Finite time observation of nonlinear time-delay systems with unknown inputs

Author manuscript, published in "NOLCOS21 (21)" Finite time observation o nonlinear time-delay systems with unknown inputs G. Zheng J.-P. Barbot, D. Boutat, T. Floquet, J.-P. Richard INRIA Lille-Nord Europe, 4 Avenue Halley, 5965 Villeneuve d Ascq, France ECS, ENSEA, 6 Av. du Ponceau, 9514 Cergy, France ENSI de Bourges, Insitut PRISME, 88 Boulevard de Lahitolle, 182 Bourges, France LAGIS UMR CNRS 8146, Ecole Centrale de Lille, BP 48, 59651 Villeneuve d Ascq, France inria-467966, version 1-29 Mar 21 Abstract: Causal non-causal observability are discussed in this paper or nonlinear timedelay systems. By extending the Lie derivative or time-delay systems in the algebraic ramework introduced by Xia et al. (22), we present a canonical orm give suicient condition in order to deal with causal non-causal observations o state unknown inputs o time-delay systems. Keywords: Time-delay systems, Observability, Causality, Canonical orm 1. INTRODUCTION Observation or estimation is one o the most important problems in control theory. For nonlinear systems without delays, the observability problem has been exhaustively studied, is characterized respectively by Hermann Krener (1977); Sontag (1984); Krener (1985) rom a dierential point o view, by Diop Fliess (1991) rom an algebraic point o view. For observable systems, many types o nonlinear observers were proposed, such as high-gain observer in Gauthier et al. (1992), algebraic observer in Barbot et al. (27), sliding mode observer in Xiong Sai (21); Floquet Barbot (27) so on. However, unlike nonlinear systems without delays, the analysis o properties or time-delay system is more complicated (see the surveys in Sename Briat (21) Richard (23)). For linear time-delay systems, various aspects o the observability problem have been studied in the literature, by dierent methods such as unctional analytic approach (Bhat Koivo (1976)), algebraic approach (Brewer et al. (1986); Sontag (1976); Fliess Mounier (1998)), so on (Przyiuski Sosnowski (1984)). The theory o non-commutative rings has been applied to analyze nonlinear time-delay systems irstly by Moog et al. (2) or the disturbance decoupling problem o nonlinear time-delay system, or observability o nonlinear timedelay systems with known inputs by Xia et al. (22), or identiiability o parameter or nonlinear time-delay systems in Zhang et al. (26), or state elimination delay identiication o nonlinear time-delay systems by Anguelova Wennberga (28). In this algebraic ramework, the let Ore ring o non-commutative polynomials deined over the ield o meromorphic unctions is used or the analysis o nonlinear time-delay systems, since the rank o a module over this ring is well deined can be used to characterize controllability, observability identiiability o nonlinear time-delay systems. Concerning observer design or linear nonlinear timedelay systems, see Boutayeb (21); Pepe (21); Germani et al. (22); Darouach (26); Seuret et al. (27); Sename Briat (27); Ibrir (29) the reerences therein. In this paper, based on the algebraic ramework proposed by Xia et al. (22), we give a general deinition o observability or time-delay systems with unknown inputs. Then we generalize the notation o the Lie derivative or nonlinear systems with time-delay, redeine the relative degree observability indices or nonlinear time-delay systems by using the theory o non-commutative rings. Ater that a canonical orm is derived suicient conditions are given to treat causal non-causal observations o states unknown inputs o time-delay systems. 2. ALGEBRAIC FRAMEWORK Denote τ the basic time delay, assume that the times delays are multiple times o τ. Consider the ollowing nonlinear time-delay system: s ẋ = (x(t iτ)) + g j (x(t iτ))u(t jτ) j= y = h(x(t iτ)) = [h 1 (x(t iτ)),..., h p (x(t iτ))] T x(t) = ψ(t), u(t) = ϕ(t), t [ sτ, ] (1) where x W R n denotes the state variables, u = [u 1,..., u m ] T R m is the unknown admissible input, y R p is the measurable output. Without loss o generality, we assume that p m. And i S = {, 1,..., s} is a inite set o constant time-delays,, g j h are meromorphic

unctions 1, (x(t iτ)) = (x, x(t τ),..., x(t sτ)) ψ : [ sτ, ] R n ϕ : [ sτ, ] R m denote unknown continuous unctions o initial conditions. In this work, it is assumed that (1) with u = is locally observable, or initial conditions ψ ϕ, (1) admits a unique solution. Based on the algebraic ramework introduced in Xia et al. (22), let K be the ield o meromorphic unctions o a inite number o the variables rom {x j (t iτ), j [1, n], i S }. With the stard dierential operator d, deine the vector space E over K: E = span K {dξ : ξ K} which is the set o linear combinations o a inite number o one-orms rom dx j (t iτ) with row vector coeicients in K. For the sake o simplicity, we introduce backward time-shit operator δ, which means δ i ξ(t) = ξ(t iτ), ξ(t) K, or i Z + (2) δ i (a(t)dξ(t)) = δ i a(t)δ i dξ(t) (3) = a(t iτ)dξ(t iτ) or a(t)dξ(t) E, i Z +. Let K(δ] denote the set o polynomials o the orm a(δ] = a (t) + a 1 (t)δ + + a ra (t)δ ra (4) where a i (t) K. The addition in K(δ] is deined as usual, but the multiplication is given as a(δ]b(δ] = r a+r b k= i r a,j r b i+j=k a i (t)b j (t iτ)δ k (5) Note that K(δ] satisies the associative law it is a noncommutative ring (see Xia et al. (22)). However, it is proved that the ring K(δ] is a let Ore ring (Ježek (1996); Xia et al. (22)), which enables to deine the rank o a module over this ring. Let M denote the let module over K(δ] M = span K(δ] {dξ, ξ K} where K(δ] acts on dξ according to (2) (3). With the deinition o K(δ], (1) can be rewritten in a more compact orm as ollows: m ẋ = (x, δ) + G i u i (t) i=1 (6) y = h(x, δ) x(t) = ψ(t), u(t) = ϕ(t), t [ sτ, ] where (x, δ) = (x(t iτ)) h(x, δ) = h(x(t iτ)) with entries belonging to K, G i = s j= gj i δj with entries belonging to K(δ]. It is assumed that rank K(δ] h = p, which implies that [h 1,..., h p ] T are independent unctions o x its backward shits. 3. NOTATIONS AND DEFINITIONS Similar to observability deinitions or nonlinear systems without delays given in Hermann Krener (1977) in Diop Fliess (1991), a deinition o observability or time-delay systems is given in Marquez-Martinez et al. (22). A more generic deinition is stated here as ollows: 1 means quotients o convergent power series with real coeicients Conte et al. (1999); Xia et al. (22). Deinition 1. System (1) is locally observable i the state x(t) W R n can be expressed as a unction o the output its derivatives with their backward orward shits as ollows: x(t) = α(y(t jτ),..., y (k) (t jτ)) (7) or j Z k Z +. Remark 1. Dierent to the deinition in Marquez-Martinez et al. (22) where only orward shits are involved in (7), Deinition 1 is more generic covers the ollowing cases: (1) Assume (7) is satisied or j = k Z +. This means that no time-delay is involved in (7) Deinition 1 is equivalent to the one or nonlinear systems without delays. (2) Assume (7) is satisied or j Z + k Z +. This implies that only backward shits can be ound in (7), then x(t) W o (1) depends only on the past values o the output their derivatives. So the observability is causal. (3) Assume (7) is satisied or j Z k Z +. This means that both backward orward shits appear in (7). So x(t) W is a unction o both the past uture values o the output their derivatives, which implies that the observability o (1) is not causal. Following the same principle o Deinition 1, we make the ollowing deinition or the unknown inputs. Deinition 2. The unknown input u(t) can be estimated i it can be written as a unction o the derivatives o the output its derivatives with backward orward shits, i.e. u(t) = β(y(t jτ),..., y (k) (t jτ)) or j Z k Z +. The ollowing example illustrates Deinition 1 2. Example 1. Consider the ollowing system ẋ 1 = x 2 + δx 1 ẋ 2 = δ 2 x 2 δx 3 ẋ 3 = δx 4 + δu 1 + δ 4 u 2 (8) ẋ 4 = δu 2 y 1 = x 1 y 2 = δx 4 A straightorward calculation gives x 1 (t) = y 1 (t) x 2 (t) = ẏ 1 (t) y 1 (t τ) x 3 (t) = ẏ 1 (t τ) y 1 (t 2τ) ÿ 1 (t + τ) + ẏ 1 (t) x 4 (t) = y 2 (t + τ) { u1 (t) = ÿ 1 (t) ẏ 1 (t τ)... y 1 (t + 2τ) + ÿ 1 (t + τ) y 2 (t + τ) ẏ 2 (t τ) u 2 (t) = ẏ(t + 2τ) It can be seen that x 1 x 2 are causally observable since they depend only on the outputs their past values. However x 3, x 4, u 1 u 2 are non causally observable since they also depend on the uture values o the outputs. So according to Deinition 1 2, system (8) is observable, but the observability is not causal. Deinition 3. (Unimodular matrix) [Marquez-Martinez et al. (22)] Matrix A K n n (δ] is said to be unimodular over

K(δ] i it has a let inverse A 1 K n n (δ], such that A 1 A = I n n. Remark 2. An algorithm to check whether a matrix o K n n (δ] is unimodular is stated in Marquez-Martinez et al. (22). Deinition 4. (Change o coordinate) [Marquez-Martinez et al. (22)] For system (1), z = φ(δ, x) K n 1 is a causal change o coordinate over K or (1) i there exist locally a unction φ 1 K n 1 some constants c 1,, c n N such that diag{δ ci }x = φ 1 (δ, z). The change o coordinate is bicausal over K i max{c i } =, i.e. x = φ 1 (δ, z). Note that the relative degree or nonlinear systems without delays is well deined via the Lie derivative (see Isidori (1995)), then many eorts have been done to extend the classical Lie derivative or nonlinear time-delay systems. In Germani et al. (21, 22), authors deined the socalled delay relative degree or a class o nonlinear timedelay systems with only single delay, by augmenting the dimension o the studied system. In Oguchi et al. (22); Oguchi Richard (26), authors introduced the delayed state derivative delayed state bracket, which are the extensions o the conventional Lie derivative. However, those deinitions are still built on the theory o commutative rings, which makes the analysis o observability or nonlinear time-delays systems still complicated. Hence in what ollows we try to characterize the relative degree observability indices or nonlinear time-delay systems by extending the Lie derivative in the algebraic ramework o Xia et al. (22), rom non-commutative rings point o view. Let (x(t jτ)) h(x(t jτ)) or j s respectively be an n p dimensional vector with entries r K or 1 r n h i K or 1 i p. Let [ = hi,, h ] i K 1 n (δ] (9) 1 n where or 1 r n: r = s j= r (t jτ) δj K(δ] then the Lie derivative or nonlinear systems without delays can be extended to nonlinear time-delay systems in the ramework o Xia et al. (22) as ollows L h i = () = n s r=1 j= r (t jτ) δj ( r ) K (1) For j =, (1) is the classical deinition o the Lie derivative o h along. For h i K, deine L Gi h i = (G i) K(δ] By the above deinition o Lie derivative, we can deine the relative degree in the ollowing way: Deinition 5. (Relative degree) System (6) has relative degree (ν 1,, ν p ) in an open set W R n i, or 1 i p, the ollowing conditions are satisied : (1) or all x W, L Gj L r h i =, or all 1 j m r < ν i 1; (2) there exists x W such that j [1, m], L Gj L νi 1 h i. I or 1 i p, (1) is satisied or all r, then we set ν i =. Remark 3. This deinition o relative degree or time-delay system is equivalent to the deinition given in Moog et al. (2). Since (6) is locally observable when u =, then one can deine, or (6), the so-called observability indices introduced in Krener (1985). Let { } F k := span K(δ] dh, dl h,, dl k 1 h or 1 k n. And it was shown that the iltration o K(δ]-module satisies then we deine or 2 k n. Let F 1 F 2 F n d 1 = rank K(δ] F 1 d k = rank K(δ] F k rank K(δ] F k 1 k i = card {d k i, 1 k n} then (k 1,, k p ) are the observability indices, p k i = i=1 n since it is assumed that (6) is observable with u =. Reordering, i necessary, the output components o (6), such that [ ] h, L h,, L n 1 T h rank K(δ] [ ] T h 1, L h 1,, L k 1 1 h 1,, h p, L h p,, L kp 1 h p = rank K(δ] = k 1 + + k p = n Since rank K(δ] h = p, then observability indices (k 1,, k p ) or (h 1,, h p ) are well deined, but the order may be not unique. 4. CANONICAL FORM AND CAUSAL OBSERVABILITY Ater having deined the relative degree observability indices via the extended Lie derivative or nonlinear timedelay systems in the ramework o non-commutative rings, we are ready to state the ollowing theorem. Theorem 1. For 1 i p, denote k i the observability indices ν i the relative degree index or y i o (6), note ρ i = min {ν i, k i }. Then there exists a change o coordinate φ(x, δ) K n 1, such that (6) can be transormed into the ollowing orm: where ż i = A i ż i + B i V i (11) ξ = α(z, ξ, δ) + β(z, ξ, δ)u (12) y i = C i z i (13)

( ) T z i = h i,, L ρi 1 h i K ρ i 1 1. A i =..... 1 Rρi ρi, B i =. Rρi 1, 1 V i = L ρi h i(x, δ) + m L Gj L ρi 1 h i (x, δ)u j K α K l 1, β K l 1 (δ] with l = n C i = (1,,, ) R 1 ρi p Moreover i k i < ν i, one has V i = L ρi h i = L ki h i. Proo 1. According to the deinition o k i, or 1 i p, one has [ rank K(δ] ] T h i, L h i,, L ki 1 h i = k i since ρ i = min{k i, ν i }, one gets [ ] T h i, L h i,, L ρi 1 h i rank K(δ] = ρ i which implies that the components o ( ) T z i = h i,, L ρi 1 h i are linearly independent over K(δ]. Denote z = ( z1 T,, zp T ) T K n l, with l = ρ j p ρ j, choose l variables ξ K l, such that ( z T, ξ T ) T = φ(x, δ) K n is a well-deined change o coordinate. Then or 1 i p 1 j ρ i 1, one has ż i,j = z i,j+1 ż i,ρi = L ρi h i + m L Gj L ρi 1 h i (x, δ)u j y i = z i,1 = C i z i ξ = α(z, ξ, δ) + β(z, ξ, δ)u which means (6) can be transormed into orm (11-13). Moreover, according to the deinition o ν i, or all 1 j m r < ν i 1, one has ( ) L r L Gj L r h i h i = (G j ) = ( ) L vi L Gj L νi 1 h i h i = (G j ) Hence, i k i < ν i, then ρ i = k i one has L Gj L ki 1 L Gj L ρi 1 h i (x, δ) =, which yields V i = L ρi h i = L ki h i. Remark 4. For (11-13), one can ind observers in the literature (Xiong Sai (21); Floquet Barbot (27)) such that z i or 1 i p can be estimated in inite time due to the triangular structure. In addition, one can also obtain the inormation V i time. For (11), note with H(x, δ) = Γ(x, δ) = = y (ρi) i in inite H(x, δ) = Ψ(x, δ) + Γ(x, δ)u (14) h (ρ1) 1. h (ρp) p h i (x, δ) = Concerning the reconstruction o unknown inputs, rewrite (14) as ollows L G1 L ρ1 1. L G1 L ρp 1, Ψ(x, δ) = L ρ1 h 1. L ρp h p h 1 L Gm L ρ1 1.... h p L Gm L ρp 1 h 1 where H(x, δ) K p 1, Ψ(x, δ) K p 1 Γ(x, δ) K p m (δ]. And or (6), let denote Φ the observable space rom its outputs: Φ = {dh 1,, dl ρ1 1 h 1,, dh p,, dl ρp 1 h p } (15) then we have the ollowing theorem. Theorem 2. For system (6) with outputs (y 1,..., y p ) corresponding (ρ 1,..., ρ p ) with ρ i = min{k i, ν i } where k i ν i are respectively the associated observability indices the relative degree, i rank K(δ] Φ = n where Φ deined in (15), then there exists a change o coordinate φ(x, δ) such that (6) can be transormed into (11-13) with ξ =. h p Moreover, i the change o coordinate is bicausal over K, then the state x(t) o (6) is causally observable. For the matrix Γ K p m (δ] where m p, i rank K(δ] Γ = m then there exists a matrix Q K p p (δ] such that [ ] Γ QΓ = where Γ K m m (δ] has ull row rank m. Moreover, i Γ K m m (δ] is also unimodular over K(δ], then the unknown input u(t) o (6) can be causally estimated as well. Proo 2. According to Theorem 1, (6) can be transormed into (11-13) by (z, ξ) = φ(x, δ). Hence, i rank K(δ] Φ = n, where Φ deined in (15), then one has p ρ j = n, which implies (6) can be transormed into (11-13) with ξ = the change o coordinate is given by z = φ(x, δ) where z = (zi T,, zt p ) T z i = (h i,, L ρi 1 h i ) T. Moreover, i φ(x, δ) K n 1 is bicausal over K, one can write x as a unction o y i, its derivative backward shit, which implies state x is causally observable. Γu = H(x, δ) Ψ(x, δ) = Υ(x, δ) (16) Since rank K(δ] Φ = n x is causally observable, then Υ(x, δ) is a vector o known meromorphic unctions belonging to K.

According to Lemma 4 in Marquez-Martinez Moog (27), or any matrix Γ K p m, i rank K(δ] Γ = m, by some necessary manipulations, one can always [ ] ind at Γ least a matrix Q K p p such that QΓ = where Γ K m m (δ] has ull row rank m. Thus, i Γ K m m (δ] is unimodular over K(δ], then there exists a matrix Γ 1 K m m (δ] such that [ Γ 1 ] QΓ = I m m u = [ Γ 1 ] QΥ Since Γ 1 K m m (δ], Q K p p Υ K p 1, then u o (6) is also causally observable. Example 2. Consider the ollowing system: ẋ 1 = δx 1 + x 2 ẋ 2 = δx 3 + u 1 ẋ 3 = δx 1 + δu 1 + u 2 ẋ 4 = x 4 + 2δx 4 /3 (17) y 1 = x 1 y 2 = x 3 y 3 = x 4 One can check that ν 1 = k 1 = 2, ν 2 = k 2 = 1, ν 3 = k 3 = 1, then one gets ρ 1 = 2, ρ 2 = 1 ρ 3 = 1. According to (15), one has Φ = {dh 1, dl h 1, dh 2, dh 3 } = {dx 1, δdx 1 + dx 2, dx 3, dx 4 } Since rank K(δ] Φ = 4, this gives the ollowing change o coordinate or (17): z = φ (x, δ) = (x 1, x 2 δx 1, x 3, x 4 ) T One can check that the change o coordinate is bicausal over K, since x can be expressed causally by z as ollows: z 1 x = φ 1 δz = 1 + z 2 z 3 z 4 Thus the state o (17) is causally observable a straightorward computation gives x 1 (t) = y 1 (t) x 2 (t) = y 1 (t τ) + ẏ 1 (t) x 3 (t) = y 2 (t) x 4 (t) = y 3 (t) Moreover, one has Γ = ( ) 1 δ 1 which gives rank K(δ] Γ = m = 2. Thus Γ is unimodular over K(δ], since one can ind ( ) Γ 1 1 = δ 1 such that Γ 1 Γ = I 2 2. According to Theorem 2, the unknown inputs u 1 u 2 are also causally observable. One can easily obtain the ollowing equations: { u1 (t) = ẏ 1 (t τ) + ÿ 1 (t) + y 2 (t τ) u 2 (t) = ẏ 2 (t) y 1 (t τ) ẏ 1 (t 2τ) ÿ 1 (t τ) y 2 (t 2τ) Remark 5. For the case where rank K(δ] Φ < n, although Theorem 2 is not valid, it is still possible to estimate the state the unknown inputs o (6) provided that some complementary conditions are satisied. In Barbot et al. (29), a constructive algorithm to solve this problem or nonlinear systems without delays has been proposed, which in act can be generalized to treat the same problem or nonlinear time-delay systems. Let us remark that it is the bicausal change o coordinate which makes the state o system causally observable. And it is the unimodular characteristic o Γ over K(δ] which guarantees the causal reconstruction o unknown inputs. The next section is devoted to dealing with the non-causal case. 5. NON-CAUSAL OBSERVABILITY In order to treat the non-causal case, let introduce the orward time-shit operator, which is similar to the backward time-shit operator δ deined in Section 2: (t) = (t + τ) i δ j (t) = δ j i (t) = (t (j i)τ) or i, j N Following the same principle o Section 2, denote K the ield o meromorphic unctions o a inite number o variables rom {x j (t iτ), j [1, n], i S} where S = { s,,,, s} is a inite set o constant one has K K. Denote K(δ, ] the set o polynomials o the orm as ollows: a(δ, ] = ā rā rā + + ā 1 +a (t) + a 1 (t)δ + + a ra (t)δ ra (18) where a i (t) ā i (t) belonging to K. Let keep the same deinition o addition or K(δ, ], the multiplication is given as ollows: r a r b r a r b a(δ, ]b(δ, ] = a i δ i b j δ i+j + a i δ i bj δ i j i= j= r b i=1 j= i= rā rā r b + ā i i b j i δ j + ā i i bj i+j i=1 (19) It is clear that K(δ] K(δ, ] the ring K(δ, ] possesses the same properties as K(δ]. Thus a module can be also deined over K(δ, ]: M = span K(δ, ] {dξ, ξ K} Given the above deinitions, Theorem 2 can then be extended as ollows in order to deal with non-causal observability or nonlinear time-delay systems. Theorem 3. For system (6) with outputs (y 1,..., y p ) corresponding (ρ 1,..., ρ p ), i rank K(δ] Φ = n where Φ deined in (15), then there exists a change o coordinate z = φ(x, δ) such that (6) can be transormed into (11-13) with ξ =. Moreover, i the change o coordinate z = φ(x, δ) is bicausal over K, then the state x(t) o (6) is at least noncausally observable.

For the deduced matrix Γ with rank K(δ] Γ = m, one can obtain a matrix Γ K m m (δ] which has ull row rank m. I Γ is unimodular over K(δ, ], then the unknown input u(t) o (6) can be at least non-causally estimated as well. Proo 3. I the change o coordinate z = φ(x, δ) K n 1 K n 1 is bicausal over K, then there exist φ 1 K n 1 certain c 1,, c n such that T x = φ 1 (z, δ) where T = diag{δ c1,, δ cn }. Thus one can deine the matrix T 1 = diag{ c1,, cn } K n n (δ, ], such that x = T 1 φ 1 n 1 (δ, z) K which means x is at least non-causally observable. For the estimation o u(t), i Γ is unimodular over K(δ, ], then ollowing the same procedure o Theorem 2, one gets u = [ Γ 1 ] QΥ In this case, since Γ 1 K(δ, ], Q K p p (δ] Υ K p 1, then u o (6) is at least non-causally observable. Remark 6. Non-causal observations o the state the unknown inputs are in some case very useul. Because o the non causality, x u are estimated with some known delays. Many proposed delay eedback control methods can then be applied or stabilizing nonlinear time-delay systems (Sename (27)). At the other h, other applications, such as cryptography based on chaotic system, need not the real time estimation, hence non-causal observations can still play an important role in those applications. 6. CONCLUSION This paper introduced a new generic deinition o observability or time-delay systems with unknown inputs, covering causal non-causal observability. Then the relative degree observability indices or nonlinear time-delay systems were deined based on the notation o the Lie derivation in the ramework o non-commutative rings. Ater that, we introduced a canonical orm or time-delay systems, suicient conditions were given to guarantee the causal non-causal observations o states unknown inputs o time-delay systems. REFERENCES M. Anguelova B. Wennberga. State elimination identiiability o the delay parameter or nonlinear timedelay systems. 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