Dual Estimation of Fractional Variable Order Based on the Unscented Fractional Order Kalman Filter for Direct and Networked Measurements
Abstract
The paper is devoted to variable order estimation process when measurements are obtained in two different ways: directly and by lossy network. Since the problem of fractional order estimation is highly nonlinear, dual estimation algorithm based on Unscented Fractional Order Kalman filter has been used. In dual estimation process, state variable and order estimation have been divided into two subprocesses. For estimation state variables and variable fractional order, the Fractional Kalman filter and the Unscented Fractional Kalman filter have been used, respectively. The order estimation algorithms were applied to numerical examples and to real fractional variable order inertial system realized as an analog circuit.
Keywords
Fractional calculus Variable order derivative Analog model Estimation Kalman filter1 Introduction
Recently, estimation problem in dynamical systems and control is widely considered. What is crucial, the order of estimated system is usually unknown and can be even fractional.
Fractional calculus is a generalization of traditional integer order integration and differentiation actions onto noninteger order. The idea of such a generalization has been mentioned in 1695 by Leibniz and L’Hospital. At the end of nineteenth century, Liouville and Riemann introduced first definition of fractional derivative. However, only just in late 60’ of the twentieth century, this idea drew attention of the engineers. Fractional calculus was found a very useful tool for modeling behavior of many materials and systems, especially those based on the diffusion processes. The description and experimental results of modeling heat transfer processes were presented in [25, 30]. Theoretical background of fractional calculus can be found in [9, 11, 12, 13, 15, 18].
When the fractional order of derivative is not constant but depends on time, the various types of fractional variable order derivatives can be distinguished. In [14], nine different variable order derivative definitions have been given, and in [8, 36], three general types of variable order definitions have been able to find, but without clear interpretation. In papers [27, 28], the explanation of two main types and two recursive types of derivatives in the form of switching schemes are given. The equivalence between particular types of definitions and appropriate switching strategies are proven by authors. Moreover, based on these strategies, analog models of proper types derivatives were build and validated according to their numerical implementations. Based on these papers, it is possible to categorize fractional order derivatives into three switching strategies. The experimental results shown high accuracy for modeling the appropriate types of variable order definitions. In [29], analog realization of variable order derivative for multipleswitching order has been introduced; however, presented model gives nonstationary (variable parameter) system. Numerical routines for simulation of variable order derivatives based on different type definitions are given in [19].
When the state vector is not available directly from measurements, the Kalman filter algorithm can be used for estimate unknown states based on measurements and system dynamics [5, 6, 17]. More practical problem occurs when the physical data of a system are measured and analyzed through a network. Therefore, one of the practical areas are communication networks, where effort in analyzing the effect of packet losses has been highly considerable. To this kind of systems, generalization of Kalman filter algorithm can be applied [10, 16, 39]. For estimation of nonlinear systems, a set of generalized algorithms like Extended Kalman filter and Unscented Kalman filter are given in the literature [4, 5, 35]; especially, interesting algorithm is the Unscented Kalman filter that, in opposition to the Extended Kalman filter, not required differentiation of nonlinear function. In [5, 17], UKF algorithm was used to teaching process of neural networks. In [1], the estimation results for fractional nonlinear systems based on Extended and Unscented Fractional Kalman filter (UFKF) were presented.
When mathematical model of dynamical systems is described by fractional order difference or derivative equations, the modified Fractional Kalman filter (FKF) algorithm should be used [23]. This algorithm has been used, for example, for estimation of state variables in the dynamical system with ultracapacitor [3], as well in a chaotic secure communication scheme [7]. In the case of systems with fractional order dynamics with data sending over lossy networks, where networkinduced packet losses can become a source of degradation in estimation performance, the improved FKF has been investigated [31]. Fractional order estimation schemes for fractional and integer order systems with constant and variable fractional order colored noise are presented in [33]. Improved FKF for variable order systems is investigated in [40].
In practical application of fractional order systems, an identification of the system order plays a very important role, especially in the case of variable order systems. Usually, the parameters of the system were obtained during offline numerical minimization routines [25, 30]. In this paper, online dual estimation algorithms for state variable and order estimation, when measurements are obtained directly and by lossy network, are presented. For estimation state variables and variable fractional order, a FKF and UFKF have been used, respectively. Moreover, the verification of the developed estimation algorithm has been performed by testing it on a real electrical circuit analog model.
The paper is organized as follows. In Sect. 2, particular types of fractional variable order derivatives are introduced. In Sect. 3, basic properties of discrete fractional variable order statespace model are recalled. In Sect. 4, analog model of fractional variable order system is presented. In Sect. 5, dual estimation schemes based on UFKF for direct and networked measurements cases are presented. In Sect. 6, numerical results of modeling are presented. Finally, in Sect. 7, order estimation for analog model is presented.
2 Fractional Variable Order Grünwald–Letnikov Type Derivatives
For the case of order changing with time (variable order case), variety of definitions can be found in the literature [8, 36]. Among them all, we present only two. The first one is obtained by replacing in (1) a constant order \(\alpha \) by variable order \(\alpha (t)\). In this approach, all coefficients for past samples are obtained for present value of the order and are given as follows:
Definition 1
The definition of dual type of variable order derivative, that is consider in this paper, is given as follows:
Definition 2
3 Discrete Variable Fractional Order StateSpace System
Remark 2
4 Analog Model of Fractional Variable Order Integral System
The circuit branches with resistors \(R_1\), \(R_2\) and capacitors \(C_1\), \(C_2\) represent an approximation of halforder (\(\alpha =0.5\)) impedance when electronic switches \(S_1\) and \(S_2\) are connected to terminals denotes as 2. The halforder impedance can be built according to algorithm described in [24] (\(R_1=2.4\,\text {k}{\Omega }\), \(R_2=8.2\,\text {k}{\Omega }\), \(C_1=330\,\text {nF}\) and \(C_2=220\,\text {nF}\)). The quantity of resistors and capacitors determines the accuracy of whole impedance. This model approximation contains 200 passive elements. The frequency response of real halforder impedance and its model are overlapping in wide range frequency. Otherwise, when switch \(S_1\) is connected to the terminal 1 and \(S_2\) is grounded, then the voltage follower \(A_3\) is charging the dominoladder branches to the value of output signal. It is a necessary condition to keep the behavior of \(\mathcal {D}\)type variable order definition. Finally, the branch with \(R_1\) and \(C_1\) elements connected to the negative input of amplifier \(A_1\) represents a firstorder impedance.
In fact, the order of system can be changed between \(0.5\) (halforder integral) and \(1\) (firstorder integral) in any time and depends on position of switches (\(S_1\), \(S_2\) and \(S_3\)). Resistors \(R_a\) and \(R_b\) allow to sustain the constant value of integrator gain. Operational amplifier \(A_2\) in configuration with resistors R gives voltage amplifier of a gain equal to \(\)1 providing reinversion of output signal (already inverted by integrator circuit).
5 Dual Estimation Based on UFKF Filter
Generally, dual estimation refers to the issue of simultaneously estimating the state of a dynamic system and its parameters. In our case, we will deal with estimation of a parameter changing in time, i.e., with estimation of the variable order. Dual estimation algorithms were already considered, e.g., in [37, 38].
5.1 Variable Order Estimation Problem
Moreover, in constant order (integer or fractional) systems the influence of the step time h can be easily incorporated into system matrices A and B. In variable order case, such incorporation leads to nonstationary system with variable in time system matrices. That is why incorporation of the step time has to be performed into the model itself, which provides the necessity for generalization of appropriate Kalman filter algorithm for this modification.
5.2 Dual Estimation Scheme
Because the state vector estimation problem (KFx filter) is linear, the fractional variable order Kalman filter, given below, has been used.
Proposition 1
Proof
Due to high nonlinearity of the order estimation problem, the Unscented Fractional Order Kalman filter is used (as the KFw filter in dual estimation scheme presented in Fig. 2).
Proposition 2
Proof
The algorithm is a generalization of the Fractional Unscented Kalman filter given in [20], while the step time h and \(\mathcal {A}\)type variable order difference definition is taken into consideration.
5.3 Dual Estimation for Networked Measurements
Analogously as in direct measurement case, for state vector estimation (KFx filter), the fractional variable order Kalman filter for the networked systems case has been used.
Proposition 3
Proof
The algorithm is a modification of the algorithm given in Proposition 1, including information about packages losing \(\gamma _k\) in last two equations and is similar to this presented in [34] with additionally including step time h. \(\square \)
For order estimation in network systems as a KFw filter the Unscented Variable Fractional Order Kalman filter has been used. For simplicity is presented the algorithm for estimation simple one order; however, it can be easily extended for multiple orders estimation. The algorithm is given in the form of following theorem:
Proposition 4
Proof
The algorithm is a modification of the algorithm given in Proposition 2, with including information about packages losing \(\gamma _k\) in last three equations. \(\square \)
6 Numerical Results
Numerical results, presented in following section, have been obtained in MATLAB/Simulink environment.
6.1 Order Estimation for Direct Measurements
Example 1
Figure 4 presents input and output of the analog system—the data for estimation process, and results of applying dual estimation algorithms to these data. As it can be seen, the order is estimated with very high accuracy, and algorithm needed very short time to adjust for order changing.
Example 2
Figure 5 presents input and output of the analog system with higher output noise than in example before. It also presents results of applying dual estimation algorithm to these data. As it can be seen, the order is estimated with quite high accuracy; however, accuracy is lower than for a case of lower noise.
Example 3
(Order estimation for direct measurements and different output noise variances) Let us consider the system from Example 1 with the same parameters, except for the variable order, which takes in the form of sinusoidal function. Below the results of series of experiments for estimation of variable order, performed for different output noise variances R, are presented. What is expected, with increasing variance noise, the accuracy of estimation error decreases (see Table 1; Fig. 6).
Comparison of order estimation error norms (where \(e_k = \alpha _k  \tilde{\alpha }_k\), \(k=200,\ldots ,T/h\)), during time T for different output noise variances R
Output noise variance R  Estimation error \(h\sum _{i=200}^{T/h}{\hbox {e}^2_i}\) 

0.00000016  0.0001 
0.00001504  0.0003 
0.00152402  0.0039 
Example 4
(Order estimation for different values of \(^*Q^w_k\)) To investigate influence of parameter \(^*Q^w_k\), which is expected variability of estimated order, into estimation accuracy, let us consider the same system as in Example 1 with different values of matrix \(^*Q^w_k\).
The comparison of order estimation results is presented in Fig. 7. As it can be noticed, the smaller value of \(^*Q^w_k\) the faster estimated order approach to the original one, however differences between results are not so significant. In order to thoroughly explain the differences, let us analyze the value of \(\sqrt{(L+\lambda )\tilde{P}_k^w}\), which define a spread of sigma points obtained in UFKF algorithm (Fig. 8).
Example 5
6.2 Order Estimation for Networked Measurements
Example 6
(Order estimation for direct measurements and low noise) Parameters of the system and filters are the same as in Example 1, and the transmission rate for measurements is \(30\,\%\).
As it can be seen in Fig. 10, accuracy of order estimation is lower than it was obtained in direct measurements case, but still it shows high accuracy of the dual estimation algorithm. The losing of accuracy is caused by losing information during transmission by the communication network.
Example 7
(Order estimation for direct measurements and high noise) Parameters of the system and filters are the same as in Example 2, and the transmission rate for measurements is 30 %.
Figure 11 presents estimation results for higher noise than in the previous example. As it can be notice, higher noise caused lower accuracy of order estimation, but it is still on the reasonable level.
7 Order Estimation for Analog Model
In order to validate proposed algorithm in real application, experimental data obtained from variable order inertial system will be used. Such a system is realized by putting fractional variable order integrator in unity feedback system, as shown in Fig. 12.
All measurement data have been gathered with time sample equals to 0.001 sec and input signal equal to \(0.5\cdot {H(t)}\), where H(t) is a Heaviside step function.
7.1 Analog Model of Variable Order System

data acquisition card dSPACE 1104;

operational amplifiers TL071;

electronic switches DG303;

passive elements such as: resistors \(R_1=2.4\,\text {k}{\Omega }\), \(R_2=8.2\,\text {k}{\Omega }\), \(R=100\,\text {k}{\Omega }\), \(R_a=43\,\text {k}{\Omega }\) and \(R_b=33\,\text {k}{\Omega }\), capacitors \(C_1=330\,\text {nF}\) and \(C_2=220\,\text {nF}\).
7.2 Order Estimation for Analog Model and Direct Measurements
7.3 Order Estimation for Analog Model and Networked Measurements
Comparison of order estimation error norms (where \(e_k = \alpha _k \tilde{\alpha }_k\), \(k=1,\ldots ,T/h\)), during time T for different step times h
Step time h (s)  Estimation error \(h\sum _{i=1}^{T/h}{\hbox {e}^2_i}\) 

0.0010  0.0113 
0.0020  0.0277 
0.0050  0.0572 
0.0080  0.0665 
0.0100  0.0505 
0.0200  0.0904 
Figure 17 shows results of applying dual estimation algorithms for data obtained by lossy network. As it can be seen, the order is estimated with quite high accuracy, however with lower accuracy than for direct measurements case.
7.4 Order Estimation for Analog Model and Networked Measurements Transmitted by Real Network
8 Conclusions
In this paper, the variable order estimation algorithms for a case when measurements are obtained directly and by lossy network have been presented. The order estimation algorithms were applied to numerical examples and to real fractional variable order inertial system. Since the problem of fractional order estimation is highly nonlinear, the dual estimation algorithm has been used. For state variables and variable fractional order estimation, the Fractional Kalman filter and the Unscented Fractional Kalman filter have been used, respectively. Numerical results shown efficiency of proposed algorithms for direct and networked measurements as well. The estimation algorithm has been also tested on a real object being electrical circuit analog model. The proposed algorithms have confirmed the possibility of further use in case of order estimation of real objects of unknown order both constant or variable.
References
 1.R. CaballeroAguila, Extended and unscented filtering algorithms in nonlinear fractional order systems with uncertain observations. Appl. Math. Sci. 6(30), 1471–1486 (2012)MathSciNetzbMATHGoogle Scholar
 2.A. Dzielinski, D. Sierociuk, Reachability, controllability and observability of the fractional order discrete statespace system, in Proceedings of the IEEE/IFAC International Conference on Methods and Models in Automation and Robotics, MMAR’2007 (Szczecin, Poland, 2007), pp. 129–134Google Scholar
 3.A. Dzielinski, D. Sierociuk, Ultracapacitor modelling and control using discrete fractional order statespace model. Acta Montan. Slovaca 13(1), 136–145 (2008)zbMATHGoogle Scholar
 4.B. Gibbs, Advanced Kalman Filtering, LeastSquares and Modeling: A Practical Handbook (Wiley, New York, 2011)CrossRefGoogle Scholar
 5.S. Haykin, Kalman Filtering and Neural Networks (Wiley, New York, 2001)CrossRefGoogle Scholar
 6.E. Kalman Rudolph, A new approach to linear filtering and prediction problems. Trans. ASME J. Basic Eng. 82(Series D), 35–45 (1960)CrossRefGoogle Scholar
 7.A. KianiB, K. Fallahi, N. Pariz, H. Leung, A chaotic secure communication scheme using fractional chaotic systems based on an extended fractional Kalman filter. Commun. Nonlinear Sci. Numer. Simul. 14(3), 863–879 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
 8.C. Lorenzo, T. Hartley, Variable order and distributed order fractional operators. Nonlinear Dyn. 29(1–4), 57–98 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
 9.K. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations (Wiley, New York, 1993)zbMATHGoogle Scholar
 10.M. Moayedi, Y. Soh, Y. Foo, Optimal Kalman filtering with random sensor delays, packet dropouts and missing measurements, in Proceedings of the American Control Conference (ACC 2009), (2009), pp. 3405–3410Google Scholar
 11.C.A. Monje, Y. Chen, B.M. Vinagre, D. Xue, V. Feliu, FractionalOrder Systems and Controls (Springer, New York, 2010)CrossRefzbMATHGoogle Scholar
 12.K.B. Oldham, J. Spanier, The Fractional Calculus (Academic Press, London, 1974)zbMATHGoogle Scholar
 13.I. Podlubny, Fractional Differential Equations (Academic Press, London, 1999)Google Scholar
 14.L.E.S. Ramirez, C.F.M. Coimbra, On the selection and meaning of variable order operators for dynamic modeling. Int. J. Differ. Equ. 2010, 846107 (2010). doi: 10.1155/2010/846107
 15.S. Samko, A. Kilbas, O. Maritchev, Fractional Integrals and Derivative. Theory and Applications (Gordon & Breach Science Publishers, New York, 1987)Google Scholar
 16.L. Schenato, Optimal estimation in networked control systems subject to random delay and packet drop. IEEE Trans. Autom. Control 53(5), 1311–1317 (2008)MathSciNetCrossRefGoogle Scholar
 17.J. Schutter, J. Geeter, T. Lefebvre, H. Bruynickx, Kalman filters: a tutorial. Journal A 40(4), 52–59 (1999)Google Scholar
 18.H. Sheng, Y. Chen, T. Qiu, Signal Processing Fractional Processes and FractionalOrder Signal Processing (Springer, London, 2012)CrossRefzbMATHGoogle Scholar
 19.D. Sierociuk, Fractional Variable Order Derivative Simulink Toolkit (2012). http://www.mathworks.com/matlabcentral/fileexchange/38801fractionalvariableorderderivativesimulinktoolkit
 20.D. Sierociuk, Uzycie ulamkowego filtru kalmana do estymacji parametrow ukladu ulamkowego rzedu (Using of Fractional Kalman Filter for parameters estimation of fractional order system), in XV Krajowa Konferencja Automatyki (Warsaw, Poland, 2005)Google Scholar
 21.D. Sierociuk, System properties of fractional variable order discrete statespace system, in Proceedings of the 13th International Carpathian Control Conference (ICCC) (2012), pp. 643–648Google Scholar
 22.D. Sierociuk, Fractional Kalman filter algorithms for correlated system and measurement noises. Control Cybern. 42(2), 471–490 (2013)MathSciNetzbMATHGoogle Scholar
 23.D. Sierociuk, A. Dzieliński, Fractional Kalman filter algorithm for states, parameters and order of fractional system estimation. Appl. Math. Comput. Sci. 16(1), 129–140 (2006)MathSciNetGoogle Scholar
 24.D. Sierociuk, A. Dzielinski, New method of fractional order integrator analog modeling for orders 0.5 and 0.25, in Proceedings of the 16th International Conference on Methods and Models in Automation and Robotics (MMAR) (Miedzyzdroje, Poland, 2011), pp. 137 –141Google Scholar
 25.D. Sierociuk, A. Dzielinski, G. Sarwas, I. Petras, I. Podlubny, T. Skovranek, Modelling heat transfer in heterogeneous media using fractional calculus. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 371(1990), 20120146 (2013). doi: 10.1098/rsta.2012.0146
 26.D. Sierociuk, M. Macias, P. Ziubinski, Experimental results of modeling variable order system based on discrete fractional variable order statespace model, in Theoretical Developments and Applications of NonInteger Order Systems, Lecture Notes in Electrical Engineering, vol. 357, eds. by S. Domek, P. Dworak (Springer International Publishing 2016), pp. 129–139Google Scholar
 27.D. Sierociuk, W. Malesza, M. Macias, Derivation, interpretation, and analog modelling of fractional variable order derivative definition. Appl. Math. Model. 39(13), 3876–3888 (2015)MathSciNetCrossRefGoogle Scholar
 28.D. Sierociuk, W. Malesza, M. Macias, On the recursive fractional variableorder derivative: equivalent switching strategy, duality, and analog modeling. Circuits Syst. Signal Process. 34(4), 1077–1113 (2015)MathSciNetCrossRefGoogle Scholar
 29.D. Sierociuk, W. Malesza, M. Macias, Practical analog realization of multiple order switching for recursive fractional variable order derivative, in Proceedings of the 20th International Conference on Methods and Models in Automation and Robotics (MMAR) (Miedzyzdroje, Poland , 2015)Google Scholar
 30.D. Sierociuk, T. Skovranek, M. Macias, I. Podlubny, I. Petras, A. Dzielinski, P. Ziubinski, Diffusion process modeling by using fractionalorder models. Appl. Math. Comput. 257, 2–11 (2015)Google Scholar
 31.D. Sierociuk, I. Tejado, B.M. Vinagre, Improved fractional Kalman filter and its application to estimation over lossy networks. Signal Process. 91(3, SI), 542–552 (2011)CrossRefzbMATHGoogle Scholar
 32.D. Sierociuk, M. Twardy, Duality of variable fractional order difference operators and its application to identification. Bull. Pol. Acad. Sci. Tech. Sci. 62(4), 809–815 (2014)Google Scholar
 33.D. Sierociuk, P. Ziubinski, Fractional order estimation schemes for fractional and integer order systems with constant and variable fractional order colored noise. Circuits Syst. Signal Process. 33(12), 3861–3882 (2014)MathSciNetCrossRefGoogle Scholar
 34.D. Sierociuk, P. Ziubinski, Variable order fractional kalman filters for estimation over lossy network, in Lecture Notes in Electrical Engineering, vol. 320 (2015), pp. 285–294Google Scholar
 35.J. Sum, C. Leung, L. Chan, Extended kalman filter in recurrent neural network training and pruning. Technical Report, Department of Computer Science and Engineering, The Chinese University of Hong Kong (1996). CSTR9605Google Scholar
 36.D. Valerio, J.S. da Costa, Variableorder fractional derivatives and their numerical approximations. Signal Process. 91(3, SI), 470–483 (2011)CrossRefzbMATHGoogle Scholar
 37.E.A. Wan, R. van der Merwe, A.T. Nelson, Dual estimation and the unscented transformation, in Advances in Neural Information Processing Systems, vol. 12 (2000), pp. 666–672Google Scholar
 38.E.A. Wan, A.T. Nelson, Dual kalman filtering methods for nonlinear prediction, smoothing, and estimation, in Advances in Neural Information Processing Systems, vol. 9 (1997)Google Scholar
 39.L. Yi, S. Hexu, L. Zhaoming, Z. Jian, Design for wireless networked control systems with timedelay and data packet dropout, in Proceedings of the 27th Chinese Control Conference (CCC 2008), vol. 4 (2008), pp. 656–659Google Scholar
 40.P. Ziubinski, D. Sierociuk, Improved fractional kalman filter for variable order systems, in Proceedings of International Conference on Fractional Differentiation and its Applications (Catania, Italy, 2014)Google Scholar
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