Performance Synthesis of Multiple Inputâ Multiple Output (MIMO

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Ind. Eng. Chem. Res. 2011, 50, 1400–1409

Performance Synthesis of Multiple Input-Multiple Output (MIMO) Exponentially Weighted Moving Average (EWMA) Run-to-Run Controllers with Metrology Delay Richard Good Aptima, Inc., Woburn, Massachusetts 01801, United States

S. Joe Qin* Department of Chemical Engineering and Materials Science, UniVersity of Southern California, Los Angeles, California 90089, United States

The increased demand on throughput in semiconductor manufacturing often leads to a lag between the processing and metrology of wafers at a manufacturing step. This metrology lag results in a detrimental impact on the performance of a process under run-to-run control. This paper analyzes the performance of processes under exponentially weighted moving average (EWMA) run-to-run control by considering timedelayed closed-loop processes with mismatch between the process and the process model. The closed-loop performance bounds for EWMA controlled processes with delays of 0-2 runs are derived using modified stability analysis tools. Several simulations are provided to illustrate the importance of considering metrology delay and plant-model mismatch explicitly in controller performance and tuning. 1. Introduction Because of tool aging and disturbances in upstream unit operations, semiconductor manufacturing processes can drift from their quality targets in the absence of corrective adjustments to tool settings. Since quality-control parameters in most semiconductor manufacturing processes cannot be measured in situ, changes in tool settings can only be made between process runs. These settings are usually set points for a system of single input-single output (SISO) proportional-integral-differential (PID) control loops on the process tool where there is little or no visibility to the real-time wafer processing. Since it is only possible to make changes to tool settings between process runs, these controllers are termed run-to-run (or run-by-run) controllers. The increased requirements on semiconductor quality has resulted in run-to-run control becoming an enabling manufacturing technology. Box and Jenkins were the first to recommend using the exponentially weighted moving average (EWMA) statistic for actively rejecting disturbances that are characterized by an IMA(1,1) process.1,2 The EWMA statistic was later applied to semiconductor manufacturing by Sachs et al.,3 where ex situ measurements from the previous batch of wafers were used to update the input recipe for the following batch of wafers. Since Sachs et al.’s pioneering work in the run-to-run control of semiconductor processes, run-to-run controllers have been successfully applied to many processes, including chemical mechanical planarization,4-6 chemical deposition,7 plasma etching,8,9 and photolithography.10,11 Like all controllers, run-to-run controllers have the potential for closed-loop instability. For this reason, many studies have been published concerning the conditions that guarantee closedloop stability of run-to-run controlled processes. Ingolfsson and Sachs first derived the closed-loop stability conditions for the SISO EWMA controller.12 The Ingolfsson and Sachs work was later extended to the multiple input-multiple output (MIMO) * To whom correspondence should be addressed. E-mail: sqin@ usc.edu.

system by Tseng, Chou, and Lee.13 The effects of metrology delay on EWMA control were first studied by Adivikolanu and Zafiriou for the SISO (and MISO) EWMA controllers, although an analytical solution was shown only for the case with a metrology delay of one run.14,15 The stability of the MIMO EWMA controller with metrology delay was studied by Good and Qin.16 Adivikolanu and Zafiriou have shown that techniques similar to those used to study closed-loop stability can be used to derive closed-loop performance criteria.14,15 However, they derived the performance boundaries for the SISO case only and did not consider the effects of metrology delay. References 17-19 show that metrology delay and measurement strategy have great impact on the performance of the control results. The aim of this paper is to derive the closed-loop performance conditions for the synthesis of the MIMO EWMA controller with metrology delay. This will be achieved using the methodology developed by Good and Qin for studying the closed-loop stability of the MIMO EWMA controller.16 The outline of the paper is as follows. Section 2 provides an introduction to the EWMA run-to-run controller. Section 3 derives the conditions for meeting closed-loop performance requirements for the SISO EWMA controller. The MIMO EWMA performance conditions are then derived in Section 4. Finally, some concluding remarks are given in Section 5. 2. The EWMA Run-to-Run Controller Because of its simplicity and robustness, the EWMA controller is the most common run-to-run controller found in semiconductor manufacturing. This controller is composed of two elementary components: an observer and a control law. The EWMA observer assumes that the process output (y) is the linear combination of the process inputs (u), a deterministic offset (ν), and uncorrelated noise (ε):20 yt ) βut + ν + εt

10.1021/ie1008053  2011 American Chemical Society Published on Web 10/01/2010

(1)

Ind. Eng. Chem. Res., Vol. 50, No. 3, 2011

where β is the process gain between the manipulated input and the measured output. The function of the EWMA filter is to estimate the process offset from process data. Since the process can drift over time, it is reasonable to assume that ν is only constant in a local sense. Therefore, ν is estimated by weighting more-recent observations heavier than older observations, i.e., older observations are exponentially discounted. The updating equation for estimating ν, according to an EWMA filter, is νˆ t ) ωνˆ t-1 + (1 - ω)(yt - but)

(2)

where νˆ t is the estimate of νt, b is the estimated process model gain (determined a priori through a designed experiment or historical data), and ω is a discount coefficient chosen by the process engineer to be within a range of 0-1. When the value of ω is close to 1, the estimate of ν is updated very slowly; conversely, when the value of ω is zero, only the most-recent process measurement is considered in estimating ν. After the EWMA filter has estimated the process offset, a control law is used to determine the tool settings (or recipe) for the following run. In the unconstrained SISO case, the recipe is determined through simple model inversion: ut+1 )

T - νˆ t b

(3)

where T is the process target. For the MIMO EWMA controller, the gain matrix (b) has the dimensions n × m, where n is the number of measured quality parameters and m is the number of inputs. When b is nonsquare (i.e., n * m) an inverse of b is not attainable. Therefore, the MIMO control law can take several forms, depending on the objective function of the optimization problem. A few of the common unconstrained MIMO control laws are listed below: (1) Minimize the sum of the manipulated variables squared, subject to the model reaching the process target: T min J ) ut+1 ut+1 ut+1

(4)

subject to T ) but+1 + νˆ t ut+1 ) bT(bbT)-1(T - νˆ t) (2) Minimize the sum of the change in the manipulated variables squared, subject to the model hitting the process target:13 T min J ) ∆ut+1 ∆ut+1 ut+1

(5)

subject to T ) but+1 + νˆ t ut+1 ) bT(bbT)-1(T - Vˆ t) + (bT(bbT)-1b - I)ut (3) Minimize the sum of squares deviation from the target: ut+1

subject to yˆt+1 ) but+1 + νˆ t ut+1 ) (bTb)-1bT(T - Vˆ t)

T T min J ) (T - yˆt+1)TQ(T - yˆt+1) + ut+1 Rut+1 + ∆ut+1 S∆ut+1 ut+1

(7) subject to yˆt+1 ) but+1 + νˆ t ut+1 ) (bTQb + R + S)-1(Sut + bTQ(T - νˆ t)) The first two control laws, shown in eqs 4 and 5, are applied when the number of outputs exceeds the number of inputs. In this case, there are an infinite number of control inputs that will bring the process to the expected target, T. It follows that an objective function must be defined to establish a criteria with which to choose the “best” controller input. The first objective function is to minimize the sum of squared controller inputs. The solution is similar to the SISO case in eq 3, where the inverse of b is replaced by the right pseudo-inverse of b, bT(bbT)-1. The second objective function is the minimization of the sum of squared changes in the controller input. The solution to this objective function (eq 5) is similar to the first solution in eq 4 with an additional term, (bT(bbT)-1b - I)ut, which penalizes the change from the previous controller input. The third control law (eq 6) is used when the number of outputs exceeds the number of inputs. In this case, there exists no controller inputs that will bring the process to the expected target. The objective function in this case is to minimize the sum of squared deviations of the expected output from the target. The solution is similar to the solution in eq 4, where the left pseudo-inverse of b, (bTb)-1bT, is used in place of the right pseudo-inverse. The final control law (eq 7) is more general, because the objective is to find the optimal balance between missing the process target, the absolute controller input, and the change in the controller input from the previous run. The control law in eq 7 can be made to return equivalent results as the other three control laws by selecting the appropriate values of Q, R, and S.6 For instance, by choosing a large Q, R ) I, and S ) 0, the solution to the MPC formulation in eq 7 is identical to the solution in eq 4. 3. Closed-Loop Performance of the SISO EWMA Controller In this section, the closed-loop performance criteria for the SISO EWMA controller is derived. Section 3.1 derives the performance boundaries for the SISO system with and without metrology delay, and a simulation illustrating these performance conditions is shown in Section 3.2. 3.1. SISO Performance Criteria. It was shown in ref 16 that the characteristic equation of the closed-loop SISO EWMA controller is f(z) ) zd+1 - zdω - (1 - ξ)(1 - ω)

21

T min J ) yˆt+1 yˆt+1

(4) Model predictive control (MPC) formulation:

(6)

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6,22

(8)

where ξ is the ratio of the true process gain (β) and the estimated gain (b): ξ)

β b

It is well-known that the closed-loop system is stable when all of the roots of the characteristic equation (eq 8) are inside the unit circle. Furthermore, requirements on closed-loop perfor-

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To determine the conditions that guarantee that all of the roots of eq 11 meet the performance requirement, the inside of the performance region is mapped to the left half plane (LHP), using the following bilinear transformation: z)F

( 11 -+ ww )

(12)

so that solving for the performance region boundaries is reduced to finding the conditions where the roots for w are in the LHP. After the conformal mapping in eq 12, the characteristic equation of a process with a delay of one run becomes Figure 1. The performance region is defined as a circle with a radius of F < 1 on the complex plane.

mance can be established by solving for the conditions where the roots of eq 8 are constrained to a region smaller than the unit circle.14,15 This region is illustrated in Figure 1 and will be referenced as the guaranteed performance region. In other words, the controller error is guaranteed to decay with the rate of Ft, where t is the number of control steps. The procedure for determining the EWMA closed-loop performance boundaries will be demonstrated by considering systems with metrology lags of zero, one, and two runs. 3.1.1. Performance Region of a System without Metrology Delay. For a process without metrology delay, eq 8 becomes f(z) ) z - ω - (1 - ξ)(1 - ω)

(9)

Since the characteristic equation in eq 9 is linear in z, all of the roots will lie on the real axis. Therefore, it is easy to see from eq 9 that the process will meet the performance criteria when -(F - 1) F+1 eξe 1-ω 1-ω

(10)

The performance boundaries for F ) 0.2, ..., 1.0 are shown in Figure 2. When F ) 1, the performance region is equivalent to the stable region, so that eq 10 is the same as the closed-loop stability boundary derived in ref 12. As expected, Figure 2 shows that, for a process with no plant-model mismatch (i.e., ξ ) 1), the highest performance run-to-run controller is achieved when the discounting factor (ω) is close to zero. Figure 2 also shows that, for an underestimated model gain (ξ g 1), the controller tends to be too aggressive. Therefore, it is necessary to increase ω to improve the controller’s performance (thus making the controller less aggressive). This is a reasonable result, since, with a large ξ, the controller will make large control moves based on an inaccurate process model, resulting in poor performance. It follows that, by increasing ω, the controller will make less aggressive control moves, which leads to better performance. Likewise, Figure 2 shows that, for a small value of ω, it is necessary to decrease the plant-model mismatch to achieve a desired closed-loop performance. For a conservative choice of ω (large ω), ξ must be large to be more aggressive. It is the value of (1 - ω)ξ that determines the performance of the controller. 3.1.2. Performance Region of a System with a Metrology Delay of One Run. For a process with a metrology delay of one run, eq 8 becomes f(z) ) z2 - zω - (1 - ξ)(1 - ω)

(11)

f(w) ) a2w2 + a1w + a0

(13)

where a2 ) F2 + ωF + ξ + ω - ξω - 1 a1 ) 2F2 + 2 - 2ξ - 2ω + 2ξω a0 ) F2 - ωF - ξω + ω + ξ - 1 The roots of eq 13 are in the LHP when all of the coefficients on w have the same sign. Solving this system of inequalities yields the performance criterion, 1-

F2 F(F - ω) eξe1+ 1-ω 1-ω

(14)

which is equivalent to the stability criterion in refs 14 and 15 when F ) 1. The performance regions for F ) 0.2, ..., 1.0 are shown in Figure 3. Figure 3 shows that, unlike the undelayed case shown in Figure 2, performance is limited when a metrology lag is introduced into the system. For the undelayed case, as long as the process gain is underestimated (i.e., ξ > 1), any performance requirement can be attained by choosing an appropriate ω. For the delayed case, on the other hand, achievable performance is limited by the plant-model mismatch. For example, when ξ ) 1.8, the process will have at least one closed-loop pole greater than 0.4, irrespective of the choice of ω. We now consider the achievable performance for a given model mismatch (ξ) and ω. Rearranging eq 14 leads to -F2 e (1 - ω)(1 - ξ) e F(F - ω)

(15)

For an overestimated gain (ξ < 1), the controller is guaranteed stable for any ω. However, eq 15 mandates that F g ω, which means that the closed-loop performance, in this case, is limited by the choice of ω. For a given ω value and ξ < 1, eq 15 requires that Fg

ω + √ω2(1 - ξ)(1 - ω) 2

(16)

which gives the exact lower bound for F. For an underestimated gain (ξ > 1), eq 15 can be rewritten as -F(F - ω) e (ξ - 1)(1 - ω) e F2

(17)

which gives a lower bound for F as

(

min √(ξ - 1)(1 - ω),

ω + √ω2 - (ξ - 1)(1 - ω) 2

)

(18)

This is the performance limitation for a given value of ω and ξ > 1.


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Figure 2. Performance region of the SISO EWMA system without metrology delay.

Figure 3. Performance region of the SISO EWMA system with a metrology delay of one run.

3.1.3. Performance Region of a System with a Metrology Delay of Two Runs. By extending the delay to two runs, the characteristic equation in eq 8 becomes

a3 ) F3 + ωF2 - ω - ξ + ξω + 1

f(z) ) z3 - z2ω - (1 - ξ)(1 - ω)

a0 ) F3 - ωF2 + ξ - ξω + ω - 1

and after mapping the inside of the performance region to the LHP, the characteristic equation becomes f(w) ) a3w3 + a2w2 + a1w + a0 where

(19)

a2 ) 3F3 + ωF2 + 3ω + 3ξ - 3ξω - 3 a1 ) 3F3 - ωF2 - 3ω - 3ξ + 3ξω + 3

The conditions where all of the roots are in the LHP are obviously more complicated with systems with a metrology lag of more than one run, because a third-order polynomial must be solved. However, several techniques exist that can be used to determine when a system has all of its roots in the LHP without actually solving for the roots of the polynomial. These

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Figure 4. Performance region of the SISO EWMA system with a metrology delay of two runs.

include Jury’s test, the Routh array technique, the Routh-Hurwitz stability criterion, and the Lienard-Chipart stability criterion.23 Although all four techniques will presumably arrive at the same performance regions, the Routh-Hurwitz stability criterion is easily extendible to higher-order systems and will be used exclusively in this paper. The Routh-Hurwitz stability criterion states that a polynomial of the form q(s) ) apsp + ap-1sp-1 + · · · + a1s + a0 and with all real coefficients, has all of its roots in the LHP when all of the principal minors of the matrix

[

ap-1 ap-3 ap ap-2 ap-1 H≡ ap

ap-5 ap-4 ap-3 ap-2

ap-7 ap-6 ap-5 ap-4

l

l

··· ··· ··· ··· ·

·. ap-2 a0

]

are greater than zero. Further details on the Routh-Hurwitz stability criterion can be found in refs 24 and 25. Applying the Routh-Hurwitz stability criterion on eq 19 yields

[ ]

a2 a0 0 H ) a3 a1 0 0 a2 a0

and solving the system of inequalities, which guarantees that all of the principal minors of H are greater than zero, results in the closed-loop performance region, 1-

1 F2(ω - √ω2 + 4F2) F2(F - ω) eξe11-ω 2 1-ω

(20) which is shown in Figure 4.

Table 1. Maximum Distance from the Origin to the Closed-Loop Poles Maximum Distance from the Origin to the Closed-Loop Poles, F delay

ω ) 0.2

ω ) 0.5

ω ) 0.8

0 1 2

0.600 0.894 0.961

0.00 0.707 0.872

0.600 0.447 0.701

By comparing Figure 4 with Figures 2 and 3, we see that, not surprisingly, increasing the metrology delay further limits the performance of the controller. This same procedure can be used to determine the performance boundaries for systems with larger metrology delays. 3.2. Simulation of an EWMA-Controlled SISO Process. As an example, consider a system with T ) 0, ν ) 0, β ) 4.0 and with normally distributed noise with unit variance. Through a designed experiment or from process data, an approximation for the model gain is determined to be b ) 2.0, corresponding to a plant-model mismatch of ξ ) 2.0. We will look at the step responses of processes with metrology delays of zero, one, and two runs, and with discounting factors of ω ) 0.2, 0.5, and 0.8. The maximum distances from the origin of the system poles for the simulations are summarized in Table 1. Figure 5 shows the impact that metrology delay has on an EWMA-controlled process. The first column in Figure 5 shows the step response of a process without delay. In this case, ω ) 0.2 (top left) and ω ) 0.8 (bottom left) are on the F ) 0.6 iso-performance line. Although the poles of these two systems are an equal distance from the origin, the system with ω ) 0.2 is underdamped, so oscillations are observed. It could be argued that, because of these oscillations, ω ) 0.2 and ω ) 0.8 do not have equivalent performance. Nevertheless, damping of the step disturbances occurs at the same rate. It may be argued that ω ) 0.8 has the best overall performance. Simulations of a step response for systems with metrology delays of one and two runs are shown in the second and third columns of Figure 5, respectively. Systems with a metrology delay have an additional pole for each lag in the process


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Figure 5. Simulations of SISO EWMA controller with ξ ) 2; ω ) 0.2, 0.5, 0.8; and delays of 0, 1, and 2 runs.

metrology. These additional poles slow the response of the controller to disturbances, so that the performance regions are more accurately described as the worse-case performance regions. That is, the performance regions shown in Figures 3 and 4 show the boundaries of the maximum system pole, but they say nothing about the remaining poles. This is illustrated in Figure 5; although the process with d ) 1 and ω ) 0.5 (the middle plot) has a larger maximum pole than the process with d ) 2 and ω ) 0.8 (lower right), the two unaccounted for poles in the later process result in a slower response to the step disturbance. 4. Closed-Loop Performance of the MIMO EWMA Controller In this section, the performance criteria for the MIMO EWMA controllers depicted in eqs 4-7 are derived. The MIMO performance conditions for processes with and without metrology delay are considered in Section 4.1 and a simulation is provided in Section 4.2. 4.1. MIMO Performance Criteria. Good and Qin show that the necessary and sufficient condition for the stability of the MIMO EWMA controller is that all of the eigenvalues of the plant-model mismatch matrix, ξ must meet the SISO stability requirements.16 This result can be extended to the performance regions of the MIMO system. That is, closed-loop performance is guaranteed for the MIMO case when all of the eigenvalues of ξ fall inside the SISO performance regions. The plantmodel mismatch matrices for the four MIMO objective functions are provided below: (1) Minimize the sum-of-squared control inputs ξ ) βbT(bbT)-1 (2) Minimize the sum-of-squared change in control input ξ ) βbT(bbT)-1 (3) Minimize the sum-of-squared deviation from the target

ξ ) β(bTb)-1bT (4) MPC formulation with S ) 0 ξ ) (β - b)(bTQb + R)-1bTQ + I The closed-loop expression under the feedback laws expressed in eqs 4-7 is16 νˆ t+d+1 ) ωνˆ t+d + (1 - ω)(1 - ξ)νˆ t + (1 - ω)ν + εt+1 (21) with a characteristic equation zd+1I - zdω - (I - ξ)(I - ω) ) 0

(22)

where ω is a diagonal matrix containing the EWMA filter constants for each disturbance channel. Using the recursive leastsquares (RLS) argument in ref 16, the diagonal elements of ω can be chosen to be identical, which corresponds to the forgetting factor of the RLS. Representing ξ in the form ξ ) PΛP-1 where Λ ) diag {λ1, λ2, ..., λn} contains the eigenvalues of ξ, and using the fact that ω ) ωI, eq 22 can be rewritten as zd+1 - zd - (1 - λj)(1 - ω) ) 0

∀j ) 1, 2, ..., n

(23) To ensure stability, Good and Qin16 used the above relationship to obtain the MIMO EWMA stability conditions. To ensure performance, we need the roots of the above equations within a radius of F < 1 for all j ) 1, 2, ..., n. Similar to the SISO performance case, the bilinear transform (eq 12) can be applied to eq 23, which gives rise to

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[F( 11 -+ ww )]

d+1

[ ( 11 -+ ww )]

d

- F

- (1 - λj)(1 - ω) ) 0

the SISO case with a delay of one run (eq 13) and substituting the values w ) iw and λ ) a + bi yields the complex polynomial

(24) Although the plant-model mismatch matrix ξ is always real, the eigenvalues (λj) can be complex. Unfortunately, the Routh-Hurwitz criterion is only valid for polynomials with all real coefficients. To determine if the roots of a polynomial with complex coefficients are all in the LHP, Frank26 and Gantmakher27 independently introduced methods to extend the Routh-Hurwitz criterion to polynomials with complex coefficients. According to Gantmakher’s Generalized RouthHurwitz stability criterion, a polynomial of the form

R{f(iw)} ) a2w2 + a1w + a0 I{f(iw)} ) b2w2 + b1w + b0 where a2 a1 a0 b2

∇2n

[

bp bp-1 · · · ap ap-1 · · · bp bp-1 ) 0 ap ap-1 0 ··· ··· ···

bo ao ··· ··· ···

0 0 bo ao ···

]

are greater than zero. The Generalized Routh-Hurwitz criterion will be used to determine the MIMO performance regions for systems with metrology delays of 0-2 runs. 4.1.1. Performance Region of a MIMO System without Metrology Delay. Starting with eq 23 for a system without metrology delay and substituting λj ) a + bi and w ) wi yield the complex polynomial R{f(wi)} ) b(1 - ω)w + F - 1 + a(1 - ω) I{f(wi)} ) (F + 1 - a(1 - ω))w + b(1 - ω) so that the ∇2n matrix is ∇2n )

(1 - a)(1 - ω) + F(F + ω) 2b(1 - ω) (1 + a)(1 - ω) + F(F - ω) -b(1 - ω)

b1 ) 2(F2 + (1 - a)(1 - ω)) b0 ) b(1 - ω)

f(iw) ) (ap + bpi)wp + (ap-1 + bp-1i)wp-1 + · · · + (a0 + b0i)

will have all of its poles in the LHP when the determinants of all of the principal minors of the matrix,

) ) ) )

and the ∇2n matrix is

∇2n

[ ]

b2 a2 ) 0 0

b1 a1 b2 a2

b0 a0 b1 a1

0 0 b0 a0

Solving the system of inequalities such that the principal minors of ∇2n are greater than zero yields the performance regions shown in Figure 7. Figure 7 shows that when eigenvalues of the plant-model mismatch matrix are complex, the closed-loop performance degrades significantly. For instance, Figure 7 shows that, for a ) 0, ..., 0.5, and with imaginary part b g 0.2, there are no values of ω in the range of 0-1 that will guarantee that all of the closed-loop poles are inside a circle with a radius of F ) 0.4. 4.1.3. Performance a MIMO System with a Metrology Delay of Two Runs. The complex characteristic equation for a process with a delay of two runs is

[ ]

R{f(iw)} ) a3w3 + a2w2 + a1w + a0

b1 b0 a1 a0

I{f(iw)} ) b3w3 + b2w2 + b1w + b0

where

where

a0 ) b(1 - ω) b0 ) F + 1 - a(1 - ω) a1 ) F - 1 + a(1 - ω) b1 ) b(1 - ω) Solving the system of inequalities such that all of the principal minors are greater than zero yields the performance region, |λj | 2 - a - √F2 |λj | 2 - b2 |λj | 2

eωe

|λj | 2 - a + √F2 |λj | 2 - b2 |λj | 2

which is equivalent to the result in ref 13 when F ) 1. The complex performance regions for the EWMA controller without metrology delay are shown in Figure 6. This figure shows that the performance regions shrink and are pushed up and to the right as the complex part of λj increases. When the real part λj is close to unity, then the imaginary part of λj becomes more dominant. It follows that the real part of λj must be increased to reach satisfactory performance. However, as with the SISO case, a larger λj means that a less-aggressive forgetting factor is necessary to achieve a desired performance. Therefore, we observe the performance regions moving up and to the right with increasing complexity in λj. 4.1.2. Performance a MIMO System with a Metrology Delay of One Run. Starting with the characteristic equation of

a3 ) -b(1 - ω) a2 ) -3(a - 1)(1 - ω) - F2(3F + ω) a1 ) 3b(1 - ω) a0 ) (a - 1)(1 - ω) + F2(F - ω) b3 ) (a - 1)(1 - ω) - F2(F + ω) b2 ) -3b(1 - ω) b1 ) -3(a - 1)(1 - ω) + F2(3F - ω) b0 ) b(1 - ω)

[ ]

so that the ∇2n matrix is

∇2n

b3 a3 0 ) 0 0 0

b2 a2 b3 a3 0 0

b1 a1 b2 a2 b3 a3

b0 a0 b1 a1 b2 a2

0 0 b0 a0 b1 a1

0 0 0 0 b0 a0

Solving this system of inequalities such that the principal minors of ∇2n are greater than zero yields the performance regions shown in Figure 8.


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Figure 6. Complex performance regions for systems without a metrology delay (b ) 0.2, ..., 0.8).

Figure 7. Complex performance regions for a system with a metrology delay of one run (b ) 0.2, ..., 0.8).

4.2. Simulation of a MIMO EWMA Controlled Process. As an example of a process under MIMO EWMA control, consider a system with T ) [0 0]T, ν ) [0 0]T, β)

[

6 -3 8 16 14 10 -1 19

]

ξ)

and

[

2.3 -2.0 4.0 7.0 b) 7.2 4.8 -0.1 10.2

The process has normally distributed noise of the appropriate dimension with unit variance. A MIMO EWMA controller will be used with an objective function of minimizing the 2-norm of the controller input, so that

]

[

2.0 0.2 -0.2 2.0

]

and the eigenvalues of ξ are λ ) 2.0 ( 0.2 i. We wish to determine the values of ω that will guarantee that the closedloop eigenvalues are all inside the circle with a radius of F )

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Figure 8. Complex performance regions for a system with a metrology delay of two runs (b ) 0.2, ..., 0.8).

Figure 9. Simulation of a closed-loop MIMO system with λ ) 2.0 ( 0.2i, ω ) 0.90, and delays of 0, 1, and 2 runs.

Table 2. Values of ω Required To Meet the Performance Requirement in the MIMO EWMA Controlled Example for Varying Lengths of Metrology Delay Value of ω delay

lower bound

upper bound

0 1 2 3 4

0.110 0.405 0.703 0.859

0.900 0.911 0.921 0.929 no solution

0.8. According to the performance regions shown in Figures 6-8, the feasible values of ω are 0.110 e ω e 0.900 for no delay, 0.405 e ω e 0.911 for a delay of one, and 0.703 e ω e 0.921 for a delay of two. The feasible regions for delays of up to four runs are summarized in Table 2. Table 2 shows that a choice of ω ) 0.90 will achieve the desired performance for processes with delays of up to three runs. By increasing the metrology delay to four runs, however, we see that there are no values of ω that will meet this performance requirement.


The simulation is shown in Figure 9, where a step disturbance is introduced in the first output at run 10. Along with the simulations, a reference performance trajectory is included, which corresponds to λ ) 0.8. Figure 9 shows that all three simulations reject the step disturbance at a rate consistent with the reference trajectory, which indicates that the performance requirement is met. 5. Conclusions In this paper, the Routh-Hurwitz stability criterion was used to determine the tuning parameters of the exponentially weighted moving average (EWMA) controller that guarantee a minimum closed-loop performance. This was accomplished by making an extension to the closed-loop stability analysis of the EWMA controller. Although stability boundaries can be determined by mapping the unit circle to the left half plane (LHP), a minimum performance boundary can be determined by mapping a circle with a radius of F < 1 to the LHP. After this mapping step, the same Routh-Hurwitz stability criterion can be used to determine the conditions where all of the closed-loop poles are inside this smaller circle. The performance regions for EWMA-controlled systems with metrology delays of zero, one, and two runs were derived. This simple technique can be used to determine the performance conditions for systems with larger delays. The performance requirements for multiple-input-multipleoutput (MIMO) systems with several different objective functions were also derived in this paper. The condition for meeting the MIMO performance requirements is that all of the eigenvalues of the plant-model mismatch matrix must fall inside the performance boundaries. However, unlike the SISO case, the eigenvalues of this matrix can have an imaginary component. Therefore, complex performance regions were derived using the Generalized Routh-Hurwitz criteria. Several simulations were provided that showed the importance of considering plant-model mismatch and metrology delay explicitly in choosing the EWMA tuning parameters. We have shown that EWMA closed-loop performance is heavily dependent on model mismatch and metrology delay. There is a need to focus future work in this field in developing tuning rules that take into account metrology delay and model mismatch. In addition, the equations that describe systems with extended metrology delays and complex eigenvalues are complicated. Therefore, there is a need to develop simplified empirical formulas that describe the performance (and stability) boundaries for these complicated systems. Literature Cited (1) Box, G.; Jenkins, G. Bull. Int. Stat. Inst. 1963, 34, 943–974.

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(2) Box, G. E. P.; Jenkins, G. M. Time Series Analysis: Forecasting and Control; Holden-Day: Oakland, CA, 1974. (3) Sachs, E.; Guo, R.; Ha, S.; Hu, A. IEEE Trans. Semicond. Manuf. 1991, 4 (2), 134–143. (4) Boning, D.; Moyne, W.; Smith, T.; Moyne, J.; Telfeyan, R.; Hurwitz, A.; Shellman, S.; Taylor, J. IEEE Trans. Semicond. Manuf. 1996, 19 (4), 307–314. (5) Chen, A.; Guo, R. IEEE Trans. Semicond. Manuf. 2001, 14 (1), 11– 19. (6) Campbell, W. Model Predictive Run-to-Run Control of Chemical Mechanical Planarization, Ph.D. Dissertation, The University of Texas at Austin, 1999. (7) Smith, T.; Boning, D.; Stefani, J.; Butler, S. IEEE Trans. Semicond. Manuf. 1998, 11 (2), 276–284. (8) Hamby, E.; Kabamba, P.; Khargonekar, P. IEEE Trans. Semicond. Manuf. 1998, 11 (4), 654–669. (9) Jowett, P.; Morozov, V. IEEE/SEMI AdV. Semicond. Manuf. Conf. 2002, 107–112. (10) Edgar, T.; Campbell, W.; Bode, C. Proc. Conf. Decision Control 1999, 38, 4185–4191. (11) Bode, C. Run-to-Run Control of Overlay and Linewidth in Semiconductor Manufacturing, Ph.D. Dissertation, The University of Texas at Austin, 2001. (12) Ingolfsson, A.; Sachs, E. J. Quality Technol. 1993, 25 (4), 271– 287. (13) Tseng, S.; Chou, R.; Lee, S. IIE Trans. 2002, 34, 541–549. (14) Adivikolanu, S.; Zafiriou, E. Proc. Am. Control Conf. 1998, 6th, 3687–3691. (15) Adivikolanu, S.; Zafiriou, E. IEEE Trans. Electron. Packag. Manuf. 2000, 23, 56–68. (16) Good, R.; Qin, S. IEEE Trans. Semicond. Manuf. 2006, 19, 78– 86. (17) Su, A.-J.; Yu, C.-C.; Ogunnaike, B. A. J. Process Control 2008, 18, 266–276. (18) Su, A.-J.; Jeng, J.-C.; Huang, H.-P.; Yu, C.-C.; Hung, S.-Y.; Chao, C.-K. Control Eng. Pract. 2007, 15, 1268–1279. (19) Wu, M.-F.; Lin, C.-H.; Wong, D. S.-H.; Jang, S.-S.; Tseng, S.-T. IEEE Trans. Semicond. Manuf. 2008, 21, 413–425. (20) Sachs, E.; Hu, A.; Ingolfsson, A. IEEE Trans. Semicond. Manuf. 1995, 8, 26–43. (21) Castillo, E. D.; Rajagopal, R. IIE Trans. 2002, 34, 1055–1068. (22) Lewis, F.; Syrmos, V. Optimal Control, 2nd ed.; John Wiley and Sons: New York, 1995. (23) Leigh, J. Applied Digital Control: Theory, Design, and Implementation; Prentice-Hall International: Englewood Cliffs, NJ, 1984. (24) Hurwitz, A. Math. Annal. 1895, 46, 273–284. (25) Routh, E. J. Dynamics of a System of Rigid Bodies; Macmillan: New York, 1892. (26) Frank, E. Bull. Am. Math. Soc. 1946, 52, 144–157. (27) Gantmakher, F. R. Applications of the Theory of Matrices; Interscience Publishers: New York, 1959.

ReceiVed for reView April 2, 2010 ReVised manuscript receiVed August 20, 2010 Accepted August 31, 2010 IE1008053

Performance Synthesis of Multiple Inputâ Multiple Output (MIMO

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