Estimation of variance reduction opportunities through cascade control ([dagger]).

Lakshminarayanan, S. ; Hermanto, M.W. ; Goradia, D.B. 等

We describe an approach that is useful in deciding if significant benefits, in terms of control loop performance index (through variability reduction), will be achieved by a change in control loop configuration from simple feedback (SFB) to cascade control. The problem is considered in a stochastic setting and solved using the variance decomposition technique. The proposed methodology requires only routine operating data from an existing simple feedback control loop and knowledge of the process delays. Several simulation examples and one experimental case study exemplify the utility of this approach.

La methode decrite dans ce travail est utile pour decider si des avantages importants en termes d'indice de performance de la boucle de controle (par reduction de la variabilite) peuvent etre obtenus par un changement de la configuration de la boucle de controle pour passer d'une retroaction simple (SFB) a un controle en cascade. Le probleme est considere dans un cadre stochastique et resolu par une technique de decomposition en variance. La methodologie proposee ne requiert que des donnees operatoires de routine pour une boucle de controle a retroaction simple et de connaitre les delais de procede. Plusieurs exemples de simulation et une etude de cas experimentale illustrent l'utilite de cette approche.

Keywords: variance decomposition, control loop performance assessment, performance monitoring, minimum variance control, cascade control

INTRODUCTION

Cascade control is probably the single most important performance enhancement strategy over simple feedback loops. The potential improvements in performance and the ease of its implementation have led to its widespread use in the chemical process industries for over five decades now. Using extra output measurement(s), in addition to the primary controlled variable, the cascade control scheme provides timely and calculated adjustment of the manipulated variable thereby decreasing the peak error as well as the integral error for disturbances affecting the process. The efficiency of the cascade control schemes in handling disturbances entering the inner loop has been well documented in several research articles and textbooks. What is relatively less appreciated is the fact that cascade control provides better performance, as compared to the single loop case, for all types of load changes. While the improvement for disturbances entering close to the process input (i.e. secondary disturbances) can be 10 to 100 fold, the improvement in performance for disturbances entering late into the process (i.e. primary disturbances) is about 2 to 5 times (Webb 1961; Harriott, 1984). The performance improvement can be in terms of metrics such as peak value of response or integral absolute error when the process is subject to step type disturbances. Marlin (2000) provides an excellent review of the principles of cascade control, details the criteria for cascade design and shows several industrial examples. It is shown that the cascade scheme provides practical benefits only if the secondary process is at least three times faster than the primary process even for disturbances entering the inner loop. Krishnaswamy et al. (1990) relate the benefits afforded by cascade control to the parameters of the primary and secondary process models in a deterministic setting. The role of integral action in the secondary (slave) controller has also been investigated by Krishnaswamy and Rangaiah (1992).

Industrial control loops are designed and implemented in order to achieve specific objectives. It is important to monitor the performance of these loops periodically and make sure they are providing the best possible performance. In this regard, the performance monitoring of control loops has received much attention in the last decade. Many researchers have used the minimum variance controller (MVC) performance as the benchmark; this benchmark is appropriate if the goal of control is the reduction of the variance in the controlled variable (the variance of the manipulated variables, the complexity of the MVC or its robustness is not of concern). Harris (1989) showed that the minimum variance achievable, with a MVC, can be computed from routine operating data if the process time delay is known. Since then, there has been a multitude of research articles that consider important extensions (Desborough and Harris, 1992; Stanfelj et al., 1993; Huang et al., 1997), alternate benchmarks (Tyler and Morari, 1996; Kendra and Cinar, 1997; Swanda and Seborg, 1999), applications (Thornhill et al., 1999) and industrial perspectives (Kozub, 1996; Desborough and Miller, 2001) on this topic. The user is also referred to the exceptional coverage provided by Huang and Shah (1999) and Qin (1998) to this topic. Recently, Agrawal and Lakshminarayanan (2003) described a method to determine the control loop performance achievable with PI type controllers, the optimal control settings that will yield the best performance and the expected robustness margins using closed loop transfer functions identified from closed loop experimental data.

Desborough and Harris (1993) devised a procedure to separate the variance contributions into components related to the controller and the disturbances by developing an analysis of variance technique. Here, the overall variance is decomposed into certain components and conclusions on the performance of the control strategy are made by analysis of the component variances. Vishnubhotla et al. (1997) applied this variance analysis method to investigate the need for feedforward control on data sets provided by Shell, USA. Ko and Edgar (2000) established the basis of performance assessment of cascade loops. In their work, multivariate time series modelling of data on primary and secondary controlled variables from a series cascade control loop was used to determine the minimum variance possible in the primary controlled variable under the series cascade control strategy. Knowledge of the time delays in the primary and secondary process is required in addition to the data on primary and secondary controlled variables. Chen et al. (2005) performed a similar computation for the parallel cascade control strategy. In addition to calculating the minimum variance for the parallel cascade configuration, they also computed the achievable minimum variance if the master and slave controllers are restricted to the PID type. Their work also showed that the parallel cascade strategy can be superior to the series cascade control strategy in terms of achieving minimum variance in the primary controlled variable.

The study here is related to series cascade loops. The scenario we consider is as follows: we have a process that is presently regulated by a simple feedback (SFB) controller. A control loop monitoring tool has flagged this loop as poorly performing when compared to MVC. We take a closer look at this loop and assess if the loop is performing to its full potential by taking into consideration factors such as the restricted structure of the controller; this is important because PID type controllers that are so common in the chemical process industries cannot provide minimum variance performance under many practical situations. Thus, any retuning exercise of the feedback controller in the SFB loop would have taken us nowhere. In such situations, better control performance can be achieved by: (i) making process modifications or (ii) changing the controller structure. Considering the second option, two obvious enhancements to the SFB scheme are feedforward and cascade control. Feedforward control is more appropriate when measured disturbances are available. Cascade control is suited when suitable secondary measurements (the secondary measurement must be influenced by the manipulated variable; it must also have a direct impact on the primary variable) are available. Feedforward scheme is not an option due to our assumption that the disturbances are not measured. Cascade control scheme would therefore be the obvious choice. The challenge taken up here is to use routine operating data from the SFB control system and estimate the extent of output variability reduction possible with the cascade control scheme. Such a scenario was mentioned in Stanfelj et al. (1993) but has remained unsolved in the literature. This work aims to resolve this gap through decomposing the overall variance into meaningful components. It must be emphasized that past performance assessment works on cascade loops have worked on a particular control structure--series cascade scheme (Ko and Edgar, 2000) or parallel cascade structure (Chen et al., 2005). We are looking at a more difficult problem--the one of assessing if migration from a SFB to series cascade scheme will bring forth a significant reduction in output variance or not. This involves using data from a simple control structure to quantify the benefits possible from a more sophisticated control structure.

This paper is structured as follows. In the next section, we outline the basics of the performance assessment for simple feedback and cascade loops. We then discuss the variance decomposition procedure as applied to a simple feedback loop and indicate the components of the variance that can be eliminated using cascade control. Several examples will be used to demonstrate the utility of the proposed methodology. We wish to emphasize that this work concerns with linear processes regulated with linear controllers only. Effects of valve and process non-linearities are not within the scope of this work.

THEORY

Consider the SFB control system shown in Figure 1. [y.sub.1] and [y.sub.2] represent the disturbance corrupted outputs of the primary and the secondary process, respectively. The primary process is denoted by [T.sub.1] = [q.sup.-d1] [[??].sub.1], where d1 denotes the number of samples of time delay in the primary process and [[??].sub.1] represents the delay free part of [T.sub.1]. Along the same lines, the secondary process [T.sub.2] is represented as [T.sub.2] = [q.sup.-d2] [[??].sub.2]. Q represents the feedback controller; [N.sub.1] and [N.sub.2] denote the disturbance transfer functions driven by zero-mean white noise sequences [a.sub.1] and [a.sub.2], respectively; disturbance [a.sub.1] is closer to the primary variable [y.sub.1] and disturbance [a.sub.2] is in proximity to the secondary variable [y.sub.2]. 'u' represents the manipulated variable.

[FIGURE 1 OMITTED]

Figure 2 shows a cascade system controlling the same process. In this case, [Q.sub.1] and [Q.sub.2] represent the primary controller and secondary controller, respectively; [u.sub.2] represents the manipulated variable that is set by the secondary controller [Q.sub.2]. The set point for [Q.sub.2] comes from the primary controller [Q.sub.1]. For a disturbance [a.sub.2] entering the system at t = 0, the output [y.sub.1] will be disturbed from time d1 onwards. The controller Q in the SFB case (Figure 1) will initiate control action at t = d1. The effect of this control action will be felt at [y.sub.1] only from (2d1 + d2) onwards. [y.sub.1] will effectively be in open loop between d1 and (2d1 + d2-1) samples. Under the cascade control system (see Figure 2), [Q.sub.2] will initiate control action at t = 0, and the output [y.sub.1] will be in open loop condition only between d1 and d1 + d2-1 samples. If d1 is large, the cascade scheme will provide better regulation of [y.sub.1] for the secondary disturbance [a.sub.2]. Next, consider the primary disturbance [a.sub.1] entering the system at t = 0. The primary controlled variable will remain in an open loop condition between t = 0 to t = d1 + d2-1 for both the SFB and the cascade control system. In going from a SFB to a cascade scheme, we can hope to eliminate the effect of [a.sub.2] between t = d1 + d2 and t = 2d1 + d2-1. This does not mean that no more reduction in variance is possible as we change from SFB to cascade control. This aspect will be elaborated upon later.

[FIGURE 2 OMITTED]

For the SFB, the closed loop relationship between the external signals and the output [y.sub.1] is given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1)

For the cascade scheme, this relationship is modified to:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2)

with [T.sup.*.sub.2] = [Q.sub.2][T.sub.2]/1 + [Q.sub.2][T.sub.2] and [N.sup.*.sub.2] = [N.sub.2]/1 + [Q.sub.2][T.sub.2]

For the SFB control scheme, the minimum variance is now computed. The disturbance and process transfer functions are expanded as follows:

[N.sub.1] = [P.sub.1] + [R.sub.1][q.sup.-(d1+d2)] (3)

[N.sub.2] = [P.sub.2] + [R.sub.2][q.sup.-(d1+d2)] (4)

[P.sub.2][[??].sub.1] = S + V[q.sup.-(d1+d2)] (5)

where [P.sub.1] and [P.sub.2] are monic polynomials (for [N.sub.1] and [N.sub.1], respectively) in [q.sup.-1] of order d1 + d2-1. In Equation (3), [N.sub.1] is expanded into two parts [P.sub.1] and [R.sub.1] [q.sup.-(d1+d2)]. When noise [a.sub.1] enters the process at time 0, the controller action would not have any effect on [y.sub.1] until time d1 + d2-1; this makes [P.sub.1] a feedback invariant term. In Equation (4), [N.sub.2] is expanded into two parts [P.sub.2] and [R.sub.2] [q.sup.-(d1+d2)]. Note that the noise [a.sub.2] entering at time 0 will upset [y.sub.2] from time 0 to d1 + d2-1 irrespective of any controller action. For our purposes, the effect of [a.sub.2] on [y.sub.1] is of interest. Therefore, in Equation (5), the product of [P.sub.2] and [[??].sub.1] is expanded into S and V[q.sup.-(d1+d2)], where S is a polynomial of order d1 + d2-1.

The closed loop transfer function shown in Equation (1) can be divided into a feedback invariant part and feedback dependent part as shown in the following:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

This can be represented as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (6)

From Equation (6), the minimum variance can be written as:

[[sigma].sup.2.sub.mv,SFB] = var([P.sub.1][a.sub.1] + S[q.sup.-d1][a.sub.2]) (7)

For the cascade control system shown in Figure 2, the minimum variance can be computed as:

[[sigma].sup.2.sub.mv,CAS] = var([P.sub.1][a.sub.1] + [S.sub.2][q.sup.-d1][a.sub.2]) (8)

with polynomial [P.sub.1] as defined in Equation (3) and [S.sub.2] being a polynomial of order d2-1 defined by Equations (9) and (10).

[N.sup.*.sub.2] = [P.sup.*.sub.2] + [R.sup.*.sub.2][q.sup.-d2] (9)

[P.sup.*.sub.2][[??].sup.1] = [S.sub.2] + [V.sub.2][q.sup.-d2] (10)

Remark 1: The only difference between [[sigma].sup.2.sub.mv,SFB] and [[sigma].sup.2.sub.mv,CAS] is in the term related to the secondary disturbance [a.sub.2].

Lemma: The 'd2-1' coefficients of the polynomial [S.sub.2] will be the same as the first 'd2-1' coefficients of the polynomial S.

Next, we seek to perform an analysis of variance for the SFB system. The variance of the primary controlled variable [y.sub.1] should be separated into an invariant component and a feedback dependent component. The result of this analysis would help in deciding if restructuring existing SFB system into cascade control system will be beneficial. In short, we are interested in predicting the cascade achievable performance.

The feedback invariant part for the SFB control system is given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (11)

If a cascade control system were to be established, the feedback invariant part would be:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (12)

In Equations (11) and (12), the [H.sub.1]'s refer to the closed loop impulse response coefficients (analytically determined or identified from routine operating data) for the primary disturbance [a.sub.1] affecting [y.sub.1]. The [H.sub.2]'s refer to the closed loop impulse response coefficients for the secondary disturbance [a.sub.2] affecting [y.sub.1]. These coefficients are estimated by performing a multivariate autoregressive (AR) modelling using [y.sub.1] and [y.sub.2] measurements. An in-house developed software implements the Yule-Walker equations based estimation procedure for determining the parameters of the vector AR model (Wei, 1990). This model is then transformed into the form:

[y.sub.1] = [H.sub.1][a.sub.1] + [H.sub.2][a.sub.2]

and forms the basis of our variance decomposition analysis. In the SFB case, the first 2d1 + d2 terms ([H.sub.2,0] to [H.sub.2,2d1 + d2-1]) are used while in the cascade case only the first d1 + d2 terms ([H.sub.2,0] to [H.sub.2,d1 + d2-1]) are used. Keeping this difference in mind, the feedback invariant for simple feedback system can be split into two parts as:

* Component (1a): SFB and Cascade invariant and

* Component (1b): Additional SFB invariant

The first part is defined as 'SFB and Cascade invariant'--this variance component cannot be altered either by a simple feedback controller or a cascade control system. The second part labelled as 'Additional SFB invariant' contains the variance contribution due to non-availability of the secondary controller. It is assumed that this contribution to overall variance can be reduced to zero if a perfect secondary controller is available. The invariant part of the SFB can hence be rearranged as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (13)

The variance contribution of the '1b' component is given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (14)

where [[sigma].sup.2.sub.a2] is the estimated variance of secondary noise [a.sub.2]. We are now ready to analyze the feedback dependent variance or remainder variance. The feedback dependent part can also be separated into two distinct parts:

* Component (2a): Variance arising due to noise sequence [a.sub.1].

* Component (2b): Variance arising due to noise sequence [a.sub.2].

For single feedback control system, the feedback dependent part is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (15)

The individual terms in Equation (15) can be used to determine the contributions to the variance in [y.sub.1] from the primary and secondary noise channels.

The multivariate AR model would become unidentifiable if: (i) there is a very strong correlation between [y.sub.1] and [y.sub.2] or (ii) if disturbances do not enter either the primary or secondary loop or (iii) if the primary and secondary disturbances are highly correlated. The failure to get an acceptable multivariate AR model (this would manifest in terms of numerical errors/ warnings during the model estimation stage) would point to one of the circumstances mentioned above. When this happens, the variance decomposition procedure outlined in this work cannot be completed. Such situations seldom occur in the analysis of experimental or industrial data.

In summary, the total variance of the primary controlled variable ([y.sub.1]) for a SFB system with an additional secondary output measurement [y.sub.2] can be split into four parts:

* Component (1a): The SFB and Cascade invariant.

* Component (1b): Additional SFB invariant.

* Component (2a): Remainder variance due to noise sequence [a.sub.1].

* Component (2b): Remainder variance due to noise sequence [a.sub.2].

Out of these four parts, the cascade scheme should ideally eliminate (or reduce considerably) the variance contribution from the (1b) and (2b) components. The cascade strategy is designed specifically to reduce the overall time constant and delay in control action to deal with a situation where the major disturbance hits the secondary process and minor stochastic disturbances hits the primary process. Hence, the reason for elimination of variance contribution arising from (1b) and (2b) components can be easily understood. In addition, the cascade control scheme can reduce a portion of the variance attributed to component (2a). The exact amount of reduction possible with component (2a) is not easy to ascertain. We have noted in our simulations that a significant decrease (of at least 50% in all of the examples we have worked on) in the variance contribution due to [a.sub.1] (the (2a) component) is also achieved along with practical elimination of the 1b and 2b components. The reduction in the variance contribution from the (2a) component in the cascade scheme could be due to one or more of the following reasons:

1. The severe control action applied by the primary controller along with the higher gain in secondary controller compared to SFB scheme effectively attenuates the primary disturbances (Harriott, 1984).

2. Webb (1961) uses alternate block diagrams to illustrate the difference in coupling between the disturbance entering the primary loop and the primary controlled variable ([y.sub.1]) for the single loop feedback system and the cascade structure. Along the same lines, we draw alternate block diagrams depicting the impact of the primary disturbance [a.sub.1] on [y.sub.1]. For this analysis, it is assumed that there are no set point changes and that [a.sub.2] is non-existent. Under these assumptions, Equations (1) and (2) reduce to:

[y.sub.1] = ([N.sub.1]/1 + Q[T.sub.1][T.sub.2])[a.sub.1] (16)

and

[y.sub.1] = ([N.sub.1]/1 + [Q.sub.1][T.sub.1][T.sup.*.sub.2])[a.sub.1] (17)

for the simple feedback and cascade loop, respectively. [T.sup.*.sub.2] is the closed loop transfer function of the secondary process. If the inner loop is tightly tuned, as is often the case, then we may use [T.sup.*.sub.2] = 1 and write the following equation for the cascade loop:

[y.sub.1] = ([N.sub.1]/1 + [Q.sub.1][T.sub.1])[a.sub.1] (18)

Equations (16) and (18) are represented in Figure 3(a and b panels), respectively. We have represented N1 as N * [T.sub.1] without any loss of generality. From Figure 3(a), it is seen that, in the single loop system, the primary controlled variable ([y.sub.1]) and the disturbance ([a.sub.1]) are more tightly coupled than is desirable. The output [y.sub.1] will follow a load change [a.sub.1] too readily because there are only two transfer functions ([T.sub.1] and N) in the forward path connecting the disturbance [a.sub.1] to [y.sub.1]. In the cascade system (Figure 3(b)), [y.sub.1] and [a.sub.1] are loosely coupled because the extra lags (the primary controller) are also included in the forward path. Also, the presence of multiple lags in the feedback path of the SFB causes the control action to be delayed. Hence, the variance of [y.sub.1] remains large. There are no lags in the feedback path for the cascade system so that any effect of [a.sub.1] on [y.sub.1] is greatly reduced (owing to lag free feedback). We direct the interested reader to Webb (1961) for parallel arguments in the context of deterministic load disturbances.

[FIGURE 3 OMITTED]

In the cascade loop, the coupling between [a.sub.1] and [y.sub.1] is loose but not zero. Thus, the cascade system cannot completely wipe out the effect of primary load disturbances. Our experience indicates that about 50% of the effect of primary disturbances can be removed through the use of cascade control.

Remark 2: With only routine closed loop data and knowledge of process delays, it is not possible to estimate the precise reduction possible in variance component (2a). Only with extra process knowledge (process and disturbance models) and/or rich process data (obtained using set point changes) can a precise quantification of the reduction in variance component (2a) be attempted. Because cascade control is known to handle secondary disturbances very well, it is safe to assume that the (2b) component will be virtually eliminated under a good cascade control scheme.

Remark 3: The variance component (1a) is the minimum variance achievable with a cascade scheme. While Ko and Edgar (2000) compute this metric using data from a cascade control loop, we are able to estimate it using data from a simple feedback loop. This presents an advantage of the variance decomposition procedure described here. Such an advantage exists as long as the correlation between the sequences [a.sub.1] and [a.sub.2] is not strong. The presented theory is not expected to account for strong correlation effects for which a multivariate analysis would be mandatory.

Before venturing into the examples, it must be pointed out that in the spirit of the Harris type performance index, this method considers only the variance of the process outputs (primary variables) and does not consider the variability of the manipulated variables. Consideration of the variance trade-offs between the outputs and the manipulated variables is outside the scope of the present work. Also, one could argue that knowledge of the time delays itself could indicate whether cascade control would be beneficial or not. While this is true to some extent, data analysis (such as what is done in this paper) would be required to quantify the possible benefits. If precise knowledge of the time delays is not available, one can use generic delay values for particular loop types as stated in Thornhill et al. (1999) to perform the variance decomposition analysis.

ILLUSTRATIVE EXAMPLES

Five simulation examples are used to demonstrate the utility of the proposed variance decomposition method for predicting the possible improvement in control loop performance if cascade control is implemented and also for choosing the secondary variable (if more than one candidate exists). An experimental proof of the concept is also provided. The choice of PI feedback controllers and PI-P cascade schemes in our examples reflects the widely adopted industrial practice (see Bissell (1994), for example). The initial PI-type feedback controllers are assumed to be tuned by the control engineer based on trial and error or by using some standard tuning rule (IMC, Cohen-Coon, ITAE, etc.) with process knowledge obtained from an open loop step test. For this work, the tuning procedure is immaterial and we assume no process knowledge beyond the time delays. We have added 5% measurement noise (Gaussian and independently identically distributed) to the outputs in all the simulations.

Example 1

The primary process ([T.sub.1]), the secondary process ([T.sub.2]), the primary noise transfer function ([N.sub.1]), and the secondary noise dynamics ([N.sub.1]) used in this example are given below. In this example, the noise dynamics affecting the primary process is purposefully kept severe compared to noise dynamics affecting the secondary process, to check the effectiveness of cascade in rejecting severe primary disturbance compared to secondary disturbance.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The white noise sequences [a.sub.1] and [a.sub.2] have a variance covariance matrix equal to:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The PI achievable performance for the SFB is computed to be 0.29. First, we optimize the values of the controller parameters ([K.sub.c] and [[tau].sub.I]) such that the least variability in the controlled variable is obtained ([[sigma].sup.2.sub.SFB]). When the theoretically obtained minimum variance of the controlled variable ([[sigma].sup.2.sub.SFB]) is divided by ([[sigma].sup.2.sub.SFB]), we get the PI achievable performance for the simple feedback controller. Note that [[sigma].sup.2.sub.SFB] can also be obtained from routine closed loop data if the delay is known (Harris, 1989). The PI achievable performance can also be obtained via iterative tuning as outlined by Goradia et al. (2005). With this optimal SFB control system, the variance of [y.sub.1] is 12.89; the break up into the 1a, 1b, 2a and 2b components is 3.71, 4.03, 0.8 and 4.36, respectively. Components 1b and 2b are substantial--they make up about 65% of the variance in [y.sub.1]. These are the components that can be targeted and reduced by the cascade control strategy. The analysis makes a strong case for implementing a cascade control scheme. When a PI-P cascade scheme is implemented, the best control loop performance index (CLPI) achieved is 0.81. With the PI-P cascade implementation, the variance in [y.sub.1] is 4.59; the break up into the SFB and Cascade invariant, 2a and 2b components is 3.71, 0.25 and 0.64, respectively. Note that there has been a substantial decrease (69%) in the 2a variance component also. The improvement in performance index is 179% (from 0.29 to 0.81) and vindicates the prediction made by the variance decomposition approach. Note that the CLPI is defined as the ratio of the achievable variance with a minimum variance controller to the current variance of the controlled variable for a given control strategy (feedback or feedback + feedforward, etc.).

Remark 4: The proposed variance decomposition approach uses only routine operating data; it cannot, therefore, predict the settings of the primary and secondary controller at which the optimal cascade loop performance is achieved. If suitable experimental data (collected either under open or closed loop conditions) are available and the process models are identified, the optimal settings of the primary and secondary controller leading to the best control loop performance can be obtained using parametric optimization

Example 2

The transfer functions used in this example are:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The white noise sequences [a.sub.1] and [a.sub.2] have a variance covariance matrix equal to:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

This process system is based on a packed bed reactor system fitted with a preheater considered by Marlin (2000) for discussing the performance of cascade control systems. The primary controlled variable is a composition variable at the exit of the packed bed reactor. The secondary variable is the reactor inlet temperature. The flow rate of the heating medium in a preheater located upstream of the reactor is the manipulated variable. Here, the primary and secondary processes have similar time constants. The primary process has a significant time delay when compared to the secondary process. The noise dynamics is considered to be non-stationary. A schematic of the SFB and cascade control schemes for this reactor system are shown in Figures 4 and 5, respectively.

[FIGURES 4-5 OMITTED]

The PI achievable performance for the SFB scheme is 0.43. At this optimal performance, the variance of [y.sub.1] is 55.52; the break up into the 1a, 1b, 2a and 2b components is 23.95, 0.16, 30.86 and 0.55, respectively. The 1b and 2b components are small indicating that the benefits from a cascade control system should mainly come from the reduction of the 2a component, which accounts for about 56% of the total variance here. On the basis of our experience, we can predict that at least 50% of component 2a will be annihilated. We would expect the 2a component with the cascade scheme to be about 15 and the variance in [y.sub.1] to reduce to around 40. A more precise answer to the expected reduction in 2a component is not possible. The control engineer should now decide if this expected decrease in variance of [y.sub.1] will justify moving to a cascade control scheme.

Let us check if our expectations turn out to be true. When a PI-P cascade scheme is implemented, the best CLPI achieved is 0.63. With the PI-P cascade implementation, the variance in [y.sub.1] is 39.71; the break up into the SFB and Cascade invariant, 2a and 2b components is 24.99, 13.94 and 0.78, respectively. Note that there has been a significant decrease in the 2a variance component as predicted. The overall increase in CLPI is 47%; this may be enough to justify the implementation of the cascade scheme.

Remark 5: Without integral action in the inner loop, offset is expected in the secondary controlled variable in the presence of non-stationary disturbances. This does not concern us presently because we are interested in reducing the variability in the primary variable only. The offset free regulation of the primary controlled variable is ensured by deploying a PI controller in the outer loop. As is well known, PI-P schemes are the most common cascade loops in the process industry.

Example 3

The system considered next is described by the following equations (time constants and delays are in minutes):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Here, [y.sub.1] is the primary controlled variable, and [y.sub.2] and [y.sub.3] represent possible secondary variables. u is the manipulated variable; [a.sub.1], [a.sub.2] and [a.sub.3] represent zero mean white noise sequences with variances [[sigma].sup.2.sub.a1], [[sigma].sup.2.sub.a2] and [[sigma].sup.2.sub.a3], respectively. The process is controlled by a PI controller with [K.sub.c] = 1 and [[tau].sub.I] = 40 min. We will examine the variance decomposition results for various combinations of the noise variances and suggest the best secondary variable in each of those cases. In each case, 5000 samples of routine closed loop data sampled at intervals of 1 min were used.

* Case 1: [[sigma].sup.2.sub.a1] = [[sigma].sup.2.sub.a2] = 1 and [[sigma].sup.2.sub.a3] = 100

If [y.sub.2] is considered as the secondary variable in the cascade scheme, the variance decomposition procedure estimates the overall variance in [y.sub.1], 1a, 1b, 2a and 2b components to be 1.38, 0.02, 0.02, 0.03 and 1.31, respectively. If [y.sub.3] were to be chosen as the secondary variable, these values are 1.40, 0.06, 0.02, 0.01 and 1.31, respectively. In this case, it does not matter whether [y.sub.2] or [y.sub.3] is chosen as the secondary variable. Since the 2b component is very strong, cascade control using either [y.sub.2] or [y.sub.3] as the secondary variable will provide a vastly improved control loop performance. Between [y.sub.2] and [y.sub.3], we can choose the one that engulfs most of the disturbances as the secondary variable (this is very likely to be [y.sub.2]).

* Case 2: [[sigma].sup.2.sub.a2] = [[sigma].sup.2.sub.a3] = 1 and [[sigma].sup.2.sub.a1] = 100

If [y.sub.2] is considered as the secondary variable in the cascade scheme, the variance decomposition procedure estimates the overall variance in [y.sub.1], 1a, 1b, 2a and 2b components to be 1.83, 0.87, 0.001, 0.94 and 0.02, respectively. If [y.sub.3] were to be chosen as the secondary variable, these values are 1.83, 0.87, 0.002, 0.92 and 0.04, respectively. The 1a and 2a components are dominant in this case. Based on our experience, we again conjecture that more than 50% of the 2a component will be consumed by the cascade scheme that could use either [y.sub.2] or [y.sub.3] as the secondary variable. Between [y.sub.2] and [y.sub.3], the choice will depend on their relative location with respect to the anticipated disturbances.

* Case 3: [[sigma].sup.2.sub.a1] = [[sigma].sup.2.sub.a3] = 1 and [[sigma].sup.2.sub.a2] = 100

If [y.sub.2] is considered as the secondary variable in the cascade scheme, the variance decomposition method estimates the overall variance in [y.sub.1], 1a, 1b, 2a and 2b components to be 0.445, 0.062, 0.014, 0.007 and 0.362, respectively. The 2b component is dominant here and a cascade control scheme with [y.sub.2] as the secondary variable can eliminate this variance component very effectively. If [y.sub.3] were to be chosen as the secondary variable, the overall variance of [y.sub.1] is 0.378 with the 1a, 1b, 2a and 2b components being 0.023, 0.000, 0.342 and 0.013, respectively. Interestingly, with [y.sub.3] as the secondary variable, the 2a component is the dominant one. With cascade control, we may not be able to eliminate this component completely (as much as we can do with the 1b or 2b component). In this case, the use of [y.sub.2] as the secondary variable would be a more prudent choice.

Example 4

This example is taken from Ko and Edgar (2000). This example is chosen because the process structure considered by them is different from the process structure assumed in our work. Their process model is given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Note that the two external noise sequences [a.sub.1] and [a.sub.2] directly affect both the primary and secondary variables [y.sub.1] and [y.sub.2]. The white noise sequences [a.sub.1] and [a.sub.2] have a variance covariance matrix equal to:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Consider the case where the process is regulated by a simple PI feedback controller, Q([q.sup.-1) = 0.144 - 0.138[q.sup.-1]/1 - [q.sup.-1]. Routine closed loop data of the primary and secondary output variables ([y.sub.1] and [y.sub.2]) were used to perform the variance decomposition calculations. The variance of [y.sub.1] is seen to be 16.38 with the variance contributions for the 1a, 1b, 2a and 2b components being 4.62, 1.63, 7.62 and 2.51, respectively. The reader may note that the 1a component (SFB and Cascade invariant part) contribution of 4.62 matches very closely with the minimum achievable variance under cascade control calculated by Ko and Edgar (2000). While Ko and Edgar (2000) calculate the minimum cascade achievable variance from data collected under cascade control, our proposed procedure estimates this value using data from a simple feedback scheme itself. This is a key feature of our work and, as mentioned in Remark 3, comes about owing to the weak correlation between [a.sub.1] and [a.sub.2]. To answer the question 'Will cascade control be useful?', we look at the other variance components. The 1b, 2a and 2b components are quite significant indicating that a cascade control system might be very successful. If we devise a good cascade control system, we should reduce the variance in [y.sub.1] by 1.63+(0.5*7.62) + 2.51 = 7.95 (the sum of 1b, 2b and 50% of the 2a component) i.e. we would expect an output variance of about 8.4 with a well designed cascade control scheme. It must be noted that even the best feedback PI controller can give an output variance of 14.31 only (the 1a, 1b, 2a and 2b components being 4.81, 2.04, 5.25 and 2.21, respectively). Clearly, the performance afforded by the simple feedback control structure that uses a PI-type controller can be improved upon particularly with a secondary measurement [y.sub.2] being available. Based upon the theory developed above, we will be able to remove the 1b and 2b components almost completely and reduce the 2a component significantly with cascade control. A significant benefit is expected with a cascade control scheme--this must be verified in any case.

A PI-P cascade control system with the primary controller as [Q.sub.1]([q.sup.-1) = 0.48 - 0.46[q.sup.-1]/1 - [q.sup.-1] and the secondary controller [Q.sub.2]([q.sup.-1]) = 0.7 is implemented. Routine closed loop data is collected from this process and a variance decomposition analysis was performed. The variance in [y.sub.1] is 7.62 with the SFB and Cascade invariant component contributing 4.84, the 2a component contributing 2.03 (more than 50% reduction in the 2a component as compared to the original SFB scheme) and the 2b component contributing 0.75 (a significant reduction from 2.51 for this component as compared to the original SFB scheme). Note that the improvement with a cascade scheme is close to what was predicted from our analysis of routine data obtained from the SFB scheme.

Example 5

This example is taken from Smith and Corripio (1997) and uses fundamental non-linear process models. Figure 6 provides a schematic of the furnace/preheater-reactor system. A first order irreversible reaction A [right arrow] B takes place in a CSTR of volume V ([m.sup.3]). The reactant is available at a low temperature [T.sub.1] (K) and must be heated in the furnace before being fed into the reactor. The reaction is exothermic with heat of reaction (-[DELTA], kJ/mole of A). The feed at concentration [C.sub.AF] (moles A/[m.sup.3]) enters the reactor with a flow rate of q (m3/min) and at temperature [T.sub.2] (K). A cooling jacket is used to remove the reaction heat and maintain the reactor at a temperature T (K). The cooling fluid is circulated at a flow rate [w.sub.C] (kg/min) through the jacket of the reactor (the coolant is heated from a temperature [T.sub.C] (K) to [T.sub.0] (K) due to heat exchange). The reactor products are withdrawn at a concentration [C.sub.A], temperature T and flow rate q. In the furnace, air is available at a flow rate [q.sub.A] ([m.sup.3]/min) and at a temperature [T.sub.A] (K) while the fuel gas is utilized at a flow rate [q.sub.F] ([m.sup.3]/min) and at a temperature [T.sub.F](K).

[FIGURE 6 OMITTED]

The detailed non-linear first principles model of a furnace provided in Doyle et al. (1999) was adopted. This model has 26 continuous states and is particularly effective in demonstrating the effect of transport delay through a distributed parameter system. Standard assumptions such as perfectly mixed reactor vessel, constant reactor volume, constant physical properties are made in order to derive the following dynamic model for the reactor.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (19)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (20)

where K = 2[C.sub.PC]/U [A.sub.H]. In deriving Equation (20), it has been assumed that the driving force for heat transfer from the reactor to the jacket is the difference between the reactor temperature T and the lumped jacket temperature [T.sub.av]. The average temperature is defined as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (21)

The following values are provided for the variables and parameters :

Furnace: Details on the geometry of the furnace (volume, tubing specifications, efficiency, physical properties, etc.) can be obtained from Doyle et al. (1999). The inlet conditions are as follows: feed inlet temperature, [T.sub.1] = 310 K; feed inlet flow rate, q = 0.035 [m.sup.3]/min; air flow rate, [q.sub.A] = 17.9 [m.sup.3]/min; inlet air temperature, [T.sub.A] = 322 K; fuel gas flow rate, [q.sub.F] = 1.211 [m.sup.3]/min; gas temperature, [T.sub.F] = 310 K; furnace temperature = 1432.9 K.

Reactor: V = 0.95 [m.sup.3]; AH (area for heat transfer provided by the jacket) = 0.68 [m.sup.2]; (-?H) = 33 000 J/mol; activation energy E = 9000 J/mol; frequency factor [k.sub.o] = 3.2208 x [10.sup.11] [min.sup.-1]; density of the feed stream and cooling fluid are 950 kg/[m.sup.3]; heat capacity of the feed stream = 350 J/kg K; heat capacity of the cooling fluid = 520 J/kg K; overall heat transfer coefficient = 5000 J/([m.sup.2] min K). The steady state values of the feed rate, q = 0.035 [m.sup.3]/min; the feed concentration, [C.sub.AF] = 1000 moles/[m.sup.3]; the feed temperature, [T.sub.2] = 611.2 K; the maximum cooling fluid flow rate, [w.sub.C] = 30 kg/min; the cooling fluid inlet temperature, [T.sub.C] = 298 K; the reactor temperature, T = 624.41 K; the average jacket temperature, [T.sub.av] = 330.07 K.

It is assumed (as mentioned in Smith and Corripio, 1997) that the cooling jacket is not able to provide the cooling capacity required. Therefore, it is decided to open the cooling valve completely and control the reactor temperature T by manipulating the fuel flow rate to the preheater. The feedback control (a PI controller with [K.sub.c] = 0.007 and [[tau].sub.I] = 22.4 min is used) schematic of the furnace/preheater and reactor process is shown in Figure 6. This process suffers from several measured and unmeasured disturbances--cooling fluid temperature, feed concentration, inlet feed temperature, inlet air temperature, fuel gas temperature, etc. It is desired to check if cascade control strategy that utilizes either the reactor feed temperature [T.sub.2] or the average jacket temperature [T.sub.av] as the secondary variable will result in better process control.

Random fluctuations were introduced in selected potential disturbance variables ([T.sub.1], [T.sub.A], [T.sub.F] , [C.sub.AF] and [T.sub.C])--this resulted in the generation of three scenarios. In Scenario 1, all of the disturbances listed above are activated while in Scenario 2, only the disturbance variables pertaining to the furnace ([T.sub.1], [T.sub.A] and [T.sub.F]) were present. In the third scenario, only the disturbance variables downstream of the furnace ([C.sub.AF] and [T.sub.C]) were activated. These scenarios were created to show that the variance decomposition results point to the right choice of secondary variables and also the correct quantitative measures. The time series of the disturbance variables (sampling interval = 0.5 min), the manipulated variables, the secondary measurements and the controlled variables for the three scenarios are shown in Figures 7, 8 and 9, respectively. The variance decomposition details presented below does not make use of any a priori knowledge (other than the estimated time delays) about the process or the location/nature of disturbances. Only measurements of the primary controlled variable (reactor temperature) and the possible secondary variables ([T.sub.2] and [T.sub.av]) will be used to judge the potential benefits possible with cascade control. Besides the closed loop data from the simple feedback control system, the a priori knowledge available are: time delay between the manipulated variable and [T.sub.2] is 10 samples, time delay between the manipulated variable and [T.sub.av] is 11 samples, one sample delay between [T.sub.2] and T and no appreciable delay between [T.sub.av] and T.

[FIGURES 7-9 OMITTED]

Scenario 1: This represents the case where all disturbances are activated. The variance in the primary output was 12.59 units. When the reactor feed temperature (furnace outlet temperature, T2) is available as the secondary variable, the 1a, 1b, 2a and 2b components are estimated as 3.12, 0.50, 2.40 and 6.57 units, respectively. The 2b component dominates and this augurs well for success with cascade control. We anticipate reduction in the output variance from 12.59 units to about 12.59-0.50-6.57-(0.50*2.40) = 4.32 units with cascade control that employs [T.sub.2] as secondary variable. Our variance decomposition procedure predicts that the 1a, 1b, 2a and 2b components are 5.26, 0, 5.92 and 1.41, respectively, when the average jacket temperature is available as the secondary variable. Component 2a dominates here and thus, in this case, we estimate the output variance with cascade control to be 8.22 units (i.e. 12.59-1.41-(0.50*5.92)). The expected reduction in output variance with the use of cascade control is more significant with [T.sub.2] as secondary variable as compared to implementing cascade control with [T.sub.av] as secondary variable. In Scenario 1, based on the data analysis procedure, we would recommend cascade control with [T.sub.2] as the secondary variable--a reduction of 66% in output variance would be possible with this implementation .

Scenario 2: The disturbances originate only in the variables related to the furnace. The 1a, 1b, 2a and 2b components are 1.88, 0.55, 0.16 and 9.68 (making total output variance = 12.27 units) if we decide to use furnace outlet temperature as the secondary variable in the cascade scheme. The huge contribution from the 2b component implies that the cascade control scheme will be highly desirable and successful in this case. The output variance is expected to come down to a meagre 2 units (84% improvement) with the cascade control scheme that employs [T.sub.2] as the secondary variable. With the average jacket temperature as the possible secondary variable, variance decomposition analysis of the data indicates that such a cascade scheme would not be beneficial at all. The closed loop data indicates a very strong correlation between [T.sub.av] and T. This is due to the lack of dynamics between these variables (see Equation (21)) and hence information on [T.sub.av] cannot be expected to contribute much to variance reduction. Our decision based on the data and knowledge of time delays will be to implement a cascade control scheme with [T.sub.2] (a clear choice) as the secondary variable.

Scenario 3: Here, the disturbances are localized to the reactor part alone. The variance in the primary output was 3.50 units. When the reactor feed temperature (furnace outlet temperature, [T.sub.2]) is available as the secondary variable, the 1a, 1b, 2a and 2b components are estimated as 1.48, 0, 2.01 and 0.01 units, respectively. Our analysis predicts the 1a, 1b, 2a and 2b components to be 1.47, 0, 1.93 and 0.1, respectively, when the average jacket temperature is available as the secondary variable. For this process, if the disturbances in the plant are as presented in this scenario (location and strength), there is very little to choose between the two secondary variables. The overwhelming domination of the 2a component also indicates that the benefits from cascade control will be rather limited (a maximum improvement of 1 unit) in any case.

In summary, based on the closed loop data and knowledge of the time delays, our analysis recommends that [T.sub.2] will be the best secondary variable and provide significant reduction in output variance under two scenarios. In the third scenario, either [T.sub.2] or [T.sub.av] is estimated to provide a moderate reduction in output variance. The results provided by our analysis make sense when one takes into consideration the location of the disturbances and the lack of dynamics between the average jacket temperature (one of the possible secondary variables) and the reactor temperature (the primary controlled variable).

Example 6

Here, we present results based on the analysis of data obtained from an experimental set-up. The schematic of a laboratory stirred tank heater system is shown in Figure 10. The level of water in the glass tank is tightly controlled at 20.5 cm (from the base) by manipulating the cold water flow rate using a PI controller. For this experiment, the hot water stream was not used. The steam flow is manipulated by a PI controller ([K.sub.c] = 0.3 and [[tau].sub.I] = 45 s) to control the temperature at location TT2 in the long winded piping section through which water exits from the tank. The temperature measurement at TT1 is available as a secondary variable for cascade control (if needed). The temperature sensors have unknown measurement dynamics. The main disturbances are the inlet temperature of the cold water stream (nominal value is about 24[degrees]C) and disturbances arising from the level loop. The steady state temperature at location TT1 is 40.3[degrees]C and 40.2[degrees]C at location TT2. It is known (through a previous laboratory experiment) that the delay between the manipulated variable ([u.sub.3]) and the primary controlled variable ([y.sub.3]) is 12 s and that between [u.sub.3] and [y.sub.2] is 8 s. The sampling interval is 1 sec. Experimental closed loop data from this feedback control loop was collected for 1.5 h (including start-up time). 4000 routine data samples are considered for the analysis.

[FIGURE 10 OMITTED]

The variance in the output [y.sub.3] was 1.32 units. Analysis of the [y.sub.3] and [y.sub.2] data using a multivariate autoregressive model provided the following split for 1a, 1b, 2a and 2b components as: 0.35, 0.15, 0.02 and 0.80. The strong contribution from the 1b and 2b components indicates that cascade control with [y.sub.2] as the secondary variable should provide a strongly enhanced control loop performance. A cascade control scheme was implemented with proportional only controller as the secondary controller and a PI controller as the primary controller. This reduced the variance of the output to 0.94 units without altering the variance of the manipulated variable by very much as compared to that observed in the simple feedback control case. When derivative action was employed in the outer controller, the output variance could be reduced to 0.64 units. However, this increased the variance of the manipulated variable considerably and the steam valve hit the physical limits (fully open or closed) several times during the run. This behaviour would be undesirable in practical applications and therefore a PI-P cascade control scheme is recommended.

CONCLUSIONS

The proposed variance decomposition method provides an estimate of the variance reduction possible by moving from a SFB scheme to a cascade scheme using only routine operating data from a process controlled by a simple feedback controller. Only routine closed loop data on the primary and secondary variables and knowledge of the process time delays is required. When certain variance components (1b and 2b) are dominant, implementing cascade control can give significant benefits. In other cases (when 2a component is dominant), considerable reduction in variance is possible. The method, therefore, provides the practitioner with a quantitative idea about what potential improvements one might expect if a SFB scheme were configured into a cascade scheme. The utility of the proposed approach has been demonstrated using six representative case studies. The results presented here establish that it is possible to attain the predicted performance by upgrading a SFB scheme to a cascade control scheme.

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([dagger]) A shorter version of this article was presented at the DYCOPS 2004 Meeting, USA, July 2004.

S. Lakshminarayanan *, M. W. Hermanto, D. B. Goradia and G. P. Rangaiah

* Author to whom correspondence may be addressed. E-mail address: chels@nus.edu.sg

Department of Chemical and Biomolecular Engineering, National University of Singapore, Singapore 117576

Manuscript received February 3, 2006; revised manuscript received July 25, 2006; accepted for publication July 25, 2006.