Further Study of an Optimum Non-linear Control System†

Global optimal feedback control for general nonlinear systems with nonquadratic performance criteria

A geometric approach to time-dependent optimal control problems is proposed. This formulation is based on the Skinner and Rusk formalism for Lagrangian and Hamiltonian systems. The corresponding unified formalism developed for optimal control systems allows us to formulate geometrically the necessary conditions given by a weak form of Pontryagin's maximum principle, provided that the differentiability with respect to controls is assumed and the space of controls is open. Furthermore, our method is also valid for implicit optimal control systems and, in particular, for the so-called descriptor systems (optimal control problems including both differential and algebraic equations).

The authors propose a method by which to design an optimum vibration control system composed of continuous body structures using modal analysis and sensitivity analysis. The mechanical structures are modeled first by FEM. This spatial model is of such a high degree-of-freedom (DOF) that a control system cannot be designed. This spatial model is transformed into a modal model in order to reduce DOF. The optimum control system is designed by applying the optimum control theory to this reduced model. However, stability of the control system is not always guaranteed because the modal model ignores high-order modes. An optimization method of the control system which guarantees stability of the actual system is proposed in this paper. We propose also an experimental optimization of the control system based on experimental modal analysis. The optimization is performed using sensitivity analysis and nonlinear programming. The effectiveness of the proposed method is verified by both simulation and experiment.

The paper deals with the implementation of optimal controller design for linear dynamic systems. The implementation is carried out with the use of quadratic cost function for both cases of continuous and discrete systems. The mathematical optimal control theory is applied to dynamical systems with considering the form of either vector differential equations in continuous time case, or difference equations in discrete time- or sampled data-case. The bulk of the work provides design techniques for synthesizing the optimal control structure by applying the Pontryagin's Maxi mum principle to linear systems with using Matrix Riccati Equation. The work is focused upon regulator problems; that is problems where the goal is to maintain the system states at zero level. The implemented techniques are tested by the application on fourth order practical example represents the linearized control model of two-reach river pollution System. Also comparative study between the continuous and discrete optimal regulator problems will be presented and discussed from a numerical point-of-view.

In this study, the problem of finding an optimal controller for nonlinear systems with one input and a reference tracking signal is approached. With the problem's formulation, any desired signal can be tracked instantly with a closed-loop controller without the need for integral terms. Presentation lies at the heart of optimal control. This study, however, does not consider the integral term, allowing tracking and stability to occur naturally. It has a broad scope with a wide range of applications, namely when dealing with affine nonlinear systems, which provide geometric control unification with asymptotic stability in some cases. A common scenario that comes from optimal control involves the minimization of integral cost functionals. Issues like asymptotic stability or even tracking to the desired reference signal have always been the main limitations. In this study, the main theorem allows the solution of optimal control problems with no-integral terms, in other words tracking problems with input/state constraints, providing closed-loop controllers. A DC motor with a pendulum in upright position is an example of an application for which singular optimal control is tested in this study. The results confirm both asymptotic stability and optimal tracking with an accuracy of 95%. The main contributions of this study include an optimal closed-loop controller with no mixed initial/final conditions, input/state constraints, asymptotic stability guarantee, a strong connection with geometric tools and finally the possibility to generalize to systems with multiple inputs. As a conclusion, general nonlinear control systems can be included in the optimal control methodology presented in this study including input/state constraints. Due to the lack of integral terms, the problem can be solved in closed form by using an optimal closed-loop controller.

Optimal control of interconnected systems with time-correlated noises: Application to vehicle platoon

Classical control methods such as Pontryagin Maximum Principle and Bang-Bang Principle and other methods are not usually useful for solving opti-mal control systems (OCS) specially optimal control of nonlinear systems (OCNS). In this paper, we introduce a new approach for solving OCNS by using some combination of atomic measures. We define a criterion for controllability of lumped nonlinear control systems and when the system is nearly null controllable, we determine controls and states. Finally we use this criterion to solve some numerical examples.

The fundamental principle of intertemporal optimization: Consumer behavior under recursive preferences

Pontryagin's Maximum Principle has been applied for optimization of secreted proteins from Pichia pastoris fed-batch fermentation. The objective of this work is to maximize the total accumulated product per unit operation time under different given conditions and system constraints. To obtain optimal solutions, an automated curve-fitting software, Table Curve 2D, was employed to construct the necessary mathematical models and solve the complicated functions. In the solution processes, the end of the glycerol batch phase was defined as the initial state of the system, the end of the methanol fed-batch phase as the final state, the cell mass produced along with product accumulated as state variables, and the specific growth rate (mu) as the control variable. Initially, a relationship between the specific production rate (rho) and mu was established. Then, according to Pontryagin's Maximum Principle, the admissible range of mu and its trajectories for the optimal operations were determined. Four representative cases with different combinations of the operation time along with the initial and final states were evaluated. A close correlation was obtained between the predicted values of the model equation with the experimental results from the Pichia pastoris fed-batch fermentations producing secreted alpha-galactosidase. The approaches proposed here greatly simplify the computational processes and validate the optimization strategy as a generalized approach to maximize the yield from fed-batch fermentations.

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.

This paper aims to apply an optimal control theory for the autonomous model of the leptospirosis epidemic to examine the effect of four time-dependent control measures on the model dynamics with cost-effectiveness. Pontryagin's Maximum Principle was used to derive the optimality system associated with the optimal control problem. Numerical simulations of the optimality system were performed for different control strategies and the results were presented graphically with and without controls. The optimality system was simulated using the Forward–Backward Sweep method in the Matlab programme. The numerical results revealed that the combination of all optimal control measures is the most effective strategy for minimizing the spread and impact of disease in the community. Furthermore, a cost-effectiveness analysis was performed to determine the most cost-effective strategy using the incremental cost-effectiveness ratio approach and we observed that the rodenticide control-only strategy is most effective to combat the spread of disease when available resources are limited.

8 - Reactors With Catalyst Decay and Regeneration

SummaryOur prime objective of the study is to exhibit the advantage to introduce a quadratic control in place of linear control in a cost function to be minimized, and that is associated to an optimal control problem that we formulate for a pre‐validated model of bacillus Calmette‐Guérin (BCG) immunotherapy in superficial bladder cancer. The compartmental model of interest is in the form of a nonlinear system of four ordinary differential equations that describe interactions between the used BCG strain, tumor cells, and immune responses. Previous studies reported that the optimal dose of BCG for treating bladder cancer is yet unknown. Hence, we aim to establish the optimization approach that can be applied for determining the values of the optimal BCG concentration along the therapy period to stimulate immune‐system cells and reduce cancer cells growth during BCG intravesical therapy. Pontryagin's maximum principle and the generalized Legendre–Clebsch condition are employed to provide the explicit formulations of the sought optimal controls. The optimality system is resolved numerically based on a fourth‐order iterative Runge–Kutta progressive‐regressive scheme, which is used to solve a two‐point boundary value problem. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, we present a global optimal and stable fuzzy controller design method for both continuous- and discrete-time fuzzy systems under both finite and infinite horizons. First, a sufficient condition is proposed which indicates that the global optimal effect can be achieved by the fuzzily combined local optimal controllers. Based on this sufficient condition, we derive a local concept approach to designing the optimal fuzzy controller by applying traditional linear optimal control theory. The stability of the entire closed-loop continuous fuzzy system can be ensured by the designed optimal fuzzy controller. The optimal feedback continuous fuzzy system can not only be guaranteed to be exponentially stable, but also be stabilized to any desired degree. Also, the total energy of system output is absolutely finite. Moreover, the resultant feedback continuous fuzzy system possesses an infinite gain margin; that is, its stability is guaranteed no matter how large the feedback gain becomes. Two examples are given to illustrate the proposed optimal fuzzy controller design approach and to demonstrate the proved stability properties.

Optimal Protection from Impact, Shock, and Vibration