There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

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.

/pdf/linear-matrix-inequalities-in-system-and-control-theory-2q5cf0pqm2.pdf

Linear Matrix Inequalities in System and Control Theory

Variable structure systems consist of a set of continuous subsystems together with suitable switching logic. Advantageous properties result from changing structures according to this switching logic. Design and analysis for this class of systems are surveyed in this paper.

Variable structure systems with sliding modes

1. Introduction. 2. Linear Algebra. 3. Linear Systems. 4. H2 and Ha Spaces. 5. Internal Stability. 6. Performance Specifications and Limitations. 7. Balanced Model Reduction. 8. Uncertainty and Robustness. 9. Linear Fractional Transformation. 10. m and m- Synthesis. 11. Controller Parameterization. 12. Algebraic Riccati Equations. 13. H2 Optimal Control. 14. Ha Control. 15. Controller Reduction. 16. Ha Loop Shaping. 17. Gap Metric and ...u- Gap Metric. 18. Miscellaneous Topics. Bibliography. Index.

/pdf/essentials-of-robust-control-3kaks6yk3h.pdf

Essentials of Robust Control

The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

A variable structure decentralized controller for dynamic routing in multi-destination large-scale data communication networks is presented. A dynamic model, which can incorporate different processing delays at different nodes, is developed to describe the network dynamics. It is shown that the proposed controller guarantees stability and clears the queues of the system in the absence of external input flows. The controller also keeps the queue lengths bounded in the presence of external flow rates, which do not exceed a certain maximum flow rate (obtained by solving a LP optimization problem - there is no solution to the routing problem if these rates are exceeded). Furthermore, the controller is also robust with respect to variations in the processing delay constants. Some simulation results are presented to illustrate the controller's performance in a number of practical cases.

https://ieeexplore.ieee.org/iel5/4790678/4790679/04790775.pdf

Decentralized Robust Control for Dynamic Routing of Large Scale Networks

Fundamental limitations for error tracking/regulation are obtained for the robust servomechanism problem (RSP) for a sampled system. In studying this problem, the cheap control problem for a multi-input/multi-output discrete time system is considered, and explicit expressions are obtained for the limiting steady state solution of its associated algebraic Riccati equation (ARE), as the weight on the control energy tends to zero. Application of these results is then made to obtain explicit expressions for the limiting performance costs associated with error tracking/regulation in the RSP. These limitations can be characterized by the system order, the dimension of the outputs, the number of the system's transmission zeros and the location of the system's nonminimum phase transmission zeros.

Performance limitations in the robust servomechanism problem for discrete-time LTI systems

Real stability radii of linear time-invariant time-delay systems

An arbitrary interconnected network of intersecting streets with one-way traffic flow is considered in which some or all intersections are assumed to be operating at maximum capacity. In this case, it is desired to find a robust decentralized controller at each traffic intersection, so that for all perturbed external traffic flow sources/sinks, the resulting perturbed queues at each intersection are asymptotically "balanced". The only local knowledge required of the controllers at each intersection is that re the queue lengths of the intersection. It is shown that the necessary and sufficient conditions for a solution to the problem to exist depend solely on the topology of the network and the percentage of vehicles turning at each intersection, and are identical to the conditions required for a solution to exist for the centralized controller case. In particular, it is shown that if turns always occur at an intersection, then a solution to the problem always exists. A characterization of controllers which solve the problem are then given. In particular, it is shown that the local controllers which solve the problem are identical to each other, which implies that if the network is expanded (assuming a solution exists) then one can use the same local controllers to solve the problem. It is also shown that the local controllers can, in principle, give "perfect control" to the problem. Some examples are included to illustrate the results.

Decentralized control of traffic networks

A decentralized controller for dynamic routing in multidestination data-communication networks is presented. A dynamic model, which can incorporate arbitrary, differing processing delays at different nodes, is developed to describe the network dynamics. The structure of the controller is motivated by an optimal control problem. It is shown that the controller has many desirable properties; it does not direct messages around a loop, and, in the absence of external input flows, it guarantees that the queues are cleared. Some simulation results are presented to illustrate the controller performance in a number of practical cases. They verify that the queue lengths remain reasonable when the input flow rates are moderate compared to the link capacities. The control strategy can also cope easily with modifications, expansions, and failures in the network. >

Edward J. Davison

Papers

Decentralized Robust Control for Dynamic Routing of Large Scale Networks

Performance limitations in the robust servomechanism problem for discrete-time LTI systems

Real stability radii of linear time-invariant time-delay systems

Decentralized control of traffic networks

A decentralized control strategy for dynamic routing