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Robust Observer Design: Logic-dynamic Approach
Proceedings of the 7th IFAC Symposium on Fault Detection, Supervision and Safety of Technical Processes Barcelona, Spain, June 30 - July 3, 2009
Robust Observer Design: Logic-dynamic Approach Alexey N. Zhirabok Institute for Marine Technology Problems, Vladivostok, Russia; e-mail: [email protected] Abstract: This paper presents a problem of robust fault detection and isolation in systems described by nonlinear dynamic models with no differentiable nonlinearities. To solve this problem, so-called logic-dynamic approach is suggested. Three existence conditions to design the observer invariant with respect to the unknown inputs are considered. A new approach to robust nonlinear observer design is suggested.
1. INTRODUCTION There are many papers and books devoted to the problem of robust fault detection and isolation (FDI) (Cassavola et al, 2008; Chen & Patton, 1999; Frank & Ding, 1997; Low et al, 1986; Shumsky, 2002). They are used for linear or nonlinear systems described by models with smooth nonlinearities. Many technical systems contain such types of no differentiable nonlinearities as Coulonm friction, backlash, saturation. Therefore the methods developed in the literature mentioned above cannot be used in this case. So-called logic-dynamic approach to fault diagnosis in systems described by nonlinear dynamic models with no differentiable nonlinearities was proposed in (Zhirabok & Usoltsev, 2002). In this paper, the logic-dynamic approach is developed for nonlinear systems described by the equations x& (t ) = Fx(t ) + Gu(t ) + С ⋅ ϕ( Ax(t ), u (t )) + Kd (t ) + Lρ(t ) , y(t ) = Hx(t ) .
(1)
1. Replacing the initial nonlinear system (1) by so-called linear logic-dynamic (LLD) system containing several linear subsystems and linear logical conditions. 2. Solving the FDI problem for the LLD system and obtaining the LLD observer. 3. Transforming the LLD observer into the nonlinear one. For detail description of the logic-dynamic approach three steps, consider the simple case with single nonlinearly (Coulomb friction) of the form С ⋅ ϕ( Ax(t ), u(t )) = G ′u (t )sign( Ax(t )) for the certain matrix G ′ and row matrix A. On the first step of this approach, the initial system Σ = ( F , A, G, H ) is replaced by certain LLD system. This system consists of three linear subsystems Σ 1 = ( F ,0, G − G ′, H ) ,
Here x, y, and u are the vectors of state, output, and control; F , G , H, and L are known matrices of appropriate dimensions; A is a row matrix; C is a column matrix: C[i ] ≠ 0 if the i-th component of the state vector contains the scalar function ϕ( Ax (t ), u (t )) , C[i ] = 0 otherwise. The term Lρ(t ) models unknown parameters and unknown inputs to the actuator and to the dynamic process; the evaluation of the vector function ρ(t ) must generally be considered unknown. The term Kd (t ) models the faults: if there are no faults, then d (t ) = 0 ; if a fault occurs, d (t ) becomes an unknown function. Denote system (1) as Σ = ( F , A, G , H ) .
Σ 2 = ( F ,0 , G , H ) ,
Σ 3 = ( F ,0, G + G ′, H )
with three linear logical conditions Ax(t ) < 0 , Ax (t ) = 0 , and Ax(t ) > 0 . If the condition Ax(t ) < 0 holds, then (in the unfaulty case and with ρ(t ) = 0 ) model (1) reduces to the subsystem Σ1 : x& (t ) = Fx (t ) + (G − G ′)u(t ) , if Ax (t ) = 0 , then to Σ 2 :
The main purpose of this paper is to use this approach to robust observers design for the class of systems with no differentiable nonlinearities.
x& (t ) = Fx (t ) + Gu (t ) , if Ax (t ) > 0 , then to Σ 3 :
2. LOGIC-DYNAMIC APPROACH The logic-dynamic approach for solving the FDI problem for system (1) includes the following three steps (Zhirabok & Usoltsev, 2002).
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x& (t ) = Fx (t ) + (G + G ′)u (t ) ; y (t ) = Hx(t ) holds for the systems Σ1 − Σ3 . It is important that these models have the identical matrices F and H.
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Φ A = A* . H
On the second step, the LLD observer has to be obtained. Assume that a structure of the LLD observer is similar to the one of the system Σ, therefore three subsystems Σ*1 , Σ*2 , Σ*3 and the row matrix A∗ exist such that the following x relationships hold (denote z = * ): y
Clearly, this equality is equivalent to the rank condition Φ Φ rank = rank H . H A
if A* z < 0 , then the LLD observer reduces to Σ *1 x& ∗ (t ) = F∗ x ∗ (t ) + (G ∗ − G *′ )u (t ) + J * y (t ) ,
(3)
(4)
This condition is an additional restriction on the matrix Φ.
if A* z = 0 , then Σ*2
3. OBSERVER DESIGN
x& ∗ (t ) = F∗ x∗ (t ) + G∗ u (t ) + J * y (t ) ,
3.1. Preliminary results
if A* z > 0 , then Σ*3 x& ∗ (t ) = F∗ x∗ (t ) + (G∗ + G*′ )u (t ) + J * y (t ) ; y∗ (t ) = H∗ x∗ (t ) holds for the systems Σ*1 − Σ*3 . Here x∗ (t ) is the state vector of the observer, y∗ (t ) is the output vector, F∗ , G∗ , J ∗ , G∗′ , and H ∗ are constant matrices to be determined. The observer generates the residual
r (t ) = Ry (t ) − y ∗ (t )
To design an observer in the linear case, there are a number of approaches, e.g., the eigenstructure assignment (Patton, 1994), the approach based on the Kronecker canonical form (Frank & Ding, 1997). Consider another linear procedure suggested in (Mironovsky, 1979; Zhirabok, 1997) also based on the Kronecker canonical form that allows taking into account condition (4) easily. According to this approach the matrices F* and H * describing the observer are represented in a canonical form as follows 0 1 0 ... 0 0 0 1 ... 0 F∗ = , . . . 0 0 0 ... 0
for certain matrix R which has to be determined. If there are no faults and ρ(t ) = 0 , then r (t ) = 0 , if a fault occurs, r (t ) ≠ 0 . It is well-known from the linear FDI theory (Frank & Ding, 1997; Chen & Patton, 1999; Zhirabok, 1997) that for the observer design, the matrix Φ is used such that
Φ1 = RH ,
in the unfaulty case after the response to unlike conditions has died out. In the absence of faults, the following wellknown set of equations holds (Chen & Patton, 1999; Zhirabok, 1997): H ∗ Φ = RH ,
G∗ = ΦG .
Φ i F = Φ i +1 + J *i H , i = 1, 2, K, k − 1,
(6)
J *k H = Φ k F ,
where Φ i and J *i are the i-th rows of the matrices Φ and J * , respectively.
(2)
To ensure the reliable fault detection, the residual r (t ) has to be sensitive to the faults and invariant with respect to the unknown inputs ρ(t ) that is ΦK ≠ 0 ,
(5)
Using matrices in (5), one can obtain from (2) the following equations:
Φ x (t ) = x ∗ (t )
ΦF = F∗ Φ + J * H ,
H ∗ = [1 0 0 ... 0].
ΦL = 0 .
To fit the logical conditions in the initial LLD system to the ones in the LLD observer, assume that the following relationships hold in the unfaulty case: if Ax (t ) < 0 , then A* z (t ) < 0 , if Ax (t ) = 0 , then A* z (t ) = 0 , if Ax (t ) > 0 , then A* z (t ) > 0 . Since x* ( t ) = Φ x( t ) and y (t ) = Hx(t ) , i.e.
3.2. Observer existence conditions There are several conditions of existence of the observer invariant with respect to ρ(t ) . Let L* be a matrix of maximal rank satisfying the equality L* L = 0 . It follows from this definition that expression ФL = 0 can be rewritten as Φ = NL* for some matrix N. Consider the equation H ∗ Φ = RH ; due to the equality Φ = NL* and (5) it can be written in the form N1 L* = RH or L [ N 1 | − R] ⋅ * = 0 H
x Φ A* z = A* * = A* x , these relationships hold if y H
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where N i is the i-th row of the matrix N. This equation has a nontrivial solution if rows of the matrices L* and H are linearly dependent that is equivalent to the rank inequality L rank * < rank ( L* ) + rank ( H ) . H
(8)
It is the first condition of existence. If it is true, the matrix R 0 of maximal rank can be obtained from (7). The matrix R can be obtained from the matrix R 0 as R = QR 0 for certain matrix Q which has to be determined. To obtain the second condition, consider the equation J *k H = Φ k F . Take into account the equality Φ = NL* that gives the equation L F [ N k | − J *k ] ⋅ * = 0 . H
As above, this equation is equivalent to the rank condition L F rank * < rank ( L* F ) + rank ( H ) . H
(9)
The third condition follows from the nonlinear feature of Φ system (1). As shown above, A = A* ; because H NL Ф = NL* , then A = A* * , i.e. the rows of the matrix A H are linear combinations of the matrices NL* and H rows. Necessary condition for existence of these combinations is as follows L* L* rank = rank H . H A
R = QR 0 ; A* is obtained from equation (3). As a result, the linear observer is described by the following equation:
x& ∗ (t ) = F∗ x∗ (t ) + G∗ u(t ) + J * y (t ) . On the third step of the suggested approach, it is necessary to transform the obtained LLD observer into the nonlinear one. According to (Zhirabok & Usoltsev, 2002) the nonlinear term C* ⋅ ϕ( A* z (t ) , u(t )) must be added to the right-hand side of the equation describing the linear observer: x&∗ (t ) = F∗ x∗ (t ) + G∗u (t ) + J * y (t ) + C* ⋅ ϕ( A* z (t ) , u (t )). 4. GENERALIZATIONS The suggested approach can be used for another types of nonlinearities such as G′u (t ) sin( Ax) , G ′u (t ) log( Ax) and so on. Actually, consider the term G′u (t ) sign( Ax) and G*′ u(t ) sign( A* z (t )) in the initial system and in the observer, respectively. Clearly, the last term is a result of certain transformation of the first one, and the LLD approach gives a rule of this transformation. Therefore, the nonlinearity G ′u (t ) sin( Ax) in the initial system gives the term G*′ u (t ) sin( A* z (t )) in the observer in spite of the fact that it is impossible to transform system (1) into any LLD system in this case. Here restriction (4) reflects not a logical condition but a condition of concordance of nonlinearities in the initial system and in the observer. If the function C ⋅ ϕ( Ax(t ), u (t )) has the form ϕ1 ( A1 x(t ), u (t )) ϕ 2 ( A2 x (t ), u (t )) C ⋅ ϕ( Ax(t ), u (t )) = C ⋅ , ... ϕ r ( Ar x(t ), u (t ))
(10)
If one of conditions (8)-(10) does not hold, the observer invariant with respect to ρ(t ) cannot be built, and the robust methods must be used. 3.3. Observer design Assume that conditions (8)-(10) hold. As it was shown in (Zhirabok & Usoltsev, 2002), equations (6) with R = QR 0 results in the single equation
then the second steps of the logic-dynamic approach is performed as well as in the above considered case with the A1 A matrix A of the form A = 2 . M Ar On the third step, the nonlinear term added to the right-hand side equation of the linear observer is of the form ϕ1 ( A*1 x (t ), u (t )) ϕ 2 ( A*2 x(t ), u (t )) C* ⋅ ... ϕ r ( A*r x (t ), u(t ))
QR 0 HF k = J *1 HF k −1 + J *2 HF k − 2 + K + J *k H .
This equation is used for finding the observer dimension k, the matrices Q , J *1 , J *2 ,…, J *k , and the matrix Φ according to (6) (Zhirabok & Usoltsev, 2002). Other matrices are obtained as follows: C* = ΦC , G* = ΦG , G*′ = ΦG ′ ,
where C* = ΦC and the row matrices A*1 , A*2 ,…, A*r are Φ obtained from the linear algebraic equation A j = A* j , H j=1,2,…,r.
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5. ROBUST OBSERVER DESIGN A canonical form of the observer is a limiting factor with respect to robustness. The way to obtain the robust (with respect to unknown parameters and unknown inputs) nonlinear observer without this factor based on the logicdynamic approach is suggested. Consider matrix equations (2) and find conditions to solve them. Rewrite the first equation as follows: ΦF = [F*
Φ J ]⋅ . H
(11)
It follows from this equation that rows of the matrix ΦF are linear combinations of the matrices Φ and H rows; this is equivalent to the rank equality Φ Φ rank = rank H . H ΦF
(12)
(14), (15), and ΦK ≠ 0 are tested again. This procedure continues until these conditions fulfill. The obtained solution gives the minimal value of the Frobenius norm Φ L
Φ − R]⋅ = 0 . H
used. This problem can be solved by known methods (Misava & Hedrick, 1989; Gauthier & Kupca, 2000). 6. EXAMPLE Consider the manipulator shown in Fig. 1 and described by the following equations:
(14)
(15)
Here x1 and x 2 are the output rotation angle and velocity at the reducer output shaft, respectively; x 3 and x 4 are the output rotation angle and velocity at the motor output shaft, respectively; x5 is the current through the servoactuator windings; H 0 and h are the components of the inertia and velocity, respectively: M d and M r are the moments of the Coulomb friction at the motor and reducer shaft output, respectively: M d = M do sign( x 4 ) , M r = M ro sign( x 2 ) ; K d and K r are the respective coefficient of viscous friction of the motor and reducer output shaft; i r is the reducing ratio of the reducer; C r is the rigidity coefficient of the mechanical reducer; J M is the moment of inertia of the electric servoactuator and of the rotating parts of the reducer; K ω and K M are the respective coefficients of the counter EMF and of the torque; R and L are the active and inductive resistances of the electric servoactuator windings, respectively; the function Bl describes the backlash: Bl ( z ) = 0.5(| z | −σ )( sign ( z + σ ) + sign ( z − σ )) ,
The main idea to design the robust observer based on the approach suggested in (Low et al, 1986) is as follows. Obtain the singular value decomposition of the matrix L when the matrix L is represented as L = U L ⋅ Σ L ⋅ V L where U L and VL are orthogonal matrices, Σ L = [diag (σ1 , σ 2 ,..., σ n ) 0] , 0 ≤ σ1 ≤ σ 2 ≤ ... ≤ σ n are singular values of the matrix L. Take the first column of the matrix U L as a row of the matrix Φ and test conditions (14), (15), and ΦK ≠ 0 for this matrix. If they fulfill, the task is solved and the matrix Φ is used to build the observer. Otherwise the second column of the matrix U L is used as a second row in Φ and conditions
ȱ
x& 4 = (1 / J M )(− K d x 4 + K M x 5 − M d − C r Bl( x3 − i r x1 )), x& 5 = (1 / L)(− K ω x 4 − Rx5 + u ).
(13)
Φ Equation A = A* is similar to (11) therefore it is H Φ Φ equivalent to the equality rank = rank H . This H A equality can be combined with (12) that gives Φ H Φ rank = rank . ΦF H A
.
x&1 = x 2 , x& 2 = (1 / H 0 )(−( K r + h) x 2 − M r + C r ir Bl( x3 − ir x1 )), x& 3 = x 4 ,
Clearly, this equation has a nontrivial solution if the rows of the matrices Φ and H are linearly dependent; this is equivalent to the inequality Φ rank < rank (Φ) + rank ( H ) . H
F
The approaches suggested in (Chen & Patton, 1999; Frank & Ding, 1997) can be used for obtaining the matrix Φ by analogy. The matrices F∗ , G∗ , J ∗ , H ∗ , R, and A∗ can be found from equations (3), (11), and (13). To design an asymptotically stable observer with the property r (t ) t → 0 , the appropriate gain matrix function must be →∞
The second equation in (2) can be rewritten as follows:
[H *
2
2σ is the backlash span, z = x3 − ir x1 .
y1 (t ) = x1 (t ) ,
y 2 (t ) = x 2 (t ) ,
y3 (t ) = x4 (t ) .
The corresponding LLD system is described by the following matrices:
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0 0 − ( K r F = 0 0 0
1 + hi ) / H ii
0
0
0
0
0 0
0 1 0 − Kd / J M
0
0
0 0 G = 0 , 0 1 / L
0 − M r 0 / H 0 C= 0 0 0
To illustrate the proposed approach to robust observer design, use the singular value decomposition of the matrix L with the matrix U L
0 , 0 KM / JM − R / L 0
− Kω / L
0 0.5 0.5 − 0.7071 0 − 0.5 − 0.5 − 0,7071 UL = 0 0,7071 0.5 − 0.5 − 0,7071 − 0.5 0.5 0 0 0 0 0
1 0 0 0 0 H = 0 1 0 0 0 , 0 0 0 1 0
0 0 0 − M d0 / JM 0
with σ1 = σ 2 = 1,4142 , σ 3 = σ 4 = σ 5 = 0 . It can be shown that last three columns of the matrix U L are linear combinations of the matrix L* rows and vice verse. Therefore one can let Φ = L* , and conditions (12) and (14) hold in this case. Equalities (11) and (13) give the following: R = (−1 1 0) , H * = (1 0 0) ,
0 C ir / H 0 , 0 −C / JM 0
0 − ( K r + h) / H 0 − 1 0 F* = 0 0 K M / J M , 0 0 − R / L
sign( A1 x ) ϕ( x, u ) = sign( A2 x), Bl ( A3 x )
0 − ( K r + h) / H 0 − 1 0 J* = 0 0 − K d / J M − 1 ; − K ω / L 0 0
A1 = [0 1 0 0 0], A2 = [0 0 0 1 0], A3 = [−i r
− M r 0 / H 0 C* = ΦC = 0 0
0 1 0 0] .
Assume that faults and unknown inputs are described as follows: 1 1 0 0 0 L= . 0 0 1 1 0
as
follows:
A*1 = [1 0 0 1 0 0] , 0 1] .
C r ir Bl( y 3 − i r y1 − x*2 )) − y 2 , &x*2 = (1 / J M )(− K d y 3 + K M x*3 − M d 0 sign( y 3 ) − C r Bl( y3 − i r y1 − x*2 )) − y 3 ,
It can be shown that conditions (8)-(10) are satisfy. The matrices R and Φ obtained from equations (7) and (6) respectively is as follows: −1 1 0 0 0 Φ= . 0 0 1 0 0
The nonlinear observer is described by the following equations:
C r i r Bl( x*2 − i r y1 )) − y 2 , &x*2 = y 3 ; y* = x*1 , r = y 2 − y1 − y* .
A*i
x&*1 = (1 / H 0 )(−( K r + h) y 2 − M r 0 sign( y 2 ) +
− 1 1 0 0 0 L* = 0 0 − 1 1 0 . 0 0 0 0 1
x&*1 = (1 / H 0 )(−( K r + h) y 2 − M r 0 sign( y 2 ) +
0
C ir / H 0 − C / J M . 0
A*2 = [0 0 0 0 0 1], A*3 = [0 −1 0 −i r As a result, the observer is described as follows:
It follows from the matrix L* definition that
R = (−1 1 0) ,
0 − M d0 / JM
Besides G* = ΦG = [0 0 1 / L ]T . Equation (3) gives the matrices
T
K = [0 1 0 0 0]T ,
0 0 0 0 1
x&*3 = (1 / L)(− K ω y 3 − Rx*3 + u ), y* = x*1 ,
r (t ) = y 2 − y1 − y* .
For simplicity, a gain matrix function to obtain the asymptotically stable observer does not used. For simulating, numerical values of the parameters are the following: K r = 0.01 Nms, K d = 10 −5 Mms, C r = 2.0 Nm, i r = 100 , σ = 1.0 rad, M r 0 = 10.0 Nmrad, M d 0 = 0.15 Nmrad, K ω = 0.02 Vs/rad, J M = 10 −4 kgm2, K M = 0.02 Nmrad/A, L=0.004 H, R=0.4 Ω, H 0 = 1 , h = 15 . In the simulation, the initial states of the system under consideration and the observer are assumed to be equal to zero. The control
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value is u=const=10. The vector function ρ(t ) is modeled by two independent values with random distribution on the interval [–0.01, +0.01]. The fault is modeled by changing the parameter M r 0 from 10.0 to 12.0 at once at t=30. Fig. 2 illustrates the first observer residual behavior. Clearly this residual is not invariant with respect to the unknown input ρ(t ) ; sensitivity to the fault is not evident. Fig. 3 illustrates the second observer residual behavior which shows that this residual is invariant with respect to ρ(t ) ; sensitivity to the fault is evident.
Shumsky, A. (2002). Robust analytical redundancy relation for fault diagnosis in nonlinear systems. Asian J. Control, 4, pp. 159-170. Zhirabok, A. (1997). Fault detection and isolation: linear and nonlinear systems. Preprints of the IFAC Symposium SAFEPROCESS’97, pp. 903-908, UK, Hall, 1997. Zhirabok, A. and S. Usoltsev (2002). Fault diagnosis for nonlinear dynamic systems via linear methods. CD ROM Proceedings of the 15th World Congress IFAC, Spain, Barcelona, 2002.
7. CONCLUSION The problem of robust fault diagnosis in nonlinear dynamic systems with no differentiable nonlinearities has been studied. To solve this problem, so-called logic-dynamic approach has been developed. This approach consists of the following main steps: replacing the initial nonlinear system by certain linear logic-dynamic system, obtaining the bank of linear logic-dynamic observers, and transforming them into the nonlinear ones. It has been shown that this approach allows taking into account different types of nonlinearities by linear methods. Three existence conditions to design the observer invariant with respect to the unknown parameters and unknown inputs have been obtained. A new approach to robust nonlinear observer design has been suggested.
Fig.1. Manipulator 0.005
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It follows from the logic-dynamic approach that it can be applied to discrete-time systems.
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Acknowledgments
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This work was supported by Russian Foundation of Basic Researches 05-08-18240 and 07-08-00102.
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REFERENCES
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Cassavola, A., Famularo, D. and G. France (2008). Robust fault detection of uncertain systems via quasi-LMIs. Automatica, 44, pp. 289-295. Chen, J. and R. Patton (1999). Robust model-based fault diagnosis. Kluwer Academic Publishers, Boston. Frank, P. and S. Ding (1997). Survey of robust residual generation and estimation methods in observer-based fault detection systems. Journal of Process Control, 7, pp. 403-424. Gauthier, J. and I. Kupca (2000). Deterministic observation theory and application. Cambridge University Press, Cambridge Lou, X., Willsky A. and G. Verghese (1986). Optimally robust redundancy relations for failure detection in uncertain systems. Automatica, 22, pp. 333-344. Mironovsky, L. (1979). Functional diagnosis of linear dynamic systems. Automation and Remote Control, No 8, pp. 120-128. Misawa, E.A. and J.K. Hedrick (1989). Nonlinear observers – a state of the art survey. Journal of dynamic systems, measurements and control, 111, pp. 344-352.
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Robust Observer Design: Logic-dynamic Approach Alexey N. Zhirabok Institute for Marine Technology Problems, Vladivostok, Russia; e-mail: [email protected] Abstract: This paper presents a problem of robust fault detection and isolation in systems described by nonlinear dynamic models with no differentiable nonlinearities. To solve this problem, so-called logic-dynamic approach is suggested. Three existence conditions to design the observer invariant with respect to the unknown inputs are considered. A new approach to robust nonlinear observer design is suggested.
1. INTRODUCTION There are many papers and books devoted to the problem of robust fault detection and isolation (FDI) (Cassavola et al, 2008; Chen & Patton, 1999; Frank & Ding, 1997; Low et al, 1986; Shumsky, 2002). They are used for linear or nonlinear systems described by models with smooth nonlinearities. Many technical systems contain such types of no differentiable nonlinearities as Coulonm friction, backlash, saturation. Therefore the methods developed in the literature mentioned above cannot be used in this case. So-called logic-dynamic approach to fault diagnosis in systems described by nonlinear dynamic models with no differentiable nonlinearities was proposed in (Zhirabok & Usoltsev, 2002). In this paper, the logic-dynamic approach is developed for nonlinear systems described by the equations x& (t ) = Fx(t ) + Gu(t ) + С ⋅ ϕ( Ax(t ), u (t )) + Kd (t ) + Lρ(t ) , y(t ) = Hx(t ) .
(1)
1. Replacing the initial nonlinear system (1) by so-called linear logic-dynamic (LLD) system containing several linear subsystems and linear logical conditions. 2. Solving the FDI problem for the LLD system and obtaining the LLD observer. 3. Transforming the LLD observer into the nonlinear one. For detail description of the logic-dynamic approach three steps, consider the simple case with single nonlinearly (Coulomb friction) of the form С ⋅ ϕ( Ax(t ), u(t )) = G ′u (t )sign( Ax(t )) for the certain matrix G ′ and row matrix A. On the first step of this approach, the initial system Σ = ( F , A, G, H ) is replaced by certain LLD system. This system consists of three linear subsystems Σ 1 = ( F ,0, G − G ′, H ) ,
Here x, y, and u are the vectors of state, output, and control; F , G , H, and L are known matrices of appropriate dimensions; A is a row matrix; C is a column matrix: C[i ] ≠ 0 if the i-th component of the state vector contains the scalar function ϕ( Ax (t ), u (t )) , C[i ] = 0 otherwise. The term Lρ(t ) models unknown parameters and unknown inputs to the actuator and to the dynamic process; the evaluation of the vector function ρ(t ) must generally be considered unknown. The term Kd (t ) models the faults: if there are no faults, then d (t ) = 0 ; if a fault occurs, d (t ) becomes an unknown function. Denote system (1) as Σ = ( F , A, G , H ) .
Σ 2 = ( F ,0 , G , H ) ,
Σ 3 = ( F ,0, G + G ′, H )
with three linear logical conditions Ax(t ) < 0 , Ax (t ) = 0 , and Ax(t ) > 0 . If the condition Ax(t ) < 0 holds, then (in the unfaulty case and with ρ(t ) = 0 ) model (1) reduces to the subsystem Σ1 : x& (t ) = Fx (t ) + (G − G ′)u(t ) , if Ax (t ) = 0 , then to Σ 2 :
The main purpose of this paper is to use this approach to robust observers design for the class of systems with no differentiable nonlinearities.
x& (t ) = Fx (t ) + Gu (t ) , if Ax (t ) > 0 , then to Σ 3 :
2. LOGIC-DYNAMIC APPROACH The logic-dynamic approach for solving the FDI problem for system (1) includes the following three steps (Zhirabok & Usoltsev, 2002).
978-3-902661-46-3/09/$20.00 © 2009 IFAC
x& (t ) = Fx (t ) + (G + G ′)u (t ) ; y (t ) = Hx(t ) holds for the systems Σ1 − Σ3 . It is important that these models have the identical matrices F and H.
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Φ A = A* . H
On the second step, the LLD observer has to be obtained. Assume that a structure of the LLD observer is similar to the one of the system Σ, therefore three subsystems Σ*1 , Σ*2 , Σ*3 and the row matrix A∗ exist such that the following x relationships hold (denote z = * ): y
Clearly, this equality is equivalent to the rank condition Φ Φ rank = rank H . H A
if A* z < 0 , then the LLD observer reduces to Σ *1 x& ∗ (t ) = F∗ x ∗ (t ) + (G ∗ − G *′ )u (t ) + J * y (t ) ,
(3)
(4)
This condition is an additional restriction on the matrix Φ.
if A* z = 0 , then Σ*2
3. OBSERVER DESIGN
x& ∗ (t ) = F∗ x∗ (t ) + G∗ u (t ) + J * y (t ) ,
3.1. Preliminary results
if A* z > 0 , then Σ*3 x& ∗ (t ) = F∗ x∗ (t ) + (G∗ + G*′ )u (t ) + J * y (t ) ; y∗ (t ) = H∗ x∗ (t ) holds for the systems Σ*1 − Σ*3 . Here x∗ (t ) is the state vector of the observer, y∗ (t ) is the output vector, F∗ , G∗ , J ∗ , G∗′ , and H ∗ are constant matrices to be determined. The observer generates the residual
r (t ) = Ry (t ) − y ∗ (t )
To design an observer in the linear case, there are a number of approaches, e.g., the eigenstructure assignment (Patton, 1994), the approach based on the Kronecker canonical form (Frank & Ding, 1997). Consider another linear procedure suggested in (Mironovsky, 1979; Zhirabok, 1997) also based on the Kronecker canonical form that allows taking into account condition (4) easily. According to this approach the matrices F* and H * describing the observer are represented in a canonical form as follows 0 1 0 ... 0 0 0 1 ... 0 F∗ = , . . . 0 0 0 ... 0
for certain matrix R which has to be determined. If there are no faults and ρ(t ) = 0 , then r (t ) = 0 , if a fault occurs, r (t ) ≠ 0 . It is well-known from the linear FDI theory (Frank & Ding, 1997; Chen & Patton, 1999; Zhirabok, 1997) that for the observer design, the matrix Φ is used such that
Φ1 = RH ,
in the unfaulty case after the response to unlike conditions has died out. In the absence of faults, the following wellknown set of equations holds (Chen & Patton, 1999; Zhirabok, 1997): H ∗ Φ = RH ,
G∗ = ΦG .
Φ i F = Φ i +1 + J *i H , i = 1, 2, K, k − 1,
(6)
J *k H = Φ k F ,
where Φ i and J *i are the i-th rows of the matrices Φ and J * , respectively.
(2)
To ensure the reliable fault detection, the residual r (t ) has to be sensitive to the faults and invariant with respect to the unknown inputs ρ(t ) that is ΦK ≠ 0 ,
(5)
Using matrices in (5), one can obtain from (2) the following equations:
Φ x (t ) = x ∗ (t )
ΦF = F∗ Φ + J * H ,
H ∗ = [1 0 0 ... 0].
ΦL = 0 .
To fit the logical conditions in the initial LLD system to the ones in the LLD observer, assume that the following relationships hold in the unfaulty case: if Ax (t ) < 0 , then A* z (t ) < 0 , if Ax (t ) = 0 , then A* z (t ) = 0 , if Ax (t ) > 0 , then A* z (t ) > 0 . Since x* ( t ) = Φ x( t ) and y (t ) = Hx(t ) , i.e.
3.2. Observer existence conditions There are several conditions of existence of the observer invariant with respect to ρ(t ) . Let L* be a matrix of maximal rank satisfying the equality L* L = 0 . It follows from this definition that expression ФL = 0 can be rewritten as Φ = NL* for some matrix N. Consider the equation H ∗ Φ = RH ; due to the equality Φ = NL* and (5) it can be written in the form N1 L* = RH or L [ N 1 | − R] ⋅ * = 0 H
x Φ A* z = A* * = A* x , these relationships hold if y H
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where N i is the i-th row of the matrix N. This equation has a nontrivial solution if rows of the matrices L* and H are linearly dependent that is equivalent to the rank inequality L rank * < rank ( L* ) + rank ( H ) . H
(8)
It is the first condition of existence. If it is true, the matrix R 0 of maximal rank can be obtained from (7). The matrix R can be obtained from the matrix R 0 as R = QR 0 for certain matrix Q which has to be determined. To obtain the second condition, consider the equation J *k H = Φ k F . Take into account the equality Φ = NL* that gives the equation L F [ N k | − J *k ] ⋅ * = 0 . H
As above, this equation is equivalent to the rank condition L F rank * < rank ( L* F ) + rank ( H ) . H
(9)
The third condition follows from the nonlinear feature of Φ system (1). As shown above, A = A* ; because H NL Ф = NL* , then A = A* * , i.e. the rows of the matrix A H are linear combinations of the matrices NL* and H rows. Necessary condition for existence of these combinations is as follows L* L* rank = rank H . H A
R = QR 0 ; A* is obtained from equation (3). As a result, the linear observer is described by the following equation:
x& ∗ (t ) = F∗ x∗ (t ) + G∗ u(t ) + J * y (t ) . On the third step of the suggested approach, it is necessary to transform the obtained LLD observer into the nonlinear one. According to (Zhirabok & Usoltsev, 2002) the nonlinear term C* ⋅ ϕ( A* z (t ) , u(t )) must be added to the right-hand side of the equation describing the linear observer: x&∗ (t ) = F∗ x∗ (t ) + G∗u (t ) + J * y (t ) + C* ⋅ ϕ( A* z (t ) , u (t )). 4. GENERALIZATIONS The suggested approach can be used for another types of nonlinearities such as G′u (t ) sin( Ax) , G ′u (t ) log( Ax) and so on. Actually, consider the term G′u (t ) sign( Ax) and G*′ u(t ) sign( A* z (t )) in the initial system and in the observer, respectively. Clearly, the last term is a result of certain transformation of the first one, and the LLD approach gives a rule of this transformation. Therefore, the nonlinearity G ′u (t ) sin( Ax) in the initial system gives the term G*′ u (t ) sin( A* z (t )) in the observer in spite of the fact that it is impossible to transform system (1) into any LLD system in this case. Here restriction (4) reflects not a logical condition but a condition of concordance of nonlinearities in the initial system and in the observer. If the function C ⋅ ϕ( Ax(t ), u (t )) has the form ϕ1 ( A1 x(t ), u (t )) ϕ 2 ( A2 x (t ), u (t )) C ⋅ ϕ( Ax(t ), u (t )) = C ⋅ , ... ϕ r ( Ar x(t ), u (t ))
(10)
If one of conditions (8)-(10) does not hold, the observer invariant with respect to ρ(t ) cannot be built, and the robust methods must be used. 3.3. Observer design Assume that conditions (8)-(10) hold. As it was shown in (Zhirabok & Usoltsev, 2002), equations (6) with R = QR 0 results in the single equation
then the second steps of the logic-dynamic approach is performed as well as in the above considered case with the A1 A matrix A of the form A = 2 . M Ar On the third step, the nonlinear term added to the right-hand side equation of the linear observer is of the form ϕ1 ( A*1 x (t ), u (t )) ϕ 2 ( A*2 x(t ), u (t )) C* ⋅ ... ϕ r ( A*r x (t ), u(t ))
QR 0 HF k = J *1 HF k −1 + J *2 HF k − 2 + K + J *k H .
This equation is used for finding the observer dimension k, the matrices Q , J *1 , J *2 ,…, J *k , and the matrix Φ according to (6) (Zhirabok & Usoltsev, 2002). Other matrices are obtained as follows: C* = ΦC , G* = ΦG , G*′ = ΦG ′ ,
where C* = ΦC and the row matrices A*1 , A*2 ,…, A*r are Φ obtained from the linear algebraic equation A j = A* j , H j=1,2,…,r.
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5. ROBUST OBSERVER DESIGN A canonical form of the observer is a limiting factor with respect to robustness. The way to obtain the robust (with respect to unknown parameters and unknown inputs) nonlinear observer without this factor based on the logicdynamic approach is suggested. Consider matrix equations (2) and find conditions to solve them. Rewrite the first equation as follows: ΦF = [F*
Φ J ]⋅ . H
(11)
It follows from this equation that rows of the matrix ΦF are linear combinations of the matrices Φ and H rows; this is equivalent to the rank equality Φ Φ rank = rank H . H ΦF
(12)
(14), (15), and ΦK ≠ 0 are tested again. This procedure continues until these conditions fulfill. The obtained solution gives the minimal value of the Frobenius norm Φ L
Φ − R]⋅ = 0 . H
used. This problem can be solved by known methods (Misava & Hedrick, 1989; Gauthier & Kupca, 2000). 6. EXAMPLE Consider the manipulator shown in Fig. 1 and described by the following equations:
(14)
(15)
Here x1 and x 2 are the output rotation angle and velocity at the reducer output shaft, respectively; x 3 and x 4 are the output rotation angle and velocity at the motor output shaft, respectively; x5 is the current through the servoactuator windings; H 0 and h are the components of the inertia and velocity, respectively: M d and M r are the moments of the Coulomb friction at the motor and reducer shaft output, respectively: M d = M do sign( x 4 ) , M r = M ro sign( x 2 ) ; K d and K r are the respective coefficient of viscous friction of the motor and reducer output shaft; i r is the reducing ratio of the reducer; C r is the rigidity coefficient of the mechanical reducer; J M is the moment of inertia of the electric servoactuator and of the rotating parts of the reducer; K ω and K M are the respective coefficients of the counter EMF and of the torque; R and L are the active and inductive resistances of the electric servoactuator windings, respectively; the function Bl describes the backlash: Bl ( z ) = 0.5(| z | −σ )( sign ( z + σ ) + sign ( z − σ )) ,
The main idea to design the robust observer based on the approach suggested in (Low et al, 1986) is as follows. Obtain the singular value decomposition of the matrix L when the matrix L is represented as L = U L ⋅ Σ L ⋅ V L where U L and VL are orthogonal matrices, Σ L = [diag (σ1 , σ 2 ,..., σ n ) 0] , 0 ≤ σ1 ≤ σ 2 ≤ ... ≤ σ n are singular values of the matrix L. Take the first column of the matrix U L as a row of the matrix Φ and test conditions (14), (15), and ΦK ≠ 0 for this matrix. If they fulfill, the task is solved and the matrix Φ is used to build the observer. Otherwise the second column of the matrix U L is used as a second row in Φ and conditions
ȱ
x& 4 = (1 / J M )(− K d x 4 + K M x 5 − M d − C r Bl( x3 − i r x1 )), x& 5 = (1 / L)(− K ω x 4 − Rx5 + u ).
(13)
Φ Equation A = A* is similar to (11) therefore it is H Φ Φ equivalent to the equality rank = rank H . This H A equality can be combined with (12) that gives Φ H Φ rank = rank . ΦF H A
.
x&1 = x 2 , x& 2 = (1 / H 0 )(−( K r + h) x 2 − M r + C r ir Bl( x3 − ir x1 )), x& 3 = x 4 ,
Clearly, this equation has a nontrivial solution if the rows of the matrices Φ and H are linearly dependent; this is equivalent to the inequality Φ rank < rank (Φ) + rank ( H ) . H
F
The approaches suggested in (Chen & Patton, 1999; Frank & Ding, 1997) can be used for obtaining the matrix Φ by analogy. The matrices F∗ , G∗ , J ∗ , H ∗ , R, and A∗ can be found from equations (3), (11), and (13). To design an asymptotically stable observer with the property r (t ) t → 0 , the appropriate gain matrix function must be →∞
The second equation in (2) can be rewritten as follows:
[H *
2
2σ is the backlash span, z = x3 − ir x1 .
y1 (t ) = x1 (t ) ,
y 2 (t ) = x 2 (t ) ,
y3 (t ) = x4 (t ) .
The corresponding LLD system is described by the following matrices:
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0 0 − ( K r F = 0 0 0
1 + hi ) / H ii
0
0
0
0
0 0
0 1 0 − Kd / J M
0
0
0 0 G = 0 , 0 1 / L
0 − M r 0 / H 0 C= 0 0 0
To illustrate the proposed approach to robust observer design, use the singular value decomposition of the matrix L with the matrix U L
0 , 0 KM / JM − R / L 0
− Kω / L
0 0.5 0.5 − 0.7071 0 − 0.5 − 0.5 − 0,7071 UL = 0 0,7071 0.5 − 0.5 − 0,7071 − 0.5 0.5 0 0 0 0 0
1 0 0 0 0 H = 0 1 0 0 0 , 0 0 0 1 0
0 0 0 − M d0 / JM 0
with σ1 = σ 2 = 1,4142 , σ 3 = σ 4 = σ 5 = 0 . It can be shown that last three columns of the matrix U L are linear combinations of the matrix L* rows and vice verse. Therefore one can let Φ = L* , and conditions (12) and (14) hold in this case. Equalities (11) and (13) give the following: R = (−1 1 0) , H * = (1 0 0) ,
0 C ir / H 0 , 0 −C / JM 0
0 − ( K r + h) / H 0 − 1 0 F* = 0 0 K M / J M , 0 0 − R / L
sign( A1 x ) ϕ( x, u ) = sign( A2 x), Bl ( A3 x )
0 − ( K r + h) / H 0 − 1 0 J* = 0 0 − K d / J M − 1 ; − K ω / L 0 0
A1 = [0 1 0 0 0], A2 = [0 0 0 1 0], A3 = [−i r
− M r 0 / H 0 C* = ΦC = 0 0
0 1 0 0] .
Assume that faults and unknown inputs are described as follows: 1 1 0 0 0 L= . 0 0 1 1 0
as
follows:
A*1 = [1 0 0 1 0 0] , 0 1] .
C r ir Bl( y 3 − i r y1 − x*2 )) − y 2 , &x*2 = (1 / J M )(− K d y 3 + K M x*3 − M d 0 sign( y 3 ) − C r Bl( y3 − i r y1 − x*2 )) − y 3 ,
It can be shown that conditions (8)-(10) are satisfy. The matrices R and Φ obtained from equations (7) and (6) respectively is as follows: −1 1 0 0 0 Φ= . 0 0 1 0 0
The nonlinear observer is described by the following equations:
C r i r Bl( x*2 − i r y1 )) − y 2 , &x*2 = y 3 ; y* = x*1 , r = y 2 − y1 − y* .
A*i
x&*1 = (1 / H 0 )(−( K r + h) y 2 − M r 0 sign( y 2 ) +
− 1 1 0 0 0 L* = 0 0 − 1 1 0 . 0 0 0 0 1
x&*1 = (1 / H 0 )(−( K r + h) y 2 − M r 0 sign( y 2 ) +
0
C ir / H 0 − C / J M . 0
A*2 = [0 0 0 0 0 1], A*3 = [0 −1 0 −i r As a result, the observer is described as follows:
It follows from the matrix L* definition that
R = (−1 1 0) ,
0 − M d0 / JM
Besides G* = ΦG = [0 0 1 / L ]T . Equation (3) gives the matrices
T
K = [0 1 0 0 0]T ,
0 0 0 0 1
x&*3 = (1 / L)(− K ω y 3 − Rx*3 + u ), y* = x*1 ,
r (t ) = y 2 − y1 − y* .
For simplicity, a gain matrix function to obtain the asymptotically stable observer does not used. For simulating, numerical values of the parameters are the following: K r = 0.01 Nms, K d = 10 −5 Mms, C r = 2.0 Nm, i r = 100 , σ = 1.0 rad, M r 0 = 10.0 Nmrad, M d 0 = 0.15 Nmrad, K ω = 0.02 Vs/rad, J M = 10 −4 kgm2, K M = 0.02 Nmrad/A, L=0.004 H, R=0.4 Ω, H 0 = 1 , h = 15 . In the simulation, the initial states of the system under consideration and the observer are assumed to be equal to zero. The control
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value is u=const=10. The vector function ρ(t ) is modeled by two independent values with random distribution on the interval [–0.01, +0.01]. The fault is modeled by changing the parameter M r 0 from 10.0 to 12.0 at once at t=30. Fig. 2 illustrates the first observer residual behavior. Clearly this residual is not invariant with respect to the unknown input ρ(t ) ; sensitivity to the fault is not evident. Fig. 3 illustrates the second observer residual behavior which shows that this residual is invariant with respect to ρ(t ) ; sensitivity to the fault is evident.
Shumsky, A. (2002). Robust analytical redundancy relation for fault diagnosis in nonlinear systems. Asian J. Control, 4, pp. 159-170. Zhirabok, A. (1997). Fault detection and isolation: linear and nonlinear systems. Preprints of the IFAC Symposium SAFEPROCESS’97, pp. 903-908, UK, Hall, 1997. Zhirabok, A. and S. Usoltsev (2002). Fault diagnosis for nonlinear dynamic systems via linear methods. CD ROM Proceedings of the 15th World Congress IFAC, Spain, Barcelona, 2002.
7. CONCLUSION The problem of robust fault diagnosis in nonlinear dynamic systems with no differentiable nonlinearities has been studied. To solve this problem, so-called logic-dynamic approach has been developed. This approach consists of the following main steps: replacing the initial nonlinear system by certain linear logic-dynamic system, obtaining the bank of linear logic-dynamic observers, and transforming them into the nonlinear ones. It has been shown that this approach allows taking into account different types of nonlinearities by linear methods. Three existence conditions to design the observer invariant with respect to the unknown parameters and unknown inputs have been obtained. A new approach to robust nonlinear observer design has been suggested.
Fig.1. Manipulator 0.005
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It follows from the logic-dynamic approach that it can be applied to discrete-time systems.
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Acknowledgments
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This work was supported by Russian Foundation of Basic Researches 05-08-18240 and 07-08-00102.
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REFERENCES
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Cassavola, A., Famularo, D. and G. France (2008). Robust fault detection of uncertain systems via quasi-LMIs. Automatica, 44, pp. 289-295. Chen, J. and R. Patton (1999). Robust model-based fault diagnosis. Kluwer Academic Publishers, Boston. Frank, P. and S. Ding (1997). Survey of robust residual generation and estimation methods in observer-based fault detection systems. Journal of Process Control, 7, pp. 403-424. Gauthier, J. and I. Kupca (2000). Deterministic observation theory and application. Cambridge University Press, Cambridge Lou, X., Willsky A. and G. Verghese (1986). Optimally robust redundancy relations for failure detection in uncertain systems. Automatica, 22, pp. 333-344. Mironovsky, L. (1979). Functional diagnosis of linear dynamic systems. Automation and Remote Control, No 8, pp. 120-128. Misawa, E.A. and J.K. Hedrick (1989). Nonlinear observers – a state of the art survey. Journal of dynamic systems, measurements and control, 111, pp. 344-352.
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