Structural Analysis and Stability Conditions of Decentralized Control

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Ind. Eng. Chem. Res. 1996, 35, 736-745

PROCESS DESIGN AND CONTROL Structural Analysis and Stability Conditions of Decentralized Control Systems Zhong-Xiang Zhu† SACDA Inc., 343 Dundas Street, London, Ontario, Canada N6B 1V5

Stability analysis in decentralized control systems relies heavily on steady state tools such as the relative gain array and the Niederlinski index. However, only necessary stability conditions are provided by these tools and their usefulness lies essentially solely in eliminating unstable pairings. In this paper, upon structurally decomposing a decentralized control system into completely equivalent individual dynamic single input-single output loops with interactions explicitly embedded, system structure and main properties, such as right half plane (RHP) zeros, RHP poles, integrity, and stability, are analyzed in a systematic and transparent way. The intrinsic connections among these properties are elucidated. New important insights into the effects of loop interaction due to the process and the controller on the closed loop system are offered. Various necessary and sufficient conditions to prevent interaction from inducing undesirable behavior, such as nonminimum phase, lack of integrity, and instability, are developed. Significant implications for variable pairing and controller tuning are presented. 1. Introduction Full scale multivariable control techniques such as state space based modern control and, more recently, model predictive control have found increasing applications in the process industries (Eaton and Rawlings, 1990; Zhu and Jutan, 1994; Zhu et al., 1995). Nevertheless, decentralized control still remains popular and often predominant, either used alone or as a lower level control on top of various model based control schemes. In decentralized control, closed loop analysis, to a large extent, relies heavily on steady state tools such as the relative gain array (RGA) (Bristol, 1966) and the Niederlinski index (NI) (Niederlinski, 1971) (e.g., Grosdidier et al., 1985; Yu and Luyben, 1987; Seborg et al., 1989; Chiu and Arkun, 1990; Yu and Fan, 1990; Zhu and Jutan, 1993a,b, 1995c). Nevertheless, the RGA and the NI, together with many other proposed steady state interaction measures (Majare et al., 1986; Chang and Yu, 1992), ignore the dynamics of the process and the controller, thus leading to some fundamental limitations (Zhu and Jutan, 1995b). Take stability analysis, for example; only necessary (not sufficient) conditions are available, and final stability usually has to resort to ad hoc approaches, typically by means of detuning the controllers independently designed on the basis of individual interaction-free loops. Numerous dynamic interaction measures have been proposed to overcome the limitations of steady state ones (Bristol, 1978; Tung and Edgar, 1981; Gagnepain and Seborg, 1982; Jensen et al., 1986; Bequette and Edgar, 1988; Balchen and Mumme, 1988; Huang et al., 1994). Unfortunately, they have little impact on the analysis and design of decentralized control systems, mainly due to their unrealistic assumptions. Seborg et al. (1989) presented some general stability conditions, but only for 2 × 2 systems. Manousiouthakis (1993) †

E-mail: [email protected]. FAX: (519) 6793977.

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attempted to parametrize all stabilizing controllers in decentralized control systems from an overall perspective. However, this approach does not provide a strong linkage between the overall system and the independent loops, so important issues, such as loop interaction, failure tolerance, and independent design, cannot be easily addressed. In this paper, upon decomposing a decentralized control system into separate but equivalent individual single input-single output (SISO) loops, dynamic analysis is systematically performed. Important insights into the effects of loop interaction due to both the process and the controller on the closed loop properties, such as right half plane (RHP) zeros and poles, integrity, and stability, are presented. New necessary and sufficient conditions for preventing interaction from inducing undesirable behavior, such as open instability, nonminimum phase, loss of integrity, and instability, are developed with direct implications for both variable pairing choices and controller tuning. 2. Structural Decomposition The following derivation follows Zhu (1993) and Zhu and Jutan (1995b). Let us examine the transmittances in a control loop with the manipulated variable uj(s) ∈ {u1(s),...,un(s)} paired with the controlled variable yi(s) ∈ {y1(s),...,yn(s)}. When the controller, cj(s) ∈ Rn, has to take action in response to setpoint changes and/or disturbances, it affects the overall system in the following sequence: the controller cj(s) attempts to bring its output yi(s) to its target (setpoint) by sending a control signal to uj(s); the control action uj(s), meanwhile, creates perturbations in all the other loops in the system through the off-diagonal elements of the plant, forcing other controllers to take actions as well; other loops further influence the original loop by adding additional dynamics to the control action uj(s) via other off-diagonal elements of the plant. This process of strong coupling in the form of action-cross action-interaction among © 1996 American Chemical Society

Ind. Eng. Chem. Res., Vol. 35, No. 3, 1996 737

The detailed expressions of aij(s) and dij(s) can be readily obtained by using the signal flow graph technique (Ogata, 1990) or by appropriately partitioning the process and the controller (see the Appendices). Take 2 × 2 systems with diagonal pairing, for example, the following terms for the two loops can be easily derived and verified:

aii(s) ) -

Figure 1. Structure of loop uj-yi by structural decomposition.

Figure 2. Structure of loop 1 after decomposition of 2 × 2 Systems.

control loops, continues throughout the whole transient until a steady state is reached. Mathematically, responses to setpoint changes are governed by the following equations: n

ym(s) )

gml(s) ul(s) ∑ l)1

∀m

ul(s) ) cl(s)[rl(s) - yl(s)]

(1)

Note that in the above equation ul is assumed to be paired with yl, however without loss of generality. Focusing on the uj-yi loop and assuming that only this loop undergoes setpoint changes, we thus have, n

yi(s) ) gij(s) uj(s) -

∑

gil(s) cl(s) yl(s)

(2)

l)1,l*j

Successively using eq 2 to express all the outputs in terms of uj(s), we find the transmittance between the control action uj(s) and its own output to be

yi(s) ) gij(s) uj(s) + aij(s) uj(s)

(3)

and the perturbations caused by uj to other loops as

yk(s) ) dkj(s) uj(s)

∀k

(4)

Structurally, aij(s) in the above equations represents the additional dynamics exerted by other loops to the uj-yi loop and is physically separated from the independent SISO loop and dij(s) represents the perturbations to other loop by the underlining loop. Apparently, aij(s) and dij(s) provide measurements of the interaction in a loop and the cross-interaction in all other loops, respectively. Consequently, a decentralized control system can be structurally decomposed into individual SISO loops with the coupling among all the loops explicitly exposed and embedded in these separate loops. Figures 1 and 2 show the physical structure of a particular loop after the structural decomposition.

dij(s) )

cj(s) gij(s) gji(s) 1 + cj(s) gjj(s) gij(s)

1 + ci(s) gii(s)

∀ i, j * i

∀ i, j * i

(5)

(6)

Similar interaction terms are also demonstrated by other researchers (Seborg et al., 1989; O’Reilly and Leithead, 1991; Shen and Yu, 1994), however, only for 2 × 2 systems. Nevertheless, the cross-interaction terms are ignored by most of the above methods and other RGA type of interaction measures (Bristol, 1978; Tung and Edgar, 1981; Gagnepain and Seborg, 1982) as well. This is the main reason why RGA can only measure the one-way interaction. Leithead and O’Reilly (1992) proposed an alternative decomposition approach, however, from a matrix algebraic perspective. Nonetheless, no implication for interaction measure is explored. We will see that the interaction defined here with physical significance has a substantial impact on the subsequent development in this paper. The structural decomposition leads to significant implications for some important issues in decentralized control, as stated in the following remarks. Remark 1. More important information about interaction measurement is provided. In particular, twoway interaction and hence the limitations of the existing methods is clearly revealed. Moreover, in contrast to existing interaction measures, the influence of both the process and the individual controllers on interaction is included. Remark 2. Substantial implications for the analysis of decentralized control systems are offered. System properties such as open loop stability, nonminimum phase behavior, integrity, and stability can be defined and studied on the basis of individual SISO loops. More importantly, the effects of interaction, variable pairing, and controller tuning can be elucidated. Remark 3. The structurally decomposed individual SISO loops are completely equivalent to the original system with no assumption made. The coupling among all the loops is distributed into separate SISO loops with direct comparison to their independent counterparts. Consequently, decentralized system design can be greatly reduced to that for SISO systems, and independent design can be performed with interactions taken into account. 3. Open Loop Structure 3.1. Loop Interaction and Connection to RGA. In the following development, diagonal pairing is assumed unless specified otherwise, without loss of generality. The Laplace variable s is omitted for simplicity. Note that nondiagonal pairings can be rearranged as diagonal ones by properly rearranging the paired elements to diagonal positions (Zhu and Jutan, 1993b). From the structural decomposition shown in Figure 1, the following remark is obvious upon associating loop interaction with the interaction-free term.

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Remark 4. The equivalent process transfer function of the ith loop, denoted by g˜ ii, is given by

g˜ ii ) gii(1 + φii)

∀i

(7)

where

∀i

φii ) aii/gii

(8)

is defined as the relative interaction in the loop. The significance of remark 4 is that it allows for the distribution of open loop poles and zeros, thus open loop stability and nonminimum phase behavior, to be examined on the basis of individual process elements and interaction in individual loops. The interaction and the relative interaction can be viewed as an addictive model deviation from the independent process due to loop interaction (see Figure 1), whereas 1 + φii can be viewed as a multiplicate model deviation. Obviously, the phase and magnitude of the relative interaction determine the feasibility and easiness of the fine-tuning process using independent design (Seborg et al., 1989). The relative interaction is also closely related to the Rijnsdorp interaction quotient (Rijnsdorp, 1965) and the RGA (Bristol, 1966) in 2 × 2 systems by

φii ) -κhj

∀ i ) 1, 2, j * i

(9)

where κ is the Rijnsdorp quotient (Rijnsdorp, 1965) defined by

κ)

g12g21 g11g22

(10)

and hj is the closed loop transfer function of the independent SISO loop defined by

hj )

cjgjj 1 + cjgjj

∀ j ) 1, 2

(11)

The Rijnsdorp quotient is related to the (1,1) element of the dynamic RGA, λ11, by

1 κ)1λ11

(12)

From eqs 9-12, it is clear that both the RGA and the Rijnsdorp quotient provide equivalent but incomplete information regarding loop interaction. Specifically, they both contain only the process dynamics and lend themselves as interaction measures only if perfect dynamic control (which never holds in reality except at steady state!) is assumed. The relative interaction defined in remark 4 shows that controllers can in fact play a significant role in loop interaction (see Zhu and Jutan, 1995b). In particular, expression 9 allows the interaction contributed by the process alone, which is referred as process interaction, and that by control loop, called control interaction, to be investigated separately. Due to its importance in this study, the relative interaction will be simply referred as loop interaction or interaction. 3.2. Open Loop Stability. The following theorem governs the distribution of open loop poles: Theorem 1. Assuming that individual elements of the process and the controller do not contain any RHP poles, all the equivalent processes in all the individual loops do not contain any RHP poles if and only if the system possesses integrity against any single loop failure.

Proof. See Appendix A. Theorem 1 gives the necessary and sufficient conditions to prevent the interaction from inducing any pole in the right half plane (RHP). For 2 × 2 systems, we have, Corollary 1. Assume that all individual elements of the process and the controller in a 2 × 2 system do not contain any RHP poles. Both equivalent processes are open loop stable if and only if both independent SISO loops are stable. Proof. The proof is straightforward from theorem 1 upon noticing that integrity is reduced to stability of the two independent loops for 2 × 2 systems, since removing one loop leaves a single independent loop. Theorem 1 and corollary 1 imply that local instability, i.e., lack of integrity, is the only source for open loop instability in a particular loop in an interactive decentralized control system. 3.3. Nonminimum Phase Behavior. It is wellknown that the existence of RHP zeros may cause inverse responses and may potentially impose limitations on the achievable dynamic performance (Seborg et al., 1989). The following remark offers a word of caution in designing decentralized control systems. Remark 5. RHP zeros may be induced in a loop by loop interaction, even though individual processes are minimum phase. Significantly, unlike methods for general multivariable systems, the existence of the RHP zeros can be determined by the RHP zeros of the individual elements of the process and the independent loops. The following theorem provides a necessary and sufficient condition to prevent interaction from inducing nonminimum phase behavior. Theorem 2. Assume that individual elements of the process and the controller do not contain any RHP poles and RHP zeros, and the system possesses integrity against failure of any single loop. The overall system remains minimum phase if and only if

N(-1,φii) ) 0

∀i

(13)

where N(-1,φii) denotes the number of clockwise encirclements of (-1,0) point by the Nyquist plot of φii. Proof. See Appendix B. For 2 × 2 systems, theorem 2 leads to the following. Corollary 2. Assuming that none of the individual elements of the process and the controller contain any RHP poles or RHP zeros, and that both independent SISO loops are stable, the overall system remains minimum phase if and only if

N(-1,-κhj) ) 0

∀j

(14)

where N(-1,-κhj) represents the number of clockwise encirclements of (-1,0) point by the Nyquist plots of -κhj. Proof. See proof of theorem 2. Corollary 2 allows for the effects of the process interaction (i.e., the Rijnsdorp quotient) and the control interaction (i.e., controller tuning) on the creation of nonminimum phase behavior to be explicitly exposed separately. Theorem 2, corollary 2, and remark 5 indicate that care should be exercised, when selecting the desirable variable pairing and performing controller tuning, to avoid creating open loop RHP zeros in any loop. The following example elucidates this. Example 1 (Zhu and Jutan, 1995a). Consider the following open loop stable, individually minimum phase process

[

1 1 S + 0.1 S + 0.1 G(s) ) 1 R S + 1 S + 0.1

]


(15)

to be controlled by the PI controller below,

C(s) )

[ ]

s + 0.1 k1 0 0 k2 s

(16)

where R is the parameter for tuning the amount of process interaction and k1 and k2 are the proportional gains of the controller for tuning the amount of control interaction. The effects of the interaction on the distribution of poles and zeros of the equivalent open loop processes are analyzed below. It can be readily verified that the closed loop transfer functions of both independent SISO loops are

hj )

kj s + kj

j ) 1, 2

Figure 3. Nonminimum phase region in a loop in example 1.

(17)

and that both independent loops are stable with any positive gains. The relative interaction and the multiplicate deviation of the independent single loop process model in each loop as defined by eq 9 are given by

φii ) -

k2 s+1 R(s + 0.1) s + k2

∀ i, j * i

(18) Figure 4. Nyquist plots as functions of interaction in example 1.

and 2

1 + φii )

s + (1 + (1 - R)kj)s + (1 - 0.1R)kj (s + 1)(s + kj) i ) 1, 2, j * i (19)

As indicated by corollary 1, eq 19 shows that the equivalent processes of both loops are also stable for any positive kj, j ) 1,2. However, nonminimum phase behavior or RHP zeros is induced if

kj >

1 R-1

j ) 1, 2, 1 < R < 10

(20)

R > 10

(21)

or

Figure 3 shows the region (shaded) for the equivalent process in a loop to maintain minimum phase. Based on theorem 2 and corollary 2, Figure 4 also clearly demonstrates the above conditions by Nyquist plots as functions of interactions due to the process and the controller tuning. The above two conditions have significant implications as discussed below. Clearly, both the process and the controller play substantial roles in creating nonmimimum phase behavior. Specifically, larger loop interaction due to a larger Rijnsdorp quotient, i.e., larger R in this example (see Figure 4c in comparison with Figure 4a), or tighter control action, i.e., large kj, j ) 1, 2 (see Figure 4b in comparison with Figure 4a), tends to drive the open loop zeros toward and eventually into the nonminimum phase region. Interestingly, if the interaction in a loop, upon closing independently designed loops, tends to create a nonminimum phase equivalent process, detuning the controller in the other loop counteracts the tendency.

Surprisingly, however, if process interaction is inherently large enough, e.g., R > 10, nonminimum phase behavior becomes inevitable regardless of controller tuning (see Figure 4d). The same situation occurs when R becomes negative, indicating that not only the magnitude but also the phase of the interaction contributes to the occurrence of undesirable behavior. It is important to point out that the limit on the process interaction for creating RHP zeros implies that the nature of the resulting nonminimum phase behavior has changed so that the inverse response becomes “permanent” and consequently instability may arise. In this case, the main system property is captured by instability, and nonminimum phase behavior becomes meaningless. In fact, the maximum process interaction can be characterized by the steady state value of the Rijnsdorp quotient, or the sign of the RGA, as stated by the following well-known result (Grosdidier et al., 1985; Chiu and Arkun, 1990; Zhu and Jutan, 1993a, 1995a,c). Theorem 3. Assume in a 2 × 2 system that the process, G(s), is stable, the controller, C(s), contains integral action, and G(s) C(s)/s is rational and proper, and that the two loops are independently stable. The overall system becomes unstable if

κ(0) > 1 or

λ11(0) < 0 where κ(0) is the Rijnsdorp quotient and λ11 (0) is the (1,1) element of the RGA, both at steady state. Proof. See Appendix C. Appendix C provides an alternative proof of theorem 3 based on the structural information directly. Theorem 3 indicates that process interaction may cause instabil-

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ity by its own (independent of controller design). Also, theorem 3 implies that nonminimum phase behavior not only causes initial inverse responses, thus imposing limitations on achievable performance, but also may create “permanent” inverse responses thus becoming a source of instability. It is worthwhile to note that, in the case of inherently large process interaction, the only choice to maintain stability by means of controller design is to reverse the control direction of one of the controllers (Zhu and Jutan, 1995a). However, this is undesirable since system integrity has to be sacrificed and, as a result, one of the control loops will become open loop unstable by theorem 1. It thus becomes clear that system properties such as RHP zeros, RHP poles, integrity, and stability are actually intrinsically related to each other. It is important to point out that the best approach to avoid undesirable system properties, such as lack of integrity, is to avoid large process interaction by means of variable pairing first and subsequently to fine tune the controller. Consequently, variable pairing and controller tuning must be jointly considered for desirable closed loop properties. 3.4. Structure for Stability Analysis. As previously noted, the cross-interaction term as shown in Figure 1 constitutes an integrated part of a loop after decomposition. However, under some circumstances, this term may be, to a large extent, ignored when performing stability analysis, as stated in the following intuitive remark. Remark 6. The cross-interaction terms contain no RHP poles if the system possesses integrity against each single loop failure. In particular, they converge to zero at steady state if individual controllers contain integral action. Further notice that the cross-interaction terms are outside the feedback loop (see Figure 1). Therefore, as far as stability is concerned, we can focus on the feedback structure composed of the equivalent process and the controller in a particular loop with the crossinteraction ignored. In fact, the effects of the crossinteraction terms on loop stability are reflected in loop interaction indirectly. This is obvious from eqs 5 and 6 and previous discussions, upon noting that the crossinteraction terms are the sources of loop interaction. 4. Stability Conditions Necessary and sufficient stability conditions can be obtained by investigating the location of the roots of the characteristic equation of each individual closed loop, as stated below. Theorem 4. A decentralized control system is stable if and only if the characteristic equation of each individual loop below,

1 + cig˜ ii ) 0

∀i

(22)

where g˜ ii is defined in eq 7, does not contain any RHP zeros. Theorem 4 imposes no assumption about the process and the system, thus representing a general stability condition. Since the interaction, φii, represents the intrinsic action of all other loops on the ith loop, the combination of interaction and direct control action in a loop should reveal the characteristics of the overall system. Hence, one could surmise that the characteristic equations of all the individual loops are actually the same, i.e., the characteristic equation of the overall

system. Indeed, for 2 × 2 systems, for which simple closed form of interaction terms are available, we have,

1 + c1g˜ 11 ) 1 + c2g˜ 22 ) 1 + c1g11 + c2g22 + c1c2(g11g22 - g12g21) ) 0 (23) Equation 23 can be easily verified from eqs 5, 7, and 8 or 7, 9, and 11. Seborg et al. (1989) and Zhu and Jutan (1995a) also demonstrated the final form of the characteristic equation, i.e., the third term in the above equation, by rather lengthy algebraic transformation of various transfer functions. In contrast, eq 23 is readily obtained directly on the basis of the structural information of the system. Applying the Nyquist stability criterion to individual loops yields the following more attractive stability conditions. Theorem 5. Assuming that individual elements of G(s) and independent SISO subsystems do not contain any RHP poles, and the system possesses integrity against any single loop failure, the decentralized control system is stable if and only if the Nyquist contour of the equivalent open loop transfer function, cigii(1 + φii), ∀ i, does not have any clockwise encirclement of the (-1,0) point, i.e.,

N(-1,cigii(1+φii)) ) 0

∀i

(24)

Proof. See Appendix D. Theorem 5 offers more insights into the intrinsic connection between the properties of various components and closed loop stability in a system. For instance, it allows for the effects of the interaction on stability to be extracted and assessed. Also, the stability robustness of a particular loop against loop interaction can be evaluated in comparison to the independent SISO subsystems. Specifically, if the independent SISO loops are designed to have a large stability margin, i.e., the Nyquist contour of cigii is far away from the (-1,0) point, the equivalent open loop transfer function in eq 24 can tolerate a “large” interaction in the loop. On the other hand, a large loop interaction may cause a deviation of the independent open loop transfer function to be large enough to move the Nyquist contour across the (-1,0) point, leading to instability of the overall system. For 2 × 2 systems, theorem 5 reduces to the following. Corollary 3. Assuming that individual elements of the process do not contain any RHP poles, and that both independent subsystems are open loop and closed loop stable, the overall 2 × 2 system is stable if and only if

∀ i, j * i

N(-1,οi(1-κhj)) ) 0

(25)

where

οi ) cigii

∀i

(26)

denotes the open loop transfer function of the ith loop, κ is the Rijnsdorp quotient, and hj represents the closed loop transfer function of the jth loop. Proof. Corollary 3 can be obtained from theorem 5 by observing that requiring integrity of the system is equivalent to requiring stability of independent SISO loops in 2 × 2 systems. Corollary 3 explicitly exposes the intrinsic connection between the stability of the overall system and the two independent loops as well as the Rijnsdorp quotient. The following important observations can be extracted from corollary 3:


1. Closed loop stability strongly depends on the interactions contributed by both the process (represented by the Rijnsdorp quotient) and the controller tuning, and their influence on stability can be examined separately. 2. For a given process, closed loop stability can usually be maintained or improved by means of tuning, usually detuning, the other controller. 3. When the process interaction itself becomes inherently large, tuning the other controller may not be sufficient enough to counteract the interaction in order to maintain stability (see theorem 3). Instead, the direction of one of the two controllers may have to be reversed, leading to the loss of integrity (Zhu and Jutan, 1995a)! 4. It is well-known that process interaction, as measured by the RGA, can be substantially decreased by means of proper variable pairing. Theorem 4 also implies this. In particular, an appropriate variable pairing can greatly facilitate controller tuning to maintain stability, especially help avoid compromising system integrity for stability during controller design. However, it is the controller tuning, not variable pairing, that takes the decisive control in maintaining stability. Consequently, it advocates a common practicesvariable pairing and controller tuning are jointly considered in the design of decentralized control systems in order to maintain stability and integrity. Theorem 5 and corollary 3 provide theoretical justifications for this common practice. 5. The overall stability also relies heavily on the stability margin of the independent SISO loops. If an independent loop has a large stability margin, i.e., cigii is far away from (-1,0) point, the loop can tolerate a large loop interaction. As a result, stability is insensitive to the tuning of the other controller and to variable pairing choices. Furthermore, the following remark offers a significant result in analyzing decentralized control systems by frequency plots. Remark 7. The encirclements of (-1,0) point by the Nyquist plots of the various components in 2 × 2 systems and their linkage to important system properties are summarized below: οi: closed loop stability of independent ith loop (overall integrity and robustness) οj, j * i: open loop stability of the independent ith loop (RHP poles) -κhj: nonminimum phase behavior of the equivalent ith loop (RHP zeros) οi(1 - κhj): closed loop stability of the ith loop with interaction (overall stability) According to remark 7, important properties of the system, such as open loop stability, existence of RHP zeros, system integrity, stability robustness, and closed loop stability, can be readily extracted and monitored by plotting the Nyquist contours of the above various quantities as functions of variable pairing choices and controller tuning. Recalling the discussions about the effects of interaction and controller tuning on the existence of RHP zeros (nonminimum phase behavior), one may obtain the following. Remark 8. A 2 × 2 system likely undergoes the following intrinsic paths from independently stable to overall unstable due to an increasing process interaction

Figure 5. Example 2: Nyquist plots with R ) 2.0, k1 ) 0.5, k2 ) 20.0.

(large RGA) and corresponding controller tuning to maintain stability: From independently stable and nonminimum phase to nonminimum phase with interaction, closed loop unstable with maximum process interaction, open loop unstable (reversing control direction), loss of integrity, and overall stable. However, undesirable behavior in remark 8 (nonminimum phase, open loop instability, and loss of integrity) can be avoided by jointly considering variable pairing and controller tuning. The above observations and remarks regarding the effects of variable pairing and controller tuning on performance limiting (nonminimum phase), system integrity, and closed loop stability are likely applicable to general n × n systems. Example 2. Consider the same process and controller given in example 1. Example 2 is intended to demonstrate the effects of loop interaction, due to both the process and controller, and variable pairing on closed loop stability as well as the RHP zeros by examining the Nyquist plots and their encirclements of (-1,0) point of various components in the first loop (the same can be performed for the second loop) in the system. Nyquist contours of the independent open loop system, c1g11, the interaction, φ11, the multiplicate model error, 1 + φ11, and the equivalent open loop system, c1g11(1 + φ11), as functions of process interaction (by adjusting R) and controller tuning (by tuning k2) are plotted in various figures. The observations from these figures can be summarized as follows. Case 1. Figure 5 shows that loop 1 is always stable (Figure 5d) with small process interaction (R ) 2.0) and large stability margin (Rk1 ) 1.0) in the independent subsystem (Figure 5a), regardless of controller tuning (even with k2 ) 20). However, loop 1, and hence the overall system, becomes nonminimum phase when the condition given by eq 20 is violated (Figure 5b,c), whereas the loop remains minimum phase when controller 2 is less tightly tuned for the same process (not shown here). Moreover, large control interaction (tight tuning of controller 2) also results in an decrease of stability robustness of the loop in the interactive system (Figure 5a,d). Case 2. When the process interaction is large (R ) 10.0), tighter controller 2 drives the loop from minimum phase and stable to stable but at the boundary of

742


Figure 6. Example 2: Nyquist plots with R ) 10.0, k1 ) 1.0, k2 ) 1/8.5.

Figure 8. Example 2: Nyquist plots with off-diagonal pairing (R ) 10, k1 ) 0.1, k2 ) 20).

Figure 7. Example 2: Nyquist plots with R ) 10.0, k1 ) 1.0, k2 ) 2.0.

Figure 9. Example 2: Nyquist plots with R ) 20, k1 ) 0.5, k2 ) 0.01.

nonminimum phase (not shown), stable but nonminimum phase (Figure 6), and eventually unstable and “nonminimum phase” (Figure 7). In contrast to the above case, the shape of the Nyquist plot of the equivalent open loop system (Figures 6d and 7d) has drastically changed from the independent open loop system due to large loop interaction (Figures 6a and 7a). Similar to case 1, the stability robustness decreases as the loop interaction increases (tuning up controller 2). Case 3. Loop 1, and hence the overall system, can be made stable by choosing a different pairing of variables. Figure 8 shows the same set of Nyquist plots in loop 1 corresponding to the off-diagonal pairing with the same stability robustness for the independent loop. It can be seen that loop 1 is stable even with a very tight tuning of the other controller. This case corresponds to case 1 with diagonal pairing. Case 4. Loop 1 will be unstable regardless of controller tuning when the process interaction becomes large enough. Figure 9 shows that loop 1 is always unstable and “nonminimum phase” even though controller 2 is drastically detuned (k2 ) 0.01), when process interaction becomes very large (R ) 20.0). The encirclement of (-1,0) point in Figure 9d could be more clearly shown by Bode plot (not presented here). Also note that the direction of the Nyquist plot of the independent open loop system is reversed due to loop interaction (Figure

9a,d). Clearly, the system can be made stable by proper variable pairing, as in case 3. Otherwise, the direction of controller 2 has to be reversed to maintain stability (see case 5 below). Case 5. Figure 10 shows the Bode plot of the equivalent open loop transfer function in loop 1. It is clear that loop 1 can be made stable by reversing the direction of controller 1 (k1 ) -0.5) and properly tuning k2. However, reversing the direction of controller 1 leads to the loss of integrity of the overall system against failure of loop 2, since a local positive feedback is formed (Zhu and Jutan, 1995a). Also, loop 2 becomes open loop unstable as a result according to theorem 1. Consequently, reversing control direction should be avoided and changing a variable pairing should be resorted. In conclusion, variable pairing and controller tuning should be jointly considered to achieve desired behavior as demonstrated in case 1 and case 3. 5. Conclusions Upon structurally decomposing an interactive multivariable control system into individual SISO loops, new important insights into interaction measurement and closed loop analysis in decentralized control systems are presented. A systematic analysis of system structure and closed loop properties such as loop interaction,


Figure 10. Example 2: Bode plot of c1g11(1 + φ11) with R ) 20, k1 ) -0.5, k2 ) 0.1.

nonminimum phase behavior, open loop stability, system integrity, and closed loop stability is performed. The intrinsic linkage among various properties and, more importantly, effects of the process and the controller on loop interaction and subsequently on system properties, particularly stability, are elucidated. In particular, various stability conditions, using the characteristic equations and the Nyquist plots directly on the basis of the individual SISO loops with interactions embedded, are provided in a transparent manner. More important insights into the significance and perspective roles of variable pairing and controller tuning in avoiding undesirable behavior, such as nonminimum phase, open loop instability, and lack of integrity, while maintaining closed loop stability, are offered. The main results are summarized as follows: 1. Both the process and the controller play an important role in contributing to loop interaction. 2. Controller tuning or design is the final decisive factor in maintaining closed loop stability. However, process interaction may impose difficulties in controller tuning to maintain integrity and stability. 3. Nonminimum phase behavior may be induced by loop interaction, thus imposing limitations on achievable performance. 4. RHP zeros not only cause performance problems but also may become a source for lack of integrity and instability. 5. Inherently severe process interaction may impose a permanent problem in maintaining integrity, in spite of controller tuning. 6. Lack of integrity is a direct source of open loop instability. 7. Variable pairing and controller tuning complement each other and consequently should be jointly considered in the design of decentralized control systems. In particular, variable pairing can be used to reduce process interaction and to alleviate difficulties in con-

troller tuning to avoid creating undesirable behavior, while controller tuning is the final tool responsible for ensuring desirable behavior, particularly system stability. Nomenclature aij ) interaction in the uj-yi loop in the addictive form C(s) ) controller transfer function (TF) matrix (diagonal) ci(s) ) controller TF in the ith loop c′i(s) ) ci(s)/s c′i(0) ) steady state value of c′i(s) dij ) perturbation term G(s) ) process TF matrix gij(s) ) the ijth element of G(s) g˜ ii(s) ) equivalent process TF with interaction included in the ith loop hi ) closed loop TF of the ith independent loop H ) closed loop TF matrix of a multivariable system ki ) controller gain in the ith loop s ) Laplace variable ui ) ith input yi ) ith output Greek Letters R ) parameter for tuning process interaction in examples φij ) relative interaction in the uj-yi loop defined by eq 12 κ ) Rijnsdorp quotient λij ) ijth element of the RGA οi ) open loop TF of the independent ith loop

Appendices A. Proof of Theorem 1. We prove the theorem for the first loop only, without loss of generality. Similar proof applies to other loops as well by properly rearranging the diagonal elements of the process and the controller to the (1,1) position respectively. Let us partition an n × n process and decentralized controller

744


as follows:

[ ] [ ] 1

· g11 · G12 · · G(s) ) ‚ ‚ ‚ · ‚ ‚ ‚ · · G21 · G22 · 1

Z(g˜ ii) ) Z(gii) + Z((1 + φii))

n-1

1

(a1)

· · 0 · · C(s) ) ‚ ‚ ‚ · ‚ ‚ ‚ · · 0 · C2 ·

Consequently, the existence of any RHP zeros of the equivalent open loop transfer function of the ith loop depends solely on (1 + φii), i.e.,

1

Z(g˜ ii) ) Z((1 + φii)) (a2)

n-1

g˜ 11 ) g11(1 + φ11)

G12G22-1H2G21 g11

H2 ) G22C2(I + G22C2)

(a4)

(a5)

is the closed loop transfer function of the system constituted by the original closed loop system with loop 1 removed. Combining eqs a3-a5 yields,

g˜ 11 ) g11 - G12C2(I + G22C2)-1G21

∀i

N(-1,φii) ) 0

(b5)

Obviously, a necessary and sufficient condition for (1 + φii) not to contain any RHP zeros is given by

is the relative interaction in the loop, and -1

(b4)

Further notice that φii can be viewed as the “open loop” transfer function of (1 + φii) in terms of the existence of RHP zeros. Consequently, the condition in theorem 2 reduces to

(a3)

where

(a6)

Clearly, from eq a6, the poles of g˜11 consist of the poles of different process and controller blocks as well as the reduced system with the first loop removed. It is known that the poles of a transfer function matrix are the poles of the least common denominator of all non-identically-zero minor of all orders of it (Postlethwaite and MacFarlane, 1979). Clearly, a transfer function matrix does not contain any RHP poles if neither of its individual elements contains any RHP poles, and vice versa. Since the individual elements of the process and the controller are assumed not to contain any RHP poles, the existence of RHP poles of the equivalent process, g˜11, is determined solely by the subsystem with the first loop removed. Clearly, the subsystem H2 contains no RHP poles if the system possesses integrity against the failure of the first loop. Likewise, g˜ 11 will inherit any RHP poles in H2, should the system lack integrity against single loop failure. Therefore, integrity against single loop failure constitutes the necessary and sufficient condition for g˜ 11 to be stable. B. Proof of Theorem 2. First notice that the overall system is minimum phase if and only if each and every individual loop is minimum phase. The open loop transfer function of the ith loop is given by

g˜ ii ) gii(1 + φii)

(b3)

n-1

The equivalent process is given by

φ11 ) -

where Z(‚) denotes the number of RHP zeros of the transfer function enclosed in the parentheses. The assumption implies that gii contains no RHP zeros, i.e.,

Z(gii) ) 0

n-1

c1

(b2)

(b1)

Apparently, the zeros of g˜ 11 are the combinations of the zeros of the independent process, gii, and the multiplicate model error, (1 + φii), thus,

∀i

N(-1,φii) ) p

(b6)

where p denotes the number of RHP poles of φii. Without loss of generality, we shall prove the condition for the first loop only. Similar proof applies to other loops as well by properly rearranging the respective diagonal elements. Upon partitioning the process and the controller as shown in eqs a1 and a2, the expression for φ11 is given by eq a4. Clearly, the poles of φii arise from the poles of various process blocks and those of the reduced subsystem with loop 1 removed. From the proof of theorem 1, one can conclude that the assumptions guarantees

p)0

(b7)

Finally, it can be concluded that the condition in theorem 2 constitutes a necessary and sufficient condition for the system to maintain minimum phase. C. Proof of Theorem 3. According to Zhu and Jutan (1995a), when the assumption in theorem 3 holds, a consistency principle for stability (necessary condition for stability) for any SISO system is given by

g(0) c′(0) > 0

(c1)

c′(s) ) (1/s)c(s)

(c2)

where

In eq c1 and c2, g(s) and c(s) denote the process and the controller transfer functions, respectively, and g(0) and c′(0) denote the steady state gain of their perspective transfer functions. Condition c1 actually specifies a feedback condition in any SISO system. In an interactive 2 × 2 system, the equivalent open loop transfer function in either loop is given by, according to eqs 7 and 9,

g˜ ) gii(1 - κhj)

∀, j * i

(c3)

Applying the necessary stability condition in eq c1 to either loop in the 2 × 2 system upon decomposition, one


has the following sufficient condition for instability,

g˜ 11(0) c′1(0) < 0

(c4)

where c′1 ) c1/s. At steady state, we have

hj ) 1

(c5)

since independent loops are assumed to be stable and integral action is used in the controller. Substituting eqs c5 and c3 into c4 yields

∀i

gii(0)(1 - κ)c′1(0) < 0

(c6)

Since independent loops are assumed stable, the following condition holds:

∀i

gii(0) ci(0) > 0

(c7)

Finally, a sufficient condition for instability is obtained as follows:

κ>1

(c8)

λ11 < 0

(c9)

By eq 12, one has

D. Proof of Theorem 5. By the Nyquist criterion, the necessary and sufficient condition for stability of the closed loop system of the ith loop is

N(-1,g°) ) z

(d1)

where N denotes the number of clockwise encirclements of the point (-1,0) by the Nyquist contour of the open loop transfer function go, which is defined by

g° ) cig˜ ii g˜ ii ) gii(1 + φii)

∀i

(d2) ∀i

(d3)

and z denotes the number of RHP poles of g°. The assumption ensures, by theorem 1,

P(g˜ ii) ) 0

(d4)

where P(‚) denotes the function to evaluate the number of RHP poles of the transfer function enclosed in the brackets, and

P(ci) ) 0

(d5)

Consequently, by eqs b2-b5 the RHP poles of the open loop system becomes

z ) P(ci) + P(g˜ ii) ) 0

(d6)

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Received for review July 24, 1995 Revised manuscript received November 14, 1995 Accepted November 29, 1995X IE950455A

Abstract published in Advance ACS Abstracts, February 1, 1996. X

Structural Analysis and Stability Conditions of Decentralized Control

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