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) Obtain optimal solution as a function of parameters Critical Region

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

Parametric Solution Only 2 optimization problems solved! Profit = 4.66 GPE KPE Crude#1 = 1.72 GPE – 7.59 KPE Crude#2 = GPE KPE if GPE KPE < < GPE < < KPE (REGION #1) Profit = 7.53 GPE Crude#1 = 1.48 GPE Crude#2 = -0.41GPE if GPE KPE > < GPE < 6000 KPE < 500 (REGION #2) GPE KPE GPE KPE Region #2 Region #1

Real Time Optimization OPTIMIZER SYSTEM System State Control Actions

Model Predictive Control (MPC) past future target output manipulated variable k k+1k+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 SYSTEM System State Control Actions Function Evaluation! OPTIMIZER SYSTEMS SYSTEMSTATE SYSTEM STATE CONTROLACTIONS CONTROL ACTIONS

Example

Explicit Solution 1 2,4

Explicit Solution 1 2, ,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 Regions where these functions are valid State-of-the-art Performance on a simple computational hardware

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

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

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

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

Control of Pilot Plant Reactor Cooling Water Product Reactor & Cooling Jacket Feed TrTr CaCa Controller FfFf TjTj

Control of Catalytic Converter 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 (Balenovic and Backx, 2001) CATALYTIC CONVERTER MODEL CAR ENGINE CATALYTIC CONVERTER Fuel Air Exhaust GasClean Gas

Parametric Control of Catalytic Converter OC: (Fractional) Oxygen Coverage EMF: Exhaust Mass Flowrate (kg/hr) AFR: (Normalised) Air to Fuel Ratio OC EMF OC – EMF <= OC + EMF <= OC + EMF <= OC – EMF <= AFR = OC EMF EMF OC AFR

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:

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, 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, 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, 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, Bemporad, A., Morari, M., Dua, V., Pistikopoulos, E.N. (2002) The explicit linear quadratic regulator for constrained systems, Automatica, 38, 3-20.