On the gap between H2 and entropy performance measures in H∞ control design

Recent papers have considered the problem of minimizing an entropy functional subject to an H_∞ performance constraint. Since the entropy is an upper bound for the H₂ cost, there remains a gap between entropy minimization and H₂ minimization. In this paper we consider a generalized cost functional involving both H₂ and entropy aspects. This approach thus provides a means for optimizing H₂ performance within H_∞ control design.     

Robust/H∞ filtering for nonlinear systems

A robust (or H_∞) approach to filtering for nonlinear systems is considered. A bound on the estimate error as a function of the disturbance energy is obtained. The corresponding dynamic programming equation is a first-order PDE. This has computational ramifications. The case where the measurements are discrete time is considered also. A numerical method is discussed.     

Robust/H∞ filtering for nonlinear systems

A robust (or H_∞) approach to filtering for nonlinear systems is considered. A bound on the estimate error as a function of the disturbance energy is obtained. The corresponding dynamic programming equation is a first-order PDE. This has computational ramifications. The case where the measurements are discrete time is considered also. A numerical method is discussed.     

Relating H2 and H∞ bounds for finite-dimensional systems

For a linear time invariant system, the infinity-norm of the transfer function can be used as a measure of the gain of the system. This notion of system gain is ideally suited to the frequency domain design techniques such as H_∞ optimal control. Another measure of the gain of a system is the H₂ norm, which is often associated with the LQG optimal control problem. The only known connection between these two norms is that, for discrete time transfer functions, the H₂ norm is bounded by the H_∞ norm. It is shown in this paper that, given precise or certain partial knowledge of the poles of the transfer function, it is possible to obtain an upper bound of the H_∞ norm as a function of the H₂ norm, both in the continuous and discrete time cases. It is also shown that, in continuous time, the H₂ norm can be bounded by a function of the H_∞ norm and the bandwidth of the system.     

An LMI approach to periodic discrete-time unbiased filtering

The problem of unbiased filtering for a discrete-time linear periodic system is faced by means of linear matrix inequality techniques. As leading case, we derive the synthesis conditions to obtain an unbiased filter and an unbiased fixed-lag smoother enforcing a bound on the H_∞ performance on the error dynamics.     

H∞ identification of “soft” uncertainty models

The paper investigates the problem of identifying uncertainty models of causal, SISO, LTI, discrete-time, BIBO stable, unknown systems, using frequency domain measurements corrupted by Gaussian noise of known covariance. Additive uncertainty models are looked for, consisting of a nominal model and an additive dynamic perturbation accounting for the modeling error. The nominal model is chosen within a class of affinely parametrized models with transfer function of given (possibly low) order. An estimate of the parameters minimizing the H_∞ modeling error is obtained by minimizing an upper bound of the worst-case (with respect to the modeling error) second moment of the estimation error. Then, a bound in the frequency domain guaranteeing to include, with probability α, the frequency response error between the estimated nominal model and the unknown system is derived.     

An algebraic approach to H∞ output feedback control problems

We derive an output feedback controller which stabilizes a system and satisfies a prescribed H_∞ norm bound of the closed loop transfer function. The proposed design method is available for any system, i.e. there are no restrictions on D₁₂ and D₂₁. This approach utilizes only algebraic operations, thus proofs are simple and clear.     

Robust H∞ observer design of linear time-delay systems with parametric uncertainty

This paper deals with the problem of H_∞ observer design for a class of uncertain linear systems with delayed state and parameter uncertainties. This problem aims at designing the linear state observers such that, for all admissible parameter uncertainties, the observation process remains robustly stable and the transfer function from exogenous disturbances to error state outputs meets the prespecified H_∞ norm upper bound constraint, independently of the time delay. The time delay is assumed to be unknown, and the parameter uncertainties are allowed to be norm-bounded and appear in all the matrices of the state-space model. An effective matrix inequality methodology is developed to solve the proposed problem. We derive the conditions for the existence of the desired robust H_∞ observers, and then characterize the analytical expression of these observers in terms of some free parameters. A numerical example demonstrates the validity and applicability of the present approach.     

H∞ filtering for a class of uncertain nonlinear systems

This paper investigates the problem of H_∞ filtering for a class of uncertain continuous-time nonlinear systems with real time-varying parameter uncertainty and unknown initial state. We develop an infinite horizon H_∞ filtering methodology which provides both robust stability and a guaranteed H_∞ performance for the filtering error irrespective of the parameter uncertainty.     

Minimum ℓ1, ℓ2, and ℓ∞ Norm Approximate Solutions to an Overdetermined System of Linear Equations

Cadzow, J. A., Minimum ℓ₁, ℓ₂, and ℓ_∞ Norm Approximate Solutions to an Overdetermined System of Linear Equations, Digital Signal Processing12 (2002) 524–560Many practical problems encountered in digital signal processing and other quantitative oriented disciplines entail finding a best approximate solution to an overdetermined system of linear equations. Invariably, the least squares error approximate solution (i.e., minimum ℓ₂ norm) is chosen for this task due primarily to the existence of a convenient closed expression for its determination. It should be noted, however, that in many applications a minimum ℓ₁ or ℓ_∞ norm approximate solution is preferable. For example, in cases where the data being analyzed contain a few data outliers a minimum ℓ₁ approximate solution is preferable since it tends to ignore bad data points. In other applications one may wish to determine an approximate solution whose largest error magnitude is the smallest possible (i.e., a minimum ℓ_∞ norm approximate solution). Unfortunately, there do not exist convenient closed form expressions for either the minimum ℓ₁ or the minimum ℓ_∞ norm approximate solution and one must resort to nonlinear programming methods for their determination. Effective algorithms for determining these two solutions are herein presented (see Cadzow, J. A., Data Analysis and Signal Processing: Theory and Applications).     

Min–max Kalman filtering

The problem of H_∞-optimal state estimation of linear continuous-time systems that are measured with an additive white noise is addressed. The relevant cost function is the expected value of the standard H_∞ performance index, with respect to the measurement noise statistics. The solution is obtained by applying the matrix version of the maximum principle to the solution of the min–max problem in which the estimator tries to minimize the mean square estimation error and the exogenous disturbance tries to maximize it while being penalized for its energy. The solution is given in terms of two coupled Riccati difference equations from which the filter gains are derived. In the case where an infinite penalty is imposed on the energy of the exogenous disturbance, the celebrated Kalman filter is recovered. In the stationary case, where all the signals are stationary, an upper-bound on the solutions of the coupled Riccati equations is obtained via a solution of coupled linear matrix inequalities. The resulting filter then guarantees a bound on the estimation error covariance matrix. An illustrative example is given where the velocity of a maneuvering target has to be estimated utilizing noisy measurements of the position.     

An equivalence between rational H2 and Hankel-norm approximations

Let H(z) be a given function in H₂ A classical problem in engineering analysis is to find a rational function G (z) ε H₂ degree M say, which is closest to H(z) in 2-norm. This problem is typically approached using the cost function |H(z) − G(z)|₂, in which G(z) is allowed to vary over the set of Mth-order rational functions in H₂ and for which stationary points are sought. We show that each stationary point of degree M of this functional coincides with a weighted Hankel-norm approximant to H(z). The weighting function derives from the outer factor of the error function H(z) − G(z) stationary point of the rational H₂ approximation problem.     

Preparation and characterization of nanocrystalline α-Fe2O3 by a sol-gel process

α-Fe₂O₃ ultra-fine powder with an average particle size of 6–26nm has been prepared by a sol-gel process. Thermal analysis, X-ray diffraction and transmission electron microscope were used to study its formation process and micro-structure. The temperature dependence of the electric conductance of the elements made of nanocrystalline α-Fe₂O₃ shows that the gas-sensing properties are strongly related to its surface. The elements exhibited good sensitivity and selectivity to ethyl alcohol, indicating it is a promising alcohol-sensing material.     

Mixed <Emphasis Type="Italic">H</Emphasis><Subscript>2</Subscript>/<Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> filtering for a class of high-speed sampling uncertain systems

The problem of mixed H2/H∞ filtering for polytopic Delta operator systems is investigated. The aim is to design a linear asymptotically stable filter which guarantees that the filtering error system has different performances in different filtering channels. Based on a parameter-dependent Lyapunov function, a new mixed H2/H∞ performance criterion is presented. Upon this performance criterion, a sufficient condition for the full-order mixed H2/H∞ filter is derived in terms of linear matrix inequalities. The filter can be obtained from the solution of a convex optimization problem. The proposed filter design procedure is less conservative than the strategy based on the quadratic stability notion. A numerical example is given to illustrate the feasibility of the proposed approach.     

P2-Packing问题参数算法的改进

P₂-Packing问题是一个典型的NP难问题.目前这个问题的最好结果是时间复杂度为O^*(2^5.301k)的参数算法,其核的大小为15k.通过对P₂-packing问题的结构作进一步分析,提出了改进的核心化算法,得到大小为7k的核,并在此基础上提出了一种时间复杂度为O^*(2^4.142k)的参数算法,大幅度改进了目前文献中的最好结果.     

Quadratic stabilisation with H∞-norm bound of non-linear discrete-time uncertain systems with bounded control

The paper addresses the problem of quadratic stabilisability with H_∞-norm bound of uncertain discrete-time control-affine systems by norm-bounded controls. Both structured parameter uncertainties and unstructured exogenous disturbances are taken into account. The given definition of quadratic stabilisability is a generalisation of that used for linear systems so far. A necessary condition of the stabilisability is formulated. A state feedback control satisfying an a priori constraint is proposed for the solution of the formulated H_∞ problem. The proposed method may be applicable even in such cases when the linearisation technique cannot be used.     

Robust H∞ estimation of stationary discrete-time linear processes with stochastic uncertainties

The problem of H_∞ filtering of stationary discrete-time linear systems with stochastic uncertainties in the state space matrices is addressed, where the uncertainties are modeled as white noise. The relevant cost function is the expected value, with respect to the uncertain parameters, of the standard H_∞ performance. A previously developed stochastic bounded real lemma is applied that results in a modified Riccati inequality. This inequality is expressed in a linear matrix inequality form whose solution provides the filter parameters. The method proposed is applied also to the case where, in addition to the stochastic uncertainty, other deterministic parameters of the system are not perfectly known and are assumed to lie in a given polytope. The problem of mixed H₂/H_∞ filtering for the above system is also treated. The theory developed is demonstrated by a simple tracking example.     

Mixed H2/H∞ multi-channel linear parameter-varying control in discrete time

This paper develops a new method for the synthesis of linear parameter-varying (LPV) controllers in discrete time. LPV plants under consideration have a linear fractional transformation (LFT) representation. In contrast to earlier results which are restricted to single-objective LPV problems, the proposed method can handle a set of H₂/H_∞ specifications that can be defined channel-wise. This practically attractive extension is derived by using specific transformations of both the Lyapunov and scaling/multiplier variables in tandem with appropriate linearizing transformations of the controller data and of the controller scheduling function. It is shown that the controller gain-scheduling function can be constructed as an affine matrix-valued function in the polytopic coordinates of the scheduled parameter, hence is easily implemented on line. Finally, these manipulations give rise to a tractable and practical LMI formulation of the multi-objective LPV control problem.     

H∞-norm and invariant manifolds of systems with state delays

The problem of finding bounds on the H_∞-norm of systems with a finite number of point delays and distributed delay is considered. Sufficient conditions for the system to possess an H_∞-norm which is less or equal to a prescribed bound are obtained in terms of Riccati partial differential equations (RPDE’s). We show that the existence of a solution to the RPDE’s is equivalent to the existence of a stable manifold of the associated Hamiltonian system. For small delays the existence of the stable manifold is equivalent to the existence of a stable manifold of the ordinary differential equations that govern the flow on the slow manifold of the Hamiltonian system. This leads to an algebraic, finite-dimensional, criterion for systems with small delays.     

On Steiner minimal trees withLp distance

LetL _p be the plane with the distanced _p (A ₁ ,A ₂ ) = (¦x ₁ –x ₂¦ ^p + ¦y₁ –y ₂¦^p)^/1p wherex _i andy _i are the cartesian coordinates of the pointA _i . LetP be a finite set of points inL _p . We consider Steiner minimal trees onP. It is proved that, for 1 <p < , each Steiner point is of degree exactly three. Define the Steiner ratio _p to be inf{L _s (P)/L _m (P)¦PL _p } whereL _s (P) andL _m (P) are lengths of the Steiner minimal tree and the minimal spanning tree onP, respectively. Hwang showed ₁ = 2/3. Chung and Graham proved ₂ > 0.842. We prove in this paper that {} = 2/3 and (2/2)_{12 p} 3/2 for anyp.This work was supported in part by the National Science Foundation of China and the President Foundation of Academia Sinica.