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A Self-Tuning Adaptive Controller for Multivariable Systems
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A SELF-TUNING ADAPTIVE CONTROLLER FOR MULTIVARIABLE SYSTEMS Deng Z-I, Guo Y-x and Du Cot IIt'i/()lIgjitlllg IlIsiiillir of Applinl MalitPllwlirs , I/arhill , '1'11.1' Pm/lip's /{I'/mlilir ojChilla
Abstract. A self-tuning adaptive controller i. desi&n.d tor multi variable systeaa with unknown par8llleters, unknown order 1Iad. tiJu delay. The.o.r.del- end t.he ti .. e delay can be determined by F-test whioh is d.sorib.d in this pap.r. A .elf-tuning controller consists of the two parts of .stimation and oontrol. Estimation of the parameter. is based on ths recursive least squares (RLS) Ilethod. The oontroller with a generalized oostfunotion is computed simply bassd on the self-tuning multi step reoursive predictor. Two sillulation examples are given to show the usefulness of the approach. Keywords. Hultivariable control systells; adaptive oontrol; self-tuning control; identification; parameter estillation. I1fTROOOCTION For multi variable systems wi~ unknown model parameters, the selt-tuning regulator and controller have besn proposed in several reterences (Keviozky and Hetthessy, 1977; Borisson. 1979; Koivo, 1980; Keviczky and Kumar. 1981). However, there are the following disadvantages I (i) Soil. a priori information such as the order and the time delay of model is still required. (11) The input (control) and output have the sne dimension.
n n y(t)2ZAiy(t-i) +:L;BiU(t-i) + e(t) is1 i-o
(1)
vhich is denoted by CAR(n). vhere y(t)(y1(t) ••••• yp(t»
T
is the PX1 output. u(t).
(u (t) •••• ,u (t»T is the qx1 input (control). q 1 e(t)-(.1(t) ••••• e (t»T is a px1 gaussian vhit. P i i noise vith zero mean, Ai-(a t .), Bi-(bts ) are i
i
(iii) If the time d.lay is large, then the identification of hiCh-order prediction model is required, which increases the computational burden.
pxp, pxq matrices vith elements a ts ' bts respectively. and symbol T d.notes the transpose.
In order to avoid above disadvantages, Bayoumi, Wong and Bagoury (1981) presented an approach which combines the determination ot the order and the identification of the proc.ss as wsll as the control ot the output vsctor. The ordsr can be det.rmined by a generalized likelihood-ratio test. The original process model is directly identified using the RLS method. The self-tuning controller with costfunction of output deviation and control energy is coaputed simply based on the recursive predictor. However, it is still assumed that the tille delay is a priori known. The determination of the time delay has significant importance. The closed loop system may be unstable, if the model time delay does not coincide with the process tille delay(Kurz and Goedecke, 1981).
If An and Bn are not siaul"-neously equal to zero Ilatrix, then n is called the .odel order. It Ao-
An approach proposed here avoids the assumption that the time delay is a priori known. The order and time delay of the model can b. determinsd by r-test vhich is described in this paper . Identification is performed by RLS method. The self-tuning controller with a generalized cost-function also is obtained based on the recursive multi step predictor, and has good aSYllptotically optimal (selftuning) behaviour.
••• ~Am+1-0' A~ or B02 ••• 2B.+1-0, B~, then the order of autoregressive part or controlled part becomes m. It Bo~ ••• sBd_1z0, B~O. then d ia called the time delay of model. Hence the order and time delay of the model can be determined by testing the so .. e sero coefticient Ilatrices. Above n,m,d, Ai and Bi are assumed to be unknown. The problem ia to c hoos e the order n,a and tne tiae delay d, to estimate the paraeeter aatrioe. Ai and B , and to find the control u(t) vhich miniaises i a generalized coat-function. PARAMETER ESTIHATIOI The model (1) can be revritten in cOllponent fora .s Yi(t)sUT(t)8i + ei(t), i=1.2 •••• ,p (~) vhere 1 n n 6 T(1 i ~ a i 1 ' ••• ,aip ' ••• , bi l' ..•• biq )
PROBLEM FORMULATION Consid.r Ilultivariable controlled autoregressive (CAR) model described by
::a
1 enp+(n+1)q) (e it ... , i
39H
D CIl !{ Z-I,
CUO
Y -x a lld DII C -I
ORDER OF PARSIMONIOUS MODEL Eq. (2) is the suo-odel of model (1), and is the n order autoregessive model with exogenous inputs for Yi(t} , but contains pn+(n+1}q parameters, and is denoted by ARX(n}. Hence the identification problem of CAR(n } is converted into the identification problem of p subaodels ARX(n}. Given (y(N},y(N-1}, ••• ,u(N},u(N-1} •••• ) the esti.at.e
8i (N)
of the vector
0i
can be computed by
using the RLS method aB
In order to obtain the parsimonious parameter models, in the obtained submodels ARX(n } , i i-1,2, •• • ,p, we must omit the some parameters which are approximately equal to zero. This is converted into a statistical problem of testing the hypothesis that some parameters in ARX(n } i are zero. It can be shown (Astrom and Wittenmark, 1971) that for subaodel ARI(n } the conditional distribution i of 0i given (y(N},y(N-1), •••• u(N),u(N-1), ••• ) is ~
(3)
gaussian distributionAwith mean ei(N) and covariance ~i(N}, where Bi(N} and Pi(N} are computed
Ki(N}=Pi(N-1}U(N}(~+UT(N}Pi(N-1}U(N}}-1
(4)
fro. Eq. (3}-(5), and estimate Ir~ of is computed by Eq. (8). In particular. the conditional distribution of the
Pi(N}=[I-Ki(N}UT(N}]Pi(N-1}/~
(5)
J th element
""
\ 9i (N}'" 0i (N-1 }+Ki (N)(y i (N}-UT(N)I 0i (N-1))
where I is the unit matrix,Oi(o}=Bio ' Pi(o}~Pio and { is the forgetting factor, 0 '" 1.. ~ 1. The residual square su. is given by
I: [-ei (t)]
Si (n}'"
ei
is ga\lssian distribution
2
e
e
Hence the 95% confidence interval for by
i is given
B
ei (N}-1. 9 6'a:i g j (N)(oi (Si (N}+1.96f"Jp~j (N)
(6)
(10)
where the residue
j=1. 2 • • •.• np+(n+1}q
/\ T BA ei(t)=Yi(t} - U (t) i(N} 2
of
with mean and variance ~~P~j(N}. where ~ i Aj i (N) is the j th element of i (N). and P jj (N) la the (jj}th diagonal element of Pi(N).
t=n+1
The variance IT" i
9i
ei(N}
(i 1,2, ••• ,p) z
N
f;
(7)
or
formally
of e (t) can be estimated by i
( 11)
sample variance as I t is obYious t.hat if
(8)
Since si(t}
is white noise, then the RLS estimation
is consistent under weak condition for closed-loop systems with unknown constant paraaetsrs (Ljung, 1976) • DETERMINATION OF SUBMODEL ORDER The order of each submodel Eq. (2) can be determined off-line by F-test. Given (u(t },y(t}), t-1,2, ••• ,N, in order to determine the order of the ith subaodel Eq. (2), we can successinly fit ARX(n} , n"1,2, .... For comparing ARI(n} and ARX(n+1}, the statistic
ence interval for
ei
ei(N}~O. then 95% confidwill contain zero, which
yields the following approach for obtaining parsimonious parameter ~odel and its t.ime delay and order: (i) First., in submodel ARI(n } we omit the parai meters whose 95% confidence intervals contain zero. then using the RLS method we rebuild a parsimonioul parameter model ARX-(n } which doea not contain i the omitted parameters. (ii) Using F-test we test whether t.he omitted parameters differ signficantly from zero. Note that the statistic (1strom. 1968) N-n i P-(n i +1}q
(9)
H.
( 12)
1
has F-distribution. 1968) •
F(p+q, N-(n+1}p-(n+2)q} (Kstrom,
has
F(M • N-n i P-(n +1 }q) distribution, where i i is the reSidual square sum of parsi.oniOUs
S~(ni }
At a risk level of a, for example a-5%, we have F(p+q, N-(n+1)p-(n+2}q}sF • a If F < Fa , then ARX(n} is suitable; If Fa ' then ARX(n} is not suitable. For
parameter modal ARX-(n }. and Hi is the nu.ber of i the omitted parameters.
n-1,2, ••• , we use successively F-test until ARX(n } i is Buitable, and n is called the order of the ith i submodel. Hence. in order to find ni we only need
case. ~n special case, i. e. N is small and / or er ~ is larger, it may be Significant. then in the omitted parameters we need retain the significant parameters which ara not equal to zero in practice. The significance of each of the omitt.ed parameters can be treated successively using F-test.
"i,
fit the n +1 ARX models. i DETERMINATION OF TIME DELAY AND
If it is not significant. then parsimonious model ARX- (n . } is accepted, which is general and usual
(iii) Let US combine the obtained p parsimonious
A Self-tuning Adaptive Controller
parameter submodels ARX-(n i ) (i a 1.2 ••••• p) in the vector form. thus we can obtain a parsimonious parameter model CAR(n). Eq. (1). It is obvious that the 1I0del order is given by (13) na aax(n ,n 2 , ••• ,np ) 1 Since SOlle of the paraaeters in sUbaodel. ARX(n i ) (i=1.2, •••• p) are oaitted. it is possible that in Eq. (1) .olle of the coefficient matrices are equal to zero matrix. Hence, If Bo= ••• aBd_1=O, B~O. then d is the time delay. If An~O or An= ••• mAm+1=O, Am~O, then the order of autoregressive part is n or 11 • If Bn~O or Bn= ••• =B m+1=O. Bm~O, then the order of the controlled part ie n or 11. Comparing generalized likelihood-ratio test (Bayouai. Wong and Bagoury. 1981) with l-test descirbed here. the former is only suitable for obtaining the model order. the latter is sutiable for obtaining both of the order and the tiae delay, and for obtaining parsimonious lIodel. Hence. the latter ha ••ignificant advantages.
are computed by the RLS algorithll Eqs. (3)-(5). The recursive predictor Eq. (18) has significant advantages for real tiae use. It can be shown (Koivo, 1980) that the problell of ainimizing J is equivalent to the problell of minimizing J*. where d-1 J*=iI Eo DiY( t+d-i/t)-w( t+d)l\~ 2
r
+11 L
i=o
Gi u( t-1)i1 R + constant
"
,,2
-IIDoBdu( t) + L( t)1i Q + 11 G u( t) + o
r 2 L Gi u( t-i)1I R +constant i 1
(1")
a
where we have froll Eq. (18) that
(20)
SELF-TUNING CONTROLLER Letting (;iJ*/8u(t)aO, the self-tuning controller is given by
Using aboveapproach!4' (1) can be reduced aa parsimonious aodel n
y(t)=
Z Aiy(t-i)
i=1
a
+2: BiU(t-i) +e(t) i=d
where now we define that n qa or a ~n,
(14)
u( t)-[ (DoEd) TQDoBd + (D
d::) 1.
The problea is to find u(t) which ainillizes a generalized coat-function.
~)Tri(t)
G~RGor1. [
T + G R2 Giu(t-iB 0 i-1
o
In particular, for cost-function J , we have 1
d-1
J. E(
(21)
(22)
II~Diy(t+d-i)-W(t+d)ll~ where from Eq. (20), (15)
which is the generalizatiOll of the cost-function in References(!oivo, 1980), and where E('/C) ie conditional expectation. condition C-(y(t).y(t-1) • .... u(t-1) .... ), sYllbol l\xll~XTAx, w(t) is known PX1 reference sequence. p~p matrix Q is non-negative definite and symaetric. qxq matrix R is positive definite and sYllmetric, and Di and Gi are P'lP. q~Q matrices reapectively. In particular (Koivo, 1980). it is usual that J 1sE( h( t+d)-w( t+d)ll~ + !lu( t)lI; and
IC)
For cost-function J2' we have u{t)=
-[B~QBd
+
RtlB~<£(t)
- Ru(t-1ij
(24)
where £(t) is cOllputed by Eq. (23). It is obvious that for D ~O, G ~O, the matrix "T " TOO (DoBd) Q DoBd + GoRG o always is a positive definite matrix, and is nonsingular for all estimated Bd matrix, which guarantees that its inverse
l
(16)
2 2 J =E( liy(t+d)-w(t+d)II Q + l\u(t)-u(t-1)IIRrc) (17)
in Eq. (21) exists. The design procedure for the self-tunin~ controller proposed here can be summarized as follows:
From Eq. (14) we obtain self-tuning multistep recursive predictor
Step 1. Determine the order n,m and the time delay by F-test method. Step 2. Estiaate parameter (18) where define
'1( t+k-i/t)ay( t+k-i),
for t+k-i,., t;
for t+k-i> t. y( t+lt-i/t) is the minimum variance prediction of y(t+k-i) given (y{t)hY(~1), ••• , u(t+k-i-d), ••• ) and the estimates Ai' Bi of Ai,Bi
matrices Ai' Si using
the RLS method. Step 3. Find the multi step prediction y(t+i/t), ia1,2, ••• ,d, using recursive equation (18). Step 4. Compute L(t) by Eq. (20). Step 5. Compute the control signal u(t) by Eq.(21)
400
Deng Z-l. Guo V-x and Du C-t
and go to step 2 for the next iteration.
11 ,1
Notice that step 1 is performed by the off-line analysis method; but the other steps operate online.
In addition, the simulation results also showed the consistency of the RLS parameter estimation.
For CAR(n) model with constant unknown parameters, it is obvious that if the RLS estimation of the parameter matrices in closed loop is consistent, then the self-tuning controller Eq. (21) will converge to the optimal controller for the model with known parameters. Thus, it has asymptotically optimal (self-tuning) behavious.
track the reference values W=(1.0 ,1.0)T.
2
Example 2. Consider the folloving model vith the number of inputa-3 while the number of outputsm2 as trull model (28)
SIMULATION EXAMPLES Consider the following CAR(3) as the true model y(t)-A y(t-1) + A2y(t-2) + B2U(t-2) 1 (25)
+ B U(t-3) + e(t) 3
0.4
-0.504
,_0 , 305],
,
0.29]
A· 2 [
A12
-0.25,
1.5
°
, _0.297]
A,-
0.9 ,
-0.,]
0.5
0.8
[
· 0.,]
(26) and e(t) is gau8sian white noise with covariance matrix given by diag (0.0025, 0.0025). The reference vector w is assumed to be w-(1.0, 1.0)T • The samples of e(t) are generated by pseudorandom generator. The input u(t) is a pseudorandom binary sequence (PRBS) vi th the values ± 1, and the output is computed by Eq. (25). The F-test algorithm was performed for the first 50 samples (N=50) of model (25). We obtain the following results: d22
m-3,
and the estimates "
"Ai
[0.407717, _0.3
28
and
682] "
Ai-
B
are
given by
[-0,513804, 0.312117J
A2= -0.249405,
"
IBi '
[1.9964
1.49662,
0
-0.55771
· -0.30""] B,,[_"806" • O"'90"J
2
2
o
,0.986553,
, 1.5
(29) and e(t) and u(t) are independent, zero-mean, gauyian vhite noise with covariances diai«(0.28' ~.2~ and diag (1,1,1). The identification is performed off-line by using the first 50 samples of output, we obtain the folloving results:
which are identical to the xorresponding true value, and the estimates 11 and B1 are given by 0.908274
n::z2,
1
-0.562
, -0.3
0.996 ,
~r
, 0.75 ,
-0.366888, -0.30528
(27) For cost-function J1 with Q=I, R-diag(0.01, 1), from the 51th sample to the 100 th sample of model (25), we use the self-tuning controller Eq. (22), instead of the PRBS input signal, and we take Eq. (27) as the initial estimation of unknown parameters, the results obtained by 50 iterations are shown in rig. 1 and Fig. 2. Fig. 1 shows that the self-tuning input signals u ' u in steady state stabilize at values 5.5 2 1 and 3 respectively. Fig. 2 shows that the corresponding output signal
~- [ .499271 B.-lO.904711
I
.893807
_0.513197]
0.782661 0.808702
0. 332033
1.55293
1.6594
J
(.30)
For oost-function J 1 with Q-I, R-diag (10 ,0.01,100) and the timevarying referenoe vector wet), whose components v (t) and v (t) assumed to . be the square 2 1 waves with amplitudes-5 and 5 and period of 40 sampling periods, respeotively, we switch to the self-tuning controller Eq. (22), instead of the white noise input u(t), and we take Eq. (.30) as the initial estimation of unknown parameters, The length of simulation is 200 sampling periods, the resultl obtained by 200 iterations are shown in Fig. 3 and 1ig. 4. Fig. 3 shaWl outputs Y1(t) and Y2(t) of self-tuniug oontrol sYltem, vhioh track w (t) and v (t) ectiyely. 1 2
resp-
Fig. 4 shows the self-tuning control signals u (t), 1 u 2 (t) and u (t) • 3 Thil example shows that self-tuning controllers can be applied to control multivariable systems in vhich the number of inputs and the naber of outputs are not equal. CONCLUSIONS For the design of the multivariable self-tuning controller, this paper presents an approach which combine the determination of the model order and the time delay and the estimation of the parameters aa well aa the control of the output vector. The approach and results proposed by Bayoumi, Wong and
A Self-luning Adaplive Controlle r
401
Bayoumi, 1981)are extended and improv.d.
oS [
The order and time delay are determined by F-test method, which provides a parsimonious parameter model and avoids the drawback that the apriori knowledge of the time delay ia required in the generalized likelihood ratio test. Original CAR(n) model are identified directly usin, the RLS method instead of the id.ntification of d-step ah.ad prediction model (Koivo, 1980; Wong and Bayoumi, 1981). The controller with a gen.raliz.d ooat-function i. designed based on self-tuning multistep recursive predictor, which can be computed simply on-lin •• HultiTariable self-tuning controller propo ••d h.r. eliminates the restrictive a8auzptions about the order, the time delay and the dimenaion of inputoutput, and haa asymptotically optimal (Belf-tuning) behaviour.
12
J o
80
160
200
1:. Fig. 3. Outputs of self-tuning control syst811 for example 2 •
REFERmCItS
Astrom, K. J. (1968). Lectures OD the identifioation problem, the least .quares method. Lund In.t. • of Technology, Rep. 6806. 0.4 Astrom , K. J., and B. Wittenmark (1971). Problem 0.3 of identification and control. J. Math. Anal. Appl1c.,~, 90-113 . 0.1 Borisaon, U. (1979). Self-tuning regulators for a class of multivariable systems. Automatica, O.O~~~~. .~~~~~~~~~~~~~O .12, 209-215. -0.1 Bayouai, H. M.,K. Y. Wong, and H. A. El-Bagoury (1981). A self-tuning regulator for multivari- -0.2 able aystems. Automatica, 11, 575-592. Ljung, L. (1976). Conaiatenoy of the l.ast-squar•• identi!1oation method. lEE!: Trans. Autom. Contr. , 21, 779-781. Koivo, H7 N. (1980). A Hultivariable a.lf-tuning controller. Automatica, 16, 351-366. Keviczky, L. and J. Hetthes.y (1977). Self-tuning minimun Tariance control of KlHO di.cret. tim. 2 ayatems. Autom. Control Theory and Appl., 2, 11-17. KeTiczky, L., and K. S. P. Kumar (1981). MultiTariable self-tuning regulator with generaliz.d cost-function. Int. J. Control, 12, 913-921. o Kurz, H. and W. Goedecke (1981). Digital parueteradaptive control of proc ••••• with unknown -1 dead time. Automatica, 17, 245-252. Wong, K. Y. and M. H. Bayoumi-r1981). Multivariable self-tuning re&Ulators. Proc. of the 20th IEEE Confereno. on Deoision and Control, 9711-983. 5 4
0.03 0.02 0.01
3 2
t
0
o
10
20
30
JIJ
50
Fig. 1. Inputs of self-tuning control sy.tem for exuole 1 • 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2
t 10 20 30 JIJ 50 Fig. 2. Outputs of self-tuning control system for example 1 •
o
200
-0.01 -0.02 -0.03 -0.04 i~g.
4. Input. of aelf-tuning control sy.tem for exup1e 2
A SELF-TUNING ADAPTIVE CONTROLLER FOR MULTIVARIABLE SYSTEMS Deng Z-I, Guo Y-x and Du Cot IIt'i/()lIgjitlllg IlIsiiillir of Applinl MalitPllwlirs , I/arhill , '1'11.1' Pm/lip's /{I'/mlilir ojChilla
Abstract. A self-tuning adaptive controller i. desi&n.d tor multi variable systeaa with unknown par8llleters, unknown order 1Iad. tiJu delay. The.o.r.del- end t.he ti .. e delay can be determined by F-test whioh is d.sorib.d in this pap.r. A .elf-tuning controller consists of the two parts of .stimation and oontrol. Estimation of the parameter. is based on ths recursive least squares (RLS) Ilethod. The oontroller with a generalized oostfunotion is computed simply bassd on the self-tuning multi step reoursive predictor. Two sillulation examples are given to show the usefulness of the approach. Keywords. Hultivariable control systells; adaptive oontrol; self-tuning control; identification; parameter estillation. I1fTROOOCTION For multi variable systems wi~ unknown model parameters, the selt-tuning regulator and controller have besn proposed in several reterences (Keviozky and Hetthessy, 1977; Borisson. 1979; Koivo, 1980; Keviczky and Kumar. 1981). However, there are the following disadvantages I (i) Soil. a priori information such as the order and the time delay of model is still required. (11) The input (control) and output have the sne dimension.
n n y(t)2ZAiy(t-i) +:L;BiU(t-i) + e(t) is1 i-o
(1)
vhich is denoted by CAR(n). vhere y(t)(y1(t) ••••• yp(t»
T
is the PX1 output. u(t).
(u (t) •••• ,u (t»T is the qx1 input (control). q 1 e(t)-(.1(t) ••••• e (t»T is a px1 gaussian vhit. P i i noise vith zero mean, Ai-(a t .), Bi-(bts ) are i
i
(iii) If the time d.lay is large, then the identification of hiCh-order prediction model is required, which increases the computational burden.
pxp, pxq matrices vith elements a ts ' bts respectively. and symbol T d.notes the transpose.
In order to avoid above disadvantages, Bayoumi, Wong and Bagoury (1981) presented an approach which combines the determination ot the order and the identification of the proc.ss as wsll as the control ot the output vsctor. The ordsr can be det.rmined by a generalized likelihood-ratio test. The original process model is directly identified using the RLS method. The self-tuning controller with costfunction of output deviation and control energy is coaputed simply based on the recursive predictor. However, it is still assumed that the tille delay is a priori known. The determination of the time delay has significant importance. The closed loop system may be unstable, if the model time delay does not coincide with the process tille delay(Kurz and Goedecke, 1981).
If An and Bn are not siaul"-neously equal to zero Ilatrix, then n is called the .odel order. It Ao-
An approach proposed here avoids the assumption that the time delay is a priori known. The order and time delay of the model can b. determinsd by r-test vhich is described in this paper . Identification is performed by RLS method. The self-tuning controller with a generalized cost-function also is obtained based on the recursive multi step predictor, and has good aSYllptotically optimal (selftuning) behaviour.
••• ~Am+1-0' A~ or B02 ••• 2B.+1-0, B~, then the order of autoregressive part or controlled part becomes m. It Bo~ ••• sBd_1z0, B~O. then d ia called the time delay of model. Hence the order and time delay of the model can be determined by testing the so .. e sero coefticient Ilatrices. Above n,m,d, Ai and Bi are assumed to be unknown. The problem ia to c hoos e the order n,a and tne tiae delay d, to estimate the paraeeter aatrioe. Ai and B , and to find the control u(t) vhich miniaises i a generalized coat-function. PARAMETER ESTIHATIOI The model (1) can be revritten in cOllponent fora .s Yi(t)sUT(t)8i + ei(t), i=1.2 •••• ,p (~) vhere 1 n n 6 T(1 i ~ a i 1 ' ••• ,aip ' ••• , bi l' ..•• biq )
PROBLEM FORMULATION Consid.r Ilultivariable controlled autoregressive (CAR) model described by
::a
1 enp+(n+1)q) (e it ... , i
39H
D CIl !{ Z-I,
CUO
Y -x a lld DII C -I
ORDER OF PARSIMONIOUS MODEL Eq. (2) is the suo-odel of model (1), and is the n order autoregessive model with exogenous inputs for Yi(t} , but contains pn+(n+1}q parameters, and is denoted by ARX(n}. Hence the identification problem of CAR(n } is converted into the identification problem of p subaodels ARX(n}. Given (y(N},y(N-1}, ••• ,u(N},u(N-1} •••• ) the esti.at.e
8i (N)
of the vector
0i
can be computed by
using the RLS method aB
In order to obtain the parsimonious parameter models, in the obtained submodels ARX(n } , i i-1,2, •• • ,p, we must omit the some parameters which are approximately equal to zero. This is converted into a statistical problem of testing the hypothesis that some parameters in ARX(n } i are zero. It can be shown (Astrom and Wittenmark, 1971) that for subaodel ARI(n } the conditional distribution i of 0i given (y(N},y(N-1), •••• u(N),u(N-1), ••• ) is ~
(3)
gaussian distributionAwith mean ei(N) and covariance ~i(N}, where Bi(N} and Pi(N} are computed
Ki(N}=Pi(N-1}U(N}(~+UT(N}Pi(N-1}U(N}}-1
(4)
fro. Eq. (3}-(5), and estimate Ir~ of is computed by Eq. (8). In particular. the conditional distribution of the
Pi(N}=[I-Ki(N}UT(N}]Pi(N-1}/~
(5)
J th element
""
\ 9i (N}'" 0i (N-1 }+Ki (N)(y i (N}-UT(N)I 0i (N-1))
where I is the unit matrix,Oi(o}=Bio ' Pi(o}~Pio and { is the forgetting factor, 0 '" 1.. ~ 1. The residual square su. is given by
I: [-ei (t)]
Si (n}'"
ei
is ga\lssian distribution
2
e
e
Hence the 95% confidence interval for by
i is given
B
ei (N}-1. 9 6'a:i g j (N)(oi (Si (N}+1.96f"Jp~j (N)
(6)
(10)
where the residue
j=1. 2 • • •.• np+(n+1}q
/\ T BA ei(t)=Yi(t} - U (t) i(N} 2
of
with mean and variance ~~P~j(N}. where ~ i Aj i (N) is the j th element of i (N). and P jj (N) la the (jj}th diagonal element of Pi(N).
t=n+1
The variance IT" i
9i
ei(N}
(i 1,2, ••• ,p) z
N
f;
(7)
or
formally
of e (t) can be estimated by i
( 11)
sample variance as I t is obYious t.hat if
(8)
Since si(t}
is white noise, then the RLS estimation
is consistent under weak condition for closed-loop systems with unknown constant paraaetsrs (Ljung, 1976) • DETERMINATION OF SUBMODEL ORDER The order of each submodel Eq. (2) can be determined off-line by F-test. Given (u(t },y(t}), t-1,2, ••• ,N, in order to determine the order of the ith subaodel Eq. (2), we can successinly fit ARX(n} , n"1,2, .... For comparing ARI(n} and ARX(n+1}, the statistic
ence interval for
ei
ei(N}~O. then 95% confidwill contain zero, which
yields the following approach for obtaining parsimonious parameter ~odel and its t.ime delay and order: (i) First., in submodel ARI(n } we omit the parai meters whose 95% confidence intervals contain zero. then using the RLS method we rebuild a parsimonioul parameter model ARX-(n } which doea not contain i the omitted parameters. (ii) Using F-test we test whether t.he omitted parameters differ signficantly from zero. Note that the statistic (1strom. 1968) N-n i P-(n i +1}q
(9)
H.
( 12)
1
has F-distribution. 1968) •
F(p+q, N-(n+1}p-(n+2)q} (Kstrom,
has
F(M • N-n i P-(n +1 }q) distribution, where i i is the reSidual square sum of parsi.oniOUs
S~(ni }
At a risk level of a, for example a-5%, we have F(p+q, N-(n+1)p-(n+2}q}sF • a If F < Fa , then ARX(n} is suitable; If Fa ' then ARX(n} is not suitable. For
parameter modal ARX-(n }. and Hi is the nu.ber of i the omitted parameters.
n-1,2, ••• , we use successively F-test until ARX(n } i is Buitable, and n is called the order of the ith i submodel. Hence. in order to find ni we only need
case. ~n special case, i. e. N is small and / or er ~ is larger, it may be Significant. then in the omitted parameters we need retain the significant parameters which ara not equal to zero in practice. The significance of each of the omitt.ed parameters can be treated successively using F-test.
"i,
fit the n +1 ARX models. i DETERMINATION OF TIME DELAY AND
If it is not significant. then parsimonious model ARX- (n . } is accepted, which is general and usual
(iii) Let US combine the obtained p parsimonious
A Self-tuning Adaptive Controller
parameter submodels ARX-(n i ) (i a 1.2 ••••• p) in the vector form. thus we can obtain a parsimonious parameter model CAR(n). Eq. (1). It is obvious that the 1I0del order is given by (13) na aax(n ,n 2 , ••• ,np ) 1 Since SOlle of the paraaeters in sUbaodel. ARX(n i ) (i=1.2, •••• p) are oaitted. it is possible that in Eq. (1) .olle of the coefficient matrices are equal to zero matrix. Hence, If Bo= ••• aBd_1=O, B~O. then d is the time delay. If An~O or An= ••• mAm+1=O, Am~O, then the order of autoregressive part is n or 11 • If Bn~O or Bn= ••• =B m+1=O. Bm~O, then the order of the controlled part ie n or 11. Comparing generalized likelihood-ratio test (Bayouai. Wong and Bagoury. 1981) with l-test descirbed here. the former is only suitable for obtaining the model order. the latter is sutiable for obtaining both of the order and the tiae delay, and for obtaining parsimonious lIodel. Hence. the latter ha ••ignificant advantages.
are computed by the RLS algorithll Eqs. (3)-(5). The recursive predictor Eq. (18) has significant advantages for real tiae use. It can be shown (Koivo, 1980) that the problell of ainimizing J is equivalent to the problell of minimizing J*. where d-1 J*=iI Eo DiY( t+d-i/t)-w( t+d)l\~ 2
r
+11 L
i=o
Gi u( t-1)i1 R + constant
"
,,2
-IIDoBdu( t) + L( t)1i Q + 11 G u( t) + o
r 2 L Gi u( t-i)1I R +constant i 1
(1")
a
where we have froll Eq. (18) that
(20)
SELF-TUNING CONTROLLER Letting (;iJ*/8u(t)aO, the self-tuning controller is given by
Using aboveapproach!4' (1) can be reduced aa parsimonious aodel n
y(t)=
Z Aiy(t-i)
i=1
a
+2: BiU(t-i) +e(t) i=d
where now we define that n qa or a ~n,
(14)
u( t)-[ (DoEd) TQDoBd + (D
d::) 1.
The problea is to find u(t) which ainillizes a generalized coat-function.
~)Tri(t)
G~RGor1. [
T + G R2 Giu(t-iB 0 i-1
o
In particular, for cost-function J , we have 1
d-1
J. E(
(21)
(22)
II~Diy(t+d-i)-W(t+d)ll~ where from Eq. (20), (15)
which is the generalizatiOll of the cost-function in References(!oivo, 1980), and where E('/C) ie conditional expectation. condition C-(y(t).y(t-1) • .... u(t-1) .... ), sYllbol l\xll~XTAx, w(t) is known PX1 reference sequence. p~p matrix Q is non-negative definite and symaetric. qxq matrix R is positive definite and sYllmetric, and Di and Gi are P'lP. q~Q matrices reapectively. In particular (Koivo, 1980). it is usual that J 1sE( h( t+d)-w( t+d)ll~ + !lu( t)lI; and
IC)
For cost-function J2' we have u{t)=
-[B~QBd
+
RtlB~<£(t)
- Ru(t-1ij
(24)
where £(t) is cOllputed by Eq. (23). It is obvious that for D ~O, G ~O, the matrix "T " TOO (DoBd) Q DoBd + GoRG o always is a positive definite matrix, and is nonsingular for all estimated Bd matrix, which guarantees that its inverse
l
(16)
2 2 J =E( liy(t+d)-w(t+d)II Q + l\u(t)-u(t-1)IIRrc) (17)
in Eq. (21) exists. The design procedure for the self-tunin~ controller proposed here can be summarized as follows:
From Eq. (14) we obtain self-tuning multistep recursive predictor
Step 1. Determine the order n,m and the time delay by F-test method. Step 2. Estiaate parameter (18) where define
'1( t+k-i/t)ay( t+k-i),
for t+k-i,., t;
for t+k-i> t. y( t+lt-i/t) is the minimum variance prediction of y(t+k-i) given (y{t)hY(~1), ••• , u(t+k-i-d), ••• ) and the estimates Ai' Bi of Ai,Bi
matrices Ai' Si using
the RLS method. Step 3. Find the multi step prediction y(t+i/t), ia1,2, ••• ,d, using recursive equation (18). Step 4. Compute L(t) by Eq. (20). Step 5. Compute the control signal u(t) by Eq.(21)
400
Deng Z-l. Guo V-x and Du C-t
and go to step 2 for the next iteration.
11 ,1
Notice that step 1 is performed by the off-line analysis method; but the other steps operate online.
In addition, the simulation results also showed the consistency of the RLS parameter estimation.
For CAR(n) model with constant unknown parameters, it is obvious that if the RLS estimation of the parameter matrices in closed loop is consistent, then the self-tuning controller Eq. (21) will converge to the optimal controller for the model with known parameters. Thus, it has asymptotically optimal (self-tuning) behavious.
track the reference values W=(1.0 ,1.0)T.
2
Example 2. Consider the folloving model vith the number of inputa-3 while the number of outputsm2 as trull model (28)
SIMULATION EXAMPLES Consider the following CAR(3) as the true model y(t)-A y(t-1) + A2y(t-2) + B2U(t-2) 1 (25)
+ B U(t-3) + e(t) 3
0.4
-0.504
,_0 , 305],
,
0.29]
A· 2 [
A12
-0.25,
1.5
°
, _0.297]
A,-
0.9 ,
-0.,]
0.5
0.8
[
· 0.,]
(26) and e(t) is gau8sian white noise with covariance matrix given by diag (0.0025, 0.0025). The reference vector w is assumed to be w-(1.0, 1.0)T • The samples of e(t) are generated by pseudorandom generator. The input u(t) is a pseudorandom binary sequence (PRBS) vi th the values ± 1, and the output is computed by Eq. (25). The F-test algorithm was performed for the first 50 samples (N=50) of model (25). We obtain the following results: d22
m-3,
and the estimates "
"Ai
[0.407717, _0.3
28
and
682] "
Ai-
B
are
given by
[-0,513804, 0.312117J
A2= -0.249405,
"
IBi '
[1.9964
1.49662,
0
-0.55771
· -0.30""] B,,[_"806" • O"'90"J
2
2
o
,0.986553,
, 1.5
(29) and e(t) and u(t) are independent, zero-mean, gauyian vhite noise with covariances diai«(0.28' ~.2~ and diag (1,1,1). The identification is performed off-line by using the first 50 samples of output, we obtain the folloving results:
which are identical to the xorresponding true value, and the estimates 11 and B1 are given by 0.908274
n::z2,
1
-0.562
, -0.3
0.996 ,
~r
, 0.75 ,
-0.366888, -0.30528
(27) For cost-function J1 with Q=I, R-diag(0.01, 1), from the 51th sample to the 100 th sample of model (25), we use the self-tuning controller Eq. (22), instead of the PRBS input signal, and we take Eq. (27) as the initial estimation of unknown parameters, the results obtained by 50 iterations are shown in rig. 1 and Fig. 2. Fig. 1 shows that the self-tuning input signals u ' u in steady state stabilize at values 5.5 2 1 and 3 respectively. Fig. 2 shows that the corresponding output signal
~- [ .499271 B.-lO.904711
I
.893807
_0.513197]
0.782661 0.808702
0. 332033
1.55293
1.6594
J
(.30)
For oost-function J 1 with Q-I, R-diag (10 ,0.01,100) and the timevarying referenoe vector wet), whose components v (t) and v (t) assumed to . be the square 2 1 waves with amplitudes-5 and 5 and period of 40 sampling periods, respeotively, we switch to the self-tuning controller Eq. (22), instead of the white noise input u(t), and we take Eq. (.30) as the initial estimation of unknown parameters, The length of simulation is 200 sampling periods, the resultl obtained by 200 iterations are shown in Fig. 3 and 1ig. 4. Fig. 3 shaWl outputs Y1(t) and Y2(t) of self-tuniug oontrol sYltem, vhioh track w (t) and v (t) ectiyely. 1 2
resp-
Fig. 4 shows the self-tuning control signals u (t), 1 u 2 (t) and u (t) • 3 Thil example shows that self-tuning controllers can be applied to control multivariable systems in vhich the number of inputs and the naber of outputs are not equal. CONCLUSIONS For the design of the multivariable self-tuning controller, this paper presents an approach which combine the determination of the model order and the time delay and the estimation of the parameters aa well aa the control of the output vector. The approach and results proposed by Bayoumi, Wong and
A Self-luning Adaplive Controlle r
401
Bayoumi, 1981)are extended and improv.d.
oS [
The order and time delay are determined by F-test method, which provides a parsimonious parameter model and avoids the drawback that the apriori knowledge of the time delay ia required in the generalized likelihood ratio test. Original CAR(n) model are identified directly usin, the RLS method instead of the id.ntification of d-step ah.ad prediction model (Koivo, 1980; Wong and Bayoumi, 1981). The controller with a gen.raliz.d ooat-function i. designed based on self-tuning multistep recursive predictor, which can be computed simply on-lin •• HultiTariable self-tuning controller propo ••d h.r. eliminates the restrictive a8auzptions about the order, the time delay and the dimenaion of inputoutput, and haa asymptotically optimal (Belf-tuning) behaviour.
12
J o
80
160
200
1:. Fig. 3. Outputs of self-tuning control syst811 for example 2 •
REFERmCItS
Astrom, K. J. (1968). Lectures OD the identifioation problem, the least .quares method. Lund In.t. • of Technology, Rep. 6806. 0.4 Astrom , K. J., and B. Wittenmark (1971). Problem 0.3 of identification and control. J. Math. Anal. Appl1c.,~, 90-113 . 0.1 Borisaon, U. (1979). Self-tuning regulators for a class of multivariable systems. Automatica, O.O~~~~. .~~~~~~~~~~~~~O .12, 209-215. -0.1 Bayouai, H. M.,K. Y. Wong, and H. A. El-Bagoury (1981). A self-tuning regulator for multivari- -0.2 able aystems. Automatica, 11, 575-592. Ljung, L. (1976). Conaiatenoy of the l.ast-squar•• identi!1oation method. lEE!: Trans. Autom. Contr. , 21, 779-781. Koivo, H7 N. (1980). A Hultivariable a.lf-tuning controller. Automatica, 16, 351-366. Keviczky, L. and J. Hetthes.y (1977). Self-tuning minimun Tariance control of KlHO di.cret. tim. 2 ayatems. Autom. Control Theory and Appl., 2, 11-17. KeTiczky, L., and K. S. P. Kumar (1981). MultiTariable self-tuning regulator with generaliz.d cost-function. Int. J. Control, 12, 913-921. o Kurz, H. and W. Goedecke (1981). Digital parueteradaptive control of proc ••••• with unknown -1 dead time. Automatica, 17, 245-252. Wong, K. Y. and M. H. Bayoumi-r1981). Multivariable self-tuning re&Ulators. Proc. of the 20th IEEE Confereno. on Deoision and Control, 9711-983. 5 4
0.03 0.02 0.01
3 2
t
0
o
10
20
30
JIJ
50
Fig. 1. Inputs of self-tuning control sy.tem for exuole 1 • 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2
t 10 20 30 JIJ 50 Fig. 2. Outputs of self-tuning control system for example 1 •
o
200
-0.01 -0.02 -0.03 -0.04 i~g.
4. Input. of aelf-tuning control sy.tem for exup1e 2