H_/H_∞ Sensor Fault Detection and Isolation of Uncertain Time-Delay Systems

Abstract

Sensor fault detection and isolation problems subject to H_/$H_{\infty}$, performance are concerned for linear time-invariant systems with time delay in a state and parametric uncertainties. To that end, a model-based observer bank approach is pursued. The design conditions for both continuous- and discrete-time cases are formulated in terms of matrix inequalities, which are then converted to the problems solvable via an algorithm involving convex optimization.

1. Introduction

Fault detection and isolation (FDI) for dynamic systems holds a major field in the modern control theory academia, due to the ever increasing demand from industry for higher safety and reliability standards [1-3]. A great deal of research works on FDI has been released from different points of view, among which a successful one is a model-based observer approach using effective design tools that have been accumulated in the field over decades. See [2] and good references therein.

When dealing with single-output systems, isolating a fault on a sensor is identical to detecting one. On the other hand, it is more complicated in multi-output systems, because constructing a non-interactive map (diagonal transfer matrix for the linear time-invariant (LTI) case) from faults to residuals is particularly challenging, even if the system is LTI [4]. A simple yet effective alternative could be a scheme via an observer bank, where as many observers as sensors are involved [5]. The residual generated by each observer in the bank should be as sensitive to the faults on all sensors except each one and as robust against disturbance, as possible, in a reasonable (for example, an H_ and an H∞ [6-10]) sense. Then the fault can be isolated based on a suitable voting scheme [5, 11]. However, time delay and uncertainty which are widely believed to exist in physical model configurations [12] could be as the major obstacles against satisfactory FDI.

Paper [5] introduces an application of the observer bank-based FDI method to load-frequency control problem without deliberating time delay, uncertainty, nor any performance index. In [11], a robust FDI for a robot manipulator is developed, where the residual sensitivity to the fault is considered in the H∞ sense. A solution to FDI is presented in the frame of H∞ model-matching in [13]. However, these referred works do not consider the time delay. Besides, although a fault detector, rather than an isolator, is designed with delays and/or uncertainties in [1, 14-16], available literatures are relatively few for the FDI of LTI systems with the time delay and the uncertainty in the H_/H∞ criterion. This motivates our present study.

In this paper, we concern design conditions of the H_/H∞ sensor FDI observer banks for continuous- and discrete-time LTI systems subject to the time delay in the state and the parametric uncertainty. The bank is composed of the sensors’ number of observers. Both the observer gain and the residual gain (that is not necessarily symmetric or triangular due to additional manipulation such as the matrix square root or a Cholesky factorization) are taken into account as the design variables. Sufficient conditions to find the gains are developed in terms of matrix inequality. An algorithm involving a convex optimization is presented based on the cone complementary linearization technique [12].

We follow standard notations: A = AT ≺ 0 is a negative definite matrix. ‖x‖ stands for a Euclidean norm while ‖x‖ℒ2 means the ℒ2 norm. Symbol * denotes a transposed element in a symmetric position. Ellipsis He{S} := S + ST is used for simplicity. For any vector y ∈ ℝm and matrix C ∈ ℝm×n , we define as follows:

 

2. Preliminaries

Consider the following uncertain time-delay LTI system:

where x ∈ ℝn is the state; y ∈ ℝm is the sensor output; d ∈ [0, du ], du ∈ ℝ>0 , is the known time-varying delay; and w ∈ ℝl and f ∈ ℝm belonging ℒ2 to are the disturbance and the sensor fault, respectively. Matrices ΔA, ΔB , and ΔC represent the parametric uncertainties that satisfy the following assumption:

Assumption 1: ΔA, ΔB, and ΔC are the real valued matrix functions fulfilling

where ■ means no restriction on that entry, F1, F2, E1, and E2 are known constant real matrices of appropriate dimensions and 𝝨 is time-varying real matrix satisfying 𝝨T𝝨≼I

Without loss of generality, assume the observability of (A, C). We introduce an m-observer bank in which the pth member should detect all faults on all sensors excluding the pth one. Such a requirement is realized if the pth observer takes the following form

where p ∈ ℐM := {1, 2,…, m}, ∈ ℝn is the estimated state; ŷ ∈ ℝm is the observer output; r ∈ ℝm−1 is the residual; and L and H are the observer and the residual gains to be designed, respectively.

Let e := x− and . The residual is then generated by the following augmented error dynamics

where and

For excellent FDI, (2) is desired to be designed so that the effect of to r is encouraged and the effect of w to r is attenuated. In connection with this, we recall the following performance measures:

Definition 1: For a map from to r in (3), the H_ performance is defined by [9]

For a map from w to r , the H∞ performance is defined by

Definition 2: Let the evaluation function be

and the threshold function be

where TW ∈ ℝ>0 is the constant time window. In cases where it is necessary to display explicitly the association of Jr and Jth with the residual from the pth observer, we write Jrp and Jthp , respectively. Define an FDI logic as Table 1.

Table 1.FDI logic

Problem 1: Given the fault sensitivity level β ∈ ℝ>0 and the disturbance attenuation level 𝛾 ∈ ℝ>0 , find L and H in (2) so as to satisfy

(C1) (2) is asymptotically stable when f = 0 and w = 0; (C2)_> β when w = 0 with the initial condition x(0) = (0) = 0; (C3)∞ < γ when f = 0 with x(0) = (0) = 0.

 

3. Main Results

We recall the following lemmas, before proceeding further.

Lemma 1: Given any matrices X = XT ≻ 0 , and M and vectors a(·) and b(·) defined on an interval 𝛀 , the following inequality holds [17]:

Lemma 2: For any compatible matrices S = ST ≺ 0, F, and E, the following equivalence holds:

for some ε ∈ ℝ>0, where 𝝨T𝝨 ≼ I .

Lemma 3: For any compatible matrices Q, R, , and Ȓ, the following inequality holds:

Proof: It is readily proved as follows:

Now, we summarize the main result on Problem 1.

Theorem 1 ((β, 𝛾)-H_H∞ FDI): Given β, 𝛾 ∈ ℝ>0 (2) has (β, 𝛾)-H_H∞ FDI performance and is asymptotically stable, if there exist , P = PT := blockdiag { P1, P2 } ≻ 0, Q=QT≻0, X = XT ≻ 0, Z = ZT ≻ 0, Ĥ = ĤT ≻ 0, , W, and Y, such that

where

Then the gains are given by ( L, H ) = ( P2−1N, Ĥ−1W ).

Proof: Denote and . Define the following positive definite function

V(z(t − α), α ∈ [0, du]) := V1 + V2 + V3

where

Using the basic calculus fact

(3) is rewritten as

Letting and b(α) := Pz(α) , and utilizing Lemma 1, we further compute

Replacing Y = X M P and assuming

yield

From (9), if = 0, it holds

Integrating the Hamilton—Jacobi—Bellman (H—J—B) inequality

from 0 to ∞, the following relation holds

Indeed (10) is necessity for (6) because for all ( z, z(t − d), w ) ∈ ℝ2n ＼ {0} × ℝ2n ＼ {0} × ℝl

where we have used Schur complement, the congruence transformation, and Lemma 2. This also implies the asymptotic stability of (2) ((C1)). The proof of (7) ⇒ (C2) is along a similar line to the proof of (6) ⇒ (C3). Next, letting w = 0 in (9) allows one to manipulate

Considering the following H—J—B inequality

it is easy to see (12) is a sufficiency for

Using (11), we know for all (z, z(t − d), )∈ ℝ2n ＼{0}×ℝ2n ＼{0}×ℝm−1, that

where we have used Schur complement, the congruence transformation, and Lemmas 2 and 3. ■

Algorithm 1 Iterative algorithm

Remark 1: The matrix inequalities in Theorem 1 are not linear due to the terms ĤĤ and − PZ−1P . It provokes a non-convex feasibility problem that is generally difficult to solve. One may simply attempt to find a feasible solution set through a tractable convex optimization algorithm after recovering their convexity. For instance, we could set Ĥ = I and P = Z, which however, may produce a quite conservative design result.

In what follows, we alternatively solve the problem to obtain a better result based on the cone complementary linearization technique [12]. Introduce V and S such that ĤĤ ≽ V and PZ−1P ≽ S or

Let (6)' and (7)' be (6) and (7) with the respective replacements of ĤĤ and − PZ−1P by V and S. Then we know that (6)' ,(7)' ,(8)⇒(6)−(8). Defining new variables (M, K, U, J, R) := (V−1, Ĥ−1, S−1, P−1, Z−1), (13) is represented as

Then, the problem concerned above is converted to the following nonlinear minimization one involving LMIs:

Algorithm 1 summarizes an iterative LMI approach to MP 1.

 

4. Paralleling to Discrete-Time Case

In this section, we discuss the FDI problem in the discrete-time domain. Consider the following discrete-time uncertain time-delay LTI system:

where wk, fk ∈ l2, and dk ∈ [0, du ] ⊂ The m- observer bank whose pth observer is in the form of

is desired to be designed to possess the ( β, 𝛾 )- H_/H∞ performance in the l2-norm sense. The augmented error dynamics is written as

where and .

Lemma 4: For any compatible matrices X = XT ≻ 0, Y, N, Z = ZT ≻ 0, following inequality is satisfied [12]:

where

Theorem 2: Given β, 𝛾 ∈ ℝ>0 , (17) has (β, 𝛾)-H_/H∞ FDI performance and is asymptotically stable, if there exist P :=blockdiag{ P1, P2 } = PT ≻ 0, Q = QT ≻ 0, X = XT ≻ 0, Ĥ = ĤT ≻ 0, Z = ZT ≻ 0, , W, and Y, such that

and (19), where

Then the gains are given by .

Proof: Define the positive definite function Vk:=V1k + V2k+V3k+V4k, where and

where Δzh := zh+1 − zh. Since , one rewrite (18) as

By assigning a := zk, b := Δzh and in Lemma 4, we obtain

Moreover, we have

From the above relations, ΔVk is majorized by

In case of is written as

Summing up the discrete-time H—J—B inequality

along (18) from 0 to ∞ yields

Inequality (21) guarantees this 𝛾-H∞ performance in the l2-norm sense, because for all (zk, zk−dk, wk) ∈ ℝ2n ＼ {0} × ℝ2n ＼ {0} × ℝl one can derive

where we have used Schur complement, the congruence transformation, and Lemma 2. This also implies the asymptotic stability of (17). Finally, it is not difficult to prove that (21) is the sufficiency for the β-H performance in the l2-norm sense when one considers (22) with wk = 0 to induce the following inductions

where we have used Schur complement, the congruence transformation, Lemma 2, and Lemma 3 with the assignments , R := H, , therein.

Though the design condition in Theorem 2 is casted in a nonlinear form, it can be efficiently solved by convex optimization algorithms in a manner similar to Algorithm 1 through the following conversion.

 

5. Example

Consider the state-space data for (1) borrowed from [15]

and du = 1. We further suppose by Assumption 1 that ΔA, ΔB, and ΔC and are decomposed as

Since y ∈ ℝ3 , a three-observer bank is employed. With a slight abuse of notations, in the sequel, the subscript p ∈ { 1, 2, 3 } denotes each observer in the bank. First, we attempt to find the gains by fixing Ĥp = I and Pp = Zp in Theorem 1, which convexifies the inequalities. However, as is concerned, it turns out to fail in finding any feasible solutions, even though we do our best in adjusting βp’s, 𝛾p’s, and εp’s. On the other hand, for the given (β1,β2,β3)=(1.1, 1.2, 1.6), and 𝛾p = 1, and , εp = 0.1, p ∈ { 1, 2, 3 }, we succeed in finding the following gains by Algorithm 1 in 8 iterations for the first and the second observer, respectively, and 16 iterations for the third one:

For simulation, we introduce a delay d ∈ ℝ≽0 and a disturbance w ∈ ℒ2 randomly varying within [0, du] and (−0.5, 0.5) , respectively. A fault f ∈ ℒ2 forms the transient pattern as

for 50 s . Set Twp = 3 , then the threshold is generally calculated as

under the zero-initial condition for both the system and the observer bank.

Three cases are simulated to investigate the validity of the designed bank.

i) In the fault-free-disturbance-activated case, the threshold is computed as Jthp = 0.5𝛾p = 0.5, p ∈ {1, 2, 3 }. As shown in Fig.1, Jrp, p ∈ { 1, 2, 3 }, does not exceed Jthp for all t ∈ [0.50], from which one judge based on Table 1 that there does not occur any fault, as it really is. The evaluation =(0.8691,0.7275,0.9093)≺(𝛾1,𝛾2,𝛾3)=(1, 1, 1) confirms that the design goal — the 𝛾-H∞ performance — is achieved.

Fig. 1.Residual evaluations when f = 0 but w ∈ ℒ2 ＼ {0}: Jrp (solid) Jthp and (dashed).

ii) Next, we simulate the fault-activated-disturbance-free case. In this case, the threshold is modified to

Fig. 2 reveals the comparison

Fig. 2.Residual evaluations when f ∈ ℒ2 ＼ {0} but : 0: Jrp (solid) and Jthp (dashed).

By the FDI logic in Table 1, the observer bank successfully declares that f1, f2, f3 are isolated for t ∈ (5.97, 16.54), t ∈ (21.73, 30.70), t ∈ (36.54, 45.96), respectively, with a retard up to 2 s from the fault-arising time. The evaluated values =(1.4049, 1.5931, 2.1226) are greater than (β1, β2, β3)=(1.1, 1.2, 1.6), entrywisely, which verifies the β-H_ performance of the observer bank.

iii) Now, we consider the fault-disturbance-activated case. The fault and the disturbance data applied in the previous runs are activated at the same time. Analyzing the result in Fig. 3 with the threshold

Fig. 3.Residual evaluation when f ≠ 0 and w ≠ 0: Jrp(solid) and Jthp (dashed).

According to the FDI logic in Table 1, we recognize that each sensor malfunctions for t ∈ (6.40, 16.10), t ∈ (22.06, 30.23), t ∈ (36.97, 45.48), in regular order. This means that the observer bank designed by the proposed method is robust against the disturbance, the uncertainties, and the time delay so that any misleading alarm is not issued, as should be expected.

iv) To highlight the benefit of the proposed method, we simulate a comparable scheme, Theorem 1 in [18] that does not consider the time-delay nor uncertainties. By this technique we compute

and apply them to the fault-disturbance-activated case. As shown in Fig. 4, the compared only isolates f1 for t ∈ [4.87, 19.89) For the rest of the simulation-run time, any isolation is not declared but detection is kept alarmed although there exist no fault for t ∈ [30.35) and t ∈ [45,50), which less accurate than ours.

Fig. 4.Residual evaluation by Theorem 1 in [18] when f ≠ 0 and w ≠ 0: Jrp(solid) and Jthp (dashed).

 

6. Conclusions

In this paper, we presented the H_/H∞ FDI observer bank design techniques for uncertain time-delay LTI systems for both continuous- and discrete-time settings. Design conditions are developed in the format of LMIs using the cone complementary linearization algorithm. Simulation results convincingly demonstrated the effectiveness of the developed methodology.