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Application of a Continuous-discrete Unknown Input Observer to Estimation in Phytoplanktonic Cultures1
8th IFAC Symposium on Advanced Control of Chemical Processes The International Federation of Automatic Control Singapore, July 10-13, 2012
Application of a Continuous-discrete Unknown Input Observer to Estimation in Phytoplanktonic Cultures ⋆ E. Rocha-C´ozatl ∗ J. A. Moreno ∗∗ A. Vande Wouwer ∗∗∗ Departamento de Ingenier´ıa Mecatr´onica, Universidad Nacional Aut´onoma de M´exico, 4510 Mexico, D.F., Mexico. (e-mail: [email protected]) ∗∗ Instituto de Ingenier´ıa. Universidad Nacional Aut´onoma de M´exico, 4510 Mexico, D.F., Mexico. (e-mail: [email protected]) ∗∗∗ Service d’Automatique, Universit´e de Mons, B-7000 Mons, Belgium. (e-mail: [email protected])
∗
Abstract: The simultaneous estimation of unmeasured state variables and unknown inputs is particularly relevant in environmental applications, such as biological wastewater treatment or the culture of microalgae in open environments. In this study, an extension of the unknown input observer (UIO) recently proposed in (7) to nonlinear continuous-time models and discrete-time measurements is presented. This UIO takes account of the process and measurement noises, and uses a linearization around the estimated trajectory, in a way similar to the extended Kalman filter. This extended continuousdiscrete UIO is applied to state and input reconstruction in continuous cultures of phytoplankton. In particular, the problem of simultaneously reconstructing the extra- and intra-cellular nitrogen concentrations together with the light intensity or the dilution rate from measurements of biomass in cultures of the marine microalgae Dunaliella tertiolecta or the cryptophyceae Rhodomonas salina are considered. Simulation results and processing of real data demonstrate the performance of the method. Keywords: State estimation; Input estimation; Robust estimation; Biotechnology. 1. INTRODUCTION State estimation in bioprocesses is particularly important for monitoring and control purposes, as hardware sensors are expensive (high acquisition and maintenance costs), are not always available for the specific measurement under consideration (for example, the internal nitrogen quota is a key variable for microalgae, especially in the case of biofuel production where a nitrogen limitation is a condition of enhanced lipid storage; this variable is however difficult to measure on-line), and have stringent operating conditions (sterilization, calibration, processing time, sample destruction, etc). In this study, attention is focused on the design of unknown input observers for monitoring cultures of microalgae in the chemostat. Unknown Input Observers (UIO) are dynamical systems that estimate the state variables of a system robustly with respect to the disturbances or unknown inputs that affect the system. Their study for linear systems has drawn continuing interest over the last three decades (10; 2; 8; 11; 4). ⋆ The authors gratefully acknowledge the support of FNRS and CONACYT in the framework of a bilateral research agreement, as well as the financial support from PAPIIT, UNAM, grant IN111012, and Fondo de Colaboraci´on del II-FI, UNAM, IISGBAS-165-2011. They also express their gratitude to Laboratoire d’Oc´eanographique in Villefranche-sur-Mer (France) for providing the experimental data. This paper presents research results of the Belgian Network DYSCO (Dynamical Systems, Control, and Optimization), funded by the Interuniversity Attraction Poles Programme, initiated by the Belgian State, Science Policy Office. The scientific responsibility rests with its authors.
978-3-902823-05-2/12/$20.00 © 2012 IFAC
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Recently, a recursive filter, optimal in a minimum-variance unbiased sense (i.e., in the spirit of Kalman filtering), has been proposed to cope with unknown inputs and process/measurement noises in (7). In this study, we propose an extension of this algorithm to address the particularly important case in bioprocess applications of a nonlinear continuous-time model associated with discrete-time (and often rare) measurements. This algorithm is applied to two case-studies (simulation and real data) related to the culture of phytoplankton in the chemostat, where it is of interest to on-line estimate component concentrations, as well as influencing inputs such as the light intensity(which varies according to day-night cycle and weather conditions) or the dilution rate (which can vary depending on the occurrence of rains or storms in uncontrolled environments). 2. PRELIMINARIES 2.1 Kalman Filter: state estimation Consider a discrete-time linear model with noise in both state and output equations x[k+1] = Ax[k] + Bu[k] + w[k]
(1a)
y[k] = Cx[k] + v[k] (1b) where x ∈ Rn is the state vector, u ∈ Rp is the input vector, y ∈ Rm is the output (measurement) vector. We assume that w[k] and v[k] are stationary zero-mean white noise processes with covariance matrices Q and R. In addition, we assume that x0 , w[k] and v[k] are uncorrelated. 10.3182/20120710-4-SG-2026.00028
8th IFAC Symposium on Advanced Control of Chemical Processes Singapore, July 10-13, 2012
• Measurement update (MU)
Kalman Filter (KF) equations are (6): • Time Update (TU)
ˆ¯[k+1|k+1] = x ˆ¯[k+1|k] + B u x ˆ[k|k+1] T ¯ [k+1] = P¯[k+1|k] C R ¯ −1 L
x ˆ[k+1|k] = Aˆ x[k|k] + Bu[k]
[k+1]
P[k+1|k] = AP[k|k] AT + Q
P¯[k+1|k+1] = P¯[k+1|k] + BPu[k|k+1] B T − ¯T ¯ [k+1] F Pu[k|k+1] B T −BPu[k|k+1] F T L −L
• Measurement Update (MU)
[k+1]
x ˆ[k+1|k+1] = x ˆ[k+1|k] + L[k+1] y[k+1] − C x ˆ[k+1|k] L[k+1] = P[k+1|k] C T R
ˆ¯[k+1|k+1] + x ˆ[k+1|k+1] = x ¯ +L[k+1] y[k+1] − C x ¯ˆ[k+1|k+1] P[k+1|k+1] = P¯[k+1|k+1] −
P[k+1|k+1] = P[k+1|k] − P[k+1|k] C T RCP[k+1|k] −1 where R = CP[k+1|k] C T + R
T ¯ ¯ [k+1] − F Pu[k|k+1] F T L ¯ [k+1] R −L [k+1]
where F = CB.
A block diagram is shown in Figure 1.
Since the input is no longer available, an estimation mechanism has been introduced between TU and MU. A block diagram is presented in Figure 2.
Fig. 1. Block diagram of Kalman Filter. If the state model is continuous in time and nonlinear, i.e.,
Fig. 2. Block diagram of unknown input observer.
x˙ = f (x, u, w)
(2a)
y[k] = Cx[k] + v[k]
(2b)
the Time Update equations are now x ˆ˙ = f (ˆ x, u) P˙ = A (t) P + P AT (t) + Q where A (t) is the Jacobian of f (x, u) with respect to x along x (t) , u). In this case the estimated trajectory, i.e., A (t) = ∂f ∂x (ˆ the estimator is called an Extended Kalman Filter (EKF) .
When it is not possible to measure the input u, the conventional KF (or EKF) must be modified in order to estimate the states as well as the input. In (7), the following algorithm is proposed for model (1a)-(1b). • Time Update (TU) x ¯ˆ[k+1|k] = Aˆ x[k|k] P¯[k+1|k] = AP[k|k] AT + Q
∂f ∂u
(3) x (t) , u[k|k] ,
δx[k+1] = A[k] δx[k] + B[k] δu[k]
(4)
where A (t) = and which reads
∂f ∂x
δ x˙ = A (t) δx + B (t) δu, x (t) , u[k|k] and B (t) =
The proposed extended continuous-discrete UIO is then: • Time Update
• Estimation of the Unknown Input (Est UI)
¯ [k+1] y[k+1] − C x ˆ u ˆ[k|k+1] = M ¯[k+1|k] −1 ¯ −1 F Pu[k|k+1] = F T R [k+1]
We consider the nonlinear continuous-time model with discretetime measurements (2a)-(2b) with a nonmeasurable input u. The proposed extension of the previous algorithm (7) is similar to a standard Time Update of a continuous-discrete EKF, i.e., the prediction is made using the original nonlinear model, starting from the last corrected state. Since u is unknown on the prediction interval, the last estimated value of this input is used instead. The propagation of the covariance matrix P , is the same as in the standard EKF. The estimation of the input u is updated at the measurement times, based on a discretized version of the linearized model
2.2 State and Input Estimation
¯ [k+1] = C P¯[k+1|k] C T + R R −1 ¯ −1 ¯ [k+1] = F T R ¯ −1 F FTR M [k+1] [k+1]
3. AN EXTENDED CONTINUOUS-DISCRETE UIO
x ˆ˙ = f x ˆ, u[k|k] P˙ = A (t) P + P AT (t) + Q with A (t) = ∂f ˆ (t) , u[k|k] . ∂x x
x ˆ[k+1|k] = x ˆ ((k + 1) T ) P[k+1|k] = P ((k + 1) T )
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8th IFAC Symposium on Advanced Control of Chemical Processes Singapore, July 10-13, 2012
corresponds to the distinguishability of two different unmeasured state/unknown input trajectories (D1 (t), Q1 (t), S1 (t)) and (D2 (t), Q2 (t), S2 (t)) from the corresponding measured variables (Sin1 (t), X1 (t)) and (Sin2 (t), X2 (t)), i.e., if Sin1 (t) = Sin2 (t), X1 (t) = X2 (t) for an interval of time t ∈ [0, T ] the unmeasured variables are identical D1 (t) = D2 (t), Q1 (t) = Q2 (t), S1 (t) = S2 (t) during the same time interval. Detectability corresponds to the validity of this property in an asymptotic manner, i.e. limt→∞ (D1 (t) − D2 (t)) = 0, limt→∞ (Q1 (t) − Q2 (t)) = 0 and limt→∞ (S1 (t) − S2 (t)) = 0 (13). Checking the observability property when all inputs are known can be done using the Observability map (or its Jacobian for the local property). However, observability with unknown inputs or detectability is a property much harder to evaluate.
• Estimation of the Unknown Input ¯ [k+1] = C P¯[k+1|k] C T + R R −1 T ¯ −1 ¯ [k+1] = F T R ¯ −1 F[k] M F[k] R[k+1] [k] [k+1] ¯ [k+1] y[k+1] − C x ˆ δu ˆ[k|k+1] = M ¯[k+1|k] −1 T ¯ −1 Pu[k|k+1] = F[k] R[k+1] F[k]
• Measurement update
ˆ¯[k+1|k] + B[k] δ u ˆ¯[k+1|k+1] = x ˆ[k|k+1] x T ¯ −1 ¯ ¯ L[k+1] = P[k+1|k] C R [k+1]
T P¯[k+1|k+1] = P¯[k+1|k] + B[k] Pu[k|k+1] B[k] − ¯T ¯ [k+1] F[k] Pu[k|k+1] B T −B[k] Pu[k|k+1] F T L −L [k]
[k+1]
[k]
¯ [k+1] y[k+1] − C x ˆ¯[k+1|k+1] + L ˆ x ˆ[k+1|k+1] = x ¯[k+1|k+1] ¯ P[k+1|k+1] = P[k+1|k+1] − ¯ [k+1] R ¯ [k+1] − F[k] Pu[k|k+1] F T L ¯T −L [k] [k+1] u ˆ[k+1|k+1] = u ˆ[k|k] + δ u ˆ[k|k+1]
(5)
Here we will use a method proposed in (13) to check state observability for a class of bioreactors when unknown inputs are present. The same method has been used in (9) to study state observability/detectability for an induction machine, when all inputs are known. The idea underlying the analysis is to consider two identical systems, one is the plant Σp (6), and a copy of it (with states x1 , x2 , x3 , unknown input d and known input u) given by
where F[k] = CB[k] . Σx
4. APPLICATION TO PHYTOPLANKTON CULTURES 4.1 Phytoplankton Culture: Droop Model Droop model (3) is a simple and widely used model of phytoplankton culture. The mass balance equations are given by X˙ (t) = −D (t) X (t) + µ (Q) X (t) (6) Σp Q˙ (t) = ρ (S) − µ (Q) Q (t) ˙ S (t) = D (t) [Sin (t) − S (t)] − ρ (S) X (t) where X is the biovolume (i.e., the volume of cells in a unit volume of culture medium), Q is the internal quota, which is defined as the quantity of nitrogen per unit of biovolume, D (t) represents the dilution rate, S is the substrate (inorganic nitrogen) concentration and Sin is the input substrate concenS(t) tration. Function ρ (S) = ρm S(t)+k is the uptake rate, and S kQ µ (Q) = µ ¯ 1 − Q(t) is the growth rate; constants kS and ρm represent a half-saturation constant for the substrate and the maximum uptake rate, respectively; constant µ ¯ is the theoretical maximum growth rate, obtained for an infinite internal quota and kQ is the minimum internal quota allowing growth. The considered biological process is the culture of the chlorophyceae Dunaliella tertiolecta in the chemostat (a continuous bioreactor operated with constant volume, which can mimic an open natural system). A more complete description of the experimental setup can be found in (1). Observability/Detectability Our objective is to construct an observer to estimate the unmeasured state variables Q, S and the unknown input D, from the knowledge of the dynamical model (6) and the measurements of the known input Sin and the output variable X. This is certainly only possible instantaneously if (state and unknown input) observability is fulfilled, or at least asymptotically if the system is (state and unknown input) detectable. (State/Unknown input) Observability 581
(
x˙ 1 (t) = −d (t) x1 (t) + µ (x2 ) x1 (t) x˙ 2 (t) = ρ (x3 ) − µ (x2 ) x2 (t) . x˙ 3 (t) = d (t) [u (t) − x3 (t)] − ρ (x3 ) x1 (t)
(7)
If we assume that the measured variables of systems Σp and Σx are identical for a time interval, i.e. Sin (t) = u(t), X(t) = x1 (t), (state and unknown input) observability is equivalent to the fact that the only possible solutions of equations (6) and (7) under these restrictions is given by Q(t) = x2 (t), S(t) = x3 (t) and D(t) = d(t), during the same time interval, whereas detectability is equivalent to satisfying these equalities asymptotically. If we introduce the state and unknown input error variables ǫ1 = X − x1 , ǫ2 = Q − x2 , ǫ3 = S − x3 , and ǫd = D − d, respectively, equation (7) can be replaced by KQ ǫ˙1 = −ǫd X − dǫ1 + µ ¯ 1− X Q KQ −¯ µ 1− (X − ǫ1 ) Q − ǫ2 S S − ǫ3 ǫ˙2 = ρm − −µ ¯ ǫ2 S + KS S − ǫ 3 + KS ǫ˙3 = ǫd [Sin − S] − dǫ3 S S − ǫ3 −ρm X− (X − ǫ1 ) S + KS S − ǫ 3 + KS
(8)
and observability requires that under the restrictions Sin (t) = u(t) and ǫ1 (t) = 0 the only solutions of system (6) and (8) are ǫ2 (t) = 0, ǫ3 (t) = 0 and ǫd (t) = 0, or asymptotic convergence in case of detectability. This Differential-Algebraic (DA) system, consisting of the differential equations (6) and (8) and the algebraic restrictions Sin (t) = u(t) and ǫ1 (t) = 0, will be next reduced to a simpler differential equation system. From ǫ1 (t) = 0 (during a time interval) it follows that also ǫ˙1 (t) = 0 and, from the first equation of (8), that ǫd = −¯ µ KQ
ǫ2 . Q (Q − ǫ2 )
Equations (8) therefore reduce to
(9)
8th IFAC Symposium on Advanced Control of Chemical Processes Singapore, July 10-13, 2012
ρm KS ǫ3 (S + KS ) (S − ǫ3 + KS ) −¯ µKQ [Sin − S] ǫ2 ǫ˙3 = Q (Q − ǫ2 ) X − d + ρm KS ǫ3 . (S + KS ) (S − ǫ3 + KS )
ǫ˙2 = −¯ µǫ2 +
Numerical Simulations Nonlinear model (6) is considered with the numerical values of the parameters reported in (1). Two measurement configurations are studied: (1) variable X is measured and variables Q and S are estimated; (2) variables X and S are measured and Q is estimated. In both cases D(t) is an unknown input (it would represent the flow in and out of the natural system) and it is estimated simultaneously with the state. In order to test the estimation scheme, the actual evolution of D(t) is defined as a sinusoid in some periods of time and as a constant in some others. Case 1. State X (biovolume) is measured once a day (T = 1) with a relative measurement error of 3%. True initial conditions are defined as X(0) = 5, Q(0) = 0.5, ˆ S(0) = 1, and for the estimator are X(0) = 5 (since it is ˆ ˆ measured), Q(0) = 0.75, S(0) = 1.5, i.e., 50% initial error in the nonmeasured states. The design parameters of the estimator are P0 = I3 , where I3 represents an identity matrix of dimension 3 × 3, Q = 1 × 10−4 I3 , and since the considered error in measurements is 2 relative, matrix R is defined as R = (0.03 ∗ X(k)) , where X(k) is the k−th measurement of biovolume X. In Fig. 3 state estimates are shown as solid lines and are compared with the real values of each variable (star markers ⋆). Only the measurements of X are used by the filter, the other values are used for validation purposes.
Substrate S [ µ mol / l]
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estimated actual UB and LB of confid. intervals
40 20 0 0
40
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40
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Fig. 3. Evolution of the real and estimated states - UIO using the measurements of X. 1.8 1.6 1.4 Dilution rate D [1/day]
It can be easily proved that ǫ2 → 0 and ǫ3 → 0, provided that the state variables are positive and the substrate concentration is lower than the inlet concentration (which is meaningful from a biological point of view). Moreover, from (9) it also follows that ǫd → 0. We conclude that the model (6) is state and unknown input detectable, and so the measurement of X and Sin provide sufficient information to estimate asymptotically the two unmeasured states Q, S and the unknown input D.
20
Internal Quota Q [ µ mol / mm3 ]
Biovolume X [mm3 / l]
40
1.2 1 0.8 0.6 estimated actual UB and LB of confidence intervals
0.4 0.2 0
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30 Time [day]
40
50
60
Fig. 4. Evolution of the dilution rate - UIO using the measurements of X. The design parametersof the estimator are P0 = I3 , Q = 1 × 2 2 10−4 I3 , and R = diag (0.03 ∗ X(k)) , (0.03 ∗ S(k)) .
Fig. 5 shows the evolution of the real and estimated state variables. The evolution of the dilution rate is presented in Fig. 6. Clearly, the availability of a second measurement improves the quality of the estimation.
States X (biovolume) and S (substrate) are measured once a day (T = 1) with a relative measurement error of 3%.
Experimental Results In order to complete the test of the proposed algorithm, real-life experimental results are considered. The experimental data were collected from a pilot plant at Laboratoire d’Oc´eanographie in Villefranche-sur-Mer (France). The dilution rate was not constant but slowly varying and the sampling period is T = 0.65(day), i.e., 1 or 2 measurements per day. In the following figures (Fig. 7 and Fig. 8), the solid lines represent again the estimated values of S, X, Q and D, and the star points are the measured values (only the measured values of X are used in the filter, the other values are used for validating the estimated values only).
True initial conditions are defined as X(0) = 5, Q(0) = 0.5, ˆ ˆ S(0) = 1, and for the estimator as X(0) = 5, S(0) = 1 ˆ (measured states) and Q(0) = 0.75 (50% initial error).
ˆ Initial conditions for the estimator are defined as X(0) = ˆ ˆ 0.157, S(0) = 55.75 and Q(0) = 5.03. The design parameters of the estimator are the same as those for Case 1 in the previous
On the other hand, the estimation of the dilution rate is presented in Fig. 4 The filter convergence is fast and the performance is quite satisfactory. 95 % confidence intervals are also drawn using the information provided by the matrices P and Pu showing the reliability of the estimates. Case 2.
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8th IFAC Symposium on Advanced Control of Chemical Processes Singapore, July 10-13, 2012
Internal Quota Q [ µ mol / mm3 ]
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1.2 1 0.8 0.6 0.4
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Fig. 5. Evolution of the real and estimated states - UIO using the measurements of X and S.
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Fig. 8. Evolution of the real and estimated dilution rate - UIO using the measurements of X - Experimental data. 4.2 A more detailed model with light influence
1.8
In (14), a more detailed model, taking the influence of the incident light intensity is developed, as
1.6
Dilution rate D [1/day]
1.4
C˙ (t) = −D (t) C (t) − λC (t) + a (I) L (t) S (t) N˙ (t) = −D (t) N (t) + ρm C (t) S (t) + KS L (t) + βL (t) −γ (I) N (t) C (t) L (t) − βL (t) L˙ (t) = −D (t) L (t) + γ (I) N (t) C (t) S (t) S˙ (t) = D (t) (Sin − S (t)) − ρm C (t) S (t) + KS
1.2 1 0.8 0.6 estimated actual UB and LB of confidence intervals
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Fig. 6. Evolution of the real and estimated dilution rate - UIO using the measurements of X and S.
Internal Quota Q [ µ mol/mm3]
Internal substrate S [ µ mol/l]
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section. The results are satisfactory since the estimation is close to the measured values of the variables.
40
Variable C represents the particulate carbon concentration (carbon biomass), N is the internal nitrogen concentration and L is the chlorophyllian nitrogen concentration. The total particulate nitrogen can be computed as N + L. S is the substrate (inorganic nitrogen) concentration and Sin is the input substrate concentration. β is the coefficient of chlorophyll degradation, λ is the factor of respiration, and functions γ and a describe the influence of the light intensity I in the process
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estimated actual UB and LB of confid. intervals 5
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a (I) =
αI (t) KI + I (t)
Observability/Detectability Considering C and L as the measured states, N and S as the non measured states, D and Sin as the known inputs, and I as the unknown input, an observability analysis leads to the logical conditions that the system is state and unknown input detectable provided that N > 0, S > 0, I > 0.
10 5 0 0
KC αKL I (t) , KI + I (t) KC + I (t)
Fig. 7. Evolution of the real and estimated states - UIO using the measurements of X - Experimental data.
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Numerical Simulations As a particular application, the culture of the cryptophyceae Rhodomonas salina is considered, that is, model (10) with the parameter values reported in (14). The state estimation objective is to reconstruct N and S, using the culture model together with on-line measurements of C and L. The dilution rate D and input concentration Sin are known,
8th IFAC Symposium on Advanced Control of Chemical Processes Singapore, July 10-13, 2012
whereas the light intensity I is the unknown input, which has to be simultaneously estimated. States C (particulate carbon) and L (chlorophyllian nitrogen) 1 are measured one time per hour (T = 24 ) with a realtive error of 1%. In this case, dilution rate D is considered to be constant and defined as D = 0.4[1/day]. True initial conditions are defined as C(0) = 3.6, L(0) = 0.03, ˆ N (0) = 0.06, S(0) = 40, and for the estimator are C(0) = 3.6, ˆ ˆ L(0) = 0.03 (since these two are measured) N (0) = 0.09, S(0) = 34, that is, 50% initial error in N and 15% in S, the nonmeasured states.
Substrate S [µ mol N/l]
Internal N Cholophyllian N Part. Carbon N [µ mol N/l] L [µ mol N/l] C [µ mol C/l]
The design parameters of the = estimator are P0 = I4 , Q 2 2 1 × 10−4 I4 , and R = diag (0.01 ∗ C(k)) , (0.01 ∗ L(k)) , where C(k), L(k) are the k−th measurements of C and L, respectively.
of unknown reaction kinetics. In this study, an extension to the minimum-variance unbiased filter developed in (7) is proposed for the case of nonlinear continuous-time systems with discretetime measurements. State and unknown input detectability conditions are also discussed. The numerical results show the generally good performance of the filter in situations where no a priori information on the unknown input is available. In situations where such information is available, it is probably interesting to incorporate it in the form of an exosystem as suggested in (12), which is the subject of future investigation. REFERENCES [1]
[2]
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[6] [7]
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Fig. 9. Evolution of the real and estimated states - UIO using the measurements of C and L.
[8]
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[10]
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[11]
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[12]
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[13]
Fig. 10. Evolution of the real and estimated light intensity - UIO using the measurements of C and L. [14]
5. CONCLUSIONS UIO observers are increasingly applied to estimation in biological systems, one prominent example being the asymptotic observer which has been used extensively to tackle the problem 584
O. Bernard, A. Sciandra, G. Sallet. A non-linear software sensor to monitor the internal nitrogen quota of phytoplanktonic cells. Oceanologica Acta, 24, 2001, pages 435442. M. Darouach, M. Zasadzinski, and S.J. Xu. Full-order observers for linear systems with unknown inputs. IEEE Transactions on Automatic Control, 39 (3), 1994, pages 606-609. M. Droop. Vitamin B12 and marine ecology IV : The kinetics of uptake growth and inhibition in Monochrisis lutheri. Journal of the Marine Biological Association, 48(3), 1968, pages 689-733. T. Floquet, J.P. Barbot. State and unknown input estimation for linear discrete-time systems. Automatica, 42, 2006, pages 1883-1889. J. P. Gauthier, A. K. Kupka. Observability and Observers for Nonlinear Systems. SIAM Journal on Control and Optimization, 32, 1994, pages 975-994. A. Gelb. Applied Optimal Estimation. MIT Press, Cambridge, 1974. S. Gillijns, B. De Moor. Unbiased minimum-variance input and state estimation for linear discrete-time systems. Automatica, 43, 2007, pages 111-116. Jiang Guo-Ping, Wang Suo-Ping, and Song Wen-Zhong. Design of observer with integrators for linear systems with unknown input disturbances. Electronics Letters, 36 (13), 2000, pages 1168-1169. Ibarra-Rojas, Sebastin, Moreno, Jaime, Espinosa-Prez, Gerardo. Global observability analysis of sensorless induction motors. Automatica, 40 (2004): 1079 - 1085. Kudva P., N. Viswanadham and A. Ramakrishna. Observers for linear systems with unknown inputs. IEEE Trans. Aut. Control, AC-25 (1), 1980, pages 113-115. J. Moreno. Existence of unknown input observers and feedback passivity for linear systems. Proceedings of the 40th IEEE Conf. on Decision and Control, Orlando, Florida, USA, 2001, pages 3366-3371. J. Moreno. Quasi-unknown input observers for linear systems. Proceedings 2001 IEEE International Conference on Control Applications, Mexico City, Mexico, 2001, pages 732–737. Moreno, J. A., Dochain, D. Global observability and detectability analysis of uncertain reaction systems and observer design. International Journal of Control. Vol. 81, No. 7 July (2008), pages 1062 - 1070. L. Pawlowski, O. Bernard, E. Le Floch, A. Sciandra. Qualitative behaviour of a phytoplankton growth model in a photobioreactor. Proceedings IFAC 15th Triennial World Congress, Barcelona, Spain, 2002.
Application of a Continuous-discrete Unknown Input Observer to Estimation in Phytoplanktonic Cultures ⋆ E. Rocha-C´ozatl ∗ J. A. Moreno ∗∗ A. Vande Wouwer ∗∗∗ Departamento de Ingenier´ıa Mecatr´onica, Universidad Nacional Aut´onoma de M´exico, 4510 Mexico, D.F., Mexico. (e-mail: [email protected]) ∗∗ Instituto de Ingenier´ıa. Universidad Nacional Aut´onoma de M´exico, 4510 Mexico, D.F., Mexico. (e-mail: [email protected]) ∗∗∗ Service d’Automatique, Universit´e de Mons, B-7000 Mons, Belgium. (e-mail: [email protected])
∗
Abstract: The simultaneous estimation of unmeasured state variables and unknown inputs is particularly relevant in environmental applications, such as biological wastewater treatment or the culture of microalgae in open environments. In this study, an extension of the unknown input observer (UIO) recently proposed in (7) to nonlinear continuous-time models and discrete-time measurements is presented. This UIO takes account of the process and measurement noises, and uses a linearization around the estimated trajectory, in a way similar to the extended Kalman filter. This extended continuousdiscrete UIO is applied to state and input reconstruction in continuous cultures of phytoplankton. In particular, the problem of simultaneously reconstructing the extra- and intra-cellular nitrogen concentrations together with the light intensity or the dilution rate from measurements of biomass in cultures of the marine microalgae Dunaliella tertiolecta or the cryptophyceae Rhodomonas salina are considered. Simulation results and processing of real data demonstrate the performance of the method. Keywords: State estimation; Input estimation; Robust estimation; Biotechnology. 1. INTRODUCTION State estimation in bioprocesses is particularly important for monitoring and control purposes, as hardware sensors are expensive (high acquisition and maintenance costs), are not always available for the specific measurement under consideration (for example, the internal nitrogen quota is a key variable for microalgae, especially in the case of biofuel production where a nitrogen limitation is a condition of enhanced lipid storage; this variable is however difficult to measure on-line), and have stringent operating conditions (sterilization, calibration, processing time, sample destruction, etc). In this study, attention is focused on the design of unknown input observers for monitoring cultures of microalgae in the chemostat. Unknown Input Observers (UIO) are dynamical systems that estimate the state variables of a system robustly with respect to the disturbances or unknown inputs that affect the system. Their study for linear systems has drawn continuing interest over the last three decades (10; 2; 8; 11; 4). ⋆ The authors gratefully acknowledge the support of FNRS and CONACYT in the framework of a bilateral research agreement, as well as the financial support from PAPIIT, UNAM, grant IN111012, and Fondo de Colaboraci´on del II-FI, UNAM, IISGBAS-165-2011. They also express their gratitude to Laboratoire d’Oc´eanographique in Villefranche-sur-Mer (France) for providing the experimental data. This paper presents research results of the Belgian Network DYSCO (Dynamical Systems, Control, and Optimization), funded by the Interuniversity Attraction Poles Programme, initiated by the Belgian State, Science Policy Office. The scientific responsibility rests with its authors.
978-3-902823-05-2/12/$20.00 © 2012 IFAC
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Recently, a recursive filter, optimal in a minimum-variance unbiased sense (i.e., in the spirit of Kalman filtering), has been proposed to cope with unknown inputs and process/measurement noises in (7). In this study, we propose an extension of this algorithm to address the particularly important case in bioprocess applications of a nonlinear continuous-time model associated with discrete-time (and often rare) measurements. This algorithm is applied to two case-studies (simulation and real data) related to the culture of phytoplankton in the chemostat, where it is of interest to on-line estimate component concentrations, as well as influencing inputs such as the light intensity(which varies according to day-night cycle and weather conditions) or the dilution rate (which can vary depending on the occurrence of rains or storms in uncontrolled environments). 2. PRELIMINARIES 2.1 Kalman Filter: state estimation Consider a discrete-time linear model with noise in both state and output equations x[k+1] = Ax[k] + Bu[k] + w[k]
(1a)
y[k] = Cx[k] + v[k] (1b) where x ∈ Rn is the state vector, u ∈ Rp is the input vector, y ∈ Rm is the output (measurement) vector. We assume that w[k] and v[k] are stationary zero-mean white noise processes with covariance matrices Q and R. In addition, we assume that x0 , w[k] and v[k] are uncorrelated. 10.3182/20120710-4-SG-2026.00028
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• Measurement update (MU)
Kalman Filter (KF) equations are (6): • Time Update (TU)
ˆ¯[k+1|k+1] = x ˆ¯[k+1|k] + B u x ˆ[k|k+1] T ¯ [k+1] = P¯[k+1|k] C R ¯ −1 L
x ˆ[k+1|k] = Aˆ x[k|k] + Bu[k]
[k+1]
P[k+1|k] = AP[k|k] AT + Q
P¯[k+1|k+1] = P¯[k+1|k] + BPu[k|k+1] B T − ¯T ¯ [k+1] F Pu[k|k+1] B T −BPu[k|k+1] F T L −L
• Measurement Update (MU)
[k+1]
x ˆ[k+1|k+1] = x ˆ[k+1|k] + L[k+1] y[k+1] − C x ˆ[k+1|k] L[k+1] = P[k+1|k] C T R
ˆ¯[k+1|k+1] + x ˆ[k+1|k+1] = x ¯ +L[k+1] y[k+1] − C x ¯ˆ[k+1|k+1] P[k+1|k+1] = P¯[k+1|k+1] −
P[k+1|k+1] = P[k+1|k] − P[k+1|k] C T RCP[k+1|k] −1 where R = CP[k+1|k] C T + R
T ¯ ¯ [k+1] − F Pu[k|k+1] F T L ¯ [k+1] R −L [k+1]
where F = CB.
A block diagram is shown in Figure 1.
Since the input is no longer available, an estimation mechanism has been introduced between TU and MU. A block diagram is presented in Figure 2.
Fig. 1. Block diagram of Kalman Filter. If the state model is continuous in time and nonlinear, i.e.,
Fig. 2. Block diagram of unknown input observer.
x˙ = f (x, u, w)
(2a)
y[k] = Cx[k] + v[k]
(2b)
the Time Update equations are now x ˆ˙ = f (ˆ x, u) P˙ = A (t) P + P AT (t) + Q where A (t) is the Jacobian of f (x, u) with respect to x along x (t) , u). In this case the estimated trajectory, i.e., A (t) = ∂f ∂x (ˆ the estimator is called an Extended Kalman Filter (EKF) .
When it is not possible to measure the input u, the conventional KF (or EKF) must be modified in order to estimate the states as well as the input. In (7), the following algorithm is proposed for model (1a)-(1b). • Time Update (TU) x ¯ˆ[k+1|k] = Aˆ x[k|k] P¯[k+1|k] = AP[k|k] AT + Q
∂f ∂u
(3) x (t) , u[k|k] ,
δx[k+1] = A[k] δx[k] + B[k] δu[k]
(4)
where A (t) = and which reads
∂f ∂x
δ x˙ = A (t) δx + B (t) δu, x (t) , u[k|k] and B (t) =
The proposed extended continuous-discrete UIO is then: • Time Update
• Estimation of the Unknown Input (Est UI)
¯ [k+1] y[k+1] − C x ˆ u ˆ[k|k+1] = M ¯[k+1|k] −1 ¯ −1 F Pu[k|k+1] = F T R [k+1]
We consider the nonlinear continuous-time model with discretetime measurements (2a)-(2b) with a nonmeasurable input u. The proposed extension of the previous algorithm (7) is similar to a standard Time Update of a continuous-discrete EKF, i.e., the prediction is made using the original nonlinear model, starting from the last corrected state. Since u is unknown on the prediction interval, the last estimated value of this input is used instead. The propagation of the covariance matrix P , is the same as in the standard EKF. The estimation of the input u is updated at the measurement times, based on a discretized version of the linearized model
2.2 State and Input Estimation
¯ [k+1] = C P¯[k+1|k] C T + R R −1 ¯ −1 ¯ [k+1] = F T R ¯ −1 F FTR M [k+1] [k+1]
3. AN EXTENDED CONTINUOUS-DISCRETE UIO
x ˆ˙ = f x ˆ, u[k|k] P˙ = A (t) P + P AT (t) + Q with A (t) = ∂f ˆ (t) , u[k|k] . ∂x x
x ˆ[k+1|k] = x ˆ ((k + 1) T ) P[k+1|k] = P ((k + 1) T )
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corresponds to the distinguishability of two different unmeasured state/unknown input trajectories (D1 (t), Q1 (t), S1 (t)) and (D2 (t), Q2 (t), S2 (t)) from the corresponding measured variables (Sin1 (t), X1 (t)) and (Sin2 (t), X2 (t)), i.e., if Sin1 (t) = Sin2 (t), X1 (t) = X2 (t) for an interval of time t ∈ [0, T ] the unmeasured variables are identical D1 (t) = D2 (t), Q1 (t) = Q2 (t), S1 (t) = S2 (t) during the same time interval. Detectability corresponds to the validity of this property in an asymptotic manner, i.e. limt→∞ (D1 (t) − D2 (t)) = 0, limt→∞ (Q1 (t) − Q2 (t)) = 0 and limt→∞ (S1 (t) − S2 (t)) = 0 (13). Checking the observability property when all inputs are known can be done using the Observability map (or its Jacobian for the local property). However, observability with unknown inputs or detectability is a property much harder to evaluate.
• Estimation of the Unknown Input ¯ [k+1] = C P¯[k+1|k] C T + R R −1 T ¯ −1 ¯ [k+1] = F T R ¯ −1 F[k] M F[k] R[k+1] [k] [k+1] ¯ [k+1] y[k+1] − C x ˆ δu ˆ[k|k+1] = M ¯[k+1|k] −1 T ¯ −1 Pu[k|k+1] = F[k] R[k+1] F[k]
• Measurement update
ˆ¯[k+1|k] + B[k] δ u ˆ¯[k+1|k+1] = x ˆ[k|k+1] x T ¯ −1 ¯ ¯ L[k+1] = P[k+1|k] C R [k+1]
T P¯[k+1|k+1] = P¯[k+1|k] + B[k] Pu[k|k+1] B[k] − ¯T ¯ [k+1] F[k] Pu[k|k+1] B T −B[k] Pu[k|k+1] F T L −L [k]
[k+1]
[k]
¯ [k+1] y[k+1] − C x ˆ¯[k+1|k+1] + L ˆ x ˆ[k+1|k+1] = x ¯[k+1|k+1] ¯ P[k+1|k+1] = P[k+1|k+1] − ¯ [k+1] R ¯ [k+1] − F[k] Pu[k|k+1] F T L ¯T −L [k] [k+1] u ˆ[k+1|k+1] = u ˆ[k|k] + δ u ˆ[k|k+1]
(5)
Here we will use a method proposed in (13) to check state observability for a class of bioreactors when unknown inputs are present. The same method has been used in (9) to study state observability/detectability for an induction machine, when all inputs are known. The idea underlying the analysis is to consider two identical systems, one is the plant Σp (6), and a copy of it (with states x1 , x2 , x3 , unknown input d and known input u) given by
where F[k] = CB[k] . Σx
4. APPLICATION TO PHYTOPLANKTON CULTURES 4.1 Phytoplankton Culture: Droop Model Droop model (3) is a simple and widely used model of phytoplankton culture. The mass balance equations are given by X˙ (t) = −D (t) X (t) + µ (Q) X (t) (6) Σp Q˙ (t) = ρ (S) − µ (Q) Q (t) ˙ S (t) = D (t) [Sin (t) − S (t)] − ρ (S) X (t) where X is the biovolume (i.e., the volume of cells in a unit volume of culture medium), Q is the internal quota, which is defined as the quantity of nitrogen per unit of biovolume, D (t) represents the dilution rate, S is the substrate (inorganic nitrogen) concentration and Sin is the input substrate concenS(t) tration. Function ρ (S) = ρm S(t)+k is the uptake rate, and S kQ µ (Q) = µ ¯ 1 − Q(t) is the growth rate; constants kS and ρm represent a half-saturation constant for the substrate and the maximum uptake rate, respectively; constant µ ¯ is the theoretical maximum growth rate, obtained for an infinite internal quota and kQ is the minimum internal quota allowing growth. The considered biological process is the culture of the chlorophyceae Dunaliella tertiolecta in the chemostat (a continuous bioreactor operated with constant volume, which can mimic an open natural system). A more complete description of the experimental setup can be found in (1). Observability/Detectability Our objective is to construct an observer to estimate the unmeasured state variables Q, S and the unknown input D, from the knowledge of the dynamical model (6) and the measurements of the known input Sin and the output variable X. This is certainly only possible instantaneously if (state and unknown input) observability is fulfilled, or at least asymptotically if the system is (state and unknown input) detectable. (State/Unknown input) Observability 581
(
x˙ 1 (t) = −d (t) x1 (t) + µ (x2 ) x1 (t) x˙ 2 (t) = ρ (x3 ) − µ (x2 ) x2 (t) . x˙ 3 (t) = d (t) [u (t) − x3 (t)] − ρ (x3 ) x1 (t)
(7)
If we assume that the measured variables of systems Σp and Σx are identical for a time interval, i.e. Sin (t) = u(t), X(t) = x1 (t), (state and unknown input) observability is equivalent to the fact that the only possible solutions of equations (6) and (7) under these restrictions is given by Q(t) = x2 (t), S(t) = x3 (t) and D(t) = d(t), during the same time interval, whereas detectability is equivalent to satisfying these equalities asymptotically. If we introduce the state and unknown input error variables ǫ1 = X − x1 , ǫ2 = Q − x2 , ǫ3 = S − x3 , and ǫd = D − d, respectively, equation (7) can be replaced by KQ ǫ˙1 = −ǫd X − dǫ1 + µ ¯ 1− X Q KQ −¯ µ 1− (X − ǫ1 ) Q − ǫ2 S S − ǫ3 ǫ˙2 = ρm − −µ ¯ ǫ2 S + KS S − ǫ 3 + KS ǫ˙3 = ǫd [Sin − S] − dǫ3 S S − ǫ3 −ρm X− (X − ǫ1 ) S + KS S − ǫ 3 + KS
(8)
and observability requires that under the restrictions Sin (t) = u(t) and ǫ1 (t) = 0 the only solutions of system (6) and (8) are ǫ2 (t) = 0, ǫ3 (t) = 0 and ǫd (t) = 0, or asymptotic convergence in case of detectability. This Differential-Algebraic (DA) system, consisting of the differential equations (6) and (8) and the algebraic restrictions Sin (t) = u(t) and ǫ1 (t) = 0, will be next reduced to a simpler differential equation system. From ǫ1 (t) = 0 (during a time interval) it follows that also ǫ˙1 (t) = 0 and, from the first equation of (8), that ǫd = −¯ µ KQ
ǫ2 . Q (Q − ǫ2 )
Equations (8) therefore reduce to
(9)
8th IFAC Symposium on Advanced Control of Chemical Processes Singapore, July 10-13, 2012
ρm KS ǫ3 (S + KS ) (S − ǫ3 + KS ) −¯ µKQ [Sin − S] ǫ2 ǫ˙3 = Q (Q − ǫ2 ) X − d + ρm KS ǫ3 . (S + KS ) (S − ǫ3 + KS )
ǫ˙2 = −¯ µǫ2 +
Numerical Simulations Nonlinear model (6) is considered with the numerical values of the parameters reported in (1). Two measurement configurations are studied: (1) variable X is measured and variables Q and S are estimated; (2) variables X and S are measured and Q is estimated. In both cases D(t) is an unknown input (it would represent the flow in and out of the natural system) and it is estimated simultaneously with the state. In order to test the estimation scheme, the actual evolution of D(t) is defined as a sinusoid in some periods of time and as a constant in some others. Case 1. State X (biovolume) is measured once a day (T = 1) with a relative measurement error of 3%. True initial conditions are defined as X(0) = 5, Q(0) = 0.5, ˆ S(0) = 1, and for the estimator are X(0) = 5 (since it is ˆ ˆ measured), Q(0) = 0.75, S(0) = 1.5, i.e., 50% initial error in the nonmeasured states. The design parameters of the estimator are P0 = I3 , where I3 represents an identity matrix of dimension 3 × 3, Q = 1 × 10−4 I3 , and since the considered error in measurements is 2 relative, matrix R is defined as R = (0.03 ∗ X(k)) , where X(k) is the k−th measurement of biovolume X. In Fig. 3 state estimates are shown as solid lines and are compared with the real values of each variable (star markers ⋆). Only the measurements of X are used by the filter, the other values are used for validation purposes.
Substrate S [ µ mol / l]
0 0 60
10
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20
30
50
60
estimated actual UB and LB of confid. intervals
40 20 0 0
40
10
20
30
40
50
60
10
20
30 Time [day]
40
50
60
5 0 0
Fig. 3. Evolution of the real and estimated states - UIO using the measurements of X. 1.8 1.6 1.4 Dilution rate D [1/day]
It can be easily proved that ǫ2 → 0 and ǫ3 → 0, provided that the state variables are positive and the substrate concentration is lower than the inlet concentration (which is meaningful from a biological point of view). Moreover, from (9) it also follows that ǫd → 0. We conclude that the model (6) is state and unknown input detectable, and so the measurement of X and Sin provide sufficient information to estimate asymptotically the two unmeasured states Q, S and the unknown input D.
20
Internal Quota Q [ µ mol / mm3 ]
Biovolume X [mm3 / l]
40
1.2 1 0.8 0.6 estimated actual UB and LB of confidence intervals
0.4 0.2 0
10
20
30 Time [day]
40
50
60
Fig. 4. Evolution of the dilution rate - UIO using the measurements of X. The design parametersof the estimator are P0 = I3 , Q = 1 × 2 2 10−4 I3 , and R = diag (0.03 ∗ X(k)) , (0.03 ∗ S(k)) .
Fig. 5 shows the evolution of the real and estimated state variables. The evolution of the dilution rate is presented in Fig. 6. Clearly, the availability of a second measurement improves the quality of the estimation.
States X (biovolume) and S (substrate) are measured once a day (T = 1) with a relative measurement error of 3%.
Experimental Results In order to complete the test of the proposed algorithm, real-life experimental results are considered. The experimental data were collected from a pilot plant at Laboratoire d’Oc´eanographie in Villefranche-sur-Mer (France). The dilution rate was not constant but slowly varying and the sampling period is T = 0.65(day), i.e., 1 or 2 measurements per day. In the following figures (Fig. 7 and Fig. 8), the solid lines represent again the estimated values of S, X, Q and D, and the star points are the measured values (only the measured values of X are used in the filter, the other values are used for validating the estimated values only).
True initial conditions are defined as X(0) = 5, Q(0) = 0.5, ˆ ˆ S(0) = 1, and for the estimator as X(0) = 5, S(0) = 1 ˆ (measured states) and Q(0) = 0.75 (50% initial error).
ˆ Initial conditions for the estimator are defined as X(0) = ˆ ˆ 0.157, S(0) = 55.75 and Q(0) = 5.03. The design parameters of the estimator are the same as those for Case 1 in the previous
On the other hand, the estimation of the dilution rate is presented in Fig. 4 The filter convergence is fast and the performance is quite satisfactory. 95 % confidence intervals are also drawn using the information provided by the matrices P and Pu showing the reliability of the estimates. Case 2.
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Internal Quota Q [ µ mol / mm3 ]
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Dilution rate D [1/day]
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Fig. 5. Evolution of the real and estimated states - UIO using the measurements of X and S.
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Fig. 8. Evolution of the real and estimated dilution rate - UIO using the measurements of X - Experimental data. 4.2 A more detailed model with light influence
1.8
In (14), a more detailed model, taking the influence of the incident light intensity is developed, as
1.6
Dilution rate D [1/day]
1.4
C˙ (t) = −D (t) C (t) − λC (t) + a (I) L (t) S (t) N˙ (t) = −D (t) N (t) + ρm C (t) S (t) + KS L (t) + βL (t) −γ (I) N (t) C (t) L (t) − βL (t) L˙ (t) = −D (t) L (t) + γ (I) N (t) C (t) S (t) S˙ (t) = D (t) (Sin − S (t)) − ρm C (t) S (t) + KS
1.2 1 0.8 0.6 estimated actual UB and LB of confidence intervals
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Fig. 6. Evolution of the real and estimated dilution rate - UIO using the measurements of X and S.
Internal Quota Q [ µ mol/mm3]
Internal substrate S [ µ mol/l]
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section. The results are satisfactory since the estimation is close to the measured values of the variables.
40
Variable C represents the particulate carbon concentration (carbon biomass), N is the internal nitrogen concentration and L is the chlorophyllian nitrogen concentration. The total particulate nitrogen can be computed as N + L. S is the substrate (inorganic nitrogen) concentration and Sin is the input substrate concentration. β is the coefficient of chlorophyll degradation, λ is the factor of respiration, and functions γ and a describe the influence of the light intensity I in the process
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a (I) =
αI (t) KI + I (t)
Observability/Detectability Considering C and L as the measured states, N and S as the non measured states, D and Sin as the known inputs, and I as the unknown input, an observability analysis leads to the logical conditions that the system is state and unknown input detectable provided that N > 0, S > 0, I > 0.
10 5 0 0
KC αKL I (t) , KI + I (t) KC + I (t)
Fig. 7. Evolution of the real and estimated states - UIO using the measurements of X - Experimental data.
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Numerical Simulations As a particular application, the culture of the cryptophyceae Rhodomonas salina is considered, that is, model (10) with the parameter values reported in (14). The state estimation objective is to reconstruct N and S, using the culture model together with on-line measurements of C and L. The dilution rate D and input concentration Sin are known,
8th IFAC Symposium on Advanced Control of Chemical Processes Singapore, July 10-13, 2012
whereas the light intensity I is the unknown input, which has to be simultaneously estimated. States C (particulate carbon) and L (chlorophyllian nitrogen) 1 are measured one time per hour (T = 24 ) with a realtive error of 1%. In this case, dilution rate D is considered to be constant and defined as D = 0.4[1/day]. True initial conditions are defined as C(0) = 3.6, L(0) = 0.03, ˆ N (0) = 0.06, S(0) = 40, and for the estimator are C(0) = 3.6, ˆ ˆ L(0) = 0.03 (since these two are measured) N (0) = 0.09, S(0) = 34, that is, 50% initial error in N and 15% in S, the nonmeasured states.
Substrate S [µ mol N/l]
Internal N Cholophyllian N Part. Carbon N [µ mol N/l] L [µ mol N/l] C [µ mol C/l]
The design parameters of the = estimator are P0 = I4 , Q 2 2 1 × 10−4 I4 , and R = diag (0.01 ∗ C(k)) , (0.01 ∗ L(k)) , where C(k), L(k) are the k−th measurements of C and L, respectively.
of unknown reaction kinetics. In this study, an extension to the minimum-variance unbiased filter developed in (7) is proposed for the case of nonlinear continuous-time systems with discretetime measurements. State and unknown input detectability conditions are also discussed. The numerical results show the generally good performance of the filter in situations where no a priori information on the unknown input is available. In situations where such information is available, it is probably interesting to incorporate it in the form of an exosystem as suggested in (12), which is the subject of future investigation. REFERENCES [1]
[2]
15 10
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[5]
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UB and LB
4
6 of Conf. Int
5
[6] [7]
6
Fig. 9. Evolution of the real and estimated states - UIO using the measurements of C and L.
[8]
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Light Intensity I [µ E/m2/d]
8
x 10
[9]
6 4
[10]
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6
[11]
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Light Intensity I [µ E/m2/d]
8
x 10
estimated actual UB and LB of confid. interval
6 4
[12]
2 0 3
3.5
4 Time [day]
4.5
5
[13]
Fig. 10. Evolution of the real and estimated light intensity - UIO using the measurements of C and L. [14]
5. CONCLUSIONS UIO observers are increasingly applied to estimation in biological systems, one prominent example being the asymptotic observer which has been used extensively to tackle the problem 584
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