Preface 1. Introduction Overview A Brief History of LMIs in Control Theory Notes on the Style of the Book Origin of the Book 2. Some Standard Problems Involving LMIs. Linear Matrix Inequalities Some Standard Problems Ellipsoid Algorithm Interior-Point Methods Strict and Nonstrict LMIs Miscellaneous Results on Matrix Inequalities Some LMI Problems with Analytic Solutions 3. Some Matrix Problems. Minimizing Condition Number by Scaling Minimizing Condition Number of a Positive-Definite Matrix Minimizing Norm by Scaling Rescaling a Matrix Positive-Definite Matrix Completion Problems Quadratic Approximation of a Polytopic Norm Ellipsoidal Approximation 4. Linear Differential Inclusions. Differential Inclusions Some Specific LDIs Nonlinear System Analysis via LDIs 5. Analysis of LDIs: State Properties. Quadratic Stability Invariant Ellipsoids 6. Analysis of LDIs: Input/Output Properties. Input-to-State Properties State-to-Output Properties Input-to-Output Properties 7. State-Feedback Synthesis for LDIs. Static State-Feedback Controllers State Properties Input-to-State Properties State-to-Output Properties Input-to-Output Properties Observer-Based Controllers for Nonlinear Systems 8. Lure and Multiplier Methods. Analysis of Lure Systems Integral Quadratic Constraints Multipliers for Systems with Unknown Parameters 9. Systems with Multiplicative Noise. Analysis of Systems with Multiplicative Noise State-Feedback Synthesis 10. Miscellaneous Problems. Optimization over an Affine Family of Linear Systems Analysis of Systems with LTI Perturbations Positive Orthant Stabilizability Linear Systems with Delays Interpolation Problems The Inverse Problem of Optimal Control System Realization Problems Multi-Criterion LQG Nonconvex Multi-Criterion Quadratic Problems Notation List of Acronyms Bibliography Index.

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Linear Matrix Inequalities in System and Control Theory

This paper provides a theoretical framework for analysis of consensus algorithms for multi-agent networked systems with an emphasis on the role of directed information flow, robustness to changes in network topology due to link/node failures, time-delays, and performance guarantees. An overview of basic concepts of information consensus in networks and methods of convergence and performance analysis for the algorithms are provided. Our analysis framework is based on tools from matrix theory, algebraic graph theory, and control theory. We discuss the connections between consensus problems in networked dynamic systems and diverse applications including synchronization of coupled oscillators, flocking, formation control, fast consensus in small-world networks, Markov processes and gossip-based algorithms, load balancing in networks, rendezvous in space, distributed sensor fusion in sensor networks, and belief propagation. We establish direct connections between spectral and structural properties of complex networks and the speed of information diffusion of consensus algorithms. A brief introduction is provided on networked systems with nonlocal information flow that are considerably faster than distributed systems with lattice-type nearest neighbor interactions. Simulation results are presented that demonstrate the role of small-world effects on the speed of consensus algorithms and cooperative control of multivehicle formations

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Consensus and Cooperation in Networked Multi-Agent Systems

We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

Algorithms that must deal with complicated global functions of many variables often exploit the manner in which the given functions factor as a product of "local" functions, each of which depends on a subset of the variables. Such a factorization can be visualized with a bipartite graph that we call a factor graph, In this tutorial paper, we present a generic message-passing algorithm, the sum-product algorithm, that operates in a factor graph. Following a single, simple computational rule, the sum-product algorithm computes-either exactly or approximately-various marginal functions derived from the global function. A wide variety of algorithms developed in artificial intelligence, signal processing, and digital communications can be derived as specific instances of the sum-product algorithm, including the forward/backward algorithm, the Viterbi algorithm, the iterative "turbo" decoding algorithm, Pearl's (1988) belief propagation algorithm for Bayesian networks, the Kalman filter, and certain fast Fourier transform (FFT) algorithms.

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Factor graphs and the sum-product algorithm

Planning algorithms are impacting technical disciplines and industries around the world, including robotics, computer-aided design, manufacturing, computer graphics, aerospace applications, drug design, and protein folding. This coherent and comprehensive book unifies material from several sources, including robotics, control theory, artificial intelligence, and algorithms. The treatment is centered on robot motion planning but integrates material on planning in discrete spaces. A major part of the book is devoted to planning under uncertainty, including decision theory, Markov decision processes, and information spaces, which are the “configuration spaces” of all sensor-based planning problems. The last part of the book delves into planning under differential constraints that arise when automating the motions of virtually any mechanical system. Developed from courses taught by the author, the book is intended for students, engineers, and researchers in robotics, artificial intelligence, and control theory as well as computer graphics, algorithms, and computational biology.

Planning Algorithms: Introductory Material

In a network of cooperating unmanned aerial vehicles (UAVs), individual UAVs usually need to localize themselves in a shared and generally global frame. This paper studies the localization problem for a group of UAVs navigating in three-dimensional space with limited shared information, viz., noisy distance measurements are the only type of interagent sensing that is available, and only one UAV knows its global coordinates, the others being GPS denied. Initially, for a two-agent problem, but easily generalized to some multiagent problems, this paper first establishes constraints on the minimum number of distance measurements required to achieve the localization. This paper then proposes a composite algorithm based on semidefinite programming (SDP) in a first step, followed by maximum likelihood estimation using gradient descent on a manifold initialized by the SDP calculation. The efficacy of the algorithm is verified with experimental noisy flight data.

3-D Relative Localization of Mobile Systems Using Distance-Only Measurements via Semidefinite Optimization

Frequency-weighted balancing related controller reduction

The exponential convergence of certain algorithms used in continuous-time adaptive identification and control is investigated by deriving persistence of excitation conditions which depend on the system inputs exclusively. Stable plants and unstable plants with external stabilizing control are considered in the identification study. For the control algorithm considered, it is shown that in the absence of foreknowledge of a gain parameter, exponential stability is impossible to claim.

Input conditions for continuous-time adaptive system problems

Persistent autonomous formations and cohesive motion control

In factor analysis, which is used for example in econometrics, by definition the number of latent variables has to exceed the number of factor variables. The associated transfer function matrix has more rows than columns, and when the factor variables are independent zero mean white noise sequences and the transfer function matrix is stable, then the output spectrum is singular. While a related paper focusses on the properties of such a nonsquare transfer function matrix, in this paper, we explore a number of properties of the spectral matrix and associated covariance sequence. In particular, a zero free minimum degree spectral factor can be computed with a finite number of rational calculations from the spectrum (in contrast to typical spectral factor calculations), assuming the spectrum fulfills a generic condition. Application of the result to Kalman filtering is indicated, and presentation of the results is also achieved using finite block Toeplitz matrices with entries obtained from the covariance of the latent variable vector.

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Brian D. O. Anderson

Papers

3-D Relative Localization of Mobile Systems Using Distance-Only Measurements via Semidefinite Optimization

Frequency-weighted balancing related controller reduction

Input conditions for continuous-time adaptive system problems

Persistent autonomous formations and cohesive motion control

Properties of Zero-Free Spectral Matrices