Title: Stabilizing LTI Systems under Partial Observability: Sample Complexity and Fundamental Limits
Venue: NeurIPS 2025
Accepted: yes
Keywords: Linear dynamical system

Learning-based control

Learn-to-stabilize

Sample complexity

Unknown system
Abstract: We study the problem of stabilizing an unknown partially observable linear time-invariant (LTI) system. For fully observable systems, leveraging an unstable/stable subspace decomposition approach, state-of-art sample complexity is independent from system dimension $n$ and only scales with respect to the dimension of the unstable subspace. However, it remains open whether such sample complexity can be achieved for partially observable systems because such systems do not admit a uniquely identifiable unstable subspace. In this paper, we propose LTS-P, a novel technique that leverages compressed singular value decomposition (SVD) on the ''lifted'' Hankel matrix to estimate the unstable subsystem up to an unknown transformation. Then, we design a stabilizing controller that integrates a robust stabilizing controller for the unstable mode and a small-gain-type assumption on the stable subspace. We show that LTS-P stabilizes unknown partially observable LTI systems with state-of-the-art sample complexity that is dimension-free and only scales with the number of unstable modes, which significantly reduces data requirements for high-dimensional systems with many stable modes.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes an approach to designing a stabilizing controller for partially observable linear time-invariant systems with disturbances. Based on a number of (partially) observed trajectories, obtained by applying random Gaussian control inputs, the unstable system components are approximated for a given rank. By raising the system matrix to a given power, the influence of the unstable eigenvalues is amplified, which is crucial for their identification. This is referred to in the paper as lifting the Hankel matrix. The resulting approximation is then used to design the stabilizing controller. The paper provides probabilistic guarantees that the derived controller actually stabilizes the system, and clearly identifies the influence of the system parameters as well as the hyperparameters of the algorithm.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n- The paper extends existing work from fully observable to partially observable systems. \n- The probabilistic guarantees are clear and expose the influence of critical parameters, both of the system and the algorithm.\n- The sample complexity is relatively low. As in prior related work, the system dimension does not directly influence it. In particular, even modest increases in the number of trajectories substantially raise the probability of success.\n\nWeaknesses:\n- The paper is somewhat technical and compact. I admit this is somewhat in the nature of the contribution and not easy to improve. For example, it took me a while to remember what m,M,p,q stand for. Since the result at the end proposes p=q=m, this could be mentioned earlier to ease some of the cognitive load. Reading Algorithm 1 requires referring to 8 equations that are listed on the next page. An intuitive explanation would be just as useful.\n- The paper is largely inspired by [50], where, thanks to full observability, only a single trajectory is required.\n\nQUESTIONS:\nIn practice, an unstable system will be only be observed or simulated over a short time horizon. This situation seems to merit some discussion. In my understanding, it suffices, roughly speaking, that m is as large as the unstable dimension k and T is larger than 3m. Then any probability can be guaranteed by making M large enough.\n\nThe claim about boundedness in line 67 could be clarified: In a strictly finite observation scenario (which is considered here), Hankel matrices are finite and bounded.\n\nLIMITATIONS:\nI did not see any discussion of numerical difficulties that might arise from observations of unstable systems, the inversion of the matrices used in the algorithm, etc. In fairness to the authors, I must admit that the paper is already heavily packed and I have no suggestions on how to make more room for such an analysis.\n\nNote that I did not check the lengthy math appendix in detail, but I have no reason to doubt its correctness.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-21"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies the problem of stabilizing unknown *partially observable* LTI systems, a natural and necessary extension of previous results in the *fully observable* setting. The authors propose **LTS-P**, a novel algorithm that efficiently stabilizes the system via isolation of  the unstable system modes using compressed SVD on a \"lifted\" Hankel matrix, and via a robust controller design under a small-gain-type assumption on the stable residue. The algorithm features a reduced sample complexity that only depends on the number of unstable modes and is thus independent of the system dimension. Preliminary simulation results confirm its outperformance over existing algorithms.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n1. The paper is overall well-written and well-organized. It provides an abundance of intuition via simplified settings and comparisons against known results in the main text, while retaining mathematical rigor in the appendix by providing additional technical details.\n2. The research question is of great theoretical interest, as it's a natural follow-up of several recent works in the fully observable setting.\n3. The algorithm design idea builds upon a solid ground that isolates the unstable modes of the system, but also provides new insights that leverages the transfer function, \"lifted\" Hankel matrix and small-gain-type arguments to properly handle partially observable systems.\n4. The theoretical results are significant, and the proofs are checked to be sound.\n\nWeaknesses:\n1. Assumption 5.2, though coming with a detailed comment on its necessity, still seems a little suspicious. Specifically, it is unclear how $\\epsilon_*$ scales with the system dimension $n$. The order of sample complexity could be made more convincing if the growth rate of $\\log \\frac{1}{\\epsilon_*}$ (and also $L$) is, at least, illustrated with a representative family of systems with increasing dimensions.\n2. The simulation results presented in Appendix A is, at best, preliminary and illustrative. Since $MT$ is selected as the complexity metric, it is natural to ask if there are any potential tradeoffs between $M$ and $T$, which is not reflected in the simulations. Instead, performance is only compared in a restrictive setting, with either $M$ or $T$ fixed. The experimental results could be more convincing through more settings and more results.\n3. Some notations, though easy to judge from context, are not defined before first use. For example, $\\hat{\\Phi}_i$ in eq. (18) seems to be the $i$th block of $\\hat{\\Phi}$.\n4. A few typos are observed, including \"ovservable\" -> \"observable\" (line 32), \"folloing\" -> \"following\" (line 175), etc.\n\nQUESTIONS:\n1. How do system parameters ($\\epsilon_*$, $L$) scale with the system dimension $n$? See above for details.\n2. Additional experimental results would be appreciated.\n\nLIMITATIONS:\nN/A", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-06-27"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper gives an algorithm for learning stabilizing controllers for an unstable LTI systems. The key insight is that the unstable dynamics can be learned via a specialized subspace identification algorithm. Then, as long as the stable part has a sufficiently small $H_{\\infty}$ norm,  a robust controller designed for the unstable component will stabilize the system with high probability.\n\nSTRENGTHS AND WEAKNESSES:\n# Strengths\n- This gives at least a partial solution to a long-standing problem in learning-based control: learning a stabilizing controller for partially observed systems.\n- The writing is reasonably clear. \n\n# Weaknesses\n- Theorem 5.3 appears to be stated incorrectly. Namely, I'm guessing (based on Lemma B.1 and Theorem 3.2) it should be that for all $\\delta\\in (0,1)$ the system is stabilized with probability at least $1-\\delta$. The theorem is currently stated as \"for some $\\delta$\" the system is stabilized with probability at least $\\delta$. \n- The work doesn't give a full solution to the stabilization problem: It really only works for sufficiently small stable parts. (I.e. Assumption 5.2 is required.) While an argument about how to extend the method is sketched, a complete solution that works for all stabilizable and detectable LTI systems would be desirable. \n- The paper is somewhat inconsistent in the treatment of process noise. Namely, process noise is included in (14) and (26), but is  not used in the (1) or the main theory.  My guess is that the problem would be substantially harder with process noise.  Namely, (31) would include terms that get amplified through the unstable dynamics, leading to substantially larger errors in the Hankel matrix estimation.\n- Similarly, the paper assumes that the state can be reset to 0 at the beginning of each trajectory. If this were not the case, similar to  the problem with process noise, unobserved, non-zero initial conditions would make the Hankel matrix errors much larger.\n- The choices of $m$, $p$, and $q$ used in the simulations are inconsistent with the theory.\n- The choice of $m$ in Theorem 5.3 requires knowledge of the eigenvalues, and properties of various Gramians. As written, this does not appear to be very useful, since it requires choosing a parameter of the algorithm, $m$, based on properties of  the unknown system. The result could be stated a bit differently, so that for a given $m$, the algorithm can stabilize the system provided the bounds on the eigenvalues and Gramians holds. I.e. for a given $m$, the algorithm would work for all systems of a particular class.\n\nQUESTIONS:\n- What happens if there are marginally stable eigenvalues?\n- Can you do the simulations with the theoretical choices for $m$, $p$, and $q$?\n- Do you run into numerical problems when $m$ is large?\n- Can Assumption 5.2 be removed completely?\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents a novel method for obtaining a stabilizing controller for a partially observed discrete-time LTI system, and analyzes the sample complexity of the resulting algorithm. The method is based on estimating the parameters of a lifted Hankel matrix for the system using multiple trajectories with iid control inputs. The estimated Hankel matrix allows for estimating the open-loop transfer function for only the unstable part of the system, which is then used to construct a stabilizing controller for the entire system. In the case where the number of unstable eigenvalues $k$ is much smaller than the state dimension $n$, this decomposition into unstable and stable parts yields better sample complexity than applying previous results on system identification to this problem. Experiments on randomly generated systems validate that the proposed method learns a stabilizing controller with less data in practice.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths: This is a strong paper with a good motivation and clear development of the technical ideas underlying the proposed method. It tackles a problem of practical interest that has not been targeted in prior works to the best of my knowledge. It is also very well-organized and relatively easy to follow. I especially appreciated the notation index that not only defines each term but cites the equation / lemma where each appears! Providing a code repository will make it easy for other researchers to use the method in their own applications. \n\nWeaknesses: One important limitation of the paper is that the method assumes $k$ is known. A procedure for estimating it is presented in the paper, but this method only works when the lifting parameters $p$, $q$, and $m$ are sufficiently large with respect to $k$, which is problematic when $k$ is truly unknown. \n\nMinor weaknesses:\n- Above equation (24), I believe “$\\bar{C}$ and $\\bar{B}$” should read “$\\hat{C}$ and $\\hat{B}$”\n- It would be helpful if the captions in figure 1 and figure 2 were more detailed, including phrases like “at different noise levels $\\sigma$” and “with rollouts of length $T=4n$” (figure 1) or “with rollouts of length $T=70$ (figure 2)\n- Line 498: trails -> trials\n\nQUESTIONS:\n- In the case when $k$ is unknown, the paper describes a procedure for finding $k$ based on the singular values of the estimated Hankel matrix which assumes that the lifting parameters $p$, $q$, and $m$ are sufficiently large with respect to $k$. What happens when this is not the case? Is it possible to mistakenly believe that the true $k$ has been identified (i.e. find a large spectral gap in the singular values that does not correspond to $k$)? I would be inclined to raise my score if I have misunderstood this limitation.\n\n- Do the simulations in appendix A use constant $B$, $C$, and $D$ matrices while randomizing $A$? \n\n- If the simulation results instead plotted the fraction of randomized systems that are successfully stabilized for a given amount of data, can we recover $\\delta$ in theorem 5.3?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}