Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Analysis and Control of Perturbed Density Systems

Igor Furtat

TL;DR

The paper studies density systems where the derivative of a positive-definite function $V(x,t)$ along trajectories depends on a density function $\rho(x,t)$, defining density systems that create (un)stable regions and forbidden domains. It generalizes classic results to densities that can be implicit on the RHS or sign-indefinite and extends the framework to systems with disturbances. A density-based adaptive control law for systems with unknown parameters is proposed, with Lyapunov-based guarantees yielding bounded signals and convergence to a density-defined target set under appropriate conditions (e.g., $|\rho|>\eta$). Numerical simulations validate the theory and illustrate density-induced region boundaries and robust tracking under disturbances.

Abstract

The paper investigates dynamical systems for which the derivative of some positive-definite function along the solutions of this system depends on so-called density function. In turn, such dynamical systems are called density systems. The density function sets the density of the space, where the system is evolved, and affects the behaviour of the original system. For example, this function can define (un)stable regions and forbidden regions where there are no system solutions. The density function can be used in the design of new adaptive control laws with the formulation of appropriate new control goals, e.g., stabilization in given bounded or semi-bounded sets. To design a novel adaptive control law that ensures the system outputs in a given set, systems with known and unknown parameters under disturbances are considered. All theoretical results and conclusions are illustrated by numerical simulations.

Analysis and Control of Perturbed Density Systems

TL;DR

The paper studies density systems where the derivative of a positive-definite function along trajectories depends on a density function , defining density systems that create (un)stable regions and forbidden domains. It generalizes classic results to densities that can be implicit on the RHS or sign-indefinite and extends the framework to systems with disturbances. A density-based adaptive control law for systems with unknown parameters is proposed, with Lyapunov-based guarantees yielding bounded signals and convergence to a density-defined target set under appropriate conditions (e.g., ). Numerical simulations validate the theory and illustrate density-induced region boundaries and robust tracking under disturbances.

Abstract

The paper investigates dynamical systems for which the derivative of some positive-definite function along the solutions of this system depends on so-called density function. In turn, such dynamical systems are called density systems. The density function sets the density of the space, where the system is evolved, and affects the behaviour of the original system. For example, this function can define (un)stable regions and forbidden regions where there are no system solutions. The density function can be used in the design of new adaptive control laws with the formulation of appropriate new control goals, e.g., stabilization in given bounded or semi-bounded sets. To design a novel adaptive control law that ensures the system outputs in a given set, systems with known and unknown parameters under disturbances are considered. All theoretical results and conclusions are illustrated by numerical simulations.

Paper Structure

This paper contains 4 sections, 2 theorems, 21 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Density systems
  3. Density adaptive control
  4. Conclusion

Key Result

Proposition 1

Let the system eq1 be strictly density in the domains $D_S$ and $D_U$. If the condition $V(x_{s},t)-V(x_{u},t)>0$ is satisfied for any $t$, where $x_{s} \in D_S$ and $x_{u} \in D_U$, then the trajectories are attracted to the region between $D_S$ and $D_U$. If the system eq1 satisfies the condition

Figures (9)

  • Figure 1: The transients for $\rho(x,t)=1$ (dotted lines) and $\rho(x,t)=10$ (solid lines) in \ref{['eq_ex1_0']} (left), as well as for $\rho(x,t)=\frac{1}{2e^{-t}+0.1-|x(t)|}$ in \ref{['eq_ex1_0']} (right) under $d(t)=0.5\textup{atan}(100\sin(0.8t))$.
  • Figure 2: The transients for $\rho(x,t)=[x-0.3-\cos(0.5t)] \textup{sign}(x)$ (left) and $\rho (x,t)=\ln \frac{2e^{-0.1t}(2+\sin(t))+0.3-x}{x-e^{-0.1t}(2+\sin(t))-0.2}$ (right) under $d(t)=0.5\textup{atan}(100\sin(0.8t))$.
  • Figure 3: The phase trajectory of \ref{['eq_ex2_1']} for $\rho(x,t)=\ln \frac{3-|x_1|-|x_2|}{|x_1|+|x_2|-2}$ (left) and $\rho(x,t)=\ln \frac{3^{0.6}-|x_1|^{0.6}-|x_2|^{0.6}}{|x_1|^{0.6}+| x_2|^{0.6}-1}$ (right).
  • Figure 4: The phase trajectories of \ref{['eq_ex2_1']} for $\rho(x,t)=1$ (dotted curve) and $\rho(x,t)=\sum_{i=1}^4\ln (|x_1-x_{1i}|^{\beta_i}+|x_2-x_{2i}|^{\beta_i}-b_i)$ (solid curve).
  • Figure 5: The phase trajectories of \ref{['eq_ex2_1']} for $\rho(x,t)=5 \ln(|x_1|+|x_2|-1)$ (left) and $\rho(x,t)=5 \ln(|x_1|^{0.5}+|x_2|^{0.5}-1)$ (right).
  • ...and 4 more figures

Theorems & Definitions (16)

  • Remark 1
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Remark 2
  • Proposition 1
  • Proof 1
  • Remark 3
  • Definition 5
  • ...and 6 more