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Local Averaging Analysis of the Incremental Pole Placement Adaptive Controller
C:OI" ri~hl
©
I F.-\C 11111 Tnt'nni,,1 \\"orld
COlIgn..'!'I!'!, Tallillll. F ... loni,t. L'SSI{, I~'~I()
LOCAL AVERAGING ANALYSIS OF THE INCREMENTAL POLE PLACEMENT ADAPTIVE CONTROLLER P. R. Barros*, I. M. Y. Mareels** and G. C. Goodwin** *011 leave from Departamento de Engenharia Eletrica, Ullivenidade FI'deral da Paraiba, Campilla Grallde, PB, Brazil (Work sponsored by C\'Pq Brazil) **Departlllellt of Electrical Ellgineerillg alld Computer Science, L'lIivn.lit)' of ,\ 'ewcastle. Newcastle, NSW 2308, Awtralia
Abstract. Incremental pole placement is an adaptive control algorithm based on the pole placement, indirect adaptive control approach. It differs from (he classical algorithm in that "the closed loop poles" are estimated directly, together with the plant poles and zeros. The deviation between the obtained "closed loop poles" and the desired ones is instrumental in the update mechanisms. Cautious control is implemented naturally as the control update and estimation algorithm are highly decoupled. This paper addresses the overall performance problem _ A local analysis is presented using averaging theory . The results of this analysis suggest some non trivial modifications to improve the performance. Keywords. Adaptive control; adaptive systems; pole placement; pole assignment; cautious control; averaging analysis.
1.
This paper is organized as follows: In section 2 we describe the closed loop system. In section 3 we introduce a simple parameter estimator for the closed loop system_ Next we present the incremental pole placement adaptation scheme. Section 5 gives the local averal;!:ing analysis of the algorit hm . In section 6 we dISCUSS the design modifications. Finally, we make some concluding remarks in section 7.
INTRODUCTION
The general aim of adaptive control is to guarantee a desired level of performance, in the face of initial uncertainty about the plant and of plant var i ations during system operation. The way in which the desired performance is specified and the way in which the adaptation is made define the different types of adaptive controllers, as surveyed, for instance, in Astrom (1987), Seborg (1986) and ortega (1988). For some of these schemes, the obtained performance is measured and appears expl ici tly in the adaptat ion scheme. In the other cases, although the desired performance is precisely stated and is used to guide the adaptat ion, the obtained performance is not used and usually is not measured_
Notation In this paper we use the 6-operator: 6y( t)
(y(t+T) -y(t))/T
°
and polynomial system descriptions for conciseness. With A, polynmials in 6, we denote by AO(=OA) the polynomial product and by A· O(_O·A) the concatenat ion of operators in 6.
The standard pole placement technique falls in the second class of adaptive controllers.
2.
The incremental pole placement adaptation strategy is based on the same concepts of closed loop pole allocation. However it estimates on line how well the closed loop poles actually approximate the desired location. Moreover it uses this information explicitly in updating the controller. In addition the adaptation and estimation procedures are highly decoupled, making cautious control possible.
TilE CLOSED LOOP SYSTEM
In this paper we consider the adaptive control system structure shown in figure 2.1
As stability is not the main concern of this paper, slow adaptat ion can be used to "finely" tune a stabilizing controller so as to obtain the desired performance.
y(l)
In this paper we present the local (performance) analy sis of this adaptive algorithm applied to linear time invariant plants in the presence of structured unmodelled dynamics and output noise. In the analysis we use the concepts of averaging theory (Anderson et aI , 1986). The analysis suggests some modifications to improve the robustness of the adaptive algorithm with respect to unmodelled dynamics.
Fig. 2.1 2.1
The adaptive control system structure
Th e Plant
The plant is assumed to belong to a class of time invariant systems described by A(6)(1+A'(6))y(t) = O(6)(1+0' (6))u(t)
405
(2 _1)
From nOli on lie shall si mplifying assumption:
Where B(6), A(6) represent the modelled part with A(6) assumed monic and aA = m, aB = m-I; B'(6), A'(6) form the unmodelled part with aB'(6) ~ aA'(6),
= Bu(t)
+
TJp(t)
3.
=
AA'y(t)
-
BB'u(t)
+
+
TJ (t)
(2.3)
3.1
Filtering all signals through a stable filter * * , yields the following set of equations to ao/A represent the plant behaviour: Ay(t)
Bu(t)
*
A*y(t) A*u(t) A*:;Jp (t)
+
:;J (t)
The Regressor Model
[A * - I'(t)]
[y (t) ] u(t)
(2.5)
+
~r(t)
The above equation can follolling regressor form
= n = 2m- 1
(3 .1 )
ao
given by ao* ' aA *
, [ [po(t)7 r (t)]
ao
(2.7)
+
~
~A
(2.6)
A*
2.2
PARAMETER ESTIMATION
(2.4)
* aou(t) * (t) aoTJp
with A* (6)
(2. 14)
From equation (2.5) (2.6) and (2. 13) we obtain
p
aoy(t)
follolling
The purpose of this section is to present a closed loop parameter estimator to be used in the local Initially lie use equation (2.11) to analysis. obtain a regressor model for the closed loop suitable for estimation. Then lie present the parameter estimator.
(2.2)
with TJp(t)
the
= po(t)
Q(t)
The plant behaviour can also be represented as Ay(t)
make
be
rellritten
The Time-varying Controller
in
the
o
The controller action is obtained from:
y(t) ] = 4> (t)T 8 (t) [ u (t) 0
u(t) = - l -[ (eo (t ) - L(t)) u(t) - P(t)y(t)
+
eo(t)
(3.2)
IIhere
(2.8) (2.9)
with L(t) aL (t)
=
monic and
(3.3)
m-I, aQ (t)
~
aP (t)
=
m-I 6
[Cor zero controller initial conditions equation (2.8) is equivalent (2.10) L(t)u(t) = Q(t)Yr(t) - P(t)y(t)
n- 1
6
v( l) no
11
-
1
v( t)
..........., ... ,'"r-'-, 6
a o
~, ... ,~, a
o
rn- 1
1
po(t) ..... y (t), ... ,p (t) A
r
0
1 "T
A
Y
r
(t)
0, . . . . . . • . . ,0
a o
2.3 The Closed Loop Eauations.
o , ... ,
The behaviour of the closed loop system, from the reference signal Yr(t) to the filtered signal s y(t)
and u(t)
AC(t) [y(t) ] lu(t)
=
can be described by
[B (t) ] A(t)
0
Q(t) y (t) r
+
TJr (t)
3.2
(2.11)
Several standard parameter estimation al~orithms, such as least squares and normalized gradIent, can be appl ied to the closed loop system model of equations 3.2 to 3.4. As far as local dynamics is concerned, least squares, normalized gradient and gradient algorithms are equivalent. In the sequel lie use a gradient algorithm:
where AC(t) = AL(t) (t) =
1)
r
+
BP(t)
L(t) :;J (t) p [-P(t):;Jp(t)
(2 .12)
+
[BQ(t) -BoQ(t)JY (t)]
+
[AQ(t) -AoQ(t)]Yr(t)
r
The Parameter Estimator
(2.13)
68 (t)
406
=
a
_e ~(t)e(t)
T
(3.5)
y(t) e(t)
in an obvious way - the coefficients are linked to respectively A(l+A'), 0(1+B'), P and L.
[ u(t)
5,1.2. Incremental pole placement equation In the local analysis we can neglect the projection operator in equations (4.2 ) and (4.3). In thIS case we can write the incremental pole placement adaptation law as
(3.6) INCREMENTAL POLE PLACEMENT
4.
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 scheme is implemented as follows:
(5.2 ) A
The controller polynomials updated by:
L(t),
S(8 A,8 B)
are
P(t)
A
A(t) AL (t) + B(t) AP (t) = aa [A * L(t+1) Proj {L(t) + AL(t)}
is the Sylvester Matrix associated with
A
A(t),B(t). (The definition of the Sylvester matrix can be found in Wolovich (1984)). USIng
- Ac (t) ] (4. 1) (4.2)
<.e
P(t+1)
Proj {P(t) + AP(t)}
(5.3)
(4.3)
8~
~
[ 8
C
where A, B, and A are the estimates for the open loop and closed loop polymonials, respectively. aa > 0 is the adaptation gain and set
Proj
and defining
-
P
(5.4 )
-
(4.4) (5,5)
is the projection operator on a known
~ ~.
we get 5.
LOCAL AVERAGING ANALYSIS
Averaging theory has been widely used to study the stability of adaptive control systems. Here we follow the analysis as outlined in the book by Anderson et al (1986). In doing so, the for the existence of some conditions approximations to be made are similar to the ones presented there in chapters 3 and 5. Initially we present the state space version for the adaptive system described in the previous We linearize this system around the sections. neighbourhood of a "tuned trajectory", associated We then perform a with the nominal system. Lyapunov transformation on the linearized system in order to separate the states into fast and slow variables. Finally, averaging results are called upon to obtain stability conditions. 5.1
5.1.3 Estimator error equation F'rom equations (3,2) to (3.5) and (3.6) we can express the par~meter estimator error 8(t) = 80 (t) - 8(t) (5.7) as
08(t)
State-Variable Equations
1 - T [8 0 (t)
In order to perform the analysis we have to express the adaptive system in a state space form.
or, using (2.13 ) and (2.3) - (2.7 )
5.1.1 Plant and controller equations From equations (2.5) and (2.11) we can write
Tip
r(t) ] ° lU(t) where Y(t)T U(t)T
114 [
~ ]
~(t) ~(t)
[0/!-1 y(t)
114,
~, ~(t)
= - ~ ~ (t) ~ (t)T 8(t) _
o
+
po(t)yr(t)
(5.1)
o o where H, given by
y(t)] u(t)]
[0/!-1 u(t)
where the matrices
lU(t)
08 (t)
(t)
o
r(t)]
and
~(t)
(5.8 )
8o (t + 1)]
~ ~(t){H(8L'8
Ta
o
..J()/
)[ p
0
(t)
~(t)
u( t)
+ Hry(8 L ,8 p)ry(t) + Hr (8 p)Yr(t) } (5.9) Hr ,Hry are time varying operators -L(t)AA ' [ P(t)AA'
are defined
+ 08
L(t)BB ' ] -P(t)BB'
o *
11 (9 ,9) L T)
[ L(t)
o
~* 1
(5.12)
+
o
-P(t) ~
p
A
BPO(t) 11 (9)
-
o
B· po(t)
=
(5_10)
[ APo(t) - A. po(t) r p Writing componentwise, we have:
*
with '!1,
~
*
*
*
corresponding to 9p '
9L.
5.2.2 . The linearized equations The resulting lineari zed equation representing the deviation of the true trajectory (given by (5. 1), (5.6) and (5.11)) in relation to the tuned trajectory is given by
+
[9
*]1
aa 0 0 - 1 0 0 L(t) ] [l'"S(9A,9B)S(9A,9B) [S(9A,9~) 9 (t) - 9A p
where we defined Z = [Y(t)TU(t)T]T and e = [BL(t)T Bp(t)T
J
o
(5.14) Bc(t )T
BB (t) T
with
BA (t)T]T (5.15)
o 5.2 Linearized Description
State-Variable
Equation
We now linearize the adapt i ve system given by equations (5.1), (5.6) and (5.11) around the neighbourhood of a trajectory associated with the tuned controller . 5.2.1 The tuned trajectory The tuned trajectory is the solution of the following system associated with the initial conditions * * l' * l' *1' X = [Y (0) U (0) (9 L) * (t) * U (t) *
'!1
*
~ ~
*
0 0
0 0 0
* (t) U* (t) *
9L(t)
0
0
0
0
9L(t)
9£ (t)
0
0
0
0
9£(t)
9c (t) * 9B(t)
0
0 0
0
9c (t) * 9B(t)
*
6
lA
*
9A(t)
0
0
0
0
o
*
0 0
0 0 0
+
*
9A(t)
o
for [ o0
(5.16) represent the sensitivity of the tuned systems with respect to the control parameters. r y , r u are representative of the unmodelled dynamics and disturbances offsetting the estimator
AL, Ap
408
along the tuned trajectory. w* is the forcing term, reflecting the fact that even at the tuned solution some disturbances are not accounted for in the model. Ac is a measure of the robustness of
the pole placement design,
f1*
iii) (Ax ,x)2 - 4(x,x)(Bx,x) > 0 Vx E 1R2,
where
A, BE 1R 2x2 ,
x -F 0,
(z,y)
= zT y .
Condition (i) is satisfied if
is the
ae
excitation (f~ = ~* ( ~* )T). A detailed description of the above matrices can be found in Barros, Mareels and Goodwin (1989).
r
*
~I]
[0
F1 > 0
(5.24)
That is, we must have a persistent excitation * and in addition that the condition for ~ (t), adapt~tion gain mu st be sufficiently small in relatIon to the product of the estimator gain by the minimum excitation level.
Lyapunov Transformation on the Linearized 5.3 System In order to separate the system into fast and slow systems we perform a Lyapunov transformation of the form
Condition (i i ) is satisfied if
F~ > ~[rrus(a~,a~)- 1 0 0]
-/I (t)] = [ I [B(t) 0
[rrus(a~,a~)-I
(5 .17) where L(t,aa,ae ) is chosen as to decouple a (the slow variable) from -/I (the fast variable).
OO]T
6[-/I(t)] = [F!! - Lo(t+l)F21 (t,a e ) B(t) F21 (t,a e )
Condition ( iii ) will be satisfied if sufficiently small in relation to a . e For sufficiently -/I (t) are stable The stability of by analysing the
5.4
aa
is
Finally, for the negative real eigenvalues to li e inside the stability circle it i s sufficient that a e be small enough.
small max(aa ,a e) the fast states as Fll is stable by assumption. the slow states can be considered matrix
= F22 (t,aa,ae)
(5 .25)
Thus, the level of excitation must be bigger than the contribution of rruS(a~,a~)- 1. The l atter is a result of the contribution of the disturbances, the level of correlation between the regressor vector and the disturbances, t he sensitivity of the closed loop in relation to the tuned controller parameters and of S(a~,a~)-I( See Barros, Mareels and Goodwin (1989)).
The above transformation yields the following state-variable system
F22 (t,aa,ae )
(5.23)
5.5
+ F21(t,ae)Lo(t,aa,ae) (5. 19)
The Main Result
We conclude theorem:
Local Averagi ng Analysi s
this
section
with
the
following
Theorem 5.1 The adaptive system given b{ equations (5.1), (5.6) and (5. 11 ) is local y stable in the neighbourhood of the tuned trajectory described by (5.12) provided that conditions (i)- (iii) are sat i sfied, a e is sufficiently smal l, and that the disturbances driving term
Assuming averages exist, stability of the unforced equation is determined by
* As in Anderson et al is of the order of a. * (1986), a is determined by a set of inequalities which makes it possible to linearize and separate slow and fast states, while retaining stabi lity properties. For stability we need that the zeros of the above polynomial must lie in a ci rcle centred at - liT and of radius liT.
Proof:
A sufficient condition for the eigenvalues to be negative real (see Bellman (1970) and Gohberg et al (1982)) is that the following inequalities are satisfied i) xTAx > 0 (5.21) ii) xTBX > 0 (5.22)
In this section we present some modifications to the adaptive control system introduced in the previous section. These modifications are based on the results of the averaging analysis as well as practical considerations, and are intended to yield a more robust adaptive system.
See Barros , Mareels and Goodwin (1989). 6.
409
DESIGN MODIFICATIONS
8.
The proposed modifications are: - Block Processing In order to improve the stability of the adaptive system, we make the adaptation gain g(t) zero for N- 1 sampling periods, and non zero at time t k , the Nth one. This procedure will also make the closed loop system block time invariant. This modification follows from the analysis. (The need for block processing in adaptive control has been argued in Shinkim and Feuer (1988) from different considerations.) - Adaptation Gain From the above averaging analysis and motivated by cautious adaptation we make the adaptation gain on the Nth sampling period proportional to the level of excitation on the previous N- 1 samples . The adaptation gain could be chosen as e.g:
Anderson, B.D.O., R.R. Bitmead, C.R. John son, P.V. Kokotovic, R.L. Kosut, I.M.Y. Mareels, L. Praly,B.D. Ri ede(1986). "Stability of adaptive systems : anaLysis". The MIT
passivity
and
averaging
Press. Astrom, K.J. (1989). "Adaptive feedback control". Proc. of IEEE , Vol. 75, No. 2, pp.185-217. Barros, P.R., I.M.Y. Mareels and G.C. Goodwin (1989) Local analysis of the incremental pole placement adapt i ve controller. Technical Report EE8909, University of Newcastle, Newcastle. Belman, R. (1980) "Introduction to matrix anaLysis". McGraw- Hill book company, Second Edition. Gohberg, I. , P. Lancaster, L. Rodman (1982). "Matrix poLynomiaLs". Academic Press. Middleton, R.H., G.C. Goodwin, D.J. Hill , D.Q. Mayne(1988). "Design issues in adaptive control". IEEE Trans. Automatic ControL, Vol. AC33, No. 1. Ortega, R. and Tang, Vu. (1987). "Theoretical results on robustness of direct adaptive controllers: a survey". Preprints 10th World Congress on Automatic ControL IFAC. Munich, pp.1-15. Seboq;;, D.E. , T.F. Edgar, S.L. Shah. (1986) "Adapti ve control strategies for process control : a survey". AICIIE Journal, Vol. 32, No. 6, pp.881-913. Shimkin, N. and A. Feuer.(1988) "On the necessity of I block invariance I for the convergence of adaptive pole-placement al~orithm with persistently exciting input. Trans . Automatic ControL, Vol. AC33, No.8. Wolovich, 'rI.A. (1984). "Linear muLtiuariabte systems". Springer-Verlag, New York,Inc.
(instead of aa) where tr A means the trace of matrix A, and Po and P(t) are covar iance matr ices of the estimator. - Pre-fi ltering We filter the signals to be used in the estimator The filter algorithm using a band- pass filter. bandw id th is chosen such as to con cent rate the modelling to fit in a frequency band of interest. This modification follows from the analysis and is similar to ideas introduced in Anderson et al. (1986). It also follows from other adaptive control analysis procedures as, for instance, the one used in Middleton et al. (1988). - Least Squares Estimator Instead of the estimator presented in section 3, we use a recursive least squares estimator. It incorporates covariance resetting to a matrix Po at time tk as well as normalization by a signal assumed to overbound the absolute value of the disturbances. The parameter estimates are also restricted to lie in a pre-defined convex set. This modification does not follow from the analysis. - Parameter Estimate Resetting At time t k , after the adaptation, the parameters ec(t) are reset to e*c . This procedure is expected to increase the robustness of the adaptive system. This is motivated by the fact that in blocks of high excitation we would be resetting the ini tial estimate for the next block to where it is expected the closed loop denominator pol)~omial would be. Also, it can be easily recognised that in blocks of low excitation resetting avoids integrator wind up in the adaptation law, consequently improving robustness. 7.
REFERENCES
COilCLUSIONS
In this paper we have presented a local averaging an adaptive control system analysis for incorporating a new adaptation strategy, namely, incremental pole placement. The main results were sufficient conditions for local stability of the adaptive system in the presence of small disturbances. Those sufficient conditions can be interpreted as the existence of enough excitation and the requirement that the adaptation gain should be small compared with th~ estimation gain weighted by the level of that eXCItatIon. We have used those results together with other practical considerations to modify the adaptive system into a more robust one.
410
©
I F.-\C 11111 Tnt'nni,,1 \\"orld
COlIgn..'!'I!'!, Tallillll. F ... loni,t. L'SSI{, I~'~I()
LOCAL AVERAGING ANALYSIS OF THE INCREMENTAL POLE PLACEMENT ADAPTIVE CONTROLLER P. R. Barros*, I. M. Y. Mareels** and G. C. Goodwin** *011 leave from Departamento de Engenharia Eletrica, Ullivenidade FI'deral da Paraiba, Campilla Grallde, PB, Brazil (Work sponsored by C\'Pq Brazil) **Departlllellt of Electrical Ellgineerillg alld Computer Science, L'lIivn.lit)' of ,\ 'ewcastle. Newcastle, NSW 2308, Awtralia
Abstract. Incremental pole placement is an adaptive control algorithm based on the pole placement, indirect adaptive control approach. It differs from (he classical algorithm in that "the closed loop poles" are estimated directly, together with the plant poles and zeros. The deviation between the obtained "closed loop poles" and the desired ones is instrumental in the update mechanisms. Cautious control is implemented naturally as the control update and estimation algorithm are highly decoupled. This paper addresses the overall performance problem _ A local analysis is presented using averaging theory . The results of this analysis suggest some non trivial modifications to improve the performance. Keywords. Adaptive control; adaptive systems; pole placement; pole assignment; cautious control; averaging analysis.
1.
This paper is organized as follows: In section 2 we describe the closed loop system. In section 3 we introduce a simple parameter estimator for the closed loop system_ Next we present the incremental pole placement adaptation scheme. Section 5 gives the local averal;!:ing analysis of the algorit hm . In section 6 we dISCUSS the design modifications. Finally, we make some concluding remarks in section 7.
INTRODUCTION
The general aim of adaptive control is to guarantee a desired level of performance, in the face of initial uncertainty about the plant and of plant var i ations during system operation. The way in which the desired performance is specified and the way in which the adaptation is made define the different types of adaptive controllers, as surveyed, for instance, in Astrom (1987), Seborg (1986) and ortega (1988). For some of these schemes, the obtained performance is measured and appears expl ici tly in the adaptat ion scheme. In the other cases, although the desired performance is precisely stated and is used to guide the adaptat ion, the obtained performance is not used and usually is not measured_
Notation In this paper we use the 6-operator: 6y( t)
(y(t+T) -y(t))/T
°
and polynomial system descriptions for conciseness. With A, polynmials in 6, we denote by AO(=OA) the polynomial product and by A· O(_O·A) the concatenat ion of operators in 6.
The standard pole placement technique falls in the second class of adaptive controllers.
2.
The incremental pole placement adaptation strategy is based on the same concepts of closed loop pole allocation. However it estimates on line how well the closed loop poles actually approximate the desired location. Moreover it uses this information explicitly in updating the controller. In addition the adaptation and estimation procedures are highly decoupled, making cautious control possible.
TilE CLOSED LOOP SYSTEM
In this paper we consider the adaptive control system structure shown in figure 2.1
As stability is not the main concern of this paper, slow adaptat ion can be used to "finely" tune a stabilizing controller so as to obtain the desired performance.
y(l)
In this paper we present the local (performance) analy sis of this adaptive algorithm applied to linear time invariant plants in the presence of structured unmodelled dynamics and output noise. In the analysis we use the concepts of averaging theory (Anderson et aI , 1986). The analysis suggests some modifications to improve the robustness of the adaptive algorithm with respect to unmodelled dynamics.
Fig. 2.1 2.1
The adaptive control system structure
Th e Plant
The plant is assumed to belong to a class of time invariant systems described by A(6)(1+A'(6))y(t) = O(6)(1+0' (6))u(t)
405
(2 _1)
From nOli on lie shall si mplifying assumption:
Where B(6), A(6) represent the modelled part with A(6) assumed monic and aA = m, aB = m-I; B'(6), A'(6) form the unmodelled part with aB'(6) ~ aA'(6),
= Bu(t)
+
TJp(t)
3.
=
AA'y(t)
-
BB'u(t)
+
+
TJ (t)
(2.3)
3.1
Filtering all signals through a stable filter * * , yields the following set of equations to ao/A represent the plant behaviour: Ay(t)
Bu(t)
*
A*y(t) A*u(t) A*:;Jp (t)
+
:;J (t)
The Regressor Model
[A * - I'(t)]
[y (t) ] u(t)
(2.5)
+
~r(t)
The above equation can follolling regressor form
= n = 2m- 1
(3 .1 )
ao
given by ao* ' aA *
, [ [po(t)7 r (t)]
ao
(2.7)
+
~
~A
(2.6)
A*
2.2
PARAMETER ESTIMATION
(2.4)
* aou(t) * (t) aoTJp
with A* (6)
(2. 14)
From equation (2.5) (2.6) and (2. 13) we obtain
p
aoy(t)
follolling
The purpose of this section is to present a closed loop parameter estimator to be used in the local Initially lie use equation (2.11) to analysis. obtain a regressor model for the closed loop suitable for estimation. Then lie present the parameter estimator.
(2.2)
with TJp(t)
the
= po(t)
Q(t)
The plant behaviour can also be represented as Ay(t)
make
be
rellritten
The Time-varying Controller
in
the
o
The controller action is obtained from:
y(t) ] = 4> (t)T 8 (t) [ u (t) 0
u(t) = - l -[ (eo (t ) - L(t)) u(t) - P(t)y(t)
+
eo(t)
(3.2)
IIhere
(2.8) (2.9)
with L(t) aL (t)
=
monic and
(3.3)
m-I, aQ (t)
~
aP (t)
=
m-I 6
[Cor zero controller initial conditions equation (2.8) is equivalent (2.10) L(t)u(t) = Q(t)Yr(t) - P(t)y(t)
n- 1
6
v( l) no
11
-
1
v( t)
..........., ... ,'"r-'-, 6
a o
~, ... ,~, a
o
rn- 1
1
po(t) ..... y (t), ... ,p (t) A
r
0
1 "T
A
Y
r
(t)
0, . . . . . . • . . ,0
a o
2.3 The Closed Loop Eauations.
o , ... ,
The behaviour of the closed loop system, from the reference signal Yr(t) to the filtered signal s y(t)
and u(t)
AC(t) [y(t) ] lu(t)
=
can be described by
[B (t) ] A(t)
0
Q(t) y (t) r
+
TJr (t)
3.2
(2.11)
Several standard parameter estimation al~orithms, such as least squares and normalized gradIent, can be appl ied to the closed loop system model of equations 3.2 to 3.4. As far as local dynamics is concerned, least squares, normalized gradient and gradient algorithms are equivalent. In the sequel lie use a gradient algorithm:
where AC(t) = AL(t) (t) =
1)
r
+
BP(t)
L(t) :;J (t) p [-P(t):;Jp(t)
(2 .12)
+
[BQ(t) -BoQ(t)JY (t)]
+
[AQ(t) -AoQ(t)]Yr(t)
r
The Parameter Estimator
(2.13)
68 (t)
406
=
a
_e ~(t)e(t)
T
(3.5)
y(t) e(t)
in an obvious way - the coefficients are linked to respectively A(l+A'), 0(1+B'), P and L.
[ u(t)
5,1.2. Incremental pole placement equation In the local analysis we can neglect the projection operator in equations (4.2 ) and (4.3). In thIS case we can write the incremental pole placement adaptation law as
(3.6) INCREMENTAL POLE PLACEMENT
4.
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 scheme is implemented as follows:
(5.2 ) A
The controller polynomials updated by:
L(t),
S(8 A,8 B)
are
P(t)
A
A(t) AL (t) + B(t) AP (t) = aa [A * L(t+1) Proj {L(t) + AL(t)}
is the Sylvester Matrix associated with
A
A(t),B(t). (The definition of the Sylvester matrix can be found in Wolovich (1984)). USIng
- Ac (t) ] (4. 1) (4.2)
<.e
P(t+1)
Proj {P(t) + AP(t)}
(5.3)
(4.3)
8~
~
[ 8
C
where A, B, and A are the estimates for the open loop and closed loop polymonials, respectively. aa > 0 is the adaptation gain and set
Proj
and defining
-
P
(5.4 )
-
(4.4) (5,5)
is the projection operator on a known
~ ~.
we get 5.
LOCAL AVERAGING ANALYSIS
Averaging theory has been widely used to study the stability of adaptive control systems. Here we follow the analysis as outlined in the book by Anderson et al (1986). In doing so, the for the existence of some conditions approximations to be made are similar to the ones presented there in chapters 3 and 5. Initially we present the state space version for the adaptive system described in the previous We linearize this system around the sections. neighbourhood of a "tuned trajectory", associated We then perform a with the nominal system. Lyapunov transformation on the linearized system in order to separate the states into fast and slow variables. Finally, averaging results are called upon to obtain stability conditions. 5.1
5.1.3 Estimator error equation F'rom equations (3,2) to (3.5) and (3.6) we can express the par~meter estimator error 8(t) = 80 (t) - 8(t) (5.7) as
08(t)
State-Variable Equations
1 - T [8 0 (t)
In order to perform the analysis we have to express the adaptive system in a state space form.
or, using (2.13 ) and (2.3) - (2.7 )
5.1.1 Plant and controller equations From equations (2.5) and (2.11) we can write
Tip
r(t) ] ° lU(t) where Y(t)T U(t)T
114 [
~ ]
~(t) ~(t)
[0/!-1 y(t)
114,
~, ~(t)
= - ~ ~ (t) ~ (t)T 8(t) _
o
+
po(t)yr(t)
(5.1)
o o where H, given by
y(t)] u(t)]
[0/!-1 u(t)
where the matrices
lU(t)
08 (t)
(t)
o
r(t)]
and
~(t)
(5.8 )
8o (t + 1)]
~ ~(t){H(8L'8
Ta
o
..J()/
)[ p
0
(t)
~(t)
u( t)
+ Hry(8 L ,8 p)ry(t) + Hr (8 p)Yr(t) } (5.9) Hr ,Hry are time varying operators -L(t)AA ' [ P(t)AA'
are defined
+ 08
L(t)BB ' ] -P(t)BB'
o *
11 (9 ,9) L T)
[ L(t)
o
~* 1
(5.12)
+
o
-P(t) ~
p
A
BPO(t) 11 (9)
-
o
B· po(t)
=
(5_10)
[ APo(t) - A. po(t) r p Writing componentwise, we have:
*
with '!1,
~
*
*
*
corresponding to 9p '
9L.
5.2.2 . The linearized equations The resulting lineari zed equation representing the deviation of the true trajectory (given by (5. 1), (5.6) and (5.11)) in relation to the tuned trajectory is given by
+
[9
*]1
aa 0 0 - 1 0 0 L(t) ] [l'"S(9A,9B)S(9A,9B) [S(9A,9~) 9 (t) - 9A p
where we defined Z = [Y(t)TU(t)T]T and e = [BL(t)T Bp(t)T
J
o
(5.14) Bc(t )T
BB (t) T
with
BA (t)T]T (5.15)
o 5.2 Linearized Description
State-Variable
Equation
We now linearize the adapt i ve system given by equations (5.1), (5.6) and (5.11) around the neighbourhood of a trajectory associated with the tuned controller . 5.2.1 The tuned trajectory The tuned trajectory is the solution of the following system associated with the initial conditions * * l' * l' *1' X = [Y (0) U (0) (9 L) * (t) * U (t) *
'!1
*
~ ~
*
0 0
0 0 0
* (t) U* (t) *
9L(t)
0
0
0
0
9L(t)
9£ (t)
0
0
0
0
9£(t)
9c (t) * 9B(t)
0
0 0
0
9c (t) * 9B(t)
*
6
lA
*
9A(t)
0
0
0
0
o
*
0 0
0 0 0
+
*
9A(t)
o
for [ o0
(5.16) represent the sensitivity of the tuned systems with respect to the control parameters. r y , r u are representative of the unmodelled dynamics and disturbances offsetting the estimator
AL, Ap
408
along the tuned trajectory. w* is the forcing term, reflecting the fact that even at the tuned solution some disturbances are not accounted for in the model. Ac is a measure of the robustness of
the pole placement design,
f1*
iii) (Ax ,x)2 - 4(x,x)(Bx,x) > 0 Vx E 1R2,
where
A, BE 1R 2x2 ,
x -F 0,
(z,y)
= zT y .
Condition (i) is satisfied if
is the
ae
excitation (f~ = ~* ( ~* )T). A detailed description of the above matrices can be found in Barros, Mareels and Goodwin (1989).
r
*
~I]
[0
F1 > 0
(5.24)
That is, we must have a persistent excitation * and in addition that the condition for ~ (t), adapt~tion gain mu st be sufficiently small in relatIon to the product of the estimator gain by the minimum excitation level.
Lyapunov Transformation on the Linearized 5.3 System In order to separate the system into fast and slow systems we perform a Lyapunov transformation of the form
Condition (i i ) is satisfied if
F~ > ~[rrus(a~,a~)- 1 0 0]
-/I (t)] = [ I [B(t) 0
[rrus(a~,a~)-I
(5 .17) where L(t,aa,ae ) is chosen as to decouple a (the slow variable) from -/I (the fast variable).
OO]T
6[-/I(t)] = [F!! - Lo(t+l)F21 (t,a e ) B(t) F21 (t,a e )
Condition ( iii ) will be satisfied if sufficiently small in relation to a . e For sufficiently -/I (t) are stable The stability of by analysing the
5.4
aa
is
Finally, for the negative real eigenvalues to li e inside the stability circle it i s sufficient that a e be small enough.
small max(aa ,a e) the fast states as Fll is stable by assumption. the slow states can be considered matrix
= F22 (t,aa,ae)
(5 .25)
Thus, the level of excitation must be bigger than the contribution of rruS(a~,a~)- 1. The l atter is a result of the contribution of the disturbances, the level of correlation between the regressor vector and the disturbances, t he sensitivity of the closed loop in relation to the tuned controller parameters and of S(a~,a~)-I( See Barros, Mareels and Goodwin (1989)).
The above transformation yields the following state-variable system
F22 (t,aa,ae )
(5.23)
5.5
+ F21(t,ae)Lo(t,aa,ae) (5. 19)
The Main Result
We conclude theorem:
Local Averagi ng Analysi s
this
section
with
the
following
Theorem 5.1 The adaptive system given b{ equations (5.1), (5.6) and (5. 11 ) is local y stable in the neighbourhood of the tuned trajectory described by (5.12) provided that conditions (i)- (iii) are sat i sfied, a e is sufficiently smal l, and that the disturbances driving term
Assuming averages exist, stability of the unforced equation is determined by
* As in Anderson et al is of the order of a. * (1986), a is determined by a set of inequalities which makes it possible to linearize and separate slow and fast states, while retaining stabi lity properties. For stability we need that the zeros of the above polynomial must lie in a ci rcle centred at - liT and of radius liT.
Proof:
A sufficient condition for the eigenvalues to be negative real (see Bellman (1970) and Gohberg et al (1982)) is that the following inequalities are satisfied i) xTAx > 0 (5.21) ii) xTBX > 0 (5.22)
In this section we present some modifications to the adaptive control system introduced in the previous section. These modifications are based on the results of the averaging analysis as well as practical considerations, and are intended to yield a more robust adaptive system.
See Barros , Mareels and Goodwin (1989). 6.
409
DESIGN MODIFICATIONS
8.
The proposed modifications are: - Block Processing In order to improve the stability of the adaptive system, we make the adaptation gain g(t) zero for N- 1 sampling periods, and non zero at time t k , the Nth one. This procedure will also make the closed loop system block time invariant. This modification follows from the analysis. (The need for block processing in adaptive control has been argued in Shinkim and Feuer (1988) from different considerations.) - Adaptation Gain From the above averaging analysis and motivated by cautious adaptation we make the adaptation gain on the Nth sampling period proportional to the level of excitation on the previous N- 1 samples . The adaptation gain could be chosen as e.g:
Anderson, B.D.O., R.R. Bitmead, C.R. John son, P.V. Kokotovic, R.L. Kosut, I.M.Y. Mareels, L. Praly,B.D. Ri ede(1986). "Stability of adaptive systems : anaLysis". The MIT
passivity
and
averaging
Press. Astrom, K.J. (1989). "Adaptive feedback control". Proc. of IEEE , Vol. 75, No. 2, pp.185-217. Barros, P.R., I.M.Y. Mareels and G.C. Goodwin (1989) Local analysis of the incremental pole placement adapt i ve controller. Technical Report EE8909, University of Newcastle, Newcastle. Belman, R. (1980) "Introduction to matrix anaLysis". McGraw- Hill book company, Second Edition. Gohberg, I. , P. Lancaster, L. Rodman (1982). "Matrix poLynomiaLs". Academic Press. Middleton, R.H., G.C. Goodwin, D.J. Hill , D.Q. Mayne(1988). "Design issues in adaptive control". IEEE Trans. Automatic ControL, Vol. AC33, No. 1. Ortega, R. and Tang, Vu. (1987). "Theoretical results on robustness of direct adaptive controllers: a survey". Preprints 10th World Congress on Automatic ControL IFAC. Munich, pp.1-15. Seboq;;, D.E. , T.F. Edgar, S.L. Shah. (1986) "Adapti ve control strategies for process control : a survey". AICIIE Journal, Vol. 32, No. 6, pp.881-913. Shimkin, N. and A. Feuer.(1988) "On the necessity of I block invariance I for the convergence of adaptive pole-placement al~orithm with persistently exciting input. Trans . Automatic ControL, Vol. AC33, No.8. Wolovich, 'rI.A. (1984). "Linear muLtiuariabte systems". Springer-Verlag, New York,Inc.
(instead of aa) where tr A means the trace of matrix A, and Po and P(t) are covar iance matr ices of the estimator. - Pre-fi ltering We filter the signals to be used in the estimator The filter algorithm using a band- pass filter. bandw id th is chosen such as to con cent rate the modelling to fit in a frequency band of interest. This modification follows from the analysis and is similar to ideas introduced in Anderson et al. (1986). It also follows from other adaptive control analysis procedures as, for instance, the one used in Middleton et al. (1988). - Least Squares Estimator Instead of the estimator presented in section 3, we use a recursive least squares estimator. It incorporates covariance resetting to a matrix Po at time tk as well as normalization by a signal assumed to overbound the absolute value of the disturbances. The parameter estimates are also restricted to lie in a pre-defined convex set. This modification does not follow from the analysis. - Parameter Estimate Resetting At time t k , after the adaptation, the parameters ec(t) are reset to e*c . This procedure is expected to increase the robustness of the adaptive system. This is motivated by the fact that in blocks of high excitation we would be resetting the ini tial estimate for the next block to where it is expected the closed loop denominator pol)~omial would be. Also, it can be easily recognised that in blocks of low excitation resetting avoids integrator wind up in the adaptation law, consequently improving robustness. 7.
REFERENCES
COilCLUSIONS
In this paper we have presented a local averaging an adaptive control system analysis for incorporating a new adaptation strategy, namely, incremental pole placement. The main results were sufficient conditions for local stability of the adaptive system in the presence of small disturbances. Those sufficient conditions can be interpreted as the existence of enough excitation and the requirement that the adaptation gain should be small compared with th~ estimation gain weighted by the level of that eXCItatIon. We have used those results together with other practical considerations to modify the adaptive system into a more robust one.
410