DETERMINANT MAXIMIZATION WITH **LINEAR** **MATRIX** **INEQUALITY** CONSTRAINTS y
LIEVEN VANDENBERGHE z , STEPHEN BOYD x , AND SHAO-PO WU x
Abstract. The problem of maximizing the determinant of a **matrix** subject to **linear** **matrix** inequalities arises in many elds, including computational geometry, statistics, system identication, experiment design, and information and communication theory. It can also be considered as a generalization of the semidenite programming problem.

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In this chapter, we presents a robust full state feedback fuzzy control law with actuator norm constraints based on Takagi-Sugeno (T-S) fuzzy model for attitude stabilization and vibra- tion suppression of a flexible spacecraft made of a rigid platform and a flexible antenna.
First, the **linear** **matrix** **inequality** conditions are derived then the parallel distributed compen- sator technique is applied to the spacecraft. The controller produces an asymptotically stable closed-loop system which is robust to external disturbances and has a simple structure which make it easy to implement. Numerical simulation is provided for performance evaluation of the proposed controller design.

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Abstract—In this paper, the technique of image noise cancellation is presented by employing cellular neural networks (CNN) and **linear** **matrix** **inequality** (LMI). The main objective is to obtain the templates of CNN by using a corrupted image and a corresponding desired image. A criterion for the uniqueness and global asymptotic stability of the equilibrium point of CNN is presented based on the Lyapunov stability theorem (i.e., the feedback template “A” of CNN is solved at this step), and the input template “B” of CNN is designed to achieve desirable output by using the property of saturation nonlinearity of CNN. It is shown that the problem of image noise cancellation can be characterized in terms of LMIs. The simulation results indicate that the proposed method is useful for practical application.

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Model Reduction-Based Control of the Buck Converter Using **Linear** **Matrix** **Inequality** and Neural Networks
Anas N. Al-Rabadi and Othman M.K. Alsmadi
Abstract - A new method to control the Buck converter using new small signal model of the pulse width modulation (PWM) switch is introduced. The new method uses recurrent supervised neural network to estimate certain parameters of the transformed system **matrix** [ A ~

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Copyright © 2013 Josep Rubi´o-Masseg´u et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
In this paper, a new strategy to design static output-feedback controllers for a class of vehicle suspension systems is presented. A theoretical background on recent advances in output-feedback control is first provided, which makes possible an effective synthesis of static output-feedback controllers by solving a single **linear** **matrix** **inequality** optimization problem. Next, a simplified model of a quarter-car suspension system is proposed, taking the ride comfort, suspension stroke, road holding ability, and control effort as the

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Abstract. Two-link flexible manipulator is a manipulator robot which at least one of its arms is made of lightweight material and not rigid. Flexible robot manipulator has some advantages over the rigid robot manipulator, such as lighter, requires less power and costs, and to result greater payload. However, suitable control algorithm to maintain the two-link flexible robot manipulator in accurate positioning is very challenging. In this study, sliding mode control (SMC) was employed as robust control algorithm due to its insensitivity on the system parameter variations and the presence of disturbances when the system states are sliding on a sliding surface. SMC algorithm was combined with **linear** **matrix** **inequality** (LMI), which aims to reduce the effects of chattering coming from the oscillation of the state during sliding on the sliding surface. Stability of the control algorithm is guaranteed by Lyapunov function candidate. Based on simulation works, SMC based LMI resulted in better performance improvements despite the disturbances with significant chattering reduction. This was evident from the decline of the sum of squared tracking error (SSTE) and the sum of squared of control input (SSCI) indexes respectively 25.4% and 19.4%.

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This chapter proposes a parametric multiplier approach to deriving parametric Lya- punov functions for robust stability analysis of linear systems involving uncertain pa- rameters.. This[r]

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One can determine deflating subspaces of a matrix pencil using numerical tools, through they are not very well developed for the singular case (see e.g. This fre[r]

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C. Constraints on eigenvalues in LMI
The Eigenvalues of the closed loop system is placed in convex region and these convex regions are formulated into LMI with the help of Lyapunov **matrix** P. LMI for any region D of the complex plane is defined as

I. I NTRODUCTION
We consider a continuous time **linear** multi–inventory sys- tem with unknown demands bounded within ellipsoids and controls bounded within ellipsoids. The system is modelled as a first order one integrating the discrepancy between controls and demands at different sites (buffers). Thus, the state represents the buffer levels. We wish to study conditions under which the state can be driven within an a-priori chosen target set through a saturated **linear** state feedback control.

6 Conclusions and future works
We have addressed the problem of ε-stabilizing the inventory of a continuous time lin- ear multi–inventory system with unknown demands bounded within ellipsoids and controls bounded within ellipsoids or polytopes. Motivations are due to the cost reduction associated with a bounded inventory. As main results we have provided certain LMIs conditions under which ε-stabilizability is possible through a saturated **linear** state feedback control. We have also exploited some recent techniques for the modeling and analysis of polytopic systems with saturations.

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f A = U f λ f λ U (2) In this paper, some new generalizations of the **matrix** form of the Brunn-Minkowski **inequality** are presented.
One of our main results is the following theorem.
Theorem 1.1. Let A , B be positive definite commuting **matrix** of order n with eigenvalues in the interval I . If f is a positive concave function on I and 0 < < λ 1 , then

which is calculated by the number of cars sold (n). the salary is given by the following equation. During the event ravi runs for r miles and cycles for c miles and his total oxygen in[r]

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Keywords: Discrete-time martingale, large deviation, probability **inequality**, random **matrix**.
Abstract
Freedman’s **inequality** is a martingale counterpart to Bernstein’s **inequality**. This result shows that the large-deviation behavior of a martingale is controlled by the predictable quadratic variation and a uniform upper bound for the martingale difference sequence. Oliveira has recently estab- lished a natural extension of Freedman’s **inequality** that provides tail bounds for the maximum singular value of a **matrix**-valued martingale. This note describes a different proof of the **matrix** Freedman **inequality** that depends on a deep theorem of Lieb from **matrix** analysis. This argument delivers sharp constants in the **matrix** Freedman **inequality**, and it also yields tail bounds for other types of **matrix** martingales. The new techniques are adapted from recent work [ Tro10b ] by the present author.

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– 3
y ≥ – 2 x + 3
F I G U R E 7 . 8
b) Because the **inequality** symbol is , every point on or above the line satisfies this **inequality**. We use the fact that the slope of this line is 2 and the y-intercept is (0, 3) to draw the graph of the line. To show that the line y 2x 3 is included in the graph, we make it a solid line and shade the region above. See Fig. 7.8.

In this paper we find exact solutions for linear ordinary differential equa- tions of any order when they are given in matrix form, as well as for classes of Riccati matrix equations wit[r]

∗ B.4 COMPUTER CONSIDERATIONS AND THE PRODUCT FORM
The revised simplex method is used in essentially all commercial computer codes for **linear** programming, both for computational and storage reasons.
For any problem of realistic size, the revised simplex method makes fewer calculations than the ordinary simplex method. This is partly due to the fact that, besides the columns corresponding to the basis inverse and the righthand side, only the column corresponding to the variable entering the basis needs to be computed at each iteration. Further, in pricing out the nonbasic columns, the method takes advantage of the low density of nonzero elements in the initial data **matrix** of most real problems, since the simplex multipliers need to be multiplied only by the nonzero coefficients in a nonbasic column. Another reason for using the revised simplex method is that roundoff error tends to accumulate in performing these algorithms. Since the revised simplex method maintains the original data, the inverse of the basis may be recomputed from this data periodically, to significantly reduce this type of error. Many large problems could not be solved without such a periodic reinversion of the basis to reduce roundoff error.

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If g is absolutely continuous and g(X) has finite variance, then
E[g 0 (X)] 2 ≥ V ar[g(X)]. (1)
Equality in (1) is achieved for **linear** functions. This **inequality** have arisen earlier, especially because of its use in variational problems. There are many papers that deal with **inequality** (1) and in many cases they relate to the single function. However, the random variables might have multivariate distributions. So, we present the study of **matrix** variance **inequality** for the normal and gamma distribution with k-functions.

Martin Tabi
October 1995
Abstract
A class of **inequality** measures that is a natural companion to the popular Lorenz curve is the class of measures that are **linear** in incomes. These measures, which include the Gini and S-Gini coefficients, can be interpreted as ethical means of relative deprivation feelings. Their change through the tax and benefit system can be decomposed simply as a sum of progressivity indices for individual taxes and benefits, minus an index of horizontal inequity measured by the extent of reranking in the population. These progressivity and horizontal inequity indices can also be interpreted as ethical means of perceptions of fiscal harshness and ill-performance. We furthermore derive the asymptotic sampling distribution of these classes of indices of redistribution, progressivity, and horizontal inequity, which enables their use with micro-data on a population. We illustrate the theoretical and statistical results through an application on the distribution and redistribution of income in Canada in 1981 and in 1990.

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1. Preliminaries and introduction
Throughout the present note, F stands for a field and N for the set of non-negative integers. The letters m and n always denote positive integers.
If W is a vector space over F , s ∈ N \ {0} and w 1 , . . . , w s ∈ W , then we define Span F (w 1 , . . . , w s ) to be the **linear** subspace of W spanned by the vectors w 1 , . . . , w s . We always consider F n with its usual structure of a vector space over F . The elements of F n will be understood as columns.

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