ECE 410H1: Linear Control Systems
This is a publicfacing page for the University of Toronto course ECE410H1 "Linear Control Systems". This page is unofficial, for informational purposes only, and is probably out of date.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 analysis of linear systems, the matrix exponential, linearization of nonlinear systems. Structural properties of linear systems: stability, controllability, observability, stabilizability, and detectability. Pole
assignment using state feedback, state estimation using observers, fullorder and reducedorder observer design, design of feedback compensators using the separation principle, control design for tracking. Control design based on optimization, linear quadratic optimal control, the algebraic Riccati equation. Laboratory experiments include computeraided design using MATLAB and the control of an inverted pendulum on a cart.
Detailed Course Description and ILOs
In a previous control course you were introduced to the socalled classical control theory, relying on frequencydomain methods to design simple feedback loops for singleinput singleoutput linear timeinvariant (LTI) control systems. In the 1960s, motivated primarily by the need to solve control problems involving many inputs and outputs, socalled statespace methods were developed. In this statespace 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 statespace models, and introduces the foundational elements of linear controller design in the statespace 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 statespace 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  Statespace and transfer functions; equilibrium points; linearization 
Week 2  The matrix exponential; solutions of statespace 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; Ainvariance 
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

By the end of this course, students will be able to:
convert between transfer function and statespace LTI model representations
linearize a nonlinear system around an equilibrium point to obtain an LTI model
quantitatively compute and qualitatively explain the solutions of statespace 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: ECE311H1.
Exclusion: ECE557H1.
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.
