Brockett openness laboratory

How much control can the openness profile support?

The open-loop field supplies a scale-dependent ball of velocities. A gain ceiling limits how far the feedback graph can reach into that field. Move the controls to compare the desired closed-loop ball with the open-loop capacity.

Graph-limited Brockett inequality

The feedback graph bounds the closed-loop profile.

Let G_u(x)=(x,u(x)) be the feedback graph and F_u=f ∘ G_u the closed-loop field. If ‖u(x)‖≤d(‖x‖), then

Ω_{F_u}(r) ≤ Ω_f(√r²+d(r)²)

Exact radial example

Resize the image, target, and gain envelope.

Here the open-loop image is an ellipse whose largest centered ball is known exactly. A green result means this necessary test does not rule the target out; it is not a general sufficiency claim.

Parameters

Open-loop openness κ = 0.80 Ω_f(s)=κs² Desired closed-loop rate c = 0.80 γ(r)=cr Allowed gain coefficient K = 0.80 d(r)=K√r

Gain-envelope shape Square root d(r)=K√r Linear d(r)=Kr

Inspection radius r = 0.10 Move the vertical cursor in the plot.

At the selected radius

The open-loop image and its centered ball

f(B_R) openness ball desired ball

Across nearby scales

Target openness versus available capacity

κ(r²+d(r)²) cr selected r

Control ceiling d(r) = 0.253

Joint radius reached R = 0.272

Desired ball cr = 0.080

Open-loop capacity κR² = 0.059

Ruled out near the equilibrium This gain envelope is too small.

Why the geometry is exact

A simple system realizes the threshold.

Open-loop field f_κ(x,u)=‖u‖A_κu

For A_κ=κ·diag(1.6,1), a joint ball of radius s maps to an ellipse with semiaxes 1.6κs² and κs². Therefore Ω_f(s)=κs² exactly.

Exact linear closed loop f_κ(x,u_c(x))=−cx

Taking y=−cA_κ⁻¹x and u_c(x)=y/√‖y‖ produces the target field. Set u_c(0)=0. Its magnitude is bounded by √c/κ√‖x‖.

Asymptotic threshold K ≥ √(c/κ) = 1.00

This is a minimum required size for an upper gain envelope—not a maximum allowed gain. A linear envelope Kr is too small near zero for every finite K.

Outside this exact example, satisfying the displayed inequality only means that this Brockett obstruction is silent. Other obstructions may still prevent stabilization.