Optimal Control theory

The status of concepts and techniques for the design of onboard flight path management systems is reviewed. Such systems are designed to increase flight efficiency and safety by automating the optimization of flight procedures onboard aircraft. After a brief review of the origins and functions of such systems, two complementary methods are described for attacking the key design problem, namely, the synthesis of efficient trajectories. One method optimizes en route, the other optimizes terminal area flight; both methods are rooted in optimal control theory. Simulation and flight test results are reviewed to illustrate the potential of these systems for fuel and cost savings.

... Calculus of variations and optimal control theory. Post a Comment. CONTRIBUTORS: Author: Hestenes, Magnus R. (b. 1906, d. ----. ... VOLUME/EDITION: PAGES (INTRO/BODY): xii, 405 p. SUBJECT(S): Calculus of variations; Control theory. DISCIPLINE: No discipline assigned. ...

A direct transcription method transforms an optimal control problem (OCP) into a nonlinear programming problem (NLP). The resulting NLP can be solved by any NLP solver, such as the Matlab's optimization toolbox, the fsqp, etc. On solving optimization problems using Matlab, the Matlab's optimization toolbox does not obtain an accurate Hessian matrix at the optimal solution due to the fact that the Hessian matrix is not being evaluated directly from the optimal solution. In this paper we compute the condition numbers associated with the optimal control computation, where the classical fourth-order Runge-Kutta method is used for the discretization of the state equations. The computations of optimal solutions are done for different numbers of switching points and quadrature points per a switching interval. A test example showed that the condition numbers of the active constraints, projected Hessian and the whole Lagrangian system are more likely to grow with the number of the switching intervals per a delay interval than by the number of the quadrature intervals per a switching interval. Also, the three medium scale optimization algorithm of the Matlab's optimization toolbox give almost similar condition numbers when used to solve an optimal control problem.

This paper studies a nonlinear SI-model featuring vertical transmission and saturated incidence terms. The model is analyzed by utilizing some dynamical system tools to assess the dynamic behavior of solutions to a system of first order nonlinear ordinary differential equations arising from the spread of a vertically transmitted disease in a population. The mathematical well-posedness of the governing model is proven through positivity and boundedness properties of solutions. Lyapunov functions with LaSalle's invariance principle are used to establish the global asymptotic dynamics of the system about the disease-free and endemic steady states. Additionally, sensitivity analysis is conducted to show how the epidemiological threshold (basic reproduction number) of the model is affected by variations in parameters. The study is extended to explore the significance of a time-dependent optimal control for blocking vertical transmission route using Pontryagin's maximum principle of the optimal control theory. The obtained qualitative results are supplemented by numerical simulations. It is shown that incorporation of the optimal control helps in preventing vertical transmission of infection, and in turn diminishes the spread of the disease in the population.

The field of Preview Control is concerned with using advanced knowledge of disturbances or references in order to improve tracking quality or disturbance rejection. Areas of application include guidance of autonomous vehicles, robotics and process control. This thesis studies Optimal Discrete-Time Preview Control, in which the controlled plant is assumed to be unconstrained,linear and discrete-time, with a preview horizon that extends for a fixed time into the future.Controller optimality is determined using either the H2 or H∞ norms.The central objective of this thesis is to produce a generic set of tools for solving a wide class of H2 and H∞Preview Control problems. To this end, a very general preview problem is introduced, which captures fixed-lag smoothing, tracking with robust performance specifications,and also preview-based mixed-sensitivity design. In principle, whilst this problem can be solved using standard generalised regulator synthesis results, such an approach is found to require the solution of unacceptably large Discrete Algebraic Riccati Equations (DAREs). A core theme of this thesis involves the efficient solution of the H2 and H∞versions of such DAREs, and also the order reduction of the resulting controllers. It is found that the dimension of all matrix equations required for controller synthesis, and also the order of the resulting controllers, is no higher than the degree of the plant plus weighting functions. Additional results are presented concerning the norm reduction that is achievable through preview action, and a discussion of the generic properties of preview controllers is provided.Preview Control appears to be well suited for application to autonomous vehicles. However,a class of nonlinearities generically associated with vehicle models prevents its immediate use.Fortunately, this nonlinearity may be accurately represented by a parameterised set of linear systems, from which a simple controller may be derived. This control scheme is successfully demonstrated on a nonlinear bicycle model.

An inverted pendulum on cart is an object
which is a nonlinear, unstable system, is used as a
standard for designing the control methods and finds
most versatile application in the field of control theory.
To achieve the stabilization of an inverted pendulum
system state observer based linear quadratic Gaussian
optimal control has been applied. As the separation
principal of LQG states, first the control law has been
obtained to design a state feedback controller and when
the system’s all states are not measurable as well as
system is affected by process and measurement noise,
Kalman Filter has been designed for the inverted
pendulum system. The results shown that the new
designed controller stabilizes the inverted pendulum
system as well as eliminates process and measurement
noise. Simulation has been carried out to show the
approach.

Unstable wheel inverted pendulum is modelled and controlled deploying Kane's method and optimal partial-state PID control scheme. A correct derivation of nonlinear mathematical model of a wheel inverted pendulum is obtained using a proper definition of the geometric context of active and inertia forces. Then the model is decoupled to two linear subsystems namely balancing and heading subsystems. Afterward partial-state PID controller is proposed and formulated to quadratic optimal regulation tuning method. It enables partial-state PID to be optimally tuned and guarantees a satisfactory level of states error and a realistic utilization of torque energy. Simulation and numerical analyses are carried out to analyse system's stability and to determine the performance of the proposed controller for mobile wheel inverted pendulum application.

Weather routing of ships is used to establish the shortest time route or the most economical route from a departure to arrival point by applying available information of the weather condition viz. wind, wave and current. Information on ship speed loss due to these effects is pre-computed using sea-keeping computing tools, which are then suitably employed in the optimum ship routing algorithm. This developed algorithm was investigated using the wave height information from GEOSAT altimeter records. Dijkstra’s path optimization scheme, which employs optimal control theory and dynamic programming technique, is used to obtain reliable optimum route in a given random sea-state.

We examine an optimal way of eradicating rabies transmission from dogs into the human population, using preexposure prophylaxis (vaccination) and postexposure prophylaxis (treatment) due to public education. We obtain the disease-free equilibrium, the endemic equilibrium, the stability, and the sensitivity analysis of the optimal control model. Using the Latin hypercube sampling (LHS), the forward-backward sweep scheme and the fourth-order Range-Kutta numerical method predict that the global alliance for rabies control’s aim of working to eliminate deaths fromcanine rabies by 2030 is attainable through mass vaccination of susceptible dogs and continuous use of pre- and postexposure prophylaxis in humans.

— This article presents a complete formulation of the challenging task of stable humanoid robot omnidirectional walk based on the Cart and Table model for approximating the robot dynamics. For the control task, we propose two novel approaches: preview control augmented with the inverse system for negotiating strong disturbances and uneven terrain and linear model-predictive control approximated by an orthonormal basis for computational efficiency coupled with constraints for stability. For the generation of smooth feet trajectories, we present a new approach based on rigid body interpolation, enhanced by adaptive step correction. Finally, we present a sensor fusion approach for sensor-based state estimation and an effective solution to sensors' noise, delay, and bias issues, as well as to errors induced by the simplified dynamics and actuation imperfections. Our formulation is applied on a real NAO humanoid robot, where it achieves real-time onboard execution and yields smooth and stable gaits.

We have previously tried to explain perceptual inference and learning under a free-energy principle that pursues Helmholtz’s agenda to understand the brain in terms of energy minimization. It is fairly easy to show that making inferences about the causes of sensory data can be cast as the minimization of a free-energy bound on the likelihood of sensory inputs, given an internal model of how they were caused. In this article, we consider what would happen if the data themselves were sampled to minimize this bound. It transpires that the ensuing active sampling or inference is mandated by ergodic arguments based on the very existence of adaptive agents. Furthermore, it accounts for many aspects of motor behavior; from retinal stabilization to goal-seeking. In particular, it suggests that motor control can be understood as fulfilling prior expectations about proprioceptive sensations. This formulation can explain why adaptive behavior emerges in biological agents and suggests a simple alternative to optimal control theory. We illustrate these points using simulations of oculomotor control and then apply to same principles to cued and goal-directed movements. In short, the free-energy formulation may provide an alternative perspective on the motor control that places it in an intimate relationship with perception.

An eco-epidemiological model representing the interactions between prey and predator populations affected by a disease in an ecosystem is presented. The model is governed by a five-dimensional nonlinear system of ordinary differential equations coupling both ecological and epidemiological features of interacting populations. The well-posedness of the model is established with respect to positivity and boundedness of solutions. Conditions for asymptotic stability of different equilibrium points are extensively investigated to determine the existence and coexistence of prey and predator species using local linearization and Lyapunov functions techniques. Additionally, the analysis of the model is extended to assess the effects of three timedependent control functions, such as disease prevention, treatment and alternative resource for predator, on the population dynamics of the prey-predator coexistence in the system.

The paper describes the design of an optimal controller, based on output feedback, to provide integrated control of the exciter and prime mover of a micro^synchronous-generator. Modelling of the control input-output measurement dynamics of the power system by an output prediction equation using least-squares analysis, is followed by multivariable control-system design based on discrete dynamic programming to minimise a quadratic performance index. This approach avoids any reference to the system state vector, which may contain states which are difficult to measure, and thus any need for state estimation techniques or the omission of unmeasurable states from the state feedback control law. The paper evaluates the effectiveness of this controller over a wide range of operating conditions, including dynamic and transient stability boundaries and 3-phase faults, with and without permanent line outage.

We present an application of optimal control theory to Dengue epidemics. This epidemiologic disease is important in tropical countries due to the growing number of infected individuals. The dynamic model is described by a set of nonlinear ordinary differential equations, that depend on the dynamics of the Dengue mosquito, the number of infected individuals, and people's motivation to combat the mosquito. The cost functional depends not only on the costs of medical treatment of the infected people but also on the ...