Collaborative operational fault tolerant control for stochastic distribution control system

Automatica 98 (2018) 141–149

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Automatica journal homepage: www.elsevier.com/locate/automatica

Brief paper

Collaborative operational fault tolerant control for stochastic distribution control system✩ Yuwei Ren a,b, *, Yixian Fang c , Aiping Wang d , Huaxiang Zhang a,b , Hong Wang e a

School of Information Science and Engineering, Shandong Normal University, Jinan 250014, Shandong Province, PR China Institute of Data Science and Technology, Shandong Normal University, Jinan 250014, Shandong Province, PR China c School of Science, Qilu University of Technology, Jinan 250353, Shandong Province, PR China d Institute of computer Science, Anhui University, Hefei 230039, Anhui Province, PR China e Pacific Northwest National Laboratory, Richland, WA 99352, USA b

article

info

Article history: Received 28 June 2017 Received in revised form 16 May 2018 Accepted 10 August 2018

Keywords: Interconnected systems Stochastic distribution control system Fault diagnosis Operational fault tolerant control

a b s t r a c t Based on a class of industrial processes, a new distributed fault diagnosis approach and a collaborative operational fault tolerant control law are proposed for an irreversible interconnected stochastic distribution control (SDC) system with boundary conditions. This control method is different from the existing collaborative fault tolerant controllers which enable the output probability density function (PDF) to track a desired PDF as close as possible. When fault occurs, a setpoint redesigned fault tolerant approach is adopted to accommodate the fault instead of reconstructing the controller. An augmented PID nominal controller and a setpoint compensation item with linear structure are used to obtain a collaborative operational fault tolerant controller via solution of linear matrix inequalities (LMIs). Simulations are included to show the effectiveness of the proposed algorithms where encouraging results have been obtained. © 2018 Elsevier Ltd. All rights reserved.

1. Introduction During the last decades there has been considerable interest in the development of modeling and control for interconnected systems (Antonelli, 2013; Patton et al., 2007; Yang, Jiang, & Zhou, 2017). So far, there are mainly two types of couplings among subsystems for the interconnected systems: Physical couplings (He, Wang, Liu, Qin, & Zhou, 2017; Yang, Jiang, Staroswiecki, & Zhang, 2015) and Network connections (Ma & Yang, 2016; Zuo, Zhang, & Wang, 2015) . For the above interconnected systems, three main control frameworks have been considered in order to compensate the coupled-dynamics: (1) Centralized framework (Zhang, Liu, & Zhang, 2005) where the whole interconnected system is supervised by one controller; (2) Decentralized framework (Li & Tong, 2017; Panagi & Polycarpou, 2011) where the stability of the ✩ The work was partially supported by the Natural Science Foundation of Shandong Province, PR China (No. ZR2015PF006), National Natural Science Foundation of China (61473176, 61573022 and 61772322). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Oswaldo Luiz V. Costa under the direction of Editor Richard Middleton. Corresponding author at: School of Information Science and Engineering, Shandong Normal University, Jinan 250014, Shandong Province, PR China. E-mail addresses: [email protected] (Y. Ren), [email protected] (Y. Fang), [email protected] (A. Wang), [email protected] (H. Zhang), [email protected] (H. Wang).

*

https://doi.org/10.1016/j.automatica.2018.09.022 0005-1098/© 2018 Elsevier Ltd. All rights reserved.

entire system is guaranteed by using only local information; and (3) Distributed framework (Keliris, Polycarpou, & Parisini, 2015; Panagi & Polycarpou, 2013) where multiple local controllers are designed for the exchange of information between subsystems. Compared with the conventional control methods (Blanke, Kinnaert, Lunze, & Staroswiecki, 2006; Shen, Shi, & Jiang, 2017), the design and analysis of interconnected systems is more complicated since the stability and performance of individual subsystem need to be addressed, in the meantime the communication with delays and loss of data packets which can have impact on other subsystems should be considered. The collaborative controller design for such systems must insure the stability of the whole system, especially ensure the ability of operating within certain performance margins in the presence of faults. Different from the prior fault tolerant controller, the impacts induced by the fault occurring in any of the subsystem or the communication channel are not only for the subsystem itself but also for the other subsystems due to the system coupling. Therefore, a collaborative fault tolerant control scheme needs to be developed in order to compensate the effects of fault on the local subsystems. The existing results of fault tolerant control for interconnected systems are mainly based on the generalized models which are not suitable for some industrial processes, such as food particles processing procedure, molecular weight distribution control process and mineral froth flotation process. These systems contain N

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Y. Ren et al. / Automatica 98 (2018) 141–149

series connected subsystems in addition the operational process is irreversible. More specifically, the output of the previous subsystem may not act as the input of the next subsystem but as a boundary condition which causes the system parameter matrices to be influenced by the previous subsystems (Ren, Wang, & Wang, 2015). Many stochastic control methods always assume that systems and variables subject to Gaussian distribution. In fact, many practical industrial processes do not meet this assumption. Motivated by this problem, Hong Wang proposed a novel approach for SDC systems which could directly control the shape of output PDF, i.e. the output SDC systems (Wang, 2000). Unlike the classical systems control problem, the objective concerned in the output SDC system is to achieve the tracking performance for the PDF of the system output, rather than the actual output values. On the basis of the SDC system, fault diagnosis and fault tolerant control methods were also developed besides a number of control algorithms. However, all the results were presented only for controlling the individual SDC system (Yao, Qin, Wang, & Jiang, 2012; Zhou, Li, & Wang, 2014). In fact, some problems are difficult or even impossible to solve by an individual system, such as multi-agent system(MAS) and the corresponding control method namely the cooperative (or collaborative) control. To address this issue, a number of profound results have been established for cooperative control of MAS (Ren & Beard, 2005). Furthermore, cooperative fault tolerant control, as an important part of cooperative control, should also be constructed when faults occur in interconnected system. The conventional FTC for SDC systems focus on designing fault tolerant controllers after fault occurs so that the closed loop system is stable and the controlled output PDFs follow the designed PDFs as close as possible, under the assumption that the designed PDF is given. However, it is difficult to give an appropriate PDF to be tracked which can not only guarantee the stability of the whole system but also keep the system operating at optimal conditions. The research on such FTC is still limited to a single control process, without considering the effect of the operational layer on the feedback control layer in case of failure. Moreover, the dynamics of fault operating conditions are different for industrial processes. It is difficult to use the existing methods for the diagnosis of fault caused by inappropriate setpoint and the self-recovery control in operational control for SDC systems. Inspired by Chai (Chai, Qin, & Wang, 2014), it is necessary to study a new diagnosis of fault operating condition as well as self-recovery control in collaborative SDC systems. In this paper, we tackle the operational fault tolerant tracking problem of collaborative SDC system with time-delays. To compensate actuator failure effects on the PDF tracking and maintain the system operation under an optimized status, an operational fault tolerant control algorithm is designed by estimating the faults and tuning the setpoint timely and appropriately. So that the overall system stability and acceptable performance can be maintained in the event of faults. The rest of this paper is organized as follows. The problem description is given in Section 2. In Section 3, fault diagnosis algorithm is proposed. Section 4 gives the design of operational fault tolerant controller for collaborative SDC system. Simulation results are included in Section 5, which is followed by some concluding remarks in Section 6. 2. System description 2.1. Preliminaries Consider a directed graph G = (V , E , A) with a nonempty finite set of N nodes V = (v1 , v2 , . . . , vN ), a set of edges or arcs E ⊂ V × V , and the adjacency matrix A = [aij ] ∈ ℜN ×N . An edge rooted at node vj and ended at node vi is denoted by (vj , vi ), which means

information can flow from node vj to node vi . aij is the weight of edge (vj , vi ) and aij = 1 if (vj , vi ) ∈ E , otherwise aij = 0 . Node vj is called a neighbor of node vi if (vj , vi ) ∈ E . The set of neighbors of node vi is denoted as Ni = {j|(vj , vi ) ∈ E } . Define ∑ the in-degree matrix as D = diag {di } ∈ ℜN ×N with di = j∈Ni aij and the Laplacian matrix as L = D − A. The edges in the form of (vi , vi ) are called loops. G = diag {gi } ∈ ℜN ×N is denoted as a loop matrix and has at least one diagonal item being 1. A graph with loops is called a multigraph, otherwise it is a simple graph. 2.2. System description Consider the dynamic collaborative stochastic distribution system which consists of N subsystems connected in the way as shown in Fig. 1 with inputs ui (t) ∈ ℜmi , (i = 1, 2, . . . , N) , respectively. Denote zi (t) ∈ [ai , bi ] as the outputs of the concerned dynamic stochastic subsystems, respectively and assume that they are uniformly bounded. In the future, fi (t) represents the unknown fault in the whole collaborative system. The output probability distribution functions of zi (t) are denoted by γi (y, ui ) which can be obtained by calculating the following probability (Wang, 2000): P {a ≤ zi (t) ≤ ξi |ui (t)} =

ξi

∫

γi (y, ui (t))dy

a

where P {a ≤ zi (t) ≤ ξi |ui (t)} is the probability of the output zi (t) lying inside the interval [a, ξi ] when ui (t) is applied to the system with ξi (t) ∈ [a, b]. As shown in Fig. 1, the considered subsystems are connected in series, the output PDF of the previous subsystem i affects the next i + 1 subsystem as a boundary condition. Denote C (y) = [b1 (y) b2 (y) Vi (t) = [vi1 (t)

vi2 (t)

··· ···

bn−1 (y)]

vi,n−1 (t)](Vi (t) ̸= 0)

Based on the well-known B-spline neural networks, the following square-root B-spline model has been used to approximate the output PDFs γi (y, ui (t)).

√

γi (y, ui ) =

n ∑

vij (ui )bj (y)

j=1

= C (y)Vi (t) + h(Vi (t))bn (y) + ωi (y, ui )

(1)

Different from the previous result for the square root B-spline models (Ren et al., 2015), model error ωi (y, ui ) is also dealt with in this paper, which can obviously make the concerned model more feasibly. Furthermore, the model uncertainty ωi (y, ui ) satisfy |ωi (y, ui )| ≤ δωi for all {y, ui } where δωi is a known positive constant. In Eq. (1), bj (y) ≥ 0, (j = 1, 2, . . . , n) are pre-specified basis B-spline functions defined on [a, b] respectively. vij (ui ) (denoted as vij (t) for simplicity) are the corresponding weights for all of the stochastic ∫distribution subsystems. The output PDFs b satisfy the condition a γi (y, ui )dy = 1. This means that only n-1 weights ∫ b T are independent ∫ b for any of the subsystem. ∫ b 2 Denote E1 = C (y)C (y)dy, E = C (y)b (y)dy, E = b (y)dy, then we 2 n 3 a a a n have √ 1 h(Vi (t)) = (−E2 Vi (t) ± ViT (t)E0 Vi (t)) E3 where E0 = E1 E3 − E2T E2 and h(Vi (t)) is a nonlinear function assumed to satisfy the following Lipschitz condition (Guo & Wang, 2005):

∥hi (V1 ) − hi (V2 )∥ ≤ ∥Ui (V1 − V2 )∥

(2)

for any V1 (t) and V2 (t) where Ui is a known matrix. Actually, many nonlinearities satisfy the Lipschitz condition, at least locally.

Y. Ren et al. / Automatica 98 (2018) 141–149

143

Fig. 1. Structure of collaborative stochastic distribution system.

Since the actual probability density function is dynamically related to the system input, after the construction of output PDFs by B-spline approximation in Eq. (1), the relationships between the system inputs and the weights can be characterized by the nonlinear square-root models. As a result, model of the subsystem i with possible faults and time-delays can be described as: V˙i (t) = Ai (Vi−1 )Vi (t) + Adi (Vi−1 )Vi (t − d)

boundary conditions in which the system matrices have strongnonlinearity affected by the previous subsystem and the residual errors are generated by the output PDFs not the output value. This constitutes the main challenge to estimate the fault. In order to estimate the fault, an observer-based fault diagnosis method will be illustrated in detail, where the fault diagnosis observer is described as:

˙

Vˆi (t) = Ai (t)Vˆi (t) + Adi (t)Vˆi (t − d) + Bi (t)ui (t)

+ Bi (Vi−1 )ui (t) + Ji fi (t)

√ γi (y, ui ) = C (y)Vi (t) + h(Vi (t))bn (y) + ωi (y, ui )

+ Ji fˆi (t) + Ksi εi (t)

(3)

where Vi (t) ∈ ℜn−1 is the weight value, ui (t) ∈ Rqi is the input, Ai (Vi−1 ), Adi (Vi−1 ) and Bi (Vi−1 ) are known parameter-varying matrices except the first subsystem. They are determined by the weights of the previous subsystems in accordance with the state vector input and the weights vector (Ai (t), Adi (t), Bi (t) for short in the following formulation). fi (t) ∈ ℜr denotes the presence of faults which is considered as an unknown time function in the ith subsystem, matrix Ji ∈ ℜ(n−1)×r is the fault distribution matrix which represents the influence of fault on the subsystem. It is assumed that |fi (t)| ≤ δfi holds, where δfi is a known positive constant. Remark 1. The parametrical matrices can be obtained either by physical modeling or by scaling estimation technique (Wang, 2000). Furthermore, time-delay d is considered in this SDC system model (3) since most of the practical complex industrial processes are nonlinear macrohysteretic system, such as flotation froth process and chemical process. The time delay d considered in (3) is a known constant which is supposed to guarantee the stability of the system. More detailed illustration about delay-dependent stability system can be found in Gu, Chen, and Kharitonov (2003) and Wang, Saberi, Stoorvogel, and Yang (2013).

√

γi (y, ui ) = C (y)Vˆi (t) + h(Vˆi (t))bn (y)

(4)

where fˆi (t) ∈ ℜr is considered as an estimation of the actuator fault fi , Ksi is the gain matrix to be determined. Vˆi (t) ∈ ℜn−1 is the estimate weight vector, εi (t) is the relative output PDF estimation error of the ith sub-system. Combined Eqs. (3) and (4), the residual error can be rewritten as:

εi (t) =

∑

b

∫

√

( γi (y) −

aij (

√

γˆi (y))dy

a

j∈Ni b

∫

( γj (y) −

√

−

√ γˆj (y))dy)

a b

∫ =

+ gi ( ∑

√

( γi (y) −

√

γˆi (y))dy)

a

aij [(Σ1 evi (t) + Σ2 ehi (t) + ∆ωi (t)) −

j∈Ni

(Σ1 evj (t) + Σ2 ehj (t) + ∆ωj (t))]

+ gi (Σ1 evi (t) + Σ2 ehi (t))

(5)

where aij and gi are entries of the adjacency matrix A and the loop ∫b ∫b matrix G, respectively. Σ1 = a C (y)dy, Σ2 = a bn (y)dy, evi (t) =

∫b

3. Fault estimation with model uncertainty Lemma 1 (Zhang, Jiang, & Cocquempot, 2016). For a symmetric ˜ the following inequality holds positive definite matrix M,

˜ + yT M ˜ −1 y, (x, y ∈ ℜn ). 2x y ≤ x Mx T

T

˜ G, ˜ N˜ are real matrices Lemma 2 (Petersen, 1987). Assuming that H, with appropriate dimension and ∥N˜ ∥ ≤ 1, then for any scalar εˆ > 0 the following inequality holds, ˜ T N˜ T H˜ T + H˜ N˜ G˜ ≤ εˆ G˜ T G˜ + εˆ −1 H˜ T H˜ G In this section a new fault estimation method is given based on the observer. There are many excellent results for fault diagnosing which can mainly divided into two groups: one is for individual system (Wang & Daley, 1996) and the other is for interconnected system (Zhang et al., 2016). The fault estimation method proposed in this paper is based on a class of SDC interconnected system with

Vˆ i (t) − Vi (t), ehi (t) = h(Vˆ i ) − h(Vi ) and ∆ωi (t) = a ωi (y, ui )dy. From the known boundary of ωi (y, ui ) we can obtain |∆ωi (t)| ≤ δ˜ ωi in which δ˜ ωi is also a known positive number. Remark 2. The residual error above is different from the centralized architecture in previous work (Ren et al., 2015). More specifically, εi (t) is not based on the relative output PDF estimation error only, but on information exchanges from neighboring subsystems.

In order to obtain the observer gain and the fault diagnosis parameters by LMI techniques, time-varying matrices Ai (t), Adi (t), and Bi (t) are supposed to satisfy Ai (t) = Ai + ∆Ai (t), Adi (t) = Adi + ∆Adi (t), Bi (t) = Bi + ∆Bi (t), in which Ai , Adi and Bi are known proper constant matrices and the pair (Ai , Σ1 ) is assumed to be observable. ∆Ai (t), ∆Adi (t) and ∆Bi (t) are variation of system matrices induced by boundary conditions which are supposed to satisfy the following equation (Zhou et al., 2014):

[∆Ai (t) ∆Adi (t) ∆Bi (t)] = Hi N˜ i (t)[G¯ i1 G¯ i2 G¯ i3 ]

144

Y. Ren et al. / Automatica 98 (2018) 141–149

¯ i1 , G¯ i2 and G¯ i3 are known constant matrices. N˜ i (t) is a where Hi , G variable matrix and satisfies N˜ i (t)N˜ iT (t) ≤ I. Denote efi (t) = fˆi (t) − fi (t), combined with Eqs. (3)–(5), the weight value estimation error ev i (t) can be expressed as:

ef (t) uniformly and ultimately bounded. The observer gain Ksi can be calculated by Ksi = P −1 R¯ i .

e˙ vi (t) = Ai (t)evi (t) + Adi (t)evi (t − d)

4. Collaborative fault tolerant controller design

+ Ji efi (t) + Ksi εi (t)

(6)

Based on the relative output estimation error, the distributed fault estimation algorithm is proposed

˙

fˆi (t) = −Γi1 fˆi (t) + Γi2 εi (t)

(7)

where Γi1 > 0, Γi2 are learning operators related to the property of the observer for fˆi (t) which can also be determined by the proposed diagnostic method. In order to consider the fault estimation problem from the overall perspective, the global error dynamics for the whole collaborative SDC system are defined as ev (t) = [eTv1 (t), eTv2 (t), . . . , eTvN (t)]T ef (t) = [eTf1 (t), eTf2 (t), . . . , eTfN (t)]T eh (t) = [eTh1 (t), eTh2 (t), . . . , eThN (t)]T f (t) = [f1T (t), f2T (t), . . . , fNT (t)]T

∆ω (t) = [∆Tω1 (t), ∆Tω2 (t), . . . , ∆TωN (t)]T Combined with the residual error in (5), then we get the global error systems e˙ v (t) = (IN

+ IN

⨂

⨂

Ai (t) − (L + G)

Adi (t)ev (t − d) + IN

−(L + G)

⨂

e˙ f (t) = −IN

+ (L + G)

⨂

⨂

⨂

Ksi Σ1 )ev (t)

⨂

The existing collaborative fault tolerant controllers designed for interconnected systems are usually given by ui = uNi + uFi , in which uNi denotes the local nominal control law to stabilize the ith faultfree subsystem. uFi is the augmented control law to compensate the system coupling and the change in dynamics due to the occurrence of a fault in the ith subsystem. Different from the previous methods (Ren et al., 2015), in this work the operational collaborative fault tolerant control law is designed not by augmenting the nominal controller but by tuning the system setpoint value. More specifically, when the operation of industrial plants become unhealthy, the system controller ui = uNi remains unchanged and the fault compensation approach which can ensure the safe operation of industrial plants is realized by changing the setpoint Vg to V¯ g . Using ∆Vg = V¯ g − Vg to compensate the fault and guarantee the output PDFs following the original setpoint values. Thus, in order to obtain the operational collaborative fault tolerant controller, the nominal tracking controller should be designed first for each subsystem without fault. In order to stabilize the healthy interconnected output SDC systems with time-delays, a generalized PID controller is given as motivated by Guo and Wang (2005). Since the output of the whole system is the output of the Nth subsystem, we can denote the tracking error of weight as e(t) = eN (t) = VN (t) − Vg

Ji (t)ef (t)

Ksi Σ2 eh (t) − (L + G)

⨂

Proof. The proof of Theorem 1 is in Appendix.

(8)

Ksi ∆ω

where Vg is the weight of given output PDF γg (y). Based on the Nth subsystem, a new state variable is introduced:

[

Γi1 (ef (t) + f (t))

t

∫

x(t) = V˙ NT (t) VNT (t)

e (τ )dτ T

]T

0

Γi2 [Σ1 ev (t) + Σ2 eh (t) + ∆ω (t)]

(9)

where L and G are the Laplace matrix and the loop matrix, respectively. ⊗ is the Kronecker product.

Then the nonlinear weight dynamics can be transformed into an equivalent form: E x˙ (t) = A(t)x(t) + Ad (t)x(t − d) + B(t)uN (t) + MVg

(11)

Remark 3. The general Kronecker product ⊗ is used for dimension expanding as usual. However, it cannot be used when the expanding matrix has subscript i, for this reason, a bold notation ⨂ of is introduced ⨂ to clarify the definition in our work. The bold notation of is also used for dimension expanding. For ⨂ example, IN ⊗ A(t) = diag {A(t), A(t), . . . , A(t)} and IN Ai (t) = diag {A1 (t), A2 (t), . . . , AN (t)} from the overall perspective.

where

Theorem 1. If there exist positive definite matrices P , Q and matrices R¯ i satisfying the following linear matrix inequality (LMI) Π < 0, in which ⎡ ⎤ ⨂ ⨂ ¯ Π IN PAdi IN PJi X1T X2T X3T X4 ⎢ ⎥ −IN ⊗ Q 0 0 0 0 X5 ⎥ ⎢∗ ⎢ ⎥ ∗ Y¯ 0 0 0 0 ⎥ ⎢∗ ⎥ Π =⎢ ∗ ∗ −ε1 I 0 0 0 ⎥ (10) ⎢∗ ⎢∗ ∗ ∗ ∗ −ε2 I 0 0 ⎥ ⎢ ⎥ ⎣∗ ∗ ∗ ∗ 0 −ε3 I 0 ⎦ ∗ ∗ ∗ ∗ ∗ ∗ −α I

With such an augmented descriptor system (11), the tracking problem can be further reduced to a stabilization control framework because the PID controller can be formulated as

¯ = IN (PAi + ATi P) − (L + G) ⊗ (R¯ Σ1 + Σ1T R¯ T ) + (ε1 + and Π ⨂ ⨂ T ⨂ ε4 )IN Ui + ⨂ IN ⊗ Q + α (IN Hi ) ×⨂(IN Hi ) + β I, R¯ i = ⨂PKsi , T T X1 = (L ⨂ + G) PKsi Σ2 , X⨂ PKs⨂ , X3T = (L + G) Γi2 , 2 = (L + G) i ¯ i1 , X5 = IN ¯ i2 , Y¯ = −2IN X4 = IN PG PG Γi1 + ε1 Y T Y + ε5 I,

⨂

E=

0 0 0

0 I 0

[ −I A(t) =

I 0

0 0 , B(t) = I

]

AN (t) 0 I

[

BN (t) 0 ,M = 0

]

0 0 , Ad (t) = 0

]

uN (t) = Kx(t), K = KD

[

KP

KI

0 0 0

[

[

0 0 , −I

Ad (t) 0 0

]

]

0 0 . 0

]

(12)

In order to maintain the system performance index (i.e., the system can follow the original setpoint Vg ), the operational collaborative fault tolerant controller should be designed when fault occurs. The setpoint of the whole system will be redesigned under the premise of remaining the controller structure unchanged. Denote the redesigned setpoint of the whole collaborative system as V¯ g = Vg + ∆Vg

∑N

in which ∆Vg = −

¯ˆ

i=1 Ki fi (t).

Then we have

N

4

Γi2 Σ2 , then the fault diagnosis algorithm (7) can Y T = (L + G) realize the state estimation error ev (t) and the fault estimation error

⨂

[

Vg = V¯ g − ∆Vg = V¯ g +

∑ i=1

K¯ i fˆi (t)

(13)

Y. Ren et al. / Automatica 98 (2018) 141–149

As a result, the nonlinear weight dynamics can be transformed into an equivalent form. In the following, we will compute gain matrices KP , KD , KI and K¯ i , such that the tracking error converges to a limited bounded. Remark 4. When fault occurs, ∆Vg is a compensator of the system setpoint which is a linear function of the estimated fault. Both linear and nonlinear functions can be chosen as compensators to tuning the value of the system setpoint. However, compared with the nonlinear structure of ∆Vg , using linear function (13) is simpler to solve the unknown function parameters by LMI toolbox. Theorem 2. For the collaborative stochastic distribution dynamic system under controller (12), the given positive constants η, κ, µ1 , µ2 and positive matrices P , S > 0, suppose that there exist matrices R and K¯ i such that the following LMI Ξ < 0 is solvable, then the collaborative stochastic distribution system is stable and the PDF tracking error is bounded with K = RT Q −1 , in which

Θ + κ HH T + ηI ⎢ ∗ ⎢ Ξ =⎢ ∗ ⎢ ⎣ ∗ ∗

Ad Q T −Sˆ

⎡

2M 0

−µ21 I ∗ ∗ [ Θ = AQ T +QAT +Sˆ +BRT +RBT , H T = HNT ¯ T + RG¯ T . and X6 = Q G ∗ ∗ ∗

P T M K¯ N 0 0 −µ22 I

∗ 0

0

⎤

X6 ¯ TN ⎥ QG 2⎥ 0 ⎥ ⎥ 0 ⎦ −κ I

(14)

]

0 , Q = (P T )−1

N3

N1

Proof. Define the following Lyapunov function

Φ (t) = xT (t)P T Ex(t) +

∫

t

xT (τ )Sx(τ )dτ

(15)

t −d

It is noted that Φ (t) ≥ 0. Differentiating at both sides of (15), we can obtain that

˙ (t) = xT (t)[P T A(t) + A(t)P + S ]x(t) Φ + 2xT (t)P T Ad (t)x(t − d) − xT (t − d)Sx(t − d) + xT (t)[P T B(t)K + K T BT (t)P ]x(t) + 2xT (t)P T M V¯ g + 2xT (t)P T M K¯ N fˆN

in which Θ1 = P T A(t) + A(t)P + S + P T B(t)K + K T B(t)T P. According to Lemma 1, for the positive scalars µ1 , µ2 and µ3 , Eq. (16) can be transformed into the following:

(17)

where Θ2 = 2 P MM P + 2 P M K¯ N K¯ NT M T P. µ µ 2

T

1

T

1

T

⎡ Ψ2 + Sˆ + Ψ3 ⎢ ∗ Ψ4 = ⎢ ⎣ ∗ ∗

P T M K¯ N 0 ⎥ ⎥<0 0 ⎦ −µ22 I

⎤

Ad (t)Q T −Sˆ

2M 0

∗ ∗

−µ21 I ∗

(18)

Obviously, this system is admissible if the LMI (18) holds. Due to AN (t) = AN + ∆AN (t), BN (t) = BN + ∆BN (t) and Ad (t) = AdN + ∆AdN (t), Ψ4 can be transformed into the following expression:

¯ Ψ4 = Γ¯ + Λ

(19)

⎡ Θ + Sˆ + BRT + RBT ⎢ ∗ Γ¯ = ⎢ ⎣ ∗ ∗

Ad Q T −Sˆ

2M 0

∗ ∗

−µ21 I ∗

⎡ Ψ5 + ∆B(t)RT + R∆BT (t) ∗ ¯ =⎢ Λ ⎣ ∗ ∗

⎤

NM K¯ i 0 ⎥ ⎥ 0 ⎦ −N µ22 I

∆Ad (t)Q T

0 0 0

0

∗ ∗

∗

⎤

0 0⎥ ⎦ 0 0

in which Ψ5 = ∆A(t)Q T + Q ∆AT (t). Using Lemma 2 and N˜ NT (t)N˜ N (t) ≤ I, for a given constant κ > 0 the following inequality can be obtained

¯ ≤ κ ΛT1 Λ1 + κ −1 ΛT2 Λ2 Λ ] [ ] [ ¯ N Q T 0 0 and in which Λ1 = HNT 0 0 0 , Λ2 = Θ3T G 2 Θ3 = Q G¯ T + RG¯ T . It is easy to see that if Ξ < 0 in Theorem 2 N1

N3

holds, Ψ < −β 2∑ I can be obtained. With inequality (14), for any ∥V¯ g ∥2 ≤ M1 and Ni=1 ∥fˆi (t)∥2 ≤ M2 , we have

Remark 5. Theorem 2 gives the solution of the fault tolerant controller parameters, it is noted that by tuning parameters {β, κ, µ1 , µ2 , µ3 }, the tracking error can be guaranteed within a satisfactory range. To solve the LMI Eq. (14) by Matlab toolbox, two steps need to be followed: (i) let K¯ i = 0 and then the controller parameters KP , KD , KI can be solved; (ii) the parameter for changing setpoint value can be obtained using the above KP , KD and KI when faults occur in the system. 5. Simulation In the output SDC systems, the output PDF γi (y, ui ) , as the variable controlled, should be measured or estimated by using instruments (for example, digital camera) or Bayesian estimation technique (Wang, 2000). In this simulation, the inputs ui and the output PDFs γi (y, ui ) are assumed to be measurable. Using the concept of PDF shaping, the B-spline functions for the triple subsystems are chosen as

2

b1 (y) =

Denote

[ Θ1 + Θ2 Ψ = ∗

where Ψ2 = A(t)Q T + QAT (t), Ψ3 = B(t)RT + RBT (t) and Sˆ = QSQ T . Moreover, based on Schur complement formula Ψ1 < 0 can be obtained from the following inequality

˙ (t) ≤ −β 2 ∥x(t)∥2 + (µ1 )2 M1 + (µ2 )2 M2 Φ

in which K¯ N = [K¯ 1 · · · K¯ N ], fˆN = [fˆ1 · · · fˆN ]. Define ξ (t) = [xT (t) xT (t − d)]. Based on Schur complement formula, we have [ ] T ˙ (t) = ξ (t) Θ1 P Ad (t) ξ T (t) Φ ∗ −S T T + 2x (t)P M V¯ g + 2xT (t)P T M K¯ N fˆN (16)

[ ] ˙ (t) ≤ ξ (t) Θ1 + Θ2 PAd (t) ξ T (t) Φ ∗ −S + µ21 ∥V¯ g ∥2 + µ22 fˆN fˆNT

145

]

PAd (t) −S

Generally, only Ψ < 0 is considered. Noted Ψ < 0 is equivalent to Ψ1 = (I2 ⊗ Q )Ψ (I2 ⊗ Q )T < 0. Therefore we have [ ] Ψ2 + Sˆ + Ψ3 + Q Θ2 Q T Ad (t)Q T <0 Ψ1 = ∗ −Sˆ

1 2 1

(y − 2)2 I1 + (−y2 + 7y −

2 39

)I2 +

1 2 1

(y − 5)2 I3

)I3 + (y − 6)2 I4 2 2 59 1 2 2 b3 (y) = (y − 4) I3 + (−y + 11y − )I4 + (y − 7)2 I5 2 2 2 b2 (y) =

{ Ij =

2 1

(y − 3)2 I2 + (−y2 + 9y −

23

1, y ∈ [j + 1, j + 2] , j = 1, 2, . . . , 5; i = 1, 2, 3. 0, other w ise

146

Y. Ren et al. / Automatica 98 (2018) 141–149

From the B-spline functions given above, we can obtain that the matrices in (5) are Σ1 = [1, 1] and Σ2 = 1. Besides, the SDC system matrices can be obtained by digital camera or Bayesian estimation technique based on the above B-spline functions given in the following:

[ −3.2 A1 = 0.3

0.9 −0.1 , Ad1 = Ad2 = −3.5 0.2

]

[

1 1 ¯ 0.2 B1 = , J1 = G = 2 2 i1 0

[ ]

¯ i2 = G

[

[ ]

0.2 0

]

0 , 0.1

0.1 0.2 , G¯ i3 = , 0.3 0.15

]

0.1V1 A2 (t) = 1

[

[

]

0.5 sin(V1 ) , 0.7V1

]

[

0.7V1 0.1V22 (t) A3 (t) = cos(V1 ) 0

[

0.5V2 (sin(V2 ))2

B2 (t) =

B3 (t) =

[

] −0.5 , −0.25

]

[

]

0 , 0.3 sin(V2 ) Fig. 2. The residual signal of subsystems.

]

where Hi = diag {0.5, 0.5}, i = 1, 2, 3. N(t) = diag {m, m} and m is a random variable subjected to uniform distribution on [−ρ0 , ρ0 ]. The initial state vector and the estimated value of the first subsystem are selected as V1 (0) = [0; 1.2], Vˆ 1 (0) = [0; 0]. The time-delay is chosen as d = 2 s, the model uncertainty is chosen as ωi (y, ui ) = 0.01sin(10π y) and Vg = [0.05; 0.1] is the desired weight of the given output PDF γg (y). For the interconnected systems considered above, we can get the sum of Laplace matrix L and loop matrix G

[ −1 L + G = −1 0

0 2 −1

0 0 . 2

]

In order to validate the proposed fault tolerant method, two types of fault, i.e. constant fault and time-varying fault, are considered in the following simulation. The constant faults are defined as f1 = 0.6 and f2 = 0.8 for the first subsystem and the second subsystem, respectively, after t > 10 s. Using the LMI toolbox of Matlab to solve the feasibility problem of (10), the fault diagnosis parameters in Theorem 1 are given as Ks1 = [0.0580 0.0179]T and Ks2 = [0.0716 0.0497]T , P and Q are second order identity matrices. The fault diagnosis parameters in (7) are chosen as Γ1 = 8.3927, Γ2 = 0.70. The small positive numbers in (10) are chosen as ε1 = 0.2, ε2 = ε3 = 0.3. The fault diagnosis parameters in Theorem 2 are KD = [−3.3194 − 6.5156], KP = [0.3597 0.2994], KI = [−0.0045 − 0.0087], µ = 0.5409 and µ2 = 0.65. The response of the residual signals for constant faults are given in Fig. 2, which can be found that the residual errors have significant change after faults occur. Fig. 3 shows the fault diagnosis results which indicate that fault estimation can track the change of fault after short transition. It manifests that the fault diagnosis algorithm is effective. On the other hand, the collaborative fault tolerant control result is shown in Fig. 4. From Fig. 4, the output PDF presents an obvious oscillation when fault occurs at t = 10 s, but it follows the given distribution after resetting the system setpoint Vg . It shows that a better tracking performance has been obtained when the collaborative fault tolerant controller (12) and (13) are applied to the SDC system. The time-varying faults are given as f1 (t) = 0.6cos(0.2t) and f2 (t) = 0.8cos(0.2t) for the first subsystem and the second subsystem, respectively, after t > 15 s. The fault diagnosis parameters

Fig. 3. The fault estimation value of subsystems.

Fig. 4. The output PDFs of the whole control process when faults are constant.

Y. Ren et al. / Automatica 98 (2018) 141–149

147

in Theorem 1 are given as Ks1 = [−1.1967 − 1.0205]T and Ks1 = [−0.8090 − 1.0290]T . In the design of the collaborative fault tolerant tracking controller, matrices KP , KI and KD are solved by (18) to obtain KD = [−0.0696, −0.9562], KP = [0.3379, 0.7911], KI = [−0.03, −0.02], then the fault estimation results and collaborative fault tolerant results are shown in Figs. 5–7. Figs. 5 and 6 that the fault estimation value can track the timevarying faults well. But the tracking errors are larger than those in Fig. 3 due to the different type of fault. Fig. 7 shows the response of fault tolerant control effect. It can been seen that the output PDFs oscillate after faults occur at t = 15 s which can be eliminated by resetting the set-point after a few seconds. 6. Conclusions

Fig. 5. The fault estimation value of subsystem (1) when fault is time-varying.

In this paper, a new type of fault diagnosis and collaborative operational fault tolerant control algorithms are investigated for the irreversible interconnected stochastic distribution control system with time-delay, which is connected in series by N squareroot B-spline approximation models. It has been shown that the tracking performance can be maintained in the event of failure not by reconstructing the system controller but by tuning the system setpoint. The gain matrices and parameters in fault estimation and collaborative fault tolerant tracking control can be obtained by solving the corresponding LMI in Theorems 1 and 2. The purpose of operational collaborative fault tolerant tracking control is to make the post-fault distribution tracking error at each time instant satisfy a certain upper bound beyond a limited time. Finally, the computer simulation results confirm the effectiveness of fault diagnosis and the collaborative operational FTC methods. Appendix. The proof of Theorem 1 Consider the following Lyapunov function

Φ (t) = eTv (t)(IN ⊗ P)ev (t) ∫ t + eTv (τ )(IN ⊗ Q )ev (τ )dτ + eTf (t)ef (t)

(A.1)

t −d

Denote L + G = L and take the derivative at the both sides of Eq. (A.1), we have Fig. 6. The fault estimation value of subsystem (2) when fault is time-varying.

⨂ ˙ (t) = eTv (t)(IN Φ (PAi (t) + ATi (t)P) ⨂ −L (PKsi Σ1 + (Ksi Σ1 )T P))ev (t) ⨂ ⨂ + 2eTv (t)IN PAdi (t)ev (t − d) + 2eTv (t)IN PJi ef (t) − 2eTv (t)X2T Σ2 eh (t) − 2eTv (t)X2T ∆ω (t) ⨂ ⨂ − 2eTf (t)IN Γ1 ef (t) − 2eTf (t)IN Γi1 f (t) ⨂ ⨂ + 2eTf (t)L Γi2 Σ1 ev (t) + 2eTf (t)L Γi2 Σ2 eh (t) ⨂ + 2eTf (t)L Γi2 ∆ω (t) + eTv (t)IN ⊗ Qev (t) − eTv (t − d)IN ⊗ Qev (t − d)

(A.2)

Using Lemma 1, for the positive scalars ε1 , ε2 , ε3 , ε4 and ε5 , it follows that

−2eTv (t)X1T eh (t) ≤ Fig. 7. The output PDFs of the whole control process when faults are time-varying.

1

ε1

eTv (t)X1T X1 ev (t) + ε1 eTv (t)IN ⊗ Ui ev (t)

2eTv (t)X2T ∆ω (t) ≤

1

ε2

eTv (t)X2T X2 ev (t) + ε2 δωT δω

(A.3) (A.4)

148

Y. Ren et al. / Automatica 98 (2018) 141–149

⨂

Γi2 ∆ω (t) ⨂ ⨂ 1 Γi2T ef (t) + ε3 δωT δω Γi2 × LT ≤ eTf (t)L ε3 ⨂ Γi2 Σ2 eh (t) 2eTf (t)L ⨂ ⨂ 1 ≤ eTf (t)L (Γi2 Σ2 )T ef (t) Γi2 Σ2 × LT ε4 ⨂ Ui ev (t) + ε4 eTv (t)IN ⨂ Γi1 f (t) −2eTf (t)IN 2eTf (t)L

≤ ε5 eTf (t)ef (t) +

1

ε5

(A.5)

(A.6)

∥f (t)∥2 ∥Γi1 ∥2

(A.7)

where Ui is the Lipschitz matrix satisfied (2). Combine Eqs. (A.3)–(A.7) and use Lemma 1, we have

˙ (t) ≤ ς T (t)Π1 ς (t) + δ Φ

(A.8)

in which ς (t) = [ev (t), ev (t − d), T

T

(ε2 + ε3 )δω δω and

] δ =

1

eTf (t) T ,

ε5

T

⎡ Π2 ⎢ ⎢ Π1 = ⎣ ∗ ∗

IN

⨂

PAdi (t)

IN

⨂

− IN ⊗ Q

∥f (t)∥ ∥Γ1 ∥2 + 2

⎤

PJi

0

−2IN

∗

⨂

Γi1 +

1

ε4

Y T Y + ε5 I

⎥ ⎥ ⎦

The signs of ∗ represent the transposition terms symmetrical with respect to the diagonal line and

⨂ Π2 = −(L + G) (PKsi Σ1 + (Ksi Σ1 )T P) ⨂ 1 T 1 T + IN (PAi (t) + ATi (t)P) + X1 X1 + X X2 ε1 ε2 2 ⨂ 1 + X3T X3 + (ε1 + ε4 )IN Ui + IN ⊗ Q ε3 It can be seen that Π1 < 0 is equivalent to the following: Π3 = Υ + Υ∆ < 0

⎡ Ω1 ⎢ ⎢∗ ⎢ ∗ Υ =⎢ ⎢∗ ⎢ ⎣∗ ∗ ⎡ Ω2 ⎢∗ ⎢ ⎢ Υ∆ = ⎢ ∗ ⎢∗ ⎣ ∗ ∗

IN

⨂

(A.9) PAdi

IN

− IN ⊗ Q ∗ ∗ ∗ ∗ IN

⨂

X1T

X2T

X3T

⎤

0 Y¯

0 0

∗ ∗ ∗

−ε1 I ∗ ∗

0 0 0

0 0 0 0

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⨂

PJi

−ε2 I ∗

P ∆Adi (t)

0

0

0

0

0

0 0

0 0 0

0 0 0 0

0⎥ ⎥ 0⎥ ⎥ 0⎥ ⎦ 0 0

∗ ∗ ∗ ∗

∗ ∗ ∗

∗ ∗

−ε3 I

⎤

∗

⨂ Γi1 + ε Y Y + ε5 I, Ω1 = IN (PAi + ATi P) − 4 ⨂ ⨂ (L ⨂ + G) (R¯ i Σ1 + Σ1T R¯ Ti ) + (ε1 + ε4 )IN Ui + IN ⊗ Q and Ω2 = T IN (P ∆Ai (t) + ∆Ai (t)P). and Y¯ = −2IN

1

⨂

T

Using Lemma 2, we can obtain

Υ∆ ≤ Υ¯ ∆ = α Υ1 Υ1T + α −1 Υ2 Υ2T

(A.10)

where

[ ⨂ Υ1T = IN HiT [ ⨂ ¯ Ti1 Υ2T = IN PG

0 IN

0

0

0

⨂

¯ i2 )T (P G

]

0 , 0

0

0

]

0 .

It is easy to see that Π3 < 0 is equivalent to Π < 0 in Theorem 1. ˙ (t) < −β∥ς (t)∥2 + δ . If condition (10) holds, one can obtain that Φ ˙ (t) < 0 for β∥ς (t)∥2 > δ , which means that ς (t) It follows that Φ converges to a small set ∥ς (t)∥2 ≤ βδ according to the Lyapunov stability theory. Moreover, ς (t) contains error vectors ev (t) and ef (t). Therefore, the weight value estimation error ev (t) and the fault estimation error ef (t) are uniformly and ultimately bounded. References Antonelli, G. (2013). Interconnected dynamic systems: An overview on distributed control. IEEE Control Systems, 33(1), 76–88. Blanke, M., Kinnaert, M., Lunze, J., & Staroswiecki, M. (2006). Diagnosis and faulttolerant control. (pp. 1379–1384). Springer Berlin Heidelberg. Chai, T., Qin, S. J., & Wang, H. (2014). Optimal operational control for complex industrial processes. Annual Reviews in Control, 38(1), 81–92. Gu, K., Chen, J., & Kharitonov, V. L. (2003). Stability of Time-Delay Systems. 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Y. Ren et al. / Automatica 98 (2018) 141–149 Yuwei Ren received the M.S. degree from Yanshan University, Qinhuangdao, China, and the Ph.D. degree from the Institute of Automation Chinese Academy of Sciences, Beijing, China, in 2007, and 2011, respectively. She is currently with the Department of Information Science and Engineering, Shandong Normal University, China. Her current research interests include Stochastic Distribution Control, collaborative fault tolerant control.

Yixian Fang is a Ph.D. candidate in the school of information and engineering, Shandong Normal University, China and a lecturer in the school of science, Qilu University of Technology, China. His research interests include machine learning, neural network, deep learning and cross-media retrieval.

Aiping Wang received the M.S. degree in computing engineering from National University of Defense Technology, Changsha, China in 1988. She was a professor with Huaibei Normal College, Huaibei, China in 1999 and is currently a Professor with the Institute of Computer Science, Anhui University, China. Her current research interests include FD for dynamic systems, SDC, applied mathematics, and computing.

149 Huaxiang Zhang received his Ph.D. from Shanghai Jiaotong University in 2004, and worked as a professor with the Department of Computer Science, Shandong Normal University now. He has authored over 170 journal and conference papers. His current research interests include machine learning, pattern recognition, evolutionary computation, cross-media retrieval, etc.

Hong Wang (FIET and FInsMC) received the M.S. and Ph.D. degrees from the Huazhong University of Science and Technology (HUST), Wuhan, China, in 1984 and 1987, respectively. He was a Research Fellow with Salford, Brunel and Southampton Universities in UK, before joining the University of Manchester Institute of Science and Technology, Manchester, U.K., in 1992, and has been a chair Professor in process control since 2002. Between 2002 and 2015 he was also a visiting professor of several distinguished titles with some Chinese universities and Chinese Academy of Sciences. In 2016, he has been with the Pacific Northwest National Laboratory, US Department of Energy, USA, as a chief scientist and laboratory fellow in controls, and is also adjunct professor with the University of Washington, Seattle, USA. His current research interests include Stochastic Distribution Control, fault detection and diagnosis, nonlinear control, and data based modeling for complex systems. Prof. Wang was an Associate Editor of the IEEE Transactions on Automatic Control and currently serves as an Associate Editor for IEEE Transactions on Control Systems Technology and IEEE Transactions on Automation Science and Engineering, and members of three IFAC technical committees.