LQR Goodwin Oscillator Control

GOODWIN OSCILLATOR DYNAMICS

NONLINEAR GOODWIN CIRCUIT MODEL

dX/dt = a/(A + Y^b) - k*X dY/dt = alpha*X - beta*Y Where: - X, Y = protein concentrations - a = maximum transcription rate - A = dissociation constant - b = Hill coefficient (nonlinearity parameter) - k = protein degradation rate - alpha, beta = translation/degradation rates

SYSTEM LINEARIZATION

δẋ = A*δx + B*δu A = [∂f/∂x |_(x₀,u₀)] (Jacobian at equilibrium) B = [∂f/∂u |_(x₀,u₀)] (Input matrix) Where: - δx = state deviation from equilibrium - δu = control input deviation - (x₀,u₀) = equilibrium operating point

LQR OPTIMAL CONTROL LAW

u*(t) = -K*x(t) K = R⁻¹*B^T*P Where P solves the Algebraic Riccati Equation: A^T*P + P*A - P*B*R⁻¹*B^T*P + Q = 0 Cost Function: J = ∫₀^∞ (x^T*Q*x + u^T*R*u) dt

CONTROL ENGINEERING CAPABILITIES

BIOLOGICAL CIRCUIT MODELING

Advanced mathematical modeling of the Goodwin biological oscillator circuit representing gene regulatory networks with inherent nonlinear dynamics, Hill-function nonlinearities, and natural oscillatory behavior characteristic of circadian rhythms and cellular processes.

NONLINEAR SYSTEM LINEARIZATION

Systematic linearization of nonlinear biological dynamics around equilibrium points using Jacobian matrix analysis. Transforms complex nonlinear differential equations into linear state-space representation suitable for classical control design methodologies.

LQR OPTIMAL CONTROLLER DESIGN

Linear Quadratic Regulator synthesis providing optimal feedback control law that minimizes quadratic cost function balancing state regulation performance and control effort. Solves Algebraic Riccati Equation for guaranteed stability and optimality.

ROBUST CONTROL PERFORMANCE

Comprehensive analysis of controller robustness against parameter uncertainty, measurement noise, and modeling errors. Demonstrates effective oscillation suppression and stable regulation of pathological biological dynamics under realistic operating conditions.

SCIENTIFIC VISUALIZATION

Publication-quality visualizations including phase portraits, time-series analysis, control effort plots, and stability analysis. Interactive plots demonstrating controller effectiveness in transforming oscillatory behavior to stable equilibrium.

THERAPEUTIC APPLICATIONS

Real-world relevance for biomedical applications including circadian rhythm disorders, pathological oscillations in metabolic networks, and synthetic biology circuit stabilization. Framework applicable to drug delivery systems and therapeutic interventions.