Fractional order unknown input filter design for fault detection of discrete fractional order linear systems

Abstract

In this study, a new method is introduced to design an estimator for discrete-time linear fractional order systems, which are affected by unknown disturbances. The main goal of this study is decoupling disturbance and uncertainties from the true states for discrete fractional order systems in noisy environment. The fractional Kalman filter framework is exploited to develop a robust estimator against unknown inputs (UIs) in noisy environment. The proposed filter is exploited to detect faults in fractional order systems. Simulation results illustrate the advantages of this robust filter for state estimation and fault detection of fractional order model of ultra-capacitor (UC). The robustness of the designed filter is shown in the sense of disturbance decoupling in the presence of noise.

Keywords

Fractional order systems fractional Kalman filter unknown input filter fault detection

Introduction

Nowadays, by development of technology and increasing the complexity of the systems, their safety and reliability are the most important concerns for engineers. Fault is one the most important issues that results in malfunctioning of the system and makes it unsafe. Accordingly, in recent decades several studies have been devoted to detect and diagnose faults in practical systems (Ahmadizadeh et al., 2014a; Chadli et al., 2013; Du, 2017; Hassani et al., 2017; Kowsari et al., 2016; Zarei and Shokri, 2014; Zhong et al., 2013).

Disturbance, which is also considered as unknown input (UI), is one of the most common problems that affect the fault detection procedure. In this regard, different approaches such as model-based fault detection (FD) (Ahmadizadeh et al., 2013, 2014b; Negash et al., 2016; Tan and Patton, 2015), data driven FD (Ma et al., 2010; Yin et al., 2014; Yao et al., 2010) and signal-based FD, have been introduced to achieve reliable FD algorithms.

Model-based FD is one of the fruitful approaches for control engineering community. These approaches commonly employ an algorithm that estimate the state of the systems, considering disturbance effects. There are different approaches to achieve this aim, for example eigen structure assignment (Patton and Chen, 2000), unknown input observer (UIO) (Gao et al., 2016), sliding mode observers (Alwi et al., 2009), fault detection filter design for nonhomogeneous Markovian jump systems by a T-S fuzzy approach (Li et al., 2016), and passive filter design for neural networks with semi-Markovian jump parameters and mixed time delays (Shi et al., 2017). In these works, no control signal is designed and the main contribution is devoted on FD filter design. However, these approaches can be used for active fault tolerant control (FTC) design, in which FD and estimation is an essential step. It should be noted that FTC performance can be influenced by FD and estimation accuracy. Moreover, H∞ approach, which uses this index to attenuate disturbance effect or uncertainty, can be used for FD purposes (Hassani et al., 2017; Zhong et al., 2013).

Monitoring the states by utilizing input, output and system model, is the first step to achieve a robust model based FD. Therefore, an appropriate observer is necessary to reach state estimation and FD simultaneously with respect to disturbance. Unknown input observer is the most applicable method for disturbance decoupling that has been attracted much attention in recent decades (Chadli et al., 2013; Gao et al., 2016; Tan and Patton, 2015). The main goal of this observer is decoupling modelling errors, disturbances or UIs from original states.

Noise effect is another issue that have been investigated for FD purposes and state estimation in several studies. In this regard, Kalman filter with an optimal gain is the best choice to cope with the noise effect. A procedure for designing UIOs by using Kalman filter algorithm, which is appropriate to estimate states with UIs for integer order systems is studied in Keller and Sauter (2013) and Chen and Patton (1996).

As mentioned earlier, a lot of researches deal with the problems of robust state estimation and FD for integer order systems, while integer order representation cannot explain all aspects of the system’s features. In other words, integer order modelling does not result in an accurate mathematical model. On the other hand, fractional calculus is an efficient way to model physical systems, which integer order fulfil it hardly (Monje et al., 2010; West et al., 2012). For instance, in the large-scale systems, identification and description of the system by integer order is almost difficult, while its fractional order counterpart can easily describe the system.

The history of fractional order calculus idea, which had been arisen from a question that Leibniz asked to L’Hospital, goes back to 1695 (Oldham and Spanier, 1974). In the first decades, the fractional calculus had been used just in theoretical calculation, while the important works on fractional calculus in control engineering were reported in the late 1960s. In Vinagre and Feliu (2002), as an application of fractional order calculus, an electromechanical system and a robot arm are modeled by fractional order equations. Fractional $P I^{λ} D^{μ}$ controller, which has more degree of freedom to control the process, is another application of fractional derivative and integration in control systems (Podlubny et al., 1997). UIO design for continuous fractional order systems, and its application in Glucose-Insulin systems in continuous form, is studied in N’Doye et al. (2013) and N’Doye et al. (2012), respectively. Nevertheless, it should be noted that noise effect has not been considered in these studies.

Owing to more complexity of systems, which are modelled by fractional order, the safety in these systems should be emphasised. Therefore, robust state estimation and FD are necessary for these systems. State estimation and FD in noisy environments for fractional order systems are an important subject that should be considered. However, few studies have been investigated FD for fractional order systems, and to the best of our knowledge, both noise and disturbances effects have not been considered for state estimation and fault detection in these systems. Since the systems is being digitalized to implement in digital computers, discrete fractional order models that is achieved from Grunwald-Letnikov (GL) definition is suitable for these applications. Therefore, filter design with GL definition is more effective and easier for practical implementation.

In Aribi et al. (2014) an approach is investigated to design a filter for FD with application in thermal systems, which is represented by fractional order model. Fractional Kalman filter (FKF) design framework is the best choice to handle the noise effect for linear fractional order systems. This approach is studied in Sierociuk and Dzielinski (2006) using GL definition, while the disturbance effect is neglected.

Motivated by these considered drawbacks, in current work, a novel fractional unknown input filter (FUIF) is proposed for state estimation with disturbance decoupling and FD of discrete-time fractional order systems in noisy environment. Obtained simulation result illustrates the efficiency of the proposed FUIF for state estimation and fault detection of an ultra-capacitor compared with the FKF in the presence of disturbances and noise. Moreover, fault estimation is another advantage of the proposed filter, which is essential for active fault tolerant control systems. In a special case, when H¹ as a parameter of the FUIF, is equal to zero, the FUIF has the same structure as the FKF. It is worth mentioning that the proposed method has the same computational efforts as the FKF.

The remainder parts of this study are organized as follows. In Section 2, fractional calculus and fractional systems in discrete-time state space are investigated. A new procedure for linear FUIF design is studied in Section 3. Simulation results for the fractional order model of an ultra-capacitor (UC) in the presence of noise, disturbance and fault are investigated in Section 4.

Fractional calculus

The aim of this section is to introduce Grunwald-Letnikov (GL) fractional derivative definition, which is a suitable form for numerical implementation and applications. This definition in discrete form is represented as (Monje et al., 2010)

Δ^{α} x (k) = lim_{h \to 0} \frac{1}{h^{α}} \sum_{i = 0}^{k} {(- 1)}^{i} (\begin{matrix} α \\ i \end{matrix}) x (k - i)

(1)

where

(\begin{matrix} α \\ i \end{matrix}) = {\begin{matrix} 1 & i = 0 \\ \frac{α (α - 1) \dots (α + 1 - i)}{i!} & i > 0 \end{matrix}

(2)

and $Δ^{α}$ denotes the GL definition of order $α$ , $h$ is sampling intervals that will be assumed 1, and $k$ is the number of samples.

Based on this definition, if the order of fractional difference be negative, it represents the integration, and for positive order, it illustrates the fractional difference. For zero order, the original equation will be deduced. According to short memory principle (Koh and Junkins, 2012), the number of samples in summation can be less than running time.

Fractional order discrete state space representation

Fractional order representation for discrete state space model with unknown inputs can be obtained from GL definition as follows

Δ^{ϒ} x_{k + 1} = A x_{k} + B u_{k} + ω_{k} + E d_{k}

(3)

x_{k + 1} = Δ^{ϒ} x_{k + 1} - \sum_{j = 1}^{k + 1} {(- 1)}^{i} ϒ_{i} x_{k - i + 1}

(4)

y_{k + 1} = C x_{k + 1} + v_{k + 1}

(5)

where $x_{k} \in R^{n}$ is the state vector, $u_{k} \in R^{r}$ and $y_{k} \in R^{m}$ are the system inputs and outputs, respectively, $d_{k} \in R^{q}$ introduce as unknown input with distribution vector $E_{k}$ . $ω_{k} ~ (0, Q_{k})$ and $ν_{k} ~ (0, R_{k})$ are the process and measurement noise with zero mean, and variances of $Q_{k}$ and $R_{k}$ , respectively. $A, B, C and E_{k}$ are known matrices with appropriate dimensions. $ϒ_{i}$ , and $Δ^{ϒ} x (k)$ are defined as follows

ϒ_{i} = diag [(\begin{matrix} α_{1} \\ i \end{matrix}), \dots, (\begin{matrix} α_{n} \\ i \end{matrix})]

(6)

Δ^{ϒ} x (k) = [\begin{matrix} Δ^{α_{1}} x_{1, k} \\ ⋮ \\ Δ^{α_{n}} x_{n, k} \end{matrix}]

(7)

$α_{1}, \dots ., α_{n}$ are different orders of the system for incommensurate order systems. Clearly, $α_{1} = α_{2} = \dots = α_{n} = α$ represent orders for commensurate form.

FUIF algorithm design

For state estimation and disturbance decoupling in noisy environment, the following fractional order filter dynamic are proposed

Δ^{ϒ} z_{k + 1} = F_{k + 1} z_{k} + T_{k + 1} B u_{k} + K_{k + 1}^{1} y_{k}

(8)

z_{k + 1} = Δ^{ϒ} z_{k + 1} - \sum_{j = 1}^{k + 1} {(- 1)}^{j} ϒ_{i} z_{k - j + 1}

(9)

{\hat{x}}_{k + 1} = z_{k + 1} + H^{1} y_{k + 1}

(10)

{\hat{x}}_{k + 1} = Δ^{ϒ} z_{k + 1} + H^{1} y_{k + 1} - \sum_{j = 1}^{k + 1} {(- 1)}^{j} ϒ_{i} z_{k - j + 1}

(11)

To achieve minimum variance for estimation error in the presence of unknown disturbance, $K_{k + 1}^{1}, T_{k + 1}, F_{k + 1}$ and $H^{1}$ should be calculated; ${\hat{x}}_{k + 1}$ is obtained from the auxiliary state $z_{k + 1}$ . The observer gain ${K^{1}}_{k + 1}$ is defined as

{K^{1}}_{k + 1} = K_{k + 1} + K_{k + 1}^{2}

(12)

The first step to design FUIF is to define estimation error as follows

e_{k + 1} = x_{k + 1} - {\hat{x}}_{k + 1}

(13)

Then, substituting $x_{k + 1}$ and ${\hat{x}}_{k + 1}$ from (4) and (11) into (13) results in

e_{k + 1} = x_{k + 1} - (z_{k + 1} + H^{1} y_{k + 1})

(14)

\begin{matrix} e_{k + 1} = x_{k + 1} - z_{k + 1} - H^{1} C x_{k + 1} - H^{1} v_{k + 1} \\ = (I - H^{1} C) x_{k + 1} - z_{k + 1} - H^{1} v_{k + 1} \end{matrix}

(15)

By using equation (10) and (14), one can obtain

\begin{matrix} z_{k + 1 - j} = {\hat{x}}_{k + 1 - j} - H^{1} y_{k + 1 - j} \\ = {\hat{x}}_{k + 1 - j} - H^{1} C x_{k + 1 - j} \end{matrix}

(16)

x_{k} = e_{k} + z_{k} + H^{1} y_{k}

(17)

To make the notation easier, the summations in (4) and (11) are defined as follows

G_{1} = \sum_{j = 1}^{k + 1} {(- 1)}^{j} ϒ_{i} x_{k - j + 1}

(18)

\begin{array}{l} G_{2} = \sum_{j = 1}^{k + 1} {(- 1)}^{j} Υ_{i} z_{k - j + 1} = \sum_{j = 1}^{k + 1} {(- 1)}^{j} Υ_{i} {\hat{x}}_{k - j + 1} - H^{1} C \\ \sum_{j = 1}^{k + 1} {(- 1)}^{j} Υ_{i} x_{k - j + 1} \end{array}

(19)

Accordingly

\begin{matrix} e_{k + 1} = (I - H^{1} C) x_{k + 1} - H^{1} v_{k + 1} \\ - [F_{k + 1} z_{k} + T_{k + 1} B u_{k} + (K_{k + 1} + {K^{2}}_{k + 1}) y_{k} - G_{2}] \\ = (I - H^{1} C) x_{k + 1} - H^{1} v_{k + 1} - T_{k + 1} B u_{k} \\ - F_{k + 1} (x_{k} - e_{k} - H^{1} y_{k}) \\ - K_{k + 1} (C x_{k} + v_{k}) - {K^{2}}_{k + 1} y_{k} + G_{2} \end{matrix}

(20)

By inserting (4) in to (20)

\begin{matrix} e_{k + 1} = F_{k + 1} e_{k} - K_{k + 1} v_{k} - H^{1} v_{k} \\ + (I - H^{1} C) ω_{k} - [F_{k + 1} - (I - H^{1} C) A + K_{k + 1} C] x_{k} \\ + (I - H^{1} C) E_{k} d_{k} - ({K^{2}}_{k + 1} - F_{k + 1} H^{1}) y_{k} \\ - (T_{k + 1} - (I - H^{1} C)) B u_{k} + H^{1} C G_{1} - G_{1} + G_{2} \end{matrix}

(21)

To achieve an appropriate estimation in stochastic systems, the state estimation error should approach to zero in mean square sense; to reach this aim, coefficients of $x_{k}, y_{k}, d_{k}, u_{k}$ must be zero, therefore estimator parameters are calculated as follows

E_{k} = H^{1} C E_{k}

(22)

T_{k + 1} = I - H^{1} C

(23)

F_{k + 1} = T_{k + 1} A - K_{k + 1} C

(24)

K_{k + 1}^{2} = F_{k + 1} H^{1}

(25)

Consequently, estimation error approaches to zero if equations (22) to (25) are satisfied and $F_{k + 1}$ be stable. Then, state estimation error is obtained as follows

\begin{matrix} e_{k + 1} = F_{k + 1} e_{k} - K_{k + 1} v_{k} - H^{1} v_{k} + ω_{k} (I - H^{1} C) \\ - \sum_{j = 1}^{k + 1} {(- 1)}^{j} ϒ_{i} e_{k - j + 1} \end{matrix}

(26)

It should be noted that to decouple disturbance effect, $H^{1}$ should be calculated such that (22) is satisfied. Decoupling disturbance effect on the state estimation, results in a robust fault detection scheme. Moreover, to reach stabilization, $F_{k + 1}$ , $K_{k + 1}$ should be designed, appropriately. The other estimator coefficients will be calculated from (23) to (25).

To draw more comparison between the FUIF and the conventional FKF, the FKF algorithm is concisely illustrated in Figure 1. More details about this filter can be found in Sierociuk and Dzielinski (2006).

Remark 1: If $H^{1} = 0_{n \times m}$ , the disturbance cannot be decoupled by FUIF and it acts closely like the FKF in the state estimation.

Remark 2: The estimated output is defined as ${\hat{y}}_{k} = C {\hat{x}}_{k}$ , and therefore, this observer is able to estimate sensor fault and output, simultaneously.

Remark 3: The proposed method has the same computational efforts as the FKF, and their computational complexities are ’n’, which is equal to the system dimension.

Remark 4: Necessary and sufficient condition to achieve disturbance decoupling is obtained as follows (Chen and Patton, 1996).

rank (C E_{k}) = rank (E_{k})

(27)

This condition express that the number of independent rows of $C$ should be less than the independent columns of $E_{k}$ . In other points of view, the disturbances that can be decoupled should be less than the independent measurements. If this condition satisfies, $H^{1}$ will be calculated as

H^{1} = E_{k} (C E_{k})^{+}

(28)

where

(C_{k + 1} E_{k})^{+} = [(C_{k + 1} E_{k})^{T} C_{k + 1} E_{k}]^{- 1} (C_{k + 1} E_{k})^{T}

(29)

As mentioned earlier, stability or convergence of estimator depends on $F_{k + 1}$ ; and with respect to $H^{1}$ , observer dynamic matrices $A^{1}$ is calculated as follows

A^{1} = A - H^{1} CA

(30)

F_{k + 1} = A^{1} - K_{k + 1} C

(31)

When the pair of $(C, A^{1})$ is observable and system is time invariant, $K_{k + 1}$ can be obtained by pole placement to reach stability. To reach the best estimation, covariance matrix is calculated as represented in the following

P_{k + 1} = E [e_{k + 1} e_{k + 1}^{T}]

(32)

where $E$ denote mean operator. Then, substituting estimation error from (26) into (32) results in

P_{k + 1} = E [(F_{k + 1} e_{k} - K_{k + 1} v_{k} - H^{1} v_{k} + ω_{k} (I - H^{1} C) - \sum_{j = 1}^{k + 1} {(- 1)}^{j} diag ϒ_{i} (e_{k - j + 1})) {(\dots)}^{T}]

(33)

Considering; $E (e_{k} e_{j}) |_{k \neq j} = 0$ $E (ω_{k} v_{j}^{T}) | = 0$ , simple calculation results in

\begin{matrix} P_{k + 1} = E [(F_{k + 1} + ϒ_{1}) e_{k} e_{k}^{T} (F_{k + 1} + ϒ_{1})^{T} \\ + K_{k + 1} v_{k} v_{k}^{T} K_{k + 1}^{T} + H^{1} v_{k} v_{k}^{T} H^{1 T} \\ + T_{k + 1} ω_{k} ω_{k}^{T} {T_{k + 1}}^{T} + \sum_{j = 2}^{k + 1} (- 1)^{j} ϒ_{j} e_{k - j + 1} e_{k - j + 1}^{T} ϒ_{j}^{T}] \\ = (A^{1} + ϒ_{1} - K_{k + 1} C) P_{k} (A^{1} + ϒ_{1} - K_{k + 1} C)^{T} + K_{k + 1} R_{k} K_{k + 1}^{T} \\ + H^{1} R_{k} H^{1 T} + T_{k + 1} Q_{k} {T_{k + 1}}^{T} + \sum_{j = 2}^{k + 1} (- 1)^{j} ϒ_{j} P_{k - j + 1} ϒ_{j}^{T} \end{matrix}

(34)

To reach the minimum square estimation error, the estimation error covariance should be minimized. The only parameter that should be determined to minimize the estimation error covariance is the filter gain $K_{k + 1}$ , to this aim

\frac{\partial Tr (P_{k + 1})}{\partial K_{k + 1}^{1}} = 2 (A^{1} + ϒ_{1} - K_{k + 1} C) P_{k} (- C)^{T} + 2 K_{k + 1} R_{k} = 0

(35)

therefore, filter gain is obtained as follows

K_{k + 1} = (A^{1} + ϒ_{1}) P_{k} (C)^{T} (C P_{k} C + R_{k})^{- 1}

(36)

FUIF algorithm

The procedure for designing robust FUIF for fractional order systems is summarized in the flowchart shown in Figure 2.

Residual generation

The aim of this section is to show how the proposed filter can be used in FD of discrete fractional order systems. To this end, a residual signal should be generated such that it is equal to zero for fault free case. In the most cases, residual is not exactly equal to zero and it cause to fault ignorance or announcing it as a false alarm even if a fault has not been occurred. The main goal of the proposed observer, which can be used for FD of fractional order systems, is to minimize estimation error in noisy environment while unknown inputs are decoupled. The residual signal can be defined as the state estimation error, that is, difference between state and its estimation as follows

r_{k} = x_{k} - {\hat{x}}_{k}

(37)

To obtain an accurate fault detection the defined residual should be zero in steady state in the fault free case.

Residual analysis

The main goal of this section is to show how one can select a proper threshold for FD purposes. Selecting a proper threshold can avoid misdetections. Therefore, in Bask (2005) some different ways are investigated to find proper threshold. One useful way to detect faults is using a binary form to make an alarm; notifying fault in binary form is an easy way to discover faults. Accordingly, if the residual be more than defined threshold, a flag as the sign of alarm will be toggled. The following analysis shows that if residual $r_{k}$ be more than relative threshold $T_{k}$ , flag $S$ as sign of fault will be as follows

\begin{matrix} | r_{k} | \leq T_{k} \to f_{k} = 0 & \to S = 0 & \to fault free case \\ | r_{k} | > T_{k} \to f_{k} \neq 0 & \to S = 1 & \to faulty case \end{matrix}

(38)

Finding proper value for thresholds as statistical form, in fault free case is defined as follows

\bar{r} = E [| r_{k} |] = \frac{1}{m} \sum_{k = 1}^{m} | r_{k} |

(39)

σ_{r}^{2} = E [{(| r_{k} | - \bar{r})}^{2}] = \frac{1}{m} \sum_{k = 1}^{m} {(| r_{k} | - \bar{r})}^{2}

(40)

To have no misinterpretation in FD, statistical residual should be between two bounds for fault free case, and beyond theses bounds, S will be 1 as the sign of alarm.

\begin{matrix} \bar{r} - λ σ_{r} \leq r_{k} \leq \bar{r} + λ σ \to f_{k} = 0 \to S = 0 \\ r_{k} < \bar{r} - λ σ_{r} or r_{k} > \bar{r} + λ σ_{r} for f_{k} \neq 0 \to S = 1 \end{matrix}

(41)

$\bar{r}, \bar{σ}$ are mean and covariance of residual in fault free case, respectively.

To avoid false alarm maximum and minimum bounds of fault can be defined by choosing proper amount of $λ$ , which $λ \geq 3$ is selected normally.

Simulation results

The aim of this section is the evaluation of the proposed method. To this end, the model of ultra-capacitor is considered. In comparison with the typical capacitor, ultra-capacitor can store more energy. This advantage distinct it from typical UC; so, it can be utilized as power saver in hybrid energy storage systems. Hybrid electrical vehicle is an example for the application of UC (Dixon et al., 2003).

In Figure 3, a bank of UC is used in electrical hybrid vehicle as an axillary energy supply and Buck-Bust converter is utilized to transfer required energy. A typical schematic of axiulary energy system used in hybrid vehicles is shown in Figure 4. True state estimation of UC with fractional model with the proposed filter will illustrate the performance and superiority of this filter in the presence of disturbances in noisy environment.

Equivalent model of the ultra-capacitor system is shown in Figure 5. With respect to this figure, the model parameters for this system that was presented by equations (3) to (5) is obtained as follows (Dzielinski and Sierociuk, 2008).

\begin{matrix} A = [\begin{matrix} 0 & 1 \\ 0.035311 & 0.001815 \end{matrix}] \\ B = [\begin{matrix} 0 \\ 1 \end{matrix}] \\ C = [\begin{matrix} - 0.018624 & 0.188432 \end{matrix}] \end{matrix}

The initial condition is considered as

x_{0} = [\begin{matrix} 1 \\ - 1 \end{matrix}], P_{0} = [\begin{matrix} 5 & 0 \\ 0 & 5 \end{matrix}]

and

α_{1} = α_{2} = 0.2

$α_{1}$ and $α_{2}$ denote the orders of the system. According to short memory principle, for simplicity in simulation, the length of summation in (4) is considered equal to 50, which is less than running time.

\begin{matrix} E [ω_{k} {ω_{k}}^{T}] = Q_{k} = [\begin{matrix} 0.01 & 0 \\ 0 & 0.01 \end{matrix}] \\ E [v_{k} {v_{k}}^{T}] = R_{k} = 0.01 \end{matrix}

In the following, the described filter is applied to the UC for the sake of FD, and disturbance decoupling in noisy environment.

Fault occurrence is an important issue that should be managed. In the first case, the effect of input disturbance, $d_{k}$ , on the residual with and without disturbance decoupling property are studied for fault free case. It is assumed that the disturbance distribution matrix $E_{k}$ , is known, and affect all states as follows

E_{k} = [\begin{matrix} 1 \\ 1 \end{matrix}], d_{k} = 1

By applying UI, which is defined in the following, simulation results illustrate the efficiency of the proposed filter for fractional order linear systems in disturbance decoupling and fault detection. Figure 6 illustrates the effectiveness of the proposed filter without disturbances in fault free case with noise effect.

In the proposed observer, occurring sensor fault on residual will be considered. Therefore, true alarm to avoid mistake is noticeable. The fault is considered as follows

f_{k} = {\begin{matrix} 0.2 & 70 \leq k \leq 150 \\ 0.2 \sin (k) & 350 \leq k \leq 430 \\ oth & 0 \end{matrix}

As shown in Figure 7 the conventional FKF cannot estimate the states when disturbance is applied to system. However, using the FUIF, not only states are estimated but also fault is detected when disturbance is applied to the system.

As shown in Figure 8 whilst disturbance is applied to the system and fault has been occurred in the fractional order system with unknown input, the FUIF residual clearly changes.

To study the ability of the proposed filter in state and fault estimation problem, the estimated output is shown in Figure 9. As this figure shows, not only the output is estimated but also sensor fault is estimated by the proposed FUIF. However, fault is not estimated by the FKF. It should be noted that disturbance is not considered in this simulation, however, if a disturbance is applied to the system, the FKF cannot estimate the output while FUIF is able to decouple disturbance and estimate the output and sensor fault.

To evaluate the performances of the proposed observer, root mean square error (RMSE) criterion as illustrative tools that illustrates the combination of bias and estimation error variance (Arasaratnam and Haykin, 2009) is computed as follows and shown in Figure 10.

RMS E_{(k)} = \sqrt{\frac{1}{N} \sum_{n = 1}^{N} (e_{k}^{n})_{i}^{2}}

(42)

Figure 10 illustrates the effect of disturbance on RMSE criterion over 50 Monte Carlo runs. The effect of disturbance on conventional FKF cause to undesired performance, but FUIF has an acceptable performance against disturbance.

Conclusion

This paper introduces a new filter to estimate states with unknown inputs in noisy environment for fractional order linear systems. Moreover, robustness against disturbances and uncertainty, which are considered as unknown inputs, are studied; hence, in the presence of disturbance in noisy environment, fault can be detected clearly. Straightforward implementation with Grunwald-Letnikov definition is one of the advantages of the proposed method. Another superiority of this filter is handling noise and disturbances that distinguish it from the FKF, whereas, the computational complexity of these two algorithms are equal. To show the advantages of this filter the proposed algorithm is applied to an ultra-capacitor in the presence of disturbances in noisy environment to detect fault. Comparisons between the conventional FKF and FUIF in fault detection and state estimation in the presence of noise and disturbance illustrate the capability of the proposed method.

In future works, the proposed fractional filter will be extended for nonlinear fractional order systems. To this end, the nonlinear fractional order system can be linearized around the operating point. Moreover, Unscented and Cubature transformations can be used in the proposed structure to achieve a derivative free filter.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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