16,823 research outputs found
Analysis and Design of Complex-Valued Linear Systems
This paper studies a class of complex-valued linear systems whose state
evolution dependents on both the state vector and its conjugate. The
complex-valued linear system comes from linear dynamical quantum control theory
and is also encountered when a normal linear system is controlled by feedback
containing both the state vector and its conjugate that can provide more design
freedom. By introducing the concept of bimatrix and its properties, the
considered system is transformed into an equivalent real-representation system
and a non-equivalent complex-lifting system, which are normal linear systems.
Some analysis and design problems including solutions, controllability,
observability, stability, eigenvalue assignment, stabilization, linear
quadratic regulation (LQR), and state observer design are then investigated.
Criterion, conditions, and algorithms are provided in terms of the coefficient
bimatrices of the original system. The developed approaches are also utilized
to investigate the so-called antilinear system which is a special case of the
considered complex-valued linear system. The existing results on this system
have been improved and some new results are established.Comment: 19 page
Exact eigenvalue assignment of linear scalar systems with single delay using Lambert W function
Eigenvalue assignment problem of a linear scalar system with a single
discrete delay is analytically and exactly solved. The existence condition of
the desired eigenvalue is established when the current and delay states are
present in the feedback loop. Design of the feedback controller is then
followed. Furthermore, eigenvalue assignment for the input-delay system is also
obtained as well. Numerical examples illustrate the procedure of assigning the
desired eigenvalue
Identification of linear systems by an asymptotically stable observer
A formulation is presented for the identification of a linear multivariable system from single or multiple sets of input-output data. The system input-output relationship is expressed in terms of an observer, which is made asymptotically stable by an embedded eigenvalue assignment procedure. The prescribed eigenvalues for the observer may be real, complex, mixed real and complex, or zero. In this formulation, the Markov parameters of the observer are identified from input-output data. The Markov parameters of the actual system are then recovered from those of the observer and used to obtain a state space model of the system by standard realization techniques. The basic mathematical formulation is derived, and extensive numerical examples using simulated noise-free data are presented to illustrate the proposed method
Tropical bounds for eigenvalues of matrices
We show that for all k = 1,...,n the absolute value of the product of the k
largest eigenvalues of an n-by-n matrix A is bounded from above by the product
of the k largest tropical eigenvalues of the matrix |A| (entrywise absolute
value), up to a combinatorial constant depending only on k and on the pattern
of the matrix. This generalizes an inequality by Friedland (1986),
corresponding to the special case k = 1.Comment: 17 pages, 1 figur
Diffusion of Context and Credit Information in Markovian Models
This paper studies the problem of ergodicity of transition probability
matrices in Markovian models, such as hidden Markov models (HMMs), and how it
makes very difficult the task of learning to represent long-term context for
sequential data. This phenomenon hurts the forward propagation of long-term
context information, as well as learning a hidden state representation to
represent long-term context, which depends on propagating credit information
backwards in time. Using results from Markov chain theory, we show that this
problem of diffusion of context and credit is reduced when the transition
probabilities approach 0 or 1, i.e., the transition probability matrices are
sparse and the model essentially deterministic. The results found in this paper
apply to learning approaches based on continuous optimization, such as gradient
descent and the Baum-Welch algorithm.Comment: See http://www.jair.org/ for any accompanying file
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