- Email: [email protected]
Robust Geometric Nonlinear Control of Process Systems
Copyright ® IF AC Advanced Control of Chemical Processes, Pisa, Italy, 2000
ROBUST GEOMETRIC NON LINEAR CONTROL OF PROCESS SYSTEMS Jesus Alvarezl,Teresa Lopez l ,2, and Eduardo Hernandez l
1Universidad
Autonoma Metropolitana - Iztapalapa, Depto. de Ingenieria de Procesos e Hidraulica, Apdo. 55534, 09340 Mexico D.F. ,Mexico E-mail: [email protected] 2 Centro de Investigacion en Polimeros, M. Achar 2, Tepexpan, 55885 Edo. Mex., MEXICO
Abstract: In this work is presented a robust nonlinear geometric control design with solvability conditions, a stability criterion, a systematic construction, and a simple tuning procedure. The measurement-driven controller is built by combining a statefeedback controller with a proportional-integral state estimator. The functioning of the controller can arbitrarily approach the one of the exact state-feedback controller by tuning the estimator sufficiently fast. The resulting controller is interpreted with the IMC and the MPC designs. An homopolymer reactor is considered as an application example, showing that the proposed controller is faster and more robust than the existing estimator-based geometric controllers. Copyright © 2000 IFA C Keywords: Nonlinear process control, measurement-feedback nonlinear control, outputfeedback nonlinear control, nonlinear detector-based control.
1. INTRODUCTION
driven nonlinear controllers in chemical engineering have been designed by combining theoretically backed nominal state-feedback controllers with ad hoc state estimators (observers or detectors), and consequently, the corresponding measurement-driven controllers have lacked closed-loop stability criteria and systematic construction-tuning procedures. Alvarez (1996) presented a proportional estimatorbased nonlinear controller with a nominal stability criterion, and a systematic construction-tuning procedure. In the existing nonIinear geometric measurement-driven control designs, the compensation of modeling errors has been accomplished by using, in ad hoc manner, integral action in the control loop, and systematic designs with robust convergence criteria are lacking.
The theory of nonlinear geometric control has reached a state of maturity by having extended to the nonlinear case the robust output regulator theory of nonlinear systems (Byrnes et aI., 1997) and having established connections with optimal control (Sepulchre et al. 1997). In chemical process control, the IM (internal model) and MP (model predictive) controllers are the most widely used advanced process control techniques, the applicability of the geometric control seems precluded by the absence of robustness, and the understanding and systematization of the model-based nonlinear IM and MP controllers constitute an important open issue. According to the linear control theory (Doyle, 1992), any robust controller can be realized as a suitable combination of a state-feedback controller with an observer that contains a representation of the modeling errors (i.e., the Internal Model Principle). On the other hand, the implementation of the modelbased nonlinear geometric (Alvarez, 1996), IM (Doyle, 1998), and MP (Phani and Bequette, 1996) controllers depends on the robust functioning of a suitable closed-loop state estimator. Until recently (see Bequette, 1991; Alvarez, 1996; Doyle, 1998; and references there in), geometric measurement-
In this work is presented a robust nonlinear geometric control design with a solvability condition, a stability criterion, a systematic construction, and a simple (conventional-type) tuning procedure. Alvarez and Lopez' (1999) PI(proportional-integral) robust estimator is employed to estimate the persistent modeling error in the input-output path of the closed-loop geometric estimator. This estimate error is then accounted for in a redesigned nonlinear geometric controller. As a
395
result, by making the estimation obselVable dynamics sufficiently fast, the functioning of the measurement-driven controller can arbitrarily approach the one of the exact state-feedback controller. The proposed controller is put in perspective with the IM and MP control approaches, and is applied to a polymer reactor that has been the subject of extensive theoretical and experimental studies. 2. CONTROL PROBLEM Let us consider nonlinear MIMO plants with n states (x), m measured outputs (y), p exogenous inputs (d), and I\ model parameters (r). x = fIx, d(t), u(t), r], y = hex, r); x(t o) = Xo
(1)
J) -- [h 1>", LK,-lh " h LKm-f 1.J'm.J h l' x,U,r f 1>", rn''' ' q>(x,a,u,r) = [L~;hl;;L K""f hml'
(4a) (4b)
where the vectors y, v and aare made of the measured outputs and inputs and of their time-derivatives: (4c)
(4d) (4e)
Definition 2. The nominal evolution E(t) (3a) of the nonlinear plant (1) has a relative (vector) degree k=(K",Krn )" K",Krn=K~n,Kj.>O (5)
x(t) is RE(robustly exponentially)-stable if there are constants a x' Ax' b u' bd' br > 0 so that the (xo,d,u)perturbed motions X(t) = 8 x[t, to' Xo' u(t), &(t), p] converge to the unperturbed one x(t) as follows
if, along E(t) (q>u = oq>/ou,lx =
o/ox):
(i) o/ou = 0, (ii) rank{lx[x(t),
~ a/Ax(t-to)IIXo_xolI + buIMt)-u(t)IIS
aCt), r]} = K
(iii) rank q>u[x(t), il(t), u(t), r] = m If, in addition, the maps and q> are continuously
+ bdll&(t)-d(t)US + brllp-rll s~p
and q>
'1'1
~ a/Ax(t-to)IIXo_xolI
where lIu(t)-u(t)IIS =
<1>1
A (
(2)
Definition J. The motion x(t) is E(exponentially)stable if there are constants a x' Ax > 0 so that, in some neighborhood ofE(t), the xo-perturbed motions X(t) = 8 x[t, to' Xo ' u(t), d(t), r] converge to the unperturbed one x(t) as follows
IIx(t) - x(t)1I
Assuming that the parameter r and the state x are known, in this section is presented the output noninteracting regulation state-feedback problem (Byrnes et at., 1995; Isidori, 1995) with the poleplacement tuning scheme presented in (Alvarez, 1996).
and introduce the nonlinear maps
This nonautonomous system represents batch and semibatch processes, as well as startups, shutdowns, and grade changes of continuous processes. A continuous process is a particular case of plant (l).
IIx(t) - x(t)1I
3. STATE-FEEDBACK CONTROL
Let L:a be the i-th directional derivative of the scalar field a(x, t) along the vector field f(x, t),
where uj(t) or d;(t) are piecewise continuous function of time, and the maps f and h are sufficiently smooth (differentiable) about the plant evolution E(t): E(t) ={ u(t),x(t),y(t)}: x(t)=8 x[t,to,xo,u(t),d(t),r]
built with an approximated parameter p, so that the resulting closed-loop plant evolution RE-converges to the nominal one (3a).
lIu(t)-u(t)lI. The evolution
differentiable along E(t), the nominal evolution Eet) is said to have a R(robust)-degree k"..
E(t) is EIRE-stable ifx(t) is EIRE-stable.• Robust stability means that a perturbed motion X(t) can stay arbitrarily close to the unperturbed one x(t) by starting sufficiently close to it and by making the exogenous and parameter errors sufficiently small. The plant (l) must operate about a prescribed
Associated to the relative degree k (5), there is a ZD(zero-dynamics) motion x*(t) satisfying the Z(zero)-dynamics (7b) in the (n-K)-dimensional timevarying surface E(t):
nominal (possibly open-loop unstable) evolution E(t) (3a), according to the nominal plant dynamics (3b):
x*(t) = 8*[t, to' x o' vet), il(t), r]
Eet) = {u(t), x(t), yet), r} :
(3a)
x = fIx, u(t), aCt), r], x(t o) = x o' y = hex, r)
(3b)
E
E(t)
(6)
x* = f{x*,d(t),q>-l[x*,il(t),v(t),r],r}, x*(to) = Xo (7a) n _ E(t) = {x E 9\ I1[X, il(t), r] = y(t)} (7b) Let us introduce the parametrized state-feedback controller (Alvarez, 1996) (se> 0):
and the problem consists in designing a measurement-driven, nonlinear, dynamic controller,
396
uO= a[x,d{t),r,t]:=
matric:s, d d d _ [ d (lvi-I] ] AiSd)-bd[.NI (Sd),··,Am(Sd)]' Aj (Sd)- -kj(Sd\O~i_1 KiSd) = bd[kf(sd) "~ disd)J
~)'
/3[x,d{t), r,t]= v(t)- Ke(se){ CPI[x,d{t),r]-CPI[x(t),a(t),r]}
k?(sd) = (Sdk1i' , Sd
Ke(se) = bd[~(se),-.,~(se)]' kr(se) = (s;~i' -.,sekrK.) 1
G and H are So-parametrized nonlinear gain matrices,
and the i-th gain vector gain kr(1) is set so that the ith polynomial (9), one for each output, has stable poles: }..K; + kC }..K; -I + + k c}.. = 0 1:::; i :::; m (9) IKi
G(X, Xd ' p, so) = [0 I(X, Xd' p)JKo(so) H(X, Xd' p, sd) = [0 I(X, Xd' p)]neKI(so)
[0 1,0 nJ(X, Xd' p) = cp~I(X, Xd ' p)
11'
The resulting closed-loop evolution (11) are given by
dynamics
(10)
Ko(so) = bd[k?(so)' , ~ O(so)J
and
k?(so) = (sok?i' , So K;~), I KI(So) -- d'lag [K,+Ikl So P
;'0 = f{xO,d(t), a[xo,d{t),r,t],r}:= fe[xo,d{t),r,t] (lOa) yO = h(xO, r), XO(to) = xo' lId{t) E~(t) = {XO(t), UO(t), yO(t)}
VI
aCt) 11:::; td (lOb)
So K",+IIrI] ~
,
the gain k?(l) and the gain pair [k:?(I), kJJ are chosen, respectively, so that the polynomials (14a) and (14b) have stable roots:
(11)
The next lemma yields the closed-loop functioning of the exact state-feedback controller (8), or equivalently, the limiting functioning attainable with any estimator-based controller-
}.. V;
+
k4. 11 }.. V;
+ .• + Ird= 0, 1 < ~jI - i < - nd
}.. K;+I + k?i}.. K; + .. + ~'i}.. + kl=O, l:::;i:::;m
(14a) (14b)
1
Lemma 1 (Proof in Appendix). Let the nominal plant (I) evolution E(t) (3a) have a R-relative degree k... with an RE-stable ZD-motion x*(t) (6). Then, the state feedback controller (8) yields a closed-loop plant (10) where: (i) The motion XO(t) RE-converges to the nominal
When So and sd are tuned sufficiently large (fast) [3 to 15 times faster than the control dynamics (14)J, X(t) RE-converges to x(t) with q(quasi)LNPA output error dynamics, and Xm is a fast convergent estimate of the persistent mode ling errors in the input-output path of the estimator.
one x(t). (ii) The output YO(t) E-converges to the nominal one
Recall the state-feedback controller (8), replace its argument (x, a., r) by (X, Xd' p), subtract the state Xm (Bb) from the term /3 in Eq. (8), get the estimatorbased feedback controller (15d), combine it with the open-loop estimator (13), and obtain the following measurement-driven controller
yet), with LNPA (linear, noninteractive, poleassignable) output error dynamics (I :::; i :::; m)
e/K,-I)+se~K.ei(K,)+ +se K;~lei = 0, ei = Ye Yi' • (12) 1 4. CONTROLLER CONSTRUCTION In this section a measurement-driven controller is constructed by combining the state-feedback controller (8) with the robust PI-estimator presented in Alvarez and Lopez (1999).
(Ba)
Xm = [KI(so)] [y(t) - heX, p )], X m(to) = 0
(Bb)
(1 Sa)
Xm = [KI(so)][y(t) - hex, p)J, Xm(to) = 0
(15b)
X = fe[X,Xd,u(t),pJ + G(X,Xd,p,so)[y - h(X,p)J (l5c) u=q> -I {X,Xd,/3[X,Xd ,p,tJxm,p,t }=r(X,Xm,Xd,p,t)(15d)
For the moment, assume the open-loop nominal evolution (3a) is RE-estimable with observability (vector) degree k... (5), and recall Alvarez and Lopez' (1999) PI (proportional-integral) estimator: Xd = [Aisd)]Xd + [Kisd)]d(t); Sd' so> 0
Xd= [Aisd)JXd + [Kisd)]d(t),sd' so> 0
where fe is defined in (lOa). The cancellation of the input-output matching term Xm in equation (15c) signifies the possibility of compensating the modeling errors in the closed-loop system. 5. CLOSED-LOOP STABILITY
X = f[X,Xd,u(t),p] + G(X,Xd'P,so)[y - h(X,p)J + [H(X, Xm' p, so)] Xm (Bc)
The interconnection of the plant (I) with the candidate controller (IS) yields the following closedloop dynamics (16) and evolution (17):
397
Xd = [Aisd)]~ + [Kisd)]d(t)
(16a)
Xm = [KI(so)][y(t) - h(X, p)], Xm(to) = 0
(16b)
inputs are: initiator feedrate u 1, coolant temperature u2, and exit flowrate U3 ' The measured outputs are: Yl = X2 (monomer), Y2 = X3 (temperature), and Y3 = X4 (volume). The exogenous input is feed temperature d.
X = fe(X, Xd' p, t) + [G(X,~,p)][h(x,r) - h(X,p)](16c) X = fIx, d(t), Y(X, Xm , Xd ' p, t), r] Ee(t) ={ u(t), x(t), y(t)}
(16d) (17)
Next is given the main robust closed-loop stability result that underlies the proposed control design.
Theorem 1 (Proof in Appendix). Let the (possibly open-loop unstable) nominal evolution (3) of the nonlinear plant (1) have a R-relative degree K. (5) with a RE-stable Z-dynamics motion x*(t) (6). Then: (i) The closed-loop evolution (17) RE-converges to the nominal one x(t) (3), with q(quasi)LNPA output error dynamics, if the parameter pair (Sd' so) is tuned sufficiently large (fast) so that the potentially destabilizing estimation error terms described in Alvarez and Lopez (1999) are dominated. (ii) By making (Sd'so) sufficiently large, the closedloop evolution (17) can arbitrarily approach the one (11) of the exact state-feedback controller (8) . • While in the previous estimator-based geometric controllers the compensation of modeling errors is done in the slow control loop, here the same compensation is done in the fast estimation loop. The proposed controller can be seen either as a robust IM controller; or a continuous-time MP controller without input constraints, and with a suitable timeinfinite objective function. The open-loop estimator (13) is a fast convergent input-output model of the plant (1), and the estimator-based controller (15d) forces the input-output plant replica to track the prescribed nominal output, with almost LNPA output dynamics and with internal (state) stability.
Initially, the reactor is close to its unstable steadystate, and then subjected to the time-varying feedtemperature of Figure 2c. The thick plots of Figures 1 and 2 show the functioning of the exact statefeedback controller (8) when the measurements are free of noise. Using the gains given in Alvarez (1996), the reactor reaches, without offset, its nominal operation with a 2% settling time of 150 min. The thin plots of Figures 1 and 2 show the functioning of the measurement-driven controller (15) with the modeling errors used in Alvarez (1996), and with the noisy measurements of Figure 1. The control gains are the ones used in the exact statefeedback controller (8), the observer gains are the ones used in the robust open-loop estimator of Alvarez and Lopez (1999), and the estimator dynamics is set to be ten times faster than the control dynamics (i.e., So = lOSe)' As it can be seen in Figures 1 and 2, the measurement-driven controller reaches asymptotically (i.e., without offset) the nominal steady-state (which is open-loop unstable) with a settling time of about 250 min, meaning that the proposed controller (with modeling error compensation in the fast estimation loop) is 2.4 times faster than the controller with modeling error compensation in the slow control loop (Alvarez, 1996). 7. CONCLUSIONS A robust nonlinear geometric control design with solvability condition, stability criterion, systematic construction, and simple tuning procedure has been presented. The tuning of gains amounted to the tuning, with conventional-type techniques, of independent single-input controllers and singleoutput linear filters. Since the rejection of the modeling errors is accomplished in the (fast) estimation dynamics, by making sufficiently fast these dynamics, the performance of the measurementdriven controller can arbitrarily approach the one of the exact state-feedback controller, with the only limitation being the measurement noise. This is different from the existing geometric estimator-based control designs in chemical engineering, where the modeling errors have been rejected via integral action in the (slow) control loop. The proposed control scheme has been put in perspective with the IM and MP control frameworks. An homopolymer reactor was considered as an application example, showing that the proposed controller is faster and more robust than the existing estimator-based geometric controllers.
6. APPLICATION EXAMPLE Let us consider the free-radical homopolymer continuous reactor used in Alvarez and Lopez (1999) to illustrate the functioning of a robust open-loop estimator, and in Alvarez (1996) to illustrate the functioning of an estimator-based geometric controller (8) with mode ling error compensation via integral action in the control-loop. A strongly exothermic reaction takes place in the reactor, with heat exchange being enabled by a heating/cooling jacket. The reactor dynamics are described by a sixstate, three-input (u), three-output (y) nonlinear dynamical system of form (1). The maps f and h are given in Alvarez and Lopez (1999). The states of the plant are: concentration of initiator Xl, dimensionless monomer concentration X2, temperature X3, volume X4, zeroth moment xs, and second moment X6' The
The nonlocally robust convergence property observed in the application example can be explained theoretically with the notions of practical (La Salle
398
and Lefschetz, 1961) or semiglobal (Isidori, 1995) stability, as it has been done in the nonlocal estimation design of Alvarez et al. (1999); and related to the results on the role of the integral action in the region of stability (Pachter et aI. , 1996). In principle, the rigorous connection of the robust estimator-based controller with the MP controller requires the consideration of saturated inputs (Alvarez et aI., 1991), the case of more inputs than outputs, and the possibility of using a finite-time controller.
(Alb) Ae is a stable matrix (Alvarez, 1996), equation (Alb) with CJz = 0 is the zero-dynamics (6), and the map CJz is L(Lipschitz)-bounded. The RE-stability of the closed-loop system (AI or 10) follows from the application of Gronwall' s lemma, Lyapunov' s direct method, and the Comparison principle (Alvarez, 1996). QED.
REFERENCES
Proof of Theorem 1. Apply the coordinate change
Alvarez, 1., J.J. Alvarez, and R. Suarez, Nonlinear Bounded Control for a Class of Continuous Agitated Nonlinear Plants, Chem. Engng. Sci., 46(2),3235 (1991). Alvarez, 1., Output Feedback Control of Nonlinear Plants, AIChEJ, 2(9), 2540 (1996). Alvarez, J., and T. Lopez, Robust Dynamic State Estimation of Nonlinear Plants, AIChEJ 45(1), 107 (1999). Alvarez, J., T. Lopez, and E. Hemandez, Robust Estimation of Free-radical Homopolymer Reactors, IFAC 1999 World Congress, N, 121 (1999). Nonlinear Control of Chemical Processes: A Review, Ind. Chem. Eng. Res., 30, 1391 (1991). Bequette, B. W. , Nonlinear Control of Chemical Processes: A Review, Ind. Chem. Eng. Res., 30, 1391 (1991). Bymes, c., F. Delli Priscoli, and A. Isidori, Output Regulation of Uncertain Systems, Birkhauser, Boston (1997). Doyle, J., B. Francis, and A.R. Tannenbaum, Feedback Control Theory, Macmillan, New York (1992). Doyle Ill, F.F. Nonlinear Inferential Control for Process Applications, J. of Process Control, 8(5,6), 339 (1998). Isidori, A., Nonlinear Control System, Springer, New York (1995). La Salle, J., and S. Lefschetz, Stability by Liapunov' s Direct Method, Academic Press (1961). Pachter, M., JJ. D' Azzo, and M . Veth, Proportional and Integral Control of Nonlinear Reactors, Int. J. ofContr. , 64(4), 679 (1996). Phani, B.S, and W. Bequette, Nonlinear ModelPredictive Control: Closed-Loop Stability Analysis, AIChEJ, 42(12), 3388 (1996). Sepulchre, R., MJankovic, and P. Kokotovic, Constructive Nonlinear Control, Springer, New York (1997).
ed = Xd-d(t), e I =I[x,d(t),r]-I[x(t),7{t),r],
en =n[x, d(t), r] - n[x(t) , 7{t) , r], em =
ed = Aded + ndSit)
[ei, .~ ' = rl A oT[ej, e~J'
(A2a) + SI(e, z, r, t)
(A2b)
~I = wz[en, r; z(t), r, t] + Sn(e, z, r, t)
(A2c)
ZI = AeZI + neqI(e, z, r, t),
(A2d)
zn = wcCzu, r, t)+CJz(Zd,Zh zn, r, t)+
(A2e)
where ne and nd are zero-one entry matrices (Alvarez, 1996), T is the identity permuted matrix (Alvarez and Lopez, 1999), and Ad and Ao are stable matrices (Alvarez and Lopez, 1999). Subsystem (A2c) with Sn = 0 is the zero-dynamics (7a) , and system (A2) with (Sd,ShSn,qhqn) = 0 is RE-stable because it is made of RE-stable decoupled subsystems. The "disturbances" Sd,ShSn,qh and
Proof of Lemma 1. Apply the coordinate change Zd = d(t) - /1(t)
Zn =n[xO, d(t), r]-u[x(t), 7{t), r], Yc=
Xm - ~
to take the closed-loop plant (16) into the estimationtracking error form
APPENDIX: PROOFS OF LEMMA AND THEOREM 1
ZI =I[Xo,d(t),r)-q)I[x(t), 7{t), r],
Zm= qm
yo - yet)
to take the closed-loop plant (16) into the form :
399
(b)
33
o
50
100
150
200
250
300
t (time, min)
t (time, min)
Fig. 1 Noiseless (- ) and noisy (-) measurements. ,....,
"0
.~
0.50
">
"
0 ~
::J ;::,
o
0
e
~
(a)
..: 0.002
:i
0
.]j
0.002
·m .Oxlo
g 0.002
.b .5XIO
...
~
0
e
0.001
"e0
?OxIO ~ ,....,8.5xIO
0
.§
"C
~
'" ,5,
8-
N
><
:0 >. "0
,e,
Q
If
!c..
0'
0.57
Q
i
"e ~ ....><
;:
e 8e .!l
310 305
"0
" ~
::J
300
"0
.;
e::I
295
"0
t (time, min)
G. ~
>< 195 0
50
100
150
200
250
Fig 2. (a) State set {Xl. X2, X3, X4}, (b) control inputs u, as weII as (c) exogenous input d and safety-quality variables {c, M n , Q} for: the exact state-feedback controller with noiseless measurements (=), and for the measurement-driven controIIer with modeling errors and noisy measurements (-).
300
t (time, min)
4()()
ROBUST GEOMETRIC NON LINEAR CONTROL OF PROCESS SYSTEMS Jesus Alvarezl,Teresa Lopez l ,2, and Eduardo Hernandez l
1Universidad
Autonoma Metropolitana - Iztapalapa, Depto. de Ingenieria de Procesos e Hidraulica, Apdo. 55534, 09340 Mexico D.F. ,Mexico E-mail: [email protected] 2 Centro de Investigacion en Polimeros, M. Achar 2, Tepexpan, 55885 Edo. Mex., MEXICO
Abstract: In this work is presented a robust nonlinear geometric control design with solvability conditions, a stability criterion, a systematic construction, and a simple tuning procedure. The measurement-driven controller is built by combining a statefeedback controller with a proportional-integral state estimator. The functioning of the controller can arbitrarily approach the one of the exact state-feedback controller by tuning the estimator sufficiently fast. The resulting controller is interpreted with the IMC and the MPC designs. An homopolymer reactor is considered as an application example, showing that the proposed controller is faster and more robust than the existing estimator-based geometric controllers. Copyright © 2000 IFA C Keywords: Nonlinear process control, measurement-feedback nonlinear control, outputfeedback nonlinear control, nonlinear detector-based control.
1. INTRODUCTION
driven nonlinear controllers in chemical engineering have been designed by combining theoretically backed nominal state-feedback controllers with ad hoc state estimators (observers or detectors), and consequently, the corresponding measurement-driven controllers have lacked closed-loop stability criteria and systematic construction-tuning procedures. Alvarez (1996) presented a proportional estimatorbased nonlinear controller with a nominal stability criterion, and a systematic construction-tuning procedure. In the existing nonIinear geometric measurement-driven control designs, the compensation of modeling errors has been accomplished by using, in ad hoc manner, integral action in the control loop, and systematic designs with robust convergence criteria are lacking.
The theory of nonlinear geometric control has reached a state of maturity by having extended to the nonlinear case the robust output regulator theory of nonlinear systems (Byrnes et aI., 1997) and having established connections with optimal control (Sepulchre et al. 1997). In chemical process control, the IM (internal model) and MP (model predictive) controllers are the most widely used advanced process control techniques, the applicability of the geometric control seems precluded by the absence of robustness, and the understanding and systematization of the model-based nonlinear IM and MP controllers constitute an important open issue. According to the linear control theory (Doyle, 1992), any robust controller can be realized as a suitable combination of a state-feedback controller with an observer that contains a representation of the modeling errors (i.e., the Internal Model Principle). On the other hand, the implementation of the modelbased nonlinear geometric (Alvarez, 1996), IM (Doyle, 1998), and MP (Phani and Bequette, 1996) controllers depends on the robust functioning of a suitable closed-loop state estimator. Until recently (see Bequette, 1991; Alvarez, 1996; Doyle, 1998; and references there in), geometric measurement-
In this work is presented a robust nonlinear geometric control design with a solvability condition, a stability criterion, a systematic construction, and a simple (conventional-type) tuning procedure. Alvarez and Lopez' (1999) PI(proportional-integral) robust estimator is employed to estimate the persistent modeling error in the input-output path of the closed-loop geometric estimator. This estimate error is then accounted for in a redesigned nonlinear geometric controller. As a
395
result, by making the estimation obselVable dynamics sufficiently fast, the functioning of the measurement-driven controller can arbitrarily approach the one of the exact state-feedback controller. The proposed controller is put in perspective with the IM and MP control approaches, and is applied to a polymer reactor that has been the subject of extensive theoretical and experimental studies. 2. CONTROL PROBLEM Let us consider nonlinear MIMO plants with n states (x), m measured outputs (y), p exogenous inputs (d), and I\ model parameters (r). x = fIx, d(t), u(t), r], y = hex, r); x(t o) = Xo
(1)
J) -- [h 1>", LK,-lh " h LKm-f 1.J'm.J h l' x,U,r f 1>", rn''' ' q>(x,a,u,r) = [L~;hl;;L K""f hml'
(4a) (4b)
where the vectors y, v and aare made of the measured outputs and inputs and of their time-derivatives: (4c)
(4d) (4e)
Definition 2. The nominal evolution E(t) (3a) of the nonlinear plant (1) has a relative (vector) degree k=(K",Krn )" K",Krn=K~n,Kj.>O (5)
x(t) is RE(robustly exponentially)-stable if there are constants a x' Ax' b u' bd' br > 0 so that the (xo,d,u)perturbed motions X(t) = 8 x[t, to' Xo' u(t), &(t), p] converge to the unperturbed one x(t) as follows
if, along E(t) (q>u = oq>/ou,
o
(i) o
~ a/Ax(t-to)IIXo_xolI + buIMt)-u(t)IIS
aCt), r]} = K
(iii) rank q>u[x(t), il(t), u(t), r] = m If, in addition, the maps and q> are continuously
+ bdll&(t)-d(t)US + brllp-rll s~p
and q>
'1'1
~ a/Ax(t-to)IIXo_xolI
where lIu(t)-u(t)IIS =
<1>1
A (
(2)
Definition J. The motion x(t) is E(exponentially)stable if there are constants a x' Ax > 0 so that, in some neighborhood ofE(t), the xo-perturbed motions X(t) = 8 x[t, to' Xo ' u(t), d(t), r] converge to the unperturbed one x(t) as follows
IIx(t) - x(t)1I
Assuming that the parameter r and the state x are known, in this section is presented the output noninteracting regulation state-feedback problem (Byrnes et at., 1995; Isidori, 1995) with the poleplacement tuning scheme presented in (Alvarez, 1996).
and introduce the nonlinear maps
This nonautonomous system represents batch and semibatch processes, as well as startups, shutdowns, and grade changes of continuous processes. A continuous process is a particular case of plant (l).
IIx(t) - x(t)1I
3. STATE-FEEDBACK CONTROL
Let L:a be the i-th directional derivative of the scalar field a(x, t) along the vector field f(x, t),
where uj(t) or d;(t) are piecewise continuous function of time, and the maps f and h are sufficiently smooth (differentiable) about the plant evolution E(t): E(t) ={ u(t),x(t),y(t)}: x(t)=8 x[t,to,xo,u(t),d(t),r]
built with an approximated parameter p, so that the resulting closed-loop plant evolution RE-converges to the nominal one (3a).
lIu(t)-u(t)lI. The evolution
differentiable along E(t), the nominal evolution Eet) is said to have a R(robust)-degree k"..
E(t) is EIRE-stable ifx(t) is EIRE-stable.• Robust stability means that a perturbed motion X(t) can stay arbitrarily close to the unperturbed one x(t) by starting sufficiently close to it and by making the exogenous and parameter errors sufficiently small. The plant (l) must operate about a prescribed
Associated to the relative degree k (5), there is a ZD(zero-dynamics) motion x*(t) satisfying the Z(zero)-dynamics (7b) in the (n-K)-dimensional timevarying surface E(t):
nominal (possibly open-loop unstable) evolution E(t) (3a), according to the nominal plant dynamics (3b):
x*(t) = 8*[t, to' x o' vet), il(t), r]
Eet) = {u(t), x(t), yet), r} :
(3a)
x = fIx, u(t), aCt), r], x(t o) = x o' y = hex, r)
(3b)
E
E(t)
(6)
x* = f{x*,d(t),q>-l[x*,il(t),v(t),r],r}, x*(to) = Xo (7a) n _ E(t) = {x E 9\ I
and the problem consists in designing a measurement-driven, nonlinear, dynamic controller,
396
uO= a[x,d{t),r,t]:=
matric:s, d d d _ [ d (lvi-I] ] AiSd)-bd[.NI (Sd),··,Am(Sd)]' Aj (Sd)- -kj(Sd\O~i_1 KiSd) = bd[kf(sd) "~ disd)J
~)'
/3[x,d{t), r,t]= v(t)- Ke(se){ CPI[x,d{t),r]-CPI[x(t),a(t),r]}
k?(sd) = (Sdk1i' , Sd
Ke(se) = bd[~(se),-.,~(se)]' kr(se) = (s;~i' -.,sekrK.) 1
G and H are So-parametrized nonlinear gain matrices,
and the i-th gain vector gain kr(1) is set so that the ith polynomial (9), one for each output, has stable poles: }..K; + kC }..K; -I + + k c}.. = 0 1:::; i :::; m (9) IKi
G(X, Xd ' p, so) = [0 I(X, Xd' p)JKo(so) H(X, Xd' p, sd) = [0 I(X, Xd' p)]neKI(so)
[0 1,0 nJ(X, Xd' p) = cp~I(X, Xd ' p)
11'
The resulting closed-loop evolution (11) are given by
dynamics
(10)
Ko(so) = bd[k?(so)' , ~ O(so)J
and
k?(so) = (sok?i' , So K;~), I KI(So) -- d'lag [K,+Ikl So P
;'0 = f{xO,d(t), a[xo,d{t),r,t],r}:= fe[xo,d{t),r,t] (lOa) yO = h(xO, r), XO(to) = xo' lId{t) E~(t) = {XO(t), UO(t), yO(t)}
VI
aCt) 11:::; td (lOb)
So K",+IIrI] ~
,
the gain k?(l) and the gain pair [k:?(I), kJJ are chosen, respectively, so that the polynomials (14a) and (14b) have stable roots:
(11)
The next lemma yields the closed-loop functioning of the exact state-feedback controller (8), or equivalently, the limiting functioning attainable with any estimator-based controller-
}.. V;
+
k4. 11 }.. V;
+ .• + Ird= 0, 1 < ~jI - i < - nd
}.. K;+I + k?i}.. K; + .. + ~'i}.. + kl=O, l:::;i:::;m
(14a) (14b)
1
Lemma 1 (Proof in Appendix). Let the nominal plant (I) evolution E(t) (3a) have a R-relative degree k... with an RE-stable ZD-motion x*(t) (6). Then, the state feedback controller (8) yields a closed-loop plant (10) where: (i) The motion XO(t) RE-converges to the nominal
When So and sd are tuned sufficiently large (fast) [3 to 15 times faster than the control dynamics (14)J, X(t) RE-converges to x(t) with q(quasi)LNPA output error dynamics, and Xm is a fast convergent estimate of the persistent mode ling errors in the input-output path of the estimator.
one x(t). (ii) The output YO(t) E-converges to the nominal one
Recall the state-feedback controller (8), replace its argument (x, a., r) by (X, Xd' p), subtract the state Xm (Bb) from the term /3 in Eq. (8), get the estimatorbased feedback controller (15d), combine it with the open-loop estimator (13), and obtain the following measurement-driven controller
yet), with LNPA (linear, noninteractive, poleassignable) output error dynamics (I :::; i :::; m)
e/K,-I)+se~K.ei(K,)+ +se K;~lei = 0, ei = Ye Yi' • (12) 1 4. CONTROLLER CONSTRUCTION In this section a measurement-driven controller is constructed by combining the state-feedback controller (8) with the robust PI-estimator presented in Alvarez and Lopez (1999).
(Ba)
Xm = [KI(so)] [y(t) - heX, p )], X m(to) = 0
(Bb)
(1 Sa)
Xm = [KI(so)][y(t) - hex, p)J, Xm(to) = 0
(15b)
X = fe[X,Xd,u(t),pJ + G(X,Xd,p,so)[y - h(X,p)J (l5c) u=q> -I {X,Xd,/3[X,Xd ,p,tJxm,p,t }=r(X,Xm,Xd,p,t)(15d)
For the moment, assume the open-loop nominal evolution (3a) is RE-estimable with observability (vector) degree k... (5), and recall Alvarez and Lopez' (1999) PI (proportional-integral) estimator: Xd = [Aisd)]Xd + [Kisd)]d(t); Sd' so> 0
Xd= [Aisd)JXd + [Kisd)]d(t),sd' so> 0
where fe is defined in (lOa). The cancellation of the input-output matching term Xm in equation (15c) signifies the possibility of compensating the modeling errors in the closed-loop system. 5. CLOSED-LOOP STABILITY
X = f[X,Xd,u(t),p] + G(X,Xd'P,so)[y - h(X,p)J + [H(X, Xm' p, so)] Xm (Bc)
The interconnection of the plant (I) with the candidate controller (IS) yields the following closedloop dynamics (16) and evolution (17):
397
Xd = [Aisd)]~ + [Kisd)]d(t)
(16a)
Xm = [KI(so)][y(t) - h(X, p)], Xm(to) = 0
(16b)
inputs are: initiator feedrate u 1, coolant temperature u2, and exit flowrate U3 ' The measured outputs are: Yl = X2 (monomer), Y2 = X3 (temperature), and Y3 = X4 (volume). The exogenous input is feed temperature d.
X = fe(X, Xd' p, t) + [G(X,~,p)][h(x,r) - h(X,p)](16c) X = fIx, d(t), Y(X, Xm , Xd ' p, t), r] Ee(t) ={ u(t), x(t), y(t)}
(16d) (17)
Next is given the main robust closed-loop stability result that underlies the proposed control design.
Theorem 1 (Proof in Appendix). Let the (possibly open-loop unstable) nominal evolution (3) of the nonlinear plant (1) have a R-relative degree K. (5) with a RE-stable Z-dynamics motion x*(t) (6). Then: (i) The closed-loop evolution (17) RE-converges to the nominal one x(t) (3), with q(quasi)LNPA output error dynamics, if the parameter pair (Sd' so) is tuned sufficiently large (fast) so that the potentially destabilizing estimation error terms described in Alvarez and Lopez (1999) are dominated. (ii) By making (Sd'so) sufficiently large, the closedloop evolution (17) can arbitrarily approach the one (11) of the exact state-feedback controller (8) . • While in the previous estimator-based geometric controllers the compensation of modeling errors is done in the slow control loop, here the same compensation is done in the fast estimation loop. The proposed controller can be seen either as a robust IM controller; or a continuous-time MP controller without input constraints, and with a suitable timeinfinite objective function. The open-loop estimator (13) is a fast convergent input-output model of the plant (1), and the estimator-based controller (15d) forces the input-output plant replica to track the prescribed nominal output, with almost LNPA output dynamics and with internal (state) stability.
Initially, the reactor is close to its unstable steadystate, and then subjected to the time-varying feedtemperature of Figure 2c. The thick plots of Figures 1 and 2 show the functioning of the exact statefeedback controller (8) when the measurements are free of noise. Using the gains given in Alvarez (1996), the reactor reaches, without offset, its nominal operation with a 2% settling time of 150 min. The thin plots of Figures 1 and 2 show the functioning of the measurement-driven controller (15) with the modeling errors used in Alvarez (1996), and with the noisy measurements of Figure 1. The control gains are the ones used in the exact statefeedback controller (8), the observer gains are the ones used in the robust open-loop estimator of Alvarez and Lopez (1999), and the estimator dynamics is set to be ten times faster than the control dynamics (i.e., So = lOSe)' As it can be seen in Figures 1 and 2, the measurement-driven controller reaches asymptotically (i.e., without offset) the nominal steady-state (which is open-loop unstable) with a settling time of about 250 min, meaning that the proposed controller (with modeling error compensation in the fast estimation loop) is 2.4 times faster than the controller with modeling error compensation in the slow control loop (Alvarez, 1996). 7. CONCLUSIONS A robust nonlinear geometric control design with solvability condition, stability criterion, systematic construction, and simple tuning procedure has been presented. The tuning of gains amounted to the tuning, with conventional-type techniques, of independent single-input controllers and singleoutput linear filters. Since the rejection of the modeling errors is accomplished in the (fast) estimation dynamics, by making sufficiently fast these dynamics, the performance of the measurementdriven controller can arbitrarily approach the one of the exact state-feedback controller, with the only limitation being the measurement noise. This is different from the existing geometric estimator-based control designs in chemical engineering, where the modeling errors have been rejected via integral action in the (slow) control loop. The proposed control scheme has been put in perspective with the IM and MP control frameworks. An homopolymer reactor was considered as an application example, showing that the proposed controller is faster and more robust than the existing estimator-based geometric controllers.
6. APPLICATION EXAMPLE Let us consider the free-radical homopolymer continuous reactor used in Alvarez and Lopez (1999) to illustrate the functioning of a robust open-loop estimator, and in Alvarez (1996) to illustrate the functioning of an estimator-based geometric controller (8) with mode ling error compensation via integral action in the control-loop. A strongly exothermic reaction takes place in the reactor, with heat exchange being enabled by a heating/cooling jacket. The reactor dynamics are described by a sixstate, three-input (u), three-output (y) nonlinear dynamical system of form (1). The maps f and h are given in Alvarez and Lopez (1999). The states of the plant are: concentration of initiator Xl, dimensionless monomer concentration X2, temperature X3, volume X4, zeroth moment xs, and second moment X6' The
The nonlocally robust convergence property observed in the application example can be explained theoretically with the notions of practical (La Salle
398
and Lefschetz, 1961) or semiglobal (Isidori, 1995) stability, as it has been done in the nonlocal estimation design of Alvarez et al. (1999); and related to the results on the role of the integral action in the region of stability (Pachter et aI. , 1996). In principle, the rigorous connection of the robust estimator-based controller with the MP controller requires the consideration of saturated inputs (Alvarez et aI., 1991), the case of more inputs than outputs, and the possibility of using a finite-time controller.
(Alb) Ae is a stable matrix (Alvarez, 1996), equation (Alb) with CJz = 0 is the zero-dynamics (6), and the map CJz is L(Lipschitz)-bounded. The RE-stability of the closed-loop system (AI or 10) follows from the application of Gronwall' s lemma, Lyapunov' s direct method, and the Comparison principle (Alvarez, 1996). QED.
REFERENCES
Proof of Theorem 1. Apply the coordinate change
Alvarez, 1., J.J. Alvarez, and R. Suarez, Nonlinear Bounded Control for a Class of Continuous Agitated Nonlinear Plants, Chem. Engng. Sci., 46(2),3235 (1991). Alvarez, 1., Output Feedback Control of Nonlinear Plants, AIChEJ, 2(9), 2540 (1996). Alvarez, J., and T. Lopez, Robust Dynamic State Estimation of Nonlinear Plants, AIChEJ 45(1), 107 (1999). Alvarez, J., T. Lopez, and E. Hemandez, Robust Estimation of Free-radical Homopolymer Reactors, IFAC 1999 World Congress, N, 121 (1999). Nonlinear Control of Chemical Processes: A Review, Ind. Chem. Eng. Res., 30, 1391 (1991). Bequette, B. W. , Nonlinear Control of Chemical Processes: A Review, Ind. Chem. Eng. Res., 30, 1391 (1991). Bymes, c., F. Delli Priscoli, and A. Isidori, Output Regulation of Uncertain Systems, Birkhauser, Boston (1997). Doyle, J., B. Francis, and A.R. Tannenbaum, Feedback Control Theory, Macmillan, New York (1992). Doyle Ill, F.F. Nonlinear Inferential Control for Process Applications, J. of Process Control, 8(5,6), 339 (1998). Isidori, A., Nonlinear Control System, Springer, New York (1995). La Salle, J., and S. Lefschetz, Stability by Liapunov' s Direct Method, Academic Press (1961). Pachter, M., JJ. D' Azzo, and M . Veth, Proportional and Integral Control of Nonlinear Reactors, Int. J. ofContr. , 64(4), 679 (1996). Phani, B.S, and W. Bequette, Nonlinear ModelPredictive Control: Closed-Loop Stability Analysis, AIChEJ, 42(12), 3388 (1996). Sepulchre, R., MJankovic, and P. Kokotovic, Constructive Nonlinear Control, Springer, New York (1997).
ed = Xd-d(t), e I =
en =
ed = Aded + ndSit)
[ei, .~ ' = rl A oT[ej, e~J'
(A2a) + SI(e, z, r, t)
(A2b)
~I = wz[en, r; z(t), r, t] + Sn(e, z, r, t)
(A2c)
ZI = AeZI + neqI(e, z, r, t),
(A2d)
zn = wcCzu, r, t)+CJz(Zd,Zh zn, r, t)+
(A2e)
where ne and nd are zero-one entry matrices (Alvarez, 1996), T is the identity permuted matrix (Alvarez and Lopez, 1999), and Ad and Ao are stable matrices (Alvarez and Lopez, 1999). Subsystem (A2c) with Sn = 0 is the zero-dynamics (7a) , and system (A2) with (Sd,ShSn,qhqn) = 0 is RE-stable because it is made of RE-stable decoupled subsystems. The "disturbances" Sd,ShSn,qh and
Proof of Lemma 1. Apply the coordinate change Zd = d(t) - /1(t)
Zn =
Xm - ~
to take the closed-loop plant (16) into the estimationtracking error form
APPENDIX: PROOFS OF LEMMA AND THEOREM 1
ZI =
Zm= qm
yo - yet)
to take the closed-loop plant (16) into the form :
399
(b)
33
o
50
100
150
200
250
300
t (time, min)
t (time, min)
Fig. 1 Noiseless (- ) and noisy (-) measurements. ,....,
"0
.~
0.50
">
"
0 ~
::J ;::,
o
0
e
~
(a)
..: 0.002
:i
0
.]j
0.002
·m .Oxlo
g 0.002
.b .5XIO
...
~
0
e
0.001
"e0
?OxIO ~ ,....,8.5xIO
0
.§
"C
~
'" ,5,
8-
N
><
:0 >. "0
,e,
Q
If
!c..
0'
0.57
Q
i
"e ~ ....><
;:
e 8e .!l
310 305
"0
" ~
::J
300
"0
.;
e::I
295
"0
t (time, min)
G. ~
>< 195 0
50
100
150
200
250
Fig 2. (a) State set {Xl. X2, X3, X4}, (b) control inputs u, as weII as (c) exogenous input d and safety-quality variables {c, M n , Q} for: the exact state-feedback controller with noiseless measurements (=), and for the measurement-driven controIIer with modeling errors and noisy measurements (-).
300
t (time, min)
4()()