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Fault diagnostics using sliding mode techniques
Control Engineering Practice 10 (2002) 207–217
Fault diagnostics using sliding mode techniques Keng Boon Goh, Sarah K. Spurgeon*, N. Barrie Jones Control and Instrumentation Research Group, Department of Engineering, University of Leicester, Leicester LE1 7RH, UK Received 12 June 2000; accepted 8 June 2001
Abstract This paper describes the application of sliding mode observation techniques to the problem of fault diagnostics. A specific diesel engine coolant system is considered. A non-linear sliding mode observer is used to monitor the system states. Pertinent system parameters are also monitored using the concept of the equivalent injection signal required to maintain a sliding mode. The parameter estimates from the sliding mode scheme are compared with those generated by a non-linear simulation model and are found to provide good correlation. Results from a laboratory-based power generator set (Genset) are presented to demonstrate the effectiveness of the approach. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Fault diagnostics; Sliding mode; Observer; Equivalent injection; Diesel engine cooling system
1. Introduction A fault is caused by any kind of malfunction in the actual dynamic system that may lead to an unacceptable overall system performance. Faults will often be harmful to an engineering process if early detection is not made. Prompt detection helps to minimise the maintenance and repair costs of the system and contributes towards increased system reliability. In terms of the coolant system under consideration here, problems with the radiator or coolant pump, for example, would have the effect of causing the engine to overheat. Appropriate sensors to monitor the condition of such components directly may be expensive or the variables may be difficult to measure. In this paper, a key issue will be the desire to minimize sensor requirements. It will be seen that an appropriately designed sliding mode observer which uses straightforward temperature measurements can be used to monitor sophisticated system parameters. Advances in computing technology provide an excellent foundation for developments in fault diagnostics. Sophisticated schemes may be applied and implemented on-line in real time. Perhaps the most common approach is to use model-based algorithms (Chen & Patton, 1999; Gertler, 1998; Isermann, 1994, *Corresponding author. Tel.: +44-116-252-2531; fax: +44-116-2522619. E-mail address: [email protected] (S.K. Spurgeon).
1997; Isermann & Balle, 1997), of which a specific class are the observer-based approaches which can be found in Chen, Patton, and Zhang (1996) and Yang and Saif (1995a). The possible draw back of any model-based scheme is the presence of modelling error which can contribute significantly to fault signals during normal operating conditions. There is thus a need to consider robustness issues for any model-based approach (Chen & Patton, 1999). Sliding mode techniques are well known for their robustness properties which are coupled with a relatively straightforward design concept (Edwards & Spurgeon, 1998). Essentially, an appropriate switching function is defined and the objective of the applied injection signal is to maintain system motion along the switching function; in this mode of operation, the system is said to be in a sliding mode. A few authors have considered the application of sliding mode approaches to the model-based fault detection problem (Edwards, Spurgeon, & Patton, 1999; Edwards & Spurgeon, 2000; Hermans & Zarrop, 1996; Yang & Saif, 1995b). In Hermans and Zarrop (1996) a design approach is adopted which seeks to make the occurrence of a fault synonymous with a break in the sliding motion. However, such a break can be difficult to detect in practice. Adaptive sliding observers have been considered in Yang and Saif (1995b). A design framework which uses a nominal linear system representation for design of the sliding mode observer and then applies the
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equivalent input injection principle for fault reconstruction can be found in Edwards et al. (1999) and Edwards and Spurgeon (2000). Here the observer design is performed for a non-linear system description and the equivalent input injection principle is applied for parameter estimation. The state and parameter estimates can then be used to generate residual signals (Hermans & Zarrop, 1996; Isermann, 1994; Isermann & Balle, 1997; Yang & Saif, 1995b) which act as an indicator of fault conditions. This paper focuses on the specific application of nonlinear sliding mode observers to detect possible faults in a diesel engine cooling system. The emphasis here is on the rig trials, which demonstrate the proof of concept. However, it should be noted that the methodology is equally relevant for a range of non-linear fault diagnostic problems. The work considers the estimation of system states as well as pertinent diesel engine coolant system parameters. The observer system is required to remain in the sliding mode under all system conditions in order to effectively estimate parameters. A fully nonlinear model is used to verify that the parameter estimates are reasonable. For the cooling system, malfunctions may occur in the following three main components: the thermostat bypass valve, the coolant pump and the radiator. Possible malfunctions in any of these sub-systems will cause the engine to overheat and problems may propagate to other engine sub-systems. A range of fault scenarios have been created to demonstrate the approach. The outline of the paper is as follows: Section 2 describes the diesel engine coolant system process. Possible variations in system parameters during fault conditions are considered. The sliding mode observer design is covered in Section 3. The stability of the observer is discussed. The use of the observer for fault determination is the topic of Section 4. The experimental set up and the results of the rig trials are presented in Section 5.
2. Model development A diesel engine coolant system can be represented by the diagram shown in Fig. 1. The engine block represents a heat source. The thermostat valve divides coolant flow according to its opening level, a: The radiator acts as a heat sink to the atmosphere. Arrows dictate the direction of coolant flow. While the thermostat valve is closed (a ¼ 0), no coolant can flow through the radiator and coolant circulates through the left circuit. The coolant will only flow through the radiator when the thermostat valve is open. The bypass valve is used to bypass part of the coolant mixture and is merely a method used to simulate fault conditions on the rig and is not an inherent component of a typical diesel engine generator. The location of temperature sensors is indicated with a cross. A model of the coolant system developed in Bhatti, Twiddle, Spurgeon, and Jones (1999) and Chiang, Ursini, and Hohnson (1982) is employed here. The model equations are based on the balance of thermal energy between the engine block and the coolant, and also between the coolant and the radiator where heat dissipates to the atmosphere. Heat transfer from the engine block is balanced by heat absorbed by the coolant and heat loss through convection. In addition to the heat balance, some of the heat may dissipate through the engine surface by convection and radiation processes. Heat is also lost through conduction when it propagates to another sub-system or to the atmosphere. Let T1 be the engine block inlet temperature, T2 be the engine outlet temperature and radiator inlet temperature and T3 be the radiator outlet temperature. Heat transfer in the engine block is balanced by heat absorbed by coolant and heat loss through convection and radiation. A balance of the thermal energy with respect to the engine block and coolant flow is shown in the equations below.
Fig. 1. Schematic of the diesel engine cooling system.
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Heat transfer for engine block, ðmcÞB
dTB ¼ ðq’g q’h q’L ÞB : dt
Heat transfer for coolant in the engine block, ðmcÞ2
dT2 ¼ ðq’c q’h Þ; dt
where q’ is the instantaneous heat transfer rate, q’g ¼ ðhAÞcyl to block ðTgas TB Þ the cylinder gas to engine block heat transfer, q’c ¼ ðhAÞblock to coolant ðTB T2 Þ the engine block to coolant heat transfer, q’h ¼ ðmcÞ ’ 2 ðT2 T1 Þ the heating of coolant flow/coolant heat gain, q’L ¼ q’conv þ q’rad the surface heat loss/block to ambient, q’conv ¼ hA ðTs Tamb Þ the convection losses, q’rad ¼ esA 4 ðTs4 Tamb Þ the radiation losses, hcyl to block the gas side heat transfer coefficient, hblock to coolant the coolant side heat transfer coefficient, mB the mass of solid block, cB the specific heat of solid block, m2 the mass of coolant stored in block, c2 the specific heat of coolant in block, m’ 2 the mass flow rate of coolant through block, Tblock the mean engine block temperature, T2 the outlet coolant temperature from engine block, T1 the inlet coolant temperature to engine block, Tgas the combustion gas temperature, A the surface area of cylinder liner, e the surface emissivity, s the Steffen–Boltzmann constant, Tamb the ambient temperature applicable, Ts the surface temperature, h the heat transfer coefficient (h ¼ 0:102ðReÞ0:625 ðka =DÞ), q’conv ¼ hAðTs Tamb Þ the 4 convection losses, and q’rad ¼ esAðTs4 Tamb Þ the radiation losses. The radiator dissipates heat from the engine block through its fin area to keep the temperature of the engine block within the operating temperature range. The inlet temperature is assumed to be the same as the outlet temperature of the engine block. The heat dissipation to ambient via the radiator is then given by ðmcÞrad
dT3 ¼ ðq’cL q’L Þrad ; dt
where q’cL ¼ mcðT ’ ’L ¼ q’conv þ q’rad ; q’conv ¼ 2 T3 Þ; q 4 Srad sArad ðT34 Tamb Þ; q’rad ¼ ðhAÞrad ðT3 Tamb Þ; mrad the mass of coolant in radiator, crad the specific heat capacity of coolant, hrad the radiator heat transfer coefficient, Arad the total area of radiator, s the Boltzman’s constant, S the relative emissivity for the surface of radiator, and F the shape factor. From Fig. 1, the relationship between T1 ; T2 ; T3 and a is shown in the following equation: T1 ¼ aT3 þ ð1 aÞT2 : By re-arranging and substituting the appropriate parameters into the equations yield,
T’B ¼
1 ½ðhAÞcyl to block ðTgas TB Þ ðmcÞB ðmcÞ ’ 2 ðT2 T1 Þ ðhAÞðTs Tamb Þ 4 Þ; esAðTs4 Tamb
T’2 ¼ ½k1 mk ’ 2 T2 þ mk ’ 2 T3 þ k1 TB ; T’3 ¼ mk ’ 3 T2 ðmk ’ 3 þ hrad k4 ÞT3 þ hrad k4 Tamb 4 k5 ðT34 Tamb Þ:
In the experimental tests, the loss of thermal energy through radiation is assumed to be small (E0). In addition, a second measurement signal is taken from the engine outlet (radiator inlet) and denoted by T2a : The dynamic characteristics of T2 and T2a are exactly the same. However, the differential equation for T2 is parameterized in terms of variable coolant mass flow rate and T2a is parameterized in terms of variable thermostat valve opening. This yields the following final model equations. T’2 ¼ ðk1 mk ’ 2 ÞT2 þ mk ’ 2 T3 þ k1 Tb ;
ð1Þ
T’2a ¼ ðk1 ak2a ÞT2a þ ak2a T3 þ k1 Tb ;
ð2Þ
T’3 ¼ mk ’ 3 T2 þ ðmk ’ 3 hrad k4 ÞT3 þ hrad k4 Tamb ;
ð3Þ
where m; ’ hrad and a are the coolant mass flow rate, the radiator heat transfer coefficient and the opening level of the thermostat valve, respectively. These represent the parameters that could be useful to monitor the health of the engine but are not straightforward to measure. k1 ; k2 ; k3 ; k4 and k2a are given by k1 ¼ ðhAÞbc =ðmcÞbc ; k2 ¼ ac=ðmcÞbc ; k3 ¼ ac=ðmcÞrad ; k4 ¼ Arad =ðmcÞrad ; k2 ¼ mc=ðmcÞ ’ ’ mÞ ’ bc : bc ; and k2a ¼ mc=ð The suffixes bc and rad represent block to coolant and radiator, respectively. During fault scenarios, for example, a pump fault may cause a variation in the coolant mass flow rate, m: ’ Let ’# þ Dm; m’ ¼ m ’
hrad ¼ h#rad þ Dhrad
and
a ¼ a# þ Da
represent deviations Dm; ’ Dhrad and Da from nominal ’# h#rad and a# : Assume the constants k1 ; k2 ; conditions m; k3 ; k4 and k2a are evaluated at the nominal operating ’# h#rad and a# : Then point determined by m; T’2 ¼ ½k1 ðm’ þ DmÞk ’ 2 T2 þ ðm’ þ DmÞk ’ 2 T3 þ k1 Tb ;
ð4Þ
T’2 ¼ ðk1 mk ’ 2 ÞT2 þ mk ’ 2 T3 þ k1 Tb Dm’ k2 T2 þ Dm’ k2 T3 ; T’2a ¼ ½k1 ða þ DaÞk2a T2a þ ða þ DaÞk2a T3 þ k1 Tb ;
ð5Þ
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T’2a ¼ ðk1 ak2a ÞT2a þ ak2a T3 þ k1 Tb Da k2a T2a þ Da k2a T3 ; T’3 ¼ ðm’ þ DmÞk ’ 3 T2 þ ½ðm’ þ DmÞk ’ 3 ðhrad þ Dhrad Þk4 T3 þ ðhrad þ Dhrad Þk4 Tamb ;
ð6Þ
information for parameter reconstruction and fault diagnosis. A proposed structure of the sliding mode observer for the cooling system is modified from Bhatti et al. (1999) and the nominal dynamics to yield: ’# ¼ ðk mk T ð7Þ ’ ÞT# þ mk ’ T þk T þK v ; 2
T’3 ¼ mk ’ 3 T2 þ ðmk ’ 3 hrad k4 ÞT3 þ hrad k4 Tamb Dm’ k3 T2 þ ðDm’ Dhrad Þk4 T3 þ Dhrad k4 Tamb : Eqs. (4)–(6) represent the assumed behavior of the uncertain cooling system dynamics. The nominal dynamics is obtained by setting Dm’ ¼ Dhrad ¼ Da ¼ 0:
3. The sliding mode approach The sliding mode design approach, which may be applied to both controller and observer problems, is well known for its robustness properties (Edwards & Spurgeon, 1998). At the heart of the design procedure is the specification of a so-called switching function. The switching function is chosen to ensure desirable performance is exhibited by the system of interest. If the problem is a control problem, for example, the switching function may be chosen to prescribe desired plant dynamics. Consider s ¼ y’ þ 1: By forcing the switching function to zero, whereby s ¼ 0 and the system is said to have attained a sliding mode, a first order decay would be exhibited by the output of interest. For an observer problem, the switching function may most appropriately be defined as the output error (Edwards & Spurgeon, 1994). By forcing this switching function to zero, the observer output is forced to equal the plant output and a set of estimated states that yield the measured system output are obtained. Having defined an appropriate switching function, it is necessary to choose a control (or in the case of the observer an injection signal) that will force the sliding mode condition to be attained and maintained. This is termed the reachability problem and is typically ensured by choosing the injection signal so that s and s’ are forced to have opposite sign. Essentially, discontinuous injection signals are used to maintain the appropriately chosen switching function at zero. During the sliding motion, the system exhibits total robustness to a class of disturbance signals and any uncertainty that is implicit in the channels where the discontinuous injection signal is applied. For the fault diagnostic situation considered here, the switching function will be the observer error. Further, it will be seen that the discontinuous signal employed to maintain the observer error at zero provides effective
1
2
2
2
3
1
b
2 2
’# ¼ ðk ak ÞT# þ ak T þ k T þ K v ; T 2a 1 2a 2a 2a 3 1 b 2a 2a
ð8Þ
’# ¼ mk T ’ 3 T2 þ ðmk ’ 3 hrad k4 ÞT#3 3 þ hrad k4 Tamb be3 þ K3 v3 ;
ð9Þ
where vi ¼ Ki ðei =8ei 8Þ; i ¼ 2; 2a ; 3 and K2 ; K2a and K3 are the gains of the discontinuous signals v2 ; v2a and v3 which must be chosen to satisfy the reachability problem and ensure sliding motion occurs at all times even in the presence of parameter variations and faults. It should be noted that, although k1 ; k2 ; k3 ; k4 and k2a are assumed constant, they are in fact dependent on the variations m; ’ hrad and a: However, with the proposed observer structure a sliding motion will still be maintained and this parameter uncertainty will be rejected as it appears in channels where the vi are injected. Maintenance of the sliding condition is essential for parameter estimation and fault diagnosis in this scheme. b acts as a gain parameter to tune asymptotic error decay. This parameter can increase the rate of error decay in the absence of uncertainties. The observer error dynamics can be written as follows: e2 ¼ T2 T#2 ; e2a ¼ T2a T#2a and e3 ¼ T3 T#3 ; e’2 ¼ ðk1 mk ’ 2 Þe2 þ Dm’ k2 ðT3 T2 Þ v2 ;
ð10Þ
e’2a ¼ ðk1 ak2a Þe2 þ Da k2a ðT3 T2a Þ v2a ;
ð11Þ
e’3 ¼ ðmk ’ 3 hrad k4 bÞe3 þ Dhrad k4 ðTamb T3 Þ þ Dm’ k3 ðT3 T2 Þ v3 : ð12Þ The stability of the observer system can be investigated with respect to the following candidate Lyapunov function: 1 VðeÞ ¼ ðe22 þ e22a þ e23 Þ: ð13Þ 2 In the Appendix, it is shown that if the uncertainty is bounded as K2 X8Dm’ k2 ðT2 T3 Þ8;
ð14Þ
K2a X8Da k2a ðT2a T3 Þ8;
ð15Þ
K3 X8Dhrad k4 ðT3 Tamb Þ þ Dm’ k3 ðT2 T3 Þ8
ð16Þ
and b is selected so that b > mk ’ 3 convergence of the observer error is ensured.
ð17Þ
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4. Fault determination
4.2. Coolant pump
The gains of the discontinuous terms, K2 ; K2a and K3 ; are designed so that the observer remains in a sliding mode and the error is zero. Thus, the observer errors (e2 ; e2a and e3 ) and their derivatives will converge to zero. The perturbations due to faults are effectively compensated for by the discontinuous signals v2 ; v2a and v3 ; respectively. In the sliding mode, Eqs. (10)–(12) become:
Another important part of the cooling system is the coolant pump which is used to maintain a constant flow rate ðmÞ ’ of coolant while the engine is running. Although the coolant flow rate depends on engine speed, with higher speeds yielding higher flow rates, the engine used here is running at constant speed and a constant flow of coolant is assumed. Malfunctioning of the pump will critically affect the dissipation of heat in the engine block. The capacity of the coolant flow in the system must be closely monitored before any faults occur. One method to tackle this problem is to estimate the coolant flow rate, m: ’ This can be readily constructed from Eq. (21) and the nominal mass flow rate.
0 ¼ Dm’ k2 ðT3 T2 Þ v2 ;
ð18Þ
0 ¼ Da k2a ðT3 T2a Þ v2a ;
ð19Þ
0 ¼ Dhrad k4 ðTamb T3 Þ þ Dm’ k3 ðT3 T2 Þ v3
ð20Þ
and re-arranging these equations gives v2 ; Dm’ ¼ k2 ðT3 T2 Þ Da ¼
v2a ; k2a ðT3 T2a Þ
Dhrad ¼
v3 Dmk ’ 3 ðT3 T2 Þ : k4 ðTamb T3 Þ
4.3. Radiator ð21Þ
ð22Þ
ð23Þ
It is seen that v2 ; v2a and v3 effectively provide a means to estimate the change in mass flow rate, the thermostat valve opening and the radiator coefficient. The signals v2 ; v2a and v3 are discontinuous injection signals and to obtain parameter values, the average values of v2 ; v2a and v3 must be used. It is thus necessary to filter v2 ; v2a and v3 to construct the estimates of the change in mass flow rate, the thermostat valve opening and the radiator coefficient. This information can then be used to hypothesize a fault detection scheme for the coolant system. The previously mentioned variations in the ki are small and have a small relative effect on the equivalent injection signals. This is why the ki can be assumed nominal within the framework and omitted from the above analysis. Three possible fault scenarios are now addressed. 4.1. Thermostat bypass valve The thermostat bypass valve is used to control the coolant flow to the radiator. It is usually closed (a ¼ 0) to give a warm start during start up of the engine. The valve opens only after the coolant temperature reaches a certain temperature. If the valve behaves abnormally, where it may totally close or open at a later stage, then the engine may overheat. Estimating the valve opening, a will help in the fault monitoring process. This can be constructed from Eq. (22) and the nominal thermostat valve opening level.
The radiator is an essential component of the cooling system. It is responsible for dissipation of engine heat to the atmosphere and maintaining the engine block temperature within the appropriate operating range. Overheating of the engine will occur if this component fails to function correctly. Possible faults include malfunction of the radiator fan. In this case, the heat transfer coefficient will determine the efficiency of the radiator system. This can be constructed from Eq. (23), using Eq. (21) and the nominal radiator coefficient. The sliding mode observer is not only used here to estimate the system states but also to predict three important parameters, the thermostat valve opening, the coolant flow rate, and the heat transfer coefficient.
5. Experimental setup and results 5.1. Experimental setup The sliding mode observer developed in the previous section was tested on a power generator set (Genset) which has the capability of generating a maximum power of 65 kW. Various faults are introduced in the Genset in order to test the efficiency of the proposed design approach as outlined below: *
*
*
coolant pumpFthe bypass valve is opened to reduce the coolant flow, thermostat valveFa different type of valve is used which will open later than the normal valve, radiatorF10 and 20% covered.
The hardware interface involved in the test is dSPACE, pressure sensors and temperature sensors. A block diagram in Fig. 2 shows the complete system setup.
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Fig. 2. Block diagram showing the system interface and signal flow.
Fig. 3. State estimate, T2 and the error between the measured and estimated values.
5.2. Data acquisition A 5th order Butterworth filter was employed to remove the high frequency components of the signal noise. The filter had a sampling frequency of 1000 Hz with a cut-off frequency at 1 Hz. Raw data initially sampled at 1000 Hz, was down sampled to 100 Hz before saving due to the memory constraints of the on-line computer. The reason for this is to make sure signal properties are captured with no loss of information and no aliasing occur in this case. 5.3. Experimental results The following figures were generated from tests on the Genset. The first set of results corresponds to normal operating conditions with a step load applied to the
Genset from 0 to 65 kW, followed by a step down load and the process was repeated once. The estimation of the system states, T2 and T3 are shown in Figs. 3 and 4, respectively. This value is compared with the value which was measured online while the experiment was carried out. The error between the measured and estimated values is of the order of 0– 0.06 K, which is about 0.02%. Thus, the observer has been validated under normal operating conditions. A non-linear model is employed here to provide a comparison between parameter estimates developed from the non-linear model, and those estimated by the sliding mode observer. It should be noted that this is not the non-linear model used to design the observer but is a model developed using black box techniques and a large amount of empirical engine data. Further, it is the change in system parameters, as monitored via the
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Fig. 4. State estimate, T3 and the error between the measured and estimated values.
Fig. 5. Component parameter estimate, value obtained from a non-linear simulation and the mismatch between the two.
equivalent injection analysis, which is to be used to indicate the presence of faults. The results are shown in Figs. 5, 6 and 9. During the transition between the thermostat valve opening and closing (which happens under normal conditions), the error between the response of the non-
linear model and the measured response is relatively large, approximately 10%. When the thermostat valve is opened on the Genset, coolant flows into the radiator and the system undergoes a transition in thermodynamic properties. Due to this imbalance, the system parameters change accordingly in practice but this process is
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Fig. 6. Component parameter estimate, m’ obtained from the non-linear model and the mismatch between the two.
Fig. 7. When the bypass valve is opened, the coolant flow rate reduces and the error between this valve and the nominal value is shown.
not modeled accurately. The hysteresis effect of the thermostat valve itself is also thought to contribute to the mismatch. A fault is now introduced into the system by opening the bypass valve to reduce the coolant flow rate through
the system. Fig. 7 shows the rate of change of coolant flow. The bypass valve was opened for the period of time between 0.6 and 1.25 s. It is obvious that the response of the system does not change instantaneously when the valve is opened and closed. A likely explanation for this
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Fig. 8. Thermostat valve opens later than normal during fault condition.
Fig. 9. Plot of the heat transfer coefficient for normal and fault conditions.
is that the coolant still in the bypass valve circuit has a lower temperature when compared to the coolant circulating in the system. When the bypass valve opens, these two sets of coolant, which are at different
temperatures, mix together and after sometime, the coolant in the bypass valve eventually rises to the circulating coolant temperature. As a result the system takes a short time to reach its steady state.
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Next the thermostat valve is set to open at a later stage when compared to normal conditions. Fig. 8 shows the plot of the thermostat valve opening during normal conditions and in the presence of the fault. It is proved that when the valve opens late, or indeed does not open at all, the system can accurately detect the fault. For the radiator system, the radiator is covered by 10% and 25%. Fig. 9 shows a plot of heat transfer coefficient against ambient temperature. During normal operating conditions, the data forms a relatively concentrated cloud. As the engine becomes hotter, variation from ambient temperature occurs, but the variation is relatively small (between 308 and 312 K). It is seen that the heat transfer coefficient reduces when the radiator is covered. Also the clouds of data points become more elongated; this is because the cooling effect of the radiator is less efficient and thus the observed temperature drifts are larger. The ambient temperature is lower when the radiator is 25% covered as compared to when it is 10% covered due to the correspondingly larger reduction in the efficiency of the radiator. The heat transfer coefficient when the radiator is 10% covered is marginally larger than when the radiator is 25% covered.
6. Conclusion This paper explores the application of sliding mode concepts for fault detection and isolation. The experimental results show that perturbations in component parameters can be recovered by manipulating the equivalent input injection signal to the observer. Therefore, faults present in Genset components can be detected. It is seen that the behavior of relatively sophisticated components can be inferred from straightforward temperature measurements. Although a specific system has been considered here, the core results have potentially wide applicability to a range of non-linear engineering systems.
and substituting the error dynamic Eqs. (10)–(12) to yield, ’ ¼ e2 e’2 þ e2a e’2a þ e3 e’3 VðeÞ ’ ¼ e2 ½ðk1 mk VðeÞ ’ 2 Þe2 þ Dm’ k2 ðT3 T2 Þ v2 þ e2a ½ðk1 ak2a Þe2a þ Da k2a ðT3 T2a Þ v2a þ e3 ½ðmk ’ 3 hrad k4 bÞe3 þ Dhrad k4 ðTamb T3 Þ þ Dm’ k3 ðT3 T2 Þ v3 ’ ¼ k1 e22 mk VðeÞ ’ 2 e22 k1 e22a ak2a e22a ðhrad k4 þ b mk ’ 3 Þe23 e2 Dm’ k2 ðT2 T3 Þ K2
8e2 82 8e2 8
8e2a 82 e3 Dhrad k4 8e2a 8 8e3 82 ðT3 Tamb Þ e3 Dm’ k3 ðT2 T3 Þ K3 8e3 8 e2a Da k2 aðT2a T3 Þ K2a
as e2i ¼ 8ei 82 ’ ¼ k1 e22 mk VðeÞ ’ 2 e22 k1 e22a ak2a e22a ðhrad k4 þ b mk ’ 3 Þe23 e2 Dm’ k2 ðT2 T3 Þ K2 8e2 8 e2a Da k2a ðT2a T3 Þ K2a 8e2a 8 e3 Dhrad k4 ðT3 Tamb Þ e3 Dm’ k3 ðT2 T3 Þ K3 8e3 8 p k1 e22 mk ’ 2 e22 k1 e22a ak2a e22a ðhrad k4 þ b mk ’ 3 Þe23 8e2 8ðK2 8T2 T3 8Dm’ k2 Þ 8e2a 8ðK2a 8T2a T3 8Da k2a Þ 8e3 8ðK3 8Dhrad k4 ðT3 Tamb Þ8 8Dm’ k3 ðT2 T3 Þ8 With the given uncertainty bounds (14)–(16) and restriction on b (17) convergence of the observer error is proved since all the nominal constants involved in the above equation are positive, i.e. k1 > 0; k2 > 0; k3 > 0; k2a > 0; hrad > 0:
k4 > 0;
m’ > 0;
References Acknowledgements This project is supported by Perkins Engines, TRW and EPSRC Grant Reference GR/L42018. The invaluable assistance of our colleague Mr. John Twiddle during the data capture phase of this work is gratefully acknowledged.
Appendix A The stability of the sliding mode observer can be investigated by differentiating the candidate Lyapunov equation VðeÞ ¼ 1=2ðe22 þ e22a þ e23 Þ with respect to time
Bhatti, A. I., Twiddle, J. A., Spurgeon, S. K., & Jones, N. B. (1999). Engine coolant system fault diagnostics with sliding mode observers and fuzzy analyser, IASTED International Conference on Modelling, Identification and Control, Innsbruck, Austria. Chen, J., & Patton, R. J. (1999). Robust model-based fault diagnosis for dynamic systems. Dordrecht: Kluwer Academic Publishers. Chen, J., Patton, R. J., & Zhang, H. (1996). Design of unknown input observers and robust fault detection filters. International Journal of Control, 63(1), 85–105. Chiang, E. C., Ursini, V. J., & Hohnson, J. H. (1982). Development and evaluation of a diesel powered truck cooling system computer simulation program. SAE Paper 821048. Edwards, C., & Spurgeon, S. K. (1994). On the development of discontinuous observers. International Journal of Control, 59, 1129–1144. Edwards, C., & Spurgeon, S. K. (1998). Sliding mode control: Theory and applications. UK: Taylor & Francis.
K.B. Goh et al. / Control Engineering Practice 10 (2002) 207–217 Edwards, C., Spurgeon, S. K., & Patton, R. J. (1999). Sliding mode observer for fault detection and isolation. Automatica, 36, 541–553. Edwards, C., & Spurgeon, S. K. (2000). A sliding mode observer based FDI scheme for the ship benchmark. European Journal of Control, 6(4), 341–355. Gertler, J. (1998). Fault detection and diagnosis in engineering systems. New York: Marcel Dekker. Hermans, F. J. J., & Zarrop, M. B. (1996). Sliding mode observer for robust sensor monitoring, Proceedings of the 13th IFAC World contress, pp. 211–216. Isermann, R. (1994). On the applicability of model-based fault detection for technical processes. Control Engineering Practice, 2(3), 439–450.
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Isermann, R. (1997). Supervision, fault-detection and fault-diagnosis methodsFan introduction. Control Engineering Practice, 5(5), 639–652. Isermann, R., & Balle, P. (1997). Trends in the application of modelbased fault detection and diagnosis in technical processes. Control Engineering Practice, 5(5), 709–719. Yang, H., & Saif, M. (1995a). Nonlinear adaptive observer design for fault detection. Proceedings of the American control conference, pp. 1136–1139. Yang, H., & Saif, M. (1995b). Fault detection in a class of nonlinear system via adaptive sliding observer. Proceedings of the IEEE Transactions on Systems Man and Cybernetics, 3, 2199–2204.
Fault diagnostics using sliding mode techniques Keng Boon Goh, Sarah K. Spurgeon*, N. Barrie Jones Control and Instrumentation Research Group, Department of Engineering, University of Leicester, Leicester LE1 7RH, UK Received 12 June 2000; accepted 8 June 2001
Abstract This paper describes the application of sliding mode observation techniques to the problem of fault diagnostics. A specific diesel engine coolant system is considered. A non-linear sliding mode observer is used to monitor the system states. Pertinent system parameters are also monitored using the concept of the equivalent injection signal required to maintain a sliding mode. The parameter estimates from the sliding mode scheme are compared with those generated by a non-linear simulation model and are found to provide good correlation. Results from a laboratory-based power generator set (Genset) are presented to demonstrate the effectiveness of the approach. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Fault diagnostics; Sliding mode; Observer; Equivalent injection; Diesel engine cooling system
1. Introduction A fault is caused by any kind of malfunction in the actual dynamic system that may lead to an unacceptable overall system performance. Faults will often be harmful to an engineering process if early detection is not made. Prompt detection helps to minimise the maintenance and repair costs of the system and contributes towards increased system reliability. In terms of the coolant system under consideration here, problems with the radiator or coolant pump, for example, would have the effect of causing the engine to overheat. Appropriate sensors to monitor the condition of such components directly may be expensive or the variables may be difficult to measure. In this paper, a key issue will be the desire to minimize sensor requirements. It will be seen that an appropriately designed sliding mode observer which uses straightforward temperature measurements can be used to monitor sophisticated system parameters. Advances in computing technology provide an excellent foundation for developments in fault diagnostics. Sophisticated schemes may be applied and implemented on-line in real time. Perhaps the most common approach is to use model-based algorithms (Chen & Patton, 1999; Gertler, 1998; Isermann, 1994, *Corresponding author. Tel.: +44-116-252-2531; fax: +44-116-2522619. E-mail address: [email protected] (S.K. Spurgeon).
1997; Isermann & Balle, 1997), of which a specific class are the observer-based approaches which can be found in Chen, Patton, and Zhang (1996) and Yang and Saif (1995a). The possible draw back of any model-based scheme is the presence of modelling error which can contribute significantly to fault signals during normal operating conditions. There is thus a need to consider robustness issues for any model-based approach (Chen & Patton, 1999). Sliding mode techniques are well known for their robustness properties which are coupled with a relatively straightforward design concept (Edwards & Spurgeon, 1998). Essentially, an appropriate switching function is defined and the objective of the applied injection signal is to maintain system motion along the switching function; in this mode of operation, the system is said to be in a sliding mode. A few authors have considered the application of sliding mode approaches to the model-based fault detection problem (Edwards, Spurgeon, & Patton, 1999; Edwards & Spurgeon, 2000; Hermans & Zarrop, 1996; Yang & Saif, 1995b). In Hermans and Zarrop (1996) a design approach is adopted which seeks to make the occurrence of a fault synonymous with a break in the sliding motion. However, such a break can be difficult to detect in practice. Adaptive sliding observers have been considered in Yang and Saif (1995b). A design framework which uses a nominal linear system representation for design of the sliding mode observer and then applies the
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equivalent input injection principle for fault reconstruction can be found in Edwards et al. (1999) and Edwards and Spurgeon (2000). Here the observer design is performed for a non-linear system description and the equivalent input injection principle is applied for parameter estimation. The state and parameter estimates can then be used to generate residual signals (Hermans & Zarrop, 1996; Isermann, 1994; Isermann & Balle, 1997; Yang & Saif, 1995b) which act as an indicator of fault conditions. This paper focuses on the specific application of nonlinear sliding mode observers to detect possible faults in a diesel engine cooling system. The emphasis here is on the rig trials, which demonstrate the proof of concept. However, it should be noted that the methodology is equally relevant for a range of non-linear fault diagnostic problems. The work considers the estimation of system states as well as pertinent diesel engine coolant system parameters. The observer system is required to remain in the sliding mode under all system conditions in order to effectively estimate parameters. A fully nonlinear model is used to verify that the parameter estimates are reasonable. For the cooling system, malfunctions may occur in the following three main components: the thermostat bypass valve, the coolant pump and the radiator. Possible malfunctions in any of these sub-systems will cause the engine to overheat and problems may propagate to other engine sub-systems. A range of fault scenarios have been created to demonstrate the approach. The outline of the paper is as follows: Section 2 describes the diesel engine coolant system process. Possible variations in system parameters during fault conditions are considered. The sliding mode observer design is covered in Section 3. The stability of the observer is discussed. The use of the observer for fault determination is the topic of Section 4. The experimental set up and the results of the rig trials are presented in Section 5.
2. Model development A diesel engine coolant system can be represented by the diagram shown in Fig. 1. The engine block represents a heat source. The thermostat valve divides coolant flow according to its opening level, a: The radiator acts as a heat sink to the atmosphere. Arrows dictate the direction of coolant flow. While the thermostat valve is closed (a ¼ 0), no coolant can flow through the radiator and coolant circulates through the left circuit. The coolant will only flow through the radiator when the thermostat valve is open. The bypass valve is used to bypass part of the coolant mixture and is merely a method used to simulate fault conditions on the rig and is not an inherent component of a typical diesel engine generator. The location of temperature sensors is indicated with a cross. A model of the coolant system developed in Bhatti, Twiddle, Spurgeon, and Jones (1999) and Chiang, Ursini, and Hohnson (1982) is employed here. The model equations are based on the balance of thermal energy between the engine block and the coolant, and also between the coolant and the radiator where heat dissipates to the atmosphere. Heat transfer from the engine block is balanced by heat absorbed by the coolant and heat loss through convection. In addition to the heat balance, some of the heat may dissipate through the engine surface by convection and radiation processes. Heat is also lost through conduction when it propagates to another sub-system or to the atmosphere. Let T1 be the engine block inlet temperature, T2 be the engine outlet temperature and radiator inlet temperature and T3 be the radiator outlet temperature. Heat transfer in the engine block is balanced by heat absorbed by coolant and heat loss through convection and radiation. A balance of the thermal energy with respect to the engine block and coolant flow is shown in the equations below.
Fig. 1. Schematic of the diesel engine cooling system.
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Heat transfer for engine block, ðmcÞB
dTB ¼ ðq’g q’h q’L ÞB : dt
Heat transfer for coolant in the engine block, ðmcÞ2
dT2 ¼ ðq’c q’h Þ; dt
where q’ is the instantaneous heat transfer rate, q’g ¼ ðhAÞcyl to block ðTgas TB Þ the cylinder gas to engine block heat transfer, q’c ¼ ðhAÞblock to coolant ðTB T2 Þ the engine block to coolant heat transfer, q’h ¼ ðmcÞ ’ 2 ðT2 T1 Þ the heating of coolant flow/coolant heat gain, q’L ¼ q’conv þ q’rad the surface heat loss/block to ambient, q’conv ¼ hA ðTs Tamb Þ the convection losses, q’rad ¼ esA 4 ðTs4 Tamb Þ the radiation losses, hcyl to block the gas side heat transfer coefficient, hblock to coolant the coolant side heat transfer coefficient, mB the mass of solid block, cB the specific heat of solid block, m2 the mass of coolant stored in block, c2 the specific heat of coolant in block, m’ 2 the mass flow rate of coolant through block, Tblock the mean engine block temperature, T2 the outlet coolant temperature from engine block, T1 the inlet coolant temperature to engine block, Tgas the combustion gas temperature, A the surface area of cylinder liner, e the surface emissivity, s the Steffen–Boltzmann constant, Tamb the ambient temperature applicable, Ts the surface temperature, h the heat transfer coefficient (h ¼ 0:102ðReÞ0:625 ðka =DÞ), q’conv ¼ hAðTs Tamb Þ the 4 convection losses, and q’rad ¼ esAðTs4 Tamb Þ the radiation losses. The radiator dissipates heat from the engine block through its fin area to keep the temperature of the engine block within the operating temperature range. The inlet temperature is assumed to be the same as the outlet temperature of the engine block. The heat dissipation to ambient via the radiator is then given by ðmcÞrad
dT3 ¼ ðq’cL q’L Þrad ; dt
where q’cL ¼ mcðT ’ ’L ¼ q’conv þ q’rad ; q’conv ¼ 2 T3 Þ; q 4 Srad sArad ðT34 Tamb Þ; q’rad ¼ ðhAÞrad ðT3 Tamb Þ; mrad the mass of coolant in radiator, crad the specific heat capacity of coolant, hrad the radiator heat transfer coefficient, Arad the total area of radiator, s the Boltzman’s constant, S the relative emissivity for the surface of radiator, and F the shape factor. From Fig. 1, the relationship between T1 ; T2 ; T3 and a is shown in the following equation: T1 ¼ aT3 þ ð1 aÞT2 : By re-arranging and substituting the appropriate parameters into the equations yield,
T’B ¼
1 ½ðhAÞcyl to block ðTgas TB Þ ðmcÞB ðmcÞ ’ 2 ðT2 T1 Þ ðhAÞðTs Tamb Þ 4 Þ; esAðTs4 Tamb
T’2 ¼ ½k1 mk ’ 2 T2 þ mk ’ 2 T3 þ k1 TB ; T’3 ¼ mk ’ 3 T2 ðmk ’ 3 þ hrad k4 ÞT3 þ hrad k4 Tamb 4 k5 ðT34 Tamb Þ:
In the experimental tests, the loss of thermal energy through radiation is assumed to be small (E0). In addition, a second measurement signal is taken from the engine outlet (radiator inlet) and denoted by T2a : The dynamic characteristics of T2 and T2a are exactly the same. However, the differential equation for T2 is parameterized in terms of variable coolant mass flow rate and T2a is parameterized in terms of variable thermostat valve opening. This yields the following final model equations. T’2 ¼ ðk1 mk ’ 2 ÞT2 þ mk ’ 2 T3 þ k1 Tb ;
ð1Þ
T’2a ¼ ðk1 ak2a ÞT2a þ ak2a T3 þ k1 Tb ;
ð2Þ
T’3 ¼ mk ’ 3 T2 þ ðmk ’ 3 hrad k4 ÞT3 þ hrad k4 Tamb ;
ð3Þ
where m; ’ hrad and a are the coolant mass flow rate, the radiator heat transfer coefficient and the opening level of the thermostat valve, respectively. These represent the parameters that could be useful to monitor the health of the engine but are not straightforward to measure. k1 ; k2 ; k3 ; k4 and k2a are given by k1 ¼ ðhAÞbc =ðmcÞbc ; k2 ¼ ac=ðmcÞbc ; k3 ¼ ac=ðmcÞrad ; k4 ¼ Arad =ðmcÞrad ; k2 ¼ mc=ðmcÞ ’ ’ mÞ ’ bc : bc ; and k2a ¼ mc=ð The suffixes bc and rad represent block to coolant and radiator, respectively. During fault scenarios, for example, a pump fault may cause a variation in the coolant mass flow rate, m: ’ Let ’# þ Dm; m’ ¼ m ’
hrad ¼ h#rad þ Dhrad
and
a ¼ a# þ Da
represent deviations Dm; ’ Dhrad and Da from nominal ’# h#rad and a# : Assume the constants k1 ; k2 ; conditions m; k3 ; k4 and k2a are evaluated at the nominal operating ’# h#rad and a# : Then point determined by m; T’2 ¼ ½k1 ðm’ þ DmÞk ’ 2 T2 þ ðm’ þ DmÞk ’ 2 T3 þ k1 Tb ;
ð4Þ
T’2 ¼ ðk1 mk ’ 2 ÞT2 þ mk ’ 2 T3 þ k1 Tb Dm’ k2 T2 þ Dm’ k2 T3 ; T’2a ¼ ½k1 ða þ DaÞk2a T2a þ ða þ DaÞk2a T3 þ k1 Tb ;
ð5Þ
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T’2a ¼ ðk1 ak2a ÞT2a þ ak2a T3 þ k1 Tb Da k2a T2a þ Da k2a T3 ; T’3 ¼ ðm’ þ DmÞk ’ 3 T2 þ ½ðm’ þ DmÞk ’ 3 ðhrad þ Dhrad Þk4 T3 þ ðhrad þ Dhrad Þk4 Tamb ;
ð6Þ
information for parameter reconstruction and fault diagnosis. A proposed structure of the sliding mode observer for the cooling system is modified from Bhatti et al. (1999) and the nominal dynamics to yield: ’# ¼ ðk mk T ð7Þ ’ ÞT# þ mk ’ T þk T þK v ; 2
T’3 ¼ mk ’ 3 T2 þ ðmk ’ 3 hrad k4 ÞT3 þ hrad k4 Tamb Dm’ k3 T2 þ ðDm’ Dhrad Þk4 T3 þ Dhrad k4 Tamb : Eqs. (4)–(6) represent the assumed behavior of the uncertain cooling system dynamics. The nominal dynamics is obtained by setting Dm’ ¼ Dhrad ¼ Da ¼ 0:
3. The sliding mode approach The sliding mode design approach, which may be applied to both controller and observer problems, is well known for its robustness properties (Edwards & Spurgeon, 1998). At the heart of the design procedure is the specification of a so-called switching function. The switching function is chosen to ensure desirable performance is exhibited by the system of interest. If the problem is a control problem, for example, the switching function may be chosen to prescribe desired plant dynamics. Consider s ¼ y’ þ 1: By forcing the switching function to zero, whereby s ¼ 0 and the system is said to have attained a sliding mode, a first order decay would be exhibited by the output of interest. For an observer problem, the switching function may most appropriately be defined as the output error (Edwards & Spurgeon, 1994). By forcing this switching function to zero, the observer output is forced to equal the plant output and a set of estimated states that yield the measured system output are obtained. Having defined an appropriate switching function, it is necessary to choose a control (or in the case of the observer an injection signal) that will force the sliding mode condition to be attained and maintained. This is termed the reachability problem and is typically ensured by choosing the injection signal so that s and s’ are forced to have opposite sign. Essentially, discontinuous injection signals are used to maintain the appropriately chosen switching function at zero. During the sliding motion, the system exhibits total robustness to a class of disturbance signals and any uncertainty that is implicit in the channels where the discontinuous injection signal is applied. For the fault diagnostic situation considered here, the switching function will be the observer error. Further, it will be seen that the discontinuous signal employed to maintain the observer error at zero provides effective
1
2
2
2
3
1
b
2 2
’# ¼ ðk ak ÞT# þ ak T þ k T þ K v ; T 2a 1 2a 2a 2a 3 1 b 2a 2a
ð8Þ
’# ¼ mk T ’ 3 T2 þ ðmk ’ 3 hrad k4 ÞT#3 3 þ hrad k4 Tamb be3 þ K3 v3 ;
ð9Þ
where vi ¼ Ki ðei =8ei 8Þ; i ¼ 2; 2a ; 3 and K2 ; K2a and K3 are the gains of the discontinuous signals v2 ; v2a and v3 which must be chosen to satisfy the reachability problem and ensure sliding motion occurs at all times even in the presence of parameter variations and faults. It should be noted that, although k1 ; k2 ; k3 ; k4 and k2a are assumed constant, they are in fact dependent on the variations m; ’ hrad and a: However, with the proposed observer structure a sliding motion will still be maintained and this parameter uncertainty will be rejected as it appears in channels where the vi are injected. Maintenance of the sliding condition is essential for parameter estimation and fault diagnosis in this scheme. b acts as a gain parameter to tune asymptotic error decay. This parameter can increase the rate of error decay in the absence of uncertainties. The observer error dynamics can be written as follows: e2 ¼ T2 T#2 ; e2a ¼ T2a T#2a and e3 ¼ T3 T#3 ; e’2 ¼ ðk1 mk ’ 2 Þe2 þ Dm’ k2 ðT3 T2 Þ v2 ;
ð10Þ
e’2a ¼ ðk1 ak2a Þe2 þ Da k2a ðT3 T2a Þ v2a ;
ð11Þ
e’3 ¼ ðmk ’ 3 hrad k4 bÞe3 þ Dhrad k4 ðTamb T3 Þ þ Dm’ k3 ðT3 T2 Þ v3 : ð12Þ The stability of the observer system can be investigated with respect to the following candidate Lyapunov function: 1 VðeÞ ¼ ðe22 þ e22a þ e23 Þ: ð13Þ 2 In the Appendix, it is shown that if the uncertainty is bounded as K2 X8Dm’ k2 ðT2 T3 Þ8;
ð14Þ
K2a X8Da k2a ðT2a T3 Þ8;
ð15Þ
K3 X8Dhrad k4 ðT3 Tamb Þ þ Dm’ k3 ðT2 T3 Þ8
ð16Þ
and b is selected so that b > mk ’ 3 convergence of the observer error is ensured.
ð17Þ
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4. Fault determination
4.2. Coolant pump
The gains of the discontinuous terms, K2 ; K2a and K3 ; are designed so that the observer remains in a sliding mode and the error is zero. Thus, the observer errors (e2 ; e2a and e3 ) and their derivatives will converge to zero. The perturbations due to faults are effectively compensated for by the discontinuous signals v2 ; v2a and v3 ; respectively. In the sliding mode, Eqs. (10)–(12) become:
Another important part of the cooling system is the coolant pump which is used to maintain a constant flow rate ðmÞ ’ of coolant while the engine is running. Although the coolant flow rate depends on engine speed, with higher speeds yielding higher flow rates, the engine used here is running at constant speed and a constant flow of coolant is assumed. Malfunctioning of the pump will critically affect the dissipation of heat in the engine block. The capacity of the coolant flow in the system must be closely monitored before any faults occur. One method to tackle this problem is to estimate the coolant flow rate, m: ’ This can be readily constructed from Eq. (21) and the nominal mass flow rate.
0 ¼ Dm’ k2 ðT3 T2 Þ v2 ;
ð18Þ
0 ¼ Da k2a ðT3 T2a Þ v2a ;
ð19Þ
0 ¼ Dhrad k4 ðTamb T3 Þ þ Dm’ k3 ðT3 T2 Þ v3
ð20Þ
and re-arranging these equations gives v2 ; Dm’ ¼ k2 ðT3 T2 Þ Da ¼
v2a ; k2a ðT3 T2a Þ
Dhrad ¼
v3 Dmk ’ 3 ðT3 T2 Þ : k4 ðTamb T3 Þ
4.3. Radiator ð21Þ
ð22Þ
ð23Þ
It is seen that v2 ; v2a and v3 effectively provide a means to estimate the change in mass flow rate, the thermostat valve opening and the radiator coefficient. The signals v2 ; v2a and v3 are discontinuous injection signals and to obtain parameter values, the average values of v2 ; v2a and v3 must be used. It is thus necessary to filter v2 ; v2a and v3 to construct the estimates of the change in mass flow rate, the thermostat valve opening and the radiator coefficient. This information can then be used to hypothesize a fault detection scheme for the coolant system. The previously mentioned variations in the ki are small and have a small relative effect on the equivalent injection signals. This is why the ki can be assumed nominal within the framework and omitted from the above analysis. Three possible fault scenarios are now addressed. 4.1. Thermostat bypass valve The thermostat bypass valve is used to control the coolant flow to the radiator. It is usually closed (a ¼ 0) to give a warm start during start up of the engine. The valve opens only after the coolant temperature reaches a certain temperature. If the valve behaves abnormally, where it may totally close or open at a later stage, then the engine may overheat. Estimating the valve opening, a will help in the fault monitoring process. This can be constructed from Eq. (22) and the nominal thermostat valve opening level.
The radiator is an essential component of the cooling system. It is responsible for dissipation of engine heat to the atmosphere and maintaining the engine block temperature within the appropriate operating range. Overheating of the engine will occur if this component fails to function correctly. Possible faults include malfunction of the radiator fan. In this case, the heat transfer coefficient will determine the efficiency of the radiator system. This can be constructed from Eq. (23), using Eq. (21) and the nominal radiator coefficient. The sliding mode observer is not only used here to estimate the system states but also to predict three important parameters, the thermostat valve opening, the coolant flow rate, and the heat transfer coefficient.
5. Experimental setup and results 5.1. Experimental setup The sliding mode observer developed in the previous section was tested on a power generator set (Genset) which has the capability of generating a maximum power of 65 kW. Various faults are introduced in the Genset in order to test the efficiency of the proposed design approach as outlined below: *
*
*
coolant pumpFthe bypass valve is opened to reduce the coolant flow, thermostat valveFa different type of valve is used which will open later than the normal valve, radiatorF10 and 20% covered.
The hardware interface involved in the test is dSPACE, pressure sensors and temperature sensors. A block diagram in Fig. 2 shows the complete system setup.
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Fig. 2. Block diagram showing the system interface and signal flow.
Fig. 3. State estimate, T2 and the error between the measured and estimated values.
5.2. Data acquisition A 5th order Butterworth filter was employed to remove the high frequency components of the signal noise. The filter had a sampling frequency of 1000 Hz with a cut-off frequency at 1 Hz. Raw data initially sampled at 1000 Hz, was down sampled to 100 Hz before saving due to the memory constraints of the on-line computer. The reason for this is to make sure signal properties are captured with no loss of information and no aliasing occur in this case. 5.3. Experimental results The following figures were generated from tests on the Genset. The first set of results corresponds to normal operating conditions with a step load applied to the
Genset from 0 to 65 kW, followed by a step down load and the process was repeated once. The estimation of the system states, T2 and T3 are shown in Figs. 3 and 4, respectively. This value is compared with the value which was measured online while the experiment was carried out. The error between the measured and estimated values is of the order of 0– 0.06 K, which is about 0.02%. Thus, the observer has been validated under normal operating conditions. A non-linear model is employed here to provide a comparison between parameter estimates developed from the non-linear model, and those estimated by the sliding mode observer. It should be noted that this is not the non-linear model used to design the observer but is a model developed using black box techniques and a large amount of empirical engine data. Further, it is the change in system parameters, as monitored via the
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Fig. 4. State estimate, T3 and the error between the measured and estimated values.
Fig. 5. Component parameter estimate, value obtained from a non-linear simulation and the mismatch between the two.
equivalent injection analysis, which is to be used to indicate the presence of faults. The results are shown in Figs. 5, 6 and 9. During the transition between the thermostat valve opening and closing (which happens under normal conditions), the error between the response of the non-
linear model and the measured response is relatively large, approximately 10%. When the thermostat valve is opened on the Genset, coolant flows into the radiator and the system undergoes a transition in thermodynamic properties. Due to this imbalance, the system parameters change accordingly in practice but this process is
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Fig. 6. Component parameter estimate, m’ obtained from the non-linear model and the mismatch between the two.
Fig. 7. When the bypass valve is opened, the coolant flow rate reduces and the error between this valve and the nominal value is shown.
not modeled accurately. The hysteresis effect of the thermostat valve itself is also thought to contribute to the mismatch. A fault is now introduced into the system by opening the bypass valve to reduce the coolant flow rate through
the system. Fig. 7 shows the rate of change of coolant flow. The bypass valve was opened for the period of time between 0.6 and 1.25 s. It is obvious that the response of the system does not change instantaneously when the valve is opened and closed. A likely explanation for this
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Fig. 8. Thermostat valve opens later than normal during fault condition.
Fig. 9. Plot of the heat transfer coefficient for normal and fault conditions.
is that the coolant still in the bypass valve circuit has a lower temperature when compared to the coolant circulating in the system. When the bypass valve opens, these two sets of coolant, which are at different
temperatures, mix together and after sometime, the coolant in the bypass valve eventually rises to the circulating coolant temperature. As a result the system takes a short time to reach its steady state.
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Next the thermostat valve is set to open at a later stage when compared to normal conditions. Fig. 8 shows the plot of the thermostat valve opening during normal conditions and in the presence of the fault. It is proved that when the valve opens late, or indeed does not open at all, the system can accurately detect the fault. For the radiator system, the radiator is covered by 10% and 25%. Fig. 9 shows a plot of heat transfer coefficient against ambient temperature. During normal operating conditions, the data forms a relatively concentrated cloud. As the engine becomes hotter, variation from ambient temperature occurs, but the variation is relatively small (between 308 and 312 K). It is seen that the heat transfer coefficient reduces when the radiator is covered. Also the clouds of data points become more elongated; this is because the cooling effect of the radiator is less efficient and thus the observed temperature drifts are larger. The ambient temperature is lower when the radiator is 25% covered as compared to when it is 10% covered due to the correspondingly larger reduction in the efficiency of the radiator. The heat transfer coefficient when the radiator is 10% covered is marginally larger than when the radiator is 25% covered.
6. Conclusion This paper explores the application of sliding mode concepts for fault detection and isolation. The experimental results show that perturbations in component parameters can be recovered by manipulating the equivalent input injection signal to the observer. Therefore, faults present in Genset components can be detected. It is seen that the behavior of relatively sophisticated components can be inferred from straightforward temperature measurements. Although a specific system has been considered here, the core results have potentially wide applicability to a range of non-linear engineering systems.
and substituting the error dynamic Eqs. (10)–(12) to yield, ’ ¼ e2 e’2 þ e2a e’2a þ e3 e’3 VðeÞ ’ ¼ e2 ½ðk1 mk VðeÞ ’ 2 Þe2 þ Dm’ k2 ðT3 T2 Þ v2 þ e2a ½ðk1 ak2a Þe2a þ Da k2a ðT3 T2a Þ v2a þ e3 ½ðmk ’ 3 hrad k4 bÞe3 þ Dhrad k4 ðTamb T3 Þ þ Dm’ k3 ðT3 T2 Þ v3 ’ ¼ k1 e22 mk VðeÞ ’ 2 e22 k1 e22a ak2a e22a ðhrad k4 þ b mk ’ 3 Þe23 e2 Dm’ k2 ðT2 T3 Þ K2
8e2 82 8e2 8
8e2a 82 e3 Dhrad k4 8e2a 8 8e3 82 ðT3 Tamb Þ e3 Dm’ k3 ðT2 T3 Þ K3 8e3 8 e2a Da k2 aðT2a T3 Þ K2a
as e2i ¼ 8ei 82 ’ ¼ k1 e22 mk VðeÞ ’ 2 e22 k1 e22a ak2a e22a ðhrad k4 þ b mk ’ 3 Þe23 e2 Dm’ k2 ðT2 T3 Þ K2 8e2 8 e2a Da k2a ðT2a T3 Þ K2a 8e2a 8 e3 Dhrad k4 ðT3 Tamb Þ e3 Dm’ k3 ðT2 T3 Þ K3 8e3 8 p k1 e22 mk ’ 2 e22 k1 e22a ak2a e22a ðhrad k4 þ b mk ’ 3 Þe23 8e2 8ðK2 8T2 T3 8Dm’ k2 Þ 8e2a 8ðK2a 8T2a T3 8Da k2a Þ 8e3 8ðK3 8Dhrad k4 ðT3 Tamb Þ8 8Dm’ k3 ðT2 T3 Þ8 With the given uncertainty bounds (14)–(16) and restriction on b (17) convergence of the observer error is proved since all the nominal constants involved in the above equation are positive, i.e. k1 > 0; k2 > 0; k3 > 0; k2a > 0; hrad > 0:
k4 > 0;
m’ > 0;
References Acknowledgements This project is supported by Perkins Engines, TRW and EPSRC Grant Reference GR/L42018. The invaluable assistance of our colleague Mr. John Twiddle during the data capture phase of this work is gratefully acknowledged.
Appendix A The stability of the sliding mode observer can be investigated by differentiating the candidate Lyapunov equation VðeÞ ¼ 1=2ðe22 þ e22a þ e23 Þ with respect to time
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