IndisputableMonolith.Robotics.PIDStabilityFromJCost
The module defines recognition descent for self-maps on positive reals and applies it to certify PID controller stability via strict J-cost decrease. Control theorists and Recognition Science researchers would cite it when linking the fundamental cost function to practical stability certificates. The structure consists of supporting definitions for trajectories and descent steps, culminating in a one-statement PID stability result.
claimA self-map $f : (0,∞) → (0,∞)$ exhibits recognition descent when $f(1)=1$ and $J(f(r)) < J(r)$ for all $r ≠ 1$, where $J$ denotes the J-cost. The module introduces trajectory sequences under such maps, proves cost descent along them, and states that PID controllers satisfy the stability certificate.
background
This module sits in the Robotics domain and imports the RS time quantum τ₀ = 1 tick from Constants together with the J-cost from the Cost module. It introduces RecognitionDescent as the property that a self-map fixes the equilibrium r = 1 while strictly decreasing J-cost off equilibrium. Additional definitions cover trajectory sequences, cost descent steps, and equilibrium conditions, all expressed in RS-native units.
proof idea
This is a definition module with supporting lemmas. It first defines RecognitionDescent, trajectory, and cost predicates, then establishes lemmas such as cost_descent_step, cost_zero_iff_equilibrium, and cost_pos_off_equilibrium before stating the PID stability result.
why it matters in Recognition Science
The module bridges the J-cost framework to control applications, supplying the RecognitionDescent property and PIDStabilityCert that realize cost minimization for robotics. It connects directly to the Recognition Composition Law and the phi-ladder structure of Recognition Science by showing how descent on trajectories yields stability at the equilibrium fixed point.
scope and limits
- Does not treat nonlinear or time-varying plants.
- Does not supply numerical simulations or experimental data.
- Does not address multi-input multi-output controllers.
- Does not derive explicit convergence rates or bounds.
depends on (2)
declarations in this module (12)
-
structure
RecognitionDescent -
def
trajectory -
theorem
trajectory_zero -
theorem
trajectory_succ -
theorem
trajectory_at_equilibrium -
theorem
trajectory_pos -
theorem
cost_descent_step -
theorem
cost_zero_iff_equilibrium -
theorem
cost_pos_off_equilibrium -
structure
PIDStabilityCert -
def
pidStabilityCert -
theorem
pid_stability_one_statement