Distributed Fault-tolerant Control Systems Design Against Actuator ...

2009 American Control Conference Hyatt Regency Riverfront, St. Louis, MO, USA June 10-12, 2009

ThB09.4

Distributed Fault-tolerant Control Systems Design Against Actuator Faults and Faulty Interconnection Links: an Adaptive Method Xiao-Zheng Jin and Guang-Hong Yang Abstract— This paper presents an adaptive method to solve the robust fault-tolerant control (FTC) problem for a class of large scale systems against actuator faults and lossy interconnection links. In terms of the special distributed architectures, adaptation laws are proposed to estimate the unknown eventual faults of actuators and interconnections, constant external disturbances, and controller parameters on-line. Then a class of distributed state feedback controllers is constructed to automatically compensate the fault and disturbance effects based on the information from adaptive schemes. On the basis of Lyapunov stability theory, it shows that the resulting adaptive closed-loop large-scale system can be guaranteed to be asymptotically stable in the presence of uncertain faults of actuators and interconnections, and constant disturbances. The proposed design technique is finally evaluated in the light of a simulation example.

I. I NTRODUCTION The problem of distributed control of large-scale interconnected systems has received considerable attention over the past few years, because there are many applications of the control design technique in a lot of practical control systems, such as the control of vehicular platoons [21], cross-directional control in the chemical industry [22], Microelectromechanical system (MEMS) and other largescale systems composed of a large number of spatially interconnected units. Therefore, many approaches have been developed to synthesize some types of distributed controllers, in which interconnections are also considered between individual controllers to make the dynamic large scale systems well-posed, stable, and contractive (see, e.g., [1]-[7], and the references therein). Many issues which always exist in interconnection channels (communication channels), such as time delays, single attenuations, bandwidth limitations (bit rate limitations) are considered by some researchers [1]-[5]. In [1], the problem of signal attenuations in interconnection channels is considered, and necessary and sufficient conditions for well-posedness, stability, and contractiveness are obtained using integral quadratic constraints (IQCs) methods. Recursive information flow [3] is proposed to compensate for the effects of bandwidth limitations. In [2], [4], [5], This work was supported in part by the Funds for Creative Research Groups of China (No. 60821063), the State Key Program of National Natural Science of China (Grant No. 60534010), National 973 Program of China (Grant No. 2009CB320604), the Funds of National Science of China (Grant No. 60674021), the 111 Project (B08015) and the Funds of PhD program of MOE, China (Grant No. 20060145019). Xiao-Zheng Jin is with the College of Information Science and Engineering, Northeastern University, Shengyang, Liaoning, 110004, China

distributed controllers are constructed using some special effective approaches for the stability and performance of closed-loop interconnected system with the problem of timedelays in channels. In this paper, we consider the problem related to the issue of single attenuations. Recently, fault-tolerant control (FTC) system design, which can make the systems operate in safety and with proper performances whenever components are healthy or faulted has received significant attention (see e.g., [6], [7] − [19]). The existing faut-tolerant design approaches can be broadly classified into two groups, namely passive approach [6], [8] − [10] and active approach [7], [11] − [19]. Since the active FTC system offers the flexibility to select different controllers, the most suitable controller can be chosen for the situation and the better performance can be obtained than the passive FTC system. There are primary two typical approaches for fault compensations in active fault-tolerant, such as adaptive approach [11] − [16] and fault detection and isolation (FDI) [17] − [19]. For the FTC system based on FDI, the controller reconfiguration or restructure is based on the fault diagnostic information, which is provided by a fault detection and isolation mechanism. However, it should be noted that the FDI mechanism might not always give the exact fault information. Different from the FDI method with the need for a mechanism to provide the exact fault information, the new proposed adaptive method is not necessary for the estimations to give the exact fault information. In this paper, the adaptive compensation design approach will be used for general actuator and interconnection fault models. Each control effectiveness and constant disturbances are assumed to be unknown. An adaptive method is proposed to solve the problem for developing some distributed state feedback controllers. Due to the distributed architectures are different from other architectures [11], [12], we first propose some adaptation laws to estimate the unknown control effectiveness, constant external disturbances, and controller parameters on-line. Then, the distributed controllers are constructed using the updated values of these estimation. Based on Lyapunov stability theory, the adaptive closed-loop largescale system can be guaranteed to be asymptotically stable in the presence of actuator and interconnection failures, and constant disturbances. The corresponding distributed fault-tolerant controller design via direct adaptive method is presented in [7]. II. P RELIMINARIES AND P ROBLEM S TATEMENT

[email protected] Guang-Hong Yang is with the College of Information Science and Engineering, Northeastern University, Shengyang, Liaoning, 110004, China

[email protected]

978-1-4244-4524-0/09/$25.00 ©2009 AACC

Notations: R stands for the set of real numbers. Given matrices Mk , k = 1, . . . , n, the notation diagnk=1 [Mk ] denotes

2910

the block-diagonal matrix with Mk along the diagonal and denoted diagk [Mk ] for brevity. For signals or vectors xk , the notation catnk=1 xk denotes the signal or vector (x1 , x2 , . . . , xn ) formed by concatenating xk . This is also usually denoted catk xk for brevity. In this paper, we consider a large-scale system G composed of N interconnected linear time-invariant continuous time subsystems Gi , i = 1, 2, . . . , N. Each subsystem is captured the following state-space equations:   xi (t) · ¸ ¸ · i  x˙i (t) AT T AiT S BiT d BiTu   vi (t)  (1) = wi (t) AiST AiSS BiSd BiSu  di  ui (t)

where xi (t) ∈ Rni is the state, ui (t) ∈ Rmi is the control input, di ∈ R pi is the constant external disturbance, and vi := cat j (vi j ), vi j ∈ Rq ji and wi := cat j (wi j ), wi j ∈ Rqi j j = 1, 2, . . . , N is the interconnection input to each subsystem and the interconnection output from each subsystem, respectively. All system matrices are known real constant matrices with appropriate dimensions. In the normal case, once the relationships between the inputs and outputs at each subsystem have been defined, the distributed system can be described by closing all loops by imposing the constraints of interconnection with the interconnection condition such that vi j (t) = w ji (t).

(2)

We assume the states of the subsystems are available at every instant, and every state has its interconnection channel interconnected with other subsystems. A state feedback controller with same interconnection structure for this system has subsystems Ki given by ¸ ¸· · ¸ · xi (t) ui (t) Ki1 Ki2 (3) = Ki3 Ki0 vKi (t) wKi (t) where vKi (t) := cat j (vKij ), wKi (t) := cat j (wKij ) and vKij (t), K

wKji (t) ∈ Rq ji also have interconnection condition with vKij (t) = wKji (t).

(4)

Then, the closed-loop system can be illustrated for example in Fig.1. We make the following assumption on the plant and controller matrices: Assumption 1. The subsystems are interconnected only through their states, that means, AiSS = 0,

BiSu = 0,

BiSd = 0,

Ki0 = 0.

Fig. 1. Example of interconnected closed-loop system with N = 3 subsystems.

disturbance does not exist in interconnection channels. Thus, we assume BiSd = 0. It’s reasonable to assume controller parameter Ki0 = 0 and design other parameters to obtain our control satisfactory objective. In this paper, we formulate the faults including loss of actuator effectiveness and outage in subsystem Gi and lossy interconnection links between communicating subsystems. hF Let uhF i f (t) and vi j (t) represent the signals from the f th actuator in Gi and jth communicating subsystem that have failed in the hth faulty mode, respectively. Then we denote the fault model as follows: h h h ¯ ihf uhF i f (t) = ρi f ui f (t), 0 < ρ i f ≤ ρi f ≤ ρ h h h hF (6) vi j (t) = σ ji vi j (t), 0 ≤ σ ji ≤ σ ji ≤ σ¯ hji i, j = 1, 2, . . . , N, f = 1, 2, . . . , mi , h = 1, 2, . . . , L where ρihf and σ hji are unknown constants, the index h denotes the hth faulty mode and L is the total faulty modes. For every faulty mode, ρ hif and ρ¯ ihf represent the lower and upper bounds of ρihf , respectively. Similarly, σ hji and σ¯ hji represent the lower and upper bounds of σ hji respectively. Note the practical case, when ρ hif = ρ¯ ihf = 1 and σ hji = σ¯ hji = I, there are no faults for the f th actuator ui f and jth interconnection links vi j . when ρ hif = ρ¯ ihf = 0 and σ hji = σ¯ hji = 0, the f th actuator ui f is outage and jth interconnection link is complete disconnection. when 0 < ρ hif ≤ ρ¯ ihf < 1 and 0 < σ hji ≤ σ¯ hji < I, that means the type of two kinds of faults is loss of effectiveness. Denote T hF hF hF h uhF i (t) = [ui1 (t), ui2 (t), · · · , uimi (t)] = ρi ui (t) T hF hF hF hF vi j (t) = [vi j1 (t), vi j2 (t), · · · , vi jq ji (t)] = σ hji vi j (t)

where ρih = diag f [ρihf ], ρihf ∈ [ρ hif , ρ¯ ihf ], and σ hji = diagk [σ hjik ], σ hjik ∈ [σ hjik , σ¯ hjik ]. Then, the sets of operators with above structures are denoted by

(5)

Remark 1. The assumption AiSS = 0 means the large-scale system must not have algebraic loops, and many mechanical systems do belong to this class of systems [1], [10]. The assumption BiSu = 0 can always be satisfied by placing lowpass filters of sufficiently high bandwidth at the control input ui as has been done in the linear parameter varying (LPV) control literature [1], [5]. In this paper, we consider external

(7)

∆ρ h = {ρih : ρih = diag f [ρihf ], ρihf ∈ [ρ hif , ρ¯ ihf ]}, i ∆σ h = {σ hji : σ hji = diagk [σ hjik ], σ hjik ∈ [σ hjik , σ¯ hjik ]}.

(8)

ji

where f = 1, 2, . . . , mi , k = 1, 2, . . . , q ji . For convenience in the following sections, for all possible faulty modes L, the following uniform actuator and interconnection links fault model is exploited:

2911

uFi (t) = ρi ui (t), ρi ∈ {ρi1 · · · ρiL } vFij (t) = σ ji vi j (t), σ ji ∈ {σ 1ji · · · σ Lji }.

(9)

Based on the above description, the equation (2) can be represented by vi j = σ ji w ji (10) for all i, j = 1 . . . N, and the dynamics with actuator fault and lossy interconnection links (1) can be rewritten by N

adaptive laws: d ρˆ i f (t) = −li f (Mi f Ri f + Mi f Si f ), f = 1, 2, . . . , mi (15) dt where the constants li f > 0; σˆ jik (t) is the estimate of the unknown σ jik which is updated according to the adaptive laws:

ji x˙i (t) = AiT T xi (t) + ∑ AiTjS σ ji AST x j (t) + BiTu ρi ui (t) + BiT d di . j=1

(11) In terms of (3) and Assumption 1, the controller form can be described as: N

ui (t) = Kˆ i1 xi (t) + ∑ Ki2 σˆ ji (t)Kˆ i3 x j (t) + Kˆ i4 dˆi (t)

(12)

j=1

where f = 1, 2, . . . , mi , k = 1, 2, . . . , q ji and g = 1, 2, . . . , pi , ρi f ∈ ∆ρ h , σ ji ∈ ∆σ h , and ρˆ i (t), σˆ ji (t), dˆi (t), Kˆ i1 (t), Kˆ i3 (t), i ji Kˆ i4 (t) are the estimate of ρi , σ ji , di , Ki1 , Ki3 , Ki4 , respectively. Ki2 is an appropriate dimensions matrix chosen by the system designer. Then, following (11) and (12), the closed-loop subsystem Gi , i ∈ {1, 2, . . . , N} is given by N

ji x˙i (t) = (AiT T + BiTu ρi Kˆ i1 (t))xi + ∑ (AiTjS σ ji AST j=1

+ BiTu ρi Ki2 σˆ ji (t)Kˆ i3 (t))x j (t) + BiTu ρi Kˆ i4 dˆi (t) + BiT d di . (13) We introduce for the system the following standard assumption denotes the internally stabilizability of each nominal isolated subsystem in actuator failure case: Assumption 2. All pairs {AiT T , BiTu ρi }, i = 1, 2, . . . , N, are uniformly completely controllable for any actuator failure mode ρi ∈ {ρi1 . . . ρiL } under consideration. Now, the main objective of this paper is to synthesize the distributed controller ui (t) given in (12) such that the closed-loop system (13) can be guaranteed to be asymptotically stable even in the cases of failures on actuators and interconnections. III. FAULT- TOLERANT C ONTROL S YSTEM D ESIGN In this section, we develop the adaptive laws to estimate the fault effect factors of actuators and interconnection links, and also to estimate the constant external disturbances. Then, a method for designing distributed adaptive fault-tolerant controllers to guarantee closed-loop system asymptotically stable via state feedback is presented in Theorem 1. Denote Mi := xiT Pi BiTu ,

N

Ri := ∑ Ki2 σˆ ji Kˆ i3 x j ,

Si := Kˆ i4 dˆi ,

j=1

(14) where i = 1, 2, . . . , N and Pi are positive symmetric matrices. Mi , Ri and Si being bound vector functions described by Mi = [Mi1 , Mi2 , · · · , Mimi ] ∈ R1×mi , Ri = [Ri1 , Ri2 , · · · , Rimi ]T ∈ Rmi , Si = [Si1 , Si2 , · · · , Simi ]T ∈ Rmi and respectively, which will be used later. In particular, for any i ∈ {1, 2, . . . , N}, ρˆ i f (t) is the estimate of the unknown ρi f which is updated by the following

d σˆ jik (t) jik x j (t), = r jik xiT (t)Pi aiTjkS aST dt

k = 1, 2, . . . , q ji (16)

jik where the constant r jik > 0, aiTjkS and aST are the kth low ij ji ˆ of AT S and AST , respectively; di (t) is the estimate of the unknown constant external disturbances di updated according to the following adaptive laws:

d dˆig (t) = sig xiT (t)Pi big Td, dt

g = 1, 2, . . . , pi

(17)

i where the constant sig > 0, and big T d is the gth column of BT d . Thus, for the fault-tolerant large-scale system described by (11), we propose the distributed adaptive state feedback controller (12) with the control gain functions Ki1 (t) = [Kˆ i1,1 (t), Kˆ i1,2 (t), . . . , Kˆ i1,mi (t)]T ∈ Rmi ×ni updated by the following adaptive laws:

d Kˆ i1, f (t) f = −Γi1, f xi xiT Pi biTu , dt

i = 1, 2, . . . , N, f = 1, 2, . . . , mi (18) where Γi1, f is any positive constant, Kˆ i1, f (t0 ) is finite, f and biTu is the f th column of BiTu ; Ki2 is an appropriate dimensions matrix chosen by the system designer; Kˆ i3 (t) = [Kˆ i3,1 (t), Kˆ i3,2 (t), . . . , Kˆ i3,mi (t)]T ∈ Rmi ×ni updated according to the adaptive law: N d Kˆ i3,k (t) = −Γi3,k ∑ x j xiT Pi BiTu ρˆ i ki2k σˆ jik dt j=1

(19)

where i, j = 1, 2, . . . , N, k = 1, 2, . . . , q ji , Γi3,k is any positive constant, Kˆ i3,k (t0 ) is finite, ki2k is the kth column of Ki2 ; Kˆ i4 (t) = [Kˆ i4,1 (t), Kˆ i4,2 (t), . . . , Kˆ i4,mi (t)]T ∈ Rmi ×ni updated according to the adaptive law: f = 1, 2, . . . , mi , g = 1, 2, . . . , pi

where Γi4,g Letting

d Kˆ i4,g (t) (20) = −Γi5,g dˆig xiT Pi BiTu ρˆ i f dt is any positive constant, Kˆ i4,g (t0 ) are finite.

ρ˜ i (t) = ρˆ i (t) − ρi , σ˜ ji (t) = σˆ ji (t) − σ ji , d˜i (t) = dˆi (t) − di ,

K˜ i1 (t) = Kˆ i1 (t) − Ki1 , K˜ i3 (t) = Kˆ i3 (t) − Ki3 , K˜ i4 (t) = Kˆ i4 (t) − Ki4 .

(21)

Due to ρi f , σ jik , dig , and Ki1 , Ki3 , Ki4 are an unknown constants, the error system can be written as following equations: ρ˙˜ i f (t) = ρ˙ˆ i f (t), σ˙˜ jik (t) = σ˙ˆ jik (t), d˙˜ig (t) = dˆ˙ig (t), (22) K˙˜ (t) = K˙ˆ (t), K˙˜ (t) = K˙ˆ (t), K˙˜ (t) = K˙ˆ (t). i1

i1

i3

i3

i4

i4

Then, the following theorem can be obtained, which shows the uniform ultimate boundedness of the closed-loop largescale system (13) and the error system (22).

2912

Theorem 1. Consider the adaptive closed-loop largescale interconnected system described by (13), satisfying Assumptions 1 and 2. Then the closed-loop fault-tolerant control system is asymptotically stable for any ρi ∈ ∆ρ h and i σ ji ∈ ∆σ h if there exist positive symmetric matrices Pi > 0, ji ρˆ i f (t), σˆ jik (t) and dˆig (t) determined according to the adaptive laws (15), (16) and (17), and control gain Kˆ i1 (t), Kˆ i3 (t), Kˆ i4 (t) updated by adaptive laws (18), (19), (20), respectively. Proof: For the adaptive closed-loop large-scale system described by (13), we first define a Lyapunov functional candidate as: N

N

V = ∑ xiT Pi xi + ∑ N

pi

2 d˜ig

N

N

q ji

N

+∑ ∑

∑

i=1 j=1 k=1

∑ ∑ ∑ 2xiT Pi BiTu ρˆi Ki2,k σˆ jik Ki3,k x j N

N

2ρ˜ i f ρ˙˜ i f ∑ li f f =1

+ 2xiT Pi BiTu ρi

2σ˜ jik σ˙˜ jik ∑ ∑ r jik j=1 k=1

∑∑

j=1 k=1

∑

m

big T d dig

+

T ˙˜ + ∑ 2K˜ i4,g Γ−1 i4,g Ki4,g }

+

T −1 ˙˜ ∑ 2ρi f K˜i1, f Γi1, f Ki1, f

f =1

pi

T ˙˜ ˜ T −1 ˙˜ Γ−1 ∑ 2K˜i3,k i3,k Ki3,k + ∑ 2Ki4,g Γi4,g Ki4,g }

(27)

g=1

(28)

N

V˙ (t) ≤ ∑ {xiT [(AiT T + BiTu ρi Ki1 )T Pi + Pi (AiT T + BiTu ρi Ki1 )]xi

mi

i=1

+ 2xiT PBiTu ρi K˜ i1 x +

q ji

˙˜ ˜ T −1 ˙˜ ∑ 2ρi f K˜i1,T f Γ−1 i1, f Ki1, f + ∑ ∑ 2Ki3,k Γi3,k Ki3,k

f =1 pi

li f

Thus, (27) can be rewritten as

g=1

N

∑

g=1

m

(AiT T + BiTu ρi Ki1 )T Pi + Pi (AiT T + BiTu ρi Ki1 ) < 0.

pi 2ρ˜ i f ρ˙˜ i f 2d˜ig d˙˜ig +∑ + 2xiT Pi BiTu ρi ∑ Kˆ i4,g dˆig + ∑ li f g=1 sig g=1 f =1 pi

j=1 k=1 mi 2ρ˜ ρ˙˜ if if

g=1 pi

Following Assumption 2, for any ρi ∈ ∆ρ h , there exist Pi > i 0 and Ki1 satisfying

q ji

Ki2,k σˆ jik Kˆ i3,k x j + 2xiT Pi

∑ 2xiT Pi BiTu ρ˜i Ki2,k σˆ jik Kˆi3,k x j − ∑ 2xiT Pi BiTu ρ˜i Kˆi4,g dˆig

j=1 k=1

i=1

pi

−∑

+∑

= ∑ {xiT [(AiT T + BiTu ρi Kˆ i1 )T Pi + Pi (AiT T + BiTu ρi Kˆ i1 )]xi

q ji

pi

∑ 2xiT Pi BiTu ρˆi Ki2,k σˆ jik K˜i3,k x j + ∑ 2xiT Pi BiTu ρˆi K˜i4,g dˆig

f =1 N q ji

g=1

N

q ji

+∑

+

N

∑∑

∑

j=1 k=1 N q ji

pi

j=1 k=1

∑

i=1

mi

T ˙˜ ˜ T −1 ˙˜ Γ−1 ∑ 2K˜i3,k i3,k Ki3,k + ∑ 2Ki4,g Γi4,g Ki4,g }

N

∑∑

N

pi m 2σ˜ jik σ˙˜ jik 2d˜ig d˙˜ig ˙˜ + ∑ r jik ∑ sig + ∑ 2ρi f K˜i1,T f Γ−1 i1, f Ki1, f j=1 k=1 g=1 f =1

jik xj + aiTjkS σ jik aST

pi

≤ ∑ {xiT [(AiT T + BiTu ρi Kˆ i1 )T Pi + Pi (AiT T + BiTu ρi Kˆ i1 )]xi

q ji

q ji

∑ 2xiT Pi BiTu ρ˜i Ki2,k σˆ jik Kˆi3,k x j + ∑ 2xiT Pi bigTd dˆig

∑∑

+∑

N

−∑

g=1

∑

ji

j=1

+ 2xiT Pi

pi

jik

∑ 2xiT Pi aT S σˆ jik aST x j − ∑ 2xiT Pi BiTu ρ˜i Kˆi4,g dˆig

∑

N

j=1 k=1

i jk

+∑

j=1 k=1 g=1 pi N q ji 2xiT Pi BiTu ρˆ i Ki2,k σˆ jik Kˆ i3,k x j 2xiT Pi BiTu ρˆ i Kˆ i4,g dˆig + + j=1 k=1 g=1 mi 2ρ˜ ρ˙˜ m if if T −1 ˙˜ 2ρi f K˜ i1, + + f Γi1, f Ki1, f l if f =1 f =1 pi N q ji T T ˙˜ ˙˜ 2K˜ i3,k Γ−1 + 2K˜ i4,g Γ−1 i4,g Ki4,g } i3,k Ki3,k + j=1 k=1 g=1

(23)

ji + 2xiT Pi [ ∑ (AiTjS σ ji AST + BiTu ρi Ki2 σˆ ji Kˆ i3 )x j ]

q ji

∑ {xiT [(AiT T + BiTu ρi Kˆi1 )T Pi + Pi (AiT T + BiTu ρi Kˆ1i )]xi

j=1 k=1 N q ji

i=1

N

(26)

i=1 g=1

i=1 N q ji

N © V˙ (t) = ∑ xiT [(AiT T + BiTu ρi Kˆ i1 )T Pi + Pi (AiT T + BiTu ρi Kˆ i1 )]xi

+∑

pi

N

Then, The time derivative of V for t > 0 associated with a certain failure mode ρi ∈ ∆ρ h and σ ji ∈ ∆σ h is

N

(25)

Then, following the above inequalities and equation (21), we can rewrite (24) as

i=1 g=1

+ 2xiT Pi BiT d di + 2xiT Pi BiTu ρi Kˆ i4 dˆi +

N

i=1 g=1

pi

i

pi

N

i=1 f =1

i=1 k=1

jik

∑ ∑ 2xiT Pi BiTu ρˆi Ki4,g dˆig ≤ − ∑ ∑ 2xiT Pi bigTd dˆig .

r jik

T ˜ ˜ T −1 ˜ Γ−1 ∑ K˜i3,k i3,k Ki3,k + ∑ ∑ Ki4,g Γi4,g Ki4,g

i jk

∑ ∑ 2xiT Pi aT S σˆ jik aST x j

i=1 j=1 k=1

mi

N

q ji

N

≤−∑

V˙ (t) ≤

σ˜ 2jik

q ji

N

i=1 j=1 k=1

˜ ∑ sig + ∑ ∑ ρi f K˜i1,T f Γ−1 i1, f Ki1, f

i=1 g=1 N q ji

+∑

∑

ρ˜ i2f

i=1 f =1 li f

i=1

+∑

mi

N

mi

j=1 k=1

m

˙˜ ∑ 2K˜i1,T f Γ−1 i1, f Ki1, f

f =1 mi

+ 2 ∑ ρ˜ i f Mi f Ri f + 2 ∑ ρ˜ i f Mi f Si f + f =1

(24)

f =1

g=1

Chosen the adaptive laws as (15)-(17), we can adjust ρˆ i , σˆ jik and dˆig to guarantee that there exist constants Ki3,k ∈ Rni and Ki4,g ∈ Rni , k = 1, 2, . . . , q ji , g = 1, 2, . . . , pi make sure the following inequalities hold true:

mi

2ρ˜ i f ρ˙˜ i f } li f f =1 (29)

∑

where Mi f , Ri f and Si f are denoted in (14). Defining

2913

Qi = −[(AiT T + BiTu ρi Ki1 )T Pi + Pi (AiT T + BiTu ρi Ki1 )] > 0. (30)

x

Then, note that the adaptive law is chosen as (15), (18), we can rewrite (27) as

x

11

0 x

N

V˙ (t) ≤ − ∑

xiT Qi xi .

1

5

12

−5

(31)

−10

i=1

0

10

20

30

40 x

Hence, it is easy see that V˙ < 0 for any x 6= 0. Thus, the state x(t) converges asymptotically to zero. Furthermore, all the signals are bounded.

50

60

70

2

5 x

21

0 x

22

−5 −10

IV. A N E XAMPLE AND S IMULATION R ESULTS

0

10

20

30

40

50

60

70

time(s)

Fig. 3. Response curves of the first and the second vehicle states x1 (t), x2 (t) with distributed controllers. The estimate of ρ

i

1

Fig. 2.

ρ

22

ρ

two cars connected by spring.

11

0.8

ρ

ρ

21

12

0

10

20

30 40 The estimate of σ

50

60

70

50

60

70

60

70

i

In this section, we will study a multiple vehicle faulttolerant control problem using distributed adaptive method. Consider a group of two vehicles connected by a spring, see Fig.2. The ith car’s dynamics is governed by:

1.5 1

σ

1

0.5

σ

0

2

0

10

20

30 40 The estimate of d

i

0

22

12

21

d

−1

mi y¨i = k(y j − yi ) + di + ui

d

d d

−0.5 11

0

10

20

30

40

50

time(s)

where yi is the position of ith vehicle from its equilibrium position. di is the constant disturbance acting on ith vehicle, such as friction. ui is its multi-input control. Then, the vehicle dynamics can be converted to the interconnected system format as:

Fig. 4. Response curves of the estimate of actuator fault effect factors, attenuation factors in interconnection links and constant disturbances with two vehicles. The estimate of K

11

2

K

11,3

·

x˙i (t) wi j (t)

¸

=

"

AiT T ji AST

AiTjS 0

= [y, ˙ y] ¨ T,

BiT d 0

BiTu 0

#





xi (t)  σ ji vi j (t)      di ρi ui (t) (32)

where i, · · and¸ ¸ · j ∈ {1, 2},¸xi 0 0 1 0 0 i j i i , BTu = , AT S = , AT T = 1 1 k − 2k 0 a i mi bi mi mi · mi ¸ £ ¤ 0 0 ji BiT d = , AST = 1 0 , and we assume that 1 0 mi m1 = 1, m2 = 1.5, k = 1.25, a1 = 1, b1 = 0.5, a2 = 1, b1 = 2. To verify the effectiveness of the proposed adaptive method, the simulations are given with the following parameters and initial conditions:

K

0

11,1

K

11,2

−2

K

11,4

−4

0

10

20

30

40

50

60

70

The estimate of K

21

1 K

0

21,3

K

21,1

−1

K

K

21,2

21,4

−2 0

10

20

30

40

50

60

70

time(s)

Fig. 5. i = 1, 2.

Response curves of the estimate of controller parameters Ki1 ,

The estimate of K

13

2 1

li f = 0.5, r jik = 10, sig = 10, ρˆ i f (0) = 1, σˆ jik (0) = 1, dˆig (0) = 0, Γi1, f = 10, Γi3,k = 10, Γ15,g = 50, Γ25,g = 100, Ki2 = [1 1]T , i = 1, 2, f , g = 1, 2, k = 1.

K

0

13,2

−1 −2 −3

K

13,1

0

10

20

30

40

50

60

70

60

70

The estimate of K

23

1

We consider the following two possible faulty modes: Normal mode 1: Both of the two subsystems’ actuators 1 = ρ1 = and interconnection links are normal, that is, ρ11 12 1 1 1 1 ρ21 = ρ22 = 1 and σ1 = σ2 = 1. Faulty mode 2: The first actuators of subsystem G1 and G2 are outage, the second actuators of them may be normal or loss of effectiveness, and the maximum loss of effectiveness

K

23,1

0.5 0 −0.5

K

23,2

0

10

20

30

40

50

time(s)

Fig. 6. i = 1, 2.

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Response curves of the estimate of controller parameters Ki3 ,

The estimate of K

14

R EFERENCES

2 K

K

14,4

14,2

0 −2

K

14,1

−4 K

14,3

−6 −8

0

10

20

30

40

50

60

70

The estimate of K

24

K

24,2

0

K

24,4

K

24,3

−5

K

24,1

−10

0

10

20

30

40

50

60

70

time(s)

Fig. 7. i = 1, 2.

Response curves of the estimate of controller parameters Ki4 ,

for the second actuators of G1 and G2 are 0.3 and 0.5, respectively. The spring may be normal or loss of effectiveness, described by a ≤ σ12 = σ22 ≤ 1, a = 0.1, which denotes the maximum loss of effectiveness for the interconnections. The following faulty case is considered in the simulations, that is, before 5 second, the interconnected systems operate in normal case, and the constant external disturbances d1 = d2 = [−5, 5]T enter into the systems at the beginning (t ≥ 0). At 5 second, some faults in interconnection channels have occur, described by σ12 = σ21 = 0.1. At 25 second, the actuator faults occur, described by ρ1 = diag[0, 0.3], ρ2 = diag[0.5, 0]. Fig.3 is the response curves of the states of subsystems G1 , G2 with adaptive state feedback controller in abovementioned faulty case, respectively. Fig.4 is the response curves of the estimation of fault effect factors ρi , σ ji , i, j ∈ {1, 2}, and constant disturbances, respectively. Fig.5Fig.7 are the response curves of the estimation of controller parameters Ki1 , Ki3 , Ki4 , i ∈ {1, 2}, respectively. It is easy to see the estimations can converge but not converge to theirs true values. In our adaptive fault-tolerant control design, there is no need for the estimations of ρi f and σi , i ∈ {1, 2} to converge to their true values. V. C ONCLUSION In this paper, an adaptive method for FTC system design against actuator faults and interconnection signal attenuations for a class of large-scale systems have been proposed. We have presented the adaptation laws to estimate the unknown eventual faults on actuators and interconnections, constant external disturbances, and controller parameters on-line. By making use of their updated values, a class of distributed state feedback controllers has been constructed for automatically compensating the fault and disturbance effects based on the information from adaptive schemes. According to the Lyapunov stability theory, it has shown that the resulting adaptive closed-loop large-scale system can be guaranteed to be asymptotically stable even in the cases of faults on actuators and interconnections, and constant disturbances. A numerical example has shown the effectiveness of the proposed adaptive method.

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