Ind. Eng. Chem. Res. 1989,28, 187-192
187
Linear Quadratic-Model Algorithmic Control Method with Manipulated Variable and Output Variable Set Points and Its Applications Chun-Min Cheng* Department of Chemical Engineering, University of Washington, Seattle, Washington 98195
T h e manipulated variable set points are incorporated into the structure of the LQ-MAC control method. Theoretical considerations and the possible applications are discussed. I t is shown that, for a system with more manipulated variables than output variables, both manipulated variable and output variable set points can be achieved. This result is useful for the coordination of sequential processes and can .help to achieve economically optimal operation condition. Furthermore, this result is applicable for time-varying set point changes, bumpless transition, and graceful controller degradation while actuator fails. T h e computer simulation results of the control of a multieffect evaporator for a pulping process were given. I t may be desirable or even essential to introduce "set points" for the manipulated variables. Such set points are used in industry for the coordination of a series of sequential processes. In Figure 1, for example, the weak black liquor (WBL) flow rate is one of the manipulated variables that can be manipulated to control the solid content of the output strong black liquor (SBL) of the evaporator. To coordinate with the upstream and downstream processes, however, the WBL flow rate cannot be manipulated without the consideration of the liquid level of the storage tank. The amount of liquor entering the storage tank varies a lot periodically because the digester is a batch reactor. The WBL storage tank may be full or nearly empty. If the tank is full, the WBL flow rate must be increased to avoid overflow. If the tank is nearly empty, the WBL flow rate must be decreased to prevent using up the WBL, which would result in a shutdown of the evaporator. As a result, there are different desirable WBL flow rates for different situations. This problem can be solved by using a liquid level controller to adjust the WBL flow rate and using another controller to control the output solid content by manipulating the steam flow rate. However, this control scheme is not as powerful as using both steam and WBL flow rates as the manipulated variables. When both steam and WBL flow rates can be manipulated simultaneously, the control on the output solid content will be more effective. Thus, our objective is to design a multivariable controller that can fulfill the desired values of the manipulated variables without sacrificing the controller performance. Let us define the desired values of the manipulated variable as the "manipulated variable set points". It must be emphasized that the main control purpose is still the regulation of the output a t its set point. The manipulated variable set points are of the second priority and should be allowed to deviate from their set points to compensate the disturbances. Otherwise, the controller will be less effective in its primary task. In addition to the coordination of the sequential processes, there is another potential application of the manipulated variable set points. For a system with more manipulated variables than output variables, a given output steady state may be achieved by employing different combinations of manipulated variable values (Koppel, 1983). But each such combination will,in general, translate into a unique cost. Therefore, it is beneficial to keep the manipulated variables near the economically optimal values without sacrificing the controller perform-
* Current address: Union Chemical Laboratories, ITRI, 321 Kuang Fu Road, Section 2, Shinchu, Taiwan, R.O.C.
ance. Cutler et al. (1983)gave a mathematic formulation to include manipulated variable set points in the QDMC method. They did not, however, elaborate on that formulation. Garcia et d. (1984)also included a manipulated variable set point in the application of QDMC to a batch reactor process. It should be pointed out that the LQ feedback control method has an inherent manipulated variable set point (zero for the normalized dimensionless form). The steady-state value is, in effect, a set point of the manipulated variable because any deviation from the nominal steady-state value of the manipulated variable is penalized in a control system with a positive definite weighting matrix for the manipulated variables. In this paper, we investigate the incorporation of explicit manipulated variable set points in the frame of LQ-MAC control (Cheng, 1989). Theoretical consideration is given, followed by discussions of its applications. Computer simulation results are demonstrated.
Formulation of LQ-MAC with Manipulated Variable and Output Variable Set Points Let the internal model be represented by x(k + 1) = Ax@) + B u ( k ) + D d ( k )
~ ( k =) Cx(k)
(1)
(2) where x is the n x 1 vector of the system state variables; u is the m X 1 vector of the manipulated variables; y is the r x 1 vector of the output variable; d is the vector of disturbances; and A, B, C, D are constant matrices of corresponding dimensions. The system is normalized at the nominal steady state. Thus, the value of the state variables is zero a t the nominal steady state. We assume that this system is controllable and m > r. Let the reference trajectory, y,, be y,(k + 1) = E Y , ( ~+ ) (1 - E ) Y , ( ~ ) (3) where y, is the r X 1 output set point vector and E is a diagonal matrix, 0 < E < I. It is essential to have 0 < E < I so that the reference trajectory can lead to the specified output set point, ys. Define (4) Yd(k) = y ( k ) - yr(k) U d ( k ) = u(k) - u,(k) (5) where u , is an m X 1 vector of the set points of the manipulated variables. Note that any element of u, can be zero should that represent the desired condition. For the evaporator example discussed previously, assume that we have three manipulated variables: u (1,l)= steam flow rate, u (2,l)= WBL flow rate, u (3,l)= cooling water flow
0 1989 American Chemical Society 0888-~885/89/2628-018~~01.50/0
188 Ind. Eng. Chem. Res., Vol. 28, No. 2, 1989
spectively. It is straightforward to prove that the following main properties of the LQ-MAC method (Cheng, 1989) are maintained.
'Aix)d Chip\
Main Properties 1. The asymptotically constant optimal control law is
l(!i
1,
'L
Figure 1. Schematic diagram of the simplified pulping process.
rate, and one output variable, y = the solid content of the output SBL. Further assume that the storage tank is full and the WBL flow set point must be changed t o us(2,1) = 0.1. When the WBL flow rate is changed, the steam flow rate and/or the cooling water flow rate must be changed so that the output solid content can be kept at its set point ys = 0. Assume that the change of WBL flow rate us(2,1) = 0.1 can be compensated by a change of steam flow rate u,(l,l) = 0.14. As a result, there is no need to change the cooling water flow rate; thus, u,(3,1) = 0. These set points, u s ( l , l ) = 0.14, us(2,1) = 0.1, us(3,1) = 0, a n d y , = 0, are "compatible" because they constitute a new steady state. If we give set point us(l,l) = 0.2 instead of u,(l,l) = 0.14, these set points are "incompatible"; i.e., given these values for the manipulated variables, the output set point y s = 0 cannot be achieved. Combining eq 1-5, we can get
2. Resetting the reference trajectory results in the feedback mechanism of the control method. Furthermore, resetting the reference trajectory introduces an integral mode to remove output offset even if the model has unknown errors and disturbances. The smaller values of the elements of E, which can be adjusted on-line, result in the stronger integral mode. 3. The reference trajectory and the manipulated variable/output variable set points can be changed without changing the control law. As a result, the LQ-MAC controller can have both functions of regulator mechanism and servomechanism. 4. A simple on-line tuning method is available to balance performance and robustness in the face of unknown modeling errors. 5. Including asymptotically constant set points of the manipulated variables in the LQ-MAC control system does not affect its stability properties. When the system has modeling errors, the stability is still determined by the eigenvalues (EVs) of A
+ B(K, + K,C) + K,C)
(B + H)(K,
I o D O CD I - E
0
E-I
B CB
(6) L
-
l
where y r is reset according to the output measurement
P ( k ) every sampling time period. The resulting reference trajectory becomes y,(k + 1) Y,@) - (1 - E)[Y(k) - y,(k)l
(7)
where P ( k ) = y ( k ) if the model is exact. Equation 6 can be written in a compact form: y,(k + 1) = Alxl(k) + B,ud(k) + D,d,(k) (8) We define the objective function as m
[xlT(k)Q1xl(k)+ UdT(k)Rud(k)l
min J = (1/2)
(9)
k=l
Note that both manipulated variable and output variable set points are included in d 2 ( k )which , is not involved in solving the Riccati equation for the control law. Therefore, both manipulated variable and output set points can be changed on-line without the requirement of changing the control law. Furthermore, eq 8 and 9 are in the standard form of the LQ-MAC method. For a system with additive modeling errors, eq 1 and 2 become *(k + 1) = (A + V ) g ( k )+ (B + H)u(k) + Dd(k) (10)
P(k) = C%(k)+ a ( k )
(11)
where % and Y are the actual state and output vectors and V and H are the linear modeling errors of A and B , re-
-
0
A +V (E - I)C
-BK, -V(B I
+ H)K,
1
Note that as E I, the closed-loop system (CLS) stability is essentially determined by the stability of the open-loop system (OLS). In this paper, we assume that the system is open-loop stable. In addition to these properties, we have the following properties for the LQ-MAC with manipulated variable and output variable set points. (1) Zero Output Offset Property. With or without modeling errors, if the CLS is stable, the CLS yields zero output offset when the manipulated variable set point change and disturbance are asymptotically constant. Proof. From eq 1, 2 , 7, and 12, we can get X ( Z ) = (21 - A - BK, - BK,C)-l(BK,(~ - l)-'(I E)[C%(z)+ &z) - Y,(z)] + Bu,(z)+ Dd(z)] (13) From eq 7 and 10-12, we can get zZ(z) = (A + V)Z(z) + (B + H)(K,x(z) + K,[Cx(z) + (1- z)-'(E - I)(Cf(z) + B(z) - Y,(z))] + u,(z)~+ Dd(z) (14) Substitute eq 13 into eq 14 for x ( z ) and rearrange to get - V - ( B + H)[(K, + K,C)(zI - A BK, - BK,C)-'B + I]K,(z - l)-'(I - E)C]-'J(B+ H)[(K, + K,C)(zI - A - BK, - BK,C)-'B + I]K,(z - l)-'(I - E)&) + (B + H)[(K, + K,C)(zI - A - BK, - BK,C)-' + I]Dd(z) + (B + H)[(K, + K,C)(zI - A - BK, - BK,C)-'B + I]u,(z) - (B + H)[(K, + K,C)(zI - A - BK, BK,C)-'B + I]K,(z - l)-'(I - E)Y,(Z)~
%(z) = {zI - A
For a step output set point change, we have
Ind. Eng. Chem. Res., Vol. 28, No. 2, 1989 189 lim p(k) = lim (1- z-l)[C&(z) +
k+-
2-1
&)I
= ys
Note that zero output offset is achieved in the face of modeling errors, even if the manipulated variable and output variable set points are not compatible, namely, ys # C(Ax, + B u s + d,). x i s the steady-state value of the state variables. d , represents the final values of the disturbances and equals zero whenever the disturbances are transient. (2) Manipulated Variable Set Point Approaching Property. For a system with or without modeling errors, if the disturbances are transient, the manipulated variables of the controlled system can approach their compatible steady-state set points. Proof. Consider a system with an exact model. Combining eq 1, 2, 7, 12 and rearranging we get
(I - [K, + K,C - K,(1 - ~ ) - ‘ ( 1- E)C] X (21 - A)-~B)u(z) = [K, + K,C - K,(1 - z)-l(I E)C](zI - A)-lDd(z) + Ky(l - z)-l(I - E)Y,(z) + u,(z) From the final value theorem, we get lim (1 - z-l)KY(l- z)-l(I - E)C(zI - A)-’Bu(z) = E-1
lim (1- z-l)KY(l- z)-l(I - E)[-C(z1 Z-1
W1Dd(z) + yS(z)l For transient disturbances, 1imz+ (1- z-l)d(z) = 0. Then lim (1- z-l)K,(l - z)-l(I - E)C(zI - A)-lBu(z) = F l
lim (1 - z-l)KY(l- z)-l(I - E)ys(z) 2-1
4.5u(l,l) + 7u(2,1) = ys (16) Therefore, under closed-loop control, any combination of u (1,l) and u (2,l) (i.e., the steam and WBL flow rates) that satisfies eq 16 can be one of the possible steady-state conditions. Assume that u,(2,1) = 0.1 is required; then we must assign 0.1 for u(2,1), so that u (1,l) has a unique solution. This solution is the corresponding compatible set point of u (1,l). Following similar procedures, one can prove that the manipulated variable set point approaching property is also valid for a system with modeling errors. A Method To Force the Manipulated Variables to Their Set Points. Various methods can be used to force the manipulated variables to approach their desired set points. In one of these methods, one can force the manipulated variables to their set points by setting
where Q is a diagonal manipulated variable set point approaching matrix, 0 < Q < I. t ~ are~ the- m~ - r manipulated variables to be forced to their final set points, Note that the purpose of this arrangement is to assign the desired set point values to the m - r manipulated variables. The values of ~ d ( ~ - r ) ( kwhich ) are calculated from the normal control law are, thereby, overidden. In the previous example, the calculated values of u (2,1)(k) are overidden, while the values of u (1,1)(k) are still obtained from the normal control law. The resulting u (m-r)(k)becomes
u(m-r)(k)= KI(m-r)xl(k)+ Aud(m-r)(k)+ Us(m-r) where
Therefore,
C(1 - A)-’Bu(l) = ~ ~ ( 1 )
(15)
Because u and ys are vectors of dimensions m X 1 and r X 1, respectively, there are r equations for m unknowns in eq 15. Therefore, for a system with m > r, there are an infinite number of combinations of the values of the manipulated variables that can satisfy the specified output set point y,(l). As a result, the final values of the manipulated variables can settle on any compatible combination which is one of the compatible steady states of the process. To obtain a specific state, it is necessary to force m - r manipulated variables to their set points. The remaining r manipulated variables will then automatically settle on their corresponding steady state through the LQ-MAC feedback control mechanism. The following example further explains this idea. The method of forcing the m - r manipulated variables to their specified set points is discussed after the example. Example. Assume that an evaporation system can be represented by the following simplified model: x d k + 1)
where u(l,l)(k) and u(2,l)(k) are the steam and WBL flow rates, respectively. We want to increase the WBL flow rate from 0 to 0.1. From eq 15 we can get
which is equivalent to
A%(m-r)(k)=
QUd(m-r)(k
- 1) - Kl(m-r)Xl(k)
Note that Aud(m-r)(k) is a transient “navigating input”. To have a unique solution for the manipulated variable vector u(1) in eq 15, the values of m - r manipulated variables need to be specified. Therefore, it is necessary to have m - r elements of A smaller than 1 to make all manipulated variables reach their compatible set points. One can give a different emphasis to each manipulated variable by adjusting the values of the corresponding A elements. The larger A element results in the slower approaching of the corresponding set points. When B = I, an overide of the normal control law is eliminated. In that case, as suggested by eq 15, the final values of the manipulated variables may not be the specified manipulated variable set points. Because of the zero output offset property and the manipulated variable set point approaching properties, we have the following conclusion. With or without modeling errors, if the compatible steady-state set points of the manipulated variable and output variable are known, both set points of the manipulated variable and output variable can be reached exactly by using B(k) < I when y ( k ) = ys. If we do not have sufficient knowledge to set up the compatible set points, B = I can be used. With B = I, zero output offset is maintained while the manipulated variables usually settle near their set points, which may be satisfactory in many cases. Examples will be demonstrated in the section of simulation results. Note that the selection of the weighting matrices affects the control law K and consequently affects the dynamic response of the manipulated variables (eq 12). However, the final values of the manipulated variables are independent of the weighting matrices (see eq 15). Therefore, one cannot force the manipulated variables to settle on their set points by manipulating the weighting matrices.
190 Ind. Eng. Chem. Res., Vol. 28, No. 2, 1989
Figure 2. Block diagram of the LQ-MAC with input/output set points.
If the disturbances are asymptotically constant, eq 15 becomes
C(I - A)-’Bu(l) = -C(I - A)-’Dd(l)
+ y,(l)
It can be verified that all the results discussed previously are still applicable. However, it must be pointed out that the compatible set points y , = C(Ax, + B u s + d , ) then contain d , , which may be difficult to identify. (3) Bumpless Transition. In industrial practice, a system may be operated at a variety of steady states, which could be different from the steady state used for the development of the internal model. To have a smooth transition to the new control system, we want the operating conditions of the process to be held constant without any interruption. Thus, the manipulated variable and output variable operating values, u and y o ,can be maintained. To have such a “bumpless” manual-automatic control transition within the frame of the LQ-MAC with manipulated variable/output variable set points, eq 5,6, and 12 need modification. Let us replace u,(k) with the difference between the set points and the operating values, i.e., us&) - u 0 . With this modification, the values of the internal model variables x ( k ) ,Yd(k),and u d ( k ) remain zero as long as u s @ )= u o ,y , ( k ) = y , ( k ) = y o and in the absence of disturbance. u ( k ) + uo,of which u (it) is computed from the modified eq 12, is then the manipulated variable of the actual process. Note that u ( k ) + u equals u for the conditions stated above. Therefore, the process can stay uon course” until disturbances perturb the output. When the output deviates from y o ,the normal LQ-MAC feedback control mechanism starts to manipulate the input, u ( k ) + u o, to compensate the disturbance. This avoids the introduction of an artificial disturbance during the transition period, while the function of the controller is fully maintained. The internal model becomes a monitor of the deviation from the operating conditions. Whenever the system deviates from the operating conditions, the LQ-MAC feedback control mechanism responds accordingly. The prediction of the internal model is now independent of the nominal steady state which is used to normalize the internal model. Consequently, the problem of starting up at different operating values is resolved. Moreover, the capability of executing manipulated variable/output variable set point changes is maintained. The block diagram in Figure 2 shows the structure of this system. (4) Time-Varying Manipulated Variable Set Point Change. The operation conditions usually have con-
straints on the rate of change of the manipulated variable because of physical limitations, safety concerns, etc. In those cases, one may wish to adjust the set points in a ramp (stairwise) fashion instead of using a step change. In LQ-MAC control system, this can be done by using a stairwise time-varying u , ( k ) in eq 6 and 12. To obtain a high performance during the adjustment of u ,(k), however, one should keep the set points of the manipulated variables in concert with each other. Referring to the previous example, in a three manipulated variable system, each unit of change of WBL flow rate, u8(2,1)(k),requires 1.4 units of change of steam flow rate, u,(l,l)(k); then one can use uS(2,l)(k)= @u,(l,l)(k)with /3 = 1.4. There is no need to change the third manipulated variable. @ is another tuning parameter, which can also be adjusted on-line. In a similar way, one can adjust the third manipulated variable instead of adjusting u,(l,l), or can adjust both u,(l,l) and u,(3,1) in concert with us(2,1). This is one type of feedforward control. @ may be obtained through step tests of the process.
Simulation Results In the following simulation, the nonlinear model of the multieffect evaporation process is used to represent the actual process. The internal model is a linear 18 x 18 state space model. It is normalized at one of the steady-state operating conditions, i.e., a steam flow rate of 104140 lb/h, a WBL flow rate of 730000 lb/h, and a output solid content of 53.93% by weight. The constant control law was obtained as described by Cheng (1989). The control law derived from the linear model is used for controlling the nonlinear process to demonstrate the situation of the control with modeling errors. To investigate the responses in the face of disturbances (which may be more realistic), all the simulations, unless indicated, include a transient disturbance: a 10% increase of the solid content of the feed WBL during the first half hour of the simulation. The set points are marked by (”. Since the simulated system is a two-input-one-output system, A < 1 is applied for only one input variable (WBL flow rate) when the use of A is indicated. Step Manipulated Variable Set Point Change. The compatible set points for the nonlinear model are 0.1445, 0.1, and 0 for the normalized steam flow rate, WBL flow rates, and output solid content, respectively. Parts a and b of Figure 3 show the manipulated variable responses for various pairs of set points. The results with incompatible set points, u,(l,l) = 0.1336 and us(l,l) = 0.152, are given to demonstrate the situation of insufficient knowledge of the compatible set points. With A = 1, the manipulated variables settle near their set points. An A < 1 can be applied to remove the manipulated variable set point offset. It should be noted that the output variable set point, 0.539, is achieved even when the manipulated variable set points are incompatible and the model has errors (Figure 3c). The prediction of the linear model in Figure 3c shows a typical closed-loop modeling error in these simulations. Time-Varying Manipulated Variable Set Point Change. Figure 4a shows the time-varying set point change of the WBL flow rate, us(2,1). The WBL flow rate is ramped up 7 70above the steady-state value, kept constant for an hour, and then ramped down to the steady state. No transient disturbance is included in this simulation. For the case of a “hard” set point change, the WBL flow rate tracks a prespecified trajectory exactly. The values of the WBL flow rate computed by the feedback control law are overridden although the value of the steam flow rate in Figure 4b is still determined by the control
Ind. Eng. Chem. Res., Vol. 28, No. 2, 1989 191
730000 2
0
0
1 19000
I
d
6
4
2
1
3
4
120000
=
1 14000
5 z
2-
118000 116000 114000 1'2000
e 0
2
1 10000
6
4
0
n "
-'--/. 5
1
2
3
4
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3
4
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!. U\( l,i)=i! l i i h 0545
-
0 65
I
063
s 5
0 540
0 61
0 535
0 57
0 59
3
-3
2
0
6
4
0 55 0
TIME ( H R S ]
Figure 3. Input and output responses with different A values for compatible or incompatible set points.
,
""""
0
.-
1'6000
=
2
3
4
2
3
4
-1
112000
a u
-
h
108000 104000
k
I
'00000
0
-
1
(I 5-
1
2
1
-
0
0
RLl 0
-A-
L O0 R1. 4
-
2
TI\fE
3
II
4
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Figure 4. Input and output responses with different 6 values.
law. In this case the controlled system is similar to a single-input (steam flow rate)-singlet-output system. In many cases, the emphasis of control is on the output variable. Exact tracking of an manipulated variable set point is, therefore, not required. This case is shown by the results with @ = 1.0 or p = 1.4. The WBL flow rate calculated from the control law is implemented to keep the output variable at its set point. The manipulated variable set point tracking is of second priority in these two cases. On the other hand, the set point of the steam flow rate, ua(l,l),is changed in concert with ua(2,1)(k)according to us(l,l)(k) = @ua(2,1)(k)(Figure 4b). Note that if a good
TI\?E ( H R S !
Figure 5. Input and output responses for starting up a t nonsteady state.
estimate of the parameter @ is available, both the regulator mechanism and the manipulated variable set point tracking can be obtained with ease (@= 1.4 in Figure 4). It should be noted that the system is quite robust to the selection of @. Figure 4c shows that, even with a hard WBL set point change and = 0, the output performance is still reasonably good (with a maximal error of about 1% of the steady-state value). Starting Up from a Nonsteady Initial Condition. When switching to automatic control, it is likely that the plant is running a t values that are different from the steady-state values of the internal model. Figure 5 is the result of starting up from a nonsteady-state situation. At the time of switching to the LQ-MAC control, the value of the output variable is increasing because of a 10% increase of the steam flow rate. Without control, the output will go higher than 64 % . The nonsteady-state WBL and steam flow rates are a t the values of zero time in parts a and b of Figure 5. There is also a transient disturbance caused by a 10% increase of the WBL solid content during the first half hour. The control objective is to stabilize the output at its current value. Following this one could specify another control goal, such as a move to a desired state. Parts a, b, and c of Figure 5 demonstrate the smooth transition to the LQ-MAC control despite the presence of modeling errors, a disturbance, and a transient initial condition. Graceful Degradation. Another potential application of the incorporation of manipulated variable set point in the control structure is a graceful degradation when actuator fails. Assume that a disturbance of a 10% increase of the WBL solid content is affecting the system; the controller is adjusting the WBL and steam flow rates to compensate it. However, the actuator controlling the WBL flow rate fails a t the time of 2 h and stays a t the smaller opening, say, 0.07. Without proper compensation, the output solid content will become too high and may plug the evaporator. For this situation, we can take the currently measured WBL flow rate as the new set point of the
192 Ind. Eng. Chem. Res., Vol. 28, No. 2, 1989 850000
--
j
750000
14
4
2
,
830OOC
-30000
1
I
.
I 2
3
4
5
2
3
4
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=7-
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9 555
II
05504 ?. -: c 515 r
-2
0540
L
0 535
-
3 530
I 3
2
4
5
TIME ( H R S )
Figure 6. Input and output responses while the WBL actuator
failed.
WBL flow rate. Then the controller will find a proper response for the steam flow rate. Parts a and b of Figure 6 showed the WBL and steam responses while the actuator failed. Figure 6c is the controlled result of the output variable. It approached the set point gracefully even though the WBL actuator was out of function. Note that the change of manipulated variable set points following an actuator failure can be an automatic procedure in the control software. When the actuator fails, a warning digital signal can be sent to the control software, or we can monitor the values of the actual inputs to see whether they are following the requested values or not. For a more complicated system, one might combine the controller with a supervisory program that would be responsible for selecting the best manipulated variable/output variable set points for different situations. A knowledge-based expert system might be set up for this purpose. Nomenclature A, B, C, D = system matrices of the state space model
Al, B1,D2 = augmented system matrices of the state space model CL_S = closed-loop system d , d = disturbances of model and actual process, respectively d 2 = augmented vector of d d , = final value of disturbances E = tuning parameter of the reference trajectory EVs = eigenvalues H = additive modeling error of B K,, K,, Kl(m-r)= subsets of the gain matrices of the control law LQ = linear quadratic feedback control method MAC = model algorithmic control method OLS = open-loop system Q1, R = weighting matrices of the state variables and manipulated variables, respectively SBL = strong black liquor u = manipulated variable u d = difference between u and u , u , = set point of the manipulated variable u d(m-r), u (m-r), u s(m-r) = subsets of u d, u , and u,,respectively u , = operating value of the manipulated variable V = additive model errors of A WBL = weak black liquor x,t = state variables of the internal model and actual process, respectively x, = steady-state value of the state variable x 1 = augmented state variable vector y , J = output variables of the model and actual process, respectively Yd = difference between the output variable and the reference trajectory y o = operating value of the output variable at the starting up time yr = reference trajectory y s = set point of the output variable & = manipulated variable set point approaching factor p = coordination factor for the manipulated variables Au d(m-r) = navigating input
Literature Cited Cheng, C. M. Ind. Eng. Chem. Res. 1989, preceding paper in this issue. Cutler, C. R.; Haydel, J. J.; Morshedi, A. M. AIChE Annual Meeting, Washington, DC, 1983; paper 44C. Garcia, C. E.; Morshedi, A. M.; Fitzpatrick,T. J. American Control Conference, San Diego, CA, 1984; paper TA4-4. Koppel, L. B. Chem. Eng. Educ. 1983, 17, 58.
Received for review December 29, 1987 Revised manuscript received July 27, 1988 Accepted August 22, 1988
