Feedback affinization of nonlinear control systems

In this paper conditions for the nonlinear control system

to have a nonlinear feedback control

u=(x,v), vΩ′R^m′, m′m, 0Ω′

such that the nonlinear system takes a form of an affine system

are presented. All results require algebraic operations and differentiation of functions only.     

A unified presentation of some classes of meromorphically multivalent functions

The authors introduce and investigate various properties of a general class

U_k[p,a,β,A,B]

p, k ε N {1, 2, 3,…,}; 0 α < p; β 0

;

−1 A < B 1; 0 < B 1)

,which unifies and extends several (known or new) subclasses of meromorphically multivalent functions. The properties and characteristics of this general class, which are presented here, include growth and distortion theorems; they also involve Hadamard products (or convolution) of functions belonging to the class U_k[p,α,β,A,B].     

Global stability for separable nonlinear delay differential equations

Consider the following separable nonlinear delay differential equation

, where we assume that, there is a strictly monotone increasing function f(x) on (−∞, +∞) such that

In this paper, to the above separable nonlinear delay differential equation, we establish conditions of global asymptotic stability for the zero solution. In particular, for a special wide class of f(x) which contains a case of f(x) = e^x−1, we give more explicit conditions. Applying these, we offer conditions of global asymptotic stability for solutions of nonautonomous logistic equations with delays.     

Stabilization and tracking in the nonholonomic integrator via sliding modes

In this paper we use an approach based on sliding mode control to design a feedback which stabilizes the origin for the so-called nonholonomic integrator or Heisenberg system, a particular case of a canonical class of nonlinear driftless control systems of the form

which fail Brockett's necessary condition for the existence of a smooth stabilizing feedback.     

Dynamics of the difference equation

We study the invariant interval, the character of semicycles, the global stability, and the boundedness of the difference equation

where the initial conditions x_−k,…,x₋₁,x₀ are nonnegative, and p,q>0.     

Dynamic behavior for a system of neutral functional differential equations

Consider the system of neutral functional differential equations

where r>0, F, . It is shown that if F is nondecreasing on , and some additional assumptions hold, then the ω limit set of every bounded solution of such a system with some initial conditions is composed of 2r-periodic solutions. Our results are new and complement some corresponding ones already known.     

On the stability of a weighted diamond of real polynomials

Recently, a subset of the robust stability literature concentrated on the so-called diamond of polynomials. In this paper, we study the weighted diamond Q_w defined as

, where the center q^*, the weights w₀, w₁, 3dot , w_n > 0 and the radius r > 0 are given. For this family, we show that an extreme point result holds if and only if a certain ‘interlacing condition’ on the weights is satisfied.     

Estimating the distance to uncontrollability: A fast method and a slow one

For a linear control system (A,B), the distance to uncontrollability is characterized by

, where σ_n([A − λI,B]) is the smallest singular value of the augmented matrix [A − λI,B]. Two methods are developed to estimate the distance to uncontrollability, giving both a lower bound and an upper bound. One method is fast, requiring only one spectral decomposition of A and computations of three smallest singular values and being used for well-conditioned A. The other is slow, requiring computations of a large number of the smallest singular values, but it produces bounds as tight as possible and also a region containing global minimizers. Newton's method can be used to compute the global minimizers if one really wants.     

Generalized dilations and the stabilization of uncertain systems via measurement feedback

We study the problem of semiglobally stabilizing uncertain nonlinear system

Full-size image

, with (A,B) in Brunowski form. We prove that if p₁(z,u,t)u and p₂(z,u,t)u are of order greater than 1 and 0, respectively, with “generalized” dilation δ_l(z,u)=(l¹⁻ⁿz₁,…,l⁻¹z_n−1,z_n,lu) and uniformly with respect to t, where z_i is the ith component of z, then we can achieve semiglobal stabilization via arbitrarily bounded linear measurement feedback.     

Oscillation theorems for certain neutral functional differential equations

In this paper, we shall offer sufficient conditions for the oscillation of all solutions to neutral functional differential equations of mixed type of the form

,here p and q are periodic functions.     

Stability and boundedness in difference systems with finite delay

In this paper, stability and boundedness theorems for delay difference systems with the condition

δV(n,x(n))≤-W₂(|x(n)|)+σ

are given, where Δ is the backward difference operator. Some known results are generalized.     

A further result on the oscillation of delay difference equations

In this paper, we are concerned with the delay difference equations of the form

(*)

y_n+1 − y_n + p_ny_n−k = 0, N = 0, 1, 2, …,

(*)where p_n ≥ 0 and k is a positive integer. We prove by using a new technique that

guarantees that all solutions of equation (*) oscillate, which improves many previous well-known results. In particular, our theorems also fit the case where Σⁿ⁻¹_i=n−kp_i ≤ k^k+1/(k + 1)^k+1. In addition, we present a nonoscillation sufficient condition for equation (*).     

Factorization of disconjugate higher-order Sturm-Liouville difference operators

Using a recently proved equivalence between disconjugacy of the 2n^th-order difference equation

and solvability of the corresponding Riccati matrix difference equation, it is shown that the equation L(y) = 0 is disconjugate on a given interval if and only if the operator L admits the factorization of the form

L(y)_k+n=M^*(c_kM(y)_k)_k+n,

where M and its adjoint M* are certain n^th-order difference operators and c_k is a sequence of positive numbers.