A review of convex approaches for control, observation and safety of linear parameter varying and Takagi-Sugeno systems

This paper provides a review about the concept of convex systems based on Takagi-Sugeno, linear parameter varying (LPV) and quasi-LPV modeling. These paradigms are capable of hiding the nonlinearities by means of an equivalent description which uses a set of linear models interpolated by appropriately defined weighing functions. Convex systems have become very popular since they allow applying extended linear techniques based on linear matrix inequalities (LMIs) to complex nonlinear systems. This survey aims at providing the reader with a significant overview of the existing LMI-based techniques for convex systems in the fields of control, observation and safety. Firstly, a detailed review of stability, feedback, tracking and model predictive control (MPC) convex controllers is considered. Secondly, the problem of state estimation is addressed through the design of proportional, proportional-integral, unknown input and descriptor observers. Finally, safety of convex systems is discussed by describing popular techniques for fault diagnosis and fault tolerant control (FTC).Peer ReviewedPostprint (published version

Control and filtering of time-varying linear systems via parameter dependent Lyapunov functions

The main contribution of this dissertation is to propose conditions for linear filter and controller design, considering both robust and parameter dependent structures, for discrete time-varying systems. The controllers, or filters, are obtained through the solution of optimization problems, formulated in terms of bilinear matrix inequalities, using a method that alternates convex optimization problems described in terms of linear matrix inequalities. Both affine and multi-affine in different instants of time (path dependent) Lyapunov functions were used to obtain the design conditions, as well as extra variables introduced by the Finsler\u27s lemma. Design problems that take into account an H-infinity guaranteed cost were investigated, providing robustness with respect to unstructured uncertainties. Numerical simulations show the efficiency of the proposed methods in terms of H-infinity performance when compared with other strategies from the literature

Simultaneous State and Unknown Input Set-Valued Observers for Some Classes of Nonlinear Dynamical Systems

In this paper, we propose fixed-order set-valued (in the form of l2-norm hyperballs) observers for some classes of nonlinear bounded-error dynamical systems with unknown input signals that simultaneously find bounded hyperballs of states and unknown inputs that include the true states and inputs. Necessary and sufficient conditions in the form of Linear Matrix Inequalities (LMIs) for the stability (in the sense of quadratic stability) of the proposed observers are derived for ($\mathcal{M},\gamma$)- Quadratically Constrained (($\mathcal{M},\gamma$)-QC) systems, which includes several classes of nonlinear systems: (I) Lipschitz continuous, (II) ($\mathcal{A},\gamma$)-QC* and (III) Linear Parameter-Varying (LPV) systems. This new quadratic constraint property is at least as general as the incremental quadratic constraint property for nonlinear systems and is proven in the paper to embody a broad range of nonlinearities. In addition, we design the optimal $\mathcal{H}_{\infty}$ observer among those that satisfy the quadratic stability conditions and show that the design results in Uniformly Bounded-Input Bounded-State (UBIBS) estimate radii/error dynamics and uniformly bounded sequences of the estimate radii. Furthermore, we provide closed-form upper bound sequences for the estimate radii and sufficient condition for their convergence to steady state. Finally, the effectiveness of the proposed set-valued observers is demonstrated through illustrative examples, where we compare the performance of our observers with some existing observers.Comment: Under review in Automatic

Observer based active fault tolerant control of descriptor systems

The active fault tolerant control (AFTC) uses the information provided by fault detection and fault diagnosis (FDD) or fault estimation (FE) systems offering an opportunity to improve the safety, reliability and survivability for complex modern systems. However, in the majority of the literature the roles of FDD/FE and reconfigurable control are described as separate design issues often using a standard state space (i.e. non-descriptor) system model approach. These separate FDD/FE and reconfigurable control designs may not achieve desired stability and robustness performance when combined within a closed-loop system.This work describes a new approach to the integration of FE and fault compensation as a form of AFTC within the context of a descriptor system rather than standard state space system. The proposed descriptor system approach has an integrated controller and observer design strategy offering better design flexibility compared with the equivalent approach using a standard state space system. An extended state observer (ESO) is developed to achieve state and fault estimation based on a joint linear matrix inequality (LMI) approach to pole-placement and H∞ optimization to minimize the effects of bounded exogenous disturbance and modelling uncertainty. A novel proportional derivative (PD)-ESO is introduced to achieve enhanced estimation performance, making use of the additional derivative gain. The proposed approaches are evaluated using a common numerical example adapted from the recent literature and the simulation results demonstrate clearly the feasibility and power of the integrated estimation and control AFTC strategy. The proposed AFTC design strategy is extended to an LPV descriptor system framework as a way of dealing with the robustness and stability of the system with bounded parameter variations arising from the non-linear system, where a numerical example demonstrates the feasibility of the use of the PD-ESO for FE and compensation integrated within the AFTC system.A non-linear offshore wind turbine benchmark system is studied as an application of the proposed design strategy. The proposed AFTC scheme uses the existing industry standard wind turbine generator angular speed reference control system as a “baseline” control within the AFTC scheme. The simulation results demonstrate the added value of the new AFTC system in terms of good fault tolerance properties, compared with the existing baseline system

Modeling and Estimation of Biological Plants

Estimating the state of a dynamic system is an essential task for achieving important objectives such as process monitoring, identification, and control. Unlike linear systems, no systematic method exists for the design of observers for nonlinear systems. Although many researchers have devoted their attention to these issues for more than 30 years, there are still many open questions. We envisage that estimation plays a crucial role in biology because of the possibility of creating new avenues for biological studies and for the development of diagnostic, management, and treatment tools. To this end, this thesis aims to address two types of nonlinear estimation techniques, namely, the high-gain observer and the moving-horizon estimator with application to three different biological plants. After recalling basic definitions of stability and observability of dynamical systems and giving a bird's-eye survey of the available state estimation techniques, we are interested in the high-gain observers. These observers may be used when the system dynamics can be expressed in specific a coordinate under the so-called observability canonical form with the possibility to assign the rate of convergence arbitrarily by acting on a single parameter called the high-gain parameter. Despite the evident benefits of this class of observers, their use in real applications is questionable due to some drawbacks: numerical problems, the peaking phenomenon, and high sensitivity to measurement noise. The first part of the thesis aims to enrich the theory of high-gain observers with novel techniques to overcome or attenuate these challenging performance issues that arise when implementing such observers. The validity and applicability of our proposed techniques have been shown firstly on a simple one-gene regulatory network, and secondly on an SI epidemic model. The second part of the thesis studies the problem of state estimation using the moving horizon approach. The main advantage of MHE is that information about the system can be explicitly considered in the form of constraints and hence improve the estimates. In this work, we focus on estimation for nonlinear plants that can be rewritten in the form of quasi-linear parameter-varying systems with bounded unknown parameters. Moving-horizon estimators are proposed to estimate the state of such systems according to two different formulations, i.e., "optimistic" and "pessimistic". In the former case, we perform estimation by minimizing the least-squares moving-horizon cost with respect to both state variables and parameters simultaneously. In the latter, we minimize such a cost with respect to the state variables after picking up the maximum of the parameters. Under suitable assumptions, the stability of the estimation error given by the exponential boundedness is proved in both scenarios. Finally, the validity of our obtained results has been demonstrated through three different examples from biological and biomedical fields, namely, an example of one gene regulatory network, a two-stage SI epidemic model, and Amnioserosa cell's mechanical behavior during Dorsal closure

Damage localization in data-driven vibration-based structural health monitoring using linear quadratic estimation theory

Vibration-based Structural Health Monitoring (SHM) is classically approached from two different directions; both involve the acquisition and processing of vibration signals. The first and most popular strategy, which is also followed in the present thesis, relies entirely on the measurements. In contrast, the second approach employs physical models such as finite element (FE) models that are designed based on mechanical principles. In times in which the real-time processing of digital twins for engineering structures becomes more and more realistic, model-based approaches for vibration-based SHM receive increasing attention. Data-driven strategies are still primarily used in vibration-based SHM, and they will remain appealing in situations where precise physical modeling appears cumbersome. Hence, the need for efficient, robust, and reliable data-driven techniques concerning all stages and hurdles of SHM that can prove themselves in practice will never vanish. In this regard, after over 25 years of research, the number of real-life validation studies is still surprisingly low. As for all SHM strategies, the difficulty concerning damage analysis increases with higher levels of realization. Beginning with the goal of detecting damage, SHM finally seeks to predict the remaining lifetime of a structure. The intermediate steps comprise the localization, classification, and assessment of damage. Without the existence of adequately calibrated physics-based models, the successful implementation of methods tackling the objectives beyond damage localization in an unsupervised data-driven scheme is questionable. The term ‘unsupervised’ refers to the fact that knowledge about the manifestation of damage is not available. Especially in civil engineering, this situation pertains in general and is considered in the present thesis. In data-driven SHM, where the area of structural alterations is narrowed down to adjacent sensors, damage localization suffers from the coarse spatial resolution of parsimonious data acquisition systems. Classical modal approaches that hold potential for damage localization require a dense sensor network or significant damage. Originating from the field of fault detection and isolation, estimator- and filter-based methods have proven to be applicable for damage identification of mechanical and civil engineering structures. Notably, they feature an enormous sensitivity towards structural changes when properly designed. Although it remains advantageous for the sake of precise damage localization, these tools such as Kalman or H-infinity filters do not exhibit the inherent demand for a dense sensor network. Consequently, they promise to be viable techniques for the application in vibration-based SHM. A central challenge of this discipline is the discrimination between the natural variability of the structure’s dynamics and the one caused by damage. The former results from varying environmental and operational conditions (EOCs). Especially highly sensitive methods for damage identification are affected by these natural changes, and thus, rely on an efficient data normalization strategy, which can prove itself in practice. In light of these challenges, this thesis provides a real-life validation for the application of quadratic estimators in data-driven vibration-based SHM. To this end, an elaborate technique for estimator-based damage localization is adapted and included in an SHM framework comprising the necessary steps of data normalization and statistical testing. The damage analysis methodology was originally designed for H-infinity filters, which seem well-suited for use in SHM, as they do not assume specific properties of the excitation acting on the structure nor of the involved disturbances. However, previous studies have shown that, in some cases, the filter performance required to achieve high levels of sensitivity towards localized damage cannot be obtained. This issue can be circumvented by employing well-tuned Kalman filters. Therefore, a novel approach for noise covariance estimation is established at first. The associated estimation scheme constitutes a parametric extension of the popular autocovariance least-squares (ALS) technique. The effectiveness of this estimation technique in the context of Kalman filter-based damage localization is studied first using simulations and laboratory experiments. The second part is dedicated to the problem of handling EOCs. This body of work proposes an identification scheme for linear parameter-varying systems based on the interpolation of linear time-invariant systems for different operating points. A simulation study demonstrates the applicability for the purpose of data normalization. Finally, real-life validation of the proposed methods for SHM is conducted. Therefore, a steel lattice mast located outdoors functions as the test object. It is naturally affected by ambient sources of excitation, variability, and uncertainty. The mast, explicitly designed for this validation purpose, is equipped with reversible damage mechanisms that may be activated or removed to reduce the stiffness at multiple locations of the structure. The investigations conducted in this part of the thesis demonstrate proper damage detection of all considered damages as well as localization for the highest degree of severity. These promising results suggest the applicability of the presented methods for Kalman filter tuning, damage localization, and data-normalization in the context of vibration-based SHM

Optimal control and robust estimation for ocean wave energy converters

This thesis deals with the optimal control of wave energy converters and some associated observer design problems. The first part of the thesis will investigate model predictive control of an ocean wave energy converter to maximize extracted power. A generic heaving converter that can have both linear dampers and active elements as a power take-off system is considered and an efficient optimal control algorithm is developed for use within a receding horizon control framework. The optimal control is also characterized analytically. A direct transcription of the optimal control problem is also considered as a general nonlinear program. A variation of the projected gradient optimization scheme is formulated and shown to be feasible and computationally inexpensive compared to a standard nonlinear program solver. Since the system model is bilinear and the cost function is not convex quadratic, the resulting optimization problem is shown not to be a quadratic program. Results are compared with other methods like optimal latching to demonstrate the improvement in absorbed power under irregular sea condition simulations. In the second part, robust estimation of the radiation forces and states inherent in the optimal control of wave energy converters is considered. Motivated by this, low order H∞ observer design for bilinear systems with input constraints is investigated and numerically tractable methods for design are developed. A bilinear Luenberger type observer is formulated and the resulting synthesis problem reformulated as that for a linear parameter varying system. A bilinear matrix inequality problem is then solved to find nominal and robust quadratically stable observers. The performance of these observers is compared with that of an extended Kalman filter. The robustness of the observers to parameter uncertainty and to variation in the radiation subsystem model order is also investigated. This thesis also explores the numerical integration of bilinear control systems with zero-order hold on the control inputs. Making use of exponential integrators, exact to high accuracy integration is proposed for such systems. New a priori bounds are derived on the computational complexity of integrating bilinear systems with a given error tolerance. Employing our new bounds on computational complexity, we propose a direct exponential integrator to solve bilinear ODEs via the solution of sparse linear systems of equations. Based on this, a novel sparse direct collocation of bilinear systems for optimal control is proposed. These integration schemes are also used within the indirect optimal control method discussed in the first part.Open Acces