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

The fundamental requirements for low-sensitivity analog filter structures are shown to be the bounded-real (BR) property of the transfer function and its implementation by a structure that "preserves" the BR property for incremental changes in the parameters of the network. Using these properties, a low-sensitivity realization of any stable BR analog transfer function is developed. The final structure is in the form of a singly constrained analog active RC "two-pair" implemented as a cascade of two-pairs where each two-pair is characterized by a lossless bounded-real (LBR) transfer matrix. The proposed theory is shown to include the class of low sensitivity wave active RC structures related to the doubly terminated lossless two-ports. The new theory has been verified by computer simulation.

A general theory and synthesis procedure for low-sensitivity active RC filters

In this paper we introduce a polynomial method for addressing sensor network localization problems when the inter-sensor measurements are noisy. We compare the results obtained applying this method with the results obtained by other localization methods in the literature. Later in the paper we propose tools from algebraic geometry to aid us solve the problem in a more computationally appealing way.

Polynomial methods in noisy network localization

Systems theory descriptions alternative to those already known are presented for transfer function matrices which are positive real. They are derived using the fact that the transpose of a positive real matrix is itself positive real.

Dual form of a positive real lemma

This paper reviews a number of very recent results in rigid graph theory and their extension for directed graphs to persistence theory, with an application focus on the cooperative control of formations. Particular attention is paid to issues related to the merging of formations, where the internal structure of each of the individually merging formations is, to the maximum extent possible, downplayed in the calculation. The meta-formation framework is then introduced in light of merging process for its construction. The ideas also have application to sensor network localization, where there is potential to make great computational saving.

Use of meta-formations for cooperative control

The main contributions presented are (i) to widen the classes of model sets for which necessary and sufficient conditions for uncertainty model unfalsification can be obtained, and (ii) to display the effect of different assumptions concerning the modeling error on the curves defining the boundary of unfalsified models (the optimal uncertainty tradeoff curve) for the same underlying data set.

Brian D. O. Anderson

Papers

A general theory and synthesis procedure for low-sensitivity active RC filters

Polynomial methods in noisy network localization

Dual form of a positive real lemma

Use of meta-formations for cooperative control

Uncertainty model unfalsification