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Ind. Eng. Chem. Res. 1998, 37, 4734-4739
Interaction Measure for the Selection of Partially Decentralized Control Structures Teck Kiang Lee and Min-Sen Chiu* Department of Chemical and Environmental Engineering, National University of Singapore, Singapore 119260, Singapore
Yaman Arkun School of Chemical Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0100
Partially decentralized controllers offer performance superior to that of decentralized controllers without the cost and complexity of full multivariable controllers. However, selection of suitable partially decentralized control structures poses a problem since there are a great number of alternatives. A method for the selection of partially decentralized control structures is proposed in this paper. Examples are given to illustrate the success of this modified interaction measure in identifying suitable partially decentralized control structures. Introduction Decentralized control structures are widely used in chemical process control. A large number of controlled variables are usually involved in complex chemical processes. Designing and maintaining full multivariable controllers for such processes would be extremely difficult. On the other hand, the use of decentralized controllers simplifies controller design greatly since they have fewer tuning parameters than full multivariable controllers. This makes it easier for relatively unspecialized personnel to understand the key concepts behind decentralized control so that they can more easily design and retune the controllers to take into account changing process conditions. It is well documented that the effectiveness of the decentralized controller depends on the magnitude of the off-diagonal elements relative to that of the diagonal elements in the process transfer function matrices. Therefore, ignoring the off-diagonal blocks in controller design can very often result in severe interactions, which can lead to poor performance or even instability. When this happens, the usual solution is to make use of more sophisticated and costly full multivariable controllers. There is, however, an alternative to the decentralized and full multivariable controllers which has largely been overlooked. Partially decentralized controllers have structures that lie in between the two extreme control structures. In decentralized controllers, different subsets of controller outputs are assigned to different subsets of controller inputs and there is no sharing of information between the different subsystems. Partially decentralized controllers, on the other hand, are characterized by the sharing of information among the different subsystems. This broad group of controllers becomes an attractive alternative when the stability or performance requirement cannot be met by decentralized controllers, and the complexity in the design and high cost in the installation of full multivariable controllers are to be avoided. * To whom all correspondence should be addressed. Telephone: (65)-874 2223. FAX: (65)-779 1936. E-mail: checms@ nus.edu.sg.
As in decentralized controller design, the inputoutput pairing problem (or control structure selection problem) plays an integral role in the design of partially decentralized controllers. More importantly, pairings for partially decentralized controllers increase exponentially with the dimension of the system as compared to the decentralized controller. For example, a 2 × 2 system offers a choice of two possible decentralized controllers given by
where
C1 )
[
c11 0 0 c22
]
[
0 c12 C2 ) c 21 0
]
(1)
When partially decentralized controllers are considered, we are no longer restricted to pairing one controller output to one controller input. Instead, a controller input (i.e., error signal) may be shared by one or two controller outputs. As such, there are four pairings possible for a 2 × 2 system:
C3 )
[
c11 c12 0 c22
]
[
c 0 C4 ) c11 c 21 22 0 c12 C5 ) c c 21 22
[
]
]
[
c c C6 ) c11 12 21 0
]
(2)
For a 3 × 3 system, there are 15 possible decentralized control structures and an overwhelming 250 possible partially decentralized control structures. Clearly, the number of alternatives increases very rapidly with the number of inputs and outputs. Selection of control structures then becomes a significant problem. To address this problem, a method of quantifying the interaction within the partially decentralized control structure is needed. Interaction measures such as the relative gain array,1 the direct Nyquist array,2 and µ interaction measure3 have proven useful for the design of decentralized controllers. In particular, the µ interaction measure is a dynamic measure which can be used
10.1021/ie980188j CCC: $15.00 © 1998 American Chemical Society Published on Web 11/17/1998
Ind. Eng. Chem. Res., Vol. 37, No. 12, 1998 4735
Figure 1. IMC structure. Figure 3. Input (error signal) sharing in partially decentralized controller.
For a square system, the pseudoinverse reduces to the conventional inverse; that is,
[G ˜ -]q ) [G ˜ -]-1
Figure 2. Feedback control structure.
to predict the stability of decentralized control systems less conservatively than other interaction measures. However, these interaction measures cannot be used to analyze partially decentralized control systems. Theories on the selection and design of partially decentralized controllers are not available due to a lack of study in this area. In this paper, a new method for the selection of partially decentralized control structures is proposed. This involves first extending the µ interaction measure to nonsquare decentralized systems within the internal model control (IMC) framework.4 Next, the partially decentralized control system is “expanded” into the nonsquare decentralized structure. The modified µ interaction measure can then be applied to this expanded system for control structure selection. Two examples are used to illustrate the use of this interaction measure in identifying suitable partially decentralized control structures.
Figure 1 shows the IMC feedback structure. Comparing this with the conventional feedback structure in Figure 2, it can be shown that the two are identical if the feedback controller C and the corresponding IMC controller Q satisfy the following relations:
C ) Q(I - G ˜ Q)-1
(3)
Q ) C(I + G ˜ C)-1
(4)
or
For square systems, Q has the general form
(5)
where G ˜ - is the minimum phase part of the process model G ˜ . This is extended to nonsquare systems by writing
Q ) [G ˜ -]qF
(6)
˜ - that satisfies where [G ˜ -]q is any pseudoinverse5 of G
G ˜ -[G ˜ -]qG ˜- ) G ˜-
For nonsquare systems, the pseudoinverse is not unique. All that is required is that eq 7 be satisfied and that Q be stable. The IMC filter F in eqs 5 and 6 has the form
F ) diag{fi}i)1-k
(9)
where fi is a low-pass filter element with steady-state gain of one, which is often expressed as
fi )
1 (is + 1)ni
∀ i ) 1-k
(10)
where ni is a positive integer that is selected to make Q realizable and i is a filter time constant which acts as a tuning parameter to achieve control objectives. Once the filter order is chosen, the only parameter that remains undetermined is the value of i. Partially Decentralized Controllers
Internal Model Control
Q ) [G ˜ -]-1F
(8)
(7)
As noted earlier, partially decentralized controllers are differentiated from decentralized controllers by the sharing of information among the different subsystems. In decentralized controllers, different subsets of controller outputs are assigned to different subsets of controller inputs without any sharing of information among the different subsystems. In partially decentralized controllers, the same subset of controller inputs can contribute to different subsets of controller outputs, resulting in the sharing of input among the different subsystems. Because of this information sharing, the number of possible input-output pairings is greatly increased. Take, for example, a 2 × 2 controller. The two decentralized structures can be denoted simply as {(1,1), (2,2)} and {(1,2), (2,1)}, respectively, where the notation is such that (1,2) indicates that the first controller output is paired to the second controller input. For partially decentralized controllers, any of the two controller outputs can have contribution from both instead of just one controller input. For example, we can have the first controller output, u1, paired to both the first and second controller inputs, e1 and e2, respectively, and the second controller output, u2, paired to the second controller input, e2, only as shown schematically in Figure 3. This pairing is denoted {(1,[1 2]), (2,2)}. In this case, (1,[1 2]) indicates that the first controller
4736 Ind. Eng. Chem. Res., Vol. 37, No. 12, 1998
output is paired to the first and second controller inputs. Likewise, there can be three other partially decentralized pairings in which one controller output is paired to both controller inputs, namely, {(1,1), (2,[1 2])}, {(1,2), (2,[1 2])}, and {(1,[1 2]), (2,1)}. The advantage of input sharing in partially decentralized controllers is that dynamic interaction can be reduced, thereby improving the performance of the system. Partially decentralized controllers can therefore be utilized to improve plant performance when decentralized controllers are deemed unsatisfactory. Expansion of Partially Decentralized Controllers We will first transform the partially decentralized control structures to nonsquare decentralized control structures. Once this is accomplished, µ interaction measure will be extended to study the nonsquare decentralized control structures. To illustrate the procedure to transform a partially decentralized control system to a nonsquare decentralized control structure, consider a 2 × 2 plant in Figure 4
[
g g G ) g11 g12 21 22
]
Figure 4. Expansion of partially decentralized system with pairing {(1,[1 2]), (2,2)}.
(11)
with a partially decentralized controller
C)
[
c11 c12 c22
]
(12)
The controller output u1 can be looked upon as being made up of two components:
u1 ) u11 + u12
(13)
u11 ) c11e1
(14)
u12 ) c12e2
(15)
where
1
2
If we consider u1 and u1 as separate signals, then the same controller can be rewritten in the nonsquare decentralized form as
[
]
[
(17)
The system is thus expanded to the nonsquare decentralized form. This is illustrated by the block diagram in Figure 4. The same plant with an alternative partially decentralized controller
[
c C ) c11 c 21 22
]
can be similarly expanded to give
g g g G′ ) g11 g12 g12 21 22 22
whereupon the plant becomes
g g g G′ ) g11 g11 g12 21 21 22
Figure 5. Expansion of partially decentralized system with pairing {(1,1), (2,[1 2])}.
(18)
]
(20)
This is shown graphically in Figure 5. Although only 2 × 2 partially decentralized control systems are considered for notational simplicity, the generalization of the “expansion” procedure to n × n partially decentralized control systems is straightforward. µ Interaction Measure In this section, a µ interaction measure for nonsquare control systems is developed. Before we proceed, let us
Ind. Eng. Chem. Res., Vol. 37, No. 12, 1998 4737
first review the earlier work on µ interaction measure for square systems3 which is based on the following theorem: Theorem 1. Consider a decentralized control system ˜ with a stable process G ) [gij]i, j)1-n, a process model G ) diag{gii}i)1-n, and a decentralized controller C ) diag{ci}i)1-n. Assume the individual control loops are stable; then the closed loop system is stable if
µ-1(E) > σ*(H ˜)
∀ ω
(21)
where µ(‚) is the structured singular value6 of (‚), σ*(‚) is the maximum singular value of (‚), and
E ) (G - G ˜ )G ˜ -1
(22)
H ˜ )G ˜ C(I + G ˜ C)-1
(23)
The µ interaction measure developed by Grosdidier and Morari3 is only applicable to square decentralized control systems. For it to be used in the selection of partially decentralized control structures, it must first be extended to nonsquare decentralized control systems. Theorem 2. Consider a nonsquare decentralized control system with a stable process G′ ) [gij]i)1-m, j)1-n, a nonsquare model G ˜ ′ ) diag{Gimi×ni}i)1-k, and a nonsquare decentralized controller C′ ) diag{Gini×mi}i)1-k k k where ∑i)1 mi ) m and ∑i)1 ni ) n. Assume the individual subsystems are stable; then the closed-loop system is stable if
µ-1(E′) > σ*(F)
∀ ω
(24)
where F is the IMC filter and
E′ ) (G′ - G ˜ ′)[G ˜ ′-]q
(25)
Proof. See Appendix. E′ can be viewed as the “relative error” arising from the difference between the plant G′ and its model G ˜ ′. The interaction measure, µ-1(E′ ), can be plotted in an amplitude-frequency diagram and serves as an upper bound for σ*(F). This interaction measure is thus easy to implement graphically. A necessary condition for eq 24 to be satisfied can be derived at steady state. Since F(0) ) I, a controller that stabilizes the system can be found by using theorem 2 only if
µ-1[E′(0)] > 1
Figure 6. µ interaction analysis for partially decentralized controller: Solid line: µ-1(E′ ). Dashed line: σ*(F) for i ) 3.5. Dotted line: σ*(F) for i ) 10. Dash-dotted line: σ*(F) for i ) 15.
[
tion matrix given by
5 2.5e-5s (2s + 1)(15s + 1) 4s + 1 G) 1 -4e-6s 3s + 1 20s + 1
Examples Example 1. Consider the 2 × 2 example studied by Grosdidier and Morari,3 with the process transfer func-
(27)
The solid line in Figure 6 shows the plot of µ-1(E′ ) for a partially decentralized control system with pairing {(1,[1 2]), (2,2)}. For this system, the plant and model are expanded to the nonsquare decentralized form given respectively by
[
]
5 2.5e-5s 2.5e-5s (2s + 1)(15s + 1) (2s + 1)(15s + 1) 4s + 1 G′ ) 1 1 -4e-6s 3s + 1 3s + 1 20s + 1 (28) and
G ˜′ )
[
2.5e-5s 0 (2s + 1)(15s + 1)
0
1 -4e-6s 3s + 1 20s + 1
0
]
(29)
Figure 6 also shows the plot of σ*(F), where F is a thirdorder IMC filter
(26)
Equation 26 is a simple condition that can be used to screen a large number of control structures without too much computational effort, thus making it useful as an initial screening tool. With the µ interaction measure extended to cover nonsquare decentralized control systems, it can now be used to predict the stability of partially decentralized control systems which have been expanded to the nonsquare decentralized form. Two examples are used in the next section to illustrate the application of theorem 2 in the selection of partially decentralized control structures.
]
F)
[
1 0 (1s + 1)3 0
1 (2s + 1)3
]
(30)
for different values of i. It can be observed that µ-1(E′ ) > σ*(F) for all frequencies when i > 3.4. According to theorem 2, stable controllers with i > 3.4 can be designed for this system. Similarly, it is possible to design a fully decentralized controller with diagonal pairings for this system.3 Using the same IMC filter in eq 30, eq 21 holds when i > 4.7. Based on theorem 1, a stable fully decentralized control system can be obtained when i > 4.7. The performances of the partially decentralized controller with 1 ) 2 ) 3.5 and the fully decentralized controller with 1 ) 2 ) 4.8 are compared. The responses of the two systems to set-point changes are
4738 Ind. Eng. Chem. Res., Vol. 37, No. 12, 1998
Figure 7. Closed-loop response of partially decentralized control system to unit step changes in (a) r1 and (b) r2. Figure 10. µ interaction analysis for partially decentralized controller: Solid line: µ-1(E′ ). Dashed line: σ*(F) for i ) 1.
Figure 8. Closed-loop response of fully decentralized control system to unit step changes in (a) r1 and (b) r2. Figure 11. Closed-loop response of 3 × 3 partially decentralized control system to unit step changes in (a) r1, (b) r2, and (c) r3.
Screening all possible partially decentralized control systems, it is found that only eight of them satisfy eq 26. Figure 10 shows the plot of µ-1(E′ ) for one of these eight partially decentralized control system with pairing {(1,2), (2,[1 3]), (3,[1 2])}. For this system, the “expanded” plant and model are given in eqs 32 and 33, respectively. A controller is designed for the expanded
Figure 9. µ interaction analysis for fully decentralized controller: Solid line: µ-1(E) for diagonal pairing. Dashed lines: µ-1(E) for other pairings.
shown in Figures 7 and 8, respectively. Evidently, the partially decentralized controller performs better than the fully decentralized controller. Example 2. Consider a 3 × 3 example with process transfer function
[
3 5 4 15s + 1 20s + 1 18s + 1 -2 5 3 G) 10s + 1 12s + 1 8s + 1 2 4 -1 5s + 1 4s + 1 5s + 1
]
(31)
For the decentralized control system with diagonal pairing, the plot of µ-1(E) is shown by the solid line in Figure 9. Since H ˜ (0) ) I for controllers with integral action, eq 21 is satisfied only if µ-1[E(0)] > 1. Consequently, a stable controller cannot be designed for this fully decentralized structure based on theorem 1 because µ-1[E(0)] < 1. The dashed lines show the plot of µ-1(E) for the other five fully decentralized systems with different pairings. All of these have µ-1[E(0)] < 1, showing that no fully decentralized controller can be designed for this system based on theorem 1.
[ [
]
5 4 5 3 3 20s + 1 18s + 1 15s + 1 18s + 1 20s + 1 5 3 5 -2 -2 G′ ) 12s + 1 8s + 1 10s + 1 8s + 1 12s + 1 4 -1 2 -1 4 4s + 1 5s + 1 5s + 1 5s + 1 4s + 1 (32) 5 3 0 20s + 1 18s + 1
0
0
G ˜′ ) 0
0
5 3 0 10s + 1 8s + 1
0
0
0
0
5 4s + 1
]
(33)
system using a second order IMC filter with i ) 1. Figure 10 shows that eq 24 is satisfied for this controller design. The servo responses of the resulting control system are shown in Figure 11. In contrast, Figure 12 shows that the fully decentralized controller with pairing {(1,2), (2,3), (3,1)} and i ) 1 produces highly oscillatory responses. Conclusions The µ interaction measure is extended to cover nonsquare decentralized control systems. By first expanding partially decentralized control systems to the nonsquare decentralized form, the modified µ interaction measure can be used to predict the stability of the
Ind. Eng. Chem. Res., Vol. 37, No. 12, 1998 4739
Also, the corresponding IMC controller is given as
Q ) [G ˜ ′-]qF ) C′S ˜′
(36)
(I + G′C′) ) (S ˜ ′ + G′C′S ˜ ′)[S ˜ ′]-1 ) (I - H ˜′+ -1 ˜ ′C′S ˜ ′ + G′C′S ˜ ′)[S ˜ ′]-1 ) G′C′S ˜ ′)[S ˜ ′] ) (I - G ˜ ′)Q][S ˜ ′]-1 ) (I - G ˜ ′Q + G′Q)[S ˜ ′]-1 ) [I + (G′ - G [I + (G′ - G ˜ ′)[G ˜ ′-]qF][S ˜ ′]-1 ) (I + E′F)[S ˜ ′]-1 (37) Figure 12. Closed-loop response of 3 × 3 fully decentralized control system to unit step changes in (a) r1, (b) r2, and (c) r3.
such systems and help identify suitable partially decentralized control structures. Application of this method to two examples shows that it can be used successfully in the selection of partially decentralized control structures.
Let N[k,ψ] denote the net number of clockwise encirclements of the point (k,0) by the image of the Nyquist D contour under ψ. Applying the multivariable Nyquist criterion7 to the system, the closed-loop system H′ is stable if and only if
N[0,det(I + G′C′)] ) -p
(38)
where G′ is stable and p is the number of unstable poles of C′. From eq 37,
N[0,det(I + E′F)] + N[0,det([S ˜ ′]-1)] ) -p (39)
Nomenclature
Since H ˜ ′ is stable by assumption, i.e.,
C ) square controller C′) nonsquare decentralized controller e ) controller input (error signal) F ) low-pass IMC filter G ) square process G ˜ ) square process model G ˜ - ) minimum phase part of G ˜ [G ˜ -]q ) pseudoinverse of G ˜G′ ) nonsquare process G ˜ ′ ) nonsquare process model H ˜ ) complementary sensitivity function Q ) IMC controller r ) reference set point S ˜ ) sensitivity function u ) controller output y ) system output
(41)
if6
µ-1(E′) > σ*(F)
∀ ω
(42)
This completes the proof. Literature Cited
Appendix. Proof of Theorem 2 For a nonsquare model G ˜ ′ controlled by a nonsquare controller C′, the complementary sensitivity function and sensitivity function are defined respectively as
S ˜ ′ ) (I + G ˜ ′C′)-1 ) I - H ˜′
N[0,det(I + E′F)] ) 0 Equation 41 is satisfied
) low-pass IMC filter time constant
H ˜′)G ˜ ′C′(I + G ˜ ′C′)
(40)
therefore
Greek Symbol
-1
N[0,det([S ˜ ′]-1)] ) -p
(1) Bristol, E. H. On a New Measure of Interaction for Multivariable Process Control. IEEE Trans. Autom. Control 1966, AC11, 133-134. (2) Rosenbrock, H. H. Computer-Aided Control System Design; Academic: London, 1974. (3) Grosdidier, P.; Morari, M. The µ Interaction Measure. Ind. Eng. Chem. Res. 1987, 26, 1193-1202. (4) Garcia, C.; Morari, M. Internal Model Controls2. Design Procedure for Multivariable Systems. Ind. Eng. Chem. Process Des. Dev. 1985, 24, 472-484. (5) Graybill, F. A. Introduction to Matrices with Applications in Statistics; Wadsworth: Belmont, CA, 1969. (6) Doyle, J. C. Analysis of Control Systems with Structured Uncertainty. IEE Proc., Part D 1982, 129, 242-250. (7) Postlethwaite, I.; MacFarlane, J. M. A Complex Variable Approach to the Analysis of Linear Multivariable Feedback Systems; Springer: Berlin, 1979.
(34)
Received for review March 25, 1998 Revised manuscript received September 14, 1998 Accepted September 15, 1998
(35)
IE980188J
