Elsevier

Signal Processing

Volume 152, November 2018, Pages 217-226
Signal Processing

The uncertainty learning filter: A revised smooth variable structure filter

https://doi.org/10.1016/j.sigpro.2018.05.025Get rights and content

Highlights

  • An advanced version of smooth variable structure filter denoted as uncertainty learning filter is proposed for state estimation of nonlinear system in case of imprecise model description.

  • Dependencies of the estimation performance on parameters can be avoided by introducing an uncertainty learning parameter which can be adapted online.

  • The uncertainty learning parameter has the capability to tune the proposed filter according to the uncertainties present in the current model description.

  • Boundedness of estimation error is proven for the adaptive filtering approach.

Abstract

In this contribution the problem of state estimation of systems with imprecise description is addressed. A revised version of smooth variable structure filter (SVSF) called uncertainty learning filter (ULF) is proposed. By introducing an uncertainty learning parameter a novel strategy to minimize a pseudo mean squared estimation error is developed. The introduced uncertainty learning parameter acts as a tuning factor controlling the influence of the imprecise process model on the estimations. Parameter dependencies can be avoided by adapting the uncertainty learning parameter from the innovation process. Consequently the proposed adaptive uncertainty learning filter (A-ULF) has the ability to tune itself according to the uncertainties of the process model. The boundedness of estimation error of A-ULF approach is proven. Simulation results are given to compare the estimation performance of the proposed method with SVSF, and the extended Kalman filter (EKF).

Introduction

Filters play an important role in the field of estimation and supervision of stochastical systems. They can be applied to state estimation, unknown input estimation, fault detection, and parameter estimation. Most estimation approaches require a certain description of the system behavior. Therefore, estimation performance decreases under uncertain conditions.

Different approaches exist to estimate system states under uncertain conditions. Using minimum variance unbiased [1] or augmented state filters [2] additive modeling variations can be considered as unknown inputs. Consequently, the effects of the erroneous model can be attenuated by unknown input estimation. An alternative approach is robust Kalman filtering [3]. By modeling parameter variations an upper bound for mean squared estimation error of Kalman filter can be found. Determining a gain that minimizes the upper bound leads to the concept of robust Kalman filtering. In H filtering energy-bounded signals which represent disturbances (model uncertainties, noise with unknown statistics) are considered [4]. The aim of H filtering is to minimize the estimation error according to worst possible disturbances. The posteriori pdf of Bayes filter can be approximated using particle filters even if the transition pdf and likelihood function are not precisely known. In [5] the true system description is considered as unknown. A set of possible models including transition pdfs and likelihood functions is considered. Using different models a bank of particle filters can be applied. By incorporating measurements the posteriori pdfs of the different models are approximated, and used to fuse state estimations of the particle filters to one global estimation based on all considered models.

Smooth variable structure filter (SVSF) firstly introduced in [6] can be applied for state estimation in case of uncertain model description. The recursive nonlinear discrete-time SVSF approach combines the discontinuous feedback of estimation error known from sliding mode observer with the prediction-correction-scheme of the Kalman filter (KF) [6]. For the purpose of robust estimation a stochastic gain (switching gain) is formulated [6]. Switching gain enforces estimation error to be bounded, even in the case of uncertain parameters. Therefore, the estimation trajectory remains in a bounded space around the true state trajectory called existence subspace. Within the existence subspace a zigzag motion of the estimation trajectory around the true state trajectory occurs (chattering) [6]. By applying a so-called smoothing boundary layer [6], chattering can be reduced and the estimation error can be decreased. Consequently, under uncertain conditions SVSF can yield more accurate state estimations compared with KF. According to [7] SVSF provides more precise state estimations of nonlinear systems compared to extended Kalman filter (EKF), unscented Kalman filter (UKF), and particle filter (PF), if the initial conditions are badly chosen. Due to the switching gain better performance of SVSF is achieved. The estimations of SVSF reach the existence subspace and estimation error stays bounded. In [8] a time continuous version of SVSF applicable to linear systems is derived.

A disadvantage of SVSF approach is that all system states have to be measured. One solution to generate artificial measurements is proposed in [6] similar to the Luenbergers method of reduced observers. Nevertheless, the proposed strategy is not suitable because it assumes that no model uncertainties exist. Further, numerical problems may occur as bijective mappings from the states to the measurements are required [9]. Another approach proposed in [10] is based on mapping states, inputs, and process noise to the measurement space. The mapping of the states is realized using the observability matrix O. The mapping of the inputs and the process noise is realized by matrices To and Tw. The matrices To, Tw are build using the description (A, B, C, D) of a linear system and have a Toeplitz structure [10], [11]. If the considered system does not have any zeros and is given in observable canonical form as defined in [10] then the introduced Toeplitz and observability matrices are independent from parameters of the system matrix A and therefore independent from model uncertainties. Consequently, additional measurements can be generated only differing from the true states by an additive term affected by process and measurement noise. The additive term itself can be considered as a zero-mean measurement noise. One problem of the Toeplitz/Observability approach is that the observable canonical form can only be obtained for systems with one measurement and one degree of freedom. Nevertheless a general observable canonical form is derived in [9].

Another problem of SVSF results from the dependency of the estimation performance on the width of the smoothing boundary layer. In [12] an estimation error model for SVSF is proposed, and in [13] the estimation error is minimized according to smoothing boundary layer width. Consequently, a smoothing boundary layer width that minimizes the estimation error can be achieved. It is worth to mention that the derived estimation error model for SVSF must be suboptimal. It can not consider model uncertainties. Furthermore switching gain is assumed to be deterministic, but in fact it is stochastic. Additionally, estimation error model derived in [12] is only applicable to linear systems.

Smooth variable structure filter can be combined with other estimators. In [14], [15] chattering process of SVSF is evaluated to determine if model uncertainties are present. In case of no model uncertainties EKF or UKF equations are used to obtain strong estimation performance. Otherwise SVSF is used to obtain stable and error-bounded estimations.

A modification of smooth variable structure filter has been introduced as second-order smooth variable structure filter (2nd-order SVSF) in [16] to reduce chattering effects. From sliding mode control it is known that chattering attenuation can be achieved if sliding variable and at least its second order derivatives converge asymptotically to zero [17]. Consequently, the gain of 2nd-order SVSF ensures that the output estimation error and its derivative converge to zero [16]. Nevertheless, in contrast to chattering known from sliding mode observer (SMO) chattering of SVSF approach is caused by measurement noise. If output estimation error and its derivative reach the zero value the remaining system state estimation error consists of unaffected measurement noise. Briefly, measurement noise can not be filtered out by 2nd-order SVSF approach.

Several applications of SVSF exist. In [6] parameters and states of an electro-hydraulic actuator (EHA) system are estimated under uncertain conditions. For a simplified linear model of EHA, containing model uncertainties, SVSF approach leads to better estimation performance in contrast to KF. In [18] a multiple-model (MM) approach based on SVSF is used to detect and isolate predefined faults e. g. leakage effects on the EHA system. It is shown that combining SVSF with the MM strategy can lead to higher probabilities in detecting faults than combining the EKF with the MM approach. The state of charge of lithium-ion batteries is estimated in [19] using SVSF, which gives better estimation results in comparison to an EKF based approach. In [20] the multiple-model SVSF approach is used for tracking of aircrafts. The dynamic behavior of aircrafts is described by different models i. e. straight flight and turning or climbing maneuvers. In comparison to EKF the SVSF approach gives more accurate state estimations, especially for the turn rate of the vehicle. Again SVSF has the lower estimation error because of the action of switching gain which enforces the trajectory of the estimated states to reach the existence subspace. In [21] SVSF is used for tracking a single object in a cluttered scene by combining SVSF with a probabilistic data association (PDA) algorithm. The SVSF approach achieves a more accurate state estimation compared to the KF.

It has to be mentioned that the calculation of the estimation error covariance matrix of SVSF used for the MM approaches and the PDA algorithm are all based on the estimation error model proposed in [12].

In this paper a revised smooth variable structure filter is developed, leading to a novel filter introduced as uncertainty learning filter. Based on a reformulation of SVSF an estimation error model is introduced. The proposed formulation of the estimation error model considers model uncertainties and is applicable to nonlinear systems. Introducing an uncertainty learning parameter gives the possibility to derive a new deterministic gain that minimizes a pseudo mean squared estimation error. Based on this novel gain design in combination with the reformulation of SVSF the new uncertainty learning filter (ULF) approach is realized. The performance of ULF depends on the introduced uncertainty learning parameters, so an adaptive uncertainty learning filter (A-ULF) approach is developed.

This paper is organized as follows. In Section 2 the original SVSF approach is briefly presented and a reformulation is detailed. In Section 3 an estimation error model for the reformulated SVSF is proposed. Based on the estimation error model the ULF and the A-ULF approaches are derived. The performance evaluation of proposed ULF and A-ULF for a linear and a nonlinear system is given in Section 4.

Section snippets

Smooth variable structure filter and reformulation

The SVSF approach and its reformulation are presented.

The uncertainty learning filter

In this section an estimation error model for SVSF is proposed. The proposed error model is applicable to nonlinear process models. Model uncertainties can be considered by introducing an uncertainty learning parameter. So a deterministic gain, minimizing a pseudo mean squared estimation error (MSE), is achieved, leading to the novel uncertainty learning filter.

Linear system behavior

As example the electrohydraulic actuator (EHA) described in [6] is considered. For sample time Ts=0.001s the system can be described as xk+1=Axk+Buk+qk,yk=Cxk+rk,withmatricesA=[10.0010010.001557.0228.6160.9418],B=[00557.02],C=diag([0.80.80.8]).Noise processes qk, rk are assumed to be uncorrelated with each other and with x0, and are simulated as white, stationary, Gaussian, zero-mean processes with covariance matrices Q=E(qkqkT)=diag([ααα]T) , and R=E(rkrkT)=diag([βββ]T). Factors α, β are

Conclusion

In this paper based on a reformulation of SVSF a new recursive filter called uncertainty learning filter (ULF) is proposed. The ULF approach can be applied to linear and nonlinear systems when the exact process model is unknown and all states are measured. By introducing an uncertainty learning parameter the influence of the process model on the estimations can be tuned. The uncertainty learning parameter can be automatically adapted from the innovation process leading to adapted uncertainty

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