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

0. Introduction Manifold topology enjoyed a golden age in the late 1950's and 1960's. Of the mysteries still remaining after that period of great success the most compelling seemed to lie in dimensions three and four. Although experience suggested that manifold theory at these dimensions has a distinct character, the dream remained since my graduate school days that some key principle from the high dimensional theory would extend, at least to dimension four, and bring with it the beautiful adherence of topology to algebra familiar in dimensions greater than or equal to five. There is such a principle. It is a homotopy theoretic criterion for imbedding (relatively) a topological 2-handle in a smooth four-dimensional manifold with boundary. The main impact, as outlined in §1, is to the classification of 1-connected 4-manifolds and topological end recognition. However, certain applications to nonsimply connected problems such as knot concordance are also obtained. The discovery of this principle was made in three stages. From 1973 to 1975 Andrew Casson developed his theory of "flexible handles". These are certain pairs having the proper homotopy type of the common place open 2-handle H = (D X D, dD X D) but "flexible" in the sense that finding imbeddings is rather easy; in fact imbedding is implied by a homotopy theoretic criterion. It was clear to Casson that: (1) no known invariant—link theoretic

/pdf/the-topology-of-four-dimensional-manifolds-1wouaj15sy.pdf

The topology of four-dimensional manifolds

4-manifolds: Introduction Surfaces in 4-manifolds Complex surfaces Kirby calculus: Handelbodies and Kirby diagrams Kirby calculus More examples Applications: Branched covers and resolutions Elliptic and Lefschetz fibrations Cobordisms, $h$-cobordisms and exotic ${\mathbb{R}}^{4,}$s Symplectic 4-manifolds Stein surfaces Appendices: Solutions Notation, important figures Bibliography Index.

4-manifolds and Kirby calculus

The analysis and design of nonlinear feedback systems has recently undergone an exceptionally rich period of progress and maturation, fueled, to a great extent, by (1) the discovery of certain basic conceptual notions, and (2) the identification of classes of systems for which systematic decomposition approaches can result in effective and easily computable control laws. These two aspects are complementary, since the latter approaches are, typically, based upon the inductive verification of the validity of the former system properties under compositions (in the terminology used in [62], the “activation” of theoretical concepts leads to “constructive” control). This expository presentation addresses the first of these aspects, and in particular the precise formulation of questions of robustness with respect to disturbances, formulated in the paradigm of input to state stability. We provide an intuitive and informal presentation of the main concepts. More precise statements, especially about older results, are given in the cited papers, as well as in several previous surveys such as [103] and [105] (of which the present paper represents an update), but we provide a little more detail about relatively recent work. Regarding applications and extensions of the basic framework, we give some pointers to the literature, but we do not focus on feedback design and specific engineering problems; for the latter we refer the reader to textbooks such as [43], [60], [58], [96], [66], [27], [44].

Input to State Stability: Basic Concepts and Results

This article has presented an introductory tutorial on discontinuous dynamical systems. Various examples illustrate the pertinence of the continuity and Lipschitzness properties that guarantee the existence and uniqueness of classical solutions to ordinary differential equations. The lack of these properties in examples drawn from various disciplines motivates the need for more general notions than the classical one. First, we introduced notions of solution for discontinuous systems. Second, we reviewed the available tools from non- smooth analysis to study the gradient information of candidate Lyapunov functions. And, third, we presented nonsmooth stability tools to characterize the asymptotic behavior of solutions.

Discontinuous dynamical systems

Convex Sets. Matrix Theory Background. Differential and Control System Preliminaries. Exponentially Nonnegative Matrices. Extended M-Matrices. Cone Reachability. Applications to Feedback Control. Controllability, Observability, and Realizability of Positive Control Systems. References. Index.

Nonnegative matrices in dynamic systems

In this paper we investigate the relationship between isotopy classes of knots and links in S and the dieomorphism types of homeomorphic smooth 4-manifolds. As a corollary of this initial investigation, we begin to uncover the surprisingly rich structure of dieomorphism types of manifolds homeomorphic to the K3 surface. In order to state our theorems we need to view the Seiberg-Witten invariant of a smooth 4-manifold as a multivariable (Laurent) polynomial. To do this, recall that the Seiberg-Witten invariant of a smooth closed oriented 4-manifold X with b2 X  > 1 is an integer valued function which is de®ned on the set of spinc structures over X , (cf. [W], [KM], [Ko1], [T1]). In case H1X ;Z has no 2-torsion (which will be the situation in this paper) there is a natural identi®cation of the spinc structures of X with the characteristic elements of H2X ;Z. In this case we view the Seiberg-Witten invariant as

Knots, links, and 4-manifolds

In this paper we introduce a surgical procedure, called a rational blowdown, for a smooth 4-manifold X and determine how this procedure affects both the Donaldson and Seiberg-Witten invariants of X.

/pdf/rational-blowdowns-of-smooth-4-manifolds-3r79troqtr.pdf

Rational blowdowns of smooth 4-manifolds

Knots, Links, and 4-Manifolds

For an oriented integral homology 3-sphere 2, A. Casson has introduced an integer invariant A(2) that is defined by using the space £%(2) of conjugacy classes of irreducible representations of ^ ( 2 ) into SU(2) (see [1]). This invariant A(2) can be computed from a surgery or Heegard description of 2 and satisfies A(2) = JU(2) (mod 2), where /i(2) is the Kervaire-Milnor-Rochlin invariant of 2. This powerful new invariant was used to settle an outstanding problem in 3-manifold topology; namely, showing that if 2 is a homotopy 3-sphere, then ju(2) = O. C. Taubes [17], utilizing gauge-theoretic considerations, has reinterpreted Casson's invariant in terms of flat connections. Refining this approach, A. Floer [13] has recently defined another invariant of 2 , its 'instanton homology', which takes the form of an abelian group /*(2) with a natural Z8-grading that is an enhancement of A(2) in that

Ronald J. Stern

Papers

Nonnegative matrices in dynamic systems

Knots, links, and 4-manifolds

Rational blowdowns of smooth 4-manifolds

Knots, Links, and 4-Manifolds

Instanton Homology of Seifert Fibred Homology Three Spheres