⚡ electrical engineering

Families of Control-Cost-Parametrized Inverse-Optimal Universal Stabilizers

This paper introduces a "half-direct-optimal" framework that generates a family of universal stabilizers by allowing users to select a control cost function, which is then transformed via a proven Lipschitz operator into a nonlinear feedback law that solves an inverse-optimal control problem with meaningful state costs, thereby enabling uniform neural operator approximation for both offline analysis and online adaptation.

Original authors: Miroslav Krstic, Luke Bhan

Published 2026-06-09

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Miroslav Krstic, Luke Bhan

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to steer a complex machine, like a self-driving car or a drone, to a safe stop. In the world of control theory, there is a famous, "one-size-fits-all" recipe called Sontag's Formula. It guarantees the machine will stop safely, no matter what. However, it's like a pre-packaged meal: it works, but you can't change the flavor. You can't make it spicier (more aggressive) or milder (more cautious). It has no knobs for you to turn.

On the other end of the spectrum, there are other methods that give you too many knobs to turn, but they are so complicated and disconnected that you have no idea which combination will actually work well.

This paper introduces a new, "Goldilocks" approach called Half-Direct-Optimal Control. Here is how it works, explained through simple analogies:

1. The Core Idea: Designing the "Cost" First

Instead of starting with a controller and asking, "What does this minimize?", the authors flip the script. They say: "You tell us how much you hate using the controls, and we will build the perfect controller for you."

The Analogy: Imagine you are hiring a chef. Instead of the chef deciding how much salt to use, you hand them a "Salt Preference Card" (a mathematical function called $\gamma$ ).
- If you write "I hate salt," the chef (the formula) creates a dish that uses very little salt but still tastes good.
- If you write "I love a strong kick," the chef creates a spicy dish.
- The paper proves that for any reasonable preference card you write, there is a specific, mathematically perfect way to cook the meal (the controller) that respects your taste while guaranteeing the food is safe to eat (the system stabilizes).

2. The Magic Machine: The "Expander"

To turn your "Salt Preference Card" into a working controller, the paper uses a three-step machine:

Differentiation: It looks at how your preference changes.
Algebraic Contraction: It squeezes that information into a specific shape.
Inversion: It flips the shape inside out to create a tool called an "Expander" ( $\kappa$ ).

The Analogy: Think of the "Expander" as a customized gear shift.
- The standard Sontag formula is like a car with a fixed gear ratio. It shifts at one specific speed.
- Your "Salt Preference Card" determines the shape of a new gear.
- The "Expander" is the mechanic who builds that custom gear. Once built, you attach it to the car, and suddenly, the car shifts gears exactly the way you wanted it to, making the ride smoother or faster depending on your choice.

3. The "Half-Direct" Name

The authors call this "Half-Direct-Optimal" because:

Direct: You get to choose the cost of the control (the "Salt") upfront.
Half: You don't get to choose the cost of the state (how the car moves) directly. That cost is a natural consequence of your choice. It's like saying, "I choose how much I want to pay for gas, and the car's speed is whatever results from that budget."

4. The AI Shortcut: Neural Operators

Calculating that custom "Expander" gear for every new preference is mathematically heavy and slow, like doing complex calculus in your head every time you want to shift gears.

The Analogy: The authors trained a Neural Network (AI) to act as a "Universal Mechanic."
- Instead of calculating the gear from scratch every time, you feed the AI your "Salt Preference Card."
- The AI instantly spits out the correct "Expander" gear.
- The paper proves this AI is reliable: if the AI is 99% accurate, the car will still stop safely, and the "mistake" in performance is tiny (mathematically, it's a "second-order" error, meaning a small input error leads to a very small output error).

5. The Result: A Toolbox for Engineers

The paper demonstrates this with a Unicycle (a robot that balances on one wheel).

They tested three controllers:
1. The "Standard" controller (Sontag's formula).
2. The "Custom" controller (based on a specific cost preference).
3. The "AI Approximated" controller (using the Neural Network).
The Outcome: All three stopped the unicycle safely. However, the "Custom" and "AI" controllers could be tuned to be much more aggressive (stopping faster) or more gentle than the standard one. The AI version was almost identical to the perfect custom version but was much faster to compute.

Summary

This paper gives engineers a universal remote control for stabilizing systems.

Old Way: You get one remote with no buttons. It works, but you can't adjust it.
New Way: You get a remote where you can program the "aggressiveness" button. The paper provides the math to ensure that whatever you program, the system stays safe.
The AI Twist: They also built a smart assistant that learns to program these remotes instantly, so you don't have to do the heavy math yourself.

The paper is dedicated to Eduardo Sontag, a giant in the field, on his 75th birthday, offering a modern evolution of his original "one-size-fits-all" idea.

Technical Summary: Families of Control-Cost-Parametrized Inverse-Optimal Universal Stabilizers

Problem Statement
The paper addresses a fundamental limitation in classical universal stabilization, specifically Sontag's formula. While Sontag's formula guarantees global asymptotic stability for any system admitting a Control Lyapunov Function (CLF), it offers no design freedom to the practitioner; it is a single, parameter-free object. Conversely, the Curtis–Beard construction provides a complete family of inverse-optimally stabilizing controllers but relies on two state-dependent functions with complex, indirect effects on closed-loop transients, making it difficult to interpret and tune. The authors aim to bridge this gap by introducing a parametrized family of stabilizing feedback laws that offers explicit design freedom while maintaining rigorous stability and inverse optimality guarantees.

Methodology
The proposed framework, termed "half-direct-optimal," reverses the traditional order of inverse-optimal design. Instead of constructing a controller and then identifying a cost functional it minimizes, the designer first prescribes a scalar function $\gamma(s)$ representing the running cost on the control input.

The methodology proceeds through a three-step construction to map the chosen cost function $\gamma$ to a nonlinear "expander" function $\kappa$ :

Differentiation: Compute the derivative $\gamma'$ .
Algebraic Contraction: Define a "contractor" function $\Theta(s) = s - \gamma(s)/\gamma'(s)$ .
Inversion: Define the expander as the inverse, $\kappa = \Theta^{-1}$ .

Given a CLF $V(x)$ and a nominal feedback law (such as the Freeman-Kokotovic minimum-norm formula or the half-Sontag formula), the final stabilizing feedback $u^*(x)$ is constructed by applying the expander $\kappa$ to the magnitude of the nominal control: $u^*(x) = \text{sgn}(u_0(x))\kappa(|u_0(x)|)$ .

To facilitate practical implementation, the paper establishes that the mapping $\mathcal{K}: \gamma \mapsto \kappa$ is Lipschitz continuous on compact subsets of the relevant function space. This regularity allows the entire family of controllers to be uniformly approximated by a single trained Neural Operator (specifically a Fourier Neural Operator). This enables offline exploration of the design space and online adaptation without solving the infinite-dimensional inversion problem at every time step.

Key Contributions

Half-Direct-Optimal Design Paradigm: The paper introduces a family of stabilizers where the control cost $\gamma$ is the primary design parameter. Every controller in this family is proven to be the unique minimizer of an infinite-horizon cost functional with the prescribed control penalty $\gamma$ and an induced state cost $q(x)$ .
Lipschitzness and Learnability: The authors prove that the operator mapping the control cost $\gamma$ to the expander $\kappa$ is Lipschitz. This theoretical result guarantees that neural operators can approximate the mapping with arbitrary accuracy, transforming the theoretical construction into a scalable engineering tool.
Stability and Near-Optimality under Approximation: The paper establishes that when the expander is approximated by a neural operator, the resulting feedback law preserves semiglobal practical asymptotic stability (SPAS). Furthermore, it quantifies the performance loss: the induced state-cost distortion is a second-order perturbation ( $O(\epsilon^2)$ ) relative to the approximation error $\epsilon$ of the neural operator. A finite-horizon optimality bound is also derived to measure the performance gap between the approximate and exact controllers.

Results
The theoretical results are illustrated numerically using a unicycle dynamics model. The study compares three controllers:

The exact inverse-optimal controller (computed via bisection).
The neural-operator approximated controller.
The nominal half-Sontag controller (quadratic cost).

The numerical results demonstrate that the neural-operator approximation is visually indistinguishable from the exact solution in closed-loop trajectories. The learned controller successfully drives the system to the origin, preserving stability. The results confirm that while the exact inverse-optimal design yields more aggressive transients and faster CLF convergence (at the cost of higher control effort), the neural approximation faithfully replicates the expander mapping, validating the feasibility of using machine learning to deploy this parametric family in real-time.

Significance
The paper claims to transform universal control from a theoretical existence result into a practical engineering tool. By allowing the practitioner to select a cost function $\gamma$ that encodes specific preferences for control aggressiveness or caution, the framework provides a systematic way to shape transient behavior. The "half-direct-optimal" nature of the design ensures that performance shaping and inverse optimality are not in tension but are intrinsic properties of the same object. The integration of neural operators allows for the "scale" usage of these controllers, enabling massive offline performance exploration and online adaptation, thereby making the theoretical elegance of universal stabilization accessible to practical application.

Note: The paper explicitly notes that the dual problem—where the state cost is arbitrary and the control cost is induced—is easier and outside the scope of this work. The current work focuses on the case where the control cost is the design input.