Multiple Model Predictive Control (MMPC) for Nonlinear Systems and - PowerPoint PPT Presentation

Multiple Model Predictive Control (MMPC) for Nonlinear Systems and Improved Disturbance Rejection • Motivation & Tutorial Overview • Multiple Model Predictive Control • Nonlinear Processes • Disturbance Rejection • Summary B. Wayne Bequette Chemical and Biological Engineering

Nonlinear Behavior: Steady-state Input Multiplicity Output Multiplicity (hysteresis) Van de Vuuse Reactor, Figure M5-2, page 609 150 140 1.2 130 1 120 conc. B, mol/liter T 110 0.8 100 0.6 90 80 0.4 70 0.2 60 50 0 0 10 20 30 40 50 60 70 80 0 1 2 3 4 5 dilution rate, 1/min F j Two different input values One input can yield three yield the same output different output values

MPC past future setpoint • Constraints y model prediction actual outputs (past) • Multivariable P • Time-delays t k Prediction current Horizon step max u min past control M • Objective function? moves Control Horizon • Optimization technique? setpoint model prediction • Model type? from k y new model prediction actual outputs (past) P • Disturbances/mismatch? t k+1 Prediction current Horizon step • Current and Future max • Initial cond./state est.? u min past control M moves Control Horizon “hidden slide” provided for additional background

Our Approaches to Nonlinear MPC • Quadratic Objective Function • Models – Fundamental: numerical integration or collocation – Fundamental with linearization at each time step – Multiple model – Artificial neural network • State Estimates/Initial Conditions – Additive output disturbance (e.g. DMC) – Estimation horizon (optimization) – Extended/appended state Kalman Filter • Importance of stochastic states

Intuitive Nonlinear Model-based Strategy Model equations Choose hypothetical set of current and future control moves  x f ( x , u ) = u , u ,  , u k k 1 k P 1 + + − ( ) y g x = Integrate model from time step k to k +P (based on hypothetical control Integrate model from time step k-1 to k moves) ˆ ( ˆ ) x F x , u = k 1 t k k ˆ ( ˆ ) x F x , u + = s k t k 1 k 1 − − ˆ ( ˆ ) s y g x d = + k 1 | k k 1 k ˆ ( ˆ ) y g x + + = k | k 1 k − to Obtain plant measurement ˆ ( ˆ ) x F x , u = y k P t k P 1 k P 1 + + − + − s k ˆ ( ˆ ) y g x d = + k P | k k P k Calculate model error + + (additive output disturbance) Evaluate objective function and repeat ˆ d y y = − k k k | k 1 until optimum is obtained −

Additive Disturbance Assumption Model converges to different steady-state than plant (compensated by additive disturbance term)

Non-Convex Problem M = 1, different values of P Sistu and Bequette, 1992 ACC

EKF-based NMPC (Lee & Ricker, 1994) • Nonlinear Model, integrate from step k-1 to k • State Estimation: Extended Kalman Filter – Linearized at each time step – Find best state estimate at time step k • Prediction – One integration of NL ODEs based on set of control moves ( unforced or “free response” ) from step k to k+P – Perturbation (linear) model - effect of changes in control moves ( forced ) • Optimization – QP, since linear model is used Can use linear state- space KF-MPC code!

Motivation for Multiple Linear Models • Development time for fundamental models – Difficulty with physiological systems • Much data required for artificial neural networks – Problems with “overfitting” and extrapolation • At particular operating points, linear models are often a good description – How to switch between models?

Multiple Model-based Control Multiple Model Adaptive Control (MMAC) Athans et al. (1977) – LQG, Jet aircraft control Roy, Kaufman - Drug infusion control Schott, Bequette (1997) – PI, CSTR Multiple Model Predictive Control (MMPC) Rao et al. (2001, 2003) Drug Infusion Control Aufderheide & Bequette (2003) Nonlinear CSTR

First, a Concise Review of Linear MPC “hidden slide” provided for additional background

Step Response & Additive Output ˆ d y y Correction term = − k k k | k 1 − measured output model predicted output The “corrected prediction” is set equal to the measured output ˆ ˆ ˆ y y y y d = = + “deadbeat” observer k | k k | k − 1 k k | k k The “corrected prediction” for jth future step is (step response form) j N 1 − ˆ ˆ y s u s u s u d ∑ ∑ = Δ + Δ + + k j | k i k i j i k i j N k N j  k j + − + − + − + +    i 1 i j 1            = = + correction term effect of future effect of past control moves control moves forced response free response ˆ ˆ ˆ  d d d y y = = = = − constant additive disturbance k j k j 1 k k k | k 1 + + − − “hidden slide” provided for additional background

Problems with “Classical MPC” (e.g. DMC) • Finite Step/Impulse Models Limited – Many parameters (~50 for each input-output relationship) – Limited to open-loop stable processes (there is no corrective feedback to model states) • Additive Output Disturbance Assumption – Poor performance for input step disturbances – No explicit measurement noise trade-off The most common criticism of MPC (Shinskey, 2002) “hidden slide” provided for additional background

State Space Models & State Estimation For perturbations to manipulated inputs d x x u d = Φ + Γ + Γ k 1 k k k + Γ d = Γ y C x = k k Assume disturbance propagation d d = k + 1 k Appended state formulation x d x Γ Φ Γ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ k 1 k + u = + ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ k d d 0 0 I ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦     k  1 k      + a a a x Γ a x Φ k k 1 + x ⎡ ⎤ k [ ] y C 0 = ⎢ ⎥    k d ⎣ ⎦  k a C “hidden slide” provided for additional background a x k

State Estimation Problem d d w Random walk = + k + 1 k k x d x x Γ 0 Φ Γ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ k 1 k k [ ] + u w y C 0 v = + + = + ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥    k k k k w d d 0 Γ d 0 I ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦      k  1 k    k      + a C a a a a x Γ w,a x a Γ x Φ k k k 1 + Kalman Filter a a a a ˆ ˆ x Φ x Γ u Prediction = + k|k 1 k 1 |k 1 k 1 − − − − ( ) ˆ a ˆ a a ˆ a x x L y C x = + − Correction k|k k|k 1 k k k|k 1 − − “hidden slide” provided for additional background

Offset-Free Performance • Next few slides present conditions for offset-free performance • Unmeasured disturbances estimated as either state or output disturbances • State observer techniques can then be used Ref: Muske & Badgwell, J. Proc. Cont., 12: 617-632 (2002) “hidden slide” provided for additional background

Disturbance Models State or Input Disturbance x d x x ⎡ ⎤ ⎡Γ ⎡ ⎤ Φ Γ ⎡ ⎤ ⎡ ⎤ ⎤ k 1 k k [ ] + u y C 0 = + = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥    k k d d d 0 0 I ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦      k  1 k k      + a C a a a a x x a x Γ Φ k k k 1 + Additive Output Disturbance x 0 x x ⎡Φ ⎡Γ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎤ ⎤ [ ] k 1 k k + u y C G = + = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ k k p      p 0 I p 0 p ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦      k  1      k k + a C a a a a a x x x Γ Φ k k k 1 + G p = I DMC: “hidden slide” provided for additional background

Disturbance Models General Input and Output Disturbances d x 0 x x ⎡ ⎤ Φ Γ ⎡Γ ⎡ ⎤ ⎡ ⎤ ⎤ ⎡ ⎤ k 1 k k + ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ [ ] ⎢ ⎥ d 0 I 0 d 0 u y C 0 G d = + = ⎢ ⎥ k 1 k k k p k ⎢ ⎥ ⎢ ⎥ ⎢ ⎥    ⎢ ⎥ + p ⎢ 0 0 I ⎥ p 0 p ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ a C ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦  k 1 k k +      a a Γ Φ Ref: Muske & Badgwell, J. Proc. Cont., 12: 617-632 (2002) “hidden slide” provided for additional background