[Paper Review] Controllability function as time of motion. I

Abdon E. Choque‐Rivero, V. I. Korobov+1 more|arXiv (Cornell University)|Sep 17, 2015

Control and Dynamics of Mobile Robots3 references16 citations

TL;DR

This paper introduces a controllability function method to solve the admissible positional control problem for canonical systems with bounded controls, where the controllability function Θ(x) explicitly represents the finite time of motion to the origin. By constructing Θ(x) such that its derivative along the system trajectory is -1, the method ensures finite-time stabilization using a feedback control law u(x) derived from Θ(x), with explicit solutions provided for a third-order system under unit control bounds.

ABSTRACT

The admissible positional control problem for the canonical system with geometrical restrictions on the control is considered. The investigation is performed with the help of the controllability function method. We obtain controllability functions which are the time of motion from an arbitrary initial condition to the origin. We also reveal a set of controls which solves this problem.

Motivation & Objective

• To address the admissible positional control problem for canonical systems under geometric control constraints.
• To construct a controllability function Θ(x) that represents the time of motion from any initial state to the origin.
• To derive a feedback control law u(x) that ensures finite-time convergence to the origin while satisfying control bounds.
• To provide explicit analytical solutions for a third-order canonical system with unit control magnitude restriction.
• To demonstrate that the derivative of Θ(x) along the closed-loop trajectory is exactly -1, ensuring finite-time reachability.

Proposed method

• The controllability function Θ(x) is constructed such that its time derivative along the closed-loop system is -1, i.e., dΘ/dt = -1.
• The control law is derived as u(x) = -6/Θ(x)·x₁ - 25/Θ²(x)·x₂ - 45/Θ³(x)·x₃ for a third-order system.
• The system is transformed via substitution t = Θ₀ - e^τ to convert the time-varying Euler-type equation into a linear ODE with constant coefficients.
• The solution is expressed in terms of exponential and trigonometric functions of τ = ln(Θ₀ - t), enabling explicit trajectory computation.
• Initial conditions are used to determine constants in the general solution, ensuring consistency with the initial state x₀.
• The method ensures that Θ(x) is continuously differentiable for x ≠ 0 and satisfies the inequality ∂Θ/∂x · f(x,u(x)) ≤ -βΘ¹⁻¹/α, with α = ∞ implying Lyapunov function behavior.

Experimental results

Research questions

• RQ1 How can a controllability function be constructed such that it represents the exact time of motion to the origin for a canonical system with bounded controls?
• RQ2 What feedback control law u(x) ensures finite-time convergence to the origin while satisfying |u| ≤ d?
• RQ3 Can the time of motion Θ(x) be explicitly computed and verified to satisfy dΘ/dt = -1 along the closed-loop trajectory?
• RQ4 How does the system trajectory behave under the derived control law, and can it be expressed in closed form?
• RQ5 What is the analytical structure of the solution when the system is transformed into a constant-coefficient ODE via time substitution?

Key findings

• For a third-order canonical system with |u| ≤ 1, the time of motion from initial state x₀ is Θ₀ = 41x₁⁰/11 when x₂⁰ = -41(x₁⁰)²/121 and x₃⁰ = 0.
• The trajectory is explicitly given as a function of time, with x(t) → 0 as t → Θ₀, confirming finite-time convergence.
• The control law on the trajectory is u(t) = -1/41(6 + 35cosα(t)), which satisfies |u(t)| ≤ 1 for all t.
• The solution is expressed in terms of α(t) = √6 ln(1 - 11t/(41x₁⁰)), showing oscillatory behavior modulated by the logarithmic time scaling.
• The constants in the solution are determined from initial conditions, with c₁, c₂, c₃ expressed in terms of x₀₁, x₀₂, x₀₃ and Θ₀.
• The derivative of Θ(x) along the trajectory is exactly -1, confirming that Θ(x) is the true time of motion to the origin.