I n d . Eng. Chem. Res. 1989,28, 178-186
178
Wu, W.-L.; Black, W. B. High Strength Polyethylene. Polym. Eng. Sci. 1979,19, 1163-1169. Wu, W.-L.; Black, W. B. Process for Producing High Tenacity Polyethylene Fibers. US Patent 4,228,118, October 14, 1980, to Monsanto Co. Wu, W.-L.; Black, W. B. High Tenacity Polyethylene Fibers and Process for Producing Same. US Patent 4,276,348, June 30, 1981, to Monsanto Co.
Zwick, M. M. Spinning of Fibers from Polymer Solutions Undergoing Phase Separation. I. Practical Considerations and Experimental Study. In Fiber Spinning and Drawing; Coplan, M. J., Ed.; Applied Polymer Symposia 6; Interscience; New York, 1967.
Received for review May 10, 1988 Revised manuscript received September 8, 1988 Accepted October 27, 1988
PROCESS ENGINEERING AND DESIGN Linear Quadratic-Model Algorithmic Control Method: A Controller Design Method Combining the Linear Quadratic Method and the Model Algorithmic Control Algorithm Chun-Min Cheng* Department of Chemical Engineering, University of Washington, Seattle, Washington 98195
The linear quadratic (LQ) feedback control method and model algorithmic control (MAC) algorithm are combined to obtain the complementary properties of these two methods. This formulation allows for a simple method of on-line controller tuning and can affect the robustness property in a direct way, while keeping the structure of the LQ optimal control method. A theoretical analysis covering closed-loop stability, convergence to the set point, and robustness properties in the case of an inexact process model is included. I t is demonstrated, for example, that zero offset is achieved even if the model is inexact. The method is used to design a controller for a simulated multieffect evaporation process. In the past decade, a few internal model-based control methods have been developed: Model Algorithmic Control (MAC) (Richalet et al., 1978; Mehra et al., 1980, 1981; Mehra and Rouhani, 1980; Rouhani and Mehra, 1982); Dynamic Matrix Control (DMC) (Cutler and Ramaker, 1980; McDonald and McAvoy, 1987); Quadratic Matrix Control (QDMC) (Garcia et al., 1984; Cutler et al., 1983); Internal Model Control (IMC) (Garcia and Morari, 1982, 1985a,b);Quadratic Programming Internal Model Control (QPIMC) (Ricker, 1985). These methods have been successful in industrial applications (e.g., Lecrique et al. (1978); Mehra et al. (1978); Mehra and Eterno (1980); Garcia et al. (1984),and Ricker et al., 1986)-because they provide a high degree of flexibility and allow for on-line tuning. For example, MAC includes an adjustable “reference trajectory” through which one can affect the robustness of the feedback system in a direct way. Garcia and Morari (1982) gave an unified review of DMC, MAC, and IMC, pointing out common features and noting analogies to classical forms of optimal control. Stephanopoulos and Huang (1986) introduced a two-port control system and demonstrated that it can be equivalent with IMC. A shortcoming of the IMC-type methods is that an open-loop stable system is required; the nonminimumphase characteristics need to be “factored out” in advance in the IMC design procedures (Garcia and Morari, 1985a,b; Holt and Morari, 1985a,b). The well-studied linear
* Current address: Union Chemical Laboratories, ITRI, 321 Kuang F u Road, Section 2, Shinchu, Taiwan, R.O.C. 08S8-5885/S9/2628-0178$01.50/0
quadratic (LQ) feedback control method (e.g., Anderson and Moore, 1971; Edgar, 1976; Astrom and Wittenmark, 1984) can stabilize the open-loop unstable process and is applicable directly for processes with nonminimum-phase characteristics. Cheng and Ricker (1986) developed a combination of the LQ method and MAC method in their controller design for a multieffect evaporator to take advantage of the potential complementary properties of the LQ method and IMC-type methods. The LQ optimization method is used to formulate a feedback control law, based on a nominal “internal model” of the process. A reference trajectory similar to that used in MAC is included to allow for on-line tuning. In this paper, we discuss the structure, formulation, properties, and applications of the LQ-MAC method. A multieffect evaporator is used as an example to demonstrate the properties and applications of this method. A comparison of the LQ-MAC method with MAC and IMC methods is also discussed.
Structure of LQ-MAC The structure of the LQ-MAC controller is similar to the MAC method that was proposed by Richalet et al. (1978) and was analyzed by Rouhani and Mehra (1982). It includes (1)an internal model for system representation and prediction, (2) a reference trajectory leading to the desired set point, (3) an optimality criterion for control law formulation, and (4) a resetting of the reference trajectory at regular intervals using the measurement of the process outputs. These components are described as follows. Internal Model. The internal model is a linear discrete-time state-space formulation: 0 1989 American Chemical Society
Ind. Eng. Chem. Res., Vol. 28, No. 2, 1989 179
x(k + 1) = Ax(k) + Bu(k) + Dd(k)
(1) Y(k) = CX(k) (2) The variables are defined as follows: x(k)is an n X 1state vector, u ( k ) is an m X 1 input vector, d ( k ) is a p X 1 disturbance vector, y ( k ) is a r X 1 output vector, and A, B, C, D are constant matrices. If the disturbances are uncharacterized, D is taken to be a null matrix. Reference Trajectory. The reference trajectory is the desired path leading to the specified set point. The trajectory is described by Yr(k + 1) = Eyr(k) + (1 - E)s(k) (3) Yr(0) = ~ ( 0 ) (4) where y r ( k ) is an r X 1 reference trajectory vector, s ( k ) is an r X 1 set point vector, E is an r X r diagonal matrix, 0 < E < I, and I is the identity matrix. This form of the reference trajectory is analogous to those used in MAC and IMC. The reference trajectory approaches the set point according to the value of E; the smaller the E value, the faster the trajectory approaches the set point. When E equals 0, the trajectory reaches the set point instantly; on the other extreme, E = I, the trajectory needs infinite time to approach the set point. In addition to effecting the speed of leading the controlled variable to the set point, and consequently the performance, we will show that the selection of E affects the robustness of the closed-loop system. Note that the diagonal elements of E must be positive and less than unity so that the reference trajectory converges to the set point. We will also show that in order to eliminate steady-state offset and to improve the transient response, it is important to “reset” the reference trajectory periodically. Control Law Formulation. The control law is an LQ feedback control type but includes a reference trajectory to make it easy to tune on-line. In the standard LQ controller, the control law is derived through the solution of a Riccati equation; modifications in its “tuning” parameters necessitate a new solution of the Riccati equation, which can be a time-consuming procedure. In the LQ-MAC method, however, the introduction of the reference trajectory does not affect the Riccati equation, which is solved initially off-line. Fine-tuning of the controller performance is then conducted by adjusting the reference trajectory on-line. The model for the formulation of the control law is the internal model described by eq 1 and 2. The variables are normalized to have zero value at a nominal steady state. The basic control objective is to force the process output, y , to follow the reference trajectory in an optimal sense, i.e., minimizing deviations from y rwhile preventing excessive movement in the manipulated variables. The controlled output variable approaches the set point a t a speed determined by E because the reference trajectory described by eq 3 is dedicated to the desired set point. The state vector is thus augmented to include the vector of the difference between the output and the desired reference trajectory. Define Yd(k) = Y(k) - yr(k) (5) From eq 1-3 and 5 , we can get the augmented form
Equation 6 can be written in a more compact form x,(k + 1) = Alxl(k) + Blu(k) + Dldi(k) (7) The standard LQ objective function is m
min J1 = ( 1 / 2 ) C [xlT(k)QIXl(k)+ uT(k>Ru(k)l k=l
(8)
with
where Q1 and R are symmetric weighting matrices. The elements of the Q1matrix for Yd, Le., Q y ,must have relatively large values compared with the values of Q , and R to force the outputs to track the reference trajectory.
Properties Inherited from the LQ Method Equations 7 and 8 are the standard form for the LQ method. Therefore, the following well-known properties of the LQ feedback control are preserved (Kuo, 1980). 1. If the system is stabilizable, positive definite Q1 and R ensure the closed-loop asymptotic stability for any kind of open-loop system, including nonminimum-phase systems and open-loop unstable systems (Astrom and Wittenmark, 1984). Throughout this paper, the stabilizability of the system is assumed. 2. The unique optimal solution is a simple linear function of the state variables, i.e.,
where K1 is found by solving the appropriate matrix Riccati equation. K,(k) is asymptotically constant (Takahashi et al., 1972; Kuo 1980). We use the steady-state gain matrix K1. 3. It is straightforward to verify that the controlled system described by eq 6 and 9 is equivalent to Z X ( Z ) = (A BK, + BK,C)x(z) - BK,y,(z) + Dd(2)
+
Z Y ~ ( Z )= Eyr(z) + (1 - E)s(z) The system described by the above two equations is stable as long as 0 < E < I and the weighting matrices are positive definite. As a result, the eigenvalues (EVs) of A + BK, + BK,C must be within a unit circle. 4. For a system with time delays, an augmented state space model can be formulated (Astrom and Wittenmark, 1984). Because the augmented form has the same format as eq 7, the standard LQ procedures discussed in this paper are still applicable for the LQ-MAC design for a system with dead time. The formulation of eq 6 provides advantages over the conventional LQ feedback control. As will be shown later, a simple on-line tuning for performance and robustness in the face of modeling errors becomes feasible. Here, the difference between the LQ-MAC method and the model following method should be pointed out. In the model following method, the “model” is included in the AI matrix (Newel1and Fisher, 1972; Markland, 1970). Therefore, the matrix Riccati equation has to be solved each time the model (corresponding t o the reference trajectory of LQMAC) is changed. In LQ-MAC, the E and s matrices appear in the D, matrix which is not involved in solving the Riccati equation. Therefore, the set point and the reference trajectory can be changed without the requirement of solving the Riccati equation again. Resetting the Reference Trajectory. Periodic resetting of the reference trajectory is very important for the
180 Ind. Eng. Chem. Res., Vol. 28, No. 2, 1989
~ , (+ k 1) = y r ( k )- ( 1 - E)[Y(K)- s(k)l
(12)
Taking z transform gives zyr(z) = Y ~ ( z-) ( 1 - E ) [ Y ( ~-) ~
>.(?I
>Y,,Z)
Figure 1. Schematic diagram of the internal model principle.
performance of the controller. In the LQ-MAC method, one must specify the frequency of resetting and the formulation of the resetting procedure. If resetting is not used, eq 3 defines the reference trajectory at each sampling point, K . Alternatively, one may reset the trajectory at every sampling point. A simple method of resetting is yr(k) = y m ( k ) (10) Then the value of y , ( k ) in the last term of eq 6 is "updated" at every sampling time point according to the output measurement, y m . Before discussing theoretical properties, it should be helpful to explain the ideas behind eq 6 and 8 and the purpose of resetting the reference trajectory. Figure 1 is the schematic diagram of the "internal model principle" discussed by Stephanopoulos and Huang (1986) and other authors. To make the controller output errors approach zero, the control loop should generate external signals Cy,(z) and d ( z ) ) at the junctions where the external signals enter the loop; i.e., the control loop must generate a y ( z ) approaching y , ( z ) and a y * ( z ) approaching - d ( z ) as indicated in Figure 1. To make y ( z ) approach y , ( z ) ,y d = y - y r is minimized and y is forced to follow y r ,which always heads for the set point. T o make y * ( z ) approach -d(z), y , ( z ) in eq 6 is updated at every sampling time. An integral mode inherent in the LQ-MAC allows y * to approach -d and achieve zero offset. It should be noted that to obtain a perfect control, equivalent to applying G, = G-' in Figure 1, one can use R = 0, Q, = 0, and E = 0 in the LQ-MAC method. However, it is well-known that a perfect control is not feasible because of practical limitations (Garcia and Morari, 1985a,b).
Theoretical Properties of Stability and Convergence to Set Point Throughout the section on theoretical properties, we assumed that the reference trajectory is reset every sampling time period. Without resetting, the controller does not receive information from the plant and it is actually under an open-loop control; therefore, the theoretical properties of the control without resetting are not given in this paper (Cheng, 1986). Their simulation results are provided, however, for comparison. The Properties of a System with an Exact Model. In this section, it is assumed that the actual process output is predicted exactly by the internal model. The disturbances are measured and its dynamic effects on the process output are represented by the model exactly. The cases of systems with unmeasurable or unmodeled disturbances are treated as systems with modeling errors and are discussed in the next section. In eq 6, when y r in the last term is reset to y ( k ) every sampling time period, we can get y d ( K + 1) = y ( k + 1) - y r ( k + 1) = C[Ax(K) + Bu(K) + D d ( k ) l - C d k ) + Y @ ) - Y @ ) + (1 - E ) [ Y ( ~-) 4 k ) l (11)
Since y(K + 1) = Cx(k + 1)and y ( k ) = Cx(k),eq 11 can be simplified to be
( ~ 1 1 (13)
With an exact model, with resetting, we have the following stability and zero offset properties. Property 1. Stability. With E = I , the controlled system is stable if the weighting matrices are symmetric positive definite and the disturbances are bounded. The open-loop system (OLS) described by eq 1 and 2 can be either stable or unstable. Proof. When the process is represented by the model exactly, the controlled system can be described as follows:
The stability of this controlled system is determined by the EVs of A2= L
A + B(K, (E - I)C
+ K,C) -BKy I
1 J
When E = I , y , ( z ) is independent of x ( z ) and s (2); it is stable as long as its initial condition is finite. In that case, the EVs of A2 are actually those of A + B(K, + K,C), which are within an unit circle as discussed previously. Therefore, the controlled system is stable. The controlled system is always stable when E = I . One can take advantage of this property for on-line tuning. Starting with E = I , then reduce the E value for a better performance. Example 1. Given a two-input-two-output system, -1.1 A = 10.2
0.1 -1.3
]
0.1 = [O
0.2 0.11
=
[i :]
The EVs of this system are -1.0268 and -1.3732. It is open-loop unstable. The weighting factors for the state variables, the augmented variables, and the input variableds are 1, 1000, and 0.1, respectively. The resulting control law is as follows.
I
22.524
K x = -1.024
-41.8051 20.781
[
K, = -8.202 -0.702
15.9871 -8.365
It can be verified that the EVs of the CLS are within an unit circle when 0 < E < I . For example, when E = I, the EVs of the CLS are -1.754 X -6.890 X and 1 of the reference trajectory; when E = 0, the EVs are 0.466 + 0.719i and 0.4999 + 0.8623 (close to the unit circle), They are all within an unit circle. When E approaches 0, however, the EVs are close to the unit circle and are vulnerable t o modeling errors. Property 2. Zero Offset. The CLS yields zero offset if the disturbance is asymptotically constant. Proof. From eq 1, 2, 9, and 13, we can get Z X ( Z ) = [A + BK, + BK,C - BK,(1 - z)-'(I E)C]X(Z)+ D d ( z ) + BK,(1 - z)-'(I - E)s(z) From the final value theorem, we can get the following result for a unit step change: lim y ( k ) = lim (1- z-')Cx(z) =
k--
P
I
lim (1 - z-')[BK,(l - z ) - l ( I -1
-
E ) ] - l [ D d ( l )+
BK,(1 - z)-'(I
-
E)s(z)] = s
Ind. Eng. Chem. Res., Vol. 28, No. 2, 1989 181
A + B(K, + K,C) (B + H)(K, + K,C)
I o
%;
(E-I)C
I
YSZ)
[ D O
.I
z x ( z ) = Ax(z)
Figure 2. Block diagram of LQ-MAC.
Resetting the reference trajectory naturally introduces an integral mode to achieve zero offset. This is shown as follows. Combining equations 9 and 13 we get
K,x(z)
+ K,+Y(z) + ~ ' ( -1z-')-'KY(I - E ) [ ~ ( z-) s ( z ) ]
where (1- Z-I)-~K~(I - E) is the integrator. Note that the integral mode can be adjusted on-line by changing E. Smaller E values result in a stronger integral mode, and the offset can be removed more rapidly. The Properties of a System without an Exact Model. In case the internal model has errors in the parameters of the A, B, C, and D matrices and have measurement errors of disturbances d , the actual system can be generally expressed as x(k 1) = Ax&) + Bu(k) + D d ( k ) + f[x(k),u(k)] (14)
+
where f [ x ( k ) , u( k ) ] represents the errors in the internal model. Equation 14 may be considered a linearized form of a general nonlinear dynamic system, x ( k + 1) = g [ x ( k ) , u( k ) ] . A and B correspond to the Jacobian of g with respect to x and u , respectively, while f corresponds to the higher order terms. Here, we assume that f[x(k),u(k)] is linear in x ( k ) and u ( k ) and can be described as
f[x(k),u(k)]= Vx(k) + Hu(k) in which V and H are n X n and n X m constant matrices, respectively. Then the actual process can be written as
f(k
+ 1) = (A + V)%(k)+ (B + H)u(k) + Dd(k) P ( k ) = Cf(k) + &k)
(15) (16)
where f and B are the "a_ctual"state vector and output vector, respectively, and d ( k ) is the unmeasured disturbance vector. The internal model is still represented by eq 1 and 2. Since the internal model has errors, the controller must keep an eye on the "real world". We reset the reference trajectory using the measured output; i.e., y , ( k ) = y , ( k ) = P ( k ) . Then following a similar procedure of eq 11 and 12, the reference trajectory is
Y A +~ 1) = Y,&) - (1 - E)[P(k) - s(k)l
+
0
Note that eq 18 is in a standard form of the system equation
U(Z)
u(z) =
]
A+v
-BK, -(B + WKy][
0
(17)
A general relationship between y and other components is shown in Figure 2. The controlled system is now described by eq l , 2,9,15,16, and 17. After rearrangement we get
+ Bu(z)
of which the stability is determined by the EVs of A. Therefore, we have the following property. Property 1. Stability. The stability of the CS with linear errors is governed by A,, where A3
I+
A + B(K,+ KyO (B H)(K, + K,C)
0
A
+V
-BK, -(B + H)K,
1
Inspecting A,, it can be shown that if E approaches I the stability of the system (eq 18) is essentially determined by the eigenvalues of A + V, i.e., the stability of the OLS, because the eigenvalues of A + B(K, + KyC) are within a unit circle. This suggests that for an open-loop stable system, which represents the majority of chemical processes, E = I is a good starting point for tuning purposes. A demonstration of tuning by adjusting E is given in the section of guidelines for on-line tuning. In addition to affecting the performance, adjusting E can affect the stability, and thus the robustness, of the controlled system. A system with a larger E usually is more robust. Examples for the cases of a system with model errors of pure gain mismatch or of pure time delay are given in a later section to demonstrate the effects of E on robustness. Property 2. Zero Offset. With additive model errors, the CLS yields zero offset if the disturbance is asymptotically constant and the CLS is stable. Proof (Similar to the Proof for the System with an Exact Model). Note that disturbances need not be modeled to preserve the zero offset property. Modeling of the measurable disturbances is helpful, however, for the performance because it provides an additional feedforward control mechanism through the internal model. It should be pointed out that the zero offset property is not restricted to the cases with additive model errors (Cheng, 1986). As will be demonstrated in the simulation result, zero offset can be achieved for a system with nonlinear model errors.
Implementation and Applications An efficient recursive method for the solution of the Riccati equation for the optimal gain was derived by Newell and Fisher (1972). The optimal gains converge to the steady-state values rapidly. The constant gain matrix is implemented for the control system. As discussed previously, the reference trajectory needs to be reset periodically. When it is reset every N sampling
182 Ind. Eng. Chem. Res., Vol. 28, No. 2, 1989
times, y , ( k ) in eq 6 is reset to a new value when k = N , 2N, 3N, etc. Between these sampling points, y,(k) follows eq 3. The choice of N affects the closed-loop performance, as will be demonstrated in the simulation section. The formula for resetting the value of y,(k) in eq 6 can combine the measured output and that predicted by the internal model
to
n
n
n
n
n
condenser
r
Y,&) = E , J ~ ( ~+ ) (1 - E,)y(k)
in which E, is an r x 1 diagonal matrix. We can assign the values of the E, matrix according to the expected relative accuracy of the measurements and the internal model. For example, if a given measurement is free of error, the corresponding element of E, should be set to one. In the case of a sensor failure, the corresponding element of E, should be set to zero. The calculation would then ignore that measurement and rely on the prediction of the internal model. In this paper, we define y , y , , and ymas r X 1 vectors. This is only for convenience. In an application, y , y,, and y mneed not have the same dimension. Also note that eq 6 can be augmented to include an integral mode and/or penalties on the rate of change of the inputs (Edger et al., 1973; Cheng, 1986). The penalties make the input variations smoother. The integral mode introduced by Johnson (1968) may also be incorporated (Cheng, 1986). However, this type of integral mode is ineffective if the model has errors. It was shown in the previous section that a more effective integral mode occurs naturally in the LQ-MAC method. Closed-Loop System with Upper Bound and Lower Bound on Inputs. Although the variation of the input variable can be affected by adjusting the weighting matrix R and the E value of the reference trajectory, the LQMAC design method does not incorporate explicit constraints of the manipulated variables. The controller thus may attempt to push one or more manipulated variables beyond their physical limits. Little and Edgar (1986) proposed a constrained optimal control method by using the LQ approach. Extensive on-line computation is required in their method. Ricker (1985) used a quadratic programming method in IMC design to address the constraint problems. An alternative method of accommodating the constraints is to set the inputs equal to the bound whenever the calculated inputs are beyond the upper or lower bounds. This method can be attractive because it is simple and effective. The controlled system then is equivalent to x,(k + 1) = (A, + BiKl)xl(k) + D,di(k) + BlAu(k) where Au ( k ) is the difference between the upper or lower bounds and the value of the input variables calculated from the control law. Imposing constraints in this way is equivalent to introduce a specific type of “disturbance” Au ( k ) to the controlled system. As discussed previously, the controlled system without input constraints is asymptotically stable. u ( k ) and consequently Au ( k ) should be bounded. From the bounded-input-bounded-state stability theorem (Kuo, 1980), we know that the system is bounded when the constraint is imposed. The possibility of an undesirable oscillation, however, cannot be eliminated if the process needs to be frequently operated at the margin of constraints. Operation a t the margin of constraints cripples any kind of control method. A process should be designed to operate well away from this region. Start-up from Various Operating Conditions. The LQ-MAC method needs a state space model, which is normalized at a nominal steady state. However, many chemical processes in industry operate at a variety of
I
product weak liquor
Figure 3. Simplified schematic diagram of the multieffect evaporation process.
steady states. If one needs to switch from manual control to automatic control while the process is operating at a steady state that is different from the nominal steady state, it is desirable to keep the process at its steady state without being disturbed. To have such a “bumpless” manual-automatic control transition during various operating situations, one can define the prestart-up operating values as zero value. Namely, we can define p ( k ) = [y,(k) y0]/yss, where y o is the operating output value at the prestarting time; y s is the nominal steady state value used to normalize the state space model; y,(k) is the measurement of the process output at time k;and 7, as defied previously, is the normalized process output and is used to update the reference trajectory. Similarly, the computed value of the manipulated variable must be converted to the implemented value u m, where u ,(k) = u ( k )u 88 + u, with u ( k ) calculated according to the control law as discussed previously. As a result, the controller can detect and respond to the disturbances which cause 9 to deviate from yo. On the other hand, when ym= yothe controlled system can stay on the course of the prestarting operating conditions. Multiple-Effect Evaporator Example. The methods discussed in this paper are demonstrated by using a multieffect, kraft-pulping evaporator as the process to be controlled. A simplified diagram of the evaporator set is shown in Figure 3. The evaporator removes water from incoming weak black liquor (WBL) to achieve a specified concentration of nonvolatile ”solids” in the strong black liquor (SBL) product. Ricker and Sewell (1984) performed dynamic response tests on a typical kraft-pulping evaporation system and developed a nonlinear, lumped-parameter model to simulate the process dynamics. The model has 12 state variables, 4 manipulated variables, and 3 disturbance variables. The state variables are the liquor temperature and weight fraction of liquor “solids” in each of the six effects. The four manipulated variables are the steam flow rate, WBL feed rate, the fraction of WBL fed to the fourth effect (the rest is fed to the fifth effect), and the condenser cooling water flow rate. The three disturbance variables are the WBL solid content, its temperature, and the inlet temperature of the cooling water. The single output variable, y , is the weight fraction “solids” in the SBL product leaving the third effect. We assume the output solid content y can be measured exactly (i.e., E, = I). Since a linear, discrete-time model is required for the control law formulation, the nonlinear model was linearized and normalized at the nominal steady state. The resulting linear, continuous-time model was then changed to a discrete-time model using the state transition method (Takahashi et al., 1972). The sampling time period was 0.01 h. The response of the linear and nonlinear models to a 30% increase in the solid content of the WBL is shown in Figure 4. Note that the qualitative features of the two
Ind. Eng. Chem. Res., Vol. 28, No. 2, 1989 183 F
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