On input-output linearization of discrete-time nonlinear systems

We consider a nonlinear discrete-time system of the form Σ: x(t+1)=f(x(t), u(t)), y(t) =h(x(t)), where x ε R^N, u ε R^m, y ε R^q and f and h are analytic. Necessary and sufficient conditions for local input-output linearizability are given. We show that these conditions are also sufficient for a formal solution to the global input-output linearization problem. Finally, we show that zeros at infinity of ε can be obtained by the structure algorithm for locally input-output linearizable systems.     

Exact finite-dimensional filters via systems realization for a class of discrete-time nonlinear systems

We obtain necessary and sufficient conditions for the existence of a finite-dimensional filter for the discrete-time nonlinear system (ε x_k+1 =φ(x_k), y_k = h(x_k)+η(x_k)ν_k, K=0, 1,…. This system is distinguished by the absence of noise in the dynamic and by the correlation between the state and the intensity of noise in the observations.The necessary and sufficient condition provides an explicit formula for the minimal filter and various system-theoretic properties of (ε) and of the minimal filter.     

Stochastic adaptive pole-zero assignment with convergence analysis

The adaptive control u_n is designed for the stochastic system A(z)y_n+1 = B(z)u_n+C(z)w_n+1 with unknown constant matrix coefficients in the polynomials A(z), B(z) and C(z) in the shift-back operator with the purposes that (1) the unknown matrices are strongly consistently estimated and (2) the poles and zeros are replaced in such a way that the system itself is transferred to A⁰(z)y_n+1 = B⁰(z)u_n⁰+_n+1 with given A⁰(z), B⁰(z) and u_n⁰ so that the pole-zero assignment error {_n+1} is minimized. The problem of adaptive pole-zero assignment combined with tracking is also considered in this paper. Conditions used are imposed only on A(z), B(z) and C(z).     

Universal stabilization of a class of nonlinear systems with homogeneous vector fields

Under relative-degree-one and minimum-phase assumptions, it is well known that the class of finite-dimensional, linear, single-input (u), single-output (y) systems (A,b,c) is universally stabilized by the feedback strategy u = Λ(λ)y, λ = y², where Λ is a function of Nussbaum type (the terminology “universal stabilization” being used in the sense of rendering /s(0/s) a global attractor for each member of the underlying class whilst assuring boundedness of the function λ(·)). A natural generalization of this result to a class ^k of nonlinear control systems (a,b,c), with positively homogeneous (of degree k 1) drift vector field a, is described. Specifically, under the relative-degree-one (cb ≠ 0) and minimum-phase hypotheses (the latter being interpreted as that of asymptotic stability of the equilibrium of the “zero dynamics”), it is shown that the strategy u = Λ(λ)/vby/vb^k−1y, assures ^k-universal stabilization. More generally, the strategy u = Λ(λ)exp(/vby/vb)y, assures -universal stabilization, where = _{k 1} ^k.     

Inclusion properties for random relations under the hypotheses of stochastic independence and non-interactivity

This paper investigates whether random set inclusion is preserved by non-interactivity and by stochastic independence. Let (𝒳₁, x ₁), (𝒳₂, x ₂) be two random sets on U ₁ and U ₂, respectively, and let (𝒴₁, y ₁), (𝒴₂, y ₂) be two consonant inclusions of theirs. Let (𝒵₁, z ₁) be the random relation on U ₁ × U ₂ obtained from (𝒳₁, x ₁) and (𝒳₂, x ₂) under the hypothesis of stochastic independence, and let (𝒵₂, z ₂) ((𝒵₃, z ₃), respectively) be the random relation on U ₁ × U ₂ obtained from (𝒴₁, y ₁), (𝒴₂, y ₂) under the hypothesis of non-interactivity (stochastic independence, respectively). We prove that these hypotheses do not imply that (𝒵₁, z ₁) ? (𝒵₂, z ₂), but imply that (𝒵₁, z ₁) ? (𝒵₃, z ₃).     

New explicit filters and smoothers for diffusions with nonlinear drift and measurements

The optimal least-squares filtering of a diffusion x(t) from its noisy measurements {y(τ); 0 τ t} is given by the conditional mean E[x(t)|y(τ); 0 τ t]. When x(t) satisfies the stochastic diffusion equation dx(t) = f(x(t)) dt + dw(t) and y(t) = ∫₀^tx(s) ds + b(t), where f(·) is a global solution of the Riccati equation /xf(x) + f(x)² = f(x)² = αx² + βx + γ, for some , and w(·), b(·) are independent Brownian motions, Benes gave an explicit formula for computing the conditional mean. This paper extends Benes results to measurements y(t) = ∫₀^tx(s) ds + ∫₀^t dx(s) + b(t) (and its multidimensional version) without imposing additional conditions on f(·). Analogous results are also derived for the optimal least-squares smoothed estimate E[x(s)|y(τ); 0 τ t], s < t. The methodology relies on Girsanov's measure transformations, gauge transformations, function space integrations, Lie algebras, and the Duncan-Mortensen-Zakai equation.     

Oscillation criteria of certain nonlinear partial difference equations

This paper is concerned with the nonlinear partial difference equation with continuous variables

,where a, σ_i, τ_i are positive numbers, h_i(x, y, u) ε C(R⁺ × R⁺ × R, R), uh_i(x, y, u) > 0 for u ≠ 0, h_i is nondecreasing in u, i = 1, …, m. Some oscillation criteria of this equation are obtained.     

Global state and feedback equivalence of nonlinear systems

The problem of finding global state space transformations and global feedback of the form u(t)= α(x) + ν(t) to transform a given nonlinear system to a controllable linear system on Rⁿ or on an open subset of Rⁿ, is considered here. We give a complete set of differential geometric conditions which are equivalent to the existence of a solution to the above problem.     

Nonlinear eigenvalue problems for quasilinear systems

The paper deals with the existence of positive solutions for the quasilinear system (Φ(u'))' + λh(t)f(u) = 0,0 < t < 1 with the boundary condition u(0) = u(1) = 0. The vector-valued function Φ is defined by Φ(u) = (q(t)(p(t)u₁), …, q(t)(p(t)u_n)), where u = (u₁, …, u_n), andcovers the two important cases (u) = u and (u) = up > 1, h(t) = diag[h₁(t), …, h_n(t)] and f(u) = (f¹(u), …, fⁿ (u)). Assume that fⁱ and h_i are nonnegative continuous. For u = (u₁, …, u_n), let

, f₀ = maxf¹₀, …, fⁿ₀ and f_∞ = maxf¹_∞, …, fⁿ_∞. We prove that the boundary value problem has a positive solution, for certain finite intervals of λ, if one of f₀ and f_∞ is large enough and the other one is small enough. Our methods employ fixed-point theorem in a cone.     

Generalized dilations and the stabilization of uncertain systems via measurement feedback

We study the problem of semiglobally stabilizing uncertain nonlinear system

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, 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.     

Filtering of some nonlinear diffusions satisfying the general Bene condition

Under some regularity assumptions and the following generalization of the well-known Bene condition [1]:

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, where F(t,z) = g⁻²(t)∫f(t,z)dz, F_t, F_z, F_zz, are partial derivatives of F, we obtain explicit formulas for the unnormalized conditional density q_t(z, x) α Px_t ε dz| y_s, 0 st, where diffusion x_t on R¹ solves x₀ = x, dx_t = [β(t) + α(t)x_t + f(t, x_t] dt + g(t) dw₁, and observation y_t = ∫_o^th(s)x_s ds + ∫_o^t(s) dw_2t, with w = (w₁, w₂) a two-dimensional Wiener process.     

Exponential stability in mean square of neutral stochastic differential functional equations

The aim of this paper is to investigate the exponential stability in mean square for a neutral stochastic differential functional equation of the form d[x(t) − G(x_t)] = [f(t,x(t)) + g(t, x_t)]dt + σ(t, x_t)dw(t), where x_t = {x(t + s): − τ s 0}, with τ > 0, is the past history of the solution. Several interesting examples are a given for illustration.     

Backstepping boundary control of Burgers’ equation with actuator dynamics

In this paper, we propose a backstepping boundary control law for Burgers’ equation with actuator dynamics. While the control law without actuator dynamics depends only on the signals u(0,t) and u(1,t), the backstepping control also depends on u_x(0,t), u_x(1,t), u_xx(0,t) and u_xx(1,t), making the regularity of the control inputs the key technical issue of the paper. With elaborate Lyapunov analysis, we prove that all these signals are sufficiently regular and the closed-loop system, including the boundary dynamics, is globally H³ stable and well posed.     

Maximum flow and critical cutset as descriptors of multi-state systems with randomly capacitated components

Let G = (V, E, s, t) denote a directed network with node set V, arc set E = {1,…, n}, source node s and sink node t. Let Γ denote the set of all minimal s−t cutsets and b₁(τ), …, B_n(τ), the random arc capacities at time τ with known joint probability distribution function. Let Λ(τ) denote the maximum s−t flow at time τ and D(τ), the corresponding critical minimal s−t cutset. Let Ω denote a set of minimal s−t cutsets. This paper describes a comprehensive Monte Carlo sampling plan for efficiently estimating the probability that D(τ)εΩ-Γ and x<λ(τ)y at time τ and the probability that D(τ) Ω given that x < Λ(τ) y at time τ. The proposed method makes use of a readily obtainable upper bound on the probability that Λ(τ) > x to gain its computational advantage. Techniques are described for computing confidence intervals and credibility measures for assessing that specified accuracies have been achieved. The paper includes an algorithm for performing the Monte Carlo sampling experiment, an example to illustrate the technique and a listing of all steps needed for implementation.     

Empirical Bayes estimation of the scale parameter in a Pareto distribution

Let f(xθ) = αθ^αx^−(α+1)I(x>θ) be the pdf of a Pareto distribution with known shape parameter α>0, and unknown scale parameter θ. Let {(X_i, θ_i)} be a sequence of independent random pairs, where X_i's are independent with pdf f(xα_i), and θ_i are iid according to an unknown distribution G in a class of distributions whose supports are included in an interval (0, m), where m is a positive finite number. Under some assumption on the class and squared error loss, at (n + 1)th stage we construct a sequence of empirical Bayes estimators of θ_n+1 based on the past n independent observations X₁,…, X_n and the present observation X_n+1. This empirical Bayes estimator is shown to be asymptotically optimal with rate of convergence O(n^−1/2). It is also exhibited that this convergence rate cannot be improved beyond n^−1/2 for the priors in class .     

Monotone properties of certain classes of solutions of second-order difference equations

The authors consider the difference equations

δ(a_nδx_n)=q_nx_n+1

and

δ(a_nδx_n)=q_nf(x_n+1),

where a_n > 0, q_n > 0, and f: R å R is continuous with uf(u) > 0 for u ≠ 0. They obtain necessary and sufficient conditions for the asymptotic behavior of certain types of nonoscillatory solutions of (*) and sufficient conditions for the asymptotic behavior of certain types of nonoscillatory solutions of (**). Sufficient conditions for the existence of these types of nonoscillatory solutions are also presented. Some examples illustrating the results and suggestions for further research are included.     

Three point discretization of order four and six for (du :dx) of the solution of non-linear singular two point boundary value problem

We present finite difference methods of order four and six for the numerical solution of (du/dx) for the non-linear differential equation u″ = f(x,u,u′), 0 < x > 1 subject to the boundary conditions u(0) = A, u(l) =B. The proposed methods require only three grid points and applicable to both singular and non-singular problems. Numerical examples are given to illustrate the methods and their convergence.     

Stochastic stabilization and destabilization

It is shown in this paper that any nonlinear systems in ^d can be stabilized by Brownian motion provided |ƒ(x,t)| ≤ K|x| for some K > 0. On the other hand, this system can also be destabilized by Brownian motion if the dimension d ≥ 2. Similar results are also obtained for any given stochastic differential equation dx(t) = ƒ(x(t), t) + g(x(t), t) dW(t).     

Stable reduction of linear discrete-time systems via multipoint squared-magnitude CFE

This paper presents an elegant and computer-oriented procedure for the stable reduction of a linear discrete-time system via a multipoint continued-fraction expansion of its z-transfer function G(z). The proposed procedure involves the following three steps: (a) transform the z-domain squared-magnitude function P(z) = G(z)G(z ^?1) of the system frequency response to P(u) by the transformation u = z + z ^?1; (b) obtain an mth-order multipoint continued-fraction approximant P_m (u) to P(u); (c) factorize P_m (u) to yield a reduced z-transfer function G_m (z). The main feature of the procedure is that it guarantees the stability as well as the minimum-phase characteristics of the system and it also gives good overall approximations to both frequency and time responses.     

On the non-existence of stationary diffusions which satisfy the Bene condition

The one-dimensional diffusion x_t satisfying dx_t = f(x_t)dt + dw_t, where w_t is a standard Brownian motion and f(x) satisfies the Bene condition f′(x) + f²(x) = ax² + bx + c for all real x, is considered. It is shown that this diffusion does not admit a stationary probability measure except for the linear case f(x) = αx + β, α < 0.