- Email: [email protected]
Adaptive output feedback control of a nonlinear chemical process
IFAC [:0[>
Copyright © IFAC Dynamics and Control of Process Systems, Jejudo Island, Korea, 200 I
Pu blications www.elsevier.com/locate/ifac
ADAPTIVE OUTPUT FEEDBACK CONTROL OF A NONLlNEAR CHEMICAL PROCESS Kap Kyun Noh, Dongil Shin, Jong Dae Lee·' and En Sup Yoon' School of Chemical Engineering, Seoul National University, Seoul 151-742, Korea. (*Email.· [email protected]) "'*Korea Institute of Industrial Technology, Korea
Abstract: In many cases, by the state transformation including an independent outputs set as a part of new states, a non linear chemical process model can be divided into two subsystems; the one is a design model and the other is a disturbance model. Design model is described only in output variables and used for control system synthesis, while disturbance model is described in the original states and its effects on the design model are shown as timevarying parameters in the design model. Method for controller synthesis is proposed as a combined form of timevariant input-output Iinearization method with parameter estimation. Since the parameter estimation method gives the parameter estimates such that the estimated outputs follow the outputs in a specified way, all the uncertainties and disturbances affecting the outputs can be reflected into the estimated parameters, thus making the adaptive linearizing controller robust. The design model in the output variables makes the proposed method suitable for the output feedback control, not requiring the state estimation. The design procedure of choosing the tuning parameters is simple and clear. For the demonstration of its applicability and validity, two control problems taking place in the chemical processes are considered; a regulation of continuous fermenter and a tracking of reference trajectory of a polymerization batch reactor. Copyright© 200 I !FA C Keywords: Input-Output Linearization, Time-Variant Nonlinear Systems, Parameter Estimation, Continuous Fermenter, Polymerization Batch Reactor. I . INTRODUCTION
(l)
Recently, there have been much developments in the nonlinear systems theory from differential geometry [7]. The theory has many applications to chemical processes. Particularly, Kravaris and Chung (1987) developed the globally Iinearizing controller (GLC) using the input-output linearization approach and Palanki and Kravaris(1997) extended the GLC to a time-varying system. These methods include an external controller with integral action such as a PI controller around the linearized input-output system to cope with uncertainty and disturbances, and used a hybrid open-loop observer for unavailable states in state feedback. In this paper, a methodology for the controller synthesis of a nonlinear chemical process is proposed. The method is based on a time-varying parametric uncertain nonlinear system described in output variables, which naturally leads to a combination of input-output linearization and uncertain parameters estimation. The systems that the proposed method is applicable will be characterized and its effectiveness will be demonstrated for two systems via simulation. 2. DESIGN MODEL FOR CONTROLLER SYNTHESIS
Consider a nonlinear system in state-space form x = f(x) + g(x)u 691
where x ErcRn,UER,YeER are the state vector, the input, and controlled output, respectively. Y m E Rq-l is secondary outputs andYE Rq is the augmented outputs containing all outputs. f(x) , g(x) are smooth vector fields, and he(x) , hm; ,i = 1, .., q -I are smooth scalar fields, respectively, on r. Here, r is a connected open set physically feasible and bounded. Subscripts c and m of y stand for 'controlled' and 'measured', respectively. Note that the system is actually a SISO (Single-Input Single Output) system since the controlled output is single. It is assumed that the system is input-output linearizable and equivalently has a well-defined relative order. It is assumed that each output in the augmented outputs is independent and allows to take a locally invertible state transformation of the form
z=[~]=q>(X)=[:(~:J
(2)
where TJ E Rn-q is new states and P(x) consists of (n-q) scalar fields satisfying nonsingularity of the transformation. The new states TJ can be simply taken as (n-q) state variables of the system of (1). In terms of new state variables, the system (1) is transformed into the form
LjH(y,T/)
z=[~]=
Jq+1(Y,T/)
procedures and their estimates be incorporated in the controller synthesis. Assumption 2: The design model can be linearly parameterized in terms of unknown parameters,
LgH(Y,T/) + gq+l(y,T/)
J. (y, T/)
u
(3)
eE RP .
g. (y,T/)
Here, Lie derivatives of a scalar field H i(x) with respect to vector fields f(x) and g(x) are used as a standard differential geometric notation. The transformed system (3) can be decomposed into two subsystems; the one is described in terms of only output variables as new state variables and the other is described in terms of original state variables. We will refer to the former subsystem as a design model and the latter subsystem as a disturbance model. The controller synthesis will be based on the design model. When outputs for the state transformation are selected, it is desired that structural characteristics of the original system such as relative order is kept in the design model. It means that the controlled output in the design model is also controllable directly by the input and the design model is input-output Iinearizable. To achieve this, some requirements will be imposed on the selection of outputs and be stated as the following assumption. It will be also assumed that the necessary outputs such that the assumption below be satisfied are available although it may be difficult to get their measurements in practice since they are related with compositions and/or physical properties. Assumption 1: a) The relative order of the augmented output vector, y , with respect to the input, u , in the design model is 1, i.e., the smallest integer, k, is 1
dyk
such that d~
:;t
0\7'0",1/
YE
(4)
b) The relative order of the controlled output, y c' with respect to the input, u , in the design model with the disturbance model excluded is well defined and is equal to that of the original system. Assumption I .a) requires that the input should appear explicitly among the first derivatives of augmented outputs which consist of the design model, and l.b) is sufficient such that the input-output Iinearization can be applied to the design model. Since the design model is a time-varying system, the relative order defined in (8) should be applied. Assumptionl.b) means that output controllability of the system (1) is carried over the newly formed, but reduced subsystem. The disturbance model will not be used directly in the controller synthesis and it will be regarded as unknown although it can be utilized in a feed forward way as the disturbance estimator driven by measured outputs and input. The effects of ignored disturbances will be treated as time-varying parameters in the design model, which are the result of the couplings between design model and disturbance model, thus making the design model nonlinear time-varying. Smce the disturbance model is an uncertain dynamics to the design model, time-varying parameters in the design model should be estimated via some
692
Y =L jH(Y,1/)+Lg H(Y,1/)u
P {PIg{(Y)8;(Y,1/) = If/(Y)ei(Y,T/)+ 1=Q
where
i =Q
iF = [1, e f
r
r
and f/ (y), g (y) are
(5) known
vector fields on Rq. By appropriately grouping the coupled as parameters, assumption 2 can be met. The relative order, r, is a useful notion to characterize internal structural characteristics of the system and thus systems can be classified according to that. Case 1: r = 1 By the definition of the relative order, the input appears in the first derivative of the controlled output. Without secondary outputs, the assumption 1 is trivially met since the original system is assumed to have the well-defined relative order. This case corresponds to the feedforward-feedback control method of Kravaris et. al. (1989), where the subsystem, 1/ , was viewed as a disturbance estimator to the design model, and to compensate for uncertainties in the disturbance model, the linearizing controller was combined with a feedback controller such as PID which results in a special case of GLC. This case also reduces to GMCCGeneric Model Control) of Lee and Sullivan (1988), in which the controller is formulated by equating the model for the derivative of the controlled output to a proportional integral term operating on the output error. Case 2: 2 :::; r < n In contrast to Case 1, (r-l) secondary outputs are necessary. Input appears in higher derivatives more than one of the controlled output, but appears explicitly among the first derivatives of r augmented outputs and the relative order of the controlled output IS well defined when only the design model is considered. Then, this case can be reduced to the Case 1. Note that the relative order of the design model is equal to the dimension of the design model. This means that the design model is linearizable in both input-state and input-controlled output relation since the disturbance model is excluded. Case 3: r = n In this case, the system (1) is both input-state and input-output linearizable. As the design model has to have the same relative order as the system, the disturbance model is not formed and n independent outputs should be available . The design model is anoth~r sy~tem equivalent to the system (1), but descnbed m the output coordinates. Controller synthesis based on the design model is the same as usual state feedback. For both Case I and Case 2, a disturbance model is formed. The disturbance model whose effects on design model are expre~sed as linear time-varying parameters m the deSIgn model is ignored as
unknown. To exclude the disturbance model in the design, it should be stable. Otherwise, it should be taken into account. Advantageous features of the proposed methodology are a) since state variables in the design model are output variables, the state estimation, which is usually needed for the state feedback control law widely used in the nonlinear systems synthesis, is not required b) the model for the design is reduced to a smaller one c) since time-varying parameters can incorporate the uncertainties from various sources such as unknown parameters and disturbances, modeling error and unmodeled dynamics as lumped forms, it is possible to design robust controller only if a proper identification method for the time-varying parameters can be designed. 3. INPUT-OUTPUT LINEARIZATION OF PARAMETRIC UNCERTAIN TIME-VARYING NONLINEAR SYSTEMS Input-output Iinearization as a controller synthesis method has been well established for the nonlinear time-invariant systems[7] and recently extended to the nonlinear time-varying systems, which finally results in an equivalent linear time-invariant system[9] . Consider a nonlinear time-varying system
x = I(x,t) + g(x ,t)u y = hex)
(6)
where x Ere Rn, U ER, Y E R are the state vector, the input and the output, respectively. f and g are smooth vector fields in r x R, and h is a smooth scalar field in r x R . The standard Lie derivatives for a time-invariant system are inadequate for a time-varying system and need to be modified to account for the explicit time dependence of the model. The modified Lie derivatives are defined as follows . k k- I aL}-l h Lf h(x,t) = +_ . -(x,t) k
at
where
=0,1 ,2,...
(7)
dL}-1h means the gradient of a scalar
. Lkf - I h an d <,> stan d s c iar pro duct f unctIOn, lor th e sca of two vectors.
Definition: For the system of (6), the relative order of the output y with respect to the input u is the smallest integer r such that
Lg(x,I)L'i(~,l)h(x,t) * 0 Vxxt E rx[to,oo)
applied and an integrator augmented is =e l
a
er =Aer+~-y;p-0h+L/;/hu) •
-
(
(r- I)
(9) t)
(10)
1]n...,. =1]n-r el +Ysp , .. ,er + Ysp
,1],
- =[e1, .. ,e ]T wtt .h where er r
(H) ei --t,·-lhf Ysp'
and
n r
1/ E R n- r is new states and ffn-r E R - is nonlinear maps. (A,B) is a controllable canonical pair, and y'~~-I) is the (i-l )th derivative of the reference output y sp ' It is assumed that the reference output is
continuously differentiable as times as necessary. The Iinearizing state feedback law applied is
-0
- Y;p h+ ±ak (L}-'h- yi;-I) +OQ1(h-Ysp)dt' u= ________~~~I------~-----------Lg r-'h 1
(11)
After the Iinearizing feedback law is applied, the resulted linear system augmented by an integrator is controllable, which is caused by the controllability of the (A,B) pair, and its dynamics are determined by the polynomial coefficients, (Xi , i=O, I , .. ,r. The nonlinear dynamics of eq . (10), called as zero dynamics when the output and the reference are constrained identically to zero, is unobservable and internal dynamics. The state feedback closed-loop system is in a cascaded form, in which the linear system is controllable such that an arbitrary pole placement is possible and the nonlinear system is driven by arbitrarily fast decaying errors, bounded reference trajectory and its derivatives as inputs. Thus, the stability of closed-loop system is guaranteed if the unobservable system is bounded input-to-state stable. This feature results from the fact that the input-output Iinearization is a nonlinear analog of pole-zero cancellation in the linear system theory. Therefore, for internal stability, the cancelled system is required to be stable and in that case, the system is called as a minimum-phase system[7]. Feedback linearizing controllers are based on the exact cancellation of nonlinearities, which is one of weak points of linearization method. Therefore, control law formulated from the model with uncertainty no longer linearizes the input-output relationship, which will cause the deterioration of control performances. In case of linearly parametric uncertainty, the estimates of I, g are obtained by using estimated parameters and the Lie derivatives can be calculated with these estimates of vector fields. For r ~ 2 , the estimates of Lie derivatives are not linear in the unknown parameters, but this can be avoided by redefining the product as another parameter. Note that derivatives of estimated parameters are included in the control law for r ~ 2 . The state feedback law using the estimated Lie derivatives is
(8)
As in the time-invariant case, the relative order is the smallest time derivative of the output which explicitly depends on the input. The existence of a finite relative order ensures that the locally invertible state transformation and the state feedback law Iinearizing input-output response exist[9] . After state transformation and state feedback, the system is in the normal form[7] . Since the error coordinate is convenient for an output tracking, the normal form in the error coordinates with the state feedback not yet
(12)
693
where bar represents estimates of corresponding Lie derivatives. When U a is applied to the system, the linear error system of eq. (9) is no longer linear and becomes er = Acer + W(e,,1],t)8 (13) The second term in the right-hand side is induced from uncertain parameters and is a nonlinear perturbing term. 8 is the parameter error vector of all multi-linear parameter products among 8 i and their derivatives. As each parameter is approaching to its true value, the parameter error vector is decaying to zero and the quasi-linear error system is asymptotically linearized.
the outputs and the number of identifiable parameters is less than or equal to the number of the outputs. We want to specify a PI control trajectory as a target trajectory for the first time derivative of the estimated output as in GMC (Generic Model Control) [8] .
y)+ Kd (y - y)dt (17) o where K" K2 are diagonal matrices whose elements are tuning parameters for each output. Because of the assumption b), the parameter estimation is achieved by combining eqs. (15) with (17), and the result is
f I
y=K1(y- y)+K2 (y- y)dt' o
4. PARAMETER ESTIMATION
kY,T])={qrqfqrk(Y-Y>+Ki.b(y-jl}it-W'cY>+g\'cY»]
The design model of section 2 is described in only output variables as state variables and states in the disturbance model are introduced in the design model as lumped time-varying parameters which should be identified through some procedure. In this section, a parameter estimation method such that the output estimated by using the parameter estimates follows asymptotically the measured output in a specified way will be proposed. Consider the design model of (5), which is in the linearly parameterized form by time-varying parameters and rearranged in the following
e=(Is 2 +K,s+K 2 tIs 2 y 2
y=Vs +K,s+K 2 t(K,s+K 2 )y
where 8 E RP is a time-varying parameter vector to be identified and 'f' (y, u) is a known parameter coefficient matrix. The estimated output is obtained from the model replaced by parameter estimates
Y-VlCY)+&\,CY)}={!t(CY) +¥CY)if,1J)
(k) :~j
"#
0
(20)
5. APPLICATIONS 5.1 Regulation ofA Continuous Fermenter Consider a continuous fermenter with a feed and a product stream. The process can be represented by the following nonlinear differential equations
(15)
The hat over the variables means estimates of the corresponding. It is assumed that a) all parameters to be identified appear explicitly in the design model and b) parameter coefficient matrix has a full column rank. As the relative order of the output with respect to the input is defined, the relative order of the outputs with respect to the parameters can be defined similarly [6]:
[[a 1 1
(19)
where e = y - y and K" K 2 are set such that the required identification performance can be obtained. Parameter estimation method proposed by Tatiraju and Soroush (1998) is such that the closed-loop servo response of the estimated output is a first-order. Since their form did not contain an integral term and the concept of parameter relative order was not utilized, the relationship with the GMC was not mentioned. The link between parameter estimation of eq. (18) and GMC is that the relative order of each parameter is one and the coefficient matrix has a full rank.
(14)
= 'f'(y,u)8(Y,l1)
(IS)
The driving inputs are the measured outputs, and the estimated outputs and outputs error have the c1osedloop servo response like a linear second-order type as
y-Vl(Y)+&\,(Y)}={¥Y(Y)+¥(y)i~'11)
rtIlJ = min k:
t ,
Y =K, (y -
X =-DX +J1X . 1 S=D(S-Sj)---tiX
YX 1S
0 . 175
~---------------
~O.165
(16)
3i E [1, .., p]
1 !!
Here,
rill;
is the relative order of output
Yi
lo·lSS
with
respect to parameter 8 j ' The parameter relative 0 . 145
degree characterizes how directly a parameter affects an output. The lower the relative order, the more direct is the effect of the parameter on the output. According to this definition, the parameters in the design model are assumed to have the relative order of 1 and assumption b) means that parameters in the design model satisfy an observability condition from
' - -_ _ _ _ _ _ _ _~_ _~_ _---.J
'0
Time{Ht\
694
7.045
r - - - - - - - - - - - - - - - - , O . 18
7 .0'0
0 . 16
g
"I
~
,.=
Batch Polymerization Reactor The proposed methodology can be applied to a tracking problem of off-line determined optimal temperature trajectory of a polymerization batch reactor, of which the reaction system consists of MMA as a monomer, Toluene as a solvent and AIBN as an initiator. In order to prevent numerical conditioning problems, all related equations are appropriately non-dimensionalized. The design model derived from energy balances has the following general form applicable to any batch reactor
~
~
is
~
7. 035
'--------~--~--~---' 0 . 14 40 20 60 30 100 Time(Hr)
Fig.l Regulation of continuous fennenter;(top) comparison of the estimated parameter and the exact parameter; (bottom) control perfonnance comparison when the exact parameter and the estimated parameter are used, respecti vel y. P = -DP+(a}.1 + fJ)X
TR
=8 1(x , fo,V,TR,19)-02(X)(TR -f.,)
~ =~2(x)(TR -TJ)+Fcw(fcw-TJ)+q/n
where
if; (t) ,82 (t)
represent the heat of reaction term
and an overall transferred heat term, respectively. y is
where ~ is the specific growth rate, Yx I S is cell-mass
10 .0
yield, and {X and p are yield parameters for the product. Any specific growth rate model can be used, but a model exhibiting both substrate and product inhibition is adopted for simulation. Note that the model can describe various fermentations. The control objective is to maintain Ye = X at the regulated point in the presence of disturbances and uncertainty by manipulating u = D. The simulation conditions and parameter values are adopted from the literature[5]. When only cell-mass concentration is available, the system has the output relative order of I and the parameter relative order of I. And, all assumptions needed to apply the proposed method are simply satisfied, and since the design model is scalar equation, the controller design and parameter estimation procedure are easy works. The design model in a linearly parameterized form is y = -p(S,P,19)y + (-y)u = ~ + (-y)u while the rest of equations are taken as disturbances model and their effects are displayed through a specific growth rate, i.e., e. 19 is the set of unknown parameters different from e. Tuning parameters are set to (1.1=21 Ec,
5.2 Tracking of Optimal Temperature Trajectory in a
695
1.0S
1'
O
°
'00
~ 0 . 9S f
SOO
with auct paramatar
}O.O
i
[~H~ ~~:=;~;I:~Hf'w~TJH:}i"
1100
1
J
y=[fR
1:
~ with parametar estimation
200
0 .&5
~
-100
0.80 .0
120
,.0
><0
300
300
Time(Min)
(d)
Fig.2 Tracking of the reference trajectory in a polymerization batch reactor; (a) comparison of the estimated and the exact heat of reaction term, 8 1 ;(b) comparison of the estimated and the exact transferred heat term, 8 2 ;(c) control performances comparison of the reactor temperature when the exact parameters and the estimated parameters are used, respectively;(d) comparison of the corresponding jacket temperature and the coolant flow rate of (c) relatively constant and x, Ao , V, tJ, gin are monomer conversion, total concentration of live polymer radicals, reaction volume, unknown process parameter set and heat input used with coolant, respectively. Subscripts R, J stand for 'reactor' and 'jacket'. Each parameter shows the dependence of states in the disturbance model and is assumed to be completely unknown. Disturbance model in this system is composed of material balance equations and constitutive equations describing gel/glass effects[2], reaction volume contraction, conversion dependence of heat transfer coefficient [10] and so on. The polymerization reaction in a batch reactor is highly exothermic, and highly nonlinear because of mainly gel effect in the course of reaction[l]. Viscouse reaction medium causes a severe heat transfer restriction and sensor problem. These effects make two parameters behave nonlinearly and be unknown. The bell shape of temperature trajectory designed to meet the quality specifications at the end of batch time also provides a challenging control problem. The design model can be described in statespace form of eq. (6) and the linearizing control method proceeds as In section 3. Since
LgL [hex) = 82(x)
'* 0,
TlT
which satisfies the conditions for the application of proposed parameter estimation. Particularly, the parameter coefficient matrix is nonsingular unless the temperature of the reactor is equal to the temperature of the cooling jacket. Tuning parameters, KI , K 2 for parameter estimation are set to
~
the relative order of the
output is 2. Note that the formula formulated by formula (11)(12) has the first order derivatives of the estimated parameters. But, actually applied control law does not necessarily include the derivatives of parameter estimates. The coefficients of the linear system in the error coordinates are selected such that the characteristic equation of the linear system is .1'2 +a2.1'+al =(£.1'+1)2
where e is a tuning parameter and its value is set to 0.002. The design model can be also expressed in the linearly parameterized form of time-varying parameters,
696
KI
=dia~/cpI
2/C p2}
K2 = dia~l/ t:;1 11 t:;2} This makes the characteristic equation of the estimated output dynamics be (t: pS + 1)2.
[0 pI'
t: p2
are set to 0.005 for both outputs. Two simulation cases are performed; a) when the exact parameters are available and they are utilized in the linearizing control law, and b) when the parameters are estimated and the estimates are utilized in the same control law. Fig.2 shows the comparative performance of the linearizing control for two cases, and comparison of estimated parameters with exact parameters for case b). As shown in Fig.2, the results are good even for parameter ~ showing the peak, and control performance and control input of case a) and b) are comparable. 6. CONCLUSIONS As demonstrated through two applications, the proposed method can be applied successfully to typical chemical processes. The resulting design model is simple in spite of complexities of the process considered. The design procedure of the method is simple and clear. And, it is also robust in sense that diverse uncertainties and disturbances can be reflected in the estimated parameters and so taken into account in the controller. The formulated controller is in the output feedback form. ACKNOWLEDGEMENT We thank the financial aid to this research from the Brain
Korea 21 Program supported by the Ministry of Education and the National Research Lab Grant of the Ministry of Science & Technology. 7. REFERENCES [I] Ballagou, P.E. and Soong, D.S. (1985), "Major Factors Contributing to the Nonlinear Kinetics of Free-Radical Polymerization," Chem. Engng. Sci., 40, 75-86 . [2] Chiu, w.Y., Carratt, G.M., and Soong, D.S. (1983), "A Computer Model for the Gel Effect in Free-Radical Polymerization," Macromolecules, 16, 348-357. [3] Kravaris, C. and Chung, C.B. (1987), "Nonlinear State Feedback Synthesis by Global Input/Output
Linearization," AIChE J., 33,592[4) Kravaris, c., Wright, RA, and Carrier, IF. (1989), "Nonlinear Controllers for Trajectory Tracking in Batch Processes," Computers Chem. Engng., 13, 73-82 . [5) Henson, MA and Seborg, D.E. (1992), "Nonlinear Control Strategies for Continuous Fermenters," Chem. Engng. Sci., 47, 821-835. [6) Hu, Q. and Rangaiah, G.P. (1999), "Adaptive Internal Model Control of Nonlinear Processes," Chem. Engng. Sci. , 54, 1205-1220. [7) Isidori, A. (1989), Nonlinear Control Systems, 2nd, Springer-VerJag. [8) Lee, P.L. and Sullivan, G.R.(1988), "Generic Model Control(GMC)," Computers Chem. Engng., 12, 573-580. [9) Palanki, S. and Kravaris, C. (1997), "Controller Synthesis for Time-Varying Systems by Input/Output Linearization," Computers Chem. Engng., 21 , 891903. [IO) Soroush, M. and Kravaris, C. (1992), "Nonlinear Control of a Batch Polymerization Reactor:an Experimental Study," AIChE J. , 39, 1429-1448. [11] Tatiraju, S. and Soroush, M. (I 998), "Parameter Estimator Design with Application to a Chemical Reactor," Ind. Eng. Chem. Res., 37, 455-463.
697
Copyright © IFAC Dynamics and Control of Process Systems, Jejudo Island, Korea, 200 I
Pu blications www.elsevier.com/locate/ifac
ADAPTIVE OUTPUT FEEDBACK CONTROL OF A NONLlNEAR CHEMICAL PROCESS Kap Kyun Noh, Dongil Shin, Jong Dae Lee·' and En Sup Yoon' School of Chemical Engineering, Seoul National University, Seoul 151-742, Korea. (*Email.· [email protected]) "'*Korea Institute of Industrial Technology, Korea
Abstract: In many cases, by the state transformation including an independent outputs set as a part of new states, a non linear chemical process model can be divided into two subsystems; the one is a design model and the other is a disturbance model. Design model is described only in output variables and used for control system synthesis, while disturbance model is described in the original states and its effects on the design model are shown as timevarying parameters in the design model. Method for controller synthesis is proposed as a combined form of timevariant input-output Iinearization method with parameter estimation. Since the parameter estimation method gives the parameter estimates such that the estimated outputs follow the outputs in a specified way, all the uncertainties and disturbances affecting the outputs can be reflected into the estimated parameters, thus making the adaptive linearizing controller robust. The design model in the output variables makes the proposed method suitable for the output feedback control, not requiring the state estimation. The design procedure of choosing the tuning parameters is simple and clear. For the demonstration of its applicability and validity, two control problems taking place in the chemical processes are considered; a regulation of continuous fermenter and a tracking of reference trajectory of a polymerization batch reactor. Copyright© 200 I !FA C Keywords: Input-Output Linearization, Time-Variant Nonlinear Systems, Parameter Estimation, Continuous Fermenter, Polymerization Batch Reactor. I . INTRODUCTION
(l)
Recently, there have been much developments in the nonlinear systems theory from differential geometry [7]. The theory has many applications to chemical processes. Particularly, Kravaris and Chung (1987) developed the globally Iinearizing controller (GLC) using the input-output linearization approach and Palanki and Kravaris(1997) extended the GLC to a time-varying system. These methods include an external controller with integral action such as a PI controller around the linearized input-output system to cope with uncertainty and disturbances, and used a hybrid open-loop observer for unavailable states in state feedback. In this paper, a methodology for the controller synthesis of a nonlinear chemical process is proposed. The method is based on a time-varying parametric uncertain nonlinear system described in output variables, which naturally leads to a combination of input-output linearization and uncertain parameters estimation. The systems that the proposed method is applicable will be characterized and its effectiveness will be demonstrated for two systems via simulation. 2. DESIGN MODEL FOR CONTROLLER SYNTHESIS
Consider a nonlinear system in state-space form x = f(x) + g(x)u 691
where x ErcRn,UER,YeER are the state vector, the input, and controlled output, respectively. Y m E Rq-l is secondary outputs andYE Rq is the augmented outputs containing all outputs. f(x) , g(x) are smooth vector fields, and he(x) , hm; ,i = 1, .., q -I are smooth scalar fields, respectively, on r. Here, r is a connected open set physically feasible and bounded. Subscripts c and m of y stand for 'controlled' and 'measured', respectively. Note that the system is actually a SISO (Single-Input Single Output) system since the controlled output is single. It is assumed that the system is input-output linearizable and equivalently has a well-defined relative order. It is assumed that each output in the augmented outputs is independent and allows to take a locally invertible state transformation of the form
z=[~]=q>(X)=[:(~:J
(2)
where TJ E Rn-q is new states and P(x) consists of (n-q) scalar fields satisfying nonsingularity of the transformation. The new states TJ can be simply taken as (n-q) state variables of the system of (1). In terms of new state variables, the system (1) is transformed into the form
LjH(y,T/)
z=[~]=
Jq+1(Y,T/)
procedures and their estimates be incorporated in the controller synthesis. Assumption 2: The design model can be linearly parameterized in terms of unknown parameters,
LgH(Y,T/) + gq+l(y,T/)
J. (y, T/)
u
(3)
eE RP .
g. (y,T/)
Here, Lie derivatives of a scalar field H i(x) with respect to vector fields f(x) and g(x) are used as a standard differential geometric notation. The transformed system (3) can be decomposed into two subsystems; the one is described in terms of only output variables as new state variables and the other is described in terms of original state variables. We will refer to the former subsystem as a design model and the latter subsystem as a disturbance model. The controller synthesis will be based on the design model. When outputs for the state transformation are selected, it is desired that structural characteristics of the original system such as relative order is kept in the design model. It means that the controlled output in the design model is also controllable directly by the input and the design model is input-output Iinearizable. To achieve this, some requirements will be imposed on the selection of outputs and be stated as the following assumption. It will be also assumed that the necessary outputs such that the assumption below be satisfied are available although it may be difficult to get their measurements in practice since they are related with compositions and/or physical properties. Assumption 1: a) The relative order of the augmented output vector, y , with respect to the input, u , in the design model is 1, i.e., the smallest integer, k, is 1
dyk
such that d~
:;t
0\7'0",1/
YE
(4)
b) The relative order of the controlled output, y c' with respect to the input, u , in the design model with the disturbance model excluded is well defined and is equal to that of the original system. Assumption I .a) requires that the input should appear explicitly among the first derivatives of augmented outputs which consist of the design model, and l.b) is sufficient such that the input-output Iinearization can be applied to the design model. Since the design model is a time-varying system, the relative order defined in (8) should be applied. Assumptionl.b) means that output controllability of the system (1) is carried over the newly formed, but reduced subsystem. The disturbance model will not be used directly in the controller synthesis and it will be regarded as unknown although it can be utilized in a feed forward way as the disturbance estimator driven by measured outputs and input. The effects of ignored disturbances will be treated as time-varying parameters in the design model, which are the result of the couplings between design model and disturbance model, thus making the design model nonlinear time-varying. Smce the disturbance model is an uncertain dynamics to the design model, time-varying parameters in the design model should be estimated via some
692
Y =L jH(Y,1/)+Lg H(Y,1/)u
P {PIg{(Y)8;(Y,1/) = If/(Y)ei(Y,T/)+ 1=Q
where
i =Q
iF = [1, e f
r
r
and f/ (y), g (y) are
(5) known
vector fields on Rq. By appropriately grouping the coupled as parameters, assumption 2 can be met. The relative order, r, is a useful notion to characterize internal structural characteristics of the system and thus systems can be classified according to that. Case 1: r = 1 By the definition of the relative order, the input appears in the first derivative of the controlled output. Without secondary outputs, the assumption 1 is trivially met since the original system is assumed to have the well-defined relative order. This case corresponds to the feedforward-feedback control method of Kravaris et. al. (1989), where the subsystem, 1/ , was viewed as a disturbance estimator to the design model, and to compensate for uncertainties in the disturbance model, the linearizing controller was combined with a feedback controller such as PID which results in a special case of GLC. This case also reduces to GMCCGeneric Model Control) of Lee and Sullivan (1988), in which the controller is formulated by equating the model for the derivative of the controlled output to a proportional integral term operating on the output error. Case 2: 2 :::; r < n In contrast to Case 1, (r-l) secondary outputs are necessary. Input appears in higher derivatives more than one of the controlled output, but appears explicitly among the first derivatives of r augmented outputs and the relative order of the controlled output IS well defined when only the design model is considered. Then, this case can be reduced to the Case 1. Note that the relative order of the design model is equal to the dimension of the design model. This means that the design model is linearizable in both input-state and input-controlled output relation since the disturbance model is excluded. Case 3: r = n In this case, the system (1) is both input-state and input-output linearizable. As the design model has to have the same relative order as the system, the disturbance model is not formed and n independent outputs should be available . The design model is anoth~r sy~tem equivalent to the system (1), but descnbed m the output coordinates. Controller synthesis based on the design model is the same as usual state feedback. For both Case I and Case 2, a disturbance model is formed. The disturbance model whose effects on design model are expre~sed as linear time-varying parameters m the deSIgn model is ignored as
unknown. To exclude the disturbance model in the design, it should be stable. Otherwise, it should be taken into account. Advantageous features of the proposed methodology are a) since state variables in the design model are output variables, the state estimation, which is usually needed for the state feedback control law widely used in the nonlinear systems synthesis, is not required b) the model for the design is reduced to a smaller one c) since time-varying parameters can incorporate the uncertainties from various sources such as unknown parameters and disturbances, modeling error and unmodeled dynamics as lumped forms, it is possible to design robust controller only if a proper identification method for the time-varying parameters can be designed. 3. INPUT-OUTPUT LINEARIZATION OF PARAMETRIC UNCERTAIN TIME-VARYING NONLINEAR SYSTEMS Input-output Iinearization as a controller synthesis method has been well established for the nonlinear time-invariant systems[7] and recently extended to the nonlinear time-varying systems, which finally results in an equivalent linear time-invariant system[9] . Consider a nonlinear time-varying system
x = I(x,t) + g(x ,t)u y = hex)
(6)
where x Ere Rn, U ER, Y E R are the state vector, the input and the output, respectively. f and g are smooth vector fields in r x R, and h is a smooth scalar field in r x R . The standard Lie derivatives for a time-invariant system are inadequate for a time-varying system and need to be modified to account for the explicit time dependence of the model. The modified Lie derivatives are defined as follows . k k- I aL}-l h Lf h(x,t) =
at
where
=0,1 ,2,...
(7)
dL}-1h means the gradient of a scalar
. Lkf - I h an d <,> stan d s c iar pro duct f unctIOn, lor th e sca of two vectors.
Definition: For the system of (6), the relative order of the output y with respect to the input u is the smallest integer r such that
Lg(x,I)L'i(~,l)h(x,t) * 0 Vxxt E rx[to,oo)
applied and an integrator augmented is =e l
a
er =Aer+~-y;p-0h+L/;/hu) •
-
(
(r- I)
(9) t)
(10)
1]n...,. =1]n-r el +Ysp , .. ,er + Ysp
,1],
- =[e1, .. ,e ]T wtt .h where er r
(H) ei --t,·-lhf Ysp'
and
n r
1/ E R n- r is new states and ffn-r E R - is nonlinear maps. (A,B) is a controllable canonical pair, and y'~~-I) is the (i-l )th derivative of the reference output y sp ' It is assumed that the reference output is
continuously differentiable as times as necessary. The Iinearizing state feedback law applied is
-0
- Y;p h+ ±ak (L}-'h- yi;-I) +OQ1(h-Ysp)dt' u= ________~~~I------~-----------Lg r-'h 1
(11)
After the Iinearizing feedback law is applied, the resulted linear system augmented by an integrator is controllable, which is caused by the controllability of the (A,B) pair, and its dynamics are determined by the polynomial coefficients, (Xi , i=O, I , .. ,r. The nonlinear dynamics of eq . (10), called as zero dynamics when the output and the reference are constrained identically to zero, is unobservable and internal dynamics. The state feedback closed-loop system is in a cascaded form, in which the linear system is controllable such that an arbitrary pole placement is possible and the nonlinear system is driven by arbitrarily fast decaying errors, bounded reference trajectory and its derivatives as inputs. Thus, the stability of closed-loop system is guaranteed if the unobservable system is bounded input-to-state stable. This feature results from the fact that the input-output Iinearization is a nonlinear analog of pole-zero cancellation in the linear system theory. Therefore, for internal stability, the cancelled system is required to be stable and in that case, the system is called as a minimum-phase system[7]. Feedback linearizing controllers are based on the exact cancellation of nonlinearities, which is one of weak points of linearization method. Therefore, control law formulated from the model with uncertainty no longer linearizes the input-output relationship, which will cause the deterioration of control performances. In case of linearly parametric uncertainty, the estimates of I, g are obtained by using estimated parameters and the Lie derivatives can be calculated with these estimates of vector fields. For r ~ 2 , the estimates of Lie derivatives are not linear in the unknown parameters, but this can be avoided by redefining the product as another parameter. Note that derivatives of estimated parameters are included in the control law for r ~ 2 . The state feedback law using the estimated Lie derivatives is
(8)
As in the time-invariant case, the relative order is the smallest time derivative of the output which explicitly depends on the input. The existence of a finite relative order ensures that the locally invertible state transformation and the state feedback law Iinearizing input-output response exist[9] . After state transformation and state feedback, the system is in the normal form[7] . Since the error coordinate is convenient for an output tracking, the normal form in the error coordinates with the state feedback not yet
(12)
693
where bar represents estimates of corresponding Lie derivatives. When U a is applied to the system, the linear error system of eq. (9) is no longer linear and becomes er = Acer + W(e,,1],t)8 (13) The second term in the right-hand side is induced from uncertain parameters and is a nonlinear perturbing term. 8 is the parameter error vector of all multi-linear parameter products among 8 i and their derivatives. As each parameter is approaching to its true value, the parameter error vector is decaying to zero and the quasi-linear error system is asymptotically linearized.
the outputs and the number of identifiable parameters is less than or equal to the number of the outputs. We want to specify a PI control trajectory as a target trajectory for the first time derivative of the estimated output as in GMC (Generic Model Control) [8] .
y)+ Kd (y - y)dt (17) o where K" K2 are diagonal matrices whose elements are tuning parameters for each output. Because of the assumption b), the parameter estimation is achieved by combining eqs. (15) with (17), and the result is
f I
y=K1(y- y)+K2 (y- y)dt' o
4. PARAMETER ESTIMATION
kY,T])={qrqfqrk(Y-Y>+Ki.b(y-jl}it-W'cY>+g\'cY»]
The design model of section 2 is described in only output variables as state variables and states in the disturbance model are introduced in the design model as lumped time-varying parameters which should be identified through some procedure. In this section, a parameter estimation method such that the output estimated by using the parameter estimates follows asymptotically the measured output in a specified way will be proposed. Consider the design model of (5), which is in the linearly parameterized form by time-varying parameters and rearranged in the following
e=(Is 2 +K,s+K 2 tIs 2 y 2
y=Vs +K,s+K 2 t(K,s+K 2 )y
where 8 E RP is a time-varying parameter vector to be identified and 'f' (y, u) is a known parameter coefficient matrix. The estimated output is obtained from the model replaced by parameter estimates
Y-VlCY)+&\,CY)}={!t(CY) +¥CY)if,1J)
(k) :~j
"#
0
(20)
5. APPLICATIONS 5.1 Regulation ofA Continuous Fermenter Consider a continuous fermenter with a feed and a product stream. The process can be represented by the following nonlinear differential equations
(15)
The hat over the variables means estimates of the corresponding. It is assumed that a) all parameters to be identified appear explicitly in the design model and b) parameter coefficient matrix has a full column rank. As the relative order of the output with respect to the input is defined, the relative order of the outputs with respect to the parameters can be defined similarly [6]:
[[a 1 1
(19)
where e = y - y and K" K 2 are set such that the required identification performance can be obtained. Parameter estimation method proposed by Tatiraju and Soroush (1998) is such that the closed-loop servo response of the estimated output is a first-order. Since their form did not contain an integral term and the concept of parameter relative order was not utilized, the relationship with the GMC was not mentioned. The link between parameter estimation of eq. (18) and GMC is that the relative order of each parameter is one and the coefficient matrix has a full rank.
(14)
= 'f'(y,u)8(Y,l1)
(IS)
The driving inputs are the measured outputs, and the estimated outputs and outputs error have the c1osedloop servo response like a linear second-order type as
y-Vl(Y)+&\,(Y)}={¥Y(Y)+¥(y)i~'11)
rtIlJ = min k:
t ,
Y =K, (y -
X =-DX +J1X . 1 S=D(S-Sj)---tiX
YX 1S
0 . 175
~---------------
~O.165
(16)
3i E [1, .., p]
1 !!
Here,
rill;
is the relative order of output
Yi
lo·lSS
with
respect to parameter 8 j ' The parameter relative 0 . 145
degree characterizes how directly a parameter affects an output. The lower the relative order, the more direct is the effect of the parameter on the output. According to this definition, the parameters in the design model are assumed to have the relative order of 1 and assumption b) means that parameters in the design model satisfy an observability condition from
' - -_ _ _ _ _ _ _ _~_ _~_ _---.J
'0
Time{Ht\
694
7.045
r - - - - - - - - - - - - - - - - , O . 18
7 .0'0
0 . 16
g
"I
~
,.=
Batch Polymerization Reactor The proposed methodology can be applied to a tracking problem of off-line determined optimal temperature trajectory of a polymerization batch reactor, of which the reaction system consists of MMA as a monomer, Toluene as a solvent and AIBN as an initiator. In order to prevent numerical conditioning problems, all related equations are appropriately non-dimensionalized. The design model derived from energy balances has the following general form applicable to any batch reactor
~
~
is
~
7. 035
'--------~--~--~---' 0 . 14 40 20 60 30 100 Time(Hr)
Fig.l Regulation of continuous fennenter;(top) comparison of the estimated parameter and the exact parameter; (bottom) control perfonnance comparison when the exact parameter and the estimated parameter are used, respecti vel y. P = -DP+(a}.1 + fJ)X
TR
=8 1(x , fo,V,TR,19)-02(X)(TR -f.,)
~ =~2(x)(TR -TJ)+Fcw(fcw-TJ)+q/n
where
if; (t) ,82 (t)
represent the heat of reaction term
and an overall transferred heat term, respectively. y is
where ~ is the specific growth rate, Yx I S is cell-mass
10 .0
yield, and {X and p are yield parameters for the product. Any specific growth rate model can be used, but a model exhibiting both substrate and product inhibition is adopted for simulation. Note that the model can describe various fermentations. The control objective is to maintain Ye = X at the regulated point in the presence of disturbances and uncertainty by manipulating u = D. The simulation conditions and parameter values are adopted from the literature[5]. When only cell-mass concentration is available, the system has the output relative order of I and the parameter relative order of I. And, all assumptions needed to apply the proposed method are simply satisfied, and since the design model is scalar equation, the controller design and parameter estimation procedure are easy works. The design model in a linearly parameterized form is y = -p(S,P,19)y + (-y)u = ~ + (-y)u while the rest of equations are taken as disturbances model and their effects are displayed through a specific growth rate, i.e., e. 19 is the set of unknown parameters different from e. Tuning parameters are set to (1.1=21 Ec,
5.2 Tracking of Optimal Temperature Trajectory in a
695
1.0S
1'
O
°
'00
~ 0 . 9S f
SOO
with auct paramatar
}O.O
i
[~H~ ~~:=;~;I:~Hf'w~TJH:}i"
1100
1
J
y=[fR
1:
~ with parametar estimation
200
0 .&5
~
-100
0.80 .0
120
,.0
><0
300
300
Time(Min)
(d)
Fig.2 Tracking of the reference trajectory in a polymerization batch reactor; (a) comparison of the estimated and the exact heat of reaction term, 8 1 ;(b) comparison of the estimated and the exact transferred heat term, 8 2 ;(c) control performances comparison of the reactor temperature when the exact parameters and the estimated parameters are used, respectively;(d) comparison of the corresponding jacket temperature and the coolant flow rate of (c) relatively constant and x, Ao , V, tJ, gin are monomer conversion, total concentration of live polymer radicals, reaction volume, unknown process parameter set and heat input used with coolant, respectively. Subscripts R, J stand for 'reactor' and 'jacket'. Each parameter shows the dependence of states in the disturbance model and is assumed to be completely unknown. Disturbance model in this system is composed of material balance equations and constitutive equations describing gel/glass effects[2], reaction volume contraction, conversion dependence of heat transfer coefficient [10] and so on. The polymerization reaction in a batch reactor is highly exothermic, and highly nonlinear because of mainly gel effect in the course of reaction[l]. Viscouse reaction medium causes a severe heat transfer restriction and sensor problem. These effects make two parameters behave nonlinearly and be unknown. The bell shape of temperature trajectory designed to meet the quality specifications at the end of batch time also provides a challenging control problem. The design model can be described in statespace form of eq. (6) and the linearizing control method proceeds as In section 3. Since
LgL [hex) = 82(x)
'* 0,
TlT
which satisfies the conditions for the application of proposed parameter estimation. Particularly, the parameter coefficient matrix is nonsingular unless the temperature of the reactor is equal to the temperature of the cooling jacket. Tuning parameters, KI , K 2 for parameter estimation are set to
~
the relative order of the
output is 2. Note that the formula formulated by formula (11)(12) has the first order derivatives of the estimated parameters. But, actually applied control law does not necessarily include the derivatives of parameter estimates. The coefficients of the linear system in the error coordinates are selected such that the characteristic equation of the linear system is .1'2 +a2.1'+al =(£.1'+1)2
where e is a tuning parameter and its value is set to 0.002. The design model can be also expressed in the linearly parameterized form of time-varying parameters,
696
KI
=dia~/cpI
2/C p2}
K2 = dia~l/ t:;1 11 t:;2} This makes the characteristic equation of the estimated output dynamics be (t: pS + 1)2.
[0 pI'
t: p2
are set to 0.005 for both outputs. Two simulation cases are performed; a) when the exact parameters are available and they are utilized in the linearizing control law, and b) when the parameters are estimated and the estimates are utilized in the same control law. Fig.2 shows the comparative performance of the linearizing control for two cases, and comparison of estimated parameters with exact parameters for case b). As shown in Fig.2, the results are good even for parameter ~ showing the peak, and control performance and control input of case a) and b) are comparable. 6. CONCLUSIONS As demonstrated through two applications, the proposed method can be applied successfully to typical chemical processes. The resulting design model is simple in spite of complexities of the process considered. The design procedure of the method is simple and clear. And, it is also robust in sense that diverse uncertainties and disturbances can be reflected in the estimated parameters and so taken into account in the controller. The formulated controller is in the output feedback form. ACKNOWLEDGEMENT We thank the financial aid to this research from the Brain
Korea 21 Program supported by the Ministry of Education and the National Research Lab Grant of the Ministry of Science & Technology. 7. REFERENCES [I] Ballagou, P.E. and Soong, D.S. (1985), "Major Factors Contributing to the Nonlinear Kinetics of Free-Radical Polymerization," Chem. Engng. Sci., 40, 75-86 . [2] Chiu, w.Y., Carratt, G.M., and Soong, D.S. (1983), "A Computer Model for the Gel Effect in Free-Radical Polymerization," Macromolecules, 16, 348-357. [3] Kravaris, C. and Chung, C.B. (1987), "Nonlinear State Feedback Synthesis by Global Input/Output
Linearization," AIChE J., 33,592[4) Kravaris, c., Wright, RA, and Carrier, IF. (1989), "Nonlinear Controllers for Trajectory Tracking in Batch Processes," Computers Chem. Engng., 13, 73-82 . [5) Henson, MA and Seborg, D.E. (1992), "Nonlinear Control Strategies for Continuous Fermenters," Chem. Engng. Sci., 47, 821-835. [6) Hu, Q. and Rangaiah, G.P. (1999), "Adaptive Internal Model Control of Nonlinear Processes," Chem. Engng. Sci. , 54, 1205-1220. [7) Isidori, A. (1989), Nonlinear Control Systems, 2nd, Springer-VerJag. [8) Lee, P.L. and Sullivan, G.R.(1988), "Generic Model Control(GMC)," Computers Chem. Engng., 12, 573-580. [9) Palanki, S. and Kravaris, C. (1997), "Controller Synthesis for Time-Varying Systems by Input/Output Linearization," Computers Chem. Engng., 21 , 891903. [IO) Soroush, M. and Kravaris, C. (1992), "Nonlinear Control of a Batch Polymerization Reactor:an Experimental Study," AIChE J. , 39, 1429-1448. [11] Tatiraju, S. and Soroush, M. (I 998), "Parameter Estimator Design with Application to a Chemical Reactor," Ind. Eng. Chem. Res., 37, 455-463.
697