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

[신간의 별자리x] 우리/미술, 그리고 ‘슬픔의 박물관’

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

Multiple-input multiple-output (MIMO) technology is maturing and is being incorporated into emerging wireless broadband standards like long-term evolution (LTE) [1]. For example, the LTE standard allows for up to eight antenna ports at the base station. Basically, the more antennas the transmitter/receiver is equipped with, and the more degrees of freedom that the propagation channel can provide, the better the performance in terms of data rate or link reliability. More precisely, on a quasi static channel where a code word spans across only one time and frequency coherence interval, the reliability of a point-to-point MIMO link scales according to Prob(link outage) ` SNR-ntnr where nt and nr are the numbers of transmit and receive antennas, respectively, and signal-to-noise ratio is denoted by SNR. On a channel that varies rapidly as a function of time and frequency, and where circumstances permit coding across many channel coherence intervals, the achievable rate scales as min(nt, nr) log(1 + SNR). The gains in multiuser systems are even more impressive, because such systems offer the possibility to transmit simultaneously to several users and the flexibility to select what users to schedule for reception at any given point in time [2].

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Scaling Up MIMO: Opportunities and Challenges with Very Large Arrays

1. Introduction. Control System Design Steps. Adaptive Control. A Brief History. 2. Models for Dynamic Systems. Introduction. State-Space Models. Input/Output Models. Plant Parametric Models. Problems. 3. Stability. Introduction. Preliminaries. Input/Output Stability. Lyapunov Stability. Positive Real Functions and Stability. Stability of LTI Feedback System. Problems. 4. On-Line Parameter Estimation. Introduction. Simple Examples. Adaptive Laws with Normalization. Adaptive Laws with Projection. Bilinear Parametric Model. Hybrid Adaptive Laws. Summary of Adaptive Laws. Parameter Convergence Proofs. Problems. 5. Parameter Identifiers and Adaptive Observers. Introduction. Parameter Identifiers. Adaptive Observers. Adaptive Observer with Auxiliary Input. Adaptive Observers for Nonminimal Plant Models. Parameter Convergence Proofs. Problems. 6. Model Reference Adaptive Control. Introduction. Simple Direct MRAC Schemes. MRC for SISO Plants. Direct MRAC with Unnormalized Adaptive Laws. Direct MRAC with Normalized Adaptive Laws. Indirect MRAC. Relaxation of Assumptions in MRAC. Stability Proofs in MRAC Schemes. Problems. 7. Adaptive Pole Placement Control. Introduction. Simple APPC Schemes. PPC: Known Plant Parameters. Indirect APPC Schemes. Hybrid APPC Schemes. Stabilizability Issues and Modified APPC. Stability Proofs. Problems. 8. Robust Adaptive Laws. Introduction. Plant Uncertainties and Robust Control. Instability Phenomena in Adaptive Systems. Modifications for Robustness: Simple Examples. Robust Adaptive Laws. Summary of Robust Adaptive Laws. Problems. 9. Robust Adaptive Control Schemes. Introduction. Robust Identifiers and Adaptive Observers. Robust MRAC. Performance Improvement of MRAC. Robust APPC Schemes. Adaptive Control of LTV Plants. Adaptive Control for Multivariable Plants. Stability Proofs of Robust MRAC Schemes. Stability Proofs of Robust APPC Schemes. Problems. Appendices. Swapping Lemmas. Optimization Techniques. Bibliography. Index. License Agreement and Limited Warranty.

Robust adaptive control

In this work we consider systems where the measurements and control signals are transmitted over networks that are affected by independent and identically distributed (i.i.d.) packet losses. In this setting, we study two separate problems. The first one is the design of an offline state estimator. The second one is the design of a linear quadratic regulator (LQR). Both designs take the statistics of the network and the system into account, and are optimal in the sense of minimizing given cost functions. It turns out that these two designs, of which each has an associated Riccati equation, are dual. We show that the convergence of the Riccati equation associated to this dual system formulation is a necessary and sufficient condition for stability of the estimator and controller. Finally, we compare the performance of the proposed offline estimator with those of other estimators available from the literature.

Unified approach to controller and MMSE estimator design with intermittent communications

In this paper, we propose a method for encoding continuous-time Gaussian signals subject to a usual data rate constraint and, more importantly, a reconstruction delay constraint. We first apply a Karhunen-Loeve decomposition to reparameterize the continuous-time signal as a discrete sequence of vectors. We then study the optimal recursive quantization of this sequence of vectors. Since the optimal scheme turns out to have a very cumbersome design, we consider a simplified method, for which a numerical example suggests that the incurred performance loss is negligible. In this simplified method, we first build a state space model for the vector sequence and then use Bayesian tracking to sequentially encode each vector. The tracking task is performed using particle filtering. Numerical experiments show that the proposed approach offers visible advantages over other available approaches, especially when the reconstruction delay is small.

A New Encoder for Continuous-Time Gaussian Signals With Fixed Rate and Reconstruction Delay

In this paper, we investigate the optimal control strategies for model‐free zero‐sum games involving the H∞$$ {H}_{\infty } $$ control. The key contribution is the development of a Q‐learning algorithm for linear quadratic games without knowing the system dynamics. The finite‐horizon setting is more practical than the infinite‐horizon setting, but it is difficult to solve the time‐varying Riccati equation associated with the finite‐horizon setting directly. The proposed algorithm is shown to solve the time‐varying Riccati equation iteratively without the use of models, and numerical experiments on aircraft dynamics demonstrate the algorithm's efficiency.

Finite‐horizon Q‐learning for discrete‐time zero‐sum games with application to 
 H∞$$ {H}_{\infty } $$ control

Shi and Gao (1986) gave a necessary and sufficient condition for the positive-definiteness of interval symmetric matrices. We point out that their result is not new and the assumptions can be less restrictive.

Comments on ‘A necessary and sufficient condition for the positive-definiteness of interval symmetric matrices’

Abstract In this paper we study the the problem of reconstructing a continuous-time model from an identified discrete-time model of a continuous-time stochastic processes. We present new necessary and sufficient condition for the existence of the soluion. It is also shown that the solution is unique if it exists. The theory is applicable to multivariable processes. The ideas are then used to develop a continuous-time ARMA (CARMA) process modelling algorithm with guaranteed solution. The theoretical results are tested using numerical simulations.

Minyue Fu

Papers

Unified approach to controller and MMSE estimator design with intermittent communications

A New Encoder for Continuous-Time Gaussian Signals With Fixed Rate and Reconstruction Delay

Finite‐horizon Q‐learning for discrete‐time zero‐sum games with application to H∞$$ {H}_{\infty } $$ control

Comments on ‘A necessary and sufficient condition for the positive-definiteness of interval symmetric matrices’

On the reconstruction of continuous-time models from estimated discrete-time models of stochastic pr