Numerical Methods for Mixed-Integer Optimal Control with Combinatorial Constraints

This thesis is concerned with numerical methods for Mixed-Integer Optimal Control Problems with Combinatorial Constraints. We establish an approximation theorem relating a Mixed-Integer Optimal Control Problem with Combinatorial Constraints to a continuous relaxed convexified Optimal Control Problems with Vanishing Constraints that provides the basis for numerical computations. We develop a a Vanishing- Constraint respecting rounding algorithm to exploit this correspondence computationally. Direct Discretization of the Optimal Control Problem with Vanishing Constraints yield a subclass of Mathematical Programs with Equilibrium Constraints. Mathematical Programs with Equilibrium Constraint constitute a class of challenging problems due to their inherent non-convexity and non-smoothness. We develop an active-set algorithm for Mathematical Programs with Equilibrium Constraints and prove global convergence to Bouligand stationary points of this algorithm under suitable technical conditions. For efficient computation of Newton-type steps of Optimal Control Problems, we establish the Generalized Lanczos Method for trust region problems in a Hilbert space context. To ensure real-time feasibility in Online Optimal Control Applications with tracking-type Lagrangian objective, we develop a Gauß-Newton preconditioner for the iterative solution method of the trust region problem. We implement the proposed methods and demonstrate their applicability and efficacy on several benchmark problems

The development of a weighted directed graph model for dynamic systems and application of Dijkstra’s algorithm to solve optimal control problems.

Master of Science (Chemical Engineering). University of KwaZulu-Natal. Durban, 2017.Optimal control problems are frequently encountered in chemical engineering process control applications as a result of the drive for more regulatory compliant, efficient and economical operation of chemical processes. Despite the significant advancements that have been made in Optimal Control Theory and the development of methods to solve this class of optimization problems, limitations in their applicability to non-linear systems inherent in chemical process unit operations still remains a challenge, particularly in determining a globally optimal solution and solutions to systems that contain state constraints. The objective of this thesis was to develop a method for modelling a chemical process based dynamic system as a graph so that an optimal control problem based on the system can be solved as a shortest path graph search problem by applying Dijkstra’s Algorithm. Dijkstra’s algorithm was selected as it is proven to be a robust and global optimal solution based algorithm for solving the shortest path graph search problem in various applications. In the developed approach, the chemical process dynamic system was modelled as a weighted directed graph and the continuous optimal control problem was reformulated as graph search problem by applying appropriate finite discretization and graph theoretic modelling techniques. The objective functional and constraints of an optimal control problem were successfully incorporated into the developed weighted directed graph model and the graph was optimized to represent the optimal transitions between the states of the dynamic system, resulting in an Optimal State Transition Graph (OST Graph). The optimal control solution for shifting the system from an initial state to every other achievable state for the dynamic system was determined by applying Dijkstra’s Algorithm to the OST Graph. The developed OST Graph-Dijkstra’s Algorithm optimal control solution approach successfully solved optimal control problems for a linear nuclear reactor system, a non-linear jacketed continuous stirred tank reactor system and a non-linear non-adiabatic batch reactor system. The optimal control solutions obtained by the developed approach were compared with solutions obtained by the variational calculus, Iterative Dynamic Programming and the globally optimal value-iteration based Dynamic Programming optimal control solution approaches. Results revealed that the developed OST Graph-Dijkstra’s Algorithm approach provided a 14.74% improvement in the optimality of the optimal control solution compared to the variational calculus solution approach, a 0.39% improvement compared to the Iterative Dynamic Programming approach and the exact same solution as the value–iteration Dynamic Programming approach. The computational runtimes for optimal control solutions determined by the OST Graph-Dijkstra’s Algorithm approach were 1 hr 58 min 33.19 s for the nuclear reactor system, 2 min 25.81s for the jacketed reactor system and 8.91s for the batch reactor system. It was concluded from this work that the proposed method is a promising approach for solving optimal control problems for chemical process-based dynamic systems

Handling Uncertainties in Process Optimization

Esta tesis doctoral presenta el estudio de técnicas que permiten manejar las incertidumbres en la optimización de procesos, desde el punto de vista del comportamiento aleatorio de las variables y de los errores en los modelos utilizados en la optimización. Para el tratamiento de las variables inciertas, se presenta la aplicación de la Programación de dos Etapas y Optimización Probabilística a un proceso de hidrodesulfuración. Los resultados permiten asegurar factibilidad en la operación, independiente del valor que tome la variable aleatoria dentro de su distribución de probabilidad. Acerca del manejo de la incertidumbre derivada del conocimiento parcial del proceso, se ha estudiado el método de Optimización en Tiempo Real con adaptación de modificadores, proponiendo mejoras que permiten: (1) evitar infactibilidades en su evolución, (2) obtener el óptimo real del proceso sin necesidad de estimar sus gradientes y (3) identificar las limitaciones para su aplicación en sistemas dinámicos de horizonteDepartamento de Ingeniería de Sistemas y Automátic

Efficient evolutionary algorithms for optimal control

If optimal control problems are solved by means of gradient based local search methods, convergence to local solutions is likely. Recently, there has been an increasing interest in the use of global optimisation algorithms to solve optimal control problems, which are expected to have local solutions. Evolutionary Algorithms (EAs) are global optimisation algorithms that have mainly been applied to solve static optimisation problems. Only rarely Evolutionary Algorithms have been used to solve optimal control problems. This may be due to the belief that their computational efficiency is insufficient to solve this type of problems. In addition, the application of Evolutionary Algorithms is a relatively young area of research. As demonstrated in this thesis, Evolutionary Algorithms exist which have significant advantages over other global optimisation methods for optimal control, while their efficiency is comparable.The purpose of this study was to investigate and search for efficient evolutionary algorithms to solve optimal control problems that are expected to have local solutions. These optimal control problems are called multi-modal. An important additional requirement for the practical application of these algorithms is that they preferably should not require any algorithm parameter tuning. Therefore algorithms with less algorithm parameters should be preferred. In addition guidelines for the choice of algorithm parameter values, and the possible development of automatic algorithm parameter adjustment strategies, are important issues.This study revealed that Differential Evolution (DE) algorithms are a class of evolutionary algorithms that do not share several theoretical and practical limitations that other Genetic Algorithms have. As a result they are significantly more efficient than other Genetic Algorithms, such as Breeder Genetic Algorithms (BGA), when applied to multi-modal optimal control problems. Their efficiency is comparable to the efficiency of Iterative Dynamic Programming (IDP), a global optimisation approach specifically designed for optimal control. Moreover the DE algorithms turned out to be significantly less sensitive to problems concerning the selection or tuning of algorithm parameters and the initialisation of the algorithm.Although it is not a DE algorithm, the GENOCOP algorithm is considered to be one of the most efficient genetic algorithms with real-valued individuals and specialized evolutionary operators. This algorithm was the starting point of our research. In Chapter 2 it was applied to some optimal control problems from chemical engineering. These problems were high dimensional, non-linear, multivariable, multi-modal and non-differentiable. Basically with GENOCOP the same solutions were obtained as with Iterative Dynamic Programming. Moreover GENOCOP is more successful in locating the global solution in comparison with other local optimisation algorithms. GENOCOP'S efficiency however is rather poor and the algorithm parameter tuning rather complicated. This motivated us to seek for more efficient evolutionary algorithms.Mathematical arguments found in the literature state that DE algorithms outperform other Evolutionary Algorithms in terms of computational efficiency. Therefore in Chapter 3, DE algorithms, generally used to solve continuous parameter optimisation problems, were used to solve two multi-modal (benchmark) optimal control problems. Also some Breeder Genetic Algorithms (BGA) were applied to solve these problems. The results obtained with these algorithms were compared to one another, and to the results obtained with IDP. The comparison confirmed that DE algorithms stand out in terms of efficiency as compared to the Breeder Genetic algorithms. Moreover, in contrast to the majority of Evolutionary Algorithms, which have many algorithm parameters that need to be selected or tuned, DE has only three algorithm parameters that have to be selected or tuned. These are the population size (µ), the crossover constant (CR) and the differential variation amplification (F). The population size plays a crucial role in solving multi-modal optimal control problems. Selecting a smaller population size enhances the computational efficiency but reduces the probability of finding the global solution. During our investigations we tried to find the best trade-off. One of the most efficient DE algorithms is denoted by DE/best/2/bin . All the investigated DE algorithms solved the two benchmark multi-modal optimal control problems properly and efficiently. The computational efficiency achieved by the DE algorithms in solving the first low multi-modal problem, was comparable to that of IDP. When applied to the second highly multi-modal problem, the computational efficiency of DE was slightly inferior to the one of IDP, after tuning of the algorithm parameters. However, the selection or tuning of the algorithm parameters for IDP is more difficult and more involved.From our investigation the following guidelines were obtained for the selection of the DE algorithm parameters. Take the population size less than or equal to two times the number of variables to be optimised that result from the control parameterisation of the original optimal control problem ( µ ≤ 2n u ). Highly multi-modal optimal control problems require a large value of the differential variation amplification ( F ≥0.9) and a very small or zero value for the crossover constant (0≤ CR ≤0.2). Low multi-modal optimal control problems need a medium value for the differential variation amplification (0.4≤ CR ≤0.6) and a large or medium value for the crossover constant (0.2≤ CR ≤0.5). In contrast to IDP, finding near-optimal values for the algorithm parameters is very simple for DE algorithms.Generally, the DE algorithm parameters are kept constant during the optimization process. A more effective and efficient algorithm may be obtained if they are adjusted on-line. In Chapter 4, a strategy that on-line adjusts the differential variation amplification ( F ) and the crossover constant ( CR ) using a measure of the diversity of the individuals in the population, was proposed. Roughly, the proposed strategy takes large values for F and small values for CR at the beginning of the optimization in order to promote a global search. When the population approaches the solution, F is decreased in order to promote a local search, and the crossover parameter CR is enlarged to increase the speed of convergence. When implemented on the DE algorithm DE/rand/1/bin and applied to the two benchmark multi-modal optimal control problems, the computational efficiency significantly improved and also the probability of locating the global solution.To judge the opportunities and advantages of using Evolutionary Algorithms to solve problems related to optimal control, in Chapter 5 several engineering applications concerning optimal greenhouse cultivation control are considered. In Chapter 5.1 genetic algorithms with binary individuals (Simple Genetic Algorithm) and floating-point representation (GENOCOP) for the individuals are used to estimate some of the parameters of a two-state dynamic model of a lettuce crop, the so-called NICOLET model. This model is intended to predict dry weight and nitrate content of lettuce at harvest time. Parameter estimation problems usually suffer from local minima. This study showed that Evolutionary Algorithms are suitable to calibrate the parameters of a dynamic model. However the required computation time is significant. Partly this is due to the high computational load of a single objective function evaluation, which for parameter optimisation problems involves a system simulation. Even though parameter optimisation is very often performed off-line, thus making computation time perhaps less important, more efficient evolutionary algorithms like DE are to be preferred.In Chapter 5.2 an optimal control problem of nitrate concentration in a lettuce crop was solved by means of two different algorithms. The ACW (Adjustable Control-variation Weight) gradient algorithm, which searches for local solutions, and the DE algorithm DE/best/2/bin that searches for a global solution. The dynamic system is a modified two-state dynamic model of a lettuce crop (NICOLET B3) and the control problem has a fixed final time and control and terminal state constraints. The DE algorithm was extended in order to deal with this.The results showed that this problem probably does not have local solutions and that the control parameterisation required by the DE algorithm causes some difficulties in accurately approximating the continuous solution obtained by the ACW algorithm. On the other hand the computational efficiency of the evolutionary algorithm turned out to be impressive. An almost natural conclusion therefore is to combine a DE algorithm with a gradient algorithm.In Chapter 5.3 the combination of a DE algorithm and a first order gradient algorithm is used to solve an optimal control problem. The DE algorithm is used to approximate the global solution sufficiently close after which the gradient algorithm can converge to it efficiently. This approach was successfully tried on the optimal control of nitrate in lettuce, which unfortunately in this case, seems to have no local solutions. Still the feasibility of this approach, which is important for all types of optimal control problems of which it is unknown a-priori whether they have local solutions, was clearly demonstrated.Finally, in Chapter six this thesis ends with an overall discussion, conclusions and suggestions for future research

Model-based approach for the plant-wide economic control of fluid catalytic cracking unit

Fluid catalytic cracking (FCC) is one of the most important processes in the petroleum refining industry for the conversion of heavy gasoil to gasoline and diesel. Furthermore, valuable gases such as ethylene, propylene and isobutylene are produced. The performance of the FCC units plays a major role on the overall economics of refinery plants. Any improvement in operation or control of FCC units will result in dramatic economic benefits. Present studies are concerned with the general behaviour of the industrial FCC plant, and have dealt with the modelling of the FCC units, which are very useful in elucidating the main characteristics of these systems for better design, operation, and control. Traditional control theory is no longer suitable for the increasingly sophisticated operating conditions and product specifications of the FCC unit. Due to the large economic benefits, these trends make the process control more challenging. There is now strong demand for advanced control strategies with higher quality to meet the challenges imposed by the growing technological and market competition. According to these highlights, the thesis objectives were to develop a new mathematical model for the FCC process, which was used to study the dynamic behaviour of the process and to demonstrate the benefits of the advanced control (particularly Model Predictive Control based on the nonlinear process model) for the FCC unit. The model describes the seven main sections of the entire FCC unit: (1) the feed and preheating system, (2) reactor, (3) regenerator, (4) air blower, (5) wet gas compressor, (6) catalyst circulation lines and (7) main fractionators. The novelty of the developed model consists in that besides the complex dynamics of the reactorregenerator system, it includes the dynamic model of the fractionator, as well as a new five lump kinetic model for the riser, which incorporates the temperature effect on the reaction kinetics; hence, it is able to predict the final production rate of the main products (gasoline and diesel), and can be used to analyze the effect of changing process conditions on the product distribution. The FCC unit model has been developed incorporating the temperature effect on reactor kinetics reference construction and operation data from an industrial unit. The resulting global model of the FCC unit is described by a complex system of partial-differential-equations, which was solved by discretising the kinetic models in the riser and regenerator on a fixed grid along the height of the units, using finite differences. The resulting model is a high order DAE, with 942 ODEs (142 from material and energy balances and 800 resulting from the discretisation of the kinetic models). The model offers the possibility of investigating the way that advanced control strategies can be implemented, while also ensuring that the operation of the unit is environmentally safe. All the investigated disturbances showed considerable influence on the products composition. Taking into account the very high volume production of an industrial FCC unit, these disturbances can have a significant economic impact. The fresh feed coke formation factor is one of the most important disturbances analysed. It shows significant effect on the process variables. The objective regarding the control of the unit has to consider not only to improve productivity by increasing the reaction temperature, but also to assure that the operation of the unit is environmentally safe, by keeping the concentration of CO in the stack gas below a certain limit. The model was used to investigate different control input-output pairing using classical controllability analysis based on relative gain array (RGA). Several multi-loop control schemes were first investigated by implementing advanced PID control using anti-windup. A tuning approach for the simultaneous tuning of multiple interacting PID controllers was proposed using a genetic algorithm based nonlinear optimisation approach. Linear model predictive control (LMPC) was investigated as a potential multi-variate control scheme applicable for the FCCU, using classical square as well as novel non-square control structures. The analysis of the LMPC control performance highlighted that although the multivariate nature of the MPC approach using manipulated and controlled outputs which satisfy controllability criteria based on RGA analysis can enhance the control performance, by decreasing the coupling between the individual low level control loops operated by the higher level MPC. However the limitations of using the linear model in the MPC scheme were also highlighted and hence a nonlinear model based predictive control scheme was developed and evaluated.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Sequential quadratic programming with indefinite Hessian approximations for nonlinear optimum experimental design for parameter estimation in differential–algebraic equations

In this thesis we develop algorithms for the numerical solution of problems from nonlinear optimum experimental design (OED) for parameter estimation in differential–algebraic equations. These OED problems can be formulated as special types of path- and control- constrained optimal control (OC) problems. The objective is to minimize a functional on the covariance matrix of the model parameters that is given by first-order sensitivities of the model equations. Additionally, the objective is nonlinearly coupled in time, which make OED problems a challenging class of OC problems. For their numerical solution, we propose a direct multiple shooting parameterization to obtain a structured nonlinear programming problem (NLP). An augmented system of nominal and variational states for the model sensitivities is parameterized on multiple shooting intervals and the objective is decoupled by means of additional variables and constraints. In the resulting NLP, we identify several structures that allow to evaluate derivatives at greatly reduced costs compared to a standard OC formulation. For the solution of the block-structured NLPs, we develop a new sequential quadratic programming (SQP) method. Therein, partitioned quasi-Newton updates are used to approximate the block-diagonal Hessian of the Lagrangian. We analyze a model problem with indefinite, block-diagonal Hessian and prove that positive definite approximations of the individual blocks prevent superlinear convergence. For an OED model problem, we show that more and more negative eigenvalues appear in the Hessian as the multiple shooting grid is refined and confirm the detrimental impact of positive definite Hessian approximations. Hence, we propose indefinite SR1 updates to guarantee fast local convergence. We develop a filter line search globalization strategy that accepts indefinite Hessians based on a new criterion derived from the proof of global convergence. BFGS updates with a scaling strategy to prevent large eigenvalues are used as fallback if the SR1 update does not promote convergence. For the solution of the arising sparse and nonconvex quadratic subproblems, a parametric active set method with inertia control within a Schur complement approach is developed. It employs a symmetric, indefinite LBL T -factorization for the large, sparse KKT matrix and maintains and updates QR-factors of a small and dense Schur complement. The new methods are complemented by two C++ implementations: muse transforms an OED or OC problem instance to a structured NLP by means of direct multiple shooting. A special feature is that fully independent grids for controls, states, path constraints, and measurements are maintained. This provides higher flexibility to adapt the NLP formulation to the characteristics of the problem at hand and facilitates comparison of different formulations in the light of the lifted Newton method. The software package blockSQP is an implementation of the new SQP method that uses a newly developed variant of the quadratic programming solver qpOASES. Numerical results are presented for a benchmark collection of OED and OC problems that show how SR1 approximations improve local convergence over BFGS. The new method is then applied to two challenging OED applications from chemical engineering. Its performance compares favorably to an available existing implementation

Strategies for Optimal Control of the Current and Rotation Profiles in the DIII-D Tokamak

The tokamak is currently the most promising device for realizing commercially-viable fusion energy production. The device uses magnetic fields to confine a circulating ring of hydrogen in the plasma state, i.e. a cloud of hydrogen ions and electrons. When sufficiently heated the hydrogen ions can overcome the electrostatic forces and fuse together, providing an overwhelmingly abundant energy source. However, stable, high-performance operation of a tokamak requires several plasma control problems to be handled simultaneously. Moreover, the complex physics which governs the tokamak plasma evolution must be studied and understood to make correct choices in controller design. In this thesis, two key control issues are studied intensely, namely the optimization and control of the plasma current profile and control of the plasma rotation (or flow).In order to maximize performance, it is preferable that tokamaks achieve advanced scenarios (AT) characterized by good plasma confinement, improved magnetohydrodynamic stability, and a largely non-inductively driven plasma current. A key element to the development of AT scenarios is the optimization of the spatial distribution of the current profile. Also, research has shown that the plasma rotation can stabilize the tokamak plasma against degradations in the desired MHD equilibrium.In this thesis, new model-based control approaches for the current profile and rotation profile are developed to allow experimental exploration of advanced tokamak scenarios. Methods for separate control of both the current profile and rotation are developed. The advanced model-based control methods presented in this thesis have contributed to theory of tokamak profile control and in some cases they have been successfully validated experimentally in the DIII-D tokamak