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ECE 682

Multivariable Control Systems

Graduate

Fall 2025

Topics

  • List:
  • Introduction to linear multivariable systems Motivation, examples.
  • Linear state-space models Deriving state models, Solution of state equation, Realizations, Poles and zeros of a multivariable system.
  • Linear algebra Vector spaces, Linear transformations, Quotient spaces, Invariant subspaces.
  • Controllability Reachable states, Properties of controllability, PBH test, Equivalence of pole placement and con- trollability, Stabilizability.
  • Observability The Kalman decomposition, Detectability, Observers, Observer based controllers.
  • Quadratic optimal control Lyapunov equation, Riccati equation, The LQR problem and its solution.
  • Stability of feedback systems Well-posedness, Feedback stability, Output feedback stabilization.
  • Regulation and tracking Output regulation problem, Solution in the case of full information, Solution in the case of mea- surement feedback, Structurally stable synthesis. 2
  • TheH 2-optimal control problem Problem setup and motivation, Standard form of the problem and its solution, Regular form of the problem, Relationship to the LQG problem. 3

Readings

  • Linear System Theory and Design
    C. T. Chen · 3rd · recommended
    Optional suggested textbook.
  • Course notes available on the course website.
    unspecified
Fall 2024

Topics

  • List:
  • Introduction to linear multivariable systems Motivation, examples.
  • Linear state-space models Deriving state models, Solution of state equation, Realizations, Poles and zeros of a multivariable system.
  • Linear algebra Vector spaces, Linear transformations, Quotient spaces, Invariant subspaces.
  • Controllability Reachable states, Properties of controllability, PBH test, Equivalence of pole placement and con- trollability, Stabilizability.
  • Observability The Kalman decomposition, Detectability, Observers, Observer based controllers.
  • Quadratic optimal control Lyapunov equation, Riccati equation, The LQR problem and its solution.
  • Stability of feedback systems Well-posedness, Feedback stability, Output feedback stabilization.
  • Regulation and tracking Output regulation problem, Solution in the case of full information, Solution in the case of mea- surement feedback, Structurally stable synthesis.
  • The H2-optimal control problem Problem setup and motivation, Standard form of the problem and its solution, Regular form of the problem, Relationship to the LQG problem.

Readings

  • Linear System Theory and Design
    C. T. Chen · 3rd · recommended
    Optional suggested textbook.
  • Course notes available on the course website.
    unspecified
Fall 2023

Topics

  • List:
  • Introduction to linear multivariable systems Motivation, examples.
  • Linear state-space models Deriving state models, Linearization, Solution of state equation, Realizations, Poles and zeros of a multivariable system.
  • Linear algebra Vector spaces, Linear transformations, Quotient spaces, Invariant subspaces.
  • Controllability Reachable states, Properties of controllability, PBH test, Equivalence of pole placement and controllability, Stabilizability.
  • Observability The Kalman decomposition, Detectability, Observers, Observer based controllers.
  • Quadratic optimal control Lyapunov equation, Riccati equation, The LQR problem and its solution.
  • Stability of feedback systems Well-posedness, Feedback stability, Output feedback stabilization.
  • Regulation and tracking Output regulation problem, Solution in the case of full information, Solution in the case of measurement feedback, Structurally stable synthesis.

Readings

  • Linear System Theory and Design
    C. T. Chen · 3rd · recommended
    Optional suggested textbook.
  • Course notes available on the course website.
    unspecified
Fall 2022

Topics

  • in There are additional lectures planned on the following Wednesdays September 28, Oct. 5 and Nov 9 from
  • Introduction to linear matrix theory. Bases and representations. Notions of range, null spaces. Introduction to Hilbert spaces.
  • Linear dynamical systems and their matrix representation. The notion of Input/Output maps and transfer functions.
  • Stability and pole placement.
  • The notions of controllability, reachability, observability and detectability of systems. Canonical forms.
  • Introduction to Lyapunov stability theory. The Lyapunov equation.
  • Linear quadratic control. Ricatti equations. Steady state theory. Applications to robust control. 1
  • Observer theory and the separation between control and observation.
  • Introduction to optimal control. Hamilton Jacobi equation. 2

Readings

  • • Weekly problem sets will be handed out. The onus is on you all to attempt them. Solutions will be posted.
    unspecified
Fall 2021

Topics

  • List:
  • Introduction to linear multivariable systems Motivation, examples.
  • Linear state-space models Deriving state models, Linearization, Solution of state equation, Realizations, Poles and zeros of a multivariable system.
  • Linear algebra Vector spaces, Linear transformations, Quotient spaces, Invariant subspaces.
  • Controllability Reachable states, Properties of controllability, PBH test, Equivalence of pole placement and controllability, Stabilizability.
  • Observability The Kalman decomposition, Detectability, Observers, Observer based controllers.
  • Quadratic optimal control Lyapunov equation, Riccati equation, The LQR problem and its solution.
  • Stability of feedback systems Well-posedness, Feedback stability, Output feedback stabilization.
  • Regulation and tracking Output regulation problem, Solution in the case of full information, Solution in the case of measurement feedback.

Readings

  • Linear System Theory and Design
    C. T. Chen · 3rd · recommended
    Optional suggested textbook.
  • Course notes available on the course website.
    unspecified
Fall 2016

Readings

  • Modern Control Theory - Brogan
    unspecified
  • Linear System Theory and Design - C.T. Chen
    unspecified
Sources