Ind. Eng. Chem. Res. 1988,27, 71-81
71
PROCESS ENGINEERING AND DESIGN Adaptive Inferential Control for Chemical Processes with Perfect Measurements Gwo-Chyau Shen a n d Won-Kyoo Lee* Department of Chemical Engineering, The Ohio State University, Columbus, Ohio 43210
A single-input/single-output(SISO) adaptive inferential control system is proposed to deal effectively with real process control problems characterized by unmeasured disturbances, nonlinearities, and/or time-varying process parameters. The basic structure of an inferential (internal model) control system is coupled with a n on-line parameter estimation method for estimation of the process model parameters and adjustment of the controller. To obtain a stable approximation to the inverse of the process model for use as the controller, the use of adaptive inverse modeling based on discrete convolution model is examined. Simulation results have shown that the adaptive inferential control system performs well with both minimum- and nonminimum-phase systems in the presence of unmeasured load changes and dead-time variations. In addition, Vogel-Edgar’s adaptive dead-time compensator and Pavlechko-Edgar’s pole-zero placement adaptive controller for convolution models are analyzed from the viewpoint of the inferential control and investigated for the adaptive inferential control. 1. Introduction Brosilow and his co-workers (Joseph and Brosilow, 1978; Brosilow and Tong, 1978) developed the inferential control for processes with meaurement limitation problems in the presence of unmeasurable disturbances. Since then, the Smith predictor has been analyzed (Brosilow, 1979) from the viewpoint of inferential control, where the effect of unmeasured disturbances on the process output is estimated and then counteracted by using a process model. Garcia and Morari (1982) adopted a similar approach in the development of Internal Model Control (IMC) and showed that a number of model-based controllers (e.g., Smith predictor, Dynamic Matrix Control, Model Algorithmic Control, etc.) are contained in the IMC control structure. In recent years, these model-based controllers have attracted considerable interest and attention as a powerful and versatile framework for process control system design. The inferential control system is designed and implemented by using a process model as shown in Figure 1. The inferential controller consists of an inverse of the process model and a fiiter with adjustable parameters. The controller filter is required to make the controller realizable and to make the closed-loop system robust to modeling errors (Brosilow, 1979; Garcia and Morari, 1982; Morari, 1983). The simplicity of the design procedure and tuning of the filter makes the inferential control system very attractive. However, the design of the controller and filter requires a good knowledge of the process. Even with an exact model of the process, the model inverse can lead to an unstable controller when the model contains unstable zeros. In this case, a factorization is needed to obtain a stable approximate inverse for the stable controller. Also, there are no practically simple filter design methods available even though Chen and Brosilow (1984) worked on the design of a robust filter to improve the control performance for uncertain processes. When the process is nonlinear and/or time-varying, the 0888-5885/88/2627-0071$01.50/0
filter dynamics has to be slow to satisfy a desired stability criterion, probably resulting in a fairly sluggish closed-loop response. Therefore, the nonlinear and often time-varying nature of a typical chemical process can degrade the performance of an inferential control system designed with a particular process model and a controller filter. The purpose of this article is to develop a robust adaptive inferential control algorithm which can cope with changing process dynamics and of which control performance is insensitive to modeling error. The approach adopted here is different from the previous works: an on-line parameter estimation method is coupled with the basic structure of an inferential control system for adaptation of the process model and the controller. In particular, the controller is designed by using a discrete deconvolution model which approximates the inverse of the process model and then adjusted through the use of an on-line identification method and the process model output. This idea is based on the “adaptive inverse controller” of Widrow et al. (1978, 1981) and Widrow and Walach (1983). In addition, Vogel-Edgar’s adaptive dead-time compensator (1980) and its modified controller for discrete convolution models, pole-zero placement adaptive controller (Pavlechko and Edgar, 1984),are analyzed from the viewpoint of adaptive inferential control and investigated since they are based on the structure of the Smith predictor and Dahlin’s controller which is equivalent to the inferential controller. Finally, simulation results of two numerical examples are presented to illustrate and compare the performance of the proposed adaptive inferential control system with those of conventional PID control, inferential control with fixed parameters, and Vogel-Edgar’s adaptive dead-time compensator. 2. Inferential Control (Internal Model Control)
The structure and design of the Smith predictor has 0 1988 American Chemical Society
72 Ind. Eng. Chem. Res., Vol. 27, No. 1, 1988 LOA0 OISTURBANCE
I+ SETPOINT
t
CONTROLLER
PROCESS
OISTURBANCE MOOEL
SETPOINT CONTROLLER
>
PROCESS
-I T I I@=
li
PROCESS MODEL
INVERSE CONVOLUTION MODEL PARAMETER
PROCESS
Figure 1. Inferential control structure (internal model control).
been analyzed from the viewpoint of an inferential control (Brosilow, 1979) as illustrated in Figure 1. This control scheme makes use of a process model to infer the effect of unmeasured disturbances on the process output and then counteracts that effect. A similar concept called Internal Model Control (IMC) was adopted by Garcia and Morari (1982) and Morari (1983). The theoretical properties of the inferential control are presented briefly in this section based on the above three works. The response of the process output y to a set point ys and a load disturbance L is given as
Figure 2. Adaptive inferential control.
Morari (1983), and Chen and Brosilow (1984) suggested adding a fiiter to the controller to compensate for expected modeling error. The design procedure is then as follows. (i) Factor the process model into two parts G,(s) = G,+(s)G;(s)
such that the inverse of G,-(s) is stable and causal and G,+(O) = 1. (ii) Combine G,-(s) with a filter to form the controller
GAS) = F(s)(G,,?s))-'
where G, is the process transfer function, G, is the process model, G , is the controller, and GL is the disturbance model. Ideally, the process model is a perfect representation of the real process and invertible, and then the controller is the inverse of the process model; i.e., G,(s) = G (s), and G,(s) = Gm-l(s). Under these unrealistic con&tions, we have, from eq 1, for all t > 0. That is, a perfect control could be obtained in the presence of any change in set point and load disturbances. However, realization of the perfect control is not likely to occur in practice for the following reasons. (i) A perfect model is seldom available, especially when the process is nonlinear and/or time-varying. Modeling error is more likely to occur in practice. (ii) The exact inverse of a process model can never be physically realizable on systems with dead-time and nonminimum-phase systems. Although the idea of perfect control cannot be realized in practice, the inferential control still has advantages. (i) If the steady-state gain of the controller is equal to the reciprocal of the steady-state gain of a process model, Gc(0) = G,-l(O)
(3)
then there will be no offset, i.e., limt-- y ( t ) = ya for any asymptotically constant disturbance and asymptotically constant set point. This can be seen easily from eq 1. (ii) The stability problem may be trivial if the process model is a sufficiently good approximation to the real process and if the controller and the process are stable. It was mentioned earlier that perfect control is not possible because of the problem of modeling error and the difficulty of realizing the exact inverse of the process model. Control performance may, therefore, deteriorate. In order to improve the performance of the inferential control system, Brosilow (1979), Garcia and Morari (1982),
(4)
(5)
Usually F(s) is of the type of a low-pass filter with an adjustable parameter. This filter plays an important role in improving the dynamic behavior of the entire control system. Methods of factorization of the process model and design of the filter were discussed in detail in the four works mentioned above and will not be given here. One important fact concerning the controller filter is that there exists a stabilizing filter with F(0) = 1 for the closed-loop system in Figure 1with the controller of eq 5 if and only if the steady-state gain of the model and the steady-state gain of the process have the same sign (Morari, 1983). Hence, this filter serves not only to compensate for modeling errors but also to provide a stabilizing element. 3. Adaptive Inferential Control As mentioned before, a controller filter is required in the inferential control system to make the controller realizable and to make the closed-loop system robust to modeling errors. In particular, in the absence of modeling errors, the filter determines the speed of response, and in the presence of modeling errors, the problem of robustness has to be addressed by adjusting the filter. The simplicity of the design procedure and tuning of the filter makes the inferential controller very attractive. However, the design of the controller and filter requires a good knowledge of the process. Even with a perfect representation of the real process, a factorization of the process model is required to obtain a stable approximate inverse for the stable controller, and there is no simple, practically efficient filter design method available. When the process is nonlinear and/or time-varying, the filter dynamics has to be slow to satisfy stability criterion, probably resulting in a fairly sluggish dynamic response. Therefore, in the presence of large modeling errors, it may not be advisable to use an inferential controller with fixed parameters. For adaptation of the inferential control system, an on-line parameter estimation method is used to estimate the process model parameters and to adjust the controller through the adaptive inverse modeling as shown in Figure 2. The adaptive inverse modeling process provides a stable approximate inverse of the process model as the
Ind. Eng. Chem. Res., Vol. 27, No. 1, 1988 73 controller by using a discrete convolution model and the process model output. Thus, the factorization of the process model is not required in the design of the inferential controller for nonminimum-phase processes. Various parameter estimation methods can be employed to update the model and controller parameters. However, the recursive least-squares method (RLS) is used because of its simplicity and reliable convergence. Also, the use of the well-known least-mean-squares algorithm (LMS) (equivalently, stochastic gradient algorithm) is examined because of its simplicity and widespread use in adaptive filtering applications. To account for expected modeling errors and improve the performance of the adaptive inferential control system, a first-order filter is added to the controller. Design details of the adaptive inferential control system are described below. 3.1. Process Model. A process model is required to estimate the effect of unmeasured disturbances on the process output and to design the inferential controller. Two types of models were used in this study, and their effects on the control performances were examined: transfer function model and convolution model. A general discrete transfer function model can be expressed by using a difference equation n
n
y ( k ) = - C a i y ( k - i) i=l
+ iC= lb , u ( k - i - m) + d
(6)
where y and u are the process output and input in terms of deviation variables, d is a bias term used to account for load disturbances, n is the model order, and m represents the dead time expressed as an integer multiple of the sampling period. If the bias term d is not included, the model parameters will be erroneously estimated when unmeasured load disturbances are present. However, special care needs to be taken to obtain correct model parameters when eq 6 is used with RLS, as to be mentioned later. To use a transfer function model, the model order has to be known a priori. In addition, if the exact dead time is not known but the minimum and maximum expected dead times are available, the model can be overparameterized to guarantee a fit by including a sufficient number of additional b terms in eq 6 (Vogel and Edgar, 1980; Chien et al., 1984). The second type of process model used is a discrete convolution model represented by 4
y ( k ) = C h ( i ) u ( k - m - i) i=l
+d
(7)
where h(i) (i = 1,2, ...,q ) represents the impulse response coefficients of the process. One advantage of the discrete convolution approach is that the model coefficients can be determined easily from simple experimental tests, such as from step-response data. The discrete convolution model has become more and more popular recently to represent process dynamics in the development of digital predictive control algorithms such as model algorithmic control, dynamic matrix control, and predictive control (Marchetti et al., 1983). The discrete convolution model is adopted in the present study for the following reasons. No assumption has to be made about the order of a process. The only knowledge required about the process is its approximate settling time which is useful in determining the number of coefficients to be used in the model. For processes with dead time, a similar modification used with the transfer function model can be employed. The disadvantage of using discrete convolution model with adaptive control is a sig-
nificant amount of computation work if the ratio of the process settling time to the sampling time is large, even when dead time is not present. 3.2. Controller. In the inferential control system, a stable approximation to the inverse of a process model is used as the controller. As a result, a factorization is sometimes required in conjunction with a filter with adjustable parameters. Factorization of the process model and design of the controller are discussed in detail elsewhere (Brosilow, 1979; Garcia and Morari, 1982; Morari, 1983; Chen and Brosilow, 1984). Our approach here is different from theirs. On the basis of the assumption that a reasonably good process model can be obtained by using the on-line parameter estimation technique, great emphasis is placed on obtaining a stable discrete inverse convolution (deconvolution) model on-line which approximates the inverse of the process model. This idea is based on the “adaptive inverse controller” of Widrow et al. (1978, 1981) and Widrow and Walach (1983). The adaptive inverse controller has the process output as an input to the discrete convolution model as s
u ( k - 6 ) = C y ( k - i)g(i)
(8)
1=1
Its weights, g’s, are adapted recursively to cause its output to be a best fit to the process input in the sense of least squares. A close fit implies that the cascade of the process and the inverse model have a “transfer function” of essentially unit value. Hereafter, the adaptive inverse controller will be called “adaptive deconvolution controller”. For nonminimum-phase process, an exact inverse can lead to an unstable controller because of the positive zeros of the process. However, since the adaptive deconvolution controller only realizes zeros, one can obtain a stable causal approximation to the inverse of a nonminimum-phase process by allowing a suitable time delay, 6, in the process input as in eq 8. Consequently, the price paid for control of a nonminimum-phase process using the adaptive deconvolution controller is a delayed control response (Widrow et al., 1978). Nevertheless, our simulation results have shown that such a time delay is not critical for obtaining a good deconvolution model for use in the adaptive inferential control system. A moderate number of parameters (approximately 5-10) coupled with a small or even zero amount of delay is shown to result in a satisfactory control performance for most of process models, including both minimum- and nonminimum-phase systems. The problem with nonminimum-phase processes provides the incentive for using discrete convolution, rather than transfer function model, for the inverse model. The adaptive inverse modeling process would, thus, eliminate the factorization of the process model required in the design of a stable controller for a nonminimum-phase process. This is also true when a convolution model is employed for the process model. Therefore, the design of an inverse controller is independent of types of the process models. To obtain and update the deconvolution model, the process model output with the bias term d being excluded (eq 6) is used instead of the process output for the following reasons. (i) In the presence of unmeasured load disturbances, the use of the process output yields an erroneous inverse which additionally accounts for a change in the process output occurring as a result of the load disturbance. (ii) It is known that offset can be eliminated if the steady-state gain of the inferential controller is the recip-
74 Ind, Eng. Chem. Res., Vol. 27, No. 1, 1988 rocal of that of the process model. Therefore, even if on-line estimation fails to provide accurate process model parameters due to load disturbances or other kinds of noises, adaptation of the deconvolution controller using the model output can still yield a zero offset in the process output. (iii) If the LMS algorithm is employed for updating the deconvolution controller, noise in the process output as an input to the controller model will cause biased estimates. Since noise does not exist in the process model output, it is definitely a better choice for adaptation for the deconvolution controller. When a sudden change in set point, unmeasured load disturbances, or process parameters occurs, estimated parameters of the process model will be in error for a period of time, which in turn causes some errors in the adaptive deconvolution controller. As a result, the control performance may be poor or even “momentarily” unstable during this transient period. To prevent such an undesirable situation, the controller is modified in two ways. Firstly, to further ensure that the steady-state gain of the deconvolution model is a reciprocal of that of the process model, the controller is modified by the following equation in 2 domain:
where G,(z-’) is the process model and GI(z-l) is the deconvolution controller. This modification guarantees a zero offset. Secondly, the controller output is passed through a simple first-order filter: u,(k + 1) = (1.0 - u ) u , ( ~+ ) uu(k + 1) (10) where u is an adjustable constant, u is the controller output, and u, is the filtered controller output which is actually used as a control input to the process. In addition to accounting for expected modeling errors, this filter also serves as a “detuner” to prevent excessive control action. These two modifications have been shown to significantly improve dynamic behavior of the adaptive inferential control system. 3.3. Parameter Estimation Algorithms. Various well-developed on-line parameter estimation methods are available (Isermann et al., 1974). However, a recursive least-squares (RLS) method is chosen here for parameter estimation because of its simplicity and reliable convergence. The equations for the recursive least squares are
covariance matrix when adaptation is required. One advantage of the random walk is that the individual elements of the covariance matrix can be separately affected by proper choice of R. Thus, the random walk method is a more selective method but requires more care in its application. Since the RLS algorithm tries to match the model output (including a bias term) with the process output by updating the process parameters and the bias term, any error in estimating the load disturbance will result in incorrect estimates of all the other parameters, which in turn cause modeling error. In the present study, emphasis is placed on the capability of the system to identify unmeasured load disturbances. A cautious use of the random walk was found helpful in estimating the load disturbance. This will be mentioned later in discussing simulation results. When a sudden change occurs in the process, e.g., setpoint changes or unmeasured load disturbances, new parameter estimates will be in error for a period of time. As a result, updating the controller based on erroneous estimates for the model parameter may lead to undesirable control performance. Therefore, Vogel and Edgar (1980) suggested passing all the model parameters through a first-order filter: e,(k + 1) = (1.0- a ) e c ( k )+ cue(k+ 1) (12) where 0 is the vector of updated process parameters and 0, is the vector of filtered process parameters which is used for the process model. Another parameter estimation method examined in this study is the LMS algorithm by Widrow et al. (1976). This LMS method is only used for adaptation of the control system with a convolution process model. The LMS algorithm is a simple form of stochastic gradient technique with nondiminishing adaptation gain:
+
eT(k + 1) = eT(k) 2 , 4 k ) z ( k ) (13) where 0 is the parameter vector and Z is the process input vector. The second term on the right-hand side is a correction term, E&) is the difference between the process and model outputs a t kth sampling time, and p , a scalar parameter, is a convergence factor which controls stability and rate of adaptation. For convergence, it is required that (14) 1,’Amax >P >0 where A, is the largest eigenvalue of the positive definite input correlation matrix, R, defined as -
1 - miu(k - 1 - m ) 2 - miu(k - 1 - m )
u(k - 1 - m)u(k - 2 u(k 2 - m ) u ( k - 2
- m) - m)
R = E u(k
where Zk is the vector of input/output data, Bk is the parameter vector, and Pk is the covariance matrix whose diagnoal elements determine the variability of the corresponding elements in the parameter vector (ek). In implementation of the RLS method, a forgetting factor p and a random walk R are utilized as shown in eq 11. The foregetting factor ( p ) is used to prevent the elements of the covariance matrix from becoming too small such that “loss of sensitivity” of parameter adaptation does not occur. This amounts to building a finite memory into the recursive least-squares algorithm. The random walk (or covariance resetting) (Wellstead and Zanker, 1982) approach adds a constant positive definite matrix R to the
I
(15)
-
n - m)uik -
rI -
m)
Because A, is rarely known in practice, eq 14 is not easy to apply. Since trR > A, as R is positive definite, a sufficient condition for convergence is l/trR > p > 0 (16)
A comparison of eq 13 to eq 11 immediately shows that much less computation effort is required with the LMS than the RLS. This advantage makes the LMS algorithm a very attractive candidate for on-line parameter estimation, particularly in the case of a convolution model with a large number of coefficients. In addition, only one parameter, the convergence factor, p , needs to be adjusted once the number of terms in the convolution model has been determined.
Ind. Eng. Chem. Res., Vol. 27, No. 1, 1988 75
CONTROLLER
PROCESS
I
Figure 4. Reconstructed Vogel-Edgar's adaptive dead-time compensator from the viewpoint of adaptive inferential control.
Figure 3. Vogel-Edgar's adaptive dead-time compensator.
However, convergence speed is a problem with the LMS algorithm. Generally speaking, large convergence factor yields fast adaptation but also leads to a large error in estimation. Small convergence factor can give more accurate estimate but has low convergence speed. Therefore, a compromise has to be made in determining the magnitude of the convergence factor. Detailed theoretical analysis of the LMS algorithm can be found in Widrow et al. (1976). Another problem with application of the LMS algorithm to the adaptive inferential control system is that an estimate for the load disturbance is usually of very low accuracy which in turn results in a poor control performance, as will be seen later in simulation results.
4. Analyses of Vogel-Edgar's Adaptive Dead-Time Compensator and Pavlechko-Edgar's Pole-Zero Placement Adaptive Controller 4.1. Vogel-Edgar's Adaptive Dead-Time Compensator. Vogel and Edgar (1980) used the basic structure of the Smith predictor (Smith, 1957) and Dahlin's controller (Dahlin, 1968) in development of the adaptive dead-time compensator to cope with the problems of changing process parameters and variable dead time, as shown in Figure 3. Later, the adaptive dead-time compensator was modified to provide effective control of processes described by discrete convolution models (Pavlechko and Edgar, 1984). Since the Smith predictor has been analyzed from the viewpoint of inferential control and Dahlin's controller uses the inverse of a process model with an adjustable filter for the desired dynamic characteristics, there should be a strong resemblance between the adaptive dead-time compensator and the adaptive inferential control system. For a second-order plus dead-time process model,
+b,~-~)z-~ 1+ + u2z-2
(biz-' G,(z-') =
u1z-1
(17)
the predictive model is employed in their structure:
where the dead-time effect has been considered implicitly by using an appropriate value for r. GD(2-l) in Figure 3 is selected in such a way that the product of G'p,(z-l) and GD(2-l) yields the correct expression of the discrete transfer function model for the process r
2 biz-' G'pr(z-')GD(z-l) = G,(z-')
=
i=l
1 + u,z-l
+ u2z-2 (19)
The process model G,(z-') is equivalent to an overparameterized discrete transfer function for a process with dead-time variations. The resulting Vogel-Edgar's adaptive dead-time compensator can be reconstructed from the viewpoint of adaptive inferential control as illustrated in Figure 4. The structure of the controller in Figure 4 is the same as that in Figure 1 if the controller and the predictive model G'pr are combined:
1+
i=l
1
+ ulz-1 + u2z-2 (20)
Since Dahlin's algorithm is employed in Gc(z-l), we have G,(z-') = (1 - exp(T/X))z-("l)
-- 1
1 - exp(-T/X)z-l - (1 - exp(-T/X))z-(N+') G 'Pr (z-l)
-
i=l
where T is the sampling period, X is the tuning constant in Dahlin's algorithm, and N , the largest integral number of sampling periods in the dead time, is zero because the predictive model GtPr is used for the process model. Substitution of eq 21 into eq 20 yields the controller 1 GI(2-I)
=
+ ulz-l + u ~ z (1 - ~- exp(-T/X))z-l
(kbiIZ-,
1 - exp(-T/X)z-'
(22)
i=l
This controller in the inferential control structure corresponds to the inverse of the predictive model of eq 18 plus a first-order filter with an adjustable constant. The controller can be also viewed as a combination of a first-order filter and a discrete convolution model, approximating the inverse of the process model of eq 19. Thus, it is clear that the adaptive inferential control system and the Vogel-Edgar's adaptive dead-time compensator are structurally equivalent to each other. However, the latter is unique in obtaining the stable approximation to the inverse of the process model and updating it: (1)the inverse of a process model as the controller can be obtained directly from updated parameters of the process model without factorization, and (2) the problem with nonminimum-phase systems is resolved since process zeros do not appear as poles of the controller.
76 Ind. Eng. Chem. Res., Vol. 27, No. 1, 1988 It is also important to point out that "from the inferential control point of view, one is estimating the effect of the disturbances on the outputs and then taking a control action to counter that effect" (Brosilow, 1979), while the viewpoint of Vogel-Edgar's dead-time compensator is to modify the structure of the Smith predictor so that the effect of variable dead time can be coped with effectively. 4.2. Pavlechko and Edgar's Pole-Zero Placement Adaptive Controller. Pavlechko and Edgar (1984) modified the Vogel-Edgar controller to provide effective control of processes described by discrete convolution models in the development of their pole-zero placement adaptive controller. This pole-zero placement controller can be analyzed in a similar way as done with Vogel-Edgar's controller. It is shown briefly below. If G,(z-') represents a discrete process model, either transfer function or convolution model, Pavlechko and Edgar's pole-zero placement controller can be represented by using Figure 4 in which
Table I time 0.0 50.0
150.0 200.0 300.0 350.0 500.0 600.0 650.0 750.0 800.0 900.0 950.0
set Doint 0.0 0.5 0.0 0.0 0.5
load disturbance 0.0 0.0 0.0 0.5 0.5
0.5 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.5 0.5 0.0 0.0
0.5 0.0
0.0 0.5 0.5 0.0
1100.0
dead time 2.0 2.0 2.0 2.0 2.0 2.0 2.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0
and the controller is
which is a modified version of Dahlin's algorithm. Combining eq 2 3 and 24, we have
xi
:I 3 3C
3
3 34
0-5,
0 68
d 85
03
I/ 20
-103
Figure 5. Closed-loop response from system 1 with PID control.
both of them can be viewed as one type of adaptive inferential control scheme. 1
(1- exp(-T/X))z-'
Gm(l)z-' 1 - exp(-T/X)z-'
y(2-I) .__
y(1)
(25)
On the basis of the inferential control structure, the first term on the right-hand side of eq 25 is an approximate inverse of the process model and the second term is a first-order filter with an adjustable constant which is the same as in eq 22. Since the approximate inverse model is oversimplified and may not yield desirable dynamic behavior, a second filter y(z-')/y(l), which is of the type of convolution model, is added to the controller to compensate for possible modeling errors and improve the control performance. Thus, the coefficients of y(2-l) serve as additional tuning parameters for the controller. It is, however, interesting to point out that eq 25 can be reduced to eq 22 by choosing y(z-') = 1 + cz,z-' + when a discrete transfer function model is used for Gm(z-l). A further comparison of eq 25 with eq 9 and 10 reveals that the y(z-') used in the pole-zero placement controller is equivalent to the deconvolution model in the proposed adaptive inferential controller and that eq 25 contains a term corresponding to the first-order filter of eq 10 used in the adaptive inferential controller. However, the parameters of y ( ~ - ' )have to be tuned off-line for a desired dynamic response, while the coefficients of the discrete deconvolution controller are updated on-line. Thus, the proposed adaptive inferential control system is more efficient than the pole-zero placement adaptive controller, especially when the process parameters are time-varying. Since both the dead-time compensator and the pole-zero placement controller use on-line parameter estimation technique to update process parameters and that they essentially have the structure of the inferential controller,
5. Simulation Results To test performance of the proposed adaptive inferential control scheme, it has been applied to many simulated processes subject to changes in set point, load disturbance, and dead time. The simulation work was performed on a VAX 111780 minicomputer using Advanced Continuous Simulation Language (ACSL) as the simulation tool. For the sake of brevity, only simulation results of two systems, one second-order nonminimum-phase system and one fourth-order minimum-phase system, are presented in this paper. 5.1. System 1. A second-order plus dead-time nonminimum-phase process with transfer function
+ 1) exp(-ks) (3s + 1)(5s + 1)
(-s
G,(s) =
with a disturbance model 0.5 exp(-ks) =
(3s
+ 1)(5s + 1)
is used. In order to compare the performance of several different controllers, step changes in set point, load, and dead time were introduced as given in Table I. The performance of a PID controller, the inferential control, the adaptive inferential control, and the VogelEdgar controller are shown in Figures 5-18. Figure 5 shows that the control performance with a well-tuned PID controller (gain of 0.4, integral time of 5.5, and derivative time of 0.01) deteriorates when the dead time changes. Since a process is nonminimum phase, the exact inverse of the process model cannot be used as a controller.
Ind. Eng. Chem. Res., Vol. 27, No. 1, 1988 77
. al-C IC
SI
i
3 3a
2-51
0 68
3 85
* c3
. 03
t.ZC
Figure 6. Closed-loop response for system 1with inferential control.
Figure 7. Closed-loop response for system 1 with adaptive inferential control with convolution model and RLS.
Through the use of a factorization of the process model and addition of a filter as given in Morari (1983), the inferential controller is designed as (-s + 1) exp(-ks) Gp(s) = (3s + 1)(5s + 1) (-s 1) exp(-ks) s+l (28) s+l (3s 1)(5s + 1) GP%) term GJs) term
+
+
Because of a change in the dead time a t t = 600, a large value of the filter time constant is used to prevent instability, even though relatively fast filter dynamics could be employed to provide a tight control before the change occurs. As shown in Figure 6, the inferential controller designed with the perfect model and a filter time constant of /3 = 5.0 performs better than the PID controller. But when the dead time changes, the dynamic response of the inferential control is degraded and shows a sluggish setpoint response. The adaptive inferential control system is investigated by using two different process models, transfer function and convolution models, and two different parameter estimation schemes, RLS and LMS methods. It should be noted that a discrete convolution model is used as the controller and its parameters are updated by using the adaptive inverse modeling regardless of the process model type. For all the simulations in this example, six coefficients are used in the controller, and a time delay of six sampling times is employed for inverse modeling with a sampling period of one time unit. Control interval is the same as the sampling period. With the convolution process model, 40 coefficients are used including the bias term. A pseudorandom binary sequence (PRBS) of a magnitude of 0.01 was added to the controller output as part of the identification process. Since the estimation of load disturbance is of significant importance to the adaptive inferential control system, relative magnitudes of the diagonal elements of the random walk matrix need to be selected based on relative variation rates of load disturbance and process parameters. If a sudden load disturbance (fast dynamics) is likely to occur, a larger magnitude of the random walk element corresponding to the bias term is preferred. Because a continuous use of the random walk causes the covariance matrix to increase indefinitely, the random walk is invoked only when the model output (including the bias term) deviates from the process output by a prespecified value
91 ?J 0
00.00
, 0.17
, 0.34
,
, 0.m
0.31 I
, 0.85
,
, 1.05
1.m
-103
Figure 8. Load estimation for system 1 with adaptive inferential control with convolution model and RLS.
“1 1
I:
0
0
21 0.00
0 17
0 34
d
0 68
51
-
3 85
I
03
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Figure 9. Comparison of bias term and load disturbance for system 1with adaptive inferential control with convolution model and RLS.
and the trace of the covariance matrix is smaller than another prespecified value. Since modeling error is inevitable, a first-order filter is added to the controller to ensure robustness of the control system. In the present study, a filter constant close to 0.2 was used in the controller filter to improve the controller performance. The simulation results are presented in Figures 7-9. Initially, all coefficients in both the convolution process model and the deconvolution controller were set to zero. After a period of learning process, all the process model and controller parameters attain “correct” values and the
78 Ind. Eng. Chem. Res., Vol. 27, No. 1, 1988 8
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8
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x, - 8-O l l t
8
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2
~
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Figure 10. Closed-loop response for system 1 with adaptive inferential control with transfer function model and RLS. 2
s
Figure 11. Load estimation for system 1 with adaptive inferential control with transfer function model and RLS. D m
O n
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d 68
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Figure 13. Closed-loop response for system 1 with adaptive inferential control with convolution model and LMS. : : 5 1
Figure 14. Load estimation for system 1 with adaptive inferential control with convolution model and LMS. 0 o
0 m
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Figure 12. Comparison of bias term and load disturbance for system 1with adaptive inferential control with transfer function model and RLS.
- -103 Figure 15. Comparison of bias term and load disturbance for system 1 with adaptive inferential control with convolution model and LMS.
control system performs very well, even in the presence of a change in the dead time, as illustrated in Figure 7. Figure 8 compares the actual load disturbance imposed on the process output to the load estimation obtained by subtracting the model output from the process output. The bias term in the convolution model provided a reasonably accurate estimate of the unmeasured load disturbance as shown in Figure 9. With the transfer function process model using two output terms and six input terms in eq 6, the adaptive inferential control yields about the same performance but has a poorer initial set-point response than with the con-
volution process model as shown in Figure 10. The difference between the process output and the transfer function model output is found to provide a very good estimate of the unmeasured load disturbance as shown in Figure 11. However, the bias term in the transfer function process poorly estimated the load as illustrated in Figure 12. The proposed adaptive inferential control with the convolution process model performs poorly as illustrated in Figures 13-15 when the LMS algorithm is used for estimating parameters of the model and controller. In implementing the LMS algorithm, off-line calculations
Ind. Eng. Chem. Res., Vol. 27, No. 1, 1988 79
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4 2 0.17
0.34
1
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0.51
7 ~ 0.85 l I ' . B
- -103 Figure 16. Closed-loop response for system 1 with Vogel-Edgar's adaptive dead-time compensator.
240.00
0.17
d.34
451 I
d.w
0.85
1.w
1.m
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Figure 17. Load estimation for system 1 with Vogel-Edgar's adaptive dead-time compensator.
were performed first to obtain relatively reasonable values for all the coefficients because of its low convergence speed. Thus, the LMS algorithm may not be suitable for use in the proposed control system. Both Vogel-Edgar's adaptive dead-time compensator and Pavlechko-Edgar's pole-zero placement adaptive controller have been analyzed here from the viewpoint of adaptive inferential control and are shown to be equivalent to the proposed adaptive inferential control system. Since the former controller is updated directly from the estimated process model parameters, it would require less computational effort. Its simulation results are shown in Figures 16-18, which are equivalent to those of the adaptive inferential control with the transfer function process model as shown in Figures 10-12. This example demonstrates that the proposed adaptive inferential control system can work quite well with nonminimum-phase processes in the presence of dead-time variations and load changes without requiring the factorization. 5.2. System 2. To further demonstrate an effective control performance of the proposed adaptive inferential control system, a fourth-order plus dead-time process is considered 2.25(s + l)(s + 2) exp(-ks) G,(s) = (30) (s + 0.5)(s + 1.5)(s2+ 2.5s + 6) and the disturbance model is assumed to be 2.25 exp(-ks) G L ~ =) (31) (s + 0.5)(s + 1.5)(s2+ 2.5s + 6)
d.17
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6.34
0.51
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Figure 18. Comparison of bias term and load disturbance for system 1 with Vogel-Edgar's adaptive dead-time compensator.
5/ 51 qi
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-103
Figure 19. Closed-loop response for system 2 with adaptive inferential control with convolution model and RLS.
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Figure 20. Closed-loop response for system 2 with adaptive inferential control with transfer function model and RLS.
This system, as demonstrated by Zafiriou and Morari (Pavlechko and Edgar, 1984), poses great difficulty for the Vogel-Edgar controller when a convolution model is used. With the same changes in set point, load, and dead time as applied to system 1,the performances of the adaptive inferential control with transfer function and convolution process models and the Vogel-Edgar controller are presented in Figures 19-21, respectively. For all the simulations in this example, all the coefficients are initially set to zero. As can be seen, they yield comparable results. However, it is further demonstrated in this example that
80 Ind. Eng. Chem. Res., Vol. 27, No. 1, 1988
vide effective control of convolution process models without having to tune additional parameters off-line, compared to Pavlechko-Edgar’s pole-zero placement adaptive controller.
Acknowledgment Partial fiiancial support from the Office of Research and Graduate Studies of The Ohio State University under the University Seed Grant Program is gratefully acknowledged.
Nomenclature
Figure 21. Closed-loopresponse for system 2 with Vogel-Edgar’s adaptive dead-time compensator.
the adaptive deconvolution controller provides effective control of processes described by the convolution model without having to tune additional parameters off-line, compared to Pavlechko-Edgar’s pole-zero placement adaptive controller. Also, the use of convolution models improves the robustness of the adaptive inferential control system as compared to the use of transfer function models since the exact order of the model of the process need not be known. 6. Conclusion Use of adaptive deconvolution controller is proposed in conjunction with on-line identification of the process model in the development of an adaptive inferential control system to improve the robustness of the inferential control. This adaptive deconvolution controller eliminates a factorization required in the design of an inferential controller. The proposed SISO adaptive inferential control is shown to work well with both high-order minimum and nonminimum systems with unknown or varying dead times, provided that the design parameters are suitably chosen. Either a transfer function or convolution model can be employed for the process model. But with the convolution process model, the LMS algorithm yields poorer control performance than the RLS method, indicating that the LMS may not be suitable for use in the proposed control system. Vogel-Edgar’s adaptive dead-time compensator and Pavlechko-Edgar’s pole-zero placement adaptive controller for convolution models are analyzed from the viewpoint of adaptive inferential control and shown to be equivalent to the proposed adaptive inferential control system. Simulation results of the two numerical examples show that the adaptive inferential control designed with a convolution process model exhibits better control performance with minimal process knowledge than the Vogel-Edgar controller. It is, however, shown that the Vogel-Edgar controller yields comparable results to those of the proposed control system with a transfer function process model. With the transfer function model, the Vogel-Edgar controller would be more computationally efficient, since the controller is updated explicitly from the estimated model parameters. The disadvantage of using the proposed control system with a convolution process model is that adaptation with the RLS scheme becomes computationally intensive because of the large number of terms in the model. Nevertheless, the adaptive deconvolution controller can pro-
a, b = coefficients for model transfer function d = bias term in discrete model transfer function F = filter transfer function G, = controller transfer function GD = part of process model used in V-E controller GL = load disturbance transfer function G , = process model transfer function G = process transfer function G$, = predictive model g = discrete deconvolution model coefficient h = impuse response coefficient L = load disturbance N = largest integral number of sampling periods in process dead time n = order of discrete model transfer function P = covariance matrix q = order of discrete convolution model R = random walk matrix; input correlation matrix r = order of overparameterized model s = order of discrete deconvolution model u = process input, controller output u, = filtered controller output y = process output ys = process output set point Z = input/output data vector
Greek Symbols a = filter constant for estimated model parameters
p = filter constant for inferential controller
6 = delay used in discrete deconvolution model e = difference between process and model outputs for LMS y = filter for Pavlechko and Edgar’s pole-zero placement
controller 8 = model parameter vector X = eigenvector of input correlation matrix R; tuning constant in Dahlin’s algorithm I.( = convergence factor for LMS p = forgetting factor for RLS = filter constant for controller output
Literature Cited Brosilow, C. B. Proceedings of Joint Automatic Control Conference, Denver, CO, 1979. Brosilow, C. B.; Tong, M. AZChE J . 1978, 24, 492-500. Chen, S. C.; Brosilow, C. B. Presented at the AIChE National Meeting, Anaheim, CA, May 1984; paper 18b. Chien, I.-L.; Seborg, D. E.; Mellichamp, D. A. Proceedings of American Control Conference, San Diego, CA, 1984; paper TP3-6. Dahlin, E. B. Znstru. Control Syst. 1968, 41, 77-83. Garcia, C. E.; Morari, M. Znd. Eng. Chem. Process Des. Deu. 1982, 21, 308-323. Isermann, R.; Baur, U.; Bamberger, W.; Kneppo, P.: Siebert, H. Automatics 1974, I O , 81-103. Joseph, B.; Brosilow, C. B. AZChE J . 1978, 24, 485-492. Marchetti, J. L.; Mellichamp, D. A.; Seborg, D. E. Znd. Eng. Chem. Process Des. Deu. 1983,22, 488-495. Morari, M. Preprints of 5th IFAC/IMEK Conference on Instrumentation and Automation in the Paper, Rubber, Plastics, and Polymerization Industries, Antwerp, Belgium, 1983. Pavlechko, P. D.; Edgar, T. F. Proceedings of American Control Conference, San Diego, CA, 1984; paper TP3-1.
Znd. Eng. Chem. Res. 1988,27, 81-85 Smith, 0. J. M. Chem. Eng. Prog. 1957,53(5), 217-219. Vogel, E. F.; Edgar, T. F. Proceedings of Joint Automatic Control Conference, San Francisco, CA, 1980; paper TP5-E. Wellstead, P. E.; Zanker, P. Opt. Control Appl. Methods 1982, 3, 305-322. Widrow, B.; McCool, J. M.; Larimore, M. G.; Johnson, C. R. Proc. IEEE 1976, 64, 1151-1162. Widrow, B.; McCool, J. M.; Madoff, B. P. Conf. Rec. of 12th Asilomar Conference on Circuits, Systems and Computers, Asilomar, NOV1978; pp 90-94.
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Widrow, B.; Shur, D.; Shaffer, S. Conf. Rec. of 15th Asilomar Conference on Circuits, Systems and Computers, Nov 1981; pp 185-189. Widrow, B.; Walach, E. Preprints of IFAC Workshop on Adaptive Systems in Control and Signal Processing, San Francisco, CA, 1983. Received for review April 17, 1986 Revised manuscript received August 24, 1987 Accepted September 24, 1987
Oxygen Enrichment by Pressure Swing Adsorption Anthony S. T. Chiang,* Mau-Yeuh Hwong, Ting-Yueh Lee, and Tsao-Wen Cheng Department of Chemical Engineering, National Central University, Chung-Li, Taiwan, R.O.C. 32054
An automated experimental apparatus was build to study the PSA oxygen enrichment process. Mass flux and concentration of all streams connected to the adsorption bed were continuously monitored so that the transient behavior of the process could be examined. In particular, a four-step process operated under various production, pressurization, and blowdown rates was studied in detail. The experimental data were then compared with the simulation results of a simple, isothermal model. The adsorption isotherm and t h e mass-transfer rate constants used in the model were taken directly from literature data. T h e simulated and the experimental transient results were in reasonable agreement when the purge step was long, but for cases with a short purge step, a slower than predicted desorption rate was observed. T h e thermal effects not considered in the model were believed t o be responsible for this difference. The principle of oxygen enrichment from air by a PSA process is relatively simple. Typical adsorbate includes zeolites A and X and mordenite. Compressed air enters the adsorption bed a t a high pressure. Since nitrogen is adsorbed appreciably more than oxygen at high pressure, oxygen-enriched air can be easily obtained at the outlet. The bed is then blown down to a lower pressure to desorb the nitrogen in zeolite. A purge step may sometimes be used to sweep the nitrogen out of the bed before a new cycle begins (Skarstrom, 1972). There has been a variety of mathematical models to describe a PSA process. Early models were mostly based on a local equilibrium assumption. For example, Shendalman and Mitchell (1972) developed a model assuming a linearly adsorbed trace component at local equilibrium conditions. Weaver and Hamrin (1974) assumed a constant separation factor and local equilibrium conditions to study the separation of hydrogen isotopes. These models were however only good for a purification process. An important difference between purification and bulk separation is that the gas velocity in bulk separation will vary along the length of the column due to the adsorption. This change of gas velocity was first considered by Fernandez and Kenney (1983). Recently, Cheng and Hill (1985) studied the separation of He and methane. A finite mass-transfer rate was considered in addition to the change of gas velocity due to adsorption. Raghavan et al. (1985) further improved the model by taking the pressure dependency of the mass transfer rate into consideration. Cen et al. (1985) and Yang et al. (1985) have developed models that considered surface diffusion or pore diffusion as the mass-transfer controlling step. Most of the above-mentioned studies compared the respective model prediction with their experimental results. However, except for the work of Yang et al. (1985), the comparison was always made based on the overall performance of the process. An overall index such as an average product concentration or a steady-state production
.
rate might be adequate for design purposes, but it could very unlikely be used to differentiate the detail mechanism of a dpamical model. Therefore, if one wanted to extend the range of applicability of a particular model, transient data must be obtained. The prediction of a model could then be examined in detail. For example, the studies by Kotsis and Argyelan (1981, 1982, 1983) and Argyelan and Kotsis (1982) found that for short cycles, the process might be limited by the macropore diffusion rate in the zeolite pellets. The adsoprtion rate of oxygen was found to be twice that of nitrogen for the 5A zeolite pellets they used. The limited experimental results of Fernandez and Kenney (1983) indicated that when the half cycle time was longer than 40 s, a isothermal local equilibrium model could give a satisfactory prediction. It should be important to define the range of operation conditions under which a simple model is adequate. It would also be useful to find the particular assumption that should be removed to extend the model ability. In this study, an apparatus was designed in which the flow rates and the concentrations of all streams connected to the adsorption bed were continuously monitored. With this apparatus, the transient behavior of a PSA oxygen enrichment process could be followed closely. These transient results were then compared with the prediction of a simple model and used to demonstrate the correctness or the inadequacy of some assumptions made to derive the model. One important application of oxygen-enriched air is in the field of combustion. Boiler feed with such air can reach a higher flame temperature and a more complete burning of the fuel. Oxygen-enriched air has also been used in the steel industry to increase the efficiency of blast furnaces. However, a feed air too high in oxygen may overincrease the furnace temperature and damage the firebrick. Therefore, only a small percentage of oxygen enrichment should be used. It was anticipated that the operation conditions of a PSA process to produce such a low en-
0888-5885/88/2627-0081$01.50/0 0 1988 American Chemical Society
