Morgan's Problem Research Articles

Solution of Static Reduced Decoupling Problem for Linear Systems

We propose a structural solution by non-regular static state feedback to the diagonal, or row by row decoupling problem for linear systems. Without being completely general, this solution concerns the Reduced Morgan's Problem, that is we do not want it increases the essential orders of the original system. The solution that we propose relies on properties between, on one hand, some partial infinite structures extracted from well chosen interactors and on the other hand, the controllability indices of a specific shifted system. To my knowledge, there was at this date only very partial solutions to this problem deemed structurally hard.

More Confessions of a Hollywood Starlet (review)

Reviewed by: More Confessions of a Hollywood Starlet Karen Coats Douglas, Lola More Confessions of a Hollywood Starlet. Razorbill, 2006 [224p] ISBN 1-59514-051-4$16.99 Reviewed from galleys Ad Gr. 7-10 Former drug addict and teen star Morgan Carter is now living "out" in Fort Wayne, Indiana, having had her identity exposed by a jealous classmate in True Confessions of a Hollywood Starlet (BCCB 2/06). She decides to stay and finish her junior year in as much obscurity as she can muster, partly to continue her so-far-successful recovery, and partly to see what, if anything, will become of her fledgling romance with the very down-to-earth Eli. The stalkerazzi are back, though, and the publicity, combined with Eli's mother's fears that Morgan's problem with drugs is not in the past, proves too much for their relationship, which founders most of the year. Meanwhile, Morgan is trying to come to terms with unfinished business, namely the unreported rape that continues to haunt her, even more so now that her agent [End Page 288] stepfather is using her to court Harlan Darly, her rapist, as a client. Morgan's stepdad orchestrates publicity that portrays actor Harlan as a scorned suitor with a broken heart, and Morgan is forced to play along, knowing that it is her fault that her stepfather's career needs the boost in the first place. Finally, though, she decides to tell her story without naming names, hoping that it will help other girls just as her story of addiction and recovery has. In general, this sequel lacks the emotional depth and complexity of its predecessor; Morgan's compelling need to get back at the girl who outed her is overdrawn nearly to the point of chick-lit banality, and the process of facing up to her rape is not as well depicted as was her rehabilitation. Readers will also be disappointed in Eli, who never emerges as the hero we want him to be, though that's probably more realistic than not. Still, Morgan continues to be an effective and believable voice for the insecurities, small victories, pain, and perseverance of those in recovery, as well as for the humanity of the celebrities teens choose to worship. Copyright © 2007 The Board of Trustees of the University of Illinois

The solution of diagonal decoupling problem by dynamic output feedback and constant precompensator: the general case

In this paper, the solution to diagonal decoupling problem via dynamic output feedback and constant precompensator is presented in the general case. The literature contains a number of particular solutions for this problem. These solutions are valid, for instance, for square transfer matrices or they employ restricted dynamic output feedback schemes. Here, we start with a nonsquare transfer matrix and consider a dynamic output feedback scheme where there is a proper compensator in the feedback loop accompanied by a constant gain (precompensator) in the feedforward path. In this respect, the problem being considered is the dynamic output feedback version of the well-known Morgan's problem. The approach taken involves the characterization of open-loop diagonalizers which admit a dynamic output feedback realization. The key concepts involved in the solution are the generalized version of diagonal causality degree dominance and the set of attainable infinite zero orders. An example is also included to illustrate the main result. >

On the solvability of Morgan's problem (II): pole assignment while decoupling by state feedback and a constant singular input transformation

On the solvability of Morgan's problem (II): pole assignment while decoupling by state feedback and a constant singular input transformation

On the solvability of Morgan's problem (I): necessary and sufficient conditions for decoupling by state feedback and a constant singular input transformation

On the solvability of Morgan's problem (I): necessary and sufficient conditions for decoupling by state feedback and a constant singular input transformation

031 Towards a structural solution of the Morgan's problem

031 Towards a structural solution of the Morgan's problem

New results about Morgan's problem

Some aspects of the nonsquare Morgan's problem are considered. In particular, it is shown that the condition given by J. Descusse et al. (1986) to solve this problem is not sufficient. It is also shown that the essential orders are not necessarily preserved in the static decoupling problem. A necessary and sufficient condition for static decoupling of a system without modifying the essential orders is given for the case k=p-1. where p is the number of outputs and k is the rank at infinity of the proper part of the interactor.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Polynominal Formulation of Morgan's Problem

The Morgan's Problem is reconsidered and new and explicit necessary conditions are established for thereto exist a solution to this problem

Towards a Structural Solution of the Morgan's Problem

Abstract The static row-by-row decoupling problem of a linear time-invariant system without affecting its essential orders is considered. A sufficient condition to solve this problem is given in terms of the controllability indices of a input-reduced system.

Block noninteracting control with (non)regular static state feedback: A complete solution

We consider linear time-invariant systems ( C, A, B) where the output map C is partitioned into k blocks C i . We assume that the system are block right invertible, i.e. the rank of ( C, A, B) equals the sum of the ranks of the subsystems ( C i , A, B). We give, for the first time, a necessary and sufficient condition for the solution of the block decoupling problem using static state feedbacks of the type u = Fx + Gv, with G possibly nonregular; for solving the decoupling problem we impose that the rank of the closed-loop system equals that of ( C, A, B). This is a structural condition in terms of invariant lists of integers: the infinite zero orders, the block essential orders and Morse's list I 2 of ( C, A, B). The main result (Theorem 3) generalizes that of our previous work for the so called Morgan's Problem, i.e. the row by row decoupling problem.

New results on the structure of non‐linear systems: Application to the (dynamic) block‐decoupling problem

AbstractIn this paper our aim is twofold. Firstly, we briefly recall the solution to the dynamic block‐decoupling problem (DBDP) we presented at ‘NOLCOS 89’. This solution generalizes the approach taken for solving Morgan's problem, i.e. the row‐by‐row decoupling problem, by Descusse and Moog. The results we develop here are strongly connected with the structure algorithm of Hirschorn. Using the latter it is possible to exhibit particular (degenerate) controlled invariant distributions which appear to be crucial in the solution of the considered problem. The solution goes through an algorithmic procedure. Secondly, we give a new structure algorithm which makes possible the computation of the aforementioned distributions as well as the differential output rank of the considered system, which we must know for solving the DBDP. It is well known that the differential output rank, and more generally the structure at infinity, can be computed from the structure algorithm of Singh but not from the structure algorithm of Hirschorn. The former provides algebraic quantities (the structure at infinity) while the latter is mainly of interest for knowing geometric quantities (controlled invariant distributions). Our structure algorithm provides both features simultaneously.

The block essential orders and the dynamic block decoupling problem

In recent papers [8], it has been pointed out that two invariant lists of integers, the ‘infinite zero orders’ and the ‘essential orders’ play a key role in the solution of the so called ‘Morgan's problem’, i.e. the row by row decoupling problem. The purpose of the present paper is to define the ‘block essential orders’ of a linear system with respect to the block decoupling problem. This is done through a new dynamic solution to the latter, based upon a procedure which extends to the block case, that previously given in [15] for Morgan's problem. This procedure is minimal in the sense that it leads to the least infinite zero structure for any decoupled system reachable from the original one.

On the Structure Algorithm, Degenerate Controlled Invariant Distributions and the Block Decoupling Problem

In this paper, we propose a solution to the dynamic block decoupling problem. We generalize the approach taken for solving Morgan's Problem, i.e. the row by row decoupling problem, in (Descusse, Moog 1985. 1987). The preliminary results, we have already obtained (Descusse 1988), are strongly connected with the structure algorithm of Hirschorn (1979). Using the latter, it is possible to exhibit particular (degenerate) controlled invariant distributions which appear to be crucial in the solution of the considered problem. The solution goes through an algorithmic procedure.

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 I/sub 4/) and the essential orders of the system, which only depend on the input-output behavior, and Morse's list I/sub 2/ of the system, which depends on the choice of a particular state realization.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Further results on Morgan's problem

This paper deals with the row-by-row decoupling problem (known as the Morgan problem) with static state feedback. The solution of this problem is well known when some artificial constraints on the feedback laws are imposed (such as regular input transformations or restricted state feedbacks). However, in its most general setting, the problem remains unsolved. We provide here a new sufficient condition of its solvability. The fact that this condition turns out to be also necessary in the particular case of systems having 2 outputs and 3 inputs enforces its interest.

On the structure at infinity of linear block-decouplable systems: The general case

The aim of this note is to show that a linear system ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C, A, B</tex> ) is block-decouplable by means of static state feedback laws ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F, G</tex> ) with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</tex> invertible if and only if the infinite structure of ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C, A, B</tex> ) equals the union of the infinite structures of the subsystems ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C_{i}, A, B</tex> ) extracted from the given output partition. This result generalizes the ones recently obtained for Morgan's problem [6] and for the particular cases of right-invertible [7] or left-invertibie [8] systems.

Generic solvability of Morgan's problem

For <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</tex> -input, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</tex> -output, linear time-invariant systems of the form <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\dot{x}(t)=Ax(t)+ Bu(t); y(t)= Cx(t)</tex> it is shown that Morgan's problem, (decoupling into an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</tex> -single input-single output subsystem) is solvable for almost all real matrix triples ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A,B,C</tex> ) consistently dimensioned.