Global nonlinear control of a continuous stirred tank reactor

Chmical Engineering Science, Vol. 44, ND. 5, pp. Il47-1160, Printed in Great Britain.

GLOBAL

1989.

NONLINEAR STIRRED

Universidad Autdnoma

0

ooo9-X09/89 53.00 + 0.00 1989 Pergamon Press plc

CONTROL OF A CONTINUOUS TANK REACTOR

JEStiS ALVAREZ + Metropolitana-Iztapalapa, Department0 de Ingenieria de Procesos, Apdo 55534, 09340 M&co, D.F., Mexico and

JAIME ALVAREZ aqd ESTELA GONZALEZ Centro de Investigacidn y Estudios Avanzados de1 IPN, Department0 de Ingenieria Ele’ctrica, Apdo 14740, 07000 Mixico, D.F., Mexico (Received 26 May

1987; accepted 8 June 1988)

Abstract-A chemical reactor, modelled by two nonlinear ordinary differential equations with a manipulable input, is controlled by means of a nonlinear control law. A differential geometric approach enabIes the characterization, in a global sense, of the structural solvability of the nonlinear control problem. The structural characterization consists in establishing a well-defined region of the state space where global controllability, stability and transformability to a linear equivalent can be assured. The nonlinear controller arises after expressing a linear feedback, for an equivalent system, in original coordinates. The tuning of the controller is done by pole placement in the equivalent system. The nonlinear controller allows for process regulation of stable stationary points and stabilization of unstable ones. Proportional and proportional-integral designs for the equivalent system are considered. By numerical simulations, the performance of the controller is favorably compared and contrasted with the performance of conventional controllers.

1. INTRODUCTION

A continuous stirred tank reactor (CSTR) is used to convert reactants into products. Usually, for exothermic reactions that are irreversible, the adiabatic operation leads to large production rates. In this mode of operation the heat generated by the reaction is removed with the flow of material. When the adiabatic operation imposes a reactor temperature above the one permitted by safety and operability specifications, the process of heat removal is aided by exchanging heat through a diathermal wall of the reactor. In fact, the reactor control is achieved by manipulation of the heat flow through the wall. Most of the industrial reactor control is done by linear proportional-integral-differential (PID) strategies (Shinskey, 1979). On the other hand, the usual approach for designing a controller based on state space multivariable control theory is to linearize, with a truncated Taylor series approximation, the reactor model (Ray, 1981) so that linear design procedures may be applied. Linear controllers should perform well when the nonlinear process operates at a point with a linearity neighborhood large enough to encompass excursions due to disturbances. Commonly, this is the result of a conservative design of the process. If the process is subjected to larger disturbances and/or if it is operated at conditions of higher state sensitivity, the state trajectory can make excursions outside the aforementioned neighborhood, and, consequently, the performance of the controller

‘To

whom correspondence should be addressed.

deteriorates. In some special situations, industrial control engineers tackle the problem with dual-mode, three-zone and variable-gain PID controllers to achieve an efficient use of the controller while maintaining speed of response and stability. However, these control designs make little use of available knowledge about the nonlinear structure of the process, and rely heavily on tuning procedures and extensive testing. As a result, in many cases the designer is forced to sacrifice efficiency to assure adequate levels of safety and operation, or, equivalently, to provide a large enough neighborhood where the linear control strategies perform appropriately. Usually, in reactor design the economic objectives draw the process design towards operation at a higher state sensitivity. As a result, the nonlinear nature of the reactor acquires more relevance and poses a more difficult control problem. In some cases the reactor must operate at an unstable stationary point or in an induced periodic transient (Padmanabham and Lapidus, 1977; Bailey, 1977; Spitz et al., 1977; Herman, 1987). The preceding observations suggest that more powerful nonlinear control techniques, based on more systematic procedures, could find application in the design and operation of CSTRs. Cebuhar and Costanza (1984) applied bilinear techniques to control showed that moving

a two-dimensional

system.

They

from linear to nonlinear reactor control is feasible and convenient. Nevertheless, the bilinear technique was still based upon an approximation to the nonlinear model, and the dimensionality of the resulting bilinear control model was doubled. Their work did not address issues related to 1147

1148

JEWS ALVAREZ

the global performance of the closed-loop scheme. Georgakis (1986) designed a reactor nonlinear controller from a geometric analysis of the first-order Taylor approximation of the process. The physics of the problem was associated with geometric properties of the linear approximation of the process. The analysis was then used to design a proportional linear feedback involving extensive variables of the process. The nonlinear controller resulted after replacing the extensive variables by the original ones. This work constitutes an interesting step towards process control design attending the physics of the process. Issues related to the functioning and performance of the controller beyond the Taylor linearization neighborhood were not addressed. As pointed out by Isidori (1985), in the last decade differential geometry has proven to be as successful for the study of nonlinear systems as Laplace transform and complex function theories were in the 1950s for the study of single-input single-output (SISO) systems and linear algebra in the 1960s for the study of multivariable linear systems. Recently, nonlinear geometric control theories for a class of nonlinear systems have been developed. These developments are applicable to the reactor considered in this work. The theories include results in global controllability (Hunt, 1980) as well as local (Su, 1982) and global (Hunt et al., 1983) transformability of the nonlinear system to a linear equivalent one. Stability has been treated with the local-transformation results in conjunction with Lyapunov’s direct method for the equivalent system (Su et al., 1983). Hoo and Kantor (1985,1986) realized that the localtransformation results of Su (1982) could be applied to the control of a two-state, one-input CSTR. Only proportional action was used for the design. From numerical simulations, they conjectured about the presence of global features in the performance of the controller. Because performance criteria, such as speed of response and damping, are not displayed by phase portraits, the work showed no advantage for the use of a nonlinear controller over a conventional one. In fact, Aris and Amundson (1958) had shown that a linear proportional controller yields phase portraits similar to the ones obtained from the nonlinear controller. The local-transformation theory of Su (1982) assures the performance of a nonlinear controller in the neighborhood of a nominal operating point. However, there is no information about the size and shape of that neighborhood_ As a consequence, the characterization of a region of validity for the controller must be obtained from numerical simulations with specific control laws and tuning parameters. The fact that a linear controller (Aris and Amundson, 1958) and a local nonlinear one (Hoo and Kantor, 1965) appear to exhibit globality in their performance suggests that there may be global geometric properties inherent to the control problem of the CSTR. Tn the present work the problem of characterizing, in a global sense, geometric structural properties that

et al.

assure solvability of the control problem of the CSTR is addressed first. The properties considered are controllability, stability and transformability to linear controllable forms. A nonlinear controller is then constructed and tuned. While local-transformability results only yield an expression for the controller, the present global treatment provides a full characterization of the control law (a functionality and a domain of validity), and, hence, the global nature of the controller is rigorously established. Finally, reference is made to the design of a bounded controller. From a practical implementation point of view, it is important to show that a nonlinear controller offers potential advantages over conventional ones, in most cases linear, SISO controllers with integral action. Using numerical simulations, performances are compared between the linear and the nonlinear designs with integral action. The result is that the nonlinear controller leads to a better performance in terms of overshoot, settling time, and control effort.

2. THE

REACTOR

CONTROL

PROBLEM

The reactor and its control scheme are represented in Fig. 1. The heat removed through the wall is proportional to the wall area and to the difference in temperature between the reactive mixture and a coolant medium circulated through a jacket. In practice the heat extraction process is regulated by manipulation of the coolant flow rate. In many cases the jacket dynamics is fast in comparison with the reactor dynamics, and, hence, the coolant flow rate is linked (via a fast SISO nested control) to the jacket temperature. Therefore, the essence of the control problem is retained if the coolant temperature, T,, is considered as the manipulated variable. A justification for the dynamic decoupling induced by the cascaded controller can be found in the singular perturbation theory (Kokotovic et al., 1976). Furthermore, the industrial control of exothermic, cooled reactors is usually done with cascaded controllers (Shinskey, 1979). The chemical reaction rate is assumed to follow first-order kinetics, and the process is modelled by the following equations (variables and parameters are

Fig. 1. Diagrammatic representation of control.

the

pair process-

Global

defined

in Appendix

nonlinear control of a continuous

Consider

A):

1149

stirred tank reactor

the following,

possibly

unstable, system:

dc Co ~=V(~,-~)-kks~e-~~‘(RT) dT

w

dt=V(~e-~)-FVT-C)

k,Aw P

a,,

P

PC, The control problem consists of maintaining the exit concentration c and temperature T around speciThe disturbances fied values in spite of disturbances. are the inlet concentration, c,, and the inlet temperature, T,. The variable available for control is T,. In a practical situation the exit concentration might not be available for measurement. In this case, a nonlinear estimator is required. For this reactor there already exist techniques to implement efficient, on-line estimators such as the one proposed by Wallman (1979) or a lumped version of the one proposed by Alvarez and Stephanopoulos (1982). Therefore, in this work the study is circumscribed to design the controller T,(c, 0 Suppose that an operating stationary point, either stable or unstable, has been chosen as the operating condition. The corresponding values of the state and input variables are denoted by C, T, C,, T= and Fc. If the following deviation variables are defined, x1 = c - C, x,=T-_, and u=T,-F=, eqs (1) and (2) can be written as i =f(x)

+ gu:

and UER

x,f,gER2

(3)

where fi(X,, f2b

~,)=8E,-~(~+x,)-~(E+x,)e-~~~~+~~)

(4)

L 9 x2)=B~=++Y=-((B+y)(~+x2) +aj?(c+x,)e-di’T+Xz) 91 =o,

(5) Y2 =y.

(6)

The parameters introduced are defined 6 = E,/R, e=W/v, B=(-AHMPc~), = h,-%l( VP cp). The objective is to find a control u =p(x),

as LX= k,, and y

(7)

where W is an open, connected set in RZ where the functioning of the controller is assured. It must be kept in mind that p and W depend on the election of a particular stationary point as the operating point.

3. A LINEAR

CONTROL

PROBLEM

In this section a control problem that resembles a linear structural version of the reactor problem is addressed. The purpose is to introduce the control design technique based on the general control canonical form (GCCF) (Kalman, 1971) and to introduce the conceptual geometric ideas that are behind the nonlinear control

problem.

(8)

where b, #O ensures the presence of control action and a,, #O allows for access of the controller to the variable n, _ In fact, a first-order Taylor linearization of the reactor model (3) has the form of system (8). The objective is to design a proportional (P) controller u=k’x,

k’=(k,,

kZ)

(9)

so that the closed-loop process has two specified eigenvalues, 1, and 1,. A ploportional-integral (PI) controller can also be considered: u=k;x+k;

f x dt.

s0

For the sake of simplicity, in this section the treatment is done only for a P controller. The design of the controller (9) can be done by choosing the gain vector k so that the closed-loop matrix (A + bk’) has R, and 1, as eigenvalues. properties 3.1. Structural The above control design overlooked certain structural properties inherent to system (8). It is well known that the linear systems theory provides tools to address relevant structural matters such as controllability, stability and transformability to equivalent forms (Kwakernaak and Sivan, 1973; Kailath, 1980; Balakrishnan, 1983). If aI2 or b2 =O, the control design runs into problems. This can be detected and explained with the concept of controllability which depends on the pair (A, b). The aforementioned problem is carried out to the stabilizability and transformability of the system because these properties depend on controllability. Suppose that A has distinct eigenvalues, the linear theory provides the following results. System (8) is:

6) Open-loop

law

p: W+R

and h,#O

asymptotically stable if and only if Re ;lAitO (i= 1, 2). (ii) Controllable if and only if rank [A, b] =2. (iii) Stabilizable if it is controllable. to the GCCF if it is con(iv) Transformable trollable. 3.2. The GCCF The GCCF is also referred to as the Brunovsky (1970) form. There are various canonical forms which display various structural properties of system (8) (Rosenbrock, 1970, Kwakernaak and Sivan, 1972; Kailath, 1980). In particular the GCCF is useful for a state feedback design which achieves a desired pole placement of eigenvalues. Suppose that system (8) has n states and m inputs: *=Ax+Bu.

1150

JESOS ALVAREZ

controllable (rank Provided the system is [B, AB, _ _ . , A”- ‘B] = n), there exists an invertible transformation that takes the system to the GCCF (Kalman, 1971): i=Gz+Qv where G, Xn= block diagonal [G,, , G,,, . . . , G,,,,] and Q,x.= block diagonal h,, q,,. . . . ,e,l. {G,,) are ri x ri matrices and {q,,} are ri x 1 vectors with the forms

Gri =

OlO---00

0

OOl---00 0 0 0

0 0.

i 0

0

0

_

_

_

0

1

-Qrt=

-

-

--

0

01

. . . , >r,>,O,

r,2r,>,

3.4. The controller The GCCF preserves the closed-loop eigenvalues (Appendix B). Hence, the assignment of eigenvalues can be done on the equivalent system (10). Consider a P controller for the GCCF (IO): u=k;z,

The equivalent closed-loop

k,,=

01

Xl

b2a12

Introduction

ITI # 0

and

a. # 0.

of the linear feedback, w = u - d’z, leads

t0 Z =

[TAT-’

-(l/a)Tbd’]z+[(l/a)Tb]o.

To find the seven scalars [a, dj and tii (i, j= 1, 2)] that define the transformation (a, d and T) the last equation is forced to adopt the GCCF (10). That is = TAT - 1 - (1 /a)Tbd’,

0 01

=(l/a)Tb.

These relationships provide six scalar independent equations. Since there are more unknowns than equations, there is an infinity of transformations. If tll = 1 is assigned arbitrarily, the following transformation is obtained z1(x,)=x1

(11)

zZ(X1,X2)=a,,x,

(12)

tix,,

+a,,x,

x2. ~)=~,,(~,,x,+~,,x,)+~,.(~,,x,+~,,x*) + Ui2U.

(13)

--1,1,,

+ k,,

k,,=;l,

+A,.

(15)

UllXl

+a,+,

b2al z

For unbounded linear systems, controllability, asymptotic stability and transformability are global properties on the entire xI-xZ plane. STRUCTURE

OF THE

CONTROL

(10)

3.3. An invertible transformation Here a transformation is obtained that takes system (8) to the GCCF (10) (Rosenbrock, 1970). Introduce invertible coordinate changes for the state and the control, z=Tx and w = au, respectively. System (8) becomes i=(TAT-l)z+(l/a)TBw,

process is

(16)

4. GLOBAL

;),LJ=(;).

(14)

k,,).

The control law is obtained by expressing eq. (14) in terms of the original coordinates. This is leads to u=k,,---

i$lri=n.

The eigenvalues of the GCCF are all zero. The control invariants are found by examining the independence of the column vectors of the controllability matrix followed by a classification, and possibly a reordering, of the control inputs (Brunovsky, 1970; Kalman, 1971). In the present case, there is only one control invariant, rl =2, and the GCCF becomes i=Ga+qv:G=(;

k;=(kpl,

The gains are tuned according to the following expressions

(ri) is an ordered set of integers denominated control invariants, or Kronecker indices, that satisfy r,>O,

et al.

NONLINEAR

REACTOR

PROBLEM

The structural characterization and the control design technique for the linear case are now generalized, in a global sense, to the nonlinear reactor control problem. In some theoretical studies, the term global is used to refer to a property over the plane. In other works(Hunt, 1980, Hunt et al., 1983), globality is also used to designate a property over a well-defined domain as contrasted to a neighborhood about a point. Since in the present case it does not make sense to study properties outside a region in the phase space delimited by fundamental physical constraints, the second meaning of globality is adopted. As mentioned in the introduction, the problem of transforming the nonlinear model of the reactor to a linear equivalent was addressed by Hoo and Kantor (1985) who utilized Su’s (1982) local theory. In the present work, the aim is to establish the existence of a global tansformation whose domain of validity is a well-defined region. This is done after a global characterization of the structure of the nonlinear reactor control problem. It is shown how the global characterization

of

controllability,

stabilizability

and

trans-

the process vector fields, f and g, independently of the control law and tuning parameters used in the feedback design.

formability

can

be done

by

examining

4.1. Preamble Here, some basic defintions and concepts mental to the differential geometric theory of are briefly presented. For further details and treatments the reader is referred to Boothby

fundacontrol formal (1975),

Global

nonlinear

control

of a continuous

Brockett (1976) and Hermann and Krener (1977). For two vector fields on an n-dimensional, smooth manifold, the Lie bracket [J g] is a smooth vector field defined by

where g, and f, are n x n Jacobian matrices. A set of independent vector fields {fl,fi, . . . ,f,} is involutiue on h4 if there exist smooth scalar fields Q(X) such that

C_L-.fjl = 5 aijtc(xlfk, x EM.

It=1

A set of independent vector fields {f, ,f2, . . _ ,.f,} is completely integrable on M if for any point x, in M, there exists a unique, maximal m-dimensional manifold N of M containing x, such that {fi , f2, _ . , f,} spans the tangent space of N at each point of M. Complete integrability means the existence of a solution for the foIlowing dynamic system 2 = F Au,(t), x(0)=x,

(17)

i=l

and it constitutes a generalization of a version of the well-known Cauchy-Lipshitz existence theorem for ordinary differential equations (Brockett, 1976). The concept of complete integrability of the dynamic control system (17) is related to the concept of involutivity of vector fields by means of Frobenius’ classical theorem (Boothby, 1975): a set of linearly independent vector fields on M is completely integrable on M if the set is involutive on M. Now look for a connection between the Lie bracket of two vector fields and the control problem by referring to a specific situation where the Lie bracket arises naturally (Brockett, 1976). Consider the problem of finding the point reached, starting from x,, by a dynamic control system (17) with two inputs (m= 2). The exogenous inputs are piece-wise constants in time: u, = 1 and u,=O from 0 to t time units, u,=O and u,=l from t to 2t, ur= -1 and u,=O from 2t to 3t, and ur = 0 and u2 = - 1 from 3t to 4t. In other words, the integration is carried out by switching vector fields and reversing their directions. A second-order Taylor series approximation leads, after time t, to the following point: x(t)=x,+tf,(x,>+(P/2)

2 (

Repeated

application

stirred

tank

reactor

Given a one-input

1151

control

system with a drift term [a =f(x) + g(x)u, x(O) = x,], x, exp {f g} denotes the reachable set attainable from x,. Under mild assumptions the reachable set of the above system contains an open subset of the manifold x,exp (A g}LA (Brockett, 1976). Finally, recall that reachability and controllability of nonlinear systems are related concepts (Hermann and Krener, 1977). By way of a summary, in the linear case, stabilizability and transformability to control canonical forms are related to controllability (Kwakernaak and Sivan, 1972). In the nonlinear case, controllability is related to Lie algebras. Therefore, one should expect Lie brackets and algebras to surface in the geometric characterization of the reactor control problem.

4.2. Global controllability For presentation purposes it was decided to work with a reactor that exhibits multiple steady states. By doing so, there is the possibility of regulating around stable stationary points and also of stabilizing the unstable mode of operation. The following parameters (Aris and Amundson, 1958) are used: Z== 1 mol/l, T,=F<=350K, a=ez5 min-‘, fi=2OOK/(mol/l), 6 = lo4 K, and 8= y = 1 min- ‘. The system has three stationary points: (C=O.963, F= 353.6 K, stable), F=400 K, (C= 0.5, unstable) and (C= 0.0885, T=441.1 K, stable). The phase flow for the autonomous system (u=O) is shown in Fig. 2. The Lie bracket forfand g is given by the expression

ay~(E+x,)e-~/(f+~*) (T+xX,)2 zzz

i

Y(Y+ @ -

af?y!s(c+x,)e-~l(~+~2~ . (18) (T+x#

Theorem (Hunt, 1980). Consider the system =f(x) + g(x)u, x(O)= x0 E M. M is a connected,

S: 1 ana-

f,(x,)+O(t3). > Xa

of the same expansion

leads to

Jc(4t)= X, + t2 [ J.1,fz] + O(P). This example shows that, at the limit, [f,, f2] represents an additional direction, different from fi and f 27 where the system can move. The involutivity requirement assures that the new direction, [f,.1;]. is still in the tangent space to the solution manifold. The Lie algebra, {S,, _f2, . . . ,fm}La, generated by a set of vector fields {fl, fi, . . . , f,} is the linear span of the vector fields generated by all linear combinations of theAs, their Lie brackets, all linear combinations of the preceding vector fields, their Lie brackets, and so forth. ces 44:5--5

l

0

’

.2

I

.4

’

I

-

.6

,

’

.0

CONCENTRATION Fig. 2.

Phase

trajectories

for the autonomous

,

1

CSTR.

1152

JES~JS ALVAREZ

lytic, two-dimensional manifold. Assume that f and g are linearly independent at some point in A4 and that g #O on M. Suppose that (i) every integral curve of g that disconnects M has a point r whereSand g are linearly dependent. (ii) g and [x g] are linearly independent at r. (iii) the dimension of (f; g}LA at x0 is 2. Then, system S is controllable from any point &EM. In the present process g #O on RZ. The vector fieldsf are linearly dependent along curve and g x,)=0, -?tx, -cEe-E} which is deC: (xER~I~(x~. scribed by the following equation (see Fig. 3): s

- T,

X2=lna-ln0-ln[-l+$_~/(~+x,)] -cc+


Since h, #O [es. (lS)] on C, it follows that g and [f, g] are linearly independent at C, and, hence, condition (ii) is satisfied. For condition (iii), the dimension of the span of the Lie algebra {J gILa must be examined point-wise. Since f and g are linearly independent for all x E R2 - C, and g and [A g] are linearly independent for all x E C, condition (iii) is satisfied for all x, E R’. Let M be a two-dimensional manifold defined as (see Fig. 3) M: (.xER’~

--c
--c~
et al.

The integral curves of g are vertical straight lines, and any of them passing through a point in M disconnects M and intersects C. Therefore,fand g on M satisfy condition (i). Concluding, reactor (3) is controllable from any x, E M. As will be seen in Section 4.3, the controller will be defined on the manifold W: M - {x E M 1x2 S - i’}. In that case, too, all conditions of the above theorem are satisfied, and reactor (3) is controllable from any X,E W. Fulfillment of condition (iii) was achieved by the independence of g and [X g]_ It must be pointed out that for the case of linear system,f= Ax and g = b, the linear independence f and [X g] becomes the controllability condition (rank [b, Ab] =2). For the nonlinear system, condition (iii) means local controllability (Hermann and Krener, 1977). The globality comes from asking condition (iii) to hold on M and, particularly, from the examination of the behavior of the vector field f along the integral curves of g [conditions (i) and (ii)]. Since the conditions in the above theorem are sufficient, it cannot be affirmed that there is noncontrollability outside M. However, the results suffice for the reactor problem because Ma@ Winclude well the region where the reactor can be encountered in practical operation. 4.3. Global transformability Hunt et al. (1983) provided sufficient conditions for the existence of a global transformation when the system is n-dimensional and there is one input_ Here the treatment is specialized to the two-dimensional case. Consider the following system dx

~=Cf’sl,xW=O,=R 550

dX

dt=g,x(s,O)=x(s),tER

(19)

500

x(s, t) is the trajectory obtained from integrating first along [f, g] from 0 to s time units, and then along g from s to t time units. The noncharacteristic matrix (Hunt et al., 1983) is defined as

460

=

/ax, 22

350

ES = & E

I-= 300 250

as ax, i -~ \ as

3x1

at ax, at

(20)

This Jacobian matrix characterizes a one-to-one (invertible) map H: R2 +R2 if there exists a constant E> 0 such that the absolute values of the leading principal minors, p1 and p2, of T(x) satisfy the ratio condition (Kou et al., 1973). This is:

.i

.e

.s

1

CO NCENTRlTlON Fig. 3. Curve C where the vector fields f and 9 are linearly dependent. Regions where the following properties hold: M for controllability, U for transformability and W for controllability and transformability.

(21) Definition. Consider the system S: 2 =f(x) + g(x)u. Let U c R2 be an open, connected subset that contains the origin. S can be globally transformed to a linear

Global nonlinear control of a continuous stirred tank reactor equivalent system if there exists a one-to-one transformation F: U x R+Z x R that maps origin to origin. 2 is the image of U under (F, , F2). The two-state version of Hunt et al’s (1983) theorem for global transformability is

Theorem. For system S, there exists a smooth transformation F on’ U if: (i) g and [f; g] are linearly independent on U. (ii) {g} is an involutive set on U. (iii) the noncharacteristic matrix (16) satisfies the ratio condition (17) on R2. It must be remarked that the involutivit3condition for the case of g # 0 on U, which is the present case, is trivially satisfied. Look at the nature of the conditions required by the theorem. Condition (i) is a point-wise test for local controllability on U (Hermann and Krener, 1967). Condition (ii) is a point-wise test for complete integrability, and condition (iii) is a test that looks at properties of curves that pass through the origin and that are generated by a composite integration along the vector fields [J g] and g. The proof of the last theorem is constructive (Hunt et al., 1983) and it provides a set of partial differential equations (PDEs) whose solutions are admissible transformations. In the case of the reactor those equations are

aF1 --0

(22)

ax2

F, =flz+f2z 1

(23) 2

F,=fl~+(f,+vu)~. 1

2

At this point it is pertinent to discuss the above transformability results in relationship to the’ local results (Su, 1980) used by Kantor and Hoo (1985) in the treatment of the reactor. The local version of the transformability theorem can be obtained after: (i) restricting the local controllability and involutivity tests to a neighborhood (undefined) about the origin in R*, (ii) eliminating the ratio condition which provided the essential nature of the invertibility (global inverse theorem). In the local version of the problem, the two left conditions are necessary and sufficient. Moreover, both the local and global treatments share the same set of PDEs, although defined on different domains. Therefore, both treatments may lead to the same expressions for the transformation. However, it must be recalled that the definition of a transformation involves both a functionality and a damain. In this regard, the essential difference between the local and global treatments, possibly sharing the s&me algebraic expressions for the transformation, lies in the characterization of the domain of validity. Proceed to establish a region for global transformability of the reactor. h,(x) #O for x E RZ

1153

-(xcR’(x,= -_)-{xER’Jx~= -?=I, and hence g and [f; g] are linearly independent on the connected set U: (xc R’fx, > -2, x2 > - F}. Hitherto, conditions (i) and (ii) of the transformability theorem are satisfied on U (see Fig. 3). The solution to system (19) is given by XI@, t)=%(s) x&, The noncharacteristic expression

t) = x2(s) + yt. matrix is then given by the

Hence, the ratio condition (18) becomes

y#O trivially satisfies the second inequality. From for x~{--ctx, c co, expression (18X h,(x)>0 -T, the horizontal component of the vector field [A g] always points towards the right. Therefore, x1(s) is a monotone increasing function on s E [0, co]. Consequently, the first inequality fulfils the ratio test and condition (iii) of the global transformability theorem is satisfied. The region Wwhere both controllability and transformability can be assured is obtained from the intersection of A4 and U (Fig. 3). This is

4.4. Global asymptotic stability Su et al. (1983) addressed the stability issue by introducing Lyapunov contours in the equivalent, closed-loop system. The Lyapunov contours expressed in the original coordinates allow for a characterization of a stability region. The region thus found depends on the choice of a particular control law as well as on the choice of a Lyapunov function (Perlniutter, 1972). Here the stability issue is addressed in terms of the global transformation, its domain, and the properties of linear systems. Under a mild assumption, it is concluded that Wis a region for stability for the nominal point as the attractor. Suppose that the equivalent system is closed with the following control law: u=H(z),

ZEZ(W)

where H is linear operator and Z is the image of W under (F,, F,), H is such that the equivalent, closedloop system is asymptotically stable, and the resulting vector field is closed on W. Since the transformation F is one-to-one, to each stable, closed-loop trajectory z(t)~Z( W) corresponds one and only one stable closed-loop trajectory x(?)E W. As a consequence, the

et al.

JESCIS ALVAREZ

1154

nonlinear system (8) closed with the nonlinear controller derived from v= H(z) possesses one asymptotically stable stationary point that is an attractor for the region IV. The last statement holds if the trajectories z(t) never leave Z( B’), or, equivalently, if the trajectories x(t) never leave W. For the reactor, this is a reasonable assumption because, as seen, the region of practical operability is more than justly included in W. 5. THE

NONLINEAR

CONTROLLER

In this section a global transformation defined on the region W, characterized in the previous section, is constructed. The nonlinear controller is obtained as the result of P and PI control designs on the equivalent system. The matter of a bounded input controller is touched upon. Lastly, it is shown how the resulting nonlinear controller can be seen as a generalization of a conventional PID controller. 5.1. Local dynamics Consider the Taylor linearization of the process whose equivalent system has been closed with a P controller (see Appendix B)

While in the linear case of Section 3, a transformation was obtained after assigning t, r = 1, here in the nonlinear case the analogous step is to assign a function, zr = xi . It can be verified that the linear transformation (11)-(13) is indeed a first-order, Taylor linear approximation of the nonlinear transformation (25H27) when the linear system (8) is the Taylor linearization of the nonlinear one (3). 5.3. The control law Here, the nonlinear controller is designed by pole placement on the equivalent linear system. 5.3.1. P control. The feedback design for the equivalent system coincides with the one already done in Section 3: given the design eigenvalues A, the vector gain for the equivalent system is evaluated according to eqs (15). In the nonlinear case, the expression for the controller is obtained by writing, using the inverse transformation of eqs (25j(27), the control law (14) for the equivalent system in terms of the original coordinates. Hence, the nonlinear controller (7) can be written as

k=Ax where A is a Jacobian matrix. Su et al. (1983) demonstrated, for a P controller in the equivalent feedback, that the eigenvalues of A and those of the equivalent system are the same. Appendix B provides this proof specifically applied to the reactor. Appendix B shows that the equivalence in the eigenvalues also holds for the case of a PI design in the equivalent feedback. The invariance of the eigenvalues between the equivalent feedback and the first-order Taylor linearization of the original closed loop constitutes the link between the dynamics of the closed-loop process and the tuning parameters of the equivalent controller. 5.2. A global transformation The existence of the global transformation has been proved in the last section. Here, the actual transformation is constructed. Equation (22) implies that F, =F,(x,). If F,(x) and F,(x) follow from eqs (23) and (24). =x1 3 F2(x), So, the transformation is given by F,(x,)=x1

%I

x2~4=fl1(x1~ +fi2h

where

p2(x)

YfiZ(X1 4

12 Xl7 4

where p,(x) (i = 1,2,3) are smooth functions on W. In original coordinates, the controller is assured to exist for {O-CC<&, O-C TC OD}. 5.3.2. PI conrroller. Consider the following control law for system (10):

, v=kpz+k;

zdt, k;=(k,,,

ufx~~.

4 x,)+2-9-,,.u

(27)

kiz)*

(2%

For simplicity the limits in the integral will be omitted. The equivalent feedback becomes 1 0 k,,

(26) X,).fi(X,Y

kp,), kf =(kil,

s Cl

(25)

F*(x, 7x,)=f1(x,,

F,(x,,

x1t =y;‘;- +)

PI(x)=

+

k,

In this case, the relation between eigenvalues tuning parameters is given by the expressions k,,=rZ,+&+&, -(&

kiI=Il,&l,,

kpl+ki2=

+ &)

and

--;1,& (30)

and the control law takes the form u=k,,p,tx)+kp2PZ(x)+ki,P3(X)+ki2P4(x)+P5(x) (31)

Global where

pi(x) (i = 1-5)

are smooth

1155

nonlinear control of a continuous stirred tank reactor functions

on IV and

they arc given by

technique based on local nonlinear results. Consider a proportional controller. The control is only allowed to take values in the following connected set

If the dynamic requirements are represented by the set of design eigenvalues 2, the controller can be represented as u=‘p(x,R), 5.4. The nonlinear controller from the perspective of a PI controller To justify the advantages of the nonlinear controller over a standard PID one, it is convenient to establish a relation between them. For illustration purposes, write the controller (31) in the following manner: u = %(x)x,

+ k:,(x)fi(x)

+kp*&M~)+~%?(x)~

x1 dt

+ Q%(x) Sfi(x) dr. This controller has state-dependent gains, and the proportional and integral operators act on functions of the states. To obtain a linear controller with the conventional form, the following simplifications must be introduced: (i) the gains are taken as constants, and (ii) fi(x) and &(x) are approximated by %I and X2. respectively. The result is u=klx,+k2jx,dt+k,~l+k,32.

(32)

On the other hand, if the design were based on a conventional PID strategy, it would not be a simple matter to arrive at the preceding control law. Assuming it were possible, there would not be straightfoward criteria to characterize, a prior-z, controllability and stability. Nor would there be any direct connection between the controller parameters and the process dynamics. The tuning procedure would therefore rely entirely on a trial-and-error procedure. To conclude, the present nonlinear control technique can be seen as a systematic way to design a twoinput one-output controller. The resulting nonlinear controller emulates a form of a PID controller with variable gains. 5.5. Bounded input control In most practical situations the controllers are bounded. At the moment, unfortunately, results for bounded control are lacking. The few applicable theories (Gutman and Hagander, 1985) are limited to linear systems. On the other hand, most control techniques (linear and nonlinear, theoretical and practical) circumvent the problem by choosing the control parameters such that the constraints are not met by the controller, and therefore the designs are done with unbounded control techniques. In view of this, the present global geometric technique assures one that there exists a large enough region in the state space where a compromise between boundedness and performance can be found. In contrast, this affirmation cannot be drawn, as a structural property, from the

XEXCW,

UEYCR

where X denotes the state set that results from the constraints on the control. The design of the bounded controller amounts to finding dynamics rl such that the control u takes only values in Y. As a faster speed of response is demanded, the size of Y increases. Conversely, the design can consist in choosing the control bounds for certain dynamics. Given a stationary nominal point, dynamics 1 and a value for the control variable u, the nonlinear control law (28) can be written as xi + a1(x2, rZ,w)xr + u2(x2. 1, u) = 0. The functions a, and a2 are defined in Appendix A. The above equation is, in the xi-+ plane, a curve with two branches that resemble a hyperbola aligned vertically. For the case of the reactor considered, the upper branch was attained at unrealistically’ large temperatures. Figure 4 shows plots of the curve for the unstable stationary point tuned with three sets of eigenvalues: (~ 1, - l), (- 2, - 2) and (- 4, - 4). The contours correspond to constant values of the controller. The graphs show that the control bounds are met more easily when the state errors coincide in sign. This is expected because both temperature and concentration increases require cooling. The graphs of Fig. 4 provide the amplitude of the initial control action required to meet a set point change in the operating point. Since a set point change is the most severe demand on a controller, the estimate is a conservative bound for load disturbances. Finally, it is interesting to notice that, about the origin, a strip delimited by a bound 1u 1
SIMULATIONS

A proportional control (28) for the equivalent linear system (10) was designed so that the closed-loop eigenvalues were (- 1.2, - 1.2) for the first stationary point, and (-4, - 4) for the second and third one. The phase flows for these cases are shown in Fig. 5. As expected, in all cases, the pair process-control exhibits a unique, stable, stationary point. The computed

1156

J~sljs ALVAREZ et al.

{cl)

aI

SPI -1.2 .-1.2

W

T

clz 3 c 0

.2

.6

.4

.a

1 a

G

Ca>

az W

420 a

410 s

400

500 * (c.1

W

390 I-

380

3+7r+rd

c

Cb)

: 300,

I

I .2

0 430

I .4

I

I .6

;

; .8

;

1

CONCENTRATION

x2

Fig. 5. Closed-loop phase trajectories corresponding to a proportional controller on the equivalent feedback at each of the stationary points (SPs). The closed-loop eigenvalues (CLEs) are indicated in each graph.

420 410

T

1

400 shows the state and control time evolutions after: (1) a step change in the concentration set point from 0.5 to

380

0.6, 380 370

(2)

a

+ 10%

tration, and temperature. .2

.4

.6

.8

1

Fig. 4. Restrictions of the phase space due to bounds in a proportional nonlinear control for the unstable stationary point. The curves represent states that lead to a control u when tuned with the following closed-loop eigenvalues for the equivalent system: (a) ( - 1, - l), (b) (-2, -2), and (c) (-4, -4)

phase portraits corroborate the global properties of the control scheme. The performance of the F controller is iilustrated with regulation about the unstable point. Figure 6

tor

temperature

the load coolant

step

change

in

the

inlet

concen-

(3) a + 10% step change in the inlet In all cases, the concentration and reacresponses

disturbances temperature,

are

smooth.

As

expected,

require smooth responses of the and the set point change de-

mands a fast and ample response. In this regard, it must be taken into account that the controller has been designed 5 times practical

to achieve

faster than the situation, and

a closed-loop uncontrolled for set point

response

about

process. In a changes, the

demand on the controller should be alleviated by designing with slower closed-loop eigenvalues (see Fig. IO). This example must be seen as a severe test of the nonlinear controller. Of course, there will be an offset in the final steady state. To test the robustness of the controller stability to model inaccuracies, the

Global

.4’ 1

1

2

nonlinear control of a continuous

3

1157

stirred tank reactor

4

410 400 390 _I___

I

_ I

I

I

I

1

2

3

4

I

TIME,min.

I

2

3

4

TIME *min.

Fig. 6. State and control responses, in time, corresponding to a proportional nonlinear controller for the unstable stationary point. The feedback was tuned with (-4, -4) as closed-loop eigenvalues for the equivalent system. The responses are caused by: (1) a concentration set point change from 0.5 to 0.6, (2) a + 10% step change in the inlet concentration, and (3) a + 10% step change in the inlet temperature.

Fig. 7. State and control responses in time to a step change in one of the parameters of the nonlinear controller for the unstable stationary point. A proportional equivalent feedback was used, and it was tuned to have (-4, -4) as eigenvalues for the equivalent system. (1) 8 is 1.1 instead of 1, (2) a is 7.9 x 10 lo instead of 7.2 x lo”‘, (3) 6 is 10,100 instead of 10,000 (4) p is 220 instead of 200, and (5) y is 1.1 instead of 1.

following simulations were done. At time t= 0, the system at the stationary point was excited by a controller with a deviation in one of its parameters. These responses are shown in Fig. 7. To reduce the offset in the terminal steady state concentration, a PI design based on the equivalent system was implemented. The nonlinear controller performance was compared to that of a SISO (coolant temperature-concentration) linear controller. As shown before, the nonlinear design accounted implicitly for a derivative-type action. The comparison seems to be fair because the derivative action is a builtin consequence of a PI design for the equivalent feedback. The tests with integral action were aimed at eliminating the offset in the stable point 3 (Fig. 2). The reactor had a time constant of about 0.5 min and an oscillation period of about 0.3 min. The nonlinear controller was designed to yield a closed-loop process with a time constant of about 0.25 min. According to this, the eigenvalues for the equivalent system were set to (-4, -4, -4). This was obtained with the following gains [eq. (30)]: k,, = - 24, k,, = - 12, ki, = - 64, and ki, = 0. To keep (as in the conventional PI design) only an integral action on the concentration, the fourth term in the PID controller (31) was eliminated.

For the conventional controller, the gains were tuned by a trial-and-error procedure aimed to obtain a reasonable compromise between settling time, damping and control effort over a range of set point and load disturbances. The tuning procedure led to the following gains [eq. (32)]: k, = - 16, and k,= -8. Figures 8 and 9 show that the performance of the nonlinear controller (measured in terms of oscillatory behavior, settling time and control effort) is superior to the one obtained from the conventional PI controller. The response of the proportional controller to a + 0.1 set point change in concentration (Fig. 6) required an extremely fast and large excursion of the coolant temperature, -2OK-~u-=z30K. This can be explained as follows. First make an estimate of the new open-loop steady-state temperature (395 K), and assume that the system is responding to an initial deviation from steady state, x0 =( -0.1, 5). Figure 4(c) shows that this requires a control deviation of - -40 K. If the requirement for speed of response is released by choosing A=( - 2, -2), the initial control deviation becomes - 15 K. Finally, if A=(1, -l), there is a design for a time response similar to the one of the opeq-loop process. In this case the initial control deviation LFig. 4(a)] is about - 6 K. Figure 10 depicts

1158

JES~JS ALVAREZ

et al.

TIME,

Fig. 8. State and control time response corresponding to a PI design on the equivalent feedback with ( -4, -4, - 4) as eigenvalues. The transient, about the third stationary point (stable), was induced by a set point change in the exit concentration from 0.088 to 0.1. (1) Response with the nonlinear controller, and (2) response with a conventional PI

controlier.

n

I

I

Fig. 10. State and control responses corresponding to a P design on the equivalent feedback. The transient was induced by a set point change in the concentration from 0.5 to 0.6, and the equivalent feedback was designed with the following sets of eigenvalues: (1) (-4, -4), (2) (-2, -2), and (3) (-1, -1).

bounds. Also, the speeds of response resemble the objectives set in the design procedure.

mol/l

1

Inin

I

I

1 ,094

,089

Fig. 9. State and control time response corresponding to a PI design on the equivalent feedback with (-4, -4, -4) as eigenvalues. The transient, about the third stationary point (stable), was induced by a step change in the inlet concentration from 1.0 to 0.9. (1) Response with the nonlinear and (2) response with a conventional PI controller, controller.

state and control evolutions for the above sets of eigenvalues. The graph shows how the control excursions were approximately within the estimated

7. CONCLUSIONS

A nonlinear controller for regulation, or stabilization, of concentration and temperature of a CSTR was constructed, with the jacket temperature as the manipulated variable. For a given stationary point, the controller was defined on a region such that global controllability and transformability of the process were guaranteed. This region could be established a priori. It encompassed, by far, the region where the reactor could be found in practical operation. The nonlinear, closed-loop process had a unique, stationary point that was stable. The geometry that underlies local and global nonlinear controllers was discussed. While a local controller is based on transformability conditions that consists of point-wise tests at the origin, the global controller required the same tests to be valid in a region. In addition, the global treatment required examination of the behavior of integral curves. It is mainly in the latter aspect where the essence of the global nature of the results resided. By solving the control problem at the structural level (independently of particular control laws and tuning parameters), one moved from earlier conjectures about globality of the control technique to establishing facts. The relationship between the nonlinear PI controller and a conventional PI controller was discussed. This connection showed that the conventional controller could be obtained as a simplification of the nonlinear one. Conversely, the latter one was a gen-

Global eralized ventional

version,

with

variable

nonlinear gains,

of

control the

of a continuous con-

controller.

As with most control techniques, the treatment for bounded controllers could be done by designing not to meet the constraints. In the present case, the globablity of the controller ensured its functioning over restrictions of the state domain that resulted from bounds on the controller. This point was illustrated with a design technique for a proportional equivalent controller. The simulations corroborated that the phase flows corresponding to the closed-loop process had a unique stationary point that was stable. When performance of the nonlinear controller, with a PI design based on the equivalent feedback, was compared with the performance of a conventional PI controller, it was observed that the first one had a superior performance. The equivalent feedback was tuned with pole placement techniques. Acknowledgements-The authors wish to thank Mr Eduardo Herna’ndez for his valuable comments. Partial support from British Council-CONACyT (Mexico) for the first author, and support from COSNET (Mexico) for a sabbatical leave, at Imperial College, for the second author, are gratefully acknowledged.

REFERENCES

Alvarez, J. and Stephanopoulos, G., 1982, An estimator for a class of non-linear distributed systems. Int. J. Control 36(5), 7X7-802. Aris, R. and Amundson, N. R., 1958, An analysis of chemical reactor stability and control. Chem. Enyny Sci. 7, 121-126. Bailey, J. E., 1977, Periodic phenomena, in Chemical Reactor Theory. A Review (Edited by L. Lapidus and N. R. Amundson), Chap. 12. Prentice-Hall, Englewood Cliffs, NJ. Balakrishnan, A. V., 1983, Elements of State Space Theory of Optimization Software, New York. Systems. Boothby, R. W., 1975, An Introduction to Differentiable Manifolds and Riemmanian Geometry. Academic Press, New York. Brockett, R. W., 1976, Nonlinear systems and differential geometry. Proc. IEEE 64(f), 61-72. Brunovsky, P., 1970, A classification of linear controllable systems. Hybernetika (Praha) 3, 173-187. Cebuhar, W. A. and Costanza, V., 1984, Nonlinear control of CSTR’S. Chem. Engng Sci. 39, 1715-1722. Georgakis, C., 1986, On the use of extensive variables in process dynamics and control. Chem. Engng Sci. 41, 1471-1484. Gutman, P. and Hagander, P., 1985, A new design of constrained controllers for linear systems. IEEE Trans. autom. Control AC-M, 22-33. Henderson, L. S., 1987, Stability analysis of polymerization in continuous, stirred-tank reactors. Chem. Engng Prog. March, 42-50. Hermann, R. and Krener, A. J., 1977, Nonlinear controllability and observability. IEEE Trans. autom. Control AC-22, 728-740. Hoo, K. A. and Kantor, J., 1985, An exothermic continuous stirred tank reactor is feedback equivalent to a linear system. Chem. Engng Commun. 37, l-10. Hoo, K. A. and Kantor, J., 1986, Linear feedback equivalence and control of an unstable biological reactor. Chem. Engng Commun. 46, 385-399. Hunt, L. R., 1980, Global controllability of nonlinear systems in two dimensions. Math. Systems Theory 13, 361-376.

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stirred tank reactor

Hunt, L. R., Su, R. and Meyer, G., 1983, Global transformations of nonlinear systems. IEEE Trans. autom. Control AC-28, 24-30. Isidori, A., 1985, Nonlinear Control Systems: an Introduction. Springer, New York. Kailath, T.,l980, Linear Systems. Prentice-Hall, Englewood Cliffs, NJ. Kalman, R. E., 1971, Kronecker invariants and feedback, in Proceedimts of Conference on Ordinary Dr@erential Equations, Nav,tl Kf~carch Laboratory, Washington. Academic Press. %cw \~>rk (1972). Kokotovic. P. V., O’Malley. R. E. and Sannuti, P., 1976, Singular perturbations and order reduction in control theory-an overview. Automatica 12, 123-132. Kou, S. R., Elliot, D. L. and Tarn, T. J., 1973, Observability of nonlinear systems. Information Control 22, 89-99. Kwakernaak, H. and Sivan, R., 1972, Linear Optimal Control Systems. Wiley, New York. Padmanabhan, L. and Lapidus L., 1977, Control of chemical reactors, in Chemical Reactor Thepry. A Review (Edited by L. Lapidus and N. R. Amundson), Chap. 13. Prentice-Hall, Englewood Cliffs, NJ. Perlmutter, D., 1972, Stability of Chemical Reactors. Prentice-Hall, Englewood Cliffs, NJ. Ray, H., 1981, Advanced Process Control. McGraw-Hill, New York. Rosenbrock, H. H., 1970, State Space and Multivariable Theory. Nelson, London. Shinskey, F. G., 1979, Process-control Systems. McGrawHill, New York. Spitz, J., Laurence, R. and Chappalear, D., 1977, An experimental study of a polymerization reactor in periodic operation. A.1.Ch.E. Symp. Ser. 72(160), 86-100. Su, R., 1982, On the linear equivalents of nonlinear systems. Systems Control Lett. 2, 48-52. Su, R., Meyer, G. and Hunt, L. R., 1983, Robustness in Geometric Control Theory nonlinear control, in Differential (Edited by R. Bracket, H. Millman and H. Sussmann). Birkhauser, New York. Wallman, P. H., 1979, Reconstruction of unmeasured quantities for nonlinear dynamic processes. Ind. Engng Chem. Fundam. 18, 327-333.

APPENDIX

A

Definition of variables and parameters for the CSTR model A, heat transfer area, cm2 c reactor concentration, mol/l c, feed concentration, mol/l c, fluid specific heat capacity, Cal/g K E, activation energy, cal h, heat transfer coefficient, Cal/cm* min K k, frequency factor for first-order kinetics, min-I R universal gas constant, Cal/K T reactor temperature, K T, feed temperature, K V reactor volume, 1 (-AH) reaction heat generation per mole reacted, cal/mol p fluid density, g/l w volumetric flow rate, I/min Functionalities a,(.+,

for

the coeficients

of eq. (33)

4 u)=c,(--~I,-cC2C3)+2C+Cq

a2(x2,

A., u)=c,(&,-cc,+,

(T+ x2)2 c,=~,E=e-6/(T+x,),c2=~1+112+e+a~ 2EZ

+E2+c4E

1160

JESCJS

ALVAREZ et al.

APPENDIX B: INVARIANCE OF EIGENVALIJES BETWEEN THE FIRST-ORDER TAYLOR LINEARIZATION OF THE NONLINEAR, CLOSED-LOOP PROCESS AND II-III LINEAR EQUIVALENT FEEDBACK Su et al. (1983) provided this proof for the general multiinput case with a P controller. Here the proof is specialized to the reactor case. Finally, it is shown that the equivalence of eigenvalues between linear forms also holds for d PI design based on the equivalent feedback. P control First, the nonlinear

controller (31). The resulting closed-loop

process is closed with the nonlinear first-order Taylor linearization of the process is

J = Ax, A = [A + h(akrT + d’)]

=1,2).

Starting from a first-order Taylor linearization of the reactor mode1 (3) and then applying the control design (16) for a linear system, the following closed-loop system is obtained: i=Lzz,

control

The first-order Taylor closed with the nonlinear

A=

I0 [ I[ d

a

The eigenvalues equation:

linearization PI controher

ab

A+InI

kbT(A+bd’)+k,T

of A satisfy

det(A-_II)=[det(A+bd’--II)] -[L~T(A+hd’)+k;Tl

of the process (33) is given by

k;Tb the

(3)

I o-1 I[ 1 d

following

a

’

characteristic

det {a&Tb--_ [A+bd’--LZ]-lab}.

The obtention of the last expression made use of the formula for the determinant of a matrix partitioned in four blocks (Kwakemaak and Sivan, 1972). Starting from a first-order Taylor linearization of the reactor mode1 (3) and then using a linear design (Section 3) leads to the following closed-loop process:

where

A=[~]~=~T=[~]._,(ii,

PI

ti=T[A+h(al$T+gd’)T-‘.

Since A is nonsingular, A and &I are related by a similar transformation.As a consequence, A and 0 have the same eigenvalues.

The eigenvalues det (a--

If)

of Cl satisfy the characteristic

= [det (A + bd’-AI)]

equation

det {ak>Tb - I

-[k;T(A+bd’)T-‘+k;][T(A+bd’)T-’ -W-‘aTb}. The identity of the above characteristic equations follows after writing [T(A+hd’)T-r ---RI]-’ as T(A+hd-_I)T-‘, and carrying out some multiplications. Consequently, A and R possess the same eigenvalues.