Application of Exact Discretization for Logistic Differential Equations to the Design of a Discrete-Time State-Observer

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1 5 Proceedings of the Interntionl Conference on Informtion nd Automtion, December 58, 5, Colombo, Sri Ln. Appliction of Exct Discretiztion for Logistic Differentil Equtions to the Design of Discrete-ime Stte-Observer Hideyui Kmei* nd Noriyui Hori + Digitl Control Lbortory Grdute School of Systems nd Informtion Engineering University of suub suub, Ibri, Jpn. *Emil: hmei@digicon-lb.esys.tsuub.c.jp +Emil: hori@digicon-lb.esys.tsuub.c.jp Abstrct In this study, it is investigted if the concept of exct discretiztion developed by the uthors for logistic differentil equtions is pplicble to the design of discrete-time stte observers. he observer is exct in the sense tht the stte of the observer mtches tht of n equivlent continuous-time observer t the smpling instnts for ny smpling intervl. he stte-observer is simulted to show tht this is indeed the cse. he proposed method is pplicble to plnts whose dynmics re governed by homogeneous logistic differentil eqution. he observer is exct lso for the non-homogeneous cse when the input is piecewise constnt, which holds true for systems under smpled-dt control. Although the system considered here is first-order nd, therefore, the stte cn be nown directly from the output, the presented pproch cn be extended to higher order cses, once the exct discretiztion method is extended to such cses. I. INRODUCION here hve been vriety of discrete-time models proposed for open- nd closed-loop systems []. While the development is firly complete for liner systems, this is not the cse for nonliner systems. One of the simplest methods tht re pplicble to liner nd nonliner systems, in the context of open- nd closed-loop configurtions, is the forwrd difference model (FDM). his is derived bsed on numericl differentition scheme the continuous-time differentitor is discretized without using the prmeters of the system to be discretized. However, this method usully requires fst smpling rte to ttin stbility of the overll discrete-time system nd n even fster rte to obtin dequte performnce. Liner multi-step methods [] cn chieve better ccurcy thn the FDM by incresing the order of the lgorithms, which mes stbility issues more involved. However, these methods do not use system prmeters either, when discretizing the continuous-time differentitor. By ming the lgorithm dependent on the system prmeters, verstility my be reduced, but ccurcy cn be incresed. Exct discrete-time models re those systems whose stte mtches exctly tht of the continuous-time originl t the smpling instnts for ny smpling intervl nd re possible when the differentitor gin is mde dependent on its system prmeters. Developing such model for clss of nonliner systems nd enlrging pplicble clsses re, therefore, importnt. Some of the most fundmentl nonliner systems, such s logistic, Bernoulli, nd Riccti equtions, cn be trnsformed into liner systems, for which ll the liner discretiztion schemes cn be pplied. A trnsformtion of logistic eqution into second-order liner differentil eqution is nown [], s well s bilineriztion with guge invrince [4]. hese methods introduce two new vribles whose rtio gives the originl vrible. Unfortuntely, the linerized system becomes higher-order thn the originl system nd usully becomes unstble, ming this model unsuitble for on-line computtion. For instnce, even if the estimted stte of the observer my converge to true stte vlue, it is obtined s the rtio of two diverging sttes nd, therefore, the computtion will stop eventully. However, trnsformtions tht cn led to stble, first-order, liner systems do exist for homogeneous logistic eqution [5], Bernoulli eqution [6], nd non-homogenous logistic eqution [7]. When such trnsformtions re used for the observer design, the order will not increse nd stbility cn be ssured. Stte observers re usully developed in continuous-time [8]. However, they re most liely implemented using digitl

2 54 Proceedings of the Interntionl Conference on Informtion nd Automtion, December 58, 5, Colombo, Sri Ln. hrdwre nd discrete-time formultion is needed. here re two pproches to designing discrete-time stte observer; digitl design nd digitl redesign. In the digitl redesign pproch, the observer is designed first in continuous-time form nd then discretized. In the digitl design pproch, the plnt is discretized first nd the observer is designed in discrete-time. In the present study, two discrete-time stte observers will be designed bsed on the digitl redesign using the FDM nd the exct discrete-time model. It will be shown tht the ltter design gives the sme observer s when the digitl design is crried out. hese observers re considered specificlly for clss of nonliner systems represented by homogenous nd non-homogeneous logistic equtions, using exct discretiztion methods reported in [5] nd [7]. he performnce of these observers will be evluted nd compred by simultions. II. CONINUOUS-IME SAE OBSERVER Consider system whose dynmics re given by the following homogeneous logistic differentil eqution s xt () = f( x) = λ x() t + xt (), () >, λ, x(), nd the prototype stte observer of the form z () t = hzv (, ) = λz () t + zt () + vt (). () he input v is the signl to be designed such tht the stte z pproches the stte x s time elpses. he observer lineriztion problem [8] is solvble for this plnt nd the continuous-time stte observer is designed in this study such tht the error system is liner nd exponentilly stble. he error system is given by ( α αλ ) ( α αλ ) x z = x x z z + v () which is to be forced to behve s liner dynmics given by x z = ϑ x z (4) ( ) with the gin stisfying ϑ < for stbility; i.e., to me z x s t. his cn be chieved by selecting the observer input s ( ) ( ) ( ) v = ϑ x z + αx αλx αz αλz. (5) he finl, stte-observer is determined, therefore, s { ( ) } z = ϑz+ αλx + α ϑ x. (6) III. DISCREE-IME SAE OBSERVER A. Digitl Redesign When digitl redesign method is used, the continuous-time stte observer (6) is designed nd then discretized using, for instnce, the FDM. ) he Forwrd Difference Model (FDM) When the FDM is used, the stte observer (6) yields the following discrete-time observer: { ( ) } δz = ϑz + αλx + α ϑ x (7) q δ = is clled the delt opertor [9], is the discretiztion intervl, nd q is the conventionl shift opertor. A few words on delt opertors re in order here. he delt form is very convenient in relting discrete-time results to continuous-time results nd hs better numericl properties thn the shift opertor form. For system given in delt opertor form s δξ = ζ, (8) the updte formul should be implemented s ξ = ξ + + ζ. (9) hese dvntges re explined in detil in [9]. It cn be seen tht, in the FDM, the delt opertor replces the differentil opertor nd ll system prmeters re the sme s those of the corresponding continuous-time originl. Since the FDM model is not exct, the error will result in estimting the true stte. A better method is to use n exct discretiztion. b) he Exct Model Since the exct discrete-time model of (6) is not nown t this stge, tht of () is used [5], s nd δ z =Φ ( z, v, ) h( z, v ) () hz (, v) = λ z + z + uv () σ Φ ( z, v, ) = σ ( σ ) + + λ z σ () σ e σ = ( ) () σ σ = + 4λv, (4) which is ssumed to be rel for ll. When σ =, defined to be unity. σ is

3 55 Proceedings of the Interntionl Conference on Informtion nd Automtion, December 58, 5, Colombo, Sri Ln. For the exct discrete-time plnt model nd the stte observer given in the previous section, the observer input v is to be designed to force z pproch x. For this purpose, consider the stte estimtion error dynmics given by (, )( ) ( z u ){ αz αλz v} δx δz = Ω x αx αλx Φ,, +. (5) If the desired dynmics of the error system re chosen to be liner s ( ) δx δz = θ x z (6) the sclr gin θ stisfies < θ < for stbility [9], the observer input v is determined s (,, ) { θ ( ) v =Φ z v x z (7) +Ω, Φ,,. ( x )( αx αλx) ( z u )( αz αλz)} herefore, the stte observer is given by { (, ) ( ) } δz = θz + Ω x f x θx. (8) It should be noted tht input (7) does not hve to be generted explicitly, but only the observer (8) need to be implemented. herefore, the gin Φ is used only in design process nd need not be computed for implementtion. It should lso be noted tht for the error system (6) to be the step-invrint-model [] of (4), the differentitor gins should ϑ e be relted such tht θ =, θ ϑ ( ). When this is the cse, the observer (8) is n exct discrete-time model of (6). For given continuous-time plnt, digitlly designed observer nd digitlly redesigned observer re different, if inexct discretiztion methods re used. However, if n exct discretiztion is used, they cn be identicl. B. Digitl Design For the nonliner plnt given by eqution (), the exct discrete-time model is given [5] by nd δ x =Ω ( x, ) f ( x ) (9) f ( x ) = λ x + x () Ω ( x, ) = () + λ x his model hs vrible gin Ω, which cn be considered s the discrete-time differentition gin. his gin depends on the plnt prmeters nd stte vrible, nd needs to be updted t every smpling instnt. In return for this complexity, the stte of this model mtches exctly tht of the continuous-time originl t the smpling instnts for ny smpling intervl. he stte observer is designed bsed on the discrete-time plnt model (9) nd by following the sme procedure s given in Section. For the plnt (9), the observer of the following form is considered: δz = Ω λz + Ω z + v. () Defining the observer input s ( ) ( ) ( ) v = θ x z +Ω αx αλx Ω αz αλz, (4) the discrete-time observer is obtined s { ( ) } δz = θz + Ω λx + x θx, (5) which is the sme s (8). IV. NON-HOMOGENEOUS LOGISIC EQUAION In the sme mnner s the homogeneous logistic plnt, the plnt with piecewise constnt input term cn be delt with. In this cse the continuous-time plnt is given s (, ) α αλ x = h x u = x x + u. (6) he observer cn be designed s in the homogeneous cse nd is given by { (, ) ϑ } z = ϑz+ h x u x, (7) ϑ <. he exct model of the plnt is given by (,, ) (, ) δ x =Φ x u f x u, (8) σ Φ ( x, u, ) = σ ( σ ) + + λ x σ. (9) he digitl design nd digitl redesign will result in the sme observer, nd is given by { (,, ) (, ) } δz = θz + Φ x u f x u θx, () ϑ e θ =. e = ( ). ()

4 56 Proceedings of the Interntionl Conference on Informtion nd Automtion, December 58, 5, Colombo, Sri Ln. 4 V. SIMULAIONS he simultion study is crried out for the following plnt: = + +, () =.5. () x x x u x Shown in Fig. to Fig. re the simultion results for the homogeneous cse ( u = ), respectively, with =.5,., nd. seconds. hey include the true stte of the continuous-time plnt () (denoted s C plnt), the estimted stte of the continuous-time observer (6) (C observer), tht of the discrete-time observer obtined using the forwrd-difference- model (7) (FDM D observer), nd tht of the proposed discrete-time observer (8) (Proposed D observer). he observer s initil stte is z () =. in ll the simultions. he liner error dynmics re chosen to be ϑ =. nd ϑ e θ =. It cn be seen from these figures tht the proposed discrete-time observer gives the stte estimte tht is the sme s tht of the continuous-time observer t the smpling instnts for ll smpling intervls tested. On the other hnd, the error tends to become lrger for the ustin s cse s the smpling intervl becomes lrger ime [sec] Fig.. Simultion results for =.s. C plnt C observer FDM D observer Proposed D observer ime [sec] Fig.. Simultion results for =.s ime [sec] Fig.. Simultion results for =.5s. Fig. 4 to Fig. 6 show the simultion results for the non-homogeneous cse, under the sme conditions s the homogeneous cse. In this cse, the continuous-time observer is given by (7) nd the discrete-time observer by (). he input is rectngulr pulse with the period of seconds, mplitude of unity, nd duty rtio of 5%. he observer still gives exct estimtes, s long s the input to the continuous-time plnt is constnt during the smpling intervl, which is the cse when the continuous-time plnt is under digitl control.

5 57 Proceedings of the Interntionl Conference on Informtion nd Automtion, December 58, 5, Colombo, Sri Ln. 5 C plnt C observer Proposed D observer FDM D observer plnt input ime [sec] Fig. 4. Simultion results for =.5s (Plnt with input). VI. CONCLUSIONS A discrete-time stte observer which gives the stte estimtes tht mtch those of the continuous-time observer t the smpling instnts for ny smpling intervl, hs been proposed nd its performnce confirmed with simultions, for homogeneous logistic systems. he observer is lso exct for the non-homogeneous cse when the plnt input is piecewise constnt, s in system under smpled-dt control. Although the exct stte observer presented in this study is pplicble only to clss of nonliner systems, the extension to higher-order systems (simultneous logistic-type differentil equtions) is underwy nd this will increse the pplicbility of this method to much wider clss of nonliner system [] ime [sec] Fig. 5. Simultion results for =.s (Plnt with input). REFERENCES [] N. Hori,. Mori nd P. N. Niiforu, Discrete-time models of continuous-time systems, Advnces in heory nd Applictions, Vol. 66, pp. -45, Acdemic Press, et. C.. Leondes, 994. [] J. D. Lmbert, Computtionl Methods in Ordinry Differentil Equtions, John Wiley & Sons, New Yor, 97. [] N. Yjim, Ordinry Differentil Equtions Introduction for Engineers, Iwnmi, 989. (in Jpnese) [4] R. Hirot, Lectures on Difference Equtions From Continuous to Discrete, Science Publishing, oyo, (in Jpnese). [5] N. Hori nd C. A. Rbbth, Invrint models of nonliner system governed by logistic eqution, SICE Annul Conference, Spporo, Jpn, pp. 66-7, 4. [6] N. Hori, C. A. Rbbth, nd J.H. Chen, Invrint discretiztion of system governed by Bernoulli s eqution, Int. Conf. Dynmics, Instrumenttion, nd Control, Nnjing, Chin, Session FE-4, Pper No. P97, 4. [7] N. Hori nd C. A. Rbbth, Appliction of digitl control theory to exct discretiztion of logistic eqution with constnt term, IEEE Conf. Control Applictions, Pper MB4.4, oronto, Cnd, 5. [8] A. Isidori, Nonliner Control Systems, Springer-Verlg, Berlin, 995. [9] R. H. Middleton nd G. C. Goodwin, Digitl Control nd Estimtion A Unified Approch, Prentice-Hll, Englewood Cliffs, N. J. 99. []. Ohtsu, Immersion of nonliner systems How fr cn model structures be simplified?, SICE Mesurement nd Control Mgzine, Vol. 4, pp. 87-8, (in Jpnese) ime [sec] Fig. 6. Simultion results for =.s (Plnt with input).