TRANSACTIONS ON AUTOMATIC CONTROL 1 A note on functional oservaility Frédéric Rotella Irène Zamettakis Astract In this note we propose an alternative to characterize the functional oservaility for linear systems The main feature is that we otain a necessary and sufficient condition for the existence of a stale multi-functional oserver of a time-invariant linear system The proof of this condition is constructive and it leads to design a stale oserver via a new procedure neither ased on the solution of a Sylvester equation nor on the use of canonical state space forms Index Terms Linear systems oserver functional oserver I INTRODUCTION SINCE Luenerger s works 21 22 23 a significant amount of research has een devoted to the prolem of oserving a linear functional of the state of a linear time-invariant system The main developments are detailed in 24 and in the recent ooks 18 31 and the reference therein The prolem can e formulated as follows For the linear state-space model ẋ(t) = Ax(t) Bu(t) y(t) = Cx(t) where for every time t in R x(t) is the n-dimensional state vector u(t) is the p-dimensional input y(t) is the m-dimensional measured output and A B and C are constant matrices of adapted dimensions the ojective is to get v(t) = Lx(t) (2) where L is a constant (l n) matrix The oservation of v(t) can e carried out with the design of a Luenerger oserver ż(t) = F z(t) Gu(t) Hy(t) w(t) = P z(t) V y(t) where z(t) is the q-dimensional state vector Constant matrices F G H P and V are determined such that lim (v(t) w(t)) = 0 t We know from 10 11 that the oservale linear functional oserver (3) exists if and only if there exists a (q n) matrix T such that G = T B and (1) (3) T A F T = HC (4) L = P T V C (5) where F is a Hurwitz matrix Namely when all the real parts of the eigenvalues of F are strictly negative When these conditions are fulfilled we have lim t (z(t) T x(t)) = 0 Moreover it is well known from 26 and 29 that the order q of the multi-functional oserver is such that q (L) and when the model (1) is detectale q < nm Indeed nm is the order of the reduced-order oserver or Cumming-Gopinath oserver 3 14 which can e uilt to oserve x(t) and consequently v(t) Among the oservers we can distinguish the minimum-order or Darouach oserver 4 where Frédéric Rotella and Irène Zamettakis are with the Laoratoire de Génie de Production 47 avenue d Azereix 65016 Tares CEDEX France e-mail: rotella@enitfrizamettakis@iut-taresfr q = l and P = I q It has een shown in 28 that the minimum-order oserver exists if and only if there exists a triplet (F M N) such that LA = F L MC NCA where F is a Hurwitz matrix In all the following we use the shorthand notation N NM O (MNk) = NM where N and M are matrices with adapted dimensions and k is an integer Recently to cope with the design prolem of a minimal order functional oserver the interesting notion of functional oservaility of the triplet (A C L) which sums up the prolem to solve has een defined in 7 6 8 Definition 1 The triplet (A C L) is functionally oservale if there exists a matrix R such that a Darouach oserver exists for the linear functional R v(t) = x(t) L Some iterative procedures have een proposed in 7 6 8 to cope with the intriguing and challenging prolem (31) to find R which leads to the minimum-order oserver A recent result ased on matrix decompositions and canonical forms to design a minimal order oserver with fixed eigenvalues at the outside is descried in 9 Nevertheless it has een proposed in (7) that the triplet (A C L) is functionally oservale if and only if ( ) O(ACn) = ( ) O (ACn) (6) O (ALn) Oviously when condition (6) is fulfilled there exist matrices L 0 L 1 L n1 such that n L = L ica i (7) Thus ( O(ACn) L ) = ( ) O (ACn) (8) Conversely let us suppose that L can e written as (7) Thus it is easy to prove y induction that for every k in {1 n 1 there exist matrices L ki such that n LA k = L ki CA i These relationships lead to (6) So we can claim the triplet (A C L) is functionally oservale if and only if (8) is fulfilled From (7) we can relate the functional oservaility notion to other oservaility notions Indeed from (1) we get for i = 0 1 2 i1 y (i) (t) = CA i x(t) CA j Bu (i1j) (t) 0000 0000/00$0000 c 2014 IEEE
TRANSACTIONS ON AUTOMATIC CONTROL 2 Thus from (7) we can write : v(t) = n1 n1 L iy (i) (t) CA j Bu (i1j) (t) L i i1 Consequently v(t) is oservale in the Fliess-Diop meaning 5 Nevertheless our aim is to propose another criterion to test functional oservaility of a triplet (A C L) which leads to a constructive procedure of functional oserver Consequently the paper is organized as follows In a first part we show that the existence of an integer matrices F L0 F L and matrices F C0 F C such that LA = F LiLA i F CiCA i (9) leads through realization theory to the design of a candidate functional oserver The proof of the sufficiency of the condition (9) is completed with the exhiition of the analytical expression of the matrix T solution of the equations (4) and (5) Let us insist here that the determination of T is not a necessary step in the design of the oserver In a second part we show that this condition is necessary as well A last part is devoted to a staility condition for the otained oserver structure II SUFFICIENCY Let us suppose here that (9) is fulfilled As we have for k = 0 1 we can write v (k) (t) = LA k x(t) LA i Bu (i) (t) v () (t) = F LiLA i x(t) A Oserver structure design F CiCA i x(t) LA i Bu (i) (t) (10) Firstly the elimination of x(t) in (10) is carried out y means of for i = 1 to 1 and for i = 1 to We get then i1 LA i x(t) = v (i) (t) LA i1j Bu (j) (t); (11) i1 CA i x(t) = y (i) (t) CA i1j Bu (j) (t) (12) v () (t) = F Liv (i) (t) F Ciy (i) (t) G iu (i) (t) (13) where the matrices G i are given y G = (L F CC) B and for 2 and j = 0 to 2 ( G j = LA j 1 1 F LiLA i1j F CiCA i1j ) B (14) Remark 2 When = 0 there exists a matrix Λ such that L = ΛC So the functional oserver ecomes w(t) = Λy(t) The case = 1 has een detailed in 27 and leads to G = (L F C1C) B Secondly the differential equation (13) is realized through the wellknown Ruffini-Horner procedure 17 Namely we write (13) as v(t) = F Cy(t) p 1 F Lv(t) F Cy(t) G u(t) p 1 F L1v(t) F C1y(t) G 1u(t) p 1 F L0v(t) F C0y(t) G 0u(t) where p stands for the continuous-time derivative operator and p 1 for the continuous-time integrator With z 0(t) = p 1 F L0v(t) F C0y(t) G 0u(t) z i(t) = p 1 F Liv(t) F Ciy(t) G iu(t) z i1(t) for i = 1 to 1 and v(t) = z (t) F Cy(t) we otain ż 0(t) = F L0z (t) H C0y(t) G 0u(t) ż 1(t) = F L1z (t) H C1y(t) G 1u(t) z 0(t) ż (t) = F Lz (t) H Cy(t) G u(t) z 2(t) where for i = 0 to 1 H Ci = F Ci F LiF C The vector z(t) = z 0 (t) z (t) is the state of the Luenerger oserver structure (3) with F L0 G 0 F L1 G 1 F = G = G 2 G H = F C0 F C1 F C2 F C F L2 F L F L0 F L1 F L2 F L F C P = 0 0 V = FC Remark 3 Notice that the realization (15) is oservale (15) Remark 4 In the case = 1 the Darouach-Luenerger oserver structure is given y 28 F = F L0 G = (L F C1C) B V = F C1 P = H = F C0 F L0F C1 B An expression for T (16) In order to complete the proof we otain here the expression of the matrix T Let us egin with the case = 1 We claim T = L F C1C Indeed from (16) we have T A F T = LA F C1CA F L0L F L0F C1C = (F C0 F L0F C1) C = HC and L = T F C1C = P T V C Now consider the case 2 Firstly let us remark that the relationship G = T B with (14) leads to the induction T = T1 T
TRANSACTIONS ON AUTOMATIC CONTROL 3 where for j = 1 to 1 T j = LA j F LiLA ij F CiCA ij and T = L F CC In the following we state that this matrix T is a solution of T A F T = HC where F and H are defined in (15) Let us denote T A = (T A) 1 (T A) and F T = (F T ) 1 (F T ) where for j = 1 to the locks (T A) j and (F T ) j have l rows On the one hand we have (T A) = LA F CCA and for j = 1 to 1 (T A) j = LA j1 F LiLA ij1 F CiCA ij1 On the other hand we have (F T ) 1 = F L0L F L0F CC and for j = 2 to (F T ) j = T j1 F Lj1T = LA 1j 1 1 For j = 2 to 1 (17) can e written as F LiLA i1j (17) F CiCA i1j F Lj1L F Lj1F CC (F T ) j = LA 1j F LiLA i1j and for j = (F Cj1 F Lj1F C) C F CiCA i1j (F T ) = LA F CCA (F C F LF C) C Let us remark that (9) leads to write LA i=1 FLiLAi i=1 FCiCAi = FLiLAi FCiCAi i=1 FLiLAi = F L0L F C0C i=1 FCiCAi Thus after some calculations (T A) j (F T ) j can e read for j = 1 to as (F Cj1 F Lj1F C) C Taking into account that for j = 0 to 1 H Cj = F Cj F LjF C we deduce that T fulfills the Sylvester equation T A F T = HC Moreover as P = 0 0 and V = FC we are led to which ends the proof P T V C = L F CC F CC = L We can then deduce the following lemma Lemma 5 If there exist an integer and matrices F L0 F L and F C0 F C such that v1 LA = F LiLA i F CiCA i then a solution (T F H P V ) of the equations T A F T = HC and P T V C = L is given y (15) and F L1 F L2 F L F L2 T = O (AL) F L F C1 F C2 F C F C F C2 O (AC) F C F C F C III NECESSITY Lemma 6 Let us suppose that the q-order asymptotic oserver (3) of Lx(t) for the system (1) is oservale then there exist matrices F Li and F Ci i = 0 to q 1 and F Cq such that LA q = F LiLA i q F CiCA i (18) Proof : From 10 11 when the linear multi-functional oserver (3) of Lx(t) exists then there exists T such that (4) and (5) are fulfilled On the one hand from (5) we can write for k N LA k = P T A k V CA k On the other hand writing (4) as T A = F T HC we can easily deduce y induction that for k N {0 T A k = F k T F i HCA i Consequently we otain for k N {0 LA k = P F k T P F i HCA i V CA k (19) Gathering (5) and the previous expressions for k = 1 to q 1 we are led to O (ALq) = O (FPq) T ΠO (ACq) (20) where Π is the matrix 0 P Π = (I q V ) P F P (I q H) P F q2 P F P 0 and stands for the Kronecker product of two matrices 15 20 As the oserver (3) is oservale we have O (FP ) = q Thus the matrix T defined y (20) is unique and is given y T = O 1 (FPq) { O(ALq) ΠO (ACq) where O (FPq) 1 stands for an aritrary generalized inverse of the oservaility matrix namely 1 O 1 (FPq) { X O (FPq) XO (FPq) = O (FPq)
TRANSACTIONS ON AUTOMATIC CONTROL 4 Consequently there exist matrices T Li and T Ci i = 0 to q 1 such that T = T LiLA i T CiCA i Let us remark that (19) gives for k = q LA q = P F q T P F i HCA i V CA q { = P F q T LiLA i T CiCA i P F i HCA i V CA q Thus there exist matrices F Li and F Ci i = 0 to q 1 and F Cq such that q LA q = F LiLA i F CiCA i which concludes the proof IV A STABILITY CONDITION The previous sections concern the design of a candidate oserver for the linear functional (2) The final step consists in finding staility conditions for F defined in (15) to ensure an asymptotic oservation A The solution set Let us consider the matrix Σ = O (AL) O (AC1) (21) The existence condition of an integer and matrices F L0 F L and F C0 F C such that (9) is fulfilled is equivalent to the consistency condition of the linear equation LA = ΦΣ (22) Namely the integer is such that ( ) LA = (Σ ) Σ From 1 when this condition is verified the solution set for the equation (22) can e written FL0 F L1 F L F C0 F C1 F C = LA Σ 1 ( Ω I ρ Σ Σ 1 ) (23) where ρ = m (m l) Ω is an aritrary (l ρ) matrix and Σ 1 is an aritrary generalized inverse of Σ Remark 7 If (Σ ) = ρ the solution LA Σ 1 is unique and independent of a particular choice for Σ 1 Remark 8 In the case where (Σ ) = r < ρ the numer of degrees of freedom for the design is reduced to the dimension of the co- of the matrix Σ namely ρ r B A stailizaility condition In the case where (Σ ) = ρ we can test if F is a Hurwitz matrix through an eigenvalues inspection or y using the Routh- Hurwitz criterion Let us suppose now that (Σ ) = r < ρ and consider the SVD decomposition 13 12 of Σ Σ = U S V where U (ρ ρ) and V (n n) are unitary matrices and S is the (ρ n)-sized diagonal matrix of the ordered singular values σ 1 σ r > 0 A particular choice for Σ 1 S = diag {σ 1 σ r 0 0 Σ 1 can e = Σ = V S U (24) where S = diag { σ 1 1 σ1 r 0 0 Thus we are led to I ρ Σ Σ Ir 0 = I ρ U U 0 0 = U 0 0 U 0 I ρr Remark 9 Two reasons motivate the proposed choice (24) for Σ 1 Firstly the pseudo-inverse Σ 1 = Σ of Σ is unique Secondly the SVD decomposition is numerically roust Let us define U2 as the matrix uilt with the ρ r last rows of U U 2 = U (: r 1 : ρ) = Υ L0 Υ L1 Υ L Υ C0 Υ C Γ 2 the ρ r last columns of the aritrary matrix Γ = ΩU and Φ = F L0 F L1 F L F C0 F C = LA Σ The previous partitions lead to the structure of F defined in (15) where for i = 0 to 1 F Li = FLi Γ 2Υ Li Due to commutativity with respect to the lock-column partition of the matrix F its eigenvalues are identical to the eigenvalues of the matrix F = F L0 F L1 F L2 F L The interest in considering F instead of F is that we have the following decomposition F = FL0 FL1 FL2 FL 0 0 Γ2 ΥL0 Υ L1 Υ L2 Υ L We can now state the following test
TRANSACTIONS ON AUTOMATIC CONTROL 5 Lemma 10 There exists a matrix Ω such that F defined in (15) is an Hurwitz matrix if and only if the system 0 η(t) = η(t) 0 ϖ(t) FL0 FL1 FL ς(t) = Υ L0 Υ L1 Υ L η(t) (25) is static output feedack stailizale Consequently any well-known static output feedack stailizaility criteria (see 19 30 2) can e used here Moreover when the system (25) is stailizale with a static output feedack we can apply for instance LMI ased methods 25 32 33 or software uiltin procedures to get a matrix Γ 2 which solves the prolem In the opposite when such a matrix cannot e found has to e increased up to a value such that the static output stailizaility prolem can e solved V CONCLUSION All these results can e summed up in the following theorem which provides a test of functional oservaility of a linear functional with respect to a given linear time-invariant system Theorem 11 The triplet (A C L) is functionally oservale if and only if there exists an integer such that ( ) LA = (Σ ) where Σ Σ = O (AL) O (AC1) has the singular value decomposition Σ = U S V and the system (25) where the essential matrices are defined y ΥL0 Υ L1 Υ L Υ C0 Υ C1 Υ C F L0 FL1 = U (r 1 : ρ :) FL FC0 FC1 FC = LA Σ is static output feedack stailizale When this theorem is fulfilled the previous sections indicate a design procedure which leads to a l-order stale Luenerger oserver When is minimal and (T ) = q < l keeping in T the linearly independent rows and eliminating the corresponding components in the state of the oserver the order of the oserver can e reduced to q Indeed another particular feature of the presented work is the closed form of the matrix T solution of the Sylvester equation (4) REFERENCES 1 Ben-Israel A Greville T N Generalized inverses Springer 2003 2 Cao Y Y Lam J Sun Y X Static output feedack stailization: an ILMI approach Automatica vol 34 n 12 pp 1641-1645 1998 3 Cumming SDG Design of oservers of reduced dynamics Electron Lett vol 5 n 10 pp 213 214 1969 4 Darouach 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