This paper aims to develop a continuous-discrete finite memory observer (CD-FMO) for a class of nonlinear dynamical systems modeled by ordinary differential equations (ODEs) with discrete measurements. The nonlinear systems under consideration are at least

Over past decades, it was noticed that state estimation, especially observer, plays an important role in the modern control theory and practice [

Observer-based method has also been widely used in fault detection and isolation (FDI) of many fields such as PEM fuel cell and heat-exchanger/reactor system [

Therefore, the main contribution of this paper is that we develop a nonlinear continuous-discrete finite memory observer (CD-FMO) for a class of nonlinear Lipschitz system. It has been proven that the designed observer has a finite-time convergence and good robustness against measurement noise. Moreover, we also perform a rapid fault detection and an accurate fault isolation to a single-link robotic arm by using the proposed nonlinear CD-FMO in the presence of measurement noise.

The work of this paper is organized as follows: Section

We consider a class of continuous-discrete nonlinear systems described by the following state-space equation:

In this section, the construction of the proposed continuous-discrete finite memory observer for nonlinear systems (

The authors in [

We are able to conclude from this remark that if we can build a finite-time observer for a nonlinear system, then this nonlinear system is observable.

Suppose that at each frozen time instant

Then, premultiplying (

Applying (

It is straightforward that the noise component

Now, the state estimation

Let

(H1). The pair

According to (

It is obvious to see from (

In order to compute

In each time window

Furthermore, the first-order Newton-Cotes formula, which yields

Let

Calculation framework of

For the sake of overall understanding, the summarized algorithm of the proposed nonlinear observer CD-FMO is shown in Algorithm

If nonlinear system (

In the case of noise-free and fault-free, according to (

In order to prove Theorem

According to (

The proof is completed.

We can see from Theorem

The proposed CD-FMO is a dead-beat observer in the case of noise-free and fault-free; the finite-time convergence is

There is no estimation when

As it is shown in (

The minimal window length

Here it should be noticed that, theoretically speaking, there is no maximum window length

In this section, we consider a nonlinear single-link robotic arm, which has an elastic joint rotating in a vertical plane [

The simulation scenario is performed according to the following parameters: elastic constant

As shown in (

Convergence of covariance

In order to perform state estimate via (

Given a definite integral

We can have the following expression of

For Step 1, according to Lemma

And from expression (

For Step 2, we can directly get the cumulative error bound

It is obvious that

It can be clearly seen from Figure

State estimation _{1}. (b) Estimation of _{2}. (c) Estimation of _{3}. (d) Estimation of _{4}.

As has been proved in Theorem

Let

Upper (lower) bound and mean of

The unbiased estimation property has also been examined by the RMSE with different

RMSE comparison between

To summarize what has been mentioned above, we have established by Monte Carlo simulation that state estimation given by the presented nonlinear observer CD-FMO in the stochastic case is also unbiased; that is,

We are going to analyze the robustness of CD-FMO against measurement noise through three scenarios shown in Table

Different scenarios of SD for measurement noise.

Measurement noise scenarios settings | CD-FMO parameter settings | |
---|---|---|

Scenario 1 | ||

Scenario 2 | ||

Scenario 3 |

By taking

Robustness analysis in different measurement noise scenarios. (a) Estimation of _{1} in different measurement noise scenarios. (b) RMSE in different measurement noise scenarios.

In this section, we are going to apply the proposed CD-FMO to perform the fault diagnosis of the considered nonlinear single-link robotic arm system. In order to deal with all faults in the same simulation launch, we suppose that each fault only occurs during certain period

Sensor bias

A bias on

A bias on

A bias on

Actuator fault

where

Actuator fault (

In this paper,

In the presence of measurement noise, CUSUM control chart can improve the performance of diagnosis. For example, in Figure

Residual _{1}. (b) CUSUM control chart of residual _{1}. (c) Residual _{2}. (d) CUSUM control chart of residual _{2}. (e) Residual _{3}. (f) CUSUM control chart of residual _{3.}

Fault signature for different faults.

Different faults | Start instant | End instant | Signature | Detect instant | ||
---|---|---|---|---|---|---|

A bias on | 1 | 0 | 1 | |||

A bias on | 0 | 1 | 1 | |||

A bias on | 1 | 0 | 1 | |||

Actuator fault ( | 1 | 0 | 1 |

It can be obviously seen from Table

GOS of CD-FMOs and residual

The results of

By comparing the fault signature obtained by CD-FMO 1 and CD-FMO 2 in Table

Fault signature by CD-FMO 1 and CD-FMO 2.

Different faults | Fault signature CD-FMO 1 | Fault signature CD-FMO 2 | ||||
---|---|---|---|---|---|---|

A bias on | 1 | 0 | 1 | 1 | 1 | 0 |

A bias on | 0 | 1 | 1 | 0 | 1 | 0 |

A bias on | 1 | 0 | 1 | 0 | 1 | 1 |

Actuator fault ( | 1 | 0 | 1 | 0 | 1 | 0 |

In this paper, a nonlinear observer has been proposed to perform state estimation and fault diagnosis for a class of continuous-discrete nonlinear dynamical systems. The performance of state estimation is great and can be significantly improved by choosing a larger window length. Also the presented approach has a finite-time convergence, which is a great advantage from the perspective of FDI. Simulations have illustrated that the proposed method provides a quite effective fault detection for sensor and actuator faults, which can also show the robustness of this nonlinear observer against the measurement noise. Meanwhile, by using the bank of observers, we are able to deal with the isolation of faults with identical fault signature. It is worth noting that the proposed observer structure can also be apply to the following cases: (1) estimation instant is not synchronized with measurement instant; that is, we are able to obtain the state estimation

The code used to support the findings of this study is available from the corresponding author upon request.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The authors would like to gratefully acknowledge the financial support of the China Scholarship Council (CSC) via the project UT-INSA.