269 research outputs found
Al'brekht's Method in Infinite Dimensions
In 1961 E. G. Albrekht presented a method for the optimal stabilization of smooth, nonlinear, finite dimensional, continuous time control systems. This method has been extended to similar systems in discrete time and to some stochastic systems in continuous and discrete time. In this paper we extend Albrekht's method to the optimal stabilization of some smooth, nonlinear, infinite dimensional, continuous time control systems whose nonlinearities are described by Fredholm integral operators
Algebraic geometric methods for the stabilizability and reliability of multivariable and of multimode systems
The extent to which feedback can alter the dynamic characteristics (e.g., instability, oscillations) of a control system, possibly operating in one or more modes (e.g., failure versus nonfailure of one or more components) is examined
Generic pure quantum states as steady states of quasi-local dissipative dynamics
We investigate whether a generic multipartite pure state can be the unique
asymptotic steady state of locality-constrained purely dissipative Markovian
dynamics. In the simplest tripartite setting, we show that the problem is
equivalent to characterizing the solution space of a set of linear equations
and establish that the set of pure states obeying the above property has either
measure zero or measure one, solely depending on the subsystems' dimension. A
complete analytical characterization is given when the central subsystem is a
qubit. In the N-partite case, we provide conditions on the subsystems' size and
the nature of the locality constraint, under which random pure states cannot be
quasi-locally stabilized generically. Beside allowing for the possibility to
approximately stabilize entangled pure states that cannot be exact steady
states in settings where stabilizability is generic, our results offer insights
into the extent to which random pure states may arise as unique ground states
of frustration free parent Hamiltonians. We further argue that, to high
probability, pure quantum states sampled from a t-design enjoy the same
stabilizability properties of Haar-random ones as long as suitable dimension
constraints are obeyed and t is sufficiently large. Lastly, we demonstrate a
connection between the tasks of quasi-local state stabilization and unique
state reconstruction from local tomographic information, and provide a
constructive procedure for determining a generic N-partite pure state based
only on knowledge of the support of any two of the reduced density matrices of
about half the parties, improving over existing results.Comment: 36 pages (including appendix), 2 figure
Lower Bounds on Complexity of Lyapunov Functions for Switched Linear Systems
We show that for any positive integer , there are families of switched
linear systems---in fixed dimension and defined by two matrices only---that are
stable under arbitrary switching but do not admit (i) a polynomial Lyapunov
function of degree , or (ii) a polytopic Lyapunov function with facets, or (iii) a piecewise quadratic Lyapunov function with
pieces. This implies that there cannot be an upper bound on the size of the
linear and semidefinite programs that search for such stability certificates.
Several constructive and non-constructive arguments are presented which connect
our problem to known (and rather classical) results in the literature regarding
the finiteness conjecture, undecidability, and non-algebraicity of the joint
spectral radius. In particular, we show that existence of an extremal piecewise
algebraic Lyapunov function implies the finiteness property of the optimal
product, generalizing a result of Lagarias and Wang. As a corollary, we prove
that the finiteness property holds for sets of matrices with an extremal
Lyapunov function belonging to some of the most popular function classes in
controls
Explicit Solutions and Stability Properties of Homogeneous Polynomial Dynamical Systems
In this paper, we provide a system-theoretic treatment of certain
continuous-time homogeneous polynomial dynamical systems (HPDS) via tensor
algebra. In particular, if a system of homogeneous polynomial differential
equations can be represented by an orthogonally decomposable (odeco) tensor, we
can construct its explicit solution by exploiting tensor Z-eigenvalues and
Z-eigenvectors. We refer to such HPDS as odeco HPDS. By utilizing the form of
the explicit solution, we are able to discuss the stability properties of an
odeco HPDS. We illustrate that the Z-eigenvalues of the corresponding dynamic
tensor can be used to establish necessary and sufficient stability conditions,
similar to these from linear systems theory. In addition, we are able to obtain
the complete solution to an odeco HPDS with constant control. Finally, we
establish results which enable one to determine if a general HPDS can be
transformed to or approximated by an odeco HPDS, where the previous results can
be applied. We demonstrate our framework with simulated and real-world
examples.Comment: 8 pages, 4 figure
Theory of nonlinear feedback under uncertainty
AbstractOur main purpose here is to demonstrate the potential of a new approach which is an important expansion of the feedback concept: we have chosen what seemed a natural way of tackling some traditional problems of the control theory and of comparing the results against those offered by conventional methods.The main problem considered is the output stabilization for uncertain plants. Using structural transformations, uncertain systems can change to the form convenient for output feedback design. Synthesis of observer-based control for asymptotical stabilization or uniform ultimate boundedness of the closed-loop system is provided.We consider the notions of asymptotic and exponential invariance of a control system implies its suboptimality.A method is described for stabilization of uncertain discrete-time plants of which only compact sets are known to which plants parameters and exogenous signals belong. New approaches for solving some central problems of mathematical control theory are considered for nonlinear dynamical systems. New criterious of local and global controllability and stabilizability are indicated and some synthesis procedures are suggested
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