Passivity-Based Control
Passivity-based control exploits energy conservation and dissipation, ensuring stability because interconnecting passive systems remains passive.
Energy as a certificate
A system is passive if it cannot produce more energy than it is supplied: the stored energy never exceeds the initial storage plus the integral of input times output power. This input-output property generalizes physical energy conservation. Passivity-based control uses it as the design principle, because passive systems have benign stability properties that survive interconnection.
The interconnection theorem
The central result is that the feedback interconnection of two passive systems is passive, and a passive system with a strictly positive-real property is stable. Consequently, if the plant is passive and the controller is designed to be passive, the closed loop is stable by construction, no matdel-precise cancellation required. This robustness to interconnection is what makes passivity attractive for uncertain and networked systems.
Shaping energy
When the plant is not naturally passive with respect to the desired output, energy shaping recasts it. Interconnection and damping assignment passivity-based control, IDA-PBC, assigns a desired energy function with its minimum at the target equilibrium and adds damping so trajectories dissipate toward it. For mechanical and electrical systems in port-Hamiltonian form, this yields controllers with clear physical interpretation as reshaping potential energy and injecting damping.
- Passive systems cannot generate energy
- Interconnection of passive systems stays passive
- Passive controller plus passive plant gives stable loop
- Energy shaping makes non-passive plants trackable
Passivity connects to the small-gain theorem through the theory of dissipative systems: both are supply-rate conditions, small gain using a gain supply rate, passivity using a power supply rate. The IQC framework unifies them.
Passivity is especially powerful for networked and teleoperated systems, where it guarantees stability despite delays that would defeat gain-based arguments. For design-stage interconnected physical subsystems, passivity-based design gives robust stability from energy structure, evaluated in simulation.