Dynamic Matrix Control - Introduction

1
Dynamic Matrix Control - Introduction

• Developed at Shell in the mid 1970s
• 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
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
Dynamic Matrix Control - Introduction

• Consider the following furnace example (Cutlet
  Ramaker)
• MV
• Fuel flow FIC
• DV
• Inlet temperatureTI
• CV
• Outlet temperature TIC

4
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
Dynamic Matrix Control - Introduction

• DMC Dynamic coefficents
• Response to step change in fuel, a
• Response to step change in inlet temperature, b

6
Dynamic Matrix Control - Introduction

• The DMC prediction may be calculated from those
  coefficients and the independent variable changes

7
Dynamic Matrix Control - Introduction

• Feedforward prediction is enabled by moving the
  DV to the left hand side

8
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
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
Dynamic Matrix Control - Introduction

• Least squares formulation including move
  suppression

Move suppression
11
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
Dynamic Matrix Control - Introduction

• Furnace Example
• Temperature Disturbance DT15 at t0
• Three Fuel Moves Calculated

13
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
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
Dynamic Matrix Control Basic Features Since 1983
pcr k gt 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

• For linear differential equations the process
  output can be given by the convolution theorem

16
Dynamic Matrix Control Basic Features Since 1983
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

• Breaking up the summation terms into past and
  future contributions

17
Dynamic Matrix Control Basic Features Since 1983

• Let Nnumber future moves, Mtime horizon to
  reach steady state, then in matrix form

18
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
19
Dynamic Matrix Control Basic Features Since 1983

• To scale the residuals, a weighted least squares
  problem is posed

pcr wi relative weighting of the ith CV which
will be repeated M times to form the diagonal
weighting matrix W

• For example, the relative weights with two CVs

20
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
Dynamic Matrix Control Basic Features Since 1983

• Subject to linear constraints
• The change in each MV is within a step bound

22
Dynamic Matrix Control Basic Features Since 1983

• Subject to linear constraints
• Size of each MV step for each time interval

23
Dynamic Matrix Control Basic Features Since 1983

• Subject to linear constraints
• MV calculated for each time interval is between
  high and low limits

24
Dynamic Matrix Control Basic Features Since 1983

• Subject to linear constraints
• CV calculated for each time interval is between
  high and low limits

25
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
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,