Robust Suboptimal Control of Nonlinear Systems

In this paper, we consider a nonlinear dynamic system with uncertain parameters. Our goal is to choose a control function for this system that balances two competing objectives: (i) the system should operate efficiently; and (ii) the system's performance should be robust with respect to changes in the uncertain parameters. With this in mind, we introduce an optimal control problem with a cost function penalizing both the system cost (a function of the final state reached by the system) and the system sensitivity (the derivative of the system cost with respect to the uncertain parameters). We then show that the system sensitivity can be computed by solving an auxiliary initial value problem. This result allows one to convert the optimal control problem into a standard Mayer problem, which can be solved directly using conventional techniques. We illustrate this approach by solving two example problems using the software MISER3

A computational method for solving time-delay optimal control problems with free terminal time

This paper considers a class of optimal control problems for general nonlinear time-delay systems with free terminal time. We first show that for this class of problems, the well-known time-scaling transformation for mapping the free time horizon into a fixed time interval yields a new time-delay system in which the time delays are variable. Then, we introduce a control parameterization scheme to approximate the control variables in the new system by piecewise-constant functions. This yields an approximate finite-dimensional optimization problem with three types of decision variables: the control heights, the control switching times, and the terminal time in the original system (which influences the variable time delays in the new system). We develop a gradient-based optimization approach for solving this approximate problem. Simulation results are also provided to demonstrate the effectiveness of the proposed approach

An Optimization Approach to State-Delay Identification

We consider a nonlinear delay-differential system with unknown state-delays. Our goal is to identify these state-delays using experimental data. To this end, we formulate a dynamic optimization problem in which the state-delays are decision variables and the cost function measures the discrepancy between predicted and observed system output. We then show that the gradient of this problem's cost function can be computed by solving an auxiliary delay-differential system. By exploiting this result, the state-delay identification problem can be solved efficiently using a gradient-based optimization method

A stochastic fleet composition problem

In this paper, we consider the problem of forming a new vehicle fleet, consisting of multiple vehicle types, to cater for uncertain future requirements. The problem is to choose the number of vehicles of each type to purchase so that the total expected cost of operating the fleet is minimized. The total expected cost includes fixed and variable costs associated with the fleet, as well as hiring costs that are incurred whenever vehicle requirements exceed fleet capacity. We develop a novel algorithm, which combines dynamic programming and the golden section method, for determining the optimal fleet composition. Numerical results show that this algorithm is highly effective, and takes just seconds to solve large-scale problems involving hundreds of different vehicle types

Guaranteed-Cost Controls of Minimal Variation: A Numerical Algorithm Based on Control Parameterization

The optimal control literature is dominated by standard problems in which the system cost functional is expressed in the well-known Bolza form. Such Bolza cost functionals consist of two terms: a Mayer term (which depends solely on the final state reached by the system) and a Lagrange integral term (which depends on the state and control values over the entire time horizon). One limitation with the standard Bolza cost functional is that it does not consider the cost of control changes. Such costs should certainly be considered when designing practical control strategies, as changing the control signal will invariably cause wear and tear on the system's acutators. Accordingly, in this paper, we propose a new optimal control formulation that balances system performance with control variation. The problem is to minimize the total variation of the control signal subject to a guaranteed-cost constraint that ensures an acceptable level of system performance (as measured by a standard Bolza cost functional). We first apply the control parameterization method to approximate this problem by a non-smooth dynamic optimization problem involving a finite number of decision variables. We then devise a novel transformation procedure for converting this non-smooth dynamic optimization problem into a smooth problem that can be solved using gradient-based optimization techniques. The paper concludes with numerical examples in fisheries and container crane control

Optimal parameter selection for nonlinear multistage systems with time-delays

In this paper, we consider a novel dynamic optimization problem for nonlinear multistage systems with time-delays. Such systems evolve over multiple stages, with the dynamics in each stage depending on both the current state of the system and the state at delayed times. The optimization problem involves choosing the values of the time-delays, as well as the values of additional parameters that influence the system dynamics, to minimize a given cost functional. We first show that the partial derivatives of the system state with respect to the time-delays and system parameters can be computed by solving a set of auxiliary dynamic systems in conjunction with the governing multistage system. On this basis, a gradient-based optimization algorithm is proposed to determine the optimal values of the delays and system parameters. Finally, two example problems, one of which involves parameter identification for a realistic fed-batch fermentation process, are solved to demonstrate the algorithm’s effectiveness

Optimal control problems with multiple characteristic time points in the objective and constraints

In this paper, we develop a computational method for a class of optimal control problems where the objective and constraint functionals depend on two or more discrete time points. These time points can be either fixed or variable. Using the control parametrization technique and a time scaling transformation, this type of optimal control problem is approximated by a sequence of approximate optimal parameter selection problems. Each of these approximate problems can be viewed as a finite dimensional optimization problem. New gradient formulae for the cost and constraint functions are derived. With these gradient formulae, standard gradient-based optimization methods can be applied to solve each approximate optimal parameter selection problem. For illustration, two numerical examples are solved

Using environmental niche models to identify Malagasy bat biodiversity hotpots and conversation priorities in the face of climate and human land use changes.

Master of Science in Biological and Life science. University of KwaZulu-Natal, Durban, 2016.Madagascar represents a global hotspot of endemism, but it faces numerous threats to its biodiversity, including impacts of climate and human land use changes. In this study, suitable habitat space of 25 Malagasy bat species was modelled under past, current and future climate projections, asking three questions: (i) Do Malagasy bat richness hotspots change over different climatic projections and human land use in the past to present to future?; (ii) Do current and future hotspots fall within Madagascar’s current protected areas (PAs)?; and (iii) Can areas be included in the current protected areas to better protect the Malagasy bat hotspots? A decline in suitable habitat space for bat species was anticipated, and, hence, decreased hotspots under warmer (Last Inter-glacial (LIG) and future) climate scenarios, particularly in combination with human land use. It was also expected that changes in climate would influence bats similarly within functional groups (FGs) and differently across functional groups. Specifically, it was predicted that bats adapted to forage in vegetation (clutter FG) should be more affected than bats adapted to hunt insects near vegetation (clutter-edge FG) and high above vegetation (open-air FG). It was further predicted bat richness hotspots would have poor coverage by Madagascar’s PA network for current and future scenarios, due to broad distribution ranges of bats and limited area coverage of PAs across the island. Suitable habitat space was modelled for bat species using environmental niche models (ENMs), taking an ensemble modelling approach to identify the most suitable ENM for each species. Species richness was quantified by stacking suitable habitat maps of individual bat species. The results showed that richness hotspots shifted in size and geographic position under different climate change scenarios. Generally, changes from warm to cold climates decreased the potential suitable habitat space of clutter bats yet increased those of clutter-edge and open-air bat species. In contrast, changes from cold to warmer climates decreased the suitable habitats of clutter bats more than those of clutter-edge and open-air bats. Null model analyses showed that under both cold and warm climatic conditions, the observed overlap in suitable habitat between clutter and clutter-edge functional groups was lower than expected, whereas overlap in suitable habitat between clutter-edge and open-air functional groups was higher than expected by chance. Further, suitable habitat space of bat species was only partially covered by the PA system for climate change and human land use scenarios (range 2 – 20%). Although most of the bat richness hotspots are in PAs, this coverage will decrease with predicted future climate change. It is suggested that an additional 58,077 km should be added to the current protected areas to ensure adequate protection under future climate and land use

Dynamic optimization for robust path planning of horizontal oil wells

This paper considers the three-dimensional path planning problem for horizontal oil wells. The decision variables in this problem are the curvature, tool-face angle and switching points for each turn segment in the path, and the optimization objective is to minimize the path length and target error. The optimal curvatures, tool-face angles and switching points can be readily determined using existing gradient-based dynamic optimization techniques. However, in a real drilling process, the actual curvatures and tool-face angles will inevitably deviate from the planned optimal values, thus causing an unexpected increase in the target error. This is a critical challenge that must be overcome for successful practical implementation. Accordingly, this paper introduces a sensitivity function that measures the rate of change in the target error with respect to the curvature and tool-face angle of each turn segment. Based on the sensitivity function, we propose a new optimization problem in which the switching points are adjusted to minimize target error sensitivity subject to continuous state inequality constraints arising from engineering specifications, and an additional constraint specifying the maximum allowable increase in the path length from the optimal value. Our main result shows that the sensitivity function can be evaluated by solving a set of auxiliary dynamic systems. By combining this result with the well-known time-scaling transformation, we obtain an equivalent transformed problem that can be solved using standard nonlinear programming algorithms. Finally, the paper concludes with a numerical example involving a practical path planning problem for a Ci-16-Cp146 well