Costate approximation in optimal control using integral Gaussian quadrature orthogonal collocation methods

Two methods are presented for approximating the costate of optimal control problems in integral form using orthogonal collocation at Legendre–Gauss (LG) and Legendre–Gauss–Radau (LGR) points. It is shown that the derivative of the costate of the continuous‐time optimal control problem is equal to the negative of the costate of the integral form of the continuous‐time optimal control problem. Using this continuous‐time relationship between the differential and integral costate, it is shown that the discrete approximations of the differential costate using LG and LGR collocation are related to the corresponding discrete approximations of the integral costate via integration matrices. The approach developed in this paper provides a way to approximate the costate of the original optimal control problem using the Lagrange multipliers of the integral form of the LG and LGR collocation methods. The methods are demonstrated on two examples where it is shown that both the differential and integral costate converge exponentially as a function of the number of LG or LGR points. Copyright © 2014 John Wiley & Sons, Ltd.     

Optimal operations in non-linear water networks

The paper introduces the practical problems associated with automated control of large-scale non-linear water networks and gives the additional theoretical requirements for optimizing the network operation. A control scheme is presented which is intended for computer controlled operations and hence requires on-line evaluation of optimal control strategies based on a simulation of the overall dynamics of the network. Realistic applications of the scheme are shown to depend on adequate simplifications of the network performance, and results are given for development of linear dynamic models together with cost criteria in linear or quadratic form. Use of a hierarchical optimization technique permits a solution to be obtained to the remaining problems of high dimensionality with constrained variables. Application of the method to a complex network leads to results which are consistent with normal operational experience and which give minimized overall costs.     

Geostationary debris mitigation using minimum time solar sail trajectories with eclipse constraints

Minimum time solar sailing trajectories are introduced using a combination of indirect and direct optimal control techniques. Here, large‐scale, multiphase optimal control problems are solved using a pseudospectral collocation technique applied to an orbital debris mitigation concept. These solutions are obtained for realistic sail dimensions, producing multirevolution, Earth‐centered trajectories while accounting for uncontrolled spacecraft dynamics in the eclipse regions. Specifically, minimum time solutions for orbit transfer and phasing maneuvers are obtained using only solar radiation pressure for propulsion and control. First, an optimal primer vector steering history is obtained through numerical approximation. Locally optimal, closed‐form solutions are then implemented based on the primer vector direction, resulting in minimum time satisfaction of desired terminal orbital conditions. In addition, a novel solution strategy is presented for multiphase optimal control problems characterized by uncontrollable dynamics with flexible phase boundaries. The maneuvers presented will be shown to enable efficient orbital debris mitigation for large‐scale debris in geostationary orbits.     

Acoustically evoked brain stem potentials. Prerequisites for clinical use, data quality and sources of error

Ninety patients were subjected to brain-stem auditory evoked potential (BAEP) measurements in the intensive care unit. The data are analyzed and discussed with respect to their quality, reliability, and reproducibility. BAEPs of a quality satisfactory for diagnosis were found in 90% of the patients. About 10% of the measurements were distorted by artifacts and could not be used for diagnostic purposes. Reasons for these artifacts and problems of interpretation are discussed. Examples of single BAEPs and on-line monitoring of BAEPs in the form of "compressed BAEPs" are shown.     

Directly Simulation of Second Order Hyperbolic Systems in Second Order Form via the Regularization Concept

We present an efficient and robust method for stress wave propagation problems (second order hyperbolic systems) having discontinuities directly in their second order form. Due to the numerical dispersion around discontinuities and lack of the inherent dissipation in hyperbolic systems, proper simulation of such problems are challenging. The proposed idea is to denoise spurious oscillations by a post-processing stage from solutions obtained from higher-order grid-based methods (e.g., high-order collocation or finite-difference schemes). The denoising is done so that the solutions remain higher-order (here, second order) around discontinuities and are still free from spurious oscillations. For this purpose, improved Tikhonov regularization approach is advised. This means to let data themselves select proper denoised solutions (since there is no pre-assumptions about regularized results). The improved approach can directly be done on uniform or non-uniform sampled data in a way that the regularized results maintenance continuous derivatives up to some desired order. It is shown how to improve the smoothing method so that it remains conservative and has local estimating feature. To confirm effectiveness of the proposed approach, finally, some one and two dimensional examples will be provided. It will be shown how both the numerical (artificial) dispersion and dissipation can be controlled around discontinuous solutions and stochastic-like results.     

Optimal control problems with delays in state and control variables subject to mixed control–state constraints

Optimal control problems with delays in state and control variables are studied. Constraints are imposed as mixed control–state inequality constraints. Necessary optimality conditions in the form of Pontryagin's minimum principle are established. The proof proceeds by augmenting the delayed control problem to a nondelayed problem with mixed terminal boundary conditions to which Pontryagin's minimum principle is applicable. Discretization methods are discussed by which the delayed optimal control problem is transformed into a large‐scale nonlinear programming problem. It is shown that the Lagrange multipliers associated with the programming problem provide a consistent discretization of the advanced adjoint equation for the delayed control problem. An analytical example and numerical examples from chemical engineering and economics illustrate the results. Copyright © 2008 John Wiley & Sons, Ltd.     

Adjuvant therapy for pituitary adenomas: the possible role of photodynamic therapy.

Although benign, pituitary adenomas may not always be curable by surgery alone. This is usually owing to dural infiltration and the technical problems of removal of involved dura in the sellar region. In such circumstances, some form of adjuvant therapy may be needed to address residual tumour and forestall recurrence. Radiotherapy and drugs, such as bromocriptine or octreotide, although effective are far from ideal adjuvant therapies and this has led to the study of the effects of photodynamic therapy on pituitary adenomas. It has been shown that photodynamic therapy using the photosensitising drugs haematoporphyrin derivative and aluminium phthalocyanine is active experimentally against these tumours. This could provide another useful weapon to use against these challenging lesions.     

A combination of total intravenous anesthesia and thoracic epidural for thymectomy in juvenile myasthenia gravis

Juvenile myasthenia gravis is the acquired form of the disease in children and presents with ocular signs, fatigability, weakness and bulbar problems. The majority of patients demonstrate thymic hyperplasia and have been shown to benefit from thymectomy. The main considerations for the anesthesiologist are the degree of muscle weakness, the muscle groups involved and sensitivity to neuromuscular blocking drugs and volatile agents. Total intravenous anesthesia (TIVA) with epidural analgesia is probably the anesthetic technique of choice, although the latter is often avoided, because of the risk of a very high block. Two cases of thymectomy are presented where anesthesia was provided using a combination of TIVA and thoracic epidural analgesia. Both patients tolerated the technique well and had an uncomplicated perioperative course.     

Physical and psychosocial correlates of head and neck cancer: a review of the literature.

This article reviews recent literature on the physical and psychosocial correlates of head and neck cancer, with a focus on quality-of-life issues, rehabilitation outcomes, and changes in the literature from the previous decade. These studies have shown that head and neck cancer has an enormous impact on the quality of life of patients. The most important physical symptoms are speech problems, dry mouth and throat, and swallowing problems. Pain is also frequently reported. Disturbances in psychosocial functioning and psychological distress are reported by a considerable number of patients; worry, anxiety, mood disorder, fatigue, and depression are the main symptoms. Cancer of the head and neck has a negative effect on social, recreational, and sexual functioning. Despite a growing number of longitudinal studies, little is known about the rehabilitation outcomes over a longer period of time. Future research is necessary to form a consensus about the further development and use of specific instruments to study patients with cancer of the head and neck, to conduct more prospective studies, and to develop programs that are aimed at maximizing rehabilitation outcomes and evaluate these programs with randomized designs.     

The private health sector in South Africa--current trends and future developments

The private health sector is experiencing a crisis of spiralling costs, with average annual cost increases of between 13% and 32% over the decade 1978-1988. This trend is partly explained by the high utilisation rates that result from the combination of the 'fee-for-service' system and the 'third-party' payment structure of the sector. Medical schemes have responded by promoting the idea of 'flexible packages', and have won the right to 'risk-rate' prospective members. It is argued that these measures will undermine the principle of equity in health care, and will not solve the problems of the private sector. Instead, a more significant restructuring of the sector is likely to emerge. This may take the form of 'managed care' structures, along the lines of the health maintenance organisation model from the USA. The principles, advantages and problems of 'managed care' structures are described. These are shown to be potentially more rational and efficient than the current structure of the private sector. Although some resistance to 'managed care' structures can be expected, the convergence of interests of large employers and trade unions in containing health care costs suggests that their emergence is a likely development.     

Generalized minimum variance control law revisited

The generalized minimum variance control law is employed in many of the most successful self-tuning control schemes. However, the control law was derived by using a single-stage cost function where a conditional expectation operator was employed. This has limited our understanding of the operation of the control law and resulted in certain operational disadvantages. By rederiving the controller using a normal LQG approach but with a cost function of the same basic form, some of these deficiencies are mitigated. Moreover, the reason that the controller has poor stability characteristics on non-minimum phase systems becomes apparent and it is shown that the magnitude of the error weighting term must be severely restricted in some problems if stability is to be maintained. Thus it may not be possible to limit error variance to the desired extent.     

The infinite-order singular problem

In singular optimal control problems, the optimal control is determined by solving the algebraic equation which results by successively differentiating the switching function until the control appears explicitly. In certain classes of problems, the control never appears, and such problems are termed infinite-order singular problems. It is shown that this class has many useful properties with respect to the theory and computation of optimal controls. In particular, it is shown that for the time-invariant, singular, linear-quadratic problem: (i) the singular order is infinity or less than or equal to the state dimension, (ii) infinite-order problems can arise only from exact differential type cost functions, (iii) the range of the second-variation operator (Hessian) is finite-dimensional, (iv) the computational method converges strongly, and (v) conjugate direction methods converge in a finite number of steps. The latter property is especially useful in the generation of test problems for optimal control computation schemes.     

Postoperative enteral feeding following resection of the colon

We tried to show that feeding via intrajejunal tube with a chemical defined diet (CDD) is an alternative to parenteral nutrition in the early postoperative period. We investigated 40 patients after colon surgery--randomized into two groups (group I: CDD, group II: control: total parenteral nutrition via central veins). It was shown that the patients in the CDD-group tolerated enteral feeding from the beginning without any problems. At the second postoperative day, the N-balance became positive in both groups. The given results proved that this form of tube feeding with a special diet is a definite alternative to parenteral nutrition.     

IgE-positive hypersensitivity V asculitis —A difficult problem for the plastic surgeon

Summary Hypersensitivity vasculitis is an autonomous form of this disease, attacking the minor vessels. It manifests itself either as an idiopathic condition or in secondary, symptomatic form following severe illnesses or as an oversensitivity reaction to certain drugs. Distinctions are drawn between a local, a systemic and a simultaneously local and systemic progression. The problems of diagnosis and therapy are described for a complicated case in the acute generalization phase. Successful treatment of this severe form of the disease requires close cooperation between the internal specialist and the plastic surgeon.     

Rückenprobleme am Arbeitsplatz

Primary prevention is very important for the avoidance of back problems. An interdisciplinary study has shown that health circles and back tutorials are suitable methods of intervention. Changes in the way work is organized so as to improve the ergonomic situation at the workplace together with ‘back-sparing behaviour’ learned in back tutorials can reduce health problems and thus also mean fewer working days are lost.     

Chaotic dynamics and convergence analysis of temporal difference algorithms with bang‐bang control

Reinforcement learning is a powerful tool used to obtain optimal control solutions for complex and difficult sequential decision making problems where only a minimal amount of a priori knowledge exists about the system dynamics. As such, it has also been used as a model of cognitive learning in humans and applied to systems, such as humanoid robots, to study embodied cognition. In this paper, a different approach is taken where a simple test problem is used to investigate issues associated with the value function's representation and parametric convergence. In particular, the terminal convergence problem is analyzed with a known optimal control policy where the aim is to accurately learn the value function. For certain initial conditions, the value function is explicitly calculated and it is shown to have a polynomial form. It is parameterized by terms that are functions of the unknown plant's parameters and the value function's discount factor, and their convergence properties are analyzed. It is shown that the temporal difference error introduces a null space associated with the finite horizon basis function during the experiment. The learning problem is only non‐singular when the experiment termination is handled correctly and a number of (equivalent) solutions are described. Finally, it is demonstrated that, in general, the test problem's dynamics are chaotic for random initial states and this causes digital offset in the value function learning. The offset is calculated, and a dead zone is defined to switch off learning in the chaotic region. Copyright © 2015 John Wiley & Sons, Ltd.     

A note on the sufficiency of the maximum principle for infinite horizon optimal control problems

This work is devoted to introduce a new sufficient optimality condition for infinite horizon optimal control problems. It is shown that normal extremal processes are optimal under this new condition, termed as maximum‐principle‐pseudo‐invexity. Such a condition is the most general possible in the sense that problems in which every normal extremal process is optimal are necessarily maximum‐principle‐pseudo‐invex.     

Optimal control problems with multiple goal objectives

Solution procedures for optimal control problems having multiple objectives are developed by using the concepts of goal programming. The solutions to optimal control problems having conflicting objectives and multidimensional decision criteria are in the form of a set of optimum levels of achievement of some target goals.     

Guaranteed cost solution of optimal control and game problems for uncertain systems

A class of systems having white noise parameter variations is considered. Solutions of the infinite-time quadratic optimal control (LQOC) and differential game (LQDC) problems are considered. It is shown that guaranteed cost solutions of these problems are possible if the effects of parameter uncertainties are first taken into account in a specified manner. The method is illustrated through a numerical example.