Energy-Based Control Strategies

• Energy-based control strategies are methodologies that derive feedback laws from a system's energy balance, dissipation, and storage to ensure stability and robustness.
• They employ techniques such as port-Hamiltonian modeling, energy shaping, and damping injection to regulate, track, and optimize complex systems.
• These strategies enable distributed and hierarchical control applications in power grids, robotics, and adaptive structures, ensuring safety and efficient performance.

Energy-based control strategies constitute a class of control methodologies in which the feedback law is derived using the physical system's energy balance, dissipation, and storage structure. These methods exploit system passivity, port-Hamiltonian modeling, energy shaping, and formal dissipation injection to achieve regulation, tracking, safety, or optimality, particularly in complex, distributed, or uncertain systems. The foundational principle is to leverage explicit or implicit Lyapunov (energy) functions and system-theoretic invariants to ensure stability, robustness, and interpretability.

1. Port-Hamiltonian and Energy-Space Frameworks

A central paradigm in energy-based control is the port-Hamiltonian (pH) system representation, applicable in both finite and infinite dimensions, and across domains (electromechanical, structural, biochemical, power systems). A pH system is typically formulated as

$$\dot x = (J(x) - R(x))\nabla H(x) + G(x)u,\qquad y = G(x)^{T}\nabla H(x)$$

where $H(x)$ is the Hamiltonian (total stored energy), $J(x)$ is a skew-symmetric structure matrix encoding interconnections, $R(x)$ is a symmetric positive semi-definite dissipation matrix, $G(x)$ maps inputs $u$ to state, and $y$ is the collocated output. This structure admits distributed-parameter extensions using jet-bundle or Stokes-Dirac frameworks for PDE systems (Malzer et al., 2018, Malzer et al., 2020).

For large-scale or networked systems, energy-based control can be systematized by lifting physical states into an "energy space" containing explicit states for stored energy $E$, its time derivative $p=\dot E$, and tangent-space energy $E_t = \frac{1}{2}\dot x^T H \dot x$ (Gada et al., 10 Jun 2025, Jaddivada et al., 2021). Port interconnections and interaction variables encode the flow of real and reactive power, forming the backbone of distributed and hierarchical control architectures.

2. Energy Shaping, Damping Injection, and Structural Invariants

Stabilization and performance enhancement are achieved by shaping the system's energy landscape such that the desired equilibrium (set-point or trajectory) is a strict minimum of a closed-loop Lyapunov functional. For linear and weakly coupled systems, energy shaping can be realized algebraically without solving PDE matching equations (Javanmardi et al., 2024). In PDE systems, Casimir functionals (structural invariants linking plant and controller energy) are constructed to constrain closed-loop dynamics (Malzer et al., 2018, Malzer et al., 2020).

Damping injection is systematically designed by introducing additional dissipative terms, either in controller dynamics or via interconnection gains, to guarantee strict energy decay: $H(x)$0 This ensures passivity and enables Lyapunov or LaSalle-based global asymptotic stability proofs, including in the infinite-dimensional setting.

3. Distributed and Hierarchical Architectures

Networked and converter-dominated power systems necessitate multilayered, physically decomposed approaches. The energy-space framework provides a component-level linear model, bypassing domain-specific nonlinearities, with interconnections enforced by generalized Tellegen’s theorem on port variables. Controllers can be distributed, using only local measurements and neighbor port-invariants, with system-wide stability ensured by compositionality (Gada et al., 10 Jun 2025, Jaddivada et al., 2021). The energy-based feedback linearizing (FBLC) and sliding-mode (SMC) controllers offer, respectively, smooth asymptotic convergence and finite-time reachability, with strong robustness against bounded disturbances.

Energy-based control also underpins hierarchical strategies for microgrid operation, integrating economic dispatch, real-time storage management, and thermal load control while respecting system power balance, state constraints, and device degradation models (Zhuo, 2019).

4. Control of Infinite-Dimensional and In-Domain Actuated Systems

Energy-based control extends to systems described by PDEs, such as flexible beams, strings, and distributed-parameter electromechanical devices. The pH framework (jet-bundle or Stokes-Dirac) provides a canonical form for modeling, stability analysis, and controller interconnection (Malzer et al., 2018, Malzer et al., 2020).

For in-domain actuation (e.g., piezo-actuated beams), the distributed input profile modifies the Hamiltonian's variational structure, enabling dynamic controllers linked by Casimir invariants. Closed-loop passivity and convergence are preserved; observer design leveraging energy-based error injection yields robust state estimation even under partial measurements.

5. Passivity, Safety Layers, and Formal Guarantees

Many energy-based strategies rely explicitly on passivity: the closed-loop system’s energy cannot increase except via external supply. Passivity ensures stability against unknown but passive environments and facilitates modular controller composition.

For physical human-robot interaction or aerial physical manipulation, energy-tank frameworks wrap the system’s non-passive behaviors in a dynamical safety envelope: the tank state represents available control energy, with energy-injection constrained by formal inequalities reflecting safety standards (e.g., ISO/TS 15066 in robotics). Adaptation policies (e.g., power-valve modulation) throttle energetic outflows, guaranteeing stability and physical constraint satisfaction, even under severe model uncertainty (Benzi et al., 2023, Brunner et al., 2022).

6. Optimization, Learning, and Inverse Control

Energy-based optimization paradigms encompass both direct optimal control (minimizing energy expenditure, maximizing harvested energy, or balancing dissipation and actuation) and inverse optimal control (inferring underlying cost/energy functions from demonstrations via maximum-likelihood estimation of energy-based models) (Schaller et al., 2023, Xu et al., 2019, Hosseinloo et al., 2015). Methods leveraging Langevin dynamics, sampling, or analytical gradient flows allow for sample-efficient and physically meaningful cost identification (analysis by synthesis).

In learning solution manifolds for high-DOF control problems, direct minimization of integrated energy across problem distributions (rather than behavioral cloning) eliminates interpolation artifacts and improves generalization, leveraging adaptive sampling and dynamic target construction based on the system’s energy landscape (Zamora et al., 2022).

7. Practical Performance and Applications

Energy-based controllers have been validated in a broad array of physical and cyber-physical systems:

• MEMS and electromechanical actuation: Enhanced damping and tracking by algebraic energy shaping and coupled-damping injection for weakly coupled devices (Javanmardi et al., 2024).
• Power grids and microgrids: Distributed, energy-based feedback stabilizes voltage and frequency, coordinates storage and load, and optimizes cost/profit under high renewable variability (Gada et al., 10 Jun 2025, Zhuo, 2019).
• Adaptive structures: Energy-optimal vibration control offers physically interpretable performance and turnpike guarantees, outperforming classical LQR approaches (Schaller et al., 2023).
• Complex networks: Targeted energy-based control reveals exponential scaling laws for regime-energy trade-offs (Klickstein et al., 2016).
• Energy harvesting: Passive optimal control exploits energy flows for maximal vibrational energy extraction (Hosseinloo et al., 2015).
• Biomolecular networks: Bond-graph energy modeling explains cyclic flow modulation and natural PI-action motifs (Gawthrop, 2020).
• Safety in robotics and aerial systems: Energy tanks enforce formal energy bounds in regulation and interaction, enabling dynamic, standards-compliant control (Benzi et al., 2023, Brunner et al., 2022).

• (Malzer et al., 2018) Energy-Based In-Domain Control of a Piezo-Actuated Euler-Bernoulli Beam
• (Javanmardi et al., 2024) Energy-Based Control Approaches for Weakly Coupled Electromechanical Systems
• (Gada et al., 10 Jun 2025) Distributed component-level modeling and control of energy dynamics in electric power systems
• (Jaddivada et al., 2021) Distributed energy control in electric energy systems
• (Gawthrop, 2020) Energy-based Modelling of the Feedback Control of Biomolecular Systems with Cyclic Flow Modulation
• (Schaller et al., 2023) Energy-optimal control of adaptive structures
• (Zhuo, 2019) Control Strategies for Microgrids with Renewable Energy Generation and Battery Energy Storage Systems
• (Hosseinloo et al., 2015) Optimal control strategies for efficient energy harvesting from ambient vibrations
• (Benzi et al., 2023) Energy Tank-based Control Framework for Satisfying the ISO/TS 15066 Constraint
• (Brunner et al., 2022) Energy Tank-Based Policies for Robust Aerial Physical Interaction with Moving Objects
• (Tafrishi et al., 2022) Discretization and Stabilization of Energy-Based Controller for Period Switching Control and Flexible Scheduling
• (Xu et al., 2019) Energy-Based Continuous Inverse Optimal Control
• (Zamora et al., 2022) Learning Solution Manifolds for Control Problems via Energy Minimization
• (Malzer et al., 2020) Energy-Based In-Domain Control and Observer Design for Infinite-Dimensional Port-Hamiltonian Systems
• (Klickstein et al., 2016) Energy Scaling of Targeted Optimal Control of Complex Networks