State-space representation for automatic control
Download in PDF- Lecturer: Mahdi Khoramshahi (mahdi.khoramshahi@isir.upmc.fr)
- Course code: UM4RBR10-Etat
- Student workload: 12h of lectures, 8h of tutorials, 4h of labs
- Credits: 3 ECTS
- Specialization tracks:
- Semester offered: S1 S2 S3 S4
- Language of instruction: French English
- Targeted audience: Eng. Sc. department Other :
- Localization : PMC Campus Other :
Course Overview
This UE provides the basis for linear automation in state representation. Starting from a general nonlinear control system, we first introduce the notion of tangent linearized system. From there, the concepts of controllability, observability, stability are presented. The last part of the course focuses on the synthesis of controllers and linear observers.
Mots-clés : linear control system, tangent linearized system, commandability, observability, stability, controller synthesis by pole placement, LQR control, observer synthesis, transfer function realization.
Prerequisites
Students should have previously acquired the following prerequisites to follow this course:
- Linear algebra (determinant and rank of a matrix)
- Differential calculus (Jacobian matrix, limited expansion)
- Ffrequency domain automatic control (Laplace transform, transfer function, PID control)
Intended Learning Outcomes
By the end of this course, students will be able to:
- Determine the equilibrium points of a non-linear control system
- Calculating a tangent linearized system
- Determine whether a linear control system is controllable or not
- Determine whether a linear control system is observable or not
- Determining the Asymptotic Stability Properties of a Linear System
- Synthesize a status feedback order by pole placement
- Summarize a state observer by pole placement
- Know the conditions of use of the LQR command
- Moving from a state representation to a frequency representation and vice versa
- Use matlab tools to synthesize state feedback control laws and observers
- Implementing Control Schematics with Simulink
Indicative Teaching Sequence and Methods
| Week | C/TD/TP* | Content | Preparation | Learn.\ outc. |
|---|---|---|---|---|
| S1 | C1 (2h) | Mathematical reminders, definition of a linear system, notion of tangent linearized system | AAV1, AAV2 | |
| S2 | C2 (2h) | Introduction of the notions of controllability, observability | AAV3, AAV4 | |
| S3 | TD1 (2h) | Covers C1 and C2 | ||
| S4 | C3 (2h) | Lyapunov stability, controller synthesis by pole placement | AAV5, AAV6 | |
| S5 | C4 (2h) | Observer Synthesis, Separation Principle, Gain Adjustment | AAV6, AAV7 | |
| S6 | TD2 (2h) | Focuses on C1-C3 | ||
| S7 | C5 (2h) | LQR Control, Matlab Tools, Command and Saturation | AAV8 | |
| S8 | TD3 (2h) | Focuses on C4-C5 | ||
| S9 | C6 (2h) | Relationships between state representation and frequency representation | AAV9 | |
| S10 | TD4 (2h) | Applies to C5-C6 | ||
| S11 | TP (4h) | Preparation of lab session | Tous |
- C/TD/TP respectively corresponds to lectures, tutorials and lab sessions.
Indicative Assessment of Intended Learning Outcomes (1st session)
| Week | Individ./group | In-person/remote | Type of exam | Evaluated outcomes | Scale % |
|---|---|---|---|---|---|
| S13 | Individual | In-person | Written | AAV1-AAV9 | 70% |
| S13 | Individual | In-person | Written | AAV10-AAV11 | 30% |
2nde session
| Session | Individ./group | In-person/remote | Type of exam | Evaluated outcomes | Scale % |
|---|---|---|---|---|---|
| 2 | Individual | In-person | Written | AAV3-AVV8 | 70% |
| 1 | Individual | In-person | Written | AAV10-AAV11 | 30% |
Bibliographic references
- B. d'Andréa-Novel, M. Cohen de Lara, "Commande linéaire des systèmes dynamiques", Presse des Mines, 2000.
- Y. Granjon, "Automatique: systèmes linéaires, non linéaires, à temps continu, à temps discret, représentation d'états", Dunod, 2021.
Date of generation of this unit description: 14/01/2026
