# Optimization in Systems and Control

## SC42055

### ECTS:

4### Responsible Instructor:

dr.ir. T.J.J. van den Boom (Ton)### Contact Hours / Week x/x/x/x:

4/0/0/0### Education Period:

1### Start Education:

1### Exam Period:

1### Course Language:

English### Course Contents:

Table of ContentsPart I: Optimization Techniques

Introduction

Mathematical framework

Unimodality and convexity

Optimization problems

Optimality conditions

Convergence and stopping criteria

Linear Programming

The linear programming problem

The simplex method

Quadratic Programming

Quadratic programming algorithm

System identification example

Nonlinear Optimization without Constraints

Newton and quasi-Newton methods

Methods with direction determination and line search

Nelder-Mead method

Constraints in Nonlinear Optimization

Equality constraints

Inequality constraints

Convex Optimization

Convex functions

Convex problems: Norm evaluation of affine functions

Convex problems: Linear matrix inequalities

Convex optimization techniques

Controller design example

Global Optimization

Local and global minima

Random search

Multi-start local optimization

Simulated annealing

Genetic algorithms

Optimization Methods: Summary

Simplification of the objective function and/or the constraints

Determination of the most efficient available algorithm

Determination of the stopping criterion

The MATLAB Optimization Toolbox

Linear programming

Quadratic programming

Unconstrained nonlinear optimization

Constrained nonlinear optimization

Multi-Objective Optimization

Problem statement

Pareto optimality

Solution methods for multi-objective optimization problems

Integer Optimization

Complexity

Search

Overview of integer optimization methods

Part II: Formulating the Controller Design Problem as an Optimization Problem

Multi-Criteria Controller Design: The LTI SISO Case

Introduction

The basic feedback loop

General formulation of the basic feedback loop

Internally stabilizing controllers

Convex Controller Design Specifications

Definition of affine and convex transfer function sets

Engineering specification with respect to overshoot

Engineering specification with respect to tracking a reference signal

Engineering specifications in terms of norms of transfer functions

Robust stability and plant uncertainty

An Example of Multi-Criteria Controller Design

The plant

Engineering specifications

General formulation of the basic feedback loop

Linear Quadratic Gaussian design

Example formulation of a robust controller design problem

Computing the noise sensitivity and its gradient

Computing the robustness constraint and its subgradient

MATLAB implementation

Discussion of the results

Appendices

Basic State Space Operations

Cascade connection or series connection

Parallel connection

Change of variables

State feedback

Output injection

Transpose

Left (right) inversion

Jury's Stability Criterion

Singular Value Decomposition

Least Squares Problems

Ordinary least squares

Total least squares

Robust least squares

### Study Goals:

Essentially, almost all engineering problems are optimization problems. If a civil engineer designs a bridge, then one of the main objectives is to obtain the cheapest design or the design that can be implemented most rapidly, where of course several specifications and constraints such as size, strength, safety, etc. have to be taken into account. When developing a new type of engine, we look for the most economical design, the cheapest design, or the design with the highest performance. A process engineer wants a production unit to deliver a final product of maximal quality, with minimal expenditure of energy or with maximal output flow. When composing a portfolio, a financial engineer tries to maximize the expected profits, subject to the given risk constraints. So we encounter optimization problems in almost every engineering field.How can we solve such an optimization problem? That is the topic that will be addressed in this course. We will consider both the transformation of real-world design problems into a more mathematical formulation, and the selection of the most efficient numerical algorithms to solve the resulting optimization problem.

The examples and case studies of this course are primarily oriented towards systems and control. In preceding courses you have already studied modeling, identification and control of systems. However, the examples in these courses were usually limited to simple or small systems, and more complex systems were often dealt with by saying that they can be tackled using optimization. And that is what we will do in this course: you will not only learn how you can identify models and design controllers for complex systems using numerical optimization, but also how this can be done in the most efficient way.

This course is divided into two parts:

1. optimization techniques

2. applications in systems and control

In the first part we study several classes of optimization problems and we discuss which algorithms are the best suited for each particular problem. In the second part we show how a controller design problem can be recast as an optimization problem and we use the results of the first part to efficiently design the controllers using numerical optimization.