The only stable normal forms of affine systems under feedback are linear

We consider the problem of existence of structurally stable normal forms of affine control systems with m inputs and n-dimensional state space, equipped with C^∞-Whitney topology and acted on by the static state feedback group. It is proved that structurally stable normal forms exist only if m = n or m =1 and n = 2, and are linear. There are no stable normal forms in any other range of dimensions (m, n).     

A partial compactification for linear systems

In this note we prove a technical result concerning the closure of finite N × N Hankel matrices of rank n. In contrast to a corresponding result of Deistler and Hannan, this partial compactification of the space of transfer functions of order n contains boundary points which cannot be interpreted as linear systems of order < n. However, the algebraic geometric structure of is rather simple and of interest for the analysis of critical points of certain distance functions which arise in model reduction.     

Simultaneously controllable fuzzy matrices

Let F be a set of n × n fuzzy matrices. F is called simultaneously controllable if there exists a permutation matrix P such that for each A ε F, C = [c_ij] = P A P^T satisfies c_ij ≥ c_ji for i > j, where is the max-min composition. In this paper, the necessary and sufficient conditions for a set of n × n fuzzy matrices to be simultaneously controllable will be established. A constructive algorithm which can determine a simultaneously controllable set of n × n fuzzy matrices is presented as well.     

An alternative solution to the H∞-optimal control problem

In this paper we present an alternative solution to the problem min _{X ε H^n×n∞} |A + BXC|_∞ where A, B, rmand C are rational matrices in H^n×n_∞. The solution circumvents the need to extract the matrix inner factors of B and C, providing a multivariable extension of Sarason's H^∞-interpolation theory [1] to the case of matrix-valued B(s) and C(s). The result has application to the diagonally-scaled optimization problem int |D(A + BXC)D⁻¹|_∞, where the infimum is over D, X εH^n×n_∞, D diagonal.     

The Bruhat decomposition of finite Hankel matrices

The Bruhat decomposition of the space of n × n Hankel matrices is derived from the Bruhat decomposition of the general linear group. Some new results concerning the Hankel minors map H → (det H₁,…,det H_n) are obtained (det H_i denotes the i-th principal minor of H). The adherence order of the Bruhat decomposition is determined in the complex case.     

Real-time solutions of the origin-crossing problem

Then-dimensional origin-crossing language,O _n, is a language each of whose words describes a walk throughn-dimensional space beginning and ending at the origin. For eachn, O _n is real-time recognizable by ann-counter machine but not by any (n — 1)-counter machine. In contrast, for alln, O _n is real-time recognizable by a one-tape Turing machine.     

Nonlinear H∞ control around periodic orbits

In this paper, we study the nonlinear H_∞ control of systems with periodic orbits. We develop the notion of an induced L₂ gain (so-called nonlinear H_∞ norm) for systems where the no-disturbance behavior of the system is a periodic orbit and provide conditions under which the induced L₂ gain of the system (around the orbit) can be made less than a specified value by state feedback. This work is a natural extension of results on nonlinear H_∞ control of nonlinear systems in a neighborhood of a stable equilibrium point to the periodic orbit case. Synthesis of a nonlinear H_∞ state feedback controller is facilitated by the use of transverse coordinates and, in particular, the transverse linearization of the system.     

A numerical method for deadbeat control of generalized state-space systems

In this paper we give a new numerical method for constructing a rank m correction BF to an n × n matrix A, such that the generalized eigenvalues of λE−(A+BF) are all at λ = 0. In the control literature, this problem is known as ‘deadbeat control’ of a generalized state-space system Ex_i+1 = Ax_i + Bu_i, whereby the matrix F is the ‘feedback matrix’ to be constructed.     

Connectivity properties of spaces of partial realizations

This paper addresses the problem of determining the number of connected components of the space of real scalar M×N Hankel matrices of rank n. For n<min{M,N} it is shown that the space has n+1 components for M+N even, and is connected for M+N odd. As an application, the number of components of sets of minimal partial realizations of McMillan degree nτ of length τ sequences is determined.     

Feedback linearization of a flexible manipulator near its rigid body manifold

It is shown that a ‘slow’ 2n-dimensional manifold exists in the 4n-dimensional state space of an n-link manipulator with n flexible joints. While the flexible manifold is not linearizable by feedback, its restriction to'the slow manifold is. A method is given for computing a feedback control which achieves an approximate linearization.     

On the Construction of Classes of Suffix Trees for Square Matrices: Algorithms and Applications

We provide a uniform framework for the study of index data structures for a two-dimensional matrixTEXT[1:n, 1:n] whose entries are drawn from an ordered alphabetΣ. An index forTEXTcan be informally seen as the two-dimensional analog of the suffix tree for a string. It allows on-line searches and statistics to be performed onTEXTby representing compactly theΘ(n³) square submatrices ofTEXTin optimalO(n²) space. We identify 4ⁿ⁻¹families of indices forTEXT, each containing ∏ⁿ_i=1 (2i−1)! isomorphic data structures. We also develop techniques leading to a single algorithm that efficiently builds any index in any family inO(n² log n) time andO(n²) space. Such an algorithm improves in various respects the algorithms for the construction of the PAT tree and the Lsuffix tree. The framework and the algorithm easily generalize tod>2 dimensions. Moreover, as part of our algorithm, we provide new algorithmic tools that yield a space-efficient implementation of the “naming scheme” of R. Karpet al.(in“Proceedings, Fourth Symposium on Theory of Computing,” pp. 125–136) for strings and matrices.     

A necessary condition for quantitative exponential stability of linear state-space systems

For an arbitrary n×n constant matrix A the two following facts are well known:

• (1/n)Re(traceA)−max_j=1,…,nRe λ_j(A)0;

• If U is a unitary matrix, one can always find a skew-Hermitian matrix A so that U=e^A.

In this note we present the extension of these two facts to the context of linear time-varying dynamical systemsAs a by-product, this result suggests that, the notion of “slowly varying state-space systems”, commonly used in literature, is mathematically not natural to the problem of exponential stability.     

Lower Bounds for Merging Networks

A lower bound theorem is established for the number of comparators in a merging network. Let M(m, n) be the least number of comparators required in the (m, n)-merging networks, and let C(m, n) be the number of comparators in Batcher's (m, n)-merging network, respectively. We prove for n≥1 that M(4, n)=C(4, n) for n≡0, 1, 3 mod 4, M(4, n)≥C(4, n)−1 for n≡2 mod 4, and M(5, n)=C(5, n) for n≡0, 1, 5 mod 8. Furthermore Batcher's (6, 8k+6)-, (7, 8k+7)-, and (8, 8k+8)-merging networks are optimal for k≥0. Our lower bound for (m, n)-merging networks, m≤n, has the same terms as C(m, n) has as far as n is concerned. Thus Batcher's (m, n)-merging network is optimal up to a constant number of comparators, where the constant depends only on m. An open problem posed by Yao and Yao (Lower bounds on merging networks, J. Assoc. Comput. Mach.23, 566–571) is solved: lim_n→∞M(m, n)/n=log m/2+m/2^{log m}.     

Optimal Finite Characterization of Linear Problems with Inexact Data

Abstract. For many linear problems, in order to check whether a certain property is true for all matrices A from an interval matrix A, it is sufficient to check this property for finitely many “vertex” matrices A ∈ A. J. Rohn has discovered that we do not need to use all 2ⁿ² vertex matrices, it is sufficient to only check these properties for 2²ⁿ⁻¹ ≪ 2ⁿ² vertex matrices of a special type A_yz. In this paper, we show that a further reduction is impossible: without checking all 2²ⁿ⁻¹ matrices A_yz, we cannot guarantee that the desired property holds for all A ϵ A. Thus, these special vertex matrices provide an optimal finite characterization of linear problems with inexact data.     

Nonnegative matrices with prescribed extremal singular values

We consider the problem of constructing nonnegative matrices with prescribed extremal singular values. In particular, given 2n−1 real numbers and , we construct an n×n nonnegative bidiagonal matrix B and an n×n nonnegative semi-bordered diagonal matrix C, such that and are, respectively, the minimal and the maximal singular values of certain submatrices B_j and C_j of B and C, respectively. By using a singular value perturbation result, we also construct an n×n nonnegative matrix with prescribed singular values σ₁≥≥σ_n.     

Variable structure exponential stabilization of chained systems based on the extended nonholonomic integrator

In this paper, a new variable structure control strategy for exponentially stabilizing chained systems is presented based on the extended nonholonomic integrator model, the discontinuous coordinate transformation and the “reaching law method” in variable structure control design. The proposed approach converts the stabilization problem of an n-dimensional chained system into the pole-assignment problem of an (n−3)-dimensional linear time-invariant system and consequently simplifies the stabilization controller design of nonholonomic chained systems.     

A note on reduced order stabilizing output feedback controllers

In this paper we show that if a certain class of nonlinear systems is globally asymptotically stabilizable through an n-dimensional output feedback controller then it can be always stabilized through an (n − p)-dimensional output feedback controller, where p is the number of outputs and n is the dimension of the state space. This result gives an alternative construction of reduced order controllers for linear systems, and recovers in a more general framework the concept of dirty derivative, used in the framework of rigid and elastic joint robots, and gives an alternative procedure for designing reduced-order controllers for nonlinear systems considered in the existing literature.     

Linear Transformation Groups and Shape Space

The problem of modelling variabilities of ensembles of objects leads to the study of the shape of point sets and of transformations between point sets. Linear models are only able to amply describe variabilities between shapes that are sufficiently close and require the computation of a mean configuration. Olsen and Nielsen (2000) introduced a Lie group model based on linear vector fields and showed that this model could describe a wider range of variabilities than linear models. The purpose of this paper is to investigate the mathematics of this Lie group model further and determine its expressibility. This is a necessary foundation for any future work on inference techniques in this model.Let ^k _m denote Kendall's shape space of sets of k points in m-dimensional Euclidean space (k > m): this consists of point sets up to equivalence under rotation, scaling and translation. Not all linear transformations on point sets give well-defined transformations of shapes. However, we show that a subgroup of transformations determined by invertible real matrices of size k – 1 does act on ^k _m. For m > 2, this group is maximal, whereas for m = 2, the maximal group consists of the invertible complex matrices. It is proved that these groups are able to transform any generic shape to any other. Moreover, we establish that for k > m + 1 this may be done via one-parameter subgroups. Each one-parameter subgroup is given by exponentiation of an arbitrary (k – 1) × (k – 1) matrix. Shape variabilities may thus be modelled by elements of a (k – 1)²-dimensional vector space.     

Representation of ℤ<Subscript>4</Subscript>-Linear Preparata Codes Using Vector Fields

A binary code is called ℤ₄-linear if its quaternary Gray map preimage is linear. We show that the set of all quaternary linear Preparata codes of length n = 2^m, m odd, m ≥ 3, is nothing more than the set of codes of the form with