Closed-Loop Disturbance Identification and Controller Tuning ... - ASQ

Closed-Loop Disturbance Identification and Controller Tuning for Discrete Manufacturing Processes Enrique del Castillo Department of Industrial & Manufacturing Engineering The Pennsylvania State University FTC 2002

Overview 1 Open-loop vs. closed-loop operation 2 Transfer function ID in open loop 3 Problems under closed-loop operation 4 Closed-loop ID approach 5 Re-tuning the controller 6 Examples 7 References

1. Open Loop vs. closed-loop operation • One “input” {Ut} and one “output” {Yt} (”SISO”). • Open loop operation:

⇒ Ut not a function of et, et−1, .... • Closed loop operation:

⇒ Ut = f (et, et−1, ...). A feedback controller.

2. Transfer Function ID in Open Loop (Box, Jenkins, and Reinsel, 1994, Ch. 11). • Ut = ∇mUt0, et = ∇me0t stationary. • Assume Box-Jenkins model form: et =

Br (B) k B Ut + Nt = H(B)Ut + Ψ(B)εt As(B)

where Nt is ARIMA(p,d,q). • H(B) = v0 +v1B+v2B2 +... is the impulse response function • Identification goal: find (r,s,k) and (p,d,q) • Problem: ID under closed-loop conditions • Benefit: ad-hoc controller can at least avoid large scrap during ID keeping a stable process.

2. Transfer Function ID in Open Loop (cont.)

• Use cross-correlation between prewhitened input (αt) and the filtered output (wt). • Prewhiten input and filter output: Aα(B) Ut = αt ( white) α B (B) Aα(B) et = wt (not necessarely white) B α(B) • Transfer function is: Aα(B) Aα(B) Aα(B) et = H(B) α Ut + α Nt B α(B) B (B) B (B) or wt = H(B)αt + ε∗t • Get impulse response weights from covariance: E[αt−j wt] = E[αt−j vj Bj αt]

+

E[αt−j ε∗t ] |

2 ⇒ v = = γαw (j) = vj σα j

{z =0

}

σw γαw (j) = ρ (j) αw 2 σα σα

3. Problems under closed-loop operation

• Suppose Ut = C(B)et. From et = H(B)Ut + Nt get Ut =

C(B) Nt 1 − H(B)C(B)

⇒ Ut strongly depends on Nt • Furthermore, αt = C(B)wt, or wt =

1 αt = (c00 + c01B + c02B2 + ...)αt C(B)

and γαw (j) = E[αt−j wt] = E[αt−j c00αt] + 2 + E[αt−j c01αt−1] + ... = c0j σα ⇒ ραw (j) = c0j σσwα (ραw (0) 6= 0 implies feedback). • We get the inverse of the controller TF, not the process TF. • Estimation problems also occur.

4. Closed loop ID approach

• Use a “dither” signal (Box and MacGregor, 1974): Ut = C(B)et + dt {dt} uncorrelated with {εt} • Look at ρde(j). How large Var(dt)? • An alternative: look at the autocorrelation function of et. • “If nothing whatever were known about H(B) or about Ψ(B), then unambiguous id would not in general be possible” (Box and MacGregor, 1974).

Assumed process and disturbance models • Consider a discrete-part manufacturing process • Assumed process is: et = βUt−1 + Nt a “responsive” process, no process dynamics, noise dynamics assumed given by Nt = δ + Nt−1 − θεt−1 + εt,

θ≤1

(a possible non-invertible, IMA(1,1) with drift process) • Particular cases:

θ=0 0 < |θ| < 1 θ=1

δ=0 Random walk (RW) IMA(1,1) White noise

δ 6= 0 RW with drift, (RWD) IMA(1,1) with drift Deterministic trend plus noise (DT)

Assumed feedback adjustment method

• ∇Ut = c1et + c2et−1, (“PI” controller) • If c1 = −λ/b = −λ/βˆ, c2 = 0 get an “EWMA” controller: at Ut = − b at = λ(et − bUt−1) + (1 − λ)at−1, (an integral (”I”) controller).

0≤λ≤1

Closed loop output description under PI adjustments • Under the assumed process and controller, closed loop output follows an ARMA(2,1): (1 − (1 + βc1)B − βc2B2)(et − µe) = (1 − θB)εt where the mean or offset of the deviations from target is given by µe =

δ . 1 − (1 + βc1) − βc2

• Reduces to an ARMA(1,1) process if an EWMA (I) controller is used instead (c2 = 0). • Unambiguous ID from: Disturbance White noise DT RW RWD IMA(1,1) IMA(1,1) with drift

θ 1 1 0 0 0 < |θ| < 1 0 < |θ| < 1

Asymp. Offset=µe 0 δ/(1 − φ1 − φ2 ) 0 δ/(1 − φ1 − φ2 ) 0 δ/(1 − φ1 − φ2 ))

In all cases, φ1 = 1 + βc1 , φ2 = βc2 .

4. Proposed closed loop ID approach

1. Use (non-optimal) PI controller with known c1, c2. 2. Fit ARMA(2,1) process and use Table 2 to ID process; watch out for θ = 1 case (cancellation of polynomials). 3. Re-tune PI controller; balance MSD of the output and variance of the input.

5. Re-tuning the PI controller Solve: min J = MSD(et)/σε2 + π Var(∇ut)/σε2 (1) c ,c 1 2

subject to

|βc2| < 1 β(c2 − c1) < 2 β(c2 + c1) < 2.

where MSD(et) =Var(et) + µ2 e and 2 Var(∇Ut) = (c2 1 + c2 )Var(et) + 2c1 c2 cov(et.et−1 )

(Box and Luce˜ no, 1995). Spreadsheet in: www.ie.psu.edu/people/faculty/castillo/research.htm

6. Examples 1. A Simulated Machining Process. True noise: δ = 1, σε2 = 1, θ = 0.5 True Process: β = 1 Controller: c1 = −0.3, c2 = 0. • Closed loop output has φ1 = 1 + βc1 = 0.7, φ2 = βc2 = 0, θ = 0.5 (ARMA(1,1)). • From first 75 parts get: ˆ1 = 0.6062(0.3229); ⇒ βˆ = 1.31; φ θˆ = 0.3829(0.3750); ⇒ try IMA(1,1) µ ˆe = 4.42(0.769); ⇒ δˆ = 1.7425 significant drift (IMA(1,1) with drift). • Optimization:

π 0.0 0.5 1.0 2.5 5.0 10

Var(∇ut)/σε2 0.31 0.22 0.185 0.135 0.1039 0.0885

MSD(et)/σε2 1.13 1.15 1.18 1.26 1.37 1.46

c1 -0.5571 -0.4588 -0.4070 -0.3285 -0.2704 -0.2391

c2 -0.0712 -0.0925 -0.0993 -0.1032 -0.1005 -0.0969

Suppose we adopt the solution for π = 5. Reduced MSD thanks to better centering.

6. Examples (cont.) 2. A semiconductor manufacturing process. • Application of EWMA (I) control to a CMP process by Sematech (Hurwitz, 1996). • Response: removal rate (target= 1900˚ A/min)); known to drift • Controllable factor: platen speed (rpm’s) • Initial control with an EWMA with λ = 0.1. • From first 50 wafers: βˆ = 145.6, θˆ = 0.5511(0.1755) ˆ1 = 0.9006(0.0921); σ φ ˆε = 31.26.. • θˆ < 1 so we infer IMA(1,1) • µe − 41.8(5.8) implies significant drift; δˆ = −4.18. • Optimize:

π × 103

λ∗

Var(∇ut) × 104

MSD(et)

0 1 10 100 1000 10000

0.5424 0.5270 0.4447 0.2809 0.1511 0.0772

0.1402 0.1320 0.0931 0.0389 0.0133 0.0051

1.0714 1.0722 1.0904 1.2848 2.0996 4.9237

σε2

σε2

• Current setting (λ = 0.1) implied too severe limitation in the variance of the adjustments; process severely off-target; drift not compensated for.

What about if there are process dynamics? • Continuous-type manufacturing • Same controller and disturbance as before • If H(B) =

Bs(B) k B Ar (B)

then for r = 1, 2, s = 1, 2, and k = 1, 2, unambiguous ID possible from autocorrelation of the output and stability conditions (Pan and Del Castillo, 2001). • Second part of this (Technometrics) paper.

7. References Box, G.E.P., and MacGregor, J.F., (1974), “The Analysis of Closed-loop Dynamic-Stochastic Systems,” Technometrics, 16,3, pp. 391-398. Box, G.E.P., and Luce˜ no, A. (1995), “Discrete Proportional Integral Adjustment with Constrained Adjustment,” The Statistician, 44, 4, pp. 479-495. Del Castillo, E., (2002), “Closed Loop Disturbance Identification and Controller Tuning for Discrete Manufacturing Processes,” Technometrics, 44, 2. Pan, R., and Del Castillo, E. (2001), “Identification and fine tuning of closed loop processed under discrete EWMA and PI adjustments,” Quality & Reliability Engineering International, 17, pp. 419-427.

Questions?