adaptive-control

name	adaptive-control
description	Adaptive control — MRAC (model reference), self-tuning regulator, MIT rule, Lyapunov-based adaptation, gain scheduling, parameter identification, stability, aerospace and process applications.
metadata	{"priority":7,"promptSignals":{"phrases":["adaptive control","model reference adaptive","MRAC","self-tuning regulator","gain scheduling","MIT rule","adaptive controller"],"minScore":3}}

Adaptive Control — Complete Skill

Motivation

Fixed-gain controllers perform poorly when plant parameters change (mass, stiffness, aerodynamics with altitude/speed, process gain with operating point) Adaptive control: controller parameters adjust automatically to maintain performance as plant changes

Types:

• MRAC (Model Reference Adaptive Control): track reference model output
• Self-Tuning Regulator (STR): estimate plant parameters online; update controller
• Gain Scheduling: lookup table of pre-computed gains vs. operating point (not truly adaptive but widely used)

MRAC — Model Reference Adaptive Control

Structure: Reference model M(s): desired closed-loop response (ideal plant + controller) Plant G(s): actual plant with unknown/varying parameters Adaptation law: adjusts controller parameters so plant output → reference model output

Error: e(t) = y_plant(t) - y_ref(t) [tracking error]

MIT Rule (gradient descent adaptation): dθ/dt = -γ × e × (∂e/∂θ) [γ = adaptation gain; θ = controller parameter vector]

Plant: G(s) = K_p / (s + a) [first order; K_p unknown] Controller: u = θ_1 × r - θ_2 × y [feedforward + feedback] Reference model: M(s) = K_m / (s + a_m) [desired response]

Adaptation law (MIT rule): dθ_1/dt = -γ × e × r × ∂y/∂θ_1 dθ_2/dt = -γ × e × y × ∂y/∂θ_2

Limitation: MIT rule can be unstable; Lyapunov-based preferred

Lyapunov-Based Adaptation (Robust)

Design via Lyapunov function V(e, θ̃): V = e²/2 + θ̃ᵀ × Γ⁻¹ × θ̃ / 2 [θ̃ = θ - θ* = parameter error; Γ = adaptation rate matrix]

Stability requirement: dV/dt ≤ 0

Adaptation law (ensures dV/dt ≤ 0): dθ/dt = -Γ × φ(t) × e(t) [φ = regressor vector; Γ > 0 = positive definite]

Result: closed-loop system stable + tracking error e → 0 (for appropriate Γ)

Key property: parameters θ converge to θ* only if φ is "persistently exciting" (PE condition) PE condition: ∫(t to t+T) φ(τ) φ(τ)ᵀ dτ ≥ α I for some T, α > 0

Direct vs. Indirect Adaptive Control

Direct MRAC: Adapt controller parameters directly: θ = [θ_ff, θ_fb, ...] No explicit parameter estimation; less computation

Indirect (STR):

1. Estimate plant parameters: â, b̂ (using RLS)
2. Calculate controller gains from estimated parameters (pole placement or LQR)
3. Apply gains to plant

RLS (Recursive Least Squares) — parameter estimation: P(k) = [P(k-1) - P(k-1)φφᵀP(k-1) / (λ + φᵀP(k-1)φ)] / λ θ̂(k) = θ̂(k-1) + P(k)φ(k)[y(k) - φ(k)ᵀθ̂(k-1)] λ = forgetting factor (0.95–0.999; accounts for parameter drift)

Gain Scheduling

Concept: precompute controller gains at multiple operating points → interpolate based on measured scheduling variable (altitude, speed, flow rate, temperature)

Scheduling variable ρ: must capture dominant plant variation Gain table: {K_p(ρ_i), K_i(ρ_i), K_d(ρ_i)} computed at each operating point ρ_i Interpolation: linear or bilinear between grid points

Aerospace example (flight control): Dynamic pressure q̄ = ½ρV² → stability derivatives scale with q̄ → gain K = K_0 / q̄

Process example (pH control): Nonlinear pH titration curve → gain proportional to titration curve slope at operating pH K_p(pH) = K_0 × [dpH/dV_add]⁻¹

Limitation: gain scheduling is NOT truly adaptive; no stability guarantee during transitions Rule: if parameter changes faster than gain schedule update rate → potential instability

Self-Tuning PID

Auto-tune methods: Relay auto-tune (Åström-Hägglund): Replace controller with relay of amplitude d; observe limit cycle K_u = 4d / (π × A_limit_cycle) [ultimate gain; A = amplitude of oscillation] P_u = period of oscillation [s] Apply Ziegler-Nichols: K_p = 0.6 K_u; T_i = P_u/2; T_d = P_u/8

Step response auto-tune: Identify K, T, L from step response → calculate PID gains

Stability Considerations

Bursting phenomenon: in MRAC, parameter estimates can drift → control input spikes Mitigation: dead zone (don't adapt when error < threshold); parameter projection (constrain θ to known bounds); σ-modification (add leakage to adaptation law)

σ-modification: dθ/dt = -Γ φ e - σ Γ θ [σ > 0 = leakage; prevents unbounded parameter drift; sacrifices perfect tracking]

Applications

Domain	Adaptive control use
Flight control	MRAC for changing aerodynamics (altitude, speed, config)
Chemical process	STR for time-varying reactor gain
CNC machine tool	Adaptive feed for variable material hardness
Robotics	MRAC for unknown payload inertia
Automotive	Adaptive suspension gain with load

Output

Provide: adaptive control type (MRAC/STR/gain scheduling), plant model structure, reference model M(s) parameters (ω_n, ζ), adaptation law (MIT/Lyapunov/RLS), adaptation gain Γ (or γ), forgetting factor λ (if RLS), scheduling variable and grid points (gain scheduling), stability proof approach, PE condition check, robustness modification (σ-mod/projection/dead zone), and expected tracking error under parameter variation.