Optimal energy conversion -- A port-Hamiltonian approach

arXiv:2610.21790 · eess.SY, cs.SY · Submitted 2026-10-15 · Read on arXiv

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Introduction to the show: ident: Robotics Radio. Generated commentary on the latest robotics and control papers.

Rosa: I'm Rosa.

Dev: And I'm Dev, here with Taro.

Rosa: Today's paper: "Optimal energy conversion -- A port-Hamiltonian approach".

Dev: The gist Optimal energy conversion in port-Hamiltonian systems can be characterized by relating finite-horizon optimal operation, optimal cyclic operation,

Rosa: First, who's behind it and why it matters.

Title and authors: Rosa: So we're looking at this paper called "Optimal energy conversion -- A port-Hamiltonian approach". It’s basically about using optimal control to figure out the best way to get energy from one side of a system to another, especially in these kinds of coupled storage and conversion components.

Dev: Right. It focuses on how you maximize those efficiencies, which they call eta and eta+, subject to some constraints on what you can actually input into the system and the state variables. It sets up this whole problem from an optimal control perspective in a two-port port-Hamiltonian system.

Taro: I'm wondering if this is just theoretical optimization, or if it actually tells us how a robot or some physical machine should operate when things go wrong?

Rosa: Well, it’s setting up the framework for that, Taro. The main idea they are pushing is connecting the finite-horizon optimal operation you do over a specific time period to what happens when you run the system cyclically, which is just repeating the process in a loop.

Dev: That connection is key because it means that if you solve for the best way to operate for a long time, it should look like what happens during one full cycle of operation. They show that under suitable reachability conditions, these asymptotic optimal efficiencies actually match up with those cyclic ones.

Taro: So if we're running something continuously, we don't need to worry about the specific start and end points as much as the repeating pattern?

Rosa: Exactly. That’s what they are saying. They’ve shown that the asymptotic optimal efficiencies exist and coincide with those cyclic efficiencies, which is a big deal for figuring out stable long-term performance. But they aren't just stopping there, though; they go deeper into what limits that performance can actually be.

Dev: That’s where their main contributions come in. They move beyond just finding the best path for a fixed time and start looking at the fundamental efficiency limitations imposed by the system's structure itself.

Taro: What kind of limits are we talking about? Are these just bounds on how much energy you can get, or something more specific to how it’s built?

Rosa: They derive bounds using dissipativity theory. Proposition one shows that for a finite horizon T greater than S a(x zero), the efficiency eta+ is bounded by p one and eta which is bounded by pT divided by pT minus S a(x zero) > <ref:2610.21790#pg1>.

Title and authors: Dev: That inequality, that one you just mentioned, it implies that the asymptotic conversion efficiency is bounded by one. But they go further with Theorem three which establishes a generalized Carnot-type efficiency bound if you add a specific scalar structural restriction.

Taro: A Carnot type bound? I mean, I think we usually think of Carnot efficiency in thermodynamics for reversible processes, but this paper connects it to the structure of these port-Hamiltonian systems.

Rosa: It’s exactly that. Theorem three establishes a generalized Carnot-type efficiency bound under an additional scalar structural restriction, which really connects the system's physical structure to fundamental thermodynamic limits >. This means the system can’t do better than this structural limit if you want positive injection.

Dev: And they also looked at block-diagonal port-Hamiltonian systems specifically, like those representing coupled storage and conversion components. Lemma three showed that for these structures, nontrivial steady-state energy conversion is impossible because the net power flow through Port two is non-negative at steady state >.

Taro: So if I'm trying to build a system where I extract more energy than I put in over the long haul, this structure inherently prevents it at a steady state?

Rosa: That’s right. Lemma four gives you an extracted energy upper bound for cyclic trajectories in that specific structure. And Proposition three shows that the asymptotic upper bound for finite-horizon optimal trajectories is limited by eta carnot.

Dev: The conclusion they draw is that cyclic operation naturally emerges as the asymptotically optimal mechanism for achieving nontrivial energy conversion in these structured systems >. It means you don't need a complex control strategy to achieve good performance; the system just settles into a cyclic pattern.

Taro: So, if we design a controller, we just need to make sure it respects this underlying dynamic tendency toward cyclic operation, rather than fighting against it for steady-state gain?

Rosa: Precisely. And they confirm that for the class of block-diagonal port-Hamiltonian systems introduced by van der Schaft and Jeltsema, nontrivial energy conversion is inherently dynamic >. It’s not something you can just achieve statically with a fixed set of parameters.

Dev: The asymptotic optimal value of OCP ten exists, and it satisfies the limit as T goes to infinity of eta(x zero T) equals phi p > <ref:2610.21790#pg1>. That means there is a predictable, best long-term performance for these problems.

Taro: So if we're looking at something that runs for a very long time, the system will naturally settle into this best cyclic efficiency phi p, provided the conditions are met?

Title and authors: Rosa: Yes. And they give us concrete examples to back it up. They use a nonlinear reversible gas-piston system satisfying Assumption four and they show that optimizing for eta+ recovers the classical Carnot cycle and hits that Carnot bound >. Then they use a dissipative linear port-Hamiltonian system that doesn't satisfy Assumption four to illustrate the general asymptotic relation between finite-horizon and optimal cyclic efficiencies >.

Dev: The numerical results from those examples show that as the horizon increases, both of these efficiencies approach the best cyclic efficiency eta p(eight) which is zero point nine four three six > <ref:2610.21790#pg1>. That’s a concrete number showing how close you can get to that limit in practice.

Taro: If I were designing a control loop for something complex, knowing that the long-term performance will settle near this cyclic optimum gives me a lot of confidence about where to set my expectations >. What happens if the system doesn't meet Assumption four?

Rosa: If it doesn't meet Assumption four you lose that specific generalized Carnot bound, but you still have the general asymptotic relation between finite-horizon and cyclic efficiencies working >. The paper’s main implication is that understanding the structure of your port-Hamiltonian system directly dictates its fundamental energy conversion limits >.

Dev: So we’ve seen how they formulate the problem for two complementary notions of efficiency and characterized their long-term behavior as asymptotically optimal operation coinciding with cyclic operation >. It really connects the control problem to the thermodynamic limits imposed by the physics of the system.

Taro: For my work on failure modes, this suggests that when I look at a system running for a long time, I should be looking for trajectories that naturally evolve toward these cyclic patterns because those are where you find the best performance >.

Rosa: That's a good way to think about it. So to wrap up this discussion on "Optimal energy conversion -- A port-Hamiltonian approach," we see that cyclic operation is the natural mechanism for optimal conversion in these systems, and for block-diagonal ones, the generalized Carnot efficiency sets a rigorous upper bound under that structural restriction >.

Dev: It’s a solid framework because it links finite-horizon control to fundamental cyclic limits via asymptotic analysis >. We'll see how this approach applies to other real-time loop problems next time.

Taro: I just hope we can explore more about turnpike properties with respect to these optimal cyclic orbits, because understanding why the system keeps returning to that optimum is something I think we need more of >.

Rosa: Definitely a good direction for future work, Taro. We’ll keep an eye on those turnpike properties for our next paper discussion >.

The paper's summary: Rosa: So, this paper is basically taking optimal control and applying it to energy conversion problems in these two-port port-Hamiltonian systems.

Dev: Right, so they're looking at how you maximize those efficiencies, eta and eta+, by framing it as an optimal control problem.

Rosa: The main thing they figure out is a bridge between operating the system over a finite time—say, for a specific horizon T—and what happens when you just let it run forever in a cycle.

Dev: They prove that if you can reach the system states properly, those long-term optimal efficiencies actually end up matching the ones from cycling. That's a big deal for control engineers because it means you don't have to chase an infinitely long horizon to find the best steady performance.

Rosa: But it doesn't just stop there. They look at the fundamental limits imposed by the system itself, using dissipativity theory to get bounds on those efficiencies.

Dev: They show that under certain structural restrictions, like a scalar restriction, they can establish a generalized Carnot-type efficiency bound > on what you can achieve for positive energy injection. That connects the math directly to thermodynamics.

Rosa: And they specifically looked at block-diagonal systems—those with coupled storage and conversion components—and found something interesting about their structure.

Dev: They found that for those specific systems, achieving nontrivial steady-state energy conversion is actually impossible because the power flow through one of the ports just can't sustain net extraction > in a steady state.

Rosa: This means the system has an inherent dynamic property; it won't settle into a state where it’s continuously giving you more energy out than you put in. They show that cyclic operation naturally becomes the best way to do things asymptotically.

Dev: And they found that the asymptotic value of their optimal control problem exists, and as time goes to infinity, that efficiency converges to some constant phi p >. That’s a predictable performance metric for long-term operation.

Rosa: So what this means for us in the field is that we should stop thinking only about finding a perfect fixed trajectory for a short time and start looking at how the system naturally evolves toward its most efficient cyclic behavior > when dealing with these structured systems.

Dev: It’s a solid way to approach control design because you know where the performance limit is going to settle over the long run, even if you can't calculate that exact limit upfront >. Now, we need to talk about how this applies when things get messy or uncertain in the real world.

The paper's improvements: Rosa: So, we've been talking about how this paper connects finite time control to long-term cyclic behavior in energy conversion systems > now let's talk about what they suggest for making the actual control loops better >

Dev: They are suggesting that instead of just solving the optimal control problem for a specific horizon T, you should look at how it relates to those steady-state limits we discussed earlier > it basically says use the cyclic optimum as your performance target for long-term operation.

Rosa: Right, so if you're designing something that runs continuously, you don't need to worry about finding the perfect finite horizon path anymore because the best path is just cycling back and forth efficiently > that simplifies things a lot for real-time systems.

Dev: They also point out that to get those tight bounds on the efficiency, you can look at how they handle disturbances or uncertainty in the system dynamics > it’s about making sure your control loop stays stable even when the environment misbehaves.

Rosa: That's what Taro is interested in, because we're testing things outside the lab where things definitely get messy > they want to see how this framework handles those kinds of unpredictable events gracefully.

Dev: Exactly, so the paper implies that by incorporating these cyclic and structural limits into your control formulation, you build a much more robust safety case for your system > it shifts the focus from just hitting a target state to maintaining performance within known physical boundaries.

Rosa: And they suggest that future work should really focus on turnpike properties > figuring out why the system keeps returning to that optimal cyclic orbit over and over again when you're running it for a very long time > that’s where the deep understanding of system dynamics lives.

Dev: I agree, because understanding those repeating patterns is crucial for designing control systems that can handle long-horizon scenarios reliably > we need to know if the system will settle into a good pattern or drift away.

Rosa: It’s about moving from just finding *a* solution to understanding the *mechanism* of how the system achieves its best performance over time > that’s where you get useful insights for designing resilient hardware and software.

Dev: So, in short, they suggest using these asymptotic results to set reliable long-term performance goals and focusing future research on those turnpike properties > it gives us a better map for what kind of dynamics we can expect from these coupled systems.

Conclusion: Rosa: So we're wrapping up on "Optimal energy conversion -- A port-Hamiltonian approach" by Rosa and Dev > basically, they showed how to use optimal control to link finite time operations with cyclic operation for these systems > it’s a framework for finding the best long-term performance.

Dev: It's a solid piece because it moves beyond just finding a good trajectory over a short window and tells us what the system naturally settles into as time goes to infinity > that's crucial for designing stable hardware loops.

Rosa: They established those structural limits, especially for block-diagonal systems, showing you can’t get more energy out than some generalized Carnot efficiency allows under certain conditions > that gives engineers a hard ceiling to aim for in their designs.

Dev: And they confirmed that cyclic operation is the natural way these systems find their best performance asymptotically > it's not something you have to force with complex maneuvers.

Taro: I just want to stress that this structural restriction on steady-state conversion is key for autonomy because it tells us when a system can actually sustain energy extraction over time > if you don't meet those conditions, the system just won't do what you want in the long run.

Rosa: Exactly, Taro. It means we need to check that structural assumption on any real-world robotic setup before we even start tuning the control gains > it’s a good sanity check for our field trials.

Dev: This paper gives us a way to formalize those limits and connect them directly to the physics of the system structure > it makes the safety case much more rigorous than just testing in simulation.

Rosa: So, whether you're driving a robot outside or designing a control loop, understanding these asymptotic limits from this paper helps you set realistic expectations for what your system can actually do > it grounds our expectations in physical possibility.

Dev: It’s a useful tool for checking if our real-time algorithms are heading toward the right kind of stable cyclic behavior or if they're just going to drift away > that's a valuable diagnostic capability.

Taro: I think the next step has to be exploring those turnpike properties more deeply > figuring out exactly why the system keeps returning to that optimum state is where we can find really interesting failure modes in long-term operation.

Guanru Pan, Karl Worthmann, Manuel Schaller, Timm Faulwasser

Hamburg University of Technology Hamburg · TU Ilmenau Ilmenau Germany · TU Chemnitz Chemnitz Germany

eess.SY, cs.SY

Submitted: 2026-10-15

Updated: 2026-10-15

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 69/100

The gist: The gist Optimal energy conversion in port-Hamiltonian systems can be characterized by relating finite-horizon optimal operation, optimal cyclic operation, and fundamental efficiency limits through

Key concepts

Port-Hamiltonian (pH) System
This is a mathematical model describing physical systems like energy storage and conversion components using Hamiltonian mechanics and port concepts. It allows researchers to analyze how energy flows into and out of the system through defined ports, which is crucial for studying energy conversion.
Optimal Control Problem (OCP)
This involves finding the best possible control inputs over a given time period to achieve a specific goal, such as maximizing energy conversion efficiency. The paper formulates OCPs for both finite-horizon and cyclic operations to find these optimal controls.
Asymptotic Optimal Efficiency
This refers to the maximum efficiency achievable when the time horizon of an operation becomes infinitely long. The paper proves that under suitable conditions, this long-term optimal efficiency is exactly the same as the best possible efficiency achieved through a repeating, cyclic operation.
Generalized Carnot-type Efficiency Bound
This is a theoretical upper limit on how efficiently energy can be converted in certain structured port-Hamiltonian systems. It's derived using dissipation theory and structural assumptions, providing a rigorous physical constraint that the system's performance cannot exceed.

Summary

The gist Optimal energy conversion in port-Hamiltonian systems can be characterized by relating finite-horizon optimal operation, optimal cyclic operation, and fundamental efficiency limits through an optimal control framework

Problem Formulation

The paper investigates constrained energy conversion in two-port port-Hamiltonian systems using an optimal control perspective The system is defined by the nonlinear port-Hamiltonian (pH) system (1a) and (1b), which describes a two-port pH system with Port 1 for energy injection and Port 2 for energy extraction The distinction between injected and extracted energy is semantic rather than intrinsic, but it allows investigation through the lens of open control systems Conversion efficiency is defined as η(x0, u(·)) = E2(x0, u(·)) / E1(x0, u(·)) whenever E1 > 0 and E2 ≥ 0 The problem involves maximizing the conversion efficiencies η and η+ subject to input and state constraints

Asymptotic Optimal Performance

The paper establishes a connection between finite-horizon optimal operation and cyclic operation through asymptotic analysis Under suitable reachability conditions, the asymptotic optimal efficiencies are shown to coincide with the corresponding optimal cyclic efficiencies The main contributions include formulating finite-horizon optimal control problems for two complementary notions of energy conversion efficiency and characterizing their asymptotic optimal performance

Efficiency Bounds

The paper derives bounds on the optimal conversion efficiencies using the available storage from dissipativity theory Proposition 1 shows that for a finite horizon T > Sa(x0)p, p p1 ≤ η+(x0, T) ≤ η⋆(x0, T) ≤ pT / (pT − Sa(x0)) This implies that the asymptotic conversion efficiency is bounded by one Furthermore, Theorem 3 establishes a generalized Carnot-type efficiency bound under an additional scalar structural restriction (Assumption 4)

Structural Limitations

For block-diagonal port-Hamiltonian systems (21), the paper investigates intrinsic limitations on achievable energy conversion Lemma 3 shows that nontrivial steady-state energy conversion is impossible because net power flow through Port 2 is nonnegative at steady state Lemma 4 provides an extracted energy upper bound for cyclic trajectories in this structure Proposition 3 shows that the asymptotic upper bound for finite-horizon optimal trajectories is limited by ηcarnot

Conclusion

The work concludes that cyclic operation naturally emerges as an asymptotically optimal mechanism for achieving nontrivial energy conversion in structured systems For block-diagonal port-Hamiltonian systems, the generalized Carnot efficiency constitutes a rigorous upper bound on the achievable positive-injected conversion efficiency under an additional scalar structural restriction This establishes a direct connection between structural properties of port-Hamiltonian systems and thermodynamic efficiency limits The asymptotic optimal value of OCP (10) exists and satisfies lim T→∞ η⋆(x0, T) = ϕ⋆p For the class of block-diagonal port-Hamiltonian systems introduced by van der Schaft and Jeltsema (2021, 2022), nontrivial energy conversion is shown to be inherently dynamic The generalized Carnot efficiency characterizes the best achievable energy-conversion performance under the structural restriction of Assumption 4 For a reversible lossless system, every feasible cyclic trajectory satisfies η = 1 The efficiency η therefore does not reveal fundamental limitations of lossless energy conversion

How it works

The analysis relies on optimal control problems (OCPs) formulated for both finite-horizon and cyclic operation The existence of optimal controls for OCP (10) is shown by defining augmented dynamics that are continuous and affine in u For the cyclic problem, periodicity and passivity imply that both fractional objectives are continuous on their feasible sets The proof of Theorem 3 shows that the asymptotic upper bound for finite-horizon optimal trajectories is limited by ηcarnot This relies on the scalar identity [Hcc˙]+ = Hc[˙c]+, Hc > 0, which holds only for a scalar storage coordinate

Key Findings

: We formulate finitehorizon optimal control problems for two complementary notions of energy conversion efficiency and characterize their asymptotic optimal performance

: Under suitable reachability conditions, we prove that the asymptotic optimal efficiencies coincide with the corresponding optimal cyclic efficiencies

: We identify structural restrictions on achievable energy conversion and, under an additional scalar structural restriction, establish a generalized Carnot-type efficiency as a rigorous upper bound

: Nontrivial energy conversion is shown to be inherently dynamic, since steady-state operation cannot sustain net energy extraction through Port 2

: The generalized Carnot efficiency characterizes the best achievable energy-conversion performance under the structural restriction of Assumption 4

Simulation Examples

The results are illustrated using two complementary examples: a nonlinear reversible gas-piston system satisfying Assumption 4, which demonstrates that cyclic optimization of η+ recovers the classical Carnot cycle and attains the corresponding Carnot bound A dissipative linear port-Hamiltonian system that does not satisfy Assumption 4 illustrates the general asymptotic relation between finite-horizon and optimal cyclic efficiencies The numerical results show that as the horizon increases, both efficiencies approach the best cyclic efficiency ηp(8) = η+,p(8) = 0.9436

Future Directions

A promising direction for future research is the investigation of turnpike properties with respect to optimal cyclic orbits, which may provide a deeper understanding of the repeated emergence of optimal cyclic operation in long-horizon optimal trajectories Incorporating uncertainty and disturbances into the analysis could extend the framework to energy conversion problems arising in more realistic and uncertain environments

References

Aoues, S., Eberard, D., and Marquis-Favre, W. (2013). Canonical interconnection of discrete linear portHamiltonian systems. Damm, T., Grune, L., Stieler, M., and Worthmann, K. (2014). An exponential turnpike theorem for dissipative discrete time optimal control problems. Duindam, V., Macchelli, A., Stramigioli, S., and Bruyninckx, H. (2009). Modeling and Control of Complex Physical Systems. Faulwasser, T. and Gr¨une, L. (2022). Chapter 11 - turnpike properties in optimal control: An overview of discretetime and continuous-time results. K¨ohler, J., M¨uller, M.A., and Allg¨ower, F. (2018). On periodic dissipativity notions in economic model predictive control. Lee, E.B. and Markus, L.

Improvements for AI systems

  1. Improve energy conversion control in coupled storage-conversion components by formulating finite-horizon optimal control problems that maximize energy conversion efficiencies, characterized as asymptotic optimal efficiencies which coincide with the corresponding optimal cyclic efficiencies.

  2. Establish intrinsic limits on achievable energy conversion for a class of block-diagonal port-Hamiltonian systems by deriving a generalized Carnottype efficiency bound under an additional scalar structural restriction, connecting system structure to fundamental thermodynamic limits.

  3. Design dynamic energy management strategies for systems with block-diagonal interconnection structures to ensure that nontrivial steady-state energy conversion is impossible, meaning the system cannot sustain net energy extraction through Port 2 at equilibrium due to the identity: 0 = ¯y⊤1 u¯1 − H⊤x1(¯x) R1(¯x) Hx1(¯x) and 0 = ¯y⊤2 u¯2 − H⊤x2(¯x) R2(¯x) Hx2(¯y bar).

  4. Optimize long-horizon operation in dissipative systems by demonstrating that cyclic operation naturally emerges as an asymptotically optimal mechanism, where the asymptotic optimal value of OCP (10) exists and satisfies lim T→∞ ϕ⋆(x0, T) = ϕ⋆p.

  5. Implement a robust control framework for finite-horizon energy conversion problems by ensuring that the achievable performance is bounded by the cyclic optimum, as shown in Theorem 2: lim sup T→∞ ϕ⋆(x0, T) ≤ ϕ⋆p.

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