Robust H-Infinity Actuator Fault Diagnosis with Neural Network
Robust H∞ actuator fault diagnosis with neural network Marcel Luzar, Marcin Witczak, Piotr Witczak Institute of Control and Computation Engineering University of Zielona G´ora ul. Podg´orna 50, 65–246 Zielona G´ora, Poland Email: {M.Luzar, M.Witczak}@issi.uz.zgora.pl, [email protected] Abstract— The paper deals with the problem of a robust actuator fault diagnosis for Linear Parameter-Varying (LPV) systems with Recurrent Neural-Network (RNN). The preliminary part of the paper describes the derivation of a discrete-time polytopic LPV model with RNN. Subsequently, a robust fault detection, isolation and identification scheme is developed, which is based on the observer and H∞ framework for a class of nonlinear systems. The proposed approach is designed in such a way that a prescribed disturbance attenuation level is achieved with respect to the actuator fault estimation error while guaranteeing the convergence of the observer.
II. D ERIVATION
OF A DISCRETE - TIME POLYTOPIC MODEL
LPV
The goal of this section is to present a neural state-space model [10] that can be represent a general class of state-space models and can be easily transformed into a LPV model. The transformation method is derived by Lachhab et. all and presented in [11]. A. Recurrent Neural-Network Topology Let us consider the following discrete-time LPV model:
I. I NTRODUCTION The problem of fault diagnosis (FD) of non-linear industrial systems [1], [2] has received considerable attention during the last three decades. Indeed, it developed from the art of designing a satisfactory performing systems into the modern theory and practice that it is today. Within the usual framework, the system being diagnosed is divided into three main components, i.e. plant (or system dynamics [3]), actuators and sensors. The paper deals with the problem of full fault diagnosis of sensors, i.e. apart from the usual two steps consisting of fault detection and isolation (FDI), the fault identification is also performed. This last step is especially important from the viewpoint of Fault-Tolerant Control (FTC) [4], which is possible if and only if there is an information about the size of the fault being a result of fault identification (or fault estimation). In this paper a robust fault estimation approach is proposed, which can be efficiently applied to realise the above-mentioned procedure. The proposed approach can be perceived as a combination of the linear-system strategies [5] and [6] for a class of non-linear systems [7] and it is based on state observers [8], [9]. The proposed approach is designed in such a way that a prescribed disturbance attenuation level is achieved with respect to the fault estimation error while guaranteeing the convergence of the observer. The paper is organised as follows. On the beginning, the derivation of a polytopic LPV model with neural network is described. Detailed design procedure of the robust observer for actuator fault diagnosis is presented in Section III. In Section IV experimental results obtained with laboratory nonlinear system are shown. Finally, Section V conclude the paper.
978-1-4673-5508-7/13/$31.00 ©2013 IEEE
xk+1 = A(θk )xk + B(θk )uk ,
(1)
y k+1 = C(θk )xk+1 ,
(2)
where A(θk ), B(θk ), C(θk ) are continuous mappings and θk is a time-varying parameter. The dependence of the A,B,C on θk represents a general LPV model; imposing that any of these matrices is parameter independent, i.e. fixed, will restrict the generality of the LPV model. The latter is indeed the case when RNN is transformed into the LPV model: A is parameter dependent but B and C are fixed. A general form of statespace neural network model is xk+1 = Axk + Buk + A1 σ(E 1 xk ) + B 1 σ(E 2 uk ),
(3)
y k+1 = Cxk+1 + C 1 σ(E 3 xk+1 ),
(4)
where x ∈ Rn denotes the state vector, y ∈ Rp the output and u ∈ Rm the input vector. A, A1 , B, B 1 , C, C 1 , E 1 , E 2 and E 3 are real valued matrices of appropriate dimensions and represent the weights which will be adjusted during the training stage of the RNN. The non-linear activation function σ(·), which is applied elementwise in (3)–(4) is taken as a continuous, differentiable and bounded function. This RNN leads to a general form of the neural state-space model in the sense that if it is transformed into an LPV model in the form (1)–(2), the matrices A,B and C will be parameter dependent. For stability and identifiability proofs of the proposed RNN reader is refereed to [11]. B. Simplification and Assumptions The Recurrent Neural-Network (RNN) model described by (3)–(4) can be simplified by removing any of the sigmoidal layers. This can be done according to a priori information
200
about the identified system. The LPV model (1)–(2) of multitank system, which was derived from a physical laboratory model and used in experiments presented in Section IV, has only matrix A parameter dependent. Thus, a specific transformation proposed by [12] is used to simplify such a model. Removing the sigmoidal layers from the input and the output paths means that the resulting LPV model has parameter dependance in matrix A only. Moreover, for a practical implementation this simplified RNN is modified as shown in Fig. 1: the outputs instead of the states are taken as input to sigmoidal layer. This modification facilities the implementation of LPV controllers designed based on this model. The modified RNN is represented as xk+1 = Axk + Buk + A1 σ(E 1 Cxk ),
(5)
y k+1 = Cxk+1 .
(6)
Multiplying (8) by H, and then substituting (7), it can be shown that f a,k = = H(y k+1 − CAxk − CBuk −
(11)
Cg (xk ) − CW 1 wk − W 2 wk+1 ).
Finally, by substituting (11) into (7) it can be shown that: xk+1 = ¯ ¯ = Axk + Buk + Gg (xk ) + ¯ k+1 + GW 1 w k − LW ¯ 2 wk+1 , Ly
(12)
¯ = GA, B ¯ = GB, L ¯ = La H. where G = (I n −La HC), A In order to estimate (11), i.e. to obtain fˆ k it is necessary ˆ k . Conseto estimate the state of the system, i.e. to obtain x quently, the fault estimate is given as follows fˆ a,k = H(y k+1 − CAˆ xk − CBuk − Cg (ˆ xk )).
(13)
The corresponding observer structure is
Fig. 1.
¯ xk + Bu ¯ k + Gg (ˆ ¯ k+1 + K(y k − C x ˆ k+1 = Aˆ ˆ k ), x xk ) + Ly (14) while the state estimation error is given by ¯ − KC ek + Gsk + ek+1 = A ¯ 2 w k+1 = (GW 1 − KW 2 )w k − LW (15) ¯ ¯ = A1 ek + Gsk + W 1 wk + W 2 wk+1 ,
Simplified state-space recurrent neural network
where III. ACTUATOR FAULT
sk = g (xk ) − g (ˆ xk ) .
DIAGNOSIS
The main objective of this section is to provide a detailed design procedure of the robust observer, which can be used for actuator fault diagnosis. In other words, the main role of this observer is to provided the information about the actuator fault. Indeed, apart from serving as a usual residual generator (see, e.g., [3]), the observer should be designed in such a way that a prescribed disturbance attenuation level is achieved with respect to the actuator fault estimation error while guaranteeing the convergence of the observer. Neural model described by (5)–(6) can be modify to following state-space form: xk+1 = Axk + Buk + g (xk ) + La f a,k + W 1 w k ,
(7)
y k+1 = Cxk+1 + W 2 wk+1 ,
(8)
n
Similarly, the fault estimation error εfa ,k can be defined εfa ,k = f a,k − fˆ a,k =
= −HC (Aek + sk + W 1 w k ) − HW 2 wk+1 .
rank(CLa ) = rank(La ) = s Under the assumption (9) it is possible to calculate −1 H = (CLa )+ = (CLa )T CLa (CLa )T .
(17)
Noth that both ek and εfa ,k are non-linear with respect to ek . To settle this problem within the framework of this paper, the following solution is proposed. Using the Differential Mean Value Theorem (DMVT) [13], it can be shown that
with
g (a) − g (b) = M x (a − b),
(18)
∂g1 ∂x (c1 ) .. , Mx = . ∂g n (cn ) ∂x
(19)
r
where xk ∈ X ⊂ R is the state vector, uk ∈ R stands for the input, y k ∈ Rm denotes the output, f a,k ∈ Rm stands for the actuator fault. While , w k ∈ Rn is a an exogenous disturbance vector with W 1 ∈ Rn×n , W 2 ∈ Rm×n being its distribution matrices. Following [3], [5], let us assume that the system is observable and the following rank condition is satisfied:
(16)
where c1 , . . . , cn ∈ Co(a, b), ci 6= a, ci 6= b, i = 1, . . . , n. Assuming that a ¯i,j ≥
(9)
∂gi ≥ ai,j , ∂xj
i = 1, . . . , n,
j = 1, . . . , n,
(20)
n o M ∈ Rn×n |¯ ai,j ≥ mx,i,j ≥ ai,j , i, j = 1, . . . , n,
(21)
it is clear that: (10)
201
Mx =
Thus, using (18), the term A1 ek + Gsk in (15) can be written as
J = −V0 +
¯ + GM x,k − KC)ek A1 ek + Gsk = (A
(22)
where M x,k ∈ Mx . From (22), it can be deduced that the state estimation error can be converted into an equivalent form ¯ 1 wk + W ¯ 2 w k+1 , ek+1 = A2 (α)ek + W ˜ A2 (α) = A(α) − KC,
˜= A
˜ A(α) :
˜ A(α) =
N X
˜ i, αi A
N X
)
αi = 1, αi ≥ 0
i=1
i=1
∞ X
2 εT fa ,k εfa ,k − µ
k=0
∞ X
2 wT k wk − µ
k=0
∞ X
wT k+1 w k+1 < 0.
k=0
µ2
∞ X
w Tk+1 w k+1 = µ2
k=0
∞ X
k=0
wTk wk − µ2 wT0 w 0 ,
(24)
,
J = −V0 +
∞ X
k=0
εTfa ,k εfa ,k − 2µ2
∞ X
wTk w k + µ2 w T0 w0 < 0.
k=0
(35)
2
Knowing that V0 = 0 for e0 = 0, (35) leads to (29) with √ ω = 2µ. Since the general framework for designing the robust observer is given, then the following form of the Lyapunov function is proposed [13]:
εfa ,k = −HC (A3 (α)ek + W 1 w k ) − HW 2 wk+1 , (25) with A3 =
A3 (α) :
A3 (α) =
N X
αi A3,i ,
i=1
N X
)
αi = 1, αi ≥ 0
i=1
.
(26)
The objective of further deliberations is to design the observer (14) in such a way that the state estimation error ek is asymptotically convergent and the following upper bound is guaranteed kεf kl2 ≤ ωkwkl2 (27) where ω > 0 is a prescribed disturbance attenuation level. Thus, on the contrary to the approaches presented in the literature, µ should be achieved with respect to the fault estimation error but not the state estimation error. Thus, the problem of H∞ observer design [15] is to determine the gain matrix K such that lim ek = 0 for wk = 0,
(28)
k→∞
kεf kl2 ≤ ωkwkl2
for wk 6= 0, e0 = 0.
(29)
In order to settle the above problem it is sufficient to find a Lyapunov function Vk such that: ∆Vk + εTfa ,k εfa ,k − µ2 w Tk wk − µ2 wTk+1 wk+1 < 0,
k = 0, . . . ∞,
(30)
where ∆Vk = Vk+1 − Vk , µ > 0. Indeed, if wk = 0, (k = 0, . . . , ∞) then (30) boils down to ∆Vk +
εTfa ,k εfa ,k
< 0, k = 0, . . . ∞,
(31)
and hence ∆Vk < 0, which leads to (28). If wk 6= 0 (k = 0, . . . , ∞) then (30) yields J=
(34)
inequality (33) can be written as
where N = 2n . Note that this is a general description, which does not take into account that some elements of M x,k maybe 2 constant. In such cases, N is given by N = 2(n−c) where c stands for the number of constant elements of M x,k . In a similar fashion, (17) can be converted into
(
(33)
Bearing in mind that
(23)
which defines an LPV polytopic system [14] with (
which can be written as
∞ X 2 T 2 T ∆Vk + εT fa ,k εfa ,k − µ w k w k − µ w k+1 w k+1 < 0,
Vk = eTk P (α)ek ,
(36)
where P (α) ≻ 0. On the contrary to the design approach presented in the literature (see, e.g. [15]) and the references therein) it is not assumed that P (α) = P is constant. Indeed, P (α) can be perceived as a parameter-depended matrix of the form (cf. [14]) P (α) =
N X
αi P i .
(37)
i=1
As a consequence: ∆Vk + εTfa ,k εfa ,k − µ2 w Tk wk − µ2 wTk+1 w k+1 =
eTk A2 (α)T P (α)A2 (α) + A3 (α)T H 1 A3 (α) − P (α) ek + ¯ 1 + A3 (α)T H 1 W 1 wk + eTk A2 (α)T P (α)W ¯ 2 + A3 (α)T H 2 wk+1 + eTk A2 (α)T P (α)W ¯ T1 P (α)A2 (α) + W T1 H 1 A3 (α) ek + w Tk W ¯ T P (α)W ¯ 1 + W T H 1 W 1 − µ2 I w k + w Tk W 1 1 ¯ T P (α)W 2 + W T H 2 w k+1 + w Tk W 1 1 T T ¯ 2 P (α)A2,k + H T2 A3 (α) ek + w k+1 W ¯ T P (α)W 1 + H T W 1 wk + w Tk+1 W 2 2 ¯ T P (α)W ¯ 2 + W T H T HW 2 − µ2 I wk+1 < 0. w Tk+1 W 2 2 with H 1 = C T H T HC and H 2 = C T H T HW 2 . By defining T v k = eTk , w Tk , wTk+1 ,
(38)
inequality (38) becomes
2 T 2 T T ∆Vk + εT fa ,k εfa ,k − µ w k wk − µ w k+1 w k+1 = v k M V v k < 0,
k=0
(32)
202
(39)
where M V is given by (40). M V = [M V1
M V1 M V2 M V3
=
=
=
M V3 ]
M V2
(40)
A2 (α)T P (α)A2 (α) + A3 (α)T H 1 A3 (α) − P (α) T ¯ 1 P (α)A2 (α) + W T H 1 A3 (α) , W 1 T ¯ T W P (α)A (α) + H A (α) 2 3 2 2 ¯ 1 + A3 (α)T H 1 W 1 A2 (α)T P (α)W T 2 ¯T ¯ , W 1 P (α)W 1 + W 1 H 1 W 1 − µ I T T ¯ W 2 P (α)W 1 + H 2 W 1 ¯ 2 + A3 (α)T H 2 A2 (α)T P (α)W T T ¯ . W 1 P (α)W 2 + W 1 H 2 ¯ T P (α)W ¯ 2 + W T H T HW 2 − µ2 I W 2 2
V (α)T X(α)V (α) − W (α) ≺ 0, ii)
The following theorem constitutes the main result of this section: Theorem 1: For a prescribed disturbance attenuation level µ > 0 for the fault estimation error (17), the H∞ observer design problem for the system (7)–(8) and the observer (14) is solvable if there exists matrices P i ≻ 0 (i = 1, . . . , N ), U and N such that the following LMIs are satisfied: Z2] ≺ 0
[Z 1
Z1 =
AT 3,i H 1 A3,i
− Pi
W T1 H 1 A3,i H T2 A3,i U A2,i
AT3,i H 1 W 1 T W 1 H 1 W 1 − µ2 I H T2 W 1 ¯1 UW
AT 3,i H 3
W T1 H 2 Z2 = T W 2 H T HW 2 − µ2 I ¯2 UW i = 1, . . . , N.
,
A2,i U T ¯ T UT W 1 ¯ T UT W 2 P i − U − UT
T
V XV − W ≺ 0
V T U T (U T + U − X)−1 U V − W ≺ 0. Substituting U = U
T
Thus, (i) implies (ii). h Multiplying (46) by T = I
,
(47)
(48)
i V T on the left and by T T on
A2 (α)T ¯T W P (α) A2 (α) 1 ¯T W 2
¯1 W
¯2 W
+
A3 (α)T H 1 A3 (α) − P (α) A3 (α)T H 1 W 1 T W 1 H 1 A3 (α) W T1 H 1 W 1 − µ2 I T H 2 A3 (α) H T2 W 1 T A3 (α) H 3 ≺ 0. W T1 H 2 T W 2 H T HW 2 − µ2 I (52)
and then applying Lemma 2 and (51) leads to (41), which completes the proof. Finally, the design procedure boils down to solving LMIs (41) and then (cf. (43)–(44)) K = U −1 N . It can be also observed that the observer design problem can be treated as an minimization task, i.e. µ∗ =
min
µ>0,P 1 ≻0,U ,N
µ
(53)
under (41).
(45)
= X yield
V T XV − W ≺ 0.
Subsequently, observing that the matrix (40) must be negative definite and writing it as
There exists X ≻ 0 such that −W V T UT ≺ 0. (46) UV X − U − UT Proof: Applying the Schur complement to (ii) gives
(50)
Proof: The proof can be realised by following the same line of reasoning as the one of Lemma 1. It is easy to show that that (50) is satisfied if there exist matrices X i ≻ 0 such that −W i V Ti U T ≺ 0, i = 1, . . . , N. (51) UV i Xi − U − UT
(42)
˜ i − KC) = U A ˜ i − N C, U A2,i = U (A (43) ¯ U W 1 = U (GW 1 − KW 2 ) = U GW 1 − N W 2 . (44) Proof: The following two lemmas can be perceived as the generalisation of those presented in [14]. Lemma 1: The following statements are equivalent i) There exists X ≻ 0 such that
(49)
There exists X(α) ≻ 0 such that −W (α) V (α)T U T ≺ 0. U V (α) X(α) − U − U T
(41)
where (cf. (24) and (15))
ii)
the left of (46) gives (45), which means that (ii) implies (i) and hence the proof is completed. Lemma 2: The following statements are equivalent i) i) There exists X(α) ≻ 0 such that
IV. E XPERIMENTAL RESULTS To verify proposed approach, a sensor fault detection was implemented for multitank system. The considered multi-tank system (Fig. 2) is designed for simulating the real industrial multi-tank system in the laboratory conditions [16]. The multitank system can be efficiently used to practically verify both linear and non-linear control, identification and diagnostics methods. The multi-tank system consists of three separate tanks placed each above other and equipped with drain valves and level sensors based on a hydraulic pressure measurement. Each of them has a different cross-section in order to reflect system nonlinearities. The lower bottom tank is a water reservoir for the system. A variable speed water pump is used
203
−4
− µ2wTw − µ2wT w
k+1 k+1
1
x 10
0
−2
k
k
−1
f,k f,k k
−3 −4 −5
∆ V + εT ε
to fill the upper tank. The water outflows the tanks due to gravity. The considered multi-tank system has been designed to operate with an external, PC-based digital controller. The control computer communicates with the level sensors, valves and a pump by a dedicated I/O board and the power interface. The I/O board is controlled by the real-time software, which operates in a Matlab/Simulink environment.
0
Fig. 3.
500
1000 1500 Discrete time
2000
2500
2 T 2 T Evolution of ∆Vk + εT fa ,k εfa ,k − µ w k w k − µ w k+1 w k+1
0.4 0.35 0.3 Fig. 2.
Multi-tank system
k
|| e ||
0.25
Let the initial condition for the system and the observer be: x0 = [0.1, 0.2, 0.3]T ,
ˆ 0 = 0.00001, x
0.2 0.15
(54) 0.1
while the input and the exogenous disturbance are: 0.05
uk = 1,
wk ∼ N (0, 0.01I).
(55) 0 0
As a result of solving the problem (41), the following couple were obtained: 0.1089 0 µ = 0.45; K = 0.0004 1.7107 . (56) 0 0.9473
Fig. 4.
The paper deals with the problem of robust sensor fault estimation with neural networks. In particular, a combination
204
15
20
Evolution of kek k (for k = 0, . . . , 20)
k
0.1
0.05
0
−0.05 0
Fig. 5.
V. C ONCLUSIONS
10 Discrete time
0.15
f
Moreover, let us consider the following actuator fault scenario 0.1, for 300 ≥ k ≥ 200, −0.2uk for 1500 ≥ k ≥ 500, f a,k = 0.0005(k − 1800) for 2100 ≥ k ≥ 1800, 0, otherwise. (57) ˆ 0 = x0 (e0 = 0). Figure First, let us consider the case when x 3 clearly indicates that condition (29) is satisfied, which means that an attenuation level µ = 0.45 is achieved. ˆ 0 6= x0 . Figure (4) Now let use assume that wk = 0 and x clearly shows that (28) is satisfied as well. Finally, figures (5-7) shows the fault and its estimate for the nominal case (ˆ x0 6= x0 and wk 6= 0). Moreover, figure 8 shows system state x3,k ˆ 3,k . and its estimate x
5
100
200 300 400 Discrete time
500
Constant bias actuator fault and its estimate
600
of the celebrated generalised observer scheme with the robust H∞ approach is proposed to settle the problem of robust fault diagnosis. The proposed approach is designed in such a way that a prescribed disturbance attenuation level is achieved with respect to the sensor fault estimation error while guaranteeing the convergence of the observer. The final part of the paper is concerned with a comprehensive case study regarding the multi-tank system. The achieved results show the performance of the proposed approach, which confirm its practical usefulness.
0.05 0
f
k
−0.05 −0.1 −0.15
R EFERENCES
−0.2 −0.25
400
Fig. 6.
600
800 1000 1200 Discrete time
1400
1600
20% decrease fault in actuator and its estimate
0.2
0.15
f
k
0.1
0.05
0
−0.05 1700
1800
Fig. 7.
1900 2000 Discrete time
2100
2200
Incipient actuator fault and its estimate
0.3
x3 x ˆ3
0.25 0.2 0.15 0.1 0.05 0 0
2
Fig. 8.
4 6 Discrete time
8
[1] M. Mrugalski, “An unscented kalman filter in designing dynamic gmdh neural networks for robust fault detection,” International Journal of Applied Mathematics and Computer Science, vol. 23, no. 1, pp. 157– 169, 2013. [2] M. Luzar, A. Czajkowski, M. Witczak, and M. Mrugalski, “Actuators ans sensors fault diagnosis with dynamic, state-space neural networks,” in Methods and Models in Automation and Robotics - MMAR 2012 : Proceedings of the 17th IEEE International Conference, 2012, pp. 196– 201. [3] M. Witczak, Modelling and Estimation Strategies for Fault Diagnosis of Non-linear Systems. Berlin: Springer-Verlag, 2007. [4] S. Montes de Oca, V. Puig, M. Witczak, and L. Dziekan, “Fault-tolerant control strategy for actuator faults using LPV techniques: Application to a two degree of freedom helicopter,” International Journal of Applied Mathematics and Computer Science, vol. 22, no. 1, pp. 161–171, 2012. [5] S. Gillijns and B. De Moor, “Unbiased minimum-variance input and state estimation for linear discrete-time systems,” Automatica, vol. 43, pp. 111–116, 2007. [6] E. Nobrega, M. Abdalla, and K. Grigoriadis, “Robust fault estimation of unceratain systems using an LMI-based approach,” International Journal of Robust and Nonlinear Control, vol. 18, no. 7, pp. 1657–1680, 2008. [7] D. Stipanovic and D. Siljak, “Robust stability and stabilization of discrete-time non-linear: the LMI approach,” International Journal of Control, vol. 74, no. 5, pp. 873–879, 2001. [8] M. Witczak and P. Pretki, “Desing of an extended unknown input observer with stochastic robustness techniques and evolutionary algorithms,” International Journal of Control, vol. 80, no. 5, pp. 749–762, 2007. [9] J. Korbicz, M. Witczak, and V. Puig, “LMI-based strategies for designing observers and unknown input observers for non-linear discrete-time systems,” Bulletin of the Polish Academy of Sciences : Technical Sciences, vol. 55, no. 1, pp. 31–42, 2007. [10] M. Mrugalski, M. Witczak, and J. Korbicz, “Confidence estimation of the multi-layer perceptron and its application in fault detection systems,” Engineering Applications of Artificial Intelligence, vol. 21, no. 8, pp. 895–906, 2008. [11] N. Lachhab, H. Abbas, and H. Werner, “A neural-network based technique for modelling and LPV control of an arm-driven inverted pendulum,” in Proceedings of the 47th IEEE Conference on Decision and Control, Cancun, Mexico, 2008, pp. 3860–3865. [12] H. Kajiwara, P. Apkarian, and P. Gahinet, “LPV techniques for control on an inverted pendulum,” IEEE Control Systems Magazine, vol. 19, no. 1, pp. 44–54, 1999. [13] A. Zemouche and M. Boutayeb, “Observer design for Lipschitz nonlinear systems: the discrete time case,” IEEE Trans. Circuits and Systems - II:Express Briefs, vol. 53, no. 8, pp. 777–781, 2006. [14] M. de Oliveira, J. Bernussou, and J. Geromel, “A new discrete-time robust stability condition,” Systems and Control Letters, vol. 37, no. 4, pp. 261–265, 1999. [15] A. Zemouche, M. Boutayeb, and G. Iulia Bara, “Observer for a class of Lipschitz systems with extension to H∞ performance analysis,” Systems and Control Letters, vol. 57, no. 1, pp. 18–27, 2008. [16] INTECO, Multitank System - User’s manual, www.inteco.com.pl, 2013.
ˆ 3,k State x3,k and its estimate x
205
II. D ERIVATION
OF A DISCRETE - TIME POLYTOPIC MODEL
LPV
The goal of this section is to present a neural state-space model [10] that can be represent a general class of state-space models and can be easily transformed into a LPV model. The transformation method is derived by Lachhab et. all and presented in [11]. A. Recurrent Neural-Network Topology Let us consider the following discrete-time LPV model:
I. I NTRODUCTION The problem of fault diagnosis (FD) of non-linear industrial systems [1], [2] has received considerable attention during the last three decades. Indeed, it developed from the art of designing a satisfactory performing systems into the modern theory and practice that it is today. Within the usual framework, the system being diagnosed is divided into three main components, i.e. plant (or system dynamics [3]), actuators and sensors. The paper deals with the problem of full fault diagnosis of sensors, i.e. apart from the usual two steps consisting of fault detection and isolation (FDI), the fault identification is also performed. This last step is especially important from the viewpoint of Fault-Tolerant Control (FTC) [4], which is possible if and only if there is an information about the size of the fault being a result of fault identification (or fault estimation). In this paper a robust fault estimation approach is proposed, which can be efficiently applied to realise the above-mentioned procedure. The proposed approach can be perceived as a combination of the linear-system strategies [5] and [6] for a class of non-linear systems [7] and it is based on state observers [8], [9]. The proposed approach is designed in such a way that a prescribed disturbance attenuation level is achieved with respect to the fault estimation error while guaranteeing the convergence of the observer. The paper is organised as follows. On the beginning, the derivation of a polytopic LPV model with neural network is described. Detailed design procedure of the robust observer for actuator fault diagnosis is presented in Section III. In Section IV experimental results obtained with laboratory nonlinear system are shown. Finally, Section V conclude the paper.
978-1-4673-5508-7/13/$31.00 ©2013 IEEE
xk+1 = A(θk )xk + B(θk )uk ,
(1)
y k+1 = C(θk )xk+1 ,
(2)
where A(θk ), B(θk ), C(θk ) are continuous mappings and θk is a time-varying parameter. The dependence of the A,B,C on θk represents a general LPV model; imposing that any of these matrices is parameter independent, i.e. fixed, will restrict the generality of the LPV model. The latter is indeed the case when RNN is transformed into the LPV model: A is parameter dependent but B and C are fixed. A general form of statespace neural network model is xk+1 = Axk + Buk + A1 σ(E 1 xk ) + B 1 σ(E 2 uk ),
(3)
y k+1 = Cxk+1 + C 1 σ(E 3 xk+1 ),
(4)
where x ∈ Rn denotes the state vector, y ∈ Rp the output and u ∈ Rm the input vector. A, A1 , B, B 1 , C, C 1 , E 1 , E 2 and E 3 are real valued matrices of appropriate dimensions and represent the weights which will be adjusted during the training stage of the RNN. The non-linear activation function σ(·), which is applied elementwise in (3)–(4) is taken as a continuous, differentiable and bounded function. This RNN leads to a general form of the neural state-space model in the sense that if it is transformed into an LPV model in the form (1)–(2), the matrices A,B and C will be parameter dependent. For stability and identifiability proofs of the proposed RNN reader is refereed to [11]. B. Simplification and Assumptions The Recurrent Neural-Network (RNN) model described by (3)–(4) can be simplified by removing any of the sigmoidal layers. This can be done according to a priori information
200
about the identified system. The LPV model (1)–(2) of multitank system, which was derived from a physical laboratory model and used in experiments presented in Section IV, has only matrix A parameter dependent. Thus, a specific transformation proposed by [12] is used to simplify such a model. Removing the sigmoidal layers from the input and the output paths means that the resulting LPV model has parameter dependance in matrix A only. Moreover, for a practical implementation this simplified RNN is modified as shown in Fig. 1: the outputs instead of the states are taken as input to sigmoidal layer. This modification facilities the implementation of LPV controllers designed based on this model. The modified RNN is represented as xk+1 = Axk + Buk + A1 σ(E 1 Cxk ),
(5)
y k+1 = Cxk+1 .
(6)
Multiplying (8) by H, and then substituting (7), it can be shown that f a,k = = H(y k+1 − CAxk − CBuk −
(11)
Cg (xk ) − CW 1 wk − W 2 wk+1 ).
Finally, by substituting (11) into (7) it can be shown that: xk+1 = ¯ ¯ = Axk + Buk + Gg (xk ) + ¯ k+1 + GW 1 w k − LW ¯ 2 wk+1 , Ly
(12)
¯ = GA, B ¯ = GB, L ¯ = La H. where G = (I n −La HC), A In order to estimate (11), i.e. to obtain fˆ k it is necessary ˆ k . Conseto estimate the state of the system, i.e. to obtain x quently, the fault estimate is given as follows fˆ a,k = H(y k+1 − CAˆ xk − CBuk − Cg (ˆ xk )).
(13)
The corresponding observer structure is
Fig. 1.
¯ xk + Bu ¯ k + Gg (ˆ ¯ k+1 + K(y k − C x ˆ k+1 = Aˆ ˆ k ), x xk ) + Ly (14) while the state estimation error is given by ¯ − KC ek + Gsk + ek+1 = A ¯ 2 w k+1 = (GW 1 − KW 2 )w k − LW (15) ¯ ¯ = A1 ek + Gsk + W 1 wk + W 2 wk+1 ,
Simplified state-space recurrent neural network
where III. ACTUATOR FAULT
sk = g (xk ) − g (ˆ xk ) .
DIAGNOSIS
The main objective of this section is to provide a detailed design procedure of the robust observer, which can be used for actuator fault diagnosis. In other words, the main role of this observer is to provided the information about the actuator fault. Indeed, apart from serving as a usual residual generator (see, e.g., [3]), the observer should be designed in such a way that a prescribed disturbance attenuation level is achieved with respect to the actuator fault estimation error while guaranteeing the convergence of the observer. Neural model described by (5)–(6) can be modify to following state-space form: xk+1 = Axk + Buk + g (xk ) + La f a,k + W 1 w k ,
(7)
y k+1 = Cxk+1 + W 2 wk+1 ,
(8)
n
Similarly, the fault estimation error εfa ,k can be defined εfa ,k = f a,k − fˆ a,k =
= −HC (Aek + sk + W 1 w k ) − HW 2 wk+1 .
rank(CLa ) = rank(La ) = s Under the assumption (9) it is possible to calculate −1 H = (CLa )+ = (CLa )T CLa (CLa )T .
(17)
Noth that both ek and εfa ,k are non-linear with respect to ek . To settle this problem within the framework of this paper, the following solution is proposed. Using the Differential Mean Value Theorem (DMVT) [13], it can be shown that
with
g (a) − g (b) = M x (a − b),
(18)
∂g1 ∂x (c1 ) .. , Mx = . ∂g n (cn ) ∂x
(19)
r
where xk ∈ X ⊂ R is the state vector, uk ∈ R stands for the input, y k ∈ Rm denotes the output, f a,k ∈ Rm stands for the actuator fault. While , w k ∈ Rn is a an exogenous disturbance vector with W 1 ∈ Rn×n , W 2 ∈ Rm×n being its distribution matrices. Following [3], [5], let us assume that the system is observable and the following rank condition is satisfied:
(16)
where c1 , . . . , cn ∈ Co(a, b), ci 6= a, ci 6= b, i = 1, . . . , n. Assuming that a ¯i,j ≥
(9)
∂gi ≥ ai,j , ∂xj
i = 1, . . . , n,
j = 1, . . . , n,
(20)
n o M ∈ Rn×n |¯ ai,j ≥ mx,i,j ≥ ai,j , i, j = 1, . . . , n,
(21)
it is clear that: (10)
201
Mx =
Thus, using (18), the term A1 ek + Gsk in (15) can be written as
J = −V0 +
¯ + GM x,k − KC)ek A1 ek + Gsk = (A
(22)
where M x,k ∈ Mx . From (22), it can be deduced that the state estimation error can be converted into an equivalent form ¯ 1 wk + W ¯ 2 w k+1 , ek+1 = A2 (α)ek + W ˜ A2 (α) = A(α) − KC,
˜= A
˜ A(α) :
˜ A(α) =
N X
˜ i, αi A
N X
)
αi = 1, αi ≥ 0
i=1
i=1
∞ X
2 εT fa ,k εfa ,k − µ
k=0
∞ X
2 wT k wk − µ
k=0
∞ X
wT k+1 w k+1 < 0.
k=0
µ2
∞ X
w Tk+1 w k+1 = µ2
k=0
∞ X
k=0
wTk wk − µ2 wT0 w 0 ,
(24)
,
J = −V0 +
∞ X
k=0
εTfa ,k εfa ,k − 2µ2
∞ X
wTk w k + µ2 w T0 w0 < 0.
k=0
(35)
2
Knowing that V0 = 0 for e0 = 0, (35) leads to (29) with √ ω = 2µ. Since the general framework for designing the robust observer is given, then the following form of the Lyapunov function is proposed [13]:
εfa ,k = −HC (A3 (α)ek + W 1 w k ) − HW 2 wk+1 , (25) with A3 =
A3 (α) :
A3 (α) =
N X
αi A3,i ,
i=1
N X
)
αi = 1, αi ≥ 0
i=1
.
(26)
The objective of further deliberations is to design the observer (14) in such a way that the state estimation error ek is asymptotically convergent and the following upper bound is guaranteed kεf kl2 ≤ ωkwkl2 (27) where ω > 0 is a prescribed disturbance attenuation level. Thus, on the contrary to the approaches presented in the literature, µ should be achieved with respect to the fault estimation error but not the state estimation error. Thus, the problem of H∞ observer design [15] is to determine the gain matrix K such that lim ek = 0 for wk = 0,
(28)
k→∞
kεf kl2 ≤ ωkwkl2
for wk 6= 0, e0 = 0.
(29)
In order to settle the above problem it is sufficient to find a Lyapunov function Vk such that: ∆Vk + εTfa ,k εfa ,k − µ2 w Tk wk − µ2 wTk+1 wk+1 < 0,
k = 0, . . . ∞,
(30)
where ∆Vk = Vk+1 − Vk , µ > 0. Indeed, if wk = 0, (k = 0, . . . , ∞) then (30) boils down to ∆Vk +
εTfa ,k εfa ,k
< 0, k = 0, . . . ∞,
(31)
and hence ∆Vk < 0, which leads to (28). If wk 6= 0 (k = 0, . . . , ∞) then (30) yields J=
(34)
inequality (33) can be written as
where N = 2n . Note that this is a general description, which does not take into account that some elements of M x,k maybe 2 constant. In such cases, N is given by N = 2(n−c) where c stands for the number of constant elements of M x,k . In a similar fashion, (17) can be converted into
(
(33)
Bearing in mind that
(23)
which defines an LPV polytopic system [14] with (
which can be written as
∞ X 2 T 2 T ∆Vk + εT fa ,k εfa ,k − µ w k w k − µ w k+1 w k+1 < 0,
Vk = eTk P (α)ek ,
(36)
where P (α) ≻ 0. On the contrary to the design approach presented in the literature (see, e.g. [15]) and the references therein) it is not assumed that P (α) = P is constant. Indeed, P (α) can be perceived as a parameter-depended matrix of the form (cf. [14]) P (α) =
N X
αi P i .
(37)
i=1
As a consequence: ∆Vk + εTfa ,k εfa ,k − µ2 w Tk wk − µ2 wTk+1 w k+1 =
eTk A2 (α)T P (α)A2 (α) + A3 (α)T H 1 A3 (α) − P (α) ek + ¯ 1 + A3 (α)T H 1 W 1 wk + eTk A2 (α)T P (α)W ¯ 2 + A3 (α)T H 2 wk+1 + eTk A2 (α)T P (α)W ¯ T1 P (α)A2 (α) + W T1 H 1 A3 (α) ek + w Tk W ¯ T P (α)W ¯ 1 + W T H 1 W 1 − µ2 I w k + w Tk W 1 1 ¯ T P (α)W 2 + W T H 2 w k+1 + w Tk W 1 1 T T ¯ 2 P (α)A2,k + H T2 A3 (α) ek + w k+1 W ¯ T P (α)W 1 + H T W 1 wk + w Tk+1 W 2 2 ¯ T P (α)W ¯ 2 + W T H T HW 2 − µ2 I wk+1 < 0. w Tk+1 W 2 2 with H 1 = C T H T HC and H 2 = C T H T HW 2 . By defining T v k = eTk , w Tk , wTk+1 ,
(38)
inequality (38) becomes
2 T 2 T T ∆Vk + εT fa ,k εfa ,k − µ w k wk − µ w k+1 w k+1 = v k M V v k < 0,
k=0
(32)
202
(39)
where M V is given by (40). M V = [M V1
M V1 M V2 M V3
=
=
=
M V3 ]
M V2
(40)
A2 (α)T P (α)A2 (α) + A3 (α)T H 1 A3 (α) − P (α) T ¯ 1 P (α)A2 (α) + W T H 1 A3 (α) , W 1 T ¯ T W P (α)A (α) + H A (α) 2 3 2 2 ¯ 1 + A3 (α)T H 1 W 1 A2 (α)T P (α)W T 2 ¯T ¯ , W 1 P (α)W 1 + W 1 H 1 W 1 − µ I T T ¯ W 2 P (α)W 1 + H 2 W 1 ¯ 2 + A3 (α)T H 2 A2 (α)T P (α)W T T ¯ . W 1 P (α)W 2 + W 1 H 2 ¯ T P (α)W ¯ 2 + W T H T HW 2 − µ2 I W 2 2
V (α)T X(α)V (α) − W (α) ≺ 0, ii)
The following theorem constitutes the main result of this section: Theorem 1: For a prescribed disturbance attenuation level µ > 0 for the fault estimation error (17), the H∞ observer design problem for the system (7)–(8) and the observer (14) is solvable if there exists matrices P i ≻ 0 (i = 1, . . . , N ), U and N such that the following LMIs are satisfied: Z2] ≺ 0
[Z 1
Z1 =
AT 3,i H 1 A3,i
− Pi
W T1 H 1 A3,i H T2 A3,i U A2,i
AT3,i H 1 W 1 T W 1 H 1 W 1 − µ2 I H T2 W 1 ¯1 UW
AT 3,i H 3
W T1 H 2 Z2 = T W 2 H T HW 2 − µ2 I ¯2 UW i = 1, . . . , N.
,
A2,i U T ¯ T UT W 1 ¯ T UT W 2 P i − U − UT
T
V XV − W ≺ 0
V T U T (U T + U − X)−1 U V − W ≺ 0. Substituting U = U
T
Thus, (i) implies (ii). h Multiplying (46) by T = I
,
(47)
(48)
i V T on the left and by T T on
A2 (α)T ¯T W P (α) A2 (α) 1 ¯T W 2
¯1 W
¯2 W
+
A3 (α)T H 1 A3 (α) − P (α) A3 (α)T H 1 W 1 T W 1 H 1 A3 (α) W T1 H 1 W 1 − µ2 I T H 2 A3 (α) H T2 W 1 T A3 (α) H 3 ≺ 0. W T1 H 2 T W 2 H T HW 2 − µ2 I (52)
and then applying Lemma 2 and (51) leads to (41), which completes the proof. Finally, the design procedure boils down to solving LMIs (41) and then (cf. (43)–(44)) K = U −1 N . It can be also observed that the observer design problem can be treated as an minimization task, i.e. µ∗ =
min
µ>0,P 1 ≻0,U ,N
µ
(53)
under (41).
(45)
= X yield
V T XV − W ≺ 0.
Subsequently, observing that the matrix (40) must be negative definite and writing it as
There exists X ≻ 0 such that −W V T UT ≺ 0. (46) UV X − U − UT Proof: Applying the Schur complement to (ii) gives
(50)
Proof: The proof can be realised by following the same line of reasoning as the one of Lemma 1. It is easy to show that that (50) is satisfied if there exist matrices X i ≻ 0 such that −W i V Ti U T ≺ 0, i = 1, . . . , N. (51) UV i Xi − U − UT
(42)
˜ i − KC) = U A ˜ i − N C, U A2,i = U (A (43) ¯ U W 1 = U (GW 1 − KW 2 ) = U GW 1 − N W 2 . (44) Proof: The following two lemmas can be perceived as the generalisation of those presented in [14]. Lemma 1: The following statements are equivalent i) There exists X ≻ 0 such that
(49)
There exists X(α) ≻ 0 such that −W (α) V (α)T U T ≺ 0. U V (α) X(α) − U − U T
(41)
where (cf. (24) and (15))
ii)
the left of (46) gives (45), which means that (ii) implies (i) and hence the proof is completed. Lemma 2: The following statements are equivalent i) i) There exists X(α) ≻ 0 such that
IV. E XPERIMENTAL RESULTS To verify proposed approach, a sensor fault detection was implemented for multitank system. The considered multi-tank system (Fig. 2) is designed for simulating the real industrial multi-tank system in the laboratory conditions [16]. The multitank system can be efficiently used to practically verify both linear and non-linear control, identification and diagnostics methods. The multi-tank system consists of three separate tanks placed each above other and equipped with drain valves and level sensors based on a hydraulic pressure measurement. Each of them has a different cross-section in order to reflect system nonlinearities. The lower bottom tank is a water reservoir for the system. A variable speed water pump is used
203
−4
− µ2wTw − µ2wT w
k+1 k+1
1
x 10
0
−2
k
k
−1
f,k f,k k
−3 −4 −5
∆ V + εT ε
to fill the upper tank. The water outflows the tanks due to gravity. The considered multi-tank system has been designed to operate with an external, PC-based digital controller. The control computer communicates with the level sensors, valves and a pump by a dedicated I/O board and the power interface. The I/O board is controlled by the real-time software, which operates in a Matlab/Simulink environment.
0
Fig. 3.
500
1000 1500 Discrete time
2000
2500
2 T 2 T Evolution of ∆Vk + εT fa ,k εfa ,k − µ w k w k − µ w k+1 w k+1
0.4 0.35 0.3 Fig. 2.
Multi-tank system
k
|| e ||
0.25
Let the initial condition for the system and the observer be: x0 = [0.1, 0.2, 0.3]T ,
ˆ 0 = 0.00001, x
0.2 0.15
(54) 0.1
while the input and the exogenous disturbance are: 0.05
uk = 1,
wk ∼ N (0, 0.01I).
(55) 0 0
As a result of solving the problem (41), the following couple were obtained: 0.1089 0 µ = 0.45; K = 0.0004 1.7107 . (56) 0 0.9473
Fig. 4.
The paper deals with the problem of robust sensor fault estimation with neural networks. In particular, a combination
204
15
20
Evolution of kek k (for k = 0, . . . , 20)
k
0.1
0.05
0
−0.05 0
Fig. 5.
V. C ONCLUSIONS
10 Discrete time
0.15
f
Moreover, let us consider the following actuator fault scenario 0.1, for 300 ≥ k ≥ 200, −0.2uk for 1500 ≥ k ≥ 500, f a,k = 0.0005(k − 1800) for 2100 ≥ k ≥ 1800, 0, otherwise. (57) ˆ 0 = x0 (e0 = 0). Figure First, let us consider the case when x 3 clearly indicates that condition (29) is satisfied, which means that an attenuation level µ = 0.45 is achieved. ˆ 0 6= x0 . Figure (4) Now let use assume that wk = 0 and x clearly shows that (28) is satisfied as well. Finally, figures (5-7) shows the fault and its estimate for the nominal case (ˆ x0 6= x0 and wk 6= 0). Moreover, figure 8 shows system state x3,k ˆ 3,k . and its estimate x
5
100
200 300 400 Discrete time
500
Constant bias actuator fault and its estimate
600
of the celebrated generalised observer scheme with the robust H∞ approach is proposed to settle the problem of robust fault diagnosis. The proposed approach is designed in such a way that a prescribed disturbance attenuation level is achieved with respect to the sensor fault estimation error while guaranteeing the convergence of the observer. The final part of the paper is concerned with a comprehensive case study regarding the multi-tank system. The achieved results show the performance of the proposed approach, which confirm its practical usefulness.
0.05 0
f
k
−0.05 −0.1 −0.15
R EFERENCES
−0.2 −0.25
400
Fig. 6.
600
800 1000 1200 Discrete time
1400
1600
20% decrease fault in actuator and its estimate
0.2
0.15
f
k
0.1
0.05
0
−0.05 1700
1800
Fig. 7.
1900 2000 Discrete time
2100
2200
Incipient actuator fault and its estimate
0.3
x3 x ˆ3
0.25 0.2 0.15 0.1 0.05 0 0
2
Fig. 8.
4 6 Discrete time
8
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ˆ 3,k State x3,k and its estimate x
205
