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DESIGN AND ANALYSIS OF A NONLINEAR OBSERVER FOR A HYDRAULIC SERVO SYSTEM
DESIGN AND ANALYSIS OF A NONLINEAR OBSERVER FOR A HYDRAULIC SERVO SYSTEM Maria Werlefors ∗ Alexander Medvedev ∗
∗
Information Technology, Uppsala University, SE-751 05 Uppsala, SWEDEN [email protected] [email protected]
Abstract: A nonlinear observer with static feedback is investigated as a tool for leakage detection in a hydraulic servo system with velocity feedback, frequently employed in vehicular technology. It is shown that, under reasonable assumptions, the estimation error dynamics have only one feasible stationary point and that is in the origin. Furthermore, the stationary point can be asymptotically stabilized by static observer feedback. Simulations indicate that the observer reliably detects and isolates hydraulic fluid leakage of two different kinds and possesses appealing convergence and parameter robustness properties. Copyright © 2007 IFAC Keywords: Hydraulic actuators, observers, nonlinear systems, fault detection
1. INTRODUCTION
Hydraulic servo systems are typically used in heavyduty, safety and mission-critical industrial and military applications, (Jelali and Kroll, 2003). They are widely appreciated due to such merits as high power, fast response and compact size. Several shortcomings are as well known, e. g. the risk of fluid leakage and contamination. Furthermore, the dynamics of hydraulic servo systems are highly nonlinear and uncertain, caused by fluid compressibility, friction, and aging. To assure service reliability and longevity of hydraulic systems, on-line monitoring of hydraulic components is desirable. This stimulates the increasing demand for fault detection and monitoring of hydraulic equipment. Hydraulic fluid leakage is the most critical fault to the hydraulic system function. Leakage detection can be addressed within the framework of model-based fault detection and isolation (FDI), see for instance (Saberi et al., 2000). Since a model of a hydraulic system is always uncertain, it is reasonable to use it together with a state estimator, suitably an observer. However, some other techniques can also work well in a harsh industrial setup, (Johansson and Medvedev, 1999). Parity space techniques, parameter estimation techniques, statistical techniques and observer-based techniques
present the most important analytical redundancy approaches to fault isolation, (Hammouri et al., 2002). Most of the available methods are well understood in case of linear time-invariant (LTI) dynamics. Nonlinear FDI is a fast developing but yet not quite established research area. Many notions and approaches developed for LTI systems are not easy to translate to nonlinear plants. Nonlinear observers have however been studied for quite a while and come in useful solving FDI problems. A geometric approach to nonlinear FDI implicitly based on nonlinear observability is, for instance, presented in (De Persis and Isidori, 2001). The field of FDI for hydraulic systems is still emerging. Some recent contributions to the area include a nonlinear analytical redundancy method applied to a tele-operated, hydraulically driven robot where a parity space method has been used (Leuschen et al., 2003). A nonlinear observer-based FDI for a hydraulic system has been performed in different ways in (Hammouri et al., 2002) and (Khan et al., 2005). The model of an electro-hydraulic servo-positioning system with velocity feedback used in (Khan et al., 2005) is typical to many vehicular applications and serves as a good starting point for developing an observer-based leakage detection and isolation system. Therefore, the model and the observer structure
suggested in (Khan et al., 2005) are followed up and studied in more detail in this paper. The paper is organized as follows. First the mathematical model of the system in hand is briefly summarized. Then stability properties of the nonlinear observer with static feedback treated in (Khan et al., 2005) are investigated analytically. Further, convergence and robustness of the observer as well as its ability to detect leakage are explored by simulation.
0 βωkd x4 g (x ) 1 2 Vi f (x, u) = βωkd x4 g (x ) − 2 3 Vo ksp u τ
2.1 Steady-state 2. HYDRAULIC SERVO SYSTEM A standard model of an electro-hydraulic servopositioning system with velocity feedback is treated in this paper. Velocity feedback is a way of reducing the influence of friction on the system performance since friction is typically modeled as a constant term plus a term proportional to velocity. The model equations are according to (Khan et al., 2005) and read as 1 (−fd x1 + Ai x2 − Ao x3 ) m β x˙ 2 = (−Ai x1 + kd ωx4 g1 (x2 )) Vi β x˙ 3 = (Ao x1 − kd ωx4 g2 (x3 )) Vo 1 x˙ 4 = (ksp u − x4 ) (1) τ where x1 is the velocity of the actuator, x2 and x3 are the input and output line pressures, respectively, x4 is the spool displacement and u is the input (position reference) signal to the servo system. For brevity of notation, the functions g1 (·) and g2 (·) are used for description of the nonlinear part of the system x˙ 1 =
(p P − x2 , g1 (x2 ) = p s x2 − Pr , (p x − Pr , g2 (x3 ) = p 3 Ps − x3 ,
x4 ≥ 0 x4 < 0
(2)
x4 ≥ 0 x4 < 0
(3)
Parameter values used in this paper are borrowed from (Khan et al., 2005) to facilitate comparison and summarized in Table 1. System (1) can be written in vector form as x˙ = Ax + f (x, u)
(4)
y = Cx where the output signal is actuator velocity yielding
In the sequel, knowledge about system (4) at steady state is utilized. Let x0 = x10 x20 x30 x40 be a steady-state value of x, i. e. for x˙ = 0. Then x40 = ksp u0 , where u0 is the corresponding steady-state value of u. The choice of equation for x20 and x30 corresponding to a steady state of (1) depends on the sign of u0 . Evaluating x10 , x20 , x30 as function of u0 gives for x40 > 0 (kd ωksp u0 )2 fd + x10 (u0 ) = − 2(Ai 3 + Ao 3 ) s 2 (kd ωksp u0 )2 fd (kd ωksp u0 )2 (Ps Ai − Pr Ao ) + 3 3 2(Ai + Ao ) Ai 3 + Ao 3 2 Ai 2 x20 (u0 ) = Ps − 2 x10 (u0 ) (kd ωksp u0 ) Ao 2 2 x30 (u0 ) = 2 x10 (u0 ) + Pr (kd ωksp u0 ) (5) Similarly, but for x40 < 0 (kd ωksp u0 )2 fd − x10 (u0 ) = 2(Ai 3 + Ao 3 ) s 2 (kd ωksp u0 )2 fd (kd ωksp u0 )2 (Ps Ao − Pr Ai ) + 2(Ai 3 + Ao 3 ) Ai 3 + Ao 3 2 Ai 2 x20 (u0 ) = 2 x10 (u0 ) + Pr (kd ωksp u0 ) Ao 2 2 x30 (u0 ) = Ps − 2 x10 (u0 ) (kd ωksp u0 ) (6) Note also that x4 = 0 implies u0 = 0 and the steady-states of (1) are obtained by solving a linear underdermined algebraic system and the steady-state values are x10 = 0; x20 =
Ao x30 Ai
(7)
for an arbitrary x30 .
C = [1 0 0 0] The linear and nonlinear terms of (4) are fd Ai Ao −m m − m 0 βAi − 0 0 0 V i A= βAo 0 0 0 Vo 1 0 0 0 − τ
3. NON-LINEAR OBSERVER Following (Khan et al., 2005), a nonlinear observer with the static feedback gain K x ˆ˙ = Aˆ x + f (ˆ x, u) + K(y − C x ˆ)
(8)
is used for the purpose of state estimation in (4). The observer equations are
1 ˆ1 + Ai x ˆ2 − Ao x ˆ3 ) + K1 (y − x ˆ1 ) x ˆ˙ 1 = (−fd x m β x ˆ˙ 2 = − Ai x ˆ1 + kd ωˆ x4 g1 (ˆ x2 ) + K2 (y − x ˆ1 ) Vi β x ˆ˙ 3 = Ao x ˆ1 − kd ωˆ x4 g2 (ˆ x3 ) + K3 (y − x ˆ1 ) Vo 1 x ˆ˙ 4 = (ksp u − x ˆ4 ) + K4 (y − x ˆ1 ) (9) τ
yields a biased estimate. Therefore, the observer feedback should be designed so that the origin is an asymptotically stable stationary point of the error dynamics and all other stationary points within the operation envelope are unstable or cannot be reached under all plausible operation conditions. The result below provides the necessary and sufficient conditions for such a design by means of the feedback gain K in observer (9).
where Ki , i = 1, . . . , 4 are the elements of K. For x4 ≥ 0 the state estimation error e = x − x ˆ obeys
To cover all possible situations, the cases u0 > 0, u0 = 0 and u0 < 0 need to be considered. For brevity, introduce the following notation
1 (−fd e1 + Ai e2 − Ao e3 ) − K1 e1 (10) m p β e˙ 2 = − Ai e1 + kd ω x4 Ps − x2 − Vi p (x4 − e4 ) Ps − x2 + e2 − K2 e1 p β e˙ 3 = Ao e1 − kd ω x4 x3 − Pr − Vo p (x4 − e4 ) x3 − e3 − Pr − K3 e1 e˙ 1 =
1 e˙ 4 = − e4 − K4 e1 τ
For x4 < 0 the error equations for e2 and e3 are exchanged for p β − Ai e1 + kd ω x4 x2 − Pr − Vi p (x4 − e4 ) x2 − e2 − Pr − K2 e1 p β e˙ 3 = Ao e1 − kd ω x4 Ps − x3 − Vo p (x4 − e4 ) Ps − x3 + e3 − K3 e1 (11)
e˙ 2 =
The residual e1 = y − C x ˆ will be used for fault detection. Therefore, it should ideally be zero under nominal process conditions and a function of the fault severity otherwise.
4. OBSERVER ANALYSIS In order to conclude on whether observer (8) can be used for FDI ends, the nonlinear dynamics of (10) and (11) have to be studied with respect to stationary points, their stability and attraction domain. Since both the plant and the observer are dynamic systems, their joint stationary points are considered, i. e. such that x˙ = e˙ = 0 (12)
4.1 Stationary points One of the problems in the design of nonlinear observers is multiple stable stationary points in the state estimation dynamics. When converged to some other point in the state space than the origin, the observer
βAi fd − K1 ; R 2 = − K2 ; R4 = −K4 τ m Vi βωkd βωkd βAo − K3 ; R 5 = ; R6 = R3 = Vo Vi Vo R1 =
Proposition 1. Consider error system (10,11) for u0 6= 0. The origin is the only stationary point if and only if 2 c b − <0 (13) 2a a where
a = −R1 R4 2 For u0 > 0, b and c take the following values R4 2 (−Ai Ps + Ai x20 − Ao x30 + m 2 Ai R2 Ao R3 2 Ao Pr ) + + mR5 2 mR6 2
b = 2R1 R4 x40 +
2x40 R4 Ai Ps − Ai x20 + Ao x30 m 2x A R √x − P o 3 30 r 40 −Ao Pr + − m R6 √ Ai R2 Ps − x20 R5
c = −R1 x240 +
while for u0 < 0
R4 2 (Ai x2 − Ai Pr − Ao x30 + m Ai R2 2 Ao R3 2 Ao Ps ) − 2 − mR5 mR6 2 2x40 R4 c = −R1 x240 + (Ai Pr − Ai x20 + Ao x30 m √ 2x40 Ai R2 x20 − Pr −Ao Ps ) + ( − m R5 √ Ao R3 Ps − x30 ) R6 b = 2R1 R4 x40 +
PROOF. The proof is straightforward by combining (10), (5), (12), and (11), (6), (12), respectively. Solving for e10 , e20 , e30 and e40 for u0 > 0 gives one solution at the origin and one solution at
b e10 = − ± 2a
r
b 2 c − 2a a √ (R5 x40 Ps − x20 + R2 e10 )2 e20 = Ps − x20 − (x40 + R4 e10 )2 R5 2 √ (R6 x40 x30 − Pr − R3 e10 )2 − x30 + Pr e30 = (x40 + R4 e10 )2 R6 2 e40 = C4 e10 For u0 < 0, the equations for e20 and e30 are
e20 = x20 − Pr −
√
(R5 x40 x20 − Pr − R2 e10 )2 (x40 − R4 e10 )2 R5 2
√ (R6 x40 Ps − x30 + R3 e10 )2 e30 = − Ps + x30 (x40 − R4 e10 )2 R6 2 The complex expressions determining the conditions on (13) are not easily analyzed analytically. Numerical analysis shows that K1 , K2 and K3 have no great effect on the condition for the origin being the only stationary point. K4 on the other hand has to take a value close to 0.01 for the parameters in Table 1 regardless of the sign of u0 . Table 1. Model parameters and their values Parameter
Value
Unit
Comment
Ps Pr fd β Ai Ao Vi Vo τ kd ω ksp m
108 0 1600 5 · 109 2.0 · 10−3 1.5 · 10−3 1.5 · 10−3 1.5 · 10−3 3.3 · 10−2 3.2 · 10−2 2 · 10−2 1.6 · 10−3 20
Pa Pa N m−1 s Pa m2 m2 m3 m3 s m3/2 kg −1/2 m mV −1 kg
Suply pressure Return pressure Force Effective bulk modulus Piston area, head side Piston area, rod side Volume, head side Volume, rod side Time constant Parameter Orifice area gradient Parameter Mass of actuator
Proposition 2. System (10,12) has stationary points only at the origin for u0 = 0 if and only if
4.2 Stability By enforcing additional limitations on the observer gain, stability of the origin and instability of other stationary points can be achieved. Linearizing (10,11) around the origin gives e˙ = (A0 − KC)e
where
fd Ai Ao − − 0 m m m βA βωk x d 40 − i 0 a24 V 2V g (x ) i i 1 20 A0= βA −βωkd x40 o a34 0 Vo 2Vo g2 (x30 ) 1 0 0 0 − τ and
a24 =
2
βωkd βωkd g1 (x20 ); a34 = − g2 (x30 ) Vi Vo
Asymptotic stability of linearized system (15) is equivalent to asymptotic stability of the origin for solutions of (10, 11). Analytical expressions for eigenvalues of A0 − KC are awkward and do not give much insight into the observer design aspects. Numerical calculations of the eigenvalues as function of K1 , K2 , K3 and K4 are more useful for observer design purpose. K2 and K3 have very small effect on the eigenvalues and the stationary points. Those are therefore set to zero. K1 and K4 have larger impact on the eigenvalues, although the simulation results are not shown here due to lack of space. To guarantee that the origin is the only stable stationary point for u0 = 0, the other stationary point has to be unstable. However, calculations show that this cannot be achieved for the considered parameter values without violating (13). Instead, the observer feedback has to be used to place both stationary points at the origin. Solving (14) gives one solution as K2 = 0, K3 = 0 and K4 = 0.0114 which is feasible. So, the choice of the feedback as K T = [ 250 0 0 0.0114 ]
2
(15)
(16)
Ai R2 Ao R3 + +Ao Pr R4 2 = Ai Ps R4 2 (14) R5 2 R6 2
satisfies (12), for all u0 , and asymptotically stabilizes linearized system (15). K1 is chosen large because it makes the observer faster and less oscillating.
PROOF. Follows by combining (10), (12) and (7) and solving for e10 .
Fig. 1 depicts the root locus for (15) when u0 rises and K is chosen as in (16). For the same choice of K the origin is a stable stationary point for −1 ≤ u0 ≤ 1.
For u0 = 0, the error system has two stationary points. One stationary point is at the origin and the location of the other, as Proposition 2 shows, can be assigned by choosing K2 , K3 and K4 . Since the origin has to be the only stable stationary point, there are two possible design approaches. Either the origin is made a stable stationary point while the other stationary point becomes unstable, or the observer feedback is chosen to assign both stationary points to the origin.
4.3 Attraction domain Once it is established that the origin can be rendered the only stationary point of the observer error dynamics and furthermore asymptotically stabilized, the attraction domain of it becomes a critical design issue. Simulations with starting points far from the origin
imaginary axis
Fig. 1. Root locus of the linearized observer error dynamics in the complex plane for the stationary input signal 0 < u0 < 1 V. 200 0
5. DETECTION OF LEAKAGE The observer under consideration is intended for detection of leakage in the system. Models for leakage in the input line and in the output line have been developed from the fluid flow equations in (Khan et al., 2005).
−200 −200
−150
−100 real axis
−50
0
are performed to illustrate its extension. The initial conditions x(0) = [ 1 105 105 0.1 ]T ; x ˆ(0) = [ 0 0 0 0 ]T are chosen. This is a physically reasonable starting point with an initial actuator velocity of 1 ms−1 , input and output line pressures equal to atmospheric pressure, and an initial spool displacement of 0.1 m. The observer converges nicely for the gain matrix as in (16) and for −1 ≤ u0 ≤ 1.
5.1 Leakage on the actuator input line A leakage on the actuator input line has been modeled as an additional flow, qli in the second state equation p qli = Ali kd x2 − Pa where Ali is the leak area and Pa is the atmospheric pressure. The state equation now reads β x˙ 2 = − Ai x1 + kd ωx4 g1 (x2 ) Vi Fig. 4 and Fig. 5 show the residual when a leakage Fig. 4. Residual when a leakage is present in the input line from time t = 0.2 to t = 0.5 for x4 ≥ 0.
Fig. 2. The residual for the nominal case and β reduced by 30%, 50% residual
0.5
0
0.05
0.1
0.15 time (s)
0.2
0.25
0.3
residual
0% 30% 50%
0
−0.5
0
0.05
0.1
0.15 time (s)
0.2
0.25
A =10
−0.4
A =10
−6
li
−5
li
0
0.2
0.4
0.6
0.8
1
time (s)
Fig. 5. Residual when a leakage is present in the input line from time t = 0.2 to t = 0.5 for x4 < 0. 6
−6
A =10
4
li
−5
A =10 li
2 0 0
0.2
0.4
0.6
0.8
1
time (s)
Fig. 3. The residual for the nominal case and τ increased by 30%, 50% 0.5
−0.2
−2
0% 30% 50%
0
−0.5
0
residual
Observer robustness against plant parameter variations has been investigated by altering the model parameters β and τ . The bulk modulus β has a great influence on the dynamics of the hydraulic system (Jelali and Kroll, 2003) and the time constant τ affects the valve dynamics. Those two parameters alter the dynamics in the hydraulic system in different ways and are therefore of interest regarding the robustness of the system. Fig. 2 shows the residual for the nominal case, as well as when β is reduced by 30% and by 50%. Similar plots for increased τ are given in Fig. 3.
residual
0.2
4.4 Robustness
0.3
occurs at time t = 0.2 and persists until t = 0.5 for x4 > 0 and x4 < 0 respectively. Observer response to leakages with the leak area of 10−6 and 10−5 is depicted. The sign of x4 determines the sign of the residual. The leakage is timely detected and clearly indicated by the steady state value of the residual. The latter can also be used for estimation of the leakage severity. Although, note that the mapping from the residual to the leakage severity differs between the two cases. The special case x4 = 0 has been handled separately. In steady state, x4 = 0 is equivalent to u = 0. Modeling a leakage occurring from t = 0.4 until t = 0.7 generates the residual shown in Fig. 6. As the leakage appears, the residual deviates from zero reacting to a change in the piston velocity, although it soon reaches a new steady state which causes the residual to return to zero. Apparently, a leakage cannot be detected from steady state residual when u = 0.
residual
Fig. 6. Residual under a leakage in the input line from time t = 0.4 to t = 0.7 for x4 = 0. 1 0 −1
0
0.2
0.4
0.6
0.8
1
time (s)
5.2 Leakage on the actuator output line The model of leakage on the actuator output line is similar to that of the input line. It has been modeled as a change in the output flow of the actuator, i. e. an additional flow, qlo in the third state equation p qlo = Alo kd x3 − Pa where Alo is the leak area and Pa is the atmospheric pressure. The state equation now reads x˙ 3 =
β (Ao x1 − kd ωx4 g2 (x3 ) − qlo ) Vo
In Fig. 7 and Fig. 8 respectively, the observer residuals Fig. 7. Residual under a leakage in the output line from time t = 0.2 to t = 0.5 for x4 ≥ 0. residual
0.4
−6
A =10 lo
−5
A =10
0.2
lo
0 −0.2
0
0.2
0.4
0.6
0.8
1
time (s)
Fig. 8. Residual under a leakage in the output line from time t = 0.2 to t = 0.5 for x4 < 0. residual
0 −6
−2
A =10 lo
−5
A =10
−4
lo
0
0.2
0.4
0.6
0.8
1
time (s)
are shown for the case of output line leakage under similar conditions as for the case of input line leakage. Once again, the leakage detection is prompt and well pronounced in the steady state response. Furthermore, it is straightforward to distinguish between the two types of leakage since the difference between input and output leakage is in the sign of the residual. Since the sign of the residual changes together with the sign of x4 , it is, therefore, necessary to know the sign of x4 , which is equivalent to keeping track of the direction of the flow. The observer residual under a leakage in the output line for x4 = 0 is similar to that for a leakage in the input line.
6. CONCLUSIONS The feasibility of a nonlinear observer for a hydraulic servo system as a fault detection tool is investigated. The state estimation error is shown to converge to zero under weak assumptions. Simulations indicate that the attraction region of the origin is sufficiently large and the convergence is robust against significant process parameter alternations. It is also shown that the same observer design is feasible for positive and negative values of the spool displacement. The observer residual timely and reliably indicates the presence of a leakage for non-zero reference signal. The leakage severity can be estimated from the stationary value of the residual while input and output line leakages can be distinguished between by means of its sign. The sign of the spool displacement toggles the sign of the residual. Since the sign of spool displacement is determined by the direction of the flow in the lines, it is sufficient to be able to distinguish between flows to and from the piston. When the reference signal is zero, the spool displacement also becomes zero at steady state. It is therefore not possible to detect leakage by the considered approach in steady state under velocity feedback and zero reference signal to the system. 7. ACKNOWLEDGMENT This work has been carried out in cooperation with Volvo Aero within the Swedish National Avionic Program. The authors would like to thank Dr. Melker H¨arefors for discussing the results of this paper. 8. REFERENCES De Persis, Claudio and Alberto Isidori (2001). A geometric approach to nonlinear fault detection and isolation. IEEE Transactions on Automatic Control 46(6), 853–865. Hammouri, H., P. Kabore, S. Othman and J. Biston (2002). Failure diagnosis and nonlinear observer. application to a hydraulic process. Journal of the Franklin Institure 339, 455–478. Jelali, Mohieddine and Andreas Kroll (2003). Hydraulic Servo-systems : Modelling, Identification and Control. Springer. Johansson, Andreas and Alexander Medvedev (1999). Model-based leakage detection in a pulverized coal injection vessel. IEEE Transactions on Control Systems Technology 7(6), 675–682. Khan, H., S. C. Abou and N. Sepehri (2005). Nonlinear observer-based fault detection technique for electro-hydraulic servo-positioning systems. Mechatronics 15, 1037–1059. Leuschen, Martin. L., Ian. D. Walker and Joseph. R. Cavallaro (2003). Nonlinear fault detection for hydraulic systems. In: Fault diagnosis and fault tolerance for mechatronic systems: recent advances (F. Caccavale and L. Villani, Eds.). Springer-Verlag. Berlin Heidelberg. Saberi, A., A. A. Stoorvogel, P. Sannuti and H. Niemann (2000). Fundamental problems in fault detection and identification. International journal of robust and nonlinear control 10, 1209–1236.
∗
Information Technology, Uppsala University, SE-751 05 Uppsala, SWEDEN [email protected] [email protected]
Abstract: A nonlinear observer with static feedback is investigated as a tool for leakage detection in a hydraulic servo system with velocity feedback, frequently employed in vehicular technology. It is shown that, under reasonable assumptions, the estimation error dynamics have only one feasible stationary point and that is in the origin. Furthermore, the stationary point can be asymptotically stabilized by static observer feedback. Simulations indicate that the observer reliably detects and isolates hydraulic fluid leakage of two different kinds and possesses appealing convergence and parameter robustness properties. Copyright © 2007 IFAC Keywords: Hydraulic actuators, observers, nonlinear systems, fault detection
1. INTRODUCTION
Hydraulic servo systems are typically used in heavyduty, safety and mission-critical industrial and military applications, (Jelali and Kroll, 2003). They are widely appreciated due to such merits as high power, fast response and compact size. Several shortcomings are as well known, e. g. the risk of fluid leakage and contamination. Furthermore, the dynamics of hydraulic servo systems are highly nonlinear and uncertain, caused by fluid compressibility, friction, and aging. To assure service reliability and longevity of hydraulic systems, on-line monitoring of hydraulic components is desirable. This stimulates the increasing demand for fault detection and monitoring of hydraulic equipment. Hydraulic fluid leakage is the most critical fault to the hydraulic system function. Leakage detection can be addressed within the framework of model-based fault detection and isolation (FDI), see for instance (Saberi et al., 2000). Since a model of a hydraulic system is always uncertain, it is reasonable to use it together with a state estimator, suitably an observer. However, some other techniques can also work well in a harsh industrial setup, (Johansson and Medvedev, 1999). Parity space techniques, parameter estimation techniques, statistical techniques and observer-based techniques
present the most important analytical redundancy approaches to fault isolation, (Hammouri et al., 2002). Most of the available methods are well understood in case of linear time-invariant (LTI) dynamics. Nonlinear FDI is a fast developing but yet not quite established research area. Many notions and approaches developed for LTI systems are not easy to translate to nonlinear plants. Nonlinear observers have however been studied for quite a while and come in useful solving FDI problems. A geometric approach to nonlinear FDI implicitly based on nonlinear observability is, for instance, presented in (De Persis and Isidori, 2001). The field of FDI for hydraulic systems is still emerging. Some recent contributions to the area include a nonlinear analytical redundancy method applied to a tele-operated, hydraulically driven robot where a parity space method has been used (Leuschen et al., 2003). A nonlinear observer-based FDI for a hydraulic system has been performed in different ways in (Hammouri et al., 2002) and (Khan et al., 2005). The model of an electro-hydraulic servo-positioning system with velocity feedback used in (Khan et al., 2005) is typical to many vehicular applications and serves as a good starting point for developing an observer-based leakage detection and isolation system. Therefore, the model and the observer structure
suggested in (Khan et al., 2005) are followed up and studied in more detail in this paper. The paper is organized as follows. First the mathematical model of the system in hand is briefly summarized. Then stability properties of the nonlinear observer with static feedback treated in (Khan et al., 2005) are investigated analytically. Further, convergence and robustness of the observer as well as its ability to detect leakage are explored by simulation.
0 βωkd x4 g (x ) 1 2 Vi f (x, u) = βωkd x4 g (x ) − 2 3 Vo ksp u τ
2.1 Steady-state 2. HYDRAULIC SERVO SYSTEM A standard model of an electro-hydraulic servopositioning system with velocity feedback is treated in this paper. Velocity feedback is a way of reducing the influence of friction on the system performance since friction is typically modeled as a constant term plus a term proportional to velocity. The model equations are according to (Khan et al., 2005) and read as 1 (−fd x1 + Ai x2 − Ao x3 ) m β x˙ 2 = (−Ai x1 + kd ωx4 g1 (x2 )) Vi β x˙ 3 = (Ao x1 − kd ωx4 g2 (x3 )) Vo 1 x˙ 4 = (ksp u − x4 ) (1) τ where x1 is the velocity of the actuator, x2 and x3 are the input and output line pressures, respectively, x4 is the spool displacement and u is the input (position reference) signal to the servo system. For brevity of notation, the functions g1 (·) and g2 (·) are used for description of the nonlinear part of the system x˙ 1 =
(p P − x2 , g1 (x2 ) = p s x2 − Pr , (p x − Pr , g2 (x3 ) = p 3 Ps − x3 ,
x4 ≥ 0 x4 < 0
(2)
x4 ≥ 0 x4 < 0
(3)
Parameter values used in this paper are borrowed from (Khan et al., 2005) to facilitate comparison and summarized in Table 1. System (1) can be written in vector form as x˙ = Ax + f (x, u)
(4)
y = Cx where the output signal is actuator velocity yielding
In the sequel, knowledge about system (4) at steady state is utilized. Let x0 = x10 x20 x30 x40 be a steady-state value of x, i. e. for x˙ = 0. Then x40 = ksp u0 , where u0 is the corresponding steady-state value of u. The choice of equation for x20 and x30 corresponding to a steady state of (1) depends on the sign of u0 . Evaluating x10 , x20 , x30 as function of u0 gives for x40 > 0 (kd ωksp u0 )2 fd + x10 (u0 ) = − 2(Ai 3 + Ao 3 ) s 2 (kd ωksp u0 )2 fd (kd ωksp u0 )2 (Ps Ai − Pr Ao ) + 3 3 2(Ai + Ao ) Ai 3 + Ao 3 2 Ai 2 x20 (u0 ) = Ps − 2 x10 (u0 ) (kd ωksp u0 ) Ao 2 2 x30 (u0 ) = 2 x10 (u0 ) + Pr (kd ωksp u0 ) (5) Similarly, but for x40 < 0 (kd ωksp u0 )2 fd − x10 (u0 ) = 2(Ai 3 + Ao 3 ) s 2 (kd ωksp u0 )2 fd (kd ωksp u0 )2 (Ps Ao − Pr Ai ) + 2(Ai 3 + Ao 3 ) Ai 3 + Ao 3 2 Ai 2 x20 (u0 ) = 2 x10 (u0 ) + Pr (kd ωksp u0 ) Ao 2 2 x30 (u0 ) = Ps − 2 x10 (u0 ) (kd ωksp u0 ) (6) Note also that x4 = 0 implies u0 = 0 and the steady-states of (1) are obtained by solving a linear underdermined algebraic system and the steady-state values are x10 = 0; x20 =
Ao x30 Ai
(7)
for an arbitrary x30 .
C = [1 0 0 0] The linear and nonlinear terms of (4) are fd Ai Ao −m m − m 0 βAi − 0 0 0 V i A= βAo 0 0 0 Vo 1 0 0 0 − τ
3. NON-LINEAR OBSERVER Following (Khan et al., 2005), a nonlinear observer with the static feedback gain K x ˆ˙ = Aˆ x + f (ˆ x, u) + K(y − C x ˆ)
(8)
is used for the purpose of state estimation in (4). The observer equations are
1 ˆ1 + Ai x ˆ2 − Ao x ˆ3 ) + K1 (y − x ˆ1 ) x ˆ˙ 1 = (−fd x m β x ˆ˙ 2 = − Ai x ˆ1 + kd ωˆ x4 g1 (ˆ x2 ) + K2 (y − x ˆ1 ) Vi β x ˆ˙ 3 = Ao x ˆ1 − kd ωˆ x4 g2 (ˆ x3 ) + K3 (y − x ˆ1 ) Vo 1 x ˆ˙ 4 = (ksp u − x ˆ4 ) + K4 (y − x ˆ1 ) (9) τ
yields a biased estimate. Therefore, the observer feedback should be designed so that the origin is an asymptotically stable stationary point of the error dynamics and all other stationary points within the operation envelope are unstable or cannot be reached under all plausible operation conditions. The result below provides the necessary and sufficient conditions for such a design by means of the feedback gain K in observer (9).
where Ki , i = 1, . . . , 4 are the elements of K. For x4 ≥ 0 the state estimation error e = x − x ˆ obeys
To cover all possible situations, the cases u0 > 0, u0 = 0 and u0 < 0 need to be considered. For brevity, introduce the following notation
1 (−fd e1 + Ai e2 − Ao e3 ) − K1 e1 (10) m p β e˙ 2 = − Ai e1 + kd ω x4 Ps − x2 − Vi p (x4 − e4 ) Ps − x2 + e2 − K2 e1 p β e˙ 3 = Ao e1 − kd ω x4 x3 − Pr − Vo p (x4 − e4 ) x3 − e3 − Pr − K3 e1 e˙ 1 =
1 e˙ 4 = − e4 − K4 e1 τ
For x4 < 0 the error equations for e2 and e3 are exchanged for p β − Ai e1 + kd ω x4 x2 − Pr − Vi p (x4 − e4 ) x2 − e2 − Pr − K2 e1 p β e˙ 3 = Ao e1 − kd ω x4 Ps − x3 − Vo p (x4 − e4 ) Ps − x3 + e3 − K3 e1 (11)
e˙ 2 =
The residual e1 = y − C x ˆ will be used for fault detection. Therefore, it should ideally be zero under nominal process conditions and a function of the fault severity otherwise.
4. OBSERVER ANALYSIS In order to conclude on whether observer (8) can be used for FDI ends, the nonlinear dynamics of (10) and (11) have to be studied with respect to stationary points, their stability and attraction domain. Since both the plant and the observer are dynamic systems, their joint stationary points are considered, i. e. such that x˙ = e˙ = 0 (12)
4.1 Stationary points One of the problems in the design of nonlinear observers is multiple stable stationary points in the state estimation dynamics. When converged to some other point in the state space than the origin, the observer
βAi fd − K1 ; R 2 = − K2 ; R4 = −K4 τ m Vi βωkd βωkd βAo − K3 ; R 5 = ; R6 = R3 = Vo Vi Vo R1 =
Proposition 1. Consider error system (10,11) for u0 6= 0. The origin is the only stationary point if and only if 2 c b − <0 (13) 2a a where
a = −R1 R4 2 For u0 > 0, b and c take the following values R4 2 (−Ai Ps + Ai x20 − Ao x30 + m 2 Ai R2 Ao R3 2 Ao Pr ) + + mR5 2 mR6 2
b = 2R1 R4 x40 +
2x40 R4 Ai Ps − Ai x20 + Ao x30 m 2x A R √x − P o 3 30 r 40 −Ao Pr + − m R6 √ Ai R2 Ps − x20 R5
c = −R1 x240 +
while for u0 < 0
R4 2 (Ai x2 − Ai Pr − Ao x30 + m Ai R2 2 Ao R3 2 Ao Ps ) − 2 − mR5 mR6 2 2x40 R4 c = −R1 x240 + (Ai Pr − Ai x20 + Ao x30 m √ 2x40 Ai R2 x20 − Pr −Ao Ps ) + ( − m R5 √ Ao R3 Ps − x30 ) R6 b = 2R1 R4 x40 +
PROOF. The proof is straightforward by combining (10), (5), (12), and (11), (6), (12), respectively. Solving for e10 , e20 , e30 and e40 for u0 > 0 gives one solution at the origin and one solution at
b e10 = − ± 2a
r
b 2 c − 2a a √ (R5 x40 Ps − x20 + R2 e10 )2 e20 = Ps − x20 − (x40 + R4 e10 )2 R5 2 √ (R6 x40 x30 − Pr − R3 e10 )2 − x30 + Pr e30 = (x40 + R4 e10 )2 R6 2 e40 = C4 e10 For u0 < 0, the equations for e20 and e30 are
e20 = x20 − Pr −
√
(R5 x40 x20 − Pr − R2 e10 )2 (x40 − R4 e10 )2 R5 2
√ (R6 x40 Ps − x30 + R3 e10 )2 e30 = − Ps + x30 (x40 − R4 e10 )2 R6 2 The complex expressions determining the conditions on (13) are not easily analyzed analytically. Numerical analysis shows that K1 , K2 and K3 have no great effect on the condition for the origin being the only stationary point. K4 on the other hand has to take a value close to 0.01 for the parameters in Table 1 regardless of the sign of u0 . Table 1. Model parameters and their values Parameter
Value
Unit
Comment
Ps Pr fd β Ai Ao Vi Vo τ kd ω ksp m
108 0 1600 5 · 109 2.0 · 10−3 1.5 · 10−3 1.5 · 10−3 1.5 · 10−3 3.3 · 10−2 3.2 · 10−2 2 · 10−2 1.6 · 10−3 20
Pa Pa N m−1 s Pa m2 m2 m3 m3 s m3/2 kg −1/2 m mV −1 kg
Suply pressure Return pressure Force Effective bulk modulus Piston area, head side Piston area, rod side Volume, head side Volume, rod side Time constant Parameter Orifice area gradient Parameter Mass of actuator
Proposition 2. System (10,12) has stationary points only at the origin for u0 = 0 if and only if
4.2 Stability By enforcing additional limitations on the observer gain, stability of the origin and instability of other stationary points can be achieved. Linearizing (10,11) around the origin gives e˙ = (A0 − KC)e
where
fd Ai Ao − − 0 m m m βA βωk x d 40 − i 0 a24 V 2V g (x ) i i 1 20 A0= βA −βωkd x40 o a34 0 Vo 2Vo g2 (x30 ) 1 0 0 0 − τ and
a24 =
2
βωkd βωkd g1 (x20 ); a34 = − g2 (x30 ) Vi Vo
Asymptotic stability of linearized system (15) is equivalent to asymptotic stability of the origin for solutions of (10, 11). Analytical expressions for eigenvalues of A0 − KC are awkward and do not give much insight into the observer design aspects. Numerical calculations of the eigenvalues as function of K1 , K2 , K3 and K4 are more useful for observer design purpose. K2 and K3 have very small effect on the eigenvalues and the stationary points. Those are therefore set to zero. K1 and K4 have larger impact on the eigenvalues, although the simulation results are not shown here due to lack of space. To guarantee that the origin is the only stable stationary point for u0 = 0, the other stationary point has to be unstable. However, calculations show that this cannot be achieved for the considered parameter values without violating (13). Instead, the observer feedback has to be used to place both stationary points at the origin. Solving (14) gives one solution as K2 = 0, K3 = 0 and K4 = 0.0114 which is feasible. So, the choice of the feedback as K T = [ 250 0 0 0.0114 ]
2
(15)
(16)
Ai R2 Ao R3 + +Ao Pr R4 2 = Ai Ps R4 2 (14) R5 2 R6 2
satisfies (12), for all u0 , and asymptotically stabilizes linearized system (15). K1 is chosen large because it makes the observer faster and less oscillating.
PROOF. Follows by combining (10), (12) and (7) and solving for e10 .
Fig. 1 depicts the root locus for (15) when u0 rises and K is chosen as in (16). For the same choice of K the origin is a stable stationary point for −1 ≤ u0 ≤ 1.
For u0 = 0, the error system has two stationary points. One stationary point is at the origin and the location of the other, as Proposition 2 shows, can be assigned by choosing K2 , K3 and K4 . Since the origin has to be the only stable stationary point, there are two possible design approaches. Either the origin is made a stable stationary point while the other stationary point becomes unstable, or the observer feedback is chosen to assign both stationary points to the origin.
4.3 Attraction domain Once it is established that the origin can be rendered the only stationary point of the observer error dynamics and furthermore asymptotically stabilized, the attraction domain of it becomes a critical design issue. Simulations with starting points far from the origin
imaginary axis
Fig. 1. Root locus of the linearized observer error dynamics in the complex plane for the stationary input signal 0 < u0 < 1 V. 200 0
5. DETECTION OF LEAKAGE The observer under consideration is intended for detection of leakage in the system. Models for leakage in the input line and in the output line have been developed from the fluid flow equations in (Khan et al., 2005).
−200 −200
−150
−100 real axis
−50
0
are performed to illustrate its extension. The initial conditions x(0) = [ 1 105 105 0.1 ]T ; x ˆ(0) = [ 0 0 0 0 ]T are chosen. This is a physically reasonable starting point with an initial actuator velocity of 1 ms−1 , input and output line pressures equal to atmospheric pressure, and an initial spool displacement of 0.1 m. The observer converges nicely for the gain matrix as in (16) and for −1 ≤ u0 ≤ 1.
5.1 Leakage on the actuator input line A leakage on the actuator input line has been modeled as an additional flow, qli in the second state equation p qli = Ali kd x2 − Pa where Ali is the leak area and Pa is the atmospheric pressure. The state equation now reads β x˙ 2 = − Ai x1 + kd ωx4 g1 (x2 ) Vi Fig. 4 and Fig. 5 show the residual when a leakage Fig. 4. Residual when a leakage is present in the input line from time t = 0.2 to t = 0.5 for x4 ≥ 0.
Fig. 2. The residual for the nominal case and β reduced by 30%, 50% residual
0.5
0
0.05
0.1
0.15 time (s)
0.2
0.25
0.3
residual
0% 30% 50%
0
−0.5
0
0.05
0.1
0.15 time (s)
0.2
0.25
A =10
−0.4
A =10
−6
li
−5
li
0
0.2
0.4
0.6
0.8
1
time (s)
Fig. 5. Residual when a leakage is present in the input line from time t = 0.2 to t = 0.5 for x4 < 0. 6
−6
A =10
4
li
−5
A =10 li
2 0 0
0.2
0.4
0.6
0.8
1
time (s)
Fig. 3. The residual for the nominal case and τ increased by 30%, 50% 0.5
−0.2
−2
0% 30% 50%
0
−0.5
0
residual
Observer robustness against plant parameter variations has been investigated by altering the model parameters β and τ . The bulk modulus β has a great influence on the dynamics of the hydraulic system (Jelali and Kroll, 2003) and the time constant τ affects the valve dynamics. Those two parameters alter the dynamics in the hydraulic system in different ways and are therefore of interest regarding the robustness of the system. Fig. 2 shows the residual for the nominal case, as well as when β is reduced by 30% and by 50%. Similar plots for increased τ are given in Fig. 3.
residual
0.2
4.4 Robustness
0.3
occurs at time t = 0.2 and persists until t = 0.5 for x4 > 0 and x4 < 0 respectively. Observer response to leakages with the leak area of 10−6 and 10−5 is depicted. The sign of x4 determines the sign of the residual. The leakage is timely detected and clearly indicated by the steady state value of the residual. The latter can also be used for estimation of the leakage severity. Although, note that the mapping from the residual to the leakage severity differs between the two cases. The special case x4 = 0 has been handled separately. In steady state, x4 = 0 is equivalent to u = 0. Modeling a leakage occurring from t = 0.4 until t = 0.7 generates the residual shown in Fig. 6. As the leakage appears, the residual deviates from zero reacting to a change in the piston velocity, although it soon reaches a new steady state which causes the residual to return to zero. Apparently, a leakage cannot be detected from steady state residual when u = 0.
residual
Fig. 6. Residual under a leakage in the input line from time t = 0.4 to t = 0.7 for x4 = 0. 1 0 −1
0
0.2
0.4
0.6
0.8
1
time (s)
5.2 Leakage on the actuator output line The model of leakage on the actuator output line is similar to that of the input line. It has been modeled as a change in the output flow of the actuator, i. e. an additional flow, qlo in the third state equation p qlo = Alo kd x3 − Pa where Alo is the leak area and Pa is the atmospheric pressure. The state equation now reads x˙ 3 =
β (Ao x1 − kd ωx4 g2 (x3 ) − qlo ) Vo
In Fig. 7 and Fig. 8 respectively, the observer residuals Fig. 7. Residual under a leakage in the output line from time t = 0.2 to t = 0.5 for x4 ≥ 0. residual
0.4
−6
A =10 lo
−5
A =10
0.2
lo
0 −0.2
0
0.2
0.4
0.6
0.8
1
time (s)
Fig. 8. Residual under a leakage in the output line from time t = 0.2 to t = 0.5 for x4 < 0. residual
0 −6
−2
A =10 lo
−5
A =10
−4
lo
0
0.2
0.4
0.6
0.8
1
time (s)
are shown for the case of output line leakage under similar conditions as for the case of input line leakage. Once again, the leakage detection is prompt and well pronounced in the steady state response. Furthermore, it is straightforward to distinguish between the two types of leakage since the difference between input and output leakage is in the sign of the residual. Since the sign of the residual changes together with the sign of x4 , it is, therefore, necessary to know the sign of x4 , which is equivalent to keeping track of the direction of the flow. The observer residual under a leakage in the output line for x4 = 0 is similar to that for a leakage in the input line.
6. CONCLUSIONS The feasibility of a nonlinear observer for a hydraulic servo system as a fault detection tool is investigated. The state estimation error is shown to converge to zero under weak assumptions. Simulations indicate that the attraction region of the origin is sufficiently large and the convergence is robust against significant process parameter alternations. It is also shown that the same observer design is feasible for positive and negative values of the spool displacement. The observer residual timely and reliably indicates the presence of a leakage for non-zero reference signal. The leakage severity can be estimated from the stationary value of the residual while input and output line leakages can be distinguished between by means of its sign. The sign of the spool displacement toggles the sign of the residual. Since the sign of spool displacement is determined by the direction of the flow in the lines, it is sufficient to be able to distinguish between flows to and from the piston. When the reference signal is zero, the spool displacement also becomes zero at steady state. It is therefore not possible to detect leakage by the considered approach in steady state under velocity feedback and zero reference signal to the system. 7. ACKNOWLEDGMENT This work has been carried out in cooperation with Volvo Aero within the Swedish National Avionic Program. The authors would like to thank Dr. Melker H¨arefors for discussing the results of this paper. 8. REFERENCES De Persis, Claudio and Alberto Isidori (2001). A geometric approach to nonlinear fault detection and isolation. IEEE Transactions on Automatic Control 46(6), 853–865. Hammouri, H., P. Kabore, S. Othman and J. Biston (2002). Failure diagnosis and nonlinear observer. application to a hydraulic process. Journal of the Franklin Institure 339, 455–478. Jelali, Mohieddine and Andreas Kroll (2003). Hydraulic Servo-systems : Modelling, Identification and Control. Springer. Johansson, Andreas and Alexander Medvedev (1999). Model-based leakage detection in a pulverized coal injection vessel. IEEE Transactions on Control Systems Technology 7(6), 675–682. Khan, H., S. C. Abou and N. Sepehri (2005). Nonlinear observer-based fault detection technique for electro-hydraulic servo-positioning systems. Mechatronics 15, 1037–1059. Leuschen, Martin. L., Ian. D. Walker and Joseph. R. Cavallaro (2003). Nonlinear fault detection for hydraulic systems. In: Fault diagnosis and fault tolerance for mechatronic systems: recent advances (F. Caccavale and L. Villani, Eds.). Springer-Verlag. Berlin Heidelberg. Saberi, A., A. A. Stoorvogel, P. Sannuti and H. Niemann (2000). Fundamental problems in fault detection and identification. International journal of robust and nonlinear control 10, 1209–1236.