The objective of this thesis is to operate membrane processes optimally in theory followed by in experiments. The research work comprises mathematical modeling, simulation, optimization, and implementation of optimal operation of batch membrane diafiltration processes.
The purpose of membrane separation is to increase the concentration of the product (macro-solute) and decrease the concentration of impurities (micro-solute). A combination of semi-permeable membrane and diluant addition (diafiltration), is used to serve the purpose.
The optimal operation implemented in this research is model based, and hence the modeling of membrane processes forms the first part of this work. Modeling of different configurations of membrane processes has been done, with some new model derivations to help the research field. The batch open-loop and closed-loop diafiltration configurations are studied. The modeling section also includes dynamically fitting the existing models to the experimental data, to obtain the optimal parameter values.
The modeling is followed by the simulation and implementation of optimal operation. Implementation involves performing the optimal operation on a laboratory scale membrane separation plant. The aim of optimization is to find analytically the addition rate of solvent (diluant) into the feed tank in order to reach the final concentrations whilst minimizing costs.
The objectives to be minimized are processing time, or diluant consumption, or both for batch open-loop diafiltration processes. Pontryagin's minimum principle is utilized to attain the analytical solution for optimal operation. The optimal operation derivation is verified experimentally on a plant using nanofiltration form of membrane separation. Case studies are implemented showing the optimal operation and its comparison with the current or traditional industrial strategies of membrane separation.
In case of batch closed-loop diafiltration processes the objectives to be minimized are time, or diluant consumption, or power, or a combination of them. The numerical methods of orthogonal collocations, and control vector parameterization are applied to obtain the optimal operation strategies. Case studies are studied in simulation. The inferences are established regarding the advantages and disadvantages of batch closed-loop over open-loop configuration.
This thesis considers a path planning problem for heterogeneous vehicles. Such vehicle consist of two parts which have the ability to move individually. One of them is faster, but has shorter range and is therefore required to keep in a close distance to the main vehicle. The objective is to devise an optimal path of minimal length under the condition that the faster part of the heterogeneous system visits all desired waypoints exactly once. Three versions of the problem are considered. One assumes that the order in which the waypoints are visited is known a-priori. In such a case we show that the optimal path can be found by solving a mixed-integer second-order cone problem. The second version assumes that the order in which the waypoints are visited is not known a-priori, but can be optimized as to shorten the length of the path. Two approaches to solve this problem are presented and evaluated with respect to computational complexity. In the third situation we consider that visited points are moving in real time along with the vehicle.
This works deals with optimal control of membrane processes. More concretely, we focus our attention to optimal operation of batch membrane diafiltration processes in the presence of fouling. Diafiltration method is used for the separation of two or more solutes based on the molecular size difference from a solution. The goal is to increase the concentration of valuable components and simultaneously decrease the concentration of impurities. This goal is achieved by determination of the addition of a solute free solvent (diluant) into the feed tank to reach the final concentration in minimum time. We use Pontryagin's minimum principle to solve the time-optimal control problem analytically. By using the analytical approach we are able to determine the control structure as a sequence of arcs. In this work two cases are discussed. In the first we consider that the fouling decreases the membrane area. In this case we are able to obtain the control structure only but the individual time durations have to be determined numerically. In the second case we assume that fouling causes a gradual decrease of the permeate flux. The optimal control structure is fully analytical with prescribed control sequences and individual time intervals. We apply the theoretical results to demonstrate applications to different classes of fouling models. We also compare the derived time-optimal operation with traditional used operations to demonstrate the benefits of the proposed methodology.
This work aims to contribute to modelling and fast predictive control of processes. It can be divided into several topics.
Process modelling is investigated and an effective approximation technique is described. It can be used to approximate an original non-linear process model as a hybrid system with piecewise affine dynamics. We discuss three different cases, how one can obtain the approximation of an arbitrary nonlinear function. The most trivial case assumes that the analytic form of the nonlinear term is already known. On the other hand, if only some set of input-output measurements are given, we employ a two-stage procedure to obtain the final approximation. This method aims to select the appropriate subset of basis functions and consecutively finding a proper linear combination of them. Once we possess the analytic formula of our approximated function, we can obtain the final PWA approximation by solving standard nonlinear programs. We show, that under mild assumptions, the task can be transformed into a series of one-dimensional problems. Finally, we demonstrate the efficiency of our technique on an illustrative example, involving a highly nonlinear reactor.
The second part of the work deals with fast model predictive control. We investigate the problem of reduction of the amount of memory needed to describe explicit MPC solutions. The main idea of explicit MPC stems from pre-computation of the optimal control action for all possible initial conditions and subsequently storing them in a form of a look-up table. On one hand, this concept allows faster implementation, but on the other, requirements for memory storage increase too. In order to eliminate this drawback, we continue with a description of an effective, three-layer compression technique, allowing fast implementation on low-cost hardware platforms. This three-layer procedure first identifies similarities between polytopic regions in form of an affine transformation. If such a mapping exists, certain regions can be represented using less data. The second layer then applies data de-duplication to identify and remove repeating sequences of data. Regions are then described by integer pointers to such a unique set. Finally Huffman encoding is applied to compress such integer pointers using prefix-free variable-length bit encoding. The chapter ends with efficiency evaluation of the proposed technique on several, randomly generated feedback law examples.
The final chapter is devoted to the so-called operator splitting methods, by means of one can solve convex optimisation problems very efficiently by simply decomposing the original possibly complex problem into a series of simple operations well known from linear algebra. Several algorithms and their range of applicability are presented.
This thesis deals with the problems of design and realization of remote laboratories, while it is particularly focused on multipurposivness and reduction of development costs for such systems. Remote laboratory is an instrumentation apparatus with space-distribution of its functional parts and it allows to control experimental devices remotely, over the computer network. Such systems apply mostly in the area of technical education and they bring the innovative approach to practical exercising.
The work is aimed on such concept of remote laboratory architectures, which allows to separate the phase of actual technological development from the process of implementation. This leads to significant savings in development time and reduces the complexity of laboratory creation process. To fulfill these requirements, the architecture has to be considered as universal since the early development phase. The universality must apply for both, the hardware resources as well as the software. Physical parts of architecture must be able to connect and serve a wide class of different experimental devices and program parts must be designed in order to mediate the remote control of such devices.
For these purposes, two different architectures have been developed in this work. First one is the multipurpose hardware and software architecture, based on industrial-aimed control and communication devices, such as programmable logic controllers and industrial network routers. The main benefit of this architecture is that it provides a ready-made hardware and also partially software solution and it is capable to serve various types of technological processes. The second architecture uses low-cost parts, such as single-board computers and programmable micro-controllers. The purpose of this approach is to demonstrate the potential of cheap components for the use in remote laboratories and process control. The practical contribution of this work is the implementation of several different process control laboratories on both types of architecture. Another topic discussed in this work are the methods of laboratory management as well as their publication for educational purposes.
This work considers the problem of finding the optimal control of batch membrane diafiltration processes. Diafiltration is known as an effective method to separate at least two solutes from given solution (liquor) at the base of their different molecular (particle) effective sizes. The goal is to concentrate (increase the concentration of) the solute(s) with bigger particle size(s) usually called macro-solute and to remove impurities, i.e. to dilute (decrease the concentration of) solute(s) with smaller particle size(s) traditionally denoted as micro-solute. The whole process is described by the set of ordinary differential equations and thus methods of dynamic optimization (open-loop optimal control) can be used to establish optimal operation of these processes.
Our task is to determine how a solute-free solvent (diluant) should be dynamically added to feed solution tank throughout the process run to achieve given separation goal in minimum time or with minimum amount of added diluant. We use analytical approach, Pontryagin’s Minimum Principle, to identify candidates for optimal control taking into account the necessary conditions for optimality. Based on these, we derive optimal operational policies for batch membrane processes of several types. Direct (discrete) numerical method of dynamic optimization, Control Vector Parameterization, is then used to confirm the theoretical findings and to obtain the optimal diluant utilization for particular process and instance.
This work is deals with optimal control of batch processes in the presence of uncertainty. An integrated two-time-scale control is proposed, whereby a run-to-run adaptation strategy with adaptation of the terminal constraints is implemented at the slow time scale, and is integrated with a neighbouring-extremal controller that operates at the fast time scale and performs further on-line corrections. This control scheme is especially suitable for repeatable batch processes with the fast changes in process dynamics. In addition, this scheme can be easily realised in real batch processes as the required computational power is low. Particularly, the only computation performed in real-time at each sampling time is a solution of a linear two-point boundary value problem. By sacrificing a bit of accuracy, all the required controlled designs and an accompanying computations might by done off-line. In the presence of uncertainty, the necessary conditions of optimality no longer hold. The core idea is to use the so called NCO-tracking approach that pushes the gradients caused by an uncertainty to zero. Neighbouring-extremal controller is approximated controller, i.e. it is based on linearisation of the nominal solution. Because of a lower performance of such control solution in chemical applications, the need for a supplementary adaptation is obvious. Our proposed control scheme thus corrects approximated control by another control. Run-to-run adaptation strategy updates the model between batches according to the latest constraints measurements and re-optimises the nominal solution. This solution then provides the reference trajectories for neighbouring-extremal controller. The thesis describes essentials to understand the basic building blocks of the proposed control scheme. In particular, the first part introduces the nominal optimisation, i.e optimisation under ideal circumstances without the influence of the uncertainty. Next part discusses the efficient algorithms that deals with the uncertainty. The proposed control is verified on real process. It is shown that the integrated two-time-scale control scheme has faster convergence rate and better performance in comparison to the other tested approaches.
The aim of the thesis is to provide techniques for computing and application of explicit model predictive control of processes in real-time. Explicit model predictive control is a method for optimal control of processes with constraints where the control law is given in an explicit form. It is an optimization based approach where a process model is used to predict the future. In the presented approach the model is derived using a hybrid modeling framework. It will be shown how that hybrid models are capable of modeling wide class of processes with a sufficient precision. Based on the hybrid model, it is possible to formulate an optimal control problem which considers varying dynamical properties of the plant and moreover, takes the operating constraints into account. If the optimal control problem is solved for the whole set of initial conditions, an explicit solution is obtained. The explicit solution characterizes the optimal control law as a function of given initial conditions. This allows the control law to be easily realized in practice, just by evaluating the function for given value of initial conditions. Therefore, the explicit model predictive controller designed in this way is especially suitable for applications with fast dynamical changes which require low implementation cost. The thesis describes essentials for understanding ingredients of explicit model predictive control. In particular, the main part concerns with construction of hybrid models and their deployment in formulation of optimization problems. Consequently, efficient algorithms for computing explicit solutions to time optimal control problems are proposed. Obtained results are applied in real-time experiments that show desired optimal performance and the controller satisfies all operating constraints.
This dissertation work deals with dynamic optimization processes which are described by sets of differential equations changed in the switching time. The switching conditions which occur depend on the process character. These systems in general are called as hybrid. A large number of processes are characterized by the change of the operational phases. The switching conditions can be given on the basis of the single logical condition, e.g. tanks with and without interaction. On the other hand, the processes with chemical reactions are more complicated and are typical e.g. for the production of fine chemicals, specialties (pharmaceuticals), polymers, biotechnology or for wastewater treatment, where nitrification/denitrification process takes place.
This dissertation work presents a numerical approach to solve dynamic optimization problems of hybrid systems. The dynamic optimization problem is transformed into the static optimization problem, which is called non-linear programming (NLP). Then it is possible to solve the problem with one of the gradient-based optimization methods. The gradient computational algorithm to obtain optimal control profiles is of a SQP (successive quadratic programming) type. The conditions of the optimality are derived in the third chapter of this work. These conditions based on the variational method consider the switching structure and they are necessary for the derivation of the adjoint equations. The advantages and drawbacks of the presented method are compared with the sensitivity equations approach.
In the next part of the dissertation work is shown the algorithm solution, which is applied on some chemical engineering processes. First, the problem of a hybrid coupled tanks in different vertical position is considered. The minimum time problem for one or more control variables and the LQ cost problem are solved. Secondly, a two stage reactor system with chemical reactions is considered. The aim of the optimization is to get a maximum amount of the desired component not later than at final time subject to a constraint that the final concentration of the desired component should be higher than, or equal to a desired value. The reactor was initially solved with the sensitivity equations approach, what gives an interesting opportunity to compare with the adjoint approach, where increased number of time intervals do not change the number of differential equations comparing to the originally applied method. It follows that the difference in computational time of the algorithm increases which is the main drawback of this method.