The problem of control of the fed-batch process for the production of yeast is further complicated by the lack of online sensors, lack of suitable models due to poorly understood dynamics. The derivation of the developed yeast model using mass balance equations and rate laws is discussed and presented in the chapter. The optimization layer problems are determined and based on the solutions from the previous upper layer, i.e.
The introduced decomposition enables the natural use of a parallel computing cluster of computers, with which subproblems of the optimal control problem are solved in parallel.
LIST OF TABLES
Statement of the problem
The problem considered in the study belongs to the category of development of industrial and scientific control technologies through the use of methods of optimal control, mathematical modelling, decomposition and calculation for different types of fermentation processes (1. Rocha and Ferreira, 2002). To develop method, algorithms and programs for optimal control of the feed-batch fermentation processes so that the maximum concentration of the biomass is produced at the end of the process. Development of method and algorithm for solving the optimal control problem based on decomposition of the initial problem.
Process simulation under optimal control developed in pg.1.4.3 to verify and test the method and programs.
Proposed strategy for control
- Layers of the control system
- Adaptation layer
- Process modeling
- Parameter estimation
- Optimization layer
7 Optimization= optimization based on model parameters from the adaptation layer above, and direct controller parameter design. Model parameters are determined based on data from process variables, control and state variables. The optimal trajectory of the physicochemical variables from the optimization layer above is passed to this layer.
The theory of variational calculus is used with, not other methods for optimal control (maximum or dynamic programming principle).
Objectives of the research project
Conclusion
We discuss the results of previous experiments to illustrate the limitations associated with the control of the studied process. Input to a process is continuously added from the initial stage to the end of the process, but the output of the process is not removed until the process is complete. The three processes mentioned above have their advantages and disadvantages, but the one that interests most control engineers is the Fed batch fermentation process.
The fed-batch processes start growing the cells for a certain amount of time under the batching motion, usually close to the end of the exponential growth phase.
Feed stream
Process modelling
To fulfill the adaptive control, an accurate mathematical model representing the reaction and reactor environment must be identified. The model is used as a tool for process analysis, for the design of control systems and for the realization of the adaptive control.
Model types
- Product mass balance equation
- Program description
For the considered case, the biological variables x,s and p are described based on the systematic approach using mass balance equations. Mx =xV, where x is the cell concentration and V is the liquid volume of the reactor vessel,. These coefficients are entered mathematically to represent the influence of physico-chemical variables on biological variables.
The optimal control problem is solved using optimal control theory, the Lagrangian functional approach. The formulation of the problem, the description of the methods, algorithms and programs developed is given in the chapter. Optimal profiles of chemical variables are the ultimate solution to the optimization problem.
Because the solution is difficult to handle analytically, decomposition in the time domain of the necessary conditions for optimality will be introduced. The obtained values of the trajectories of the state and control variables are a function of the coordinating variables. Maximum number of steps for state and control variable gradient procedure - Ml. b) Steps in the gradient procedures.
The initial trajectories of the control variables are substituted into the model equations and the state trajectories are calculated. me). If the error is smaller, the first-level calculations stop and the current values of the state and control variables are optimal. If the conditions are met, we obtain an optimal solution to the coordination and overall problem.
If the conditions are not met, new values of the coordination variables are calculated.
Results from the calculations
Conclusion
Of the mechanisms that have been generally adopted to increase the speed of performance, parallel processing is the most worthy of mention. Parallel processing can be simply defined as completing a large task more quickly and efficiently by dividing it into several smaller tasks and executing them simultaneously using more resources. However, when applying parallel processing to a process, a number of factors need to be considered, for example, whether the task in question can be performed in parallel, its cost effectiveness, synchronization of subtasks, and communication between resources.
To understand parallel processing at the implementation level, one needs to know some basic terms such as task, process, processors and the fundamental characteristics of parallel processing. If the number of processes is less than the number of processors, some processors will not be assigned processes and will remain idle during program execution. 4 A parallel system performs tasks or programs faster than sequential processing of the same tasks or programs.
Parallel processing reduces the execution time of the considered program and thus increases the speed or improves the response time. A parallel processing system is more likely to be fault tolerant than a single processor system, since parallel systems offer the flexibility to add and remove processors or otherwise. If the problem tends to grow over time, parallel processing can handle it by adding extra hardware.
Parallel processing has a wide range of applications in various fields ranging from weather forecasting to robotics. In order to get the maximum benefit from a parallel solution, different issues need to be considered. It is always necessary to do some kind of background investigation and investigation based on the factors involved to find out if parallel treatment is worthwhile for a particular case.
INTERCONNECT
Input
The From Workspace functions are used in the program to introduce time and input variables from the Matlab workspace. The To Workspace functions are used in the program to send the trajectories of these variables to the Matlab workspace. The kinetic coefficients Jl_, Yxis, Yph<, Ks as functions of the control variables T, pH, DOz, F are calculated in the film programming block.
There are some function blocks in the program: Fen, Unit Delay, Mux, Sum and Product. This block can add or subtract scalar, vector or matrix inputs. It can also sum the elements of a single input vector. The number of inputs is 2, the multiplications are element wise (.*) and the sample time is -1 in the program.
Script file for the input and output parameters
The optconfer2.m program implements a sequential program for optimal control calculation with special functions and commands from the Matlab distributed computing toolbox. Since parallel computing is based on shared resources and memory, the global variable definition is used with the command: .. global par T pH D02 F kd m pi n si K MI epsl mu I x s p v el conmin conmax statemin statemax astate acon dt yO xl sI pI vI. Tl pH D021 Ft. The steps of the gradient procedures to calculate the state control and conjugate state variables and penalty coefficient values are combined into vectors. astate=[ax as ap av)'; . acon=[aTaFapH3002)';.
The same is done for the min and max values of the state and control variables: .. statemin=[xmin smin pmin vrnin]'; .. statemax=[xmax smax pmax vmax)';. These vectors are global, the same for the main program given by a script file optconferl(2).m and for the parallel subprogram given by function file subprogl(2).m. Two variants of the program were developed using the subprogram subprogl.m is used to organize the calculations in order in the optconferl.m script file.
In the second case, the subprogram is used to organize parallel computation in the cluster. In this way, the user has three steps in the calculation. These three steps of organizing the computation help to overcome many problems if the program is written directly for parallel computation.
Organizing ofthe parallel computation
The part of the program implemented in parallel is described by the function file subprog2.m. The subprogram 2.m starts with the calculation of the start and stop points of the optimization horizon for each employee. Calculation of the expressions Iv(k), k=O,K-I, v = x, s, p, v is followed by calculation of the errors and the values of the state and control variables.
The sequential calculation program is used to calculate the optimal trajectories of the physiochemical (control) and biological (condition) variables of the fed-batch fermentation process. The purpose of producing yeast is to reduce the production of the product of fermentation, ethanol. The product inhibits the production of biomass as its quantity is very high towards the end of the fennentation.
The optimal trajectories of the product show lower values during fennentation, allowing more biomass to be produced, as shown in Figure 6.14(b). Table 6.2 examines the dependence of the time used for parallel computing on the number of workers used. The time used for each iteration for the corresponding number of employees is given in the rows of the table.
It can be seen from table 6.2 that the calculation time is reduced with the increase in the number of workers. In addition, the control of the process is mandatory, which would otherwise make the complexity of the mode of operation useless.
Problems solved in the dissertation
CLAES J., VAN IMPE J., 1998 (a): Online monitoring and optimal adaptive control of the fed-batch baker's yeast fermentation.
