INRIA Rocquencourt, Le Chesnay Cedex (France) y Dept. of Mathematics, North Carolina State University, Raleigh NC USA

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1 Nonlinear Observer Design using Implicit System Descriptions D. von Wissel, R. Nikoukhah, S. L. Campbell y and F. Delebecque INRIA Rocquencourt, 78 Le Chesnay Cedex (France) y Dept. of Mathematics, North Carolina State University, Raleigh NC USA Abstract Observers are usually formulated as explicit systems of dierential equations and implemented using standard ODE solvers. In this paper, we show that there can be advantages in formulating the observer as a DAE (Dierential-Algebraic Equation). We review the general idea of DAE observer design of [0]. We give two special normal forms for which DAE observer design yields an observer with linear error dynamics. The idea to use DAE observer normal forms is introduced on index one DAE observers and then extended to index two Hessenberg DAEs. This allows us to enlarge the class of nonlinear systems for which linear observer error dynamics can be achieved.. Introduction Consider the nonlinear systems _x f(x u) () y h(x): () where f and h are smooth vector elds on IR n and IR p, and the p measurements y in () are independent. The problem of observer design consist in nding a nonlinear system 0 f(_! b! u y) bx bg(! u y) () that generates an estimate bx(t) of the true value x(t). Denition System () is an observer for System (), if bx(0) x(0) implies bx(t) x(t) for all t>0 and, for bx(0) 6 x(0) we have that bx(t)! x(t) as t goes to innity. The problem of how to design system () to be an observer for () has been extensively studied in the past. There are essentially two approaches. The rst approach is a natural extension of linear observers and is very commonly adopted (for example see the techniques presented in the comparative study of []). The other approach to observer design is to work directly with system equations ()-(), either formulating the estimation problem as a nonlinear algebraic system of equations which must be solved periodically using for example Newton's method (see for example [9]), or formulating it as an optimization problem over some sliding nite horizon which is again solved periodically [8]. In this paper, we present an alternative to these two approaches. We show that there can be advantages in formulating the observer as an implicit (descriptor) system which can then be solved using adae (Dierential-Algebraic Equation) solver. For index one DAEs this can for example be done by the Dierential-Algebraic-System-Solver (DASSL)[, ]. More importantly, if ()-() has a special form and veries some algebraic conditions we can easily construct an DAE index one observer that has linear invariant observer error dynamics. The class of nonlinear systems is even larger if we allow () to be an index two Hessenberg DAE []. Index two Hessenberg DAEs are of particular interest since this type of DAEs can also be safely solved by dierential algebraic system solver (for instance with DASSL with xed stepsize []). The usefulness of this approach will be shown on a simple example. For a more detailed analysis of DAE index two design and its application to mechanical type problems see [].. Index one DAE observer System ()-() is a DAE in x (u and y are supposed to be known). This over-determined (n unknowns, n + p equations) DAE describes all the constraints (information) that we have for constructing ^x. To make this DAE integrable, we can do relaxation by introducing a p-dimensional vector (t) into this DAE. One way to introduce in the algebraic part (the observation ) () _^x f(^x u)+g() () 0 y ; h(^x)+ () which means that we relax the entire estimate bx. It easy to see that introducing as in ()-() leads to the usual explicit formulation of the observer.

2 The other way is to introduce in the dierential part (). In this case we constrain partially the estimate bx through h(bx) by the observation y and relax only the remaining part. This is the way index one DAE observer design is introduced in [0]. Throughout the remainder of this paper we use the following denition. Denition The DAE of the form _bx +(bx) _ f(bx u)+;(bx u) (6) 0 y ; h(bx) (7) is called canonical observer DAE for System ()-(). For ;h x (x) ;@h(x)@x ( ; is any matrix of appropriate dimensions depending on bx and u) we recover the canonical index one DAE observer form of [0]. Note that we obtain the original system equations ()-() for 0. Under appropriate observability conditions, (t)! 0 implies that bx(t)! x(t) for bx(0) 6 x(0). ; has to be chosen such that (t)! 0. To this end we observe that for suciently small observation error ex bx ; x we have I ;(bx) 0 0 _ ex_ fx (bx);(bx u) h x (bx) 0 ex +O(kexk ) (8) (where f System (8) has an equilibrium point at(ex ) (0 0). For xed (bx u) wemaychose ; such that this equilibrium point is stable for all xed (bx u). If (bx u)varies slowly against ( ex) stabilityof (8) for xed (bx u) implies stability for varying (bx u) (see extended linearization []). In this case we may use Ackermann's formula (see for example []) to compute a ;(bx u)... Exact linearization of the error equation If System ()-() is in the special form _x f (x u)+f (x u)x _x f (x u)+f (x u)x (9) y x (we will refer to as index one DAE observer normal form) and we chose ; (y) ; (y) ;(y) ; (y) ; (y) (6)-(7) is an index one DAE (which will be referred to as index one canonical DAE observer form) and (8) becomes the linear varying system ex ; F ; ; ; F ; F ex (0) There exist a number of methodologies to stabilize (0) by use of and ; (extended linearization, Lyapunov design, etc.). Here we consider the case where (0) can be made invariant. For that we need that the matrices F and F have a particular structure. Theorem Let F (bx u)f + e F (bx u) where F is a constant matrix. Then, if. there exists an invertible matrix such that F is constant. F can be chosen such that the matrix pair ff F g is observable and there exists a matrix F such that e F F F the error equation (0) can be made invariant and its modes can arbitrarily be placed by proper choice of; and ; for F. Proof If () has Properties and, the error equation (9) is ex ; F ; ; F ; F ex () Since F is constant and is invertible, () is a linear invariant system for ; ; K and ; K + F ;, where K and K are constant matrices. The modes of () can arbitrarily be set by a proper choice of K and K if the matrix pair 0 F I 0 0 F is observable which is the case if the matrix pair f F F g is observable.. Index two DAE observer The idea to use DAEs as observer for ()-() can be extended to the case where (6)-(7) is in Hessenberg semi explicit index two form. Denition ADAE is in Hessenberg semi explicit index two form if it has the structure _! f(t b!! ) () 0 bg(t! ) where (@bg@! )(@ b f@! ) is nonsingular. We use the following result from [], Theorem.. : Lemma Suppose the nonlinear semi-explicit index two system (), is to be solved numerically be the k-step BDF method (k <7) with xed step size h, the errors in the initial values are ke 0 k O(h k ),and the errors in terminating the Newton iteration satisfy O(h k+ ). Then the k-step BDF method isconvergent and globally accurate to O(h k ), after k +steps. The generalization of DAE index one observe design is for example of interest in the case where ()-() can be put in the form _x x _x f (x x u)+f (x u)x () _x f (x x u)+f (x u)x y x It can easily be veried that two dierentiations of the algebraic equation 0 bg(!) suce to transform () into an ODE. Backward Dierential Formula, see for example [].

3 This particular form is frequently encountered in mechanical type systems (exible joint robots are for example in this form []) and will be refered to as index two DAE observer normal form.the observer is the canonical observer DAE (6)-(7) with 0 ; ; (y) (y) (y) ; 0 ; (y bx u) ; (y bx u) which we will refer to as index two canonical DAE observer form. To insure integrability, we need to show that the index two canonical DAE observer form can be transformed into Hessenberg form. Lemma The change of coordinates! bx! bx ()! bx ; (bx )bx! + (bx )bx transforms () into the Hessenberg semi explicit index two form where _!! _! f b (!!! u)+ F b (!! u)! () _! f b (!!! u)+ F b (!! u)! 0 y ;! bf f ; f ; (; ; ; ) _ ; F + F! + bf f + _ + F! + ; bf F ; F b F F (6) and! ;!. Proof We have _! _ bx ; _ bx ; _ bx (7) _! _ + _ bx + _ bx (8) Elimination of bx _ in (7) by use of (8) and the left multiplication of (8) by ; yields n _! + ; _! _bx + ; _ o ; _ ; ; _ bx n ; _! _bx + ; _ o + ; _ bx The sums in the curly brackets at the right hand side are the left hand side of the of the third and second block row of the index two DAE observer form. Together with () and () we get 0 f b + ; bf + f ; F bx ; ; + 0 ; ; bf + ; _ bx bf + ; bf (bx ; bx ); _ ; ; _ bx bf (bx ; bx ) ; f ; F bx ; ; ; which yield the denitions (6)... Exact linearization of the error equation We proceed like in the index one case: due to the assumption that () is in index two DAE observer normal form (), the error equation is the linear varying system. ex ; F ; ; ; F ; F ex (9) To get a linear error equation we need that F and F have a particular structure. Theorem Let F (bx bx u) F + e F (bx bx u) where F is a constant matrix. Then, if. there exists an invertible matrix such that F is constant. F can be chosen such that the matrix pair ff F g is observable and there exists a matrix F such that e F F F, the error equation (9) can be made invariant and its modes can arbitrarily be placed by proper choice of; and ; for F. Proof The proof is similar to that of Theorem.. Example We use as example the model of a three-phase current motor, which is also used in[] and [0]. We show on this model how to apply index one DAE observer design if ()-() is in index one DAE normal form. Furthermore, we show that DAE index two observer design may yield linear error dynamics even if DAE index one observer design does not. This shows that the extension of DAE observer design to index two DAEs allows us to enlarge the class of nonlinear systems with linearizable error dynamics. We consider the following system. _x x _x B ; A x ; A x sin(x )+ sin(x ) _x u ; D x + D cos(x ) (0) where x (x x x ) T is the state, u a control input and B, A, A, D and D are constants... Index one observer For the index one DAE observer design we use the output y (x x ) T () It can easily be seen that System (0) is in index one DAE normal form (9). We have f (y u) F (y u) x B ; A x + sin(x ) 0 ;A sin(x ) f (y u) u + D cos(x ) F (y u) ;D

4 To see that (0) has for the output () a linear invariant error equation we need the show that F and F verify Theorem. As F (y) F and for 0 0 ; k A sin(x) () (k is a constant scalar) the product F (0 k ) T is a constant matrix, the observer error admits a linear invariant error equation. To see that the poles of the error equation can be placed at arbitrary locations we need to show that the matrix pair ff F g is observable, that is, we need to show that F 6 rank F F. 7 which is the case since F ;D is a constant scalar. The result is consistent with[] where it is shown that System (0) admits a linear invariant error equation for the output (). For ; k 0 0 ; k k A sin(x ) ; (k k ) (0 0) and () we obtain the DAE index one observer _bx bx ; _ + k _bx B ; A bx ; A bx sin(bx )+ sin(bx )+ A sin(bx ) k ( _ ; k ) _bx u ; D bx + D cos(bx )+k + k y bx y bx which has the linear invariant error equation _ex k k k k k ;D ex Now, instead of two observations, we take just one y x () It can easily be seen that for this observation System (0) is still in index one DAE observer normal form (9) but Theorem does not hold any longer. More importantly, the conditions of nonlinear observer design of [] or[7](or[6] for the single output case) applied on System (0) show that System (0) cannot be transformed into nonlinear observer form and, consequently, that it has no linear error equation for the output (). [6] gives sucient conditions for the single output case. Let System (0) dene _x f(x), where x (x x x ) T and y h(x) be the system output. By Proposition of [6] the auxiliary vector eld g(x) dened by 0 0 k n ; g(x) L g L k f(h) k n ; is g(x) T (0 0 ;(A sin(x )) ; ) T. To admit a linear error equation the brackets [g ad (f)g] and [g ad (f)g] must be zero. We obtain that [g ad (f)g] 0,but[g ad (f)g] 6 0 hence, system (0) admits no linear error equation... Index two observer If we use DAE index two observer design for System (0) with the output () we obtain a linear invariant observer error equation (x(),x()) and estimation of (x(),x()) Figure Without perturbation the estimate bx and x (solid line), and bx and x (dotted line) are identical for all t>0. However, bx mayhaveajumpat t 0if bx (0) 6 y(0). bx is impulsive at t 0if bx (0) 6 y(0) and has a jump if bx (0) 6 _y(0). In fact, System (0) is in index two DAE observer normal form (), where f (y _y u) B ; A x + sin(x ) F (y _y u) ;A sin(x ) f (y _y u) D sin(x )+u F (y _y u) ;D Furthermore, F and F verify Theorem since we have F F, the choice ; (A sin(x )) () yields F and the pair ff F g is observable as F D const.. For ; ; A sin(x ), ; and 0theDAE index two observer is _bx bx _bx B ; A bx ; A bx sin(bx )+ sin(bx )+ A sin(x ) ( _ ; ) _bx u ; D bx + D cos(bx )+ y bx which is an index two DAE provide bx 6 k k 0. The error equation is ~x ;D

5 Error: e()x()-xhat() and output perturbation v(t).. Numeric Simulation: For the following computer simulation we use the same values for the model parameters as in [0] and []: Figure The estimation error for x is zero for all t>0 (dotted line). With perturbation (solid line) the estimation error is of the size of the perturbation v Error: e()x()-xhat() and output perturbation v(t) Figure The estimation error for x is zero for all t>0 (dashed line). With perturbation (solid line) the estimation error is the rst derivative of the perturabation v (dotted line). A A B B D D The index two DAE observer includes a sort of numeric dierentiation. To show the impact of a perturbation in the observation we have perturbed the constraint by v(t) 0:00 cos(0t), i.e., the perturbed constraint is0^x ;y+v. The perturbation on bx is obviously v(t) and that on bx is _v(t) ;0:0 cos(0t). If the observation is not perturbed bx jumps immediately to its true value x. If bx (0) 6 y(0) the observation bx is impulsive att 0. The simulation Simulation Result x() and the estimation of x() Figure Independently of the perturbation the estimate bx (dotted line) converges to the true value x (solid line). If the observation is perturbed bx remains in the neighborhood ofx. We note that the index two DAE observer for System (0) is obtained from the index one DAE observer if we eliminate _ in the rst line, set, k 0, k, k k 0 and k. To insure integrability we need to transform the index two DAE into Hessenberg form, that is we needthat the observer () be in the special form (). We have for the observer () Error: e()x()-xhat() _! ;! _! ; b f (!!! u) ; b F (!! u)! _! ; f b (!!! u) ; F b (!! u)! y ;! (bx bx bx )(!!! ) Figure Estimation error on x with (solid line) and without perturbation (dotted line). where b f D sin(! )+u +, bf ; A sin(!) B ;A! + sin(! )+! cos! A sin (!) + bf ;A sin(! ), b F and! +! (A sin(! ). The resulting DAE is in Hessenberg semi explicit index two form, which can safely be integrated by BDF integration schemes with xed stepsize. shows that the additional state in the observer is a relaxation for hidden constraints. The constraint includes two hidden constraints: 0y ; bx () 0 _y ; bx (6) 0 y ; b f (bx bx ) (7)

6 Relaxation Figure 6 Additional state Relaxation with and without perturbed observation Figure 7 Without perturbation (t) goes to zero for growing. If the observation is perturbed stays in the neighborhood of zero. is a measurement of the violation of the hidden constraint (7) and the observation error ex. where f b (bx bx )(B ; A ^x ; A ^x sin(^x )+ sin(^x )+ ; _+; ). The integration by BDF integration schemes assures that the hidden constraint (6) is veried for all t>0. The hidden constraint (7) is relaxed by the supplementary state. The constraint error y ; f b goes asymptoticly to zero if! 0.. Conclusion We have shown that by a slight generalization of index one DAE observer design of [0] and the introduction of a normal form, DAE index one observer design yields, applied on a special class of nonlinear systems (which isdenedby the normal form) observers with linear error dynamics. We have given easy to test conditions for which the resulting error equation can be made, in addition, invariant. If these conditions are veried the construction of an index one DAE observer with linear invariant error dynamics is straightforward. We have shown that the idea of DAE observer design can be extended to the case where the DAE observer is in Hessenberg semi explicit index two form. Wehave given a second normal form for which index two DAE observer design yields linear error dynamics. The conditions for which wehave a linear invariant error equation are similar to that of the index one case. We have shown on a simple example that index two DAE observer design can yield a linear error equation even if index one DAE observer design fails. References [] W. T. Baumann., \Feedback control of multiinput nonlinear systems by extended linearization", IEEE Trans. on Automat. Contr., Vol., No., 988. [] J. Birk and M. Zeitz, \ Extended Luenberger observer for non-linear multivariable systems", Int. J. Contr., Vol 7, No. 6, pp. 8-86, 988. [] K. E. Brenan, S. L. Campbell and L. R. Petzold, \Numerical Solution of Initial-Value Problems in Dierential-Algebraic Equations", SIAM Publications, 99. [] C. W. Gear, \Simultaneous numerical solution of Dierential-Algebraic Equations", IEEE Trans. on Circuit Theory, Vol. 8, No., pp. 89-9, 97. [] T. Kailath, \Linear Systems", Prentice-Hall, 980. [6] A. J. Krener and A. Isidori, \Linearization by output injection and nonlinear observers", Sys. Contr. Letters, No., pp. 7-, 98. [7] A. J. Krener and W. Respondek, \Nonlinear observers with linearizable error dynamics", SIAM J. Contr. Opt., Vol., No., pp. 97-6, 98. [8] H. Michalska and D. Q. Mayne, \Moving horizon observers and observer-based control", IEEE Trans. on Automat. Contr., Vol. 0, No. 6, 99. [9] P. E. Moraal and J. W. Grizzle, \Observer design for nonlinear systems with discrete- measurements", IEEE Trans. on Automat. Contr., Vol. 0, No., 99. [0] R. Nikoukhah, \A new methodology for observer design and implementation", INRIA report, No. 677, 99. [] L. R Petzold, \A description of DASSL: a differential/algebraic system solver", in Scientic Computing, eds. R. S. Stepleman et al., North-Holland, 98. [] D. von Wissel, PhD Thesis, Ecole des Mines de Paris, 996. [] B. L. Walcott, M. J. Corless and S. H. Zak, \Comparative study of non-linear state-observation techniques", Int. J. Contr., Vol., No. 6, 987.

Auxiliary signal design for failure detection in uncertain systems

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