arXiv subjects

Mehran Mesbahi

Publications and source records attributed to Mehran Mesbahi.

At least 73 records · Page 4Linked to original sources

arXiv2019

Dual Quaternion Based Powered Descent Guidance with State-Triggered Constraints

This paper presents a numerical algorithm for computing 6-degree-of-freedom free-final-time powered descent guidance trajectories. The trajectory generation problem is formulated using a unit dual quaternion representation of the rigid body dynamics, and several standard path constraints. Our formulation also includes a special line of sight constraints that is enforced only within a specified band of slant ranges relative to the landing site, a novel feature that is especially relevant to Terrain and Hazard Relative Navigation. We use the newly introduced state-triggered constraints to formulate these range constraints in a manner that is amenable to real-time implementations. The resulting non-convex optimal control problem is solved iteratively as a sequence of convex second-order cone programs that locally approximate the non-convex problem. Each second-order cone program is solved using a customizable interior point method solver. Also introduced are a scaling method and a new heuristic technique that guide the convergence process towards dynamic feasibility. To demonstrate the capabilities of our algorithm, two numerical case studies are presented. The first studies the effect of including a slant-range-triggered line of sight constraint on the resulting trajectories. The second study performs a Monte Carlo analysis to assess the algorithm's robustness to initial conditions and real-time performance.

math.OC↗

arXiv2019

On Topological Properties of the Set of Stabilizing Feedback Gains

This work presents a fairly complete account on various topological and metrical aspects of feedback stabilization for single-input-single-output (SISO) continuous and discrete time linear-time-invariant (LTI) systems. In particular, we prove that the set of stabilizing output feedback gains for a SISO system with n states has at most $\lceil{\frac{n}{2}}\rceil$ connected components. Furthermore, our analysis yields an algorithm for determining intervals of stabilizing gains for general continuous and discrete LIT systems; the proposed algorithm also computes the number of unstable roots in each unstable interval. Along the way, we also make a number of observations on the set of stabilizing state feedback gains for MIMO systems.

eess.SY↗

arXiv2019

On Topological and Metrical Properties of Stabilizing Feedback Gains: the MIMO Case

In this paper, we discuss various topological and metrical aspects of the set of stabilizing static feedback gains for multiple-input-multiple-output (MIMO) linear-time-invariant (LTI) systems, in both continuous and discrete-time. Recently, connectivity properties of this set (for continuous time) have been reported in the literature, along with a discussion on how this connectivity is affected by restricting the feedback gain to linear subspaces. We show that analogous to the continuous-time case, one can construct instances where the set of stabilizing feedback gains for discrete time LTI systems has exponentially many connected components.

eess.SY↗

arXiv2019

Global Convergence of Policy Gradient Methods for the Linear Quadratic Regulator

Direct policy gradient methods for reinforcement learning and continuous control problems are a popular approach for a variety of reasons: 1) they are easy to implement without explicit knowledge of the underlying model 2) they are an "end-to-end" approach, directly optimizing the performance metric of interest 3) they inherently allow for richly parameterized policies. A notable drawback is that even in the most basic continuous control problem (that of linear quadratic regulators), these methods must solve a non-convex optimization problem, where little is understood about their efficiency from both computational and statistical perspectives. In contrast, system identification and model based planning in optimal control theory have a much more solid theoretical footing, where much is known with regards to their computational and statistical properties. This work bridges this gap showing that (model free) policy gradient methods globally converge to the optimal solution and are efficient (polynomially so in relevant problem dependent quantities) with regards to their sample and computational complexities.

cs.LG↗

arXiv2019

Distributed Q-Learning for Dynamically Decoupled Systems

Control of large-scale networked systems often necessitates the availability of complex models for the interactions amongst the agents. However in many applications, building accurate models of agents or interactions amongst them might be infeasible or computationally prohibitive due to the curse of dimensionality or the complexity of these interactions. In the meantime, data-guided control methods can circumvent model complexity by directly synthesizing the controller from the observed data. In this paper, we propose a distributed Q-learning algorithm to design a feedback mechanism based on a given underlying graph structure parameterizing the agents' interaction network. We assume that the distributed nature of the system arises from the cost function of the corresponding control problem and show that for the specific case of identical dynamically decoupled systems, the learned controller converges to the optimal Linear Quadratic Regulator (LQR) controller for each subsystem. We provide a convergence analysis and verify the result with an example.

math.OC↗

arXiv2019

Successive Convexification for 6-DoF Powered Descent Guidance with Compound State-Triggered Constraints

This paper introduces a continuous formulation for compound state-triggered constraints, which are generalizations of the recently introduced state-triggered constraints. State-triggered constraints are different from ordinary constraints found in optimal control in that they use a state-dependent trigger condition to enable or disable a constraint condition, and can be expressed as continuous functions that are readily handled by successive convexification. Compound state-triggered constraints go a step further, giving designers the ability to compose trigger and constraint conditions using Boolean and and or operations. Simulations of the 6-degree-of-freedom (DoF) powered descent guidance problem obtained using successive convexification are presented to illustrate the utility of state-triggered and compound state-triggered constraints. The examples employ a velocity-triggered angle of attack constraint to alleviate aerodynamic loads, and a collision avoidance constraint to avoid large geological formations. In particular, the velocity-triggered angle of attack constraint demonstrates the ability of state-triggered constraints to introduce new constraint phases to the solution without resorting to combinatorial techniques.

math.OC↗

arXiv2018

Linear Model Regression on Time-series Data: Non-asymptotic Error Bounds and Applications

Data-driven methods for modeling dynamic systems have received considerable attention as they provide a mechanism for control synthesis directly from the observed time-series data. In the absence of prior assumptions on how the time-series had been generated, regression on the system model has been particularly popular. In the linear case, the resulting least squares setup for model regression, not only provides a computationally viable method to fit a model to the data, but also provides useful insights into the modal properties of the underlying dynamics. Although probabilistic estimates for this model regression have been reported, deterministic error bounds have not been examined in the literature, particularly as they pertain to the properties of the underlying system. In this paper, we provide deterministic non-asymptotic error bounds for fitting a linear model to the observed time-series data, with a particular attention to the role of symmetry and eigenvalue multiplicity in the underlying system matrix.

math.OC↗

arXiv2018

Influence Models on Layered Uncertain Networks: A Guaranteed-Cost Design Perspective

Control and estimation on large-scale social networks often necessitate the availability of models for the interactions amongst the agents. However characterizing accurate models of social interactions pose new challenges due to inherent complexity and unpredictability. Moreover, model uncertainty becomes more pronounced for large-scale networks. For certain classes of social networks, the layering structure allows a compositional approach. In this paper, we present such an approach to determine performance guarantees on layered networks with inherent model uncertainties. A factorization method is used to determine robust stability and performance and this is accomplished by a layered cost-guaranteed design via a layered Riccati-type solver, mirroring the network structure. We provide an example of the proposed methodology in the context of opinion dynamics on large-scale social networks.

math.OC↗

arXiv2018

Optimal mass transport and kernel density estimation for state-dependent networked dynamic systems

State-dependent networked dynamical systems are ones where the interconnections between agents change as a function of the states of the agents. Such systems are highly nonlinear, and a cohesive strategy for their control is lacking in the literature. In this paper, we present two techniques pertaining to the density control of such systems. Agent states are initially distributed according to some density, and a feedback law is designed to move the agents to a target density profile. We use optimal mass transport to design a feedforward control law propelling the agents towards this target density. Kernel density estimation, with constraints imposed by the state-dependent dynamics, is then used to allow each agent to estimate the local density of the agents.

math.OC↗

arXiv2018

Bregman Parallel Direction Method of Multipliers for Distributed Optimization via Mirror Averaging

Distributed optimization aims to optimize a global objective formed by a sum of coupled local convex functions over a graph via only local computation and communication. In this paper, we propose the Bregman parallel direction method of multipliers (PDMM) based on a generalized averaging step named mirror averaging. We establish the global convergence and $O(1/T)$ convergence rate of the Bregman PDMM, along with its $O(n/\ln n)$ improvement over existing PDMM, where $T$ denotes the number of iterations and $n$ the dimension of solution variable. In addition, we can enhance its performance by optimizing the spectral gap of the averaging matrix. We demonstrate our results via a numerical example.

math.OC↗

arXiv2018

Controllability Analysis of Threshold Graphs and Cographs

In this paper, we investigate the controllability of a linear time-invariant network following a Laplacian dynamics defined on a threshold graph. In this direction, an algorithm for deriving the modal matrix associated with the Laplacian matrix for this class of graphs is presented. Then, based on the Popov-Belevitch-Hautus criteria, a procedure for the selection of control nodes is proposed. The procedure involves partitioning the nodes of the graph into cells with the same degree; one node from each cell is then selected. We show that the remaining nodes can be chosen as the control nodes rendering the network controllable. Finally, we consider a wider class of graphs, namely cographs, and examine their controllability properties.

math.OC↗

arXiv2017

Controllability and data-driven identification of bipartite consensus on nonlinear signed networks

Nonlinear networked systems are of interest in several areas of research, such as multi-agent systems and social networks. In this paper, we examine the controllability of several classes of nonlinear networked dynamics on which the underlying graph admits negative weights. Such signed networks exhibit bipartite clustering when the underlying graph is structurally balanced. We show that structural balance is the key ingredient inducing uncontrollability when combined with a leader-node symmetry and a certain type of dynamical symmetry. We then examine the problem of extracting the bipartite structure of such graphs from data using Extended Dynamic Mode Decomposition to approximate the corresponding Koopman operator.

math.OC↗

arXiv2017

Optimal Control with Limited Sensing via Empirical Gramians and Piecewise Linear Feedback

This paper is concerned with the design of optimal control for finite-dimensional control-affine nonlinear dynamical systems. We introduce an optimal control problem that specifically optimizes nonlinear observability in addition to ensuring stability of the closed loop system. A recursive algorithm is then proposed to obtain an optimal state feedback controller to maximize the resulting non-quadratic cost functional. The main contribution of the paper is presenting a control synthesis procedure that provides closed loop asymptotic stability, on one hand, and empirical observability of the system, as a transient performance criteria, on the other.

eess.SY↗

arXiv2017

Controllability and Stabilizability Analysis of Signed Consensus Networks

Signed networks have been a topic of recent interest in the network control community as they allow studying antagonistic interactions in multi-agent systems. Although dynamical characteristics of signed networks have been well-studied, notions such as controllability and stabilizability for signed networks for protocols such as consensus are missing in the literature. Classically, graph automorphisms with respect to the input nodes have been used to characterize uncontrollability of consensus networks. In this paper, we show that in addition to the graph symmetry, the topological property of structural balance facilitates the derivation of analogous sufficient conditions for uncontrollability for signed networks. In particular, we provide an analysis which shows that a gauge transformation induced by structural balance allows symmetry arguments to hold for signed consensus networks. Lastly, we use fractional automorphisms to extend our observations to output controllability and stabilizability of signed networks.

math.OC↗

arXiv2017

Large-scale distributed Kalman filtering via an optimization approach

Large-scale distributed systems such as sensor networks, often need to achieve filtering and consensus on an estimated parameter from high-dimensional measurements. Running a Kalman filter on every node in such a network is computationally intensive; in particular the matrix inversion in the Kalman gain update step is expensive. In this paper, we extend previous results in distributed Kalman filtering and large-scale machine learning to propose a gradient descent step for updating an estimate of the error covariance matrix; this is then embedded and analyzed in the context of distributed Kalman filtering. We provide properties of the resulting filters, in addition to a number of applications throughout the paper.

math.OC↗

arXiv2017

Adaptive Communication Networks with Privacy Guarantees

Utilizing the concept of observability, in conjunction with tools from graph theory and optimization, this paper develops an algorithm for network synthesis with privacy guarantees. In particular, we propose an algorithm for the selection of optimal weights for the communication graph in order to maximize the privacy of nodes in the network, from a control theoretic perspective. In this direction, we propose an observability-based design of the communication topology that improves the privacy of the network in presence of an intruder. The resulting adaptive network responds to the intrusion by changing the topology of the network-in an online manner- in order to reduce the information exposed to the intruder.

eess.SY↗

arXiv2016

Growing Controllable Networks via Whiskering and Submodular Optimization

The topology of a network directly influences the behaviour and controllability of dynamical processes on that network. Therefore, the design of network topologies is an important area of research when examining the control of distributed systems. We discuss a method for growing networks known as whiskering, as well as generalizations of this process, and prove that they preserve controllability. We then use techniques from submodular optimization to analyze optimization algorithms for adding new nodes to a network to optimize certain objectives, such as graph connectivity.

math.OC↗

arXiv2016

Patterns of patterns of synchronization: Noise induced attractor switching in rings of coupled nonlinear oscillators

Following the long-lived qualitative-dynamics tradition of explaining behavior in complex systems via the architecture of their attractors and basins, we investigate the patterns of switching between qualitatively distinct trajectories in a network of synchronized oscillators. Our system, consisting of nonlinear amplitude-phase oscillators arranged in a ring topology with reactive nearest neighbor coupling, is simple and connects directly to experimental realizations. We seek to understand how the multiple stable synchronized states connect to each other in state space by applying Gaussian white noise to each of the oscillators' phases. To do this, we first identify a set of locally stable limit cycles at any given coupling strength. For each of these attracting states, we analyze the effect of weak noise via the covariance matrix of deviations around those attractors. We then explore the noise-induced attractor switching behavior via numerical investigations. For a ring of three oscillators we find that an attractor-switching event is always accompanied by the crossing of two adjacent oscillators' phases. For larger numbers of oscillators we find that the distribution of times required to stochastically leave a given state falls off exponentially, and we build an attractor switching network out of the destination states as a coarse-grained description of the high-dimensional attractor-basin architecture.

nlin.AO↗