Estimation of Unmeasured States in a Bioreactor under Unknown

Subscriber access provided by MIDWESTERN UNIVERSITY

Process Systems Engineering

Estimation of Unmeasured States in a Bioreactor under Unknown Disturbances Xinghua Pan, Jonathan P. Raftery, Chiranjivi Botre, Melanie R. DeSessa, Tejasvi Jaladi, and Muhammad Karim Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b02235 • Publication Date (Web): 17 Jan 2019 Downloaded from http://pubs.acs.org on January 18, 2019

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Industrial & Engineering Chemistry Research

Estimation of Unmeasured States in a Bioreactor under Unknown Disturbances Xinghua Pan, Jonathan P. Raftery, Chiranjivi Botre, Melanie R. DeSessa, Tejasvi Jaladi, M. Nazmul Karim* Artie McFerrin Department of Chemical Engineering, Texas A&M University, 3122 TAMU, College Station, Texas 77843, United States *Corresponding Author: [email protected]; Phone: +1 979 845 9806; Fax: +1 979 845 3266

1 ACS Paragon Plus Environment

Industrial & Engineering Chemistry Research 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 2 of 36

Abstract The measurement or estimation of bioprocess states is critical for process control and optimization applications. However, certain disturbances or unknown inputs can generate significant model-plant mismatch if not considered by the process models. To better estimate the process states in the presence of these disturbances or unknown inputs, a nonlinear unknown input observer is applied. Experimental studies of batch and fed-batch operations of a bioreactor were performed using a recombinant Saccharomyces cerevisiae to produce β-carotene. Previously developed kinetic models produce model-plant mismatch with changes to the initial conditions or operating mode of the bioreactor. The observer is applied to the bioreactor system to estimate the batch and fed-batch state variables. State estimates from the designed observer are compared to model predictions and experimental measurements. Results show improved state estimation over first-principles model predictions when applying the unknown input observer to the nonlinear dynamic process with unknown disturbances. Keywords: bioreactor, nonlinear process, state estimation, uncertain disturbances, unknown input observer


Page 3 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Introduction Knowledge of process states is key for process monitoring, real-time optimization, and advanced process control applications.1,2 If these process states cannot be directly measured in real-time, state estimation techniques can be applied to predict the state information. Model-based state estimation is a widely applied approach but requires a high-fidelity model which can describe the chemical process precisely. However, extensive knowledge and effort are required to build a reliable model to describe complex processes, including those with biological applications. Disturbances and unknown inputs which are not captured by the model will lead to significant model-plant mismatch, resulting in inaccurate state predictions that are unfit for use in optimization and control applications. Biological systems have inherent uncertainties that are often not well understood, including the stochastic nature which often serves as an obstacle for process modeling and optimization. Inherent variation of the dynamics in biological systems has been briefly summarized by Toni and Tidor.3 Intrinsic and extrinsic sources of variations include the probabilistic nature of the timing of collision events among biomolecules and the effects of varying components upstream of the system of interest. Brown and Sethna explained the ‘sloppy’ nature of the biological models with poorly known parameters, simplified dynamics, and uncertain connectivity.4,5 To identify the source of the variations, Bowsher and Swain analyzed biochemical networks to identify the effects of various components. A method was proposed to predict the magnitude of the components from the models.6 In the case of bioprocessing, several situations that generate model-plant mismatch can be treated as unknown inputs or disturbances, such as the effect of nutrient limitation, oxygen delivery at high cell density, and carbon dioxide stripping.7–9



Page 4 of 36

Different solutions have been proposed to address the model-plant mismatch for process monitoring. One solution is to apply parameter estimation for each set of experiments.10,11 However, parameter estimation requires extensive measurement of the states and any resulting first-principles models cannot be reliably extrapolated to future experiments outside of the conditions used for initial data collection without re-estimation of the parameters. Disturbance and fault observers offer alternative approaches for state estimation. Researchers have developed different disturbance and fault detection observers for the estimation of disturbances and states. Wei et al. developed a disturbance-observer-based disturbance attenuation control method for a class of stochastic systems with multiple disturbances.12 Chen et al. applied a disturbance observerbased control system for a process with time-varying parameters and time delays.13 Stobart et al. developed a robust uncertainty and disturbance estimator-based control strategy for uncertain linear time-invariant systems with state delays.14 Zhong and Rees proposed an effective uncertainty and disturbance estimator for linear systems with uncertainties and disturbances.15 Li et al. proposed an extended state observer-based control method for non-integral-chain systems with mismatched uncertainties.16 Applications of disturbance and fault detection observers for bioreactors have been demonstrated using simulation or experimental validation. Rocha-Cózatl and Vouwer applied a linear quasi-unknown input observer to estimate concentration, flow rates, and light intensity in phytoplankton cultures. The authors linearized the nonlinear process model of the chemostat to apply the linear quasi-unknown input observer.17 Lemesle and Gouzé developed a bounded error observer for partially known bioreactor models. The hybrid bounded observer incorporated a high gain asymptotic observer to improve the error convergence rate, which is dependent on knowledge of the process model. A simulation study of a bioreactor model was provided.18 Moisan et al.


Page 5 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


extended the bounded error observers to further improve the convergence properties.19 Farze et al. proposed an adaptive high gain observer for state and parameter estimation for a class of uniformly observable nonlinear systems with nonlinear parametrization and sampled outputs.20 Ghanmi et al. extended Farze’s work on an adaptive observer for state and parameter estimation of a nonlinear system based on a high gain adaptive observer.21 Kravaris et al. proposed a systematic framework for designing a nonlinear observer to estimate process state variables with unknown process or sensor disturbances.22 A simulation study was applied to a bioreactor model in these last two cases. An unknown input observer was developed to estimate the states when the process is operating with certain types of faults and disturbance.23–25 Compared to other observers, unknown input observers can eliminate the effect of certain disturbances or faults despite their size. The design of a linear unknown input observer with both full and reduced order has been provided, and the sufficient and necessary conditions for such an observer have also been discussed.25–29 However, for most chemical processes, inherent nonlinearity hinders the application of such linear observers. Ding and Frank demonstrated design of nonlinear observer using extended linearization of the observer error.30 Chen and Saif designed nonlinear unknown input observer for Lipschitz nonlinear system using linear matrix inequality and H∞ solving method.31 Pertew et al. applied the concept of dynamic observer to design unknown input observer for nonlinear system.32 The method required problem regularization to apply the proposed framework. Chakrabarty et al. designed nonlinear unknown input observer for system with bounded exogenous inputs, which requires prior estimation of the inputs.33

Design of a class of unknown input observers for a

Lipschitz system has been demonstrated by Chen and Saif through a linear matrix inequality method.31 Application of nonlinear observer such as extended Kalman filter has been evaluated by



Page 6 of 36

experimental bioreactor.34 The application of nonlinear observer for bioreactors with unknown inputs has not been demonstrated experimentally. In this paper, we modify the work of Chen and Saif and develop a nonlinear unknown input observer using a more general nonlinear format, which opens opportunities for more general applications.31 This work extends the work done by Chen and Saif, who presented an unknown input observer for nonlinear systems with linear state and linear input terms.31 In the current work, elimination of linear state and linear input terms opens opportunities for many other nonlinear systems with nonlinear input terms. Estimation of unmeasured states in biological systems experimentally is a challenge when disturbances are present in these systems. To demonstrate the applicability of the proposed observer for biological applications, validation of the unknown input observer is demonstrated using an experimental bioreactor case study of a recombinant strain of Saccharomyces cerevisiae. Online measurements of the biomass concentration are used to facilitate the prediction of glucose utilization and product formation using the designed observer. Sensors are currently available for the online quantification of biomass, and it is assumed that these measurements can be used alongside the implementation of the observer developed here.

Design of a nonlinear unknown input observer In this work, an unknown input observer is designed for general nonlinear systems defined as 𝑓(𝑥,𝑦) in the presence of linear unknown inputs. To design this nonlinear unknown input observer extension, a nonlinear system with unknown inputs is written in the format of Equation (1), in which 𝑥 ∈ 𝑅𝑛, 𝑦 ∈ 𝑅𝑘, and 𝑢 ∈ 𝑅𝑝 are the state vector, output vector, and known input vector, respectively. The vector 𝑑 ∈ 𝑅𝑑 is the timevariant unknown input in the system and is linearly coupled with the nonlinear system 𝑓(𝑥,𝑢) using a constant parameter matrix E that is assumed to have full column rank. The matrix 𝐶 linearly 6 ACS Paragon Plus Environment

Page 7 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


relating the process states to the process outputs is constant. Current work assumes no process noise and measurement noise. 𝑥 = 𝑓(𝑥,𝑢) + 𝐸𝑑 𝑦 = 𝐶𝑥

(1)

An unknown input observer is designed in the following form to estimate the process states under a time-varying unknown input: 𝑧 = (𝐼 ― 𝐻𝐶)𝑓(𝑧 + 𝐻𝑦,𝑢) ― 𝐾(𝑦 ― 𝑦) 𝑥 = 𝑧 + 𝐻𝑦

(2)

The vector 𝑥 is an estimation of the state vector 𝑥. To design an unknown input observer, estimation error from the observer needs to be asymptotically stable. The parameters 𝐻 and 𝐾 are the tuning parameters for the observer, and must be calculated to ensure stability. The proof of existence and stability is similar to the work of Chen and Saif with certain modifications to account for the nonlinearity of 𝑓(𝑥,𝑢).31 The full proof is shown in Appendix A and uses a proposed Lyapunov function of 𝑉 = 𝑒𝑇𝑃𝑒 and the Shur complement method to ensure the resulting linear matrix inequality is strictly negative.37 The calculation of H and K are also discussed in Appendix A. The parameter matrix H in Equation (2) was determined similarly to a previous publication by Pan et al.38 To solve for the parameter matrix H, the following matrix equation in Equation (3) is defined using the third condition of the Existence Condition, or Equation (A1.3). The matrix H

[

]

𝐼 𝐸 can then be determined by using Equation (4) below, in which the matrix 𝐶 0

[

+

is a generalized

]

𝐼 𝐸 inverse of 𝐶 0 and 𝜃 is a parameter matrix with appropriate dimension, which needs to be determined. The matrix inequality can be solved using the linear matrix inequality toolbox in MATLAB®. 7 ACS Paragon Plus Environment


[

Page 8 of 36

]

𝐼 𝐸 [𝐼 ― 𝐻𝐶 𝐻] 𝐶 0 = [𝐼 0]

[

]

𝐼 𝐸 𝐻 = [𝐼 0] 𝐶 0

+

(

[

][

(3)

]

𝐼 𝐸 𝐼 𝐸 [0 𝐼] + 𝜃 𝐼 ― 𝐶 0 𝐶 0

+

)[ 0

𝐼]

(4)

Experimental section A bioreactor example is used to demonstrate the application of the nonlinear unknown input observer defined in the previous section. The cultivation of Saccharomyces cerevisiae strain mutant SM14 was performed in both batch and fed-batch modes to produce the intracellular product β-carotene. The details of the bioprocessing system were described in our previous studies.39,40 Operation and modeling of a bioreactor The S. cerevisiae strain mutant SM14 engineered to produced β-carotene was used in this study. The yeast strain was stored in frozen vials at -80 ̊C, and in plates at 4 ̊C, which were subcultured every three weeks for maintenance. The cells were grown in fresh Yeast Nitrogen Base (YNB) media in all the experiments with supplemented D-glucose. The inoculum for the bioreactor and shake-flask cultures in the following experiments were prepared from single colonies to inoculate 50×10-3 L of YNB media (20 g/L glucose) and incubated at 30 ̊C for 72 hours with constant agitation at 200 rpm. The bioreactor studies were carried out in a glass, autoclavable bioreactor with a 3 L working volume (Applikon®, Foster City, CA). The bioreactor was inoculated with the entire seed culture. The temperature, pH, agitation speed, and airflow were set at 30 ̊C, 4, 800 rpm, and 0.1 L/s, respectively.


Page 9 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Kinetic modeling and parameter estimation studies of the bioreactor were performed in our previous work.39,40 The kinetic model begins by describing the cell growth, which is shown in Equation (5), where X is the biomass concentration and 𝜇 is the overall specific growth rate. The overall specific growth rate is defined as the summation of the specific growth rates on all three carbon sources 𝜇𝐺, 𝜇𝐸𝑡, and 𝜇𝐴 and is given in Equation (6a). In the individual growth rate equations, defined in Equations (6b), (6c), and (6d), the variable 𝑎𝑖𝑗 represents the inhibition effect of the 𝑗th substrate on the utilization of the 𝑖th substrate by the microorganism. The glucose, ethanol and acetic acid concentrations are given by 𝐺, 𝐸𝑡, and 𝐴, respectively. The parameters 𝜇𝑚𝑎𝑥,𝐺, 𝜇𝑚𝑎𝑥,𝐸𝑡, and 𝜇𝑚𝑎𝑥,𝐴 are the maximum specific growth rates on glucose, ethanol and acetic acid, respectively. The variables 𝜒𝐸𝑡 and 𝜒𝐴 denote functions describing the inhibition effects of the ethanol and acetic acid compounds, respectively, and their specific functional forms can be found in our previous work.39,40 𝑑𝑋 = (𝜇𝐺 + 𝜇𝐸𝑡 + 𝜇𝐴) 𝑋 𝑑𝑡

(5)

𝜇 = 𝜇𝐺 + 𝜇𝐸𝑡 + 𝜇𝐴

(6a)

𝜇𝐺 =

𝜇𝐸𝑡 =

𝜇𝐴 =

( ( (

𝜇𝑚𝑎𝑥,𝐺 ⋅ 𝜒𝐸𝑡 ⋅ 𝜒𝐴 ⋅ 𝐺

) ) )

𝐾𝑆𝐺 + 𝐺 + 𝑎𝑔𝑒𝐸𝑡 + 𝑎𝑔𝑎𝐴 𝜇𝑚𝑎𝑥,𝐸𝑡𝐸𝑡

𝐾𝑆𝐸 + 𝐸𝑡 + 𝑎𝑒𝑔𝐺 + 𝑎𝑒𝑎 𝐴 𝜇𝑚𝑎𝑥,𝐴𝐴

𝐾𝑆𝐴 + 𝐴 + 𝑎𝑎𝑔𝐺 + 𝑎𝑎𝑒𝐸𝑡

(6b)

(6c)

(6d)

The glucose (G), ethanol (Et), acetic acid (𝐴), and β-carotene (P) concentrations are modeled using Equations (7) through (10) below. Here, the variable 𝑌𝑋 𝐺 is the biomass yield coefficient on glucose, 𝑌𝑋

𝐸𝑡

is the biomass yield coefficient on ethanol, and 𝑌𝑋 𝐴 is the biomass 9 ACS Paragon Plus Environment


Page 10 of 36

yield coefficient on acetic acid. The variable 𝛼𝑖 represents the coefficients for growth-associated product formation related to the yeast growth on each substrate, and 𝛽 is the coefficient for nongrowth-associated carotenoid production. Estimation of process parameters is demonstrated in our previous work. The state vector is [G, X, P, Et, A]. The values for the kinetic model parameters in Equations (7) through (10) are given in our previous paper.39 𝜇𝐺𝑋 𝑑𝐺 =― 𝑑𝑡 𝑌𝑋 𝐺 𝑑𝐸𝑡 𝑑𝑡

= 𝑘1𝜇𝐺𝑋 ―

𝜇𝐸𝑡𝑋 𝑌𝑋 𝐸𝑡

(7)

(8)

𝜇 𝐴𝑋 𝑑𝐴 = (𝑘2𝜇𝐺 + 𝑘3𝜇𝐸𝑡)𝑋 ― 𝑑𝑡 𝑌𝑋 𝐴

(9)

𝑑𝑃 = (𝛼1𝜇𝐺 + 𝛼2𝜇𝐸𝑡 + 𝛼3𝜇𝐴)𝑋 + 𝛽𝑋 𝑑𝑡

(10)

Validation experiments Two batch and two fed-batch experiments were performed to demonstrate the application of the unknown input observer. The batch experiments started with different inoculum sizes, with slightly different levels of initial glucose, ethanol, β-carotene, biomass, and acetic acid. Fed-batch experiments were performed by feeding glucose with set concentrations of 200 g/L or 20 g/L of glucose in YNB medium for a defined length of time. The fed-batch model, shown as Equation (11), was derived as an extension of the batch model and uses the same model parameters. In the fed-batch model, 𝐹𝑔 represents the feeding flow rate of glucose and 𝐺𝑖𝑛 is the concentration of glucose in the feed stock solution. Details of the batch and fed-batch validation experiments are listed in Table 1, including initial concentrations, feed flowrates, and feed glucose concentration. The variable V is the liquid reactor volume measured in liters.


Page 11 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


𝜇𝐺𝑋 𝐹𝑔 𝐹𝑔 𝑑𝐺 ― 𝐺 ⋅ + 𝐺𝑖𝑛 ⋅ =― 𝑉 𝑉 𝑑𝑡 𝑌𝑋 𝐺 𝐹𝑔 𝑑𝑋 = (𝜇𝐺 + 𝜇𝐸𝑡 + 𝜇𝐴) 𝑋 ― 𝑋 ⋅ 𝑑𝑡 𝑉 𝐹𝑔 𝑑𝑃 = (𝛼1𝜇𝐺 + 𝛼2𝜇𝐸𝑡 + 𝛼3𝜇𝐴)𝑋 + 𝛽𝑋 ― 𝑃 ⋅ 𝑑𝑡 𝑉 𝑑𝐸𝑡 𝑑𝑡


𝜇𝐸𝑡𝑋 𝑌𝑋 𝐸𝑡

― 𝐸𝑡 ⋅

𝐹𝑔

(11)

𝑉

𝜇 𝐴𝑋 𝐹𝑔 𝑑𝐴 ―𝐴⋅ = (𝑘2𝜇𝐺 + 𝑘3𝜇𝐸𝑡)𝑋 ― 𝑉 𝑑𝑡 𝑌𝑋 𝐴 𝑑𝑉 = 𝐹𝑔 𝑑𝑡

Observability analysis To overcome the computational and scalability limitations encountered by traditional methods to determine the system observability, Liu et al. have proposed the use of an inference diagram to perform an observability analysis of complex systems.41 An inference diagram is constructed based on analyzing the differential equations governing the states of a given process. A link is drawn in the form of an arrow from 𝑥𝑖 to 𝑥𝑗 for all j components on the right hand side of differential equation

𝑑𝑥𝑖

( ). The main advantage of an inference diagram is that it can be applied to 𝑑𝑡

complex processes to identify the minimum number of sensors required for the system to be observable. This method gives a graphical representation of interdependence between the state variables of the system to identify the components that are well connected. An inference diagram can be decomposed into two main parts, strongly connected components (SCC) and sensor nodes. SCCs are well connected states, and all the SCC components within a loop can be observed by



Page 12 of 36

knowing any one of the contained states. In contrast, sensor nodes of the system are the minimum number of the sensors required for the complete set of states to be observable. Using the constructed inference diagrams for the batch and continuous bioreactor processes shown in Figure 1, glucose (G), biomass (X), acetic acid (A) and ethanol (𝐸𝑡) are determined to be SCC nodes and are observable from knowledge of the biomass measurement. From the observability analysis, we can conclude that while P can be used to estimate X, G, A, and 𝐸𝑡, no other measurement can be used to estimate P. The product P represents an intracellular product, for which no direct online measurement is available. Hence, an observable subsystem is selected that omits the product concentration P and only contains X, G, A, and 𝐸𝑡. This subsystem, shown in Equation (12), is observable from only the online measurement of X. This observable system is utilized in both the batch and fed-batch case studies. 𝜇𝐺𝑋 𝐹𝑔 𝐹𝑔 𝑑𝐺 ― 𝐺 ⋅ + 𝐺𝑖𝑛 ⋅ =― 𝑉 𝑉 𝑑𝑡 𝑌𝑋 𝐺 𝐹𝑔 𝑑𝑋 = (𝜇𝐺 + 𝜇𝐸𝑡 + 𝜇𝐴) 𝑋 ― 𝑋 ⋅ 𝑑𝑡 𝑉 𝑑𝐸𝑡 𝑑𝑡


𝜇𝐸𝑡𝑋 𝑌𝑋 𝐸𝑜

― 𝐸𝑡 ⋅

𝐹𝑔 𝑉

(12)

𝜇 𝐴𝑋 𝐹𝑔 𝑑𝐴 ―𝐴⋅ = (𝑘2𝜇𝐺 + 𝑘3𝜇𝐸𝑡)𝑋 ― 𝑉 𝑑𝑡 𝑌𝑋 𝐴 𝑑𝑉 = 𝐹𝑔 𝑑𝑡

Application of the unknown input observer Validation of Existence Condition and estimation of unknown input parameter matrix It is necessary to check whether the Lipschitz condition of the kinetic bioreactor model, defined in general by Equation (A1.4), is satisfied. The feed of glucose, 𝐹𝑔, is assumed to have an


Page 13 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


upper limit, with |𝐹𝑔| < 𝐹𝑟. In the kinetic model of Equation (12), the parameters such as 𝑘𝑖𝑗, 𝑎𝑖𝑗, 𝜇𝐴 , 𝜇𝐸𝑜, and 𝜇𝐺 are all positive. The states are all non-negative. It is easy to verify that, in the operating range, the function

∂f(x) ∂x

is continuous and bounded, and therefore the local Lipschitz condition is

met. In the design of the unknown input observer in Equation (2), and application of unknown input observer for the bioprocess, the unknown input parameter matrix 𝐸 is assumed to be constant. The magnitude of the unknown inputs (d in Equation (1)) for each state may be time-varying during the whole bioprocess. The stability of the observer is guaranteed when the unknown input parameter matrix is kept constant, ensuring minimal error between the predicted process states and actual process states. From the theoretical part of the observer design, the accuracy of state estimation depends on the unknown input parameter matrix E in Equation (1). The E matrix is estimated using a leastsquares curve fitting approach described by our previous work to fit Equation (1), with f(x) defined by Equation (12), to batch experimental data (batch experiment 1).39,40 The best-fit value of the parameter matrix E is used for all validation experiments. Disturbances are assumed to affect two inputs, the biomass and ethanol concentrations, and as such they are the only elements considered in the unknown input matrix E. Estimation of unknown input matrix To design the unknown input observer and solve the parameters for the observer, the first step is to obtain the unknown input matrix E in Equation (1). Equation (1) is a combination of a known nonlinear model and linear unknown inputs. It is still an open question whether all nonlinear systems can be described with such a combination especially for the linear unknown inputs. Some



Page 14 of 36

unknown inputs can be nonlinear. However, it is difficult to configure the exact effect of the unknown inputs due to the possible combination of many different inputs. In this paper, we assume that the overall effect of unknown inputs is linear. A line fitting procedure is used to estimate the parameter matrix of the biomass and ethanol concentrations.40 Experimental data from batch experiment 1 is used for estimating the unknown input matrix 𝐸.

[

The unknown input matrix E is obtained with the values of 0

1

0.35

― 1000

1000

]

𝑇

0 0 . The

process model in the format of Equation (1) is shown in Equation (13). The other parameters for the observer in Equation (2), specifically the matrices 𝐶, 𝐻, and 𝐾, are shown below. 𝑇

[] [

0 1 𝐶= 0 ; 𝐻= 0 0

0 0 0 0 0

0 1.0 ―6.6 ―0.35 0

0 0 0 0 0

0 0 0 0 0

] [

0 0 0 ; 𝐾= 0 0

0 0 0 0 0

0 ―9.8374 68.6454 ―3.1358 0

0 0 0 0 0

0 0 0 0 0

]

0 0 0 0 0

𝜇𝐺𝑋 𝑑𝐺 =― 𝑑𝑡 𝑌𝑋 𝐺 𝑑𝑋 1 = (𝜇𝐺 + 𝜇𝐸𝑡 + 𝜇𝐴) 𝑋 ― 𝑑 𝑑𝑡 1000 𝑑𝑃 𝑑𝑡

= (𝛼1𝜇𝐺 + 𝛼2𝜇𝐸𝑡 + 𝛼3𝜇𝐴)𝑋 + 𝛽𝑋 + 𝑑𝐸𝑡 𝑑𝑡 𝑑𝐴 𝑑𝑡


0.35

𝑑 1000

𝜇𝐸𝑡𝑋

(13)

𝑌𝑋 𝐸𝑡

= (𝑘2𝜇𝐺 + 𝑘3𝜇𝐸𝑡)𝑋 ―

𝜇 𝐴𝑋 𝑌𝑋 𝐴


Page 15 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Implementation of unknown input observer for a bioreactor Three validation experiments, including a second batch experiment and both fed-batch experiments, were performed to test the unknown input observer and provide an estimate of the process states. The unknown input matrix 𝐸 was determined from batch experiment 1 and was kept constant for all other experiments. Prior to applying the unknown input observer, the unknown input matrix ‘E’ needs to be estimated. The goal of the unknown input observer is to use the on-line measurement to estimate others states in presence of unknown inputs. The only online measurement for the observer is the biomass concentration, which can be performed using either hardware or software sensors.42 The application of the unknown input observer in the current study is to estimate the concentration of glucose, ethanol, and acetic acid, by measuring the biomass concentration. In the current experiment measurement setup, there is 10 mins of time-delay in the measurement of biomass. However, this time-delay was not considered in the model and observer. The biomass concentration is a discrete measurement of every 4 hours. To use the discrete measurement in a continuous system, the continuous observer was applied recursively with a time span of 4 hours. The state estimation from previous observer was used as initial condition for the next observer estimation every 4 hours. The different initial conditions for the validation experiments are listed in Table 1. To quantify the difference between the original kinetic model prediction and the observer estimation, the normalized root mean square (RMS) error is calculated using Equation (14), in which 𝑋𝑚𝑒𝑎. is the value of states from both online and offline measurement and 𝑋𝑒𝑠𝑡. is the estimated value of states from either the original kinetic model prediction or the estimation from unknown input



Page 16 of 36

observer. The variable n is the number of measurements in an experiment and 𝑋𝑚𝑎𝑥 is the highest measurement value of the state recorded in each experiment.

(𝑋𝑒𝑠𝑡. ― 𝑋𝑚𝑒𝑎.)2

𝑛 ∑1

𝑅𝑀𝑆 𝑒𝑟𝑟𝑜𝑟 =

𝑛

(14)

𝑋𝑚𝑎𝑥

Results and Discussion Comparison between original kinetic model and validation experiments The parameters for the kinetic model were obtained from batch experimental runs as demonstrated in our previous papers.39,40 Bioprocesses are typically more complex systems than other chemical processes due to many unknown factors in the microbial system. While the growth and inhibition rates of ethanol are considered in the presented 𝛽-carotene model, there are many factors not considered, including the effects of high cell density, potential nutrient limitations, oxygen and carbon dioxide effects, and chemical inhibition from other byproducts. To fully understand and model these individual effects, extensive scale-down experiments are required. Due to certain unexpected inputs or uncertainty in the bioprocess, model-plant mismatch is observed when comparing the original kinetic model to the results of the validation experiments. This model-plant mismatch will affect the ability to use state estimation in process optimization and control applications. Figure 2 shows the mismatch by comparing the predictions of the original kinetic model (Equations 7-10) to two additional batch experiments. As the parameters in the process model are obtained from a similar but separate set of experiments, the model-plant mismatch in these similar batch experiments is expected to be small. Alternatively, Figure 3 demonstrates the predictions from the extended version of the original kinetic model (Equation 11) to two fed-batch experiments. Each of the two experiments uses a glucose feed to the bioreactor that differs in the 16 ACS Paragon Plus Environment

Page 17 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


flowrate, concentration, and duration of the glucose feeding. The details of each feeding strategy are shown in Table 1. The feeding strategy used in fed-batch experiment 1 (Figure 3a and Figure 3b) leads to a high glucose concentration during the middle of the exponential growth phase; alternatively, the glucose concentration in the reactor during fed-batch experiment 2 (Figure 3c and Figure 3d) results in a small amount of glucose in the bioreactor for a period of time after the initial glucose has been nearly depleted. Figure 3a and Figure 3b show significant model-plant mismatch is observed for the biomass, glucose, and ethanol concentrations for fed-batch experiment 1, which utilized the high glucose feed rate and concentration. The extended kinetic model predicts the biomass concentration to increase to a very high level due to the feeding of additional glucose for a prolonged period of time. The lower level of biomass observed during the experiment may be a result of nutrient or oxygen limitations. Comparison between the model prediction and fed-batch experiment 2 shows a smaller level of model-plant mismatch, as shown in Figure 3c and Figure 3d. This can be attributed to batch-like nature when using a lower glucose flowrate and concentration. The higher level of model-plant mismatch, especially in the fed-batch experiments, exemplifies the need for a better method of state estimation that can incorporate disturbance rejection and lead to better predictive capabilities. Figure 3 showed significant discrepancy between biomass and ethanol concentration, which can be used to determine the parameter in the unknown input matrix E. The discrepancy indicated the unknown input in the glucose to biomass yield, which will eventually impact the product formation. So at least two unknown terms need to be added in the unknown input matrix. The unknown input term in ethanol is optional, because it is correlated with the biomass production and glucose to biomass yield. Comparison between observer estimation and experiments in batch operation 17 ACS Paragon Plus Environment


Page 18 of 36

With the determination of the unknown input matrix E, the unknown input observer can be applied to the other batch experiment to validate its capability for state estimation as well as demonstrate its ability to be extended to fed-batch experiments. Figure 4 shows the state estimation results from the unknown input observer and its comparison to the offline measurements for both batch experiments. It is assumed that the online measurement of biomass concentration is available for the unknown input observer. In Figure 4, the system states show a similar or improved estimation performance when compared to only using the model predictions in Figure 2. The calculated RMS error for both batch experiments using the predictions from both the original model and the unknown input observer also shows this trend, as depicted in Table 2. As the batch experiment is the most studied and best understood operation mode, and the estimation of the original model parameters and the unknown input parameter matrix are both performed in the batch mode, the good performance of the unknown input observer in the batch mode is expected. Comparison between observer estimation and experiments in fed-batch operation In an ideal case of process modeling, the batch model can be directly modified into a fedbatch model through the application of a mass balance. However, due to some uncertainties or disturbances, significant model-plant mismatch can exist, as shown in Figure 3. The significant mismatch limits the application of the model for fed-batch or continuous operation. The use of an unknown input observer with a fed-batch system can significantly reduce the model-plant mismatch when estimating fed-batch reactor states. The results of applying the unknown input observer developed in this work to the state estimation of the fed-batch production of 𝛽-carotene are shown in Figure 5 in the case of the high glucose addition (Figure 5a and Figure 5b) and low glucose addition (Figure 5c and Figure 5d).


Page 19 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


The difference in the results shown in Figure 5 for the two fed-batch experiments is largely in part to the feeding strategy of glucose. The use of the unknown input observer leads to an improved state estimation when compared to the original kinetic model prediction for all system states. This is verified by calculated RMS error for fed-batch experiment 1 and fed-batch experiment 2 summarized in Table 3. In the case of the high glucose flowrate, shown in Figure 5a and Figure 5b, the unknown input observer overestimated the production of ethanol and acetic acid as a result of a higher estimated rate of glucose utilization. However, it can be noted that the unknown input observer results in a greatly reduced error in the case of fed-batch experiment 2, which utilizes a small glucose addition. The significant error reduction is due to the similar nature of this glucose feeding strategy to that of a traditional batch process. The real unknown inputs are nonlinear in nature and should come from multiple sources. As indicated in the results, the linear approximation of unknown inputs indeed significantly reduce the estimation error. But there are still estimation errors which indicate higher order term of the unknown inputs. The structure of the unknown inputs is not clear. The estimation error from the original model and experiment data such as Figure 3 can indicate the unknown inputs in terms of states and magnitude. For our current example, the results indicated that the yield of glucose to biomass is inhibited due to certain fermentation condition, according to the parameter of the unknown input matrix. This can be one way to trouble shot the unusual bioreactor. However, the reason for decreased biomass yield can be various, which will not be shown in the observer or the model. Comparison between model with updated parameters and unknown input observer An ideal process model has a fixed set of parameters for different operations. However, the biological experiments discussed in this work demonstrate that the development of such a process 19 ACS Paragon Plus Environment


Page 20 of 36

model is a challenge. One method to improve the process model is to update the parameters for each experimental run. However, typical parameter estimation is an offline procedure done after the completion of an experiment that requires the measurements of most, if not all, system states to be available. In this work, the designed unknown input observer is an online estimation tool that can predict the system states in real time based only on the previous model parameters and the availability of the biomass concentration measurement. To show the efficacy of the designed observer, the prediction of the batch model with updated parameter values and the predictions of the unknown input observer are compared with the experimental measurements for both batch experiments. As shown in Figure 6, both the model with updated parameter values and the unknown input observer can predict the batch experiments well, which indicates the existence of only slight disturbances, most likely in the form of batch-to-batch variability that is unaccounted for in the model. In the comparison, the parameters were re-estimated for batch experiment 1 using the same method in our previous study (Figure 6a and 6b), and then re-estimated parameters were applied for batch experiment 2 (Figure 6c and 6d), fed-batch experiment 1 (Figure 7a and 7b), and fedbatch experiment 2 (Figure 7c and 7d).39,40 As shown in Figure 6, the updated model and the unknown input observer can predict the batch experiments with a similar accuracy. This is not the case for the fed-batch experiments, as the higher error in the model predictions compared to the observer predictions indicates the existence of disturbances unaccounted for in the model. To compare the effectiveness of the proposed observer with commonly used state observer, extended Kalman filter was applied to the fed-batch experiment 1 in Figure 7 a&b. The extended Kalman filter uses biomass concentration as the online measurement for feedback estimation.


Page 21 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


The algorithm and results from extended Kalman filter were shown in Appendix B. As can be seen in Figure B1 in Appendix B, the extended Kalman filter has significant estimation error in glucose and ethanol. The performance of extended Kalman filter and proposed observer can be compared in Figure 7 a&b and Figure B1. The result indicates that extended Kalman filter was not able to estimate the current fed-batch experiment with unknown inputs. The RMS error from the kinetic model prediction with updated parameters and the unknown input observer is shown in Table 4 and Table 5. The ‘updated model’ in the table indicates the model with re-estimated parameters. As indicated in Table 4, both batch experiments exhibit a comparable amount of estimation error when comparing the model with the updated parameters and the unknown input observer. The kinetic models with updated parameters also results in a similar level of error for the fed-batch experiment 2 when a small amount of glucose is fed for a short period of time, as shown in Table 5. This is a result of the batch-like nature of a low glucose feed rate, resulting in only a small disturbance to the system described by the original batch models. Alternatively, the updated model predictions for a fed-batch case with a high glucose feeding rate, such as fed-batch experiment 1, indicate that using parameters estimated from a batch experiment is not reliable under a high deviation from batch behavior. Meanwhile, the application of the observer formulated in this work only requires online measurement of a single state, the biomass concentration, and shows the ability to estimate both batch and fed-batch systems well, as seen in Tables 4 and 5. Certain widely-applied advanced control applications involve online optimization and require measurements of the process states in real-time, e.g. model predictive control. Due to the laborious procedure of measuring process states in a bioprocess, the implantation of a real-time optimization-based control system becomes impractical. Furthermore, it is usually subjected to



Page 22 of 36

certain measurement delays. The observer-based state estimation offers an alternative way for state measurement in a real-time manner without frequent online sampling and measurement. In this proposed method, only an online biomass concentration measurement is required, which can be built into the bioreactor. Other state variables can be estimated from the observer without frequent sampling. Comparison of three carbon sources As is shown in Tables 2 through 5, state estimation for acetic acid concentration in both the original model and the observer has a larger normalized RMS error than any of the other states. In this bioprocess, acetic acid is utilized as a third carbon source for β-carotene production after glucose and ethanol are depleted. To determine the contributions of glucose, ethanol, and acetic acid as a carbon sources, the molar contribution of carbon from each source is calculated. It is assumed that ethanol begins to be consumed as a carbon source after all glucose is consumed; similarly, it is assumed acetic acid acts as a carbon source after the ethanol is entirely consumed. These trends can be seen in the state profiles shown in Figure 2a. The inhibitory effect of ethanol on cell growth and the toxic effect of acetic acid may lead to this prioritization of the carbon source.43–45 The molar contribution of carbon from glucose is calculated using its initial concentration, while the quantities of ethanol and acetic acid are calculated by their maximal concentration. The molar carbon composition of the added glucose substrate is also included for the fed-batch experiments. Advanced scheduling of the feeding source can be an alternative solution for obtaining higher production.46 The comparison results are shown in Figure 8. As can be seen in the figure, acetic acid contributes less than 6% of the carbon source except for fed-batch experiment 2. This is significantly lower than the glucose and ethanol contributions. Due to the low impact of acetic 22 ACS Paragon Plus Environment

Page 23 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


acid, the relatively large estimation error for acetic acid discussed in the previous sections is acceptable. Conclusions A nonlinear unknown input observer was modified and implemented using a bioreactor case study. To the best of our knowledge, it’s the first time a nonlinear unknown input observer was applied to a dynamic bioreactor system, experimentally. It has been shown here that the method is effective and easy to implement for biological systems while decreasing the dependence on first-principles process models. By using the biomass concentration measurement to estimate the other states, this work has also demonstrated the ability to utilize an unknown input observer to extend knowledge from batch experiment to differing batch experiments and a fed-batch mode of operation. Future work will need to consider methods for estimation beyond scalar unknown inputs, methods for adaptively updating the unknown input matrix for dynamically changing systems, and implementations to handle process and measurement noise in the process data. Acknowledgements The study was financially supported by the Michael O’Connor Chair II from Texas A&M University and Texas A&M Energy Institute Graduate Fellowships. We would like to thank Dr. Katy Kao for providing the Saccharomyces cerevisiae strain mutant SM14 used in this work.

Appendix A – Proof of Observer Existence and Stability To design an unknown input observer, the estimation error from the observer needs to be proven to be asymptotically stable. The estimation error of the observer is defined as 𝑒 = 𝑥 ― 𝑥, and Equation (1) and Equation (2) can be used to obtain Equation (A1) below.



Page 24 of 36

𝑥 = 𝑧 +𝐻𝑦 = 𝑧 +𝐻𝐶𝑥 = (𝐼 ― 𝐻𝐶)𝑓(𝑥,𝑢) ―𝐾𝐶(𝑥 ― 𝑥) +𝐻𝐶𝑥 (A1)

𝑒 = 𝑥 ― 𝑥 = (𝐼 ― 𝐻𝐶)(𝑓(𝑥,𝑢) ― 𝑓(𝑥,𝑢)) + (𝐼 ― 𝐻𝐶)𝐸𝑑 + 𝐾𝐶𝑒

The following Existence Condition and Theorem is used to simplify the dynamics of the estimation error: Existence Condition: The existence of an unknown input observer for system in Equation (1) needs to satisfy the following conditions: 1. 2. 3. 4.

System in Equation (1) is observable E = HCE rank (CE) = rank (E) Lipschitz condition: |𝑓(𝑥,𝑢) ― 𝑓(𝑥,𝑢)| ≤ 𝛾|𝑥 ― 𝑥|, 𝛾 is a positive constant number 5. 𝐾 satisfies Theorem 1

(A2.1) (A2.2) (A2.3) (A2.4) (A2.5)

Theorem 1: If there exists a symmetric matrix 𝑃 > 0 satisfying the following matrix inequality, then the observer is asymptotically stable if (𝐾𝐶)𝑇𝑃 + 𝑃𝐾𝐶 + 𝛾𝑃(𝐼 ― 𝐻𝐶)(𝐼 ― 𝐻𝐶)𝑇𝑃 + 𝛾𝐼 < 0

(A3)

By applying Equation (A2.2), the estimation error in Equation (A1) can be reduced to Equation (A4). 𝑒 = (𝐼 ― 𝐻𝐶)(𝑓(𝑥,𝑢) ― 𝑓(𝑥,𝑢)) + 𝐾𝐶𝑒

(A4)

The asymptotic stability of the estimation error is proven using a Lyapunov function. Theorem 1 is provided to guarantee the existence of such a Lyapunov function. A quadratic Lyapunov function is chosen as 𝑉 = 𝑒𝑇 𝑃𝑒, and the derivative of the Lyapunov function can be written as Equation (A5) below, which is modified from Chen and Saif’s paper.30 The current observer is modified of from Chen and Saif’s work by elimating of the linear term Ax in the nonlinear model, which fits into more nonlinear systems. 𝑉 = 𝑒𝑇𝑃𝑒 + 𝑒𝑇𝑃𝑒 = = (𝐾𝐶𝑒 + (𝐼 ― 𝐻𝐶)(𝑓(𝑥,𝑢) ― 𝑓(𝑥,𝑢)))𝑇𝑃𝑒 + 𝑒𝑇𝑃(𝐾𝐶𝑒 + (𝐼 ― 𝐻𝐶)(𝑓(𝑥,𝑢) ― 𝑓(𝑥,𝑢)))

(A5)

= 𝑒𝑇((𝐾𝐶)𝑇𝑃 + 𝑃𝐾𝐶)𝑒 + (𝑓(𝑥,𝑢) ― 𝑓(𝑥,𝑢))𝑇(𝐼 ― 𝐻𝐶)𝑇𝑃𝑒 + 𝑒𝑇𝑃(𝐼 ― 𝐻𝐶)(𝑓(𝑥,𝑢) ― 𝑓(𝑥,𝑢)) The derivative of the Lyapunov function needs to be negative for asymptotic stability. By applying Existence Condition (A2.4) to Equation (A5), the following inequality can be obtained: 𝑉 ≤ 𝑒𝑇((𝐾𝐶)𝑇𝑃 + 𝑃𝐾𝐶)𝑒 + 2‖𝑒𝑇𝑃(𝐼 ― 𝐻𝐶)‖‖𝑓(𝑥,𝑢) ― 𝑓(𝑥,𝑢)‖ ≤ 𝑒𝑇((𝐾𝐶)𝑇𝑃 + 𝑃𝐾𝐶)𝑒 + 2‖𝑒𝑇𝑃(𝐼 ― 𝐻𝐶)‖𝛾‖𝑒‖

(A6)


Page 25 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


2

≤ 𝑒𝑇((𝐾𝐶)𝑇𝑃 + 𝑃𝐾𝐶)𝑒 + 𝛾(‖𝑒𝑇𝑃(𝐼 ― 𝐻𝐶)‖ + ‖𝑒‖2) = 𝑒𝑇((𝐾𝐶)𝑇𝑃 + 𝑃𝐾𝐶 + 𝛾𝑃(𝐼 ― 𝐻𝐶)(𝐼 ― 𝐻𝐶)𝑇𝑃𝑇 + 𝛾𝐼)𝑒 = 𝑒𝑇((𝐾𝐶)𝑇𝑃 + 𝑃𝐾𝐶 + 𝛾𝑃(𝐼 ― 𝐻𝐶)(𝐼 ― 𝐻𝐶)𝑇𝑃 + 𝛾𝐼)𝑒 (because P is a symmetric matrix) Thus we can have: (𝐾𝐶)𝑇𝑃 + 𝑃𝐾𝐶 + 𝛾𝑃(𝐼 ― 𝐻𝐶)(𝐼 ― 𝐻𝐶)𝑇𝑃 + 𝛾𝐼 < 0

(A7)

As this result is identical to Equation (A3) in Theorem 1, the stability of the observer has been proven. To solve for the parameters using the Equation (A7), which has already been proven to be identical to the condition defined in Theorem 1, the Schur complement condition for positive definiteness can be applied:

[

]

𝐴 𝐵 Schur Complement Condition: If X is a symmetric matrix given by X = 𝐵𝑇 𝐶 then X is positive definite if and only if A and X/A are both positive definite, or X > 0↔A > 0 & C ― 𝐵𝑇𝐴 ―1𝐵 > 0 Equation (A7) can be rewritten as the linear matrix inequality shown in Equation (A8.1).35 Through the multiplication of both side of (A8.1) by -1, we obtain Equation (A8.2). ―𝐼 [(𝐼 ― 𝐻𝐶)

𝑇

𝐼 [ ―(𝐼 ― 𝐻𝐶)

𝑇

]

𝑟𝑃(𝐼 ― 𝐻𝐶) 𝑟𝑃 𝑃𝐾𝐶 + 𝐶𝑇𝐾𝑇𝑃 + 𝛾𝐼 < 0

]

― 𝑟𝑃(𝐼 ― 𝐻𝐶) >0 ―(𝑃𝐾𝐶 + 𝐶𝑇𝐾𝑇𝑃 + 𝛾𝐼)

𝑟𝑃

(A8.1)

(A8.2)

Using the Schur complement condition with Equation (A8.2), we can reach the conclusions below: Equation (A8.2) > 0 ↔ ― (𝑃𝐾𝐶 + 𝐶𝑇𝐾𝑇𝑃 + 𝛾𝐼) ―𝛾𝑃(𝐼 ― 𝐻𝐶)(𝐼 ― 𝐻𝐶)𝑇𝑃 > 0

(A8.3)

To solve this nonlinear matrix problem of Equation (A5.2), we introduce new variables 𝑌 = 𝑃𝐾𝐶 and 𝑊 = 𝑃𝐻, so the nonlinear matrix inequality in Equation (A8.1) can be transformed into Equation (A9).

[ 𝑟𝑃 ――𝐼𝑟𝐶 𝑊 𝑇

𝑇

]

𝑟𝑃 ― 𝑟𝑊𝐶)

Estimation of Unmeasured States in a Bioreactor under Unknown

Recommend Documents