Solving the Hamilton-Jacobi-Bellman Equation for a Stochastic System with State Constraints

We present a method for solving the Hamilton-Jacobi-Bellman (HJB) equation for a stochastic system with state constraints. A variable transformation is introduced which turns the HJB equation into a combination of a linear eigenvalue problem, a set of partial differential equations (PDE:s), and a point-wise equation. For a fixed solution to the eigenvalue problem, the PDE:s are linear and the point-wise equation is quadratic, indicating that the problem can be solved efficiently using an iterative scheme. As an example, we numerically solve for the optimal control of a Linear Quadratic Gaussian (LQG) system with state constraints. A reasonably accurate solution is obtained even with a very small number of collocation points (three in each dimension), which suggests that the method could be used on high order systems, mitigating the curse of dimensionality

Optimal Control of a Batch Reactor Using the Linearized Hamilton-Jacobi-Bellman Equation

AbstractIn this work we present an efficient method for solving an optimal control problem for a batch reactor, where a temperature dependent exothermic reaction takes place within a preset duration and within specified temperature bounds. The Hamilton-Jacobi-Bellman (HJB) equation corresponding to the optimal control problem is nonlinear and has infinite boundary conditions due to the state constraints (bounds on temperature and concentration), which makes it troublesome to solve. However, using a logarithmic transformation, the HJB-equation is transformed into a linear partial differential equation with zero boundary conditions. Furthermore, the problem can then be solved using variable separation such that the time- dependent part has an analytical solution and the state dependent part becomes a linear eigenvalue problem which can readily be solved using standard software

Finite-time state-constrained optimal control for input-affine systems with actuator noise

Abstract: We show that a linearizing transformation of the Hamilton-Jacobi-Bellman (HJB) equation can be applied to certain finite-time problem such that the time dependence can be separated and also has a simple analytical solution. The remaining state dependence is the solution to a linear eigenvalue problem that may have an analytical solution or is readily solved numerically. The efficiency of the method is illustrated by an inventory control problem

Remote light stress detection for greenhouse LED lighting control

The illumination in greenhouses is in general still controlled manually by on/off control because of the type of lamps (High Pressure Sodium) that are traditionally used. With High Brightness LEDs being introduced on the market today, sufficiently high power for greenhouse grown crops can be achieved, which opens up for advanced lighting control since both light spectrum and intensity can be controlled then. For the growers, maximizing production in order to meet customer demand and economically optimize the production, often imply a high light intensity and a high level of artificial light complementing the natural sunlight. However, a too high intensity causes light stress and a photo inhibition that can significantly reduce the photosynthetic yield and hence, production. A key issue to address is therefore to detect when this level is reached. Here we present new results on how to diagnose the plants remotely based on transient and frequency analysis, system identification and frequency function properties

Integrated Dynamic Aquaculture and Wastewater Treatment Modelling for Recirculating Aquaculture Systems

Recirculating aquaculture systems (RAS) in land based fish tanks, where the fish tank effluent is biologically treated and then recirculated back to the fish tanks, offers a possibility for large scale ecologically sustainable fish production. In order to fully exploit the advantages of RAS, however, the water exchange should be as small as possible. This implies strong demands on the water treatment, e.g. the maintenance of an efficient nitrification, denitrification and organic removal. Because of the RAS complexity, though, dynamic simulations are required to analyze and optimize a plant with respect to effluent water quality, production and robustness. Here, we present a framework for integrated dynamic aquaculture and wastewater treatment modelling. It provides means to analyze, predict and explain RAS performance. Using this framework we demonstrate how a new and improved RAS configurations is identified

Using Fluorescence as Control Parameter to Decide Optimal Light Spectrum for Plant Growth

Modern greenhouses having lighting systems are large consumers of electricity. In Europe alone, the lighting consumption is estimated to 150 TWh per year. High pressure sodium (HPS) lamps are still dominating and the illumination is in general controlled manually by on/off control. Changing to light emitting diodes (LED) gives the possibility of adapting the spectrum (i.e. changing the power split to diodes of different colours) and to gradually changing the intensity, which implies an energy saving potential. The optimal spectrum might depend on a number of factors, for example plant species, required characteristics and energy use effciency on the diodes. Using LEDs with different blue to red (B:R) ratios, as a supplement to sunlight, have been investigated for growing of cucumber seedling [Hernández and Kubota, 2014] and tomato seedling [Hernández and Kubota, 2012]. Their conclusion was that 100% red LED is preferred, indicating that the blue light in the sunlight is sufficient (B:R in sunlight is about 4:3 on photons=m2=s basis [ASTM, 2012]

Stochastic optimal control over unreliable communication links

In this paper LQG control over unreliable communication links is derived. That is to say, the communication channels between the controller and the actuators and between the sensors and the controller are unreliable. This is of growing importance as networked control systems and use of wireless communication in control are becoming increasingly common. The problem of how to optimize LQG control in this case is examined in the situation where communication between the components is done with acknowledgments. Previous solutions to finite horizon discrete time hold-input LQG control for this case do not fully utilize the available information. Here a new solution is presented which resolves this limitation. The solution is linear and covers communication channels subject to both packet losses and delays. The new control scheme is compared with previous solutions for LQG control in simulations, which demonstrates that a significant improvement in the cost can be achieved by fully utilizing the available information

Finding feedforward configurations using gramian based interaction measures

A sparse control structure can be seen as a decentralised controller that is expanded to include feedforward or MIMO blocks. Here, use of the gramian based interaction measures to determine a sparse control structure with feedforward is examined. A modification to the method used today is proposed and it is demonstrated that it results in a considerable improvement. Furthermore, recently proposed modifications to scaling gramian based measures are expanded to also cover sparse control structures. We show that the method that yields the best result is when two different scaling methods are combined, using one to design a decentralized controller and another to find feedforward connections