Ind. Eng. Chem. Process Des. Dev. 1986, 25, 403-411
403
Internal Model Control. 5. Extension to Nonlinear Systems Constantln 0. Economou and Manfred Morarl’ Chemical Englneering, 206-4 1, California Institute of Technology, Pasadena, California 9 1 125
Bernhard 0. Palsson Department of Chemical Engineering, University of Michigan, Ann Arbor, Michigan 48 109
An operator formalism is used to extend the Internal Model Control (IMC) design approach to nonlinear lumped parameter systems. It is proven that the attractive properties of IMC-as studied in linear control system design-carry over to the general nonlinear case: for open-loop stable systems, the feedback controller can be designed like a feedforward controller; the structure of the nonlinear controller is determined by the physical system, while the stabilty issue can be disregarded during the inltlal stages of the design procedure: robustness, i.e., stability in the presence of plant/model mismatch, c a n be addressed directly. Simulation examples demonstrate the simplicity
of the design procedure and the good performance characteristics of IMC for nonlinear lumped parameter systems and show that IMC works well even in the case where no linear controller can yield stable behavior. These promising results are an incentive to both solve some of the theoretical questions which had to be left unanswered and to pursue experimental testing of the scheme.
I. Introduction All physical systems are nonlinear. Often the linear models employed for control system design are only very poor approximations of the real behavior. While it is generally feasible to deal with mild nonlinearities just by using detuned linear controllers, in the presence of strong nonlinearities nonlinear controllers can offer distinct advantages. The design of open-loop nonlinear controllers is a well-established practice. When vpiational methods have been used, virtually every conceivable problem has been tackled. On the other hand, a general theory for the design of nonlinear feedback controllers does not exist at present for all practical purposes. One of the very few exceptions is the work of Frank (1974) who was probably the first one to use the advantageous features of the IMC structure for the design of linear and nonlinhar control systems. Most of the available nonlinear feedback control literature concentrates on stability analysis. The recent results of Safonov (1980) should be mentioned here, who extended the pioneering work of Zames (1966). He again put the work of Popov (1964) in a general context. Although Safanov’s formulation is very broad in scope, in practice, all these results are limited to feedback sys&fns consisting of a linear dynamic part and a nonlinear hbmoryless element whose characteristics can be bounded by a conic sector. In this work, a first step toward a practical approach to the synthesis of nonlinear feedback controllers is attempted. The IMC structure is particularly well suited to this effort. On one hand, it reveals the nature of the nonlinear controller; on the other, it offers powerful features for assessing and adjusting the performance of a closed-loop system. This scheme is therefore adopted and is used throughout this paper. The basic idea of the IMC structure has been around for some years-at least implicitly. It appeared as a result of the dissatisfaction with the ability of the available control system design methods to deal effectively with process control applications (Morari, 1983a) and the in-
creased power or readily available computer hardware. The most prominent IMC-type schemes have become known as Model Algorithmic Control (Richalet et al., 1978), Dynamic Matrix Control (Cutler and Ramaker, 1980), and Inferential Control (Brosilow, 1979). Zames (1981) used the inherent analytical powers of the IMC approach (Q-parameterization in his terminology) as a vehicle to define optimal sensitivity to disturbances and modeling errors and to design optimal compensators in the frequency domain. Although not immediately evident from the specific formulations employed by the respective authors, the principle features which gave these methods their power are identical. The key issue is the capability of the new techniques to combine the advantages of open-loop (feedforward) and feedback control. The linear IMC structure is briefly presented and its features are elucidated in the next section. Some results from nonlinear operator theory are sketched out in section 111, and subsequently they are used to develop the nonlinear control structure, derive its properties, and discuss its implications (section IV). From this discussion, the critical importance of the inverse of the nonlinear operator modeling the plant is evident, and section V is devoted to its study. The stability of the IMC structure is briefly studied in section VI. Finally, simulation examples demonstrate the promising performance of the suggested design procedure. It should be emphasized that we do not consider this work a complete solution to the nonlinear feedback controller design problem. It is well-understood that at present, it is intangible to address all the issues and implications of nonlinear control system design. What is sought in this first step is a working approximation to assess potential advantages and pinpoint inherent restrictions and/or problems of nonlinear controllers. The theoretical results summarized in this paper as well as the promising examples are amenable to refinement and enhancement, thus becoming motivation for future research in this field. 11. Linear IMC A unifying review of the IMC-type schemes was first presented by Garcia and Morari (1982). The following
* To whom all correspondence should be addressed. 0
1986 American Chemical Society
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Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 2, 1986
C(0) = m , which implies integral control action as expected.
The advantage of the IMC structure is twofold. In simplified terms, we can say the larger the “gain”, the better the performance. In the classical structure, the objective is to make the “gain” as large as possible without causing instability. Simultaneously attention is to be paid to other criteria-like robustness to modeling errors and to constraints like input saturation. According to P2, with IMC, we can start with a stable closed-loop system with perfect control. Thus, the first design problem is eliminated altogether and full attention can be devoted to the additional criteria and constraints. The second advantage of IMC is that the design philosophy can be extended to nonlinear systems in an orderly fashion as will be shown next.
U
G
GC
Y
l
yss
Figure 1. Linear IMC structure.
Figure 2. Classical feedback structure.
brief introduction to IMC derives from this work. It is intended to be elucidating rather than complete. The interested reader will find all the details in the original presentation (Garcia and Morari, 1982, 1985a and b). The IMC control structure is shown in Figure 1. From this block diagram, follow the relationships
u = [ I + G,(G - G)]-’G,(y, - d )
(1)
and y = GII
+ G,(G - Q1-l
G,(y, - d ) + d
(2)
The advantages of the IMC structure, discussed qualitatively in the Introduction section, can be summarized in three properties which can be proven easily from (1)and (2). Property Pl (Dual Stability). Assume the model is perfect (G = G). Then, the closed-loop system in Figure 1is stable if the controller G, and the plant G are stable. Property P2 (Perfect Control). Assume that the controller is equal to the model inverse (G, = G-l) and that the closed-loop system in Figure 1 is stable. The y ( t ) = y,(t) for all t > 0 and all disturbances d ( t ) . Property P3 (Zero Offset). Assume that the steadystate gain of the con_trolleris equal to the inverse of the model gain (G,(O)= G(O)-l)and that the closed-loopsystem in Figure 1is stable. Then for asymptotically constant set points and disturbances, there will be no offset (lim t-m Y ( t ) = Ys). P1 implies that unless there are modeling errors and as long as the open-loop system is stable, the stability issue is trivial. P2 reasserts that the ideal open-loop controller leads to perfect closed-loop performance when the IMC structure is employed. P3 states that integral-type control action can be built into the structure without the need of additional tuning parameters. Superficially these properties seem too good to be true. However, it should be emphasized that the IMC structure is equivalent to a classical feedback structure (Figure 2) if the IMC controller is related to the classical controller through the equations
G, = C(Z +
&)-l
c = ( I - G,@-~G,
(3)
(4)
and therefore the properties can be easily explained: Whatever is possible with the classical structure is possible with the IMC structure and vice versa. We know intuitively that P 2 requires an infinite co_ntrollergain, and this is confirmed by substituJing G, = G-’ in (4). In a similar fashion, setting G,(O) = G(O)-l as postulated in P3, we find
111. Mathematical Preliminaries The emphasis in this paper is to present a practical method for the control of nonlinear systems and not a mathematical treatise. However, in order to deal with nonlinear systems in a general manner, it is unavoidable to establish first a mathematical framework to describe some basic notions. Toward this objective, we will make use of resulb from both operator and control theories. The works of Rall(1979), Stakgold (1979), and Zames (1966) provide a more extensive treatment of the material in this section. Norms. The notion of a “norm” in a function space is instrumental. For a space of real functions f ( t ) defined over the interval (0, m), a p norm is defined for 1 I p I by (5)
the
m
norm being simply
llfll-
= max
JIf(t)l
(6)
tE Operators. An operator M is generally defined on a subspace of an input space U and maps this subspace of U into a subspace of an output space Y. These subspaces and range (RM)of the operator will be called domain (DM) M, respectively. An operator K is called linear if it has the following two properties: K ( u , + u,) = Ku1 + Ku, u,,u2EU (7) K(Xu) = X ( u ) u E U , XfR (8) An operator M is called nonlinear when it is not linear. A nonlinear operator of particular importance for control systems is the operator N , which maps u to y (=Nu) through the relations rl = f(x,u)
x(0) =
Y = g(x,u)
x,ER“ (9)
where the dot notation indicates partial differentiation with respect to time, and f and g are real analytic vector valued functions. A special case of the operator N is when f and g are linear with respect to both u and x. The result is a linear operator L i = Ax + Bu x ( 0 ) = xo y = Cx
+ Du
(10)
which is the familiar state-space description of a linear system, where A , B, C, and D are n x n, n X m, m x n, and m x m constant matrices, respectively.
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 2, 1986 405
Operator Properties. The addition and product of operators are defined (Mi + M J u = Miu + M ~ u (11)
(MiMAu = Mi(M2u) (12) The following axioms hold for the operators M1,M2,and M3 in the proper domains: (Mi + MJM3 = MiM3 + MzM3 (13) (MiMAM3 = Mi(M2M3) (14) On the other hand, in general, MlM2 f M2Ml (15) Mi(M2 + M3) # MiMz + MiM3 (16) (Note that when Mi, M2,and M3 are linear operators, (16) becomes an equality, while (15) generally holds.) For an operator M , it is often possible to find operators M1 and M with the property M'Mu=u
uEDM
(17)
MMY = Y YERM (18) Operators M' and M are called left and right inverse operators of M , respectively. Operator Gain. Let the operator M map the domain D, into the range RM. The gain g(M) of the operator is defined as the supremum over all uEDM of the ratios of the norm of the operator output to the norm of the associated input:
For nonlinear dynamic operators, the gain can be computed conveniently only in special cases. For linear systems, the supremum can be found by searching only over all inputs of the type eiUt,which can be easily done. On the other hand, for nonlinear operators, the search has to extend over all possible inputs, a task which is not feasible in general. Steady-State Operators. Let M be an input-output stable operator in DM (Zames, 1966) and uEDM with lim t-- u ( t ) = u, < m. Letting y , = lim t-m M u ( t ) (y, < m because of the stability assumption), the steady-state operator M , is defined by y m = M,u, (20) It should be pointed out that M , is in general a function of u,. Although M generally maps function spaces into function spaces, M , is mapping vectors from R" into R". As an example of a steady-state operator, consider the operator in (9) where N is assumed input-output stable. For this operator, N , is given by the system of algebraic equations 0 = f(x,u,) Y m = g(x,um)
In the case of the operator L in steady-state or dc "gainn: L , = -CA-lB
(lo), L ,
(21)
is the familiar
+D
IV. IMC Structure for Nonlinear Systems and Its Properties In analogy to the linear case, we postulate the nonlinear IMC structure in Figure 3. Here the nonlinear operators P, M , and C denote the plant, model, and controller, respectively. Following Frank (1974), blocks with double lines are used to emphasize that the operators are non-
I
h
Figure 3. Nonlinear IMC structure.
linear and that the usual block diagram manipulations do not hold. The functions e, u, d , etc., shown in Figure 3 are generally vector valued functions of time t. A very important difference between linear and nonlinear systems should be emphasized here. For linear systems, disturbances can be assumed to act additively on the output (Figure 3, d = 0, d ' # 0 ) without loss of generality because of the superposition principle: P(u + d ) = Pu + Pd = Pu + d', with d' = Pd. For nonlinear systems by definition, this principle does not hold. Disturbances usually have to be represented by d as shown in Figure 3. This is to indicate that contrary to linear systems, unmeasured disturbances lead to differences between the model and the plant. In a discussion of robustness, not only modeling errors but also the expected disturbances have to be considered. In the following, we will assume the symbol Pd for the plant operator to signify the effect of disturbances ( P will denote the plant when no disturbances are present). The following relationships can be deduced directly from the diagram: e = Ys - Y + Y M YM
= MCe
h = (Pd - M)Ce
(22) (23)
+ d'
(24)
The attractive characteristics of IMC are the consequence of three properties which we will now state and prove. Property P1 (Stability). Assume that c and Pd are input-output-stable and that a perfect model of the plant is available, Le., M = Pd. Then the closed-loop system is input-output-stable. Proof. P1 follows trivially because the feedback path is interrupted when M = Pd ( h in (24) is not a function of the plant input) and the system is effectively open loop. Property P2 (Perfect Control). Assume that the right inverse of the model operator M exists, that C = M, and that the closed-loop system is input-output-stable with this controller. Then the control will be perfect, i.e., y = ys. Proof. P2 follows directly from (22) and (23). Property P3 (Zero Offset). Assume that the right inverse of the steady-state model operator M f , exists, that the controller satisfies C, = Mf,, and that the closed-loop system is input-output-stable with this controller. Then offset free control is attained for asymptotically constant inputs. Proof. P3 follows directly from (22) and (23) by taking the limit as t a. Some remarks concerning the significance of these results are in order: (1) For nonlinear systems, general guidelines are not available on how to design a feedback controller for which the closed-loop system is stable, even less for which the overall control structure has some desired performance characteristics. The IMC formalism is aimed at alleviating this problem a t least for systems which are input-out-
-
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Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 2, 1986
we will try to summarize the relevant results, as well as add some new. This is done in an effort to establish a rigorous and practically applicable framework for studying the most important questions of invertibility, construction, and stability of inverse operators. 1. Invertibility Conditions. Existence and construction of left inverses for the class of operators described by
x = f ( x ) + Cu,h,(x) x ( 0 ) = xg Figure 4. Complete nonlinear IMC structure.
Y = g(x)
(25)
put-stable (or stabilizable by output feedback) and systems which do not exhibit multiple output steady states. Under these assumptions, if a good model of the plant is available, P2 prescribes exactly the structure and parameters of the controller which will result in “perfect control”, i.e., exact set point following despite unmeasured disturbances. Moreover, P1 guarantees the stability of the closed-loop nonlinear control system. The underlying idea is that as far as the design is concerned, IMC transforms the problem into a feedforward control problem, which can be solved even for nonlinear systems. But on the other hand, IMC preserves all the important characteristics of feedback control, in particular the suppression of unmeasured plant disturbances as properties P2 and P3 show. 2. It is intuitively obvious that a feedback controller with infinite gain is necessary to achieve the performance stipulated in P2. From this interpretation, it is evident that the control system suggested in P2 will suffer from stability/sensitivity problems: If the compensator is not correct (the model of the plant is not exact), the closed-loop system can be unstable. A t this point in the IMC design procedure, we can back off from the “perfect” controller in an orderly fashion and reduce the gain to improve the robustness characteristics. This is accomplished by employing the IMC filter F in series with the controller C; see Figure 4. 3. The primary reason for including this filter is to introduce robustness in the IMC structure in the face of modeling errors, by appropriately reducing the loop gain. A t the same time, it serves a number of other functions: By definition of the inverse operator iW, its input space is the range of the operator M, consequently the operator M is not defined for inputs outside the range of M . However, there is no assertion that the error signal e will belong to this space. In this case, the filter is used to project e to the appropriate space. Finally, the filter smoothes out noisy and/or rapidly changing signals in order to reduce the transient response of the IMC controller (or, using linear systems terminology, it makes the controller “proper”; compare with the notion of the inherent integration constant to be introduced later in section V). The preceding discussion points from every direction to the crucial importance of the inversion operator. Therefore, a thorough study of both the theoretical and the practical implementation aspects of operator inversion is in order.
have been studied extensively in the literature. Sain and Massey (1969) gave necessary and sufficient invertibility conditions for the special case of a linear operator, establishing at the same time the notion of the inherent integration constant. This constant directly reflects the “improperness”notion for linear systems in the frequency domain (for a single-inputsingle-outputsystem, it is equal to the difference between the numerator and denominator degrees of the transfer function; for a multiple-inputmultiple-output system, it is greater or equal to the largest such difference in the transfer matrix); it proves to be very useful in the design of the IMC filter. Silverman (1969) treated the more general case of the linear time-varying operator that results from the linearization of the operator in (25). Hirschorn (1979) extended Silverman’s approach to the general nonlinear operator (25) and was able to derive sufficient invertibility conditions. These conditions become necessary and sufficient in the single-input case. Hirschorn’s approach consists of an algorithm which recursively generates a sequence of operators (SI),(SJ, ..., (S,) from (25) by differentiating the output map y = g(x). The sequence (S,)terminates when the output map for Sk can be solved for ul, up,...,u, (nis the number of system inputs) in terms of dly/dtL,i = 0, 1, ..., k. The value k is the system inherent integration constant. This method is the only available theoretical tool for judging the invertibility of a nonlinear system. However, the algorithm is rather involved, and we will not attempt to present it here‘in detail. The interested reader is referred to the original paper. An illustrative application of the method is included in section VII. 2. Construction of the Inverse. A. Analytic Construction. The only available technique for the inversion of operators of the class (25) is again by Hirschorn (1979): The inverse operator maps derivatives of y with respect to time, up to kth order, to ul, u2,..., u,. Though mathematically rigorous, the method implies use of higher order derivatives and is therefore sensitive to noise and/or numerical errors. In all practical cases considered, it failed to produce a satisfactory approximation to the exact inverse. Therefore, its use was not recommended. B. Numerical Computation. The inversion problem can be formulated as the problem of solving an operator equation. Given the operator M , mapping u* to y* Mu* = y*
V. Nonlinear Inverses Nonlinear operator inversion has received considerable attention in the past. Powerful results have been reported by mathematicians and numerical analysts, concentrating on sufficient conditions for invertibility and numerical computation of inverse operators. From a control theory point of view, Hirschorn (1979) addressed the problem of invertibility conditions and analytical construction of inverses for a class of nonlinear operators. In the following,
compute u* given y*, i.e., solve the operator equation Mu* - Y * (26) -0 In this context, results from numerical analysis can be employed directly. Two basic techniques for the solution of nonlinear operator equations are the Contraction Principle (successive substitution) and Newton’s method. To simplify the computational problem, it will be assumed that the system is sampled at a constant rate T. A t time
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 2, 1986
put-output-stable if the product of the operator gains in the loop is less than 1: 11111 111 - MI1 < 1 111 - MI1 < 1 (32)
lmt Figure 5. Block diagram for the inversion of the operator M . I is the identity operator.
t , the objective is to find an input u* which, if held constant from t to t T, will produce an output equal to y* a t t + T. Contraction Principle Method. Equation 26 can be cast in the form Fu* = u*
+
where
Fu* A U* + y* - MU*
(27)
and u*, the fixed point of the operator F , can be computed by successive substitution: Un+l = un + y* - MU“ (28) Equation 28 can be represented by the simple block diagram in Figure 5 where I represents the identity operator. This method of successive substitutions will converge to the fixed point u* if the conditions of the Contraction Principle Theorem (Stakgold, 1979) hold. Newton’s Method. For our specific needs, the operator in (9) will be studied. The solution of (26) by Newton’s method is the limit of the sequence (u”)defined by Un+l
=
[ aau
- (y*
Un-
- Nun)
407
l1
(y* - Nu”)
The operator derivative in (29) is calculated next: (30) where agfax = [dgi/dxj]and agfdu = [agi/duk]are the derivatives of the vector function g(x,u) with respect to vectors x and u and can be explicitly calculated, while r [axjfauk]is the solution a t time t + T of the linear variational problem (see: Keller, 1976)
Finally, Nu“ is the system output at t + T , if un is the input to the system from t to t + T. The Kantorovic theorem (Kantorovic and Akilow, 1959) establishes the sufficient conditions for convergence of the method. Implementation of the method is exemplified in section VII. Newton’s method is the most powerful technique for the solution of operator equations. It should be noted here that if the user is willing to sacrifice some of these advantages in order to avoid solving the variational problem (31), the derivative axfau can be approximated by finite differences. Finally, it is unlikely that convergence problems will appear in practice for these computational methods: The inverse is computed a t each sampling interval, and therefore a very good initial estimate is available for the next step. 3. Stability of the Inverse. The block diagram representation (Figure 5) of the inverse is well-suited for stability analysis, employing the methods available in the literature. Zames’ (1966) small gain theorem is the most useful criterion. According to it, the inverse operator W is in-
This type of stability analysis method is “positive” in the sense that it guarantees stability of the inverse when sufficient conditions hold. The following theorem establishes a sufficient “negative” criterion for inverse stability. Theorem 1. Let M be the operator, mapping D M into R M , where D M and R M are subspaces of appropriate normed linear spaces. If M has a right inverse itP, then M’ is input-output-unstable on R M if there exists a family u, in RWCDM such that lim lluall = m and lim llMuall < m (33) a-a
a-a
Proof. See Appendix. This theorem is an extension of the boundedness away from zero notion for linear operators (Stakgold, 1979). However, in the transition from linear to nonlinear, the necessity of the condition is lost. The following example illustrates an application of the theorem. The operator (10) with
[ ] 0
A = -2
1 -3
B=[:]
C = [-2
11
maps u = exp(2t) to y = Lu = e-t - e-2t. It is obvious that llull = m while llyll = - e-2tll < m. We conclude that L-l is unstable. This is a t once verified by taking the Laplace transform of the operator s-2 L(s) = (34) (s + l)(s + 2 )
L-l(s) = ( s + l)(s + 2)/s - 2 is clearly unstable because of the right-half plane pole at s = +2. VI. Stability of the IMC Structure So far the stability of the closed-loop system was discussed only under the assumption that a perfect model is available (section IC, property 1). However, in practice, there is always plant/model mismatch ( M # pd). Therefore, it is essential to know how the controller should be modified for closed-loop stability. However, at present, a good synthesis procedure for robust controller is not established even for linear systems. Accordingly, we will only introduce the general idea and show that a “robustness filter” might be as useful for nonlinear systems as it has proven for linear ones. Zames (1966) showed that a general feedback system is input-output-stable if the open-loop gain is less than unity. For the IMC structure, this translates to g((pd - M)c)< 1 (35) or the sufficient condition implying (35) g(c)g(Pd- M) < 1 (36) When Pd = M and the IMC controller C is input-output-stable, this inequality is trivially satisfied. When the model/plant mismatch is large in the sense of g(Pd - M) being large, the controller gain g ( C ) has to be small and usually (35) will not be satisfied if the perfect controller C = M’ is chosen. Therefore the adjustable filter F is introduced: C = WF. The stability condition (36) becomes g(wF)g(pd - M) < 1 (37) or the sufficient condition implying (37) (38) g(F)g(M%(Pd - M) < 1 which shows that as long as W and pd - M are stable, there will always be an F satisfying this inequality. The objective
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Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 2, 1986
i
F/i
L
I
Ao,Ro,To c
Figure 6. Continuous stirred tank reactor for the reaction A
R.
is to select F in the least conservative manner such that the performance does not deteriorate more than necessary. This last fact is apparent from (22) and (23) which become for Pd = M (nominal-transfer function) Y Fb, - d ) (39) A t present, guidelines for the optimal filter selection or existence conditions for a stabilizing zero-offset filter (F, = I) are not available. It is to be expected that nonlinear filters can lead to distinct advantages. VII. Example Though the specific example used might seem simplistic, the underlying dynamic characteristics which give rise to the control difficulties are common. The objective is to exemplify the implementation and the potential advantages of a nonlinear controller over a linear one. It turns out that even when dealing with processes with relatively mild nonlinearities, no linear controller can match the performance and robustness characteristics of a rationally designed nonlinear controller. The process consists of an ideal continuous stirred tank reactor (Figure 6), where the reversible exothermic reaction k
A Z R k-i
is carried out. A system of coupled ordinary differential equations models the process: (a) reactant mass balance
(b) product mass balance
(c) energy balance
f3(Ao,Ro,To,Ti) (42) where k1 = C, exp(-Q1/.72To)and k-, = C-, exp(-Q-l/WTo) and the specific parameters used in our example are shown in Table I. The reactor equilibrium conversion as a function of temperature has a well-defined maximum (solid line, Figure 7 ) . The control objective is to operate the reactor tightly near this optimal point, maintaining at the same time stability of the closed-loop system in the face of disturbances and/or unmodeled system dynamics. The inlet stream feed temperature Ti and the desired product output concentration Ro are selected as manipulated ( u ) and controlled (y) variables, respectively. This together with (57)-(59) comprises the specific nonlinear operator M of the form (9) to be treated in the remainder of this section. Invertibility. Hirschorn's conditions are applied to the system model: After differentiating the output map Cy =
Table I. Reaction Design: Constants and Steady-State Operating Conditions T = 60 s A, = 1.0 mol/L C1 = 5 x 103 ~1 R, = 0.0 mol/L = 1 x 106 s-1 A0 = 0.492 C-I QI = 10 OOO cal mol-' Ro = 0.508 Q-I = 15000 cal mol-' T, = 427 K R = 1.987 cal mol-' K-' To = 430 K -A% = 5000 cal molc1 = 0.5085 = 434 K = 1 kg/L = 1000 cal kg-l K-'
xopt
Topt
P
CP
'. '..
0 .8
C
1
nfi !
I
... I..
1
I
I
I
I
I
0 H V E R
s I 0 N
REACTOR TMVEPATURF,
K
Figure 7. Equilibrium diagram (solid line) and invertibility condition (dashed line) for the
CSTR of the example.
x 2 ) twice with respect to time, it is found that u can be recovered from d2y/dt2when the quantity kiQi-40 - k-iQ-iRo
(43)
T,2 is non-zero. This shows that the inverse ib$ does not exist when kiQiAo = k-iQ-iRo (44) which-for Ai + Ri = 1-in the phase plane of the reactor takes the form
represented by the dashed line in Figure 7 . Condition (45) is necessary and sufficient for this single-input system as Hirschorn has shown. It follows that the operator M has a unique inverse at every point in the phase plane that does not belong to the set (45). Notice that the system is not invertible at the maximum conversion point as would be expected, because at this point the system steady-state gain is zero. Another result obtained by the same method is the value for the system inherent integration constant: Because the output map had to be differentiated twice in order to recover the system inputs, the inherent integration constant is 2, a result that will be used in the filter design. Inversion of the Model Operator M . 1. Hirschorn's method results in the following representation for the inverse operator:
u = C*(t)
+ D*(z)
1:3
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 2, 1986 409
where A*, B*, C*, and D* are given in Table 11. Application of the method led to highly sensitive inversion results. 2. Among the numerical inversion methods, the contraction principle was found to be reliable but very slow. Its on-line implementation is not efficient. On the contrary, Newton's method and its approximation were both reliable and efficient, converging at most after three iterations. Newton's method was used for the simulation runs. To illustrate its implementation, its specific form for the studied one-input, one-output, three-states system is presented. At the beginning of each sampling interval ts,the IMC filter supplies a "target" value e, which is the difference between the set point and the "disturbance effect" h (Figure 3). The IMC controller hP computes the input ut that generates a system output equal to et at t s + T, T being the sampling rate. ut is approximated by the sequence U" generated through the recursive relationship Un+l = U"
- x2 +et
where x 2 and r, = ax,/au are computed by integrating the model ODES over the time interval ( t , t + At): i , = f2(x1,x,,x3,u) i 3
=f
360
380
400
420
440
REACTOR TFMF'CMTWRC. K
Figure 8. Reactor trajectory in the temperature/conversion plane [(-I equilibrium curve, (*) reactor] for startup. IMC controller, exponential filter, time constant 7 = 1 min.
-'-
= Ao(t = t s )
x,(tS) = R,(t =
3 ( ~ 1 , ~ , , ~ 3 , x~3 )( t S )
340
(47)
r2
il = ~ ~ ( X ~ , X Z , X ~ , U )x 1 ( t s )
320
tS)
= To(t = t s )
. = -rl atl + -rz atl + -r3 afl + -,af, rl au ax, ax, ax, . = -rl af3 + -r, af3 + -r3 af3 + -,af3 r3 ax, au
rl(ts) = 0
r3(ts) = o ax1 ax, where x are the states of the controller and f l , f 2 , and f 3 are defined in (401, (41), and (42). Filter Design. For all case studies, a simple exponential filter gave satisfactory results in terms of robustness and performance. This filter was appropriately modified to project the error signal to the domain RM of the inverse operator. For piecewise constant inputs, the model outputs are smooth continuous functions of time, which do not exceed the maximum conversion limit. As a consequence,the inverse operator is not defined except for smooth continuous inputs which do not exceed the maximum conversion. The IMC filter f is used to compensate for this existence problem by processing the error signal e to a smooth continuous function with no maxima beyond the maximum conversion value. Simulations. Computer simulations of the reactor using this nonlinear IMC structure were carried out with excellent regulatory and servobehavior for all reasonable disturbances and/or modeling errors. Note that because of the conversion maximum, the system gain changes sign as the operating point changes from one side of the maximum to the other. Systems of this type are not ''integral controllable" (Morari, 1983b); i.e., they cannot be stable over the entire operating region when controllers with integral action are employed. To make stability possible, the "no offset requirement" has to be abandoned when linear controllers are employed. Therefore, all the comparisons in this paper are q a d e between linear proportional controllers and nonlinear offset-free IMC controllers.
3
2
8
3
4
8
3
6
8
3
8
8
4
8
8
4
2
0
4
u
KEACTOR TEMPERATURE. X
Figure 9. Reactor trajectory for shut down [(-I equilibrium curve, (*) reactor]. IMC controller, exponential filter, second-order il = 7* = 1 min.
0
1
2
i
i
2
3
4
5
6
4
5
6
T I M E , S I N
Figure 10. Reactor input (Tfeed)and output (Gout) response with linear controller under a temporary 2% disturbance in reactant input concentration [(- - -) set point, (-) output]. Proportional controller, gain k = 10.
Figures 8 and 9 show the reactor start up and shut down to a point within 0.1% of the optimal conversion. Figures 10 and 11present the process behavior under proportional feedback (gain K = 10) when a temporary disturbance
410
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 2, 1986
Table 11. Inverse Operator 1
L C*(z) =
[-;]
r
p
0
J
D * ( z ) = O[
i]
le
20
38
40
58
1
1
1
I
1
38
40
se
432
0
I8
28 Y I N
T I M E ,
Figure 12. Reactor input and output response with nonlinear IMC controller under the same conditions as in Figure 10 [(- - -) set point, (-) output]. Exponential filter, second-order T * = r2 = 1 min. 0.s10
i
T I Y E ,
Y I N
Figure 13. Reactor input and output response with linear controller when a disturbance initially pushes the operating temperature beyond Topt[(- - -1 set point, (-) output]. Proportional controller, k = 10.
A
8.584 425.0
I
427.5
438.0
432.5
435.0
437.5
440.9
442.5
RFACTOR TLXPERATURE. K
Figure 14. Reactor trajectory with nonlinear IMC controller when a disturbance initially pushes the operating temperature beyond To,, and a +lo% error in K1 and K2 [(-) “true” equilibrium curve, ( - - -) model equilibrium curve, (*) reactor, (t) reactor model]. Exponential filter, second-order T~ = r2 = 1 min.
the merits of the nonlinear controller, a modeling error of +lo% in 12, and k , in (54)-(56) is introduced. The re-
Ind. Eng. Chem. Process Des. Dev. 1986, 25, 41 1-419
411
sulting response (shown in Figure 14) is again satisfactory. These conclusions do not hold only for the particular linear controller used here but have general validity for any linear feedback controller. As was mentioned previously, the gain of the r,eactor changes sign when passing through the optimal point and significant offset and/or instability occurs with linear controllers. For any linear controller, there is a disturbance (or modeling error) of sufficient magnitude to push the system to a point where the stability/performance problems discussed above arise.
Then,
VIII. Conclusions An extension of the Internal Model Control (IMC) design procedure to stationary lumped parameter multipleinput-multiple-output nonlinear systems was presented. It was found that the attractive features of linear IMC carry over naturally to the nonlinear case. A first step toward a practically applicable nonlinear feedback controller design technique was proposed. It was pointed out that under mild restrictions, nonlinear IMC possesses all the indispensible qualities of a reliable control scheme, namely set point following, disturbance rejection, no offset, etc. Throughout the formulation, the inverse of the nonlinear operator model appeared to be of crucial importance. Consequently,a thorough study of inversion operators was included, where a synopsis of the available work on this subject and some new results were presented. The issues of existence, uniqueness, construction, and stability were addressed. Extensive simulation testing of the IMC structure showed ease of implementation with very promising results. Some theoretical issues, among them output multiplicity and existence of an IMC filter with F , = I , had to be left unanswered. The promising results are a motivation to pursue a deeper understanding of this nonlinear IMC controller design technique. Proof of Theorem 1. For the family (u,) in (56), there is a corresponding (u,) in RM such that
This implies that 1 1 i W 1 1> K , V K Literature Cited
u, = M'u,
lim 11M11 = lim lluall = a-a
a-a
a
(A3
Also, lim lluall = lim IIMu,ll < a-a a-a
(A3)
(A2) and (A3) show that for every positive k,there exists an cy close enough to a with
> 0.
Brosiiow, C. B. Paper presented at the Joint Automatic Control Conference Proceedings, Denver, CO, 1979. Cutler, C. R.; Ramaker, 8. L. Paper presented at the AIChE 86 National Meeting, April, 1979. Frank, P. M. "Entwurf von Regelkreisen mit Vorgeschirebenem Berhalten": G. Braun Verlag; Karlsruhe, 1974. Garcia, C. E.; Morari, M. Ind. Eng. Chem. Process Des. Dev. 1982,21, 308. Garcia, C. E.; Morari, M. Ind. Eng. Chem. Process Des. Dev. 1985a,2 4 , 472. Garcia, C. E.; Morari, M. Ind. Eng. Chem. Process Des. Dev. 1985b,2 4 , 485. Hlrschorn, R. M. IEEE Trans. Autom. Control 1979,AC-24, 6 . Kantorovic, L.; Akilow, G. "Functional Analysis in Normed Spaces"; MoscowFizmatgiz; Moscow 1959; German transl. Berlin Akademie-Verlag: Berlin, 1964. Keiler, H. B.; Paper presented at the Regional Conference Series in Applied Mathematics, Philadelphia, 1976. Morari, M. Paper presented at the 5th International IFACIIMEKO, AntwerpBelgium, 1983a. Morari, M. Paper presented at the Proceedings of the IEEE Conference on Decomposition and Control, San Antonio, TX, 1983 (1983b). Morari, M.; Stephanopouios, G. AIChE J. 1980,2 6 . 247-260. Popov, V. M. Autom. Remote Control 1964,2 2 , 857-875. Rali, L. 8. "Computational Solution of Nonlinear Operator Equations"; Krieger: New York. 1979. Richalet, J. A.; Rault, A.; Testtid, J. D.; Papon, J. Automatica, 1978, 14, 413. Safonov, M. G. "Stability and Robustness of Multivariable Feedback Systems": MIT Press: Cambridge, MA, 1980. Sain, M. K.; Massey. J. L. IEEE Trans. Autom. Control 1969, AC-14, 1. Siiverman, L. M. IEEE Trans. Autom. Control 1969,AC- 14, !. Stakgold, I."Green's Functions and Boundary Value Problems": Wiiey Interscience: New York, 1979. Zames, G. IEEE Trans. Autom. Control 1966,AC- 1 1 , 228-238, 465-476. Zames, G. IEEE Trans. Autom. Control 1981,AC-26, 301-320.
Received for review June 18, 1984 Accepted July 29, 1985
(AI)
Internal Model Control. 6. Multiloop Design Constantln G. Economou and Manfred Morari" Chemical Engineering, 206-4 1, California Institute of Technology, Pasadena, California 9 1 125
A multiloop structure derived from Internal Model Control Principles is presented in an effort to simplify the design and implementation tasks associated with full multivariable controller design. Interactions are treated as perturbations on an underlying single-input-single-output (SISO) structure and a series of SISO controllers is designed with the objective of robustly stable control against such perturbations. As part of the development, the IMC interaction measure surfaces as a practical tool to assess the potential value of the multiloop design. The main questions arising in a multiloop environment are discussed in detail: what is sacrificed in terms of closed-loop performance by using a series of SISO controllers instead of a full multivariable controller; how should the SISO controllers be tuned to account for interactions; and, finally, which are the best input-output pairings.
the design of robust controllers for multivariable systems. However, as system dimensions increase, the design and implementation tasks can become unnecessarily complex in many cases. A multiloop structure encompassing the IMC principles is presented here, to alleviate this problem; interactions are treated as perturbations on an underlying
1. Introduction Internal Model Control (IMC) has been established as a general design concept offering distinct advantages for
* To whom correspondence should be addressed. 0196-4305/86/1125-0411$01.50/0
0
1986
American Chemical Society
