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Non-Linear Discrete Time Modelling and Control of a Pilot Neutralization Chemical Reactor
Copnighl © IFAC Conlrol of Di slillalit>ll Columns alld Chemical Reactors. Bournemoulh. l" K 19Hti
NON-LINEAR DISCRETE TIME MODELLING AND CONTROL OF A PILOT NEUTRALIZATION CHEMICAL REACTOR G. Gilles, N. Laggoune, B. Neyran and D. Thomasset C.R.A.P. (Croupe de Recherche RhOlle-Alpill en Autumatique et ses ApplicatiullS 1, -13 boulevard du 11 Ilovembre 1918, 69622 Villeurballne, Frallce alld Laboratoire d'Ellergetique et d'Automatique, INSA de LyulI 20 avenue Albert EillSteill, 69621 Villeurballlle, Frallce
a la ProductiulI), Laboratuire d'Autumatique, L'lIi,'ersite L)'oll
Abstract. Many continuous industrial processes working inside a large operating range can be modelled by variable parameter and/or bilinear continuous state equations. In the case of processes with variable parameters, the state affine models allow to get a discrete approximated representation. An original direct identification method, involving a variable parameter filter, is processed inside the whole operating range in order to estimate the model coefficients . A dual approach leads to state affine control algorithms by gathering a family of linear controllers into a unique control law. In the case of bilinear continuous processes, their discrete model is of linear time-varying type. A way for digitally controlling them can be a closed loop linearization which leads to an implicit control law. An improvement can be obtained by adding integrators. Experimental results got, in each operating case (variable parameter or bilinear system), from a pilot neutralization reactor show the applicability of the methods. Keywords. Digital control control ; optimal control
non-linear control systems ; time-varying systems pH control.
bilinear
INTRODUCTION The continuous industrial p lant s dynamically working inside a large operating range involve non-linear phenomena that mostly cannot be suitably approached by linear models. In order to describe their beha viour, it is necessary to use non-linear models and/ or variable parameter models . The control of such systems can be managed from adaptive methods or mul ti-model techniques but it may be preferable to try to find a global non-linear model correctly describing the system behaviour in a large operating range and a unique control law. l1oreover, the goal being the control by means of one or several digital processors, it is necessary to build a discrete non-linear and/or time-varying parameter model. In this paper, two methods for discrete modelling and control are presented in the domain of conti nuous industrial processes which are linear with time-varying parameters or bilinear. In the case of continuous processes with variable parameters, the state affine models , linear in the state and polynomial with respect to the extended inputs allow to get a discrete approximated representation. An original direct identification method operating inside the whole range is used in order to find the model coefficients : it involves a least square method and a variable parameter filter. A dual approach allowed us to propose a synthesis method for state affine control algorithms : the technique consists in gathering a family of linear controllers into a unique control law described by a state affine sys tem for which some closed loop characteristics can be considered as quasi-invariant. In the case of
through experimental results got from a pilot neutralization reactor working in each operating case:
variable parameter system or bilinear system. PRESENTATION OF THE CHEMICAL PILOT PLANT The chemical pilot plant (Fig. 1) is a continuous neutralization reactor involving a strong base (Na OH) and a strong acid (HCl). For a constant acid flow-rate QA' the base flow-rate QB(t) is chosen as control variable, the output variable being concentration C(t) of H+ ions obtained by means of a pH-meter.
COIl P L: T ( R
v c
continuous bilinear processes, we get a discrete
model which is linear with variable parameters depending upon the inputs. By applying the idea of closed-loop linearization, we get a linearizing control via non- linear state feedback . The corres ponding equation is implicitly solved during each sampling period by means of numerical analysis methods. It is shown how an improvement of the control law performances with respect to modelling errors can be obtained by adding an integrator. The applicability of those methods is illustrated
v
Fig. 1
129
Chemical pilot plant
z
(;. Gilles 1'1 al.
1:~()
In order to get a model independant from the expe-
Control
rimental conditions, we consider:
c (t) =
-C (t)
with :
Co
initial value of C(t) bas e concentra tion
Around a working point defined by QBO and c(o), the infinitesimal mass balance leads to a dynamic model, between the base flow-rate deviation q(t) and the variation of the output normalized concentration c( t), under the following form:
v QO with
dc(t) dt
q(t) _ c(t) _ q(t).c(t) QO QO
(1)
QO QA + QBO V reactor volume
Moreover, we must consider a time- delay (30 sec) on the output concentration measurement because of the pH-meter location. Equation (1) represents a bilinear system with time-varying parameters depending on the reactor volume (as a rule, QO is constant) . In this paper, we consider two different approaches of this process corresponding with two different working conditions : - in the case of small deviations of the control variable around the operating point and when the volume is time-varying, the system is linear with variable parameters. Then, the use of state - affine model leads to a unique control law available for al l reactor volumes (section 3) - with a constant volume working condition and for large deviations of the control variable, the system must be considered as a bilinear one and section 4 presents a linearizing control via nonlinear state feedback. IDENTIFICATI0!l AND OPTIMAL CONTROL FOR TIME-VARYING VOLlJllE WORKING CONDITION In order to get a global model suitable for all reactor volumes and to desi gn a unique control law, we use the theoretical results on approxima tion of non- linear discrete time systems by state-affine models (Fliess and Normand-Cyrot, 1980)
For each value of the reactor volume, the sys t em is linear. Then, the control law of the state- affine model is necessarily a linear one depending on the measurable parameter V and it seems very difficult to build directly this control law. Therefore, we can app r oximate this law by using the same approach in control than Dang Van Mien and Normand-Cyrot used in identification (Du four and Thomasset, 1984) . For N values of V, denoted V , chosen in (VlUN ' VMAX )' the chemical pilot pl~nt has a linear behaviour and there are many methods to design a linear control law . We chose one of the most efficien t , the optima l control with integrator and reference model (Foulard and co - workers, 1977) . For each value, V ' a the control law q(k) is given by : q(k) = - L(V ) .w(k) - H(Va).r
(3)
where: {W(k) r(k) s(k) z(k)
is the state of the process is the output of the integrato r is the state of the reference model is the reference input L~Va) , .M(V a ) , P(V a ), N(V a ) are appropriate dlmenSlonal matrlces .
Just as we built the state- affine model of the process and the state-affine reference model, we can get, by means of multilinear regressions computed from the values of the control matrices L, M, P, N at different working points V , the variation of these matrices with respe c t tg the variable reactor volume V. So, we can write the £, ,1T1.. , 9 , J'(' matrices under the following forms : °1
et = 1:
j=O
0 1'
~ .
J
v j ;'T7l=
.° 2 , ° 3 , ° 4 ,
°2 1: j=O
j m. v ; J
cP
°3 j = 1: p. V j=O J
regressions orders, are chosen to
mlnlmlze a performance criterion.
Identification The searched state- affine model is on the following form : 3 x(k+ l ) = (A +A ·V(k)+ . . . +A · V (k)+B ·q(k)+B .q(k) 1 o o 1 3 3 + ... +B .q(k) .V (k») .x(k) (2) 3
The following figure describ es the complete struc re of the control.
;---------~
<;(k)
1
y(k) = C. x(k)
with q(k) control variable and V(k) measurable vo lume which is an extended input. The parameter estimation of th i s state-affine model can be achieved by means of two different approa ches. The first developped by Dang Van Hien and Normand - Cyrot (Dang Van Mien and Normand - Cyrot, 1980, 1984, Neyran and co-workers, 1984) consists in gathering in the state- affine system by means of multilinear regressions a set of linear identi fications obtained for N values of V belonging to (VMIN' V11AX )· An other method presented by the authors (Dadugineto, 1985) is a more direct approach. It allows to build the state - affine model from an input-output sequence obtained du r ing a variation of V belonging to (VHIN , VMAX ). The least square estimation is biased in the case of an additive noise on the output . The use of a time-varying li near filter which has the same structure as the state-affine model, allows to reach and unbiased estimation . Applied to the chemical process, the direct approach gives better results than the othe r (Neyran and co - workers 1985) .
1.01.)+
Fig. 2
Structure of
the state-affine control
Notice that the estimated state of the process is obtained by means of the internal state-affine mo del issued from the identifica tion. In the same way, the reference model is a state affine one with res pect to the extended inputs (z(k) - Ea(k») and V(k). Application to the chemical pilot pl ant Fig. 3 shows the output concentration evolution with
1:\1
Pilot :--leutralization Chemica l Reactor respect t o a serie of reference input steps during a linear time variation of the reactor volume. In order to improve the robustness of the stat e-affine control, we compare three control laws: two linear optimal controls computer for V = 4 £ and V = 12 £ and the state- affine one. Let us notice that only this control law giv es good results for all reactor capacities.
sed loop system, we want to prescribe the system performances by setting a closed loop sampled data transfert function under the fo ll owing form : X(z) H(z) = E(z)
(l-d)z-l -1 1-dz in order to reach the step input during the steadystate working, the d pole setting the response time. The corresponding closed loop state model is
LINEARIZING CONTROL FOR BILINEAR WORKING CONDITION When the neutralization reactor is operating under a constant volume, and when the input range is large, it has been shown that the model (1) becomes bilinear and can be written under the following form : (4) x = ax + bu + nxu Numerous works have been devoted to bilinear sys tems but they have been more related to the analy sis of their properties than to their control. Several types of control have been proposed: "bang - bang' (Mohler, 1973), stabilization by Lyapunov functions (Gauthier and Bornard, 1980), immersion into another systems (Claude, Fliess and Isidori, 1983), model matching (Di Benedetto and Isidori, 1984), pseudo-l inearization (Mouyon, Champetier and Reboul let, 1984). Very rare are the papers dealing with concrete applications : let us mention a recent paper (Cebuhar and Castanza, 19 84) related to the control of chemical reactors by minimizing quadratic criterions. A linearizing control, which minimizes a quadratic criterion with variable weighting matrices (Gilles, 1983), involving a non-linear state feedback, has been presented by the authors and applied to this neutralization reactor (Gilles and Laggoune, 198sa). Much rarer are the works related to digital computer control of continuous bilinear processes. Nevertheless, we can mention the
paper of Alvarez Gallegos brothers (1982), upon digital control of fermentation processes modelled by a larger class of systems : models linear in control ; the criterion which is minimized by the con trol law is still a classical quadratic criterion. More recently, the authors (Gilles and Laggoune, 198sb)proposed an implicit digital linearizing control law for multivariable continuous bilinear systems. Based on a discrete time variable parameter model of bilinear systems, the method is recalled for the monovariable model (4) in the next two paragraphs and then twice extended. Discrete model of the continuous bilinea r system Systems modelled by equation (4) and controlled by a digital computer with a zero hold have their evo lution described by :
~(t) = (a + nu ) x(t) + bU k k with u = const. inside the sampling period k kT ::: t < (k+llT.
(6)
and, comparing to the open loop model (5), we get the equation : f(u ) k
this scalar problem and if the sampling period is sma 11 enough . P.I. Discrete linearizing control Due to the closed loop s tat e model form, the static error corresponding to a step input is theoretically equal to zero. Nevertheless, in practice, because of modell ing errors, the "perfect model following" is not guarante ed and static errors can occur. That is why it is preferable to systematically introduce an integrator in the control law in order to cancel static errors .
The introduction of integrators can be managed in the c lassical way, like in linear quadratic optimal problems (Foulard and co - workers, 1977) by augmenting the dimension of the state vector -+ ~k = (x ' v ) t with v + = v + (e - x k ) k k k k l k k and recomputing the control law in terms of the procedin ~ one . Nevertheless, by applying Zdan's method, we notice that we are not able t o independantly set the overshoot time and magnitude of the closed loop second order system. It is much more preferable t o control by means of a P.I. controller, within a se cond loop, the preceding linearized system (Eq . 6); we can then apply the classical linear system theory after the linearizing control law. Inserting the P.I. transfer function: K + [Ki/( l-z-l ) ] leads to the overall closed loop transfer function
x
(l -d )(K>'i.)z -l [1- z-l /CK+K ) ] i l+ [( l-d) (K+K. )-(l+d) ] z -l+[d-(l-d)K] z 2 1
which must be compa r ed with Zdan ' s form:
(5)
Discrete linearizing control By means of thi s discrete model, it is possible to find a control law which allows the continuous bilinear system controlled by a digital computer to get the pleasant property of global linearity. Remaining the same state variable xk=sk for the clo -
(7)
ton-Raphson's one, is not necessary in the case of
H(z) = Az
which is linear with variable parameters depending upon the control variable setting the operating point.
+[
If we can get the solution uk of this equation, we obtain a non- l inear s tate f eedback contr o l law . But, contrary to the continuous linearizing control case, equation (7) does not lead to an explicit control law. It has to be solved, during each sampling period, by nwr.er ical ana ly si s procedures. For example, the successive substitutions method a ll ows to solve equation (7) by means of the iterative al gorithm [d- H uk,i)] x k +(l -d )e k (8) u k , i + 1 = F (u k , i) = Y (u , i) k The use of a faster convergence algorithm, like New-
H(z)= The system is then piecewise linear and its solution involves the transition matrix: b[~ (uk,T)-l]
= Y(u k ) .uk
with
-1
l ( 1 - Bz- )
(sine 0 A=p l---8--- Log p - cos e + o 0 0 0 Po
1,
C
sin8
B=
--e;-o Log Po
+cos 80+po
sin8 o Log p0 -co s e + ~ ---8--o 0 Po
, D
the conjugate poles poexp( :j 80) being related to the overshoot magnitude D and the overshoot time NT by the relationships : (9) TT /N Po = NvfD and 8 o
G. (;illes
132
o
and N being chosen, the following formulas (got from the comparison) allow to tune the controller parameters : 2 sin e d- p o 0 d= Po(---e--Log Po+Cos eo),k= 1-d o
We can notice that, not only k and k. are set, but d is also constrained to a numerical'value. Linearizing control of the delayed bilinear system The neutralization reactor involving, as many industrial processes, a time delay, it is important
to use a model taking into account this time delay for a more precise linearization.
In the case of the reactor delay equal to 2 sampling periods, the state dimension is augmented to 3 and the discrete model becomes
o The prescribed closed loop transfer function being now: H(z)= 1(1-d)z- 3 1/ ! 1-dz- 1 I, the corresponding state model is :
o o
~J ~k
d
+ [
~
1-d
J e k ; sk
pi
Ill.
linear systems. They have been applied on a neutralization pilot reactor for the two classes of operating conditions and the results can be considered as rather satisfactory. In the case of time varying working condition, it has bee n shown that a state-affine control lalv allOlols to keep a desirable closed-loop behaviour all over the working volume. The extension of this method to a multivariable approach of the pilot plant is under investigation. In the case of the bilinear working condition, it has been shown that a digital closed loop linearization leads to a state feedback control law which, if the model is exact, guarantees the same dynamic behaviour inside the whole operating range. In the practical case of modelling errors, improvements have been pointed out by designing a digital P.I. non-linear controller. In the case of processes involving time delays, it has been shown how the linearizing control law and the Smith controller can be mixed in order to solve the problem. In order to improve the performances, and specially the robustness, it would be interesting to use a reference model following structure and to study the static error cancellation by introducing a parallel internal model. Finally, theses works could be fruitfully extended for systems presenting both a bilinear structure and a parametric variation in their operating condition. Again, the pilot reactor can be, in this case,
and, by identification, we get: ( 11) Y(uk)·u k + ~(uk)-dJx3(k)-(1-d)ek = 0 The implicit equation (7) is just slightly modified, x1(k) being 2T delayed: x3(k) = s(k-2). The controller which is obtained is an extension for bilinear systems of the well-known Smith's controller.
Experimental results All experiments are processed for an initial total flow-rate Qo=QAo+QBo=55Q./h and for a pH around 2.5. A reference corresponding to xc=0.1 is equivalent to a pH ref erence of 0.3. The sampl ing period is T=15sec. A first feedback yields to a linear system from a bilinear one, and a second feedback realizes a P.I control, allowing a cancellation of static errors. Moreover, we master the response time by specifing the overshoot magnitude D(%) and the first overshoot time NT. Fig. 4 shows the normalized concentration evolution for two desired response times (0=5%, N=20 or N=15). The static error is perfectly cancelled. The obtained overshoot is greater than the desired one, but the first overshoot time is rather good. Fig. 5 shows the evolution of the normalized concentration and the base flow-rate, with respect to a greater step magnitude. The same remarks, as in Fig. 4, can be mentioned. The overshoot differences with respect to the theory are due to the fact that, the tuning parameters (d, K, Ki) are calculated from the system model mich involve some identification errors. Bilinear delayed system In Fig. 6, we implement, the "Smith non-linear algorithm" (Eq. 11) on the process. The desired close:! loop response is a two sampling periods delayed first order one. The result is rather satisfactory, but the delay is too small (10% of the approximated time constant) to point out a real improvement of the closed-loop linearization with respect to the non-delayed system. CONCLUSION AND PROSPECTS Several digital control methods have been proposed for two classes of non-linear continuous processes: linear system with time-varying parameters and bi-
a very good testing device. REFERENCES Alvarez Gallegos, J. (1982). Optimal control of a class of discrete multivariable non-linear systems. Application to a fermentation process, Journal of Dynamic Systems, Measurement and Control, vol. 104, pp. 212-217. Cebuha~Costanza, V. (1984). Non-linear control of CSTR's, Chemical Engineering Science, vol. 39, 12, pp. 1715-1722. Claude, D., Fliess, M., Isidori, A. (1983). Inunersion directe et par bouclage d'un systeme nonlineaire dans un lineaire, Compte-rendu a l'Academie des Sciences, serie 1, t. 296, pp.237240. Oadugineto, (1985). Identification de differents modeles di sc rets non-lineaires continus. Applicati on aux procedes industri e ls pilotes, Conf e rence internationale CNRS Automatique n0n= lineaire, p,qris. Oang Van Mien, H., Normand-Cyrot, O. (1984). Nonlinear state-affine identification methods. Application to electrical power plants, I.F . A.C. symposium an Aut. Control on Power generation, Distribution and Protection, Pretoria,
Automatica, vol. 20, n° 2. Dufour, J., Thomasset, D. (1984). Definition and synthesis of state-affine control algorithms. Application to vary-linear systems, IEEE Conf. on Computers, Systems and signal Processy, Bangalore, India. Fliess, H., Normand-Cyrot, D. (1980). Vers une approche algebrique des systemes non-lineaires en temps discret, Analysis and optimization of systems, Versailles, Lecture notes control Inform. Sciences, n° 28, Springer-Verlag. Gauthier, J-P., Bornard, G. (1980). Stabilization of bilinear systems. Performance specification and optimality, 4th INRIA international conf. on Control and Optimization, Versailles, France. Gilles, G., Laggoune, N. (1985a). Linearizing control of a class of non-linear continuous processes,
IFAC conf. CSTO'85, Beijing, China. Gilles, G., Laggoune, N. (1985b). Digital control of bilinear continuous processes. Application to a chemical pilot plant, IFAC conf . on Digital Computer Application to Process Control, Vienna, Austria.
133
Pilot Neutralization Chemical Reactor Mohler, R. (1973). Bilinear control processes, Academic press. -Neyran, B., Saade-Castro, J., Thomasset, D., Reynaud, R. (1984). Modelisation dynamique non-l ineaire d'un procede chimique de neutralisation, Congres international S.E.E., Nice.
Neyran, B., Thomasset, D., Dufour, J. (1985). Direct identification of a class of nonlinear systems. Application to a neutralization process. IFAC Conf. CSTD'8S, Beijing, China.
, Cone
~:
v~
linear control computed at
*: linear control computed at
4
V~12
.: state-affine control
.1t---J--------'O"'~~~
.05
1000
0
T i m e (sec::)
V(I)
12 8 4
1000
2000 T i m e (sec::)
Fig.3
Comparison between linear and state-affine control
x
STEP REFERENCE MEASURED OUTPUT (N=20) ~ffiASURED
OUTPUT (N=1S)
O~~4-~----~--------~-------.--------~~ I.
O.
Fig. 4. P-I. linearizing control
600·· C
two different dynamics
G. Gi lles et al.
134
.
x
.
r - - - - - - - - - - - - '- .....- ... ,-" - - ~ ~.- ....... - - ' T
0.1
... -
STEP REFERENCE MEASURED OUTPUT
a) O~~+-------,-------.--------,-------r--~ 600 •• e o
60 I/h
Q.
40
b) 20~L--.--------r--------.--------r--------r~ 600·· e o
Fig. 5 . a) Normalized concentration for gr ea t er step magnitude b) Con trol variable (base flow-rate) evo luti on
x 0 .1
,.. - - - - - - - - - -
- - .-. - - ;.- -. -
~.~
: .-., ..
-
-: -
-
,-.-.-.
STEP REFERENCE MEASURED OUTPUT
O~-r-r~------'---------r--------,---------,~_ o 600 •• e:
Fig . 6. Cl osed loop linearization with de layed response (smith non-linear con troller)
NON-LINEAR DISCRETE TIME MODELLING AND CONTROL OF A PILOT NEUTRALIZATION CHEMICAL REACTOR G. Gilles, N. Laggoune, B. Neyran and D. Thomasset C.R.A.P. (Croupe de Recherche RhOlle-Alpill en Autumatique et ses ApplicatiullS 1, -13 boulevard du 11 Ilovembre 1918, 69622 Villeurballne, Frallce alld Laboratoire d'Ellergetique et d'Automatique, INSA de LyulI 20 avenue Albert EillSteill, 69621 Villeurballlle, Frallce
a la ProductiulI), Laboratuire d'Autumatique, L'lIi,'ersite L)'oll
Abstract. Many continuous industrial processes working inside a large operating range can be modelled by variable parameter and/or bilinear continuous state equations. In the case of processes with variable parameters, the state affine models allow to get a discrete approximated representation. An original direct identification method, involving a variable parameter filter, is processed inside the whole operating range in order to estimate the model coefficients . A dual approach leads to state affine control algorithms by gathering a family of linear controllers into a unique control law. In the case of bilinear continuous processes, their discrete model is of linear time-varying type. A way for digitally controlling them can be a closed loop linearization which leads to an implicit control law. An improvement can be obtained by adding integrators. Experimental results got, in each operating case (variable parameter or bilinear system), from a pilot neutralization reactor show the applicability of the methods. Keywords. Digital control control ; optimal control
non-linear control systems ; time-varying systems pH control.
bilinear
INTRODUCTION The continuous industrial p lant s dynamically working inside a large operating range involve non-linear phenomena that mostly cannot be suitably approached by linear models. In order to describe their beha viour, it is necessary to use non-linear models and/ or variable parameter models . The control of such systems can be managed from adaptive methods or mul ti-model techniques but it may be preferable to try to find a global non-linear model correctly describing the system behaviour in a large operating range and a unique control law. l1oreover, the goal being the control by means of one or several digital processors, it is necessary to build a discrete non-linear and/or time-varying parameter model. In this paper, two methods for discrete modelling and control are presented in the domain of conti nuous industrial processes which are linear with time-varying parameters or bilinear. In the case of continuous processes with variable parameters, the state affine models , linear in the state and polynomial with respect to the extended inputs allow to get a discrete approximated representation. An original direct identification method operating inside the whole range is used in order to find the model coefficients : it involves a least square method and a variable parameter filter. A dual approach allowed us to propose a synthesis method for state affine control algorithms : the technique consists in gathering a family of linear controllers into a unique control law described by a state affine sys tem for which some closed loop characteristics can be considered as quasi-invariant. In the case of
through experimental results got from a pilot neutralization reactor working in each operating case:
variable parameter system or bilinear system. PRESENTATION OF THE CHEMICAL PILOT PLANT The chemical pilot plant (Fig. 1) is a continuous neutralization reactor involving a strong base (Na OH) and a strong acid (HCl). For a constant acid flow-rate QA' the base flow-rate QB(t) is chosen as control variable, the output variable being concentration C(t) of H+ ions obtained by means of a pH-meter.
COIl P L: T ( R
v c
continuous bilinear processes, we get a discrete
model which is linear with variable parameters depending upon the inputs. By applying the idea of closed-loop linearization, we get a linearizing control via non- linear state feedback . The corres ponding equation is implicitly solved during each sampling period by means of numerical analysis methods. It is shown how an improvement of the control law performances with respect to modelling errors can be obtained by adding an integrator. The applicability of those methods is illustrated
v
Fig. 1
129
Chemical pilot plant
z
(;. Gilles 1'1 al.
1:~()
In order to get a model independant from the expe-
Control
rimental conditions, we consider:
c (t) =
-C (t)
with :
Co
initial value of C(t) bas e concentra tion
Around a working point defined by QBO and c(o), the infinitesimal mass balance leads to a dynamic model, between the base flow-rate deviation q(t) and the variation of the output normalized concentration c( t), under the following form:
v QO with
dc(t) dt
q(t) _ c(t) _ q(t).c(t) QO QO
(1)
QO QA + QBO V reactor volume
Moreover, we must consider a time- delay (30 sec) on the output concentration measurement because of the pH-meter location. Equation (1) represents a bilinear system with time-varying parameters depending on the reactor volume (as a rule, QO is constant) . In this paper, we consider two different approaches of this process corresponding with two different working conditions : - in the case of small deviations of the control variable around the operating point and when the volume is time-varying, the system is linear with variable parameters. Then, the use of state - affine model leads to a unique control law available for al l reactor volumes (section 3) - with a constant volume working condition and for large deviations of the control variable, the system must be considered as a bilinear one and section 4 presents a linearizing control via nonlinear state feedback. IDENTIFICATI0!l AND OPTIMAL CONTROL FOR TIME-VARYING VOLlJllE WORKING CONDITION In order to get a global model suitable for all reactor volumes and to desi gn a unique control law, we use the theoretical results on approxima tion of non- linear discrete time systems by state-affine models (Fliess and Normand-Cyrot, 1980)
For each value of the reactor volume, the sys t em is linear. Then, the control law of the state- affine model is necessarily a linear one depending on the measurable parameter V and it seems very difficult to build directly this control law. Therefore, we can app r oximate this law by using the same approach in control than Dang Van Mien and Normand-Cyrot used in identification (Du four and Thomasset, 1984) . For N values of V, denoted V , chosen in (VlUN ' VMAX )' the chemical pilot pl~nt has a linear behaviour and there are many methods to design a linear control law . We chose one of the most efficien t , the optima l control with integrator and reference model (Foulard and co - workers, 1977) . For each value, V ' a the control law q(k) is given by : q(k) = - L(V ) .w(k) - H(Va).r
(3)
where: {W(k) r(k) s(k) z(k)
is the state of the process is the output of the integrato r is the state of the reference model is the reference input L~Va) , .M(V a ) , P(V a ), N(V a ) are appropriate dlmenSlonal matrlces .
Just as we built the state- affine model of the process and the state-affine reference model, we can get, by means of multilinear regressions computed from the values of the control matrices L, M, P, N at different working points V , the variation of these matrices with respe c t tg the variable reactor volume V. So, we can write the £, ,1T1.. , 9 , J'(' matrices under the following forms : °1
et = 1:
j=O
0 1'
~ .
J
v j ;'T7l=
.° 2 , ° 3 , ° 4 ,
°2 1: j=O
j m. v ; J
cP
°3 j = 1: p. V j=O J
regressions orders, are chosen to
mlnlmlze a performance criterion.
Identification The searched state- affine model is on the following form : 3 x(k+ l ) = (A +A ·V(k)+ . . . +A · V (k)+B ·q(k)+B .q(k) 1 o o 1 3 3 + ... +B .q(k) .V (k») .x(k) (2) 3
The following figure describ es the complete struc re of the control.
;---------~
<;(k)
1
y(k) = C. x(k)
with q(k) control variable and V(k) measurable vo lume which is an extended input. The parameter estimation of th i s state-affine model can be achieved by means of two different approa ches. The first developped by Dang Van Hien and Normand - Cyrot (Dang Van Mien and Normand - Cyrot, 1980, 1984, Neyran and co-workers, 1984) consists in gathering in the state- affine system by means of multilinear regressions a set of linear identi fications obtained for N values of V belonging to (VMIN' V11AX )· An other method presented by the authors (Dadugineto, 1985) is a more direct approach. It allows to build the state - affine model from an input-output sequence obtained du r ing a variation of V belonging to (VHIN , VMAX ). The least square estimation is biased in the case of an additive noise on the output . The use of a time-varying li near filter which has the same structure as the state-affine model, allows to reach and unbiased estimation . Applied to the chemical process, the direct approach gives better results than the othe r (Neyran and co - workers 1985) .
1.01.)+
Fig. 2
Structure of
the state-affine control
Notice that the estimated state of the process is obtained by means of the internal state-affine mo del issued from the identifica tion. In the same way, the reference model is a state affine one with res pect to the extended inputs (z(k) - Ea(k») and V(k). Application to the chemical pilot pl ant Fig. 3 shows the output concentration evolution with
1:\1
Pilot :--leutralization Chemica l Reactor respect t o a serie of reference input steps during a linear time variation of the reactor volume. In order to improve the robustness of the stat e-affine control, we compare three control laws: two linear optimal controls computer for V = 4 £ and V = 12 £ and the state- affine one. Let us notice that only this control law giv es good results for all reactor capacities.
sed loop system, we want to prescribe the system performances by setting a closed loop sampled data transfert function under the fo ll owing form : X(z) H(z) = E(z)
(l-d)z-l -1 1-dz in order to reach the step input during the steadystate working, the d pole setting the response time. The corresponding closed loop state model is
LINEARIZING CONTROL FOR BILINEAR WORKING CONDITION When the neutralization reactor is operating under a constant volume, and when the input range is large, it has been shown that the model (1) becomes bilinear and can be written under the following form : (4) x = ax + bu + nxu Numerous works have been devoted to bilinear sys tems but they have been more related to the analy sis of their properties than to their control. Several types of control have been proposed: "bang - bang' (Mohler, 1973), stabilization by Lyapunov functions (Gauthier and Bornard, 1980), immersion into another systems (Claude, Fliess and Isidori, 1983), model matching (Di Benedetto and Isidori, 1984), pseudo-l inearization (Mouyon, Champetier and Reboul let, 1984). Very rare are the papers dealing with concrete applications : let us mention a recent paper (Cebuhar and Castanza, 19 84) related to the control of chemical reactors by minimizing quadratic criterions. A linearizing control, which minimizes a quadratic criterion with variable weighting matrices (Gilles, 1983), involving a non-linear state feedback, has been presented by the authors and applied to this neutralization reactor (Gilles and Laggoune, 198sa). Much rarer are the works related to digital computer control of continuous bilinear processes. Nevertheless, we can mention the
paper of Alvarez Gallegos brothers (1982), upon digital control of fermentation processes modelled by a larger class of systems : models linear in control ; the criterion which is minimized by the con trol law is still a classical quadratic criterion. More recently, the authors (Gilles and Laggoune, 198sb)proposed an implicit digital linearizing control law for multivariable continuous bilinear systems. Based on a discrete time variable parameter model of bilinear systems, the method is recalled for the monovariable model (4) in the next two paragraphs and then twice extended. Discrete model of the continuous bilinea r system Systems modelled by equation (4) and controlled by a digital computer with a zero hold have their evo lution described by :
~(t) = (a + nu ) x(t) + bU k k with u = const. inside the sampling period k kT ::: t < (k+llT.
(6)
and, comparing to the open loop model (5), we get the equation : f(u ) k
this scalar problem and if the sampling period is sma 11 enough . P.I. Discrete linearizing control Due to the closed loop s tat e model form, the static error corresponding to a step input is theoretically equal to zero. Nevertheless, in practice, because of modell ing errors, the "perfect model following" is not guarante ed and static errors can occur. That is why it is preferable to systematically introduce an integrator in the control law in order to cancel static errors .
The introduction of integrators can be managed in the c lassical way, like in linear quadratic optimal problems (Foulard and co - workers, 1977) by augmenting the dimension of the state vector -+ ~k = (x ' v ) t with v + = v + (e - x k ) k k k k l k k and recomputing the control law in terms of the procedin ~ one . Nevertheless, by applying Zdan's method, we notice that we are not able t o independantly set the overshoot time and magnitude of the closed loop second order system. It is much more preferable t o control by means of a P.I. controller, within a se cond loop, the preceding linearized system (Eq . 6); we can then apply the classical linear system theory after the linearizing control law. Inserting the P.I. transfer function: K + [Ki/( l-z-l ) ] leads to the overall closed loop transfer function
x
(l -d )(K>'i.)z -l [1- z-l /CK+K ) ] i l+ [( l-d) (K+K. )-(l+d) ] z -l+[d-(l-d)K] z 2 1
which must be compa r ed with Zdan ' s form:
(5)
Discrete linearizing control By means of thi s discrete model, it is possible to find a control law which allows the continuous bilinear system controlled by a digital computer to get the pleasant property of global linearity. Remaining the same state variable xk=sk for the clo -
(7)
ton-Raphson's one, is not necessary in the case of
H(z) = Az
which is linear with variable parameters depending upon the control variable setting the operating point.
+[
If we can get the solution uk of this equation, we obtain a non- l inear s tate f eedback contr o l law . But, contrary to the continuous linearizing control case, equation (7) does not lead to an explicit control law. It has to be solved, during each sampling period, by nwr.er ical ana ly si s procedures. For example, the successive substitutions method a ll ows to solve equation (7) by means of the iterative al gorithm [d- H uk,i)] x k +(l -d )e k (8) u k , i + 1 = F (u k , i) = Y (u , i) k The use of a faster convergence algorithm, like New-
H(z)= The system is then piecewise linear and its solution involves the transition matrix: b[~ (uk,T)-l]
= Y(u k ) .uk
with
-1
l ( 1 - Bz- )
(sine 0 A=p l---8--- Log p - cos e + o 0 0 0 Po
1,
C
sin8
B=
--e;-o Log Po
+cos 80+po
sin8 o Log p0 -co s e + ~ ---8--o 0 Po
, D
the conjugate poles poexp( :j 80) being related to the overshoot magnitude D and the overshoot time NT by the relationships : (9) TT /N Po = NvfD and 8 o
G. (;illes
132
o
and N being chosen, the following formulas (got from the comparison) allow to tune the controller parameters : 2 sin e d- p o 0 d= Po(---e--Log Po+Cos eo),k= 1-d o
We can notice that, not only k and k. are set, but d is also constrained to a numerical'value. Linearizing control of the delayed bilinear system The neutralization reactor involving, as many industrial processes, a time delay, it is important
to use a model taking into account this time delay for a more precise linearization.
In the case of the reactor delay equal to 2 sampling periods, the state dimension is augmented to 3 and the discrete model becomes
o The prescribed closed loop transfer function being now: H(z)= 1(1-d)z- 3 1/ ! 1-dz- 1 I, the corresponding state model is :
o o
~J ~k
d
+ [
~
1-d
J e k ; sk
pi
Ill.
linear systems. They have been applied on a neutralization pilot reactor for the two classes of operating conditions and the results can be considered as rather satisfactory. In the case of time varying working condition, it has bee n shown that a state-affine control lalv allOlols to keep a desirable closed-loop behaviour all over the working volume. The extension of this method to a multivariable approach of the pilot plant is under investigation. In the case of the bilinear working condition, it has been shown that a digital closed loop linearization leads to a state feedback control law which, if the model is exact, guarantees the same dynamic behaviour inside the whole operating range. In the practical case of modelling errors, improvements have been pointed out by designing a digital P.I. non-linear controller. In the case of processes involving time delays, it has been shown how the linearizing control law and the Smith controller can be mixed in order to solve the problem. In order to improve the performances, and specially the robustness, it would be interesting to use a reference model following structure and to study the static error cancellation by introducing a parallel internal model. Finally, theses works could be fruitfully extended for systems presenting both a bilinear structure and a parametric variation in their operating condition. Again, the pilot reactor can be, in this case,
and, by identification, we get: ( 11) Y(uk)·u k + ~(uk)-dJx3(k)-(1-d)ek = 0 The implicit equation (7) is just slightly modified, x1(k) being 2T delayed: x3(k) = s(k-2). The controller which is obtained is an extension for bilinear systems of the well-known Smith's controller.
Experimental results All experiments are processed for an initial total flow-rate Qo=QAo+QBo=55Q./h and for a pH around 2.5. A reference corresponding to xc=0.1 is equivalent to a pH ref erence of 0.3. The sampl ing period is T=15sec. A first feedback yields to a linear system from a bilinear one, and a second feedback realizes a P.I control, allowing a cancellation of static errors. Moreover, we master the response time by specifing the overshoot magnitude D(%) and the first overshoot time NT. Fig. 4 shows the normalized concentration evolution for two desired response times (0=5%, N=20 or N=15). The static error is perfectly cancelled. The obtained overshoot is greater than the desired one, but the first overshoot time is rather good. Fig. 5 shows the evolution of the normalized concentration and the base flow-rate, with respect to a greater step magnitude. The same remarks, as in Fig. 4, can be mentioned. The overshoot differences with respect to the theory are due to the fact that, the tuning parameters (d, K, Ki) are calculated from the system model mich involve some identification errors. Bilinear delayed system In Fig. 6, we implement, the "Smith non-linear algorithm" (Eq. 11) on the process. The desired close:! loop response is a two sampling periods delayed first order one. The result is rather satisfactory, but the delay is too small (10% of the approximated time constant) to point out a real improvement of the closed-loop linearization with respect to the non-delayed system. CONCLUSION AND PROSPECTS Several digital control methods have been proposed for two classes of non-linear continuous processes: linear system with time-varying parameters and bi-
a very good testing device. REFERENCES Alvarez Gallegos, J. (1982). Optimal control of a class of discrete multivariable non-linear systems. Application to a fermentation process, Journal of Dynamic Systems, Measurement and Control, vol. 104, pp. 212-217. Cebuha~Costanza, V. (1984). Non-linear control of CSTR's, Chemical Engineering Science, vol. 39, 12, pp. 1715-1722. Claude, D., Fliess, M., Isidori, A. (1983). Inunersion directe et par bouclage d'un systeme nonlineaire dans un lineaire, Compte-rendu a l'Academie des Sciences, serie 1, t. 296, pp.237240. Oadugineto, (1985). Identification de differents modeles di sc rets non-lineaires continus. Applicati on aux procedes industri e ls pilotes, Conf e rence internationale CNRS Automatique n0n= lineaire, p,qris. Oang Van Mien, H., Normand-Cyrot, O. (1984). Nonlinear state-affine identification methods. Application to electrical power plants, I.F . A.C. symposium an Aut. Control on Power generation, Distribution and Protection, Pretoria,
Automatica, vol. 20, n° 2. Dufour, J., Thomasset, D. (1984). Definition and synthesis of state-affine control algorithms. Application to vary-linear systems, IEEE Conf. on Computers, Systems and signal Processy, Bangalore, India. Fliess, H., Normand-Cyrot, D. (1980). Vers une approche algebrique des systemes non-lineaires en temps discret, Analysis and optimization of systems, Versailles, Lecture notes control Inform. Sciences, n° 28, Springer-Verlag. Gauthier, J-P., Bornard, G. (1980). Stabilization of bilinear systems. Performance specification and optimality, 4th INRIA international conf. on Control and Optimization, Versailles, France. Gilles, G., Laggoune, N. (1985a). Linearizing control of a class of non-linear continuous processes,
IFAC conf. CSTO'85, Beijing, China. Gilles, G., Laggoune, N. (1985b). Digital control of bilinear continuous processes. Application to a chemical pilot plant, IFAC conf . on Digital Computer Application to Process Control, Vienna, Austria.
133
Pilot Neutralization Chemical Reactor Mohler, R. (1973). Bilinear control processes, Academic press. -Neyran, B., Saade-Castro, J., Thomasset, D., Reynaud, R. (1984). Modelisation dynamique non-l ineaire d'un procede chimique de neutralisation, Congres international S.E.E., Nice.
Neyran, B., Thomasset, D., Dufour, J. (1985). Direct identification of a class of nonlinear systems. Application to a neutralization process. IFAC Conf. CSTD'8S, Beijing, China.
, Cone
~:
v~
linear control computed at
*: linear control computed at
4
V~12
.: state-affine control
.1t---J--------'O"'~~~
.05
1000
0
T i m e (sec::)
V(I)
12 8 4
1000
2000 T i m e (sec::)
Fig.3
Comparison between linear and state-affine control
x
STEP REFERENCE MEASURED OUTPUT (N=20) ~ffiASURED
OUTPUT (N=1S)
O~~4-~----~--------~-------.--------~~ I.
O.
Fig. 4. P-I. linearizing control
600·· C
two different dynamics
G. Gi lles et al.
134
.
x
.
r - - - - - - - - - - - - '- .....- ... ,-" - - ~ ~.- ....... - - ' T
0.1
... -
STEP REFERENCE MEASURED OUTPUT
a) O~~+-------,-------.--------,-------r--~ 600 •• e o
60 I/h
Q.
40
b) 20~L--.--------r--------.--------r--------r~ 600·· e o
Fig. 5 . a) Normalized concentration for gr ea t er step magnitude b) Con trol variable (base flow-rate) evo luti on
x 0 .1
,.. - - - - - - - - - -
- - .-. - - ;.- -. -
~.~
: .-., ..
-
-: -
-
,-.-.-.
STEP REFERENCE MEASURED OUTPUT
O~-r-r~------'---------r--------,---------,~_ o 600 •• e:
Fig . 6. Cl osed loop linearization with de layed response (smith non-linear con troller)