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Ind. Eng. Chem. Res. 2002, 41, 1242-1248
PROCESS DESIGN AND CONTROL Optimal Smith-Predictor Design Based on a GPC Approach K. K. Tan,* T. H. Lee, and F. M. Leu Department of Electrical and Computer Engineering, National University of Singapore, 4 Engineering Drive 3, Singapore 117576, Singapore
In this paper, an optimal Smith-predictor design based on a generalized predictive control (GPC) approach is developed. The basic idea is to back-calculate the Smith control parameters based on the GPC control law, which has been designed to minimize a cost function. The design has turned out to be very simple to use. With a second-order model, the primary controller is simply a PID control. With a first-order model, it is a PI controller. The relationship between the control parameters obtained via the GPC approach and a pole placement approach is derived so that the latter approach may be used to derive initial GPC weights for further fine-tuning. Simulation and a real-time experiment illustrate the performance of the thus-designed optimal Smith control system. 1. Introduction The presence of considerable time delays in many industrial processes is a well recognized phenomenon. The achievable performance of conventional unity feedback control systems can be significantly degraded if a process has a relatively large time delay as compared to the dominant time constant.1 In this case, dead time compensation may be necessary in order to enhance the performance. The most popular scheme for such compensation is possibly the Smith control,2,3 first proposed in the late 1950s. Since then, the scheme has also been extended to multivariable processes.4,5 Different ways of designing the Smith control system have been reported. A number of researchers have proposed modifications of the basic Smith structure which are motivated toward improving various capabilities of the Smith predictor. In ref 6, a modified plant model was used in the inner feedback loop to improve the Smith regulatory function, because the Smith control is known to exhibit poor regulatory performance under load disturbances. A predictor approximating the inverse of dead time has been suggested in ref 7. An observer-predictor (OP) structure that employed staticstate feedback to improve the stability properties of the system has been reported in ref 8. An adaptive Smith control based on an online parameter identification algorithm was used during the closed loop operations in ref 9. In this paper, the design of an optimal Smith controller is addressed. The control law is derived to yield an optimum control performance in terms of a specified cost function. It is based on the application of the generalized predictive control (GPC) methodology, while still retaining the popular Smith structure. When a general second-order model is used, the primary controller reduces to a PID control. One problem with a GPC * Corresponding author. Phone: (65) 8742110. Fax: (65) 7791103. E-mail:
[email protected].
Figure 1. Smith control.
design may be the difficulty it poses to practitioners in commissioning and maintenance. To this end, the initial specifications can also be in terms of a desired damping ratio and natural frequency of the closed loop. These specifications are more intuitively inclined to control engineers and practitioners. The relationship between the control parameters obtained via the pole placement method and the GPC approach is derived and provided. Thus, the initial pole placement parameters (which maybe more intuitive to the practitioners) can be translated into an equivalent set of GPC parameters for subsequent fine-tuning purposes. Furthermore, with this model, the method is directly amenable to the implementation of automatic tuning techniques.10,11 Simulation examples and a real-time experiment are given to illustrate the superior performance of the optimal Smith control scheme. 2. Smith Predictor Control: a Review Consider a time-delay system described by eq 1 and under Smith control, as shown in Figure 1.
X(k + 1) ) FX(k) + Bu(k - h)
(1)
A model for the system is assumed to be available and described by eq 2
X ˜ (k + 1) ) F ˜X ˜ (k) + B ˜ u(k - h ˜)
(2)
Under the Smith control configuration, the control effort is computed on the basis of the predicted system states
10.1021/ie000498d CCC: $22.00 © 2002 American Chemical Society Published on Web 02/05/2002
Ind. Eng. Chem. Res., Vol. 41, No. 5, 2002 1243
and current prediction error (eq 3)
()
Y ) (XT(k + 1), XT(k + 2), ‚‚‚, XT(k + p))T
u(k) ) -DX ˜ (k + h ˜) - D h [X(k) - X ˜ (k)]
(3)
I F G) l FP-1
If there is no modeling error (i.e., F ˜ ) F, B ˜ ) B, h ˜ ) h), then X(k) ) X ˜ (k) and
{
u(k) ) -DX(k + h) u(k - h) ) -DX(k)
}
(
(4)
According to eq 1, the closed loop is thus given by
X(k + 1) ) (F - BD)X(k)
(5)
The closed loop characteristic equation, given by det(zI - (F - BD)) ) 0, is free of the delay term. Thus, the controller may be designed with respect to the states of the time-delay free portion of the model. However, caution must be applied to this conclusion. It has been shown12 that the Smith control scheme can be unstable under infinitesimal perturbation in the model, even if it is nominally stable. Furthermore, the response to disturbances can be unacceptably poor, when the actual process exhibit sluggish, poorly damped, and unstable dynamics. However, in practice, modeling errors will always exist so that some robustness considerations are necessary. In this paper, we use a separate set of gains (D and D h , respectively) for the predicted set point deviation (X ˜ (k + h ˜ )) and the current model deviation [X(k) - X ˜ (k)]. D can be designed for performance while D h can be designed for robustness. In what follows, we will develop a framework for the design of state feedback gains in D to achieve optimal control in terms of a specified cost function. 3. Formulation of Optimal Smith Control In this section, the formulation of the proposed optimal Smith control is given. The basic idea is to utilize the GPC design methodology toward the design of the control gains. Using a general second-order model, the primary controller can be simplified to a PID control acting on the deviation between the set point and the actual process variable. With a first-order model, it further reduces to a PI control. 3.1. Expanded Smith Prediction Horizon. Consider a general time-delay system given in eq 1. Using the m control and p output prediction horizon, the future outputs X(k + l) can be obtained recursively as follows (where p g m):
X(k + 1) ) FX(k) + Bu(k - h)
‚‚‚ ‚‚‚ · · · Fp-1B Fp-2B ‚‚‚
B FB A) l
0 B l
0 0 0 Fp-mB
)
U ) (u(k - h), u(k + 1 - h), ‚‚‚, u(k + m - 1 - h))T 3.2. Optimal Control Design. Consider the quadratic cost function given in eq 7, where x and u are the states of the system and control effort, respectively. p
J)
∑ l)1
m
2 ||x(k + l)||Q(l) +
2 ||u(k + j - 1)||R(j) ∑ j)1
The main idea of the optimal control design is to derive a series of m control u(k), ..., u(k + m - 1) at each sample time so that the cost function is minimized. When eq 6 is substituted into eq 7, the cost function can be rewritten as follows:
J ) [GFX(k) + AU]TQ[GFX(k) + AU] + UTRU (8) The solution that minimizes the cost function J can be obtained by solving eq 9.
∂J ) 2ATQ[GFX(k) + AU] + 2RU ) 0 ∂U
X(k + 2) ) F X(k) + FBu(k - h) + Bu(k + 1 - h)
(9)
Thus, the optimal control sequence is given as
U ) -[ATQA + R]-1[ATQGF]X(k)
(10)
Under the principle of receding horizon, only the first value of the optimal control sequence is output at each sampling time steps. Thus, eq 10 can be rewritten as eq 11.
u(k) ) -H[ATQA + R]-1[ATQGF]X(k + h)
2
(7)
(11)
) -DX(k + h)
l p-1
X(k + p - 1) ) F
X(k) + Fp-2Bu(k - h) + ‚‚‚ + Fp-m-1Bu(k + m - 2 - h)
X(k + p) ) FpX(k) + Fp-1Bu(k - h) + ‚‚‚ + Fp-mBu(k + m - 1 - h) When the equations are combined, we have the following augmented matrix description:
Y ) GFX(k) + AU where
(6)
where D ) -H[ATQA + R]-1[ATQGF] and H ) [1, 0, ‚‚‚, 0]. In eq 11, the current control effort is a linear combination of the future predicted states in X(k + h). In this way, the optimal D to minimize the cost function (eq 7) can be derived. 3.2.1. Single Step Control Horizon, m ) 1. A onestep control horizon, that is u(k + 1) ) u(k + 2) ) ... ) u(k + m), allows (m - 1) steps of infinite weighting on the changes of the control action ahead of one step of finite weight. It is naturally robust against unmodeled dynamics and attractive for control applications. The matrices A are simply replaced by the following relation in eq 12:
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( )
Ind. Eng. Chem. Res., Vol. 41, No. 5, 2002
From eq 14, shifting the time origin results in eq 17.
B FB A) l Fp-1B
e(k + 1) ) -a1′e(k) - a2′e(k - 1) - b1′ u˜ (k - h) (17) where
) GB
(12) u˜ (k - h) ) u(k - h) +
3.3. Optimal Smith Control Using a SecondOrder Model. Given the general formulation described in Sections 3.1 and 3.2, we now consider the specific case with a commonly used model. Consider a general second-order discrete time transfer function.
G′(z) )
b1′z + b2′
z-h
(13)
z + a1′z + a2′ 2
Letting e ) ysp - y, the equivalent time domain difference equation of eq 13 is given in eq 14.
e(k + 2) ) -a1′e(k + 1) - a2′e(k) - b1′u(k + 1 - h) b2′u(k - h) + ysp(1 + a1′ + a2′) (14) The model described by eq 14 encompasses a rich and general class of time-continuous linear systems given by
G(s) )
ds + c e-sL (s + a)(s + b)
(15)
In time-discrete form, the pulse-transfer function of eq 15 is adequately represented by the generic form of eq 13, where the parameters are related by the following equations: -aTs
a1′ ) -(e b1′ )
[
c ae ab
-bTs
+e
- be b-a
-bTs
-aTs
-(a+b)Ts
), a2′ ) e
] [
+1 +d
e
-aTs
]
-bTs
-e b-a
b2 ′ ) c(a - b)e
-(a+b)Ts
-aTs
+ a(bd - c)e ab(a - b) h)
L Ts
+ b(c - ad)e
-bTs
(16)
Ts is the sampling interval. Although the difference equation model in eq 14 is constrained with a maximum order of 2, it is now wellknown and generally acknowledged, especially in the process control community, that a second-order model is sufficiently adequate for most applications, including those involving high-order process dynamics. Even if a higher-order model can be available, the second-order one is usually still preferred because it is much simpler structurally and has fewer parameters to be estimated. Controllers of low orders are also more easily designed based on this model. State feedback control for secondorder processes augmented with an integrator readily leads to a PID controller. As such, we have in eq 14 a process model which can adequately capture the dynamics of a rich and general class of linear systems encountered in the industry.
b2′ u(k - 1 - h) b1′ (1 + a1′ + a2′) ysp (18) b1′
In what follows, however, we will adopt a general formulation. When eq 17 is formulated using a statespace description, it follows
( )(
e(k) 0 1 0 e(k + 1) ) -a2′ -a1′ 0 0 1 1 θ(k + 1)
Defining
(
)( ) ( )
e(k - 1) 0 e(k) + -b1′ u˜ (k - h) 0 θ(k)
) ( )
( )
e(k - 1) 0 1 0 0 F ) -a2′ -a1′ 0 , B ) -b1′ , X(k) ) e(k) 0 1 1 0 θ(k) it follows
X(k + 1) ) FX(k) + Bu˜ (k - h)
(19)
Following eq 11, we obtain the control law to yield optimum performance.
u˜ (k) ) -DX(k + h)
[
e(k - 1 + h) ) -[d1 d2 d3] e(k + h) θ(k + h)
]
) -[d1e(k - 1 + h) + d2e(k + h) + d3θ(k + h)] ) -(d1 + d2)e(k + h) + d1(e(k + h) e(k + h - 1)) - d3θ(k + h) (20) where D ) -H[ATQA + R]-1[ATQGF] ) -[d1, d2, d3]. When combined with eq 18, the control law is thus given by eq 21.
b2 ′ 1 + a1′ + a2′ u(k) ) - u(k - 1) + ysp b1 ′ b1 ′ e(k - 1 + h) [d1, d2, d3] e(k + h) θ(k + h)
[
]
(21)
This is equivalent to a PID control on the set point error with the PID gains given by
kp ) -(d1 + d2) ki ) - d3 kd ) d 1 Therefore, we have the final control law in eq 22
Ind. Eng. Chem. Res., Vol. 41, No. 5, 2002 1245
b2 ′ 1 + a1′ + a2′ u(k) ) - uk-1 + ysp + kpek+h + b1 ′ b1′ kd(ek+h - ek+h-1) + kiθk+h (22) This derivation of the control algorithm is applicable to first-order plant models as well, with the control law simplifying further to one with a PI control structure. 3.3.1. Selection of Weighting Matrices: Definitions of D. The selection of weighting matrices Q and R has a direct and significant effect on the system performance. The overall performance of the proposed control system may, therefore, be quite sensitive to the knowledge of the process operator or control engineer who may not appreciate of the implications of these weighting matrices. On the other hand, classical control specifications, such as the damping ratio and natural frequency, are already well-known and common concepts. To this end, it is useful to link the selection of weighting matrices to the desirable classical specifications (i.e., a direct relationship between the weighting matrices Q and R) and classical specifications such as the damping ratio ζ and natural frequency ωn of the closed loop system. These relationships are provided in this section. Consider eqs 1 and 11. When k g h, the closed loop equation is given as follows:
X(k + 1) ) (F - BD)X(k)
(23)
The closed loop characteristic equation is thus given by
det[zI - (F - BD)]
(
z -1 0 ) det a2′ - b1′d1 z - (b1′d2 - a1′) -b1′ 0 -1 z-1
)
) z + (a1′ - 1 - b1′d2)z + [a2′ - a1′ b1′(d1 - d2 + d3)]z + b1′d1 - a2′ (24) 3
2
D can be chosen to achieve a desired closed loop pole constellation. Under a dominant pole placement design, the third pole can be placed more than 4 times deeper in the left-hand plane so that the response is dominated by the two complex poles. The desired characteristic equation in s domain is thus given by
(s2 + 2ζω0s + ω02)(s + p1) ) 0
(25)
where ζ and ωn are the damping ratio and natural frequency specifications of the closed loop and p1 corresponds to the third pole. This is equivalent to eq 26 in the z domain.
z3 + (a1 - e-p1Ts)z2 + (a2 - a1e-p1Ts)z - a2e-p1Ts ) 0 (26) where
ω ) ω0x(1 - ζ2), R ) e-ζω0Ts, β ) cos(ωTs), γ ) sin (ωTs) a1 ) -2Rβ, a2 ) R2 Equating the coefficients of eqs 25 and 26, the elements of D may be obtained as follows in eq 27:
d1 )
a2′ - a2e-p1Ts a1′ - a1 - 1 + e-p1Ts , d2 ) , b1 ′ b1 ′ (1 + a1 + a2)(e-p1Ts - 1) (27) d3 ) b1 ′
3.3.2. Choice of ζ, ωn, and P1. The specification ζ relates to the damping ratio of the closed loop. For a well-damped response, generally ζ > 0.7 is desired. With ζ fixed, the choice of natural frequency ωn will determine the closed loop response speed. The following approximate relationship, between ζ, ωn, and the desired closed loop time constant τcl, provides a good guideline for the choice of ωn.
1 ≈ τcl ζωn The choice of ζ and ωn is essentially to achieve a compromise between the often conflicting control objectives of speed and stability. Finally, the third pole may be placed at least 4 times deeper into the left-hand plane so there is no dominant effect on the closed loop dynamics. P1 can thus be specified automatically according to
1 P1 ) r , r>4 ζωn 3.3.3. Q-D Relationship. With D computed according to eq 27, it is possible to obtain the equivalent GPC weighting matrix Q. This can serve as an initial weight set from which finer adjustments can be made for enhanced performance. The relationship between D, Q, and R is given as
()(
q1 β1 β2 β3 q2 ) λ1 λ2 λ3 q3 µ1 µ2 µ3
)( -1
d1
)
b1′(a2′ - b1′d1) R -d2 -d3
(28)
where the proof and various notations are given in Appendix A. 3.4. Robust Stability. The development so far has been based on a nominal scenario. Under modeling errors, the closed loop performance may degrade to an unacceptable level, and a robust design is needed which usually involves detuning of the control gains. Necessary and sufficient conditions for robust stability and performance of the Smith control systems can be found in refs 12 and 13. These conditions are similarly applicable to the proposed design in this paper. In this paper, we use a separate set of gains D h acting on the difference between current process variable and model prediction. D h may be designed subject to the robust stability and performance conditions.12,13 In this paper, we adopt a simple way where we simply detune the gain for the nominal case (i.e., D h ) γD, where γ < 1). The robust performance of the control scheme will also be illustrated via simulation and experiment on systems with a dissimilar structure compared to the model so that both parametric as well as nonparametric errors are present. 4. Simulation In this section, several simulation examples, involving processes of different types of dynamics, are given to
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Ind. Eng. Chem. Res., Vol. 41, No. 5, 2002
Figure 2. Simulation model.
Figure 4. Control performance for second-order system: (1) 10% overestimate in deadtime, (2) 10% underestimate in deadtime.
Figure 3. Control performance for second-order system: (1) proposed method, (2) the method of ref 14, and (3) the method of ref 15.
demonstrate the applicability of the optimal Smith control system. Simulation is done in Matlab (R11) Simulink. The generic simulation model is given in Figure 2. System nonlinearities, such as actuator saturation, have also been applied to the model. In the simulations, the user specifications r and γ are fixed at r ) 10 and γ ) 0.1. 4.1. Second-Order Process Model. In this example, we will compare the closed loop performance of three Smith predictor controller systems, tuned using the proposed method and the methods documented in refs 14 and 15. Consider a second-order system described by eq 29.
P(s) )
1 e-10s s2 + 2s + 1
(29)
The step response, and a subsequent 10% input load response, of the system is given in Figure 3. The specifications are ωn ) 1 and ξ ) 2, and a mismatched deadtime of 10.1 is simulated in all three cases. The superior performance of the proposed design method is clearly evident. It is known that the Smith predictor is most sensitive to mismatch in the deadtime of the process. To illustrate the robust performance, a 10% deadtime mismatch is simulated next. Figure 4 shows the simulation results for a 10% deviation on both sides of the nominal values. 4.2. High-Order Characteristics. Consider a highorder process (eq 30)
P(s) )
1 (s + 1)n
Figure 5. Control performance for high-order system.
(30)
with n ) 10. The parameters ω0 and ζ were set to be 2
Figure 6. Real-time response.
and 1, respectively. The control performance is shown in Figure 5. 5. Real-Time Application A real time experiment, with the developed control system, is applied to a time-delay dynamics emulator (hardware). This emulator is designed to simulate a variety of process dynamics (selectable by the user) encountered in the industry, including stable, unstable, and time-delay dynamics. The input and output of the emulator are interfaced to the PC via a standard ADDA data acquisition card. The user specifications for the experiment are r ) 10, ωn ) 1, and ξ ) 1. The real-time step response with a step disturbance is shown in Figure 6.
Ind. Eng. Chem. Res., Vol. 41, No. 5, 2002 1247
6. Conclusions An optimal Smith predictor has been developed based on a GPC approach. The basic idea is to back-calculate the Smith control parameters based on the GPC control law, which has been designed to minimize a cost function. The design has turned out to be very simple to use. With a second-order model, the primary controller is simply a PID control. With a first-order model, it is a PI controller. The relationship between the proposed GPC design and one based on a pole-placement approach is derived and provided. Simulation and a real time experiment have verified the good performance of the thus-designed optimal Smith control system. Appendix A (Proof of Q-D Relationship) In eq 29, D is given in terms of matrices G, F, A, R, and Q. The state weighting matrix Q has a form given in eq 31. The size of Q depends on the choice of prediction horizon, p.
( )
q1 0 0 Q) 0 l 0 0
(
Q0 0 ) l 0
0 0 ‚‚‚ 0 q2 0 0 q3 l · · · q l l 1 0 0 0 ‚‚‚ 0
0 Q0 l ‚‚‚
‚‚‚ ‚‚‚ · · · ‚‚‚
0 0 l Q0
)
( ) n
p-1
(FT)nQ0kFn ∑ n)1
(
)
(31)
‚‚‚ 0 l · l · · ‚‚‚ Q0
][ ]
(F )
(n) (n) f k2 f k3 ∑ n)1
×
)
×
(35)
(
k)1
×
3
(n) (n) k2 f k1 )
k
n)1
p-1
∑ q (∑f k
k)1
×
3
(n) (n) k2 f k2 )
+ q2
n)1
p-1
∑q ( ∑ f k
k)1
×
(n) (n) k2 f k3 )
n)1
×
)
(36)
( )
× × × ) R β γ × × ×
3
p-1
∑ n)1 ∑ k)1 qk(
3
β)
p-1
(n) (n) f k2 f k1 ) ) q1
p-1
∑ n)1 ∑ k)1 qk (
p-1
q2(1 +
Q0 F
T 2
∑ n)1
p-1
(n) f (n) 12 f 11 + q2
(n) f (n) ∑ 22 f 21 + n)1
(n) f (n) ∑ 32 f 31 ) R1q1 + R2q2 + R3q3 n)1 p-1
(n) (n) f k2 f k2 ) + q2 ) q1
p-1
∑ n)1
3
2
+ Q02 + F Q02F + (F ) Q02F + Q02Fp-1 + Q03 + FTQ03F +
(FT)2Q03F2 + ‚‚‚ + (FT)p-1Q03Fp-1 (33) where
p-1
(n) (n) f k2 f k2
p-1
) Q01 + FTQ01F + (FT)2Q01F2 + ‚‚‚ + ‚‚‚ + (F )
∑ n)1
q3
I F l Fp-1
T p-1
T p-1
×
p-1
∑q ( ∑ f
R)
) Q0 + FTQ0F + (FT)2Q0F2 + ‚‚‚ +
Q01F
∑ n)1
(n) (n) f k2 f k1
×
×
3
(32)
Expanding the common factor of ATQA, it follows
(F )
p-1
where
) -[BTATQAB + R]-1[BTATQAF]
T
×
p-1
×
D ) -[(AB)TQ(AB) + R]-1[(AB)TQGF]
p-1
×
ATQA )
G, F, and A are constant matrices in terms of only process parameters. Consider D in eq 32. The first part is a scalar, and the second part is a 1 × 3 vector.
T p-1
(
) FTQ0kF + (FT)2Q0kF2 + ‚‚‚ + (FT)p-1Q0kFp-1
Equation 33 may thus be expressed as
q1 0 0 where Q0 ) 0 q2 0 0 0 q3
[
)
In eq 34, f (n) ij is the ith row and j th column element of Fn. Therefore, it follows
) qk
3p×3p
Q0 0 0 Q0 ATQA ) [I, FT, ‚‚‚, (FT)P-1] l 9 ‚‚‚
(
× × × (n) (n) (n) (n) (n) (n) (F ) Q0kF ) qk f k2 f k1 f k2 f k2 f k2 f k3 , × × × for k ) 1, 2, 3 (34) T n
l
0 0 q2 0 0 q3
( )
Consider a general term
0 0 l
( )
q1 0 0 0 0 0 0 0 0 q Q01 ) 0 0 0 , Q02 ) 0 2 0 , Q03 ) 0 0 0 0 0 q3 0 0 0 0 0 0
γ)
p-1
(n) f (n) 22 f 22 ) + q3
(n) f (n) ∑ 32 f 32 ) β1q1 + β2q2 + β3q3 n)1
p-1
∑ n)1 ∑ k)1 qk (
(n) f (n) ∑ 12 f 12 + n)1
p-1
(n) (n) f k2 f k3 ) ) q1
∑ n)1
p-1
(n) f (n) 12 f 13 + q2
(n) f (n) ∑ 22 f 23 + n)1
p-1
q3
(n) f (n) ∑ 32 f 33 ) γ1q1 + γ2q2 + γ3q3 n)1
Hence, the two components in D may be expressed as those in eq 37a,b.
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Ind. Eng. Chem. Res., Vol. 41, No. 5, 2002
[ ][
1 0 × × × 0 BTATQAF ) [0, - b1′, 0] R β γ -a2′ -a1′ 0 × × × 0 1 1 ) -b1′(-a2′R, R - a1′β + γ, γ)
[ ][ ]
]
(37a)
× × × 0 BTATQAB ) [0, -b1′, 0] R β γ -b1′ × × × 0 ) b1′ 2β (37b) When eqs 37a and 37b are combined, it follows
b1′[-a2′β, R - a1′β + γ, γ] ) (d1, d2, d3) (38) D)b1′ 2β + R Equation 38 expressed in terms of q1, q2, and q3 gives
β1q1 + β2q2 + β3q3 )
d1R b1′(a2′ - b1′d1)
[b1′(R1 + γ1) + (d2b1′ 2 - a1′)β1]q1 + [b1′(R2 + γ2) + (d2b1′ 2 - a1′)β2]q2 + [b1′(R3 + γ3) + (d2b1′ 2 - a1′)β3]q3 ) -d2R (b1′γ1 + d3b1′ 2β1)q1 + (b1′γ2 + d3b1′ 2β2)q2 + (b1′γ3 + d3b1′2β3)q3 ) -d3R Using a matrix formulation, we have
(
)( )
(
d1R β1 β2 β3 q1 λ1 λ2 λ3 q2 ) b1′(a2′ - b1′d1) -d2R µ1 µ2 µ3 q3 -d3R
)
where λi ) b1′(Ri + γi) + (d2b1′ 2 - a1′)βi and µj ) b1′γj + d3b1′ 2βj for i, j ) 1, 2, 3. Thus, the weighting factors are given by
()(
q1 β1 β2 β3 q2 ) λ1 λ2 λ3 q3 µ1 µ2 µ3
)(
The proof is completed.
-1
d1
)
b1′(a2′ - b1′d1) R -d2 -d3
(39)
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Received for review May 17, 2000 Revised manuscript received May 31, 2001 Accepted November 27, 2001 IE000498D
