arXiv subjects

Bassam Bamieh

Publications and source records attributed to Bassam Bamieh.

At least 37 records · Page 2Linked to original sources

arXiv2024

A Convex Parameterization of Controllers Constrained to use only Relative Measurements

The optimal controller design problem for systems equipped with sensors that measure only relative, rather than absolute, quantities is considered. This relative measurement structure is formulated as a design constraint; it is demonstrated that the resulting constrained controller design problem can be written as a convex program. Certain additional network structural constraints can be incorporated into this formulation, making it especially useful in distributed or networked settings. An illustrative example highlights the advantage of the proposed methodology over the standard formulation of the output feedback controller design problem. A numerical example is provided.

eess.SY↗

arXiv2024

Implicit Boundary Conditions in Partial Differential Equations Discretizations: Identifying Spurious Modes and Model Reduction

We revisit the problem of spurious modes that are sometimes encountered in partial differential equations discretizations. It is generally suspected that one of the causes for spurious modes is due to how boundary conditions are treated, and we use this as the starting point of our investigations. By regarding boundary conditions as algebraic constraints on a differential equation, we point out that any differential equation with homogeneous boundary conditions also admits a typically infinite number of hidden or implicit boundary conditions. In most discretization schemes, these additional implicit boundary conditions are violated, and we argue that this is what leads to the emergence of spurious modes. These observations motivate two definitions of the quality of computed eigenvalues based on violations of derivatives of boundary conditions on the one hand, and on the Grassmann distance between subspaces associated with computed eigenspaces on the other. Both of these tests are based on a standardized treatment of boundary conditions and do not require a priori knowledge of eigenvalue locations. The effectiveness of these tests is demonstrated on several examples known to have spurious modes. In addition, these quality tests show that in most problems, about half the computed spectrum of a differential operator is of low quality. The tests also specifically identify the low accuracy modes, which can then be projected out as a type of model reduction scheme.

math.NA↗

arXiv2024

Linear-Quadratic Problems in Systems and Controls via Covariance Representations and Linear-Conic Duality: Finite-Horizon Case

Linear-Quadratic (LQ) problems that arise in systems and controls include the classical optimal control problems of the Linear Quadratic Regulator (LQR) in both its deterministic and stochastic forms, as well as $H^\infty$-analysis (the Bounded Real Lemma), the Positive Real Lemma, and general Integral Quadratic Constraints (IQCs) tests. We present a unified treatment of all of these problems using an approach which converts linear-quadratic problems to matrix-valued linear-linear problems with a positivity constraint. This is done through a system representation where the joint state/input covariance (the outer product in the deterministic case) matrix is the fundamental object. LQ problems then become infinite-dimensional semidefinite programs, and the key tool used is that of linear-conic duality. Linear Matrix Inequalities (LMIs) emerge naturally as conal constraints on dual problems. Riccati equations characterize extrema of these special LMIs, and therefore provide solutions to the dual problems. The state-feedback structure of all optimal signals in these problems emerge out of alignment (complementary slackness) conditions between primal and dual problems. Perhaps the new insight gained from this approach is that first LMIs, and then second, Riccati equations arise naturally in dual, rather than primal problems. Furthermore, while traditional LQ problems are set up in $L^2$ spaces of signals, their equivalent covariance-representation problems are most naturally set up in $L^1$ spaces of matrix-valued signals.

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arXiv2023

An Adaptation of the AAA-Interpolation Algorithm for Model Reduction of MIMO Systems

We consider the Adaptive Antoulas-Anderson (AAA) rational interpolation algorithm recently developed by Trefethen and co-authors, which can be viewed as a type of moment-matching technique for system realization and approximation. We consider variations on this algorithm that are suitable for model reduction of linear time invariant systems while addressing some of the shortcomings of the block-AAA variant of the algorithm for MIMO systems. In particular, we develop state-space formulas and keep track of the state-space dimension at every step of the adaptive block-AAA algorithm, showing an unfavorable increase of the state dimension. We propose a new low-rank adaptive interpolation algorithm that addresses this shortcoming. Comparative computational results are included for the algorithms above, together with comparisons to balanced reduction.

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arXiv2023

Continuum Swarm Tracking Control: A Geometric Perspective in Wasserstein Space

We consider a setting in which one swarm of agents is to service or track a second swarm, and formulate an optimal control problem which trades off between the competing objectives of servicing and motion costs. We consider the continuum limit where large-scale swarms are modeled in terms of their time-varying densities, and where the Wasserstein distance between two densities captures the servicing cost. We show how this non-linear infinite-dimensional optimal control problem is intimately related to the geometry of Wasserstein space, and provide new results in the case of absolutely continuous densities and constant-in-time references. Specifically, we show that optimal swarm trajectories follow Wasserstein geodesics, while the optimal control tradeoff determines the time-schedule of travel along these geodesics. We briefly describe how this solution provides a basis for a model-predictive control scheme for tracking time-varying and real-time reference trajectories as well.

eess.SY↗

arXiv2023

Scale Fragilities in Localized Consensus Dynamics

We consider distributed consensus in networks where the agents have integrator dynamics of order two or higher ($n\ge 2$). We assume all feedback to be localized in the sense that each agent has a bounded number of neighbors and consider a scaling of the network through the addition of agents in a modular manner, i.e., without re-tuning controller gains upon addition. We show that standard consensus algorithms, which rely on relative state feedback, are subject to what we term scale fragilities, meaning that stability is lost as the network scales. For high-order agents ($n\ge 3$), we prove that no consensus algorithm with fixed gains can achieve consensus in networks of any size. That is, while a given algorithm may allow a small network to converge, it causes instability if the network grows beyond a certain finite size. This holds in families of network graphs whose algebraic connectivity, that is, the smallest non-zero Laplacian eigenvalue, is decreasing towards zero in network size (e.g. all planar graphs). For second-order consensus ($n = 2$) we prove that the same scale fragility applies to directed graphs that have a complex Laplacian eigenvalue approaching the origin (e.g. directed ring graphs). The proofs for both results rely on Routh-Hurwitz criteria for complex-valued polynomials and hold true for general directed network graphs. We survey classes of graphs subject to these scale fragilities, discuss their scaling constants, and finally prove that a sub-linear scaling of nodal neighborhoods can suffice to overcome the issue.

math.OC↗

arXiv2022

A Short Introduction to the Koopman Representation of Dynamical Systems

The Koopman representation is an infinite dimensional linear representation of linear or nonlinear dynamical systems. It represents the dynamics of output maps (aka observables), which are functions on the state space whose evaluation is interpreted as an output. Conceptually simple derivations and commentary on the Koopman representation are given. We emphasize an important duality between initial conditions and output maps of the original system, and those of the Koopman representation. This duality is an important consideration when this representation is used in data-driven applications such as the Dynamic Mode Decomposition (DMD) and its variants. The adjoint relation between the Koopman representation and the transfer operator of mass transport is also shown.

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arXiv2022

Discovering Transforms: A Tutorial on Circulant Matrices, Circular Convolution, and the Discrete Fourier Transform

How could the Fourier and other transforms be naturally discovered if one didn't know how to postulate them? In the case of the Discrete Fourier Transform (DFT), we show how it arises naturally out of analysis of circulant matrices. In particular, the DFT can be derived as the change of basis that simultaneously diagonalizes all circulant matrices. In this way, the DFT arises naturally from a linear algebra question about a set of matrices. Rather than thinking of the DFT as a signal transform, it is more natural to think of it as a single change of basis that renders an entire set of mutually-commuting matrices into simple, diagonal forms. The DFT can then be "discovered" by solving the eigenvalue/eigenvector problem for a special element in that set. A brief outline is given of how this line of thinking can be generalized to families of linear operators, leading to the discovery of the other common Fourier-type transforms, as well as its connections with group representations theory.

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arXiv2022

A Tutorial on Matrix Perturbation Theory (using compact matrix notation)

Analytic perturbation theory for matrices and operators is an immensely useful mathematical technique. Most elementary introductions to this method have their background in the physics literature, and quantum mechanics in particular. In this note, we give an introduction to this method that is independent of any physics notions, and relies purely on concepts from linear algebra. An additional feature of this presentation is that matrix notation and methods are used throughout. In particular, we formulate the equations for each term of the analytic expansions of eigenvalues and eigenvectors as {\em matrix equations}, namely Sylvester equations in particular. Solvability conditions and explicit expressions for solutions of such matrix equations are given, and expressions for each term in the analytic expansions are given in terms of those solutions. This unified treatment simplifies somewhat the complex notation that is commonly seen in the literature, and in particular, provides relatively compact expressions for the non-Hermitian and degenerate cases, as well as for higher order terms.

math.SP↗

arXiv2022

A Tutorial on Solution Properties of State Space Models of Dynamical Systems

The starting point of analysis of state space models is investigating existence, uniqueness and solution properties such as the semigroup property, and various formulas for the solutions. Several concepts such as the state transition matrix, the matrix exponential, the variations of constants formula (the Cauchy formula), the Peano-Baker series, and the Picard iteration are used to characterize solutions. In this note, a tutorial treatment is given where all of these concepts are shown to be various manifestations of a single abstract method, namely solving equations using an operator Neumann series involving the Volterra operator of forward integration. The matrix exponential, the Peano-Baker series, the Picard iteration, and the Cauchy formula can be "discovered" naturally from this Neumann series. The convergence of the series and iterations is a consequence of the key property of asymptotic nilpotence of the Volterra operator. This property is an asymptotic version of the nilpotence property of a strictly-lower-triangular matrix.

eess.SY↗

arXiv2022

On Structured-Closed-Loop versus Structured-Controller Design: the Case of Relative Measurement Feedback

We consider the optimal distributed controller design problem subject to two structural requirements: locality, i.e. available measurements and sub-controllers' interactions are governed by a graph structure, and relative feedback, i.e. only differences of measurements are available to the controller. We formalize controller locality in terms of the controller's transfer function, state-space realization, or resulting closed-loop mapping. We demonstrate that the relative feedback requirement can be written as a convex constraint on the controller and (in special cases) on the resulting closed-loop, and we characterize the allowable structures of relative feedback controllers. We prove that sparse closed-loop design is a convex relaxation of structured controller state-space design, even in the continuous time IIR setting. This formalizes and extends results of the recently developed System Level Synthesis framework. We take a first step toward quantifying the performance gap associated with this convex relaxation by constructing a class of examples (based on relative feedback requirements) for which the difference in performance, measured by an H2 norm, is infinite. The results presented are used to contrast several issues of structural constraints in distributed control design that remain as open problems.

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arXiv2022

An Explicit Parametrization of Closed Loops for Spatially Distributed Controllers with Sparsity Constraints

In this article, we study the linear time-invariant state-feedback controller design problem for distributed systems. We follow the recently developed system level synthesis (SLS) approach and impose locality structure on the resulting closed-loop mappings; the corresponding controller implementation inherits this prescribed structure. In contrast to existing SLS results, we derive an explicit (rather than implicit) parameterization of all achievable stabilized closed-loops. This admits more efficient IIR representations of the temporal part of the closed-loop dynamics, and it allows for the H2 design problem with closed-loop spatial sparsity constraints to be converted to a standard model matching problem, with the number of transfer function parameters scaling linearly with the closed-loop spatial extent constraint. We illustrate our results with two applications: consensus of first-order subsystems and the vehicular platoons problem. In the case of first-order consensus, we provide analytic solutions and further analyze the architecture of the resulting controller implementation. Results for infinite extent spatially invariant systems are presented to provide insight to the case of a large but finite number of subsystems.

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arXiv2020

Diffusion and Consensus in a Weakly Coupled Network of Networks

We study diffusion and consensus dynamics in a Network of Networks model. In this model, there is a collection of sub-networks, connected to one another using a small number of links. We consider a setting where the links between networks have small weights, or are used less frequently than links within each sub-network. Using spectral perturbation theory, we analyze the diffusion rate and convergence rate of the investigated systems. Our analysis shows that the first order approximation of the diffusion and convergence rates is independent of the topologies of the individual graphs; the rates depend only on the number of nodes in each graph and the topology of the connecting edges. The second order analysis shows a relationship between the diffusion and convergence rates and the information centrality of the connecting nodes within each sub-network. We further highlight these theoretical results through numerical examples.

cs.SI↗

arXiv2019

Stochasticity in Feedback Loops; Great Expectations and Guaranteed Ruin

Stochastic feedback systems give rise to a variety of notions of stability. The conditions for the stability of the median, mean, and variance stability conditions differ. These conditions can be stated explicitly for scalar discrete-time systems with (almost) arbitrary distributions of the stochastic feedback gain. The state variable in such systems evolves towards a heavy-tailed distribution and exhibits some non-intuitive characteristics. For example, one can use stochastic feedback to stabilise unstable systems where one does not know the sign of the unstable pole or the sign of the system gain. A more dramatic example is an investment scheme which simultaneously yields unbounded expected profit and almost certain bankruptcy to every investor.

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arXiv2019

Localized high-order consensus destabilizes large-scale networks

We study the problem of distributed consensus in networks where the local agents have high-order ($n\ge 3$) integrator dynamics, and where all feedback is localized in that each agent has a bounded number of neighbors. We prove that no consensus algorithm based on relative differences between states of neighboring agents can then achieve consensus in networks of any size. That is, while a given algorithm may allow a small network to converge to consensus, the same algorithm will lead to instability if agents are added to the network so that it grows beyond a certain finite size. This holds in classes of network graphs whose algebraic connectivity, that is, the smallest non-zero Laplacian eigenvalue, is decreasing towards zero in network size. This applies, for example, to all planar graphs. Our proof, which relies on Routh-Hurwitz criteria for complex-valued polynomials, holds true for directed graphs with normal graph Laplacians. We survey classes of graphs where this issue arises, and also discuss leader-follower consensus, where instability will arise in any growing, undirected network as long as the feedback is localized.

math.OC↗

arXiv2019

On Fundamental Limitations of Dynamic Feedback Control in Regular Large-Scale Networks

We study fundamental performance limitations of distributed feedback control in large-scale networked dynamical systems. Specifically, we address the question of whether dynamic feedback controllers perform better than static (memoryless) ones when subject to locality constraints. We consider distributed linear consensus and vehicular formation control problems modeled over toric lattice networks. For the resulting spatially invariant systems we study the large-scale asymptotics (in network size) of global performance metrics that quantify the level of network coherence. With static feedback from relative state measurements, such metrics are known to scale unfavorably in lattices of low spatial dimensions, preventing, for example, a 1-dimensional string of vehicles to move like a rigid object. We show that the same limitations in general apply also to dynamic feedback control that is locally of first order. This means that the addition of one local state to the controller gives a similar asymptotic performance to the memoryless case. This holds unless the controller can access noiseless measurements of its local state with respect to an absolute reference frame, in which case the addition of controller memory may fundamentally improve performance. In simulations of platoons with 20-200 vehicles we show that the performance limitations we derive manifest as unwanted accordion-like motions. Similar behaviors are to be expected in any network that is embeddable in a low-dimensional toric lattice, and the same fundamental limitations would apply. To derive our results, we present a general technical framework for the analysis of stability and performance of spatially invariant systems in the limit of large networks.

math.OC↗

arXiv2018

Stochastic Models for Cochlear Instabilities

We investigate instabilities in a stochastic mathematical model of cochlear dynamics. The cochlea is modeled as a spatio-temporal dynamical system made up of a spatially distributed array of coupled oscillators, together with the cochlear amplification mechanism. We consider a setting where the cochlear amplifier has relatively small spatially and temporally varying stochastic parameters. It is shown that relatively small parameter variations (five to four orders of magnitude smaller than the nominal values) are sufficient to destabilize the dynamics and induce spontaneous oscillations. This extreme sensitivity of the cochlear dynamics appears to be due to a combination of the local cochlear amplification mechanism, as well as the spatial coupling of the distributed resonators. We use an analysis technique which allows for a simulation-free prediction of the stability thresholds, as well as the statistics of the spontaneous oscillations. We verify our theoretical predictions using full non- linear stochastic simulations which appear to be in very good agreement with our theory.

physics.bio-ph↗

arXiv2018

An Input-Output Approach to Structured Stochastic Uncertainty in Continuous Time

We consider the continuous-time setting of linear time-invariant (LTI) systems in feedback with multiplicative stochastic uncertainties. The objective of the paper is to characterize the conditions of Mean-Square Stability (MSS) using a purely input-output approach, i.e. without having to resort to state space realizations. This has the advantage of encompassing a wider class of models (such as infinite dimensional systems and systems with delays). The input-output approach leads to uncovering new tools such as stochastic block diagrams that have an intimate connection with the more general Stochastic Integral Equations (SIE), rather than Stochastic Differential Equations (SDE). Various stochastic interpretations are considered, such as Itō and Stratonovich, and block diagram conversion schemes between different interpretations are devised. The MSS conditions are given in terms of the spectral radius of a matrix operator that takes different forms when different stochastic interpretations are considered.

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