There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

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

Forecasting with artificial neural networks: the state of the art

Being a motion on a discontinuity set of a dynamic system, sliding mode is used to keep accurately a given constraint and features theoretically-infinite-frequency switching. Standard sliding modes provide for finite-time convergence, precise keeping of the constraint and robustness with respect to internal and external disturbances. Yet the relative degree of the constraint has to be 1 and a dangerous chattering effect is possible. Higher-order sliding modes preserve or generalize the main properties of the standard sliding mode and remove the above restrictions. r-Sliding mode realization provides for up to the rth order of sliding precision with respect to the sampling interval compared with the first order of the standard sliding mode. Such controllers require higher-order real-time derivatives of the outputs to be available. The lacking information is achieved by means of proposed arbitrary-order robust exact differentiators with finite-time convergence. These differentiators feature optimal asymptot...

/pdf/higher-order-sliding-modes-differentiation-and-output-438zdzb0ld.pdf

Higher-order sliding modes, differentiation and output-feedback control

关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

We study a trajectory tracking problem for a multi-rotor in the presence of modeling error and external disturbances. The desired trajectory is unknown and generated from a reference system with unknown or partially known dynamics. We assume that only position and orientation measurements for the multi-rotor and position measurements for the reference system can be accessed. We adopt an extended high-gain observer (EHGO) estimation framework to estimate the feed-forward term required for trajectory tracking, the multi-rotor states, as well as modeling error and external disturbances. We design an output feedback controller for trajectory tracking that comprises a feedback linearizing controller and the EHGO. We rigorously analyze the proposed controller and establish its stability properties. Finally, we numerically illustrate our theoretical results using the example of a multi-rotor landing on a ground vehicle.

Robust tracking of an unknown trajectory with a multi-rotor UAV: A high-gain observer approach

INTRODUCTION Table 1: Beam loading requirements for 400 kW, 400 MeV/u uranium in RIA elliptical cavities. Superconducting radio frequency (SRF) linacs with light beam loading, such as the Rare Isotope Accelerator (RIA), S-DALINAC, CEBAF upgrade, and energy recovery linacs, operate more efficiently with loaded-Q, QL, values greater than 10. The resulting narrow bandwidth puts stringent limits on acceptable levels of vibration, also called microphonics, which detune the SRF cavities and require additional rf power to maintain amplitude and phase. Type 6-cell 6-cell 6-cell βg 0.47 0.61 0.81 Va(MV) 5.12 8.17 13.46 Pbeam(W) 166

/pdf/measurement-and-control-of-microphonics-in-high-loaded-q-44pjm2xm9y.pdf

Measurement and Control of Microphonics in High Loaded-Q Superconducting RF Cavities

Precise spatiotemporal control of the output of a network of intricately connected neurons through microstimulation is highly desirable in many neural prosthetic applications. This control, however, is challenging, in part due to the large number of unobserved variables in the system under consideration, the complexity underlying the local mechanisms of microstimulation, and the interplay between the intrinsic network structure and its dynamic response to external stimulation. In this work we use a simplified firing rate model, identified from a network of Hodgkin-Huxley (HH) type spiking Basal Ganglia (BG) neurons, to study the response of the network to patterned microstimulation, and to design effective feedback control laws to approximate a desired spatiotemporal pattern. Mathematical analysis of the simplified model using Singular Value Decomposition (SVD) suggests that the BG neural circuit under study exhibits strong spatiotemporal selectivity and only responds strongly to a range of specific spatiotemporal stimulation patterns. We use the concept of functional controllability based on SVD to evaluate the effectiveness of various combinations of stimulation sites for a given set of neurons to be controlled. The results suggest that the functional controllability is largely decided by the network connectivity and the connection strength. Finally, we demonstrate that the controller design based on the simplified model is indeed effective in driving the output neurons to follow a prescribed spatiotemporal firing pattern in the network output.

Model-based spatiotemporal analysis and control of a network of spiking Basal Ganglia neurons

A study is made of the problem of output feedback control of nonlinear systems that are left-invertible, minimum-phase, and fully linearizable (by static state feedback). The proposed control is an observer-based control in which the state feedback component is a robust, possibly nonlinear, stabilizing state-feedback control law. The observer is designed so that the closed loop system is a two-time-scale singularly perturbed system. However, due to the epsilon -dependent scaling of the state variables, some of the standard singular perturbation results do not hold for the closed-loop system, in general. The authors present new singular perturbation results on the behavior of the closed-loop system in the presence of such scaling. In particular, they argue that the undesirable effects of such scaling would not arise if the state-feedback control were globally bounded. >

Observer-based control of fully-linearizable nonlinear systems

The ability to give a prognosis for failure of a system is an invaluable tool and can be applied to electric motors. In this paper, three wavelet based methods have been developed that achieve this goal. Wavelet and filter bank theory, the nearest neighbor rule, and linear discriminant functions are reviewed. A framework for the development of a fault detection and classification algorithm based on the coefficients calculated from the discrete wavelet transform and using clustering is described. An experimental setup based on RT-Linux is described and results from testing are presented, verifying the analysis.

Hassan K. Khalil

Papers

Robust tracking of an unknown trajectory with a multi-rotor UAV: A high-gain observer approach

Measurement and Control of Microphonics in High Loaded-Q Superconducting RF Cavities

Model-based spatiotemporal analysis and control of a network of spiking Basal Ganglia neurons

Observer-based control of fully-linearizable nonlinear systems

Comparison of wavelet-based methods for the prognosis of failures in electric motors