Estimation of time varying parameters in nonlinear systems by using dynamic optimization

Unlike steady-state optimization and linear model predictive control, dynamic optimization and nonlinear model predictive control is underway to industry application. This paper discusses the implementation issues of two-level control strategy integrated dynamic real-time optimization and nonlinear model predictive control of a heat-integrated distillation column system. To realize this scheme, four nonlinear programming problems: economic operation optimization, data reconciliation and state estimation, parameter estimation and nonlinear model predictive control, are considered based on system rigorous dynamic model. The economic operation optimization is to minimize the cost of the plant. The unmeasured states are estimated by measurements of temperatures, pressures and flow rates. Time-variant parameters such as tray efficiencies and feed composition are updated on-line. The sequential quadratic programming and orthogonal collocation are used to solve the dynamic nonlinear optimization problems.

This thesis aims at developing efficient mathematical methods for solving hierarchical dynamic optimization problems. The main motivation is to model processes in nature, for which there is evidence to assume that they run optimally. We describe models of such processes by optimal control problems (called optimal control models (OCMs)). However, an OCM typically includes unknown parameters that cannot be derived entirely on a theoretical basis, which is in particular the case for the cost function. Therefore, we develop parameter estimation techniques to estimate the unknowns in an OCM from observation data of the process. Mathematically, this leads to a hierarchical dynamic optimization problem with a parameter estimation problem on the upper level and an optimal control problem on the lower level. We focus on multi-stage equality and inequality constrained optimal control problems based on nonlinear ordinary differential equations. The main goal of this thesis is to derive numerically efficient mathematical methods for solving hierarchical dynamic optimization problems, and to use these methods to estimate parameters in high-dimensional OCMs from real-world measurement data. We develop parameter-dependent OCMs for the gait of cerebral palsy patients and able-bodied subjects. The unknown parameters in the OCMs are then estimated from real-world motion capture data provided by the Heidelberg MotionLab of the Orthopedic University Clinic Heidelberg by using the mathematical methods developed within this work. The main novelties and contributions of this thesis to the field of hierarchical dynamic optimization are summarized herein. - We establish a novel mathematical method, a so-called direct all-at-once approach, for solving hierarchical dynamic optimization problems based on the direct multiple shooting method and first-order optimality conditions. - Furthermore, we propose an efficient numerical algorithm for large-scale hierarchical dynamic optimization problems, which fully exploits the structures inherited from both the hierarchical setting and the discretization. - Pontryagin's maximum principle is used to analyze solution properties of hierarchical dynamic optimization problems like second-order optimality conditions of the lower-level problem. - In addition, we propose and discuss alternative methods for hierarchical dynamic optimization that are based on derivative-free optimization and a bundle approach. These methods keep the hierarchical problem setting and do not reformulate the lower-level problem using first-order optimality conditions. - We establish a novel lifting method for regularizing mathematical programs with complementarity constraints, which is discussed and numerically investigated by means of a well-known collection of benchmark problems. - Proofs of regularity and convergence results for sequential quadratic programming methods applied to lifted mathematical programs with complementarity constraints are provided. - Efficient state-of-the-art implementations of all mathematical methods derived in this thesis, as well as a benchmark collection of hierarchical dynamic optimization problems are presented. - High-dimensional optimal control gait models for cerebral palsy patients and able-bodied subjects are developed. The mathematical methods derived in this thesis are used to estimate the unknown model parameters from real-world motion capture data provided by the Heidelberg MotionLab of the Orthopedic University Clinic Heidelberg. The theoretical and practical results presented in this thesis can be considered an initial motivating step towards answering open questions in current medical research in fields like treatment planning, classification of gaits, or the evaluation of surgeries by means of hierarchical dynamic optimization.

Aiming at the problems such as low computational efficiency and long process in dynamic topology optimization, combined with bi-directional evolutional structure optimization, the dynamic optimization problem is transformed into the static optimization problem in multiple load cases. Newmark method is used to analyze the structure dynamics load. The topology optimization model based on equivalent static load is established. On the basis of the equivalent static load, multi-objective dynamics evolutionary structure optimization is studied. And the mathematical model of the dynamic stiffness and natural frequency about the multi-objective optimization is established. The mathematical formula of the sensitivity is deduced. Then a complete optimization iterative process is given. The numerical example illustrates that it is feasible to solve the multi-objective dynamic topology optimization with equivalent static loads. Modal tracking technique is introduced to the multi-objective dynamic topology optimization. The numerical example shows that modal tracking technology can track the required modal in the optimization process.

In silico experimentation has opened new ways to analyze biological systems behavior under different conditions. The incorporation of an outer optimization loop may help to find the right operation conditions to achieve specific goals (maximization of a given product concentration, minimization of process energy/time, etc.). Mathematically, this is stated as a dynamic optimization problem being particularly challenging when the system is described by nonlinear sets of partial differential equations as well as when constraints are considered. These issues impose a number of difficulties, such as the presence of suboptimal solutions, which call for robust and efficient numerical techniques. In this work, the control vector parametrization approach is combined with reduced order methods and suitable hybrid global optimization methods to overcome such difficulties. The capabilities of this strategy are illustrated considering the solution of two challenging problems: bacterial chemotaxis and the FitzHugh-Nagumo model. The presented methodology can be used for the efficient dynamic optimization of generic distributed biological systems.

Dynamic multiobjective optimization problem (DMOP) is a class of complex dynamic optimization problem (DOP), its widely exists in many real-world problems. Firstly, an core estimation of distribution model which used to approximately estimate the Pareto solution in the next environment is given, when a change in the environment is detected, the algorithm uses the collected information from the previous searching environments to predict the location of the Pareto solution in the next environment. Furthermore, a dynamic multiobjective optimization evolutionary algorithm with core estimation of distribution is proposed. The computer simulations are made on two dynamic multiobjective optimization problems, and the results indicate the proposed algorithm is effective.

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> High purity distillation columns are critical unit operations in cryogenic air separation plants that supply purified gases to a number of industries. We have developed a nonlinear model predictive control (NMPC) strategy based on the assumption of full-state feedback for a prototypical cryogenic distillation column to allow effective operation over a wide range of plant production rates. The controller design was based on a reduced-order compartmental model derived from detailed mass and energy balances by exploiting time-scale separations. Temporal discretization of the compartmental model produced a very large set of nonlinear differential and algebraic equations with advantageous sparsity properties, enabling online solution of the NMPC problem. The synergistic combination of several real-time implementation techniques were found to be essential for further reducing computation time and allowing reliable solution within the 2-min controller sampling interval. Closed-loop simulation studies demonstrated the performance advantages of NMPC compared to linear model predictive control technology currently used in the air separation industry. </para>

This paper aimed to apply artificial neural network proportional, plus integral, PI controlled superconducting magnetic energy storage SMES to improve the transient stability of power systems. The PI controller parameters is firstly determined based on eigenvalue assignment approach. The artificial neural network, ANN is used to determine the optimum gains of the PI controller at different load values. The ANN is trained off line using Matlab software to obtain the optimum parameters of the PI controller. The speed deviation, Deltaomega and load angle deviation Deltadelta are used as input signal to the PI controller. The studied power system consists of single machine connected to an infinite bus via double transmission lines. The studied system is modeled by a set of nonlinear differential and algebraic equations and simulated by the Matlab software. The simulation results indicates the effect of the proposed ANN PI controlled SMES.

In this study, a single-degree-of-freedom structure with multiple nonlinear tuned mass dampers is investigated. A sinusoidal excitation is assumed to be applied to the structure. In order to obtain the results, a nonlinear mathematical model of the general system is obtained in the form of a set of nonlinear ordinary differential equations. Dry friction and cubic stiffness nonlinearities are considered for the nonlinear terms. To obtain steady-state frequency responses of the system, these nonlinear ordinary differential equations are transformed into a set of nonlinear algebraic equations by using Harmonic Balance Method. Fourier transformation is used in the calculation of the Fourier coefficients of the nonlinear forcing terms. Then, numerical solutions to the resulting nonlinear algebraic equations are obtained using Newton’s Method with Arc Length Continuation. The effect of the number of tuned mass dampers and their arrangement, i.e. in series or in parallel, on vibration reduction is observed by performing optimization on several cases and comparing them. In order to observe the separate and combined effects of the nonlinearities, and the number of tuned mass dampers used, several case studies are carried out using optimized case parameters with different combinations of the nonlinearities. The results of these case studies and the linear model are compared with each other to see the effect of the nonlinearities. Additionally, results of the system with optimum nonlinear tuned mass dampers are compared with that of the optimum linear tuned mass dampers and the effect of nonlinearities are discussed.

ABSTRACTThis article considers a dynamic optimization problem arising in a one-dimensional magnetohydrodynamic flow, which can be modelled by coupled partial differential equations, where the external control input (external induction of magnetic field) takes a multiplicative effect on the system states (momentum and magnetic components). The aim is to drive the flow velocity to within close proximity of a desired target of flow velocity at the pre-indicated terminal time. First, the Galerkin method is utilized to reduce the original dynamic optimization problem to a lower finite-dimensional dynamic optimization problem governed by a set of ordinary differential equations. Then the control parameterization method is employed to parameterize the finite-dimensional optimization problem and thus obtain an approximate optimal parameter selection problem, which can be solved using gradient-based optimization techniques, such as sequential quadratic programming. The exact gradients of the cost functional with respect to the decision parameters, which are the key advantages of this approach, are computed using the analytical equations. Finally, numerical examples are illustrated to verify the effectiveness of the proposed method. The novel scheme of this article is to develop an effective computational optimal control method for realizing the optimal tracking control of flow velocity in a one-dimensional magnetohydrodynamic flow.

Wind-resistant design of tall buildings has been traditionally treated using the equivalent static load approach. In order to account for the uncertainties in random wind excitation, it is necessary to develop a comprehensive and reliable dynamic optimisation technique in the time domain. The optimal lateral stiffness design problem of wind-excited tall buildings consists of (1) identifying the critical dynamic drift response in the time domain and (2) searching for the optimal distribution of element stiffness of the building subject to multiple drift design constraints. The critical time-history drift constraints of a wind-excited building are first treated by the worst-case formulation and then explicitly expressed in terms of element sizing variables using the principle of virtual work. The extreme value distribution and the Gaussian assumption are employed to formulate and simplify the probabilistic drift constraints, which are explicitly considered in the dynamic optimisation problem. The system reliability associated with the inter-story drift is estimated approximately by the bound approach to ensure that the most cost-efficient solution also attains an acceptable reliability level. A full-scale 45-story building example under wind tunnel derived time history wind loading is presented to illustrate the effectiveness and practicality of the reliability-based dynamic optimisation technique.

In this paper, a real-time implementation of selective harmonic elimination pulsewidth modulation (SHEPWM) using generalized Hopfield neural network (GHNN) as applied to the single-phase inverter is designed and discussed. Finding the location of the switching instants in the case of SHEPWM involves the solution of a set of nonlinear algebraic transcendental equations. The problem is redrafted as an optimization problem, and it is solved by using GHNN. Though, in principle, then number of harmonics can be eliminated using the presented methodology, this paper tries for the case of the elimination of the 5th-, 7th-, 11th-, and 13th-order harmonics while retaining the desired fundamental. An energy function is formulated for this problem and the set of ordinary differential equations (ODEs) describing the behavior of GHNN is obtained. A program in KEIL C was developed to solve these systems of ODEs by the Runge-Kutta fourth-order method to give the switching instants for continuously varying modulation indexes (M). A MATLAB simulation was carried out, and an experimental setup was also constructed in order to validate the simulated results. The fast Fourier transform analysis of the simulated output voltage waveform and the experimental output voltage waveform confirms the effectiveness of the proposed method. Hence, the proposed method proves that it is much applicable in the industrial applications by virtue of its suitability in real-time applications.

We present modeling and analysis for the static behavior and collapse instabilities of doubly-clamped and cantilever microbeams subjected to capillary forces. These forces can be as a result of a volume of liquid trapped underneath the microbeam during the rinsing and drying process in fabrication. The model considers the microbeam as a continuous medium, the capillary force as a nonlinear function of displacement, and accounts for the mid-plane stretching nonlinearity. The capillary force is assumed to be distributed over a specific length underneath the microbeam. The Galerkin procedure is used to derive a reduced-order model consisting of a set of nonlinear algebraic and differential equations that describe the microbeams static and dynamic behaviors. We study the collapse instability, which brings the microbeam from its unstuck configuration to touch the substrate and gets stuck in the so-called pinned configuration. We calculate the pull-in length that distinguishes the free from the pinned configurations as a function of the beam thickness and gap width for both microbeams. Comparisons are made with analytical results reported in the literature based on the Ritz method for linear and nonlinear beam models. The instability problem, which brings the microbeam from a pinned to adhered configuration is also investigated. For this case, we use a shooting technique to solve the boundary-value problem governing the deflection of the microbeams. The critical microbeam length for this second instability is also calculated.

General linear continuous stochastic systems are considered with multiplicative noise in the control and state channels and stabilizing optimal inputs are synthesized for both time-varying and time-invariant situations. In the nonstationary case, a set of nonlinear matrix differential equations has to be solved. In the stationary problem, a set of nonlinear matrix algebraic equations provides the solution. The separation principle is not valid and both the state filtering and the control synthesis problem should be treated simultaneously.

It is important to design and operate energy conversion systems such as gas turbine cogeneration ones optimally from the thermoeconomic viewpoint. However, an energy conversion system has a complex network structure, and it takes much time to create its model for the thermoeconomic analysis and optimization. In this paper, a systems approach is presented for the performance analysis and optimization of mechanical systems with network structures, and it is applied to the thermoeconomic analysis and optimization of a gas turbine cogeneration unit. The system modeling for the performance analysis is conducted by a building block approach. Static and dynamic problems for the performance analysis are formulated as sets of nonlinear algebraic and differential algebraic equations, and are solved by the Newton-Raphson method and a hierarchical combination of the Runge-Kutta and Newton-Raphson methods, respectively. The performance optimization is conducted to determine design and operation conditions which optimize performance criteria. This problem is formulated as a nonlinear programming one and is solved by a global optimization method. In the application, the cycle analysis is conducted to determine mass flow rates, pressures, and temperatures, which is followed by the exergy and cost analyses to determine exergy flow rates and efficiencies, and capital costs, respectively. In addition, design and operation conditions are determined to maximize the exergy efficiency or minimize the annual total cost based on the results of the cycle, exergy, and cost analyses. Through a numerical study, it turns out that the proposed systems approach enables one to conduct the thermoeconomic analysis and optimization efficiently.

Dynamic multi-objective optimization problems (DMOPs) have become a research hotspot in engineering optimization, because their objective functions, constraints, or parameters may change over time, while quickly and accurately tracking the changing Pareto optimal set (POS) during the optimization process. Therefore, solving dynamic multi-objective optimization problems presents great challenges. In recent years, transfer learning has been proved to be one of the effective means to solve dynamic multi-objective optimization problems. However, this paper proposes a new transfer learning method based on clustering difference to solve DMOPs (TCD-DMOEA). Different from the existing methods, it uses the clustering difference strategy to optimize the population quality and reduce the data difference between the target domain and the source domain. On this basis, transfer learning technology is used to accelerate the construction of initialization population. The advantage of the TCD-DMOEA method is that it reduces the possibility of negative transfer and improves the performance of the algorithm by improving the similarity between the source domain and the target domain. Experimental results show that compared with several advanced dynamic multi-objective optimization algorithms based on different benchmark problems, the proposed TCD-DMOEA method can significantly improve the quality of the solution and the convergence speed.