B2378 - Theory of Systems and Controls for Automation

Learning outcomes

At the end of the course the student handles Linear Time-Invariant systems and their representation through Laplace transform and inverse transform and transfer functions, the basic principles of linear-system stability and the response modes (1st and 2nd order elementary systems and their composition to implemented higher-order systems). Students know how to use Bode and Nyquist diagrams and root locus to describe structural properties of linear systems, and how to design controllers for linear systems, lead and lag networks, PID controllers, cascade controllers. Nonlinear system control, with linearization-based control will also be introduced.

Course contents

1. Introduction to Control Systems. Examples of control systems. Definition of system: control and disturbance inputs, outputs, state variables. Principles of control systems design.
2. Mathematical Models of Systems. Differential equation models of physical systems; Linear approximations of nonlinear models. Modeling principles: Electric systems; Mechanical systems; Flow systems.
3. Input-Output Models. Laplace transform. Transfer functions. Forced and free response. Block diagrams. Signal-flow graphs. Examples.
4. State Variable Models. State-space models. Realization from differential equation models: Canonical forms. Solution of the state equation. Relation with input-output models.
   
5. Stability of Systems. Definitions. External stability (BIBO stability). Routh-Hurwitz criterion. Stability of state-space models: Internal stability and external stability. Transient and steady-state response.
6. Feedback Systems. Introduction. Error-signal Analysis. Sensitivity Analysis. Test input signals. Time-response of first- and second-order systems. Higher-order models. Transient response. Performance specification in time-domain and relation with pole location. Steady-state error of feedback control systems. Design examples.
   
7. Root Locus. Definition and procedure. Parameter design via the root locus method. Introduction to PID controllers. Anti-windup control. Design examples. Design using computer software.
8. Frequency Response Methods. Frequency response. Polar plots and Bode plots. Performance specifications in the frequency domain. Design Examples.
9. Feedback Control Systems Design. Cascade compensators. Phase-lead compensator networks: Frequency domain and root-locus design methods. Phase-lag design: Frequency domain and root-locus design methods. Lead-lag controllers. Pre-filters. Design examples.
10. Design Using State-Space Methods. Controllability and observability. Full-state feedback design. State-observer design. Output-feedback design using state-observers. Internal model design. Design examples.

Readings/Bibliography

• Main Textbook: Modern Control Systems, R. Dorf and R. Bishop, Prentice Hall, Global Edition, 2021.
  
• Auxiliary Textbook: Control Systems: An Introduction, H. Khalil, Michigan Publishing, 2023. Note: this textbook is freely available in electronic form at https://control.eecs.umich.edu
  
  Copies of the slides used in the lectures will be made available to the students. These are by no means to be considered exhaustive to achieve proficiency in the subject matter of the course.

Teaching methods

In-class lectures. Occasional remote lectures may be delivered in both synchronous and asynchronous mode for make-ups.

Assessment methods

Comprehensive written and oral exam.

The assessment of the student’s proficiency consists of an individual final written examination followed by an individual oral examination. There are no intermediate examinations.

The written examination is comprehensive and is intended to assess the student’s ability to analyze and design linear control systems using the methods presented in the course. It may include theoretical questions, numerical exercises, and design problems. Typically, students may be asked to:

• Derive mathematical models of physical systems using differential equations.
• Obtain linear approximations of nonlinear models around an equilibrium condition.
• Derive state-space and input-output representations of dynamical systems.
• Compute system responses in the time domain using Laplace-transform methods and partial-fraction expansions.
• Analyze the internal and external stability of a system.
• Apply the Routh–Hurwitz criterion.
• Analyze feedback systems in terms of transient response, steady-state error, sensitivity, and disturbance rejection.
• Determine the principal time-domain and frequency-domain performance indices.
• Construct and interpret root loci, Bode diagrams, and polar plots.
• Design PID controllers and phase-lead, phase-lag, or lead-lag compensators.
• Select controller parameters to meet prescribed stability and performance requirements.
• Interpret or complete block diagrams and relate input-output models to state-space representations.
  

The oral examination is based partly on the student’s written work and partly on the general course content. During the first part of the oral examination, the student may be asked to explain, justify, correct, or further develop the solutions presented in the written examination. The purpose of this discussion is to verify the student’s understanding of the methods used and to distinguish conceptual understanding from the mere application of standard procedures. The written examination and the corresponding portion of the oral examination are evaluated jointly and carry a maximum score of 30/30.

The oral examination also includes at least one additional question on a topic that was not covered in the written examination. This question is intended to verify the breadth of the student’s preparation and the ability to establish connections among different parts of the course. The additional oral question carries a maximum weight of 10/30, which is added to the score obtained in the first part of the examination.

In the students's evaluation, particular emphasis is placed on the correctness of the methodology, the clarity and rigor of the reasoning, the ability to interpret the results from a control-systems perspective, and the appropriate use of mathematical notation and technical terminology. This latter point is not negotiable: Points will be taken out for sloppiness and/or carelessness.

Class participation is considered very important for understanding the course material, although attendance does not directly contribute to the final grade.

Teaching tools

• Examples and recitation sessions.
• Computer-aided design tools (MATLAB & SIMULINK)

Office hours

See the website of Andrea Serrani