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Projection Operator in Adaptive Systems
Eugene Lavretsky, Travis E. Gibson
TL;DR
The paper examines properties of the projection algorithm used in adaptive control. It defines projection forms and analyzes their behavior; under stated initialization conditions, parameter norms remain bounded by specified radii and parameters stay within a convex set.
Problem
The paper examines properties of the projection algorithm, which is frequently used in adaptive control.
Source: §1 · p. 1Method
The paper defines vector and matrix projection operators and analyzes their behavior with convex functions and adaptive-control laws.
Source: §3 · pp. 2, 4, 5Results
For parameters initialized with f_i(θ_i) ≤ 0, ∥θ_i(t)∥ ≤ ϑ_i + ε_i for all t ≥ 0, and θ initialized in Ω₀ never leaves Ω₁.
Source: §3 · p. 5Takeaways & Limitations
Under the stated initialization, projected adaptive parameters remain within the componentwise radius bounds ϑ_i + ε_i.
Source: §3 · p. 5
Abstract
from arXiv · showhide
The projection algorithm is frequently used in adaptive control and this note presents a detailed analysis of its properties.
1 Introduction
The introduction traces these notes to a personal communication and says they summarize earlier results on the projection operator, whose properties are explored in the following section.
2 Properties of Convex Sets and Functions
This section states a defining property of convex sets and gives a convexity result for sublevel sets, then frames a lemma using interior and boundary points.
2 Properties of Convex Sets and Functions
- A set is convex when the line segment joining any two of its points lies entirely within the set.Source: §2 · p. 1
- For convex f, the sublevel set Ωδ = {θ ∈ R^k | f(θ) ≤ δ} is convex when δ > 0.Source: §2 · p. 2
- Lemma 4 considers a continuously differentiable convex function and an interior point θ* and boundary point θb of Ωδ.Source: §2 · p. 2
3 Projection
The section defines vector and matrix forms of the projection operator and analyzes its behavior in adaptive control. Under stated initialization conditions, the projected parameters remain bounded and within the specified convex set.
3 Projection
- Within the annulus, projection subtracts a scaled component of y normal to the boundary; at λ = 1 it removes that component entirely, making the result tangent.Source: §3 · p. 3
- Adding f(θ) to the active modification makes the projection definition continuous, and Proj(θ, y) is shorthand when the structure of f is unimportant.Source: §3 · pp. 2–3
- The general projection operator extends the vector definition to an n × m matrix, with Θ and Y as n × m matrices and F as an m-vector.Source: §3 · pp. 2, 4
- Starting from θ₀ ∈ Ω₀, θ may reach the boundary f(θ) = 1 but never leaves the convex set Ω₁.The stated derivative behavior is positive for 0 < f(θ) < 1 under an outward update, zero at f(θ) = 1, and negative when f(θ) ≤ 0 or the update is not outward.Source: §3 · p. 5
- The adaptive-control example treats Θ(t) as a time-varying feedback gain, adjusted using an adaptive law with error, Lyapunov-derived P, and input Jacobian B.Source: §3 · p. 5
- ∥θ_i(t)∥ ≤ ϑ_i + ε_i for all t ≥ 0 when the initial parameters satisfy f_i(θ_i) ≤ 0.The component functions equal 0 at radius ϑ_i and 1 at radius ϑ_i + ε_i.Source: §3 · p. 5
- Γ-projection is a variant that updates the parameter along a symmetric positive definite gain; it was first introduced in.Source: §4 · p. 6