Observers design for unknown input nonlinear descriptor systems via convex optimization

Transcription

1 Observers design for unknown input nonlinear descriptor systems via convex optimization Damien Koenig To cite this version: Damien Koenig. Observers design for unknown input nonlinear descriptor systems via convex optimization. IEEE Transactions on Automatic Control, Institute of Electrical and Electronics Engineers, 6, AC-5 (6), pp <hal-3594> HAL Id: hal Submitted on 9 Mar 7 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 Observers design for unknown input nonlinear descriptor systems via convex optimization D. KOENIG Abstract This paper treats the design problem of full-order observers for nonlinear descriptor systems with unknown input (UI). Depending on the available knowledge on the UI dynamics, two cases are considered. First, an unknown input proportional observer (UIPO) is proposed when the spectral domain of the UI is unknown. Second, a proportional integral observer (PIO) is proposed when the spectral domain of the UI is in the low frequency range. Sufficient conditions for the existence and stability of such observers are given and proved. Based on the linear matrix inequality (LMI) approach, an algorithm is presented to compute the observer gain matrix that achieves the asymptotic stability objective. An example is included to illustrate the method. Index Terms Lipschitz nonlinear descriptor systems, proportional integral observers, unknown input observers, linear matrix inequalities. I. INTRODUCTION Observer design for linear systems has received great attention in the literature and some extensions have been proposed to the case of unknown inputs 9 and descriptor systems. For physical processes that are described by nonlinear models, three approaches can be distinguished for the design of nonlinear observers. The first one is based on a nonlinear transformation using Lie algebra that brings the system into a canonical form and then uses linear techniques to design state observers. Necessary and sufficient conditions for a nonlinear system to be equivalent to the canonical form have been established in 3 and 4 but this approach necessitates conservative conditions. The ond approach is based on the linearized model. In spite of the local convergence of this method, it is widely used in practice and generally gives better results under less restrictive conditions than the first approach. In 6, the authors have established a necessary condition for the existence of a local exponential observer for nonlinear systems. The third approach treats the observer design problem for a class of nonlinear systems which are composed of a linear part and a vector of nonlinear functions. It was developed by 3, 8 and completed by 7, 7, 5, where sufficient conditions for global stability of the observer were established. Damien Koenig is with Laboratoire d Automatique de Grenoble (UMR CNRS-INPG-UJF), France ( Damien.Koenig@inpg.fr). BP 46, 384 Saint Martin d Hères, Cedex, June 6, 5

3 However, few works have been done to extend the methods mentioned above to the general representation of nonlinear descriptor systems. In and 5, linearization is used to design a state observer for nonlinear descriptor systems without unknown inputs (UI) with application to AC/DC converters. The work presented here considers a general class of nonlinear descriptor systems subject to UI and unknown measurement disturbances where nonlinearities are assumed to be Lipschitz. Before presenting the main results, a brief review of the PIO is presented. PIO are used to attenuate the effect of UI, nonlinearities and uncertain parameters. PIO have been applied in many applications such as robust controller design 3, fault diagnosis 5, loop transfert recovery design 6, parameter estimation, state and fault estimation. In this paper, two rigorous estimation algorithms that are robust to both process and sensor noise are proposed for a class of UI nonlinear descriptor systems. The first one consists in designing a UI observer which gives a perfect UI decoupled state estimation, while the ond one consists in designing a PIO which attenuates the impact of disturbances in the low and high spectral domains. Notation: (.) T is the transpose matrix and ( ) the transconjugate. (.) > denotes symmetric positive definite matrices. σ denotes singular values with σ the smallest and σ the largest singular values. (.) + is the generalized inverse matrix. II. PROBLEM FORMULATION Consider the nonlinear system of the form Eẋ = Ax + Fw + Hφ (x,u,t) () y = Cx + Gw where E may be rank deficient, x R n, u R nu, w R nw, φ : R n R nu R R n and y R m denote respectively the state, the known input, the UI, the nonlinearity and the output vectors. E, A, H R n n, F R n nw,g R m nw and C R m n are known constant matrices. Before giving the main results, let us make the following well-known assumptions. A A A3a A3b A4a A4b The nonlinearity φ(x, u, t) is globally Lipschitz in x with Lipschitz constant γ, i.e., φ(x,u,t) φ(ˆx,u,t) γ x ˆx, u R nu,t R rank F = n w and rank C G = m G E F rank G = n + rank F + rankg G C G rank E = n C rank pe A C rank pe A F G F pi nw C G = n + rank F G = n + rank F G R (p) R (p) June 6, 5

4 3 Remark : ) The system () is singular and is affected by Lipschitz nonlinearity and UI. If we consider the system ( ) T ( ) T Eẋ = Ax + f (x,t) where x = x x, f (x,t) = tx (t).3sin x (t), it is clear that the nonlinear function of this example is not fully Lipschitz due to presence of the term tx (t). However, T the nonlinear function of this example can be expressed as (), where F =, w (t) = tx (t), T H =, φ(x,u,t) =.3sin x (t) and where γ =.3 is the Lipschitz constant. It is thus clear that the class of nonlinear systems considered in this paper is more general than those reported in the literature 7, 7, 5, 5. ) If the system is globally Lipschitz (see the definition in 8), observer proposed produces global convergence of the observer error. The assumption that φ is Lipschitz globally may be relaxed to assume that φ is only locally Lipschitz. All the results in the ensuing tions will then be valid in some local neighborhood around a nominal point. In that case the proposed observer, produces local convergence of the observer error, the region of stability can be computed and its computation is shown in the last tion of 7. 3) Consider the general nonlinear system Eẋ = f (x) + g (x)u + Fw () y = Cx + Gw where f (.), g (.), are continuously differentiable function, with f() =. Let us denote A = f, x= B = g(). Then the given system () can be expanded as () where φ(x,u,t) = Bu + f (x) + g (x)u, H = I n, and where f (x) (resp. g (x)) is obtained from expanding f (x) (resp. g (x)) in a Taylor series about x =. 4) A3a is necessary for the UIPO design while A3b is necessary for the PIO design. More precisely, for F = G =, A3a is equivalent to A3b. For E = I n and G =, A3a is equivalent to the UI decoupled condition needed in the standard UIO 9 (i.e.rank In F = n + rankf rankcf = rankf = nw). For a C full row rank E, A3a is equivalent to the generalized impulse observability (ii) given in. 5) A4a is necessary for the UIPO design while A4b is necessary for the PIO design. More precisely, for F = G =, E = I n, assumption A4a is equivalent to the detectability of the pair(a,c). Assumptions A4a, A4b can often be satisfied, for engineering processes, by a preliminary control. Like in, the measurement y is time integrated (i.e., y I = t ydυ Rm ) in order to attenuate the noise impact in the estimation error (see the discussions in and ). Therefore () is transformed to the restricted system equivalence (r.s.e) Ē x = Ā x + Fw + Hφ (x,u,t) y I = C I x, y = C x + Gw, y = C x + Ğw (3) x June 6, 5

5 where C I = Ē = E m n I m Objectives: m n I m, C = n m, Ā = A C C m m, y T = n m m m, Ğ = m n w G yi T y T, x = x R n+m, F = F y I G H m n, C = C I C and H = ) If any knowledge about the spectral domain of the UI w is given, then under A,A, A3a and A4a the following UIPO is proposed ż = πz + K p y I + K p y + T Hφ (ˆx,u,t) ˆ x = z + N y, ˆx = I n ˆ x where π, K p, K p, T and N are determined such that ˆx asymptotically converge to x for any w and any initial condition (eventually in a given set if it consists of local convergence). ) If the spectral domain of the UI w is in the low frequency range, then under A,A, A3b and A4b the following PIO is proposed. 4, (4) ż = πz + K p y I + K p y + T Fŵ + T Hφ (ˆx,u,t) ( ŵ = K I yi C I ˆ x ) ˆ x = z + N y, ˆx = I n ˆ x (5) where z, x, ˆ x R n+m, ˆx R n, ŵ R nw and π, K p, K p,k I, T, N are unknown matrices which must be determined such that ˆ x, ˆx and ŵ asymptotically converge to x, x and w respectively for any initial condition (eventually in a given set if it consists of local convergence). 3) Find the largest Lipschitz constant γ in the nonlinearity for which the observer (4) or (5) exists for system (3) (r.s.e. ()) 4) Find the observer gain such that the asymptotic convergence to zero of the estimation error is satisfied. III. OBSERVERS DESIGN In this tion, a new method is presented to design both UIPO and PIO for (). June 6, 5

6 5 A. UIPO Before giving the main results, we introduce the following notations to clarify and simplify the presentation: φ = φ(x,u,t) φ(ˆx,u,t) α = Ψ Θ + ϕ e x = x ˆ x R n+m α = Ψ Θ + ϕ U = P Z Ψ = I n+m (n+m) (n+m+nw) χ = α T P β T U T + P α U β + I n+m + χ χ = γ (P α U β ) ( ) α T P β T U T β = ( I n+5m Θ Θ + ) ϕ α = σ ( χ ) β = ( I n+5m Θ Θ + ) ϕ β = σ (P ) Ē Ā F β = σ (P ) C Ğ Θ = C I I n+m Ā H ϕ = m (n+m) ϕ = m n C I m n (n+m) (n+m) (n+m) n Theorem : If ) there exist matrices T, N, K p, π such that TĒ + N C = I n+m (6) π = TĀ K p C I (7) K p = πn (8) NĞ = (9) T F = () ) there exists a solution P, U to the following convex optimization problem max γ subject to P > and (,) γ P α γ U β I n < () June 6, 5

7 6 where (,) = P α U β + α T P β T U T + I n+m then the objectives, 3, 4 hold and the UIPO (4) is a global observer (i.e., asymptotically estimates x for any w and any initial estimate error). Moreover the resulting observer gain Z = P U,ensures that the estimation error is exponentially stable, i.e., e x β V (e x ())exp α β t Proof part ) Suppose that (6) and (9) hold, then the state estimation error e x becomes e x = TĒ x z. In this case, the dynamics of the estimation error e x is described by () ė x = πe x ) + (TĀ πtē K p C Kp C I x ( + T F ) K p Ğ w + T H φ It follows from (6-) that ė x = ( TĀ K pc I ) e x + T H φ (3) Rewriting (3) and (6,7,9,) respectively as ė x = T N K p π + T N K p π T N K p π ϕ e x ϕ φ (4) Θ = Ψ (5) The solution of (5) depends on the rank of matrix Θ. A solution exists if and only if (iff) 9 rank Θ Ψ = rankθ (6) Using relation (6) and the definition of matrix Θ and Ψ, the necessary and sufficient condition for the existence of a solution to equations (6,7,9,) of theorem, or equivalently, to matrix equation (5) is A3a. Therefore, under assumption A3a, the general solution of (5) is T N K p π = Ψ Θ + Z ( In+5m Θ Θ + ) where Z is an arbitrary matrix of appropriate dimension. Substituting (7) into (4) gives (7) ė x = (α Z β )e x + (α Z β ) φ (8) Proof part ) Consider the quadratic Lyapunov function candidate V (e x ) = e T x P e x with P >. The time derivative of V (e x ) along system trajectories of (8) is V (e x ) = e T x ( α T P β T U T ) + P α U β e x +e T x (P α U β ) φ June 6, 5

8 7 From assumption A, we have e T x (P α U β ) φ φ ( α T P β T U T ) e x γ e x ( α T P β T U T ) e x e T x χ e x + e T x e x and thus V (e x ) e T x χ e x. The inequality e T x χ e x < holds for all e x if there exists a solution P, U to the optimization problem defined in Theorem. In addition, since V (e x ) β e x and V (e x ) e T x ( χ )e x α e x then e x β V (e x ) and V (e x ) α β V (e x ) which implies V (e x (t)) < exp α β t V (e x ()). Finally, since β e x V (e x ), we deduce (). Remark : ) The convex, nonlinear inequality χ < is converted to a convex, linear inequality using the Schur complement. Note that for a fixed γ the inequality (), is linear and convex with respect to its variables P and U. ) For a fixed γ, the existence of a solution on P >, U of the LMI () needs that the matrix α Z β is Hurwitz, since the element (,) in () implies P (α Z β ) + (α Z β ) T P <. Let us recall ) that α Z β can be stabilisable iff the pair (α β is detectable. Now we can establish the necessary conditions for the existence of the proposed observer (4). Lemma : The necessary conditions for the existence of the observer (4) for system () are: ) ) A4a which is equivalent to the detectability of the pair (α β is detectable, i.e. ) A3a rank pi n+m α = n + m, R(p) (9) Proof of part ) is done in the appendix while the proof of part ) is done above (see (6)). The following algorithm summarizes the design procedure of the UIPO (4) for system (). β Algorithm : Assume that lemma is satisfied. Solve the convex optimization problem defined in theorem and deduce Z = P U. Matrices T, N, K p, π and K p are computed from (7) and (8) respectively. B. PIO Before giving the main results, we introduce the following notations to clarify and simplify the presentation: A e = TĀ T F Θ = Ē nw (n+m) nw n w C T e = T H nw n K e = K p K I Ψ = I n+m C e = C I m nw U = P K e e w = w ŵ June 6, 5

9 8 e T = e T x e T w χ = P T e T T e P + γ I n+m+nw χ = A T e P + P A e U C e Ce T U T + χ Remark 3: If the spectral domain of the UI w is in low frequency range, a general approach is possible by assuming the disturbance as piecewise constant. See the remarks on PIO design in tion 3. and remark in. Theorem : Under ẇ =, if ) there exist matrices T, N, K p, π such that (6-8) hold ) there exists a solution P, U to the following convex optimization problem max γ subject to P > and (,) γ P T e < () I n (,) = P A e U C e + A T e P C T e U T + I n+m+nw then the objectives, 3, 4 hold and the PIO (4) is a global observer (i.e., asymptotically estimates x and w for any initial estimate error). Moreover, as in the previous tion, the resulting observer gain K e = P U,ensures that the estimation error e (i.e., e x and e w ) is exponentially stable. Proof part ) Suppose that (6) holds, then the state estimation error e x becomes e x = TĒ x z NĞw. The dynamics of the estimation errors e x and e w become respectively since ẇ =. It follows from (6-8) that ė x = πe x + T H φ ( ) + TĀ πtē K p C I K p C x ( + T F ) + πnğ K p Ğ w T Fŵ () ė w = K I C I e x () ė x = ( TĀ K pc I ) e x + T Fe w + T H φ (3) Rewriting (6) as T N Θ = Ψ (4) The solution of (4) depends on the rank of matrix Θ. A solution exists iff 9 rank Θ Ψ = rankθ (5) June 6, 5

10 9 which is obviously equivalent to the assumption A3b. Then, under A3b, the general solution of (5) is T N = Ψ Θ + + Z ( In+3m Θ Θ + ) (6) where Z is an arbitrary matrix, fixed by the designer such that the matrix T is of maximal rank (i.e. n+m, see the discussion in 9). Using the definition of A e,k e,c e,t e and e, the relations (3) and () become ė = (A e K e C e )e + T e φ (7) Proof part ) Consider the quadratic Lyapunov function candidate V (e) = e T P e with P >. From assumption A, the time derivative of V (e) along systems trajectories (7) gives V (e) < e T χ e. Using the Schur complement formula, V (e) < for all e if there exists a solution on P, U to the optimization problem defined in Theorem. Remark 4: For a fixed γ, the existence of a solution on P >, U of the LMI () needs that the matrix A e K e C e is Hurwitz, since the element (,) in () implies P (A e K e C e ) + (A e K e C e ) T P <. Now we can establish the necessary conditions for the existence of the proposed observer (5). Lemma : The necessary conditions for the existence of the observer (5) for system () are: ) A4b which (under rankt = m + n) is equivalent to the detectability of the pair(a e,c e ), i.e., pin+m+nw Ae rank = n + m + nw, R (p) (8) C e ) A3b Proof of part ) is done in the appendix while the proof of part ) is done above (see (5)). Remark 5: The assumption A3b is same as the assumption b) given in 5. A3b can be relaxed to the impulse observability condition i.e., E A rank E = n + ranke (9) C if rank E H = ranke (see remark and proposition in and 5 respectively) or if the nonlinear algebraic constraints obtained after the transformation P T = P T P T can be rewritten with the known inputs 5, i.e., P Hφ (x,u,t) = f (y,u). Obviously A3b and (9) are less restrictive than A3a. The following algorithm summarizes the design procedure of the PIO (5) for system (). Algorithm : Assume that lemma is satisfied. From (6), fixed Z such that the matrix T is of maximal rank (i.e. n+m) and deduce T, N. Solve the convex optimization problem defined in theorem and deduce Matrices π and K p are deduced from (7) and (8) respectively. K p K I = K e. June 6, 5

11 IV. DISCUSSION Since E is singular, system () can be rewritten as E ẋ = A x + F w + Hφ (x,u,t) (3) y = C x E ẋ = A x + Hφ (x,u,t) or (3) y = C x where x = x ζ R n+nw x = x w R n+nw A = A n m m n I m F = F E = G E = E n n w nw n nw n w E p nw C = C I m ) Let Θ 3 = iff 9 E F C m nw A = A F C = C G. An UIPO can be designed for (3) which satisfy constraint T rank Θ3 Ψ = rankθ 3 n + rank F G = rank E F G N Θ 3 = Ψ This implies that ranke = n (i.e. no descriptor system) which is more restrictive compared to A3a. ) An UIPO can be designed for (3) iff rank E = n+n w, or equivalently iff rank E = n+n w, C C G which is also more restrictive than A3a. June 6, 5

12 V. NUMERICAL EXAMPLE The following example illustrates respectively the UIPO (4) and PIO (5) estimation performance. Consider () described by: E = F = C =,A =,,φ = u(t) + γ sin(x 3 ),G = where the Lipschitz constant is γ, fixed to.5 (< γ ). In order to illustrate the robustness of each observer with respect to the noise and UI, we disturb the process by w = (w,w ) T, bu = (bu,bu ) T and by = (by,by ) T which represent respectively the UI, actuator and sensor noises. The components of the actuator and sensor noises are described in fig..b. The known input is u = (u,u ) T where u =.7sin.5t+bu and u = sin.t+bu. A. UIPO The lemma is satisfied for all p, then we can arbitrary fixe the eigenvalues of (α Z β ). For a good estimation performance and a maximal γ solution to the convex optimization problem defined in theorem, we propose to set the eigenvalues of the observer in a specified LMI region D 6. Matrix (α Z β ) has all its eigenvalues in the vertical strip defined by D = {x + jy C : h < x < h },h,h R (3) iff there exists P > and U, such that P α + α T P U β β T U T + h P < P α + α T P U β β T U T + h P > (33) Therefore, the convex optimization problem defined in theorem consists in finding P, U and the maximal γ subject to P >, (33) and (). After some iterations, we find h = 5.5, h =.3 and γ =.49. Due to space limitation, the matrices U, P,Z,T, N, K p, π and K p are omitted. A satisfactory estimation is obtained for any UI and normally distributed random actuator and sensor noises. Fig..a shows that the state is well filtered. June 6, 5

13 B. PIO The lemma 3 is satisfied for all p. In order to compare the estimation performances for both observers, the same LMI region D is defined (with h = 5.5, h =.3). There are several solutions Z and we choose Z = I 6 since it gives both maximal rank T (i.e. n+m = 6) and γ. Matrices T and N are deduced from (6). The convex optimization problem defined in theorem 4 consists in finding P, U and the maximal γ subject to P >, () and P A e + A T e P U C e C T e U T + h P < P A e + A T e P U C e C T e U T + h P > After some iterations, we find γ =.57. Due to space limitation, the matrices U, P,K I,T, N, K p, π and K p are omitted. Contrary to the UIPO, the matrix Z is not optimal. In fact, the designer must test different values for Z until maximal rank T (i.e. n + m = 6) and γ are obtained. Satisfactory estimation is obtained for normally distributed random actuator and sensor noises. The observer gives a good UI estimation and Fig..d and.e show that the state and UI are well filtered. More precisely, the UI attenuation properties can clearly be observed in the bode transfer function (i.e. w to e x ) given in fig..f while the transfer w to ŵ shows that the UI estimation error decreases at low frequencies. Fig..c shows a poor state estimation performance since the impact of the UI w = sin t is not attenuated in this spectral domain (see fig..f) although the UIPO presents a good estimation performance (see fig. a). Obviously, if we increase h and h, we increase the bandwidth but we decrease the maximal Lipschitz constant γ. For example with h =,h = and h = 5, h =, we find respectively γ =.7 and γ =.49. For h > 4 the LMI constraints are infeasible. Example : Consider the system described by 7 where E = I, A =,F = G =,H = I,C = and h = 7, h = 6. The convex optimization problem defined for the UIPO gives γ =.989 although the Rajamani algorithm 7 gives only γ =.49. Example 3: Consider the system described by 5 where H = I 4,F = G =, and h =.5, h =.95, we obtain γ =.393. Then, for γ =.333, our observer (4) is guaranteed to be exponentially stable since the Lipschitz constant is less than γ. VI. CONCLUSION We have presented a rigorous method for the design of observers for nonlinear descriptor systems in presence of UI and noise. Depending on the available knowledge on the dynamics of the UI, two cases were considered. First, for any knowledge about the dynamics of the UI, an UIPO was proposed. Second, for UI with low frequencies, a PIO was proposed. Existence conditions of such observers have been given and proved with a strict LMI formulation. June 6, 5

14 3 VII. APPENDIX Proof of lemma. Define the following nonsingular matrices V, V and the full-column rank matrix V 3 V = In+m I n+m Ψ Θ +, V3 Θ + ϕ I = I n+5m Θ Θ + (n+m+nw) Since V = rank I n+m pi n+m I n+m I n+m I nw pi nw I nw pi n+m Ψ ϕ Θ rank A4a pi n+m Ψ ϕ Θ Θ Θ + n 3m rankg = n + n w (34) V n 3m rankg = n + n w the problem of proving that A4a (9) is equivalent to prove that (34) (9). Proof of (34) (9). From rank Θ Ψ = rankθ A3a, we obtain (34) rankv 3 pi n+m Ψ V n 3m rankg ϕ Θ n w + rank = n + n w, R(p) (9) Proof of lemma. Define the following full rank matrix V 4 = pi m pi n+m Ψ Θ + ϕ ( In+5m Θ Θ + ) ϕ m T N I nw I m Im pi m I m pim where V 4 R (n+m+nw+3m) (n+m+nw+3m), T R (n+m) (n+m), rankt = n+m and rankv 4 = n+m+n w +3m since T N is of full row rank i.e. n + m. In addition, since pē Ā F pi rank nw p C pğ (35) C I = n + m + rank F, R(p) A4b June 6, 5

15 4 the problem of proving that A4b (8) is equivalent to prove that (35) (8). We obtain pē Ā F (35) rankv pi nw 4 p C pğ C I = n + m + n w, R(p) pi TĀ T F + pnğ pi nw C I = n + m + n w, R(p) (8) Note that all the above equivalences hold using the Sylvester s inequality, Ò where à Rn m, B R m p. rankã + rank B m rankã B min rankã, rank B Ó REFERENCES C. Aboki, G. Sallet and J.-C. Vivalda, Obsevers for Lipschitz nonlinear systems, Int. J. Contr., vol. 75, no. 3, pp. 4-,. K. K. Busawon and P. Kabore, Disturbance attenuation using integral observers, Int. J. Contr., vol. 74, no. 6, pp ,. 3 S. R., Beale and B. Shafai, Robust control system design with a proportional integral observer, Int. J. Contr., vol. 5, no., pp. 97-, M. Boutayeb, M. Darouach, H. Rafaralahy and G. Krazakala, Asymptotic observers for a class of non linear singular systems, Proc. of IEEE ACC, Baltimore, Maryland, M. Boutayeb and M. Darouach, Observers design for non linear descriptor systems, Proc. of IEEE CDC, New Orleans, LA, M. Chilali and P. Gahinet, H Design with pole placement constraints: an LMI approach, IEEE Trans. Automat. Contr., vol. 4, no. 3, pp , Y. M. Cho and R. Rajamani, A systematic approach to adaptive observer synthesis for nonlinear systems, IEEE Trans. Automat. Contr., vol. 4, no. 4, pp , S.R. Cou, D. L. Elliot and T.J. Tarn, Exponential observers for non linear systems, Info. Control, vol. 9, pp. 4-6, M. Darouach, M. Zasadzinski and S. J. Xu, Full-order observers for linear systems with unknown inputs, IEEE Trans. Automat. Contr., vol. AC-39, no. 3, pp , 994. M. Darouach, M. Zasadzinski and M. Hayar, Reduced-order observer design for descriptor systems with unknown inputs, IEEE Trans. Automat. Contr., AC-4, no. 7, pp. 68-7, 996. S. Kaprielian and J. Turi, An observer for a non linear descriptor system, proc. of IEEE CDC, Tucson, Arizona, pp , 99. D. Koenig and S. Mammar, Design of Proportional-Integral Observer for Unknown Input Descriptor Systems, IEEE Trans. Automat. Contr., Vol 47, no., pp. 57-6,. 3 A. J. Krener and A. Isidori, Linearization by output injection and non linear observers, System Control Letters, Vol. 3, pp. 47-5, A. J. Krener and W. Respondek, Non linear observers with linearizable error dynamics, Siam. J. Control Optim, Vol. 3, pp. 97-6, B. Marx, D. Koenig and D. Georges, Robust fault diagnosis for linear descriptor systems using proportional integral observers, Proc. of IEEE CDC, pp. 457, 46 Hawaii, USA, 3. June 6, 5

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