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Stability of an Input Amplitude Constrained Predictive Control System
Copyright © 1996 IFAC 13th Triennial World Congress, San Francisco, USA
2b-15 5
STABILITY OF AN INPUT AMPLITUDE CONSTRAINED PREDICTIVE CONTROL SYSTEM K. Y. Zhu and A. C. Koh
School of Electrical and Electronic Engineering Nanyang Technological UnivC1"sity Singapore 2263 Abstract: This paper aIlalyzes stability for a discrete time model-based predictive coutroi system HuiJject to an input amplitude constraint. Stabilitv resultH in the seW'I~ of BIBO a.re established for a cla.<.;s of the systems poles of wbich COllsist of multiple integrators and a Htaule polynomial. It is HllOwn that an adequate choice of t.he coutrol parameter:-> in the cost function would be important for system stabilitv. Propert.ies Rlld generalized deHcriptioIlH for Diophatiue equatioIls are ~lso addressed'. Keywords:
Predictive control, Quadratic performance indiceH, PolynomialH, Satmatioll cOlltrol, Stability analysis.
all been estahlished for linear HYHtel1lS, in which HO COllstraintl'! have heen consiuereu. III practice almoRt all ;;;yH-tems would be subject to COllHtraints due to hard limits Oil the inpnts, onLputs and staLe;;; of t.he HYHtem. For example, a control system may be Hubject to ::>aturatiOlI const.raint.s. Then st.ability analysis for constrained COlltrol systems would be much more difficult beca1l:->e that Ilonlillearities are involved in the closed-loop contl'olsystem and thus there does not exist. a linear relationsbip between the system 01ltput and control input or reference signal. III (Fcng d al., 1994), Rtabili1.y in thf! ;;;CTlse of BlBO has been analyzed for a system with Illultiple integ;rators and a stable polynomial tlsiug a constrained pole-placement control method. A similar Hystern Lut based on state-Hpace equatiOIll-l ha.'! al;;;o been di;;;cllHSCJ ill (SllssmaHll et (J.i., 1994: Tee!. 1992) uHing a part.icular control Rclteme. Sta.bility lIaR Leen addressed in (Glattfelder aHd Schanfelberger, 1983; KrikeliH and Barka.'!, 1981; Pa.ytle.1086) foJ' the Ht.rictly stable systems. The main motivation of the paper iH to investigate stability for a model-based predictive control system Hub.iect to an inpllt amplitude const.raint. Stability reslllts are e:-iLablisbed for a class of the ;;;YHtems wit.h multiple int~ gratOl'H atHJ a I'1tahl~ polYllomial, out it might be either llIillimlllll or IlOll-minimUIll phase Hystem:-i. In sectioll 2 the Hystem to be cout.rolleu iH deRcl'ibed and a modelbased predictive controller with and wit.hout cOllHtraiHts is pre.seIlted. Stability is analyzed ill section 3 and Himu-
1. INTRODUCTION Model-based predictive control is oue of the well known predictive cOlltrol lIIetbods and ha,., received a wide range of applicatioIlH in syst.eIlls cOIJtrol dlle to its good performance (Clarke et al., 1087; Clarke and Scattolini, 1991; Gorez et al., 1987; Gl'imble, 1992; Mo~ca and Zhang, 1992). HoweveL oue drawba.ck widl this method i;;; that it iH difficlllt ill gellera.l to yield a stability guaranteed controller becallHP of limiting r:oIltroi horizons adopt.ed ill the derivation of the control law for reducing cOIllplltatiolls. Recently. the auove problem has been inveHtigated and some st.alJilit:v guaranteed coutrollers }lave heen presented, Hee f01' exalllpleH, (Demircioglu and Clarke, 1993; Grimble. 19!J2; KWOTl and Byull, 1989). These tncthoth; are mainl.v based ou the iutrodllct.ion of filters or choice of a partic1Ilar set of the coutrol paraIUet.ers. ht (Delllircioghl and Clarke. 1093), the prediction and control horizons have been cilOSClI greater than the Hystelll order am] a\Ho a cOllstraiut at the eud Htate of the SYHt.CIll bas been introdllced slldl tbat the result jug closed-loop control Hyst.cm has all poles within the ullit circle of the z-plane. Allother reslll1. has Leen presellteJ by (GrimLle, 1992) which is ba<;eJ on :tero;;; caIlcella· tion and thus a dead-beat coutrol a.pproach ha.., ueell resulted. It.
j;;;
[Jot iced however t.hat the above ;;;tability results have
2353
ahead ontput prediction of the Hystem y(t achieved by l1!'iillg Diophautinc eqnation
latioIlf> are also c8rrieu ant in section 4. conclusions arc given ill f>cctioTl 5.
+ .j).
That is
2. CONTROL LAW k = 1,2,···, m
COIH;ider a discrete time siugle-inpnt single-ontput system described b.y the lllodel
where a general ~ k is adopted for the further use in the later part of the paper, and
(1)
Ej(z -" k) = 1 + ", (l: )z-I
where y(t) and 'flU) are the Hystem output and input reHpectively. A(z-l) and n(z-l) are PO~YIloJl1iall'l in the backward shift. operator z-l with order of n. It is asI'I1l1fled that polynomial A (z -1) COIl sistI' of multiple integraton; and a stable polynomial, such that it can also be rewritten as
+ ... + e ,_ , (k)z -.i+1
Fj(z-I,k) = f,,(k)+.f,,(k)z-l +"'+.f,,_,(k)z-k+l
Choosing k = one has
?J(t+.iJ =
(2)
HI.
ill (6) and then applying it into (1).
B(z-I)E,(z-l. m.) 1 Al(Z I) n(t+j-l)+Fj(z- .",.)//(t) (7)
where Ll. = 1 ~ z-l. m is all illteger (;:::: 1) and A 1(z-l) is of order n - m awl is :.;('ric.t.ly stable. The following are HOIlle propertieH of the Hystelll modeL
For further manipulation, the Diophantine eqnation B(Z-I)Ej(Z-l,k) ~ Gj(z-l.k)AI(z-I)+z-jH,(z-l. k ) (8) is applied to (7) with k = 'm, where
PI: The fo:iYfo:item polynomials A(z-l) and B(z-l) are relatively prillle, P2: 6,t11lz-+1 prime.
=
0 awl .8(1)
-f.
O.
Gj(Z-I. k)
for t.hC!.Y are relatively
H,(Z-I, k) = h" (k)
P3: Adl} > 0, for A 1 (z-1) is a strictly stable polynomiaL It is knowu that IUodel-ba.<;eu preiuctive control is a receding llOriwB control system based on a quadratic cost function N2
.I =
y(t
+ .i)]' + A 2: ,,2(t +.i -
+ h"
(k)z - I
+ ... + h,,,
., (k)z -"+1
That. gives
Nu
2)"(1, +:i) )=1
= 9o(k) + 9, (k)z-I + ... + Y j _, (k)z-;+I
+
1) (3)
j=l
Hj(Z-I, m)
(9)
(I) u(t-l) .
A, z
Equation (9) is one of the descriptions of the .1-l'Itep ahead output predictions of the system and will be used in minimization of the cost functioIl. A sequence of the output preciictionH obtained by increaHillg .i from 1 to N2 sllccesHively il'l given by
where w(t + j) is the reference Hig,11al which the system output wonl(1 follow. N2 and N'II are the prediction a.wl cOlltrol horizons, respectively. >.. is a weighting factor lmed to avoid an exceed control action. A choice of Nu :::; N2 lHeans that the cOllstraint of u(t + j) = 0 for :i = N'If,· .. N2 - 1 is illtrodnced illtu the cost fUllctiou for reducing complltaLiow.; aHd also for improving performallce. The cost fUllctioll eau alHo he rewrittell as
Y
(10)
= GU+Fy+Fu
where Y and U are the vetO[H of the outp1lt predictions amI future inputs as defined in (6), G is the (N2)x(Nu.) 1beplitz Illatrix and
(4)
(11)
where
w ~ [",(t + 1) .... w(t + N2J]T } Y = [y(t + 1) ... ,,(t + N2W . U-[!I(t) !llt+N,,-IW
I
HN2(Z- ,m)
(5 )
jTU(t-1)
(I)'
(12)
A1z
The control law minimizing the cost fUIlctioll is given by
Control law is thCll derived through minimization of the COl'lt fUllction (4) with r~spcct to U. However. to minimize the cost fll11Ctioll it is required to compute a ,i-step
N2
u(t) =
2: "J"'U + j) j=l
2354
Pj (z -I. m )y(t)
- [fj(z
-1
, "')
a(1 - 1) 1)1
A I (z
llIilleJ from (6) and has the coefficients
( 13)
whCt'e it . are the coetficiel.lt~ of the first row of (OT G + A!)- IC I , It is shown (see. Gorez. d a.l. , 1987) t.hat Lhe result,jug closp.(I-looJ.> control s'ystem ha"> the characteristic polynomial A c{.z -l ). So if an poles of A r.{z -l) lie st.rictly withill the uuit ci rcle of the z-plauc, the control system is Htable. Note that. A . Jz - 1 ) is a [HllctjOll of the control parameter!>; , N2, :Vu awl'x. It is then possible to have a set of the coateol parameters slu~h tlla1. A c(z-l) i." stable. However, title to t he maill purpose of the paper is to im n"" if 1"'( /,)1::; u.' if u(f) < - u"
" .(k) ,
= Ci + k -
I )! j! (k - I)!
(15)
and lilll Ej(z- lk) HI
= (j + k -I)'
(16)
(j - I)!k!
.
k = 1,2,··, !tn.:i = 1,2,··
Pmoj: ( 16 ) will be verified first. It is kuown that Ej(z- l, ~:) is wmally oompllted lL'Iing aH iterative method aud then all a lgel>raic s um of its coefficients ma.\, g'iv~ Ei(l , k) . Here a dilferellt method it; lls~d. Mllltiplying by zi+k - 1 botb liideli of DiophantilLe eqllatiolL (6) y ields ·+k 1 z1 =6.- k Ej(z ,k)+Fj(z,k) (1 7) wbere l:. = z6. = z - 1, Ej (z, k) = z,-IE,(z-l,k) ami Fj ( z l ~~ );;;;;: zk-1Fi (Z- 1, k ). Tbell ~i[Jce
(14)
(j +k-I)! U - I)!
if -
1; \Vhen a ~at nratioI1 cOIllpoueut a') (14) is introduced iuto tlie Itlouel- uaHed predictive co utrol !'i..,.·Stem, uo a liHear c.:out rol relatioll~hip exists even if the control law is JcriveJ La.'leJ OH a liuear eqllaLioll. Of COlll':iC staLility 811alysi:-; ah,Q ltecOlll es lIIuch l!Jore Jiffic lIlt. RC1fW1'k
zlilll __ ] ,,-;;F;(z, (jz
k)
( 18)
=0
(10)
where Fj(z, k) is of order k - 1, ami Uk _ lilll:-;;6. k-E;(z , k ) = k !Ej(l , k)
z--->l
nl~ ma1'k 2; The constraint considered here is a symmetrical :mturat.ioll. III practice, the maxiUllwl ami l1liniIlHlIll saturation value:-; may not have the same magnitude dne t.o physical limitations, For example, a valve actllator-based <':olltroi system may bavc differeut setting for m6xillL1lI1I anJ IlL ill ilUUIU , However, the s taLility res ults pre:-;enteJ ill the papel' woulJ a lso /.le a pplicable to that case.
(20)
iJ z
it follow, immeuiately Ej(l , k) of (16), "j(k) i, outained lI!iiug the following method. If oue choos~ k ""'* k - l in (6). it beCOIllp.~
Usillg (16 ) olle outaills lilU:t -ol Ej( z - l .k -
J)
U+k-2}!
b - l )l(k I )! '
Suutracting (21) frol11 (6) yield,
3, STABILITY RESULTS This section is cO llcer Jl(~ d witb st.ability analysis when the inpnt atllplitude cOlls t rained lllodel-bft."ie
u=
6. k [Ej( Z-I ,k) _
+ z -j IFj(z - l , k)
It is showll that Diophalltillc cqllatioml uefinei.l lJy (6) aud (8) ill the prc vio u ~ sectiOH a llow OHP. to derive the ont.pllt predictiollS of t he !';,Ystelll , sHch that the minimizatioIl of t.he cost fnuctiou caIl ue easily realized. SiIlce the re~mltillg CO lIl.rollaw (13) cOlltaius polYllomial:-; of Diophantine equatiolls. one should l:olllp1lte tIte equations. The followillg are :-;01ll P. rcsllits of the polYllomial
Ej(Z-~k' -1)]
- Fj(z-I , k - 1)1
(22)
a mI then arranging (22 ) and equating botlt si(l P..s leads to (23)
Thus, (15) is verified too. This has completed the proof. Lemma 2: PolYIlornial~ Fj (z -l,k) which is Ilniquely det.el'luinCd from (6) , ha."i
('.oI11pIltatioIts.
-
Fj(z, k) = 1 +
Lemma 1: PolYllomials E;.i( z - l. ~: ) is IIl1iq ltely deter-
2355
-
-
-
-
- "2
I" (1)6. + 1,,(1)6.
+ ... + f
.1~
.
k (k)L5. .
- t
1
(24)
where (32)
where
-
U+k-l)! f" (k) ~ (j + !. _ 1 _. 'i.)'"
(25) (33)
i. = 1,2, ... ,k - 1.
where Dl(Z-l) IH a polynomial of order n - HI and B2(Z-I) iH of order m with B,(l) ~ B(I)/Adl). It
and
lilll F,k A)
.:--+ 1
.
~
can be seen that Yt (t) is hOllwJeJ sillce u"'{t} iH bounded and A 1(z-1) is .'itrictly stable. Next one proves by COlltradiction that lJ(t) lllllst abo he bOlllllled. It call Heen that .6.""11 (t) is LOlluded, ASH lime that Ll TI1-k+l1J( t) is bonnded, l)11t f).711--ky(t) is nnhollllued, sHcb that.
(26)
1
Proof: (26) can ue ea!iily verified by applyillg z ---+ 1 to both sides of (6). Now wc prove (24). For (17), compute . ijt J+k-l hm-. -.2 = J;---->
I I)z"
where fiince 1
.
ij'
lUll -. .- .
;;:--+1 dz 1
:s: .i. :S A: -
[-k':.. . -, 1 .6. Ej(z,k)+fj(z,k) (27)
1.
OHe
This meam; that. at any time t = tj (0 "S t I < 00) we call always find aT (tt < T < (0), ~mch that
has
(28)
(30 )
Thwi, slIbstitnting (2-1) illto (27) awl computillg it yielJ
(j+k-l)!
~ilf-(kJ
7'U"'+'-:-j-_-=-j-_'-ci-;-;)1 "
..
~"'_l'+l
(.yftJ (t )
III thi,'i case since ~ ... 0 for t" approaching infinity from lemma 3, it can Ge assllmed, without loss of generality. that {.6.m-k1J(t~)} with (t2 ~ r < 00) is a positivE' Hllbsp.qllClJCC. Then (34) anti (35) hecome, respectively lilll Ll m-k"u,) ~ 00 (36)
(29)
This has verified correctll€SS of (24). Remark:1: Lemma 1 shows that. Ej{z-l,k} ill (6) cau Jp~..,cribcJ ill El. geIleral uescriptioll aad unth ('j (k) and E j ( L k) are all positive vallles iIlcreasilLg with respect to j or k. Pmpose of the dCTIHitioll of j;j(z-l, k:) as (24) call be seen ill the proof of theorelll 1.
f- k
--+{)O
ue
The inpllt amplitude constrained lUouel-ua."ied predictive cOlltrollaw (13) ami (14) at time t. 1. is
The followilLg lelllllla is also l1seflli ill stability allai.ysis. N2
.,,'(t,,) ~
Lemma 3: For the iIlPllt alHplitlltie cOlI.'itrailled modelhased predictive control systpm (1), (1.1) and (14), if ,,'v(l) 1·Illlt-H)O 1 ..tI (t ) 1~ 00, t I, tUt I'llHt.oo .6,1. Iy(t) = 0 : k =
(38)
j=l
where
1,2,···,lII.
1 .,,'(i, - 1) +,i)=",(i, +,i)-Hj(z- ,m) Al(Z 1)
Proof: See (Feng do al., 1904). Now we pretient the satbility reslllis ill the followillg theorelUK,
m~l
-L
ij,(m)f).i!I(tI. -·m
+ 1 + -i).
(39)
i=l
Theorem 1: The input aUlp1itlld~ constrained lllodelLaKed predictive cont.rol sysl,elll (1), (13) aHt! (14) iK stabIt' ill the BIBO sellse if
(36) aud (37) show '!I{tJ
---'> DC for t l , cc, which aIKo means that '1/.* UJ should Le saturated. Then, acconliug to the reslllt of lelllma 3, oue can have
N2
L",B(l) > O.
""'{L ", [-,(t, + j)- !lU, - m. + Ill}
(30)
N'
u."'(t,,)
)=1
= -sgn{L:>~/.II(t~, -
m
+ l)}n-i'
'j=1
Proof: Finit, we rn8uiplllate the system Illodel (1) sllch that. (31)
N2
=
-sg·o·{LOj}U*. )=1
2356
(40)
Mult iply ing Loth s ill cs of (3 1) by ~ 1ll -1.: and choosing t = T y ield
Theorem 2: The ilJpllt amplit1lde constrained Dlodelba~ed pl'eJictive cOlltl'ol :-iY:-iteul (1), (13) amI (14) i~ ~ta ble ill the BIBO sellse if Nu = 1 alllI N2 00, ------j.
P1'(jof: If Nu = 1, olte ha'! (sce, Core-.l: et al.,19B?) F'nrther m:illg Dioph a llLillc cqllatioH (6) lea ds to A m -k
L.l.
(I =u. Am -
?I T
k
( I + F(' - 1 Am-k ( ,j Z , k )u. !J2T
',111 r
I ',j ( z -1 .h.'')'/I. O( 'r n (1 +( -IE Al 1
1)
(46)
.
- :I)
and t!IllS
(411
-.
N2
L
Applying (40) illtO (41) a lld using the result of lemm a 2 yiel,1 A m L.l.
- k !I (T) =
A
L.l.
(
'"
( I-
m - k 'Il l T
A L.l
m - k NI ( T -' :I - k
H'.J
j= l
GN2 I ,m) = ;:::;NNo;2-"':2~-'--:'-' Li=l Yi-l (711) ..... ..\
(47)
FrotH (8), oue oLt ailis
+ 1I
B( l)G N2(1, mlA 1 (J) k- I
+ .6. tn.- kYJ (T _ j_k+ 1)+
L J ;{J.:}6. j
w-k+iYz(r_ j
= B'(l )EN2 (L m) - B(l )H N2( 1, 1nl
- k+ l+i )
i= l
2: B'(1IEN20 ," " )-1 B (l)HN2(l,ml
(421
I.
(48)
It cau be verified throngh compnt atioll of H)( z - l . m.}
where
N'
B(l )
J= l
.
frolll (8 ) that I,here p.xi~t.s cOll~l,aJlt (' . .'111<..: 11 that
u.{"n . J-(- . . ~ ' A l JI
It =
.'4 Q
IH"I2(1,lIIli $
C l B(l ) 1 "N' _'(1I/.).
(49)
SuLsLi llltillg (49) illto (48) aud llsiug
Sin c<1
Ej(l,q
i,,(q
Ci+k-l-i)!i! Ci - 11!k!
j! > U - l )!k!
71/.
(N2
.i
= k!
(43)
B( I)G ",(1, rH )A l(l)
2: B'(1)E ,., (1,1I/.) - CD'(l)"NH
= n2 ( I )EN,(l , m) +L
f:i(r.~)6. m -k +;'lJ2(T - J - ~:
(,0)
1)
one has
ami .6,. m - k+i.'!l z (t£) is LoullJeu , there lllllst exist a ~11tti cielltly large 1 but .still iiIlit,e .J 1 sHell that.
-':-1
+ '" -
+ 1+n
Thlls ~
if lV 2
[
1-
cm 1
N2 + m - 1
.
(,1)
------'> 00,
; ':"01
1J(1 )GN2(1,m )
(44)
> O.
(52)
This Ita..; ver ilip-d tl. at (30) is satisfieJ.
TIi1U;, if It > 0 which is eqnival«;':lIt to (30), t heIl olle obtains
4, SIMULATION
(4'1
This :-;ec:t. ioll presents tile simulation.'I re~mlts for the iJ.pllt amplituue constrai ued pret.lict.ive cO lltrol systelll. Two sys te m moJe[~ are taken in t.he :-;imulatioIl.
Tllis cOllt raLiicts (37) . This JIleallS t hat o ue call alwayse fillt! r i ' sHe ll Lhat 110 T satis fyiIlg (37 ). T hus, Lhe sltuseJOE'_'Illot exq11euee .6,.m- k u ( f,,) Hat isfyillg ( :W ) n!lJ i.'it anu .6,. ",,-k y(t.) 1IJ1lst be bOlluueu. By iIlduct.ioll, u(O IIIllSt. abo be bOllnJ ~u, This Ilfl.'! cOlllp ie teu the proof.
(an
~'"(I -
0.5z - 1 )(1 - 0.8z- 1 ) 1 - 0.2z- 1 L ~, 3
It call \)e seell that. illp.
(,3)
Fig, 4.1 - 4.3 show the simulatiowi re:mlts for the "hove ",ystClllS with 1/1. = 112 and 3 l'esvec tively. Not ~ that cOllstrai lled prp.(IicLive cOlltl'ol system give stable results. However, the :wsteIlIs rE'~'Ipt)m;p.s sllollld be slower t ha u
2357
the unconstrained control system with same control parameters setting.
eters for a class of the systems poles of which consist of multiple integrators and a stable polynomial, but zeros of which could be either stable or unstable. The simulation results show that the system output converges to the reference signal without steady-state error/:). That means that for a given system there exists a set that if the initial conditions are within the set, then the system may converge to a reference.
REFERENCES
.:~ so
-0.20
Fig.4.1, m
100
= I,N,
1SO
~
3,N.
200
250
lOO
= 1 and A = 0.1
MIrimom PhIo .. T~2 PioII't
.'~ 1
•
~-
0 . 5 ,
o
'-.
-0.50
so
Fig.4.2,
m
lOO
lSO
200
250
300
= 2,Nl = 5,Nu = 2 and A = 0.1
MiIiIIun Pto... Typw Plant
"~ . . .
o.~o
-0.50
50
Fig.4.3, m
100
ISO
200
2SO
300
= 3,N, = 5,Nu = 2 and A = 0.1
5. CONCLUSIONS This paper presents stability re."iults for a discrete time model-based predictive control system subject to an input amplitude constraint. BIBO stability may be assured through an adequate choice of the control paraIll-
Clarke,D.W., C.Mohtadi and P.S.Tuff, (1987). Generalized predictive control part L The basic algorithm, Part n. extension and interpretations. A utomatica, 23, pp.137-160. Clarke,D.W. and R.Scattolini (1991). Constrained receding-horizon predictive control. Prof'!. lEE, 140, pp.347-354. Demircioglu,H. and D.W.Clarke (1993). Generalized predictive control with eIld~point state weighting. Pmc lEE, 140, pp.275-282. Feng,G .• C.Zhaug and M.Palanisawami (1994). StaLility of input amplitude constrained adaptive pole placement control sYHtems. A1Ltomatica, 30, pp.1065-1070. Glattfelder,A.H. and W.Schaufelberger (1983). Stability analysis of single loop control system with saturation and antireset-windup circuits. IEEE. Trans. Autmnat. Contr., 28, pp.l074-1080. Gorez,R., V.Wertz and K.Y.Zhu (1987). On a generalized predictive control algorithm. Syst. Cantr. Lett., 9, pp.369-377. Grimble,M.J. (1992). Long-range predictive optimal control law with guaranteed stability for process control applications. Proc. of Amer"ican Control Conference, pp.2022-2026, USA. Kwon,W.H. and D.G.BYllIl (1989). Receding horizon tracking control aH a predictive control and its stability properties. bd. J. Control, 50, pp.1807-1824_ Krikelis,N ..l. and S.K.Barkas (1984). Design of tracking system subject to actuator saturation and integrator wind-lip. Int._ .I. Cmdml, 39, pp.667-682_ Mosca,E. and J.Zhang (1992). Stable redesign of predictive control," Automatica, 28, pp.1229-1233. Payne,A.N. (1986). Adaptive one step ahead control subject to an input amplitude constraint. Int. .I. Control, 43, pp.19U-207. Sussmann,H.J., E.D.Sontag and Y.Yang (1994). A general ref'iUlt on the stabilization of linear system using bounded controls. IEEE. Tr·a1l8. Automat. Contr., 39, pp.2411-2425. Teel,A.R. (1992). Global stabilization and restricted tracking for multiple integrators with bounded controb. Syst.. COIltl'. Lett., 18, pp.165-171.
2358
2b-15 5
STABILITY OF AN INPUT AMPLITUDE CONSTRAINED PREDICTIVE CONTROL SYSTEM K. Y. Zhu and A. C. Koh
School of Electrical and Electronic Engineering Nanyang Technological UnivC1"sity Singapore 2263 Abstract: This paper aIlalyzes stability for a discrete time model-based predictive coutroi system HuiJject to an input amplitude constraint. Stabilitv resultH in the seW'I~ of BIBO a.re established for a cla.<.;s of the systems poles of wbich COllsist of multiple integrators and a Htaule polynomial. It is HllOwn that an adequate choice of t.he coutrol parameter:-> in the cost function would be important for system stabilitv. Propert.ies Rlld generalized deHcriptioIlH for Diophatiue equatioIls are ~lso addressed'. Keywords:
Predictive control, Quadratic performance indiceH, PolynomialH, Satmatioll cOlltrol, Stability analysis.
all been estahlished for linear HYHtel1lS, in which HO COllstraintl'! have heen consiuereu. III practice almoRt all ;;;yH-tems would be subject to COllHtraints due to hard limits Oil the inpnts, onLputs and staLe;;; of t.he HYHtem. For example, a control system may be Hubject to ::>aturatiOlI const.raint.s. Then st.ability analysis for constrained COlltrol systems would be much more difficult beca1l:->e that Ilonlillearities are involved in the closed-loop contl'olsystem and thus there does not exist. a linear relationsbip between the system 01ltput and control input or reference signal. III (Fcng d al., 1994), Rtabili1.y in thf! ;;;CTlse of BlBO has been analyzed for a system with Illultiple integ;rators and a stable polynomial tlsiug a constrained pole-placement control method. A similar Hystern Lut based on state-Hpace equatiOIll-l ha.'! al;;;o been di;;;cllHSCJ ill (SllssmaHll et (J.i., 1994: Tee!. 1992) uHing a part.icular control Rclteme. Sta.bility lIaR Leen addressed in (Glattfelder aHd Schanfelberger, 1983; KrikeliH and Barka.'!, 1981; Pa.ytle.1086) foJ' the Ht.rictly stable systems. The main motivation of the paper iH to investigate stability for a model-based predictive control system Hub.iect to an inpllt amplitude const.raint. Stability reslllts are e:-iLablisbed for a class of the ;;;YHtems wit.h multiple int~ gratOl'H atHJ a I'1tahl~ polYllomial, out it might be either llIillimlllll or IlOll-minimUIll phase Hystem:-i. In sectioll 2 the Hystem to be cout.rolleu iH deRcl'ibed and a modelbased predictive controller with and wit.hout cOllHtraiHts is pre.seIlted. Stability is analyzed ill section 3 and Himu-
1. INTRODUCTION Model-based predictive control is oue of the well known predictive cOlltrol lIIetbods and ha,., received a wide range of applicatioIlH in syst.eIlls cOIJtrol dlle to its good performance (Clarke et al., 1087; Clarke and Scattolini, 1991; Gorez et al., 1987; Gl'imble, 1992; Mo~ca and Zhang, 1992). HoweveL oue drawba.ck widl this method i;;; that it iH difficlllt ill gellera.l to yield a stability guaranteed controller becallHP of limiting r:oIltroi horizons adopt.ed ill the derivation of the control law for reducing cOIllplltatiolls. Recently. the auove problem has been inveHtigated and some st.alJilit:v guaranteed coutrollers }lave heen presented, Hee f01' exalllpleH, (Demircioglu and Clarke, 1993; Grimble. 19!J2; KWOTl and Byull, 1989). These tncthoth; are mainl.v based ou the iutrodllct.ion of filters or choice of a partic1Ilar set of the coutrol paraIUet.ers. ht (Delllircioghl and Clarke. 1093), the prediction and control horizons have been cilOSClI greater than the Hystelll order am] a\Ho a cOllstraiut at the eud Htate of the SYHt.CIll bas been introdllced slldl tbat the result jug closed-loop control Hyst.cm has all poles within the ullit circle of the z-plane. Allother reslll1. has Leen presellteJ by (GrimLle, 1992) which is ba<;eJ on :tero;;; caIlcella· tion and thus a dead-beat coutrol a.pproach ha.., ueell resulted. It.
j;;;
[Jot iced however t.hat the above ;;;tability results have
2353
ahead ontput prediction of the Hystem y(t achieved by l1!'iillg Diophautinc eqnation
latioIlf> are also c8rrieu ant in section 4. conclusions arc given ill f>cctioTl 5.
+ .j).
That is
2. CONTROL LAW k = 1,2,···, m
COIH;ider a discrete time siugle-inpnt single-ontput system described b.y the lllodel
where a general ~ k is adopted for the further use in the later part of the paper, and
(1)
Ej(z -" k) = 1 + ", (l: )z-I
where y(t) and 'flU) are the Hystem output and input reHpectively. A(z-l) and n(z-l) are PO~YIloJl1iall'l in the backward shift. operator z-l with order of n. It is asI'I1l1fled that polynomial A (z -1) COIl sistI' of multiple integraton; and a stable polynomial, such that it can also be rewritten as
+ ... + e ,_ , (k)z -.i+1
Fj(z-I,k) = f,,(k)+.f,,(k)z-l +"'+.f,,_,(k)z-k+l
Choosing k = one has
?J(t+.iJ =
(2)
HI.
ill (6) and then applying it into (1).
B(z-I)E,(z-l. m.) 1 Al(Z I) n(t+j-l)+Fj(z- .",.)//(t) (7)
where Ll. = 1 ~ z-l. m is all illteger (;:::: 1) and A 1(z-l) is of order n - m awl is :.;('ric.t.ly stable. The following are HOIlle propertieH of the Hystelll modeL
For further manipulation, the Diophantine eqnation B(Z-I)Ej(Z-l,k) ~ Gj(z-l.k)AI(z-I)+z-jH,(z-l. k ) (8) is applied to (7) with k = 'm, where
PI: The fo:iYfo:item polynomials A(z-l) and B(z-l) are relatively prillle, P2: 6,t11lz-+1 prime.
=
0 awl .8(1)
-f.
O.
Gj(Z-I. k)
for t.hC!.Y are relatively
H,(Z-I, k) = h" (k)
P3: Adl} > 0, for A 1 (z-1) is a strictly stable polynomiaL It is knowu that IUodel-ba.<;eu preiuctive control is a receding llOriwB control system based on a quadratic cost function N2
.I =
y(t
+ .i)]' + A 2: ,,2(t +.i -
+ h"
(k)z - I
+ ... + h,,,
., (k)z -"+1
That. gives
Nu
2)"(1, +:i) )=1
= 9o(k) + 9, (k)z-I + ... + Y j _, (k)z-;+I
+
1) (3)
j=l
Hj(Z-I, m)
(9)
(I) u(t-l) .
A, z
Equation (9) is one of the descriptions of the .1-l'Itep ahead output predictions of the system and will be used in minimization of the cost functioIl. A sequence of the output preciictionH obtained by increaHillg .i from 1 to N2 sllccesHively il'l given by
where w(t + j) is the reference Hig,11al which the system output wonl(1 follow. N2 and N'II are the prediction a.wl cOlltrol horizons, respectively. >.. is a weighting factor lmed to avoid an exceed control action. A choice of Nu :::; N2 lHeans that the cOllstraint of u(t + j) = 0 for :i = N'If,· .. N2 - 1 is illtrodnced illtu the cost fUllctiou for reducing complltaLiow.; aHd also for improving performallce. The cost fUllctioll eau alHo he rewrittell as
Y
(10)
= GU+Fy+Fu
where Y and U are the vetO[H of the outp1lt predictions amI future inputs as defined in (6), G is the (N2)x(Nu.) 1beplitz Illatrix and
(4)
(11)
where
w ~ [",(t + 1) .... w(t + N2J]T } Y = [y(t + 1) ... ,,(t + N2W . U-[!I(t) !llt+N,,-IW
I
HN2(Z- ,m)
(5 )
jTU(t-1)
(I)'
(12)
A1z
The control law minimizing the cost fUIlctioll is given by
Control law is thCll derived through minimization of the COl'lt fUllction (4) with r~spcct to U. However. to minimize the cost fll11Ctioll it is required to compute a ,i-step
N2
u(t) =
2: "J"'U + j) j=l
2354
Pj (z -I. m )y(t)
- [fj(z
-1
, "')
a(1 - 1) 1)1
A I (z
llIilleJ from (6) and has the coefficients
( 13)
whCt'e it . are the coetficiel.lt~ of the first row of (OT G + A!)- IC I , It is shown (see. Gorez. d a.l. , 1987) t.hat Lhe result,jug closp.(I-looJ.> control s'ystem ha"> the characteristic polynomial A c{.z -l ). So if an poles of A r.{z -l) lie st.rictly withill the uuit ci rcle of the z-plauc, the control system is Htable. Note that. A . Jz - 1 ) is a [HllctjOll of the control parameter!>; , N2, :Vu awl'x. It is then possible to have a set of the coateol parameters slu~h tlla1. A c(z-l) i." stable. However, title to t he maill purpose of the paper is to im
" .(k) ,
= Ci + k -
I )! j! (k - I)!
(15)
and lilll Ej(z- lk) HI
= (j + k -I)'
(16)
(j - I)!k!
.
k = 1,2,··, !tn.:i = 1,2,··
Pmoj: ( 16 ) will be verified first. It is kuown that Ej(z- l, ~:) is wmally oompllted lL'Iing aH iterative method aud then all a lgel>raic s um of its coefficients ma.\, g'iv~ Ei(l , k) . Here a dilferellt method it; lls~d. Mllltiplying by zi+k - 1 botb liideli of DiophantilLe eqllatiolL (6) y ields ·+k 1 z1 =6.- k Ej(z ,k)+Fj(z,k) (1 7) wbere l:. = z6. = z - 1, Ej (z, k) = z,-IE,(z-l,k) ami Fj ( z l ~~ );;;;;: zk-1Fi (Z- 1, k ). Tbell ~i[Jce
(14)
(j +k-I)! U - I)!
if -
1; \Vhen a ~at nratioI1 cOIllpoueut a') (14) is introduced iuto tlie Itlouel- uaHed predictive co utrol !'i..,.·Stem, uo a liHear c.:out rol relatioll~hip exists even if the control law is JcriveJ La.'leJ OH a liuear eqllaLioll. Of COlll':iC staLility 811alysi:-; ah,Q ltecOlll es lIIuch l!Jore Jiffic lIlt. RC1fW1'k
zlilll __ ] ,,-;;F;(z, (jz
k)
( 18)
=0
(10)
where Fj(z, k) is of order k - 1, ami Uk _ lilll:-;;6. k-E;(z , k ) = k !Ej(l , k)
z--->l
nl~ ma1'k 2; The constraint considered here is a symmetrical :mturat.ioll. III practice, the maxiUllwl ami l1liniIlHlIll saturation value:-; may not have the same magnitude dne t.o physical limitations, For example, a valve actllator-based <':olltroi system may bavc differeut setting for m6xillL1lI1I anJ IlL ill ilUUIU , However, the s taLility res ults pre:-;enteJ ill the papel' woulJ a lso /.le a pplicable to that case.
(20)
iJ z
it follow, immeuiately Ej(l , k) of (16), "j(k) i, outained lI!iiug the following method. If oue choos~ k ""'* k - l in (6). it beCOIllp.~
Usillg (16 ) olle outaills lilU:t -ol Ej( z - l .k -
J)
U+k-2}!
b - l )l(k I )! '
Suutracting (21) frol11 (6) yield,
3, STABILITY RESULTS This section is cO llcer Jl(~ d witb st.ability analysis when the inpnt atllplitude cOlls t rained lllodel-bft."ie
u=
6. k [Ej( Z-I ,k) _
+ z -j IFj(z - l , k)
It is showll that Diophalltillc cqllatioml uefinei.l lJy (6) aud (8) ill the prc vio u ~ sectiOH a llow OHP. to derive the ont.pllt predictiollS of t he !';,Ystelll , sHch that the minimizatioIl of t.he cost fnuctiou caIl ue easily realized. SiIlce the re~mltillg CO lIl.rollaw (13) cOlltaius polYllomial:-; of Diophantine equatiolls. one should l:olllp1lte tIte equations. The followillg are :-;01ll P. rcsllits of the polYllomial
Ej(Z-~k' -1)]
- Fj(z-I , k - 1)1
(22)
a mI then arranging (22 ) and equating botlt si(l P..s leads to (23)
Thus, (15) is verified too. This has completed the proof. Lemma 2: PolYIlornial~ Fj (z -l,k) which is Ilniquely det.el'luinCd from (6) , ha."i
('.oI11pIltatioIts.
-
Fj(z, k) = 1 +
Lemma 1: PolYllomials E;.i( z - l. ~: ) is IIl1iq ltely deter-
2355
-
-
-
-
- "2
I" (1)6. + 1,,(1)6.
+ ... + f
.1~
.
k (k)L5. .
- t
1
(24)
where (32)
where
-
U+k-l)! f" (k) ~ (j + !. _ 1 _. 'i.)'"
(25) (33)
i. = 1,2, ... ,k - 1.
where Dl(Z-l) IH a polynomial of order n - HI and B2(Z-I) iH of order m with B,(l) ~ B(I)/Adl). It
and
lilll F,k A)
.:--+ 1
.
~
can be seen that Yt (t) is hOllwJeJ sillce u"'{t} iH bounded and A 1(z-1) is .'itrictly stable. Next one proves by COlltradiction that lJ(t) lllllst abo he bOlllllled. It call Heen that .6.""11 (t) is LOlluded, ASH lime that Ll TI1-k+l1J( t) is bonnded, l)11t f).711--ky(t) is nnhollllued, sHcb that.
(26)
1
Proof: (26) can ue ea!iily verified by applyillg z ---+ 1 to both sides of (6). Now wc prove (24). For (17), compute . ijt J+k-l hm-. -.2 = J;---->
I I)z"
where fiince 1
.
ij'
lUll -. .- .
;;:--+1 dz 1
:s: .i. :S A: -
[-k':.. . -, 1 .6. Ej(z,k)+fj(z,k) (27)
1.
OHe
This meam; that. at any time t = tj (0 "S t I < 00) we call always find aT (tt < T < (0), ~mch that
has
(28)
(30 )
Thwi, slIbstitnting (2-1) illto (27) awl computillg it yielJ
(j+k-l)!
~ilf-(kJ
7'U"'+'-:-j-_-=-j-_'-ci-;-;)1 "
..
~"'_l'+l
(.yftJ (t )
III thi,'i case since ~ ... 0 for t" approaching infinity from lemma 3, it can Ge assllmed, without loss of generality. that {.6.m-k1J(t~)} with (t2 ~ r < 00) is a positivE' Hllbsp.qllClJCC. Then (34) anti (35) hecome, respectively lilll Ll m-k"u,) ~ 00 (36)
(29)
This has verified correctll€SS of (24). Remark:1: Lemma 1 shows that. Ej{z-l,k} ill (6) cau Jp~..,cribcJ ill El. geIleral uescriptioll aad unth ('j (k) and E j ( L k) are all positive vallles iIlcreasilLg with respect to j or k. Pmpose of the dCTIHitioll of j;j(z-l, k:) as (24) call be seen ill the proof of theorelll 1.
f- k
--+{)O
ue
The inpllt amplitude constrained lUouel-ua."ied predictive cOlltrollaw (13) ami (14) at time t. 1. is
The followilLg lelllllla is also l1seflli ill stability allai.ysis. N2
.,,'(t,,) ~
Lemma 3: For the iIlPllt alHplitlltie cOlI.'itrailled modelhased predictive control systpm (1), (1.1) and (14), if ,,'v(l) 1·Illlt-H)O 1 ..tI (t ) 1~ 00, t I, tUt I'llHt.oo .6,1. Iy(t) = 0 : k =
(38)
j=l
where
1,2,···,lII.
1 .,,'(i, - 1) +,i)=",(i, +,i)-Hj(z- ,m) Al(Z 1)
Proof: See (Feng do al., 1904). Now we pretient the satbility reslllis ill the followillg theorelUK,
m~l
-L
ij,(m)f).i!I(tI. -·m
+ 1 + -i).
(39)
i=l
Theorem 1: The input aUlp1itlld~ constrained lllodelLaKed predictive cont.rol sysl,elll (1), (13) aHt! (14) iK stabIt' ill the BIBO sellse if
(36) aud (37) show '!I{tJ
---'> DC for t l , cc, which aIKo means that '1/.* UJ should Le saturated. Then, acconliug to the reslllt of lelllma 3, oue can have
N2
L",B(l) > O.
""'{L ", [-,(t, + j)- !lU, - m. + Ill}
(30)
N'
u."'(t,,)
)=1
= -sgn{L:>~/.II(t~, -
m
+ l)}n-i'
'j=1
Proof: Finit, we rn8uiplllate the system Illodel (1) sllch that. (31)
N2
=
-sg·o·{LOj}U*. )=1
2356
(40)
Mult iply ing Loth s ill cs of (3 1) by ~ 1ll -1.: and choosing t = T y ield
Theorem 2: The ilJpllt amplit1lde constrained Dlodelba~ed pl'eJictive cOlltl'ol :-iY:-iteul (1), (13) amI (14) i~ ~ta ble ill the BIBO sellse if Nu = 1 alllI N2 00, ------j.
P1'(jof: If Nu = 1, olte ha'! (sce, Core-.l: et al.,19B?) F'nrther m:illg Dioph a llLillc cqllatioH (6) lea ds to A m -k
L.l.
(I =u. Am -
?I T
k
( I + F(' - 1 Am-k ( ,j Z , k )u. !J2T
',111 r
I ',j ( z -1 .h.'')'/I. O( 'r n (1 +( -IE Al 1
1)
(46)
.
- :I)
and t!IllS
(411
-.
N2
L
Applying (40) illtO (41) a lld using the result of lemm a 2 yiel,1 A m L.l.
- k !I (T) =
A
L.l.
(
'"
( I-
m - k 'Il l T
A L.l
m - k NI ( T -' :I - k
H'.J
j= l
GN2 I ,m) = ;:::;NNo;2-"':2~-'--:'-' Li=l Yi-l (711) ..... ..\
(47)
FrotH (8), oue oLt ailis
+ 1I
B( l)G N2(1, mlA 1 (J) k- I
+ .6. tn.- kYJ (T _ j_k+ 1)+
L J ;{J.:}6. j
w-k+iYz(r_ j
= B'(l )EN2 (L m) - B(l )H N2( 1, 1nl
- k+ l+i )
i= l
2: B'(1IEN20 ," " )-1 B (l)HN2(l,ml
(421
I.
(48)
It cau be verified throngh compnt atioll of H)( z - l . m.}
where
N'
B(l )
J= l
.
frolll (8 ) that I,here p.xi~t.s cOll~l,aJlt (' . .'111<..: 11 that
u.{"n . J-(- . . ~ ' A l JI
It =
.'4 Q
IH"I2(1,lIIli $
C l B(l ) 1 "N' _'(1I/.).
(49)
SuLsLi llltillg (49) illto (48) aud llsiug
Sin c<1
Ej(l,q
i,,(q
Ci+k-l-i)!i! Ci - 11!k!
j! > U - l )!k!
71/.
(N2
.i
= k!
(43)
B( I)G ",(1, rH )A l(l)
2: B'(1)E ,., (1,1I/.) - CD'(l)"NH
= n2 ( I )EN,(l , m) +L
f:i(r.~)6. m -k +;'lJ2(T - J - ~:
(,0)
1)
one has
ami .6,. m - k+i.'!l z (t£) is LoullJeu , there lllllst exist a ~11tti cielltly large 1 but .still iiIlit,e .J 1 sHell that.
-':-1
+ '" -
+ 1+n
Thlls ~
if lV 2
[
1-
cm 1
N2 + m - 1
.
(,1)
------'> 00,
; ':"01
1J(1 )GN2(1,m )
(44)
> O.
(52)
This Ita..; ver ilip-d tl. at (30) is satisfieJ.
TIi1U;, if It > 0 which is eqnival«;':lIt to (30), t heIl olle obtains
4, SIMULATION
(4'1
This :-;ec:t. ioll presents tile simulation.'I re~mlts for the iJ.pllt amplituue constrai ued pret.lict.ive cO lltrol systelll. Two sys te m moJe[~ are taken in t.he :-;imulatioIl.
Tllis cOllt raLiicts (37) . This JIleallS t hat o ue call alwayse fillt! r i ' sHe ll Lhat 110 T satis fyiIlg (37 ). T hus, Lhe sltuseJOE'_'Illot exq11euee .6,.m- k u ( f,,) Hat isfyillg ( :W ) n!lJ i.'it anu .6,. ",,-k y(t.) 1IJ1lst be bOlluueu. By iIlduct.ioll, u(O IIIllSt. abo be bOllnJ ~u, This Ilfl.'! cOlllp ie teu the proof.
(an
~'"(I -
0.5z - 1 )(1 - 0.8z- 1 ) 1 - 0.2z- 1 L ~, 3
It call \)e seell that. illp.
(,3)
Fig, 4.1 - 4.3 show the simulatiowi re:mlts for the "hove ",ystClllS with 1/1. = 112 and 3 l'esvec tively. Not ~ that cOllstrai lled prp.(IicLive cOlltl'ol system give stable results. However, the :wsteIlIs rE'~'Ipt)m;p.s sllollld be slower t ha u
2357
the unconstrained control system with same control parameters setting.
eters for a class of the systems poles of which consist of multiple integrators and a stable polynomial, but zeros of which could be either stable or unstable. The simulation results show that the system output converges to the reference signal without steady-state error/:). That means that for a given system there exists a set that if the initial conditions are within the set, then the system may converge to a reference.
REFERENCES
.:~ so
-0.20
Fig.4.1, m
100
= I,N,
1SO
~
3,N.
200
250
lOO
= 1 and A = 0.1
MIrimom PhIo .. T~2 PioII't
.'~ 1
•
~-
0 . 5 ,
o
'-.
-0.50
so
Fig.4.2,
m
lOO
lSO
200
250
300
= 2,Nl = 5,Nu = 2 and A = 0.1
MiIiIIun Pto... Typw Plant
"~ . . .
o.~o
-0.50
50
Fig.4.3, m
100
ISO
200
2SO
300
= 3,N, = 5,Nu = 2 and A = 0.1
5. CONCLUSIONS This paper presents stability re."iults for a discrete time model-based predictive control system subject to an input amplitude constraint. BIBO stability may be assured through an adequate choice of the control paraIll-
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