Disturbance modelling

Transcription

1 Lecture 3 Disturbance modelling This section reviews the main aspects in disturbance modelling and the corresponding relations of descriptions in the time and frequency domain, respectively. We will also consider the two related questions illustrated in Fig.3.; (i) Givenaknowninputspectraandknowntransferfunction,whatisthespectral density of the output (ii) Givenaknownspectraldensityforasignal,findastablelinearsystemwith white noise input which gives the same spectral density on its output. The latter problem is called the spectral factorization problem and will be used to rewrite systems with coloured disturbances to an equivalent system with white noise input, which will be used as a standard form for different estimation and prediction problems later on in course. G(iω)known? {U, Φ uu }? "white noise" {Y, Φ yy } Figure 3. Illustration of two main questions of disturbance modelling in this chapter;(left) Whatistheoutputspectraldensity Φ yy giventhatweknowtheinputspectraldensity Φ uu and and the linear filter G(iω)?(right) Knowing the output spectral density, find a stable linear filter which gives the same output spectral density if fed by white noise. 3. Disturbances In the basic control diagram of Fig. 3. we consider load disturbances d and measurement noise n r d F e u Σ C Σ P z Σ n y c u d m d s g z + y n Figure 3. (Left) The basic control loop with load disturbances d and measurement noise n. (Right)Loaddisturbanceswhichcanbemeasured,d m,e.g.,changesinoutertemperature,can be(partially) compensated for by feedforward to the control signal.

2 Lecture 3. Disturbance modelling The load disturbance d drives the system from its desired state whereas the measurement noise n corrupts the feedback information about z. Load disturbancescanbedividedintomeasurableloaddisturbances,d m,whichpartiallycan be compensated by feedforward, and load disturbances which can t be measured. Evenifwecan tmeasured s infig.3.,statisticalinformationlikecovarianceor spectral density will help us to design controllers which reduces/supresses the effect of the disturbances with respect to e.g., average and variance of the control objective z. Example In paper production there are a lot of disturbances which affect the paper quality andthepaperthickness.oneobjectiveistokeepdownthevariationinthepaper thickness, see Fig Setpoint for controller with good tuning Distribution Test limit Setpoint for controller with bad tuning (Relative) paper thickness, test limit at. Figure 3.3 To be of acceptable quality, products must exceed a certain threshold. By minimizingthevarianceofthethicknessweseethattheaverageofthepaperthicknesscanbe reducedsignificantly(wecomeclosertothetestlimit)forthesameyield.thismaysavea lot in production costs regarding both energy and raw material. All paper production below the test limit is wasted. Good control allows for lower setpoint with the same yield. By having a lower variance of the production, the average paper thickness can thus also be lower, which saves significant costs inbothenergyandrawmaterial.keepingdownthevarianceoftheoutputwillbe an important control objective for us in this course. A first glimpse at linear stochastic control Inthepreviousexamplewesawthatoneobjectivecouldbetominimizethevarianceoftheoutputorofastate.Inamoregeneralsettingonecanchooseatrade-off withrespecttohowmuchcontrolactiononewillusebyintroducingacostforthis as well. Consider a system with state-disturbances w and measurement-disturbances v. The LQ-problem(Linear system, Quadratic cost function) is then described as follows: Minimize subjectto (x T Q x+x T Q u+u T Q u ) dt ẋ=ax+bu+w y=cx+du+v wherevandwiswhitenoisewithintensityr andr respectively.

3 3. Disturbances ξ Arealization x(, ω ) F( ξ,t ) x(, ω ) x(, ω 3 ) x(, ω 4 ) t t Figure3.4 Astochasticprocess:Forafixed ω wecallitarealization,forfixedtimet it willcorrespondtoarandomvariablewithadistributionf(ξ,t )=Prob{x(t ) ξ}. Aswewillseelateroninthiscourse,onecansolvethisastwoindependent sub-problems thanks to the separation principle by considering Controller design for full state information, u = Lx Optimal estimation of states(kalman filter), ˆx=Aˆx+Bu+K(y ŷ) combination= Outputfeedbackusingobserveru= Lˆx... Beforewedothiswewillhaveacloserlookonhowtodescribedisturbancesusing statistical properties. Stochastic processes A stochastic process(random process, random function) is a family of stochastic variables{x(t),t T}wheretrepresentstime.Thestocasticprocesscanbeviewed asafunctionoftwovariablesx(t,ω).forafixed ω= ω itgivesatimefunction x(,ω ),oftencalledarealization,whereasifwefixthetimet=t itgivesa randomvariablex(t, )withacertaindistribution,seefig.3.4. For a zero-mean stationary stochastic processes the distribution is independent oft.werefertothebasiccourseinstatisticsformoredetailsonthefollowingconcepts: Mean-value function Ex(t) Covariance function. A zero mean Gaussian process x is completely determined by its covariance function: Cross-covariance function R x (τ)=ex(t+ τ)x(t) T R xy (τ)=ex(t+ τ)y(t) T Spectral density(defined for(weakly) stationary processes). The spectral density is the Fourier transform of the covariance function Φ xy (ω)= R xy (t)e itω dt 3

4 Lecture 3. Disturbance modelling and R xy (t)= e itω Φ xy (ω)dω π In particular, we get the following expressions for the covariance matrix: Exx T =R x ()= π Φ xx (ω)dω Whenxisscalar,thisissimplythevarianceofx.(Notation:Wewilluse Φ y as shortfor Φ yy.) For relations between covariance function, spectral density and a typical realization,seefig.3.5,whereonemaynoticethattherealizationsseemtobe"more random" the flatter the spectra is(over a larger frequency range) while peaks in the spectral density corresponds to periodic covariance functions. Covariance Spectrum Output τ frequency ω time - Figure 3.5 Relations between covariance function, spectral density and a typical realization. Correction: The spectra should be divided by π White noise Aparticulardisturbanceisso-calledwhitenoiseewithintensityR.HereRisa constant matrix, which corresponds to a constant spectrum, totally flat and equal for all frequencies: Φ e (ω)=r One effect of this definition is that the continuous-time version of white noise has infinite energy, and causes some issues to be handled mathematically rigorously, butwewillnotgointothesedetailshere. 4

5 3. Disturbances Themostimportantpropertyofwhitenoisewhichwewillusebelow,isthat it can not be predicted; based on previous measurements there is no information about future values(infinite variance). From transform theory we also have that thefouriertransformofthediracpulse δ(t),isconstant,whichcorrespondstoan alternative interpretation: by applying a Dirac impulse as input to a linear system, the spectral density of the corresponding output(i.e., of the impulse response), will be like a finger-print of the system s frequency properties. ThetwoproblemsrelatedtoFig.3.canbeformulatedas. Determine the covariance function and spectral density of y when white noise u is filtered through the linear system ẋ=ax+u(k) y=cx. Conversely,findfilterparametersforastablelinearfilter,AandC,togive the output y a desired spectral density. u G(s) y Whatistheoutputspectraldensityforyiftheinputuhasspectraldensity Φ u (ω)? We use the transfer function representation Y(iω)=G(iω)U(iω) wherey=f{y},u=f{u}arethefouriertransforms.accordingtothedefinition,weget Φ y (ω)ˆ=φ yy (ω)=y(iω)y(iω) =G(iω)U(iω)U(iω) G(iω) wherewecanidentifythespectraldensityoftheoutputas Φ yy (ω)=g(iω)φ uu (ω)g(iω) u G(s) y In similar way we find the cross-spectral density Spectral factorization Φ yu (ω)=g(iω)φ uu (ω) Thenextquestionisthenhowwego"backwards"accordingtoFig.3.right,to find what linear filter which will do. Assumethatthedisturbancewhasspectrum Φ w (ω) (Spectral factorization) Assume that we can find a transfer function G(s) suchthatg(iω)rg(iω) = Φ w (ω)foraconstantr. InthatcasewecanconsiderwasanoutputfromthelinearsystemGwithwhite noiseasinput, Φ v (ω)=r(equalenergyforallfrequencies/flatspectrum). 5

6 Lecture 3. Disturbance modelling THEOREM 3. Spectral factorization [G&L 5.3] Assume that the real, scalar valued function Φ w (ω) isarationalfunctionof ω.thenthereexistsarationalfunction G(s)withallpolesstrictlyinthelefthalfplaneandallzerosinthelefthalfplane orontheimaginaryaxissuchthat Φ w (ω)= G(iω) =G(iω)G( iω) Ifvandwarescalarvaluedand Φ w (ω)isarationalfunctionof ω itiseasyto follow the proof in[glad& Ljung] and factorize to first or second order polynomials of ω inboththenumeratorandthedenominator.thesecanthenbesplitinstable and unstable poles, respectively, and comes from the fact that if the characteristic polynomialfor G(iω)is Π n k= (iω λ k)then G = G( iω)willhaveitspoles mirroredinthetheimaginaryaxis.thisisdoneintransferfunctionform,andin the next section we will see how this looks in state-space representation. Assume we have state-space model with disturbances where ẋ(t)=ax(t)+bu(t)+nw (t) z(t)=mx(t)+d z u(t) y(t)=cx(t)+d y u(t)+w (t) w iscalledstate-orsystemnoise w iscalledmeasurement-oroutputnoise The question is how to handle coloured noise? Ifw andw iscolourednoisewithknownorestimatedspectraldensitythen re-writew andw asoutputsignalsfromlinearsystemswithwhitenoiseinputs v andv. w (t)=g (p)v (t), w (t)=g (p)v (t) wherep= d dt (correspondingtothelaplacesinfrequencydomain). MakeastatespacerealizationofG andg andextendthesystemdescription with these states ẋ(t)=ax(t)+bu(t)+nv (t) z(t)=mx(t)+d z u(t) y(t)=cx(t)+d y u(t)+v (t) wheretheextendedstatexconsistsofthestatexandthestatesfromthestatespacerealizationsofg andg. A is the corresponding system matrix for the extended system etc. We illustrate this procedure with an example. Example Consider the system ẋ = 7x +u+w y=x +w wherew iscolourednoisewithspectraldensity Φ w = 9 ω +4 = (spectralfactorization)= 3 3 (iω+)( iω+) 6

7 3. Disturbances We can then introduce a state-space form of this transfer function, representing thecolourednoisew as ẋ = x +3v w =x wherev iswhitenoisewithintensity.thesystemcannowbewrittenas ẋ = 7x +u+x ẋ x +3v y=x +w andwecanproceedinthesamewaywiththecolourednoisew Covariance and spectral density for a state vector Consider the linear system ẋ=ax+bv, Φ v (ω)=r v We can calculate the transfer function from noise to state as G v x (s)=(si A) B andthespectraldensityforxwillthusbe Φ x (ω)=(iωi A) BR v B ( iωi A) T }{{} ((iωi A) B) ẋ=ax+bv, Φ v (ω)=r v Onewaytocalculatethecovariancematrixforstatexis Π x =R x = Φ x (ω)dω π Howeverthereisanalternativewayofcalculating Π x THEOREM 3. [GLAD&LJUNG 5.3] Ifalleigenvaluesof Aarestrictlyinthelefthalfplane(i.e.Re{λ k }<)then thereexistsauniquematrix Π x = Π T x >whichisthesolutiontothematrix equation AΠ x + Π x A T +BR v B T = WewillseethatasimilarformulacanbeusedtocalculatetheoptimalgainK in the Kalman filter with respect to measurement and state noise covariances. Anintuitiveinterpretationhowlargethegain K shouldbeisthatifwehave much state noise but little output noise(i.e, reliable measurements), then the optimization chooses a large gain which"trusts" the measurements. With very large measurement noise, it will choose a low gain which means that the observer will almost run in open-loop; trusting the model and gaining very little information from the measurements. 7

8 Lecture 3. Disturbance modelling Example 3 Consider the system [ ẋ=ax+bv = whereviswhitenoisewithvariancer v =. Whatisthecovarianceforx? ][ x x ] + FirstchecktheeigenvaluesofA:λ= ±i 7 LHP. OK! Solve the Lyapunov equation Find Π x : [ AΠ x + Π x A T +BRB T = [ ][ ] [ ][ ] [ ] Π Π Π Π + + [ ]= Π Π Π Π [ ] [ ] ( Π +Π + Π +Π Π = = Π +Π Π Π ] v Solvingfor Π, Π and Π gives [ ] [ ] Π Π / = Π x = = > Π Π /4 Inmatlab: lyap([- ; - ],[ ; ]*[ ]) 8