Academia.edu no longer supports Internet Explorer.
To browse Academia.edu and the wider internet faster and more securely, please take a few seconds to upgrade your browser.
2002, Journal of the Franklin Institute
…
8 pages
1 file
Basic properties of a new class of strictly positive real (SPR) functions are stated. Four problems are studied. The first deals with SPR preservation of transfer functions, obtained under the composition of polynomials with SPR0 functions. The second deals with Hurwitz stability preservation of the numerator of transfer functions, obtained under the composition of polynomials with SPR0 functions. The third deals with making a Hurwitz closed-loop plant, an SPR0 function by substituting s by SPR0 functions. The four deals with the synthesis of simultaneous SPR feedback plants. For the new class of SPR0 functions, a characterization is presented. For the first and second problems, sufficient conditions are presented using the new class of SPR0 functions. For the third and four problems, two examples are presented, the first being for simultaneous SPR closed-loop systems via constant controllers. The second is for simultaneous stabilization via universal feedback adaptive control.
Strict positive realness is an important concept in many fields such as adaptive control, nonlinear control, network theory, filtering theory, absolute stability and others [1,2,[4][5][6]8,9,11]. The properties of strictly positive real (SPR) functions has *Corresponding author. Tel.: +1-5-267-4085; fax: +1-5-723-1104. E-mail address: guillermo.fernandez@uia.mx (G. Fern! andez-Anaya).
been intensively studied by several researchers [10,11,13]. Recently, Huang et al. [6] establish necessary and sufficient conditions for the existence of constant output feedback such that the closed-loop system is SPR. This result is used to develop an adaptive control scheme that can stabilize a system of unknown order with unknown parameters. Transfer functions obtained under the composition of polynomials with real rational functions have been studied in [12] where a general criterion has been established for the robust stability of pðGðsÞÞ where pðsÞ is a polynomial and GðsÞ is an uncertain transfer function. In [3], sufficient conditions to preserve stability in rational function under substitutions by SPR functions of zero relative degree (SPR0 functions) were presented and the closure of SPR0 functions under the composition of functions was proved.
In this note, the set of SPR0 functions (called SPR0 n functions) such that powers of these functions are SPR0 functions is characterized. These SPR0 n functions are used for give some solutions to several specific problems.
First we would like to find conditions under which pðGðsÞÞ is an SPR0 function, where pðsÞ is a given nontrivial polynomial with nonnegative coefficients and GðsÞ is an SPR0 function.
This will allow us to derive conditions under which the numerator of pðGðsÞÞ is a Hurwitz polynomial.
We also will treat some particular cases of the following stabilization problems: First two problems are of interest in the so-called uniform systems [7,12], these systems appears in many practical problems, uniform systems can be regarded as consisting of identical elements, for instance, identical regulators or sensors or servos, etc.
The solution of the last two problems are important in many contexts, for instance in adaptive control, network theory, filtering theory, absolute stability, adaptive estimation, synthesis of robust SPR systems and uniform systems [1,2,[4][5][6]9,[11][12][13].
Finally, an application to simultaneous stabilization via universal feedback adaptive control is also presented.
In this section, some basic definitions and a strict formulation of problems that we are going to study in this work are given.
The following definition is well known.
Definition 1 (Narendra and Annaswamy [11]). A rational function GðsÞ of zero relative degree is an SPR0 function if:
* GðsÞ is analytic in Re½sX0; * Re½GðjoÞ > 0 for all oAR with j ¼ ffiffiffiffiffiffi ffi À1 p :
The following definition plays the key role for the note.
Definition 2. Define the following set of transfer functions SPR0 n ¼ fGðsÞ j G n ðsÞASPR0g:
In particular SPR0 ¼ SPR0 1 :
It is known that SPR0 functions enjoy the properties:
1. If GðsÞ is SPR0 function, then 1=GðsÞ is also SPR0 function; 2. If G 1 ðsÞ and G 2 ðsÞ are SPR0 functions, then aG 1 ðsÞ þ bG 2 ðsÞ is an SPR0 function for a; bX0 (see [11]).
Uniform systems are systems constructed of identical components and ideal amplifiers [7,12]. The transfer functions of such a system is
where
and GðsÞ is a proper rational function.
The following lemma is a direct consequence of Theorem 9.9 from [2] and establishes the characterization of the SPR0 n functions. The following theorem is the main contribution of the note, and establishes necessary and sufficient coefficient type conditions that a linear fractional transform is an element of SPR0 n : Now, we make the following substitution u ¼ o 2 in sin 2 fðoÞ; then
We are looking for the max uX0 F ðuÞ: To this end we calculate dF du ¼ ½ðdbÞ 2 À ðacuÞ 2 ðad À bcÞ 2 ðd 2 þ c 2 uÞ 2 ðb 2 þ a 2 uÞ 2 :
If dF =du ¼ 0 then u ¼ 7bd=ac: So the max uX0 F ðuÞ is reached in # u ¼ bd=ac; and F ð # uÞ ¼ ððad À bcÞ=ðad þ bcÞÞ 2 : This means that sin fð # uÞ ¼ jðad À bcÞ=ðad þ bcÞj: We note that aðsÞASPR0 n if and only if Re½a n ðjoÞ > 0 for all oAR: Observe that cos j > 0 if and only if Àp=2ojop=2 module 2p: Then, aðsÞASPR0 n if and only if jðad À bcÞ=ðad þ bcÞjosinðp=2nÞ: & Corollary 6. If aðsÞ fulfills the conditions of the Theorem 5, then pðaðsÞÞASPR0 for each nontrivial polynomial pðsÞ with nonnegative coefficients of 0odeg ppn:
In this section, we present two examples which are related with the problem of synthesis of simultaneous SPR feedback plants.
The first example treats the problem for the class of all-pole plants, see [16]. In the second example the problem of simultaneous SPR feedback is studied via universal feedback control, for uniform systems, see [6,8,12]. Example 1. Consider the class of all-pole plants (see [16]) G p ðsÞ ¼ b 0 =pðsÞ; where b 0 > 0 is a real constant and pðsÞ is a real polynomial with deg pðsÞpm:
where k n ðtÞ is the estimate of k at time t [6,8]. By Theorem 2 in [3], the closed-loop systems G ki ðsÞ are SPR0 for any H i ðsÞASPR0 and by Theorem 1 in [6] there exists an adaptive control law u ¼ Àk n ðtÞy with ' k n ðtÞ ¼ Gy 2 ðtÞ and G > 0 an arbitrary positive real number, that can simultaneously stabilize each system GðH i ðsÞÞ of unknown parameters and drive y; x to zero in each one of them for any H i ðsÞASPR0 (see [8]).
Notice that if GðsÞ is minimum-phase and zero relative degree, by the lemma in [14], always there exists a constant controller k > 0; such that GðsÞ=ð1 þ GðsÞkÞ is an SPR0 function.
This example establishes that if a constant controller k makes a closed-loop plant an SPR0 function, then there exist an adaptive control law, which can simultaneously stabilize each system GðH i ðsÞÞ of unknown parameters and drive y;
x to zero in each one of them for any H i ðsÞ in SPR0.
In this paper a new class of SPR functions was defined. Using this new class of SPR functions, simple solutions were given for the four problems formulated in the introduction. The basic property studied is when a power of an SPR function is an SPR function again, with this property we obtain preservation of stability and preservation of SPR functions. This methodology was used for to establish conditions under which a controller simultaneously and robustly makes the closed-loop plant an SPR function. The study of powers of SPR matrix functions is under research.
IEEE Transactions on Automatic Control, 1999
International Journal of Systems Science, 2010
IEE Proceedings - Control Theory and Applications, 1996
IFAC Proceedings Volumes, 2004
IET Control Theory & Applications, 2008
2008 47th IEEE Conference on Decision and Control, 2008
IFAC Proceedings Volumes, 1995
IEEE Transactions on Automatic Control, 2003
3. Bilsel Uluslararasi Truva Bilimsel Araştirmalar ve İnovasyon Kongresi, 25-26 Mayıs 2024. Kongre Kitabı. Editors Emil Raul oğlu Ağayev, Doç. Dr. İlyas Erpay. -Çanakkale / Türkiye. ISBN: 978-625-6501-82-9 Astana Publications: Release Date: 07 June 2024. -p.1917-1931., 2024
Memórias da Academia de Marinha, 2021
Chronique d'Égypte, 2018
Iberia Judaica
Population & Avenir, 2006
2019
Russian Journal of Linguistics, 2021
Kadim, 2024
DPCE online, 2016
HISTORY OF ECONOMIC THOUGHT AND POLICY, 2023
Hepatology, 2009
IMAPS symposia and conferences, 2015
Jurnal Cakrawala Ilmiah, 2023
Biomacromolecules, 2015
Journal of materials chemistry. A, Materials for energy and sustainability, 2017
Maǧallaẗ Buḥūṯ Al-Šarq Al-Awsaṭ (Print), 2023
The Journal of Physical Chemistry B, 2012