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e-mail: eric.campos@ipicyt.edu.mx''' | e-mail: eric.campos@ipicyt.edu.mx''' | ||
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==Resumen== | ==Resumen== | ||
Latest revision as of 13:15, 7 March 2018
CÁLCULO DE COTAS INFERIORES DE LA ABSCISA DE ESTABILIDAD Y ALGUNAS APLICACIONES
Edgar Díaz, Baltazar Aguirre, Carlos Loredo
Departamento de Matemáticas
Universidad Autónoma Metropolitana
Iztapalapa, San Rafael Atlixco 186, 09340, CDMX México
e-mail: dige_uam@yahoo.com.mx, bahe@xanum.uam.mx, carlos.loredo@ipicyt.edu.mx Jorge López
Departamento de Matemáticas
Universidad de Sonora
Boulevard Rosales y Luis Encinas s/n. Col. Centro, 83000, Hermosillo, Son, México
e-mail: conacyt.pnc01@ibero.mx
Eric Campos
División de Matemáticas
IPICyT, SLP
Camino a la Presa San José 2055, Col. Lomas 4a Sección, 78216 SLP, México
e-mail: eric.campos@ipicyt.edu.mx
Resumen
Si Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): p(t)
es un polinomio Hurwitz la mayor parte real de sus raíces es llamada la abscisa de estabilidad de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): p(t)
- este parámetro nos permite analizar la estabilidad de un sistema. En este artículo se presentan diversas cotas inferiores de la abscisa de estabilidad así como también una relación entre el Teorema de Bialas Generalizado y la abscisa de estabilidad que nos permite analizar la estabilidad robusta de un tipo de Familia de Polinomios.
Palabras Clave. polinomios Hurwitz, cotas inferiores, abscisa de estabilidad, estabilidad robusta.
COMPUTATION OF LOWER BOUNDS OF THE ABSCISSA OF STABILITY AND SOME APPLICATIONS
Abstract
If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): p(t)
is a Hurwitz polynomial, the largest real part of its roots is named the abscissa of stability of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): p(t)
, a parameter which allows us to analyze the stability of a system. In this paper we present several lower bounds of the stability abscissa as well as a relationship between the Generalization of the Bialas Theorem and the abscissa of stability that allows us to analyze the robust stability of a Family of Polynomials.
1 Introducción
Para tener estabilidad asintótica en un sistema continuo es necesario que todas las raíces de su polinomio característico asociado se encuentren en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbb{C}^{-}} , donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbb{C}^{-}}
es el conjunto de números complejos que tienen parte real negativa. En el caso en que un polinomio tenga todas sus raíces en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbb{C}^{-}}
se dice que es un polinomio Hurwitz. Routh (1887) y Hurwitz (1895) demostraron que la estabilidad se puede determinar directamente de los coeficientes del polinomio característico asociado. Los trabajos de Routh y Hurwitz dieron lugar al Criterio de Routh-Hurwitz, probablemente el criterio más popular para determinar si un polinomio es o no polinomio Hurwitz. Múltiples problemas de la mecánica, de la física y de la teoría del control se reducen al problema de las raíces de los polinomios, esto motivo numerosas investigaciones que tenían por objeto conocer la posición de las raíces en el plano complejo sin conocerlas. Para polinomios de coeficientes reales se buscaban cotas entre las que podían estar estas raíces, así, se elaboraban métodos para la búsqueda de cotas entre las que están comprendidas las raíces reales de un polinomios de coeficientes reales, obteniéndose diversas cotas para las raíces reales. A continuación damos la siguiente definición de abscisa de estabilidad.
Definición 1: Si Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p(t)}
es un polinomio estable de grado Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n}
con ceros Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle w_1}
, …, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle w_n} , el número negativo
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es definido como la abscisa de estabilidad de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p(t)} . El siguiente teorema nos ayudara a comprender la importancia de conocer la abscisa de estabilidad de un sistemas de ecuaciones diferenciales lineales.
Teorema 1: Consideremos el sistema de ecuaciones diferenciales:
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Las siguientes afirmaciones son equivalentes:
- Todos los valores propios de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle A}
tienen parte real negativa.
- Existen constantes positivas Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle a}
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle c} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle m}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle M}
, tales que, para todo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle x_{0} \in \mathbb{R}^{n}}
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Conociendo la abscisa de estabilidad tenemos que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle |e^{At} x_{0}| \leq M e^{\sigma _{p}t} |x_{0}|} . Entonces si Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma _{p}}
es la abscisa de estabilidad de un polinomio Hurwitz Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p(t)}
podemos predecir una cierta tasa mínima de decaimiento, por otra parte, el cálculo de la abscisa de estabilidad a sido requerido en el diseño de sistemas dinámicos y de control (ver [20]- [22]).
En este artículo presentamos diversas cotas inferiores de la abscisa de estabilidad primero mediante funciones simétricas y después en un segundo enfoque a partir de una relación entre las abscisas de estabilidad de un polinomio Hurwitz Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p(t)}
y su derivada Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p'(t)}
, esta desigualdad la usaremos para obtener cotas inferiores de la abscisa de estabilidad de un polinomio estable.
Como aplicación de la abscisa de estabilidad estudiamos su relación con el Máximo Intervalo de Estabilidad: Dado un polinomio Hurwitz Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p(t)}
de grado Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n}
y un polinomio arbitrario Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle q(t)}
consideremos la Familia Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p(t) + k q(t)}
donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k \in \mathbb{R}}
. El Máximo Intervalo de Estabilidad es denotado por Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (k_{\min }, k_{\max })}
de modo que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p(t) + k q(t)}
resulta ser Hurwitz para todo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k \in (k_{\min }, k_{\max })}
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k_{\min }}
es conocido como el Mínimo Extremo Izquierdo y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k_{\max }}
como el Máximo Extremo Derecho que resulta ser la abscisa de estabilidad de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p(t)}
.
2 Desarrollo
2.1 Cotas inferiores
Ahora presentamos algunas cotas inferiores para la abscisa de estabilidad de un polinomio utilizando funciones simétricas de los ceros. El parámetro abscisa de estabilidad se denomina en ocasiones como el grado de estabilidad. Aunque el término abscisa de estabilidad tiene preferencia ya que grado implica una cierta cantidad entera. Los índices Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle j}
se usaran de manera habitual para los siguientes rangos: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n= 2, 3, \ldots}
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i=1, 2, \ldots, n}
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle j=0, 1, \ldots, n}
.
2.2 Cotas inferiores a partir de funciones simétricas elementales
Las cotas inferiores del Teorema 2 son estructuralmente más simples que las de los siguientes teoremas, estas cotas son funciones de las funciones simétricas elementales de los ceros de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p(t)} .
Teorema 2: Sea Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p(t)}
un polinomio real estable y sea Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma }
su abscisa de estabilidad. Entonces las constantes
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(1) |
son cotas inferiores de la abscisa de estabilidad de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p(t)} , es decir,
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La igualdad se tiene si todos los ceros de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p(t)}
son reales e iguales, excepto para Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle S_{n-1, n}}
para la cual la igualdad se tiene si todos los ceros tiene igual parte real. Ver [17] para una demostración. El Teorema 2 define Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \binom{n+1}{2}}
cotas inferiores. El siguiente teorema muestra la mejor cota inferior de este conjunto.
Teorema 3: Sea Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p(t)}
un polinomio estable real y sean Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle S_{mn}}
cotas inferiores de la abscisa de estabilidad como fueron definidas en (1). Entonces la mejor cota inferior estará entre las cotas siguientes
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Ver [17] para una demostración. A partir de este teorema se desprende el siguiente corolario.
Corolario 1: Sea Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p(t)}
un polinomio real estable y sea Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma }
su abscisa de estabilidad, además, sea Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mu \leq 0}
un número dado y sea Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle q(t)}
un polinomio con abscisa de estabilidad Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mu }
. Una condición necesaria para que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p(t)}
sea al menos estable como Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle q(t)}
es que las siguientes desigualdades
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(2) |
se satisfagan. Este corolario es una generalización de la siguiente condición necesaria: Un polinomio estable tiene coeficientes positivos y de hecho se reduce a esta condición para Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mu=0} .
2.3 Cotas inferiores a partir de sumas de potencias
Diferentes cotas inferiores de la abscisa de estabilidad pueden ser obtenidas a partir de funciones simétricas de los recíprocos de los ceros de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p(t)} . Consideremos el polinomio siguiente:
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cuyos ceros son Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t_{i}^{-1}} . Consideremos ahora las sumas de potencias de los recíprocos de los ceros de un polinomio Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p(t)} . Denotemos la k-ésima suma de potencias por Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle s_{k}} , es decir,
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(3) |
Específicamente, las primeras cuatro sumas de potencias como funciones de los coeficientes de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p(t)}
son:
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Tales sumas de potencias aparecerán en el siguiente teorema.
Teorema 4: Sean Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p(t)}
un polinomio real estable, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma }
su abscisa de estabilidad y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle s_{k}}
, (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k=1, 2, \cdots ,} ) las sumas de potencias de los recíprocos de los ceros de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p(t)}
definidos como en (3). Entonces, para Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle s_{k} \neq 0} las constantes
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donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle c_{k} = \cos ^{k+1}{\left(\frac{\pi }{k+1}\right)}}
son cotas inferiores de la abscisa de estabilidad de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p(t)}
, es decir,
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La igualdad se tiene si todos los ceros de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p(t)}
son reales e iguales. Ver [17] para una demostración.
2.4 Cotas inferiores a partir de otras funciones simétricas
Adicionalmente diversas cotas inferiores de la abscisa de estabilidad pueden ser obtenidas a partir de otras funciones simétricas elementales.
Consideremos la siguiente función simétrica de los recíprocos de los ceros de un polinomio Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p(t)}
definida por:
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En términos de los coeficientes de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p(t)} ,
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Obtenemos la siguiente cota inferior de la abscisa de estabilidad de un polinomio real estable Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p(t)}
para Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h(t) \neq 0}
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con
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La igualdad se obtiene si todos los ceros de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p(t)}
son reales e iguales.
2.5 Segundo enfoque
Ahora presentamos algunas cotas inferiores de la abscisa de estabilidad como una consecuencia del Teorema de Gauss-Lucas. A partir de un polinomio Hurwitz de orden Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n}
mediante una sucesivo descenso del orden del polinomio obtenemos una sucesión de cotas inferiores que dependen de los coeficientes del polinomio de grado Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n}
.
Primero presentamos una desigualdad entre las abscisas de estabilidad de un polinomio Hurwitz Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p(t)}
y su derivada Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p'(t)}
. Sin pérdida de generalidad, vamos a considerar Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p(t)}
de la forma siguiente:
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A partir de esta desigualdad podemos obtener una cota inferior para la abscisa de estabilidad de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p(t)} . Usando la notación dada anteriormente vamos a denotar la abscisa de estabilidad de un polinomio Hurwitz Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p(t)}
por Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma _{p}}
y la abscisa de estabilidad de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p'(t)}
por Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma _{p'}}
. Esta relación es la desigualdad Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma _{p'} \leq \sigma _{p} } . Después utilizaremos esta desigualdad para obtener una cota inferior para la abscisa de estabilidad de un polinomio.
2.5.1 Abscisa de polinomios Hurwitz: Una desigualdad entre Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \sigma _ {p}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \sigma _{p'}
2.5.1 Abscisa de polinomios Hurwitz: Una desigualdad entre Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \sigma {p}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \sigma {p'}
La prueba del Teorema 6 está basada en el siguiente resultado de Gauss-Lucas.
Teorema 5: Cualquier semiplano cerrado que contiene todos los ceros de un polinomio Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p(t)}
también contiene todos los ceros de su derivada, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p'(t)}
. Ver [10, pág. 463], [14, pág. 11 y 12] y [19, pág. 84] para una demostración. Consideremos un polinomio Hurwitz Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p(t)}
de la forma indicada; entonces tenemos que
|
Así tenemos el siguiente resultado.
Teorema 6: Si Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p(t)}
es un polinomio Hurwitz y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma _{p}}
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma _{p'}}
son las abscisas de estabilidad de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p'}
respectivamente, entonces tenemos que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma _{p'} \leq \sigma _{p}}
. Ahora vamos a considerar dos ejemplos.
Ejemplo 1: Sea Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p(t) = t^{3} + \frac{19}{6} t^{2} + \frac{8}{3} t + \frac{2}{3}} . La abscisa de estabilidad de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p(t)}
es Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma _{p} = -0{.}5}
y la abscisa de estabilidad de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p'(t) = 3 t^{2} + \frac{19}{3} t + \frac{8}{3}}
es Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma _{p'} \approx - 0{.}58}
. Así tenemos que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma _{p'} < \sigma _{p}} .
Ejemplo 2: Sea Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p (t) = t^{3} + 4 t^{2} + 5 t + 2 } . La abscisa de estabilidad de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p(t)}
es Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma _{p} = -1}
y la abscisa de estabilidad de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p'(t) = 3 t^{2} + 8 t + 5}
es Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma _{p'} = -1}
. En este caso tenemos que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma _{p'} = \sigma _{p}} . El Teorema 6 nos plantea el siguiente problema abierto: Si Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p(t)}
es un polinomio Hurwitz encontrar condiciones necesarias y suficientes para tener la igualdad entre Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma _{p'}}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma _{p}}
. Relacionado con este problema, de manera inmediata se pueden dar los siguientes resultados.
Teorema 7: Si Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p(t) = a_{n} (t - \alpha )^{n}}
es un polinomio Hurwitz entonces se tiene que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma _{p'} = \sigma _{p}}
.
Teorema 8: Si Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p(t) = a_{n} (t-\alpha _{1}) (t- \alpha _{2}) \cdots (t - \alpha _{n})}
es un polinomio Hurwitz donde sus raíces son reales y distintas entonces Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma _{p'} < \sigma _{p}}
.
2.5.2 Una cota inferior de la abscisa de estabilidad de un polinomio
A partir de esta desigualdad podemos obtener el siguiente resultado que nos da una cota inferior para la abscisa de estabilidad de un polinomio estable.
Teorema 9: Sea Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p(t) = a_{n} t^{n} + a_{n-1} t^{n-1} + \cdots + a_{1} t + a_{0}}
un polinomio Hurwitz con coeficientes positivos, entonces tenemos que
(a) Si Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle [2 (n-1) a_ {n-1}]^{2} - 8 n (n-1) a_ {n} a_ {n-2} \geq 0} , entonces
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. (b) Si Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle [2 (n-1) a_{n-1}]^{2} - 8 n (n-1) a_{n} a_{n-2} < 0} , entonces
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.
A continuación presentamos algunos ejemplos los cuales ilustran el teorema dado.
Ejemplo 3: Consideremos el polinomio Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p(t) = 6 t^{5} + 43 t^{4} + 110 t^{3} + 125 t^{2} + 64 t + 12}
para este polinomio tenemos que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle [2 (n-1) a_{n-1}]^{2} - 8 n (n-1) a_{n} a_{n-2} = 12736 \geq 0}
. De la parte Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (a)}
del teorema 9 tenemos que
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es una cota inferior de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma _{p}} , esto es, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle -0{.}96 \leq {-1 / 2} = \sigma _{p}} .
Ejemplo 4: Sea el polinomio Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p(t) = t^{4} + 3 t^{3} + 5 t^{2} + 4 t + 2} . Entonces tenemos que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle [2 (n-1) a_{n -1}]^{2} - 8 n (n-1) a_{n} a_{n-2} = - 156 \leq 0 } . Por la parte Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (b)}
del teorema 9 tenemos que
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es una cota inferior de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma _{p}} , esto es, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle - 3/4 \leq - 1/2 = \sigma _{p}} . Realizando las siguientes sustituciones Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle m = n-1} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i = n}
en las cotas inferiores dadas en [16] y [17].
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(4) |
obtenemos que
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el cual es el mismo resultado obtenido en la parte Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (b)}
del Teorema 9. Del Teorema 9 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (a)} tenemos una cota para la abscisa de estabilidad de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p(z)} la cual depende de tres coeficientes mientras que las cotas inferiores de (4) sólo dependen de dos coeficientes de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p(t)} así en algunos casos esta cota es mejor que las dadas en (4) como se ilustra en los siguientes ejemplos.
Ejemplo 5: Sea Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p(t) = 6 t^{5} + 43 t^{4} + 110 t^{3} + 125 t^{2} + 64 t + 12} . Por la parte Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (a)}
del Teorema 9 tenemos que
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es una cota inferior de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma _{p} = - 1/2}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle - \frac{a_{n-1}}{n a_{n}} = -1{.}43 < - 0{.}96 < - \frac{1}{2} = \sigma _{p}}
.
Ejemplo 6: Sea Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p(t) = 6 t^{3} + 19 t^{2} + 16 t + 4} . Por la parte Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (a)}
del Teorema 9 tenemos que
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es una cota inferior de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma _{p} = - 1/2}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle S_{mi} = - \left[{\binom{n}{m}}^{-1} \binom{n}{i} \left(\frac{a_{m}}{a_{i}} \right)\right]^{1/(i-m)} < - 0{.}58 < \sigma _{p}, \forall m = 0, 1, \ldots, i-1}
.
2.6 La abscisa de estabilidad y Familias de Polinomios
Consideremos el siguiente polinomio Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p(t)= t^{n} + a_{1} t^{n-1} + a_{2} t^{n-2} + \cdots + a_{n-2} t^{2} + a_{n-1} t + a_{n}}
estable, podemos escribir la familia Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f_{r}(t)}
de la forma siguiente:
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donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f_{0}(t)= p(t)} , el cual es estable. Entonces tenemos que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f_{r}(t)}
es una familia de polinomios cuyos coeficientes son polinomios en la variable Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r}
. Con este planteamiento podemos relacionar el estudio de la abscisa de estabilidad de un polinomio con el estudio de la estabilidad de familias de polinomios. Existe una gran cantidad de literatura acerca de la Teoría de la Estabilidad de Familias de Polinomios ver, por ejemplo, [1],[2], [4] y [13].
2.6.1 El Teorema de Bialas Generalizado
El Teorema de Bialas fue publicado en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 1985} [6], ahora vamos a utilizar una generalización de este Teorema para calcular la abscisa de estabilidad de un polinomio.
Consideremos el polinomio de la forma
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donde
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle P_{0}(t)}
es un polinomio estable con coeficientes positivos,
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (2) \textrm{gr} < \textrm{gr}, i= 1, \ldots, m}
.
Además, sea Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H_{i}}
la matriz Hurwitz de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle P_{i}(t) \forall i, i = 1, \ldots, m}
.
El mayor intervalo de estabilidad Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (r_{\textrm{min}}, r_{\textrm{max}}) }
, tal que, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle P(t, r)}
es Hurwitz Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \forall r \in (r_{\textrm{min}}, r_{\textrm{max}})}
esta dado por
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r_{\textrm{max}} = 1 / \lambda _{\textrm{max}}^{+} (M)}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r_{\textrm{min}} = 1 / \lambda _{\textrm{min}}^{-} (M)}
donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \lambda _{\textrm{max}}^{+} (M)}
es el máximo valor propio positivo de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle M}
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \lambda _{\textrm{min}}^{-} (M)}
es el mínimo valor propio negativo de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle M}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle M}
esta dada por
|
(5) |
La prueba de este teorema es similar a la del Teorema Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 17{.}8{.}1}
[4] (ver también [9] y [15]. Ideas relacionadas fueron expuestas en el trabajo de J. Chen [7]. Esta generalización ha sido ampliamente divulgada por D. Henrion, ver [12].
2.7 La abscisa de estabilidad y el Teorema de Bialas Generalizado
Podemos escribir la familia de polinomios Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f_{r}(t)}
de la forma siguiente
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Mediante el resultado dado en 2.6.1 podemos calcular el Mínimo Extremo Izquierdo (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r_{\textrm{min}}} ) y el Máximo Extremo Derecho (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r_{\textrm{max}}} ). En este caso, la abscisa de estabilidad coincide con el Máximo Extremo Derecho, ya que
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Teorema 10: Sea p(t) un polinomio Hurwitz y sea la familia de polinomios Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f_{r}(t) = p_{0}(t) + r p_{1}(t) + r^{2} p_{2}(t) + \cdots + r^{n} p_{n}(t)}
con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p_{0}(t) = p(t)}
,
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y sean Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H_{0}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H_{1}} , …, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H_{n}}
las matrices Hurwitz de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p_{i}(t)}
para Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i = 0, 1, \ldots, n}
. Definamos Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle M}
como 5, entonces la abscisa de estabilidad esta dada por Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma = - r_{\textrm{max}}}
.
Sea Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p(t)}
un polinomio Hurwitz de grado Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n}
, obtenemos la familia de polinomios Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f_{r}(t) = p(t-r)}
la cual podemos escribir como
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A partir del Teorema de Bialas Generalizado, tomando Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle P(t, r) =f_{r}(t)} , el Teorema se tiene ya que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma = - r_{\textrm{max}}} .
El Teorema de Bialas generalizado se obtiene directamente de las ideas de Saydy, Tits, Abed y Barmish (ver [4] y [15]). A continuación presentamos un ejemplo el cual ilustra el teorema dado.
Ejemplo 7: Consideremos el polinomio Hurwitz Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p(t) = t^{3} + 6 t^{2} + 11 t + 6} , entonces Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f_{r}(t)}
es una familia de polinomios, tal que,
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con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H_{0}} , …, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H_{3}}
las matrices Hurwitz de los polinomios Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle P_{0}}
, …, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle P_{3}}
respectivamente.
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y
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construyendo la matriz Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle M}
y calculando sus valores propios Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \lambda }
tenemos que
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Así Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \lambda _{\max }^{+}{(M)} = 1} , entonces por el Teorema 10 tenemos que
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Es fácil verificar que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p(t)}
puede ser factorizado como Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p(t) = (t+1)(t+2)(t+3)}
.
3 Conclusiones
En el análisis de la estabilidad de un sistema de ecuaciones diferenciales a partir de la abscisa de estabilidad asociada al sistema se obtuvieron diversas cotas inferiores de la abscisa primero mediante la utilización de funciones simétricas y después mediante el Teorema de Gauss-Lucas se obtuvieron otras cotas inferiores que en algunos casos son mejores que las obtenidas por el primer método. Como aplicación se presenta una relación entre la abscisa de estabilidad de un polinomio Hurwitz Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p(t)}
y el Teorema de Bialas Generalizado que nos permite analizar la estabilidad robusta de un tipo de Familia de polinomios.
Referencias
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[5] S.P. Bhattacharyya, H. Chapellat y L.H. Keel, Robust Control: The Parametric Approach, Prentice-Hall, Upper Saddle River, NJ, (1995).
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[10] P. Henrici, Applied and Computational Complex Analysis, Vol. 1, John Wiley Sons, (1974).
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[14] C.A. Loredo-Villalobos, Criterios para determinar si un polinomio es un polinomio Hurwitz, Reporte de los Seminarios de Investigación I y II, UAM-I, (2005).
[15] L.Saydy, A.L. Tits y E.H Abed, Guardian maps and the generalized stability of parametrized families of matrices and polynomials, Mathematics of Control, Signal and Systems, Vol. 3, pp. 345-371, (1990).
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[17] G.F. Schrack, Lower bounds to the abscissa of stability of a stable polynomial from symmetric functions, SIAM J. Appl. Math. Vol. 21, No. 3, pp. 373-379, (1971).
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[22] V. Zakian, Computation of the Abscissa of Stability by Repeated Use of the Routh Test, IEEE Transactions on Automatic Control, Vol. AC-24, No. 4, pp. 604-606, (1979).
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Published on 11/12/17
Submitted on 21/11/17
Volume 1, 2017
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