Does anyone have information on publications, research, or data on the theroritical and practical stability analysis of the extended Kalman filter? Most of the stuff I've found seems to be a bit *off the wall*. Many thanks, Maurice
Extended Kalman Filter
Started by ●July 28, 2011
Reply by ●July 28, 20112011-07-28
On Thu, 28 Jul 2011 10:00:30 -0700, maury wrote:> Does anyone have information on publications, research, or data on the > theroritical and practical stability analysis of the extended Kalman > filter? Most of the stuff I've found seems to be a bit *off the wall*.Most of the discussions that I've seen relating to nonlinear systems stability in general is either off the wall or really difficult. My favorite book on Kalman filtering mentions stability for both continuous-time and discrete-time EKF. "Optimal State Estimation", Dan Simon, Wiley. He doesn't devote much space to it, but he does give some references -- it may be something that you'll need to dig up by spending some time at a Uni library. -- www.wescottdesign.com
Reply by ●July 30, 20112011-07-30
On Jul 28, 1:00�pm, maury <maury...@core.com> wrote:> Does anyone have information on publications, research, or data on the > theroritical and practical stability analysis of the extended Kalman > filter? Most of the stuff I've found seems to be a bit *off the wall*.Maurice, Part of the problem with analyzing the stability of the EKF is that it's really just an approximation to the non-linear system that you're trying to do something with. As such, the stability is hugely dependent on 1) how accurate your model is and 2) how accurate your initial state estimate is. Because of this, asking for stability analysis of "the EKF" without specifying a particular problem (or putting analytic bounds on 1) and 2)) is like asking the question "how long is a piece of string?". The best analysis that I know about is by La Scala, Bitmead and James: B. F. La Scala, R. R. Bitmead, and M. R. James, �Conditions for stability of the extended Kalman filter and their application to the frequency tracking problem,� Math. Control, Signals Syst., vol. 8, no. 1, pp. 1-27, 1995. There may be better, more recent stuff... I'm a little out of touch. Their work is based (partly) on that of Song & Grizzle: http://www.eecs.umich.edu/~grizzle/papers/ekf.pdf Ciao, Peter K.
Reply by ●July 31, 20112011-07-31
On Jul 28, 1:00�pm, maury <maury...@core.com> wrote:> Does anyone have information on publications, research, or data on the > theroritical and practical stability analysis of the extended Kalman > filter? Most of the stuff I've found seems to be a bit *off the wall*.Apparently, these guys are also worth reading: Reif, K. & Unbehauen, R. (1999). Stochastic stability of the discrete- time extended Kalman filter, IEEE Transactions on Automatic Control, Vol. 44, pp. 714-728 Reif, K.; Gunther, S. & Unbehauen, R. (2000). Stochastic stability of the continuous-time extended Kalman filter, IEE Proc.-Control Theory Appl., Vol. 147, pp. 45-52 Ciao, Peter K.
Reply by ●August 1, 20112011-08-01
On Jul 28, 12:15�pm, Tim Wescott <t...@seemywebsite.com> wrote:> On Thu, 28 Jul 2011 10:00:30 -0700, maury wrote: > > Does anyone have information on publications, research, or data on the > > theroritical and practical stability analysis of the extended Kalman > > filter? Most of the stuff I've found seems to be a bit *off the wall*. > > Most of the discussions that I've seen relating to nonlinear systems > stability in general is either off the wall or really difficult. > > My favorite book on Kalman filtering mentions stability for both > continuous-time and discrete-time EKF. �"Optimal State Estimation", Dan > Simon, Wiley. �He doesn't devote much space to it, but he does give some > references -- it may be something that you'll need to dig up by spending > some time at a Uni library. > > --www.wescottdesign.comThanks Tim. I will look into it.
Reply by ●August 1, 20112011-08-01
On Jul 28, 7:00�pm, maury <maury...@core.com> wrote:> Does anyone have information on publications, research, or data on the > theroritical and practical stability analysis of the extended Kalman > filter? Most of the stuff I've found seems to be a bit *off the wall*. > > Many thanks, > > MauriceThe problem with nonlinear filters is that their only common denominator is that they are not linear. To study any specific filter means to study the specific kind of nonlinearity. Which means that little can be said that is both general and meaningful. Rune
Reply by ●August 1, 20112011-08-01
On Mon, 01 Aug 2011 13:08:22 -0700, Rune Allnor wrote:> On Jul 28, 7:00 pm, maury <maury...@core.com> wrote: >> Does anyone have information on publications, research, or data on the >> theroritical and practical stability analysis of the extended Kalman >> filter? Most of the stuff I've found seems to be a bit *off the wall*. >> >> Many thanks, >> >> Maurice > > The problem with nonlinear filters is that their only common denominator > is that they are not linear. To study any specific filter means to study > the specific kind of nonlinearity. Which means that little can be said > that is both general and meaningful.Yup. It would be nice if some smart cookie would go to school and get a PhD centered around making it easy for ordinary mortals to extract and understand meaningful results from nonlinear systems, rather than investigating yet another corner of the nonlinear systems space. Sort of like Bode, Nyquist and Nichols, only for nonlinear systems. -- www.wescottdesign.com
Reply by ●August 2, 20112011-08-02
On Aug 1, 3:26�pm, Tim Wescott <t...@seemywebsite.com> wrote:> On Mon, 01 Aug 2011 13:08:22 -0700, Rune Allnor wrote: > > On Jul 28, 7:00�pm, maury <maury...@core.com> wrote: > >> Does anyone have information on publications, research, or data on the > >> theroritical and practical stability analysis of the extended Kalman > >> filter? Most of the stuff I've found seems to be a bit *off the wall*. > > >> Many thanks, > > >> Maurice > > > The problem with nonlinear filters is that their only common denominator > > is that they are not linear. To study any specific filter means to study > > the specific kind of nonlinearity. Which means that little can be said > > that is both general and meaningful. > > Yup. �It would be nice if some smart cookie would go to school and get a > PhD centered around making it easy for ordinary mortals to extract and > understand meaningful results from nonlinear systems, rather than > investigating yet another corner of the nonlinear systems space. > > Sort of like Bode, Nyquist and Nichols, only for nonlinear systems. > > --www.wescottdesign.comI guess what I was looking into was a measurement/parameter distinction/detection scheme that would indicate an onset of stability. Something that would give an improvement of the reliability of the EKF and minimize the risk of divergence of the state estimates. The EKF may well be the best tool we have for non-linear state estimation, but there is no theoritical proof of stability or convergence that I know of (which prompted the original question). So, in the spirit of Tim's suggestion, it seems as if this would be a nice, and needed, research project. Maurice p.s. sorry for the delay in responding. I am reduced to Google groups, and it seems to be about 2 days behind real time, again.
Reply by ●August 2, 20112011-08-02
On Tue, 02 Aug 2011 06:41:21 -0700, maury wrote:> On Aug 1, 3:26 pm, Tim Wescott <t...@seemywebsite.com> wrote: >> On Mon, 01 Aug 2011 13:08:22 -0700, Rune Allnor wrote: >> > On Jul 28, 7:00 pm, maury <maury...@core.com> wrote: >> >> Does anyone have information on publications, research, or data on >> >> the theroritical and practical stability analysis of the extended >> >> Kalman filter? Most of the stuff I've found seems to be a bit *off >> >> the wall*. >> >> >> Many thanks, >> >> >> Maurice >> >> > The problem with nonlinear filters is that their only common >> > denominator is that they are not linear. To study any specific filter >> > means to study the specific kind of nonlinearity. Which means that >> > little can be said that is both general and meaningful. >> >> Yup. It would be nice if some smart cookie would go to school and get >> a PhD centered around making it easy for ordinary mortals to extract >> and understand meaningful results from nonlinear systems, rather than >> investigating yet another corner of the nonlinear systems space. >> >> Sort of like Bode, Nyquist and Nichols, only for nonlinear systems. >> >> --www.wescottdesign.com > > I guess what I was looking into was a measurement/parameter > distinction/detection scheme that would indicate an onset of stability. > Something that would give an improvement of the reliability of the EKF > and minimize the risk of divergence of the state estimates. The EKF may > well be the best tool we have for non-linear state estimation, but there > is no theoritical proof of stability or convergence that I know of > (which prompted the original question). So, in the spirit of Tim's > suggestion, it seems as if this would be a nice, and needed, research > project.One of the old methods for analyzing nonlinear systems is to look at the stability of the linearized system as the state varies over its space. While you're not guaranteed that a nonlinear system which can be everywhere linearized to a stable system is, itself, stable (look at subharmonic oscillations, for example), you _are_ guaranteed that if the linearized system _isn't_ stable at some point, then the system in all its nonlinear glory is certainly not globally stable. -- www.wescottdesign.com
