ECE 557H1: Linear Control Theory
This is a public-facing page for the University of Toronto course ECE557H1 "Linear Control Theory". This page is unofficial, for informational purposes only, and is probably out of date. Enrolled students should consult the course website hosted on Quercus for the official course information sheet.
Course Information
Calendar Description
State-space approach to linear system theory. Mathematical background in linear algebra, state space equations vs. transfer functions, solutions of linear ODE?s, state transition matrix, Jordan form, controllability, eigenvalue assignment using state feedback, observability, designing observers, separation principle, Kalman filters, tracking and the regulator problem, linear quadratic optimal control, stability. Laboratories cover the state space control design methodology.
Detailed Course Description and ILOs
In a previous control course you were introduced to the so-called classical control theory, relying on frequency-domain methods to design simple feedback loops for single-input single-output linear time-invariant (LTI) control systems. In the 1960s, motivated primarily by the need to solve control problems involving many inputs and outputs, so-called state-space methods were developed. In this state-space setting, new kinds of control problems were formulated, and new design tools applicable to multivariable control problems were developed. This course covers the fundamental properties of LTI state-space models, and introduces the foundational elements of linear controller design in the state-space setting. Our progression will be as follows.
Stabilization of equilibria via state feedback: When does there exist a state feedback controller capable of stabilizing the origin of an LTI control system? The answer to this question will take us through a journey through controllability and Kalman decompositions of control systems. In the process, we will learn how to synthesize state feedback controllers.
State estimation: Suppose we have sensors which record measurements from a LTI state-space model. Under what conditions do the sensors contain enough information to reconstruct the state? We will answer this question completely and, using the insight we gain in the process, we will learn how to design an algorithm for state estimation.
Output feedback stabilization of equilibria and tracking: Using state feedback controllers and state estimators, we will design a controller using sensor feedback to stabilize the origin an LTI control system. We will then extend this foundational design to enable reference tracking via integral control.
The table below shows an approximate timeline for course topics.
Week 1 | State-space and transfer functions; equilibrium points; linearization |
Week 2 | The matrix exponential; solutions of state-space models |
Week 3 | Modes and qualitative behaviour; phase portraits; stability |
Week 4 | Stability, asymptotic stability, BIBO and I2O stability of LTI systems |
Week 5 | Linear algebra review and advanced concepts; A-invariance |
Week 6 | Controllability; the reachable set; Kalman's test and decomposition |
Week 7 | Controllable form; state feedback; the Wonham pole placement theorem |
Week 8 | The PBH test; linear quadratic control |
Week 9 | Fall Break |
Week 10 | Observability and detectability; duality of control and state estimation |
Week 11 | Observer design; The separation theorem for output feedback stabilization; Kalman's decomposition |
Week 12 | internal vs. I/O stability; integral tracking control
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By the end of this course, students will be able to:
convert between transfer function and state-space LTI model representations
linearize a nonlinear system around an equilibrium point to obtain an LTI model
quantitatively compute and qualitatively explain the solutions of state-space LTI models
assess stability of LTI models, explain relationships and distinctions between stability concepts
explain the concepts of controllability/stabilizability and assess controllability/stabilizability of LTI systems
explain the concepts of observability/detectability and assess observability/detectability of LTI systems
design stabilizing state feedback controllers and state observers via eigenvalue assignment and optimal control methods
Prerequisites
Prerequisite: ECE356H1 or AER372H1.
Exclusion: ECE410H1.
Course Notes
M. Maggiore, Foundations of Linear Control Theory, available on Quercus.
Supplementary References
S. Axler, Linear Algebra Done Right, 3rd edition, Springer, 2015.
J.P. Hespanha, Linear Systems Theory, 2nd edition, Princeton University Press, 2018.
C.-T. Chen, Linear Systems Theory and Design, 3rd edition, Oxford University Press, 1999.
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