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

Real and Complex Analysis

Stochastic programming can effectively describe many decision-making problems in uncertain environments. Unfortunately, such programs are often computationally demanding to solve. In addition, their solution can be misleading when there is ambiguity in the choice of a distribution for the random parameters. In this paper, we propose a model that describes uncertainty in both the distribution form (discrete, Gaussian, exponential, etc.) and moments (mean and covariance matrix). We demonstrate that for a wide range of cost functions the associated distributionally robust (or min-max) stochastic program can be solved efficiently. Furthermore, by deriving a new confidence region for the mean and the covariance matrix of a random vector, we provide probabilistic arguments for using our model in problems that rely heavily on historical data. These arguments are confirmed in a practical example of portfolio selection, where our framework leads to better-performing policies on the “true” distribution underlying the daily returns of financial assets.

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Distributionally Robust Optimization Under Moment Uncertainty with Application to Data-Driven Problems

The complex structured singular value

The problem of robustly stabilizing a linear uncertain system is considered with emphasis on the interplay between the time-domain results on the quadratic stabilization of uncertain systems and the frequency-domain results on H/sup infinity / optimization. A complete solution to a certain quadratic stabilization problem in which uncertainty enters both the state and the input matrices of the system is given. Relations between these robust stabilization problems and H/sup infinity / control theory are explored. It is also shown that in a number of cases, if a robust stabilization problem can be solved via Lyapunov methods, then it can be also be solved via H/sup infinity / control theory-based methods. >

Robust stabilization of uncertain linear systems: quadratic stabilizability and H/sup infinity / control theory

In this paper, we develop a method for adaptive stabilization without a minimum-phase assumption and without knowledge of the sign of the high-frequency gain. In contrast to recent work by Martensson [8], we include a compactness requirement on the set of possible plants and assume that an upper bound on the order of the plant is known. Under these additional hypotheses, we generate a piecewise linear time-invariant switching control law which leads to a guarantee of Lyapunov stability and an exponential rate of convergence for the state. One of the main objectives in this paper is to eliminate the possibility of "large state deviations" associated with a search Over the space of gain matrices which is required in [8].

Adaptive stabilization of linear systems via switching control

Kharitonov's four-polynomial concept is generalized to the case of linearly dependent coefficient perturbations and more general zero location regions. To this end, a specially constructed scalar function of a scalar variable is instrumental to the robustness analysis. The present work is motivated by two fundamental limitations of Kharitonov's theorem, namely: (1) the theorem only applies to polynomials with independent coefficient perturbations and (2) it only applies to zeros in the left-hand plane. >

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A generalization of Kharitonov's four-polynomial concept for robust stability problems with linearly dependent coefficient perturbations

Consider a control system with admissible uncertain parameter values which exceed the bounds specified by classical robustness theory. The article concerns trade-offs between performance degradation risk and uncertainty tolerance. If a large increase in the uncertainty bound can be established, an acceptably small risk may be justified. Since robustness formulations do not include statistical descriptions of the uncertainty, the question arises whether it is possible to provide such assurances for any distribution. The paper concentrates on problems associated with Kharitonov's theorem and the edge theorem. If p(s, q) denote the uncertain polynomial under consideration and /spl Pscr/(/spl omega/) a frequency dependent convex target set for the uncertain values p(j/spl omega/, q), /spl Pscr/(/spl omega/) symmetric with respect to the nominal p(j/spl omega/, 0), the uncertain parameters q/sub i/ zero-mean independent random variables with known support interval, and for each uncertainty /spl Fscr/ consists of symmetric nonincreasing density functions, then, for fixed frequency /spl omega/, the first theorem indicates that the probability that p(j/spl omega/, q) is in /spl Pscr/(/spl omega/) is minimized by the uniform distribution for q. The second theorem, a generalization of the first, indicates that the same result holds uniformly with respect to frequency. Then, probabilistic guarantees for robust stability are given in the third theorem. In many cases, one can far exceed classical robustness margins while keeping the risk of instability small.

The uniform distribution: rigorous justification for its use in robustness analysis

It has been shown previously that a first-order compensator robustly stabilizes an internal plant family if and only if it stabilizes all of the extreme plants. These extreme plants are obtained by considering all possible combinations for the extreme values of the numerator and denominator coefficients. In this work, the authors prove a stronger result, namely, that it is necessary and sufficient to stabilize only sixteen of the extreme plants. These sixteen plants are generated using the Kharitonov polynomials associated with the numerator and denominator. Furthermore, when additional information about the compensator is specified (sign of the gain and signs and relative magnitudes of the pole and zero), then, in some cases, it is necessary and sufficient to stabilize eight critical plants, while, in other cases, it is necessary and sufficient to stabilize twelve critical plants. >

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Extreme point results for robust stabilization of interval plants with first-order compensators

Given a dynamical system whose description includes time-varying uncertain parameters, it is often desirable to design an output feedback controller leading to asymptotic stability of a given equilibrium point. When designing such a controller, one may consider static (i.e., memoryless) or dynamic compensation. In this paper, we show that solvability of various output feedback design problems is implied by existence of a solution to a certain constrained Lyapunov problem (CLP). The CLP can be stated in purely algebraic terms. Once the CLP is described, we provide necessary and sufficient conditions for its solution to exist. Subsequently, we consider application of the CLP to a number of robust stabilization problems involving static output feedback and observer-based feedback.

B.R. Barmish

Papers

Adaptive stabilization of linear systems via switching control

A generalization of Kharitonov's four-polynomial concept for robust stability problems with linearly dependent coefficient perturbations

The uniform distribution: rigorous justification for its use in robustness analysis

Extreme point results for robust stabilization of interval plants with first-order compensators

The constrained Lyapunov problem and its application to robust output feedback stabilization