# LQG and Gaussian process techniques

LQG and Gaussian process techniques: For fixed-structure wind turbine control

AuthorBijl, H.J. (TU Delft Numerics for Control & Identification)

Delft University of Technology

Date2018-10-17

AbstractWind turbines are growing bigger to becomemore cost-efficient. This does increase the severity of the vibrations that are present in the turbine blades, both due to predictable effects like wind shear and tower shadow, and due to less predictable effects like turbulence and flutter. If wind turbines are to become bigger and more cost-efficient, these vibrations need to be reduced. This can be done by installing trailing-edge flaps to the blades. Because of the variety of circumstances which the turbine should operate in, this results in large uncertainties. As such, we need methods that can take stochastic effects into account. Preferably we develop an algorithmthat can learn from online data how the flaps affect the wind turbine and how to optimally control them. A simple prior analysis can be done using a linearized version of the system. In this case it is important to know not only the expected cost (damage) that will be incurred by the wind turbine in various situations, but also the spread of this cost. This can for instance be done by looking at the variance of the cost function. Various expressions are available to analytically calculate this variance. Alternatively, we can prescribe a degree of stability for the system. Due to the limitations of linear approximations of systems, it is more effective to apply nonlinear regression methods. A promising one is Gaussian Process (GP) regression. Given a training set (X, y) it can predict function values f (x¤) for test points x¤. It has its basis in Bayesian probability theory, which allows it to not only make this prediction, but also give information (the variance) about its accuracy. The usual way in which GP regression is applied has a few important limitations. Most importantly, it is computationally intensive, especially when applied to constantly growing data sets. In addition, it has difficulties dealing with noise present in the training input points x. There are methods to solve either of these issues, but these tricks generally do not work well together, or their combination requires many computational resources. However, by making the right approximations, like Taylor expansions and at times even linearizations, Gaussian process regression can be applied efficiently, in an online way, to data sets with noisy input points. This enables GP regression to be used for system identification problems like online non-linear black-box modeling. Another limitation is that it can be difficult to find the optimum of a Gaussian process. The reason is that the optimum of a Gaussian process is not a fixed point but a random variable. The distribution of this optimum cannot be calculated analytically, but we can use particle methods to approximate it. We can subsequently use this principle to efficiently explore an unknown nonlinear function, trying to locate its optimum. To do so, we sample a point x from the optimum distribution, measure what the function value f (x) at this point is, update the Gaussian process approximation of the function, update the optimum distribution and repeat this process until the distribution has converged. Finding the optimum of a function like this has shown to have competitive performance at keeping the cumulative regret low, compared to similar algorithms. In addition, it allows wind turbines to tune the gains of a fixed-structure controller so as to optimize a nonlinear cost function like the damage equivalent load. All these improvements are a step forward in the application of Gaussian process regression to wind turbine applications. But as is always the case with research, there are still many things left to improve further.

SubjectGaussian processes

regression

machine learning

optimization

system identification

automatic control

wind energy

smart rotor

https://doi.org/10.4233/uuid:f613079c-90a1-47dc-afcb-f6833646ca5a

ISBN978-94-6299-501-7

Part of collectionInstitutional Repository

Document typedoctoral thesis

Rights© 2018 H.J. Bijl