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Linear system

About: Linear system is a research topic. Over the lifetime, 59550 publications have been published within this topic receiving 1432557 citations.

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01 Jan 1994
TL;DR: In this paper, the authors present a brief history of LMIs in control theory and discuss some of the standard problems involved in LMIs, such as linear matrix inequalities, linear differential inequalities, and matrix problems with analytic solutions.
Abstract: 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.

11,085 citations

Journal ArticleDOI
TL;DR: An iterative method for solving linear systems, which has the property of minimizing at every step the norm of the residual vector over a Krylov subspace.
Abstract: We present an iterative method for solving linear systems, which has the property of minimizing at every step the norm of the residual vector over a Krylov subspace. The algorithm is derived from t...

10,907 citations

Journal ArticleDOI
TL;DR: An iterative algorithm is given for solving a system Ax=k of n linear equations in n unknowns and it is shown that this method is a special case of a very general method which also includes Gaussian elimination.
Abstract: An iterative algorithm is given for solving a system Ax=k of n linear equations in n unknowns. The solution is given in n steps. It is shown that this method is a special case of a very general method which also includes Gaussian elimination. These general algorithms are essentially algorithms for finding an n dimensional ellipsoid. Connections are made with the theory of orthogonal polynomials and continued fractions.

7,598 citations

01 Jan 1980

6,837 citations

05 Apr 1977
TL;DR: In this paper, Liapunov functional for autonomous systems is used to define the saddle point property near equilibrium and periodic orbits for linear systems, which is a generalization of the notion of stable D operators.
Abstract: 1 Linear differential difference equations.- 1.1 Differential and difference equations.- 1.2 Retarded differential difference equations.- 1.3 Exponential estimates of x(?, f).- 1.4 The characteristic equation.- 1.5 The fundamental solution.- 1.6 The variation-of-constants formula.- 1.7 Neutral differential difference equations.- 1.8 Supplementary remarks.- 2 Retarded functional differential equations : basic theory.- 2.1 Definition.- 2.2 Existence, uniqueness, and continuous dependence.- 2.3 Continuation of solutions.- 2.4 Differentiability of solutions.- 2.5 Backward continuation.- 2.6 Caratheodory conditions.- 2.7 Supplementary remarks.- 3 Properties of the solution map.- 3.1 Finite- or infinite-dimensional problem?.- 3.2 Equivalence classes of solutions.- 3.3 Exponential decrease for linear systems.- 3.4 Unique backward extensions.- 3.5 Range in ?n.- 3.6 Compactness and representation.- 3.7 Supplementary remarks.- 4 Autonomous and periodic processes.- 4.1 Processes.- 4.2 Invariance.- 4.3 Discrete systems-maximal compact invariant sets.- 4.4 Fixed points of discrete dissipative processes.- 4.5 Stability and maximal invariant sets in processes.- 4.6 Periodic trajectories of ?-periodic processes.- 4.7 Convergent systems.- 4.8 Supplementary remarks.- 5 Stability theory.- 5.1 Definitions.- 5.2 The method of Liapunov functional.- 5.3 Liapunov functional for autonomous systems.- 5.4 Razumikhin-type theorems.- 5.5 Supplementary remarks.- 6 General linear systems.- 6.1 Global existence and exponential estimates.- 6.2 Variation-of-constants formula.- 6.3 The formal adjoint equation.- 6.4 The true adjoint.- 6.5 Boundary-value problems.- 6.6 Stability and boundedness.- 6.7 Supplementary remarks.- 7 Linear autonomous equations.- 7.1 The semigroup and infinitesimal generator.- 7.2 Spectrum of the generator-decomposition of C.- 7.3 Decomposing C with the formal adjoint equation.- 7.4 Estimates on the complementary subspace.- 7.5 An example.- 7.6 The decomposition in the variation-of-constants formula.- 7.7 Supplementary remarks.- 8 Linear periodic systems.- 8.1 General theory.- 8.2 Decomposition.- 8.3 Supplementary remarks.- 9 Perturbed linear systems.- 9.1 Forced linear systems.- 9.2 Bounded, almost-periodic, and periodic solutions stable and unstable manifolds.- 9.3 Periodic solutions-critical cases.- 9.4 Averaging.- 9.5 Asymptotic behavior.- 9.6 Boundary-value problems.- 9.7 Supplementary remarks.- 10 Behavior near equilibrium and periodic orbits for autonomous equations.- 10.1 The saddle-point property near equilibrium.- 10.2 Nondegenerate periodic orbits.- 10.3 Hyperbolic periodic orbits.- 10.4 Supplementary remarks.- 11 Periodic solutions of autonomous equations.- 11.1 Hopf bifurcation.- 11.2 A periodicity theorem.- 11.3 Range of the period.- 11.4 The equation $$\dot x(t) = - \alpha x(t - 1)[1 + x(t)]$$.- 11.5 The equation $$\dot x(t) = - \alpha x(t - 1)[1 - {x^2}(t)]$$.- 11.6 The equation $$\ddot x(t) + f(x(t))\dot x(t) + g(x(t - r)) = 0$$.- 11.7 Supplementary remarks.- 12 Equations of neutral type.- 12.1 Definition of a neutral equation.- 12.2 Fundamental properties.- 12.3 Linear autonomous D operators.- 12.4 Stable D operators.- 12.5 Strongly stable D operators.- 12.6 Properties of equations with stable D operators.- 12.7 Stability theory.- 12.8 General linear equations.- 12.9 Stability of autonomous perturbed linear systems.- 12.10 Linear autonomous and periodic equations.- 12.11 Nonhomogeneous linear equations.- 12.12 Supplementary remarks.- 13 Global theory.- 13.1 Generic properties of retarded equations.- 13.2 The set of global solutions.- 13.3 Equations on manifolds : definitions.- 13.4 Retraded equations on compact manifolds.- 13.5 Further properties of the attractor.- 13.6 Supplementary remarks.- Appendix Stability of characteristic equations.

5,799 citations

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