Turbulence, Coherent Structures, Dynamical Systems and SymmetryP. Holmes, J. L. Lumley, G. Berkooz, and C. W. Rowley, 2nd ed., Cambridge University Press, Cambridge, England, U.K., 2012, 386 pp., $90

Data-driven discovery is revolutionizing the modeling, prediction, and control of complex systems. This textbook brings together machine learning, engineering mathematics, and mathematical physics to integrate modeling and control of dynamical systems with modern methods in data science. It highlights many of the recent advances in scientific computing that enable data-driven methods to be applied to a diverse range of complex systems, such as turbulence, the brain, climate, epidemiology, finance, robotics, and autonomy. Aimed at advanced undergraduate and beginning graduate students in the engineering and physical sciences, the text presents a range of topics and methods from introductory to state of the art.

Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control

This paper studies the problem of control for discrete time delay linear systems with Markovian jump parameters. The system under consideration is subjected to both time-varying norm-bounded parameter uncertainty and unknown time delay in the state, and Markovian jump parameters in all system matrices. We address the problem of robust state feedback control in which both robust stochastic stability and a prescribed H/sub /spl infin// performance are required to be achieved irrespective of the uncertainty and time delay. It is shown that the above problem can be solved if a set of coupled linear matrix inequalities has a solution.

Control of Markovian jump discrete-time systems with norm bounded uncertainty and unknown delay

Multi-agent differential graphical games: Online adaptive learning solution for synchronization with optimality

Optimal State Estimation

1 Basic results for linear equations.- 1.1 Linear differential equations and linear algebraic equations.- 1.2 Exponential dichotomy and L2evolutions.- 2 Hamiltonian Matrices and Algebraic Riccati equations.- 2.1 Solutions of algebraic Riccati equations and graph subspaces.- 2.2 Indefinite scalar products and a canonical form of Hamiltonian matrices.- 2.3 Hermitian algebraic Riccati equations.- 2.4 Positive semi-definite solutions of standard algebraic Riccati equations.- 2.5 Hermitian discrete-time algebraic Riccati equations.- 3 Global aspects of Riccati differential and difference equations.- 3.1 Riccati differential equations and associated linear systems.- 3.1.1 Riccati differential equations, Riccati-transformation and spectral factorization.- 3.1.2 Riccati differential equations and linear boundary value problems.- 3.2 A representation formula.- 3.3 Flows on Grassmann manifolds: The extended Riccati differential equation.- 3.4 General representation formulae for solutions of RDE and PRDE, the time-continuous and periodic Riccati differential equation, and dichotomy.- 3.4.1 A general representation formula for solutions of RDE.- 3.4.2 A representation formula for solutions of the periodic Riccati differential equation PRDE.- 3.5 A representation formula for solutions of the discrete time Riccati equation.- 3.5.1 Properties of the solutions to DARE.- 3.5.2 Properties of the solutions to DRDE.- 3.6 Global existence results.- 4 Hermitian Riccati differential equations.- 4.1 Comparison results for HRDE.- 4.1.1 Arbitrary coefficients.- 4.1.2 Periodic coefficients.- 4.1.3 Constant coefficients.- 4.1.4 Riccati inequalities.- 4.2 Monotonicity and convexity results: A Fre1 Basic results for linear equations.- 1.1 Linear differential equations and linear algebraic equations.- 1.2 Exponential dichotomy and L2evolutions.- 2 Hamiltonian Matrices and Algebraic Riccati equations.- 2.1 Solutions of algebraic Riccati equations and graph subspaces.- 2.2 Indefinite scalar products and a canonical form of Hamiltonian matrices.- 2.3 Hermitian algebraic Riccati equations.- 2.4 Positive semi-definite solutions of standard algebraic Riccati equations.- 2.5 Hermitian discrete-time algebraic Riccati equations.- 3 Global aspects of Riccati differential and difference equations.- 3.1 Riccati differential equations and associated linear systems.- 3.1.1 Riccati differential equations, Riccati-transformation and spectral factorization.- 3.1.2 Riccati differential equations and linear boundary value problems.- 3.2 A representation formula.- 3.3 Flows on Grassmann manifolds: The extended Riccati differential equation.- 3.4 General representation formulae for solutions of RDE and PRDE, the time-continuous and periodic Riccati differential equation, and dichotomy.- 3.4.1 A general representation formula for solutions of RDE.- 3.4.2 A representation formula for solutions of the periodic Riccati differential equation PRDE.- 3.5 A representation formula for solutions of the discrete time Riccati equation.- 3.5.1 Properties of the solutions to DARE.- 3.5.2 Properties of the solutions to DRDE.- 3.6 Global existence results.- 4 Hermitian Riccati differential equations.- 4.1 Comparison results for HRDE.- 4.1.1 Arbitrary coefficients.- 4.1.2 Periodic coefficients.- 4.1.3 Constant coefficients.- 4.1.4 Riccati inequalities.- 4.2 Monotonicity and convexity results: A Frechet derivative based approach.- 4.2.1 Notation and preliminaries.- 4.2.2 Results for HARE.- 4.2.3 Results for HDARE.- 4.2.4 Results for HRDE.- 4.3 Convergence to the semi-stabilizing solution.- 4.4 Dependence of HRDE on a parameter.- 4.5 An existence theorem for general HRDE.- 4.6 A special property of HRDE.- 5 The periodic Riccati equation.- 5.1 Linear periodic differential equations.- 5.2 Preliminary notation and results for linear periodic systems.- 5.3 Existence results for periodic Hermitian Riccati equations.- 5.4 Positive semi-definite periodic equilibria of PRDE.- 6 Coupled and generalized Riccati equations.- 6.1 Some basic concepts in dynamic games.- 6.2 Non-symmetric Riccati equations in open loop Nash differential games.- 6.3 Discrete-time open loop Nash Riccati equations.- 6.4 Non-symmetric Riccati equations in open loop Stackelberg differential games.- 6.5 Discrete-time open loop Stackelberg equations.- 6.6 Coupled Riccati equations in closed loop Nash differential games.- 6.7 Rational matrix differential equations arising in stochastic control.- 6.8 Rational matrix difference equations arising in stochastic control.- 6.9 Coupled Riccati equations in Markovian jump systems.- 7 Symmetric differential Riccati equations: an operator based approach.- 7.1 Popov triplets: definition and equivalence.- 7.2 Associated objects.- 7.3 Associated operators.- 7.4 Existence of the stabilizing solution.- 7.5 Positivity theory and applications.- 7.6 Differential Riccati inequalities.- 7.7 The signature condition.- 7.8 Differential Riccati theory: A Hamiltonian descriptor operator approach.- 7.8.1 Descriptors and dichotomy.- 7.8.2 Hamiltonian descriptors.- 7.8.3 The stabilizing (anti-stabilizing) solution.- 8 Applications to Robust Control Systems.- 8.1 The Four Block Nehari Problem.- 8.1.1 Problem Statement.- 8.1.2 A characterization of all solutions.- 8.1.3 Main Result.- 8.2 Disturbance Attenuation.- 8.2.1 Problem statement.- 8.2.2 A necessary condition.- 8.2.3 The Disturbance Feedforward Problem.- 8.2.4 The least achievable tolerance of the DF problem.- 9 Non-symmetric Riccati theory and applications.- 9.1 Non-symmetric Riccati theory.- 9.1.1 Basic notions and preliminary results.- 9.1.2 Toeplitz operators and Riccati equations.- 9.2 Application to open loop Nash games.- 9.2.1 Definitions and Hilbert space.- 9.2.2 Unique Nash equilibria.- 9.2.3 The general case.- 9.2.4 If any, then one or infinitely many.- 9.3 Application to open loop Stackelberg games.- 9.3.1 Characterization in Hilbert space.- 9.3.2 Unique Stackelberg equilibria.- 9.3.3 A value function type approach.- A Appendix.- A.1 Basic facts from control theory.- A.2 The implicit function theorem.- References.- List of Figures.

Matrix Riccati Equations in Control and Systems Theory

A method for optimal positioning of piezoelectric actuators and sensors on a flexible structure is presented. First, a two-dimensional (2-D) model of a piezoelectric actuator bonded to a plate is obtained. Then, a Ritz formulation is used to find a state model of the system in view of its control. To define an optimal positioning strategy, an energy based approach is developed. This leads quite naturally to the study of controllability and observability properties of the overall dynamical model. A new criterion based on energy assessment is proposed to locate actuators and sensors.

Piezoelectric actuators and sensors location for active control of flexible structures

Presents comparison and global existence results for solutions of coupled matrix Riccati differential equations appearing in closed loop Nash games and in mixed H/sub 2//H/sub /spl infin//-type problems. Convergence of solutions is established for the diagonal case, solutions of the corresponding algebraic equations are discussed using numerical examples.

Hisham Abou-Kandil

Papers

Matrix Riccati Equations in Control and Systems Theory

Piezoelectric actuators and sensors location for active control of flexible structures

On global existence of solutions to coupled matrix Riccati equations in closed-loop Nash games

On the solution of discrete-time Markovian jump linear quadratic control problems

Generalized Riccati difference and differential equations