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Ind. Eng. Chem. Res. 2005, 44, 7428-7434
Continuation Method for the Modified Ziegler-Nichols Tuning of Multiloop Control Systems Jietae Lee Department of Chemical Engineering, Kyungpook National University, Taegu 702-701, Korea
Thomas F. Edgar* Department of Chemical Engineering, University of Texas, Austin, TX 78712
Multiloop control systems are often used for industrial multivariable processes because of their simplicity. To design multiloop control systems, single-input single-output (SISO) methods that guarantee specified closed-loop characteristics can be applied. Because the design equations are nonlinear due to loop interactions and may cause computational difficulties, a continuation method is proposed to obtain the solutions. By choosing the interaction level as a continuation parameter, we can design multiloop control systems without worrying about numerical difficulties such as divergence. To illustrate the approach, the modified Ziegler-Nichols method is applied to design multiloop control systems. Introduction Industrial processes often have many control loops and may suffer from loop interactions. Multivariable controllers can reduce interactions, but they are complex and costly to implement. Hence, multiloop controllers are frequently utilized to control multivariable processes in the chemical industry. Many methods to design multiloop controllers are available. The independent design method1 designs a separate controller for each paired, open-loop transfer function and then detunes to meet the stability robustness constraints. Related methods include the Nyquist array method,2 the biggest log-modulus tuning (BLT) method,3 and the µ-interaction measure method.4,5 Because the process interactions cannot be described in a precise manner, methods in this class take a conservative approach. For example, loops designed by the BLT method can be very sluggish due to excessive detuning. The sequential loop closing method6 designs loops one after the other. Single-input single-output (SISO) design methods can be applied to design each controller, and SISO autotuning methods can also be applied.7,8 One of the drawbacks of this design method is that the closed-loop performance depends on the design sequence and how the first controller is designed. To reduce this dependence, Shen and Yu8 suggested repeating the design sequence. Design methods extending SISO methods that guarantee some closed-loop characteristics have been studied recently. The modified Ziegler-Nichols (MZN) method,9 the desired closed-loop response (DCLR) method,10 and the dominant pole (DP) method11 can be included in this class of methods. For each method, closed-loop performance can be guaranteed up to a certain point. The main drawback of these methods is that the design equations are nonlinear due to loop interactions9 and, thus, iterative methods are required to solve them. Here, to reduce the computational difficulties, the * Corresponding author. Tel.: (512) 471-3080. Fax: (512) 471-7060. E-mail:
[email protected].
continuation method12 is applied to solve the design equations given by the MZN method or the DP method. Continuation Method Many engineering problems require iterative methods to solve nonlinear equations, but methods such as Newton’s method can diverge depending on the starting point. Often the difficulties of iterative methods can be resolved by using the continuation method. Consider the problem
f(x,η) ) 0
(1)
where x and f(x,η) are n-dimensional vectors and η is a real parameter. It is assumed that f(x,η) is sufficiently smooth. The above equations define a family of curves, and continuation means tracing these curves. The solution of f(x,η) ) 0 can be traced from the known solution xo for η0. Since f(x,η) ) 0 is maintained, the total differential of f(x,η) should satisfy
df )
∂f ∂f dx + dη ) 0 ∂x ∂η
(2)
and, consequently, we have
∂f -1 ∂f dx )dη ∂x ∂η
[ ]
(3)
By integrating the ordinary differential equation (eq 3), starting from an initial value (xo, ηo), we can trace the solution of eq 1 as η varies. This ordinary differential equation method fails at points with a singular matrix of ∂f/∂x. Procedures to resolve these difficulties in the continuation method are available in ref 12. Derivatives for eq 3 can be obtained by applying the perturbation technique to eq 1. Let η ) η j + η˜ and x ) xj + x˜ . Then
f(x,η) ) f(xj+x˜ ,η j +η˜ ) ∂f(xj,η j) ∂f(xj,η j) x˜ + η˜ + O(2) ) f(xj,η j) + ∂x ∂η
(
10.1021/ie0501531 CCC: $30.25 © 2005 American Chemical Society Published on Web 08/10/2005
)
(4)
Ind. Eng. Chem. Res., Vol. 44, No. 19, 2005 7429
Figure 1. Continuation model for designing a multiloop control system.
where O(2) means terms of 2 and higher order. From the equation
∂f(xj,η j) ∂f(xj,η j) x˜ + η˜ ) 0 ∂x ∂η
(5)
which eliminates terms of in eq 4, we can obtain x˜ ) dx/dη with η˜ ) 1. These derivatives in eq 4 can also be used in a Newton iteration to solve the nonlinear equation for a fixed η. Here, to integrate eq 3, the Euler method with the Newton correction12 is used. Application of Continuation to Multiloop Control System Design. Consider a multivariable process
G(s) ) {gij(s); i ) 1, 2, ..., n; j ) 1, 2, ..., n}
(6)
with n inputs and n outputs. For the process, assuming that appropriate pairings have been selected, we design a diagonal controller,
C(s) ) diag{ci(s); i ) 1, 2, ..., n}
(7)
The closed-loop transfer-function matrix becomes
H(s) ) (I + G(s)C(s))-1G(s)C(s)
(8)
Since the multiloop controller C(s) has n elements, it has sufficient degrees of freedom to specify n properties in the closed-loop transfer-function matrix. The modified Ziegler-Nichols (MZN) method,9 the desired closed-loop response (DCLR) method,10 and the dominant pole (DP) method11 may be included in this class of design methods which specify n diagonal elements of H(s). These methods will guarantee certain closed-loop characteristics. However, because C(s) occurs nonlinearly in the closed-loop transfer-function matrix of eq 8, an iterative solution is required. To solve this nonlinear problem, the continuation method is applied. For this, we decompose G(s) as follows (see Figure 1)
G(s,η) ) Gd(s) + ηGo(s)
(9)
where Gd(s) and Go(s) are the diagonal and off-diagonal parts of G(s), respectively. When η ) 0, G(s,η) is diagonal and we can design each element of the controller C(s) easily by a SISO method. By applying the continuation method and increasing η to 1, we can obtain the controller C(s) for the process G(s,η)|η)1 ) G(s). In the following sections, derivatives needed to apply a continuation method are derived by the pertur-
Figure 2. Effective process-transfer function for the MZN method.
bation technique for a specific design method of the MZN method. Modified Ziegler-Nichols Method. The ZieglerNichols method can be interpreted as finding controller parameters such that the critical point where the Nyquist diagram intersects the negative real axis is moved to the point of -(0.6 + 0.28j).13,14 The modified ZieglerNichols (MZN) method generalizes this concept in such a way that a point on the Nyquist curve of the process is moved to a new point by applying the controller. This MZN method was applied to design multiloop control systems for two-input two-output processes by Wang et al.9 However, because of the computational difficulty of solving nonlinear design equations, its extension to higher dimensional processes has not yet been presented. When loops are closed, the effective process-transfer function for the ith loop (Figure 2) is quite different from the open-loop process transfer function gii(s). Let qi(s) be the effective process transfer function for the ith loop while other loops are closed. The MZN method finds ci(s) such that
qi(jωi) ) rai exp(j(-π + ξai)), i ) 1, 2, ..., n
(10)
qi(jωi)ci(jωi) ) rbi exp(j(-π + ξbi)), i)1, 2, ..., n (11) for a given rbi, ξai, and ξbi. Since qi(s) contains controllers of other loops, eqs 10 and 11 are nonlinear. From eqs 10 and 11, we have ∠ci(jωi) ) ∠qi(jωi)ci(jωi) - ∠qi(jωi) ) ξbi - ξai, and the proportional-integral (PI) controller ci(s) can be expressed as follows:
(
ci(s) ) Kci 1 +
) (
)
1 1 1 + ) βi τIis ωitan (ξai - ξbi) s
(12)
The closed-loop transfer function between the set point and output of the ith loop is hii(s) ) qi(s)ci(s)/(1+qi(s)ci(s)), and hence, eq 11 becomes
hii(jωi) )
rbi exp(j(- π + ξbi)) 1 + rbi exp(j(- π + ξbi))
≡ φi
(13)
Now the design equation is hii(jωi) - φi ) 0, i ) 1, 2, ..., n, where hii(s) is the ith diagonal element of H(s) ) (I + G(s)C(s))-1G(s)C(s). The continuation method is applied to this nonlinear equation with the PI control-
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Ind. Eng. Chem. Res., Vol. 44, No. 19, 2005
lers of eq 12. First, we calculate derivatives to trace solutions of hii(jωi) - φi ) 0 as η increases from 0 to 1. For this, we perturb design variables
η)η j + η˜
˜ n, β˜ 1, ‚‚‚, β˜ n)T and φio and φi1 are where σ ) (ω ˜ 1, ‚‚‚, ω corresponding constants calculated from eq 18. From φi0Tσ + φi1η˜ ) 0 (i ) 1, 2, ‚‚‚, n), which eliminates terms of order in eq 19, we can obtain
[ ][ ] Re(φ10T) Im(φ10T) σ)‚‚‚ T Re(φn0 ) Im(φn0T)
j i + ω ˜i ωi ) ω βi ) β h i + β˜ i i ) 1, 2, ..., n
(14)
and apply them to eq 13. Then we have
j i + jω ˜ i) + (η j + η˜ )Go(jω j i + jω ˜ i) G(jωi,η) ) Gd(jω
((
j i) + η j Go(jω j i) + ) Gd(jω
d j )+ G (jω ds d i
)
)
η j Go(jω j i) jω ˜ i + Go(jω j i)η˜ + O(2) ≡G h + G ˜ + O(2)
(15)
and
C(jωi) ) diag
) diag
{(
{(
(
{(
diag -
)
1 1 β , ‚‚‚, + ω1tan (ξa1 - ξb1) jωi 1 1 1 β + ωn tan(ξan - ξbn) jωi n
} ) ( ) ) } )
1 1 β h , ‚‚‚ + ji 1 ω j 1 tan(ξa1 - ξb1) jω β h1 2
ω j 1 tan(ξa1 - ξb1)
(
ω ˜1 + -
β h1
jω j i2
1 1 β˜ , ‚‚‚ + jω ji 1 ω j 1 tan(ξa1 - ξb1)
≡C h + C ˜ + O(2)
)}
+
ω ˜i + + O(2)
qi(jωu)ci(jωu) ) qi(jωu)
H(jωi,η) ) [I + G(jωi,η)C(jωi)]-1G(jωi,η)C(jωi)
Hence,
) [I + G hC h ]-1[I - (G hC ˜ +G ˜C h )(I + G hC h )-1 + hC h + (G hC ˜ +G ˜C h ) + O(2)] O(2)][G 2
)H h + [S hG hC ˜S h +S hG ˜C hS h ] + O( )
h ) + [(C hS h )T X (S h )]vec(G ˜) + vec(H(jωi,η)) ) vec(H hG h )]vec(C ˜ ) + O(2) (18) [S h T X (S where vec(A) is a vector such that [a11, a21, ..., an1, a12, a22, ..., ann]T and X means the Kronecker product. Arranging eq 18 for the unknown variables,
hii(jωi,η) ) h h ii + (φi0Tσ + φi1η˜ ) + O(2)
rbi )
(19)
) )
(
0.45 x1 + 1/(1.666Fπ)2 F
ξbi ) -arctan(1/(1.666Fπ))
(22)
On the basis of the tuning rule for the SISO process of Zhuang and Atherton,16 Wang et al.9 recommended design parameters rbi and ξbi as
ξbi ) -arctan
(17)
where H h ) [I + G hC h ]-1G hC h and S h ) [I + G hC h ]-1. Equation 17 is linear for C ˜ and G ˜ . An explicit equation can be obtained as follows15
(
0.45Ku 1 1+ ) F 0.833jPuFωu 0.45 1 j (21) 1F 1.666Fπ
Here, G ˜ contains variables of ω ˜ i and η˜ linearly and C ˜ contains variables of ω ˜ i (i ) 1, 2, ‚‚‚, n) and β˜ i (i ) 1, 2, ‚‚‚, n). The closed-loop transfer-function matrix becomes
) [I + G hC h + (G hC ˜ +G ˜C h ) + O(2)]-1[G hC h + (G hC ˜ +G ˜C h ) + O(2)]
(20)
where Re( ) and Im( ) denote the real and imaginary parts of a complex number, respectively. As shown in eq 4, the solution of eq 20 with η˜ ) 1 and σ ) (ω ˜ 1, ‚‚‚, ω ˜ n, β˜ 1, ‚‚‚, β˜ n)T is such that dωi/dη ) ω ˜ i and dβi/dη ) β˜ i for continuation. Integrating dωi/dη ) ω ˜ i and dβi/dη ) β˜ i for i ) 1, 2, ..., n from η ) 0 to 1, we can obtain multiloop PI controllers which satisfy the closed-loop MZN conditions of eqs 10 and 11. Initial values for η ) 0 can be easily obtained by applying the MZN conditions to the openloop process-transfer functions of gii(s). Design Parameters. The design parameters of ξai, ξbi, and rbi are very important. The parameter ξai for the process Nyquist point is usually set to zero when gii(0) > 0 and ξai ) π otherwise. For the parameters of ξbi and rbi, the Ziegler-Nichols tuning rule (Kc ) 0.45Ku and τI ) 0.833Pu, where Ku and Pu are the ultimate gain and period, respectively) can be used. However, these controller parameters should be detuned for multiloop control systems. The biggest log-modulus tuning (BLT) type detuning, where the controller gain is divided by a given factor F and the integral time is multiplied by F, can be used for eq 11.
(16)
h + C ˜ + O(2))]-1[ ) [I + (G h + G ˜ + O(2))(C 2 h + C ˜ + O(2))] (G h + G ˜ + O( ))(C
Re(φ11) Im(φ11) η˜ ‚‚‚ Re(φn1) Im(φn1)
(
1 0.166π(1.935κi + 1)
)
rbi ) 0.361/cos(ξbi)
(23)
where κi ) |1/([G-1(0)]iigii(jωuii))| and ωuii is the critical frequency of gii(s). Here, parameters of eq 23 are further detuned as
ξbi ) -arctan
(
1 0.166π(1.935κi + 1)
rbi )
0.361 Q cos(ξbi)
where Q is a given detuning constant.
) (24)
Ind. Eng. Chem. Res., Vol. 44, No. 19, 2005 7431 Table 1. Design Results for Various Example Processes biggest log-modulus tuning
modified Ziegler-Nichols (eq 22 with F ) 2)
modified Ziegler-Nichols (eq 24 with Q ) n)
Kc ) [0.375, -0.075] τI ) [8.29, 23.6] Kc ) [-1.07, 1.97] τI ) [7.1, 2.58] Kc ) [27.4, -13.3] τI ) [41.4, 52.9] Kc ) [1.51, -0.295, 2.63] τI ) [16.4, 18, 6.61] Kc ) [-11.26, -3.52, -0.182] τI ) [7.09, 14.5, 15.1] Kc ) [-16.61, 0.701, -0.239] τI ) [4.762, 5.94, 68.89] Kc ) [-0.118, -7.26, 0.429, 0.743] τI ) [23.5, 11, 12.1, 7.94] Kc ) [-0.243, -16.86, 0.711, 0.0329] τI ) [67.9, 4.693, 5.853, 5.14] Kc ) [2.28, 2.94, 1.18, 2.02] τI ) [72.2, 7.48, 7.39, 27.8]
Kc ) [0.4597, -0.0583] τI ) [6.534, 22.45] Kc ) [-0.927, 2.143] τI ) [6.177, 2.301] Kc ) [29.61, -14.24] τI ) [38.31, 49.76] Kc ) [2.044, -0.2585, 2.764] τI ) [14.59, 20.57, 6.23] Kc ) [-16.39, -4.65, -0.2017] τI ) [4.724, 9.879, 10.29] Kc ) [-16.79, 0.711, -0.413] τI ) [4.714, 5.837, 38.65] Kc ) [-0.1954, -16.41, 0.3114, -5.035] τI ) [10.12, 4.724, 7.474, 1.154] Kc ) [-0.397, -16.79, 0.709, 0.0327] τI ) [38.41, 4.714, 5.841, 5.161] Kc ) [0.8063, 3.254, 1.327, 3.31] τI ) [100.56, 6.694, 6.559, 17.245]
Kc ) [0.2476, -0.0439] τI ) [8.745, 9.369] Kc ) [-0.562, 1.157] τI ) [4.669, 5.834] Kc ) [15.72, -7.592] τI ) [24.43, 15.78] Kc ) [0.979, -0.1882, 1.492] τI ) [4.049, 3.769, 4.722] Kc ) [-8.851, -2.662, -0.1266] τI ) [61.9, 69.59, 119.5] Kc ) [-8.978, 0.3795, -0.1907] τI ) [61.83, 5.044, 54.77] Kc ) [-0.0948, -6.663, 0.4705, 0.6266] τI ) [95.9, 61.8, 6.6, 1129] Kc ) [-0.1055, -6.732, 0.2843, 0.0131] τI ) [67.55, 61.78, 6.123, 1.553] Kc ) [1.982, 1.311, 0.5297, 1.086] τI ) [33.55, 12.02, 7.552, 66.81]
process WB VL WW OR T4 T4a DL DLb A1 a
1-1/2-3/3-2 pairing. b 1-4/2-2/3-1/4-3 pairing.
Multiloop PID Controller Design. Because only the critical point is moved to a certain Nyquist point in the Ziegler-Nichols method, an additional rule such as the derivative time of a quarter of the integral time is needed to design the proportional-integral-derivative (PID) controller. Wang et al.9 used
(
ci(s) ) Kci 1 + γi )
1 + γiτIis τIis
)
0.413 3.302κi + 1
(25)
Here, κi is the process normalized gain, as in eq 23. From eqs 10 and 11, we have
(
ci(s) ) βi 1 +
mi )
ωi 1 γimi s + mi s ωi
)
tan(ξbi - ξai) + x4γi + tan2(ξbi - ξai) (26) 2γi
Applying the perturbation technique as above, we can obtain the continuation equation about dωi/dη ) ω ˜ i and dβi/dη ) β˜ i for i ) 1, 2, ..., n. Other SISO Design Methods. SISO control system design methods such as the dominant pole method and the frequency response method14 can be applied to design multiloop control systems. A continuation scheme for the dominant pole method is given in the Appendix. For given dominant poles, multiloop control systems can be easily calculated via the continuation scheme. However, it is uncertain how the dominant poles should be chosen. For example, undesirable controller parameters with negative values often result. Simulations. a. Example 1. Consider the column model of Wood and Berry:17
[
12.8 exp(-s) -18.9 exp(-3s) 21s + 1 G(s) ) 16.7s + 1 6.6 exp(-7s) -19.4 exp(-3s) 10.9s + 1 14.4s + 1 al.9
]
-0.213, rb1 ) 0.362, and rb2 ) 0.369. We solved the design problem of the MZN method by the continuation method. The integration step size for the continuation method was set to 0.1. The controller parameters obtained are Kc ) [0.699, -0.0895] and τI ) [8.42, 9.52] by Wang et al.9 and Kc ) [0.730, -0.0875] and τI ) [8.42, 9.50] by the proposed modified Ziegler-Nichols method. The proposed continuation method shows slightly different results. Wang et al.9 did not try to find the true solutions, because iterative refinements gave little improvement and both results will be the same if iterations are applied. The dominant pole method was applied to this process by Zhang et al.11 They used p1 ) -0.198 + 0.198j and p2 ) -0.0707 + 0.0707j for the desired closed-loop poles. For the same closed-loop poles, we applied the dominant pole method. The controller parameters obtained are Kc ) [0.776, -0.0288] and τI ) [8.75, 2.04] by Zhang et al.11 and Kc ) [1.029, -0.0295] and τI ) [9.48, 1.926] by the dominant pole method in the Appendix. Because of the imaginary parts of the desired poles, control performances were somewhat oscillatory. When p1 ) -0.2 and p2 ) -0.06 were used, less oscillatory responses were obtained with the multiloop control system designed by the dominant pole method (Kc ) [0.35, -0.0164] and τI
(27)
Wang et obtained multiloop PI controllers by the MZN method with ξa1 ) 0, ξa2 ) π, ξb1 ) -0.0741, ξb2 )
Figure 3. Step set point responses for the Wood and Berry column: dotted line ) BLT method; solid line ) MZN method with Q ) 2; dashed line ) method by Wang et al.;9 and dash-dotted line ) DP method with poles at -0.2 and -0.06.
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Ind. Eng. Chem. Res., Vol. 44, No. 19, 2005
Figure 4. Step set point responses for the OR column: dotted line ) BLT method and solid line ) MZN method with Q ) 3.
Figure 5. Step set point responses for the T4 column: dotted line ) BLT method and solid line ) MZN method with Q ) 3.
) [7.26, 3.354]). However, applications of the dominant pole method to other processes were not easy because selections of appropriate dominant poles were hard to determine. Figure 3 shows closed-loop step responses for control systems designed by BLT, modified Ziegler-Nichols, and dominant pole methods. b. Example 2. The modified Ziegler-Nichols method is applied to several example processes used by Luyben.3 Among them, the pairing for the DL column example in Luyben3 does not satisfy the necessary condition of decen-
tralized integral controllability and, hence, can suffer from the integrity problem.18 So, we correct the pairing, and the following discussions are valid for the corrected pairing. The pairing for the T4 column is not the best one in the sense of the relative gain array and the dynamic interaction measure, as discussed by Lee and Edgar.19 Hence, the T4 column with a different pairing is also included. Tuning results are shown in Table 1. We can see that the controller parameters of the modified ZieglerNichols method with the design parameters of eq 22 are
Ind. Eng. Chem. Res., Vol. 44, No. 19, 2005 7433
Figure 6. Step set point responses for the A1 column: dotted line ) BLT method and solid line ) MZN method with Q ) 4.
very similar to those of the BLT method. At high frequencies, we have a small ||G(jω)C(jω)|| and
This is equivalent to the condition that at least one eigenvalue of G(pi)C(pi) is -1. We trace multiloop PI controllers satisfying eq A1. For perturbed variables
H(jω) ) [I + G(jω)C(jω)]-1G(jω)C(jω) ≈ G(jω)C(jω) (28)
η)η j + η˜
Hence, if the same design parameters are used, MZN designs based on H(s) might be similar to BLT designs based on gii(s)ci(s). Better controller parameters are obtained from the design parameters of eq 24. Wang et al.9 used eq 24 with Q ) 1, but it provides somewhat aggressive control parameters as shown in Figure 3. From various simulations for processes up to 4 × 4, Q was set to the process order. Figures 3-6 show simulations for set point changes. We can see that closed-loop responses of the proposed MZN method are better than those of the BLT method. The detuning parameter Q can also be used to optimize a different criterion for improving closed-loop performance further.
Ri ) R j i + R˜ i βi ) β h i + β˜ i i ) 1, 2, ‚‚‚, n we have
j Go(pi) + G(pi,η) ) Gd(pi) + ηGo(pi) ) Gd(pi) + η h + G ˜ (A3) η˜ Go(pi)) G and
( (
C(pi) ) diag R1 +
The continuation method is applied to solve nonlinear equations arising in designing multiloop control systems. It can resolve numerical difficulties such as divergence, enabling many SISO design methods to be applicable to multiloop control systems. Here, the modified Ziegler-Nichols method is solved through the continuation method. Appendix (Continuation Method for the Dominant Pole Tuning)
β h1 β hn , ‚‚‚, R jn + + pi pi β˜ 1 β˜ n diag R˜ 1 + , ‚‚‚, R˜ n + pi pi
(
)C h + C ˜
)
(A4)
If the eigenvalue of G hC h at -1 is algebraically simple, the perturbation theorem of an eigenvalue20 shows that
h + G ˜ )(C h + C ˜ )) eig(G(pi,η)C(pi)) ) eig((G
Let pi be a closed loop pole of the multiloop control system. It satisfies the characteristic equation
det(I + G(pi)C(pi)) ) 0
) )
β1 βn , ‚‚‚, Rn + pi pi
) diag R j1 +
Conclusion
(A2)
(A1)
) eig(G hC h + (G hC ˜ +G ˜C h )) u*(G hC ˜ +G ˜C h )v + u*v O(2) (A5)
) eig(G hC h) +
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Ind. Eng. Chem. Res., Vol. 44, No. 19, 2005
where u and v are the left and right eigenvectors of G hC h for the eigenvalue of -1, respectively, and the notation * means the conjugate transpose of a complex vector. The continuation method solves
u*(G hC ˜ +G ˜C h )v ) 0
(A6)
φi0Tσ + φi1η˜ ) 0, i ) 1, 2, ‚‚‚, n
(A7)
or equivalently
where φi1η˜ ) u*G ˜C h v, σ ) (R˜ 1, ‚‚‚, R˜ n, β˜ 1, ‚‚‚, β˜ n), and φi0 are corresponding constants calculated from eq A6. For multiloop PI controllers, we can specify n different closed-loop poles of pi and, given n closed-loop poles, we can obtain dRi/dη ) R˜ i and dβi/dη ) β˜ i by solving
[ ][ ] Re(φ10T) Im(φ10T) σ)‚‚‚ Re(φn0T) Im(φn0T)
Re(φ11) Im(φ11) η˜ ‚‚‚ Re(φn1) Im(φn1)
(A8)
with setting η˜ ) 1. How to choose the design parameters of pi is not well-established. Literature Cited (1) Skogestad, S.; Morari, M. Robust Performance of Decentralized Control Systems by Independent Design. Automatica 1989, 25, 119-125. (2) Rosenbrock, H. H. Computer-Aided Control System Design; Wiley: New York, 1974. (3) Luyben, W. L. Simple Method for Tuning SISO Controllers in Multivariable Systems. Ind. Eng. Chem. Process Des. Dev. 1986, 25, 654-660. (4) Grosdidier, P.; Morari, M. Interaction Measures under Decentralized Control. Automatica 1986, 22, 309-319. (5) Lee, J.; Edgar, T. F. Phase Conditions for Stability of Multiloop Control Systems. Comput. Chem. Eng. 2000, 23, 16231630.
(6) Mayne, D. Q. The Design of Linear Multivariable Systems. Automatica 1973, 9, 201-207. (7) Loh, A. P.; Hang, C. C.; Quek, C. K.; Vasnani, V. U. Autotuning of Multivariable Proportional-Integral Controllers Using Relay Feedback. Ind. Eng. Chem. Res. 1993, 32, 1002-1007. (8) Shen, S. H.; Yu, C. C. Use of Relay-Feedback Test for Autotuning of Multivariable Systems. AIChE J. 1994, 40, 627646. (9) Wang, Q. G.; Lee, T. H.; Zhang, Y. Multiloop Version of the Modified Ziegler-Nichols Method for Two Input Two Output Processes. Ind. Eng. Chem. Res. 1998, 37, 4725-4733. (10) Jung, J.; Choi, J. Y.; Lee, J. One-Parameter Design Method for a Multiloop Control System Design. Ind. Eng. Chem. Res. 1999, 38, 1580-1588. (11) Zhang, Y.; Wang, Q. G.; Astrom, K. J. Dominant Pole Placement for Multi-Loop Control Systems. Proc. ACC, Chicago, 2000. (12) Seydel, R.; Hlavacek, V. Role of Continuation in Engineering Analysis. Chem. Eng. Sci. 1987, 42, 1281-1295. (13) A° stro¨m, K. J.; Ha¨gglund, T. Automatic Tuning of Simple Regulators with Specifications on Phase and Amplitude Margins. Automatica 1984, 20, 645-651. (14) A° stro¨m, K. J.; Ha¨gglund, T. Automatic Tuning of PID Controllers; Instrument Society of America: Research Triangle Park, NC, 1988. (15) Horn, R. A.; Johnson, C. R. Topics in Matrix Analysis; Cambridge University Press: New York, 1991. (16) Zhuang, M.; Atherton, D. P. Automatic Tuning of Optimum PID Controllers. Proc. IEE Part D 1993, 140, 216-224. (17) Wood, R. K.; Berry, M. W. Terminal Composition Control of a Binary Distillation Column. Chem. Eng. Sci. 1973, 28, 17071717. (18) Campo, P. J.; Morari, M. Achievable Closed-Loop Properties of Systems Under Decentralized Control: Conditions Involving the Steady-State Gain. IEEE Trans. Autom. Control 1994, AC39, 932-943. (19) Lee, J.; Edgar, T. F. Dynamic Interaction Measures for Decentralized Control of Multivariable Processes. Ind. Eng. Chem. Res. 2004, 43, 283-287. (20) Horn, R. A.; Johnson, C. R. Matrix Analysis; Cambridge University Press: New York, 1985.
Received for review February 7, 2005 Revised manuscript received June 23, 2005 Accepted June 27, 2005 IE0501531
