1. Environmentally Conscious Design & Manufacturing (ME592) Date: May 8, 2000 Slide:1 Environmentally Conscious Design & Manufacturing Class 26: Data Dependent Systems Prof. S. M. Pandit

2. Environmentally Conscious Design & Manufacturing (ME592) Date: May 8, 2000 Slide:2 Autoregressive Model - AR(1) AR(1) model: Estimation of parameters:

3. Environmentally Conscious Design & Manufacturing (ME592) Date: May 8, 2000 Slide:3 Checks of Adequacy To check the dependence of a t on a t-1

4. Environmentally Conscious Design & Manufacturing (ME592) Date: May 8, 2000 Slide:4 Prediction or Forecasting One step ahead prediction at time t-1: One step ahead prediction error: Conditional distribution: The probability limits (e.g. 95 %):

5. Environmentally Conscious Design & Manufacturing (ME592) Date: May 8, 2000 Slide:5 Conditional Distribution of X t

6. Environmentally Conscious Design & Manufacturing (ME592) Date: May 8, 2000 Slide:6 Autoregressive Model - AR(2) AR(2) model: or

7. Environmentally Conscious Design & Manufacturing (ME592) Date: May 8, 2000 Slide:7 Estimate of Parameters Use the collected data: Then,

8. Environmentally Conscious Design & Manufacturing (ME592) Date: May 8, 2000 Slide:8 Estimate of Parameters (cont.) The conditional least square estimates of

9. Environmentally Conscious Design & Manufacturing (ME592) Date: May 8, 2000 Slide:9 Extension of AR (2) In practice, it is enough to use AR model in many applications. The AR model can also be extended to multiple inputs single output. For one input- one output, model takes the form In general form:

10. Environmentally Conscious Design & Manufacturing (ME592) Date: May 8, 2000 Slide:10 Estimates of Parameters Use the collected data: The conditional least square estimates of

11. Environmentally Conscious Design & Manufacturing (ME592) Date: May 8, 2000 Slide:11 Estimates of Parameters (cont.) Use the collected data:

12. Environmentally Conscious Design & Manufacturing (ME592) Date: May 8, 2000 Slide:12 Estimates of Parameters (cont.) The conditional least square estimates of

13. Environmentally Conscious Design & Manufacturing (ME592) Date: May 8, 2000 Slide:13 Example A model fitted to the data of system A for the purpose of developing a maintenance strategy is given below. where X 2t is the measured pH and X 1t is the caustic added. a). Express the model in transfer function form

14. Environmentally Conscious Design & Manufacturing (ME592) Date: May 8, 2000 Slide:14 b). Show a block diagram of the system including the above transfer functions? c). Determine the ratio of steady state output to steady state input. d).Given the following observations, determine the 95% probability limit on the 5th observation.

15. Environmentally Conscious Design & Manufacturing (ME592) Date: May 8, 2000 Slide:15 Backshift operator is defined by In general, Backshift Operator

16. Environmentally Conscious Design & Manufacturing (ME592) Date: May 8, 2000 Slide:16 Transfer Function Using B operator It follows

17. Environmentally Conscious Design & Manufacturing (ME592) Date: May 8, 2000 Slide:17 Block Diagram X 1t X 2t a 2t

18. Environmentally Conscious Design & Manufacturing (ME592) Date: May 8, 2000 Slide:18 Steady State When a system reaches a STEADY state, EX 2t =EX 2t-1 =…=EX 2t-n =say X 2s and EX 1t =EX 1t-1 =…=EX 1t-n =say X 1s i.e. B=1, and Ea t =0 since a t has zero mean. Therefore, taking the mean or expectation on both sides the model becomes i.e., same as obtained before substituting B=1 in the transfer function.

19. Environmentally Conscious Design & Manufacturing (ME592) Date: May 8, 2000 Slide:19 Prediction The prediction of X 2t at t=5 is given by

20. Environmentally Conscious Design & Manufacturing (ME592) Date: May 8, 2000 Slide:20 Probability Interval 95% probability interval on the observation That is

21. Environmentally Conscious Design & Manufacturing (ME592) Date: May 8, 2000 Slide:21 How to decide n? Increase n until does not reduce significantly.