On the Separation Theorem of Stochastic Systems in the Case Of Continuous Observation Channels with Memory

Transcription

1 Journal of Physics: Conference eries PAPER OPEN ACCE On he eparaion heorem of ochasic ysems in he Case Of Coninuous Observaion Channels wih Memory o cie his aricle: V Rozhova e al 15 J. Phys.: Conf. er View he aricle online for updaes and enhancemens. Relaed conen - On he eparaion heorem of ochasic ysems in he Case Of Coninuous- Discree Observaion Channels wih Memory V Rozhova, V I Rozhova, V V Lasuov e al. - AN ABRAC CHEME FOR HE MEHOD OF VARIAIONAND NECEARY CONDIION FOR AN EXREMUM V I Kazimirov, V I Ploniov and I M arobines - Geomeric properies of he se of Banach limis E M emenov, F A uochev and A Usachev his conen was downloaded from IP address on 19/9/18 a :1

2 1h European Worshop on Advanced Conrol and Diagnosis (ACD 15) Journal of Physics: Conference eries 659 (15) 14 doi:1.188/ /659/1/14 On he eparaion heorem of ochasic ysems in he Case Of Coninuous Observaion Channels wih Memory V Rozhova 1, V I Rozhova, V V Lasuov 3 and E V Devyashina 4 1 Professor, Deparmen of Higher Mahemaics, Insiue of Physics and echnology, Naional Research oms Polyechnic Universiy, Lenina 3, oms 6345, Russia enior eacher, Deparmen of Higher Mahemaics, Insiue of Physics and echnology, Naional Research oms Polyechnic Universiy, Lenina 3, oms 6345, Russia 3 Assisan Professor, Deparmen of Higher Mahemaics, Insiue of Physics and echnology, Naional Research oms Polyechnic Universiy, Lenina 3, oms 6345, Russia 4 uden, Elie echnical Educaion, Naional Research oms Polyechnic Universiy, Lenina 3, oms 6345, Russia rozhova@pu.ru Absrac. he aricle proves he separaion heorem for opimal conrol of sochasic sysems in he case when an observed coninuous-ime process possess memory of arbirary raion relaing o a sae vecor. 1. Inroducion he separaion heorem [1] is he basis for a heory of opimal conrol by incompleely observable sochasic sysems. Being fundamenal heoreical resuls, i ensured solving of a number of imporan pracical problems in a wide range of areas [, 3]. Wih he resuls [4, 5], we provided a generalized separaion heorem in he case when observaions have memory of arbirary raio, i. e. hey depend no only on curren values bu also on an arbirary number of previous values of he sysem sae vecor. Used noaions: P is even probabiliy; E is he mahemaical expecaion; normal (Gaussian) densiy wih parameers a and b by N {a; b}; and r are deerminan and race of marix; D he inversion marix of D ; D denoes ranspose of a marix or a vecor; (non-negaive) definie marix. 1 D is D is posiive. he problem saemen On а cerain probabiliy space, F,F F, P [6] he unobservable n -dimensional process x (sae vecor of sysem) and he observable l - dimensional process z wih coninuous ime are deermined by he sochasic differenial equaions (Io s differenial rule), x, u d 1, x dw,,, dx f (1) dz h, x, x,, x, zd dv, () 1 N Conen from his wor may be used under he erms of he Creaive Commons Aribuion 3. licence. Any furher disribuion of his wor mus mainain aribuion o he auhor(s) and he ile of he wor, journal ciaion and DOI. Published under licence by Ld 1

3 1h European Worshop on Advanced Conrol and Diagnosis (ACD 15) Journal of Physics: Conference eries 659 (15) 14 doi:1.188/ /659/1/14 where u is I is assumed: 1) ) x, w, r dimensional conrol vecor, N N 1 1 m, and cons, 1; N. w and v are r1 dimensional and r dimensional sandard Wiener processes; v are joinly independen; 3) are coninuous funcions for all f, h, 1, argumens; 4) Q 1 1, p x Px x x. as: o find conrol funcional, where u z u, R ; 5) original densiy is se u ha provides opimum condiion, on he class of, z z ; s s J Eb x s, x, u ds p x min, u, s s F z, -measurable where 1 u us; s, b,. o solve he se as, le us apply he mehod of sufficien coordinaes [7], assuming ha here is F - measurable process z ha fully characerizes poserior densiy z p x Px x z x (3) (4) of x sysem sae vecor, on he one hand. On he oher hand, i can be found by means of p x. Remar 1. We consider ha he opimal conrol daes from momen 1. An arbirary measurable process is used as u on he inerval,. z F - 3. Preliminary resuls In accordance wih he sufficien coordinaes mehod, we inroduce Bellman funcion where, min Eb, x u /, x /, u / d. heorem 1. Le he following condiions be me: 1) process is Marov diffusion process wih characerisics. ) Le process x p a 1, lim E 1, lim E D hen, Bellman equaion for, has he form ; be Marov process wih ransiion probabiliy densiy (5), (6), (7) / / / /, x,, Px x, / x x, * min L u /. (8),, E, x, u, E b, x,, (1) (9)

4 1h European Worshop on Advanced Conrol and Diagnosis (ACD 15) Journal of Physics: Conference eries 659 (15) 14 doi:1.188/ /659/1/14 * where L, denoes inverse Kolmogorov operaor ha corresponds o process [6], i.e.,, *, 1 L,, a, rd, (11) and he smalles value of performance crierion has he form, Proof. Le hen, by expanding operaor min p u p J. x Px x x. (1) E ino (), and considering condiion), we obain, / / / / / x,,,,,,,,,,, x u p x x dx d d b x p x x dx d dx, where u u s s;. Lemma 1. Accurae o o, he funcion, saisfies - recurrence equaion, min,,, p d, up x dx o, (14) u (13) where,, P p (15) is ransiion probabiliy densiy of Marov process. Proof. Expressing inerval, as,,,, we obain from (13) p, min min 1,,, u u (16) / /, p x, x, up, x,, dx d d d 1 (17) / / / / / x,,,,,,,,,,, x u p x x dx d d b x p x x dx d d / / o far as, x,, x, p, when, i follows from (17), ha,, up x dx o. 1 (18) (19) Marov process ransiion probabiliy densiyx, (see condiion ) saisfies he Chapman- Kolmogorov equaion [8] Wih regard o (1), (15), x,, p, x,, x, p, x,, dx d. p () p, x,, p x dx p x p,,, (1) 3

5 1h European Worshop on Advanced Conrol and Diagnosis (ACD 15) Journal of Physics: Conference eries 659 (15) 14 doi:1.188/ /659/1/14 hen, i follows from (18), (), (1) ha p / /,, p x, x, up, x,, b / / /, x p, x,, x, dx d dx d., / dx d d / / () Wih regard o (19), i follows from (16), ha, minmin,, up x dx o. (3) u u ubsiuing () in (3) wih regard o (13) leads o (14). Lemma 1 is proved. We expand,,, 1, hen, wih regard o (6), (7), (15) i follows, ha, p,, d,, o. (4) (5),, p,, d, a o,,, p,, d r r, D p,, d E o E o, E o. (6) (7) o (8) ubsiuion of (4) in (14) wih regard o he laer formulae and (11) leads o he relaion ( b g g b, rbd rdb) *, min, L,, up x dx o. (9) u, x In (9) proceeding o he limi wih and considering (1), we obain equaion (9). Boundary condiion (1) and he expression for J arise from (3), (5). heorem 1 is proved. Affirmaion 1. Le f F x B,, x Nx; u 1 1 p, (3), h H x H N x, (31) 1 4

6 1h European Worshop on Advanced Conrol and Diagnosis (ACD 15) Journal of Physics: Conference eries 659 (15) 14 doi:1.188/ /659/1/14 Ex z Ex z, (3), 1; N, (33) hen, for a poseriori densiy (4) propery, x x E z, (34) x, E x, he densiy parameers are specified by he equaions where d, z, (35), Ex x, z, (36),,,, i i E x x i z, (37) i 1; N, i ; N, i. (38) p x x;, N (39) 1 F B u d H R d z, d (4) d (41) 1, H R d z, 1 d F F H R H Q, d his relaion arises from [4]. 1, d H R H, (4) d (43) (44) 1, d F, H R H, dz d i 1,, d H R H i (45) dz H H,, H j1 i N j j d (46) N H H j j j, j1 N H, H, j j, (47) H,. (48) Lemma. A sufficien coordinaes vecor is an opimal in a mean-roo-square sense esimae of he process (9)) x, i.e. z, which is a Marov diffusion process wih he characerisics (see a 1, F B u, D, H R H j j, (49) 5

7 1h European Worshop on Advanced Conrol and Diagnosis (ACD 15) Journal of Physics: Conference eries 659 (15) 14 doi:1.188/ /659/1/14 Proof. ince, according o (4), does no depend on z, hen i follows from (39), ha. According o [6], process z which Z z, F, is he Wiener process wih hen, Marov propery z, a differenial of which is given by (46), and for z z z F Rs E ds. (5) and formulae (49) arise from (41), (5). Affirmaion. he coupled processes x ; are a Marov diffusion process. Proof. he saemen is proved by (1), (31), (41) wih regard o Lemma. Remar. ince he condiions of heorem 1 are saisfied for, hen,,, and i follows from (3)-(5), ha where, a and, ubsiuion of E z (46), (5)). 4. Main resuls heorem. Le L, z,, * min L,, E, x, u (51) u Eb, x z, (5),,, *, 1 L,, a, rd, (53) D have he form (49). for E resuls from z F measurabiliy of he process x L x u N u, b x x where N, are symmeric marices, and L, equaion (51) and boundary condiion (5) ae he following form, min F u L B u, 1 rh u N u r L, N 1 R H,, r. (see (41),, (54). Hence, he Bellman, (55) (56) Proof. From (54), wih regard o (37) and Analogous E x, u z E x L x u L z F -measurabiliy u, we obain u z Ex L x z u L, L r L u L u. u (57) 6

8 1h European Worshop on Advanced Conrol and Diagnosis (ACD 15) Journal of Physics: Conference eries 659 (15) 14 doi:1.188/ /659/1/14 x z Ex x z r E b. (58), ubsiuion of (49), (53), (57), in (51) resuls in (55), and subsiuion of (58) in (5) leads o (56). he heorem is proved. From now on, he -derivaive will be idenified by he poin a he op. heorem 3 (he separaion heorem). he opimal conrol u has he form u 1 where an opimal in a mean-roo-square sense esimae equaions (4)-(48), where wih boundary equaion and he smalles value u u, he marix N B, (59) of he sae vecor x is defined by filer is defined by marix differenial Riccai equaion 1 F F B N B L, (6), J of qualiy crierion has he following form (61) 1 J r rl d r H R H d (6) Proof. aing u -derivaive from he lef-hand side of (55), we obain he equaion o compue he opimal conrol B, N u. Hence, we obain he expression for opimal conrol by means of Bellman funcion in he following form 1 1 N B, (63) u. (64) ubsiuing (64) in (55), we obain a second-order parial equaion for Bellman funcion in he following form, F L r, 1, N B 1 L r H R H B,,. We solve he equaion (65) by using he separaion of variables mehod in he form, l, where l -unnown scalar funcion, and - unnown marix n n symmery condiion. hereafer (65) (66), l,,,, -funcion wih imposed. (67) 7

9 1h European Worshop on Advanced Conrol and Diagnosis (ACD 15) Journal of Physics: Conference eries 659 (15) 14 doi:1.188/ /659/1/14 As long as is imposed wih symmery condiion, hen, wih regard o (67) and he fac ha b b b b 1 is rue for scalar b, we obain F, F ubsiuing (67), (68) in (65), we have he following formula F F. 1 F F B N B 1 L rl r H R H l Ne according o he mehod of separaion of variables, we se he coefficiens in (69) equal, provided powers are equal. hen, we have he equaion (6) for, and for l we obain he following equaion According o (66). 1 L r H R H (68) (69) l r (7), l. (71) he boundary condiion (61) for equaion (6) follows from comparison (56) and (71), while he boundary condiion for (7) has he form According o heorem 1, J l r. (7),, and, hus, i follows from (66), ha J. l (73) he soluion o equaion (7) wih boundary condiion (7) has he form 1 l r rl d r H R H d. (74) When, subsiuion of (74) in (73) resuls in (6). Applying (67) in (64), we ge (59). 5. Conclusions Comparing resuls of heorem 3 wih separaion heorem for memoryless sources in he classical case [1], we come o he conclusion ha he expression of opimal conrol has he same form (59). he difference is he following: in he classical case, Kalman filer esimaes he sae vecor, while in he case considered above, i is he filer (4)-(48), ha esimaes no only filering parameers for he curren value of sae vecor, for previous values of he sae vecor x, bu also esimaes inerpolaion x, 1; N. hus, he expression for smalles values of qualiy crierion J changes. Acnowledgmens All commens and suggesions by criical readers and ediorial assisance are acnowledged wih graiude. References [1] Wonham W M 1968 On he separaion heorem of sochasic conrol IAM J. Conrol

10 1h European Worshop on Advanced Conrol and Diagnosis (ACD 15) Journal of Physics: Conference eries 659 (15) 14 doi:1.188/ /659/1/14 [] Bryson A E, Yu-Chi-Ho 1969 Applied opimal conrol (London: Blaisdell publishing) [3] Boguslavsii A I 197 Navigaion and conrol mehods by incomplee saisical informaion (Moscow: Mashinosroenie) [4] Abaumova O L, Demin N, usho V 1995 Filraion of sochasic processes for coninuous and discree observaions wih memory. II. ynhesis of filers Auomaion and Remoe Conrol 56:1, [5] Demin N, Rozhova V Coninuously discree esimaion of sochasic processes for observaions wih memory under anomalous noise: ynhesis Journal of Compuer and ysems ciences Inernaional 39:3, [6] Lipser R, hiryaev A N 1 aisics of Random Processes (Berlin: pringer-verlag Berlin Heidelberg) [7] raonovich R L 1968 Condiional Marov processes and heir applicaion o he heory of opimal conrol (New Yor: American Elsevier Publishing Company) [8] Gihman I I, orohod AV 4 he heory of ochasic Processes (Berlin: pringer-verlag Berlin Heidelberg) 9