Linearization by input output injections on homogeneous time scales

Transcription

1 Proceedings of the Estonian Academy of Sciences doi: 0376/proc Avaiabe onine at wwweapee/proceedings Linearization by input output injections on homogeneous time scaes Monika Ciukin a Vadim Kaparin b Üe Kotta b and Ewa Pawłuszewicz a a Facuty of Mechanica Engineering Biaystok University of Technoogy u Wiejska 45C 5-35 Białystok Poand; mciukin@doktorancipbedup epawuszewicz@pbedup b Institute of Cybernetics at Tainn University of Technoogy Akadeemia tee Tainn Estonia; vkaparin@cciocee kotta@cciocee Received February 204 revised 7 May 204 accepted 6 June 204 avaiabe onine 20 November 204 Abstract The probem of inearization by input output i/o injections is addressed for noninear singe-input singe-output systems defined on a homogeneous time scae The paper provides conditions for the existence of a state transformation bringing state equations into the observer form which is inear up to some noninear input- and output-dependent functions caed i/o injections These conditions are based on differentia one-forms associated with the i/o equation of the system Key words: noninear contro system time scae observer form differentia one-forms INTRODUCTION From a modeing point of view dynamica systems on time scaes incorporate both continuous- and discrete-time systems as specia cases aowing us to unify the study and consider the cassica resuts as specia cases from the new theory On the other hand the study of dynamica systems on time scaes heps to revea and expain discrepancies occasionay appearing between the resuts obtained for continuoustime systems and their discrete-time counterparts see eg [238202] However it is important to note that the discrete-time mode in the time scae formaism is given in terms of the difference operator and not in terms of the more conventiona shift operator The difference-based modes often referred to as deta-domain modes are not competey new for the description of discrete-time systems They have been promoted during the ast decades as the modes cosey inked to continuous-time systems being ess sensitive to round-off errors at higher samping rates eg [425] More information on noninear contro systems on time scaes is avaiabe in [ ] The method of inearization of the noninear contro systems by input output i/o injections aternativey transformation of the state equations into the observer form is the intermediate step in the observer construction Design of the noninear observer for the system in the observer form inear up to i/o injections is reativey easy [5] aowing one to construct the noninear observer in such a way that the dynamics of the estimation error are inear making it simpe to guarantee that the error converges asymptoticay to zero The purpose of this paper is to present necessary and sufficient conditions for the existence of the state transformation aowing transformation of the singe-input singe-output SISO state equations defined on a homogeneous time scae into the observer form The conditions are formuated within the agebraic Corresponding author vkaparin@cciocee

2 388 Proceedings of the Estonian Academy of Sciences framework based on differentia one-forms and can be considered as the extension of the resut from [] to the case of a homogeneous time scae The computation of one-forms is based on the i/o equation of the system and the conditions can be easiy verified whenever the i/o equation is found The paper is organized as foows In Section 2 we give a brief exposition of the basic notions from the time scae cacuus and an overview of the agebraic framework of differentia forms on a homogeneous time scae The probem of inearization by i/o injections is formuated in Section 3 Section 4 presents first a direct formua for computation of the differentia one-forms in terms of which the main resut is formuated and then provides necessary and sufficient conditions The theory is iustrated by an exampe Concusions are drawn in Section 5 2 PRELIMINARIES 2 Time scae cacuus In this subsection we reca ony those facts that we need in this paper For a genera introduction to the time scae cacuus see [9] A time scae T is an arbitrary nonempty cosed subset of the rea numbers This paper is focused on the two most important for contro theory instances of time scae ie the continuous-time case T = R and the discrete-time case T = τz := {τk : k Z} for τ > 0 The exampes of the other type of time scaes incuding the non-uniformy samped time can be found for instance in [6] The most important notions of time scae cacuus are the forward jump operator σ the backward jump operator ρ the deta derivative and the graininess function µ Appications of σ ρ and to the function ξ : T R as we as the vaues of µ are presented in Tabe for two specia cases T = R and T = τz A time scae T is caed homogeneous if µ const and as can be seen from Tabe both time scaes T = R and T = τz possess this property Hereinafter we eave out the time argument t in order to simpify the exposition so ξ := ξ t We denote by ξ i the deta derivative of an arbitrary order i Moreover for notationa convenience denote ξ in := ξ i ξ n for 0 i n and ξ 0 := ξ 22 Agebraic framework Consider the noninear SISO contro system defined on a homogeneous time scae T and described either by the state equations x = f xu y = hx or by the higher-order i/o deta-differentia equation y n = yy y n uu u n 2 where x : T X R n is an n-dimensiona state vector u : T U R is an input and y : T Y R is an output Moreover f : X U X h : X Y and : Y n U n R are assumed to be rea anaytic functions Tabe Basic types of operators/functions T ξ σ t ξ ρ t ξ t µ R ξ t ξ t τz ξ t + τ ξ t τ dξ t dt 0 ξ t+τ ξ t τ τ

3 M Ciukin et a: Linearization on time scaes 389 The foowing assumption is specification of Theorem 3 from [23] where the muti-input muti-output case was considered Assumption System 2 is submersive ie the function in 2 satisfies the condition rank + n k=0 n k µ n k y k n j=0 n j µ n j+ u k = 3 In this subsection we reca some facts from [5 7] focusing on equation 2 Let K denote the fied of { meromorphic functions in a finite number } of independent system variabes from the infinite set C = yy y n ;u k k 0;υ ρ where ρ denotes the -fod appication of the backward jump operator ρ and the variabe υ can be chosen either to be y or u The choice can be briefy described as foows If the first eement of the matrix in 3 is not identicay equa to zero then one can choose υ = u In this case using the i/o equation 2 the variabes y ρ can be expressed through the independent variabes from C If the second eement of the matrix in 3 is not identicay equa to zero then one can set υ = y and consider the variabes u ρ as dependent If both eements of the matrix in 3 are not identicay equa to zero then one has freedom of choice For F y 0n u 0k K the forward-shift operator σ : K K is defined by F σ y 0n u 0k+ := F y 0n σ u 0k σ where y 0n σ = y + µy y n 2 + µy n y n + µ u 0k σ = u 0k +µu k+ and is determined by 2 Hereinafter we denote the n-fod appication of the forward-shift operator by F σ n := F σ n σ The backward-shift operator ρ : K K is defined as the inverse of σ ie ρ := σ Thus denoting by ρn the n-fod appication of the backward-shift operator we have F = Fρ σ σ and F ρn = F ρn+ The deta derivative operator : K K is defined by F y 0n u 0k+ := µ Fσ F if µ 0 n =0 F F y y + + k 0 u k u k+ if µ = 0 where according to 2 y n shoud be repaced by whenever it appears Observe that for µ 0 the nth deta derivative can be computed by the formua F n = n µ n k CnF k σ n k 4 k=0 where C k n is a binomia coefficient ie C k n = n!/n k!k! Proposition 2 [5] For FG K the deta derivative and forward-shift operators satisfy the foowing properties: i F σ = F + µf ii αf + βg = αf + βg for αβ R iii FG = F σ G + F G iv on a homogeneous time scae operators and σ commute ie F σ = F σ

4 390 Proceedings of the Estonian Academy of Sciences Generaization of i in Proposition 2 yieds the n-fod appication of operator σ as F σ n = n s=0 C s nµ s F s 5 Consider the infinite set of symbos dc = { dydy dy n du k k 0 } and define E = span K dc The eements of E are caed the differentia one-forms Any eement of E has the form n ω = =0 A dy + B k du k k 0 where A B k K and ony a finite number of coefficients B k are nonzero For F y 0n u 0k K define the operator d : K E by n df := =0 F F y dy + u k du k Starting from the space E it is possibe to buid up the structures used in exterior differentia cacuus We refer to [7] for detais whereas here we just reca some basic notions Define the set dc = {dζ dη : dζ dη dc } where denotes the wedge product with the standard properties dζ dη = dη dζ and dζ dζ = 0 for dζ dη dc Introduce the space E 2 = span K dc of two-forms The operator d : E E 2 caed exterior derivative operator is defined for ω = k = a ζ ζ k dζ E where ζ ζ k C by the rue dω := a / ζ dζ dζ The notion of two-form is generaized to the p-form and wedge product is defined for arbitrary p-forms One says that ω E is an exact one-form if ω = df for some F K A one-form ω for which dω = 0 is said to be cosed It is we known that an exact one-form is cosed whereas a cosed one-form is ony ocay exact k 0 3 PROBLEM STATEMENT Our purpose is to find the conditions under which there exists a oca state transformation ie diffeomorphism ψ : X X defined by z = ψx 6 such that in the new state coordinates the state equations are in the observer form z = z 2 + ϕ yu z n = z n + ϕ n yu z n = ϕ n yu y = z 7 which is inear up to noninear functions ϕ yuϕ n yu caed i/o injections State eimination System is caed genericay singe-experiment observabe if the rank of the observabiity matrix is genericay equa to n [9] ie if [ hh h n ] rank K = n 8 x

5 M Ciukin et a: Linearization on time scaes 39 Given a SISO observabe noninear contro system defined on a homogeneous time scae the probem is to find the higher-order i/o deta-differentia equation 2 corresponding to In genera the representation 2 is vaid ony ocay Compute y = hx y = h xu y n = h n xuu u n 2 9 The set of equations 9 can be soved under the observabiity assumption 8 with respect to the state variabes x = ζ yy y n uu u n 2 0 Next compute y n and substitute x from 0 to get y n = h n ζ yy y n uu u n 2 uu u n Note that the state equations can be transformed into the observer form 7 with the state transformation 6 if the i/o equation 2 corresponding to can be rewritten in the form y n = ϕ yu n + + ϕ n yu + ϕ n yu for some functions ϕ yuϕ n yu The converse hods too since is aways reaizabe into the extended observer form 7 Indeed if 2 has the form one can define the new state variabes as z = y z 2 = y ϕ z 3 = y 2 ϕ ϕ 2 z n = y n ϕ n 2 ϕ n 2 ϕ n yieding the state equations in the observer form 7 Note that using the state equations one can substitute the variabes yy y n in such a way that the right-hand side of equation 2 depends ony on x meaning that 2 is the state transformation NECESSARY AND SUFFICIENT CONDITIONS For i = n define the differentia one-forms i ω i = j C j n i+ j j=0 y n i+ j n i+ j j ρ dy + u n i+ j n i+ j j ρ du 3 The proof of Theorem 4 beow is based on the foowing Proposition which extends the resuts of [7] to the case of a homogeneous time scae and the proof of which is given in Appendix

6 392 Proceedings of the Estonian Academy of Sciences Proposition 3 Let Φξ tξ 2 tξ r t be a composite function for which deta derivatives up to order a + b where a and b are nonnegative integers are defined Then on a homogeneous time scae T for = 2r the foowing hods: [ Φξ tξ 2 tξ r t a+b ] Φξ ξ a = Ca+b b tξ 2 tξ r t b σ a 4 t ξ t Theorem 4 The observabe system of the form can be transformed by the state transformation 6 into the observer form 7 if and ony if for i = n where the one-forms ω i are defined by 3 dω i = 0 5 Proof Necessity: Assume that system is transformabe into the observer form 7 Consequenty the i/o equation 2 corresponding to the state equations can be rewritten in the form yieding For the compactness of the proof denote ω iy := ω iu := i j=0 i j=0 = j C j n i+ j j C j n i+ j n k= ϕ n k k 6 y n i+ j u n i+ j n i+ j j ρ n i+ j j ρ such that 3 may be rewritten as ω i = ω iy dy + ω iu du 7 First consider ω iy Using 6 one obtains ω iy = i n j=0 k= n k j C j ϕ k n i+ j y n i+ j ρ n i+ j Observe that if k > i j then n k < n i + j and so ϕ n k k / y n i+ j = 0 Therefore instead of taking k = n we can take k = i j Moreover by Proposition 3 for r = 2 a = n i+ j and b = i j k we have ϕ n k ϕk n i+ j i j k σ k j k = Ci y n i+ j n k y j Thus one can write ω iy = i j=0 i j k= i k j C j j k ϕk n i+ jci n k y

7 M Ciukin et a: Linearization on time scaes 393 Changing the summation order i j=0 i j k= a jk = i k= i k+ j= a j k and taking into account C j j k+ n i+ j Ci n k = Cn k i k C j i k one obtains ω iy = i k= C i k n k i k i k+ ϕk y j C j i k j= Note that for i = the above formua yieds ω y = ϕ / y In the case i 2 one can separate the ast addend of the sum ω iy eading to ω iy = ϕ i i y + k= C i k n k i k i k+ ϕk y In [6] Lemma says that for k = i and i 2 j C j i k j= i k+ j= j C j i k = 0 8 Then by 8 ω iy = ϕ i / y In the same manner we get ω iu = ϕ i / u for i = n Finay from 7 we obtain ω i = dϕ i 9 yieding 5 Sufficiency: Assume that the conditions 5 are satisfied Then ocay there exist functions ϕ i yu satisfying 9 Integrating the one-forms ω i the corresponding functions ϕ i can be found such that 6 hods and as a consequence the state equations in the observer form 7 can be constructed Exampe 5 Consider the system x = u + x 2 + x 3 + x + u x 3 u x 2 = u 2 x x 3 + ux 3 + u + x 2 + 2x 3 x 2 x 3 = x + ux 2 + u x 3 y = x 2 + x 3 20 The i/o equation corresponding to 20 is y 3 = µ 2 y 2 2u 2 + y 2 + 4µy 2 u + y + 2y 2 u + y + µy u + 4u 2 + 2y y + 2u + y u + u + 2u 2 + u 2 2u + 4µu + µ 2 u 2 + 2u 2 + uy + Compute according to 3 ω = 2u + ydy + 2u + ydu ω 2 = udy + ydu ω 3 = u dy + y u 2 du

8 394 Proceedings of the Estonian Academy of Sciences which eads to dω = 2dy dy + 2du dy 2du dy + 2du du = 0 dω 2 = du dy du dy = 0 dω 3 = /u 2 du dy + /u 2 du dy + 2/u 3 du du = 0 meaning that the conditions of Theorem 4 are satisfied Since in this case 9 hods whenever u 0 integration of one-forms ω i eads to ϕ yu = y + u 2 Using 2 one obtains the state transformation as yieding the state equations in the observer form ϕ 2 yu = yu ϕ 3 yu = y u + u z = x 2 + x 3 z 2 = x 3 z 3 = x x 3 z = z 2 + y + u 2 z 2 = z 3 + yu z 3 = y u + u y = z 5 CONCLUSIONS In the paper necessary and sufficient conditions for inearization of the noninear state equations defined on a homogeneous time scae by input-output i/o injections are given For this aim the state transformation is used The conditions are formuated in terms of differentia one-forms directy computabe from the i/o equation of the given system The main theorem states that the probem is sovabe if and ony if the exterior derivatives of these one-forms are equa to zero Note that our conditions are simpe and transparent but require that first the i/o equation of the contro system has to be found This can be done using the extension of the state eimination agorithm from [] to the systems defined on a homogeneous time scae provided the contro system is observabe Simiary to the continuous-time case the i/o equations can aways be found at east ocay Nevertheess it can sometimes be a difficut task impying that the i/o equation cannot aways be represented in terms of the eementary functions In order to provide the proof of the main theorem the supporting proposition was stated and proved The proposition is the extension of the theorem from [7] and it shows how the partia derivative of the tota deta derivative of the composite function with a vector argument can be expressed through the tota deta derivative of the partia derivative of the composite function The proposition can aso serve as a usefu too for research in the area of studying systems on homogeneous time scaes It shoud be mentioned that though the resuts obtained in this paper basicay recover those for the continuous-time case in [] for the discrete-time systems given in terms of the difference operator the resuts are competey new Moreover unike [] where the step-by-step agorithm requiring integration of one-forms was empoyed we suggested the direct formua for computation of the one-forms necessary for conditions To concude the theoretica resuts were obtained in a constructive way such that they can be impemented ater in the software package NLContro Mathematica-based package deveoped in the Institute of Cybernetics see [8]

9 M Ciukin et a: Linearization on time scaes 395 Regarding the future extension of the resuts of this paper one may address the construction of the observer Moreover using both the state and the output transformations ike in [6] one may reax the conditions of Theorem 4 Note that concerning this probem in the discrete-time case simpe necessary and sufficient conditions exist that are directy computabe from the i/o equation and do not depend on an unknown singe-variabe output-dependent function [26] whereas in the continuous-time case derivation of simiar conditions seems to be a difficut task We expect that the unified formaism of time scae cacuus wi hep to understand the reasons for this discrepancy and suggest a soution Finay one may empoy the toos of differentia geometry ike in [24] and [27] to derive the aternative conditions which do not rey on the i/o equation of the system These conditions woud be preferabe in the situations when the i/o equation is difficut to find ACKNOWLEDGEMENTS The research of M Ciukin was supported by European Socia Funds Doctora Studies and Internationaisation Programme DoRa carried out by Foundation Archimedes The work of V Kaparin and Ü Kotta was supported by the European Union through the European Regiona Deveopment Fund and the Estonian Research Counci persona research funding grant PUT48 The work of E Pawłuszewicz was supported by Białystok University of Technoogy grant No S/WM//202 APPENDIX PROOF OF PROPOSITION 3 Proof Note that in [7] the formua 4 was proved for the case T = R Therefore we consider here ony the case T = τz τ > 0 First consider the eft-hand side of equation 4 Using 4 for n = a + b the definition of the operator σ and the chain rue for the partia derivative with respect to ξ a one obtains [ Φξ ξ 2 ξ r a+b ] ξ a = a+b µ a+b k=0 which according to 5 for n = a + b k yieds [ Φξ ξ 2 ξ r a+b ] ξ a = a+b µ a+b k=0 k C k a+b k C k a+b Φ ξ σ a+b k Φ ξ σ a+b k 2 ξ σ a+b k r ξ σ a+b k ξ σ a+b k ξ σ a+b k 2 ξ σ a+b k r ξ σ a+b k a+b k s=0 C s a+b k ξ σ a+b k ξ a ξ s µs ξ a Note that ξ s / ξ a = 0 for every s except for s = a when it equas Furthermore s = a occurs ony when a + b k a impying k b Thus one can write [ Φξ ξ 2 ξ r a+b ] ξ a = b Φ ξ σ a+b k µ b k=0 k Ca+b k ξ σ a+b k 2 ξ σ a+b k r Ca a+b k ξ σ a+b k

10 396 Proceedings of the Estonian Academy of Sciences Taking into account that by direct computations Ca+b k Ca a+b k = Cb a+b Ck b and using the properties one obtains F ξ σ i ξ σ i Fξ = ξ [ Φξ ξ 2 ξ r a+b ] = Ca+b b ξ a µ b σ i and F σ i+ j = F σ i j σ b k=0 k C k b Φξ ξ 2 ξ r which according to 4 for n = b confirms 4 This competes the proof ξ σ b k σ a REFERENCES Aranda-Bricaire E and Moog C H Linearization of discrete-time systems by exogenous dynamic feedback Automatica Bartosiewicz Z and Pawłuszewicz E Reaizations of noninear contro systems on time scaes IEEE Trans Autom Contr Bartosiewicz Z and Pawłuszewicz E Externa dynamica equivaence of time-varying noninear contro systems on time scaes Int J Contr Bartosiewicz Z and Piotrowska E On stabiisabiity of noninear systems on time scaes Int J Contr Bartosiewicz Z Kotta Ü Pawłuszewicz E and Wyrwas M Agebraic formaism of differentia one-forms for noninear contro systems on time scaes Proc Estonian Acad Sci Phys Math Bartosiewicz Z Kotta Ü Pawłuszewicz E and Wyrwas M Contro systems on reguar time scaes and their differentia rings Math Contro Signas Syst Bartosiewicz Z Kotta Ü Pawłuszewicz E Tõnso M and Wyrwas M Agebraic formaism of differentia p-forms and vector fieds for noninear contro systems on homogeneous time scaes Proc Estonian Acad Sci Beikov J Kotta Ü and Tõnso M NLContro: Symboic package for study of noninear contro systems In IEEE Muti- Conference on Systems and Contro Hyderabad India August Bohner M and Peterson A Dynamic Equations on Time Scaes Birkhäuser Boston MA USA Casagrande D Kotta Ü Tõnso M and Wyrwas M Transfer equivaence and reaization of noninear input-output detadifferentia equations on homogeneous time scaes IEEE Trans Autom Contr Conte G Moog C H and Perdon A M Agebraic Methods for Noninear Contro Systems Theory and Appications Springer-Verag London UK 2nd edition Fiess M Reversibe inear and noninear discrete-time dynamics IEEE Trans Autom Contr Funahashi Y An observabe canonica form of discrete-time biinear systems IEEE Trans Autom Contr Goodwin G C Graebe S F and Sagado M E Contro System Design Prentice Ha Upper Sadde River NJ USA Huijberts H J C On existence of extended observers for noninear discrete-time systems In New Directions in Noninear Observer Design Nijmeijer H and Fossen T I eds Lecture Notes Contro Inform Sci Kaparin V and Kotta Ü Necessary and sufficient conditions in terms of differentia-forms for inearization of the state equations up to input-output injections In UKACC Internationa Conference on CONTROL Coventry UK Kaparin V and Kotta Ü Theorem on the differentiation of a composite function with a vector argument Proc Estonian Acad Sci Kaparin V and Kotta Ü Shumsky A Ye and Zhirabok A N A note on the reationship between singe- and mutiexperiment observabiity for discrete-time noninear contro systems Proc Estonian Acad Sci Kaparin V Kotta Ü and Wyrwas M Observabe space of noninear contro system on a homogeneous time scae Proc Estonian Acad Sci Kotta Ü Decomposition of discrete-time noninear contro systems Proc Estonian Acad Sci Phys Math Kotta Ü and Schacher K Possibe non-integrabiity of observabe space for discrete-time noninear contro systems In The 7th IFAC Word Congress Seou Korea Kotta Ü Bartosiewicz Z Pawłuszewicz E and Wyrwas M Irreducibiity reduction and transfer equivaence of noninear input-output equations on homogeneous time scaes Syst & Contr Lett

11 M Ciukin et a: Linearization on time scaes Kotta Ü Rehák B and Wyrwas M On submersivity assumption for noninear contro systems on homogeneous time scaes Proc Estonian Acad Sci Krener A J and Respondek W Noninear observers with inearizabe error dynamics SIAM J Contr Optim Middeton R H and Goodwin G C Digita Contro and Estimation: A Unified Approach Prentice Ha Engewood Ciffs NJ USA Muari T and Kotta Ü Transformation the noninear system into the observer form: simpification and extension European J Contr Respondek W Pogromsky A and Nijmeijer H Time scaing for observer design with inearizabe error dynamics Automatica Oekuvõrrandite ineariseerimine sisend-väjund-injektsioonide kaudu homogeense ajaskaaa Monika Ciukin Vadim Kaparin Üe Kotta ja Ewa Pawłuszewicz On uuritud homogeense ajaskaaa defineeritud mitteineaarse juhtimissüsteemi oekuvõrrandite oekutaastaja kujue teisendamist Aternatiivset nimetatakse antud kujue teisendamist süsteemi ineariseerimiseks sisend-väjund-injektsioonide abi sest seise kuju on süsteemi võrrandid ineaarsed kuni mõnede mitteineaarsete ainut sisenditest ja väjunditest sõtuvate iidetavateni; viimaseid nimetatakse sisendväjund-injektsioonideks Oekutaastaja kuju esitatud süsteemi oekuid saab ihtsat hinnata seiset et hindamisviga äheneb asümptootiiset nuie Oekute hindamine on ouine oukorras kus need poe vahetut mõõdetavad või on mõõtmine iiga kais Anaüüs ajaskaaa võimadab ühidada pidevate ja diskreetsete dünaamiiste süsteemide uurimise On eitud tarviikud ja piisavad tingimused süsteemi võrrandite oekuteisendustega oekutaastaja kujue teisendamiseks Tingimused on formueeritud teatud otseset süsteemi sisend-väjund-võrrandist eitavate diferentsiaasete üks-vormide eksaktsuse kaudu Kui tingimused on rahudatud siis üks-vormide integreerimise eitakse funktsioonid sisend-väjundinjektsioonid mie abi saab oekuvõrrandite oekutaastaja kuju ihtsat konstrueerida