MacMillan Degree Research Articles

Realization of multi-input/multi-output switched linear systems from Markov parameters

This paper presents a four-stage algorithm for the realization of multi-input/multi-output (MIMO) switched linear systems (SLSs) from Markov parameters. In the first stage, a linear time-varying (LTV) realization that is topologically equivalent to the true SLS is derived from the Markov parameters assuming that the discrete states have a common MacMillan degree and a mild condition on their dwell times holds. In the second stage, stationary point set of a Hankel matrix with fixed dimensions built from the Markov parameters is examined. Splitting of this set into disjoint intervals and complements reveals linear time-invariant dynamics prevailing on these intervals. Clustering over a feature space permits recovery of the discrete states up to similarity transformations which is complete if a unimodality assumption holds and the discrete states satisfy a residence requirement. In the third stage, the switching sequence is estimated by three schemes. The first scheme is non-iterative in time. The second scheme is based on matching the estimated and the true Markov parameters of the SLS system over segments. The third scheme works also on the same principle, but it is a discrete optimization/hypothesis testing algorithm. The three schemes operate on different dwell time and model structure requirements, but the dwell time requirements are weaker than that needed to recover the discrete states. In the fourth stage, the discrete state estimates are brought to a common basis by a novel basis transformation which is necessary for predicting outputs to prescribed inputs. Robustness of the four-stage algorithm to amplitude bounded noise is studied and it is shown that small perturbations may only produce small deviations in the estimates vanishing as noise amplitude diminishes. Time complexities of the stages are also studied. A numerical example illustrates the derived results.

The notion of almost zeros and randomness

Abstract We investigate the problem of almost zeros of polynomial matrices as used in system theory. It is related to the controllability and observability notion of systems as well as the determination of Macmillan degree and complexity of systems. We also present some new results on this important invariant in the light of randomness and we prove an uncertainty type relation appearing in such ensembles of operators.

Equivalence of a behavioral distance and the gap metric

In this article it is shown that for the class of stable linear state space systems with a fixed MacMillan degree, the topology of the gap between the graphs of the systems is equivalent to the topology of the gap between the extended graphs of the systems.

The 2-block superoptimalAAK problem

This paper presents a solution to a superoptimal version of the 2-blockAAK problem: Given a rational and antistable matrix functionR(s)=[R11(s)R12(s)] and a nonnegative integerk, find all superoptimal approximationsQ(s), with no more thank poles in the right-half complex plane, that minimize the supremum, over the imaginary axis, of the singular values of the error functionE(s)=[R11(s) R12(s)+Q(s)], with respect to lexicographic ordering. Conditions are given for which the superoptimal approximation is unique. In addition, ana priori upper bound on the MacMillan degree of the approximation is provided. The algorithm may be stopped after minimizing a given number of the singular values. This premature termination carries with it a predictable reduction in the MacMillan degree of the approximation. The algorithm only requires standard linear algebraic computations, and is therefore easily implemented.

Linear dynamical systems: compactness theorem

We define a natural compact topology on the set of all behaviours with a fixed input number and MacMillan degree contained in a given behaviour, and obtain a continuous bejective parametrization of this set. By a behaviour, we mean a certain generalization of what Willems terms by the same word.

A State-Space Algorithm for the Solution of the 2-Block Superoptimal Distance Problem

A state-space algorithm for computing the solution of the 2-block superoptimal distance problem (SODP) is presented. Given a rational and antistable matrix function $R(s) = [R_{11} (s)R_{12} (s)]$, find all stable approximations $Q(s)$ that lexicographically minimize the singular values of the error function $E(s) = [R_{11} (s)R_{12} (s) + Q(s)]$. Conditions are given for which the superoptimal approximation is unique. In addition, an a priori upper bound on the MacMillan degree of the approximation is given. The algorithm may be stopped after minimizing a given number of singular values. This premature termination of the algorithm carries with it an expected saving in the computational effort and a predictable reduction in the MacMillan degree of the approximation. The algorithm only requires standard linear algebraic computations and is, therefore, easily implemented.

Robust Output Regulation using Reduced-Order Controllers

In previous articles, robust regulator problem has been studied. It has been shown that an optimal robust controller (a solution to regulator problem) can be approached by using H ∞ -optimization and jω -axis shifting techniques. A problem with this approach is that the controller might have a high MacMillan degree which is n + 2pl – q , where n is the degree of the nominal plant P 0 (s) є (R(s)) p×m , l is the number of the unstable modes contained in the external reference input and the disturbance and q is one in general. In (Wu and Mansour 1990b) it was shown that there are always stable cancellations between the two parts of the controller: the servo-and the stabilizing compensators and the value n + pl – q is a degree bound for the controller. In this paper, the problem of using reduced-order controllers to achieve robust output regulation will be discussed.

A Degree Bound of the H∞-Optimal Solutions of the Robust Regulator Problem

In previous articles, the robust regulator problem has been studied. It has been shown that an optimal robust controller (a solution of the regulator problem) can be approached by using H∞-optimization and jw-axis shifting techniques. A problem with this approach is that the controller might have a high MacMillan degree which is n+2pl − r, where n is the degree of the nominal plant p0(s) ∊ (R(s))p × m, l is the number of the unstable modes contained in the external reference input and the disturbance and r is one in general. A few examples have suggested that a degree bound might be n + pi − r, because there are cancellations between the two parts of the controller: the servo- and the stabilizing compensators. In this paper, we will show that the value n + pl − r is indeed a degree bound provided the optimal stabilizing compensator is obtained by using a special computation method.

Optimal Hankel-norm approximation of stable systems with first-order stable weighting functions

Given a stable rational transfer function G( s) and weighting function W( s), the problem of finding G ̂ (s) of MacMillan degree k so as to minimise ∥W(G - G ̂ ∥ H is considered. This problem is solved for W( s) =( s- β)/( s- α) with no assumptions on the signs of α and β. This gives rise to approximations where ∥W(G - G ̂ )∥ ∞ can be accurately bounded from above and below in terms of the Hankel singular values of WG (when α, β > 0).

Proper and stable, minimal MacMillan degrees bases of rational vector spaces

The structure of proper and stable bases of rational vector spaces is investigated. We prove that if t(s) is a rational vector space, then among the proper bases of 3(s) there is a subfamily of proper bases which are 1) stable, 2) have no zeros in C\bigcup \{\infty\} and therefore are column (row) reduced at infinity, and 3) their MacMillan degree is minimum among the MacMillan degrees of all other proper bases of 3(s) and it is given by the sum of the MacMillan degrees of their columns taken separately. It is shown that this notion is the counterpart of Forney's concept of a minimal polynomial basis for the family of proper and stable bases of 3(s).

Classification of proper bases of rational vector spaces: minimal MacMillian degree bases

Abstract The algebraic structure of the get of all proper rational vectors contained in a given rational vector space 𝒯(s) is shown to be that of a noothorian ℝpr (s) -motiulo M*. ( Ropf;pr(a) is the ring of proper rational functions.) The proper submodules Mt of M* form an ascending chain of submodules partially ordered by an invariant of Mt defined as the valuation at s= ∞ ofMt , Tho various bases of Mt are examined and classified according to their property of ‘column reduceness at s = ∞’The concept of a prime column reduced at basis of Mt is introduced. It is shown that tho prime bases of Mt can be further classified by their MacMillan degrees and the existence of minimal MacMillan degree bases for Mt is established. A prime and minimal MacMillan degree basis ofMt extends Forney's concept of a minimal polynomial basis of ∞ (s) for the Rpr Mt -module The MacMillan degrees of the columns of such bases form a set of invariants Mt for which are defined as the 𝒯(s) generalized invariant dynamicol indices of Mt and a simple relation is established between (I) the generalized invariant dynamical indices Mt (ii) the orders of zeros at = ∞ s and (iii) the Forney invariant dynamical indices of Finally those results are specialized to the (maximal) noetherian ( Rpr(a)smodule M* it is shown that in this case the ' generalized invariant dynamical indices ' of M*. coincide with the invariant dynamical indices of Forney for 𝒯(s) thus providing an alternative interpretation of the Forney ' invariant dynamical order ' of 𝒯(s) as an absolute minimum of the MacMillan degree of any proper basis for 𝒯(s)