Resistance to Noise of Non-linear Observers in Canonical Form Application to a Sludge Activation Model

The aim of this study was to increase the resistance to noise of an observer of a non-linear MISO system transformed into canonical regulation form of order n. For this, the principle idea was to add n observers on the output equations of the main observer. By adjusting the time scale of the output observers, the resistance to noise of the final estimates is considerably increased. The proposed method is illustrated by model simulations based on a non-linear Sludge Activation Model (SAM)


Introduction
State observers have been intensely exploited since (Luenberger, 1966), to model, control or identify linear and non-linear systems, including the studies of (Krener & Isidori, 1983;Zheng, Boutat, & Barbot, 2009) relating to non-linear systems transformable into a canonical form.The key idea in such approaches is to produce approximate measures of non-linearity of order 1, as in Extended Luenberger Observers (ELO) (Ciccarella, Mora, & Germani, 1993).Approximations of nonlinearities in the canonical form (which results in ELO) have already been suggested (Bestle & Zeitz, 1983), and this approach can be extended to higher order approximations (Röbenack & Lynch, 2004).An observer using a partial nonlinear observer canonical form (POCF) (Röbenack & Lynch, 2006) has weaker observability and integrability existence conditions than the well-established non-linear observer canonical form (OCF). Non-linear sliding mode observers use a quasi-Newtonian approach, applied after pseudo-derivations of the output signal (Efimov & Fridman, 2011).State observers using Extended Kalman Filters (EKF) provide another method of transforming non-linear systems (Boker & Khalil, 2013), (Rauh, Butt, & Aschemann, 2013).Finding an appropriate method for parameter synthesis remains one of the major difficulties with state observers for non-linear systems.(Tornambè, 1992), (Mobki, Sadeghia, & Rezazadehb, 2015) proposed high-gain state observers to deal with this problem.High-gain state observers reduce observation errors for a range of predetermined amplitudes or fluctuations by making the observations independent of parameters.The weak point of this method is its sensitivity to noise and uncertainty.In network identification and encryption, observers with delays are used to synchronize chaotic oscillators, as shown in several studies (Ibrir, 2009;Martínez-Guerra, et al., 2011).Noise and uncertainty are not critical factors in such a context.This can be very different in the case of industrial processes, as shown in a recent study (Bodizs, 2011), where the performances of observers using ELO, EKF or Integrated Kalman Filters (IKF) are compared.The influence of noise and uncertainty on these observer types was emphasized, with more reliable results produced by ELO observers, which permit the exact state reconstruction of highly perturbed systems.For PI and ELO observer classes, (Söffker, et al., 2002) demonstrated a compensation effect on measurement errors ; (Khalifa & Mabrouk, 2015) addressed the problem of uncertainty of non-linear models.One way of overcoming the problem of parametric uncertainty is to use adaptive observers (Tyukina, et al., 2013;Farza, et al., 2014) in the particular case where the measurements are only available at discrete instants and have disturbances.Another approach (Mazenc & Dinh, 2014;Thabet, et al., 2014) consists of defining interval observers.Modeling observer systems by Takagi-Sugeno decomposition (Bezzaoucha, et al., 2013;Guerra, et al., 2015) is another possibility, as is the use of models using symmetries and semi-invariants (Menini & Tornambè, 2011), or the use of immersible techniques for systems transformed into a non-linear observer form (Back & Seo, 2008).
A large number of non-linear MISO systems with multiple inputs and a single output can be transformed into state equations using the form : with the following definitions : n : the order of the system of non-linear differential equations m : number of independant inputs u 1 ptq T : the vector ru 11 ptq, . . .u m1 ptqs of the m independent inputs yptq : the measurable output variable z T ptq : the state vector r z 1 ptq . . .z n ptq s d T : the vector of the output parameters of the system Φptq : the non-linear function of vector u 1 ptq of the inputs s i " zptq, u 1 ptq ‰ : one of the n non-linear functions of the state vector sptq.
Such systems are often found in nonlinear robotic systems in the form of trigonometric functions.Other systems contain non-linear polynomials (strange attractors, Bernouilli functions, non-linear springs), polynomial fractions, or various common simple functions . . .The n non-linear functions of vector sptq employ a vector of m independent inputs u 1 ptq, as well as the state vector zptq as input variables.Such a procedure allows amongst other possibilities the description of bi-linear systems.We limited ourselves in this study to continuous functions in all points of type C 1 .One considers that the measurable output is a linear combination of zptq, superimposed on a non-linear function Φ " u 1 ptq ‰ .For an engineer or physicist, many applications have such a form.Often, non-linear systems (1) are transformable in a regulation canonical form concieved by (Fliess, 1990), and are written : with the following definitions : u i ptq : the pi ´1q ´th temporal derivative of the vector u 1 ptq, either u i ptq T " r u 1i ptq, . . .u mi ptq s i " 2 . . .n Uptq " " u 1 ptq . . .u n ptq ‰ : the n ˆm input matrix, with the group of n vectors u i ptq x i ptq : pi ´1q th temporal derivative of x 1 ptq x T ptq : state vector r x 1 ptq, . . .x n ptq s c : the output parameters vector of the transformed system θ ď n : index of last coefficient c i ‰ 0 Ψ r xptq, Uptq s : a scalar non-linear C 1 function A : the n ˆn matrix of which the last line is zero.
Conversion of the transformed version (2) to the initial representation (1) is performed using : with gptq : the vector of n non-linear inverted transformation functions g i r xptq, Uptq s which link xptq to zptq.
In (Schwaller, Ensminger, Dresp-Langley, & Ragot, 2016) a new observer was proposed which was adapted to this transformed form, and which provided non-biased robust estimates of xptq.This is not always the case for estimates of zptq.Functions g i rxptqs (1b) permit linking xptq to zptq (2c) and are called inverted transformations.Because of the nonlinearity of gptq, small perturbations of estimates of xptq may be considerably increased and strongly disturb estimates of zptq.The main aim of this study was to solve this type of situation, by introducing the inverted observer transform functions g i rxptqs.Doing this, the resistance to observer noise is affected (Bodizs et al., 2011), and one obtains a tool capable of limiting its impact on estimates of zptq.
Definition 1 Let us define, for the moment, a normalised pulse ω o " 2π{T o , which introduces a new time scale τ for the representation of the transformed state of the system : and for the inverse transformation system : zpτq " gpτq (5a) with : τ " ω o t, 9 x n ptq " 9 x n pτq ω o n (6a) f pτq and gpτq are vectors with dimension n.In (4b), Φpτq " Φ " u 1 pτq ‰ " Φ " u 1 ptq ‰ .Equations ( 6) define time dilatation or retraction of the state representation and its new parameters, without changing the pattern of the signal x i pτq.For the function Ψ, this is translated by the relation of changing the following scale representation : The function r Ψ r xpτq, Upτq s is obtained by replacing every state or command variable by the corresponding one in (6) and dividing everything by ω o n .
Afterwards, the procedure can be separated into several steps: in section 2, the estimation of the state of the transformed system (4) is dealt with ; in section 3 a new observation method of the inverse transformation functions which permit estimation of state variables (1) is presented ; in section 4 this new approach is applied to observe a system of management of activated sludge in a purification station ; the study is concluded in section 5.

Structure of the Observer in Canonical Form
To begin with, let us isolate the componant x 1 pτq of (4b) which will subsequently serve to determine the observation error.
To obtain y 1 pτq, the estimation of variable x 1 pτq, three cases may be distinguished.For θ " 1 : For θ " 2, it becomes : In the most general case where θ ą 2, ypτq ´Φpτq is filtered by : To analyze the effect of the filter, we rewrite (4b) in scalar form, ignoring r c θ`1 . . .r c n , which are all zero : If ( 11) is inserted in (10a), ( 9) or ( 8) as a function of θ, it becomes : The Laplace transformation of (12a) gives the transfer function : To develop the rest, y 1 pτq is used to determine the observer error.
Definition 2 To generate state estimates vpτq for the system (4), a PI observer structure is defined in (Schwaller, Ensminger, Dresp-Langley, & Ragot, 2016) with : with q xpτq (14g) and p xpτq (14h) as two distinct state vectors of dimension n ´1, coupled using the matrices A (14n) and q A (14o) of dimension pn ´1q ˆpn ´1q.The vectors q h and p h are also of dimension n ´1.The matrix A is constructed using the Kronecker operator which puts the upper diagonal at 1.The parameters h i , i " 0 . . .n are the gains of the observer.
Figure 1 illustrates the functional diagram of such an observer of third order.
The augmented vector vpτq (14i),(14h) and ( 14g) is used as estimation of xpτq and as variable of the function r Ψ r vpτq, Upτq s (14f).The state p xpτq (14b) is an observer exploiting the observation error ∆y 1 pτq (14c) via the gains h i (14m) serving to correct the state distances between the system and its observer.
In figure 1, for example, we have : The choice of using two state variables p xpτq and q xpτq is motivated by the n ´1 successive integrations of 9 q x n pτq in which no p h ∆y 1 pτq re-injection error is involved.This allows an increase in the robustness of the estimations to the measurement noise, which in general affects the variable y 1 pτq.One thus overcomes a common weak point of high gain observations, i.e. their sensitivity to measurement noise.The second advantage comes from the non-linear function r Ψ rvpτq, Upτqs which is no longer subjected to the restrictive conditions used in (Schwaller, Ensminger, Dresp-Langley, & Ragot, 2013), and covers the ensemble of the systems described by (Fliess, 1990).The vector r f pτq (14d), of dimension n ´1, compensates the effects of f pτq, and of possible external exogenous disturbance of (2) using the integral component I 0 pτq (14e).One notes that at the second order, for a gain h 0 " 0 inhibiting the integrator I 0 , the observer becomes similar to that proposed by (Gauthier, Hammouri, & Othman, 1992) for a SISO system.
In (Schwaller, Ensminger, Dresp-Langley, & Ragot, 2016), a full analysis was performed in order to determine the dynamics of the observation error ∆y 1 pτq (14c) and its successive derivatives, to characterise stability conditions and also the exponential convergent nature of estimates vpτq.A mthod to synthesize parameters h 0 . . .h n was also proposed.

New observers definitions
In (5b), the inverted transform functions gpτq allow converting the system in the canonical form of regulation back to the original form (1). Using the estimates vpτq reconstructed by the observer ( 14), it is possible to define : (15) One thus obtains estimates p zptq of zptq (1).If the stability conditions (Theorem 1 of ( The n estimates p z i ptq can be used as reference inputs to observe n state variables q z i ptq which tend towards (15a).Their temporal derivatives tend towards 9 p zptq, which themselves tend towards 9 zptq (16).With the model ( 14), one defines n first order observers.Each is normalised by a pulse ω i which leads to its dimensionless time definition (17e), possesses its own Lipschitz constant, and its specific stability conditions that we have to find.Synthesising the gains h i (subsection 2.4 of (Schwaller, Ensminger, Dresp-Langley, & Ragot, 2016)) gives h 0 " 1 and h 1 " 2. The n observers are written : 9 q z i pτ i q " I i pτ i q `2 ∆z i pτ i q `q s i pτ i q (17a) 9 I i pτ i q " ∆z i pτ i q i " 1 . . .n (17b) ∆z i pτ i q " p z i pτ i q ´q z i pτ i q (17c) with q zptq " " q z 1 ptq . . .q z n ptq ‰ the vector of the estimations of p zptq ; q s i pτ i q is the normalised non-linear function of 9 q z i pτ i q. Figure 2 illustrates (15) and ( 17).The general calculation procedure is as follows : • estimation of vpτq (14i) after treatment of (14); • estimation of p zptq (15) ; • estimation of the n state distances (17c) ; • determination of the n non-linear functions q s i pτ i q (17d) to access the n terms 9 q z i pτ i q (17a) and 9 I i pτ i q (17b) ; • integration of the n equations (17a) to obtain q zptq.
The temporal derivative of (17c) and inserting (17a) in the rest obtained enables one to obtain the expression of ∆9 z i pτq : ∆9 z i pτ i q " 9 p z i pτ i q ´9 q z i pτ i q i " 1 . . .n (18a) " ∆ r Ψ i pτ i q ´Ii pτ i q ´2 ∆z i pτ i q (18b) ∆ r Ψ i pτ i q " p s i pτ i q ´q s i pτ i q (18c)

Dynamics of the Observer Errors
We now characterise the dynamics of the observer errors by searching the n differential equations of the distances ∆z i pτ i q.
Due to the presence of integrators I i pτ i q, an extra temporal derivative is necessary to obtain the differential equation of the distances ∆z i pτ i q.To do this, it is necessary to define the following augmented vectors : q z i pτ i q T " " q z i pτ i q 9 q z i pτ i q ı (19c) The temporal derivative of (18b) is written : and gives the scalar expression of the differential equations of the observation errors.Using notations (19) gives the matricial writing of (20a) in the form of state equations : Assuming that the non-linear functions 9 s i are at least locally Lipschitz in Zpτ i q, and uniformly bounded in Υpτ i q in an invariant set, they are associated with a Lipschitz constant L i : Applying the Lipschitz inequality to (20b) permits reduction to ∆Zpτ i q the number of useful variables to characterise the perturbing difference ∆ 9 r Ψ i pτ i q.For many systems, if functions 9 s i are not globally of a Lipschitz type, they can be locally or be transformed adequately into the Lipschitz type.

Convergence of State Observations
Now let us try to analyse the globally asymptotic development of the observation errors and to characterise the limiting stability conditions of each observer (17).
Theorem 1 Let us consider a MISO system decomposable as described in (4), for which the observer structures ( 14) and (17) are used, and related to each other by the inverted transform function (15a).If the system function 9 s i " p Zpτ i q, Υpτ i q ı is locally of the Lipschitz type in p Zpτ i q and uniformly bounded in Υpτ i q in an invariant set, with a Lipschitz constant L i (22), then the observer (17) will be locally stable if the Lipschitz constant L i satisfies the following conditions : If the system function 9 s i " p Zpτ i q, Υpτ i q ı is globally of the Lipschitz type, and if the Lipschitz constant L i satisfy (23), then the observers (17) will be globally asymptotically stable.
Proof.The proof of theorem 1 can be demonstrated by proving the stability of (21a) using an appropriate positive Lyapunov function, like the following quadratic function : v i pτ i q " ∆z i pτ i q T P i ∆z i pτ i q (24b) The P i lower triangular matrix are defined as positive and satisfying the Sylvester criteria, with (24c).The proof of convergence is linked to the study of the sign of the derivative of the candidate for a Lyapunov function.This is obtained after temporal derivation of (24a), and after placing (21a) in the result obtained for terms ∆9 z i pτ i q : An appropriate choice of ϕ i1 , ϕ i2 can provide negative diagonal coefficients for Q i .The criterion of semi-negativity of Sylvester is then respected, and the successive minors of Q i will be of opposite sign, ensuring the semi-negativity of the first member on the right of (25b).Verifying the sign of the second member on the right of (25b) involves increasing N i pτ i q using the inequalities of Schwartz and Lipschitz (22) : To determine the sign of 9 v i pτ i q function, one applies the following inequality : to (26c) to obtain the desired increase of N i pτ i q : In (28a) yields a positive lower triangular matrix R i (28b), the diagonal elements of which are written : With negative functions 9 v i pτ i q, adding together the diagonal terms of (25c) and (29), and imposing Q i `Ri ď 0, one obtains the conditions (23).The sum Q i `Ri yields an inferior triangular matrix that satisfies Sylvester criteria of seminegativity if inequalities (23a) and (23b) are satisfied.Then, if ∆ 9 r Ψ i pτ i q (20b) is Lipschitz (22), 9 v i pτ i q is semi-negative and (21a) is globally and asymptotically stable ; (21a) is locally stable if ( 22) is locally Lipschitz Using the (theorem 2, section 2.3, (Schwaller, Ensminger, Dresp-Langley, & Ragot, 2016)), it is easy to demonstrate that the observers (17) will be exponentially convergent.

Application to a Sludge Activation Model
Let us now illustrate the proposed observation method by applying it to a non-linear example with multiple inputs.
The variables z 1 ptq, z 2 ptq z 3 ptq represent the state of the reactor (figures 3(e),(f),(g)), respectively the concentration of rapidly biodegradable substrate, the concentration of dissolved oxygen, the particle concentration of biomass, with (34) its parameters, all known, and z 1 p0q " 4.1, z 2 p0q " 3.0, z 3 p0q " 867 the initial conditions.The sizes y 1 ptq, y 2 ptq (31b) represent the measurable outputs.As this application of the general procedure of transformation permits passage from systems (1) to (2) (Fliess, 1990) it will permit the use of an observer similar to that proposed in ( 14), associated with inverted transformation (15) and with observers (17).

Observation of the Inverted Transformation System
The inverted transformation system (43) serves to form the errors (14c) of three first order observers of the same type as those defined in (17), in order to estimate p zptq.
We now try to determine the Lipschitz constants that subsequently will allow defining the stability conditions of each observer.We thus start by looking for L in (36) using the same calculation method as that explained in ((Schwaller, Ensminger, Dresp-Langley, & Ragot, 2016), section 3.1).
Figure 3. Input variables and state variables of the bioreactor