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Automatic Control, IEEE Transactions on

Issue 5 • Date May 2009

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Displaying Results 1 - 25 of 43
  • Table of contents

    Page(s): C1 - C4
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    Freely Available from IEEE
  • IEEE Transactions on Automatic Control publication information

    Page(s): C2
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    Freely Available from IEEE
  • Editorial Quantity vs Quality

    Page(s): 933 - 934
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    Freely Available from IEEE
  • Guest Editorial: Special Issue on Positive Polynomials in Control

    Page(s): 935 - 936
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    Freely Available from IEEE
  • Scanning the issue

    Page(s): 937
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    Freely Available from IEEE
  • Robust Performance Analysis of Uncertain LTI Systems: Dual LMI Approach and Verifications for Exactness

    Page(s): 938 - 951
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (574 KB) |  | HTML iconHTML  

    This paper addresses robust performance analysis problems of linear time-invariant (LTI) systems affected by real parametric uncertainties. These problems, known also as a special class of structured singular value computation problems, are inherently intractable (NP-hard problems). As such intensive research effort has been made to obtain computationally tractable and less conservative analysis conditions, where linear matrix inequality (LMI) plays an important. Nevertheless, since LMI-based conditions are expected to be conservative in general, it is often the case that we cannot conclude anything if the LMI at hand turns out to be infeasible. This motivates us to consider the dual of the LMI and examine the structure of the dual solution. By pursuing this direction, in this paper, we provide rank conditions on the dual solution matrix under which we can conclude that the underlying robust performance is never attained. In particular, a set of uncertain parameters that violates the specified performance can be computed. These results come from block-moment matrix structure of the dual variable, which is consistent with the recent results on polynomial optimization. This particular structure enables us to make good use of simultaneous diagonalizability property of commuting diagonalizable matrices so that the sound rank conditions for the exactness verification can be obtained. View full abstract»

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  • Convex Matrix Inequalities Versus Linear Matrix Inequalities

    Page(s): 952 - 964
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (303 KB) |  | HTML iconHTML  

    Most linear control problems lead directly to matrix inequalities (MIs). Many of these are badly behaved but a classical core of problems are expressible as linear matrix inequalities (LMIs). In many engineering systems problems convexity has all of the advantages of a LMI. Since LMIs have a structure which is seemingly much more rigid than convex MIs, there is the hope that a convexity based theory will be less restrictive than LMIs. How much more restrictive are LMIs than convex MIs? There are two fundamentally different classes of linear systems problems: dimension free and dimension dependent. A dimension free MI is a MI where the unknowns are g -tuples of matrices and appear in the formulas in a manner which respects matrix multiplication. Most of the classic Mis of control theory are dimension free. Dimension dependent Mis have unknowns which are tuples of numbers. The two classes behave very differently and this survey describes what is known in each case about the relation between convex Mis and LMIs. The proof techniques involve and necessitate new developments in the field of semialgebraic geometry. View full abstract»

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  • On Positivity of Polynomials: The Dilation Integral Method

    Page(s): 965 - 978
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (765 KB) |  | HTML iconHTML  

    The focal point of this paper is the well known problem of polynomial positivity over a given domain. More specifically, we consider a multivariate polynomial f(x) with parameter vector x restricted to a hypercube X sub R n. The objective is to determine if f(x) > 0 for all x isin X. Motivated by NP-Hardness considerations, we introduce the so-called dilation integral method. Using this method, a ldquosofteningrdquo of this problem is described. That is, rather than insisting that f(x) be positive for all x isin X, we consider the notions of practical positivity and practical non-positivity. As explained in the paper, these notions involve the calculation of a quantity epsiv > 0 which serves as an upper bound on the percentage volume of violation in parameter space where f(x) les 0 . Whereas checking the polynomial positivity requirement may be computationally prohibitive, using our epsiv-softening and associated dilation integrals, computations are typically straightforward. One highlight of this paper is that we obtain a sequence of upper bounds epsivk which are shown to be ldquosharprdquo in the sense that they converge to zero whenever the positivity requirement is satisfied. Since for fixed n , computational difficulties generally increase with k, this paper also focuses on results which reduce the size of the required k in order to achieve an acceptable percentage volume certification level. For large classes of problems, as the dimension of parameter space n grows, the required k value for acceptable percentage volume violation may be quite low. In fact, it is often the case that low volumes of violation can be achieved with values as low as k=2. View full abstract»

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  • Exponentially Stable Nonlinear Systems Have Polynomial Lyapunov Functions on Bounded Regions

    Page(s): 979 - 987
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (271 KB) |  | HTML iconHTML  

    This paper presents a proof that existence of a polynomial Lyapunov function is necessary and sufficient for exponential stability of a sufficiently smooth nonlinear vector field on a bounded set. The main result states that if there exists an n -times continuously differentiable Lyapunov function which proves exponential stability on a bounded subset of Rn, then there exists a polynomial Lyapunov function which proves exponential stability on the same region. Such a continuous Lyapunov function will exist if, for example, the vector field is at least n-times continuously differentiable. The proof is based on a generalization of the Weierstrass approximation theorem to differentiable functions in several variables. Specifically, polynomials can be used to approximate a differentiable function, using the Sobolev norm W 1,infin to any desired accuracy. This approximation result is combined with the second-order Taylor series expansion to show that polynomial Lyapunov functions can approximate continuous Lyapunov functions arbitrarily well on bounded sets. The investigation is motivated by the use of polynomial optimization algorithms to construct polynomial Lyapunov functions. View full abstract»

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  • Positive Polynomial Constraints for POD-based Model Predictive Controllers

    Page(s): 988 - 999
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (788 KB) |  | HTML iconHTML  

    This paper presents an application of positive polynomials to the reduction of the number of temperature constraints of a proper orthogonal decomposition (POD)-based predictive controller for a non-isothermal tubular reactor. The objective of the controller is to maintain the reactor at a desired operating condition in spite of disturbances in the feed flow, while keeping the maximum temperature low enough to avoid the formation of undesired byproducts. The controller is based on a model derived by means of POD, which reduces the high dimensionality of the discretized system used to approximate the partial differential equations that model the reactor. However, POD does not lead to a reduction in the number of temperature constraints which is typically very large. If we use univariate polynomials to approximate part of the basis vectors derived with the POD technique, it is possible to apply the theory of positive polynomials to find good approximations of the temperature constraints by linear matrix inequalities and to get a reduction in their number. This is the approach that is followed in this paper. The simulation results show that the predictive controller presented a good behavior and that it dealt with the temperature constraints very well. View full abstract»

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  • An Asymptotically Exact Approach to Robust Semidefinite Programming Problems with Function Variables

    Page(s): 1000 - 1006
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (330 KB) |  | HTML iconHTML  

    This technical note provides an approximate approach to a semidefinite programming problem with a parameter-dependent constraint and a function variable. This problem covers a variety of control problems including a robust stability/performance analysis with a parameter-dependent Lyapunov function. In the proposed approach, the original problem is approximated by a standard semidefinite programming problem through two steps: first, the function variable is approximated by a finite-dimensional variable; second, the parameter-dependent constraint is approximated by a finite number of parameter-independent constraints. Both steps produce approximation error. On the sum of these approximation errors, this technical note provides an upper bound. This bound enables quantitative analysis of the approach and gives an efficient way to reduction of the approximation error. Moreover, this technical note discusses how to verify that an optimal solution of the approximate problem is actually optimal also for the original problem. View full abstract»

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  • Pre- and Post-Processing Sum-of-Squares Programs in Practice

    Page(s): 1007 - 1011
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (193 KB) |  | HTML iconHTML  

    Checking non-negativity of polynomials using sum-of-squares has recently been popularized and found many applications in control. Although the method is based on convex programming, the optimization problems rapidly grow and result in huge semidefinite programs. Additionally, they often become increasingly ill-conditioned. To alleviate these problems, it is important to exploit properties of the analyzed polynomial, and post-process the obtained solution. This technical note describes how the sum-of-squares module in the MATLAB toolbox YALMIP handles these issues. View full abstract»

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  • Stability Analysis of Implicit Polynomial Systems

    Page(s): 1012 - 1018
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (427 KB) |  | HTML iconHTML  

    This technical note develops a robust stability analysis method for a class of implicit polynomial systems subject to uncertain constant parameters. A polynomial (with respect to the state and uncertain parameters) Lyapunov function is computed to assess the robust local stability of the system and to provide a robust estimate of the domain of attraction. The results are given in terms of linear matrix inequalities that are affinely dependent on the system state and uncertain parameters. Numerical examples illustrate the effectiveness of the developed approach. View full abstract»

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  • Stability Analysis of Time-Delay Systems With Incommensurate Delays Using Positive Polynomials

    Page(s): 1019 - 1024
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (342 KB) |  | HTML iconHTML  

    We present a new stability condition for linear time-delay systems (TDS) with multiple incommensurate delays based on the Rekasius substitution and positive polynomials. The condition is checked by testing the positivity of multivariate polynomials in certain domains. For this purpose, we propose two alternative algorithms based on linear programming and sum of squares methods, respectively. The efficiency and accuracy of the algorithms is compared in an example to alternative stability conditions taken from the literature. View full abstract»

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  • Sum-of-Squares Decomposition via Generalized KYP Lemma

    Page(s): 1025 - 1029
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (246 KB) |  | HTML iconHTML  

    The Kalman-Yakubovich-Popov (KYP) lemma establishes the equivalence between a frequency domain inequality (FDI) of a proper rational function and a linear matrix inequality (LMI). A recent result generalized the KYP lemma to characterize an FDI of a possibly nonproper rational function on a portion of a curve on the complex plane. This note examines implications of the generalized KYP result to sum-of-squares (SOS) decompositions of matrix-valued nonnegative polynomials of a single complex variable on a curve in the complex plane. Our result generalizes and unifies some existing SOS results, and also establishes equivalences among FDI, LMI, and SOS. View full abstract»

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  • A Region-Dividing Technique for Constructing the Sum-of-Squares Approximations to Robust Semidefinite Programs

    Page(s): 1029 - 1035
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (299 KB) |  | HTML iconHTML  

    In this technical note, we present a novel approach to robust semidefinite programs, of which coefficient matrices depend polynomially on uncertain parameters. The approach is based on approximation with the sum-of-squares polynomials, but, in contrast to the conventional sum-of-squares approach, the quality of approximation is improved by dividing the parameter region into several subregions. The optimal value of the approximate problem converges to that of the original problem as the resolution of the division becomes finer. An advantage of this approach is that an upper bound on the approximation error can be explicitly obtained in terms of the resolution of the division. A numerical example on polynomial optimization is presented to show usefulness of the present approach. View full abstract»

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  • Robust Stability Analysis of Nonlinear Hybrid Systems

    Page(s): 1035 - 1041
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (527 KB) |  | HTML iconHTML  

    We present a methodology for robust stability analysis of nonlinear hybrid systems, through the algorithmic construction of polynomial and piecewise polynomial Lyapunov-like functions using convex optimization and in particular the sum of squares decomposition of multivariate polynomials. Several improvements compared to previous approaches are discussed, such as treating in a unified way polynomial switching surfaces and robust stability analysis for nonlinear hybrid systems. View full abstract»

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  • Local Stability Analysis for Uncertain Nonlinear Systems

    Page(s): 1042 - 1047
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (331 KB) |  | HTML iconHTML  

    We propose a method to compute provably invariant subsets of the region-of-attraction for the asymptotically stable equilibrium points of uncertain nonlinear dynamical systems. We consider polynomial dynamics with perturbations that either obey local polynomial bounds or are described by uncertain parameters multiplying polynomial terms in the vector field. This uncertainty description is motivated by both incapabilities in modeling, as well as bilinearity and dimension of the sum-of-squares programming problems whose solutions provide invariant subsets of the region-of-attraction. We demonstrate the method on three examples from the literature and a controlled short period aircraft dynamics example. View full abstract»

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  • Optimal Control for Polynomial Systems Using Matrix Sum of Squares Relaxations

    Page(s): 1048 - 1053
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (380 KB) |  | HTML iconHTML  

    This note deals with a computational approach to an optimal control problem for input-affine polynomial systems based on a state-dependent linear matrix inequality (SDLMI) from the Hamilton-Jacobi inequality. The design follows a two-step procedure to obtain an upper bound on the optimal value and a state feedback law. In the first step, a direct usage of the matrix sum of squares relaxations and semidefinite programming gives a feasible solution to the SDLMI. In the second step, two kinds of polynomial annihilators decrease the conservativeness of the first design. The note also deals with a control-oriented structural reduction method to reduce the computational effort. Numerical examples illustrate the resulting design method. View full abstract»

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  • Positivity, Complete Positivity, and SOS Representation

    Page(s): 1054 - 1057
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (152 KB) |  | HTML iconHTML  

    The gap between positive multivariate polynomials and the sum-of-squares (SOS) representation has resurfaced as a problem of importance and interest in control theory. A representation theorem for a bihermitian form is proved. A linear map and its dual are associated with this unique form. Complete positivity and not just positivity of the linear maps is necessary as well as sufficient for SOS representation of the form. View full abstract»

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  • Analysis of Polynomial Systems With Time Delays via the Sum of Squares Decomposition

    Page(s): 1058 - 1064
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (270 KB) |  | HTML iconHTML  

    We present a methodology for analyzing robust independent-of-delay and delay-dependent stability of equilibria of systems described by nonlinear Delay Differential Equations by algorithmically constructing appropriate Lyapunov-Krasovskii functionals using the sum of squares decomposition of multivariate polynomials and semidefinite programming. We illustrate the methodology using an example from population dynamics. View full abstract»

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  • Exact Discretization of a Matrix Differential Riccati Equation With Constant Coefficients

    Page(s): 1065 - 1068
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (161 KB) |  | HTML iconHTML  

    An exact method is presented for discretizing a constant-coefficient, non-square, matrix differential Riccati equation, whose solution is assumed to exist. The resulting discrete-time equation gives the values that have no error at discrete-time instants for any discrete-time interval. The method is based on a matrix fractional transformation, which is more general than existing ones, for linearizing the differential Riccati equation. A numerical example is presented to compare the proposed method with that based on gage invariance and bilinearization, which has better performances than the conventional forward-difference method. View full abstract»

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  • A Polynomial Approach to Bias Aware Fixed-Lag Smoothing Problem

    Page(s): 1068 - 1072
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (276 KB) |  | HTML iconHTML  

    In the present technical note, a solution to the problem of fixed-lag smoothing of SISO systems in presence of a dynamical bias is presented in a polynomial framework. The bias aware smoothing problem is solved in three steps, namely, the design of the general structure of the smoother, the estimation of the dynamical bias by means of a deconvolution technique and, then, the combination of the previous results to obtain the bias aware fixed-lag smoother. Applied to an example, this approach shows its efficiency. View full abstract»

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  • State Feedback Control of Asynchronous Machines with Nondeterministic Models

    Page(s): 1072 - 1076
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (233 KB) |  | HTML iconHTML  

    This note is concerned with control of input/state asynchronous sequential machines with nondeterministic models. The objective is to use state feedback for controlling the machine so as to match any possible behavior of the model. When exact model matching is unsolvable, the model matching inclusion problem is considered in which we find the supremal controllable sub-model, or the largest sub-model of a given nondeterministic model that can be matched by the closed-loop system. View full abstract»

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  • Fundamental Limitations on the Variance of Estimated Parametric Models

    Page(s): 1077 - 1081
    Save to Project icon | Request Permissions | Click to expandQuick Abstract | PDF file iconPDF (232 KB) |  | HTML iconHTML  

    In this technical note fundamental integral limitations are derived on the variance of estimated parametric models, for both open and closed loop identification. As an application of these results we show that, for multisine inputs, a well known asymptotic (in model order) variance expression provides upper bounds on the actual variance of the estimated models for finite model orders. The fundamental limitations established here give rise to a dasiawater-bedpsila effect, which is illustrated in an example. View full abstract»

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Aims & Scope

In the IEEE Transactions on Automatic Control, the IEEE Control Systems Society publishes high-quality papers on the theory, design, and applications of control engineering.  Two types of contributions are regularly considered

Full Aims & Scope

Meet Our Editors

Editor-in-Chief
P. J. Antsaklis
Dept. Electrical Engineering
University of Notre Dame