Nonlinear Control of Heartbeat Models

Transcription

1 Nonlinea Contol of Heatbeat Moels Witt THANOM Robet N. K. LOH Depatment of Electical an Compte Engineeing Cente fo Robotics an Avance Atomation Oaklan Univesity Rocheste Michigan 489 U. S. A. ABSTRACT This pape pesents a novel application of nonlinea contol theoy to heatbeat moels. Existing heatbeat moels ae investigate an moifie by incopoating the contol inpt as a pacemake to povie the contol channel. A nonlinea feeback lineaization techniqe is applie to foce the otpt of the systems to geneate atificial electocaiogam (ECG) signal sing iscete ata as the efeence inpts. The synthetic ECG may seve as a flexible signal soce to assess the effectiveness of a iagnostic ECG signal-pocessing evice. Keywos: Heatbeat moel Electocaiogam Nonlinea contol Feeback lineaization Phase potait analysis.. INTRODUCTION The hman heat is a complex an yet obst system. One of the most impotant signals that elates to hman heat opeation is the ECG signal. It is a time-vaying signal epesenting the electical potential geneate by the electical activity in the caiac tisse. A single cycle of the ECG eflects the contaction an elaxation of the heat leaing to the heat s pmping action. The ECG can be mease by ecoing the potential between two electoes place on the sface of the skin at some pe-etemine points. Chaacteistic infomation extacte fom the ECG signal can be se to inicate the state of caiac health as well as a potential heat poblem []. Mch effot has been investe into the evelopment of mathematical moels that escibe the opeation of the hman heat. One of the ccial evelopments is by Zeeman [] whee he evelope a mathematical moel that capte thee impotant qalities of caiac chaacteistics: (i) stable eqilibim; (ii) theshol fo tiggeing an action potential; an (iii) etn to eqilibim. The eslting moels ae a n -oe nonlinea iffeential eqation epesenting the heatbeat system an a -oe nonlinea iffeential eqation that can be applie to the neve implse. Othe inteesting an elate moels wee pesente in [ - 6]. Reseach that focses on geneating the ECG signal is also vey active. In [7] the -imensional nonlinea moel fom [] was moifie by aing a contol vaiable in oe to contol the heat ate vaiability an to poce the ECG sing a neal netwok. In [8] the athos moifie the n -oe nonlinea heatbeat system in [] by aing an on-off type contol vaiable epesenting the pacemake fo flfilling the mechanism of contaction-elaxation of the heat. In [9] a ynamical moel that geneates a synthetic ECG signal by specifying the mean an the stana eviation of the heat ate an the powe spectm of the RR tachogam was popose. The moel oes not aess how the heat woks bt athe tilizes the statistical infomation of the ECG as a pioi ata to geneate a signal. This pape pesents a novel application of nonlinea contol system theoy feeback lineaization to the heatbeat systems oiginate fom []. The systems wee moifie by aing a pacemake to povie the contol channel. One of the objectives is to ceate a synthetic ECG signal base on existing ECG ata whee the ata is se as the efeence signal fo a tacking contol poblem. The synthetic ECG signal can be se as a flexible signal soce to assess the effectiveness of a iagnostic ECG signal-pocessing evice [7]. The pape is oganize as follows. In Section the moel ynamics an its chaacteistics ae investigate. Phase potait an stability analysis ae concte. The nonlinea feeback lineaization contol theoy is evelope in Section an applie to the heatbeat systems to ceate a synthetic ECG in Section 4. Lastly the conclsion is pesente in Section 5.. HEARTBEAT MATHEMATICAL MODELS Thee ae two states of the heat in a cycle of the heatbeat: iastole which is the elaxe state an systole which is the contacte state. The cycle stats when the heat is in the iastolic state. The pacemake which is locate at the top of the ight atim one of the ppe chambes of the heat tigges an electochemical wave that speas slowly ove the atim. This electochemical wave cases the mscle fibes to contact an psh the bloo into the venticles the lowe chambes of the heat. The same electochemical wave then speas apily ove the venticles casing the whole venticle to contact into the systolic state an pmping the bloo into the lng an the ateies. Immeiately following the systolic state the mscle fibes qickly elax an etn the heat to the iastolic state to complete one cycle of the heatbeat [8]. ISSN: SYSTEMICS CYBERNETICS AND INFORMATICS VOLUME 9 - NUMBER - YEAR

2 Secon-Oe Nonlinea Heatbeat Moel A mathematical moel that escibes the behavio of the heatbeat was evelope in [] whee it was sggeste that sch a moel shol contain thee basic feates: (i) a stable eqilibim state epesenting iastole; (ii) the theshol fo tiggeing the electochemical wave casing the heat to go into systole; an (iii) the etn of the heat into the iastolic state. The eslting moel is given by ( x Tx + x ) x = T > () x = x x () whee x () t epesents the length of the mscle fibe x () t is a vaiable elate to electochemical activity; the paamete is a small positive constant associate with the fast eigenvale of the system x is a scala qantity epesenting a typical length of mscle fibe in the iastolic state an T epesents tension in the mscle fibe. Fig. illstates the phase potait of Eqs. () an () with the initial conitions along the left an ight iagonals acoss the x x plane. The paamete vales se to poce the phase potait ae =. T = an x =. Heat mscle fibe length x A - - C Limit cycle B x - x + x = Electochemical activity x Fige. Phase potait of the n -oe heatbeat system. D T fx ( x T) A = = = () x x= whee x= fx = x Tx+ x x x. The eigenvales of A ae given by λ =.6 an λ =.8 fo T = an =.. Theefoe the oigin is nstable since both eigenvales ae eal an positive. In Fig. since the vecto fiel aon the segment AB an CD always points towa the cbic line an away fom the cbic line in the BC potion any point along the cbic line in the AB an CD segments is consiee to be stable wheeas points along the BC section ae nstable. The points B an C ae impotant as they specify the theshol the secon basic feate (ii) of the heatbeat moel mentione ealie. These points can be obtaine easily by consieing the eigenvale of the matix A in Eq. () λ = ( x T) ± ( x T) 4. (4) The conition that the eal pat of the eigenvales is negative is x T >. Theefoe the system is stable if x T / which efes to the section AB an x T / which escibes the section CD. In othe wos the theshols fo switching between the iastolic an the systolic states at point B is x = T / an x = T / at point C. The stable eqilibim point that epesents the state of iastole can be etemine by changing the vale of x in Eq. () sch that it satisfies the stability conition above. Fig. isplays the phase potait of the system with x =.4. The eqilibim point is stable at ( ) an qalifies to be the iastolic eqilibim state that is satisfies the fist feate (i): a stable eqilibim. T In Fig. the cbic line (ashe cve) epesents the steaystate of Eq. (). When x = in Eq. () the eqilibim point of the system is at the oigin. All tajectoies initiate above the cbic line that is x Tx + x > iect ownwa towa the oigin along the cbic line. Likewise all tajectoies state below the cbic line i.e. x Tx + x < iect pwa towa the oigin along the cbic line. All tajectoies en p at the limit cycle aon the eqilibim point. It is obvios that the eqilibim point is nstable as the vecto fiel insie the limit cycle iects away fom the point. This conclsion can be confime by analyzing the stability of the eqilibim point sing the well-known Lyapnov iniect stability theoem []. Fo this ppose let A be the constant Jacobian matix of Eqs. () an () at the oigin it follows that Heat mscle fibe length x - - Diastole Eqilibim x - x + x = Electochemical activity x Fige. Phase potait of the n -oe heatbeat system ing iastole. SYSTEMICS CYBERNETICS AND INFORMATICS VOLUME 9 - NUMBER - YEAR ISSN:

3 In Fig. all of the tajectoies egaless of thei initial conitions en p at the iastolic eqilibim point. Since the eqilibim point is stable the system will stay at this point foeve nless thee is an extenal excitation that foces the system to a new eqilibim point. In [8] the athos sggest moifying the system by aing a contol inpt as ( x Tx + x ) x = T > (5) x = x x + x x (6) s whee the aitional paamete x s epesents a typical fibe length when the heat is in the systolic state an epesents caiac pacemake contol mechanism that iects the heat into the iastolic an the systolic states. By poposing the caiac pacemake contol signal in the fom of an (on-off contol) the eqilibim point of the system can be change between the iastolic an the systolic states. In this pape we will attempt to geneate atificial o synthetic ECG signals by sing the nonlinea feeback contol stategy inpt-otpt feeback lineaization techniqe an methoology to contol the n -oe heatbeat moel given by Eqs. (5) - (6) an the -oe heatbeat moel given by Eqs. (7) - (9).. NONLINEAR FEEDBACK LINEARIZATION Consie a contol-affine single-inpt single-otpt (SISO) nonlinea system escibe by n n x = f( x) + g( x) fg : D () n y = h( x ) h: D () n whee x is the state vecto y ae the contol an otpt signals espectively; f g ae smooth vecto fiels in a omain D an h a smooth fnction in D whee D is an open set n in. Heat mscle fibe length x - - x - x + x = Systole Eqilibim Given the nonlinea system of Eq. () an the measement of Eq. () o goal is to fin a iffeomophism o nonlinea tansfomation of the fom z = T( x ) with T = that tansfoms the nonlinea system in the x-cooinates to a linea system in the z-cooinates. One of the most impotant easons fo fining the tansfomation is that the powefl linea system theoy an methoologies can be applie once a nonlinea system has been lineaize. Diffeentiating the otpt y with espect to t yiels y = L f h( x) + L g h( x) () Electochemical activity x Fige. Phase potait of the n -oe heatbeat system ing systole. Fig. isplays the phase potait of Eqs. (5) an (6) with t () = an x s =.84. The stable eqilibim point is locate at (-.84.5). Theefoe the on-off contol scheme is sccessfl in moeling the state changes fom iastole to systole epening on the contol signal. Thi-Oe Nonlinea Heatbeat Moel The -oe nonlinea heatbeat moel is given by [] ( x x x x ) x = + + (7) x = x x (8) x = x + (9) whee x () t epesents the length of the mscle fibe x () t epesents tension in the mscle fibe x () t is elate to electochemical activity is a small positive constant an epesents caiac pacemake contol signal which iects the heat into the iastolic an the systolic states. The ynamics of the -oe system is simila to that of the n -oe system except that the ynamics of the mscle fibe tension is taken into consieation. In the n -oe system this qantity is consiee as a constant paamete. whee L f h( x ) an Lh g ( x ) enote the Lie eivatives of h( x ) with espect to fx an gx espectively. If Lh g ( x ) = then yt is inepenent of. Contining sccessive iffeentiation ρ times ntil appeas explicitly we obtain y ρ = L ρ h L L ρ f x + g f h( x) b( x) D( x). () The smallest intege ρ fo which appeas is efee to as the elative egee. The nonlinea system in Eqs. () - () is sai to have a well-efine elative egee ρ in a egion D D if k ρ LLh g f ( x ) = k k< ρ ; an LL g f h( x ) (4) fo all x D. Note that ρ n. Fom Eq. () efine v y ρ = b( x) + D( x) (5) whee v is a one-imensional tansfome inpt ceate by the feeback lineaization pocess; b(x) is calle the nonlineaity cancellation facto an D(x) the ecopling matix (a scala in the pesent SISO system). Eqation (5) yiels the lineaizing feeback contol law [ ]: = D ( x) b( x ) + v (6) povie D( x ) is nonsingla (invetible). ISSN: SYSTEMICS CYBERNETICS AND INFORMATICS VOLUME 9 - NUMBER - YEAR

4 To evelop an oveall epesentation of the system fo the case with elative egee ρ < n the iffeomophism z = T( x ) can be expesse as h( x) ρ ξ L h f x z = T( x) = (7) η φ ( x) φ n ρ ( x) ρ n ρ whee ξ η ; an φi ( x ) i =... n ρ ae chosen sch that Tx is a iffeomophism on a omain D D that is the Jacobian matix associate with Tx is nonsingla an φi Lgφi ( x) = g( x) = i n ρ x fo all x D. The iffeomophism in Eq. (7) leas to the nomal fom (8) ξ =A v ξξ + B ξ (9) η =f o(ξη ) () y = h( x ) = ξ. () Setting ξ ( t ) = in Eq. () fo all t yiels η =f ( η) () o which epesents the zeo ynamics fo Eqs. () an (). The stability of the zeo ynamics in Eq. () is an impotant isse in esigning a contolle. The system whose zeo ynamics ae asymptotically stable in the omain of inteest is calle a minimm phase system. The local asymptotic stability of the zeo ynamics is clealy the necessay an sfficient conitions fo the local asymptotic stability of the feeback lineaize system in Eqs. (9) - () [ ]. In the case that the zeo ynamics ae nstable in the egion of inteest the system is known as a non-minimm phase system. Geneally a system of this type cannot be se fo state-feeback contol system esign becase some of the state vaiables will ivege to infinity. In this case the stabilization of the nstable zeo ynamics nee to be consiee if possible. Asymptotic Otpt Tacking Let the contol objective be steeing the otpt y to a esie efeence y () t. This gives ise to an otpt tacking contol poblem. Define the otpt tacking eo as e y y. () The main objective is to foce et sch that yt y ( t) as t. It follows that e = y y ( ρ) ( ρ) ( ρ) ( ρ) e = y y = v y. (4) A sitable tacking contol law fo the tansfome inpt v is given by whee v= Ke + (5) ξ y ρ T = e e e e ρ ρ ξ e an the constant feeback gain K is etemine sch that Acl = Aξ BξK ξ is Hwitz that is all eigenvales of A cl lie in the open left-haft complex plane. Finally the oveall close-loop nonlinea system in the x-cooinates is given by [ ] x = f( x) + g( x) D ( x) b( x) + v (6) whee v is given by Eq. (5). 4. APPLICATION TO HEARTBEAT MODELS Contolling the Secon-Oe Nonlinea Heatbeat System Consie the n -oe nonlinea heatbeat moel given by Eqs. (5) - (6). Fist we consie choosing the otpt measement to be yt () = x() t. This selection is easonable in the physical viewpoint since the electochemical activity can be mease as the potential acoss the membane of the mscle fibe []. Diffeentiating yt with espect to t yiels y = x x + x x (7) s whee appeas which shows that the elative egee is ρ =. Ths the heatbeat system has both extenal an intenal ynamics. The iffeomophism is given by h( x) ξ x z = Tx = = = φ η x. (8) x The Jacobian matix associate with Tx is given by Tx = (9) x which is nonsingla fo all x ; theefoe Tx is a global iffeomophism fo Eqs. (5) an (6). Eqation (8) shows that the oiginal system is aleay in a nomal fom when the otpt is chosen as yt () = x() t. Howeve one of the benefits of eiving (8) is that it eveals x () t as the intenal ynamics an x () t as the extenal ynamics of the system. The eslting system in nomal fom is obtaine as ξ = η x + x xs () η = ( η Tη + ξ) () y = ξ. () Next consie the stability of the intenal ynamics in Eq. (). The zeo ynamics satisfy 4 SYSTEMICS CYBERNETICS AND INFORMATICS VOLUME 9 - NUMBER - YEAR ISSN:

5 η = f( ξ η) = ( η Tη ξ = ). () Thee ae thee eqilibim points fo Eq. (): η = ± T. We applie the Lyapnov iniect stability theoem [] to analyze the stability of the eqilibim points. Fist at the oigin T λ = ( η T ) =. (4) η = Since T an ae positive constants it follows that λ > ; hence the eqilibim at the oigin is nstable. Consie the othe eqilibim points T λ = ( η T ) =. (5) η =± T Heat mscle fibe length x Fige 4. Simlation eslt of x () t of the n -oe heatbeat system. It is clea that λ < an λ < fo all T > an > ths the eqilibim points at η =± T ae asymptotically stable. In othe wos egaless of the nstable eqilibim at the oigin the steay-state of the zeo ynamics will en p at eithe the point η = T o η = T epening on the initial conition. As a eslt the zeo ynamics ae asymptotically stable. We concle that the system is a minimm-phase system. To pocee to the otpt tacking contol esign task efine the tacking eo as e y y whee y () t is the efeence inpt. It follows that Electochemical activity x Physionet ECG ata x e = x x + x x y v y. (6) s whee v is the tansfome inpt. Let the tacking contol law fo the tansfome inpt v be given by ( x y ) v= Ke+ y = K + y (7) whee K = is obtaine by placing the eal pole at - of the complex plane. We obtain the eslting lineaizing feeback contol law = K( x y ) y ( x x). (8) x x + s Finally sbstitting Eq. (8) into Eqs. (5) - (6) an sing Eq. (6) yiels the oveall feeback contol system in the x- cooinates x = x Tx + x T > (9) ( x y ) x = K + y (4) y x. (4) Fig. 4 to 6 show the eslts of tacking eal iscete ECG ata obtaine fom the PhysioNet atabase [4]. In Fig. 4 the initial conition of x () is. an the steay state conveges to T = as expecte. The otpt x () t tacks the iscete ECG ata vey nicely as shown in Fig. 5. The contol signal o pacemake in Eq. (8) which is se to geneate an to tack the ECG signal is shown in Fig Fige 5. Simlation eslt of x () t of the n -oe heatbeat system. Pacemake Fige 6. Simlation eslt of t () of the n -oe heatbeat system. Contolling the Thi-Oe Nonlinea Heatbeat System Consie the -oe heatbeat system given by Eqs. (7) - (9) an choosing the otpt as yt () = x() t. Diffeentiating the otpt with espect to t yiels y = x + (4) ISSN: SYSTEMICS CYBERNETICS AND INFORMATICS VOLUME 9 - NUMBER - YEAR 5

6 which shows that the elative egee is ρ =. The iffeomophism is obtaine as h( x) ξ x z = T( x) = x = =. (4) φ η x φ ( x) η x It can be shown easily that the Jacobian matix associate with Tx is nonsingla fo all x ; ths Tx is a global iffeomophism fo Eqs. (7) - (9). The eslting system in nomal fom is escibe by ξ = η + (44) η = ( η + ηη + ξ) (45) η = η η (46) y = ξ. (47) The zeo ynamics ae given by η = ( η ηη ) + η = η η. (48) Thee ae two eqilibim points associate with Eq. (48): the oigin an ( η η ) = ( ). Applying the Lyapnov iniect stability theoem [] to analyze the stability of each eqilibim point yiels ( η + η) η A = =. (49) It follows that ( ) Re λ i < i = ; whee λ epesents the eigenvale. Theefoe the matix A is Hwitz an the eqilibim point at (-) is asymptotically stable. Next consie the eqilibim point at the oigin ( η + η ) η A. (5) = = () The eigenvales of A ae an -. Since one of the eigenvale is zeo we cannot aw the stability conclsion by the Lyapnov iniect theoem. Howeve sing the ece system theoem [] it can be shown that the zeo ynamics in Eq. (48) ae asymptotically stable. This conclsion is illstate by the phase potait of the zeo ynamics itself as shown in Fig. 7. All tajectoies with initial conitions η convege to the oigin. With the stability analysis eslts we concle that the nomal fom system in Eqs. (44) - (47) is a minimm-phase system. To pocee on the otpt tacking contol esign task efine the tacking eo as e y y whee y () t is the efeence inpt. It follows that η - - () (-) η Fige 7. Phase potait of zeo ynamics. e = x y + y (5) = x y v whee v is the tansfome inpt. Let the tacking contol law fo the tansfome inpt v be given by v= Ke+ y = K x y + y (5) whee K = is obtaine by placing the eal pole at -. The eslting lineaizing feeback contol law is obtaine as = K x y + y + x + (5). The simlation eslts fo the final feeback contol system in the x-cooinates ae shown in Fig. 8-. The state tajectoies of the mscle fibe length x () t an the mscle fibe tension x () t ae isplaye in Fig. 8 an 9 espectively. Fig. emonstates the eslt of the otpt y() t = x() t that tacks the ECG ata obtaine fom the William Beamont Hospitals. Fig. illstates the pacemake of Eq. (5) se to geneate the eslts of the -oe heatbeat contol system. Mscle fibe length x Fige 8. Simlation eslt of x () t of the -oe heatbeat system. 6 SYSTEMICS CYBERNETICS AND INFORMATICS VOLUME 9 - NUMBER - YEAR ISSN:

7 Mscle fibe tension x a contol inpt into the system theeby ceating two inteesting contol-affine SISO nonlinea systems. We showe that the eslting heatbeat moels ae minimm-phase systems sitable fo the esign of otpt tacking contol laws; these otpt tacking contol laws wee se to geneate synthetic ECG signals. The simlation eslts show that the systems can be foce to tack the ECG ata obtaine fom the William Beamont Hospitals an the PhysioNet atabase [4] satisfactoily. Othe biomeical applications of Zeeman's moels sing the nonlinea contol techniqe evelope in this pape ae ne consieation Fige 9. Simlation eslt of x () t of the -oe heatbeat system. Electochemcal activity x WBH ECG Data x Fige 9. Simlation eslt of x () t of the -oe heatbeat system. Pacemake Fige. Simlation eslt of t () of the -oe heatbeat system. 5. CONCLUSION We pesente the application of nonlinea contol system theoy base on inpt-otpt feeback lineaization to biological heatbeat systems. Seveal caiac elate mathematical moels have been investigate an two moels evelope by Zeeman wee chosen in this sty. The moels wee moifie by aing ACKNOWLEDGMENT This eseach was sppote by Oaklan Univesity-Beamont Mltiisciplinay Reseach Awa ne fn #979. We wol also like to thank D. Robet Hammon of the William Beamont Hospitals Royal Oak Michigan fo poviing a set of ECG ata se in the simlation sties. REFERENCES [] N. Kannathal C. M. Lim U. Rajena Achaya an P. K. Saasivan "Caiac state iagnosis sing aaptive neo-fzzy techniqe" Meical Engineeing & Physics Vol. 8 6 pp [] E. C. Zeeman "Diffeential eqations fo the heatbeat an neve implse" Towas a Theoetical Biology Vol pp [] F. A. Robege P. Bhee an R. A. Naea "A caiac pacemake moel" Meical & Biology Engineeing Vol pp. -. [4] D. S. Beitenstein "Caiovascla moeling: the mathematical expession of bloo ciclation" Maste's thesis Univesity of Pittsbgh PA 99. [5] Y. C. Y J. Boston M. Simaan an J. Antaki "Estimation of systemic vascla be paametes fo atificial heat contol" IEEE Tansactions on Atomatic Contol Vol. 4 Jne 998 pp [6] A. Feeia S. Chen M. Simaan J. Boston an J. Antaki "A nonlinea state-space moel of a combine caiovascla system an a otay pmp" Poceeings of the 44th IEEE Confeence on Decision an Contol an the Eopean Contol confeence 5 Seville Spain 5. [7] N. Jafania-Dabanloo D. C. McLenon H. Zhang A. Ayatollahi an V. Johai-Maj "A moifie Zeeman moel fo pocing HRV signals an its application to ECG signal geneation" Jonal of Theoetical Biology Vol pp [8] D. S. Jones an B. D. Sleeman Diffeential Eqations an Mathematical Biology Chapman & Hall/CRC UK. [9] P. E. McShay G. D. Cliffo L. Taassenko an L. A. Smith "A Dynamical Moel fo Geneating Synthetic Electocaiogam Signals" IEEE Tansactions on Biomeical Engineeing Vol. 5 No. Mach pp [] H. K. Khalil Nonlinea Systems eition New Jesey: Pentice Hall. [] A. Isioi Nonlinea Contol Systems New Yok: Spinge- Velag 995. [] M. A. Henson an D. E. Sebog Nonlinea Pocess Contol New Jesey: Pentice Hall 997. [] C. I. Bynes an A. Isioi "Asymptotic Stabilization of Minimm Phase Nonlinea Systems" IEEE Tansaction on Atomatic Contol Vol. 6 No. Octobe 99 pp [4] A. L. Golbege L. A. N. Amaal L. Glass J. M. Hasoff P. Ch. Ivanov R. G. Mak J. E. Miets G. B. Mooy C. K. Peng H. E. Stanley "PhysioBank PhysioToolkit an PhysioNet: Components of a New Reseach Resoce fo Complex Physiologic Signals" Ciclation Vol. No. pp. e5- e. ISSN: SYSTEMICS CYBERNETICS AND INFORMATICS VOLUME 9 - NUMBER - YEAR 7