Finite Horizon Control Design for Optimal Model Discrimination

Transcription

1 Finie Horizon Conrol Design for Opimal Model Discriminaion Lars Blackmore and Brian Williams Absrac In many faul deecion and sysem idenificaion problems, i is essenial o be able o discriminae beween a number of compeing models of a sysem based on observed sysem oupus. For example, in a faul deecion scenario we may wish o deermine wheher a sysem is bes modeled by a known nominal model, or a known failure model. The probabiliy of deecing he rue sysem model can be enhanced by design of he conrol inpus applied o he sysem. In his paper we presen a mehod by which a finie sequence of conrol inpus is designed auomaically in order o minimize an upper bound on he probabiliy of model selecion error beween any wo linear, discree-ime sysems. We are able o solve his problem efficienly by showing ha i is an insance of a Quadraic Program. In addiion, linear equaliy and inequaliy consrains can be applied o he conrol inpus and expeced sysem sae. These consrains can be used o ensure ha a cerain ask is fulfilled, make sure he sysem says wihin a valid linearizaion region, or o guaranee safe operaion. Experimenal resuls for he case of an aircraf acuaor failure scenario show ha he mehod significanly reduces he upper bound on he probabiliy of model selecion error when compared o a manually generaed sequence and a fuel-opimal sequence. I. INTRODUCTION In muliple-model (MM) faul deecion i is necessary o selec he mos likely model from a finie se, given observaions [1[2. For example Hanlon e al. invesigaed he deecion of an aircraf fligh conrol acuaor failure [3. In his case deecing a faul becomes a problem of deciding wheher he dynamic model describing he nominal behavior of he aircraf, or he dynamic model describing he acuaor failure is mos likely. Previous auhors have developed mehods for deciding beween models given a se of observaions [3[4. For his paper we assume a Bayesian decision rule, which selecs he mos likely model given he observaions and a prior disribuion over he models. The abiliy of a deecion mehod o discriminae beween differen compeing dynamic sysem models is highly dependen on he conrol inpus applied o he sysem. For example, in he conrol acuaor failure case, if no conrol inpus are applied o he acuaor, he responses of boh he fauly and he nominal sysem model will be idenical. A new mehod for designing sysem inpus ha discriminae beween he possible models in an opimal sense is presened in his paper. Esposio e al. creaed a persisen exciaion soluion o he problem of model discriminaion for wo linear filers [5. This work is suppored by NASA Award NNA4CK91A Lars Blackmore is a PhD suden, Massachuses Insiue of Technology, Cambridge, MA 2139 larsb@mi.edu Brian Williams is an associae professor, Massachuses Insiue of Technology, Cambridge, MA 2139 williams@mi.edu However, as Prasanh e al. [6 noed for he case of model parameer esimaion, a persisen exciaion approach [7[8 may no be feasible for a physical sysem such as a saellie, because conrol inpus ypically need o be designed subjec o sae and conrol consrains. They noed ha a finie horizon opimizaion approach is more appropriae, and pose he problem as a Model Predicive Conrol (MPC) problem. We exend he work of [5 and [6 by describing a mehod by which finie, opimized sequences of conrol inpus can be designed, subjec o conrol and sae consrains, in order o discriminae beween wo discree-ime dynamic linear models. While our mehod is limied o he case of wo models, discriminaion beween wo models is useful for binary hypoheses such as wheher he sysem is in eiher a nominal or a failure mode, and is also a sep owards he design of conrol inpus for discriminaion beween more han wo models. Prior work in he field of experimen design [9[1 has suggesed a number of differen crieria for he design of experimens for model discriminaion. In his paper, consisen wih a Bayesian approach o model selecion, he aim is o minimize he probabiliy of model selecion error by he Bayes-opimal decision rule, known as he Bayes Risk [11. Assuming a -1 loss funcion for he model selecion ask, he decision-heoreic opimal design is he one ha minimizes he probabiliy of model selecion error [12. We use an upper bound on he probabiliy of model selecion error, he Baacharyya bound, o creae a racable opimizaion crierion for model discriminaion. Then we pose he problem of designing a finie sequence of conrol inpus o minimize his bound, subjec o consrains, as a finie horizon rajecory design problem. Lasly, we show ha in he case of linear consrains his is an example of a concave Quadraic Program. Prior work in he field of opimizaion allows concave Quadraic Programs o be solved efficienly [13[14[15[16. The resul of his work is a new algorihm ha generaes a finie sequence of conrol inpus ha minimize an upper bound on he probabiliy of model selecion error. These sequences are designed subjec o sae and conrol consrains. The algorihm can be used o ensure ha a given ask, defined in erms of consrains on he expeced sae, is fulfilled while opimally deecing failures. We presen resuls for he example of deermining wheher an aircraf s elevaor acuaor has failed. Compared o a ypical sequence designed by a human, and a sequence opimized o minimize fuel consumpion, he mehod dramaically reduces he upper bound on he probabiliy of model selecion error.

2 A. Aircraf Faul Deecion Scenario In his paper, he deecion of elevaor acuaor failure for an aircraf is used as a moivaing example. The discree ime approximaion o he longiudinal dynamics of he aircraf, linearized abou he rim sae, is shown in Fig. 1. In he model selecion ask, we mus deermine which y H ) H ) Bayes Risk Bayes-opimal classificaion hreshold y H1) H1) Vx Vy x = θ V y θ x + 1 y + 1 = Ax + Bu + w = Cx + 1 V x + Du + v Fig. 1. Discree-ime aircraf model linearized abou he rim sae. Here x is he sae of he sysem a ime sep while y is he observed oupu of he sysem a ime, which is aken o be he pich rae θ. The inpu is denoed u, and is aken o be he elevaor angle. The erms w and v are he process noise and observaion noise. The noise a any ime sep is assumed o be independen of he noise a any oher ime sep, and w and v are independen of each oher wih normal disribuions N (, Q) and N (, R) respecively. The iniial sae of he sysem is also assumed o be normally disribued. sysem described by he sysem marices {A, B, C, D} bes models he daa. Here he siuaion under consideraion has wo candidae models. Under Hypohesis, he sysem is described by {A,B,C,D } while under Hypohesis 1, he sysem is described by {A 1,B 1,C 1,D 1 }. For he aircraf in Fig. 1, we may wish o deermine wheher he elevaor acuaor is fauly. In he fauly case, he B marix is zero, indicaing ha he inpu (commanded elevaor angle) has no effec on he sysem. Inuiively, one migh carry ou an experimen where, a he rim sae, a large elevaor angle would be commanded. In he case of a working acuaor he effec on he sysem would be significan, while in he case of a fauly acuaor he commanded inpu would have no effec. The resuling observaions would herefore reveal which of he wo hypoheses is correc. This paper presens an online, opimized echnique for designing experimens of his ype. II. HYPOTHESIS SELECTION AND BAYES RISK Here we assume models are seleced by Bayesian hypohesis selecion. Resricing our aenion o selecion beween wo models, Bayesian hypohesis selecion can be expressed as follows: Selec H if H y, u) > H 1 y, u), else selec H 1. Using Bayes rule, his selecion is given by: Selec H if y H, u)h ) > y H 1, u)h 1 ), else selec H 1. The erms H ) and H 1 ) correspond o prior probabiliies of he wo hypoheses. These can be calculaed in a number of differen ways: here may be explici knowledge abou how a priori likely he differen hypoheses are, or he prior can represen he belief sae creaed by an esimaor. R R 1 Fig. 2. Selecion beween wo models given an observaion y and a prior. In general Bayesian selecion beween wo hypoheses yields a hreshold ha splis he possible observaions ino wo ses. If he observaion y falls ino se R hen he classifier selecs H, whereas if he observaion falls ino se R 1 hen he classifier selecs H 1. Even wih Bayes opimal selecion here is a finie probabiliy of error given by he Bayes Risk, denoed by he shaded region. The Bayesian selecion rule minimizes he likelihood of selecing an incorrec hypohesis given he available informaion. As shown in Fig. 2, he Bayesian opimal classifier has a finie probabiliy of selecing he incorrec hypohesis, known as he Bayes Risk. The Bayes risk is given by: P (error) = P (y R 1, H u) + P (y R, H 1 u) = P (y R 1 H, u)p (H ) + P (y R H 1, u)p (H 1 ) = y H, u)p (H )dy + y H 1, u)p (H 1 )dy (1) R 1 R Since he Bayes Risk is he probabiliy of error when using he opimal classifier, we would like o opimize our conrol inpus o he sysem o minimize his measure. III. THE ALGORITHM A. Minimizing he Bayes Risk The idea behind he design of conrol inpus for model selecion is ha while he probabiliy of error canno be reduced beyond he Bayes Risk by selecion of he classificaion hreshold, he Bayes Risk iself is affeced by he conrol inpus o he sysem. Hence by selecing he inpus o he sysem, we can reduce he Bayes Risk and herefore significanly reduce he probabiliy of error of he Bayesian classifier. The effec of inpu choice on he Bayes Risk is illusraed in Fig. 3. B. Inpu Design as Trajecory Opimizaion A key observaion is ha he design of conrol inpus o minimize he Bayes risk is in fac a problem of opimal rajecory design for dynamic sysems. In ypical opimal rajecory design problems, an opimized sequence of conrol inpus is designed so ha a sysem passes hrough a sequence of prediced saes ha minimize some cos funcion (for example he ime aken o reach a goal sae). In he model discriminaion problem, we would like o design an opimal sequence of conrol inpus so ha he sysem passes hrough y

3 y H, u) H ) Significan Bayes Risk where: y H1, u) H1) k = 1 4 (µ 1 µ ) T [Σ + Σ 1 1 (µ 1 µ ) + 1 Σ +Σ 1 2 ln 2 Σ Σ 1. (3) y H, u) H ) choice of conrol inpus Bayes Risk y H1, u) H1) y Since he logarihm is a monoonically increasing funcion, he value of x ha opimizes f(x) is also he value ha opimizes ln[f(x). We herefore ake he logarihm of he Baacharyya bound for Gaussian disribuions o yield he following cos funcion: J = 1 2 ln[p (H )P (H 1 ) 1 Σ +Σ 1 2 ln 2 Σ Σ 1 y 1 4 (µ 1 µ ) T [Σ + Σ 1 1 (µ 1 µ ) (4) Fig. 3. Graph showing y H, u) and y H 1, u) for wo differen choices of u. In he upper figure, he prediced disribuions overlap significanly, leading o a large Bayes risk. In he lower figure, a differen selecion of u has separaed he wo disribuions, meaning ha when he observaion y is made, he correc hypohesis can be seleced wih high confidence. The Bayes risk is very low, meaning ha he probabiliy of error is very low. a rajecory of sae disribuions ha minimize he probabiliy of model selecion error (Bayes Risk). Opimal rajecory design for linear, discree-ime sysems can be posed as a quadraic program in he case of a cos funcion ha is quadraic in he decision variables. A quadraic program involves he minimizaion of a quadraic funcion subjec o linear consrains. For he rajecory design problem, he linear naure of he sae updae equaion shown in Fig. 1 means ha linear sae consrains correspond o linear consrains on he decision variables. Efficien algorihms exis for he soluion of quadraic programs. The main conribuion of his projec is o show ha he model discriminaion problem can be posed as an opimal rajecory design problem and solved using a quadraic program. This is described in he following secions. 1) Cos Funcion for Opimal Inpu Design: In he model discriminaion problem, he objecive is o minimize he probabiliy of error of Bayesian model selecion, known as he Bayes Risk. However he Bayes Risk given in (1) is no in closed form and can only be calculaed using numerical inegraion. I is, however, possible o bound he Bayes Risk in closed form, and doing so allows he use of Quadraic Programming which is reliable and efficien. The Baacharyya Bound [11 provides an upper bound on he Bayes Risk and is given by he inegral: P (error) P (H )P (H 1 ) y H )y H 1 )dy (2) If he disribuions are Gaussian such ha y H ) has mean µ and variance Σ, and y H 1 ) has mean µ 1 and variance Σ 1, he above value can be calculaed wihou he need for inegraion: P (error) P (H )P (H 1 ) exp{ k}, Minimizing his cos funcion will herefore minimize an upper bound on he probabiliy of error when using Bayesian selecion o decide beween wo models based on a vecor of observaions y. 2) Error Probabiliy for a Finie Horizon: Raher han selecing inpus a one momen in ime, we would like o plan a finie sequence of inpus in order o minimize he upper bound on he probabiliy of error. This form of planning is known as finie horizon planning. In his case, if he horizon is of lengh k, we are concerned wih a sequence of observaions y +1...y +k and a sequence of inpus u...u +k 1. The derivaion in III-B.1 exends readily o a finie horizon. Defining: y = [ y T +1 y T y T +k u = [ u T u T u T +k 1 T T (5) The new objecive is o minimize an upper bound on he probabiliy of error of a Bayes opimal classifier ha makes is decision based on all of he observaions y +1...y +k, by designing he sequence of conrol inpus u...u +k 1. Due o uncerainy in he iniial sae and noise, he fuure observaions y +1...y +k are random variables, which we denoe Y +1...Y +k. Under he assumpions in Secion I-A, Y +i is normally disribued given a sequence of inpus u and given a model (H or H 1 ). We now define µ +i,h and Σ +i,h for ime seps i = 1,..., k and hypoheses h =, 1 such ha: p Y+i (y H, u) = N (µ +i,, Σ +i, ) p Y+i (y H 1, u) = N (µ +i,1, Σ +i,1 ) (6) Then he vecor of all observaions Y = [Y T +1...Y T +k T is a vecor of normally disribued random variables given a sequence of inpus and a hypohesis. We define µ h and Σ h for h =, 1 o be he mean and covariance of he vecor of all observaions such ha: p Y (y H, u) = N (µ, Σ ) p Y (y H 1, u) = N (µ 1, Σ 1 ) (7) From he above definiions he disribuion of Y h is given by: µ h = [ µ T +1,h...µ T +k,h T (8)

4 [ ([Y )( ) [Σ h i,j = E i [µ h i [Y j [µ h Hh j Here [ i denoes he value a index i ino he vecor, and similarly [ i,j denoes he value a index (i, j) ino he marix. Having defined µ, µ 1, Σ and Σ 1, he Baacharyya bound given in (3) provides an upper bound for he probabiliy of error when using he enire sequence of observaions from ime + 1 o ime + k. Hence he cos funcion in (4) applies o he finie horizon formulaion given in his secion. The problem of designing a sequence of inpus o minimize his cos funcion is addressed in he following secions. 3) Predicing he Disribuion of Fuure Observaions: Given a cerain hypohesis, he sysem equaions shown in Fig. 1 are fully known. Hence he disribuions y H, u) and y H 1, u) can be calculaed. This secion gives he resul for y H, u); an idenical mehod applies for y H 1, u). For some sequence of inpus u...u +k 1 he sysem will pass hrough a sequence of saes x...x +k. In order o minimize (2), we mus deermine how he disribuion of x...x +k, and hence y H, u), depends on his sequence of inpus. Given a sequence of inpus, he iniial sae x and a sysem model, he sysem equaions can be applied recursively o derive he following equaion relaing he observaion a ime sep + i o he iniial sae, he inpus, and he noise: i 1 y +i, = C A i l 1 (B u +l + w +l ) l= (9) + C A i x + D u +i 1 + v +i 1 (1) Given a disribuion for he iniial sae of he sysem N (ˆx, P ), he mean µ and covariance Σ as defined in (8) and (9) can be calculaed. The resuls are given here for a sysem where y R n. Define: i = n(p 1) + q, (11) where: 1 q n 1 p k p, q Z. (12) Then following from (5) and (1): [µ i = [µ +p, q [ p 1 = C A p ˆx + C A p l 1 B u +l + D u +p 1 l= q (13) Defining j = n(r 1) + s in he same manner as (11), he expression for he covariance is: [Σ i,j = [ R (p, r) + C A p P A T r CT q,s [ m 1 + C A (p l 1) QA T (r l 1) C T (14) l= q,s where m = min{i, j} and: R (p, r) = { R p = r p r. (15) Under he assumpions menioned in Secion I-A, hese equaions give an expression for he belief sae y H, u) in erms of a sufficien saisic for Y, namely he mean and variance of he disribuion. These equaions have wo imporan properies: 1) The equaion for he mean of he prediced disribuion of Y is linear in he conrol inpus u. 2) The covariance of he prediced disribuion of Y is no a funcion of he conrol inpus u. These wo properies are imporan because hey mean ha he cos funcion is a quadraic funcion of he conrol inpus as will be shown in Secion III-C. C. Cos as a Quadraic Funcion of Conrol Variables Examining (4) i can be seen ha he only erm ha depends on he inpus u is he erm involving µ 1 and µ, since neiher he prior nor he covariance depend on he inpu. The oher erms need herefore no be considered when opimizing J wih respec o he inpus. The erm o be minimized is shown here: J = (µ 1 µ ) T [Σ + Σ 1 1 (µ 1 µ ) (16) Now ha we have explici expressions for Σ and Σ 1 from (14) ha do no depend on he conrol inpus u, and since he means µ 1 and µ are linear in he conrol inpus, his equaion can be wrien in erms of a quadraic wih regard o he inpu variables: J = u T Hu + f T u + g (17) The quaniies H, f T and g can be calculaed explicily. The Baacharyya bound on he Bayes Risk herefore yields a cos funcion ha is quadraic in he conrol inpus. Since he covariance marices Σ and Σ 1 are posiive definie, he cos funcion given by (16) is a concave funcion of (µ 1 µ ). From (13), boh µ 1 and µ are linear funcions of u, and hence he cos in (17) is a concave funcion of he conrol inpus u. This concaviy makes he cos funcion paricularly racable for opimizaion and guaranees ha a global opimum can be found in bounded ime [13. Alhough he wo erms in (4) ha do no involve µ and µ 1 can be negleced when opimizing (4), hese erms do affec he value of he opimum ha opimizaion can achieve. A rivial example is he case where he prior for one of he hypoheses is zero. In his case he upper bound on he Bayes Risk is zero regardless of he observaions. D. Linear Consrains As noed by Prasanh e al., a powerful aspec of he finie horizon opimizaion formulaion is ha boh equaliy and inequaliy consrains can be placed on he rajecory design [6. In he model discriminaion problem here are a number of consrains on he design ha can be expressed as linear

5 funcions of he conrol inpus; for example, consraining he expeced sae of he sysem a any poin in he horizon for a paricular hypohesis. This follows from he following equaion (shown for Hypohesis ): i 1 x +i, = A i l 1 (B u +l + w +l ) + A i x l= i 1 E[x +i H = A i l 1 (B u +l ) + A i ˆx (18) l= Consrains on he expeced mean of he sysem sae can be used o: 1) Ensure ha a cerain ask, defined in erms of he expeced sysem sae, is fulfilled 2) Ensure ha he mean of sysem says wihin a safe operaing region or wihin a valid linearizaion region 3) Ensure ha he sysem ends he experimen in he same region as i sared In addiion linear consrains can be placed on he conrol inpus direcly, for example u min < u +i < u max modeling acuaor limis, or hose of he ype i u +i < fuel ha limi oal conrol effor over he horizon. Such consrains allow he user o rade off he cos of he conrol effor agains he corresponding reducion in he probabiliy of model selecion error. E. Summary We have shown ha he problem of designing a sequence of conrol inpus in order o minimize an upper bound on he probabiliy of error of Bayesian model selecion can be posed as a finie-horizon rajecory design problem, and have shown ha his problem has a cos funcion ha is quadraic in he decision variables. In addiion, we are able o place a number of linear consrains on he conrol variables in order o model conrol or sae consrains. Hence he acive model discriminaion problem can be posed as a Quadraic Program and can be solved efficienly using exising mehods. IV. SIMULATION This secion describes resuls from a number of rajecory design asks for he aircraf elevaor failure scenario. In his scenario he abiliy o deec a faul is criical, and depends heavily on he conrol inpus issued. In addiion, acuaor sauraion, linearizaion abou a rim sae and safey consideraions mean ha he abiliy o consrain he designed inpus and expeced sae is criical. Alhough he designed rajecories would ideally be compared in erms of he rue Bayes Risk, he cos of compuing his value hrough numerical inegraion means ha here he designed rajecories are compared in erms of he upper bound on he probabiliy of model selecion error, he Baacharyya bound. A. Consrained Inpus The algorihm was used o design a sequence of conrol inpus in order o choose beween wo models for he aircraf shown in Fig. 1. According o Hypohesis he aircraf has a working elevaor acuaor, and according o Hypohesis 1 his acuaor is broken. The inpus in he laer case have no effec on he sysem, as described in secion I-A. The maximum allowable elevaor angle is ±.25rad and he horizon lengh is 4 ime seps, or 2 seconds. Fig. 4 shows he sequence of inpus designed by he algorihm along wih he rajecories of he expeced observaions condiioned on Hypohesis and Hypohesis 1. The Baacharyya bound for he generaed sequence is.29, meaning ha he probabiliy of model selecion error is a mos.29%. B. Consrained Sae and Inpus Alhough he conrol sequence for his scenario produces a low bound on he Bayes Risk, he resuling moion of he aircraf closely resembles an unsable oscillaion, wih seadily increasing magniude. This is readily addressed by placing consrains on he expeced mean of he sysem sae x. Fig. 5 shows resuls for an idenical scenario, excep wih he addiional consrains ha a he end of he horizon, E[ θ.25rad/s and E[θ.25rad for boh H and H 1. This ime he generaed conrol sequence induces a rajecory for he mean of he sysem sae ha ends up wihin hese bounds a he end of he horizon; and wih a Baacharyya bound ha is only slighly greaer han for he unconsrained case a a value of.31. C. Manually Generaed Idenificaion Sequence When performing sysem idenificaion for he longiudinal dynamics of an aircraf, a pilo ofen issues elevaor inpus ha form a pulse or double paern [17. Fig. 6 shows he expeced observaions for such a sequence wih he same acuaor limis as for he opimized sequences. The Baacharyya bound for his sequence is.73, more han weny imes he bound for he opimized sequence in Fig. 5. Noe ha he human-generaed double sequence is similar o he firs porion of he sequence opimized for model discriminaion. However he opimized sequence in Fig. 5 is, by comparison, able o reduce he bound on he probabiliy of error dramaically while guaraneeing ha he final sae of he sysem is bounded. D. Model Discriminaion during Aliude Change Maneuver The abiliy o consrain he sae of he sysem means ha his mehod can be used o opimize model discriminaion during a specified maneuver. Fig. 7 shows resuls for a maneuver where he aircraf performs a change in aliude. The wo plos compare he maneuver designed o minimize he probabiliy of model selecion error o ha designed o minimize fuel consumpion. A fuel-opimal design is ypical for finie horizon pah planning wih unmanned air vehicles. The Baacharyya bound for he fuel-opimal case is.13 whereas he bound for he model discriminaion opimized case is.53. This example demonsraes he significan improvemen in faul deecion ha can be achieved by using conrol inpus designed specifically for model discriminaion, raher han hose opimized wih regard o some oher parameer and employing only passive model selecion.

6 1.5 1 E[y H E[y H 1 Elevaor Angle.3.2 Observed Pich Rae (rad/s) Commanded Elevaor Angle (rad) Time(s) Fig. 4. Opimized conrol inpus and prediced observaion means for aircraf wih u max =.25rad. The algorihm has arrived a a soluion ha gives a sequence of inpus a he furhes exremes of he allowable range, alernaing beween periods of u =.25rad and periods of u =.25rad. Noice also ha he period of his sequence is close o ha of he shor period oscillaion mode of he aircraf seen in he unforced sequence µ 1. Hence he conrol sequence effecively drives he aircraf as far as possible over he ime horizon in order o reduce he ambiguiy beween he wo hypoheses. Observed Pich Rae (rad/s) Expeced Pich Angle (rad) E[y H E[y H 1 Elevaor Angle E[pich angle H Time(s) E[pich rae H E[pich angle H E[pich rae H Time(s) Fig. 5. Opimized conrol inpus, expeced observaion, pich angle and pich rae for aircraf wih expeced final sae consrained by E[ θ.25rad/s and E[θ.25rad for boh H and H Commanded Elevaor Angle (rad) Expeced Pich Rae (rad/s)

7 Observed Pich Rae (rad/s) E[y H E[y H 1 Elevaor Angle Commanded Elevaor Angle (rad) Time(s) Fig. 6. Conrol inpu, expeced observaion, pich angle and pich rae for manually generaed double sequence..4 Discriminaion Opimal Fuel Opimal Commanded Elevaor Angle (rad) Aliude(m) Discriminaion Opimal(working acuaor) Fuel Opimal(working acuaor) Failed Acuaor(boh cases) Time(s) Fig. 7. Comparison of aliude increase maneuvers opimized for fuel consumpion and model discriminaion. Condiioned on a working acuaor, he end sae of he aircraf is consrained o have aliude=15m and zero verical velociy. No consrains are placed on he sae given a broken acuaor.

8 V. DISCUSSION In his secion, some properies of he new mehod are discussed. Firsly, he new mehod has an addiional capabiliy ha was no demonsraed in his paper. The expeced sysem sae can be consrained condiioned on eiher model being he rue one. This means ha, for example, conrol inpus can be consrained so as o guaranee safe operaion or ask compleion under eiher hypohesis. Noe ha he abiliy o apply hese consrains does no guaranee ha a feasible soluion exiss. In addiion he mehod can be applied o a slighly differen problem formulaion han ha described in he previous secions. In his alernaive formulaion, we would like o consrain he probabiliy of model selecion error o be below an accepable hreshold while minimizing some oher cos, such as fuel consumpion. In his formulaion he consrain on he probabiliy of error gives rise o quadraic consrains, and he new problem can be solved using Quadraically Consrained Quadraic Programming. This formulaion is, however, less racable han he simpler Quadraic Programming formulaion. The algorihm presened in his paper has hree main limiaions. Firsly, while he mehod aims o reduce he Bayes Risk for model selecion, in fac i is only an upper bound on his value ha is minimized. Furhermore, here are no guaranees of he ighness of his bound. In many cases, however, an opimized rajecory ha yields a low upper bound on he Bayes Risk will be an accepable soluion. Second, he mehod is limied o he case of discriminaion beween wo models. In a faul deecion scenario, for example, here may be more han wo compeing models, corresponding o differen failure modes of he sysem. Fuure work will exend he mehod o he case of muliple models. Lasly, he mehod is resriced o linear sysems. While linearizaion may be used o solve his problem in cerain cases, fuure work will invesigae model discriminaion for non-linear dynamic sysems. VI. CONCLUSION This paper presens a new algorihm for model discriminaion ha poses he problem as a finie horizon rajecory design problem and minimizes an upper bound on he probabiliy of model selecion error. This problem is an example of a concave Quadraic Program, and hence can be solved efficienly by mehods ha are guaraneed o converge o he global opimum in bounded ime. The Quadraic Programming formulaion allows arbirary linear consrains o be placed on he conrol inpus and expeced sysem sae. Simulaion resuls showed ha compared o a ypical human-generaed conrol sequence, and a conrol sequence opimized wih regard o fuel consumpion, he new mehod can significanly reduce he upper bound on he probabiliy of model selecion error. REFERENCES [1 A. Pouliezos and G. Savrakakis, Real Time Faul Monioring of Indusrial Processes. Boson: Kluwer Academic, [2 J. Chen and R. Paon, Robus Model-Based Faul Diagnosis for Dynamic Sysems. Boson: Kluwer Academic, [3 P. Hanlon and P. Maybeck, Muliple-model adapive esimaion using a residual correlaion kalman filer bank, IEEE Transacions on Aerospace and Elecronic Sysems, vol. 36, pp , Apr. 2. [4 A. Willsky, A survey of design mehods for failure deecion in dynamic sysems, Auomaica, vol. 12, pp , [5 R. Esposio and M. Schumer, Probing linear filers - signal design for he deecion problem, IEEE Transacions on Informaion Theory, vol. IT-16, 197. [6 R. Prasanh, J. Amin, K. J, S. Seereeram, R. Mehra, D. Bayard, and F. Hadaegh, Predicive conrol approach o maneuver design for auonomous formaion flying sensor calibraion, in Proc. AIAA Conference and Exhibi on Guidance, Navigaion and Conrol, Rhode Island, Aug. 24. [7 R. Mehra, Opimal inpu signals for parameer esimaion in dynamic sysems: Survey and new resuls, IEEE Transacions on Auomaic Conrol, vol. 19, Dec [8 L. Ljung, Sysem Idenificaion: Theory for he user. New York: Prenice Hall, [9 V. Fedorov, Theory of Opimal Experimens. New York: Academic Press, [1 P. Hill, A review of experimenal design procedures for regression model discriminaion, Technomerics, vol. 2, Feb [11 R. Duda, P. Har, and D. Sork, Paern Classificaion (2nd ed.). Wiley Inerscience, 2. [12 K. Felsensein, Opimal bayesian design for discriminaion among rival models, Compuaional Saisics and Daa Analysis, vol. 14, pp , [13 P. Pardalos and J. Rosen, Mehods for global concave minimizaion: A bibliographic survey, Sociey for Indusrial and Applied Mahemaics, SIAM Review, Sep [14 J. Rosen, Global minimizaion of large-scale consrained concave quadraic problems by separable programming, Mahemaical Programming, vol. 34, pp , [15 I. Bomze, A global opimizaion algorihm for concave quadraic programming problems, SIAM Journal on Opimizaion, vol. 3, pp , [16 B. Kalanari, Large scale global minimzaion of linearly consrained concave quadraic funcions and relaed problems, Ph.D. disseraion, Univ. of Minnesoa, Minneapolis, Minnesoa, [17 J. Jang and C. Tomlin, Longiudinal sabiliy augmenaion sysem design of he sanford dragonfly uav using a single gps receiver, in Proceedings of he AIAA GNC conference, Ausin, Texas, Aug. 23.