumption J(f) given by (4.11).
This sensitivity synthesis problem has an optimal solution if the combined system is controllable at time ti.
In the case of e being not necessarily small, two-level optimization techniques or
game theoretic approaches can be applied.
13 Sensitivity Synthesis for the Systems with Moving Boundaries
Most of the existing works are devoted to systems which are defined on fixed spatial domains. In many physical situations such as in the melting of solids or solidification of liquids, the systems spatial domains vary with time. The mathe- matical description of these systems' generally leads to partial differential equations involving moving boundaries. For the system with moving boundaries, the direct computation of an optimal control from the optimality condition is also difficult in practice. Therefore in this subsection we consider the sensitivity synthesis of optimal control for the moving boundary systems.
Let us consider the following system:
=a-=Årjt-::-L-U(ttX)=agt!Z!1:itS!L)-U(i,X)inQ=:(t,x);oÅqtÅqt,,oÅqxÅqs(t): (4.23) '
u(O, x) ==uo(x) OSx$s(O)=so (4.24)
:O:.::-,sLt-::-L-u(t.x)=o, a-=a-::-L:si-y-uu(t,,s(t))+dsitt) u(t,s(t))=-f(t). (4.2s)
For this system, synthesize such a control f"(t) that minimizes
J(f)=pf,S(`"u2( t,, x)dx +7f,t'f2( t)dt BÅrÅr7 (4.26)
where B and 7 are positive constants.
In this case the optimality condition is for each f 3f,S(t') u(t,, x; f") [u (t,, x;f) -u(ti, x; f") ] dx+ 7f,t' f'(t) [f (t) -f'(t)
Introduce the adjoint state defined by
!22g-Ztit!-)=-aopax inQ (4.2s)
P(t,, X) =-BU(t,, x) OS xS s(t,) (4.29)
EaZiLlisi!-2.(t.O)=o, a!a22S-Sie9SL!-22-(t.(t))+alsitt)p(t,s(t))==o (4.3o) Multiply (4.28) by u(t, x; f) - u(t, x; f") and integrate over Q. Then we obtain
f,t' [7f'(t) +p(t, s(t))][f(t) -f"(t)]dt-O (4•31)
from which an optimal control is given by
f*(t) =- -jli- p(t, s(t)). (4.32)
The two point boundary value problem(4.23) -- (4.25)and(4.28) -- (4.30)is very
difficult to solve, since the spatial domain varies with time. Therefore we consider
the sensitivity synthesis of optimal control in the, case of s(t) =l+et (E:
small(unknown)parameter). As e is small, the conventional synthesis is stated as
follows :
For the simplified model system :
=0- =}3";-::-`-U(tt X) =a it!Z!{Ili3I!-)-U(,t,X) in Q' =={ (t, x) ; OÅqtÅq tb OÅqxÅq II (4•33)
u(O, x) =uo(x) OS xl (4.34)
-0u(tO)=o, ,.a-,,-.-,-..U(t1)=f(t), (4.3s)
0x ax
synthesize the optimal control such that (i) satisfies the terminal constraint
and
(ii) minimizes the energy consumption
J(f) == 7f,`i f2(t) dt. (4'37)
Even if, by this conventional synthesis, the computational effort is greatly decreased, the control objects are not enough satisfied because of e being not zero.
The sensitivity synthesis is stated as follows:
For the combined system:
-0.-.bi-L:2u(tx).,a!t21-!iiiiltE-gL)-USt,X)inQt
(4.38)
!aÅrliS2-i!L2(tt)=.!a2LISf!E2-2(E,)i. Q•
u(O, x) =uo(x),y(O, x) ==O OSxSl, (4.39)
=a=bi;-LuU(,,tO)=o, .=O-:bi-L=-t"(,,t1)=f(t)
ga:\2iiLyz(EO)-o, .:az2i}L=-Ly(E1)+.tgtE!4Siti!)-u(i,1)+u(t,i)-o,
synthesize the sensitivity optimal control such that (i) satisfies the terminal constraint
U(tb X) =O and
(ii) minimizes the new performance index
Js (f) =r f,t' f2 (t) dt +k f,' y2 (ti, x) dx. (4•41)
Keast solved the system with such boundary conditions as (4.40) by finite difference methods.
In the case of s(t)being not necessarily of the form 1+st, we can apply two-level
optimization techniques or game theoretic approaches.
5. NUMERICAL RESULTS
We shall show that the sensitivity synthesis is superior to the conventional one by giving several numerical examples.
(Example 5.1) (a-Variation) Consider a simple heating system
.a=-::-}st-=-L.u(ttx)=agt!2u3SÅr!L)-u(i,x)+f(t,x) oÅqxÅqi
u(t, O) =u(t, 1) =O, u(O, x) =:O.
For this system, we consider energy consumption J(f) =' l}-f,' f,' f2(t, x) dtdx'
Suppose a is a variable parameter with its nominal value a.=O.1 and then the combined system is controllable at time 1.
Numerical results are shown in Fig.1 and Table 1, where ud(x)=1-l2x-1l, ti=
1 and v(ti,X)=O. Fig.1 shows the state trajectories at x=O.5 of both the actual
system( Aa SO.Ol) and the model one (Aa==O) corresponding to the conventional synthesis and the sensitivity one. In Table 1, the values of the terminal state error
are listed up for the demonstration of how the object u(ti,x) =ud(x)is satisfied.
In Table 2, where v(ti,x) is free and P(x) =p in J. given by(4.12), the values of the terminal state error are listed up for the various values of p.
(Example 5.2) (S-variation)
Let us consider a time delay system
ia:=:-Årsii-::-LU(ttX)=a!t2e-!{:lri!-l!U(i,X)+u(t-e,x)+f(t,x) oÅqxÅqi,a==o.oi
u(t, O) =u(t, 1) =O, u(t, x) =O (-eStSO) .
For this system, we consider the energy consumption J(f) = Sf,'f,' f2(t• x) dtdx.
Suppose E is a small time delay parameter with its nominal value e.=O, and then the combined system is controllable at time 1.
Numerical results are shown in Fig.2, where ud(x) =sinrrx, ti=1 and z(ti,x)=O- Fig.2 shows the state trajectories of both the actual system(e==O.1)and the model one (e=O) corresponding to the conventional synthesis and sensitivity one.
(Example 5.3) (a-variation)
Consider a moving boundary system
=a-:=}s;i==-u(ttx).,,!t2e-z{Il}lfN)-u(tt,x) oÅqxÅq1+,t
u(O, x) =O
au(i,O)=o, :a:"=:-Lbi-L-=-!-L-u(t.1+et)+,.(t,1+,t)=f(t).
Then the system has the boundary condition
:a:-t=isi}t-=L-"(t.O)==o, -O:-::-}sLt-x"(t.i)=f(t)
which is obtained by letting E=O.
Numerical results are shown inFig. 3, where ud(x) ==1, ti=1, y(ti,x) =O and J(f) ='5f,'f2(t) dt'
The equation
sazzssif!I-L(ttx)..ga;}lisy-z-2(Åí,) ,y(o,x)=O
-0:=Lxs}=LY(t.O)==o, -:a:s"3tiL-Y(t.i)+u(t,i)=o
is used in place of the sensitivity equation.
Fig.3 shows the state trajectories of both the actual system(e=o.1)and the model one (e==O) corresponding to the conventional sythesis and sensitivity one.
The numerical results of above three example clearly show that the superiority of the sensitivvty synthesis to the conventional one from the fact that the object u(ti, x)=ud(x)of the synthesis problems is satisfied in spite of the parameter variations The numerical computations are done by using the eigenfunction expansion. In the case of e being negative, we cannot obtain a desirable result in Example 5.3, because the value of the first eigenvalue is O for the model system and about n for the actual system.
G 2- g
1.0
O.5
Fig.1 Comparison of trajectories by conv. syn. and by sens. syn against a-variation Table 1 Comparison of 'terminal error due to a-variation
(u(1, x) =1.0- l 2x-1.01 , v(1, x) =O, u(O,x) =O, t,=1.0)
parameter value
synthesis method
terminalerror nu(1,O.5)da==O.Ol
energyJ(f).consumptlon
