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.

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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...

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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

One of the fundamental objectives in systems neuroscience is to precisely control the spatiotemporal firing patterns of cortical neurons to elicit a desired pattern of activity. In this work, we study the effects of intracortical micro-stimulation on the dynamics of a basal ganglia microcircuit model, and explore the feasibility of controlling the spatiotemporal firing patterns of the modeled population in the presence of unobserved inputs. Results from the simulation study suggest that properly designed Multiple-Input-Multiple-Output (MIMO) feedback control paradigm can force a subpopulation of observed output neurons to follow a prescribed spatiotemporal firing pattern despite the presence of unobserved inputs. The accuracy of the spike timing of the controlled neural firing with respect to the reference spike trains is in the order of tens of milliseconds.

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Feedback control of the spatiotemporal firing pattern of a basal ganglia microcircuit model

A non-linear robust output feedback control is designed for a sixth-order model of an induction motor. The control uses only measurement of the rotor position and stator currents. It contains two observers, a second-order observer to estimate the rotor flux from the stator current and a third-order high-gain observer to estimate the rotor speed and acceleration from its position. The control is robust to uncertainties in the rotor and stator resistances, and a bounded time-varying load torque. It guarantees that the speed tracking error can be made arbitrarily small by choice of certain design parameters. Moreover, when the speed reference and load torque and constant, it ensures asymptotic regulation. The controller is applied, via simulation, to a benchmark example. Copyright © 2000 John Wiley & Sons, Ltd.

Robust speed control of induction motors: application to a benchmark example

This paper proves a new property of the nonlinear regulators of [4] and [7]. In both papers the steady-state control is immersed into a linear internal model. In general, the model produces the sinusoidal signals generated by an exosystem, as well as a number of their harmonics, which are induced by the system's nonlinearities. When the internal model does not account for all harmonics, there will be an error between the steady-state control needed to achieve zero steady-state regulation error and the steady-state control produced by the internal model. If the norm of this error is bounded by a constant δ, it is shown in [4] and [7] that the steady-state regulation error will be of the order O(δ). In this paper we prove a shaper result where the steady-state regulation error is shown to be of the order O(μδ), where μ is a design parameter of the continuously-implemented sliding mode controller of [4] and [7]. Therefore, the regulation error can be reduced by decreasing μ. This result allows us to tradeoff the dimension of the internal model versus the value of μ as means of reducing the regulation error.

On the steady-state error of a nonlinear regulator

This paper studies sampled-data output feedback control of a class of nonlinear systems. It is shown that the performance of a stabilizing continuous-time state feedback controller can be recovered by a sampled-data output feedback controller when the sampling period is sufficiently small. The output feedback controller uses a deadbeat discrete-time observer to estimate the unmeasured states. Two schemes are proposed to overcome large initial transients when the controller is switched on.

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Output feedback sampled-data stabilization of nonlinear systems

The dynamics and control of a strongly nonlinear 3-DOF model for ship motion are investigated. The model describes the roll, sway, and heave motions occurring in a vertical plane when the vessel is subjected to beam seas. The ship is installed with active antiroll tanks as a means of preventing large amplitude roll motions. A robust state feedback controller for the pumps is designed that can handle model uncertainties, which arise primarily from unknown hydrodynamic loads. The approach for the controller design is a combination of sliding mode control and composite control for singularly perturbed systems, with the help of the backstepping technique. It is shown that this design can effectively control roll motions of large amplitude, including capsize prevention. Numerical simulation results for an existing fishing vessel, the twice-capsized Patti-B, are used to verify the analysis. @S0022-0434~00!02701-5#

Hassan K. Khalil

Papers

Feedback control of the spatiotemporal firing pattern of a basal ganglia microcircuit model

Robust speed control of induction motors: application to a benchmark example

On the steady-state error of a nonlinear regulator

Output feedback sampled-data stabilization of nonlinear systems

Robust stabilization of large amplitude ship rolling in beam seas