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

This paper designs a high-gain predictor (HGP) for the output feedback control of nonlinear systems in the presence of control, output, and state delays. The high-gain predictor realizes the states appearing in the output feedback control in terms of predictive state, delayed state, and current state. The system is combined by internal and external dynamics, and the closed-loop system with time-delays is supposed globally asymptotically stable and locally exponentially stable. Positively invariant sets are found to verify boundedness, ultimate boundedness, and exponential stability of the closed-loop system. In the simulation, a bounded sliding mode control is applied to demonstrate the performance recovery of the closed-loop system, and the fact that the high-gain-predictor parameter has a lower bound related to the time-delays.

High-gain-predictor-based output feedback control for time-delay nonlinear systems

In this paper robustness of sampled-data control designs to unmodeled high frequency dynamics is studied, using singular perturbation theory. It is argued that when the plant is preceded by a zero order hold, the direct transmission term of the reduced-order model should be modeled as a delay element in order to ensure robustness.

On the robustness of sampled-data control to unmodeled high frequency dynamics

High-gain observers proved to be a useful tool in the design of output feedback control of nonlinear systems. However, the observer faces a numerical challenge when its dimension is high. For an observer of dimension ρ and a high-gain parameter k, the observer gain is of the order of kρ and the observer variables could be of the order of kρ−1 during the transient period. This paper presents a new high-gain observer that is based on cascading lower-dimensional observers with saturation functions in between them. In the new observer, the gains and variables are of the order of k. It is shown that the cascade observer has properties similar to the standard one.

Cascade high-gain observer for high-dimensional systems

In, Serrani, Isidori, and Marconi design an output feedback controller that achieves semiglobal output regulation for a class of minimum phase nonlinear systems. The controller has three components: an internal model to ensure asymptotic regulation, a high-gain partial state feedback controller to stabilize the augmented system of the plant and internal model, and a high-gain observer to implement the partial state feedback. In this note we show a property of the transient response of this controller. We consider an alternate controller that shares with the controller of the structure of high-gain partial state feedback and high-gain observer, but without internal model. Such controller can only achieve practical regulation. We show that when the controller and observer gains are large enough, the trajectories under the two controllers will be close to each other. Hence, the transient response of the controller of is not degraded by the inclusion of internal model. A similar transient response property is shown by Seshagiri and Khalil for the conditional servocompensator. Similarities and differences between the two designs are pointed out.

On the Transient Response of a Nonlinear Output Regulator

Lyapunov’s theory for characterizing and studying the stability of equilibrium points is presented for time-invariant and time-varying systems modeled by ordinary differential equations.

Hassan K. Khalil

Papers

High-gain-predictor-based output feedback control for time-delay nonlinear systems

On the robustness of sampled-data control to unmodeled high frequency dynamics

Cascade high-gain observer for high-dimensional systems

On the Transient Response of a Nonlinear Output Regulator

Lyapunov's Stability Theory.