Distributed Fuzzy and Stochastic Observers for Nonlinear Systems

Abstract

Many problems in decision making, control, and monitoring require that all variables of interest, usually states and parameters of the system, are known at all times. However, in practical situations, not all variables are measurable or they are not measured due to technical or economical reasons. Therefore, these variables need to be estimated using an observer, based on a model of the system and measured data. For such a purpose, dynamic systems are often modeled in the state space framework, using a state transition model, which describes the evolution of the states over time; and a measurement model, which relates the measurements to the states. In some cases, these models also consider random external disturbances influencing the process. While for linear systems several solutions to estimate the unknown variables exist, state estimation for general nonlinear systems still represents a challenge. This thesis develops efficient observer design methods for nonlinear systems. Two types of systems are considered: deterministic nonlinear systems, represented by Takagi-Sugeno (TS) fuzzy models, and stochastic systems. For a large-scale or time-varying system, the design and tuning of an observer may be complicated and may involve large computational costs. By taking into account the specific properties of the system (such as cascaded, distributed, or time-varying), the observer design becomes easier and the computational costs are reduced. In the first part of the thesis, we consider nonlinear systems represented by TS fuzzy models, and investigate three system structures: cascaded systems, distributed systems, and systems affected by unknown disturbances. The motivation for investigating the cascaded and distributed structures comes from large scale systems. Many large-scale systems, such as power networks, material processing systems, communication and transportation networks are composed of interconnected lower dimensional subsystems. An important class of these systems can be represented as a cascade of subsystems. We study the cascade of nonlinear systems represented by TS fuzzy models. For cascaded TS systems with normalized membership functions we prove that the stability of the subsystems implies the stability of the cascade. Therefore, the stability analysis of a cascaded TS system may be performed by analyzing the individual subsystems. This approach is also extended to observer design. In order to design observers for the cascaded TS system it is sufficient to design observers for the subsystems. We also show that a cascaded design does not lead to the loss of performance in the terms of the estimation error decay rate. Therefore, the cascaded approach reduces the computational costs, while preserving the performance of the observer. In order to determine whether a nonlinear system is a cascade of subsystems, we give an algorithm that partitions a nonlinear system into cascaded subsystems. However, large-scale systems are in general not cascaded, but distributed, i.e., the influence among the subsystems is not unidirectional. In addition, the structure is often not fixed, i.e., subsystems may be added or removed on-line. For such systems, decentralized analysis and design present several advantages, such as flexibility and easier analysis. Therefore, we consider the stability analysis and observer design for distributed systems where each subsystem is represented by a TS fuzzy model. The conditions previously obtained for cascaded TS systems are extended to distributed TS systems. We analyze the stability of the overall TS system based on the stability of the subsystems, allowing that new subsystems may be added on-line. When the structure of the system is not fixed, the influence of the interconnection terms due to the addition of a new subsystem is not known before the subsystem is actually added. Moreover, even though the new subsystem is stable, the interconnection terms may have a destabilizing effect. Therefore, we derive conditions on the strength of the interconnection terms so that the stability of the overall system is maintained. Next, the approach is extended to observer design. We assume that a fuzzy observer is already designed for an existing subsystem or a collection of subsystems. When a new subsystem, together with the interconnection terms is added, a new observer is designed only for this subsystem. Since the already analyzed parts of the system or designed observers do not need to be analyzed or designed again, the computational costs are reduced. We also study TS systems that are influenced by unknown inputs (disturbances) or that change over time. The design of observers in the presence of unknown inputs is an important problem, since in many cases not all the inputs are known. The unknown inputs may also represent effects of actuator or plant component failures. Two types of inputs are considered in this thesis: model-plant mismatch and time-varying disturbances that can be represented as or approximated by polynomial functions of time. Based on the known part of the fuzzy model, we design observers that simultaneously estimate both the states and the unknown inputs. In case of a polynomial input, the observer guarantees an exponential convergence of the error to zero. When the input is only approximated by a polynomial function of time, a bound on the estimation error is derived. If the disturbance is due to a model mismatch, the true model is estimated, with an asymptotic convergence of the error to zero. In the second part of the thesis, we consider stochastic systems, and investigate the combination of different observers for cascaded stochastic systems. In many applications, in order to efficiently analyze the process or to efficiently design observers, one also has to consider the noise that is affecting the states or the measurements. In such cases, probabilistic estimation methods have to be used. The most well-known of these are the Kalman filter (KF), its nonlinear variants, the extended and unscented KF, and particle filters (PFs). We consider combinations of KFs for stochastic systems that are cascades of subsystems. We compare cascaded and centralized KFs both from a theoretical point of view and on simulation examples. If the KFs are designed independently for the subsystems, the individual KFs are optimal for the subsystems. Our theoretical results show that the cascaded KFs are jointly optimal and therefore have the same performance as a centralized KF for all possible inputs and outputs if and only if the subsystems are decoupled. However, simulation results indicate that for practical purposes, the performance of the centralized and cascaded KFs is comparable. We also compare cascaded and centralized stochastic observers on two real-world applications, namely the estimation of the overflow losses in a hopper dredger and the estimation of the model parameters in a water treatment plant. In both cases, the models are nonlinear and non-Gaussian, and the states of interest are not measurable. By employing the cascaded approach, an unscented KF and a PF are combined to obtain a better estimate of the overflow losses in a hopper dredger. In the second application, PFs are used in cascade to estimate the model parameters in a water treatment plant. In both cases, the cascaded filters are easier to tune and yield better estimation results than a centralized filter, with reduced computational costs. The thesis closes with some concluding remarks and a discussion on important open issues regarding the approaches studied. Additionally, some fundamental unsolved issues in state estimation are discussed, and promising research directions to address these issues are suggested.