- Email: [email protected]
A Quasi-Infinite Horizon Nonlinear Predictive Control Scheme for Stable Systems: Application to a CSTR
Copyright , II :AC Advanced (ontrol"fChcl11ical Proce"es. Banf!~ Canada. 1997
A QUASI-INFINITE HORIZON NONLINEAR PREDICTIVE CONTROL SCHEME FOR STABLE SYSTEMS: APPLICATION TO A CSTR H. Chen' F. Allgower"
• Institut fur Systemdynamik und Regelungstechnik. UniveT'sitiit Stuttgart. 70550 Stuttgart , Germany email: chen @isr.uni-stuttgart.de •• Institut fur A utomatik, Eidg. Techn. Hochschule Zurich , CH-8092 Zurich , Switzerland, email: [email protected] .ethz.ch
Abstract: We introduce in this paper a nonlinear model predictive control scheme for stable systems with input constraints. Closed-loop stability is guaranteed by an appropriate choice of the finite prediction horizon, independent of the specification of the desired control performance. In addition , this control scheme is likely to allow "real time" implementation, because of its computational attractiveness. The theoretical results will be demonstrated and discussed with a CSTR control application. Keywords: Nonlinear predictive control , reactor control , constraints, stability. terminal conditions.
1. INTRODUCTION
in the cost functional , a number of suggestions have been made in the literature: A terminal equality constraint forcing the states at the end of a finite prediction horizon to be zero is introduced into the open-loop optimal control problem (Genceli and ~ikolaou. 1993; Mayne and Michalska, 1990; Rawlings and Muske , 1993) . Due to the equality constraint, this approach is however computationally extremely expensive for nonlinear systems. Thus, a terminal inequalit~· constraint that requires Illerd~' the terminal states to lie in a prescribed terminal region is suggested and a dual-mode receding horizon control scheme with a local linear statf' feedback controller inside the terminal region and a receding horizon controller outside the terminal region is proposed (Michalska and i\layne, 1993). The authors of this paper extend the idea of the terminal inequality constraint and introduce a terminal cost into the finite horizon cost functional (Chen and Allgower, 1997). The~' suggest a quasi-infinitf' horizon nonlinear MPC scheme in which the local linear feedback needs not be directly appliecl to the system , thus. eliminating the need to swit(,h between controllers.
Most reactors exhibit nonlinear dynamics , for example due to some nonlinear reaction mechanism , and/or when they are operated at or near optimal operating points that are desired for economic considerations (for example. optimal yield) (Klatt et al. , 1995). In addition , in many cases the inputs can only vary in some region, for example. due to actuator saturation. In order to achiew satisfying control performance, control approaches developed for those systems are required to be able to handle nonlinearity and constraints . Due to its ability to handle constraints and nonlinearity in an optimal way, model predictive control (:\1 PC) has become a favorable candidate for control of nonlinear systems subject to constraints. In general, model predictive control is formulated as solving on-line a finite horizon constrained openloop optimal control problem. From a theoretical point of view , a fini te horizon cost functional cannot deliver asymptotic stability (Bitmead et al. , 1990) . In order to achieve guaranteed stability, i. e.stahility independent of the choice of the design paramet.ers 529
For non linear systems. the additional inequality terminal constraint. that guarantees closed-loop stability, is implicitly nonlinear. For numerical optimizers, nonlinear constraints are Yery difficult to handle and lead. in general , to the increase of on-line computation time. In addition , it eventually gives rise to infeasibility of the optimization problem. All these make the ';real time" implementation of the suggested approaches difficult. Thus , it is usually beneficial to remove the terminal constraint from the formulation of the problem , but without violating the stability conditions. This motivates the work of this paper. We will show that an appropriate choice of the prediction horizon can guarantee the satisfaction of the terminal inequality constraint for stable systems. Thus . we can remove the unfavorable nonlinear terminal constraint from the formulation of the open-loop optimal control problem without loss of t.he guaranteed stability.
1+1',.
-
+ TT': x(t). t)11/'2
x = I(x, u).
x(t: x(t). t)
U(T) E [;, T E [t ot U(T)
(3 )
= x(t)
(·b )
+ Tc)
(4b )
= 0 , T E [t + Tc.t + Tp).
(4c)
where Tp and Tc are the finite prediction and control horizons with Tp 2: Tc: Q, Rand Pan' pusitive definite , symmetric weighting matrices. In order to distinguish clearly between the system and the system model used to predict the future "\\'ithin" the controller, we denote the internal yariables in the controller by a bar (x , u) to indicate that tlH' predicted values need not be the same as the actual yalues. Thus, xC; x(t) . t) denotes the predicted trajectory of the system model (4a) driven b)' u ( ·) It, t + Tp) -+ U. Note the initial condition in (.ta ) The system model in the controller is init.ialized by the actual , measured system states at "real" time' t. We assume that all states are measurable . In this setup, Q and R in (3) are design parameters that can be chosen freely to specif)' t.he desired rontrol performance. However . the weighting matrix P in the additional terminal penalty term is not an additional design parameter that can be chosen freely. We will refer to the additional term Ilx (t + Tp ; x(t), t)ll~ as terminal cost and to P as te1'1ninal penalty matrix. The idea to guarant.ee closedloop stability independent of the specification of the desired control performance is as follows: o\'cr a finite horizon, an optimal input profile found b~' solving the optimization problem drives the nonlinear system into a prescribed neighborhood of thc origin, called here terminal region: after that , the uncontrolled syst.em asymptotically would deca~' to the origin.
\\-'e introduce in this section a nonlinear model predictive control (l'\MPC) scheme for asymptotically stable systems described by (1)
subject to hard input constraints u(t) E U, where x E IRn and u E IRm denote the vectors of states and inputs respectively, and 1 : IRn x /Rm -+ IR n is assumed to be continuously differentiable. Without loss of generality, we also assume that 1(0,0) = O. Thus. 0 E /Rn is an equilibrium of the system. In order to ensure the feasibility of the origin, 0 E U has to hold.
The terminal penalty matrix P , the terminal region n and the prediction horizon Tp can be df'termined off-line by the following procedure: Step 1: Sol\'(> the Lyapuno\' equation (A
In the following , Ilxll denotes the 2-norm of a vector x E IRn , Ilxll? is the weighted norm defined by I lxll~ := x T Px, where P is an arbitrary Hermitian , positive definite matrix. To the general framework of I\1PC (Garda et al., 1989) we add an additional term that penalizes the states at the end of the prediction horizon to the standard finite horizon cost functional and formulate the open-loop optimal control problem at time t as follows : J!1in J (x(t) , u)
riT
subject to
2. PREDICTIVE CONTROL SCHEME FOR STABLE NONLINEAR SYSTEMS
= 1 (x(t) , u(t))
+ Il u ( T)II~)
1
+llx(t
The paper is structured as follows: Section 2 describes the problem setup of the proposed nonlinear !\lPC scheme for stable systems subject to input constraints. A procedure to determine some controller parameters is outlined. In Section 3, sufficient conditions for guaranteed closed-loop stability are given . In Section 4, we discuss the control of a realistic continuous stirred tank reactor that is operated at the point of optimal yield .
x(t)
(1Ix(T: x(t) , t)llb
J (x(t). u):= /
with
K
E
+ I\.I)T P + P
(.4
+ K.J) = -Q
(5 )
[0, -.A ma .r(.4)) to get a positiw definit!'
and symmetric P. where A := ~ (0.0) and .Amax(A) is the largest eigenvalue of A. Step 2: Find the largest possible 0 E (0. x ) sppcifying a terminal region defined by
such that the optimal value of the following optimization problem is nonpositive:
(2)
u ( )
max{xTP
x
with 530
-
n is invariant for the uncontrolled nonlinear system, (b) the infinite horizon cost of the uncontrolled system st.arting from any Xl E n is hounded above as follows:
Step 3: The prediction horizon is chosen to meet
(a)
(8)
where Ts is the time that the uncontrolled system needs to reach the terminal region n.
x
The optimal solution of (2) with (3) subject to (4) is denoted by u' (-; x (t) , t) : [t. t + Tp) -+ U. The corresponding optimal open-loop trajectory and optimal value are denoted by x' (-; x ( t) . t) and J* (t) := J(x(t). u'), respectively. According to the principle of r--IPC , the optimal input profile found is applied to the system onl:; until the next measurement becomes available. With the new measurement. the optimization problem will be solved again. \Ve assume that this will be the case every <5 time-units. So tl denotes the "sampling time" and the closed-loop control u (-) is defined by U(T) := U'(T ; x(t) , t), T E
It, t + <5).
Xi"PXI 2: / x(t)1Qx(t)dt. I,
Remark 2. l7sing the notation introduced in Section 2 for internal variables in the controller. provided that x(t + Tp; x(t) , t) E n, then . ( 11) leads t.o
= Ilx(t + Tp ; x(t). t)lIt 2: /
Substituting the above inequality into (3) , wC' call conclude that the finite horizon cost functional to be minimized bounds the infinite horizon cost defined by
(9)
QC
.I"" (x(t), u)
Ilu(T)II~) liT.
It is clear that J(x(t),u) -+ J OO( x(t),u), as TT' -+ 00. Thus , as the finite prediction horizon becomes longer, the control performance achieved with the proposed quasi-infinite horizon NMPC will become closer and closer to the one with an infinite horizon NMPC. This will also be demonstrated in the CSTR control application in Section 4. In the formulation of the optimization problem. we do not have any terminal inequality constraint that forces the terminal states to lie in the terminal n 'gion. However, ill order to show closed-loop stability. we do make use of the fact that the states at the end of the prediction horizon lie in the terminal region. The following lemma shows that an appropriate choice of the prediction horizon guarantees the terminal states to lie in the terminal region.
Lemma 2. Suppose that the uncontrolled nonlillear system is globally as~'mptotically stable and that the prediction horizon is chosen according to (8). then , for any initial condition x(O) = Xo E Ul." . the optimization problem (2) - (4) is feasible and the terminal state x' (t + Tp; x(t), t) is guaranteed to li(~ in the terminal region at each time t 2: O.
is determined
by' the procedure in Section 2, then , along any
x = !(x , O)
Proof: Due to the asymptotic stability of t he uncontrolled nonlinear system and the choice of the prediction horizon according to (8) , u(·) : [0. Tr) -+ 0 is a feasible solution to the optimization problplll at time t = 0 and x(Tp: Xo , 0) E n. r--loreoycr. WE' ha\'!' also x' (TI' ; Xo. 0 ) E n by' (8).
(10)
Pmof: See (Chen and AlIgower , 1997).
(1Ix(T: x(t), t)llb +
where U(T) = 0 for T 2: t + Tp. Thus. J (x(t), u ) 2: JOO(x(t). u). In this sense, the prediction horizon in the proposed NMPC scheme can be thought of as expanding quasi to infinity.
In order to address closed-loop stability of the proposed :,{MPC scheme. we first give some preliminary results. The following lemma shows the properties of the terminal region.
trajectory of the uncontrolled system st arting from n we have
:= / 1
3. CLOSED-LOOP STABILITY
n
Ilx (r: x(t). t)11~(fT.
1+1"
Remark 1. Since Q is positive definite and symmetric. by the general conditions for the solvability of Lyapunov equations, the solution of (5) is positive definite and symmetric for any constant K. E [0 , - Amax(.4)). Because of the freedom in the choice of K.. following the above procedure does not yield a unique terminal region for a given nonlinear system. For the sake of reducing the on-line computational burden (if n is large, Ts will be small and thus Tp will be small), we are of course interested in the largest possible terminal region, which may be found with a constant K. that is close to -Amax(.4) (Chen and Allgower, 1997) . However, a constant K. close to - Am ax (A) implies a large sol u tion P of the Lyapunov equation (5). From the structure of the cost functional , we know that a very large terminal penalty matrix P may have a bad influence on the achievement of the control performance that is specified by the weighting matrices Q and R. Thus , choosing the constant K. requires a trade-off between a large t.erminal region and good achievement of the desired control performance.
Lemma 1. If the terminal region
(11)
o
From Lemma 1. it follows directly that 531
Based on component and energy balances. the non linear model
:'\ow suppose that there exits a feasible solution to the optimization problem at time t that drives the non linear system into the terminal region. In terms of I\IPC, the closed-loop control u' (.) is given by' (9). For the nominal sy'stem without disturbances, the system state at time t + J is x(t + 6) = x· (t + b; x (t ), t). For solving the optimization problem at time t + cS, a candidate input profile u(·) OIl [t + b. t + b + Tp) may be chosen as
_
U(T)
=
{U'(T: x(t). t) 0
T
E
T
E
CA
.
V
. ih
CO~TROL
OF A CSTR AT THE OPTIMAL YIELD
1 p
=
-
00) v
pCr'R 1 ( QK . + ku·.4 (v - ih) ) R
nth'CPh-
kiO·exp (V+2~'31 5 ) , i = l. 2. 3. In this model , the concentrations of the initial reactant A and the product B in the reactor are denoted as CA and CB: the temperatures in the reactor and in the cooling jacket are represented by V and VK , respectivel:.; : the normalized flow rate ~\ to the reactor and the heat removal power QK from the cooling jacket are considered as control inputs. A more detailed description , together with values for the physical and chemical parameters in the model, are given in (Chen et al. , 1995: Klatt et al., 1995). The chosen inputs can only vary within the following bounds :
1 iT 1 kJ . I,;,] 3 - < - < 35 - -9000 - < QK <0 -.(13) h - 1/R h' It h For economic reasons, the reactor is operated at the point of maximal ~.. ield , that has some unfavorabk features (strong non linearity and unstable zero dynamics ) leading to a difficult control problem (C hen et al. , 1995; Klatt et al. , 1995 ). The new states and the new inputs are df'finpd as XI =CA -
eAls
X2= CB -
cB ls
, ' - "R "1 X4=lJK - vKls Ul="R
o POI~T
()
to describe the dynamics in the reactor. The rpaction rates are assumed to depend on the reaction temperature via the Arrhenius la,,' k ;(v ) =
then , in the absence of disturbances , the closedloop system with the model predictive control (9) is nominally asymptotically stable.
-t .
Cn ~
- v) - pC (kl (v)cA i::..H RAll
ku·A.R (oa +--.VK
f : /Rn X /Rm -+ /Rn is twice continuously differentiable , U c /Rm is compact, convex and 0 is contained in the interior of U, the system (1) has a unique solution for any initial value Xo E /Rn and any piecewise continuous u(-) : [0 ,(0) -+ U, the prediction horizon Tr is chosen to meet (8) , for x(t ) = 0 , the optimization problem (2) - (4) can be globally optimally solved,
Proof: See (Chen and Allgower, 1997).
- k~ (v) r. D .
C"
Theorem 1. Suppose that
(d) (e)
kl (v)e ..\ - k3(iJ)C A:!. Cl ~ 0
+ k 2(v)CB6.H RR + k 3 (tJ) CA 2 !:lHRA I»
In the following, we give the stability conditions of the proposed quasi-infinite horizon I\MPC scheme for any Q > 0, R > 0, and T c ~ b.
(c)
,.
= "R (vo
[t + b, t + Tp) . (12) [t + Tp, t + b + Tp)
R emark 3. The requirement of global asymptotic stability of the uncontrolled nonlinear system can he relaxed (see (Chen , n.d.)) .
(b)
,.
CA) -
eB = - "R CB + k 1 (iJ) ("A
that clearly satisfies the input constraints and thus ff'asible. Furthermore, from Lemma 1, we have x'(t + Tp:x(t) , t) E n implies x (t + b + Tp ;x(t + 6) , t + b) E n with the feasible solution (12). The result follows then by indu ction . 0
(a)
= "R " (eAO -
obtain
\\T
X3 =1? -
1715
. 112=QK
, Ch,'
I., .
s where 'Is denotes the steady state value at tlH' optimal operating poi nt. The uncontroll ed plant is asymptotically stable, therefore. we can use thp proposed quasi-infinite horizon r\l\lPC scheme. The weighting matrices in the cost fun ctional. that specify the desired control performance, are chosen as
OF
To illustrate the theoretical results we consider the control of a realistic continuous stirred tank reactor (CSTR) with cooling jacket, in which the exot herm ic Van der Vuss e reaction
° oo °0) R = (0 .5 ° - ° ° 0.5° 0 ' ° ° the terminal penalty mat.rix °determine In order to 0.2
1.0
Q-
(
A ~B~C 2A ~ D
0 .2
P and the prediction horizon Tp . wp follow s t hp procedure in Section 2. The largest. pigemalup of the Jacobi matrix of the CSTR model is -().()(J38~ Thus. any constant h E [0 , 0.0038) yirlds a positin'
t.akes place to produce cyclopentenol from cyc!opent.adiene by acid-catalyzed electrophyli c hydration in aqueous solution (Klatt et al. , 1995) . 532
definite and sy'mmetric P that can be us('d as terminal penalty matrix. We choose K. = 0.0015~ to obtain an appropriate terminal penalty matrix P
=
3278.78 1677.31 ( 681.02 271.50
1677.31 919.78 344.19 137.27
681.02 34-U9 172.45 65 .53
271.50 ) 137.27 04) 65.53 29.28
h:~~', . ~ . v~
From the simple optimization (7), wc find the corresponding terminal region
------1
o
100
200
300
400
soo
•
600
700
BOO
900
1000
h~L2:~ · .· · ----------;~-i
The terminal region is small, because the CSTR is operated at the point of optimal yield and is thus strongly non linear. The CSTR in the neighborhood of the optimal operating point has unstable zero dynamics. therefore , we choose the control horizon as Tc = 20s. We find that the uncontrolled plant is driven to the terminal region described by (15) from far awa:·; from the operating point in at most Ts = 13808. Thus. the prediction horizon is chosen as Tp = 14008 according to (8).
100
~
~~
20
o
"
'
- 20
-::: , o-=-o---:BC:-:OO--:::90:~' - --;;00
_~ ... .. ~
' , .. . ..
~~ ~-- - - -
0;
~
-4~00:---:C50:-:-0---:.-<:oo-
300
~'.
..---
\..J ...:!::. ". §";;:
~
200
•
__ _ _ -
0~-'~ 00:--~200 ---3~ 00:--~400 ---:5C:-: OO--::: .o-:o ---:'~ 00-C:-: BO-:-O---:9~ O O~'00C
'~'1
.~~ ::n"I--' ~
The optimization problem (2) - (4) is solved in discrete time with a sampling period of 6 = 208. Time profiles for the closed-loop system emanating from some different initial conditions are shown in Fig. 1 with different line ty·pes. Note that large deviations in the initial conditions from the steady state values are chosen. We clearly see that the input constraints (13) are not violated.
-I...c:::
,
1 ; \) -,of 0
x
1~3
'\ " .. ':':
'r ;'"
1
' ~ .... ' .
100
.:= .. .,. := 1
200
300
400
500
600
700
800
900
lODe
500
600
700
8 00
900
, 000
1~~fr= l~ ,: ~
It should be emphasized that the optimization problem does not include a terminal inequality constraint. The terminal region described by (15) is only used to determine Ts.
o
, 00
200
3 00
40 0
Time [s]
Fig. 1. Profiles for the CSTR cont.rolled b~' the proposed non linear predictive controller. Different line types correspond to different initial conditions.
The computational burden is a very important criterion for "real time" implementation . Since there is no nonlinear constraint in the optimization problem and the control horizon can be chosen considerably shorter than the prediction horizon, the proposed quasi-infinite horizon T\l\lPC scheme has advantages from a computational point of view . To show this, we compare two controllers: Controller A is designed with the proposed ~r\'lPC scheme. while controller B is based on the control scheme suggested in (Chen and Allgower, 1997) that involves a terminal inequality constraint . With bot.h controllers the closed-loop system has guaranteed stability.
implementation of the proposed quasi-infinite horizon NMPC scheme with guaranteed closed-loop stability is possible. Table 1. Comparison of elapsed
IdO ) -:2.14 2.96 - 2.14 296
A : P given by (14) , no terminal inequality con-
Initial state J'4(0 ) I2 (0) Il(0 ) -1.09 -29.20 - 27 .91 20.0 20.0 l.41 20 .0 1.41 -29.20 20.0 - 27.91 -1.09
cpe
time
j
Elapsed (,Pt· timp (s i Contr . A Cont r . B (jOOfi .H7 ,5 02:12 .',·Bl .:'!(j .:;02 .0:2 ()OO .,):.!7 500.69 ',6:20:.!!) ';25 64
The terminal cost Ilx(t+ TI'; x(t). t)l l ~ is introducpd to achieve closed-loop stability and has also some "performance meaning" in the sense that it is linked to the infinite horizon cost. In order to show the influence of the terminal cost term on the achievement of the desired control performance. \\'p compare controller A with different prediction hOlizons Tp = 500s , TI' = 700s and Tp = 900s to a controlln C with "infinite" (i. c .. IIcry long) prediction horizon that "guarant.ees" also closed-loop stabilit.\,:
straint, Tc = 208 , Tp = 14008, B : P given by (14). terminal inequality constraint x(t + Tp: x(t), t) E l! with l! given by (15) , Tc = Tp = 440s. For a total simulation time of 2000 seconds. the elapsed CPU times for some arbitrarily chosen initial st.ates are shown in Table 1. It is clear that controller A needs significantly less cpe time than controller B. For the CSTR model. the "real time" 533
C : P
TI )
=
0, no terminal constraint. Te
= 2800s ,
=
Since we need not add any terminal equality or inequality constraints to the optimization problem in order to guarantee closed-loop stability allcl a yer~' short control horizon (for example ol1e step) can be chosen, the on-line computational burdpII can be significantly reduced. Thus, "real t.ime" implementability of the proposed quasi-infinite horizon Nt..IPC is much more likely when compared to other Nt..IPC schemes including the Olle in (Chen and Allgower, 1997). This was also shown in simulation for the CSTR control problem.
20s
Time profiles of the closed-loop CSTR are shown in Fig 2. The initial condition of XI (0) = -0.64 "'tl. :r3(0) = O.07°C, :r2(0) = 0.22 "~Ol , and X4(0) = 0.39 QC is outside the terminal region (15). But from this initial state the uncontrolled plant needs only' 7" = 460s to reach the terminal region (15). It
The achieved control performance does not differ Significantly from the ol1e achieved hy an :\'!\IPC with infinite prediction horizon. The rea..<;on is that the choice of prediction horizon according to the st.ability condition is very conservative and we haw an additional terminal cost in the cost functional. that. bounds the infinite horizon cost of the uncont.rolled system in a prescribed terminal region. Thus. thp prediction horizon in the proposed N:\lPC schenlf' is quasi infinite.
6. REFERENCES Bitmead, R .R., rvl. Gevers and Y. Wertz (1990 ). Adaptive Optimal Control - The Thinking Man's GPe. Prentice Hall. New York . Chen , H. (n.d .). Stability and Robustness Considerations in Nonlinear Model Predictive Control. Dissertation, Universitiit Stuttgart. Chen, H., A. Kremling and F. Allgower (1995). Nonlinear predictive control of a benchmark CSTR. In: Proc. 3rd European Control Confer'ence ECC'9S. Rome. pp. 3247- 3252. Chen , H. and F. Allgower (1997). A quasi-infinit.e horizon nonlinear model predictive cont.rol scheme with guaranteed stabilit~;. accepted for Automatica. Garda, C.E., D.:\1. Prett and M. l\-Iorari (1989). Model Predictive Control: Theor~' and pract.ice - A survey. Automatica 25. 335- 347. Genceli, H. and M. Nikolaou (1993) . Robust stability analysis of constrained L I-norm model predictive control. AIChE 1. 39(12). 19541965. Klatt, K.-C ., S. Engell, A. Kremling and F . Allgower (1995). Testbeispiel: Riihrkessdrpaktor mit Parallel- und Folgereaktion. In: Entwurf Nichtlinearer Rege/ungen (S. Engell, Ed.). pp. 425- 432. Oldenbourg Yerlag. l\-1iinchen. !\layne, D.Q. and H. Michalska (1990). Receding horizon control of non linear s~·stems. IEEE Tmns. Automat. Contr. AC-35(7) , 814 -824. I\lichalska, H. and D.Q. Mayne (1993). Robust receding horizon control of constrained Ilonlinear systems. IEEE Tmns. Automat. Contr. AC38(11) , 1623- 1633. Rawlings, J.B. and K.R . I\Iuske (1993). The stability of constrained receding horizon control. IEEE Tmns. Automat. Coutr. AC-38(1O ). 1512 1516.
Time [s]
Fig. 2. Profiles for the CSTR model controlled by controller C (-) and controller A with Tp = 500s (--), Tp = 700s (_. -), Tp = 900s (- .. ). is clearly seen that the t.rajectories of the CSTR model controlled by controller A are getting closer to that by the "infinite" horizon controller C, as the prediction horizon in controller A grows. In general, the difference of the achieved control performance between the quasi-infinite horizon Nl\IPC and the infinite horizon NMPC will be not big. Indeed , the choice of the prediction horizon according to (8) is wry conservative, since the uncontrolled system is in general slower than the controlled system.
5. CONCLL'SIO:"J"S For asymptotically stable systems subject to input constraints, a quasi-infinite horizon NMPC scheme with guaranteed closed-loop stability was proposed. The open-loop optimal control problem is formulated as minimizing a finite horizon cost plus a terminal cost subject to non linear system dynamics and input constraints in which the inputs are set to lw zero beyond the control horizon Te. An appropriate choice of the finite prediction horizon guarantees dosed-loop stability for any Q > 0, R > 0 and Te 2 J. Thus , we can choose Q and R freely to specify the desired control performance. 534
A QUASI-INFINITE HORIZON NONLINEAR PREDICTIVE CONTROL SCHEME FOR STABLE SYSTEMS: APPLICATION TO A CSTR H. Chen' F. Allgower"
• Institut fur Systemdynamik und Regelungstechnik. UniveT'sitiit Stuttgart. 70550 Stuttgart , Germany email: chen @isr.uni-stuttgart.de •• Institut fur A utomatik, Eidg. Techn. Hochschule Zurich , CH-8092 Zurich , Switzerland, email: [email protected] .ethz.ch
Abstract: We introduce in this paper a nonlinear model predictive control scheme for stable systems with input constraints. Closed-loop stability is guaranteed by an appropriate choice of the finite prediction horizon, independent of the specification of the desired control performance. In addition , this control scheme is likely to allow "real time" implementation, because of its computational attractiveness. The theoretical results will be demonstrated and discussed with a CSTR control application. Keywords: Nonlinear predictive control , reactor control , constraints, stability. terminal conditions.
1. INTRODUCTION
in the cost functional , a number of suggestions have been made in the literature: A terminal equality constraint forcing the states at the end of a finite prediction horizon to be zero is introduced into the open-loop optimal control problem (Genceli and ~ikolaou. 1993; Mayne and Michalska, 1990; Rawlings and Muske , 1993) . Due to the equality constraint, this approach is however computationally extremely expensive for nonlinear systems. Thus, a terminal inequalit~· constraint that requires Illerd~' the terminal states to lie in a prescribed terminal region is suggested and a dual-mode receding horizon control scheme with a local linear statf' feedback controller inside the terminal region and a receding horizon controller outside the terminal region is proposed (Michalska and i\layne, 1993). The authors of this paper extend the idea of the terminal inequality constraint and introduce a terminal cost into the finite horizon cost functional (Chen and Allgower, 1997). The~' suggest a quasi-infinitf' horizon nonlinear MPC scheme in which the local linear feedback needs not be directly appliecl to the system , thus. eliminating the need to swit(,h between controllers.
Most reactors exhibit nonlinear dynamics , for example due to some nonlinear reaction mechanism , and/or when they are operated at or near optimal operating points that are desired for economic considerations (for example. optimal yield) (Klatt et al. , 1995). In addition , in many cases the inputs can only vary in some region, for example. due to actuator saturation. In order to achiew satisfying control performance, control approaches developed for those systems are required to be able to handle nonlinearity and constraints . Due to its ability to handle constraints and nonlinearity in an optimal way, model predictive control (:\1 PC) has become a favorable candidate for control of nonlinear systems subject to constraints. In general, model predictive control is formulated as solving on-line a finite horizon constrained openloop optimal control problem. From a theoretical point of view , a fini te horizon cost functional cannot deliver asymptotic stability (Bitmead et al. , 1990) . In order to achieve guaranteed stability, i. e.stahility independent of the choice of the design paramet.ers 529
For non linear systems. the additional inequality terminal constraint. that guarantees closed-loop stability, is implicitly nonlinear. For numerical optimizers, nonlinear constraints are Yery difficult to handle and lead. in general , to the increase of on-line computation time. In addition , it eventually gives rise to infeasibility of the optimization problem. All these make the ';real time" implementation of the suggested approaches difficult. Thus , it is usually beneficial to remove the terminal constraint from the formulation of the problem , but without violating the stability conditions. This motivates the work of this paper. We will show that an appropriate choice of the prediction horizon can guarantee the satisfaction of the terminal inequality constraint for stable systems. Thus . we can remove the unfavorable nonlinear terminal constraint from the formulation of the open-loop optimal control problem without loss of t.he guaranteed stability.
1+1',.
-
+ TT': x(t). t)11/'2
x = I(x, u).
x(t: x(t). t)
U(T) E [;, T E [t ot U(T)
(3 )
= x(t)
(·b )
+ Tc)
(4b )
= 0 , T E [t + Tc.t + Tp).
(4c)
where Tp and Tc are the finite prediction and control horizons with Tp 2: Tc: Q, Rand Pan' pusitive definite , symmetric weighting matrices. In order to distinguish clearly between the system and the system model used to predict the future "\\'ithin" the controller, we denote the internal yariables in the controller by a bar (x , u) to indicate that tlH' predicted values need not be the same as the actual yalues. Thus, xC; x(t) . t) denotes the predicted trajectory of the system model (4a) driven b)' u ( ·) It, t + Tp) -+ U. Note the initial condition in (.ta ) The system model in the controller is init.ialized by the actual , measured system states at "real" time' t. We assume that all states are measurable . In this setup, Q and R in (3) are design parameters that can be chosen freely to specif)' t.he desired rontrol performance. However . the weighting matrix P in the additional terminal penalty term is not an additional design parameter that can be chosen freely. We will refer to the additional term Ilx (t + Tp ; x(t), t)ll~ as terminal cost and to P as te1'1ninal penalty matrix. The idea to guarant.ee closedloop stability independent of the specification of the desired control performance is as follows: o\'cr a finite horizon, an optimal input profile found b~' solving the optimization problem drives the nonlinear system into a prescribed neighborhood of thc origin, called here terminal region: after that , the uncontrolled syst.em asymptotically would deca~' to the origin.
\\-'e introduce in this section a nonlinear model predictive control (l'\MPC) scheme for asymptotically stable systems described by (1)
subject to hard input constraints u(t) E U, where x E IRn and u E IRm denote the vectors of states and inputs respectively, and 1 : IRn x /Rm -+ IR n is assumed to be continuously differentiable. Without loss of generality, we also assume that 1(0,0) = O. Thus. 0 E /Rn is an equilibrium of the system. In order to ensure the feasibility of the origin, 0 E U has to hold.
The terminal penalty matrix P , the terminal region n and the prediction horizon Tp can be df'termined off-line by the following procedure: Step 1: Sol\'(> the Lyapuno\' equation (A
In the following , Ilxll denotes the 2-norm of a vector x E IRn , Ilxll? is the weighted norm defined by I lxll~ := x T Px, where P is an arbitrary Hermitian , positive definite matrix. To the general framework of I\1PC (Garda et al., 1989) we add an additional term that penalizes the states at the end of the prediction horizon to the standard finite horizon cost functional and formulate the open-loop optimal control problem at time t as follows : J!1in J (x(t) , u)
riT
subject to
2. PREDICTIVE CONTROL SCHEME FOR STABLE NONLINEAR SYSTEMS
= 1 (x(t) , u(t))
+ Il u ( T)II~)
1
+llx(t
The paper is structured as follows: Section 2 describes the problem setup of the proposed nonlinear !\lPC scheme for stable systems subject to input constraints. A procedure to determine some controller parameters is outlined. In Section 3, sufficient conditions for guaranteed closed-loop stability are given . In Section 4, we discuss the control of a realistic continuous stirred tank reactor that is operated at the point of optimal yield .
x(t)
(1Ix(T: x(t) , t)llb
J (x(t). u):= /
with
K
E
+ I\.I)T P + P
(.4
+ K.J) = -Q
(5 )
[0, -.A ma .r(.4)) to get a positiw definit!'
and symmetric P. where A := ~ (0.0) and .Amax(A) is the largest eigenvalue of A. Step 2: Find the largest possible 0 E (0. x ) sppcifying a terminal region defined by
such that the optimal value of the following optimization problem is nonpositive:
(2)
u ( )
max{xTP
x
with 530
-
n is invariant for the uncontrolled nonlinear system, (b) the infinite horizon cost of the uncontrolled system st.arting from any Xl E n is hounded above as follows:
Step 3: The prediction horizon is chosen to meet
(a)
(8)
where Ts is the time that the uncontrolled system needs to reach the terminal region n.
x
The optimal solution of (2) with (3) subject to (4) is denoted by u' (-; x (t) , t) : [t. t + Tp) -+ U. The corresponding optimal open-loop trajectory and optimal value are denoted by x' (-; x ( t) . t) and J* (t) := J(x(t). u'), respectively. According to the principle of r--IPC , the optimal input profile found is applied to the system onl:; until the next measurement becomes available. With the new measurement. the optimization problem will be solved again. \Ve assume that this will be the case every <5 time-units. So tl denotes the "sampling time" and the closed-loop control u (-) is defined by U(T) := U'(T ; x(t) , t), T E
It, t + <5).
Xi"PXI 2: / x(t)1Qx(t)dt. I,
Remark 2. l7sing the notation introduced in Section 2 for internal variables in the controller. provided that x(t + Tp; x(t) , t) E n, then . ( 11) leads t.o
= Ilx(t + Tp ; x(t). t)lIt 2: /
Substituting the above inequality into (3) , wC' call conclude that the finite horizon cost functional to be minimized bounds the infinite horizon cost defined by
(9)
QC
.I"" (x(t), u)
Ilu(T)II~) liT.
It is clear that J(x(t),u) -+ J OO( x(t),u), as TT' -+ 00. Thus , as the finite prediction horizon becomes longer, the control performance achieved with the proposed quasi-infinite horizon NMPC will become closer and closer to the one with an infinite horizon NMPC. This will also be demonstrated in the CSTR control application in Section 4. In the formulation of the optimization problem. we do not have any terminal inequality constraint that forces the terminal states to lie in the terminal n 'gion. However, ill order to show closed-loop stability. we do make use of the fact that the states at the end of the prediction horizon lie in the terminal region. The following lemma shows that an appropriate choice of the prediction horizon guarantees the terminal states to lie in the terminal region.
Lemma 2. Suppose that the uncontrolled nonlillear system is globally as~'mptotically stable and that the prediction horizon is chosen according to (8). then , for any initial condition x(O) = Xo E Ul." . the optimization problem (2) - (4) is feasible and the terminal state x' (t + Tp; x(t), t) is guaranteed to li(~ in the terminal region at each time t 2: O.
is determined
by' the procedure in Section 2, then , along any
x = !(x , O)
Proof: Due to the asymptotic stability of t he uncontrolled nonlinear system and the choice of the prediction horizon according to (8) , u(·) : [0. Tr) -+ 0 is a feasible solution to the optimization problplll at time t = 0 and x(Tp: Xo , 0) E n. r--loreoycr. WE' ha\'!' also x' (TI' ; Xo. 0 ) E n by' (8).
(10)
Pmof: See (Chen and AlIgower , 1997).
(1Ix(T: x(t), t)llb +
where U(T) = 0 for T 2: t + Tp. Thus. J (x(t), u ) 2: JOO(x(t). u). In this sense, the prediction horizon in the proposed NMPC scheme can be thought of as expanding quasi to infinity.
In order to address closed-loop stability of the proposed :,{MPC scheme. we first give some preliminary results. The following lemma shows the properties of the terminal region.
trajectory of the uncontrolled system st arting from n we have
:= / 1
3. CLOSED-LOOP STABILITY
n
Ilx (r: x(t). t)11~(fT.
1+1"
Remark 1. Since Q is positive definite and symmetric. by the general conditions for the solvability of Lyapunov equations, the solution of (5) is positive definite and symmetric for any constant K. E [0 , - Amax(.4)). Because of the freedom in the choice of K.. following the above procedure does not yield a unique terminal region for a given nonlinear system. For the sake of reducing the on-line computational burden (if n is large, Ts will be small and thus Tp will be small), we are of course interested in the largest possible terminal region, which may be found with a constant K. that is close to -Amax(.4) (Chen and Allgower, 1997) . However, a constant K. close to - Am ax (A) implies a large sol u tion P of the Lyapunov equation (5). From the structure of the cost functional , we know that a very large terminal penalty matrix P may have a bad influence on the achievement of the control performance that is specified by the weighting matrices Q and R. Thus , choosing the constant K. requires a trade-off between a large t.erminal region and good achievement of the desired control performance.
Lemma 1. If the terminal region
(11)
o
From Lemma 1. it follows directly that 531
Based on component and energy balances. the non linear model
:'\ow suppose that there exits a feasible solution to the optimization problem at time t that drives the non linear system into the terminal region. In terms of I\IPC, the closed-loop control u' (.) is given by' (9). For the nominal sy'stem without disturbances, the system state at time t + J is x(t + 6) = x· (t + b; x (t ), t). For solving the optimization problem at time t + cS, a candidate input profile u(·) OIl [t + b. t + b + Tp) may be chosen as
_
U(T)
=
{U'(T: x(t). t) 0
T
E
T
E
CA
.
V
. ih
CO~TROL
OF A CSTR AT THE OPTIMAL YIELD
1 p
=
-
00) v
pCr'R 1 ( QK . + ku·.4 (v - ih) ) R
nth'CPh-
kiO·exp (V+2~'31 5 ) , i = l. 2. 3. In this model , the concentrations of the initial reactant A and the product B in the reactor are denoted as CA and CB: the temperatures in the reactor and in the cooling jacket are represented by V and VK , respectivel:.; : the normalized flow rate ~\ to the reactor and the heat removal power QK from the cooling jacket are considered as control inputs. A more detailed description , together with values for the physical and chemical parameters in the model, are given in (Chen et al. , 1995: Klatt et al., 1995). The chosen inputs can only vary within the following bounds :
1 iT 1 kJ . I,;,] 3 - < - < 35 - -9000 - < QK <0 -.(13) h - 1/R h' It h For economic reasons, the reactor is operated at the point of maximal ~.. ield , that has some unfavorabk features (strong non linearity and unstable zero dynamics ) leading to a difficult control problem (C hen et al. , 1995; Klatt et al. , 1995 ). The new states and the new inputs are df'finpd as XI =CA -
eAls
X2= CB -
cB ls
, ' - "R "1 X4=lJK - vKls Ul="R
o POI~T
()
to describe the dynamics in the reactor. The rpaction rates are assumed to depend on the reaction temperature via the Arrhenius la,,' k ;(v ) =
then , in the absence of disturbances , the closedloop system with the model predictive control (9) is nominally asymptotically stable.
-t .
Cn ~
- v) - pC (kl (v)cA i::..H RAll
ku·A.R (oa +--.VK
f : /Rn X /Rm -+ /Rn is twice continuously differentiable , U c /Rm is compact, convex and 0 is contained in the interior of U, the system (1) has a unique solution for any initial value Xo E /Rn and any piecewise continuous u(-) : [0 ,(0) -+ U, the prediction horizon Tr is chosen to meet (8) , for x(t ) = 0 , the optimization problem (2) - (4) can be globally optimally solved,
Proof: See (Chen and Allgower, 1997).
- k~ (v) r. D .
C"
Theorem 1. Suppose that
(d) (e)
kl (v)e ..\ - k3(iJ)C A:!. Cl ~ 0
+ k 2(v)CB6.H RR + k 3 (tJ) CA 2 !:lHRA I»
In the following, we give the stability conditions of the proposed quasi-infinite horizon I\MPC scheme for any Q > 0, R > 0, and T c ~ b.
(c)
,.
= "R (vo
[t + b, t + Tp) . (12) [t + Tp, t + b + Tp)
R emark 3. The requirement of global asymptotic stability of the uncontrolled nonlinear system can he relaxed (see (Chen , n.d.)) .
(b)
,.
CA) -
eB = - "R CB + k 1 (iJ) ("A
that clearly satisfies the input constraints and thus ff'asible. Furthermore, from Lemma 1, we have x'(t + Tp:x(t) , t) E n implies x (t + b + Tp ;x(t + 6) , t + b) E n with the feasible solution (12). The result follows then by indu ction . 0
(a)
= "R " (eAO -
obtain
\\T
X3 =1? -
1715
. 112=QK
, Ch,'
I., .
s where 'Is denotes the steady state value at tlH' optimal operating poi nt. The uncontroll ed plant is asymptotically stable, therefore. we can use thp proposed quasi-infinite horizon r\l\lPC scheme. The weighting matrices in the cost fun ctional. that specify the desired control performance, are chosen as
OF
To illustrate the theoretical results we consider the control of a realistic continuous stirred tank reactor (CSTR) with cooling jacket, in which the exot herm ic Van der Vuss e reaction
° oo °0) R = (0 .5 ° - ° ° 0.5° 0 ' ° ° the terminal penalty mat.rix °determine In order to 0.2
1.0
Q-
(
A ~B~C 2A ~ D
0 .2
P and the prediction horizon Tp . wp follow s t hp procedure in Section 2. The largest. pigemalup of the Jacobi matrix of the CSTR model is -().()(J38~ Thus. any constant h E [0 , 0.0038) yirlds a positin'
t.akes place to produce cyclopentenol from cyc!opent.adiene by acid-catalyzed electrophyli c hydration in aqueous solution (Klatt et al. , 1995) . 532
definite and sy'mmetric P that can be us('d as terminal penalty matrix. We choose K. = 0.0015~ to obtain an appropriate terminal penalty matrix P
=
3278.78 1677.31 ( 681.02 271.50
1677.31 919.78 344.19 137.27
681.02 34-U9 172.45 65 .53
271.50 ) 137.27 04) 65.53 29.28
h:~~', . ~ . v~
From the simple optimization (7), wc find the corresponding terminal region
------1
o
100
200
300
400
soo
•
600
700
BOO
900
1000
h~L2:~ · .· · ----------;~-i
The terminal region is small, because the CSTR is operated at the point of optimal yield and is thus strongly non linear. The CSTR in the neighborhood of the optimal operating point has unstable zero dynamics. therefore , we choose the control horizon as Tc = 20s. We find that the uncontrolled plant is driven to the terminal region described by (15) from far awa:·; from the operating point in at most Ts = 13808. Thus. the prediction horizon is chosen as Tp = 14008 according to (8).
100
~
~~
20
o
"
'
- 20
-::: , o-=-o---:BC:-:OO--:::90:~' - --;;00
_~ ... .. ~
' , .. . ..
~~ ~-- - - -
0;
~
-4~00:---:C50:-:-0---:.-<:oo-
300
~'.
..---
\..J ...:!::. ". §";;:
~
200
•
__ _ _ -
0~-'~ 00:--~200 ---3~ 00:--~400 ---:5C:-: OO--::: .o-:o ---:'~ 00-C:-: BO-:-O---:9~ O O~'00C
'~'1
.~~ ::n"I--' ~
The optimization problem (2) - (4) is solved in discrete time with a sampling period of 6 = 208. Time profiles for the closed-loop system emanating from some different initial conditions are shown in Fig. 1 with different line ty·pes. Note that large deviations in the initial conditions from the steady state values are chosen. We clearly see that the input constraints (13) are not violated.
-I...c:::
,
1 ; \) -,of 0
x
1~3
'\ " .. ':':
'r ;'"
1
' ~ .... ' .
100
.:= .. .,. := 1
200
300
400
500
600
700
800
900
lODe
500
600
700
8 00
900
, 000
1~~fr= l~ ,: ~
It should be emphasized that the optimization problem does not include a terminal inequality constraint. The terminal region described by (15) is only used to determine Ts.
o
, 00
200
3 00
40 0
Time [s]
Fig. 1. Profiles for the CSTR cont.rolled b~' the proposed non linear predictive controller. Different line types correspond to different initial conditions.
The computational burden is a very important criterion for "real time" implementation . Since there is no nonlinear constraint in the optimization problem and the control horizon can be chosen considerably shorter than the prediction horizon, the proposed quasi-infinite horizon T\l\lPC scheme has advantages from a computational point of view . To show this, we compare two controllers: Controller A is designed with the proposed ~r\'lPC scheme. while controller B is based on the control scheme suggested in (Chen and Allgower, 1997) that involves a terminal inequality constraint . With bot.h controllers the closed-loop system has guaranteed stability.
implementation of the proposed quasi-infinite horizon NMPC scheme with guaranteed closed-loop stability is possible. Table 1. Comparison of elapsed
IdO ) -:2.14 2.96 - 2.14 296
A : P given by (14) , no terminal inequality con-
Initial state J'4(0 ) I2 (0) Il(0 ) -1.09 -29.20 - 27 .91 20.0 20.0 l.41 20 .0 1.41 -29.20 20.0 - 27.91 -1.09
cpe
time
j
Elapsed (,Pt· timp (s i Contr . A Cont r . B (jOOfi .H7 ,5 02:12 .',·Bl .:'!(j .:;02 .0:2 ()OO .,):.!7 500.69 ',6:20:.!!) ';25 64
The terminal cost Ilx(t+ TI'; x(t). t)l l ~ is introducpd to achieve closed-loop stability and has also some "performance meaning" in the sense that it is linked to the infinite horizon cost. In order to show the influence of the terminal cost term on the achievement of the desired control performance. \\'p compare controller A with different prediction hOlizons Tp = 500s , TI' = 700s and Tp = 900s to a controlln C with "infinite" (i. c .. IIcry long) prediction horizon that "guarant.ees" also closed-loop stabilit.\,:
straint, Tc = 208 , Tp = 14008, B : P given by (14). terminal inequality constraint x(t + Tp: x(t), t) E l! with l! given by (15) , Tc = Tp = 440s. For a total simulation time of 2000 seconds. the elapsed CPU times for some arbitrarily chosen initial st.ates are shown in Table 1. It is clear that controller A needs significantly less cpe time than controller B. For the CSTR model. the "real time" 533
C : P
TI )
=
0, no terminal constraint. Te
= 2800s ,
=
Since we need not add any terminal equality or inequality constraints to the optimization problem in order to guarantee closed-loop stability allcl a yer~' short control horizon (for example ol1e step) can be chosen, the on-line computational burdpII can be significantly reduced. Thus, "real t.ime" implementability of the proposed quasi-infinite horizon Nt..IPC is much more likely when compared to other Nt..IPC schemes including the Olle in (Chen and Allgower, 1997). This was also shown in simulation for the CSTR control problem.
20s
Time profiles of the closed-loop CSTR are shown in Fig 2. The initial condition of XI (0) = -0.64 "'tl. :r3(0) = O.07°C, :r2(0) = 0.22 "~Ol , and X4(0) = 0.39 QC is outside the terminal region (15). But from this initial state the uncontrolled plant needs only' 7" = 460s to reach the terminal region (15). It
The achieved control performance does not differ Significantly from the ol1e achieved hy an :\'!\IPC with infinite prediction horizon. The rea..<;on is that the choice of prediction horizon according to the st.ability condition is very conservative and we haw an additional terminal cost in the cost functional. that. bounds the infinite horizon cost of the uncont.rolled system in a prescribed terminal region. Thus. thp prediction horizon in the proposed N:\lPC schenlf' is quasi infinite.
6. REFERENCES Bitmead, R .R., rvl. Gevers and Y. Wertz (1990 ). Adaptive Optimal Control - The Thinking Man's GPe. Prentice Hall. New York . Chen , H. (n.d .). Stability and Robustness Considerations in Nonlinear Model Predictive Control. Dissertation, Universitiit Stuttgart. Chen, H., A. Kremling and F. Allgower (1995). Nonlinear predictive control of a benchmark CSTR. In: Proc. 3rd European Control Confer'ence ECC'9S. Rome. pp. 3247- 3252. Chen , H. and F. Allgower (1997). A quasi-infinit.e horizon nonlinear model predictive cont.rol scheme with guaranteed stabilit~;. accepted for Automatica. Garda, C.E., D.:\1. Prett and M. l\-Iorari (1989). Model Predictive Control: Theor~' and pract.ice - A survey. Automatica 25. 335- 347. Genceli, H. and M. Nikolaou (1993) . Robust stability analysis of constrained L I-norm model predictive control. AIChE 1. 39(12). 19541965. Klatt, K.-C ., S. Engell, A. Kremling and F . Allgower (1995). Testbeispiel: Riihrkessdrpaktor mit Parallel- und Folgereaktion. In: Entwurf Nichtlinearer Rege/ungen (S. Engell, Ed.). pp. 425- 432. Oldenbourg Yerlag. l\-1iinchen. !\layne, D.Q. and H. Michalska (1990). Receding horizon control of non linear s~·stems. IEEE Tmns. Automat. Contr. AC-35(7) , 814 -824. I\lichalska, H. and D.Q. Mayne (1993). Robust receding horizon control of constrained Ilonlinear systems. IEEE Tmns. Automat. Contr. AC38(11) , 1623- 1633. Rawlings, J.B. and K.R . I\Iuske (1993). The stability of constrained receding horizon control. IEEE Tmns. Automat. Coutr. AC-38(1O ). 1512 1516.
Time [s]
Fig. 2. Profiles for the CSTR model controlled by controller C (-) and controller A with Tp = 500s (--), Tp = 700s (_. -), Tp = 900s (- .. ). is clearly seen that the t.rajectories of the CSTR model controlled by controller A are getting closer to that by the "infinite" horizon controller C, as the prediction horizon in controller A grows. In general, the difference of the achieved control performance between the quasi-infinite horizon Nl\IPC and the infinite horizon NMPC will be not big. Indeed , the choice of the prediction horizon according to (8) is wry conservative, since the uncontrolled system is in general slower than the controlled system.
5. CONCLL'SIO:"J"S For asymptotically stable systems subject to input constraints, a quasi-infinite horizon NMPC scheme with guaranteed closed-loop stability was proposed. The open-loop optimal control problem is formulated as minimizing a finite horizon cost plus a terminal cost subject to non linear system dynamics and input constraints in which the inputs are set to lw zero beyond the control horizon Te. An appropriate choice of the finite prediction horizon guarantees dosed-loop stability for any Q > 0, R > 0 and Te 2 J. Thus , we can choose Q and R freely to specify the desired control performance. 534