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

Given a real square matrix A we address the question: Does there exist a positive diagonal matrix P such that PA + A^{T}P is negative definite? We present an algorithm which when applied to A gives a definite yes or no answer to this question. If the answer is yes, the algorithm will provide one such P .

On the existence of positive diagonal P such that PA + A^{T}P l 0

A steady-state optimal control problem is considered for nearly completely decomposable Markov chains. In order to apply the policy iteration method of R.A. Howard (Dynamic Programming and Markov Processes, Cambridge, MA, MIT Press, 1960), a high-dimensional ill-conditioned system of algebraic equations must be solved in the value-determination step. Although algorithms exist for aggregation of the steady-state probability distribution problem, they only provide methods for computing the cost, not the dual variables. Using a singular perturbation approach, an aggregation method for the value-determination equation is developed. The aggregation method is developed in three steps. First, a class of similarity transformations that transform the system into a singularly perturbed form is developed. Second, an aggregation method to compute the steady-state probability distribution is derived. Third, this aggregation method is applied to the value determination step of Howard's method (1960). >

Aggregation of the policy iteration method for nearly completely decomposable Markov chains

Asymptotic stability of nonlinear multiparameter singularly perturbed systems is analyzed. Sufficient conditions for existence of a Lyapunov function and uniform asymptotic stability are derived. The new feature of these conditions over earlier results is that there is no restriction on the relative magnitudes of the small singular perturbation parameters. Moreover, the class of systems under consideration can be nonlinear in both the slow and fast variables, while earlier results were limited to systems linear in the fast variables.

Stability analysis of nonlinear multiparameter singularly perturbed systems

This paper studies a high-gain observer with a nonlinear gain. The nonlinearity is chosen to have a higher observer gain during the transient period and a lower gain afterwards, thus overcoming the tradeoff between fast state reconstruction and measurement noise attenuation. The observer is designed such that the behavior of the innovation process can be controlled separately from the other system states. This is accomplished by assigning one fast eigenvalue, with the remaining eigenvalues chosen relatively slow. Without this key step, the stability analysis for the proposed observer is unattainable. Recently, a switched observer approach has been investigated, but the nonlinear gain approach bypasses the complications associated with switching, with little to no appreciable degradation in performance. This paper presents a new model for the nonlinear observer, accompanied by a discussion focusing on the main ideas behind the proof.

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Hassan K. Khalil

Papers

On the existence of positive diagonal P such that PA + A^{T}P l 0

Aggregation of the policy iteration method for nearly completely decomposable Markov chains

Stability analysis of nonlinear multiparameter singularly perturbed systems

High-gain observers in the presence of measurement noise: A nonlinear gain approach

Funnel control for nonlinear systems with arbitrary relative degree using high-gain observers