Non-Asymptotic State and Disturbance Estimation for a Class of Triangular Nonlinear Systems using Modulating
Functions
Yasmine Marani
1, Ibrahima N’Doye
1, and Taous-Meriem Laleg-Kirati
2,11Computer, Electrical and Mathematical Science and Engineering Divi- sion (CEMSE), King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia (e-mail: [email protected];
[email protected]; [email protected])
2National Institute for Research in Digital Science and Technology, Paris- Saclay, France.
Abstract
Dynamical models are often corrupted by model uncertainties, ex- ternal disturbances, and measurement noise. These factors affect the performance of model-based observers and as a result, affect the closed- loop performance. Therefore, it is critical to develop robust model- based estimators that reconstruct both the states and the model dis- turbances while mitigating the effect of measurement noise in order to ensure good system monitoring and closed-loop performance when de- signing controllers. In this article, a robust step by step non-asymptotic observer for triangular nonlinear systems for the joint estimation of the state and the disturbance is developed. The proposed approach provides a sequential estimation of the states and the disturbance in finite time using smooth modulating functions. The robustness of the proposed observer is both in the sense of model disturbances and mea- surement noise. In fact, the structure of triangular systems combined with the modulating function-based method allows the estimation of the states independently of model disturbances and the integral oper- ator involved in the modulating function-based method mitigates the noise. Additionally, the modulating function method shifts the deriva- tive from the noisy output to the smooth modulating function which strengthens its robustness properties. The applicability of the proposed modulating function-based estimator is illustrated in numerical simu- lations and compared to a second-order sliding mode super twisting observer under different measurement noise levels.
Key words Modulating functions, non-asymptotic estimation, nonlinear systems, disturbance estimation.
arXiv:2306.07620v1 [eess.SY] 13 Jun 2023
1 Introduction
Physical systems can be described by mathematical models that are often subject to disturbances caused by modeling uncertainties, faults, or muta- tions in the system’s behavior [1]. In bioreactors, the mathematical model is subject to model uncertainty in the form of a partially known nonlinear function of the state called the reaction rate [2]. Another example is the binary distillation column systems in which the dynamics of the feed com- positions is unknown [3]. The presence of unknown model disturbances and measurement noise affect the state estimation and make the model based observer design task challenging. As a result, the closed-loop performance of the system is affected. That being the case, a robust estimation algorithm that jointly estimates the states and the disturbance is required to ensure good closed-loop performance.
The simultaneous estimation of the states and the disturbance for nonlin- ear systems affected by model disturbances has been addressed in a number of papers. For instance, an Extended Kalman Filter with Unknown Input (EKF-UI) is developed in [4] and in [5] an adaptive observer for a nonlin- ear system with Lipschitz non-linearities is proposed. Both these observers impose strong assumptions on the disturbance (unknown input), where it is assumed to be piecewise constant. In addition the EKF-UI does not provide global convergence guarantees. Other works take another direction when dealing with such systems by using learning-based methods. Neural networks are used to approximate the model disturbance [6], [7] [8]. How- ever these learning based observers require training of the neural network which prevents their applicability in online estimation and control frame- work. In addition, these neuro-based observers suffer from all the drawbacks of learning based approaches such as generalization and sensitivity to noise.
However, most of the above-mentioned methods provide an asymptotic es- timation of the state and often do not reconstruct the disturbance term. In most cases, non-asymptotic or finite time estimation is important, especially for nonlinear systems as it allows to satisfy the separation principle.
A number of finite time observers for nonlinear systems subject to model dis- turbance have been reported in the literature. The most common one is the sliding mode observer [9]. For instance, a sliding mode observer for a class of uncertain triangular systems was developed in [10]. In [11], a second-order sliding mode super twisting observer was proposed for mechanical systems which was generalized to a larger class of systems in [12]. Despite providing
finite time convergence properties of the states in presence of uncertainties and disturbances, these observers do not allow the estimation of the distur- bance term.
Recently, a new approach for non-asymptotic state estimation called Mod- ulating Function Based Method (MFBM) was proposed (see, for instance [13–17]). MFBM has desired properties of common interest that include non-asymptotic convergence and robustness features. Fortunately, it has the capability to transform the ordinary differential equations into a set of algebraic equations by means of an integral operator. Solving this alge- braic system prevents solving the direct problem which allows reconstruct- ing the states in finite time without requiring initial conditions. MFBM was originally used in the fifties for parameter identification problems of Ordi- nary Differential Equations (ODEs) [18]. The method was then extended to source and parameter estimation of one-dimensional PDEs [19], [20], [21]
and fractional order differential equations [22–25]. The MFBM was used for state estimation for the first time in [13] for linear ODEs. Later, it was extended to linear PDEs [26] and more recently to nonlinear PDEs [17]. The modulating function-based approach was also used to estimate the pseudo- state of a fractional linear differential equation [27]. Nevertheless, the design of modulating function-based estimators for nonlinear ODEs is still an active research area and few works exist in the literature. For instance, in [14] a linearization-based modulating function observer for a class of nonlinear sys- tems is proposed. In [28], the authors propose a coordinate transformation based on MFBM that transforms the original system into an observer canon- ical form. There is, however, a strong dependence on the initial conditions when using this approach. Moreover, all the above-mentioned modulating function-based observers do not take into account model disturbances. Re- cently, in our previous work [15], an extension of the MFBM to triangular nonlinear systems subject to model disturbances was proposed to jointly estimate the states and the disturbance.
In the present article, we propose a robust non-asymptotic observer based on modulating functions by extending the results of our previous work in [15] to a larger class of nonlinear systems. The class of systems considered in this work is the triangular nonlinear system with multi-non-linearities, where the last equation is affected by model disturbances that encompass model un- certainties and external disturbances. The proposed observer estimates the states and the disturbance term in a step-by-step approach. The proposed modulating function-based estimator will consider both offline and online
frameworks.
The present article is organized as follows. Section 2 introduces the modulat- ing function-based method as well as its main properties. Section 3 presents the class of systems considered for the estimation as well as the main results for the non-asymptotic estimation of the state and disturbance. The appli- cability of the proposed robust modulating function estimator is illustrated in section 4 through numerical simulations. It is first applied to an academic example, then compared to the second-order sliding mode super twisting ob- server under different levels of measurement noise to assess its robustness.
Finally, concluding remarks and future work directions are given in section 5
2 Modulating Function Based Estimation Method
In this Section, the definition of modulating functions is provided as well as its main property. The offline and online estimation schemes of the modu- lation function-based method are also provided.
2.1 Definition and property
Definition 1 [23] (Modulating Function) A non-zero functionϕ(t) : [a, b]→ Ris said to be a modulating function of order k, withk∈N∗, if it satisfies the following
(P1): ϕ∈ Ck([a, b])
(P2): ϕ(i)(a) =ϕ(i)(b) = 0, i= 0,1, . . . , k−1.
Definition 2 [23] (Modulation operator) The modulation operator asso- ciated to the modulating function ϕ(t) ∈ Ck([a, b]) applied to an integrable signal y : [a, b]⊂R+ →R is given by the following inner product over the interval I = [a, b]:
⟨ϕ, y⟩I = Z b
a
ϕ(t)y(t)dt.
Property 1 The main property of the modulating function is derived using integration by parts and the boundary conditions (P2) in Definition 1
⟨ϕ, y(i)⟩I = Z b
a
ϕ(t)y(t)(i)dt= (−1)i Z b
a
ϕ(t)(i)y(t)dt. (1)
In several dynamical systems, the derivatives of the output are rarely mea- sured. However, their knowledge is necessary to get an idea about the state of the system. The Modulating Function-based method (MFBM) consists of multiplying both sides of the model differential equation by a set of mod- ulating functions, allowing to shift the derivatives from the unknown, and possible noisy signal to the smooth modulating function which allows avoid- ing the amplification of noise. Additionally, the use of the integral operator will mitigate the effect of noise on the estimation. One of the main ad- vantages provided by the MFBM is that the estimation of the variable of interest involves transforming the ordinary differential equations into a set of algebraic equations, therefore, the direct problem does not have to be solved, and the initial condition is no longer needed.
2.2 Offline and online settings
MFBM allows estimating the variable or parameter of interest both offline and online. The offline estimation is achieved by applying the modulation operator over the intervalI = [0;T] where T is the final time.
⟨ϕ, y(i)⟩I = Z T
0
ϕ(τ)y(τ)(i)dτ = (−1)(i) Z T
0
ϕ(i)(τ)y(τ)dτ. (2) The online estimation, on the other hand, is obtained by using a sliding integration window [29]
⟨ϕ, y(i)⟩I = Z t
t−h
ϕ(τ −t+h)y(τ)(i)dτ
⟨ϕ, y(i)⟩I = (−1)(i) Z t
t−h
ϕ(i)(τ −t+h)y(τ)dτ. (3) The modulation operator is applied over the intervalI = [t−τ;t],∀t∈[h, T], and h∈[0, T] representing the sliding integration window.
In the rest of the paper, the modulation operator⟨ϕ, y(i)⟩I represents, with- out loss of generality, both the offline and online estimation schemes. How- ever, in the numerical simulation, the offline and online estimation setups will be distinguished and analyzed separately.
3 Non-asymptotic State and Disturbance Estima- tion
3.1 Problem Formulation
We consider the following nonlinear system in the triangular canonical form with multi non-linearities
˙ x=
˙ x1
...
˙ xn−1
˙ xn
=
x2+f1(x1, u) ...
xn+fn−1(x1, ..., xn−1, u) fn(x, u) +d(t)
y=x1,
(4)
wherex∈Rnis the system’s state vector,u∈Ris the input bounded signal, andy∈Ris the measured output. fi(x1, ..., xi, u),i= 1,2, .., n are continu- ous nonlinear functions. d(t) is a bounded unknown disturbance term that comprises model uncertainties and external disturbances. The functions fi
are assumed to be locally Lipschitz on their arguments for all bounded x and u.
The goal is to estimate the statesxk,k= 2, ..., n, and the disturbance term d(t) using the Modulating Function-Based Method (MFBM). The triangu- lar structure of system 4, when combined with the modulation operator and property 1, offers the advantage to decouple the estimation of the states from the estimation of the disturbance d(t). Assuming that u(t) and y(t) are persistently exciting, the states and the disturbance can be estimated using the MFBM in a two-step framework. First, given the input u and measured output y=x1, the statesxk,k= 2, ..., n are estimated using the first n−1 equations of (4). Once the states are estimated, the disturbance term can be estimated using the last equation of system 4.
The estimation of the disturbance term d(t) is crucial when designing con- trollers, as it affects the performance of the closed-loop system. In addi- tion, estimating the disturbance provides transient performance guarantees, which is one of the main challenges in adaptive control.
3.2 Non-asymptotic state estimation
Sincexk,k= 2, ..., nare time-varying, we decompose each state in the space spanned by a set of unknown coefficients aj,k and known basis functions
αj,k(t) as follows
xk(t) =
+∞
X
j=1
aj,kαj,k(t), ∀k= 2, ..., n, (5) We usually truncate the decomposition to the firstMkterms, fork= 2, ..., n.
xk(t)≈
Mk
X
j=1
aj,kαj,k(t), ∀k= 2, ..., n, (6) Indeed, each state xk is a linear combination of Mk basis functions where the coefficient aj,k of these basis functions are estimated using the MFBM, which leads to the following proposition.
Proposition 1 Let αj,k(t) be the known basis functions and aˆj,k the corre- sponding unknown coefficients∀k= 2, ..., nand letu(t)y(t)be the input and the output of the nonlinear system defined in (4), respectively. Consider a set of modulating function {ϕi}i=Si=1 of order l⩾1 satisfying (P1) and (P2).
Then, the estimation of the coefficientsaj,k is given by the following closed- form solution
ˆ a1,k
... ˆ aMk,k
=−Θ−1k
⟨ϕ˙1,xˆk−1⟩+⟨ϕ1, fk−1(y, ..,xˆk−1, u)⟩I ...
⟨ϕ˙S,xˆk−1⟩+⟨ϕS, fk−1(y, ..,xˆk−1, u)⟩I
(7) where
Θk =
⟨ϕ1(τ), α1(τ)⟩I· · · ⟨ϕ1(τ), αM(τ)⟩I ...
⟨ϕS(τ), α1(τ)⟩I· · · ⟨ϕS(τ), αM(τ)⟩I
,
andS, Mk∈N∗ withS ⩾Mk.
proof We start by multiplying the first (n−1) equations of (4) with the modulating function ϕ
ϕx˙1(t) = ϕx2(t) +ϕf1(x1(t), u(t)) ϕx˙2(t) = ϕx3(t) +ϕf2(x1(t), x2(t), u(t))
... ...
ϕx˙n−1(t) = ϕxn(t) +ϕfn−1(x1(t), ..., xn−1(t), u(t))
(8)
Applying the modulation operator, one obtains
⟨ϕ,x˙1⟩I = ⟨ϕ, x2⟩I+⟨ϕ, f1(x1, u)⟩I
⟨ϕ,x˙2⟩I = ⟨ϕ, x3⟩I+⟨ϕ, f2(x1, x2, u)⟩I
... ...
⟨ϕ,x˙n−1⟩I = ⟨ϕ, xn⟩I+⟨ϕ, fn−1(x1, ..., xn−1, u)⟩I
(9)
Using Property 1 shifts the derivative from ˙xk, ∀k = 1, .., n −1 to the modulating function ϕ
−⟨ϕ, x˙ 1⟩I = ⟨ϕ, x2⟩I+⟨ϕ, f1(x1, u)⟩I
−⟨ϕ, x˙ 2⟩I = ⟨ϕ, x3⟩I+⟨ϕ, f2(x1, x2, u)⟩I
... ...
−⟨ϕ, x˙ n−1⟩I = ⟨ϕ, xn⟩I+⟨ϕ, fn−1(x1, ..., xn−1, u)⟩I
(10)
Sincexk,∀k= 2, ..., nis time-varying, it is decomposed into a space spanned byMkchosen basis functions αj,k(t) multiplied by unknown coefficientsaj,k as in equation (6), and then substituted into (10)
⟨ϕ, xk⟩I =
Mk
X
j=1
aj,k⟨ϕ, αj,k⟩I
=−⟨ϕ, x˙ k−1⟩I− ⟨ϕ, fk−1(x1, ..., xk−1, u)⟩I, (11) which is equivalent in vector notations to
⟨ϕ, α1,k⟩I. . .⟨ϕ, αMk,k⟩I
a1,k
... aMk,k
=− ⟨ϕ, x˙ k−1⟩I− ⟨ϕ, fk−1⟩I. (12) To estimate the coefficients aj,k, j = 1, . . . , Mk, at least Mk linearly inde- pendent equations are required. To that end,S ⩾Mk different modulating functionsϕi,i= 1, . . . , Sof orderl⩾1 are used, which leads to the following algebraic system
Θk
ˆ a1,k
... ˆ aMk,k
=−
⟨ϕ˙1,xˆk−1⟩+⟨ϕ1, fk−1(y, ..,xˆk−1, u)⟩I ...
⟨ϕ˙S,xˆk−1⟩+⟨ϕS, fk−1(y, ..,xˆk−1, u)⟩I
(13) where ˆaj,k are estimates ofaj,k, the statesx2, ..., xk−1have been substituted by their estimates ˆx2, ...,xˆk−1, for k = 2, ..., n, and the output y was used instead of the statex1 . Θk is anS×Mk matrix given by
Θk=
⟨ϕ1(τ), α1,k(τ)⟩I· · · ⟨ϕ1(τ), αMk,k(τ)⟩I ...
⟨ϕS(τ), α1,k(τ)⟩I· · · ⟨ϕS(τ), αMk,k(τ)⟩I
.
Finally, the parameters ˆaj,k,∀j = 1, . . . , Mk, are obtained by solving (13), and the estimated states ˆxk is given by
ˆ xk(t) =
Mk
X
j=1
ˆ
aj,kαj,k(t), ∀k= 2, ..., n (14) Lemma 1 The algebraic system (7) is equivalent to the following system for a set of modulating functions {ϕi}i=Si=1 of orderl⩾k satisfying (P1) and (P2), for k= 2, ..., n
ˆ a1,k
... ˆ aMk,k
= Θ−1k Ψk (15) where
Ψk =
ψk1 ψk2 ... ψkST
(16) withψik for k= 2, ..., nand i= 1, ..., S given by
ψik= (−1)(k−1)⟨ϕi(k−1), y⟩I−
k−1
X
r=1
(−1)(k−1−r)D
ϕ(k−1−r)i , fr
E
I (17)
proof Step 1: We start by proving by induction that
˙
xk(t) =x(k)1 (t)−
k−1
X
j=1
fi(k−i)(x1(t), .., xi(t), u(t)),∀k= 2, ..., n. (18) For k = 2, the result is straightforward. We consider the first equation of system (4) and differentiate with respect to time, which leads to the following equation
˙
x2(t) = ¨x1(t)−f˙1(x1(t), u(t)). (19) Now assume that the following hold true fork
˙
xk(t) =x(k)1 (t)−
k−1
X
r=1
fi(k−r)(x1(t), .., xr(t), u(t)),∀k= 2, ..., n (20)
and prove that the above equality holds true fork+ 1.
Considering the (k)-th equation of system (4) and differentiating it with respect to time
¨
xk(t) = ˙xk+1(t) + ˙fk(x1(t), ..., xk(t), u(t)). (21) Differentiating (20) w.r.t time
¨
xk(t) =x(k+1)1 (t)−
k−1
X
r=1
fr(k+1−r)(x1(t), .., xr(t), u(t)), (22) Substituting (22) into (21)
˙
xk+1(t) =x(k+1)1 (t)
−
k−1
X
r=1
fr(k+1−r)(x1(t), .., xr(t), u(t))−f˙k(x1(t), ..., xk(t), u(t)) (23) Finally, one obtains
˙
xk+1(t) =x(k+1)1 (t)−
k
X
r=1
fr(k+1−r)(x1(t), .., xr(t), u(t)) (24) Step 2: Recall now equation (9)
⟨ϕ,x˙1⟩I = ⟨ϕ, x2⟩I+⟨ϕ, f1(x1, u)⟩I
⟨ϕ,x˙2⟩I = ⟨ϕ, x3⟩I+⟨ϕ, f2(x1, x2, u)⟩I ... ...
⟨ϕ,x˙n−1⟩I = ⟨ϕ, xn⟩I+⟨ϕ, fn−1(x1, ..., xn−1, u)⟩I
(25)
Substituting ˙xk, for k= 2, .., n−1, by their expression in (20) in the left- hand-side of equation (25)
⟨ϕ, x2⟩I =⟨ϕ,x˙1⟩I− ⟨ϕ, f1(x1, u)⟩I
⟨ϕ, x3⟩I =⟨ϕ,x¨1⟩I− ⟨ϕ,f˙1(x1, u)⟩I− ⟨ϕ, f2(x1, x2, u)⟩I
⟨ϕ, x4⟩I =⟨ϕ, x(3)1 ⟩I− ⟨ϕ,f¨1(x1, u)⟩I− ⟨ϕ,f˙2(x1, x2, u)⟩I
−⟨ϕ, f3(x1, x2, x3, u)⟩I ... ...
⟨ϕ, xn⟩I =⟨ϕ, x(n−1)1 ⟩I−
n−1
X
r=1
⟨ϕ, frn−r−1(x1, ..., xr, u)⟩I
Applying Property 1, one obtains for the statesxk,k= 2, ..., n
⟨ϕ, xk⟩= (−1)(k−1)⟨ϕ(k−1), x1⟩I−
k−1
X
r=1
(−1)(k−1−r)D
ϕ(k−1−r), frE
I
Substituting xk by its decomposition (equation (6)) leads to the following algebraic equation
⟨ϕ, α1,k⟩I. . .⟨ϕ, αMk,k⟩I
ak1
... akM
=ψk (26) where
ψk= (−1)(k−1)⟨ϕ(k−1), x1⟩I−
k−1
X
r=1
(−1)(k−1−r)D
ϕ(k−1−r), frE
I
To solve system (26), one needs S ⩾ M equations obtained by using S modulating functions of order l⩾k, which leads to
Θk
ˆ a1,k
... ˆ aMk,k
= Ψk (27) where ˆaj,k are the estimated values of the unknown coefficientsaj,k, and the vector Ψk is given by
Ψk =
ψk1 ψk2 ... ψkST
withψki fork= 2, ..., nandi= 1, ..., Sgiven by, wherex1is substituted byy, and the statesx2, ..., xrby their estimated values ˆx2, ...,xˆr, forr= 1, ..., k−1, and k= 2, ..., n.
ψki = (−1)(k−1)⟨ϕi(k−1), y⟩I−
k−1
X
r=1
(−1)(k−1−r)D
ϕ(k−1−r)i , fr
E
I
Therefore the coefficient ˆaj,k, for j = 1, .., M, k = 2, ..., n are obtained by solving either system (13) or system (27), which concludes the proof.
3.3 Disturbance estimation
In most cases, the disturbanced(t) is time-varying and can be in the form of model uncertainties or external disturbances. Following the same reasoning as the state estimation in the previous subsection, the disturbance d(t) can be decomposed into a sum of known basis functions βj(t) and unknown coefficients bj, which is truncated to the first N terms
d(t) =
∞
X
j=1
bjβj(t)≈
N
X
j=1
bjβj (28)
Given the estimated states ˆxk, ∀k = 2, ..., n and the measured input u(t) and outputy(t) =x1(t), The estimation of the disturbance term d(t) using the MFBM can be achieved using the last equation of system (4). The estimation of the disturbance is given in the following proposition.
Proposition 2 Let u(t) andy(t) =x1(t)be respectively the measured input and output of system (4), and xˆk the estimated states ∀k = 2, ..., n. Let {ϕi}i=Di=1 be a set of modulating functions of orderl⩾1 satisfying (P1) and (P2). Then an estimate of the disturbance termd(t) is given by
d(t) =ˆ
N
X
j=1
ˆbjβj(t) (29)
where N, D ∈ N∗, with D ⩾ N, βj(t) are known basis functions and ˆbj are estimates of the unknown coefficients given by the following closed-form solution
ˆb1
... ˆbN
=−Θ−1d
⟨ϕ˙1,xˆn⟩I+⟨ϕ1, fn(y, ..,xˆn, u)⟩I ...
⟨ϕ˙D,xˆn⟩I+⟨ϕD, fn(y, ..,xˆn, u)⟩I
(30) where Φd is given by
Θd=
⟨ϕ1, β1(t)⟩I . . . ⟨ϕ1, βN(t)⟩I ... . .. ...
⟨ϕD, β1(t)⟩I . . . ⟨ϕD, βN(t)⟩I
(31)
proof Considering the last equation of system (4) and applying the mod- ulation operator
⟨ϕ,x˙n⟩I =⟨ϕ, fn(x, u)⟩I+⟨ϕ, d(t)⟩I (32) Using Property 1, the derivative is shifted fromxn to the modulating func- tion
−⟨ϕ, x˙ n⟩I =⟨ϕ, fn(x, u)⟩I+⟨ϕ, d(t)⟩I (33) Given that the disturbance termd(t) is time-varying, it is decomposed into the space spanned by a set of known basis functions βj(t) multiplied by unknown coefficients bj, forj= 1, ..., N, as given by equation (28).
Rearranging (33), and substitutingdby (28)
⟨ϕ, d⟩I =
N
X
j=1
bj⟨ϕ, βj⟩I=−⟨ϕ, x˙ n⟩I− ⟨ϕ, fn(x, u)⟩I, which is equivalent to
⟨ϕ, β1⟩I. . .⟨ϕ, βN⟩I
ˆb1
... ˆbN
=− ⟨ϕ,˙ xˆn⟩I− ⟨ϕ, fn(ˆx, u)⟩I. (34) Where ˆbj are the estimated values of the unknown coefficients ˆbj,j= 1, ..N, and the states xk are substituted by their estimates ˆxk, ∀k = 2, ..., n, and thex1 is substituted by the measured outputy.
To solve for ˆbj,j= 1, ..N, we need Dequations, whereD⩾N, which leads to the following system of algebraic equation
Θd
ˆb1
... ˆbN
=−
⟨ϕ˙1,xˆn⟩I+⟨ϕ1, fn(ˆx, u)⟩I ...
⟨ϕ˙D,xˆn⟩I+⟨ϕD, fn(ˆx, u)⟩I
(35) where Θdis a D×N matrix
Θd=
⟨ϕ1, β1(t)⟩I . . . ⟨ϕ1, βN(t)⟩I ... . .. ...
⟨ϕD, β1(t)⟩I . . . ⟨ϕD, βN(t)⟩I
(36)
Finally, the coefficients ˆbj, forj= 1, ..., N are obtained by solving (35), and the estimated disturbance is given by
d(t) =ˆ
N
X
j=1
ˆbjβj(t) (37)
Lemma 2 For a set of modulating functions {ϕi}i=Di=1 of orderl⩾n satisfy- ing (P1) and (P2), the algebraic system (30) is equivalent to the following system
ˆb1
... ˆbN
= Θ−1d
(−1)n⟨ϕ˙1, y⟩I−
n
X
j=1
(−1)(n−i)D
ϕ(n−i)1 , fi
E
I
... (−1)n⟨ϕ˙D, y⟩I−
n
X
j=1
(−1)(n−i)D
ϕ(n−i)D , fiE
I
Then, the estimate of the disturbance is given as follows d(t) =ˆ
N
X
j=1
ˆbjβj(t).
proof Considering the last equation of system (4)
˙
xn=fn(ˆx, u) +d(t), (38)
Recall that
˙
xn=x(n)1 −
n−1
X
r=1
fr(n−r)(x1, .., xr, u). (39) Equation (38) becomes
d(t) =x(n)1 −
n
X
r=1
fr(n−r)(x1, .., xr, u). (40)
Applying the modulation operator and Property 1
⟨ϕ, d⟩I = (−1)n⟨ϕ(n), x1⟩I−
n
X
r=1
(−1)n−r⟨ϕ(n−r), fr(x, u)⟩I, (41)
Substituting (29) into (41) and writing it in a vector form, one obtains ⟨ϕ, β1⟩I. . .⟨ϕ, βN⟩I
ˆb1
... ˆbN
= (−1)n⟨ϕ(n), y⟩I
−
n
X
r=1
(−1)n−r⟨ϕ(n−r), fr(x, u)⟩I. Where ˆbj are the estimated values of the unknown coefficients ˆbj,j= 1, ..N, and the states xk are substituted by their estimates ˆxk, ∀k = 2, ..., n, and thex1 is substituted by the measured outputy.
To solve for ˆbj,j = 1, .., N, one needsDequations, whereD≥N, obtained by usingD different modulating functions of orderk≥n, which leads
Θd
ˆb1
... ˆbN
=
(−1)n⟨ϕ˙1, y⟩I−
n
X
j=1
(−1)(n−i)D
ϕ(n−i)1 , fi
E
I
... (−1)n⟨ϕ˙D, y⟩I−
n
X
j=1
(−1)(n−i)D
ϕ(n−i)D , fiE
I
where Θdis given as (31).
4 Numerical simulation
To illustrate the performance of the proposed Modulating function Based estimation method, we consider two numerical examples. First, the proposed estimator is applied to an academic example of a third-order system. Then, the MFBM is compared to the second-order super-twisting observer [11].
Finally, a robustness analysis against the measurement noise is performed.
4.1 Academic example
Consider the following third-order nonlinear triangular system
˙ x=
˙ x1
˙ x2
˙ x3
=
x2−x21 x3−x1x2
−x23
1+x21 +d(t)
y=x1,
(42)
In this example, a general form of the disturbance term d(t) that is time and state-dependent is considered.
d(t) = 0.1 cos 2π
5t−π 4
×(1 + 0.1t)×x1×x2×x3.
For this example, both offline and online schemes are considered. The states x2 andx3, and the disturbanced(t) are decomposed respectively into poly- nomial basis functions αj,k(t) = tj−1 , for k = 2,3, and βj(t) = tj−1 as follows
x2(t)≈
M2
X
j=1
aj,2αj,2(t), x3(t)≈
M3
X
j=1
aj,3αj,3(t),
d(t)≈
N
X
j=1
bjβj(t).
4.1.1 Offline estimation
The selected modulating functions are normalized polynomial modulating functions [22] given by
ϕi(t) =
ϕ¯i(t)
∥ϕ¯i(t)∥L2, (43) where∥.∥L2 is theL2-norm and the modulating functions ¯ϕi(t) for the state and disturbance estimation are given by
ϕ¯i(t) = (T−t)(px,k+i)t(px,k+Sk+1−i),∀i= 1,· · · , Sk, k= 2,3, ϕ¯i(t) = (T −t)(pd+i)t(px,k+D+1−i), i= 1,2,· · · , D.
where T is the simulation final time, S and D represent the number of modulating functions for the state and disturbance estimation, respectively.
px,k, pd ∈ N∗ are degrees of freedom. The parameters of the modulating functions for the simulation are given as follows S2=M2= 12, S3 =M3 = 10, andD=N = 9. px,k = 2,k= 2,3 and pd= 3.
The parameter vectors ˆa2, ˆa3, and ˆb obtained by the modulating function- based method for ˆx2, ˆx3 and ˆd, respectively, are given by :
ˆ
a2 =[1.98 −1.82 2.97 −3.36 2.55 −1.37 0.52 −0.14 0.02 −3.26×10−3 2.35×10−4 −7.55×10−6]T
ˆ
a3 =[2.01 −0.27 −0.789 0.92 −0.59 0.24 −0.06 9.43×10−3 −8.07×10−4 2.95×10−5]T
ˆb=[0.52 −0.24 0.15 −0.39 0.29 −0.10 0.02 −1.5×10−3 4.92×10−5]T
The choice of the parameters M2, M3, and N is crucial to obtain a good estimation. The bigger the coefficient the better the approximation of the states and disturbance is. However, the computation burden increases which may lead to numerical instabilities. Therefore, there is a trade-off between the accuracy of the approximation and the computation complexity. It can be noticed from the obtained parameters that the last values decrease fast to zero. Therefore, one can deduce that the choice of the truncation parameters was suitable for our problem.
Figure 1 shows that the modulating function-based estimator was able to accurately estimate the states and the disturbance. However, some esti- mation errors at the boundary can be noticed for the estimation of d(t), which is common when using a modulating function with polynomial basis functions [29].
4.1.2 Online estimation
The normalized polynomial modulating functions are adapted to the moving horizon scheme by considering
ϕ¯i(τ −t+h) = (t−τ)(px,k+i)(τ−t+h)(px,k+Sk+1−i), (44)
∀i= 1,· · · , Sk, k= 1,2,
ϕ¯i(t) = (t−τ)(pd+i)(τ−t+h)(px,k+D+1−i), i= 1,2,· · ·, D. (45) The moving integration window size was taken ash= 1s, and the following parameters for the modulating function-based estimation were considered:
S2=M2= 5, S3=M3 = 4, D=N = 2,pd=px,1= 2, px,2= 3.
Figure 2 shows that modulating function-based estimator provides an accu- rate online estimation of both the states and the disturbance, which offers the potential of good closed-loop performance in case it is combined with a controller.
0 1 2 3 4 5 Time (s)
0.5 1 1.5 2
0 0.01 0.02 0.03 0.04 0.05
1.91 1.92 1.93 1.94 1.95 1.96 1.97 1.98 1.99
0 1 2 3 4 5
Time (s) 0
0.5 1 1.5 2
0 0.05 0.1 0.15 0.2 0.25 0.3
1.88 1.9 1.92 1.94 1.96 1.98 2
0 1 2 3 4 5
Time (s) -0.1
0 0.1 0.2 0.3 0.4 0.5 0.6
Figure 1: Offline estimation of the states and the disturbance term.
4.2 Comparison with a second order sliding mode super twist- ing observer
We consider an unforced pendulum subject to Coulomb friction and external disturbance given by the following second-order system [11]
˙ x1 =x2
˙
x2 =−g
Lsinx1− Vs
J x2−Ps
J tanhx2
0.1
+d(t) y=x1
The pendulum parameters have the following valuesM = 1.1, g= 9.815, L= 0.9, J = M L2 = 0.891, VS = 0.18, Ps = 0.45 and the disturbance term is given by
d(t) = 0.5 sin(t) + 0.5 cos(2t).
The state x2 and the disturbance term d are estimated online using the modulating functions in (44) and (45), and polynomial basis functions, where S2 =M2 = 7, D=N = 3,pd=px,1 = 2, and h= 1s. The state estimation
0 1 2 3 4 5 Time (s)
0.5 1 1.5 2
0 0.02 0.04 0.06 0.08
1.85 1.9 1.95 2
0 1 2 3 4 5
Time (s) 0
0.5 1 1.5 2
0 0.05 0.1 0.15 0.2 0.25
1.85 1.9 1.95 2 2.05
0 1 2 3 4 5
Time (s) -0.1
0 0.1 0.2 0.3 0.4 0.5 0.6
Time (s)
Figure 2: Online estimation of the states and the disturbance term.
result is compared with the state estimation obtained by the second order sliding mode super twisting observer (STO) in [11] given by
˙ˆ
x1 = ˆx2+ 1.5 f+1/2
|x1−xˆ1|1/2sign (x1−xˆ1) x˙ˆ2 =− g
Ln
sinx1−Vsn
Jn
ˆ
x2+ 1.1f+sign (x1−xˆ1)
wheref+= 6 is the double maximal possible acceleration of the system [11].
The initial conditions are taken as ˆx1(0) =y(0) =x1(0) and ˆx2(0) = 0.
The estimated state using Modulating Function Based Method (MFBM) and the Super Twisting observer (STO) are illustrated in Figure 3a). Both observers provide an accurate estimation of the state. Unlike the super twisting observer, the modulating function estimator does not require the initial condition which results in a faster convergence and a smaller relative estimation error as displayed in Table 1. In addition to the state estima- tion, the modulating function estimator allows estimating the disturbance d. In Figure 3b), one can see that the MFBM reconstructs accurately the disturbance.
0 1 2 3 4 5 Time (s)
-1 -0.5 0 0.5 1 1.5
a)
0 0.005 0.01 0.015
Time (s) 0.995
1 1.005 1.01 1.015 1.02 1.025 1.03 1.035
0 1 2 3 4 5
Time (s) -1
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6
b)
0 0.05 0.1 0.15 0.2
0.48 0.5 0.52 0.54 0.56 0.58
Figure 3: a) Online estimation of the state x2 using MFBM and STO; b) online estimation of the disturbancedusing MF.
4.3 Robustness analysis
To evaluate the robustness of the proposed estimator against measurement noise, different levels of white Gaussian noise were added to the output y.
Figure 4a) shows the state estimation with modulating function estimator and super twisting observer in presence of 1% level of Gaussian noise. The state estimation with the MFBM is smoother and less affected by measure- ment noise. This is due mainly to two factors: the result of Property 1 and the integral operator that reduce the effect of measurement noise. In Figure 4b), one can see that the MFBM reconstructs accurately the dis- turbance in presence of noise. Table 1 shows the state relative estimation error for different levels of noise. One can see the relative estimation error is smaller with the MFBM than with STO.
5 Conclusion
The present paper proposed a robust step-by-step non-asymptotic estimator based on modulating functions to estimate the states and disturbance term of nonlinear triangular systems. The robustness properties of the modulat- ing function-based estimator are the result of the modulating operator and the main property of modulating functions that shifts the derivatives of the input-output towards the smooth modulating function which makes the pro-
0 1 2 3 4 5 Time (s)
-1 -0.5 0 0.5 1 1.5
a)
0 1 2 3 4 5
Time (s) -1
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6
b)
Figure 4: a) Online estimation of the statex2using MF and STO in presence of noise; b) online estimation of the disturbancedin presence of noise.
Table 1: Relative estimation error w.r.t different noise levels in % Noise level |x2|x−ˆx2|
2| with STO |x2|x−ˆx2|
2| with MF
0 % 10.43 1.41
1 % 28.13 1.81
3 % 33.29 2.07
5 % 45.58 2.92
10 % 49.67 7.55
posed non-asymptotic estimator input-output derivative-free. Additionally, the modulating function-based method transforms the estimation problem from solving ordinary differential equations to solving a set of algebraic equations. Therefore the initial condition is not required which results in a faster convergence and a smaller estimation error. The proposed observer was applied to an academic example of a third-order nonlinear triangular system to evaluate its performance where both offline and online estimation schemes were considered. Moreover, it was compared to the second-order super twisting sliding mode observer under different levels of measurement noise. While the sliding mode observer is more robust to model uncertain- ties, the modulating function estimator is more robust against measurement noise. Future work will focus on extending the modulating function-based
estimator to a more general class of nonlinear uncertain systems.
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