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

Linear closed-loop Stackelberg strategies for systems with slow and fast modes are considered. The asymptotic behavior of the solution is studied and used to derive near optimal strategies which does not require the knowledge of the small singular perturbation parameter.

Closed-loop stackelberg strategies for singularly perturbed linear quadratic problems

Partly motivated by the nanopositioning application in AFM and SPM systems, we consider the problem of tracking periodic signals for a class of systems consisting of linear dynamics preceded by a hysteresis operator, where uncertainties exist in both the dynamics and the hysteresis. A servocompensator is proposed, in combination with an approximate hysteresis inverse, to achieve high-precision tracking. The servocompensator accommodates the internal model of the reference signal and a finite number of harmonic terms. It is shown that, with a Prandtl-Ishlinskii (PI) hysteresis operator, the closed-loop system admits a unique, asymptotically stable, periodic solution, which justifies treating the inversion error as an exogenous periodic disturbance. Consequently, the asymptotic tracking error can be made arbitrarily small when the servocompensator accommodates a sufficient number of harmonic terms. Robustness with respect to uncertainties in the dynamics is also established. Simulation and experimental results are presented to validate the approach and evaluate the role of hysteresis inversion. In particular, experiments on a nanopositioner show that, with the proposed method, tracking can be achieved for a 200 Hz reference signal with 0.6% mean error and 2.3% peak error for a travel range of 20 µm.

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Control of systems with hysteresis via servocompensation and its application to nanopositioning

The design of stabilizing feedback control of singularly perturbed diserete-time systems is decomposed into the design of slow and fast controllers which are combined to form the composite control. Composite control strategies are developed for the case of single rate measurements (all variables are measured at the same rate) as well as for the case of multirate measurements (slow variables are measured at a rate slower than that of fast variables).

Multirate and composite control of two-time-scale discrete-time systems

In this paper we propose a robust control scheme for a piezo-actuated nanopositioner to track arbitrary references. The positioner is represented as a linear system preceded by hysteresis, which is modeled with a Prandtl-Ishlinskii (PI) operator. In order to reduce the hysteresis effect, an approximate operator is used as a feedforward compensator. A sliding mode controller is then used to mitigate the effect of inversion error. In existing work on sliding mode control of piezo-actuated systems, the coefficient of the switching component of the control is typically chosen by trial and error. In contrast, in this work we analytically derive an upper bound on the inversion error using the hysteresis model. This bound is a function of the state and time, which is much less conservative than constant bounds. The stability of the closed-loop system is established by Lyapnuov analysis. Simulation and experimental results are presented to demonstrate the effectiveness of the proposed method.

Sliding-mode tracking control of piezo-actuated nanopositioners

A novel nonlinear control scheme is applied to the TORA benchmark system. This approach incorporates a continuous implementation of sliding mode control and an extended high-gain observer, and is based on previous work on the stabilization of a non-minimum phase nonlinear system, under the assumption that its associated auxiliary system has a stabilizing controller. The rotor angle is the only measurement required by this controller, and the closed-loop system thus obtained has quite interesting properties. The performance characteristics of a full-order observer-based second order linear system were recovered through an iterative tuning procedure. The final design provides good transient performance and some robustness to perturbations in the masses of the cart and rotor.

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Hassan K. Khalil

Papers

Closed-loop stackelberg strategies for singularly perturbed linear quadratic problems

Control of systems with hysteresis via servocompensation and its application to nanopositioning

Multirate and composite control of two-time-scale discrete-time systems

Sliding-mode tracking control of piezo-actuated nanopositioners

A novel nonlinear output feedback control applied to the TORA benchmark system