1. 2/25/2001Industrial Process Control1 Dynamic Matrix Control - Introduction Developed at Shell in the mid 1970’s Evolved from representing process dynamics with a set of numerical coefficients Uses a least square formulation to minimize the integral of the error/time curve

2. 2/25/2001Industrial Process Control2 Dynamic Matrix Control - Introduction DMC algorithm incorporates feedforward and multivariable control Incorporation of the process dynamics makes it possible to consider deadtime and unusual dynamic behavior Using the least square formulation made it possible to solve complex multivariable control calculations quickly

3. 2/25/2001Industrial Process Control3 Dynamic Matrix Control - Introduction Consider the following furnace example (Cutlet & Ramaker) MV –Fuel flow FIC DV –Inlet temperatureTI CV –Outlet temperature TIC

4. 2/25/2001Industrial Process Control4 Dynamic Matrix Control - Introduction The furnace DMC model is defined by its dynamic coefficients Response to step change in fuel, a Response to step change in inlet temperature, b

5. 2/25/2001Industrial Process Control5 Dynamic Matrix Control - Introduction DMC Dynamic coefficents Response to step change in fuel, a Response to step change in inlet temperature, b

6. 2/25/2001Industrial Process Control6 Dynamic Matrix Control - Introduction The DMC prediction may be calculated from those coefficients and the independent variable changes

7. 2/25/2001Industrial Process Control7 Dynamic Matrix Control - Introduction Feedforward prediction is enabled by moving the DV to the left hand side

8. 2/25/2001Industrial Process Control8 Dynamic Matrix Control - Introduction Predicted response determined from current outlet temperature and predicted changes and past history of the MVs and DVs Desired response is determined by subtracting the predicted response from the setpoint Solve for future MVs -- Overdetermined system –Least square criteria (L2 norm) –Very large changes in MVs not physically realizable –Solved by introduction of move suppression

9. 2/25/2001Industrial Process Control9 Dynamic Matrix Control - Introduction Controller definition –Prediction horizon = 30 time steps –Control horizon = 10 time steps Initialization –Set CV prediction vector to current outlet temperature –Calculate error vector

10. 2/25/2001Industrial Process Control10 Dynamic Matrix Control - Introduction Least squares formulation including move suppression Move suppression

11. 2/25/2001Industrial Process Control11 Dynamic Matrix Control - Introduction DMC Controller cycle –Calculate moves using least square solution –Use predicted fuel moves to calculate changes to outlet temperature and update predictions –Shift prediction forward one unit in time –Compare current predicted with actual and adjust all 30 predictions (accounts for unmeasured disturbances) –Calculate feedforward effect using inlet temperature –Solve for another 10 moves and add to previously calculated moves

12. 2/25/2001Industrial Process Control12 Dynamic Matrix Control - Introduction Furnace Example Temperature Disturbance DT=15 at t=0 Three Fuel Moves Calculated

13. 2/25/2001Industrial Process Control13 Dynamic Matrix Control Basic Features Since 1983 Constrain max MV movements during each time interval Constrain min/max MV values at all times Constrain min/max CV values at all times Drive to economic optimum Allow for feedforward disturbances

14. 2/25/2001Industrial Process Control14 Dynamic Matrix Control Basic Features Since 1983 Restrict computed MV move sizes (move suppression) Relative weighting of MV moves Relative weighting of CV errors (equal concern errors) Minimize control effort

15. 2/25/2001Industrial Process Control15 Dynamic Matrix Control Basic Features Since 1983 For linear differential equations the process output can be given by the convolution theorem pcr: k > M M = number of time intervals required for CV to reach steady- state j = index on time starting at the initial time d = unmeasured disturbance pcr: k > M M = number of time intervals required for CV to reach steady- state j = index on time starting at the initial time d = unmeasured disturbance

16. 2/25/2001Industrial Process Control16 Dynamic Matrix Control Basic Features Since 1983 Breaking up the summation terms into past and future contributions pcr: Where, N = number of future moves M = time horizon required to reach steady state Note: the estimated outputs depend only on the N computed future inputs pcr: Where, N = number of future moves M = time horizon required to reach steady state Note: the estimated outputs depend only on the N computed future inputs

17. 2/25/2001Industrial Process Control17 Dynamic Matrix Control Basic Features Since 1983 Let N=number future moves, M=time horizon to reach steady state, then in matrix form

18. 2/25/2001Industrial Process Control18 Dynamic Matrix Control Basic Features Since 1983 Setting the predicted CV value to its setpoint and subtracting the past contributions, the “simple” DMC equation results pcr: Dynamic matrix A is size MxN where M is the number of points required to reach steady-state and N is the number of future moves pcr: Dynamic matrix A is size MxN where M is the number of points required to reach steady-state and N is the number of future moves

19. 2/25/2001Industrial Process Control19 Dynamic Matrix Control Basic Features Since 1983 To scale the residuals, a weighted least squares problem is posed For example, the relative weights with two CVs pcr: wi = relative weighting of the ith CV which will be repeated M times to form the diagonal weighting matrix W pcr: wi = relative weighting of the ith CV which will be repeated M times to form the diagonal weighting matrix W

20. 2/25/2001Industrial Process Control20 Dynamic Matrix Control Basic Features Since 1983 To restrict the size of calculated moves a relative weight for each of the MVs is imposed

21. 2/25/2001Industrial Process Control21 Dynamic Matrix Control Basic Features Since 1983 Subject to linear constraints –The change in each MV is within a “step” bound

22. 2/25/2001Industrial Process Control22 Dynamic Matrix Control Basic Features Since 1983 Subject to linear constraints –Size of each MV step for each time interval

23. 2/25/2001Industrial Process Control23 Dynamic Matrix Control Basic Features Since 1983 Subject to linear constraints –MV calculated for each time interval is between high and low limits

24. 2/25/2001Industrial Process Control24 Dynamic Matrix Control Basic Features Since 1983 Subject to linear constraints –CV calculated for each time interval is between high and low limits

25. 2/25/2001Industrial Process Control25 Dynamic Matrix Control Basic Features Since 1983 The following LP subproblem is solved –where the economic weights are know a priori Minimize Subject to:

26. 2/25/2001Industrial Process Control26 Dynamic Matrix Control Basic Features Since 1983 The original dynamic matrix is modified –Aij is the dynamic matrix of the ith CV with respect to the jth MV,