Left Inversion of Nonlinear Time Delay System
53rd IEEE Conference on Decision and Control December 15-17, 2014. Los Angeles, California, USA
Left inversion of nonlinear time delay system Zohra Kader, Gang Zheng and Jean-Pierre Barbot
and estimate its state, one solution is to rewrite this internal dynamics such that its dynamic becomes independent of the unknown input [1]. Thus in the paper we will adapt this new way to determine the internal dynamics for the nonlinear systems with time delay. This method is based on the finite time convergence by using the existing observer for time delay system in the literature [5], [7], [8], [10], [2] and the high order sliding mode proposed in [12]is applied in this paper. The paper is organized as follows: In the next section the algebraic framework and some notations are recalled. In section III, we firstly propose a canonical form based on the previously proposed algorithm in [21]. After that the inverse dynamics is computed and sufficient condition for causal and non causal left invertibility are discussed. This section ends with an observer scheme dedicated to the proposed canonical form. In section IV two examples and numerical results highlight the feasibility and the efficiency of the proposed approach.
Abstract— This paper investigates the left invertibility for nonlinear time delay system with internal dynamics. Under the assumption imposed on the internal dynamics, it has been shown that the unknown inputs can be estimated. Causal and non causal estimation of the unknown inputs are respectively discussed, and the high-order sliding mode observer is used to estimate the observable states.
I. I NTRODUCTION In many applications, recovering of unknown inputs from the outputs is crucial. This may occur for example in data secure transmission, where the unknown input is the message or in fault detection and isolation where the fault is the unknown input. It is the reason why this problem was studied since at least forty five years ago in linear control theory [16], [18] and thirty five year ago in nonlinear control theory [6], [17]. Most of those works are in the context of nonlinear systems without time delays. Considering nonlinear systems with time delays, an important tool based on noncommutative ring was introduced in [19]. This tool of non commutative ring [9] is used to model nonlinear time delayed system in algebraic framework, and many results are already obtained. The notions of Lie derivatives and relative degree are defined, and the differences between the causal and noncausal invertibility are clarified in [22]. The canonical form of invertibility is also given in [21], and in [20] a method for estimating of the unknown inputs is proposed. However, the algorithm for left invertibility proposed in [21] is only for the system without internal dynamics. For system with or without time delay, the main difficulty when the internal dynamics (or inverse dynamics [3]) occurs is to estimate the state of such dynamics. One interesting solution in order to overcome such a difficulty is to allow the derivative of the unknown inputs [15] with a geometrical approach and with an algebraical one. If however the input derivatives are not possible, then it is necessary to compute and analyze the internal dynamics. For nonlinear systems without delay, if the vector fields associated to the inputs verify some involutivity properties, then the internal dynamic does not depend on the unknown input. However, this rule is not valid for nonlinear systems with time delay. In order to analyze the internal dynamics
II. A LGEBRAIC
Consider the following class of nonlinear time delay system: m X gi (x(t − jτ ))uj (t) x ˙ = f (x(t − jτ )) + i=1
(1)
x = ψ(t), u(t) = ϕ(t) t ∈ −sτ, 0
with x ∈ W ⊂ ℜn is the state vector of the system, u(t) ∈ ℜm is the vector of its unknown input and y ∈ ℜp is the output. τ represents the basic commensurate time delay. In [19], an algebraic framework is developed, defining the field K of a finite number of the variables from: {xj (t − iτ ), j ∈ [1, n] , i ∈ [0, s] ⊂ N }
(2)
such that E represents the vector space over K : E = spanK {dξ : ξ ∈ K} and τ represents the basic commensurate time delay. Denote the operator δ as a backward shift operator, which means δ i ξ(t) = ξ(t − iτ ) and δ i (a(t)dξ(t)) = a(t − iτ )dξ(t − iτ ). With this operator, we can then define the following set of polynomials K(δ]: a(δ] = a0 (t) + a1 (t)δ + · · · + ara (t)δ ra , ai (t) ∈ K
This paper was supported by Ministry of Higher Education and Research, Nord-Pas de Calais Regional Council and FEDER through the Contrat de Projets Etat Region (CPER) CIA 2007-2013, and also supported by ARCIR Project ESTIREZ, Nord-Pas de Calais Regional Council. Zohra Kader and Gang Zheng are with Project Non-A, INRIA Lille-Nord Europe, 40 avenue Halley and LAGIS UMR CNRS 8219, Ecole Centrale de Lille, 59650 Villeneuve d’Ascq, France. Jean-Pierre Barbot is with ECSLab EA 3649, ENSEA, Cergy-Pontoise, and Non-A Team INRIA, France
(3)
The addition for the entries in K(δ] is defined as usual, but its multiplication is given by the following criteria: a(δ]b(δ] =
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FRAMEWORK AND NOTATIONS
rX a ,j≤rb a +rb i≤rX k=0
469
i+j=k
ai (t)bj (t − iτ )δ k
(4)
With the standard differential operator d, denote by M the left module over K(δ]: M = spanK(δ] {dξ, ξ ∈ K}
Based on the above notations, the relative degree can be defined in the following way. Definition 3: (Relative degree) System (6) has the relative degree (ν1 , · · · , νp ) in an open set W ⊆ Rn if the following conditions are satisfied for 1 ≤ i ≤ p: 1) for all x ∈ W , LGj Lrf hi = 0 for all 1 ≤ j ≤ m and 0 ≤ r < νi − 1; 2) there exists x ∈ W such that ∃j ∈ {1, · · · , m}, LGj Lνfi −1 hi 6= 0. If the first condition is satisfied for all r ≥ 0 and some i ∈ {1, · · · , p}, we set νi = ∞. Moreover, for system (6), one can also define observability indices [11] over non-commutative rings. For 1 ≤ k ≤ n, let Fk be the following left module over K(δ]: n o Fk := spanK(δ] dh, dLf h, · · · , dLfk−1 h .
(5)
Hence, K(δ] is a non-commutative ring satisfying the associative law, and it has been proved in [9] and [19] that it is a left Ore ring, which enables us to define the rank conception. Then, for any matrix with entries belonging to K(δ], we can introduce the following definition. Definition 1: (Unimodular matrix) A matrix A ∈ Kn×n (δ] is said to be unimodular over K(δ] if it has a left inversion A−1 ∈ Kn×n (δ], such that A−1 A = In×n . Based on the above algebraic framework a nonlinear time delay system can be represented in a compact algebraic form as follows: m X Gi (x, δ)ui (t) x˙ = f (x, δ) + i=1 (6) y = h(x, δ) x = ψ(t), u(t) = ϕ(t) t ∈ −sτ, 0
It was shown that the filtration of K(δ]-module satisfies F1 ⊂ F2 ⊂ · · · ⊂ Fn , then define d1 = rankK(δ] F1 , and dk = rankK(δ] Fk − rankK(δ] Fk−1 for 2 ≤ k ≤ n. Let ki = card {dk ≥ i, 1 ≤ k ≤ n}, then (k1 , · · · , kp ) are the observability indices. Reorder, if necessary, the output components of (6) so that
where the notation f (x, δ) means f (x, δ) = f (x, x(t − τ ), . . . , x(t − sτ ))) and the same is considered for G(x, δ) s P gij δ j with entries belonging to the and h(x(t), δ), Gi =
1 rankK(δ] { ∂h ∂x , · · · , = k1 + · · · + kp .
j=0
ring K(δ]. Since this paper studies the left invertibility of the nonlinear time-delay systems, the following definition will be used. Definition 2: (Invertibility) The unknown inputs u(t) can be estimated if it can be written as follows: u(t) = ϕu (δ j y, · · · , δ j y (k) )
,
∂hp ∂x , · · ·
k −1
,
∂Lf p hp } ∂x
Definition 4: (Change of coordinates) If there exists φ−1 ∈ Kn×1 and some constants c1 , · · · , cn ∈ N such that diag {δ ci } x = φ−1 (δ, z) then the change of coordinates z = φ(δ, x) ∈ Kn×1 is a causal change of coordinates over Kn×1 . The change of coordinate is bicausal over K if max = {ci } = 0, i.e x = φ−1 (δ, z).
(7)
for j ∈ Z and k ∈ Z + . It can be locally causally estimated if (7) is satisfied for j ∈ Z + and k ∈ Z + , and be locally noncausally estimated if (7) is satisfied for j ∈ Z and k ∈ Z + . The conventional concepts like Lie derivative and relative degree which are widely used in nonlinear systems without delay, are introduced in the case of nonlinear time delay systems by [4], [14], [13] and we can extend them by using the above algebraic framework. Let f (x(t − jτ )) and h(x(t − jτ )) for 0 ≤ j ≤ m be an n and p dimensional vector field respectively, with fr the entries of f belonging to K for 1 ≤ r ≤ n and hi ∈ K for 1 ≤ i ≤ p, with p ≥ m, and then ∂hi ∂hi ∂hi ,··· , = ∈ K1×n (δ] (8) ∂x ∂x1 ∂xn
Based on the above definition, the following result has been reported in [22], [20]. Theorem 1: For 1 ≤ i ≤ p, denote by ki the observability indices and νi the relative degree index for yi of (6), and note ρi = min(ki , νi ). Then there exists a causal change of coordinates φ(x, δ) ∈ Kn×1 , such that (6) can be transformed into the following form: ˙ ξi = Ai ξi + Bi µi (10) ζ˙ = α(ξi , ζ, δ) + β(ξi , ζ, δ)u y = Ci ξi where Ai and Bi arei in the Brunovsky form. ξi = h T T = ξi,1 , ξi,2 , . . . , ξi,ρi hi , Lf hi , . . . Lρfi −1 hi , such that Lf hi is the Lie derivative of the output hi along the vector field f , and m P LGj Lρfi −1 hi (x, δ)uj ∈ K µi = Lρfi hi (x, δ) +
Where for 1 ≤ r ≤ n: s
X ∂hi ∂hi = δ j ∈ K(δ] ∂xr ∂x (t − jτ ) r j=0 n X s X ∂hi ∂hi (f ) = ( δ j ) (fr ) L f hi = ∂x ∂x (t − jτ ) r r=1 j=0
k −1
∂Lf 1 h1 ,··· ∂x
j=1
α∈K
l×1
,β∈K
l×1
where l = n −
p P
ρi and the change
j=1
of coordinates is given as: φT (x, δ) = (φT1 (x, δ), φT2 (x, δ)), with
(9)
φT1 (x, δ) =
and in the same way we can define LGi hi 470
ρ
h1 , Lf h1 , . . . , Lρf1 h1 , . . . hp , Lf hp , . . . , Lfp hp
and φ2 (x, δ) is an (n−l) dimensional complementary change of coordinates of φ1 (x, δ). Consider now the last dynamics of ξ˙i and the one of ζ˙ in (10), one can obtain the following dynamics: ( ˙ = Ψ(ξ, ζ, δ) + Γ(ξ, ζ, δ)u H(ξ) (11) ˙ζ = f¯(ζ, ξ, δ, u)
III. L EFT INVERTIBILITY
This section will firstly discuss how to iteratively obtain a new canonical form for the non-trivial case, and secondly study the left invertibility of the unknown inputs with internal dynamics in the deduced canonical form. Finally a high order sliding mode observer is designed to estimate the state variables and their derivatives in order to estimate the unknown inputs.
where
A. Canonical form
Γ(ξ, ζ, δ) =
ρ −1 LG L 1 h1 1 f . . . ρp −1 LG L hp 1 f
Lρ1 h
Ψ=
WITH INTERNAL DYNAMICS
.
.
.
···
ρ −1 LG L 1 h1 m f . . . ρp −1 LG L hp m f
1 f ρ L 2 h2 f
. . . ρp hp L f
···
|x=Φ−1 (ξ,ζ,δ)
,H =
(ρ ) ξ1 1 (ρ ) ξ2 2 . . . (ρp ) ξp
|x=Φ−1 (ξ,ζ,δ)
ρ −1
Φ = {dh1 , · · · , dLfρ1 −1 h1 , · · · , dhp , · · · , dLfp
Remark 1: Passing from (10) to (11) , the substitution of x = Φ−1 (ξ, δ) must be done. Indeed if the change of coordinates is not bicausal, this may generate non causal terms in the equation (11). Let us consider the following system: x˙ 1 = δx2
hp } (13)
and n o ρ −1 £ = spanR[δ] h1 , · · · , Lfρ1 −1 h1 , · · · , hp , · · · , Lfp hp (14) where R[δ] is the commutative ring of polynomials in δ with coefficients belonging to the field R, and let £(δ] be the set of polynomials in δ with coefficients over £. The module spanned by element of Φ over £(δ] is defined as follows:
x˙ 2 = a1 x1 + a2 x2 + a3 x3 + x1 u
x˙ = b x + b x + b3 x3 y 3 = x1 , 1y =2x 2 1 1 2 3
choosing the change of coordinates as: z1 = x1 , z2 = Lf h1 = δx2 , and z3 = x3 then we obtain: z˙1 = z2 −1
Ω = span£(δ] {ξ, ξ ∈ Φ} .
(15)
G = spanR[δ] {G1 , . . . , Gm }
(16)
Define where Gi is given in (6), and its left annihilator:
z˙2 = δ(a1 z1 + a2 δ z2 + a3 z3 + z1 u) −1 z˙3 = b1 z1 + b2 δ z2 + b3 z3 y1 = z 1 , y2 = z 3
G ⊥ = span£(δ] {ω ∈ M | ωβ = 0, ∀β ∈ G},
(17)
where M is defined in (5). Then the new outputs can be virtually obtained if the sufficient condition rankK (Hγ ) = lγ is satisfied, where /£ (18) Hγ = spanR[δ] ω ∈ G ⊥ ∩ Ω|ωf ∈
and the obtained system is not causal. Noticing that the system (10) contains an internal dy˙ For the trivial case where dim ζ = 0, i.e. namics ζ. rankK(δ] φ1 (x, δ) = n, if rankK(δ] Γ(ξ, δ) = m, then there ¯ ∈ always exists a matrix Q ∈ Km×p (δ] such that QΓ = Γ ¯ = m. In this case, according to Km×m (δ] with rankK(δ] Γ (11), one can obtain the estimation of u as follows: ˆ δ) − Ψ(ξ, ˆ δ)). ¯ −1 Q(H(ξ, u ˆ=Γ
Considering system (6), if rankK(δ] φ1 (x, δ) < n and rankK(δ] Γ(ξ, δ) < m, an algorithm was developed in [22] to generate lγ new outputs of the systems, which are combinations of y, its derivatives and their Pp backward shifts. Without loss of generality, suppose i=1 ρi = j, ρ −1 thus {dh1 , · · · , dLfρ1 −1 h1 , · · · , dhp , · · · , dLfp hp } are j linearly independent vectors over K(δ]. Then note:
More explicitly, the new generated outputs are noted as: y¯i = ωi f mod £, for 1 ≤ i ≤ l − γ and ωi ∈ Hγ . The above algorithm will be iterated until rankK(δ] φ1 (x, δ) = n or rankK(δ] Γγ (z, δ) = m. For the final obtained canonical form, if rankK(δ] φ1 (x, δ) = n and rankK(δ] Γγ (z, δ) < m, then the problem of left invertibility for the unknown inputs cannot be solved. For the second case where rankK(δ] φ1 (x, δ) < n and rankK(δ] Γγ (z, δ) = m, after lγ iterations, the obtained canonical form can be written as follows:
(12)
where ξˆ represents the estimation of ξ, which in fact can be estimated by using many existing methods, like sliding mode observer, algebraic observer and so on. Moreover, if ¯ is unimodular over K(δ], the above estimation the matrix Γ of u is obviously causal. Considering the non-trivial case, where rankK(δ] φ1 (x, δ) < n and rankK(δ] Γ(ξ, δ) < m, the internal dynamics ζ˙ in (10) is not vanished, thus the invertibility of the unknown inputs depends on the internal dynamics. The following section is devoted to treating this non-trivial case.
8 z˙i,j = zi,j+1 , for 1 ≤ i ≤ p + lγ and 1 ≤ j ≤ θi − 1 > > m > X < z˙i,θi = bi (z, η, δ) + ai,j (z, η, δ)uj , for 1 ≤ i ≤ p + lγ > > j=1 > : η˙ = f¯γ (η, z, δ, u) (19)
with ai,j = Lgj Lfθi −1 hi and bi (z, δ, η) = Lθfi hi , where lγ is the number of new outputs, θi is the minimum value 471
of the observability index and relative degree for the ith output. Then the second and the third equation in (19) can be rewritten in the following compact way:
defined dynamics (23) can asymptotically estimate η. This is due to the existence of a Lyapounov function V (e) such that V˙ (e) < 0, imposed by Assumption 1.
Hγ (z) ˙ = Ψγ (z, η, δ) + Γγ (z, η, δ)u η˙ = f¯γ (η, z, u, δ)
After the convergence of ηˆ to η, the unknown input u ¯γ ∈ of (6) can then be calculated via (28). Moreover, if Γ m×m −1 m×m ¯ K (δ] is unimodular over K(δ], i.e. Γγ ∈ K (δ], since Qγ ∈ K(δ]m×(p+lγ ) , Hγ (z, η, δ) ∈ K(δ](p+lγ )×1 and Ψγ (z, η, δ) ∈ K(δ](p+lγ )×1 , then from (28) we have u ∈ Km×1 (δ], which implies the causal estimation of the unknown inputs.
(20)
where = Γγ (z, η, δ) θ1 −1
LG L h1 1 f . . . θp −1 LG L hp 1 f θp+1 −1 LG L hp+1 1 f θp+2 −1 LG L hp+2 1 f
··· . . . ··· ··· ···
θ −1 LG L 1 h1 m f . . . θp −1 LG L hp m f θp +1 LG L hp+1 m f θp+2 −1 LG L hp+2 m f
. . . . . . . . . θp+l −1 θp+l −1 γ γ ··· LG L hp+lγ hp+l LG L m f γ 1 f (θ1 ) z 1,θ1 (θ ) z 2 2,θ2 . . . (θp ) γ z p,θp . . . (θp+l ) γ z p+lγ ,θp+l γ 1 0 θ L 1 h1 f C B θ2 C B L h2 C B f C B C B C B . C B . C B . C B C B θp C B γ L hp C B f C B C B C B . C B . C B C . B A @ θ p+lγ L hp+l f γ |x=φ−1 (z,δ)
Theorem 2 indicates that the causality of the estimation for ¯ γ ∈ Km×m (δ]. the unknown inputs depends on the matrix Γ If this matrix is not unimodular over K(δ], then the estimation of u might be non causal. In order to take into account the non causal estimation, let us introduce the forward shift operator ∇ such that ∇j x(t) = x(t + jτ ). ¯ as the field of functions of finite number of Denote K variables from {xi (t − jτ ), i ∈ [1, n] , j ∈ [−s, s]}, then one ¯ n×1 (δ, ∇) whose entry is of the following can define the set K form:
|x=φ−1 (z,δ)
,
H (z) ˙ =
a1 (t)∇+a0 (t)+a1 (t)δ · · ·+ara (t)δ ra a(δ, ∇] = a ¯ra¯ (t)+· · ·+¯ (25) ¯ Then, for K(δ, ∇], where all the coefficients belong to K. the addition is as usual, but the multiplication is given by the following relation:
Ψ (z, η, δ) =
a(δ, ∇]b(δ, ∇]
B. Left inversion For the iteratively obtained canonical form (19) (or the compact form (20)), if rankK(δ] Γγ = m, then there always ¯γ ∈ exists a matrix Qγ ∈ K(δ]m×(p+lγ ) such that Qγ Γγ = Γ ¯ γ = m. Then we can obtain: Km×m (δ] with rankK(δ] Γ ¯ −1 u=Γ ˙ − Ψγ (z, η, δ)) γ Qγ (Hγ (z)
+
i=0 j=0 rb ra ¯ P P
ai δ i bj δ i+j +
r¯ ra P b P
ai δ i¯bj δ i ∇j
i=0 j=1 r¯ ra ¯ P b P
a ¯ i ∇i b j ∇i δ j +
a ¯i ∇i¯bj ∇i+j .
i=1 j=1
(26)
Finally, we have the following result. ¯ γ ∈ Km×m (δ] is Lemma 1: Under Assumption 1, if Γ unimodular over K(δ, ∇], then the unknown input u of the system (6) can be at least non-casually estimated.
(21)
The proof of this lemma is quite similar to that of Theorem 2, thus the following will just explain the main differ¯ γ is unimodular over K(δ, ∇], i.e. ence. If the matrix Γ ¯ m×m (δ, ∇], then the causality of the estimation ¯ −1 ∈ K Γ γ for the unknown input u depends on the power of the terms ¯ −1 Γ ˙ Ψγ (z, η, δ) in (20). γ (z, η, δ, ∇), Qγ (z, η, δ, ∇), Hγ (z), However, the unknown input u of the system (6) can be at least non-causally estimated.
(22)
for which we can propose the following estimator: η , zˆ, δ, zˆp+lγ ,θp +lγ ) ηˆ˙ = f¯γ (ˆ
rb ra P P
i=1 j=0
Thus, the dynamics of η in (20) can be rewritten in the following form: η˙ = f¯γ (η, z, δ, z˙ p+lγ ,θp+lγ )
=
(23)
where zˆ represents the estimation of z. Assumption 1: It is assumed that there exists a Lyapunov function V (e) with e = η − ηˆ such that for all δ and z: ∂V ¯ η , z, δ)] < 0 (24) [fγ (η, z, δ) − f¯γ (ˆ V˙ = ∂e Then we are able to state the following result. ¯ γ ∈ Km×m (δ] is Theorem 2: Under Assumption 1, if Γ unimodular over K(δ], then the unknown input u of the system (6) can be casually estimated. For system (6), since we can use the existing method to estimate z and its derivative in a finite time T , then the
C. Observer design As we have explained, the estimation of the unknown input is based on the finite-time estimation of the state variable z and their derivatives, which is also used to estimate the internal dynamics η. In what follows, we use the high order 472
sliding mode observer [12] to estimate z and its derivatives:
z˙1 = z2 z˙2 = −z2 − δz1 + u1 δz1 z˙3 = −z3 + z4 + cos(100)u1 1 z˙4 = −z1 z2 + ( )u2 1 − δη η˙ = −η + sin(δz1 )u 2 50 y1 = z1 , y2 = z3 , y¯ = z4 , δη 6= 1.
8 θi > > zi,j − zi,j ) zi,j − zi,j |θi +1 sign(ˆ >zˆ˙i,j = zˆi,j+1 − λi,j |ˆ > > > ˙ > zˆi,j+1 = zˆi,j+2 > > > θi −1 > > > > −λi,j+1 |ˆ zi,j+1 − zˆ˙i,j | θi sign(ˆ zi,j+1 − zˆ˙i,j ) < .. .. > .=. > > > > > z ˆ˙i,θi −1 = zˆi,θi > > > 1 > > −λi,θi −1 |ˆ zi,θi −1 − zˆ˙i,θi −2 |2 sign(ˆ zi,θi −1 − zˆ˙i,θi −2 ) > > > :˙ zi,θi − zˆ˙i,θi −1 ) zˆi,θi = −λi,θi sign(ˆ (27)
The matrix Γγ is directly obtained as:
The convergence of the above observer has already been demonstrated in [12]. It has been shown that, if the rth (r is the relative degree) derivative of the output has a Lipschitz constant L > 0 and by choosing λ > L, then we have zˆ = z after a finite time T . Then, the internal dynamics η can be estimated by using the estimator (23). After η was converged, the unknown input can be estimated from the following algebraic equation: ˆ ¯ γ )−1 Q ˆ γ (Hγ (z) ˆ˙ − Ψγ (ˆ uˆ = (Γ z , ηˆ, δ)).
1 cos( δz1 ) 100 Γγ = 0
(28)
this gives the unimodular matrix: δz1 ) cos( 100 ¯ Γγ = 0
EXAMPLE
Let us consider the nonlinear time delay system: ξ˙1 = ξ2 ξ˙2 = −ξ2 − δξ1 + u1 δξ1 ξ˙3 = −ξ3 + ξ4 + cos( )u1 100 1 ˙ ξ4 = −ξ1 ξ2 + ( )u2 1 − δξ5 δξ1 ξ˙5 = −ξ5 + sin( )u2 50 y1 = ξ1 , y2 = ξ3 , with δξ5 6= 1. j δ ξi = ξi (t − jτ ) where τ = 0.5s
0 1 1 − δη
0
1 1 − δη
Then, the unknown inputs are given by: u1 = ( 1 )[z˙3 + z3 − z4 ] δz1 cos(100 ) u2 = (1 − δη)[z˙4 + z1 z2 ]
(29)
(31)
(33)
(34)
where u2 is, unfortunately, a function of the internal dynamics governed by: η˙ = −η + sin(
δz1 )[(1 − δη)(z˙4 + z1 z2 )] 50
(35)
Consequently, u can be estimated only after that ηˆ converges to η. As it can be seen the internal dynamic is a function of the states and their derivatives, which are estimated using the following High-Order Sliding Mode observer: 2 ˙ z1,0 − y1 ) z1,0 − y1 |3 sign(ˆ zˆ1,0 = zˆ1,1 − λ1,0 |ˆ 2 z1,1 − zˆ˙1,0 ) zˆ˙1,1 = zˆ1,2 − λ1,1 |ˆ z1,1 − zˆ˙1,0 |3 sign(ˆ ˙ z1,2 − zˆ˙1,1 ) zˆ1,2 = −λ1,2 sign(ˆ 1 (36) zˆ˙2,0 = zˆ2,1 − λ2,0 |ˆ z2,0 − y2 |2 sign(ˆ z2,0 − y2 ) ˙ ˙ zˆ2,1 = −λ2,1 sign(ˆ z2,1 − zˆ2,0 ) 1 zˆ˙ = zˆ − λ |ˆ z3,0 − y3 ) 3,0 3,1 3,0 z3,0 − y3 |2 sign(ˆ ˙zˆ3,1 = −λ3,1 sign(ˆ ˙ z3,1 − zˆ3,0 )
Following the first step of the algorithm proposed in [22], we obtain ! 1 0 δξ1 Γγ = ) 0 cos( 100 and this matrix is not full rank (rankK(δ] Γγ = 1 < m), then the unknown input u2 cannot be estimated. To solve this problem, we have to used one more step the algorithm given in [22] to generate a new virtual output. Thus, we find y¯3 = ξ4 = y2 + y˙ 2 − (¨ y1 + y˙ 1 + δy1 )cos(
0
It can be clearly seen that rankK(δ] Γγ = m = 2 then we can choose the matrix Qγ as 0 1 0 Qγ = (32) 0 0 1
ˆ ¯γ = Γ ¯ γ (ˆ ˆ γ = Qγ (ˆ z , ηˆ, δ) and Q η , zˆ, δ) and the where Γ matrices Hγ , Ψγ are given in the equation (20). IV. N UMERICAL
(30)
δy1 ) 100
which is a combination of the outputs and their derivatives and gives the reconstruction of ξ4 , only with known variables at the previous step. Thus, the internal dynamics is reduced to ξ5 . From section III, the system is rewritten as follows:
As (36) is a finite time observer, there exists a finite time T such that for all t > T we have zˆ1,0 = y1 , zˆ1,1 = y˙ 1 , zˆ1,2 = y¨1 , zˆ2,0 = y2 , and zˆ2,1 = y˙ 2 thus yˆ3 = z4 = zˆ3,0 .
473
Now, we design the following estimator for the internal dynamic: δˆ z1 )[(1 − δ ηˆ)(zˆ˙4 + zˆ1 zˆ2 )] ηˆ˙ = −ˆ η + sin( 50
R EFERENCES [1] Barbot J.-P., Boutat D., Busawon K., and Mangal K. 2014. High Order Sliding Mode Differentiator for Dynamical Inversion of Non-Involutive Systems. IFAC World Congress. [2] Bhat K. and Koivo, H., 1976. An observer theory for time delay systems, IEEE Trans. Automat. Control, vol. 21, pp. 266-269. [3] Daoutidis P., and Kravaris C., 1991. Inversion and zero dynamics in nonlinear multivariable control, AICHE J, vol. 527, pp. 583-603. [4] Germani A., Manes C., and Pepe P., 1996. Linearization of input-output mapping for nonlinear delay systems via static state feedback, CESA, IMACS multiconference. ive, Int. J. Control, vol. 75, pp. 582-590. [5] Germani A., Manes C., and Pepe P., 2001. An asymptotic state observer for a class of Nonlinear Delay Systems. Kybernetika, vol. 37, pp. 459478. [6] Hirschorn K.M., 1979, Invertibility of multivariable nonlinear control systems, IEEE Trans. Automat. Control, vol. 24, pp. 855-865. [7] Hou M., and Patton R. T., 2002. An observer design for linear timedelay systems. IEEE Trans. Automat. Control, vol. 47, pp. 121-125. [8] Ibrir S., 2009. Adaptative observers for time delay systems in traingular form. Automatica, vol. 45, pp. 2392-2399. [9] Jeˇzek J., 1996. Rings of skew polynomials in algebraical approach to control theory. Kybernetika, vol. 32, pp. 63-80. [10] Kharitonov V. L., and Hinrichsen D., 2004. Exponential estimates for time delay systems. Syst. Control Lett, vol. 53, pp. 395-405. [11] Krener A., 1985. (Adf,g ), (adf,g ) and locally (adf,g ) invariant and controllability distributions. SIAM Journal on Control and Optimization, vol. 23, pp. 523-549. [12] Levant A., 1998. Robust Exact Differentiation via Sliding Mode Technique. Automatica, vol. 34, pp. 379-941. [13] Oguchi T., 2007. A finite spectrum assignment for retarded non-linear systems and its solvability condition, Int. J. Control, vol. 80, pp. 898907. [14] Oguchi T., Watanabe A., and Nakamizo T., 2002, Input-output linearization of retarded non-linear systems by using an extension of Lie derivat [15] Respondek W., 1990.Right and left Invertibility of Nonlinear Control Systems, in Nonlinear Controlability and Optimal control, ed Sussmunn (Mancel Dekker, New Yourk), pp. 133-176. [16] Sain M. K. and Massey J.L., 1969, Invertibility of linear timeinvariant dynamical systems of multivariable linear systems, IEEE Trans. Automat. Control, vol. 14, pp. 141-149. [17] Singh S. N., 1981, A modified algorithm for invertibility in nonlinear systems, IEEE Trans. Automat. Control, vol. 26, pp. 595-598. [18] Silverman L.M., 1969, Inversion of multivariable linear systems, IEEE Trans. Automat. Control, vol. 14, pp. 270-276. [19] Xia X., Marquez L., Zagalak P., and Moog C., 2002. Analysis of nonlinear time-delay systems using modules over non-commutative rings. Automatica, vol. 38, pp. 1549-1555. [20] Zheng G., Barbot J.-P., Boutat D., 2013. Identification of the delay parameter for nonlinear time-delay systems with unknown inputs, Automatica, vol. 49, pp. 1755-1760. [21] Zheng G., Barbot J.-P., Boutat D., Floquet T., and Richard J.,P., 2010. Finite time observation of nonlinear time-delay systems with unknown inputs. IFAC on Nonlinear Control Systems. [22] Zheng G., Barbot J.-P., Floquet T., Boutat D., and Richard, J.-P., 2011. On observation of time-delay systems with unknown inputs. IEEE Trans. Automat. Control, vol. 56, pp. 1973-1978.
(37)
Choosing a Lyapunov function V (eη ) =
1 T e eη 2 η
(38)
with eη = η − ηˆ, we obtain: V˙ = −(η − ηˆ)2 − (η − ηˆ)[δ(η − ηˆ)f (e) + f (e)]
(39)
δ zˆ1 1 with f (e) = sin( δz ˆ˙4 + zˆ1 zˆ2 )]. 50 )(z˙ 4 + z1 z2 ) − sin( 50 )(z When t > T we have e = 0, i.e. f (e) = 0, so V˙ = −(η − ηˆ)2 < 0, thus Assumption 1 is verified. Then the unknown inputs can be causally estimated from the following algebraic relation: 1 u )[zˆ˙3 + zˆ3 − zˆ4 ] ˆ1 = ( δ zˆ1 ) cos(100 (40) ˙ u ˆ2 = (1 − δ ηˆ)[zˆ4 + zˆ1 zˆ2 ]
control u2 and its estimate
control u1 and its estimate
For the simulation, the initial conditions for the studied example are chosen as z0 = (1, 1, 3, 5, 1)T , and the initial conditions for the observer are fixed to zero. The gains for the high-order sliding mode observer are : λ1,0 = 5, λ1,1 = 2, λ1,2 = 1.9, λ2,0 = 2.5, λ2,1 = 3, λ3,0 = 6, λ3,1 = 5. The simulation results are depicted in Fig. 1, where it clearly shows the convergence of the estimation of the unknown input. 1 actual estimated
0.5 0 −0.5 −1
0
5
10
15 Time(s)
20
25
30
1 actual estimated
0.5 0 −0.5 −1
0
5
Fig. 1.
10
15 Time(s)
20
25
30
Estimation of unknown inputs u1 and u2 .
V. C ONCLUSION This paper proposed a solution based on high-order sliding mode observer to solve the left inversion problem of nonlinear time delay system with unknown inputs. The delay is considered as constant, and under the assumption on the internal dynamics, the left invertibility problem can be solved. The problem of causality and non causality for the estimation of the unknown inputs are discussed. 474
Left inversion of nonlinear time delay system Zohra Kader, Gang Zheng and Jean-Pierre Barbot
and estimate its state, one solution is to rewrite this internal dynamics such that its dynamic becomes independent of the unknown input [1]. Thus in the paper we will adapt this new way to determine the internal dynamics for the nonlinear systems with time delay. This method is based on the finite time convergence by using the existing observer for time delay system in the literature [5], [7], [8], [10], [2] and the high order sliding mode proposed in [12]is applied in this paper. The paper is organized as follows: In the next section the algebraic framework and some notations are recalled. In section III, we firstly propose a canonical form based on the previously proposed algorithm in [21]. After that the inverse dynamics is computed and sufficient condition for causal and non causal left invertibility are discussed. This section ends with an observer scheme dedicated to the proposed canonical form. In section IV two examples and numerical results highlight the feasibility and the efficiency of the proposed approach.
Abstract— This paper investigates the left invertibility for nonlinear time delay system with internal dynamics. Under the assumption imposed on the internal dynamics, it has been shown that the unknown inputs can be estimated. Causal and non causal estimation of the unknown inputs are respectively discussed, and the high-order sliding mode observer is used to estimate the observable states.
I. I NTRODUCTION In many applications, recovering of unknown inputs from the outputs is crucial. This may occur for example in data secure transmission, where the unknown input is the message or in fault detection and isolation where the fault is the unknown input. It is the reason why this problem was studied since at least forty five years ago in linear control theory [16], [18] and thirty five year ago in nonlinear control theory [6], [17]. Most of those works are in the context of nonlinear systems without time delays. Considering nonlinear systems with time delays, an important tool based on noncommutative ring was introduced in [19]. This tool of non commutative ring [9] is used to model nonlinear time delayed system in algebraic framework, and many results are already obtained. The notions of Lie derivatives and relative degree are defined, and the differences between the causal and noncausal invertibility are clarified in [22]. The canonical form of invertibility is also given in [21], and in [20] a method for estimating of the unknown inputs is proposed. However, the algorithm for left invertibility proposed in [21] is only for the system without internal dynamics. For system with or without time delay, the main difficulty when the internal dynamics (or inverse dynamics [3]) occurs is to estimate the state of such dynamics. One interesting solution in order to overcome such a difficulty is to allow the derivative of the unknown inputs [15] with a geometrical approach and with an algebraical one. If however the input derivatives are not possible, then it is necessary to compute and analyze the internal dynamics. For nonlinear systems without delay, if the vector fields associated to the inputs verify some involutivity properties, then the internal dynamic does not depend on the unknown input. However, this rule is not valid for nonlinear systems with time delay. In order to analyze the internal dynamics
II. A LGEBRAIC
Consider the following class of nonlinear time delay system: m X gi (x(t − jτ ))uj (t) x ˙ = f (x(t − jτ )) + i=1
(1)
x = ψ(t), u(t) = ϕ(t) t ∈ −sτ, 0
with x ∈ W ⊂ ℜn is the state vector of the system, u(t) ∈ ℜm is the vector of its unknown input and y ∈ ℜp is the output. τ represents the basic commensurate time delay. In [19], an algebraic framework is developed, defining the field K of a finite number of the variables from: {xj (t − iτ ), j ∈ [1, n] , i ∈ [0, s] ⊂ N }
(2)
such that E represents the vector space over K : E = spanK {dξ : ξ ∈ K} and τ represents the basic commensurate time delay. Denote the operator δ as a backward shift operator, which means δ i ξ(t) = ξ(t − iτ ) and δ i (a(t)dξ(t)) = a(t − iτ )dξ(t − iτ ). With this operator, we can then define the following set of polynomials K(δ]: a(δ] = a0 (t) + a1 (t)δ + · · · + ara (t)δ ra , ai (t) ∈ K
This paper was supported by Ministry of Higher Education and Research, Nord-Pas de Calais Regional Council and FEDER through the Contrat de Projets Etat Region (CPER) CIA 2007-2013, and also supported by ARCIR Project ESTIREZ, Nord-Pas de Calais Regional Council. Zohra Kader and Gang Zheng are with Project Non-A, INRIA Lille-Nord Europe, 40 avenue Halley and LAGIS UMR CNRS 8219, Ecole Centrale de Lille, 59650 Villeneuve d’Ascq, France. Jean-Pierre Barbot is with ECSLab EA 3649, ENSEA, Cergy-Pontoise, and Non-A Team INRIA, France
(3)
The addition for the entries in K(δ] is defined as usual, but its multiplication is given by the following criteria: a(δ]b(δ] =
[email protected] 978-1-4673-6088-3/14/$31.00 ©2014 IEEE
FRAMEWORK AND NOTATIONS
rX a ,j≤rb a +rb i≤rX k=0
469
i+j=k
ai (t)bj (t − iτ )δ k
(4)
With the standard differential operator d, denote by M the left module over K(δ]: M = spanK(δ] {dξ, ξ ∈ K}
Based on the above notations, the relative degree can be defined in the following way. Definition 3: (Relative degree) System (6) has the relative degree (ν1 , · · · , νp ) in an open set W ⊆ Rn if the following conditions are satisfied for 1 ≤ i ≤ p: 1) for all x ∈ W , LGj Lrf hi = 0 for all 1 ≤ j ≤ m and 0 ≤ r < νi − 1; 2) there exists x ∈ W such that ∃j ∈ {1, · · · , m}, LGj Lνfi −1 hi 6= 0. If the first condition is satisfied for all r ≥ 0 and some i ∈ {1, · · · , p}, we set νi = ∞. Moreover, for system (6), one can also define observability indices [11] over non-commutative rings. For 1 ≤ k ≤ n, let Fk be the following left module over K(δ]: n o Fk := spanK(δ] dh, dLf h, · · · , dLfk−1 h .
(5)
Hence, K(δ] is a non-commutative ring satisfying the associative law, and it has been proved in [9] and [19] that it is a left Ore ring, which enables us to define the rank conception. Then, for any matrix with entries belonging to K(δ], we can introduce the following definition. Definition 1: (Unimodular matrix) A matrix A ∈ Kn×n (δ] is said to be unimodular over K(δ] if it has a left inversion A−1 ∈ Kn×n (δ], such that A−1 A = In×n . Based on the above algebraic framework a nonlinear time delay system can be represented in a compact algebraic form as follows: m X Gi (x, δ)ui (t) x˙ = f (x, δ) + i=1 (6) y = h(x, δ) x = ψ(t), u(t) = ϕ(t) t ∈ −sτ, 0
It was shown that the filtration of K(δ]-module satisfies F1 ⊂ F2 ⊂ · · · ⊂ Fn , then define d1 = rankK(δ] F1 , and dk = rankK(δ] Fk − rankK(δ] Fk−1 for 2 ≤ k ≤ n. Let ki = card {dk ≥ i, 1 ≤ k ≤ n}, then (k1 , · · · , kp ) are the observability indices. Reorder, if necessary, the output components of (6) so that
where the notation f (x, δ) means f (x, δ) = f (x, x(t − τ ), . . . , x(t − sτ ))) and the same is considered for G(x, δ) s P gij δ j with entries belonging to the and h(x(t), δ), Gi =
1 rankK(δ] { ∂h ∂x , · · · , = k1 + · · · + kp .
j=0
ring K(δ]. Since this paper studies the left invertibility of the nonlinear time-delay systems, the following definition will be used. Definition 2: (Invertibility) The unknown inputs u(t) can be estimated if it can be written as follows: u(t) = ϕu (δ j y, · · · , δ j y (k) )
,
∂hp ∂x , · · ·
k −1
,
∂Lf p hp } ∂x
Definition 4: (Change of coordinates) If there exists φ−1 ∈ Kn×1 and some constants c1 , · · · , cn ∈ N such that diag {δ ci } x = φ−1 (δ, z) then the change of coordinates z = φ(δ, x) ∈ Kn×1 is a causal change of coordinates over Kn×1 . The change of coordinate is bicausal over K if max = {ci } = 0, i.e x = φ−1 (δ, z).
(7)
for j ∈ Z and k ∈ Z + . It can be locally causally estimated if (7) is satisfied for j ∈ Z + and k ∈ Z + , and be locally noncausally estimated if (7) is satisfied for j ∈ Z and k ∈ Z + . The conventional concepts like Lie derivative and relative degree which are widely used in nonlinear systems without delay, are introduced in the case of nonlinear time delay systems by [4], [14], [13] and we can extend them by using the above algebraic framework. Let f (x(t − jτ )) and h(x(t − jτ )) for 0 ≤ j ≤ m be an n and p dimensional vector field respectively, with fr the entries of f belonging to K for 1 ≤ r ≤ n and hi ∈ K for 1 ≤ i ≤ p, with p ≥ m, and then ∂hi ∂hi ∂hi ,··· , = ∈ K1×n (δ] (8) ∂x ∂x1 ∂xn
Based on the above definition, the following result has been reported in [22], [20]. Theorem 1: For 1 ≤ i ≤ p, denote by ki the observability indices and νi the relative degree index for yi of (6), and note ρi = min(ki , νi ). Then there exists a causal change of coordinates φ(x, δ) ∈ Kn×1 , such that (6) can be transformed into the following form: ˙ ξi = Ai ξi + Bi µi (10) ζ˙ = α(ξi , ζ, δ) + β(ξi , ζ, δ)u y = Ci ξi where Ai and Bi arei in the Brunovsky form. ξi = h T T = ξi,1 , ξi,2 , . . . , ξi,ρi hi , Lf hi , . . . Lρfi −1 hi , such that Lf hi is the Lie derivative of the output hi along the vector field f , and m P LGj Lρfi −1 hi (x, δ)uj ∈ K µi = Lρfi hi (x, δ) +
Where for 1 ≤ r ≤ n: s
X ∂hi ∂hi = δ j ∈ K(δ] ∂xr ∂x (t − jτ ) r j=0 n X s X ∂hi ∂hi (f ) = ( δ j ) (fr ) L f hi = ∂x ∂x (t − jτ ) r r=1 j=0
k −1
∂Lf 1 h1 ,··· ∂x
j=1
α∈K
l×1
,β∈K
l×1
where l = n −
p P
ρi and the change
j=1
of coordinates is given as: φT (x, δ) = (φT1 (x, δ), φT2 (x, δ)), with
(9)
φT1 (x, δ) =
and in the same way we can define LGi hi 470
ρ
h1 , Lf h1 , . . . , Lρf1 h1 , . . . hp , Lf hp , . . . , Lfp hp
and φ2 (x, δ) is an (n−l) dimensional complementary change of coordinates of φ1 (x, δ). Consider now the last dynamics of ξ˙i and the one of ζ˙ in (10), one can obtain the following dynamics: ( ˙ = Ψ(ξ, ζ, δ) + Γ(ξ, ζ, δ)u H(ξ) (11) ˙ζ = f¯(ζ, ξ, δ, u)
III. L EFT INVERTIBILITY
This section will firstly discuss how to iteratively obtain a new canonical form for the non-trivial case, and secondly study the left invertibility of the unknown inputs with internal dynamics in the deduced canonical form. Finally a high order sliding mode observer is designed to estimate the state variables and their derivatives in order to estimate the unknown inputs.
where
A. Canonical form
Γ(ξ, ζ, δ) =
ρ −1 LG L 1 h1 1 f . . . ρp −1 LG L hp 1 f
Lρ1 h
Ψ=
WITH INTERNAL DYNAMICS
.
.
.
···
ρ −1 LG L 1 h1 m f . . . ρp −1 LG L hp m f
1 f ρ L 2 h2 f
. . . ρp hp L f
···
|x=Φ−1 (ξ,ζ,δ)
,H =
(ρ ) ξ1 1 (ρ ) ξ2 2 . . . (ρp ) ξp
|x=Φ−1 (ξ,ζ,δ)
ρ −1
Φ = {dh1 , · · · , dLfρ1 −1 h1 , · · · , dhp , · · · , dLfp
Remark 1: Passing from (10) to (11) , the substitution of x = Φ−1 (ξ, δ) must be done. Indeed if the change of coordinates is not bicausal, this may generate non causal terms in the equation (11). Let us consider the following system: x˙ 1 = δx2
hp } (13)
and n o ρ −1 £ = spanR[δ] h1 , · · · , Lfρ1 −1 h1 , · · · , hp , · · · , Lfp hp (14) where R[δ] is the commutative ring of polynomials in δ with coefficients belonging to the field R, and let £(δ] be the set of polynomials in δ with coefficients over £. The module spanned by element of Φ over £(δ] is defined as follows:
x˙ 2 = a1 x1 + a2 x2 + a3 x3 + x1 u
x˙ = b x + b x + b3 x3 y 3 = x1 , 1y =2x 2 1 1 2 3
choosing the change of coordinates as: z1 = x1 , z2 = Lf h1 = δx2 , and z3 = x3 then we obtain: z˙1 = z2 −1
Ω = span£(δ] {ξ, ξ ∈ Φ} .
(15)
G = spanR[δ] {G1 , . . . , Gm }
(16)
Define where Gi is given in (6), and its left annihilator:
z˙2 = δ(a1 z1 + a2 δ z2 + a3 z3 + z1 u) −1 z˙3 = b1 z1 + b2 δ z2 + b3 z3 y1 = z 1 , y2 = z 3
G ⊥ = span£(δ] {ω ∈ M | ωβ = 0, ∀β ∈ G},
(17)
where M is defined in (5). Then the new outputs can be virtually obtained if the sufficient condition rankK (Hγ ) = lγ is satisfied, where /£ (18) Hγ = spanR[δ] ω ∈ G ⊥ ∩ Ω|ωf ∈
and the obtained system is not causal. Noticing that the system (10) contains an internal dy˙ For the trivial case where dim ζ = 0, i.e. namics ζ. rankK(δ] φ1 (x, δ) = n, if rankK(δ] Γ(ξ, δ) = m, then there ¯ ∈ always exists a matrix Q ∈ Km×p (δ] such that QΓ = Γ ¯ = m. In this case, according to Km×m (δ] with rankK(δ] Γ (11), one can obtain the estimation of u as follows: ˆ δ) − Ψ(ξ, ˆ δ)). ¯ −1 Q(H(ξ, u ˆ=Γ
Considering system (6), if rankK(δ] φ1 (x, δ) < n and rankK(δ] Γ(ξ, δ) < m, an algorithm was developed in [22] to generate lγ new outputs of the systems, which are combinations of y, its derivatives and their Pp backward shifts. Without loss of generality, suppose i=1 ρi = j, ρ −1 thus {dh1 , · · · , dLfρ1 −1 h1 , · · · , dhp , · · · , dLfp hp } are j linearly independent vectors over K(δ]. Then note:
More explicitly, the new generated outputs are noted as: y¯i = ωi f mod £, for 1 ≤ i ≤ l − γ and ωi ∈ Hγ . The above algorithm will be iterated until rankK(δ] φ1 (x, δ) = n or rankK(δ] Γγ (z, δ) = m. For the final obtained canonical form, if rankK(δ] φ1 (x, δ) = n and rankK(δ] Γγ (z, δ) < m, then the problem of left invertibility for the unknown inputs cannot be solved. For the second case where rankK(δ] φ1 (x, δ) < n and rankK(δ] Γγ (z, δ) = m, after lγ iterations, the obtained canonical form can be written as follows:
(12)
where ξˆ represents the estimation of ξ, which in fact can be estimated by using many existing methods, like sliding mode observer, algebraic observer and so on. Moreover, if ¯ is unimodular over K(δ], the above estimation the matrix Γ of u is obviously causal. Considering the non-trivial case, where rankK(δ] φ1 (x, δ) < n and rankK(δ] Γ(ξ, δ) < m, the internal dynamics ζ˙ in (10) is not vanished, thus the invertibility of the unknown inputs depends on the internal dynamics. The following section is devoted to treating this non-trivial case.
8 z˙i,j = zi,j+1 , for 1 ≤ i ≤ p + lγ and 1 ≤ j ≤ θi − 1 > > m > X < z˙i,θi = bi (z, η, δ) + ai,j (z, η, δ)uj , for 1 ≤ i ≤ p + lγ > > j=1 > : η˙ = f¯γ (η, z, δ, u) (19)
with ai,j = Lgj Lfθi −1 hi and bi (z, δ, η) = Lθfi hi , where lγ is the number of new outputs, θi is the minimum value 471
of the observability index and relative degree for the ith output. Then the second and the third equation in (19) can be rewritten in the following compact way:
defined dynamics (23) can asymptotically estimate η. This is due to the existence of a Lyapounov function V (e) such that V˙ (e) < 0, imposed by Assumption 1.
Hγ (z) ˙ = Ψγ (z, η, δ) + Γγ (z, η, δ)u η˙ = f¯γ (η, z, u, δ)
After the convergence of ηˆ to η, the unknown input u ¯γ ∈ of (6) can then be calculated via (28). Moreover, if Γ m×m −1 m×m ¯ K (δ] is unimodular over K(δ], i.e. Γγ ∈ K (δ], since Qγ ∈ K(δ]m×(p+lγ ) , Hγ (z, η, δ) ∈ K(δ](p+lγ )×1 and Ψγ (z, η, δ) ∈ K(δ](p+lγ )×1 , then from (28) we have u ∈ Km×1 (δ], which implies the causal estimation of the unknown inputs.
(20)
where = Γγ (z, η, δ) θ1 −1
LG L h1 1 f . . . θp −1 LG L hp 1 f θp+1 −1 LG L hp+1 1 f θp+2 −1 LG L hp+2 1 f
··· . . . ··· ··· ···
θ −1 LG L 1 h1 m f . . . θp −1 LG L hp m f θp +1 LG L hp+1 m f θp+2 −1 LG L hp+2 m f
. . . . . . . . . θp+l −1 θp+l −1 γ γ ··· LG L hp+lγ hp+l LG L m f γ 1 f (θ1 ) z 1,θ1 (θ ) z 2 2,θ2 . . . (θp ) γ z p,θp . . . (θp+l ) γ z p+lγ ,θp+l γ 1 0 θ L 1 h1 f C B θ2 C B L h2 C B f C B C B C B . C B . C B . C B C B θp C B γ L hp C B f C B C B C B . C B . C B C . B A @ θ p+lγ L hp+l f γ |x=φ−1 (z,δ)
Theorem 2 indicates that the causality of the estimation for ¯ γ ∈ Km×m (δ]. the unknown inputs depends on the matrix Γ If this matrix is not unimodular over K(δ], then the estimation of u might be non causal. In order to take into account the non causal estimation, let us introduce the forward shift operator ∇ such that ∇j x(t) = x(t + jτ ). ¯ as the field of functions of finite number of Denote K variables from {xi (t − jτ ), i ∈ [1, n] , j ∈ [−s, s]}, then one ¯ n×1 (δ, ∇) whose entry is of the following can define the set K form:
|x=φ−1 (z,δ)
,
H (z) ˙ =
a1 (t)∇+a0 (t)+a1 (t)δ · · ·+ara (t)δ ra a(δ, ∇] = a ¯ra¯ (t)+· · ·+¯ (25) ¯ Then, for K(δ, ∇], where all the coefficients belong to K. the addition is as usual, but the multiplication is given by the following relation:
Ψ (z, η, δ) =
a(δ, ∇]b(δ, ∇]
B. Left inversion For the iteratively obtained canonical form (19) (or the compact form (20)), if rankK(δ] Γγ = m, then there always ¯γ ∈ exists a matrix Qγ ∈ K(δ]m×(p+lγ ) such that Qγ Γγ = Γ ¯ γ = m. Then we can obtain: Km×m (δ] with rankK(δ] Γ ¯ −1 u=Γ ˙ − Ψγ (z, η, δ)) γ Qγ (Hγ (z)
+
i=0 j=0 rb ra ¯ P P
ai δ i bj δ i+j +
r¯ ra P b P
ai δ i¯bj δ i ∇j
i=0 j=1 r¯ ra ¯ P b P
a ¯ i ∇i b j ∇i δ j +
a ¯i ∇i¯bj ∇i+j .
i=1 j=1
(26)
Finally, we have the following result. ¯ γ ∈ Km×m (δ] is Lemma 1: Under Assumption 1, if Γ unimodular over K(δ, ∇], then the unknown input u of the system (6) can be at least non-casually estimated.
(21)
The proof of this lemma is quite similar to that of Theorem 2, thus the following will just explain the main differ¯ γ is unimodular over K(δ, ∇], i.e. ence. If the matrix Γ ¯ m×m (δ, ∇], then the causality of the estimation ¯ −1 ∈ K Γ γ for the unknown input u depends on the power of the terms ¯ −1 Γ ˙ Ψγ (z, η, δ) in (20). γ (z, η, δ, ∇), Qγ (z, η, δ, ∇), Hγ (z), However, the unknown input u of the system (6) can be at least non-causally estimated.
(22)
for which we can propose the following estimator: η , zˆ, δ, zˆp+lγ ,θp +lγ ) ηˆ˙ = f¯γ (ˆ
rb ra P P
i=1 j=0
Thus, the dynamics of η in (20) can be rewritten in the following form: η˙ = f¯γ (η, z, δ, z˙ p+lγ ,θp+lγ )
=
(23)
where zˆ represents the estimation of z. Assumption 1: It is assumed that there exists a Lyapunov function V (e) with e = η − ηˆ such that for all δ and z: ∂V ¯ η , z, δ)] < 0 (24) [fγ (η, z, δ) − f¯γ (ˆ V˙ = ∂e Then we are able to state the following result. ¯ γ ∈ Km×m (δ] is Theorem 2: Under Assumption 1, if Γ unimodular over K(δ], then the unknown input u of the system (6) can be casually estimated. For system (6), since we can use the existing method to estimate z and its derivative in a finite time T , then the
C. Observer design As we have explained, the estimation of the unknown input is based on the finite-time estimation of the state variable z and their derivatives, which is also used to estimate the internal dynamics η. In what follows, we use the high order 472
sliding mode observer [12] to estimate z and its derivatives:
z˙1 = z2 z˙2 = −z2 − δz1 + u1 δz1 z˙3 = −z3 + z4 + cos(100)u1 1 z˙4 = −z1 z2 + ( )u2 1 − δη η˙ = −η + sin(δz1 )u 2 50 y1 = z1 , y2 = z3 , y¯ = z4 , δη 6= 1.
8 θi > > zi,j − zi,j ) zi,j − zi,j |θi +1 sign(ˆ >zˆ˙i,j = zˆi,j+1 − λi,j |ˆ > > > ˙ > zˆi,j+1 = zˆi,j+2 > > > θi −1 > > > > −λi,j+1 |ˆ zi,j+1 − zˆ˙i,j | θi sign(ˆ zi,j+1 − zˆ˙i,j ) < .. .. > .=. > > > > > z ˆ˙i,θi −1 = zˆi,θi > > > 1 > > −λi,θi −1 |ˆ zi,θi −1 − zˆ˙i,θi −2 |2 sign(ˆ zi,θi −1 − zˆ˙i,θi −2 ) > > > :˙ zi,θi − zˆ˙i,θi −1 ) zˆi,θi = −λi,θi sign(ˆ (27)
The matrix Γγ is directly obtained as:
The convergence of the above observer has already been demonstrated in [12]. It has been shown that, if the rth (r is the relative degree) derivative of the output has a Lipschitz constant L > 0 and by choosing λ > L, then we have zˆ = z after a finite time T . Then, the internal dynamics η can be estimated by using the estimator (23). After η was converged, the unknown input can be estimated from the following algebraic equation: ˆ ¯ γ )−1 Q ˆ γ (Hγ (z) ˆ˙ − Ψγ (ˆ uˆ = (Γ z , ηˆ, δ)).
1 cos( δz1 ) 100 Γγ = 0
(28)
this gives the unimodular matrix: δz1 ) cos( 100 ¯ Γγ = 0
EXAMPLE
Let us consider the nonlinear time delay system: ξ˙1 = ξ2 ξ˙2 = −ξ2 − δξ1 + u1 δξ1 ξ˙3 = −ξ3 + ξ4 + cos( )u1 100 1 ˙ ξ4 = −ξ1 ξ2 + ( )u2 1 − δξ5 δξ1 ξ˙5 = −ξ5 + sin( )u2 50 y1 = ξ1 , y2 = ξ3 , with δξ5 6= 1. j δ ξi = ξi (t − jτ ) where τ = 0.5s
0 1 1 − δη
0
1 1 − δη
Then, the unknown inputs are given by: u1 = ( 1 )[z˙3 + z3 − z4 ] δz1 cos(100 ) u2 = (1 − δη)[z˙4 + z1 z2 ]
(29)
(31)
(33)
(34)
where u2 is, unfortunately, a function of the internal dynamics governed by: η˙ = −η + sin(
δz1 )[(1 − δη)(z˙4 + z1 z2 )] 50
(35)
Consequently, u can be estimated only after that ηˆ converges to η. As it can be seen the internal dynamic is a function of the states and their derivatives, which are estimated using the following High-Order Sliding Mode observer: 2 ˙ z1,0 − y1 ) z1,0 − y1 |3 sign(ˆ zˆ1,0 = zˆ1,1 − λ1,0 |ˆ 2 z1,1 − zˆ˙1,0 ) zˆ˙1,1 = zˆ1,2 − λ1,1 |ˆ z1,1 − zˆ˙1,0 |3 sign(ˆ ˙ z1,2 − zˆ˙1,1 ) zˆ1,2 = −λ1,2 sign(ˆ 1 (36) zˆ˙2,0 = zˆ2,1 − λ2,0 |ˆ z2,0 − y2 |2 sign(ˆ z2,0 − y2 ) ˙ ˙ zˆ2,1 = −λ2,1 sign(ˆ z2,1 − zˆ2,0 ) 1 zˆ˙ = zˆ − λ |ˆ z3,0 − y3 ) 3,0 3,1 3,0 z3,0 − y3 |2 sign(ˆ ˙zˆ3,1 = −λ3,1 sign(ˆ ˙ z3,1 − zˆ3,0 )
Following the first step of the algorithm proposed in [22], we obtain ! 1 0 δξ1 Γγ = ) 0 cos( 100 and this matrix is not full rank (rankK(δ] Γγ = 1 < m), then the unknown input u2 cannot be estimated. To solve this problem, we have to used one more step the algorithm given in [22] to generate a new virtual output. Thus, we find y¯3 = ξ4 = y2 + y˙ 2 − (¨ y1 + y˙ 1 + δy1 )cos(
0
It can be clearly seen that rankK(δ] Γγ = m = 2 then we can choose the matrix Qγ as 0 1 0 Qγ = (32) 0 0 1
ˆ ¯γ = Γ ¯ γ (ˆ ˆ γ = Qγ (ˆ z , ηˆ, δ) and Q η , zˆ, δ) and the where Γ matrices Hγ , Ψγ are given in the equation (20). IV. N UMERICAL
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δy1 ) 100
which is a combination of the outputs and their derivatives and gives the reconstruction of ξ4 , only with known variables at the previous step. Thus, the internal dynamics is reduced to ξ5 . From section III, the system is rewritten as follows:
As (36) is a finite time observer, there exists a finite time T such that for all t > T we have zˆ1,0 = y1 , zˆ1,1 = y˙ 1 , zˆ1,2 = y¨1 , zˆ2,0 = y2 , and zˆ2,1 = y˙ 2 thus yˆ3 = z4 = zˆ3,0 .
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Now, we design the following estimator for the internal dynamic: δˆ z1 )[(1 − δ ηˆ)(zˆ˙4 + zˆ1 zˆ2 )] ηˆ˙ = −ˆ η + sin( 50
R EFERENCES [1] Barbot J.-P., Boutat D., Busawon K., and Mangal K. 2014. High Order Sliding Mode Differentiator for Dynamical Inversion of Non-Involutive Systems. IFAC World Congress. [2] Bhat K. and Koivo, H., 1976. An observer theory for time delay systems, IEEE Trans. Automat. Control, vol. 21, pp. 266-269. [3] Daoutidis P., and Kravaris C., 1991. Inversion and zero dynamics in nonlinear multivariable control, AICHE J, vol. 527, pp. 583-603. [4] Germani A., Manes C., and Pepe P., 1996. Linearization of input-output mapping for nonlinear delay systems via static state feedback, CESA, IMACS multiconference. ive, Int. J. Control, vol. 75, pp. 582-590. [5] Germani A., Manes C., and Pepe P., 2001. An asymptotic state observer for a class of Nonlinear Delay Systems. Kybernetika, vol. 37, pp. 459478. [6] Hirschorn K.M., 1979, Invertibility of multivariable nonlinear control systems, IEEE Trans. Automat. Control, vol. 24, pp. 855-865. [7] Hou M., and Patton R. T., 2002. An observer design for linear timedelay systems. IEEE Trans. Automat. Control, vol. 47, pp. 121-125. [8] Ibrir S., 2009. Adaptative observers for time delay systems in traingular form. Automatica, vol. 45, pp. 2392-2399. [9] Jeˇzek J., 1996. Rings of skew polynomials in algebraical approach to control theory. Kybernetika, vol. 32, pp. 63-80. [10] Kharitonov V. L., and Hinrichsen D., 2004. Exponential estimates for time delay systems. Syst. Control Lett, vol. 53, pp. 395-405. [11] Krener A., 1985. (Adf,g ), (adf,g ) and locally (adf,g ) invariant and controllability distributions. SIAM Journal on Control and Optimization, vol. 23, pp. 523-549. [12] Levant A., 1998. Robust Exact Differentiation via Sliding Mode Technique. Automatica, vol. 34, pp. 379-941. [13] Oguchi T., 2007. A finite spectrum assignment for retarded non-linear systems and its solvability condition, Int. J. Control, vol. 80, pp. 898907. [14] Oguchi T., Watanabe A., and Nakamizo T., 2002, Input-output linearization of retarded non-linear systems by using an extension of Lie derivat [15] Respondek W., 1990.Right and left Invertibility of Nonlinear Control Systems, in Nonlinear Controlability and Optimal control, ed Sussmunn (Mancel Dekker, New Yourk), pp. 133-176. [16] Sain M. K. and Massey J.L., 1969, Invertibility of linear timeinvariant dynamical systems of multivariable linear systems, IEEE Trans. Automat. Control, vol. 14, pp. 141-149. [17] Singh S. N., 1981, A modified algorithm for invertibility in nonlinear systems, IEEE Trans. Automat. Control, vol. 26, pp. 595-598. [18] Silverman L.M., 1969, Inversion of multivariable linear systems, IEEE Trans. Automat. Control, vol. 14, pp. 270-276. [19] Xia X., Marquez L., Zagalak P., and Moog C., 2002. Analysis of nonlinear time-delay systems using modules over non-commutative rings. Automatica, vol. 38, pp. 1549-1555. [20] Zheng G., Barbot J.-P., Boutat D., 2013. Identification of the delay parameter for nonlinear time-delay systems with unknown inputs, Automatica, vol. 49, pp. 1755-1760. [21] Zheng G., Barbot J.-P., Boutat D., Floquet T., and Richard J.,P., 2010. Finite time observation of nonlinear time-delay systems with unknown inputs. IFAC on Nonlinear Control Systems. [22] Zheng G., Barbot J.-P., Floquet T., Boutat D., and Richard, J.-P., 2011. On observation of time-delay systems with unknown inputs. IEEE Trans. Automat. Control, vol. 56, pp. 1973-1978.
(37)
Choosing a Lyapunov function V (eη ) =
1 T e eη 2 η
(38)
with eη = η − ηˆ, we obtain: V˙ = −(η − ηˆ)2 − (η − ηˆ)[δ(η − ηˆ)f (e) + f (e)]
(39)
δ zˆ1 1 with f (e) = sin( δz ˆ˙4 + zˆ1 zˆ2 )]. 50 )(z˙ 4 + z1 z2 ) − sin( 50 )(z When t > T we have e = 0, i.e. f (e) = 0, so V˙ = −(η − ηˆ)2 < 0, thus Assumption 1 is verified. Then the unknown inputs can be causally estimated from the following algebraic relation: 1 u )[zˆ˙3 + zˆ3 − zˆ4 ] ˆ1 = ( δ zˆ1 ) cos(100 (40) ˙ u ˆ2 = (1 − δ ηˆ)[zˆ4 + zˆ1 zˆ2 ]
control u2 and its estimate
control u1 and its estimate
For the simulation, the initial conditions for the studied example are chosen as z0 = (1, 1, 3, 5, 1)T , and the initial conditions for the observer are fixed to zero. The gains for the high-order sliding mode observer are : λ1,0 = 5, λ1,1 = 2, λ1,2 = 1.9, λ2,0 = 2.5, λ2,1 = 3, λ3,0 = 6, λ3,1 = 5. The simulation results are depicted in Fig. 1, where it clearly shows the convergence of the estimation of the unknown input. 1 actual estimated
0.5 0 −0.5 −1
0
5
10
15 Time(s)
20
25
30
1 actual estimated
0.5 0 −0.5 −1
0
5
Fig. 1.
10
15 Time(s)
20
25
30
Estimation of unknown inputs u1 and u2 .
V. C ONCLUSION This paper proposed a solution based on high-order sliding mode observer to solve the left inversion problem of nonlinear time delay system with unknown inputs. The delay is considered as constant, and under the assumption on the internal dynamics, the left invertibility problem can be solved. The problem of causality and non causality for the estimation of the unknown inputs are discussed. 474
