A design framework for event-triggered active fault-tolerant control systems

Abstract

This article proposes a design framework for an event-triggered active fault-tolerant control system. The considered control system is modelled with an injected actuator fault into the plant system, and it includes: (i) two event detectors to determine when information is to be sampled or updated, (ii) a fault diagnosis observer to provide the fault and state information, and (iii) a fault-tolerant controller to compensate the fault. The proposed design framework allows for the separation of the designs of the fault diagnosis observer and the fault-tolerant controller, to mitigate the coupling effect caused by the event-triggered mechanisms and avoid the computations of a higher order system. Also, a design procedure is presented to deliver the computation of the design variables to reduce the information transfer, in both centralised and decentralised cases. Finally, a batch reactor benchmark system is adopted to demonstrate the applicability and superiority of the proposed design framework.

Keywords:

• Fault-tolerant control
• event-triggered control
• fault estimation
• design separation

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 Throughout this article, the notation $\Vert .\Vert$ denotes the 2-norm of vectors and matrices, and the notation $\Vert .{\Vert }_{{\ell }_2}$ denotes the ${\ell }_2$-norm of matrix and vector sequences.

2 In general, a larger value of the triggering parameter ${\delta }_y$ may lead to a lower frequency of information transfer (Zou et al., 2017); and if ${\delta }_y=0$, the event-triggered sampling mechanism becomes a periodic time-triggered sampling mechanism.

3 It should be noted that the condition given in (6(6) $\Vert u(r_k)-u(r_k+i)\Vert \le {\delta }_x\Vert \hat{x}(r_k+i)\Vert +{\delta }_f\Vert \mathrm{\Delta }\hat{f}(r_k+i)\Vert ,$(6) ) differs from that given in (2(2) $\Vert y(t_k)-y(t_k+i)\Vert \le {\delta }_y\Vert y(t_k+i)\Vert ,$(2) ) in the two aspects: (i) using $\Vert \hat{x}\Vert$ rather than $\Vert u\Vert$ eliminates the negative impact of the injected fault on the plant system, and (ii) introducing an additional term to account for the injected fault in order to improve the communication performance of the closed-loop control system.

4 As the control input vector $\tilde{u}$ is specified by the event-triggered updating detector ${\mathrm{\Sigma }}_{\mathcal{U}}$ that resides on the controller's side, the fault diagnosis observer has access to the transferred input vector $u(r_k)$ as well as $\tilde{u}(k)$.

5 The proposed approach is iterative such that one triggering parameter is maximized at each iteration, instead of attempting to maximise the combination of the triggering parameters directly (namely, ${\delta }_y^2+{\delta }_x^2+{\delta }_f^2$), as the latter approach may lead to a higher frequency of information transfer.

6 For example, the triggering parameter ${\delta }_{yi}$ can be assigned with a weight $q_{ni}$ based on its importance or the condition of a communication channel, namely, ${\delta }_{y1}:{\delta }_{y2}:\cdots :{\delta }_{y{n}_m}=q_1:q_2:\cdots :q_{n_m}$. A similar argument also applies to the triggering parameter ${\delta }_{ui}$.

7 If the event-triggered updating mechanism in (6(6) $\Vert u(r_k)-u(r_k+i)\Vert \le {\delta }_x\Vert \hat{x}(r_k+i)\Vert +{\delta }_f\Vert \mathrm{\Delta }\hat{f}(r_k+i)\Vert ,$(6) ) is also used in the simultaneous design, we obtain several complex nonlinear matrix inequalities which are very difficult to solve. Therefore, a comparison with the simultaneous design approach using event-triggered mechanisms (2(2) $\Vert y(t_k)-y(t_k+i)\Vert \le {\delta }_y\Vert y(t_k+i)\Vert ,$(2) ) and (6(6) $\Vert u(r_k)-u(r_k+i)\Vert \le {\delta }_x\Vert \hat{x}(r_k+i)\Vert +{\delta }_f\Vert \mathrm{\Delta }\hat{f}(r_k+i)\Vert ,$(6) ) is not provided.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [grant numbers 61473159, 61374136, 61104028], the Six talent peaks project in Jiangsu Province [grant number XYDXX-091] and Natural Sciences and Engineering Research Council of Canada.