- Email: [email protected]

An economic measure for comparing dynamic robustness R. Di Mascio* School of Marketing and International Business, University of Western Sydney, Locked Bag 1797, Penrith South DC NSW 1797, Australia

Abstract This paper develops a procedure that enables an engineer to present a more compelling argument for process control investment. The procedure calculates an economic measure of the dynamic robustness of plant control systems in the face of plant-based and external uncertainties. It involves: (a) characterising each type of plant-based or external uncertainty that aﬀects operations and proﬁtability; (b) computer-simulating a plant-control system model to show how it responds to diﬀerent combinations of uncertainties; (c) calculating an economic index of control quality for each combination of uncertainties; and (d) plotting a frequency distribution of the economic indices. The tighter the distribution, the more robust the plant-control system. The procedure is demonstrated by comparing the robustness of a distillation column control system, with and without decoupling. This procedure has three advantages. Firstly, it presents a more realistic picture of robustness because it characterises uncertainty by a range of probable values, rather than a single value. Secondly, the procedure can be applied to SISO, MIMO, linear and nonlinear systems alike without requiring model simpliﬁcation. Thirdly, it presents results in economic terms using a graphical format that is more meaningful to business managers than an abstract mathematical quantity, such as a structured singular value. # 2002 Elsevier Science Ltd. All rights reserved. Keywords: Taguchi; Process economics; Robustness; Control quality

1. Introduction Consider the following situation: a methanol water distillation column currently uses single-loop PI controllers to regulate top and bottom composition by manipulating reﬂux ﬂowrate and steam ﬂowrate respectively. As both loops are interacting strongly, the control engineer is considering connecting a decoupler to each loop to reduce the interaction. To ensure that this enhancement is consistent with the plant’s overall objective, ‘‘to convert certain raw materials (input feedstock) into desired products using available sources of energy, in the most economical way’’ [1], the engineer must also evaluate its economic beneﬁt. However, while it is straightforward to estimate the costs of improving dynamic robustness (e.g. installation and operating costs), currently no generally applicable method exists to estimate the economic beneﬁt.

* Tel.: +61-2-9685-9458; fax: +61-2-9685-9612. E-mail address: [email protected]

There are two reasons for this. First, many of the methods used to evaluate dynamic robustness express results in mathematical rather than economic terms. These include frequency domain measures for robustness towards unstructured and structured uncertainties in linear systems [2–5]. Measures for nonlinear systems include those produced by operator equations [6,7], Nyquist plots [8], Monte Carlo simulation [9] and Lyapunov’s stability theorem [10]. The second reason is that many existing robustness methods consider only plant-based uncertainty (i.e. occurring within the plant’s physical environment, such as feed disturbances and measurement noise), and ignore external uncertainty (i.e. arising outside the plant’s physical environment, such as market forces aﬀecting prices of raw materials and ﬁnished products, and government regulation of waste emissions). Both types of uncertainty can cause deviation from the desired operating point, and external uncertainty can change the cost of that deviation. Just as operators and engineers desire steady plant behaviour irrespective of physical perturbations, managers desire consistent economic yield irrespective of changing external conditions. Thus, an

0959-1524/02/$ - see front matter # 2002 Elsevier Science Ltd. All rights reserved. PII: S0959-1524(01)00052-X

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economic measure of robustness should consider the impact of both types of uncertainties. This paper develops such a measure. This measure is generated by a four-step procedure: (a) characterise each type of plant-based or external uncertainty that aﬀects operations and proﬁtability; (b) conduct computer simulations to show how the plant responds to diﬀerent combinations of uncertainties; (c) calculate an economic index of control quality for each combination of uncertainties; and (d) plot a frequency distribution of the economic indices. The tighter the resulting distribution, the more robust the plant-control system. This procedure is built on two streams of research. The ﬁrst is in the area of robust plant design, which for the past ﬁve years or so has characterised uncertainties by their probability distributions, and used stochastic simulation to determine their eﬀects on plant economics. For example, stochastic optimisation has been used to ﬁnd a plant’s best steady-state operating region [11– 13] but this work did not consider dynamic behaviour. The work of Samsatli et al. [14] is based on dynamic behaviour and considers plant economics, but it was applied to plant design, not control system design. This paper adapts work from that research to evaluate the dynamic robustness of any process control design. The second stream of research is in the area of Taguchi-based measurement of control quality. This approach is ideal for determining robustness because it provides a way of determining the eﬀects on process performance of uncontrollable variations in the environment. This approach has been applied to tune controllers to cope with plant-based uncertainty, such as measurement noise and valve chatter [15]. However, this work did not use economic indices of control quality. An economic index of control quality was developed using Taguchi principles [16], but this index can only be used to assess controller performance under a single perturbation. This paper adapts this economic index to handle multiple stochastic perturbations. The advantages of this procedure are three-fold. Firstly, it presents a more realistic picture of robustness because it characterises uncertainty by a range of probable values, rather than a single value. Second, the use of computer-based simulation enables the procedure to be applied to SISO (single-input, single-output), MIMO (multiple-input, multiple-output), linear and nonlinear systems alike without requiring simpliﬁcation, such as model linearisation. Thirdly, the presentation of results graphically and in economic terms, rather than by a mathematical quantity (such as the structured singular value [4]) allows meaningful communication between engineers, operators and managers. The following sections of this paper describe the procedure in detail, apply the procedure to evaluate the dynamic robustness of two alternate control system designs, and discuss the implications of this work.

2. The robustness-assessment procedure The procedure uses a four-step process to evaluate the economic dynamic robustness of process-control systems. The steps are: 1. characterise each type of plant-based or external uncertainty that aﬀects operations and proﬁtability; 2. conduct simulations to show how the plant responds to diﬀerent combinations of uncertainties; 3. calculate an economic index of control quality for each combination of uncertainties; 4. plot a frequency distribution of the economic indices. The following sections describe each step in more detail. 2.1. Step 1. Characterise each type of plant-based or external uncertainty that aﬀects operations and proﬁtability This paper divides uncertainties into those occurring inside (plant-based) and outside (external) the plant. Plant-based uncertainties arise from lack of knowledge about: a plant model; valve and measurement devices; and disturbances such as upstream upsets. This type of uncertainty aﬀects how closely the plant is to its optimal operating point. Information about these uncertainties can be obtained from experts such as operators, engineers, and consultants; historical plant data; and manufacturers’ speciﬁcations (if available). Uncertainty in the external environment includes lack of knowledge about economic conditions aﬀecting demand for a plant’s products and supply of raw materials (e.g. inﬂation, disposable income, business investment and interest rates); regulatory trends (e.g. waste management legislation); customer attitudes to product quality; environmental factors aﬀecting the cost and availability of energy and raw materials (e.g. uranium mining and forest conservation); and industrial disruption. This type of uncertainty aﬀects the plant’s optimal operating point as well as the cost of deviating from that point. Information about these uncertainties can be obtained from buyer intention surveys, the opinions of sales representatives and other experts such as suppliers, marketing consultants and trade associations and lead indicators that move in advance of sales [17]; input/ output methods [18] describing patterns of interrelationships of industries; and scenario planning to derive the possible futures of the business environment [19]. Uncertainties can be characterised by its probability distribution or a scenario. A probability distribution

R. Di Mascio / Journal of Process Control 12 (2002) 745–751

is the range of continuous values an uncertain variable can take. Two of the many types of probability distribution are: empirical derived from historical data (if available); and triangular distributions derived from expert opinion (if available), in which case the distribution is deﬁned by the minimum, most likely and maximum values [20]. For example, in a temperature measurement, random electrostatic noise may cause uncertainty in the temperature variable. For example, proposed environmental legislation would, if passed, double waste disposal costs. Two scenarios are: (a) that the legislation is passed, in which case waste disposal costs will double; and (b) that the legislation is defeated, in which case costs will be unchanged. Although this paper divides uncertainties into plantbased and external sources, this is only to help identify uncertainties. The type of uncertainty does not aﬀect the application of the procedure. In practice, it can be diﬃcult to place uncertainties in either category if the boundary between is blurred. For example, a new chief executive oﬃcer may have a diﬀerent view of inventory policy than his/her predecessor (aﬀecting the operating point); the appointment of a new chief executive can be considered both plant-based and external. Also, plant-based and external uncertainties can aﬀect the same plant variable. For example, a setpoint change can result from an upstream disturbance (plant-based) or a market change (external). 2.2. Step 2. Conduct simulations to show how the plant responds to diﬀerent combinations of uncertainties To evaluate the plant’s dynamic robustness, it is necessary to determine how the plant behaves under uncertainty. Computer simulation (using a plant model) is appropriate for this because it can be applied to virtually any type of model (e.g. SISO, MIMO, linear or nonlinear). Monte Carlo simulation is the most appropriate method for simulating systems with random parameters. It can simulate many types of uncertainties simultaneously to produce a range of probable outcomes [20]. Thus, this procedure uses Monte Carlo simulation to show how the plant responds to diﬀerent combinations of uncertainties (e.g. random disturbances, measurement noise, valve chatter and setpoint changes) that were identiﬁed and characterised in Step 1. Monte Carlo analysis simulates the plant-control system model repeatedly. Each run of the simulation, called a replication, begins with the plant-control system at steady state. (If the system has several steady states, then the computer chooses one at random.) The computer samples all sources of uncertainty from their probability distributions or scenarios, and simulates the system’s behaviour under this set of uncertainties. After each replication, the computer records the value of system

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variables, such as the manipulated variables, controlled outputs and states (required in Step 3). Because the precision of the probable outcomes increases with the number of replications [20], a simulation can comprise hundreds or even thousands of replications depending on the precision required. A drawback of this type of simulation, however, is that the computing power required increases with system complexity and replication numbers. Over time, this will become less of a problem (for a given system) with the development of more powerful computers and more eﬃcient sampling of probabilities, such as Latin– Hypercube sampling [20]. 2.3. Step 3. Calculate an economic index of control quality for each combination of uncertainties This paper uses Taguchi-based indices of control quality because they express control quality in monetary terms and provide a way of handling plant-based and external uncertainties. These indices are based on the Taguchi quality loss approach [21] used in production engineering, in which the quality of a product is estimated by how closely product characteristics remain to nominal values. A quality loss arises when the characteristics depart from nominal values even if they remain within speciﬁcation. Separate Taguchi-based indices have been developed for stability and performance quality [16]. The stability quality loss captures the deviation of manipulated variables from their nominal values and their excursion beyond upper and lower control limits. A stability quality loss can be calculated in each replication by: ð Tf

wu uðtÞ uss þ wc ðmaxð0; ucalc ðtÞ uu Þ 0 dt þ max 0; u1 ucalc ðtÞ

Ls;r ¼

ð1Þ

where Ls,r is the stability quality loss in monetary units (mu) for the rth replication, Tf is the settling time, u(t) is the value of the manipulated variable at time t, uss is the nominal steady state value, uu is the upper limit, ul is the lower limit, ucalc(t) is the calculated value of u(t) that would occur if there were no constraints, wu is the cost of deviation of uðtÞ from uss; and wc is the cost of excursion beyond limits. wu can be derived from Lagrange multipliers where proﬁt optimisation data is available, or, if this data is unavailable, from other information such as the cost of reworking oﬀ-spec material. (To simplify the presentation, Ls,r for a single input system has been shown and the value of wu in Eq. (1) is the same for positive and negative deviations from uss. However, the equation can be modiﬁed to accommodate multiple inputs and diﬀerent values of wu.)

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The performance quality loss for each replication is expressed by: ð Tf

wy yðtÞ ysp þ wyc ðmaxð0; yðtÞ yu Þ 0 þ max 0; yl yðtÞ dt

Lp;r ¼

ð2Þ

where Lp,r is the performance quality loss in monetary units, y(t) is the output variable, wy is the cost of deviation of y(t) from its desired value ysp, wyc is the cost of y(t)’s excursion beyond the upper limit yu or lower limit yl. The excursions of other variables beyond constraints that aﬀect performance can also be incorporated into Eq. (2). The sum of the quality losses for stability and performance in each replication give the total quality loss: Ltot;r ¼ Ls;r þ Lp;r

ð3Þ

Another beneﬁt of using this method for calculating quality loss is that it is suitable when a variable violates one-sided soft constraints, a situation that some [14] say is problematic in control quality evaluation. 2.4. Step 4. Plot a frequency distribution of the economic indices A frequency diagram of Ltot values enables visual comparison of the robustness of diﬀerent control systems. The tighter the distribution (or the smaller the standard deviation) the more robust the system. This is in line with Samsatli et al.’s [14] assertion that it is important to consider quality indices’ variability, not just their average values. Alternatively, a cumulative frequency diagram can be plotted, and the steeper the distribution the more robust is the system. A scatterplot using Ls and Lp values as axes also gives an indication of the stability and performance robustness. The next section illustrates how this technique can be used to compare the dynamic robustness of two alternate control schemes.

3. Distillation column example 3.1. The process The example system considered here was an experimental distillation column separating methanol from water into methanol distillate and a water bottoms stream. This system is often used in the literature for comparison of control schemes [22–24] because it captures the interactions and time delays common to a distillation column. A model of the system is described below [22]:

3 12:8es 18:9e3s 2 u 3 1 16:7s þ 1 21:0s þ 1 7 6 7 6 7 76 4 5¼6 4 5 4 6:6e7s 19:4e3s 5 y2 u2 10:9s þ 1 14:4s þ 1 3 2 3:8e8s 6 14:9s þ 1 7 7 þ6 4 4:9e3s 5d 13:2s þ 1 2

y1

3

2

ð4Þ

The controlled variables are the overheads composition (y1) and bottoms composition (y2), while the manipulated variables are the reﬂux ﬂowrate (u1) and steam ﬂowrate (u2). The main disturbance to the column is a change in feed ﬂow rate (d). Table 1 shows the operating parameters at one steady-state. Constraints on some variables were incorporated for illustrative purposes: y1 should not fall below 0.5 mol% methanol from setpoint; y2, whose setpoint was ﬁxed at 0.5 mol%, had an upper limit of 1.0 mol% methanol; and u2 had an upper limit of 1.8 kg/min. 3.2. Alternate control systems The two control systems whose robustness was compared were the same as those used by Ogunnaike and Ray [23] to compare the performance of a delay compensator they had developed. The ﬁrst system (System A) comprised two SISO PI controllers, and the second system (System B) comprised two SISO PI controllers with steady-state decouplers, calculated as the steadystate inverse of the plant model. The gains and integral times of each controller are shown in Table 2. A sample time of 1 min was used for both controllers. 3.3. Application of the robustness-testing procedure Step 1 characterised each type of plant-based and external uncertainty that aﬀects operations and proﬁtability. Uncertainty in setpoint and load variables results from changing economic conditions and upstream perturbations, the size of which were assumed to follow a triangular distribution deﬁned by minimum, average and maximum values. The economic weights for deviations from nominal values for manipulated and controlled variables and for violation of constraints also Table 1 Operating parameters at one steady-state Variable

Values

y1 y2 u1 u2 d

96.25 mol% methanol 0.5 mol% methanol 0.88 kg/min 0.77 kg/min 1.11 kg/min

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R. Di Mascio / Journal of Process Control 12 (2002) 745–751 Table 2 Gains and integral times of the SISO PI controller in Systems A and B [23]

System A Overheads controller Bottoms controller System B Overheads controller Bottoms controller

Gain

Integral time (min)

0.091 0.018

4.5 2.7

0.227 0.272

1.0 6.0

NB: (The change in sign bottoms controller gain is as reported in [23]).

vary according to economic conditions. These weights were assumed to be distributed triangularly. Uncertainty in the plant model gains, which aﬀects the gains in the steady-state decoupler, was also distributed triangularly. Noise was assumed to occur in the valve and measurement devices. Table 3 summarises these uncertainties. Step 2 conducted Monte Carlo simulations to show how each plant-control system responded to various combinations of uncertainties. Steady-state conditions prevailed at time zero in each replication. Setpoint, disturbance and model error uncertainties were sampled at time zero according to their distributions in Table 3. Valve and measurement uncertainties were sampled at every sampling interval. Each replication simulated 120 min of real-time, which was deemed suﬃcient for all disturbance and setpoint changes to settle out. At the end of each replication, system variables were recorded. Two thousand replications were conducted for each control system, as this number provided similar results to 2500 replications. Step 3 calculated the economic index of control quality for each replication. The Taguchi-based stability and performance indices, shown in Eqs. (1) and (2), were

constructed using the economic weights shown in Table 3. The calculations were performed using an Excel1 spreadsheet. Step 4 plotted frequency distributions of the Ltot values for each control system in Fig. 1. Most of the Ltot values for System A ranged from approximately 100 to 300 mu while most of the values for System B ranged from approximately 20 to 200 mu. From cumulative frequency data, 80% of the Ltot values for Systems A and B were under 260 and 150 mu respectively. As the tightness of a system’s distribution reﬂects its robustness, System B was more robust than System A. Further, the Ls Lp plots for the two systems (Figs. 2 and 3) revealed smaller absolute Ls and Lp values for System B, indicating superior economic stability and performance properties. It should be borne in mind that this procedure is limited to comparing the economic robustness of control systems. It does not determine the acceptable level of robustness, which requires engineering and managerial judgement. In fact, in practice either system may be acceptable or unacceptable. Also, as with other robustness

Fig. 1. Frequency distribution of Ltot values for System A (- - - -) and System B (—).

Table 3 Summary of uncertainties and their characteristics Uncertainty type

Parameters

Setpoint and load changes—triangularly distributed Minimum, average, maximum ysp 1 (mol%) Minimum, average, maximum d (kg/min)

95.75, 96.25, 97.0 0.88, 1.11, 1.34

Economic weights—triangularly distributed Minimum, average, maximum wu1 (mm/kg/min) Minimum, average, maximum wu2 (mm/kg/min) Minimum, average, maximum wuc1 (mm/kg/min) Minimum, average, maximum wy1 (mm/mol%) Minimum, average, maximum wy2 (mm/mol%) Minimum, average, maximum time wyc1 (mm/mol%) Minimum, average, maximum time wyc2 (mm/mol%)

5, 10, 15 15, 20, 30 45, 50, 60 3, 10, 15 5,15,20 12, 15, 18 17, 20, 22

Noise—uniformly distributed Measurement noise Valve noise

1.5% of reading 1.5% of reading

Model errors—uniformly distributed Gain error Time constant error

10% 20%

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Fig. 2. LsLp scatterplot for System A.

valve dynamic. For example, if an engineer has to decide whether to upgrade either measurement or valve devices to ones with faster dynamics, then this procedure can be used to compare the two systems’ degrees of robustness. Furthermore, the combinations of uncertainties producing high Ltot values can be investigated so that the control system can be programmed to trigger an alarm for operators to intervene. For example, the combination of a y1 setpoint of 96.47 mol%, a disturbance of 1.07 kg/min, and economic weights at their maximum values, results in the quality loss represented by point X in Fig. 1. Most importantly, the procedure provides results in economic terms using a graphical format that is more meaningful to business managers than an abstract mathematical quantity, such as the structured singular value [4]. By using this procedure, an engineer can present a more compelling argument for process control investment.

Acknowledgements

Fig. 3. LsLp scatterplot for System B.

assessment techniques, this procedure relies on information about a plant model, its operation and economic environment. Hence, the validity of the results depends on the quality of this information.

4. Conclusion This paper developed a procedure to estimate, in economic terms, the dynamic robustness of control systems in the face of plant-based and external uncertainties. The procedure involved identifying the types of uncertainty that aﬀect a plant’s operation and proﬁtability, and characterising those uncertainties by probability distribution or scenarios. It then conducted repeated computer simulations to show how the plant responded to various combinations of uncertainties, which enabled the procedure to be applied to SISO, MIMO, linear and nonlinear systems alike. The next step calculated Taguchi-based economic indices of control quality for each replication, and plotted a frequency distribution of those indices. The tighter the distribution, the more robust the plant-control system. This procedure produces a more realistic estimate of robustness because it characterises uncertainty by a range of probable values, rather than a single value. It can be used to compare the dynamic robustness of plant-control systems that diﬀer in any aspect, such as plant design, control algorithm, model type, measurement device or

I thank Michael Jacobson and two anonymous reviewers for helpful comments on earlier versions of this paper.

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