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Performance limitations in decentralized control
Journal of Process Control 12 (2002) 485–494 www.elsevier.com/locate/jprocont
Performance limitations in decentralized control Hong Cui, Elling W. Jacobsen* Signals, Sensors and Systems Process Control, Royal Institute of Technology-KTH, S-100 44 Stockholm, Sweden
Abstract In decentralized control of multivariable systems, the system is decomposed into a number of subsystems and individual controllers are designed for each subsystem. Advantages of such decomposition include reduced modelling requirements and ease of implementation. However, a potential disadvantage is a reduction in achievable control performance due to restricted controller structure. In this paper we consider performance limitations from non-minimum phase transmission zeros in decentralized control. In particular, we derive conditions on when closing the loop around one subsystem moves transmission zeros of other subsystems across the imaginary axis. Such zero crossings may occur regardless of the existence of non-minimum phase behavior in the openloop system, and may, therefore, represent performance limitations specific to the use of decentralized controllers. # 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction We consider in this paper decentralized control of a general square multivariable nn system G(s), i.e. having n inputs and n outputs. Decentralized control implies that the overall system is decomposed into a number of (interacting) subsystems for which individual controllers are designed, i.e. the overall controller may be written on a block-diagonal form. Such a decomposition of the control problem is often preferred due to increased robustness, both with respect to model uncertainty (reduced modelling cost) and sensor/actuator failures, and ease of implementation. However, the potential cost is reduced closed-loop performance due to the presence of interactions among the subsystems. There exist a number of tools for analyzing the effect of interactions in decentralized control systems, of which the most widespread is the relative gain array (RGA) [1]. However, most of the theoretical results related to these tools concern (conditional) stability only. For instance, the common rule that one should not close loops around scalar subsystems corresponding to negative steady-state RGA elements, e.g. [2], is related to integrity (conditional stability) only. Hovd and Skogestad [3] have derived conditions, in terms of required minimum loop gains for the individual subsystems, in order for a decentralized controller to satisfy given specifications on the overall closed-loop * Corresponding author. E-mail address: [email protected] (E.W. Jacobsen).
performance. However, whether the required loop gains are achievable, under the constraint of closed-loop stability, depends on the presence of control limitations in the system under consideration. For the case of full multivariable controllers, a number of results on performance limitations are available. For instance, if the system G(s) has a real Right Half Plane (RHP) transmission zero at z, the bandwidth of the closed-loop system is limited by !B < z. See e.g. Skogestad and Postlethwaite [2]. Since a decentralized, or block-diagonal, controller is just a special case of a full controller, it is clear that the same limitations apply in decentralized control. However, it seems reasonable that a limited control structure may impose additional performance limitations. In this paper we address performance limitations from RHP zeros in decentralized control. In particular, we address the problem: under which conditions will closing the loop around one subsystem introduce RHP transmission zeros in other subsystems? If such zeros exist, and they are more severe than the RHP zeros of the overall system, they will represent performance limitations specific to the use of decentralized control. The problem of zero crossings under loop closure has been addressed by other authors previously. Bristol [1], in his original work on the relative gain array (RGA), proposes that a scalar loop corresponding to a negative steady-state RGA-element will have a RHP zero or pole when all the other outputs are controlled with integral action. Shinskey [4] argues, using heuristic arguments, that closing the loop around a negative steady-state
0959-1524/02/$ - see front matter # 2002 Elsevier Science Ltd. All rights reserved. PII: S0959-1524(01)00015-4
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RGA-element in a 22 system will introduce an inverse response, corresponding to a RHP zero, from the remaining input to the remaining output. Grosdidier et al. [5] show that the proposal put forward by Bristol holds under certain assumptions on the open-loop system GðsÞ Finally, Rosenbrock [6] has shown that, if all outputs but one of a multivariable system are controlled perfectly, then the remaining open-loop transfer-function will have the transmission zeros of the multivariable system as its zeros. In this work we derive more general conditions for RHP zeros induced by closing the loops around subsystems. In particular, we consider multivariable subsystems, real as well as complex zero crossings, finite bandwidth, general type controllers and relax many of the assumptions employed previously. Furthermore, we show that zero crossings may be from the LHP to the RHP as well as vice versa. Parts of this work have previously been presented in Jacobsen and Cui [7].
2. Effect of loop closure on subsystems external to the loop We consider a general square system y ¼ GðsÞu and want to analyze the effect of closing the loop between a subset of y and u on the transfer-matrix between the remaining inputs and outputs. We will first limit ourselves to consider single loop controllers, and then extend the results to multivariable controllers towards the end of the paper. Thus, we first decompose GðsÞ according to y1 ¼ g11 ðsÞu1 þ G12 ðsÞu2
ð1Þ
y2 ¼ G21 ðsÞu1 þ G22 ðsÞu2
ð2Þ
where y1 and u1 are scalars and y2 and u2 are vectors of dimension n1, and consider the effect of closing the loop between y1 and u1 on the transfer-matrix between u2 and y2. Closing the loop between y1 and u1 using the feedback law u1 ¼ c1 ðsÞy1
ð5Þ
Here s=zi and s=pj are the zeros and poles, respectively, of G(s). Note that, in general, there may be poles and zeros at the same location without canceling. From (4) we derive the following expression for the determinant of G^ 22 ðsÞ (see Appendix) detG22 ðsÞ þ c1 ðsÞdetGðsÞ detG^ 22 ðsÞ ¼ 1 þ c1 ðsÞg11 ðsÞ
ð6Þ
It is clear from (6) that the zeros as well as the poles of the subsystem G^ 22 ðsÞ are moved by the feedback (3). Our aim is, based on (6), to derive conditions for when there is a different number of RHP zeros in the two systems G22 (s) and G^ 22 ðsÞ, or equivalently, for when one or more zeros of the subsystem involving y2 and u2 are moved between the complex LHP and RHP as the loop (3) is closed. To do this we will first employ the simplifying assumption that the loop (3) is closed perfectly, i.e. with infinite bandwidth, and then consider the more general case where the loop is closed with finite bandwidth.
3. Zero crossings under perfect control We here assume that control of output y1 is perfect over all frequencies, i.e. infinite controller gain. Perfect control of y1 corresponds to zero sensitivity, i.e. 1 ¼0 1 þ c1 ðsÞg11 ðsÞ
ð7Þ
which is equivalent to c1 ð s Þ 1 ¼ 1 þ c1 ðsÞg11 ðsÞ g11 ðsÞ
ð8Þ
With (7) and (8), Eq. (6) becomes ð3Þ
yields for the transfer-matrix G^ 22 ðsÞ, between inputs u2 and outputs y2, c1 ð s Þ G^ 22 ðsÞ ¼ G22 ðsÞ G21 ðsÞG12 ðsÞ 1 þ c1 ðsÞg11 ðsÞ
Q i¼1;nz ðs zi Þ detGðsÞ ¼ K Q j¼1;np s pj
ð4Þ
Thus, the feedback (3) affects the transfer-matrix from u2 to y2, provided G21(s)G12(s) is not identically zero. We are mainly interested in the transmission zeros of G^ 22 ðsÞ. As defined here, the transmission zeros of a general square transfer-matrix G(s) can be expressed in terms of the determinant of the matrix
detGðsÞ detG^ 22 ðsÞ ¼ g11 ðsÞ
ð9Þ
From (9) it is clear that the only RHP zeros of G^ 22 ðsÞ will be those of G(s) which are not zeros of g11(s).1 Thus, if the multivariable system is minimum phase there will be no limitations in the remaining subsystem G^ 22 ðsÞ after closing the first loop perfectly. Note that this applies regardless of whether G22(s) has RHP zeros or not.
1
Note that any RHP poles in g11(s) will be cancelled by their equivalents in detG(s), and will hence not appear as RHP zeros in G^ 22 ðsÞ.
H. Cui, E.W. Jacobsen / Journal of Process Control 12 (2002) 485–494
Rosenbrock [6] shows that if all outputs but one of a multivariable system are perfectly controlled, then the remaining scalar subsystem will have the zeros of G(s) as its zeros. The result presented above shows that the subsystem remaining after closing any number of loops perfectly will have the zeros of G(s) as its zeros, and is thus a generalization of Rosenbrock’s result. Example 1: Consider the 33 system 0 1 1 s 1 1 @4 GðsÞ ¼ 1 1 A ðs þ 1Þ2 1 2 3
1þ
c1 ðsÞdetGðsÞ ¼ 0; detG22 ðsÞ
ReðsÞ > 0
487
ð13Þ
The following theorem provides necessary and sufficient conditions for when G^ 22 ðsÞ and G22(s) has a different number of transmission zeros in the RHP. Theorem 1. Assume G(s) and c1(s) stable and G21(s)G12(s)6 0. Then the image of
ð10Þ
which has a transmission zero at s=0.77. The transfermatrix G22(s) between (u2, u3) and (y2, y3) is minimum phase. Closing the loop around g11(s) using a proportional controller with gain k=1 yields 1 4s 1 3 ð11Þ G^ 22 ðsÞ ðs þ 1Þ2 ðs þ 2Þ 4 which has a zero at s=0.77, corresponding to the RHP zero of G(s). Also note that the transfer-function from u2 to y2 now has a RHP zero at s=0.25, implying that the original open-loop subsystem with y3 and u3 removed has this zero. Infinite bandwidth can of course never be achieved in practice, but the results derived above may serve as asymptotic results. An illustration of their usefulness as such will be given below. We consider next the case where the assumption of perfect control is relaxed.
4. Zero crossings under finite bandwidth When output y1 is controlled with a finite bandwidth, the determinant for the remaining subsystem G^ 22 ðsÞ is given by (6), i.e. detG22 ðsÞ þ c1 ðsÞdetGðsÞ detG^ 22 ðsÞ ¼ 1 þ c1 ðsÞg11 ðsÞ
ðsÞ ¼
c1 ðsÞdetGðsÞ detG22 ðsÞ
ð14Þ
as s follows clockwise around the Nyquist D-contour will encircle the point 1, in the complex plane, N=ZˆZ times, where Zˆ is the number of RHP zeros of Gˆ22(s) and Z is the number of RHP zeros of G22(s). Proof. The result follows directly from application of the argument principle to (13) using the fact that, when G(s) and c1(s) are stable, the RHP zeros and poles of 1+(s) are equivalent to the RHP zeros of G^ 22 ðsÞ and G22(s), respectively. If G21(s)G12(s)=0,8s, then from (4) G^ 22 ðsÞ G22(s). In this case the RHP zeros of 1+(s)= 1+c1(s)g11(s) will be exactly canceled by equivalent poles. & Theorem 1 states that if the image of (s) encircles the 1 point n times, for s=(j1, j1), then n zeros of the subsystem involving y2 and u2 cross between the complex LHP and RHP as the loop (3) is closed. If the encirclements are clockwise, as s goes clockwise around the Nyquist D-contour, the crossings are from the LHP to the RHP, while they are from the RHP to the LHP if the encirciements are anticlockwise. If (s) is strictly proper all zero crossings are through the imaginary axis, while they may be through 1 if (s) is semi- or improper. This is easily understood by considering (13) which can not have roots at s=1 if (s) is strictly proper. For systems in which the Nyquist curve (j!), for !>0, crosses the negative real axis once only, a simpler criterion can be employed.
In the following we assume stability of the open loop systems G(s) and c1(s). With this assumption, G^ 22 ðsÞ will have zeros in the RHP if and only if
Corollary 1. Assume that G(s) and c1(s) are stable, that (s) is strictly proper and that (j!) is negative real for one frequency ! ¼ !18050 only. Then G22(s) and Gˆ22(s) will have a different number of RHP zeros if
detG22 ðsÞ þ c1 ðsÞdetGðsÞ ¼ 0;
ðj!180 Þ > 1;
ReðsÞ > 0
ð12Þ
Thus, to check for RHP zeros in G^ 22 ðsÞ one can for instance apply the argument principle to (12). Since we are here mainly interested in the case when there is a difference in the number of RHP zeros between G^ 22 ðsÞ and G22(s), we divide (12) by detG22(s) to obtain
ffðj!180 Þ ¼ 180
ð15Þ
Proof. The condition follows directly from Theorem 1 since it ensures at least one encirclement of the point 1. The assumption that (s) is strictly proper ensures that (j!)! 0 as !!1. &
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Note that the zero crossing will be real if !180=0, while it will be complex conjugate if !180>0. Thus, the simplified condition for a real zero crossing can be written c1 ð0ÞdetGð0Þ < 1 detG22 ð0Þ
ð16Þ
The next result concerns the case in which the function (s) contains a pure integrator, which for instance is the case if the controller c1(s) contains integration. Corollary 2. Assume that G(s) and c1(s) are stable and that (s) is strictly proper with one pole at s=0, i.e. (s)=0 (s) 1s, where 0 (s) is analytic with no zeros at s=0. Then G22(s) and Gˆ22(s) will have a different number of RHP zeros provided 0 (0)< 0. Proof. The result follows from considering the image of (s) for the small semicircular detour of s=0 in the Nyquist D-contour, i.e. s ¼ re j ; jrj ! 0; 2 ½=2; =2. Then the image of (s) makes an infinite semi-circle around the LHP if it contains an integrator and 0 (0) < 0. The assumption of strictly proper (s) ensures that the image approaches the origin for !!1, and hence an encirclement of the 1 point. & Note that Corollary 2 provides a sufficient condition only. The function (s) is closely related to the relative gain array (RGA). In particular, the 1,1-element of the RGA of G(s) is given by l11 ¼
g11 detG22 detG
ð17Þ
g11 ð0Þc1 ð0Þ < 1 l11 ð0Þ
ð20Þ
Grosdidier et al. [5] have previously shown that, under certain assumptions, a SISO loop corresponding to a negative steady-state RGA element will have an odd number of RHP zeros, or a RHP pole, when all other loops are closed with integral action. The results presented above show that closing the loop around an element corresponding to a negative steady-state RGA element, using negative feedback with integral action, is a sufficient condition for moving a zero of the remaining subsystem across the imaginary axis, and is thus in some sense dual to the previous result. However, we stress that it is a sufficient condition only, and by no means necessary for zero crossings. Below we illustrate the results derived above on a number of examples. Example 2: Consider 0 4 1 @ GðsÞ ¼ 0:9 10s þ 1 1
the minimum phase 33 system 1 4 2 1 1A ð21Þ 2 3
in which all subsystems are also minimum phase. Apply negative proportional feedback around g11(s), i.e. u1 ¼ ky1
ð22Þ
The corresponding RGA element is l11(0)=3.33. Thus, condition (20) for a real zero crossing becomes 4k < 1 ) k > 0:833 3:33
ð23Þ
and thus we get that c1 ðsÞg11 ðsÞ ðsÞ ¼ l11 ðsÞ
ð18Þ
In terms of the RGA, the condition (13) for a different number of RHP zeros in Gˆ22(s) and G22 ðsÞ can then be written c1 ðsÞg11 ðsÞ ¼ 0; 1þ l11 ðsÞ
ReðsÞ > 0
ð19Þ
It is interesting to note that condition (19) for RHP zeros in G^ 22 ðsÞ under finite bandwidth control of y1 corresponds to the condition for RHP poles, i.e. instability, in loop 1 under perfect control of y2. This follows from the fact that the loop gain for loop 1 will be g11 c1 =l11 when output y2 is controlled with infinite bandwidth. In terms of the RGA, the sufficient condition (16) for a real zero crossing becomes
Since G22(s) is minimum phase, a zero crossing must be from the LHP to the RHP. Indeed, with k=1, corresponding to a bandwidth of approx. 0.5 for y1, we get 1 2s þ 0:28 2s þ 0:64 G^ 22 ðsÞ ¼ 6s þ 2:6 20s2 þ 12s þ 1 4s þ 1:2 which has a zero at s=0.02, which implies that the bandwidth for this subsystem is limited by !B < 0.02. Note that although one chooses to only pair on positive RGA elements for an nn system, the pairing may correspond to pairing on a negative RGA for some subsystems. We illustrate this on the example below. Example 3. Consider the 33 system 0 1 0:3 1:8 0:9 1 @ 1:7 GðsÞ ¼ 0:32 1:4 A 100s þ 1 0:50 0:50 0:63
ð24Þ
H. Cui, E.W. Jacobsen / Journal of Process Control 12 (2002) 485–494
which is minimum phase. The RGA is, at all frequencies, 0 1 0:76 3:40 3:16 ¼ @ 5:92 0:42 5:35 A ð25Þ 5:68 2:83 9:51 The common pairing rules based on the RGA, i.e. prefer pairings for which the RGA-elements are closest to 1 and avoid pairing on negative RGA-elements, suggest pairing on the diagonal. However, the RGA for the 22 subsystems obtained by removing (y3, u3) and (y2, u2), respectively, both have negative RGA-elements on the diagonal. Thus, according to the results above, closing the loop around g11(s), with a controller containing integral action, will introduce RHP zeros in the transfer-functions from u2 to y2 and from u3 to y3, respectively. It is easy to show that these zeros cannot be moved back to the LHP by closing further loops. As stated above, a common rule in decentralized control is to avoid pairings on negative steady-state RGAelements. However, this is mainly related to integrity, i.e. the system should remain stable as loops are taken in and out of service. See e.g. Skogestad and Postlethwaite [2]. Furthermore, the rule is in general only necessary, and not sufficient. On the other hand, if some elements contain RHP zeros, then pairing on negative RGA elements may be advantageous from a performance point of view, as illustrated by the next example.
Example 4. Consider the minimum phase system 1 s þ 0:1 s þ 0:1 GðsÞ ¼ ð5s þ 1Þ2 s 0:05 s 0:025
489
ð26Þ
Both transfer-functions for y2 contain severe RHP zeros, and the worst appears to be from input u2. However, from the results above it is clear that these zeros will be moved as soon as the loop for y1 is closed. Furthermore, since the overall system is minimum phase, closing the loop for y1 with infinite bandwidth will remove all zeros in the RHP for y2, independent of which pairing that is used. Thus, perfect control is, in theory, possible with decentralized control, using either pairing. However, with finite bandwidth control of y1, the results derived above tells us that the RHP zero for y2 can only be removed by pairing on a negative steady-state RGA. The steady-state RGA is 1 2 ð27Þ ð 0Þ ¼ 2 1 Consider choosing either input u1 or input u2 to control y1 with the PI-controller ui ¼ k
5s þ 1 ðr1 y1 Þ; 5s
i ¼ 1; 2
ð28Þ
Fig. 1 shows the zero with maximum real part for output y2 as a function of the controller gain k when u1 and
Fig. 1. Example 4. Maximum real part of zeros of transfer-functions for output y2 when y1 is controlled with the controller (28) and input u1 and u2, respectively.
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H. Cui, E.W. Jacobsen / Journal of Process Control 12 (2002) 485–494
u2 are used for control of y1, respectively. As seen from the figure, with y1 controlled using u1 (negative RGA), the RHP zero for y2 moves towards the LHP for increasing k and crosses the imaginary axis for k=11.5, corresponding to a bandwidth !B=0.3 for output y1. However, with y1 controlled using u2 (positive RGA), the RHP zero for y2 moves further into the RHP as the gain k is increased. Since, for a real RHP zero at s=z, the bandwidth is limited by !B < z, we thus have a bandwidth limitation for output y2 if we choose to pair on positive RGA. With k=25, corresponding to a bandwidth of 1 for y1, we get a zero at s=0.16 for y2 and thus a maximum bandwidth of 0.16 for this loop. If we want to achieve a bandwidth of 1 for output y2, then we need to increase k above 970, corresponding to a bandwidth exceeding 45 for y1, if y1 is controlled using u2. Note that for the above example, integrity can not be satisfied for either pairing if the bandwidth for y2 is to exceed 0.05. Thus, if a bandwidth higher than 0.05 is required then one should in this case pair on negative RGA elements since this most likely will provide the best performance. One might argue that if integrity can not be incorporated, then one might as well employ full multivariable control. However, there are a number of factors, apart from integrity, such as robustness (reduced modelling requirements), which motivates the use of decentralized control. In all examples considered so far, the zero crossings have been due to pairing on negative steady-state RGA
elements for the overall system, or for some subsystem. However, zero crossings may occur also when pairing on positive RGA-elements. In this case the zero crossings will be complex conjugate, and the simplified criterion (15) can be written
log g11 c1 logjl11 j > 0; ffg11 c1 ffl11 ¼ ð29Þ From this expression we see that complex zero crossings are most likely to occur for small values of jl11 j and positive phase ffl11 . Example 5. Consider the minimum phase system ! 0:1 1 0:05 GðsÞ ¼ 30s þ 1 ð10s þ 1Þ3 0:1 0:1
ð30Þ
The plots of jl11 j and ffl11 are shown as functions of frequency in Fig. 2. Using the usual rules for selecting pairings, i.e. to pair on the RGA for which lij is closest to 1 around the expected bandwidth, we see that we should pair on the diagonal provided the desired bandwidth !B exceeds 0.04. We note that jl11 j < 1 and ffl11 >0 for all frequencies, and according to the above this increases the probability of a complex zero crossing. Indeed, with the controller c1 ð s Þ ¼ k
10s þ 1 10s
Fig. 2. Amplitude and phase of 1,1-element of RGA in Example 5.
ð31Þ
H. Cui, E.W. Jacobsen / Journal of Process Control 12 (2002) 485–494
0
and k=1 we obtain the Bode plot of ðsÞ ¼
g11 ðsÞc1 ðsÞ l11 ðsÞ
ð32Þ
shown in Fig. 3. Employing criterion (15), i.e.
ð!180 Þ > 1; ffðj!180 Þ ¼ 180 we find from Fig. 3 that g^ 22 ðsÞ will have RHP zeros if k>8.33, corresponding to a bandwidth !B>0.058 in the first loop. For example, with k=10 we find that the transfer-function from u2 to y2 contains zeros at s= 0.0013 j 0.061, imposing a severe performance limitation for this loop. Note that one situation in which zero crossings in G^ 22 ðsÞ is expected as the bandwidth for y1 is increased, is when the overall system G(s) is non-minimum phase with more RHP zeros than G22(s). In this case, the asymptotic result derived using infinite bandwidth for y1, guarantees that zeros of G^ 22 ðsÞ will cross from the LHP to the RHP as the bandwidth of y1 is increased beyond a critical value. If the pairing for y1 corresponds to a positive steady-state RGA-element, the crossing must be in the form of complex conjugate zeros (assuming the zeros can not cross through 1, i.e. (s) strictly proper). Example 6. Consider the non-minimum phase system (taken from Example 2.7 in Rosenbrock [6]).
1 1 2 B 1 s þ 3C GðsÞ ¼ @ s þ 1 1 A sþ1 sþ1
491
ð33Þ
G(s) has a transmission zero at s=1. Consider applying the feedback u1=ky1, for which l11 (0)=3. According to the results for perfect control derived above, the transfer-function g^ 22 ðsÞ will have one, and only one, RHP zero at s=1 as k!1. Thus, there will be a zero crossing the imaginary axis for some value of k>0. Since the loop is closed around a positive RGA, the zero crossing of g^ 22 ðsÞ must be complex. Thus, as k is increased, a complex conjugate pair of zeros will cross the imaginary axis, which then become real and one moves to s=1 and one toward s=1 as k!1. For instance, for k=10, we find a pair of complex zeros at s=3 j 2 in g^ 22 ðsÞ. The conditions derived above can be used to determine whether closing loops in a decentralized control system may impose inherent bandwidth limitations in some subsystems. The specific limitations imposed will depend on the location of the zeros in the RHP. In general, the location will depend on the system properties as well as the controllers employed and it is hence difficult to derive any general results. However, one specific situation in which the zero location can be easily predicted is when the dynamics in all elements of G(s) are equivalent, so that the RGA is independent of frequency, and the controller c1(s) is based on model inversion (direct synthesis) such that
Fig. 3. Amplitude and phase of (s) with diagonal pairing and k=1 in Example 5.
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H. Cui, E.W. Jacobsen / Journal of Process Control 12 (2002) 485–494
c1 ð s Þ ¼
!c 1 s g11 ðsÞ
ð34Þ
From (19) we then find that G^ 22 ðsÞ will get exactly one extra RHP zero at s¼
!c l11
ð35Þ
provided l11 < 0. Note that the RHP zero in this case is directly proportional to the bandwidth !c and inversely proportional to the magnitude of the RGA-element. Thus, the performance limitation will in this case decrease with increasing bandwidth !c, while becoming more severe with a larger magnitude of the RGA.
The next theorem provides a necessary and sufficient condition for when closing the loop (36) causes zeros in the sub-system involving y2 and u2 to cross between the LHP and RHP. Theorem 2. Assume G(s) and C1(s) stable and G21(s) and G12(s) not identically 0. Define ð41Þ ðsÞ ¼ C1 ðsÞ G11 ðsÞ G12 ðsÞG1 22 ðsÞG21 ðsÞ Then the image of det(I+D(s)), as s follows clockwise around the Nyquist D-contour, will encircle the origin N= ZˆZ times, where Zˆ and Z are the number of RHP zeros of Gˆ22(s) and G22(s), respectively. Proof. Define
5. Block-diagonal controllers
fðsÞ ¼ detðI þ ðsÞÞ ¼
We have so far considered closing SISO loops only, and the derived results thus apply to diagonal controllers only. However, block-diagonal controllers, i.e. containing multivariable sub-controllers, are also commonly used. We here derive conditions on when closing a multivariable controller around a subsystem drives zeros of other subsystems between the LHP and RHP. Consider decomposing the overall nn system G(s) as in (1) and (2), but let now y1 and u1 be vectors of any dimension less than n. The corresponding square transfer-matrix is written G11(s). Closing the loop u1 ¼ C1 ðsÞy1
ð36Þ
detG^ 22 ðsÞ detðI þ C1 ðsÞG11 ðsÞÞ detG22 ðsÞ ð42Þ
and apply the argument principle to f(s). With G(s) and C1(s) stable, I+C1(s)C11(s) is stable. Furthermore, any RHP zeros of I+C1(s)G11(s) (RHP poles of the closed loop) are canceled in f(s) by equivalent RHP poles in Gˆ22(s) provided neither G12(s) nor G21(s) are identically zero. Thus, the RHP zeros and poles of I+(s) are equivalent to the RHP zeros of G^ 22 ðsÞ and G22(s), respectively. & Similar to the SISO case, we propose a simple sufficient condition for checking the existence of zeros crossing the imaginary axis.
yields for the remaining subsystem G^ 22 ðsÞ ¼ G22 ðsÞ G21 ðsÞ ðI þ C1 ðsÞG11 ðsÞÞ1 C1 ðsÞG12 ðsÞ
ð37Þ
Corollary 3. Assume G(s), C1(s) stable, G21(s) and G12(s) not identically 0 and D(s) strictly proper. Assume furthermore that det(I+D(j!)) is negative real for one frequency !18050 only. Then there is a difference of at least one RHP zero between Gˆ22(s)and G22(s).
and the determinant can be written (see Appendix) detG^ 22 ¼ detG22 detðI þ C1 G11 Þ1 det I þ C1 G11 G12 G1 22 G21
ð38Þ
With perfect control of y1, i.e. when SðsÞ ¼ ðI þ C1 ðsÞG11 ðsÞÞ1 ¼ 0
ð39Þ
(38) can be simplified to (see Appendix) detGðsÞ detG^ 22 ðsÞ ¼ detG11 ðsÞ
ð40Þ
which is a generalization of the SISO result in Eq. (9). A similar result can be found in Johansson and Rantzer [8].
Proof. The corollary follows from Theorem 2 since the assumptions ensure that the image of detðI þ ðsÞÞ encircles the origin at least once. & As for the SISO case, it is easily shown that the zero crossing is real if o180=0 and complex conjugate otherwise. Thus, for a real zero crossing, the simplified condition (Corollary 3) becomes that an odd number of eigenvalues of (0) should be real and less than 1. Finally, similar to Corollary 2 for SISO loop closure, it is easily shown that if a strictly proper (s) can be written as 0 (s)/s where 0 (0) has full rank, e.g. when C1(s) contains integration, then a sufficient condition for a different number of RHP zeros between G^ 22 ðsÞ and G22(s), is that an odd number of eigenvalues of 0 (0) are negative real.
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Example 7. Consider the minimum phase system 0 1 0:3 1:9 0 0:5 1 B 1:8 0:1 0:8 C B 0:1 C GðsÞ ¼ @ 10s þ 1 0:4 0:7 0:8 0:7 A 0:5 0:7 1:8 0:3
ð43Þ
for which all subsystems are also minimum phase. Consider controlling the two first outputs with the two first inputs using the IMC controller C1 ðsÞ ¼
!c 1 G ð sÞ s 11
ð44Þ
which provides a bandwidth !c for this subsystem. The corresponding function is ðsÞ ¼
!c s
0:70 0:56
1:36 1:40
ð45Þ
and the eigenvalues l(0 (0)), i.e. after removing the integrator, become l=0.23!c, 0.93!c. Thus, since one eigenvalue is negative, we should expect a RHP zero in the remaining subsystem G^ 22 ðsÞ involving the two last inputs and outputs. Indeed, with !c=1, the system has a zero s=0.042, imposing a severe performance limitation for this subsystem. The limitation can be relaxed by increasing the bandwidth !c for the other subsystem.
Finally, complex zero crossings are found to be most likely when closing a loop around an element corresponding to a RGA with small magnitude and large positive phase. Thus, the phase information of the RGA proves useful for predicting zero crossings.
Acknowledgements This work is a part of the ‘‘Ecocyclic Pulp Mill’’ research programme financed by MISTRA, the Swedish Foundation for Strategic Environmental Research.
Appendix. Derivation of Eqs. (6), (38) and (40) Consider a square multivariable system G, and apply negative feedback control around a subsystem G11 with inputs u1 and outputs y1, u1=C1y1. Then the remaining subsystem G^ 22 ðsÞ, with inputs u2 and outputs y2, is given by G^ 22 ¼ G22 G21 ðI þ C1 G11 Þ1 C1 G12
ð46Þ
Taking the determinant on both sides yields detG^ 22 ¼ det G22 G21 ðI þ C1 G11 Þ1 C1 G12 0
B C G ðI þ C1 G11 Þ1 C1 G12 A ¼ detG22 det@I G1 |fflfflffl22 ffl{zfflfflffl21 ffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} A1
A2
6. Conclusions We have in this paper shown that closing loops in a decentralized control system may move zeros of subsystems external to the loop between the complex LHP and RHP. Such zero crossings may occur regardless of the existence of non-minimum phase behavior in the open-loop system, and may therefore represent performance limitations specific to the use of decentralized control. Necessary and sufficient conditions for zero crossings as a result of loop closure, considering diagonal as well as block-diagonal controllers, have been derived. The conditions show that the crossings between the LHP and RHP may be in either direction, and may be real or complex. For diagonal controllers, real zero crossings can only occur when the pairing, for the subsystems in question, corresponds to a negative steady-state RGA element. It is stressed that this does not necessarily imply that the pairing corresponds to a negative RGA for the overall system. Also, for systems in which some subsystems contain a real RHP zero, it may be advantageous to pair on negative steady-state RGA elements since loop closure then may move the zero into the LHP.
1
ð47Þ If A1 and A2 are matrices of compatible dimensions such that both matrices A1A2 and A2A1 are square, then (e.g. Skogestad and Postlethwaite [2]) detðI þ A1 A2 Þ ¼ detðI þ A2 A1 Þ
ð48Þ
Using this result, we have detG^ 22 ¼ detG22 det I ðI þ C1 G11 Þ1 C1 G12 G1 22 G21 ¼ detG22 detðI þ C1 G11 Þ1 det I þ C1 G11 C1 G12 G1 22 G21 ¼ detG22 detðI þ C1 G11 Þ1 det I þ C1 G11 G12 G1 22 G21
ð49Þ
which is the desired form of the determinant of Gˆ22, i.e. (38). With perfect control of y1, i.e. when S ¼ ðI þ C1 G11 Þ1 ¼ 0
ð50Þ
or T ¼ I S ¼ ðI þ C1 G11 Þ1 C1 G11 ¼ I
ð51Þ
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(46) becomes
Inserting (56) into (58) yields
G^ 22 ¼ G22 G21 ðI þ C1 G11 Þ1 C1 G12
1 þ c1 ðdetG=detG22 Þ detG^ 22 ¼ detG22 1 þ c1 g11 detG22 þ c1 detG ¼ 1 þ c1 g11
¼ G22 G21 ðI þ C1 G11 Þ1 C1 G11 G1 11 G12 ¼ G22
ð52Þ
G21 G1 11 G12
ð59Þ
This implies that the determinant of Gˆ22(s) becomes detG^ 22 ðsÞ ¼ det G22 G21 G1 11 G12 Apply Schur’s formula G12 G detG ¼ det 11 G21 G22
ð53Þ
ð54Þ
¼ detG11 det G22 G21 G1 11 G12
ð55Þ
¼ detG22 det G11 G12 G1 22 G21
ð56Þ
Thus, (53) is equivalent to detG detG^ 22 ¼ detG11
ð57Þ
which is (40). When the feedback is monovariable, i.e. u1 and y1 are both scalars, (49) becomes detG^ 22 ¼ detG22 detð1 þ c1 g11 Þ1 det 1 þ c1 g11 G12 G1 22 G21
ð58Þ
This gives the result for the case under monovariable feedback, i.e. (6).
References [1] E.H. Bristol, On a new measure of interactions for multivariable process control, IEEE Trans. AC, AC- 11 (1966) 133–134. [2] S. Skogestad, I. Postlethwaite, Multivariable Feedback Control, Wiley, Chichester, 1996. [3] M. Hovd, S. Skogestad, Simple frequency-dependent tools for control-system analysis, structure selection and design, Automatica 28 (5) (1992) 989–996. [4] F.G. Shinskey, Process Control Systems, 3rd Edition, McGrawHill, New York, 1988. [5] P. Grosdidier, M. Morari, R. Holt, Closed-loop properties from steady-state gain information, Ind. Eng. Chem. Fund. 24 (1985) 221–235. [6] H.H. Rosenbrock, State-Space and Multivariable Theory, Nelson, London, 1970. [7] E.W. Jacobsen, H. Cui, Zero crossings due to loop closure in decentralized control systems. 1998 AIChE Annual Meeting, paper 233e, 1998. [8] K.H. Johansson, A. Rantzer, Decentralized control of sequentially minimum phase systems, IEEE Trans. AC 44 (10) (1999).
Performance limitations in decentralized control Hong Cui, Elling W. Jacobsen* Signals, Sensors and Systems Process Control, Royal Institute of Technology-KTH, S-100 44 Stockholm, Sweden
Abstract In decentralized control of multivariable systems, the system is decomposed into a number of subsystems and individual controllers are designed for each subsystem. Advantages of such decomposition include reduced modelling requirements and ease of implementation. However, a potential disadvantage is a reduction in achievable control performance due to restricted controller structure. In this paper we consider performance limitations from non-minimum phase transmission zeros in decentralized control. In particular, we derive conditions on when closing the loop around one subsystem moves transmission zeros of other subsystems across the imaginary axis. Such zero crossings may occur regardless of the existence of non-minimum phase behavior in the openloop system, and may, therefore, represent performance limitations specific to the use of decentralized controllers. # 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction We consider in this paper decentralized control of a general square multivariable nn system G(s), i.e. having n inputs and n outputs. Decentralized control implies that the overall system is decomposed into a number of (interacting) subsystems for which individual controllers are designed, i.e. the overall controller may be written on a block-diagonal form. Such a decomposition of the control problem is often preferred due to increased robustness, both with respect to model uncertainty (reduced modelling cost) and sensor/actuator failures, and ease of implementation. However, the potential cost is reduced closed-loop performance due to the presence of interactions among the subsystems. There exist a number of tools for analyzing the effect of interactions in decentralized control systems, of which the most widespread is the relative gain array (RGA) [1]. However, most of the theoretical results related to these tools concern (conditional) stability only. For instance, the common rule that one should not close loops around scalar subsystems corresponding to negative steady-state RGA elements, e.g. [2], is related to integrity (conditional stability) only. Hovd and Skogestad [3] have derived conditions, in terms of required minimum loop gains for the individual subsystems, in order for a decentralized controller to satisfy given specifications on the overall closed-loop * Corresponding author. E-mail address: [email protected] (E.W. Jacobsen).
performance. However, whether the required loop gains are achievable, under the constraint of closed-loop stability, depends on the presence of control limitations in the system under consideration. For the case of full multivariable controllers, a number of results on performance limitations are available. For instance, if the system G(s) has a real Right Half Plane (RHP) transmission zero at z, the bandwidth of the closed-loop system is limited by !B < z. See e.g. Skogestad and Postlethwaite [2]. Since a decentralized, or block-diagonal, controller is just a special case of a full controller, it is clear that the same limitations apply in decentralized control. However, it seems reasonable that a limited control structure may impose additional performance limitations. In this paper we address performance limitations from RHP zeros in decentralized control. In particular, we address the problem: under which conditions will closing the loop around one subsystem introduce RHP transmission zeros in other subsystems? If such zeros exist, and they are more severe than the RHP zeros of the overall system, they will represent performance limitations specific to the use of decentralized control. The problem of zero crossings under loop closure has been addressed by other authors previously. Bristol [1], in his original work on the relative gain array (RGA), proposes that a scalar loop corresponding to a negative steady-state RGA-element will have a RHP zero or pole when all the other outputs are controlled with integral action. Shinskey [4] argues, using heuristic arguments, that closing the loop around a negative steady-state
0959-1524/02/$ - see front matter # 2002 Elsevier Science Ltd. All rights reserved. PII: S0959-1524(01)00015-4
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RGA-element in a 22 system will introduce an inverse response, corresponding to a RHP zero, from the remaining input to the remaining output. Grosdidier et al. [5] show that the proposal put forward by Bristol holds under certain assumptions on the open-loop system GðsÞ Finally, Rosenbrock [6] has shown that, if all outputs but one of a multivariable system are controlled perfectly, then the remaining open-loop transfer-function will have the transmission zeros of the multivariable system as its zeros. In this work we derive more general conditions for RHP zeros induced by closing the loops around subsystems. In particular, we consider multivariable subsystems, real as well as complex zero crossings, finite bandwidth, general type controllers and relax many of the assumptions employed previously. Furthermore, we show that zero crossings may be from the LHP to the RHP as well as vice versa. Parts of this work have previously been presented in Jacobsen and Cui [7].
2. Effect of loop closure on subsystems external to the loop We consider a general square system y ¼ GðsÞu and want to analyze the effect of closing the loop between a subset of y and u on the transfer-matrix between the remaining inputs and outputs. We will first limit ourselves to consider single loop controllers, and then extend the results to multivariable controllers towards the end of the paper. Thus, we first decompose GðsÞ according to y1 ¼ g11 ðsÞu1 þ G12 ðsÞu2
ð1Þ
y2 ¼ G21 ðsÞu1 þ G22 ðsÞu2
ð2Þ
where y1 and u1 are scalars and y2 and u2 are vectors of dimension n1, and consider the effect of closing the loop between y1 and u1 on the transfer-matrix between u2 and y2. Closing the loop between y1 and u1 using the feedback law u1 ¼ c1 ðsÞy1
ð5Þ
Here s=zi and s=pj are the zeros and poles, respectively, of G(s). Note that, in general, there may be poles and zeros at the same location without canceling. From (4) we derive the following expression for the determinant of G^ 22 ðsÞ (see Appendix) detG22 ðsÞ þ c1 ðsÞdetGðsÞ detG^ 22 ðsÞ ¼ 1 þ c1 ðsÞg11 ðsÞ
ð6Þ
It is clear from (6) that the zeros as well as the poles of the subsystem G^ 22 ðsÞ are moved by the feedback (3). Our aim is, based on (6), to derive conditions for when there is a different number of RHP zeros in the two systems G22 (s) and G^ 22 ðsÞ, or equivalently, for when one or more zeros of the subsystem involving y2 and u2 are moved between the complex LHP and RHP as the loop (3) is closed. To do this we will first employ the simplifying assumption that the loop (3) is closed perfectly, i.e. with infinite bandwidth, and then consider the more general case where the loop is closed with finite bandwidth.
3. Zero crossings under perfect control We here assume that control of output y1 is perfect over all frequencies, i.e. infinite controller gain. Perfect control of y1 corresponds to zero sensitivity, i.e. 1 ¼0 1 þ c1 ðsÞg11 ðsÞ
ð7Þ
which is equivalent to c1 ð s Þ 1 ¼ 1 þ c1 ðsÞg11 ðsÞ g11 ðsÞ
ð8Þ
With (7) and (8), Eq. (6) becomes ð3Þ
yields for the transfer-matrix G^ 22 ðsÞ, between inputs u2 and outputs y2, c1 ð s Þ G^ 22 ðsÞ ¼ G22 ðsÞ G21 ðsÞG12 ðsÞ 1 þ c1 ðsÞg11 ðsÞ
Q i¼1;nz ðs zi Þ detGðsÞ ¼ K Q j¼1;np s pj
ð4Þ
Thus, the feedback (3) affects the transfer-matrix from u2 to y2, provided G21(s)G12(s) is not identically zero. We are mainly interested in the transmission zeros of G^ 22 ðsÞ. As defined here, the transmission zeros of a general square transfer-matrix G(s) can be expressed in terms of the determinant of the matrix
detGðsÞ detG^ 22 ðsÞ ¼ g11 ðsÞ
ð9Þ
From (9) it is clear that the only RHP zeros of G^ 22 ðsÞ will be those of G(s) which are not zeros of g11(s).1 Thus, if the multivariable system is minimum phase there will be no limitations in the remaining subsystem G^ 22 ðsÞ after closing the first loop perfectly. Note that this applies regardless of whether G22(s) has RHP zeros or not.
1
Note that any RHP poles in g11(s) will be cancelled by their equivalents in detG(s), and will hence not appear as RHP zeros in G^ 22 ðsÞ.
H. Cui, E.W. Jacobsen / Journal of Process Control 12 (2002) 485–494
Rosenbrock [6] shows that if all outputs but one of a multivariable system are perfectly controlled, then the remaining scalar subsystem will have the zeros of G(s) as its zeros. The result presented above shows that the subsystem remaining after closing any number of loops perfectly will have the zeros of G(s) as its zeros, and is thus a generalization of Rosenbrock’s result. Example 1: Consider the 33 system 0 1 1 s 1 1 @4 GðsÞ ¼ 1 1 A ðs þ 1Þ2 1 2 3
1þ
c1 ðsÞdetGðsÞ ¼ 0; detG22 ðsÞ
ReðsÞ > 0
487
ð13Þ
The following theorem provides necessary and sufficient conditions for when G^ 22 ðsÞ and G22(s) has a different number of transmission zeros in the RHP. Theorem 1. Assume G(s) and c1(s) stable and G21(s)G12(s)6 0. Then the image of
ð10Þ
which has a transmission zero at s=0.77. The transfermatrix G22(s) between (u2, u3) and (y2, y3) is minimum phase. Closing the loop around g11(s) using a proportional controller with gain k=1 yields 1 4s 1 3 ð11Þ G^ 22 ðsÞ ðs þ 1Þ2 ðs þ 2Þ 4 which has a zero at s=0.77, corresponding to the RHP zero of G(s). Also note that the transfer-function from u2 to y2 now has a RHP zero at s=0.25, implying that the original open-loop subsystem with y3 and u3 removed has this zero. Infinite bandwidth can of course never be achieved in practice, but the results derived above may serve as asymptotic results. An illustration of their usefulness as such will be given below. We consider next the case where the assumption of perfect control is relaxed.
4. Zero crossings under finite bandwidth When output y1 is controlled with a finite bandwidth, the determinant for the remaining subsystem G^ 22 ðsÞ is given by (6), i.e. detG22 ðsÞ þ c1 ðsÞdetGðsÞ detG^ 22 ðsÞ ¼ 1 þ c1 ðsÞg11 ðsÞ
ðsÞ ¼
c1 ðsÞdetGðsÞ detG22 ðsÞ
ð14Þ
as s follows clockwise around the Nyquist D-contour will encircle the point 1, in the complex plane, N=ZˆZ times, where Zˆ is the number of RHP zeros of Gˆ22(s) and Z is the number of RHP zeros of G22(s). Proof. The result follows directly from application of the argument principle to (13) using the fact that, when G(s) and c1(s) are stable, the RHP zeros and poles of 1+(s) are equivalent to the RHP zeros of G^ 22 ðsÞ and G22(s), respectively. If G21(s)G12(s)=0,8s, then from (4) G^ 22 ðsÞ G22(s). In this case the RHP zeros of 1+(s)= 1+c1(s)g11(s) will be exactly canceled by equivalent poles. & Theorem 1 states that if the image of (s) encircles the 1 point n times, for s=(j1, j1), then n zeros of the subsystem involving y2 and u2 cross between the complex LHP and RHP as the loop (3) is closed. If the encirclements are clockwise, as s goes clockwise around the Nyquist D-contour, the crossings are from the LHP to the RHP, while they are from the RHP to the LHP if the encirciements are anticlockwise. If (s) is strictly proper all zero crossings are through the imaginary axis, while they may be through 1 if (s) is semi- or improper. This is easily understood by considering (13) which can not have roots at s=1 if (s) is strictly proper. For systems in which the Nyquist curve (j!), for !>0, crosses the negative real axis once only, a simpler criterion can be employed.
In the following we assume stability of the open loop systems G(s) and c1(s). With this assumption, G^ 22 ðsÞ will have zeros in the RHP if and only if
Corollary 1. Assume that G(s) and c1(s) are stable, that (s) is strictly proper and that (j!) is negative real for one frequency ! ¼ !18050 only. Then G22(s) and Gˆ22(s) will have a different number of RHP zeros if
detG22 ðsÞ þ c1 ðsÞdetGðsÞ ¼ 0;
ðj!180 Þ > 1;
ReðsÞ > 0
ð12Þ
Thus, to check for RHP zeros in G^ 22 ðsÞ one can for instance apply the argument principle to (12). Since we are here mainly interested in the case when there is a difference in the number of RHP zeros between G^ 22 ðsÞ and G22(s), we divide (12) by detG22(s) to obtain
ffðj!180 Þ ¼ 180
ð15Þ
Proof. The condition follows directly from Theorem 1 since it ensures at least one encirclement of the point 1. The assumption that (s) is strictly proper ensures that (j!)! 0 as !!1. &
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Note that the zero crossing will be real if !180=0, while it will be complex conjugate if !180>0. Thus, the simplified condition for a real zero crossing can be written c1 ð0ÞdetGð0Þ < 1 detG22 ð0Þ
ð16Þ
The next result concerns the case in which the function (s) contains a pure integrator, which for instance is the case if the controller c1(s) contains integration. Corollary 2. Assume that G(s) and c1(s) are stable and that (s) is strictly proper with one pole at s=0, i.e. (s)=0 (s) 1s, where 0 (s) is analytic with no zeros at s=0. Then G22(s) and Gˆ22(s) will have a different number of RHP zeros provided 0 (0)< 0. Proof. The result follows from considering the image of (s) for the small semicircular detour of s=0 in the Nyquist D-contour, i.e. s ¼ re j ; jrj ! 0; 2 ½=2; =2. Then the image of (s) makes an infinite semi-circle around the LHP if it contains an integrator and 0 (0) < 0. The assumption of strictly proper (s) ensures that the image approaches the origin for !!1, and hence an encirclement of the 1 point. & Note that Corollary 2 provides a sufficient condition only. The function (s) is closely related to the relative gain array (RGA). In particular, the 1,1-element of the RGA of G(s) is given by l11 ¼
g11 detG22 detG
ð17Þ
g11 ð0Þc1 ð0Þ < 1 l11 ð0Þ
ð20Þ
Grosdidier et al. [5] have previously shown that, under certain assumptions, a SISO loop corresponding to a negative steady-state RGA element will have an odd number of RHP zeros, or a RHP pole, when all other loops are closed with integral action. The results presented above show that closing the loop around an element corresponding to a negative steady-state RGA element, using negative feedback with integral action, is a sufficient condition for moving a zero of the remaining subsystem across the imaginary axis, and is thus in some sense dual to the previous result. However, we stress that it is a sufficient condition only, and by no means necessary for zero crossings. Below we illustrate the results derived above on a number of examples. Example 2: Consider 0 4 1 @ GðsÞ ¼ 0:9 10s þ 1 1
the minimum phase 33 system 1 4 2 1 1A ð21Þ 2 3
in which all subsystems are also minimum phase. Apply negative proportional feedback around g11(s), i.e. u1 ¼ ky1
ð22Þ
The corresponding RGA element is l11(0)=3.33. Thus, condition (20) for a real zero crossing becomes 4k < 1 ) k > 0:833 3:33
ð23Þ
and thus we get that c1 ðsÞg11 ðsÞ ðsÞ ¼ l11 ðsÞ
ð18Þ
In terms of the RGA, the condition (13) for a different number of RHP zeros in Gˆ22(s) and G22 ðsÞ can then be written c1 ðsÞg11 ðsÞ ¼ 0; 1þ l11 ðsÞ
ReðsÞ > 0
ð19Þ
It is interesting to note that condition (19) for RHP zeros in G^ 22 ðsÞ under finite bandwidth control of y1 corresponds to the condition for RHP poles, i.e. instability, in loop 1 under perfect control of y2. This follows from the fact that the loop gain for loop 1 will be g11 c1 =l11 when output y2 is controlled with infinite bandwidth. In terms of the RGA, the sufficient condition (16) for a real zero crossing becomes
Since G22(s) is minimum phase, a zero crossing must be from the LHP to the RHP. Indeed, with k=1, corresponding to a bandwidth of approx. 0.5 for y1, we get 1 2s þ 0:28 2s þ 0:64 G^ 22 ðsÞ ¼ 6s þ 2:6 20s2 þ 12s þ 1 4s þ 1:2 which has a zero at s=0.02, which implies that the bandwidth for this subsystem is limited by !B < 0.02. Note that although one chooses to only pair on positive RGA elements for an nn system, the pairing may correspond to pairing on a negative RGA for some subsystems. We illustrate this on the example below. Example 3. Consider the 33 system 0 1 0:3 1:8 0:9 1 @ 1:7 GðsÞ ¼ 0:32 1:4 A 100s þ 1 0:50 0:50 0:63
ð24Þ
H. Cui, E.W. Jacobsen / Journal of Process Control 12 (2002) 485–494
which is minimum phase. The RGA is, at all frequencies, 0 1 0:76 3:40 3:16 ¼ @ 5:92 0:42 5:35 A ð25Þ 5:68 2:83 9:51 The common pairing rules based on the RGA, i.e. prefer pairings for which the RGA-elements are closest to 1 and avoid pairing on negative RGA-elements, suggest pairing on the diagonal. However, the RGA for the 22 subsystems obtained by removing (y3, u3) and (y2, u2), respectively, both have negative RGA-elements on the diagonal. Thus, according to the results above, closing the loop around g11(s), with a controller containing integral action, will introduce RHP zeros in the transfer-functions from u2 to y2 and from u3 to y3, respectively. It is easy to show that these zeros cannot be moved back to the LHP by closing further loops. As stated above, a common rule in decentralized control is to avoid pairings on negative steady-state RGAelements. However, this is mainly related to integrity, i.e. the system should remain stable as loops are taken in and out of service. See e.g. Skogestad and Postlethwaite [2]. Furthermore, the rule is in general only necessary, and not sufficient. On the other hand, if some elements contain RHP zeros, then pairing on negative RGA elements may be advantageous from a performance point of view, as illustrated by the next example.
Example 4. Consider the minimum phase system 1 s þ 0:1 s þ 0:1 GðsÞ ¼ ð5s þ 1Þ2 s 0:05 s 0:025
489
ð26Þ
Both transfer-functions for y2 contain severe RHP zeros, and the worst appears to be from input u2. However, from the results above it is clear that these zeros will be moved as soon as the loop for y1 is closed. Furthermore, since the overall system is minimum phase, closing the loop for y1 with infinite bandwidth will remove all zeros in the RHP for y2, independent of which pairing that is used. Thus, perfect control is, in theory, possible with decentralized control, using either pairing. However, with finite bandwidth control of y1, the results derived above tells us that the RHP zero for y2 can only be removed by pairing on a negative steady-state RGA. The steady-state RGA is 1 2 ð27Þ ð 0Þ ¼ 2 1 Consider choosing either input u1 or input u2 to control y1 with the PI-controller ui ¼ k
5s þ 1 ðr1 y1 Þ; 5s
i ¼ 1; 2
ð28Þ
Fig. 1 shows the zero with maximum real part for output y2 as a function of the controller gain k when u1 and
Fig. 1. Example 4. Maximum real part of zeros of transfer-functions for output y2 when y1 is controlled with the controller (28) and input u1 and u2, respectively.
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u2 are used for control of y1, respectively. As seen from the figure, with y1 controlled using u1 (negative RGA), the RHP zero for y2 moves towards the LHP for increasing k and crosses the imaginary axis for k=11.5, corresponding to a bandwidth !B=0.3 for output y1. However, with y1 controlled using u2 (positive RGA), the RHP zero for y2 moves further into the RHP as the gain k is increased. Since, for a real RHP zero at s=z, the bandwidth is limited by !B < z, we thus have a bandwidth limitation for output y2 if we choose to pair on positive RGA. With k=25, corresponding to a bandwidth of 1 for y1, we get a zero at s=0.16 for y2 and thus a maximum bandwidth of 0.16 for this loop. If we want to achieve a bandwidth of 1 for output y2, then we need to increase k above 970, corresponding to a bandwidth exceeding 45 for y1, if y1 is controlled using u2. Note that for the above example, integrity can not be satisfied for either pairing if the bandwidth for y2 is to exceed 0.05. Thus, if a bandwidth higher than 0.05 is required then one should in this case pair on negative RGA elements since this most likely will provide the best performance. One might argue that if integrity can not be incorporated, then one might as well employ full multivariable control. However, there are a number of factors, apart from integrity, such as robustness (reduced modelling requirements), which motivates the use of decentralized control. In all examples considered so far, the zero crossings have been due to pairing on negative steady-state RGA
elements for the overall system, or for some subsystem. However, zero crossings may occur also when pairing on positive RGA-elements. In this case the zero crossings will be complex conjugate, and the simplified criterion (15) can be written
log g11 c1 logjl11 j > 0; ffg11 c1 ffl11 ¼ ð29Þ From this expression we see that complex zero crossings are most likely to occur for small values of jl11 j and positive phase ffl11 . Example 5. Consider the minimum phase system ! 0:1 1 0:05 GðsÞ ¼ 30s þ 1 ð10s þ 1Þ3 0:1 0:1
ð30Þ
The plots of jl11 j and ffl11 are shown as functions of frequency in Fig. 2. Using the usual rules for selecting pairings, i.e. to pair on the RGA for which lij is closest to 1 around the expected bandwidth, we see that we should pair on the diagonal provided the desired bandwidth !B exceeds 0.04. We note that jl11 j < 1 and ffl11 >0 for all frequencies, and according to the above this increases the probability of a complex zero crossing. Indeed, with the controller c1 ð s Þ ¼ k
10s þ 1 10s
Fig. 2. Amplitude and phase of 1,1-element of RGA in Example 5.
ð31Þ
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0
and k=1 we obtain the Bode plot of ðsÞ ¼
g11 ðsÞc1 ðsÞ l11 ðsÞ
ð32Þ
shown in Fig. 3. Employing criterion (15), i.e.
ð!180 Þ > 1; ffðj!180 Þ ¼ 180 we find from Fig. 3 that g^ 22 ðsÞ will have RHP zeros if k>8.33, corresponding to a bandwidth !B>0.058 in the first loop. For example, with k=10 we find that the transfer-function from u2 to y2 contains zeros at s= 0.0013 j 0.061, imposing a severe performance limitation for this loop. Note that one situation in which zero crossings in G^ 22 ðsÞ is expected as the bandwidth for y1 is increased, is when the overall system G(s) is non-minimum phase with more RHP zeros than G22(s). In this case, the asymptotic result derived using infinite bandwidth for y1, guarantees that zeros of G^ 22 ðsÞ will cross from the LHP to the RHP as the bandwidth of y1 is increased beyond a critical value. If the pairing for y1 corresponds to a positive steady-state RGA-element, the crossing must be in the form of complex conjugate zeros (assuming the zeros can not cross through 1, i.e. (s) strictly proper). Example 6. Consider the non-minimum phase system (taken from Example 2.7 in Rosenbrock [6]).
1 1 2 B 1 s þ 3C GðsÞ ¼ @ s þ 1 1 A sþ1 sþ1
491
ð33Þ
G(s) has a transmission zero at s=1. Consider applying the feedback u1=ky1, for which l11 (0)=3. According to the results for perfect control derived above, the transfer-function g^ 22 ðsÞ will have one, and only one, RHP zero at s=1 as k!1. Thus, there will be a zero crossing the imaginary axis for some value of k>0. Since the loop is closed around a positive RGA, the zero crossing of g^ 22 ðsÞ must be complex. Thus, as k is increased, a complex conjugate pair of zeros will cross the imaginary axis, which then become real and one moves to s=1 and one toward s=1 as k!1. For instance, for k=10, we find a pair of complex zeros at s=3 j 2 in g^ 22 ðsÞ. The conditions derived above can be used to determine whether closing loops in a decentralized control system may impose inherent bandwidth limitations in some subsystems. The specific limitations imposed will depend on the location of the zeros in the RHP. In general, the location will depend on the system properties as well as the controllers employed and it is hence difficult to derive any general results. However, one specific situation in which the zero location can be easily predicted is when the dynamics in all elements of G(s) are equivalent, so that the RGA is independent of frequency, and the controller c1(s) is based on model inversion (direct synthesis) such that
Fig. 3. Amplitude and phase of (s) with diagonal pairing and k=1 in Example 5.
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c1 ð s Þ ¼
!c 1 s g11 ðsÞ
ð34Þ
From (19) we then find that G^ 22 ðsÞ will get exactly one extra RHP zero at s¼
!c l11
ð35Þ
provided l11 < 0. Note that the RHP zero in this case is directly proportional to the bandwidth !c and inversely proportional to the magnitude of the RGA-element. Thus, the performance limitation will in this case decrease with increasing bandwidth !c, while becoming more severe with a larger magnitude of the RGA.
The next theorem provides a necessary and sufficient condition for when closing the loop (36) causes zeros in the sub-system involving y2 and u2 to cross between the LHP and RHP. Theorem 2. Assume G(s) and C1(s) stable and G21(s) and G12(s) not identically 0. Define ð41Þ ðsÞ ¼ C1 ðsÞ G11 ðsÞ G12 ðsÞG1 22 ðsÞG21 ðsÞ Then the image of det(I+D(s)), as s follows clockwise around the Nyquist D-contour, will encircle the origin N= ZˆZ times, where Zˆ and Z are the number of RHP zeros of Gˆ22(s) and G22(s), respectively. Proof. Define
5. Block-diagonal controllers
fðsÞ ¼ detðI þ ðsÞÞ ¼
We have so far considered closing SISO loops only, and the derived results thus apply to diagonal controllers only. However, block-diagonal controllers, i.e. containing multivariable sub-controllers, are also commonly used. We here derive conditions on when closing a multivariable controller around a subsystem drives zeros of other subsystems between the LHP and RHP. Consider decomposing the overall nn system G(s) as in (1) and (2), but let now y1 and u1 be vectors of any dimension less than n. The corresponding square transfer-matrix is written G11(s). Closing the loop u1 ¼ C1 ðsÞy1
ð36Þ
detG^ 22 ðsÞ detðI þ C1 ðsÞG11 ðsÞÞ detG22 ðsÞ ð42Þ
and apply the argument principle to f(s). With G(s) and C1(s) stable, I+C1(s)C11(s) is stable. Furthermore, any RHP zeros of I+C1(s)G11(s) (RHP poles of the closed loop) are canceled in f(s) by equivalent RHP poles in Gˆ22(s) provided neither G12(s) nor G21(s) are identically zero. Thus, the RHP zeros and poles of I+(s) are equivalent to the RHP zeros of G^ 22 ðsÞ and G22(s), respectively. & Similar to the SISO case, we propose a simple sufficient condition for checking the existence of zeros crossing the imaginary axis.
yields for the remaining subsystem G^ 22 ðsÞ ¼ G22 ðsÞ G21 ðsÞ ðI þ C1 ðsÞG11 ðsÞÞ1 C1 ðsÞG12 ðsÞ
ð37Þ
Corollary 3. Assume G(s), C1(s) stable, G21(s) and G12(s) not identically 0 and D(s) strictly proper. Assume furthermore that det(I+D(j!)) is negative real for one frequency !18050 only. Then there is a difference of at least one RHP zero between Gˆ22(s)and G22(s).
and the determinant can be written (see Appendix) detG^ 22 ¼ detG22 detðI þ C1 G11 Þ1 det I þ C1 G11 G12 G1 22 G21
ð38Þ
With perfect control of y1, i.e. when SðsÞ ¼ ðI þ C1 ðsÞG11 ðsÞÞ1 ¼ 0
ð39Þ
(38) can be simplified to (see Appendix) detGðsÞ detG^ 22 ðsÞ ¼ detG11 ðsÞ
ð40Þ
which is a generalization of the SISO result in Eq. (9). A similar result can be found in Johansson and Rantzer [8].
Proof. The corollary follows from Theorem 2 since the assumptions ensure that the image of detðI þ ðsÞÞ encircles the origin at least once. & As for the SISO case, it is easily shown that the zero crossing is real if o180=0 and complex conjugate otherwise. Thus, for a real zero crossing, the simplified condition (Corollary 3) becomes that an odd number of eigenvalues of (0) should be real and less than 1. Finally, similar to Corollary 2 for SISO loop closure, it is easily shown that if a strictly proper (s) can be written as 0 (s)/s where 0 (0) has full rank, e.g. when C1(s) contains integration, then a sufficient condition for a different number of RHP zeros between G^ 22 ðsÞ and G22(s), is that an odd number of eigenvalues of 0 (0) are negative real.
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Example 7. Consider the minimum phase system 0 1 0:3 1:9 0 0:5 1 B 1:8 0:1 0:8 C B 0:1 C GðsÞ ¼ @ 10s þ 1 0:4 0:7 0:8 0:7 A 0:5 0:7 1:8 0:3
ð43Þ
for which all subsystems are also minimum phase. Consider controlling the two first outputs with the two first inputs using the IMC controller C1 ðsÞ ¼
!c 1 G ð sÞ s 11
ð44Þ
which provides a bandwidth !c for this subsystem. The corresponding function is ðsÞ ¼
!c s
0:70 0:56
1:36 1:40
ð45Þ
and the eigenvalues l(0 (0)), i.e. after removing the integrator, become l=0.23!c, 0.93!c. Thus, since one eigenvalue is negative, we should expect a RHP zero in the remaining subsystem G^ 22 ðsÞ involving the two last inputs and outputs. Indeed, with !c=1, the system has a zero s=0.042, imposing a severe performance limitation for this subsystem. The limitation can be relaxed by increasing the bandwidth !c for the other subsystem.
Finally, complex zero crossings are found to be most likely when closing a loop around an element corresponding to a RGA with small magnitude and large positive phase. Thus, the phase information of the RGA proves useful for predicting zero crossings.
Acknowledgements This work is a part of the ‘‘Ecocyclic Pulp Mill’’ research programme financed by MISTRA, the Swedish Foundation for Strategic Environmental Research.
Appendix. Derivation of Eqs. (6), (38) and (40) Consider a square multivariable system G, and apply negative feedback control around a subsystem G11 with inputs u1 and outputs y1, u1=C1y1. Then the remaining subsystem G^ 22 ðsÞ, with inputs u2 and outputs y2, is given by G^ 22 ¼ G22 G21 ðI þ C1 G11 Þ1 C1 G12
ð46Þ
Taking the determinant on both sides yields detG^ 22 ¼ det G22 G21 ðI þ C1 G11 Þ1 C1 G12 0
B C G ðI þ C1 G11 Þ1 C1 G12 A ¼ detG22 det@I G1 |fflfflffl22 ffl{zfflfflffl21 ffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} A1
A2
6. Conclusions We have in this paper shown that closing loops in a decentralized control system may move zeros of subsystems external to the loop between the complex LHP and RHP. Such zero crossings may occur regardless of the existence of non-minimum phase behavior in the open-loop system, and may therefore represent performance limitations specific to the use of decentralized control. Necessary and sufficient conditions for zero crossings as a result of loop closure, considering diagonal as well as block-diagonal controllers, have been derived. The conditions show that the crossings between the LHP and RHP may be in either direction, and may be real or complex. For diagonal controllers, real zero crossings can only occur when the pairing, for the subsystems in question, corresponds to a negative steady-state RGA element. It is stressed that this does not necessarily imply that the pairing corresponds to a negative RGA for the overall system. Also, for systems in which some subsystems contain a real RHP zero, it may be advantageous to pair on negative steady-state RGA elements since loop closure then may move the zero into the LHP.
1
ð47Þ If A1 and A2 are matrices of compatible dimensions such that both matrices A1A2 and A2A1 are square, then (e.g. Skogestad and Postlethwaite [2]) detðI þ A1 A2 Þ ¼ detðI þ A2 A1 Þ
ð48Þ
Using this result, we have detG^ 22 ¼ detG22 det I ðI þ C1 G11 Þ1 C1 G12 G1 22 G21 ¼ detG22 detðI þ C1 G11 Þ1 det I þ C1 G11 C1 G12 G1 22 G21 ¼ detG22 detðI þ C1 G11 Þ1 det I þ C1 G11 G12 G1 22 G21
ð49Þ
which is the desired form of the determinant of Gˆ22, i.e. (38). With perfect control of y1, i.e. when S ¼ ðI þ C1 G11 Þ1 ¼ 0
ð50Þ
or T ¼ I S ¼ ðI þ C1 G11 Þ1 C1 G11 ¼ I
ð51Þ
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(46) becomes
Inserting (56) into (58) yields
G^ 22 ¼ G22 G21 ðI þ C1 G11 Þ1 C1 G12
1 þ c1 ðdetG=detG22 Þ detG^ 22 ¼ detG22 1 þ c1 g11 detG22 þ c1 detG ¼ 1 þ c1 g11
¼ G22 G21 ðI þ C1 G11 Þ1 C1 G11 G1 11 G12 ¼ G22
ð52Þ
G21 G1 11 G12
ð59Þ
This implies that the determinant of Gˆ22(s) becomes detG^ 22 ðsÞ ¼ det G22 G21 G1 11 G12 Apply Schur’s formula G12 G detG ¼ det 11 G21 G22
ð53Þ
ð54Þ
¼ detG11 det G22 G21 G1 11 G12
ð55Þ
¼ detG22 det G11 G12 G1 22 G21
ð56Þ
Thus, (53) is equivalent to detG detG^ 22 ¼ detG11
ð57Þ
which is (40). When the feedback is monovariable, i.e. u1 and y1 are both scalars, (49) becomes detG^ 22 ¼ detG22 detð1 þ c1 g11 Þ1 det 1 þ c1 g11 G12 G1 22 G21
ð58Þ
This gives the result for the case under monovariable feedback, i.e. (6).
References [1] E.H. Bristol, On a new measure of interactions for multivariable process control, IEEE Trans. AC, AC- 11 (1966) 133–134. [2] S. Skogestad, I. Postlethwaite, Multivariable Feedback Control, Wiley, Chichester, 1996. [3] M. Hovd, S. Skogestad, Simple frequency-dependent tools for control-system analysis, structure selection and design, Automatica 28 (5) (1992) 989–996. [4] F.G. Shinskey, Process Control Systems, 3rd Edition, McGrawHill, New York, 1988. [5] P. Grosdidier, M. Morari, R. Holt, Closed-loop properties from steady-state gain information, Ind. Eng. Chem. Fund. 24 (1985) 221–235. [6] H.H. Rosenbrock, State-Space and Multivariable Theory, Nelson, London, 1970. [7] E.W. Jacobsen, H. Cui, Zero crossings due to loop closure in decentralized control systems. 1998 AIChE Annual Meeting, paper 233e, 1998. [8] K.H. Johansson, A. Rantzer, Decentralized control of sequentially minimum phase systems, IEEE Trans. AC 44 (10) (1999).