Reconfiguration Criterion for Fault-Tolerant Control

Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2015, Article ID 218384, 11 pages http://dx.doi.org/10.1155/2015/218384 Research Article Reconfiguration Criterion for Fault-Tolerant Control Inseok Yang,1 Dongik Lee,2 and Dong Seog Han2 1 Center for ICT & Automobile Convergence, Kyungpook National University, Daegu 702-701, Republic of Korea School of Electronics Engineering, Kyungpook National University, Daegu 702-701, Republic of Korea 2 Correspondence should be addressed to Dong Seog Han; Received 31 July 2014; Accepted 19 December 2014 Academic Editor: Qingling Zhang Copyright © 2015 Inseok Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The reconfiguration criterion for tolerating actuator fault is proposed. The proposed reconfiguration criterion analyzes the relationship between normal actuators and the system states that are directly affected by faulty actuators. So the proposed criterion provides the possibilities of fault-tolerance. Moreover, it also provides the required number of redundant normal actuators. 1. Introduction Faults occurring in systems such as automotive vehicles and aircrafts cause a catastrophic accident that leads to loss of property, life, and so forth. To avoid an accident caused by faults, many systems require a high level of dependability. Adopting redundant actuators has been considered as an efficient method of achieving the required dependability. Conventionally, redundant actuators are considered as backup systems if the primary ones operate normally. But if faults occur in the primary actuators, then redundant ones are activated as main actuators. However, adopting redundant actuators leads to the losses of fuel, space, cost, and weight during normal operation. In contrast to adopting hardware redundancies such as secondary actuators, software based fault accommodation methods have been proposed for the last 30 years. The goal of these methods is to provide the feasible control input in order to maintain the normal performance. For this reason, these methods are defined as fault-tolerant control (FTC) or reconfiguration. There have been proposed various fault-tolerant control techniques: pseudoinverse [1], model reference adaptive control [2], sliding mode control [3–5], multiple model switching and tuning [6], control allocation [3, 7–10], and so forth. However, most of the proposed methods shown above consider the reconfiguration ability. This means that although the proposed reconfiguration methods can accommodate faults theoretically and practically, there are some faulty systems that cannot be tolerated. Generally, reconfiguration possibility highly depends on the relationship between the faulty states and controllable normal inputs. If a faulty system cannot take sufficient controllable inputs related to the faulty states, then the goal of reconfiguration must be changed to achieve stabilization of the faulty system in order to avoid structural damage. This paper proposes the reconfiguration condition that provides the possibility of fault-tolerance. By explicitly analyzing the relationship between the faulty states and normal inputs, the condition also proposes the required number of redundant actuators that can achieve fault-tolerance. 2. General Dynamic Model of Actuator Faulty System The response of a faulty actuator can be categorized into one of four types: Lock-in-Place (LiP), Hardover, Float, and Loss of Effectiveness (LoE) [11]. Figure 1 shows the typical examples of these fault types. In this figure, faults such as LiP, Hardover, and Float lead an actuator to stopping at one position or diverging to upper-/lower-saturation position without any consideration of the input commands. So these faults are defined as total faults. In contrast to total faults, the response of LoE fault degrades the performance relative to 2 Mathematical Problems in Engineering Desired Desired Trim point Actual Actual tf tf Time Time (a) Lock-in-Place (b) Float Maximum limit Actual Desired y(t) x(t) tf Time (d) Loss of Effectiveness (c) Hardover Figure 1: Typical failures of actuator [11]. its desired (normal) output. Hence, the general response of a faulty actuator can be represented as follows: NORMAL (1 − ⟦𝛾𝑖 ⟧) 𝑢𝑖TOTAL , 𝑢𝑖 = ⟦𝛾 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟𝑖 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ + ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝑖 ⟧ 𝛾𝑖 ⏟𝑢 LoE fault term (1) Total fault term where the subscript 𝑖 indicates the 𝑖th actuator and ⟦ ⋅ ⟧ is the smallest integer greater than or equal to ⋅. And 𝛾𝑖 is the performance degradation factor represented by a quantitative value in [0, 1] according to the degraded performance: 0, if total fault occurs { { 𝛾𝑖 = {𝜀 (0 < 𝜀 < 1) , if LoE fault occurs { if no fault occurs. {1, For example, if the 𝑖th actuator is operated normally, then 𝛾𝑖 = 1, so 𝛿𝑖 = 𝑢𝑖NORMAL . And if 50% of LoE failure occurs on the 𝑖th actuator, then 𝛾𝑖 = 0.5, so 𝑢𝑖 = 0.5𝑢𝑖NORMAL . For the total number of actuators 𝑚, suppose faults are occurring on 𝑘 (1 ≤ 𝑘 ≤ 𝑚) actuators; then, the general dynamics of faulty actuators yields 𝑢1NORMAL [ NORMAL ] ] [𝑢 ] ][ 2 ] ][ ] .. ][ ] ] [ . ⋅ ⋅ ⋅ ⟦𝛾𝑘 ⟧ 𝛾𝑘 ] 𝑢NORMAL ] [ 𝑘 0 ⋅⋅⋅ ⟦𝛾1 ⟧ 𝛾1 𝑢1 [𝑢2 ] [ 0 ⟦𝛾 ⟧ 𝛾 ⋅ 2 2 ⋅⋅ [ ] [ [ .. ] = [ .. .. [.] [ . . d [𝑢𝑘 ] [ 0 0 (2) And 𝑢𝑖NORMAL denotes the desired normal position shown as a dotted red line in Figure 1(d), and 𝑢𝑖TOTAL denotes the faulty position depicted as a blue line in Figures 1(a), 1(b), and 1(c). 0 0 .. . 𝑢1TOTAL [ TOTAL ] ] [𝑢 ] ][ 2 ] ][ . ] ][ . ] [ . ] ⋅ ⋅ ⋅ 1 − ⟦𝛾𝑘 ⟧] 𝑢TOTAL ] [ 𝑘 0 ⋅⋅⋅ 1 − ⟦𝛾1 ⟧ [ 0 1 − ⟦𝛾 ⟧ ⋅ ⋅⋅ 2 [ +[ .. .. [ . . d [ 0 0 0 0 .. . Mathematical Problems in Engineering 3 0𝑘×(𝑚−𝑘) I𝑘 ⟦Γ𝐹 ⟧ 0𝑘×(𝑚−𝑘) ]−[ ]} + {[ 0(𝑚−𝑘)×𝑘 I𝑚−𝑘 0(𝑚−𝑘)×𝑘 ⟦I𝑚−𝑘 ⟧ ⟦𝛾1 ⟧ 0 ⋅ ⋅ ⋅ 0 𝛾1 0 ⋅ ⋅ ⋅ 0 [ 0 ⟦𝛾2 ⟧ ⋅ ⋅ ⋅ 0 ] [ 0 𝛾2 ⋅ ⋅ ⋅ 0 ] ][ ] [ =[ . .. ] .. .. ] [ .. .. ] [ .. [ . . d .] . d . 0 ⋅ ⋅ ⋅ ⟦𝛾𝑘 ⟧] [ 0 0 ⋅ ⋅ ⋅ 𝛾𝑘 ] [ 0 ×[ 𝑢1NORMAL [ NORMAL ] ] [𝑢 ] [ ×[ 2 . ] ] [ .. ] [ 󳨐⇒ u = ⟦Γ⟧ ΓuNORMAL + (I𝑚 − ⟦Γ⟧) uTOTAL , Let (5) 𝑚−𝑘 is a set of normal actuators u𝑁,NORMAL and where u𝑁 2 ∈R I𝑚 is an (𝑚 × 𝑚)-identity matrix. And NORMAL ] [𝑢𝑘 ⟦𝛾1 ⟧ 0 ⋅ ⋅ ⋅ 0 } 1 0 ⋅⋅⋅ 0 { { } { ] [ 0 ⟦𝛾2 ⟧ ⋅ ⋅ ⋅ 0 ]} 0 1 ⋅ ⋅ ⋅ 0 {[ ]} [ ] [ + {[ .. .. .. ] − [ .. .. .. ]} { . d . ]} } {[ . . d . ] [ . } { 0 0 ⋅ ⋅ ⋅ 1 0 ⋅ ⋅ ⋅ ⟦𝛾𝑘 ⟧]} {[ ] [ 0 u𝐹1 [ ], u= u𝑁 [ 2] uTOTAL = [ 𝑢1TOTAL [ TOTAL ] ] [𝑢 ] [ ×[ 2 . ] [ . ] [ . ] [𝑢𝑘 TOTAL u𝐹,TOTAL ] 0(𝑚−𝑘)×1 u𝐹,NORMAL uNORMAL = [ 𝑁,NORMAL ] , u u𝐹,TOTAL ], 0(𝑚−𝑘)×1 Γ𝐹 0𝑘×(𝑚−𝑘) Γ=[ ], 0(𝑚−𝑘)×𝑘 I𝑚−𝑘 ⟦Γ𝐹 ⟧ ⟦0𝑘×(𝑚−𝑘) ⟧ ⟦Γ𝐹 ⟧ 0𝑘×(𝑚−𝑘) ⟦Γ⟧ = [ ]. ]=[ 0(𝑚−𝑘)×𝑘 I𝑚−𝑘 ⟦0(𝑚−𝑘)×𝑘 ⟧ ⟦I𝑚−𝑘 ⟧ (6) ] 󳨐⇒ u𝐹1 = ⟦Γ𝐹 ⟧ Γ𝐹 u𝐹,NORMAL + (I𝑘 − ⟦Γ𝐹 ⟧) u𝐹,TOTAL , Let (3) where u𝐹,NORMAL = [𝑢1NORMAL , 𝑢2NORMAL , . . . , 𝑢𝑘NORMAL ]𝑇 and u𝐹,TOTAL = [𝑢1TOTAL , 𝑢2TOTAL , . . . , 𝑢𝑘TOTAL ]𝑇 . And I𝑘 is a (𝑘 × 𝑘)identity matrix. Moreover, 𝛾 (...truncated)