Particle filters for state and parameter estimation in batch processes

In process engineering, on-line state and parameter estimation is a key component in the modelling of batch processes. However, when state and/or measurement functions are highly non-linear and the posterior probability of the state is non-Gaussian, conventional filters, such as the extended Kalman filter, do not provide satisfactory results. This paper proposes an alternative approach whereby particle filters based on the sequential Monte Carlo method are used for the estimation task. Particle filters are initially described prior to discussing some implementation issues, including degeneracy, the selection of the importance density and the number of particles. A kernel smoothing approach is introduced for the robust estimation of unknown and time-varying model parameters. The effectiveness of particle filters is demonstrated through application to a benchmark batch polymerization process and the results are compared with the extended Kalman filter.     

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 hybrid data assimilation scheme for model parameter estimation: Application to morphodynamic modelling

We present a novel algorithm for joint state-parameter estimation using sequential three dimensional variational data assimilation (3D Var) and demonstrate its application in the context of morphodynamic modelling using an idealised two parameter 1D sediment transport model. The new scheme combines a static representation of the state background error covariances with a flow dependent approximation of the state-parameter cross-covariances. For the case presented here, this involves calculating a local finite difference approximation of the gradient of the model with respect to the parameters. The new method is easy to implement and computationally inexpensive to run. Experimental results are positive with the scheme able to recover the model parameters to a high level of accuracy. We expect that there is potential for successful application of this new methodology to larger, more realistic models with more complex parameterisations.     

Maximum a posteriori estimation of activation energies that control silicon self-diffusion

Self-diffusion in crystalline silicon is controlled by a network of elementary steps whose activation energies are important to know in a variety of applications in microelectronic fabrication. The present work employs maximum a posteriori (MAP) estimation to improve existing values for these activation energies, based on self-diffusion data collected at different values of the loss rates for interstitial atoms to the surface. Parameter sensitivity analysis shows that for high surface loss fluxes, the energy for exchange between an interstitial and the lattice plays the leading role in determining the shape of diffusion profiles. At low surface loss fluxes, the dissociation energy of large-atom clusters plays a more important role. Subsequent MAP analysis provides significantly improved values for these parameters.     

Adaptive divided difference filtering for simultaneous state and parameter estimation

A novel adaptive version of the divided difference filter (DDF) applicable to non-linear systems with a linear output equation is presented in this work. In order to make the filter robust to modeling errors, upper bounds on the state covariance matrix are derived. The parameters of this upper bound are then estimated using a combination of offline tuning and online optimization with a linear matrix inequality (LMI) constraint, which ensures that the predicted output error covariance is larger than the observed output error covariance. The resulting sub-optimal, high-gain filter is applied to the problem of joint state and parameter estimation. Simulation results demonstrate the superior performance of the proposed filter as compared to the standard DDF.     

Bootstrap prediction mean squared errors of unobserved states based on the Kalman filter with estimated parameters

In the context of linear state space models with known parameters, the Kalman filter (KF) generates best linear unbiased predictions of the underlying states together with their corresponding Prediction Mean Square Errors (PMSE). However, in practice, when the filter is run with the parameters substituted by consistent estimates, the corresponding PMSE do not take into account the parameter uncertainty. Consequently, they underestimate their true counterparts. In this paper, we propose two new bootstrap procedures to obtain PMSE of the unobserved states designed to incorporate this latter uncertainty. We show that the new bootstrap procedures have better finite sample properties than bootstrap alternatives and than procedures based on the asymptotic approximation of the parameter distribution. The proposed procedures are implemented for estimating the PMSE of several key unobservable US macroeconomic variables as the output gap, the Non-accelerating Inflation Rate of Unemployment (NAIRU), the long-run investment rate and the core inflation. We show that taking into account the parameter uncertainty may change their prediction intervals and, consequently, the conclusions about the utility of the NAIRU as a macroeconomic indicator for expansions and recessions.     

On asymptotic behaviors of a sensitivity penalization based robust state estimator

Asymptotic properties are investigated in this paper for the robust state estimator derived by Zhou (2008) [11]. A new formula is derived for the update of the pseudo-covariance matrix of estimation errors. In the case where plant nominal parameters are time invariant, it is shown that, in order to guarantee that this pseudo-covariance matrix converges to a constant positive definite matrix, it is necessary and sufficient that some stabilizability and detectability conditions are satisfied. It is also proved that when these conditions are satisfied, the robust estimator converges to a stable time-invariant system. Moreover, when the system is exponentially stable, this estimate is asymptotically unbiased and its estimation errors are upper bounded.     

State estimation for nonlinear systems under model uncertainties: a class of sliding-mode observers

This paper deals with the classical problem of state estimation, considering partially unknown, nonlinear systems with noise measurements. Estimation of both, state variables and unstructured uncertain term, are performed simultaneously. In order to transform the measured disturbance into system disturbance, an alternative system representation is proposed, which lead a more advantageous observer structure. The observer proposed contains a proportional-type contribution and a sliding term for the measurement of error, which provides robustness against noisy measurements and model uncertainties. Convergence analysis of the estimation methodology proposed is performed, analysing the equation of the dynamics of the estimation error; it is shown that the observer exhibits asymptotic convergence. Estimation of monomer concentration, average molecular weight, polydispersity and filtering of temperature in a batch stirred polymerization reactor illustrates the good performance of the observer proposed.     

Distributed moving horizon state estimation for nonlinear systems with bounded uncertainties

In this work, we propose a distributed moving horizon state estimation (DMHE) design for a class of nonlinear systems with bounded output measurement noise and process disturbances. Specifically, we consider a class of nonlinear systems that are composed of several subsystems and the subsystems interact with each other via their subsystem states. First, a distributed estimation algorithm is designed which specifies the information exchange protocol between the subsystems and the implementation strategy of the DMHE. Subsequently, a local moving horizon estimation (MHE) scheme is designed for each subsystem. In the design of each subsystem MHE, an auxiliary nonlinear deterministic observer that can asymptotically track the corresponding nominal subsystem state when the subsystem interactions are absent is taken advantage of. For each subsystem, the nonlinear deterministic observer together with an error correction term is used to calculate a confidence region for the subsystem state every sampling time. Within the confidence region, the subsystem MHE is allowed to optimize its estimate. The proposed DMHE scheme is proved to give bounded estimation errors. It is also possible to tune the convergence rate of the state estimate given by the DMHE to the actual system state. The performance of the proposed DMHE is illustrated via the application to a reactor-separator process example.     

Application and roll-out of infinite horizon MPC employing a nonlinear mechanistic model to paper machines

A mechanistic nonlinear model of the wet end of paper machine 6 (PM6) at Norske Skog Saugbrugs, Norway has been developed, and used in an industrial MPC implementation. The MPC uses an infinite horizon criterion, successive linearization of the model, and estimation of states and parameters by an augmented Kalman filter. Variation in important quality variables and consistencies in the wet end have been reduced substantially, compared to the variation prior to the MPC implementation. The MPC also provides better efficiency through faster grade changes, control during sheet breaks and start ups, and better control during periods of poor measurements. From May 2002 of the MPC has been the preferred controller choice for the process operators at PM6.In this paper we also study if the model can be applied to PM4 at Norske Skog Saugbrugs, and PM3 at Norske Skog Skogn, Norway. No structural changes were necessary in order to fit the model to PM4 and PM3 data. The time spent on fitting the PM6 model to PM4 and PM3 is approximately 1% of the time spent on developing the original model. This should be a strong incentive for focusing on mechanistic modeling in industries were there are many similar production lines or units.     

硬件在环系统中电机参数辨识及状态估计研究

矢量控制电机中的敏感参数转子时间常数的实时辨识和电机状态实时估计是电机高性能运行的保证,首先讨论了转子时间常数参数辨识,其次讨论了滑模观测器并给出了观测器稳定性分析,通过观测器得到感应电机磁链状态估计值;将得到的转子磁链值用于MRAS状态估计,得到实时的电机转速.设计了硬件在环仿真系统验证电机状态估计方法及参数辨识方法;硬件在环试验系统包含感应电机,电气测功器以及实时仿真器等硬件,通过硬件在环实时试验得到了实时的电机状态估计及参数辨识结果,试验结果验证了参数辨识及状态估计方法的有效性和实时性.     

多变量系统状态空间模型的递阶辨识

研究多变量系统状态空间模型的递阶辨识问题,推广了作者提出的标量系统状态和参数联合辨识算法.当状态可量测时,利用最小二乘原理直接辨识状态空间模型的参数矩阵;当状态不可测时,利用递阶辨识原理提出了状态空间模型递阶辨识方法,使用系统输入输出数据来估计系统的未知状态和参数.状态空间模型递阶辨识方法分为两步:首先假设系统状态是已知的(即参数估计算法中的未知系统状态用其估计代替),基于状态估计和系统输入输出数据递归计算系统参数估计;然后基于系统输入输出数据和获得的参数估计,递归计算系统的状态估计.