On functional observers for linear time-varying systems

Transcription

1 IEEE TRANSATIONS ON AUTOMATI ONTROL SUBMISSION FOR A TEHNIAL NOTE 1 On functional observers for linear time-varying systems F. Rotella I. Zambettakis Abstract The paper deals with existence conditions of a functional observer for linear time-varying systems in the case where the order of the observer is equal to the number of observed variables. onstructive procedures for the design of such a linear functional observer are deduced from the existence conditions. As a specific feature the proposed procedures do not require the solution of a differential Sylvester equation. Some examples illustrate the presented results. Index Terms Linear time-varying system functional observer Luenberger observer. I. INTRODUTION The interest to consider linear time-varying systems is twofold : on the one hand as general models of linear behaviour for a plant on the other hand as linearized models of non linear systems about a given trajectory. For state feedback control or fault diagnosis purposes the need of asymptotic observers of a given linear functional is of primary importance. Therefore we consider the problem of observing a linear functional v(t) = L(t)x(t) (1) where for every time t in R + L(t) is a constant full row (l n) differentiable matrix and x(t) is the n-dimensional state vector of the state space system ẋ(t) = A(t)x(t) + B(t)u(t) y(t) = (t)x(t) where u(t) is the p-dimensional control and y(t) is the m- dimensional output. For every t in R + A(t) B(t) and (t) are known matrices of appropriate dimensions. To avoid tedious counts and distracting lists of differentiability requirements we assume every time-varying matrices are such that all derivatives that appear are continuous for all t. Without loss of generality and in order to (t) avoid useless dynamic parts in the observer we suppose L(t) has full row for all t. Indeed the rows of an arbitrary given L(t) which are linearly dependant of the rows of (t) induce obvious estimation of the corresponding components of v(t) from the available informations. The observability matrix of (2) is defined by Ω(t) = Ω (t) Ω 1(t). Ω n 1(t) where Ω (t) = (t) and Ω j(t) = Ω j 1(t)A(t) + Ω j 1(t) for j = n 1. System (2) or shortly (A(t) (t)) is completely observable if (Ω(t)) = n for some t in R +. It is uniformly observable if (Ω(t)) = n for every t in R Following 8 if r states of (2) are not observable there exists a transformation which induces the following partitions for A(t) and (t) A(t) = A11(t) A 12(t) (n r) r A 22(t) (t) = m r 2(t) F. Rotella and I. Zambettakis are with the Laboratoire de Génie de Production Ecole Nationale d Ingénieurs de Tarbes France rotella@enit.fr izambettakis@iut-tarbes.fr. (2) where (A 22(t) 2(t)) is completely observable. Then the system is detectable when A 11(t) is a Hurwitz matrix. A matrix F (t) is said to be a Hurwitz (convergent in 25) matrix if every solution x(t t x ) of the differential system ẋ(t) = F (t)x(t) x(t ) = x is such that lim t x(t t x ) = for every t and x. Since Luenberger s seminal work 12 a significant amount of research is devoted to the problem of observing a linear functional in a time-invariant setting see for instance and the references therein. Whereas unlike the time-invariant counterpart since 28 there are few papers dealing with the observer design for time-varying systems. The main part of the proposed developments are limited to the case of state observers design (see and the references therein). For the functional observer case we can mention 18 for the design of a single-functional observer while 25 gives a necessary and sufficient existence condition of a linear functional observer. Now it is well known that the observation of v(t) can be carried out with the design of the Luenberger observer ż(t) = F (t)z(t) + G(t)u(t) + H(t)y(t) w(t) = P (t)z(t) + V (t)y(t) where z(t) is a q-dimensional state vector. The time-varying matrices F (t) G(t) H(t) P (t) and V (t) must be determined such that (3) is an asymptotic observer of (1) for the system (2). Namely they have to ensure lim (v(t) w(t)) =. t It has been shown in 25 (theorem 3.9) that the completely observable system ż(t) = F (t)z(t) + G(t)u(t) + H(t)y(t) w(t) = P (t)z(t) is an asymptotic observer for system (2) and linear functional M(t)x(t) if and only if there exists a continuously differentiable matrix T (t) such that G(t) = T (t)b(t) T (t)a(t) F (t)t (t) + T (t) = H(t)(t) (5) and F (t) is a Hurwitz matrix. (3) (4) M(t) = P (t)t (t) (6) So for any matrix V (t) when we choose M(t) = L(t) V (t)(t) the theorem stated in 25 can be read as Theorem 1. The completely observable system (3) is an asymptotic observer of linear functional (1) for system (2) if and only if there exists a continuously differentiable solution T (t) of equations G(t) = T (t)b(t) T (t)a(t) F (t)t (t) + T (t) = H(t)(t) (7) and F (t) is a Hurwitz matrix. L(t) = P (t)t (t) + V (t)(t) (8) Due to linear independence of (t) and L(t) we know that as in the time-invariant case (15 22) l is a lower bound for the order of observer (4). In functional observer literature the words minimal and minimum are both used to denote a lower order stable observer (see for instance 18 26) nevertheless minimal is more frequently used. Moreover all these uses are mainly subject to the restriction that the observer eigenvalues are freely assignable. In

2 IEEE TRANSATIONS ON AUTOMATI ONTROL SUBMISSION FOR A TEHNIAL NOTE 2 22 it has been underlined that minimality must be considered with respect to two different problems depending whether the observer poles are arbitrarily specified at the outset (the fixed poles observer problem) or are permitted to lie anywhere on the left half plane (the stable observer problem). So in this paper we call minimum observer the stable observer for which the order is equal to the size of the functional to be observed while we imply that minimal observer characterizes the other cases. Therefore our main motivation is to analyze the existence conditions and to propose the design of a minimum asymptotic observer ż(t) = F (t)z(t) + G(t)u(t) + H(t)y(t) w(t) = z(t) + V (t)y(t) where w(t) R l is an asymptotic estimate for v(t). When this linear functional observer exists its order namely l according to our notations is much less than the order (n m) of a reduced-order state observer. As a consequence observing a linear function of the state may afford a significant reduction in observer order compared to observing the entire state vector. A second motivation of our purpose is to circumvent the determination of T (t) as solution of the differential equation (7). The material in this paper is organized as follows. The first section deals with the analysis of existence conditions of a minimum observer. The second section is concerned with the proposal of design procedures. In the third section the linear time-invariant case is revisited. Indeed the existence conditions of Darouach 6 for time-invariant systems are based on Hautus-type criteria and can t be directly applied in the time-varying case. Theorem 3 applied to linear time-invariant systems leads to the Darouach conditions which appear then as a particular case of ours. Thus the extension of the Darouach conditions to linear time-varying systems is an additionnal motivation of our paper. Finally two generic examples are detailed. II. EXISTENE ONDITIONS The application of theorem 1 leads to the following result. orollary 2. The asymptotic observer (9) exists if and only if there exists a (l n) matrix T (t) such that for all t. G(t) = T (t)b(t) T (t)a(t) F (t)t (t) + T (t) = H(t)(t) (1) (9) L(t) = T (t) + V (t)(t) (11) F (t) is a Hurwitz matrix Theorem 1 and corollary 2 require the solution T (t) of a differential Sylvester equation in which F (t) is unknown as well as the initial conditions of T (t). In order to overcome this difficulty we propose to eliminate T (t) in conditions of corollary 2. From (11) we get T (t) = L(t) V (t)(t) so from (1) we obtain L(t)A(t) V (t)(t)a(t) F (t)l(t) + F (t)v (t)(t) + L(t) V (t)(t) V (t)ċ(t) = H(t)(t). (12) (12) can be written L(t) + L(t)A(t) = H(t) + V (t) F (t)v (t) F (t) V (t) Σ(t). (14) Theorem 3. The minimum asymptotic observer (9) of linear functional (1) for system (2) exists if and only if the factorization of L(t) + L(t)A(t) L(t) + L(t)A(t) = M(t) M L(t) M 1(t) Σ(t) (15) is such that M (t) M L(t) and M 1(t) have m l and m columns respectively the derivative of M 1(t) exists and M L(t) is a Hurwitz matrix. Proof: The if part follows from (14). The only if part is established by looking for F (t) H(t) V (t) such that in (15) : M (t) = H(t) + V (t) F (t)v (t) M L(t) = F (t) M 1(t) = V (t). It is obvious that if M 1(t) has a derivative the solution of this system is F (t) = M L(t) V (t) = M 1(t) H(t) = M (t) + M L(t)M 1(t) Ṁ1(t). Let T (t) = L(t) M 1(t)(t). This matrix fulfills the differential equation (1) and since F (t) is a Hurwitz matrix this ends the proof. Let us underline here that the existence conditions of theorem 3 do not require the determination of T (t). Nevertheless we can get matrix T (t) as a consequence of factorization (15) without solving equation (1) in the sense of the solution of a differential equation. This standpoint has already been used in 16 to design a minimal order single functional stable observer for linear time-invariant systems. III. MINIMUM ORDER OBSERVER DESIGNS From theorem 3 two observer design methods are described in the following. However the last step of these algorithms needs uniform observability to be assumed. A. Generalized inverse based design The first design method lays on the solution of the linear equation L(t) + L(t)A(t) = X(t)Σ(t) (16) for all t. As the solution X(t) exists if and only if Σ(t) Σ(t) = L(t) + L(t)A(t) equation (16) can be solved by means of time-varying generalized inverses 11. For example a generalized inverse 2 1 Σ {1} (t) for Σ(t) can be obtained from the time-varying singular value factorization of Σ(t) or from its QR factorization 7 4. The solution set of (16) can then be expressed a Denoting Σ(t) = (t) L(t) Ċ(t) + (t)a(t) (13) X(t) = M (t) M L(t) M 1(t) = ( L(t) + L(t)A(t))Σ {1} (t) +W (t)(i n Σ(t)Σ {1} (t))

3 IEEE TRANSATIONS ON AUTOMATI ONTROL SUBMISSION FOR A TEHNIAL NOTE 3 where W (t) is an arbitrary (l n) matrix. When for every t Σ(t) = 2m + l the solution set is reduced to the unique element ( L(t) + L(t)A(t))Σ {1} (t). Otherwise a first design can be considered with the partitions and ( L(t) + L(t)A(t))Σ {1} (t) = S(t) S L(t) S 1(t) I n Σ(t)Σ {1} (t) = R(t) R L(t) R 1(t) where the number of columns is l for S L(t) and for R L(t) while S (t) R (t) S 1(t) and R 1(t) have m columns. It yields F (t) = M L(t) = S L(t) + W (t)r L(t) V (t) = M 1(t) = S 1(t) + W (t)r 1(t) H(t) = M (t) + M L(t)M 1(t) Ṁ1(t) G(t) = (L(t) M 1(t)(t)) B(t). When the pair (S L(t) R L(t)) is uniformly observable W (t) can be chosen via eigenvalues assignement techniques (2 13 3) to yield uniform exponential stability at any desired rate5 17 for the observation error system η(t) = F (t)η(t) where F (t) = S L(t) + W (t)r L(t). When the pair (S L(t) R L(t)) is only detectable we can just obtain W (t) such that F (t) is a Hurwitz matrix. B. Direct design The previous design procedure needs to choose a generalized inverse for Σ(t). To avoid computational burden with respect to this choice we propose to extend the design procedure used recently to determine the minimal order single functional stable observer for linear timeinvariant systems 16. When factorization (15) exists we distinguish two cases. 1) Σ(t) has full row : Namely we are in the case where Σ(t) = 2m + l for every t except for some isolated instants τ. When Σ(t) < 2m + l at some instants t i we have to split the time domain in successive time domains bounded by these isolated values t i of t where the procedure can be applied. Then matrices M (t) M L(t) and M 1(t) are unique. Moreover a derivation of (1) yields v(t) = ( L(t) + L(t)A(t))x(t) + L(t)B(t)u(t) ) = M 1(t) (Ċ(t) + (t)a(t) x(t) +M L(t)L(t)x(t) + M (t)(t)x(t) +L(t)B(t)u(t) = M 1(t)Ċ(t)x(t) + M1(t)(t)ẋ(t) M 1(t)(t)B(t)u(t) + L(t)B(t)u(t) +M L(t)v(t) + M (t)y(t) = M 1(t)ẏ(t) + M L(t)v(t) + M (t)y(t) +(L(t) M 1(t)(t))B(t)u(t). To eliminate the derivative of y(t) we define z(t) = v(t) M 1(t)y(t) or v(t) = z(t) + M 1(t)y(t). (17) Then ż(t) = M L(t)z(t) + (M (t) Ṁ1(t) +M L(t)M 1(t))y(t) (18) +(L(t) M 1(t)(t))B(t)u(t). Since factorization (15) is unique when M L(t) is a Hurwitz matrix the minimum stable observer problem is solved. However the rate of convergence for the uniform exponential stability of the observation error system η(t) = M L(t)η(t) cannot be arbitrarily fixed. 2) Σ < 2m + l: Let us introduce up to a possible permutation in the outputs the partition 1(t) = (19) 2(t) where 1(t)(m 1 n) and 2(t)(m 2 n) verify (t) L(t) 1(t) + 1(t)A(t) = m + l + m 1 = Σ(t). In this case in factorization (15) M (t) M L(t) and M 1(t) are not unique. However we can write Ċ 2(t) + 2(t)A(t) = N (t)(t)+ N L(t)L(t) + N 1(t) 1(t) + 1(t)A(t) (2) where matrices N (t) N L(t) and N 1(t) are unique. Moreover we have ) L(t) + L(t)A(t) = M 1(t) (Ċ(t) + (t)a(t) + M L(t)L(t) + M (t)(t) = M 11(t) 1(t) + 1(t)A(t) + M L(t)L(t) + M (t)(t) + M 12(t) 2(t) + 2(t)A(t) = P 11(t) 1(t) + 1(t)A(t) + P L(t)L(t) + P (t)(t) (21) where matrices P (t) P L(t) and P 11(t) are unique and given by P 11(t) = M 11(t) + M 12(t)N 1(t) P L(t) = M L(t) + M 12(t)N L(t) (22) P (t) = M (t) + M 12(t)N (t). As previously providing that the pair (P L(t) N L(t)) is detectable the stability of the observer can be ensured. Nevertheless when the pair (P L(t) N L(t)) is uniformly observable we can obtain M 12(t) to yield uniform exponential stability at any desired rate for the observation error system η(t) = M L(t)η(t) where M L(t) = P L(t) M 12(t)N L(t). Theorem 4. When in the unique factorizations (21) and (2) the pair (P L(t) N L(t)) is detectable there exists a matrix M 12(t) such that the Luenberger observer (9) with F (t) = P L(t) M 12(t)N L(t) G(t) = (L(t) M 1(t)(t))B(t) H(t) = M (t) + F (t)m 1(t) Ṁ1(t) V (t) = M 1(t) (23)

4 IEEE TRANSATIONS ON AUTOMATI ONTROL SUBMISSION FOR A TEHNIAL NOTE 4 where M 1(t) = P11(t) M 12(t)N 1(t) M 12(t) M (t) = P (t) M 12(t)N (t) is a minimum asymptotic observer of (1) for system (2). IV. THE TIME-INVARIANT ASE REVISITED In the time-invariant case theorem 3 can be read as orollary 5. The time-invariant system ż(t) = F z(t) + Gu(t) + Hy(t) w(t) = z(t) + V y(t) (24) is a minimum asymptotic observer of the linear functional Lx(t) for the state space system if and only if we can write ẋ(t) = Ax(t) + Bu(t) y(t) = x(t) (25) LA = M M L M 1 Σ (26) where M L is a Hurwitz matrix and Σ = L A The necessary and sufficient existence condition proposed in 6 for a minimum asymptotic observer (24) can be written. s Re(s) A L(sI n A) = = L A L A LA. (27) ondition (27) can be split in twofold. On the one hand the existence of the minimum observer structure is ensured with Σ Σ = (28) LA and on the other hand an asymptotic tracking condition is given by s Re(s) A L(sI n A) = Σ. (29) When condition (28) is fulfilled there exist matrices M M 1 and M L such that (26) is satisfied. It yields A L(sI n A) = A sl (M 1A + M LL + M ) = A sl M LL I m = si l M L Σ. ondition (29) can be read I m s Re(s) I m si l M L Σ = Σ I m which means that M L is a Hurwitz matrix. Thus the Darouach result appears as a particular case for linear timeinvariant systems of our main result stated in the linear time-varying case. V. EXAMPLES A. Scalar linear functional example Let us consider the observable single-output system (2) with a 1(t) 1. a 2(t) A(t) = a n(t) B(t) = b 1(t) b 2(t) b n(t) T = 1 where n > 2 and the linear functional (1) defined by L = l 2(t) l n 1(t). (3) Our purpose is to find the conditions in terms of L that ensure the existence of a first-order observer for v(t). With we get Σ(t) = 1 l 2(t) l n 1(t) 1 a n(t) L(t)A(t) + L(t) = l 2(t) + l n 1(t) + l n 2(t) ln 1(t) α(t) where α(t) = n 1 i=1 ai(t)li(t). According to the previous notations we denote X(t) = m (t) m L(t) m 1(t). An examination of factorization (15) leads to express the existence conditions for a first-order observer by the following relationships

5 IEEE TRANSATIONS ON AUTOMATI ONTROL SUBMISSION FOR A TEHNIAL NOTE 5 for i = 1 to n 2 : l i+1(t) + l i(t) = l i(t)m L(t); l n 1(t) = l n 1(t)m L(t)+m 1(t); α(t) = m (t) a n(t)m 1(t). Let us notice that when for every time t in R + k 1 n 1 l k (t) = we get by induction on i k for every time t in R + j k n 1 l j(t) =. Thus when for every time t in R + = we get L = and it does not exist a first-order observer for the linear functional v(t). Assuming does not vanish in R + and there exists some k 2 such that t R + l k (t) = we get m (t) = α(t) and m 1(t) =. Therefore the existence condition for a first-order minimum functional observer is for every t with l 2(t) + = = l k 1(t) + l k 2 (t) l k 2 (t) = l k 1 (t) l k 1 (t) = ml(t) ˆ t lim m t L(τ)dτ =. t Otherwise when none of the coefficients l i(t) vanish we get m (t) = α(t) + a n(t)m 1(t) m 1(t) = l n 1(t) l n 1(t)m L(t) and the existence condition for a first-order minimum functional observer becomes for every t with l 2(t) + = l3(t) + l 2(t) = l 2(t) = ln 1(t) + l n 2(t) l n 2(t) ˆ t lim m t L(τ)dτ =. t = m L(t) Anyway assuming does not vanish in R + and the existence condition for m L(t) is fulfilled it yields T (t) = L(t) m 1(t)(t) = l n 1(t) l n 1(t)m L(t) l n 1(t) g(t) = T (t)b(t) h(t) = m (t) m L(t)m 1(t) ṁ 1(t) = α(t) + a n(t) l n 1(t) l n 1(t)m 2 L(t) + (ṁ 1(t) a n(t)l n 1(t)) m L(t) l n 1(t) l n 1(t)ṁ L(t). Parametrized with m L(t) the asymptotic observer is given by ż(t) = m L(t)z(t) + g(t)u(t) + h(t)y(t) ) w(t) = z(t) + ( ln 1(t) l n 1(t)m L(t) y(t). These results allow to detect the scalar functional for a single-input single-output system that can be observed with a first-order system. B. Multiple linear functional example Let us consider the observable single-output system (2) with a 1(t) b 1(t) A(t) = 1 a 2(t) 1 a 3(t) B(t) = b 2(t) b 3(t) 1 a 4(t) b 4(t) = 1 and linear functional (1) defined by l1(t) L(t) = l 2(t) with for every t l 2(t). (31) Following the given procedure we get the nonsingular matrix 1 Σ(t) = l 2(t). 1 a 4(t) As and we obtain L(t)A(t) + L(t) = Σ(t) = (L(t)A(t) + L(t))Σ 1 (t) = l1(t) a 1(t) l 2(t) l2(t) a 2(t)l 2(t) 1 l 2(t) 1 a 4(t) a 1(t) a 2(t)l 2(t) l 2(t) l 2(t) l 2(t). onsequently matrices M (t) M L(t) and M 1(t) can be read as a1(t) M (t) = M a 1(t) = 2(t)l 2(t) M L(t) = l 2(t) l 2(t). l 2(t) Let us suppose that the system ż(t) = M L(t)z(t) is uniformly asymptotically stable (for instance if we have = exp( t) and l 2(t) = exp( 2t) the stability conditions are fulfilled) we can implement following (18) the second-order observer for L(t)x(t) ż(t) = M L(t)z(t) + M (t)y(t) + L(t)B(t)u(t). v(t) = z(t). VI. ONLUSION In the time-varying case we propose existence conditions for a minimum functional linear observer. When it exists we describe a design procedure. Our algorithm uses two unique matrix factorizations based on linearly independent rows of a time-varying matrix. With respect to other procedures 18 our design method is carried out without needing to solve a differential Sylvester equation. Moreover the proposed algorithm points out whether we can fix

6 IEEE TRANSATIONS ON AUTOMATI ONTROL SUBMISSION FOR A TEHNIAL NOTE 6 at any desired rate the convergence of the observation error. In addition if there exists a Lyapunov transform P (t) 5 such that F (t) = P (t)φp (t) 1 + P (t)p (t) 1 where Φ is a constant Hurwitz matrix this can be performed by means of eigenvalues of Φ. The main result can be considered as an extension to the time-varying case of a necessary and sufficient existence condition of a minimum functional observer established previously in the time-invariant case 6. Since such a minimum functional observer may not exist for a given linear functional the presented results as underlined in 9 for timeinvariant systems can be the basis of an algorithm for the design of a minimal-order observer. This development will be the subject of a next paper. 24 Trinh H. Fernando T. Functional Observers for Dynamical Systems Springer Trumpf J. Observers for Linear Time-Varying Systems Linear Algebra and Appl. vol. 425 pp Tsui.. What is the Minimum Function Observer Order J. Flin Inst. vol. 335B n. 4 pp Tsui.. An Overview of the Applications and Solution of a Fundamental Matrix Equation Pair Jour. of Flin Institute vol. 341 pp Wolowich W.A. On the State Estimation of Observable Systems Proc Joint Automat. ontr. onf. pp Yüksel Y.Ö Bongiorno J.J. Observers for Linear Multivariable Systems with Applications IEEE Trans. Aut. ontrol A-16 n. 6 pp Zhang Q. Adaptive Observer for Multiple-Input-Multiple-Output (MIMO) Linear Time-Varying Systems IEEE Trans. Aut. ontrol vol. A-47 n. 3 pp REFERENES 1 Aldeen M. Trinh H. Reduced-Order Linear Functional Observers for Linear Systems IEE proc. control theory appl. vol. 146 n. 5 pp Ben-Israel A. Greville T.N.E. Generalized Inverses : Theory and Applications John Wiley & Sons hai W. Loh N.K. Hu H. Observer Design for Time-Varying Systems Int. J. Systems Sci. vol. 22 n. 7 pp iampa M. Volpi A. A Note on Smooth Matrices of onstant Rank Rend. Istit. Mat. Univ. Trieste vol. XXXVII pp hen.-t. Linear System Theory and Design Holt Rinehart and Winston 2nd ed Darouach M. Existence and Design of Functional Observers for Linear Systems IEEE Trans. Aut. ontrol A 45 n. 5 pp Dieci L. Eirola T. On Smooth Decompositions of Matrices SIAM J. Matrix Anal. Appl. vol. 2 n. 3 pp D Angelo H. Linear Time-Varying Systems Allyn and Bacon Fernando T. Trinh H. Jennings L. Functional Observability and the Design of Minimum Order Linear Functional Observers for Linear Systems IEEE Trans. Aut. ontrol A-55 n. 5 pp Lovass-Nagy V. Miller R.J. Powers D.L. An Introduction to the Application of the Simplest Matrix-Generalized Inverse in Systems Science IEEE Trans. ircuits and Systems vol. AS 25 n. 9 pp Lovass-Nagy V. Miller R.J. Mukundan R. On the Application of Matrix Generalized Inverses to the Design of Observers for Time- Varying and Time-Invariant Linear Systems IEEE Trans. Aut. ontrol vol. A-25 n. 6 pp Luenberger D.G. Observers for Multivariable Systems IEEE Trans. Aut. ontrol A-11 n. 2 pp Nguyen. Lee T.N. Design of a State Estimator for a lass of Time- Varying Multivariable Systems IEEE Trans. Aut. ontrol A 3 n. 2 pp O Reilly J. Observers for Linear Systems Academic Press Roman J.R. Bullock T.E. Design of Minimal Orders Stable Observers for Linear Functions of the State via Realization Theory IEEE Trans. Aut. ontrol vol. A 2 n. 5 pp Rotella F. Zambettakis I. Minimal Single Linear Functional Observers for Linear Systems Automatica vol. 47 n. 1 pp Rugh W.J. Linear System Theory Prentice Hall 2nd ed Shafai B. arroll R.L. Minimal-Order Observer Designs for Linear Time-Varying Multivariable Systems IEEE Trans. Aut. ontrol vol. A- 31 n. 8 pp Shafai B. Design of Single-Functional Observers for Linear Time- Varying Multivariable Systems Int. J. Systems Sci. vol. 2 n. 11 pp Silverman L.M. Transformation of Time-variable Systems to anonical (phase-variable) Form IEEE Trans. Aut. ontrol A-11 pp Silverman L.M. Meadows H.E. ontrollability and Observability in Time-variable Linear Systems J. SIAM. ontrol vol. 5 n. 1 pp Sirisena H.R. Minimal Order Observers for Linear Functions of a State Vector Int. Jour. of ontrol vol. 29 n.2 pp Tian Y. Floquet T. Perruquetti W. Fast State Estimation in Linear Time-Varying Systems : an Algebraic Approach Proc. D Mexico 28.