Interaction between process design and process control: the impact of disturbances and uncertainty on estimates of achievable economic performance

~

UTTE I N

RWQRTH E M A N

,L Proc. Cont. Vol, 5, No 1, pp. 49~62, 1995

Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0959-1524/95 $10.00 + 0.00

a

Interaction between process design and process control: the impact of disturbances and uncertainty on estimates of achievable economic performance J. B. Lear*, G. W. Barton t and J. D. Perkins* *ICl Australia Engineering, 16-20 Beauchamp Road, Matraville, NSW 2036, Australia tDepartment of Chemical Engineering, University of Sydney, NSW 2006, Australia ~Centre for Process Systems Engineering, Imperial College, London SW7 2BY, UK Received 4 September 1993; revised 16 May 1994 Model-based controllers permit existing processes to be operated close to their economically optimal conditions at all times. At the process design stage, however, a choice must often be made between alternatives with little information available as to the final form (or likely performance) of any control scheme. In this paper, a quick methodology is presented for examining the likely economic impact of disturbances and model uncertainty on the achievable optimal performance of a process. The resulting algorithm is based on a consideration of the amount the optimal operating point has to be 'backed-oft" from the active constraint set to ensure no operational constraints are violated. This algorithm was implemented within Matlab, a commercial numerical analysis package. The algorithm is applied to an illustrative single-input single-output example. Extensions to multivariable systems are discussed.

Keywords: interaction; design; uncertainty The fact that failure to consider simultaneously both steady-state and dynamic behaviour for any new process design can lead to operational problems has been appreciated, at least qualitatively, for a long time I. One of the major reasons why such a simultaneous assessment is not routinely performed is the lack of tools to calculate the economic cost of poor dynamic behaviour. Barton et al. 2 considered a set of 12 alternative mineral flotation circuits and showed via steadystate optimization, a range of open-loop controllability indicators and full dynamic simulations, that the 'best' steady-state circuits were different from the 'best' dynamic circuits. Using several of these flotation circuits to provide an illustrative example, Narraway et al. 3 presented a design stage method for quantitatively assessing the impact of disturbances on a plant's economic performance. In recent times, the examination of the interactions between process design and process control has assumed such importance that it now forms the subject of a dedicated conference series4. In the flotation circuit example cited above 2, it was assumed that each circuit would be operated as close to its optimal steady-state operating point as possible. However, without simultaneously considering the interactions between circuit dynamics, disturbances and process variations, the question of how close to the steady-state optimum it might be possible to operate

cannot be answered, and thus it is not possible to compare alternative flowsheet designs on the common basis of their optimal achievable economic performance. In this paper, a first-pass design stage technique is presented which allows an operating point to be determined such that under all expected process variations and disturbances, input and output constraints will not be violated. The operating point so determined should provide a reliable indication of the economic performance of the plant once built. Alternative flowsheets can thus be compared on a common economic basis. The basic algorithm is described in the next section. Constraint-handling model-based control schemes 5 have shown themselves capable of providing tight multivariable control 6. Indeed, commercial implementations of such controllers are now readily available and are regularly used in the process industries, particularly in cases where multiple quality specifications must be met simultaneously. However, despite the availability of a variety of model-based control schemes which can be used to ensure optimal performance from an operational plant, their use does not seem practical in assessing alternative process flowsheets at the design stage. The question of determining the 'best' possible control for a process at the design stage is addressed in the section on 'perfect' controller design.

49

Interaction between process design and control: d. B. Lear et al.

50

An illustrative single-input single-output example is presented in the following section. Application of the methodology to choose between a number of alternative flotation circuit designs will be presented in a later paper.

Optimal operation algorithm

Input

Outlmt

u

•\",,\NN_\\\\\\": y., Changes ..................."ii ................ related ' u°t-~ ............ by G(0)

yL

,,L

The steady-state operating point for a process is determined to ensure that all input and output variables lie within upper and lower bounds. In a dynamic sense, an operating point must be selected to ensure that under disturbance conditions and with process variation (equivalent to model uncertainty), the input and output constraints are not violated. As described below, this requirement may be analysed by modifying the process constraints. It is also necessary to check whether optimal economic operation occurs under open or closed-loop conditions. This question is discussed in the sub-section on partial control.

Figure 1 Nominal and optimal operating points - no disturbance. Nominal operating point -Y0,U0; optimal operating point - y * , , , *

Output

Input

uu

yu Buffer

o

UO*u.

.L

y,.

Figure 2 Open-loop disturbance - optimal operating point. Nominal operating point - y0,u0; optimal operating point - y*,u*; open-loop output magnitude - d

Constraint modification

The optimal operating point y* (that is, the setpoint values for the closed-loop controllers or the mean openloop measurement values) can be found by performing an optimization to maximize (or minimize) an objective function (qb) using a model of the process, expressed as equality constaints (h --- 0), such that the operational inequality constraints (g > 0) are not exceeded in the face of disturbances. In the most general case, this may be expressed mathematically in terms of the plant inputs (u), the plant measurements, or setpoints, (y), the disturbances (d) and other process model variables (z). Maximize ~(u,y,d,z) y

(1)

such that g(u,y,d,z) > 0 h(u,y,d,z) = 0

However, to consider the dynamic performance, some quantitative description of how disturbance dynamics affect the inequality constraints must be included in the above steady-state formulation. That is, how far from the active inequality constraints must the operating point be 'backed-off' to ensure constraints are not violated due to the dynamic effect of disturbances on the input and output variables. To develop the necessary concepts of constraint modification, a simple linear example will be used where the single output y and single input u must lie between upper (Yu, Uu) and lower bounds (YL, UL). YL -< Y -< Yu UL ~ U _< UU

(2)

This is shown diagrammatically in Figure 1. The positions represented by Y0 and u0 can be considered as the nominal operating point for this system. The relationship between changes in these steady-state values is therefore given by the steady-state model G(0). (Y - Yo) = G(O)(u - Uo)

(3)

If we optimize a linear objective function ~(y,u), then the optimum will occur when one of the input or output constraints is active, that is, when either y and/or u lie on an upper or lower bound. This is also shown in Figure 1 where the optimum is at y = y* = Yu and u -U* 5/= UL.

Returning to the nominal operating point (Y0, u0), we may now consider that a sinusoidal disturbance enters the system causing the output y to vary. The magnitude of the variation induced in y by the disturbance is d. If we were now to optimize this system, we must ensure that y does not approach either of its bounds closer than d. This is shown in Figure 2, where the optimal operating point is now at y* = Yu - d. Also shown in the diagram are 'buffers' which have been added to YL and Yu. Possible operating points must lie in the region between the buffers to ensure that the output bounds are not violated due to a disturbance. Note that for this open-loop situation, there is no variation in the input u due to the presence of the disturbance, although u has moved to a new u* due to the change in y Another situation that should be considered is the closed-loop case, with a controller present to reduce the effect of the disturbance by manipulating the process input uo Depending on the process and the controller, there may still be some output variation despite the controller's manipulation of the input. However, in our

Interaction between process design and control: J. B. Lear et al. Output

Input U/]

y*. Buffer

.,

YL < Y <- Y u

yu

Yo o

/

gOO

d' ¢a,-A4%/'-~Ar Pua

Y ,.

Figure 3 Closed-loop disturbance - optimal operating point. Nominal operating point - y0,u0; optimal operating point - y*,u*; dosed-loop input magnitude- ud

search for the best peformance that a process may ultimately achieve, we will initially consider the case of 'perfect' control. That is, some unspecified controller has been implemented which ensures that there is no output variation when the disturbance occurs. For the case above where the disturbance causes an open-loop variation in y of magnitude d, let the magnitude of the input manipulation necessary to achieve perfect control be u a. If this perfectly controlled process were to be optimized, we would now have to ensure that the value of u at the optimal operating point was at least u a away from its bounds. This is shown in Figure 3, where at the optimum u* = uL + u a. Also shown in this figure are the buffers of size u a which must be added to the upper and lower bounds for u when performing the optimization, so that the optimal value of u lies between these modified constraints. F o r this simple system, we now have two possible achievable optimal operating points. The first is when the system is open-loop which has an objective function value ~OL, and the second is for closed-loop perfect control where the objective function is qbCL. The 'better' of these two cases will define the achievable optimal performance for the process (qbA). Mathematically this can be described by Equations (4)-(6): (IDA = M a x i m u m

((I)oL,(I)cL)

(4)

C~(u,y)

(5)

~(u,y)

(6)

where (I)oL = Maximum Y

such that yL + d < - y < - Y u - d u L <_ u < u U

and

q)CL = Maximum Y

such that

51

U L + U d <-- U ~ blU -- U d

A useful extension here is the determination of the maximum disturbance that the system can handle. Considering first the open-loop case, the maximum disturbance that the system can handle without exceeding the output constraints is of magnitude l/2(Yu - Y L ) " Similarly, there will be a disturbance that under perfect closed-loop control will use all of the input magnitude available, that is, l/2(uL. - UL). If the maximum size of a disturbance entering the plant is greater than the maximum that can be handled in either the open or closed-loop cases, then feasible operation of the plant may not be possible. Partial control

It should be noted that it may be possible for a process to handle a larger disturbance than calculated in either the maximum open or closed-loop cases. This turns out to be a useful concept when determining the achievable optimum when process dynamics are taken into account. Consider the case where a 'less than perfect' controller is being used. That is, despite manipulation of the input, some of the disturbance still appears at the output. If this controller were designed so that when a disturbance enters the system despite using the maximum available input magnitude, some of the disturbance appears at the output, the capacity of the system to handle disturbances may be increased. Looking at this another way, some of the disturbance is being attenuated by input manipulation while the rest is allowed to appear at the output. In this paper, the case where a controller is designed so that only part of the disturbance is attenuated will be called 'partial control'. Although details of the implementation and calculational procedures involving partial control are given later, it is useful at this point to introduce the concept of the partial control factor, k. This can be thought of as a scalar which modifies the error signal being sent to the controller which is in a sense detuned. A value of k = 0 indicates that no error is being sent to the controller, which is the open-loop case. A value of k = 1 indicates that all the error is being sent to the controller, which is the normal closed-loop situation. We can now postulate two functions, Ay(k) and Au(k), which give the magnitude of the disturbance appearing at the output and the magnitude of the input required by a fixed controller, each as a function of the partial control factor k. The delta indicates that we are considering movement of the variable away from the nominal steady-state value. In general, we might expect that these two functions would show the following trends: Ay(k):

As k increases from 0 to 1, Ay(k) should decrease from the open-loop disturbance value

Interaction between process design and control: J. B. Lear et al.

52

Au(k):

(d), down to the minimum disturbance appearing at the output, being zero for a perfect controller. As k increases from 0 to 1, Au(k) should increase from zero for the open-loop case to the maximum input requirement for closed-loop control.

Later, we will see that the shape of these two functions is not always in agreement with simple intuition. It is worth emphasizing again that we are not concerned here with designing real control systems for real operational plants. Our fundamental problem is to estimate the best achievable economic performance of a process at the design stage, so that alternative flowsheets may be meaningfully compared on a common basis. The best achievable performance may correspond to the open-loop case or to the case where some measure of closed-loop control is employed. The concept of partial control has been introduced to provide us with a simple means of spanning between these two extreme possibilities.

Algorithm implementation It is now possible to set up an optimization problem which will find the optimal achievable operating point, such that disturbances which enter the system will not cause the operational inequality constraints to be violated. The result of this optimization will also indicate the value of the partial control factor, and thus whether it is optimal to operate the plant open-loop, closed-loop or with partial feedback control. Equation (7) gives the achievable optimal operating point for a plant with a linear steady-state model and objective function, and for linear (or linearized) dynamics. The change in the process economics from the nominal steady-state value is taken to be due solely to the modification of the input and output constraints (Au, Ay) to account for process dynamics. Such an essentially linear analysis is an adequate tool to assist in the ranking of flowsheet alternatives at an early stage in the design procedure. Although only limited process model and controller information may be available, a design stage estimate of achievable economic performance can be made and used in early comparative studies.

Optimal operation algorithm Maximize ~(u,y) y$

y = G(0)u

uL+An_
(7)

Au = f(k) : Au(0) = 0, Au(1) = max(AucL ) Ay = g(k) : Ay(0) = max(AYoL), Ay(1) = max(AYcL)

The algorithm as given in Equation (7) has been expressed in terms of input (u) and output (y) vectors. This is an extension of the single-input single-output example used earlier to the multivariable case. The constraint modification procedure is unaltered, with buffers being added to each input and output constraint. The size of the buffer will be different for each variable, as given by the vector functions Au(k) and Ay(k). Note, however, that for simplicity the partial control factor k has been retained as a scalar, modifying the error inputs to the 'fixed' multivariable controller equally. Operation of the process with the calculated setpoints y* and with a controller modified by k*, ensures that despite the worst possible disturbance entering the system, the input and output constraints will not be violated and the objective function calculated will be both optimal and feasible. In order to use this algorithm, information relevant to both the optimization problem (that is, an objective function, a steady-state model and the necessary inequality constraints) and dynamic behaviour (that is, Ay(k) and Au(k)) is required. To calculate Ay(k) and Au(k) a dynamic model (together with an uncertainty description) and disturbance information are required, so that a controller can be designed, to allow the bounds on the input and output variations to be determined. Note that the optimization problem to be solved is in a more restrictive form than the general problem posed in Equation (1). In practice, both the objective function and the inequality constraints may include unmeasured variables (z) which are neither inputs nor outputs. In addition, the inequality constraints shown in Equation (7) are upper and lower bounds on each individual input and output variable, rather than mixed (linear) functions of the input and output variables. 'Dummy' variables can be introduced to extend Equation (7) to the more general case, as demonstrated by Lear 7.

'Perfect' controller design The solution to the optimization problem given by Equation (7) is the optimal operating point at which a given process design can operate without any inequality constraint being violated. To determine the 'best' achievable operating point, the controller used needs to be the 'best' available, in some sense. One possibility for this is a 'perfect' controller, whereby all disturbances entering the system are completely attenuated, with the measured variables never deviating from their setpoint values. However, as will be discussed later, while a perfect controller is useful in that it can provide a quick estimate of the upper bound on the achievable performance for a given process design, such controllers are frequently unrealizable. For this reason, a realizable controller that incorporates model uncertainty information (in our case, an H . controller) would be expected

Interaction between process design and control: J. B. Lear et al. to provide a more meaningful estimate of the achievable performance for a given process. Closed-loop performance measures

Given a fixed controller C, our optimization problem requires estimates of the maximum input and output magnitudes as a function of the partial control factor k (that is, Ay(k) and Au(k)). To calculate these, we will need the sensitivity (E) and complementary sensitivity (H) functions. E=(I+GC)

1 =Y-d

(8)

H = (I + GC)-~GC = y

(9)

r

These functions are written in terms of the plant (G) and controller (C) transfer function matrices. From a performance point of view (that is, for disturbance rejection), it is desirable to make E as small as possible. Note that for real processes, the most desirable value for E (that is, zero) can generally only be approached over a limited frequency range. Partial control

As noted previously, the concept of partial control was introduced to allow varying degrees of control from open-loop through to closed-loop to be incorporated into the optimal operation algorithm. An output of the Equation (7) optimization is the partial control factor k*. This may be thought of at the design stage as indicating whether tight closed-loop control is likely to be necessary to ensure optimal economic performance, or whether it is better to operate with somewhat looser control or in the extreme case with no control at all. For simplicity, the effective controller C' is taken to be klC, whereby each error signal is scaled equally. For a simple design stage screening tool, it was felt that the additional computational effort incurred by permitting each error signal to be scaled independently was not justified. For the closed-loop system with a partial controller, the relationships between the disturbance at the output (d), the measurements (y) and the controller outputs (u) are thus given by: y = (I + GCkl) ld

(10)

53

y = (1 + GCk)-ld y =d

(if

(12)

IGCkl < <

1)

This means that the magnitude of the disturbance measured at (or reaching) the output is independent of k, and is in fact much the same as in the open-loop case (as might be expected for a 'small' controller term). Alternatively, if GCk is 'large' then Equation (10) becomes: y = (1 + GCk)-ld Y = 1 (GC)-ld g

(13) (if I G C k l I

>> 1

In this case, the disturbance appearing at the output is inversely proportional to k. Although this change is in the expected direction, intuitively the relationship might have been expected to be linear. For the controller output, when GCk is 'small', Equation (11) becomes: u = Ck(1 + GCk)-ld u --- k C d

(if

Iafkl

(14)

< < 1)

which means that the controller output is proportional to the partial control factor k, a relationship that is in agreement with simple intuitive expectations. If GCk is 'large', then Equation (11) becomes: u = Ck(1 + GCk)-ld u=Gd

(15)

( i f l G C k ] > > 1)

Here, the input to the plant generated by the controller is independent of k. This is the least expected result, that under certain circumstances, as the controller is effectively 'turned down', the magnitude of the controller output remains unchanged. Given the above results for the single-input singleoutput case, it is difficult to make any general predictions about the variation of input and output magnitudes with k for the multivariable case. Therefore, in all cases a numerical approach was adopted, where for a given value of k, each of the two maximum magnitudes was determined, as shown in Equation (16), consistent with the disturbance description. Ay,(k) = M a x i m u m [y = (I + GCkI) ld]i for any d

(16) u = CkI(I + GCkl)-td

(11) While Equations (10) and (11) are applicable to the multivariable case, it is instructive to consider the single-input single-output case, and see how k might be expected to affect the measurement and the controller output for a given disturbance. Looking first at the measurement, if the term GCk is 'small' then Equation (10) becomes:

Aui(k) -- Maximum [u = C k I ( I + GCkl)-ld]j for any d

where i, j represent the ith and jth components of vectors y and u. Details of examples involving multiple disturbances are given in Lear 7. However, for the single-input singleoutput case, the disturbance d may be described in terms of the magnitude of a bounding function w:

Interaction between process design and control: J. B. Lear et al.

54

[dlto < ]w(s)lto

(17)

Im l G ( i t o ) ] ~

Re [G(R0)I v

The magnitude of the output variation is given by: [Yl = l(1 + GCk)-ad[

(18)

= I(1 + aCk)-llldl Substituting for the disturbance description gives: [Yl-< I(1 + GCk)-llfw]

(19)

= 1(1 + GCk)qwl The partial control function Ay(k) is therefore given by: Ay(k) = Maximum 1(1 + Gfk)qwl

Figure 4

Uncertainty regions on the Nyquist plane

(20)

for any to

zero. This leads to the definition of the perfect IMC controller (Q).

A similar development gives the result for Au(k): Au(k) : Maximum ICk(1 + GCk)qw[

(21)

for any

Q=G q

These maxima may be determined by plotting the magnitude of the right-hand side of Equations (20) and (21) against frequency over the required range of k values. For a given controller and disturbance description, the partial control functions Ay(k) and Au(k) can be generated using the technique described above. It should be noted that if the description of the disturbance entering the system is not in terms of its direct effect at the output, but rather first passes through a disturbance transfer function (Gd), then the bounding function term w in the above expressions should be replaced by GdW.

Perfect control

In order to consider 'perfect' control, it is convenient to use the concept of internal model control (IMC), whereby a model of the plant (G) is introduced into the controller. The sensitivity function (E) and the complementary sensitivity function (H) can be determined for the IMC structure 8. For the nominal case, where the model is the same as the process (that is, G -- G'), these functions simplify significantly. E = (I - G a)(I + (G - G ) O)-' ~E=I-GQ

(22)

(ifG = G)

n = GQ(I + (G - G)Q)-' H = GQ

E = I-

(23)

(ifG=G)

A perfect controller should ensure that whatever disturbance enters the system, the outputs should not change. This requires that the sensitivity function should be

GQ

(24)

(if E = 0 )

This simple form for the perfect controller could give the impression that perfect control can be readily achieved. However, limitations for real systems, including dead-times and right-half plane zeros, mean that it is generally not possible to invert the process model. In addition, model uncertainty can also limit the achievable performance if we are to ensure closed-loop stability. Model uncertainty descriptions

To extend the concept of perfect control into a more useful tool for the examination of the optimal achievable performance of alternative processes, the uncertainty associated with the models used must be considered. The linear models used here will generally be derived by linearizing a nonlinear dynamic model. Any deviation of the process operating conditions away from the point of linearization gives rise to model inaccuracy. In addition, model uncertainty can arise from a variety of other sources including changes in process conditions (for example, catalyst activity changing with time) or deliberate model simplification (such as neglecting high-order dynamics). The method that will be adopted here to describe model uncertainty is based in the frequency domain. At a given frequency (to), the nominal value of the model transfer function is represented by the point G(ito) on a Nyquist plot. As a result of uncertainty, there will be a region around this point in which the actual (or 'true') transfer function will lie. Extending this representation over a given frequency range will map out a region on the Nyquist plane which contains the actual transfer function. Figure 4 shows an example for two frequencies, where there is uncertainty in both the gain and phase shift of the transfer function.

Interaction between process design and control: J. B. Lear et al.

Im [ ; ( i t ~

I lm(iO)) [ < ]m(t

Re [G(i~0)l

(

,

w

(30)

A multiplicative uncertainty is employed in the illustrative example presented in the next section.

H~ controllers Original Unce

~ /

~..Additive Uncertainty

Approximation Figure 5 Diskapproximationto uncertaintyregion

For mathematical convenience, the actual uncertainty is often approximated by a series of discs, as shown in Figure 5. Depending on the original uncertainty description, this may be a very conservative approximation, adding considerably more uncertainty than may actually occur. For the disc shown in Figure 5, with a radius [2 the family of all possible plants H can be described mathematically as: H = {G" IG(ito) - G(ito) I < 1, (to)}

(25)

As before, G(io) is the nominal plant, now at the centre of the disc, and any plant within the family H satisfies the following relationship: G(ko) = G (ion) + la (ko)

(26)

where II.(ito) I---7a(¢0)

(27)

Equation (26) is an additive uncertainty description, where the bound on the uncertainty, Equation (27), is a function of frequency. This description allows the flexibility of having a smaller uncertainty at low frequencies (that is, near steady-state conditions) where the transfer function is better known, and a larger value at high frequencies where the uncertainty is generally greater. Another method of describing the uncertainty is to use a multiplicative description, limiting the relative error by an upper bound. The family of plants II in this case is given by:

FI={ G''G(im)-~(i°D'[ G(ic0) [

<-- im(O))}

GOoD = G(im)(1 + Im(ioD)

Although we wish to use the best possible controller in any design stage comparison between alternative processes, perfect control generally cannot be achieved due to certain inherent characteristics of the process itself and the model being used. Consequently, the H~ design procedure was adopted as a consistent method for designing the 'best' controller, as it explicitly incorporates model uncertainty and performance requirements in its design procedure. Simply put, the objective of the H= design procedure is to minimize the sensitivity function (that is, maximize disturbance rejection) over a specified frequency range. In addition, the controller must ensure closed-loop stability and meet a specified performance requirement for all processes which may occur within the given uncertainty description. These requirements can be formalized as:

Nominal stability.

The closed-loolL system with the process model at its nominal value ((7) must be stable.

Nominal performance.

The performance of the nominal closed-loop system, as measured by the sensitivity function, must always be within the designer's specification.

Robust stability.

The closed-loop system, for all possible process models included within the uncertainty description, must be stable.

Robust performance.

The performance of the closedloop system, for all possible process models included within the uncertainty description, must always be within the designer's specification. These requirements are increasingly demanding, with nominal stability and performance each being a subset of the robust case. The use of an H~ controller design procedure allows each of these criteria to be achieved. In the following example, the IMC based procedure of Morari and Zafiriou 8 was used to design an H= controller. Basically their approach consists of two steps. First, a controller (Q) is designed t o achieve nominal performance. Then a filter (F) is added to the controller to ensure robust stability and performance.

(28)

Single-input single-output example

In this case, each member of II satisfies:

where

55

(29)

Before presenting this example, it is worth reminding ourselves that the traditional approach to process design has generally used the optimal steady-state

Interaction between process design and control: J. B. Lear et al.

56

performance as the sole economic indicator of performance.

subject to the steady-state model and a set of input and output inequality constraints:

Process description

Y = 3ul + 4u 2 + d

In this example, let us assume that we are being asked to choose not between two different process designs, but between two different plant inputs in a conventional SISO control scheme. The input which is not to be used .for control will be set to a calculated optimal value. To further simplify matters, we will only consider the extreme cases of open-loop (k = 0) and closed-loop (k -1) control - the partial control factor will not be calculated. A more comprehensive example involving the choice between a set of alternative flotation circuits, with the partial control option duly considered, is given elsewhere 7 and will be described in a later paper, The state-space description of our 'process' is given by: dx d t = 3ul + 4u2 + d ' - x (31)

Y < 10 u1 > 0

y=x

where u~ and u2 are the two possible inputs, x is the single state variable, y is the output and d' is the disturbance entering the process. An equivalent transfer function relationship is given by Equation (32). Note that here the disturbance is given as d, the value which actually appears at the output.

1

s+l

(32) d'(s)

The disturbance expected to enter the system (d') has a zero mean value, and a frequency content described by the following transfer function:

Id'r--- lO0-d This allows a maximum magnitude of 1 at low frequencies, dropping to 0.01 at a frequency of 1 radian per unit time. The frequency range that will be considered throughout this analysis is l0 -3 to 10 radians per unit time, as this essentially covers steady-state (to-~ 0) through to ten times the frequency corresponding to the system time constant. The objective of 'operating' this process wili be taken as the maximization of the following (linear) objective function: max • Ul,U2

= - 5 u l - 6u 2 + 7 y

u2 > 0 Optimization of this system without allowing for the impact of the disturbance (that is, for d' = d -- 0), process dynamics and model uncertainty gives the following result: •

-- 55

y -- 10 u1 = 0

(36)

u2 = 2.5 with both the output y and the input u~ at their constraint limits. A number of different options will now be investigated in determining the optimal achievable performance from this process. Firstly, the consequences of running the plant open-loop will be considered. Then, the closed-loop case will be analysed using both perfect and Ha control. For both closed-loop controllers, two cases will be considered corresponding to the use of u 1 and u 2 as the input variable. Open loop

3 4 uz(s ) + d ( s ) y(s) = -u 1(s) + s+l s+l d(s) =

(35)

(34)

To operate the plant without control requires an operating point to be selected which is sufficiently removed from the output constraint to ensure that the disturbance will not cause constraint violation. For this openloop case, input constraints need not be modified as there is no input manipulation. Therefore, as a first step the maximum open-loop magnitude of the disturbance appearing at the output of the plant (AYoL) needs to be evaluated. Using the bounded description of the disturbance entering the plant (d') and the transfer function relating this to the disturbance as it appears at the plant output, gives the following: [d(s)[

< s 1 1 1 + 1 100s + (37)

AYoL = m a x s l 1+1 o~ + 1 lOOs The magnitude of AyOL is shown in Figure 6. The maximum value of this function (and others throughout the paper) was found using standard routines within the commercial numerical analysis package Matlab. For this simple example, the maximum value is I (as to -~ 0) and, therefore, the modified output constraint becomes:

Interaction between process design and control: J. B. Lear et al.

57

to achieve this, and consequently the appropriate input constraint might need to be modified to ensure that for optimal operating conditions sufficient input action is available. The modification will be by the maximum amount of input manipulation required by the controller, AucL. For a conventional feedback structure, the relationship between a disturbance appearing at the output ( d ) a n d an input to the process is given by Equation (11). Substitution of the equivalent IMC controller Q into this relationship gives:

101

10-1

10-a

10-3

u -- Q(1 + ( G - G ) Q ) - ' d 104

10-3

10-2

10-1

10o

10!

F,requemy Figure 6

Magnitude

of

AyoLv e r s u s

(42)

which for the case where the model is known perfectly (that is, G = G) simplifies to:

frequency

(43)

u = Qd

Y - (Yu - AYoL) = 9

(38)

The result of performing the optimization again but with this modified constraint is: =49.5 y

= 9

The maximum magnitude of the input required may be determined using the transfer function description of the bounded disturbance entering the plant, as shown in Equation (44), and then plotting the modulus of the right-hand side of this equation against frequency. The results are given in Equation (45).

(39) l u(s) l

U I =0

<-IQ(s)

I

1 1 s + 1 100s + 1 (44)

u 2 -- 2.25 AUcL = max Q(s) - - 1 1 1 ,o s + 1 100s +

The change in the objective function is: AdO = ~(with disturbance) - cI) (no disturbance) = 49.5 55 (40) = -5.5 Therefore, ensuring that the disturbance does not cause the output constraint to be violated in the open-loop case requires changing the operating point, with the result that the mean value of the objective function decreases by 5.5 units to 49.5. The disturbance may thus be viewed as 'costing us money'.

Case 1 -

-

(u 1 - y )

Case 1

Q = 0 -1

(u 1 - y)

Case 2

Q=0

l

(u2 - y) s+1 3

s+1 4

(41)

To calculate the optimal achievable performance under perfect control, the output constraint does not require modification. However, input manipulation is required

~

U -> u L + A U c L

y)

(45)

u, -- 0.25

Performing the process optimization with the modified input constraints gives the results shown in Equation (46).

do = 54.8333

(u 1 - y )

Perfect control means that despite the presence of the disturbance d', the measurement remains constant due to manipulation of one of the inputs. Using the IMC configuration, the perfect controller is obtained as the inverse of the plant model G.

~

(u2

u 2 >_ 0.25

Case 1 Perfect control

Case 2 U _> U L + A / , / C L

Case 2

do= 55

(u 2 - y )

y=10 uj = 0.3333 u 2 = 2.25 A'/) = -0.1667

v

=

10

UI =0

(46)

u 2 = 2.5 Ado=0

Use of the alternative employing the manipulation of u~ gives a slight decrease in the objective function, whereas there is no decrease at all in the case where u 2 is used to maintain the output. This is because at the nominal optimum (see Equation (36)), u 2 was sufficiently removed from its lower bound that it did not have to be moved to allow for its use for control. As a result there is no cost associated with the use of u 2 as the control input. Therefore, if economic criteria were being strictly employed, at this stage a process designer would be

Interaction between process design and control: J. B. Lear et al.

58

inclined to choose a closed-loop arrangement rather than open-loop, with u z being used in preference to u~ as the means of controlling y. The final stage in our illustrative example is to consider the impact that model uncertainty and the requirement that a realizable control system be employed have on the process economics.

(47)

where lot[ < 0.6, 1131~ 0.6. These uncertain parameters mean that the value of the transfer function at any given frequency is imprecisely known, so that the real plant is just one member of a family of possible plants. To design a controller that will give both robust stability and performance, the parametric uncertainty must be approximated using an uncertainty description. For this example, an unstructured multiplicative uncertainty description (see Equations (28)-(30)) was chosen. Equation (48) gives an expression for l m as a function of ot and [3.

lm= G(s)- ~(s)

&s)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

..-° 100

(i) .........

A perfect controller can be used to give a theoretical upper limit to how well a given process design might be controlled. However, such perfect controllers generally cannot be used in practice. Consequently, to provide a consistent estimate of the 'best' achievable control, an H= controller was designed for the two cases. To add some realism to the example, parametric uncertainty in the transfer function dynamic model was considered. Uncertainty in both the gains and time constants was included using the ot and [3 terms in the following equation: 3(1 + a) 4(1 + a) Ul(S) + UE(S) (1 + fl)s + 1 (1 + fl)s + 1

,

lOq

1-1= controllers

y(s) =

lOt

(48)

(a-#)s +a (1 + fl)s + 1 Note that as the steady-state gain from the original transfer function does not appear in this general formula, Equation (48) is, in fact, valid for both choices of input variable (that is, u I - y and u 2 - y). To find the upper bound on the multiplicative uncertainty/,,(to ), the relative error lm was calculated for the four cases corresponding to the extreme variations of the ot and [3 parameters - (i) a = 0.6, [3 = 0.6 (ii) ot = 0.6, [3 = -0.6 (iii) ot = -0.6, [3 = 0.6 (iv) ot = -0.6, [3 = -0.6. Figure 7 shows that combination (ii) has the largest magnitude. Therefore, the multiplicative uncertainty bound should have a magnitude equal to this case, which leads to the following frequency domain representation:

I~o-3

(ii)

.................

(iii)

.............

Ov)

. . . . . . . . .i~2 . . . . . . . . . . . . . .m-, ...........

loo

lot

V.,mt~r, ey Figure 7

Multiplicative

uncertainty

for four parameter

im(~O) = 1.2i09 + 1 0.4i0 + 1

extremes

(49)

As discussed earlier, the IMC procedure was adopted for the design of an H= controller to achieve both robust stability and peformance. In this approach, a filter time constant (0) must be found to meet the following robust performance (and hence robust stability) design criterionS:

I~/)ml+

[EWll < 1

V¢o

(50)

Here w~ is a weighting on the sensitivity function that allows the performance of the controller to be selected within the design procedure. The weighting function chosen: Wl =

1 1 - s + 1 lOOs+ 1

(51)

was selected to give higher performance at low frequencies where disturbance effects are large with a decrease in performance at higher frequencies. Substituting the sensitivity and complementary sensitivity functions with their IMC controller relationship (that is, using the process model inverse together with a first-order filter) into Equation (50), gives the following robust performance bound: [GG 1Fim [ +1 (1-GG-IF)wl [ < 1 1 "Ws+l

1.2Sl+(1 0.4s+

1 ) 1 Ws+l s+l

1 1 (52) 100s+

By trial and error, a filter time constant ~ = 0.9 was found to just satisfy this bound, as shown in Figure 8. With an H= controller in use, the impact of a disturbance will result in both input manipulation and output variation. However, when the model of the process is

Interaction between process design and control: J. B. Lear et al.

59

I0e

101

-4

10-1

l

10-2

i

10-3 " ' " ' " " "

1.1 1.0 0.9 0.8

......... ......

10-4

3

IO'l~s

Input

10-2

10-1

IDa

101

........

,

,

,

,,,nil

I0-i0-3

Output

i

i

a

A

i

i

l i , i H

. . . . . .

I0o

101

Frequency

Robust performance bound (differentfilter time constants)

uncertain, the calculation of the maximum possible input and output changes is not straightforward, with all possible plants within the multiplicative uncertainty description needing to be considered, over all possible frequencies. Fortunately for the SISO case, it is possible to determine analytically the maximum input (AUcL) and output (AYcL) magnitudes. The derivations o f the formulae in Equation (53) are given in the Appendix. AUcL=

......

10-1

r~mu~ney Figure 8

h

i0 a

Figure 9

H = c o n t r o l l e r , case 1 i n p u t a n d o u t p u t m a g n i t u d e s

Table 1

Illustrative SISO example results

Controller Case Base case None Perfect Perfect H~ H=

1 2 1 2

u1 0 0 0.3333 0 0.8333 0

u2 2.5 2.25 2.25 2.5 1.8513 2.4763

y

qb

Aqb

10 9 10 10 9.8995 9.8995

55 49.5 54.833 55 54.031 54.447

0 -5.5 -0.1667 0 -0.969 4).553

max I u l GeYl,t0

=

[G-'Fdl 1

max

Optimization using these modified constraints yields the results shown in Equation (56).

(1-I F I i~)

tO

(53) AYcL=

max [ y l

Case 1

GeH,t0

(u I - y)

(t(1-1Fllm))q

=max

Case 1 =

0.8333

(Ul - Y)

Case 2 ~ A U C L

=

0.6250

(u2 - Y) AYcL = 0.1005

(54) AYcL = 0.1005

In each case, both the input and output constraints require modification, as shown in Equation (55). Case 1

Case 2 U

~

UL q - A U c L

(u I - y) u1 -> 0.8333 Y "~ Y u - AycL y < 9.8995

~

U ~ U L -b A U c L

(u2 - y) ~ u2 >_ 0.6250 Y -< Yu - AYcL =~ y _< 9.8995

(55)

• = 54.447

(u2 - y) y = 9.8995 u I =0.8333 u 2 = 1.8513 A@ = -0.969

Figure 9 is a plot of these two magnitudes as functions o f frequency for Case 1. The maximum values calculated for AucL and AycL for each of the two cases are given in Equation (54): ~ A U c L

Case 2

@ = 54.031

y = 9.8995 uI = 0 u 2 = 2.4763 AiD = 4).553

(56)

Although the use of u2 is still the more profitable option, the mean objective function value in this case is significantly less than if perfect control were employed. The inclusion of model uncertainty considerations, and the subsequent use of an H~ controller to ensure robust stability and performance, have moved the operating conditions, and thus the process economics, away from the 'traditional' optimization result given in Equation (36).

Discussionof results Table 1 shows that for this

example there is a considerable economic incentive to employ closed-loop rather than open-loop control. However, this result will not always be true. Application of this methodology to the ranking of alternative flotation circuits 7 showed that in some cases (particularly at high levels of uncertainty), there was a distinct economic advantage in running some circuits open-loop.

Interaction between process design and centre# J. B. Lear et al.

60

NyquistPlot

NyquistPlot

:2 ~

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

6.~

-2

i

-',

tiplicative

~

;

~

.

;

~

~

;

6

Figure 10 Uncertaintyregions,frequency= 0.1 radiansper unit time

Figure11 Uncertaintyregions,frequency= 1 radianper unit tir,le

For this simple example, although the closed-loop ranking is the same for both perfect and H~ control (that is, Case 2 is always the more profitable option), there is no reason to believe that in the general case it would be possible simply to consider perfect control alone as a means of ranking candidate designs. Certainly, with the commercial software available today, the design of an H® controller does not significantly complicate this design-stage tool, while it certainly provides a better indication of the expected economic performance of the alternative processes. As might have been expected, as more 'realism' is incorporated (perfect then H® control), the controller becomes less effective with a greater amount of the disturbance's effect remaining unattenuated at the output. This illustrates the importance of having a methodology that considers the control of plants different from the nominal model, a s there may well be plants contained in the uncertainty description which are considerably more difficult to control than is the nominal process. Finally, it is worth examining the use of the unstructured multiplicative uncertainty description in the H~ controller design and subsequent input and output magnitude analysis. It is well known that such (approximate) uncertainty descriptions can be very conservative, including in the analysis many plants (possibly significantly 'worse' plants) that were not in the original uncertainty description. Figures 10 and 11 are Nyquist plots showing at two frequencies, the nominal plant, the parameter (a and 13) based uncertainty region and the multiplicative uncertainty region. It is clear that in both cases the multiplicative uncertainty region is much larger than the region describing plants based on known parameter variations in the model. Also shown on each plot is the 'worst' plant, that is, the plant corresponding to the largest input and output magnitudes for the given disturbance description. The technique used to determine the 'worst' plant is given in the Appendix. At the lower frequency, the worst plant is actually quite close to the parametric uncertainty region. However, at the

higher frequency, the worst plant is well away from those which could possibly ocur due to parameter variation. Although, it would undoubtedly be possible to reduce the level of conservativeness employed by using structured uncertainty descriptions, such an approach did not seem warranted in a methodology targeted at the early design stage, when available information is limited and a quick ranking of alternatives is required.

Conclusions At the process design stage, it is important to be able to compare alternatives on a common economic basis. To date, steady-state optimization has been used for such comparisons, with sensitivity analysis (to consider the effects of disturbances and model variations) carried out via perturbation studies. No approach seems to be available that permits the calculation of the simultaneous impact of process dynamics, model uncertainty and controller complexity on the process economics. With this general aim of the ranking of alternatives in mind, a method has been proposed for estimating the best performance that a plant can achieve for given descriptions of likely disturbances and model uncertainty. The technique modifies input and output constraints to ensure that the optimal operating point is sufficiently removed from these bounds so that input manipulation and output variation due to the impact of disturbances do not cause the constraints to be violated. In the illustrative example presented, three levels of control (open-loop, perfect and H~ control) were considered. It was found that for this example, closedloop control improved the economic performance of the plant. However, as the closed-loop control was made more realistic, the value of the achievable objective function fell. In terms of realistic control, H~ control provides an upper bound on what may be implemented, and consequently an upper bound on what might be achieved for the process design under study. Here the difference in the objective function between the open-

Interaction between process design and control: J. B. Lear et al. loop and H~ closed-loop control cases provides a measure of the economic benefit achievable through tight control of the process. Importantly, the techniques used here were implemented within commercially available software (Matlab), readily available to any design engineer. Finally, the optimal operation algorithm can be applied to more complex multivariable systems 7 and, indeed, can be readily modified to include meeting specified requirements on output variance. Thus, it is possible to incorporate the requirements of the recent drive for improved product quality at the early stages of flowsheet development and evaluation. Application of this methodology to the selection of the 'best' flotation circuit from a range of possible alternatives will be presented in a later paper.

1 Anderson, J. S. Chem. Eng. 1966, 97 2 Barton, G. W., Chan, W. K. and Perkins, J. D. J. Proc. Cont. 1991, 1, 161 3 Narraway, L. T., Perkins, J. D. and Barton, G. W. J. Proc. Cont. 1991, 1,243 4 Perkins, J. D. (Ed), 'Interactions Between Process Design and Process Control', IFAC Workshop, London, 7-8 September 1992 5 Garcia, C. E., Prett, D. M. and Morari, M. Automatica, 1989, 25, 335 6 Richalet, J. Automatica 1993, 29, 1251 7 Lear, J. B. 'The Effects of Uncertainty on the Economics of Optimising Control' PhD Thesis, University of Sydney, 1993 8 MoraN, M. and Zafiriou, E. 'Robust Process Control' PrenticeHall, London, 1989

Appendix: Analytical derivation of maximum input and output magnitudes for the SISO case The sub-section on perfect control uses the maximum input and output magnitudes for an uncertain SISO system to determine the optimal achievable operating point. The derivation of these formulae (see Equation (53)) is given below. Previously a relationship was given for the input in terms of the IMC controller Q, the uncertain process transfer function G, the perfect process model G and the disturbance appearing at the output d: u = Q(1 + ( G - G)Q)-Id

Imaginary ,.........- ....

...... • ..,..

Real

i

...-

RaOius= FIi. -''I!'''/...

q Figure A.1 SISO multiplicative uncertainty magnitudes

Max au t = Max Q a,~o lm,~O 1 + QlmO

References

61

d

(A.2)

The IMC controller is given by Q = G-IF and thus:

Max Au = Max f f -1F d G,,o lmxo + El m

(A.3)

which may be written as:

(pl./ Max Au[ = Max - axo ,m,O~ [ l + F l m J

(A.4)

At a given frequency, the values of G, d and F are known, the only variable being lm. This therefore fixes the value of the numerator on the right-hand side of Equation (A.4). The expression can therefore be maximized at a given frequency, by minimizing the value of the denominator, [l+Flm[. The minimum value of this term can be found geometrically as shown in Figure A. 1. This Nyquist plot shows the vector representations o f F , Flm,1 + 0 i a n d l + F l m. The only restriction on lm is an upper bound on its magnitude. Therefore, the magnitude of Fl m is limited by:

(42) IFZ~l = IFI Ilml

For the SISO case, this equation can be rearranged and used to define the maximum input magnitude for any possible process transfer function G and frequency to;

MaxAu = M a x . _ ~ - ~ .~ Q d ~,~o o,,o ~ . , % . _..:

(A.1)

-
In addition, the argument (Arg) of Fl m, or its rotational position, can be shown to be unrestricted: Arg(Flm) = Arg(F) + Arg(lm) with

Substituting G with the multiplicative uncertainty defined by Equations (29) and (30) gives:

(A.5)

-2~r < Arg(lm) < 2~r

(A.6)

62

Interaction between process design and control: J. B. Lear et al.

Imaginary °°°.....°' .....

tude for any possible process transfer function G and frequency to:

............:..,,Min (1 + FI.)

.°°

MaxAyc:,
# ~ • ",°°°

(A.7)

1-." _

;

V

,%

v

,,,l ,.' t,

ee .,

m l woelt

Real

I\ ,

,.

|

i

,."

FI.

i,\ .''" ~

AyCL ----- m a x ~v GeYI,m

,"

..................... 1 "S"Fl "

Once again, this expression is maximized by minimizing the denominator, giving the second of the results presented in Equation (53):

1

= max [ [ (_1_F.)d[ /

<,, I,,(1-I

(53)

F I lm))

Figure A.2 Multiplicative uncertainty, maximum magnitudes

Therefore, the vector Flm can lie anywhere within a circle with centre the origin and radius IFllm. However, a limitation on the size of IFIlm is set by the robust performance specification, given by Equation (50). In effect, this requires the magnitude of IFIIm to be less than unity. From Figure A.2, the minimum value for 1 + FIm is clearly 1 - IFIIm. This may now be substituted into Equation (A.4) to give the first expression in Equation (53):

The final analytical result used in the sub-section on perfect control is the actual value of G which maximizes the input and output magnitudes, that is, an expression that defines 'the worst possible plant'. Looking at Figure A.2, the vector which causes this maximization is -IFllm + 0i. We may therefore write the following equality:

-IFIlm = Elm

(A.8)

which may be rearranged to give: lm =

- I F I im

1

(A.9)

AUcL = m a x ul GeFl,to

= max

I G-1Fd[

<,, (1-1 FI

(53)

ira)

For the SISO case, a similar line of reasoning to that employed for calculating the maximum input magnitude leads to an expression for the maximum output magni-

This in turn may be substituted into the definition of the multiplicative uncertainty (Equation (29)) to give the value of the 'worst' process transfer function as a function of frequency: