Contents:
An overview of optimization problem, some examples of optimum design problem – Concepts and terms related to optimization problem, necessary and sufficient conditions for a multivariable function.

Effects of scaling or adding a constant to an objective function and understanding of constrained and unconstrained optimization problems. Concept of Lagrange multipliers and its application to unconstrained optimization problem.

Solution of unconstrained minimization problem using : Gradient descent method,Steepest descent method,Newton’s method,Davison-Fletcher-Powell method,Exterior point method.

Numerical examples are considered to illustrate the algorithmic steps of the above methods – Solution of constrained minimization problems using Karush-Kuhn-Tucker (KKT) necessary and sufficient conditions – Numerical examples are considered to illustrate the technique.

Understanding the following terms : convex sets, convex and concave functions,properties of convex function,definiteness of a matrix and test for concavity of function,Explain with numerical examples.
Problem statement of : convex optimization,quadratic optimization,quadratically constrained quadratic optimization,local and global optima.

Solution of quadratic programming problems using KKT necessary condition – Basic concept of interior penalties and solution of convex optimization problem via interior point method – Numerical examples are considered to illustrate the techniques mentioned in Lec.-11 and Lec.12 – Linear programming: Simple method; matrix form of the simplex method – Illustrate the solution of linear programming problems in tabular form via simplex method – Two-phase simplex method – Primal and dual problem: Determination of primal solution from its dual form solution and vice-versa – Properties of dual problems and sensitivity analysis – Basic concept of multi-objective optimization problem and some definitions – Solution of multi-objective optimization problem and illustrate the methodoly with numerical examples.

Concept of functional, variational problems and performance indices – Euler-Lagrange equation to find the extremal of a functional. Transversality condition – Application of variation approach to control problems – Statement of Linear quadratic regulator (LQR) problem and establish a mathematical framework to solve this problem – Optimal solution of LQR problem – Different techniques for solution of algebraic Riccati equation.

Numerical problem are considered to illustrate the LQR design procedures and discussed the role of state and input weighting matrices on the system performance – Frequency domain interpretation of LQR problem – Stability and robustness properties of LQR design – Optimal control with constraints on input – Optimal saturating controllers – Dynamic programming principle of optimality – Concept of time optimal control problem and mathematical formulation of problem.

Solution of time-optimal control problem and explained with a numerical example – Concept of system and signal norms. Small-gain theorem, physical interpretation of H∞ norm – Computation of H∞ Norm, statement of H∞ control problem – H∞ control problem: Synthesis – Illustrative example – Discussion on stability margin and performance of H∞ based controlled systems.

Other Resources

Course Curriculum

Mod-01 Lec-01 Introduction to Optimization Problem: Some Examples Details
Mod-01 Lec-02 Introduction to Optimization Problem: Some Examples (Contd.) Details
Mod-01 Lec-03 Lecture-03-Optimality Conditions for Function of Several Variables Details
Mod-01 Lec-04 Lecture-04-Optimality Conditions for Function of Several Variables (Contd.) Details
Mod-01 Lec-05 Lecture-05-Unconstrained Optimization Problem (Numerical Techniques) Details
Mod-01 Lec-06 Solution of Unconstrained Optimization Problem Using Conjugate Quadient Method Details
Mod-01 Lec-07 Solution of Unconstrained Optimization Problem Using Conjugate Quadient Details
Mod-01 Lec-08 Solution of Constraint Optimization Problem-Karush-Kuhn Tucker (KKT) Conditions Details
Mod-01 Lec-09 Lecture-09-Solution of Constraint Optimization Problem Details
Mod-01 Lec-10 Lecture-10-Problem and Solution Session Details
Mod-01 Lec-11 Post Optimality Analysis, Convex Function and its Properties Details
Mod-01 Lec-12 Post Optimality Analysis, Convex Function and its Properties (Contd.) Details
Mod-01 Lec-13 Quadratic Optimization Problem Using Linear Programming Details
Mod-01 Lec-14 Lecture-14-Matrix form of the Simplex Method Details
Mod-01 Lec-15 Lecture-15-Matrix form of the Simplex Method (Contd.) Details
Mod-01 Lec-16 Solution of Linear Programming Using Simplex Method:- Algebraic Approach Details
Mod-01 Lec-17 Solution of Linear Programming Using Simplex Method:- Algebraic Approach (Contd.) Details
Mod-01 Lec-18 Solution of LP Problems with Two Phase Method Details
Mod-01 Lec-19 Lecture-19-Solution of LP Problems with Two Phase Method (Contd.) Details
Mod-01 Lec-20 Lecture-20-Standard Primal and Dual Problems Details
Mod-01 Lec-21 Relationship Between Primal and Dual Variables Details
Mod-01 Lec-22 Solution of Quadratic Programming Problem Using Simplex Method Details
Mod-01 Lec-23 Interior Point Method for Solving Optimization Problems Details
Mod-01 Lec-24 Lecture-24-Interior Point Method for Solving Optimization Problems (Contd.) Details
Mod-01 Lec-25 Lecture-25-Solution of Nonlinear Programming Problem Details
Mod-01 Lec-26 Solution of Nonlinear Programming Problem Using Exterior Penalty Details
Mod-01 Lec-27 Solution of Nonlinear Programming Problem Using Interior Penalty Function Method Details
Mod-01 Lec-28 Solution of Nonlinear Programming Problem Using Interior Penalty Fun Mthd (Contd.) Details
Mod-01 Lec-29 Lecture-29-Multiobjective Optimization Problem Details
Mod-01 Lec-30 Lecture-30-Dynamic Optimization Problem: Details
Mod-01 Lec-31 Dynamic Optimization Problem: Basic Concepts and Necessary (cont.,) Details
Mod-01 Lec-32 Lecture-32-Dynamic Optimization Problem: Details
Mod-01 Lec-33 Lecture-33-Numerical Example and Solution of Optimal Control Problem Details
Mod-01 Lec-34 Lecture-34-Numerical Example and Solution of Optimal Control Problem Details
Mod-01 Lec-35 Lecture-35-Hamiltonian Formulation for solution of optimal Control problem Details
Mod-01 Lec-36 Hamiltonian Formulation for solution of optimal Control problem(Contd.) Details
Mod-01 Lec-37 Lecture-37-Performance Indices and Linear Quadratic Regulator Problem Details
Mod-01 Lec-38 Lecture-38-Performance Indices and Linear Quadratic Regulator Problem (Contd.) Details
Mod-01 Lec-39 Lecture-39-Solution and Stability Analysis of Finite – time LQR Problem: Details
Mod-01 Lec-40 Lecture-40-Solution and Infinite – time LQR Problem and Stability Analysis Details
Mod-01 Lec-41 Numerical Example and Methods for Solution of A.R.E. Details
Mod-01 Lec-42 Lecture-42-Numerical Example and Methods for Solution of A.R.E. (Contd.) Details
Mod-01 Lec-43 Lecture-43-Frequency Domain Interpretation of LQR Controlled System Details
Mod-01 Lec-44 Lecture-44-Gain and Phase Margin of LQR Controlled System Details
Mod-01 Lec-45 The Linear Quadratic Gaussian Problem Details
Mod-01 Lec-46 Loop-Transfer Recovery Details
Mod-01 Lec-47 Lecture-47-Dynamic Programming for Discrete Time Systems Details
Mod-01 Lec-48 Lecture-48-Minimum — Time Control of a Linear Time Invariant System Details
Mod-01 Lec-49 Lecture-49-Solution of Minimum — Time Control Problem with an Example Details

These video tutorials are delivered by IIT Kharagpur as part of NPTEL online courses program.

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