Ind. Eng. Chem. Res. 2009, 48, 9175–9185
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Multivariable Control with Generalized Decoupling for Disturbance Rejection Feng-Yi Lin,† Jyh-Cheng Jeng,‡ and Hsiao-Ping Huang*,† Department of Chemical Engineering, National Taiwan UniVersity, Taipei 106-17, Taiwan, Department of Chemical Engineering and Biotechnology, National Taipei UniVersity of Technology, Taipei 106, Taiwan
In this paper, a systematic procedure to design multivariable controllers that have options for selective decoupling of different structures (e.g., full or partial decoupler) is proposed. The controller structure is determined based on an analysis of relative load gain (RLG). For controller design, the adjoint of a template matrix is provided to design the MIMO decoupler that is needed for a given process. In the case where decoupling is desirable, according to the number of zero deficiency that the adjoint of the template matrix has, a pole-zero compensator is incorporated to preserve the properness of the decoupling controller and to define the decoupled open-loop dynamics. By incorporating the decoupler, the controller in the main loop is synthesized for disturbance rejection. Stability robustness of the system is tuned using measures of modeling errors in the decoupled open-loop process. Simulation examples are used to illustrate the proposed method and show its effectiveness in disturbance rejection in MIMO systems. 1. Introduction Most chemical processes have multi-inputs and multi-outputs and are known as MIMO processes. One of the main characteristics of a MIMO process is the presence of interactions between the inputs and outputs that makes conventional multiloop control difficult. Because of this, many methods for multivariable control (MVC) have been reported. These methods include inverse Nyquist array,1 characteristic-locus,2 inversebased decoupling controls,3-6 internal model controls,7-9 multivariable predictive control,10-13 dynamic matrix control (DMC),14 and quadratic dynamic matrix control (QDMC).15 Many of the above-mentioned methods tried to design the openloop transfer function to be strictly or roughly diagonally dominant, so as to reduce the effect of the interaction. Diagonal decoupling6 is also one approach toward this end. In general, the decoupling MIMO controllers are considered to have better control than multiloop SISO controllers. However, Niederlinski16 indicated that, for some cases, multiloop SISO controllers may have better performance for load rejection than the inverse-based multivariable ones. To analyze the differences of load responses between multiloop SISO controllers and inverse-based multivariable controllers, Stanley et al.17 proposed a relative disturbance gain (RDG) which is defined as a ratio of the manipulated variable under a perfect single-loop control at steady-state. Actually, the control structure that has superior ability for disturbance rejection may not be categorized as multiloop SISO control or a full inverse-based multivariable control.18,19 It can be any in between, for example, the partial decoupling. Some forms of partial decoupling have been proposed, such as block diagonal decoupling20,21 and triangular decoupling.22,23 But, most of the above-mentioned works discussed the delay-free systems which, in fact, seldom exist in real chemical processes. Although some one-way decoupling methods19,24 can be easily applied to TITO systems that have multiple delays, they are difficult to extend to higher dimensional systems. Besides, most of the methods lack a criterion to select a structure in between the fully decoupled and multiloop control. * To whom correspondence should be addressed. E-mail: huanghpc@ ntu.edu.tw. Tel.: 886-2-23638999. Fax: 886-2-2362-3935. † National Taiwan University. ‡ National Taipei University of Technology.
In this paper, a systematic procedure is proposed to design the multivariable controller with structure of different options to perform good disturbance rejection. A relative load gain (RLG), which has explicit physical meaning and direct connection to the control performance, is defined. By making use of this RLG, a template matrix for selective decoupling is proposed. By incorporating the adjoint matrix of this template matrix, a generalized decoupling control, ranging from multiloop SISO to the multivariable with options from full or partial decoupling can be developed. Together with the decoupling element, the multivariable system is decomposed into several equivalent single loops for controller design. On the basis of these equivalent single loop systems, controllers are designed individually and independently to achieve better disturbance rejection and maintain the robust stability. Furthermore, measures of modeling error are given to facilitate the analysis of system robustness. Simulation examples show that the proposed method can be applied to design controllers for MIMO control with many structure options, and the system can reject disturbance effectively. 2. Illustrative Comparison of Decoupling and Nondecoupling A multivariable control scheme with unity feedback loop is presented in Figure 1. Usually, to control this MIMO system, two common approaches are adopted. One of the approaches uses decoupling MVC that consists of a diagonal controller C(s) and an inverse-based decoupler D(s). The other uses multiple single loop controllers that consist of a diagonal C(s) and an identity matrix D(s). Nevertheless, depending on cases, researches found that control system with another structure in between the above-mentioned two (e.g., one-way decoupling)
Figure 1. A multivariable control scheme with unity feedback loop.
10.1021/ie801477z CCC: $40.75 2009 American Chemical Society Published on Web 09/29/2009
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G(s) is the process transfer function matrix (TFM). Both G(s) and GL(s) are open-loop stable. The objective of generalized decoupling is to remove interactions from some outputs but to keep them in others. In general, G(s) is first factorized into two parts:
where θi ) min {θi1, θi2, ..., θin}. For explicit explanation, Go(s) is permuted to the following form; that is,
Figure 2. Load responses and ISE values for the example in section 2.
may have better ability on disturbance rejection.19 To illustrate, the following example is used to address this issue. Consider the following transfer function matrix for the process and transfer function vector for load.
[
-5s
-5s
] [ ] -5s
4e 7e 5e 10s + 1 20s + 1 30s + 1 G(s) ) and GL(s) ) 4e-10s -6e-10s 4e-10s 10s + 1 20s + 1 30s + 1
If there are m out of n outputs need to be decoupled from the other n-m outputs, a template matrix for selective decoupling is in the following:
(1)
In the decoupling MVC, the process is decoupled using the method of Huang and Lin6 and, then, optimal PID-type controllers are assigned to each individual decoupled loop by minimizing the integral of squared error (ISE). In the multiloop control, optimal PID controllers are tuned sequentially and iteratively to achieve the minimum summation of ISE values from all loops. For exploring the best performance for comparison, both MVC and MLC demonstrated in this example are tuned very tightly to achieve minimal ISEs without considering stability robustness. The responses and their ISE values from the two systems to a unit step load are shown in Figure 2. The complete decoupling MVC has better control performance in the first output but not in the second. Later, we will show that only the first output is preferred to be decoupled using an MISO controller, so, a partial decoupling is more suitable for this case. 3. Generalized Decoupling Aside from the issue of should or should not decouple the outputs, the concept of full decoupling and partial decoupling have been studied and illustrated in open literature and text books.6,25-28 The problem is tackled by multiplying the inverse of process transfer function with a diagonal or upper/lower triangle matrix. The difficulties encountered in this approach have been addressed in the book of Skogestad and Postlethwaite.26 In literature, not many papers presented methods for decoupling a MIMO process that has multiple time delays. In light of the fact of example 2, the selection of outputs to be decoupled from others is also an important issue in design. To accommodate the demand for selective decoupling, a method to provide the design of generalize decoupler is desirable. In the following, we will present a template matrix for this purpose. Consider a n × n system as the following: Y(s) ) G(s) U(s) + GL(s) l(s)
where Go11 ∈ Rm×m, Go12 ∈ Rm×(n - m), Go21 ∈ R(n - m)×m and Go22 ∈ R(n - m)×(n - m). The upper m rows correspond to those outputs that need to be decoupled from others.
(2)
where, Y(s) and U(s) designate the output and input vectors, l(s) and GL(s) represent the load and its transfer function vector (TFV),
where
The upper part of A(s) is directly the same as the first m rows of Go(s), which is associate to the m outputs to be decoupled from the remaining n-m outputs. The lower part of A(s) is designed by I(n-m)×(n-m) and O(n-m)×m with proper dimensions as indicated. On the basis of the template matrix A(s), an effective decoupler is proposed as the following: D ) adj{A}Z ) A-1det{A}Z G-1 -G-1 o11Go12 ) o11 det{Go11}Z O(n-m)×m I(n-m)×(n-m) adj{Go11} -adj{Go11}Go12 Z ) O(n-m)×m det{Go11}I(n-m)×(n-m)
[ [
]
]
(6)
where Z(s) is a diagonal matrix with elements designated as zi(s), (i.e., Z(s) ) diag{zi(s)}). Notice that each diagonal element zi(s) is given as a simple and stable transfer function. The adj{A(s)} designates the adjoint matrix of A(s). In other words, adj{A(s)} ) [Aji(s); i,j ) 1, 2, ..., n], where Aij(s) is the cofactor of the (i, j) element, aij(s) of A(s). The decoupler designed in this way will always give stable controller, provided that G(s) is open-loop stable.
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As a result, the decoupled open-loop process Q(s) is given as Q ) ΘGoD Im×mdet{Go11} Om×(n-m) Z )Θ -1 Go21Go11 det{Go11} {Go22 - Go21Go11-1Go12}det{Go11} Im×mdet{Go11} Om×(n-m) Z )Θ Go21adj{Go11} Go22det{Go11} - Go21adj{Go11}Go12 Qo11 Om×(n-m) )Θ Qo21 Qo22
[ [ [
]
]
]
(7)
From eq 7, a partially decoupled and open-loop stable Q(s) is obtained. If A(s) is taken identically to Go(s), then eq 7 results in a full decoupling MVC. In other words, Q becomes Θ det{Go}Z. Therefore, with the template matrix, the resulting decoupled openloop transfer function can be structurally represented by a spatial matrix as shown in eq 7, which means partial decoupling for m outputs. 4. Index for Selection of Decoupling Options As the example demonstrated earlier, the effect of load change can be suppressed or amplified due to process interactions. If interactions amplify the load effects, the decoupling control will be required. On the other hand, if interaction favors the system response toward rejecting the load, decoupling becomes not necessary. Therefore, a measure for evaluating the decoupling for disturbance rejection is desirable. By the change of manipulation responding to a load input, Stanley et al.17 proposed a relative disturbance gain (RDG) as an interaction measure to justify the control types between a multiloop one and a fully decoupled multivariable one. Since RDG was derived from the IE (integration of errors) of multiloop type in comparing to fully decoupled one, it has limited use to other control types, for example, the partial decoupling.18 Theoretically, errors caused by the disturbances can only be eliminated after one dead-time, so the error magnitude in the output is proportional to the load gain of the system during this period. In this paper, a relative load gain (RLG) is thus defined in the following:29
γi )
|
( ) ( ) ∂yi ∂l
all loops except i closed
|
∂yi ∂l all loops open Thus, RLG is closely linked to the control performance. From the definition in eq 8, RLG can be computed as γi )
|
| | |
gE,Li(0) kE,Li ) gLi(0) kLi
(8)
(9)
where, kE,Li and kLi are the gains of gE,Li(s) and gLi(s), respectively. The gE,Li(s) is defined as the effective disturbance that means the open-loop effect of load input to the ith output, when the other outputs are under closed-loop control. To derive the gE,Li(s), the matrices in eq 2 are first permuted and partitioned into the following forms: G(i) )
[
gii
G(i) 12
G(i) 21
G(i) 22
]
;
G(i) C )
[
]
gCi 0 ; 0 G(i) C2
G(i) L )
[ ] gLi
G(i) L2
(10)
Then, the effective disturbance of the ith loop is given as (i) (i) (i) (i) -1 -1 gE,Li(s) ) gLi(s) - G(i) 12(s)[G22(s)] {I - (I + G22GC2(s)) }GL2(s)
(11)
which results in an equivalent load gain of (i) -1 (i) gE,Li(0) ) kL,i(0) - K(i) 12[K22] KL2(0)
Notice that gE,Li(0) contains no gains and structural information of controller GC(i). As a result, the equivalent load added to the ith output will be uniquely determined for each i, regardless the structural decoupling applied on the remaining part G(i) 22. It is independent (i) (i) of the structural decoupling on [G22 , G22 ]. In the following, we shall illustrate with two extreme cases and show the equivalent load gains in these two cases are identically equal. Consider a process where G(s) is first factorized into the following two parts, according to the requirement for decoupling m outputs:
Then, the following cases are considered: (a) When the upper part of G(s) in eq 12 is decoupled, the decoupled open-loop process can be obtained as
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Because the RLG is calculated only for those not to be decoupled, we are focusing on the outputs of the second part, that is, Y2(s), when the outputs from the first part is closed for control. On the basis of eq 13, when the upper part is closed, Y2(s) can be derived as -1 -1 (a) ˜ (a) ˜ (a) Y(a) 2 ) [(G22 - G21G11 G12) det{Go11}Z22]U2 + [GL2 - G21G11 H1 GL1]l ) G2 U2 + GL,2l
(14)
where (a) -1 (a) H(a) 1 ) (I + Θ11I det{Go11}GC1) Θ11I det{Go11}GC1
According to eq 14, and the fact that H1(a)(0) ) I, the load gains for Y2 are found as
( )
∂Y(a) 2 -1 ˜ (a) )G L,2(0) ) KL2 - K21K11 KL1 ∂l s)0 where KL1, KL2, K11 and K21 are the gain matrices of GL1, GL2, G11 and G21, respectively. (b) When the upper part of G(s) in eq 12 is not decoupled and is closed, the outputs of the lower part become
(15)
(b) -1 -1 (b) (b) ˜ (b) Y(b) 2 ) (G22 - G21H1 G11 G12)U2 + (GL2 - G21G11 H1 GL1)l ) G2 (s)U2 + GL,2l
(16)
where (b) -1 (b) H(b) 1 ) (I + G11GC1) G11GC1
Again, from eq 16 and H1(b)(0) ) I, the load gains of Y2 are
( ) ∂Y(b) 2 ∂l
s)0
-1 ˜ (b) )G L,2(0) ) KL2 - K21K11 KL1
(17)
(a) (b) ˜ L,2 ˜ L,2 (0) and G (0) being equal, the relative load gain to each output in the systems which are represented by eq 14 and In light of G eq 16 will be equal. In other words, the RLG of a MIMO process does not depend on any specific partial decoupling. Furthermore, the output that has corresponding value of γi g 1 can be considered to be decoupled, and an MISO (i.e., multiple input single output) controller will be used there. On the other hand, the output having γi e 1 favors a SISO controller. The selection of controllers can be based on the following criterion:
{
γi > 1; MISO controller is preferred for yi γi e 1; SISO controller is preferred for yi
Notice that, if both SISO and MISO controllers are needed in an MIMO process, the controller needed will be a partial decoupling controller. n
5. General Multivariable Controller with Partial Decoupling
kAe-δs
A general multivariable controller K(s) can be regarded as combination of a decentralized controller C(s) and a decoupler D(s), as shown in Figure 1. The design can be conducted by two parts: the designs of C(s) and the design of D(s). The Design of D(s). As the generalized decoupling mentioned, it needs to specify the template matrix first in order to design the decoupler in eq 6. According to the RLG in eq 8, this design matrix can be specified by the following criteria:
{
Ai,•(s) ) Goi,•(s) ∀i ∈ {i|γi > 1} Ai,•(s) ) Ii,•(s) ∀i ∈ {i|γi e 1}
(18)
where Ai,•(s), Goi,•(s) and Ii,•(s) designate the ith rows of A(s), Go(s), and a unit matrix I, respectively. To implement D(s) of eq 6, each Aij(s) can be reduced to a simpler transfer function; that is,
A r,is
+ 1)
i)1
φji(s) ) Aˆij(s) )
p
(τAp,1s2 + τAp,2s + 1)
∏ (τ
A g,is
+ 1)
i)1
(19) where Aˆij(s) designates the approximation of Aij(s), n and p are the numbers of first order leads and lags, respectively, and they obey the inequality of p + 2 - n > 0. The parameters in the model of eq 19 can be obtained by solving the following optimization problem: P ) arg min P
A(s) )
∏ (τ
∫
ωf
0
|Aˆij(jω) - Aij(jω)| 2 dω
(20)
where, P ) [kA,δ,τAg,i, τAr,i,τAp,1, and τAp,2], ωf is a frequency band which is chosen as 10 times frequency bandwidth of Aij(s). To make each element of D(s) realizable, a number of excess zeros of zi(s) is given as the following: Nez[zi(s)] ) min{Nep[φji(s)], j ) 1, 2, ..., n} ep
(21)
where N [φji(s)] is the number of excess poles in φji(s) [i.e., Aˆij(s)].
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From eq 7, the decoupled parts in Q(s) are dominated by det{Go11}. According to that, the decoupled loop in the proposed design can be obtained by a simpler expression as the following: n
w(s) )
∑g A
∀i ∈ {i|γi > 1}
ij
ij
(22)
j)1
The w(s) can be further implemented by a reduced order form of the following:
To do so, the decoupled process is first found according to the proposed method; that is, Q(s) ) G(s) adj{A(s)}Z(s)
(26)
Next, the matrices are permuted and partitioned into the following forms: Q(i) )
[
qii
Q(i) 12
Q(i) 21
Q(i) 22
]
; C(i) )
[ ]
ci 0 ; 0 C(i) 2
G(i) L )
[ ] gLi
G(i) L2
n
D -θexs
k e
∏
+ 1)
D (τr,i s
i)1
wˆ(s) )
) wˆ (s)e o
p
(τDp,1s2
+
τDp,2s
+ 1)
∏
(τDg,is
-θexs
(23) where θex is the extra delay resulted from det{Go11}. Similarly, the parameters in eq 23 can be obtained by solving the optimization problem as in eq 20 except that wˆ(s) and w(s) are used instead of Aˆij and Aij. The ωf is chosen as 10 times the frequency bandwidth of w(s). Then, by reallocate the pole(s) and zero(s) in wˆ(s), zi(s) provides the availability to modify undesirable dynamic characteristics in w(s), and thus can improve the dynamics resulting from some large time constants or excessive lags. The decoupler D(s) is thus implemented via the transfer functions of the following: ∀i, j ∈ n
(24)
An index is defined to indicate the effectiveness of decoupling,
|
qˆji(jω) - qji(jω) εji ) max ω qii(jω)
|
∀ω ∈ (0, ωg,i]
(25)
n
∑g A z ik
jk
i
k)1
n
qˆji )
∑ k)1
n
gjkφkizi )
∑
Then, the equivalent single-loop system for the ith loop is presented as
(i) -1 (i) (i) -1 (i) gE,Li ) gLi - Q(i) 12[Q22] {I - (I + Q22C2 ) }GL2
(28) 30
According to Huang et al., the equivalent loop and disturbance can be approximated as the following forms; that is, (i) -1 (i) (i) q*E,i ) qii - Q(i) 12[Q22] Q21 X H* (i) -1 (i) (i) g*E,Li ) gLi - Q(i) 12[Q22] GL2 X H*
(29)
T and each h*i is where H*(i) ) [h*1 ,h*2 , · · · ,h*i-1,h*i+1, · · · ,h*] n designed for qii(s) and q*E,oi(s). Then, the reduced models of q*E,i and g*E,Li can be found by fitting their frequency responses as mention earlier. After these procedures, the controller design for ci(s) becomes one SISO control problem. The process output in response to a load l(s) can be regarded as
yi(s) )
g*E,Li(s) ) g*E,Li(s)[1 - hE,i(s)] 1 + q*E,i(s)ci(s)
(30)
where hE,i(s) is the equivalent complementary sensitivity function and is designed for the equivalent system. By Huang and Lin,6 the hE,i(s) can be found to minimize an integral of the absolute error (IAE) constrained by an assigned peak value of sensitivity function. Then, ci(s) can be synthesized by
where qji )
(27)
(i) -1 (i) (i) -1 (i) qE,i ) qii - Q(i) 12[Q22] {I - (I + Q22C2 ) }Q21
+ 1)
i)1
dij(s) ) φij(s) zj(s) ) Aˆji(s) zj(s),
9179
gjkAˆikzi
k)1
and ωg,i is the frequency bandwidth of qii(s). The index in eq 25 means the relative discrepancy between qji(s) and qˆji(s). If this value is too large to be satisfactory, the model orders of φ(s) need to be increased. In other words, εji serves as a tuning factor to improve the stability robustness of the system. For good stability robustness, it is recommended that εji be less than 0.1. The Design of C(s). As the multivariable control scheme in Figure 1, after the decoupling, a decentralized controller C(s) is designed for a new open-loop process Q(s) that is presented as the dotted block in Figure 1. For an inverse-based multivariable controller or a multivariable controller with complete decoupling, the process is decoupled into several individual open-loop processes qii(s) so the design of each decentralized controller ci(s) can be simplified as the design in single-loop system with each open-loop process qi(s). However, the generalized decoupling may give partial results of complete decoupling as the upper part of Q(s) in eq 7 and the other results of nondecoupling as the lower part of Q(s) in eq 7. To simplify the design problem, this decoupled process is decomposed into several effective processes.26 Furthermore, the effective disturbance to each effective process can be derived as in eq 11.
ci(s) )
hE,i(s) hE,oi(s) 1 1 ) q*E,i(s) 1 - hE,i(s) q*E,oi(s) 1 - h (s)e-θi*s E,oi (31)
where θ*i is the delay time of q*E,i(s) and hE,i(s), and q*E,oi(s) and hE,oi(s) are the delay-free part of q*E,i(s) and hE,i(s). By applying * the first order Pade’s approximation for e-θi s in eq 31, the controller ci(s) can be given as ci(s) )
hE,oi(s)(1 + θ*s/2) gf,i(s) i q*E,oi(s) (1 + θ*s/2) h (s)(1 - θ*s/2) i E,oi i (32) gf,i(s) )
1 (τf,is + 1)n
(33)
where τf,i is the filter time constant and has a default value as 0.05θ*. i Finally, the generalized multivariable controller can be obtained by kij(s) ) φij(s) zj(s) cj(s)
∀i, j ∈ n
(34)
6. Stability and Robustness Assume that m rows in G(s) with a dimension n × n are decoupled. For convenience, the row to be decoupled are
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σ¯ (∆(jω)) e |l (jω)|
(37)
where the perturbation ∆(s) is bounded on l(jω). And, the system will be robust stable if and only if σ¯ [M(jω)]|l(jω)| < 1,
ω ∈ [0, ∞)
(38)
where M(s) ) -D(s) C(s) [I + G(s) D(s) C(s)]-1 Figure 3. An equivalent multivariable control scheme in the generalized decoupling.
permuted and moved to the forehead part, hence the open-loop process Q(s) and the controller C(s) are rewritten as the following:
where C1 ) diag[c1,ii], C2 ) diag[c2,jj], Q11 ) diag[q11,ii], Q21 ) [q21,ji] and Q22 ) [q22,jj] for all i ∈ [1, 2, ..., m] and j ∈ [m + 1, m + 2, ..., n]. According to the proposed design, the control scheme in Figure 1 can be regarded as an equivalent multivariable control scheme as shown in Figure 3 that conjugates a multivariable decentralized control system with some singleloop control systems. Because each element of the process G(s) in eq 2 is an open-loop stable function, the decoupled openloop process Q(s) in eq 26 and the generalized decoupler in eq 24 are designed to be open-loop stable. Under the conjunctive framework in Figure 3, the stability of the system can be individually discussed by two steps: one is C1(s) stabilizes a diagonal system Q11(s) and the other is C2(s) stabilizes a full system Q22(s). However, the approximation of D(s) in eq 24 leads to the existence of modeling error in the desired process Q(s). Thus, the nominal stability of the proposed control scheme in Figure 1 is guaranteed by designing C(s) to satisfy the following conditions: 1. C(s) stabilizes Q(s) in a simple closed loop. a. c1,ii(s) stabilizes q11,ii(s) for all i ∈[1,2,.. .,m] b. 1 + c2,ii(s)q22E,i(s) ) 0 has no RHP zero, and q22E,i(s) has no RHP pole for all i ∈[m + 1,m + 2,.. .,n]. 2. σ¯ {-C(jω)[I + Q(jω) C(jω)]-1} e 1 ; ∀ω ∈ [0, ∞) max{σ¯ [∆Q(jω)]} ω
where q22E,i is the effective process of Q22. σ j denotes the largest singular value. Owing to an approximation made in eq 19, G(s) D(s) may not equal to Q(s) exactly. As a result, a model error (i.e., ∆Q(s)) originating from this approximation can be estimated by the index εij of eq 25 in the frequency range of concerned for nominal stability, and then the second condition can easily be satisfied. As for stability robustness to modeling error of G(s), consider that the control system has an additive uncertainty, where the real process is presented as ˜ (s) ) G(s) + ∆(s) G and
(39)
Thus, by selecting an adequate hE,i(s), the controller ci(s) is synthesized also to satisfy the robust stability in eq 38. Typically, the peak value of sensitivity function, that is, maxω |1 - hE,i(jω)|, is assigned in the range of 1.2-2.0 for stability robustness. 7. Illustrative Examples In this section, systems of 2 × 2, 3 × 3, and 4 × 4 dimensions are considered for illustration. Based on the RLG, controllers with different partial decoupling are suggested in each different case. Then, controllers with proper template matrix are designed. Together with this, controllers with fully decoupling are also given for comparison purpose. It is to demonstrate that the use of RLG can identify correctly the structure for partial decoupling, and, the design method, indeed, could provide controllers that have better performances as expected. Example 1. Consider the same 2 × 2 process as presented in section 2. First, the process TFM is factorized into two parts; that is,
[
e-5s 0 G) 0 e-10s
][
4 7 10s + 1 20s + 1 -6 4 10s + 1 20s + 1
]
(40)
According to the definition of RLG in eq 8, the values of RLG are computed as γ1 ) 1.53 and γ2 ) 0.29. These results indicate that the first row needs to be decoupled but the second row does not. Thus, by eq 18, the template design matrix is given as A1,• )
[ 10s7+ 1
4 20s + 1
]
A2,• ) [0 1 ] and the adjoint matrix ofA(s) is
[ ]
-4 20s + 1 adj{A(s)} ) 7 0 10s + 1 1
(41)
According to eq 21, a number of excess zeros of z1(s) and z2(s) are 0 and 1, respectively. Because the first loop has γi > 1, eq 22 gives the following relation: w(s) )
7 10s + 1
(42)
On the basis of Nez[z1(s)] ) 0 and Nez[z2(s)] ) 1, z1 is specified as one and z2(s) is selected as (10s + 1) to compensate the undesired pole of w(s) in eq 42. By eq 24, the decoupler is determined as
(36) D)
[
-4(10s + 1) 20s + 1 0 7 1
]
(43)
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Table 1. The Equivalent Single-Loop Control Systems for Example 1 loop 1 q*E,i(s) g*E,Li(s) hE,i(s) reduced PID form of ci(s)
loop 2
-5s
-10s
(7e )/(10s + 1) (5e-5s)/(30s + 1) (13.25s + 1)e-5s/(55.63s2 + 12.06s + 1) 0.0375(135.4s2 + 18.37s + 1)/(s(22.8s + 1))
Then, the reduced order forms of the equivalent process to the G adj{A}Z, is shown in Table 1. Then, the equivalent complementary sensitivity functions are found by the method of Huang and Lin,6 which is constrained from above by a magnitude of 1.7. Next, controllers can be synthesized by eq 31 and the results are further reduced to the PID form as shown in Table 1. For comparison, two extreme control systems (i.e., fully decoupling and without decoupling) are also given under the same robust condition. Simulation results for a unit-step load input and their ISE values are given in Figure 4. These results indicate that the proposed system gives better load responses than two others. For system robustness to modeling error, simulations for (20% changes in process gains, time constants, and time delays of g11(s) and g22(s) are shown in Figure 5. Furthermore, the maximum singular value of M(s) in eq 39 is plotted in Figure 6, and the value of σ j [M(jω)]σ j [∆(jω)] for modeling error of +20% changes in process gains is shown in Figure 7. These results
Figure 4. Load responses and ISE values for example 1.
Figure 5. Load responses with 20% errors of process gains, time constant, and time delay in g11(s) and g22(s) for example 1.
(-58e )/(20s + 1) (1.14e-10.3s)/(5.14s + 1) e-10s/(4.67s + 1) -0.00118(100s2 + 25s + 1))/(s(1.592s + 1)
clearly show that the proposed design indeed meets the robust stability condition of eq 38 for the assumed modeling errors. Example 2. The Tyreus31 3 × 3 process as the following is considered.
[
]
-5.24e-60s -5.984e-2.24s 1.986e-0.71 66.7s + 1 400s + 1 14.29s + 1 -2.38e-0.42 -0.0204e-0.59s 0.33e-0.68s G(s) ) (7.14s + 1)2 (2.38s + 1)2 (1.43s + 1)2 11.3e-3.79s -0.374e-7.75s 9.811e-1.59 2 22.22s + 1 (21.74s + 1) 11.36s + 1 (44)
Assume disturbance models having the following form:
Figure 6. Maximum singular value of M(s) (eq 39) for example 1.
Figure 7. Value of σ j [M(jω)]σ j [∆(jω)] for modeling error of +20% changes in process gains for example 1.
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[
-1.5s 0.5e-s 4e-0.5s GL(s) ) -2e 15s + 1 20s + 1 25s + 1
]
T
(45)
The RLGs are computed as γ1 ) 0.176, γ2 ) 3.00, and γ3 ) 3.57. First, G(s) is factorized into Go(s) and Θ(s) by eq 3. According to eq 18, the design matrix is specified as
[
1 0 0 -0.26s -0.17s -2.38 0.33e -0.0204e 2 2 (1.43s + 1)2 A(s) ) (7.14s + 1) (2.38s + 1) -2.2s -6.16s 11.3e 9.811 -0.374e 2 11.36s + 1 22.22s + 1 (21.74s + 1)
]
(46)
Following the design procedures from eq 19 to eq 24, zi(s) is specified as z1 ) 19.77s2 + 38.06s + 1, z2 ) 24.78s + 1, and z3 ) 20.55s2 + 20.1s + 1, and the generalized decoupler dij(s) can be found as the following forms. d11(s) )
GL )
d13(s) ) 0
2e-0.5s 18e-0.3s 1e-0.2s 25e-0.5s 40s + 1 20s + 1 30s + 1 25s + 1
]
T
(47)
The RLGs in the system are computed as γ1 ) 1.4, γ2 ) 1.86, γ3 ) 2.48, and γ4 ) 0.279. Notice that the values of γ1, γ2, and γ3 are greater than 1, but the value of γ4 is less than 1. According to eq 18, the template design matrix is determined as the following:
By following the design procedures, the approximation models of Aji and w are given as the following:
φ21 )
-0.1071(-33.21s + 1)(19.77s2 + 38.06s + 1)e-7.51s (6.183s + 1)(22.11s + 1)2
φ32 d33(s) )
0.33(20.55s2 + 20.1s + 1)e-0.26s (2.38s + 1)2
26.64 (16s + 8s + 1)(16s + 1)(13s + 1) 16.02 φ22 ) 2 (16s + 8s + 1)(14s + 1)(13s + 1) 19.109(-0.06481s + 1) ) 2 (36.44s + 15.84s + 1)(13.05s + 1)(5.409s + 1) φ42 ) 0
933.5s2 + 62.66s + 1 s(2.576s + 1)
c2(s) ) 0.0617
5.368s2 + 5.119s + 1 s(0.0421s + 1)
c3(s) ) 0.0127
7.361s2 + 10.52s + 1 s(0.5471s + 1)
φ23 ) 0 φ33 )
-54.43 (71.76s2 + 26.56s + 1)(17.26s + 1) φ43 ) 0
-193.43(-1.752s + 1) (124.1s2 + 23.26s + 1)(14.97s + 1)2 -125.59(-1.734s + 1) φ24 ) (87.2s2 + 18.39s + 1)(16.95s + 1)2 -148.46(-1.663s + 1) φ34 ) (82.13s2 + 17.07s + 1)(17.78s + 1)2 -195.95(-1.311s + 1) φ44 ) (72.93s2 + 17.13s + 1)(21.19s + 1)2
φ14 )
and
Simulation results for a unit-step input of load and their ISE values are given in Figure 8. For comparison, the responses of multivariable controllers based on fully decoupling and multiloop
2
φ13 ) 0
Then, with the partial decoupling, the equivalent SISO openloops for design are as shown in Table 2. To satisfy robust stability, the maximum peak of sensitivity function is confined to be 1.3. The elements ci(s) for each equivalent open-loop are reduced to the following PID forms. c1(s) ) 0.0086
-71.598(-0.7505s + 1) (1.978s2 + 4.717s + 1)(14.41s + 1)2 φ41 ) 0
φ12 )
-11.3(24.78s + 1)e-2.2s (21.74s + 1)2
and
-62.28(-0.5675s + 1) (14.76s + 17.85s + 1)(13.04s + 1)(3.95s + 1) 2
φ31 )
2.38(20.55s2 + 20.1s + 1) d23(s) ) (1.43s + 1)2
d32(s) )
-147.6 (84s2 + 25s + 1)(13s + 1)(s + 1)
φ11 )
1.090(19.77s2 + 38.06s + 1)e-6.04s (20.74s + 1)(3.443s + 1) 9.811(24.78s + 1) d22(s) ) 11.36s + 1
d31(s) )
[
30.13(19.77s2 + 38.06s + 1)e-1.319s (24.71s + 1)(19.13s + 1) d12(s) ) 0
d21(s) )
controllers are also given. Compared with the latter two systems, the results of partial decoupling are as expected. Example 3. Consider a heat-integrated distillation column (CL column) studied by Chiang and Luyben.32 The process TFM is given in Table 3 and the disturbance transfer function vector is assumed to have the following form:
and wˆ )
-195.95 (54.88s + 16.25s + 1)(20.98s + 1)2(2.582s + 1) 2
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Figure 8. Load responses and ISE values for example 2. Table 2. The Equivalent Single-Loop Control Systems for Example 2 (Tyreus Process) loop 1
loop 2
loop 3
Equivalent Open-Loop Process (54.77(598.8s + 1)e
-0.97s
(30.13e-0.68s)/(20.55s2 + 20.1s + 1)
)/((9227s + 576.9s + 1)(51.5s + 1)) 2
(30.13e-1.85s)/(24.78s + 1)
Equivalent Disturbances (-0.3513(3286s + 1)e-1.6s)/(10450s2 + 399s + 1)
(0.5e-s)/(20s + 1)
(4e-0.5s)/(25s + 1)
Equivalent Complementary Sensitivity Functions (6.869s + 1)e-0.97s/17.49s2 + 6.541s + 1
(4.363s + 1)e-0.68s/8.05s2 + 4.221s + 1
(9.543s + 1)e-1.85s/47.32s2 + 10.31s + 1
Table 3. The Process Transfer Function Matrix for CL Column G(s) (4.45)/((14s + 1)(4s + 1)) (17.3e-0.9s)/((17s + 1)(0.5s + 1)) (0.22e-1.2s)/((17.5s + 1)(4s + 1)) (1.82e-s)/((21s + 1)(s + 1))
(-7.4)/((16s + 1)(4s + 1)) (-41)/((21s + 1)(s + 1)) (-4.66)/((13s + 1)(4s + 1)) (-34.5)/((20s + 1)(s + 1))
0 0 (3.6)/((13s + 1)(4s + 1)) (12.2e-0.9s)/((18.5s + 1)(s + 1))
Because all the values of Nez[z1(s)], Nez[z2(s)], Nez[z3(s)] and Nez[z4(s)] are 3, Z(s) is designed as
(0.35)/((25.7s + 1)(2s + 1)) (9.2e-0.3s)/(20s + 1) (0.042(78.7s + 1))/((21s + 1)(11.6s + 1)(3s + 1)) (-6.92e-0.6s)/(20s + 1)
c1 ) -0.0005103
54.88s2 + 16.25s + 1 s(2.5s + 1)
c2 ) -0.0005103
54.88s2 + 16.25s + 1 s(2.5s + 1)
c3 ) -0.0005103
54.88s2 + 16.25s + 1 s(2.5s + 1)
z1 ) z2 ) z3 ) z4 ) (20.98s + 1)2(2.582s + 1) To check the decoupling performance, the performance indices in eq 25 are computed as: ε11 ) 0.1%, ε21 ) 0.51%, ε31 ) 0.39%, ε41 ) 1.3%, ε12 ) 0%, ε22 ) 0%, ε32 ) 0.059%, ε42 ) 0.17%, ε13 ) 0%, ε23 ) 0%, ε33 ) 0.28%, ε43 ) 0.82%, ε14 ) 0.056%, ε24 ) 0.18%, ε34 ) 0.016%, and ε44 ) 0.21%. All of their values are less than 0.1 so the model of Aji is reliable. Then, the decoupler is obtained by eq 24. Next, this decoupled open-loop process is decomposed into four equivalent singleloop control systems as shown in Table 4. On the basis of these, the decentralized controller can be synthesized by eq 32 and is reduced to the simple PID forms as the following:
c4 ) 2.837 × 10-5
290s2 + 30.71s + 1 s(0.05s + 1)
According to eq 34, the proposed generalized multivariable controller can be obtained by the combination of generalized decoupler and decentralized controller. Simulation results for a unit-step input of load and their ISE values are given in Figure 9. Similarly, the multivariable controllers based on the fully decoupling and nondecoupling are also given for comparison. In Figure 9, the control of the
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Figure 9. Load responses and ISE values for example 3 (CL column). Table 4. The Equivalent Single-Loop Control Systems for CL Column loop 1
loop 2
loop 3
(-195.95)/ (54.88s2 + 16.25s + 1)
(-195.95)/(54.88s2 + 16.25s + 1)
(2e-0.5s)/(40s + 1)
(18e-0.3s)/(10s + 1)
(1)/((5s + 1)2)
(1)/((5s + 1)2)
loop 4
Equivalent Open-Loop Processes (-195.95)/(54.88s2 + 16.25s + 1)
(3525.5)/ ((145.2s2 + 25.84s + 1)(8.264s + 1))
Equivalent Disturbances (e-0.2s)/(30s + 1)
(6.9833(35.8s + 1)2e-s)/ ((824.9s2 + 129.6s + 1)(6.71s + 1))
Equivalent Complementary Sensitivity Functions (1)/((5s + 1)2)
4th output with partial decoupling outperforms the one with fully decoupling. This result consists with that predicted by the RLG. 8. Conclusion The inverse-based multivariable controller and decentralized controller are two conventional choices for the control of multivariable processes. However, in some cases, control structure (e.g., partial decoupling) other than the two conventional ones may have better performance for load rejection. To enhance the use of decoupling control, a general approach is proposed to design controllers that may have various decoupling options. An index of RLG is proposed to select the structure among these options. By a template matrix, a systematic method is proposed to design a multivariable controller to achieve better disturbance rejection. Furthermore, this method can applied to more complex processes which have higher dimensions and multiple time delays. The stability and robustness of the system is also included to take account of modeling errors and process errors. Simulation examples have been illustrated to show that
(1)/((5s + 1)2)
the proposed method can provide a controller with proper partial decoupling in each various case, which is more effective for disturbance rejection than the conventional choices. Literature Cited (1) Rosenbrock, H. H. Computer-Aided Control System Design; Academic Press: New York, 1974. (2) MacFarlane, A. G. J.; Kouvaritakis.; B. A Design Technique for Linear Multivariable Feedback Systems. Int. J. Control 1977, 25, 837. (3) Falb, P. L.; Wolovich, W. A. Decoupling in Design and Synthesis of Multivariable Control Systems. IEEE Trans. Autom. Control. 1967, 12, 651. (4) Agamennoni, O. E.; Desages, A. C.; Romagnoli, J. A. Robust Controller Design Methodology for Multivariable Chemical Processes. Chem. Eng. Sci. 1988, 43, 2937. (5) Wang, Q. G.; Zhang, Y.; Chiu, M. S. Non-interacting Control Design for Multivariable Industrial Processes. J. Process Control 2003, 13, 253. (6) Huang, H. P.; Lin, F. Y. Decoupling Multivariable Control with Two Degrees of Freedom. Ind. Eng. Chem. Res. 2006, 45, 3161. (7) Garcia, C. E.; Morari, M. Internal Model Control. 2. Design Procedure for Multivariable Systems. Ind. Eng. Chem. Process Des. DeV. 1985, 24, 472.
Ind. Eng. Chem. Res., Vol. 48, No. 20, 2009 (8) Morari, M.; Zafiriou, E. Robust Process Control; Prentice-Hall: Englewood Cliffs, NJ, 1989. (9) Wang, Q. G.; Hang, C. C.; Yang, X. P. IMC-Based Controller Design for MIMO Systems. J. Chem. Eng. Jpn. 2002, 35, 1231. (10) Ogunnaike, B. A.; Ray, W. H. Multivariable Controller Design for Linear Systems Having Multiple Time Delays. AIChE J. 1979, 25, 1043. (11) Jerome, N. F.; Ray, W. H. High Performance Multivariable Strategies for Systems Having Time Delay. AIChE J. 1986, 32, 914. (12) Agamennoni, O. E.; Desages, A. C.; Romagnoli, J. A. A Multivariable Delay Compensator Scheme. Chem. Eng. Sci. 1992, 47, 1173. (13) Wang, Q. G.; Zou, B.; Zhang, Y. Decoupling Smith Predictor Design for Multivariable Systems with Multiple Time Delays. IChemE 2000, 78, 565. (14) Cutler, C. R.; Ramaker, B. L. Dynamic Matrix Control-A Computer Control Algorithm. In Proceedings of the Joint Automatic Control Conference, San Francisco, CA, 1980. (15) Garcia, C. E.; Morshedi, A. M. Quadratic Programming Solution of Dynamic Matrix Control (QDMC). Chem. Eng. Commun. 1986, 46, 73. (16) Niederlinski, A. Two-Variable Distillation Control: Decouple or Not Decouple. AIChE J. 1971, 17, 1261. (17) Stanley, G.; Marino-Galarraga, M.; McAvoy, T. J. Short-Cut Operability Analysis: 1. The Relative Disturbance Gain. Ind. Eng. Chem. Process Des. DeV. 1985, 24, 1181. (18) Chang, J. W.; Yu, C. C. Relative Disturbance Gain Array. AIChE J. 1992, 4, 521. (19) Fagervik, K. C.; Waller, K. V.; Hammarstro¨m, L. G. Two-Way or One-Way Decoupling in Distillation. Chem. Eng. Commun. 1983, 21, 235. (20) Hautus, M. L. J.; Heymann, M. Linear Feedback DecouplingTransfer Function Analysis. IEEE Trans. Autom. Control 1983, 28, 823. (21) Linneman, A.; Wang, Q. G. Block Decoupling with Stability by Unity Output Feedback-Solution and Performance Limitations. Automatica 1993, 29, 735.
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(22) Hung, N. T.; Anderson, B. D. O. Triangularization Technique for the Design of Multivariable Control Systems. IEEE Trans. Autom. Control 1979, 24, 455. (23) Go´mez, G. I.; Goodwin, G. C. An Algebraic Approach to Decoupling in Linear Multivariable Systems. Int. J. Control 2000, 73, 582. (24) Arkun, Y.; Manouslouthakis, B.; Palazogˇlu, A. Robustness Analysis of Process Control Systems. A Case Study of Decoupling Control in Distillation. Ind. Eng. Chem. Process Des. DeV. 1984, 23, 93. (25) Ogunnaike, B. A.; Ray, W. H. Process Dynamics, Modeling, and Control, Oxford University Press, 1994, p. 776. (26) Skogestad, S.; Postlethwaite, I. MultiVariable Feedback Controls Analysis and Design; JohnWiley & Sons: New York, 1996; p 80. (27) Garelli, F.; Mantz, R. J.; De.Battista, H. Partial Decoupling of Nonminmum Phase Processes with Bounds on the Remaining Coupling. Chem. Eng. Sci. 2006, 61, 7706. (28) Liu, T.; Zhang, W.; Gao, F. Analytical Decoupling Control Strategy Using Feedback Control Structure for MIMO Process with Time Delays. J. Process Control 2007, 17, 173. (29) Huang, H. P.; Lin, F. Y.; Jeng, J. C. Control Structure Selection and Performance Assessment for Disturbance Rejection in MIMO Processes. Ind. Eng. Chem. Res. 2007, 46, 9170. (30) Huang, H. P.; Jeng, J. C.; Chiang, C. H.; Pan, W. A Direct Method for Multi-loop PI/PID Controller Design. J. Process Control 2003, 13, 769. (31) Tyreus, B. D. Paper presented at the Lehigh University Distillation Control Short Course, Lehigh University, Bethlehem, PA, 1982. (32) Chiang, T. P.; Luyben, W. L. Comparison of the Dynamic Performances of Three Heat-Integrated Distillation Configurations. Ind. Eng. Chem. Res. 1988, 27, 99.
ReceiVed for reView October 1, 2008 ReVised manuscript receiVed August 13, 2009 Accepted August 14, 2009 IE801477Z
