Observer/Kalman Filter Time Varying System Identification Manoranjan Majji Texas A&M University, College Station, Texas, USA Jer-Nan Juang 2 National Cheng Kung University, Tainan, Taiwan and John L. Junins 3 Texas A&M University, College Station, Texas, USA An algorithm for comutation of the generalized Marov arameters of an observer or Kalman filter for discrete time varying systems from inut-outut exerimental data is resented. Relationshis between the generalized observer Marov arameters and the system Marov arameters are derived for the time varying case. A systematic rocedure to comute the time varying sequence of system Marov arameters and the time varying observer (or Kalman) gain Marov arameter sequence from the generalized time varying observer Marov arameters is resented. This rocedure is shown to be a time varying generalization of the recursive relations develoed for the time invariant case using an Autoregressive model with an exogenous inut (ARX model in a rocedure nown as Observer/Kalman filter identification, OKID). These generalized time varying inut-outut relations with the time varying observer in the loo are referred to in the aer as the Generalized Time Varying Autoregressive model with an exogenous inut (GTV-ARX model). The generalized system Marov arameters thus derived are used by the Time Varying Eigensystem Realization Algorithm (TVERA) develoed by the authors, to obtain a time varying discrete time state sace model. Qualitative relationshi of the time varying observer with the Kalman observer in the stochastic environment and an asymtotically Post Doctoral Research Associate, Aerosace Engineering Deartment, 66 D, 34 TAMU, College Station, Texas, 77843-34, majji@tamu.edu, Student Member, AIAA. 2 Professor, National Cheng Kung University, Tainan, Taiwan; Former President, National Alied Research Laboratories, Taiei, Taiwan; Adjunct Professor, Aerosace Engineering Deartment, 34 TAMU, College Station, Texas, 77843-34, jjuang@cox.net, Fellow, AIAA. 3 Distinguished Professor, Regents Professor, Royce E. Wisenbaer Chair, Aerosace Engineering Deartment, 722 B, 34 TAMU, College Station, Texas, 77843-34, junins@aeromail.tamu.edu, Fellow, AIAA.
stable realized observer are discussed briefly to develo insights for the analyst. The minimum number of reeated exeriments for accurate recovery of the system Marov arameters is derived from these develoments, which is vital for the racticing engineer to design multile exeriments before analysis and model comutations. The time varying observer gains realized in the rocess are subsequently shown to be in consistent coordinate systems for closed loo state roagation. It is also demonstrated that the observer gain sequence realized in case of the minimum number of exeriments corresonds naturally to a time varying deadbeat observer. Numerical examles demonstrate the utility of the concets develoed in the aer. S I. Introduction YSTEM identification has emerged as an imortant toic of research over the ast few decades owing to the advancements of model based modern guidance, navigation and control. Eigensystem Realization Algorithm[] (ERA) is widely acnowledged as a ey contribution from the aerosace engineering community to this dynamic research toic. The system identification methods for time invariant systems have seen efforts from various researchers. The methods are now well understood for continuous and discrete time systems including the relationshis between the continuous and discrete time system models. On the other hand, discrete time varying system identification methods are comaratively oorly understood. Several ast efforts by researchers have documented the develoments in the identification of discrete time varying models. Cho et al.[2] exlored the dislacement structure in the Hanel matrices to obtain time invariant models from instantaneous inut-outut data. Shooohi and Silverman [3] and Dewilde and Van der Veen[4], generalized several concets of the classical linear time invariant system theory to include the time varying effects. Verhaegen and coworers [5, 6] subsequently introduced the idea of reeated exeriments (termed ensemble i/o data), enabling further research in the develoment of methods for identification of time varying systems. Liu [7] develoed a methodology for develoing time varying model sequences from free resonse data (for systems with an asymtotically stable origin) and made initial contributions to the develoment of time varying modal arameters and their identification[8]. An imortant concet of inematic similarity among linear discrete time varying system models concerns certain time varying transformations involved in the state transition matrices. Gohberg et al.[9] 2
discuss fundamental develoments of this theory using a difference equation oerator theoretic aroach. In comanion aers, Majji et al.[0, ] extend the ERA, a classical algorithm for system identification of linear time invariant systems to realize discrete time varying models from inut-outut data following the framewor and conventions of the above aers. The time varying eigensystem realization algorithm (TVERA) resented in the comanion aers [0, ] uses the generalized Marov arameters to realize time varying system descritions by maniulations on Hanel matrix sequences of finite size. The realized discrete time varying models are shown to be in time varying coordinate systems and a method is outlined to transform all the time varying models to a single (observable or controllable subsace) coordinate system at a given time ste. However the algorithm develoed there-in requires the determination of the generalized Marov arameters from sets of inut-outut exerimental data. Therefore we need a ractical method to calculate them without resorting to a high dimensioned calculation. This calculation becomes further comounded in systems where stability of the origin cannot be ascertained, since the number of otentially significant generalized Marov arameters grows raidly. In other words, in case of the roblems with an unstable origin, the outut at every time ste in the time varying case deends on the linear combinations of the (normalized) ulse resonse functions of all the inuts alied until that instant (causal inuts). Therefore the number of unnowns increase by m * r for each time ste in the model sequence and consequently, the analyst is required to erform more exeriments if a refined discrete time model is sought. In other words, the number of reeated exeriments is roortional to the resolution of the model sequence desired by the analyst. This comutational challenge has been among the main reasons for the lac of readyadotion of the time varying system identification methods. In this aer, we use an asymtotically stable observer to remedy this roblem of unbounded growth in the number of exeriments. The algorithm develoed as a consequence is called the time varying observer/kalman filter system identification (TOKID). In addition, the tools systematically resented in this aer give an estimate on the minimum number of exeriments one needs to erform for identification and/or recovery of all the Marov arameters of interest until that time instant. Thus, the central result of the current is to mae the number of reeated exeriments indeendent of the desired resolution of the model. Furthermore, since the frequency resonse functions for time varying systems are not well nown, the method outlined seems to be the one of the first ractical ways to obtain the generalized Marov arameters bringing most of the generalized Marov arameter based discrete time varying identification methods to the table of the racticing engineer. 3
Novel models relating inut-outut data are develoed in this aer and are found to be elegant extensions of the ARX models well nown in the analysis of time invariant models (cf. Juang et al., [2]). This generalization of the classical ARX model to the time varying case admits analogous recursive relations with the system Marov arameters as was develoed in the time invariant case. This analogy is shown to go even further and enable us to realize a deadbeat observer gain sequence for time varying systems. The generalization of this deadbeat definition is rather unique and general for the time varying systems as it is shown that not all the closed loo time varying eigenvalues need to be zero for the time varying observer gain sequence to be called dead beat. Further, it is demonstrated that the time varying observer sequence (deadbeat or otherwise) comuted from the GTV-ARX model is realized in a comatible coordinate system with the identified lant model sequence. Relations with the time varying Kalman filter are made comaring features of the arameters of the Kalman filter gains with the time varying observer gains realized from the generalized OKID rocedure resented in the aer. II. Basic Formulation We start by revisiting the relations between the inut outut sets of vectors via the system Marov arameters as develoed in the theory concerning the Time Varying Eigensystem Realization Algorithm (TVERA, refer to a comanion aer [0] based on [] and the references therein). The fundamental difference equations governing the evolution of a linear system in discrete time are given by x = A x + + B u () together with the measurement equations y = C x + D u (2) with the state, outut, and inut dimensions x R n, y R m, u R r and the system matrices to be of comatible dimensions Z, an index set. The solution of the state evolution (the linear time varying discrete time difference equation solution) is given by (, 0) 0 (, ) x =Φ x + Φ i+ B u (3) i i i= 0 0 +, where the state transition matrix, Φ(.,.) is defined as 4
A A... A, > Φ (, 0) = I, = 0 undefined, < 0 2 0 0 (4) Using the definition of the comound state transition matrix, the inut-outut relationshi is given by y = C Φ (,0 ) x + C Φ (, i+ ) B u + D u (5) 0 i i i= 0 This enables us to define the inut-outut relationshi in terms of the two index coefficients as where the generalized Marov arameters are defined as y = C Φ (,0) x + h u + D u (6) 0, i i i= 0 ( ) CΦ, i+ Bi, i< h, i = C B, i= 0, i> (7) From now on, we try to use the exanded form of the state transition matrix Φ(.,.) to imrove the clarity of the resentation. Thus the outut at any general time ste t is related to the initial conditions and all the inuts as y u = C A... A A + D C B C A... A B u x u 0+ 0 0 0+ 0 0 (8) where 0 can denote any general time ste rior to (in articular let us assume that it denotes the initial time such that 0 = 0 ). As was ointed out in the comanion aer, such a relationshi between the inut and outut leads to a roblem that increases by m * r arameters for every time ste considered. Thus it becomes difficult to comute the increasing number of unnown arameters. In the secial case of systems whose oen loo is asymtotically stable, this is not a roblem. However, frequently, one tries to use identification in roblems which do not have a stable origin for control and estimation uroses. In such roblems, the analyst may be required to comute time varying model sequences with higher resolution. Hence we need to exlore alternative methods in which lant arameter models can be realized from inut-outut data. A viable alternative to this roblem useful to the racticing engineer is develoed in the following section. 5
The central assumtion involved in the develoments of this aer is that (in order to obtain the generalized system and observer gain Marov arameters for all time stes involved), one should start the exeriments from zero initial conditions or from the same initial conditions each time the exeriment is erformed. The more general case deals with the resence of initial condition resonse in the outut data. In the hysical situation of unnown initial conditions, this roblem is comounded and the searation of zero inut resonse from the outut data becomes an involved rocedure. We do not discuss this general situation in the resent aer. Most imortantly since the connections between time varying ARX model and the state sace model, and a discussion on the associated observer are comlicated by themselves, we roceed with the resentation of the algorithm under the assumtion that each exeriment can be erformed with zero initial conditions. III. Inut Outut Reresentations: Observer Marov Parameters The inut-outut reresentations for the time varying systems are quite similar to the inut outut model estimation of a lightly damed flexible sacecraft structure in the time invariant case. In the identification roblem involving a lightly damed structure, one has to trac a large number of Marov arameters to obtain reasonable accuracy in comutation of the modal arameters involved. An effective method for comressing exerimental inut-outut data, called observer/kalman filter Marov arameter identification theory (OKID) was develoed by Juang et al. [, 2, 3]. In this section, we generalize these classical observer based schemes for determination of generalized Marov arameters. The concet of frequency resonse functions that enables the determination of system Marov arameters for time invariant system identification does not have a clear analogous theory in case of the time varying systems. Therefore, the method described here-in constitutes one of the first efforts to efficiently comute the generalized Marov arameters from exerimental data. Imortantly, for the first time, we are also able to isolate a minimum number of reeated exeriments to hel the racticing engineer to lan the exeriments required for identification a riori. Following the observations of the revious researchers, consider the use of a time varying outut feedbac style gain sequence in the difference equation model Eq. () governing the linear lant, given by 6
x = A x + B u + + G y G y ( ) ( ) = A + G C x + B + G D u G y u = A x + ( B G D) G + y = A x + B v (9) with the definitions A = A + G C v B = B + G D G u = y (0) and no change in the measurement equations at the time ste t y = C x + D u () The oututs at the consecutive time stes, starting from the initial time ste t 0 (denoted by 0 = 0) are therefore written as y = C x + D u 0 0 0 0 0 y = C A x + D u + C B v 0+ 0+ 0 0 0+ 0+ 0+ 0 0 = C A x + D u + h v 0+ 0 0 0 0 0+, 0 0 y = C A A x + D u + C B v + C A B v... 0+ 2 0+ 2 0+ 0 0 0+ 2 0+ 2 0+ 2 0+ 0+ 0+ 2 0+ 0 0 = C A A x + D u + h 0+ 2 0+ 0 0 0+ 2 0+ 2 0+ 2, 0+ 0+ 0+ 2, 0 0 v + h v (2) with the definition of generalized observer Marov arameters C A A 2... Ai+ Bi, > i+ h, i = C B, = i+ 0, < i+ (3) we arrive at the general inut-outut relationshi 0 y = C A... A x + D u + h v (4), j j 0 0 j= We oint out that the generalized observer Marov arameters have two bloc comonents similar to the linear time invariant case shown in the artitions to be 7
h = C A... A B, j j+ j ( ) ( 2), j h, j ( ) = C A... A B + G C C A... A G j+ j j j j+ j = h (5) where the artitions h ( ) ( 2), j, h, j are used in the calculations of the generalized system Marov arameters and the time varying observer gain sequence in the subsequent develoments of the aer. The closed loo thus constructed, is now forced to have an asymtotically stable origin by the observer design rocess. The goal of an observer constructed in this fashion is to enforce certain desirable (stabilizing) characteristics into the closed loo (e.g., deadbeat-lie stabilization, etc.). stes The first ste involved in achieving this goal of closed loo asymtotic stability is to choose the number of time (variable each time in general) sufficiently large so that the outut of the lant (at t + ) strictly deends on only the + revious augmented control inuts { v }, u + j j= + and indeendent of the state at every time ste t. Therefore by writing y = C A... A x + D u + h v +, + j + + + + + j + j= D u + h +, j + + + j + j= v (6) we have set C A... A x + + 0 (with exact equality assignable i.e., C A... A x + + = 0, in the absence of measurement noise = 0,,..., f ). This leads to the construction of a generalized time varying autoregressive with exogenous inut (GTV-ARX) model at every time ste. Note that the order of the GTV-ARX model can also change with time (we coin the term generalized to describe this variability in the order). This variation and comlexity rovides a large number of observer gains at the disosal of the analyst under the time varying OKID framewor. In using this inut-outut relationshi (Eq.(6)) instead of the exact relationshi given in Eq.(8), we introduce daming into the closed loo. For simlicity and ease in imlementation and understanding, we set the generally variable order to remain fixed and minimum (time varying deadbeat) at each time ste. That is to say, = = min where min is the smallest ositive integer such that min mn. This restriction (albeit unnecessary) forces a time varying deadbeat observer at every time, roviding ease in calculations by requiring 8
minimum number of reeated exeriments. The deadbeat conditions are different in the case of time varying systems, due to the transition matrix roduct conditions (Eq. (6)) that are set to zero. This situation is in contrast with (and is a modest generalization of the situation in) the time invariant systems where higher owers of the observer system matrix give sufficient conditions to lace all the closed loo system oles at the origin (deadbeat). The nature and roerties of the time varying deadbeat condition are briefly summarized in the Aendix B, along with an examle roblem. Considerations of the time varying deadbeat condition aear sarse (Minamide et al.,[4] and Hostetter[5] resent some fundamental results on the design of time varying deadbeat observers), if not comletely heretofore unnown in modern literature. Therefore the connections made here-in esecially in the context of system identification are quite unique in nature. If the reeated exeriments (as derived and resented in [0, ]) are erformed so as to comute a least-squares solution to the inut-outut behavior conjectured in Eq.(6), we have identified the system (together with the observer-in-the-loo) such that the outut y + does not deend on the state matrix form, for any time ste t (denoted by and > ) we have that x. Stating the same in a vector y u v = D h h h v v,, 2, 2 (7) This reresents a set of m equations in m ( r *( r m) ) + + unnowns. In contrast to the develoments using the generalized system Marov arameters, (to relate the inut-outut data sets; refer Eq. (8) in the comanion aer [0, ] and the references there-in for more information) the number of unnowns remains constant in this case. This maes the comutation of observer Marov arameters ossible in ractice since the number of reeated exeriments required to comute these arameters is now constant (derived below) and does not change with the discrete time ste t (resolution of the model sequence desired by the analyst). This is an imortant result of the ( ) current aer. In fact, it is observed that a minimum of N min r ( r m ) = + + exeriments are necessary to ex min * determine the observer Marov arameters uniquely. From the develoments of the subsequent sections, this is the minimum number of reeated exeriments one should erform in order to realize the time varying system models desired from the TVERA. Equations (7) with N reeated exeriments yields 9
( ) ( 2) ( N ) Y = y y y ( ) ( 2) ( N ) u u u ( ) ( 2) ( N) v v v ( ) ( 2) ( N) = D h, h, 2 h, v 2 v 2 v 2 ( ) ( 2) ( N ) v v v = M V (8) >. Therefore the least squares solution for the generalized observer Marov arameters is given for each time ste as ˆ = M Y V (9) where (). denotes the seudo inverse of a matrix [6, 7]. The calculation of the system Marov arameters and observer gain Marov arameters is detailed in the next section. IV. Comutation of Generalized System Marov Parameters and Observer Gain Sequence We first outline a rocess for the determination of system Marov arameter sequence from the observer Marov arameter sequence calculated in the revious section. A recursive relationshi is then given to obtain the system Marov arameters with the index difference of greater than time stes. Similar rocedures are set u for observer gain Marov arameter sequences. A. Comutation of System Marov Parameters from Observer Marov Parameters Considering the definition of the generalized observer Marov arameters, we write h = C B, ( ) = C B + G D G ( ) ( 2), h, = h (20) where the suerscrits () and (2) are used to distinguish between the Marov arameter sequences useful to comute the system arameters and the observer gains resectively. Consider the following maniulation written as ( ) ( 2),, = h h D C B = h, (2) 0
where the unadorned hi, j are used to denote the generalized system Marov arameters, following the conventions and notations set u in the comanion aers[0, ]. A similar exression for Marov arameters with two time stes between them yields ( ) ( 2), 2, 2 2 = 2 2 2 h h D C A B C A G D ( ) = C A B + G D C A G D 2 2 2 2 2 = C A B 2 ( ) = C A + G C B 2 ( 2) 2, 2 = C A B + h C B ( 2), 2,, 2 = h + h h (22) This elegant maniulation leads to an exression for the generalized system Marov arameter h, 2 to be calculated from observer Marov arameters at the time ste t and the system Marov arameters at revious time stes. This recursive relationshi was found to hold in general and enables the calculation of the system Marov arameters from the observer Marov arameters h, h. () (2) i, j i, j To show this holds in general, consider the induction ste with observer Marov arameters (with time ste searation) given by ( ) ( 2) ( ) h h D = C A A... A B + G D C A A... A G D,, 2 + 2 + = C A A... A B 2 + ( ) = C A A... A A + G C B 2 + 2 + + + = C A A... A A B + C A A... A G C B 2 + 2 + 2 + 2 + + ( 2) 2... + 2 +, + +, = C A A A A B + h h (23) Careful examination reveals that the term C A A 2... A + 2 A + B can be written as ( ) C A A... A A B = C A... A A + G C A B 2 + 2 + + 3 + 2 + 2 + 2 + = C A... A A B + C A... G C A B + 2 + + 2 + 2 + = C A... A A B + h ( ) =... 2 + 2 +, + 2 + 2, ( 2) ( 2) ( 2)... +,,, 2 2,..., + 2 + 2, = C A A B + h h + h h + + h h ( 2) ( 2),,,, 2 2,... 2, + h 2 + 2, = h + h h + h h + + h ( ) h (24) This maniulation enables us to write
( ) ( 2) ( 2) ( 2) ( 2),, =, +,, +, 2 2, +... +, + +, h h D h h h h h h h ( 2),, j j, j= = + h h h (25) Writing the derived relationshis between the system and observer Marov arameters yields the following set of equations ( ) ( 2), =,, h h h D ( 2) ( ) ( 2), 2 +,, 2 =, 2, 2 2 h h h h h D... ( 2) ( 2) ( ) ( 2), +,, +... +, + +, =,, h h h h h h h D (26) Defining ( ) ( 2) i, j : i, j i, j j r = h h D, we obtain the system of linear equations relating the system and observer Marov arameters as ( 2) ( 2) ( 2) m,, 2, +,, 2, ( 2) ( 2) I 0 m h, 2 h h, 2, 2 h +, I h h h h h h 0 0 0 0 I 0 0 h m +, r r r 0 r r 0 0 r,, 2,, 2, = +, (27) We note the striing similarity of this equation to the relation between observer Marov arameters and the system Marov arameters in the classical OKID algorithm for time invariant systems (comare coefficient matrix of Eq. (27) with equation (6.8) of Juang[]). Considering the exressions for h, : = C A... A + B and choosing sufficiently large, we have that owing to the asymtotic stability of the closed loo (including the observer) h, 0. This fact enables us to establish recursive relationshis for the calculation of the system Marov arameters h,, γ >. Generalizing Eq. (25) (to introduce the variability of the order of the GTV-ARX model, as roosed else-where in the aer, i.e., γ setting = ) roduces γ ( ) ( 2) ( 2), γ, γ, γ γ, j j, γ j= h = h h D h h (28) 2
γ >. Then based on the constraint imosed in Eq. (7) for the calculation of the generalized observer Marov arameters, all the terms with time ste searation greater than vanish identically, and we obtain the relationshi h = h h (29) ( 2), γ, j j, γ j= For maintaining the simlicity of the resentations here-in, we will not mae any more references to the variable order otion in the subsequent develoments of the aer. That is to say that the variable order of the GTV-ARX model at each time ste is set to realize the time varying deadbeat observer, i.e., = ( = to further clarify the min develoments). It also imlies that only a minimum number of reeated exeriments needs to be erformed. Insight into the flexibility offered by the variable order of the GTV-ARX model is rovided by aealing to the relations of the identified observer with a linear time varying Kalman filter in the next section of this aer. B. Comutation of Observer Gain Marov Parameters from the Observer Marov Parameters Consider the generalized observer gain Marov arameters defined as C A A... A G, > i+ 2 i+ i o, i h = C G, = i+ 0, < i+ (30) We will now derive the relationshi between these arameters and the GTV-ARX model coefficients h ( 2), j. These arameters will be used in the calculation of the observer gain sequence from the inut-outut data in the next subsection, a generalization of the time invariant relations obtained in [, 2] similar to Eq. (27). From their corresonding definitions, note that Similarly ( 2) o,, h = C G = h (3) ( 2), 2 = 2 h C A G o ( 2) ( ) = C A + G C G = h + h h o 2, 2,, 2 (32) In general, an induction ste similar to Eq. (23) holds and is given by, 3
( 2), = 2... + h C A A A G ( ) = C A A... A A + G C G 2 + 2 + + + = C A A... A A G + C A A... A G C G 2 + 2 + 2 + 2 + + = C A A... A A G + h ( ) 2 o 2 + 2 +, + +, o ( 2) o ( 2) o,,,..., + +, = h + h h + + h h h (33) Where the identity derived in Eq. (24) (relace B in favor of G ) is used. This enables us to write the general relationshi, γ ( 2) o ( 2) o, γ, γ, i i, γ i= h = h + h h (34) + γ Z analogous to Eq.(28) in case of the system Marov arameters. Also, similar to Eq.(29) we have the aroriate recursive relationshi for the observer gain Marov arameters searated by more than time stes for each given as o ( 2) o, γ, i i, γ i= h = h h (35) γ >. Therefore to calculate the observer gain Marov arameters we have a similar uer bloc triangular system of linear equations which can be written as ( 2) ( 2) ( 2) m,, 2, +,, 2, ( 2) ( 2) I 0 m h, 2 h h, 2, 2 h +, I h h h h h h 0 0 0 0 I 0 0 h m +, ( 2) ( 2) ( 2),, 2, ( 2) ( 2) 0 h, 2 h, h h h = ( 2) 0 0 h +, (36) to be solved at each time ste. Having outlined a method to comute the observer gain Marov arameters, let us now roceed to loo at the rocedure to extract the observer gain sequence from them. C. Calculation of the Realized Time Varying Observer Gain Sequence From the definition of the observer gain Marov arameters, (recall equation (30)) we can stac the first few arameters in a tall matrix and observe that 4
P + o h +, o h+ 2, : = o h+ m, C G C + + C+ 2 A+ G C+ 2 A+ = = G C A... A G C A... A = + m + m + + m + m + ( m) O+ G (37) such that a least squares solution for the gain matrix at each time ste is given by ( m) + G = O P (38) However from the develoments of the comanion aer, we find that it is, in general, imossible to determine the observability grammian in the true coordinate system[0], as suggested by Eq.(38) above. This is because the comuted observability grammian is, in general in a time varying and unnown coordinate system denoted by, ( m) O + at the time ste t +. We will now show that the gain comuted from this time varying observability grammian is consistent with the time varying coordinates of the lant model comuted by the Time Varying Eigensystem Realization Algorithm (TVERA) resented in the comanion aer. Therefore uon using the comuted observability grammian (in its own time varying coordinate system) and roceeding with the gain calculation as indicated by Eq.(38) above, we arrive at a consistent comuted gain matrix. That is to say that, given a transformation matrix T +, ( m) + = + ( m) = O+ T+ T + G ˆ( m) = O+ T+ G ˆ( m) = O ˆ + G P O G (39) such that ( ) ( ˆ m ) + + + Gˆ = T G = O P (40) therefore, with no exlicit intervention by the analyst, the realized gains are automatically in the right coordinate system for roducing the aroriate time varying OKID closed loo. For consistency, it is often convenient, if one obtains the first few time ste models as included in the develoments of the comanion aer. This automatically gives the observability grammians for the first few time stes to calculate the corresonding observer gain matrix 5
values. To see that the gain sequence comuted by the algorithm is indeed in consistent coordinate systems, recall the identified system, control influence and the measurement sensitivity matrices in the time varying coordinate systems, to be derived as (cf. comanion aer[0]) : Aˆ = T A T + Bˆ = T B Cˆ + = C T (4) The time varying OKID closed loo system matrix, with the realized gain matrix sequence is seen to be consistently given as ( ) Aˆ + Gˆ Cˆ = T A + G C T (42) + in a inematically similar fashion to the true time varying OKID closed loo. The nature of the comuted stabilizing (deadbeat or near deadbeat) gain sequence are best viewed from a reference coordinate system as oosed to the time varying coordinate systems comuted by the algorithm. The rojection based transformations can be used for this urose and are discussed in detail in the comanion aer. V. Relationshi between the Identified Observer and a Kalman Filter We now qualitatively discuss several features of the observer realized from the algorithm resented in the aer. Constructing the closed loo of the observer dynamics, it can be found to be asymtotically stable as urorted at the design stage. Following the develoments of the time invariant OKID aer, we use the well understood time varying Kalman filter theory to mae some intuitive observations. These observations hel us qualitatively address the imortant issue: Variable order GTV-ARX model fitting of inut-outut data what it all means?. Insight is also obtained as to what haens in the resence of measurement noise. In the ractical situation where there is the resence of rocess and measurement noise in the data the GTV-ARX model becomes a moving average model that can be termed as the GTV-ARMAX (Generalized time varying autoregressive moving average with exogenous inut) model (generalized is used to indicate variable order at each time ste). A detailed quantitative examination of this situation is beyond the scoe of the current aer. Hence the authors limit the discussions to qualitative relations. The Kalman filter equations for a truth model given in Eq.(54) of the aendix are given by xˆ = + ˆ A x + + B u (43) 6
or together with the roagated outut equation where the gain xˆ ˆ + = A I K C x + B u + A K y (44) yˆ = C xˆ + D u (45) K is otimal (exression in Eq.(69)). As documented in the standard estimation theory textboos, otimality translates to any one of the equivalent necessary conditions of minimum variance, maximum lielihood, orthogonality or Bayesian schemes. A brief review of the exressions for the otimal gain sequence is derived in the Aendix A which also rovides an insight into the useful notion of orthogonality of the discrete innovations rocess, in addition to deriving an exression for the otimal gain matrix sequence (See Eq. (69) in Aendix A for an exression for the otimal gain). From an inut-outut stand oint, the innovations aroach rovides the most insight for analysis and is used in this section. Using the definition of the innovations rocess ε : = y y ˆ, the measurement equation of the estimator shown in Eq.(45) can be written in favor of the system oututs as given by y = C xˆ + D u + ε (46) Rearranging the state roagation shown in Eq.(43), we arrive at a form given by xˆ = A I K C xˆ + B u + A K y = Aɶ xˆ + Bɶ v + (47) with the definitions Aɶ = A I KC Bɶ = B A K (48) v u = y Notice the structural similarity in the layout of the rearranged equations to the time varying OKID equations in section III. This rearrangement hels in maing comarisons and observations as to what are the conditions in which we actually manage to obtain the Kalman filter gain sequence. Starting from the initial condition, the inut-outut relation of the Kalman filter equations can be written as 7
y0 = C0 xˆ 0 + D0 u0+ ε0 y ˆ = C Aɶ 0 x0 + D u + C Bɶ 0 v0+ ε y = C Aɶ Aɶ xˆ + D u + C Bɶ v + C Aɶ Bɶ v + ε...... 2 2 0 0 2 2 2 2 0 0 2 ˆ = C Aɶ... Aɶ 0 0 + D + hɶ, j j + j= y x u v ε (49) suggesting the general relationshi + ˆ + = C+ A +... A0 0 + D+ + + h ɶ +, + j + j + + j= y ɶ ɶ x u v ε (50) with the Kalman filter Marov arameters h ɶ, i being defined by C Aɶ A ɶ 2... Aɶ i B ɶ + i, > i+ hɶ, i = C Bɶ, = i+ (5) 0, < i+ Comaring the Eqs.(4) and (50) we conclude that their inut-outut reresentations are identical for a suitable choice of (i.e., > ), if G = A K together with the additional condition that ε = 0, >. In the resence of noise in the outut data, the additional requirement is to satisfy the orthogonality (innovations roerty) of the residual sequence, as derived in the Aendix A. Therefore under these conditions, (more secifically the innovations roerty) our algorithm is exected to roduce a gain sequence that is otimal. However, we roceeded to enforce the (in general = was set) term deendence in Eq.(6) using the min additional freedom obtained due to the variability of the time varying observer gains. This enabled us to minimize the number of reeated exeriments and the number of comutations while also arriving at the fastest observer gain sequence owing to the definitions of time varying deadbeat observer notions set u in this aer (following the classical develoments of Minamide et al.[4], and Hostetter[5] discussed in Aendix B). Notice that the Kalman filter equations are in general not truncated to the first ( ) terms. Furthermore, the observer realized using the otimality condition (minimum variance for examle) are seldom the fastest observers. An immediate question arises as to whether we can ever obtain the otimal gain sequence using the truncated reresentation for gain calculation. 8
To answer this question qualitatively, we consider the inut-outut behavior of the true Kalman filter in Eq.(50). Observe that Kalman gains can indeed be constructed so as to obtain matching truncated reresentations as the GTV-ARX (more recisely GTV ARMAX) model as in equation (6) via the aroriate choice of the tuning arameters P 0, Q. In the GTV-ARMAX arlance using a lower order for (at any given time ste) means the incororation of a forgetting factor which in the Kalman filter framewor is tantamount to using larger values for the rocess noise arameter Q (at the same time instant). Therefore, the generalized time varying ARX and ARMA models used for the observer gain sequence and the system Marov arameter sequence in the algorithmic develoments of this aer are intimately tied into the tuning arameters of the Kalman filter and reresent the fundamental balance existing in statistical learning theory between ignorance of the model for the dynamical system and incororation of new information from measurements. Further research is required to develo a more quantitative relation between the observer identified using the develoments of the aer and the time varying Kalman filter gain sequence. VI. Numerical Examle We now detail the roblem of comuting the generalized system Marov arameters from the comuted observer Marov arameters as outlined in the revious sections. Consider the same system as resented in an examle of the comanion aer. It has an oscillatory nature and does not have a stable origin. In case of the time invariant systems, systems of oscillatory nature are characterized by oles on the unit circle and the origin is said to be marginally stable[8, 9]. However, since the system under consideration is not autonomous, the origin is said to be unstable in the sense of Lyaunov[6, 20]. A searate classification has been rovided in the theory of nonlinear systems for systems with origin of this tye. That is called orbital stability or stability in the sense of Poincare (cf. Meirovitch [2]). We follow the convention of Lyaunov and term the system under consideration unstable. In this case the lant system matrix was calculated as 9
A = ex Ac t 0 0 0.2 B =, C =, 0 0 0.5 0 0 D = 0. 0 (52) where the matrix is given by A c 0 I 2 2 2 2 = Kt 02 2 (53) with K t 4+ 3τ = and, 7+ 3τ τ τ are defined as τ sin( 0 t), τ : cos( 0t) = =. The time varying OKID algorithm, as described in the main body of the aer, is now alied to this examle to calculate the system Marov arameters and the observer gain Marov arameters from the simulated reeated exerimental data. The system Marov arameters thus comuted are used by the TVERA algorithm of the comanion aer to realize system model sequence for all the time stes for which exerimental data is available. We demonstrate the comutation of the deadbeat lie observer where the smallest order for the GTV-ARX model is chosen throughout the time history of the identification rocess. Aendix B details the definition of time varying deadbeat observer, for the convenience of the readers along with a reresentative closed-loo sequence result using this examle roblem. Relating to the discussions of the revious section, this imlies that the rocess noise is set very high as the forgetting factor of the GTV-ARX model is imlied to be largest ossible for unique identification of the coefficients. In this case we were able to realize an asymtotically stable closed loo for the observer equation with OKID. In fact two of the closed loo eigenvalues could be assigned to zero at each time ste and there is a certain distribution of closed loo eigensaces such that the roduct of any two consecutive closed loo matrices has all the oles at origin. This time varying deadbeat condition realized is demonstrated using the same examle in the Aendix B. The time history of the oen loo and the closed loo eigenvalues as viewed from the coordinate system of the initial condition resonse decomosition is lotted in the Figure. 20
Figure. Case : Plant Oen Loo Vs OKID Closed Loo Pole Locations ( Minimum No of Reeated Exeriments)\ The error incurred in the identification of the system Marov arameters is in two arts. The system Marov arameters for the significant number of stes in the GTV-ARX model are in general comuted exactly. However, we would still need extra system Marov arameters to assemble the generalized Hanel matrix sequence for the TV ERA algorithm. These are comuted using the recursive relationshis. Since the truncation of the inut-outut relationshi even with the observer in the loo is an aroximation for the time varying case, we incur some error. The worst case error although it is sufficiently small is incurred in the situation when minimum number of exeriments is erformed. This is lotted in Figure 2. The comarison in this figure is made with error in system Marov arameters comuted from the full inut-outut relationshi of the observer (shown to have the same structure as the Kalman Filter in the no noise case). Performing larger number of exeriments in general leads to better accuracy as shown in Figure 3. Note that more exeriments give a better condition number for maing the seudo inverse of the matrix terms (er time ste) in the inut-outut ma. V shown in Eq.(8). The accuracy is also imroved by retaining larger number of 2
Figure 2. Case : Error in System Marov Parameters (Minimum No of Reeated Exeriments = 0) 22
Figure 3. Case 2: Error in Marov Parameters Comutations (Non-Minimum Number of Reeated Exeriments) The error incurred in the system Marov arameter comutation is directly reflected in the outut error between the comuted and true system resonse to test functions. It was found to be of the same order of magnitude (and never greater) in several reresentative situations incororating various test cases. The corresonding outut error lots for Figures 2 and 3 are shown in Figure 4 and Figure 5. 23
Figure 4. Case : Error in Oututs (Minimum No of Reeated Exeriments) Figure 5. Case 2: Error in Oututs for Test Functions (True Vs. Identified Plant Model) 24
Because the considered system is unstable (oscillatory) in nature, the initial condition resonse was used to chec the nature of state-error decay of the system in the resence of the identified observer. The oen loo resonse of the system (with no observer in the loo) and the closed loo state-error resonse including the realized observer are lotted in Figure 6. The lot reresents the errors of convergence of a time varying deadbeat observer to the true states of the system. The comuted states converge to the true states in recisely two time ( min = 2 ) stes to zero resonse. An imortant comment is due at this stage regarding the convergence of the observer in min time stes. Since the observer gain Marov arameters for the first few time stes cannot be calculated (because the free decay exeriments do not yield any information for the gain calculations - cf. comanion aer [0] for details) the corresonding observer gain sequence cannot be determined uniquely. Hence the min time stes in imlementation imlies after the first few time stes this translates to around 2 time stes in most imlementations. In other words the decay of the deadbeat closed loo starts after the correct determination of unique gains from the time min varying OKID rocedure. In the examle roblem, this number 2 = 4 can be clearly seen from the nonzero min outut error time stes in Figure 6. This decay to zero was exonential and too stee to lot for the (time varying) deadbeat case. However when the order was chosen to be slightly higher (near deadbeat observer is realized in this case and therefore it taes more than two stes for the resonse to decay to zero). The gain history of the realized observer as seen in the initial condition coordinate system is lotted as Figure 7. 25
Figure 6. Case : Oen Loo Vs OKID Closed Loo Resonse to Initial Conditions Figure 7. Case : Gain History ( Minimum No of Reeated Exeriments) 26
VII. Conclusion The aer rovides an algorithm for efficient comutation of system Marov arameters for use in time varying system identification. An observer is inserted in the inut outut relations and this leads to effective utilization of the data in comutation of the system Marov arameters. As a byroduct one obtains an observer gain sequence in the same coordinate system as the system models realized by the time varying system identification algorithm. The efficiency of the method in bringing down the number of exeriments and comutations involved is imroved further by truncation of the number of significant terms in the inut-outut descrition of the closed loo observer. In addition to the flexibility achieved in using a time varying ARX model, it is shown that one could indeed use models with variable order. Relationshi with a Kalman filter is detailed from an inut-outut stand oint. It is shown that the flexibility of variable order moving average model realized in the time varying OKID comutations is related to the forgetting factor introduced by the rocess noise tuning arameter of the Kalman filter. The woring of the algorithm is demonstrated using a simle examle roblem. Aendix A Linear Estimators of the Kalman Tye: A Review of the Structure and Proerties We review the structure and roerties of the state estimators for linear discrete time varying dynamical systems (Kalman Filter Theory[22, 23]) using the innovations aroach roounded by Kailath[24] and Mehra[25]. The most commonly used truth model for the linear time varying filtering roblem is given by x = A x + B u +Γ w (54) + together with the measurement equations given by y = C x + D u + υ (55) The rocess noise sequence is assumed to be a Gaussian random sequence with zero mean E( w ) 0, variance sequence ( w w T ) δ,, i j i ij i = i and a E = Q i j having an uncorrelated rofile in time (with itself, as shown by the variance exression) and no correlation with the measurement noise sequence E( w υ T ) 0,, i j = i j. Similarly, the measurement noise sequence is assumed to be a zero mean Gaussian random vector with covariance sequence given 27
by E( υ υ T ) = R δ. We the Kronecer delta is denoted asδ = 0, i j and δ =, i= j along with the usual i j i ij notation E(). for the exectation oerator of random vectors. A tyical estimator of the Kalman tye (otimal) assumes the structure (following the notations of [2]) ij ij + xˆ ˆ ˆ = x + K y y : = xˆ + K ε (56) where the term ε : = y yˆ reresents the so called innovations rocess. In classical estimation theory, this innovations rocess is defined to reresent the new information brought into the estimator dynamics through the measurements made at each time instant. The state transition equations and the corresonding roagated measurements (most often used to comute the innovations rocess) of the estimator are given by xˆ = A xˆ + B u + + = A I K C xˆ + B u + A K y (57) and yˆ = C xˆ + D u (58) Defining the state estimation error to be given by, e : = x x ˆ (for analysis urose), the innovations rocess is related to the state estimation error as ε = C e + υ (59) while the roagation of the estimation error dynamics (estimator in the loo, similar to the time varying OKID develoments of the aer) is governed by e + = A I K C e A K υ +Γ w : = Aɶ e A K υ +Γ w (60) Defining the uncertainty associated by the state estimation rocess, quantified by the covariance to be T P : = E e e, covariance roagation equations are given by T T T T P = Aɶ P Aɶ + A K R K A +Γ + Q Γ (6) Instead of the usual, minimum variance aroach in develoing the Kalman recursions for the discrete time varying linear estimator, let us use the orthogonality of the innovations rocess, a necessary condition for otimality and to obtain the Kalman filter recursions. This roerty is usually called the innovations roerty is the concetual basis 28
for rojection methods[24] in a Hilbert sace setting. As a consequence of this roerty we have the following condition. If the gain in the observer gain is otimal, then the resulting recursions should render the innovations rocess orthogonal (uncorrelated) with resect to all other terms of the sequence. That is to say that for any time ste t i and a time ste t ( i ) i denoted as, > 0 stes behind the ith ste, we have that T E εi ε i = 0 (62) Using the definitions for the innovations rocess and the state estimation error, we use the relationshi between them to arrive at the following exression for the necessary condition that T T T T E ε ε i i = Ci E ei e i Ci + Ci E ei υ i = 0 (63) T T where the two terms E υi e i = E υi υ i = 0 dro out because of the lac of correlation, in lieu of the standard assumtions of the Kalman filter theory. For the case of = 0, it is easy to see that Eq. (63) becomes T T T T E ε ε i i = Ci E ei e i Ci + E υi υ i = C P C + R T i i i i (64) Alying the evolution equation for the estimation error dynamics for time stes bacward in time from t i, we have that e = Aɶ Aɶ... Aɶ Aɶ e Aɶ... Aɶ A K υ +... + Aɶ A K υ + A K υ + Aɶ i... Aɶ i + Γ i wi +... + Aɶ i Γ i 2wi 2+Γi w i i i i 2 i + i i i i + i i i i i 2 i 2 i 2 i i i (65) We obtain exressions for E e e T i i and E ei υt i by oerating equation (65) on both sides with et i and υt, i and taing the exectation oerator T T E ei e i = Aɶ i A ɶ i 2... Aɶ i A ɶ + i E e i ei = Aɶ Aɶ... Aɶ Aɶ P i i 2 i + i i T ( ) E e υ = Aɶ... Aɶ A K E υ υ = Aɶ... Aɶ A K R T i i i i + i i i i i i + i i i (66) (67) Substituting Eqs. (67) and (66) in to the exression for the inner roduct shown in Eq. (63), we arrive at the exressions for Kalman gain sequence as a function of the statistics of the state estimation error dynamics for all time instances u to ti as 29
E ε ε = C Aɶ Aɶ... Aɶ Aɶ P C C Aɶ... Aɶ A K R T = Ci Aɶ i A ɶ i 2... Aɶ i + Aɶ i Pi Ci Ai Ki R i T T i i i i i 2 i + i i i i i i + i i i ɶ ɶ ɶ T i ( i i i i ) T = Ci Ai Ai 2... Ai A + i Pi Ci K R + C P C = 0 (68) which is necessary to hold for all Kalman tye estimators with the familiar udate structure, > 0 T ( ) K = P C R + C P C (69) T i i i i i i i because of the innovations roerty involved. Qualitative relationshi between the identified observer realized from the time varying OKID calculations (GTV-ARX model) and the classical Kalman filter is exlained in the main body of the aer using the innovations roerty of the otimal filter develoed above. Aendix B Time Varying Deadbeat Observers It was shown in the aer that the generalization of the ARX model in the time varying case gives rise to an observer that could be set to a deadbeat condition that has different roerties and structure when comared to its linear time invariant counterart. The toic of extension of the deadbeat observer design to time varying systems has not been ursued aggressively in the literature and only scattered results exist in this context. Paer by Minamide et. al.[4], develos a similar definition of the time varying deadbeat condition and resent an algorithm to systematically assign the observer gain sequence to achieve the generalized condition thus derived. In contrast, through the definition of the time varying ARX model we arrive at this definition quite naturally and we further develo lant models and corresonding deadbeat observer models directly from inut-outut data. First we recall the definition of a deadbeat observer in case of the linear time invariant system and resent a simle examle to illustrate the central ideas. Following the conventions of Juang[] and Kailath[9], if a linear discrete time dynamical system is characterized by the evolution equations given by with the measurement equations (assuming that ( C, A ) is an observable air) x = A x + + B u (70) y = Cx + Du (7) 30
where the usual assumtions on the dimensionality of the state sace are made, x R n, m y R, u R r and C, A, B are matrices of comatible dimensions. Then the gain matrix G is said to roduce a deadbeat observer, if and only if the following condition is satisfied (the so-called deadbeat condition): ( ) [ 0] A + GC = (72) n n where is the smallest integer such that m * n and [ 0] n n is an n n matrix of zeros. D. Examle of a Time Invariant Deadbeat Observer: Let us consider the following simle linear time invariant examle to fix the ideas. 0 A= 2 C = [ 0 ] Now the necessary and sufficient conditions for a deadbeat observer design give rise to a gain matrix G (73) g = g2 such that ( A GC) ( ) ( ) + g g 3+ g 2 2 + = 2 3+ g2 g + 2+ g 2 0 0 = 0 0 (74) giving rise to the gain matrix G = 3 (it is easy to see that = 2 for this roblem). The closed loo can be verified to be given by + = ( A GC) (75) which can be verified to be a singular, defective (reeated roots at the origin) and nilotent matrix. Therefore the deadbeat observer is the fastest observer that could ossibly be achieved, since in the time invariant case, it designs the observer feedbac such that the closed loo oles are laced at the origin. However, it is quite interesting to note that the necessary conditions, albeit redundant nonlinear functions in fact have a solution that exists (one tyically does not have to resort to least squares solutions) since some of the conditions are deendent on each other (not necessarily linear deendence). This nonlinear structure of the necessary conditions to realize a deadbeat observer 3