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On the Estimation Problem for One Class of Nonlinear Systems
Copyright © IFAC Nonsmooth and Discontinuous Problems of Control and Optimization, Chelyabinsk, Russia, 1998
ON THE ESTIMATION PROBLEM FOR ONE CLASS OF NONLINEAR SYSTEMS Kayumov R.I.
*
*
Ural State University, Lenina st., 51, Ekaterinburg, 620083, Russia, E-mail: [email protected]
Abstract: This paper considers the estimation problem for the state of one class of nonlinear multistage and continuous systems under uncertainty and with the quantities observed described by linear equations. It is assumed that the system's initial state and the disturbances in measurement equations are inexact, and that the allowable a priori information on these two is limited to their range of variation. The solution of the estimation problem relies on describing the evolution of the system's state information sets which are consistent with both the measurement results and the a priori restrictions on uncertain quantities. An information set consists of all trajectories that are possible in the system and which, together with certain admissible disturbances in the observation, determine the given realization of the observed signal. A cross-section, at some time-instant, of information sets generated by the signal observation prior to that instant, is a multiple-valued analog of the system's current phase condition. Information sets provide guaranteed estimates of the system's state based on measurement results. Copyright @1998 IFAC Keywords: differential inclusions, dynamic system, state estimation, attainability set, nonlinear systems, uncertain system.
1. INTRODUCTION
2. A SINGLE-STAGE SYSTEM Consider a single-stage process where the transition of state is described by equation
In this paper, the solutions of determinate estimation problems for one class of nonlinear multistage systems with prior restrictions rest on approximation procedures that are based on optimizing the solutions of specially constructed problems with quadratic limitations. The same approach was used in works (Kurzhanski, 1980; Kurzhanski, 1981; Koscheev and Kurzhanski, 1983). Also, a transition from a nonlinear multistage system to a nonlinear continuous system is considered. For continuous nonlinear systems under uncertainty, a parameterized family of differential inclusions with no restrictions on the system's phase condition is introduced. The intersection of the attainability domains for the differential inclusions over the variety of all functional parameters yields the solution to the original problem (Kayumov, 1994).
Xl
= Xo + 6f(to, xo),
(1)
where X is an n-dimensional system state vector, 15 - a certain positive number, f(t, x) - a given mapping from RI X Rn into Rn. The initial state Xo is subject to constraint Xo E X o ,
(2)
where X o is a compact in Rn. The measurement device equation is given by (3) where y is an m-dimensional output signal, G I a matrix of correspondent dimensions with a full rank, 6 - a noise subject to constraint
6 E:=:, 121
(4)
where =: is a convex compact in Rm. Let, by virtue of system (1), (3), Yl be the realized signal.
3. A MULTISTAGE SYSTEM Consider the following multistage process described by equations
Definition. Informational domain (Kurzhanski, 1977) Xl of the states of system (1) that are compatible with signal Yl under restrictions (2), (4) is defined as the set of such and only such vectors x E Rn that for each of them there exists a pair Xo E X o, E =: such that the solution to the system (1), (3), found under Xo = xo, 6 = ~l' satisfies the conditions Xl = X, Yl = Yl'
Xi
€l
°
i
If(t, x) [ ~ k l (t) [xl + k 2 (t),
Consider the following modified system. Let the pair {xo, 6} be fixed, and the actual realization generated not by the original system, but by the modified system
where =(ti) are convex compacts in
Definition. Informational domain X(s, t, M) is defined as the set of such and only such vectors x E Rn that, on step s, the trajectories of system (12), induced by the condition Xt = X, constructed on interval It, s], and generating the realized signal Yt,s, under constraints
(7)
where No and L l are matrices of appropriate dimensions, and also No > 0, L l > (positively defined matrices). In restriction (7) the number J1, can be chosen in such a way that the domain compatible with signal Yl by virtue of the modified system (5), (6) is not empty.
°
i
l
From this definition one easily obtains
Lemma. For arbitrary s, t, r (s
+ 8f(to, zo))+
+K1lG"[ Ll(Yl - 6),
= t + 1, ... , s,
- pass through these vectors.
The informational domain of the modified system is an ellipsoid. Its center is given by = K 1 812 No(zo
Rm .
In what follows, the symbol Yt,s = {Yt, Yt+l, ... ,Ys} will be used to denote the set of signals received by virtue of equation (13) for steps indexed t, t + 1, ... , s.
Yl=G l Zl +6+Wl' (6) where Zo E X o, ho and Wl are additional fictitious noises subject to quadratic restriction
Zl
(13)
(14)
(5)
(ho,Noh o) + (wl,LlWl) ~ jJ,z,
= 1,2, ... , 2k ,
where Y is an m-dimensional output signal, G i known matrices of appropriate dimension having a full rank. Disturbances ~i are subject to geometrical restrictions
where kl (t) and k 2 (t) are functions summable on any finite time interval.
= Zo + 8f(to, zo) + 8ho,
(12)
where x is an n-dimensional system state vector, 8 - partition step of segment [to, 'l9], f(t, x) - a given mapping from RI X Rn into Rn. The initial state Xo is not known in advance but is subject to geometrical restriction (2). Information on the system's state is provided by a measuring device described by equation
Assumption 1. f(t,x) is a continuous in (t,x) function satisfying the Lipschitz condition in X with constant K > and the solution extendibility condition
Zl
= Xi-l + 8f(ti-l,Xi-d, i = 1,2, ... , 2\
~
t
~
r) equality
X(s,t,M) = X(s,r,X(r,t,M)).
(8)
(15)
is fulfilled.
where
1
K l = 82 No
T
+ Gl
This equality means that the information sets possess semigroup properties. A description of the information sets' X(s, 0, X o) (0 ~ s ~ 2k ) evolution is then required. Though unknown, the system's actual condition on step s by virtue of the information set's definition is in the set X(s,O,X o). This means that the proposed estimation procedure for the unknown system state under uncertainty is "guaranteed".
LlGl ·
Denote D l to be the attainability domain of system (1), namely
D l = {did = Zo
+ 8f(to,zo), Zo E X o}.
(9)
Consider the sum of the centers of ellipsoids over all Zo E X o and 6 E
=:
z?
0(,...,) -1 1 H = K l 82NoDl+
Zl
+K1lG"[ Ll(Yl - =), where D = {No,Ld.
Denote X(s,O,X o) = X s , s E [0,2 k ]. Consider a modified system
(10)
= Zi-l + 8f(ti-l,Zi-l) + 8h i- l = GiZi + ~i + Wi, i = 1,2, ... , 2k .
Zi
Theorem. The following equality is fulfilled Xl
= n{Zp(D)jD},
(16)
Yi (17) Let, in what follows, the pair {xo, 0 be fixed. Here the following denotation is used ~ = {6, ..., {zk }.
(11)
where D = {En,€Em },€ > 0. 122
tube is a generalization of the notion of an Euler polygon for ordinary differential equations. Let us do the following conversion
Noises hi- 1 and Wi are on every step subject to the following quadratic restriction
(hi-l,Ni-lhi-l) + (wi,LiWi) :::; k
i = 1,2, ... , 2
/-l;,
K i- 1 = 62 (Ni-=-ll
(18)
,
+84 Ni-=-ll
where N i - 1 and L i are positively defined matrices of appropriate dimensions.
°
where K i
(A
(19)
X(Z?_l
1
+6
+ 6j(ti-l,Z?_1))+
When assumptions 1, 2 and conditions (2) and (23) are true, the properties of R-solutions to differential inclusions (Panasyuk and Panasyuk, 1977; Tolstonogov, 1986) yield the following proposition:
(20)
zg -0 = {nZ °I(Nj-1,L j ) = (En,cjEm),cj > 0, Zi i
where
j = 1, ... , i}.
Theorem. There exists an R-solution to differential inclusion
(21)
Assumption 2. The mapping j (t, x) is homeomor-
iO
phic.
= Xi,
i
j(t, zo) - N- 1 (t)G T (t)L(t) x
= 0, ... , 2 k .
(22)
Let t E [to, 'l9], G i - values at instants to + i6 of a continuous matrix G(t), ~i - values at instants to + i8 of function ~(t), which at every instant is subject to geometric restriction
';(t) E 3(t),
z(to) E Xo.
(27)
The solution is defined on the entire interval [to,'l9]. For every t E [to,'l9] an R-solution is an attainability set of inclusion (27) at instant t from the initial set XO. In Hausdorff metric, an R solution can be approximated by a polygon tube of inclusion (24).
Theorem. Under assumptions 1 and 2 the following equation is fulfilled: -0
E
x(G(t)zO - y(t) + 2(t)),
The following assertion is true:
Zi
= A-I - 8A- 1 BA- 1+
x (G(t + 8)zO(t) - y(t + 8) + 2(t + 8))}+
612 N i-I X
+Ki-1G; Li(Yi - ~i), Z?_l E Z?_l' ~i E 2(ti)}, = X o. Denote
+ 6B)-1
ZO(t) + 8{f(t, zO(t)) - N-1(t)G T (t + 8)L(t + 8)x
Consider the following recurrent procedure. Denote Z? to be the following set i
(25)
which is true for an invertible matrix A, an arbitrary matrix B, and sufficiently small 6. The righthand side of inclusion (24) takes the following form
= bNi-1 + G; LiG i .
°= K-
2
+8 2 A-I B(A + 6B)-1 BA- 1,
° ))
Z io = {ol Zi zi
G; LiGi(Ni- 1 + 8 G; LiGi)-l x
The above conversion uses the equality
Zi - K i-1 812 Ni-dzi_l + 6j(ti-l,zi_1 + +Ki-1G; Li(fj; - ~i),
62 Ni-=-ll G; LiGiNi-=-ll +
xG; LiGiNi-=-ll)'
On every step, the domains that are compatible with the realized signal by virtue of system (16), (17) are ellipsoids. An ellipsoid's center satisfies relation
°_
-
4. A CONTINUOUS SYSTEM
(23) Let the system's dynamics be expressed by equation (28) x = j(t,x), t E T = [to, 'l9],
where 2(t) is a convex compact in Rm, varying continuously in t. Let us presuppose Ni = 6N(to+ i6), L i = L(to + i8), where N(t) and L(t) are matrices of correspondent dimensions continuous on the interval [to, 'l9].
where x is an n-dimensional system state vector. Assumption 1 is true of j(t, x). The initial state x( to) = xO satisfies restriction (2). The measurement equation is
Consider the following inclusion 1
zO(t + 6) E zO(t) + {K-1(t + 6)JN(t)(zO(t)+
y
= G(t)x +~,
(29)
where y is an m-dimensional vector, G(t) - a continuous matrix of appropriate dimensions. Functions ~(t) that represent disturbances are measurable. The noises are subject to geometric qualification (23). Let measurement y*(t) have realized on closed interval T by virtue of system (28) (29).
+8j(t,zO(t)) + K-1(t + 8)G T (t + 8)L(t + 6)x x (y(t + 8) - 3(t + 8)) - zO(t)}, (24) t = to, to + 6, ... , to + (2 k -1)6. The assembly of all polygonal curves whose legs' ends satisfy inclusion (24) at instant t+8 defines a polygonal tube whose cross-section at instant t is set Z? A polygonal 123
Definition. Informational domain X (1J, y* (.» = X (1J, .) of the states of system (28) at time-instant 1J that are consistent with function y*(t) measured on closed interval T, is defined as the set of all vectors x E Rn such that for each of them, at instant 1J, a trajectory of system (28) inducing, with noise ~(t) satisfying condition (23), by virtue of equation (29) a realization y(t) such that y(t) = y*(t) almost everywhere on T, - passes through it. A description of informational sets X(1J,·) provides guaranteed estimates for the system's unknown actual phase states, and no other information on the actual state can possibly be derived. It is therefore only natural to accept informational sets as the system's current set-valued phase state.
5. EXAMPLE
Consider a nonlinear system of the following type
Consider the following parameterized family of differential inclusions with no restrictions on the system's phase state i E f(t, z) - N(t)G T (t)L(t)x
0,
= xi,
t ET
The initial state x(O) =
Xo
= [0, 1].
(34)
is subject to condition
xo E XO = {x : Ixl ~ I}.
(35)
The measurement equation is given by y
= X2 +~.
(36)
Noise in the measurement device is subject to geometric qualification ~(t)
E 2 = {~: I~I ~ I}.
(37)
Let signal y*(t) == 0, t E T have realized (e.g. with the initial condition being xO = (0,0) T and the noise being C (t) == 0, t ET).
Requirement 1. Suppose there exist matrix functions N(t) and L(t) such that differential inclusion dz/dT E -N(t)G T (t)L(t) x
The compatible domain is given by the inequality
viI - xr(l) ~ x2(1) ~ 1.
xi(l) -
x(G(t)z - y(t) - 2(t», (31) where T 2: 0 and t is a parameter, has an asymptotically stable set
The attainability domain Z (1, c) for differential inclusion
(:J
o E -N(t)GT(t)L(t)x
E [( zr _ is given by
x (G(t)z - y(t) - 2(t».
(32) Inclusion (31) is analogous to the so-called adjoined system for ordinary differential equations containing a small parameter multiplying the derivative, and inclusion (32) - to a confluent equation (Tikhonov, 1952; Vasilieva and Butuzov, 1978). Denote f! = {N(·), L(·)}.
~Z2
E
c) ;(zr _ COZ + c)] 2
-
(38)
zl(l) = z?, 1) ((z?)2 _ c) e + e- ozg ~ z2(1) ~ -c (-0 1) ~((z?)2+e)e +e-ozg. -e (
-0
Letting e --+ 0, one gets set Zl
Theorem. The following equality is fulfilled
= n{D(1J, f!)If!},
=
Xl { X2
x(G(t)z - y(t) + 2(t», (30) where matrices N(t) and L(t) are positively defined for all t. Elements of matrices N(t) and L(t) are summable on T. The initial condition is z (to) = zo E Xo. A Caratheodory solution to differential inclusion (30) is defined as an absolutely continuous function z(t) defined on closed interval T and almost everywhere satisfying inclusion (30).
X(1J,·)
general case, this extension becomes possible if requirement 1 is verified (Kayumov, 1994). Requirement 1 will be verified if, for example, the product of matrices -N(t)G T (t)L(t)G(t) is a diagonal matrix with identical negative eigenvalues or a diagonal matrix such that some of its eigenvalues are negative and equal and the rest of them identically equal zero when t E T. Because one is free to choose the form of matrices N(t) and L(t), this can be done, for instance, for coefficient matrices G(t) invertible for every t E T or for a m x n constant matrix G = G(t), t E T, with rank m.
zl(l) = z?, (33)
Letting c --+
where D(1J, f!) is the attainability set at instant t = 1J of differential inclusion (30) with y(t) = y' (t).
00,
zl(l)
z2(1) = (z?)2
+ zg.
(39)
one gets set Z2
= z?,
-1 ~ z2(1) ~ 1.
(40)
In view of (39) and (40) one observes that
Mention should be made of the fact that though, as demonstrated in (Donchev, 1987), the fundamental Tikhonov theorem (Tikhonov, 1952) can not be extended to differential inclusions in the
zl(l)
= z?,
zf(l) -
)1- zr(l) ~ z2(1) ~ 1,
i.e.
X(I,·) 124
= n{Z(I,e)le > O}.
6. CONCLUSION
Vasilieva, A.B. and V.F. Butuzov (1978). Singularly disturbed equations in critical cases. Moscow Univ. Press. Moscow (in Russian).
In this paper, exact solutions of the initial nonlinear multistage and continuous systems are obtained. The employed analytical methods for the description of the said sets (proposed in (Kurzhanski and Filippova, 1986; Kurzhanski and Filippova, 1987) enable the development of computational procedure algorithms for the construction of these sets.
7. REFERENCES Donchev, A. (1987). Optimal control systems: Perturbations, approximations, and sensitivity analysis. Mir. Moskow (in Russian). Filippov, A.F. (1979). Stability for differential equations with discontinuous and multiplevalued right-hand sides. Differentsialnye Uravneniya 15(6), 1018-1027. Kayumov, R.I. (1994). Exact solution of the problem of estimation for one class of the nonlinear systems. Izvestia RAN, Tehnicheskaya Kibemetika (2), 177-182. Koscheev, A.S. and A.B. Kurzhanski (1983). Adaptive estimation of multistage systems' evolution under uncertainty. Izvestia AN SSSR, Tehnicheskaya Kibemetika (3), 72-93. Kurzhanski, A.B. (1977). Control and observation under conditions of uncertainty. Nauka. Moskow (in Russian). Kurzhanski, A.B. (1980). Dynamic control system estimation under uncertainty conditions, part i. Problems of Control and Information 9(6), 395-406. Kurzhanski, A.B. (1981). Dynamic control system estimation under uncertainty conditions, part ii. Problems of Control and Information 10(1), 33-42. Kurzhanski, A.B. and T.F. Filippova (1986). On the description of the set of viable trajectories of a differential inclusion. Doklady AN SSSR 289(1),38-41. Kurzhanski, A.B. and T.F. Filippova (1987). On the description of the set of viable trajectories of a control system. Differentsialnye Uravneniya 23(1), 1303-1315. Panasyuk, A.I. and V.I. Panasyuk (1977). Asymptotic optimization of non-linear control systems. Belorussian Univ. Press. Minsk (in Russian) . Tikhonov, A.N. (1952). Systems of differential equations containing a small parameter multiplying the derivatives. Matematicheskiy Sbomik 31(3), 575-586. Tolstonogov, A.A. (1986). Differential inclusions in Banach space. Nauka. Novosibirsk (in Russian). 125
ON THE ESTIMATION PROBLEM FOR ONE CLASS OF NONLINEAR SYSTEMS Kayumov R.I.
*
*
Ural State University, Lenina st., 51, Ekaterinburg, 620083, Russia, E-mail: [email protected]
Abstract: This paper considers the estimation problem for the state of one class of nonlinear multistage and continuous systems under uncertainty and with the quantities observed described by linear equations. It is assumed that the system's initial state and the disturbances in measurement equations are inexact, and that the allowable a priori information on these two is limited to their range of variation. The solution of the estimation problem relies on describing the evolution of the system's state information sets which are consistent with both the measurement results and the a priori restrictions on uncertain quantities. An information set consists of all trajectories that are possible in the system and which, together with certain admissible disturbances in the observation, determine the given realization of the observed signal. A cross-section, at some time-instant, of information sets generated by the signal observation prior to that instant, is a multiple-valued analog of the system's current phase condition. Information sets provide guaranteed estimates of the system's state based on measurement results. Copyright @1998 IFAC Keywords: differential inclusions, dynamic system, state estimation, attainability set, nonlinear systems, uncertain system.
1. INTRODUCTION
2. A SINGLE-STAGE SYSTEM Consider a single-stage process where the transition of state is described by equation
In this paper, the solutions of determinate estimation problems for one class of nonlinear multistage systems with prior restrictions rest on approximation procedures that are based on optimizing the solutions of specially constructed problems with quadratic limitations. The same approach was used in works (Kurzhanski, 1980; Kurzhanski, 1981; Koscheev and Kurzhanski, 1983). Also, a transition from a nonlinear multistage system to a nonlinear continuous system is considered. For continuous nonlinear systems under uncertainty, a parameterized family of differential inclusions with no restrictions on the system's phase condition is introduced. The intersection of the attainability domains for the differential inclusions over the variety of all functional parameters yields the solution to the original problem (Kayumov, 1994).
Xl
= Xo + 6f(to, xo),
(1)
where X is an n-dimensional system state vector, 15 - a certain positive number, f(t, x) - a given mapping from RI X Rn into Rn. The initial state Xo is subject to constraint Xo E X o ,
(2)
where X o is a compact in Rn. The measurement device equation is given by (3) where y is an m-dimensional output signal, G I a matrix of correspondent dimensions with a full rank, 6 - a noise subject to constraint
6 E:=:, 121
(4)
where =: is a convex compact in Rm. Let, by virtue of system (1), (3), Yl be the realized signal.
3. A MULTISTAGE SYSTEM Consider the following multistage process described by equations
Definition. Informational domain (Kurzhanski, 1977) Xl of the states of system (1) that are compatible with signal Yl under restrictions (2), (4) is defined as the set of such and only such vectors x E Rn that for each of them there exists a pair Xo E X o, E =: such that the solution to the system (1), (3), found under Xo = xo, 6 = ~l' satisfies the conditions Xl = X, Yl = Yl'
Xi
€l
°
i
If(t, x) [ ~ k l (t) [xl + k 2 (t),
Consider the following modified system. Let the pair {xo, 6} be fixed, and the actual realization generated not by the original system, but by the modified system
where =(ti) are convex compacts in
Definition. Informational domain X(s, t, M) is defined as the set of such and only such vectors x E Rn that, on step s, the trajectories of system (12), induced by the condition Xt = X, constructed on interval It, s], and generating the realized signal Yt,s, under constraints
(7)
where No and L l are matrices of appropriate dimensions, and also No > 0, L l > (positively defined matrices). In restriction (7) the number J1, can be chosen in such a way that the domain compatible with signal Yl by virtue of the modified system (5), (6) is not empty.
°
i
l
From this definition one easily obtains
Lemma. For arbitrary s, t, r (s
+ 8f(to, zo))+
+K1lG"[ Ll(Yl - 6),
= t + 1, ... , s,
- pass through these vectors.
The informational domain of the modified system is an ellipsoid. Its center is given by = K 1 812 No(zo
Rm .
In what follows, the symbol Yt,s = {Yt, Yt+l, ... ,Ys} will be used to denote the set of signals received by virtue of equation (13) for steps indexed t, t + 1, ... , s.
Yl=G l Zl +6+Wl' (6) where Zo E X o, ho and Wl are additional fictitious noises subject to quadratic restriction
Zl
(13)
(14)
(5)
(ho,Noh o) + (wl,LlWl) ~ jJ,z,
= 1,2, ... , 2k ,
where Y is an m-dimensional output signal, G i known matrices of appropriate dimension having a full rank. Disturbances ~i are subject to geometrical restrictions
where kl (t) and k 2 (t) are functions summable on any finite time interval.
= Zo + 8f(to, zo) + 8ho,
(12)
where x is an n-dimensional system state vector, 8 - partition step of segment [to, 'l9], f(t, x) - a given mapping from RI X Rn into Rn. The initial state Xo is not known in advance but is subject to geometrical restriction (2). Information on the system's state is provided by a measuring device described by equation
Assumption 1. f(t,x) is a continuous in (t,x) function satisfying the Lipschitz condition in X with constant K > and the solution extendibility condition
Zl
= Xi-l + 8f(ti-l,Xi-d, i = 1,2, ... , 2\
~
t
~
r) equality
X(s,t,M) = X(s,r,X(r,t,M)).
(8)
(15)
is fulfilled.
where
1
K l = 82 No
T
+ Gl
This equality means that the information sets possess semigroup properties. A description of the information sets' X(s, 0, X o) (0 ~ s ~ 2k ) evolution is then required. Though unknown, the system's actual condition on step s by virtue of the information set's definition is in the set X(s,O,X o). This means that the proposed estimation procedure for the unknown system state under uncertainty is "guaranteed".
LlGl ·
Denote D l to be the attainability domain of system (1), namely
D l = {did = Zo
+ 8f(to,zo), Zo E X o}.
(9)
Consider the sum of the centers of ellipsoids over all Zo E X o and 6 E
=:
z?
0(,...,) -1 1 H = K l 82NoDl+
Zl
+K1lG"[ Ll(Yl - =), where D = {No,Ld.
Denote X(s,O,X o) = X s , s E [0,2 k ]. Consider a modified system
(10)
= Zi-l + 8f(ti-l,Zi-l) + 8h i- l = GiZi + ~i + Wi, i = 1,2, ... , 2k .
Zi
Theorem. The following equality is fulfilled Xl
= n{Zp(D)jD},
(16)
Yi (17) Let, in what follows, the pair {xo, 0 be fixed. Here the following denotation is used ~ = {6, ..., {zk }.
(11)
where D = {En,€Em },€ > 0. 122
tube is a generalization of the notion of an Euler polygon for ordinary differential equations. Let us do the following conversion
Noises hi- 1 and Wi are on every step subject to the following quadratic restriction
(hi-l,Ni-lhi-l) + (wi,LiWi) :::; k
i = 1,2, ... , 2
/-l;,
K i- 1 = 62 (Ni-=-ll
(18)
,
+84 Ni-=-ll
where N i - 1 and L i are positively defined matrices of appropriate dimensions.
°
where K i
(A
(19)
X(Z?_l
1
+6
+ 6j(ti-l,Z?_1))+
When assumptions 1, 2 and conditions (2) and (23) are true, the properties of R-solutions to differential inclusions (Panasyuk and Panasyuk, 1977; Tolstonogov, 1986) yield the following proposition:
(20)
zg -0 = {nZ °I(Nj-1,L j ) = (En,cjEm),cj > 0, Zi i
where
j = 1, ... , i}.
Theorem. There exists an R-solution to differential inclusion
(21)
Assumption 2. The mapping j (t, x) is homeomor-
iO
phic.
= Xi,
i
j(t, zo) - N- 1 (t)G T (t)L(t) x
= 0, ... , 2 k .
(22)
Let t E [to, 'l9], G i - values at instants to + i6 of a continuous matrix G(t), ~i - values at instants to + i8 of function ~(t), which at every instant is subject to geometric restriction
';(t) E 3(t),
z(to) E Xo.
(27)
The solution is defined on the entire interval [to,'l9]. For every t E [to,'l9] an R-solution is an attainability set of inclusion (27) at instant t from the initial set XO. In Hausdorff metric, an R solution can be approximated by a polygon tube of inclusion (24).
Theorem. Under assumptions 1 and 2 the following equation is fulfilled: -0
E
x(G(t)zO - y(t) + 2(t)),
The following assertion is true:
Zi
= A-I - 8A- 1 BA- 1+
x (G(t + 8)zO(t) - y(t + 8) + 2(t + 8))}+
612 N i-I X
+Ki-1G; Li(Yi - ~i), Z?_l E Z?_l' ~i E 2(ti)}, = X o. Denote
+ 6B)-1
ZO(t) + 8{f(t, zO(t)) - N-1(t)G T (t + 8)L(t + 8)x
Consider the following recurrent procedure. Denote Z? to be the following set i
(25)
which is true for an invertible matrix A, an arbitrary matrix B, and sufficiently small 6. The righthand side of inclusion (24) takes the following form
= bNi-1 + G; LiG i .
°= K-
2
+8 2 A-I B(A + 6B)-1 BA- 1,
° ))
Z io = {ol Zi zi
G; LiGi(Ni- 1 + 8 G; LiGi)-l x
The above conversion uses the equality
Zi - K i-1 812 Ni-dzi_l + 6j(ti-l,zi_1 + +Ki-1G; Li(fj; - ~i),
62 Ni-=-ll G; LiGiNi-=-ll +
xG; LiGiNi-=-ll)'
On every step, the domains that are compatible with the realized signal by virtue of system (16), (17) are ellipsoids. An ellipsoid's center satisfies relation
°_
-
4. A CONTINUOUS SYSTEM
(23) Let the system's dynamics be expressed by equation (28) x = j(t,x), t E T = [to, 'l9],
where 2(t) is a convex compact in Rm, varying continuously in t. Let us presuppose Ni = 6N(to+ i6), L i = L(to + i8), where N(t) and L(t) are matrices of correspondent dimensions continuous on the interval [to, 'l9].
where x is an n-dimensional system state vector. Assumption 1 is true of j(t, x). The initial state x( to) = xO satisfies restriction (2). The measurement equation is
Consider the following inclusion 1
zO(t + 6) E zO(t) + {K-1(t + 6)JN(t)(zO(t)+
y
= G(t)x +~,
(29)
where y is an m-dimensional vector, G(t) - a continuous matrix of appropriate dimensions. Functions ~(t) that represent disturbances are measurable. The noises are subject to geometric qualification (23). Let measurement y*(t) have realized on closed interval T by virtue of system (28) (29).
+8j(t,zO(t)) + K-1(t + 8)G T (t + 8)L(t + 6)x x (y(t + 8) - 3(t + 8)) - zO(t)}, (24) t = to, to + 6, ... , to + (2 k -1)6. The assembly of all polygonal curves whose legs' ends satisfy inclusion (24) at instant t+8 defines a polygonal tube whose cross-section at instant t is set Z? A polygonal 123
Definition. Informational domain X (1J, y* (.» = X (1J, .) of the states of system (28) at time-instant 1J that are consistent with function y*(t) measured on closed interval T, is defined as the set of all vectors x E Rn such that for each of them, at instant 1J, a trajectory of system (28) inducing, with noise ~(t) satisfying condition (23), by virtue of equation (29) a realization y(t) such that y(t) = y*(t) almost everywhere on T, - passes through it. A description of informational sets X(1J,·) provides guaranteed estimates for the system's unknown actual phase states, and no other information on the actual state can possibly be derived. It is therefore only natural to accept informational sets as the system's current set-valued phase state.
5. EXAMPLE
Consider a nonlinear system of the following type
Consider the following parameterized family of differential inclusions with no restrictions on the system's phase state i E f(t, z) - N(t)G T (t)L(t)x
0,
= xi,
t ET
The initial state x(O) =
Xo
= [0, 1].
(34)
is subject to condition
xo E XO = {x : Ixl ~ I}.
(35)
The measurement equation is given by y
= X2 +~.
(36)
Noise in the measurement device is subject to geometric qualification ~(t)
E 2 = {~: I~I ~ I}.
(37)
Let signal y*(t) == 0, t E T have realized (e.g. with the initial condition being xO = (0,0) T and the noise being C (t) == 0, t ET).
Requirement 1. Suppose there exist matrix functions N(t) and L(t) such that differential inclusion dz/dT E -N(t)G T (t)L(t) x
The compatible domain is given by the inequality
viI - xr(l) ~ x2(1) ~ 1.
xi(l) -
x(G(t)z - y(t) - 2(t», (31) where T 2: 0 and t is a parameter, has an asymptotically stable set
The attainability domain Z (1, c) for differential inclusion
(:J
o E -N(t)GT(t)L(t)x
E [( zr _ is given by
x (G(t)z - y(t) - 2(t».
(32) Inclusion (31) is analogous to the so-called adjoined system for ordinary differential equations containing a small parameter multiplying the derivative, and inclusion (32) - to a confluent equation (Tikhonov, 1952; Vasilieva and Butuzov, 1978). Denote f! = {N(·), L(·)}.
~Z2
E
c) ;(zr _ COZ + c)] 2
-
(38)
zl(l) = z?, 1) ((z?)2 _ c) e + e- ozg ~ z2(1) ~ -c (-0 1) ~((z?)2+e)e +e-ozg. -e (
-0
Letting e --+ 0, one gets set Zl
Theorem. The following equality is fulfilled
= n{D(1J, f!)If!},
=
Xl { X2
x(G(t)z - y(t) + 2(t», (30) where matrices N(t) and L(t) are positively defined for all t. Elements of matrices N(t) and L(t) are summable on T. The initial condition is z (to) = zo E Xo. A Caratheodory solution to differential inclusion (30) is defined as an absolutely continuous function z(t) defined on closed interval T and almost everywhere satisfying inclusion (30).
X(1J,·)
general case, this extension becomes possible if requirement 1 is verified (Kayumov, 1994). Requirement 1 will be verified if, for example, the product of matrices -N(t)G T (t)L(t)G(t) is a diagonal matrix with identical negative eigenvalues or a diagonal matrix such that some of its eigenvalues are negative and equal and the rest of them identically equal zero when t E T. Because one is free to choose the form of matrices N(t) and L(t), this can be done, for instance, for coefficient matrices G(t) invertible for every t E T or for a m x n constant matrix G = G(t), t E T, with rank m.
zl(l) = z?, (33)
Letting c --+
where D(1J, f!) is the attainability set at instant t = 1J of differential inclusion (30) with y(t) = y' (t).
00,
zl(l)
z2(1) = (z?)2
+ zg.
(39)
one gets set Z2
= z?,
-1 ~ z2(1) ~ 1.
(40)
In view of (39) and (40) one observes that
Mention should be made of the fact that though, as demonstrated in (Donchev, 1987), the fundamental Tikhonov theorem (Tikhonov, 1952) can not be extended to differential inclusions in the
zl(l)
= z?,
zf(l) -
)1- zr(l) ~ z2(1) ~ 1,
i.e.
X(I,·) 124
= n{Z(I,e)le > O}.
6. CONCLUSION
Vasilieva, A.B. and V.F. Butuzov (1978). Singularly disturbed equations in critical cases. Moscow Univ. Press. Moscow (in Russian).
In this paper, exact solutions of the initial nonlinear multistage and continuous systems are obtained. The employed analytical methods for the description of the said sets (proposed in (Kurzhanski and Filippova, 1986; Kurzhanski and Filippova, 1987) enable the development of computational procedure algorithms for the construction of these sets.
7. REFERENCES Donchev, A. (1987). Optimal control systems: Perturbations, approximations, and sensitivity analysis. Mir. Moskow (in Russian). Filippov, A.F. (1979). Stability for differential equations with discontinuous and multiplevalued right-hand sides. Differentsialnye Uravneniya 15(6), 1018-1027. Kayumov, R.I. (1994). Exact solution of the problem of estimation for one class of the nonlinear systems. Izvestia RAN, Tehnicheskaya Kibemetika (2), 177-182. Koscheev, A.S. and A.B. Kurzhanski (1983). Adaptive estimation of multistage systems' evolution under uncertainty. Izvestia AN SSSR, Tehnicheskaya Kibemetika (3), 72-93. Kurzhanski, A.B. (1977). Control and observation under conditions of uncertainty. Nauka. Moskow (in Russian). Kurzhanski, A.B. (1980). Dynamic control system estimation under uncertainty conditions, part i. Problems of Control and Information 9(6), 395-406. Kurzhanski, A.B. (1981). Dynamic control system estimation under uncertainty conditions, part ii. Problems of Control and Information 10(1), 33-42. Kurzhanski, A.B. and T.F. Filippova (1986). On the description of the set of viable trajectories of a differential inclusion. Doklady AN SSSR 289(1),38-41. Kurzhanski, A.B. and T.F. Filippova (1987). On the description of the set of viable trajectories of a control system. Differentsialnye Uravneniya 23(1), 1303-1315. Panasyuk, A.I. and V.I. Panasyuk (1977). Asymptotic optimization of non-linear control systems. Belorussian Univ. Press. Minsk (in Russian) . Tikhonov, A.N. (1952). Systems of differential equations containing a small parameter multiplying the derivatives. Matematicheskiy Sbomik 31(3), 575-586. Tolstonogov, A.A. (1986). Differential inclusions in Banach space. Nauka. Novosibirsk (in Russian). 125