Optimal Control theory

A multigrid and sparse-grid computational approach to solving nonlinear elliptic optimal control problems with random coefficients is presented. The proposed scheme combines multigrid methods with sparse-grids collocation techniques. Within this framework the influence of randomness of problem's coefficients on the control provided by the optimal control theory is investigated. Numerical results of computation of stochastic optimal control solutions and formulation of mean control functions are presented.

Practical requirements on the design of control systems, especially process control systems, are usually specified in terms of time-domain response, such as overshoot and rise time, or frequency-domain response, such as resonance peak and stability margin. Although numerous methods have been developed for the design of the proportional-integralderivative ͑PID͒ controller, little work has been done in relation to the quantitative time-domain and frequency-domain responses. In this paper, we study the following problem: Given a nominal stable process with time delay, we design a suboptimal PID controller to achieve the required time-domain response or frequency-domain response for the nominal system or the uncertain system. An H ϱ PID controller is developed based on optimal control theory and the parameters are derived analytically. Its properties are investigated and compared with that of two developed suboptimal controllers: an H 2 PID controller and a Maclaurin PID controller. It is shown that all three controllers can provide the quantitative time-domain and frequency-domain responses.

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.

A new optimal control law for direct yaw moment control, to improve the vehicle handling, is developed. Although, this can be considered as part of a multi-layer system, for the traction control of a motorized wheels electric vehicle, but the results of this study are quite general and can be applied to other types of vehicles. The dynamic model of the vehicle system is initially developed and, using the well-known optimal control theory, an optimal controller is designed. Two different versions of control laws are considered here and the performance of each version of the control law is compared with the other one. The numerical simulation of the vehicle handling with and without the use of the optimal yaw moment controller, and assuming a comprehensive non-linear vehicle dynamic model, has been carried out. Simulation results obtained indicate that considerable improvements in the vehicle handling can be achieved whenever the vehicle is governed by the optimal yaw moment control.

... 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. ...

This paper records the development of a state variable model of the megawatt-frequency control problem of multiarea electric energy systems. The model is in a mathematical form necessary for application of theorems of modem optimal control theory.

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.

The ultimate goal of this paper is to control the angular speed ω , in a model of a DC motor driving an inertial load has the angular speed, ω , as the output and applied voltage, app v , as the input, by varying the applied voltage using different control strategies for comparison purpose. The comparison is made between the proportional controller, integral controller, proportional and integral controller, phase lag compensator, derivative controller, lead integral compensator, lead lag compensator, PID controller and the linear quadratic tracker design based on the optimal control theory. It has been realized that the design based on the linear quadratic tracker will give the best steady state and transient system behavior, mainly because, the other compensator designs are mostly based on trial and error while the linear quadratic tracker design is based on the optimal control theory which can give best dynamic performance for the controlled system.

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.

Control strategies for hybrid powertrains are algorithms that choose the power split between the engine and motor of a hybrid vehicle in order to minimize the fuel consumption and/or emissions. The goal of this paper is to propose an efficient tool to evaluate minimal fuel consumption that is achievable in simulation. Several approaches have been proposed, using heuristics (Delprat et al. 1999) or dynamic programming . One drawback of these approaches is the huge amount of time required to obtain solutions. The approach described here is based on optimal control theory (Lewis & Syrmos, 1995) and avoids this drawback. Moreover, it can be easily applied to a large family of parallel arrangements.

Ahtract-This paper is concerned with the problem of estimating the state of a linear dynamic system using noise-corrupted observations, when input disturbances and observation errors are onknown except for the fact that they belong to given bounded sets. The cases of both energy constraints and individual instantaneous constraints for the uncertain quantities are considered. In the former case, the set of possible system states compatible with the observations received is shown to be an ellipsoid, and equations for its center and weighting matrix are given, while in the latter case, equations describing a boonding ellipsoid to the set of possible states are derived. All three problems of filtering, prediction, and smoothing are examined by relating them to standard tracking problems of optimal control theory. The resulting estimators are similar in strnctnre and comparable in simplicity to the corresponding stochastic linear minimum-variance estimators, and it is shown that they provide distinct advantages over existing schemes for recursive estimation with a set-membership description of uncertainty.

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.

V. [Risk intermediation in supply chains. IIE Transactions, 32, 819-831] considered a problem in which 7 a single risk neutral distributor supplies a short-lifecycle, long-leadtime product to several retailers that are identical except in their attitudes towards risk. They proved that the distributor should not offer the same terms to every retailer but instead offer less risky (from the demand 9 risk perspective) contracts to more risk averse retailers. They did not prove the optimality of their menu.

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.

The focus of this research is in the area of controllability and observability properties of a system to determine the quality of input and output structures using energy requirements at a fixed distance R from the origin. Such a study is a novice one, and involves many interesting problems yet to be investigated.

Hybrid vehicles use two energy sources for their propelling. Usually an Internal Combustion Engine (ICE) is used with one or more electric machine(s) (EM). The problem is then to split the driver power demand between the ICE and the EM in order to minimize a criterion, usually the fuel consumption. A global optimization algorithm based on optimal control theory is recalled. The obtained results are optimal but can only be obtained in simulation. For real time control purpose, this optimization algorithm is applied on a receding horizon. The main problem is then to choose the variables to be predicted on this horizon. By analyzing the optimization algorithm, it is shown that the prediction of the future driving conditions (vehicle speed and driver torque demand) is not necessary. Therefore, under some assumptions, a real time control is possible.

In the framework of stochastic processes, the connection between the dynamic programming scheme given by the Hamilton-Jacobi-Bellman equation and a recently proposed control approach based on the Fokker-Planck equation is discussed. Under appropriate assumptions it is shown that the two strategies are equivalent in the case of expected cost functionals, while the Fokker-Planck formalism allows considering a larger class of objectives. To illustrate the connection between the two control strategies, the cases of an Itō stochastic process and of a piecewise-deterministic process are considered.

We apply optimal control theory to a tuberculosis model given by a system of ordinary differential equations. Optimal control strategies are proposed to minimize the cost of interventions, considering reinfection and post-exposure interventions. They depend on the parameters of the model and reduce effectively the number of active infectious and persistent latent individuals. The time that the optimal controls are at the upper bound increase with the transmission coefficient. A general explicit expression for the basic reproduction number is obtained and its sensitivity with respect to the model parameters is discussed. Numerical results show the usefulness of the optimization strategies.

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.

An important issue in motor control is understanding the basic principles underlying the accomplishment of natural movements. According to optimal control theory, the problem can be stated in these terms: what cost function do we optimize to coordinate the many more degrees of freedom than necessary to fulfill a specific motor goal? This question has not received a final answer yet, since what is optimized partly depends on the requirements of the task. Many cost functions were proposed in the past, and most of them were found to be in agreement with experimental data. Therefore, the actual principles on which the brain relies to achieve a certain motor behavior are still unclear. Existing results might suggest that movements are not the results of the minimization of single but rather of composite cost functions. In order to better clarify this last point, we consider an innovative experimental paradigm characterized by arm reaching with target redundancy. Within this framework, we make use of an inverse optimal control technique to automatically infer the (combination of) optimality criteria that best fit the experimental data. Results show that the subjects exhibited a consistent behavior during each experimental condition, even though the target point was not prescribed in advance. Inverse and direct optimal control together reveal that the average arm trajectories were best replicated when optimizing the combination of two cost functions, nominally a mix between the absolute work of torques and the integrated squared joint acceleration. Our results thus support the cost combination hypothesis and demonstrate that the recorded movements were closely linked to the combination of two complementary functions related to mechanical energy expenditure and joint-level smoothness.

The aim of the paper is to investigate how environmental policies affect the production-inventory decisions of a firm. Optimal control theory is applied to analyze optimal production policies. The environmental policy investigated in the paper is a combined emission charges and limits policy. The paper modifies the results of Wirl (European Journal of Operational Research 55 (1991) 191-200 [1]) for the case of this combined policy.

The problem of the brachistochronic motion of a variable mass particle is considered. The particle moves through a resistant medium in the field of arbitrary active forces. Beginning from these general assumptions, and applying Pontryagin's minimum principle along with singular optimal control theory, a corresponding two-point boundary value problem is obtained and solved. The solution proposed involves an appropriate numerical procedure based upon the shooting method. In this numerical procedure, the evaluation of ranges for unknown values of costate variables is avoided by the choice of a corresponding Cartesian coordinate of the particle as an independent variable. A numerical example assuming the resistance force proportional to the square of the particle speed is presented. A review of existing results for related problems is provided, and it can be shown that these problems may be regarded as special cases of the brachistochrone problem formulated and solved in this paper under very general assumptions by means of optimal control theory.

The Lagrangian-Hamiltonian unified formalism of R. Skinner and R. Rusk was originally stated for autonomous dynamical systems in classical mechanics. It has been generalized for non-autonomous first-order mechanical systems, as well as for first-order and higher-order field theories. However, a complete generalization to higher-order mechanical systems has yet to be described. In this work, after reviewing the natural geometrical setting and the Lagrangian and Hamiltonian formalisms for higher-order autonomous mechanical systems, we develop a complete generalization of the Lagrangian-Hamiltonian unified formalism for these kinds of systems, and we use it to analyze some physical models from this new point of view.

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 ...

In this paper, we intro duce optimal control-relat ed gradient-based numeri cal algorithms for the design of pulse sequences in NMR spectroscopy. This methodology is used for designing pulse sequences that maximize the coherence tran sfer between coupled spins in a given spe ed time, minimize the relaxation e ects in a given coherence tr ansfer step or minimize the time required to produce a given unitar y propagator, as desir ed. T he applicati on of these gradient ascent pulse engineering (GRAPE) methods to design pulse sequences that are robust to experimentally important parameter variation s, such as chemical shift dispersion or rf variations due to imperfecti ons such as rf-inhomogeneity is also explain ed.

Motivated by the complexity of spatio-temporal patterns of interconnected human processes (e.g., crowds, car traffic, social networks), this paper sets forth the fractal dynamic games as an analytical tool for modeling and predicting human dynamics. Starting from a statistical physics description of interactions between agents and from the observed statistical properties of economic measures, we construct a master equation characterizing the dynamics of cost functionals as stochastic variables affected by additive and multiplicative noise forces. Given the significance of human behavior, we allow the cost distribution to depend on the evolution of agents density. By coupling the description of agent dynamics through a fractal structure with a generic stochastic utility function, we formulate a new dynamic game. Employing optimal control theory concepts, we derive a continuum formulation of the car traffic dynamics optimization resulting in a nonlinear fractional partial differential equation.

key words: partial differential equations, finite element methods, discrete systems, preconditioning SUMMARY This survey paper is based on three talks given by the second author at the London Mathematical Society Durham Symposium on Computational Linear Algebra for Partial Differential Equations in the summer of 2008. The main focus will be on an abstract approach to the construction of preconditioners for symmetric linear systems in a Hilbert space setting. Typical examples which are covered by this theory are systems of partial differential equations which correspond to saddle point problems. We will argue that the mapping properties of the coefficient operators suggest that block diagonal preconditioners are natural choices for these systems. To illustrate our approach a number of examples will be considered. In particular, parameter-dependent systems arising in areas like incompressible flow, linear elasticity, and optimal control theory will be studied. The paper contains analysis of several models which have previously been discussed in the literature . However, here each example is discussed with reference to a more unified abstract approach.