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Fault detection for switched systems with all modes unstable based on interval observer
Fault Detection for Switched Systems with All Modes Unstable based on Interval Observer
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Fault Detection for Switched Systems with All Modes Unstable based on Interval Observer Qingyu Su, Zhongxin Fan, Tong Lu, Yue Long, Jian Li PII: DOI: Reference:
S0020-0255(19)31197-1 https://doi.org/10.1016/j.ins.2019.12.071 INS 15114
To appear in:
Information Sciences
Received date: Revised date: Accepted date:
20 July 2019 6 December 2019 27 December 2019
Please cite this article as: Qingyu Su, Zhongxin Fan, Tong Lu, Yue Long, Jian Li, Fault Detection for Switched Systems with All Modes Unstable based on Interval Observer, Information Sciences (2019), doi: https://doi.org/10.1016/j.ins.2019.12.071
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Fault Detection for Switched Systems with All Modes Unstable based on Interval Observer Qingyu Sua , Zhongxin Fana , Tong Lua , Yue Longb , Jian Lia,∗ a School
of Automation Engineering, Northeast Electric Power University, Jilin, Jilin, 132012, China. b School of Physics, Liaoning University, Shenyang, Liaoning, 110036, China.
Abstract This paper concentrates on observer-based fault detection (FD) for switched systems. By the introduction of the interval observer, the error dynamic system is proposed. In light of the switching logic, exponential stability of the augmented system is firstly achieved. Then, to ensure the fault sensitivity as well as disturbance robustness, H∞ /H− performance analysis is employed. One thing has to be mentioned is that we investigate the unobservable condition of (Ai ; Ci ). Moreover, the instability of the error dynamic system brought by the unobservability is eliminated by the switching strategy using average dwell time (ADT) method. Afterwards, sufficient conditions guaranteing the H∞ /H− performance level are obtained , which is divided into disturbance attenuation and fault sensitivity analysis. Finally, examples demonstrating the effectiveness of the provided method are provided. Keywords: average dwell time, unstable subsystems, interval observer, discretized Lyapunov function, mixed H∞ /H− performance. 1. Introduction Switched systems, being an of importance part of hybrid systems, compose several subsystems and a switching logic rule indicating which subsystem is ∗ Corresponding
author Email address: [email protected] (Jian Li) URL: [email protected] (Qingyu Su), [email protected] (Zhongxin Fan), [email protected] (Tong Lu), [email protected] (Yue Long)
Preprint submitted to Journal of Information Sciences
December 28, 2019
chosen at a certain time instant. Since the proposition of switched systems, its 5
importance has been widely recognized. In recent years, it has been applied to many areas, such as [1, 2, 3, 4]. In addition, the switching method is also wildly used[5, 6]. Due to the diversity of its subsystems, the researches of switched systems are different from the other systems. Take the stability as an example, the stability of the subsystem does not necessarily bring the stability
10
of the entire system due to the improper switching signal. Conversely, the unstable subsystems may not influent the stability of the whole system if the switching rule can be chosen properly. Therefore, the stability can be viewed as the combination results of the switching rule and the subsystems, which is the essential difference between switched systems and other systems.
15
According to the property of the subsystem, we can divide the literatures into three categories. (1). Switched system consisting of all stable subsystems. event-triggered filter is designed for switched discrete-time systems [7]. [8] realizes the fault detection and control when actuator lose efficacy. (2). Switched system including unstable modes. [9] investigates the stable property with L2 -
20
gain for switched time-delay systems. [10] designs the H∞ filters for nonlinear systems while [11] researches the nonlinear switched systems. (3). Switched systems with all modes unstable. [12] employs the dwell time to finish the stabilization of continuous systems. Stability and L2 -gain are analyzed in [13] when all modes unstable for switched systems. Discrete-time systems’ stability
25
is analyzed in [14]. Under asynchronous switching, [15] studies the nonlinear systems with time-delays. Sliding mode control, memory filter design, stabilization analysis and adaptive control are investigated in [16, 17, 18, 19] Taking switched system with all modes unstable into consideration, as discussed in [12], we find that a proper switching signal can make the switched
30
systems stable even though all the subsystems are unstable. Hence, this paper attempts to explore an appropriate switching rule. As far as we know, the dwell time method is frequently used in the analysis of the property of switched systems. [20] uses the dwell time (DT) approach to accomplish the H∞ filtering for discrete-time switched systems under missing measurements. In [21], the DT 2
35
method is employed to analyze the stability, L1 -gain and asynchronous L1 -gain control for uncertain switched positive systems. On the basis of the DT, average dwell time (ADT) method is proposed for its small conservation and less computation resource usage. In [22], ADT is used to design the estimator for discrete-time switched systems. Asynchronous H∞ control for switched delay
40
systems is completed with ADT [23]. Additionally, stability and L2 -gain are analyzed for switched systems in [24] and [25], respectively. Many external interference signals including known control inputs, unknown disturbance signals, fault signals and malicious attacks may be applied during system operation.[26]. In [27] and [28], fault tolerant issues are involved for
45
polynominal-fuzzy-model-based systems and multi-agent systems, respectively. Sampled-data control is researched in [29, 30] for nonlinear switched system. Considering quantization effects, the FD (fault detection) filter is designed [31]. [32] studies finite-time FD problem for interconnected systems while [33] researches the FD for linear-parameter-varying (LPV) system. In the above lit-
50
eratures, FD is finished depending on the residual signal generator and the threshold function[34, 35, 36]. The former is used to produce the residual signal. The latter is used to evaluate the appearance of the fault. The FD logic can be stated as, the fault detection is proceeded depending on whether the residual evaluation function surpasses the threshold. However, as pointed in
55
[37], the improper choose of the threshold may greatly influent the results of the FD, which bring a lot of uncertainty. Thence, in order to get rid of the uncertainty and get more degree of freedom, this paper employs the interval observer which is wildly used in [37, 38, 39, 40]. In [37], FD in finite-frequency domain is completed via interval observer. [38] constructs the hybrid interval
60
observers for linear switched system. Similarly, interval observers are designed in [39] and [40] for discrete-time systems, respectively. When fault occurs in a modern power grid system, it has been studied and demonstrated that the system has a great potential to go from observable to unobservable. However, there are few researches on observer design and fault detection for unobservable power
65
grids. This paper provides a research idea, i. e. we convert the unobservable 3
condition to a instability analysis, which is of great research significance. The contributions of the paper are summarized in this part. • An ADT approach is employed to generate the switching logic, which is used to ensure the stability. • From the theoretical aspect, this paper solves the observer design problem
70
under the unobservable condition, which is rarely discussed in other literature. In addition, the introduction of the interval observer improves the accuracy of fault detection and reduces computing resources. • In practical aspect, this paper provides another method of fault detection, 75
which enables users to have a better comparison in actual operation. Additionally, the H∞ /H− ensures a better fault detection effect is guaranteed under a reasonable anti-interference condition. Main structure is arranged as below. Section 2 provides the systems model while Section 3 concludes the sufficient conditions ensuring the exponential sta-
80
bility and H∞ /H− performance. Also, the FD decision scheme is provided. Then, in Section 4, an example is raised to finish the illustration of the validity of the interval observer. Section 5 gives the conclusions and acknowledgments. Notations: The notation k ∗ k is the Euclidean norm for vectors. He(A) =
A +A. The inverse of a matrix is stated by ∗−1 and the transpose one is denoted T
85
by A∗ . Matrix A > 0(A < 0) represents it is positive definite (negative definite). In addition, the minimum and maximum eigenvalues of A are expressed as (λmin (A)) and λmax (A), respectively. R is the set of the whole real number.
Rn indicates the n-dim Euclidean space. Without loss of generality, I is used as the identity matrix. ∗ stands for the main-diagonal-under elements. 90
2. Preliminaries and System Model This section gives the system description and some necessary definitions. Consider the following switched systems with all modes unstable, x(t) ˙ = Aσ(t) x(t) + Bσ(t) d(t) + Eσ(t) f (t), y(t) = C
σ(t) x(t) + Dσ(t) d(t) + Fσ(t) f (t)
4
(1)
where x(t) ∈ Rx , y(t) ∈ Ry represent the unstable system state and system
output, respectively. f (t) ∈ Rf indicates the fault signal. d(t) ∈ Rd denotes the 95
disturbance input. The function σ(t) : [0, ∞) → N = {1, 2, · · · , n} denotes the switching rule indicating which mode is activated at time t, which is a piecewise constant function from the right. In addition, the matrices A, B, C, D, E, F are known constant matrices with appropriate dimensions. For switched system (1), the interval observer is obtained as follows, ˙ x(t) = (A − LC)x(t) + LCx(t) + LDd + LF f − B − d + B + d − LD+ d + LD− d x(t) ˙ = (A − LC)x(t) + LCx(t) + LDd + LF f − B − d + B + d − LD+ d + LD− d + − y(t) = Cσ(t) x(t) − Cσ(t) x(t) y(t) = C + x(t) − C − x(t) σ(t) σ(t)
100
(2)
where D+ = max{0, D}, D− = D+ − D, x(t) ∈ Rx , x(t) ∈ Rx are the upper
and lower estimates of state x(t). y(t) ∈ Ry , y(t) ∈ Ry denote the upper and lower estimates of output y(t), respectively. r(t) ∈ Rr and r(t) ∈ Rr represent the lower and upper bound. L ∈ Rl is the parameter to be determined.
105
If we denote e(t) = x(t) − x(t), e(t) = x(t) − x(t), r(t) = y(t) − y(t), e(t) r(t) d−d , r(t) = , d˜ = , we can r(t) = y(t) − y(t), e(t) = e(t) r(t) d−d gain ˜ +E ˜σ(t) d(t) ˜σ(t) f (t), e(t) ˙ = A˜σ(t) e(t) + B r(t) = C˜ e(t) + D ˜ + F˜σ(t) f (t). ˜ σ(t) d(t) σ(t)
where
A˜σ(t) =
Aσ(t) − Lσ(t) Cσ(t)
0
0
Aσ(t) − Lσ(t) Cσ(t)
˜σ(t) = B
+ + Bσ(t) − Lσ(t) Dσ(t)
− − Bσ(t) − Lσ(t) Dσ(t)
, C˜σ(t) =
− − Bσ(t) − Lσ(t) Dσ(t) + + Bσ(t) − Lσ(t) Dσ(t)
5
(3)
− −Cσ(t) + Cσ(t)
, F˜σ(t) =
Fσ(t) Fσ(t)
+ −Cσ(t) − Cσ(t)
,
˜σ(t) = E
Eσ(t) − Lσ(t) Fσ(t) Lσ(t) Fσ(t) − Eσ(t)
˜ σ(t) = ,D
+ −Dσ(t) − Dσ(t)
− −Dσ(t) + Dσ(t)
,
To conduct the following work, some useful Assumptions and Lemmas are 110
needed. Assumption 1. There are known bound function d(t) and d(t) such that d(t) ≤ d(t) ≤ d(t) Lemma 1. [39] Given x(t) ∈ Rn and x(t) ≤ x(t) ≤ x(t) for some x(t) ∈ Rn ,
x(t) ∈ Rn , we have
+ − + − Bσ(t) d(t) − Bσ(t) d(t) ≤ Bσ(t) d(t) ≤ Bσ(t) d(t) − Bσ(t) d(t) + − + − Cσ(t) x(t) − Cσ(t) x(t) ≤ Cσ(t) x(t) ≤ Cσ(t) x(t) − Cσ(t) x(t)
Remark 1. The results of fault detection using interval observer have been a lot, such as [37, 39, 40]. It should be mentioned that switched system is rarely related. As in [37], the system matrix A is required to be Schur stable. However, in this paper, the restriction of stability of A is relaxed for that the whole 115
stability can be reached via proper switching strategy though all the subsystems are unstable, which is hardly involved in the previous literatures. Definition 1. [23] When there is no external input, k e(t) k=
sup {k e(t +
−τ ≤θ≤0
θ) k, k e(t ˙ + θ) k}, if k e(t) k≤ o expς(t−t0 ) k e(t0 ) k when t ≥ t0 , o > 1, ς < 0, then, under the switching logic σ(t), system (3) is exponentially stable. Definition 2. [23] Let Nσ (t, T ) denotes the number of switching of σ(t) on time interval (t, T ). If it has N0 ≥ 0, σ(t) is viewed as having average dwell time τa and Nσ (t, T ) ≤ N0 + 120
T −t , ∀T ≥ t ≥ 0. τa
(4)
then, τa is called as ADT and N0 is chatter bound. We set the chatter bound N0 = 0. 6
With the assistance of [41, 42], we give the following definition.
Definition 3. [43] System (3) is said to have weighted H∞ /H− performance if Z ∞ Z ∞ ˜ d˜T (s)d(s)ds (5) exp−ηs rT (s)r(s)ds ≤ γ 2 0
0
Z
0
∞
rT (s)r(s)ds ≥ β 2
Z
∞
exp−ηs f T (s)f (s)ds
(6)
0
Under the circumstance that (Ai ; Ci ) is not observable, the main idea of this 125
paper is find the proper observer gain such that 1. Ai − Li Ci and Ai − Li Ci are metzler. 2. condition (5) is satisfied, i. e., the effects of disturbance on residual can be minimize. 3. condition (6) is satisfied, i. e., the effects of fault on residual can be
130
maximize.
3. Main results This section talks about the main results. To be clear, this part is divided into five subsections in which the exponential stability, disturbance attenuation, fault sensitivity, Metzler condition and fault detection scheme are involved, re135
spectively. Taking the inexistence of stable subsystems into consideration, the traditional Lyapunov function method is not applicable any more. In light of [12], the discretized Lyapunov function method is introduced. The construction method is given below.
140
Given time period [tk , tk + τa ), firstly we divide it into Q segments which can be denoted by lk,m = [tk + θm , tk + θm+1 ), where θm =
mτa Q ,
m = 0, 1, · · · Q − 1.
Afterwards, choose the continuous matrix function Pi (t) as linear on lk,m and define Pi,m = Pi (tk + θm ), we have Pi (t) = Pi (tk + θm + αd) = (1 − α)Pi,k + (k)
αPi,k+1 = Pi (α), where α = (t − tk − θm )/d, d = τa /Q. Since Pi (t) is 7
145
piecewise linear on [tk , tk + τa ), utilizing the linear interpolation formula, it can be described using the values at dividing points. Additionally, we find that function Pi (t) is a constant matrix Pi,L . Hence, the discretized function Pi (t) has the following shape,
Pi (t) =
Pi(k) (α), P
t ∈ lk,m , m = 0, 1, 2, · · · , Q − 1.
(7)
t ∈ [tk + τa , tk+1 ).
i,Q ,
The corresponding discretized Lyapunov function can be obtained as
Vi (t) =
150
xT (t)Pi(k) (α)x(t), xT (t)P
i,Q x(t),
t ∈ lk,m , m = 0, 1, 2, · · · , Q − 1.
(8)
t ∈ [tk + τa , tk+1 ).
Remark 2. From the definition of the discretized Lyapunov function, it is clear that the greater the value of Q, the more segments are obtained on time interval [tk , tk +τa ), which may reduce the conservativeness obviously. The related results have been displayed in [10]. To be brief and save computational resource, this paper sets Q as 1.
155
3.1. Exponential Stability Analysis When disturbance signal d(t) and fault signal f (t) are set as 0, we have the ˜ system dynamic e(t) ˙ = Ae(t). Under this assumption, the following part is the analysis on exponential stability. Theorem 1. For 0 < µ < 1, any i, j ∈ N, i 6= j, η > 0, matrix Pi,m > 0, m = 0, 1, · · · Q − 1, τa > 0, if ln µ + ητa ≤ 0
(9)
Pi,0 − µPj,Q ≤ 0
(10)
˜ φ1i = A˜T i Pi,m + Pi,m Ai + ηPi,m + κp ≤ 0,
(11)
8
˜ φ2i = A˜T i Pi,m+1 + Pi,m+1 Ai + ηPi,m+1 + κp ≤ 0.
(12)
where κp = Q(Pi,m+1 − Pi,m )/τa , in that way, system (3) has the property of 160
exponential stability. ˜ = 0 and f (t) = 0, we have Proof. With the assumption that d(t) ˜ e(t) ˙ = Ae(t) Selecting the Lyapunov function as (8), through derivative technology, the following equation can be derived . V˙ i (t) = e˙ T (t)Pi e(t) + eT (t)P˙i e(t) + eT (t)Pi e(t) ˙
(13)
Thus, it has V˙ i (t) − ηVi (t) = e˙ T (t)Pi e(t) + eT (t)P˙i e(t) + eT (t)Pi e(t) ˙ − ηeT (t)Pi e(t) ≤ eT (t)φi e(t) where φi (t) follows the below logic,
φi =
φ1i ,
φ2 , i
t ∈ lk,m , m = 0, 1, 2, · · · , Q − 1.
(14)
t ∈ [tk + τa , tk+1 ).
(11) establishes V˙ i (t) − ηVi (t) ≤ 0 when t ∈ lk,m . Additionally, on [tk +
τa , tk+1 ), it can be concluded that V˙ i (t) − ηVi (t) ≤ 0 due to (12). Combing with
(11),(12) and (14), it has V˙ i (t) − ηVi (t) ≤ 0, ∀t ∈ [tk , tk+1 ). 165
˙ From (10), we acquire Vi (tk ) ≤ µVj (t− k ). Then, combine it with Vi (t) −
ηVi (t) ≤ 0, the below is established. Vi (t) ≤ expη(t−tk ) Vi (tk ) ≤ µ expη(t−tk ) Vi (t− k) ≤ µ expη(t−tk )+η(tk −tk−1 ) Vi (tk−1 ) ≤ µ2 expη(t−tk )+η(tk −tk−1 ) Vi (t− k−1 ) ≤ ··· ≤ µk expη(t−t0 ) Vi (t0 ) 9
(15)
Depending on the construct way of the function, we draw the conclusion that Vi (t0 ) ≤ [(1 − ρ)λmax (Pi,m ) + ρλmax (Pi,m+1 )] k e(t0 ) k2 . Let k e(t) kc = sup {k e(t) k, k e(t) ˙ k}, F = (1−ρ)λmax (Pi,m )+ρλmax (Pi,m+1 ), [−ι,0]
170
υ = min{λmin (Pi,m ), λmin (Pi,L )}. Vi (t) ≤ µk expη(t−t0 ) Vi (t0 ) can be transformed into the below form, r
k ln h+η(t−t0 ) F 2 k e(t0 ) kc exp υ q ηTd +ln h F 0 , δ = , it has Set k = Nσ (t0 , t) ≤ t−t Td υ, ς = 2Td
k e(t) k≤
where δ =
q
k e(t) k≤ δ expς(t−t0 ) k e(t0 ) kc F υ
≥ 1, ς =
ηTd +ln h 2Td
≤ 0. With the help of Definition 1, the
exponential stability proof is accomplished. 175
3.2. Fault Sensitivity Condition Lemma 2 in this subsection is provided to illustrate the establishment of (6). Lemma 2. Given constants 0 < µ < 1, τa > 0, matrices Pi,m > 0, m = 0, 1, · · · , Q − 1, η > 0. if Lyapunov function (8) meets V˙ i (t) + ηVi (t) − Γ1 (t) ≤ 0,
(16)
Vi (tk ) ≤ µVj (t− k ), i, j ∈ N, i 6= j
(17)
where Γ1 (t) = −β02 f T (t)f (t) + rT (t)r(t), then system (3) is exponentially stable with H− performance with ADT τa ≤
− ln µ η
Proof. To testify the sensitiveness to the fault signal, we assume disturbance d(t) = 0. (3) can be rewritten as ˜ ˜ (t), e(t) ˙ = Ae(t) + Ef
r(t) = Ce(t) ˜ + F˜ f (t), 10
(18)
Pick Lyapunov function (8) for (3) and the derivative is V˙ i (t) = e˙ T (t)Pi e(t)+ 180
eT (t)P˙i e(t) + eT (t)Pi e(t). ˙ Given scalar η > 0, Γ1 (t) = −β02 f T (t)f (t) + rT (t)r(t),
if we have V˙ i (t) + ηVi (t) − Γ1 (t) ≤ 0, then, employing differential inequality
theory, it can be transferred as
Vi (t) ≤ Vi (tm ) exp−η(t−tm ) +
Z
t
exp−η(t−s) Γ1 (s)ds.
(19)
tm
Combing with (10), we can acquire the following Vi (t) ≤ exp
−η(t−tm )
Vi (tm ) +
Z
t
exp−η(t−s) Γ1 (s)ds
tm
≤ µ exp−η(t−tm ) Vi (t− m) +
Z
t
exp−η(t−s) Γ1 (s)ds
tm
≤ ··· ≤ µk exp−η(t−t0 ) Vi (t0 ) Z + µk exp−η(t−t1 )
t1
exp−η(t−s) Γ1 (s)ds
t0
+ µk−1 exp−η(t−t2 ) +
Z
t2
t1
Z
t
(20)
exp−η(t−s) Γ1 (s)ds + · · ·
exp−η(t−s) Γ1 (s)ds
tk
≤ exp−ηt+Nσ (t0 ,t) ln µ Vi (t0 ) Z t + exp−η(t−s)+Nσ (s,t) ln µ Γ1 (s)ds. 0
With the zero-initial condition, the following inequality is concluded. Z
t
0
185
exp−η(t−s)+Nσ (s,t) ln h Γ1 (s)ds ≥ 0.
(21)
Then, taking Γ1 (t) = −β02 f T (t)f (t) + rT (t)r(t) into (21), we obtain
β02
Z
0
t
exp−η(t−s)+Nσ (s,t) ln µ f T (s)f (s)ds ≤ Z t exp−η(t−s)+Nσ (s,t) ln µ rT (s)r(s)ds, 0
11
(22)
both sides multiply exp−Nσ (0,t) ln µ , it draws that
β02
Z
t
exp−η(t−s)−Nσ (0,s) ln µ f T (s)f (s)ds ≤ 0 Z t exp−η(t−s)−Nσ (0,s) ln µ rT (s)r(s)ds,
(23)
0
It derives that Nσ (0, t) ln µ < −Nσ (0, s) ln µ < −Nσ (0, t) ln µ from 0 < µ < yields ln µNσ (0, s) ≥ ln µ Tsd ≥ −ηs. Hence, combining
s Td
1, Besides, Nσ (0, s) ≤
the above issues, we can conclude
β02 β02
Z
t
Z
0
t
exp−ηt f T (s)f (s)ds ≤
−η(t−s)−Nσ (0,s) ln µ
exp
0
Z
f T (s)f (s)ds ≤
t
(24)
exp−η(t−s)−Nσ (0,s) ln µ rT (s)r(s)ds ≤ 0 Z t exp−η(t−s)−Nσ (0,t) ln µ rT (s)r(s)ds 0
190
Via integrating approach on both sides, we have
β2
Z
0
∞
exp−ηs f T (s)f (s)ds ≤
Z
∞
rT (s)r(s)ds,
(25)
0
where β 2 = µk β02 . Therefore, if V˙ i (t) + ηVi (t) − Γ1 (t) ≤ 0, then system (3) is sensitive to the fault signal f (t) with H− performance index β. Inherited from Lemma 2, the below Theorem proposes the H− performance.
195
Theorem 2. For any i, j ∈ N, i 6= j, 0 < µ1 < 1, η1 > 0, matrix Pi,m = Pi,m,1 0 > 0, m = 0, 1, · · · Q − 1, given matrices Hi , i = 1, 2, 3, 4, 0 Pi,m,2 ε1 , ε4 , ε5 , Xi,m,1 , Xi,m+1,1 , Yi,m,2 , Yi,m+1,2 constants β0 , τa , if
ln µ + ητa ≤ 0 12
(26)
Pi,0 − µPj,L ≤ 0
1 φi =
φ1i =
(27)
−He(Wi1 )
0
Ω13
0
Ω15
∗
−He(Wi2 )
0
Ω24
Ω25
∗
∗
Ω33
ε12
Ω35
∗
∗
∗
Ω44
Ω45
∗
∗
∗
∗
β 2 I − FiT Fi − FiT Fi
≤ 0.
(28)
−He(Wi1 )
0
Λ13
0
Ω15
∗
−He(Wi2 )
0
Λ24
Ω25
∗
∗
Λ33
ε12
Ω35
∗
∗
∗
Λ44
Ω45
∗
∗
∗
∗
β 2 I − FiT Fi − FiT Fi
≤ 0.
(29)
where T T Ω13 = Pi,m,1 − aWi1 + Wi1 Ai − Xi1 Ci , Ω15 = Wi1 Ei − Xi1 Fi T T Ω24 = Pi,m,2 − aWi2 + Wi2 Ai − Xi2 Ci , Ω25 = Wi2 Ei − Xi2 Fi T Ω33 = aHe(Wi1 Ai − Xi1 Ci ) + ηPi,m,1 + κp1 + ε1 T Ω35 = aWi1 Ei − Xi1 Fi + ε4 , ε1 = −Ci−T Ci− − Ci+T Ci+ T Ω44 = aHe(Wi2 Ai − Xi2 Ci ) + ηPi,m,1 + κp2 + ε1 T Ω45 = aWi1 Ei − Xi1 Fi + ε5 , ε5 = −Ci−T Fi + Ci+T Fi T Λ13 = Pi,m+1,1 − aWi1 + Wi1 Ai − Xi1 Ci T Λ24 = Pi,m+1,2 − aWi2 + Wi2 Ai − Xi2 Ci T Λ33 = aHe(Wi1 Ai − Xi1 Ci ) + ηPi,m+1,1 + κp1 + ε1 T Λ44 = aHe(Wi2 Ai − Xi2 Ci ) + ηPi,m+1,1 + κp2 + ε1
13
κp1 = L(Pi,m+1,1 − Pi,m,1 )/τa , κp2 = L(Pi,m+1,2 − Pi,m,2 )/τa , Γ1 (t) = −β02 f T (t)f (t) + rT (t)r(t), ε4 = Ci−T Fi − Ci+T Fi then, system (18) is considered to have H− performance. Proof. Pick Lyapunov function as the one in Lemma 2. And by the derivative approach, we gain V˙ i (t) + ηVi (t) − Γ1 (t) = e˙ T (t)Pi e(t) + eT (t)P˙i e(t) +eT (t)Pi e(t) ˙ + ηeT (t)Pi e(t) − eT (t)C˜iT C˜i e(t) − eT (t)C˜iT F˜i f (t) − f T (t)F˜iT C˜i e(t) − f T (t)F˜iT F˜i f (t) + β02 rT (t)r(t) ≤ ξ T (t)φi ξ(t) where ξ T (t) = [eT (t) f T (t)] and φi has the following rule,
φi =
φ1i ,
φ2 , i
t ∈ lk,m , m = 0, 1, 2, · · · , Q − 1.
(30)
t ∈ [tk + τa , tk+1 ).
When t ∈ lk,m , (28) indicates V˙ i (t) + ηVi (t) − Γ1 (t) ≤ 0. (29) means V˙ i (t) +
200
ηVi (t)−Γ1 (t) ≤ 0 on time interval [tk +τa , tk+1 ). Hence, V˙ i (t)+ηVi (t)−Γ1 (t) ≤ 0 on [tk , tk+1 ) is guaranteed. Also, φ1i and φ2i are obtained as
φ1i =
φ2i =
He(A˜i Pi,m ) + κp − C˜iT C˜i ∗
He(A˜i Pi,m+1 ) + κp −C˜iT C˜i ∗
˜i − C˜ T F˜i Pi,m E i β02 I − F˜iT F˜i
≤0.
˜i −C˜ T F˜i Pi,m+1 E i β02 I − F˜iT F˜i
≤0.
(31)
(32)
˜i and F˜i into account, (28) and (29) can be acquired. With Taking A˜i , C˜i , E Lemma 2, the H− performance is proved. 14
205
3.3. Disturbance Attenuation Conditions To get the condition (5) established, we have the following Lemma 3. Lemma 3. Given matrices Pi,m > 0, m = 0, 1, · · · , Q − 1, constants 0 < µ < 1, η > 0, τa > 0. If there exists Lyapunov function (8) satisfying V˙ i (t) + ηVi (t) − Γ2 (t) ≤ 0,
(33)
Vi (tk ) ≤ µVj (t− k ), i, j ∈ N, i 6= j
(34)
˜ − rT (t)r(t), then system (3) is exponentially stable where Γ2 (t) = γ02 d˜T (t)d(t) with H∞ performance with ADT τa ≤
− ln µ η
Proof. Under the assumption that fault signal f (t) equals to 0, system (3) can be re-given as ˜ ˜ ˜ d(t), e(t) ˙ = Ae(t) +B
r(t) = Ce(t) ˜ ˜ ˜ d(t), +D
(35)
T ˜ Given scalar η > 0, Γ2 (t) = γ02 d˜T (t)d(t)−r (t)r(t), if we have V˙ i (t)+ηVi (t)− 210
Γ2 (t) ≤ 0, then, employing differential inequality theory, it can be transferred as
Vi (t) ≤ Vi (tm ) exp−η(t−tm ) +
Z
t
exp−η(t−s) Γ2 (s)ds.
(36)
tm
For the similarity with Lemma 2, here omits the following proof process of Lemma 3. As depicted in Lemma 3, we have the following Theorem to guarantee the 215
disturbance attenuation conditions. Theorem 3. For any i, j ∈ N, i 6= j, 0 < µ < 1, η > 0, matrix Pi,m = Pi,m,1 0 > 0, m = 0, 1, · · · Q − 1, given matrices Hi , i = 1, . . . , 8, ε1 , 0 Pi,m,2 ε2 , ε3 , Xi,m,1 , Xi,m+1,1 , Yi,m,2 , Yi,m+1,2 constants β0 , τa , scalar γ > 0 if 15
1 ϕi =
ϕ1i =
ln µ + ητa ≤ 0
(37)
Pi,0 − µPj,L ≤ 0
(38)
−He(Wi1 )
0
Y13
0
Y15
Y16
∗
−He(Wi2 )
0
Y24
Y25
Y26
∗
∗
Y33
ε12
Y35
Y36
∗
∗
∗
Y44
Y45
Y46
∗
∗
∗
∗
−γ 2 I + ε4
ε5
∗
∗
∗
∗
∗
−γ 2 I + ε4
≤0
(39)
−He(Wi1 )
0
S13
0
Y15
Y16
∗
−He(Wi2 )
0
S24
Y25
Y26
∗
∗
S33
ε12
Y35
Y36
∗
∗
∗
S44
Y45
Y46
∗
∗
∗
∗
−γ 2 I + ε4
ε5
∗
∗
∗
∗
∗
−γ 2 I + ε4
≤0
(40)
where T + T − Y15 = Wi1 Bi − Xi1 Di+ , Y16 = Wi1 Bi − Xi1 Di− , ε2 = Ci−T Di+ + Ci+T Di− T + T − Y25 = Wi2 Bi − Xi2 Di+ , Y26 = Wi2 Bi − Xi2 Di− , ε1 = −Ci−T Ci− − Ci+T Ci+ T + T − Y35 = aWi1 Bi − aXi1 Di+ + ε2 , Y36 = aWi1 Bi − aXi1 Di− + ε3 T + T − Y45 = aWi2 Bi − aXi2 Di+ + ε3 , Y46 = aWi2 Bi − aXi2 Di− + ε2 T S13 = Pi,m+1,1 − aWi1 + Wi1 Ai − Xi1 Ci , ε4 = Di+T Di+ + Di−T Di−
16
T S24 = Pi,m+1,2 − aWi2 + Wi2 Ai − Xi2 Ci , ε5 = Di+T Di− + Di−T Di+ T Y13 = Pi,m,1 − aWi1 + Wi1 Ai − Xi1 Ci , ε3 = Ci−T Di− + Ci+T Di+ T Y24 = Pi,m,2 − aWi2 + Wi2 Ai − Xi2 Ci , ε12 = −Ci−T Ci+ − Ci+T Ci− T Y33 = aHe(Wi1 Ai − Xi1 Ci ) + ηPi,m,1 + κp1 − ε1 T Y44 = aHe(Wi2 Ai − Xi2 Ci ) + ηPi,m,1 + κp2 − ε1
κp1 = L(Pi,m+1 − Pi,m,1 )/τa , κp2 = L(Pi,m+1,2 − Pi,m,2 )/τa , ˜ − rT (t)r(t) Γ2 (t) = γ02 d˜T (t)d(t) T Ai − Xi1 Ci ) + ηPi,m+1,1 + κp1 − ε1 S33 = aHe(Wi1 T S44 = aHe(Wi2 Ai − Xi2 Ci ) + ηPi,m+1,1 + κp2 − ε1
then, system (35) is considered to have H∞ performance. Proof. Choosing the Lyapunov function as (8), through the derivative technique, we can gain V˙ i (t) + ηVi (t) − Γ2 (t) = e˙ T (t)Pi e(t) + eT (t)P˙i e(t) + eT (t)Pi e(t) ˙ + ηeT (t)Pi e(t) ˜ + d˜T (t)F˜ T C˜i e(t) + eT (t)C˜iT C˜i e(t) + eT (t)C˜iT F˜i d(t) i ˜ − γ 2 rT (t)r(t) + d˜T (t)F˜iT F˜i d(t) 0 ≤ ζ T (t)ϕi ζ(t) 220
where ζ T (t) = [eT (t) d˜T (t)] and φi has the following rule,
ϕi =
ϕ1i ,
t ∈ Ik,m , m = 0, 1, 2, · · · , Q − 1.
ϕ2 ,
(41)
t ∈ [tk + τa , tk+1 ).
i
When t ∈ Ik,m , (39) indicates V˙ i (t) + ηVi (t) − Γ2 (t) ≤ 0. Similarly, (40)
means V˙ i (t) + ηVi (t) − Γ2 (t) ≤ 0 on time interval [tk + τa , tk+1 ). Hence, the establishment of V˙ i (t) + ηVi (t) − Γ2 (t) ≤ 0 on [tk , tk+1 ) is guaranteed. Also, ϕ1i and ϕ2i are obtained as
ϕ1i =
˜ ˜T ˜ A˜T i Pi,m + Pi,m Ai + κp + Ci Ci
˜i + C˜ T D ˜i Pi,m B i
∗
˜ TD ˜i −γ02 I + D i 17
≤ 0.
(42)
ϕ2i = 225
˜ ˜T ˜ A˜T i Pi,m+1 + Pi,m+1 Ai + κp + Ci Ci
˜i + C˜ T D ˜i Pi,m+1 B i
∗
˜ TD ˜i −γ02 I + D i
≤ 0. (43)
˜i and D ˜ i into account, (39) and (40) can be acquired. With Taking A˜i , C˜i , B
the above Lemma 3, the H∞ performance can be ensured. Remark 3. As investigated in [42] the introduction of H∞ performance index can reduce the effect of the exoteric disturbance while H− performance can be used to evaluate the sensitivity level of residual on fault signal. In our paper, 230
we utilize the mixed index which are mentioned in Theorems 2 and 3. Remark 4. In the past few years, few studies deal with the switched systems with unstable systems. Even in [9], stable subsystems are required to exist. In [9], a special condition to ensure stability is proposed. That is
T − (pk ,pk+1 ) T + (pk ,pk+1 )
≥
β+χ α−χ ,
where [pk , pk+1 ) denote the time interval, T − (T1 , T2 )(T + (T1 , T2 )) represents the 235
total running time of stable subsystems (unstable subsystems). In this paper, according to [12], we deal with the switched system with all modes unstable. With the switching strategy created by the time-dependent method, we realize the fault detection and ensure the exponential stability with H∞ /H− performance. Theorems 2 and 3 give the mixed H∞ /H− performance analysis. The fol-
240
lowing task is considering the Metzler property. To ensure the Metzler property of Ai − Li Ci and Ai − Li Ci , we have the following subsection. 3.4. Metzler condition Consider a second-order switched system Ai = [aigh ]2∗2 , Li = [li1 Li = [li1
li2 ]T , also,
Ai − Li Ci =
ai11 − l1 c1
ai12 − l1 c2
ai21 − l2 c1
ai22 − l2 c2
18
,
li2 ]T ,
Ai − Li Ci =
ai11 − l1 c1
ai12 − l1 c2
ai21 − l2 c1
ai22 − l2 c2
Recall the definition of metzler matrix, it has
245
.
aigh − lg ch ≥ 0, aigh − lg ch ≥ 0, i, g, h = 1, 2, g 6= h. From Theorems 1-3, we findthat Ai − Li Ci and Ai − Li Ciare coupled with Pi,m,1 0 Pi,m+1,1 0 , Pi,m+1 = Pi,m , Pi,m+1 . Choosing Pi,m = 0 Pi,m,2 0 Pi,m+1,2 T
T and considering Xi,m = LT i Pi,m , Yi,m = Li Pi,m , Xi,m+1 = Li Pi,m+1 , Yi,m+1 = T
Li Pi,m+1 , we have
T Xi,m =
T Xi,m+1 =
0
0
Pi,m,2
Pi,m+1,1
0
0
Pi,m+1,2
T Yi,m =
T Yi,m+1 = 250
Pi,m,1
Pi,m,1
0
0
Pi,m,2
Pi,m+1,1
0
0
Pi,m+1,2
In addition, we get
Pi,m (Ai − Li Ci ) =
li1
li2
Pi,m,1 li1 Pi,m,2 li2
=
li2
li1
=
li2
li1
= li1 li2
=
Pi,m+1,1 li1 Pi,m+1,2 li2
Pi,m,1 li1 Pi,m,2 li2
=
=
Pi,m,2 li2
Xim2
=
Pi,m,1 li1
Xim1
Xim+12
Yim2
=
Xim+11
Yim1
Yim+11 Yim+12
Pi,m,1 ai11 − Xi,m,1 c1
Pi,m,1 ai12 − Xi,m,1 c2
Pi,m,2 ai21 − Xi,m,2 c1
Pi,m,2 ai22 − Xi,m,2 c2
19
Pi,m,1 ai11 − Xi,m+1,1 c1
Pi,m,1 ai12 − Xi,m+1,1 c2
Pi,m,2 ai21 − Xi,m+1,2 c1
Pi,m,2 ai22 − Xi,m+1,2 c2
Pi,m+1,1 ai11 − Yi,m,1 c1
Pi,m+1,1 ai12 − Yi,m,1 c2
Pi,m+1,2 ai21 − Yi,m,2 c1
Pi,m+1,2 ai22 − Yi,m,2 c2
Pi,m+1 (Ai − Li Ci ) = Pi,m (Ai − Li Ci ) =
Pi,m+1 (Ai − Li Ci ) =
Pi,m+1,1 ai11 − Yi,m+1,1 c1
Pi,m+1,1 ai12 − Yi,m+1,1 c2
Pi,m+1,2 ai21 − Yi,m+1,2 c1
Pi,m+1,2 ai22 − Yi,m+1,2 c2
Hence, the metzler condition can be transferred into the below equivalent
inequalities,
Pi,m,g aigh − Xi,m,g ch ≥ 0, Pi,m+1,g aigh − Yi,m+1,g ch ≥ 0, Pi,m,g aigh − Yi,m,g ch ≥ 0, Pi,m+1,g aigh − Xi,m+1,g ch ≥ 0, g, h = 1, 2, g 6= h. (44) Obviously, the condition (44) can be solved through the LMI toolbox. In like manner, the metzler condition for 2-nd order system can be gener255
alized to the n-th order switched systems, with the help of [37]. The metzler condition can be summarized as
Pi,m,g aigh − Xi,m,g ch ≥ 0, Pi,m+1,g aigh − Yi,m+1,g ch ≥ 0, Pi,m,g aigh − Yi,m,g ch ≥ 0, Pi,m+1,g aigh − Xi,m+1,g ch ≥ 0, g, h = 1, 2, . . . , n, g 6= h. (45) Based on Theorems 1-3, the following Theorem are given to obtain the interval observer. Theorem 4. Given scalar τa > 0, γ > 0, β > 0, 0 < µ < 1, η > 0, matrices 260
εx , x = 1, 2, 3, 4, 5, Xi,m,1 , Yi,m,2 , the observer gain Li and Li can be obtained −1 −1 by Li = Pi,m,1 Xi,m,1 , Li = Pi,m,2 Yi,m,2 if (26)-(29), (39), (40) and (45) are
satisfied. 20
As discussed in Theorems 2, 3 and 4, the disturbance attenuation (5), fault sensitivity (6) and observer gain have been realized. The following subpart 265
concludes the FD decision scheme as resulted in [37]. 3.5. Fault detection decision scheme To show the method more clearly, the fault detection logic is depicted in Fig. 1. As shown in the below picture, a disturbance with known bound and a fault with known expression are applied to the switched system. Then, the system
270
output is transferred to the interval observer. Through the transformation of the residual generator, the output of the interval observer is divided into the upper bound y(t) and the lower bound y(t). By subtracting with y(t), the upper and lower residual signal bound are obtained. Afterwards, the fault detection decision scheme gives the alarm rule depending on the logic which is given
275
later. Hence, the fault detection for switched systems with all modes unstable is finished.
Figure 1: Architecture of the FD Scheme
Here gives the FD decision scheme. As shown in Fig. 1, the residual signal is divided into the upper bound r(t) and the lower bound r(t) according to the rule of r(t) = y(t) − y(t), r(t) = y(t) − y(t). When there is no fault, 0 should be 21
280
bounded by r(t) and r(t), i. e., 0 ∈ [r(t), r(t)]. On the contrary, if 0 6∈ [r(t), r(t)], i. e., 0 cannot be bounded by the upper and lower bound, we can obtain that the fault appears in the switched system. By discriminating the membership of 0 and interval [r(t), r(t)], the FD can be accomplished. Through the discussion above, the following FD decision scheme is given,
Alarm
285
activated,
others
not activated,
(46) 0 ∈ [r(t), r(t)]
Remark 5. This paper employs the interval observer-based fault detection method, which is different from [31]-[33]. As pointed out in [37], the most obvious advantage lies in the fact the design of extra threshold generator can be removed due to that r(t) and r(t) can be used for FD directly. This may avoid the uncertainty brought by the improper threshold. Here gives the algorithm of this paper. ALGORITHM begin given µ, η, L Step 1
calculate ADT τa
Step 2
if (26)-(29), (37)-(40), and β >100 then output µ, η, τa , Li , Li else µ=µ + 0.05, η=η − 0.05 Step 1 end
290
Remark 6. Compared with the traditional fault detection method[31, 32, 33], the method used in this paper reduce the conservativeness and improve the accuracy to some extent. In [32], the threshold is set as Jth = sup Jr (t) with f (t)=0
22
Jr (t) = 295
q R 1 t t
0
rT (s)r(s)ds, which relies heavily on r(t). When a small fault oc-
curs, it may not be detected. Instead, in this paper, the upper and lower bound of the interval observer can be chosen artificially. The judgment logic of whether a fault occurs is to judge whether 0 is included in (r(t), r(t)). Obviously, this method saves computing resources than the traditional method.
300
4. Examples In this section, two examples are given to illustrate the validity of the interval observer. Pick the system dynamic in (1), some necessary parameters are given as
A1 C1
A2 C2
B1 D1
B2 D2
= F1
E1
0.35
0.6
0.69
−0.1
−0.1
−0.9439 0.3302
= F2
E2
−1.95
−0.6
−0.1
0.5
0.1
0.4
−1.99
0.5
0.9804 0.1970
−0.7
−0.1
−0.1
0.1 , −2
0.2 1
The eigenvalues of A1 and A2 are λ1,1 = −2.0724, λ1,2 = 0.0224, λ2,1 = 0.0005, λ2,2 = −2.0905, respectively. It is obvious that A1 and A2 are all unstable. In addition, from the parameters matrices, it has (C, A) is unobservable. Then, set the bounded disturbance as d(t) = 0.1 ∗ cos(0.5 ∗ t) ∗ exp(−0.1 ∗ t) d(t) = 0.1 ∗ cos(0.5 ∗ t) ∗ exp(−0.1 ∗ t) + 0.1 d(t) = 0.1 ∗ cos(0.5 ∗ t) ∗ exp(−0.1 ∗ t) − 0.1
23
2.5
2
1.5
1
0.5
0
10
20
30
40
50
60
70
80
90
100
Figure 2: Switching Signal
According to (9), if we choose η = 0.03, µ = 0.5, the average dwell time τa 305
can be acquired as τa = 2 < 23. Hence, we get the switching signal as Fig. 2. Choose the interval observer as (2). To meet the disturbance attenuation and fault sensitivity condition, (37)-(40), (28), (29) must be feasible. Using the LMI toolbox in MATLAB, we obtain the observer 0.3102 L L2 −0.5047 1 = 0.3102 L1 L2 −0.5047
gain as 0.0807
−1.0033 0.0807 −1.0033
Set H3 = A1 −L1 C1 , H4 = A2 −L2 C2 , H5 = A1 −L1 C1 and H6 = A2 −L2 C2 . It is clear that both of them are metzler from the below matrices.
H3 H5
−1.6572 0.2476
0.2136 0.0667 = −1.6572 0.2476 H6 0.2136 0.0667 H4
−0.1791 1.3836 −0.1791 1.3836
0.4841
−1.7923 0.4841 −1.7923
Here gives the eigenvalues of H3 , H4 , H5 and H6 . Obviously, due to the
24
unobservable condition of (Ai , Ci ), H3 , H4 , H5 , H6 are all unstable. 310
λ(H3 ) λ(H5 )
−1.6873
0.0968 = −1.6873 λ(H6 ) 0.0968 λ(H4 )
0.1634
−2.1348 0.1634 −2.1348
(47)
Remark 7. (47) implies that the unobservability of (Ai ; Ci ) result in the instability of the error dynamic system. Taking the error system (3) as a switched system with two unstable modes, the interval observer gains are obtained.Figs 2-7 indicates the effectiveness of the proposed method. Remark 8. In [37, 39], the interval observer is designed under the assump-
315
tion that (Ai ; Ci ) is observable. However, to our best knowledge, there is little literatures related to the interval observer design for switched system with all modes unstable, especially under the unobservable condition. To indicate the effectiveness of the designed observer, in Example 2, we discuss the multi-fault circumstance.
320
To test the actual effectiveness of the interval observer, we take the following two examples. 4.1. Example 1 When there is no fault, the results are shown in Fig. 3. Fig. 3(a) shows the trajectories of output y(t), upper output estimation y(t) and lower output
325
estimation y(t). The residual signal r(t), upper residual bound r(t) and lower residual bound r(t) are depicted in Fig. 3(b). Obviously, the output y(t) is bounded by the upper and lower bound when there is no fault. Similarly, 0 ∈ [r(t), r(t)] is always hold in this fault-free case. Then, the fault is given in subsystem 1 in the shape of (48) and in subsystem
330
2 in the shape of (49)
f1 (t) =
1,
0,
t ∈ [20, 25) others. 25
(48)
0.5
0.5
0
0
-0.5
0
20
40
60
80
100
-0.5
0
20
40
60
80
Figure 3: Residual signal and output without fault
f2 (t) =
1,
t ∈ [40, 45)
0,
(49)
others.
Main results are shown in Figs. 4 and 5. Fig. 4 describes the system output and residual signal when fault occurs in subsystem 1. As shown in the picture, output y(t) and 0 are bounded by the upper and lower bound when t 6∈ [20, 25). On the time interval t ∈ [20, 25), 335
y(t) is not constrained by the bounds and 0 6∈ [r(t), r(t)] is established. Under this situation, the fault can be detected. In Fig. 5, the residual signal and output with fault in subsystem 2 are provided. At the tome period [40, 45], we can clearly find that the residual signal with fault is not in the residual signal bound. In addition, the outputs is not belong to the upper and lower bound,
340
which means that fault is happening. The detailed description is similar with the one of Fig. 4 and here omits it. 4.2. Example 2 In Example 1, the fault detection of the interval observer on the single subsystem is accomplished. Next, we want to test the ability of the interval observer
345
on the multi-fault and multi-type fault. Figs. 6-7 display the effectiveness and the validity. 26
100
1
1
0.5
0.5
0
0
-0.5
-0.5
-1 -1
-1.5
-1.5
-2
-2
-2.5 -3
0
20
40
60
80
100
-2.5
0
20
40
60
80
100
Figure 4: Residual signal and output with fault in subsystem 1
2
2
1.5
1.5
1
1
0.5
0.5
0
0
-0.5
-0.5
-1
0
20
40
60
80
100
-1
0
20
40
Figure 5: Residual signal and output with fault in subsystem 2
27
60
80
100
Firstly, we assume that the fault occurs in subsystem s 1 and 2 in the following shape.
f (t) =
1,
t ∈ [45, 50)
0,
(50)
others.
Utilizing the parameters in Example 1, we obtain the results as Fig. 6. As 350
is displayed, we find that the residual signal is in the upper and lower bound of the residual signal, expect when t ∈ [45, 50). Similarly, y(t) is not restricted to be in [y(t), y(t)]. 1.5
2
1 1
0.5 0
0
-0.5 -1
-1 -1.5
-2
-2 -3
0
20
40
60
80
100
-2.5
0
20
40
60
80
Figure 6: Residual signal and output with fault in subsystems 1 and 2
f (t) =
1,
0,
t ∈ [35, 40) others.
[
[55, 60) (51)
In Fig. 7, we consider the multi-fault circumstance. First, we give the fault signal as (51). As shown in Fig. 7, the residual signal is beyond the upper and 355
lower bound of r(t) when t is in the fault time period. At the same time, the S outputs is not in [y(t), y(t)] when t ∈ [35, 40) [55, 60).
From the two examples above, it can be concluded that y(t) is bounded by
y(t) and y(t) and 0 ∈ [r(t), r(t)] holds in the fault-free case. On the other hands, when fault appears, the output y(t) increases sharply leading to 0 6∈ [r(t), r(t)]. 28
100
2
1
1
Outputs
Residual Signal
2
0
0
-1
-1
-2
-2
-3
0
20
40
60
80
100
-3
0
20
40
Time (s)
60
80
Time (s)
Figure 7: Residual signal and output under multi-fault circumstance
360
According to the FD decision scheme in Section 3, the fault can be detected. This example verifies the effectiveness of the proposed method.
5. Conclusions In this paper, observer-based fault detection is accomplished for switched system when all modes unstable. Compared with the literature related to tra365
ditional FD scheme, i. e., the appearance of a fault is evaluated by the threshold function, this paper employs the interval observer-based FD scheme, which reduces the conservativeness and add the degree of freedom. With the consideration of unstable modes, a special switching rule is given utilizing average dwell time to realize the exponential stability. One special thing of this paper
370
is that the interval observer is designed when (Ai ; Ci ) are unobservable and we believe that the unobservability is the root of the instability. Utilizing LMIs, sufficient conditions making sure the H∞ /H− performance are derived. In the end, examples are listed to verify the detect property of the interval observer.
Conicts of Interest 375
The authors declare that they have no conflicts of interest.
29
100
6. Acknowledgments This work is supported by the National Natural Science Foundation of China (61873057, 61773187), Jilin City Science and Technology Bureau (201831727, 201831731), Natural Science Foundation of Jilin Province (20180520211JH) and 380
Education Department of Jilin Province (201693, JJKH20170106KJ).
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Journal Pre-proof
Fault Detection for Switched Systems with All Modes Unstable based on Interval Observer Qingyu Su, Zhongxin Fan, Tong Lu, Yue Long, Jian Li PII: DOI: Reference:
S0020-0255(19)31197-1 https://doi.org/10.1016/j.ins.2019.12.071 INS 15114
To appear in:
Information Sciences
Received date: Revised date: Accepted date:
20 July 2019 6 December 2019 27 December 2019
Please cite this article as: Qingyu Su, Zhongxin Fan, Tong Lu, Yue Long, Jian Li, Fault Detection for Switched Systems with All Modes Unstable based on Interval Observer, Information Sciences (2019), doi: https://doi.org/10.1016/j.ins.2019.12.071
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Fault Detection for Switched Systems with All Modes Unstable based on Interval Observer Qingyu Sua , Zhongxin Fana , Tong Lua , Yue Longb , Jian Lia,∗ a School
of Automation Engineering, Northeast Electric Power University, Jilin, Jilin, 132012, China. b School of Physics, Liaoning University, Shenyang, Liaoning, 110036, China.
Abstract This paper concentrates on observer-based fault detection (FD) for switched systems. By the introduction of the interval observer, the error dynamic system is proposed. In light of the switching logic, exponential stability of the augmented system is firstly achieved. Then, to ensure the fault sensitivity as well as disturbance robustness, H∞ /H− performance analysis is employed. One thing has to be mentioned is that we investigate the unobservable condition of (Ai ; Ci ). Moreover, the instability of the error dynamic system brought by the unobservability is eliminated by the switching strategy using average dwell time (ADT) method. Afterwards, sufficient conditions guaranteing the H∞ /H− performance level are obtained , which is divided into disturbance attenuation and fault sensitivity analysis. Finally, examples demonstrating the effectiveness of the provided method are provided. Keywords: average dwell time, unstable subsystems, interval observer, discretized Lyapunov function, mixed H∞ /H− performance. 1. Introduction Switched systems, being an of importance part of hybrid systems, compose several subsystems and a switching logic rule indicating which subsystem is ∗ Corresponding
author Email address: [email protected] (Jian Li) URL: [email protected] (Qingyu Su), [email protected] (Zhongxin Fan), [email protected] (Tong Lu), [email protected] (Yue Long)
Preprint submitted to Journal of Information Sciences
December 28, 2019
chosen at a certain time instant. Since the proposition of switched systems, its 5
importance has been widely recognized. In recent years, it has been applied to many areas, such as [1, 2, 3, 4]. In addition, the switching method is also wildly used[5, 6]. Due to the diversity of its subsystems, the researches of switched systems are different from the other systems. Take the stability as an example, the stability of the subsystem does not necessarily bring the stability
10
of the entire system due to the improper switching signal. Conversely, the unstable subsystems may not influent the stability of the whole system if the switching rule can be chosen properly. Therefore, the stability can be viewed as the combination results of the switching rule and the subsystems, which is the essential difference between switched systems and other systems.
15
According to the property of the subsystem, we can divide the literatures into three categories. (1). Switched system consisting of all stable subsystems. event-triggered filter is designed for switched discrete-time systems [7]. [8] realizes the fault detection and control when actuator lose efficacy. (2). Switched system including unstable modes. [9] investigates the stable property with L2 -
20
gain for switched time-delay systems. [10] designs the H∞ filters for nonlinear systems while [11] researches the nonlinear switched systems. (3). Switched systems with all modes unstable. [12] employs the dwell time to finish the stabilization of continuous systems. Stability and L2 -gain are analyzed in [13] when all modes unstable for switched systems. Discrete-time systems’ stability
25
is analyzed in [14]. Under asynchronous switching, [15] studies the nonlinear systems with time-delays. Sliding mode control, memory filter design, stabilization analysis and adaptive control are investigated in [16, 17, 18, 19] Taking switched system with all modes unstable into consideration, as discussed in [12], we find that a proper switching signal can make the switched
30
systems stable even though all the subsystems are unstable. Hence, this paper attempts to explore an appropriate switching rule. As far as we know, the dwell time method is frequently used in the analysis of the property of switched systems. [20] uses the dwell time (DT) approach to accomplish the H∞ filtering for discrete-time switched systems under missing measurements. In [21], the DT 2
35
method is employed to analyze the stability, L1 -gain and asynchronous L1 -gain control for uncertain switched positive systems. On the basis of the DT, average dwell time (ADT) method is proposed for its small conservation and less computation resource usage. In [22], ADT is used to design the estimator for discrete-time switched systems. Asynchronous H∞ control for switched delay
40
systems is completed with ADT [23]. Additionally, stability and L2 -gain are analyzed for switched systems in [24] and [25], respectively. Many external interference signals including known control inputs, unknown disturbance signals, fault signals and malicious attacks may be applied during system operation.[26]. In [27] and [28], fault tolerant issues are involved for
45
polynominal-fuzzy-model-based systems and multi-agent systems, respectively. Sampled-data control is researched in [29, 30] for nonlinear switched system. Considering quantization effects, the FD (fault detection) filter is designed [31]. [32] studies finite-time FD problem for interconnected systems while [33] researches the FD for linear-parameter-varying (LPV) system. In the above lit-
50
eratures, FD is finished depending on the residual signal generator and the threshold function[34, 35, 36]. The former is used to produce the residual signal. The latter is used to evaluate the appearance of the fault. The FD logic can be stated as, the fault detection is proceeded depending on whether the residual evaluation function surpasses the threshold. However, as pointed in
55
[37], the improper choose of the threshold may greatly influent the results of the FD, which bring a lot of uncertainty. Thence, in order to get rid of the uncertainty and get more degree of freedom, this paper employs the interval observer which is wildly used in [37, 38, 39, 40]. In [37], FD in finite-frequency domain is completed via interval observer. [38] constructs the hybrid interval
60
observers for linear switched system. Similarly, interval observers are designed in [39] and [40] for discrete-time systems, respectively. When fault occurs in a modern power grid system, it has been studied and demonstrated that the system has a great potential to go from observable to unobservable. However, there are few researches on observer design and fault detection for unobservable power
65
grids. This paper provides a research idea, i. e. we convert the unobservable 3
condition to a instability analysis, which is of great research significance. The contributions of the paper are summarized in this part. • An ADT approach is employed to generate the switching logic, which is used to ensure the stability. • From the theoretical aspect, this paper solves the observer design problem
70
under the unobservable condition, which is rarely discussed in other literature. In addition, the introduction of the interval observer improves the accuracy of fault detection and reduces computing resources. • In practical aspect, this paper provides another method of fault detection, 75
which enables users to have a better comparison in actual operation. Additionally, the H∞ /H− ensures a better fault detection effect is guaranteed under a reasonable anti-interference condition. Main structure is arranged as below. Section 2 provides the systems model while Section 3 concludes the sufficient conditions ensuring the exponential sta-
80
bility and H∞ /H− performance. Also, the FD decision scheme is provided. Then, in Section 4, an example is raised to finish the illustration of the validity of the interval observer. Section 5 gives the conclusions and acknowledgments. Notations: The notation k ∗ k is the Euclidean norm for vectors. He(A) =
A +A. The inverse of a matrix is stated by ∗−1 and the transpose one is denoted T
85
by A∗ . Matrix A > 0(A < 0) represents it is positive definite (negative definite). In addition, the minimum and maximum eigenvalues of A are expressed as (λmin (A)) and λmax (A), respectively. R is the set of the whole real number.
Rn indicates the n-dim Euclidean space. Without loss of generality, I is used as the identity matrix. ∗ stands for the main-diagonal-under elements. 90
2. Preliminaries and System Model This section gives the system description and some necessary definitions. Consider the following switched systems with all modes unstable, x(t) ˙ = Aσ(t) x(t) + Bσ(t) d(t) + Eσ(t) f (t), y(t) = C
σ(t) x(t) + Dσ(t) d(t) + Fσ(t) f (t)
4
(1)
where x(t) ∈ Rx , y(t) ∈ Ry represent the unstable system state and system
output, respectively. f (t) ∈ Rf indicates the fault signal. d(t) ∈ Rd denotes the 95
disturbance input. The function σ(t) : [0, ∞) → N = {1, 2, · · · , n} denotes the switching rule indicating which mode is activated at time t, which is a piecewise constant function from the right. In addition, the matrices A, B, C, D, E, F are known constant matrices with appropriate dimensions. For switched system (1), the interval observer is obtained as follows, ˙ x(t) = (A − LC)x(t) + LCx(t) + LDd + LF f − B − d + B + d − LD+ d + LD− d x(t) ˙ = (A − LC)x(t) + LCx(t) + LDd + LF f − B − d + B + d − LD+ d + LD− d + − y(t) = Cσ(t) x(t) − Cσ(t) x(t) y(t) = C + x(t) − C − x(t) σ(t) σ(t)
100
(2)
where D+ = max{0, D}, D− = D+ − D, x(t) ∈ Rx , x(t) ∈ Rx are the upper
and lower estimates of state x(t). y(t) ∈ Ry , y(t) ∈ Ry denote the upper and lower estimates of output y(t), respectively. r(t) ∈ Rr and r(t) ∈ Rr represent the lower and upper bound. L ∈ Rl is the parameter to be determined.
105
If we denote e(t) = x(t) − x(t), e(t) = x(t) − x(t), r(t) = y(t) − y(t), e(t) r(t) d−d , r(t) = , d˜ = , we can r(t) = y(t) − y(t), e(t) = e(t) r(t) d−d gain ˜ +E ˜σ(t) d(t) ˜σ(t) f (t), e(t) ˙ = A˜σ(t) e(t) + B r(t) = C˜ e(t) + D ˜ + F˜σ(t) f (t). ˜ σ(t) d(t) σ(t)
where
A˜σ(t) =
Aσ(t) − Lσ(t) Cσ(t)
0
0
Aσ(t) − Lσ(t) Cσ(t)
˜σ(t) = B
+ + Bσ(t) − Lσ(t) Dσ(t)
− − Bσ(t) − Lσ(t) Dσ(t)
, C˜σ(t) =
− − Bσ(t) − Lσ(t) Dσ(t) + + Bσ(t) − Lσ(t) Dσ(t)
5
(3)
− −Cσ(t) + Cσ(t)
, F˜σ(t) =
Fσ(t) Fσ(t)
+ −Cσ(t) − Cσ(t)
,
˜σ(t) = E
Eσ(t) − Lσ(t) Fσ(t) Lσ(t) Fσ(t) − Eσ(t)
˜ σ(t) = ,D
+ −Dσ(t) − Dσ(t)
− −Dσ(t) + Dσ(t)
,
To conduct the following work, some useful Assumptions and Lemmas are 110
needed. Assumption 1. There are known bound function d(t) and d(t) such that d(t) ≤ d(t) ≤ d(t) Lemma 1. [39] Given x(t) ∈ Rn and x(t) ≤ x(t) ≤ x(t) for some x(t) ∈ Rn ,
x(t) ∈ Rn , we have
+ − + − Bσ(t) d(t) − Bσ(t) d(t) ≤ Bσ(t) d(t) ≤ Bσ(t) d(t) − Bσ(t) d(t) + − + − Cσ(t) x(t) − Cσ(t) x(t) ≤ Cσ(t) x(t) ≤ Cσ(t) x(t) − Cσ(t) x(t)
Remark 1. The results of fault detection using interval observer have been a lot, such as [37, 39, 40]. It should be mentioned that switched system is rarely related. As in [37], the system matrix A is required to be Schur stable. However, in this paper, the restriction of stability of A is relaxed for that the whole 115
stability can be reached via proper switching strategy though all the subsystems are unstable, which is hardly involved in the previous literatures. Definition 1. [23] When there is no external input, k e(t) k=
sup {k e(t +
−τ ≤θ≤0
θ) k, k e(t ˙ + θ) k}, if k e(t) k≤ o expς(t−t0 ) k e(t0 ) k when t ≥ t0 , o > 1, ς < 0, then, under the switching logic σ(t), system (3) is exponentially stable. Definition 2. [23] Let Nσ (t, T ) denotes the number of switching of σ(t) on time interval (t, T ). If it has N0 ≥ 0, σ(t) is viewed as having average dwell time τa and Nσ (t, T ) ≤ N0 + 120
T −t , ∀T ≥ t ≥ 0. τa
(4)
then, τa is called as ADT and N0 is chatter bound. We set the chatter bound N0 = 0. 6
With the assistance of [41, 42], we give the following definition.
Definition 3. [43] System (3) is said to have weighted H∞ /H− performance if Z ∞ Z ∞ ˜ d˜T (s)d(s)ds (5) exp−ηs rT (s)r(s)ds ≤ γ 2 0
0
Z
0
∞
rT (s)r(s)ds ≥ β 2
Z
∞
exp−ηs f T (s)f (s)ds
(6)
0
Under the circumstance that (Ai ; Ci ) is not observable, the main idea of this 125
paper is find the proper observer gain such that 1. Ai − Li Ci and Ai − Li Ci are metzler. 2. condition (5) is satisfied, i. e., the effects of disturbance on residual can be minimize. 3. condition (6) is satisfied, i. e., the effects of fault on residual can be
130
maximize.
3. Main results This section talks about the main results. To be clear, this part is divided into five subsections in which the exponential stability, disturbance attenuation, fault sensitivity, Metzler condition and fault detection scheme are involved, re135
spectively. Taking the inexistence of stable subsystems into consideration, the traditional Lyapunov function method is not applicable any more. In light of [12], the discretized Lyapunov function method is introduced. The construction method is given below.
140
Given time period [tk , tk + τa ), firstly we divide it into Q segments which can be denoted by lk,m = [tk + θm , tk + θm+1 ), where θm =
mτa Q ,
m = 0, 1, · · · Q − 1.
Afterwards, choose the continuous matrix function Pi (t) as linear on lk,m and define Pi,m = Pi (tk + θm ), we have Pi (t) = Pi (tk + θm + αd) = (1 − α)Pi,k + (k)
αPi,k+1 = Pi (α), where α = (t − tk − θm )/d, d = τa /Q. Since Pi (t) is 7
145
piecewise linear on [tk , tk + τa ), utilizing the linear interpolation formula, it can be described using the values at dividing points. Additionally, we find that function Pi (t) is a constant matrix Pi,L . Hence, the discretized function Pi (t) has the following shape,
Pi (t) =
Pi(k) (α), P
t ∈ lk,m , m = 0, 1, 2, · · · , Q − 1.
(7)
t ∈ [tk + τa , tk+1 ).
i,Q ,
The corresponding discretized Lyapunov function can be obtained as
Vi (t) =
150
xT (t)Pi(k) (α)x(t), xT (t)P
i,Q x(t),
t ∈ lk,m , m = 0, 1, 2, · · · , Q − 1.
(8)
t ∈ [tk + τa , tk+1 ).
Remark 2. From the definition of the discretized Lyapunov function, it is clear that the greater the value of Q, the more segments are obtained on time interval [tk , tk +τa ), which may reduce the conservativeness obviously. The related results have been displayed in [10]. To be brief and save computational resource, this paper sets Q as 1.
155
3.1. Exponential Stability Analysis When disturbance signal d(t) and fault signal f (t) are set as 0, we have the ˜ system dynamic e(t) ˙ = Ae(t). Under this assumption, the following part is the analysis on exponential stability. Theorem 1. For 0 < µ < 1, any i, j ∈ N, i 6= j, η > 0, matrix Pi,m > 0, m = 0, 1, · · · Q − 1, τa > 0, if ln µ + ητa ≤ 0
(9)
Pi,0 − µPj,Q ≤ 0
(10)
˜ φ1i = A˜T i Pi,m + Pi,m Ai + ηPi,m + κp ≤ 0,
(11)
8
˜ φ2i = A˜T i Pi,m+1 + Pi,m+1 Ai + ηPi,m+1 + κp ≤ 0.
(12)
where κp = Q(Pi,m+1 − Pi,m )/τa , in that way, system (3) has the property of 160
exponential stability. ˜ = 0 and f (t) = 0, we have Proof. With the assumption that d(t) ˜ e(t) ˙ = Ae(t) Selecting the Lyapunov function as (8), through derivative technology, the following equation can be derived . V˙ i (t) = e˙ T (t)Pi e(t) + eT (t)P˙i e(t) + eT (t)Pi e(t) ˙
(13)
Thus, it has V˙ i (t) − ηVi (t) = e˙ T (t)Pi e(t) + eT (t)P˙i e(t) + eT (t)Pi e(t) ˙ − ηeT (t)Pi e(t) ≤ eT (t)φi e(t) where φi (t) follows the below logic,
φi =
φ1i ,
φ2 , i
t ∈ lk,m , m = 0, 1, 2, · · · , Q − 1.
(14)
t ∈ [tk + τa , tk+1 ).
(11) establishes V˙ i (t) − ηVi (t) ≤ 0 when t ∈ lk,m . Additionally, on [tk +
τa , tk+1 ), it can be concluded that V˙ i (t) − ηVi (t) ≤ 0 due to (12). Combing with
(11),(12) and (14), it has V˙ i (t) − ηVi (t) ≤ 0, ∀t ∈ [tk , tk+1 ). 165
˙ From (10), we acquire Vi (tk ) ≤ µVj (t− k ). Then, combine it with Vi (t) −
ηVi (t) ≤ 0, the below is established. Vi (t) ≤ expη(t−tk ) Vi (tk ) ≤ µ expη(t−tk ) Vi (t− k) ≤ µ expη(t−tk )+η(tk −tk−1 ) Vi (tk−1 ) ≤ µ2 expη(t−tk )+η(tk −tk−1 ) Vi (t− k−1 ) ≤ ··· ≤ µk expη(t−t0 ) Vi (t0 ) 9
(15)
Depending on the construct way of the function, we draw the conclusion that Vi (t0 ) ≤ [(1 − ρ)λmax (Pi,m ) + ρλmax (Pi,m+1 )] k e(t0 ) k2 . Let k e(t) kc = sup {k e(t) k, k e(t) ˙ k}, F = (1−ρ)λmax (Pi,m )+ρλmax (Pi,m+1 ), [−ι,0]
170
υ = min{λmin (Pi,m ), λmin (Pi,L )}. Vi (t) ≤ µk expη(t−t0 ) Vi (t0 ) can be transformed into the below form, r
k ln h+η(t−t0 ) F 2 k e(t0 ) kc exp υ q ηTd +ln h F 0 , δ = , it has Set k = Nσ (t0 , t) ≤ t−t Td υ, ς = 2Td
k e(t) k≤
where δ =
q
k e(t) k≤ δ expς(t−t0 ) k e(t0 ) kc F υ
≥ 1, ς =
ηTd +ln h 2Td
≤ 0. With the help of Definition 1, the
exponential stability proof is accomplished. 175
3.2. Fault Sensitivity Condition Lemma 2 in this subsection is provided to illustrate the establishment of (6). Lemma 2. Given constants 0 < µ < 1, τa > 0, matrices Pi,m > 0, m = 0, 1, · · · , Q − 1, η > 0. if Lyapunov function (8) meets V˙ i (t) + ηVi (t) − Γ1 (t) ≤ 0,
(16)
Vi (tk ) ≤ µVj (t− k ), i, j ∈ N, i 6= j
(17)
where Γ1 (t) = −β02 f T (t)f (t) + rT (t)r(t), then system (3) is exponentially stable with H− performance with ADT τa ≤
− ln µ η
Proof. To testify the sensitiveness to the fault signal, we assume disturbance d(t) = 0. (3) can be rewritten as ˜ ˜ (t), e(t) ˙ = Ae(t) + Ef
r(t) = Ce(t) ˜ + F˜ f (t), 10
(18)
Pick Lyapunov function (8) for (3) and the derivative is V˙ i (t) = e˙ T (t)Pi e(t)+ 180
eT (t)P˙i e(t) + eT (t)Pi e(t). ˙ Given scalar η > 0, Γ1 (t) = −β02 f T (t)f (t) + rT (t)r(t),
if we have V˙ i (t) + ηVi (t) − Γ1 (t) ≤ 0, then, employing differential inequality
theory, it can be transferred as
Vi (t) ≤ Vi (tm ) exp−η(t−tm ) +
Z
t
exp−η(t−s) Γ1 (s)ds.
(19)
tm
Combing with (10), we can acquire the following Vi (t) ≤ exp
−η(t−tm )
Vi (tm ) +
Z
t
exp−η(t−s) Γ1 (s)ds
tm
≤ µ exp−η(t−tm ) Vi (t− m) +
Z
t
exp−η(t−s) Γ1 (s)ds
tm
≤ ··· ≤ µk exp−η(t−t0 ) Vi (t0 ) Z + µk exp−η(t−t1 )
t1
exp−η(t−s) Γ1 (s)ds
t0
+ µk−1 exp−η(t−t2 ) +
Z
t2
t1
Z
t
(20)
exp−η(t−s) Γ1 (s)ds + · · ·
exp−η(t−s) Γ1 (s)ds
tk
≤ exp−ηt+Nσ (t0 ,t) ln µ Vi (t0 ) Z t + exp−η(t−s)+Nσ (s,t) ln µ Γ1 (s)ds. 0
With the zero-initial condition, the following inequality is concluded. Z
t
0
185
exp−η(t−s)+Nσ (s,t) ln h Γ1 (s)ds ≥ 0.
(21)
Then, taking Γ1 (t) = −β02 f T (t)f (t) + rT (t)r(t) into (21), we obtain
β02
Z
0
t
exp−η(t−s)+Nσ (s,t) ln µ f T (s)f (s)ds ≤ Z t exp−η(t−s)+Nσ (s,t) ln µ rT (s)r(s)ds, 0
11
(22)
both sides multiply exp−Nσ (0,t) ln µ , it draws that
β02
Z
t
exp−η(t−s)−Nσ (0,s) ln µ f T (s)f (s)ds ≤ 0 Z t exp−η(t−s)−Nσ (0,s) ln µ rT (s)r(s)ds,
(23)
0
It derives that Nσ (0, t) ln µ < −Nσ (0, s) ln µ < −Nσ (0, t) ln µ from 0 < µ < yields ln µNσ (0, s) ≥ ln µ Tsd ≥ −ηs. Hence, combining
s Td
1, Besides, Nσ (0, s) ≤
the above issues, we can conclude
β02 β02
Z
t
Z
0
t
exp−ηt f T (s)f (s)ds ≤
−η(t−s)−Nσ (0,s) ln µ
exp
0
Z
f T (s)f (s)ds ≤
t
(24)
exp−η(t−s)−Nσ (0,s) ln µ rT (s)r(s)ds ≤ 0 Z t exp−η(t−s)−Nσ (0,t) ln µ rT (s)r(s)ds 0
190
Via integrating approach on both sides, we have
β2
Z
0
∞
exp−ηs f T (s)f (s)ds ≤
Z
∞
rT (s)r(s)ds,
(25)
0
where β 2 = µk β02 . Therefore, if V˙ i (t) + ηVi (t) − Γ1 (t) ≤ 0, then system (3) is sensitive to the fault signal f (t) with H− performance index β. Inherited from Lemma 2, the below Theorem proposes the H− performance.
195
Theorem 2. For any i, j ∈ N, i 6= j, 0 < µ1 < 1, η1 > 0, matrix Pi,m = Pi,m,1 0 > 0, m = 0, 1, · · · Q − 1, given matrices Hi , i = 1, 2, 3, 4, 0 Pi,m,2 ε1 , ε4 , ε5 , Xi,m,1 , Xi,m+1,1 , Yi,m,2 , Yi,m+1,2 constants β0 , τa , if
ln µ + ητa ≤ 0 12
(26)
Pi,0 − µPj,L ≤ 0
1 φi =
φ1i =
(27)
−He(Wi1 )
0
Ω13
0
Ω15
∗
−He(Wi2 )
0
Ω24
Ω25
∗
∗
Ω33
ε12
Ω35
∗
∗
∗
Ω44
Ω45
∗
∗
∗
∗
β 2 I − FiT Fi − FiT Fi
≤ 0.
(28)
−He(Wi1 )
0
Λ13
0
Ω15
∗
−He(Wi2 )
0
Λ24
Ω25
∗
∗
Λ33
ε12
Ω35
∗
∗
∗
Λ44
Ω45
∗
∗
∗
∗
β 2 I − FiT Fi − FiT Fi
≤ 0.
(29)
where T T Ω13 = Pi,m,1 − aWi1 + Wi1 Ai − Xi1 Ci , Ω15 = Wi1 Ei − Xi1 Fi T T Ω24 = Pi,m,2 − aWi2 + Wi2 Ai − Xi2 Ci , Ω25 = Wi2 Ei − Xi2 Fi T Ω33 = aHe(Wi1 Ai − Xi1 Ci ) + ηPi,m,1 + κp1 + ε1 T Ω35 = aWi1 Ei − Xi1 Fi + ε4 , ε1 = −Ci−T Ci− − Ci+T Ci+ T Ω44 = aHe(Wi2 Ai − Xi2 Ci ) + ηPi,m,1 + κp2 + ε1 T Ω45 = aWi1 Ei − Xi1 Fi + ε5 , ε5 = −Ci−T Fi + Ci+T Fi T Λ13 = Pi,m+1,1 − aWi1 + Wi1 Ai − Xi1 Ci T Λ24 = Pi,m+1,2 − aWi2 + Wi2 Ai − Xi2 Ci T Λ33 = aHe(Wi1 Ai − Xi1 Ci ) + ηPi,m+1,1 + κp1 + ε1 T Λ44 = aHe(Wi2 Ai − Xi2 Ci ) + ηPi,m+1,1 + κp2 + ε1
13
κp1 = L(Pi,m+1,1 − Pi,m,1 )/τa , κp2 = L(Pi,m+1,2 − Pi,m,2 )/τa , Γ1 (t) = −β02 f T (t)f (t) + rT (t)r(t), ε4 = Ci−T Fi − Ci+T Fi then, system (18) is considered to have H− performance. Proof. Pick Lyapunov function as the one in Lemma 2. And by the derivative approach, we gain V˙ i (t) + ηVi (t) − Γ1 (t) = e˙ T (t)Pi e(t) + eT (t)P˙i e(t) +eT (t)Pi e(t) ˙ + ηeT (t)Pi e(t) − eT (t)C˜iT C˜i e(t) − eT (t)C˜iT F˜i f (t) − f T (t)F˜iT C˜i e(t) − f T (t)F˜iT F˜i f (t) + β02 rT (t)r(t) ≤ ξ T (t)φi ξ(t) where ξ T (t) = [eT (t) f T (t)] and φi has the following rule,
φi =
φ1i ,
φ2 , i
t ∈ lk,m , m = 0, 1, 2, · · · , Q − 1.
(30)
t ∈ [tk + τa , tk+1 ).
When t ∈ lk,m , (28) indicates V˙ i (t) + ηVi (t) − Γ1 (t) ≤ 0. (29) means V˙ i (t) +
200
ηVi (t)−Γ1 (t) ≤ 0 on time interval [tk +τa , tk+1 ). Hence, V˙ i (t)+ηVi (t)−Γ1 (t) ≤ 0 on [tk , tk+1 ) is guaranteed. Also, φ1i and φ2i are obtained as
φ1i =
φ2i =
He(A˜i Pi,m ) + κp − C˜iT C˜i ∗
He(A˜i Pi,m+1 ) + κp −C˜iT C˜i ∗
˜i − C˜ T F˜i Pi,m E i β02 I − F˜iT F˜i
≤0.
˜i −C˜ T F˜i Pi,m+1 E i β02 I − F˜iT F˜i
≤0.
(31)
(32)
˜i and F˜i into account, (28) and (29) can be acquired. With Taking A˜i , C˜i , E Lemma 2, the H− performance is proved. 14
205
3.3. Disturbance Attenuation Conditions To get the condition (5) established, we have the following Lemma 3. Lemma 3. Given matrices Pi,m > 0, m = 0, 1, · · · , Q − 1, constants 0 < µ < 1, η > 0, τa > 0. If there exists Lyapunov function (8) satisfying V˙ i (t) + ηVi (t) − Γ2 (t) ≤ 0,
(33)
Vi (tk ) ≤ µVj (t− k ), i, j ∈ N, i 6= j
(34)
˜ − rT (t)r(t), then system (3) is exponentially stable where Γ2 (t) = γ02 d˜T (t)d(t) with H∞ performance with ADT τa ≤
− ln µ η
Proof. Under the assumption that fault signal f (t) equals to 0, system (3) can be re-given as ˜ ˜ ˜ d(t), e(t) ˙ = Ae(t) +B
r(t) = Ce(t) ˜ ˜ ˜ d(t), +D
(35)
T ˜ Given scalar η > 0, Γ2 (t) = γ02 d˜T (t)d(t)−r (t)r(t), if we have V˙ i (t)+ηVi (t)− 210
Γ2 (t) ≤ 0, then, employing differential inequality theory, it can be transferred as
Vi (t) ≤ Vi (tm ) exp−η(t−tm ) +
Z
t
exp−η(t−s) Γ2 (s)ds.
(36)
tm
For the similarity with Lemma 2, here omits the following proof process of Lemma 3. As depicted in Lemma 3, we have the following Theorem to guarantee the 215
disturbance attenuation conditions. Theorem 3. For any i, j ∈ N, i 6= j, 0 < µ < 1, η > 0, matrix Pi,m = Pi,m,1 0 > 0, m = 0, 1, · · · Q − 1, given matrices Hi , i = 1, . . . , 8, ε1 , 0 Pi,m,2 ε2 , ε3 , Xi,m,1 , Xi,m+1,1 , Yi,m,2 , Yi,m+1,2 constants β0 , τa , scalar γ > 0 if 15
1 ϕi =
ϕ1i =
ln µ + ητa ≤ 0
(37)
Pi,0 − µPj,L ≤ 0
(38)
−He(Wi1 )
0
Y13
0
Y15
Y16
∗
−He(Wi2 )
0
Y24
Y25
Y26
∗
∗
Y33
ε12
Y35
Y36
∗
∗
∗
Y44
Y45
Y46
∗
∗
∗
∗
−γ 2 I + ε4
ε5
∗
∗
∗
∗
∗
−γ 2 I + ε4
≤0
(39)
−He(Wi1 )
0
S13
0
Y15
Y16
∗
−He(Wi2 )
0
S24
Y25
Y26
∗
∗
S33
ε12
Y35
Y36
∗
∗
∗
S44
Y45
Y46
∗
∗
∗
∗
−γ 2 I + ε4
ε5
∗
∗
∗
∗
∗
−γ 2 I + ε4
≤0
(40)
where T + T − Y15 = Wi1 Bi − Xi1 Di+ , Y16 = Wi1 Bi − Xi1 Di− , ε2 = Ci−T Di+ + Ci+T Di− T + T − Y25 = Wi2 Bi − Xi2 Di+ , Y26 = Wi2 Bi − Xi2 Di− , ε1 = −Ci−T Ci− − Ci+T Ci+ T + T − Y35 = aWi1 Bi − aXi1 Di+ + ε2 , Y36 = aWi1 Bi − aXi1 Di− + ε3 T + T − Y45 = aWi2 Bi − aXi2 Di+ + ε3 , Y46 = aWi2 Bi − aXi2 Di− + ε2 T S13 = Pi,m+1,1 − aWi1 + Wi1 Ai − Xi1 Ci , ε4 = Di+T Di+ + Di−T Di−
16
T S24 = Pi,m+1,2 − aWi2 + Wi2 Ai − Xi2 Ci , ε5 = Di+T Di− + Di−T Di+ T Y13 = Pi,m,1 − aWi1 + Wi1 Ai − Xi1 Ci , ε3 = Ci−T Di− + Ci+T Di+ T Y24 = Pi,m,2 − aWi2 + Wi2 Ai − Xi2 Ci , ε12 = −Ci−T Ci+ − Ci+T Ci− T Y33 = aHe(Wi1 Ai − Xi1 Ci ) + ηPi,m,1 + κp1 − ε1 T Y44 = aHe(Wi2 Ai − Xi2 Ci ) + ηPi,m,1 + κp2 − ε1
κp1 = L(Pi,m+1 − Pi,m,1 )/τa , κp2 = L(Pi,m+1,2 − Pi,m,2 )/τa , ˜ − rT (t)r(t) Γ2 (t) = γ02 d˜T (t)d(t) T Ai − Xi1 Ci ) + ηPi,m+1,1 + κp1 − ε1 S33 = aHe(Wi1 T S44 = aHe(Wi2 Ai − Xi2 Ci ) + ηPi,m+1,1 + κp2 − ε1
then, system (35) is considered to have H∞ performance. Proof. Choosing the Lyapunov function as (8), through the derivative technique, we can gain V˙ i (t) + ηVi (t) − Γ2 (t) = e˙ T (t)Pi e(t) + eT (t)P˙i e(t) + eT (t)Pi e(t) ˙ + ηeT (t)Pi e(t) ˜ + d˜T (t)F˜ T C˜i e(t) + eT (t)C˜iT C˜i e(t) + eT (t)C˜iT F˜i d(t) i ˜ − γ 2 rT (t)r(t) + d˜T (t)F˜iT F˜i d(t) 0 ≤ ζ T (t)ϕi ζ(t) 220
where ζ T (t) = [eT (t) d˜T (t)] and φi has the following rule,
ϕi =
ϕ1i ,
t ∈ Ik,m , m = 0, 1, 2, · · · , Q − 1.
ϕ2 ,
(41)
t ∈ [tk + τa , tk+1 ).
i
When t ∈ Ik,m , (39) indicates V˙ i (t) + ηVi (t) − Γ2 (t) ≤ 0. Similarly, (40)
means V˙ i (t) + ηVi (t) − Γ2 (t) ≤ 0 on time interval [tk + τa , tk+1 ). Hence, the establishment of V˙ i (t) + ηVi (t) − Γ2 (t) ≤ 0 on [tk , tk+1 ) is guaranteed. Also, ϕ1i and ϕ2i are obtained as
ϕ1i =
˜ ˜T ˜ A˜T i Pi,m + Pi,m Ai + κp + Ci Ci
˜i + C˜ T D ˜i Pi,m B i
∗
˜ TD ˜i −γ02 I + D i 17
≤ 0.
(42)
ϕ2i = 225
˜ ˜T ˜ A˜T i Pi,m+1 + Pi,m+1 Ai + κp + Ci Ci
˜i + C˜ T D ˜i Pi,m+1 B i
∗
˜ TD ˜i −γ02 I + D i
≤ 0. (43)
˜i and D ˜ i into account, (39) and (40) can be acquired. With Taking A˜i , C˜i , B
the above Lemma 3, the H∞ performance can be ensured. Remark 3. As investigated in [42] the introduction of H∞ performance index can reduce the effect of the exoteric disturbance while H− performance can be used to evaluate the sensitivity level of residual on fault signal. In our paper, 230
we utilize the mixed index which are mentioned in Theorems 2 and 3. Remark 4. In the past few years, few studies deal with the switched systems with unstable systems. Even in [9], stable subsystems are required to exist. In [9], a special condition to ensure stability is proposed. That is
T − (pk ,pk+1 ) T + (pk ,pk+1 )
≥
β+χ α−χ ,
where [pk , pk+1 ) denote the time interval, T − (T1 , T2 )(T + (T1 , T2 )) represents the 235
total running time of stable subsystems (unstable subsystems). In this paper, according to [12], we deal with the switched system with all modes unstable. With the switching strategy created by the time-dependent method, we realize the fault detection and ensure the exponential stability with H∞ /H− performance. Theorems 2 and 3 give the mixed H∞ /H− performance analysis. The fol-
240
lowing task is considering the Metzler property. To ensure the Metzler property of Ai − Li Ci and Ai − Li Ci , we have the following subsection. 3.4. Metzler condition Consider a second-order switched system Ai = [aigh ]2∗2 , Li = [li1 Li = [li1
li2 ]T , also,
Ai − Li Ci =
ai11 − l1 c1
ai12 − l1 c2
ai21 − l2 c1
ai22 − l2 c2
18
,
li2 ]T ,
Ai − Li Ci =
ai11 − l1 c1
ai12 − l1 c2
ai21 − l2 c1
ai22 − l2 c2
Recall the definition of metzler matrix, it has
245
.
aigh − lg ch ≥ 0, aigh − lg ch ≥ 0, i, g, h = 1, 2, g 6= h. From Theorems 1-3, we findthat Ai − Li Ci and Ai − Li Ciare coupled with Pi,m,1 0 Pi,m+1,1 0 , Pi,m+1 = Pi,m , Pi,m+1 . Choosing Pi,m = 0 Pi,m,2 0 Pi,m+1,2 T
T and considering Xi,m = LT i Pi,m , Yi,m = Li Pi,m , Xi,m+1 = Li Pi,m+1 , Yi,m+1 = T
Li Pi,m+1 , we have
T Xi,m =
T Xi,m+1 =
0
0
Pi,m,2
Pi,m+1,1
0
0
Pi,m+1,2
T Yi,m =
T Yi,m+1 = 250
Pi,m,1
Pi,m,1
0
0
Pi,m,2
Pi,m+1,1
0
0
Pi,m+1,2
In addition, we get
Pi,m (Ai − Li Ci ) =
li1
li2
Pi,m,1 li1 Pi,m,2 li2
=
li2
li1
=
li2
li1
= li1 li2
=
Pi,m+1,1 li1 Pi,m+1,2 li2
Pi,m,1 li1 Pi,m,2 li2
=
=
Pi,m,2 li2
Xim2
=
Pi,m,1 li1
Xim1
Xim+12
Yim2
=
Xim+11
Yim1
Yim+11 Yim+12
Pi,m,1 ai11 − Xi,m,1 c1
Pi,m,1 ai12 − Xi,m,1 c2
Pi,m,2 ai21 − Xi,m,2 c1
Pi,m,2 ai22 − Xi,m,2 c2
19
Pi,m,1 ai11 − Xi,m+1,1 c1
Pi,m,1 ai12 − Xi,m+1,1 c2
Pi,m,2 ai21 − Xi,m+1,2 c1
Pi,m,2 ai22 − Xi,m+1,2 c2
Pi,m+1,1 ai11 − Yi,m,1 c1
Pi,m+1,1 ai12 − Yi,m,1 c2
Pi,m+1,2 ai21 − Yi,m,2 c1
Pi,m+1,2 ai22 − Yi,m,2 c2
Pi,m+1 (Ai − Li Ci ) = Pi,m (Ai − Li Ci ) =
Pi,m+1 (Ai − Li Ci ) =
Pi,m+1,1 ai11 − Yi,m+1,1 c1
Pi,m+1,1 ai12 − Yi,m+1,1 c2
Pi,m+1,2 ai21 − Yi,m+1,2 c1
Pi,m+1,2 ai22 − Yi,m+1,2 c2
Hence, the metzler condition can be transferred into the below equivalent
inequalities,
Pi,m,g aigh − Xi,m,g ch ≥ 0, Pi,m+1,g aigh − Yi,m+1,g ch ≥ 0, Pi,m,g aigh − Yi,m,g ch ≥ 0, Pi,m+1,g aigh − Xi,m+1,g ch ≥ 0, g, h = 1, 2, g 6= h. (44) Obviously, the condition (44) can be solved through the LMI toolbox. In like manner, the metzler condition for 2-nd order system can be gener255
alized to the n-th order switched systems, with the help of [37]. The metzler condition can be summarized as
Pi,m,g aigh − Xi,m,g ch ≥ 0, Pi,m+1,g aigh − Yi,m+1,g ch ≥ 0, Pi,m,g aigh − Yi,m,g ch ≥ 0, Pi,m+1,g aigh − Xi,m+1,g ch ≥ 0, g, h = 1, 2, . . . , n, g 6= h. (45) Based on Theorems 1-3, the following Theorem are given to obtain the interval observer. Theorem 4. Given scalar τa > 0, γ > 0, β > 0, 0 < µ < 1, η > 0, matrices 260
εx , x = 1, 2, 3, 4, 5, Xi,m,1 , Yi,m,2 , the observer gain Li and Li can be obtained −1 −1 by Li = Pi,m,1 Xi,m,1 , Li = Pi,m,2 Yi,m,2 if (26)-(29), (39), (40) and (45) are
satisfied. 20
As discussed in Theorems 2, 3 and 4, the disturbance attenuation (5), fault sensitivity (6) and observer gain have been realized. The following subpart 265
concludes the FD decision scheme as resulted in [37]. 3.5. Fault detection decision scheme To show the method more clearly, the fault detection logic is depicted in Fig. 1. As shown in the below picture, a disturbance with known bound and a fault with known expression are applied to the switched system. Then, the system
270
output is transferred to the interval observer. Through the transformation of the residual generator, the output of the interval observer is divided into the upper bound y(t) and the lower bound y(t). By subtracting with y(t), the upper and lower residual signal bound are obtained. Afterwards, the fault detection decision scheme gives the alarm rule depending on the logic which is given
275
later. Hence, the fault detection for switched systems with all modes unstable is finished.
Figure 1: Architecture of the FD Scheme
Here gives the FD decision scheme. As shown in Fig. 1, the residual signal is divided into the upper bound r(t) and the lower bound r(t) according to the rule of r(t) = y(t) − y(t), r(t) = y(t) − y(t). When there is no fault, 0 should be 21
280
bounded by r(t) and r(t), i. e., 0 ∈ [r(t), r(t)]. On the contrary, if 0 6∈ [r(t), r(t)], i. e., 0 cannot be bounded by the upper and lower bound, we can obtain that the fault appears in the switched system. By discriminating the membership of 0 and interval [r(t), r(t)], the FD can be accomplished. Through the discussion above, the following FD decision scheme is given,
Alarm
285
activated,
others
not activated,
(46) 0 ∈ [r(t), r(t)]
Remark 5. This paper employs the interval observer-based fault detection method, which is different from [31]-[33]. As pointed out in [37], the most obvious advantage lies in the fact the design of extra threshold generator can be removed due to that r(t) and r(t) can be used for FD directly. This may avoid the uncertainty brought by the improper threshold. Here gives the algorithm of this paper. ALGORITHM begin given µ, η, L Step 1
calculate ADT τa
Step 2
if (26)-(29), (37)-(40), and β >100 then output µ, η, τa , Li , Li else µ=µ + 0.05, η=η − 0.05 Step 1 end
290
Remark 6. Compared with the traditional fault detection method[31, 32, 33], the method used in this paper reduce the conservativeness and improve the accuracy to some extent. In [32], the threshold is set as Jth = sup Jr (t) with f (t)=0
22
Jr (t) = 295
q R 1 t t
0
rT (s)r(s)ds, which relies heavily on r(t). When a small fault oc-
curs, it may not be detected. Instead, in this paper, the upper and lower bound of the interval observer can be chosen artificially. The judgment logic of whether a fault occurs is to judge whether 0 is included in (r(t), r(t)). Obviously, this method saves computing resources than the traditional method.
300
4. Examples In this section, two examples are given to illustrate the validity of the interval observer. Pick the system dynamic in (1), some necessary parameters are given as
A1 C1
A2 C2
B1 D1
B2 D2
= F1
E1
0.35
0.6
0.69
−0.1
−0.1
−0.9439 0.3302
= F2
E2
−1.95
−0.6
−0.1
0.5
0.1
0.4
−1.99
0.5
0.9804 0.1970
−0.7
−0.1
−0.1
0.1 , −2
0.2 1
The eigenvalues of A1 and A2 are λ1,1 = −2.0724, λ1,2 = 0.0224, λ2,1 = 0.0005, λ2,2 = −2.0905, respectively. It is obvious that A1 and A2 are all unstable. In addition, from the parameters matrices, it has (C, A) is unobservable. Then, set the bounded disturbance as d(t) = 0.1 ∗ cos(0.5 ∗ t) ∗ exp(−0.1 ∗ t) d(t) = 0.1 ∗ cos(0.5 ∗ t) ∗ exp(−0.1 ∗ t) + 0.1 d(t) = 0.1 ∗ cos(0.5 ∗ t) ∗ exp(−0.1 ∗ t) − 0.1
23
2.5
2
1.5
1
0.5
0
10
20
30
40
50
60
70
80
90
100
Figure 2: Switching Signal
According to (9), if we choose η = 0.03, µ = 0.5, the average dwell time τa 305
can be acquired as τa = 2 < 23. Hence, we get the switching signal as Fig. 2. Choose the interval observer as (2). To meet the disturbance attenuation and fault sensitivity condition, (37)-(40), (28), (29) must be feasible. Using the LMI toolbox in MATLAB, we obtain the observer 0.3102 L L2 −0.5047 1 = 0.3102 L1 L2 −0.5047
gain as 0.0807
−1.0033 0.0807 −1.0033
Set H3 = A1 −L1 C1 , H4 = A2 −L2 C2 , H5 = A1 −L1 C1 and H6 = A2 −L2 C2 . It is clear that both of them are metzler from the below matrices.
H3 H5
−1.6572 0.2476
0.2136 0.0667 = −1.6572 0.2476 H6 0.2136 0.0667 H4
−0.1791 1.3836 −0.1791 1.3836
0.4841
−1.7923 0.4841 −1.7923
Here gives the eigenvalues of H3 , H4 , H5 and H6 . Obviously, due to the
24
unobservable condition of (Ai , Ci ), H3 , H4 , H5 , H6 are all unstable. 310
λ(H3 ) λ(H5 )
−1.6873
0.0968 = −1.6873 λ(H6 ) 0.0968 λ(H4 )
0.1634
−2.1348 0.1634 −2.1348
(47)
Remark 7. (47) implies that the unobservability of (Ai ; Ci ) result in the instability of the error dynamic system. Taking the error system (3) as a switched system with two unstable modes, the interval observer gains are obtained.Figs 2-7 indicates the effectiveness of the proposed method. Remark 8. In [37, 39], the interval observer is designed under the assump-
315
tion that (Ai ; Ci ) is observable. However, to our best knowledge, there is little literatures related to the interval observer design for switched system with all modes unstable, especially under the unobservable condition. To indicate the effectiveness of the designed observer, in Example 2, we discuss the multi-fault circumstance.
320
To test the actual effectiveness of the interval observer, we take the following two examples. 4.1. Example 1 When there is no fault, the results are shown in Fig. 3. Fig. 3(a) shows the trajectories of output y(t), upper output estimation y(t) and lower output
325
estimation y(t). The residual signal r(t), upper residual bound r(t) and lower residual bound r(t) are depicted in Fig. 3(b). Obviously, the output y(t) is bounded by the upper and lower bound when there is no fault. Similarly, 0 ∈ [r(t), r(t)] is always hold in this fault-free case. Then, the fault is given in subsystem 1 in the shape of (48) and in subsystem
330
2 in the shape of (49)
f1 (t) =
1,
0,
t ∈ [20, 25) others. 25
(48)
0.5
0.5
0
0
-0.5
0
20
40
60
80
100
-0.5
0
20
40
60
80
Figure 3: Residual signal and output without fault
f2 (t) =
1,
t ∈ [40, 45)
0,
(49)
others.
Main results are shown in Figs. 4 and 5. Fig. 4 describes the system output and residual signal when fault occurs in subsystem 1. As shown in the picture, output y(t) and 0 are bounded by the upper and lower bound when t 6∈ [20, 25). On the time interval t ∈ [20, 25), 335
y(t) is not constrained by the bounds and 0 6∈ [r(t), r(t)] is established. Under this situation, the fault can be detected. In Fig. 5, the residual signal and output with fault in subsystem 2 are provided. At the tome period [40, 45], we can clearly find that the residual signal with fault is not in the residual signal bound. In addition, the outputs is not belong to the upper and lower bound,
340
which means that fault is happening. The detailed description is similar with the one of Fig. 4 and here omits it. 4.2. Example 2 In Example 1, the fault detection of the interval observer on the single subsystem is accomplished. Next, we want to test the ability of the interval observer
345
on the multi-fault and multi-type fault. Figs. 6-7 display the effectiveness and the validity. 26
100
1
1
0.5
0.5
0
0
-0.5
-0.5
-1 -1
-1.5
-1.5
-2
-2
-2.5 -3
0
20
40
60
80
100
-2.5
0
20
40
60
80
100
Figure 4: Residual signal and output with fault in subsystem 1
2
2
1.5
1.5
1
1
0.5
0.5
0
0
-0.5
-0.5
-1
0
20
40
60
80
100
-1
0
20
40
Figure 5: Residual signal and output with fault in subsystem 2
27
60
80
100
Firstly, we assume that the fault occurs in subsystem s 1 and 2 in the following shape.
f (t) =
1,
t ∈ [45, 50)
0,
(50)
others.
Utilizing the parameters in Example 1, we obtain the results as Fig. 6. As 350
is displayed, we find that the residual signal is in the upper and lower bound of the residual signal, expect when t ∈ [45, 50). Similarly, y(t) is not restricted to be in [y(t), y(t)]. 1.5
2
1 1
0.5 0
0
-0.5 -1
-1 -1.5
-2
-2 -3
0
20
40
60
80
100
-2.5
0
20
40
60
80
Figure 6: Residual signal and output with fault in subsystems 1 and 2
f (t) =
1,
0,
t ∈ [35, 40) others.
[
[55, 60) (51)
In Fig. 7, we consider the multi-fault circumstance. First, we give the fault signal as (51). As shown in Fig. 7, the residual signal is beyond the upper and 355
lower bound of r(t) when t is in the fault time period. At the same time, the S outputs is not in [y(t), y(t)] when t ∈ [35, 40) [55, 60).
From the two examples above, it can be concluded that y(t) is bounded by
y(t) and y(t) and 0 ∈ [r(t), r(t)] holds in the fault-free case. On the other hands, when fault appears, the output y(t) increases sharply leading to 0 6∈ [r(t), r(t)]. 28
100
2
1
1
Outputs
Residual Signal
2
0
0
-1
-1
-2
-2
-3
0
20
40
60
80
100
-3
0
20
40
Time (s)
60
80
Time (s)
Figure 7: Residual signal and output under multi-fault circumstance
360
According to the FD decision scheme in Section 3, the fault can be detected. This example verifies the effectiveness of the proposed method.
5. Conclusions In this paper, observer-based fault detection is accomplished for switched system when all modes unstable. Compared with the literature related to tra365
ditional FD scheme, i. e., the appearance of a fault is evaluated by the threshold function, this paper employs the interval observer-based FD scheme, which reduces the conservativeness and add the degree of freedom. With the consideration of unstable modes, a special switching rule is given utilizing average dwell time to realize the exponential stability. One special thing of this paper
370
is that the interval observer is designed when (Ai ; Ci ) are unobservable and we believe that the unobservability is the root of the instability. Utilizing LMIs, sufficient conditions making sure the H∞ /H− performance are derived. In the end, examples are listed to verify the detect property of the interval observer.
Conicts of Interest 375
The authors declare that they have no conflicts of interest.
29
100
6. Acknowledgments This work is supported by the National Natural Science Foundation of China (61873057, 61773187), Jilin City Science and Technology Bureau (201831727, 201831731), Natural Science Foundation of Jilin Province (20180520211JH) and 380
Education Department of Jilin Province (201693, JJKH20170106KJ).
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