Optimal Norm Research Articles

Optimal controllability norm for the power cost function

Consideration was given to a model of the multilevel hierarchical structure controlling a "production line" (successive business process) where the manager's costs are defined by a power function depending on the intensity of the subprocess controlled by this manager. For the model under study, the optimal hierarchy of control was determined, and the dependence of the optimal controllability norm (the number of "deputies" of one manager) on the model parameters (for example, the power which may be interpreted as the degree of environmental instability) was analyzed.

A new approach on designing l 1 optimal regulator with minimum order for SISO linear systems

For a SISO linear discrete-time system with a specified input signal, a novel method to realize optimal l1 regulation control is presented. Utilizing the technique of converting a polynomial equation to its corresponding matrix equation, a linear programming problem to get an optimal l1 norm of the system output error map is developed which includes the first term and the last term of the map sequence in the objective function and the right vector of its constraint matrix equation, respectively. The adjustability for the width of the constraint matrix makes the trade-off between the order of the optimal regulator and the value of the minimum objective norm become possible, especially for achieving the optimal regulator with minimum order. By norm scaling rules for the constraint matrix equation, the optimal solution can be scaled directly or be obtained by solving a linear programming problem with l1 norm objective.

Robust and guaranteed-cost filter designs of uncertain time-varying delay systems

In this paper, we consider the problems of robust filter design and guaranteed-cost filter design for uncertain time-varying delay systems. Moreover, norm bounded parameter uncertainties and convex bounded uncertainties are investigated to find an admissible robust filter which guarantees the asymptotic stability and optimal norm bound of filtering-error dynamics. The appropriate condition for the existence of filter, the robust filter-design method, and the optimal upper bound are proposed by the linear matrix inequality approach. Furthermore, the robust guaranteed-cost filter-design problem can be solved using the proposed method and proper manipulations. In this case, we present an optimization problem to obtain the optimal upper bound of the guaranteed cost and the guaranteed-cost filter-design method.

Norm bounds for summation of two normal matrices

A sharp upper bound is obtained for ∥ A+i B∥, where A and B are n× n Hermitian matrices satisfying a 1 I⩽ A⩽ a 2 I and b 1 I⩽ B⩽ b 2 I. Similarly, an optimal bound is obtained for ∥ U+ V∥, where U and V are n× n unitary matrices with any specified spectra; the study leads to some surprising phenomena of discontinuity concerning the spectral variation of unitary matrices. Moreover, it is proven that for two (non-commuting) normal matrices A and B with spectra σ( A) and σ( B), the optimal norm bound for A+ B equals min λ∈ C max α∈σ(A)|α+λ|+ max β∈σ(B)|β−λ| . Extensionsof the results to infinite dimensional cases are also considered.

Symmetric modified finite volume element methods for self-adjoint elliptic and parabolic problems

The finite volume element method is a discretization technique for partial differential equations, but in general case the coefficient matrix of its linear system is not symmetric, even for the self-adjoint continuous problem. In this paper we develop a kind of symmetric modified finite volume element methods both for general self-adjoint elliptic and for parabolic problems on general discretization, their coefficient matrix are symmetric. We give the optimal order energy norm error estimates. We also prove that the difference between the solutions of the finite volume element method and symmetric modified finite volume element method is a high order term.

A Gradient Flow Approach to Optimal Model Reduction of Discrete-Time Periodic Systems

This paper is concerned with the optimal model reduction for linear discrete periodic time-varying systems and digital filters. Specifically, for a given stable periodic time-varying model, we shall seek a lower order periodic time-varying model to approximate the original model in an optimal H2 norm sense. By orthogonal projections of the original model, we convert the optimal periodic model reduction problem into an unconstrained optimization problem. Two effective algorithms are then developed to solve the optimization problem. The algorithms ensure that the H2 cost decreases monotonically and converges to an optimal (local) solution. Numerical examples are given to demonstrate the computational efficiency of the proposed method. The present paper extends the optimal model reduction for linear time invariant systems to linear periodic discrete time-varying systems.

ON CONTROL PERFORMANCE OF MULTI-D.O.F. ELASTIC-PLASTIC STRUCTURES UNDER EARTHQUAKES

In this paper, a control algorithm developed for a single-d.o.f. elastic-plastic structural system subjected to a dynamic excitation is presented. The algorithm is based on evaluation of the bounds for plastic displacement occurring during the motion1–7and it can be regarded as an extension of the optimal norm control8,9which works for structures with no plastic excursion. Analytical foundations and numerical tests are carried out, providing some general conclusions on the advantage of adopting an ad hoc non-linear control model and on its applicability to elastic-plastic structures.

Optimal Hankel norm approximation for the Pritchard-Salamon class of infinite-dimensional systems

The optimal Hankel norm approximation problem is solved under the assumptions that the system Σ(A, B, C) is an exponentially stable, regular Pritchard-Salamon infinite-dimensional system. An explicit parameterization of all solutions is obtained in terms of the system parametersA, B, C.

Adaptation to Spatially Heterogeneous Modifying and Adaptive Environments

The development of an individual's phenotype is influenced by environmental factors (the modifying environment) which may differ from those factors (the adaptive environment) that decide on the adaptational value of the developed phenotype. The shapes of adaptationally optimal norms of reaction are therefore essentially determined by associations between these two environmental components together with the degree of adaptational sensitivity of the developed phenotypes. Two complementary aspects of optimality are accounted for: (a) environments can be optimal for a given norm of reaction and (b) norms of reaction can be optimal for a given environment. The results are obtained for random distribution of genotypes over environmental conditions and under the physiologically reasonable premise that fitness is a function of the costs of modification and adaptation. It turned out that the associations of adaptive and modifying environments are the primary sources of adaptational optimization. More specifically, it is shown that (i) independence between the two environmental components constitutes an adaptationally optimal environment only for norms of reaction in which all phenotypes are adaptively insensitive; (ii) if costs of modification do not depend on the environment, and if the two environmental components are not associated, adaptationally optimal norms of reaction can always be realized through phenogenetic invariance; (iii) as a rule, adaptively sensitive phenotypes developed under strong environmental associations necessitate phenogenetic plasticity for the optimal norm of reaction; (iv) a norm of reaction which is adaptationally optimal in its adaptationally optimal environment can always be realized through phenogenetic invariance, if costs of modification do not vary with the environment. These results reveal an important role of patterns of adaptive sensitivity of phenotypes, which may even surpass that of shapes of norms of reaction in adaptational processes.

Adaptive plasticity in mate preference linked to differences in reproductive effort.

There is abundant evidence for the existence of marked mate preferences in natural populations, but the occurrence of within-population variation in mate preferences has received little attention and is often regarded as nonadaptive deviation from the optimal norm. Here we show experimentally that the preference of female collared flycatchers Ficedula albicollis for male forehead patch size, a sexually selected trait, varies with the time of breeding, an environmental factor with strong effects on reproductive success. Contrary to expectations based on time-constrained choice models, only late-breeding females prefer males with a large patch size. The variation in mate preference matches a seasonal change in female reproductive success: long-term data reveal a positive relationship between female reproductive success and male patch size exclusively in late breeders. In addition, female reproductive effort, as assessed by clutch size, appears to be adjusted relative to both timing of breeding and male phenotype. We conclude that not only can mate preferences display adaptive plasticity within populations, but this plasticity can also be linked to differences in reproductive investment.

A quantitative genetic analysis of phenotypic plasticity of diapause induction in the cricket Allonemobius socius.

Although numerous studies have indicated that diapause is heritable and phenotypically plastic, none of them has examined the quantitative genetic basis of this plasticity. In this paper we report such an analysis for egg diapause in the cricket Allonemobius socius, the induction of which appears to be largely determined by the mother. We analysed the quantitative genetic basis of the phenotypically plastic response of female A. socius to age and environmental conditions. We measured the production of diapause eggs on four occasions over a 16-day period, and in two environments; one mimicking an 'early' period of the year and another mimicking a 'late' period. We analysed genetic variation in phenotypic plasticity using the character-state approach. Diapause proportion was heritable (h2 ranged from 0.17 to 0.49, being larger in the 'early' environment), and the genetic correlation between ages in proportion of diapausing eggs was close to 1 but showed a decrease with increased difference between ages. There were significant genetic correlations between environments for all ages. Because of the reduction in genetic correlation as the difference in ages increases, selection will be more effective at changing the overall shape of the reaction norm than causing local changes. Furthermore, the high genetic correlations may constrain the evolution of the reaction norm. When the two environments are converted into the estimated days in the year the two reaction norms form approximately a single curve as predicted from previous theoretical analysis of the optimal reaction norm.

L ∞-convergence of mixed finite element method for Laplacian operator

In this paper two so-called regularizedGreen’s functions are introduced to derive the optimal maximum norm error estimates for the unknown function and the adjoint vector-valued function for mixed finite element methods of Laplacian operator. One contribution of the paper is a demonstration of how the boundedness of L1—norm estimate for the secondGreen’s function λ2 and the optimal maximum norm error estimate for the adjoint vector-valued function are proved. These results are seemed to be new in the literature of the mixed finite element methods.

How Close Can the Logarithmic Norm of a Matrix Pencil Come to the Spectral Abscissa?

Given a least upper bound norm, the usefulness of the concept of logarithmic norm depends on how closely the logarithmic norm approximates the spectral abscissa. To study this problem, Strom introduced in 1975 the concepts of logarithmically optimal norm and $\varepsilon$-logarithmically optimal norm with respect to a matrix A. Recently Higueras and Garcia-Celayeta have done an extension of the concept of logarithmic norm for matrix pencils and, in a similar way, the usefulness of the concept depends on how closely the logarithmic norm of the pencil approximates the spectral abscissa. In this paper we study this problem and extend the concepts by Strom to matrix pencils.

H2 optimal envelope-constrained FIR filter design: An LMI approach

This paper is concerned with an H 2 optimal envelope-constrained (EC) finite impulse response (FIR) filtering problem. The aim is to design an FIR filter such that the H 2 norm of the combined filter and fixed noise shaping filter is minimized subject to the constraint that for a given input to the filter, its noiseless output is contained or bounded in a prescribed envelope. The filter design problem is formulated as a standard optimization with linear matrix inequality (LMI) constraints. Futhermore, by relaxing the H 2 optimal norm constraint, we propose a robust ECFIR filter design algorithm based on the LMI approach.

State space identification of stable systems

This paper studies a two stage approach to state space identification with certain optimality properties. The first stage is essentially a linear correlation stage via orthonormal filtering to reduce the problem to a setup with small equivalent noise. The second stage is an optimal Hankel norm model reduction step. This two stage algorithm is shown to be consistent for a large class of stable, possibly infinite dimensional, linear time-invariant systems under a very mild condition on the additive noise. The consistency analysis is based on a non-probabilistic framework for noise which allows the inclusion of large classes of non-ergodic and non-stationary noise, possibly having deterministic components, with maximum transparency. Furthermore, l and H convergence results are derived for a popular class of ortho1 normal basis expansions based on shift operators. These bases include, e.g. Laguerre bases.

Individual optimization: Mechanisms shaping the optimal reaction norm

In general, optimal reaction norms in heterogeneous populations can be obtained only by iterative numerical procedures (McNamara, 1991; Kawecki and Stearns, 1993). We consider two particular, but biologically plausible and analytically tractable cases of individual optimization to gain insight into the mechanisms which shape the optimal reaction norm of fecundity in relation to an environmental variable or an individual trait. In the first case, we assume that the quality of the environment (e.g. food abundance) or the quality of the individual (e.g. body size) is fixed during its entire life; it may also be a heritable individual trait. In the second case, individual quality is assumed to change randomly such that the probability distribution of quality in the next year is the same for the parent and for her offspring. For these two cases, we obtain analytical expressions for the shape of the optimal reaction norm, which are heuristically interpretable in terms of underlying selective mechanisms. It is shown that better quality may reduce the optimal fecundity. This outcome is particularly likely if better quality increases a fecundity-independent factor of parental survival in a long-lived species with fixed quality.

Robust regulation with [formula omitted]2 performance

We consider the problem of minimizing the H 2 norm of a feedback system subject to the constraint that the controller robustly regulate against constants and sinusoids. We show that, for any ϵ > 0, there exists a controller that both achieves robust regulation and renders the closed-loop H 2 norm within ϵ of the optimal norm (the norm achievable without the robust regulation constraint). We also provide conditions that are both necessary and sufficient for there to exist a robustly regulating controller that achieves the optimal H 2 norm. All proofs are constructive.

MIMOl 1-optimization with a scalar control

In Ref. 1, the author showed that some scalar mixed sensitivity minimization problems have rationall1-optimal solutions and computed these solutions. Here, a similar approach is applied to a multi-input multi-output (MIMO) plant. The general flavor of the solution is the same as in Ref. 1, but there is one distinct new feature. We prove that some of the outputs may be ignored, in the sense that they have no influence on the optimal solution and that all the operators corresponding to the remaining outputs have the same optimal norm. This fact suggests alternative methods for solving the problem and it is employed in the construction of an exact rationall1-optimal solution of a particular problem.

Discussion of the Paper by Goodall

Discussion of the Paper by Goodall

An algebraic Riccati equation approach to H∞ optimization

This paper considers the general (so-called four block) H ∞ optimal control problem with the assumption that system states are available for feedback. It is shown that infimization of the H ∞ norm of the closed loop transfer function over all linear constant, i.e., nondynamic, stabilizing state feedback laws can be completely characterized via an algebraic Riccati equation. It is further shown that the optimal norm is not improved by allowing feedback to be dynamic. Thus, the general state-feedback H ∞ optimal control problem can be solved by iteratively solving one ARE and the controller can be chosen to be static gain.