# LINEAR MULTIVARIABLE SYSTEMS

## Learning outcomes of the course unit

Knowledge and understanding:

- Understanding the state-space representation of linear systems.

- Understanding the method for analitically solving linear dynamic systems.

- Understanding issues of reachability and observability in control systems.

- Understanding basic elements of optimal control theory and Kalman filtering.

Applying knowledge and understanding:

- Represent a linear systems in state-space form.

- Compute forced and free evolution of a linear system.

- Decompose a systems in its reachable and observable parts.

- Design a regulator based on state-to-input feedback, both with pole placement and with optimal control methods.

- Design an asymptotic state observerm both with pole placement and with Kalman filtering.

## Prerequisites

Automatic control fundamentals.

Geometry.

## Course contents summary

1) Elements of linear systems modelling.

2) Review of linear algebra.

3) Continuous-time systems.

4) Discrete-time systems.

5) Reachability and controllability for discrete-time systems.

6) Reachability and controllability for continuous-time systems.

7) Observability and reconstructability for discrete-time systems.

8) Observability and reconstructability for continuous-time systems.

9) Kalman decomposition.

10) Stability.

11) Stabilization with state-input feedback.

12) Observers

13) Optimal control.

13) Kalman filter.

## Course contents

1) Elements of linear systems modelling. (2 hrs)

2) Review of linear algebra. (8 hrs)

-Vector spaces, subspaces, linear applications, determinant, eigenvalues and eigenvectors.

-Generalized eigenvalues and eigenvectors, primary decomposition theorem, Hamilton-Cayley theorem.

3) Continuous-time systems. (8 hrs)

-The fundamental solution matrix and its properties.

-Matrix exponential: definition and computation.

-Modes of a system.

-Total evolution.

-Impulse response and transfer function.

4) Discrete-time systems. (4 hrs)

-The fundamental solution matrix and its properties.

-Computation of the matrix power.

-Modes of a system.

-Total evolution.

-Impulse response and transfer function.

-Sampling of continuous-time systems.

5) Reachability and controllability for discrete-time systems. (4 hrs)

-Definitions.

-Reachability matrix.

-Properties.

6) Reachability and controllability for continuous-time systems. (6 hrs)

-Definitions.

-Reachability Gramian.

-Properties.

-Standard form for non-completely reachable systems.

-PBH test for reachability.

7) Observability and reconstructability for discrete-time systems. (4 hrs)

-Definitions.

-Observability matrix.

-Properties.

8) Observability and reconstructability for continuous-time systems. (8 hrs)

-Definitions .

-Observability Gramian.

-Properties.

-Standard form for non-completely observable systems.

9) Kalman decomposition. (4 hrs)

10) Stability. (4 hrs)

-Equilibria.

-Simple and asymtptic stability.

-BIBO stability.

11) Stabilization with state-input feedback. (8 hrs)

-Stabilizability.

-The companion form and its properties.

-The canonic control form.

-Ackermann's formula.

-The pole placement theorem for multi-input systems.

12) Observers (6 hrs)

-Open-loop observer.

-Luenberger observer.

-Detectability.

-Dual system.

-Conditions for detectability.

-Separation principle.

13) Optimal control. (6 hrs)

-Riccati equation.

-Hamiltonian matrix.

-Conditions for the existence of a solution of the Riccati equation.

13) Kalman filter.

-Review of random variables and stochastic processes.

-Evolution of linear systems affected by white gaussian noise.

-Riccati equation for the synthesis of the Kalman optimal observer.

## Recommended readings

For consultation:

-A Linear Systems Primer, author: Antsaklis, Michel, editor: Birkhauser.

## Teaching methods

Lectures and exercises given by the teacher on the blackboard or on a tablet screen. The lectures will be delivered online.

## Assessment methods and criteria

Written and oral exam. In the written exam the student will solve problems related to the analysis of linear system and to the design of regulators and observers.

The oral exams will include theoretical questions and exercises.

The written exam can be substituted by a midterm exam on the first part of the course and a final exam on the second part of the course.

The final mark will be given for 2/3 by the written exam and for 1/3 by the oral exam.