7.2 Passivity and hyperstability concepts

Passivity and hyperstability are key concepts in adaptive control, ensuring systems consume more energy than they produce and maintain bounded input-output behavior. These principles are crucial for designing robust controllers that can handle parameter variations and uncertainties in various applications.

Adaptive controllers using hyperstability techniques employ energy-based Lyapunov functions, implement damping injection, and satisfy Popov integral inequality. These methods are applied in diverse fields like robotics, flexible structures, and power systems, enhancing stability and performance in complex, uncertain environments.

Passivity and Hyperstability in Adaptive Systems

Passivity and hyperstability concepts

• Passivity energy-based concept in control theory relates input-output behavior of a system consumes more energy than it produces (electrical circuits, mechanical systems)
• Hyperstability generalizes absolute stability applies to wider class of nonlinear systems ensures bounded input-output behavior (robot control, aircraft stabilization)
• Relevance to adaptive systems provides robust stability guarantees allows for parameter variations and uncertainties facilitates design of adaptive controllers (self-tuning regulators, model reference adaptive control)

Conditions for passive systems

• Input-output relationship: $\int_0^T y^T(t)u(t)dt \geq -\beta$ where $\beta$ is finite constant $u(t)$ is input $y(t)$ is output
• State-space representation uses positive real lemma for passive systems defines conditions on system matrices (A, B, C, D)
• Physical interpretation system dissipates or stores energy cannot generate energy (RC circuits, mass-spring-damper systems)

Stability analysis with passivity theory

• Lyapunov stability analysis constructs appropriate Lyapunov function proves negative definiteness of its derivative
• Passivity-based stability analysis:
  1. Decompose system into feedforward and feedback paths
  2. Show passivity of individual components
  3. Apply passivity theorem
• Hyperstability-based analysis uses Popov criterion or Integral Quadratic Constraint (IQC) verifies conditions for bounded input-output behavior
• Robustness analysis investigates stability margins assesses performance under parameter variations (gain margins, phase margins)

Adaptive controllers using hyperstability

• Passive adaptive control ensures controller maintains passivity uses energy-based Lyapunov functions implements damping injection for improved performance (adaptive impedance control)
• Hyperstable adaptive schemes design adaptation laws satisfying Popov integral inequality employ switching σ-modification or e-modification (model reference adaptive control)
• Robust adaptive control incorporates projection algorithms implements dead-zone modifications (adaptive flight control)
• Passivity-based adaptive control (PBAC) exploits natural energy-dissipation properties designs energy-shaping controllers (robot manipulators, flexible structures)
• Applications include robot manipulators flexible structures power systems (adaptive force control, vibration suppression, power system stabilizers)