Learning objectives
Provide basic concepts on linear systems theory.
Prerequisites
Automatic control fundamentals.
Geometry.
Course unit content
Elements of linear algebra.
State space description of a linear system.
Solution of continuous and discrete-time linear systems.
Controllability and observability for continuous and discrete-time linear systems.
State-feedback stabilization.
Asymptotic observers.
Elements of LQR control and Kalman filtering.
Full programme
1) Elements of linear systems modelling.
2) Review of linear algebra.
-Vector spaces, subspaces, linear applications, determinant, eigenvalues and eigenvectors.
-Generalized eigenvalues and eigenvectors, primary decomposition theorem, Hamilton-Cayley theorem.
3) Continuous-time systems.
-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.
-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.
-Definitions.
-Reachability matrix.
-Properties.
6) Reachability and controllability for continuous-time systems.
-Definitions.
-Reachability Gramian.
-Properties.
-Standard form for non-completely reachable systems.
-PBH test for reachability.
7) Observability and reconstructability for discrete-time systems.
-Definitions.
-Observability matrix.
-Properties.
8) Observability and reconstructability for continuous-time systems.
-Definitions .
-Observability Gramian.
-Properties.
-Standard form for non-completely observable systems.
9) Kalman decomposition.
10) Stability.
-Equilibria.
-Simple and asymtptic stability.
-BIBO stability.
11) Stabilization with state-input feedback.
-Stabilizability.
-The companion form and its properties.
-The canonic control form.
-Ackermann's formula.
-The pole placement theorem for multi-input systems.
12) Observers
-Open-loop observer.
-Luenberger observer.
-Detectability.
-Dual system.
-Conditions for detectability.
-Separation principle.
13) Optimal control.
-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.
Bibliography
For consultation:
-A Linear Systems Primer, authors:Antsaklis, Michel, editor: Birkhauser
Teaching methods
Lectures at the blackboard.
Assessment methods and criteria
Written and oral exam.
Other information
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