# ECE 410H1: Linear Control Systems

This is the public-facing website for the University of Toronto course ECE410H1 "Linear Control Systems". This website is unofficial and for informational purposes only. 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, full-order and reduced-order 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 computer-aided 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 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.

• Basic properties: We will review ways to represent LTI control systems, and how to linearize a nonlinear control system about an equilibrium in order to get an LTI model. We will investigate how to predict the qualitative properties of solutions of an LTI system without computing its solutions.

• 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.

• Quadratic optimal control: We will learn how to design a state feedback controller that minimizes a quadratic cost function. In the process, we will be introduced to the principles of optimal control: dynamic programming and the Hamilton Jacobi Bellman (HJB) equation.

• 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

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: 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.