REAL TIME OPTIMIZATION: A Parametric Programming Approach

REAL TIME OPTIMIZATION:A Parametric Programming Approach Vivek Dua You Only Solve Once

Parametric Programming • Given: • a performance criterion to minimize/maximize • a vector of constraints • a vector of parameters • Obtain: • the performance criterion and the optimization variables as a function of the parameters • the regions in the space of parameters where these functions remain valid

Parametric Optimization (POP) Critical Region Obtain optimal solution as a function of parameters

An Example – Linear Model (Edgar and Himmelblau, 1989) Current Max. Prod. Gasoline Kerosene Fuel Oil Residual 24,000 bbl/day 2,000 bbl/day 6,000 bbl/day Crude Oil # 1 REFINERY Crude Oil # 2 KPE Objective:MaximizeProfit Parameters: Gasoline Prod. Expansion (GPE) Kerosene Prod. Expansion (KPE) GPE Solve optimization problems at many points?

Parametric Solution KPE KPE Region #1 Region #2 GPE GPE Only 2 optimization problems solved!

Real Time Optimization OPTIMIZER Control Actions System State SYSTEM

Model Predictive Control (MPC) target future past output manipulated variable k k+1 k+p Prediction Horizon

Model Predictive Control • Solve an optimization problem at each time interval k

Model Predictive Control min A quadratic and convex function of discretized state and control variables s.t. 1. Constraints linear in discretized state and control variables 2. Lower and upper bounds on state and control variables Solve a QP at each time interval

Parametric Programming Approach • State variables Parameters • Control variables Optimization variables • MPC Parametric Optimization problem • Control variables = F(State variables)

Multi-parametric Quadratic Programs Theorem 1: Theorem 2:

Critical Region (CR) • CR: the region where a solution remains optimal • Feasibility Condition: • Optimality Condition: • CR: • A polyhedron • Obtain: CR

Real Time Optimization POP PARAMETRIC PROFILE OPTIMIZER CONTROLACTIONS SYSTEMSTATE Control Actions SYSTEMS System State SYSTEM Function Evaluation!

Example

Explicit Solution 1 2,4

Explicit Solution 1 2,4 3 5 6 7,8 9

Parametric Programming Approach Model Predictive Control Real Time Optimization Problem Off-line Parametric Optimization Problem Measurements as Parameters Control Variables as Optimization variables Obtain Explicit Control Law (a) Explicit functions of measurements (b) Critical Regionswhere these functions are valid State-of-the-art Performance on a simple computational hardware

Blood Glucose Control Exogenous Insulin U(t) Clearance Plasma Insulin I(t) Effective Insulin X(t) Plasma Glucose G(t) Liver Exercise, Meals D(t) Tissue • State variables: G(t), I(t), X(t) • Control variable: U(t) • Parameters: Pi, n (Bergman et al., 1981)

Parametric Control of Blood Glucose Meals, Exercise Parametric Glucose Control (Off-line) Reference Mechanical Pump Glucose Sensor Patient Insulin • an in-vivo glucose sensor • a parametric ‘look-up function’ to manipulate the insulin delivery rate given a sensor measurement • a mechanical pump

Isoflurane uptake 1 2 5 3 4 Control of Anesthesia RESPIRATORY SYSTEM • Lungs and Heart • Vessel rich organs (e.g. liver) • Muscles • Others • Fat DP, SNP Injection Pharmacokinetic aspect Pharmacodynamic aspect

Surgery under Anesthesia 0.6% Isoflurane 0.3 g/kg/min SNP 4.5 g/kg/min DP 20 mmHg MAP drop Isoflurane, SNP stopped DP stopped

Tr Ca Control of Pilot Plant Reactor Ff Product Controller Feed Tj Reactor & Cooling Jacket Cooling Water

Control of Catalytic Converter (Balenovic and Backx, 2001) CATALYTIC CONVERTER MODEL Air CATALYTIC CONVERTER CAR ENGINE Exhaust Gas Clean Gas Fuel • Clean Exhaust Gas • Control the amount of Oxygen stored on the Catalyst to an Optimal amount • Use Converter Model as an inferential sensor • Ensuring Minimum Energy Consumption and Maximum Emissions Reduction

Parametric Control of Catalytic Converter AFR = -0.68 OC - 0.0059 EMF + 0.60 -115.58 OC – EMF <= -84.77 -60.88 OC + EMF <= 33.67 70.90 OC + EMF <= 85.10 186.70 OC – EMF <= 44.10 AFR EMF EMF OC OC: (Fractional) Oxygen Coverage EMF: Exhaust Mass Flowrate (kg/hr) AFR: (Normalised) Air to Fuel Ratio OC

Concluding Remarks • Real Time Optimization • Solve optimization problem at regular time intervals • Parametric Programming Approach • Obtain optimal solution as a set of functions of state variables • Optimality and satisfaction of constraints are guaranteed • Function Evaluations! • PAROS plc: www.parostech.com

References • Dua, P., Doyle III, F.J., Pistikopoulos, E.N. (2006) Model based blood glucose control for type 1 diabetes via parametric programming, accepted for publication in IEEE Transactions on Biomedical Engineering. • Dua, P., Dua, V., Pistikopoulos, E.N. (2005) Model based drug delivery for anesthesia, Proceedings of the 16th IFAC World Congress, Prague, 2005. • Sakizlis, V., Kakalis, N.M.P., Dua, V., Perkins, J.D.,Pistikopoulos, E.N. (2004) Design of robust model-based controllers via parametric programming, Automatica, 40, 189-201. • Dua, V., Bozinis, N. A., Pistikopoulos, E.N. (2002) A multiparametric programming approach for mixed-integer and quadratic process engineering problems, Computers & Chemical Engineering, 26, 715-733. • Pistikopoulos, E.N., Dua, V., Bozinis, N. A., Bemporad, A., Morari, M. (2002) On-line optimization via off-line parametric optimization tools, Computers & Chemical Engineering, 26, 175-185. • Bemporad, A., Morari, M., Dua, V., Pistikopoulos, E.N. (2002) The explicit linear quadratic regulator for constrained systems, Automatica, 38, 3-20.

REAL TIME OPTIMIZATION: A Parametric Programming Approach