H0K03a : Advanced Process Control Model-based Predictive Control 2 : Dynamic Optimization Bert Pluymers Prof. Bart De Moor Katholieke Universiteit Leuven, Belgium Faculty of Engineering Sciences Department of Electrical Engineering (ESAT) Research Group SCD-SISTA H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be
Overview • Overview • Optimization Basics • Convex Optimization • Dynamic Optimization • Optimization Algorithms Lesson 2 : Dynamic Optimization • Optimization basics • Convex Optimization • Dynamic Optimization • Optimization Algorithms S ignal processing I dentification S ystem T heory 1 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be
Notation • Overview • Optimization Basics • Convex Optimization General form : • Dynamic Optimization • Optimization Algorithms Legend : • : vector of optimization variables • : objective function / cost function • : equality constraints • : inequality constraints • : solution to optimization problem S ignal processing I dentification • : optimal function value S ystem T heory 2 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be
Gradient & Hessian • Overview • Optimization Basics • Convex Optimization Gradient : • Dynamic Optimization • Optimization Algorithms (points in direction of steepest ascent) Hessian : (gives information about local curvature of ) S ignal processing I dentification S ystem T heory 3 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be
Gradient & Hessian • Overview • Optimization Basics • Convex Optimization Example : • Dynamic Optimization • Optimization Algorithms Gradients for different Eigenvectors of hessian at the origin ( ) S ignal processing I dentification S ystem T heory 4 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be
Unconstrained Optimality Conditions • Overview • Optimization Basics • Convex Optimization Necessary condition for optimality of • Dynamic Optimization • Optimization Algorithms Sufficient conditions for minimum positive definite Classification of optima : positive definite minimum indefinite saddle point S ignal processing I dentification negative definite maximum S ystem T heory 5 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be
Lagrange Multipliers • Overview • Optimization Basics • Convex Optimization • Dynamic Optimization Introduction of Lagrange multipliers leads to Lagrangian : • Optimization Algorithms with Lagrange multipliers of the ineq. constraints Lagrange multipliers of the eq. constriants Constrained optimum can be found as S ignal processing I dentification Minimization over but Maximization over !!!! S ystem T heory 6 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be
Lagrange Multipliers • Overview • Optimization Basics • Convex Optimization • Dynamic Optimization Constrained optimum can be found as • Optimization Algorithms First-order optimality conditions in Gradient of Gradient of Gradient of eq. ineq. S ignal processing I dentification Interpretation ??? S ystem T heory 7 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be
Lagrange Multipliers • Overview • Optimization Basics • Convex Optimization • Dynamic Optimization • Optimization Algorithms S ignal processing I dentification S ystem T heory 8 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be
Lagrange Multipliers • Overview • Optimization Basics • Convex Optimization • Dynamic Optimization • Optimization Algorithms S ignal processing I dentification S ystem T heory 9 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be
Lagrange Multipliers • Overview • Optimization Basics • Convex Optimization • Dynamic Optimization • Optimization Algorithms S ignal processing I dentification S ystem T heory 10 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be
Lagrange Duality • Overview • Optimization Basics • Convex Optimization • Dynamic Optimization • Optimization Algorithms S ignal processing I dentification S ystem T heory 11 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be
Karush-Kuhn-Tucker Conditions • Overview • Optimization Basics • Convex Optimization • Dynamic Optimization From previous considerations we can now state • Optimization Algorithms necessary conditions for constrained optimality : These are called the KKT conditions . S ignal processing I dentification S ystem T heory 12 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be
Optimization Tree • Overview • Optimization Basics • Convex Optimization Optimization • Dynamic Optimization • Optimization Algorithms discrete continuous unconstrained constrained non-convex convex optimization optimization S ignal processing I dentification NLP QP SOCP SDP LP S ystem T heory 13 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be
Convex Optimization • Overview • Optimization Basics • Convex Optimization An optimization problem of the form • Dynamic Optimization • Optimization Algorithms is convex iff for any two feasible points : • is feasible • This is satisfied iff • `the cost function is a convex function • the equality constraints or linear or absent S ignal processing I dentification • the inequality constraints define a convex region S ystem T heory 14 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be
Convex Optimization • Overview • Optimization Basics • Convex Optimization • Dynamic Optimization Importance of convexity : • Optimization Algorithms • no local minima, one global optimum • under certain conditions, primal and dual have same solution • efficient solvers exist • polynomial worst-case execution time • guaranteed precision S ignal processing I dentification S ystem T heory 15 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be
From LP to SDP • Overview generality • Optimization Basics • Convex Optimization • Dynamic Optimization • Optimization Algorithms SDP Semi-Definite Programming SOCP Second Order Cone Progr. QP Quadratic Programming LP Linear Programming computational efficiency S ignal processing I dentification S ystem T heory 16 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be
Linear Programming (LP) • Overview • Optimization Basics • Convex Optimization General form : • Dynamic Optimization • Optimization Algorithms Remarks : • always convex • optimal solution always at a corner of ineq. constraints S ignal processing I dentification • typically used in finance / economics / management S ystem T heory 17 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be
Linear Programming (LP) • Overview • Optimization Basics • Convex Optimization Eliminating equality constraints : • Dynamic Optimization • Optimization Algorithms Reparametrize optimization vector : Leading to S ignal processing I dentification S ystem T heory 18 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be
Quadratic Programming (QP) • Overview • Optimization Basics • Convex Optimization General form : • Dynamic Optimization • Optimization Algorithms Remarks : • convex iff • LP is a special case of QP (imagine ) • Used in all domains of engineering S ignal processing I dentification S ystem T heory 19 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be
Second-Order Cone Programming (SOCP) • Overview • Optimization Basics • Convex Optimization General form : • Dynamic Optimization • Optimization Algorithms SOC constraint Remarks : • Always convex • Second-Order, Ice-Cream, Lorentz cone : • S ignal processing I dentification • Engineering applications with sum-of-squares, S ystem T heory robust LP, robust QP 20 A utomation H0k03a : Advanced Process Control – Model-based Predictive Control 2 : Dynamic Optimization bert.pluymers@esat.kuleuven.be
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