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Ind. Eng. Chem. Res. 2003, 42, 5152-5156
PROCESS DESIGN AND CONTROL Control Structure Selection Based on Block-Decentralized Integral Controllability Wen Z. Zhang,† Jie Bao,*,† and Peter L. Lee‡ School of Chemical Engineering & Industrial Chemistry, The University of New South Wales, Sydney, NSW 2052, Australia, and Division of Engineering & Science, Curtin University of Technology, G.P.O. Box U1987, Perth, WA 6845, Australia
Block-decentralized controllers are of great importance in control applications when processes cannot be well controlled by multiloop controllers. In this paper, a new concept, blockdecentralized integral controllability is defined and sufficient conditions based on the passivity theorem and the small gain theorem are derived. The numerical methods to examine these conditions are also presented. 1. Introduction Despite the advance in multivariable control theory, fully decentralized control (multiloop) still remains the most widely used control technique in the chemical process industry because of its design and hardware simplicity.1 A major task involved in decentralized control system design is to select an appropriate control structure in which different inputs (manipulated variables) are assigned to different outputs (controlled variables). It is desirable to choose input/output pairing schemes such that the resulting system is decentralized integral controllable (DIC). That is, it can be stabilized by a multiloop controller with integral action, of which any one or more loops can be arbitrarily detuned without causing closed-loop instability.2 Conditions for DIC and DIC-based pairing methods have been an active research area, with a number of necessary or sufficient conditions developed.3-7 For highly coupled multivariable processes, multiloop control structures may lead to degradation of the control performance or even instability.8-10 When process interactions are severe, processes may not be DIC. Moreover, some DIC pairings may not be feasible because of physical limitations in interconnecting parts of the system. Under the above circumstances, block-decentralized controllers (with block diagonal controller structure) are often considered because they may provide effective control without incurring the complexity of multivariable designs.11 Similar to the multiloop design, it is preferable that the block diagonal controller has integral action to achieve offset free control (block integral controllability), and any one or more controller blocks can be arbitrarily detuned without endangering closed-loop stability (block detunability). In this paper, processes that can be stabilized by controllers with the above desirable features are defined as block-decentral* To whom correspondence should be addressed. Tel.: +61 (2) 9385-6755. Fax: +61 (2) 9385-5966. E-mail: j.bao@ unsw.edu.au. † The University of New South Wales. ‡ Curtin University of Technology.
Figure 1. Feedback system under block-decentralized control.
ized integral controllable (BDIC). It is therefore useful to study how to assign different subsets of inputs to different subsets of outputs such that the paired process is BDIC. There are very few pairing methods for processes under block-decentralized control available in the literature.12-14 Block relative gain (BRG), although not a good interaction measure, is often used to find pairing schemes that lead to less interaction in block diagonal controller structure selection.12 However, the BRGbased pairing rule only provides an engineering rule of thumb and cannot guarantee closed-loop stability. Chiu and Arkun introduced the concept of decentralized closed-loop integrity (DCLI) for both fully and blockdecentralized control, which is relevant to BDIC. A number of necessary conditions for DCLI were also developed.13 The contributions in this paper are twofold. First, a new concept, BDIC is defined, which represents certain preferable properties (block integral controllability and block detunability) of a process under block-decentralized control. Second, two sufficient conditions for BDIC, based on the passivity theorem and the small gain theorem, are presented. These conditions are easy to use, and their applications in block-decentralized control structure selection are demonstrated by examples. 2. BDIC 2.1. Definition. Consider a feedback system under block-decentralized control as shown in Figure 1. The n × n process transfer function matrix G(s) is parti-
10.1021/ie020924q CCC: $25.00 © 2003 American Chemical Society Published on Web 09/06/2003
Ind. Eng. Chem. Res., Vol. 42, No. 21, 2003 5153
into k × k block subsystems, G(s) possesses DCLI only if
tioned in the following block form:
G(s) ) [Gij(s)]i,j)1,...,k
(1)
where each block Gij(s) is an ni × nj transfer function submatrix. Define
Gb(s) ) diag[Gii(s)]i)1,...,k
(2)
Under block-decentralized control, the diagonal blocks Gii(s) are controlled by multivariable subcontrollers, which form the overall controller C(s) with the following block diagonal structure:
C(s) ) diag[Ci(s)]i)1,...,k
(3)
where Ci(s) is an ni × ni multivariable controller for the ith block Gii(s). Definition 1. BDIC. For a given n × n stable process G(s) in the block partition form of eq 1, if there exists a corresponding block diagonal controller in the form of eq 3 with integral action, i.e.,
1 C(s) ) diag[Ci(s)]i)1,...,k ) diag[Ki(s)]i)1,...,k s
(4)
such that the feedback system is stable and such that each individual controller block Ci(s) (i ) 1, ..., k) may be detuned independently by a factor ξi (0 e ξi e 1, ∀ i ) 1, ..., k) without introducing instability, this process is said to be BDIC with respect to the prespecified controller structure. Note that each controller block Ci(s) is a multivariable controller. The concept of BDIC is illustrated in Figure 1. The BDIC represents some preferable features of a process with respect to the prespecified controller structure. First, integral action can be used in each subcontrol system to achieve offset free control. Second, each controller block can be detuned or switched off independently without endangering the closed-loop stability. A concept relevant to BDIC is DCLI: Definition 2. DCLI.13 An n × n stable process G(s) is said to possess DCLI if it can be stabilized by a stable block diagonal controller C(s) which contains integral action and if it remains stable after failure occurs in one or more control blocks. DCLI only addresses the situation of control subsystem failures; i.e., the process can be stabilized by a controller C ˆ (s) ) ΞC(s), where
Ξ ) diag{ξ1, ..., ξi, ..., ξk}, ξi ∈ {0, 1}, ∀ i ) 1, ..., k (5) which is a special case of BDIC. For BDIC processes, the closed-loop stability remains when controller blocks are arbitrarily detuned. Therefore, BDIC is a more appealing property than DCLI because BDIC processes are easier to tune online by field control engineers. 3. Stability Conditions for BDIC 3.1. Existing Decentralized Stability Conditions. There are a few existing stability conditions for decentralized control systems. A necessary condition for DCLI is given by the following theorems: Theorem 1. Necessary Condition for DCLI.13 Given an n × n stable process G(s), which can be partitioned
det[Λi(G(0))] > 0, ∀ i ) 1, ..., k
(6)
where Λi(G(0)) is the BRG of the ith block Gii(s).12 Theorem 2. Necessary Condition for DCLI.13 Given an n × n stable process G(s), which can be partitioned into k × k block subsystems, G(s) possesses DCLI only if its Niederinski index (NI)15 is positive, i.e.
NI[G(0)] )
det[G(0)] >0 det[Gb(0)]
(7)
Obviously, the above theorems can be directly used as necessary conditions for BDIC. It is observed that the combination of theorems 1 and 2 can provide a more useful method to screen out unworkable pairings. The concept of BDIC is a direct extension of DIC to block-decentralized control. Some sufficient conditions for DIC are given below: Theorem 3. Passivity-Based Sufficient Condition for DIC.4 A stable linear multiple input-multiple output (MIMO) process G(s) with nonsingular steady-state gain matrix G(0) ∈ Rn×n is DIC if a real diagonal matrix D ) diag{d1, ..., dj, ..., dn} (dj * 0, ∀ j ) 1, ..., n) can be found such that
G(0) D + DGT(0) g 0
(8)
Theorem 4. Small Gain-Based Sufficient Condition for DIC.16 A stable process G(s) is DIC if it is generalized diagonally dominant at steady state, i.e.
µ(Ed(0)) < 1
(9)
where Ed(s) ) [G(s) - Gd(s)]Gd-1(s), Gd(s) consists of the diagonal transfer functions of G(s), and µ(Ed(0)) is the structured singular value of Ed(0). 3.2. Sufficient Condition for BDIC on the Basis of the Passivity Theorem. Because the passivity theorem can deal with systems with integral action explicitly, it is a useful tool to study closed-loop stability in an open-loop fashion. Because the passivity theorem is applicable to both single input-single output and MIMO processes, the passivity-based DIC condition given in theorem 3 can be extended to BDIC as follows. Theorem 5. An n × n stable linear MIMO process G(s) with nonsingular steady-state matrix G(0) ∈ Rn×n is BDIC with respect to the prespecified controller structure in eq 3 if a nonsingular real block diagonal matrix
W ) diag{W1, ..., Wi, ..., Wk} ∈ Rn×n
(10)
where Wi ∈ Rni×ni can be found such that
G(0) W + WTGT(0) g 0
(11)
Proof. For any nonsingular block diagonal matrix W given above, it can always be factorized into two nonsingular, square matrices M and N with the same block diagonal structure of W, i.e.
W ) MN-1
(12)
where M ) diag{M1, ..., Mi, ..., Mk} ∈ Rn×n, N ) diag{N1, ..., Ni, ..., Nk} ∈ Rn×n, Mi, Ni ∈ Rni×ni. Inequality
5154 Ind. Eng. Chem. Res., Vol. 42, No. 21, 2003
(11) thus can be written as
G(0) MN-1 + N-TMTGT(0) g 0
(13)
which is equivalent to
NT[G(0) MN-1 + N-TMTGT(0)]N g 0
(14)
NTG(0) M + MTGT(0) N g 0
(15)
i.e.
As shown in Figure 2, define a virtual process G′(s) ) NTG(s) M. It is easy to see that G′(0) is nonsingular. Inequality (15) indicates that G′(0) satisfies the following inequality: Figure 2. Stability analysis using the passivity theorem.
G′(0) In + InG′(0) g 0
(16)
where In is an n × n identity matrix. From theorem 3, the virtual process G′(s) is DIC. As a result, there exists a multiloop controller
1 Cd(s) ) diag{Kd1(s), ..., Kdj(s), ..., Kdn(s)} s
(17)
which stabilizes G′(s) and maintains closed-loop stability when each diagonal subcontroller is independently detuned by an arbitrary factor of ξdj ∈ [0, 1], ∀ j ) 1, ..., n. When the same detuning factor ξi is chosen for the ith controller block, it is seen that process G′(s) can also be stabilized by
diagonal structure is represented by a series of matrices Uj (j ) 1, ..., p). The LMI problem formulated above, with the structural constraint on decision variable W, is convex and can be solved by using any semidefinite programming technique. 3.3. Small Gain-Based Sufficient Condition for BDIC. The frequently used small gain-based sufficient condition for DIC (theorem 4) can also be extended to BDIC as follows: Theorem 6. An n × n stable process G(s) is BDIC if Eb(s) ) [G(s) - Gb(s)]Gb-1(s) is stable, and a nonsingular matrix D can be found such that
min σ j (D-1Eb(0) D) < 1 D
C ˆ d(s) ) ΞCd(s)
(18)
where
Ξ ) diag{ξ1I1, ..., ξiIi, ..., ξkIk}, Ii ∈ Rni×ni, ξi ∈ [0, 1], ∀ i ) 1, ..., k (19) Therefore, the real controller “seen” by the original process G(s) is
1 C(s) ) MCd(s) NT ) diag{K1(s), ..., Ki(s), ..., Kk(s)} s (20) which can be detuned to
1 C ˆ (s) ) ΞC(s) ) diag{ξ1K1(s), ..., ξiKi(s), ..., ξkKk(s)} s (21) without causing closed-loop instability. Therefore, process G(s) is BDIC. 9 For a given process G(s), inequality (11) becomes a linear matrix inequality (LMI) in decision variable matrix W, which can be rewritten in the following form:17 p
W ) U0 +
∑ j)1
k
xiUj, p )
ni2 ∑ i)1
(22)
where xj ∈ R and Uj ∈ Rn×n (j ) 1, ..., p) are sparse constant matrices with only one nonzero element “1” located at the position corresponding to one nonzero element of W. Therefore, decision variable W can be written as an affine function of a number of scalar variables xj (j ) 1, ..., p), while its prespecified block
(23)
where σ j (‚) denotes the maximum singular value and Gb(s) consists of the block diagonal transfer function submatrices of G(s) as given in eq 2. Proof. The closed-loop system of process G(s) and the block diagonal controller C(s) can be rearranged in Figure 3a and further represented in Figure 3b, where
G(s) ) [I + Eb(s)]Gb(s)
(24)
Hb(s) ) Gb(s) C(s) [I + Gb(s) C(s)]-1
(25)
It is noted that, at steady state, the block diagonal transfer function matrix Hb(s) becomes an n × n identity matrix In, because of the integral action in the controller C(s). As a result, the stability of the system in Figure 3b is equivalent to that of the system in Figure 3c with an arbitrary nonsingular matrix D. Because Eb(s) is stable, according to the small gain theorem, the closedloop system is stable if Hb(s) is stable and
j (D-1Hb(jω) D) < 1 σ j (D-1Eb(jω) D)‚σ
(26)
Because Gb(s) is stable, a controller C(s) always exists such that Hb(s) is stable and has an arbitrary small gain at non-steady-state by rolling off C(s) fast enough. Therefore, only the steady-state condition needs to be addressed. Note that Hb(0) ) D-1Hb(0) D ) In. Condition (26) is reduced to
σ j (D-1Eb(0) D) < 1
(27)
9 When an optimal D matrix is chosen, the conservativeness of this condition can be significantly reduced. Note that the lower bound of σ j (D-1Eb(0) D) can be
Ind. Eng. Chem. Res., Vol. 42, No. 21, 2003 5155 Table 1. BDIC Tests for Different Pairing Schemes example 1
2 3
pairing scheme
control structure
cond. 1a holds?
cond. 2b holds?
cond. 3c holds?
cond. 4d holds?
BDIC
1 2 3 4 5 6 1 2 1 2
[1-1]/[2,3,4-2,3,4] [1,2-1,2]/[3,4-3,4] [1,2,3-1,2,3]/[4-4] [1-1]/[2,3-2,3]/[4-4] [1-1]/[2-2]/[3,4-3,4] [1,2-1,2]/[3-3]/[4-4] [1-1]/[2,3-2,3] [1,2-1,2]/[3-3] [1-1]/[2,3-2,3] [1,2-1,2]/[3-3]
yes yes yes no yes no no yes no yes
no no no no no no no yes no yes
yes yes yes yes yes yes no yes no yes
yes yes yes no yes no no yes no yes
yes yes yes no yes no no yes no yes
a Condition 1: passivity-based sufficient condition for BDIC (theorem 5). b Condition 2: small gain-based sufficient condition for BDIC (theorem 6). c Condition 3: BRG-based necessary condition for BDIC (theorem 1). d Condition 4: NI-based necessary condition for BDIC (theorem 2).
the small gain-based condition relates to the size of interactions, the passivity-based condition is concerned with the destabilizing effect of interactions.20 4. BDIC-Based Control Structure Selection
Figure 3. Small gain-based stability analysis.
represented by the structured singular value of Eb(0). Therefore, condition (23) can be tested using the existing µ-analysis methods, which are available in some commercial software, such as Matlab (a product of Mathworks Inc.).18 3.4. Remarks. The concept of BDIC has the following interesting properties: (1) BDIC considers the existence of a desirable controller. So, BDIC depends only on the process model and the chosen pairings; i.e., BDIC is controllerindependent. (2) BDIC can be tested by using only the steady-state information of process models. (3) BDIC is a less restrictive condition than DIC but a more restrictive condition than DCLI. (4) Unstable processes are not BDIC. According to our case studies, it is observed that the passivity-based condition (theorem 5) is often less conservative than the small gain-based condition (theorem 6). It has been found that interactions may not always destabilize decentralized control systems.19 While
After introducing the definition of BDIC and deriving sufficient and necessary conditions for processes to be BDIC, we are now at the position to present a new pairing rule for block-decentralized control. Based on the above work for BDIC, the control structure selection procedure leading to appropriate block diagonal pairing schemes is as follows: (1) Determine the size of each controller block Ki(s) (i ) 1, ..., k). (2) Determine pairing for each block diagonal subsystem Gii(s) (i ) 1, ..., k) by swapping the rows (columns) of the process transfer function matrix G(s). (3) Use theorems 1 and 2 to screen out unworkable pairings. (4) Examine the conditions given in theorems 5 and 6 to determine the BDIC property of the process with the block diagonal pairing. Like DIC, BDIC is a qualitative property of a process with a certain control structure. For a given process, it is possible to have multiple pairing schemes that all satisfy the BDIC conditions. Under this circumstance, the best pairing scheme should be determined in conjunction with other pairing criteria. One of the major considerations is resilience, which is the quality of regulatory and servo behaviors that can be obtained by feedback control.21 This can be quantified by using the minimum singular values (at steady state or in a frequency band) of each subsystem of Gb(s). The larger the minimum singular values are, the more resilient the subsystems are because larger disturbances can be handled by the controller for given constraints on the manipulated variables. 5. Illustrative Examples 5.1. BDIC Analysis for Control Structure Selection. Three non-DIC processes from ref 16 are analyzed in this section. The steady-state transfer function matrices of these processes are given below: Example 1.
[
8.72 6.54 G1(0) ) -5.82 -7.23
2.81 -2.92 0.99 2.92
2.98 2.50 -1.48 3.11
-15.80 -20.79 -7.51 7.86
]
(28)
5156 Ind. Eng. Chem. Res., Vol. 42, No. 21, 2003
Example 2.
[
0.5 0.5 -0.005 2 -0.01 G2(0) ) 1 -30 -250 1 Example 3.
[
1 1 -0.1 G3(0) ) 0.1 2 -1 -2 -3 1
]
]
ξ ) detuning factor Λi(G(0)) ) block relative gain of the ith block Gii(s)
(29)
(30)
We now demonstrate how controller structures affect the processes’ BDIC property. The processes with different controller structures were tested for BDIC by solving the passivity-based condition (11) and the small gain-based condition (23). The necessary conditions for BDIC in theorems 1 and 2 were also examined. The analysis results are shown in Table 1. The best controller structures for the three processes appear to be pairing 5, pairing 2, and pairing 2 because they all possess BDIC and have the smallest dimensionality of individual controllers (i.e., 6, 5, and 5 subcontrollers, respectively). It can also be observed that compared with the small gain-based sufficient condition, the passivitybased sufficient condition for BDIC is less conservative. 6. Conclusion Block-decentralized controllers are significant in control performance improvement. In this paper, the concept of DIC is extended to block-decentralized control systems, namely, BDIC. Similar to DIC, BDIC is shown to be the property of the process model only and thus offers a useful pairing criterion for control structure selection. On the basis of both the passivity theorem and the small gain theorem, two sufficient BDIC conditions are proposed. Their applications are illustrated using examples. Acknowledgment The authors gratefully acknowledge the financial support of the Australian Research Council (Grant A00104473). Nomenclature G(s) ) process transfer function matrix Gd(s) ) diagonal transfer function submatrix of G(s) Gb(s) ) block diagonal transfer function submatrix of G(s) C(s) ) block diagonal controller transfer function matrix Cd(s) ) diagonal controller transfer function matrix Ed(s) ) relative error defined by Ed(s) ) [G(s) - Gd(s)]Gd-1(s) Eb(s) ) relative error defined by Eb(s) ) [G(s) - Gb(s)]Gb-1(s) NT ) transpose of matrix N N-1 ) inverse of matrix N N-T ) transpose of the inverse of matrix N det(A) ) determinant of matrix A σ j (A) ) maximum singular value of matrix A µ(A) ) structured singular value of matrix A
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Received for review November 18, 2002 Revised manuscript received May 12, 2003 Accepted August 5, 2003 IE020924Q
