- Email: [email protected]
Actuator Fault Compensation Control for Nonlinear Systems
Proceedings of the 17th World Congress The International Federation of Automatic Control Seoul, Korea, July 6-11, 2008
Actuator Fault Compensation Control for Nonlinear Systems S. Joe Qin * and Yingwei Zhang
**
*The Mork Family Department of Chemical Engineering and Materials Science, Ming Hsieh Department of Electrical Engineering, Daniel J. Epstein Department of Industrial and Systems Engineering, University of Southern California, 925 Bloom Walk, HED 210, Los Angeles, CA 90089-121190 (email:sqin @usc.edu) **Key Laboratory of Integrated Automation of Process Industry, Ministry of Education , Northeastern University, Shenyang, Liaoning 110004, P. R. China ([email protected]) Abstract: The partial loss of actuator effectiveness is considered for the control of a class of nonlinear processes in the presence of a fault. Not only unmatched uncertainties, but also matched uncertainties are discussed in Lyapunov stability sense. The partial loss of actuator effectiveness is approximated by high order neural networks. Application of the proposed design indicates that the fault compensation control law is effective for a nonlinear fermentation system.
1. INTRODUCTION The study of fault diagnosis and fault-tolerant control has attracted much attention (Patton et al. (2000) and Zhang et al. (2004)), due to the industrial demands for safety and efficiency. For certain processes, it is important not only to detect (and identify) but also to accommodate any faults quickly. Fault-tolerant controls have been developed to keep such processes in control, in spite of the occurrence of a fault. For most control systems the faults can be seen in their actuators, sensors, and systems themselves. The purpose of fault detection and diagnosis (FDD) is to use available inputs, outputs, and operating points of the systems to detect faults. Once the faults are detected, fault diagnosis should be performed so as to locate the fault and also give a good estimation of fault sizes. (Zhang et al. (2002)). In terms of fault-tolerant control, various techniques have been developed which can be roughly classified into passive and active fault-tolerant control schemes. The former is ‘similar’ to robust control while the latter can be regarded as a reinforced nonlinear control in the sense that control structures and parameters are improved so as to realize a reliable operation of concerned closed-loop systems before a scheduled repair is made. (El-Farra et al. (2005)). In an active fault-tolerant control, faults are accommodated, typically by a reconfiguration of the feedback control law. Faults are typically associated with sensors and actuators failures; accordingly, respective compensation strategies can be designed. For example, sensor fault compensations for MIMO systems have been discussed by Tortora et al. (2002).
control. For example, an adaptive compensation method for actuator faults with known plant dynamics has been formulated by Zhang et al. (2004) and Sastry and Isidori (1989); Idan (2001) presents an intelligent fault tolerant flight control system design method that blends aerodynamic and propulsion actuation for safe flight operation in the presence of actuator failures by utilizing redundancy. The actuator and component faults are combined and denoted as a nonlinear function (Zhang and Qin (2007)). The partial loss of actuator effectiveness are considered and represented as ( I − Γ)u with the fault parameter Γ in this paper. Specially, not only unmatched uncertainties, but also matched uncertainties are discussed. Based on neural networks which are capable of approximating, with arbitrary accuracy, any real continuous function on a compact set, the partial loss of actuator effectiveness is approximated by the neural networks. Then an adaptive compensation control law is formulated to ensure the system stability. The remainder of the paper is organized as follows. The problem statement and its assumptions are given in section 2, followed by the formulation of our controller and its relevant proofs in section 3. An illustrative example is given in section 4 to demonstrate the effectiveness of the proposed method. Finally, conclusions are drawn in section 5. 2. PROBLEM STATEMENT The following actuator fault model is adopted in this paper (Veillette 1995): i = 1, " , m ， j = 1, " , L
Adaptive approaches have also been used in fault tolerant
978-3-902661-00-5/08/$20.00 © 2008 IFAC
11160
uijC = (1 − Γ ij )ui (t ), 0 ≤ Γ ij ≤ Γ ij ≤ Γ ij ≤ 1
(1)
10.3182/20080706-5-KR-1001.0604
17th IFAC World Congress (IFAC'08) Seoul, Korea, July 6-11, 2008
⎧0 if t < T β i (t − T ) = ⎨ , i = 1, 2," , m , ⎩1 if t ≥ T
where Γ ij is an unknown constant. Here, the index j denotes the j th fault mode and L is the total fault modes. Let u
C ij
represent the signal from the i th actuator that has failed in the j th fault mode. For every fault mode, Γi and Γi j
j
represent the lower and upper bounds of Γ ij , respectively. Note that, when Γi = Γi =0, there is no fault for the i th j
j
actuator ui in the j th fault mode. When Γi = Γi =0=1, the j
j
where T is the fault occurrence time. The problem considered is as follows: Fault compensation (FC) problem: Given system (1), design a control u N for the normal system, and an additional control
uF for fault compensation, so that u = u N + uF as the new
i th actuator ui is outage in the j th fault mode. When
control after the occurrence of a fault can guarantee the resulted closed-loop nonlinear system to be stable.
0 ≤ Γi ≤ ρ i ≤ Γi ≤ 1 , in the j th fault mode the type of
The following assumptions are used.
actuator faults is loss of effectiveness. Denote
Assumption 1: There exists u = u ( x ) and Lyapunov
j
j
j
a
function V ( x ) , such that
C uijC = [u1Cj , u2Cj , " , umj ] = ( I − Γ j )u (t )
2
where
Γ = diag [Γ1 , Γ 2 , " , Γ m ] j
j
j
j
j = 1, 2, " , L
,
Considering the lower and upper bounds ( Γi , Γi ) , the j
j
∂V ( x ) ∂x
following set can be defined j
j
j 1
j 2
(6)
T
(2) ρ ( x)
where T
x = ζ ( x ) + Δζ ( x) + G ( x)[( I − β (t − T )Γ )u + Δg ( x)] (3) where x ∈ R , u ∈ R are the state and input of the system, respectively, Δζ ( x ) and Δg ( x ) are the model uncertainty in the normal operation. The normal system, in the absence of any faults, is described by m
G ( x)
is continuous, ξ (•) and ρ (•) are
( ) ∂V ( x ) ∂x
known and continuous. Remark 1: From Assumption 2, we have ρ ( x ) = 0 , if
⎛ ∂V ( x) ⎞ ⎟=0. ⎝ ∂x ⎠
G T ( x) ⎜
(4)
The fault matrix Γ is multiplied by a switching function β (t − T ) ,
where
(5)
⎛ ∂V ( x) ⎞ ⎜ ⎟ Δζ ( x) ≤ ρ ( x ) ⎝ ∂x ⎠
Consider a system described as:
β (t − T ) = diag ( β1 (t − T ), β 2 (t − T ), " , β m (t − T ))
∂x
Δg ( x ) ≤ ξ ( x )
where Γ can be described by
x = ζ ( x ) + Δζ ( x ) + G ( x )[u + Δg ( x )]
2
Assumption 2: For system (1)
L , the uniform actuator fault model
Γ = diag[Γ1 , Γ 2 ," , Γ m ] .
∂V ( x )
where k1 , k 2 , k3 , k4 are positive constants.
u F = ( I − Γ)u , Γ ∈ {Γ1 , Γ 2 ," , Γ L }
n
)
( x ) ≤ − k3
j m
Γ ij = Γ ij or Γ ij = Γ ij } For all fault modes
( ζ ( x ) + G ( x )u
a
≤ − k 4V ( x)
N ρ = {Γ Γ = diag[Γ , Γ , " , Γ ], j
2
k1 x ≤ V ( x ) ≤ k 2 x ,
.
3. FAULT COMPENSATION The partial loss of actuator effectiveness can be approximated by the neural networks. Then system (3) can be rewritten as:
x = ζ ( x ) + Δζ ( x ) + G ( x )[u + W * S ( x ) + ε ( x ) + Δg ( x )]
11161
(7)
17th IFAC World Congress (IFAC'08) Seoul, Korea, July 6-11, 2008
where, ε ( x ) = f ( x ) − W S ( x ) ≤ ε is the estimation error. If *
with the following adaptive weight update law
we denote W as the estimate of the uncertain weight matrix W * , then
⎧ ∂v0 T ⎪⎪ 2k0 ∂x S ( x ) if W < M W W = ⎨ ⎪ − β W + 2k ∂v0 S T ( x ) if W ≥ M 0 W ⎪⎩ ∂x ∂v θ = −γ 1θ + k0 0 ∂x
( x) + WS ( x) + x = ζ ( x ) + Δζ ( x ) + G ( x )[u − WS
ε ( x) + Δg ( x)] (8) where W = W − W * and it has the appropriate dimension. S ( x ) is a vector with S i ( x ) , i = 1, 2, " , L . Here, Si ( x) =
∏
lj (i )
j∈J i
[ s ( x j )]
with J i , i = 1, 2, " , L and l j (i ) is
nonnegative integers. Because W * is bounded, we assume W
*
< MW
.
Theorem 1: Under Assumptions 1 and 2, we can design a controller in the form of the following:
u = uN + uF
(9)
uN = u a + u b + u c where u N is the control law of “healthy” systems (4) and
as in the proof. The novelty of this paper is that fault compensation controller is clearly given after faults occur. The main contributions of this paper are the design of the corrective control law of nonlinear system with unmatched and matched uncertainties, and the stability analysis of the closed-loop system in the presence of the fault modeling errors is proposed. Proof: step 1 Substituting the controller Eqs. (9-12) into system (1), we have:
x = ζ ( x) + Δζ ( x) + G ( x)[u a + u b + u c + Δg ( x)]
Assumption 1, and let
∂V ( x) ⎧ ⎫ G ( x )T = 0 ⎬, ∂x ⎩ ⎭ ∂ V ( x ) ⎧ GT ( x) ⎪ ∂x ξ x , x ∉ E ( ) ⎪− b ∂V ( x ) , u =⎨ T G ( x) ⎪ ∂x ⎪ 0, x ∈ E ⎩
Define a positive function v0 ( x ) = V ( x ) , then we have:
E=⎨ x
∂V ( x) ⎧ T ⎪ G ( x) ∂x ρ ( x) , x ∉ E ⎪− 2 ∂ ( ) V x , uc = ⎨ G T ( x) ⎪ ∂x ⎪ 0, x ∈ E ⎩ uF =
WS ( x)u N 2
+
v0 ( x ) = ( (10)
⎧ ⎩
)T (ζ ( x ) + G ( x )u a ) +
∂x ∂V ( x ) T ( ) G ( x )(u b + Δg ( x )) + ∂x ∂V ( x ) T ( ) ( Δζ ( x ) + G ( x )u c ) ∂x
From Assumption 1, we have (11)
(
2
(12).
∂V ( x ) ∂x
)T (ζ ( x ) + G ( x )u a ) ≤ − k3
∂V ( x )
have
(
Where Θ ∈ R and Θ = [θ , 0, " , 0] . Then, the state x is ultimately consistently bounded by the set:
D = ⎨ x ∈ R n : v0 ( x ) ≤
∂V ( x )
∂x
2
(16)
From Assumption 2 and the structure of u b ( x) , we
Θ uN
λ[1 + G ( x) ] λ1[1 + G ( x) ] n× L
(15)
The parameters of λ , λ1 , k1 , k 2 , α , μ can be determined
u a is given by
uF is the fault compensation. And
(14)
T
⎫ μ k2 , ≤ k0 ≤ 1⎬ , k0α k1 ⎭
∂V ( x ) ∂x
(G T ( x ) (13)
)T G ( x )(u b + Δg ( x )) =
∂V ( x ) ∂xi
when x ∈ E , and
11162
)T (u b + Δg ( x )) = 0
17th IFAC World Congress (IFAC'08) Seoul, Korea, July 6-11, 2008
(
∂V ( x ) ∂x
=(
Proof: step 2:
) G ( x )(u + Δg ( x )) T
∂V ( x ) ∂x
b
T
∂V ( x )
T
∂x ξ ( x ) + Δg ( x )) ∂V ( x )
G ( x) ) G ( x )( − T
G ( x)
∂V ( x )
= − G T ( x )(
∂x ∂V ( x )
≤ − G ( x )( T
∂x
Define a Lyapunov function for system (1) of the following form:
1 1 V ( x, W , θ ) = k0 v0 ( x ) + tr W T W + θ 2 2 2
{
∂x
∂V ( x )
)T ξ ( x ) + (
∂x
) ξ ( x) + ( T
with θ = θ − ε , then the derivatives of V is
∂v a b c V = k0 0 { f ( x ) + Δf ( x ) + G ( x )[u + u + u + ∂x
) G ( x ) Δg ( x ) T
≤0 ,
Δg ( x )]} + k0
when x ∉ E . Hence (
∂V ( x ) ∂x
) G ( x )(u + Δg ( x )) ≤ 0 T
(20)
)T G ( x ) Δg ( x )
∂V ( x ) ∂x
}
b
k
(17)
∂v0
∂v0 ∂x
G ( x )u − k 0 F
∂v0
G ( x )WS ( x ) + k0
∂x T tr W W + θθ
{
From Assumption 2 and structure of u c (x) , we have
∂x
}
∂v0
( x) + G ( x )WS
∂x
G ( x )ε ( x ) +
(21) (
∂V ( x ) ∂x
=(
Using (14), we obtain ) ( Δζ ( x ) + G ( x )u ) T
∂V ( x ) ∂x
≤ (
c
) Δζ ( x ) + ( G ( x ) T
∂V ( x ) ∂x
T
∂V ( x ) ∂xi
T
) u
∂v ∂v V = k0 v0 + k0 0 G ( x)u F + k0 0 G ( x)WS ( x ) ∂x ∂x ∂v0 + k0 G ( x)ε ( x ) − β IW tr W T W + θθ ∂x
c
{
) Δζ ( x ) ≤ ρ ( x ) = 0 , T
}
where IW is the indicator function of W , and it satisfies
when x ∈ E , and (
∂V ( x )
=(
∂x
) ( Δζ ( x ) + G ( x )u ) T
∂V ( x )
uF =
⎧1 if W ≥ M W IW = ⎨ ⎩0 if W < M W
∂x
c
Choosing
) Δζ ( x ) T
WS ( x ) 2
λ [1 + G ( x ) ]
+
2
λ1 ≥
where Θ ∈ R and Θ = [θ , 0, " , 0] . When x ∉ E . Hence
∂x
2k 2 β − sk0
λ1 [1 + G ( x ) ] T
∂V ( x )
k0 s
λ≥
Θ
n
(
(22)
) T ( Δζ ( x ) + G ( x ) u c ) ≤ 0
(18)
β>
,
k0 2k0 k 2 γ 1 − k0 s 2 k02 2k 2
(23)
,
,
(24)
(25)
Thus, we obtain v0 ( x ) ≤ − k3
∂V ( x )
γ1 >
2
∂x From (19), the stability of the normal system is proven.
(19)
k0 2k 2
By using Eq. (5) from Zhang and Qin (2007) we have
11163
(26)
17th IFAC World Congress (IFAC'08) Seoul, Korea, July 6-11, 2008
2 kk k β γ V ≤ − 0 3 4 v0 ( x ) − W ξ 2 − 1 θ 2ξ 2 + 2 2 k3
2 β M W2 +
γ1 2
(27)
. The parameters of y , q, M and V are given in Table 1 for the process.
Table 1: Fermentation model parameters
ε 2ξ 2
⎧k k ⎫ therefore V ≤ −αV + μ , with α = min ⎨ 3 4 , β , γ 1 ⎬ , ⎩ k3 ⎭ μ = 2β M + 2 W
γ1 2
ε
2
(28)
Integrating both sides of (36) yields V (t ) ≤
μ
⎡ α ⎢⎣
+ V ( 0) −
Volume Constant Constant Toxin production constant
μ⎤
−α t e , ∀t ≥ 0 ⎥ α⎦
(29)
Due to (27), it can be deduced that x, W ( x ) , θ ( x ) are bounded consistently. From (20), we have
k0 v0 ( x ) ≤ V Therefore,
y M
q
200[l] 0.417 0.0196 0.0296[l/h(g/l)2/3
The possible fault modes considered are once one of the actuator is outage, the maximum loss of effectiveness of the actuator is 80%. The fault case here is that at the beginning, the first actuator becomes loss of effectiveness of 70%. The main contribution of the paper is theoretical in nature. In the
(30)
V
fermentation
example,
let
⎡ θ1x1x2e x2 ⎤ ⎢ ⎥ Δg ( x) = ⎢2 x22e x2 sin θ 2 ⎥ , ⎢ ⎥ x1 ⎣⎢ θ 3 x1e ⎦⎥
⎡θ 2 x12 cosθ1 ⎤ ⎢ ⎥ Δζ ( x) = ⎢ x12 sin θ 2 ⎥ , θ1 ∈ (−2, 2) and θ 2 ,θ 3 ∈ (−1, 1) are the ⎢ 2 ⎥ ⎢⎣ θ 3 x1 ⎥⎦
(31)
uncertainty parameters. In this example, a radial basis function (RBF) network is chosen to represent the dynamic changes after the fault occurrence, with 10 hidden nodes and 10 centers that are distributed uniformly in region [-1,1].
The above completes the proof that x is ultimately consistently bounded by the set D .
Choose ξ ( x) = 2 x e x , ρ ( x) = 2 x12 , v0 = xT x = x . Then the
v0 ( x ) ≤
μ k 0α
+
1 ⎡ μ ⎤ −α t V ( 0) − e , ∀t ≥ 0. ⎢ α ⎥⎦ k0 ⎣
control input is:
u a = −0.4 x12 / 3 + 0.9 x2
4. ILLUSTRATION EXAMPLE
⎧⎪−2 x 2 e x x1 ≠ 0 and x2 ≠ 0 and x3 ≠ 0
The fermentation process is assumed to operate at a constant volume V , with the dynamics of biomass X , substrate S , and toxin concentration Ct , described by the follows (Zhang and Qin (2007)): Defining the state as x = [ X
S
ub = ⎨
⎪⎩0 otherwise ⎧ 2 x12 − ⎪ ( x 2 + x 2 + x 2 )1/ 2 x1 ≠ 0 and x2 ≠ 0 2 3 ⎪⎪ 1 c u =⎨ and x3 ≠ 0 ⎪0 otherwise ⎪ ⎪⎩ ⎡θ ⎤ ⎢ ⎥ WS ( x ) ⎣ 0 ⎦ uF = + ,
Ct ] , and the input T
u = F / V , the state equations become:
⎡ dX ⎤ ⎢ dt ⎥ μX ⎤ ⎡−X ⎤ ⎢ ⎥ ⎡ dS ⎢ ⎢ ⎥ = −( M + μ / y ) X ⎥ + ⎢ − S ⎥ u ⎥ ⎢ ⎥ ⎢ dt ⎥ ⎢ 1/ 3 qX ⎢ dC ⎥ ⎣⎢ ⎦⎥ ⎣⎢ −Ct ⎦⎥ ⎢ t⎥ ⎣ dt ⎦
(32)
0.005
0.005
In the simulation, taking the weight adaptive law:
Using the data in Table 1, we can find:
⎡ 0.5 x1 ⎤ ⎡ − x1 ⎤ ζ ( x) = ⎢ −1.4 x1 ⎥ , G ( x) = ⎢ − x2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢⎣ 0.6 x11/ 3 ⎥⎦ ⎢⎣ − x3 ⎥⎦
2
2
(33)
11164
∂v W = 2k0 0 S T ( x ) , ∂x ∂v θ = −0.0025θ + k0 0 , ∂x
17th IFAC World Congress (IFAC'08) Seoul, Korea, July 6-11, 2008
and the set D = ⎧⎨ x ∈ R n : v 0 (x ) ≤ 1.6 , 0.5 ≤ k 0 ≤ 1⎫⎬ k0
⎩
⎭
We choose k0 = 0.7 , the control result is shown in Fig 1. From Fig. 1, the three states of systems converge to zero, which indicates that the fault compensation control law is effective. 2
1.5
x1
1
x2
x3
0.5
0
-0.5
-1
0
1
2
3
4
5
6
7
8
9
10
Fig. 1 State responses of systems
5. CONCLUSION This paper presents a fault compensation scheme for a class of uncertain nonlinear systems with actuator faults. The proposed scheme combines an adaptive estimator with an adaptive feedback control law that provides corrective action following faults. An active fault-compensation control law has been developed to ensure the closed-loop stability for a class of nonlinear systems. The partial loss of actuator effectiveness are considered and represented as ( I − Γ)u with the fault parameter. Lyapunov techniques are used to analyze closed-loop stability, and the results are applied to a bioreactor example.
Yang, G., Zhang, S., Lam, J., and Wang, J. (1998), Reliable control using redundant controllers”, IEEE Transactions on Automatic Control, 43, 1588-1593. Chu, D., and Malabre, M. (2002). Numerically reliable design for proportional and derivative state-feedback decoupling controller, 15th Triennial world congress of the international federation of automatic control, Barcelona, pp: 599-604. Veillette, J. R. (1995). Reliable linear-quadratic state feedback control. Automatica, 31, 137-143. Tortora, G., Kouvaritakis, B., and Clarke, D. W. (2002) Simultaneous optimization of tracking performance and accommodation of sensor faults. International Journal of Control, 75, 163-176. Zhang, Y., Wang, F. and Yu, G. (2004) Fault Accommodation of a Class of Nonlinear Systems Using a Learning Approach. ACTA Automatica Sinica. vol. 30, no. 5, pp: 757-762. Idan, M., Johnson, M., Calise, J. A., and Kaneshge, J. (2001). Intelligent aerodynamic /propulsion flight control for flight safety: A nonlinear adaptive approach, proceedings of the American Control conference, pp: 2918-2923. Sastry, S. S., and Isidori, A. (1989). Adaptive control of linearizable systems. IEEE Transactions on Automatic Control, 34, 1123-1131. Wang, L. –X. (1997). A Course in Fuzzy Systems and Control. NJ: Prentice-Hall. Slotine, J. -J. E., and Li, W. (1991). Applied Nonlinear control. NJ: Prentice-Hall. Kim, S. W. (1995). Fuzzy logic based sliding mode control. Fuzzy Set and Systems, 71, 359-367. Zhang, Y., and Qin, S. J. (2007). Adaptive Actuator/Components Fault Compensation for Nonlinear systems. Revised for AIChE J. El-Farra, N. H., Gani, A., Christofides, P. D. (2005) Faulttolerant control of process systems using communication networks, AIChE J., vol. 51, pp. 1665–1682.
REFERENCES Patton, J. R., Frank, P., and Clark, R. (2000). Issues of Fault diagnosis for dynamic systems. London: Spring-Verlag. Zhang, X., Parisini, T., Polycarpou, M.M. (2004). Adaptive fault-tolerant control of nonlinear uncertain systems: An information-based diagnostic approach, IEEE Trans. Autom. Control, vol. 49, pp. 1259–1274. El-Farra, N. H., Gani, A., Christofides, P. D. (2005). Faulttolerant control of process systems using communication networks, AIChE J., vol. 51, pp. 1665–1682. Dunia R., Qin S. J. (1998) Subspace approach to multidimensional fault identification and reconstruction. AIChE J., vol. 44, pp. 1813–1831. Zhang, Y., Zhang, Y., and Zhang, S. (2002). Fuzzy adaptive sliding mode decentralized control for interconnected systems. 15th Triennial world congress of the international federation of automatic control, Barcelona, Spain, 322-326.
11165
Actuator Fault Compensation Control for Nonlinear Systems S. Joe Qin * and Yingwei Zhang
**
*The Mork Family Department of Chemical Engineering and Materials Science, Ming Hsieh Department of Electrical Engineering, Daniel J. Epstein Department of Industrial and Systems Engineering, University of Southern California, 925 Bloom Walk, HED 210, Los Angeles, CA 90089-121190 (email:sqin @usc.edu) **Key Laboratory of Integrated Automation of Process Industry, Ministry of Education , Northeastern University, Shenyang, Liaoning 110004, P. R. China ([email protected]) Abstract: The partial loss of actuator effectiveness is considered for the control of a class of nonlinear processes in the presence of a fault. Not only unmatched uncertainties, but also matched uncertainties are discussed in Lyapunov stability sense. The partial loss of actuator effectiveness is approximated by high order neural networks. Application of the proposed design indicates that the fault compensation control law is effective for a nonlinear fermentation system.
1. INTRODUCTION The study of fault diagnosis and fault-tolerant control has attracted much attention (Patton et al. (2000) and Zhang et al. (2004)), due to the industrial demands for safety and efficiency. For certain processes, it is important not only to detect (and identify) but also to accommodate any faults quickly. Fault-tolerant controls have been developed to keep such processes in control, in spite of the occurrence of a fault. For most control systems the faults can be seen in their actuators, sensors, and systems themselves. The purpose of fault detection and diagnosis (FDD) is to use available inputs, outputs, and operating points of the systems to detect faults. Once the faults are detected, fault diagnosis should be performed so as to locate the fault and also give a good estimation of fault sizes. (Zhang et al. (2002)). In terms of fault-tolerant control, various techniques have been developed which can be roughly classified into passive and active fault-tolerant control schemes. The former is ‘similar’ to robust control while the latter can be regarded as a reinforced nonlinear control in the sense that control structures and parameters are improved so as to realize a reliable operation of concerned closed-loop systems before a scheduled repair is made. (El-Farra et al. (2005)). In an active fault-tolerant control, faults are accommodated, typically by a reconfiguration of the feedback control law. Faults are typically associated with sensors and actuators failures; accordingly, respective compensation strategies can be designed. For example, sensor fault compensations for MIMO systems have been discussed by Tortora et al. (2002).
control. For example, an adaptive compensation method for actuator faults with known plant dynamics has been formulated by Zhang et al. (2004) and Sastry and Isidori (1989); Idan (2001) presents an intelligent fault tolerant flight control system design method that blends aerodynamic and propulsion actuation for safe flight operation in the presence of actuator failures by utilizing redundancy. The actuator and component faults are combined and denoted as a nonlinear function (Zhang and Qin (2007)). The partial loss of actuator effectiveness are considered and represented as ( I − Γ)u with the fault parameter Γ in this paper. Specially, not only unmatched uncertainties, but also matched uncertainties are discussed. Based on neural networks which are capable of approximating, with arbitrary accuracy, any real continuous function on a compact set, the partial loss of actuator effectiveness is approximated by the neural networks. Then an adaptive compensation control law is formulated to ensure the system stability. The remainder of the paper is organized as follows. The problem statement and its assumptions are given in section 2, followed by the formulation of our controller and its relevant proofs in section 3. An illustrative example is given in section 4 to demonstrate the effectiveness of the proposed method. Finally, conclusions are drawn in section 5. 2. PROBLEM STATEMENT The following actuator fault model is adopted in this paper (Veillette 1995): i = 1, " , m ， j = 1, " , L
Adaptive approaches have also been used in fault tolerant
978-3-902661-00-5/08/$20.00 © 2008 IFAC
11160
uijC = (1 − Γ ij )ui (t ), 0 ≤ Γ ij ≤ Γ ij ≤ Γ ij ≤ 1
(1)
10.3182/20080706-5-KR-1001.0604
17th IFAC World Congress (IFAC'08) Seoul, Korea, July 6-11, 2008
⎧0 if t < T β i (t − T ) = ⎨ , i = 1, 2," , m , ⎩1 if t ≥ T
where Γ ij is an unknown constant. Here, the index j denotes the j th fault mode and L is the total fault modes. Let u
C ij
represent the signal from the i th actuator that has failed in the j th fault mode. For every fault mode, Γi and Γi j
j
represent the lower and upper bounds of Γ ij , respectively. Note that, when Γi = Γi =0, there is no fault for the i th j
j
actuator ui in the j th fault mode. When Γi = Γi =0=1, the j
j
where T is the fault occurrence time. The problem considered is as follows: Fault compensation (FC) problem: Given system (1), design a control u N for the normal system, and an additional control
uF for fault compensation, so that u = u N + uF as the new
i th actuator ui is outage in the j th fault mode. When
control after the occurrence of a fault can guarantee the resulted closed-loop nonlinear system to be stable.
0 ≤ Γi ≤ ρ i ≤ Γi ≤ 1 , in the j th fault mode the type of
The following assumptions are used.
actuator faults is loss of effectiveness. Denote
Assumption 1: There exists u = u ( x ) and Lyapunov
j
j
j
a
function V ( x ) , such that
C uijC = [u1Cj , u2Cj , " , umj ] = ( I − Γ j )u (t )
2
where
Γ = diag [Γ1 , Γ 2 , " , Γ m ] j
j
j
j
j = 1, 2, " , L
,
Considering the lower and upper bounds ( Γi , Γi ) , the j
j
∂V ( x ) ∂x
following set can be defined j
j
j 1
j 2
(6)
T
(2) ρ ( x)
where T
x = ζ ( x ) + Δζ ( x) + G ( x)[( I − β (t − T )Γ )u + Δg ( x)] (3) where x ∈ R , u ∈ R are the state and input of the system, respectively, Δζ ( x ) and Δg ( x ) are the model uncertainty in the normal operation. The normal system, in the absence of any faults, is described by m
G ( x)
is continuous, ξ (•) and ρ (•) are
( ) ∂V ( x ) ∂x
known and continuous. Remark 1: From Assumption 2, we have ρ ( x ) = 0 , if
⎛ ∂V ( x) ⎞ ⎟=0. ⎝ ∂x ⎠
G T ( x) ⎜
(4)
The fault matrix Γ is multiplied by a switching function β (t − T ) ,
where
(5)
⎛ ∂V ( x) ⎞ ⎜ ⎟ Δζ ( x) ≤ ρ ( x ) ⎝ ∂x ⎠
Consider a system described as:
β (t − T ) = diag ( β1 (t − T ), β 2 (t − T ), " , β m (t − T ))
∂x
Δg ( x ) ≤ ξ ( x )
where Γ can be described by
x = ζ ( x ) + Δζ ( x ) + G ( x )[u + Δg ( x )]
2
Assumption 2: For system (1)
L , the uniform actuator fault model
Γ = diag[Γ1 , Γ 2 ," , Γ m ] .
∂V ( x )
where k1 , k 2 , k3 , k4 are positive constants.
u F = ( I − Γ)u , Γ ∈ {Γ1 , Γ 2 ," , Γ L }
n
)
( x ) ≤ − k3
j m
Γ ij = Γ ij or Γ ij = Γ ij } For all fault modes
( ζ ( x ) + G ( x )u
a
≤ − k 4V ( x)
N ρ = {Γ Γ = diag[Γ , Γ , " , Γ ], j
2
k1 x ≤ V ( x ) ≤ k 2 x ,
.
3. FAULT COMPENSATION The partial loss of actuator effectiveness can be approximated by the neural networks. Then system (3) can be rewritten as:
x = ζ ( x ) + Δζ ( x ) + G ( x )[u + W * S ( x ) + ε ( x ) + Δg ( x )]
11161
(7)
17th IFAC World Congress (IFAC'08) Seoul, Korea, July 6-11, 2008
where, ε ( x ) = f ( x ) − W S ( x ) ≤ ε is the estimation error. If *
with the following adaptive weight update law
we denote W as the estimate of the uncertain weight matrix W * , then
⎧ ∂v0 T ⎪⎪ 2k0 ∂x S ( x ) if W < M W W = ⎨ ⎪ − β W + 2k ∂v0 S T ( x ) if W ≥ M 0 W ⎪⎩ ∂x ∂v θ = −γ 1θ + k0 0 ∂x
( x) + WS ( x) + x = ζ ( x ) + Δζ ( x ) + G ( x )[u − WS
ε ( x) + Δg ( x)] (8) where W = W − W * and it has the appropriate dimension. S ( x ) is a vector with S i ( x ) , i = 1, 2, " , L . Here, Si ( x) =
∏
lj (i )
j∈J i
[ s ( x j )]
with J i , i = 1, 2, " , L and l j (i ) is
nonnegative integers. Because W * is bounded, we assume W
*
< MW
.
Theorem 1: Under Assumptions 1 and 2, we can design a controller in the form of the following:
u = uN + uF
(9)
uN = u a + u b + u c where u N is the control law of “healthy” systems (4) and
as in the proof. The novelty of this paper is that fault compensation controller is clearly given after faults occur. The main contributions of this paper are the design of the corrective control law of nonlinear system with unmatched and matched uncertainties, and the stability analysis of the closed-loop system in the presence of the fault modeling errors is proposed. Proof: step 1 Substituting the controller Eqs. (9-12) into system (1), we have:
x = ζ ( x) + Δζ ( x) + G ( x)[u a + u b + u c + Δg ( x)]
Assumption 1, and let
∂V ( x) ⎧ ⎫ G ( x )T = 0 ⎬, ∂x ⎩ ⎭ ∂ V ( x ) ⎧ GT ( x) ⎪ ∂x ξ x , x ∉ E ( ) ⎪− b ∂V ( x ) , u =⎨ T G ( x) ⎪ ∂x ⎪ 0, x ∈ E ⎩
Define a positive function v0 ( x ) = V ( x ) , then we have:
E=⎨ x
∂V ( x) ⎧ T ⎪ G ( x) ∂x ρ ( x) , x ∉ E ⎪− 2 ∂ ( ) V x , uc = ⎨ G T ( x) ⎪ ∂x ⎪ 0, x ∈ E ⎩ uF =
WS ( x)u N 2
+
v0 ( x ) = ( (10)
⎧ ⎩
)T (ζ ( x ) + G ( x )u a ) +
∂x ∂V ( x ) T ( ) G ( x )(u b + Δg ( x )) + ∂x ∂V ( x ) T ( ) ( Δζ ( x ) + G ( x )u c ) ∂x
From Assumption 1, we have (11)
(
2
(12).
∂V ( x ) ∂x
)T (ζ ( x ) + G ( x )u a ) ≤ − k3
∂V ( x )
have
(
Where Θ ∈ R and Θ = [θ , 0, " , 0] . Then, the state x is ultimately consistently bounded by the set:
D = ⎨ x ∈ R n : v0 ( x ) ≤
∂V ( x )
∂x
2
(16)
From Assumption 2 and the structure of u b ( x) , we
Θ uN
λ[1 + G ( x) ] λ1[1 + G ( x) ] n× L
(15)
The parameters of λ , λ1 , k1 , k 2 , α , μ can be determined
u a is given by
uF is the fault compensation. And
(14)
T
⎫ μ k2 , ≤ k0 ≤ 1⎬ , k0α k1 ⎭
∂V ( x ) ∂x
(G T ( x ) (13)
)T G ( x )(u b + Δg ( x )) =
∂V ( x ) ∂xi
when x ∈ E , and
11162
)T (u b + Δg ( x )) = 0
17th IFAC World Congress (IFAC'08) Seoul, Korea, July 6-11, 2008
(
∂V ( x ) ∂x
=(
Proof: step 2:
) G ( x )(u + Δg ( x )) T
∂V ( x ) ∂x
b
T
∂V ( x )
T
∂x ξ ( x ) + Δg ( x )) ∂V ( x )
G ( x) ) G ( x )( − T
G ( x)
∂V ( x )
= − G T ( x )(
∂x ∂V ( x )
≤ − G ( x )( T
∂x
Define a Lyapunov function for system (1) of the following form:
1 1 V ( x, W , θ ) = k0 v0 ( x ) + tr W T W + θ 2 2 2
{
∂x
∂V ( x )
)T ξ ( x ) + (
∂x
) ξ ( x) + ( T
with θ = θ − ε , then the derivatives of V is
∂v a b c V = k0 0 { f ( x ) + Δf ( x ) + G ( x )[u + u + u + ∂x
) G ( x ) Δg ( x ) T
≤0 ,
Δg ( x )]} + k0
when x ∉ E . Hence (
∂V ( x ) ∂x
) G ( x )(u + Δg ( x )) ≤ 0 T
(20)
)T G ( x ) Δg ( x )
∂V ( x ) ∂x
}
b
k
(17)
∂v0
∂v0 ∂x
G ( x )u − k 0 F
∂v0
G ( x )WS ( x ) + k0
∂x T tr W W + θθ
{
From Assumption 2 and structure of u c (x) , we have
∂x
}
∂v0
( x) + G ( x )WS
∂x
G ( x )ε ( x ) +
(21) (
∂V ( x ) ∂x
=(
Using (14), we obtain ) ( Δζ ( x ) + G ( x )u ) T
∂V ( x ) ∂x
≤ (
c
) Δζ ( x ) + ( G ( x ) T
∂V ( x ) ∂x
T
∂V ( x ) ∂xi
T
) u
∂v ∂v V = k0 v0 + k0 0 G ( x)u F + k0 0 G ( x)WS ( x ) ∂x ∂x ∂v0 + k0 G ( x)ε ( x ) − β IW tr W T W + θθ ∂x
c
{
) Δζ ( x ) ≤ ρ ( x ) = 0 , T
}
where IW is the indicator function of W , and it satisfies
when x ∈ E , and (
∂V ( x )
=(
∂x
) ( Δζ ( x ) + G ( x )u ) T
∂V ( x )
uF =
⎧1 if W ≥ M W IW = ⎨ ⎩0 if W < M W
∂x
c
Choosing
) Δζ ( x ) T
WS ( x ) 2
λ [1 + G ( x ) ]
+
2
λ1 ≥
where Θ ∈ R and Θ = [θ , 0, " , 0] . When x ∉ E . Hence
∂x
2k 2 β − sk0
λ1 [1 + G ( x ) ] T
∂V ( x )
k0 s
λ≥
Θ
n
(
(22)
) T ( Δζ ( x ) + G ( x ) u c ) ≤ 0
(18)
β>
,
k0 2k0 k 2 γ 1 − k0 s 2 k02 2k 2
(23)
,
,
(24)
(25)
Thus, we obtain v0 ( x ) ≤ − k3
∂V ( x )
γ1 >
2
∂x From (19), the stability of the normal system is proven.
(19)
k0 2k 2
By using Eq. (5) from Zhang and Qin (2007) we have
11163
(26)
17th IFAC World Congress (IFAC'08) Seoul, Korea, July 6-11, 2008
2 kk k β γ V ≤ − 0 3 4 v0 ( x ) − W ξ 2 − 1 θ 2ξ 2 + 2 2 k3
2 β M W2 +
γ1 2
(27)
. The parameters of y , q, M and V are given in Table 1 for the process.
Table 1: Fermentation model parameters
ε 2ξ 2
⎧k k ⎫ therefore V ≤ −αV + μ , with α = min ⎨ 3 4 , β , γ 1 ⎬ , ⎩ k3 ⎭ μ = 2β M + 2 W
γ1 2
ε
2
(28)
Integrating both sides of (36) yields V (t ) ≤
μ
⎡ α ⎢⎣
+ V ( 0) −
Volume Constant Constant Toxin production constant
μ⎤
−α t e , ∀t ≥ 0 ⎥ α⎦
(29)
Due to (27), it can be deduced that x, W ( x ) , θ ( x ) are bounded consistently. From (20), we have
k0 v0 ( x ) ≤ V Therefore,
y M
q
200[l] 0.417 0.0196 0.0296[l/h(g/l)2/3
The possible fault modes considered are once one of the actuator is outage, the maximum loss of effectiveness of the actuator is 80%. The fault case here is that at the beginning, the first actuator becomes loss of effectiveness of 70%. The main contribution of the paper is theoretical in nature. In the
(30)
V
fermentation
example,
let
⎡ θ1x1x2e x2 ⎤ ⎢ ⎥ Δg ( x) = ⎢2 x22e x2 sin θ 2 ⎥ , ⎢ ⎥ x1 ⎣⎢ θ 3 x1e ⎦⎥
⎡θ 2 x12 cosθ1 ⎤ ⎢ ⎥ Δζ ( x) = ⎢ x12 sin θ 2 ⎥ , θ1 ∈ (−2, 2) and θ 2 ,θ 3 ∈ (−1, 1) are the ⎢ 2 ⎥ ⎢⎣ θ 3 x1 ⎥⎦
(31)
uncertainty parameters. In this example, a radial basis function (RBF) network is chosen to represent the dynamic changes after the fault occurrence, with 10 hidden nodes and 10 centers that are distributed uniformly in region [-1,1].
The above completes the proof that x is ultimately consistently bounded by the set D .
Choose ξ ( x) = 2 x e x , ρ ( x) = 2 x12 , v0 = xT x = x . Then the
v0 ( x ) ≤
μ k 0α
+
1 ⎡ μ ⎤ −α t V ( 0) − e , ∀t ≥ 0. ⎢ α ⎥⎦ k0 ⎣
control input is:
u a = −0.4 x12 / 3 + 0.9 x2
4. ILLUSTRATION EXAMPLE
⎧⎪−2 x 2 e x x1 ≠ 0 and x2 ≠ 0 and x3 ≠ 0
The fermentation process is assumed to operate at a constant volume V , with the dynamics of biomass X , substrate S , and toxin concentration Ct , described by the follows (Zhang and Qin (2007)): Defining the state as x = [ X
S
ub = ⎨
⎪⎩0 otherwise ⎧ 2 x12 − ⎪ ( x 2 + x 2 + x 2 )1/ 2 x1 ≠ 0 and x2 ≠ 0 2 3 ⎪⎪ 1 c u =⎨ and x3 ≠ 0 ⎪0 otherwise ⎪ ⎪⎩ ⎡θ ⎤ ⎢ ⎥ WS ( x ) ⎣ 0 ⎦ uF = + ,
Ct ] , and the input T
u = F / V , the state equations become:
⎡ dX ⎤ ⎢ dt ⎥ μX ⎤ ⎡−X ⎤ ⎢ ⎥ ⎡ dS ⎢ ⎢ ⎥ = −( M + μ / y ) X ⎥ + ⎢ − S ⎥ u ⎥ ⎢ ⎥ ⎢ dt ⎥ ⎢ 1/ 3 qX ⎢ dC ⎥ ⎣⎢ ⎦⎥ ⎣⎢ −Ct ⎦⎥ ⎢ t⎥ ⎣ dt ⎦
(32)
0.005
0.005
In the simulation, taking the weight adaptive law:
Using the data in Table 1, we can find:
⎡ 0.5 x1 ⎤ ⎡ − x1 ⎤ ζ ( x) = ⎢ −1.4 x1 ⎥ , G ( x) = ⎢ − x2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢⎣ 0.6 x11/ 3 ⎥⎦ ⎢⎣ − x3 ⎥⎦
2
2
(33)
11164
∂v W = 2k0 0 S T ( x ) , ∂x ∂v θ = −0.0025θ + k0 0 , ∂x
17th IFAC World Congress (IFAC'08) Seoul, Korea, July 6-11, 2008
and the set D = ⎧⎨ x ∈ R n : v 0 (x ) ≤ 1.6 , 0.5 ≤ k 0 ≤ 1⎫⎬ k0
⎩
⎭
We choose k0 = 0.7 , the control result is shown in Fig 1. From Fig. 1, the three states of systems converge to zero, which indicates that the fault compensation control law is effective. 2
1.5
x1
1
x2
x3
0.5
0
-0.5
-1
0
1
2
3
4
5
6
7
8
9
10
Fig. 1 State responses of systems
5. CONCLUSION This paper presents a fault compensation scheme for a class of uncertain nonlinear systems with actuator faults. The proposed scheme combines an adaptive estimator with an adaptive feedback control law that provides corrective action following faults. An active fault-compensation control law has been developed to ensure the closed-loop stability for a class of nonlinear systems. The partial loss of actuator effectiveness are considered and represented as ( I − Γ)u with the fault parameter. Lyapunov techniques are used to analyze closed-loop stability, and the results are applied to a bioreactor example.
Yang, G., Zhang, S., Lam, J., and Wang, J. (1998), Reliable control using redundant controllers”, IEEE Transactions on Automatic Control, 43, 1588-1593. Chu, D., and Malabre, M. (2002). Numerically reliable design for proportional and derivative state-feedback decoupling controller, 15th Triennial world congress of the international federation of automatic control, Barcelona, pp: 599-604. Veillette, J. R. (1995). Reliable linear-quadratic state feedback control. Automatica, 31, 137-143. Tortora, G., Kouvaritakis, B., and Clarke, D. W. (2002) Simultaneous optimization of tracking performance and accommodation of sensor faults. International Journal of Control, 75, 163-176. Zhang, Y., Wang, F. and Yu, G. (2004) Fault Accommodation of a Class of Nonlinear Systems Using a Learning Approach. ACTA Automatica Sinica. vol. 30, no. 5, pp: 757-762. Idan, M., Johnson, M., Calise, J. A., and Kaneshge, J. (2001). Intelligent aerodynamic /propulsion flight control for flight safety: A nonlinear adaptive approach, proceedings of the American Control conference, pp: 2918-2923. Sastry, S. S., and Isidori, A. (1989). Adaptive control of linearizable systems. IEEE Transactions on Automatic Control, 34, 1123-1131. Wang, L. –X. (1997). A Course in Fuzzy Systems and Control. NJ: Prentice-Hall. Slotine, J. -J. E., and Li, W. (1991). Applied Nonlinear control. NJ: Prentice-Hall. Kim, S. W. (1995). Fuzzy logic based sliding mode control. Fuzzy Set and Systems, 71, 359-367. Zhang, Y., and Qin, S. J. (2007). Adaptive Actuator/Components Fault Compensation for Nonlinear systems. Revised for AIChE J. El-Farra, N. H., Gani, A., Christofides, P. D. (2005) Faulttolerant control of process systems using communication networks, AIChE J., vol. 51, pp. 1665–1682.
REFERENCES Patton, J. R., Frank, P., and Clark, R. (2000). Issues of Fault diagnosis for dynamic systems. London: Spring-Verlag. Zhang, X., Parisini, T., Polycarpou, M.M. (2004). Adaptive fault-tolerant control of nonlinear uncertain systems: An information-based diagnostic approach, IEEE Trans. Autom. Control, vol. 49, pp. 1259–1274. El-Farra, N. H., Gani, A., Christofides, P. D. (2005). Faulttolerant control of process systems using communication networks, AIChE J., vol. 51, pp. 1665–1682. Dunia R., Qin S. J. (1998) Subspace approach to multidimensional fault identification and reconstruction. AIChE J., vol. 44, pp. 1813–1831. Zhang, Y., Zhang, Y., and Zhang, S. (2002). Fuzzy adaptive sliding mode decentralized control for interconnected systems. 15th Triennial world congress of the international federation of automatic control, Barcelona, Spain, 322-326.
11165