- Email: [email protected]
On a problem of stochastic stabilization
Copyrlab'
ON A PROBLEM OF STOCHASTIC STABILIZATION A.N. Kotel'nikova, N.N. Krasovski Inst. of MathematiC3 and Mechanics of RAS, Ekaterinburg, Russia E-mail: [email protected]
Abstract: Considered here is the problem of stable tracking and leading the motion of a controlled plant with after-effect, under uncertainty or conflict [1-5]. This is done through a computer model in the closed feedback loop. The basis for solving this problem is Lyapunov's theory of stability of motion modified for hereditary and stochastic systems [6-16]. A probabilistic algorithm of control within a discrete-time scheme is provided.
1. STATEMENT OF THE PROBLEM
z",[t]
+ The motion of the x-plant is described by the differential equation
en
x",[t] = FI%I(t, x",[t, + BI%I(t)· u~"J[tl+ + fl%"l(t, ut,ui[t], vt,VI[t], x",[t - h~,"l[t]], x",[t - h~,uJ[t]])
+ A.,[t], (1.1)
= Flzl(t, z..,[t, en + Blzl(t) . u~"l[t]+
J Jf[·,·l(t, ulz
,,,!, vlz,v l, x:[t - hl.,..l],
Mt-] NI-]
x:[t - hl.,vID x xjL(du{·,"I, dh[·,ul j t) . f)(dv l,.uJ , dhl•.v1j t)
(1.2)
to + h ~ t~%l ~ t~'1 ~ t < 00, h> 0 - const, z={.z;,i=I, ... ,n}, z[t, e] = (z[t + flj: - h ~ fl ~ 0).
to + h ~ t~%J ~ t < 00, h > 0 - const, x = {xi,i = 1, ... ,n}, x[t, ej = (x[t + 19]: - h ~ fl ~ 0).
The motion of the z-model is described by the differential equation
The index w denotes an elementary event; x, z are vector columns. The time to and the constant h > 0 are unconditionally fixed. The initial instants of time t~%I, t~'1 ~ t.[x] determine a certain motion x[t], z[t]; x[t, e], z[t, ej are the continuous histories of
the motion. These histories are estimated by the following nonns: IIw(e)lIc
=
to the feedback principle using the Dext infonnation image being realized: Y
max Iw(19)1,
-h~~~O
= {x·(e),z(e)}jY[t] = {x'[t, e). z[t,e]},
1
II w (e)IIL2' = (IW(O)\2
Iwl=
t.
+1,W(19) ,2 d19) 2
(tw;)t, W=X,z.
(1.3)
1;:;1
u[x,u] E Ulx,u l, vlz,v] E V(I,V], ulz,u! E Ull,u], vlx,v] E V[x,vI, (1.4)
<
00.
(1.6)
Here the continuous function x·[t,e] is the corrupted result of observation for the actual history x[t, ej :
x·[t,e] Operators F(e), f(e) are uniformly continuous in all variables, with the history variable estimated by norm IIw[e]lIc (1.3). These operators uniformly map closed bounded sets of variables into bounded sets of images and satisfy the Lipschitz condition in the history variables. The matrix functions B(t) = {bij }, i = 1, ... , 1Ij j = 1, ... , m are unifonnly continuous and bounded for t E [to, (0). The following constraints are given:
~ t
= x[t, e] + ilIx][t, e].
(1.7)
The disturbances vlx,v], hlx,.I, il, il lx] are endogeneous. Operators FI
hlx,u] E Hlx,u!, hlx,v! E Hlx,v!,
hI' ,vi E H["v!, hII,u! E HI',u], (1.5)
o
where U, V, H are compacts of finite dimensions.
yw[t] =
J~Glx](t, 19) . yw[t, e]+ -h
+ R1XI(t, yw[t, e). zw[t,e]) + F1x](t, zw[t, e])- F[I](t, zw[t, e]) + BIx](t) . u~,·I[t]- B['](t) . ut..l[t] + .£lw[t]+
In (1.2) tl( ej t), ,.,( e; t) are probability measures on the sets of admissible values for u[<,u l, vl"vl, hl',u], hlz,vl. These measures depend on time t in such a way that there exists a solution z,.,[t]. The external disturbance il[t] is uniformly bounded and piecewise continuous. Symbol 0 further stands for the vector with zero coordinates.
fIx'·!(t, u~,uJ[t). vt,v![t). xw[t - h~,uI[tlJ, xw[t - h~,v![t]])-
J J fl<"I(t, u[·,uI, V(I,V!, x:[t - hl<,ul). M!'] NI'!
x:[t - h[I,vID x Xtl(du(I,U j , dh("u]; t) . ,.,(dv["v], dhl"v]j t)
The motion x[t] should be stably tracked by the z-motion. This z-motion could be used as a guide [5] in order to ensure the desired quality of x-motion.
(1.8) to
Inputs u[x,.j ulz,.] ulx,u] ul"ul v["vl hlx,u! hl',u], hl<,v] ~e ge~erated in the co~trolle; U within a discrete-time scheme with small step t, ~ t ~ t'+ll I = 1,2, ... , due
+ h ~ t~x]
~ t~'1 ~ t < 00, h> 0 - const,
where Glx](t, 19) is a bounded matrix function with bounded variation, unifonn in 150
t E [to,oo) and -h ~ 19 ~ O. A long as
{x",[t], z..,[t], x:[t]} may be visualized as an assembly of realizations which corresponds to the set n of simple ~ w.
motions remain in the bounded domain
Ixl
~
Izl ~ H·,
H·,
H· - const, (1.9) the operator from (1.8) defined by the Stieltjes integral is uniformly continuous in t E [to, 00) for a fixed history y[t, e].
2. THE FORMATION OF THE PROCESS We suppose that the linearized information image of equation (1.8) is stabilized [12-16] in the form o ri[t] = d,a[z!(t, 19) . y[t + 19]+
The parameters of equation (1.1) are represented by means of their information images:
a 12OI (t,19) -h
-7
~
J
a[zl(t, 19); B(2OI(t) + B[zJ(t); f[2O,o!(e) + f12O,.I(e) (1.10) 19
~
0,
~ t
to
-h
< 00.
+ B[zl(t) . u[zl[t]
=
o
Jd,G[zJ(t, 19) . y[t + 19]+ + B[z)(t) . Jd"U.(t, 19) . y[t + 19], =
The histories x[t, e], z[t, ej satisfy the Lipschitz conditions
-h
o
jw[t + 19[1 1] - w[t + 19[2!] 1 ~ L[UlI
.1 19[1) _
19[2]
I'
(2.1)
-h
- h ~ 19[;1 ~ 0, t ~ t~z), w = x, z (1.11)
~ 15(01,
(1.12)
IIBI2OI(t) - B[zl(t)1I ~ c5[BI,
(1.13)
where Uo(t,19) is a uniformly bounded in t E [to, 00) matrix function with uniformly bounded variation for -h ~ 19 ~ 0; the second operator in the right-hand side of (2.1) is uniformly continuous in t E [to, 00) for a fixed history y[t, e] from domain (1.9). It is also assumed that the resulting asymptotic stability of the unperturbed motion y[t] == 0, tfl ~ t < 00 relative to the perturbed motions y[t] (2.1) is uniform with respect to the initial time t~zJ and the phase variable y.
(1.14)
A Lyapunov functional V(t,y(e)) [10, 12, 14, 15) is constructed:
as long as the motions remain in domain (1.9). At the same time, the following conditions are true: o
Jd,(Gl2OJ(t, 19) - G[zJ(t, 19)) . y[t + 19] ~ -h
IRi2O!(t, y[t,
e], z[t, en! ~
~ c5 lRJ ·lIy[t, eHle,
V(t, y(e)) = V(t, (y(19), -h ~ 19 ~ 0)) = y'(O) . A[OI(t) . y(O)+ o +y'(0) . A[IJ(t, 19) . y(19) . d19+
If[2O,OI(e)1 ~ D[/I, D[/1 > 0 - const, If[2O,·I(e) - f12O,01(e)1 ~ 15[/·1
(1.15)
=
J
(1.16)
o
-h 0
JJy'(r) . (t, r, 19) . y(19) . dr . d19+ + Jy'(19) . (t, 19) . y(19) . d19,
under the same values of the independent variables in operators f[2O,ol(e) and f12O,.I(e), 15 are small positive constants.
+
A[21
-h·-h
o
A[3J
The probabilistic proc~ combining xmotion and z-motion is generated by di&crete scheme (in time t). Thus the details concerning the stochastic differential equations are not actual here. And the process
-h
(2.2) t~zl ~ t < 00.
Here the prime denotes transposition. 151
The functional (2.2) satisfies the conditions:
19 ~ 0)) ~ w.(Iy(O)I), w.(r) > 0, r > 0; w.(O) = 0; W.(r2) > w.(rd, r2 > rh (2.3)
be realized. Denote s~[t] = groo.w[l] V(t,
~
V(t, (y(19), -h
~ 19 ~ 0)) ~ w·(lIy(O)IIL,.l, w'(r) > 0, r > 0; w'(O)
s:'[t]
The small games [5J min
l'(du1z •• I,dhlz.-I)
)(2.1)
max
~(dvlz.·j,dhlz"J)
s:'[t,j·
.! ! f!:t,-I(t/,u[:t,v],v[:t,v
= 0;
W'(r2) > w'(rd, r2 > ri, (2.4)
en
en,
y:[t] = x:[t] - Zw[tJ. (2.9)
V(t, (y(19) , -h
(V+(t, y[t,
y",[t,
= groo.~[I] V(t, y:[t, en,
l,
MI-JNlzl
X:[tl - h[:t,v)], X:[tl - hl:t,v]n x x ~(du[:t,vl, dhl:t,v]) . !)(dv[:t,v] , dh[:t,v)) ,
~ - lIy[t,elll~,., (2.5)
(2.10)
where w.(r), w'(r) are continuous functions, (V+(t, y[t, denotes the right
en)
min
(2.1)
V(t, (y(19) , -h
~
19
~
MI·)NI·j
x:[t/ - hiz,u]], x:[t, - hIz,vl]) x
x ~(du[z,uJ, dhlz,u)) . !)(dv~z,vl, dh[z,VI)
Assume that the control inputs U are generated in the controller through a discrete time scheme with step tl+ - tl = 8!11(1), 8(1 )(1) > 0,
(2.11) have [17-21] the following values and saddle points:
1
1= 1,2, ...; t l = t~zl. (2.6)
The continuous disturbance Ll~I[tl and the piecewise continuous disturbance Ll..,[t] satisfy the conditions:
= Ix:[t] -
s:'[tl!'
.! ! f[z"](t/,u[z,v),vlz,v],
0))
along motion y[tJ (2.1).
jLltJ[tJI
max
,,(dul·.-j ,dhl•.•j) ~(dvl·.-I,dhl·,.I)
upper derivative [12-14J of functional
d:t,·)
=
J4.:t"J
=
d:t"I(tz, s:' [tl!, x:[t" en, (2.12)
= J4.:t"I(du[:t,U1, dh[:t,uJjt"
x..,[t]l ~ C[:t!,
s:'[t,], x:[t/,
en,
E{IILltl[t,e]llc} = = E{lIx:[t, e] - x..,[t, ellld ~ ~ 8[:t) (I, x.., [tl! , x:[t,n, 8[:tI (I, x",[tl!, x: [tl!) > 0, tl ~ t ~ tl+ h (2.7)
ILlw[tjl ~ CI!>I,
E{ILl",[tjl} ~ o[!>], 8[!» > 0 - const. (2.8)
t4',.j = = J4.z,·J(du[z,u], dh[Z'U!ltt, s:'[tl!, x:[t"
Here E{ ... } is the mathematical expectation. Let the triplet of histories
{x",[t/, eJ, z",[t/, e], x:[t/, en 152
en,
III (2.10), (2.11) M and N are the sets of admissible values for the variables u, v, h. We assume that che process
And the measures Il(du[z,u l, dh[z,u); t), t, < t < t'+1 are assigned on the basis of
{x",[t], z..,[t], x:[t]}
additional considerations on the desired quality of the process {x",[tj, z..,[t]}.
is generated in such a way that for all the anticipated situations the following inequality is true:
Thus, the bias y",[t] is determined by equation:
(~"J(t" s:'[t,], x:[t" e])- dZ"I(t" S:'[t,], x:[t" e]) ~ et> 0 -
o
y",[t] = / d"G[:tI(t, 19) . y",[t + 19]+ -et
·Is:'[t,jl '
-h
o
const. (2.16)
+ BI:t!(t) . / d"U.(t, 19) . x:[t, + 19]+
In the controller U at the time t = t l a probabilistic ~ takes place which is to choose the optimal inputs U",[t,] [oJ, htl[t,jIO] with probabilities determined by measure IlbZ " ] (dul:t,uI, dh[:t,uJlt" s:'[t,], x:[t" e]).
-h
+ R[:tl(t, y",[t, e], z",[t, e]) + F!:tl(t, z..,[t, e])- Flzj (t, z",[t, e]) + L'l",[tJo
- B[zl(t) . / d"U.(t, 19) . z",[t + 19]+ -h
Let U~,UI[tl = u~,ul[t,j[OI
+ j[:t"I(t, u~,ul[t,jlol, vt'UI[t],
=
x",[t - h~,UI[t,j[Oll, x",[t - ht,U1[tJ])-
u~,u)[t" x:[t" e], S:'[t,]],
ht,UI[t] =
- J / (jlz"I(t" ut,u!, vt,ul,
h~,ul[t,j[Ol =
h~,uJ[t" x:[t" el, s:'[td], t, < t < tHI, 1= 1,2, ... ; t l = t~zl. (2.17)
M[·)/il·j
x:[t,- ht,UI],x:[t, - ht,ul])x x ll(du 1z ,u1, dh[z,u!; t)·
Simultaneously, the disturbances vt,uI[t], h~,uJ[t] arrive. These are determined by an admissible probability ~ f/ = f/(dv, dhluqt). In particular, v~,ul[t], h~,ul[t] can be deterministic functions of time for t, < t < tHI. It is assumed that the pair of controls u~,uJ[t], h~,ul[tJ is stochasticalIy independent of the pair of disturbances v~,uj[t], ht,ul[tj. The random disturbances L'lt![t] = x:[t]- x",[t] and L'l",[t] are generated by the environment independently of other circumstances. The controls for the z-model abilistic measures ll(du1z ,ul, dh[z,u l; t)
. f/bz"I(dv[z,ul, dh1z,ullt" s:'[t,j, x:[t" e])) (2.19)
Let V+(t, y",[t, e]) be the averaged right upper derivative which is an infinitesimal generator [5, 9] for the functional V(t, y",[t, e]) along motion y",[t] on the stochastic process {x",[tj, z..,[t], x:[t]}
{y",[t], z",[t], x:[t]} :
the prob-
( V+(t, y",[t, =
and
e]))
= (2.19)
Hm sup(E{V(r, y",[r, e])lt, y",[t, e]}-
~-+I+O
-V(t,y",[t,en)/(r-t) (2.20)
v(dv1z,UJ, dh[z,ul; t) - are assigned by the controller U at times t = t,. Here we assume f/(dv1z,u 1, dh[z,uI; t) =
Here E{... I...} denotes the conditional mathematical expectation.
= f/~"J(dvlz,ul, dh[z,u]!t" s:'[t,j, x:[t" e]),
The realized triplet of histories
t, ~ t < t'+I' (2.18)
{y",[t" e], z",[t" el, x:[t" e]} 153
shall determine the further realizations y..,[t, e], which extend this triplet, and the value
3. THE STABLE TRACKING OF X-MOTION BY Z··MOTION
E{V+ (t, y..,[t, e]) It" y..,[t" e], z..,[t" e], x:[t" en = = E{ (V+(t, y..,[t, e))) It t, y.,[t" el,
Suppose differential equation (2.19) and functional V(t, y.,[t, e]) (2.2) satisfy the following conditions. Choose in (2.6) and (2.7) the following estimates:
1
(2. 19
Z.,[t" e], x:[t" e]), (2.21 )
Using the estimates (1.12)-(1.16) of proximity between the parameters of equations and the phase variables on one hand and their information images on the other, and also the estimates (2.6)-(2.8) for the disturbances Ll.,[t], Ll!;![t] and the step Ll[tl = t'+ 1 -t" while assuming the values of /j > 0 in these estimates to be sufficiently small, we come, in view of the properties of the operators I( e), to the next estimates:
E{V+(t, y..,[t, eDltt, y..,[t" e], Z.,[t" e], x:[t" en ~ ~ C[I] . /j[II(I) + c[z! . II Ll!:l [t, e]II~,
I Ll!:][t, e]ll~ =
sup(IILl!:I[t,e]11 ' t , ~ t ~ tt+d, c lll > 0 , c z > 0 - const , t, ~ t ~ tl+1> (2.22)
?
E{V+(t, y.,[t, e])\t" y..,[t" en = . = E z:.!I...I......{I..·I
{E{\\(t, y..,[t, e])lt" y.,[tl> e], z..,[t" e], X:[t" en}
(2.23) C[IJ . /j[II(I)+ +c[z! . /j[z1 (I, x.., [td, x: [tl))' c!t] > 0, C!zl > 0 - const, tt ~ t < t'+ 1 for realizations {y..,[ttl, z..,[td, x:[td} which satisfy conditions: ~
z",[td E ?i['l, lIy..,[tj + "lIl c < Hlrl,
/j[II(I) =
<[IJ T'
/j[z} /j['!(I, X.,lt,], x:[td) = /-1' 'Y
> 0, 0111 > 0, /j1.1 > 0 - const. (3.1)
Suppose that by time t = t, the histories of motion z..,[t" e] and y..,[t" ej which satisfy conditions (2.24) and (2.25) have realized. Consider realizations Z.,[t, e] and y..,[t, e] which extend these realizations z.,[tl> e] and y.,[tl> ej for t, ~ t ~ t,-H , We assume that inputs u!;'''![ttl , h!;·..I[ttl , of the xplant and inputs '1~ •• l(dv[···J, dh[···qtj, s:'[tt], x:[tl> e])
of the z-model are generated in the controller U optimally on the basis of the s0lutions to games (2.10), (2.11). Suppose in addition that input J.l(du[··..I, dh[··..l; t), t, ~ t ~ tt+i> of the z-model may be generated in the controller U such that condition z..,[tt+d E ?i 1•1 is guaranteed. And let the following inequality be true for the functional Y[t, y.,[t, e]) (2.2)on the set of realizations z.,[t, e] and y.,[t, e]:
E{V+(t, y.,[t, eDlt/ y.,[t" en ~ /j[I}
/jlz}
I1 < - C[II . - 1 + C • . - 11 ' 1 = 1"2 ... , 'Y > 0, C!II > 0, C[·I > 0, /j[11 > 0, /j['] > 0 - const, t, ~ t < tl+ I '
(3.2)
In addition, we assume that conditions mentioned above to be true for some fixed pair {?i~'l, H?l} imply these conditions to be true for all pairs {?i~'I, HUtI}, 0 <
(2.24)
-h ~ " ~ 0, (2.25)
H[rl ~ H!r].
where ?i 1•1 is a bounded domain selected in advance; H!rl > 0 is a number selected in advance.
The following statement is true. 154
Lemma 1. Let conditions mentioned above on equation (2.19) and functional
V(t, y",[t,
en
formed due to the discrete-time scheme of the above with step (3.1) and generated by initial history V",(t., e), z",(t., e), which would satisfy condition
(2.2), be satisfied for some pair {1£~zl, H?I}.
z",[t.]
E 1£(zl, lIy",(t.,e)lIc ~ 8.,
Let the initial history z",[t., ej satisfy condition (2.24), where 1£[zl = 1£~zl for t, = t1
= t •.
would satisfy condition P(ly",[tJl ~
Then for any £ > 0, £ < Hr~l, 13 < 1 there exist a o. > 0, o[tl > 0 and a o(zJ > 0, such that any motion y",[t] (2.19), formed due to the discrete time scheme of the above, with step (3.1) and generated by initial history y",(t., e) that satisfies inequality
lIy",(t., e)lI c
~
0.,
(3.3)
~ £, t. ~
t < 00) > {3.
(3.4)
t < 00) > {3.
E{IIAtJ[tlll c } ~ 8[zl . Ix",[t] 1
2
(3.8)
,
8[zl > 0 - const. (3.9)
In many cases it is possible on the basis of an appropriate Lyapunov function v(t, y) = y'. A1oJ(t) . Y to construct a functional V(t, y[t, (2.2) of specific type
en
Suppose conditions for equation (2.19) and for functional V(t, y",[t, (2.2) considered in Section 2 and leading to estimate (2.25),(that is to estimate (3.2) for sufficiently small estimates of 0 in (1.12)(1.16), (2.6)-(2.8)), to be fulfilled. Then estimate (2.25) and Lemma 1 yield the following assertion.
en
V(t, y(e))
= V(t, (y(19), -h ~ 19 ~ 0)) =
= y'(O) . A[OI(t) . y(O)+ o
+
Jy(19) .
A[31(t,
19) . y(19) . d19, (3.10)
-h
which satisfies conditions (2.3)-(2.5). For such functional (3.10) we have
Theorem 2. Suppose the control inputs of the x-plant u~,ul[t,](oJ, h~,uJ[tl](O] and the control inputs of the z-model 1Jhz,·J(dv[z,vJ, dh[z,v1Itl, S:'[tl], [tl,
= gradv(tIV(t, y[t, e]) = 2y'[tJ· AIOI(t).
s[t]
(3.11 ) And constraint (2.7) with estimate (3.1) could be substituted by estimate
x: en
E{IIAtl[tlllc} ~ o[zl ·jx:[t,ll.
are generated optimally on the basis of solutions to small games (2.10), (2.11). Assume under conditions (2.24), (2.25) that it is possible to generate control inputs J.I(du1z,ul, dh[z,u]; t), t, ~ t < t'+ ll of the zmodel which ensure condition z",[t,+d E 1£[zl.
£, t. ~
Remark 1. A similar result may obtained if constraint (2.7) with estimate (3.1) is replaced by the next one:
satisfies the condition P(ly",[tJl
(3.6) (3.7)
(3.12)
The constraint on step A(t1(1) = t'+l-t" at which tHI - tl ~ 0 as t ~ 00, is essential. Indeed, consider equation .
(3.5)
1
y[t] = -y[t]+2·y[t-l]+ludtj- vdtJl- 1 + tt2
Then it is possible to choose sufficiently small estimates 8[·1 > 0 in conditions (1.12)-(1.16), (2.8) such that for any £ > o £ < H(~I, (3 < 1 there exist such 8ft ] > 0 and o(zl > 0 in estimates (3.1) and such 8. > 0 that any solution y",[t] (2.19)
+ U2
(3.13)
= {U\IJ = 1; tt\21 = -I}, VI = {v[11 = 1; V[2] = -I}, 1tl21 ~ 0.2, 1V21 ~ 0.1,
(3.14)
under constraints ttl
155
where y, ui' vi' j = 1,2 are scalars.
=
For the functional 1
V(y(.)) = "2 . y2(0)+ 1
+ 4.
f (1J + 2) . y2(1J) . d1J, D
(3.15)
-I
which satisfies conditions (2.3), (2.4) we come, due to the equation,
lilt] = -y[t] + ! . y[t - 1]
(3.16)
2
to the inequality .
(V+(y[t, ·]))(3.16)
1
2
•
(3.17) And due to equation (3.13) for optimal controls u we come to the inequality
E{ (v+(y",[t, .])) 1
(3.13) 2 L"
Indeed, by direct calculations based on standard formulas for conditional and total probability it could be checked that condition
P(ly",[tll ::; c, t. ::; t ::; t,.) > f3
-4 ·lly[t, ·lII L"
::;
This is insufficient to prove the desired stability of motion y[tJ 0 due to probability measure, on the half-line t, ~ t < 00, no matter how small are lIy[t., -lIl e and 6[IJ > O. In fact, the desired stability does not exist.
does not hold for small c as I' ~ 00 whatever be the positive number f3 < 1 selected in advance and whatever small are 6[IJ > 0 and 6, > 0, which restrict the initial history
ily",(t., .) lie ::; 6,.
Iy",[t".n ::;
::; E{ -4 . 11 y",[t, ·lII + +y",[t]· (lul",[t,j[DI - VI",[tJ!- 1+ +u2loI[t,jID\ + U2loI[t])ly",[t".n, t, ::; t < tl+ l •
Various methods to construct appropriate quadratic Lyapunov functionals for problems of stability and stabilization with after-effect are known [10-16]. effective construction of such functionals requires to take in account the specific conditions for the considered system. At the same time, in many cases the following computational procedure is satisfactory. This procedure is realized under the assumption that the asymptotic stability of the considered system is somehow established. For an actual time t' :::: to the actual history y[t',.] = y(.) is taken to be fixed. This history initiates a motion y[t] constructed by numerical integration of the corresponding equation of perturbed motion on a large time interval t' ::; t ::; t' + T. Using the variable Iy[t]l, the value of the functional V (t, y(.)) is approximately computed according to the equality
Here the value ([DJ of the corresponding small game and the saddle point {pD, q'}; D1 , v~DI are as follows: --
uk
2
2
= (L L IUlil -
vlill'
.plil~~~tl~_ -1.1 . !y",[t,jl) = -0.1 . !y",[t,jl,
PlilD
=!. 2'
qUID =! 2' i = 1 2' J' = 1 2 Dr uk = -0.2 . sign y",[t,] , v~DI = = 0.1 . sign y",[t,j. I)
,
,
(3.19) For constant step tl+ 1 - t, the inequality
E{(V+(y",[t, .])) (3.13)
1
= 6[tl we obtain
Iy",[t".n ::;
2
::; -4 .lIy",[t,-]II L "
-
0.1 . !y",[t,] I+
I"+T
V(t',y(.))
+ Iy",[t] - y",[t,jl· e[ll ::; ~
2 -41 . 11 y",[t, ·lIlL,. -
(3.22)
4. THE COMPUTATIONAL SCHEME
(3.18)
(IDI(y",[t,])
(3.21)
=
f
lIy[t,.11~2' ·dt. (4.1)
I"
0.1 . Iy",[t,]I +
Thus, when solving the problem of stabilization by using Lyapunov functional,
+ e[11 . LII ] . 6[11. (3.20) 156
one may obtain the value of the funcfor any appearing histional V (t, y[t*, tory y[t*, e]. Moreover, for the gradient gT(ut~t{t.] V( t, y[t*, the partial derivatives 8V(t, y[t*,e]) /8Yi[t*j , i = 1, ... , n which define this gradient can be obtained as follows. For a chosen history y[t*, e] a corrupted history (yli][t* + t?], -h ~ t? ~ 0) = (y[t* + t?], -h ~ t? < 0, y[t*J + ~[i]) is constructed, where ~[il is the vector 6,[il = (~~i) =I- 0; ~~i] = 0, i =I- j). For a fixed index i of initial history (yli] [to + t?j, -h ~ t? ~ 0) one may use a numerical procedure to get the value of functional V(yli][t* + t?j, -h ~ t? ~ 0) which is constructed due to equality (4.1) by through history (y[t* + t?], -h ~ t? ~ 0) by corrupted history (ylil[t* + t?], -h ~ t? ~ 0). The value obtained this way
en en
+ t?], -h ~ t? ~ 0)- V(y[t* + t?], -h ~ t? ~ 0)) / ~!iJ
(V(YliJ[t*
represents precisely an approximate value of the corresponding partial derivative i = 1, ... ,no 8V(t, y[t*,.)) /8yi[t*j ,
REFERENCES [1] Kurzhanski A.B. Control and Obeervation under Uncertainty. Nauka, Moscow, 1977 (in Russian). [2] Osipov Yu.S., Kryazhimskii A.V. Inverse problem of ordinary differential equations. Dynamical solutions. Gordon and Breach, 1995 [3J Subbotin A.I., Chent80V A.G. Optimization of Guarantees in Problems of Control. Nauka, Moscow, 1981 (in Russian). [4] Kleimyonov A.F. Nonantogonistic Positional Differential Games. Nauka, Russian Academy of Sciences (Ural Branch), Ekaterinburg, 1993 (in Russian). [5J Krasovskii N.N., Subbotin A.I. Game-Theoretical Control Problems. Springer-Verlag, 1988. [6J Kats LYa. The Method of Lyapunov FUnctions in Problems of Stability and Stabilization of Systems With Random Structure. Ekaterinburg, 1998 (in Russian). [7J Kolmanovm V.B., Nosov V.R. Stability and Periodic Modes of Controllable Systems with Post-Effect. Nauka, Moscow, 1981 (in Russian).
[8] Kushner H.J. Stochastic stability and control. Academic Press, New York, 1967. [9J Khasminski R.Z. Stability of System of Differential Equations under Random Perturbations of their Parameters. Nauka, Moscow, 1969 (in Russian). [10] Kim A.V. i-Smooth analysis and functional differential equations. Russian Academy of Sciences (Ural Branch), Ekaterinburg, 1996 (In Russian). [11] Razumikhin B.S. On the stability of systems with a delay / / PrikI. Mat. Meh. 1956. V.20. P.500-512. [12] Hale J.K. Theory of Functional Differential Equations. Springer-Verlag, New York-Heidelberg-Berlin, 1977. [13} Yoshizawa T. Stability theory by Liapunov's Second Method. Math.Soc. Japan, 1966. [14J Krasovski N.N. Some Problems of the Theory of Stability of Motion. Gostechizdat, Moscow, 1959 (in Russian). [15J Osipov Yu.S. On stabilization of control systems with time-delay / / Differential Equations. 1965. V.I. P.605~18. [16J Shimanov S.N. On theory of linear differential equations with post-effect / / Differential Equations. 1965. V.l, NI. P.I02-116. [17} von Neumann J., Morgenstern O. Theory of games and economic behavior. Princeton, 1944 (I 00.), 1947 (2 00.). [18J GnOOenko B.V. The theory of probability, 4th edition. Nauka, Moscow, 1988; English transl. of 3rd edition, Chelsea, New York, 1984. [19] Kantorovich L.V., Akilov G.P. FUnctional Analysis. Pergamon Press, Oxford, second edition, 1982. [20] Parthasarathy T., Raghavan T. Some Topics in Two-Person Games, Vol.22 of Modern Analytic and Computational Methods in Science and Mathematics. American Elsevier, New York,197I. [21] Fan K. Minimax theorems / / Proc. Nat. Acad. Sci. USA. 1953. V.39, NI. P.42-47.
157
ON A PROBLEM OF STOCHASTIC STABILIZATION A.N. Kotel'nikova, N.N. Krasovski Inst. of MathematiC3 and Mechanics of RAS, Ekaterinburg, Russia E-mail: [email protected]
Abstract: Considered here is the problem of stable tracking and leading the motion of a controlled plant with after-effect, under uncertainty or conflict [1-5]. This is done through a computer model in the closed feedback loop. The basis for solving this problem is Lyapunov's theory of stability of motion modified for hereditary and stochastic systems [6-16]. A probabilistic algorithm of control within a discrete-time scheme is provided.
1. STATEMENT OF THE PROBLEM
z",[t]
+ The motion of the x-plant is described by the differential equation
en
x",[t] = FI%I(t, x",[t, + BI%I(t)· u~"J[tl+ + fl%"l(t, ut,ui[t], vt,VI[t], x",[t - h~,"l[t]], x",[t - h~,uJ[t]])
+ A.,[t], (1.1)
= Flzl(t, z..,[t, en + Blzl(t) . u~"l[t]+
J Jf[·,·l(t, ulz
,,,!, vlz,v l, x:[t - hl.,..l],
Mt-] NI-]
x:[t - hl.,vID x xjL(du{·,"I, dh[·,ul j t) . f)(dv l,.uJ , dhl•.v1j t)
(1.2)
to + h ~ t~%l ~ t~'1 ~ t < 00, h> 0 - const, z={.z;,i=I, ... ,n}, z[t, e] = (z[t + flj: - h ~ fl ~ 0).
to + h ~ t~%J ~ t < 00, h > 0 - const, x = {xi,i = 1, ... ,n}, x[t, ej = (x[t + 19]: - h ~ fl ~ 0).
The motion of the z-model is described by the differential equation
The index w denotes an elementary event; x, z are vector columns. The time to and the constant h > 0 are unconditionally fixed. The initial instants of time t~%I, t~'1 ~ t.[x] determine a certain motion x[t], z[t]; x[t, e], z[t, ej are the continuous histories of
the motion. These histories are estimated by the following nonns: IIw(e)lIc
=
to the feedback principle using the Dext infonnation image being realized: Y
max Iw(19)1,
-h~~~O
= {x·(e),z(e)}jY[t] = {x'[t, e). z[t,e]},
1
II w (e)IIL2' = (IW(O)\2
Iwl=
t.
+1,W(19) ,2 d19) 2
(tw;)t, W=X,z.
(1.3)
1;:;1
u[x,u] E Ulx,u l, vlz,v] E V(I,V], ulz,u! E Ull,u], vlx,v] E V[x,vI, (1.4)
<
00.
(1.6)
Here the continuous function x·[t,e] is the corrupted result of observation for the actual history x[t, ej :
x·[t,e] Operators F(e), f(e) are uniformly continuous in all variables, with the history variable estimated by norm IIw[e]lIc (1.3). These operators uniformly map closed bounded sets of variables into bounded sets of images and satisfy the Lipschitz condition in the history variables. The matrix functions B(t) = {bij }, i = 1, ... , 1Ij j = 1, ... , m are unifonnly continuous and bounded for t E [to, (0). The following constraints are given:
~ t
= x[t, e] + ilIx][t, e].
(1.7)
The disturbances vlx,v], hlx,.I, il, il lx] are endogeneous. Operators FI
hlx,u] E Hlx,u!, hlx,v! E Hlx,v!,
hI' ,vi E H["v!, hII,u! E HI',u], (1.5)
o
where U, V, H are compacts of finite dimensions.
yw[t] =
J~Glx](t, 19) . yw[t, e]+ -h
+ R1XI(t, yw[t, e). zw[t,e]) + F1x](t, zw[t, e])- F[I](t, zw[t, e]) + BIx](t) . u~,·I[t]- B['](t) . ut..l[t] + .£lw[t]+
In (1.2) tl( ej t), ,.,( e; t) are probability measures on the sets of admissible values for u[<,u l, vl"vl, hl',u], hlz,vl. These measures depend on time t in such a way that there exists a solution z,.,[t]. The external disturbance il[t] is uniformly bounded and piecewise continuous. Symbol 0 further stands for the vector with zero coordinates.
fIx'·!(t, u~,uJ[t). vt,v![t). xw[t - h~,uI[tlJ, xw[t - h~,v![t]])-
J J fl<"I(t, u[·,uI, V(I,V!, x:[t - hl<,ul). M!'] NI'!
x:[t - h[I,vID x Xtl(du(I,U j , dh("u]; t) . ,.,(dv["v], dhl"v]j t)
The motion x[t] should be stably tracked by the z-motion. This z-motion could be used as a guide [5] in order to ensure the desired quality of x-motion.
(1.8) to
Inputs u[x,.j ulz,.] ulx,u] ul"ul v["vl hlx,u! hl',u], hl<,v] ~e ge~erated in the co~trolle; U within a discrete-time scheme with small step t, ~ t ~ t'+ll I = 1,2, ... , due
+ h ~ t~x]
~ t~'1 ~ t < 00, h> 0 - const,
where Glx](t, 19) is a bounded matrix function with bounded variation, unifonn in 150
t E [to,oo) and -h ~ 19 ~ O. A long as
{x",[t], z..,[t], x:[t]} may be visualized as an assembly of realizations which corresponds to the set n of simple ~ w.
motions remain in the bounded domain
Ixl
~
Izl ~ H·,
H·,
H· - const, (1.9) the operator from (1.8) defined by the Stieltjes integral is uniformly continuous in t E [to, 00) for a fixed history y[t, e].
2. THE FORMATION OF THE PROCESS We suppose that the linearized information image of equation (1.8) is stabilized [12-16] in the form o ri[t] = d,a[z!(t, 19) . y[t + 19]+
The parameters of equation (1.1) are represented by means of their information images:
a 12OI (t,19) -h
-7
~
J
a[zl(t, 19); B(2OI(t) + B[zJ(t); f[2O,o!(e) + f12O,.I(e) (1.10) 19
~
0,
~ t
to
-h
< 00.
+ B[zl(t) . u[zl[t]
=
o
Jd,G[zJ(t, 19) . y[t + 19]+ + B[z)(t) . Jd"U.(t, 19) . y[t + 19], =
The histories x[t, e], z[t, ej satisfy the Lipschitz conditions
-h
o
jw[t + 19[1 1] - w[t + 19[2!] 1 ~ L[UlI
.1 19[1) _
19[2]
I'
(2.1)
-h
- h ~ 19[;1 ~ 0, t ~ t~z), w = x, z (1.11)
~ 15(01,
(1.12)
IIBI2OI(t) - B[zl(t)1I ~ c5[BI,
(1.13)
where Uo(t,19) is a uniformly bounded in t E [to, 00) matrix function with uniformly bounded variation for -h ~ 19 ~ 0; the second operator in the right-hand side of (2.1) is uniformly continuous in t E [to, 00) for a fixed history y[t, e] from domain (1.9). It is also assumed that the resulting asymptotic stability of the unperturbed motion y[t] == 0, tfl ~ t < 00 relative to the perturbed motions y[t] (2.1) is uniform with respect to the initial time t~zJ and the phase variable y.
(1.14)
A Lyapunov functional V(t,y(e)) [10, 12, 14, 15) is constructed:
as long as the motions remain in domain (1.9). At the same time, the following conditions are true: o
Jd,(Gl2OJ(t, 19) - G[zJ(t, 19)) . y[t + 19] ~ -h
IRi2O!(t, y[t,
e], z[t, en! ~
~ c5 lRJ ·lIy[t, eHle,
V(t, y(e)) = V(t, (y(19), -h ~ 19 ~ 0)) = y'(O) . A[OI(t) . y(O)+ o +y'(0) . A[IJ(t, 19) . y(19) . d19+
If[2O,OI(e)1 ~ D[/I, D[/1 > 0 - const, If[2O,·I(e) - f12O,01(e)1 ~ 15[/·1
(1.15)
=
J
(1.16)
o
-h 0
JJy'(r) . (t, r, 19) . y(19) . dr . d19+ + Jy'(19) . (t, 19) . y(19) . d19,
under the same values of the independent variables in operators f[2O,ol(e) and f12O,.I(e), 15 are small positive constants.
+
A[21
-h·-h
o
A[3J
The probabilistic proc~ combining xmotion and z-motion is generated by di&crete scheme (in time t). Thus the details concerning the stochastic differential equations are not actual here. And the process
-h
(2.2) t~zl ~ t < 00.
Here the prime denotes transposition. 151
The functional (2.2) satisfies the conditions:
19 ~ 0)) ~ w.(Iy(O)I), w.(r) > 0, r > 0; w.(O) = 0; W.(r2) > w.(rd, r2 > rh (2.3)
be realized. Denote s~[t] = groo.w[l] V(t,
~
V(t, (y(19), -h
~ 19 ~ 0)) ~ w·(lIy(O)IIL,.l, w'(r) > 0, r > 0; w'(O)
s:'[t]
The small games [5J min
l'(du1z •• I,dhlz.-I)
)(2.1)
max
~(dvlz.·j,dhlz"J)
s:'[t,j·
.! ! f!:t,-I(t/,u[:t,v],v[:t,v
= 0;
W'(r2) > w'(rd, r2 > ri, (2.4)
en
en,
y:[t] = x:[t] - Zw[tJ. (2.9)
V(t, (y(19) , -h
(V+(t, y[t,
y",[t,
= groo.~[I] V(t, y:[t, en,
l,
MI-JNlzl
X:[tl - h[:t,v)], X:[tl - hl:t,v]n x x ~(du[:t,vl, dhl:t,v]) . !)(dv[:t,v] , dh[:t,v)) ,
~ - lIy[t,elll~,., (2.5)
(2.10)
where w.(r), w'(r) are continuous functions, (V+(t, y[t, denotes the right
en)
min
(2.1)
V(t, (y(19) , -h
~
19
~
MI·)NI·j
x:[t/ - hiz,u]], x:[t, - hIz,vl]) x
x ~(du[z,uJ, dhlz,u)) . !)(dv~z,vl, dh[z,VI)
Assume that the control inputs U are generated in the controller through a discrete time scheme with step tl+ - tl = 8!11(1), 8(1 )(1) > 0,
(2.11) have [17-21] the following values and saddle points:
1
1= 1,2, ...; t l = t~zl. (2.6)
The continuous disturbance Ll~I[tl and the piecewise continuous disturbance Ll..,[t] satisfy the conditions:
= Ix:[t] -
s:'[tl!'
.! ! f[z"](t/,u[z,v),vlz,v],
0))
along motion y[tJ (2.1).
jLltJ[tJI
max
,,(dul·.-j ,dhl•.•j) ~(dvl·.-I,dhl·,.I)
upper derivative [12-14J of functional
d:t,·)
=
J4.:t"J
=
d:t"I(tz, s:' [tl!, x:[t" en, (2.12)
= J4.:t"I(du[:t,U1, dh[:t,uJjt"
x..,[t]l ~ C[:t!,
s:'[t,], x:[t/,
en,
E{IILltl[t,e]llc} = = E{lIx:[t, e] - x..,[t, ellld ~ ~ 8[:t) (I, x.., [tl! , x:[t,n, 8[:tI (I, x",[tl!, x: [tl!) > 0, tl ~ t ~ tl+ h (2.7)
ILlw[tjl ~ CI!>I,
E{ILl",[tjl} ~ o[!>], 8[!» > 0 - const. (2.8)
t4',.j = = J4.z,·J(du[z,u], dh[Z'U!ltt, s:'[tl!, x:[t"
Here E{ ... } is the mathematical expectation. Let the triplet of histories
{x",[t/, eJ, z",[t/, e], x:[t/, en 152
en,
III (2.10), (2.11) M and N are the sets of admissible values for the variables u, v, h. We assume that che process
And the measures Il(du[z,u l, dh[z,u); t), t, < t < t'+1 are assigned on the basis of
{x",[t], z..,[t], x:[t]}
additional considerations on the desired quality of the process {x",[tj, z..,[t]}.
is generated in such a way that for all the anticipated situations the following inequality is true:
Thus, the bias y",[t] is determined by equation:
(~"J(t" s:'[t,], x:[t" e])- dZ"I(t" S:'[t,], x:[t" e]) ~ et> 0 -
o
y",[t] = / d"G[:tI(t, 19) . y",[t + 19]+ -et
·Is:'[t,jl '
-h
o
const. (2.16)
+ BI:t!(t) . / d"U.(t, 19) . x:[t, + 19]+
In the controller U at the time t = t l a probabilistic ~ takes place which is to choose the optimal inputs U",[t,] [oJ, htl[t,jIO] with probabilities determined by measure IlbZ " ] (dul:t,uI, dh[:t,uJlt" s:'[t,], x:[t" e]).
-h
+ R[:tl(t, y",[t, e], z",[t, e]) + F!:tl(t, z..,[t, e])- Flzj (t, z",[t, e]) + L'l",[tJo
- B[zl(t) . / d"U.(t, 19) . z",[t + 19]+ -h
Let U~,UI[tl = u~,ul[t,j[OI
+ j[:t"I(t, u~,ul[t,jlol, vt'UI[t],
=
x",[t - h~,UI[t,j[Oll, x",[t - ht,U1[tJ])-
u~,u)[t" x:[t" e], S:'[t,]],
ht,UI[t] =
- J / (jlz"I(t" ut,u!, vt,ul,
h~,ul[t,j[Ol =
h~,uJ[t" x:[t" el, s:'[td], t, < t < tHI, 1= 1,2, ... ; t l = t~zl. (2.17)
M[·)/il·j
x:[t,- ht,UI],x:[t, - ht,ul])x x ll(du 1z ,u1, dh[z,u!; t)·
Simultaneously, the disturbances vt,uI[t], h~,uJ[t] arrive. These are determined by an admissible probability ~ f/ = f/(dv, dhluqt). In particular, v~,ul[t], h~,ul[t] can be deterministic functions of time for t, < t < tHI. It is assumed that the pair of controls u~,uJ[t], h~,ul[tJ is stochasticalIy independent of the pair of disturbances v~,uj[t], ht,ul[tj. The random disturbances L'lt![t] = x:[t]- x",[t] and L'l",[t] are generated by the environment independently of other circumstances. The controls for the z-model abilistic measures ll(du1z ,ul, dh[z,u l; t)
. f/bz"I(dv[z,ul, dh1z,ullt" s:'[t,j, x:[t" e])) (2.19)
Let V+(t, y",[t, e]) be the averaged right upper derivative which is an infinitesimal generator [5, 9] for the functional V(t, y",[t, e]) along motion y",[t] on the stochastic process {x",[tj, z..,[t], x:[t]}
{y",[t], z",[t], x:[t]} :
the prob-
( V+(t, y",[t, =
and
e]))
= (2.19)
Hm sup(E{V(r, y",[r, e])lt, y",[t, e]}-
~-+I+O
-V(t,y",[t,en)/(r-t) (2.20)
v(dv1z,UJ, dh[z,ul; t) - are assigned by the controller U at times t = t,. Here we assume f/(dv1z,u 1, dh[z,uI; t) =
Here E{... I...} denotes the conditional mathematical expectation.
= f/~"J(dvlz,ul, dh[z,u]!t" s:'[t,j, x:[t" e]),
The realized triplet of histories
t, ~ t < t'+I' (2.18)
{y",[t" e], z",[t" el, x:[t" e]} 153
shall determine the further realizations y..,[t, e], which extend this triplet, and the value
3. THE STABLE TRACKING OF X-MOTION BY Z··MOTION
E{V+ (t, y..,[t, e]) It" y..,[t" e], z..,[t" e], x:[t" en = = E{ (V+(t, y..,[t, e))) It t, y.,[t" el,
Suppose differential equation (2.19) and functional V(t, y.,[t, e]) (2.2) satisfy the following conditions. Choose in (2.6) and (2.7) the following estimates:
1
(2. 19
Z.,[t" e], x:[t" e]), (2.21 )
Using the estimates (1.12)-(1.16) of proximity between the parameters of equations and the phase variables on one hand and their information images on the other, and also the estimates (2.6)-(2.8) for the disturbances Ll.,[t], Ll!;![t] and the step Ll[tl = t'+ 1 -t" while assuming the values of /j > 0 in these estimates to be sufficiently small, we come, in view of the properties of the operators I( e), to the next estimates:
E{V+(t, y..,[t, eDltt, y..,[t" e], Z.,[t" e], x:[t" en ~ ~ C[I] . /j[II(I) + c[z! . II Ll!:l [t, e]II~,
I Ll!:][t, e]ll~ =
sup(IILl!:I[t,e]11 ' t , ~ t ~ tt+d, c lll > 0 , c z > 0 - const , t, ~ t ~ tl+1> (2.22)
?
E{V+(t, y.,[t, e])\t" y..,[t" en = . = E z:.!I...I......{I..·I
{E{\\(t, y..,[t, e])lt" y.,[tl> e], z..,[t" e], X:[t" en}
(2.23) C[IJ . /j[II(I)+ +c[z! . /j[z1 (I, x.., [td, x: [tl))' c!t] > 0, C!zl > 0 - const, tt ~ t < t'+ 1 for realizations {y..,[ttl, z..,[td, x:[td} which satisfy conditions: ~
z",[td E ?i['l, lIy..,[tj + "lIl c < Hlrl,
/j[II(I) =
<[IJ T'
/j[z} /j['!(I, X.,lt,], x:[td) = /-1' 'Y
> 0, 0111 > 0, /j1.1 > 0 - const. (3.1)
Suppose that by time t = t, the histories of motion z..,[t" e] and y..,[t" ej which satisfy conditions (2.24) and (2.25) have realized. Consider realizations Z.,[t, e] and y..,[t, e] which extend these realizations z.,[tl> e] and y.,[tl> ej for t, ~ t ~ t,-H , We assume that inputs u!;'''![ttl , h!;·..I[ttl , of the xplant and inputs '1~ •• l(dv[···J, dh[···qtj, s:'[tt], x:[tl> e])
of the z-model are generated in the controller U optimally on the basis of the s0lutions to games (2.10), (2.11). Suppose in addition that input J.l(du[··..I, dh[··..l; t), t, ~ t ~ tt+i> of the z-model may be generated in the controller U such that condition z..,[tt+d E ?i 1•1 is guaranteed. And let the following inequality be true for the functional Y[t, y.,[t, e]) (2.2)on the set of realizations z.,[t, e] and y.,[t, e]:
E{V+(t, y.,[t, eDlt/ y.,[t" en ~ /j[I}
/jlz}
I1 < - C[II . - 1 + C • . - 11 ' 1 = 1"2 ... , 'Y > 0, C!II > 0, C[·I > 0, /j[11 > 0, /j['] > 0 - const, t, ~ t < tl+ I '
(3.2)
In addition, we assume that conditions mentioned above to be true for some fixed pair {?i~'l, H?l} imply these conditions to be true for all pairs {?i~'I, HUtI}, 0 <
(2.24)
-h ~ " ~ 0, (2.25)
H[rl ~ H!r].
where ?i 1•1 is a bounded domain selected in advance; H!rl > 0 is a number selected in advance.
The following statement is true. 154
Lemma 1. Let conditions mentioned above on equation (2.19) and functional
V(t, y",[t,
en
formed due to the discrete-time scheme of the above with step (3.1) and generated by initial history V",(t., e), z",(t., e), which would satisfy condition
(2.2), be satisfied for some pair {1£~zl, H?I}.
z",[t.]
E 1£(zl, lIy",(t.,e)lIc ~ 8.,
Let the initial history z",[t., ej satisfy condition (2.24), where 1£[zl = 1£~zl for t, = t1
= t •.
would satisfy condition P(ly",[tJl ~
Then for any £ > 0, £ < Hr~l, 13 < 1 there exist a o. > 0, o[tl > 0 and a o(zJ > 0, such that any motion y",[t] (2.19), formed due to the discrete time scheme of the above, with step (3.1) and generated by initial history y",(t., e) that satisfies inequality
lIy",(t., e)lI c
~
0.,
(3.3)
~ £, t. ~
t < 00) > {3.
(3.4)
t < 00) > {3.
E{IIAtJ[tlll c } ~ 8[zl . Ix",[t] 1
2
(3.8)
,
8[zl > 0 - const. (3.9)
In many cases it is possible on the basis of an appropriate Lyapunov function v(t, y) = y'. A1oJ(t) . Y to construct a functional V(t, y[t, (2.2) of specific type
en
Suppose conditions for equation (2.19) and for functional V(t, y",[t, (2.2) considered in Section 2 and leading to estimate (2.25),(that is to estimate (3.2) for sufficiently small estimates of 0 in (1.12)(1.16), (2.6)-(2.8)), to be fulfilled. Then estimate (2.25) and Lemma 1 yield the following assertion.
en
V(t, y(e))
= V(t, (y(19), -h ~ 19 ~ 0)) =
= y'(O) . A[OI(t) . y(O)+ o
+
Jy(19) .
A[31(t,
19) . y(19) . d19, (3.10)
-h
which satisfies conditions (2.3)-(2.5). For such functional (3.10) we have
Theorem 2. Suppose the control inputs of the x-plant u~,ul[t,](oJ, h~,uJ[tl](O] and the control inputs of the z-model 1Jhz,·J(dv[z,vJ, dh[z,v1Itl, S:'[tl], [tl,
= gradv(tIV(t, y[t, e]) = 2y'[tJ· AIOI(t).
s[t]
(3.11 ) And constraint (2.7) with estimate (3.1) could be substituted by estimate
x: en
E{IIAtl[tlllc} ~ o[zl ·jx:[t,ll.
are generated optimally on the basis of solutions to small games (2.10), (2.11). Assume under conditions (2.24), (2.25) that it is possible to generate control inputs J.I(du1z,ul, dh[z,u]; t), t, ~ t < t'+ ll of the zmodel which ensure condition z",[t,+d E 1£[zl.
£, t. ~
Remark 1. A similar result may obtained if constraint (2.7) with estimate (3.1) is replaced by the next one:
satisfies the condition P(ly",[tJl
(3.6) (3.7)
(3.12)
The constraint on step A(t1(1) = t'+l-t" at which tHI - tl ~ 0 as t ~ 00, is essential. Indeed, consider equation .
(3.5)
1
y[t] = -y[t]+2·y[t-l]+ludtj- vdtJl- 1 + tt2
Then it is possible to choose sufficiently small estimates 8[·1 > 0 in conditions (1.12)-(1.16), (2.8) such that for any £ > o £ < H(~I, (3 < 1 there exist such 8ft ] > 0 and o(zl > 0 in estimates (3.1) and such 8. > 0 that any solution y",[t] (2.19)
+ U2
(3.13)
= {U\IJ = 1; tt\21 = -I}, VI = {v[11 = 1; V[2] = -I}, 1tl21 ~ 0.2, 1V21 ~ 0.1,
(3.14)
under constraints ttl
155
where y, ui' vi' j = 1,2 are scalars.
=
For the functional 1
V(y(.)) = "2 . y2(0)+ 1
+ 4.
f (1J + 2) . y2(1J) . d1J, D
(3.15)
-I
which satisfies conditions (2.3), (2.4) we come, due to the equation,
lilt] = -y[t] + ! . y[t - 1]
(3.16)
2
to the inequality .
(V+(y[t, ·]))(3.16)
1
2
•
(3.17) And due to equation (3.13) for optimal controls u we come to the inequality
E{ (v+(y",[t, .])) 1
(3.13) 2 L"
Indeed, by direct calculations based on standard formulas for conditional and total probability it could be checked that condition
P(ly",[tll ::; c, t. ::; t ::; t,.) > f3
-4 ·lly[t, ·lII L"
::;
This is insufficient to prove the desired stability of motion y[tJ 0 due to probability measure, on the half-line t, ~ t < 00, no matter how small are lIy[t., -lIl e and 6[IJ > O. In fact, the desired stability does not exist.
does not hold for small c as I' ~ 00 whatever be the positive number f3 < 1 selected in advance and whatever small are 6[IJ > 0 and 6, > 0, which restrict the initial history
ily",(t., .) lie ::; 6,.
Iy",[t".n ::;
::; E{ -4 . 11 y",[t, ·lII + +y",[t]· (lul",[t,j[DI - VI",[tJ!- 1+ +u2loI[t,jID\ + U2loI[t])ly",[t".n, t, ::; t < tl+ l •
Various methods to construct appropriate quadratic Lyapunov functionals for problems of stability and stabilization with after-effect are known [10-16]. effective construction of such functionals requires to take in account the specific conditions for the considered system. At the same time, in many cases the following computational procedure is satisfactory. This procedure is realized under the assumption that the asymptotic stability of the considered system is somehow established. For an actual time t' :::: to the actual history y[t',.] = y(.) is taken to be fixed. This history initiates a motion y[t] constructed by numerical integration of the corresponding equation of perturbed motion on a large time interval t' ::; t ::; t' + T. Using the variable Iy[t]l, the value of the functional V (t, y(.)) is approximately computed according to the equality
Here the value ([DJ of the corresponding small game and the saddle point {pD, q'}; D1 , v~DI are as follows: --
uk
2
2
= (L L IUlil -
vlill'
.plil~~~tl~_ -1.1 . !y",[t,jl) = -0.1 . !y",[t,jl,
PlilD
=!. 2'
qUID =! 2' i = 1 2' J' = 1 2 Dr uk = -0.2 . sign y",[t,] , v~DI = = 0.1 . sign y",[t,j. I)
,
,
(3.19) For constant step tl+ 1 - t, the inequality
E{(V+(y",[t, .])) (3.13)
1
= 6[tl we obtain
Iy",[t".n ::;
2
::; -4 .lIy",[t,-]II L "
-
0.1 . !y",[t,] I+
I"+T
V(t',y(.))
+ Iy",[t] - y",[t,jl· e[ll ::; ~
2 -41 . 11 y",[t, ·lIlL,. -
(3.22)
4. THE COMPUTATIONAL SCHEME
(3.18)
(IDI(y",[t,])
(3.21)
=
f
lIy[t,.11~2' ·dt. (4.1)
I"
0.1 . Iy",[t,]I +
Thus, when solving the problem of stabilization by using Lyapunov functional,
+ e[11 . LII ] . 6[11. (3.20) 156
one may obtain the value of the funcfor any appearing histional V (t, y[t*, tory y[t*, e]. Moreover, for the gradient gT(ut~t{t.] V( t, y[t*, the partial derivatives 8V(t, y[t*,e]) /8Yi[t*j , i = 1, ... , n which define this gradient can be obtained as follows. For a chosen history y[t*, e] a corrupted history (yli][t* + t?], -h ~ t? ~ 0) = (y[t* + t?], -h ~ t? < 0, y[t*J + ~[i]) is constructed, where ~[il is the vector 6,[il = (~~i) =I- 0; ~~i] = 0, i =I- j). For a fixed index i of initial history (yli] [to + t?j, -h ~ t? ~ 0) one may use a numerical procedure to get the value of functional V(yli][t* + t?j, -h ~ t? ~ 0) which is constructed due to equality (4.1) by through history (y[t* + t?], -h ~ t? ~ 0) by corrupted history (ylil[t* + t?], -h ~ t? ~ 0). The value obtained this way
en en
+ t?], -h ~ t? ~ 0)- V(y[t* + t?], -h ~ t? ~ 0)) / ~!iJ
(V(YliJ[t*
represents precisely an approximate value of the corresponding partial derivative i = 1, ... ,no 8V(t, y[t*,.)) /8yi[t*j ,
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