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     

Balanced realization of stable transfer function matrices

Simple formulas are presented to compute the internally balanced minimal realization and the singular decomposition of the Hankel operator of a given continuous-time p×m stable transfer function matrix E(s)/d(s). The proposed formulas involve the Schwarz numbers of d(s) and the singular eigenvalues-eigenmatrices of a suitable finite matrix. Similar results are also obtained for a given discrete-time transfer function matrix     

Vibrational feedback control in the problem of acoustic stability

The problem of absolute stability in a vibrational feedback controller is introduced and discussed. It is shown that for any rational G(s)=n(s)/d(s ) with d(s) Hurwitz and deg d(s) -deg n(s)=1 there exists a linear dynamic periodic controller that ensures, in a certain sense, the infinite sector of absolute stability. This implies that an additional dynamical element, inserted in the feedback loop, may lead to improvements in the robustness of nonlinear systems     

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     

Comments on `Conditions for stable zeros of sampled systems' by M.Ishitobi

The commenter argues that the result of the above-titled work (see ibid., vol.37, no.10, p.1558-1561, Oct. 1992) is incorrect. It is pointed out that when sampling a continuous-time system G(s ) using zero-order hold, the zeros of the resulting discrete-time system H(z) become complicated functions of the sampling interval T. The system G(s) has unstable continuous-time zeros, s=0.1±i. The zeros of the corresponding sampled system start for small T from a double zero at z=1 as exp(T(0.1±i )), i.e., on the unstable side. For T>1.067 . . . the zeros become stable. The criterion function of the above-titled work, F(T)=G*(jω_s/2)= H(-1)T/2, is, however, positive for all T, indicating only stable zeros. The zero-locus crosses the unit circle at complex values     

Systolic computation of multivariable frequency response

An algorithm intended for software implementation on a programmable systolic/wavefront computer is presented for the computation of a complex-valued frequency-response matrix G. Typically, real-valued state-space model matrices are given and the calculation of G must be performed for a very large number of values of the scalar frequency parameter. The algorithm is an orthogonal version of an algorithm described previously by A.J. Laub (ibid., vol.26, no.4, p.407-8, 1981). The system matrix A is reduced initially to an upper Hessenberg form which is preserved as the frequency varies subsequently. A systolic QR factorization of a certain complex-valued matrix is then implemented for effecting the necessary linear system solution (inversion). The critical computational component is the back solve. This computational component's process dependency graph is embedded optimally in space and time through the use of a nonlinear spacetime transformation. The computational period of the algorithm is O(n) where n is the order of the matrix A     

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     

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     

Computational aspects of the bilinear transformation basedalgorithm for S-plane to Z-plane mapping

The author analyzes the computational complexity of an algorithm by F.D. Groutage et al. (ibid., vol.AC-32, no.7, p.635-7, July 1987) for performing the transformation of a continuous transfer function to a discrete equivalent by a bilinear transformation. Groutage et al. defend their method by noting that their technique is not limited to the bilinear transformation. Rather, it can be extended to any higher-order integration rule (Simpson, Runge-Kutta, etc.), or to any higher-order expansion of the ln function. In general, using the method, s can be any appropriate mapping function s=f (z)     

Balancing transformations for unstable nonminimal linear systems

It is shown that an unstable nonminimal continuous (discrete) realization (A, B, C) can be transformed via a similarity transformation into a balanced one if and only if the product of the controllability, observability Gramians is similar to a real diagonal matrix Λ. If, in addition, the eigenvalues of A , say λ, satisfy the relation λ_i+λ _j≠0(λ_iλ_j≠1) then the matrix Λ will always be positive semidefinite, and a balanced realization with its minimal part in the internally balanced form can always be obtained     

Out-of-roundness problem revisited

The properties and computation of the minimum radial separation (MRS) standard for out-of-roundness are discussed. Another standard out-of-roundness measurement called the minimum area difference (MAD) center is introduced. Although the two centers have different characteristics, the approach to finding both centers shares many commonalities. An O(n log n+k) time algorithm which is used to compute the MRS center is presented. It also computes the MAD center of a simple polygon G, where n is the number of vertices of G, and k is the number of intersection points of the medial axis and the farthest-neighbor Voronoi diagram of G. The relationship between MRS and MAD is discussed     

Adaptive structuring of binary search trees using conditionalrotations

Consider a set A={A₁,A₂ ,. . ., A_n} of records, where each record is identified by a unique key. The records are accessed based on a set of access probabilities S=[s₁,s₂ ,. . ., s_N] and are to be arranged lexicographically using a binary search tree (BST). If S is known a priori, it is well known that an optimal BST may be constructed using A and S. The case when S is not known a priori is considered. A new restructuring heuristic is introduced that requires three extra integer memory locations per record. In this scheme, the restructuring is performed only if it decreases the weighted path length (WPL) of the overall resultant tree. An optimized version of the latter method, which requires only one extra integer field per record has, is presented. Initial simulation results comparing this algorithm with various other static and dynamic schemes indicates that this scheme asymptotically produces trees which are an order of magnitude closer to the optimal one than those produced by many of the other BST schemes reported in the literature     

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)     

On a weaker notion of controllability of a language K withrespect to a language L

A necessary and sufficient condition for a prefix-closed language K⊆Σ* to be controllable with respect to another prefix-closed language L⊆Σ* is that K⊆ L. A weaker notion of controllability where it is not required that K⊆L is considered here. If L is the prefix-closed language generated by a plant automaton G, then essentially there exists a supervisor Θ that is complete with respect to G such that L(Θ|G)=K ∩ L if and only if K is weakly controllable with respect to L. For an arbitrary modeling formalism it is shown that the inclusion problem is reducible to the problem of deciding the weaker notion of controllability. Therefore, removing the requirement that K ⊆ L from the original definition of controllability does not help the situation from a decidability viewpoint. This observation is then used to identify modeling formalisms that are not viable for supervisory control of the untimed behaviors of discrete-event dynamic systems     

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     

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     

The LQG optimal regulation problem for systems with perfectmeasurements: explicit solution, properties, and application topractical designs

The continuous-time stationary linear-quadratic-Gaussian (LQG) optimal regulation problem is considered where the measurements are free of white noise components. A simple direct solution in the s-domain is derived for the optimal controller for general linear, time-variant right-invertible or left-invertible systems. The explicit expressions that are found for the controller transference and for the regulator return-ratio matrix can be used to obtain a practical suboptimal design in the s-domain. These expressions are applied to derive simple conditions for the precise recovery of the return-ratio matrix of the LQG regulator     

CAREL: computer aided reliability evaluator for distributedcomputing networks

An efficient method to compute the terminal reliability (the probability of communication between a pair of nodes) of a distributed computing system (DCS) is presented. It is assumed that the graph model G(V,E) for DCS is given and that the path and/or cut information for the network G(V,E) is available. Boolean algebraic concepts are used to define four operators: compare, reduce, combine, and generate. The proposed method, called CAREL, uses the four operators to generate exclusive and mutually disjoint events. CAREL has been implemented using bit vector representation on an Encore MULTIMAX 320 system. It is shown that CAREL solves large DCS networks (having a pathset on the order of 780 and a cutset on the order of 7300 or more) with a reasonable memory requirement. A comparison with other algorithms reveals the computational efficiency of the method. The proof of correctness of CAREL is included     

Descent algorithms for optimal periodic output feedback control

Periodic output feedback is investigated in the context of linear-quadratic regulation for finite-dimensional time-invariant linear systems. Discrete output samples are multiplied by a periodic gain function to generate a continuous feedback control. The optimal solution is obtained in two steps by separating the continuous-time from the discrete-time structure. First, the optimal pole placement problem under periodic output feedback is solved explicitly under the assumption that the behavior at the sample times has been specified in terms of a gain matrix G. Then the minimum value, which depends on G, is substituted into the overall objective. This results in a finite-dimensional nonlinear programming problem over all admissible gain matrices G. The solution defines the optimal periodic output feedback control via the formulas of the optimal pole placement problem. A steepest descent and a direct iterative method for solving this problem are formulated and compared. Numerical examples show that the performance using periodic output feedback is almost equivalent to that using optimal continuous-state feedback