Control in Shared Environments

All controllers begin with zero output and reference 1.5. At each step, perceptions are evaluated from the current outputs, then all outputs are updated simultaneously:

At S = 0, a = 1. In the replay model, pᵢ = oᵢ + dᵢ(t), with dᵢ(t) = pᵢ(recording) − oᵢ(recording). This stored sequence is independent of all outputs during replay. Its construction from a training trajectory is disclosed; it is not independently measured physical evidence.

N counts controllers; rᵢ is the reference, oᵢ the output, pᵢ the perception, C the shared capacity, and S the total output. All use arbitrary signal units. G has units s⁻¹; dt = 0.01 s; oₘₐₓ is the output bound. The 60 s run contains 6,000 updates and 6,001 samples. The model identifies environmental output with perception through an identity perceptual function.

The integrator state itself is clipped. There is no hidden integral state accumulating above the actuator limit: the observed effect is output saturation with residual error. No non-negative clipping is applied to replayed perceptions; for some settings, additive replay can produce negative perceptions.

Above capacity, the local derivatives with other outputs held fixed are:

Below capacity the derivative matrix is the identity. At S = C it is generally not differentiable. The local experiment reports finite differences, which can cross this boundary. An independent additive replay has own-channel derivative 1 and cross-channel derivatives 0.

Shared capacity: record a baseline with all references fixed at 1.5, then apply the same test intervention to the shared model and the replay. At default settings the baseline disturbance is exactly zero because total baseline demand stays below C = 8.

Frozen replay: record a shared-capacity training run with a +3 reference step at 30 s on controller 0. Freeze its entire disturbance sequence. Adjust only the test step, time or receiving controller to assess prediction. Changing C, N, G or the output limit creates a new training recording and starts a new comparison.

Local intervention: hold a shared-run output vector fixed and perturb one component. This is an intervention on the environmental map, not the closed-loop response of a person or a Test for the Controlled Variable.

The default shared-model means over 50–60 s are p₀ = 2.00185, o₀ = 10, other p = 1.49954, other o = 7.49075. Baseline replay gives 4.5, 4.5, 1.5 and 1.5 respectively. The others’ output ratio is 4.99384 versus 1. A training-matched full replay reproduces the trajectory to floating-point precision.

The sum of default post-step references is 10.5, greater than capacity 8. Not all references can be met simultaneously. At the limiting shared state, ∂p₀/∂o₀ = 0.15: the environmental connection has not vanished. Good perceptual tracking can also hide substantial differences in output.

A discriminating test: use the same fixed recording and controller parameters across predeclared interventions; compare full output and perception trajectories. A replay fitted separately to each outcome cannot establish predictive equivalence. Numerical consistency is distinct from empirical validation against independently measured behavior.

This laboratory develops the shared-capacity comparison in the forum experiment. The distinction between general PCT and a fixed linear environmental approximation matters: post 102 allows nonlinear, varying feedback functions, and post 104 explicitly allows gain to depend on another controller’s output.

The maintenance of shared means of control discussed in post 123 is a further question. Here capacity is fixed; no controller maintains it. Neither queues nor hierarchical changes in goals are included. The earlier priority-allocation example and this proportional-allocation rule are different environmental models.