Přepis řeči - CONVERGENCE RESULTS IN DISTRIBUTED KALMAN FILTERING

0:00:14	i
0:00:18	i
0:00:28	okay okay
0:00:29	i will catch up
0:00:31	yeah
0:00:32	oh
0:00:33	um
0:00:34	so
0:00:35	you had to tie them a simple uh
0:00:38	we talk about this really the common filtering for caminfo to really the well known
0:00:42	but it is utilised
0:00:43	no i to our problem
0:00:45	okay so we establish some asymptotic
0:00:48	uh commanders results
0:00:50	uh
0:00:51	before i
0:00:52	stuff i wanted you most credit to uh sumatra and the clusters
0:00:56	so i do the leader in this
0:00:58	oh less mode you it's
0:01:00	the talk a little bit
0:01:01	so that's yeah random filter uh where things going on
0:01:06	i the linear regression model
0:01:08	so i R W is noise such that the whole lecture twice to is a lecture run them
0:01:15	process that's okay
0:01:16	so each sensor here
0:01:18	how a hard i but of the we score a actually in the sense that this uh of the which
0:01:23	a matrix is you really or
0:01:25	i matrix
0:01:26	okay
0:01:26	so is that it is that um can out
0:01:29	basically i how a good estimate of the unknown vector
0:01:33	we got
0:01:35	so
0:01:35	what to do so we should allow the sensors to collaborate
0:01:39	know this addition get the way you know other two
0:01:42	achieve the past
0:01:43	if possible i sufficient uh i estimation performance
0:01:47	but we don't know why they're we can achieve the same performance i L
0:01:50	centralized counterpart well for the example you've we have a few says are we we're you have would be a
0:01:55	use a matrix to stack all the most image X okay the coming or team the same performance as as
0:02:01	and that's as the solution
0:02:03	so that's or go to use that is
0:02:06	okay
0:02:07	so of course this kind of application uh okay be uh
0:02:10	uh applied to many and new uh
0:02:13	uh us such as the smart grid you do state estimation for all the
0:02:17	pass is
0:02:18	uh social networks so
0:02:20	okay
0:02:22	so for this problem might if people are uh out
0:02:25	where you interested and D the law of good work
0:02:28	uh
0:02:29	but you're are it this kind of a uh
0:02:31	these to build you guys to me G is it can there is based L
0:02:35	some of good out of addition based okay so i the limitation
0:02:39	you early the have the ability is that a gun it even worse soon
0:02:43	that
0:02:46	at each sensor
0:02:47	the vocal cortical billy a defect ability to not enough
0:02:50	and we are you assume that if we are kind it's that all the C matrix to the right you
0:02:54	can how certain
0:02:56	uh
0:02:57	these like really T so that so called a we call it can mobile
0:03:00	gullible able uh
0:03:02	defect really okay
0:03:03	basically assume the to i as a central an okay but we now it do things a locally
0:03:09	in that case
0:03:10	the you this a results
0:03:11	uh you dense and see this kind of iris the ability to K so called defined that will show you
0:03:16	remains see room later wrong okay
0:03:18	so
0:03:20	just so of this special case cases
0:03:22	the so some have to go that's the believe he was established but under why we stick condition
0:03:26	so in this work we show you were that's we just as a uh i need some very weak assumption
0:03:32	on the system
0:03:33	so we can prove
0:03:34	asymptotic stability for the error covariance matrix
0:03:37	okay
0:03:39	so a recently i read this is uh a previous work go for this one so same mouse there if
0:03:44	we same for author
0:03:45	assume my
0:03:46	so
0:03:47	basically you that work we assume that a model the sensors will allow
0:03:53	ish i each peer rate
0:03:54	will a a lot of one round of estimate selecting
0:03:58	between one pair of neighbours
0:04:01	okay
0:04:02	and and is that kind of a it were uh approach we i
0:04:06	prove certain
0:04:07	uh that's them at all because the ability for the data
0:04:10	okay
0:04:11	so a the may and needs the them then a mix system a result
0:04:16	okay in of or a node
0:04:18	so
0:04:19	how are do that work
0:04:21	we can then they use for one not a free them well he's
0:04:24	in addition to a lot of i C meets my be we can also a lot of the and
0:04:29	so okay
0:04:31	so you will do that
0:04:32	how much for we aim from and no can have
0:04:35	and if we do that
0:04:37	how fast
0:04:38	we can approach the central like the performance
0:04:41	okay
0:04:42	actually we we found that with this new free them
0:04:45	the whole analysis to we use of year cannot be applied anymore sorry
0:04:50	so we have to redefine re introduce there to new to to analyse the performance
0:04:55	yeah so has started with the side up
0:04:58	so let's say oh
0:05:00	we have a collect are you didn't or scissors okay
0:05:03	so we estimate of lecture
0:05:06	random process okay a lot of time
0:05:09	so we allow communication between sensors
0:05:11	okay
0:05:12	so here the ratings a
0:05:14	i don't want to a of and the the comedy re guy but he are really totally only uh
0:05:18	you value when the two to here rate
0:05:20	so that here defined i the link i actually
0:05:23	i Q we should rate so per second
0:05:25	how many in it
0:05:27	is i i okay so that's the mean may link
0:05:30	i'm also so the them's the come to get an scheme a long time is random
0:05:34	okay
0:05:36	okay yeah
0:05:38	we assume that
0:05:40	we we a constant uh them largest communicate a rate you can assume K so
0:05:46	i should be sure that i any tell that he would really easy enough
0:05:50	okay
0:05:52	so that's the main in a approach at the each moment uh P read the K okay so here's a
0:05:57	K K plus one so first so we do have the okay you she's as or do uh of the
0:06:01	region that we do communication or model scissors so the communication
0:06:06	you divided into two parts for us so we do i estimates well
0:06:10	so you each week peak run them if you go parents of neighbours to strap of their previous estimate
0:06:16	okay
0:06:16	after that
0:06:17	we allow multiple round of all observation
0:06:21	uh
0:06:22	we call obligation okay
0:06:24	so so that means you don't keep yourself
0:06:26	uh you always i C you only get the a neighbours but i vision means
0:06:31	you also keep yourself your all alone
0:06:34	all the we a along with neighbours okay for different that's different
0:06:38	a a a a and we go to do it's not total communication rate why a third an upper about
0:06:43	okay
0:06:44	so we're going to a
0:06:46	sure you that such kind of an estimate swiping easy enough
0:06:49	to guarantee a some talk "'cause" the ability of the i-th meeting error
0:06:53	okay
0:06:53	and the extra of the vision
0:06:55	aggregation process
0:06:57	can
0:06:58	dramatically improve
0:07:00	so i estimation performance in term of
0:07:02	the speed to converge to that's centralized
0:07:05	formants
0:07:06	okay
0:07:07	so
0:07:08	in this for local we call it's a modified
0:07:11	gauss safe
0:07:12	uh based interactive a common filter okay so um do yeah K F
0:07:17	so we have a certain us
0:07:19	us
0:07:20	or or side tops okay
0:07:22	so i that's that i want to do a a total rate of communication lies that's to upper bound
0:07:26	so that how many coming occasions we have we have a two times why is uh
0:07:31	uh uh i C made the being
0:07:33	okay another another E is uh all the which should exchange so the error rate just um of this the
0:07:39	kind of what you who's should be limit to but this are about so that's all read constraints
0:07:44	okay
0:07:46	and uh
0:07:48	now we a to be we use sort and uh you tell us about that is to kind of a
0:07:53	communication
0:07:54	uh in the network wise i se means way if you another ease
0:07:58	uh a of the which an exchange
0:08:00	uh or aggregation
0:08:01	so the first of why is ice mates so might be an
0:08:04	in this case we the we do is
0:08:07	we
0:08:07	we use to Q is is simple scheme okay he's not an not meeting self to this
0:08:13	one okay they we'll going summarise
0:08:15	what's a requirement for scheme to would be a reasonable you you order two
0:08:19	uh guarantee over result
0:08:21	okay
0:08:22	so that's a would have to go a simple scheme which all
0:08:25	uh
0:08:26	each time we actually with the link
0:08:29	according to our a distance metric
0:08:31	K is the success of a distance metrics we see that ideally drawing from all even distribution
0:08:37	okay so on each time of that
0:08:38	only one in P the active
0:08:40	i
0:08:41	if that i this is a matrix happen to be i didn't uh identity matrix which means
0:08:46	no node
0:08:47	we also have
0:08:48	they just keep their own i submit so that's also allowed
0:08:51	okay
0:08:52	so that's kind of what is to be a D you find
0:08:55	to guarantee the
0:08:57	for volume property okay
0:08:59	so if it this
0:09:01	if this uh
0:09:04	if this kind of it you to be defined
0:09:07	and and
0:09:08	this as some frames hold well basically was to that
0:09:11	so so called a maximum of which me
0:09:14	although each time a only a one to be active body we use that all the possible activity
0:09:19	or a long time
0:09:20	so that's not work to be connected of that's all row uh only requirement okay so we that
0:09:26	we have a fairly in fact
0:09:28	with a for cool okay
0:09:30	was as that i guess as met you matrix sequence
0:09:33	K
0:09:34	it is john and from this division D and wish discussed
0:09:38	okay as a result
0:09:39	so i
0:09:40	so called mean in is matrix
0:09:43	i about bar
0:09:44	is already reduce more small and uh of your all tick matrix
0:09:47	okay so that's all or requirement on the
0:09:50	properties of we do since may she sequence
0:09:52	okay
0:09:53	yeah are we have we also
0:09:55	can guarantee that
0:09:57	so rate we consume in this phase the i mates what you i i can face is lies that well
0:10:03	half the it remember with of how come about not we we consumed
0:10:08	as most come borrow two
0:10:10	okay
0:10:11	now the rest phase is the so called of their which and
0:10:15	uh aggregation okay
0:10:17	so this kind of a long words but
0:10:20	the message here is
0:10:22	when not have
0:10:24	oh about a lot too but it full communication between nodes okay
0:10:28	that we assume that the whole he we shall for in the four million a some for size
0:10:32	with a M to come borrow it two
0:10:34	but you this since the average communicate read consumption here will be less than come about or what to okay
0:10:40	so we don't
0:10:41	okay i'm cool to details but that's what do we did in the second phase
0:10:45	of communication
0:10:47	okay
0:10:47	so we is that of a local we can also a of the following fact
0:10:51	basically it for small the average
0:10:53	uh committee can a that we consume a the color about what to that's good because we are use of
0:10:58	half an now we use other half the photo total you less then
0:11:01	come up bar that's all that's what we were okay
0:11:04	i say we have a "'cause" of uh
0:11:08	in that
0:11:09	okay so this base each a use the collection be index
0:11:12	you know think that
0:11:13	i've know the and uh time K
0:11:16	so
0:11:17	what cat which a node that do you are you the regions
0:11:20	remember at each P are rate we all are much more round of
0:11:24	uh duration
0:11:25	exchange okay is i the results at the end of this period is should know the will how much pull
0:11:31	nodes was centre there
0:11:33	of the to to know then
0:11:35	okay so we you know
0:11:36	such a seconds of index
0:11:38	i as i K and
0:11:39	okay
0:11:40	actually we can show that
0:11:42	at the end of each period
0:11:44	although
0:11:45	we have some up to
0:11:46	somehow probability that you should know that we all receive the observations from other node but that probability the strictly
0:11:52	positive
0:11:53	okay so that's also critical to reestablish the our results with raw
0:11:59	but this is also the top apart because this sequence is a random i actually is the nonstationary
0:12:04	so that's give us trouble to prove things
0:12:08	so that's my downfall protocols for
0:12:11	estimator swiping and
0:12:13	a observation
0:12:15	aggregation
0:12:16	but when only met also have to that because
0:12:19	when he found out scheme to side of atherton three conditions
0:12:22	it's good enough
0:12:23	"'kay" the three conditions basically saying that
0:12:26	the
0:12:27	sec "'cause" uh of a is matrices do we used to decide activation of what links should be i D
0:12:32	O okay and the mean and their chicks to the beat
0:12:34	probably score so can guy uh reduce one of your T
0:12:38	right that's first requirement
0:12:39	okay K i can't is this sequence you know thing where you get the or a a a you want
0:12:44	guide observations from
0:12:46	should if i this uh probability constraint we should have a lot
0:12:49	uh up that you've probability okay to get a all other nodes of the region okay
0:12:53	you can be a small
0:12:55	so i kind of course
0:12:56	to
0:12:57	total committee can rate had to side i this constraints
0:13:00	you with this is a to with the uh protocol you I D then you are happy
0:13:04	because
0:13:05	then the the to all the results so we have a
0:13:07	i the in
0:13:08	okay
0:13:10	so no finish the
0:13:12	i a order
0:13:13	make up their wishes we finish
0:13:15	the estimate is we finish the observation aggregation now is time to update the or i right
0:13:23	so
0:13:23	yeah we assume that's
0:13:25	in the estimates to i been face what we slap a is a prediction okay based on the previous uh
0:13:31	uh read out to get and uh the the recall covariance matrix
0:13:35	so use to have this with the a well the a neighbour
0:13:37	okay
0:13:39	so which an mlp can we you know it's for the uh uh no then we need we you know
0:13:44	that that's than than a appear gaze and K or bar i'd
0:13:47	time K
0:13:48	okay so this is a face
0:13:50	then the observation aggregation give we you that
0:13:52	i'd and of each uh time i okay
0:13:55	so as and we will have a vector
0:13:57	oh of
0:13:58	all their wishes
0:13:59	from a a from other nodes that you have talk to
0:14:02	okay
0:14:03	so maybe from a hope to talk of multiple hot talk
0:14:06	you to this kind of exchange change okay
0:14:08	the best have this result we take the
0:14:11	or dicked are to one step
0:14:13	for there okay
0:14:14	so to minimize the mean square error so there's a well-known of
0:14:17	uh for for that
0:14:19	i i i i i same time we have a date data the error cornfields crimes matrix
0:14:23	so if i guide it's wide of the got are so we are ready for the next round iteration
0:14:28	right so used time we do this
0:14:29	you can that you can see that with this kind of predictor
0:14:32	we can see what we are doing in does a special case of common filter
0:14:37	okay
0:14:37	so right at we can you elements
0:14:39	what about doing a use the lazy in the results
0:14:42	you in common filter
0:14:43	uh
0:14:44	you each
0:14:45	okay
0:14:45	so
0:14:47	you limit the shape is not a for chorus all a "'cause" it if we don't that
0:14:50	what kind of probably we have
0:14:52	asymptotically or the i requires matrix
0:14:55	okay
0:14:56	so can show that at each node
0:14:57	and
0:14:58	okay so i i requires the matrix you malls according to a this T the person
0:15:03	"'kay"
0:15:04	it's quite complicated okay is a is itself is a random so is a random sequence of the matrices
0:15:10	okay
0:15:11	so now we about the no
0:15:12	you which says
0:15:13	asymptotically hardly this guy's table
0:15:16	okay
0:15:16	so what can we
0:15:17	establish regarding that
0:15:20	oh two
0:15:21	you know to do that a an a it is uh rather than they share to be the clean in
0:15:24	there so okay
0:15:25	so basically we define are are required to be a reader is the basically function being okay based on other
0:15:31	you used version
0:15:32	so
0:15:33	with this that each of this a particle a function we can rewrite
0:15:37	the a lotion of our requires a matrix of log do this or simple form
0:15:42	okay
0:15:42	so you can say that
0:15:46	there are the modulation factor here basically is this "'cause" of the indexes
0:15:50	we use you where you get
0:15:52	new uh where you get the of the regions from okay
0:15:55	and because of this segment C is analysis the and the rate this whole thing is a non-stationary
0:16:01	okay so that's really bad
0:16:02	so the result we had before kind to be used the here
0:16:05	okay
0:16:06	so
0:16:07	what that we can get here
0:16:10	so we need to make to weak assumptions
0:16:12	okay
0:16:13	yeah other to establish a main results the first why is we assume that in the uh i your regression
0:16:19	model for the uh a actor
0:16:21	so this pair of matrix
0:16:22	so this the the uh i've metrics remember member that's a linear regression matrix
0:16:27	we and of the
0:16:28	what noise
0:16:29	for matrix
0:16:30	so this pair is this
0:16:32	that uh uh
0:16:34	that that lights will okay
0:16:35	so
0:16:37	i course the plot that you uh that when is of this go uh are are are of noise
0:16:41	course to easy enough to use for that so it's not that strong
0:16:45	here
0:16:45	the second or is also not strong because for you know for central the scheme
0:16:50	we required this
0:16:51	condition K is so called a global to detect ability
0:16:55	so this pair of matrix
0:16:56	say that stack of all the small all but we should make use this
0:16:59	okay
0:17:00	so this we require
0:17:01	to be you tractable
0:17:03	"'kay"
0:17:03	the is enough
0:17:05	we only require this
0:17:06	similarly i that's and right the scheme to the main result okay establish is
0:17:11	for each node and
0:17:13	okay remember we are we are how we a distributed i-th emission scheme so we have to guarantee that ice
0:17:18	is good
0:17:19	at each node
0:17:20	okay
0:17:21	so for a guy show that
0:17:23	the error is the matrix is the stock as you body
0:17:26	okay so here a can be come and you know
0:17:28	yeah
0:17:30	and that there out is
0:17:31	i say we a but if you got a node
0:17:34	okay so it
0:17:35	can be this that can be model as the
0:17:37	drawing from a fusion uniformly
0:17:39	okay from the index
0:17:42	the core of the matrix
0:17:44	at that particular node will come or or to a description
0:17:47	okay
0:17:48	but that as well uh
0:17:51	that's basically come try threshold us that's we can try to
0:17:55	i pretty we can come roach to the theme perform as i doesn't centralized
0:18:00	scheme case the same this region as of less game
0:18:02	okay
0:18:03	the set the not the result is a fast we um are to centre as scheme we sure that
0:18:08	the screen of
0:18:09	a in less scheme
0:18:11	is the exponential or or the rate
0:18:13	come of K so that is a it is a full any find that even come or bar but if
0:18:17	we can is this one the
0:18:19	three at approaches and of in can be spanish exponentially fast
0:18:23	okay
0:18:24	so how do we prove this of
0:18:28	i mean it i just
0:18:29	sketch it
0:18:30	remember that use this guy's an null decision a so what do we do is
0:18:34	for one we construct a
0:18:36	oh as
0:18:37	a a to cool
0:18:39	process a a colour of course by modifying in this process
0:18:42	and that process is a stationary
0:18:45	okay
0:18:45	and with that have set go process
0:18:48	we can
0:18:50	apply apply the out you know a a a T S run them that exist systems okay
0:18:54	so we have a lot of an interesting without there okay
0:18:57	and i also show that the sole constructed
0:18:59	a us gender process the
0:19:02	which you to this R T S
0:19:03	the are to say R T I self
0:19:05	can be shown to would be so called all other pretty the ravine and a strong solve a linear
0:19:10	those that terms use the in the R T S you feature
0:19:12	okay
0:19:14	so
0:19:15	that was it
0:19:17	oh that some shall we have
0:19:18	such a at the colour about you can't really D and the the connection
0:19:21	a the can connectivity "'cause" that some for we have a word the network
0:19:26	the commercial read out for this i think a process
0:19:29	can be established
0:19:30	a a what this is not of for the or an no that right
0:19:33	so not the G
0:19:34	so basically by of uniting the fact
0:19:39	uh does suppose that is not station or
0:19:42	okay
0:19:43	but at the a magically a gandhi
0:19:45	so that's the key for beauty for us to build a connection at pretty step by step a connection between
0:19:51	the have a setting a process and of the or no process
0:19:55	then the
0:19:56	come as a result we got a for of this is a uh
0:19:59	so their or process can be you applied
0:20:03	so the or don't only not stationary process
0:20:05	okay
0:20:06	so
0:20:07	i'm collusion
0:20:08	basically we establish uh
0:20:11	it's but results
0:20:13	for a common three in uh
0:20:16	where are we only assume a week
0:20:18	assumptions
0:20:19	on global jack ability and a connective use of network
0:20:22	okay so this one i but that is required or a totally new approach okay
0:20:27	to solve this uh it hardly recursion system okay which is a stationary
0:20:32	okay
0:20:33	uh
0:20:35	we have some or all results on weird is established well of show that
0:20:39	the are
0:20:40	performance can approach the central the
0:20:43	uh from as which is optimal you can do okay
0:20:46	is but usually fast or or the oral communication read in the network
0:20:51	thank you
0:20:53	a
0:20:58	yeah
0:21:03	uh
0:21:10	yes
0:21:11	yes yes
0:21:12	i for here we uh
0:21:15	this for either are yeah it is a well the monophone
0:21:19	that
0:21:19	or oh
0:21:21	we we would like
0:21:22	oh
0:21:24	we don't call it uh a model one is to collect all that
0:21:37	um a a question but as concept have estimates and come from i mean
0:21:41	when you just keep your an estimate
0:21:43	oh
0:21:44	you mean why why uh we do
0:21:46	for that of
0:21:48	i
0:21:49	no you have to estimate swapping so each symbol not doesn't keep put own estimate you just get it of
0:21:54	your own you it basically
0:21:55	that
0:21:58	oh
0:21:59	hmmm
0:22:01	for that or
0:22:03	or
0:22:04	and
0:22:05	a
0:22:06	i
0:22:07	oh
0:22:08	and i C made we are able to come uh
0:22:11	is that it
0:22:12	i
0:22:13	so basis in a i don't on a method which would keep you own estimate
0:22:17	uh not know
0:22:19	we we
0:22:21	for
0:22:23	a
0:22:24	we
0:22:26	i
0:22:27	that's that
0:22:28	that
0:23:09	i

CONVERGENCE RESULTS IN DISTRIBUTED KALMAN FILTERING

Sensor Networks

Přednášející: Shuguang Cui, Autoři: Soummya Kar, Princeton University, United States; Shuguang Cui, Texam A&M University, United States; H. Vincent Poor, Princeton University, United States; José M.F. Moura, Carnegie Mellon University, United States