# Modern Control Theory

Control Theory 4

**Prerequisites:**CT 3

## Introduction

While classical control theory involves solving differential equations in a frequency domain (Laplace/Fourier/Z-transform), modern control theory solves in state-space (time domain). In classical control, we analyze SISO (single input, single output) systems. In addition to these, modern control theory can handle MIMO (multiple input, multiple output) as well as non-linear and time-variant systems (which describe most practical problems).

The state-space approach (known as modern control theory) is not something relatively new; it gained popularity after the 60's because it involved numerically (and computer) friendly matrix operations.

The need to move further from simple controllers was first posed by aerospace researchers due to the MIMO systems that they wanted to control. For space applications like orbital maneuvers, attitude control or terminal rendezvous (Hill-Clohessy-Wiltshire equations), state-space approaches might be best suited. In robotics, reasonably accurate control can be obtained via classical controllers applied to SISO sytem or decoupled MIMO sytems. However, most of the optimal control applications such as legged locomotion (hybrid systems), aerial acrobatics, swarm control, industrial robots and other highly articulated robots use state-space.

## Linear Algebra

Modern control theory applies tools from linear algebra to model and control physical systems. We recommend watching the Intro to Linear Algebra video series by 3Blue1Brown, which provides students with a geometrical understanding of linear algebra topics. Python will be used for performing most of the matrix operations for our purposes.

## Submodules

Covers design of linear estimators in the context of modern control.

## State-Space Equations

Read this article for an introduction to state-space notation and how it relates to classical control approaches.

## Standard Forms

Read this article for a description of controllable and observable canonical forms.