Solving optimal control problems with state constraints using nonlinear programming and simulation tools

This paper illustrates how nonlinear programming and simulation tools, which are available in packages such as MATLAB and SIMULINK, can easily be used to solve optimal control problems with state- and/or input-dependent inequality constraints. The method presented is illustrated with a model of a single-link manipulator. The method is suitable to be taught to advanced undergraduate and Master's level students in control engineering.

Unravelling the rationale of 'overgrazing' and stocking rates in the beef production systems of Central Brazil using a bi-criteria compromise programming model

In Central Brazil, the long-term sustainability of beef cattle systems is under threat over vast tracts of farming areas, as more than half of the 50 million hectares of sown pastures are suffering from degradation. Overgrazing practised to maintain high stocking rates is regarded as one of the main causes. High stocking rates are deliberate and crucial decisions taken by the farmers, which appear paradoxical, even irrational given the state of knowledge regarding the consequences of overgrazing. The phenomenon however appears inextricably linked with the objectives that farmers hold. In this research those objectives were elicited first and from their ranking two, ‘asset value of cattle (representing cattle ownership)' and ‘present value of economic returns', were chosen to develop an original bi-criteria Compromise Programming model to test various hypotheses postulated to explain the overgrazing behaviour. As part of the model a pasture productivity index is derived to estimate the pasture recovery cost. Different scenarios based on farmers' attitudes towards overgrazing, pasture costs and capital availability were analysed. The results of the model runs show that benefits from holding more cattle can outweigh the increased pasture recovery and maintenance costs. This result undermines the hypothesis that farmers practise overgrazing because they are unaware or uncaring about overgrazing costs. An appropriate approach to the problem of pasture degradation requires information on the economics, and its interplay with farmers' objectives, for a wide range of pasture recovery and maintenance methods. Seen within the context of farmers' objectives, some level of overgrazing appears rational. Advocacy of the simple ‘no overgrazing' rule is an insufficient strategy to maintain the long-term sustainability of the beef production systems in Central Brazil.

Unravelling the rationale of 'overgrazing' and stocking rates in the beef production systems of Central Brazil using a bi-criteria compromise programming model

In Central Brazil, the long-term, sustainability of beef cattle systems is under threat over vast tracts of farming areas, as more than half of the 50 million hectares of sown pastures are suffering from. degradation. Overgrazing practised to maintain high stocking rates is regarded as one of the main causes. High stocking rates are deliberate and crucial decisions taken by the farmers, which appear paradoxical, even irrational given the state of knowledge regarding the consequences of overgrazing. The phenomenon however appears inextricably linked with the objectives that farmers hold. In this research those objectives were elicited first and from their ranking two, 'asset value of cattle (representing cattle ownership and 'present value of economic returns', were chosen to develop an original bi-criteria Compromise Programming model to test various hypotheses postulated to explain the overgrazing behaviour. As part of the model a pasture productivity index is derived to estimate the pasture recovery cost. Different scenarios based on farmers' attitudes towards overgrazing, pasture costs and capital availability were analysed. The results of the model runs show that benefits from holding more cattle can outweigh the increased pasture recovery and maintenance costs. This result undermines the hypothesis that farmers practise overgrazing because they are unaware or uncaring caring about overgrazing costs. An appropriate approach to the problem of pasture degradation requires information on the economics,and its interplay with farmers' objectives, for a wide range of pasture recovery and maintenance methods. Seen within the context of farmers' objectives, some level of overgrazing appears rational. Advocacy of the simple 'no overgrazing' rule is an insufficient strategy to maintain the long-term sustainability of the beef production systems in Central Brazil. (C) 2004 Elsevier Ltd. All rights reserved.

Automatic nonlinear predictive model-construction algorithm using forward regression and the PRESS statistic

An automatic nonlinear predictive model-construction algorithm is introduced based on forward regression and the predicted-residual-sums-of-squares (PRESS) statistic. The proposed algorithm is based on the fundamental concept of evaluating a model's generalisation capability through crossvalidation. This is achieved by using the PRESS statistic as a cost function to optimise model structure. In particular, the proposed algorithm is developed with the aim of achieving computational efficiency, such that the computational effort, which would usually be extensive in the computation of the PRESS statistic, is reduced or minimised. The computation of PRESS is simplified by avoiding a matrix inversion through the use of the orthogonalisation procedure inherent in forward regression, and is further reduced significantly by the introduction of a forward-recursive formula. Based on the properties of the PRESS statistic, the proposed algorithm can achieve a fully automated procedure without resort to any other validation data set for iterative model evaluation. Numerical examples are used to demonstrate the efficacy of the algorithm.

Bifurcation Analysis of Eigenstructure Assignment Control in a Simple Nonlinear Aircraft Model

Aircraft systems are highly nonlinear and time varying. High-performance aircraft at high angles of incidence experience undesired coupling of the lateral and longitudinal variables, resulting in departure from normal controlled � ight. The construction of a robust closed-loop control that extends the stable and decoupled � ight envelope as far as possible is pursued. For the study of these systems, nonlinear analysis methods are needed. Previously, bifurcation techniques have been used mainly to analyze open-loop nonlinear aircraft models and to investigate control effects on dynamic behavior. Linear feedback control designs constructed by eigenstructure assignment methods at a � xed � ight condition are investigated for a simple nonlinear aircraft model. Bifurcation analysis, in conjunction with linear control design methods, is shown to aid control law design for the nonlinear system.

Closed loop design for a simple nonlinear aircraft model using eigenstructure assignment

Aircraft systems are highly nonlinear and time varying. High-performance aircraft at high angles of incidence experience undesired coupling of the lateral and longitudinal variables, resulting in departure from normal controlled flight. The aim of this work is to construct a robust closed-loop control that optimally extends the stable and decoupled flight envelope. For the study of these systems nonlinear analysis methods are needed. Previously, bifurcation techniques have been used mainly to analyze open-loop nonlinear aircraft models and investigate control effects on dynamic behavior. In this work linear feedback control designs calculated by eigenstructure assignment methods are investigated for a simple aircraft model at a fixed flight condition. Bifurcation analysis in conjunction with linear control design methods is shown to aid control law design for the nonlinear system.

Inclusion of nonlinear demand-supply relationships within large-scale partial equilibrium linear programming models

This paper presents a new method for the inclusion of nonlinear demand and supply relationships within a linear programming model. An existing method for this purpose is described first and its shortcomings are pointed out before showing how the new approach overcomes those difficulties and how it provides a more accurate and 'smooth' (rather than a kinked) approximation of the nonlinear functions as well as dealing with equilibrium under perfect competition instead of handling just the monopolistic situation. The workings of the proposed method are illustrated by extending a previously available sectoral model for the UK agriculture.

A simple theoretical model for the intensification of tropical cyclones and polar lows

A simple theoretical model for the intensification of tropical cyclones and polar lows is developed using a minimal set of physical assumptions. These disturbances are assumed to be balanced systems intensifying through the WISHE (Wind-Induced Surface Heat Exchange) intensification mechanism, driven by surface fluxes of heat and moisture into an atmosphere which is neutral to moist convection. The equation set is linearized about a resting basic state and solved as an initial-value problem. A system is predicted to intensify with an exponential perturbation growth rate scaled by the radial gradient of an efficiency parameter which crudely represents the effects of unsaturated processes. The form of this efficiency parameter is assumed to be defined by initial conditions, dependent on the nature of a pre-existing vortex required to precondition the atmosphere to a state in which the vortex can intensify. Evaluation of the simple model using a primitive-equation, nonlinear numerical model provides support for the prediction of exponential perturbation growth. Good agreement is found between the simple and numerical models for the sensitivities of the measured growth rate to various parameters, including surface roughness, the rate of transfer of heat and moisture from the ocean surface, and the scale for the growing vortex.

From a discrete to a continuum model of cell dynamics in one dimension

Multiscale modeling is emerging as one of the key challenges in mathematical biology. However, the recent rapid increase in the number of modeling methodologies being used to describe cell populations has raised a number of interesting questions. For example, at the cellular scale, how can the appropriate discrete cell-level model be identified in a given context? Additionally, how can the many phenomenological assumptions used in the derivation of models at the continuum scale be related to individual cell behavior? In order to begin to address such questions, we consider a discrete one-dimensional cell-based model in which cells are assumed to interact via linear springs. From the discrete equations of motion, the continuous Rouse [P. E. Rouse, J. Chem. Phys. 21, 1272 (1953)] model is obtained. This formalism readily allows the definition of a cell number density for which a nonlinear "fast" diffusion equation is derived. Excellent agreement is demonstrated between the continuum and discrete models. Subsequently, via the incorporation of cell division, we demonstrate that the derived nonlinear diffusion model is robust to the inclusion of more realistic biological detail. In the limit of stiff springs, where cells can be considered to be incompressible, we show that cell velocity can be directly related to cell production. This assumption is frequently made in the literature but our derivation places limits on its validity. Finally, the model is compared with a model of a similar form recently derived for a different discrete cell-based model and it is shown how the different diffusion coefficients can be understood in terms of the underlying assumptions about cell behavior in the respective discrete models.

A linear model of gravity wave drag for hydrostatic sheared flow over elliptical mountains

An analytical model of orographic gravity wave drag due to sheared flow past elliptical mountains is developed. The model extends the domain of applicability of the well-known Phillips model to wind profiles that vary relatively slowly in the vertical, so that they may be treated using a WKB approximation. The model illustrates how linear processes associated with wind profile shear and curvature affect the drag force exerted by the airflow on mountains, and how it is crucial to extend the WKB approximation to second order in the small perturbation parameter for these effects to be taken into account. For the simplest wind profiles, the normalized drag depends only on the Richardson number, Ri, of the flow at the surface and on the aspect ratio, γ, of the mountain. For a linear wind profile, the drag decreases as Ri decreases, and this variation is faster when the wind is across the mountain than when it is along the mountain. For a wind that rotates with height maintaining its magnitude, the drag generally increases as Ri decreases, by an amount depending on γ and on the incidence angle. The results from WKB theory are compared with exact linear results and also with results from a non-hydrostatic nonlinear numerical model, showing in general encouraging agreement, down to values of Ri of order one.

Farming systems methodology for efficient resource management at the farm level: a review from an Indian perspective

Farming systems research is a multi-disciplinary holistic approach to solve the problems of small farms. Small and marginal farmers are the core of the Indian rural economy Constituting 0.80 of the total farming community but possessing only 0.36 of the total operational land. The declining trend of per capita land availability poses a serious challenge to the sustainability and profitability of farming. Under such conditions, it is appropriate to integrate land-based enterprises such as dairy, fishery, poultry, duckery, apiary, field and horticultural cropping within the farm, with the objective of generating adequate income and employment for these small and marginal farmers Under a set of farm constraints and varying levels of resource availability and Opportunity. The integration of different farm enterprises can be achieved with the help of a linear programming model. For the current review, integrated farming systems models were developed, by Way Of illustration, for the marginal, small, medium and large farms of eastern India using linear programming. Risk analyses were carried out for different levels of income and enterprise combinations. The fishery enterprise was shown to be less risk-prone whereas the crop enterprise involved greater risk. In general, the degree of risk increased with the increasing level of income. With increase in farm income and risk level, the resource use efficiency increased. Medium and large farms proved to be more profitable than small and marginal farms with higher level of resource use efficiency and return per Indian rupee (Rs) invested. Among the different enterprises of integrated farming systems, a chain of interaction and resource flow was observed. In order to make fanning profitable and improve resource use efficiency at the farm level, the synergy among interacting components of farming systems should be exploited. In the process of technology generation, transfer and other developmental efforts at the farm level (contrary to the discipline and commodity-based approaches which have a tendency to be piecemeal and in isolation), it is desirable to place a whole-farm scenario before the farmers to enhance their farm income, thereby motivating them towards more efficient and sustainable fanning.

Analysis of the calving pattern of herds in small-scale dairy systems in central Mexico

This study sets out to find the best calving pattern for small-scale dairy systems in Michoacan State, central Mexico. Two models were built. First, a linear programming model was constructed to optimize calving pattern and herd structure according to metabolizable energy availability. Second, a Markov chain model was built to investigate three reproductive scenarios (good, average and poor) in order to suggest factors that maintain the calving pattern given by the linear programming model. Though it was not possible to maintain the optimal linear programming pattern, the Markov chain model suggested adopting different reproduction strategies according to period of the year that the cow is expected to calve. Comparing different scenarios, the Markov model indicated the effect of calving interval on calving pattern and herd structure.

Autonomous intelligent cruise control using a novel multiple-controller framework incorporating fuzzy-logic-based switching and tuning

This paper presents a novel intelligent multiple-controller framework incorporating a fuzzy-logic-based switching and tuning supervisor along with a generalised learning model (GLM) for an autonomous cruise control application. The proposed methodology combines the benefits of a conventional proportional-integral-derivative (PID) controller, and a PID structure-based (simultaneous) zero and pole placement controller. The switching decision between the two nonlinear fixed structure controllers is made on the basis of the required performance measure using a fuzzy-logic-based supervisor, operating at the highest level of the system. The supervisor is also employed to adaptively tune the parameters of the multiple controllers in order to achieve the desired closed-loop system performance. The intelligent multiple-controller framework is applied to the autonomous cruise control problem in order to maintain a desired vehicle speed by controlling the throttle plate angle in an electronic throttle control (ETC) system. Sample simulation results using a validated nonlinear vehicle model are used to demonstrate the effectiveness of the multiple-controller with respect to adaptively tracking the desired vehicle speed changes and achieving the desired speed of response, whilst penalising excessive control action. Crown Copyright (C) 2008 Published by Elsevier B.V. All rights reserved.

Solving complex optimal control problems at no cost with PSOPT

This paper introduces PSOPT, an open source optimal control solver written in C++. PSOPT uses pseudospectral and local discretizations, sparse nonlinear programming, automatic differentiation, and it incorporates automatic scaling and mesh refinement facilities. The software is able to solve complex optimal control problems including multiple phases, delayed differential equations, nonlinear path constraints, interior point constraints, integral constraints, and free initial and/or final times. The software does not require any non-free platform to run, not even the operating system, as it is able to run under Linux. Additionally, the software generates plots as well as LATEX code so that its results can easily be included in publications. An illustrative example is provided.

A tutorial on pseudospectral methods for computational optimal control

[English] This paper is a tutorial introduction to pseudospectral optimal control. With pseudospectral methods, a function is approximated as a linear combination of smooth basis functions, which are often chosen to be Legendre or Chebyshev polynomials. Collocation of the differential-algebraic equations is performed at orthogonal collocation points, which are selected to yield interpolation of high accuracy. Pseudospectral methods directly discretize the original optimal control problem to recast it into a nonlinear programming format. A numerical optimizer is then employed to find approximate local optimal solutions. The paper also briefly describes the functionality and implementation of PSOPT, an open source software package written in C++ that employs pseudospectral discretization methods to solve multi-phase optimal control problems. The software implements the Legendre and Chebyshev pseudospectral methods, and it has useful features such as automatic differentiation, sparsity detection, and automatic scaling. The use of pseudospectral methods is illustrated in two problems taken from the literature on computational optimal control. [Portuguese] Este artigo e um tutorial introdutorio sobre controle otimo pseudo-espectral. Em metodos pseudo-espectrais, uma funcao e aproximada como uma combinacao linear de funcoes de base suaves, tipicamente escolhidas como polinomios de Legendre ou Chebyshev. A colocacao de equacoes algebrico-diferenciais e realizada em pontos de colocacao ortogonal, que sao selecionados de modo a minimizar o erro de interpolacao. Metodos pseudoespectrais discretizam o problema de controle otimo original de modo a converte-lo em um problema de programa cao nao-linear. Um otimizador numerico e entao empregado para obter solucoes localmente otimas. Este artigo tambem descreve sucintamente a funcionalidade e a implementacao de um pacote computacional de codigo aberto escrito em C++ chamado PSOPT. Tal pacote emprega metodos de discretizacao pseudo-spectrais para resolver problemas de controle otimo com multiplas fase. O PSOPT permite a utilizacao de metodos de Legendre ou Chebyshev, e possui caractersticas uteis tais como diferenciacao automatica, deteccao de esparsidade e escalonamento automatico. O uso de metodos pseudo-espectrais e ilustrado em dois problemas retirados da literatura de controle otimo computacional.