5.2 Minimum variance and generalized minimum variance control

Minimum variance control aims to reduce output fluctuations in systems. It uses an ARMAX model to represent system dynamics and derives a control law that minimizes output variance. The approach balances performance with stability and causality constraints.

Generalized minimum variance control expands on this concept. It introduces a control effort weighting factor, allowing engineers to balance output tracking with energy use. This modification improves robustness and reduces noise sensitivity, offering more flexible control strategies.

Minimum Variance Control

Objective function and constraints

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• Minimize output variance $J = E[y^2(t+d)]$ optimizes control performance by reducing output fluctuations
• ARMAX model $A(q^{-1})y(t) = B(q^{-1})u(t-1) + C(q^{-1})e(t)$ represents system dynamics with input, output, and noise terms
• Causality ensures control decisions based only on available information prevents impossible future predictions
• Stability requirement maintains system equilibrium avoids unstable oscillations or divergence

Control law derivation

• Certainty equivalence assumes estimated parameters are true simplifies controller design process
• Express future output $y(t+d)$ using past inputs/outputs separates controllable and uncontrollable terms
• Set predictor of $y(t+d)$ to zero minimizes expected future output variance
• Solve for current input $u(t)$ yields control law $u(t) = -\frac{F(q^{-1})}{E(q^{-1})B(q^{-1})}y(t) - \frac{G(q^{-1})}{E(q^{-1})B(q^{-1})}u(t-1)$
• Polynomial functions $F(q^{-1})$ and $G(q^{-1})$ depend on system model parameters shape control response

Generalized Minimum Variance Control

Generalized minimum variance control

• Modified objective $J = E[(y(t+d) - w(t))^2 + \lambda u^2(t)]$ balances output tracking and control effort
• $\lambda$ factor weights control effort importance allows tuning between performance and energy use
• Control law incorporates $\lambda$ adjusts control actions based on effort weighting
• Improved robustness handles model uncertainties better (parameter variations, unmodeled dynamics)
• Reduced noise sensitivity decreases impact of measurement errors on control decisions
• Prevents excessive control limits actuator wear and energy consumption

Impact of control effort weighting

• Higher $\lambda$ reduces control effort conserves energy, increases output variance allows more fluctuations
• Lower $\lambda$ increases control effort more aggressive control, decreases output variance tighter regulation
• Larger $\lambda$ improves stability margins enhances system robustness (gain margin, phase margin)
• Smaller $\lambda$ decreases stability margins reduces tolerance to modeling errors or disturbances
• Performance vs robustness trade-off requires balancing control objectives and system constraints
• $\lambda$ selection considers system requirements (energy limitations, output precision, disturbance rejection)
• Closed-loop pole analysis reveals $\lambda$ impact on system dynamics (damping, natural frequency)
• Disturbance rejection affected by $\lambda$ influences system's ability to counteract external perturbations