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.

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Linear Matrix Inequalities in System and Control Theory

Many problems in signal processing and statistical inference involve finding sparse solutions to under-determined, or ill-conditioned, linear systems of equations. A standard approach consists in minimizing an objective function which includes a quadratic (squared ) error term combined with a sparseness-inducing regularization term. Basis pursuit, the least absolute shrinkage and selection operator (LASSO), wavelet-based deconvolution, and compressed sensing are a few well-known examples of this approach. This paper proposes gradient projection (GP) algorithms for the bound-constrained quadratic programming (BCQP) formulation of these problems. We test variants of this approach that select the line search parameters in different ways, including techniques based on the Barzilai-Borwein method. Computational experiments show that these GP approaches perform well in a wide range of applications, often being significantly faster (in terms of computation time) than competing methods. Although the performance of GP methods tends to degrade as the regularization term is de-emphasized, we show how they can be embedded in a continuation scheme to recover their efficient practical performance.

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Gradient Projection for Sparse Reconstruction: Application to Compressed Sensing and Other Inverse Problems

In the first part of this thesis, we introduce a specific class of Linear Matrix Inequalities (LMI) whose optimal solution can be characterized exactly. This family corresponds to the case where the associated linear operator maps the cone of positive semidefinite matrices onto itself. In this case, the optimal value equals the spectral radius of the operator. It is shown that some rank minimization problems, as well as generalizations of the structured singular value ($mu$) LMIs, have exactly this property.

In the same spirit of exploiting structure to achieve computational efficiency, an algorithm for the numerical solution of a special class of frequency-dependent LMIs is presented. These optimization problems arise from robustness analysis questions, via the Kalman-Yakubovich-Popov lemma. The procedure is an outer approximation method based on the algorithms used in the computation of hinf norms for linear, time invariant systems. The result is especially useful for systems with large state dimension.

The other main contribution in this thesis is the formulation of a convex optimization framework for semialgebraic problems, i.e., those that can be expressed by polynomial equalities and inequalities. The key element is the interaction of concepts in real algebraic geometry (Positivstellensatz) and semidefinite programming.

To this end, an LMI formulation for the sums of squares decomposition for multivariable polynomials is presented. Based on this, it is shown how to construct sufficient Positivstellensatz-based convex tests to prove that certain sets are empty. Among other applications, this leads to a nonlinear extension of many LMI based results in uncertain linear system analysis.

Within the same framework, we develop stronger criteria for matrix copositivity, and generalizations of the well-known standard semidefinite relaxations for quadratic programming.

Some applications to new and previously studied problems are presented. A few examples are Lyapunov function computation, robust bifurcation analysis, structured singular values, etc. It is shown that the proposed methods allow for improved solutions for very diverse questions in continuous and combinatorial optimization.

Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization

A hierarchy of convex relaxations for semialgebraic problems is introduced. For questions reducible to a finite number of polynomial equalities and inequalities, it is shown how to construct a complete family of polynomially sized semidefinite programming conditions that prove infeasibility. The main tools employed are a semidefinite programming formulation of the sum of squares decomposition for multivariate polynomials, and some results from real algebraic geometry. The techniques provide a constructive approach for finding bounded degree solutions to the Positivstellensatz, and are illustrated with examples from diverse application fields.

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Semidefinite programming relaxations for semialgebraic problems

If the Internet is the next great subject for Theoretical Computer Science to model and illuminate mathematically, then Game Theory, and Mathematical Economics more generally, are likely to prove useful tools. In this talk I survey some opportunities and challenges in this important frontier.

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Algorithms, games, and the internet

Introduction. Background. Existence and Multiplicity. Pivoting Methods. Iterative Methods. Geometry and Degree Theory. Sensitivity and Stability Analysis. Chapter Notes and References. Bibliography. Index.

The Linear Complementarity Problem

Fundamental operating principles early internal combustion engine development, characteristics of internal combusion engines, additional types of internal combustion engine, prospects for internal combustion engines, thermodynamic principles introduction and definitions of efficience, ideal air standard cycles, comparison between thermodynamic and mechanical cycles, additional performance parameters for internal combustion engines, fuel-air cycle, computer models, combustion and fuels combustion chemistry and fuel chemistry, combustion thermodynamics, dissociation, combustion in spark ignition engines, combustion in compression ignition engines, fuels and additives, engine emissions, combustion modelling, spark ignition engines combustion chambers, catalysts and emissions from spark ignition engines, mixture preparation, electronic control engines, compression ignition engines direct injection (DI) systems, indirect injection (IDI) systems, cold starting of compression ignition engines, fuel injection equipment, diesel engine emissions, induction and exhaust processes valve gear, flow characteristics of poppet valves, valve timing, unsteady compressible fluid flow, manifold, silencing, two stroke engines two stroke gas flow performance parameters, scavenging systems, scavenge modelling, experimental techniques for evaluating scavenge and results for port flow co-efficients, engine performance and technology, in-cylinder motion flow measurement techniques, turbulence, turbocharging radial flow and axial flow machines, turbocharging the compression ignition engine, turbocharging the spark ignition engine, engine modelling zero-dimensional modelling, application of modelling to a turbocharged medium speed diesel engine, mechanical design considerations the disposition and number of the cylinders, cylinder block and head materials, the piston and rings, the connecting-rod, crankshaft, camshaft and valves, lubrication and bearings, advanced design concepts, heat transfer in internal combustion engines engine cooling, liquid coolant systems, experimental facilities quasi-steady engine instrumentation, experimental accuracy, measurement of exhaust emissions, computer based combustion analysis, advanced test systems, case studies Jaguar V12 HE engine, Chrysler 2.2 litre spark ignition engine, Ford 2.5 litre DI diesel engine. Appendices: the use of SI units answers to numerical problems engine specifications stratified charge engines engine tuning.

Introduction to Internal Combustion Engines

Correlations for the Laminar-Burning Velocity of Methane/Diluent/Air Mixtures Obtained in Free-Fall Experiments

Currently 99.8% of global transport is powered by internal combustion engines (ICEs) and 95% of transport energy comes from liquid fuels made from petroleum. Many alternatives including battery electric vehicles (BEVs) and other fuels like biofuels and hydrogen are being considered. However, all these alternatives start from a very low base and face very significant barriers to unlimited expansion so that 85–90% of transport energy is expected to come from conventional liquid fuels powering combustion engines even by 2040. Hence it is imperative that ICEs are improved in order to reduce the local and global environmental impact of transport. This paper considers the scope for such improvement after discussing the basic principles that govern engine efficiency and the technologies to control exhaust pollution. The great scope for such improvement is illustrated by considering various practical approaches already in the market. For instance, the best in class SI engines in the U.S. have 14% lower fuel consumption compared to the average. Engine and conventional powertrain developments alone could reduce the fuel consumption by over 30% for light duty vehicles (LDVs). Implementing other technologies such as hybridisation and light-weighting could reduce fuel consumption by 50% compared to the current average for LDVs. Current after-treatment technology can ensure that the exhaust pollutant levels meet the most stringent current emissions requirements. Indeed, with the most modern diesel vehicles, the exhaust can be cleaner than the intake air in urban centres. The implications for transport policy, particularly where there are plans to ban ICEs, are considered in the final discussion. All available technologies need to be deployed to mitigate the environmental impact of transport and it would be extremely short-sighted to discourage further development of ICEs by limiting their sales.

Richard Stone

Papers

The Linear Complementarity Problem

Introduction to Internal Combustion Engines

Correlations for the Laminar-Burning Velocity of Methane/Diluent/Air Mixtures Obtained in Free-Fall Experiments

The scope for improving the efficiency and environmental impact of internal combustion engines

A study of mixture preparation and PM emissions using a direct injection engine fuelled with stoichiometric gasoline/ethanol blends