Control Theory for Linear Systems deals with the mathematical theory of feedback control of linear systems. It treats a wide range of control synthesis problems for linear state space systems with inputs and outputs. The book provides a treatment of these problems using state space methods, often with a geometric flavour. Its subject matter ranges from controllability and observability, stabilization, disturbance decoupling, and tracking and regulation, to linear quadratic regulation, $H_2$ and $H_\infty$ control, and robust stabilization. Each chapter of the book contains a series of exercises, intended to increase the reader's understanding of the material. Often, these exercises generalize and extend the material treated in the regular text.

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Control theory for linear systems

Given a network of interconnected nodes, each with its own value (such as a measurement, position, vote, or other data), we develop a distributed strategy that enables some or all of the nodes to calculate any arbitrary function of the node values, despite the actions of malicious nodes in the network. Our scheme assumes a broadcast model of communication (where all nodes transmit the same value to all of their neighbors) and utilizes a linear iteration where, at each time-step, each node updates its value to be a weighted average of its own previous value and those of its neighbors. We consider a node to be malicious or faulty if, instead of following the predefined linear strategy, it updates its value arbitrarily at each time-step (perhaps conspiring with other malicious nodes in the process). We show that the topology of the network completely characterizes the resilience of linear iterative strategies to this kind of malicious behavior. First, when the network contains 2f or fewer vertex-disjoint paths from some node xj to another node xi , we provide an explicit strategy for f malicious nodes to follow in order to prevent node xi from receiving any information about xj's value. Next, if node xi has at least 2f+1 vertex-disjoint paths from every other (non-neighboring) node, we show that xi is guaranteed to be able to calculate any arbitrary function of all node values when the number of malicious nodes is f or less. Furthermore, we show that this function can be calculated after running the linear iteration for a finite number of time-steps (upper bounded by the number of nodes in the network) with almost any set of weights (i.e., for all weights except for a set of measure zero).

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Distributed Function Calculation via Linear Iterative Strategies in the Presence of Malicious Agents

In part I of this paper [S. Chen, X. Li, and X. Zhou, SIAM J. Control Optim., 36 (1998), pp. 1685--1702], an optimization model of stochastic linear quadratic regulators (LQRs) with indefinite control cost weighting matrices is proposed and studied. In this sequel, the problem of solving LQR models with system diffusions dependent on both state and control variables, which is left open in part I, is tackled. First, the solvability of the associated stochastic Riccati equations (SREs) is studied in the normal case (namely, all the state and control weighting matrices and the terminal matrix in the cost functional are nonnegative definite, with at least one positive definite), which in turn leads to an optimal state feedback control of the LQR problem. In the general indefinite case, the problem is decomposed into two optimal LQR problems, one with a forward dynamics and the other with a backward dynamics. The well-posedness and solvability of the original LQR problem are then obtained by solving these two subproblems, and an optimal control is explicitly constructed. Examples are presented to illustrate the results.

Stochastic Linear Quadratic Regulators with Indefinite Control Weight Costs. II

This report contains a survey on recent results concerning solutions of the second order nonlinear differential equation $x'' + a(t)|x|^\gamma \operatorname{sgn} x = 0$, $\gamma > 0$, where $a(t)$ is continuous and nonnegative. Specifically, the questions concerning continuability, boundedness, stability, oscillation, asymptotic growth and boundary value problems are discussed. An extensive list of related references is also included.

On the Generalized Emden–Fowler Equation

Strong detectability and observers

System structure and singular control

The output-stabilizable subspace and linear optimal control A.H. W. Geerts & M.L.J. Hautus Properties of a certain subspace are linked to well-known problems in system theory.

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The output-stabilizable subspace and linear optimal control

In this paper realization algorithms for systems over a principal ideal domain are described This is done using the Smith form or a modified Hermite form for matrices over a principal ideal domain It is shown that Ho's algorithm and an algorithm due to Zeiger can be generalized to the ring case Also a recursive realization algorithm, including some results concerning the partial realization problem, is presented Applications to systems over the integers, delay differential systems and two-dimensional systems are discussed

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Realization algorithms for systems over a principal ideal domain

https://pure.tue.nl/ws/files/1812961/339165.pdf

Mlj Malo Hautus

Papers

Strong detectability and observers

System structure and singular control

The output-stabilizable subspace and linear optimal control

Realization algorithms for systems over a principal ideal domain

Asymptotic properties of matrix differential operators