Advancements in real-time state estimation optimized observers and enhanced MHE

Dynamic systems can be described as anything evolving in time, making them a constant presence in our daily life, in any process, machine, or natural phenomenon we deal with. The description and manipulation of dynamic systems led to remarkable results in the history of science and engineering, from the ancient such as the Eolipila, invented by Erone, and considered as the ancestor of the steam engine, or again the Centrifugal Governor from Christiaan Huygens, which played a fundamental role in the 17th-century industrial revolution. However, a dedicated research field, Systems and Control Theory, only developed in the last decades, addressing the structural properties of dynamic systems and their main division into classes. Indeed, once a system is well described and analyzed, it must be appropriately sensed and observed in real-world scenarios to get to the next step: control it. Observation Theory (OT) is a research branch in Systems and Control Theory addressing the design of dynamic systems capable of fully reconstructing the information describing a generic plant, starting from a limited set of signals. A clear example of an observation problem can be found in our daily routine; at any time we locate ourselves in the environment, processing all the information coming from our senses and providing ourselves with an estimation of our position, orientation, and current interaction with obstacles and features in the surroundings. From a more technical point of view, knowing the current state of a dynamic system is crucial for controlling it or obtaining real-time information for decision-making or monitoring purposes. The dynamic systems developed by the Observation Theory to solve these kinds of tasks are commonly called observers. As we will explain through the Thesis, the main issues addressed in Observation Theory are the correct plant estimation and the computational burden required to achieve it successfully. In this Thesis, we focus on the latter, asking ourselves how to speed up current state-of-the-art estimation algorithms. Specifically, we will address optimization-based algorithms; indeed, with current processors’ continuously growing computational power, optimization is increasingly regarded as an interesting approach to solving complex problems. This background influences the research approaches, which constantly slide towards optimization, functional approximation, etc. Nevertheless, the computational burden still represents the main bottleneck of these approaches. However, in the Observation Theory background, the main efforts in introducing optimization in estimation problems have focused List of Tables 2 on guaranteeing the correct reconstruction of plants rather than on the speed of the online implementation. Hence, this Thesis aims to develop mathematical tools to design fast and effective nonlinear adaptive observers using optimization-based approaches. The development of these methods is supported by specific case studies showing their performance and robustness. To provide a quick overview, we anticipate widespread estimation problems such as the reconstruction of a battery state-of-charge and robot localization in GPS-denied environments will be considered. This being said, the main topics covered by the Thesis are the following: Moving Horizon Estimation: MHE consists of an observer design technique based on solving an optimization problem on the plant state vector and over a collection of sensor measurements, called measurements buffer. MHE is often called the dual of Model Predictive Control (MPC) for the observation problem. Indeed, the solution to this optimization problem becomes more computationally demanding as the complexity of the considered system grows. This usually makes this solution impracticable for online implementation. In this regard, we aim to propose solutions capable of speeding up the execution and convergence of MHE-based observers. More specifically, we propose the following solutions: 1. Filtered MHE: the number of necessary measurements required for the feasibility of the MHE problem is lowered by exploiting filters on the output measurements provided by the original plant, thus requiring fewer model integrations. 2. Adaptive MHE: starting from a general definition of signal richness related to the derivative of the output measurements, we investigate the role of the down-sampling between the elements of the measurement buffer in the MHE, showing its crucial role in the correct convergence of the observer. Furthermore, we exploit the wavelet framework to adaptively select the best samples to be considered in such measurement buffers. 3. Multi scale MHE: we continue investigating the role of the down-sampling in MHE-based observers, extending the proposed Adaptive MHE. Specifically, we aim to estimate state variables with different frequency characteristics correctly. Trajectory Based Observer Design: we propose an offline (non-realtime) observer design methodology inspired by the MHE approach. The proposed method considers a set of output trajectories from the target system and exploits them to define and tune the observer dynamics through optimization. Hence, we will refer to this methodology as Trajectory Based Optimization Design (TBOD). Indeed, this approach allows us to directly code the observer dynamics in the online implementation, removing the inherent computational burden of MHE observers. Decentralized observer: to conclude, we tried to move towards observer design for multi-agent systems; in the last years, Observation Theory has opened to collaborative robotics and distributed systems. Thus, we propose preliminary work on a decentralized observer for a fleet of satellites with a predefined communication topology. List of Tables 3 From this framework, we first developed a decentralized solution where each agent estimates its state, and we did so by exploiting a standard Kalman-based solution. Indeed, the future goal would be to scale up the optimization-based methodologies developed in the previous chapters of the Thesis, exploiting them for multi-agent observation problems. All proposed methods have been tested and validated on specific benchmarks and real scenarios. The code used to run all the simulations presented in this Thesis is available at this GitHub page. We are currently working to clean and simplify its structure. We aim to provide the Community with a simple, easy-to-use tool for implementing MHE observers. The Thesis is divided into the following chapters: Introduction: this chapter recalls the fundamentals of System Theory, providing the general concepts and notations that will be used over the entire Thesis. Furthermore, Observation Theory is also presented and described through a general review of the most important results developed in the last decades. Lastly, more specific system representation techniques will be detailed, as they will be useful when dealing with optimization-based observers. Moving Horizon Estimators - An introduction: this chapter goes more in detail in the description of MHE, namely the optimizationbased observers that will be addressed in the first set of solutions proposed in the Thesis. This chapter presents the main MHE formalizations, implementation, and links to Kalman Filtering Theory. Furthermore, an interpretation ofMHE as a hybrid system is also proposed, namely as a system characterized by both continuous and discrete-time dynamics. Moving Horizon Estimators - Towards online implementation: this chapter presents the main results proposed by the Author to speed up the MHE implementation. All the proposed solutions are tested on several case studies, from a simple Van Der Pol oscillator to a battery state-of-charge estimation problem. Trajectory Based Observer Design: this chapter presents the TBOD observer design methodology, namely an optimizationbased approach to design and tune observers for general nonlinear systems. The proposed method is thoroughly discussed and tested on a localization problem in a GPS-denied environment, also addressing the stability of the estimation error dynamics. Lastly, a sensor-fusion-related interpretation of the TBOD method is also discussed, providing interesting directions for future work. Decentralized optimized observation - A case study: this chapter presents a preliminary work on decentralized observation solutions exploiting optimization in the tuning process. We considered a localization problem on a swarm of nanosatellites, and we solved it by extending a standard Kalman-based solution to achieve a decentralized solution and improving the performance by optimizing the observer parameters. This work aimed to provide an initial feasibility study later to exploit MHE and TBOD solutions on multi-agent systems. List of Tables 4 Conclusions: this short chapter aims to wrap up the main concepts, solutions, and results developed throughout the Thesis. Indeed, some comments are also presented, remarking on the improvements concerning the current state-of-the-art, future goals, and research directions. Concluding this short abstract, this Thesis results from three years of constant and relentless work, often filled with doubts but always engraved with sincere curiosity. Interestingly, in hindsight, the convoluted paths tend to straighten out just as we reach the end. As we delve deeper into the subsequent chapters, we really hope the Reader will enjoy the main steps of this journey with us.