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Asymptotically Exact Linearization of Chemical Processes
Copyright © IFAC Dynamics and Control of Chemical Reactors (DYCORD+'92), Maryland, USA, 1992
ASYMPTOTICALL Y EXACT LINEARIZATION OF CHEMICAL PROCESSES M. Friedrich, M. Slorz and E.D. Gilles instilUljUr Syslemdynamik und Regelungstechnik, Universitiit StulIgart , Pfaffenwaldring 9, 7000 SIUllgart 80, Germany
Abstract Nonlinear control of chemical processes using methods based on geometric theory has been proposed by a number of researchers during the last years. The most important approach for practical applications is exact linearization of the 1/0behavior of nonlinear systems. However, there are a number of problems associated with this approach. In this paper, an extension to exact I/O-linearization for SISOsystems is proposed to obtain output feedback instead of state feedback as well as improved robustness properties. The results of the approach are illustrated by simulation examples. Keywords chemical variables control, nonlinear control systems, quality control, state feedback, state estimation, robustness
treated. (2) The whole system state has to be known since state feedback is used. (3) The robustness of geometric methods in the presence of modeling errors is very limited, and (4) constraints on manipulated variables can not be taken into account explicitly.
INTRODUCTION Due to the nonlinear nature of chemical processes, nonlinear control techniques appear to be especially well suited for applications in chemical process control. This is the reason why there has been a growing interest in this area during the last years. Beside methods that are mainly based on intuitive concepts there are basically two different approaches . The first makes use of optimization techniques (e.g. Brengel and Seider, 1989; Eaton and Rawlings, 1990), whereas the other is based on differential geometry (Isidori, 1989). A review of geometric control methods that may be applied to chemical processes was given by Kravaris and Kantor (1990) . For practical purposes one of the most important geometric control methods is exact input/output linearization (EIOL) of multivariable nonlinear systems. Under certain restrictions it is thereby possible to find a state feedback which makes a nonlinear system I/O equivalent to a linear system, that may be controlled by linear methods to achieve desired performance of the control loop. Also, in the control of interconnected processes, e.g. chemical plants, an exactly linearized process model may be included into a larger, possibly plant wide, control structure (Friedrich and Gilles, 1991a,b).
The focus of this paper is on the use of 1/0linearization for chemical processes with special reference to output feedback and improvement of robustness properties. This results eventually in a concept called asymptotically exact I/O-linearization (AEIOL), which achieves 1/0linearity in a less rigorous sense than methods already known. The systems considered in this paper are slightly more general than common singleinput-single-output (SISO) systems in the following sense: In chemical process control one of the most important objectives is to keep certain product properties within specified bounds. These properties of the stream leaving a process unit are the variables to be controlled and should therefore be included explicitly in a process control concept. Usually it is not possible, however, to measure such a product property online. In many practical cases the value of a product property is inferred from variables that may be measured easily. Therefore, we allow one controlled variable w and one measured output y which may not necessarily coincide. The class of models treated in this paper can therefore be described by the following set of equations:
However, with respect to practical applications of feedback linearization there are problems to be dealt with. (1) The basic problem with I/O-linearization is the fact that only systems with asymptotically stable zero dynamics can be 137
t(~)
y
h(~)
w
q(~)
+ ~(~)u
w
• (1) Y I>.y
r-
Here, u, y, and ware scalars, and ~ is a vector of dimension n. In our context this implies the simplifying assumption that the state vector ~ of the process contains the same physical variables as the vector characterizing the outflowing material stream. Obviously, model (1) reduces to the common SISO case if y coincides with w.
+
I U
In the following part of the paper general ideas of the concept will be outlined. Then different ways are proposed how the concept may be realized, and simulation examples are presented.
" Y
" w
Fig. 1: Block diagram of asymptotically exact IjO-linearization with state estimation (solid lines: classical structure; whole diagram: new structure)
used for state estimation. The difference to the classical approach is represented by the dashed line. As an advantage of this procedure the states ~ of M can be determined by an online simulation of the model equations. Therefore, we can always achieve exact linear behavior between v and w provided the model has asymptotically stable zero dynamics and the I/O-behavior of the model can be decoupled from the second input which is the correction term of the state estimation scheme. The latter is possible for a limited class of systems as will be discussed later. Then, if model M is a perfect description of process P, exact 1/0linearization of P is also achieved after the effects of differences in the initial conditions of M and P have been eliminated by the state estimator. This can be seen as a new approach to AEIOL using model correction. Since the output w of process P is forced to follow w it can also be viewed as the solution to a model matching problem (Isidori, 1989).
GENERAL IDEAS In this section, we present the basic ideas of the concept of asymptotically exact linearization for model (1). If the system with w as the controlled output is minimum phase, its I/O-behavior can be made linear by static state feedback . Otherwise, an internal instability arises in the closed loop, which renders the whole system unstable in a practical application. It is not necessary to present details about the derivation of a feedback law at this point, we rather look at its structure. The physical input u of process P is calculated by the controller C in such a way that the dynamic behavior between a new input v and the output w is linear. It is then possible to use another feedback loop to control the output w of the resulting linear system. For practical applications this method can only be used if the process states can be measured, if there are no constraints on the manipulated input u, and if the process model describes the dynamic behavior of the process with high accuracy.
Commonly, model M will not be perfect. In this case, one will encounter a deviation of the output w from so that AEIOL of the process can not be achieved. However, the extent of this deviation is determined by the performance of the estimator alone so that one has a means to act on it. It is important to note that this is not the case if an observer or filter is used only to estimate the system state (classical structure) since in that case the deviation of w from the desired value is determined by the robustness properties of both, estimation and linearization technique.
w,
In many cases it is not possible to measure the complete system state. It is then common practice to use a Kalman filter or an observer to obtain an estimate of the actual state. Exact linearization of the process is now achieved only in an asymptotic sense, that is after convergence of the state estimation technique. However, unlike in the linear case where a separation theorem holds, it is generally not possible to predict the effect of an estimation technique on controller performance if nonlinear systems are treated . Especially in connection with exact I/O-linearization, where robustness is a critical issue, the effects of errors in the state estimates can hardly be foreseen. The block diagram of this classical approach to asymptotically exact I/O-linearization (AEIOL) is given by the solid lines in Fig. 1.
Finally, there is a third AEIOL procedure besides those we have outlined so far. Again, only the I/O-behavior of model M is linearized, and the linearizing input u is also used as an input for the process P. But in this case the correction for deviations of w from tU is done in a different way. Fig. 2 shows that in this structure the difference between the measurable outputs y and y is used to calculate an additional correction Au which is added to u to form the input u for process P.
We propose now a slightly modified approach in which it is not attempted to achieve exact linearization of the process P itself but rather of the process model M including the correction term E
Up to now, nothing has been said about the way 138
linear part using the condition
r:-l-----w L....:_I-----,
(5)
Y
u
where v(t) is an arbitrary function of time. Eq. (5) is a static state feedback law that can be expressed in original coordinates and that achieves linear behavior between the new input v and the output w in an exact way. The linear system is given by a chain of p integrators. It is said to be in Brunovsky canonical form. Its dynamic behavior can be changed by an additional linear feedback from w to v.
Fig. 2: Block diagram of asymptotically exact I/O-linearization with additional input au
In particular, such a linear feedback can be used to force the output w to follow a reference output WR produced by the linear system
the state estimator for the AEIOL structure with model correction or the additional input ~u for the AEIOL structure in Fig. 2 are designed . This issue will be treated in the sequel after a brief review of exact I/O-linearization.
A~ +!lv fT ~
WR
One first requires that the relative degree of (6) be greater or equal to the relative degree p of process (1). If this were not the case the resulting control law would not be realizable. The first p equations of system (6) in normal form are:
EXACT INPUT/OUTPUT LINEARIZATION Exact linearization of the I/O-behavior of the nonlinear process (1) by means of coordinates change and static state feedback is well understood (Isidori, 1989). If W is the output to be controlled, the following local coordinates transform is used, whose Jacobian matrix is nonsingular in a neighborhood of a certain point ~o:
Cf-lq(~)
(6)
(7)
Z;
fT AP (
+ fT AP-l!lv
provided that (6) has relative degree p. If the output w has to converge to WR the p-th equation of (4) and (7) must be identical except for an error that vanishes as time proceeds. If we require that the error e = W - W R vanishes according to the linear dynamics
(2)
IlIp+1(~)
where p is the relative degree of the system if W = q(~) is taken to be the output and C H(~) is the derivative of q(~) along f(~). The specific form of III p+ 1 (~), •.. , III n (~) is -not important at this point. If we denote the first p variables of the transformed system by the vector and the remaining n - p variables by '!1., -
the appropriate feedback u is determined from the following condition:
e
p ".
~II-l
((i-I) _ W
(i-I))
WR
, (9)
i=1
(3)
so that we obtain in original coordinates: u
we obtain the following transformed system:
ZI
Z2
Zp-l
zp
'!1.
b({, '!1.) + a({, '!1.)u E({,'!1.)
W
Zl
zp
(4)
A block diagram of this dynamic feedback law is shown in Fig. 3, were L5 represents the linear system (6). ASYMPTOTICALLY EXACT 1/0LINEARIZATION
If the zero dynamics of the system is stable one can achieve exactly linear I/O-behavior by decoupling the nonlinear part of the system from the
In most applications of chemical process control state feedback can only be used together with 139
the performance of the observer even in the case of modeling errors. This is not true if the classical structure is used. In that case the IjO-behavior of the process depends again on the observer performance but also on the robustness properties of the linearization procedure.
w
x
Apparently, there is no restriction to the choice of the state estimation technique for this configuration . Although no general solution to the problem of observer design for nonlinear systems is known, there is still a number of approaches that lead to acceptable results for practical applications (see e.g. Birk and Zeitz, 1988, and the references therein). Furthermore, observers and Kalman filters have been designed with great success for chemical processes using physical insight.
Fig. 3: Block diagram of exact model matching
state estimation because the system state ~ is not completely measurable. The classical structure of state feedback with state estimation has been given by the solid lines in Fig. 1. The state ~ of an observer or filter is used in the feedback law for the process, in which case one obtains AEIOL. However, it is well known that exact linearization is quite sensitive to modeling and measurement errors, and we have therefore proposed two alternative schemes for AEIOL with improved robustness properties.
The capabilities of AEIOL are illustrated by the following example. The IjO-behavior of a continuous stirred tank reactor (CSTR) with consecutive reaction A -+ B -+ C is to be linearized. Cooling temperature Tc is used as the manipulated variable 1£, the outputs wand y coincide and are represented by the reactor temperature T since p = 1 for this output. The reactions are first order, total system mass m is taken to be constant, and the molar masses Mi (i = A, B, C) are equal. The process equations are then given in dimensionless form by:
AEIOL with model correction We are now first concerned with the new AEIOL structure using model correction. The process we want to control is again given by equations (1) . Appropriate observer equations are (Zeitz, 1977): ~
f.{~)
y
h(~)
W
q(~)
+ fl.(~)1£ + k(~, 1£)(Y - y) (11)
The difference between the classical structure and the new configuration, both given in Fig. 1, is the fact that in the latter case we do not try to linearize the process. Instead, the IjO-behavior of the observer is exactly linearized. If model matching with the linear model (6) is desired the derived feedback law (10) has to be modified to compensate for the observer correction. To be able to compensate for this term model (11) must have relative degree p = 1. Otherwise the compensation could only be done if time derivatives of y up to the order p - 1 were known. This is usually not the case so that this method is restricted to the class of systems with p = 1. The appropriate feedback law is then:
~2Z2ezp ( 112 - ~:) 1 - (Z3/ - Z3) + {3 (1£
dZ 3
dt
T
y
(13) -
Z3)
+
~lzlezp (111 - ~J (-D.ht)
+
~2Z2ezp (112 - ~:) (-D.h2) w
= Z3
Here, ZI and Z2 are the mass fractions of A and B, Z3 is the dimensionless temperature, and 1£ the dimensionless cooling temperature. ~i, 11i, and (-D.hi) are frequency factors, activation energies, and heats of reaction, respectively, whereas {3 denotes the dimensionless heat transfer coefficient . Inlet streams are marked by index I and are taken to be constant. Fig. 4 shows simulation results for the classical structure (upper diagram) and the AEIOL structure with model correction (lower diagram). A 5% error in 111 is used in both cases. The observer corrections are constant and the same and have been determined from the approximate linear model at the nominal steady state. In each case three trajectories are shown. The solid line represents the process, observer and
1£
+ where £, and £g denote the Lie derivative operators with respect to~. Since the model and the states ~ of the observer are always exactly known and the process output y can be measured one can always exactly achieve linear IjO-behavior of the observer. The deviation of the process from the exactly linearized observer depends only on 140
linearizing input u for model (14) using a static state feedback law of the type (10). In the second step an additional feedback ~1L is calculated and added to u to result in the manipulated input 1L for process P. To calculate ~ 1L the equations of both, plant and model, are transformed into generalized controller canonical form (Rudolph, Birk, and Zeitz, 1991). Here, y is the output of interest because ~1L is calculated using the difference of y and y:
desired reference are given by the dashed and the dotted lines, respectively. The reference trajectory is produced by applying a step input to the reference model which is the reactor model approximately linearized at the operating point. It is clear from the derivation that the observer output w in the AEIOL structure coincides with the reference WR. Due to the modelling error there remains a certain deviation between wand W R at steady state. With the classical structure, there is about the same steady state error between process and model. However, since w does not coincide with WR there is a larger deviation of W from the desired trajectory WR. This deviation is due to the sensitivity of the feedback law to modelling errors which can not compensate for the difference in steady states of process and model.
=
= h(~)
Y y
= y(r-l)
= CJ
+ Cg
= CJ
~r-__~c~la~s~s:~w~w~m~w~r____~
(15)
390
=
y(n-l) =CJ
+
+
~~--------~--------~
o
~r-
390
500 Time
1000
•
~,U,1L,
n-r-l '"' 8
L...J 8u(i-l)1L
•••
,u(n-r-2») U
(i)
i=1
____n~e~w~:~w~w~nt~w~r~__- .
::c ~O~--------~------~
500 Time
(
In these equations T is the relative degree of the system with y as the output. An equivalent transformation can be used to transform the model equations into generalized controller canonical form. It is then possible to define an error f = ~ - i. between the state ~ of the transformed plant equations and the state i. of the transformed model equations. Its dynamic behavior is given by
.-••• •..•• "'" '"
o
A"..
;0
L..g~n-l
1000
(16) Fig. 4: Simulation results for CSTR (explanation in text)
CJ
" " u ~ ( CJ
t;t
We turn now to the third approach to AEIOL, which is not restricted by a certain value of the relative degree . Instead of using an observer correction to compensate for plant/model mismatch, we introduce an additional feedback ~1L which is calculated from the difference in the outputs of plant and model (see Fig. 2) . The model that is used has the same form as the process equations (1), but its state variables are denoted by ~ and its input by U, so that we have: h(~)
8
(i)
81L(i-l) 1L
i=1
AEIOL with additional input
t(~)
~
+ Cg
8~n
8u(i-l) u
Cl) I
We require now that the error f vanishes with given dynamics f = Lf, where L is
o o
1
o
o 1
o o
L
o -h This can be achieved if a solution for the compensation condition
+ ~(~)u (14)
+
q(~)
The procedure for the calculation of an asymptotically exact linearization feedback law consists of two steps. The first step is to calculate an exactly 141
n
- L'; (~; - ~;)
(18)
0.7 r.-_ _----'w:!J...!w!..!,m~=:.!w!..!.r_ _ _~
;=1
can be found. But this equation depends on the system state ~ which is unknown. To obtain an estimate of ~ we set
= %1 €2 = %2 €1
/:::;.y
0.3
i2
€2*
0. 2~------'--------l
o
(19) €n
= %n -
-
i1
in
* €n
where f* are the estimated derivatives of tained by the following linear system:
500
1000
Time 310 r-_ _ _~u~u~m!!..__ _ _~
/:::;.y
ob300
(20) i~
-11 /:::;.y - 12€; - ... - 'n€~
~O~----~----~
o
If we express €1, " " €n in original coordinates, eqns. (18), (19), and (20) form a system of n - 1 first order differential equations for €;, ... , €~, n - 1 algebraic equations to determine /:::;.y, . .. , /:::;'y(n-1), and one differential equation of order n - r for u. These equations are nonlinear so that an analytic solution can usually not be obtained. However, an extended linearization around the estimated state ~, u can be used to yield a linear system of equations for /:::;.X x - Z and /:::;.U u-u. This system can be solv-;d fo-; /:::;.~ so that the manipulated input u for the process can be calculated.
=
500
1000
Time
Fig. 5: Simulation results for CSTR
so that the robustness of the whole structure is considerably improved. REFERENCES Birk, J., and M. Zeitz. (1988) Extended Luenberger observer for nonlinear multi variable systems. Int. J. Cont., 47, 1823-1836. Brengel D. D., and W. D. Seider. (1989) Multistep nonlinear predictive controller. Ind. Eng. Chem. Res., 28, 1812-1822. Eaton, J. W ., and J. Rawlings. (1990) Feedback control of nonlinear processes using on-line optimization techniques. Comp. Chem. Eng., 14,469-479. Fried rich , M., and E. D. GiIles. (199Ia) Plantwide process control for quality assurance. Proc. World Congo Chem. Eng., Karlsruhe. Friedrich, M., and E. D. Gilles. (1991b) On a process control framework for quality assurance. Prepr. Int. Symp. Adv. Cont. Chem. Proc. ADCHEM '91, 235-240, Toulouse, IFAC. Isidori, A. (1989) Nonlinear Control Systems . Springer, Berlin, 2 ed. Kravaris, C., and J. C. Kantor. (1990) Geometric methods for nonlinear process control. Ind. Eng. Chem. Res ., 29, 2295-2323. Rudolph, J ., J . Birk, and M. Zeitz . (1991) Dependence on time derivatives of the input in the nonlinear controller canonical form. In G. Conte, A. M. Perdon, and B. Wyman, eds, New Trends in Systems Theory, 636643. Birkhiiuser, Boston. Zeitz, M. (1977) Nichtlineare Beobachter fiir chemische Reaktoren. VDI-Verlag, Diisseldorf.
=
As a simulation example we consider the same stirred tank reactor as before. However, since we can allow the output to have p > 1, we treat now the mass fraction X2 of substance B as the controlled output wand the reactor temperature X3 as the measured output y. This is a more realistic example than the one before. The upper diagram in Fig. 5 shows the convergence of the process trajectory (solid line) to the desired behavior (broken line) for the case of an error in the initial condition of Xl. In the lower diagram the manipulated inputs u (broken line) and u (solid line) for model and process, respectively, are plotted. The additional input /:::;.U is given by the difference between these trajectories. CONCLUSIONS An extension to exact I/O-linearization of nonlinear processes has been presented that provides output feedback as well as improved robustness properties. With this approach asymptotically exact I/O-linearization is achieved by an exact linearization of the model equations instead of the process. There are then two possibilities to obtain convergence between process and model so that the process is exactly linearized in an asymptotic sense. With this procedure the performance of the structure is completely independent of the robustness properties of the linearization technique, 142
ASYMPTOTICALL Y EXACT LINEARIZATION OF CHEMICAL PROCESSES M. Friedrich, M. Slorz and E.D. Gilles instilUljUr Syslemdynamik und Regelungstechnik, Universitiit StulIgart , Pfaffenwaldring 9, 7000 SIUllgart 80, Germany
Abstract Nonlinear control of chemical processes using methods based on geometric theory has been proposed by a number of researchers during the last years. The most important approach for practical applications is exact linearization of the 1/0behavior of nonlinear systems. However, there are a number of problems associated with this approach. In this paper, an extension to exact I/O-linearization for SISOsystems is proposed to obtain output feedback instead of state feedback as well as improved robustness properties. The results of the approach are illustrated by simulation examples. Keywords chemical variables control, nonlinear control systems, quality control, state feedback, state estimation, robustness
treated. (2) The whole system state has to be known since state feedback is used. (3) The robustness of geometric methods in the presence of modeling errors is very limited, and (4) constraints on manipulated variables can not be taken into account explicitly.
INTRODUCTION Due to the nonlinear nature of chemical processes, nonlinear control techniques appear to be especially well suited for applications in chemical process control. This is the reason why there has been a growing interest in this area during the last years. Beside methods that are mainly based on intuitive concepts there are basically two different approaches . The first makes use of optimization techniques (e.g. Brengel and Seider, 1989; Eaton and Rawlings, 1990), whereas the other is based on differential geometry (Isidori, 1989). A review of geometric control methods that may be applied to chemical processes was given by Kravaris and Kantor (1990) . For practical purposes one of the most important geometric control methods is exact input/output linearization (EIOL) of multivariable nonlinear systems. Under certain restrictions it is thereby possible to find a state feedback which makes a nonlinear system I/O equivalent to a linear system, that may be controlled by linear methods to achieve desired performance of the control loop. Also, in the control of interconnected processes, e.g. chemical plants, an exactly linearized process model may be included into a larger, possibly plant wide, control structure (Friedrich and Gilles, 1991a,b).
The focus of this paper is on the use of 1/0linearization for chemical processes with special reference to output feedback and improvement of robustness properties. This results eventually in a concept called asymptotically exact I/O-linearization (AEIOL), which achieves 1/0linearity in a less rigorous sense than methods already known. The systems considered in this paper are slightly more general than common singleinput-single-output (SISO) systems in the following sense: In chemical process control one of the most important objectives is to keep certain product properties within specified bounds. These properties of the stream leaving a process unit are the variables to be controlled and should therefore be included explicitly in a process control concept. Usually it is not possible, however, to measure such a product property online. In many practical cases the value of a product property is inferred from variables that may be measured easily. Therefore, we allow one controlled variable w and one measured output y which may not necessarily coincide. The class of models treated in this paper can therefore be described by the following set of equations:
However, with respect to practical applications of feedback linearization there are problems to be dealt with. (1) The basic problem with I/O-linearization is the fact that only systems with asymptotically stable zero dynamics can be 137
t(~)
y
h(~)
w
q(~)
+ ~(~)u
w
• (1) Y I>.y
r-
Here, u, y, and ware scalars, and ~ is a vector of dimension n. In our context this implies the simplifying assumption that the state vector ~ of the process contains the same physical variables as the vector characterizing the outflowing material stream. Obviously, model (1) reduces to the common SISO case if y coincides with w.
+
I U
In the following part of the paper general ideas of the concept will be outlined. Then different ways are proposed how the concept may be realized, and simulation examples are presented.
" Y
" w
Fig. 1: Block diagram of asymptotically exact IjO-linearization with state estimation (solid lines: classical structure; whole diagram: new structure)
used for state estimation. The difference to the classical approach is represented by the dashed line. As an advantage of this procedure the states ~ of M can be determined by an online simulation of the model equations. Therefore, we can always achieve exact linear behavior between v and w provided the model has asymptotically stable zero dynamics and the I/O-behavior of the model can be decoupled from the second input which is the correction term of the state estimation scheme. The latter is possible for a limited class of systems as will be discussed later. Then, if model M is a perfect description of process P, exact 1/0linearization of P is also achieved after the effects of differences in the initial conditions of M and P have been eliminated by the state estimator. This can be seen as a new approach to AEIOL using model correction. Since the output w of process P is forced to follow w it can also be viewed as the solution to a model matching problem (Isidori, 1989).
GENERAL IDEAS In this section, we present the basic ideas of the concept of asymptotically exact linearization for model (1). If the system with w as the controlled output is minimum phase, its I/O-behavior can be made linear by static state feedback . Otherwise, an internal instability arises in the closed loop, which renders the whole system unstable in a practical application. It is not necessary to present details about the derivation of a feedback law at this point, we rather look at its structure. The physical input u of process P is calculated by the controller C in such a way that the dynamic behavior between a new input v and the output w is linear. It is then possible to use another feedback loop to control the output w of the resulting linear system. For practical applications this method can only be used if the process states can be measured, if there are no constraints on the manipulated input u, and if the process model describes the dynamic behavior of the process with high accuracy.
Commonly, model M will not be perfect. In this case, one will encounter a deviation of the output w from so that AEIOL of the process can not be achieved. However, the extent of this deviation is determined by the performance of the estimator alone so that one has a means to act on it. It is important to note that this is not the case if an observer or filter is used only to estimate the system state (classical structure) since in that case the deviation of w from the desired value is determined by the robustness properties of both, estimation and linearization technique.
w,
In many cases it is not possible to measure the complete system state. It is then common practice to use a Kalman filter or an observer to obtain an estimate of the actual state. Exact linearization of the process is now achieved only in an asymptotic sense, that is after convergence of the state estimation technique. However, unlike in the linear case where a separation theorem holds, it is generally not possible to predict the effect of an estimation technique on controller performance if nonlinear systems are treated . Especially in connection with exact I/O-linearization, where robustness is a critical issue, the effects of errors in the state estimates can hardly be foreseen. The block diagram of this classical approach to asymptotically exact I/O-linearization (AEIOL) is given by the solid lines in Fig. 1.
Finally, there is a third AEIOL procedure besides those we have outlined so far. Again, only the I/O-behavior of model M is linearized, and the linearizing input u is also used as an input for the process P. But in this case the correction for deviations of w from tU is done in a different way. Fig. 2 shows that in this structure the difference between the measurable outputs y and y is used to calculate an additional correction Au which is added to u to form the input u for process P.
We propose now a slightly modified approach in which it is not attempted to achieve exact linearization of the process P itself but rather of the process model M including the correction term E
Up to now, nothing has been said about the way 138
linear part using the condition
r:-l-----w L....:_I-----,
(5)
Y
u
where v(t) is an arbitrary function of time. Eq. (5) is a static state feedback law that can be expressed in original coordinates and that achieves linear behavior between the new input v and the output w in an exact way. The linear system is given by a chain of p integrators. It is said to be in Brunovsky canonical form. Its dynamic behavior can be changed by an additional linear feedback from w to v.
Fig. 2: Block diagram of asymptotically exact I/O-linearization with additional input au
In particular, such a linear feedback can be used to force the output w to follow a reference output WR produced by the linear system
the state estimator for the AEIOL structure with model correction or the additional input ~u for the AEIOL structure in Fig. 2 are designed . This issue will be treated in the sequel after a brief review of exact I/O-linearization.
A~ +!lv fT ~
WR
One first requires that the relative degree of (6) be greater or equal to the relative degree p of process (1). If this were not the case the resulting control law would not be realizable. The first p equations of system (6) in normal form are:
EXACT INPUT/OUTPUT LINEARIZATION Exact linearization of the I/O-behavior of the nonlinear process (1) by means of coordinates change and static state feedback is well understood (Isidori, 1989). If W is the output to be controlled, the following local coordinates transform is used, whose Jacobian matrix is nonsingular in a neighborhood of a certain point ~o:
Cf-lq(~)
(6)
(7)
Z;
fT AP (
+ fT AP-l!lv
provided that (6) has relative degree p. If the output w has to converge to WR the p-th equation of (4) and (7) must be identical except for an error that vanishes as time proceeds. If we require that the error e = W - W R vanishes according to the linear dynamics
(2)
IlIp+1(~)
where p is the relative degree of the system if W = q(~) is taken to be the output and C H(~) is the derivative of q(~) along f(~). The specific form of III p+ 1 (~), •.. , III n (~) is -not important at this point. If we denote the first p variables of the transformed system by the vector and the remaining n - p variables by '!1., -
the appropriate feedback u is determined from the following condition:
e
p ".
~II-l
((i-I) _ W
(i-I))
WR
, (9)
i=1
(3)
so that we obtain in original coordinates: u
we obtain the following transformed system:
ZI
Z2
Zp-l
zp
'!1.
b({, '!1.) + a({, '!1.)u E({,'!1.)
W
Zl
zp
(4)
A block diagram of this dynamic feedback law is shown in Fig. 3, were L5 represents the linear system (6). ASYMPTOTICALLY EXACT 1/0LINEARIZATION
If the zero dynamics of the system is stable one can achieve exactly linear I/O-behavior by decoupling the nonlinear part of the system from the
In most applications of chemical process control state feedback can only be used together with 139
the performance of the observer even in the case of modeling errors. This is not true if the classical structure is used. In that case the IjO-behavior of the process depends again on the observer performance but also on the robustness properties of the linearization procedure.
w
x
Apparently, there is no restriction to the choice of the state estimation technique for this configuration . Although no general solution to the problem of observer design for nonlinear systems is known, there is still a number of approaches that lead to acceptable results for practical applications (see e.g. Birk and Zeitz, 1988, and the references therein). Furthermore, observers and Kalman filters have been designed with great success for chemical processes using physical insight.
Fig. 3: Block diagram of exact model matching
state estimation because the system state ~ is not completely measurable. The classical structure of state feedback with state estimation has been given by the solid lines in Fig. 1. The state ~ of an observer or filter is used in the feedback law for the process, in which case one obtains AEIOL. However, it is well known that exact linearization is quite sensitive to modeling and measurement errors, and we have therefore proposed two alternative schemes for AEIOL with improved robustness properties.
The capabilities of AEIOL are illustrated by the following example. The IjO-behavior of a continuous stirred tank reactor (CSTR) with consecutive reaction A -+ B -+ C is to be linearized. Cooling temperature Tc is used as the manipulated variable 1£, the outputs wand y coincide and are represented by the reactor temperature T since p = 1 for this output. The reactions are first order, total system mass m is taken to be constant, and the molar masses Mi (i = A, B, C) are equal. The process equations are then given in dimensionless form by:
AEIOL with model correction We are now first concerned with the new AEIOL structure using model correction. The process we want to control is again given by equations (1) . Appropriate observer equations are (Zeitz, 1977): ~
f.{~)
y
h(~)
W
q(~)
+ fl.(~)1£ + k(~, 1£)(Y - y) (11)
The difference between the classical structure and the new configuration, both given in Fig. 1, is the fact that in the latter case we do not try to linearize the process. Instead, the IjO-behavior of the observer is exactly linearized. If model matching with the linear model (6) is desired the derived feedback law (10) has to be modified to compensate for the observer correction. To be able to compensate for this term model (11) must have relative degree p = 1. Otherwise the compensation could only be done if time derivatives of y up to the order p - 1 were known. This is usually not the case so that this method is restricted to the class of systems with p = 1. The appropriate feedback law is then:
~2Z2ezp ( 112 - ~:) 1 - (Z3/ - Z3) + {3 (1£
dZ 3
dt
T
y
(13) -
Z3)
+
~lzlezp (111 - ~J (-D.ht)
+
~2Z2ezp (112 - ~:) (-D.h2) w
= Z3
Here, ZI and Z2 are the mass fractions of A and B, Z3 is the dimensionless temperature, and 1£ the dimensionless cooling temperature. ~i, 11i, and (-D.hi) are frequency factors, activation energies, and heats of reaction, respectively, whereas {3 denotes the dimensionless heat transfer coefficient . Inlet streams are marked by index I and are taken to be constant. Fig. 4 shows simulation results for the classical structure (upper diagram) and the AEIOL structure with model correction (lower diagram). A 5% error in 111 is used in both cases. The observer corrections are constant and the same and have been determined from the approximate linear model at the nominal steady state. In each case three trajectories are shown. The solid line represents the process, observer and
1£
+ where £, and £g denote the Lie derivative operators with respect to~. Since the model and the states ~ of the observer are always exactly known and the process output y can be measured one can always exactly achieve linear IjO-behavior of the observer. The deviation of the process from the exactly linearized observer depends only on 140
linearizing input u for model (14) using a static state feedback law of the type (10). In the second step an additional feedback ~1L is calculated and added to u to result in the manipulated input 1L for process P. To calculate ~ 1L the equations of both, plant and model, are transformed into generalized controller canonical form (Rudolph, Birk, and Zeitz, 1991). Here, y is the output of interest because ~1L is calculated using the difference of y and y:
desired reference are given by the dashed and the dotted lines, respectively. The reference trajectory is produced by applying a step input to the reference model which is the reactor model approximately linearized at the operating point. It is clear from the derivation that the observer output w in the AEIOL structure coincides with the reference WR. Due to the modelling error there remains a certain deviation between wand W R at steady state. With the classical structure, there is about the same steady state error between process and model. However, since w does not coincide with WR there is a larger deviation of W from the desired trajectory WR. This deviation is due to the sensitivity of the feedback law to modelling errors which can not compensate for the difference in steady states of process and model.
=
= h(~)
Y y
= y(r-l)
= CJ
+ Cg
= CJ
~r-__~c~la~s~s:~w~w~m~w~r____~
(15)
390
=
y(n-l) =CJ
+
+
~~--------~--------~
o
~r-
390
500 Time
1000
•
~,U,1L,
n-r-l '"' 8
L...J 8u(i-l)1L
•••
,u(n-r-2») U
(i)
i=1
____n~e~w~:~w~w~nt~w~r~__- .
::c ~O~--------~------~
500 Time
(
In these equations T is the relative degree of the system with y as the output. An equivalent transformation can be used to transform the model equations into generalized controller canonical form. It is then possible to define an error f = ~ - i. between the state ~ of the transformed plant equations and the state i. of the transformed model equations. Its dynamic behavior is given by
.-••• •..•• "'" '"
o
A"..
;0
L..g~n-l
1000
(16) Fig. 4: Simulation results for CSTR (explanation in text)
CJ
" " u ~ ( CJ
t;t
We turn now to the third approach to AEIOL, which is not restricted by a certain value of the relative degree . Instead of using an observer correction to compensate for plant/model mismatch, we introduce an additional feedback ~1L which is calculated from the difference in the outputs of plant and model (see Fig. 2) . The model that is used has the same form as the process equations (1), but its state variables are denoted by ~ and its input by U, so that we have: h(~)
8
(i)
81L(i-l) 1L
i=1
AEIOL with additional input
t(~)
~
+ Cg
8~n
8u(i-l) u
Cl) I
We require now that the error f vanishes with given dynamics f = Lf, where L is
o o
1
o
o 1
o o
L
o -h This can be achieved if a solution for the compensation condition
+ ~(~)u (14)
+
q(~)
The procedure for the calculation of an asymptotically exact linearization feedback law consists of two steps. The first step is to calculate an exactly 141
n
- L'; (~; - ~;)
(18)
0.7 r.-_ _----'w:!J...!w!..!,m~=:.!w!..!.r_ _ _~
;=1
can be found. But this equation depends on the system state ~ which is unknown. To obtain an estimate of ~ we set
= %1 €2 = %2 €1
/:::;.y
0.3
i2
€2*
0. 2~------'--------l
o
(19) €n
= %n -
-
i1
in
* €n
where f* are the estimated derivatives of tained by the following linear system:
500
1000
Time 310 r-_ _ _~u~u~m!!..__ _ _~
/:::;.y
ob300
(20) i~
-11 /:::;.y - 12€; - ... - 'n€~
~O~----~----~
o
If we express €1, " " €n in original coordinates, eqns. (18), (19), and (20) form a system of n - 1 first order differential equations for €;, ... , €~, n - 1 algebraic equations to determine /:::;.y, . .. , /:::;'y(n-1), and one differential equation of order n - r for u. These equations are nonlinear so that an analytic solution can usually not be obtained. However, an extended linearization around the estimated state ~, u can be used to yield a linear system of equations for /:::;.X x - Z and /:::;.U u-u. This system can be solv-;d fo-; /:::;.~ so that the manipulated input u for the process can be calculated.
=
500
1000
Time
Fig. 5: Simulation results for CSTR
so that the robustness of the whole structure is considerably improved. REFERENCES Birk, J., and M. Zeitz. (1988) Extended Luenberger observer for nonlinear multi variable systems. Int. J. Cont., 47, 1823-1836. Brengel D. D., and W. D. Seider. (1989) Multistep nonlinear predictive controller. Ind. Eng. Chem. Res., 28, 1812-1822. Eaton, J. W ., and J. Rawlings. (1990) Feedback control of nonlinear processes using on-line optimization techniques. Comp. Chem. Eng., 14,469-479. Fried rich , M., and E. D. GiIles. (199Ia) Plantwide process control for quality assurance. Proc. World Congo Chem. Eng., Karlsruhe. Friedrich, M., and E. D. Gilles. (1991b) On a process control framework for quality assurance. Prepr. Int. Symp. Adv. Cont. Chem. Proc. ADCHEM '91, 235-240, Toulouse, IFAC. Isidori, A. (1989) Nonlinear Control Systems . Springer, Berlin, 2 ed. Kravaris, C., and J. C. Kantor. (1990) Geometric methods for nonlinear process control. Ind. Eng. Chem. Res ., 29, 2295-2323. Rudolph, J ., J . Birk, and M. Zeitz . (1991) Dependence on time derivatives of the input in the nonlinear controller canonical form. In G. Conte, A. M. Perdon, and B. Wyman, eds, New Trends in Systems Theory, 636643. Birkhiiuser, Boston. Zeitz, M. (1977) Nichtlineare Beobachter fiir chemische Reaktoren. VDI-Verlag, Diisseldorf.
=
As a simulation example we consider the same stirred tank reactor as before. However, since we can allow the output to have p > 1, we treat now the mass fraction X2 of substance B as the controlled output wand the reactor temperature X3 as the measured output y. This is a more realistic example than the one before. The upper diagram in Fig. 5 shows the convergence of the process trajectory (solid line) to the desired behavior (broken line) for the case of an error in the initial condition of Xl. In the lower diagram the manipulated inputs u (broken line) and u (solid line) for model and process, respectively, are plotted. The additional input /:::;.U is given by the difference between these trajectories. CONCLUSIONS An extension to exact I/O-linearization of nonlinear processes has been presented that provides output feedback as well as improved robustness properties. With this approach asymptotically exact I/O-linearization is achieved by an exact linearization of the model equations instead of the process. There are then two possibilities to obtain convergence between process and model so that the process is exactly linearized in an asymptotic sense. With this procedure the performance of the structure is completely independent of the robustness properties of the linearization technique, 142