# Optimization in Systems and Control

## SC42055

4

### Responsible Instructor:

dr.ir. T.J.J. van den Boom (Ton)

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English

### Course Contents:

Part 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
System identification example
Nonlinear Optimization without Constraints
Newton and quasi-Newton methods
Methods with direction determination and line search
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
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.

lectures

### Assessment:

written examination (closed book, no calculators)

### Remarks:

Old course code: SC4091

### Department:

3mE Department Delft Center for Systems and Control
© Copyright Delft Center for Systems and Control, Delft University of Technology, 2017.