Damping margins of polynomials with perturbed coefficients

Considers the polynomial P(s)=t₀ Sⁿ+t₁ S^n-1 +···+t_n where 0<a _j⩽t_j⩽b_j. Recently, V.L. Kharitonov (1978) derived a necessary and sufficient condition for this polynomial to have only zeros in the open left-half plane. Two lemmas are derived to investigate the existence of theorems similar to the theorem of Kharitonov. Using these lemmas, the theorem of Kharitonov is generalized for P(s) to have only zeros within a sector in the complex plane. The aperiodic case is also considered     

A qualitative bound in robustness of stabilization by statefeedback

It is proved that placing the poles of a linear time-invariant system arbitrarily far to the left of the imaginary axis is not possible if small perturbations in the model coefficients are taken into account. Given a nominal controllable system (A₀, B ₀) with one input and at least two states and an open ball around B₀ (no matter how small), there exists a real number γ and a perturbation B within that ball such that for any feedback matrix K placing the eigenvalues of A ₀+B₀K to the left of Res=γ, there is an eigenvalue of A₀+BK with real part not less than γ     

Distributed detection with consulting sensors and communicationcost

The problem of distributed detection with consulting sensors in the presence of communication cost associated with any exchange of information (consultation) between sensors is considered. The system considered has two sensors, S1 and S2; S1 is the primary sensor responsible for the final decision u₀ , and S2 is a consulting sensor capable of relaying its decision u₂ to S1 when requested by S 1. The final decision u₀ is either based on the raw data available to S1 only, or, under certain request conditions, also takes into account the decision u₂ of sensor S2. Random and nonrandom request schemes are analyzed and numerical results are presented and compared for Gaussian and slow-fading Rayleigh channels. For each decision-making scheme, an associated optimization problem is formulated whose solution is shown to satisfy certain set design criteria that the authors consider essential for sensor fusion     

Damping ratio of polynomials with perturbed coefficients

Let a family of polynomials be P(s)=t ₀Sⁿ+t₁s ^n-1 . . .+t_n where O<α _j⩽t_j⩽β. Recently, C.B. Soh and C.S. Berger have shown that a necessary and sufficient condition for this equation to have a damping ratio of φ is that the 2ⁿ⁺¹ polynomials in it which have t_k=α_k or t_k=β_k have a damping ratio of φ. The authors derive a more powerful result requiring only eight polynomials to be Hurwitz for the equation to have a damping ratio of φ using Kharitonov's theorem for complex polynomials     

The stability of a family of polynomials can be deduced from afinite number 0(k3) of frequency checks

Let φ(s,a)=φ₀(s,a)+ a₁φ₁(s)+a₂ φ₂(s)+ . . .+a_kφ _k(s)=φ₀(s)-q(s, a) be a family of real polynomials in s, with coefficients that depend linearly on parameters a_i which are confined in a k-dimensional hypercube Ω_a . Let φ₀(s) be stable of degree n and the φ_i(s) polynomials (i⩾1) of degree less than n. A Nyquist argument shows that the family φ(s) is stable if and only if the complex number φ₀(jω) lies outside the set of complex points -q(jω,Ω_a) for every real ω. In a previous paper (Automat. Contr. Conf., Atlanta, GA, 1988) the authors have shown that -q(jω,Ω_a ), the so-called `-q locus', is a 2k convex parpolygon. The regularity of this figure simplifies the stability test. In the present paper they again exploit this shape and show that to test for stability only a finite number of frequency checks need to be done; this number is polynomial in k, 0(k³), and these critical frequencies correspond to the real nonnegative roots of some polynomials     

An H∞ approach to feedback design withtwo objective functions

The problem of tightly bounding and shaping the frequency responses of two objective functions T_i(s)( i=1,2) associated with a closed-loop system is considered. It is proposed that an effective way of doing this is to minimize (or bound) the function max {∥T₁(s)∥ _∞, ∥T₂(s)∥_∞} subject to internal stability of the closed-loop system. The problem is formulated as an H^∞ control problem, and an iterative solution is given     

M-dimensional Cayley-Hamilton theorem

The theorem states that every block square matrix satisfies its own m-D (m-dimensional, m⩾1) matrix characteristic polynomial. The exact statement and a simple proof of this theorem are given. The theorem refers to a matrix A subdivided into m blocks, and hence having dimension at least m. The conclusion is that every square matrix A with dimension M satisfies several m-D characteristic matrix polynomials with degrees N₁ . . ., N _m, such that N₁+ . . . +N_m ⩽M     

Strict aperiodic property of polynomials with perturbedcoefficients

Let a family of polynomials be P(s)=t ₀sⁿ+t₁s ⁿ±¹ + . . . + t_n where 0<a_j⩽t_j⩽b _j. V.L. Kharitonov (1978) derived a necessary and sufficient condition for the above equation to have only zeros in the open left-half plane. The present authors derive some similar results for the equation to be strictly aperiodic (distinct real roots)     

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, ∞)     

Explicit expressions for cascade factorization of generalnonminimum phase systems

Explicit expressions for two different cascade factorizations of any detectable left invertible nonminimum phase systems are given. The first one is a well known minimum phase/all-pass factorization by which all nonminimum phase zeros of a transfer function G(s) are collected into an all-pass factor V(s), and G (s) is written G_m(s)V$ where G_ms is considered as a minimum phase image of G(s). The second one is a new cascade factorization by which G(s) is rewritten as G_M( s)U(s) where U(s) collects all `awkward' zeros including all nonminimum phase zeros of G( s). Both G_m(s) and G_M(s) retain the given infinite zero structure of G(s). Further properties of G _m(s), G_M(s), and U (s) are discussed. These factorizations are useful in several applications including loop transfer recovery     

Decomposition of 2-D linear systems into 1-D systems in thefrequency domain

A method is presented for the decomposition of the frequency domain of 2-D linear systems into two equivalent 1-D systems having dynamics in different directions and connected by a feedback system. It is shown that under some assumptions the decomposition problem can be reduced to finding a realizable solution to the matrix polynomial equation X(z₁)P(z₂ )+Q(z₁)Y(z₂ )=D(z₁, z₂). A procedure for finding a realizable solution X(z₁ ), Y(z₂) to the equation is given     

A note on a classical bound for the moduli of all zeros of apolynomial

Considers the monic polynomial f(z):=z ⁿ+a_n-1z^n-1+. . .+a₀ in the complex variable z with complex coefficients. Under the assumption that the nonleading coefficients of f lie in the disk |z|⩽A the authors give an estimate for the smallest disk |z|⩽R containing all zeros of f. The estimate has a guaranteed precision of a few percent. They proceed similarly to obtain a zero-free disk |z |⩽r     

Bicoprime factorizations of the plant and their relation to right and left-coprime factorizations

In a general algebraic framework, starting with a bicoprime factorization P=N_prD^-1 N_pl, a right-coprime factorization N_p D_p^-1, a left-coprime factorization D^-1_pN_p, and the generalized Bezout identities associated with the pairs (N_p, D_p) and (D˜ _p, N˜_p) are obtained. The set of all H-stabilizing compensators for P in the unity-feedback configuration S(P, C) are expressed in terms of (N_pr, D, N _pt) and the elements of the Bezout identity. The state-space representation P=C(sI-A)^-1B is included as an example     

A formal analysis of the fault-detecting ability of testing methods

Several relationships between software testing criteria, each induced by a relation between the corresponding multisets of subdomains, are examined. The authors discuss whether for each relation R and each pair of criteria, C₁ and C₂ , R(C₁, C₂) guarantees that C₁ is better at detecting faults than C₂ according to various probabilistic measures of fault-detecting ability. It is shown that the fact that C ₁ subsumes C₂ does not guarantee that C₁ is better at detecting faults. Relations that strengthen the subsumption relation and that have more bearing on fault-detecting ability are introduced     

Solution to Morgan's problem

A necessary and sufficient condition is presented for the solution of the row-by-row decoupling problem (known as Morgan's problem) in the general case, that is, without any restrictive assumption added to the system to the feedback law u=Fx+Gy (G may be noninvertible). This is a structural condition in terms of invariant lists of integers which are easily computable from a given state realization of the system. These integers are the infinite zero orders (Morse's list I₄) and the essential orders of the system, which only depend on the input-output behavior, and Morse's list I₂ of the system, which depends on the choice of a particular state realization     

A sufficient condition for simultaneous stabilization

The condition under which it is possible to find a single controller that stabilizes k single-input single-output linear time-invariant systems p_i(s) (i=1,. . .,k) is investigated. The concept of avoidance in the complex plane is introduced and used to derive a sufficient condition for k systems to be simultaneously stabilizable. A method for constructing a simultaneous stabilizing controller is also provided and is illustrated by an example     

Performance evaluation of dynamic sharing of processors intwo-stage parallel processing systems

The performance of job scheduling is studied in a large parallel processing system where a job is modeled as a concatenation of two stages which must be processed in sequence. P_i is the number of processors required by stage P as the total number of processors in the system. A large parallel computing system is considered where Max(P₁, P₂)⩾P≫1 and Max(P₁ , P₂)≫Min(P₁, P₂). For such systems, exact expressions for the mean system delay are obtained for various job models and disciplines. The results show that the priority should be given to jobs working on the stage which requires fewer processors. The large parallel system (i.e. P≫1) condition is then relaxed to obtain the mean system time for two job models when the priority is given to the second stage. Moreover, a scale-up rule is introduced to obtain the approximated delay performance when the system provides more processors than the maximum number of processors required by both stages (i.e. P>Max(P₁, P₂)). An approximation model is given for jobs with more than two stages     

The design of a precompensator for multivariable adaptive control-anetwork-theoretic approach

A network-theoretic approach to the design of a dynamic precompensator C(s) for a multiinput, multioutput plant T(s) is considered. The design is based on the relative degree of each element of T(s). Specifically, an efficient algorithm is presented for determining whether a given plant T(s) has a diagonal precompensator C( s) such that, for almost all cases, T(s)C (s) has a diagonal interactor. The algorithm also finds any optimal precompensator, in the sense that the total relative degree is minimal. The algorithm can be easily modified to work even when a T(s) represented by a nonsquare matrix is given     

Simultaneous controller design for linear time-invariant systems

The use of generalized sampled-data hold functions (GSHF) in the problem of simultaneous controller design for linear time-invariant plants is discussed. This problem can be stated as follows: given plants P₁, P₂, . . ., P_N , find a controller C which achieves not only simultaneous stability, but also simultaneous optimal performance in the N given systems. By this, it is meant that C must optimize an overall cost function reflecting the closed-loop performance of each plant when it is regulated by C. The problem is solved in three aspects: simultaneous stabilization, simultaneous optimal quadratic performance, and simultaneous pole assignment in combination with simultaneous intersampling performance     

A frame approach to the H∞ superoptimal solution

A frame approach to the H_∞ superoptimal solution which offers computational improvements over existing algorithms is given. The approach is based on interpreting s numbers as the largest gains between appropriately defined spaces. Some useful bounds on Hankel singular values and s numbers are derived