Description Of Model Uncertainty Research Articles

Learning‐based adaptive‐scenario‐tree model predictive control with improved probabilistic safety using robust Bayesian neural networks

SummaryScenario‐based model predictive control (MPC) methods can mitigate the conservativeness inherent to open‐loop robust MPC. Yet, the scenarios are often generated offline based on worst‐case uncertainty descriptions obtained a priori, which can in turn limit the improvements in the robust control performance. To this end, this paper presents a learning‐based, adaptive‐scenario‐tree model predictive control approach for uncertain nonlinear systems with time‐varying and/or hard‐to‐model dynamics. Bayesian neural networks (BNNs) are used to learn a state‐ and input‐dependent description of model uncertainty, namely the mismatch between a nominal (physics‐based or data‐driven) model of a system and its actual dynamics. We first present a new approach for training robust BNNs (RBNNs) using probabilistic Lipschitz bounds to provide a less conservative uncertainty quantification. Then, we present an approach to evaluate the credible intervals of RBNN predictions and determine the number of samples required for estimating the credible intervals given a credible level. The performance of RBNNs is evaluated with respect to that of standard BNNs and Gaussian process (GP) as a basis of comparison. The RBNN description of plant‐model mismatch with verified accurate credible intervals is employed to generate adaptive scenarios online for scenario‐based MPC (sMPC). The proposed sMPC approach with adaptive scenario tree can improve the robust control performance with respect to sMPC with a fixed, worst‐case scenario tree and with respect to an adaptive‐scenario‐based MPC (asMPC) using GP regression on a cold atmospheric plasma system. Furthermore, closed‐loop simulation results illustrate that robust model uncertainty learning via RBNNs can enhance the probability of constraint satisfaction of asMPC.

Engineering analysis with probability boxes: A review on computational methods

The consideration of imprecise probability in engineering analysis to account for missing, vague or incomplete data in the description of model uncertainties is a fast-growing field of research. Probability-boxes (p-boxes) are of particular interest in an engineering context, since they offer a mathematically straightforward description of imprecise probabilities, as well as allow for an intuitive visualisation. In essence, p-boxes are defined via lower and upper bounds on the cumulative distribution function of a random variable whose exact probability distribution is unknown. However, the propagation of p-boxes on model inputs towards bounds on probabilistic measures describing the uncertainty on the model responses is numerically still very demanding, and hence is subject of intensive research. In order to provide an overview on the available methods, this paper gives a state-of-the art review for the modelling and propagation of p-boxes with a special focus on structural reliability analysis.

Probabilistic Studies on the Shear Strength of Slender Steel Fiber Reinforced Concrete Structures

Shear failure is a brittle and undesirable mode of failure in reinforced concrete structures. Many of the existing shear design equations for steel fiber reinforced concrete (SFRC) beams include significant uncertainty due to the failure in accurately predicting the true shear capacity. Given these, adequate quantification and description of model uncertainties considering the systematic variation in the model prediction and measured shear capacity is crucial for reliability-based investigation. Reliability analysis must account for model uncertainties in order to predict the probability of failure under prescribed limit states. This study focuses on the quantification and description of model uncertainty related to the current shear resistance predictive models for SFRC beams without shear reinforcement. The German (DAfStB) model displayed the lowest bias and dispersion, whereas the fib Model 2010 and the Bernat et al., model displayed the highest bias and dispersion. The inconsistencies observed in the resistance model uncertainties at the variation of shear span to effective depth ratio are a major cause for concern, and differentiation with respect to this parameter is advised. Finally, in line with the EN 1990 semi-probabilistic approach for reliability-based design, the global partial safety factors related to model uncertainties in the shear resistance prediction of SFRC beams are proposed.

Structural vibration control using delayed state feedback via LMI approach: with application to chatter stability problems

This paper considers the problem of active vibration control in machining processes to prevent dynamic instability and chatter. A method for robust delayed-state feedback control is presented based on Lyapunov–Krasovskii functionals (LKFs). For fast computation and realizability, a partial state feedback approach based on a reduced order model of structural vibration is considered that matches available sensors and captures the main vibratory modes that impact on stability limits. Robustness to neglected dynamics and other model errors are accounted for by a frequency domain description of model uncertainty, augmented with the basic LKF stability condition. The controller synthesis problem is formulated as a set of linear matrix inequality equations that can be solved numerically using convex optimization methods. Evaluations on a hardware-in-the-loop flexible structure test-bed show that feedback controllers optimized via the LKF stability condition provide significant improvements in stability regions compared with optimized PD control and quadratic regulation. The results confirm the methodology as a useful approach for robust chatter control based on combined flexible structure and cutting process models.

Falsified Model-Invariant Safety-Preserving Control With Application to Closed-Loop Anesthesia

This brief introduces a novel safety-preserving control scheme with minimal conservatism for uncertain systems. We have recently introduced the model-invariant safety verification technique that provides a formal guarantee of safety for systems with multiplicative model uncertainty. This approach requires a multi-model description of model uncertainty. The resulting safety system may be conservative for systems that do not exhibit the worst case dynamical response. In this brief, we employ model falsification to reduce conservatism of the model-invariant safety verification technique. Members of a model set that characterizes model uncertainty are falsified if discrepancy between predictions of those models and measured responses of the uncertain system is established, thereby reducing model uncertainty. To demonstrate the effectiveness of the proposed technique, we formalize a model-invariant safety system for closed-loop propofol anesthesia. The safety system maintains predicted propofol concentration in plasma as well as the patient’s blood pressure within safety bounds despite uncertainty in patient responses to propofol.

Fast Stochastic Model Predictive Control of Unstable Dynamical Systems

Fast stochastic model predictive control (FSMPC) is a multivariable control algorithm that explicitly takes constraints and probabilistic parametric uncertainties into account while having low online computational cost for dynamical systems of high state dimension. This article extends FSMPC to be applicable to model uncertainty descriptions that include unstable dynamical systems. The proposed control structure, which embeds output feedback into past FSMPC formulations, is illustrated in a numerical example. Two different options for designing the embedded output feedback are compared and discussed.

Stochastic NMPC/DRTO of batch operations: Batch-to-batch dynamic identification of the optimal description of model uncertainty

The effectiveness of stochastic online process optimization strongly depends on the choice of the uncertain parameters, which are used to characterize the uncertainty embedded in the process model. This contribution presents a framework for rapid identification of the optimal set of uncertain parameters, needed for the formulation of stochastic online optimization problems. This algorithm relies on a combination of approximate statistical analysis, multi-point/global sensitivity analysis and ad-hoc ranking indices, and is tailored for applications in the field of stochastic dynamic optimization/optimal control of campaigns of batch cycles. To demonstrate the potential of the proposed approach, we apply it within the optimization of a batch campaign, in the presence of equipment fouling and of dynamic variations in the campaign targets. The process model, utilized in all of these studies, is a batch adaptation of the Tennessee Eastman Challenge problem.

Assessment of model uncertainties for structural resistance

Uncertainties in resistance models play a significant role in the reliability analysis of structures and code calibration of partial factors for semi-probabilistic design. In spite of this importance, existing knowledge concerning model uncertainties and their characteristics seems to suffer from imprecise definitions and lack of available experimental data. Currently used probabilistic models and derived factors for model uncertainties are mostly based on intuitive judgements and limited data. This often leads to an unrealistic description of model uncertainties. The present study attempts to improve definitions of model uncertainties and proposes a general methodology for their quantification by comparing experimental and model results. It appears that model uncertainty should be always related to a specific model and scope of its application. The proposed approach seems to offer better understanding of model uncertainties and enables the specification of their real characteristics.

Impact of uncertainty description on assimilating hydraulic head in the MIKE SHE distributed hydrological model

The ensemble Kalman filter (EnKF) is a popular data assimilation (DA) technique that has been extensively used in environmental sciences for combining complementary information from model predictions and observations. One of the major challenges in EnKF applications is the description of model uncertainty. In most hydrological EnKF applications, an ad hoc model uncertainty is defined with the aim of avoiding a collapse of the filter. The present work provides a systematic assessment of model uncertainty in DA applications based on combinations of forcing, model parameters, and state uncertainties. This is tested in a case where groundwater hydraulic heads are assimilated into a distributed and integrated catchment-scale model of the Karup catchment in Denmark. A series of synthetic data assimilation experiments are carried out to analyse the impact of different model uncertainty assumptions on the feasibility and efficiency of the assimilation. The synthetic data used in the assimilation study makes it possible to diagnose model uncertainty assumptions statistically. Besides the model uncertainty, other factors such as observation error, observation locations, and ensemble size are also analysed with respect to performance and sensitivity. Results show that inappropriate definition of model uncertainty can greatly degrade the assimilation performance, and an appropriate combination of different model uncertainty sources is advised.

A formal statistical approach to representing uncertainty in rainfall–runoff modelling with focus on residual analysis and probabilistic output evaluation – Distinguishing simulation and prediction

Summary While there seems to be consensus that hydrological model outputs should be accompanied with an uncertainty estimate the appropriate method for uncertainty estimation is not agreed upon and a debate is ongoing between advocators of formal statistical methods who consider errors as stochastic and GLUE advocators who consider errors as epistemic, arguing that the basis of formal statistical approaches that requires the residuals to be stationary and conform to a statistical distribution is unrealistic. In this paper we take a formal frequentist approach to parameter estimation and uncertainty evaluation of the modelled output, and we attach particular importance to inspecting the residuals of the model outputs and improving the model uncertainty description. We also introduce the probabilistic performance measures sharpness, reliability and interval skill score for model comparison and for checking the reliability of the confidence bounds. Using point rainfall and evaporation data as input and flow measurements from a sewer system for model conditioning, a state space model is formulated that accounts for three different flow contributions: wastewater from households, and fast rainfall–runoff from paved areas and slow rainfall-dependent infiltration–inflow from unknown sources. We consider two different approaches to evaluate the model output uncertainty, the output error method that lumps all uncertainty into the observation noise term, and a method based on Stochastic Differential Equations (SDEs) that separates input and model structure uncertainty from observation uncertainty and allows updating of model states in real-time. The results show that the optimal simulation (off-line) model is based on the output error method whereas the optimal prediction (on-line) model is based on the SDE method and the skill scoring criterion proved that significant predictive improvements of the output can be gained from updating the states continuously. In an effort to attain residual stationarity for both the output error method and the SDE method transformation of the observations were necessary but the statistical assumptions were nevertheless not 100% justified. The residual analysis showed that significant autocorrelation was present for all simulation models. We believe users of formal approaches to uncertainty evaluation within hydrology and within environmental modelling in general can benefit significantly from adopting the evaluation measures applied here, so the probabilistic performance of their models can be assessed properly.

State-Space GMDH Neural Networks for Actuator Robust Fault Diagnosis

1 Abstract—Most fault diagnosis methods focus on the fault detection of the system or sensors and do not take into account the problem of the fault detection and isolation of the actuators, which are an important part of the contemporary industrial systems. To solve such a problem, the system outputs and inputs estimator based on a dynamic Group Method of Data Handling neural network in the state-space representation is proposed. In particular, the methodology of the adaptive thresholds calculation for system inputs and outputs is presented. The approach is based on the application of the Unscented Kalman Filter and Unknown Input Filter is presented. This result enables performing robust fault detection and isolation of the actuators. The final part of the paper presents an application study, which confirms the effectiveness of the proposed approach. The m ost desirable feature of the neural models applied to the FDI and FTC schemes is a small modeling uncertainty, which is defined as a mismatch between the model and the system being considered (3). Therefore, it is important to use an approach reducing the contribution of the structure errors and the parameter estimation inaccuracy of the neural model uncertainty. To solve such a challenging problem, a Group Method of Data Handling (GMDH) neural network (10), (12), (20), (13), (18) have been proposed. The concept of the GM DH approach relies on replacing the complex neural model by the set of the hierarchically connected partial models, which can be chosen with the application of appropriate selection methods. Besides the reduction of the neural m odel inaccuracy it is important to calculate the model uncertainty in the form of mathematical description allowing performing the robust fault diagnosis. In this paper, a new approach to the actuator and sensors fault detection and isolation is proposed. To solve such a challenging problem, the system outputs and inputs estimator based on the dynamic GMDH neural network is employed. In order to achieve this goal a new structure of the dynamic neuron in the state-space representation is proposed. This description enables to obtain constraints of the parameter estimates which warranty the stability of dynamic GMDH neural model. In order to obtain the constrained parameter estimates and the neural model uncertainty description the Unscented Kalman Filter (UKF) (7), (27), (28) was used. This knowledge enables to calculate the output adaptive threshold. Moreover, the methodology of estimation of the GMDH neural model inputs and the calculation of the input adaptive threshold with the application of the Unknown Input Filter (UIF) is proposed. The obtained input and output adaptive thresholds allow to perform the robust fault diagnosis and actuators isolation of the dynamic non-linear systems. The paper consists of six sections. After the introduction, presenting the subject of this paper, in Section II a concept of robust fault detection and isolation of sensors and actuators is presented. Section III presents the process of synthesis of GMDH neural model. In section IV, the estimation of neural model inputs is realized with the UIF. The experimental results of the robust fault diagnosis are shown in section V. Finally, section VI concludes the paper.

Application of reduced models for robust control and state estimation of a distributed parameter system

This paper studies the application of reduced models of a distributed parameter system for robust process control and state estimation. We take the approach of integrating model reduction, parameter identification, and model uncertainty analysis, in purpose to find an appropriate trade-off between complexity and robust performance. The application example is the temperature system in a continuous paper pulp digester. Physical modeling of this process results in coupled linearized partial differential equations which are then reduced into low-order nominal process models using an orthogonal collocation approximation method. Two different approaches to obtaining a model uncertainty description are adapted for use on a distributed parameter system with low-order nominal model and shown to produce similar results when tested with measurement data. It is also demonstrated how this uncertainty description, in combination with the reduced model, may be used for robust control design and verification of the control performance on the distributed parameter system. Finally, the possibility of estimating the distributed process state using a state observer for the reduced process is demonstrated. Measurements of the process state in a certain position is available and is shown to agree with the estimated state at the same position.

Robust model predictive control design using a generalized objective function

An alternative robust model predictive control (MPC) design method is proposed. This approach is based on the use of a generalized objective function. It exploits the similarity between the parameter estimation problem and MPC. Robustness is achieved by choosing the objective function according to the error probability density function, as in robust system identification. The stability of the controller is demonstrated through the nominal asymptotic stability and that the plant cost can be located in the demanded vicinity of the non-increasing nominal cost, given a suitable objective function. One advantage of the proposed method is its relative simplicity regarding to the problem of choosing suitable model uncertainty descriptions in other robust approaches. The performance of the proposed method is illustrated by a chemical engineering example.

An efficient off-line formulation of robust model predictive control using linear matrix inequalities

The practicality of model predictive control (MPC) is partially limited by its ability to solve optimization problems in real time. Moreover, on-line computational demand for synthesizing a robust MPC algorithm will likely grow significantly with the problem size. In this paper, we use the concept of an asymptotically stable invariant ellipsoid to develop a robust constrained MPC algorithm which gives a sequence of explicit control laws corresponding to a sequence of asymptotically stable invariant ellipsoids constructed off-line one within another in state space. This off-line approach can address a broad class of model uncertainty descriptions with guaranteed robust stability of the closed-loop system and substantial reduction of the on-line MPC computation. The controller design is illustrated with two examples.

Application of moving horizon estimation based fault detection to cold tandem steel mill

For a class of model uncertainty descriptions, plant/model mismatch can be directly incorporated into a model based fault detection scheme using Moving Horizon Estimation. The model uncertainty is represented by a set of bounded parameters which can be used to alter the dynamics of the model through injection of the measured output as well as inputs. The uncertainty class includes gain uncertainty as well as uncertainty in pole locations. Using a bank of filters, detection of exclusive fault scenarios can be accomplished. The proposed method is compared to other methods employing an adaptive threshold, and is demonstrated on a simulation example of a cold tandem steel mill.

Constrained H∞ Control of an Experimental Flexible Link

A new approach is presented to design robust controllers for an experimental flexible beam with noncolocated sensors-actuator. The objective is to position the free end of the beam in the specified time under tight bounds on the trajectory of the tip and control action. We derive a transfer function nominal model for the beam and a quantitative description of model uncertainties based on experimentally obtained frequency response data. Robust controllers are designed by applying the recently developed Constrained H∞ control approach (Sideris and Rotstein, 1993, 1994), in which time-domain constraints are treated directly without translation to the frequency-domain. Constrained H∞ controllers are compared with standard H∞ and μ-synthesis designs in terms of both simulation and experimental results and are found to be superior in the considered case of stringent time-domain specifications.

Modeling parallel operating PWM DC/DC power supplies

The control objective of a single operating PWM (pulse-width-modulated) DC/DC (direct current-to-direct current) power supply is to maintain the output voltage close to the reference. In the case of parallel operating power supplies, the control objective is enriched with the demand of keeping the power distribution between the units close to a specified pattern. The paper suggests a model for parallel operating DC/DC converters. The physical meaning of the parameters of the model are discussed. Finally, the paper suggests a structured model uncertainty description based on measurements, simulations, and "experience".< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Identification of chemical processes using canonical variate analysis

A method of identification of linear input-output models using canonical variate analysis (CVA) is developed for application to chemical processes. This approach yields both a process model and a nonparametric description of model uncertainty, utilizing CVA for selection of a state coordinate system that optimally relates past inputs and outputs to future outputs. Regression procedures are then used for estimation of the state-space model parameters, and the Akaike Information Criterion (AIC) is used to determine the model order. The primary computations involve singular value decompositions which are numerically stable and accurate. The effectiveness of the CVA approach is first evaluated with simulated chemical processes that exhibit most of the practical problems encountered by existing system identification methods: nonlinear dynamics, unknown model orders and time delays, nonminimum phase dynamics, partial stiffness (requiring two time-scale approaches when other identification methods are used), low input excitation in some frequency bands, and measurement and process noise. The CVA methodology is then applied to the identification of models for a pilot-scale distillation column.

Robust Controllers for Pulse-Width-Modulated D.C./D.C. Converters Using Internal-Model-Control Design

A linear feedback controller for pulse-width-modulated d.c./d.c. regulator is designed using a frequency domain optimization method based on internal-model-control theory. This method aims to produce suboptimal low-order controllers which are ‘robust’, in the sense that the closed-loop system is guaranteed to meet stability objectives in the presence of model uncertainty. The small-signal model of a d.c./d.c. converter is used for the controller design. The model uncertainty description derived here is based on experiments and non-linear modelling. The result of the synthesis is a family of controllers, and each member of this family satisfies the robust control objectives. All controllers have a multi-loop structure including two feedback loops and one feedforward loop. A detailed design of the controller, including experimental results, is presented.

Impact of model uncertainty descriptions for high‐purity distillation control

AbstractThe ways in which modeling uncertainties are described for a particular process critically affects the results obtained in robustness studies. In this paper, four multivariable robust stability methodologies are used to characterize and analyze the effects of model inaccuracy due to non‐linearity in high‐purity distillation processes. The unstructured and structured singular value, numerical range, and a mapping of det(l + GPGc) are compared in terms of their ability to predict the stability of the dual‐composition control system over a wide composition range. The importance of using uncertainty descriptions that include a realistic representation of the phase‐magnitude relationship as well as the corrections between uncertainties in each element of the model is demonstrated. The conservation associated with norm‐bounded uncertainty descriptions reveals itself by the extent of detuning needed to insure stability and the subsequent degradation in control performance.