Using hybrid automata for set-membership state estimation with uncertain nonlinear continuous-time systems

This paper deals with set-membership state estimation for continuous-time systems from discrete-time measurements, in the unknown but bounded error framework. The classical predictor–corrector approach to state estimation uses interval Taylor methods for solving the prediction phase, which are known to have poor performance in presence of large model or input uncertainty. In this paper, we show how to derive more efficient predictors by using a nonlinear hybridization method which builds hybrid automata to characterize the boundaries of reachable sets. The derived continuous–discrete set-membership predictor–corrector estimator is then tested with simulated data from a bioreactor. Our method is compared to classical continuous-time interval observers and is shown to have promising performance.     

Set-membership identification and fault detection using a Bayesian framework

This paper deals with the problem of set-membership identification and fault detection using a Bayesian framework. The paper presents how the set-membership model estimation problem can be reformulated from the Bayesian viewpoint in order to, first, determine the feasible parameter set in the identification stage and, second, check the consistency between the measurement data and the model in the fault-detection stage. The paper shows that, assuming uniform distributed measurement noise and uniform model prior probability distributions, the Bayesian approach leads to the same feasible parameter set than the well-known set-membership technique based on approximating the feasible parameter set using sets. Additionally, it can deal with models that are nonlinear in the parameters. The single-output and multiple-output cases are addressed as well. The procedure and results are illustrated by means of the application to a quadruple-tank process.     

Set-membership approach for identification of parameter and prediction uncertainty in power-law relationships: The case of sediment yield

Power laws are used to describe a large variety of natural and industrial phenomena. Consequently, they are used in a wide range of scientific research and management applications. This paper focuses on the identification of bounds on the parameter and prediction uncertainty in a power-law relation from experimental data, assuming known bounds on the error between model output and observations. The prediction uncertainty bounds can subsequently be used as constraints, for example in optimisation and scenario studies. The set-membership approach involves identification and removal of outliers, estimation of the feasible parameter set, evaluation of the feasible model-output set and tuning of the specified bounds on model-output error. As an example the procedure is applied to data of scattered sediment yield versus catchment area (Wasson, 1994). The key result is an un-falsified relationship between sediment yield and catchment area with uncertainty bounds on its parameters. The set-membership results are compared with the results from a conventional least-squares approach with first-order variance propagation, assuming a zero-mean, symmetrical error distribution.     

The detection mechanism for false data injection attack via the ellipsoidal set-membership approach

The false data injection (FDI) attack detection problem in cyber-physical systems (CPSs) is investigated in this paper. A novel attack detection algorithm is proposed based on the ellipsoidal set-membership approach. In comparison to the existing FDI attack detection methods, the developed attack detection approach in this paper neither requires predefined thresholds nor specific statistical characteristics of the attacks. In order to guarantee that the estimation ellipsoid contains normal states despite the unknown but bounded (UBB) process and measurement noises, the one-step ellipsoidal set-membership estimation method is put forward. In addition, a convex optimization algorithm is introduced to calculate the gain matrix of the observer recursively. Moreover, with the help of the state estimation ellipsoid, the residual ellipsoid can be obtained for attack detection. Whether a detector can detect the FDI attack depends on the relationship between the residual value and residual ellipsoidal set. Finally, the effectiveness of the proposed method is demonstrated by a numerical simulation example.     

Determination of stoichiometric association constants by a non-iterative computational method

Computational methods for the estimation of stoichiometric association constants for multiple-ligand binding systems are currently based on non-linear least-squares regression analysis. These computational methods require sophisticated, iterative algorithms to assure convergence to a solution, as well as initial parameter and error estimates. A simple procedure, called lambda-invariance testing (LIT), was developed that provides a single-pass (non-iterative) estimation of stoichiometric association constants. The LIT method was applied to simulated binding data containing Gaussian error and to real data drawn from the literature. This method provided parameter estimates essentially equivalent to those obtained by least-squares regression analysis, with no initial parameter or error estimates required.     

基于超平行空间集员滤波的非线性系统状态估计方法

针对噪声有界但未知条件下的非线性系统状态估计问题,提出基于超平行空间集员滤波算法.利用Stirling矩阵将模型进行一阶展开,基于凸差规划完成线性化误差定界,采用超平行空间表示误差边界和状态可行集,求解下一时刻预测状态可行集超平行体.在更新步将观测值分解为多个带,融入观测值的线性化误差并将带依次与超平行体相交,得到该时刻超平行空间描述下的状态可行集更新情况.所提出算法能够避免在求解线性化误差过程中外包误差集合带来的体积扩充,降低非线性集员滤波算法的保守性,仿真示例验证了所提出算法的可行性和有效性.     

Guaranteed non-linear estimation using constraint propagation on sets

Bounded-error estimation is the estimation of the parameter or state vector of a model from experimental data, under the assumption that some suitably defined errors should belong to some prior feasible sets. When the model outputs are linear in the vector to be estimated, a number of methods are available to contain all estimates that are consistent with the data within simple sets such as ellipsoids, orthotopes or parallelotopes, thereby providing guaranteed set estimates. In the non-linear case, the situation is much less developed and there are very few methods that produce such guaranteed estimates. In this paper, the problem of characterizing the set of all state vectors that are consistent with all data in the case of non-linear discrete-time systems is cast into the more general framework of constraint satisfaction problems. The state vector at time k should be estimated either on-line from past measurement only or off-line from a series of measurements that may include measurements posterior to k . Even in the causal case, prior information on the future value of the state and output vectors, due for instance to physical constraints, is readily taken into account. Algorithms taken from the literature of interval constraint propagation are extended by replacing intervals by more general subsets of real vector spaces. This makes it possible to propose a new algorithm that contracts the feasible domain for each uncertain variable optimally (i.e. no smaller domain could be obtained) and efficiently.     

Robust fault detection of non-linear systems using set-membership state estimation based on constraint satisfaction

In this paper, the robust fault detection problem for non-linear systems considering both bounded parametric modelling errors and measurement noises is addressed. The non-linear system is monitored by using a state estimator with bounded modelling uncertainty and bounded process and measurement noises. Additionally, time-variant and time-invariant system models are taken into account. Fault detection is formulated as a set-membership state estimation problem, which is implemented by means of constraint satisfaction techniques. Two solutions are presented: the first one solves the general case while the second solves the time-variant case, being this latter a relaxed solution of the first one. The performance of the time-variant approach is tested in two applications: the well-known quadruple-tank benchmark and the dynamic model of a representative portion of the Barcelona's sewer network. In both applications, different scenarios are presented: a faultless situation and some faulty situations. All considered scenarios are intended to show the effectiveness of the presented approach.     

Parameter set estimation for non-linear systems

Models of dynamical systems are instrumental for many purposes: prediction, control, simulation, tracking and so on. In this paper, we will show how parameter set estimation (PSE) can be applied to non-linear systems. Parameter set estimation identifies a set of estimates which are feasible with respect to the measured data and a priori information. This set of parameters, feasible for the given model structure, can then be used for system tracking or robust control designs. For application to robust control, it is important that the size of this set be as small as possible. In order to apply parameter set estimation techniques to a non-linear system, the system function is expressed in a tensor parameterization which is linear in the parameters (LP). Then it is shown how an optimum volume ellipsoid strategy for linear time invariant systems can be extended to this tensor parameterization of a non-linear system. The methodology is illustrated on two examples, the second of which uses data obtained from an operating glass furnace.     

Analysis and identification of non-linear systems via shifted Jacobi series

Based on shifted Jacobi series, algorithms have been established for the analysis and identification of non-linear systems described by a Hammerstein model consisting of a single-valued non-linearity followed by a linear plant. Moreover, the parameter estimation of a certain non-linear lumped system is also presented. By using the shifted Jacobi expansion for the analysis, the solution of a non-linear state equation is reduced to the solution of a linear algebraic matrix equation. For the identification, by expanding the measured input-output data into the shifted Jacobi series the unknown parameters of the non-linear system are estimated through the least-squares method. Examples are given to demonstrate the usefulness of this approach.     

基于GA-LSSVR的渭河水质参数遥感反演研究

针对渭河水质参数遥感反演这一典型的非线性、小样本回归估计问题,引入最小二乘支持向量回归（LSSVR）方法来解决,它将SVR中的二次规划问题转化为线性方程组求解,在保证精度的同时极大地降低了计算复杂性,加快了求解速度;针对其参数难以选择的问题,利用遗传算法(GA)来优选模型参数。采用提出的方法对标准数据集进行了实验,并建模对渭河的4种水质参数CODmn(高锰酸盐指数)、NH₃-N(氨氮)、 DO(溶解氧)、COD(化学需氧量)进行了遥感反演,结果表明GA-LSSVR模型可用于解决复杂的回归问题并具有较好的预测性能。     

Output-feedback predictive control of constrained linear systems via set-membership state estimation

This paper combines model predictive control (MPC) and set-membership (SM) state estimation techniques for controlling systems subject to hard input and state constraints. Linear systems with unknown but bounded disturbances and partial state information are considered. The adopted approach guarantees that the constraints are satisfied for all the states which are compatible with the available information and for all the disturbances within given bounds. Properties of the proposed MPC-SM algorithm and simulation studies are reported.     

Identification for systems with bounded noise

In this paper, we present a set-membership identification algorithm for systems with unknown but bounded disturbance. The algorithm contains a weighting factor which is selected according to whether the new observed data contains sufficient information. The proposed approach ensures that the estimation error is bounded and nonincreasing. Furthermore, it is shown that the parameter estimates provided by the algorithm will converge to a region containing the true parameters, and its upper bound is also given     

Ellipsoidal parameter or state estimation under model uncertainty

Ellipsoidal outer-bounding of the set of all feasible state vectors under model uncertainty is a natural extension of state estimation for deterministic models with unknown-but-bounded state perturbations and measurement noise. The technique described in this paper applies to linear discrete-time dynamic systems; it can also be applied to weakly non-linear systems if non-linearity is replaced by uncertainty. Many difficulties arise because of the non-convexity of feasible sets. Combined quadratic constraints on model uncertainty and additive disturbances are considered in order to simplify the analysis. Analytical optimal or suboptimal solutions of the basic problems involved in parameter or state estimation are presented, which are counterparts in this context of uncertain models to classical approximations of the sum and intersection of ellipsoids. The results obtained for combined quadratic constraints are extended to other types of model uncertainty.     

A bounded-error approach to simultaneous state and actuator fault estimation for a class of nonlinear systems

This paper proposes an approach for the joint state and fault estimation for a class of uncertain nonlinear systems with simultaneous unknown input and actuator faults. This is achieved by designing an unknown input observer combined with a set-membership estimation in the presence of disturbances and measurement noise. The observer is designed using quadratic boundedness approach that is used to overbound the estimation error. Sufficient conditions for the existence and stability of the proposed state and actuator fault estimator are expressed in the form of linear matrix inequalities (LMIs). Simulation results for a quadruple-tank system show the effectiveness of the proposed approach.     

Fault detection and identification relying on set-membership identifiability

Identifiability is the property that a mathematical model must satisfy to guarantee an unambiguous mapping between its parameters and the output trajectories. It is of prime importance when parameters must be estimated from experimental data representing input–output behavior and clearly when parameter estimation is used for fault detection and identification. Definitions of identifiability and methods for checking this property for linear and nonlinear systems are now well established and, interestingly, some scarce works (Braems et al., 2001, Jauberthie et al., 2011) have provided identifiability definitions and numerical tests in a bounded-error context. This paper resumes and better formalizes the two complementary definitions of set-membership identifiability and μ-set-membership identifiability of Jauberthie et al. (2011) and presents a method applicable to nonlinear systems for checking them. This method is based on differential algebra and makes use of relations linking the observations, the inputs and the unknown parameters of the system. Using these results, a method for fault detection and identification is proposed. The relations mentioned above are used to estimate the uncertain parameters of the model. By building the parameter estimation scheme on the analysis of identifiability, the solution set is guaranteed to reduce to one connected set, avoiding this way the pessimism of classical set-membership estimation methods. Fault detection and identification are performed at once by checking the estimated values against the parameter nominal ranges. The method is illustrated with an example describing the capacity of a macrophage mannose receptor to endocytose a specific soluble macromolecule.     

Parameter estimation of continuous-time bilinear systems based on numerical integration and separable non-linear least-squares

This paper considers the problem of estimating the parameters in continuous-time bilinear systems. The system identification approach is based on numerical integration and separable non-linear least-squares. The situation where the output signal is contaminated with noise is also discussed. The suggested estimation method uses a bias compensating approach and the model parameters together with parameters associated with the noise are estimated by solving an overdetermined system of equation in a least-squares sense.     

A unified wavelet-based modelling framework for non-linear system identification: the WANARX model structure

A new unified modelling framework based on the superposition of additive submodels, functional components, and wavelet decompositions is proposed for non-linear system identification. A non-linear model, which is often represented using a multivariate non-linear function, is initially decomposed into a number of functional components via the well-known analysis of variance (ANOVA) expression, which can be viewed as a special form of the NARX (non-linear autoregressive with exogenous inputs) model for representing dynamic input–output systems. By expanding each functional component using wavelet decompositions including the regular lattice frame decomposition, wavelet series and multiresolution wavelet decompositions, the multivariate non-linear model can then be converted into a linear-in-the-parameters problem, which can be solved using least-squares type methods. An efficient model structure determination approach based upon a forward orthogonal least squares (OLS) algorithm, which involves a stepwise orthogonalization of the regressors and a forward selection of the relevant model terms based on the error reduction ratio (ERR), is employed to solve the linear-in-the-parameters problem in the present study. The new modelling structure is referred to as a wavelet-based ANOVA decomposition of the NARX model or simply WANARX model, and can be applied to represent high-order and high dimensional non-linear systems.     

Efficient uncertainty quantification with the polynomial chaos method for stiff systems

The polynomial chaos (PC) method has been widely adopted as a computationally feasible approach for uncertainty quantification (UQ). Most studies to date have focused on non-stiff systems. When stiff systems are considered, implicit numerical integration requires the solution of a non-linear system of equations at every time step. Using the Galerkin approach the size of the system state increases from n to S × n, where S is the number of PC basis functions. Solving such systems with full linear algebra causes the computational cost to increase from O(n³) to O(S³n³). The S³-fold increase can make the computation prohibitive. This paper explores computationally efficient UQ techniques for stiff systems using the PC Galerkin, collocation, and collocation least-squares (LS) formulations. In the Galerkin approach, we propose a modification in the implicit time stepping process using an approximation of the Jacobian matrix to reduce the computational cost. The numerical results show a run time reduction with no negative impact on accuracy. In the stochastic collocation formulation, we propose a least-squares approach based on collocation at a low-discrepancy set of points. Numerical experiments illustrate that the collocation least-squares approach for UQ has similar accuracy with the Galerkin approach, is more efficient, and does not require any modification of the original code.