Stability radii of infinite-dimensional systems subjected to unbounded stochastic perturbations

In this paper, we use the framework of stability radii to study the robust stability of linear deterministic systems on real Hilbert spaces which are subjected to unbounded stochastic perturbations. First, we establish an existence and uniqueness theorem of the solution of the abstract equation describing the system. Then we characterize the stability radius in terms of a Lyapunov equation or equivalently in terms of the norm of an input-output operator.     

Linear state-space systems in infinite-dimensional space: the roleand characterization of joint stabilizability/detectability

Several fundamental results from the theory of linear state-space systems in finite-dimensional space are extended to encompass a class of linear state-space systems in infinite-dimensional space. The results treated are those pertaining to the relationship between input-output and internal stability, the problem of dynamic output feedback stabilization, and the concept of joint stabilizability/detectability. A complete structural characterization of jointly stabilizable/detectable systems is obtained. The generalized theory applies to a large class of linear state-space systems, assuming only that: (i) the evolution of the state is governed by a strongly continuous semigroup of bounded linear operators; (ii) the state space is Hilbert space; (iii) the input and output spaces are finite-dimensional; and (iv) the sensing and control operators are bounded. General conclusions regarding the fundamental structure of control-theoretic problems in infinite-dimensional space can be drawn from these results     

Approximation-theoretic methods for nonlinear deconvolution and inversion

Nonlinear deconvolution and nonlinear inversion are cast as inverse problems in generalized Fock spaces. Generalized Fock spaces, introduced by de Figueiredo and Dwyer in [1], are reproducing kernel Hilbert spaces (RKHSs) of input-output maps represented by Volterra series equipped with an appropriately weighted inner product, the choice of the weights in the inner product depending on the particular problem under consideration. The solution to the nonlinear deconvolution problem presented here is the same as the one obtained previously for the nonlinear system identification problem [1–4]. However, the present solution to the nonlinear inversion problem consists of a new approach, whereby the unknown samples of the input are obtained from the given samples of the output by means of an efficient sequential algorithm. The algorithm is based on a framework which interpolates the input samples by an appropriate spline, and its sequential nature is elicited by the use of a truncated function basis to represent the spline.     

Sensitivity and robust stability of general input-output systems

The author considers a general model of an input-output system that is governed by nonlinear operator equations which relate the input, the state, and the output of the system. This model encompasses feedback systems as a special case. Assuming that the governing equations depend on a parameter A which is allowed to vary in a neighborhood of a nominal value A₀ in a linear space, the author studies the dependence of the system behavior on A. A system is considered insensitive if, for any fixed input, the output depends continuously on A. Similarly, the system is robust if it is stable for each A in a neighborhood of A₀. Stability is defined as an appropriate continuity of the input-output operator. The results give various sufficient conditions for insensitivity and robustness. Applications of the theory are discussed, including the estimation of the difference of operator inverses, and the insensitivity and robust stability of a Hilbert network, a feedback-feedforward system, a traditional feedback system, and a time-varying dynamical system described by a linear vector differential equation on (0, ∞)     

On controllability of diagonal systems with one-dimensional input space

This paper deals with diagonal systems on a Hilbert state space with a one-dimensional input space and a (possibly unbounded) control operator. A priori it is not assumed that the input operator is admissible. Necessary and sufficient conditions for different notions of controllability such as null-controllability, exact controllability and approximate controllability are presented. These conditions, which are given in terms of the eigenvalues of the diagonal operator and in terms of the control operator, are linked with the theory of interpolation in Hardy spaces.     

Interpolation and identification problems

The paper refines the relationship between the method of orthogonal moments for the identification of polynomial systems and the method of Lagrange-and Hermite-type operator interpolation in Hilbert space. The identification accuracy is estimated by the interpolation method and the minimum number of input signals that guarantee the prescribed accuracy is determined. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 100–107, May–June 2006.     

On the additive decomposition for polynomic maps

A polynomic operator can be represented asT= Ltauwhere τ is an imbedding of the input Hilbert resolution space into the appropriate Hilbert scale, andLis a mapping of the Hilbert resolution space that results from closing the span of the Hilbert scale with respect to a naturally defined inner product, into the output Hilbert resolution space. This paper demonstrates the relationship between the strongly causal (strongly anticausal, memoryless) components ofLandT.     

On the distributed stable full information H∞ minimax problem

We study the distributed parameter suboptimal full information H^∞ problem for a stable well-posed linear system with control u, disturbance w, state x, and output y. Here u, w, and y are L²-signals on (0,∞) with values in the Hilbert spaces U, W, and Y, and the state x is a continuous function of time with values in the Hilbert space H. The problem is to determine if there exists a (dynamic) γ-suboptimal feedforward compensator, i.e., a compensator 𝒰 such that the choice u = 𝒰w makes the norm of the input/output map from w to y less than a given constant γ. A sufficient condition for the existence of a γ-suboptimal compensator is that an appropriately extended input/output map of the system has a (J,S)-inner–outer factorization of a special type, and if the control and disturbance spaces are finite-dimensional and the system has an L¹ impulse response, then this condition is also necessary. Moreover, in this case there exists a central state feedback/feedforward controller, which can be used to give a simple parameterization of the set of all γ-suboptimal compensators. Our proof use a game theory approach. © 1998 John Wiley & Sons, Ltd.     

Adaptive control for infinite-dimensional systems

This paper deals with the design of a model reference adaptive controller for a distributed parameter system generated by a self-adjoint operator. The space of inputs and the space of outputs are both one dimensional. After introducing input-output representations, an adaptive control system is designed for the system with unknown input function and unknown output function. The input-output equations can be expressed using filtered values generated from the inputs and outputs     

Dynamical indices and order of delay-operator models

It is shown how to determine, just by inspection, the input and output dynamical indices and hence the order of (any minimal realization of) input-output discrete-time models involving a delay operator.     

Nonlinear input-normal realizations based on the differential eigenstructure of Hankel operators

This paper investigates the differential eigenstructure of Hankel operators for nonlinear systems. First, it is proven that the variational system and the Hamiltonian extension with extended input and output spaces can be interpreted as the Ga/spl circ/teaux differential and its adjoint of a dynamical input-output system, respectively. Second, the Ga/spl circ/teaux differential is utilized to clarify the main result the differential eigenstructure of the nonlinear Hankel operator which is closely related to the Hankel norm of the original system. Third, a new characterization of the nonlinear extension of Hankel singular values are given based on the differential eigenstructure. Finally, a balancing procedure to obtain a new input-normal/output-diagonal realization is derived. The results in this paper thus provide new insights to the realization and balancing theory for nonlinear systems.     

Input-output finite-time stability of time-varying linear singular systems

This paper studies the input-output finite-time stabilization problem for time-varying linear singular systems. The output and the input refer to the controlled output and the disturbance input,respectively.Two classes of disturbance inputs are considered,which belong to L-two and L-infinity.Sufficient conditions are firstly provided which guarantee the input-output finite-time stability.Based on this,state feedback controllers are designed such that the resultant closed-loop systems are input-output finite-time stable.The conditions are presented in terms of differential linear matrix inequalities.Finally,an example is presented to show the validity of the proposed results.     

On Covering Problems of Rado

T. Rado conjectured in 1928 that if ℱ is a finite set of axis-parallel squares in the plane, then there exists an independent subset ℐ⊆ℱ of pairwise disjoint squares, such that ℐ covers at least 1/4 of the area covered by ℱ. He also showed that the greedy algorithm (repeatedly choose the largest square disjoint from those previously selected) finds an independent set of area at least 1/9 of the area covered by ℱ. The analogous question for other shapes and many similar problems have been considered by R. Rado in his three papers (in Proc. Lond. Math. Soc. 51:232–264, 1949; 53:243–267, 1951; and J. Lond. Math. Soc. 42:127–130, 1968) on this subject. After 45 years, Ajtai (in Bull. Acad. Polon. Sci. Sér. Sci. Math. Astron. Phys. 21:61–63, 1973) came up with a surprising example disproving T. Rado’s conjecture. We revisit Rado’s problem and present improved upper and lower bounds for squares, disks, convex bodies, centrally symmetric convex bodies, and others, as well as algorithmic solutions to these variants of the problem.     

Optimal Test Signals as a Solution of the Generalized Bulgakov Problem

The Bulgakov problem on the accumulation of perturbations at the output of an object when disturbances of bounded amplitude, along with useful signals, act at the input is generalized. The optimal test signal is determined as the worst input disturbance for which the output error is maximal, using estimates of operator norms and eigenfunctions of operators associated with a linear stationary dynamic system. Norms for input and output signals are taken to be modular Hilbert and Chebyshev norms in different combinations. Nine problems thus obtained are classified by the type of constraints for the amplitude, area, and energy of the test signal. The results are applicable in diagnostics, metrology, and identification.     

Statistical properties of kernel principal component analysis

The main goal of this paper is to prove inequalities on the reconstruction error for kernel principal component analysis. With respect to previous work on this topic, our contribution is twofold: (1) we give bounds that explicitly take into account the empirical centering step in this algorithm, and (2) we show that a “localized” approach allows to obtain more accurate bounds. In particular, we show faster rates of convergence towards the minimum reconstruction error; more precisely, we prove that the convergence rate can typically be faster than n ^−1/2. We also obtain a new relative bound on the error. A secondary goal, for which we present similar contributions, is to obtain convergence bounds for the partial sums of the biggest or smallest eigenvalues of the kernel Gram matrix towards eigenvalues of the corresponding kernel operator. These quantities are naturally linked to the KPCA procedure; furthermore these results can have applications to the study of various other kernel algorithms. The results are presented in a functional analytic framework, which is suited to deal rigorously with reproducing kernel Hilbert spaces of infinite dimension. Editor: Nicolo Cesa-Bianchi An erratum to this article is available at .     

Generalized linear systems with lumped or distributed parameters and differential vector spaces

By employing differential vector spaces, we obtain results for linear systems with constant or varying coefficients and with lumped or distributed parameters which were only known for constant linear systems described by rational transfer matrices. Questions such as the characterization of input and output, input-output equivalence, invertibility, feedback decoupling and realization are investigated.     

On strong stabilization of asymptotically time-invariant linear time-varying systems

This paper considers the strong stabilization problem: given a linear time-varying system which is stabilizable by dynamic feedback, when can the stabilizer be chosen to be itself stable? We consider here the case of algebras of discrete time, time-varying systems which are asymptotically time-invariant, in the sense that as time evolves the time-varying transfer operator converges to a time-invariant transfer operator. Convergence here is in the sense of uniform or strong convergence of sequences of operators on an appropriate Hilbert space of input–output signals.     

广义时变脉冲系统的输入输出时域稳定

研究了广义时变脉冲系统的输入输出时域稳定问题.基于矩阵微分不等式(Differential matrix inequalities,DMI),给出了两个上述系统输入输出时域稳定的充分条件分别对应 L2干扰输入和 L∞干扰输入.这样的条件要求矩阵微分不等式解的存在性.接下来根据给出的充分条件设计了控制器,使得闭环系统输入输出时域稳定.本文的结果对于一般情况下的广义时变系统同样适用.最后,给出了两个算例来验证结果的有效性.