Input selection in observer design for non-uniformly observable systems

Transcription

1 9th IAC Symposim on Nonlinear Control Systems Tolose, rance, September -6, ra. selection in observer design or non-niormly observable systems Gildas Besançon, Ignacio Rbio Scola Didier Georges Control Systems Dept., Gipsa-lab, Grenoble Institte o Technology, Ense BP 6, re des Math. 8 Saint-Martin d Hères, rance ( {gildas.besancon;ignacio.rbioscola;didier.georges}@gipsa-lab.r) Institt Universitaire de rance. Abstract: In this paper the problem o inpts in observer design or systems which are not niormly observable is considered. It is emphasized how it amonts to a control problem, which can be solved in a general way by some appropriate optimization approach. This is illstrated on the basis o a qite general Kalman-like observer orm - possibly with high gain, as well as related simlation reslts on an application example. Keywords: Non-niormly observable systems, inpt condition, optimization, leak detection.. INTRODUCTION The problem o observer design or nonlinear systems has been more and more stdied over the last two decades, bt still remains challenging. In particlar, the reqired observability property or sch an observer design may depend on the applied inpt or a nonlinear system (see eg Besançon (7)), which makes it a speciic problem. The case o observability or any inpt [Gathier and Bornard (98)] and the related amos high gain observer have been very largely stdied since early reslts o Gathier et al. (99); Tornambe (99), while less eorts have been dedicated to systems or which this is not tre. In act, or this latter sitation, appropriate inpts have been well characterized or some classes o systems (eg state aine systems [Bornard et al. (988); Besançon et al. (996)]), bt this characterization is not directly helpl or a practical observer design. Usally in practice, inpts are heristically designed (eg as in Torres et al. (9) or a pipeline application example), and the observer convergence is checked a posteriori. In the present paper, the prpose is to ormally address this problem, at least on the basis o a signiicant class o reslts on observer design relying on appropriate inpt excitation: the main point is to propose a way to bild sch inpts, meaning that nlike in standard control-oriented problems, where the observer is designed or the prpose o applying an inpt to a system, here instead, the inpt is designed or the prpose o its application to an observer. This is obviosly o interest or observer applications which are not control ones - sch as alt detection, or parameter estimation, bt can also be sel or control prposes. The problem is qite natrally related to identiication isses (mch concerned abot excitation), and or preliminary stdy o Rbio-Scola et al. () in that direction was based on a Gramian characterization o the appropriate excitation, and limited to state aine systems. Here instead, the problem is stated directly on some observer eqations aected by the inpt, meaning that the inpt selection amonts to some extent to a control problem, and a direct way to address it is optimization. This observeroriented optimal control problem is settled or a qite general class o nonlinear systems, discssed, and illstrated by its application to a alt-detection example, or which simlation reslts are provided. The paper is ths organized as ollows: section irst ormally presents the problem nder consideration, and then ormlates its soltion as a control optimization one. Section then discsses some isses related to the practical implementation o sch an approach, and section sbseqently proposes an illstrative example, with corresponding simlation reslts. Section 5 inally gathers some conclsions and perspectives.. PROBLEM STATEMENT AND OPTIMAL SOLUTION Let s consider systems o the ollowing orm: ẋ(t) = A((t), v(t))x(t) + (x(t), (t), v(t)) y(t) = Cx(t) where x IR n denotes the state vector, IR m some control inpt vector which can be sed in the observer design, y IR the measred otpt, and v IR q gathering some known signals which can be injected in the observer (possibly inclding t or y(t) or instance). Let s assme that A(, v) a() or any, v, and some smooth a(). Let s rther assme that is globally Lipschitz in x, niormly in, v, with a constant γ. The irst point to be nderlined is that sch a system may a priori admit inpts or which observability is lost (ie is not niormly observable [Besançon (7)]). We also remind the reader o earlier reslts on inpt conditions to garantee a possible observer design, depending on the system strctre (eg as in Besançon and Ţiclea (7); Torres et al. (); Dor et al. ()). () Copyright IAC 66

2 Under sch conditions, an observer can typically be written in the ollowing orm: ˆx(t) = A((t), v(t))ˆx(t) + (ˆx(t), (t), v(t)) + +Λ(λ)P (t)c T (y(t) C ˆx(t)); () P (t) = λ[σp (t) + P (t)a T ((t), v(t)) + P () ; +A((t), v(t))p (t) P (t)c T CP (t)]; () or some appropriate positive tning parameters λ, σ, and related matrix Λ and meaning positive deinite. In the case when (x,, v) = (, v) or instance, λ can be set to, Λ to the identity matrix (I), and one gets the Kalman-like observer (with orgetting actor) [Besançon et al. (996); Ţiclea and Besançon (9)]. I A, satisy the high gain strctre [Gathier et al. (99)], system ()-() can become an observer or () provided that λ is large enogh - and Λ a diagonal matrix with increasing powers o λ as diagonal entries [Besançon and Ţiclea (7)]. It can also be an observer i λ =, Λ = I and σ is chosen large enogh [Dor et al. ()]. In any o those cases, the observer convergence is associated with a Lyapnov nction deined rom P, and the existence o the latter is related to appropriate inpt excitation. Looking then at () as some state eqation driven by inpt, the observer problem trns into a control problem, in the sense that is to be designed so that P remains deined. I sch an inpt exists (corresponding to some observability o the system), it can even be looked or via some optimal control approach, by minimizing P. This indeed will maximize P, and can even be done by taking into accont the inpt energy at the same time, typically via some qadratic cost nction. Some additional constraints on can also be taken into accont, as well as speciic constraints on P i reqired (or observer stability or instance). In this way, the inpt selection amonts to a nonlinear optimal control problem, which allows to rely on available optimization tools in that respect. Notice that this generalizes or ormer approach o [Rbio- Scola et al. ()] on this topic, which was limited to systems () with (x,, v) = (, v), and based on the optimization o some ad-hoc criterion related to observability characterization in terms o Gramian instead o P. In order to smmarize, let s consider that ()-() is an observer or () i the soltion o () admits niorm positive lower and pper bonds and can be sed to characterize the observer stability in a qadratic Lyapnov nction, and let s denote by P the set o sch appropriate positive deinite matrices. Then we can claim the ollowing: Proposition.. Consider a system o the orm (), and assme that or given λ, σ there exists some admissible sch that ()-() is an observer or it (with a related set P), then sch an appropriate inpt can be ond on some time interval [, T ], by solving a problem o the orm: T [ min ρ (s) + P (s) ] ds nder (), P P, and U with U the set o admissible inpts, and ρ >. In the case when (x,, v) = (, v), the constraint on P (set P) redces to an pper bond, since in that case λ can be set to, and it is known that a lower bond on P is garanteed or a σ large enogh whenever a() is bonded (see e.g. Besançon et al. (996)). Hence, the sal stability analysis or the corresponding observer holds (c eg Hammori and de Leon Morales (99)). In the case when depends on x in a γ Lipschitz way, with A, satisying the high gain strctre, one needs to consider an additional constraint on P to ensre the observer stability, which can or instance be o the orm: γ P λσp + I <, (5) with λσ > γ or the easibility o the problem, and P () to be chosen satisying (5). In that case indeed, the lower bond on P is again garanteed by appropriate choice o σ, while an pper bond is garanteed by (5), and one can then consider V (e) = e T Λ P Λ e as a candidate Lyapnov nction or the error dynamics in e := ˆx x. rom direct comptations, it reslts that: V λσv + e T Λ P Λ where = (ˆx,, v) (x,, v). rom this, we can get: V λσv + e T Λ P P Λ e + T Λ and sing the analysis o the high gain observer [Gathier et al. (99)], we obtain: V λσv + e T Λ P P Λ e + γ Λ e. inally, (5) implies that: λσp + P P + γ I < which gives the convergence to zero o the estimation error e by standard Lyapnov argments. Notice that the choice o the constraints on P (that is the stability characterization) may aect the comptational brden o the optimization, bt we present in section an example where the approach is sccessl, and we leave improvements in that respect or tre stdies. In particlar, in the above approach, observer parameters λ and σ are pre-speciied, bt in the spirit o recent reslts on adaptive high-gain observers (Ahrens and Khalil (9); Andrie et al. (9) or Boizot et al. () or instance), one cold instead simltaneosly look or and some related λ, σ.. PRACTICAL IMPLEMENTATION In order to se proposition. in practice, the inpt which is looked or can be approached by piecewise constant nctions, pdated with a period (say T ), which can be chosen rom a trade-o between approximation accracy and reqency constraints on inpt variations. This means that an appropriate inpt seqence can be bilt by solving, or every k : (k+)t [ min k ρ + P (s) ] ds (6) kt nder (), P P, and U () Copyright IAC 665

3 Considering that possible inpt magnitdes may additionally be limited, the set o admissible inpts U may jst be deined by some constraint o the orm i U imax or any inpt i o vector. Abot constraints on P (set P), they will depend on the speciic class o system which is considered as discssed in the previos section. In the same way, the choice o λ depends on the class o system, while σ is to be chosen large enogh with respect to an pper bond on a(). Parameter ρ in the criterion is a sal weighting coeicient, which can be chosen in practice very low so as to emphasize the minimization o P. Coming now to the possible presence o some variable v(t) in the dynamics o P, the implementation o the above optimization procedre reqires that at time kt v(s) be known over s [kt, (k + )T ], or any k. I this is not the case, the variable is to be predicted. I v = y or instance, it can be predicted by a simple zeroorder or irst-order hold. Notice inally that in practice, the observer gain eqation () can also be enhanced with some additional positive (resp. deinite) matrix Q (resp. R), which may be chosen according to the noise level in the state eqation (resp. otpt eqation). This in particlar makes sense in the case o Kalman eqations, when system () redces to a state aine one, and in that case, the proposed inpt selection procedre reslts in improving iltering properties o the observer as well.. ILLUSTRATIVE EXAMPLE In order to illstrate the approach discssed above, let s consider an example o observer application to leak detection in pipelines.this problem indeed has attracted a lot o attention over the last decades (see eg (Billman and Isermann, 98; Brnone and errante, ; Shields et al., ; Verde, 5; Torres et al., 9,...) and reerences therein). We propose here to look at the simple example o a pmp-pipe-tank system recently stdied in Besançon et al. (): the system is made o a pipeline connected to a pmp at one end, and to a water tank at the other end, as depicted by ig.. Also, we assme that the pipe can be sbject to leaks. Pmp H in ig.. Pmp-pipe-tank system. L Q in Q ot Water tank H ot The dynamics o sch a system can simply be described as: d dt H in = c agl Q ot c Hot + c agl agl Q in d dt Q ot = ag L (H ot H in ) (7) Da Q ot Q ot where H in, H ot denote the inpt, otpt pressres (m), Q in and Q ot are the inpt and the otpt low rates in the pipeline (m /s), and denotes the magnitde coeicient o the pipeline leak, i any. Notice that the inpt low Q in in act reslts rom the pmp eect, which can be modeled as a nonlinear nction o the pmp piezometric head H p and the inpt pressre head H in [Besançon et al. ()]: ( ) Q in = θ θ θ B ( A + H in H p ) or constant parameters A, B, θ characterizing the pmp. In this example, the pipe parameters are taken rom an experimental prototype or instance described in [Padilla and Begovich ()], and all nmerical vales are smmarized in table. Table. Constant parameters o the system Letters Vales Units Description a. m Cross sectional area g 9.8 m/s Gravitational acceleration L 85 m Length o the pipe c m/s Wave speed in the lid.89 Nominal riction coeicient D.66 m Diameter o the pipe θ.9 Pmp parameter A 7. Pmp parameter B 7 Pmp parameter Let s then assme that we only measre y = Q ot, and that the otpt pressre H ot can be modiied as an inpt = H ot. Let s also assme that the pmp pressre H p is ixed (H p = 6.5m in the simlations), and known, as well as all other constant parameters, except the riction coeicient: the latter indeed in general depends on the low rate in the pipe [Chadry (979)], which means in particlar that when a leak occrs, it changes. An observer can then be sed to monitor the system and detect leaks taking into accont this riction change, by directly estimating the leak coeicient together with the riction one. To that end, one jst needs to rewrite eqations (7) as a state-space representation, by considering an extended state vector as ollows: x = (Q ot H in ) with approximations =, =. (8) Copyright IAC 666

4 Let s then illstrate observer reslts in two sitations: (i) the case when the inpt low o the pipe (Q in ) is known; (ii) the case when Q in is nknown (bt its model (8) is). x 5.5 In case (i), the state space model can be written in the orm: ẋ = A (, y)x + B (, y, v) y = Cx ag L y y Da with v = Q in and A = c, agl B = ag L c (v y) agl, C = ( ). (9) Hence an observer via classical Kalman-like eqations can be obtained as in ()-(), ie with λ =, Λ = I. In case (ii), the model becomes: ẋ = A (, y)x + B (, y, x) y = Cx with A = A, B = ag L c (Q in (, ) y) agl () and C as beore, with Q in given by (8). In that case, the observer needs to be o some high gain type, either with λ large enogh, as in Besançon and Ţiclea (7), or with λ = and σ large enogh, as in Dor et al. (). In both cases (i) and (ii), it is clear that observability depends on the inpt, and even i one cold ind some candidate inpt via trial and error approach, or point here is to show what can be obtained via the proposed optimization approach. Varios simlation reslts are ths provided hereater, either related to case (i) or to case (ii). In both cases, simlations are started in nominal operation conditions (with a low rate arond. m /s), and a leak eect is added in the system at time t = 5, modiying both and coeicients, while the observer is to recover those vales. In all observers here, σ =., while λ is set to in case (i), and to.5 in case (ii). irst o all, igres and show the obtained estimation errors in case (i) or both states, and both parameters respectively: it can be checked that convergence is indeed achieved. The corresponding inpt obtained rom the optimization procedre can be seen in igre (constrained between. and.7). 5 5 ig.. State estimation errors in case (i). x Estimation o and x 5 x xe xe 5 5 ig.. Parameter estimations in case (i) ig.. in case (i). igres 5 and 6 then show similar reslts or case (ii): the state and parameter estimation can again be checked to be sccessl. The corresponding inpt is presented in igre 7. In order to better evalate the observer perormances, let s also present estimation reslts when the measrement is corrpted by some noise: here a band-limited white noise has been simlated corresponding to magnitdes o a ew percents o the otpt nominal vale. x Copyright IAC 667

5 x 5 5 x ig. 5. State estimation errors in case (ii). Estimation o and. x. xe. 5 5 x 5 ig. 8. State estimation errors in case (i) with noise x 5 Estimation o and x xe x xe x xe 5 5 ig. 6. Parameter estimations in case (ii). 5 5 ig. 9. Parameter estimations in case (i) with noise ig. 7. in case (ii). Notice that observer eqations inclde Q, R matrices in a similar way as in the standard Kalman ormlation: P = λ[σp + P A T (, y) + A(, y)p (t) P C T R CP + Q] with R = 9, Q = I d ig.. in case (i) with noise. orms pretty well. rther improvements can be expected via more adaptive versions - as ormerly mentioned, and this is let or tre stdies. The related inpt is shown in igre. x 5 igres 8 and 9 again show estimation errors or both states and parameters in case (i): it can be seen that the actal vales are indeed recovered in spite o the noise. igre shows the corresponding inpt As or case (ii), similar reslts are presented in igres and, where it can be seen how the high gain still per- ig.. State estimation errors in case (ii) with noise. Copyright IAC 668

6 .. Estimation o and 5 5 x 5 x xe 5 5 ig.. Parameter estimation errors in case (ii) with noise ig.. in case (ii) with noise. 5. CONCLUSIONS AND PERSPECTIVES In this paper, we have emphasized how appropriate inpts or observers can be chosen in a systematic way or systems which are not niormly observable, and the method has been illstrated with a simple example. Reining optimization criterion and procedre will be part o tre stdies, in particlar with the prpose o noise attenation. REERENCES Ahrens, J.H. and Khalil, H.K. (9). High-gain observers in the presence o measrement noise: A switched-gain aaproach. Atomatica, 5(), Andrie, V., Praly, L., and Astoli, A. (9). High gain observers with pdated gain and homogenes correction term. Atomatica, 5(), 8. Besançon, G. (7). Nonlinear Observers and applications. Springer. Besançon, G., Bornard, G., and Hammori, H. (996). Observer synthesis or a class o nonlinear systems. Eropean Jornal o Control,. x xe Besançon, G. and Ţiclea, A. (7). An immersion-based observer design or rank-observable nonlinear systems. IEEE Transactions on Atomatic Control, 5(), Besançon, G., Gillen, M., Dlhoste, J., Santos, R., and Georges, D. (). inite-dierence modeling improvement or alt detection in pipelines. In Proceedings 8th IAC Symp. alt Detection, Spervision and Saety o Technical Processes, SaeProcess. Mexico City, Mex. Billman, L. and Isermann, R. (98). Leak detection methods or pipelines. In Proceeding o the 8th IAC Congresse, Bdapest, Hngary. Boizot, N., Bsvelle, E., and Gathier, J.P. (). An adaptive high-gain observer or nonlinear systems. Atomatica, 6(9), Bornard, G., Coenne, N., and Celle,. (988). LNCIS, chapter Reglarly persistent observers or bilinear systems,. Springer. Brnone, B. and errante, M. (). Detecting leaks in pressrised pipes by means o transients. Jornal o Hydralic Research, 9(5), Chadry, M.H. (979). Applied Hydralic Transients. Van. Dor, P., lila, S., and Hammori, H. (). Observer design or mimo non-niormly observable systems. IEEE Trans. Atomat. Contr., 57(), Gathier, J.P., Hammori, H., and Othman, S. (99). A simple observer or nonlinear systems-applications to bioreactors. IIIE Transactions on Atomatic Control, 7(6), Gathier, J. and Bornard, G. (98). Observability or any (t) o a class o nonlinear systems. IEEE Transactions on Atomatic Control, 6(5), Hammori, H. and de Leon Morales, J. (99). On systems eqivalence and observer synthesis. In G. Conte, A.M. Perdon, and B. Wyman (eds.), New Trends in Systems Theory, volme 7 o Progress in Systems and Control Theory, 7. Birkhäser. Padilla, E.A. and Begovich, O. (). Real-time leak isolation based on a alt model approach algorithm in a water pipeline prototypes. In Proceedings 8th IAC Symp. alt Detection, Spervision and Saety o Technical Processes, SaeProcess. Mexico City, Mex. Rbio-Scola, I., Besançon, G., and Georges, D. (). optimization or observability o state aine systems. In IAC Symp. Syst Strctre and Control, Grenoble, rance. Shields, D., Ashton, S., and Daley, S. (). Design o nonlinear observers or detecting alts in hydralic sbsea pipelines. Control Eng. Practice, 9, 97. Ţiclea, A. and Besançon, G. (9). State and parameter estimation via discrete-time exponential orgetting actor observer. In 5th IAC Symposim on System Identiication. St Malo, rance. Tornambe, A. (99). High-gain observers or non-linear systems. International Jornal o System Science,. Torres, L., Besançon, G., and Georges, D. (9). Mltileak estimator or pipelines based on an orthogonal collocation model. In Proceedings o the 8th IEEE Conerence on Decision and Control. Shanghai, China. Torres, L., Besançon, G., and Georges, D. (). Ek-like observer with stability or a class o nonlinear systems. IEEE Trans. Atomat. Contr., 57(6), Verde, C. (5). Accommodation o mlti-leak location in a pipeline. Control Engineering Practice,, Copyright IAC 669