4436
Ind. Eng. Chem. Res. 1995,34, 4436-4450
Graphical Interpretations of Steady-State Interaction Measures R. Piette? T. J. Harris, and P. J. McLellan* Department of Chemical Engineering, Queen’s University, Kingston, Ontario, Canada K7L 3N6
The relative gain, the relative disturbance gain, and the singular value decomposition are commonly used methods to analyze multivariable linear processes. In this paper, graphical interpretations of these interaction measures are described, and additional graphical techniques for visualizing steady-state process interactions are proposed. For systems with two inputs and two outputs, these interaction measures are directly related to the angle of intersection and degree of rotation of constant output contours, or, in the case of singular value decomposition, constant sums of squares contours. Extensions to larger systems and the relative disturbance gain are identified in terms of projections between one-dimensional subspaces. The relationship between these interaction measures and measures of association in multivariate statistical analysis is established. The impact of process nonlinearities on interaction measures is identified graphically as deformations of the constant output norm contours. The use of graphical methods is demonstrated for mixing and high-purity distillation examples.
Introduction Stability and performance of closed-loopprocesses are affected by interactions between controllers in multiloop configurations. The most serious problems occur when transmission interaction is present in which the effects of one control loop “loop back” through interaction and the action of the other controllers (McAvoy, 1983). The relative gain and relative gain array (RGA, Bristol, 1966) were developed as empirical measures of the extent of this interaction and represent the ratio of open and effective closed-loop gains. The RGA is a valuable tool for the screening of preliminary control loop pairings, providing a quantitative measure of the extent of controller interaction using steady-state gain information. Guidelines for the interpretation of the relative gains are provided in many references, including McAvoy (1983) and Shinskey (1984). The relationship between the RGA and robust stability continues to be a major research topic (e.g., Grosdidier and Morari, 1987; Skogestad and Morari, 1987b; Campo and Morari, 1994). In particular, the relationship between the extent of interaction and the conditioning of the steady-state gain matrix has been a focus of research (Nett and Manousiouthakis, 1987;Skogestad and Morari, 1987b). The condition number of the gain matrix provides a set of bounds for the norm of the relative gain array, which in turn implies bounds on the range of relative gains for the control problem. The relative gain approach has been extended to the assessment of disturbance effects on the closed-loop process using the relative disturbance gain (Stanley et al., 19851, and the disturbance condition number (Skogestad and Morari, 1987a). In particular, the direction of the disturbance relative to the input-output gain structure has a significant impact on the ease with which a given loop configuration can reject disturbance effects. This suggests an underlying geometric character to the interaction and disturbance rejection problem, which is the focus of this paper. The objective of this paper is to propose graphical methods for interpreting process interaction, and, in particular, the RGA and SVD (singular value decomposition) analyses for linear and nonlinear process Current address: Treiber Controls, Inc., Toronto, Ontario, Canada. +
0888-5885/95/2634-4436$09.00/0
control problems. To date, the RGA and SVD approaches have primarily constituted quantitative approaches for analyzing interaction, and the graphical measures proposed in this paper provide a visual means of summarizing interaction information. This graphical approach can be particularly useful when studying the interaction behavior of processes using complex nonlinear models (e.g., using design simulators). Preliminary results for the graphical interpretation of the relative gain for 2 x 2 problems were presented by Koung and Harris (1989). The graphical interpretation for 2 x 2 problems was also discussed briefly in terms of gain vectors in the output space by Koung and MacGregor (1991). The current paper provides a detailed graphical interpetation in terms of constant output contours in the input space and the gradients of these outputs and provides an extensive summary of the range of graphs that can arise over the range of possible process interactions. The results are extended to problems involving more than two inputs and outputs. Graphical displays are also developed in terms of constant output norm contours; the ellipsoids associated with these contours are used t o relate interaction measures to multivariate statistical measures of association (specifically,principal component analysis). These graphical measures are also readily extended to nonlinear processes, typically using process simulations to generate the necessary information. The resulting displays indicate both the extent of process interaction and the extent of process nonlinearity. The empirical RGA provides a local measure of interaction; variation of this interaction is summarized in terms of a “relative gain surface plot” which enables the control engineer to readily identify regions of significant interaction and changes in the interaction structure. The graphical measures are extended to include the impact of disturbance direction. The relative disturbance gain is identified directly from the constant output contour plots for 2 x 2 problems, and the extension to larger systems is also described. The direction of the disturbances relative to the singular vectors of the gain matrix can be seen on plots of constant output norms in which the disturbance is represented as an “equivalent input vector”. Skogestad and Morari (1987a) have identified that the direction of the disturbance relative t o the output singular vectors of the process gain matrix is a strong indicator of the 0 1995 American Chemical Society
Ind. Eng. Chem. Res., Vol. 34,No. 12, 1995 4437 ease with which disturbances can be rejected when a multiloop controller configuration is used. The contributions of this paper consist of the detailed graphical interpretation of the relative gain in terms of intersections of constant output contours, development and interpretation of the constant output norm contours, establishment of the relationship between the interaction measures and measures of association in multivariate statistics, extension of the graphical measures to nonlinear processes, and the graphical assessment of disturbance effects. The paper proceeds as follows. The definition of the RGA and the SVD are briefly summarized for completeness, following which the graphical interpretation of the RGA for linear processes is presented. The graphical interpretation for the SVD is then proposed, and extensions to nonlinear processes are described. Graphical interpretations for the relative disturbance gain are developed in terms of constant output and constant output norm contours. The use of the graphical measures is illustrated using mixing and high-purity distillation examples.
Numerical Measures of Interaction Relative Gain Array. The relative gain, first proposed by Bristol (1966) as an empirical measure of process interaction, is defined as:
-
I
auj lYjr,
The output derivatives are obtained from the steadystate gain matrix. The ratio indicates the degree t o which the open-loop gain associated with a specific input-output pair is amplified or attenuated when the other control loops are closed. This information can be summarized for all possible input-output pairings in the RGA:
A = [A,]
(2)
The RGA can also be calculated in matrix form as the Hadamard (or Schur) product (McAvoy, 1983; Horn and Johnson, 1991):
A =GoG-~
(3)
The Hadamard product denotes elementwise multiplication. The definition of the relative gain implies certain properties, including the fact that the sum of the elements in a given row or column is unity. The relative gain is also independent of the scaling of the process variables. Additional properties of the RGA are summarized in McAvoy (1983). The RGA is primarily a tool for identifylng significant transmission interaction (also referred to as “Figure 8” interaction). Recall that transmission interaction exists between two loop pairings when the action of one controller on its output occurs via a path that passes through the other controller in addition to the direct loop gain (e.g., Marlin, 1995). In order for transmission interaction to exist, interaction paths must exist from each manipulated variable to the other output. Triangular systems do not exhibit transmission interaction but can exhibit substantial one-way interaction. A number of different “pairing rules” for selecting control
loop configurations have been developed using the RGA and are based primarily on experience with various problems (e.g., McAvoy, 1983; Shinskey, 1984). A relative gain of 1indicates no transmission interaction; however, the process may still exhibit significant oneway interaction. Low relative gains imply significant amplification of the input-output gain, and high relative gains indicate significant attenuation of the inputoutput gain. Negative relative gains indicate that the effective gain of the input-output pair will change direction when the other controllers are closed and is generally an undesirable pairing. Regardless of the value of the relative gain, the presence of one-way interaction should also be assessed in case one-way decoupling is required. Finally, the RGA has been extended to the dynamic case as a function of frequency (e.g., McAvoy, 1983; Tung and Edgar, 1981; Hovd and Skogestad, 1992). Singular Value Decomposition. The SVD has played a major role in robust linear control theory (e.g., Morari and Zafiriou, 1989) and multivariate statistical analysis (e.g., Gittens, 1985; Jackson, 1991). Although defined, in general, for complex, nonsquare matrices, the SVD for square, real matrices is (Golub and van Loan, 1989):
G = uCVT
(4)
where U is the output rotation matrix and V is the input rotation matrix. U and V are orthonormal matrices (unitary in the complex case). is a diagonal matrix containing the singular values arranged in descending order. The singular values are real and nonnegative. The number of nonzero singular values indicates the rank of the matrix G. The SVD is scale dependent, and thus the issue of scaling of the process variables must be addressed when the SVD is used for analyzing process dynamics. The SVD has been used t o assess process structure, providing, in particular, information concerning the “degree of independence” in the process. This in turn provides insight into the difficulty of the control problem. This analysis is performed using the condition number, which is defined as the ratio of the maximum singular value to the minimum singular value (Golub and van Loan, 1989). Large condition numbers indicate near singularity of the matrix, and thus a loss of “independence”. The condition number of the process has important consequences for robust stability (Morari and Zafiriou, 1989; references contained in Braatz and Morari, 1994). Several researchers have attempted to relate the condition number t o the extent of interaction in the process (e.g., Grosdidier et al., 1985). Nett and Manousiouthakis (1987) have obtained bounds on the RGA in terms of the minimum condition number of the process gain matrix over all proper scalings. The bound must be calculated over all scalings because the condition number is scale dependent. These bounds represent limits on various norms of the RGA, which in turn imply limits o n the specific elements. Thus, large condition numbers suggest difficulty in control without identifying specific elements. Furthermore, the condition number bounds are upper bounds and thus represent necessary but not sufficient conditions for significant interaction. Grosdidier et al. (1985)have presented upper and lower bounds for 2 x 2 problems, which imply necessary and sufficient conditions for a large RGA in terms of the minimum condition number of the gain matrix.
4438 Ind. Eng. Chem. Res., Vol. 34, No. 12, 1995
The SVD has also been used to assess the impact of disturbances, and, specifically, disturbance direction, on multiloop control of chemical processes. Skogestad and Morari (1987a) have identified that the direction of the disturbance relative to the output singular vectors (the columns of U) indicates the degree of difficulty in rejecting the disturbance. Disturbances entering in directions close to the output singular vector associated with the maximum singular value pose the least difficulty, whereas disturbances entering in directions close t o the output singular vector associated with the minimum singular value result in difficult disturbance rejection problems. Skogestad and Morari quantify this result by defining a “disturbance condition number” representing the ratio of the 2-norms of the corrective action and the disturbance divided by the reciprocal of the maximum singular value. The reciprocal of the maximum singular value represents the least disturbance impact and is used t o provide an appropriate scaling relative t o the best disturbance case. The disturbance condition number is bounded by the reciprocals of the maximum and minimum singular values of the process gain matrix and represents a quantitative test for assessing the impact of disturbance direction on closed-loop performance. Additional results are presented in Grosdidier (1990)using an argument based on the sensitivity function for the closed-loop process. The SVD results, in particular, suggest a geometric character to the problem of process interaction. In the following sections, graphical interpretations are developed for both the RGA and SVD approaches which illustrate this geometric character and can be used to visualize the extent of interaction in a given process.
$ YlC
Ul
a
b
C
Graphical Interpretations of the Relative Gain Constant Output Contours. The conditioning of a gain matrix can be assessed visually by examining contours of constant outputs in the space of input variables. When the matrix is nearly singular, the contours will be nearly parallel, and this suggests a link between the intersection of the contours and the extent of process interaction. Preliminary results relating the RGA to the intersection of constant output contours were presented for 2 x 2 processes by Koung and Harris (1989). This interpretation was described briefly by Koung and MacGregor (1991) in terms of the angle of rotation of the gain vectors in the output space. By focusing on the gain vectors, Koung and MacGregor chose to view the interaction in terms of the constant input contours in the output space, as opposed to constant output contours in the input space. They identified a relationship between the angles of intersection and rotation of the gain vectors for 2 x 2 systems. A brief interpretation was provided for the cases in which the gain vectors intersect orthogonally and when the gain vectors are collinear. Although their approach provides an indication of process interaction, the use of constant output contours in the input space provides a more direct representation since it identifies directly the control moves required t o maintain a constant output. Thus, given a product or process specification,the values and behavior of the manipulated variables associated with these specifications can be readily determined. The contour orientation depends on the choice of scaling for the input and output variables. For a graphical analysis, the scalings should be chosen so that the variables have physical meaning. It is also desirable to make the process outputs dimensionless. These
VY
I
Figure 1. (a) Constant output contours in input space. (b) Openloop gain for a unit step in u1. (c) Closed-loop gain for a unit step in u1.
features are accomplished by defining a standardized variable:
where Y, is the ith process output, Y,,,o,is the nominal or reference value, which might be chosen as the nominal operating point or setpoint, and Y,,, is a standardizing value. This might be chosen as the distance from the nominal operating point to a specification limit, or the standard deviation of Y,. The input variables are scaled in a similar fashion. Given a process gain matrix, we can draw a diagram of constant y1 and y2 lines in the input space for a 2 x 2 system as shown in parts a-c of Figure 1. Note that the contours associated with a given output me parallel and equispaced for equal sized increments in the output. The graphical interpretation is developed by examining the open-loop and closed-loopcomponents of the relative gain defined in eq 1. The relative gain A11 is chosen to illustrate the development. The numerator of the relative gain represents the open-loop gain of the process and can be related to the gradient of y1 as follows. A unit step in u1 represents a change in this input in the direction of the unit vector el shown in Figure lb. The change in y1 associated with this step is obtained as the projection of el onto the gradient of y1 and represents the open-loop gain of the ul-yl pair. The expression for the open-loop gain becomes:
Ind. Eng. Chem. Res., Vol. 34, No. 12, 1995 4439
= Ivy,[ sin 6,
(6)
where (,) denotes the inner product, and Vyl is the gradient of y1 as a function of u1 and uz. Unless otherwise noted, all positive angles represent counterclockwise rotation from the positive u1 axis. The closed-loop gain for the ul-yl pair is the gain when the second control loop is closed; i.e.,yz is constant. The process is constrained to move along the constant yz contour, and the unit step in u1 must be related t o a vector along this contour. The resulting direction is the vector v shown in Figure ICand represents the vector on the constant yz contour which has a horizontal component of el. The 'change in y1 in the direction v is now obtained as the projection ofv onto the gradient of y1, agshown in Figure IC.Note that the direction of v is orthogonal t o the gradient ofyz. The two constraints on v require that the magnitude of v be l/cos(8z). The resulting expression for the closed-loop gain becomes: (7) with 7c
6, = - - 8, 2
+ 6,
Note that ea is the positive rotation angle from the gradient t o the constant y~ contour as shown in Figure IC.
Relative Gain Angle Condition. The relative gain is the ratio of the open- and closed-loop gains, yielding the following expression:
4
sin el cos 6, 1 = cos 6,
-
1
t a n 8, l-tan 6,
with angles as indicated in Figure la. Note that eq 8 has 2 degrees of freedom; i.e., in order to specify the interaction structure, two angles must be specified. For example, the interaction structure can be specified in terms of the angles between the output contours and the input axes, or as the angle between the y1 contour and the u1 axis and the angle of intersection between the two constant output contours. Thus, the nature of the intersection of the constant output contours is an important indication of the extent of interaction; however, the position of one of the output contours relative to a coordinate axis must also be specified. The angle between a given output contour and a given coordinate axis is an indication of the degree of one-way interaction between the two input variables and the specified output variable. Alternatively, this information can also be obtained by examining the angle between the gradient and the coordinate axis. For example, if y1 is affected only by changes in u1, the resulting constant y1 output contours will be parallel to the uz axis. The behavior of the constant output contours for the range of possible relative gains is summarized quantitatively in Table 1. The angles of the constant output
contours need only be specified in the first and fourth 8i n12 rad) as the mirror images quadrants ( - d 2 in other quadrants yield the same conclusions. The magnitudes of the gains are preserved; however, the signs of the gains change when the angle is changed by n rad. However, since the relative gain is the ratio of two gains each with changed sign, the relative gain will remain the same. The following qualitative observations can be made for the entries in Table 1. Rows 1 and 2. These cases represent diagonal systems in which the output contours are aligned with the input axes and hence intersect orthogonally. Thus, a given input variable affects only one output variable, and the relative gain is 1 or 0. Row 3. The output contours in this process are nearly collinear. Since 81 is nonzero, the contours are rotated relative to the u1 axis and thus this input affects more than one output. The process is highly interactive and ill-conditioned. The condition number of the gain matrix will be large, indicating near-singularity. Row 4. The output contour lines are reflections through the u1 axis, and the process has a relative gain of 0.5. The relative gain may be misleading in this case, since if p approaches zero, the relative gain will remain at 0.5; however, the process will be ill-conditioned. Decoupling in this instance will be difficult, and the impact of model uncertainty will be a concern. This illustrates why the condition number should always be checked regardless of the value of the relative gain. The RGA is always 0.5 for this process as both outputs are equally affected by both inputs; thus, there is no preference for pairings. The 2 x 2 case presented by Skogestad and Morari (1987b) with gain matrix rows [l -0.011 and [ l 0.011 is an example of such a problem. Rows 5 and 6. These cases represent processes with one-way interaction. One of the constant output contours is aligned with an input axis, while the remaining angle is between 0 and n12 rad, indicating that both inputs affect this output. Note that, although the constant output contours do not intersect orthogonally, there is no transmission interaction present. The relative gains of these processes will be either 0 or 1 depending on the loop pairing. As in the previous case, it is possible for the contours t o be nearly collinear, and thus the condition number should be verified in addition to the relative gain. Row 7. This process has output contours that are orthogonal;however, they are not aligned with the input axes and thus transmission interaction exists. This process is well-conditioned and can be easily decoupled. The range of relative gains identified for this case is consistent with those noted by Koung and MacGregor (1991). Row 8. Under the angle condition stated, the relative gain will always be negative. Moreover, the gain associated with the ul-yz pair is larger than the gain of the ul-yl pair. This can be interpreted graphically by noting that, as the angle between the constant y1 contour and the u1 axis increases, the projection of the gradient of y1 along the u1 axis increases. Recall that the projection of el onto the gradient represents the open-loop gain; thus, as the gradient approaches the coordinate axis (Le., the contour is almost orthogonal to the coordinate axis), the open-loop gain associated with the input-output pair increases. Row 9. Under the angle condition stated, the relative gain will always be positive.
4440 Ind. Eng. Chem. Res., Vol. 34, No. 12, 1995 Table 1. Relationship between Relative Gains and Constant Contour Lines
e2
el
row
comments
).11
noninteracting system
1 2
3
highly interactive (ill-conditioned system)
4
high interaction" one-way interaction"
5
6 7 8b
interaction present despite orthogonal outputs
o o
sgn(el) sgn(e2) > sgn(el) sgn(e2)> o sgn(el) sgn(e2) < sgn(el) S g n m < o
9b 1Ob 1l b a
Decoupling should be used when appropriate. For -7~12 < 81, 0 2
In order t o determine whether the relative gain will be positive or negative, two alternative angle conditions can be used. If the angles 81 and 8 2 are always restricted to be positive rotations from the positive u1 axis, the condition 8 2 > 81 implies that the relative gain ill1 will be negative. Alternatively, if 1821 > 1811 and sgn(82) sgn(81) > 0, A11 will be negative. The angle conditions are summarized in Table 1in rows 8-11. When analyzing a constant output contour diagram, the following rules should be applied. R1. If the contours for a given output are parallel to one of the input coordinate axes, the gain associated with this input-output pair is zero. Thus, the process will exhibit, at most, one-way interaction. R2. If the angle of intersection between the output contours is small (close to zero), the process exhibits significant interaction and is close to being singular. If condition R1 does not hold, the process exhibits significant transmission interaction. The use of this graphical interpretation is illustrated with the following examples. Example 1: Mixer. Two streams are combined in a simple mixer. The first stream contains a stock concentration (c1) of a reactant required downstream, and the second stream is a solvent used to adjust the concentration of the reactor feed to the desired value. Heat effects are neglected, and a static (pure gain) model with no deadtime is considered. The two outputs of the mixing system are the total flow rate Foutand the exit concentration tout and are denoted Y1 and Y2, respectively. The flow rates are manipulated by two PI flow controllers, and perfect control is assumed at steady state since integral action is present. The input variables U1 and U2 are the setpoints to the flow controllers for F1 and F2, respectively. The steady-state model for this process is:
YI = F o u t = q
s p
+ F 2 , s p = Ul + U2
and is nonlinear in the two inputs. Since the inputs and outputs are in similar numerical ranges, a simple scaling will be used to make the variables dimensionless:
yl=
u1 =
Yl - 0.5 US 1US ;
Y2 - 0.15 m o m
U, - 0.15 U S 1Us
lmol/L
Y2=
;
u2
=
U,a - 0.35 U S 1 Us
(10)
