# CDS 101/110 - Linear Systems

CDS 101/110a | ← Schedule → | Recitations | FAQ | () |

## Overview

**Monday:** Linear Time-Invariant Systems (Slides, MP3)

This lecture gives an introduction to linear input/output systems. The main properties of linear systems are given and the matrix exponential is used to provide a formula for the output response given an initial condition and input signal. Linearization of nonlinear systems as an approximation of the dynamics is also introduced.

- Lecture handout
- MATLAB code: L4_1_linsys.m

**Wednesday:** Linear Systems Analysis (Notes, MP3)

Further analysis of linear systems, including a derivation of the convolution integral and the use of Jordan form. This lecture also covers the use of linearization to approximate the dynamics of a nonlinear system by a linear system.

- Lecture notes

**Friday:** recitations

## Reading

- K. J. Åström and R. M. Murray,, Princeton University Press, 2008..

## Homework

This homework set covers linear control systems. The first problem asks for stability, step and frequency response for some common examples of linear systems. The second problem considers stabilization of an inverted pendulum on a cart, using the local linearization. The remaining problems (for CDS 110 students) include derivation of discrete time linear systems response functions.

- hw3 - 101
- hw3 - 110
- hw3 - 210
- balance_simple.mdl - SIMULINK model of a balance system
- ambode.m - Bode plot with AM unit choices

## FAQ

**Monday**

**Wednesday**

- Could you please make it clear what is an 'A' and what is a 'Lambda' (matrix)?
- Do complex matrices also have a Jordan canonical form?
- How do you actually find the Jordan canonical form of a matrix?
- I keep getting mixed up on whether the diagonalized form of A is T^(-1)AT or TAT^(-1). Is there an easy way to remember the correct form?
- What does "up to permutations" mean?

**Friday**

**Homework**