arXiv subjects

Aleksandr Talitckii

Publications and source records attributed to Aleksandr Talitckii.

6 recordsLinked to original sources

arXiv2026

Picard Iteration for Parameter Estimation in Nonlinear Dynamic Models of Aircraft and Spacecraft

The attitude dynamics of aircraft and spacecraft exhibit significantly nonlinear behaviour. In spacecraft, torque is generated through reaction wheels and control moment gyros. In aircraft, torque is generated using lift on control surfaces. In both cases, complex geometries, unique configurations, and internal/environmental changes imply that models must be identified, verified, and updated using in-flight experimental data. However, this data is often noisy, sparsely sampled, and partial in that modeled states may not be directly measurable. In this paper, we propose a method for estimating key parameters in realistic Ordinary Differential Equation (ODE) models of both spacecraft and aircraft dynamics. This method avoids the need to directly measure state derivatives by coupling sampled outputs using the Picard mapping -- an integral constraint on the solution of the parameterized ODE. This constraint is then enforced, and optimal parameter estimates are found using a gradient contraction algorithm. This algorithm is applied to well-studied models of spacecraft and aircraft motion. First, the algorithm is used to estimate the inertia tensor in a 4 control-moment gyro (CMG) model of spacecraft motion. Second, we estimate the 28 higher-order control surface coefficients in a model of the F/A-18 aircraft.

math.DS↗

arXiv2025

A $μ$-Analysis and Synthesis Framework for Partial Integral Equations using IQCs

We develop a $μ$-analysis and synthesis framework for infinite-dimensional systems that leverages the Integral Quadratic Constraints (IQCs) to compute the structured singular value's upper bound. The methodology formulates robust stability and performance conditions jointly as Linear Partial Integral Inequalities within the Partial Integral Equation framework, establishing connections between IQC multipliers and $μ$-theory. Computational implementation via PIETOOLS enables computational tools that practically applicable to spatially distributed infinite dimensional systems. Illustrations with the help of Partial and Delay Differential Equations validate the effectiveness of the framework, showing a significant reduction in conservatism compared to unstructured methods and providing systematic tools for stability-performance trade-off analysis.

eess.SY↗

arXiv2024

Picard Iteration for Parameter Estimation in Nonlinear Ordinary Differential Equations using Low-Quality Data

We consider the problem of using experimental time-series data for parameter estimation in nonlinear ordinary differential equations, focusing on the case where the data is noisy, sparse, irregularly sampled, includes multiple experiments, and does not directly measure the system state or its time derivative. To account for such low-quality data, we propose a new framework for gradient-based parameter estimation which uses the Picard operator to reformulate the problem as constrained optimization with infinite-dimensional variables and constraints. We then formulate the Karush-Kuhn-Tucker (KKT) conditions necessary for optimality and define a convergent sequence of approximations to these KKT conditions obtained by replacing the solution map by the n-th order Picard iterate. Then, for any element of this sequence, and by exploiting the contractive properties of the Picard operator, we propose a gradient-contractive algorithm which (under regularity and convexity assumptions) is guaranteed to converge to a solution of these approximated KKT conditions. Finally, the algorithms are then tested on a battery of models and a variety of datasets in order to demonstrate robustness and improvement over alternative approaches.

math.OC↗

arXiv2024

Efficient Convex Algorithms for Universal Kernel Learning

The accuracy and complexity of machine learning algorithms based on kernel optimization are determined by the set of kernels over which they are able to optimize. An ideal set of kernels should: admit a linear parameterization (for tractability); be dense in the set of all kernels (for robustness); be universal (for accuracy). Recently, a framework was proposed for using positive matrices to parameterize a class of positive semi-separable kernels. Although this class can be shown to meet all three criteria, previous algorithms for optimization of such kernels were limited to classification and furthermore relied on computationally complex Semidefinite Programming (SDP) algorithms. In this paper, we pose the problem of learning semiseparable kernels as a minimax optimization problem and propose a SVD-QCQP primal-dual algorithm which dramatically reduces the computational complexity as compared with previous SDP-based approaches. Furthermore, we provide an efficient implementation of this algorithm for both classification and regression -- an implementation which enables us to solve problems with 100 features and up to 30,000 datums. Finally, when applied to benchmark data, the algorithm demonstrates the potential for significant improvement in accuracy over typical (but non-convex) approaches such as Neural Nets and Random Forest with similar or better computation time.

stat.ML↗

arXiv2023

Employing Feature Selection Algorithms to Determine the Immune State of a Mouse Model of Rheumatoid Arthritis

The immune response is a dynamic process by which the body determines whether an antigen is self or nonself. The state of this dynamic process is defined by the relative balance and population of inflammatory and regulatory actors which comprise this decision making process. The goal of immunotherapy as applied to, e.g. Rheumatoid Arthritis (RA), then, is to bias the immune state in favor of the regulatory actors - thereby shutting down autoimmune pathways in the response. While there are several known approaches to immunotherapy, the effectiveness of the therapy will depend on how this intervention alters the evolution of this state. Unfortunately, this process is determined not only by the dynamics of the process, but the state of the system at the time of intervention - a state which is difficult if not impossible to determine prior to application of the therapy. To identify such states we consider a mouse model of RA (Collagen-Induced Arthritis (CIA)) immunotherapy; collect high dimensional data on T cell markers and populations of mice after treatment with a recently developed immunotherapy for CIA; and use feature selection algorithms in order to select a lower dimensional subset of this data which can be used to predict both the full set of T cell markers and populations, along with the efficacy of immunotherapy treatment.

stat.ML↗

arXiv2023

Integral Quadratic Constraints with Infinite-Dimensional Channels

Modern control theory provides us with a spectrum of methods for studying the interconnection of dynamic systems using input-output properties of the interconnected subsystems. Perhaps the most advanced framework for such input-output analysis is the use of Integral Quadratic Constraints (IQCs), which considers the interconnection of a nominal linear system with an unmodelled nonlinear or uncertain subsystem with known input-output properties. Although these methods are widely used for Ordinary Differential Equations (ODEs), there have been fewer attempts to extend IQCs to infinite-dimensional systems. In this paper, we present an IQC-based framework for Partial Differential Equations (PDEs) and Delay Differential Equations (DDEs). First, we introduce infinite-dimensional signal spaces, operators, and feedback interconnections. Next, in the main result, we propose a formulation of hard IQC-based input-output stability conditions, allowing for infinite-dimensional multipliers. We then show how to test hard IQC conditions with infinite-dimensional multipliers on a nominal linear PDE or DDE system via the Partial Integral Equation (PIE) state-space representation using a sufficient version of the Kalman-Yakubovich-Popov lemma (KYP). The results are then illustrated using four example problems with uncertainty and nonlinearity.

math.OC↗