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

It is shown that the performance of a globally bounded partial state feedback control of an input-output linearizable system can be recovered by a sufficiently fast high-gain observer. The performance recovery includes recovery of asymptotic stability of the origin, the region of attraction. and trajectories.

A separation principle for the stabilization of a class of nonlinear systems

Separation results for the stabilization of nonlinear systems using different high-gain observer designs

We consider a single-input-single-output nonlinear system which can be represented globally by an explicit input-output model. The system is input-output linearizable by feedback and is required to satisfy a minimum phase condition. The nonlinearities are not required to satisfy any global growth condition. The model depends linearly on unknown parameters which belong to a known compact, convex set. We design a semiglobal adaptive output feedback controller which ensures that the output of the system tracks any given reference signal which is bounded and has bounded derivatives up to the (n+l)th order, where n is the order of the system. The reference signal and its derivatives are assumed to belong to a known compact set. It is also assumed to be sufficiently rich to satisfy a persistence of excitation condition. The design process is simple. First we assume that the output and its derivatives are available for feedback and design the adaptive controller as a state feedback controller in appropriate coordinates. Then we saturate the controller outside a domain of interest and use a high-gain observer to estimate the derivatives of the output. We prove, via asymptotic analysis, that when the speed of the high-gain observer is sufficiently high, the adaptive output feedback controller recovers the performance achieved under the state feedback one. >

Adaptive output feedback control of nonlinear systems represented by input-output models

The theory of high-gain observers has been developed for about twenty years. This talk is a brief introduction to high-gain observers in nonlinear feedback control, with emphasis on the peaking phenomenon and the role of control saturation in dealing with it. The talk surveys recent results on the nonlinear separation principle, extended high-gain observers, performance in the presence of measurement noise, sampled-data control, and experimental testbeds.

High-gain observers in nonlinear feedback control

We address the problem of state estimation for a class of nonlinear systems with measurement noise in the context of feedback control. It is well-known that high-gain observers are robust against model uncertainty and disturbances, but sensitive to measurement noise when implemented in a feedback loop. This work presents the benefits of a nonlinear-gain structure in the innovation process of the high-gain observer, in order to overcome the tradeoff between fast state reconstruction and measurement noise attenuation. The goal is to generate a larger observer gain during the transient response than in the steady-state response. Thus, by reducing the observer gain after achieving satisfactory state estimates, the effect of noise on the steady-state performance is reduced. Moreover, the nonlinear-gain observer presented in this paper is shown to surpass the system performance achieved when using comparable linear-gain observers. The proof argues boundedness and ultimate boundedness of the closed-loop system under the proposed output feedback.

Hassan K. Khalil

Papers

A separation principle for the stabilization of a class of nonlinear systems

Separation results for the stabilization of nonlinear systems using different high-gain observer designs

Adaptive output feedback control of nonlinear systems represented by input-output models

High-gain observers in nonlinear feedback control

A Nonlinear High-Gain Observer for Systems With Measurement Noise in a Feedback Control Framework