Closed-Loop Disturbance Identification and Controller Tuning for Discrete Manufacturing Processes. Overview

Transcription

1 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

2 1. Open Loop vs. closed-loop operation One input {U t } and one output {Y t } ( SISO ). Open loop operation: U t not a function of e t, e t 1,... Closed loop operation: U t = f(e t, e t 1,...). A feedback controller.

3 2. Transfer Function ID in Open Loop (Box, Jenkins, and Reinsel, 1994, Ch. 11). U t = m U t, e t = m e t stationary. Assume Box-Jenkins model form: e t = B r(b) A s (B) Bk U t + N t = H(B)U t + Ψ(B)ε t where N t is ARIMA(p,d,q). H(B) = v 0 +v 1 B+v 2 B 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.

4 2. Transfer Function ID in Open Loop (cont.) Use cross-correlation between prewhitened input (α t ) and the filtered output (w t ). Prewhiten input and filter output: A α (B) B α (B) U t = α t ( white) A α (B) B α (B) e t = w t (not necessarely white) Transfer function is: or A α (B) B α (B) e t = H(B) Aα (B) B α (B) U t + Aα (B) B α (B) N t w t = H(B)α t + ε t Get impulse response weights from covariance: E[α t j w t ] = E[α t j v j B j α t ] + E[α t j ε t ] }{{} =0 = γ αw (j) = v j σ 2 α v j = γ αw(j) σ 2 α = ρ αw (j) σ w σ α

5 3. Problems under closed-loop operation Suppose U t = C(B)e t. From e t = H(B)U t + N t get U t = C(B) 1 H(B)C(B) N t U t strongly depends on N t Furthermore, α t = C(B)w t, or and w t = 1 C(B) α t = (c 0 + c 1 B + c 2 B2 +...)α t γ αw (j) = E[α t j w t ] = E[α t j c 0 α t] + + E[α t j c 1 α t 1] +... = c j σ2 α ρ αw (j) = c j σ α σ w (ρ αw (0) 0 implies feedback). We get the inverse of the controller TF, not the process TF. Estimation problems also occur.

6 4. Closed loop ID approach Use a dither signal (Box and MacGregor, 1974): U t = C(B)e t + d t {d t } uncorrelated with {ε t } Look at ρ de (j). How large Var(d t )? An alternative: look at the autocorrelation function of e t. 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).

7 Assumed process and disturbance models Consider a discrete-part manufacturing process Assumed process is: e t = βu t 1 + N t a responsive process, no process dynamics, noise dynamics assumed given by N t = δ + N t 1 θε t 1 + ε t, θ 1 (a possible non-invertible, IMA(1,1) with drift process) Particular cases: δ = 0 δ 0 θ = 0 Random walk (RW) RW with drift, (RWD) 0 < θ < 1 IMA(1,1) IMA(1,1) with drift θ = 1 White noise Deterministic trend plus noise (DT)

8 Assumed feedback adjustment method U t = c 1 e t + c 2 e t 1, ( PI controller) If c 1 = λ/b = λ/ˆβ, c 2 = 0 get an EWMA controller: U t = a t b a t = λ(e t bu t 1 ) + (1 λ)a t 1, 0 λ 1 (an integral ( I ) controller).

9 Closed loop output description under PI adjustments Under the assumed process and controller, closed loop output follows an ARMA(2,1): (1 (1 + βc 1 )B βc 2 B 2 )(e t µ e ) = (1 θb)ε t where the mean or offset of the deviations from target is given by µ e = δ 1 (1 + βc 1 ) βc 2. Reduces to an ARMA(1,1) process if an EWMA (I) controller is used instead (c 2 = 0). Unambiguous ID from: Disturbance θ Asymp. Offset=µ e White noise 1 0 DT 1 δ/(1 φ 1 φ 2 ) RW 0 0 RWD 0 δ/(1 φ 1 φ 2 ) IMA(1,1) 0 < θ < 1 0 IMA(1,1) with drift 0 < θ < 1 δ/(1 φ 1 φ 2 )) In all cases, φ 1 = 1 + βc 1, φ 2 = βc 2.

10 4. Proposed closed loop ID approach 1. Use (non-optimal) PI controller with known c 1, c 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.

11 Solve: 5. Re-tuning the PI controller min J = MSD(e c 1,c t )/σε 2 + π Var( u t )/σε 2 (1) 2 subject to βc 2 < 1 β(c 2 c 1 ) < 2 β(c 2 + c 1 ) < 2. where MSD(e t ) =Var(e t ) + µ 2 e and Var( U t ) = (c c2 2 )Var(e t) + 2c 1 c 2 cov(e t.e t 1 ) (Box and Luceño, 1995). Spreadsheet in:

12 6. Examples 1. A Simulated Machining Process. True noise: δ = 1, σ 2 ε = 1, θ = 0.5 True Process: β = 1 Controller: c 1 = 0.3, c 2 = 0. Closed loop output has φ 1 = 1 + βc 1 = 0.7, φ 2 = βc 2 = 0, θ = 0.5 (ARMA(1,1)). From first 75 parts get: ˆφ 1 = (0.3229); ˆβ = 1.31; ˆθ = (0.3750); try IMA(1,1) ˆµ e = 4.42(0.769); ˆδ = significant drift (IMA(1,1) with drift). Optimization:

13 π Var( u t )/σε 2 MSD(e t )/σε 2 c 1 c Suppose we adopt the solution for π = 5. MSD thanks to better centering. Reduced

14 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= 1900Å/min)); known to drift Controllable factor: platen speed (rpm s) Initial control with an EWMA with λ = 0.1. From first 50 wafers: ˆβ = 145.6, ˆθ = (0.1755) ˆφ 1 = (0.0921); ˆσ ε = ˆθ < 1 so we infer IMA(1,1) µ e 41.8(5.8) implies significant drift; ˆδ = Optimize:

15 π 10 3 λ Var( u t ) σ 2 ε 10 4 MSD(e t ) σ 2 ε Current setting (λ = 0.1) implied too severe limitation in the variance of the adjustments; process severely off-target; drift not compensated for.

16 What about if there are process dynamics? Continuous-type manufacturing Same controller and disturbance as before If H(B) = B s(b) A r (B) Bk 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.

17 7. References Box, G.E.P., and MacGregor, J.F., (1974), The Analysis of Closed-loop Dynamic-Stochastic Systems, Technometrics, 16,3, pp Box, G.E.P., and Luceño, A. (1995), Discrete Proportional Integral Adjustment with Constrained Adjustment, The Statistician, 44, 4, pp 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 Questions?