MECH 513 - Control Systems

State-space modelling and related linear algebra. Controllability and observability of linear time-invariant systems and corresponding tests, system realizations. Stability including bounded-Input-Bounded-Output (BIBO), internal, and Lyapunov stability. Linear state feedback control, state observers, optimal control, optimal estimation, and robust control.

Instructor: Prof. James Richard Forbes

Course Overview

MECH 513 introduces modern control theory using state-space methods. Students develop mathematical tools for the analysis and synthesis of dynamical systems, with particular emphasis on controllability, observability, stability analysis, state feedback control, state estimation, optimal control, and robust control.

Topics include:

  • State-space modelling and system realizations
  • Lyapunov stability theory
  • Linear matrix inequalities (LMIs)
  • Controllability and observability
  • State-feedback control and pole placement
  • State observers
  • Optimal control and estimation
  • H₂ and H∞ control
  • Robust stability and robust performance

Prerequisite and Corequisite Courses

Students are expected to be comfortable with:

  • Calculus
  • Differential equations
  • Linear algebra
  • Python programming
Undergraduate Prerequisite Courses
  • MECH 412 or MECH 419
Graduate Students

No formal prerequisite.

Python Resources

The course makes extensive use of Python, the python-control package, and cvxpy.

Useful resources:

Learning Outcomes

By the end of the course, students will be able to:

  • Use linear algebra tools to manipulate and analyze linear dynamical systems.
  • Assess whether a system is controllable and observable, and understand the implications of such assessments.
  • Design state-feedback controllers and state observers using state-space methods.
  • Formulate and solve optimal control and estimation problems.
  • Analyze uncertainty, robustness, and performance of feedback control systems.
  • Apply modern computational tools to control-system analysis and design.

Textbooks

There is no required textbook. However, lectures are based on the following references.

  • S. Skogestad and I. Postlethwaite, Multivariable Feedback Control: Analysis and Design, 2nd ed. Hoboken, NJ: John Wiley & Sons, Inc., 2005.
  • J. Hespanha, Linear Systems Theory. Princeton, NJ: Princeton University Press, 2009.
  • R. L. Williams and D. A. Lawrence, Linear State-Space Control Systems. Hoboken, NJ: John Wiley & Sons, Inc., 2007.
  • K. Zhou and J. C. Doyle, Essentials of Robust Control. Upper Saddle River, NJ: Prentice Hall, 1998.

Additional Resources

Schedule

Week Date Topic Materials
1 Sep/Jan State-space and Transfer Matrix Representations

State-space and transfer function/matrix forms, linearization, and frequency response.
2 Sep/Jan Lyapunov Stability Theory

Lyapunov and asymptotic stability.
3 Sep/Jan Lyapunov Equations

Lyapunov equations and their application to stability analysis.
4 Sep/Jan Linear Matrix Inequalities

Introduction to LMIs and control-system applications.
5 Oct/Feb Controllability

Controllability concepts, tests, and implications.
6 Oct/Feb Observability

Observability concepts, tests, and system realizations.
7 Oct/Feb State Feedback Control

Static full-state feedback and pole placement.
8 Oct/Feb State Observers

Full-order and reduced-order observer design.
9 Nov/Mar Observer-Based Controllers

Separation principle and output-feedback control.
10 Nov/Mar The Generalized Plant

Motivation, performance channel, exogenous input channel, examples.
11 Nov/Mar Optimal Control

Problem formulation, linear quadratic regulator (LQR) design.
12 Nov/Mar H₂ and H∞ Control

Norms, full-state feedback control, and estimation.
13 Dec/Apr Robust Control

Uncertainty modelling, robust stability, and robust performance.