History; examples; future directions.

# Feedback systems

Often we are interested in finding and characterizing luginattern 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:** Fridays 11:00am – 1:00pm

**Level:** undergraduate

**TA:** Javier Alejandro Velasco

## Schedule

Week starting on… | Lectures (Mondays) | Recitations (Wednesdays) | Labs (Fridays) |
---|---|---|---|

July 24 | Lec 1 (intro) | ||

July 31 | Lec 2 (modeling) | ||

Aug. 7 | - | Lec 3 (stability) | Lab 1 |

Aug. 14 | Lec 4 (Lyapunov) | Quiz 1 | |

Aug. 21 | Lec 5 (Lasalle) | - | |

Aug. 28 | Lec 6 (linearization) | Project | Lab 2 |

Sept. 4 | Lec 7 (reachability) | Quiz 2 | |

Sept. 11 | Lec 8 (observability) | Quiz 3 | |

Sept. 18 | Lec 9 (transfer functions) | Midterm review | Lab 3 |

Sept. 25 | Lec 10 (transfer functions) | Midterm I | |

Oct. 2 | Lec 11 (Nyquist) | First report | |

Oct. 9 | Lec 12 (PID) | Quiz 4 | Lab 4 |

Oct. 16 | - | Lec 13 (PID) | |

Oct. 23 | Lec 14 | Project | |

Oct. 30 | Last class | - | Lab 5 |

Nov. 6 | Final review | - |

## Assignments

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

**Homework sets**

HW1 – due August 18th (submit)

HW2 – due August 25th (submit)

HW3 – due September 1st (submit)

HW4 – due September 8th (submit)

HW5 – due September 15th (submit)

HW6 – due September 22nd (submit)

HW7 – due September 29th (submit)

HW8 – due October 6th (submit)

HW9 – due October 13th (submit)

HW10 – due October 27th (submit)

**Project**

**Homework sets**

HW1 – due August 18th (submit)

HW2 – due August 25th (submit)

HW3 – due September 1st (submit)

HW4 – due September 8th (submit)

HW5 – due September 15th (submit)

HW6 – due September 22nd (submit)

HW7 – due September 29th (submit)

HW8 – due October 6th (submit)

HW9 – due October 13th (submit)

HW10 – due October 27th (submit)

**Project**

**Lab assignments**

Lab1 – due September 1st

Lab2 – due September 22nd

Lab3 – due October 13th

Lab4 – due November 3rd

Lab5 – due November 17th

**Past midterm and final exams**

**Midterm solutions**

**Lab assignments**

Lab1 – due September 1st

Lab2 – due September 22nd

Lab3 – due October 13th

Lab4 – due November 3rd

Lab5 – due November 17th

**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