arXiv subjects

Mehran Mesbahi

Publications and source records attributed to Mehran Mesbahi.

At least 55 records · Page 3Linked to original sources

arXiv2020

Graph-theoretic optimization for edge consensus

We consider network structures that optimize the $\mathcal{H}_2$ norm of weighted, time scaled consensus networks, under a minimal representation of such consensus networks described by the edge Laplacian. We show that a greedy algorithm can be used to find the minimum-$\mathcal{H}_2$ norm spanning tree, as well as how to choose edges to optimize the $\mathcal{H}_2$ norm when edges are added back to a spanning tree. In the case of edge consensus with a measurement model considering all edges in the graph, we show that adding edges between slow nodes in the graph provides the smallest increase in the $\mathcal{H}_2$ norm.

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arXiv2020

A Note on Nesterov's Accelerated Method in Nonconvex Optimization: a Weak Estimate Sequence Approach

We present a variant of accelerated gradient descent algorithms, adapted from Nesterov's optimal first-order methods, for weakly-quasi-convex and weakly-quasi-strongly-convex functions. We show that by tweaking the so-called estimate sequence method, the derived algorithm achieves optimal convergence rate for weakly-quasi-convex and weakly-quasi-strongly-convex in terms of oracle complexity. In particular, for a weakly-quasi-convex function with Lipschitz continuous gradient, we require $O(\frac{1}{\sqrt{\varepsilon}})$ iterations to acquire an $\varepsilon$-solution; for weakly-quasi-strongly-convex functions, the iteration complexity is $O\left( \ln\left(\frac{1}{\varepsilon}\right) \right)$. Furthermore, we discuss the implications of these algorithms for linear quadratic optimal control problem.

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arXiv2020

Policy Gradient-based Algorithms for Continuous-time Linear Quadratic Control

We consider the continuous-time Linear-Quadratic-Regulator (LQR) problem in terms of optimizing a real-valued matrix function over the set of feedback gains. The results developed are in parallel to those in Bu et al. [1] for discrete-time LTI systems. In this direction, we characterize several analytical properties (smoothness, coerciveness, quadratic growth) that are crucial in the analysis of gradient-based algorithms. We also point out similarities and distinctive features of the continuous time setup in comparison with its discrete time analogue. First, we examine three types of well-posed flows direct policy update for LQR: gradient flow, natural gradient flow and the quasi-Newton flow. The coercive property of the corresponding cost function suggests that these flows admit unique solutions while the gradient dominated property indicates that the underling Lyapunov functionals decay at an exponential rate; quadratic growth on the other hand guarantees that the trajectories of these flows are exponentially stable in the sense of Lyapunov. We then discuss the forward Euler discretization of these flows, realized as gradient descent, natural gradient descent and quasi-Newton iteration. We present stepsize criteria for gradient descent and natural gradient descent, guaranteeing that both algorithms converge linearly to the global optima. An optimal stepsize for the quasi-Newton iteration is also proposed, guaranteeing a $Q$-quadratic convergence rate--and in the meantime--recovering the Kleinman-Newton iteration. Lastly, we examine LQR state feedback synthesis with a sparsity pattern. In this case, we develop the necessary formalism and insights for projected gradient descent, allowing us to guarantee a sublinear rate of convergence to a first-order stationary point.

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arXiv2020

Strong Structural Controllability of Networks under Time-Invariant and Time-Varying Topological Perturbations

This paper investigates the robustness of strong structural controllability for linear time-invariant and linear time-varying directed networks with respect to structural perturbations, including edge deletions and additions. In this direction, we introduce a new construct referred to as a perfect graph associated with a network with a given set of control nodes. The tight upper bounds on the number of edges that can be added to, or removed from a network, while ensuring strong structural controllability, are then derived. Moreover, we obtain a characterization of critical edge-sets, the maximal sets of edges whose any subset can be respectively added to, or removed from a network, while preserving strong structural controllability. In addition, procedures for combining networks to obtain strongly structurally controllable network-of-networks are proposed. Finally, controllability conditions are proposed for networks whose edge weights, as well as their structures, can vary over time.

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arXiv2020

Laplacian Dynamics on Cographs: Controllability Analysis through Joins and Unions

In this paper, we examine the controllability of Laplacian dynamic networks on cographs. Cographs appear in modeling a wide range of networks and include as special instances, the threshold graphs. In this work, we present necessary and sufficient conditions for the controllability of cographs, and provide an efficient method for selecting a minimal set of input nodes from which the network is controllable. In particular, we define a sibling partition in a cograph and show that the network is controllable if all nodes of any cell of this partition except one are chosen as control nodes. The key ingredient for such characterizations is the intricate connection between the modularity of cographs and their modal properties. Finally, we use these results to characterize the controllability conditions for certain subclasses of cographs.

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arXiv2020

Data-driven parameterizations of suboptimal LQR and H2 controllers

In this paper we design suboptimal control laws for an unknown linear system on the basis of measured data. We focus on the suboptimal linear quadratic regulator problem and the suboptimal H2 control problem. For both problems, we establish conditions under which a given data set contains sufficient information for controller design. We follow up by providing a data-driven parameterization of all suboptimal controllers. We will illustrate our results by numerical simulations, which will reveal an interesting trade-off between the number of collected data samples and the achieved controller performance.

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arXiv2020

Efficient Computation of H2 Performance on Series-Parallel Networks

Series-parallel networks are a class of graphs on which many NP-hard problems have tractable solutions. In this paper, we examine performance measures on leader-follower consensus on series-parallel networks. We show that a distributed computation of the $\mathcal{H}_2$ norm can be done efficiently on this system by exploiting a decomposition of the network into atomic elements and composition rules. Lastly, we examine the problem of adaptively re-weighting the network to optimize the $\mathcal{H}_2$ norm, and show that it can be done with similar complexity.

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arXiv2020

Nonlinear Observability via Koopman Analysis: Characterizing the Role of Symmetry

This paper considers the observability of nonlinear systems from a Koopman operator theoretic perspective--and in particular--the effect of symmetry on observability. We first examine an infinite-dimensional linear system (constructed using independent Koopman eigenfunctions) such that its observability is equivalent to the observability of the original nonlinear system. Next, we derive an analytic relation between symmetry and nonlinear observability; it is shown that symmetry in the nonlinear dynamics is reflected in the symmetry of the corresponding Koopman eigenfunctions, as well as presence of repeated Koopman eigenvalues. We then proceed to show that the loss of observability in symmetric nonlinear systems can be traced back to the presence of these repeated eigenvalues. In the case where we have a sufficient number of measurements, the nonlinear system remains unobservable when these functions have symmetries that mirror those of the dynamics. The proposed observability framework provides insights into the minimum number of the measurements needed to make an unobservable nonlinear system, observable. The proposed results are then applied to a network of nano-electromechanical oscillators coupled via a symmetric interaction topology.

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arXiv2020

Global Convergence of Policy Gradient Algorithms for Indefinite Least Squares Stationary Optimal Control

We consider policy gradient algorithms for the indefinite least squares stationary optimal control, e.g., linear-quadratic-regulator (LQR) with indefinite state and input penalization matrices. Such a setup has important applications in control design with conflicting objectives, such as linear quadratic dynamic games. We show the global convergence of gradient, natural gradient and quasi-Newton policies for this class of indefinite least squares problems.

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arXiv2020

Consensus on Matrix-weighted Time-varying Networks

This paper examines the consensus problem on time-varying matrix-weighed undirected networks. First, we introduce the matrix-weighted integral network for the analysis of such networks. Under mild assumptions on the switching pattern of the time-varying network, necessary and/or sufficient conditions for which average consensus can be achieved are then provided in terms of the null space of matrix-valued Laplacian of the corresponding integral network. In particular, for periodic matrix-weighted time-varying networks, necessary and sufficient conditions for reaching average consensus is obtained from an algebraic perspective. Moreover, we show that if the integral network with period $T>0$ has a positive spanning tree over the time span $[0,T)$, average consensus for the node states is achieved. Simulation results are provided to demonstrate the theoretical analysis.

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arXiv2020

On the Controllability of Matrix-weighted Networks

This letter examines the controllability of consensus dynamics on matrix-weighed networks from a graph-theoretic perspective. Unlike the scalar-weighted networks, the rank of weight matrix introduces additional intricacies into characterizing the dimension of controllable subspace for such networks. Specifically, we investigate how the definiteness of weight matrices influences the dimension of the controllable subspace. In this direction, graph-theoretic characterizations of the lower and upper bounds on the dimension of the controllable subspace are provided by employing, respectively, distance partition and almost equitable partition of matrix-weighted networks. Furthermore, the structure of an uncontrollable input for such networks is examined. Examples are then provided to demonstrate the theoretical results.

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arXiv2019

Global Convergence of Policy Gradient for Sequential Zero-Sum Linear Quadratic Dynamic Games

We propose projection-free sequential algorithms for linear-quadratic dynamics games. These policy gradient based algorithms are akin to Stackelberg leadership model and can be extended to model-free settings. We show that if the leader performs natural gradient descent/ascent, then the proposed algorithm has a global sublinear convergence to the Nash equilibrium. Moreover, if the leader adopts a quasi-Newton policy, the algorithm enjoys a global quadratic convergence. Along the way, we examine and clarify the intricacies of adopting sequential policy updates for LQ games, namely, issues pertaining to stabilization, indefinite cost structure, and circumventing projection steps.

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arXiv2019

Strong Structural Controllability of Signed Networks

In this paper, we discuss the controllability of a family of linear time-invariant (LTI) networks defined on a signed graph. In this direction, we introduce the notion of positive and negative signed zero forcing sets for the controllability analysis of positive and negative eigenvalues of system matrices with the same sign pattern. A sufficient combinatorial condition that ensures the strong structural controllability of signed networks is then proposed. Moreover, an upper bound on the maximum multiplicity of positive and negative eigenvalues associated with a signed graph is provided.

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arXiv2019

Time scale design for network resilience

In this paper we consider the $\mathcal{H}_2$-norm of networked systems with multi-time scale consensus dynamics. We develop a general framework for such systems that allows for edge weighting, independent agent-based time scales, as well as measurement and process noise. From this general system description, we highlight an interesting case where the influences of the weighting and scaling can be separated in the design problem. We then consider the design of the time scale parameters for minimizing the $\mathcal{H}_2$-norm for the purpose of network resilience.

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arXiv2019

Augmented State Feedback for Improving Observability of Linear Systems with Nonlinear Measurements

This paper is concerned with the design of an augmented state feedback controller for finite-dimensional linear systems with nonlinear observation dynamics. Most of the theoretical results in the area of (optimal) feedback design are based on the assumption that the state is available for measurement. In this paper, we focus on finding a feedback control that avoids state trajectories with undesirable observability properties. In particular, we introduce an optimal control problem that specifically considers an index of observability in the control synthesis. The resulting cost functional is a combination of LQR-like quadratic terms and an index of observability. The main contribution of the paper is presenting a control synthesis procedure that on one hand, provides closed loop asymptotic stability, and addresses the observability of the system--as a transient performance criteria--on the other.

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arXiv2019

Mass-spring-damper Networks for Distributed Optimization in Non-Euclidean Spaces

We consider the problem of minimizing the sum of non-smooth convex functions in non-Euclidean spaces, e.g., probability simplex, via only local computation and communication on an undirected graph. We propose two algorithms motivated by mass-spring-damper dynamics. The first algorithm uses an explicit update that computes subgradients and Bregman projections only, and matches the convergence behavior of centralized mirror descent. The second algorithm uses an implicit update that solves a penalized subproblem at each iteration, and achieves the iteration complexity of $\mathcal{O}(1/K)$. The results are also demonstrated via numerical examples.

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arXiv2019

LQR through the Lens of First Order Methods: Discrete-time Case

We consider the Linear-Quadratic-Regulator (LQR) problem in terms of optimizing a real-valued matrix function over the set of feedback gains. Such a setup facilitates examining the implications of a natural initial-state independent formulation of LQR in designing first order algorithms. It is shown that this cost function is smooth and coercive, and provide an alternate means of noting its gradient dominated property. In the process, we provide a number of analytic observations on the LQR cost when directly analyzed in terms of the feedback gain. We then examine three types of well-posed flows for LQR: gradient flow, natural gradient flow and the quasi-Newton flow. The coercive property suggests that these flows admit unique solutions while gradient dominated property indicates that the corresponding Lyapunov functionals decay at an exponential rate; we also prove that these flows are exponentially stable in the sense of Lyapunov. We then discuss the forward Euler discretization of these flows, realized as gradient descent, natural gradient descent and the quasi-Newton iteration. We present stepsize criteria for gradient descent and natural gradient descent, guaranteeing that both algorithms converge linearly to the global optima. An optimal stepsize for the quasi-Newton iteration is also proposed, guaranteeing a $Q$-quadratic convergence rate--and in the meantime--recovering the Hewer algorithm.

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arXiv2019

Fast Trajectory Optimization via Successive Convexification for Spacecraft Rendezvous with Integer Constraints

In this paper we present a fast method based on successive convexification for generating fuel-optimized spacecraft rendezvous trajectories in the presence of mixed-integer constraints. A recently developed paradigm of state-triggered constraints allows to efficiently embed a subset of discrete decision constraints into the continuous optimization framework of successive convexification. As a result, we are able to solve difficult trajectory optimization problems at interactive speeds, as opposed to a mixed-integer programming approach that would require significantly more solution time and computing power. Our method is applied to the real problem of transposition and docking of the Apollo command and service module with the lunar module. We demonstrate that, within seconds, we are able to obtain trajectories that are up to 90 percent more fuel efficient (saving up to 45 kg of fuel) than non-optimization based Apollo-era design targets. Our trajectories take explicit account of minimum thrust pulse width and plume impingement constraints. Both of these constraints are naturally mixed-integer, but we handle them as state-triggered constraints. In its current state, our algorithm will serve as a useful off-line design tool for rapid trajectory trade studies.

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