Fault detection for discrete-time Lipschitz nonlinear systems with signal-to-noise ratio constrained channels

Author’s Accepted Manuscript Fault detection for discrete-time Lipschitz nonlinear systems with signal-to-noise ratio constrained channels Fumin Guo, Xuemei Ren, Zhijun Li, Cunwu Han www.elsevier.com/locate/neucom

PII: DOI: Reference:

S0925-2312(16)00295-2 http://dx.doi.org/10.1016/j.neucom.2016.02.048 NEUCOM16799

To appear in: Neurocomputing Received date: 11 July 2015 Revised date: 10 December 2015 Accepted date: 15 February 2016 Cite this article as: Fumin Guo, Xuemei Ren, Zhijun Li and Cunwu Han, Fault detection for discrete-time Lipschitz nonlinear systems with signal-to-noise ratio constrained channels, Neurocomputing, http://dx.doi.org/10.1016/j.neucom.2016.02.048 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Fault detection for discrete-time Lipschitz nonlinear systems with signal-to-noise ratio constrained channels Fumin Guoa,∗, Xuemei Rena , Zhijun Lib , Cunwu Hanb b The

a School

of Automation, Beijing Institute of Technology, Beijing 100081, China Key Lab of Fieldbus and Automation Technology of Beijing, North China University of Technology, Beijing 100144, China

Abstract In this paper, the problem of fault detection for discrete-time Lipschitz nonlinear systems with additive white Gaussian noise channels subject to signal-to-noise ratio constraints is investigated. An optimal residual generator based on the mixed H− /H∞ performance index is designed to generate the so-called residual signal, and the H− -index is used to measure the minimum effect of faults on the residual signal, while the influence of unknown disturbances and channel noise on the residual signal is maximized by the means of the H∞ -index. Then, in order to detect the occurrence of faults, a norm-based residual evaluation function is provided, and a dynamic threshold including upper bounds on the modulus of the solution of Lipschitz nonlinear systems and the stochastic properties of channel noise is also constructed. Finally, a simulated example is presented to demonstrate the effectiveness of the proposed approach. Keywords: fault detetion, Lipschitz nonlinear systems, signal-to-noise ratio, H− /H∞ , threshold

1. Introduction Due to the ever-increasing demands for higher safety and reliability of modern complex systems, in recent years, fault detection (FD) has attracted consid∗ Corresponding

author Email address: [email protected] (Fumin Guo)

Preprint submitted to Journal of LATEX Templates

March 3, 2016

erable research attention. In general, FD techniques can be classified into datadriven multivariate statistical process monitoring (MSPM) and model-based approaches. Because of less requirements on the design and simple forms, the data-driven MSPM methods, such as principle component analysis (PCA) and partial least squares (PLS), are widely used in large-scale industrial processes [1,2]. With the increasing complication of operation conditions and control objectives, nowadays the complicated dynamical multimode process has become a research focus, and some advanced MSPM approaches have been developed for the process; for example, the work in [3] proposed a new model migration method for the multimode non-Gaussian batch processes, and the common and specific variations were separated for the multimode problems. In addition, a novel manifold learning method was presented in [4] for the analysis of multimode batches in electro-fused magnesia furnace, and the proposed method showed superior monitoring and FD ability. More results of this research line can be found in [5,6]. Compared with the data-based MSPM techniques, the model-based FD methods can be successfully applied in the framework of modern control theory if the mathematical and physical knowledge of industrial processes are available. The core of model-based FD schemes is to detect the fault from the presence of unknown inputs on the basis of mathematical models, and the observer-based approach is the common one in these schemes. Generally, a typical observerbased FD system consists of a residual generator and a residual evaluator. The purpose of the residual generator is to produce a so-called residual signal, and then the generated residual signal is compared with a predefined threshold in the residual evaluator stage. A fault will be detected and an alarm will be released when the residual signal value exceeds the threshold [7-9]. Lipschitz nonlinear systems contain a large range of nonlinear systems, and any nonlinear functions whose arguments are smooth can be transformed into Lipschitz nonlinear [10-15]. Recently, some scholars have worked on FD for Lipschitz nonlinear systems and received some results. For a class of switched Lipschitz nonlinear systems with asynchronous switching surface, a FD filter 2

was designed in [16] by using the average dwell time approach. Based on the results of [16], the authors in [17] further discussed FD for uncertain switched Lipschitz nonlinear systems, and a new hybrid FD filter with state update was constructed. Moreover, the works in [18] and [19] used different approaches to investigate the FD problem for uncertain Lipschitz nonlinear systems, and a novel observer-based FD method based on adaptive estimation techniques was presented in [18], while two nonlinear observer design strategies, i.e., Thau’s observer and sliding-mode observer were proposed in [19]. In addition, reference [20] designed a nonlinear neural network FD observer to detect the fault for the non-Gaussian non-linear stochastic systems. On the other hand, with the widespread application of network and communication technology, the merger between control and information theory has received more and more attentions [21]. When networked communication channels take place in feedback control systems, the systems are inevitably subject to some constraints, such as transmission data rate, power (or variance) constraints, or signal-to-noise ratio (SNR) constraints, so one line of research proposes a framework to study stability of feedback control systems with SNR constraints. As the first step in the study of SNR constrained control systems, [22] investigated stabilization of multiple feedback systems subject to SNR constrained channels, and a minimal SNR was obtained in the discrete-time case for static feedback controllers to stabilize an unstable plant. However, [22] did not provide the robustness performance guarantee. Then, the authors in [23] considered the robustness related issue, and a closed-form characterization of SNR was achieved to satisfy the required robustness. In addition, the work in [24] investigated the required SNR for stabilization of linear invariant time (LTI) systems with additive colored Gaussian noise (ACGN) channels subject to channel input quantization, and two different infimum of SNR were gained for the logarithmic and uniform quantization. More research results can be found in [25-27]. It should be noticed that all the aforementioned Lipschitz results [16-20] were obtained in the continuous-time cases, and little attention has been paid 3

to FD for discrete-time Lipschitz nonlinear systems. Furthermore, to the best of the authors’ knowledge, most of the existing works on SNR constraints have considered merely the stabilization or performance issues, and no results have been obtained on FD for Lipschitz nonlinear systems with SNR constraints. Under this motivation, this paper investigates the problem of FD for discrete-time Lipschitz nonlinear systems subject to SNR constrained channels. Our aim is to design an FD filter to determine whether faults occur within the discrete-time nonlinear system. Since noise and disturbances may lead to significant changes in the residual, FD filters have to remain robust for the purpose of escaping from false alarms [8]. Different from the concept in robust control, the robustness of an FD system includes not only robustness against disturbances and channel noise but also sensitivity to the possible faults, i.e., the designed FD system ought to guarantee a suitable compromise between sensitivity to faults and robustness against noise and disturbances. Thus, an optimal residual generator is constructed in the mixed H− /H∞ framework to achieve the compromise; then, in order to detect faults, a norm-based residual evaluation function and a dynamic threshold based on the modulus of the solution of Lipschitz nonlinear systems and the stochastic properties of channel noise are designed. The remainder of the paper is organized as follows: Section 2 introduces system model and some assumptions. Section 3 discusses the design procedures of an optimal residual generator and a residual evaluator which includes a residual evaluation function and a dynamic threshold. A simulation example and a conclusion are presented in Section 4 and Section 5, respectively. 2. System model and assumptions Consider the feedback control system with additive white Gaussian noise (AWGN) channels subject to SNR constraints shown in Fig.1. The discretetime Lipschitz nonlinear system can be described as: x (k + 1) = Ax (k) + ϕ (x (k) , u (k)) + Bu (k) + Ed d (k) + Ef f (k) y (k) = Cx (k) + Fd d (k) + Ff f (k) 4

(1)

where x (k) ∈ Rn is the state vector, y (k) ∈ Rm is the plant output, u (k) ∈ Rp is the plant input, d (k) ∈ Rnd denotes the unknown disturbance, and f (k) ∈ Rnf is the fault to be detected. A, B, C, Ed , Ef , Fd , Ff are known matrices with appropriate dimensions. In addition, the following assumptions should be made throughout the paper. (1) The pair (A, C) is detectable. (2) The nonlinear function ϕ (x (k) , u (k)) is assumed to be a known nonlinear function satisfying the following local Lipschitz condition on a set M ⊂ Rn with respect to x: ϕ (x1 , u (k)) − ϕ (x2 , u (k)) ≤ γ x1 − x2  ,

∀x1, x2 ∈ M

(2)

where γ > 0 is a Lipschitz constant. Remark 1. If M = Rn , the nonlinear ϕ (x, u) is said to be globally Lipschitz. Lipschitz systems constitute a very important class; for example, the sinusoidal terms encountered in robotics are usually termed as globally Lipschitz, while the square or cubic nonlinearities in nature are regarded as locally Lipschitz [28]. (3) Each system output is transmitted to the FD filter through an AWGN channel, i.e., wi = yi + ni , where ni is independent of yi and is a zero mean   Gaussian white noise sequence with variance σn2 i 0 < σn2 i < ∞ . Moreover, each AWGN channel is subject to SNR constraint [29]: σy2i = Si , i = 1, · · · , m σn2 i

(3)

where Si is SNR of the ith channel, and σy2i is the stationary variance of the ith channel input yi . From [29], it can be seen that σn2 i is not a given constant and is proportional to σy2i . Then, under the above assumptions, the main objective of the paper is to design a FD system which includes a residual generator and a residual evaluator for the nonlinear system (1) to detect the system faults.

5

n1 (k )  d (k ) Sensor 1

Sensor m

y m (k ) 

ĂĂ

f k

ĂĂ

ĂĂ

Plant

w1 ( k )

y1 ( k ) 

wm (k )

Fault Detection Filter

r k



nm (k ) AWGN

Figure 1: Fault detection for the Lipschitz nonlinear systems subject to SNR constrained channels.

3. Main results In this section, in order to deal with the above FD problem, a residual generator based on the mixed H− /H∞ performance index and a residual evaluator including a norm-based residual evaluation function and a dynamic threshold are designed. 3.1. Optimal residual generator In this subsection, for the system (1), we will construct an optimal residual generator on the basis of the mixed H− /H∞ index to generate the residual signal. In the H− /H∞ index, H− -norm is used to measure the sensitivity to faults, and it represents the minimum effect of faults on the residual signal; similarly, H∞ -norm measures the robustness against the unknown disturbances, and it can be regarded as the maximum influence of unknown disturbances on the residual signal [30-34]. As in most contributions, a residual generator can be constructed as:   ∧  ∧ ∧ ∧ y x (k + 1) = A x (k) + Bu (k) + ϕ x (k) , u (k) + L w (k) − (k) ∧

∧

y (k) = C x (k)

(4)

w (k) = y (k) + n (k)   ∧ y r (k) = R w (k) − (k)

6

∧

∧

where x(k) ∈ Rn is the estimated state vector; y (k) ∈ Rm is the estimated output vector; r(k) is the residual signal; L and R are design parameters which should guarantee a suitable compromise between sensitivity to faults and robustness against unknown disturbances as well as channel noise. ∧

Define e (k) = x (k) − x (k), the dynamics of (4) can be given by:  ∧ e (k + 1) = (A − LC) e (k) + ϕ (x (k) , u (k)) − ϕ x (k) , u (k)   + E d − LF d d (k) + (Ef − LFf ) f (k)

(5)

r (k) = RCe (k) + RF d d (k) + RFf f (k)  T where E d = [Ed , 0], F d = [Fd , I], d (k) = dT (k) , nT (k) . Then, the system (5) can be rewritten as: xe,k+1 = Ae xe,k + Φk + Ee,d dk + Ee,f fk rk = Ce xe,k + Fe,d dk + Fe,f fk

(6)

 ∧ where xe,k = e (k), dk = d (k), fk = f (k), Φk = ϕ (x (k) , u (k))−ϕ x (k) − u (k) , Ae = A − LC, Ee,d = E d − LF d , Ee,f = Ef − LFf , Ce = RC, Fe,d = RF d , Fe,f = RFf . In order to analyze the influence of unknown disturbances and channel noise on the residual signal, the fault-free case is assumed, i.e., fk = 0. The corresponding model of (6) can be expressed as: xe,k+1 = Ae xe,k + Φk + Ee,d dk rk = Ce xe,k + Fe,d dk

(7)

and the associated H∞ -norm index is:



rk 2 ≤ α dk 2 .

(8)

Similarly, for the purpose of investigating the effect of faults on the residual signal, let dk = 0. Then, the corresponding model of (6) can be given by: xe,k+1 = Ae xe,k + Φk + Ee,f fk rk = Ce xe,k + Fe,f fk

7

(9)

and the associated H− -norm index is: rk 2 ≥ βfk 2 .

(10)

Remark 2. The H− -index is initially introduced by [30] and is widely used to measure the sensitivity of the residual signal to faults. Several attempts have been proposed to characterize the performance of FD filter, for example, H∞ /H∞ , H− /H2 , H− /H∞ trade-off design, all of which are treated as the multiple objective optimization problems [36]. The underlying idea of solving such optimization problems is to reduce them to a single optimization problem with constraints. In this paper, the H− /H∞ index is adopted, and other design strategies can be discussed in a similar manner. To design the aforementioned residual generator, the following lemma should be introduced first. Lemma 1. For any given constant λ > 0, define   1 1 1 1 Θ = λ 2 xT (k) AT P 2 − λ− 2 φT (k) P 2 , and the inequality ΘΘT ≥ 0 ⇒ it turns out that xT (k) AT P φ (k) + φT (k) P Ax (k) ≤ λxT (k) AT P Ax (k) + λ−1 φT (k) P φ (k) . (11) Remark 3. Lemma 1 can be obtained by the direct multiplication, so the proof is omitted. Based on Lemma 1, the following Theorem 1 can be obtained. Theorem 1. Consider the residual generator (4) with the gain matrices L and R, and the given scalars α > 0 and β > 0. Suppose that if there exist symmetric positive definite matrices P and Q, and scalars εi |i=1,···,4 > 0, ηi |i=1,2 > 0 such that

⎤

⎡ ⎣

η1 I

P

∗

η1 I 8

⎦>0

(12)

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

− ε12 P

0

0

0

P E d − HF d

∗

−P

0

P A − HC

P E d − HF d

∗

∗

− ε11 P

P A − HC

0

∗

∗

∗

Ω1

C T KF d

∗

∗

∗

∗

−α2 I + F d KF d

⎡ ⎣ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

T

⎤ η2 I

Q

∗

η2 I

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥<0 ⎥ ⎥ ⎥ ⎦

⎦>0

(13)

(14)

− ε14 Q

0

0

0

QEf − GFf

∗

−Q

0

QA − GC

QEf − GFf

∗

∗

∗

∗

∗

Ω2

−C T KFf

∗

∗

∗

∗

β 2 I − FfT KFf

− ε13 Q QA − GC

0

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥<0 ⎥ ⎥ ⎥ ⎦

(15)

  hold, where P L = H, RT R = K, QL = G, Ω1 = −P +C T KC+ 1 + ε11 + ε12 γ 2 η1 I,   Ω2 = −Q−C T KC + 1 + ε13 + ε14 γ 2 η2 I. Then, the residual generator is stable and satisfies the following constraints simultaneously:



rk 2 ≤ α dk 2 andrk 2 ≥ βfk 2 . Proof. Constructing a Lyapunov function Vk = xTe,k P xe,k , P = P T > 0 and calculating the difference of the Lyapunov function Vk , we have: ΔVk = Vk+1 − Vk = xTe,k+1 P xe,k+1 − xTe,k P xe,k T    = Ae xe,k + Φk + Ee,d dk P Ae xe,k + Φk + Ee,d dk − xTe,k P xe,k (16) T    = Ae xe,k + Ee,d dk P Ae xe,k + Ee,d dk − xTe,k P xe,k + ΦTk P Φk T

T + 2xTe,k ATe P Φk + 2dk Ee,d P Φk .

9

According to Lemma 1, the above last two nonlinear terms of (16) can be expanded as: 2xTe,k ATe P Φk ≤ xTe,k ε1 ATe P Ae xe,k + T T P Φk 2dk Ee,d

≤

T T dk ε2 Ee,d P Ee,d dk

+

1 T ε1 Φk P Φk , 1 T ε2 Φk P Φk ,

using Cauchy-Schwarz inequality, we obtain: ΦTk P Φk ≤ γ 2 σ (P ) xTe,k xe,k , where σ (·) is the maximum singular value, then we have: 2xTe,k ATe P Φk ≤ xTe,k ε1 ATe P Ae xe,k + T T P Φk 2dk Ee,d

≤

T T dk ε2 Ee,d P Ee,d dk

+

1 2 T ε1 γ σ (P ) xe,k xe,k , 1 2 ε2 γ σ

(P ) xTe,k xe,k ,

Thus, (16) can be represented as: T    ΔVk ≤ Ae xe,k + Ee,d dk P Ae xe,k + Ee,d dk − xTe,k P xe,k   T T +xTe,k ε1 ATe P Ae xe,k + dk ε2 Ee,d P Ee,d dk + 1 + ε11 + ε12 γ 2 σ (P ) xTe,k xe,k . (17) Let σ (P ) < η1 (η1 > 0), it is equivalent to 0 < η1 I −

1 T η1 P P

which can be

represented by (12) of Theorem 1. Notice that the inequality (17) implies: T    ΔVk ≤ Ae xe,k + Ee,d dk P Ae xe,k + Ee,d dk − xTe,k P xe,k   T T +xTe,k ε1 ATe P Ae xe,k + dk ε2 Ee,d P Ee,d dk + 1 + ε11 + ε12 γ 2 η1 IxTe,k xe,k . Considering ∞   



T rkT rk − α2 dk dk ≤ 0, rk 2 ≤ α dk 2 ⇒ k=0

and denoting M1 =

∞  

 T rkT rk − α2 dk dk ,

k=0

under the zero-initial condition: M1 <

∞  

 T rkT rk − α2 dk dk + ΔVk .

k=0

Then, a sufficient condition for M1 ≤ 0 can be described as: T

rkT rk − α2 dk dk + ΔVk < 0 ⇒ 10

it turns out that T

rkT rk − α2 dk dk + ΔVk T    T Ce xe,k + Fe,d dk − α2 dk dk + ΔVk ≤ Ce xe,k + Fe,d dk T    T Ce xe,k + Fe,d dk − α2 dk dk ≤ Ce xe,k + Fe,d dk T    + Ae xe,k + Ee,d dk P Ae xe,k + Ee,d dk − xTe,k P xe,k   T T +xTe,k ε1 ATe P Ae xe,k + dk ε2 Ee,d P Ee,d dk + 1 + ε11 + ε12 γ 2 η1 IxTe,k xe,k (18) and in order to ensure M1 ≤ 0, the right hand side of the inequality (18) should be negative, i.e., ⎤ ⎛⎡ ⎡ ⎣

xe,k dk

⎤T ⎡

T

Ae Ee,d P 0 A Ee,d ⎦ ⎜ ⎦ ⎣ ⎦⎣ e ⎝⎣ Ce Fe,d Ce Fe,d 0 I

where ζ1 = −P +

ε1 ATe P Ae



+ 1+

1 ε1

⎤⎞ ⎡ ⎤ ζ1 0 ⎟ xe,k ⎦+⎣ ⎦⎠ ⎣ ⎦<0 dk 0 ζ2 ⎤

⎤⎡

+

1 ε2



⎡

(19) 2

γ η1 I, ζ2 =

T ε2 Ee,d P Ee,d

− α2 I.

Using Schur complement, it follows that inequality (19) can be transformed into (13) of Theorem 1. Due to the fact that (14) and (15) can be obtained by a similar way as (12) and (13), we omit the proofs of (14) and (15). T

For stability analysis, the condition rkT rk − α2 dk dk + ΔVk < 0 can be rewritten as: T

ΔVk < −rkT rk + α2 dk dk ,

(20)

which implies the system (7) is dissipative with V (·), and the supply rate is T

κ = −rkT rk + α2 dk dk . Substituting dk = 0 to the inequality (20), we obtain: ΔVk < −rkT rk ,

(21)

which shows the asymptotic stability of the origin, i.e., xe,k = 0 of the nominal form of the system (7). Then, the inequality (21) can be described as: ⎡ ⎤ η1 I P ⎣ ⎦ > 0, ∗ η1 I

11

(22)

⎡ ⎢ ⎢ ⎢ ⎣

−P

0

∗

− ε11 P

∗

∗

⎤

P Ae P Ae  −P + CeT Ce + 1 +

1 ε1



γ 2 η1 I

⎥ ⎥ ⎥ < 0. ⎦

(23)

Note that (22) and (23) can also be gained from (12) and (13) by substituting dk = 0. Therefore, the feasibility of (12) and (13) implies the feasibility of (22) and (23), it means that the stability of the residual generator can be ensured from the feasibility of (12) and (13), thus it is no need to satisfy (22) and (23) additionally. This completes the proof. Remark 4. In the process of calculating matrices L and R, we can set α to some values and choose P = Q such that β is maximized for some scalars εi |i=1,···,4 > 0 and ηi |i=1,2 > 0, and once the problem is solved, the desired gain √ matrices can be obtained by L = P −1 H and R = K. 3.2. Residual evaluator In the residual evaluator stage, a norm-based residual evaluation function and a dynamic threshold which includes upper bounds on the modulus of the solution of Lipschitz nonlinear systems and the stochastic properties of channel noise will be provided to detect the occurrence of faults. In order to evaluate the generated residual signal, one widely approach is to choose a threshold Jth (Jth > 0), and the threshold Jth can be adopted based on the following FD logic relationship: ⎧ ⎨ r (k)2 < J , e th ρ 2 ⎩ r (k) ≥ J , e

ρ

th

no f ault alarm f or f ault

(24)

2

and the residual evaluation function re (k)ρ is defined as: 2

re (k)ρ =

k 

rT (k) r (k),

i=k−ρ+1

where ρ is the length of the evaluation window.

12

(25)

In the following text, we will design the dynamic threshold for the nonlinear systems (1). Recall the designed residual generator x (k + 1) = Ax (k) + Bu (k) + ϕ (x (k) , u (k)) + Ed d (k) + Ef f (k)  ∧ e (k + 1) = (A − LC) e (k) + ϕ (x (k) , u (k)) − ϕ x (k) , u (k) + (Ed − LFd ) d (k) − Ln (k) + (Ef − LFf ) f (k)

(26)

r (k) = RCe (k) + RFd d (k) + Rn (k) + RFf f (k) Based on the defined residual evaluation function (25), we have r (k)2ρ = rd,u (k) + rn (k) + rf (k)2ρ ,

(27)

where rd,u (k) = r (k) |n=0,f =0 , rn (k) = r (k) |d=0,u=0,f =0 , rf (k) = r (k) |d=0,u=0,n=0 . Thus, the fault-free scenario residual evaluation function can be determined by: 2

2

2

rd,u (k) + rn (k)ρ ≤ rd,u (k)ρ + rn (k)ρ ≤ Jth,d,u + Jth,n , where 2

Jth,d,u = sup rd,u (k)ρ , d,u

2

Jth,n = sup rn (k)ρ . n

Consequently, we can choose the threshold Jth as: Jth = Jth,d,u + Jth,n .

(28)

From the equation (28), it can be seen that in order to obtain Jth , we should calculate Jth,d,u and Jth,n , respectively. Rewrite the fault-free case residual generator system (26) as: x0 (k + 1) = A0 x0 (k) + B0 u (k) + Ψ (x0 (k) , u (k)) + E0 d (k) − E0,n n (k) r (k) = C0 x0 (k) + F0 d (k) + F0,n n (k) (29) 13

where

⎡

x0 (k) = ⎣

⎡

⎤ x (k) e (k)

⎦ , A0 = ⎣

⎡ E0 = ⎣

⎤ A

0

0

A − LC ⎡

⎤ Ed Ed − LFd

⎦ , E0,n = ⎣ ⎡

⎡

⎦ , B0 = ⎣

⎤ B 0

⎦ , C0 = RC,

⎤ 0 L

⎦ , F0 = RFd , F0,n = R,

⎤ ϕ (x (k) , u (k))  ⎦. ∧ Ψ (x0 (k) , u (k)) = ⎣ ϕ (x (k) , u (k)) − ϕ x (k) , u (k) For the purpose of calculating Jth,d,u , the channel noise is set equal to be zero, i.e., n (k) = 0, and the system (29) can be rewritten as: x0 (k + 1) = A0 x0 (k) + B0 u (k) + Ψ (x0 (k) , u (k)) + E0 d (k)

(30)

rd,u (k) = C0 x0 (k) + F0 d (k)

Before computing Jth,d,u , the following lemmas and property should be stated. and M, N ∈ Lm×r Lemma 2 [35]. Let F ∈ Ln×m pe qe , 1 ≤ p ≤ ∞ and

1 p

+

1 q

= 1.

Moreover, let G ∈ Lr×s ∞e and define G (k) = sup |G (k)| , k∈[0,k]

then (1) If F (k) ≥ 0 for all k and N ≥ M then F ∗ N ≥ F ∗ M . (2) |F ∗ M | ≤ |F | ∗ |M |. (3) If F (k) ≥ 0 for all k, then F ∗ |M G| ≤ (F ∗ |M |) G. and the above convolutions denoted by  ∗ are finite for all K ≥ 0. denotes a set of functions such that Γτ Xp < ∞ Remark 5. The space Ln×n pe for all τ ≥ 0, and the truncation operator Γτ is defined as: ⎧ ⎨ X (k) , K ≤ τ (Γτ X) (k) = ⎩ 0 otherwise. Remark 6. For matrices X = [xij ]m×n and Y = [yij ]m×n , X ≤ Y means that xij ≤ yij for all i ∈ 1, · · ·, n and j ∈ 1, · · ·, m. The notation |·| represents 14

the matrix modulus function, i.e., the element-wise absolute value, so |X| = [|xij |]m×n . for some p, 1 ≤ p < ∞, and define linear Lemma 3 [35]. Let G ∈ Ln×n p Δ

Δ

operator Δ for ∀F : ΔF = G ∗ F . Let H = (I − Δ)−1 − I and the function T is Δ

defined such that HF = T ∗ F . If Δp < 1 and G (k) ≥ 0 for all k ≥ 0, then Hp ≤ 1 and T (k) ≥ 0 for all k ≥ 0. Property 1 [35]. Let A, B, and C be matrices with compatible dimensions and t be an arbitrary scalar, then (1) If A ≥ 0 and B ≥ C, then AB ≥ AC and BA ≥ CA. (2) |A + B| ≤ |A| + |B|. (3) |AC| ≤ |A| |C|. (4) |tA| = |t| |A|. Based on the above lemmas and property, Theorem 2 can be established. Theorem 2. Consider the system (30), and A0 is supposed to be invertible. Let θ (k) = Ak0 and there exist a matrix T such that |θ (k)| ≤ T, then the modulus of the residual rd,u (k) is: |rd,u (k)| ≤ |C0 | (I − Υγ)

−1

(T |ξ0 | + Υ (|B0 u (k)| + |E0 | δ0,d )) + |F0 | δ0,d

   (M × I + I), |d (k)| ≤ δ0,d , ξ0 is the initial condition of x. where Υ = A−1 0 Proof. Consider the first equation in (30): x0 (k + 1) = A0 x0 (k) + B0 u (k) + Ψ (x0 (k) , u (k)) + E0 d (k) ,

(31)

and the solution of (31) can be given by: k−1  k−1−i Ak−1−i Ψ (x0 (i) , u (i)) + A0 (B0 u (i) + E0 d (i)) 0 i=0  i=0 k k   = Ak0 ξ0 + A−1 Ak−i Ak−i (B0 u (i) + E0 d (i)) 0 0 Ψ (x0 (i) , u (i)) + 0

x0 (k) = Ak0 ξ0 +

k−1 

i=0

i=0

−Ψ (x0 (k) , u (k)) − B0 u (k) − E0 d (k)} , where ξ0 is the initial condition of x, Ak0 is the state transition matrix. Let

15

θ (k) = Ak0 and Ψ (k) = Ψ (x0 (k) , u (k)), we have:  k k   θ (k − i) Ψ (i) + θ (k − i) (B0 u (i) + E0 d (i)) x0 (k) = θ (k) ξ0 + A−1 0 i=0

i=0

−Ψ (k) − B0 u (k) − E0 d (k)} = θ (k) ξ0 + A−1 0 {θ (k) ∗ Ψ (k) + θ (k) ∗ B0 u (k) + θ (k) ∗ E0 d (k) −Ψ (k) − B0 u (k) − E0 d (k)} , (32) where  ∗ denotes the convolution operator. Taking the modulus of equation (32), we obtain:  |x0 (k)| = θ (k) ξ0 + A−1 0 {θ (k) ∗ Ψ (k) + θ (k) ∗ B0 u (k) + θ (k) ∗ E0 d (k) −Ψ (k) − B0 u (k) − E0 d (k)| . Then, according to property 1, the upper bound on the modulus of x0 (k) is:    {|θ (k) ∗ Ψ (k)| + |θ (k) ∗ B0 u (k)| + |θ (k) ∗ E0 d (k)| |x0 (k)| ≤ |θ (k) ξ0 | + A−1 0 + |Ψ (k)| + |B0 u (k) + E0 d (k)|}    {T ∗ |Ψ (k)| + T ∗ |B0 u (k)| + T ∗ |E0 d (k)| ≤ T |ξ0 | + A−1 0 + |Ψ (k)| + |B0 u (k) + E0 d (k)|}    {T ∗ γI |x0 (k)| + T ∗ |B0 u (k)| + T ∗ |E0 d (k)| ≤ T |ξ0 | + A−1 0 +γI |x0 (k)| + |B0 u (k) + E0 d (k)|}    {T ∗ γI |x0 (k)| + T ∗ |B0 u (k) + E0 d (k)| ≤ T |ξ0 | + A−1 0 +γI |x0 (k)| + |B0 u (k) + E0 d (k)|}    {(T ∗ γI + γI) |x0 (k)| + (T ∗ I + I) |B0 u (k) + E0 d (k)|} ≤ T |ξ0 | + A−1 0  −1  ≤ T |ξ0 | + A0  {(T ∗ I + I) γ |x0 (k)| + (T ∗ I + I) |B0 u (k) + E0 d (k)|} ≤ T |ξ0 | + Υγ |x0 (k)| + Υ (|B0 u (k)| + |E0 d (k)|) ,    (T ∗ I + I). Then, based on where |θ (k)| ≤ T, |Ψ (k)| ≤ γ |x0 (k)|, Υ = A−1 0 the Lemmas 2 and 3, after some mathematical manipulations, we can obtain: −1

(T |ξ0 | + Υ (|B0 u (k)| + |E0 d (k)|))

−1

(T |ξ0 | + Υ (|B0 u (k)| + |E0 | δ0,d )) ,

|x0 (k)| ≤ (I − Υγ) ≤ (I − Υγ) where |d (k)| ≤ δ0,d .

16

(33)

Considering the second equation in (30) and the inequality (33), the modulus of the residual can be given by: |rd,u (k)| ≤ |C0 | (I − Υγ)−1 (T |ξ0 | + Υ (|B0 u (k)| + |E0 | δ0,d )) + |F0 | δ0,d . This completes the proof. Therefore, Jth,d,u can be expressed as: Jth,d,u = |C0 | (I − Υγ)−1 (T |ξ0 | + Υ (|B0 u (k)| + |E0 | δ0,d )) + |F0 | δ0,d .

(34)

However, it should be noted that the matrix T is unknown. Thus, the following lemma should be introduced to obtain T. Lemma 4. Consider a matrix U (k) = ζ k , where ζ has complex eigenvalues, and whose magnitudes are less than unity. Assume that matrix ζ = ΣΛΣ−1 is diagonalizable, where Λ(|Λ| =

2

2

|Λ|real + |Λ|imag , Λreal and Λimag are diagonal

matrices represent the real and imaginary part of Λ, respectively)is a diagonal matrix which contains the eigenvalues of ζ at the diagonal, and the matrix Σ contains the eigenvector corresponding to the eigenvalues of ζ. Then  k |U (k)| ≤ |Σ| |Λ| Σ−1  .

(35)

Proof. Let ζ = ΣΛΣ−1 be diagonalizable, we have: k       U (k) = ζ k = ΣΛΣ−1 = ΣΛΣ−1 ΣΛΣ−1 · · · ΣΛΣ−1       = ΣΛ ΣΣ−1 Λ ΣΣ−1 · · · ΣΣ−1 ΛΣ−1 = ΣΛk Σ−1 , and the modulus of ζ k is:  k   k −1     ζ  = ΣΛ Σ  ≤ |Σ| Λk  Σ−1  . Considering a complex vector x = a+bi, and its polar coordinate form is x = √   |c| (cos (υ) + sin (υ) i), where |c| = a2 + b2 , υ = tan−1 ab . According to the n

n

De Moivre’s formula, we have xn = |c| (cos (nυ) + sin (nυ) i) and |cn | = |c| . Using these arguments, the absolute value of the diagonal matrix is |Λn | = |Λ|n . Thus,

      k |U (k)| = ζ k  ≤ |Σ| Λk  Σ−1  = |Σ| |Λ| Σ−1  . 17

This completes the proof. Next, we will calculate Jth,n . Since n (k) is a stochastic process, it is reasonable to choose the threshold Jth,n based on the mathematical expectation and variance of n (k). Because we cannot obtain the exact values of these two parameters, we use their upper bounds instead: " " ! ! Jth,n = sup E n (k)2ρ + sup σ n (k)2ρ , n

n

(36)

where σ (·) is standard deviation. In equation (36), the first term is used to express the bound on the mean 2

value of n (k)ρ , meanwhile the second term represents the expected deviation 2

of n (k)ρ from its mean value [36]. Then, the above two terms can be gained as(D(·) is variance): ⎧ ⎫ k ⎨  ⎬ " ! 2 nT (i) n (i) E n (k)ρ = E ⎩ ⎭ i=k−ρ+1

=

k 

& ' E nT (i) n (i)

(37)

i=k−ρ+1

=

k 



 σn2 1 (i) + σn2 2 (i) + · · · + σn2 m (i) ,

i=k−ρ+1

⎛ D⎝

k 

⎞

k 

nT (i) n (i)⎠ =

i=k−ρ+1



  D nT (i) n (i)

i=k−ρ+1 k 

=

  D n21 (i) + n22 (i) + · · · + n2m (i)

(38)

i=k−ρ+1

=2

k 



 σn4 1 (i) + σn4 2 (i) + · · · + σn4 m (i) .

i=k−ρ+1

On the other hand, the FD filter cannot obtain y (i) at each sample time i since n (i) is unknown, but we can estimate y (i) using the following inequality: 2

2

2 2 yj2 (i) ≤ γju u (i)2 + γjd d (i)2 ,

according to Theorem 1, we have: min γju 18

(39)

s.t.

⎤

⎡ ⎣ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

ηI

Xu

∗

ηI

− υ11 Xu

0

0

∗

−Xu

0

∗

∗

− υ12 Xu

∗

∗

∗

∗

∗  where Θ = −Xu + C T C + 1 +

∗ 1 υ1

1 υ2

+

⎦>0 ⎤ 0

Xu B

⎥ ⎥ Xu B ⎥ ⎥ ⎥ ⎥<0 Xu A 0 ⎥ ⎥ ⎥ Θ 0 ⎦ 2 I ∗ −γju Xu A



γ 2 ηI, and

min γjd s.t.

⎤

⎡ ⎣ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

μI

Xd

∗

μI

⎦>0 ⎤

− ϕ11 Xd

0

0

0

Xd Ed

∗

−Xd

0

Xd A

Xd Ed

∗

∗

− ϕ12 Xd

Xd A

0

∗

∗

∗

Θ

C T Fd

∗

∗

∗

∗ 

2 −γjd I + FdT Fd

 where Θ = −Xd + C T C + 1 +

1 ϕ1

+

1 ϕ2

⎥ ⎥ ⎥ ⎥ ⎥ ⎥<0 ⎥ ⎥ ⎥ ⎦

γ 2 μI, Cj and Fdj are the jth line

elements of C and Fd , respectively. Assume that sup dT (i)d(i) = δd , i

and let Pj (i) =

2 γju

2 u (i)2

2 + γjd δd .

tion (3), we obtain:

Then, based on the SNR constraint condi-

Pj (i) ≥ Sj , σn2 j

and then we have: " ! 2 E n (i)ρ =



 σn2 1 (i) + σn2 2 (i) + · · · + σn2 m (i) i=k−ρ+1   + PS2 (i) + · · · + PSmm(i) , ≤ ρ PS1 (i) 1 2 k 

19

!

σ n (i)2ρ

"

( + ) 2  2 2 ,   ) P1 (i) P2 (i) Pm (i) * . ≤ 2ρ + +···+ S1 S2 Sm

Thus, Jth,n can be rewritten as: Jth,n = ρ +



P1 (i) S1

-

2ρ

 + · · · + PSmm(i) 2  2 2   P1 (i) P2 (i) Pm (i) . + + · · · + S1 S2 Sm

+ 

P2 (i) S2

(40)

It is worth to notice that the equation (40) is a dynamic threshold, for Pj (i) can be calculated online. Therefore, Jth = Jth,d,u + Jth,n can be gained by equations (34), (35), and (40). 4. Simulation example In this section, in order to illustrate the effectiveness of the proposed method, a DC motor system with the following equations is given [37]: Ra ia + La didta + Ce w = e J dw dt + f w − CM ia = 0 where Ra is the armature resistance; ia is the armature current; La is the armature inductance; w is the shaft angular velocity; J is the motor inertia; f is the viscous friction coefficient; Ce and CM represent the back electromotiveforce coefficient and electromagnetic torque coefficient, respectively. Let x1 = w, x2 = ia , u = e, y = w, and consider the disturbances, faults, and nonlinearities existed in this system, then the dynamic nonlinear model is obtained as: ⎡ −f · x (t) = ⎣ J Ce −L a

CM J a −R La

⎤

⎡

⎦ x (t) + ⎣

⎤ 0 1 La

⎦ u (t) + ϕ (x, u) + Ed d (t) + Ef f (t)

y (t) = [1 0] x (t) + Fd d (t) + Ff f (t) Considering the parameters La = 50mH, Ra = 2Ω, J = 0.003kg · m2 , f = 0.015N m·s/rad, Ce = 67.2V /KRP M , CM = 1.066N ·m/A, Ed = [−1; 0.1], 20

Ef = [0.3; −0.5], Fd = 2.3, Ff = −1.2, ϕ (x, u) = 0.1tan(x), and supposing that the sample time h = 20ms, S = 45dB, ρ = 20. The disturbance d (k) is assumed to be a random signal uniformly distributed between [0, 0.16] and is shown in Fig. 2. In this simulation, two different faults are considered: ⎧ ⎧ ⎨ 0, 0 < t ≤ 10 ⎨ 0, 0 < t ≤ 10 , f2 = f1 = ⎩ 1, 10 < t ≤ 50 ⎩ 1.2 cos(2t), 10 < t ≤ 50. According to Theorem 1, for the given disturbance attenuation level α = 0.8233, the optimal gain matrix L and weighting matrix R are: ⎡ ⎤ 0.0002 ⎦ , R = 0.2366. L=⎣ 0.0012

0.16 0.14 0.12

Amplitude

0.1 0.08 0.06 0.04 0.02 0

0

5

10

15

20

25 Time(s)

30

35

40

45

50

Figure 2: Random disturbance d (k).

Figs. 3 and 4 depict the generated residual r (k) for these two faults f (k), respectively.

21

0.1 residual signal 0.05 0

Amplitude

−0.05 −0.1 −0.15 −0.2 −0.25 −0.3 0

5

10

15

20

25 Time(s)

30

35

40

45

50

Figure 3: Generated residual signal r(k) for f1 .

0.5 residual signal 0.4 0.3

Amplitude

0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 0

5

10

15

20

25 Time(s)

30

35

40

45

50

Figure 4: Generated residual signal r(k) for f2 .

The simulation results are shown in Figs. 5 and 6. From Figs. 5 and 6, it can be seen that, whether the constant fault f1 or the time-varying fault f2 , the residual evaluation value is less than the threshold in the absence of faults, while the residual evaluation value exceeds the threshold immediately after the fault occurs, that is, the residual evaluation value does not surpass the threshold until the occurrence of the faults. By calculation, in Fig. 5, it is obtained that re (10.05) = 0.1497 > Jth = 0.1232, i.e., the fault can be detected when k = 201, which indicated the fault can be detected one time step after its occurrence.

22

1.4

1.2

Amplitude

1

0.8

0.6

0.4

residual evaluation Jth

0.2

0

0

5

10

15

20

25 Time(s)

30

35

40

45

50

Figure 5: Residual evaluation and threshold for f1 under SNR=45dB.

2.5

Amplitude

2

1.5

1

0.5

0

residual evaluation Jth

0

5

10

15

20

25 Time(s)

30

35

40

45

50

Figure 6: Residual evaluation and threshold for f2 under SNR=45dB.

Remark 7. Some comparisons between this paper and the reference [7] are given in Table 1. From the comparison results, it can be seen that the method proposed in this paper has a better detection performance than the approach presented in [7]. 5. Conclusion In this paper, the FD problem for discrete-time Lipschitz nonlinear systems with AWGN channels subject to SNR constraints is investigated. An FD system 23

Table 1: Comparisons of the existing results

Reference

method

Steps of detection time

[7]

H∞ filtering method

12

This paper

H− /H∞ trade-off method

1

including a residual generator and a residual evaluator is designed. First, an optimal residual generator based on the mixed H− /H∞ index is constructed to guarantee a suitable compromise between sensitivity to faults and robustness against system disturbances as well as channel noise. Then, for the purpose of detecting system faults, a norm-based residual evaluation function is provided, and a dynamic threshold is also designed based on the modulus of the solution of Lipschitz nonlinear systems and the stochastic properties of channel noise. Finally, a simulation example is given to show the effectiveness of the proposed approach. Acknowledgments This work was supported by the National Natural Science Foundation of China under Grants 61433003, 61174116, 61273150, 61573024, the Research Fund for the Doctoral Program of Higher Education of China under Grant 20121101110029, Beijing Natural Science Foundation under Grant 4142014, Funding Project for Academic Human Resources Development in Institutions of Higher Learning Under the Jurisdiction of Beijing Municipality under Grant CIT&TCD201304007.

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