Learning objectives
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 observer with pole placement.
Prerequisites
- Automatic control fundamentals
- Linear algebra
- Geometry
Course unit content
1) Overview of linear system modeling.
2) Review of linear algebra.
3) Continuous-time autonomous systems.
4) Discrete-time autonomous systems.
5) Non-autonomous systems.
6) Reachability and controllability for discrete-time systems.
7) Reachability and controllability for continuous-time systems.
8) Stability.
9) Stabilization with state-input feedback.
10) Observability and reconstructibility for discrete-time systems.
11) Observability and reconstructibility for continuous-time systems.
12) Observers.
Full programme
1) Overview of linear system modeling.
- Examples of electrical circuits and definition of control system
2) Review of linear algebra.
- Vector spaces, Gauss reduction, matrix inversion calculation, eigenvalues, and eigenvectors.
- Invariant subspaces, generalized eigenvalues and eigenvectors, primary decomposition theorem, Hamilton-Cayley theorem.
3) Continuous-time autonomous systems.
- Fundamental matrix of the system and its properties.
- Matrix exponential: definition and computation.
- Laplace transform.
4) Discrete-time autonomous systems.
- Fundamental matrix of the system and its properties.
- Matrix power: definition and computation.
- Z-transform.
5) Non-autonomous systems.
- Impulse response and transfer function.
- Sampling of continuous-time systems.
- Internal and external equivalence of systems.
6) Reachability and controllability for discrete-time systems.
- Definitions.
- Reachability matrix.
- Properties.
7) Reachability and controllability for continuous-time systems.
- Definitions.
- Reachability Gramian.
- Properties.
- Standard form for non-completely reachable systems.
- PBH test.
8) Stability.
- Equilibrium states.
- Simple and asymptotic stability.
- BIBO stability.
9) Stabilization with state-input feedback.
- Definition of stabilizable pair.
- Companion form and its properties.
- Canonical control form.
- Ackermann’s formula.
- Eigenvalue assignment for systems with more than one input.
- Pole assignment theorem.
10) Observability and reconstructibility for discrete-time systems.
- Definitions.
- Observability matrix.
- Properties.
11) Observability and reconstructibility for continuous-time systems.
- Definitions.
- Observability Gramian.
- Properties.
- Standard form for partially unobservable systems.
12) Observers.
- Open-loop observer.
- Luenberger observer.
- Definition of detectable pair, conditions, and duality properties.
- Separation principle.
Bibliography
"A Linear Systems Primer", author: Antsaklis, Michel, editor: Birkhauser.
Teaching methods
Lectures conducted on the blackboard by the teacher and exercises carried out on the blackboard by the students, guided by the teacher.
The teaching materials used to support the lectures will be uploaded to the Elly platform during the course.
To download the teaching materials, it is necessary to enroll in the online course.
Non-attending students are reminded to check the available teaching materials and the instructions provided by the teacher through the Elly platform.
Assessment methods and criteria
The summative assessment of learning includes two phases:
1) A written exam consisting of six exercises in which the student must solve problems related to the analysis of linear systems and the design of controllers and observers.
The duration of the written exam is 3 hours.
The written exam is graded on a scale of 0-30.
The complete written exam can be replaced by two partial in-term written exams. Each partial exam lasts 2 hours.
The partial exams are graded on a scale of 0-30.
The arithmetic mean of the grades from the two partial exams equates to the grade of the complete written exam.
The grade of the complete written exam and the partial written exams will be communicated within two weeks after the exams, through publication on the Elly platform.
2) An oral exam consisting of the discussion of two theoretical topics covered during the course. During the oral exam, it will be verified that the student can present the concepts, definitions, and results of the required topics in a structured manner, presenting formal proofs of properties and showing mastery of the necessary logical-mathematical tools.
The oral exam is graded on a scale of 0-30.
To obtain the final grade, it is necessary to calculate the arithmetic mean of the grades from the oral exam and the complete written exam (both on a scale of 0-30). The teacher reserves the right to assign additional points for particularly outstanding oral exams or for achieving honors.
The final grade is communicated immediately at the end of the oral exam.
To be eligible for the oral exam, the grade of the complete written exam or each individual partial written exam must be at least 18 out of 30.
One must take the oral exam in the same exam session as the complete written exam. In case of successfully completing the two partial written exams, it is possible to take the oral exam during one of the first two exam sessions of 2025.
It is reminded that online registration for the exam is MANDATORY both for the written exams and for the oral exam.
On the Elly platform, past written exam texts and solutions will be available, which are formative and useful for monitoring the achievement of learning objectives during the course and providing feedback to students before the official exams.
The dates of the partial written exams will be communicated by the teacher both during the lessons and on the Elly platform.
Other information
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