Time-domain input-output representations of linear systems

A non-oriented matrix pencil model containing internal and external variables is shown to possess properties useful for modelling and design of linear dynamical systems. Three of these properties are trivial inversion, invariance under a useful class of operations, and simple decomposition using easily-implemented row and column operations. Realizations of state-space and polynomial operator models are given, then a definition of system dimension and a minimal-reduction algorithm. A new time-domain canonical form compatible with polynomial operator systems is described. Examples of the use of the model in algebraic design problems are given.     

Persistence of equilibria for locally asymptotically stable systems

In this paper, we derive a necessary condition for local asymptotic stability of equilibria of nonlinear systems with parameters. As a corollary of our general result, we deduce Brockett's necessary condition for local asymptotic stability of equilibria of nonlinear autonomous systems. The proof we give, however, is quite different from the existing proofs in the literature. We show by way of an example that our new necessary condition is stronger than Brockett's necessary condition, but retains much of the user‐friendly appeal of Brockett's criterion. We further illustrate our result with more examples. Copyright © 2001 John Wiley & Sons, Ltd.     

A controller perturbation technique for transferring closed-loop stability between systems

This brief paper presents a perturbation technique with which a controller stabilizing one plant may be modified so that it stabilizes a second, related plant. The proposed technique produces an internally stable loop for a broad class of linear systems without requiring any further calculations on the part of the designer. Four seemingly different examples are described in terms of this result.     

An analysis of the parametrization by data driven local coordinates for multivariable linear systems

In this paper, we study a novel parametrization for state-space systems, namely data driven local coordinates (DDLC) which have recently been introduced and applied. Even though DDLC has meanwhile become the default parametrization used in the system identification toolbox of the software package MATLAB, an analysis of properties of DDLC, which are relevant to identification, has not been performed up to now. In this paper, we provide insights into the geometry and topology of the DDLC construction and show a number of results which are important for actual identification such as maximum likelihood-type estimation.     

Input-output finite time stabilization of linear systems

Bounded Input Bounded Output (BIBO) stability is usually studied when only the input-output behavior of a dynamical system is of concern. The present paper investigates the analogous concept in the framework of Finite Time Stability (FTS), namely the Input-Output FTS (IO-FTS). FTS has been already investigated in several papers in terms of state boundedness, whereas in this work we deal with the characterization of the input-output behavior. Sufficient conditions are given, concerning the class of L₂ and L_∞ input signals, for the analysis of IO-FTS and for the design of a static state feedback controller, guaranteeing IO-FTS of the closed-loop system. The effectiveness of the proposed results is eventually illustrated by means of some numerical examples.     

On the stability of slowly time-varying linear systems

New conditions are given in both deterministic and stochastic settings for the stability of the system x=A(t)x when A(t) is slowly varying. Roughly speaking, the eigenvalues of A(t) are allowed to wander into the right half-plane as long as on average they are strictly in the left half-plane.This work was funded by the NSF under Grant ECS-8806063, and was completed while the author was with the Department of Electrical and Computer Engineering, Johns Hopkins University, Baltimore, MD 21218, U.S.A.     

Feedback equivalence of constant linear systems

Two constant linear systems are said to be feedback equivalent if one can be transformed into the other via an element of the “feedback group”, which acts by state space feedback and by change of basis in the state and input spaces. Let C_n,m be the space of n-dimensional completely reachable systems with m-dimensional input (pairs of matrices, n × n and n × m). The action of the feedback group partitions the space C_n,m into finitely many orbits (equivalence classes), and the closure of each orbit is a union of orbits. If one views orbit closure as ‘deformation’, then orbit closure may be considered in terms of perturbations or system failure. In this paper we determine: (1) a classification of the orbits, and (2) the orbits contained in the closure of a given orbit. Both of these problems have been solved previously (see [1,4,6,3]); here we present simple proofs and point out a connection between this problem and the analogous problem for nilpotent matrices.     

Uniquely identifiable state-space and ARMA parametrizations for multivariable linear systems

Multivariable systems can be represented, in a uniquely identifiable way, either by canonical forms or by so-called overlapping forms. The advantage of the latter is that they do not require the a priori estimation of a set of structural invariants (e.g. Kronecker invariants). We show here how to define uniquely identifiable overlapping parametrizations for state-space and ARMA models. We show that these parametrizations are all related to a set of intrinsic invariants, which are obtained from the Markov parameters of the system. Different forms of overlapping ARMA parametrizations are derived and their properties discussed.     

Numerical analysis of the reduction of linear systems into orthogonal canonical form

An efficient numerically stable computational algorithm for reduction of linear systems into orthogonal canonical form is described. The algorithm is based on QR decomposition with column pivoting. Exact error bounds and operation count for the algorithm are derived.     

A stability result for linear parameter-varying systems

A negative-real-part eigenvalue assumption on a linear parameter-varying system implies exponential stability under a condition that includes either slowly varying parameters, or slowly accelerating parameters.     

Fast diffeomorphic matching to learn globally asymptotically stable nonlinear dynamical systems

We propose a new diffeomorphic matching algorithm and use it to learn nonlinear dynamical systems with the guarantee that the learned systems have global asymptotic stability. For a given set of demonstration trajectories, and a reference globally asymptotically stable time-invariant system, we compute a diffeomorphism that maps forward orbits of the reference system onto the demonstrations. The same diffeomorphism deforms the whole reference system into one that reproduces the demonstrations, and is still globally asymptotically stable.     

一类线性时变系统的输出反馈控制

基于线性时不变系统能控能观标准型变换及非线性系统高增益观测器方法,本文研究了一类线性时变系统 的输出反馈控制问题. 通过引入时变的状态变量坐标变换,分别设计了线性时变系统的状态反馈控制器、状态观测器以及基于 状态观测器的输出反馈控制器. 进一步地,本文分别证明了观测器动态误差是渐近收敛于零的,而状态反馈控制器以及输出反馈控制器可以 保证闭环系统的渐近稳定性.     

Generation of stable limit cycles in controllable linear systems

In this paper we prove that any controllable linear systems , admits a polynomial feedback u= u(x) such that the closed-loop system admits an orbitally asymptotically stable limit cycle.Moreover, we prove that for any positive integer n, there exists an nth-order polynomial, autonomous, ordinary differential equation with a unique limit cycle.     

Invertibility of switched linear systems

We address the invertibility problem for switched systems, which is the problem of recovering the switching signal and the input uniquely given an output and an initial state. In the context of hybrid systems, this corresponds to recovering the discrete state and the input from partial measurements of the continuous state. In solving the invertibility problem, we introduce the concept of singular pairs for two systems. We give a necessary and sufficient condition for a switched system to be invertible, which says that the individual subsystems should be invertible and there should be no singular pairs. When the individual subsystems are invertible, we present an algorithm for finding switching signals and inputs that generate a given output in a finite interval when there is a finite number of such switching signals and inputs. Detailed examples are included.     

Local parametrizations of ARMAX systems with nonlinear restrictions

A necessary and sufficient condition of local identifiability is derived by using the well-known Rank Theorem. Using this condition a suitable local parametrization is defined. The topological and geometrical properties of this parametrization are investigated. The results obtained generalize the corresponding results of Deistler and Wang (1989), which deal with essentially the systems with linear restrictions. The treatment covers also the systems without a priori restrictions.     

Constructive algebra methods for the L2-problem for stable linear systems

We investigate the feasibility of using computer algebra for solving L₂ system approximation problems. The first-order optimality conditions yield a set of polynomial equations, which can, in principle, be solved using Gröbner basis methods. A general solution along these lines would be tremendously useful, although it does not appear to be feasible at present, except for rather low McMillan degrees. We demonstrate that it can be feasible for specific examples; in such cases global optima can be found reliably. We show that the use of a Schwarz-like canonical form simplifies the structure of the problem.