Zeros and Zero Dynamics for Linear, Time-delay System
|
|
- Jennifer Hudson
- 6 years ago
- Views:
Transcription
1 UNIVERSITA POLITECNICA DELLE MARCHE - FACOLTA DI INGEGNERIA Dpartmento d Ingegnerua Informatca, Gestonale e dell Automazone LabMACS Laboratory of Modelng, Analyss and Control of Dynamcal System Zeros and Zero Dynamcs for Lnear, Tme-delay System
2 Tme delay systems are nterestng n connecton wth: ndustral applcatons (where delays are unavodable effects of the transportaton of materals) tele-operated systems, networked systems, large Integrated Communcaton Control Systems or ICCS (where delays orgnates from dspatchng nformaton through slow or very long communcaton lnes). In the last years, a great research effort has been devoted to the development of analyss and synthess technques for tme delay systems (Proc. IFAC Workshop on LTDS 2, 21, 23, 25).
3 The noton of ZERO and of ZERO DYNAMICS play an mportant role n several control problems, especally when solutons requre some sort of nverson. For classcal lnear systems, ZERO and ZERO DYNAMICS can be characterzed n abstract algebrac terms by the noton of ZERO MODULE (Wyman,San-1981). The noton of ZERO MODULE can be generalzed to other classes of dynamcal systems, notably to that of systems wth coeffcents n a rng. By explotng the relatons between systems wth coeffcents n a rng and tme-delay systems, sutable notons of ZERO and ZERO DYNAMICS can be defned these latter.
4 WS Zero Module for Lnear Systems Zero Module and Zero Dynamcs for Systems over Rngs Tme-delay Lnear Systems Zero Module and Zero Dynamcs for Tme-delay Lnear Systems Applcaton to nverson and trackng problems
5 Tme delay system wth uncommensurable delays: Σ d x& (t) y(t) k a 1 j k c 1 j A C j j x(t x(t jh jh ) + ) k b 1 j B j u(t jh ) + t R, contnuous tme axs x X R n, state value space {statesfunctons x: [T-ah,T) R n } ( -dm. R-vector space) u U R m, nput value space (m-dm. R-vector space) y Y R p, output value space (p-dm. R-vector space) A j, B j, C j real matrces of sutable dmensons h for 1,...,k are fxed tme delays
6 Gven a rng R, a system Σ wth coeffcents n R s a quadruple (A,B,C,X) where A, B, C are matrces of dmensons n n, n m, p n wth entres n R and X R n. Dynamcal nterpretaton: Σ x(t + 1) Ax(t) y(t) Cx(t) + Bu(t) t Z, ordered set of nteger numbers (dscrete tme) x X R n, state module (n-dm. free R-module) u U R m, nput module (m-dm. free R-module) y Y R p, output module (p-dm. free R-module)
7 Tme delay system Σ d x&(t) y(t) δ δ Σ d x& (t) y(t) a c A x(t h) + C x(t h) a c A C δ δ x(t) + x(t) A B C b B u(t a A b B c C b B δ δ δ δ u(t) h) Substtute δ wth the ndetermnate Δ System w. c.. the rng R R[Δ] Σ x(t + 1) Ax(t) y(t) Cx(t) + Bu(t)
8 Tme delay system Σ d x&(t) y(t) a c A x(t C x(t h) h) + b B u(t h) t, x, u, and y have dfferent meanngs n the two frameworks: e.g. x(t) R n n the tme delay fremework, x(t) R n n the rng framework; notatons are kept equal by abuse. System w. c. n the rng R R[Δ] Σ x(t + 1) Ax(t) y(t) Cx(t) + Bu(t)
9 x(t + 1) Ax(t) + Bu(t) Σ y(t) Cx(t) R[z] rng of polynomals n the ndetermnate z wth coeffcents n R R(z) S -1 R[z] localzaton at the multplcatve set S of all monc polynomals Transfer Functon Matrx G Σ C(zI-A) -1 B (entres n R(z)) R(z)-morphsm C(zI-A) -1 B : U R(z) Y R(z) t u(z) U R(z), u(z) u t z, u t U (nput sequence) y(z) Y R(z), y(z) y t z, y t Y (output sequence) t t t
10 Σ x(t + 1) Ax(t) + Bu(t) y(t) Cx(t) G Σ C(zI-A) -1 B : U R(z) Y R(z) R[z]-modules ΩU U R[z] U R(z) ΩY Y R[z] Y R(z) Defnton (CP-1983, after WS-1981) Gven the system Σ (A,B,C,X) wth coeffcents n the rng R and transfer functon matrx G Σ, the Zero Module of Σ s the R[z]-module Z Σ defned by Z Σ (G -1 Σ (ΩY) + ΩU)/(KerG + ΩU)
11 Σ x(t + 1) Ax(t) + Bu(t) G Σ C(zI-A) -1 B : U R(z) Y R(z) y(t) Cx(t) (G -1 (ΩY) + ΩU)/(KerG + ΩU) Z Σ Proposton (CP-1983) Let G Σ D(z) -1 N(Z) be a coprme factorzaton. The canoncal projecton p N : ΩY ΩY/NΩU nduces an njectve R[z]-homomorphsm α: Z Σ Tor(ΩY/NΩU). As a consequence of the above Proposton we have the followng foundamental result Z Σ s a fntely generated, torson R[z]-module
12 Tme delay system Σ d x&(t) y(t) δ δ Σ d x& (t) y(t) a c A x(t h) + C x(t h) a c A C δ δ x(t) + x(t) A B C b B u(t a A b B c C b B δ δ δ δ u(t) h) Substtute δ wth the ndetermnate Δ System w. c. n the rng R R[Δ] Σ x(t + 1) Ax(t) y(t) Cx(t) + Bu(t)
13 Tme delay system Σ d x&(t) y(t) a c A x(t C x(t h) h) + b B u(t h) Z Σ (G -1 (ΩY) + ΩU)/(KerG + ΩU) Z Σd ZERO MODULE System w. c. n the rng R R[Δ] Σ x(t + 1) Ax(t) y(t) Cx(t) + Bu(t)
14 Tme delay system Σ d x&(t) y(t) a c A x(t C x(t h) h) + b B u(t h) Defnton Gven a tme-delay system Σ d, let Σ be the assocated system wth coeffcents the rng R. The Zero Module Z Σd of Σ d s the Zero Module Z Σ of Σ. System w. c. n the rng R R[Δ] Σ x(t + 1) Ax(t) y(t) Cx(t) + Bu(t)
15 If the zero module Z Σ of Σ s free, over the rng R, then t can be represented as Z Σ (R m, D), where D: R m R m s an R- homomorphsm. Then, we can consder the followng noton. Defnton Gven a system Σ wth coeffcents n the rng R, whose zero module Z Σ can be represented as the par (R m,d), the Zero Dynamcs of Σ s the dynamcs nduced on R m by D, that s by the dynamc equaton z(t+1) Dz(t), for z R m. Remark that n case Z Σ cannot be represented as a par (R m,d), the Zero Dynamcs s not defned.
16 Σ (A,B,C,X) wth coeffcents n R: a controlled nvarant submodule (c..s.) of X s a submodule V X such that A(V) V + ImB feedback property: there exsts an R-morphsm F: X U such that (A+BF)V V (F s called a frend of V). V* maxmum c..s. contaned n KerC R* mnmum c..s. contanng ImB V* Proposton Gven Σ (A,B,C,X), w.c.n R and C(zI-A) -1 B D -1 N coprme, let N(U R(z)) be a drect summand of Y R(z) and let V* have the feedback property wth a frend F. Then, V*/R* endowed wth the R[z] structure nduced by (A+BF) s somorphc to Z Σ.
17 If V* has the feedback property wth a frend F and V*/R* s a free R-module, say V*/R* R m, lettng D be a matrx that represents (A + BF) V*/R* wth respect to the canoncal bass of R m, t s be possble to represent the Zero Dynamcs of Σ as the dynamcs nduced on R m by D: z(t+1) Dz(t), for z R m. The above characterzaton of Zero Dynamcs allows us to analyse t n a smple, practcal way, avodng the necessty of workng wth R[z]-modules and of nvolved computatons. Unfortunately, t holds only f V* has the feedback property (strong requrement).
18 Proposton Gven Σ (A,B,C,X), w.c.n R, t s possble to construct, n a canonc way, a dynamcal extenson Σ e of Σ such that V e * has the feedback property wth a frend F e. Then, f V e */ R e * s free, t s somorphc to the largest free submodule of V*/ R*. Proposton In the above context and wth the above notatons, assumng that the Zero Dynamcs of Σ s defned, let V e */ R e * be a free R-module of dmenson m and let D be a matrx representng the R-morphsm (A e + B e F e ) Ve*/ Re* wth respect to the canoncal bass of R m. Then, the ZeroDynamcs of Σ s the dynamcs nduced on R m by D z(t+1) Dz(t), for z R m.
19 Defnton Gven a tme-delay system Σ d, let Σ be the assocated system wth coeffcents the rng R. The Zero Dynamcs of Σ d s that of Σ, f the latter s defned. Proposton Gven the tme-delay system Σ d, wth commensurable delays, let Σ be the assocated system wth coeffcents the rng R. Then, f Σ s left nvertble and V* s free, the Zero Dynamcs of Σ s defned and so s that of Σ d.
20 Hurwtz set: a set H of monc polynomals n R[z] H contans at least one lnear monomal z + a wth a R; H s multplcatvely closed; any factor of an element n H belongs to H. Defnton A system Σ (A,B,C,X) w.c.. R s sad to be H-stable f det(zi-a) belongs to H H-mnmum phase f ts zero dynamcs s defned and H- stable. For systems assocated to tme-delay ones, R R[Δ] H {p(z,δ) R[z, Δ], such that p(s,e -hs ) for all s C wth Re(s) } H-stablty (phase mn.) n the rng framework Asymptotc stablty (phase mn.) n the tme-delay framework
21 Proposton Gven a left (respectvely, rght) nvertble system Σ (A,B,C,X) wth coeffcents n the rng R and transfer functon G, let G nv denote a left (respectvely, rght) nverse of G and let Σ nv (A nv, B nv, C nv, X nv ) be ts canoncal realzaton. Then, the relaton G nv G Identty nduces an njectve R[z]-homomorphsm ψ : Z Σ X nv (respectvely, the relaton G G nv Identty nduces a surjectve R[z]-homomorphsm. ϕ: X nv Z Σ ) between the Zero Module of Σ and the state module X nv of the canoncalrealzaton of G nv. In case the Zero Dynamcs of Σ s defned, but not mnmum phase, the above Proposton allows us to say that Σ has no H- stable nverses.
22 Problem Gven a SISO tme-delay system Σ d and the correspondng system Σ w.c.n R R[Δ], consder the problem of desgnng a compensator whch forces Σ d to track a reference sgnal r(t). Consder the extended system Σ E. x(t + 1) Ax(t) + e(t) cx(t) r(t) bu(t) whose output s the trackng error and apply the Slverman Inverson Algorthm.
23 Ths gves the relaton k. k 1 e(t + k ) ca x(t) + ca bu(t) r(t + k ) Then, choosng a real polynomal p(z) z + az n such a way that t s n the Hurwtz set H, we can construct the compensator Σ C z(t + 1) Az(t) + bu(t). u(t) (ca (ca k whose acton on Σ causes the error to evolve accordng to the equaton k e(t k ) 1 k + + a e(t + ) ca (x(t) z(t)) k k 1 1 b) b) 1 1 (ca k k 1 k 1 z(t) r(t a e(t + ) + k )) +
24 The compensator Σ C z(t + 1) u(t) (ca solves the trackng problem. Az(t) + bu(t) (ca k k 1 1 b) b) 1 1 (ca k k 1 z(t) r(t a e(t + ) + k )) + H-stablty of the compensator s a key ssue and, snce ts. constructon s based on nverson, t can be dealt wth by usng phase mnmalty. If Σ d and Σ, and hence Σ E, are H-mnmum phase (that s: ther zero dynamcs s H-stabe), an H-stabe compensator s obtaned.
25 Example. Consder the tme-delay system and the assocated system Σ (A,B,C,X) wth coeffcents n R R[Δ] and matrces V* span (Δ ) T R 3, V* s not of feedback type (because t s not closed), but t s free. R* {}. The Zero Dynamcs s defned and t can be represented as a sutable par (R m,z) n order to check, for nstance, phase mnmalty.
26 Example (contnued). To analyze the Zero Dynamcs, we consder the extenson Σ e (A e,b e,c e,x e ), wth V e * span (Δ 1) T R 4 s of feedback type, F e The dynamc matrx A c (A e + B e F) of the compensated system s A c
27 Example (contnued). The Zero Dynamcs turns out to be gven by (R, [ 1]) or, n other terms, by the dynamc equaton. ξ(t + 1) ξ(t) ξ(t) ξ(t) n the tme-delay framework. We can conclude that the system s mnmum phase.
28 The notons of Zero Module and of Zero Dynamcs have been ntroduced n the tme-delay framework, by explotng the correspondence between system wth coeffcents n a rng and tme-delay systems and the algebrac characterzaton of Zeros. Stablty of the Zero Dynamcs and Phase Mnmalty can then be defned and used for the constructon of stable nverses and of stable solutons to trackng problems n the tme delay framework.
APPENDIX A Some Linear Algebra
APPENDIX A Some Lnear Algebra The collecton of m, n matrces A.1 Matrces a 1,1,..., a 1,n A = a m,1,..., a m,n wth real elements a,j s denoted by R m,n. If n = 1 then A s called a column vector. Smlarly,
More informationMTH 819 Algebra I S13. Homework 1/ Solutions. 1 if p n b and p n+1 b 0 otherwise ) = 0 if p q or n m. W i = rw i
MTH 819 Algebra I S13 Homework 1/ Solutons Defnton A. Let R be PID and V a untary R-module. Let p be a prme n R and n Z +. Then d p,n (V) = dm R/Rp p n 1 Ann V (p n )/p n Ann V (p n+1 ) Note here that
More informationLectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix
Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could
More information5 The Rational Canonical Form
5 The Ratonal Canoncal Form Here p s a monc rreducble factor of the mnmum polynomal m T and s not necessarly of degree one Let F p denote the feld constructed earler n the course, consstng of all matrces
More informationInner Product. Euclidean Space. Orthonormal Basis. Orthogonal
Inner Product Defnton 1 () A Eucldean space s a fnte-dmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,
More information12 MATH 101A: ALGEBRA I, PART C: MULTILINEAR ALGEBRA. 4. Tensor product
12 MATH 101A: ALGEBRA I, PART C: MULTILINEAR ALGEBRA Here s an outlne of what I dd: (1) categorcal defnton (2) constructon (3) lst of basc propertes (4) dstrbutve property (5) rght exactness (6) localzaton
More informationFINITELY-GENERATED MODULES OVER A PRINCIPAL IDEAL DOMAIN
FINITELY-GENERTED MODULES OVER PRINCIPL IDEL DOMIN EMMNUEL KOWLSKI Throughout ths note, s a prncpal deal doman. We recall the classfcaton theorem: Theorem 1. Let M be a fntely-generated -module. (1) There
More informationMATH 241B FUNCTIONAL ANALYSIS - NOTES EXAMPLES OF C ALGEBRAS
MATH 241B FUNCTIONAL ANALYSIS - NOTES EXAMPLES OF C ALGEBRAS These are nformal notes whch cover some of the materal whch s not n the course book. The man purpose s to gve a number of nontrval examples
More informationChapter 5. Solution of System of Linear Equations. Module No. 6. Solution of Inconsistent and Ill Conditioned Systems
Numercal Analyss by Dr. Anta Pal Assstant Professor Department of Mathematcs Natonal Insttute of Technology Durgapur Durgapur-713209 emal: anta.bue@gmal.com 1 . Chapter 5 Soluton of System of Lnear Equatons
More informationExample: (13320, 22140) =? Solution #1: The divisors of are 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 27, 30, 36, 41,
The greatest common dvsor of two ntegers a and b (not both zero) s the largest nteger whch s a common factor of both a and b. We denote ths number by gcd(a, b), or smply (a, b) when there s no confuson
More informationRepresentation theory and quantum mechanics tutorial Representation theory and quantum conservation laws
Representaton theory and quantum mechancs tutoral Representaton theory and quantum conservaton laws Justn Campbell August 1, 2017 1 Generaltes on representaton theory 1.1 Let G GL m (R) be a real algebrac
More informationProblem Set 9 Solutions
Desgn and Analyss of Algorthms May 4, 2015 Massachusetts Insttute of Technology 6.046J/18.410J Profs. Erk Demane, Srn Devadas, and Nancy Lynch Problem Set 9 Solutons Problem Set 9 Solutons Ths problem
More informationLecture Notes on Linear Regression
Lecture Notes on Lnear Regresson Feng L fl@sdueducn Shandong Unversty, Chna Lnear Regresson Problem In regresson problem, we am at predct a contnuous target value gven an nput feature vector We assume
More informationAffine transformations and convexity
Affne transformatons and convexty The purpose of ths document s to prove some basc propertes of affne transformatons nvolvng convex sets. Here are a few onlne references for background nformaton: http://math.ucr.edu/
More information9 Characteristic classes
THEODORE VORONOV DIFFERENTIAL GEOMETRY. Sprng 2009 [under constructon] 9 Characterstc classes 9.1 The frst Chern class of a lne bundle Consder a complex vector bundle E B of rank p. We shall construct
More information2.3 Nilpotent endomorphisms
s a block dagonal matrx, wth A Mat dm U (C) In fact, we can assume that B = B 1 B k, wth B an ordered bass of U, and that A = [f U ] B, where f U : U U s the restrcton of f to U 40 23 Nlpotent endomorphsms
More informationNeuro-Adaptive Design - I:
Lecture 36 Neuro-Adaptve Desgn - I: A Robustfyng ool for Dynamc Inverson Desgn Dr. Radhakant Padh Asst. Professor Dept. of Aerospace Engneerng Indan Insttute of Scence - Bangalore Motvaton Perfect system
More informationAn Introduction to Morita Theory
An Introducton to Morta Theory Matt Booth October 2015 Nov. 2017: made a few revsons. Thanks to Nng Shan for catchng a typo. My man reference for these notes was Chapter II of Bass s book Algebrac K-Theory
More informationMAXIMAL INVARIANT SUBSPACES AND OBSERVABILITY OF MULTIDIMENSIONAL SYSTEMS. PART 2: THE ALGORITHM
U.P.B. Sc. Bull., Seres A, Vol. 80, Iss. 1, 2018 ISSN 1223-7027 MAXIMAL INVARIANT SUBSPACES AND OBSERVABILITY OF MULTIDIMENSIONAL SYSTEMS. PART 2: THE ALGORITHM Valeru Prepelţă 1, Tberu Vaslache 2 The
More informationFACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP
C O L L O Q U I U M M A T H E M A T I C U M VOL. 80 1999 NO. 1 FACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP BY FLORIAN K A I N R A T H (GRAZ) Abstract. Let H be a Krull monod wth nfnte class
More informationwhere a is any ideal of R. Lemma Let R be a ring. Then X = Spec R is a topological space. Moreover the open sets
11. Schemes To defne schemes, just as wth algebrac varetes, the dea s to frst defne what an affne scheme s, and then realse an arbtrary scheme, as somethng whch s locally an affne scheme. The defnton of
More information8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS
SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 493 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces you have studed thus far n the text are real vector spaces because the scalars
More informationA property of the elementary symmetric functions
Calcolo manuscrpt No. (wll be nserted by the edtor) A property of the elementary symmetrc functons A. Esnberg, G. Fedele Dp. Elettronca Informatca e Sstemstca, Unverstà degl Stud della Calabra, 87036,
More informationErrata to Invariant Theory with Applications January 28, 2017
Invarant Theory wth Applcatons Jan Drasma and Don Gjswjt http: //www.wn.tue.nl/~jdrasma/teachng/nvtheory0910/lecturenotes12.pdf verson of 7 December 2009 Errata and addenda by Darj Grnberg The followng
More informationNP-Completeness : Proofs
NP-Completeness : Proofs Proof Methods A method to show a decson problem Π NP-complete s as follows. (1) Show Π NP. (2) Choose an NP-complete problem Π. (3) Show Π Π. A method to show an optmzaton problem
More informationDIFFERENTIAL FORMS BRIAN OSSERMAN
DIFFERENTIAL FORMS BRIAN OSSERMAN Dfferentals are an mportant topc n algebrac geometry, allowng the use of some classcal geometrc arguments n the context of varetes over any feld. We wll use them to defne
More informationMEM633 Lectures 7&8. Chapter 4. Descriptions of MIMO Systems 4-1 Direct Realizations. (i) x u. y x
MEM633 Lectures 7&8 Chapter 4 Descrptons of MIMO Systems 4- Drect ealzatons y() s s su() s y () s u () s ( s)( s) s y() s u (), s y() s u() s s s y() s u(), s y() s u() s ( s)( s) s () ( s ) y ( s) u (
More informationINVARIANT STABLY COMPLEX STRUCTURES ON TOPOLOGICAL TORIC MANIFOLDS
INVARIANT STABLY COMPLEX STRUCTURES ON TOPOLOGICAL TORIC MANIFOLDS HIROAKI ISHIDA Abstract We show that any (C ) n -nvarant stably complex structure on a topologcal torc manfold of dmenson 2n s ntegrable
More informationLECTURE V. 1. More on the Chinese Remainder Theorem We begin by recalling this theorem, proven in the preceeding lecture.
LECTURE V EDWIN SPARK 1. More on the Chnese Remander Theorem We begn by recallng ths theorem, proven n the preceedng lecture. Theorem 1.1 (Chnese Remander Theorem). Let R be a rng wth deals I 1, I 2,...,
More informationDynamic Systems on Graphs
Prepared by F.L. Lews Updated: Saturday, February 06, 200 Dynamc Systems on Graphs Control Graphs and Consensus A network s a set of nodes that collaborates to acheve what each cannot acheve alone. A network,
More informationConvexity preserving interpolation by splines of arbitrary degree
Computer Scence Journal of Moldova, vol.18, no.1(52), 2010 Convexty preservng nterpolaton by splnes of arbtrary degree Igor Verlan Abstract In the present paper an algorthm of C 2 nterpolaton of dscrete
More informationLECTURE 9 CANONICAL CORRELATION ANALYSIS
LECURE 9 CANONICAL CORRELAION ANALYSIS Introducton he concept of canoncal correlaton arses when we want to quantfy the assocatons between two sets of varables. For example, suppose that the frst set of
More informationALGEBRA MID-TERM. 1 Suppose I is a principal ideal of the integral domain R. Prove that the R-module I R I has no non-zero torsion elements.
ALGEBRA MID-TERM CLAY SHONKWILER 1 Suppose I s a prncpal deal of the ntegral doman R. Prove that the R-module I R I has no non-zero torson elements. Proof. Note, frst, that f I R I has no non-zero torson
More informationALGEBRA SCHEMES AND THEIR REPRESENTATIONS
ALGEBRA SCHEMES AND THEIR REPRESENTATIONS AMELIA ÁLVAREZ, CARLOS SANCHO, AND PEDRO SANCHO Introducton The equvalence (Carter dualty) between the category of topologcally flat formal k-groups and the category
More informationwhere a is any ideal of R. Lemma 5.4. Let R be a ring. Then X = Spec R is a topological space Moreover the open sets
5. Schemes To defne schemes, just as wth algebrac varetes, the dea s to frst defne what an affne scheme s, and then realse an arbtrary scheme, as somethng whch s locally an affne scheme. The defnton of
More informationarxiv: v1 [math.ac] 5 Jun 2013
On the K-theory of feedback actons on lnear systems arxv:1306.1021v1 [math.ac] 5 Jun 2013 Mguel V. Carregos,, Ángel Lus Muñoz Castañeda Departamento de Matemátcas. Unversdad de León Abstract A categorcal
More informationDeterminants Containing Powers of Generalized Fibonacci Numbers
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol 19 (2016), Artcle 1671 Determnants Contanng Powers of Generalzed Fbonacc Numbers Aram Tangboonduangjt and Thotsaporn Thanatpanonda Mahdol Unversty Internatonal
More informationTime-Varying Systems and Computations Lecture 6
Tme-Varyng Systems and Computatons Lecture 6 Klaus Depold 14. Januar 2014 The Kalman Flter The Kalman estmaton flter attempts to estmate the actual state of an unknown dscrete dynamcal system, gven nosy
More informationSL n (F ) Equals its Own Derived Group
Internatonal Journal of Algebra, Vol. 2, 2008, no. 12, 585-594 SL n (F ) Equals ts Own Derved Group Jorge Macel BMCC-The Cty Unversty of New York, CUNY 199 Chambers street, New York, NY 10007, USA macel@cms.nyu.edu
More informationALGEBRA SCHEMES AND THEIR REPRESENTATIONS
ALGEBRA SCHEMES AND THEIR REPRESENTATIONS AMELIA ÁLVAREZ, CARLOS SANCHO, AND PEDRO SANCHO Introducton The equvalence (Carter dualty) between the category of topologcally flat formal k-groups and the category
More informationFixed points of IA-endomorphisms of a free metabelian Lie algebra
Proc. Indan Acad. Sc. (Math. Sc.) Vol. 121, No. 4, November 2011, pp. 405 416. c Indan Academy of Scences Fxed ponts of IA-endomorphsms of a free metabelan Le algebra NAIME EKICI 1 and DEMET PARLAK SÖNMEZ
More informationTransfer Functions. Convenient representation of a linear, dynamic model. A transfer function (TF) relates one input and one output: ( ) system
Transfer Functons Convenent representaton of a lnear, dynamc model. A transfer functon (TF) relates one nput and one output: x t X s y t system Y s The followng termnology s used: x y nput output forcng
More informationDIFFERENTIAL SCHEMES
DIFFERENTIAL SCHEMES RAYMOND T. HOOBLER Dedcated to the memory o Jerry Kovacc 1. schemes All rngs contan Q and are commutatve. We x a d erental rng A throughout ths secton. 1.1. The topologcal space. Let
More informationOn a direct solver for linear least squares problems
ISSN 2066-6594 Ann. Acad. Rom. Sc. Ser. Math. Appl. Vol. 8, No. 2/2016 On a drect solver for lnear least squares problems Constantn Popa Abstract The Null Space (NS) algorthm s a drect solver for lnear
More informationMath 101 Fall 2013 Homework #7 Due Friday, November 15, 2013
Math 101 Fall 2013 Homework #7 Due Frday, November 15, 2013 1. Let R be a untal subrng of E. Show that E R R s somorphc to E. ANS: The map (s,r) sr s a R-balanced map of E R to E. Hence there s a group
More informationDISCRIMINANTS AND RAMIFIED PRIMES. 1. Introduction A prime number p is said to be ramified in a number field K if the prime ideal factorization
DISCRIMINANTS AND RAMIFIED PRIMES KEITH CONRAD 1. Introducton A prme number p s sad to be ramfed n a number feld K f the prme deal factorzaton (1.1) (p) = po K = p e 1 1 peg g has some e greater than 1.
More informationApplication of Generalized Polynomials to the Decoupling of Linear Multivariable Systems
Applcaton of Generalzed Polynomals to the Decouplng of Lnear Multvarable Systems,,3 J. Ruz-León and D. Henron. CINVESTAV-IPN, Undad Guadalajara P.O. Box 3-438, Plaza la Luna, 4455 Guadalajara, Jalsco,
More informationn α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0
MODULE 2 Topcs: Lnear ndependence, bass and dmenson We have seen that f n a set of vectors one vector s a lnear combnaton of the remanng vectors n the set then the span of the set s unchanged f that vector
More informationNOTES ON SIMPLIFICATION OF MATRICES
NOTES ON SIMPLIFICATION OF MATRICES JONATHAN LUK These notes dscuss how to smplfy an (n n) matrx In partcular, we expand on some of the materal from the textbook (wth some repetton) Part of the exposton
More informationA Note on \Modules, Comodules, and Cotensor Products over Frobenius Algebras"
Chn. Ann. Math. 27B(4), 2006, 419{424 DOI: 10.1007/s11401-005-0025-z Chnese Annals of Mathematcs, Seres B c The Edtoral Oce of CAM and Sprnger-Verlag Berln Hedelberg 2006 A Note on \Modules, Comodules,
More informationAli Omer Alattass Department of Mathematics, Faculty of Science, Hadramout University of science and Technology, P. O. Box 50663, Mukalla, Yemen
Journal of athematcs and Statstcs 7 (): 4448, 0 ISSN 5493644 00 Scence Publcatons odules n σ[] wth Chan Condtons on Small Submodules Al Omer Alattass Department of athematcs, Faculty of Scence, Hadramout
More informationLinear Approximation with Regularization and Moving Least Squares
Lnear Approxmaton wth Regularzaton and Movng Least Squares Igor Grešovn May 007 Revson 4.6 (Revson : March 004). 5 4 3 0.5 3 3.5 4 Contents: Lnear Fttng...4. Weghted Least Squares n Functon Approxmaton...
More informationNon-negative Matrices and Distributed Control
Non-negatve Matrces an Dstrbute Control Yln Mo July 2, 2015 We moel a network compose of m agents as a graph G = {V, E}. V = {1, 2,..., m} s the set of vertces representng the agents. E V V s the set of
More informationPolynomials. 1 More properties of polynomials
Polynomals 1 More propertes of polynomals Recall that, for R a commutatve rng wth unty (as wth all rngs n ths course unless otherwse noted), we defne R[x] to be the set of expressons n =0 a x, where a
More informationNOTES FOR QUANTUM GROUPS, CRYSTAL BASES AND REALIZATION OF ŝl(n)-modules
NOTES FOR QUANTUM GROUPS, CRYSTAL BASES AND REALIZATION OF ŝl(n)-modules EVAN WILSON Quantum groups Consder the Le algebra sl(n), whch s the Le algebra over C of n n trace matrces together wth the commutator
More informationThe Pseudoblocks of Endomorphism Algebras
Internatonal Mathematcal Forum, 4, 009, no. 48, 363-368 The Pseudoblocks of Endomorphsm Algebras Ahmed A. Khammash Department of Mathematcal Scences, Umm Al-Qura Unversty P.O.Box 796, Makkah, Saud Araba
More informationYong Joon Ryang. 1. Introduction Consider the multicommodity transportation problem with convex quadratic cost function. 1 2 (x x0 ) T Q(x x 0 )
Kangweon-Kyungk Math. Jour. 4 1996), No. 1, pp. 7 16 AN ITERATIVE ROW-ACTION METHOD FOR MULTICOMMODITY TRANSPORTATION PROBLEMS Yong Joon Ryang Abstract. The optmzaton problems wth quadratc constrants often
More information= = = (a) Use the MATLAB command rref to solve the system. (b) Let A be the coefficient matrix and B be the right-hand side of the system.
Chapter Matlab Exercses Chapter Matlab Exercses. Consder the lnear system of Example n Secton.. x x x y z y y z (a) Use the MATLAB command rref to solve the system. (b) Let A be the coeffcent matrx and
More informationGELFAND-TSETLIN BASIS FOR THE REPRESENTATIONS OF gl n
GELFAND-TSETLIN BASIS FOR THE REPRESENTATIONS OF gl n KANG LU FINITE DIMENSIONAL REPRESENTATIONS OF gl n Let e j,, j =,, n denote the standard bass of the general lnear Le algebra gl n over the feld of
More informationErrors for Linear Systems
Errors for Lnear Systems When we solve a lnear system Ax b we often do not know A and b exactly, but have only approxmatons  and ˆb avalable. Then the best thng we can do s to solve ˆx ˆb exactly whch
More informationModule 3 LOSSY IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module 3 LOSSY IMAGE COMPRESSION SYSTEMS Verson ECE IIT, Kharagpur Lesson 6 Theory of Quantzaton Verson ECE IIT, Kharagpur Instructonal Objectves At the end of ths lesson, the students should be able to:
More information2 More examples with details
Physcs 129b Lecture 3 Caltech, 01/15/19 2 More examples wth detals 2.3 The permutaton group n = 4 S 4 contans 4! = 24 elements. One s the dentty e. Sx of them are exchange of two objects (, j) ( to j and
More informationThe Geometry of Logit and Probit
The Geometry of Logt and Probt Ths short note s meant as a supplement to Chapters and 3 of Spatal Models of Parlamentary Votng and the notaton and reference to fgures n the text below s to those two chapters.
More informationMore metrics on cartesian products
More metrcs on cartesan products If (X, d ) are metrc spaces for 1 n, then n Secton II4 of the lecture notes we defned three metrcs on X whose underlyng topologes are the product topology The purpose of
More information= s j Ui U j. i, j, then s F(U) with s Ui F(U) G(U) F(V ) G(V )
1 Lecture 2 Recap Last tme we talked about presheaves and sheaves. Preshea: F on a topologcal space X, wth groups (resp. rngs, sets, etc.) F(U) or each open set U X, wth restrcton homs ρ UV : F(U) F(V
More informationRandom Walks on Digraphs
Random Walks on Dgraphs J. J. P. Veerman October 23, 27 Introducton Let V = {, n} be a vertex set and S a non-negatve row-stochastc matrx (.e. rows sum to ). V and S defne a dgraph G = G(V, S) and a drected
More informationp-adic Galois representations of G E with Char(E) = p > 0 and the ring R
p-adc Galos representatons of G E wth Char(E) = p > 0 and the rng R Gebhard Böckle December 11, 2008 1 A short revew Let E be a feld of characterstc p > 0 and denote by σ : E E the absolute Frobenus endomorphsm
More informationDECOUPLING OF LINEAR TIME-VARYING SYSTEMS WITH A BOND GRAPH APPROACH
DECOUPLING OF LINEAR IME-VARYING SYSEMS WIH A BOND GRAPH APPROACH Stefan Lchardopol Chrstophe Sueur L.A.G.I.S., UMR 846 CNRS Ecole Centrale de Llle, Cté Scentfque, BP48 5965 Vlleneuve d'ascq Cedex, France
More information= z 20 z n. (k 20) + 4 z k = 4
Problem Set #7 solutons 7.2.. (a Fnd the coeffcent of z k n (z + z 5 + z 6 + z 7 + 5, k 20. We use the known seres expanson ( n+l ( z l l z n below: (z + z 5 + z 6 + z 7 + 5 (z 5 ( + z + z 2 + z + 5 5
More informationTHE VIBRATIONS OF MOLECULES II THE CARBON DIOXIDE MOLECULE Student Instructions
THE VIBRATIONS OF MOLECULES II THE CARBON DIOXIDE MOLECULE Student Instructons by George Hardgrove Chemstry Department St. Olaf College Northfeld, MN 55057 hardgrov@lars.acc.stolaf.edu Copyrght George
More informationOn a ρ n -Dilation of Operator in Hilbert Spaces
E extracta mathematcae Vol. 31, Núm. 1, 11 23 (2016) On a ρ n -Dlaton of Operator n Hlbert Spaces A. Salh, H. Zeroual PB 1014, Departement of Mathematcs, Sences Faculty, Mohamed V Unversty n Rabat, Rabat,
More information17. Coordinate-Free Projective Geometry for Computer Vision
17. Coordnate-Free Projectve Geometry for Computer Vson Hongbo L and Gerald Sommer Insttute of Computer Scence and Appled Mathematcs, Chrstan-Albrechts-Unversty of Kel 17.1 Introducton How to represent
More informationA Notion of Zero Dynamics for Linear, Time-delay System
Proceedings of the 17th World Congress The International Federation of Automatic Control A Notion of Zero Dynamics for Linear, Time-delay System G. Conte A. M. Perdon DIIGA, Università Politecnica delle
More informationCSci 6974 and ECSE 6966 Math. Tech. for Vision, Graphics and Robotics Lecture 21, April 17, 2006 Estimating A Plane Homography
CSc 6974 and ECSE 6966 Math. Tech. for Vson, Graphcs and Robotcs Lecture 21, Aprl 17, 2006 Estmatng A Plane Homography Overvew We contnue wth a dscusson of the major ssues, usng estmaton of plane projectve
More informationSUPER PRINCIPAL FIBER BUNDLE WITH SUPER ACTION
talan journal of pure appled mathematcs n. 33 2014 (63 70) 63 SUPER PRINCIPAL FIBER BUNDLE WITH SUPER ACTION M.R. Farhangdoost Department of Mathematcs College of Scences Shraz Unversty Shraz, 71457-44776
More informationCalculation of time complexity (3%)
Problem 1. (30%) Calculaton of tme complexty (3%) Gven n ctes, usng exhaust search to see every result takes O(n!). Calculaton of tme needed to solve the problem (2%) 40 ctes:40! dfferent tours 40 add
More informationZ d -TOEPLITZ ARRAYS
Z d -TOEPLITZ ARRAYS MARIA ISABEL CORTEZ Abstract In ths paper we gve a defnton of Toepltz sequences and odometers for Z d actons for d whch generalzes that n dmenson one For these new concepts we study
More informationρ some λ THE INVERSE POWER METHOD (or INVERSE ITERATION) , for , or (more usually) to
THE INVERSE POWER METHOD (or INVERSE ITERATION) -- applcaton of the Power method to A some fxed constant ρ (whch s called a shft), x λ ρ If the egenpars of A are { ( λ, x ) } ( ), or (more usually) to,
More informationHOPF ALGEBRAS WITH TRACE AND CLEBSCH-GORDAN COEFFICIENTS. 1. Recollections and the problem
HOPF ALGEBRAS WITH TRACE AND CLEBSCH-GORDAN COEFFICIENTS CORRADO DE CONCINI Abstract. In ths lecture I shall report on some jont work wth Proces, Reshetkhn and Rosso [1]. 1. Recollectons and the problem
More informationp 1 c 2 + p 2 c 2 + p 3 c p m c 2
Where to put a faclty? Gven locatons p 1,..., p m n R n of m houses, want to choose a locaton c n R n for the fre staton. Want c to be as close as possble to all the house. We know how to measure dstance
More informationDynamic Programming. Preview. Dynamic Programming. Dynamic Programming. Dynamic Programming (Example: Fibonacci Sequence)
/24/27 Prevew Fbonacc Sequence Longest Common Subsequence Dynamc programmng s a method for solvng complex problems by breakng them down nto smpler sub-problems. It s applcable to problems exhbtng the propertes
More informationCounterexamples to the Connectivity Conjecture of the Mixed Cells
Dscrete Comput Geom 2:55 52 998 Dscrete & Computatonal Geometry 998 Sprnger-Verlag New York Inc. Counterexamples to the Connectvty Conjecture of the Mxed Cells T. Y. L and X. Wang 2 Department of Mathematcs
More informationLecture 7: Gluing prevarieties; products
Lecture 7: Glung prevaretes; products 1 The category of algebrac prevaretes Proposton 1. Let (f,ϕ) : (X,O X ) (Y,O Y ) be a morphsm of algebrac prevaretes. If U X and V Y are affne open subvaretes wth
More information763622S ADVANCED QUANTUM MECHANICS Solution Set 1 Spring c n a n. c n 2 = 1.
7636S ADVANCED QUANTUM MECHANICS Soluton Set 1 Sprng 013 1 Warm-up Show that the egenvalues of a Hermtan operator  are real and that the egenkets correspondng to dfferent egenvalues are orthogonal (b)
More informationZeros and Zero Dynamics for Linear, Time-delay System
Zeros and Zero Dynamics for Linear, Time-delay System Giuseppe Conte Anna Maria Perdon Abstract The aim of this paper is to discuss a notion of Zero Module and Zero Dynamics for linear, time-delay systems.
More informationU.C. Berkeley CS294: Beyond Worst-Case Analysis Luca Trevisan September 5, 2017
U.C. Berkeley CS94: Beyond Worst-Case Analyss Handout 4s Luca Trevsan September 5, 07 Summary of Lecture 4 In whch we ntroduce semdefnte programmng and apply t to Max Cut. Semdefnte Programmng Recall that
More informationHidden Markov Models & The Multivariate Gaussian (10/26/04)
CS281A/Stat241A: Statstcal Learnng Theory Hdden Markov Models & The Multvarate Gaussan (10/26/04) Lecturer: Mchael I. Jordan Scrbes: Jonathan W. Hu 1 Hdden Markov Models As a bref revew, hdden Markov models
More informationThe equation of motion of a dynamical system is given by a set of differential equations. That is (1)
Dynamcal Systems Many engneerng and natural systems are dynamcal systems. For example a pendulum s a dynamcal system. State l The state of the dynamcal system specfes t condtons. For a pendulum n the absence
More informationLecture Notes Introduction to Cluster Algebra
Lecture Notes Introducton to Cluster Algebra Ivan C.H. Ip Updated: Ma 7, 2017 3 Defnton and Examples of Cluster algebra 3.1 Quvers We frst revst the noton of a quver. Defnton 3.1. A quver s a fnte orented
More informationAppendix B. The Finite Difference Scheme
140 APPENDIXES Appendx B. The Fnte Dfference Scheme In ths appendx we present numercal technques whch are used to approxmate solutons of system 3.1 3.3. A comprehensve treatment of theoretcal and mplementaton
More informationSubset Topological Spaces and Kakutani s Theorem
MOD Natural Neutrosophc Subset Topologcal Spaces and Kakutan s Theorem W. B. Vasantha Kandasamy lanthenral K Florentn Smarandache 1 Copyrght 1 by EuropaNova ASBL and the Authors Ths book can be ordered
More informationPHYS 705: Classical Mechanics. Canonical Transformation II
1 PHYS 705: Classcal Mechancs Canoncal Transformaton II Example: Harmonc Oscllator f ( x) x m 0 x U( x) x mx x LT U m Defne or L p p mx x x m mx x H px L px p m p x m m H p 1 x m p m 1 m H x p m x m m
More informationCoordinate-Free Projective Geometry for Computer Vision
MM Research Preprnts,131 165 No. 18, Dec. 1999. Beng 131 Coordnate-Free Proectve Geometry for Computer Vson Hongbo L, Gerald Sommer 1. Introducton How to represent an mage pont algebracally? Gven a Cartesan
More informationNotes on Frequency Estimation in Data Streams
Notes on Frequency Estmaton n Data Streams In (one of) the data streamng model(s), the data s a sequence of arrvals a 1, a 2,..., a m of the form a j = (, v) where s the dentty of the tem and belongs to
More information7. Products and matrix elements
7. Products and matrx elements 1 7. Products and matrx elements Based on the propertes of group representatons, a number of useful results can be derved. Consder a vector space V wth an nner product ψ
More informationThe Prncpal Component Transform The Prncpal Component Transform s also called Karhunen-Loeve Transform (KLT, Hotellng Transform, oregenvector Transfor
Prncpal Component Transform Multvarate Random Sgnals A real tme sgnal x(t can be consdered as a random process and ts samples x m (m =0; ;N, 1 a random vector: The mean vector of X s X =[x0; ;x N,1] T
More informationLINEAR REGRESSION ANALYSIS. MODULE IX Lecture Multicollinearity
LINEAR REGRESSION ANALYSIS MODULE IX Lecture - 31 Multcollnearty Dr. Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur 6. Rdge regresson The OLSE s the best lnear unbased
More informationDeriving the X-Z Identity from Auxiliary Space Method
Dervng the X-Z Identty from Auxlary Space Method Long Chen Department of Mathematcs, Unversty of Calforna at Irvne, Irvne, CA 92697 chenlong@math.uc.edu 1 Iteratve Methods In ths paper we dscuss teratve
More informationCHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE
CHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE Analytcal soluton s usually not possble when exctaton vares arbtrarly wth tme or f the system s nonlnear. Such problems can be solved by numercal tmesteppng
More informationσ τ τ τ σ τ τ τ σ Review Chapter Four States of Stress Part Three Review Review
Chapter Four States of Stress Part Three When makng your choce n lfe, do not neglect to lve. Samuel Johnson Revew When we use matrx notaton to show the stresses on an element The rows represent the axs
More information