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Optimal control in unobservable integral Volterra systems
Journal of the Franklin Institute 339 (2002) 13–27
Optimal control in unobservable integral Volterra systems Michael V. Basin*, Irma R. Valadez Guzman Department of Physical and Mathematical Sciences, Autonomous University of Nuevo Leon, Adpo postal 144-F, C.P. 66450, San Nicolas de los Garza, Nuevo Leon, Mexico Received 22 May 2000; accepted 20 October 2001
Abstract This paper presents solution of the optimal linear-quadratic controller problem for unobservable integral Volterra systems with continuous/discontinuous states under deterministic uncertainties, over continuous/discontinuous observations. Due to the separation principle for integral systems, the initial continuous problem is split into the optimal minmax filtering problem for integral Volterra systems with deterministic uncertainties over continuous/discontinuous observations and the optimal linear-quadratic control (regulator) problem for observable deterministic integral Volterra systems with continuous/discontinuous states. As a result, the system of the optimal controller equations are obtained, including the linear equation for the optimally controlled minmax estimate and two Riccati equations for its ellipsoid matrix (optimal gain matrix of the filter) and the optimal regulator gain matrix. Then, in the discontinuous problems, the equation for the optimal controller and the equations for the optimal filter and regulator gain matrices are obtained using the filtering procedure for deriving the filtering equations over discontinuous observations proceeding from the known filtering equations over continuous ones and the dual results in the optimal control problem for integral systems. The technical example illustrating application of the obtained results is finally given. r 2002 The Franklin Institute. Published by Elsevier Science Ltd. All rights reserved. Keywords: Linear Volterra system; Deterministic uncertainties; Control; Filtering
*Corresponding author. Tel.: +52-8-329-4030; fax: +52-8-352-2954. E-mail address: [email protected] (M.V. Basin). 0016-0032/02/$22.00 r 2002 The Franklin Institute. Published by Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 6 - 0 0 3 2 ( 0 1 ) 0 0 0 5 4 - 0
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1. Introduction The optimal control and filtering problems for dynamic systems with delays, which represent a particular case of discontinuous integral systems (see [1] for substantiation), have been studied in a number of publications from various viewpoints (see, for example, [2–4] for dynamic systems and [5] for a particular case of integral Volterra ones). This attention is directly related to common use of dynamic systems with delays in global economy concepts [6], marketing models [7], technical systems [8], and others. Since the class of integral Volterra systems includes the class of retarded dynamic ones, studies of integral systems, such as solutions of the optimal LQR problem [9,10], minmax filtering problem [9,10], and the optimal controller (this paper), become a significant part of the control theory. Nevertheless, the integral Volterra systems have been of independent interest in the deterministic environment, as well as in the stochastic one (see [11]). This paper presents solution to the optimal linear–quadratic controller problem for unobservable integral Volterra systems with continuous/discontinuous states under deterministic uncertainties, over continuous/discontinuous observations. Due to the separation principle for integral systems, which is stated analogously to that for dynamic differential ones [12], the initial continuous problem is split into the optimal minmax filtering problem for integral Volterra systems with deterministic uncertainties over continuous observations and the optimal linear–quadratic control (regulator) problem for observable deterministic integral Volterra systems with continuous states. There are a number of papers investigating these problems for system state and observations given by differential equations [13,14] or bivariate Volterra ones [5], and the problems have been solved in the general case of integral governing equations in the previous authors’ papers [9,10]. Based on those results, the system of the optimal controller equations are obtained, including the linear equation for the optimally controlled minmax estimate and two Riccati equations for its ellipsoid matrix (optimal gain matrix of the filter) and the optimal regulator gain matrix. Note that it is possible to form closed systems of the minmax filtering and optimal regulator equations for an integral system state over integral observations, using only two filtering variables, the optimal estimate (state) and its ellipsoid matrix (gain matrix of the regulator), although the analogous result cannot be reached in stochastic systems (see [15,16]). Thus, the complete system of the obtained controller equations consists of only three variables, instead of six expected in the stochastic case. This encouraging result may be explained by the fact that deterministic uncertainties in the minmax filtering problem satisfy energy-type integral restrictions, i.e., belong to certain ellipsoids with fixed boundaries in functional spaces, and cannot take any possible values as stochastic Gaussian disturbances can. The optimal controller equations for integral systems with discontinuous states over discontinuous observations are then obtained using the filtering procedure [17,18] for deriving the filtering equations over discontinuous observations proceeding from the known filtering equations over continuous ones, which have already been obtained in the paper, and the dual results in the optimal control problem for integral systems [9,10]. In particular, the obtained results enable
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15
one to compute jumps of the optimal filtering and control parameters that can appear due to discontinuities in observations and states, respectively, and this is actually done in the technical example related to the process of missile launching. The paper is organized as follows. In Section 2, the optimal controller problem is stated and then solved for unobservable continuous integral Volterra systems. The separation principle for integral Volterra systems is substantiated there as well. Section 3 generalizes the mentioned results to discontinuous unobservable integral Volterra systems. Finally, Section 4 presents the technical example illustrating application of the obtained results to solution of the optimal controller problem of launching a missile with continuous and impulsive jet motors and unobservable velocity to the maximal possible altitude with the minimal fuel consumption.
2. Optimal controller for unobservable continuous integral systems 2.1. Problem statement Let us consider an unobservable linear system given by the state equation Z t Z t Z t xðtÞ ¼ xðt0 Þ þ Aðt; sÞxðsÞ ds þ Bðt; sÞuðt; sÞ ds þ Dðt; sÞrðt; sÞ ds t0
t0
and the output (observation) one Z t Z t yðtÞ ¼ Cðt; sÞxðsÞ ds þ Gðt; sÞqðt; sÞ ds; t0
ð1Þ
t0
ð2Þ
t0
where both state and observation equations are integral equations of the Volterra type, xðtÞARn is the unobservable system state, yðtÞARm is the observation vector, rðt; sÞARr is the input uncertainty, uðt; sÞARp is the input control, and qðt; sÞARm is the observation disturbance. Both uncertainties are considered unknown deterministic ones and satisfy the following energy-type restriction: Z 1 1 t T T 1 ½xðt0 Þ x0 C ½xðt0 Þ x0 þ r ðt; sÞRðt; sÞrðt; sÞ ds 2 2 t0 Z t 1 þ qT ðt; sÞG T ðt; sÞQðt; sÞGðt; sÞqðt; sÞ dsp1; ð3Þ 2 t0 where x0 is a given vector, and C; R; Q are positive (nonnegative) definite symmetric matrices. The symbol aT denotes transpose of a vector (matrix) a: In addition, the quadratic cost function J is defined as follows 1 J ¼ ½xðTÞ z0 T F1 ½xðTÞ z0 2 Z Z 1 T T 1 T T þ u ðt; sÞKðt; sÞuðt; sÞ ds þ x ðsÞLðt; sÞxðsÞ ds; ð4Þ 2 t0 2 t0 where z0 is a given vector, T > t0 is a certain time moment, and F; K; L are positive (nonnegative) definite symmetric matrices.
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The optimal control problem for the unobservable system state xðtÞ is to find the control un ðtÞ; tA½t0 ; T; that minimizes the criterion J along with the trajectory xn ðtÞ; tA½t0 ; T; generated upon substituting un ðtÞ into the state equation (1). 2.2. Separation principle in integral systems As well as in linear dynamic systems governed by differential equations, the separation principle remains valid in linear integral systems governed by Volterra equations with deterministic disturbances. Indeed, let us replace the unobservable # given by the equation (see [9] system state xðtÞ by its optimal minmax estimate xðtÞ for statement and derivation) Z t Z t # ds þ # ¼ x0 þ Aðt; sÞxðsÞ Bðt; sÞuðt; sÞ ds xðtÞ þ
Z
t0 t
t0
# SðsÞC T ðt; sÞQðt; sÞ½yðsÞ ds; ’ Cðt; sÞxðsÞ
ð5Þ
t0
where the matrix SðtÞ of the encircling ellipsoid # T S 1 ðtÞ½xðtÞ xðtÞp1 # X ðtÞ ¼ fxðtÞ: ½xðtÞ xðtÞ b2 ðtÞg; satisfies the integral Riccati equation Z t SðtÞ ¼ C þ ½Dðt; sÞR1 ðt; sÞDT ðt; sÞ þ Aðt; sÞSðsÞ ds þ
Z
t0 t
½SðsÞAT ðt; sÞ SðsÞC T ðt; sÞQðt; sÞCðt; sÞSðsÞ ds
ð6Þ
t0
with the initial condition Sðt0 Þ ¼ C; and the positive real number b2 is determined as follows: Z 1 t b2 ðtÞ ¼ ½yðsÞ ’ Cðt; sÞxðsÞT Qðt; sÞ½yðsÞ ’ Cðt; sÞxðsÞ ds: 2 t0 It is readily verified that the optimal control problem for the system state (1) and cost function (4) is equivalent to the optimal control problem for the optimal minmax estimate (5) and the cost function J represented as 1 # # z0 T F1 ½xðTÞ J ¼ ½xðTÞ z0 2 Z T Z 1 1 T T T # ds þ u ðt; sÞKðt; sÞuðt; sÞ ds þ x# ðsÞLðt; sÞxðsÞ 2 t0 2 t0 Z 1 T tr½SðsÞLðt; sÞds þ tr½SðTÞF1 ; þ 2 t0
ð7Þ
where tr½A denotes trace of a matrix A: Since the latter part of M is independent of control uðtÞ or state xðtÞ; the reduced effective cost function M to be minimized takes
M.V. Basin, I.R. Valadez Guzman / Journal of the Franklin Institute 339 (2002) 13–27
17
the form 1 # # z0 T F1 ½xðTÞ M ¼ ½xðTÞ z0 2 Z T Z 1 1 T T # ds: þ uT ðt; sÞKðt; sÞuðt; sÞ ds þ x# ðsÞLðt; sÞxðsÞ 2 t0 2 t0
ð8Þ
Thus, the solution for the optimal control problem specified by (1), (4) can be found solving the optimal control problem given by (5), (8). However, the minimal value of the criterion J should be determined using (7). This conclusion presents the separation principle in integral deterministic Volterra systems. 2.3. Optimal control problem solution Based on the solution of the optimal control problem obtained in [9] in the case of an observable system state, the following results are valid for the optimal control # problem (5), (8), where the system state (the minmax estimate xðtÞ) is completely available and, therefore, observable. The optimal control law is given by # un ðt; sÞ ¼ K 1 ðt; sÞBT ðt; sÞPðsÞxðsÞ;
ð9Þ
where PðtÞ is the solution of the integral Riccati equation Z
PðtÞ ¼ Pðt0 Þ þ
t
½Lðt; sÞ AT ðt; sÞPðsÞ PðsÞAðt; sÞ
t0
PðsÞBðt; sÞK 1 ðt; sÞBT ðt; sÞPðsÞ ds;
ð10Þ
with the terminal condition PðTÞ ¼ F1 : Upon substituting the optimal control (9) into the Eq. (5) for the reconstructed # the following optimally controlled state estimate equation is obtained: system state x; # ¼ x0 þ xðtÞ þ
Z
Z
t t0
t
# ds þ Aðt; sÞxðsÞ
Z
t
# ds Bðt; sÞK 1 ðt; sÞBT ðt; sÞPðsÞxðsÞ
t0
# SðsÞC T ðt; sÞQðt; sÞ½yðsÞ ds: ’ Cðt; sÞxðsÞ
ð11Þ
t0
Thus, the optimally controlled state estimate equation (11), the regulator gain matrix equation (10), the optimal control law (9), and the filter gain matrix equation (6) give the complete solution to the optimal controller problem for unobservable states of continuous integral systems governed by Volterra equations with deterministic disturbances.
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3. Optimal controller for unobservable discontinuous integral systems 3.1. Problem statement Let us now consider an unobservable linear system given by the state equation Z t Z t xðtÞ ¼ xðt0 Þ þ Aðt; sÞxðsÞ ds þ Bðt; sÞuðt; sÞ dvðsÞ þ
Z
t0
t0
t
ð12Þ
Dðt; sÞrðt; sÞ ds t0
and the output (observation) one Z t Z t yðtÞ ¼ Cðt; sÞxðsÞ dwðsÞ þ Gðt; sÞqðt; sÞ dwðsÞ; t0
ð13Þ
t0
where both state and observation equations are integral equations of the Volterra type with integration w.r.t. discontinuous measures, xðtÞARn is the unobservable system state, yðtÞARm is the observation vector, uðt; sÞARp is the input control, rðt; sÞARr is the input uncertainty, and qðt; sÞARm is the observation disturbance. The discontinuous measures in the state and observation equations are generated by bounded variation functions vðtÞ and wðtÞ; which can of course coincide or have discontinuities (jumps) at the same points. Therefore, the observation function yðtÞ may be discontinuous due to discontinuity of the integral with discontinuous measure dwðtÞ in the right-hand side of Eq. (13). This model of observations enables one to consider continuous and discrete observations in the common form: continuous observations correspond to the continuous component of a bounded variation function wðtÞ; and discrete observations correspond to its function of jumps. Both uncertainties are considered unknown deterministic ones and satisfy the following energy-type restriction: Z 1 1 t T T 1 ½xðt0 Þ x0 C ½xðt0 Þ x0 þ r ðt; sÞRðt; sÞrðt; sÞ ds 2 2 t0 Z 1 t T þ q ðt; sÞG T ðt; sÞQðt; sÞGðt; sÞqðt; sÞ dwðsÞp1; ð14Þ 2 t0 where x0 is a given vector, and C; R; Q are positive (nonnegative) definite symmetric matrices. In addition, the quadratic cost function J is defined as follows 1 J ¼ ½xðTÞ z0 T F1 ½xðTÞ z0 2 Z Z 1 T T 1 T T þ u ðt; sÞKðt; sÞuðt; sÞ dvðsÞ þ x ðsÞLðt; sÞxðsÞ ds; 2 t0 2 t0
ð15Þ
where z0 is a given vector, T > t0 is a certain time moment, and F; K; L are positive (nonnegative) definite symmetric matrices.
M.V. Basin, I.R. Valadez Guzman / Journal of the Franklin Institute 339 (2002) 13–27
19
The optimal control problem for the unobservable system state xðtÞ is to find the control un ðtÞ; tA½t0 ; T; that minimizes the criterion J along with the trajectory xn ðtÞ; tA½t0 ; T; generated upon substituting un ðtÞ into the state equation (12). The state trajectory xðtÞ may also be discontinuous due to discontinuity of the integral with discontinuous function vðtÞ in the right-hand side of (12). This model of system states enables one to consider sharp changes (jumps) in system position, as well as its gradual continuous movement. Modeling discontinuous unobservable system states of an integral Volterra system with deterministic disturbances along with discontinuous observations of the Volterra type enables one to consider linear continuous, discrete, and delayed systems in the unique general form given by Eqs. (12), (13), as it was done for stochastic systems in [1]. 3.2. Separation principle in discontinuous integral systems The separation principle for discontinuous system states (12) and discontinuous observations (13) is based on the separation principle for continuous states and observations (5), (6). Actually, the corresponding filtering procedure was suggested [17] to obtain filtering equations over discontinuous observations proceeding from the known filtering equations over continuous ones. In the examined case, the following actions substantiated in [17] should be performed: *
*
assuming functions vðtÞ and wðtÞ in state and observation equations (12) and (13) to be absolutely continuous, write out the separation principle given by the modified optimal control problem (5), (8), the ellipsoid matrix equation (6), and the criterion (7) obtained in Section 2.2 for continuous systems; in thus obtained optimal control problem, assume the functions vðtÞ and wðtÞ to be arbitrary bounded variation ones again, keeping in mind that their derivative v’ðtÞ and wðtÞ can be generalized functions of zero singularity order (for example, ’ d-functions), generating integration with the discontinuous measures dvðtÞ and dwðtÞ:
As a result, the unobservable system state xðtÞ of system (12) is replaced by its # optimal minmax estimate xðtÞ given by the equation (see [10] for statement, and derivation) # ¼ x0 þ xðtÞ þ
Z
Z
t t0
t
# ds þ Aðt; sÞxðsÞ
Z
t
Bðt; sÞuðt; sÞ dvðsÞ t0
# SðsÞC T ðt; sÞQðt; sÞ½yðsÞ dwðsÞ; ’ Cðt; sÞxðsÞ
t0
where the matrix SðtÞ of the encircling ellipsoid # T S 1 ðtÞ½xðtÞ xðtÞp1 # X ðtÞ ¼ f½xðtÞ: xðtÞ xðtÞ b2 ðtÞg;
ð16Þ
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M.V. Basin, I.R. Valadez Guzman / Journal of the Franklin Institute 339 (2002) 13–27
satisfies the integral Riccati equation SðtÞ ¼ C þ þ
Z
Z
t 1
T
Dðt; sÞR ðt; sÞD ðt; sÞ ds þ t0 t
SðsÞAT ðt; sÞ ds
t0
Z
Z
t
Aðt; sÞSðsÞ ds t0
t
SðsÞC T ðt; sÞQðt; sÞCðt; sÞSðsÞ dwðsÞ;
ð17Þ
t0
with the initial condition Sðt0 Þ ¼ C; and the positive real number b2 is determined as follows b2 ðtÞ ¼
1 2
Z
t
½yðsÞ ’ Cðt; sÞxðsÞT Qðt; sÞ½yðsÞ ’ Cðt; sÞxðsÞ dwðsÞ: t0
Furthermore, the optimal control problem for the system state (12) and cost function (15) is equivalent to the optimal control problem for the optimal minmax estimate (16) and the cost function J represented as Z 1 1 T T # # J ¼ ½xðTÞ z0 T F1 ½xðTÞ z0 þ u ðt; sÞKðt; sÞuðt; sÞ dvðsÞ 2 2 t0 Z Z 1 T T 1 T # ds þ tr½SðsÞLðt; sÞds þ tr½SðTÞF1 ; þ x# ðsÞLðt; sÞxðsÞ 2 t0 2 t0
ð18Þ
which can be reduced to the effective cost function M 1 # # z0 T F1 ½xðTÞ z0 M ¼ ½xðTÞ 2 Z T Z 1 1 T T T # ds: þ u ðt; sÞKðt; sÞuðt; sÞ dvðsÞ þ x# ðsÞLðt; sÞxðsÞ 2 t0 2 t0
ð19Þ
Thus, the solution for the optimal control problem specified by (12), (15) can be found solving the optimal control problem given by (16), (19) and using (18) for the minimal value of the criterion J: 3.3. Optimal control problem solution for discontinuous systems Based on the solution of the optimal control problem obtained in [10] in the case of an observable discontinuous system state, the following results are valid for the # is optimal control problem (16), (19), where the system state (the minmax estimate x) completely available and, therefore, observable. The optimal control law is given by # un ðt; sÞ ¼ K 1 ðt; sÞBT ðt; sÞPðsÞxðsÞ;
ð20Þ
M.V. Basin, I.R. Valadez Guzman / Journal of the Franklin Institute 339 (2002) 13–27
where PðtÞ is the solution of the integral Riccati equation Z t PðtÞ ¼ Pðt0 Þ þ ½Qðt; sÞ AT ðt; sÞPðsÞ PðsÞAðt; sÞ ds 0 Z t ½PðsÞBðt; sÞK 1 ðt; sÞBT ðt; sÞPðsÞ dvðsÞ;
21
ð21Þ
0
with the terminal condition PðTÞ ¼ F1 : Upon substituting the optimal control (20) into Eq. (12) for the reconstructed # the following optimally controlled state estimate equation is obtained: system state x; Z t Z t # ds þ # dvðsÞ # ¼ x0 þ Aðt; sÞxðsÞ Bðt; sÞK 1 ðt; sÞBT ðt; sÞPðsÞxðsÞ xðtÞ þ
Z
t0 t
t0
# SðsÞC T ðt; sÞQðt; sÞ½yðsÞ dwðsÞ: ’ Cðt; sÞxðsÞ
ð22Þ
t0
The obtained Eqs. (21) and (22), as well as Eq. (17) for the estimate variance SðtÞ; are integral equations with integration w.r.t. discontinuous measures generated by bounded variation functions vðtÞ and wðtÞ; which do not tell us how to compute # and the gain matrices jumps of the controller and filtering variables (the estimate xðtÞ PðtÞ and SðtÞ) at the discontinuity points of the functions vðtÞ and wðtÞ; corresponding to discontinuities in the system states xðtÞ and the observation process yðtÞ: Nevertheless, in accordance with Theorem 3 in [18], the jumps can be computed solving the following system of differential equations: dx# dv # ¼ Bðt; tÞK 1 ðt; tÞBT ðt; tÞPðwÞxðwÞ dw dw dy T # # # þ SðwÞC ðt; tÞQðt; tÞ Cðt; tÞxðwÞ ; xðwðtÞÞ ¼ xðtÞ; dw dS ¼ SðwÞC T ðt; tÞQðt; tÞCðt; tÞSðwÞ; dw
SðwðtÞÞ ¼ SðtÞ;
dP dv ¼ PðwÞBðt; tÞK 1 ðt; tÞBT ðt; tÞPðwÞ ; dw dw which yields the following jump expressions:
PðwðtÞÞ ¼ PðtÞ;
# ¼ SðtÞfI þ C T ðt; tÞQðt; tÞCðt; tÞSðtÞDwðtÞg1 DxðtÞ #
C T ðt; tÞQðt; tÞ½DyðtÞ Cðt; tÞxðtÞDwðtÞ þ Bðt; tÞK 1 ðt; tÞBT ðt; tÞfI þ C T ðt; tÞQðt; tÞCðt; tÞ #
SðtÞDwðtÞg1 PðtÞxðtÞDvðtÞ; DSðtÞ ¼ SðtÞfI þ C T ðt; tÞQðt; tÞCðt; tÞSðtÞDwðtÞg1 SðtÞ; DPðtÞ ¼ PðtÞfI þ Bðt; tÞK 1 ðt; tÞBT ðt; tÞPðtÞDvðtÞg1 PðtÞ: Following [18], the obtained jump expressions can be incorporated into the controller and filtering equations (21), (22) and (17) using the form of the equivalent
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equations with a measure # ¼ x0 xðtÞ
Z
t
# ds þ Aðt; sÞxðsÞ
Z
t0
þ
t
Bðt; sÞuðt; sÞ ds t0
Z
t
Bðt; sÞK 1 ðt; sÞBT ðt; sÞfI þ C T ðt; sÞQðt; sÞCðt; sÞ t0
#
SðsÞDwðsÞg1 PðsÞxðsÞ dvðsÞ Z t SðsÞfI þ C T ðt; sÞQðt; sÞCðt; sÞSðsÞDwðsÞg1 þ t0
#
C T ðt; sÞQðt; sÞ½dyðsÞ Cðt; sÞxðsÞ dwðsÞ;
SðtÞ ¼ C þ þ
Z
t
Dðt; sÞR1 ðt; sÞDT ðt; sÞ ds
t0 t
Z
ð23Þ
Aðt; sÞSðsÞ ds þ
t0 Z t
Z
t
SðsÞAT ðt; sÞ ds
t0
SðsÞfI þ C T ðt; sÞQðt; sÞCðt; sÞSðsÞDwðsÞg1
t0
C T ðt; sÞQðt; sÞCðt; sÞSðsÞ dwðsÞ;
ð24Þ
with the initial condition Sðt0 Þ ¼ C; and PðtÞ ¼ Pðt0 Þ þ
Z
Z
t
½Qðt; sÞ AT ðt; sÞPðsÞ PðsÞAðt; sÞ ds
t0 t
PðsÞfI þ Bðt; sÞK 1 ðt; sÞBT ðt; sÞPðsÞDvðsÞg1
t0
Bðt; sÞK 1 ðt; sÞBT ðt; sÞPðsÞ dvðsÞ;
ð25Þ
with the terminal condition PðTÞ ¼ F1 ; where DwðtÞ; DvðtÞ; and DyðtÞ are the jumps of the bounded variation functions wðtÞ; vðtÞ; and the observation process yðtÞ at a # point t; respectively, and xðtÞ; SðtÞ; and PðtÞ are the values of the discontinuous # controller and filtering parameters (the estimate xðtÞ; and the gain matrices SðtÞ and PðtÞ) at a point t from the left. The optimally controlled state estimate equation (23), the regulator gain matrix equation (25), filter gain matrix equation (24), and the optimal control law (20) give the complete solution to the optimal controller problem for unobservable states of discontinuous integral systems governed by Volterra equations with deterministic disturbances, including analytic expressions for jumps of the controller and filtering variables at the discontinuity points of the real system state xðtÞ and the observation process yðtÞ:
M.V. Basin, I.R. Valadez Guzman / Journal of the Franklin Institute 339 (2002) 13–27
23
4. Movement of missile with impulsive and jet motors and unobservable velocity Let us consider the optimal control problem for movement of a missile with two motors, impulsive and jet (continuous), whose task is to reach the maximal possible altitude at a certain time moment T > 0 with the minimal possible fuel consumption. The missile movement is considered governed by the state equations (cf. [8]) Z t Z t Pp ðsÞ hðtÞ ¼ h0 þ ds; vðsÞ ds; mðtÞ ¼ m0 þ 0 0 Cðt; sÞ vðtÞ ¼
Z
t
0
Pp ðsÞ Qðh; vÞ dwðsÞ mðsÞ
Z 0
t
g ds þ
Z
t
rðsÞ ds; 0
where t0 ¼ 0; vðtÞ is the missile velocity, v0 ¼ vð0Þ ¼ 0; and information on the missile velocity is collected using the velocity measurer (observation) equation Z t Z t yðtÞ ¼ Hðt; sÞvðsÞ dwðsÞ þ F ðt; sÞqðsÞ dwðsÞ: 0
0
Here, h0 ¼ hð0Þ > 0 is the initial adjusted altitude corresponding the missile position on the earth surface, hðtÞ is the current adjusted altitude; mðsÞ is the missile and fuel mass, m0 b0; Pp ðtÞ is the propulsion force; Cðt; sÞo0 is the difference factor of the ideal velocities of the missile at time t and the outflowed fuel at time s; which is varying with change of altitude and, consequently, temperature, pressure, gravity acceleration, etc.; g is the gravity acceleration; rðsÞ is the input disturbance due to inaccuracy of modeling velocity dynamics; wðsÞ is a bounded variation function which represents functioning of two missile motors, impulsive and jet (continuous): the jet motor expels fuel gradually and the impulsive one does this instantaneously at a certain time moment. t1 ; 0pt1 pT: Thus, the motors functioning is described using decomposition of wðtÞ into its continuous component wc ðtÞ (continuous jet) and the Heaviside function wðt t1 Þ with jump at the moment t1 (impulsive motor), i.e., wðtÞ ¼ wc ðtÞ þ wðt t1 Þ: Due to inaccuracy of measuring devices and natural reasons (such as the Doppler effect), the velocity measurer collects information on the velocity values not only at the current time t but as a summation with the values at some previous time moments, R t presenting a classical case of data fusion. This effect is modeled by the term 0 Hðt; sÞvðsÞ dwðsÞ in the observation equation, whereas the last term Rt F ðt; sÞqðsÞ dwðsÞ takes into account influence of disturbances qðsÞ affecting 0 measurements of the velocities vðsÞ at the observation moment t: Integration with the discontinuous measure dwðtÞ in the observation equation enables one to correctly incorporate the jump in the system state (missile velocity) into the model of observations. The total energy of the disturbances (total error of modeling) satisfies the restriction Z Z 1 t T 1 t T r ðt; sÞrðt; sÞ ds þ q ðt; sÞF ðt; sÞF ðt; sÞqðt; sÞ dwðsÞp1: 2 0 2 0 It is assumed that the atmosphere resistance force is absent: Qðh; vÞ ¼ 0:
M.V. Basin, I.R. Valadez Guzman / Journal of the Franklin Institute 339 (2002) 13–27
24
Upon selecting the mass outflow function uðsÞ ¼ ðmðsÞ=mðsÞÞ ¼ ðd=dsÞ½lnðmðsÞÞ as ’ control, the optimal control problems is completely stated for the system state xðtÞ ¼ ½hðtÞ; vðtÞ governed by the equation Z t Z 1 Z t Z t xðtÞ ¼ x0 þ AxðsÞ ds þ Bðt; sÞuðsÞ dwðsÞ þ G ds þ DrðsÞ ds; 0
where
" xðtÞ ¼
hðsÞ
0
#
"
0
0
1
#
"
0
0
#
; A¼ ; Bðt; sÞ ¼ ; vðsÞ 0 0 Cðt; sÞ " # " # 0 0 mðsÞ d ’ ¼ ½lnðmðsÞÞ; D¼ ; G¼ ; uðsÞ ¼ mðsÞ ds 1 g
x0 ¼ ½h0 ; 0; and the cost function to be minimized " " ##T " " ## hn hn 1 J ¼ xðTÞ c xðTÞ 2 0 0 Z T 1 þ uðsÞuðsÞ dwðsÞ- min; uð Þ 2 0 where c¼
"
1 0
# 0 ; 0
hn bh0
and T > 0 is certain time moment. The observation equation takes the form " # Z t Z t hðsÞ yðtÞ ¼ ½0 Hðt; sÞ dwðsÞ þ F ðt; sÞqðsÞ dwðsÞ: vðsÞ 0 0 In accordance with (20), the optimal control is given by " # hðsÞ n u ðt; sÞ ¼ ½0 Cðt; sÞPðsÞ : vðsÞ Note that the initial adjusted altitude h0 > 0 is determined from the conditions vð0Þ ¼ 0 and v’ð0Þ ¼ 0 (there is equilibrium of the missile on the earth surface at the initial time moment), which, upon substituting the optimal control un ðt; sÞ into the velocity equation, yield 0 ¼ Cðt0 ; t0 Þun ðt0 ; t0 Þ g ¼ Cð0; 0Þun ð0; 0Þ g: Thus, the initial adjusted altitude h0 > 0 is determined from the equation " # h0 g ¼ Cð0; 0Þ½0 Cð0; 0ÞPð0Þ : 0 In accordance with (23)–(25), the equations for the controlled optimal estimate # trajectory xðtÞ; the filter gain matrix SðtÞ; and the regulator gain matrix PðtÞ take
M.V. Basin, I.R. Valadez Guzman / Journal of the Franklin Institute 339 (2002) 13–27
the forms SðtÞ ¼
Z
"
1 0
Z
1
0
0
# SðsÞ ds þ ("
t
0
0 Hðt; sÞ
Z t" # 0 0
SðsÞ "
0
#
1 0
1
½0 1 ds þ
Z
"
t
SðsÞ 0
0
0
1
0
25
# ds
# " # )1 0 0 þ ½0 Hðt; sÞSðsÞDwðsÞ 1 Hðt; sÞ
½0 Hðt; sÞSðsÞ dwðsÞ
with the initial condition Sð0Þ ¼ 0; " # # Z t Z t " 0 1 0 0 PðsÞ ds PðsÞ ds PðtÞ ¼ Pð0Þ 0 0 1 0 0 0 (" # " # )1 Z t 1 0 0 PðsÞ þ ½0 Cðt; sÞ PðsÞDwðsÞ 0 1 Cðt; sÞ 0 " # 0
½0 Cðt; sÞPðsÞ dwðsÞ; Cðt; sÞ with the terminal condition PðTÞ ¼ c; and # ) # Z t (" Z t " 0 1 0 # þ G ds þ # ¼ x0 þ ½0 Cðt; sÞ xðsÞ xðtÞ 0 0 Cðt; sÞ 0 0 (" # " # )1 1 0 0
þ ½0 Hðt; sÞ SðsÞDwðsÞ 0 1 Hðt; sÞ (" # Z t 1 0 # dwðsÞ þ
PðsÞxðsÞ SðsÞ 0 1 0 " # )1 0 þ ½0 Hðt; sÞ SðsÞDwðsÞ Hðt; sÞ " # 0 #
½dyðsÞ ½0 Hðt; sÞxðsÞ dwðsÞ; Hðt; sÞ # with the initial condition xð0Þ ¼ ½h00 ; and their jumps at the point t1 ; where the impulsive motor is applied, are equal to (" # " )1 # 1 0 0 DSðt1 Þ ¼ Sðt1 Þ þ ½0 Hðt1 ; t1 ÞSðt1 ÞDwðt1 Þ 0 1 Hðt1 ; t1 Þ " # 0
½0 Hðt1 ; t1 Þ Sðt1 ÞDwðt1 Þ; Hðt1 ; t1 Þ
26
M.V. Basin, I.R. Valadez Guzman / Journal of the Franklin Institute 339 (2002) 13–27
(" DPðt1 Þ ¼ Pðt1 Þ "
1 0
0 Cðt1 ; t1 Þ
# " )1 # 0 0 þ ½0 Cðt1 ; t1 ÞPðt1 ÞDwðt1 Þ 1 Cðt1 ; t1 Þ # ½0 Cðt1 ; t1 Þ Pðt1 ÞDwðt1 Þ;
(" # # 0 1 0 # 1Þ ¼ Dxðt ½0 Cðt1 ; t1 Þ Cðt1 ; t1 Þ 0 1 )1 " # 0 # 1 ÞDwðt1 Þ þ Pðt1 Þxðt ½0 Hðt1 ; t1 Þ Sðt1 ÞDwðt1 Þ Hðt1 ; t1 Þ # " # (" )1 0 1 0 þ ½0 Hðt1 ; t1 Þ Sðt1 ÞDwðt1 Þ þ Sðt1 Þ Hðt1 ; t1 Þ 0 1 " # 0 # 1 ÞDwðt1 Þ:
½Dyðt1 Þ ½0 Hðt1 ; t1 Þxðt Hðt1 ; t1 Þ "
Thus, the complete algorithm for solving this optimal control problem is described as follows: *
*
* * *
*
the equation for the filter gain matrix SðtÞ with the initial condition Sð0Þ ¼ 0 and the jump DSðt1 Þ at the point t1 is solved; the equation for the regulator gain matrix PðtÞ with the terminal condition PðTÞ ¼ c and the jump DPðt1 Þ at the point t1 is solved; the initial condition Pð0Þ is thus determined; the initial adjusted altitude h0 is calculated; substituting un ðt; sÞ into the optimal estimate equations and solving them with # ¼ h0 and v#ð0Þ ¼ 0 yields the optimally controlled estimate initial conditions hð0Þ # v#ðtÞ ¼ xðtÞ; # trajectories ½hðtÞ; where the optimally controlled velocity estimate v#ðtÞ has a jump at the point t1 ; and the optimally controlled estimate of the adjusted # is continuous; altitude hðtÞ the optimally controlled estimate of the desirable maximal altitude is determined # as hðTÞ h0 :
References [1] M.V. Basin, M.A. Villanueva-Llanes, I.R. Valadez-Guzman, On filtering problems over observations with delays, Proceedings of the 38th IEEE Conference on Decision and Control, 1999, pp. 4572–4577. [2] D. Chyung, E.B. Lee, Linear optimal systems with delays, SIAM J. Control 3 (1996) 548–575. [3] M.C. Delfour, The linear-quadratic optimal control problem with delays in state and control variables: a state space approach, SIAM J. Control 24 (1986) 835–883. [4] R.L. Alford, E.B. Lee, Sampled data hereditary systems: linear quadratic theory, IEEE Trans. Automat. Control 31 (1986) 60–65.
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27
[5] E.B. Lee, Y.C. You, Optimal control of bivariate linear Volterra integral type systems, Proceedings of the 26th IEEE Conference on Decision and Control 1987, pp. 721–726. [6] G.L. Fonseca, Keynesian Business Cycle Theory, Department of Economics, Baltimore University, 1998. [7] R.F. Hartl, Optimal dynamic advertising policies for hereditary processes, J. Optim. Theory Appl. 43 (1984) 51–72. [8] A.E. Bryson, Y.C. Ho, Applied Optimal Control, Hemisphere Publishing Company, New York, 1979. [9] M.V. Basin, I.R. Valadez-Guzman, Minmax filtering in Volterra systems, Proceedings of the American Control Conference, Chicago, IL, 2000, pp. 1380–1385. [10] M.V. Basin, I.R. Valadez-Guzman, Optimal minmax filtering and control in discontinuous Volterra systems, Proceedings of the American Control Conference, Chicago, IL, 2000, pp. 904–909. [11] A.V. Balakrishnan, On stochastic bang–bang control, Lecture Notes on Control Information Sciences, Vol. 25, Springer, New York, 1980, pp. 221–238. [12] H. Kwakernaak, R. Sivan, Linear Optimal Control Systems, Wiley, New York, 1972. [13] D.P. Bertsekas, I.B. Rhodes, Recursive state estimation for a setmembership description of uncertainty, IEEE Trans. Automat. Control AC-16 (1971) 117–128. [14] A.J. Krener, Kalman-Bucy and minmax filtering, IEEE Trans. Automat. Contr. AC-25 (1980) 291–292. [15] M.L. Kleptsina, A.Yu. Veretennikov, On filtering and properties of conditional laws of Ito–Volterra processes, Statistics and control of stochastic processes. Steklov seminar, 1984, Optimization Software Inc., Publication Division, New York, 1985, pp. 179–196. [16] L.E. Shaikhet, On an optimal control problem of partly observable stochastic Volterra’s process, Probl. Control Inf. Theory 16 (1987) 439–448. [17] Yu.V. Orlov, M.V. Basin, On minmax filtering over discrete-continuous observations, IEEE Trans. Automat. Control AC-40 (1995) 1623–1626. [18] M.V. Basin, M.A. Villanueva-Llanes, On filtering problems over Ito–Volterra observations, Proceedings of the American Control Conference, San Diego, CA, 1999, pp. 3407–3412.
Optimal control in unobservable integral Volterra systems Michael V. Basin*, Irma R. Valadez Guzman Department of Physical and Mathematical Sciences, Autonomous University of Nuevo Leon, Adpo postal 144-F, C.P. 66450, San Nicolas de los Garza, Nuevo Leon, Mexico Received 22 May 2000; accepted 20 October 2001
Abstract This paper presents solution of the optimal linear-quadratic controller problem for unobservable integral Volterra systems with continuous/discontinuous states under deterministic uncertainties, over continuous/discontinuous observations. Due to the separation principle for integral systems, the initial continuous problem is split into the optimal minmax filtering problem for integral Volterra systems with deterministic uncertainties over continuous/discontinuous observations and the optimal linear-quadratic control (regulator) problem for observable deterministic integral Volterra systems with continuous/discontinuous states. As a result, the system of the optimal controller equations are obtained, including the linear equation for the optimally controlled minmax estimate and two Riccati equations for its ellipsoid matrix (optimal gain matrix of the filter) and the optimal regulator gain matrix. Then, in the discontinuous problems, the equation for the optimal controller and the equations for the optimal filter and regulator gain matrices are obtained using the filtering procedure for deriving the filtering equations over discontinuous observations proceeding from the known filtering equations over continuous ones and the dual results in the optimal control problem for integral systems. The technical example illustrating application of the obtained results is finally given. r 2002 The Franklin Institute. Published by Elsevier Science Ltd. All rights reserved. Keywords: Linear Volterra system; Deterministic uncertainties; Control; Filtering
*Corresponding author. Tel.: +52-8-329-4030; fax: +52-8-352-2954. E-mail address: [email protected] (M.V. Basin). 0016-0032/02/$22.00 r 2002 The Franklin Institute. Published by Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 6 - 0 0 3 2 ( 0 1 ) 0 0 0 5 4 - 0
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M.V. Basin, I.R. Valadez Guzman / Journal of the Franklin Institute 339 (2002) 13–27
1. Introduction The optimal control and filtering problems for dynamic systems with delays, which represent a particular case of discontinuous integral systems (see [1] for substantiation), have been studied in a number of publications from various viewpoints (see, for example, [2–4] for dynamic systems and [5] for a particular case of integral Volterra ones). This attention is directly related to common use of dynamic systems with delays in global economy concepts [6], marketing models [7], technical systems [8], and others. Since the class of integral Volterra systems includes the class of retarded dynamic ones, studies of integral systems, such as solutions of the optimal LQR problem [9,10], minmax filtering problem [9,10], and the optimal controller (this paper), become a significant part of the control theory. Nevertheless, the integral Volterra systems have been of independent interest in the deterministic environment, as well as in the stochastic one (see [11]). This paper presents solution to the optimal linear–quadratic controller problem for unobservable integral Volterra systems with continuous/discontinuous states under deterministic uncertainties, over continuous/discontinuous observations. Due to the separation principle for integral systems, which is stated analogously to that for dynamic differential ones [12], the initial continuous problem is split into the optimal minmax filtering problem for integral Volterra systems with deterministic uncertainties over continuous observations and the optimal linear–quadratic control (regulator) problem for observable deterministic integral Volterra systems with continuous states. There are a number of papers investigating these problems for system state and observations given by differential equations [13,14] or bivariate Volterra ones [5], and the problems have been solved in the general case of integral governing equations in the previous authors’ papers [9,10]. Based on those results, the system of the optimal controller equations are obtained, including the linear equation for the optimally controlled minmax estimate and two Riccati equations for its ellipsoid matrix (optimal gain matrix of the filter) and the optimal regulator gain matrix. Note that it is possible to form closed systems of the minmax filtering and optimal regulator equations for an integral system state over integral observations, using only two filtering variables, the optimal estimate (state) and its ellipsoid matrix (gain matrix of the regulator), although the analogous result cannot be reached in stochastic systems (see [15,16]). Thus, the complete system of the obtained controller equations consists of only three variables, instead of six expected in the stochastic case. This encouraging result may be explained by the fact that deterministic uncertainties in the minmax filtering problem satisfy energy-type integral restrictions, i.e., belong to certain ellipsoids with fixed boundaries in functional spaces, and cannot take any possible values as stochastic Gaussian disturbances can. The optimal controller equations for integral systems with discontinuous states over discontinuous observations are then obtained using the filtering procedure [17,18] for deriving the filtering equations over discontinuous observations proceeding from the known filtering equations over continuous ones, which have already been obtained in the paper, and the dual results in the optimal control problem for integral systems [9,10]. In particular, the obtained results enable
M.V. Basin, I.R. Valadez Guzman / Journal of the Franklin Institute 339 (2002) 13–27
15
one to compute jumps of the optimal filtering and control parameters that can appear due to discontinuities in observations and states, respectively, and this is actually done in the technical example related to the process of missile launching. The paper is organized as follows. In Section 2, the optimal controller problem is stated and then solved for unobservable continuous integral Volterra systems. The separation principle for integral Volterra systems is substantiated there as well. Section 3 generalizes the mentioned results to discontinuous unobservable integral Volterra systems. Finally, Section 4 presents the technical example illustrating application of the obtained results to solution of the optimal controller problem of launching a missile with continuous and impulsive jet motors and unobservable velocity to the maximal possible altitude with the minimal fuel consumption.
2. Optimal controller for unobservable continuous integral systems 2.1. Problem statement Let us consider an unobservable linear system given by the state equation Z t Z t Z t xðtÞ ¼ xðt0 Þ þ Aðt; sÞxðsÞ ds þ Bðt; sÞuðt; sÞ ds þ Dðt; sÞrðt; sÞ ds t0
t0
and the output (observation) one Z t Z t yðtÞ ¼ Cðt; sÞxðsÞ ds þ Gðt; sÞqðt; sÞ ds; t0
ð1Þ
t0
ð2Þ
t0
where both state and observation equations are integral equations of the Volterra type, xðtÞARn is the unobservable system state, yðtÞARm is the observation vector, rðt; sÞARr is the input uncertainty, uðt; sÞARp is the input control, and qðt; sÞARm is the observation disturbance. Both uncertainties are considered unknown deterministic ones and satisfy the following energy-type restriction: Z 1 1 t T T 1 ½xðt0 Þ x0 C ½xðt0 Þ x0 þ r ðt; sÞRðt; sÞrðt; sÞ ds 2 2 t0 Z t 1 þ qT ðt; sÞG T ðt; sÞQðt; sÞGðt; sÞqðt; sÞ dsp1; ð3Þ 2 t0 where x0 is a given vector, and C; R; Q are positive (nonnegative) definite symmetric matrices. The symbol aT denotes transpose of a vector (matrix) a: In addition, the quadratic cost function J is defined as follows 1 J ¼ ½xðTÞ z0 T F1 ½xðTÞ z0 2 Z Z 1 T T 1 T T þ u ðt; sÞKðt; sÞuðt; sÞ ds þ x ðsÞLðt; sÞxðsÞ ds; ð4Þ 2 t0 2 t0 where z0 is a given vector, T > t0 is a certain time moment, and F; K; L are positive (nonnegative) definite symmetric matrices.
16
M.V. Basin, I.R. Valadez Guzman / Journal of the Franklin Institute 339 (2002) 13–27
The optimal control problem for the unobservable system state xðtÞ is to find the control un ðtÞ; tA½t0 ; T; that minimizes the criterion J along with the trajectory xn ðtÞ; tA½t0 ; T; generated upon substituting un ðtÞ into the state equation (1). 2.2. Separation principle in integral systems As well as in linear dynamic systems governed by differential equations, the separation principle remains valid in linear integral systems governed by Volterra equations with deterministic disturbances. Indeed, let us replace the unobservable # given by the equation (see [9] system state xðtÞ by its optimal minmax estimate xðtÞ for statement and derivation) Z t Z t # ds þ # ¼ x0 þ Aðt; sÞxðsÞ Bðt; sÞuðt; sÞ ds xðtÞ þ
Z
t0 t
t0
# SðsÞC T ðt; sÞQðt; sÞ½yðsÞ ds; ’ Cðt; sÞxðsÞ
ð5Þ
t0
where the matrix SðtÞ of the encircling ellipsoid # T S 1 ðtÞ½xðtÞ xðtÞp1 # X ðtÞ ¼ fxðtÞ: ½xðtÞ xðtÞ b2 ðtÞg; satisfies the integral Riccati equation Z t SðtÞ ¼ C þ ½Dðt; sÞR1 ðt; sÞDT ðt; sÞ þ Aðt; sÞSðsÞ ds þ
Z
t0 t
½SðsÞAT ðt; sÞ SðsÞC T ðt; sÞQðt; sÞCðt; sÞSðsÞ ds
ð6Þ
t0
with the initial condition Sðt0 Þ ¼ C; and the positive real number b2 is determined as follows: Z 1 t b2 ðtÞ ¼ ½yðsÞ ’ Cðt; sÞxðsÞT Qðt; sÞ½yðsÞ ’ Cðt; sÞxðsÞ ds: 2 t0 It is readily verified that the optimal control problem for the system state (1) and cost function (4) is equivalent to the optimal control problem for the optimal minmax estimate (5) and the cost function J represented as 1 # # z0 T F1 ½xðTÞ J ¼ ½xðTÞ z0 2 Z T Z 1 1 T T T # ds þ u ðt; sÞKðt; sÞuðt; sÞ ds þ x# ðsÞLðt; sÞxðsÞ 2 t0 2 t0 Z 1 T tr½SðsÞLðt; sÞds þ tr½SðTÞF1 ; þ 2 t0
ð7Þ
where tr½A denotes trace of a matrix A: Since the latter part of M is independent of control uðtÞ or state xðtÞ; the reduced effective cost function M to be minimized takes
M.V. Basin, I.R. Valadez Guzman / Journal of the Franklin Institute 339 (2002) 13–27
17
the form 1 # # z0 T F1 ½xðTÞ M ¼ ½xðTÞ z0 2 Z T Z 1 1 T T # ds: þ uT ðt; sÞKðt; sÞuðt; sÞ ds þ x# ðsÞLðt; sÞxðsÞ 2 t0 2 t0
ð8Þ
Thus, the solution for the optimal control problem specified by (1), (4) can be found solving the optimal control problem given by (5), (8). However, the minimal value of the criterion J should be determined using (7). This conclusion presents the separation principle in integral deterministic Volterra systems. 2.3. Optimal control problem solution Based on the solution of the optimal control problem obtained in [9] in the case of an observable system state, the following results are valid for the optimal control # problem (5), (8), where the system state (the minmax estimate xðtÞ) is completely available and, therefore, observable. The optimal control law is given by # un ðt; sÞ ¼ K 1 ðt; sÞBT ðt; sÞPðsÞxðsÞ;
ð9Þ
where PðtÞ is the solution of the integral Riccati equation Z
PðtÞ ¼ Pðt0 Þ þ
t
½Lðt; sÞ AT ðt; sÞPðsÞ PðsÞAðt; sÞ
t0
PðsÞBðt; sÞK 1 ðt; sÞBT ðt; sÞPðsÞ ds;
ð10Þ
with the terminal condition PðTÞ ¼ F1 : Upon substituting the optimal control (9) into the Eq. (5) for the reconstructed # the following optimally controlled state estimate equation is obtained: system state x; # ¼ x0 þ xðtÞ þ
Z
Z
t t0
t
# ds þ Aðt; sÞxðsÞ
Z
t
# ds Bðt; sÞK 1 ðt; sÞBT ðt; sÞPðsÞxðsÞ
t0
# SðsÞC T ðt; sÞQðt; sÞ½yðsÞ ds: ’ Cðt; sÞxðsÞ
ð11Þ
t0
Thus, the optimally controlled state estimate equation (11), the regulator gain matrix equation (10), the optimal control law (9), and the filter gain matrix equation (6) give the complete solution to the optimal controller problem for unobservable states of continuous integral systems governed by Volterra equations with deterministic disturbances.
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3. Optimal controller for unobservable discontinuous integral systems 3.1. Problem statement Let us now consider an unobservable linear system given by the state equation Z t Z t xðtÞ ¼ xðt0 Þ þ Aðt; sÞxðsÞ ds þ Bðt; sÞuðt; sÞ dvðsÞ þ
Z
t0
t0
t
ð12Þ
Dðt; sÞrðt; sÞ ds t0
and the output (observation) one Z t Z t yðtÞ ¼ Cðt; sÞxðsÞ dwðsÞ þ Gðt; sÞqðt; sÞ dwðsÞ; t0
ð13Þ
t0
where both state and observation equations are integral equations of the Volterra type with integration w.r.t. discontinuous measures, xðtÞARn is the unobservable system state, yðtÞARm is the observation vector, uðt; sÞARp is the input control, rðt; sÞARr is the input uncertainty, and qðt; sÞARm is the observation disturbance. The discontinuous measures in the state and observation equations are generated by bounded variation functions vðtÞ and wðtÞ; which can of course coincide or have discontinuities (jumps) at the same points. Therefore, the observation function yðtÞ may be discontinuous due to discontinuity of the integral with discontinuous measure dwðtÞ in the right-hand side of Eq. (13). This model of observations enables one to consider continuous and discrete observations in the common form: continuous observations correspond to the continuous component of a bounded variation function wðtÞ; and discrete observations correspond to its function of jumps. Both uncertainties are considered unknown deterministic ones and satisfy the following energy-type restriction: Z 1 1 t T T 1 ½xðt0 Þ x0 C ½xðt0 Þ x0 þ r ðt; sÞRðt; sÞrðt; sÞ ds 2 2 t0 Z 1 t T þ q ðt; sÞG T ðt; sÞQðt; sÞGðt; sÞqðt; sÞ dwðsÞp1; ð14Þ 2 t0 where x0 is a given vector, and C; R; Q are positive (nonnegative) definite symmetric matrices. In addition, the quadratic cost function J is defined as follows 1 J ¼ ½xðTÞ z0 T F1 ½xðTÞ z0 2 Z Z 1 T T 1 T T þ u ðt; sÞKðt; sÞuðt; sÞ dvðsÞ þ x ðsÞLðt; sÞxðsÞ ds; 2 t0 2 t0
ð15Þ
where z0 is a given vector, T > t0 is a certain time moment, and F; K; L are positive (nonnegative) definite symmetric matrices.
M.V. Basin, I.R. Valadez Guzman / Journal of the Franklin Institute 339 (2002) 13–27
19
The optimal control problem for the unobservable system state xðtÞ is to find the control un ðtÞ; tA½t0 ; T; that minimizes the criterion J along with the trajectory xn ðtÞ; tA½t0 ; T; generated upon substituting un ðtÞ into the state equation (12). The state trajectory xðtÞ may also be discontinuous due to discontinuity of the integral with discontinuous function vðtÞ in the right-hand side of (12). This model of system states enables one to consider sharp changes (jumps) in system position, as well as its gradual continuous movement. Modeling discontinuous unobservable system states of an integral Volterra system with deterministic disturbances along with discontinuous observations of the Volterra type enables one to consider linear continuous, discrete, and delayed systems in the unique general form given by Eqs. (12), (13), as it was done for stochastic systems in [1]. 3.2. Separation principle in discontinuous integral systems The separation principle for discontinuous system states (12) and discontinuous observations (13) is based on the separation principle for continuous states and observations (5), (6). Actually, the corresponding filtering procedure was suggested [17] to obtain filtering equations over discontinuous observations proceeding from the known filtering equations over continuous ones. In the examined case, the following actions substantiated in [17] should be performed: *
*
assuming functions vðtÞ and wðtÞ in state and observation equations (12) and (13) to be absolutely continuous, write out the separation principle given by the modified optimal control problem (5), (8), the ellipsoid matrix equation (6), and the criterion (7) obtained in Section 2.2 for continuous systems; in thus obtained optimal control problem, assume the functions vðtÞ and wðtÞ to be arbitrary bounded variation ones again, keeping in mind that their derivative v’ðtÞ and wðtÞ can be generalized functions of zero singularity order (for example, ’ d-functions), generating integration with the discontinuous measures dvðtÞ and dwðtÞ:
As a result, the unobservable system state xðtÞ of system (12) is replaced by its # optimal minmax estimate xðtÞ given by the equation (see [10] for statement, and derivation) # ¼ x0 þ xðtÞ þ
Z
Z
t t0
t
# ds þ Aðt; sÞxðsÞ
Z
t
Bðt; sÞuðt; sÞ dvðsÞ t0
# SðsÞC T ðt; sÞQðt; sÞ½yðsÞ dwðsÞ; ’ Cðt; sÞxðsÞ
t0
where the matrix SðtÞ of the encircling ellipsoid # T S 1 ðtÞ½xðtÞ xðtÞp1 # X ðtÞ ¼ f½xðtÞ: xðtÞ xðtÞ b2 ðtÞg;
ð16Þ
20
M.V. Basin, I.R. Valadez Guzman / Journal of the Franklin Institute 339 (2002) 13–27
satisfies the integral Riccati equation SðtÞ ¼ C þ þ
Z
Z
t 1
T
Dðt; sÞR ðt; sÞD ðt; sÞ ds þ t0 t
SðsÞAT ðt; sÞ ds
t0
Z
Z
t
Aðt; sÞSðsÞ ds t0
t
SðsÞC T ðt; sÞQðt; sÞCðt; sÞSðsÞ dwðsÞ;
ð17Þ
t0
with the initial condition Sðt0 Þ ¼ C; and the positive real number b2 is determined as follows b2 ðtÞ ¼
1 2
Z
t
½yðsÞ ’ Cðt; sÞxðsÞT Qðt; sÞ½yðsÞ ’ Cðt; sÞxðsÞ dwðsÞ: t0
Furthermore, the optimal control problem for the system state (12) and cost function (15) is equivalent to the optimal control problem for the optimal minmax estimate (16) and the cost function J represented as Z 1 1 T T # # J ¼ ½xðTÞ z0 T F1 ½xðTÞ z0 þ u ðt; sÞKðt; sÞuðt; sÞ dvðsÞ 2 2 t0 Z Z 1 T T 1 T # ds þ tr½SðsÞLðt; sÞds þ tr½SðTÞF1 ; þ x# ðsÞLðt; sÞxðsÞ 2 t0 2 t0
ð18Þ
which can be reduced to the effective cost function M 1 # # z0 T F1 ½xðTÞ z0 M ¼ ½xðTÞ 2 Z T Z 1 1 T T T # ds: þ u ðt; sÞKðt; sÞuðt; sÞ dvðsÞ þ x# ðsÞLðt; sÞxðsÞ 2 t0 2 t0
ð19Þ
Thus, the solution for the optimal control problem specified by (12), (15) can be found solving the optimal control problem given by (16), (19) and using (18) for the minimal value of the criterion J: 3.3. Optimal control problem solution for discontinuous systems Based on the solution of the optimal control problem obtained in [10] in the case of an observable discontinuous system state, the following results are valid for the # is optimal control problem (16), (19), where the system state (the minmax estimate x) completely available and, therefore, observable. The optimal control law is given by # un ðt; sÞ ¼ K 1 ðt; sÞBT ðt; sÞPðsÞxðsÞ;
ð20Þ
M.V. Basin, I.R. Valadez Guzman / Journal of the Franklin Institute 339 (2002) 13–27
where PðtÞ is the solution of the integral Riccati equation Z t PðtÞ ¼ Pðt0 Þ þ ½Qðt; sÞ AT ðt; sÞPðsÞ PðsÞAðt; sÞ ds 0 Z t ½PðsÞBðt; sÞK 1 ðt; sÞBT ðt; sÞPðsÞ dvðsÞ;
21
ð21Þ
0
with the terminal condition PðTÞ ¼ F1 : Upon substituting the optimal control (20) into Eq. (12) for the reconstructed # the following optimally controlled state estimate equation is obtained: system state x; Z t Z t # ds þ # dvðsÞ # ¼ x0 þ Aðt; sÞxðsÞ Bðt; sÞK 1 ðt; sÞBT ðt; sÞPðsÞxðsÞ xðtÞ þ
Z
t0 t
t0
# SðsÞC T ðt; sÞQðt; sÞ½yðsÞ dwðsÞ: ’ Cðt; sÞxðsÞ
ð22Þ
t0
The obtained Eqs. (21) and (22), as well as Eq. (17) for the estimate variance SðtÞ; are integral equations with integration w.r.t. discontinuous measures generated by bounded variation functions vðtÞ and wðtÞ; which do not tell us how to compute # and the gain matrices jumps of the controller and filtering variables (the estimate xðtÞ PðtÞ and SðtÞ) at the discontinuity points of the functions vðtÞ and wðtÞ; corresponding to discontinuities in the system states xðtÞ and the observation process yðtÞ: Nevertheless, in accordance with Theorem 3 in [18], the jumps can be computed solving the following system of differential equations: dx# dv # ¼ Bðt; tÞK 1 ðt; tÞBT ðt; tÞPðwÞxðwÞ dw dw dy T # # # þ SðwÞC ðt; tÞQðt; tÞ Cðt; tÞxðwÞ ; xðwðtÞÞ ¼ xðtÞ; dw dS ¼ SðwÞC T ðt; tÞQðt; tÞCðt; tÞSðwÞ; dw
SðwðtÞÞ ¼ SðtÞ;
dP dv ¼ PðwÞBðt; tÞK 1 ðt; tÞBT ðt; tÞPðwÞ ; dw dw which yields the following jump expressions:
PðwðtÞÞ ¼ PðtÞ;
# ¼ SðtÞfI þ C T ðt; tÞQðt; tÞCðt; tÞSðtÞDwðtÞg1 DxðtÞ #
C T ðt; tÞQðt; tÞ½DyðtÞ Cðt; tÞxðtÞDwðtÞ þ Bðt; tÞK 1 ðt; tÞBT ðt; tÞfI þ C T ðt; tÞQðt; tÞCðt; tÞ #
SðtÞDwðtÞg1 PðtÞxðtÞDvðtÞ; DSðtÞ ¼ SðtÞfI þ C T ðt; tÞQðt; tÞCðt; tÞSðtÞDwðtÞg1 SðtÞ; DPðtÞ ¼ PðtÞfI þ Bðt; tÞK 1 ðt; tÞBT ðt; tÞPðtÞDvðtÞg1 PðtÞ: Following [18], the obtained jump expressions can be incorporated into the controller and filtering equations (21), (22) and (17) using the form of the equivalent
22
M.V. Basin, I.R. Valadez Guzman / Journal of the Franklin Institute 339 (2002) 13–27
equations with a measure # ¼ x0 xðtÞ
Z
t
# ds þ Aðt; sÞxðsÞ
Z
t0
þ
t
Bðt; sÞuðt; sÞ ds t0
Z
t
Bðt; sÞK 1 ðt; sÞBT ðt; sÞfI þ C T ðt; sÞQðt; sÞCðt; sÞ t0
#
SðsÞDwðsÞg1 PðsÞxðsÞ dvðsÞ Z t SðsÞfI þ C T ðt; sÞQðt; sÞCðt; sÞSðsÞDwðsÞg1 þ t0
#
C T ðt; sÞQðt; sÞ½dyðsÞ Cðt; sÞxðsÞ dwðsÞ;
SðtÞ ¼ C þ þ
Z
t
Dðt; sÞR1 ðt; sÞDT ðt; sÞ ds
t0 t
Z
ð23Þ
Aðt; sÞSðsÞ ds þ
t0 Z t
Z
t
SðsÞAT ðt; sÞ ds
t0
SðsÞfI þ C T ðt; sÞQðt; sÞCðt; sÞSðsÞDwðsÞg1
t0
C T ðt; sÞQðt; sÞCðt; sÞSðsÞ dwðsÞ;
ð24Þ
with the initial condition Sðt0 Þ ¼ C; and PðtÞ ¼ Pðt0 Þ þ
Z
Z
t
½Qðt; sÞ AT ðt; sÞPðsÞ PðsÞAðt; sÞ ds
t0 t
PðsÞfI þ Bðt; sÞK 1 ðt; sÞBT ðt; sÞPðsÞDvðsÞg1
t0
Bðt; sÞK 1 ðt; sÞBT ðt; sÞPðsÞ dvðsÞ;
ð25Þ
with the terminal condition PðTÞ ¼ F1 ; where DwðtÞ; DvðtÞ; and DyðtÞ are the jumps of the bounded variation functions wðtÞ; vðtÞ; and the observation process yðtÞ at a # point t; respectively, and xðtÞ; SðtÞ; and PðtÞ are the values of the discontinuous # controller and filtering parameters (the estimate xðtÞ; and the gain matrices SðtÞ and PðtÞ) at a point t from the left. The optimally controlled state estimate equation (23), the regulator gain matrix equation (25), filter gain matrix equation (24), and the optimal control law (20) give the complete solution to the optimal controller problem for unobservable states of discontinuous integral systems governed by Volterra equations with deterministic disturbances, including analytic expressions for jumps of the controller and filtering variables at the discontinuity points of the real system state xðtÞ and the observation process yðtÞ:
M.V. Basin, I.R. Valadez Guzman / Journal of the Franklin Institute 339 (2002) 13–27
23
4. Movement of missile with impulsive and jet motors and unobservable velocity Let us consider the optimal control problem for movement of a missile with two motors, impulsive and jet (continuous), whose task is to reach the maximal possible altitude at a certain time moment T > 0 with the minimal possible fuel consumption. The missile movement is considered governed by the state equations (cf. [8]) Z t Z t Pp ðsÞ hðtÞ ¼ h0 þ ds; vðsÞ ds; mðtÞ ¼ m0 þ 0 0 Cðt; sÞ vðtÞ ¼
Z
t
0
Pp ðsÞ Qðh; vÞ dwðsÞ mðsÞ
Z 0
t
g ds þ
Z
t
rðsÞ ds; 0
where t0 ¼ 0; vðtÞ is the missile velocity, v0 ¼ vð0Þ ¼ 0; and information on the missile velocity is collected using the velocity measurer (observation) equation Z t Z t yðtÞ ¼ Hðt; sÞvðsÞ dwðsÞ þ F ðt; sÞqðsÞ dwðsÞ: 0
0
Here, h0 ¼ hð0Þ > 0 is the initial adjusted altitude corresponding the missile position on the earth surface, hðtÞ is the current adjusted altitude; mðsÞ is the missile and fuel mass, m0 b0; Pp ðtÞ is the propulsion force; Cðt; sÞo0 is the difference factor of the ideal velocities of the missile at time t and the outflowed fuel at time s; which is varying with change of altitude and, consequently, temperature, pressure, gravity acceleration, etc.; g is the gravity acceleration; rðsÞ is the input disturbance due to inaccuracy of modeling velocity dynamics; wðsÞ is a bounded variation function which represents functioning of two missile motors, impulsive and jet (continuous): the jet motor expels fuel gradually and the impulsive one does this instantaneously at a certain time moment. t1 ; 0pt1 pT: Thus, the motors functioning is described using decomposition of wðtÞ into its continuous component wc ðtÞ (continuous jet) and the Heaviside function wðt t1 Þ with jump at the moment t1 (impulsive motor), i.e., wðtÞ ¼ wc ðtÞ þ wðt t1 Þ: Due to inaccuracy of measuring devices and natural reasons (such as the Doppler effect), the velocity measurer collects information on the velocity values not only at the current time t but as a summation with the values at some previous time moments, R t presenting a classical case of data fusion. This effect is modeled by the term 0 Hðt; sÞvðsÞ dwðsÞ in the observation equation, whereas the last term Rt F ðt; sÞqðsÞ dwðsÞ takes into account influence of disturbances qðsÞ affecting 0 measurements of the velocities vðsÞ at the observation moment t: Integration with the discontinuous measure dwðtÞ in the observation equation enables one to correctly incorporate the jump in the system state (missile velocity) into the model of observations. The total energy of the disturbances (total error of modeling) satisfies the restriction Z Z 1 t T 1 t T r ðt; sÞrðt; sÞ ds þ q ðt; sÞF ðt; sÞF ðt; sÞqðt; sÞ dwðsÞp1: 2 0 2 0 It is assumed that the atmosphere resistance force is absent: Qðh; vÞ ¼ 0:
M.V. Basin, I.R. Valadez Guzman / Journal of the Franklin Institute 339 (2002) 13–27
24
Upon selecting the mass outflow function uðsÞ ¼ ðmðsÞ=mðsÞÞ ¼ ðd=dsÞ½lnðmðsÞÞ as ’ control, the optimal control problems is completely stated for the system state xðtÞ ¼ ½hðtÞ; vðtÞ governed by the equation Z t Z 1 Z t Z t xðtÞ ¼ x0 þ AxðsÞ ds þ Bðt; sÞuðsÞ dwðsÞ þ G ds þ DrðsÞ ds; 0
where
" xðtÞ ¼
hðsÞ
0
#
"
0
0
1
#
"
0
0
#
; A¼ ; Bðt; sÞ ¼ ; vðsÞ 0 0 Cðt; sÞ " # " # 0 0 mðsÞ d ’ ¼ ½lnðmðsÞÞ; D¼ ; G¼ ; uðsÞ ¼ mðsÞ ds 1 g
x0 ¼ ½h0 ; 0; and the cost function to be minimized " " ##T " " ## hn hn 1 J ¼ xðTÞ c xðTÞ 2 0 0 Z T 1 þ uðsÞuðsÞ dwðsÞ- min; uð Þ 2 0 where c¼
"
1 0
# 0 ; 0
hn bh0
and T > 0 is certain time moment. The observation equation takes the form " # Z t Z t hðsÞ yðtÞ ¼ ½0 Hðt; sÞ dwðsÞ þ F ðt; sÞqðsÞ dwðsÞ: vðsÞ 0 0 In accordance with (20), the optimal control is given by " # hðsÞ n u ðt; sÞ ¼ ½0 Cðt; sÞPðsÞ : vðsÞ Note that the initial adjusted altitude h0 > 0 is determined from the conditions vð0Þ ¼ 0 and v’ð0Þ ¼ 0 (there is equilibrium of the missile on the earth surface at the initial time moment), which, upon substituting the optimal control un ðt; sÞ into the velocity equation, yield 0 ¼ Cðt0 ; t0 Þun ðt0 ; t0 Þ g ¼ Cð0; 0Þun ð0; 0Þ g: Thus, the initial adjusted altitude h0 > 0 is determined from the equation " # h0 g ¼ Cð0; 0Þ½0 Cð0; 0ÞPð0Þ : 0 In accordance with (23)–(25), the equations for the controlled optimal estimate # trajectory xðtÞ; the filter gain matrix SðtÞ; and the regulator gain matrix PðtÞ take
M.V. Basin, I.R. Valadez Guzman / Journal of the Franklin Institute 339 (2002) 13–27
the forms SðtÞ ¼
Z
"
1 0
Z
1
0
0
# SðsÞ ds þ ("
t
0
0 Hðt; sÞ
Z t" # 0 0
SðsÞ "
0
#
1 0
1
½0 1 ds þ
Z
"
t
SðsÞ 0
0
0
1
0
25
# ds
# " # )1 0 0 þ ½0 Hðt; sÞSðsÞDwðsÞ 1 Hðt; sÞ
½0 Hðt; sÞSðsÞ dwðsÞ
with the initial condition Sð0Þ ¼ 0; " # # Z t Z t " 0 1 0 0 PðsÞ ds PðsÞ ds PðtÞ ¼ Pð0Þ 0 0 1 0 0 0 (" # " # )1 Z t 1 0 0 PðsÞ þ ½0 Cðt; sÞ PðsÞDwðsÞ 0 1 Cðt; sÞ 0 " # 0
½0 Cðt; sÞPðsÞ dwðsÞ; Cðt; sÞ with the terminal condition PðTÞ ¼ c; and # ) # Z t (" Z t " 0 1 0 # þ G ds þ # ¼ x0 þ ½0 Cðt; sÞ xðsÞ xðtÞ 0 0 Cðt; sÞ 0 0 (" # " # )1 1 0 0
þ ½0 Hðt; sÞ SðsÞDwðsÞ 0 1 Hðt; sÞ (" # Z t 1 0 # dwðsÞ þ
PðsÞxðsÞ SðsÞ 0 1 0 " # )1 0 þ ½0 Hðt; sÞ SðsÞDwðsÞ Hðt; sÞ " # 0 #
½dyðsÞ ½0 Hðt; sÞxðsÞ dwðsÞ; Hðt; sÞ # with the initial condition xð0Þ ¼ ½h00 ; and their jumps at the point t1 ; where the impulsive motor is applied, are equal to (" # " )1 # 1 0 0 DSðt1 Þ ¼ Sðt1 Þ þ ½0 Hðt1 ; t1 ÞSðt1 ÞDwðt1 Þ 0 1 Hðt1 ; t1 Þ " # 0
½0 Hðt1 ; t1 Þ Sðt1 ÞDwðt1 Þ; Hðt1 ; t1 Þ
26
M.V. Basin, I.R. Valadez Guzman / Journal of the Franklin Institute 339 (2002) 13–27
(" DPðt1 Þ ¼ Pðt1 Þ "
1 0
0 Cðt1 ; t1 Þ
# " )1 # 0 0 þ ½0 Cðt1 ; t1 ÞPðt1 ÞDwðt1 Þ 1 Cðt1 ; t1 Þ # ½0 Cðt1 ; t1 Þ Pðt1 ÞDwðt1 Þ;
(" # # 0 1 0 # 1Þ ¼ Dxðt ½0 Cðt1 ; t1 Þ Cðt1 ; t1 Þ 0 1 )1 " # 0 # 1 ÞDwðt1 Þ þ Pðt1 Þxðt ½0 Hðt1 ; t1 Þ Sðt1 ÞDwðt1 Þ Hðt1 ; t1 Þ # " # (" )1 0 1 0 þ ½0 Hðt1 ; t1 Þ Sðt1 ÞDwðt1 Þ þ Sðt1 Þ Hðt1 ; t1 Þ 0 1 " # 0 # 1 ÞDwðt1 Þ:
½Dyðt1 Þ ½0 Hðt1 ; t1 Þxðt Hðt1 ; t1 Þ "
Thus, the complete algorithm for solving this optimal control problem is described as follows: *
*
* * *
*
the equation for the filter gain matrix SðtÞ with the initial condition Sð0Þ ¼ 0 and the jump DSðt1 Þ at the point t1 is solved; the equation for the regulator gain matrix PðtÞ with the terminal condition PðTÞ ¼ c and the jump DPðt1 Þ at the point t1 is solved; the initial condition Pð0Þ is thus determined; the initial adjusted altitude h0 is calculated; substituting un ðt; sÞ into the optimal estimate equations and solving them with # ¼ h0 and v#ð0Þ ¼ 0 yields the optimally controlled estimate initial conditions hð0Þ # v#ðtÞ ¼ xðtÞ; # trajectories ½hðtÞ; where the optimally controlled velocity estimate v#ðtÞ has a jump at the point t1 ; and the optimally controlled estimate of the adjusted # is continuous; altitude hðtÞ the optimally controlled estimate of the desirable maximal altitude is determined # as hðTÞ h0 :
References [1] M.V. Basin, M.A. Villanueva-Llanes, I.R. Valadez-Guzman, On filtering problems over observations with delays, Proceedings of the 38th IEEE Conference on Decision and Control, 1999, pp. 4572–4577. [2] D. Chyung, E.B. Lee, Linear optimal systems with delays, SIAM J. Control 3 (1996) 548–575. [3] M.C. Delfour, The linear-quadratic optimal control problem with delays in state and control variables: a state space approach, SIAM J. Control 24 (1986) 835–883. [4] R.L. Alford, E.B. Lee, Sampled data hereditary systems: linear quadratic theory, IEEE Trans. Automat. Control 31 (1986) 60–65.
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[5] E.B. Lee, Y.C. You, Optimal control of bivariate linear Volterra integral type systems, Proceedings of the 26th IEEE Conference on Decision and Control 1987, pp. 721–726. [6] G.L. Fonseca, Keynesian Business Cycle Theory, Department of Economics, Baltimore University, 1998. [7] R.F. Hartl, Optimal dynamic advertising policies for hereditary processes, J. Optim. Theory Appl. 43 (1984) 51–72. [8] A.E. Bryson, Y.C. Ho, Applied Optimal Control, Hemisphere Publishing Company, New York, 1979. [9] M.V. Basin, I.R. Valadez-Guzman, Minmax filtering in Volterra systems, Proceedings of the American Control Conference, Chicago, IL, 2000, pp. 1380–1385. [10] M.V. Basin, I.R. Valadez-Guzman, Optimal minmax filtering and control in discontinuous Volterra systems, Proceedings of the American Control Conference, Chicago, IL, 2000, pp. 904–909. [11] A.V. Balakrishnan, On stochastic bang–bang control, Lecture Notes on Control Information Sciences, Vol. 25, Springer, New York, 1980, pp. 221–238. [12] H. Kwakernaak, R. Sivan, Linear Optimal Control Systems, Wiley, New York, 1972. [13] D.P. Bertsekas, I.B. Rhodes, Recursive state estimation for a setmembership description of uncertainty, IEEE Trans. Automat. Control AC-16 (1971) 117–128. [14] A.J. Krener, Kalman-Bucy and minmax filtering, IEEE Trans. Automat. Contr. AC-25 (1980) 291–292. [15] M.L. Kleptsina, A.Yu. Veretennikov, On filtering and properties of conditional laws of Ito–Volterra processes, Statistics and control of stochastic processes. Steklov seminar, 1984, Optimization Software Inc., Publication Division, New York, 1985, pp. 179–196. [16] L.E. Shaikhet, On an optimal control problem of partly observable stochastic Volterra’s process, Probl. Control Inf. Theory 16 (1987) 439–448. [17] Yu.V. Orlov, M.V. Basin, On minmax filtering over discrete-continuous observations, IEEE Trans. Automat. Control AC-40 (1995) 1623–1626. [18] M.V. Basin, M.A. Villanueva-Llanes, On filtering problems over Ito–Volterra observations, Proceedings of the American Control Conference, San Diego, CA, 1999, pp. 3407–3412.