Unmeasured State Variables Research Articles

Observer-based non-linear control of a fed-batch autoinductive fermentation process

Optimal regulatory control of an autoinductive recombinant culture in a fed-batch reactor is considered. End point optimization results in a three-stage process: biomass growth, inducer synthesis and product synthesis. It is shown that in the last stage the substrate concentration should be maintained constant. This is achieved using an input—output linearizing controller accompanied by a novel non-linear state observer for the estimation of unmeasured state variables on the basis of on-line off-gas carbon dioxide concentration measurements. Experimental runs of luminous recombinant E. coli strain in a laboratory fermenter demonstrate the rapid convergence of the observer estimates as well as the effectiveness and robustness of the overall control system.

Process Identification in On-line Optimizing Control, an Application to a Heat Pump

The objective of this paper is to focus on on-line state and parameter estimation in connection with on-line model-based optimizing control of continuous processes. A nonlinear programming approach is used to estimate unmeasured state variables and parameters in systems modelled by nonlinear differential-algebraic equations. The nonlinear dynamic model is discretized by orthogonal collocation on finite elements, and the moving-horizon approach is used to reduce the dimension of the final optimization problem. A priori parameter information is included in the minimization criterion (Bayes estimation), and this makes the estimation problem more robust with respect to missing process excitations and over-parameterization. The a priori parameter covariance matrix is treated as a tuning matrix, where the diagonal elements can be set acording to the amount of information in the measurements. The updated steady-state part of the process model is used to optimize the economic performance of the process, where new optimum set-points are calculated for the regulatory control system. The sequential quadratic programming method (SQP) is used to solve this nonlinear optimization problem, where the objective function, the model equations, and the operational feasibility constraints are solved simultaneously in an infeasible path approach. The identification and on-line optimizing control approach are illustrated with an example from an experimental water-to-water heat pump unit.

Fault and contamination detection in a continuous bakers yeast fermentation

Biological systems exhibit different phases or biological states during the cultivation process. It is important to be able to identify these phases as secondary metabolite productions are dependant upon different phases of the fermentation. It allows one to determine better feeding or dosing strategies to obtain optimum productivity of the desired products. Since the secondary metabolites are not measurable, a major requirement is a technique to predict the different concentrations of the unmeasured state variables based upon measured on-line variables such as dissolved oxygen, carbon dioxide evolution rate, pH, acid/base additions, etc. However, cultivation processes also suffer from contamination problems which result in the loss of a batch. These also suffer from “biological fault” conditions which can be brought about by under or over feeding, or early or late feeding resulting in substrate related malfunctions giving rise to low productivity. Biological faults can occur also due to oxygen or nutrient limitations, or due to production of excessive inhibitory products (toxins). Although all these factors may not lead to the loss of a batch, they can result in a loss of productivity in terms of quality and quantity of primary or secondary metabolites produced and contribute to the frustrations of the industrial plant operators. Other obvious faults occur when some equipment (pump failure, motor malfunction, etc) or sensors (DO probe, rpm sensors, pH probe, air plugging, etc.) fail to perform satisfactory giving false information to the operator which may result in erroneous decisions regarding process operation. In this study, a comprehensive simulation of a yeast fermentation is used to study different aspects of biological faults and contamination problems. A Fuzzy-Neural Network (FNN) methodology is used for fault diagnosis.

Feedback Linearizing Controller Design for Chemical Processes: Challenges and Recent Advances

Many chemical processes are difficult to control due to their inherent nonlinear characteristics. Feedback linearization is a promising approach for controlling such highly nonlinear processes. In this paper, we discuss a few problems which are encountered when feedback linearization is applied to chemical processes and provide an overview of our attempts to address these problems. More specifically, we focus on several issues not adequately addressed by existing feedback linearization theory including: (i) unmeasured state variables; (ii) unmeasured disturbances; (iii) time delays; and (iv) process constaints. The concepts are illustrated using a simple chemical reactor model which exhibits many of the nonlinear characteristics observed in industrial processes.

Ljapunow-Synthese nichtlinearer Beobachter durch Gütemaßangleichung

For the special class of nonlinear systems, which are linear in the inputs and linear in the unmeasurable state variables, a nonlinear observer design method based on the theory of Lyapunov is proposed. Besides enlarging the operating region in which global convergence is guaranteed by use of a quadratic Lyapunov function, the design method leads to a given dynamical behaviour for the observer error system via a cost functional assimilation

Issues in the Modelling and Control of Fed Batch Fermentation Processes

In this paper alternative approaches to modelling and state estimation of a typical, industrial fermentation process will be described and comparative results of the application of the Kalman filter estimator to provide on-line estimation of the unmeasurable state variables will be given. Different techniques to control the product concentration of the process around a desired trajectory will be described. The several results are tabulated and compared, using a least squares criterion. A self organising system based on fuzzy logic control and model reference variable structure control will be illustrated as a possible candidate strategy to control the process.

A new approach to robust fault detection and identification

A methodology for instrument fault detection and identification (FDI) in linear dynamical systems subject to plant parameter variations or uncertainties is presented. At the heart of this approach is a robust estimator for which the necessary and sufficient conditions to its existence are outlined. The robust estimator can simultaneously estimate the unmeasurable state variables of the system for the purpose of control and provide necessary information for FDI purposes. A novel feature of this approach is that it can actually identify the shape and magnitude of the failures. The scheme allows for fast and accurate FDI, and can account for structural uncertainties and variations in the parameters of the dynamical model of the system. The overall fault tolerant control system strategy proposed is verified through simulation studies performed on the control of a vertical takeoff and landing (VTOL) aircraft in the vertical plane.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Robust servo design with applications

An optimal multivariable proportional-integral (PI) controller with preassigned eigenspectrum is designed so that the system output would track constant reference inputs in spite of constant but unknown plant disturbances. A sequential strategy for modal servomechanism design is given for the above problem. In addition, since the proposed PI controller is of state feedback type, an observer capable of estimating the unmeasurable state variables in spite of the presence of unknown disturbances is introduced, thus resulting in a suboptimal overall system. The separation property for the overall servo/observer system is illustrated. The design strategy is successfully used for position control of a brushless DC motor, and in a regulator mode for controlling a headbox which is an integral part of the modern paper machines used in the pulp and paper industry. >

Bilinear Modelling and State-Feedback Control of an Electro-Hydraulic Drive

A continuous-time bilinear model is identified for an Electro-Hydraulic Servo Drive. Based on this identified model a state-feedback controller is designed. The controller consists of a feedback linearizing control law and a tracking control algorithm. The unmeasurable state variables are determined with a bilinear observer from the position measurements of the drive. A new method is introduced to design observers for bilinear systems.

CONTROL AND ESTIMATION FOR PERTURBED NONLINEAR PROCESSES WITH DEADTIME

Feedback linearization techniques are used to deal with the nonlinear controller designs which have attracted many researchers' attention in recent years. The approach has been applied successfully to solve a number of practical nonlinear control problems, but typically requires on-line full state measurement which is usually not the case in real chemical process industries. In this paper, we address the problem of synthesizing nonlinear state feedback controllers for time-delay nonlinear systems which are perturbed by disturbances. On-line estimation of the unmeasurable disturbances and unavailable state variables is introduced to facilitate the implementation of coordinate transformations and state feedback and prediction. Two kinds of dynamic compensators are then proposed to handle the process deadtime. Finally numerical simulations in a CSTR example demonstrate the promising performance of the overall nonlinear control structure in disturbance rejection.

Estimation of Neutron Flux and Xenon Distributions via Observer-Based Control Theory

State feedback control provides many advantages, such as stabilization and improved transient response. However, when state feedback control is considered for spatial control of a nuclear reactor, it requires complete knowledge of the distributions of the system state variables. Also, if the reactor is in a transient, flux mapping systems that are based on steady-state conditions are not appropriate for an accurate representation of the operating state of the reactor. In this paper a method is described for reconstructing the measurable and unmeasurable state variables in a nuclear reactor from output measurement data, which can be used to generate input for a feedback control system or serve as a core observer (estimator) in a reactor transient. The method is based on a Luenberger-type observer theory that is extended to infinite-dimensional distributed parameter systems. The method was applied to a simple reactor model in one spatial dimension and one energy group with xenon dynamics that exhibited spatial oscillations. The resulting observer was tested by using model-based data for measurement output. The results show that the spatial distributions of iodine, xenon, and neutron flux are estimated very well be the observer using information from a finite number of sensors.

Identification and Modelling of Fermentation Process using MATLAB: A Case Study

The paper will illustrate the application of selected MATLAB packages to the modelling of a Cythus Striatus fermentation process and estimation of unmeasurable state variables. Two approaches for modelling the fermentation process are adopted. In one approach, a linear model has been fitted to experimentally obtained data, and the Kalman filter has been used in conjunction with this model to estimate the state variables. In the second techniques, kinetic model based on the Monod equation is derived. To overcome limitations of the MATLAB packages, a software tool has been developed to translate MATLAB into C source code. A general description of the software tools is given.

Nonlinear predictive control using multi‐rate sampling

AbstractA nonlinear predictive control (NLPC) strategy based on a nonlinear, lumped parameter model of the process is developed in this paper. A constrained optimization approach is used to estimate unmeasured state variables and load disturbances. Additional model/process mismatch is handled by using an additive output term which is equivalent to the Internal Model Control approach. Similar to linear predictive control methods, an optimal sequence of future control moves is determined in order to minimize an objective function based on a desired output trajectory, subject to manipulated variable constraints (absolute and velocity).Deadtime is explicitly included in the model formulation, giving NLPC the same deadtime compensation feature of linear model‐predictive techniques. The multi‐rate sampling nature of most chemical processes is also used to improve estimates of process disturbances. Infrequent composition measurements in conjunction with frequent temperature measurements are used to improve the “inferential” control of the composition in a continuous flow stirred tank reactor (CSTR).

Decentralized observer-based stabilization for a class of interconnected systems

A decentralized output feedback law for a class of interconnected systems is described. The proposed output feedback scheme is based on the decentralized reconstruction of the unmeasurable subsystem state variables and interacting state variables. A sufficient condition is given for the overall system to be stabilizable by using the proposed output feedback control law.

Robust Optimal Tracking Controller for a Robotic Manipulator

Abstract In this paper, the experimental results of an actual implementation of a robust tracking controller on a robotic manipulator are presented. The objective is to follow a ramp reference input with zero steady state error in the presence of a disturbance and system parameter variations. The main difficulties are caused by the Coulomb frictions, the disturbance due to the gravitational pull. the spring effect of the link between the drive motor and the manipulator arm, and the unmeasurable state variables. The controller is based on feeding back delayed values of the observable variables and augmenting the system by additional integrators. The feedback gain parameters are obtained by first applying the quadratic optimal control theory and then readjusting the feedback parameters to eliminate the limit cycle by using the describing function method

LINEAR FEEDBACK EQUIVALENCE AND CONTROL OF AN UNSTABLE BIOLOGICAL REACTOR

A novel method for the feedback control of an unstable, continuous stirred tank bioreactor is described in this paper. The models considered describe the growth of a single microorganism on a single, rate limiting substrate. A theoretical discussion demonstrates that for models of this type, a global feedback transformation exists which linearizes the state response. Such a transformation is computed using the methods of Hunt, Su and Meyer (1983). A controller is computed using linear state feedback control, and an extended Kalman filter is used to estimate unmeasured state variables. Simulation results demonstrate satisfactory control behavior over a wide region of the phase plane.

Optimization of a Fed-batch Culture by Statistical Data Analysis

An algorithm for the optimization of a fermentation process was investigated using the combination of dynamic programming and a linear predictive procedure by regression analysis. It was applied to the fed-batch culture for glutamic acid production with ethanol feeding and was proved to be effective for solving the optimization problems including online adaptive control. The effect of linear predictive error on the optimum policy was discussed by use of Fishman and Biryukov’s model of penicillin production. The prediction method was modified for application to processes including unmeasurable state variables. The use of time-series data was proved to be effective for prediction of a fed-batch culture of baker’s yeast.

An algorithm for non-linear space—Time nuclear reactor control

A novel multivariable control algorithm for non-linear space-time nuclear reactor dynamics is proposed in this paper. The multivariable control algorithm is based on a mathematical model of the nuclear reactor which includes: a single energy group of neutrons, delayed neutron precursors, iodine, xenon and thermal-hydraulic feedback. The multivariable control algorithm is composed of non-linear time-varying feedforward and feedback control signals, a reference model of the nuclear reactor and a dynamic observer. The non-linear proportional plus integral feedback controller forces the nuclear reactor to follow the response of the reference model. The dynamic observer estimates the unmeasurable state variables. The feedforward and feedback control signals are determined in a novel approach by specifying the form of the closed-loop response of the neutron density variables. By virtue of the multivariable control algorithm the closed-loop differential equations are linear and time-varying. A linear stability analysis for base-load and load-cycle operation indicates that the closed-loop system is stable provided that the thermal-hydraulic subsystem is inherently stable. The simulated dynamic response indicates that the multivariable control algorithm provides excellent response characteristics.

Stabilization of Controlled Current Induction Motor Drive Systems via New Nonlinear State Observer

In this paper a new bilinear and nonlinear second-order state observer which is globally asymptotically stable is constructed to estimate the unmeasurable state variables. Then an unstable controlled current induction motor drive system which cannot be stabilized by a linear state observer is successfully stabilized by feeding the reconstructed state variables from the nonlinear observer to the frequency control channel. This scheme is easily implementable, resulting in an asymptotically stable overall closed-loop system. These results are numerically verified by simulation.

Existence of limit cycle and stabilization of induction motor via new nonlinear state observer

In this paper the boundedness of the solutious of the bilinear and nonlinear differential equations, which describe the dynamic behavior of an ideal three-phase squirrel cage induction motor, is shown using a Lyapunov function. It is then proved by sampling combined with a digital simulation that an unstable machine has limit cycle. Utilizing these results a new bilinear and nonlinear reduced-order state observer, which is globally asymptotically stable, is constructed to estimate the unmeasurable state variables. By using this observer a new two-step procedure for stabilizing an unstable machine, which has a limit cycle, is proposed. This scheme can be easily implemented resulting in an asymptotically stable overall system. These results are numerically verified by simulation.