13th IEEE Workshop on Variable Structure Systems, VSS’14, June 29 -July 2, 2014, Nantes, France.
Application of Super-Twisting-Like Observers for Bioprocesses Jaime A. Moreno
Ismael Mendoza
Coordinación Eléctrica y Computación, Instituto de Ingeniería, Universidad Nacional Autónoma de México, 04510 México D.F., Mexico Email:
[email protected],
[email protected]
Abstract— State and reaction rate estimation in highly uncertain bioprocess models is an important topic for a successful control of these processes. Discontinuous observers, as for example the super-twisting estimator, offer unique properties as convergence in finite time and insensitivity with respect to non-vanishing perturbations. We present a generalization of the classical super-twisting observer and illustrate its use in the estimation of variables in a bioprocess. Keywords: Discontinuous Observers, Higher Order Sliding Modes, Bioprocesses, Nonlinear Estimation.
I. I NTRODUCTION Biotechnological processes, including also the biodegradation used in the wastewater treatment, have become an important part of modern life, and its appropriate control, optimization and supervision is clearly a fundamental issue. These tasks are particularly challenging since the dynamical models of bioprocesses are highly uncertain and there is a lack of reliable and/or economical sensors for key variables. This explains the interest in the last decades for developing estimation strategies for the states and the (specific) reaction rates in bioprocess models [4]. Asymptotic Observers (AO) [1], [4] are robust, since they are able to estimate the states of a bioreactor without knowledge of the reaction rates. However, this requires the measurement of at least as many state variables as the number of reaction rates and usually the convergence of the observer cannot be assigned. Properties of AOs can be explained using the theory of Unknown Input Observers [18]. For the estimation of the (specific) reaction rates, considered as unknown inputs, High-Gain Observers (HGO) [11], [9] have been successfully used [5], [6], [8]. A drawback of HGOs, and in general of any continuous observer, is that they are unable to estimate without error the reaction rates, even in the absence of measurement noise. This is due to the lack of knowledge of the velocity of variation of the reaction rate, and this uncertainty cannot be completely compensated by continuous observers. In order to reduce the estimation error the High Gain of a HGO has to be increased, but this increases the sensitivity to measurement noise of the estimator [19]. An important feature of discontinuous observers, and in particular of Higher Order Sliding Mode Observers (HOSMO) [12], [13], [22] is that they are able to estimate an unknown input exactly and in finite time despite of the lack of knowledge of the velocity of change of the signal. In particular, for the reaction rate estimation in bioreactors
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observers based on the Super-Twisting Algorithm (STA), a Second Order Sliding Mode Algorithm, have been used recently in the literature. In [24] a Super-Twisting Observer (STO) is used to estimate the reaction rate in a chemical reactor to improve the control. The authors of [3], [2], [23] provide a modification of the STO to estimate the specific reaction rate in a bioreactor, which is multiplied by a measured signal. Proofs are based on [14], [15]. In this paper we extend the results of [14], [16], [20], [17], [15], [21], [22] and provide a Generalized SuperTwisting Observer (GSTO) which is able to estimate an unmeasured state exactly and in finite time despite of the lack of knowledge of its velocity of variation, when this state is multiplied by a known time varying signal which does not change sign. Using the GSTO it is possible not only to generalize and to extend, but also to unify, many results in the literature for the estimation of the reaction rates in bioprocesses, including the STO in [3], [2], [23] and the HGO in [5], [6], [8]. We show that in this case a quadratic Lyapunov function with matrix P constant allows us to assure the convergence of the observer in finite time and despite of the perturbations using time-varying gains. We provide in the paper a study example to illustrate the application of the GSTO for the estimation of reaction rates and states in a particular bioprocess. II. T HE G ENERALIZED S UPER -T WISTING O BSERVER (GSTO) We will consider a second order systems in the observability form x˙ 1 = f1 (y, u) + b (t, u, y) x2 , x˙ 2 = f2 (y, u) + δ (t, x, u, w) , y = x1
(1)
where x1 ∈ R, x2 ∈ R are the states, u ∈ Rm is a known input, w ∈ Rr represents an unknown input and y ∈ R is the measured output. f1 is a known continuous function and f2 corresponds to a known possibly discontinuous or multivalued function. δ represents and uncertain term. The measured variables are x1 and the known input u. b (t, u, y) is a known function which is lower and upper bounded, i.e. 0 ≤ bm ≤ b (t, u, y) ≤ bM .
(2)
For simplicity we consider here only the case when b (·) is positive, but the case when b (·) is negative can be treated in
the same manner. It is assumed that system (1) has solutions in the sense of Filippov [10]. Our aim is to propose an observer that is able to estimate the unmeasured state x2 from the measurement of x1 . Technically, our main result is to extend the results presented in [14], [15], [20], [21], [17], [22], which consider the coefficient b (t, u, y) in (1) to be constant, to the case when this coefficient is time-varying but it does not change its sign. We propose a (discontinuous) observer, named Generalized Super-Twisting Observer (GSTO), for the plant (1) with the form x ˆ˙ 1 = −l1 (t)φ1 (e1 ) + f1 (y, u) + b (t, u, y) x ˆ2 , (3) ˙x ˆ2 = −l2 (t)φ2 (e1 ) + f2 (y, u) , ˆ1 −x1 , and e2 = x ˆ2 −x2 are the state estimation where e1 = x errors. l1 (t) and l2 (t) are time-varying observer gains, that have to be selected to assure the convergence of the observer. The injection nonlinearities φ1 and φ2 are of the form 1
q
(4) φ1 (e1 ) = μ1 |e1 | 2 sign (e1 ) + μ2 |e1 | sign (e1 ) , 2 μ 1 q− 1 φ2 (e1 ) = 1 sign (e1 ) + μ1 μ2 q + |e1 | 2 sign (e1 ) + 2 2 + μ22 |e1 |
2q−1
sign (e1 ) ,
(5)
where μ1 and μ2 are non negative constants, not both zero, and q > 12 is a real number. Note that φ1 and φ2 are related, since φ2 (e1 ) = φ1 (e1 ) φ1 (e1 ), that they are both monotone increasing functions of e1 and φ1 is continuous while φ2 is discontinuous at e1 = 0 (when μ1 > 0). Solutions of the observer (3) are also understood in the sense of Filippov [10]. The state estimation errors (i.e. the estimation error vector T e = [e1 , e2 ] ) satisfy the differential equation e˙ 1 = −l1 (t)φ1 (e1 ) + b (t, u, y) e2 e˙ 2 = −l2 (t)φ2 (e1 ) − δ (t, x, u, w) ,
(6)
We impose the following growth conditions on the perturbation term: For every value of (t, x, u, w) |δ (t, x, u, w)| ≤ Δ(t) |φ2 (e1 )| ,
(7)
n
for every x ∈ R and t ≥ 0. where Δ(t) ≥ 0 is a known time function (or constant). Note that when μ1 > 0 the discontinuity of function φ2 allows in (7) non vanishing (or persistent) perturbations δ, i.e. |δ (t, x, u, w)| ≤ Γ, with Γ > 0. If μ1 = 0 the class of perturbations described by (7) vanishes when e1 = 0. It will be shown by means of a time-invariant, strong, quadratic Lyapunov function (10), that an appropriate selection of the gains, according to the variation of the perturbations and of the coefficient b (t, u, y), renders the origin globally and robustly stable. Moreover, it is exactly stable. Theorem 1. Consider the GSTO (3), and suppose that the perturbation satisfies (7) for some known function (or constant) Δ(t) ≥ 0. Assume further that b (t, u, y) is bounded as in (2). Choose the gains as l1 (t) =
1 β k1 (t)
+ β k2 (t)
l2 (t) =
β k1 (t)
+
β+2 β k2 (t)
(8) ,
where β > 0, > 0 are arbitrary positive constants, k2 (t) is an arbitrary time function and 2 Δ(t) k2 (t) 2 + − β + k1 (t) > Δ(t) + 4 2 b(t,u,y) + b(t,u,y) 4
k2 (t) b(t,u,y)
− β+
2
2
(9) .
Then e = 0 is a globally, robustly stable point, so that all trajectories of system (6) converge asymptotically to the origin despite of the perturbations, and the quadratic form VQ (e) = ζ T P ζ , where
ζ T = ΦT (e) = φ1 (e1 ) ,
(10)
e2 ,
(11)
and P is a constant, positive definite matrix P = P T > 0, given by p1 p 3 β + 2 , − P = = , (12) p3 p 2 − 1 is a strong, robust Lyapunov function. Moreover, if μ1 > 0 or 1/2 < q < 1 convergence is in Finite-Time. When μ2 > 0 system (6) is Input-to-State Stable (ISS) with input δ. For lack of space the proof is not presented here. But it is an extension of previous results. The distinguishing properties of the proposed observer are: 1) It is able to estimate exactly the state x2 after a finite time and robustly with respect to uncertainties/perturbations, represented by δ (t, x, u, w) in (1), that are persistent, i.e. |δ (t, x, u, w)| ≤ Γ, with Γ > 0. This feature is provided to our observer by the discontinuous injection term φ2 (e1 ), when μ1 > 0. It is important to note that finite time convergence can be achieved without discontinuous injection terms (just with continuous but not locally Lipschitz continuous ones in the neighborhood of the origin), but only if the perturbation δ vanishes when e1 = 0. 2) All proofs are based on Lyapunov’s method. The Lyapunov functions used here are of quadratic type, so that the mathematical machinery required is very similar to what is needed for linear systems. 3) The proposed observer is a generalization and improvement of several observers available in the literature. In particular, it includes a linear observer with variablegains. The GSTO (3) has several design parameters: two positive constants β > 0 and > 0; two functions k1 (t) and k2 (t); two non negative constants μ1 ≥ 0 and μ2 ≥ 0 and the power 1/2 < q. It is possible to show that if Δ is constant, and bm and bM are known, then it is possible to select constant observer gains l1 and l2 . Moreover, when b(·) = 1 is constant, the previous Super-Twisting-like observers, reported in [22], are recovered. In particular, with μ1 = 1 and μ2 = 0 the classical Super-Twisting Observer (STO) is obtained. In contrast, with μ1 = 0, μ2 = 1 and q = 1 a Linear Observer is recovered,
which has the structure of a High-Gain Observer (HGO) when l1 = α1 θ and l2 = α2 θ2 , with constant positive values of α1 , α2 and the positive high gain parameter θ. An alternative HGO with varying gains proposed in [5] can be recovered selecting in (3) l1 (t) = α1 b(t)θ and l2 (t) = α2 b(t)θ2 , with constant positive values of α1 , α2 and the positive high gain parameter θ. Note, however, that the HGOs are not able to estimate exactly the state of the plant in the presence of non vanishing perturbations, whole the observers with μ1 > 0 are able to achieve this. If the bound of the perturbation Δ(t) is time-varying, the GSTO (3) can "adapt" its gains to the the varying perturbation, in the same manner as the ST Controller reported in [21]. III. E STIMATION OF REACTION RATES IN BIOPROCESSES USING THE GSTO Consider the general dynamical model of a (Bio)chemical processes [1], [4], obtained from mass and energy balances, in stirred tank reactors: ξ˙ = Kϕ(ξ, t) − Dξ − Q(ξ) + F
(13)
where ξ ∈ Rn represents the vector of concentrations of the process components (reactants, products, biomass) and possibly temperature. The matrix K ∈ Rn×q is the stoichiometric coefficient matrix, ϕ ∈ Rq is the reaction rate vector, D is the dilution rate, Q is the gaseous outflow rate vector, and F is the feedrate vector. Note that ξi represents the i-th component of ξ ∈ Rn . Let us consider that there are q independent reactions, and that there are q measured components. We partition the state in measured ξ a and non measured states ξ b a ξ ξ= . (14) ξb With the corresponding partition of matrices and vectors the dynamical model (13) can be written as follows: ξ˙a ξ˙b
= =
K a ϕ(ξ, t) − Dξ a − Qa (ξ) + F a b
b
b
b
K ϕ(ξ, t) − Dξ − Q (ξ) + F .
(15) (16)
If the matrix K a is full rank (which holds true if the reactions and the measurements are independent), then the variable z = ξ b − K b (K a )−1 ξ a has a dynamic behavior, given by z˙ = −Dz − Qb + F b − K b (K a )−1 (−Qa + F a ) ,
(17)
which is independent of the reaction kinetics. An Asymptotic Observer (AO) is given by zˆ˙ ξˆb
= −Dˆ z − Qb + F b − K b (K a )−1 (−Qa + F a ) , = zˆ + K b (K a )−1 ξ a . (18) This AO estimates the unmeasured states ξ b asymptotically if the stoichiometric coefficient matrix K, the feedrates F , the dilution rate D and the gaseous outflow rate Q are known. However, nothing has to be known about the reaction rates ϕ. An important observation here, is that under the same hypothesis for the construction of an AO it is possible to
estimate all the reaction rates exactly and in finite time using a GSTO (3). To see this consider the variable y given by y = (K a )−1 ξ a
(19)
that has the dynamic behavior y˙ = ϕ(t) − (K a )−1 (Da ξ a + Qa − F a ) ϕ˙ = Δ ,
(20)
T
where Δ = [ϕ˙ 1 , . . . , ϕ˙ q ] is the uncertain vector of velocities of change of the reaction rates ϕ. It is easy to see that the use of a GSTO (3) for each channel provides an estimation of the reaction rates if a bound on Δ(t) is known. Sometimes it is of interest to estimate the specific reaction rate. In this case it is necessary to use the version of the GSTO with varying coefficient b(·). Instead of going into the details using the general model, we illustrate this in the following section by means of an example. IV. C ASE S TUDY: P HYTOPLANKTON G ROWTH M ODEL In this section we use a biotechnological example to study the estimation of different variables, including the states and the reaction rates, under several measurement scenarios. We compare the results of the GSTO with a High-Gain Observer (HGO) for the estimation of the reaction rates and with the Asymptotic Observer (AO) for the state estimation. A simplified growth model of the phytoplankton in a photobioreactor has been presented in [7], and is given by the four DE ⎧ ˙ S S = −ρm S+K C + D[Sin − S] ⎪ s ⎪ ⎨ ˙ S L N = ρm S+Ks C − γ(I)N C − DN + βL (21) ΣP L ˙ ⎪ L = γ(I)N − DL − βL ⎪ C ⎩ C˙ = a(I)L − DC − λC , where S is the substrate (inorganic nitrogen) concentration, N is the internal nitrogen concentration, L is the chlorophyllian nitrogen concentration and C represents the particulate carbon concentration; D represents the (scalar) dilution rate (scalar), Sin is the input substrate concentration, λ is the respiration factor, β is the coefficient of chlorophyl degradation; finally I is the light intensity, functions a(I) and γ(I) describe the influence of light intensity I in the process αI (22) a(I) = kI + I γ(I) = a(I)
K L KC KC + I
(23)
For the simulation study a 25 days period was considered, the dilution rate was zero during the first 3 days (batch period) and then it has taken a constant value of D = 0.4[d− 1]. The light intensity varied in time according to day and night periods, and for the final days, from day 19 to day 25, it takes a constant value. The used parameters and initial conditions considered are shown in Table I. In the following paragraphs we show 4 observation scenarios, in which different assumptions about the knowledge of the dynamical model (21) and the observation objectives are
TABLE I I NITIAL C ONDITIONS AND VALUES OF THE PARAMETERS IN THE MODEL
S(0) = 10(μmol N.L−1 ) N (0) = 0.3(μmol N.L−1 ) L(0) = 10(μmol N.L−1 ) C(0) = 50(μmol C.L−1 )
Parameter Values ρm = 0.5(μ molN.molC−1 .d−1 ) KS = 0.43(μ M N) KL = 6.59(n.d) KC = 33(μ mol quanta.m−2 .s−1 ) α = 24.1(d−1 ) KI = 208.5(μ mol quanta.m−2 .s−1 ) λ = 0.054(d−1 ) β = 0.345(d−1 ) Sin = 40(μmol N.L−1 )
done. In the simulations we show results without and with measurement noise. In the latter case white Gaussian noise is added to the measured signals (L, C) with an amplitude of 5% of the amplitude of the signal. We suppose for system (21) that the variables (C, L) are measured and that the specific reaction rate a(I) is unknown. The objective is the online estimation of the photosynthesis rate a(I). Consider the photosynthesis rate ϕ a(I)L as an unknown variable. From the dynamics of the particulate carbon C in (21) we can write C˙ = −(D + λ)C + ϕ ΣP,1 ϕ˙ = δ(t) , where δ(t), the velocity of variation of ϕ, is an unknown signal, and we assume that it is bounded. This is a reasonable hypothesis for a bioreactor. Finite-time estimation of ϕ, in spite of the uncertainty in the knowledge of its time variation δ, can be achieved by the GSTO ˙ ˜ − (D + λ)C + ϕˆ Cˆ = −l1 φ1 (C) ΣΩ,1 (24) ˙ ˜ ϕˆ = −l2 φ2 (C) , where C˜ Cˆ − C and ϕ˜ ϕˆ − ϕ are the estimation errors for C and ϕ, respectively. To estimate the specific reaction rate a(I) we divide ϕˆ by the measured value of L, i.e. ϕˆ . L An alternative GSTO is given by ˙ ˜ − (D + λ)C + Lˆ a Cˆ = −l1 φ1 (C) ΣΩ,2 ˜ , a ˆ˙ = −l2 φ2 (C)
(25)
where the gains l1 and l2 can possibly depend on b(·). An advantage of the latter observer is that it avoids the division in (25) by a noisy and perhaps small signal L. We compare the results of the (discontinuous) GSTO in (24) and the relation (25), with the continuous (linear) HGO given by Cˆ = −(D + λ)C + a ˆL − 2θL[Cˆ − C] ΩHG (26) 2 ˙a ˆ ˆ = −θ L[C − C] , where the High-Gain is set as θ = 7.
Parameters μ1 μ2 l1 4 1 6 8 2 1
20
a ˆ(I)OAG
l2 1 2 a ˆ(I)GST O
a(I)
15 18
10
14
5
0 0
8 2 17
5
17.5
10
18
15
18.5
20
19
25
tiempo (d)
A. Scenario 1: Reaction rate estimation
a ˆ(I) =
Observer used for a ˆ, Iˆ ˆ N
tasa de fotosíntesis (d−1)
Initial Conditions
TABLE II O BSERVER PARAMETERS
Fig. 1. Estimation of the specific reaction rate a ˆ without measurement noise: GSTO (green), HGO (gray)
The parameters and the gains for GSTO ΣΩ,1 are shown in Table II. Figures 1 and 2 show the simulation results without measurement noise. Note that the GSTO converges in finite time to the true value while the HGO is not able to estimate exactly the value of a(I), but the error remains bounded around the zero value. Although the gain θ of the HGO can be increased so that the estimation error remains in an arbitrary neighborhood of zero, the amplification of noise in the noisy case counteracts this positive effect. The trade off between convergence velocity and noise amplification is also valid for the GSTO but a good selection of the gains can produce a better result than for the HGO. This is illustrated in Figure 3, where the simulation results with measurement noise are presented. In this case the GSTO overperforms the HGO in convergence velocity and noise rejection in the reconstruction of a(I). This is true for both the day/night cycles and for the constant light period at the end of the simulation. B. Scenario 2: Finite-Time Estimation of N We assume that estimate N in finite (21) one obtains ΣP,3
(L, C) are measured and we want to time. From the dynamics of L in model L N − dL − βL L˙ = γ(I) C ˙ N = δ2 (t) ,
(27)
L where γ(I) C is considered as known due to the estimation of γ(I) in (30). This corresponds to the plant (1), where N corresponds to the state to be estimated, whose velocity of L correvariation δ2 (t) is uncertain but bounded, and γ(I) C sponds to the known time-varying coefficient b(·). Using the GSTO (3) the following observer ˆ − DL − βL − l1 φ1 (L) ˆ˙ = γ(I) ˆ LN ˜ L C ΩP,3 (28) ˙ˆ L ˜ I) ˆ , N = −l2 φ2 (L)γ( C
0 −2 0
5
Fig. 2. 25
10 15 tiempo (d)
20
25
Estimation Error of a ˜: GSTO (green), HGO (gray)
a ˆ(I)OAG
a ˆ(I)GST O
(μmol N.l−1)
error
30
a ˜(I)OAG a ˜(I)GST O
2
20
10 ˆesc4 N
0 0
5
a(I)
20
Fig. 4.
ˆesc3 N
10 15 tiempo (d)
ˆesc2 N
N
20
25
Estimation of N without measurement noise
15
a(I) 10
ˆ is given by (30). The estimation error, N ˜ = where γ(I) ˆ − N , satisfies the differential equation N
30 20
5 10 17
0 0
17.5
18
18.5
19
19.5
20
0
5
10
15
20
25
timH (d
Fig. 3. Estimation of the specific reaction rate a ˆ with measurement noise: GSTO (green), HGO (gray)
˜ = −[ˆ ˆ − γN ] L − D[N ˆ − N] . N γN C Since γˆ = γ after a finite time, this observer converges asymptotically whenever D(t) is persistently exciting. Simulation results are presented in subsection IV-E below. D. Scenario 4: Asymptotic Observer (AO) for N estimation
˜ = L ˆ − L, is able to estimate N exactly in finite where L time. Note that, in contrast to the following scenario 3 (subsection IV-C), the measurement of S is unnecessary for the estimation of N , and the convergence is in finitetime and not asymptotic. The results of this estimator for N are presented in subsection IV-E below, together with the estimators in the following two subsections. C. Scenario 3: Estimation of I, γ(I), N We consider the variables L, C, S to be measured and we use the estimation of a ˆ obtained previously with (25). First, an estimation of the light intensity I is obtained, which is used to estimate the specific reaction rate γ(I) (23). Finally, the internal nitrogen concentration N is estimated. We assume also that the functions a(I) and γ(I) are known, with known values of the parameters. The light intensity can be obtained from (22), and an estimation is given by ˆ KI a , (29) Iˆ = α−a ˆ where a ˆ is the estimation given previously. To avoid the singularity in (29), a ˆ is saturated at the 95% of its maximal value, given by α. Using a ˆ and Iˆ the specific reaction rate γ(I) can be estimated as K L KC ˆ =a γ(I) ˆ . (30) KC + Iˆ Since the dynamics of N in (21) is given by S L C − γ(I)N − DN + βL . S + Ks C an estimation of N can be obtained by the observer N˙ = ρm
ˆ˙ = ρm N
S ˆ + βL , ˆN ˆ L − DN C − γ(I) S + KS C
(31)
We assume again that (L, C, S) are measured, but that the reaction rates a(I) and γ(I) are completely unknown. We design an Asymptotic Observer (AO) [1] to estimate N . Defining the auxiliar variable ζ = N + L, which satisfies ζ˙ = −Dζ + ρm
S C, S + KS
the AO is given by ζˆ = −Dζˆ + ρm
S C. S + KS
Since the dynamics of the estimation error ξ˜ = ξˆ − ξ is ˙ ζ˜ = −Dζ˜ , ξˆ converges asymptotically to ξ whenever the dilution rate D(t) is persistently exciting. The internal nitrogen concentration N can be estimated through ˆAO = ζˆ − L . N
(32)
This AO is able to estimate N without the knowledge of the reaction rates a(I) and γ(i), but the convergence rate is asymptotic and its velocity is not assignable and depends on the dilution rate D(t). E. Estimation of N : Simulation Results We compare in simultation the estimation results of N provided by the three observers (28), (31) and (32)) designed in the previous subsections IV-B, IV-C and IV-D, respectively. Figure 4 presents the simulation results without measurement noise for the three observers, while Figure 5 presents the simulation results with measurement noise. The initial conditions of all observers are the same. The asymptotic convergent observer (31) provides the best performance of the three observers, with good convergence
(μmol N.l−1)
30
R EFERENCES
20
10 ˆesc4 N
0 0
5 Fig. 5.
ˆesc3 N
10 15 tiempo (d)
ˆesc2 N 20
N 25
Estimation of N with measurement noise
rate and noise rejection. This is probably due to the fact that this observer takes into account the dynamic behavior of N . The finite-time convergent observer (28) provides the fastes convergent velocity, but is sensitive to measurement noise. It is remarkable that the AO (32) has a poor noise rejection. This is due to the fact that the measurement noise in L passes unfiltered to the estimation of N . Note that during the first days the reactor operates in batch (i.e. D(t) = 0). This explains the constant estimation error at the beginning of the simulation for the AO. V. C ONCLUSIONS Discontinuous observers, and in particular second order sliding mode observers based on the super-twisting algorithm, provide an attractive alternative for the estimation of reaction rates and states in highly uncertain bioreactors. In this paper we provide a generalization and extension of the classical Super-Twisting Observer (STO), that we call Generalized Supert-Twisting Observer (GSTO) in several directions: 1) it allows known time-varying coefficients of the unmeasured states. 2) It includes continuous and discontinuous injection terms, so that the performance of the observer can be improved. 3) It allows time varying observer gains and time varying perturbation bounds. 4) It includes as special cases many previous observation schemes, as the High-Gain Observer or the STO. 5) The convergence and performance proofs are all unified and are simple, since they use the linear-like framework orginaly presented in [16], [17]. We show and illustrate by a worked example that the GSTO can be used for the estimation of reaction rates and states in bioreactors, which can be done usually in finite time and exactly despite of the lack of knowledge of the analytical form and the time variation of the reaction rate. The use of GSTOs can improve the performance of controllers of this kind of bioprocesses. Preliminary work on this topic has been presented in [24]. ACKNOWLEDGMENTS The authors gratefully acknowledge the financial support from PAPIIT, UNAM, grant IN113614, Fondo de Colaboración del II-FI, UNAM, IISGBAS-144-2012.
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