Inventory Control Structure Independence of the Process Operability

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Ind. Eng. Chem. Res. 2002, 41, 3970-3983

Inventory Control Structure Independence of the Process Operability Index David R. Vinson*,† and Christos Georgakis‡ Chemical Process Modeling and Control Research Center and Department of Chemical Engineering, Lehigh University, Bethlehem, Pennsylvania 18015

A new measure of process operability has been developed by the authors. This steady-state, multivariable, and nonlinear measurescalled the operability index, or OIsevaluates the effectiveness of a given design by quantifying the ability of the process inputs to drive the outputs over the desired output space while rejecting expected disturbances. The OI is independent of the inventory control structure, a very important property. This property enables the OI to be applied during the process synthesis phase prior to control structure selection, thus permitting the comparison of operability for competing designs. This paper provides proof of the property of inventory control structure independence for linear process models (low and high dimension, square, more inputs than outputs, and more outputs than inputs) and the more general nonlinear processes. 1. Introduction The output controllability index (OCI) was introduced1,2 in order to provide a method to evaluate the operability of specific process designs. The OCI is designated the operability index, or OI, in this paper to avoid confusion with well-established controllability indices. Emphasis was placed on the steady-state operability of a process, a necessary condition for overall process operability. It was shown to overcome the inability of existing measures of controllability and operability, such as the relative gain array,3 relative disturbance gain,4 and closed-loop disturbance gain,5 to accurately represent the operability of a process over its full required range of operation. See work by Vinson6 for an in-depth literature review of steady-state and dynamic controllability and operability measures. In the domain of output variables, the OI is defined by the relationship

OI )

µ(AOS ∩ DOS) µ(DOS)

(1.1)

where DOS, the desired output space, represents the multidimensional space incorporating the complete range of operation over which the controlled process outputs are required to vary. The function µ(‚) represents a measure, such as volume, of the corresponding space. For the OI to accurately distinguish between operable and nonoperable processes, the entire set of constraint-limited outputs must be included. The DOS can be mathematically described by

{∀ yD,i ∈ DOS|yDmin,i e yi e yDmax,i, 1 e i e c} (1.2) * To whom all correspondence should be addressed. Phone: (610) 481-8496. Fax: (610) 481-2177. E-mail: vinsondr@ apci.com. † Current address: Air Products and Chemicals, Inc., 7201 Hamilton Blvd., Allentown, PA 18195. ‡ Current address: Polytechnic University, 728 Rodgers Hall, Six Metrotech Center, Brooklyn, NY 11201. E-mail: [email protected]

where c represents the number of constraint-limited outputs. The DIS, or desired input space, is the set of values obtained by translating the DOS into the input space via the inverse of the process model.

DIS ) f-1(DOS)

(1.3)

The AOS, the achievable output space, represents the set of values the process outputs can actually achieve given the available range of input variables defined in the available input space AIS.

AOS ) f(AIS)

(1.4)

where

{∀ uA,i ∈ AIS|uAmin,i e ui e uAmax,i, 1 e i e c} Therefore, the OI is a measure of the ratio of the intersection of the available and desired output spaces to the DOS. In other words, it represents the fraction of the DOS that can be reached with the existing design. In general, good designs will have an OI approaching 1.0, and poor designs will have an OI significantly less than 1.0. Because the OI can also be represented graphically, visual inspection of low-dimensional intersections can guide decisions to improve the controllability of the design. A representation of the spaces involved in the OI calculation for a dual-composition distillation column in the LV configuration is shown in Figure 1. The compositions of the top and bottoms products are the important outputs to be controlled. This configuration uses the liquid reflux to control the overhead product composition, the boilup vapor to control the bottoms product composition, the overhead product flow to control the condenser level, and the bottoms product flow to control the reboiler level. Control of the pressure at a constant value is assumed. The DOS is the rectangular space bounded by 0 e y1 e 0.1 and 0.8 e y2 e 1.0, and the AOS is the large striped space generated by varying the inputs by (50% around the design point. The AOS covers an area much larger than the DOS, but it fails to cover the super-pure regions, indicating that the process operability is less

10.1021/ie0109814 CCC: $22.00 © 2002 American Chemical Society Published on Web 07/31/2002

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Figure 1. OI for a dual-composition control distillation column.

than 100%. One way to overcome this operability limitation is to consider an enlargement of the AOS by an increase in the number of trays or other design modifications. Alternately, it could also be that the definition of the DOS is inappropriate because it is difficult or impossible to reach some regions of its space. In this situation, the DOS should be redefined. The control scheme implicit in the calculations supporting Figure 1 is the LV configuration, where the reflux flow L and the boilup flow V are used to control the distillate and bottoms product compositions; the level in the condenser is controlled by the distillate flow D, and the reboiler level is controlled by bottoms flow B. However, there are a number of alternate control schemes that could be selected, such as using the distillate product flow and boilup flow (the DV scheme) or the reflux liquid flow and bottoms product flow (the LB scheme) for composition control. At first glance, it would appear that the value of OI would be dependent on the control scheme selected, but for a fixed process design, the steady-state operability of the distillation column given above should be the same regardless of the inventory control scheme. Therefore, usefulness of the OI is predicated on its independence from the selected inventory control configuration comprised of the level controllers. This paper will show the independence of the OI from the choice of the inventory control structure for the distillation column described above, and it will extend the proof to any process that can be described by heat and material balance equations. Consequently, the value of the OI expresses an inherent operability characteristic of the process and is of significantly greater value than inventory control structure dependent measures. The following sections will prove first the independence of the OI from the inventory control structure for a 2 × 2 linear process and then for nonsquare linear processes. After the level independence property of the OI for linear processes is established, it will be proven for a 2 × 2 nonlinear process and then for n × n nonlinear processes. Finally, some application examples will be demonstrated.

2. Linear Square Processes The proof that the OI is independent of the regulatory level control structure is presented in this section for the general linear n × n case. To motivate the general case, we will first consider an example 2 × 2 case. Two different approaches will be used. The first will prove that the maximum achievable output value is the same regardless of the level control pairing, and the second will prove that the vertexes of the AOS are identical regardless of the input selected for level control. 2.1. A 2 × 2 Example Process. Given a simple linear 2 × 2 process, the steady-state gain matrix G0 can be used to calculate the steady-state process outputs from the inputs via

() (

y a y ) G0u, or y1 ) a11 2 21

)( )

a12 u1 a22 u2 , where -uim e ui e uim (2.1)

In eq 2.1 the ui’s and yi’s are deviation variables from the reference steady-state point of (0, 0). It is also assumed that the input range for ui, (-uim, uim), is symmetric about (0, 0) and the same for all ui. This assumption is made to simplify the calculations presented for this example and to make the idea clear. This limitation will be removed in the following section under the more general proof. The steady-state gains between inputs and integrating variables such as level are infinite, but it has been shown that the rate of change of level can be substituted for the absolute level for steady-state analysis.7,8 Here y1 represents the nonintegrating output and y2 the rate of change of the integrating, or level, output. Control of y2 will be assigned to a single-input, single-output (SISO) pair prior to evaluating the steady-state operability. Initially, it will be assumed that u2 will be the input assigned to control the rate of change of level y2. At steady state,

a21u1 + a22u2 ) 0, or u2 ) -(a21/a22)u1 (2.2) This relationship eliminates a degree of freedom for

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Figure 2. Excess input range available for cases A and B.

control of y1 and forces u2 to respond to changes in u1 so that the rate of change in the level is maintained at steady state. Note that if |a21| > |a22| (case A), the range that u1 achieves is less than the maximum (-u1m, u1m) because u2 reaches its limits in the input space first. When u2 reaches its limit, then the corresponding value for the upper limit of u1 is u1* ) |a22/a21|u2m and the ranges (-u1m, -u1*) and (u1*, u1m) are not useful for controlling the output y1. Similarly, if |a21| < |a22| (case B), the range of u2 will be less than (-u2m, u2m) because u1 reaches its limit first. These two cases are schematically depicted in Figure 2. The relationship between the remaining input and output is now simplified to be

y1 ) (a11 - a12a21/a22)u1

(2.3)

This expression leads to the calculation of the AOS with the range of y1 given as (-|y1m|, |y1m|). For case A, y1 is calculated as

y1m ) (a11 - a12a21/a22)(-a22/a21)u2m ) (a12 - a11a22/a21)u2m

(2.4)

For case B, y1 is calculated as

y1m ) (a11 - a12a21/a22)u1m

(2.5)

Now, the alternate level control pairing will be examined, namely, controlling the rate of change of level y2 with u1. This level controller introduces the same interdependence between u1 and u2 given by eq 2.2 and leading to the following steady-state gain relationship between y1 and u2:

y1 ) (-a11a22/a21 + a12)u2

(2.6)

Cases A and B of Figure 2 apply equally well for this control structure if the inverse ratio a22/a21 is substituted for a21/a22 and the AOS for y1, (-|y1m|, |y1m|), is calculated identically to be those given in eqs 2.4 and 2.5. For case A

y1m ) (a12 - a11a22/a21)(-a21/a22)u1m ) (a11 - a12a21/a22)u1m

(2.7)

which is identical to eq 2.5, and for case B

y1m ) (a12 - a11a22/a21)u2m

(2.8)

which is identical to eq 2.4. Consequently, for case A or B, the AOS (-|y1m|, |y1m|) is independent of the choice of the variable that controls the level. It should be noted that these cases characterize not only the control structure but also the process characteristics, making this conclusion quite general. The trivial case of a21 ) a22 leads immediately to the conclusion that the AOSs are identical regardless of the level control structure. Appendix A contains an alternate proof based on the vertexes of the linear spaces being identical. To facilitate the more general discussion that is given in the following section, it is instructive to examine the saddle change in the AIS that is caused by the inventory control structure. Before the inventory control structure is selected, the AIS appears to be a two-dimensional one, namely,

{(u1, u2)|-uim < ui < uim, i ) 1 and 2}

(2.9)

When the level control structure is selected, the loss of 1 degree of freedom forces the AIS to be governed by one input {u1|-u1* < u1 < u1*}, or equivalently {u2|u2* < u2 < u2*}. For case A maximum values of the inputs are u1* ) |a22/a21|u2m and u2* ) u2m; for case B the corresponding maximum input values are u1* ) u1m and u2* ) |a21/a22|u1m. In addition, after the level controller is selected the AIS becomes a constrained onedimensional subset of the original AIS, and it is the same one for any of the level controller choices. Furthermore, the relationship of eq 2.3 represents an overall material balance for the process which limits the independent movement of the u1 and u2 input variables. This AIS depends on the process design characteristics distinguishing case A from case B, but it is not affected by the choice of level controller. Half of the AIS is depicted in the thick diagonal line in either the left of right part of Figure 2. 2.2. General Square Case. Moving now to the general square linear case, the output will be divided into integrating outputs yI and nonintegrating outputs yN. The inputs available for the control of the integrating and nonintegrating outputs will be divided into four


groups, u1, u2, u3, and u4. It is necessary to designate the inputs into four classes as specified in order to prove that two arbitrary selections of input variables for control of the integrating output variables will lead to the identical calculation of the OI. The manner in which these four sets of input variables are selected will be described shortly. It will be assumed that yI ∈ RnI and yN ∈ RnN. First, the case in which two arbitrary sets of input variables I ) {ui1, ui2, ..., uinI} and K ) {uk1, uk2, ..., uknI} are selected as alternate choices for the control of the yI variables is examined. In these assignments, ij, kj ∈ {1, 2, ..., n} for j ) 1, 2, ..., nI where n ) nI + nN is the total number of outputs or inputs. The two sets of input variables selected above might have p(0 e p e nI - 1) common input variables. These common variables are designated as u3, where u3 ∈ Rp. The input variables that are in set I but not in set K are denoted by u2, where u2 ∈ RnI-p, and the input variables that are in set K but not in set I are denoted by u4, where u4 ∈ RnI-p. The input variables that are not included in either set K or set I will be denoted by u1, where u1 ∈ Rn-2nI+p. Consequently, the set I of input variables is identical to the union of the inputs in u2 and u3, and the set K is identical to the union of the set of input variables in u3 and u4. These definitions lead to an expression for the steadystate gain matrix as

( ) (

yN A11 yL ) A21

A12 A22

A13 A23

()

u1 A14 u2 A24 u3 u4

)

(2.10)

where u1 ∈Rn-2nI-p, u2 ∈RnI-p, u3 ∈Rp, and u4 ∈RnI-p. The shapes of the submatrices are governed by the number of integrating variables and the possible pairings of inputs with integrating outputs. For example, A22 is a nI × (nI - p) matrix. Two cases of input variable selection for control of the integrating variables will now be examined. Case I: Integrating Variables Controlled by u3 and u4. Just as in the previous section, the rate of change of the integrating outputs is zero, which leads to the relationship between the inputs u4 assigned to control the integrating variables yL and the outputs as

( )

( )

u u 0 ) -(A21|A22) u1 + (A23|A24) u3 2 4

(2.11)

Notice that both the (A21|A22) and (A23|A24) matrices are square and will be assumed to be nonsingular. When (u3T, u4T)T is solved for, the variables for integrating output control can be found as

( )

( )

u3 u1 -1 u4 ) (A23|A24) (A21|A22) u2

(2.12)

The steady-state matrix for control of the yN outputs is given by

[ ]

u yN ) [(A11|A12) - (A13|A14)(A23|A24)-1(A21|A22)] u1 2 (2.13) For the AOS to be independent of the inputs used to control the integrating outputs, the vector yN must be shown to be the same for all pairings. For this case, the

inputs (u1T, u2T)T must be selected inside the AIS which also satisfy the relationship in eq 2.11. Case II: Integrating Variables Controlled by u2 and u3. Solving eq 2.10 for the case where u2 and u3 control the integrating variables instead of u3 and u4 leads to

( )

( )

u u 0 ) (A21|A24) u1 + (A22|A23) u2 4 3

(2.14)

and just as in case I, the expression for the variables for integrating output control can be found as

( )

( )

u2 u1 -1 u3 ) -(A22|A23) (A21|A24) u4

(2.15)

This yields the following expression for the nonintegrating outputs:

( )

u yN ) [(A11|A14) - (A12|A13)(A22|A23)-1(A21|A24)] u1 4 (2.16) The AOS can now be calculated from the above relationship and the values of (u1T, u4T)T that are in the AIS and also satisfy eq 2.14. The AISI of case I will now be proven to be identical to the AISII of case II. If there are no common input variables in the two input sets I and K (p ) 0), then vector u3 is nonexistent and the above relationships reduce to those below. Case I. The inputs used to control the integrating outputs, expressed in terms of u1 and u2, are given by

( )

u u4 ) -A24-1(A21 A22 ) u1 2

(2.17)

and the nonintegrating outputs are

( )

u yIN ) [(A11 A12 ) - A14A24-1(A21 A22 )] u1 2

(2.18)

Case II. Similarly,

( )

u u2 ) -A22-1(A12 A24 ) u1 4

(2.19)

and

( )

u A A14 ) - A12A22-1(A21 A24 )] 1 yII N ) [( 11 u4

(2.20)

When eq 2.17 is substituted into eq 2.20, the following expression is derived: -1 -1 yII N ) (A11 - A12A22 A21)u1 + (A14 - A12A22 A24)u4

) (A11 - A12A22-1A21)u1 + (A14 A12A22-1A24)[-A24-1(A21u1 + A22u2)] ) (A11 - A14A24-1A21)u1 + [(A14 - A12A22-1A24)(-A24-1)A22]u2 ) (A11 - A14A24-1A21)u1 + (A12 + A12A24-1A14)u2 (2.21)

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which is identical to eq 2.18 and proves that yIN ) yII N for all values of the AIS that satisfy eq 2.11. If p > 0, case I is described by

[ ]

u yIN ) [(A11|A12) - (A13|A14)(A23|A24)-1(A21|A22)] u1 2 (2.22) and for case II

[ ]

u1 -1 yII N ) [(A11|A14) - (A12|A13)(A22|A23) (A21|A24)] u 4 (2.23) both under the constraint that

0 ) A21u1 + A22u2 + A23u3 + A24u4

(2.24)

It will now be shown that yIN ) yII N for all of the allowed values of (u1T, u2T)T or (u1T, u4T)T in the AIS. This can I be demonstrated by transforming yII N to yN. -1 yII N ) (A11u1 + A14u4) - (A12|A13)(A22|A23) (A21u1 + A24u4)

) (A11u1 + A14u4) + (A12|A13)(A22|A23)-1(A22u2 +

( )

A23u3)

u ) (A11u1 + A14u4) + (A12|A13) u2 3

) A11u1 + A12u2 + A13u3 + A14u4

( )

u ) (A11|A12) u1 2

( )

u (A13|A14)(A23|A24)-1(A21|A22) u1 2 )

yIN

(2.25)

Having labored through the foregoing proof, there is an alternative proof that is equivalent but more compact and abstract. It can be argued that the AOS is identical regardless of the inputs assigned to control the integrating outputs in the following manner: Regardless of the level control structure selected, the AIS is the same. It actually is the intersection of the set defined by the customary initial constraints on all of the inputs (e.g., ui,min e ui e ui,max, i ) 1, 2, ..., n) and the nN-dimensional subspace defined by the constraining relationships for the material balances, namely

0 ) A21u1 + A22u2 + A23u3 + A24u4 For any choice of the level control structures, the AOS is the transformation of the AIS defined above by the following nN × n matrix: (A1|A2|A3|A4). The proof is then complete. 3. Nonsquare Linear Systems Nonsquare systems are typically encountered in industrial practice and lead to situations where there are either excess degrees of freedom (more inputs than outputs) or too few degrees of freedom (fewer inputs than outputs). In these cases, it might be suspected that the assignment of input variables to control the inte-

Figure 3. Block diagram of a 2 × 3 process with SISO level control.

grating outputs would influence operability calculations. The discussion below will show that this is not the case and that the OI remains independent of the level control structure. The case of more inputs than outputs will be dealt with first by examining an example process. 3.1. More Inputs than Outputs: A 2 × 3 Linear Example. The following steady-state gain matrix describes the example 2 × 3 process:

() (

y a y ) Gu, or y1 ) a11 2 21

a12 a22

)(

u1 a13 u2 a23 u3

)

(3.1)

with the AIS defined by the following boundaries on the inputs:

-uim e ui e uim, i ) 1-3 Here again the second output is defined at the rate of change of a level. It will be shown that the AOS, and hence the OI, is not affected by the choice of the variable that controls the level. Assuming that the level output y2 is at steady state, the three input variables will need to satisfy a relationship similar to that of eq 2.2 but for three inputs.

a21u1 + a22u2 + a23u3 ) 0

(3.2)

An enumerative approach to proving independence of the OI for this 2 × 3 linear example, similar to that used in section 2.1, is presented in appendix B. A more general discussion follows. A block diagram of the process with the integrating output y2 controlled by the input u3 is shown in Figure 3. It can be seen that at steady state the derivative of the output will be zero and hence the pairing of u3-y2 reduces the effective dimension of the problem to 2 × 1. Similar dimensional reduction occurs regardless of the input assigned for level control. Figure 4 presents a geometrical perspective of the dimensional reduction of the AIS for gain values a21 ) 1, a22 ) 0.75, and a23 ) -1.5, which are representative of a material balance equation. The cube represented by the graph axes in the threedimensional view of Figure 4 corresponds to the initial AIS for the three input variables, but the requirement to control the level y2 in a SISO pairing produces a linear constraint (eq 3.2) that is imposed on the three input variables. The resulting AIS is represented in the three-dimensional view by the shaded hyperplane. The three three-dimensional views clarify that the shape of the hyperplane is a hexagon, which is the general case for the 2 × 3 case. Selection of different gains can reduce the shape of the hyperplane to a parallelogram (a21 ) 4, a22 ) 5, and a23 ) 6 will do this).


Figure 4. (a) Three-dimensional view of AIS for controlling y1. (b) Effect of three SISO level controllers on the active projections of the AIS for a 2 × 3 process.

The requirement to control the level y2 in a SISO pairing also produces an AOS (-y1m e y1 e y1m and y2 ) 0) that is represented by a line rather than an area. Operability for the output y2 with respect to only one input is calculated to be equal to C1 ) min(1, a11) with respect to u1, C2 ) min(1, a12) with respect to u2, and with respect to u1 and u2 is C12 ) min(1, a11 + a12), where C1, C2, and C12 represent reduced forms of the OI. For example, if a11 ) 0.4 and a12 ) 0.7, then C1 ) 0.4, C2 ) 0.7, and C12 ) 1. The true controllability of the process is 1 because two inputs are available to control the outputs y1 and y2. The preceding discussion leads to the conclusion that the calculation of the AOS is not dependent on the level controller used. 3.2. More Outputs than Inputs: A Linear 2 × 1 Problem. The special case of more outputs than inputs requires a new approach to defining operability because all outputs cannot be simultaneously controlled at independent setpoints. Therefore, a modified definition of the AOS will be presented first, followed by application to a specific example. 3.2.1. Definition of the AOS. The discussion to follow will be based on a simple 2 × 1 example. It should be noted that the discussion in this section is based on nonintegrating outputs (the next section deals specifically with the presence of integrating outputs).

() ( )

y1 a11 y2 ) a21 u1

(3.3)

where it is also assumed that -1 e u1 e 1, -1 e y1 e 1, and -1 e y2 e 1 and that a11 and a12 > 0. It is obvious that we cannot simultaneously control both y1 and y2

Figure 5. Range of y1 and y2 values for two outputs and one input.

at independent setpoints. The independence of the two outputs is schematically denoted in Figure 5. If there are no disturbances, the range of y2 would be max(-1,-a21/a11) e y2 e min(1, a21/a11), forcing a fixed ratio y1/y2 ) a11/a21. Clearly, the possible range of y2 values is set by the intersection of the line y2 ) a21/a11y1 with the limits of y1. Using the definition of DOS and AOS developed for the square case leads to a definition of poor controllability because the OI is defined by the narrow intersection of the AOS (line) with the DOS (square). Because the nonsquare case with excess outputs occurs frequently in industry when a multivariable model predictive controller is utilized, a new definition of the AOS that appropriately indicates operability is needed. The achievable output values for a given input u1 will be not be defined as a single point in the AOS (y1, y2) but as a larger set that has a nonzero measure. This provides more freedom of movement for the input variable (larger AIS) to achieve the DOS and to reject disturbances, leading to a larger OI. For example, the

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y3, the following equation shows that the AOS is equal to that produced from the y1-u2 pairing:

y2 )

Figure 6. DOS and AOS defined for two outputs and one input.

first output can be fixed at y1s, and the second output range can be defined as a range dependent on y1s:

y2-(y1s) e y2 e y2+(y1s)

(3.4)

In the framework of a multivariable model predictive controller, this amounts to fixing critical outputs at constraints and permitting the other outputs to vary within their minimum to maximum values. Another way to describe the practical impact of this definition is that the inputs are not required to keep y2 at a fixed value when affected by a disturbance, but rather y2 can be anywhere between the minimum and maximum value for a given y1. This leads to a definition of the AOS as shown in Figure 6. The AOS space is striped because it consists of lines, not points, whereas the AOS as defined by the square definition is simply represented by the line a21/a11. By definition, the OI is calculated as

OI )

µ(AOS ∩ DOS) µ(DOS)

(3.5)

which results in a reasonable value of OI for the defined y2 range. This indicates that this process controllability depends on the range of y1. 3.2.2. A 3 × 2 Linear Process. The argument that the OI is independent of level control pairing is virtually identical to the square case. For completeness, the initial calculations for a 3 × 2 process with one integrating variable is described.

( ) ( )( )

y1 a11 a12 u y y ) Gu, or 2 ) a21 a22 u1 , 2 y3 a31 a32 -uim e ui e uim, i ) 1 and 2 (3.6) where y3 will be designated the integrating variable. If u2 is assigned to control the level, the relationship between y1 and y2 becomes

y2 )

a21a32 - a22a31 a˜ 2 y1 ) y1 a11a32 - a12a31 a˜ 1

(3.7)

Comparison with the relationship between y2 and y1 in the previous section indicates that the AOS as defined there is valid in this case. If alternately u1 is paired with

a22 - a21a32/a31 a22a31 - a21a32 y1 ) y ) a12 - a11a32/a31 a12a31 - a11a32 1 a˜ 2 a21a32 - a22a31 y1 ) y1 (3.8) a11a32 - a12a31 a˜ 1

Therefore, the choice of input to control the integrating variable does not affect the calculation of the AOS, and hence the operability measure OI. 3.3. General Nonsquare Linear Case. The arguments in sections 3.1 and 3.2 clearly showed that a specific choice of input(s) assigned to control the integrating output(s) has no impact on the operability measure OI or the underlying input and output spaces. Proof that this conclusion applies to the general nonsquare case would follow the same logic as that for the general square case in section 2.2. Once the input, output, and gain matrices are divided into nonintegrating and integrating parts, the same logic would be applied to prove that the choice of input to control the integrating variables does not affect the operability measure OI. 4. Nonlinear Processes It is necessary to consider whether the conclusion that the operability measure OI is independent of the assignment of inputs to control integrating outputs for linear processes is also valid for nonlinear processes. This will be shown to be the case by first extending the proofs of level independence from linear to nonlinear processes for a 2 × 2 process. Subsequently, it will be shown that the OI is also independent of the level control structure for a n × n process regardless of the number of level control SISO pairings. 4.1. A 2 × 2 Nonlinear Process. The nonlinear form of eq 2.1 is

y1 ) f1(u1,u2) y2 ) f2(u1,u2), with -uim e ui e uim, i ) 1 and 2 (4.1) and f1 and f2 are the nonlinear functions describing the relationship between process inputs ui and outputs yi. Just as in the linear case, it is assumed that y2 represents the rate of change of the level variable. In first case (case A) to be examined, u2 will be the input assigned to the SISO pair to control y2. At steady state y2 ) 0 for all steady-state values of y1. The relationship for y1 in eq 4.1 can then be solved either analytically or numerically to give

u2 ) hA(u1)

(4.2)

where the function hA is to ensure that f2[u1,hA(u1)] ) 0. The subscript A indicates the first of the two cases to be considered here. In physical terms the function hA indicates how the input u2 needs to change to ensure that the level variable is at steady state as the remaining degree of freedom u1 is moved over its allowable values. This leads to the following expression indicating


From eqs 4.6 and 4.7, the relationship between the two AISs is given by

AISB ) hA(AISA)

and

AISA ) hB(AISB) (4.9)

Substitution of the second of these relationships into eq 4.8 arrives at a relationship between the input space for the y2-u2 pairing and the output space for the y2u1 pairing of

AOSB ) gB[hA(AISA)]

(4.10)

It can also be argued that the functions gA(‚) and gB[hA(‚)] are identical by careful inspection of their definitions. Consequently, it has been proven that

AOSA ≡ AOSB Figure 7. Description of AISA and AISB for a 2 × 2 nonlinear process.

how the other output y1 now depends on the as yet free input variable u1:

y1 ) f1[u1,hA(u1)] ≡ gA(u1)

(4.3)

Because u1 ∈ AISA (the subscript on AIS specifies the case under consideration), it can be seen by examining eq 4.3 that the function gA maps the AIS to the AOS. So, the following expression can be written for the transformation of the spaces:

AOSA ) gA(AISA)

(4.4)

The AISA in eq 4.4 is graphically described in Figure 7. Curve 2 represents the material balance equation f2(u1,u2) ) 0 influencing the level variable. The corresponding AISA and AISB are one-dimensional intervals in the u1 and u2 axes, respectively. However, only those values of u2 in AISA that lead via the hA(‚) function to u1 values inside the original two-dimensional AIS are allowed. Similarly, for case B, only those values of u1 that lead to u2 values inside the original two-dimensional AIS are allowed. Case B will now be addressed, where u1 is used to control the level variable y2. Using logic similar to that of case A, u1 ) hB(u2) so that f1[hB(u2),u2] ) 0 for all available values of u2. This yields y1 as a function of u2:

y1 ) f1[hB(u2),u2] ≡ gB(u2)

(4.5)

It is critical to observe that hA(‚) is the inverse function of hB(‚):

u1 ) hB(u2) ) hB[hA(u1)], u1 ∈ AISA

(4.6)

u2 ) hA(u1) ) hA[hB(u2)], u2 ∈ AISB

(4.7)

The importance of this relationship is that the two AOSs for different level control structures are identical, and thus the operability characteristics of the 2 × 2 process are not affected by the choice of the level control structure. 4.2. Nonlinear n × n Processes. The foundation for the n × n case will be developed by first examining a 3 × 3 process

y1 ) f1(u1,u2,u3) y2 ) f2(u1,u2,u3) y3 ) f3(u1,u2,u3), where -uim e ui e uim

(4.12)

In a manner very similar to that of the previous section, it can be shown that the AOS is identical regardless of the SISO pairing for level control. The y3-u3 level pairing yields the following results:

y3 ) 0 ) f3(u1,u2,u3), or u3 ) f3-1(u1,u2) ) h33(u1,u2) (4.13) Substituting into eq 4.12

y1 ) f1[u1,u2,h33(u1,u2)] ≡ g1(u1,u2) y2 ) f2[u1,u2,h33(u1,u2)] ≡ g2(u1,u2)

(4.14)

and expressing in terms of the input and output spaces

AOS33 ) g33(AIS33)

(4.15)

When the expression for the y3-u2 SISO pairing is developed in a similar manner

y3 ) 0 ) f3(u1,u2,u3), or u2 ) f3-1(u1,u3) ) h32(u1,u3), or u3 ) h32-1(u1,u2) ) h33(u1,u2) (4.16)

and

This is simply because both function hA and hB are solutions to the same equation constraining the respective values of u1 and u2. As above, this expression can be written in terms of the input and output spaces.

AOSB ) gB(AISB)

(4.11)

(4.8)

The input-output relationship is

AOS32 ) g32(AIS32)

(4.17)

and because g33(u1,u2) ) g32(u1,u3), the output and input spaces are equivalent. The 3 × 3 case will now be discussed for the specific case where both y2 and y3 are levels. Looking first at the direct y2-u2 and y3-u3 pairing, logic from the

3978


previous cases yields

u2 ) f2-1(u1,u3) ) h22[u1,h33(u1,u2)] ) hˆ 22(u1) u3 ) f3-1(u1,u2) ) h33[u1,h22(u1,u3)] ) hˆ 33(u1) (4.18) y1 ) f1[u1,hˆ 22(u1),hˆ 33(u1)] ) g1(u1)

(4.19)

AOSI ) g1(AISI)

(4.20)

When the pairing for level control is reversed

u3 ) f2-1(u1,u2) ) h23(u1,u2) ) h23[u1,h32(u1,u3)] ) hˆ 23(u1) ) hˆ 33(u1) u2 ) f3-1(u1,u3) ) h32[u1,h23(u1,u2)] ) hˆ 32(u1) ) hˆ 22(u1) (4.21) which again proves that the AOS is identical for different level control pairings. The concept can now be extended to the full multivariable notation. The system is expressed in vector notation as

y ) f(u)

(4.22)

Breaking the input and output vectors into dedicated level control pairings versus nonlevel process inputs and outputs yields

Figure 8. Dual-composition control distillation column.

with the letters indicating the inputs used to control the distillate and bottoms composition, respectively. The LV control structure is shown in Figure 8. In this scheme, the liquid reflux L is used to control the overhead composition xD and the vapor boilup is used to control the bottoms composition xB; the column levels are controlled by the product off-takes D and B. The material balance equations are

F)D+B

yN ) f(uN,uL), 1 e N e η1 yL ) f(uN,uL), 0 e L e η2

xFF ) xDD + xBB (4.23)

where uN represents the inputs available for control of the process variables yN and uL represents the inputs dedicated to control of the levels yL. Because the vector yL ) 0,

uL ) fL-1(uN) ) h(uN)

(4.24)

B + V ) qF + L V + F(1 - q) ) L + D

where q is the fraction liquid in the feed flow F, which has a composition xF. For a fixed feed flow, the LV structure constraint equations are

V e L + qF, B g 0

which leads to this relationship between the inputs and outputs

yN ) f[uN,h(uN)] ≡ g(uN)

(5.1)

V g L - F(1 - q), D g 0

(4.25)

0 e V e 2Vs

or

(5.2)

0 e L e 2Ls

AOS ) g(AIS), where AOS ∈ DIM(uN), AIS ∈ DIM(uN) (4.26) so

y ) g(u), u ∈ AIS

(4.27)

It is not required that yN and uN be the same dimension, so extension to the general nonsquare case would follow a similar logic. 5. Distillation Column Example A binary distillation column with control of both the distillate and bottoms composition will be used as a process example to demonstrate the inventory control structure independence principle for the OI. To accomplish our purpose, the LV, DV, and LB control structures will be considered. The naming of these control structures follows conventions in the literature,

where Vs and Ls are the steady-state design conditions for the reflux and boilup flows and 2Vs and 2Ls are selected for this example as the maximum steady-state reflux and boilup flows. These constraint equations produce an input map as seen in Figure 9. The polytope that bounds the input space is labeled at its vertexes v1-v6. The coordinates based on the design variables (Ls, Vs) are determined from the intersection of the two applicable active constraint equations. For a linear model the translation of the input space vertexes to output space vertexes is calculated by

∆Y ) G∆U so for

[] [ ] y1 xD y2 ) xB

and

[ ] [] u1 L u2 ) V


Figure 10. AIS for the DV configuration.

Figure 9. AIS for the LV configuration.

Table 1. Output Space Vertexes for the LV Structure

then

[ ] [

δxD g11 δxB ) g21

vertex

][ ]

g12 δL g22 δV

(5.3)

v1

For example, the v3 coordinates are calculated as

v2

[

v3

][

]

g11 g12 2Vs - qF - Ls ) g21 g22 Vs g11(2Vs - qF - Ls) + g12Vs g21(2Vs - qF - Ls) + g22Vs

[

]

(5.4)

v4 v5

The output space vertexes for the LV structure are seen in the table below. If it can be shown that the vertexes for the output spaces of the LV, DV, and LB control structures are identical, then the output spaces will be proven identical, and hence the OI for each structure would be identical. Transformation between different control structures has been addressed in several papers,9-11 and it is sufficient to show that the material balance equations in eq 5.1 yield

v6

input

[ ] [ ] [ [ ] [ [

output

[

-Ls -Vs

-g11Ls - g12Vs -g21Ls - g22Vs

-Ls Vs

2Vs - qF - Ls Vs

[ [

-g11Ls - g12(Fq - Vs) -g21Ls - g22(Fq - Vs)

]

]

g11(2Vs - qF - Ls) + g12Vs g21(2Vs - qF - Ls) + g22Vs

[

g11Ls + g12Vs g21Ls + g22Vs

Ls Vs

]

][ ] [

]

]

Ls 2Ls - F(1 - q) - Vs

g11Ls + g12[2Ls - F(1 - q) - Vs] g21Ls + g22[2Ls - F(1 - q) - Vs]

F(1 - q) - Ls -Vs

g11[-Ls + F(1 - q)] + g12Vs g21[-Ls + F(1 - q)] + g22Vs

]

]

so

( ) (

δxD g11 + g12 δxB ) g21 + g22

)( )

-g12 δD -g22 δV

(5.6)

The constraint equations for the DV structure are

0eDeF

δD ) δV - δL

V g D - F(1 - q)

or

( ) (

δD -1 ) δV 0

V e D + 2Ls - F(1 - q)

)( )

1 δL 1 δV

0 e V e 2Vs

so

( ) (

δxD -g11 δxB ) -g21

)( )

g11 + g12 δD g21 + g22 δV

for the DV structure. For the LB structure

δB ) δL - δV or

( ) (

δD 1 ) δV 1

)( )

0 δL -1 δV

(5.5)

(5.7)

The input map for the DV structure is found in Figure 10, and the calculated output space vertexes are shown in Table 3. Substitution of Ds ) Vs + F(1 - q) - Ls permits comparison of the vertexes in Table 2 with those of Table 1. When Tables 1 and 2 are compared, it can be seen that the vertexes for the LV and DV structures are rotated but equal. The same is true for the LB structure. The equivalencies between vertexes are shown in Table 3. Therefore, the OI calculated for each of the inventory control structures will be identical. This allows use of the OI to compare competing process designs regardless of the inventory control structure implemented. It is

3980


Table 2. Output Space Vertexes for the DV Structure vertex v1 v2 v3 v4 v5

v6

input

[ ] [ [ ] [ [ ] [ [ ] [ [ ] [ [ ] [

g11Ds - Vs(g11 + g12) g21 - Vs(g21 + g22)

-Ds 2Ls - F(1 - q) - Vs

g11Ds + (g11 + g12)[2Ls - F(1 - q) - Vs] g21Ds + (g21 + g22)[2Ls - F(1 - q) - Vs]

]

g11[Ds - F(1 - q) + 2Ls - Vs] + g12Vs g21[Ds - F(1 - q) + 2Ls - Vs] + g22Vs

F - Ds Vs

g11(Ds - F + Vs) + g12Vs g21(Ds - F - Vs) + g22Vs

F - Ds qF - Vs

g11[Ds - F(1 - Fq) - Vs] + g12(Fq - Vs) g21[Ds - F(1 - Fq) - Vs] + g22(Fq - Vs)

F(1 - q) - Ds -Vs

g11[Ds - F(1 - Fq) - Vs] - g12Vs g21[Ds - F(1 - Fq) - Vs] - g22Vs

]

g11Ls + g12[2Ls - F(1 - q) - Vs] g21Ls + g22[2Ls - F(1 - q) - Vs]

[

g11Ls + g12Vs g21Ls + g22Vs

[ ] [

]

g11(2Vs - qF - Ls) + g12Vs g21(2Vs - qF - Ls) + g22Vs

]

]

-g11Ls + g12(Fq - Vs) -g21Ls + g22(Fq - Vs)

[

-g11Ls - g12Vs -g21Ls - g22Vs

]

]

]

]

The same is true for the nonlinear distillation column model represented by the equation

LV

DV

LB

LV

DV

LB

v1 v2 v3

v6 v5 v4

v2 v1 v6

v4 v5 v6

v3 v2 v1

v5 v4 v3

-1

∆u ) (g11g22 - g12g21) -g22(δd1gd11 + δd2gd12) + g12(δd1gd21 + δd2gd22) g21(δd1gd11 + δd2gd12) - g11(δd1gd21 + δd2gd22)

xD ) f1(L,V,D,B) xB ) f2(L,V,D,B)

]

(5.8)

and for the DV structure

LD ) f3(L,V,D,B) LB ) f4(L,V,D,B)

(5.11)

Using the LV structure pairs LD-D and LB-B for level control and following the logic in previous sections, the problem reduces to

xD ) h1(L,V) xB ) h2(L,V)

(5.12)

At v5 vertex, the relationship is

xD ) h1[2Ls,2Ls-F(1-q)] xB ) h2[2Ls,2Ls-F(1-q)]

(5.13)

Similarly, the DV structure produces the following system:

xD ) g1(D,V)

-1

∆u ) (g11g22 - g12g21) × (g21 + g22)(δd1gd11 + δd2gd12) - (g11 + g12)(δd1gd21 + δd2gd22) g21(δd1gd11 + δd2gd12) - g11(δd1gd21 + δd2gd22)

[ ] [

g11[-Ls + F(1 - q)] - g12Vs g21[-Ls + F(1 - q)] - g22Vs

2Vs - 2Ls + F(1 - q) - Ds Vs

important, however, to identify the acceptable range over which the outputs should vary in order to calculate a useful OI, but this should not be difficult because product specifications are necessary for completing the process design. When attention is turned to the input space, careful inspection of the input maps of Figures 9 and 10 reveals that the vertexes are also equivalent but rotated. For example, v6 in Figure 9 has the coordinates (F(1-q), 0), and the active constraints are D ) 0 and V ) 0. The intersection of those two constraints in Figure 10 occurs at v1 (0, 0) and from eq 5.7 L ) F(1-q). Therefore, if the vertexes of the desired input space are different by the gain translations for different control structures, then the DISs will also be equal. In the LV structure the input move required to counteract ∆d ) (δd1, δd2) is

[

]

-Ds -Vs

Table 3. Vertex Equivalencies for Different Control Structures

[

output with Ds ) Vs + F(1 - q) - Ls

output

]

xB ) g2(D,V)

(5.14)

By material balance (eq 5.1)

(5.9)

These moves are different simply by the transformation required between the two cases. This can be seen by using the relationship ∆D ) ∆V - ∆L from eq 5.5. Subtracting δu1 from δu2 in eq 5.8 yields

xD ) g1(D,V)

δu ) (g21 + g22)(δd1gd11 + δd2gd12) - (g11 + g12)(δd1gd21 + δd2gd22)

At the v5 vertex, D ) 0 and V ) 2Ls - F(1-q), which means that the input points in the AIS are equivalent and therefore the AOS points will be equivalent. In general, if a process is represented by material balance equations, the input spaces are the same except for rotation.

(g11g22 - g12g21)

(5.10) which is the same as δu1 in eq 5.9.

xB ) g2(D,V)

(5.15)


Figure 11. AIS, DOS, AOS, and OI for the LV configuration.

Figure 12. AIS, DOS, AOS, and OI for the DV configuration.

Figure 13. AIS for a methanol distillation column. Table 4. AOS Vertexes When a y2-u3 SISO Loop Exists gain ratio

AOS vertexes

|a˜ 1| > |a˜ 2| and a21/a23 < 1 |a˜ 1| > |a˜ 2| and a21/a23 > 1 |a˜ 1| < |a˜ 2| and a22/a23 < 1 |a˜ 1| < |a˜ 2| and a22/a23 > 1

(a˜ 11u1m + a˜ 13u3m, 0), (a˜ 11u1m - a˜ 13u3m, 0), (-a˜ 11u1m + a˜ 13u3m, 0), (-a˜ 11u1m - a˜ 13u3m, 0) (a˜ 11u1* + a˜ 13u3m, 0), (a˜ 11u1* - a˜ 13u3m, 0), (-a˜ 11u1* + a˜ 13u3m, 0), (-a˜ 11u1* - a˜ 13u3m, 0) (a˜ 12u2m + a˜ 13u3m, 0), (a˜ 12u2m - a˜ 13u3m, 0), (-a˜ 12u2m + a˜ 13u3m, 0), (- a˜ 12u2m - a˜ 13u3m, 0) (a˜ 11u1* + a˜ 13u2*, 0), (a˜ 11u1* - a˜ 13u2*, 0), (-a˜ 11u1* + a˜ 13u2*, 0), (- a˜ 11u1* - a˜ 13u2*, 0)

As an example, a dual-composition control distillation column is represented by the model:

[ ] [

xD 0.0147 xB ) 0.00374

][ ] [ ] [] []

u1 -0.0150 , -0.003 38 u2

u L D or where u1 ) V V 2

The AIS, DOS, AOS, and OI are shown in Figure 11 for the LV control scheme and in Figure 12 for the DV control scheme. The shapes of the AIS for the two cases are different, but the AOS and resulting OI are identical. This is also true for nonlinear distillation models

which can be seen with the simple example of a methanol-water column. The column design specifics are 14 stages, a total condenser at 45 psig, a kettle reboiler, a feed stream of a 50/50% mix of methanol and water at 150 °F and 50 psig fed above tray 10, and a stage-to-stage pressure drop of 0.5 psi. The column was simulated using Aspen Plus, and the LV and DV input spaces are shown in Figure 13. The AISs are represented here by the individually simulated conditions. The AOS and OI are the same for both input spaces as seen in Figure 14. Because the DOS would be the same for either control configuration, the OIs will be identical.

3982


A similar analysis for a21 < a22 yields AIS vertexes of

(

u1m, -

)(

a21 a21 u1m , -u1m, u a22 a22 1m

)

(A.4)

and AOS vertexes

[(

-a11 +

Figure 14. AOS for methanol distillation.

) ] [(

a12a21 u1m, 0 , a22

a11 -

) ]

a12a21 u1m, 0 a22

Reversal of the pairing of the variables in order to consider the case when u1 is used to control the level y2 and first looking at the case where a22 > a21, the vertexes of the AIS are

6. Conclusion During the process synthesis stage, an inventory control structure is inherently selected by the process engineer as the steady-state simulation is configured. During the procedure of process design, different process designs may be completed and evaluated against each other to decide on a final optimized process. Evaluation of the operability of each of these designs is important in making the best selection. The OI is a measure that quantifies the operability of a given design, and it has the important property of being independent of inventory control structure. Proof that the OI is independent of the inventory control structure has been presented for square and nonsquare linear processes and general nonlinear processes. In addition, this independence has been demonstrated by application to a dual-composition distillation column with different level control structures.

(

u1m, -

)(

a21 a21 u1m , -u1m, u a22 a22 1m

)

(A.6)

These vertexes are identical to those of eq A.4 and yield the AOS vertexes

[(

-

)( ) ] )( ) ] [( ) ] [( [(

a11a22 -a21 + a12 u ,0 , a21 a22 1m -

a11a22 a21 + a12 u ,0 a21 a22 1m

a11 -

a12a21 u1m, 0 , a22

-a11 +

or

) ]

a12a21 u1m, 0 a22

(A.7)

Appendix A: Linear 2 × 2 Proof Using Identical Vertexes For a linear mapping of two input spaces to two output spaces, it is sufficient to prove that the two output spaces have identical vertexes for the output spaces to be equal. Therefore, if it can be proved that the AOS vertexes generated from the AIS based on the y2-u2 level control pairing are identical to the AOS vertexes generated by the y2-u1 AIS, then the OI can be proven to be independent of the regulatory level control structure. Referring to eq 2.2 and first looking at the case where a21 > a22, the vertexes of the AIS are

(

)(

)

a22 a22 u2m, -u2m , u ,u a21 a21 2m 2m

a11 -

)(

)]

a12a21 a22 u ,0 , a22 a21 2m a12a21 a22 a11 u , 0 (A.2) a22 a21 2m

[(

)( ) ]

or

[(

-

) ] [(

a11a22 + a12 u2m, 0 , a21

) ]

a11a22 - a12 u2m, 0 a21

which are identical to the AOS vertexes in eq A.5. Following similar logic for a22 < a21 yields AIS vertexes of

(

)(

)

a22 a22 u2m, -u2m , u ,u a21 a21 2m 2m

(A.8)

and AOS vertexes of

[(

-

) ] [(

a11a22 + a12 u2m, 0 , a21

) ]

a11a22 - a12 u2m, 0 a21

(A.9)

(A.1)

There are only two vertexes because of the restriction imposed by the dedicated level control. The corresponding vertexes of the AOS are

[(

(A.5)

(A.3)

which are identical to the AOS vertexes in eq A.3. So, the AOS is independent of the choice of the variables to control level. Appendix B: Linear 2 × 3 Proof When the input u3 is selected to control the level y2, the relationship of the inputs assigned to the nonintegrating outputs to u3 is

u3 ) -

1 (a u + a22u2), a23 * 0 a23 21 1

(B.1)

The relationship between the maximum values of the inputs as described in eq B.1 must be analyzed to demonstrate level independence, but the relationship


of the inputs to the remaining output y1 must first be considered:

y1 ) a11u1 + a12u2 + a13

( )

-1 (a u + a22u2) a23 21 1

) (a11 - a13a23-1a21)u1 + (a12 - a13a23-1a22)u2 ) a˜ 1u1 + a˜ 2u2

(B.2)

a˜ 1 ≡ a11 - a13a23-1a21 and a˜ 2 ≡ a12 - a13a23-1a22 (B.3) Assuming that the remaining inputs, u1 and u2, are available for control, the y1 maximum value y1m is dependent upon the values of a˜ 1 and a˜ 2 and upon the gain ratios in eq B.1. Specifically, if |a˜ 1| > |a˜ 2|, then increasing values of u1 yield a larger AOS. Conversely, if |a˜ 1| < |a˜ 2|, then the opposite is true. For the case where |a˜ 1| > |a˜ 2|, the limiting value of u1, identified here as u1*, is determined by the ratio a21/a23. If a21/a23 < 1, then u1* ) u1m and u2* is equal to the value that sets u3* ) u3m. If a21/a23 > 1, then u1* < u1m unless the signs of the gain ratios are opposite. Looking at the case where |a˜ 1| > |a˜ 2| and a21/a23 < 1, the vertexes of the AIS are

u1m, -

]

1 (a u + a21u1m), u3m , a22 23 3m 1 u1m, (-a23u3m + a21u1m), -u3m , a22 1 -u1m, (a u - a21u1m), u3m , a22 23 3m 1 -u1m, (a u + a21u1m), -u3m (B.4) a22 23 3m

[ [

[

] ]

]

The dedicated level relationship restricts the number of vertexes in eq B.4 to 4. These vertexes are translated to the AOS as

[(

)

(

) ] ) ( ) ] ) ( ) ] ) ( ) ]

a12a21 a12a23 u1m + a13 u3m, 0 , a22 a22 a12a21 a12a23 a11 u1m - a13 u3m, 0 , a22 a22 a12a21 a12a23 - a11 u1m + a13 u3m, 0 , a22 a22 a12a21 a12a23 - a11 u1m - a13 u3m, 0 (B.5) a22 a22

a11 -

u2 ) -

1 (a u + a23u3) a22 21 1

(B.6)

and that for y1

where

[

wherever the AOS vertex limit is imposed by an input restricted to a nonmaximum value ui*, that input has an unusable available range. This implies that the process is overdesigned with respect to that input. The case in which the u2 variable is used to control level y2 will now be considered. The equation for u2 is

[( [( [(

Similar analyses for the remaining cases produce the matrix of AOS vertexes shown in Table 4, where a˜ 11 ) a11 - a12a21a22-1, a˜ 13 ) a13 - a12a23a22-1, a˜ 12 ) a12 a11a22a21-1, and a˜ 13 ) a13 - a11a23a21-1. Note that

y1 ) a˜ 11u1 + a˜ 13u3

(B.7)

For |a˜ 11| > |a˜ 13| and a21/a23 > 1, the value of y1 at the maximum AOS size is then calculated by

y1m ) a˜ 11u1m + a˜ 13u3m

(B.8)

and by comparison to the AOS vertexes generated from eq B.5, it can be shown that it is identical to the value calculated when u3 was used for control (|a˜ 1| > |a˜ 2| and a21/a23 < 1). This implies that the AOS remains unchanged in the case that either u3 or u2 is used to control the level. The same can be shown to be true if u1 is used to control the level. Literature Cited (1) Vinson, D. R.; Georgakis, C. 5th IFAC Symposium on Dynamics and Control of Process Systems (DYCOPS-5); IFAC: Corfu, Greece, 1998. (2) Vinson, D. R.; Georgakis, C. A New Measure of Process Ouput Controllability. J. Process Control 2000, 10, 185-194. (3) Bristol, E. H. On a New Measure of Interaction for Multivariable Control. IEEE Trans. Autom. Control 1966, AC-11. (4) Stanley, G.; Marino-Galarranga, M.; McAvoy, T. J. Shortcut Operability Analysis. 1. The Relative Disturbance Gain. Ind. Eng. Chem. Process Des. Dev. 1985, 24, 1181-1188. (5) Hovd, M.; Skogestad, S. Simple Frequency-Dependent Tools for Control System Analysis, Structure Selection and Design. Automatica 1992, 28, 989-996. (6) Vinson, D. R. A New Measure of Process Operability for Improved Steady-State Design of Chemical Processes. Ph.D. Dissertation, Lehigh University, Bethlehem, PA, 2000. (7) Arkun, Y.; Downs, J. A General Method to Calculate InputOutput Gains and the Relative Gain Array for Integrating Processes. Comput. Chem. Eng. 1990, 14, 1101-1110. (8) McAvoy, T. J. Interaction Analysis: Principles and Applications; Instrument Society of America: Research Triangle Park, NC, 1983; Vol. 6. (9) Ha¨ggblom, K. E.; Waller, K. V. Transformations and Consistency Relations of Distillation Control Structures. AIChE J. 1988, 34, 1634-1648. (10) Skogestad, S.; Jacobsen, E. W.; Lundstro¨m, P. IFAC Symposium on the Dynamics and Control of Chemical Reactors, Distillation Columns, and Batch Processes (DYCORD+ ’89); IFAC: Maastricht, The Netherlands, 1989; pp 315-322. (11) Ha¨ggblom, K. E.; Waller, K. V. AIChE J. 1990, 36, 11071113.

Received for review December 5, 2001 Revised manuscript received May 30, 2002 Accepted June 3, 2002 IE0109814

Inventory Control Structure Independence of the Process Operability

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