Observability Criteria and Estimator Design for Stochastic Linear Hybrid Systems

Transcription

1 Observablty Crtera and Estmator Desgn for Stochastc Lnear Hybrd Systems Inseok Hwang, Hamsa Balakrshnan, Clare Tomln Hybrd Systems Laboratory Department of Aeronautcs and Astronautcs Stanford Unversty, Stanford, CA 9435, U.S.A. shwang, hamsa, Abstract A stochastc lnear hybrd system s sad to be observable f the hybrd state of the system s unquely determned from the output. In ths paper, we derve the condtons for the observablty of stochastc lnear hybrd systems by explotng the nformaton obtaned from system nose characterstcs. Havng establshed the necessary crtera for observablty, we study the effect of these condtons on estmator desgn, and also fnd bounds on the swtchng tmes of the system to acheve guaranteed estmator performance. We then apply these results to the estmaton of a two-mode arcraft trajectory. 1 Introducton The trackng of arcraft trajectores s a problem that has been approached wth some success usng hybrd systems models [1]. Related problems of nterest to us are the ablty to estmate the hybrd states of such systems from ther outputs, and also the desgn of estmators for such systems. The problem of observablty, namely, the ablty to estmate or reconstruct the actual state of a system gven ts output, s well-known and has been studed extensvely, both for contnuous systems [2] as well as for dscrete ones [3, 4]. More recently, several researchers have approached the problem of observablty of hybrd systems. A practcal problem that has receved ncreasng research attenton recently s the extenson of these concepts to stochastc hybrd systems. In ths paper, we address the ssue of observablty of a class of stochastc hybrd systems systems where the contnuous dynamcs are affected by whte Gaussan nose. Alessandr and Coletta [5] proposed a Luenberger observer desgn methodology for determnstc lnear hybrd systems, and proved that the error converges f the dscrete state evoluton s known. Balluch et al. [6] developed a method of combnng locaton observers for dscrete state estmaton wth Luenberger observers for contnuous state estmaton for lnear systems, such that they can guarantee the exponental convergence of the estmaton error. Bemporad et al. [7] defned the concept of ncremental observablty of contnuous-tme lnear hybrd systems, usng the solutons of a mxed-nteger lnear program. Recently, Vdal et al. [8] derved observablty condtons for lnear hybrd systems wth contnuous-tme contnuous-state dynamcs, gven n the form of rank condtons smlar to those for contnuous-tme lnear system observablty. For stochastc systems, the defnton of observablty n ts classcal form, as proposed by Kalman for systems wth no nose, fals; we therefore need to fnd a meanngful nterpretaton of observablty for systems wth random nose. Baram and Kalath [9] proposed the concept of estmablty as a better crteron to gauge stochastc lnear systems. Whle ths s one way of approachng the problem, we try to extend the defnton of observablty to nclude stochastc hybrd systems. An mportant class of problems assocated wth applcatons n mult-target trackng [1] and speech recognton [1] pertans to the estmaton of dscrete-tme Markov jump lnear systems. Cost and do Val [11] analyzed such systems wth fnte Markov states and determnstc contnuous dynamcs, and derved the observablty condton that the soluton to the coupled Rccat equaton assocated wth the quadratc control problem has 1

2 a stablzng soluton. Vdal et al. [12] derved observablty condtons for jump lnear systems based on rank tests smlar to those of determnstc lnear hybrd systems. The frst part of ths study s motvated by the results of Vdal et al. [12]. They proposed the noton of ndstngushablty as: two ntal states are ndstngushable f the correspondng outputs n free evoluton are equal. Ths approach results n elegant rank tests for the observablty of stochastc jump lnear systems. Snce n the desgn of estmators for arcraft trackng we have knowledge of not just the system dynamcs, but also the nose covarances, we try to explot ths addtonal knowledge to mprove our ablty to dfferentate between state trajectores. Snce the output sequences of stochastc systems mght be dfferent from the same ntal condton, we extend the noton of ndstngushablty [12] for such systems, and based on our defnton, we derve condtons for the observablty of dscrete-tme stochastc lnear hybrd systems. The latter part of ths paper apples the approach of Balluch et al. [6], so far used n the desgn of hybrd observers for determnstc hybrd systems wth contnuous-tme state evoluton, to dscrete-tme stochastc hybrd systems and estmator desgn. Ths paper s organzed as follows: Secton 2 presents the observablty condtons of dscrete-tme stochastc jump lnear systems. In Secton 3, we obtan condtons on the system parameters that would guarantee the exponental convergence of hybrd estmators. Examples and conclusons are presented n Sectons 4 and 5 respectvely. 2 Observablty of dscrete-tme stochastc lnear hybrd systems In ths secton, nspred by Vdal et al. [12], we extend the concepts of ndstngushablty, observablty of the hybrd ntal state, and dscrete transton tmes as defned n [12] and derve more general observablty condtons for dscrete-tme stochastc lnear hybrd systems usng the knowledge of nose covarances. We consder a dscrete-tme stochastc lnear hybrd system x k+1 = A(q k )x k + w k (q k ) H : y k = C(q k )x k + v k (q k ) q k+1 = δ(q k, γ k ), k {, 1, } (1) where k s a non-negatve nteger (k N); x k R n and y k R p are the contnuous state and output varables respectvely; q k {1, 2,, N} s the dscrete state, γ k {γ 1,, γ m } s a dscrete control nput, and δ(, ) s a determnstc dscrete transton relaton whch governs the dscrete state evoluton. We assume the event tme at whch a dscrete transton occurs s unknown. The system parameters A(q k ) R n n and C(q k ) R p n for q k {1, 2,, N} are real matrces. We assume that the ntal state x k s an unknown, zero-mean whte Gaussan random varable wth covarance E[x k x T k ] = Π and that the process nose w k (q k ) and the measurement nose v k (q k ) are uncorrelated, zero-mean whte Gaussan sequences wth the covarance matrces E[w k (q k )w k (q k ) T ] = ρ(q k )I and E[v k (q k )v k (q k ) T ] = σ(q k )I respectvely. These random sequences are assumed to be uncorrelated wth the ntal state,.e., E[x k w k (q k ) T ] = E[x k v k (q k ) T ] =. I denotes the dentty matrx. Snce the state evoluton of a hybrd system has contnuous trajectores as well as dscrete jumps, we defne a hybrd tme trajectory: Defnton 1 (Hybrd tme trajectory) A hybrd tme trajectory s a sequence of ntervals [k, k 1 1][k 1, k 2 1] [k, k +1 1] where k ( 1) s the tme at whch -th dscrete state transton occurs. Before dervng the observablty condtons, we revew the defnton of observablty for dscrete-tme stochastc lnear hybrd systems [12]: Defnton 2 (Observablty of dscrete-tme stochastc lnear hybrd systems) A dscrete-tme lnear hybrd system H s observable on [k, k + K] f the hybrd state (q k, x k ) for k [k, k + K] s unquely determned from the output sequence Y K = [y T k y T k +K ]T, where K N. 2

3 Vdal et al. [12] developed rank tests for the observablty of stochastc jump lnear systems of the form descrbed by H (Eq.(1)) usng the noton of ndstngushablty. Snce we know the nose covarances as well as the system dynamcs for a stochastc system, we use ths addtonal knowledge to obtan a more general condton. Snce the output sequences of stochastc systems could be dfferent from the same ntal condton, we extend the noton of ndstngushablty [12] as follows: Defnton 3 (Indstngushablty of dscrete-tme stochastc lnear hybrd systems) A dscrete-tme lnear hybrd system H s ndstngushable on [k, k + K] f there exst output sequences Y K and Y K on k [k, k + K] startng from any two dfferent hybrd states (q k, x k ) and (q k, x k ), whose covarances are equal. 2.1 Observablty of the hybrd ntal state In ths secton, usng a procedure smlar to that n [12], we derve the condtons under whch the hybrd ntal state (q k, x k ) can be unquely determned from the output sequence {y k } on [k, k 1 1] (k 1 1 k + K),.e., before the frst dscrete transton occurs. We defne κ := k +1 k ( ) as the sojourn tme, whch denotes how long the system stays n a dscrete state after the -th dscrete transton. Based on Defnton 2 and Defnton 3, we get the followng lemma: Lemma 1 The hybrd ntal state of a dscrete-tme lnear hybrd system H s observable f and only f t s dstngushable. Proof: The proof follows drectly from Defnton 2 and Defnton 3. In order to check f the hybrd ntal state s ndstngushable, we have to compute the covarance of output sequence Y κ on [k, k 1 1]. The output sequence startng from the hybrd ntal state (q k, x k ) on [k, k 1 1] s Y κ (q k ) = O κ (q k )x k + T κ (q k )W κ (q k ) + V κ (q k ) (2) where O κ (q k ) = [C(q k ) T (C(q k )A(q k )) T ((C(q k )A(q k )) k1 1 ) T ] T C(q k ) T κ (q k ) = C(q k )A(q k ) C(q k ).. C(q k )A(q k ) k1 k 2 C(q k )A(q k ) k1 k 3 C(q k ) W κ (q k ) = [w k (q k ) T w k +1(q k ) T w k1 1(q k ) T ] T V κ (q k ) = [v k (q k ) T v k+1(q k ) T v k1 1(q k ) T ] T O κ (q k ) R pκ n s the extended observablty matrx for the lnear system n Eq.(1) [12] and T κ (q k ) s a Toepltz matrx. If rank[o κ (q k )] = n,.e., the lnear system (A(q k ), C(q k )) s observable and κ n, then a least-squares soluton (whch we denote by ˆx k (q k )) to Eq.(2) can be determned unquely. ˆx k (q k ) = O κ (q k )Y κ (q k ) = x k + O κ (q k )T κ (q k )W κ (q k ) + O κ (q k )V κ (q k ) (3) where O κ (q k ) = (O T κ (q k )O κ (q k )) 1 O T κ (q k ). The last two terms on the rght hand sde of Eq.(3) represent the estmaton error due to the process nose and the measurement nose. Smlarly, the output sequence from another hybrd ntal state (q k, x k ) over [k, k 1 1] s Y κ (q k ) = O κ (q k )x k + T κ (q k )W κ (q k ) + V κ (q k ) (4) From Lemma 1, n order that the hybrd ntal state of a dscrete-tme stochastc lnear hybrd system be observable, t should be dstngushable,.e., the covarances of Y κ (q k ) and Y κ (q k ) satsfy: E[Y κ (q k )Y κ (q k ) T ] E[Y κ (q k )Y κ (q k ) T ] (5) 3

4 where E[Y κ (q k )Y κ (q k ) T ] = O κ (q k )Π O κ (q k ) T + ρ(q k )T κ (q k )Tκ T (q k ) + σ(q k )I E[Y κ (q )Y κ (q ) T ] = O κ (q )Π O κ (q ) T + ρ(q )T κ (q )Tκ T (q ) + σ(q k (6) )I Then, the dscrete ntal state can be unquely determned from the covarance of the output sequence and the contnuous ntal state can also be unquely determned usng Eq.(3). In order to reduce the requred κ for observablty (the sojourn tme n the dscrete state q k requred for observablty of the hybrd ntal state), we defne τ as the mnmum nteger whch satsfes rank[o τ (q k )] = n( q k {1, 2,, N}), and τ = max τ (smlar to the jont observablty ndex used n [12]). Then, we have the followng condton for the observablty of the hybrd ntal state: Lemma 2 (Observablty of the hybrd ntal state) If (A(q k ), C(q k )) are observable for each q k {1,, N} and κ τ, the hybrd ntal state (q k, x k ) s observable f and only f for all q k q k {1,, N}. O τ (q k )Π O τ (q k ) T + ρ(q k )T τ (q k )T τ T (q k ) + σ(q k )I O τ (q )Π O τ (q ) T + ρ(q )T τ (q )T τ T (q ) + σ(q )I Proof: Snce the lnear system n each dscrete state s observable and κ τ, the ntal contnuous state can be unquely determned usng Eq.(3) f the ntal dscrete state s dentfed. (f) Snce the covarances of the output sequences for each dscrete state are dstnct, the ntal dscrete state s unquely determned by checkng the covarance of the output sequence. (only f) The proof follows drectly from Defnton 3. We show through the followng smple example how a nose free unobservable dscrete-tme lnear hybrd system may be rendered observable, f each dscrete state has dfferent measurement nose covarances. Example: Consder a dscrete-tme lnear hybrd system wth two dscrete states { { xk+1 = x q 1 : k xk+1 = x q y k = c 1 x k + v 2 : k 1 y k = c 2 x k + v 2 where c 1, c 2, and c 1 c 2. The covarance of the ntal state s E[x x T ] = π R +. v 1 and v 2 are uncorrelated, zero-mean whte Gaussan sequences wth covarances E[v 1 v1 T ] = σ 1 and E[v 2 v2 T ] = σ 2 respectvely. If v 1 = v 2 =, the hybrd system s unobservable because two dfferent hybrd ntal states (q 1, x ) and (q 2, c 1 c 2 x ) generate the same output sequences [8]. However, f v 1 and v 2 are not dentcally zero and have dfferent covarances, then we can unquely determne the hybrd ntal state. If we consder the case n whch the actual hybrd ntal state s (q 1, x ), the output and ts covarance are y = c 1 x + v 1, E[yy T ] = π c 1 c T 1 + σ 1 (7) Next, f the actual hybrd ntal state s (q 2, c1 c 2 x ), the output and ts covarance are y = c 2 ( c 1 c 2 x ) + v 2, E[yy T ] = π c 1 c T 1 + σ 2 (8) Snce σ 1 σ 2, we can determne the dscrete ntal state unquely. For nstance, f the output comes from q 1, then the estmate of the ntal state s ˆx = x + v1 c Observablty of the dscrete transton tmes Lemma 2 gves the condton for the hybrd ntal state to be observable, over a tme nterval up to, but not ncludng the frst transton. In ths secton, we focus wthout loss of generalty on dervng the condtons under whch the frst dscrete transton tme k 1 can be unquely determned from the output sequence Y K on [k, k + K]; the tmes of the ensung transtons k ( {2,... }) can be computed n the same way [12]. We defne observablty of the frst dscrete transton tme as follows: Defnton 4 (Observablty of the frst dscrete transton tme) The frst dscrete transton tme of a dscretetme lnear hybrd system H s observable on [k, k + K] f t can be determned unquely from the output sequence Y K = [y T k y T k +K ]T. 4

5 If there s a dscrete transton at tme k 1, the output at tme k 1 and ts covarance are y k1 = C(q k1 )A(q k ) k 1 k x k + C(q k1 )F κ (q k )W κ (q k ) + v k1 (q k1 ) E[y k1 yk T 1 ] = C(q k1 )A(q k ) k 1 k Π (A(q k ) k 1 k ) T C(q k1 ) T +ρ(q k )C(q k1 )F κ (q k )F κ (q k ) T C(q k1 ) T + σ(q k1 )I (9) where F κ (q k ) := [A(q k ) k1 k 1 A(q k ) k1 k 2 I]. If there s no state transton at tme k 1, the output at tme k 1 and ts covarance are y k1 = C(q k )A(q k ) k 1 k x k + C(q k )F κ (q k )W κ (q k ) + v k1 (q k ) E[y k1 yk T 1 ] = C(q k )A(q k ) k1 k Π (A(q k ) k1 k ) T C(q k ) T +ρ(q k )C(q k )F κ (q k )F κ (q k ) T C(q k ) T + σ(q k )I (1) In order that the transton at tme k 1 be observable, the covarances of y k1 s n Eq.(9) and Eq.(1) should be dfferent. Thus, the observablty condton of the frst dscrete transton tme s: Lemma 3 (Observablty of the frst dscrete transton tme) The frst dscrete transton tme s observable f and only f C(q k1 )A(q k ) k 1 k Π (A(q k ) k 1 k ) T C(q k1 ) T +ρ(q k )C(q k1 )F κ (q k )F κ (q k ) T C(q k1 ) T + σ(q k1 )I C(q k )A(q k ) k 1 k Π (A(q k ) k 1 k ) T C(q k ) T +ρ(q k )C(q k )F κ (q k )F κ (q k ) T C(q k ) T + σ(q k )I for all q k q k {1,, N}. Proof: The proof follows by constructon. Therefore, from Lemma 2 and Lemma 3, the hybrd ntal state and the frst dscrete transton tme can be unquely determned. The remanng state trajectores can be determned by repeatng the procedure. For k ( 1), the ˆx k wll be gven from the ntal state estmate [12]. Thus, we have the followng observablty condton: Theorem 1 A dscrete-tme lnear hybrd system H s observable f and only f t satsfes Lemma 2 and Lemma 3. Proof: The proof follows drectly from Lemma 2 and Lemma 3. Ths test needs the operatons of multplcaton and addton of matrces whch are system parameters and nose covarances, the computaton s straghtforward wth computatonal complexty dependng on data sze. 3 Desgn of estmators for stochastc hybrd systems Havng establshed condtons for the observablty of stochastc lnear hybrd systems, we would lke to desgn estmators for those observable systems, and also quantfy values of system parameters that would guarantee performance (exponental convergence, n our case). We extend the desgn methods proposed by Balluch et al. [6] for hybrd systems wth contnuous-tme, contnuous state dynamcs to encompass dscrete-tme stochastc hybrd systems. A hybrd estmator fnds estmates ˆq and ˆx for the current dscrete state q and the contnuous state x respectvely. In ths secton, we frst descrbe the structure of the hybrd estmator, and then analyze the contnuous component of the estmator n detal to obtan bounds on the tme between dscrete transtons of state whch would guarantee exponental convergence of our hybrd estmator. Throughout ths paper, all norms, unless specfed otherwse, are 2-norms. Defnton 5 (Exponental convergence of a hybrd estmator) Gven a hybrd system H wth N dscrete modes, we say that a hybrd estmator s exponentally convergent f ts dscrete state estmate ˆq exhbts correct dentfcaton of the dscrete-state transton sequence of the orgnal system after a fnte number of steps; the 5

6 contnuous state estmate at any nstant has a unque mean and convergent covarance; and the mean of the estmaton error, ζ = E[ˆx x] converges exponentally to the set ζ M wth a rate of convergence µ, where M s the steady-state error bound, and µ < 1. In other words, the estmator s convergent f, for any swtchng tme k, ˆq k = q k, k > K, K N + (11) ζ k µ (k k ) ζ k + M, k > k (12) 3.1 Structure of the hybrd estmator We desgn the hybrd estmator as a combnaton of a dscrete observer to detect the dscrete state swtches, and an estmator to estmate the contnuous dynamcs, as proposed n [6]. In the rest of ths paper, we assume that we have a dscrete observer that correctly dentfes the dscrete state, ether mmedately after a swtch takes place, or wth a known tme delay after a dscrete transton. A dscrete observer could be constructed usng a bank of N estmators as a resdual generator [1, 6] even n ths case, we could further ncrease the probablty of correct dscrete-state dentfcaton by enforcng a decson tme delay on the dscrete observer. Ths would be possble only f the system were observable n the sense of a stochastc hybrd system, as explaned earler. In ths secton, we desgn a least-squares estmator n the form of N Kalman flters for the contnuous state estmate. Although the underlyng system n [6] s contnuous-tme and determnstc, the desgn methodology of [6] adapts well to dscrete-tme stochastc hybrd systems, as we show here. 3.2 Dscrete-tme Kalman flter We consder a hybrd system of the form descrbed n Eq.(1). For the sake of smplcty of notaton, we replace A(q k ) and C(q k ) wth A l and C l, where l {1... N}. We can then wrte the equatons for the least-square estmator of a lnear stochastc system as ˆx k+1 = (A l K l C l )ˆx k + K P,k,l y k, k (13) where l s the estmated dscrete state, and K P,k,l s the optmal Kalman flter gan for mode l, gven by and P k satsfes the dscrete Rccat recurson K P,k,l = A l P k C T l (R l + C l P k C T l ) 1 (14) P k+1 = A l P k A T l + Q l K P,k,l (R l + C l P k C T l )K T P,k,l, P () Π (15) Theorem 2 ([13]): The Dscrete Algebrac Rccat Equaton (DARE) has a stablzng soluton that s unque f and only f Any such soluton s postve defnte. {A l, C l } s detectable (16) {A l, Q 1/2 l } s controllable on the unt crcle. (17) If these condtons are satsfed for every dscrete state {1... N}, we can desgn a bank of N steady-state, exponentally convergent Kalman flters to estmate the contnuous state of the system. We can then show that, for a gven dscrete state, correctly dentfed, ˆx k+1 = (A l K l C l )ˆx k + K l y k (18) ˆζ k+1 = (A l K l C l )ˆζ k (19) Clearly, ˆζ s exponentally convergent f (A l K l C l ) s stable (2) 6

7 3.3 Error dynamcs In ths secton, we follow the methodology of [6] to determne the evoluton of the estmaton error across the dscrete transton sequence. Let us consder two consequent dscrete transtons of H, occurrng at tmes k and k +1. Suppose the transton at tme k +1 was from dscrete state m to l, and was detected at tme k +1 such that k +1 k +1. Smlarly, k k. Ths s llustrated n Fg.(1). swtch detect swtch detect k k k k q = m q = l q = l q = l q ~ = m q ~ = m ~ q = l Fgure 1: Illustraton of the transton sequence We are nterested n the regon k {k, k + 1,..., k +1 }. Snce we assume that by tme-step k the dscrete state has been dentfed correctly, for the exponental convergence of the estmaton error on k to k +1 : 1. The error converges exponentally between k and k +1; and 2. The error dvergence between k +1 and k +1 due to wrong dscrete state estmaton does not upset the exponental convergence of the error on k to k +1. Followng the methodology of [6], dvdng the tme nterval between k and k +1 nto two regons, we get error dynamcs of the form ζ k+1 = (A m K m C m ) ζ k k {k,..., k +1 1} (21) ζ k+1 = (A m K m C m ) ζ k + [(A m A l ) K m (C m C l )] x k k {k +1,..., k +1 1} (22) where x = E[x]. The second term n Eq.(22) arses because a Kalman flter desgned for the dscrete state m s beng used to estmate the dynamcs of the dscrete state l. Combnng Eqs. (21) and (22), we express the error dynamcs by ζ k+1 = (A m K m C m ) ζ k + u k, k {k,..., k +1 1} (23) where From ths, we get u k = {, k {k,..., k +1 1} ((A m A l ) K m (C m C l )) x k, k {k +1,..., k +1 1} (24) ζ k+1 = (A m K m C m ) k+1 k ζk + [(A m K m C m ) k k :... I] ζk+1 = (A m K m C m ) k+1 k ζk + [(A m K m C m ) k k :... I] u k.. u k u k. u k (25) (26) 7

8 Ths gves us ζk+1 = k k (A m K m C m ) k+1 k ζk + (A m K m C m ) k k l u k +l l=, k {k +1,..., k +1 1} (27) k k (A m K m C m ) k+1 k ζk + (A m K m C m ) k k l u k +l, k {k +1,..., k +1 1}(28) l= Lemma 4 Gven a matrx A R n n wth all dstnct egenvalues, A t k(a)α t (A), t (29) where α(a) s the maxmal absolute value of the egenvalues of A, and k(a) = Q Q 1, the condton number of A under the nverse, where Q 1 AQ = J, the Jordan canoncal form. Proof: The proof follows that for the contnuous-tme case ( [6], [14]). From ths we can show that for t, f m s the sze of the largest Jordan block of A, When A has all dstnct egenvalues, ths reduces to Eq.(29). tr A t mk(a)α t (A) max α r, r m 1, (3) (A) Further smplfcaton of Eq.(28) usng Lemma 4 gves us ζk+1 k(a m C m )[α(a m K m C m )] k+1 k ζk Snce k +1 k +1, f + k(a m C m )max u k (k k +1 ), k {k +1,..., k +1} (31) u k U = max (A m A l ) K m (C m C l ) 1 X (32) such that X x, X >, we can wrte ζk+1 k(am C m )[α(a m K m C m )] k+1 k ζk + nu k(am C m ) (33) Lemma 5 Consder a hybrd system wth a sngle dscrete state, n whch the dscrete-tme evoluton of the contnuous state varable s gven by x k+1 = ηx k, η < 1 (34) Suppose the state x s subject to resets x(t s ) = aηx(t s 1) + b, occurrng at swtchng tmes {t s }, wth a 1 and b. Then the evoluton of x can be descrbed by x k = η k t s 1 x ts 1, k {t s 1,..., t s 1} (35) x ts = aη t s t s 1 x ts 1 + b (36) Let us also assume there exsts a lower bound β on the tme between resets,.e., t s t s 1 β 1, for all s > 1. Then, f x t > and µ = η ( log η a β +1) such that µ < 1, then x(k) converges exponentally to the set b [, ] wth a rate of convergence greater than or equal to µ. 1 η β Proof: The proof s smlar to the proof of Lemma 3 n Balluch et al.([6]). We can show that f the above condtons are satsfed, then x ts < µ (ts t) x t + b 1 η β (37) Ths mples that the state x ts after every reset s bounded above exponentally by rate µ, and converges to b the set [, ]. Snce the nter-reset dynamcs decays exponentally wth rate η, and η µ, the evoluton 1 η β between resets s also bounded above by an exponental wth rate µ. Usng Eqs.(16), (17), (2), (32) and (33) wth Lemma 4 and Lemma 5, we arrve at the followng theorem: 8

9 Theorem 3 Consder a stochastc lnear hybrd system of the form n Eq.(1), a steady-state error bound M and rate of convergence µ, µ < 1, α(a m K m C m ) µ for all m = 1... N, where α(a) s the maxmal absolute value of the egenvalues of A. Let k(a) = Q Q 1, the condton number of A under the nverse, where Q 1 AQ = J, the Jordan canoncal form. Then f the followng seven condtons are satsfed: 1. The system s observable under the defnton n Secton 2 2. {A m, C m } couples are observable for all m = 1... N 3. {A m, Q 1/2 m } couples are controllable for all m = 1... N 4. (A m K m C m ) s stable for all m = 1... N wth all dstnct egenvalues 5. There exsts X > such that x k X, k = 1, 2,... such that 6. The dscrete decson tme, satsfes the relaton u k U = max (A m A l ) K m (C m C l ) 1 X (38) M numax[k(am K m C m )] (39) 7. The tme between swtchng events, β satsfes the condtons β > β mn +, where (4) ( ) 1 β mn > max[ log µ log nu k(am K m C m ) 1, M max log[k(a m K m C m )] log[α(a m K m C m )] ] (41) we can desgn a hybrd estmator that converges to wthn the steady-state bound M wth a rate of convergence greater than or equal to µ. Proof: The proof follows drectly from the fact that the error dynamcs are bounded by Eq.(33), whch s n the form of Eq.(36) n Lemma 5. Applyng Lemma 5 for the approprate values of a, b and η, we can prove Theorem 3. Condtons (2)-(4) are needed for convergence of the estmators, whle condton (1) s needed for the detecton of the swtch and for the desgn of the dscrete observer. Corollary 1 If Condtons (1)-(6) of Theorem 3 are satsfed, then, gven a steady-state error bound M and a rate of convergence µ, we can desgn an estmator that converges exponentally to M wth a rate of at least µ f the tme between swtchng events s at least β = β mn +, where [ ( ) 1 β mn = max log µ log nu k(am K m C m ) 1, max log[k(a ] m K m C m )] (42) log[α(a m K m C m )] M Remark 1 : An mportant dfference between the contnuous-tme hybrd systems analyzed n [6] and the dscrete-tme hybrd systems that we consder s that we can no longer make M arbtrarly small by smply changng the value of such that Eq.(39) s stll satsfed - the dscrete nature of the system restrcts to values n N. 4 Example: Arcraft Trajectory We apply the above desgn crtera to the desgn of an estmator for the swtched, lnearzed trajectory of an arcraft. We consder two dscrete states, both coordnated turns, but wth dfferent angular veloctes, one 9

10 wth a turn rate of 2 per second, and the other wth a turn rate of 5 per second, whch represent arcraft trajectores composed of slow turns and sharp turns. For brevty, we only nclude two dscrete state example n ths paper but we have successfully desgned hybrd estmators for arcraft trajectory trackng and conflct detecton and resoluton problems wth multple dscrete states such as constant velocty straght flght modes wth dfferent nose characterstcs and coordnated turn modes wth varous angular veloctes. The dynamcs of a coordnated turn s gven by x k = y k = 1 sn ωt 1 cos ωt ω ω cos ωt sn ωt 1 cos ωt sn ωt ω 1 ω 1 sn ωt cos ωt [ 1 1 x k 1 + T 2 2 T T 2 2 T u k 1 + w k (43) ] x k + v k (44) where x = [ x 1 ẋ 1 x 2 ẋ 2 ] where x 1 and x 2 are the poston coordnates, u = [ u 1 u 2 ] T where u 1 and u 2 are the velocty components, ω s the turn rate, T s the samplng nterval, w s the process nose, and v s the sensor nose. We choose an operatng velocty of 15 knots. We fnd that for an nstantaneous dscrete decson tme, the tme between dscrete transtons should be at least 8 seconds to guarantee exponental convergence wth a rate of.99. The comparson of the bounds s shown n Fgure (2(a)). We also note that by Lemma (5) the norm of the mean error does not have to be monotonc, but f the condtons explaned above are satsfed, t wll be bounded by an exponental of rate µ. Ths s also seen n the example. 9 8 error norm exponental bound dscrete transton 9 8 error norm exponental bound dscrete transton norm of mean error 5 4 norm of mean error t (a) t (b) Fgure 2: (a) Exponental convergence of error. (b) Convergence of error when modes have same dynamcs but dfferent nose characterstcs. The trangles denote dscrete transton tmes (µ =.99, M =, and T = 2sec). As explaned n the Secton 2, Lemma 2, dentcal dynamcs wth dfferent nose characterstcs n each dscrete state mght stll make the system observable n the stochastc hybrd context. We demonstrate ths by desgnng an exponentally convergent hybrd estmator for a swtched arcraft trajectory - the two dscrete states correspond to 2 per second turns wth dfferent process nose covarances. Ths s shown n Fgure (2(b)). 5 Conclusons In ths paper, we have extended the defnton of observablty to nclude stochastc lnear hybrd systems, and have used pror knowledge of system nose characterstcs to mprove the observablty condtons for a dscretetme stochastc lnear hybrd system. We have also found bounds on the tme between dscrete transtons to guarantee the exponental convergence of hybrd estmators for such systems. An nterestng drecton for future work would be the extenson of these results to hybrd systems wth contnuous state resets. 1

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