Ind. Eng. Chem. Res. 2002, 41, 2013-2022
2013
PID Control Design Based on a GPC Approach K. K. Tan,* T. H. Lee, S. N. Huang, and F. M. Leu Department of Electrical and Computer Engineering, National University of Singapore, 4 Engineering Drive 3, Singapore 117576
In this paper, we present the design of a PID controller that is applicable to commonly encountered process dynamics and that yields an optimum control performance according to a specified performance index. Generalized predictive control (GPC) design principles are used to develop the PID control with predictive capabilities. The user can specify an optimal performance index directly or, alternatively, classical performance measures in terms of a closed-loop damping ratio and natural frequency. Simulation examples are presented to highlight the principles and effectiveness of this control scheme. 1. Introduction Proportional-integral-derivative (PID) controllers have remained as the most commonly used controllers in industrial process control for more than 50 years, despite the advances in mathematical control theory. The main reason is that such controllers have a simple structure that is easily understood by engineers, and under practical conditions, they perform more reliably than more advanced and complex controllers. Over the years, numerous techniques have been suggested for tuning of the PID parameters. However, many of these techniques are applicable only to a limited class of processes. In ref 1, tuning rules are developed for processes with monotonic step responses that can be adequately described using a first-order-with-deadtime model. Most literature reports2-4 are also based on this model. A comparison of popular controller tuning techniques is given in ref 5. In ref 4, PID tuning for underdamped processes is considered. Independently, processes with integrating action are addressed in refs 6-9, and those exhibiting nonminimum phase characteristics are addressed in ref 10. Separate attention is focused on unstable processes with and without dead time in refs 11-19. PID control for servomechanisms are considered in ref 20, with a distinct difference from the same PID applied to process control. Furthermore, different variations of the PID control structure have to be configured for different processes. For example, for time-delay processes, usually only a PI controller structure is used, as the derivative action invokes an undesirable response. For processes with long time constants or integrating action, a PD structure can be used instead. With the many classes of PID designs and structures available, it has become necessary to determine the characteristics of the process/system before the appropriate class of design rules can be selected and applied. Furthermore, apart from the difference in the associated model structure, the basis and nature on which these tuning rules are derived also differ from one approach to another. The prespecifications required for commissioning the control system are accordingly also different.
Another well-known constraint associated with fixedgain conventional PID control is that the control structure rapidly loses its effectiveness when applied to processes with more complex dynamics, such as those with long dead times or poorly damped or unstable dynamics. Under these circumstances, more complex control structures become necessary, such as using a dead-time compensator for time-delay processes. This paper presents a new PID control design based on the application of generalized predictive control (GPC) to a general linear process model that is representative of many classes of processes encountered in the industry. The proposed PID control design, with the same tuning rules and same prespecifications, can be used and applied to low-order processes, high-order processes, underdamped processes, unstable processes, inverse-responding processes, integrating processes, time-delay processes, and various other combinations of the aforementioned characteristics. In addition, the design is intended to yield an optimum tracking control performance according to a performance index. The main idea is based on back-calculating an equivalent set of PID parameters from a GPC control law derived using a second-order general process model. In this way, an optimal controller is developed that has a simple and desirable PID structure but that can yield a level of performance expected from GPC. One less-desirable prerequisite and possible impedance to the application of a GPC solution is the need to select the weight matrices, which might not be easily and effectively accomplished by the average control engineer. To overcome this difficulty, relationships are derived that map the weighting parameters to the classical pole-placement type of control specifications in terms of desired natural frequency and damping ratio of the closed-loop system. Thus, specifications can be made in terms of more intuitive specifications, and an initial set of equivalent GPC weights can be derived for either further fine-tuning or direct use. The paper also discusses the disturbance rejection and regulation aspects of the proposed PID control design. Simulation examples are given to highlight the general applicability of the design method. 2. General Process Model
* To whom correspondence should be addressed. Tel.: 658742110. Fax: 65-7791103. E-mail:
[email protected].
Consider the general second-order discrete-time transfer function
10.1021/ie010480i CCC: $22.00 © 2002 American Chemical Society Published on Web 03/23/2002
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Ind. Eng. Chem. Res., Vol. 41, No. 8, 2002
G′(z) )
b′1z + b′2 2
z-h
z + a′1z + a′2
(1)
Letting e ) ysp - y, the equivalent time-domain difference equation of eq 1 is given in eq 2 with ysp ) 0
e(k+2) ) -a′1e(k+1) - a′2e(k) - b′1u(k+1-h) b′2u(k-h) (2) The model described by eq 2 encompasses a rich and general class of time-continuous linear systems given by
ds + c G(s) ) e-sL (s + a)(s + b)
(3)
The relationships between the time-discrete and timecontinuous model parameters can be obtained using various conversion techniques (e.g., Tustin with or without frequency prewarping, zero-pole mapping, etc.). In particular, we assume the relationship between L and h to be h ) L/Ts. We will subsequently explore, using the simulations in section 6, the sensitivity of this assumption to an over- or underestimation in the amount of actual time delay present. Note that the two poles of G(s) can, in general, be complex, so that the model can adequately describe processes with the following characteristics and their combinations: overdamped dynamics, underdamped dynamics, time-delay dynamics, minimum and nonminimum phase dynamics, unstable dynamics, and integrating dynamics. Although the difference equation model in eq 2 is constrained with a maximum order of two, it is now well-known and generally acknowledged, especially in the process-control community, that a second-order model is adequate for most applications, including those involving high-order process dynamics. Even if a higherorder model is available, the second-order one is usually still preferred as it is much simpler structurally and has fewer parameters to be estimated. Controllers of low orders are also more easily designed using this model. As such, we have in eq 2 a process model that can adequately capture the dynamics of a rich and general class of linear systems encountered in the industry. From eq 2, a shift in the time origin results in
e(k+1) ) -a′1e(k) - a′2e(k-1) - b′1u˜ (k-h)
(4)
where
u˜ (k-h) ) u(k-h) +
b′2 u(k-1-h) b′1
We will apply a GPC design approach based on this process model to derive the general control law. Then, using this GPC control law, we will back-calculate the equivalent set of PID control parameters. In this way, we aim to incorporate certain advanced control aspects related to GPC into a PID control structure. Many references on GPC are available. Interested readers might want to refer to refs 21-27.
Formulating eq 4 using a state-space description, one obtains
( )(
)( ) ( )
e(k) 0 1 0 e(k-1) 0 e(k+1) ) -a′2 -a′1 0 e(k) + -b′1 u˜ (k-h) 0 0 1 1 θ(k) θ(k+1)
where
k-1
θ(k) ) Defining
(
e(i) ∑ i)0
) ( )
0 1 0 0 F ) -a′2 -a′1 0 , B ) -b′1 , 0 0 1 1
( )
e(k-1) and X(k) ) e(k) θ(k)
it follows that
X(k+1) ) FX(k) + Bu˜ (k-h)
(5)
Using m controls and p output prediction horizons, the future outputs X(k+l) can be obtained recursively as follows (where p g m)
X(k+1) ) FX(k) + Bu˜ (k-h) X(k+2) ) F2X(k) + FBu˜ (k-h) + Bu˜(k+1-h) l X(k+p-1) ) Fp-1X(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) Combining the equations, we have the following augmented matrix description
X h ) GFX(k) + AU ˜
(6)
where
X h ) (XT(k+1) XT(k+2) ‚‚‚ XT(k+p) )T
() (
I F G) l Fp-1
B FB A) l Fp-1B
0 B l Fp-2B
‚‚‚ ‚‚‚ ‚‚‚ ‚‚‚
0 0 0 Fp-mB
)
U ˜ ) (u˜ (k-h) u˜ (k+1-h) ‚‚‚ u˜ (k+m-1-h) )T 3. GPC Control Design Consider the quadratic cost function p
J)
m
2 2 ||X(k+l)||Q(l) + ∑||u˜ (k+j-1)||R(j) ∑ l)1 j)1
(7)
Ind. Eng. Chem. Res., Vol. 41, No. 8, 2002 2015
where Q is the state weighting matrix and R is the control weighting matrix. The main idea of a GPC design is to derive a series of m controls u˜ (k), ..., u˜ (k+m-l) at each sample time so that the cost function is minimized. Substituting eq 6 into eq 7, the cost function can be rewritten as
J ) [GFX(k) + AU ˜ ]TQ[GFX(k) + AU ˜] + U ˜ TRU ˜
(8)
Substituting X(h) ) F h-1X(1) according to eq 12a, we have
X(1+h) ) (F - BD)Fh-1X(1) which proves eq 13a for k ) 0, 1. We assume next that eq 13a holds for k ) j < h - 1, i.e.
X(j+h) ) (F - BD)jFh-jX(j)
The solution that minimizes the cost function J can be obtained by solving
∂J ˜ ] + 2RU ˜ )0 ) 2ATQ[GFX(k) + AU ∂U ˜
(9)
X(j+1+h) ) (F - BD)X(j+h) ) (F - BD)(F - BD)jFh-jX(j)
Thus, the optimal control sequence is given as
U ˜ ) -[ATQA + R]-1[ATQGF]X(k)
(10)
Under the principle of receding horizons, only the first value of the optimal control sequence is output at each sampling time step. Thus, eq 10 can be rewritten as T
-1
We will prove that eq 13a then also holds for k ) j + 1 < h.
T
u˜ (k-h) ) -H[A QA + R] [A QGF]X(k) ) -DX(k) (11)
Now, X(j) ) F-1X(j + 1) according to eq 12a. Therefore
X(j+1+h) ) (F - BD)j+1Fh-(j+1)X(j+1) which proves eq 13a for k ) j + 1. Thus, the lemma is proved by induction. Denoting
[K1(k) K2(k) K3(k) ] ) D(F - BD)kFh-k
where
D ) -H[ATQA + R]-1[ATQGF] and H ) [1 0 ‚‚‚ 0 ]
the final control laws are given by
u˜ (k) ) -D(F - BD)kFh-kX(k) for 0 e k < h
It follows that
) -[K1(k) K2(k) K3(k) ]X(k) u˜ (k) ) -DX(k+h)
In eq 11, the current control value is a linear combination of the future predicted states in X(k+h). We need to predict the future states from the present ones to have a causal control law. From eqs 11 and 5, we have
X(k+1) ) FX(k), 0 e k < h
(12a)
X(k+1) ) (F - BD)X(k), k g h
(12b)
Lemma 1 extends the prediction to h steps. Lemma 1. The future states of the system described by eq 5 are given by
X(k+h) ) (F - BD)kFh-kX(k), 0 e k < h h
X(k+h) ) (F - BD) X(k), k g h
(13a)
) -[(K1(k) + K2(k))e(k) - K1(k)(e(k) e(k-1)) + K3(k)θ(k)] (14a) u˜ (k) ) -D(F - BD)hX(k) for k g h ) -[(K1(h) + K2(h))e(k) - K1(h)(e(k) e(k-1)) + K3(h) θ(k)] (14b) Thus, we have the following theorem: Theorem 1. For the system described by eq 2, the GPC control is given by
u˜ (k) ) KP(k) e(k) + KI(k) θ(k) + KD(k)(e(k) - e(k-1)) (15) where
(13b)
Proof. For k g h, one can obtain directly from eq 12b by recursion that
X(k+h) ) (F - BD)hX(k) which proves eq 13b. Equation 13a can be proved for 0 e k < h using an induction method. For k ) 0, from eq 12a, we have
X(h) ) FhX(0) For k ) 1, from eq 12b, we have
X(1+h) ) (F - BD)X(h)
KP(k) ) -(K1(k) + K2(k)) KI(k) ) -K3(k) KD(k) ) K1(k)
}
KP(k) ) -(K1(h) + K2(h)) KI(k) ) -K3(h) KD(k) ) K1(h)
}
and
0ek 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 closedloop 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 can 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 4.2. Q-D Relationship. With D thus computed according to the classical specifications, it will be interesting to know the equivalent GPC weights in Q. This relationship can yield an initial weight set from which finer adjustments to the GPC parameters can be made for enhanced performance. The relationship between D, Q, and R0 (from eq 18) is given in Theorem 2. Theorem 2. The equivalent GPC Q matrix (for the single-step control horizon case) is related to d1, d2, d3, and R by
()(
q1 κ1 κ2 κ3 q2 ) λ1 λ2 λ3 q3 µ1 µ2 µ3
)( ) -1
d1 -d2 R -d3
(24)
where the various notations are defined in Appendix A. Proof. See Appendix A Remark 5. The user can thus choose to provide classical control specifications in terms of ζ, ωn, and P1, and the equivalent weighting matrices can be computed according to eqs 23 and 24. The equivalent weights can be used directly to commission the PID control, or they can serve as initial values for subsequent direct finetuning of the weights by the operator. A summary of the PID control design is as follows: (1) The process model is known or identified, i.e., F, B, and h are known. (2) (a) The user can specify GPC weights and horizons Q, R, p, and m from which A and G can be computed. D can be directly computed from eq 11 or 18 (single-step control horizon) (b) Alternatively, the classical specifications ζ, ωo, and P1 can be used. D can be computed directly from eq 23. The equivalent GPC weights can be obtained, if desired, for fine-tuning via Theorem 2. (3) Given D from either step 2a or step 2b, the set of PID parameters can be computed from Theorem 1. 5. Disturbance Attenuation Disturbance signals are usually treated as extraneous signals in the input to the process. For time-delay processes, these signals are not evident in the process variable until at least h sampling intervals after the point at which they occur. If the disturbances are measurable, feed-forward control can be used to eliminate/reduce them directly. Otherwise, their effects are inevitable and unavoidable during this period of time. However, after the latent period, feedback control alone still might not respond sufficiently fast to quench the nearly full-blown effects of the disturbances, especially if the time delay is large. This usually results in a large overshoot/undershoot being incurred before rectification can even begin. Conventional GPC has a disturbance-tailoring filter that can be used as a second degree of freedom for tuning the controller, especially when the process is not
2018
Ind. Eng. Chem. Res., Vol. 41, No. 8, 2002
well-modeled. This filter is not included in the proposed PID control design, which is optimized mainly for setpoint tracking. With one degree of freedom in the control, a compromise in the performance of the trackingand regulatory functions of the PID can be achieved by adjusting the user specifications, ζ and ωo. Different and independent sets of specifications can also be used separately for the tracking and regulation purposes. Alternatively, a simple disturbance observer can be used. For example, if the disturbance signals do not change during the latent period, tracking and directly compensating for the disturbances can yield further performance improvement. From eq 4, considering an additional input disturbance signal d(k), we have
e(k+1) ) -a′1e(k) - a′2e(k-1) - b′1[u˜ (k-h) - d(k-h)] Figure 1. Simulated response for L ) 1 s with with ζ ) 2 and ω0 ) 8.
It follows that
d(k-h) )
e(k+1) + a′1e(k) + a′2e(k) + u˜ (k-h) b′
Thus, if d(k) ≈ d(k-h), then according to our assumption, the control law can be directly modified (u j (k) f u j m(k)) to better compensate for the disturbance on the basis of only the error and control signals, i.e., u j m(k) ) u j (k) - d(k-h). 6. Simulation In this section, several simulation examples involving processes of different dynamics are discussed to demonstrate the wide applicability of the control algorithm. Specifications are made in terms of the closed-loop damping factor ζ and the natural frequency ωn, which is more familiar to practising engineers. The equivalent GPC weights can be obtained via Theorem 2 if necessary. The common specification P1 ) 20 was assumed. In addition, the sampling time used was 1 ms. Additive white noise, with 0 mean and 0.1 variance, was also included to simulate measurement noise. 6.1. First-Order with Delay Characteristics. Consider an actual process given by
G(s) )
110.6315 -s e s + 105.2216
(25)
The closed-loop response and the control effort are shown in Figure 1. The response is compared to Ho’s gain phase margin method (GPM),33 with the gain and phase margins set to recommended values of Am ) 3 and φm ) 45°. A 10% input step disturbance was applied at time t ) 10 s. The variations of KP and KI are displayed in Figure 2. The next figure, Figure 3, shows the system response with a 10% underestimation in the dead time. Figure 4 shows the response with a 10% overestimation of the delay. These figures show the performance robustness to modeling error in the time delay. 6.2. Underdamped and Overdamped Characteristics. Consider the second-order process
G(s) )
ωp2 s2 + 2ζpωps + ωp2
Figure 2. Variation of KP and KI.
(26)
Figure 3. Response with 10% underestimation in plant delay.
with ωp ) 10 and ζp ) 0.5 and 5 to provide both underdamped and overdamped characteristics for simulation. The specifications ζ ) 2 and ωn ) 10 are assumed for both simulations. The responses are shown in Figure 5 and Figure 6. Note that the two closedloop responses are rather similar, although the open-
Ind. Eng. Chem. Res., Vol. 41, No. 8, 2002 2019
Figure 4. Response with 10% overestimation in plant delay.
Figure 7. Response and control effort for n ) 10 (ζ ) 5, ω0 ) 2).
respectively. The performance is compared to a PID controller tuned using the Ziegler-Nichols (ZN) formula. (GPM is applicable to first-order-with-dead-time models.) 6.3. High-Order Characteristics. Thus far, we have simulated for processes with a structure that is contained completely in the process model of eq 2. It is interesting to see how the control system with the same model as in eq 2 performs when faced with processes of dissimilar structures. This simulation will also show the robustness of the controller in the face of nonparametric modeling uncertainties. Consider the higher-order process
G(s) ) Figure 5. Simulated response for an underdamped system with ζp ) 0.5, ζ ) 2, and ωn ) 10, with the dotted line being the counterpart of the ZN-tuned controller.
1 (s + 1)n
(27)
with n ) 10. A system identification experiment is carried out to obtain the parameters of eq 2. The response and control effort are given in Figure 7. A 10% input step disturbance is added at time of 30 s in the simulation using the proposed control method. A comparison with the results from the application of the method of Smith and Corripio34 (suited for time-delay or high-order systems) is provided. 7. Conclusions
Figure 6. Simulated response for an overdamped system with ζp ) 5, ζ ) 2, and ωn ) 10, with the dotted line being the counterpart of the ZN-tuned controller.
loop damping ratio differs by a factor of 10. The PID gains for the underdamped and overdamped plants are (7.788, 0.01951, 485.3) and (8.819, 0.0204, -388),
This paper has addressed the design of a PID controller that is applicable to commonly encountered process dynamics and that yields an optimum control performance according to a specified performance index. Generalized predictive control (GPC) design principles were used to develop the PID control with predictive capabilities. The user needs to specify directly only an optimal performance index or, alternatively, classical performance measures in terms of closed-loop damping ratio and natural frequency. Simulation examples were presented to highlight the principles and effectiveness of this control scheme. Appendix A. Proof of Q-D Relationship In eq 11, D is given in terms of the matrices G, F, A, R, and Q. The state weighting matrix, Q, has the form given in eq 28. The size of Q depends on the
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Ind. Eng. Chem. Res., Vol. 41, No. 8, 2002
( )
choice of prediction horizon, p, via
0 0 ‚‚‚ 0 q2 0 0 q3 l ‚‚‚ q1 l l 0 0 0 ‚‚‚ 0
q1 0 0 Q) 0 l 0 0
(
Q0 0 ) l 0 where
‚‚‚ ‚‚‚ ‚‚‚ ‚‚‚
0 Q0 l ‚‚‚
0 0 l Q0
)
Consider a general term
0 0 l
l
0 0 q2 0 0 q3
(28)
(F T + ... + I)nQ0k(F n + ... + I) ) × × × (n) (n) (n) (n) (n) (n) qk f k2 for k ) 1, 2, 3 (31) f k1 f k2 f k2 f k2 f k3 × × ×
(
)
In eq 31, f (n) ij represents the element in the ith row and jth column of Fn + ... + I. Therefore, it follows that
3pX3p
p-1
(F p + ... + I)TQ0k(F n + ... + I) ) ∑ n)1
(
q1 0 0 Q 0 ) 0 q2 0 0 0 q3
(
)
qk
D ) -[(A0B)TQ(A0B) + R]-1[(A0B)TQGF] ) -[BTA0TQA0B + R]-1[BTA0TQGF]
(
p-1
∑ ∑ qk (
k)1
(n) (n) f k2 f k1 )
[
T
][
I F+I l F P-1 + ‚‚‚ + I 2
]
(29)
( )
∑ n)1 ∑ k)1
p-1
∑ ∑f qk (
k)1
(n) k2
(n) f k3 )
n)1
×
)
(33)
p-1
qk(
3
β)
T
p-1
(n) (n) f k2 gk1 ) ) q1
∑ n)1
p-1
(n) f (n) 12 g11 + q2
(n) f (n) ∑ 22 g21 + n)1
∑ n∑ k)1 )1 ∑ n)1
(30)
p-1
(n) (n) f k2 gk2 ) + q2 ) q1
+ ... + I) Q01
(F p-1 + ... + I) where
( )
q1 0 0 0 0 0 0 0 0 Q01 ) 0 0 0 , Q02 ) 0 q2 0 , Q03 ) 0 0 0 0 0 q3 0 0 0 0 0 0
g(n) 12 + q2(1 +
∑ n)1 ∑ k)1
g(n) 32 ) β1q1 + β2q2 + β3q3
p-1
(n) (n) f k2 gk3 ) ) q1
∑ n)1
p-1
(n) f (n) 12 g13 + q2
(n) f (n) ∑ 22 g23 + n)1
p-1
q3
+ Q02 + (F + I)TQ02(F + I) + (F 2 + F + (F 2 + F + I) + ‚‚‚ + (F p-1 + ... + I)TQ03
f (n) ∑ 32 n)1
p-1
qk (
(Fp-1+ ... + I)
+ Q03 + (F + I)TQ03(F + I) + (F 2 + F + I)TQ03
f (n) ∑ 12 n)1
p-1
(n) f (n) 22 g22 ) + q3
3
γ)
(n) f (n) ∑ 32 g31 ) R1q1 + R2q2 + R3q3 n)1
p-1
qk (
p-1
T
( )
(32)
p-1
) Q01 + (F + I)TQ01(F + I) + (F 2 + F + I)TQ01
( )
3
(n) (n) f k2 f k2 ) + q2
n)1
q3
(Fp-1 + ... + I) (F + F + I) + ‚‚‚ + (F
∑ ∑ qk (
× × × ) R β γ × × ×
(F 2 + F + I) + ‚‚‚ + (Fp-1 + ... + I)TQ0
p-1
)
×
×
) Q0 + (F + I) Q0(F + I) + (F + F + I) Q0
2
×
p-1
k)1
3
Q0 0 ‚‚‚ 0 0 Q0 l l ‚‚‚ l 0 ‚‚‚ ‚‚‚ Q0
×
3
n)1
×
R)
+ ‚‚‚ + I) ]
×
×
3
A0TQA0 ) [I(F + I) ‚‚‚ (F
× p-1
∑ f k2(n) f k1(n) n)1 ∑ f k2(n) f k2(n) n)1 ∑ f k2(n) f k3(n)
n)1
×
where
T
p-1
A0TQA0 )
Expanding the common factor of A0TQA0, it follows that
P-1
×
p-1
Equation 30 can thus be expressed as
G, F, and A are constant matrices in terms of only process parameters. Consider D in eq 29. The first part is a scalar, and the second part is a 1 × 3 vector.
T
×
(n) f (n) ∑ 32 g33 ) γ1q1 + γ2q2 + γ3q3 n)1
Similarly, A0TQG can be expressed as
(
A0TQG ) ×
×
3
p-1
∑q ( ∑ f k
k)1
×
n)1
×
3
(n) k2
(n) gk1 )
p-1
∑q ( ∑ f k
k)1
×
( )
× × × j β h γ j ) R × × ×
n)1
3
(n) k2
(n) gk2 ) + q2
p-1
∑q ( ∑ f k
k)1
×
n)1
(n) k2
(n) gk3 )
)
Ind. Eng. Chem. Res., Vol. 41, No. 8, 2002 2021
where g(n) ij represents the element in the ith row and jth column of Fn and 3
R j)
p-1
p-1
where
κi ) b′1a′2β h i - d1b′12βi
p-1
(n) (n) (n) ∑ f (n) ∑ qk(n)1 ∑ f k2(n) gk1(n)) ) q1n)1 12 g11 + q2 ∑ f 22 g21 + n)1
k)1
λi ) b′1(R ji + γ j i + d2b′1β h i - a′1βi)
p-1
q3 3
β h)
(n) f (n) j 1q1 + R j 2 q2 + R j 3q3 ∑ 32 g31 ) R n)1
p-1
p-1
(n) ∑ qk(n)1 ∑ f k2(n) gk2(n)) + q2 ) q1n)1 ∑ f (n) 12 g12 + q2(1 + k)1 p-1
p-1
(n) (n) (n) f (n) h 1q1 + β h 2 q2 + β h 3q3 ∑ 22 g22 ) + q3 ∑ f 32 g32 ) β n)1 n)1 3
γ j)
p-1
p-1
p-1
(n) f (n) j 1q1 + γ j 2q2 + γ j 3q3 ∑ 32 g33 ) γ n)1
Hence, the two components of D can be expressed as
[ ][
1 0 × × × 0 BTA0TQGF ) [0 -b′1 0 ] R j β h γ j -a′2 -a′1 0 × × × 0 1 1
]
(34a)
j R j - a′1β h+γ j γ ) -b′1(-a′2R j)
[ ][ ]
× × × 0 BTATQAB ) [0 -b′1 0 ] R β γ -b′1 × × × 0 2 ) b′1 β
(34b)
Combining eqs 34a and 34b, it follows that
h R j - a′1β h+γ j γ j] b′1[-a′2β ) (d1 d2 d3 ) D)2 b′1 β + R
(35)
Expressing eq 35 in terms of q1, q2, and q3, one obtains
h 1 - d1b′12β1)q1 + (b′1a′2β h 2 - d1b′12β2)β2q2 + (b′1a′2β h 3 - d1b1′2β3)q3 ) d1R (b′1a′2β [b′1(R j1 + γ j 1 + d2b′1β1 - a′1β h 1)]q1 + [b′1(R j2 + γ j2 + d2b′1β2 - a′1β h 2)]q2 + [b′1(R j3 + γ j 3 + d2b′12β3 h 3)]q3 ) -d2R a′1β j 1 + d3b′12β1)q1 + (b′1γ j 2 + d3b′12β2)q2 + (b′1γ j3 + (b′1γ d3b′12β3)q3 ) -d3R Using a matrix formulation, we have
(
for i ) 1, 2, 3. Thus, the weighting factors are given by
()(
q1 κ1 κ2 κ3 q2 ) λ1 λ2 λ3 q3 µ1 µ2 µ3
)( ) -1
d1 -d2 R -d3
(36)
p-1
(n) (n) (n) ∑ qk(n)1 ∑ f k2(n) gk3(n)) ) q1n)1 ∑ f (n) 12 g13 + q2 ∑ f 22 g23 + k)1 n)1
q3
µi ) b′1γ j i + d3b′12βi
)( ) ( )
κ1 κ2 κ3 q1 d1R λ1 λ2 λ3 q2 ) -d2R -d3R µ1 µ2 µ3 q3
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Received for review May 31, 2001 Revised manuscript received January 9, 2002 Accepted January 29, 2002 IE010480I
