Inversion of linear dynamical systems in the presence of additive noise

We consider the problem of inversion of a stationary dynamical system (i.e., the generation of an estimate of the unknown output signal on the basis of the known output signal) in the presence of an unknown perturbation acting “not in the same channel” as the input to be estimated. We present necessary and sufficient conditions for the inversion of such systems and suggest inversion algorithms.     

Robust stabilization of uncertain linear dynamical systems with time-varying delay

In this paper, the problem of the robust stabilization for a class of uncertain linear dynamical systems with time-varying delay is considered. By making use of an algebraic Riccati equation, we derive some sufficient conditions for robust stability of time-varying delay dynamical systems with unstructured or structured uncertainties. In our approach, the only restriction on the delay functionh(t) is the knowledge of its upper boundh ^–. Some analytical methods are employed to investigate these stability conditions. Since these conditions are independent of the delay, our results are also applicable to systems with perturbed time delay. Finally, a numerical example is given to illustrate the use of the sufficient conditions developed in this paper.     

Inversion of vector delay systems

We consider stationary linear vector systems with commensurable delays. We obtain sufficient conditions for the reducibility of such systems to canonical form with the extraction of null dynamics. A constructive algorithm for the reduction of a system to that form is presented. We suggest a method for estimating the unknown input for vector delay systems with given accuracy.     

Perturbed linear stochastic dynamical systems

This paper is concerned with a class of stochastic differential equations which arises by adding a nonlinear term involving a small parameter δ to the drift coefficient of a linear stochastic system. First, a stochastic representation of the solutions of a certain class of Cauchy problems is studied. Second, a finite time expansion in powers of δ of the expectations of functions is established. Third, the corresponding ergodic expansions are sought with additional assumptions regarding the existence of a unique ergodic measure.     

Minimal stabilizers for linear dynamical systems

Doklady Mathematics -     

Fuzzy linear dynamical systems over fields

We investigate the problem of formalization of ways of describing fuzzy systems. The concept of fuzzy linearity for the class of dynamical systems is analyzed. Also, we consider some properties of the fuzzy dynamical systems that are linear over fields. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 3, pp. 147–155, 2007.     

Finite-time stabilization of impulsive dynamical linear systems

Finite-time stabilization of a special class of hybrid systems, namely impulsive dynamical linear systems (IDLS), is tackled in this paper. IDLS exhibit jumps in the state trajectory which can be either time-driven (time-dependent IDLS) or subordinate to specific state values (state-dependent IDLS). Sufficient conditions for finite-time stabilization of IDLS are provided. Such results require solving feasibility problems which involve Differential–Difference Linear Matrix Inequalities (D/DLMIs), which can be numerically solved in an efficient way, as illustrated by the proposed examples.     

On stabilizing uncertain linear delay systems

This note concerns the stabilization problem for linear delay systems containing uncertain elements. If the so-called matching conditions are satisfied and the uncertainties in the control matrix are not too large, there exist linear current-response feedback controls which guarantee asymptotic stability of the zero response of the system, no matter what the uncertainties and initial conditions are.     

Long-run behavior of constrained discrete-time linear dynamical systems

We consider a system of linear equations whose variablesy(t) E ^m describe an economy or process which is observed in succeeding discrete time periodst. With forms given for the equations relating the values of the state vectory(t) in succeeding periods and limiting the set of feasible choices ofy(t) in each period, conditions which determine the nature of feasible sequences of state vectors are given, and the characteristics ofy(t) ast are analyzed. Some of the results are studied in the framework of specified infinite horizon optimization problems.Research for this paper was partially supported by the Office of Naval Research, Contract N00014-69-A-0200-1055 at the Operations Research Center of the University of California, Berkeley.     

All-time Morse decompositions of linear nonautonomous dynamical systems

Morse decompositions provide inside information about the global asymptotic behavior of dynamical systems on compact metric spaces. Recently, the existence of Morse decompositions for nonautonomous dynamical systems was proved by restricting attention to the past or the future of the system, but in general, such a construction is not realizable for the entire time. In this article, it is shown that all-time Morse decompositions can be defined for linear systems on the projective space. Moreover, the dynamical properties are discussed and an analogue to the Theorem of Selgrade is proved.

     

Invariant cones and stability of linear dynamical systems

We present a method for the investigation of the stability and positivity of systems of linear differential equations of arbitrary order. Conditions for the invariance of classes of cones of circular and ellipsoidal types are established. We propose algebraic conditions for the exponential stability of linear positive systems based on the notion of maximal eigenpairs of a matrix polynomial. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 11, pp. 1446–1461, November, 2006.     

Stabilization of linear dynamical systems with respect to arbitrary linear subspaces

The classical notion of stabilizing a controlled dynamical system to some specified equilibrium point is extended to include stabilization to a specified linear subspace. Necessary and sufficient conditions for existence of a solution are derived and an explicit solution recipe is given for one special case.     

LDG method for reaction-diffusion dynamical systems with time delay

In this paper we introduce a local discontinuous Galerkin method to solve nonlinear reaction-diffusion dynamical systems with time delay. Stability and convergence of the schemes are obtained. Finally, numerical examples on two biologic models are shown to demonstrate the accuracy and stability of the method.     

Finite-dimensional time-invariant linear dynamical systems: Algebraic theory

Willems' approach to dynamical systems without a priori distinguishing between inputs and outputs is accepted, and with this as a starting point, new linear dynamical systems are introduced and studied. It is proved in particular that (in the complex case) the set of isomorphism classes of completely observable (or completely reachable) linear systems with given input and output numbers and McMillan degree, has a natural structure of a compact algebraic variety. This variety is closely connected to the one constructed by Hazewinkel using the Rosenbrock linear systems % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabeiEayaaca% aaaa!35DB!\[{\rm{\dot x}}\]=Ax+Bu, v=Cx+D(·)u, where D is a polynomial matrix, and may be regarded as the most natural compactification of it. (The connection is very similar to that of Grass_m,mx+p() and Mat_m.p(). Input/output linear systems, i.e. linear systems equipped with an extra structure which distinguishes input and output signals, are also considered. It is shown that each of them may be represented by the equations K% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabeiEayaaca% aaaa!35DB!\[{\rm{\dot x}}\]+L% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabeyDayaaca% aaaa!35D8!\[{\rm{\dot u}}\]=Fx+Gu, v=Hx+Ju (det(K–sF)0). Such systems clearly contain the so-called generalized linear systems. They also contain the Rosenbrock linear systems mentioned above and essentially coincide with them.     

A dynamical systems result on asymptotic integration of linear differential systems

We are interested in the asymptotic integration of linear differential systems of the form x′=[Λ(t)+R(t)]x, where Λ is diagonal and R∈L^p[t₀,∞) for p∈[1,2]. Our dichotomy condition is in terms of the spectrum of the omega-limit set ω_Λ. Our results include examples that are not covered by the Hartman-Wintner theorem.