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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 2/25/2001 Industrial Process Control 1 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 2/25/2001 Industrial Process Control 2 Dynamic Matrix Control - Introduction • Consider the following Process furnace example Heater (Cutlet & Ramaker) Process Flow • MV – Fuel flow FIC Ti Fp • DV FIC TIC – Inlet temperatureTI • CV Fuel – Outlet temperature TIC 2/25/2001 Industrial Process Control 3 Dynamic Matrix Control - Introduction • The furnace DMC 0.014 0.0 0.086 0.240 model is defined by its 0.214 0.340 dynamic coefficients 0.414 0.465 • Response to step 0.600 0.540 change in fuel, a a , b 0.736 0.590 • Response to step 0.836 0.622 change in inlet 0.904 0.640 temperature, b 0.949 0.653 0.986 0.658 2/25/2001 Industrial Process Control 4 Dynamic Matrix Control - Introduction Fuel Coefficients ai • DMC Dynamic 1.5 coefficents 1 Fuel Coefficients ai 0.5 • Response to step 0 0 5 10 change in fuel, a • Response to step Inlet Temperature Coefficients bi change in inlet 1 Inlet Temperature 0.5 temperature, b Coefficients bi 0 0 5 10 2/25/2001 Industrial Process Control 5 Dynamic Matrix Control - Introduction • The DMC prediction may be calculated from those coefficients and the independent variable changes MV 1 CV 1 a1 0 0 b1 0 0 2 a MV 2 CV 2 a1 0 b2 b1 0 MV 3 CV 3 a3 a2 a1 b3 b2 b1 1 DV DV 2 CV i ai bi 2 ai 1 ai 2 bi bi 1 DV 3 2/25/2001 Industrial Process Control 6 Dynamic Matrix Control - Introduction • Feedforward prediction is enabled by moving the DV to the left hand side CV 1 b1 a1 0 0 2 a CV b2 2 a1 0 MV 1 2 CV 3 b3 DV 1 a3 a2 a1 MV MV 3 CV i bi ai ai 2 ai 1 2/25/2001 Industrial Process Control 7 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 2/25/2001 Industrial Process Control 8 Dynamic Matrix Control - Introduction • Controller definition CV 1 CV 0 – Prediction horizon = 30 time steps CV 30 CV 0 – Control horizon = 10 time steps • Initialization e1 SP1 CV 1 – Set CV prediction vector to current outlet e 30 SP 30 CV 30 temperature – Calculate error vector 2/25/2001 Industrial Process Control 9 Dynamic Matrix Control - Introduction • Least squares formulation including move suppression a1 0 0 0 a a1 0 2 a3 a2 a1 0 e1 0 2 a1 MV 1 e MV 2 10 e a a29 a28 a21 30 0 1,1 0 0 0 MV 30 0 Move 2 , 2 0 0 0 suppression 9,9 0 0 0 0 10,10 2/25/2001 Industrial Process Control 10 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 2/25/2001 Industrial Process Control 11 Dynamic Matrix Control - Introduction • Furnace Example Furnace Temperature Response • Temperature 15 625 Disturbance 10 Temperature DT=15 at t=0 5 • Three Fuel 0 600 -5 0 5 10 15 Moves -10 Fuel_0 Temp_0 Calculated -15 Fuel_1 575 Temp_1 Time Fuel_2 Prediction 2/25/2001 Industrial Process Control 12 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 2/25/2001 Industrial Process Control 13 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 2/25/2001 Industrial Process Control 14 pcr: k>M Dynamic Matrix Control M = number of time intervals required for CV Basic Features Since 1983 to reach steady- state • For linear differential equations the process outputon time j = index starting at the can be given by the convolution theorem initial time k d = unmeasured CV k 1 CV 0 ak j 1 MV j d k disturbance j 0 CV 1 CV 0 a1 MV 0 d 0 CV 2 CV 0 a2 MV 0 a1 MV 1 d 1 CV 3 CV 0 a3 MV 0 a2 MV 1 a1 MV 2 d 2 2/25/2001 Industrial Process Control 15 pcr: Where, Dynamic Matrix Control N = number of future moves Basic Features Since 1983 M = time horizon required to reach steady state • Breaking up the summation terms into past and the Note: estimated outputs future contributions depend only on k N 1 the N computed k l CV k l CV ak l j MV j k j future inputs k 1 k 1 CV CV a1 MV k k 2 k 2 CV CV a2 MV k a1 MV k 1 k 3 k 3 CV CV a3 MV k a2 MV k 1 a3 MV k 2 2/25/2001 Industrial Process Control 16 Dynamic Matrix Control Basic Features Since 1983 • Let N=number future moves, M=time horizon to reach steady state, then in matrix form a1 0 0 a a1 2 CV k 1 CV k 1 0 MV k k 2 k 2 CV CV a N a N 1 a1 MV k 1 k M k M CV CV aM aM 1 aM N 1 MV k N 1 aM aM aM 2/25/2001 Industrial Process Control 17 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 CV k 1 CV k 1 M is the number of points required to reach steady-state and N is the S k 2 k 2 number of future moves CVS CV e A ΔMV k M CVSk M CV 2/25/2001 Industrial Process Control 18 Dynamic Matrix Control Basic Features Since 1983 • To scale the residuals, a weighted least squares problem is posed 1 pcr: Min W 2 A ΔMV e 1 wi = relative weighting of 2 2 the ith CV which will be repeated M times to form w1 the diagonal weighting matrix W • For example, w1 0 the relative 1 w1 weights with W2 two CVs w2 0 w2 w2 2/25/2001 Industrial Process Control 19 Dynamic Matrix Control Basic Features Since 1983 • To restrict the size of r1 calculated moves a r1 0 relative weight for each of the MVs is r1 R imposed r2 1 A 1 e Min W ΔMV R 2 0 r2 2 0 2 r2 2/25/2001 Industrial Process Control 20 Dynamic Matrix Control Basic Features Since 1983 • Subject to linear constraints – The change in each MV is within a “step” bound MV LO MV k MV HI LO k 1 HI MV I MV MV LO k N 1 HI MV MV MV 2/25/2001 Industrial Process Control 21 Dynamic Matrix Control Basic Features Since 1983 • Subject to linear constraints IL – Size of each MV step for each time interval MV k MV 1 0 0 MV k k 1 k 1 MV MV 1 1 MV MV 0 k N 1 k N 1 MV MV 1 1 1 MV 2/25/2001 Industrial Process Control 22 Dynamic Matrix Control Basic Features Since 1983 • Subject to linear constraints – MV calculated for each time interval is between high and low limits MV LO MV MV k MV HI MV LO k 1 HI MV MV I MV MV MV L LO k N 1 HI MV MV MV MV MV 2/25/2001 Industrial Process Control 23 Dynamic Matrix Control Basic Features Since 1983 • Subject to linear constraints – CV calculated for each time interval is between high and low limits CV CV k 1 LO MV k CV HI LO k 2 k 1 HI CV CV A MV CV LO k M k N 1 HI CV CV MV CV 2/25/2001 Industrial Process Control 24 Dynamic Matrix Control Basic Features Since 1983 • The following LP subproblem is solved – where the economic weights are know a priori Minimize 1MV1* 2 MV2* 3 MV3* Subject to: CV1LO g11MV1* g12 MV2* g13 MV3* CV1HI CV2LO g 21MV1* g 22 MV2* g 23 MV3* CV2HI 2/25/2001 Industrial Process Control 25 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, 1t 1,1, ,1 A11 A12 A13 e1 A A 22 A 23 e2 21 A 1t 0 0 , e MV1* MV 1 MV2 MV 2 * 0 1t 0 0 0 1t MV3* MV 3 2/25/2001 Industrial Process Control 26

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posted: | 4/20/2011 |

language: | English |

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