MOST engineering systems are bounded input-bounded

Transcription

1 This artile has been aepted for pbliation in a ftre isse of this jornal, bt has not been flly edited Content may hange prior to final pbliation Citation information: DOI 9/TAC655978, IEEE Bonded Integral Control of Inpt-to-State Pratially Stable Non-linear Systems to Garantee Closed-loop Stability G C Konstantopolos, Member, IEEE, Q-C Zhong, Senior Member, IEEE, B Ren, Member, IEEE and M Krsti, Fello, IEEE Abstrat A fndamental problem in ontrol systems theory is that stability is not alays garanteed for a losed-loop system even if the plant is open-loop stable With the only knoledge of the inpt-to-state (pratial stability (ISpS of the plant, in this note, a bonded integral ontroller (BIC is proposed hih generates a bonded ontrol otpt independently from the plant parameters and states and garantees losed-loop system stability in the sense of bondedness When a given bond is reqired for the ontrol otpt, an analyti seletion of the BIC parameters is proposed and its performane is investigated sing Lyapnov methods, extending the reslt for loally ISpS plant systems Additionally, it is shon that the BIC an replae the traditional integral ontroller (IC and garantee asymptoti stability of the desired eqilibrim point nder ertain onditions, ith a garanteed bond for the soltion of the losed-loop system Simlation reslts of a d/d bk-boost poer onverter system are provided to ompare the BIC ith the IC operation Index Terms Integral ontrol, non-linear systems, inpt-tostate stability, bonded inpt, small-gain theorem I INTRODUCTION MOST engineering systems are bonded inpt-bonded otpt stable (BIBO For this type of systems, an open-loop ontroller an easily bring the system in a desirable and stable operation Hoever, it is idely knon that, hen external distrbanes or parameter variations or, feedbak is essential to ahieve a desired performane [], [] By losing the loop, stability is no longer garanteed even for BIBO plants Many researhers have fosed on solving the stability problem of a losed-loop system, espeially for the most ommon senario, ie reglation Dring the last 4 years, integral ontrol (IC has been extensively sed in ontrol systems for ahieving asymptoti reglation and distrbane rejetion for systems ith inherent parameter variations The addition of the integrator dynamis G C Konstantopolos is ith the Department of Atomati Control and Systems Engineering, The University of Sheffield, Sheffield, S 3JD, UK, tel: , fax: ( gkonstantopolos@sheffieldak Q-C Zhong is ith the Department of Eletrial and Compter Engineering, Illinois Institte of Tehnology, Chiago, IL 666, USA, and also ith the Department of Atomati Control and Systems Engineering, The University of Sheffield, Sheffield, S 3JD, UK ( zhongq@ieeeorg B Ren is ith the Department of Mehanial Engineering, Texas Teh University, Lbbok, TX 7949, USA ( beibeiren@tted M Krsti is ith the Department of Mehanial and Aerospae Engineering, University of California, San Diego, 95 Gilman Drive, La Jolla, CA 993-4, USA ( krsti@sded The finanial spport from the EPSRC, UK nder Grant No EP/J558X/ is greatly appreiated reslts in an agmented system, here traditional state feedbak tehniqes an be applied [], [3], [4] Hoever, even for linear systems, losed-loop system stability ith an integral ontrol ation is only garanteed ith sffiiently small integral gain and nder neessary and sffiient onditions on the plant [5], [6] Partilarly, an analyti allation of the maximm integral gain for garanteeing losed-loop system stability of finite-dimensional linear systems an be fond in [7] The appliation of IC as extended to non-linear systems [], [8], [9] ith loal losed-loop stability reslts Semi-global reslts ere provided in [] [] for minimm-phase systems sing otpt feedbak ontrol and high-gain observers The idea as to transform the system into the normal form [3] and apply a satrating ontroller otside a ompat set of interest These reslts ere frther extended in [4] here a robst integral ontroller as designed aording to the relative degree of the non-linear plant Reently, onditional integrators ere proposed in [5], [6], hih provide the integral ation inside a bondary layer and at as a stable system otside of it In many of these orks, some of the assmptions mentioned for the plant are diretly related to the inpt-to-state stability (ISS property [7], [8], hile in [4], the generalised small-gain theorem as sed [9] [], hih represents a fndamental tool for robst stability A different approah of IC in port-hamiltonian systems for distrbane rejetion an be also fond in reent orks [], [3], here the port-hamiltonian form is maintained and losed-loop system stability an be proven for systems ith relative degree higher than one As demonstrated in the previos orks [], [8] [4], the IC design for non-linear systems depends on the strtre of the system (relative-degree et and often reslts in a very ompliated ontrol sheme that reqires a satration nit to garantee a bonded area for the ontroller otpt The satration nit is often applied to the traditional IC leading to a simple strtre and easy implementation, bt it is diffilt to rigorosly prove the stability, sine it often leads to integrator indp and ndesired osillations Antiindp tehniqes an be sed to ope ith this problem, bt the knoledge of the plant is often reqired to garantee losed-loop system stability [4] [8] The proof of stability beomes even more diffilt if the plant is loally ISS ith the plant inpt onsidered in general as nonstrained Therefore, the existene of a generi ontroller ithot satration nits ( 5 IEEE Personal se is permitted, bt repbliation/redistribtion reqires IEEE permission See for more information

2 This artile has been aepted for pbliation in a ftre isse of this jornal, bt has not been flly edited Content may hange prior to final pbliation Citation information: DOI 9/TAC655978, IEEE that operates similar to the traditional IC independently from the non-linear plant strtre and parameters, and garantees losed-loop system stability is of signifiane In this note, a bonded integral ontroller (BIC is proposed to garantee the non-linear losed-loop stability for globally or loally inpt-to-state (pratially stable (ISpS plant systems With the only knoledge of the inpt-to-state pratial stability (ISpS property of the plant [7], [8], it is proven that the proposed BIC garantees losed-loop system stability in the sense of bondedness sing the generalised small-gain theorem [9], [] It shold be noted that the system dynamis and/or parameters an be ompletely nknon Althogh the plant inpt is onsidered nonstrained, often a given bond is introded for stability reasons, sh as for loally ISS systems Therefore, an analyti seletion of the ontroller parameters is presented to ahieve a bonded ontroller otpt ithin a given range, ths extending the stability analysis to loally ISpS plant systems Additionally, it is proven that the BIC maintains the performane of the traditional IC near the eqilibrim point nder some onditions Partilarly, if linearisation arond an eqilibrim point of an ISpS plant operating ith the traditional IC reslts in asymptoti stability, then asymptoti stability is still maintained if the IC is replaed by the BIC Moreover, the BIC garantees that the soltion ill remain bonded, ie instability is avoided, even if the eqilibrim point or the system parameters hange The bondedness of the soltion is garanteed in some systems even hen the eqilibrim point is shifted otside of the bonded range or the eqilibrim point is nstable This approah does not obsolete the IC methods proposed in the literatre; in ontrary, it an be easily ombined ith many of them to simplify the stability analysis and garantee a given bond for the ontrol otpt Note that the proposed BIC does not se a satration nit and as it is proven, it does not sffer from integrator indp problems In fat, it is shon that the integration slos don near the limits ithot reqiring any sithes or knoledge of the plant parameters Ths, the proposed BIC is expeted to solve many pratial and indstrial problems here the traditional IC is sed ithot any rigoros stability proof Sh an example is a d/d bkboost onverter system, hih is simlated to verify the BIC method ompared to the traditionally sed IC and provide the theory that is rrently missing II PROBLEM FORMULATION Many engineering systems are BIBO stable de to their inherent dissipative strtre In the ideal ase, simple openloop ontrol strategies that generate a bonded ontrol otpt an reglate the system otpt to its desired vale ithot affeting the system stability Hoever, in a real environment, there are external distrbanes and parameter variations that old onsiderably degrade the system performane As a reslt, it is essential to lose the loop by sing feedbak ontrol, as shon in Fig, in order to ahieve desired performane, eg zero steady-state error, even hen there are distrbanes, parameter variations and nertainties The problem is that a feedbak ontroller no longer garantees a bonded ontrol otpt, hih may ase instability In other ords, the stability of the system is no longer preserved Developing feedbak ontrol strategies that preserve the BIBO stability of the system is of signifiane referene + - error Figre Closing the loop Controller bonded? ontroller otpt Sensor BIBO Plant plant otpt Partilarly, hen a reglation problem is onsidered, hih is the most ommon ontrol objetive, an IC is sed to ahieve zero steady-state error Consider a general non-linear system ẋ = f (x,, ( here f : D D R n is loally Lipshitz in x and and D, D are open neighborhoods of the origin for x and, respetively For simpliity, onsider a single-inpt system in the form of ( and assme that the ontrol task is the reglation of a salar fntion g(x to zero This assmption also inldes the ommon reglation senario of a state variable x i to a desired level x ref i The traditional IC that ahieves this task is given as (t = t, ie g(x = x ref i x i k I g (x(τ dτ, ( here k I > represents the integral gain Then, the IC introdes a dynami ontroller that an be ritten as = (3 ẇ = k I g(x (4 Hoever, losed-loop system stability is not alays garanteed even if the plant ( is BIBO Note that for a non-linear system, the BIBO or inpt-to-otpt stability is garanteed if the plant is inpt-to-state stable (ISS and the otpt fntion is K bonded [7] A generi ontroller that garantees the stability of the losed-loop system ill be developed in this paper III MAIN RESULT In this setion, the main task is to design a ontroller that operates similarly to the traditional IC (3-(4 and generates a bonded otpt This ontroller is alled Bonded Integral Controller (BIC and introdes a seond ontroller state as shon belo: = (5 ( [ ] ẇ k + (q b k = max Ig(x ] ( ẇ q [ k Ig(x k q + (q b q ( ( 5 IEEE Personal se is permitted, bt repbliation/redistribtion reqires IEEE permission See for more information

3 This artile has been aepted for pbliation in a ftre isse of this jornal, bt has not been flly edited Content may hange prior to final pbliation Citation information: DOI 9/TAC655978, IEEE 3 here and q are the ontroller state variables, b is a nonnegative onstant and max, k, k q,, are positive onstants Consider, no, the plant system dynamis ẋ = f(x,, (7 here desribes the ontrol inpt and is a vetor of external nontrolled inpts After applying the BIC into the general plant, the losedloop system is desribed in Fig, hih is a omposite feedbak interonnetion form Here, it is assmed that the fntion g(x is loally Lipshitz, hih is tre in most ontrol appliations Additionally, the plant system is assmed to possess the ISpS (or ISS property hih holds for most engineering systems Then, the folloing theorem garantees the ISpS property of the losed-loop system BIC plant Figre Closed-loop system ith BIC k I g(x Theorem The feedbak interonnetion of plant system (7 ith the proposed BIC (5-(6 is ISpS ith respet to inpt, hen the plant system (7 is ISpS ith respet to both inpts and Proof: For the ontroller dynamis (6, onsider the folloing Lyapnov fntion andidate V = Taking the time derivative of V, it yields V = ẇ = ( + qẇ q + ( q b x + q (8 ( k q + k q max Its sign is related to an ellipse at the point (, b defined by C = {, q R : + ( q b } = ( The derivative of the Lyapnov fntion V is negative otside of the ellipse C and positive inside of the ellipse exept from the origin here it is zero Note that the Lyapnov fntion is defined as an ellipsoid strtre arond the origin, hile C represents a given ellipse arond (, b Defining { B = + ( q b (9 ( + δ }, here δ is an arbitrary positive onstant, from (9 it is holds that V < otside and on the bondary of B exept from the origin Consider no a losed set Ω s = {V (, q s} One an find the vale of s sh that B Ω s and the bondaries of B and Ω s interset at point (, b + ( + δ, as shon in Fig 3, ie this point shold satisfy Therefore s = (b+(+δ and S =, q R: ( (b+(+δmax + q = s ( q + (b + ( + δ = ( desribes the bondary of Ω s Hene, V < otside and on the bondary of Ω s, hih garantees that the ontroller states and q introde an ltimate bond As a reslt, it is proven that for any initial onditions ( and q (, there exists a lass KL fntion β and a ftre time instant T sh that [] q and q ( β ( q (, t, t T (3 ( max + (b + ( + δ, t T (4 Ineqality ( (4 reslts from the norm properties x p x, p and taking into aont from ( that for all t T, ie after the time instant that and q enter ellipse S, it holds tre that (b+(+δ max and q b + ( + δ hih yield that ( max+(b+(+δ q Hene, the ontrol states soltion an be ritten in the form: ( ( β q q (, t + d (5 here d = ( max+(b+(+δ is a positive onstant Sine ineqality (5 is satisfied independently from any bonded inpt g(x of the ontroller, the ontroller states an be ritten in the general ISpS form ( β ( x (, t+γ ontrol sp k I g (x (τ +d q τ t (6 ith zero gain, ie γ ontrol =, regardless of the seletion of the initial onditions, q and the parameters k, k q, max, b and Sine the losed-loop system, as shon in Fig, is given in the omposite feedbak interonnetion form, the small-gain theorem given in [9], [] an be applied Partilarly, given that the ontroller gain is zero, then the small-gain ondition is obviosly satisfied Therefore, the losed-loop system is ISpS ith respet to the external inpt vetor The speial strtre of the BIC provides the opportnity of proving the ISpS property for a ide lass of non-linear systems It is obvios that if the external inpt of the plant is zero, the losed-loop system soltion is bonded It is also orth noting that Theorem holds independently from the plant strtre, the ontroller parameters b and max, k, k q, > or the initial onditions of the BIC states ( 5 IEEE Personal se is permitted, bt repbliation/redistribtion reqires IEEE permission See for more information

4 This artile has been aepted for pbliation in a ftre isse of this jornal, bt has not been flly edited Content may hange prior to final pbliation Citation information: DOI 9/TAC655978, IEEE 4 q (,b+ε(+δ Ο B max (+δ S Ω s that the same analysis holds for any initial onditions ith q > and defined on W By onsidering the folloing transformation = max sinθ q = osθ, it yields from the BIC dynamis (8 that (3 θ = k Ig(x q max (4 Figre 3 Bondedness of states and q Hene, BIC provides a generi ontroller for non-linear ISpS systems here the plant dynamis and parameters may be nknon or hange dring the operation IV BIC WITH A GIVEN OUTPUT BOUND A Controller design Althogh the BIC otpt is proven to remain bonded, a given bond is not garanteed in general In order for the ontrol signal to remain inside a given bond [ max, max ], here max denotes the maximm absolte vale of the ontroller otpt, the BIC parameters an be seleted as b =, =, = q, k =, k q >, (7 here ( max, max is onstant Aording to this seletion, the BIC dynamis (6 beome [ ] ẇ k = I g(x q [ ] ( ẇ q k I g(x q k q + q q (8 here the initial onditions are hosen = (initial ondition of the IC, sally zero and q = No, onsidering the Lyapnov fntion andidate W = its derivative yields ( Ẇ = + q, (9 + q k q q ( hih implies that the BIC states are on the ellipse { } W =, q R : + q = ( as shon in Fig 4 This is de to the fat that the initial onditions are defined on W, here obviosly Ẇ = W (t = W ( =, t ( proving that the BIC states ill start and remain ( at all times on the ellipse W, ie the diagonal term k q + q max ill be zero This term is only sed to inrease the robstness ith respet to external distrbanes or allation errors in the dynamis of q dring a pratial implementation Note hih proves that and q ill move on the ellipse W ith anglar veloity θ (Fig 4 Therefore, it is garanteed that [ max, max ] for all t and as a reslt it extends the BIC operation to garantee stability for loally ISpS systems It shold be noted that de to the seletion of the initial onditions, the desired operation of the ontroller states on the ellipse is garanteed even if k = and is varying sh as in the present ase If it is assmed that there exists a desired eqilibrim point x = x e for the plant ith = e ( max, max, for hih g(x e =, this implies that and q an stop at the desired eqilibrim, orresponding to ( e, qe on q plane, at hih θ = k Ig(x e qe max = The onditions nder hih a possible onvergene to the desired eqilibrim exists are investigated in the next sbsetion W q qe Ο Figre 4 BIC states on q plane e max B Ahieving bondedness hile preserving the stability of the system ith the traditional IC Consider the non-linear ISpS system of the form of ( ith the proposed BIC ith the given bond (8 Sine no other external inpts are present, the losed-loop system soltion x BIC (t ill be bonded, here x BIC = [ x T q ] T is the state vetor of the losed-loop system Hoever, sine in this note the BIC is sed to perform similarly to the traditional IC for ahieving a desired reglation senario, it is important to prove that the BIC does not hange the behavior of the IC near the desired eqilibrim point Consider an ISpS plant ontrolled by the traditional IC (, (3, (4 In this ase assme that both f and g are ontinosly differentiable fntions The losed-loop system an be ritten in the form ẋ IC = f IC (x IC (5 here x IC = [ x T ] T is the state vetor Assme that x ICe = [ x T e e ] T is an eqilibrim point here g(xe = θ ɺ ( 5 IEEE Personal se is permitted, bt repbliation/redistribtion reqires IEEE permission See for more information

5 This artile has been aepted for pbliation in a ftre isse of this jornal, bt has not been flly edited Content may hange prior to final pbliation Citation information: DOI 9/TAC655978, IEEE 5 If linearisation arond the eqilibrim point reslts in a Jaobian matrix A IC = IC(x IC x IC xic =x ICe ith Reλ i < for all eigenvales of A IC, then the eqilibrim point of (5 ill be asymptotially stable Hoever, it is not garanteed that the soltion of the losed-loop system ill not esape to infinity, eg if initial onditions are defined aay from the eqilibrim point As it is shon in the seqel, the BIC maintains the asymptoti stability of the eqilibrim point and aording to the previos analysis, the proposed ontrol method additionally garantees a maximm bond for the losed-loop soltion and a given bond for the ontroller otpt, leading to a sperior performane and more rigoros theoretial analysis ompared to the traditional IC In this frameork, onsider the folloing onditions: x ICe = [ ] e e is an eqilibrim point of (5 ith e ( max, max Reλ i < for all eigenvales of A IC and for any < k I < k Imax 3 The BIC parameter satisfies max k I < < max k I k Imax k Imax (6 Then the folloing proposition an be formlated: Proposition If Conditions -3 above are satisfied, then the losed-loop system reslting from the feedbak interonnetion of the ISpS plant ( and the BIC (5, (8 has an asymptotially stable eqilibrim point [ ] e e qe ith qe = ± e Proof: Based on the analysis of the previos sbsetion, the eqilibrim point x ICe = [ ] e e of (5, here e ( max, max, ill orrespond to an eqilibrim point x BICe = [ ] e e qe of the feedbak interonnetion of the ISpS plant ( and the BIC (5, (8, here qe = ± e for hih < qe, sine it is defined on max W ith e ( max, max Aording to Condition all eigenvales of x A IC = (xe, e (xe, e g k I x (xe, e of A BIC ill have negative real parts if matrix x A BIC = (xe, e g k I x (xe, e qe max is Hritz Sine < qe, then < k I qe (xe, e k I, qe max < k I < k Imax taking into aont (6 from Condition 3 Therefore, all eigenvales of A BIC have negative real parts sine the eigenvales of A IC are loated at the left half plane for any < k I < k Imax As a reslt, the eqilibrim point of the losed-loop system ith the BIC is asymptotially stable It shold be noted that if Condition is satisfied for any k I >, then the desired eqilibrim of the losed-loop system ith the BIC is asymptotially stable for any ( max, max Frthermore, even if the ontrol otpt tries to reah the limits dring transients, ie ± max, then q and the first eqation of (8 reslts in ẇ independently from the fntion g(x This means that the integration slos don near the limits preventing an integration indp problem Opposed to the traditional anti-indp strtres, the BIC does no stop the integration bt smoothly slos it don near the limits ithot additional sithes; hene the plant inpt remains a ontinos-time signal, hih proves the losed-loop system stability Additionally, and q stay exlsively in the first qadrants in Fig 4 for initial onditions defined on the pper semi-ellipse of W, and therefore they annot move arond W, hih exldes an osillating behavior of the ontroller state dynamis arond the ellipse The losed-loop system stability in the sense of bondedness and the given bond for the ontroller otpt have been proven in this note independently from the existene of an eqilibrim point or its stability properties (stable or nstable Therefore, if the eqilibrim point hanges from a stable to an nstable mode (eg hange of gain k I or is shifted otside the bonded range, losed-loop stability in the sense of bondedness is still maintained, opposed to the traditional IC or the IC ith a satration nit have negative real parts for any < k I < k Imax In the same frameork, linearisation arond x BICe = [ x T e e qe ] T for the losed-loop system ith the BIC reslts in the Jaobian x A BIC = g k I x (xe, e (xe, e g k I x (xe, e qe max e qe (xe, e n kqeqe k q qe, here qe sine e ( max, max and e and qe are defined on W Sine k q qe <, then all eigenvales V PRACTICAL EXAMPLE In order to verify the proposed BIC in omparison to the traditional IC, the d/d bk-boost onverter, shon in Fig 5, is simlated This poer onverter system is idely sed in poer appliations (photovoltai, energy storage systems, et sine it an reglate the d otpt voltage to a higher or loer level than the d inpt voltage by sitably ontrolling the sithing element of the devie Using average analysis [9], it has been proven that the ontinos-time non-linear dynamis of the onverter are ( 5 IEEE Personal se is permitted, bt repbliation/redistribtion reqires IEEE permission See for more information

6 This artile has been aepted for pbliation in a ftre isse of this jornal, bt has not been flly edited Content may hange prior to final pbliation Citation information: DOI 9/TAC655978, IEEE 6 E + - = = Figre 5 The d/d bk-boost onverter given as L C v R i L di = ( v + E (7 dt C dv = ( i v dt R, (8 here L and C are the onverter indtane and apaitane, respetively, R is the load resistor and E is the d inpt voltage The system states are the indtor rrent i and the apaitor voltage v, hile the ontrol inpt is the dty-ratio, hih is a ontinos-time signal in the range [, ] It shold be noted that the system states are bonded for any [, γ], here < γ, hile the pper limit of the inpt = leads the indtor rrent to instability The main task is to reglate the otpt voltage v to a given d referene vale v ref Althogh several ontrol shemes have been developed in the literatre, sh as traditional or asaded PI ontrollers [3], passivity-based ontrollers [9], et, in the indstry, traditional or asaded PI ontrollers are ommonly sed de to their simple strtre and implementation This is also de to the fat that the system dynamis an hange (eg if a ompliated load is added in the otpt and the system parameters an be nknon or hange dring the operation Even thogh, in these ases, stability may not be garanteed, traditional ontrollers are still sed for simpliity and are sally tned in an empirial manner In this example, a traditional voltage IC ith g(x = v ref v is investigated and ompared ith the BIC ith a given bond The system parameters are L = mh, C = 3µF, R = 5Ω and E = 5V Initially the referene otpt voltage is set to v ref = 3V For stability reasons, in pratie, it is often reqired the dty-ratio to be limited belo, sally 8 (ie, γ = to avoid a high indtor rrent Sine the BIC maintains the ontroller otpt in the range [ max, max ], one an set = and max = 8 In this ay the reqired range [, 8] for the ontrol otpt an be ahieved ith the BIC If the losed-loop system ith the orresponding IC is linearised arond the desired eqilibrim point, it an be obtained (eg sing root los that the eqilibrim is asymptotially stable for all < k I < k Imax, here k Imax Ths, the integral gain an be hosen k I = for both the IC and the BIC, here additionally to different hoies of are tested = 65 8 and = 5 7 that satisfy (6 Note that if the system parameters are nknon in pratie, k I and k Imax are sally hosen based on experiene and observation The onverter is simlated ith the traditional IC and the IC ith a satration nit in the otpt at [, 8], and is ompared - + to the BIC ith to different vales of Starting ith zero initial onditions for the system states and the ontrol otpt, the otpt voltage referene is set to v ref = 3V at t = 5s At time instant t = s, v ref sddenly inreases to 7V and drops bak to 3V at t = s Finally, at t = 3s, v ref is set to 5V The time response of the system is shon in Fig 6 Initially, both the IC ith and ithot the satration nit and the BIC reglate the otpt voltage at the desired level Hoever, hen v ref is set to 7V, the traditional IC leads the indtor rrent to instability The dty-ratio of the IC ith the satration nit satrates at the pper limit 8, hile the BIC ith either seletion of smoothly onverges to the pper limit In this ase, the desired eqilibrim is shifted otside the bonded range and the IC ith the satration sffers from integrator indp, opposed to the BIC hih atomatially slos don the integration This is observed hen v ref hanges bak to 3V and the IC ith satration reslts in a larger transient Finally, hen v ref is set to 5V, the BIC onverges to the desired eqilibrim hile the IC ith satration sffers again from integrator indp and reslts in an osillatory response Note that the different hoie of in the BIC design ill reslt into slightly different transient response, sine this parameter affets the anglar veloity (4 of the BIC states on the desired ellipse W The operation on the ellipse is illstrated in Fig 7, here it is lear that the ontroller states remain on the pper semi-ellipse of W as reqired It shold be nderlined that if the system parameters are ompletely nknon or hange dring the system operation, neither the IC or the BIC an garantee asymptoti stability of the desired eqilibrim Hoever, the BIC an still garantee an ltimate bond for the losed-loop system, a given bond for the ontrol otpt and the fat that it ill not sffer from integrator indp isses This is the main reslt of the rrent note hih offers a replaement of the traditional IC ith the BIC and an be applied in many engineering systems here the IC is sed ithot a rigoros proof of stability VI CONCLUSIONS In this note, a bonded integral ontrol (BIC as proposed to garantee losed-loop stability for a ide lass of openloop stable non-linear systems Bondedness of the ontroller otpt signal has been ahieved sing the generalised smallgain theorem independently from the plant otpt and ithot external satration nits or sithes, ths solving the losedloop stability problem of many engineering systems ithot reqiring knoledge of the plant strtre or parameters By sitably hoosing the BIC parameters, a given bond for the ontroller otpt an be obtained to garantee stability of loally ISpS plants Therefore, for systems operating ith the traditional IC, the same reglation senario an be ahieved by replaing the IC ith the BIC and reslt in a garanteed bonded response The bondedness of the losed-loop system soltion ith the BIC is maintained even hen the eqilibrim point hanges or beomes nstable Simlation reslts of a d/d bk-boost onverter system sitably verified the proposed BIC in omparison to the traditional IC ( 5 IEEE Personal se is permitted, bt repbliation/redistribtion reqires IEEE permission See for more information

7 This artile has been aepted for pbliation in a ftre isse of this jornal, bt has not been flly edited Content may hange prior to final pbliation Citation information: DOI 9/TAC655978, IEEE 7 IC ith and ithot satration v/v i/a IC IC+sat IC IC v/v (a otpt voltage v IC+sat 3 4 i/a BIC BIC ( =8 BIC ( = (b indtor rrent i IC+sat 5 BIC ( =8 BIC ( = ( dty-ratio BIC ( =8 BIC ( =7 3 4 Figre 6 Simlation reslts of the bk-boost onverter ith the IC and the BIC q 5 5 W 5 5 Figre 7 q plane for the BIC REFERENCES [] H K Khalil, Nonlinear Systems Prentie Hall, [] Q-C Zhong and T Hornik, Control of Poer Inverters in Reneable Energy and Smart Grid Integration Wiley-IEEE Press, 3 [3] M Corless and G Leitmann, Continos state feedbak garanteeing niform ltimate bondedness for nertain dynami systems, IEEE Trans Atom Control, vol 6, no 5, pp 39 44, 98 [4] J Tsinias, Sffiient Lyapnov-like onditions for stabilization, Mathematis of ontrol, Signals and Systems, vol, no 4, pp , 989 [5] M Morari, Robst stability of systems ith integral ontrol, IEEE Trans Atom Control, vol 3, no 6, pp , 985 [6] T Fliegner, H Logemann, and E P Ryan, Lo-gain integral ontrol of ontinos-time linear systems sbjet to inpt and otpt nonlinearities, Atomatia, vol 39, no 3, pp , 3 [7] D Mstafa, Ho mh integral ation an a ontrol system tolerate? Linear algebra and its appliations, vol 5, pp , 994 [8] A Isidori and C I Byrnes, Otpt reglation of nonlinear systems, IEEE Trans Atom Control, vol 35, no, pp 3 4, 99 [9] J Hang and W J Rgh, On a nonlinear mltivariable servomehanism problem, Atomatia, vol 6, no 6, pp , 99 [] N A Mahmod and H K Khalil, Asymptoti reglation of minimm phase nonlinear systems sing otpt feedbak, IEEE Trans Atom Control, vol 4, no, pp 4 4, 996 [] A Isidori, A remark on the problem of semiglobal nonlinear otpt reglation, IEEE Trans Atom Control, vol 4, no, pp , 997 [] H K Khalil, Universal integral ontrollers for minimm-phase nonlinear systems, IEEE Trans Atom Control, vol 45, no 3, pp , [3] C I Byrnes and A Isidori, Asymptoti stabilization of minimm phase nonlinear systems, IEEE Trans Atom Control, vol 36, no, pp 37, 99 [4] Z-P Jiang and I Mareels, Robst nonlinear integral ontrol, IEEE Trans Atom Control, vol 46, no 8, pp , [5] A Singh and H K Khalil, State feedbak reglation of nonlinear systems sing onditional integrators, in 43rd IEEE Conferene on Deision and Control (CDC, vol 5 IEEE, 4, pp [6] R Li and H K Khalil, Conditional integrator for non-minimm phase nonlinear systems, in 5st IEEE Conferene on Deision and Control (CDC,, pp [7] E D Sontag, Smooth stabilization implies oprime fatorization, IEEE Trans Atom Control, vol 34, no 4, pp , 989 [8] E D Sontag and Y Wang, Ne haraterizations of inpt-to-state stability, IEEE Trans Atom Control, vol 4, no 9, pp 83 94, 996 [9] Z-P Jiang, A R Teel, and L Praly, Small-gain theorem for ISS systems and appliations, Mathematis of Control, Signals and Systems, vol 7, no, pp 95, 994 [] Z-P Jiang and I M Y Mareels, A small-gain ontrol method for nonlinear asaded systems ith dynami nertainties, IEEE Trans Atom Control, vol 4, no 3, pp 9 38, 997 [] A R Teel, A nonlinear small gain theorem for the analysis of ontrol systems ith satration, IEEE Trans Atom Control, vol 4, no 9, pp 56 7, 996 [] A Donaire and S Jno, On the addition of integral ation to portontrolled Hamiltonian systems, Atomatia, vol 45, no 8, pp 9 96, 9 [3] R Ortega and J G Romero, Robst integral ontrol of port- Hamiltonian systems: The ase of non-passive otpts ith nmathed distrbanes, Systems & Control Letters, vol 6, no, pp 7, [4] L Zaarian and A R Teel, Nonlinear shedled anti-indp design for linear systems, IEEE Trans Atom Control, vol 49, no, pp 55 6, 4 [5] S Galeani, S Onori, and L Zaarian, Nonlinear shedled ontrol for linear systems sbjet to satration ith appliation to anti-indp ontrol, in 46th IEEE Conferene on Deision and Control, 7, pp [6] Y Peng, D Vrani, and R Hans, Anti-indp, bmpless, and onditioned transfer tehniqes for PID ontrollers, IEEE Control Syst Mag, vol 6, no 4, pp 48 57, 996 [7] C Bohn and D P Atherton, An analysis pakage omparing PID antiindp strategies, IEEE Control Syst Mag, vol 5, no, pp 34 4, 995 [8] S Tarborieh and M Trner, Anti-indp design: an overvie of some reent advanes and open problems, IET Control Theory & Appliations, vol 3, no, pp 9, 9 [9] R Ortega, A Loria, P J Niklasson, and H Sira-Ramirez, Passivitybased Control of Eler-Lagrange Systems, Mehanial, Eletrial and Eletromehanial Appliations Springer-Verlag Great Britain, 998 [3] S Verma, S K Singh, and A G Rao, Overvie of ontrol Tehniqes for DC-DC onverters, Res J Engineering Si, vol, no 8, pp 8, Agst ( 5 IEEE Personal se is permitted, bt repbliation/redistribtion reqires IEEE permission See for more information