Robust stabilization of jacketed chemical reactors by PID controllers

Chemical Engineering Science 56(2001)2775}2787
Robust stabilization of jacketed chemical reactors by PID controllers
Jose Alvarez-Ram m rez *,Ame rica Morales
Departamento de Ingenieria de Procesos e Hidraulica,Uni v ersidad Autonoma Metroplitana-Iztapalapa,Apartado Postal 55-534,
Mexico D.F.09340,Mexico
Received 24January 2000;received in revised form 4August 2000
Abstract
This paper presents a proof for the semiglobal asymptotic stabilization of stirred jacketed chemical reactors under PID control,which is based on results from nonlinear singularly perturbed systems.A novel PID control con "guration is developed in terms of a parameter that is directly related to the size of the region of attraction and the closed-loop performance.The only prior information required is an upper bound of the heat transfer coe $cient.It is shown that,in the high-gain limit,the proposed PID control approaches the performance of the inverse dynamics control law.Numerical simulations are provided to support our theoretical "ndings. 2001Elsevier Science Ltd.All rights reserved.
Keywords:Jacketed chemical reactors;PID control;Output feedback;Semiglobal stability
1.Introduction
Most stirred-jacketed-tank chemical reactors (JCR)employed in the industrial operations are controlled by PID algorithms.There are many reasons for this,includ-ing their long history of proven operation,the fact that they are well understood by many industrial,operational,technical,and maintenance individuals,and,in many applications,the fact that properly designed and well-tuned PID controllers meets or exceeds the control ob-jectives.
While industrial practice witnesses the e !ectiveness of PID control for complex nonlinear systems as JCR,it is claimed by some researchers that PID control is inad-equate to cope with highly nonlinear systems,since the design of the control law is based solely on local argu-ments worked out on the linearized system (Kravaris &Kantor,1990).Besides the basic stabilization problem,the issues of transient performance of the controller,robustness in the face of time-varying modeling errors (e.g.,variations in reactor feed conditions),and e $cient handling of constraints are central.In this way,several control design strategies,most of them based on inverse dynamics control (Isidori,1989),have been proposed
*Corresponding author.Tel.:#52-5-8044649;fax:#52-5-8044900.E-mail address:jjar @xanum.uam.mx (J.Alvarez-Ram m rez).
(Kravaris &Kantor,1990;Bequette,1991;Russo &Be-quette,1997).Although these control design methodolo-gies have a strong theoretical support with guaranteed robustness margin,the resulting controllers present a complex structure and their impact and implementa-tion under actual industrial conditions is still under dis-cussion (Soroush &Kravaris,1992,1994).
From a practical viewpoint,it is desirable to operate the JCR operation with PID control.There are many reasons for this,including their long history of proven operation,the fact that they are well understood by many industrial operational,technical,and maintenance prac-titioners.Besides,in many applications,the fact that a properly designed and well-tuned PID controller meets or exceeds the control objectives.An industrial PI con-troller has many additional advantages that make it practical for operating a CSTR.For example,it has automatic and manual switching,set-point tracking and emergency manual modes (see,for instance,Rivera &Jun,2000).
Despite its widespread usage in industrial applications,PID control for JCR su !ers from a surprising lackof theoretical support in view of modern theory of nonlin-ear control systems.In fact,the vast majority of the stability results are obtained from linear control theory,so that these results are of local nature.In this way,the stability results presented in the literature are far from being conclusive (see,for instance,Russo &Bequette,1997)since the nonlinear nature of the JCR dynamics are
0009-2509/01/$-see front matter 2001Elsevier Science Ltd.All rights reserved.PII:S 0009-2509(00)00532-7
not accounted for.Moreover,since nonlinear models of chemical reactors are highly uncertain,mainly due to imperfect modeling of chemical reaction kinetics and heat transport phenomena,it is not clear to date how these uncertainties a!ects the nonlocal performance and stability of PID controllers.
On the other hand,the stability proof is actually ham-pered by two factors:(a)the di$culty to bound nonlinear terms under a linear function of the norm of the state,or, in other words,to de"ne sector conditions for non nom-inal terms of the system,and(b)the di$culty to accom-modate the PID control structure within a modern nonlinear control theory framework.These facts imply the di$culty to derive nonlocal stability results(e.g., either global or semiglobal stability and performance results).
Russo and Bequette(1995,1997)performed a system-atic study on control of JCRs.They showed that,under full-state measurements,cascade control is equivalent to inverse dynamics control.IMC-based control designs proposed for linear control of JCRs.Although their stab-ility results are of local nature,the resulting linear con-trollers provided acceptable performance in a wide range of operating conditions.Under the assumption that the volume of the reactor is much larger than the volume of the jacket,Christo"des,Teel and Daoutidis(1996)used nonlinear singular perturbations techniques to obtain an open-loop reduced system and subsequently,to design state-feedbacklinearizing-lik e control laws.Their results are a very elegant nonlinear extension to the common practice in linear control designs where fast jacket dy-namics are not taken into account,so obtaining a simple-st feedbackcontrol con"guration with guaranteed robust stability.
As in previously reported theoretical results(Russo &Bequette,1997,1995;Christo"des et al.,1996),this paper takes its motivation from the need of stability results really exploitable and consistent with engineering intuition and industrial practice.The main contribution of the paper is a proof for semiglobal asymptotic stabiliz-ation of JCR under PID control.This is,we prove that for each compact set of initial conditions,there exist PID control gains,such that a given physically realizable equilibrium point of the closed-loop system is asymp-totically stable with basin of attraction containing .To prove the result,we proceed along the following steps:(1) build an output-feedbackcontroller based on modeling compensation ideas;(2)show that the resulting controller has the structure of PID controller;(3)exploit the pro-posed PID control con"guration to write the closed-loop system as a nonlinear singularly perturbed system;and (4)use standard results on nonlinear singularly perturbed systems(Hoppensteadt,1974;Esfandiari&Khalil,1992; Christo"des&Teel,1996)to prove semiglobal asymp-totic stability.In this way,our theoretical results provide a support for PID industrial applications and recently reported linear control designs(Russo&Bequette,1997). It should be stressed that our results are based on the actual nonlinear model of JCR,so that the existing PID stability results derived from a linear model about can be obtained as a corollary of our results.A feature of the proposed control design is that a natural antireset wind up scheme is obtained to handle e$ciently the input constraints(see,for instance,Kothare,Campo,Morari, &Nett,1994).
A further conclusion from the(nonlinear)stability analysis is that the traditional PID control can recover the transient performance of the inverse dynamics con-trol law.This result reveals that traditional PID control of JCR is able to induce acceptable transient perfor-mance,including exponential stability and quasilinear closed-loop convergence.This clari"es the role of PID control within modern control theory of JCR.
The paper is organized as follows:Section2describes the dynamic model of a JCR and states the control problem and the main assumptions;Section3presents several preliminary results on input}output linearizing feedbackcontrol;Section4proposes a novel PID control con"guration and presents the stability analysis;Section 5illustrates the application of our results to a simple simulation example;and Section6proposes some con-cluding remarks.
2.System description and control problem statement Consider the dynamic model of a JCR described by the following di!erential equations:
c" (c !c)#Er(c,¹),
¹" (¹ !¹)#H2r(c,¹)#
(¹H!¹),
¹
H"(u/<H)(¹ H!¹H)! (¹H!¹),(1) where c31L is the vector of reactor concentrations of the chemical species,¹31is the reactor temperature,¹H31 is the jacket temperature,`in a denotes inlet conditions, r(c,¹)31K is the vector of nonnegative reaction kinetics, E31L"K is the stoichiometric matrix,H31K is the vector of reaction heats, is the dilution rate, '0and '0 are,respectively,the heat transfer coe$cient in the reac-tor and jacket sides,and<H is the jacket volume.The jacket#ow rate u that will be taken as the manipulated input for control.
To ensure that the JCR model(1)is well posed with respect to mass conservation,the following assumption is made(Gavalas,1968;Feinberg,1987).
Assumption1(Principle of mass conservation).There exists a positive vector 31L,such that 2E"0.
The equality 2E"0states that what is produced by the reaction system is not larger than what is consumed.
2776J.Al v arez-Ram n&rez,A.Morales/Chemical Engineering Science56(2001)2775}2787
As a consequence of Assumption1,it can be shown that the polyhedral set C"+c31L: 2(c !c))0,c G*0, 1)i)n,is positively invariant under the dynamics of the JCR(Viel,Jacob,&Bastin,1997a).As a consequence, throughout the rest of the paper the vector of concentra-tions c will be restricted to the bounded set C.
Assumption2.(i)r(c,0)"0.This means that there is not reaction activity at the absolute zero(K)temperature.(ii) ""r(c,¹)""( (R,for some ,for all c3C and
¹31
V
"+¹31:¹*0,.
This is a very mild assumption which is satis"ed by most plausible models of chemical reaction kinetics,in particular by Arrhenius-type laws.
The natural domain of the JRC model(1)is the set D N"C;1 V L1L> .In actual industrial applications, initial conditions that are required to be attracted to the operating point are contained in a neighborhood L D N of the operating point.In this way,stabilization of indus-trial chemical reactors is actually a problem of feedback stabilization on compacta.
The aim of this paper is to prove that a PID controller can lead to nonlocal stabilization results.Speci"cally, `our control problem is to prove that,given any operat-ing point(c,¹,¹H)3D N and any given set of initial condi-tions L D N,there exist PID control gains such that (c,¹,¹H)3D N is an asymptotically stable equilibrium point of the controlled JCR(1)and L D N is contained in the basin of attraction a.
Let us introduce the following assumptions regarding the reactor system(1).
Assumption3(Minimum-phase assumption).The isother-mal dynamics c" (c !c)#Er(c,¹)are globally asymptotically stable at the single equilibrium point c3C.Several industrial chemical reaction systems satisfy this assumption(Feinberg,1987),including catalytic re-actions.
Assumption4.Flow rates( ,u)and system temperatures (¹,¹ ,¹H,¹ H)are available for measurements.
Assumption5.The jacket#ow rate is restricted to take values into the domain[u ,u ],where0)u ( u(u and u " <H(¹H!¹)/(¹ H!¹H).
It is noted that the map u P¹is relative degree two. By virtue of Assumption3,the problem of stabilization of the JCR model(1)can be reformulated as follows:`Un-der Assumptions1}5,design a PID control law to regu-late the reactor temperature¹(t)about the setpoint¹via manipulations of the jacket#ow rate u(t).Moreover, since control inputs are subjected to unavoidable phys-ical constraints,the PID control system must be en-dowed with a suitable antireset windup scheme for an e$cient handling of control input saturations a.
3.Preliminaries
In this section,we recall some results on the stability of the JCR model(1)under saturated inverse dynamics control,called input}output linearizing control as well (see,for instances,Kravaris&Kantor,1990;Alvarez, Alvarez-Ramirez,&Suarez,1991,Soroush&Kravaris, 1992,1994).These stability results will be instrumental for the stability analysis of PID-controlled JCRs in Sec-tion4.
Consider the change of coordinates( ,z)" (c,¹,¹H), where 31L,z31 and
"c!c,
z "¹!¹,
z " (c,¹,¹H) " (¹ !¹)#H2r(c,¹)
#
(¹H!¹).(2) In this coordinates frame,the JCR model(1)can be written in the canonical form
z "z ,
z "f(z, )#g(z, )u,
" (z, ),(3) where
(z, )" (c ! !c)#ER( ,z ),(4) R( ,z )
"r( #c,z
#¹)
and
f(z, )"!( # ) (z, )#H2[* R( ,z) (z, )
#*
X R( ,z)z ]
!
[ \ ( ,z)!z #¹],
g(z, )"( /<H)[¹ H! \ ( ,z)](5) and
(z, )" (¹ !z !¹)#H2R( ,z)
#
[ \ ( ,z)!z !¹].(6) We have that (0,0)"0.In fact, (0,0)" (c !c)# ER(0,0)" (c !c)#Er(c,¹)"0.
The physical domain D N is mapped to the translated physical domain D2."C ;(1V !¹);( \ 1V ),
J.Al v arez-Ram n&rez,A.Morales/Chemical Engineering Science56(2001)2775}27872777
where C "C!c.It is noted that the origin is an equilib-rium point of(6)with u"u.Then,the control problem is equivalent to design a control law to stabilize the(devi-ation)reactor temperature z about the origin. Consider now the feedbackfunction
u "Sat[ (z, )],(7)
where (z, )is the linearizing feedbackcontroller(also called inverse dynamics feedbackcontrol)given by
(z, )"[!f(z, )#Kz]/g(z, ).(8) Sat:1P[u ,u ]is a C -saturation function and K"(K ,K )31 is the inner-loop control gain such that the polynomial s !K s!K "0is Hurwitz.In a neighborhood B of the origin,we have that u " (z, ), for all(z, )3B .Hence,the closed-loop system(3),(7) becomes
z"A A z,
" (z, )(9) for all(z, )3B ,where
A A" 01K K 31 .(10) For a more compact notation,introduce the state vector x "(z, )231L> ,so that the closed-loop system(3),(7) is written as
x"F (x),(11) where
F (x)" z f(x)#g(x)Sat[ (x)] (x) .(12)
Lemma A.1in Viel,Jadot,and Bastin(1997b)implies that the origin is an asymptotically stable equilibrium point of system(3)under the control action(7),(8).It is likely that in some cases,control input saturations might defeat global asymptotic stabilization(Alvarez et al., 1991;Viel et al.,1997a).Necessary conditions to achieve global stabilization under saturated control have been described by Alvarez et al.(1991).On the other hand, a stability analysis to derive su$cient conditions for global asymptotic stability can be found in Viel et al. (1997b),which depend strongly on the control input limit u .Anyway,the most relevant result for our workis that asymptotic stability of the origin in(11)implies the existence of a,global or not,basin of attraction of the origin.From Lyapunov converse theorems(Massera, 1956),there exists a maximal domain containing the origin V-D2.and a Lyapunov function<(x): V P 1V ,such that<(0)"0and1*V<,F (x)2is negative,for all x3 V!+0,.This means that V-D2.is the basin of attraction of the origin of the JCR(1)under the saturated inverse dynamics control law(7),(8).Then,all trajectories of the closed-loop system(3),(7)starting into the domain V converge asymptotically to the origin.If D2.- V, we will say that the controlled JCR is globally asymp-totically stable with respect to the physical domain D N.
4.Robust PID control of jacketed chemical reactors
In actual industrial applications,the feedbackfunction (7),(8)cannot be implemented because of strong model uncertainties in the chemical reaction kinetics r(c,¹)and heat transfer parameters and ,hence the necessity of robust control design methodologies.In the present sec-tion,we are going to demonstrate that a traditional PID controller with a suitable antireset windup(ARW) scheme can recover the performance achieved by the inverse dynamics control law(7),(8).This recovery in-cludes the basin of attraction V and the trajectories achieved under(7),(8).To this end,we will provide a novel robust PID control con"guration based on modeling error compensation ideas(Alvarez-Ramirez, 1999).
Let '0and '0be estimates of '0and '0,respectively.Since¹H" \ ( ,z),we have that g(x),( /<H)(¹ H!¹H).Then,g(x),g(¹H)"( /<H) (¹ H!¹H)can be taken as an estimate of g(x).On the other hand,since chemical species concentration are rarely measured in industrial applications,we consider the worst-case control design where the reaction rates vector R( ,z)is unknown.Obviously,if a nominal model R( ,z)of the chemical kinetics is known and the chemical species concentration is available via either direct measurements or state observers,R( ,z)can be incorpor-ated into the robust control design.
The function f(x)can be separated into two parts: f(x)"f(¹,¹H)# f(x),where
f(z, )"! (z, )#H2[* R( ,z) (z, )
#*
X R( ,z)z ]! [ \ ( ,z)!z #¹] is the unknown part and
f(¹,¹H)"!( # )[ (¹ !z !¹)
#
( \ ( ,z)!z !¹)]
!
( \ ( ,z)!z #¹)
"!( #
)[ (¹ !¹)# (¹H!¹)]
!
(¹H!¹)
2778J.Al v arez-Ram n&rez,A.Morales/Chemical Engineering Science56(2001)2775}2787
is the known(nominal)part.Here " ! and " ! .In this way,it makes sense to de"ne a modeling error function as
(x,u) " f(x)# g(x)u,(13)
where g(x)"g(x)!g(¹H).
System(3)can be rewritten as
z "z ,
z "f(¹,¹H)# (x,u)#g(¹H)u,(14) " (z, ).
¹he modeling error function (x,u)contains all the uncer-tainties related to the input-output map u P¹!¹.On the other hand,since z " (¹ !¹)#H2r(c,¹)
#
(¹H!¹),the coordinates transformation ( ,z)" (c,¹,¹H)that leads to the canonical representa-tion(3),involves uncertain parameters and functions.In this way,we can see z as an unmeasured state.Hence, the fact that (x,u)is unknown and the fact that z is unmeasured are the main obstructions to implement an inverse dynamics control law like(7),(8).In what follows, we proceed to design a robust feedbackcontrol law.
It is not hard to see that the state z (t)and the modeling error signal (x(t),u(t))are observable.This means that the signals z (t)and (x(t),u(t))can be recon-structed from the actual measurements( ,u)and (¹,¹ ,¹H,¹ H).We exploit this key property to construct an observer to approximate the signals z (t)and (x(t),u(t))and use these observations z (t)and (t)to design an observer-based inverse dynamics control law.
We choose the following observer(where ,unknown, is not introduced):
z "f(¹,¹H)# #g(¹H)u#2¸(z !z ),
"¸ (z !z ),(15) with¸a design parameter to be tuned.Based on the observations z and ,our choice of controller is the (certainty equivalence)inverse dynamics feedbackfunc-tion:
u"Sat[F(z, )],(16)
where
F(z, )"[!f(¹,¹H)! #Kz]/g(¹H)(17) and z ,z .
The observer(15)plays the role of a reduced-order observer for the estimation of the state z and the term . It is noted that the observer(15)depends on the unmeasured signal z (t).To implement the observer(15), we proceed as follows:since z "z ,system(15)is written as
z "f(¹,¹H)# #g(¹H)u#2¸(z !z ),
(18) "¸ (z !z ).
Introduce the variables w "z !2¸z and w " !¸ z .Then,the observations z and are obtained from the second-order"lter
w "f(¹,¹H)#w #¸ z #g(¹H)u!2¸(w #2¸z ),
w "!¸ (w #2¸z ),
(19) z "w #2¸z ,
"w #¸ z ,
which,since z "¹!¹P,depends only on measured signals(¹,¹ ,¹H,¹ H,u).The variables w and w are initialized as follows:since z and are unknown,we can take w (0)"!2¸z (0)and w (0)"!¸ z (0). Discussion:(1)The controller(15)}(17)is a kind of adaptive control law.Contrary to traditional adaptive control laws where adaptation is induced by parameters estimation,in the proposed control law adaptation is induced by the estimation of the signal (t).
(2)As the developments before show,the controller con"guration is derived essentially on the basis of geo-metric control techniques.Contrary to state-feedback inverse dynamics controllers,as the given by Eqs.(7),(8), the implementation of the proposed PID control con"g-uration is quite simple.In fact,since such controller is linear and depends only on easily measured signals,its implementation can be made on actual hardware tech-nologies(e.g.,PLCs)with exact discretization strategies.
(3)The controller's robustness stems primarily from the cancellation of the estimate of the matched uncertain term (x,u).Although we have focused on the time-invariant case(i.e., (x,u)does not depends explicitly on time),the proposed control design can used without any modi"cation to the time-varying case by taking the modeling error function as
(x,u,t)" f(x,t)# g(x,t)u.
This is the case where the inlet conditions(c ,¹ ,¹ H) have sustained perturbations.
(4)To clarify the structure of the proposed controller (15)}(17),let us compute its`transfer function a.To this end,let u A "F(z, )be the computed control input.From (17)and(19),we get
"¸ (p !K p!K )
p(p!K #2¸)
x !
g(¹H)¸
p(p!K #2¸)
(u!u A) and
z "
K #2¸p
p!K #2¸
x #
g(¹H)
p!K #2¸
(u!u A),
J.Al v arez-Ram n&rez,A.Morales/Chemical Engineering Science56(2001)2775}27872779
where p"d/d ing these expressions in Eq.(17),it is not hard to show that the computed control input can be written as follows:
u A"!g(¹H)\ f(¹,¹H)#g(¹H)\ C.'"(p)F (p)[!z ] #G
05(p)[u!u A],(20) where C.'"(p)is a classical PID control action with proportional,integral and derivative gains given by K."!(2¸K #¸ K )/(2¸!K ),
K'"!K ¸ /(2¸!K ),(21) K""!(K !¸ #2¸K )/(2¸!K ).
F (p)"1/( D p#1)is a"rst-order"lter with time con-stant D"1/(2¸!K )'0,and
G 05(p)is the antireset windup(ARW)operator
G 05(p)"K p#¸
p(p!K #2¸)
(22)
This shows that the proposed controller is basically a PID control law with dc-bias!f(¹,¹H)/g(¹H)and endowed with a natural ARW scheme of feedbacknature G 05(p)[u!u A].In this way,when the control input is saturated,the feedbacksignal G 05(p)[u!u A]drives the error u!u A to zero by recomputing u A such that the controller output u A attains exactly the saturation limit. This prevents the controller from winding up(Kothare et al.,1994).When the control input is not saturated,u"u A and the ARW action has no e!ect on the control loop.
(5)We have shown that the proposed controller is basically a classical PID control law with time-varying `high-frequency a gain g(¹H)and dc-bias!g(¹H)\ f(¹,¹H).It is noted that if we take f(¹,¹H)"0and g(¹H)"constant,this leads to a classical linear PID con-trol law.Hence,robust stability of the linear PID control law g&$C.'"(p)F (p)[!z ],g&$'0,will follow as a co-rollary of our results.
In the next part of the paper,we will show that non-local stabilization results of the JCR(1)can be obtained via the robust PID control law(15)}(17),hence the title of the paper.
4.1.Stability analysis
We"rst write out the closed-loop dynamics via appro-priate coordinates.Introduce the scaled estimation errors
e "¸(z !z ),
(23)
e " ! .
¹hen,using(13),(16)and(17)we get
u"Sat[ (x,e,u)],(24)where
(x,e,u)
"(!f(x)! g(x)u#Kz#¸\ K
e #e )/g(¹H). Unfortunately,when working with this system o
f coordi-nates,we"nd that the control input u is now implicitly de"ned.This problem can be overcome if the saturation function Sat is chosen appropriately,as shown in the followin
g result.
Lemma4.1.Let
be an upperbound of the heat transfer parameter .Assume that * .For any function Sat whose deri v ati v e is dominated by one,i.e.,
d Sat(s)d s )1for all s31(25) ther
e exists a C function 1 (x,e)sol v ing Eq.(24). Proof.See the appendix.ᮀ
By virtue of this lemma,the feedbackfunction(16),(17) can be rewritten as u" 1 (x,e).In the next result,we will establish a connection between the PID control con"guration(15),(17)and the saturated inverse dynam-ics control law(7),(8).In fact,this result shows that the PID control law(15),(17)is an alternative representation of the inverse dynamics control law(7),(8)as P .This property will be crucial to prove our stability results. Lemma4.2. 1 (x,0)"Sat[ (x)],for all x31L> . Proof.See the appendix.ᮀ
It can be shown that the closed-loop dynamics can be described by(see the appendix)
x"F (x)#B [ 1 (x,e)! 1 (x,0)],
e"¸A C( (x,e))e#B (x,e),(26) where (x,e) "Sat ( (x,e))( / )and
A C( (x,e))" !21
! (x,e)0
.
We recall that we have to study the solutions of the closed-loop system with initial conditions x(0)31L> in the given compact set V-D2..Since we have w (0)"!2¸z (0)and w (0)"!¸ z (0),initial condi-tions e(0)31 are also contained in a compact set C31 .Hence,we have to study the solutions of the system(26)in the given compact set V; C.The idea is
2780J.Al v arez-Ram n&rez,A.Morales/Chemical Engineering Science56(2001)2775}2787
to choose¸large enough so that the e!ect of the un-known function (x,e)can be neglected.In this way, system(26)can be seen as a singularly perturbed nonlin-ear system(Hoppensteadt,1974)with¸\ as the small parameter,x31L> as the slow variable and e31 as the fast variable.The slow model is the system(11)(i.e., x"F (x)),which is asymptotically stable with basin of attraction V L1L> .On the other hand,the fast model is
e "A C( (x,e))e,(27) where e is the derivative o
f e with respect to the scaled time t "¸t.
Lemma4.3.Assume that ' .The fast model(27)is quadratically asymptotically stable.This means that there exists a quadratic function<C(e)"e2P C e,P'0,such that <(e)is a Lyapuno v function for the fast model(27). Proof.See the appendix.ᮀ
As a consequence of Lemmas4.1}4.3,our main result can be proven as a direct application of Hoppensteadt's Theorem2(Hoppensteadt,1974).
Theorem4.4.Under Assumptions1}5,for any compact set contained in the interior of V,there exists a positi v e number¸ such that,for all¸'¸ ,the closed-loop system(1)and(15)}(17)admits the operating point (c,¹,¹H)3D N as an asymptotically stable equilibrium point with basin of attraction containing .
A second consequence of Hoppensteadt's Theorem2 (1974)(see also Esfandiari&Khalil,1992)is related to the recovery of the transient performance induced by the saturated inverse dynamics control law(7),(8). Corollary4.5.For all initial states in V,
x(t,¸)P x H(t)as¸PR(28)
uniformly in t and¸,for all t'0,where x H(t)is the solution of the slow model x"F (x)and x(t,¸)is the solution of the fully perturbed system(26).
Discussion:(1)Roughly speaking,the result in the above theorem states that the robust PID controller (15)}(17)can recover the basin of attraction V induced by the inverse dynamics control law(7),(8).This means that all trajectories starting in any compact domain contained in V can be stabilized via PID control.In modern control theory terminology(Christo"des et al., 1996;Teel&Praly,1995),it is said that the PID control-ler(15)}(17)is a robust semiglobal asymptotic stabilizer for the JCR(1).
(2)Recall that if f(¹,¹H)"0and g(¹H)"constant lead to a classical linear PID control law.Robust stability of the classical linear PID control law g&$C.'"(p)F (p) [!z ],g&$'0can be established following the same steps of our proof,hence it follows as a corollary of our results.This is a quite interesting result since it is a com-mon belief that PID control is inadequate to cope with highly nonlinear systems,since the design of the control law is based solely on local arguments worked out on the linearized system(Kravaris&Kantor,1990).
(3)Corollary4.5states that the performance induced by the ideal control law(7),(8)is recovered by the PID control law(15)}(17)as the unmeasured state z and the modeling error term are estimated quickly.Such recov-ery includes the basin of attraction V(Theorem4.4)and the trajectories induced by the exact inverse dynamics control law(7),(8),x H(t).
(4)In the proposed control design framework,the saturation function Sat:1P[u ,u ]was included to satisfy the physical constraints.It is clear from the ob-server design(15)that the eigenvalues of the fast system e"¸A C( (x,e))e are located at O(¸\ )-locations so that they approach in"nity as¸P0.In general,placing poles far in the left-half plane causes an impulsive-like behavior (overshoot)known as the(high-gain)peaking phenom-enon(Sussman&Kokotovic,1989).In our case,the peaking phenomenon can be described in terms of the¸-dependent scaling of variables in Eq.(23)(Esfan-diari&Khalil,1992).Most of the undesirable features of peaking will not exist in a globally bounded control. Global boundedness of the control input is achieved by virtue of the saturation functions(i.e.,the term B [ 1 (x,e)! 1 (x,0)]is globally bounded with re-spect to the scaled-estimation error e).In this paper,we have assumed that the control input is subjected to hard saturations,which introduces in a natural way the global boundedness property discussed above.It should be stressed that,if the control input is not subjected to hard constraints,saturation of the control input is not the only procedure to reduce peaking phenomenon e!ects.In fact, an e$cient procedure suggested by Christo"des(2000) consists in the saturation of the estimated states,which eliminates wrong estimates for short times.This ap-proach leads to the same semiglobal stabilization as described by Theorem4.4.
(5)In general,the JCR parameters such as¹ ,c , , and are of time-varying nature.Here we can use the results reported by Christo"des and Teel(1996)to state the stability properties of the PID-controlled JCR. In this case,it can be concluded that the stability prop-erty stated by Theorem4.4continue to hold,up to an arbitrarily small o!set,for initial disturbances and their derivatives in an arbitrarily large compact set as long as the parameter¸'0is su$ciently large.This property is a type of total stability of PID control with respect to time-varying disturbances.
J.Al v arez-Ram n&rez,A.Morales/Chemical Engineering Science56(2001)2775}27872781。