⚡ electrical engineering

Recursive Experiment Design for Closed-Loop Identification of ARMAX Systems with Output Perturbation Limits

This paper proposes a recursive, closed-form method for designing informative probing signals in closed-loop ARMAX system identification that respects user-specified output perturbation limits while ensuring identifiability and feasibility.

Original authors: Jingwei Hu, Dave Zachariah, Torbjörn Wigren, Petre Stoica

Published 2026-06-08

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Jingwei Hu, Dave Zachariah, Torbjörn Wigren, Petre Stoica

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 learn the exact recipe for a complex dish (the "system") while you are already cooking it for a hungry customer. You can't just stop cooking to taste-test every ingredient because the customer is waiting, and if you mess up the flavor too much, the customer might get angry or the dish could burn. This is the challenge of closed-loop system identification: you need to learn how the system works without shutting it down or causing dangerous disruptions.

Here is how the paper by Hu, Zachariah, Wigren, and Stoica solves this problem, explained through simple analogies.

The Problem: The Silent Chef

Usually, to learn a recipe, you might add a pinch of salt here and a dash of pepper there to see how the taste changes. In engineering terms, this is called "probing."

However, in many real-world systems (like a power grid or a chemical plant), there is already a "Chef" (a feedback controller) in charge. This Chef is very good at keeping the temperature or pressure steady. If the Chef is doing their job perfectly, the system stays so stable that you can't tell much about how it works just by watching it. It's like trying to learn the recipe of a cake by only watching a machine that keeps the oven at a constant 350°F; nothing changes, so you learn nothing.

To learn the recipe, you need to nudge the system. But you can't just throw in random spices; you have to stay within strict safety limits so the customer (the system's operator) doesn't get upset.

The Solution: The "Smart Nudge"

The authors propose a clever way to add a probing signal (a tiny, calculated nudge) to the input. Think of this as the Chef whispering a tiny suggestion to the machine: "Hey, let's try turning the heat up just a tiny bit for a second, then back down."

The goal is to make these nudges big enough to reveal the system's secrets (so we can build a better model) but small enough that the final output (the temperature, speed, or pressure) never strays outside the "safe zone" the user has set.

How It Works: The Adaptive Dance

The paper introduces a method that is recursive, meaning it learns and adjusts on the fly, step-by-step.

The Guess: The system starts with a rough guess of the recipe (the model parameters).
The Calculation: Based on this guess, the algorithm calculates the perfect amount of nudge to apply right now. It asks: "If I nudge the input by this specific amount, will I learn the most about the system while keeping the output safe?"
The Safety Net: The algorithm has a strict rule: The output cannot go above or below a certain limit (like a thermostat that won't let the room get too hot or too cold). The math ensures the nudge respects this limit, even accounting for the fact that the system might react a few seconds later (a "delay").
The Update: After the nudge, the system measures the result, updates its guess of the recipe, and immediately calculates the next perfect nudge.

The "Secret Sauce" of the Math

The paper claims to have found a closed-form solution. In plain English, this means they didn't have to use a slow, trial-and-error computer search to find the right nudge. Instead, they derived a direct formula (like a simple equation) that instantly tells you exactly what to do.

Analogy: Imagine trying to find the best angle to throw a ball into a basket. Most methods would try throwing it at 10 degrees, then 11, then 12, checking if it goes in. This paper gives you a formula that instantly calculates the exact angle you need based on where the basket is right now.

Why It's Better Than the Old Ways

The authors tested their method against two other approaches:

Doing Nothing: Just letting the system run. (Result: You learn almost nothing because the system is too stable).
Random Noise: Throwing in random, loud signals (like a "Pseudo-Random Binary Signal"). (Result: You learn the recipe quickly, but you risk blowing the system out of its safe zone or annoying the customer).

The Result: Their "Smart Nudge" method learns the recipe just as accurately as the loud, random noise method, but it keeps the system's behavior smooth and strictly within the safety limits. It's like a master chef who knows exactly how much to stir to perfect the sauce without ever splashing a drop on the counter.

Key Takeaways

Safety First: You can learn a system while it's running, as long as you respect strict limits on how much the output can wiggle.
Adaptability: The method gets smarter as it goes. As it learns more about the system, it adjusts its nudges to be even more efficient.
Efficiency: It uses a direct mathematical formula to make decisions instantly, rather than guessing and checking.
Feasibility: The authors proved mathematically that if you set your safety limits correctly, you can keep nudging the system forever without ever breaking the rules.

In short, this paper provides a mathematical "playbook" for engineers to safely and efficiently learn how complex machines work while they are already in operation, ensuring the machine never gets out of control during the learning process.

Technical Summary: Recursive Experiment Design for Closed-Loop Identification of ARMAX Systems with Output Perturbation Limits

1. Problem Formulation

The paper addresses the challenge of performing system identification for linear dynamical systems operating in a closed-loop configuration. In many practical scenarios, experiments must be conducted while the system is under a known, fixed output feedback controller to ensure safety or maintain operation. The system is modeled as an ARMAX (AutoRegressive Moving Average with eXogenous input) structure:
$y_t = G(q; \theta)u_t + H(q; \theta)e_t$
where $G$ and $H$ are unknown transfer functions parameterized by $\theta$ , and $e_t$ is a white noise innovation process.

The core problem arises when the existing closed-loop dynamics, driven by a constant reference signal, do not provide sufficient excitation to identify the system parameters (a condition known as lack of informativeness). To remedy this, a probing signal $d_t$ is added to the control input ( $\tilde{u}_t = u_t + d_t$ ). However, injecting this signal inevitably perturbs the system output ( $\tilde{y}_t = y_t + \delta_t$ ). The objective is to recursively design the probing signal $d_t$ such that:

The resulting input-output data is sufficiently informative for identifying the ARMAX model.
The output perturbation $\delta_t$ remains within user-specified bounds ( $\delta_{\min} \le \delta_t \le \delta_{\max}$ ).
The design adapts to sequentially estimated model parameters.

2. Methodology

2.1 Identifiability Analysis

The authors first establish conditions under which the closed-loop data is informative.

Structural Identifiability: They derive a condition (Theorem 1) based on the orders of the plant, controller, and noise models. If the controller dynamics are sufficiently rich (specifically, if a calculated integer $\gamma < 0$ ), the system is identifiable even without a probing signal.
Probing Signal Requirements: If structural richness is insufficient ( $\gamma \ge 0$ ), the paper proves (Theorem 2) that a bounded, discrete-valued probing signal that is persistently exciting of order $n_e \ge \gamma$ is sufficient to ensure identifiability.

2.2 Feasibility of Perturbation Limits

A critical contribution is the analysis of "recursive feasibility." The authors analyze whether satisfying the perturbation limits at time $t$ guarantees that a feasible solution exists for time $t+1$ .

They model the perturbation $\delta_t$ as the output of a sensitivity function driven by the probing signal.
Proposition 1 provides a sufficient condition on the user-specified limits ( $\delta_{\min}, \delta_{\max}$ ) relative to the maximum allowable probing amplitude ( $d_{\max}$ ) and the system's impulse response. If this condition is met, the multi-step constraint problem can be relaxed to a single-step constraint, significantly simplifying the design.

2.3 Recursive Design Algorithm

The paper proposes a computationally efficient, closed-form solution for the probing signal design using a Recursive Prediction Error Method (RPEM).

Parameterization: An expanded parameter vector is used to handle unknown input delays.
Design Criterion: The design minimizes the estimation error covariance. Two criteria are considered: the determinant of the covariance matrix (D-criterion) and the trace of the weighted covariance (L-criterion). Both reduce to maximizing a quadratic form involving the gradient of the prediction error.
Closed-Form Solution (Theorem 3): By partitioning the covariance matrix and the gradient vector, the optimization problem is reduced to maximizing a convex quadratic function subject to linear constraints. The optimal probing signal $d^*_t$ is shown to be a binary signal (taking values at the boundaries of the feasible interval) determined by a simple threshold comparison.
Adaptivity: The constraints and the signal values are updated recursively using the current parameter estimates $\hat{\theta}_t$ . The authors also provide a pragmatic method to estimate the unknown input delay online.

3. Key Contributions

Adaptive Recursive Design: The paper introduces a method to design a probing signal that adapts to sequentially estimated model parameters, unlike fixed-frequency or fixed-time designs that require a good initial model estimate.
Closed-Form Solution: A computationally efficient, closed-form expression for the optimal probing signal is derived. This signal is discrete-valued and bounded, ensuring it satisfies the theoretical conditions for persistent excitation.
Feasibility Analysis: The paper provides rigorous conditions ensuring that user-specified output perturbation limits are recursively feasible, allowing the complex multi-step optimization to be solved as a one-step problem.
Handling Unknown Delays: The methodology incorporates an expanded parameterization and an estimator to handle unknown input delays within the recursive framework.

4. Numerical Results

The method was evaluated using Monte Carlo simulations on two ARMAX systems with different characteristics (including cases with and without input delays).

Performance vs. Baselines: The proposed adaptive design was compared against a "zero probing" baseline (no excitation) and a Pseudo-Random Binary Sequence (PRBS) baseline.
- The zero baseline resulted in very slow or non-existent convergence due to insufficient excitation.
- The PRBS baseline provided fast convergence but caused large output perturbations, often exceeding 25% of the reference signal.
- The proposed method achieved convergence rates comparable to PRBS while strictly adhering to user-defined perturbation limits (e.g., keeping perturbations below 10-20% of the reference).
Trade-off: The results demonstrate a clear trade-off: tighter perturbation limits ( $\delta_{\max}$ ) result in slightly slower convergence but ensure safety. The adaptive nature of the design allows the system to respect these limits even as the model estimates improve.
Robustness: The method successfully handled systems with unknown input delays, correctly estimating the delay and adjusting the probing signal accordingly.

5. Significance and Claims

The paper claims that its primary significance lies in enabling safe, informative closed-loop identification for systems where a fixed controller is already in place. By adding a bounded, adaptive probing signal, the method allows for the improvement of existing controllers without violating operational safety constraints.

The authors emphasize that their approach:

Does not require redesigning the controller itself.
Adapts to the information gained during the experiment, unlike static designs.
Provides a mathematically rigorous guarantee of identifiability and recursive feasibility.
Offers a practical, low-computational-cost solution suitable for real-time implementation.

The work positions itself as a bridge between theoretical experiment design and practical constraints, specifically addressing the need to balance identification accuracy with output perturbation limits in industrial control settings.