arXiv subjects

Milan Korda

Publications and source records attributed to Milan Korda.

At least 37 records · Page 2Linked to original sources

arXiv2024

Sparse decompositions of nonlinear dynamical systems and applications to moment-sum-of-squares relaxations

In this paper, we propose a general sparse decomposition of dynamical systems provided that the vector field and constraint set possess certain sparse structures, which we call subsystems. This notion is based on causal dependence in the dynamics between the different states. This results in sparse descriptions for fundamental problems from nonlinear dynamical systems: region of attraction, maximum positively invariant set, and global attractor. The decompositions can be paired with any method for computing (outer) approximations of these sets to reduce the computation to lower dimensional systems. This is illustrated by methods from previous work based on infinite-dimensional linear programming. This exhibits one example where the curse of dimensionality is present and hence dimension reduction is crucial. In this context, for polynomial dynamics, we show that these problems admit a sparse sum-of-squares (SOS) approximation with guaranteed convergence such that the number of variables in the largest SOS multiplier is given by the dimension of the largest subsystem appearing in the decomposition. The dimension of such subsystems depends on the sparse structure of the vector field and the constraint set; if the dimension of the largest subsystem is small compared to the ambient dimension, this allows for a significant reduction in the computation time of the SOS approximations. Numerical examples accompany the approach.

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arXiv2023

Invariant Consistent Dynamic Mode Decomposition

Any deterministic autonomous dynamical system may be globally linearized by its' Koopman operator. This object is typically infinite-dimensional and can be approximated by the so-called Dynamic Mode Decomposition (DMD). In DMD, the central idea is to preserve a fundamental property of the Koopman operator: linearity. This work augments DMD by preserving additional properties like functional relationships between observables and consistency along geometric invariants. The first set of constraints provides a framework for understanding DMD variants like Higher-order DMD and Affine DMD. The latter set guarantees the estimation of Koopman eigen-functions with eigen-value 1, whose level sets are known to delineate invariant sets. These benefits are realized with only a minimal increase in computational cost, primarily due to the linearity of constraints.

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arXiv2023

Exploiting Term Sparsity in Moment-SOS hierarchy for Dynamical Systems

In this paper, we develop a dynamical system counterpart to the term sparsity sum-of-squares (TSSOS) algorithm proposed for static polynomial optimization. This allows for computational savings and improved scalability while preserving convergence guarantees when sum-of-squares methods are applied to problems from dynamical systems, including the problems of approximating region of attraction, the maximum positively invariant set, and the global attractor. At its core, the method exploits the algebraic structure of the data, thereby complementing existing methods that exploit causality relations among the states of the dynamical system. The procedure encompasses sign symmetries of the dynamical system as was already revealed for polynomial optimization. Numerical examples demonstrate the efficiency of the approach in the presence of this type of sparsity.

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arXiv2023

Converging outer approximations to global attractors using semidefinite programming

This paper develops a method for obtaining guaranteed outer approximations for global attractors of continuous and discrete time nonlinear dynamical systems. The method is based on a hierarchy of semidefinite programming problems of increasing size with guaranteed convergence to the global attractor. The approach taken follows an established line of reasoning, where we first characterize the global attractor via an infinite dimensional linear programming problem (LP) in the space of Borel measures. The dual to this LP is in the space of continuous functions and its feasible solutions provide guaranteed outer approximations to the global attractor. For systems with polynomial dynamics, a hierarchy of finite-dimensional sum-of-squares tightenings of the dual LP provides a sequence of outer approximations to the global attractor with guaranteed convergence in the sense of volume discrepancy tending to zero. The method is very simple to use and based purely on convex optimization. Numerical examples with the code available online demonstrate the method.

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arXiv2023

Dictionary-free Koopman model predictive control with nonlinear input transformation

This paper introduces a method for data-driven control based on the Koopman operator model predictive control. Unlike exiting approaches, the method does not require a dictionary and incorporates a nonlinear input transformation, thereby allowing for more accurate predictions with less ad hoc tuning. In addition to this, the method allows for input quantization and exploits symmetries, thereby reducing computational cost, both offline and online. Importantly, the method retains convexity of the optimization problem solved within the model predictive control online. Numerical examples demonstrate superior performance compared to existing methods as well as the capacity to learn discontinuous lifting functions.

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arXiv2023

Exploiting ideal-sparsity in the generalized moment problem with application to matrix factorization ranks

We explore a new type of sparsity for the generalized moment problem (GMP) that we call ideal-sparsity. This sparsity exploits the presence of equality constraints requiring the measure to be supported on the variety of an ideal generated by bilinear monomials modeled by an associated graph. We show that this enables an equivalent sparse reformulation of the GMP, where the single (high dimensional) measure variable is replaced by several (lower-dimensional) measure variables supported on the maximal cliques of the graph. We explore the resulting hierarchies of moment-based relaxations for the original dense formulation of GMP and this new, equivalent ideal-sparse reformulation, when applied to the problem of bounding nonnegative- and completely positive matrix factorization ranks. We show that the ideal-sparse hierarchies provide bounds that are at least as good (and often tighter) as those obtained from the dense hierarchy. This is in sharp contrast to the situation when exploiting correlative sparsity, as is most common in the literature, where the resulting bounds are weaker than the dense bounds. Moreover, while correlative sparsity requires the underlying graph to be chordal, no such assumption is needed for ideal-sparsity. Numerical results show that the ideal-sparse bounds are often tighter and much faster to compute than their dense analogs.

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arXiv2023

Symmetry reduction and recovery of trajectories of optimal control problems via measure relaxations

We address the problem of symmetry reduction of optimal control problems under the action of a finite group from a measure relaxation viewpoint. We propose a method based on the moment-SOS aka Lasserre hierarchy which allows one to significantly reduce the computation time and memory requirements compared to the case without symmetry reduction. We show that the recovery of optimal trajectories boils down to solving a symmetric parametric polynomial system. Then we illustrate our method on the symmetric integrator and the time-optimal inversion of qubits.

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arXiv2023

Peak Estimation of Time Delay Systems using Occupation Measures

This work proposes a method to compute the maximum value obtained by a state function along trajectories of a Delay Differential Equation (DDE). An example of this task is finding the maximum number of infected people in an epidemic model with a nonzero incubation period. The variables of this peak estimation problem include the stopping time and the original history (restricted to a class of admissible histories). The original nonconvex DDE peak estimation problem is approximated by an infinite-dimensional Linear Program (LP) in occupation measures, inspired by existing measure-based methods in peak estimation and optimal control. This LP is approximated from above by a sequence of Semidefinite Programs (SDPs) through the moment-Sum of Squares (SOS) hierarchy. Effectiveness of this scheme in providing peak estimates for DDEs is demonstrated with provided examples

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arXiv2023

Convergence rates for sums-of-squares hierarchies with correlative sparsity

This work derives upper bounds on the convergence rate of the moment-sum-of-squares hierarchy with correlative sparsity for global minimization of polynomials on compact basic semialgebraic sets. The main conclusion is that both sparse hierarchies based on the Schmüdgen and Putinar Positivstellensätze enjoy a polynomial rate of convergence that depends on the size of the largest clique in the sparsity graph but not on the ambient dimension. Interestingly, the sparse bounds outperform the best currently available bounds for the dense hierarchy when the maximum clique size is sufficiently small compared to the ambient dimension and the performance is measured by the running time of an interior point method required to obtain a bound on the global minimum of a given accuracy.

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arXiv2023

Occupation measure relaxations in variational problems: the role of convexity

This work addresses the occupation measure relaxation of calculus of variations problems, which is an infinite-dimensional linear programming relaxation amenable to numerical approximation by a hierarchy of semidefinite optimization problems. We address the problem of equivalence of this relaxation to the original problem. Our main result provides sufficient conditions for this equivalence. These conditions, revolving around the convexity of the data, are simple and apply in very general settings that may be of arbitrary dimensions and may include pointwise and integral constraints, thereby considerably strengthening the existing results. Our conditions are also extended to optimal control problems. In addition, we demonstrate how these results can be applied in non-convex settings, showing that the occupation measure relaxation is at least as strong as the convexification using the convex envelope; in doing so, we prove that a certain weakening of the occupation measure relaxation is equivalent to the convex envelope. This opens the way to application of the occupation measure relaxation in situations where the convex envelope relaxation is known to be equivalent to the original problem, which includes problems in magnetism and elasticity.

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arXiv2022

Controlled Synchronization of Coupled Pendulums by Koopman Model Predictive Control

We propose and experimentally demonstrate a feedback control method that allows synchronizing the motion of a chain of several coupled nonlinear oscillators actuated through one end of the chain. The chain considered in this work is a one-dimensional array of pendulums pivoting around a single axis and interacting with adjacent pendulums through torsion springs; the array is actuated using a single torque motor attached to one of the two boundary pendulums. This represents a mechanical realization of the Frenkel-Kontorova model { a spatially discrete version of a sine-Gordon equation describing (nonlinear) waves. The main challenges of controlling these systems are: high order (the number of pendulums can be high), nonlinear dynamics, and (as we set the problem here) only one actuator. The presented problem of synchronization of motion is a special case of the problem of reference tracking, where all pendulums reach a common point or a trajectory. In particular, we demonstrate synchronization to a stable equilibrium (all pendulums downward), unstable equilibrium (all pendulums upward), and a periodic orbit (all pendulums revolving). We use the Koopman Model Predictive Control (KMPC) that constructs a linear predictor of the nonlinear system in a higher-dimensional lifted space and uses the predictor within a classical linear MPC, thereby maintaining low computational cost that allows for a real-time implementation, while taking into account the complex nonlinear dynamics.

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arXiv2022

Stability and performance verification of dynamical systems controlled by neural networks: algorithms and complexity

This work makes several contributions on stability and performance verification of nonlinear dynamical systems controlled by neural networks. First, we show that the stability and performance of a polynomial dynamical system controlled by a neural network with semialgebraically representable activation functions (e.g., ReLU) can be certified by convex semidefinite programming. The result is based on the fact that the semialgebraic representation of the activation functions and polynomial dynamics allows one to search for a Lyapunov function using polynomial sum-of-squares methods. Second, we remark that even in the case of a linear system controlled by a neural network with ReLU activation functions, the problem of verifying asymptotic stability is undecidable. Finally, under additional assumptions, we establish a converse result on the existence of a polynomial Lyapunov function for this class of systems. Numerical results with code available online on examples of state-space dimension up to 50 and neural networks with several hundred neurons and up to 30 layers demonstrate the method.

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arXiv2022

Urysohn in action: separating semialgebraic sets by polynomials

A classical result from topology called Uryshon's lemma asserts the existence of a continuous separator of two disjoint closed sets in a sufficiently regular topological space. In this work we make a search for this separator constructive and efficient in the context of real algebraic geometry. Namely, given two compact disjoint basic semialgebraic sets which are contained in an $n$-dimensional box, we provide an algorithm that computes a separating polynomial greater than or equal to 1 on the first set and less than or equal to 0 on the second one.

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arXiv2022

Revisiting semidefinite programming approaches to options pricing: complexity and computational perspectives

In this paper we consider the problem of finding bounds on the prices of options depending on multiple assets without assuming any underlying model on the price dynamics, but only the absence of arbitrage opportunities. We formulate this as a generalized moment problem and utilize the well-known Moment-Sum-of-Squares (SOS) hierarchy of Lasserre to obtain bounds on the range of the possible prices. A complementary approach (also due to Lasserre) is employed for comparison. We present several numerical examples to demonstrate the viability of our approach. The framework we consider makes it possible to incorporate different kinds of observable data, such as moment information, as well as observable prices of options on the assets of interest.

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arXiv2021

Sparsity structures for Koopman operators

We present a decomposition of the Koopman operator based on the sparse structure of the underlying dynamical system, allowing one to consider the system as a family of subsystems interconnected by a graph. Using the intrinsic properties of the Koopman operator, we show that eigenfunctions for the subsystems induce eigenfunctions for the whole system. The use of principal eigenfunctions allows to reverse this result. Similarly for the adjoint operator, the Perron-Frobenius operator, invariant measures for the dynamical system induce invariant measures of the subsystems, while constructing invariant measures from invariant measures of the subsystems is less straightforward. We address this question and show that under necessary compatibility assumptions such an invariant measure exists. Based on these results we demonstrate that the a-priori knowledge of a decomposition of a dynamical system allows for a reduction of the computational cost for data driven approaches on the example of the dynamic mode decomposition.

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arXiv2021

Peak Estimation for Uncertain and Switched Systems

Peak estimation bounds extreme values of a function of state along trajectories of a dynamical system. This paper focuses on extending peak estimation to continuous and discrete settings with time-independent and time-dependent uncertainty. Techniques from optimal control are used to incorporate uncertainty into an existing occupation measure-based peak estimation framework, which includes special consideration for handling switching uncertainties. The resulting infinite-dimensional linear programs can be solved approximately with Linear Matrix Inequalities arising from the moment-SOS hierarchy.

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arXiv2021

Model Predictive Control of a Vehicle using Koopman Operator

This paper continues in the work from arXiv:1903.06103 [math.OC] where a nonlinear vehicle model was approximated in a purely data-driven manner by a linear predictor of higher order, namely the Koopman operator. The vehicle system typically features a lot of nonlinearities such as rigid-body dynamics, coordinate system transformations and most importantly the tire. These nonlinearities are approximated in a predefined subset of the state-space by the linear Koopman operator and used for a linear Model Predictive Control (MPC) design in the high-dimension state space where the nonlinear system dynamics evolve linearly. The result is a nonlinear MPC designed by linear methodologies. It is demonstrated that the Koopman-based controller is able to recover from a very unusual state of the vehicle where all the aforementioned nonlinearities are dominant. The controller is compared with a controller based on a classic local linearization and shortcomings of this approach are discussed.

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arXiv2021

Spatio-Temporal Decomposition of Sum-of-Squares Programs for the Region of Attraction and Reachability

This paper presents a method for calculating Region of Attraction of a target set (not necessarily an equilibrium) for controlled polynomial dynamical systems, using a hierarchy of semidefinite programming problems (SDPs). Our approach builds on previous work and addresses its main issue, the fast-growing memory demands for solving large-scale SDPs. The main idea in this work is in dissecting the original resource-demanding problem into multiple smaller, interconnected, and easier to solve problems. This is achieved by spatio-temporal splitting akin to methods based on partial differential equations. We show that the splitting procedure retains the convergence and outer-approximation guarantees of the previous work, while achieving higher precision in less time and with smaller memory footprint.

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