Total Energy Shaping for Dynamical Systems

Updated 7 February 2026

Total energy shaping is a control method that modifies both kinetic and potential energy functions to stabilize dynamical systems.
It integrates port-Hamiltonian and Euler-Lagrange frameworks by solving matching PDEs to design feedback laws for desired closed-loop dynamics.
The approach finds broad applications from robotics and power electronics to communication systems, with extensions to distributed and soft continuum models.

Total energy shaping is a systematic methodology for synthesizing stabilizing controllers by modifying both kinetic and potential energy functions of a dynamical system, typically formulated within port-Hamiltonian or Euler-Lagrange frameworks. It is central to Interconnection and Damping Assignment Passivity-Based Control (IDA-PBC) in finite- and infinite-dimensional underactuated mechanical systems, with rigorous extensions to networked, distributed, and soft continuum models. The total energy shaping paradigm involves solving associated matching partial differential equations (PDEs) to design feedback laws, and has applications ranging from robotics and exoskeletons to power electronics and communication systems.

1. Core Principles and Problem Statement

Total energy shaping aims to assign a new total energy function—Hamiltonian or Lagrangian—by reshaping both kinetic and potential terms through feedback, thereby generating desired closed-loop stability properties. For mechanical systems in port-Hamiltonian form,

$\begin{pmatrix}\dot{q}\\dot{p}\end{pmatrix} = \begin{pmatrix} 0 & I \ -I & 0 \end{pmatrix} \begin{pmatrix} \nabla_q H \ \nabla_p H \end{pmatrix} + \begin{pmatrix} 0 \ G(q) \end{pmatrix} u,$

the total energy shaping objective is to design a new storage function,

$H_d(q,p) = \frac{1}{2}p^\top M_d^{-1}(q) p + V_d(q),$

and enforce closed-loop dynamics of passive port-Hamiltonian type. The approach splits into kinetic energy shaping (modifying the inertia tensor $M \to M_d$ ) and potential energy shaping ( $V \to V_d$ ). Achieving this requires solving the so-called matching PDEs, which express compatibility between the plant's structure and desired closed-loop energy function.

The methodology generalizes to continuum (distributed parameter) systems, where the state includes spatially distributed variables and the energy functionals are defined over fields $x(\zeta,t)$ with spatial variable $\zeta$ (Liu et al., 2023), and to specialized contexts such as constrained communication channels (Valenti et al., 2012).

2. Structure of the Matching Conditions

The central step in total energy shaping is solving for $M_d(q)$ and $V_d(q)$ :

Finite-Dimensional Systems

The matching PDEs for underactuated mechanical systems are typically:

Kinetic energy PDE (inertia shaping):

$G^\perp \left\{ \nabla_q(p^T M^{-1}p) - M_d M^{-1} \nabla_q(p^T M_d^{-1}p) + 2 J_2 M_d^{-1}p \right\} = 0$

where $G^\perp$ spans the left annihilator of the input distribution, and $J_2$ is a skew-symmetric interconnection matrix.

Potential energy PDE (potential shaping):

$G^\perp \left\{ \nabla_q V - M_d M^{-1} \nabla_q V_d \right\} = 0.$

Solving these PDEs is generally challenging. For systems with a single unactuated degree of freedom, the kinetic energy PDE reduces to a single ODE, and general existence/solvability theorems apply (Harandi et al., 2020, Gharesifard, 2010).

Infinite-Dimensional and Distributed Systems

For distributed parameter systems, early lumping or discretization is performed, and the matching reduces to algebraic equations on the discretized states (Liu et al., 2023). Casimir-based control-by-interconnection methods can further recast the energy shaping constraint as algebraic or ODE conditions on the added energy (Ferguson, 2023).

3. Controller Synthesis and Feedback Law

After determining the shaped energy components $(M_d, V_d)$ and optionally the interconnection ( $J_2$ ) and damping blocks, one synthesizes a feedback law: $u(q,p) = (G^T G)^{-1} G^T \left\{ \nabla_q H - M_d M^{-1} \nabla_q H_d + J_2 M_d^{-1} p \right\} - K_v G^T M_d^{-1} p,$ where $K_v$ injects additional damping. For distributed or infinite-dimensional settings, the feedback interconnection is constructed to preserve the extended Hamiltonian structure on the plant–controller composite (Liu et al., 2023). In power electronics, the controller often incorporates additional integral states for output regulation and disturbance rejection (He et al., 2024).

In modern approaches, the PDE constraint can be solved approximately, e.g., by interpolating $M_d$ over a finite grid and enforcing potential-matching only at sample points, with asymptotic convergence guarantees provided the residual is sufficiently small (Yildiz et al., 2020).

4. Extensions: Robustness, Actuator Constraints, and Computation

Robustness to Disturbance

Adaptation and observer design have been integrated for robust energy shaping in presence of unknown disturbances, with explicit separation of matched and mismatched channels, often leveraging Lyapunov-based or cascaded-system analysis (Harandi et al., 31 Jan 2026, He et al., 2024).

Actuator Saturation and Control-Effort Reduction

The control law’s velocity-dependent (kinetic-shaping) terms may provoke large or unbounded inputs. Approaches to bound control effort include:

Lyapunov-based a priori estimation of velocity given closed-loop energy (Harandi et al., 2021),
Design of bounded homogeneous (nullspace) contributions to $V_d$ ,
Conditional switching to avoid increase in effort from velocity-shaping attenuation (Harandi et al., 31 Jan 2026). An explicit $\ell_\infty$ -norm optimization admits analytic solutions for the free component of the generalized velocity-dependent forces, suppressing kinetic-shaping contributions within feasible bounds even in single-actuator cases (Harandi et al., 31 Jan 2026).

Data-Driven and Neural Synthesis

Universal approximation properties of neural networks (NNs) have enabled physics-informed learning architectures for total energy shaping, where neural networks approximate the desired energy functions and damping matrices while encoding Lyapunov and structural constraints into the loss via collocation and physics-informed regularization (Sanchez-Escalonilla et al., 2021).

5. Applications Across Domains

Mechanical and Robotics Systems

Pedagogical and research benchmarks include cart-pole, acrobot, pendubot, and rotary inverted pendulum (Harandi et al., 2020, Harandi et al., 31 Jan 2026). Backdrivable hip exoskeleton control employs total energy shaping with passivity-preserving basis function fitting to normative human torques, yielding task-invariant assistance across gait activities (Zhang et al., 2022).

Flexible and Soft Robotics

Total energy shaping generalizes to continuum Cosserat rod models for bio-inspired soft arms, with distributed feedback enforcing desired equilibrium shapes and penalizing deviations in intrinsic strain variables (Chang et al., 2020). Stability and real-time implementation are verified numerically with forward–backward sweeps and fast time-integration.

Power Electronics

Total energy shaping governs robust voltage regulation in power converters with mixed ZIP loads, utilizing port-Hamiltonian state modeling, energy-based PI-like control, integral compensation for reference tracking, and observer-based disturbance rejection (He et al., 2024).

Communication Systems

A distinct variant of "total energy shaping" (Editor’s term: "probabilistic energy shaping"), focuses on reducing average symbol energy in coded modulation by probabilistically shaping input distributions (via biased codewords) and jointly optimizing constellation parameters. Simulation gains exceed 1 dB at 3 bit/symbol in bit-interleaved LDPC coded APSK over iterative demod–decoding protocols (Valenti et al., 2012).

Distributed Parameter Systems

Port-Hamiltonian control-by-interconnection with early lumping extends total energy shaping to distributed transport and wave systems, providing both theoretical stability (via semigroup methods) and practical approximation schemes for underactuated actuator layouts (Liu et al., 2023).

6. Structural and Computational Innovations

Formal geometric integrability, such as Goldschmidt–Spencer theory for jet bundles, guarantees the solvability of the full matching PDE system for simple mechanical systems with one degree of underactuation, allowing indefinite (not necessarily positive-definite) closed-loop metrics if required for stabilization (Gharesifard, 2010). Control-by-interconnection recasts energy shaping as a passive two-port network interconnection, yielding alternative algebraic matching criteria on the added energy that can simplify controller synthesis especially for low-dimensional unactuated motion (Ferguson, 2023).

Reformulations of the potential matching equation highlight that, under mild positivity (LMI) conditions, it suffices to solve only the homogeneous PDE portion, replacing a nonhomogeneous PDE with a tractable algebraic gain selection problem (Harandi et al., 2021).

Various algorithmic approaches realize controller design:

ODE or algebraic reduction for $n-m=1$ ,
grid-based or basis-function parametrization for underactuated nonlinear systems,
data-driven or learning-based architectures for multi-DOF systems (Sanchez-Escalonilla et al., 2021, Zhang et al., 2022).

7. Performance, Limitations, and Comparative Analysis

The total energy shaping approach offers a powerful, physically interpretable, and stability-guaranteed paradigm for control of complex mechanical, electrical, and robotic systems, robust to model uncertainty and disturbances under appropriate extensions (Ferguson, 2023, Harandi et al., 31 Jan 2026). It enables integration of energy-based feedback, data-driven fitting, and passivity guarantees within a unified framework.

Solving the matching PDEs remains a significant barrier for high-dimensional or strongly nonlinear underactuated systems, though geometric, computational, and approximation advances are broadening practicality. Kinetic energy shaping, while enabling unprecedented flexibility (e.g., stabilization via indefinite metrics), can induce challenging velocity-dependent control actions and risk actuator saturation unless carefully managed (Harandi et al., 2021, Harandi et al., 31 Jan 2026).

Comparison with potential-only shaping or classic IDA-PBC approaches reveals that total (kinetic plus potential) energy shaping, especially when harnessing the full formal integrability of the matching equations, substantially enlarges the design space for closed-loop dynamics, often providing stabilization guarantees not possible in the restricted setting (Gharesifard, 2010).

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