Dissipativity Theory in Nonlinear Dynamical Systems

Dissipativity theory provides a unifying framework for analysing the stability and performance of nonlinear dynamical systems by quantifying the exchange of stored and supplied energy. At its core is the concept of a storage function, which generalises classical Lyapunov functions, together with a supply rate that captures how external inputs deliver or extract energy from the system. When the dissipation inequality holds, indicating that the rate of change of stored energy does not exceed the supplied energy, one can infer stability, robustness and performance bounds under interconnections and feedback. Originally developed to extend the ideas of positive realness and bounded realness to the nonlinear regime, dissipativity has since become central to control design, system identification and networked systems analysis.

This theory underpins a broad array of practical applications. In power systems it guides the design of controllers that reshape energy to damp oscillations; in biochemical networks it ensures robust behaviour despite uncertainty in reaction rates; in robotics it informs force–motion interaction for safety; and in emerging neural dynamical models it exposes the energetic properties of deep-learning architectures. Across these domains, dissipativity offers a powerful lens for guaranteeing stability, ensuring passivity when interconnecting subsystems, and deriving computationally tractable conditions—often in the form of linear matrix inequalities—for synthesis and verification.

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Dissipativity Theory in Nonlinear Dynamical Systems publication trend

The graph below shows the total number of articles in dissipativity theory in nonlinear dynamical systems across all publications each year (not limited to Nature Index journals).

Technical terms

Dissipativity: A property whereby the system’s rate of change of stored energy never exceeds the supplied energy, ensuring bounded responses and facilitating stability analysis.

Storage function: A scalar function, analogous to energy, that measures the internal “storage” of a system and is central to formulating the dissipation inequality.

Supply rate: A bilinear form that quantifies how external inputs add to or subtract from the system’s stored energy over time.

Passivity: A special case of dissipativity characterised by a non-negative supply rate, often interpreted as the system not generating energy on its own.

References

1. A Generalized Passivity Theory Over Abstract Time Domains. IEEE Transactions on Automatic Control (2024).
2. Nonlinear Observer-Based Robust Passive Control of Doubly-Fed Induction Generators for Power System Stability Enhancement via Energy Reshaping. Energies (2017).
3. Exponentially dissipative nonlinear dynamical systems: a nonlinear extension of strict positive realness. Mathematical Problems in Engineering (2003).
4. A passivity-based stability criterion for a class of biochemical reaction networks. Mathematical Biosciences and Engineering (2008).
5. Dissipativity Analysis for a Class of Discrete‐Time Neutral Stochastic Nonlinear Systems with Time Delay. Discrete Dynamics in Nature and Society (2021).
6. Dissipative Deep Neural Dynamical Systems. IEEE Open Journal of Control Systems (2022).