Multiobjective performance-based designs in fault estimation and isolation for discrete-time systems and its application to wind turbines

ABSTRACT

In this work, we develop a performance-based design of model-based observes and statistical-based decision mechanisms for achieving fault estimation and fault isolation in systems affected by unknown inputs and stochastic noises. First, through semidefinite programming, we design the observers considering different estimation performance indices as the covariance of the estimation errors, the fault tracking delays and the degree of decoupling from unknown inputs and from faults in other channels. Second, we perform a co-design of the observers and decision mechanisms for satisfying certain trade-off between different isolation performance indices: the false isolation rates, the isolation times and the minimum size of the isolable faults. Finally, we extend these results to a scheme based on a bank of observers for the case where multiple faults affect the system and isolability conditions are not verified. To show the effectiveness of the results, we apply these design strategies to a well-known benchmark of wind turbines which considers multiple faults and has explicit requirements over isolation times and false isolation rates.

Keywords:

• Fault estimation
• fault diagnosis
• multiobjective optimisation
• wind turbines

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Ester Sales-Setién http://orcid.org/0000-0003-3310-7528

Ignacio Peñarrocha-Alós http://orcid.org/0000-0002-7502-2787

Notes

1 The proposed method entails a more general approach compared with some other existing works that only consider either actuator faults (Rodrigues et al., 2015) or sensor faults (Aouaouda et al., 2015; Liu & Shi, 2013).

2 In practice, FE is performed in stable (controlled or uncontrolled) systems. Hence, if $\mathrm{\Delta }A$ is bounded, the uncertainty $d=\mathrm{\Delta }Ax$ is bounded as x is also bounded.

3 It is also bounded as $\parallel \tilde{f}{\parallel }_{\mathrm{RMS}}^2\le {\gamma }_d{n}_d^2{\overline{d}}^2$ because $\parallel d{\parallel }_{\mathrm{RMS}}\le n_d\parallel d{\parallel }_{\mathrm{\infty }}$ and $\parallel d{\parallel }_{\mathrm{\infty }}\le \overline{d}$.

4 It is also bounded as $\parallel {\tilde{f}}_i{\parallel }_{\mathrm{RMS}}^2\le \sum _{j=1}^{n_f}{\mathrm{\Gamma }}_{jj}^i{\overline{\delta }}^2$ because $\parallel {\delta }_j{\parallel }_{\mathrm{RMS}}\le \parallel \delta {\parallel }_{\mathrm{\infty }}$ and $\parallel \delta {\parallel }_{\mathrm{\infty }}\le \overline{\delta }$.

5 A constraint regarding the fault tracking speed w.r.t. ramp faults is conservative because it covers the worst-case fault signal form considered in Assumption 2.4 (i.e. a ramp fault of slope ${\mathrm{\Delta }}_i=\overline{\delta }$).

6 If a>0 and x is a random variable of mean μ and variance σ, then $\mathbb{P}\{|x-\mu |>a\sigma \}\le 1/a^2$.

7 Note that the threshold $J_i$ defined as (33(33) $J_i:={\mathrm{\Phi }}_Z^{-1}(1-{\phi }_i^{\ast }/2)\sqrt{{\mathrm{\Sigma }}_{ii}},$(33) ) ensures ${\phi }_i\equiv {\phi }_i^{\ast }$ in the case of Gaussian noises while the threshold $J_i$ defined as (32(32) $J_i:=\sqrt{{\mathrm{\Sigma }}_{ii}/{\phi }_i^{\ast }}.$(32) ) ensures the bound ${\phi }_i\le {\phi }_i^{\ast }$ regardless of the statical distribution of the noises.

8 Strategy 3.2 guarantees practical UI and interfault decoupling and thus, in the fault-free scenarios, signal ${\hat{f}}_i$ is zero-mean and its variance is given by the marginal variance ${\mathrm{\Sigma }}_{ii}$.

9 Once a fault is accommodated, the observers in the bank must be reset to avoid the existence of wrong initial conditions derived from the previous presence of ignored faults.

10 The problems are set up in YALMIP (Lofberg, 2004) and we successfully solve them with the PENBMI solver (Henrion et al., 2005). For sake of brevity, we do not include the value of the obtained gain matrices.

11 If the uncertainties regarding parameter changes lead to poor estimation results, these uncertainties must be modelled as UIs (see Remark 2.3) and certain degree of UI decoupling (i.e. constraint (19(19) ${\gamma }_d\le {\overline{\gamma }}_d.$(19) )) must be introduced as an additional requirement in the observer design.

Additional information

Funding

This work has been supported by the Spanish Ministry of Education, Culture and Sports [grant number FPU14/01592], the Spanish Ministry of Economy, Industry and Competitiveness [Project no. TEC2015-69155-R] and the Universitat Jaume I of Castelló [Project no. P11B2015-42].