This repository contains the project developed for the Nonlinear Dynamics and Control exam. The goal of the project is the study and control of a nonlinear dynamical model, specifically the Zeeman Heartbeat Model.
The assignment is structured into three main phases:
-
Analysis
This phase provides an overview of the application field, introduces the mathematical model, explains the physical meaning of the state variables and parameters, and describes the real system behavior. -
Synthesis
This phase focuses on defining a meaningful control problem, oriented toward practical objectives rather than purely academic goals.
A linear control technique and at least two nonlinear control techniques are implemented.
The aim is to solve the control problem and analyze robustness with respect to disturbances. -
Conclusion
This phase discusses the obtained results and outlines possible future developments of the work.
A fundamental aspect of the entire project is that all observations and design choices are justified by their physical interpretation.
- MATLAB
- Simulink
- PhasePlane8
- MATCONT
The first step consists of characterizing the real system and its behavior.
In this case, the heartbeat is a system that switches between two different states: diastole and systole.
The mathematical model is then introduced, and all state variables and parameters are described together with their physical meaning.
Phase plane analysis is performed to identify the equilibrium points of the system through linearization.
In particular, two equilibrium points are found, corresponding to the two physiological states of the heartbeat.
The second step is to verify the presence of possible bifurcations by varying one or more parameters.
It is observed that variations in the parameter representing the transition velocity between the two states do not lead to any bifurcation.
Conversely, variations in the parameter associated with the muscle fiber tension lead to a Hopf bifurcation, resulting in the appearance of a limit cycle with increasing amplitude.
The final step of the analysis phase is the stability analysis, carried out using Lyapunov theory.
In this case, it was necessary to apply LaSalle’s invariance principle to construct a suitable Lyapunov function and verify Global Asymptotic Stability (GAS).
The synthesis phase begins with the formulation of the control problem, distinguishing between regulation and tracking objectives.
In this project, the chosen objective is to ensure that the ECG signal of the heartbeat model (one of the state variables) tracks a reference ECG corresponding to a healthy patient, similarly to the behavior of an intelligent pacemaker.
Control specifications such as maximum tracking error, maximum control effort, and robustness with respect to disturbances are defined.
The control objectives are first addressed using a linear control technique and then using at least two nonlinear control techniques.
