Fundamentals of linear time-invariant control systems. State space modeling and control design, controllability, stabilization, pole placement controllers, observability, Kalman filters, observer design, optimal control, tracking controllers. Labs: real-time control experiments on design techniques.
Prerequisite: ECE356H1.

The course presents a more advanced treatment of linear control theory via the geometric approach. The coverage roughly corresponds to the first six chapters of “Linear Multivariable Control: A Geometric Approach”, by W.M. Wonham. We adopt the abstract algebra approach of the text to study controllability, observability, controlled invariant subspaces, controllability subspaces, and controllability indices. These concepts are applied to solve the problems of stabilization, output stabilization, disturbance decoupling, and the restricted regulator problem. Areas of current research in linear geometric control will also be discussed.

Prerequisite: ECE410H1 or ECE557H1 or an equivalent State Space Control course.

This course presents recent developments on control of underactuated mechanical systems, focusing on the notion of virtual constraint.

Traditionally, motion control problems in robotics are partitioned in two parts: motion planning and trajectory tracking. The motion planning algorithm converts the motion specification into reference signals for the robot joints. The trajectory tracker uses feedback control to make the robot joints track the reference signals. There is an emerging consensus in the academic community that this approach is inadequate for sophisticated motion control problems, in that reference signals impose a timing on the control loop which is unnatural and inherently non robust.

The virtual constraint technique does not rely on any reference signal, and does not impose any timing in the feedback loop. Motions are characterized implicitly through constraints that are enforced via feedback. Through judicious choice of the constraints, one may induce motions that are surprisingly natural and biologically plausible. For this reason, the virtual constraints technique has become a dominant paradigm in bipedal robot locomotion, and has the potential of becoming even more widespread in other area of robot locomotion.

The virtual constraint approach is geometric in nature. This course presents the required mathematical tools from differential geometry and surveys the basic results in this emergent research area. Topics covered will include:

– Differentiable manifolds and basic operations.
– Controlled invariant manifolds and zero dynamics of nonlinear control systems
– Euler-Lagrange robot models and models of impulsive impacts
– Virtual holonomic constraints (VHCs)
– Constrained dynamics resulting from VHCs, and conditions for existence of a Lagrangian structure
– Virtual constraint generators
– Stabilization of periodic orbits on the constraint manifold.
– Virtual constraints for walking robots

Required backgound: This course has no formal prerequisites, but assumes knowledge of vector calculus and linear algebra. Ideally, the student taking this course will have taken an introductory course on nonlinear control theory, such as ECE1647F, and be familiar with the Lagrangian modelling of robots from a course like ECE470.

Prerequisites: ECE410 or ECE557 or equivalent.

This course is an introduction to the control of discrete, asynchronous, nondeterministic systems like manufacturing systems, traffic systems, and certain communication systems. Architectural issues (modular, decentralized and hierarchical control) are emphasized. The theory is developed in an elementary framework of automata and formal languages, and is supported by a software package for creating applications. There are no special prerequisites.

This course is a mathematical introduction to nonlinear control theory, a subject with roots in dynamical systems theory, mechanics, and differential geometry. The focus of this course is on the dynamical systems perspective. The material covered in this course finds application in fields as diverse as orbital mechanics and aerospace engineering, circuit theory, power systems, robotics, and mathematical biology, to name a few. The course is organized in four chapters, as follows.

1. Vector Fields and Dynamical Systems: Finite dimensional dynamical systems, vector fields, and their equivalence. Existence and uniqueness of solutions of ODEs.
2. Foundations of Dynamical Systems Theory: Invariant sets and their characterization by the Nagumo theorem. Limit sets as a tool to characterize the asymptotic behaviour of bounded orbits. Limit sets of two-dimensional systems: the Poincaré-Bendixson theorem. Poincaré theory of stability of closed orbits. Linearization of vector fields about equilibria. Linearization of vector fields about closed orbits.
3. Foundations of Stability Theory: Equilibrium stability and its characterization by means of Lyapunov’s theorem. Domain of attraction of an equilibrium. The Krasovskii-LaSalle invariance principle. Stability of LTI systems, and exponential stability of equilibria. Converse stability theorems.
4. Introduction to Nonlinear Stabilization: Control-Lyapunov functions. Parametrization of equilibrium stabilizers by CLFs (Artstein-Sontag theorem). Passive systems and passivity-based equilibrium stabilization. Passivity  of mechanical control systems. Port-Hamiltonian systems.

The course explores design of control systems that achieve complex specifications. This is an emerging area in control theory that contrasts with traditional control design focused on stabilization and tracking. We introduce linear temporal logic (LTL) and show how LTL specifications capture a rich class of transient and steady-state behaviours of control systems. The LTL control problem is reduced to a hybrid control problem using ideas from computer science to obtain a design that includes high-level discrete algorithms and low-level continuous time controllers. The course covers the most important techniques and tools that come to play in this methodology, including triangulation, behaviour of affine systems on polytopes, the Reach Control Problem (RCP), and flow functions. We explore in depth control synthesis methods for the RCP based on affine, continuous state, and piecewise affine feedbacks. The techniques studied in the course come together to solve the problem of motion planning for a group of quadrocopters.
Prerequisites: ECE410 or ECE557 or equivalent.

This course presents a mathematical treatment of classical and evolutionary game theory. Topics covered in classical game theory: matrix games, continuous games, Nash equilibrium (NE) solution, existence and uniqueness, best-response correspondence. Topics covered in evolutionary games:  evolutionary stable strategy concept, population games, replicator dynamics, relation to dynamic asymptotic stability. Learning in games: imitation dynamics, fictitious play and their relation to replicator dynamics. Applications to engineering: communication networks, multi-agent learning. There is no required textbook.  PDF course notes are available; the notes are self-contained and serve as a textbook. Weekly formal lectures based on the course notes.

The main purpose of this course is to introduce a series of distinct topics in control to students who have not seen control system design beyond a first course in control, which includes classical methods such as root locus, and Bode design, for example. The topics discussed in MIE 1064 F are selected to give students a broad overview of a variety of control design methods and concepts in stability.

Prerequisite: At least one introductory course in control is required and a Mechanical Engineering course in one of either Mechanisms or Vibrations.

This course provides an introduction to the main concepts in nonlinear control system design. The
emphasis in this class is on issues encountered in application to physical systems. The first portion of this course reviews stability analysis tools for nonlinear systems. The second portion of this class focuses on control design using methods such as feedback linearization, sliding mode control, and adaptive control. The last portion of the class looks at practical implementation for applications of interest such as robotics, flight control, and process control.