Model-Free Output Feedback Stabilization via Policy Gradient Methods

arXiv:2601.19284 · eess.SY, cs.LG, cs.SY, math.OC · Submitted 2026-01-27 · Read on arXiv

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

Rosa: Today's paper: "Model-Free Output Feedback Stabilization via Policy Gradient Methods".

Dev: Stabilizing dynamical systems is a fundamental problem in control theory,

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

Paper summary: Rosa: So we're diving into "Model-Free Output Feedback Stabilization via Policy Gradient Methods," which is a paper that tackles stabilizing dynamical systems when you don't have a system model, focusing specifically on output feedback for discrete-time linear systems. What I want to know is if this stuff actually works outside of some perfectly controlled lab environment, and realistically, how long can we expect this kind of learning policy to maintain stability in a real-world application?

Dev: That's a fair question, Rosa; from my side, I'm thinking about the loop rate and any potential latency issues that might arise when deploying such an algorithm. If the system dynamics change slightly due to noise or unmodeled disturbances, how robust is this output feedback policy going to be under those real-world conditions?

Taro: From an autonomy research angle, I'm curious about what happens when the world misbehaves; if this framework is learning a policy based on trajectories, how does it handle unexpected events or situations where the environment suddenly changes its rules?

Rosa: That brings us to what the paper claims: they’re proposing an algorithmic framework that pushes the boundaries of policy gradient methods into this partially observable scenario without guaranteeing global convergence. They suggest that by using zeroth-order PG updates based on system trajectories and observing their convergence toward stationary points, these algorithms actually manage to return a stabilizing output feedback policy for discrete-time linear dynamical systems.

Dev: That sounds like a significant step because most existing research on PG methods for unknown linear systems assumes full-state feedback, which we know isn't always feasible in practical control setups. The paper seems to address the challenge of learning controllers when you only have access to the system output, which usually lacks that gradient dominance we need for guaranteed convergence.

Taro: I see why that's important for autonomy; if we can stabilize something using just the output, without knowing the internal state dynamics, it opens up a lot more possibilities for robots operating in complex, partially observable environments. It moves beyond needing an initial stabilizing controller to start with.

Rosa: Exactly, and what really interests me is the mechanism they use to make this work: they introduce a discount mechanism that transforms the stabilization of the original system into policy learning problems for a sequence of discounted partially observable systems with carefully chosen discount factors. This effectively allows them to start from a trivial initial policy and slowly bring it toward stability as those discount factors approach one.

Paper summary: Dev: The paper describes the gradient estimation using two-point methods, where they simulate the system from time zero to a specific time tau e to get cost functions like J tau e, gamma,x zero(K + rU i) and J tau e, gamma,x zero(K - rU i), which is how they estimate the gradient grad b J gamma(K). That seems like a specific way to handle the model-free aspect without needing explicit system knowledge.

Taro: From a theoretical standpoint, it’s intriguing how they manage to get these zeroth-order PG updates to converge toward regions where the cost function gradient is small enough, specifically showing that for a desired accuracy epsilon > zero they find a policy K j such that grad J gamma(K j) F at most epsilon within at most M at least 9J gamma(K zero) eta epsilon squared iterations, given certain parameter bounds.

Rosa: So they’re establishing a concrete condition for when the policy will stop improving, which is a key part of the stability analysis in their paper. They show that by controlling the gradient estimation error to be less than two epsilon/three the cost function keeps decreasing monotonically, which ensures that any policy generated by this zeroth-order PG method also has a small gradient norm and is stabilizing.

Dev: That monotonic decrease in the cost function is reassuring because it gives us a clear path to convergence, even though they admit they aren't proving global convergence for the original problem immediately. The framework also includes an adaptive updating rule for the discount factor gamma k+one = (one + zeta alpha k) gamma k with a specific update formula involving J tau k,N gamma k(K k+one) and zero.

Taro: That adaptive discount factor evolution is what allows the system to transition from being stabilized by a heavily discounted problem to finally stabilizing the original system as that factor gradually approaches one, which is a clever way to bridge the gap. It suggests a pathway for learning stabilization even when starting from nothing.

Rosa: And looking at their complexity analysis, they characterize the total sample complexity as N total = (one/gamma zero) sum' k=zero M k N e k + k'N(a) at most (one/gamma zero) (3J - zero) (two rho(A) two)/zeta zero. This final complexity is expressed in a compact form involving (rho(A)), O m 2p two epsilon four and polynomial factors related to the system matrices and cost parameters.

Dev: Those bounds are pretty telling about the computational burden, especially how it scales with the dimensions of the system and how much accuracy you demand from epsilon. If we're looking at high-dimensional systems, that polynomial scaling might become a real constraint for real-time execution on embedded hardware.

Paper summary: Taro: Considering what they've shown, the implication is that we might be able to deploy model-free stabilization policies in scenarios where the system dynamics are only partially known, which is a huge step for real-world autonomous navigation where you can't always get a perfect map of everything.

Rosa: Thinking about the broader impact, if this technique translates well outside the lab, it could allow robots to maintain stability in environments where they encounter novel or unexpected dynamics without needing extensive pre-training on those specific conditions.

Dev: I agree, but we have to keep in mind that the paper explicitly states its limitation: it doesn't provide global convergence guarantees for the original problem, relying instead on reaching a region of stabilizing policies with a small gradient norm. So, if the system trajectory leads it away from that region before stabilization is achieved, the algorithm might fail to converge to a stable policy.

Taro: That limitation is important because it sets expectations; we can't assume perfect stability from the start without further checks. But as an autonomy researcher, I see this as a powerful tool for exploration; maybe we use it to find *some* stabilizing behavior even if the system misbehaves initially.

Rosa: So, to wrap up what we've heard on "Model-Free Output Feedback Stabilization via Policy Gradient Methods," the authors present a method using zeroth-order PG updates and a discount mechanism to learn stabilizing output feedback for unknown linear systems without needing a system model.

Dev: That’s right, and it’s built on simulating the system trajectories to estimate gradients through two-point methods, which helps guide the learning process toward policies with small cost gradients.

Taro: The authors are demonstrating how to find a stabilizing policy by relaxing the original stabilization problem into a sequence of discounted problems, which is an interesting conceptual move for tackling these complex systems.

Rosa: Ultimately, the implication is that we can learn control policies for partially observable systems using PG methods focused on output feedback, even without a full system model.

Dev: And while it's a solid framework for learning stabilizing controllers from trajectories, the paper’s limitation is that it doesn't guarantee global convergence to a globally stabilizing policy, only to one where the gradient norm is small.

Taro: I think this work has serious implications because it shows how we can make autonomous systems more resilient by learning from experience in situations where the underlying physics are just not fully defined yet.

Rosa: It’s definitely a piece of research that pushes the boundaries of what policy gradient methods can achieve for output feedback control, even under model-free constraints.

Conclusion: Rosa: So, to wrap up our discussion on "Model-Free Output Feedback Stabilization via Policy Gradient Methods," we’ve looked at how this paper tackles learning controllers for unstable systems without needing a system model.

Dev: Yeah, and the authors are using a specific framework that relies on policy gradient methods operating under certain conditions, which is pretty interesting for a control engineer to hear about.

Taro: From an autonomy standpoint, what I find most compelling is their approach of transforming the stabilization problem into learning policies for a sequence of discounted systems; that sounds like a way to build up stability gradually.

Rosa: Exactly, it seems they’re showing us a method that lets us learn how to stabilize something just by observing its output over time, which is huge for field robotics.

Dev: I'm still thinking about the practical loop rate and latency issues; if this policy is being deployed on a real system, we gotta worry about how fast these updates can actually run without introducing instability.

Taro: And when we consider what happens when things get messy in the real world, like unexpected disturbances or changes in the environment, how robust is this learned policy going to be?

Rosa: That’s the core question for me; does this method hold up when it’s not running on a perfectly controlled lab bench?

Dev: It seems they are focusing on reaching a region where the cost function gradient is small, but I wonder if that region is large enough to handle real-world uncertainties.

Taro: If the system misbehaves, does this policy have any inherent mechanism to recover or adapt in a meaningful way?

Rosa: The paper suggests it’s about finding *some* stabilizing behavior even when the underlying physics aren't perfectly defined yet, which is a significant step for exploration.

Dev: That sounds like promising work for scenarios where we can't rely on perfect modeling, but I need to see concrete data on how quickly it converges in practice.

Taro: It’s definitely something worth watching because if this concept scales, it could enable autonomous systems to function in environments with very little prior knowledge of the system dynamics.

School of Artificial Intelligence and Automation, Huazhong University of Science and Technology · College of Automation, Nanjing University of Posts and Telecommunications

eess.SY, cs.LG, cs.SY, math.OC

Submitted: 2026-01-27

Updated: 2026-09-28

Comments: 39 pages, 2 figures

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

Importance score: 83/100

The gist: Stabilizing dynamical systems is a fundamental problem in control theory, and this work proposes an algorithmic framework for learning stabilizing static output feedback (SOF) controllers for

Key concepts

Model-Free SOF Synthesis: This is the core goal: creating a controller that stabilizes an unstable system using only input/output data, without needing the system's mathematical equations. It achieves this by using 'zeroth-order PG methods,' which update the policy based on observed system behavior.
Discount Mechanism for Partially Observable System: Since only the output is known, the problem is treated as a sequence of discounted systems. This mechanism introduces a discount factor ($\gamma$) that helps guide the learning process from an initial unstable state toward a stable region by transforming it into a policy learning task.
Gradient Estimation via Two-Point Methods: The method estimates the necessary gradient for policy improvement using two-point methods. This involves simulating the system forward and backward over specific time intervals to calculate cost functions, which are then used to determine the direction in which the controller parameters should be adjusted.

Terminology

Sample Complexity — 22× in this paper · explained in 15 episodes
Convergence Guarantees — 9× in this paper · explained in 3 episodes
Accuracy — 7× in this paper · explained in 1 episode
Static Output Feedback — 6× in this paper
Reinforcement Learning — 4× in this paper · explained in 4 episodes
System Identification — 4× in this paper · explained in 3 episodes
Adam — 2× in this paper · explained in 2 episodes
Among the Reinforcement Learning — 2× in this paper
Closed-Loop System — 2× in this paper · explained in 4 episodes
Core — 2× in this paper · explained in 2 episodes
Regret — 2× in this paper · explained in 3 episodes
Reinforcement Learning (RL) — 2× in this paper · explained in 25 episodes
Trace — 2× in this paper · explained in 2 episodes
Homogeneity — 1× in this paper · explained in 2 episodes
Lyapunov function — 1× in this paper · explained in 2 episodes
Objective Function — 1× in this paper · explained in 2 episodes
Sample Efficiency — 1× in this paper · explained in 4 episodes
Theoretical Guarantees — 1× in this paper · explained in 2 episodes
alpha — 1× in this paper · explained in 1 episode

Summary

Stabilizing dynamical systems is a fundamental problem in control theory, and this work proposes an algorithmic framework for learning stabilizing static output feedback (SOF) controllers for open-loop unstable discrete-time linear systems without requiring a system model. The gist: "We propose an algorithmic framework that stretches the boundary of PG methods to the problem without global convergence guarantees. We show that by leveraging zeroth-order PG update based on system trajectories and its convergence to stationary points, the proposed algorithms return a stabilizing output feedback policy for discrete-time linear dynamical systems."

Problem Formulation and Motivation

The paper addresses the challenge of designing a model-free algorithm for learning SOF controllers where system matrices are unknown. While policy gradient (PG) methods are effective in model-free control, existing research often assumes full-state feedback (SF). The authors focus on the partially observable scenario where only the output is available, which lacks gradient dominance and may have disconnected stabilizing policies. The motivation is to move beyond relying on an initial stabilizing controller by introducing a discount mechanism that transforms the problem into learning policy problems for a sequence of discounted partially observable systems.

Proposed Algorithmic Framework

The proposed framework integrates several key components:

Model-Free SOF Synthesis: A model-free algorithmic framework is developed for learning SOF controllers from system trajectories, built upon zeroth-order PG methods with only local convergence guarantees.
Discount Mechanism for Partially Observable System: This mechanism transforms the stabilization of the original system into policy learning problems for a sequence of discounted partially observable systems with properly chosen discount factors.
Gradient Estimation via Two-Point Methods: The gradient estimate, denoted as gradient estimate ∇b Jγ(K), is obtained using two-point methods in Algorithm 1, which simulates the system from time zero to time τe to obtain cost functions like Jτeγ,x0(K + rUi) and Jτeγ,x0(K - rUi).

Convergence Guarantees and Stability Analysis

The convergence analysis relies on establishing conditions for the PG method to reach a region of SOFs with small gradient norm. Key results include:

"Theorem 1. Suppose that K0 ∈ Sγ(ν). For a desired accuracy ϵ > 0, let τe, Ne, r, and η satisfy [specific parameter bounds]. Then... the zeroth-order PG method in (13) based on ∇b Jγk(Kj) obtained by Algorithm 1 will return a policy Kj satisfying∥∇Jγ(Kj)∥F ≤ ϵ within at most M ≥ 9Jγ(K0)ηϵ2 iterations."

The proof demonstrates that by controlling the gradient estimation error, ∥∇b Jγ(Kj − Kj)∥F ≤ 2ϵ/3, the cost function remains monotonically decreasing along the way (Jγ(Kj+1) − Jγ(Kj) ≤ 0), ensuring that KM generated by the zeroth-order PG method (13) is also a stabilizing policy with∥∇Jγ(KM)∥F ≤ ϵ.

Discount Factor Evolution and Sample Complexity

The framework incorporates an adaptive updating rule for the discount factor:

Update γk+1 = (1 + ζαk)γk with αk = l0 / (2Jbτk,Nγk(Kk+1) − l0).

This update ensures that the policy navigates from the initial discounted regime toward the stability region S1 of the original system. The overall sample complexity is characterized by Algorithm 2:

**"Ntotal = 2(k) Σ' k=0 MkNe k + k'N(a) ≤ log(1/γ0) log(1 + ζl0/(3J − l0)) **

**(b) ≤ (3J − l0) log(2ρ(A) 2)/ζl0 **

This complexity is expressed in a compact form as: Ntotal = log(ρ(A)) · O m 2p 2ϵ 4 · poly(A, B, Q, R, J).

Numerical Validation

The effectiveness of the framework is validated through numerical experiments. The paper presents two simulations:

An example involving an unstable discrete-time system (31), showing the evolution of the closed-loop spectral radius and the convergence process of the discount factor γ over iterations.
An application to a linearized cart-pole system (32), demonstrating that "the discount factor increases from its initial value γ0 = 0.1 to 1 within 150 iterations.

Improvements for AI systems

As a fastidious and diligent AI researcher, I have thoroughly reviewed the provided scientific paper, Model-Free Output Feedback Stabilization via Policy Gradient, dated January 30, 2026. This work proposes a novel algorithmic framework for learning stabilizing static output feedback (SOF) policies for unknown partially observable discrete-time linear systems using zeroth-order Policy Gradient (PG) methods combined with a discount mechanism.

Here are the specific improvements and capabilities this framework enables in AI systems:

Improving the Robustness and Applicability of Model-Free Control in Partially Observable Systems:

The core contribution is developing a model-free learning method for stabilizing partially observable linear dynamical systems using only system trajectories (zeroth-order PG). This directly addresses the model-free gap, which is critical when system dynamics are unknown or too complex to model explicitly.

Enabling Real-World Control in Unknown Environments:

The framework allows AI agents operating in environments where they only receive noisy or incomplete output measurements (rather than full state feedback) to learn a stabilizing controller without needing an explicit system model. This is invaluable for robotics, autonomous navigation, and industrial control systems where obtaining complete state information is prohibitive or impossible.

Facilitating Online Adaptation via Discounting:

The introduction of the Discount Mechanism allows the agent to transition from learning stabilization on the original unstable system to learning on a sequence of discounted systems. This enables an iterative approach where the agent starts from a trivial policy (e.g., zero policy) and gradually refines its behavior toward stability, providing a structured path for complex control tasks that require long-term planning.

Providing Sample Complexity Guarantees for Model-Free RL:

The paper rigorously characterizes the total sample complexity required to learn the stabilizing SOF controller (Theorem 2). This quantification is vital for practical deployment, as it allows engineers to estimate the necessary amount of interaction data (system rollouts) needed to achieve a desired level of stabilization accuracy and reliability.

Achieving Performance in Non-Convex Control Landscapes:

The algorithm is specifically designed to navigate the challenging, non-convex optimization landscape associated with output feedback control, where gradient dominance does not hold. By leveraging zeroth-order PG methods that converge to stationary points (rather than requiring global cost minimization), the system can find stabilizing policies even when traditional gradient descent methods fail due to saddle points or disconnected solution sets.

Adaptive Policy Refinement via Dynamic Discount Factors:

The framework incorporates an adaptive update rule for the discount factor, which is informed by empirical cost estimators derived from system rollouts. This means the AI agent dynamically adjusts its horizon of planning based on the observed performance, allowing it to focus computational resources where they are most needed—i.e., refining the control policy when instability is greatest.

In summary, this research provides a rigorous mathematical foundation for building resilient, model-free controllers that can function effectively in real-world scenarios characterized by partial observability and unknown dynamics. It moves beyond full state feedback limitations to offer a practical, data-driven solution for stabilizing complex systems.