Observer Design with Reduced Measurement Information

Transcription

1 Observer Desig with Redued Measuremet Iformatio I pratie all the states aot be measured so that SVF aot be used Istead oly a redued set of measuremets give by y = x + Du p is available where y( R We assume that the diret feed matrix D is zero though the followig developmet a be modified if it is ot We would like to build a dyamial system kow as a observer that a estimate the iteral state x( give kowledge of the otrol iputs u( ad the outputs y( his a be aomplished usig the sheme show i the figure u (t ) x (t ) y (t ) Plat Dyamis L ~ y ( t ) Dyamial Observer x ˆ ( t ) yˆ ( t ) Full-Order Observer ( he figure shows the plat dyamis with iteral state x ( iput u ( ad output y ( lso show is a proposed "dyamial observer" that has two portios: - a exat model of the plat dyamis () - ad a error orretig part L( y( yˆ( ) whih should take are of all the errors determied by iorret iitial iformatio o the states of the system he p matrix L is alled the observer gai he observer has iteral states x ˆ( ad two m p iputs u( R ad y( R First we will show that x ˆ( provides a estimate of the full state x ( if L is orretly hose he we make the output of the observer to be the state estimate x ˆ( 1

2 he equatio of the observer is = + u + L( y yˆ) or x ˆ = ( L) + u + Ly his is a -th order dyamial system with iitial state ˆx (0) equal to the iitial estimate of the state he quatity ~ y( = y( yˆ( t ) is alled the output estimatio error How to hoose L? he observer gai matrix L must be seleted so that eve though the iitial estimate ˆx (0) is ot equal to the atual iitial state x (0) as time passes the state estimate x ˆ( overges to the atual state x ( hus we defie the state estimatio error ~ x( = x( ( t ) ad write its dyamis as ~ x = x = x + u ( + u + L( y yˆ)) = ( x ) + L( y yˆ) or ~ x = ( L) ~ x o~ x Note that the otrol iput does ot appear sie it aels out his is beause the iput is fed diretly ito the observer through the matrix his equatio is kow as the error dyamis From this equatio easy to see that as log as we selet the observer gai L so that the losed-loop observer matrix o = L is asymptotially stable the estimatio error ~ x ( t ) will go to zero asymptotially whatever the iitial estimatio error ~ x(0) = x(0) x ˆ(0) happes to be It is ot diffiult to selet L so that ( L) is asymptotially stable ompare this problem to that of seletig the SVF gai K so that = K is S I the observer desig problem the desig matrix L is o the left while i the SVF problem the desig matrix K is o the right Now we a make the former look like the latter by matrix traspositio: = ( L) = L o Now this looks the same as the SVF problem sie the desig matrix L is o the right Note however that SVF desig used () while observer desig uses () I fat the two problems are the same if oe equates ( K) i SVF desig with ( L ) i observer desig 2

3 herefore to desig a stabilizig observer oe may proeed as follows: 1 Reame ( ) to ( ) 2 Use ay SVF desig tehique you wish to determie a stabilizig gai K (eg kerma s formula) [Note: We will disuss i the ext leture a method whih allows alulatio of a state feedbak gai suh that a ost futio quadrati with respet to the values of the states ad the otrol iput is miimized ie LQR] 3 Reame K to L kerma Desig for Observers Whe there is oly oe output so that p = 1 oe may use kerma's formula hus selet the desired observer polyomial (s od ) ad replae ( ) i K = e U ( ) od by ( ) the set L = K We a maipulate this equatio ito its dual form usig matrix traspositio to write L = e ( V ) od ( ) or L = od ( ) V e whih is kerma's formula for observer desig We have speifially writte the desired observer polyomial as (s od ) (whih depeds o L ) to distiguish it from the desired losed-loop plat polyomial (s D ) (whih depeds o K ) Notie that i this ase the plae of the otrollability matrix U is take by the observability matrix V hus if the system is observable the the observability matrix V is osigular ad the observer poles a be plaed aywhere oe desires whe p = 1 usig kerma's formula he L is a good eough gai for the observer system whih will produe estimates of the states of the system whih (as time passes) will ome lose to the real values of the states of the system Notie that the amout of time required for overgee of the estimated state to the real state of the system depeds o the values of the poles of the observer system ie the values of the eigevalues of the matrix = L o t this poit oe woders: - If we desire to otrol our system usig a state feedbak otroller - but we a ot diretly measure all the states that we osidered i the state spae represetatio hose for our system - ad we deide to itrodue ad desig a state estimator - suh that we a use the estimated state istead of the measured state i the implemetatio of our state feedbak tehique he - what sort of relatio should we have betwee the dyamis of the otroller give by the eigevalues of -K ad the dyamis of the state estimator give by the eigevalues of -L? 3

4 Oe sees ow that this questio is i fat askig: How should we hoose (s od ) with respet to ()(whih we have previously hose)? he aswer is as you have expeted he observer poles should be seleted muh faster (about 10 times faster) tha the desired losed-loop poles of ( K) he the effets of iaurately kow iitial states will die out quikly ad ot iterplay with the iput/output dyamis D s Dyami Regulator Desig (state variable feedbak based o the estimated state) he followig blok diagram provides a dyami regulator for the plat based o the estimated state v( u ( x (t ) y (t ) Plat Dyamis K L ~ y ( t ) x ˆ ( t ) yˆ ( t ) Dyamial Observer ( Dyami Feedbak Regulator he losed-loop system is desribed by the equatios PLN OSERVER ONROL x = x + u x ˆ = ( L) + u + Ly u = K + v herefore the regulator has dyamis provided by the observer plus a feedbak gai portio from the SVF he regulator is formally speified by the pair of matries ( K L) he proposed regulator oly eeds to kow the iputs u ( ad the measured outputs y ( ot the full state vetor x ( he feedbak used here is alled state estimate feedbak 4

5 he losed-loop dyamis of the overall feedbak system are give by x = x K + v = ( K) x + Kx ~ + v (isertig the otrol i the plat dyamis) ~ x = ( L ) ~ x (estimatio error dyamis) Defie the augmeted system state as x x whih has 2 ompoets he the losed-loop dyamis may be writte as d x K K x v dt ~ + x = L x 0 ~ 0 x y = [ 0 ] ~ x his otais the dyamis of the plat plus the observer Note that the observer dyamis is writte i terms of the estimatio error for oveiee i the upomig developmet he losed-loop harateristi polyomial is give by K K si ( K) K ( s) = si 2 0 L = 0 si ( L) where I is the idetity matrix Sie this is a blok triagular matrix the determiat is the produt of the determiats of the diagoal matries herefore ( s) = si ( K) si ( L) his shows that the 2 losed-loop poles usig the regulator desiged based o the observer are the uio of the poles assumig full state feedbak ad the observer poles his is kow as the Separatio Priiple he separatio priiple implies the followig two-step desig proedure for dyami regulators: Use ay tehique to selet a feedbak matrix K assumig that full state feedbak a be used Desig a observer L he dyami regulator is the give i the figure above ordig to the augmeted dyamis the losed-loop trasfer futio is give by or H ( s) = si ( K) K ( L) 0 [ 0] H ( s) = ( si ( K)) 0 si his ovies us that the losed-loop trasfer futio is the same as if full SVF had bee used meaig that the observer dyamis do ot appear i the iput-output ouplig of the losed-loop system 5