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VIRTUAL REFERENCE^ FEEDBACK^ TUNING (VRET)
- to^ easy presentation (^) , online^ version^ can^ be^ conceived -off-line
S
adaptive scheme
~ (^) direct main feature Setup :^ output feedback^ control^ of^ an^ uncertain^ linear^ system (discrete^ time^ and^ deterministic)^ represented by a^ transfer^ function tunable control action · (^) uncertain linear system y(t) = P(z) u(t) · (^) for the moment we are (discrete time) observable discrete^ not (^) considering anydisturbance output time^ t. f. P(t) is (^) regarded as un certain (^) , can be (^) any rational +f. uncertainty we^ want^ to counteract (^) by means of (^) adaptation Control (^) problem : we want to (^) regulate y(t) (^) tracking rit) (^) via an (^) 1-degree of freedom control scheme deciding based (^) on the ult tracking error r(t)te(t)& ,^ (1z^ , G) u^ PC^ -y(t) Controller is (^) a linear (^) dyn. System controller chosen among classe of controllers (^) parametrized (^) by a certain (^) parameter vector (^) o
· linear controller <> ((z) +f.
C = [C(z , 8)^ :^ ge-I admissible domain^ for^ o
- (^) · jo & In VRFT is the (^) parameter vector for^ the^ controller NOT of models for (^) P(z) lalso because there^ won't be (^) models-direct) & :^ the coefficients of numerator and denominator the controller tot. depend on VRFT (^) applies for (^) every type of^ parametrization leither linear (^) or non (^) linear) But (^) when (12, 8) is (^) linearly parametrized in o additional (^) features and benefits (^) are achieved o (
m-dimensional rector
C(z, 8) is (^) linearly parametrized in o^ if^ : ((z, 8)^ =^01. B i (z) +^02. B2(z) +...^ +^ Om^ · pm(z)
linear combination of
given pi(t)^ transfer^ functions BASES TRANSFER FUNCTION ((z, 8) =^ OT.^ B(z) = B(z)T.^0 O · ( PIE) = (BR) vector of bass^ oin linear e = (OTB(z) :^ de^
- > - (^) class offcontrollers linearly parametrized in^ o VRFT (^) gives its^ best^ in^ this^ set^ up Example :^ FIR^ (Finite^ Impulse Response) Bi(z) =^1 p2(z) = z (^) ... pi(z) =^ z" (^) ... (^) Bm(z) = z-m
basis t.^ f. (^) represent (^) delays C(70) =^ a +^ 02zt... + 8mz^ MH (^) = Maz-^ = zm, + 827m (^) +... Om i = 1 zm poles all in (^) zeros FIR is^ dense^ within^ all linear controller^. (^) Every linear^ controller^ is well (^) approximated (^) by
a FIR controller^ as
long as (^) m is large (^) enough order · (^) cons : (^) m in application (^) typically hat^ to^ be^ too^ large "there are ather basis tot (^) (Laguerre polynomials) that (^) are still dense but (^) requires still (^) order 6 with (^) poles to
Example :^ PID
Bi(z) =^1 B2(t)^ = B3(z) = z - 1 Z
C(z,d) =^0 , + 02 Z + 83z
discrete time
proportional discretetime^ integrator differentiation e = (c(z ,8) (^) : (^) GEIR3] = (^) PID controller VRFT (^) optimize the^ controller^ according control (^) design
JMRCO) cannot be even (^) computed -^ Gir (^) Is not (^) achievable when P(z) is un certain Idea : (^) an (^) experiment on the (^) plant can be (^) performed -^ >^ data^ driven Approximate the^ MR^ controller^ using a^ raw^ source^ of^ knowledge-data experiment on^ P(z) ult) & P(z)^ [(t)^ i^ injected,^ j is (1) (^) y(1) measured
ü(z) (^) y(z) i (^) i (^) approximate MR (^) control by (N) (^) y(N) using data (^) only DV =^ I/0 data (^) collected from our (^) experiment DN = (^) [ü(D (^) , (^) y()) , ...,IN) (^) y(N] VRFT -^ enforce MR^ goals by (^) relying on^ DN We could use and^ indirect (^) approach to^ solve the (^) problem : Di P = (^) P(z) model for Plz) P(z) can^ be^ any system ~ min IlPCO0)
- M/ but when^ P(z) is (^) complex , the result^ can be (^) poor and^ it^ is even difficult to^ identify how (^) poor it is.
Drowbacks : difficult to tune identification to the final^ objective i . e.
obtaining a^ a^ good controller for^ P(z) Ilpo-MJMR(
Difficult to acea
(model (^) order selection-huge input on the mismatch (^) ( For this^ reason we^ will^ use a direct^ approach (much wiser (^) usage of^ data) VRFT : data preprocessing (set point) VIRTUAL REFERENCE = signal ebtained (^) by processing measured (^) output (t) with inverse of Model Reference /whole batch of data is available ( r(i) = M(z)"y(i) i^ = (^1) , ....^ N^ always well^ defined^ in^ an^ off-line^ setup (whole batch^ of^ data^ is available)
M(z) desired (^) closed-loop I (^). f (^).^ +^ often (^) strictly proper e (^). g. (^) M(z) =^11
(z - p)t
p =^ real^ pole 1pk1^ as^ Stability shift (^) operator
M(z)
= ~ (^) r(i) =^ z -py(i) = (py(i
1 - P^ ppy(i) no issue (^) , off-line set^ up i (^) y(i)(i) data all (^) available (^1) y(1) = (1) (^2) y(z) = (^) (2) uli) (^) can be (^) computed and this is the^ case 3 i whatever (^) M(z) is i N (^) y(N) VIRTUAL ERROR =^ difference between^ virtual^ reference and the measured (^) output e (i) =^ v(i) - y(i) i^ =^1 , ... N rii) eli (^) ult) (^) y(t)
G = P(z)^ g
(1) (^) y(1) & ü(z) (^) y(z) i (^) i (N) (^) y(N) M(z) OBS :(i) has^ the (^) property that^ if^ it^ is^ used as^ set-point the^ desired (^) response for the (^) class (^) loop coincides with the measured Output (^) gli) , vi)^ set-point^ , desired^ output :^ M(z)^ (i)^ =^ M(z)^ (M(z)"gli)] = Gli virtual (^) , not used in (^) practice key Fact^ : A (^) good controller (i (^). e. that achieves the MR (^) design (^) goals Plz)CIz,^ d)^ = M1z)) is such that when^ it^ is fed^ by the virtual^ error eli)^1 +^ P(z)((z,^ 8) it must return(i) as control action to^ be (^) injected into^ the (^) system V
-y
good controller
" = (^) C(z, 8) " (^) -P(z)
In (^) general (^) any notion^ of^ distance^ can be^ used Fur(e)
=di)(((i)) not^ necessary^
for He It can be used (^) pre-filtering before minimizing (i) (^) = ((z) uu(i) = (^) ((z)u(i) & (i) (^) y(i) = ((z)y(i) pre-filter : [il any user^ chosen^ ↑L^ (i)^ = L(z)(i) e(i) (^) digital filter (^) et(i) = ((z)e(i) by linearity ,^ are^ obtained^ via URFT (^) signal construction over il and (^) il Jur(0) =li)-Cl VRFT cost function^ defined over (^) pre-filtered (^) signal &^ N
I E
OVR = (in(i) -^ C(z^ , 8) er(i)) argmin (^) Ni ((z) = 1 - original algorithm L(z) =^ additional^ degree of freedom (^) that (^) we are
going
to (^) use to tune VRFT 2 JvR(d) =^ Jurlo) how (^) good is the^ =P)Mw
approximation?
Right now^ ,^ at^ an^ intuitive^ level^ an^ approximation^ ok^ but^ not^ perfect Next (^) goal : understand the level of (^) approximation exatly and^ use^ (12)^ to^ reduce^ it as much^ as (^) possible -^ optimal selection of^ the^ prefilter - significant to^ improve VRFT^ to a (^) huge extent We need to Introduce (^) FREQUENCY INTERPRETATION OF PEM IDENTIFICATION OBS :^ VRFT^ and C1z (^) , 0)^ linearly parametrized Ind (17,0) =^ o^ + B(z) o B1z) : ( rector of (^) basis function I Jur(0) (^) -i) Cz,ili) B)
Bilz)e(i)
YL(i) BL(z)ec(i)
- : Blz)ez(i) I : - I Bm(z)er(i) vector^ of^ outputs of^ basis^ tof.'s^ fed^ with^ virtual^ error^ (pufilt) computable from^ data
Jur(o) = ( üli)
- (i)) vr = argmini) -T(i) Least (^) Squares (^) probe e exact solution can be^ computed at (^) extremely small^ computational cost ladvantage of^ linearly parametrized^ controller^ ( vr = [Silecist]" (^) [blisanilI Frequency Interpretation^ of^ PEM^ Identification
Now we'd like to^ answare the question : how close are Jur(0) and JMR(0) , and OvR^ and OR^?
Is (^) CIz (^) , OvR) (vRFT (^) controller) a (^) good MR controller? vr = argmin(i) C,^ il^ Fur(0) prefitered (^) prefiltered
input virtual^ error
or =
argmin/1(PM)l JMR(0) same formulation as a PEM^ Identification problem what's the (^) relationship between Jur(0)^ and (^) JMR(0) and ovr^ and^ Our^? Moreover (^) ,^ tune^ (12)^ so^ that^ the^ match^ is as^ good as (^) possible We can answare to^ these (^) questions Via PEM IDENTIFICATION Frequency interpretation^ of^ PEM^ identification arginally developed^ in^ the^ context^ of^ identification and (^) we will (^) initially study it in this context llater we will use (^) acquired (^) knowledge in (^) VRFT) Data (^) generating system. Let's consider a (^) generic system observable &
any portion^ of^ reality^ that input (^) output connets^ variables^ of^ interest u(i) (^) y(1) u(N) (^) yin) N Experiment : Do(u(i) , y(1)^. (^) ..., u(N)^ , y(N)] collection of^110 pairs
We want to write (^) M10) in (^) prediction form (^) lisolating the (^) predictable part (^) up to +-1) : M(0) :^ y(t) =^ G(z, d)u(t - d) +^ H(z (^) , (^) b)z(t) y(t)
- y(t) + H(z,g)y(t) =^ G(z,^ 0)u(t - d) + z(t) H(z (^) , 0) y(t) = ( Hiz)(y(t)
- G(z^ ,^ d)^ ult -d) + z(t) un predictable I H(z,^ 0)^ I^ (^11) at t - 1 predictable at^ -1^ (function^ of^ y(t -1)^ , y(t -^ 2)^ ...^ ) a (^1) since His (^) canonical H(0) canonical ↓ Hiz,) = (^) 1thiz"Thaz" (^) ... > (^1) - 1 = (^) X-X-hiz" Azz... H(z, 8) function of (^) y(t-1) , y(t- 2),.^ ·^ predictable at t (^) - M(0) :^ y(tt - (^1) , 0) = (1 - p,^ 0))2(t)^
G
u(t - d) (^1) step predictor for (^) generic M(0) Asymtotic (^) theory of^ PEM^ identification^ : almost (^) surely
Tr(8) = 1 y(i) -^ gli^1.^ e^.^ for^ all^ possible
relations of^ the^ 1/0 data
No J (^) (0) = 1((y(t)y(t( + - (^1). (^) 0))] stationarity implies that stationary
It does not depend on t
· (^) Almost surely =^ convergence^ arises^ Irrespective^ of^ the^10 data ~ asymptotically all^ experiments^ lead^ to the (^) same result · uniformly in^ SN(0)-510KEN)^ vo N relations of JN(8) (for different realizations)
w
small N
z ↓
the bondle concentrates on
V-o
a (^) unique curve (^) 510) N (^) convergence of^ and^ the^ biggest error^ over ~ (^) different realization o goes to (^) zero
W
of (^) In(0) Vo ↓ unique rate^ of^ convergence
V -o big N^ for^ all^ o
The convergence happen^ with (^) probability I ( = (^) for all (^) possible experiment) and (^) uniformly in o
I convergence with the same rate to)
Uniform convergence in
property to^ have^ convergence -of minimizers Ev = argmin SN(0) vargmin()
=:^0
O minimizer minimizer^ of^ asymptotic
of empirical cost^ function
cost function In (^) practice it^ is often the case that (^) Nis (^) large (^) enough so that mismatch between In(d)^ and "old and between^ or^ and 8 is (^) negligible /we work^ under this (^) condition) -study properties^ of^ Flo)^ and these will (^) apply approximately toÖN^ andV(0) (^) irrespective of the collected data with^ an error that (^) is (^) negligible The result of^ PEM Identification (^) is the same (^) irrespective of the (^) particular dataset D" as (^) long as the number of data (^) points is (^) big enough Hp :^ from^ now^ on^ we^ will^ consider^ N^ big enough /200^ -^300 points) y The mismatch (^) between M104) and Mon) (^) is (^) neghigible and (^) so if we characterize the (^) mismatch between (^) M(84) and S (^) , we are also (^) characterizing the mismatch (^) between MION) and S. Frequency interpretation^
- (^) representation of 5(8) in the (^) frequency domain (^) (depends on the Identification setup) Open (^) Loop Closed^ Loop (^) enWN enWN J S J S Ho Ho U
= Go
- r + = = (^) c = (^) ge + - + Y U is user chosen ↑ u(t)^ correlated^ with^ elt)
ult) uncorrelated with^ e(t)
We will have different conclusions for the 2 setup
5(0) (^) = Gew)G, ~ (^) w(W) I (^) #le5w (^) , O) (^) dwHesw, o) du -T fundamental tool to (^) evaluate Melw) =^ XVw^ white (^) nolse model mismatch^ frequency by (^) frequency
This expression reveals the importance of the mismatch between Glz) and Glz, 8) and
between (^) Ho(z) and H(z,8) in the construction (^) of the cost function (^) frequency (^) by frequency
- (^) we can draw conclusions between S and M10 % ) in the (^) domain which is most useful for control (^) design frequency domain^ representation^ of^ 510) extremely important^ to^ understand^ ident.^ behavior^ and^ Ident.^ error^ frequency by frequency Example S : (^) y(t) = u(t) +^ elt) Co^ :^ Output Era u(t) -^ wN(0, 1)^ e(t)^ VWN(0, 1)^ uncorrelated (^1).^ MOT^ E model^ class MOF (^) (0) : (^) y(t) =
bz
0 - 5 wwN10, 02) 1 - az - 1u(t) +^ z(t) (^2).^ MARY^ Ary^ model^ class MARY (0) : (^) y(t) = ay(t - 1) + bu(t - 1) + (^) z(t) o = [) can (^) you (^) guess what's^ going on^ with^ PEM^ ident^ in the^ two^ cases^?
- DE S :^ y(t) = 42z"
u(t) +^ e(t) ~^ G(z)^ =^
Y2z (^1) - z I 1 - 42z-
ult) -WN(0, 1) e(t) ~Wi(0, 1) Ho(z) =^1
↑u (w) =^3 Melw) M°:^ y(t) =bu(t)^ & & G (^) (z (^) ,0) =b (t) ~^ wi^ (0^ ,^ 52)^ H(z^ ,^ 8)^ =^1
(0)-bd the minimum is for Alo-o^ for^ ofi]
Glz (^) , EN) (^) NoGlz, % (^) = G^ primary result from a central (^) perspective
H(z,8n)^ =^1 =^ Ho (z)
> PEM is consistent :
asymptotically Sis^ perfectly^
reconstrated
2) ARX
MARY (^) : y(t) = ay(t - 1) + bu(t- 1) + (^) z(t) (^) zwN(0 , +2)o = [] y(t) = az^ - y(t) +^ bz (^) u(t) + (^) z(t) (1-az^ (y(t) =^ bz^ u(t)^ +^ z(t) y(t) = bz- 1 - az^ -^ u(t) + 1 - az
=(t)^ G(z,^ 8)^ =b H(z,^ 0)^
(^1) az -^1
-youcannot achive (^) G =^ Gand H^ simultanosse a partial (^) consistency?^ Glow)^ G^
Tu (^) bejw 2 5(0) & = (^) 1- (^) Yeju 1-asu 1 Gesw^. du a A(0)^0 B(0) 1 = X -[] A() =^0 minimum for^ A^ 8 -^ [3] -B(82) =^1 J (^) (0) = Alo) + (^) B(0) no (^) minimum trade off *^ SIE^ (le(t)+^ vielt- 1) + Velt2)+.. (^) )
between and 02 ~ (^88) + G(EN) v GG() = no (^) partial (^) consistency HON) H( % ) #Ho no consistency
2B) (Case 2 :^70 ° - : Glz (^) , 8 % = (^) G(t) but Hz8) + Ho(t) vo)
- (^) G(z (^) , 8) and (^) H(z, 8) share (^) the same parameters (or^ part of^ them) 5(0)Gd A(0) =^0 B(O) everything evoluted^ for^ z-est A(0) =^ o -^ minimum =^ (0) =^ A(0) + B(0) ( but^80 does^ not^ min^ B(0)) trade^ off^ between^ minimizing B(0) minimized (^) by some ·^ &^ A and B -8 (^) +8 ° 8
8 minimizers of 5(0)
GlzEN) (^) NoGlz , 0 % G(z, 8) = Gi(z)
- > (^) No (^) partial consistency :no^ consistency
- lunder (^) parametrization) G(z (^) , 8)^ + Gi(z)o^ and H1z (^) , 8) Hz) Vo no (^) enough (^) degrees of^ freedom^ to^ describe^ S > (^) no (^) consistency (not even (^) partial consistency) No (^) (partial) (^) consistency - (^) frequency identification (^) is of the most (^) important to understand what's^ going on^ between Glon) and (^) Ge It (EN) and Ho T 510-GdIH(0)/
dwz = esa
1G -^ Glo)"u-to-y model mismatch (^) weighed by Hopta
allows (^) us to understand the (^) impact of the (^) -to-y model mismatch in^ 5(0) Pu(w) X2^ -high signal to^ noise^ ratio^ -^ contribution^ of^ the^ disturbance = (0) (^) = 16 · Glo incondw in (^) J(0) can^ be (^) neglected -T at 80 EN (^510) =G - G10 % Ho (^) prada -T (^) choice of o allows (^) us to (^) tune underparametrization, mismatch^ mismatch^ over^ frequency
Mu(w) (^) frequency (^) by frequency weight to^ the model mismatch (Hew, 84)/2^ Lindication^ when^ the^ mismatch^ is^ small^ or^ large ( large ~^ mismatch^ at^ that^ frequency has^ huge importance !
- > chosen (^) so that (^) mismatch is small small"mismatch at that (^) freg has small (^) impact
- (^) mismatch is large Weight is^ not^ a^ priori known^ (depend)^ but^ it^ is^ a^ posteriori^ revealed Ta(w) (^) F (^) perform ident first (^) and draw conclusions by IHesw,^ del ur)/^ inspecting weight PEM Ident^ with^ pre-filtered data un(i) =^ ((z)u(i) % yy(i) =^ L(z)y(i)^ any digital filter u(i) (^) y(1) (^) prefilter i :^ pre-filtered data u(N) (^) y(N) vargmin (i)- il by linearity I((i(i +^1 , 8)^ = predictor fed^ by Unli)^ and^ y, (i)^ =^ ((z)^ glili +,^0
(mo](i)
((z) Gli(i) y(i)
- argmin (2)^ · (y(i)ili)argmin(((z)(y(i)gili same calculation^ as^ before^ , we^ always have^ an^ L(z) (^) upfront (^) every +^.^ F^. 's T = (^) (0) = Gle)Glee^
- T
° (eswy IHesw,)(u(w)dw^
- (^) c) Inesw, 0) 12/Llew)l^ *da Prefiltered effect negligible if^ signal^ to^ noise^ ratio^ is^ high Model mismatch^ IG ° -G(8) 1 is^ now^ weighted by (Llew)12Nn(w) known^ but^ not^ tunable^ after^ collecting data 1H(e-w (^) , 0)/^ 2
- a (^) posteriori revealed (^) - > functional to^ tune^ mismatch known (^) , and moreover is tunable between Gand^ G(04) after (^) collecting the data (^) frequency by frequency
07 : (^) y(t) = -2u(t) + 1 - e(t) biz" + b2z^
- o. Ja 1 + a (^) , z- + azZ
G(0) H(0) =^1
ARX :^ y(t) =
biz "^ + baz^1
1 + a (^) , z^ +^ azz
-^ 24(t)^ + 1 +^ aiz"^ +^ azz 2 e(t) G(0) H(0)
The models have^ the^ same descriptive capability as for the u-to-y t.^ f^.^ but^ with
different (^) error models. In (^) any case (^) we know that none of the models are able^ to^ fully describe Go^ since it^ is a4th order^ +.^ f^. del ARX model How (^) to decide the model to (^) use : Now we'll (^) try to (^) see in which (^) freg. the error is localed Frequency interpretation^ : X 16-Glöi)) (^) weighted (^) by model mismatch IHIEN)/ 1 DE ~ weight 1
= 1 constant Vu
~ (^) G-GlNO) (^) is uniform (^) over all w freg. we (^) expect (^) I 1 big^ error^ here^ small ARX" (^) weight 1 =^1 + a, ARXy
- (^) Jw +aARX - 50 ↓ (^) errorhere 1 +^ aiz"^ +^ azz-^ inspectable GO-G(ONAR) much smaller at (^) high frequencies that low^ frequency
103z" (^) (1 + (^0). 432" + (^0). 05622 + (^0). 00232-3) 1 - (^3). 17" +^3.^5802
- (^2) - (^1). 82482 - (^3) + (^0). 3462z - 4
OE seems to have a huge
mismatch wrt the system
but (^) it's due to (^) logaritmic DE scale (^) (the error is still constant) ARY SYSTEM linear scale ARx is bad at low freq.^ but^ much^ closer then of at^ high freq. ARY SYSTEM DE We can not^ say which^ model^ is best^ , it^ depend on the^ objective, and^ on the^ We of^ the^ system We want make ARX (^) good at^ low^ frequencies (^) by using prefiltering GARY-ARX, (^1) R = L(z) =^ H(z, VARY ,
- to^ compensate the^ weight and^ hopefully have (^) a constant (^) weight EvARY,^ L ARy with (^) prefilt weight IL IHvARX,^1112 With (^) prefilter
