Kalman Assumption
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From: HardySpicer on 27 Apr 2010 11:54 On Apr 27, 2:53 pm, Tim Wescott <t...(a)seemywebsite.now> wrote: > HardySpicer wrote: > > On Apr 27, 4:40 am, Tim Wescott <t...(a)seemywebsite.now> wrote: > >> Cagdas Ozgenc wrote: > >>> Hello, > >>> In Kalman filtering does the process noise have to be Gaussian or > >>> would any uncorrelated covariance stationary noise satisfy the > >>> requirements? > >>> When I follow the derivations of the filter I haven't encountered any > >>> requirements on Gaussian distribution, but in many sources Gaussian > >>> tag seems to go together. > >> The Kalman filter is only guaranteed to be optimal when: > > >> * The modeled system is linear. > >> * Any time-varying behavior of the system is known. > >> * The noise (process and measurement) is Gaussian. > >> * The noise's time-dependent behavior is known > >> (note that this means the noise doesn't have to be stationary -- > >> just that it's time-dependent behavior is known). > >> * The model exactly matches reality. > > >> None of these requirements can be met in reality, but the math is at its > >> most tractable when you assume them. Often the Gaussian noise > >> assumption comes the closest to being true -- but not always. > > >> If your system matches all of the above assumptions _except_ the > >> Gaussian noise assumption, then the Kalman filter that you design will > >> have the lowest error variance of any possible _linear_ filter, but > >> there may be nonlinear filters with better (perhaps significantly > >> better) performance. > > > Don't think so. You can design an H infinity linear Kalman filter > > which is only a slight modification and you don't even need to know > > what the covariance matrices are at all. > > H infinity will give you the minimum of the maximum error. > > But strictly speaking the H-infinity filter isn't a Kalman filter. It's > certainly not what Rudi Kalman cooked up. It is a state-space state > estimator, and is one of the broader family of "Kalmanesque" filters, > however. > > And the H-infinity filter won't minimize the error variance -- it > minimizes the min-max error, by definition. > > -- > Tim Wescott > Control system and signal processing consultingwww.wescottdesign.com Who says that minimum mean-square error is the best? That's just one convenient criterion. For example, the optimal control problem with a Kalman filter is pretty bad. It doesn't even have integral action. Simple PID gives better results for many occasions. Hardy
From: Vladimir Vassilevsky on 27 Apr 2010 12:00 HardySpicer wrote: > Who says that minimum mean-square error is the best? That's just one > convenient criterion. > For example, the optimal control problem with a Kalman filter is > pretty bad. It doesn't even have integral action. > Simple PID gives better results for many occasions. "Of all the idiots the most insufferable are those who are not completely deprived mind" Francois de La Rochefoucauld
From: Rune Allnor on 27 Apr 2010 12:06 On 27 apr, 17:25, Tim Wescott <t...(a)seemywebsite.now> wrote: > Rune Allnor wrote: > > On 27 apr, 04:54, Tim Wescott <t...(a)seemywebsite.now> wrote: > > >> Indeed, the first step .... > > Ehh... I would rate that as the *second* step. .... > That doesn't sound as nifty. Suffice it to say that rethorics has never been among my fortes. Rune
From: Tim Wescott on 27 Apr 2010 14:43 HardySpicer wrote: > On Apr 27, 2:53 pm, Tim Wescott <t...(a)seemywebsite.now> wrote: >> HardySpicer wrote: >>> On Apr 27, 4:40 am, Tim Wescott <t...(a)seemywebsite.now> wrote: >>>> Cagdas Ozgenc wrote: >>>>> Hello, >>>>> In Kalman filtering does the process noise have to be Gaussian or >>>>> would any uncorrelated covariance stationary noise satisfy the >>>>> requirements? >>>>> When I follow the derivations of the filter I haven't encountered any >>>>> requirements on Gaussian distribution, but in many sources Gaussian >>>>> tag seems to go together. >>>> The Kalman filter is only guaranteed to be optimal when: >>>> * The modeled system is linear. >>>> * Any time-varying behavior of the system is known. >>>> * The noise (process and measurement) is Gaussian. >>>> * The noise's time-dependent behavior is known >>>> (note that this means the noise doesn't have to be stationary -- >>>> just that it's time-dependent behavior is known). >>>> * The model exactly matches reality. >>>> None of these requirements can be met in reality, but the math is at its >>>> most tractable when you assume them. Often the Gaussian noise >>>> assumption comes the closest to being true -- but not always. >>>> If your system matches all of the above assumptions _except_ the >>>> Gaussian noise assumption, then the Kalman filter that you design will >>>> have the lowest error variance of any possible _linear_ filter, but >>>> there may be nonlinear filters with better (perhaps significantly >>>> better) performance. >>> Don't think so. You can design an H infinity linear Kalman filter >>> which is only a slight modification and you don't even need to know >>> what the covariance matrices are at all. >>> H infinity will give you the minimum of the maximum error. >> But strictly speaking the H-infinity filter isn't a Kalman filter. It's >> certainly not what Rudi Kalman cooked up. It is a state-space state >> estimator, and is one of the broader family of "Kalmanesque" filters, >> however. >> >> And the H-infinity filter won't minimize the error variance -- it >> minimizes the min-max error, by definition. >> >> -- >> Tim Wescott >> Control system and signal processing consultingwww.wescottdesign.com > > Who says that minimum mean-square error is the best? That's just one > convenient criterion. Not me! I made the point in another branch of this thread -- my "optimum" may well not be your "optimum". Indeed, my "optimum" may be a horrendous failure to fall inside the bounds of your "good enough". Minimum mean-square error certainly makes the math easy, though. > For example, the optimal control problem with a Kalman filter is > pretty bad. It doesn't even have integral action. > Simple PID gives better results for many occasions. OTOH, if you model the plant as having an uncontrolled integrator and you track that integrator with your Kalman, you suddenly have an 'I' term. -- Tim Wescott Control system and signal processing consulting www.wescottdesign.com
From: HardySpicer on 27 Apr 2010 15:25
On Apr 28, 6:43 am, Tim Wescott <t...(a)seemywebsite.now> wrote: > HardySpicer wrote: > > On Apr 27, 2:53 pm, Tim Wescott <t...(a)seemywebsite.now> wrote: > >> HardySpicer wrote: > >>> On Apr 27, 4:40 am, Tim Wescott <t...(a)seemywebsite.now> wrote: > >>>> Cagdas Ozgenc wrote: > >>>>> Hello, > >>>>> In Kalman filtering does the process noise have to be Gaussian or > >>>>> would any uncorrelated covariance stationary noise satisfy the > >>>>> requirements? > >>>>> When I follow the derivations of the filter I haven't encountered any > >>>>> requirements on Gaussian distribution, but in many sources Gaussian > >>>>> tag seems to go together. > >>>> The Kalman filter is only guaranteed to be optimal when: > >>>> * The modeled system is linear. > >>>> * Any time-varying behavior of the system is known. > >>>> * The noise (process and measurement) is Gaussian. > >>>> * The noise's time-dependent behavior is known > >>>> (note that this means the noise doesn't have to be stationary -- > >>>> just that it's time-dependent behavior is known). > >>>> * The model exactly matches reality. > >>>> None of these requirements can be met in reality, but the math is at its > >>>> most tractable when you assume them. Often the Gaussian noise > >>>> assumption comes the closest to being true -- but not always. > >>>> If your system matches all of the above assumptions _except_ the > >>>> Gaussian noise assumption, then the Kalman filter that you design will > >>>> have the lowest error variance of any possible _linear_ filter, but > >>>> there may be nonlinear filters with better (perhaps significantly > >>>> better) performance. > >>> Don't think so. You can design an H infinity linear Kalman filter > >>> which is only a slight modification and you don't even need to know > >>> what the covariance matrices are at all. > >>> H infinity will give you the minimum of the maximum error. > >> But strictly speaking the H-infinity filter isn't a Kalman filter. It's > >> certainly not what Rudi Kalman cooked up. It is a state-space state > >> estimator, and is one of the broader family of "Kalmanesque" filters, > >> however. > > >> And the H-infinity filter won't minimize the error variance -- it > >> minimizes the min-max error, by definition. > > >> -- > >> Tim Wescott > >> Control system and signal processing consultingwww.wescottdesign.com > > > Who says that minimum mean-square error is the best? That's just one > > convenient criterion. > > Not me! I made the point in another branch of this thread -- my > "optimum" may well not be your "optimum". Indeed, my "optimum" may be a > horrendous failure to fall inside the bounds of your "good enough". > > Minimum mean-square error certainly makes the math easy, though. > > > For example, the optimal control problem with a Kalman filter is > > pretty bad. It doesn't even have integral action. > > Simple PID gives better results for many occasions. > > OTOH, if you model the plant as having an uncontrolled integrator and > you track that integrator with your Kalman, you suddenly have an 'I' term.. > > -- > Tim Wescott > Control system and signal processing consultingwww.wescottdesign.com That's right and what people did, but it doesn't come out naturally, whereas it does in H infinity control. Kalman filters are not robust to changes in the plant either. Hardy |