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Dynamic Matrix control
- 1. Model Predictive Control Dynamic Matrix Control Process Dynamics and Control Supplementary Material Indian Institute of Technology Kanpur Modeule 6.3
- 2. Process Control Notes 2 The MPC Paradigm ySP y past . . . u past . . . u future y future (predicted) y reference time -4 -3 -2 -1 0 1 2 3 4 M P . . . . . . . . . . . . GIVEN A predictive model that predicts y for candidate control move sequences QUESTION Of all the candidate future control move sequences, which one causes the predicted y to best follow the reference trajectory SOLUTION Usually solved as an optimization problem MPC PROCEDURE 1. Obtain current y measurement 2. Calculate reference trajectory 3. Calculate uoptimum 4. Implement current move 5. Repeat at next sampling instant M: Control Horizon P: Prediction Horizon
- 3. Process Control Notes 3 Dynamic Matrix Control Unit Step Response Coefficients g = [g1 g2 g3 … gN-1 gN] g1 g2 g3 gN-1 gN . . . . . . . . . . . . time 0 1 2 3 N-1 N u y Unit Step Process Reaction Curve time 0 1 2 3 u0 u1 u2 u1 u0 u2 u = u0 + u1 + u2 Input Change Sequence time 0 1 2 3 u Any general input change sequence is a superposition of appropriately delayed steps Predicted output response is superposition of appropriately delayed step responses
- 4. Process Control Notes 4 SISO DMC: Predictive Model Process d y u + + ySP y past . . . u past . . . u future y future (predicted) y reference time -4 -3 -2 -1 0 1 2 3 4 M P . . . . . . . . . . . . k k 𝑦𝑘 = 𝑖=−∞ min(𝑘−1,𝑀) 𝑔𝑘−𝑖𝑢𝑖 + 𝑑𝑘 𝑦𝑘 = 𝑖=−∞ −1 𝑔𝑘−𝑖𝑢𝑖 + 𝑖=0 min(k−1,M) 𝑔𝑘−𝑖𝑢𝑖 + 𝑑𝑘 𝑑𝑘 = 𝑑𝑘 = 𝑑0 = 𝑦0 − 𝑦0 𝑦0 = 𝑖=−∞ −1 𝑔−𝑖𝑢𝑖 𝑦𝑘 = 𝑖=0 min(k−1,M) 𝑔𝑘−𝑖𝑢𝑖 + 𝑖=1 ∞ 𝑔𝑘+𝑖 − 𝑔𝑖 𝑢−𝑖 + 𝑦0 Effect of control moves Effect of past moves 𝑦𝑘 = 𝑖=0 min(k−1,M) 𝑔𝑘−𝑖𝑢𝑖 + 𝑖=1 N 𝑔𝑘+𝑖 − 𝑔𝑖 𝑢−𝑖 + 𝑦0 Forced response Free response
- 5. Process Control Notes 5 DMC Predictive Model 𝑦1 = 𝑔1𝑢0 + 𝑓1 𝑦2 = 𝑔2𝑢0 + 𝑔1𝑢1 + 𝑓2 𝑦𝑘 = 𝑔𝑘𝑢0 + 𝑔𝑘−1𝑢1 + ⋯ + 𝑔𝑘−𝑀𝑢𝑀 + 𝑓𝑀 𝑦𝑃 = 𝑔𝑃𝑢0 + 𝑔𝑃−1𝑢1 + ⋯ + 𝑔𝑃−𝑀𝑢𝑀 + 𝑓𝑃 . . . . . . 𝑦1 𝑦2 . . . 𝑦𝑘 . . . 𝑦𝑃 𝑓1 𝑓2 . . . 𝑓𝑘 . . . 𝑓𝑃 𝑔1 𝑔2 . . . 𝑔𝑘 . . . 𝑔𝑃 0 𝑔1 𝑔2 . . 𝑔𝑘−1 . . . 𝑔𝑃−1 0 0 𝑔1 . . 𝑔𝑘−2 . . . 𝑔𝑃−2 … … … . . ... . . . … 0 0 . . . 𝑔𝑘−𝑀 . . . 𝑔𝑃−𝑀 𝑢0 𝑢1 𝑢2 . . . 𝑢𝑀 = + 𝐲 = 𝐆𝐮 + 𝐟 𝑦𝑘 = 𝑖=0 min(k−1,M) 𝑔𝑘−𝑖𝑢𝑖 + 𝑖=1 N 𝑔𝑘+𝑖 − 𝑔𝑖 𝑢−𝑖 + 𝑦0
- 6. Process Control Notes 6 DMC Optimum Control Move Sequence 𝐽 = 𝑖=1 𝑃 (𝑦𝑖 − 𝑟𝑖)2+ 𝜆 𝑖=1 𝑃 𝑢𝑖 2 𝐽 = 𝐲 − 𝐫 𝐓 𝐲 − 𝐫 + 𝜆𝐮𝐓𝐮 = 𝐆𝐮 + 𝐟 − 𝐫 𝐓 𝐆𝐮 + 𝐟 − 𝐫 + 𝜆𝐮𝐓𝐮 min 𝐮 𝐽 Put 𝐲 = 𝐆𝐮 + 𝐟 𝐽 = 𝐮𝐓 𝐆𝐓 𝐆𝐮 + 𝐮𝐓 𝐆𝐓 𝐟 − 𝐫 + (𝐟 − 𝐫)𝐓 𝐆𝐮 + 𝐟 − 𝐫 𝐓 𝐟 − 𝐫 + 𝜆𝐮𝐓 𝐮 For optimality 𝜕𝐽 𝜕𝐮 = 0 2𝐆𝐓𝐆𝐮∗ + 2𝐆𝐓 𝐟 − 𝐫 + 2𝜆𝐮∗ = 𝟎 (𝐆𝐓𝐆 + 𝜆𝐈)𝐮∗ = 𝐆𝐓 𝐫 − 𝐟 𝐮∗ = (𝐆𝐓𝐆 + 𝜆𝐈)−𝟏𝐆𝐓 𝐫 − 𝐟 Implement first element of u* and repeat procedure at next instant
- 7. Process Control Notes 7 Constrained DMC 𝐽 = 𝑖=1 𝑃 (𝑦𝑖 − 𝑟𝑖)2+ 𝜆 𝑖=1 𝑃 𝑢𝑖 2 min 𝐮 𝐽 𝐲 = 𝐆𝐮 + 𝐟 Rate Constraint −𝑢𝑀𝐴𝑋 ≤ 𝑢𝑖≤ 𝑢𝑀𝐴𝑋 Valve Saturation Constraint 0% ≤ 𝑖=0 𝑀 𝑢𝑖 − 𝑢𝑐𝑢𝑟𝑟 ≤ 100% Easily solved using quadratic programming Subroutine quadprog in Matlab
- 8. Process Control Notes 8 Reference Trajectory and Tuning Reference Trajectory 𝑟𝑖+1 = 𝛼𝑟𝑖 + 1 − 𝛼 𝑦𝑠𝑝 0 ≤ 𝛼 ≤ 1 Initial Condition 𝑟0 = 𝑦0 v v v α↑ ySP y0 time CONTROLLER TUNING • M, P, α, λ: Tuning parameters • Usually M, P and α are chosen to reasonable values and λ adjusted for stable and tight closed loop control • No standard tuning procedures for MPC (Hit-and-trial)
- 9. Process Control Notes 9 Multivariable DMC DMC Controller ⋮ ⋮ u1 u2 uN y1 y2 yN PROCESS ⋮ ⋮ 𝐲𝟏 𝐲𝟐 ⋮ 𝐲𝐣 ⋮ 𝐲𝐍 G11 G12 … G1k … G1N G21 G22 … G2k … G2N ⋮ Gj1 Gj2 … Gjk … GNN ⋮ GN1 GN2 … GNk … GNN u1 u2 ⋮ uk ⋮ uN f1 f2 ⋮ fj ⋮ fN = + 𝐲 = 𝐆𝐮 + 𝐟 𝐽 = 𝑗=1 𝑁 𝑖=1 𝑃𝑗 (𝑦𝑖𝑗 − 𝑟𝑗)2 + 𝑘=1 𝑁 𝑖=0 𝑀𝑖 𝜆𝑘𝑢𝑖𝑘 2 𝐽 = 𝐲 − 𝐫 𝐓 𝐲 − 𝐫 + 𝛌𝐮𝐓𝐮 𝐮∗ = (𝐆𝐓 𝐆 + 𝛌𝐈)−𝟏 𝐆𝐓 𝐫 − 𝐟 CONSTRAINED MULTIVARIABLE DMC Rate Constraint −𝑢𝑘 𝑀𝐴𝑋 ≤ 𝑢𝑖𝑘≤ 𝑢𝑘 𝑀𝐴𝑋 for k = 1 to N Valve Saturation Constraint 0% ≤ 𝑖=0 𝑀𝑘 𝑢𝑖𝑘 − 𝑢𝑐𝑢𝑟𝑟,𝑘 ≤ 100% for k = 1 to N Solve using a quadratic program (quadprog in Matlab)
- 10. Process Control Notes 10 Some Comments • MPC does not address stability directly. Classical methods do. • Stability addressed indirectly by adjusting the objective function using primarily λ (move suppression factor) • Since objective function itself is getting tuned to ensure stability, it makes little sense to claim “optimality” of control moves compared to classical control methods • The (pseudo)inversion of the dynamic matrix makes the technique truly multivariable • Same formalism can handle SISO, MIMO (square and non-square) systems • MPC performance degrades significantly with increasing plant model mismatch. Models and tuning must be periodically updated • No systematic MPC tuning procedures (unlike classical methods)