arXiv subjects

Kazuhiro Sato

Publications and source records attributed to Kazuhiro Sato.

At least 19 recordsLinked to original sources

arXiv2026

Bilinear Koopman-Based Robust Model Predictive Control for Unknown Nonlinear Systems via Contraction Metrics

Data-driven model predictive control (MPC) using Koopman operator theory is a promising approach for constrained control of unknown nonlinear systems. While linear Koopman realizations are commonly used due to their simplicity, bilinear Koopman realizations can provide significantly higher approximation accuracy for nonlinear control systems. However, robust MPC (RMPC) formulations that account for modeling errors in bilinear Koopman realizations remain limited. This paper proposes a RMPC framework for unknown nonlinear systems with general nonlinear constraints based on data-driven bilinear Koopman realizations. A central difficulty is that finite-dimensional Koopman predictors need not preserve the manifold of valid lifted states, so multi-step prediction in lifted coordinates may leave the region where one-step error certificates apply. We address this issue by reprojecting each predicted lifted state back onto the manifold, thereby obtaining an error-aware discrete-time control-affine predictor in the original state space without impractical assumptions. For this predictor, we develop a discrete-time robust control contraction metric based homothetic tube construction, and then formulate a tube-based RMPC problem with terminal ingredients. Under the proposed formulation, we prove robust satisfaction of the original nonlinear constraints by the true closed-loop trajectory, recursive feasibility, and convergence to a neighborhood of the target state. Numerical experiments demonstrate robust stabilization of nonlinear systems and the advantages of the proposed method over existing Koopman-based RMPC approaches in terms of performance.

math.OC↗

arXiv2026

Projected-Gradient Analysis for Open-Domain Convex Optimization under Boundary Blow-Up:Application to Controllability Scoring

We study convex optimization over a compact convex set when the objective is smooth and convex only on an open domain. Under a boundary-blow-up condition, every feasible initialization yields a compact invariant sublevel set separated from the complement of the objective domain, and an optimal solution exists. For a domain-aware Armijo projected-gradient method, a safe-neighborhood analysis establishes well-defined objective evaluations, finite backtracking, sufficient decrease, and a run-specific but iteration-independent positive lower bound on the accepted step sizes. These properties yield explicit sublinear objective and stationarity guarantees, together with convergence of the full iterate sequence. Positive curvature restricted to feasible displacement directions further guarantees uniqueness and linear convergence. We apply the framework to controllability scoring with prescribed input directions and compact convex allocation constraints. Feasibility is characterized exactly by controllability of the input directions eligible for positive allocation, while restricted injectivity of the Gramian map guarantees uniqueness of the optimal allocation and provides explicit strong-convexity bounds. A directed-network example illustrates how candidate exclusion can preserve or destroy feasibility and alter the optimal allocation in a criterion- and horizon-dependent manner.

math.OC↗

arXiv2026

Relationship Between Controllability Scoring and Optimal Experimental Design

Controllability scores provide control-theoretic centrality measures that quantify the relative importance of state nodes in networked dynamical systems. We establish a structural connection between finite-time controllability scoring and approximate optimal experimental design (OED): the finite-time controllability Gramian decomposes additively across nodes, yielding an affine matrix model of the same form as the information-matrix model in OED. This yields a direct correspondence between the volumetric controllability score (VCS) and D-optimality, and between the average energy controllability score (AECS) and A-optimality, implying that the classical D/A invariance gap has a direct analogue in controllability scoring. By contrast, we point out that controllability scoring generically admits a unique optimizer, unlike approximate-OED formulations. Finally, we uncover a long-horizon phenomenon with no OED counterpart: source-like state nodes without a negative self-loop can be increasingly downweighted by AECS as the horizon grows. Two numerical examples corroborate this long-horizon downweighting behavior.

math.OC↗

arXiv2026

Data-Driven Regularized Time-Limited h2 Model Reduction from Noisy Impulse Responses

This paper develops a data-driven time-limited h2 model reduction method for discrete-time linear time-invariant systems. Specifically, we formulate and solve a regularized time-limited h2 model reduction problem using only noisy impulse response data. Furthermore, we show that the objective function and its gradient can be represented using only noisy impulse response data. Numerical experiments using SLICOT benchmarks demonstrate that the proposed regularized method achieves lower relative time-limited h2 errors than the tested alternatives and is effective in situations where the unregularized method may deteriorate under noise.

eess.SY↗

arXiv2026

Target Controllability Scores for Actuation-Constrained Network Intervention

We introduce the target controllability score (TCS), a concept for evaluating node importance under actuator constraints and designated target objectives, formulated within a virtual system setting. The TCS consists of the target volumetric controllability score (VCS) and the target average energy controllability score (AECS), each defined as an optimal solution to a convex optimization problem associated with the output controllability Gramian. We establish existence and uniqueness (for almost all time horizons), develop a projected gradient method for computation, and show that target VCS/AECS can behave qualitatively differently from their standard full-state counterparts because projection onto the target nodes changes the underlying Gramian structure. To enable scalability, we construct a target-only reduced virtual system and derive non-asymptotic bounds showing that weak cross-coupling and a low or negative logarithmic norm of the system matrix yield accurate approximations of target VCS/AECS, particularly over short or moderate time horizons. Experiments on human brain networks reveal a clear trade-off: at short horizons, both target VCS and target AECS are well approximated by their reduced formulations, while at long horizons, target AECS remains robust but target VCS deteriorates.

math.OC↗

arXiv2026

Generalized Resistance Geometry from Kron Reduction and Effective Resistance

We develop a generalized resistance geometry for unsigned directed graphs and signed undirected graphs based on Kron reduction and effective resistance. For unsigned strongly connected weight-balanced directed graphs, we establish a generalized Fiedler--Bapat identity involving an associated signed undirected Laplacian constructed from the symmetrized pseudoinverse. We identify this Laplacian as the sum of the symmetrized directed Laplacian and a positive-semidefinite correction induced by graph asymmetry. This decomposition yields an effective-resistance comparison that extends a previous normality-based result, and we prove that the construction commutes with Kron reduction. For general unsigned strongly connected directed graphs satisfying a positivity condition, we define resistance curvature and resistance radius through a weight-balanced representation. Motivated by the associated signed undirected graphs, we introduce a class whose effective resistances form a metric and show that the resulting resistance matrices are precisely the strict negative type metric matrices. Within this framework, we characterize the unique solution of the maximum graph-variance problem and develop generalized resistive embeddings into Euclidean space.

cs.DM↗

arXiv2026

Task-Dependent Weighted Average Energy Controllability Score for Network Intervention

Controllability scores provide principled information on where intervention should be applied in large-scale network systems when explicit control design is difficult. Two representative controllability scores are the volumetric controllability score (VCS) and the average energy controllability score (AECS). While both are important, the standard AECS treats all state-transition directions uniformly. In this paper, we propose the weighted average energy controllability score (W-AECS), a task-dependent extension of AECS that incorporates a prescribed transition of interest through a weighting matrix. We show that the proposed formulation admits a control-theoretic interpretation via expected minimum-energy steering, and establish strict convexity and generic uniqueness. These results support the interpretation of W-AECS as a well-defined node-wise task-dependent intervention score. We also illustrate the proposed method on a structural brain-network dataset, where transition-dependent weighting reshapes the scoring pattern, yielding a VCS-like preference among the highest-ranked regions while preserving an overall structure distinct from both standard AECS and VCS.

math.OC↗

arXiv2026

Performance bound analysis of linear consensus algorithm on strongly connected graphs using effective resistance and reversiblization

We study the performance of the linear consensus algorithm on strongly connected directed graphs using the linear quadratic (LQ) cost as a performance measure. In particular, we derive bounds on the LQ cost by leveraging effective resistance and reversiblization. Our results extend previous analyses-which were limited to reversible cases-to the nonreversible setting. To facilitate this generalization, we introduce novel concepts, termed the back-and-forth path and the pivot node, which serve as effective alternatives to traditional techniques that require reversibility. Moreover, we apply our approach to Cayley graphs and random geometric graphs to estimate the LQ cost without the reversibility assumption. The proposed approach provides a framework that can be adapted to other contexts where reversibility is typically assumed.

math.OC↗

arXiv2026

NOMADS: Non-Markovian Optimization-based Modeling for Approximate Dynamics with Spatially-homogeneous Memory

We propose a system identification method, Non-Markovian Optimization-based Modeling for Approximate Dynamics with Spatially-homogeneous memory (NOMADS), for identifying linear dynamical systems from a set of multi-dimensional time-series data obtained through multiple partially excited experiments. NOMADS formulates model identification as a convex optimization problem, in which the state-space coefficient matrices and a memory kernel are estimated jointly under physically motivated constraints using projected gradient descent. The proposed framework models memory effects through a spatially homogeneous kernel, enabling scalable identification of non-Markovian dynamics while keeping the number of free parameters moderate. This structure allows NOMADS to integrate information from multiple multi-dimensional time-series data even when no single experiment provides full excitation. In the Markovian setting, physical constraints can be incorporated to enforce conservation laws. Numerical experiments on synthetic data demonstrate that NOMADS achieves substantially improved generalization accuracy compared to existing DMD-based methods even for noisy train data, and reproduces energy conservation in the Markovian case.

math.OC↗

arXiv2026

Infinite-horizon controllability scores for linear time-invariant systems

We formulate infinite-horizon controllability scores for linear time-invariant systems whose conventional controllability Gramians may diverge. A mode-dependent scaling yields finite-horizon volumetric controllability score (VCS) and average energy controllability score (AECS) problems that are exactly equivalent to the original ones, while the scaled Gramians converge as the horizon tends to infinity. Using epi-convergence, we establish convergence of the finite-horizon optimal allocations to the limiting solution set and obtain convergence to a unique limiting score under explicit rank conditions. The limiting VCS retains the stable, center, and unstable components and preserves full-system controllability, whereas the limiting AECS depends only on the stable component and may neglect an input required for a center or unstable mode. The VCS theory allows arbitrary spectral structures, whereas the AECS limit requires a nonempty stable component. The additional restriction on the center spectrum is imposed only for the real-Schur implementation. A directed-Laplacian example verifies the convergence and uniqueness conditions and illustrates the controllability distinction between the two limiting scores.

math.OC↗

arXiv2026

Model-Free Design and Analysis of 2DOF PI Controller for Noisy LTI Systems

Set-point tracking for systems with unknown model parameters is a fundamental problem in control, and two-degree-of-freedom (2DOF) Proportional-Integral (PI) controllers -- consisting of a feedforward controller and PI controller -- are widely employed for this task. In this paper, we propose a model-free design of 2DOF PI controllers, establish its theoretical properties, and compare them with a model-based method from both theoretical and numerical perspectives. For the feedforward design, we extend an existing model-free algorithm to systems subject to Gaussian process and measurement noises. We derive a nonasymptotic lower bound on the required control input/output data length and characterize the resulting estimation error. For PI gain tuning, we formulate a constrained optimization problem and establish sample complexity of a zeroth-order optimization method. Moreover, we quantify how inaccuracies in the feedforward design propagate to the performance of the PI controller, highlighting an interaction that has not been examined in prior work. We further provide a theoretical comparison between the proposed method and the model-based method. In particular, for PI gain tuning, the proposed method is computationally more efficient by avoiding explicit gradient computations. Numerical experiments demonstrate that the 2DOF PI controller designed by the proposed method exhibits better control performance than the model-based method.

math.OC↗

arXiv2025

A Deep State-Space Model Compression Method using Upper Bound on Output Error

We study deep state-space models (Deep SSMs) that contain linear quadratic-output (LQO) systems as internal blocks and present a compression method with a provable output error guarantee. We first derive an upper bound on the output error between two Deep SSMs and show that the bound can be expressed in terms of the $h^2$-error norms between the layerwise LQO systems. In particular, we show that reducing the $h^2$ approximation errors of the LQO systems placed in shallow layers is effective in reducing the derived upper bound on the output error. Next, we formulate an optimization problem for the derived upper bound and develop a gradient-based MOR method. In the numerical experiments, using the IMDb task from the LRA benchmark, we demonstrate the effectiveness of the proposed upper-bound-based compression method. In particular, we show that the number of trainable parameters can be reduced by approximately 60\% without retraining while maintaining the performance of the original model.

eess.SY↗

arXiv2025

Controllability scores of linear time-varying network systems

For large-scale network systems, network centrality based on control theory plays a crucial role in understanding their properties and controlling them efficiently. The controllability score is such a centrality index and can give a physically meaningful measure. It is originally proposed for linear time-invariant (LTI) systems, and we extend it to linear time-varying (LTV) systems in this paper. Since the controllability score is defined as an optimal solution to some optimization problem, it is not necessarily uniquely determined. Its uniqueness must be guaranteed for reproducibility and interpretability. In this paper, we show its uniqueness in almost all cases, which guarantees its use as a network centrality measure. We also prove its continuity with respect to the time parameters. In addition, we propose a data-driven method to compute it. Finally, we verify the effectiveness of the extension and examine the performance of the data-driven method through numerical experiments.

math.OC↗

arXiv2025

Data-driven h2 model reduction for linear discrete-time systems

We present a data-driven framework for $h^{2}$-optimal model reduction for linear discrete-time systems. Our main contribution is to create optimal reduced-order models in the $h^{2}$-norm sense directly from the measurement data alone, without using the information about the original system. In particular, we focus on the fact that the gradients of the $h^{2}$ model reduction problem are expressed using the discrete-time Lyapunov equation and the discrete-time Sylvester equation, and derive the data-driven gradients. The proposed algorithm uses the output of an existing MOR as the initial point, and convergence to a stationary point is guaranteed under certain assumptions. In numerical experiments, we demonstrate that, for a modeling task in neuroscience, our method constructs a reduced-order model that outperforms DMDc in terms of the $h^{2}$-norm.

math.OC↗

arXiv2025

Compression Method for Deep Diagonal State Space Model Based on $H^2$ Optimal Reduction

Deep learning models incorporating linear SSMs have gained attention for capturing long-range dependencies in sequential data. However, their large parameter sizes pose challenges for deployment on resource-constrained devices. In this study, we propose an efficient parameter reduction method for these models by applying $H^{2}$ model order reduction techniques from control theory to their linear SSM components. In experiments, the LRA benchmark results show that the model compression based on our proposed method outperforms an existing method using the Balanced Truncation, while successfully reducing the number of parameters in the SSMs to $1/32$ without sacrificing the performance of the original models.

cs.LG↗

arXiv2025

Uniqueness Analysis of Controllability Scores and Their Application to Brain Networks

Assessing centrality in network systems is critical for understanding node importance and guiding decision-making processes. In dynamic networks, incorporating a controllability perspective is essential for identifying key nodes. In this paper, we study two control theoretic centrality measures -- the Volumetric Controllability Score (VCS) and Average Energy Controllability Score (AECS) -- to quantify node importance in linear time-invariant network systems. We prove the uniqueness of VCS and AECS for almost all specified terminal times, thereby enhancing their applicability beyond previously recognized cases. This ensures their interpretability, comparability, and reproducibility. Our analysis reveals substantial differences between VCS and AECS in linear systems with symmetric and skew-symmetric transition matrices. We also investigate the dependence of VCS and AECS on the terminal time and prove that when this parameter is extremely small, both scores become essentially uniform. Additionally, we prove that a sequence generated by a projected gradient method for computing VCS and AECS converges linearly to both measures under several assumptions. Finally, evaluations on brain networks modeled via Laplacian dynamics using real data reveal contrasting evaluation tendencies and correlations for VCS and AECS, with AECS favoring brain regions associated with cognitive and motor functions, while VCS emphasizes sensory and emotional regions.

math.OC↗

arXiv2025

Extension of Controllability Score to Infinite-Dimensional Systems

Centrality analysis in dynamical network systems is essential for understanding system behavior. In finite-dimensional settings, controllability scores -- namely, the Volumetric Controllability Score (VCS) and the Average Energy Controllability Score (AECS) -- are defined as the unique solutions of specific optimization problems. In this work, we extend these concepts to infinite-dimensional systems by formulating analogous optimization problems. Moreover, we prove that these optimization problems have optimal solutions under weak assumptions, and that both VCS and AECS remain unique in the infinite-dimensional context under appropriate assumptions. The uniqueness of the controllability scores is essential to use them as a centrality measure, since it not only reflects the importance of each state in the dynamical network but also provides a consistent basis for interpretation and comparison across different researchers. Finally, we illustrate the behavior of VCS and AECS with a numerical experiment based on the heat equation.

math.OC↗

arXiv2024

Model Compression Method for S4 with Diagonal State Space Layers using Balanced Truncation

To implement deep learning models on edge devices, model compression methods have been widely recognized as useful. However, it remains unclear which model compression methods are effective for Structured State Space Sequence (S4) models incorporating Diagonal State Space (DSS) layers, tailored for processing long-sequence data. In this paper, we propose to use the balanced truncation, a prevalent model reduction technique in control theory, applied specifically to DSS layers in pre-trained S4 model as a novel model compression method. Moreover, we propose using the reduced model parameters obtained by the balanced truncation as initial parameters of S4 models with DSS layers during the main training process. Numerical experiments demonstrate that our trained models combined with the balanced truncation surpass conventionally trained models with Skew-HiPPO initialization in accuracy, even with fewer parameters. Furthermore, our observations reveal a positive correlation: higher accuracy in the original model consistently leads to increased accuracy in models trained using our model compression method, suggesting that our approach effectively leverages the strengths of the original model.

cs.LG↗