We derive stability conditions for Model Predictive Control (MPC) with hard constraints on the inputs and "soft" constraints on the outputs for an infinitely long output horizon. We show that with state feedback MPC is globally asymptotically stabilizing if and only if all the eigenvalues of the open loop system are in the closed unit disk. With output feedback the eigenvalues must be strictly inside the unit circle. The on-line optimization problem defining MPC can be posed as a finite dimensional quadratic program even though the output constraints are specified over an infinite horizon

Morari, M.

Zheng, A.

English

Caltech Authors - Main

TECHNICAL MEMORANDUM NO. CIT-CDS 93-025 
December 14,1993 
revised January 11,1994 
"Stability of Model Predictive Control 
with Soft Constraints" 
A. Zheng and M. Morari 
Control and Dynamical Systems 
California Institute of Technology 
Pasadena, CA 91125 
Stability of Model Predictive Coi-~trol with Soft Constraints 
Ales Zheng Ma.llfred Mora,ri * 
Chelllica,l Engiaeering, 210-41 
Ca.lifornia Institute of Techi~ology 
Pasa,dena,, CA 91 125 
April 26, 1994 
Abstrac t  
We derive stabilit,y condit.ions for Model Predictjive Control (MPC) with ha.rd constraints on the 
input,s and "soft" constra.int,s on the outputs for a.n infinitely long output horizon. 1% show that with 
state feedback MPC is globa.11~. a.symptot,icallp sta.bilizing if a.nd only if all the eigellvadues of the open 
loop syst,em a.re in t,he closed unit disk. With output feedba.ck t,he eigenvadnes must be strict,ly inside 
the unit circle. The on-line opt,imiza.t.ion problem defining MPC can be posed as a finite dimensional 
qna.dra.t,ic progra.111 even t,llongl~ t9he output const,rabnt,s are specified over a.n infiilit,e liorizol~. 
1 Introduction 
Many pra~t~ica.1 coiltrol problems a.re ~lolnillat~ed by constra.ints. There are genera.lly two types of constra.ints 
- illput constra,iilts a,nd output  coast,ra.ints. The  illput constra.ints a.re always present and are ilnposed by 
physica.1 liinitatioas of the  a.ctua.tors which ca.ilnot be exceeded under a.ny circuinstances. Often, i t  is also 
desirable t o  keep specific outputs  within certain liniits for rea.soas related to  plant opera.tion, e.g. safety, 
material constra.iilts, etc. I t  is usually unavoidahle t o  exceed the  output  coi~stradnt~s, a t  least temporarily, 
for example, when the  systeill is sulsjected to  ui~expected disturbances. 
Industry 11a.s ein11ra.cecl Model Predictive Control (MPC), a,lso referred to  as moving horizon control a,nd 
receding llorizoil control, a,s a, po\verful feedback stra.tegy t,o coiltrol systems with constraints. The  ba.sic 
idea behind MPC is a.s follo~vs: At sampling time k ,  711. future control moves are calculated such tha t  a n  
objective fullction over some (output,) horizon is iniiliinized subject to  constra.ints. Oilly the first one of the 
in computed coi~t~rol inoves is iinpleinent,ed. i l t  t he  next sainpling t ime, the  mea.suremeat is used t o  update 
the  sta.te estima.te a.nd the  sa.me calcula,tions a.re repeated. 
Rawlings a.nd Muske [2] sl~owied lha t  globa,l asymptotic stability of the coilstrained systeill call be guar- 
anteed by ma.lcing the horizon infinite, provided that  the optimizatiol~ problem defitliilg the  R4PC coiltroller 
is fea.sible. However, output  ~oilst~rail l ts  call 1ea.d to  a a  infea.sible optilniza.tion problel~l. They proposed to  
remove the  output coilstraints during the z i l i t~n l  portzoii of the infinite horizon to  inalte the optilllization 
problem fea.sihle. Unfort,una.t~el~v, this ca.n result in poor performance: the  viola.tioi1 of the output constraints 
*Author to whom all corrrsponcler~~:e should be adclressed: phone (818)395-4186, fax (818)568-8743, e-nail: 
MMQIA/IC.CALTECH.E;Il~.~ 
during this initial port(ioi1 of the  iilfiilit,e horizon ca.11 be very la,rge in order to  sa.tisfy the constra.ints dur- 
ing the rest. Thus, large constraint violations 1na.y be experienced, n ~ h e n  the conlputed control actions a re  
implemented. 
An a,lterna.taive wa3i t o  haadle t,he fea~ibilit~y problem is t o  relax the  infeasible sta.te constraints for t h e  
elitire horizon and to peaa,lize the  extent of the viola.tioa. This technique is referred to  a.s "constra.int 
softening" [3]. The problem is t11a.t g1oba.l stal~il i ty inay not be guara.nteed. Zafiriou a,nd Chiou [7] have 
derived some conditions for stability. However, these conditions a.re generally conserva.tive and  difficult t o  
check. 
In this note, we shorn? tha.t global a.sjrmptotic sta,bilit,y ca.11 be gua.ranteed for systems with nlixed h a d  a.nd 
soft constra.ints. Furthermore, in t,he ca.se tI1a.t the sta.te nlust be estima.ted, we show tha t  global stability is 
preserved by using an asymptotic observer. Finally, are show that  the optii~lization can be ca.st as a finite 
diinensio11a.l quadra.tic program eve11 though the output  constraiilts a.re specified over the  infinite horizon. 
2 State Feedback 
Consider the  systei11 
z ( k  + 1 )  = A z ( k )  + B u ( k )  
y ( k )  = C z ( k )  
where x ( k )  E % n z ,  n j k )  E 3?n*r and y ( k )  E !I?'". Define the  objective fiiilction as 
00 n z -  1 
@k = z ~ ( k  + i ~ k ) ~ ~ o : ( k  + i l k ) + z [ ~ ( k  + i l k ) T ~ ? n ( k  + i l k )  + A n ( k  + i l k j T p A u , ( k  + i l k ) ]  
i = l  i = O  
( 2 )  
where R > 0, S > 0 ,  P 2 0 ,  ancl 111 is finite. R, S, aad  P  a.re symmetric. ( . ) ( k  + i l k )  denotes the  va.riable (.) 
a.t sainpling time k  + i precli~t~ecl a.t. saillplii~g time k .  T h e  coiltrol a.ctions a.re genera.tec1 by Co~etrol ler  A f P C  
which is defined a,s follows. 
Defiiiitioii 1 Co~itroller MPC: .4t sainpling t ime  k ,  tI1,e colrlrol moue u ( k )  epuuls the first element u , ( k l k )  
of tlze sequence { . ~ l , ( k l k ) ,  26(k + I l k ) ,  . . . ,21,(k + 112 - I l k ) }  which is  the I ~ ~ z I ~ z I I ~ . ~ ~ ~ I -  of th.e optzmzzutzon problem 
jAu(k  + i l k ) /  5 i = 0 , 1 , . . . , 1 1 a -  1  
Zb71Z%" 5 u ( k +  i l k )  _< unzaz i x 0 , 1 , .  . . , m  - 1  
subject t,o A u ( k  + i l k )  = 0  i = ~ n , m + l , . . . , o o  
Go:(k+ i l k )  < g + e ( k )  i = 0, 1 , . . . , 0 0  
~ ( k )  > 0 
where (_: E % n ~ X 1 a ~  u~zd  Q > O i s  dingonol. 
The  input constra.intss reisresent physical limita.t,ions 011 t,he actua.tors which ca.nnot be viola.t,ed. The  output 
constra.ints a.re softelred by the sla.cli variables € ( I ; ) .  They caa be viola.tec1 temporarily, if necessa,ry. I11 the 
long term,  the  pena.lt,y tern1 ~ ( k ) ~ Q t ( k )  in t,he objective fui~ction ki~ill drive the  sla.cli va.ria.bles t o  zero. T h e  
optiiniza.tion problem (3) can be ca.st a.s a, qua.dratic pr0gra.m. 
T h e  control prolslem is t,o bring tlie st,a.te to t,lle origin. To  malte i t  well posed, t,he fea.sible region for 
illust coiltaiil ,u(k + ilk) = 0,  i = 0, 1 ,  . . . ,172 - 1,  a.s a n  i ~ i t e r i o r  point. T h e  feasible rea,gioil for 
Gz(k  + ilk) 5 g + ~ ( k )  i = 0 , 1 , .  . .,a 
€ ( k )  = 0 
contains z (k  + ilk) = 0, ,i = 0, 1, . . . , m, as a.11 i . w t e ~ i o r  point. Note tha t  this implies g > 0. Then we have the  
followiilg tlleorein which extends the  results in [2] for ~ ( k )  = 0 V k 2 0. 
Theoreill 1 T h e  closed-loop systena w i t h  C o ~ l t r o l l e r  MPC i s  globally asymptot ical ly  stable i f  and oldy  i f  the 
opt im. izat ion problem (3) i s  feasible. 
Proof: If t he  optimiza.tion problem is not feasible, the coiltroller is not defiaed. Fea,sibility of the  optiinization 
problem iinplies t11a.t J1 is finite. At sa.inpling t ime k + 1 ,  let* 
u*(k + ilk + 1) = u(k + ilk) i = 1 , 2 , .  . . ,  nz 
€*(k + 1) = t (k) 
Thus ,  (u*, e*)  is a fea.silsle solut8ion but  inay not be optima.1. Ale ha.ve 
which yields 
Note tha,t we repla.cec1 :c(k + I lk)  with n:(k + 1) since z ( k  + 1) = n:(k + I lk).  This toget,l.ler with R, S > 0 
implies tha.t z (k)  --+ 0 and ,rr(k) - 0 a.s k - cm. 
Reinark 1 T11,is theore171 ul.so holds i f  S > 0 PI-ovided tha t  at steady s ta te  x = 0 i f  a~ad olaly i f  u = 0.  B y  
a l l o w i ~ t g  S = 0 ,  7cie can inti-odtice integral co~ztrol  t o  ob ta i~z  q f se t - f ree  tracking.  
Reinark 2 I f  Q = oo, t h e n  t h e  ovtp7rt c o ~ i s t r a i ~ i t s  becon.le hard and the  optinxization problenz m a y  no t  be 
feasible. 
T h e  following theoren1 states tlra,t for Q < m fea.sibility of the  optiiniza~tion pro11len-1 (3) is gua.ra.nteed 
for stable s~lst~eins. 
Theore111 2 If  A i s  s fah le ,  i.e. all eigenvnl7ie.s of A (ire stl-ictly i11.side the  7i11.if circle , tke12. the  o p t i ~ i ~ . i z a t i o ~ ~ .  
pro0lem (3) i s  feasiOle 'd 177. 2 1 n ~ ~ d  V Q < m. 
Proof: All we ha.ve to  clo is t,o prove the  feasibility of the optiinizatioil problem a.t the first sampling time. We 
will prove this tlleorenl by construct,ion. Since A is stahle, x(k) is bounded V k 2 0 for any initia.1 condition. 
Then 
,u* ( i / l )  = 0 i =  1 , 2 , . . , , 1 n  
~ * ( l )  = 111a.x /Gz(i11)lm < m 
i>l 
satisfies all tlie constra.ints aad results in J1  < a. Thus i t  is a feasible solutSion. 
In light of results by Tsiruliis and Rlora.ri [ti], Bala.krishna.11 et a1 [I], a.nd Zheng and Mora.ri [ S ] ,  we ca.n 
show tha t  Theorein 2 also holds for sta,biliza.ble systems with poles in the closed unit disk provided that  m 
is sufficiently la,rge. This is sta,t,ed in the  follo\ving theorem. 
Theorein 3 A s s u m e  tha t  {A, B) i s  stabilizable a18d tha t  all eigelevalt~es of .4 are i n  t h e  closed u n i t  disk. 
T h e n  for a s u f i c i e ~ z t l y  lctrge btit j%rite ,rialt~e of 171 t he  op t imiza t ion  problem (3) i s  feasible V Q < m. 
Proof: See, for example, [ 5 ] .  
We have shown, t,ha.t \i,ith ,117. properly chosen, Controller M P C  globa.lly a.sympt~otica1ly stabilizes a n y  
constra.ined st,a~biliza.ble syst,em wit<h poles in t,lle closed unit disk, using state feedba.ck. IVhen the  inputs 
a.re constra.ined, i .  e. tr""" < u(k) 5 ern'"" b' k, Sontag [4] showed that  there does n o t  exist a coiltroller 
tha t  globa.lly sta.bilizes ally systein wit,h poles outside the unit circle.' Thus,  the  XiIPC controller globally 
stabilizes nll constra,ined systems for which a global sta.biliza,tion is possible. 
Reinark 3 T l ~ e o r e m s  1, 2 and 3 hold as ,cue11 if  o ther  uornzs  for  s o f t e n i ~ ~ g  t h e  o,c~tp,ut collstraints are used. 
3 Output Feedback 
I11 the  previous section, we a.ssumed tha t  the state is mea.sured. Since the  closed loop systein inay be 
non1inea.r beca.use of the  const,ra.ints, we ca,nilot apply the Sepa.ratioi1 Principle to  prove stability for the  
outsput feeclba.ck ca.se. Denot,e t,lle st,a.t,e (output,) a t  sa,mpling t,iine X: + i estfima.tsecl a.t sa.inplii~g t4iine k by 
i ( k  + ilk) ($(k + i lk)) .  We replace the sta.te z in the objective function (2) by i ts  estima.te 2 .  T h e  state is 
estimated a.s f o l l o ~ ~ s .  
i ( k l k )  = A?(k - 1 / k  - I )  + Bu(k  - 1) + L(y(k) - jj(klk - 1)) 
i ( k  + i lk)  = . h ( k  + i  - I lk)  + Bu(k  + i  - 1) i 2 1 (4) 
where L is the  observer gain. C~oml~ining this ecluation with equation (1) yields 
where e(k) = z (k )  - ~ ( k l k ) .  Thus equation (4) can be written as 
which yields 
t(k1k) = LCAe(k - 1) 
( ( k  + ilk) = AE(k + i - I lk)  i > 1 
where [(k + ilk) = x(k + ilk) - x(k + ilk - 1).  Then we have the  following lemina 
Leinma 1 Assanae tha t  A and ( I  - L C ) A  are sta~ble, i.e. all t h e  e igenva l t~es  are s tr ic t ly  a~eside t h e  u n i t  
circle. De f ine  
Wit,h const,raints on At': \ve can also sho\v that t,liere does not, esist a controller which globally stabilizes an unstable system. 
4 
where 0 < R, & < no. 
Proof: Froin equations (5) and (7), tile lmve 
and 
1[(k + ilk)[:! 5 5 ~ ~ ~ ~ i ~ ~ ~ k ~ ~ - ~ ~ ~ ~ ~ l e ( ~ ) l ~  
where p l  = X,,,((I - LC)A) ancl pz = X,,,,,(A); c l  a a d  cz are consta.nt; a1 and a? are the  multiplicies 
associa.ted with the  la.rgest eigeilva.lues ' of ( I -  LC)A and A, respectively. Here X,,,,(A) denotes the spectral 
ra.dius of A. Stability of A and (I - LC)A iinplies t l n t  p l ,  pa < 1. Thus,  
T h e  other two expressioils can be proven simila.rly. 
Reillark 4 If A i s  urlslal~le o r  has  poles o n  t1l.e un i t  circle, Lenanaa 1 clearly does n o t  I~old .  
T h e  followiilg theoreill states t,llat glol3al asgillptotic stability ~ v i t h  output feedback call be guarailteed for 
stable syst,eins. 
Theorein 4 A s s u m e  lhn t  A a,nrl ( I -  LC)A a,re sta,ble, i.e. (dl eigenvrrltres of A a~ad  ( I -  LC)A are s tr ic t ly  in- 
side tlre 7rnit circle. Tlzen the  over t~ l l  systena wi th  Con t ro l l e r  M P C  and o b s e ~ v e r  (A)  i s  globally nsynaptotically 
stable. 
Proof: Denote tohe vrieiglited 2-i~orin d n  13y 12IR2. Let 
Thus ,  (u*, E * )  is a, feasible solut,ioil but  111a,j7 not be opt,iiua,l. Define 
a ( k )  = 1 4 k  + 11k)1i2 + le~(k)I& + l A ~ ( k ) l ; ~  
We have 
2 T l ~ e  largest eigenvalue is clefinecl t,c> Le the eigenvalue with the largest absolute value. 
5 
Ta,kilig squa.re root both sides yields 
which i a  turn  yields 
&< 
By Lelnina 1, the  secoild term on the  right-hand-side is bouiided for all k .  Therefore, we 1ia;ve 
J k  < J n a a B  < W  v b > O  
Froin before, we have 
which yields 
By Lemma. 1 and boundness of Jn'"", t,he sec,oad t,erin is bounded for a,ll k .  Thus,  
Following a, simi1a.r a.rgument. as in tjhe proof of Theorem 1, we can therefore conclude t,lla.t z ( k )  + 0 a.nd 
,u(k)  i 0 as b  cxl. 
Tile follo~ving theorern sho~vs  t1la.t the output const,ra.ints over the  i12,finite horizoil call be repla.ced by the 
output  constraillt,s over the , f i l~ i t ,~  horiz011. A silnilar result wa.s clerivecl by Rawlings a.nd R/luslie [2]. 
Theorein 5 4sszrin.e that A is .st~.61e. Givela a,ny z (k1k)  and t ( b )  > 0, there exists a fi~aite N such that 
G i ( b  + i l k )  5 g + r ( k )  V i > M 
Proof: We need o~r ly  prove this theorem for ~ ( k )  = 0: since ~ ( k )  2 0 'd k ,  G?(k + i l k )  5 g 'd i  2 N implies 
G 2 ( k  + i l k )  5 g + ~ ( k )  'd i 2 N. MILOG, assuine t,hat A is i ~ o i ~ s i n ~ u l a r . ~  Coilsider a zero input, 2.e. 
u ( k  + i l k )  = 0, i = 0 ,  . . . ,171 - 1, and cleilote the value of the objective fulictioil for this  iilput sequence by 
J;  . Theil, 
05 
Jk 5 Jt = . k ( k ~ k ) ~  C ( A ~ ) W A ~ E ( ~ ~ ~ )  z j . ( k J k ) T l 1 2 ( k J k )  
? = I  
where IT is positive definite and bounded since A is nonsingu1a.r and  sttable. Also nJe ha.ve 
05 117 - 1 
jk = e j k  + i k ) ' ~ i ( k  + i l k )  + [a,(k + i l k ) T . ~ u ( k  + i l k )  + A u ( k  + ~ I ~ - ) ~ P A U ( I L -  + i l k ) ]  + r ( ~ ) ~ ~ r ( k )  
2 i ( k  + i ~ k ) ~ R i ( k  + i l k )  
Coinbiiiiilg these two inequa.lities, we obta,in 
which yields 
I?(k + 11?./k)1? 5 ~(n )1? ( /1"1k )1~  
where r ; ( l T )  < oo deilot,es the c~ildit~ioil  nurnber of II. Finally, 
t,heil 
G2(b + i l k )  5 g V i 2 N + 7n 
mili(gj) > 0 and st;a.bilit,y of A i r l l ~ l y  t1ia.t a. finite N exist,s. 
3 
4 Exarnple 
Consider the system 
"If A is singular, we can write .'I = T-I [ "' ] T where S1 > 0 and Zz is nilpotent. Define ( k )  = T z ( k )  and 0 5 2  
El ( k  + 1) c I .El ( k )  [ E 2 ( k +  1) ] = [ (I i 2  ] [ i2(P) ] . Tlten, after a finit.e number of sampling t,imes, Ez becornes iclent,ically zero since C2 
is nilpotent. Thus it, st~ffices t,o coiisidel- t,lle reduced system \vit,h 51 as its st,ates. 
which is obta.inec1 from t,he cont,innous-time tra.nsfer function $@ with a, sa.mpling time of 0.2. The initial 
coildition is x(0)  = 11.5 1.5IT. The output, is tonstra.ined between &I.  Since t,he system exhibits inverse 
response beha.vior, 11a.rcl output con~traiilt~s can ca.use stability problems ( [ G I ) .  To use the a.pproa,ch proposed 
in [2], the output const,ra.int at  the first sa.lnpling time lnust be ignored to ma.ke the optilni~at~ion problem 
fea.sible. We can a.lso use the a.pproac11 presented in this note ancl softsen the output constra.ints over the 
illfinite horizon. The following pa.ra.meter d u e s  are used: 
Using the a.rguinents leading to Theorem 5 one caa show t11a.t the sta,te constra,iilts will be sa.tisfied automat- 
i d l y  after 35 time steps. Thus, the output constra.ints must be enforced only over a finite horizon of length 
3 5 .  The responses for the two a.pproa,ches are depicted in Figure 1. A very la,rge overshoot is observed for 
the controller designed via, t,he a.pproa,ch proposed in [2]. 
Solid: hard constraints ([2]) 
Dashed: soft constraints (Q = I) 
-121 I I I I I I I I 
0 1 2 3 4 5 6 7 8 
Time 
5 Conclusions 
We ana,lyzed the closed loop st,a.bilit,y for a.n infinit<e horizon MPC algorithm with soft ancl hard constra.ints. 
We showed t,ha.t globa.1 st,a.bilit,y call guara.nteed for both sta.te feedba,ck and output, feeclbaclr. The on-line 
optirniza.tio11 pl.oblem ca.n be ca.st as a. ,finite cliineosiolza.1 qua.dra,tic program. 
References 
[I] V.  Ba~la1trishnan, A .  Zheng, a.ncl M.  R/Iora.ri. ~onstra.ined stabilization of discrete-time linear systems. 
I11 Proc. T/TforRshop O I L  Adva~rces ill Model-Based Predictive Control, University of Oxford, 1993. Oxford 
University Press. 
[2] J .  Rawlings and I<. R. A'luslie. The sta,bility of constra.ined receding horizon cont,rol. IEEE Transactions 
on A.uiomatic Control, 38:1512-1516, 1993. 
[3] N .  L. Ricker, T. Subrahmaaian, aad T. Sim. Ca.se st,udies of model-predictive control in pulp a,nd paper 
production. I11 T. J .  McAvoy and E. Za.firiou, editors, Model Based Process Coi~trol - Proc. of the 1988 
IFAC Workskop, Oxforcl, 1'389. Perga.mon Press. 
[4] E. Sontra,g. A11 a1gebra.i~ a.pproa,c,h to bouncled ~ont~rollability of 1inea.r systems. I~ater~antional Jolrl-na.1 of 
Control, 39:181-188, 1984. 
[5] A. Tsiruliis and hiI Rforari. Coiltroller design with actuator constraints. I11 Conf. on Deczston and 
Control, pages 2623-2628, 1992. 
[GI E. Za.firiou. Robust inodel predictive control of processes with 11a.rd constra.ints. Conap. Clhem. Engmg., 
14(4/5):359-371, 1990. 
[7] E. Za.firiou and H.-W. Chiou. Output coiistraint softening for SISO i1iodel predict-ive control. Proceedangs 
of Amel-icnn Control Coii,f., pa,ges 372-276, 1993. 
[8] A. Zheiig and A/I. Morari. Global sta.bilizat,ion of linea,r discrete-time systeins with bouiidecl controls - 
a model predictive coiltrol approa.ch. PI-oceedings of American Control Co~zf., 1994. Submitted. 


Stability of Model Predictive Control with Soft Constraints

TECHNICAL MEMORANDUM NO. CIT-CDS 93-025 
December 14,1993 
revised January 11,1994 
"Stability of Model Predictive Control 
with Soft Constraints" 
A. Zheng and M. Morari 
Control and Dynamical Systems 
California Institute of Technology 
Pasadena, CA 91125 
Stability of Model Predictive Coi-~trol with Soft Constraints 
Ales Zheng Ma.llfred Mora,ri * 
Chelllica,l Engiaeering, 210-41 
Ca.lifornia Institute of Techi~ology 
Pasa,dena,, CA 91 125 
April 26, 1994 
Abstrac t  
We derive stabilit,y condit.ions for Model Predictjive Control (MPC) with ha.rd constraints on the 
input,s and "soft" constra.int,s on the outputs for a.n infinitely long output horizon. 1% show that with 
state feedback MPC is globa.11~. a.symptot,icallp sta.bilizing if a.nd only if all the eigellvadues of the open 
loop syst,em a.re in t,he closed unit disk. With output feedba.ck t,he eigenvadnes must be strict,ly inside 
the unit circle. The on-line opt,imiza.t.ion problem defining MPC can be posed as a finite dimensional 
qna.dra.t,ic progra.111 even t,llongl~ t9he output const,rabnt,s are specified over a.n infiilit,e liorizol~. 
1 Introduction 
Many pra~t~ica.1 coiltrol problems a.re ~lolnillat~ed by constra.ints. There are genera.lly two types of constra.ints 
- illput constra,iilts a,nd output  coast,ra.ints. The  illput constra.ints a.re always present and are ilnposed by 
physica.1 liinitatioas of the  a.ctua.tors which ca.ilnot be exceeded under a.ny circuinstances. Often, i t  is also 
desirable t o  keep specific outputs  within certain liniits for rea.soas related to  plant opera.tion, e.g. safety, 
material constra.iilts, etc. I t  is usually unavoidahle t o  exceed the  output  coi~stradnt~s, a t  least temporarily, 
for example, when the  systeill is sulsjected to  ui~expected disturbances. 
Industry 11a.s ein11ra.cecl Model Predictive Control (MPC), a,lso referred to  as moving horizon control a,nd 
receding llorizoil control, a,s a, po\verful feedback stra.tegy t,o coiltrol systems with constraints. The  ba.sic 
idea behind MPC is a.s follo~vs: At sampling time k ,  711. future control moves are calculated such tha t  a n  
objective fullction over some (output,) horizon is iniiliinized subject to  constra.ints. Oilly the first one of the 
in computed coi~t~rol inoves is iinpleinent,ed. i l t  t he  next sainpling t ime, the  mea.suremeat is used t o  update 
the  sta.te estima.te a.nd the  sa.me calcula,tions a.re repeated. 
Rawlings a.nd Muske [2] sl~owied lha t  globa,l asymptotic stability of the coilstrained systeill call be guar- 
anteed by ma.lcing the horizon infinite, provided that  the optimizatiol~ problem defitliilg the  R4PC coiltroller 
is fea.sible. However, output  ~oilst~rail l ts  call 1ea.d to  a a  infea.sible optilniza.tion problel~l. They proposed to  
remove the  output coilstraints during the z i l i t~n l  portzoii of the infinite horizon to  inalte the optilllization 
problem fea.sihle. Unfort,una.t~el~v, this ca.n result in poor performance: the  viola.tioi1 of the output constraints 
*Author to whom all corrrsponcler~~:e should be adclressed: phone (818)395-4186, fax (818)568-8743, e-nail: 
MMQIA/IC.CALTECH.E;Il~.~ 
during this initial port(ioi1 of the  iilfiilit,e horizon ca.11 be very la,rge in order to  sa.tisfy the constra.ints dur- 
ing the rest. Thus, large constraint violations 1na.y be experienced, n ~ h e n  the conlputed control actions a re  
implemented. 
An a,lterna.taive wa3i t o  haadle t,he fea~ibilit~y problem is t o  relax the  infeasible sta.te constraints for t h e  
elitire horizon and to peaa,lize the  extent of the viola.tioa. This technique is referred to  a.s "constra.int 
softening" [3]. The problem is t11a.t g1oba.l stal~il i ty inay not be guara.nteed. Zafiriou a,nd Chiou [7] have 
derived some conditions for stability. However, these conditions a.re generally conserva.tive and  difficult t o  
check. 
In this note, we shorn? tha.t global a.sjrmptotic sta,bilit,y ca.11 be gua.ranteed for systems with nlixed h a d  a.nd 
soft constra.ints. Furthermore, in t,he ca.se tI1a.t the sta.te nlust be estima.ted, we show tha t  global stability is 
preserved by using an asymptotic observer. Finally, are show that  the optii~lization can be ca.st as a finite 
diinensio11a.l quadra.tic program eve11 though the output  constraiilts a.re specified over the  infinite horizon. 
2 State Feedback 
Consider the  systei11 
z ( k  + 1 )  = A z ( k )  + B u ( k )  
y ( k )  = C z ( k )  
where x ( k )  E % n z ,  n j k )  E 3?n*r and y ( k )  E !I?'". Define the  objective fiiilction as 
00 n z -  1 
@k = z ~ ( k  + i ~ k ) ~ ~ o : ( k  + i l k ) + z [ ~ ( k  + i l k ) T ~ ? n ( k  + i l k )  + A n ( k  + i l k j T p A u , ( k  + i l k ) ]  
i = l  i = O  
( 2 )  
where R > 0, S > 0 ,  P 2 0 ,  ancl 111 is finite. R, S, aad  P  a.re symmetric. ( . ) ( k  + i l k )  denotes the  va.riable (.) 
a.t sainpling time k  + i precli~t~ecl a.t. saillplii~g time k .  T h e  coiltrol a.ctions a.re genera.tec1 by Co~etrol ler  A f P C  
which is defined a,s follows. 
Defiiiitioii 1 Co~itroller MPC: .4t sainpling t ime  k ,  tI1,e colrlrol moue u ( k )  epuuls the first element u , ( k l k )  
of tlze sequence { . ~ l , ( k l k ) ,  26(k + I l k ) ,  . . . ,21,(k + 112 - I l k ) }  which is  the I ~ ~ z I ~ z I I ~ . ~ ~ ~ I -  of th.e optzmzzutzon problem 
jAu(k  + i l k ) /  5 i = 0 , 1 , . . . , 1 1 a -  1  
Zb71Z%" 5 u ( k +  i l k )  _< unzaz i x 0 , 1 , .  . . , m  - 1  
subject t,o A u ( k  + i l k )  = 0  i = ~ n , m + l , . . . , o o  
Go:(k+ i l k )  < g + e ( k )  i = 0, 1 , . . . , 0 0  
~ ( k )  > 0 
where (_: E % n ~ X 1 a ~  u~zd  Q > O i s  dingonol. 
The  input constra.intss reisresent physical limita.t,ions 011 t,he actua.tors which ca.nnot be viola.t,ed. The  output 
constra.ints a.re softelred by the sla.cli variables € ( I ; ) .  They caa be viola.tec1 temporarily, if necessa,ry. I11 the 
long term,  the  pena.lt,y tern1 ~ ( k ) ~ Q t ( k )  in t,he objective fui~ction ki~ill drive the  sla.cli va.ria.bles t o  zero. T h e  
optiiniza.tion problem (3) can be ca.st a.s a, qua.dratic pr0gra.m. 
T h e  control prolslem is t,o bring tlie st,a.te to t,lle origin. To  malte i t  well posed, t,he fea.sible region for 
illust coiltaiil ,u(k + ilk) = 0,  i = 0, 1 ,  . . . ,172 - 1,  a.s a n  i ~ i t e r i o r  point. T h e  feasible rea,gioil for 
Gz(k  + ilk) 5 g + ~ ( k )  i = 0 , 1 , .  . .,a 
€ ( k )  = 0 
contains z (k  + ilk) = 0, ,i = 0, 1, . . . , m, as a.11 i . w t e ~ i o r  point. Note tha t  this implies g > 0. Then we have the  
followiilg tlleorein which extends the  results in [2] for ~ ( k )  = 0 V k 2 0. 
Theoreill 1 T h e  closed-loop systena w i t h  C o ~ l t r o l l e r  MPC i s  globally asymptot ical ly  stable i f  and oldy  i f  the 
opt im. izat ion problem (3) i s  feasible. 
Proof: If t he  optimiza.tion problem is not feasible, the coiltroller is not defiaed. Fea,sibility of the  optiinization 
problem iinplies t11a.t J1 is finite. At sa.inpling t ime k + 1 ,  let* 
u*(k + ilk + 1) = u(k + ilk) i = 1 , 2 , .  . . ,  nz 
€*(k + 1) = t (k) 
Thus ,  (u*, e*)  is a fea.silsle solut8ion but  inay not be optima.1. Ale ha.ve 
which yields 
Note tha,t we repla.cec1 :c(k + I lk)  with n:(k + 1) since z ( k  + 1) = n:(k + I lk).  This toget,l.ler with R, S > 0 
implies tha.t z (k)  --+ 0 and ,rr(k) - 0 a.s k - cm. 
Reinark 1 T11,is theore171 ul.so holds i f  S > 0 PI-ovided tha t  at steady s ta te  x = 0 i f  a~ad olaly i f  u = 0.  B y  
a l l o w i ~ t g  S = 0 ,  7cie can inti-odtice integral co~ztrol  t o  ob ta i~z  q f se t - f ree  tracking.  
Reinark 2 I f  Q = oo, t h e n  t h e  ovtp7rt c o ~ i s t r a i ~ i t s  becon.le hard and the  optinxization problenz m a y  no t  be 
feasible. 
T h e  following theoren1 states tlra,t for Q < m fea.sibility of the  optiiniza~tion pro11len-1 (3) is gua.ra.nteed 
for stable s~lst~eins. 
Theore111 2 If  A i s  s fah le ,  i.e. all eigenvnl7ie.s of A (ire stl-ictly i11.side the  7i11.if circle , tke12. the  o p t i ~ i ~ . i z a t i o ~ ~ .  
pro0lem (3) i s  feasiOle 'd 177. 2 1 n ~ ~ d  V Q < m. 
Proof: All we ha.ve to  clo is t,o prove the  feasibility of the optiinizatioil problem a.t the first sampling time. We 
will prove this tlleorenl by construct,ion. Since A is stahle, x(k) is bounded V k 2 0 for any initia.1 condition. 
Then 
,u* ( i / l )  = 0 i =  1 , 2 , . . , , 1 n  
~ * ( l )  = 111a.x /Gz(i11)lm < m 
i>l 
satisfies all tlie constra.ints aad results in J1  < a. Thus i t  is a feasible solutSion. 
In light of results by Tsiruliis and Rlora.ri [ti], Bala.krishna.11 et a1 [I], a.nd Zheng and Mora.ri [ S ] ,  we ca.n 
show tha t  Theorein 2 also holds for sta,biliza.ble systems with poles in the closed unit disk provided that  m 
is sufficiently la,rge. This is sta,t,ed in the  follo\ving theorem. 
Theorein 3 A s s u m e  tha t  {A, B) i s  stabilizable a18d tha t  all eigelevalt~es of .4 are i n  t h e  closed u n i t  disk. 
T h e n  for a s u f i c i e ~ z t l y  lctrge btit j%rite ,rialt~e of 171 t he  op t imiza t ion  problem (3) i s  feasible V Q < m. 
Proof: See, for example, [ 5 ] .  
We have shown, t,ha.t \i,ith ,117. properly chosen, Controller M P C  globa.lly a.sympt~otica1ly stabilizes a n y  
constra.ined st,a~biliza.ble syst,em wit<h poles in t,lle closed unit disk, using state feedba.ck. IVhen the  inputs 
a.re constra.ined, i .  e. tr""" < u(k) 5 ern'"" b' k, Sontag [4] showed that  there does n o t  exist a coiltroller 
tha t  globa.lly sta.bilizes ally systein wit,h poles outside the unit circle.' Thus,  the  XiIPC controller globally 
stabilizes nll constra,ined systems for which a global sta.biliza,tion is possible. 
Reinark 3 T l ~ e o r e m s  1, 2 and 3 hold as ,cue11 if  o ther  uornzs  for  s o f t e n i ~ ~ g  t h e  o,c~tp,ut collstraints are used. 
3 Output Feedback 
I11 the  previous section, we a.ssumed tha t  the state is mea.sured. Since the  closed loop systein inay be 
non1inea.r beca.use of the  const,ra.ints, we ca,nilot apply the Sepa.ratioi1 Principle to  prove stability for the  
outsput feeclba.ck ca.se. Denot,e t,lle st,a.t,e (output,) a t  sa,mpling t,iine X: + i estfima.tsecl a.t sa.inplii~g t4iine k by 
i ( k  + ilk) ($(k + i lk)) .  We replace the sta.te z in the objective function (2) by i ts  estima.te 2 .  T h e  state is 
estimated a.s f o l l o ~ ~ s .  
i ( k l k )  = A?(k - 1 / k  - I )  + Bu(k  - 1) + L(y(k) - jj(klk - 1)) 
i ( k  + i lk)  = . h ( k  + i  - I lk)  + Bu(k  + i  - 1) i 2 1 (4) 
where L is the  observer gain. C~oml~ining this ecluation with equation (1) yields 
where e(k) = z (k )  - ~ ( k l k ) .  Thus equation (4) can be written as 
which yields 
t(k1k) = LCAe(k - 1) 
( ( k  + ilk) = AE(k + i - I lk)  i > 1 
where [(k + ilk) = x(k + ilk) - x(k + ilk - 1).  Then we have the  following lemina 
Leinma 1 Assanae tha t  A and ( I  - L C ) A  are sta~ble, i.e. all t h e  e igenva l t~es  are s tr ic t ly  a~eside t h e  u n i t  
circle. De f ine  
Wit,h const,raints on At': \ve can also sho\v that t,liere does not, esist a controller which globally stabilizes an unstable system. 
4 
where 0 < R, & < no. 
Proof: Froin equations (5) and (7), tile lmve 
and 
1[(k + ilk)[:! 5 5 ~ ~ ~ ~ i ~ ~ ~ k ~ ~ - ~ ~ ~ ~ ~ l e ( ~ ) l ~  
where p l  = X,,,((I - LC)A) ancl pz = X,,,,,(A); c l  a a d  cz are consta.nt; a1 and a? are the  multiplicies 
associa.ted with the  la.rgest eigeilva.lues ' of ( I -  LC)A and A, respectively. Here X,,,,(A) denotes the spectral 
ra.dius of A. Stability of A and (I - LC)A iinplies t l n t  p l ,  pa < 1. Thus,  
T h e  other two expressioils can be proven simila.rly. 
Reillark 4 If A i s  urlslal~le o r  has  poles o n  t1l.e un i t  circle, Lenanaa 1 clearly does n o t  I~old .  
T h e  followiilg theoreill states t,llat glol3al asgillptotic stability ~ v i t h  output feedback call be guarailteed for 
stable syst,eins. 
Theorein 4 A s s u m e  lhn t  A a,nrl ( I -  LC)A a,re sta,ble, i.e. (dl eigenvrrltres of A a~ad  ( I -  LC)A are s tr ic t ly  in- 
side tlre 7rnit circle. Tlzen the  over t~ l l  systena wi th  Con t ro l l e r  M P C  and o b s e ~ v e r  (A)  i s  globally nsynaptotically 
stable. 
Proof: Denote tohe vrieiglited 2-i~orin d n  13y 12IR2. Let 
Thus ,  (u*, E * )  is a, feasible solut,ioil but  111a,j7 not be opt,iiua,l. Define 
a ( k )  = 1 4 k  + 11k)1i2 + le~(k)I& + l A ~ ( k ) l ; ~  
We have 
2 T l ~ e  largest eigenvalue is clefinecl t,c> Le the eigenvalue with the largest absolute value. 
5 
Ta,kilig squa.re root both sides yields 
which i a  turn  yields 
&< 
By Lelnina 1, the  secoild term on the  right-hand-side is bouiided for all k .  Therefore, we 1ia;ve 
J k  < J n a a B  < W  v b > O  
Froin before, we have 
which yields 
By Lemma. 1 and boundness of Jn'"", t,he sec,oad t,erin is bounded for a,ll k .  Thus,  
Following a, simi1a.r a.rgument. as in tjhe proof of Theorem 1, we can therefore conclude t,lla.t z ( k )  + 0 a.nd 
,u(k)  i 0 as b  cxl. 
Tile follo~ving theorern sho~vs  t1la.t the output const,ra.ints over the  i12,finite horizoil call be repla.ced by the 
output  constraillt,s over the , f i l~ i t ,~  horiz011. A silnilar result wa.s clerivecl by Rawlings a.nd R/luslie [2]. 
Theorein 5 4sszrin.e that A is .st~.61e. Givela a,ny z (k1k)  and t ( b )  > 0, there exists a fi~aite N such that 
G i ( b  + i l k )  5 g + r ( k )  V i > M 
Proof: We need o~r ly  prove this theorem for ~ ( k )  = 0: since ~ ( k )  2 0 'd k ,  G?(k + i l k )  5 g 'd i  2 N implies 
G 2 ( k  + i l k )  5 g + ~ ( k )  'd i 2 N. MILOG, assuine t,hat A is i ~ o i ~ s i n ~ u l a r . ~  Coilsider a zero input, 2.e. 
u ( k  + i l k )  = 0, i = 0 ,  . . . ,171 - 1, and cleilote the value of the objective fulictioil for this  iilput sequence by 
J;  . Theil, 
05 
Jk 5 Jt = . k ( k ~ k ) ~  C ( A ~ ) W A ~ E ( ~ ~ ~ )  z j . ( k J k ) T l 1 2 ( k J k )  
? = I  
where IT is positive definite and bounded since A is nonsingu1a.r and  sttable. Also nJe ha.ve 
05 117 - 1 
jk = e j k  + i k ) ' ~ i ( k  + i l k )  + [a,(k + i l k ) T . ~ u ( k  + i l k )  + A u ( k  + ~ I ~ - ) ~ P A U ( I L -  + i l k ) ]  + r ( ~ ) ~ ~ r ( k )  
2 i ( k  + i ~ k ) ~ R i ( k  + i l k )  
Coinbiiiiilg these two inequa.lities, we obta,in 
which yields 
I?(k + 11?./k)1? 5 ~(n )1? ( /1"1k )1~  
where r ; ( l T )  < oo deilot,es the c~ildit~ioil  nurnber of II. Finally, 
t,heil 
G2(b + i l k )  5 g V i 2 N + 7n 
mili(gj) > 0 and st;a.bilit,y of A i r l l ~ l y  t1ia.t a. finite N exist,s. 
3 
4 Exarnple 
Consider the system 
"If A is singular, we can write .'I = T-I [ "' ] T where S1 > 0 and Zz is nilpotent. Define ( k )  = T z ( k )  and 
0 5 2  
El ( k  + 1) c I .El ( k )  [ E 2 ( k +  1) ] = [ (I i 2  ] [ i2(P) ] . Tlten, after a finit.e number of sampling t,imes, Ez becornes iclent,ically zero since C2 
is nilpotent. Thus it, st~ffices t,o coiisidel- t,lle reduced system \vit,h 51 as its st,ates. 
which is obta.inec1 from t,he cont,innous-time tra.nsfer function $@ with a, sa.mpling time of 0.2. The initial 
coildition is x(0)  = 11.5 1.5IT. The output, is tonstra.ined between &I.  Since t,he system exhibits inverse 
response beha.vior, 11a.rcl output con~traiilt~s can ca.use stability problems ( [ G I ) .  To use the a.pproa,ch proposed 
in [2], the output const,ra.int at  the first sa.lnpling time lnust be ignored to ma.ke the optilni~at~ion problem 
fea.sible. We can a.lso use the a.pproac11 presented in this note ancl softsen the output constra.ints over the 
illfinite horizon. The following pa.ra.meter d u e s  are used: 
Using the a.rguinents leading to Theorem 5 one caa show t11a.t the sta,te constra,iilts will be sa.tisfied automat- 
i d l y  after 35 time steps. Thus, the output constra.ints must be enforced only over a finite horizon of length 
3 5 .  The responses for the two a.pproa,ches are depicted in Figure 1. A very la,rge overshoot is observed for 
the controller designed via, t,he a.pproa,ch proposed in [2]. 
Solid: hard constraints ([2]) 
Dashed: soft constraints (Q = I) 
-121 I I I I I I I I 
0 1 2 3 4 5 6 7 8 
Time 
5 Conclusions 
We ana,lyzed the closed loop st,a.bilit,y for a.n infinit<e horizon MPC algorithm with soft ancl hard constra.ints. 
We showed t,ha.t globa.1 st,a.bilit,y call guara.nteed for both sta.te feedba,ck and output, feeclbaclr. The on-line 
optirniza.tio11 pl.oblem ca.n be ca.st as a. ,finite cliineosiolza.1 qua.dra,tic program. 
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Stability of Model Predictive Control with Soft Constraints

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