⚡ electrical engineering

Towards Co-Designed Event-Triggered Extremum Seeking

This paper proposes a co-design framework that jointly synthesizes full or diagonal controller gains and an event-triggering mechanism for multivariable extremum seeking under Hessian uncertainty, formulating the problem as a convex optimization to guarantee exponential stability while maximizing communication efficiency.

Original authors: Roberto Luo, Pedro Henrique Silva Coutinho, Victor Hugo Pereira Rodrigues, Tiago Roux Oliveira, Miroslav Krstic

Published 2026-08-12

📖 3 min read☕ Coffee break read

CC BY 4.0

Original authors: Roberto Luo, Pedro Henrique Silva Coutinho, Victor Hugo Pereira Rodrigues, Tiago Roux Oliveira, Miroslav Krstic

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 find the deepest point in a vast, foggy valley. You can't see the whole landscape, and you don't have a map. All you have is a small, wiggly probe that you can stick into the ground to feel if the slope is going up or down. This is the world of Extremum Seeking, a clever trick engineers use to find the best settings for a machine without needing to know exactly how the machine works. It's like tuning a radio by slowly turning the dial until the static clears, except the "static" is a complex mathematical function, and the "dial" is a control knob.

Now, imagine you are the robot doing this tuning, but you have a strict rule: you can only shout your findings to your brain (the controller) when you are absolutely sure the signal is worth sending. This is Event-Triggered Control. Instead of constantly screaming updates every second (which wastes energy and clogs the network), you only speak up when something important changes. This saves battery and bandwidth, which is crucial for things like self-driving cars or smart grids. But here's the tricky part: if you wait too long to shout, you might miss a turn and crash; if you shout too often, you waste all your energy. The big question scientists have been asking is: how do we design the robot's "brain" and its "shouting rules" at the same time to make sure it finds the bottom of the valley fast, without talking too much?

This paper, titled "Towards Co-Designed Event-Triggered Extremum Seeking," tackles that exact puzzle. The authors, a team from Brazil and the US, propose a new way to build these systems. Instead of designing the robot's brain first and then trying to figure out when it should shout (a method they call "emulation"), they design both the brain and the shouting rules together in a single, unified plan. They call this Co-Design.

The researchers discovered that by letting the robot's brain use a "full" map of the terrain—meaning it pays attention to how different parts of the valley slope relate to each other, rather than just looking at one direction at a time—it can be much smarter. In their simulations, they tested this against the old way of doing things, where the brain only looked at each direction independently (like checking the slope North, then South, then East, then West, without connecting the dots).

The results were clear. When the robot used the "full" brain that understood the connections between directions, it needed to shout far fewer times to find the perfect spot. In fact, the old "independent" brain simply couldn't match the performance of the new one; it either took too long to find the bottom or had to shout way too often to stay safe. The paper proves mathematically that this new co-designed approach is stable and won't get stuck in an infinite loop of shouting (a problem known as "Zeno behavior"). Through computer simulations with a two-dimensional valley, they showed that the new method successfully guides the system to the optimal point while drastically cutting down on unnecessary communication, proving that understanding the "shape" of the problem helps you talk less and achieve more.

Technical Summary: Towards Co-Designed Event-Triggered Extremum Seeking

Problem Formulation
This paper addresses the control of nonlinear static maps with polytopic Hessian uncertainty using a model-free extremum seeking (ES) approach. The objective is to drive the system output to its extremum while minimizing resource usage (computation and communication) through event-triggered control (ETC).

Existing event-triggered ES methods typically rely on an "emulation-based" design where the controller gain is fixed (often as a diagonal matrix) first, and the event-triggering mechanism (ETM) is synthesized subsequently. This sequential approach restricts the achievable closed-loop performance because the controller and triggering law cannot be optimized simultaneously. Furthermore, restricting the controller gain to a diagonal structure ignores the coupling between optimization variables induced by the Hessian matrix, leading to conservative triggering conditions and unnecessary communication.

The paper specifically targets multivariable quadratic static maps where the Hessian matrix $H^*$ is unknown but belongs to a known polytopic domain defined by vertices $\{H_1, \dots, H_N\}$ . The system operates under a zero-order hold (ZOH) with discrete control updates triggered by a condition dependent on the gradient estimate and transmission error.

Methodology
The proposed approach introduces a co-design framework that jointly synthesizes the controller gain matrix $K$ and the event-triggering mechanism (ETM). Unlike previous works, this framework allows for full-matrix controller gains, thereby exploiting Hessian-induced cross-couplings, rather than being restricted to diagonal gains.

The technical methodology proceeds as follows:

System Modeling: The ES system is modeled using probing and demodulation signals to estimate the gradient. The closed-loop dynamics are expressed in terms of the estimation error and transmission error.
Averaging Analysis: To handle the periodic perturbation signals and the discontinuous nature of event-triggered updates, the authors employ averaging theory. This transforms the original non-autonomous, discontinuous system into an autonomous average system. The stability of the original system is then inferred from the stability of this average system.
Co-Design Condition: The joint synthesis problem is formulated as a convex optimization problem subject to Linear Matrix Inequality (LMI) constraints.
- Stability: A Lyapunov-based analysis establishes that if specific LMIs (involving decision variables for the gain $K$ and triggering matrices $Q_G, Q_e$ ) are satisfied, the average closed-loop system is exponentially stable with a prescribed decay rate $\eta$ .
- Optimization: To reduce communication, the authors formulate an optimization problem that minimizes the trace of the decision variables (related to the triggering thresholds). This maximizes the admissible transmission error, thereby increasing the inter-event time intervals and reducing the number of triggers.
Zeno-Freeness: The paper proves that the inter-event intervals are bounded from below by a positive value, ensuring the system is free from Zeno behavior (infinite triggers in finite time).

Key Contributions

Unified Co-Design Framework: The paper proposes a method to simultaneously design the controller gain and the triggering mechanism, moving beyond the limitations of sequential emulation-based designs.
Full-Matrix Gain Utilization: The framework explicitly supports full-matrix controller gains. The authors demonstrate that utilizing off-diagonal elements (coupling information) allows for larger admissible triggering thresholds compared to diagonal designs.
Polytopic Uncertainty Handling: The approach accommodates Hessian uncertainty modeled as a polytopic domain, ensuring robustness across the range of possible curvature values.
Theoretical Guarantees: The paper provides rigorous proofs for the exponential stability of the average closed-loop system, the convergence of the original system to the optimum (within $O(a + 1/\omega)$ bounds), and the exclusion of Zeno behavior.

Results
Numerical simulations were conducted on a two-dimensional nonlinear map with a Hessian uncertainty bound $\bar{\sigma}$ .

Comparison of Gain Structures: When solving the co-design optimization problem for various uncertainty levels, the full-matrix gain design consistently yielded lower objective function values (representing fewer transmissions) compared to the diagonal gain design.
Feasibility: In specific test cases (e.g., $\bar{\sigma}=0.6$ , decay rate $\eta=1$ ), the co-design with a full-matrix gain was feasible, whereas the diagonal gain design failed to satisfy the same decay rate and triggering threshold constraints simultaneously.
Performance: The full-matrix co-designed controller achieved the prescribed exponential convergence rate with significantly fewer triggering events (e.g., 38 updates vs. 104 updates for different Hessian vertices) compared to scenarios where diagonal gains would be required to meet similar stability criteria.
Uncertainty Impact: The results illustrated that as the uncertainty polytope size increases, the number of required transmissions increases, highlighting the value of accurate prior curvature information.

Significance
The paper claims that its primary significance lies in demonstrating that prior information about the objective function's curvature (specifically the Hessian structure) is not only useful for guaranteeing stability but is also critical for reducing communication loads. By exploiting the coupling between optimization variables through full-matrix gains within a co-design framework, the proposed method achieves superior triggering performance. The results indicate that diagonal gains are insufficient to achieve the same decay rates and triggering thresholds as full-matrix gains, even when both are synthesized using the same co-design methodology. This suggests that the "extra degrees of freedom" provided by full-matrix structures are essential for optimizing the trade-off between convergence speed and communication efficiency in event-triggered extremum seeking.

More like this