History; examples; future directions.

# Feedback systems

Often we are interested in finding and characterizing pattern which appear over time. This lecture introduces formal tools to design, model, and analyze evolving socio-technological systems in which information, data sensing, and decision-making mechanisms strongly couple two or more systems. Formal techniques include frequency-domain and time-domain methods.

## Overview

**Instructor: **Jorge Finke

**Office hours:** Tu 11:00am – 1:00pm

**Level:** undergraduate

**TA:** Danny Cartagena

## Schedule

Week | Lectures | Recitations | Labs |
---|---|---|---|

Jan 23 | Lec 1 - Intro to feedback systems | ||

Jan 30 | Lec 2 - Dynamics and modeling | Lec 3 - Stability and performance / HW 1 | |

Feb 6 | Lec 4 - Lyapunov stability theory | Quiz 1 / HW 2 | |

Feb 13 | Lec 5 - Lasalle theorem | HW 3 / Project | |

Feb 20 | Lec 6 - Linearization | HW 4 | Lab 1 |

Feb 27 | Lec 7 - State feedback | HW 5 | |

Mar 6 | Lec 8 - Output feedback | HW 6 | |

Mar 13 | Review | Quiz 2 / HW 7 | Lab 2 |

Mar 20 | — | Midterm / HW 8 | |

Mar 27 | spring break | ||

Apr 3 | Lec 9 - Transfer functions | Project | |

Apr 10 | Lec 10 - Frequency domain design | HW 9 | Lab 3 |

Apr 17 | Lec 11 - Nyquist plots | Quiz 3 | |

Apr 24 | Lec 12 - PID controlers | Lab 4 | |

May 1 | Lec 13 - PID controlers | Quiz 4 | |

May 8 | Lec 14 - Model validation | HW 10 | Lab 5 |

May 15 | Project presentations | Review |

## Assignments

To access the problem sets and lab assignments please use the password provided in class.

**Homework sets**

**Project**

**Homework sets**

**Project**

**Lab assignments**

Lab1 – due March 3rd

Lab2 – due September 4th

Lab3 – due September 18th

Lab4 – due October 30th

Lab5 – due November 18th

**Past midterm and final exams**

**Midterm solutions**

**Lab assignments**

Lab1 – due March 3rd

Lab2 – due September 4th

Lab3 – due September 18th

Lab4 – due October 30th

Lab5 – due November 18th

**Past midterm and final exams**

**Midterm solutions**

**Notebooks (how to..)**

Create state space representations and phase diagrams

Find Lyapunov functions for linear modes

Test reachability and observability

Create transfer function representations and root locus diagrams

**Notebooks (how to..)**

Create state space representations and phase diagrams

Find Lyapunov functions for linear modes

Test reachability and observability

Create transfer function representations and root locus diagrams

## Lessons

**List of lectures**

### 1. Intro to feedback systems

### 2. Dynamics and modeling

What a is a dynamic model? What does a model say about a system? Define concepts of state, dynamics, inputs and outputs. Non-linear vs. linear systems; overview modeling techniques.

### 3. Stability and performance

Lyapunov stability for time-invariant systems.

### 4. Lyapunov stability theory

Lyapunov stability for time-invariant systems.

### 5. Lasalle theorem

Krasovskii-Lassale Invariance Principle; Lyapunov functions for linear systems.

### 6. Linearization

Compute linearization of a nonlinear system around an equilibrium point; Lyapunov Indirect Method; Examples.

### 7. State feedback

Define reachability of a system; test for reachability of linear systems; state feedback for linear systems

### 8. Output feedback

Define observability; conditions for linear systems; state estimation; examples.

### 9. Transfer functions

Content: Transfer function; Routh-Hurwitz criterion; Canonical form realizations

### 10. Frequency domain design

Bode plot; Sketching Bode Plots; Block algebra.

### 11. Nyquist plots

Loop transfer function; Nyquist plots; Stability margins.

### 12. PID

PID (proportional–integral–derivative controller); basic properties; PID implementation.

### 13. PID

PID implementation; Empirical tuning methods.

### 14. System identification

Identification of linear time invariant systems

### 15. Model validation

Validation of linear time-invariant models using barrier certificates