Date: August 30 - September 3, 2021

The aim of this condensed 5-day PhD course is to introduce basic notions in nonlinear systems and control to interested students. The studied topics will cover the classical theory of input-affine systems (analysis of basic properties such as reachability, observability, stability; linearization-based controller design), passivity, Hamiltonian system representation, model based predictive control, as well as the control of multi-agent systems. For each day, 4x45 mins of theoretical lectures and tutorials are planned.

The course includes the following topics:

- Introduction: mathematical models of dynamical systems, linear and nonlinear models, control goals and constraints, summary of LTI systems
- Mathematical tools: Vector fields, diffeomorphisms, tangent vectors, Lie derivative, Lie product, distributions, co-distributions
- Most important special nonlinear system classes: positive systems, polynomial systems, QP-systems, CRNs, Hamiltonian systems.
- The underlying physical picture behind these classes
- Summary of the theory of nonlinear autonomous systems: continuity, existence and unicity of solutions, equilibrium points, stability
- Nonlinear control systems in ODE-form, steady states, linearization around steady states/solutions
- Analysis of input-affine systems: reachability and controllability
- Classical control of input-affine systems: exact linearization, zero dynamics, input-output linearization (SISO and MIMO cases)
- Dissipative and passive systems: storage functions, dissipativity, passivity, Lq-gain. Control Lyapunov functions, passivation, feedback equivalence, minimum-phase nonlinear systems.
- Hamiltonian systems: system interconnections and Hamiltonian functions, control by feedback and interconnection, generalized Hamiltonian systems
- Model-based predictive controllers and their application for traffic systems.
- Analysis and control of multi-agent systems.

## Program

#### August 30

Time | Room | Topic | Lecturer | Notes |

9:00-10:30 | I122 | Nonlinear state space models (lecture) Review of LTI state space and input-output models; Input-affine state space models; Local linearization |
Katalin Hangos | presentation |

11:00-12:30 | I122 | Basic tools (computer tutorial) Review of MATLAB and its important toolboxes |
Péter Polcz | materials |

13:30-14:30 | I122 | Model transformations and properties (lecture, blackboard tutorial) Review of LTI state transformations, canonical model forms; Lie derivative, Lie brackets, relative degree; Nonlinear state transformations |
Katalin Hangos | presentation |

14:45-15:45 | I122 | Model transformations II. (lecture, blackboard tut.) Nonlinear feedback form for general and input affine systems, the notion of feedback equivalence to ensure a required property; the notion of relative degree; normal form of SISO nonlinear systems; the zero dynamics; exact linearization and I/O linearization via feedback linearization in the MIMO case |
Gábor Szederkényi | presentation |

#### August 31

Time | Room | Topic | Lecturer | Notes |

9:00-10:30 | I122 | Distributions and co-distributions; Reachability analysis, local and global reachability, controllability distribution; Observability analysis, local and global observability, observability co-distribution; Minimal realizations | Gábor Szederkényi | presentation |

11:00-12:30 | I122 | Stability (Lecture and tutorial) BIBO and asymptotic stability of LTI systems, Lyapunov stability, Lq stability for nonlinear systems, Lq-gain |
Attila Magyar | presentation |

13:30-14:30 | I122 | Stabilizing controller design (Lecture and tutorial) LTI state feedback controllers (pole placement and LQ),Control Lyapunov functions; Passivity, passivity by feedback |
Attila Magyar | presentation |

14:45-15:45 | I122 | Model analysis and control (computer and blackboard tutorial) basic and advanced | Attila Magyar |

#### September 1

Time | Room | Topic | Lecturer | Notes |

9:00-9:45 | I122 | Basics of system identification, Gaussian Processes, GP-based identification, theory part 1 | Roland Tóth | materials |

10:00-10:45 | I122 | Basics of system identification, Gaussian Processes, GP-based identification, theory part 2 | Roland Tóth | |

11:00-11:45 | I122 | Basics of system identification, Gaussian Processes, GP-based identification, theory part 3 | Roland Tóth | |

13:00-14:30 | I122 | Basics of system identification, Gaussian Processes, GP-based identification, tutorial | Roland Tóth | |

#### September 2

Time | Room | Topic | Lecturer | Notes |

9:00-10:30 | I122 | Analysis and control of Multi-Agent Systems Multi-Agent Systems: Basic notions. Consensus over undirected graphs. Weak formation control |
Lőrinc Márton | presentation |

11:00-12:30 | I122 | Consensus over directed graphs. Strong formation control. Synchronization of nonlinear passive systems. Application example: Lotka-Volterra networks. | Lőrinc Márton | materials |

13:30-14:30 | I122 | MATLAB Tutorial: The consensus protocol | Áron Fehér | |

14:45-15:45 | I122 | MATLAB Tutorial: Formation control | Áron Fehér |

#### September 3

Time | Room | Topic | Lecturer | Notes |

9:00-10:30 | I122 | Special nonlinear model classes: QP and CRN models (lecture, blackboard) Positive polynomial system classes; Special nonlinear transformations of QP and CRN models Stability analysis, structural (robust) stability |
Katalin Hangos | presentation |

11:00-12:30 | I122 | Stabilyzing controller design for QP and CRN models (lecture) LQ controllers for LV systems; (Robust) stabilyzing controllers based on CRN models |
Gábor Szederkényi | |