Riccati equations and normalized coprime factorizations for strongly stabilizable infinite-dimensional systems

The first part of the paper concerns the existence of strongly stabilizing solutions to the standard algebraic Riccati equation for a class of infinite-dimensional systems of the form Σ(A,B,S^−1/2B^*,D), where A is dissipative and all the other operators are bounded. These systems are not exponentially stabilizable and so the standard theory is not applicable. The second part uses the Riccati equation results to give formulas for normalized coprime factorizations over H_∞ for positive real transfer functions of the form D+S^−1/2B^*(author−A)⁻¹,B.     

On solving the Lyapunov and Stein equations for a companion matrix

When the matrix A is in companion form, the essential step in solving the Lyapunov equation PA + A^TP = −Q involves a linear n × n system for the first column of the solution matrix P. The complex dependence on the data matrices A and Q renders this system unsuitable for actual computation. In this paper we derive an equivalent system which exhibits simpler dependence on A and Q as well as improved complexity and robustness characteristics. A similar results is obtained also for the Stein equation P − A^TPA = Q.     

Two-step compensation of nonlinear systems

We study the problem of stabilization of nonlinear plants. We show that given a nonlinear plant P, if there exists a (nonlinear) compensator F, possibly unstable, which stabilizes P, then, with P₁: = P(I − F(− P))⁻¹, any C defined by C:= F + Q(I − P₁Q)⁻¹ for some finite-gain stable Q will stabilize P.     

Semigroup generation and stabilization by A-bounded feedback perturbations

Let A be a generator of a strongly continuous semigroup of operators, and assume that C and H are operators such that A + CH generates a strongly continuous semigroup S_H(t) on X. Let λ₀ be a real number in the resolvent set of A, and let ε [−1, 1]. Then there are some fairly unrestrictive conditions under which A+(λ₀ − A)CH(λ₀ − A)⁻ also generates a strongly continuous semigroup S_K(t) on X which has the same exponential growth rate as S_H(t). Given an input operator B, we can use this to identify a class of feedback perturbations K such that A + BK generates a strongly continuous semigroup. We can also use this result to identify classes of feedbacks which can and cannot uniformly stabilize a system. For example, we show that if the control on a cantilever beam in the state space H₀²[0, 1] × L²[0, 1] is a moment force on the free end, then we cannot stabilize the beam with an A^−1/2-bounded feedback, but we can find an A^−1/4-bounded feedback, for any > 0, which does stabilize the beam.     

Stability radii of differential algebraic equations with structured perturbations

This paper deals with a formula for computing stability radii of a differential algebraic equation of the form AX^′(t)−BX(t)=0, where A,B are constant matrices. A computable formula for the complex stability radius is given and a key difference between the ordinary differential equation (ODEs for short) and the differential algebraic equation (DAEs for short) is pointed out. A special case where the real stability radius and the complex one are equal is considered.     

Some investigation on Hermitian positive-definite solutions of a nonlinear matrix equation

In this paper, the Hermitian positive-definite solutions of the matrix equation X^s+A*X^?tA=Q are considered. New necessary and sufficient conditions for the equation to have a Hermitian positive-definite solution are derived. In particular, when A is singular, a new estimate of Hermitian positive-definite solutions is obtained. In the end, based on the fixed point theorem, an iterative algorithm for obtaining the positive-definite solutions of the equation with Q=I is discussed. The error estimations are found.     

Two iteration processes for computing positive definite solutions of the equation X - A*X−nA = Q

In the present paper, we suggest two iteration methods for obtaining positive definite solutions of nonlinear matrix equation X - A^*X⁻ⁿA = Q, for the integer n ≥ 1. We obtain sufficient conditions for existence of the solutions for the matrix equation. Finally, some numerical examples to illustrate the effectiveness of the algorithms and some remarks.     

Characterizations of distributions of exponential or geometric type by the integrated lack of memory property and record values

Let be a sequence of i.i.d. random variables, the sequence of its upper record values (i.e. L(0) = 1, L(n) = inf{jX_j>X_L(n−1} for n≥1). Without any assumptions to the support of P^X₁ the equidistribution of X₁ and a record increment X_L(n − X_L(n−1), n ≥ 1 yields X₁ to be either exponentially or geometrically distributed according to whether the additive subgroup generated by the support of P^X₁ is dense or a lattice in . The integrated lack of memory property can easily be reduced to the above problem for the case n = 1. Similarly the independence of X_L(n−1) and X_L(n) − X_L(n−1) for some n>1 characterizes X₁ to have e exponential or a geometric tail provided that the support of P^X₁ is bounded to the left and its right extremity no atom. Hence, if also its left extremity is no atom the independence of X_L(n−1) and X_L(n−1) − X_L(n−1) characterizes X₁ to be exponentially distributed.     

QFT template generation for time-delay plants based on zero-inclusion test

Plant template generation is the key step in applying quantitative feedback theory (QFT) to design robust control for uncertain systems. In this paper we propose a technique for generating plant templates for a class of linear systems with an uncertain time delay and affine parameter perturbations in coefficients. The main contribution lies in presenting a necessary and sufficient condition for the zero inclusion of the value set f(T,Q)={f(τ,q): τT[τ⁻,τ⁺], qQ∏_k=0^m−1[q_k⁻,q_k⁺]}, where f(τ,q)=g(q)+h(q)e^−jτω*, g(q) and h(q) are both complex-valued affine functions of the m-dimensional real vector q, and ω^* is a fixed frequency. Based on this condition, an efficient algorithm which involves, in the worst case, evaluation of m algebraic inequalities and solution of m2^m−1 one-variable quadratic equations, is developed for testing the zero inclusion of the value set f(T,Q). This zero-inclusion test algorithm allows one to utilize a pivoting procedure to generate the outer boundary of a plant template with a prescribed accuracy or resolution. The proposed template generation technique has a linear computational complexity in resolution and is, therefore, more efficient than the parameter gridding and interval methods. A numerical example illustrating the proposed technique and its computational superiority over the interval method is included.     

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.     

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 .     

Necessary and sufficient conditions for the existence of a positive definite solution of a nonlinear matrix equation

Some necessary and sufficient conditions for the existence of a positive definite solution of the nonlinear matrix equation X+A*X ^?α A=Q (0<α≤1) are given. By the way an iterative method is presented. Furthermore, the convergence and error estimation of the iterative algorithm are derived. The illustrative numerical examples due to Peng are worked out.     

Monoid-labeled transition systems

Given a -complete (semi)lattice , we consider -labeled transition systems as coalgebras of a functor ⁽⁻⁾, associating with a set X the set ^X of all -fuzzy subsets. We describe simulations and bisimulations of -coalgebras to show that L⁽⁻⁾ weakly preserves nonempty kernel pairs iff it weakly preserves nonempty pullbacks iff L is join infinitely distributive (JID).Exchanging for a commutative monoid , we consider the functor ⁽⁻⁾_ω which associates with a set X all finite multisets containing elements of X with multiplicities m M. The corresponding functor weakly preserves nonempty pullbacks along injectives iff 0 is the only invertible element of , and it preserves nonempty kernel pairs iff is refinable, in the sense that two sum representations of the same value, r₁ + … + r_m = c₁ + … + c_n, have a common refinement matrix (m_{(i, j)}) whose k-th row sums to r_k and whose l-th column sums to c_l for any 1≤ k ≤ m and 1 ≤ l ≤ n.     

On the connection between the exponential stability of C0-semigroups and the admissibility of a certain Sobolev space

Let T be a strongly continuous semigroup on a Banach space X and A its infinitesimal generator. We will prove that T is exponentially stable, if and only if, there exist p[1,∞) such that the space is admissible to the system Σ(A,I,I), defined below (i.e for all f belonging to the Sobolev space the convolution T*f lies in .     

Left-noncounting languages

LetX be a finite alphabet and letX ^* be the free monoid generated byX. A languageA X ^* is called left-noncounting if there existsk 0 such that forx,y X ^*,x ^k y A if and only ifx ^k+i y A. The class of all left-noncounting languages overX forms a Boolean algebra which generally contains properly the class of all noncounting languages overX and is properly contained in the class of all power-separating languages overX. In this paper, we discuss some relations among these three classes of languages and we characterize the automata accepting the left-noncounting languages and the syn tactic monoids of the left-noncounting languages.This research has been supported by Grant A7877 of the National Research Council of Canada.     

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.     

Solutions to the matrix equation

An explicit solution to the generalized Sylvester matrix equation AX−EXF=BY, with the matrix F being a companion matrix, is given. This solution is represented in terms of the R-controllability matrix of (E,A,B), generalized symmetric operator and a Hankel matrix. Moreover, several equivalent forms of this solution are presented. The obtained results may provide great convenience for many analysis and design problems. A numerical example is used to illustrate the effectiveness of the proposed approach.     

A generalization of concavity for finite differences

The concept of concavity is generalized to discrete functions, u, satisfying the n^th-order difference inequality, (−1)^n−kΔⁿu(m) ≥ 0, M = 0, 1,..., N and the homogeneous boundary conditions, u(0) = … = u(k−1) = 0, u(N + k + 1) = … = u(N + n) = 0 for some k “1, …, n − 1”. A piecewise polynomial is constructed which bounds u below. The piecewise polynomial is employed to obtain a positive lower bound on u(m) for m = k, …, N + k, where the lower bound is proportional to the supremum of u. An analogous bound is obtained for a related Green's function.     

Involution codes: with application to DNA coded languages

For an involution θ : Σ^* → Σ^* over a finite alphabet Σ we consider involution codes: θ-infix, θ-comma-free, θ-k -codes and θ-subword-k-codes. These codes arise from questions on DNA strand design. We investigate conditions under which both X and X⁺ are same type of involution codes. General methods for generating such involution codes are given. The information capacity of these codes show to be optimized in most cases. A specific set of these codes was chosen for experimental testing and the results of these experiments are presented.     

Optimal diagonal scaling for infinity-norm optimization

A solution is presented for the previously unsolved diagonally scaled multivariable infinity-norm optimization problem of minimizing D(s)(A(s) + Ψ(s) X(s))D⁻¹(s)_∞ over the set of stable minimum-phase diagonal D(s) and stable X(s). This problem is of central importance in the synthesis of feedback control laws for robust stability and insensitivity in the presence of ‘structured’ plant uncertainty. The result facilitates the design of feedback controllers which optimize the ‘excess stability margin’ [3] (or, equivalently, the ‘structured singular value μ’ [4]) of diagonally perturbed feedback systems.