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Analysis of the Incremental Tuner
Copyright © IFAC Adaptive Systems in Control and Signal Processing, Grenoble, France, 1992
ANALYSIS OF THE INCREMENTAL TUNER P.R. Barros DeparlamenlO de Engenharia Erelrica, Universidade Federal da Paratba, Campina Grande, Para{ba, Brazil
Abstract. The tuning of PI(D) controllers is an issue that is always returning to the attention of researchers due to the wide spread use of those controllers. In a previous paper an incremental tuner was presented, based on the incremental pole placement adaptive controller. The resulting tuner corrects for the difference between the estimated and desired closed loop polynomials, in the frequency domain and restricted to the variation of the controller numerator polynomial. Cautious adaptation is performed by making the adaptation gain proportional to the reference excitation level in the different frequency points. In this paper an averaging analysis is performed on a particular case of the incremental tuner. Keywords. Adaptive control; pole placement; PID control; adaptive systems; nonlinear systems averaging analysis. 1.
INTRODUCTION
controllers tuning;
cautious
procedure to the tuning of the numerator polynomial, while keeping the denominator polynomial fixed. For that we have reformulated the IPP pole allocation procedure. The proposed incremental tuner also corrects for the difference between the estimated and the desired closed loop denominator polynomials, in the frequency domain and restricted to the variation of the controller numerator polynomial. The resulting cautious adaptation gain is proportional to the level of excitation of the reference signal in the corresponding frequency points.
The incremental pole placement adaptation technique (IPP), presented in Barros(1990), Barros, Mareels and Goodwin (1990) and Barros and Mareel s( 1991) is a strateg y to allocate the closed loop poles by correcting for the difference between the estimated closed loop denominator polynomial and a polynomial expressing the desired closed loop pole allocation. The existence of an adaptation gain independent of the estimator gain allows the introduction of cautious control. Analyses of the adaptive system using the IPP strategy have been performed and are found in the above references. The advantage of this strategy over the conventional one (usi ng the Diophanti ne equation) is that the correction takes into consideration an estimate of the obtained performance. That is important in the presence of noise and unmodelled dynamics.
As mentioned in Barros (1991), analysis of the incremental tuner is a complex task, as it involves the combination of time and frequency domain concepts. Nevertheless it would be interesting to find out some properties of the incremental tuner. In this paper we make an averaging analysis on a p articu lar case of the incremental tuner for which the frequency domai n components of the incr'emental tuner are not present.
The tuning and adaptation of PI and PID controllers is an issue that has r eceived a great deal of attention in last decades, as su rveyed in c 0 Astrom and Wi ttenmark (1989) and Astrom and Haglund (1989). Such an interest is caused by the fact that these controllers are exte nsively used in practice, as they are simple and work reasonably well for a large va riety of plants. As a result, there exist already many commercial PI(D) tuners and adaptive controllers in the market. Many of these tuners are based on the application of an adaptive controller to a reduced order plant. In a similar way, the above mentioned incremental pole placement appl ied to fi rst and second order systems can be used to tune PI(D) controllers.
In section 2 we formalize the problem, by presenting the plant and controller structures. Next we discuss the consequences of restricting the tuning in the final closed loop. In sections 4 and 5 we present the estimation and incremental tuner. In section 6 we perform an analysis on a par' ticular case of the tuning system. 1.1 Notation In this paper we use the following notation : 1) Given the signal yet) and polynomials A and B in the (, operator, with time varying coefficients, B(t) = B(t, ') = b m(t)(Jl1 + .•. b1(t)o + bott) , and
In control systems where PI(D) controllers are used, the controller denominator is basically formed by the integrator (and a wide bandwidth pole). The traditional tuning is done by varying the numerator polynomial. Nevertheless, when the strategies are derived from adaptive controllers the resulting tuning procedures are made by varying both numerator and denominator polynomials. That also happens to when using the IPP strategy in the above mention~d way. In Barros (1991) we have taken an alternative
A(t) = A(t,o) = a n (t) ,lIl + ... a1 (t)" + sort), then: i) A(t).B(t) yet) = A(t) z(t) where z(t) = B(t) yet) ii) A(t)B(t)y(t)=C(t)y(t)
n m
where C(t)=
r r ak(t)bl(t)ok+1
, k=OI=O and " is the delta operator (Middleton Goodwin (1990», defined as ''x (t) = [x(t+1) - x(t»)/T,
157
and
where the integrator part is given by S(,;):,;, the with T the sampling period.
"numerator" polynomial P(t'';)=Pn(t)on+ ••• +Pl (t)o + po(t),
2) We define the order function 0(11) for £--i() as: y(e) = 0(.) for. -+0 If there exists a constant k, Independent of
l2iill •
. , such that
A*(';)
the
L( ,;)u'(t):PO(t)E( O)y
eO adaptive
L(';):on+
u'(t): ~ u'(t).
(2.9)
LEMMA 2.1: The can be rewritten as
PROBLEM STATEMENT
In this paper we consider system shown in figure 2.1.
Is
... +110+10 ' E(';):enon+ ... +elO+eO ' Is the observer part of the desl red closed loop polynomial, and
S k for I: -+ O.
3) We define the time scale as: Y(I:)=O(I:) for .-+0 on the time scale 1/. If the estimate is valid for o S f1: S L, for some constant L, Independent of E:. 2.
the "denominator" polynomial
controller
equation
(2.7)
r(t)-P(t,,;)y(t)+~A*( ,;)-~)u'(t). "0
(2.10) Furthermore, for zero' initial controller conditions the complete controller equation Is given by
control
L( ,;) S( o) u(t)
:
po(t)~ eO
Yr(t)
-
P(t, b) yet) ( 2.11)
PROOF :
Immediate using operator properties.
2.2 The Closed Loop Equations LEMMA 2.2: The behavior of the closed loop system,
,(I) - --)0
from
the
filtered
reference
signal
Yr(t)
to
the
filtered plant output y et) is desc ribed by Acl(t" I) ye t) : B( ,I) . [poet) Fi g. 2. 1 The adapti v e control system
~ eO
Yr(t)] + dc(t)
(2.1 2 )
where 2.1 The Closed Loop System
Acl(t ,O): ASL+BP(t) :
(2. 13)
The Plant: The plant behavior is assumed to be described by a discrete time system model of the form _B( ';)[1+B'(,;)] )+d() (2.1) yet) - A( 0) [1+A'( ,;)] u(t t
and
u(t) is the plant input, yet) is the plant d(t) denotes output noi se, B(';) = b m 8Tl + ... bl'; + bO'
ao,
n E{1,2}, m
, and
L( o)S( O)A( o) y (t):L( o)S( O)B( o)u(t) + L( o)S( o)dp(t). (2.15)
the unmodelled dynamics ( deg(B'( ,;» ~ deg ( A'( o))).
Substituti ng the controller equation on the abo v e equation and rearranging the terms we have
The plant behavior can also be represented as A( ,;) yet) = B( o) u(t) + dp(t), (2.2)
[L( o)S( o)A( o)+B( 0). P(t" I)] y (t)
with d p ( t) =A( ,,)[1 +A'( ,;)]d(t)-A( ,;)A '( c5 )y(t)+B( ,;)B' (,; ) )u(t). (2.3 )
oty r(t)]+L( o)S( old pet).
:B( 0). [po(t)E( eO
(2.16) Finall y , introduci ng the swapp i ng term we get the desi red result.
Filtering all signals through a stable filter ~ / A* (M , y ields the filtered plant equati ons :
y
(t) =
B( 6)
u (t)
(2.4)
+ dp(t) ,
LS dp(t)+[BP(t)-B.P(t)] yet) .
f ix ed , the swapping term [BP(t)-B.P(t)] y et) is equal to zero. PROOF: Operating with L(O)S( O) on the filtered plant equation we get
A'(o) and B'( ';) are polynomials describing
A( O)
dc(t):
(2.14) Furthermore, in the time interval in which pet) is
where output,
A(,;) = anon + .. . al'; +
,~ n+l + ai~(t) ~n + ... + a8 ' (t)
3.
DESIGN MOTIVATION
where In control s y stems where PI and PID control lers are used, the denom i nator pol y nomial is basical l y formed b y the integrator part. In order t o i mp lement the deri v ati v e part, a ( h i gh frequen cy) po le i s added t o the ex isting integrator. The controller tuning i s normall y don e by v arying the numerator pol y nomial , for the different combinations of the proportional, der ivati v e and integral gains. It seems also reasonable to ex pect that it is po ssible to extend the same procedure f o r the controller structure of section 2. Ev en more, we want t o keep the L polynomial fi xed and vary the P polynomial in order to get a desired performance in a frequency band of interest.
A*( o)u ( t) = ~u ( t ) ;
A*( ,I)dp(t ) =~dp(t ),
( 2. 5)
with A* O ) gi v en b y A*( '-'l : Ad( O)E( o) : a* 2n+l
~ n+l
+ ... +
at
,I + ao,
wh ose ze ros f o rm the desi red closed loop all ocat ion. It is fac t o rized i n a controll er Ad (o)
(of degree n+1) and an obser v er part
(2 .6) pol e p art E(o)
(of deg ree n). The Controller: The controller action is i mplemented as
To see the consequence of fixing the L polynomial in the conventional pole placement context, consider the Diophanti ne equation A* : ASL + BP. (3.1) Let us use the factorization of polynomial A*( c5) of eq. (2.6) into a controller and an observer form, i. e., A*: Ad E . We can rewrite eq (3.1) as ASL : AdE + (-B)P. (3.2)
u'(t) : ;-'i,l (lO-L( ,;)]u'(t) + Po(t)E( O) Yr(t) - P(t, o) y (t)} o eO
+~A*(O)-~)U'(t) S( ,I) u(t):
u'(t)
(2.7) (2.8 )
158
We use the standard least squares estimation algorithm with covariance resetting applied to the block of data from times t = tk-l up to time t =
Here, for A,S,L, Ad and 8 given, one can always find unique E and P polynomials, provided Ad and 8 are coprime. Thus, the effect of fixing L is that the polynomials E and P will vary for different A and 8 polynomials. Thus, an adaptive controller can be implemented u si ng L fixed, provided that the variation of the E polynomial is negligible in the frequency range of interest defined by polynomial Ad . In 8arros (1991) we have analyzed the influence of freezing the controller polynomial L for a few types of plants of industrial use. In general the results showe d that the variation of the E polynomial was negl igi ble in the frequency range of interest. (A conventional pole placement adaptive controller incorporating equation (3.2) is presented in Barros and Lima (1991». 4.
tk-1. For
time tk -1 ~ t
< tk
make
-Predicted output: yet) = q,(t)T 3(t) -Prediction error : e(t) = y(t) - yet) - Standard least squares: - Parameter update _ ae P(t)1(t) .; !J(t) T 1 + q,(t )P(t)q,( t) e(t)
(4.11 ) (4.12)
( 4.13)
- Covariance update
o pet)
ae P(t)4.(tHCtl2.eru
=
T
(4 .14)
1 + ·p( t)P(t j.f.(t)
where e(t) T =[8c (t)T eB(t)T] d "c d "c d "c" =[(an-an(t» ... (a l-al (t» (ao-an(t» bm(t)
CLOSED LOOP ESTIMATION
bott)] (4.16)
In this section we present the closed loop estimator. Initially we o btain the regression model for the closed loop system equation. Then we present the least squares estimator used in the implementati on.
- Covariance and Parameter Resetting: At time t = tk make P( t k) = p' I, p' > 0
(4.17)
A(tk' o) = Ad ( 0)
(4.18)
4.1 The Regression Model
REMARK: The covariance and parameter resetting is i ncl uded to speed up the estimator convergence in the case of block processing, as discussed in Barros (1990). This ad hoc procedure is moti va ted by the fact that after the adaptation the initia l parameter estimates for the next bloc k of data should be close to the expected closed loop parameters.
From the closed loop equation AcI(t,';) ye t) = B( b) . [poet)
g£L eO
Yr(t) ] + dc(t)
(4.1)
we can factorize Acl(t,!;) as ACI(t,';) = AC(t,';) EC(t,';) in order to obtai n
(4.2)
AC(t,';) E(!;) yet) = B( o) . [poet) with
E.ili. eO
5.
Yr(t) ]+d'(t) (4.3)
In this section we present the incremental tuning procedure for the (PI and PID) controllers described in the preceding sections. We start by briefly discussing the incremental pole placement procedure. Then we present the incremental PI( D) tuning procedure.
d'(t) = AC(t,';)[E(c5) - EC(t, ';)] y(t)+d c(t). (4.4)
Rearranging eq. (4.3) we get yet) =
[Ad( O)-A C(t, O)] E ('i)~( t)
5.1 Incremental Pole Placement
+B( " ).[pO(t)E( O)y r(t)]+~'(t). eO "0
The main concept behind the incremental pole placement adaptation scheme is to correct the existing controller based on the difference between the estimated closed loop poles and the desired ones. The correction is made cautiousl y, by conditioning the adaptation gain to be propo r·ti onal to the le v el of excitation. The scheme is implemented as foll o ws: The controller polynom ials L(t), pet) are updated by: A(tk)SLl.L(tk) + B(tk) Ll.P(tk) = g(t k) [A* -AC I( tk l ]
(4.5) The above equation can be reduced to y(t)=[A d( o)-A C(t,O)]? '(t)+B( o). [PO (t)y ;.(t)]+~ '(t), aO aO (4.6) with
Ad( oh;'(t)=agy(t) and
INCREMENTAL CONTROLLER TUNING
Ad(t)y;'(t)=agYr(t), (4.7)
and where we cancelled the E( c5) filter operations and neglected the effects of fi Iters initial conditions.
(5. 1 ) (5.2) (5.3)
We can express the above equation in the following reg ression form where y(t) = q,(t) T 8( t) + d(t) where q,(t)T=
[onTY'(t) aO
(4.8)
b-:!.--Y' (t ) d aO
A(tk),
TY'(t) aO
8TlPo(t)y;'(t) ... po(t) y;'(t)] (4.9) 8(t)T = [ 8c ( t)T 88 T ] = [(ed - OC(t»T liB T ] d c d c d c =[(an-an(t» ... (al-al(t» (ao-an(t» b m '" and
d(t) =
B(tk),
ACI(tk)
are
the
estimates
at
time tk for the open loop and closed loop pol ynomials, respectivel y, g(tk) ~ 0 is the adaptation gain, expressing the desi red adaptation rate as well as the confidence level one have on the estimate s. and Proj is the projection operator on a known
fP
bo], (4.10)
set
~'(t).
fP.
5.2 Incremental PHD) Controller Tuning
aO
In this work we difference between
4.2 The Parameter Estimator
allocation estimated
159
want to compensate for the the desired closed loop pole
given by polynomial Ad(o) and the closed loop denominator polynomial
initial controller design, by the choice of E(o), as well as the space in which P(t) is allowed to vary. For that reason we restrict our main objective to the study of the local properties of the adaptive system when the unmodelled part due to the difference (E(o) - EC(t,o» Is small. Note that stability Is not our main concern, but performance.
AC(tk,O), by tuning the controller polynomial P(t,o) at times tk+1. From equation (5.1), with L( 0) (I.e., t.L = 0) we
fixed
compute P(tk+1,0) by solving
B(tk) t.p(tk) = g(tk) [Ad - AC(tk)]E.
(5.4)
In general the above equation does not always have a solution. An approximate solution is to restrict the controller update to correct for the difference In the frequency domain, for a frequency band of interest. That is made performing the following steps at times tk:
Averaging analysis is a mature field (see Sanders and Verhulst (1985» and has been extensively used to study the characteristics of adaptive systems (Astrom and Wittenmark (1989), Sastry and Bodson (1990), among others). In this paper the averaging analysis is made assuming three different time scales, as in Barros(1990) and Barros and Mareels (1991). In the fast time scale the trajectories of the estimation and adaptation equations are almost constant. Assuming stability for the Initial closed loop, we obtai n approximations for the plant and controller trajectories. These trajectories are then substituted in the estimator equation. In the intermediate time scale, the adaptation trajectory Is almost constant, and the fast trajectory is averaged out. Exponential stability is then establ ished for the averaged estimation equation. Now, the approximated estimator trajectory is obtained and used in the adaptation equation. In the slow time scale the approximated estimator trajectory is averaged out and conditions for exponential stability of the adaptation equation are obtained. Finally, stability conditions for the adaptive system are obtained.
i) Compute the adaptation gain g(tk, Yi) which express the level of excitation of the reference signal at a delta frequency point
=
ej "H
in T the frequency range of interest (i.e., bandwidth of 1/Ad( Yi», where (.vi is the associated continuous time frequency point. Yi
ii) Compute the controller increment by an approximate solution to the equation B(tk' Yi )P'(tk+ 1,Yi)
finding
=B(tk, ' i )P(tk' Vi )+g(tk' Yi )[A d (Yi )-A C(tk' 'i )]E( Yi) (5.5) by maki ng a least squares fitting at M frequency points Yi' i=1, ... ,M.
The assumptions, lemmas and theorem presented in this section follow closely those presented in Barros(1990) and Barros and Mareels (1991). The proofs will be omitted for the sake of brevity.
iii) Update the controller with a projection scheme P(tk+1) Proj P'(tk+1 ). (5.6)
0'
System Description
6.2 The adaptation Is made cautiously, considering the level of excitation of the reference signal in the time interval from tk-1 to tk at pre-defined delta
Jl1e_ p.lant: The plant (2.4), with B( o) = bOo
frequency points Yi, i=1, ... ,M. In this work we consider the average energy level of the reference signal in a few frequency bands. The average energy of the band pass filtered reference signal is computed using tk (> 1 or (5.7) l i(tk) = tk- t k-1 ~ tk-1 where Wire) is a band pass filter with pass band
I.h!LJt§!i!T)J'llQ.r:::
centered at Yi
The gain
Ci a
if
o
(,i ~ (,min otherwise
where 0 < O:a S 1, averaged energy le v el occur. ~
6.1
g(tk,Yi)
,) '"t'(t)
is
descri bed
by
The gradient estimator
O:e = TNt) e(t)
(6.1)
is used instead of the least squares equations (4.13), (4.14), (4.17) and (4.18). The adaptation: The adaptation simplified to eO d ' t.,P(t) = Cia -,- [AO - AC(t)]. bO
is given by
The above equation can be operator form as Cia eO d ' ,'.p(t) = T ;::-- [AO - AC(t)]. bO
(5.8)
f min > 0 is the minimum for which adaptation can
AVERAGING ANALYSIS
equation
given
equation
(5.5)
is
(6.2)
rewritten
in
delta (6.3)
6.3
State Equations
6.3.1
The Plant and Controller Equations
From the plant and controller equations write : oX( t) = F( ,Op(t» X( t ) + H( l'p(t»w(t)
Introduction
by
we
can (6.4)
where
Due to the frequency domain approximati o n in the controller redesign procedure of section 6. it is very hard to analyze the full adaptive system presented in the previous sections. For that reason, in this section we analyze a particular case. More specifically, we restrict ourselves to a particular class of plants, namely those with no zeros in the modelled part. We also slightly simplify the adaptation equation, substituting the observer polynomial E by its D.C. gain. In addition we assume adaptation occurring for each time t. Finally, we use a gradient estimator.
Y(t») X( t)= ( U(t) , U(t)T =[ ,ik- 1 u(t) F(8p(t»
A=
-a'k-1 1 0 0
As discussed in section 2, the contribution of the unmodelled term E( ,5) - EC(t,';) will depend on the
160
=
[(}J'(~) -a 'k-2 0 1 0
u(t)], (6.5) (6.6)
mJsJ -ai'
-aQ
0 0
0 0 0
rbk~l0 IE=
- b"1 0 0
(6.7)
1
0
0
~[
rP'(t)
respectl vel y, for Bp in ascont' with Xs (0 ,"p
0 0
0 0
-Pn(t) 0 0
0
0
0
0
0
6.3.2
k~[ 0:
0 0
-In 1 0
0
0
and
Mareels
Cl
~
-10
or
(4.11)
and
Cl
~
e
(4.8),
e
oa(t)=- T'""(X(t), Y r(t» ,9 (t)+T'""(X(t), Y r(t»8(t)
1
:.
(1991)
Estimator Equation
The estimator equations (6.1), (4.12) can be rewritten as (6.8)
0
Barros
and
Cl
_
+ TeA(X(t),y r(t),d(t»
(6.9'
where
(6.14)
r(X(t),Yr(t»= : ; :}o(t)T, A(X(t), Y r(t),d(t) )=q,(t)d(t),
H( ap(t» T = [d'(t) 0 0 po(t)y r(t) .•. 0] (6.10) with a'( and b'j' coefficients of A"=A(1+A')S and B"=B(1+B'), respectively, d'(t)=A(1+A')Sd(t) and 8p (t) T = [Pn(t) ..• poet)].
Estimator Assumptions:
System Assumptions: A 1. The si gnals Y r(t) and cS i d(t) , i=O, .•. , 2n+1, are
Consider the bounded set Best such that, for all t, e(t) E Best. We assume the following:
bounded, Consider the bounded set
A9. For all
t, the A2. A3. the lie
,9cont
t-
t>
for some
0'
such that, for all
0'
>0
obtained
by
setting
=
0,
Yr(t)
is sufficientl y
rich
LEMMA 6.2: assumptions sufficiently satisfyi ng approximated
is
such
exciting (P.E.), i. e., for all s ? 0 there exists
0:0
1I
> 0 and N ? dim ~
1 s+N-1
N
L
Cl e
with radius
-
Cl e
A(X~(t,ep)'Yr(t),d(t»
~ %2 1• with es(O,X~(O, l'p(O»,8(O»=O, and X~(t, ,cp(t» = Xs (t, 8p (t» + Z'(t, 8p (t». PROOF; See Barros and Mareels (1991) Bar rose 1990).
E 8scont the difference ::[E(o) -
,J
-
(6. 17)
EC(t)]y(t):: is small. A7. For all ,'F- such that li ed - eCII ~ J , we have that the true closed loop system matrix F has its 1 eigenvalue in the circle centered at (- 1',0) and
l'1 -
,
+Tf(XS(t, ep),Yr(t»a + T
0'01,
q,(t), such that
~o( k) 1>0( k) T
k=s A6. For all 8p (t)
For the adaptive system satisfying A 1, A3, A4, A5 and A 10, and for O'a small, the estimates trajectory equations (6.12)-(6.15) can be by
'$(t)=es(t,x~(t,ep(t»,8(t»+e~(t,X~(t,l)p(t») + 0 (Cl a 1+Y) (6.16) with
that the associated regressor
Cla Y = ((e'
8p (t) and 8(t) are almost constant. Also, by construction the homogeneous part of the estimator equation (6.14) is exponentially stable on the time scale 0(1/Cla). From that we can state the following lemmas.
asymptotically stable. A5. For the frozen closed loop system (CIa = 0) satisfyi ng assumptions A 1, A2 and A4, the reference si gnal
bo(t)? bOmi n'
On the intermediate time scale
i.e., for a
Cia
e(t) E Best we have
A 1O. There exists Y E (0,1) such that
fixed F the closed loop system is stable with stability margin 0'). A4. ep(o) E 8scont. This implies that the frozen system,
PO(t)Y~(t)]. (6.15)
"p(t) ~ ilcont . We assume the following about plant and controller equations bO ? bOmin > 0, with bOmin known. For all 8p (t) E escont C 8cont we have that ei genval ues of the closed loop system matrix F 1 in the circle centered at (- T'O) and with
radius
Y r(t)=[ 8TlpO(t)y~(t)
and
for some
1 T >
0
?
(1'
>
LEMMA 6.3: Let us assume that the average of equation (6.17) exists and is given by
0
./'
-
Q
'
e
-
"
-
'-"~sav(t" e ) = - Trav(Xs(t, 8p),Yr(t» 8sav(t, ,9 )
i.e., the closed loop system is stable with stability margin e ).
ct e , , - ~e +Trav( Xs(t, p ), Y r(t» ,o +T'\av( Xs(t, ep )' Y r(t),d(t»
e
For the plant and controller signals we have the following lemma:
(6.19)
O'a sufficiently small, the plant LEMMA 6.1: For signals satisfying equation (6.4) and assumptions A3, A4 can be approximated as o ~ t ~ 0(1/Cl a ) X(t)=xs(t,ep(t) )+Z'(t,8p (t) )+O( CIa) (6.11) where Xs and Z' satisfy OXs(t,ep ) = F(ep ) Xs(t,e p ) + H(ep)W(t) oZ'(t,ilp(t)) = F(ap(t» Z'(t,ap(t»
or
with
esav(O,X~(O,tl»=es(O,X~(O,(l),e)).
es(t,X~(t,ap(t»,.9(t»
-
Then we have
esav(t,X~(t,8p(t),8(t» on a time scale
= O(Cl e ) 0(1/ Cl a ).
(6.20) Proof: Similar to Barros and Mareels (1991) or Barros (1990). LEMMA 6.4: The equilibrium trajectory of equation (6.19) is given by
(6.12) (6.13)
161
Bsav (t,8)=8 + Bd(8) +
exponentlally decaying term, (6.21 )
=
where
•
-
,-
Bd( B)=rav(Xs(t,B p )' Y r(t))-
Is the noise, EC(t,.5). Proof. Barros
1
Barros' and
Mareels
(1991)
e
for some
Il f (t)-{
for
= -
with
~\:~t)S(A.B)[ e:(t)]'
substituting the corresponding part of equilibrium trajectory from lemma (6.4) in above equation we get - -'! a _0_ e S(A B) [ T ·b ott) • Ilc (t)
°
• + f'dc( 8( t»
7.
(6.25) the the
Now . as we can write Ac l(t) = AC(t)E C(t) = D
]
° + AC(t)EC(t)
for some polynomial D coprime with EC(t) (and i s the zero pol y nomial) • then we have or
(ec~d=S(D.EC(t))-l
8.
°
o
As to ~
o
/lC1(t).
subst ituting eq. (6.27) i nto eq . (6.26) we get _ Cia ~ S(A.B)S(D .EC(tW1 ,9C I ( t ) T b ott)
~a.b e0ott)-S(A.B)[8dc (o"""\ t » ]+O(0'a H
),)
+ ex po dec. term s.
( 6. 28)
We can no w state the foll o wing : !...1;_MMU-,.2.; For Cl: a sufficiently small, the av eraged equati o n correspo nd i ng to equati on (6.28 ), g iven b y ci ( t )uf Oav -
, Cl() - Cia -=r=s ( A.B )eO ~(. -1 - S ( D.E c (t »)-1 ~)av "av t b ott)
~a
S(A.B)eo
~-.b_1_ )[-;, (0.Qj ))] ~av' ott) " dc '.'\ t
(6. 34 )
S t
s
O(l /c..a ).
CONCLUSION
REFERENCES K.J. and. Hig l und (1989) . ' Tunin g of Pl O Cont rol/ ers·. Instrument Soci ety of America.
Aston K.J. and B. W i tten~ark ( 1989). ' Adaptive Cont rol', Add ison We sl ey. Barros P.R. (1990) . ' Robust Perfornance In Adap tiv e Cortrol·. Ph.D. Th eSIS. UniverSi t y of Nevcastl e, Apr il. Barros P.R. ( 199 1). ' Inc remental PIIO) Tu ne r". Technical Rep ort, Unive rs Idade FederaI da Para iba , Camp ir a Grande -erw ; . September. Submitted for public ation. Barros P.R. and A.M.N. Li ma ( 1991) . 'Rest rict ed Complexity P~le P l a ce~ent Adaot ation ·. Techn'cal Repo r t, UniverSldade Fedefll da Parafba, Ca ~Pln a Grande -Bra: I I , in preparation. Barros P.R . and I.M .Y. Maree l s (1 99 1) . ' Nonllnear A ~ e r a ging AnalYSIS of the Incre~ental Pal e Pl acement Adl Ptlve Cont roller'. Ac~e o t9d for publication i n AUTO~.HICA . Barros P.R . I.H .Y. Mareels and G.C. Goodv i n (1990). 'Loc al Aver l ginl Analysis of the Incre~9rt a l Pol. Pl a c e ~ent Adaptiv e Controller. Proceedings of the /FiC 11th ~or ld Congress, Ta ll,"n, URSS. Hi dd l enton R. and G.C. Goodw i n( 1990) . Digi ta l Con tro l and Estlftlt lon: l Unifi ed Approach ' . Pr entlce 4all-lnternatlonal, Engl ewo01 Cl I ffs. Sanders J.A. and F. Verhu l st 11985) . 'A vera gin g Met hods In Nonlin ear Dynamical Systems ' . Ao oi. ~a t~ . Se r i es, Vol 59, So r in ger Verl ag, 8e rlin. Sastry S. and H. Bodson 11989) . 'Adap t i ve ControI Stab il l ty , Con vergence and Robust ness' . Pre nt i ce Ha II Inter nat Iona I. EngI ew oods C! i If s. N. J.
(6.27)
_
°
). 1'( t) )+tl;(t.X~(t. ,9p(t) )}II=O( a e ) (6.33)
In this paper we have analyzed the incremental PI(D) tuner. This tuner is based on the incremental pole placement adaptive controller. Averaging analysis for a particular case is presented. with the result showing local ex ponential stability . provided noise. unmodelled dynamics are small. and good a-priori information is known about the plant.
(6.26)
&:c l ( t )
esav(t. x~(t. ep(t)
Mo reov er. provided ll tl~bll is suffi c iently small. (i.e. the contri butions of noise. unmodelled dynamics and the term E(o)-EC(t ,o». the averaged systems are locally ex ponentially stable and the above approx imations are valid on an infinite time scale.
(6.24 )
From equations (6.23) and (6.24) we can write
08~(t)]
0(..1...). aa Mareels (1991) and
there exists an ~>o such that for all eta in (O,~) the adaptive system response can be approximated by the averaged system of equations (6.12), (6.13), (6.17). (6.18), (6.19) and (6.29) with IIX(t) - {Xs(t. £lp(t» + Z'(t. 8p (t))}1I = O(et a ) (6.32)
bo]
- S(A,B)[ e;I(t)]'
~1(t)=S(D. EC(t))(ec~t)]
on a time scale
(ar-a~l(t» (~-a~(t»
S(A,B) the Sylvester matrix associated with polynomials A and B. and III = [ In ... 10 ].
c5/lC I (t) =
°
Theorem 6.1: Consider the adaptive system of section 6 satisfying assumptions A1-Al0. Then.
(6.23)
bm
,w:;1(t)= - S(A.B)(
>
PROOF: Similar to Barros and Barros (1990).
and
We are interested in the nomi nal closed loop polynomial Acl(t). Thus, instead of considering equation (6.23) let us consider the deviation of the desired closed loop polynomial (A*(t) - Acl(t». Rewriting this deviation in the associated state vector form we get
~
E:
(6.31 )
We can now state the main result:
=
...
e
6.4 The Main Result
Equation (6.3) can be rewritten as aa eO • Ulp(t) T -.- - 8c (t). b ott)
~1(t)T =[(a~-a~l(t»
~ -d
1 (. 0 ] ~av?-'--) (BCI ) av' b ott) dc av
Furthermore, cl ,..,1 , 0- (t) av(t) = O(aa)
Adaptation Equation
6.3.3
-1
(6.30)
Aav(X~(t,8p),Y r(t),d(t»
(6.22) estimator bias term, due to the presence of unmodelled dynamics and the term E(.5)Slmi lar to (1990).
~ 1 S(D,E c (t» -?-.-b ott)
(6.29)
is locally exponentially attractive to the equilibrium points satisfying cI fl cl BaV' eq
162
ANALYSIS OF THE INCREMENTAL TUNER P.R. Barros DeparlamenlO de Engenharia Erelrica, Universidade Federal da Paratba, Campina Grande, Para{ba, Brazil
Abstract. The tuning of PI(D) controllers is an issue that is always returning to the attention of researchers due to the wide spread use of those controllers. In a previous paper an incremental tuner was presented, based on the incremental pole placement adaptive controller. The resulting tuner corrects for the difference between the estimated and desired closed loop polynomials, in the frequency domain and restricted to the variation of the controller numerator polynomial. Cautious adaptation is performed by making the adaptation gain proportional to the reference excitation level in the different frequency points. In this paper an averaging analysis is performed on a particular case of the incremental tuner. Keywords. Adaptive control; pole placement; PID control; adaptive systems; nonlinear systems averaging analysis. 1.
INTRODUCTION
controllers tuning;
cautious
procedure to the tuning of the numerator polynomial, while keeping the denominator polynomial fixed. For that we have reformulated the IPP pole allocation procedure. The proposed incremental tuner also corrects for the difference between the estimated and the desired closed loop denominator polynomials, in the frequency domain and restricted to the variation of the controller numerator polynomial. The resulting cautious adaptation gain is proportional to the level of excitation of the reference signal in the corresponding frequency points.
The incremental pole placement adaptation technique (IPP), presented in Barros(1990), Barros, Mareels and Goodwin (1990) and Barros and Mareel s( 1991) is a strateg y to allocate the closed loop poles by correcting for the difference between the estimated closed loop denominator polynomial and a polynomial expressing the desired closed loop pole allocation. The existence of an adaptation gain independent of the estimator gain allows the introduction of cautious control. Analyses of the adaptive system using the IPP strategy have been performed and are found in the above references. The advantage of this strategy over the conventional one (usi ng the Diophanti ne equation) is that the correction takes into consideration an estimate of the obtained performance. That is important in the presence of noise and unmodelled dynamics.
As mentioned in Barros (1991), analysis of the incremental tuner is a complex task, as it involves the combination of time and frequency domain concepts. Nevertheless it would be interesting to find out some properties of the incremental tuner. In this paper we make an averaging analysis on a p articu lar case of the incremental tuner for which the frequency domai n components of the incr'emental tuner are not present.
The tuning and adaptation of PI and PID controllers is an issue that has r eceived a great deal of attention in last decades, as su rveyed in c 0 Astrom and Wi ttenmark (1989) and Astrom and Haglund (1989). Such an interest is caused by the fact that these controllers are exte nsively used in practice, as they are simple and work reasonably well for a large va riety of plants. As a result, there exist already many commercial PI(D) tuners and adaptive controllers in the market. Many of these tuners are based on the application of an adaptive controller to a reduced order plant. In a similar way, the above mentioned incremental pole placement appl ied to fi rst and second order systems can be used to tune PI(D) controllers.
In section 2 we formalize the problem, by presenting the plant and controller structures. Next we discuss the consequences of restricting the tuning in the final closed loop. In sections 4 and 5 we present the estimation and incremental tuner. In section 6 we perform an analysis on a par' ticular case of the tuning system. 1.1 Notation In this paper we use the following notation : 1) Given the signal yet) and polynomials A and B in the (, operator, with time varying coefficients, B(t) = B(t, ') = b m(t)(Jl1 + .•. b1(t)o + bott) , and
In control systems where PI(D) controllers are used, the controller denominator is basically formed by the integrator (and a wide bandwidth pole). The traditional tuning is done by varying the numerator polynomial. Nevertheless, when the strategies are derived from adaptive controllers the resulting tuning procedures are made by varying both numerator and denominator polynomials. That also happens to when using the IPP strategy in the above mention~d way. In Barros (1991) we have taken an alternative
A(t) = A(t,o) = a n (t) ,lIl + ... a1 (t)" + sort), then: i) A(t).B(t) yet) = A(t) z(t) where z(t) = B(t) yet) ii) A(t)B(t)y(t)=C(t)y(t)
n m
where C(t)=
r r ak(t)bl(t)ok+1
, k=OI=O and " is the delta operator (Middleton Goodwin (1990», defined as ''x (t) = [x(t+1) - x(t»)/T,
157
and
where the integrator part is given by S(,;):,;, the with T the sampling period.
"numerator" polynomial P(t'';)=Pn(t)on+ ••• +Pl (t)o + po(t),
2) We define the order function 0(11) for £--i() as: y(e) = 0(.) for. -+0 If there exists a constant k, Independent of
l2iill •
. , such that
A*(';)
the
L( ,;)u'(t):PO(t)E( O)y
eO adaptive
L(';):on+
u'(t): ~ u'(t).
(2.9)
LEMMA 2.1: The can be rewritten as
PROBLEM STATEMENT
In this paper we consider system shown in figure 2.1.
Is
... +110+10 ' E(';):enon+ ... +elO+eO ' Is the observer part of the desl red closed loop polynomial, and
S k for I: -+ O.
3) We define the time scale as: Y(I:)=O(I:) for .-+0 on the time scale 1/. If the estimate is valid for o S f1: S L, for some constant L, Independent of E:. 2.
the "denominator" polynomial
controller
equation
(2.7)
r(t)-P(t,,;)y(t)+~A*( ,;)-~)u'(t). "0
(2.10) Furthermore, for zero' initial controller conditions the complete controller equation Is given by
control
L( ,;) S( o) u(t)
:
po(t)~ eO
Yr(t)
-
P(t, b) yet) ( 2.11)
PROOF :
Immediate using operator properties.
2.2 The Closed Loop Equations LEMMA 2.2: The behavior of the closed loop system,
,(I) - --)0
from
the
filtered
reference
signal
Yr(t)
to
the
filtered plant output y et) is desc ribed by Acl(t" I) ye t) : B( ,I) . [poet) Fi g. 2. 1 The adapti v e control system
~ eO
Yr(t)] + dc(t)
(2.1 2 )
where 2.1 The Closed Loop System
Acl(t ,O): ASL+BP(t) :
(2. 13)
The Plant: The plant behavior is assumed to be described by a discrete time system model of the form _B( ';)[1+B'(,;)] )+d() (2.1) yet) - A( 0) [1+A'( ,;)] u(t t
and
u(t) is the plant input, yet) is the plant d(t) denotes output noi se, B(';) = b m 8Tl + ... bl'; + bO'
ao,
n E{1,2}, m
, and
L( o)S( O)A( o) y (t):L( o)S( O)B( o)u(t) + L( o)S( o)dp(t). (2.15)
the unmodelled dynamics ( deg(B'( ,;» ~ deg ( A'( o))).
Substituti ng the controller equation on the abo v e equation and rearranging the terms we have
The plant behavior can also be represented as A( ,;) yet) = B( o) u(t) + dp(t), (2.2)
[L( o)S( o)A( o)+B( 0). P(t" I)] y (t)
with d p ( t) =A( ,,)[1 +A'( ,;)]d(t)-A( ,;)A '( c5 )y(t)+B( ,;)B' (,; ) )u(t). (2.3 )
oty r(t)]+L( o)S( old pet).
:B( 0). [po(t)E( eO
(2.16) Finall y , introduci ng the swapp i ng term we get the desi red result.
Filtering all signals through a stable filter ~ / A* (M , y ields the filtered plant equati ons :
y
(t) =
B( 6)
u (t)
(2.4)
+ dp(t) ,
LS dp(t)+[BP(t)-B.P(t)] yet) .
f ix ed , the swapping term [BP(t)-B.P(t)] y et) is equal to zero. PROOF: Operating with L(O)S( O) on the filtered plant equation we get
A'(o) and B'( ';) are polynomials describing
A( O)
dc(t):
(2.14) Furthermore, in the time interval in which pet) is
where output,
A(,;) = anon + .. . al'; +
,~ n+l + ai~(t) ~n + ... + a8 ' (t)
3.
DESIGN MOTIVATION
where In control s y stems where PI and PID control lers are used, the denom i nator pol y nomial is basical l y formed b y the integrator part. In order t o i mp lement the deri v ati v e part, a ( h i gh frequen cy) po le i s added t o the ex isting integrator. The controller tuning i s normall y don e by v arying the numerator pol y nomial , for the different combinations of the proportional, der ivati v e and integral gains. It seems also reasonable to ex pect that it is po ssible to extend the same procedure f o r the controller structure of section 2. Ev en more, we want t o keep the L polynomial fi xed and vary the P polynomial in order to get a desired performance in a frequency band of interest.
A*( o)u ( t) = ~u ( t ) ;
A*( ,I)dp(t ) =~dp(t ),
( 2. 5)
with A* O ) gi v en b y A*( '-'l : Ad( O)E( o) : a* 2n+l
~ n+l
+ ... +
at
,I + ao,
wh ose ze ros f o rm the desi red closed loop all ocat ion. It is fac t o rized i n a controll er Ad (o)
(of degree n+1) and an obser v er part
(2 .6) pol e p art E(o)
(of deg ree n). The Controller: The controller action is i mplemented as
To see the consequence of fixing the L polynomial in the conventional pole placement context, consider the Diophanti ne equation A* : ASL + BP. (3.1) Let us use the factorization of polynomial A*( c5) of eq. (2.6) into a controller and an observer form, i. e., A*: Ad E . We can rewrite eq (3.1) as ASL : AdE + (-B)P. (3.2)
u'(t) : ;-'i,l (lO-L( ,;)]u'(t) + Po(t)E( O) Yr(t) - P(t, o) y (t)} o eO
+~A*(O)-~)U'(t) S( ,I) u(t):
u'(t)
(2.7) (2.8 )
158
We use the standard least squares estimation algorithm with covariance resetting applied to the block of data from times t = tk-l up to time t =
Here, for A,S,L, Ad and 8 given, one can always find unique E and P polynomials, provided Ad and 8 are coprime. Thus, the effect of fixing L is that the polynomials E and P will vary for different A and 8 polynomials. Thus, an adaptive controller can be implemented u si ng L fixed, provided that the variation of the E polynomial is negligible in the frequency range of interest defined by polynomial Ad . In 8arros (1991) we have analyzed the influence of freezing the controller polynomial L for a few types of plants of industrial use. In general the results showe d that the variation of the E polynomial was negl igi ble in the frequency range of interest. (A conventional pole placement adaptive controller incorporating equation (3.2) is presented in Barros and Lima (1991». 4.
tk-1. For
time tk -1 ~ t
< tk
make
-Predicted output: yet) = q,(t)T 3(t) -Prediction error : e(t) = y(t) - yet) - Standard least squares: - Parameter update _ ae P(t)1(t) .; !J(t) T 1 + q,(t )P(t)q,( t) e(t)
(4.11 ) (4.12)
( 4.13)
- Covariance update
o pet)
ae P(t)4.(tHCtl2.eru
=
T
(4 .14)
1 + ·p( t)P(t j.f.(t)
where e(t) T =[8c (t)T eB(t)T] d "c d "c d "c" =[(an-an(t» ... (a l-al (t» (ao-an(t» bm(t)
CLOSED LOOP ESTIMATION
bott)] (4.16)
In this section we present the closed loop estimator. Initially we o btain the regression model for the closed loop system equation. Then we present the least squares estimator used in the implementati on.
- Covariance and Parameter Resetting: At time t = tk make P( t k) = p' I, p' > 0
(4.17)
A(tk' o) = Ad ( 0)
(4.18)
4.1 The Regression Model
REMARK: The covariance and parameter resetting is i ncl uded to speed up the estimator convergence in the case of block processing, as discussed in Barros (1990). This ad hoc procedure is moti va ted by the fact that after the adaptation the initia l parameter estimates for the next bloc k of data should be close to the expected closed loop parameters.
From the closed loop equation AcI(t,';) ye t) = B( b) . [poet)
g£L eO
Yr(t) ] + dc(t)
(4.1)
we can factorize Acl(t,!;) as ACI(t,';) = AC(t,';) EC(t,';) in order to obtai n
(4.2)
AC(t,';) E(!;) yet) = B( o) . [poet) with
E.ili. eO
5.
Yr(t) ]+d'(t) (4.3)
In this section we present the incremental tuning procedure for the (PI and PID) controllers described in the preceding sections. We start by briefly discussing the incremental pole placement procedure. Then we present the incremental PI( D) tuning procedure.
d'(t) = AC(t,';)[E(c5) - EC(t, ';)] y(t)+d c(t). (4.4)
Rearranging eq. (4.3) we get yet) =
[Ad( O)-A C(t, O)] E ('i)~( t)
5.1 Incremental Pole Placement
+B( " ).[pO(t)E( O)y r(t)]+~'(t). eO "0
The main concept behind the incremental pole placement adaptation scheme is to correct the existing controller based on the difference between the estimated closed loop poles and the desired ones. The correction is made cautiousl y, by conditioning the adaptation gain to be propo r·ti onal to the le v el of excitation. The scheme is implemented as foll o ws: The controller polynom ials L(t), pet) are updated by: A(tk)SLl.L(tk) + B(tk) Ll.P(tk) = g(t k) [A* -AC I( tk l ]
(4.5) The above equation can be reduced to y(t)=[A d( o)-A C(t,O)]? '(t)+B( o). [PO (t)y ;.(t)]+~ '(t), aO aO (4.6) with
Ad( oh;'(t)=agy(t) and
INCREMENTAL CONTROLLER TUNING
Ad(t)y;'(t)=agYr(t), (4.7)
and where we cancelled the E( c5) filter operations and neglected the effects of fi Iters initial conditions.
(5. 1 ) (5.2) (5.3)
We can express the above equation in the following reg ression form where y(t) = q,(t) T 8( t) + d(t) where q,(t)T=
[onTY'(t) aO
(4.8)
b-:!.--Y' (t ) d aO
A(tk),
TY'(t) aO
8TlPo(t)y;'(t) ... po(t) y;'(t)] (4.9) 8(t)T = [ 8c ( t)T 88 T ] = [(ed - OC(t»T liB T ] d c d c d c =[(an-an(t» ... (al-al(t» (ao-an(t» b m '" and
d(t) =
B(tk),
ACI(tk)
are
the
estimates
at
time tk for the open loop and closed loop pol ynomials, respectivel y, g(tk) ~ 0 is the adaptation gain, expressing the desi red adaptation rate as well as the confidence level one have on the estimate s. and Proj is the projection operator on a known
fP
bo], (4.10)
set
~'(t).
fP.
5.2 Incremental PHD) Controller Tuning
aO
In this work we difference between
4.2 The Parameter Estimator
allocation estimated
159
want to compensate for the the desired closed loop pole
given by polynomial Ad(o) and the closed loop denominator polynomial
initial controller design, by the choice of E(o), as well as the space in which P(t) is allowed to vary. For that reason we restrict our main objective to the study of the local properties of the adaptive system when the unmodelled part due to the difference (E(o) - EC(t,o» Is small. Note that stability Is not our main concern, but performance.
AC(tk,O), by tuning the controller polynomial P(t,o) at times tk+1. From equation (5.1), with L( 0) (I.e., t.L = 0) we
fixed
compute P(tk+1,0) by solving
B(tk) t.p(tk) = g(tk) [Ad - AC(tk)]E.
(5.4)
In general the above equation does not always have a solution. An approximate solution is to restrict the controller update to correct for the difference In the frequency domain, for a frequency band of interest. That is made performing the following steps at times tk:
Averaging analysis is a mature field (see Sanders and Verhulst (1985» and has been extensively used to study the characteristics of adaptive systems (Astrom and Wittenmark (1989), Sastry and Bodson (1990), among others). In this paper the averaging analysis is made assuming three different time scales, as in Barros(1990) and Barros and Mareels (1991). In the fast time scale the trajectories of the estimation and adaptation equations are almost constant. Assuming stability for the Initial closed loop, we obtai n approximations for the plant and controller trajectories. These trajectories are then substituted in the estimator equation. In the intermediate time scale, the adaptation trajectory Is almost constant, and the fast trajectory is averaged out. Exponential stability is then establ ished for the averaged estimation equation. Now, the approximated estimator trajectory is obtained and used in the adaptation equation. In the slow time scale the approximated estimator trajectory is averaged out and conditions for exponential stability of the adaptation equation are obtained. Finally, stability conditions for the adaptive system are obtained.
i) Compute the adaptation gain g(tk, Yi) which express the level of excitation of the reference signal at a delta frequency point
=
ej "H
in T the frequency range of interest (i.e., bandwidth of 1/Ad( Yi», where (.vi is the associated continuous time frequency point. Yi
ii) Compute the controller increment by an approximate solution to the equation B(tk' Yi )P'(tk+ 1,Yi)
finding
=B(tk, ' i )P(tk' Vi )+g(tk' Yi )[A d (Yi )-A C(tk' 'i )]E( Yi) (5.5) by maki ng a least squares fitting at M frequency points Yi' i=1, ... ,M.
The assumptions, lemmas and theorem presented in this section follow closely those presented in Barros(1990) and Barros and Mareels (1991). The proofs will be omitted for the sake of brevity.
iii) Update the controller with a projection scheme P(tk+1) Proj P'(tk+1 ). (5.6)
0'
System Description
6.2 The adaptation Is made cautiously, considering the level of excitation of the reference signal in the time interval from tk-1 to tk at pre-defined delta
Jl1e_ p.lant: The plant (2.4), with B( o) = bOo
frequency points Yi, i=1, ... ,M. In this work we consider the average energy level of the reference signal in a few frequency bands. The average energy of the band pass filtered reference signal is computed using tk (> 1 or (5.7) l i(tk) = tk- t k-1 ~ tk-1 where Wire) is a band pass filter with pass band
I.h!LJt§!i!T)J'llQ.r:::
centered at Yi
The gain
Ci a
if
o
(,i ~ (,min otherwise
where 0 < O:a S 1, averaged energy le v el occur. ~
6.1
g(tk,Yi)
,) '"t'(t)
is
descri bed
by
The gradient estimator
O:e = TNt) e(t)
(6.1)
is used instead of the least squares equations (4.13), (4.14), (4.17) and (4.18). The adaptation: The adaptation simplified to eO d ' t.,P(t) = Cia -,- [AO - AC(t)]. bO
is given by
The above equation can be operator form as Cia eO d ' ,'.p(t) = T ;::-- [AO - AC(t)]. bO
(5.8)
f min > 0 is the minimum for which adaptation can
AVERAGING ANALYSIS
equation
given
equation
(5.5)
is
(6.2)
rewritten
in
delta (6.3)
6.3
State Equations
6.3.1
The Plant and Controller Equations
From the plant and controller equations write : oX( t) = F( ,Op(t» X( t ) + H( l'p(t»w(t)
Introduction
by
we
can (6.4)
where
Due to the frequency domain approximati o n in the controller redesign procedure of section 6. it is very hard to analyze the full adaptive system presented in the previous sections. For that reason, in this section we analyze a particular case. More specifically, we restrict ourselves to a particular class of plants, namely those with no zeros in the modelled part. We also slightly simplify the adaptation equation, substituting the observer polynomial E by its D.C. gain. In addition we assume adaptation occurring for each time t. Finally, we use a gradient estimator.
Y(t») X( t)= ( U(t) , U(t)T =[ ,ik- 1 u(t) F(8p(t»
A=
-a'k-1 1 0 0
As discussed in section 2, the contribution of the unmodelled term E( ,5) - EC(t,';) will depend on the
160
=
[(}J'(~) -a 'k-2 0 1 0
u(t)], (6.5) (6.6)
mJsJ -ai'
-aQ
0 0
0 0 0
rbk~l0 IE=
- b"1 0 0
(6.7)
1
0
0
~[
rP'(t)
respectl vel y, for Bp in ascont' with Xs (0 ,"p
0 0
0 0
-Pn(t) 0 0
0
0
0
0
0
6.3.2
k~[ 0:
0 0
-In 1 0
0
0
and
Mareels
Cl
~
-10
or
(4.11)
and
Cl
~
e
(4.8),
e
oa(t)=- T'""(X(t), Y r(t» ,9 (t)+T'""(X(t), Y r(t»8(t)
1
:.
(1991)
Estimator Equation
The estimator equations (6.1), (4.12) can be rewritten as (6.8)
0
Barros
and
Cl
_
+ TeA(X(t),y r(t),d(t»
(6.9'
where
(6.14)
r(X(t),Yr(t»= : ; :}o(t)T, A(X(t), Y r(t),d(t) )=q,(t)d(t),
H( ap(t» T = [d'(t) 0 0 po(t)y r(t) .•. 0] (6.10) with a'( and b'j' coefficients of A"=A(1+A')S and B"=B(1+B'), respectively, d'(t)=A(1+A')Sd(t) and 8p (t) T = [Pn(t) ..• poet)].
Estimator Assumptions:
System Assumptions: A 1. The si gnals Y r(t) and cS i d(t) , i=O, .•. , 2n+1, are
Consider the bounded set Best such that, for all t, e(t) E Best. We assume the following:
bounded, Consider the bounded set
A9. For all
t, the A2. A3. the lie
,9cont
t-
t>
for some
0'
such that, for all
0'
>0
obtained
by
setting
=
0,
Yr(t)
is sufficientl y
rich
LEMMA 6.2: assumptions sufficiently satisfyi ng approximated
is
such
exciting (P.E.), i. e., for all s ? 0 there exists
0:0
1I
> 0 and N ? dim ~
1 s+N-1
N
L
Cl e
with radius
-
Cl e
A(X~(t,ep)'Yr(t),d(t»
~ %2 1• with es(O,X~(O, l'p(O»,8(O»=O, and X~(t, ,cp(t» = Xs (t, 8p (t» + Z'(t, 8p (t». PROOF; See Barros and Mareels (1991) Bar rose 1990).
E 8scont the difference ::[E(o) -
,J
-
(6. 17)
EC(t)]y(t):: is small. A7. For all ,'F- such that li ed - eCII ~ J , we have that the true closed loop system matrix F has its 1 eigenvalue in the circle centered at (- 1',0) and
l'1 -
,
+Tf(XS(t, ep),Yr(t»a + T
0'01,
q,(t), such that
~o( k) 1>0( k) T
k=s A6. For all 8p (t)
For the adaptive system satisfying A 1, A3, A4, A5 and A 10, and for O'a small, the estimates trajectory equations (6.12)-(6.15) can be by
'$(t)=es(t,x~(t,ep(t»,8(t»+e~(t,X~(t,l)p(t») + 0 (Cl a 1+Y) (6.16) with
that the associated regressor
Cla Y = ((e'
8p (t) and 8(t) are almost constant. Also, by construction the homogeneous part of the estimator equation (6.14) is exponentially stable on the time scale 0(1/Cla). From that we can state the following lemmas.
asymptotically stable. A5. For the frozen closed loop system (CIa = 0) satisfyi ng assumptions A 1, A2 and A4, the reference si gnal
bo(t)? bOmi n'
On the intermediate time scale
i.e., for a
Cia
e(t) E Best we have
A 1O. There exists Y E (0,1) such that
fixed F the closed loop system is stable with stability margin 0'). A4. ep(o) E 8scont. This implies that the frozen system,
PO(t)Y~(t)]. (6.15)
"p(t) ~ ilcont . We assume the following about plant and controller equations bO ? bOmin > 0, with bOmin known. For all 8p (t) E escont C 8cont we have that ei genval ues of the closed loop system matrix F 1 in the circle centered at (- T'O) and with
radius
Y r(t)=[ 8TlpO(t)y~(t)
and
for some
1 T >
0
?
(1'
>
LEMMA 6.3: Let us assume that the average of equation (6.17) exists and is given by
0
./'
-
Q
'
e
-
"
-
'-"~sav(t" e ) = - Trav(Xs(t, 8p),Yr(t» 8sav(t, ,9 )
i.e., the closed loop system is stable with stability margin e ).
ct e , , - ~e +Trav( Xs(t, p ), Y r(t» ,o +T'\av( Xs(t, ep )' Y r(t),d(t»
e
For the plant and controller signals we have the following lemma:
(6.19)
O'a sufficiently small, the plant LEMMA 6.1: For signals satisfying equation (6.4) and assumptions A3, A4 can be approximated as o ~ t ~ 0(1/Cl a ) X(t)=xs(t,ep(t) )+Z'(t,8p (t) )+O( CIa) (6.11) where Xs and Z' satisfy OXs(t,ep ) = F(ep ) Xs(t,e p ) + H(ep)W(t) oZ'(t,ilp(t)) = F(ap(t» Z'(t,ap(t»
or
with
esav(O,X~(O,tl»=es(O,X~(O,(l),e)).
es(t,X~(t,ap(t»,.9(t»
-
Then we have
esav(t,X~(t,8p(t),8(t» on a time scale
= O(Cl e ) 0(1/ Cl a ).
(6.20) Proof: Similar to Barros and Mareels (1991) or Barros (1990). LEMMA 6.4: The equilibrium trajectory of equation (6.19) is given by
(6.12) (6.13)
161
Bsav (t,8)=8 + Bd(8) +
exponentlally decaying term, (6.21 )
=
where
•
-
,-
Bd( B)=rav(Xs(t,B p )' Y r(t))-
Is the noise, EC(t,.5). Proof. Barros
1
Barros' and
Mareels
(1991)
e
for some
Il f (t)-{
for
= -
with
~\:~t)S(A.B)[ e:(t)]'
substituting the corresponding part of equilibrium trajectory from lemma (6.4) in above equation we get - -'! a _0_ e S(A B) [ T ·b ott) • Ilc (t)
°
• + f'dc( 8( t»
7.
(6.25) the the
Now . as we can write Ac l(t) = AC(t)E C(t) = D
]
° + AC(t)EC(t)
for some polynomial D coprime with EC(t) (and i s the zero pol y nomial) • then we have or
(ec~d=S(D.EC(t))-l
8.
°
o
As to ~
o
/lC1(t).
subst ituting eq. (6.27) i nto eq . (6.26) we get _ Cia ~ S(A.B)S(D .EC(tW1 ,9C I ( t ) T b ott)
~a.b e0ott)-S(A.B)[8dc (o"""\ t » ]+O(0'a H
),)
+ ex po dec. term s.
( 6. 28)
We can no w state the foll o wing : !...1;_MMU-,.2.; For Cl: a sufficiently small, the av eraged equati o n correspo nd i ng to equati on (6.28 ), g iven b y ci ( t )uf Oav -
, Cl() - Cia -=r=s ( A.B )eO ~(. -1 - S ( D.E c (t »)-1 ~)av "av t b ott)
~a
S(A.B)eo
~-.b_1_ )[-;, (0.Qj ))] ~av' ott) " dc '.'\ t
(6. 34 )
S t
s
O(l /c..a ).
CONCLUSION
REFERENCES K.J. and. Hig l und (1989) . ' Tunin g of Pl O Cont rol/ ers·. Instrument Soci ety of America.
Aston K.J. and B. W i tten~ark ( 1989). ' Adaptive Cont rol', Add ison We sl ey. Barros P.R. (1990) . ' Robust Perfornance In Adap tiv e Cortrol·. Ph.D. Th eSIS. UniverSi t y of Nevcastl e, Apr il. Barros P.R. ( 199 1). ' Inc remental PIIO) Tu ne r". Technical Rep ort, Unive rs Idade FederaI da Para iba , Camp ir a Grande -erw ; . September. Submitted for public ation. Barros P.R. and A.M.N. Li ma ( 1991) . 'Rest rict ed Complexity P~le P l a ce~ent Adaot ation ·. Techn'cal Repo r t, UniverSldade Fedefll da Parafba, Ca ~Pln a Grande -Bra: I I , in preparation. Barros P.R . and I.M .Y. Maree l s (1 99 1) . ' Nonllnear A ~ e r a ging AnalYSIS of the Incre~ental Pal e Pl acement Adl Ptlve Cont roller'. Ac~e o t9d for publication i n AUTO~.HICA . Barros P.R . I.H .Y. Mareels and G.C. Goodv i n (1990). 'Loc al Aver l ginl Analysis of the Incre~9rt a l Pol. Pl a c e ~ent Adaptiv e Controller. Proceedings of the /FiC 11th ~or ld Congress, Ta ll,"n, URSS. Hi dd l enton R. and G.C. Goodw i n( 1990) . Digi ta l Con tro l and Estlftlt lon: l Unifi ed Approach ' . Pr entlce 4all-lnternatlonal, Engl ewo01 Cl I ffs. Sanders J.A. and F. Verhu l st 11985) . 'A vera gin g Met hods In Nonlin ear Dynamical Systems ' . Ao oi. ~a t~ . Se r i es, Vol 59, So r in ger Verl ag, 8e rlin. Sastry S. and H. Bodson 11989) . 'Adap t i ve ControI Stab il l ty , Con vergence and Robust ness' . Pre nt i ce Ha II Inter nat Iona I. EngI ew oods C! i If s. N. J.
(6.27)
_
°
). 1'( t) )+tl;(t.X~(t. ,9p(t) )}II=O( a e ) (6.33)
In this paper we have analyzed the incremental PI(D) tuner. This tuner is based on the incremental pole placement adaptive controller. Averaging analysis for a particular case is presented. with the result showing local ex ponential stability . provided noise. unmodelled dynamics are small. and good a-priori information is known about the plant.
(6.26)
&:c l ( t )
esav(t. x~(t. ep(t)
Mo reov er. provided ll tl~bll is suffi c iently small. (i.e. the contri butions of noise. unmodelled dynamics and the term E(o)-EC(t ,o». the averaged systems are locally ex ponentially stable and the above approx imations are valid on an infinite time scale.
(6.24 )
From equations (6.23) and (6.24) we can write
08~(t)]
0(..1...). aa Mareels (1991) and
there exists an ~>o such that for all eta in (O,~) the adaptive system response can be approximated by the averaged system of equations (6.12), (6.13), (6.17). (6.18), (6.19) and (6.29) with IIX(t) - {Xs(t. £lp(t» + Z'(t. 8p (t))}1I = O(et a ) (6.32)
bo]
- S(A,B)[ e;I(t)]'
~1(t)=S(D. EC(t))(ec~t)]
on a time scale
(ar-a~l(t» (~-a~(t»
S(A,B) the Sylvester matrix associated with polynomials A and B. and III = [ In ... 10 ].
c5/lC I (t) =
°
Theorem 6.1: Consider the adaptive system of section 6 satisfying assumptions A1-Al0. Then.
(6.23)
bm
,w:;1(t)= - S(A.B)(
>
PROOF: Similar to Barros and Barros (1990).
and
We are interested in the nomi nal closed loop polynomial Acl(t). Thus, instead of considering equation (6.23) let us consider the deviation of the desired closed loop polynomial (A*(t) - Acl(t». Rewriting this deviation in the associated state vector form we get
~
E:
(6.31 )
We can now state the main result:
=
...
e
6.4 The Main Result
Equation (6.3) can be rewritten as aa eO • Ulp(t) T -.- - 8c (t). b ott)
~1(t)T =[(a~-a~l(t»
~ -d
1 (. 0 ] ~av?-'--) (BCI ) av' b ott) dc av
Furthermore, cl ,..,1 , 0- (t) av(t) = O(aa)
Adaptation Equation
6.3.3
-1
(6.30)
Aav(X~(t,8p),Y r(t),d(t»
(6.22) estimator bias term, due to the presence of unmodelled dynamics and the term E(.5)Slmi lar to (1990).
~ 1 S(D,E c (t» -?-.-b ott)
(6.29)
is locally exponentially attractive to the equilibrium points satisfying cI fl cl BaV' eq
162