ARR Based System Mode Identification

Abstract

In the previous chapters, the focus has been on ARR-based FDI of systems represented by a hybrid model. ARRs derived from a DBG of a hybrid system model depend on the discrete states of the switches used in the hybrid model. In order to avoid reporting non-existent faults to a supervisor system, it is essential to detect discrete mode changes. The system mode must be known to make sure that the correct ARRs valid for the current system mode are evaluated. This chapter shows that ARRs obtained from a DBG with non-ideal switches and thus with fixed, mode independent causalities can not only be used for FDI in systems represented by a hybrid model but also for system mode identification.

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In the previous chapters, the focus has been on ARR-based FDI of systems represented by a hybrid model. ARRs derived from a DBG of a hybrid system model depend on the discrete states of the switches used in the hybrid model. The values of all switch states at a time instant $t > 0$ define the system mode that lasts until at least one switch changes its state giving rise to a new system mode. If ARRs valid for the current system mode are used beyond a mode change, their evaluation may give residuals that exceed current thresholds although no parametric fault has happened. In order to avoid reporting non-existent faults to a supervisor system, it is essential to detect discrete mode changes. The system mode must be known to make sure that the correct ARRs valid for the current system mode are evaluated. This chapter shows that ARRs obtained from a DBG with non-ideal switches and thus with fixed, mode independent causalities can not only be used for FDI in systems represented by a hybrid model but also for system mode identification. In [1], the method has been formulated for hybrid LTI systems and has been applied to a DC–DC buck converter. The concept, however, is not limited to linear systems. Another recent bond graph approach makes use of controlled junctions and a sequential causality assignment procedure (SCAP) adapted for FDI [2, 3].

7.1 Bond Graph Based System Mode Identification Using ARRs

The time instances of discrete mode changes may be either known or may be given by contraints. For instance, the pass transistor in a buck power converter is switched on and off by an external control signal, while the diode switches autonomously conversely to the pass transistor. Once conditions for controlled and autonomous switching have been formulated, e.g. for the bouncing ball problem, a Petri net (PN) or a finite state machine can be set up that determines the new mode after a mode change. For each mode, a DBG model can be developed. Its evaluation provides the ARR residuals valid for that mode and the PN governs the transition from the current DBG to the one valid for the next mode. This corresponds to the presentation of hybrid models by a combination of a PN and a set of bond graphs as has been outlined in Sect. 2.2.

7.1.1 Identifying a Set of ARR Residuals Close to Zero

Alternatively, a DBG with non-ideal switches and mode invariant causalities can be developed that holds for all system modes. It is assumed that the system under consideration is healthy and that no faults occur during system mode identification. For simplicity it is assumed that mode dependent ARRs in closed symbolic form can be deduced from the DBG. Let $s$ be the number of switches in the model and $n_{\textit{f}} \le 2^s$ the number of physically feasible switch state combinations, i.e. $n_{\textit{f}}$ denotes the number of system modes. Furthermore, let $\varvec{u} = (u_1 \, u_2 \cdots u_N)^T$ be the vector of $N$ known system inputs and $\varvec{y} = (y_1 \, y_2 \cdots y_M)^T$ the vector of $M$ inputs into a DBG either obtained by measurements from the real system or by evaluating a behavioural model of the real system. Then each ARR residual $r_i(t)$, $i = 1, \ldots s$, is the weighted sum of known system inputs, known measurements and derivatives of measurements.

$$\begin{aligned} r_i^{(\sigma )}(t)&= \sum _{\nu = 1}^N \, c_{i \nu }^{(1)}(\varvec{\varTheta }, \varvec{m}^{(\sigma )}(t)) u_\nu (t) \, + \, \sum _{\upmu = 1}^M c_{i \upmu }^{(2)}(\varvec{\varTheta }, \varvec{m}^{(\sigma )}(t)) y_\upmu (t)\nonumber \\&\quad + \, \sum _{\upmu = 1}^M c_{i \upmu }^{(3)}(\varvec{\varTheta }, \varvec{m}^{(\sigma )}(t)) \dot{y}_\upmu (t) \end{aligned}$$

(7.1)

where $\varvec{m}^{(\sigma )}(t) = (m_1^{(\sigma )}(t) \, m_2^{(\sigma )}(t) \ldots m_s^{(\sigma )}(t))^T$ with $m_j^{(\sigma )}(t) \in {0,1}$, $j = 1, \ldots , s$ and $1 \le \sigma \le n_{\textit{f}}$ represents the system mode at time instant t and $\varvec{\varTheta }$ denotes the vector of all system parameters.

For each of the $n_{\textit{f}}$ technically feasible system modes, there is a set of $n_s^{(\sigma )}$ ARRs. Some of them may be system mode independent and can be discarded with regard to system mode identification. Furthermore, for each sampled time instant $t$, the weighting factors $c_{i \nu }^{(1)}(\varvec{\varTheta }, \varvec{m}^{(\sigma )}(t))$, $c_{i \upmu }^{(2)}(\varvec{\varTheta }, \varvec{m}^{(\sigma )}(t))$, and $c_{i \upmu }^{(3)}(\varvec{\varTheta }, \varvec{m}^{(\sigma )}(t))$ are constants. That is, the ARRs for system mode $\varvec{m}^{(\sigma )}(t)$ are a weighted sum of $\varvec{u}(t)$, $\varvec{y}(t)$, and $\dot{\varvec{y}}(t)$. Their number is $n_r^{(\sigma )} \le n_s^{(\sigma )}$.

Now, for each of the $n_{\textit{f}}$ feasible system modes, the set of $n_r^{(\sigma )} \le n_s^{(\sigma )}$ ARRs is evaluated for discrete time instances $t$ yielding ARR residuals $\varvec{r}^{(\sigma )}(t)$, $\sigma = 1, \ldots , n_{\textit{f}}$. The resulting ARR residuals may be arranged into a matrix $(\varvec{r}^{(\sigma )}(t))$ with $s_f$ columns and $n_s$ rows. For $n_r^{(\sigma )} < n^{(\sigma )}_s$ rows with an index $n^{(\sigma )}_r \le i \le n^{(\sigma )}_s$ are filled up with zeros so that all residuals build a $n_s \times n_{\textit{f}}$ matrix. The current system mode is identified by searching for a column of which all entries have values close to zero. As there are no faults, only one out of the $n_{\textit{f}}$ sets of ARRs will give residuals close to zero within given bounds because the discrete state switch variables used in this set of ARRs are the ones that reflect the current system mode. Once some discrete switch states change, the system mode changes. Accordingly, output variables, obtained by measurement of a real system or by a numerical evaluation of a behavioural model that may encompass models of the sensors, change significantly. Accordingly, all entries of another column in the matrix become close to zero indicating the new system mode.

In conclusion, there is a unique set of ARRs for each system mode. A system mode is characterised by a set of discrete switch states that are variables in the ARRs. When switches change their state, the system mode of operation changes. As a result, residuals of the current set of ARRs may take significantly different values invalidating the set of ARRs that characterises the current system mode. Sets of ARRs can be used to identify system modes and changes of system modes.

This system mode identification may require considerable computational costs especially if ARRs cannot be deduced in closed symbolic form so that the entire DBG model is to be evaluated to obtain numerically the time history of ARR residuals. However, as it is one and same the problem with different sets of discrete switch state variables, this computation can be easily and efficiently performed in parallel on multicore processors or multiprocessor computers.

Measurements from the real system are overloaded with noise. In order to be able to clearly identify the set of ARRs with residuals close to zero, measured data should undergo appropriate filtering before it is used in the evaluation of ARRs. Moreover, some of the sensors providing measured values may operate in a faulty mode due to external disturbances caused by changes in the ambient or by internal parametric faults. If details of the internal build-up of a sensor are not fully known so that a bond graph model cannot be developed, then for small changes, its dynamic behaviour may be approximately captured by a transfer function that, at least, accounts for the sensor’s delay and its gain.

After all, it should be kept in mind that ARRs are deduced from a bond graph that is the outcome of a modelling process based on physical first principles. In this process, modelling assumptions have been made, effects have been conceptually captured in an idealised manner or have been neglected. Accordingly, because of these modelling simplifications, because of a possible linearisation of model equations or a reduction of the model order and because of model parameter uncertainties some ARR residuals may deviate from zero even if no fault has happened.

7.1.2 Identification of System Mode Changes in a Healthy System

Once an initial system mode is known, this knowledge can be used to identify candidate sets of ARRs for the next system mode and only these sets of equations need to evaluated to identify the next system mode [3]. To that end, similar to a FSM, a so-called Mode-change Signature Matrix (MCSM) is introduced in [3] with the assumption that any mode change is caused by the change of one single discrete state variable. In this matrix, there is a row for each discrete state variable $q_i$ and a column for each ARR of index $j$. If $q_i$ occurs in an ARR of index $j$, the matrix entry in place $(i,j)$ is equal to one and zero otherwise. That is, a mode change represented by a change of $q_i$ can be detected, if at least one entry is equal to one in that row. Moreover, if the row signature is unique, the mode change can be isolated. System mode changes may be viewed as a special case of switch faults . The value of a discrete switch state may indicate that an instantaneous mode change has taken place but that the system is still healthy or that an open circuit or short circuit fault has occurred. Discrete switch states in mode dependent ARRs may therefore be included in a FSM in the same way as continuous component parameters. In this book, the MCSM is part of the all-mode FSM.

If a system under consideration is faulty and if the evaluation of ARR gives residuals that are outside some given bounds then this could be due to a fault in some component, or the values of the discrete state variables used in the set of ARRs are not valid for the current mode. Changing the set of discrete state variables, i.e. accounting for mode changes affects ARR residuals differently. Some ARRs in a set of mode dependent ARRs may be even independent (cf. the example network in Sect. 4.2, ARR 4.8). If ARR based fault parameter estimation is used for fault isolation, sampled data would be collected while the system is faulty and the ARRs are no longer valid. That is, parameter estimation is likely to give wrong and misleading results.

For these reasons, in the following, it is assumed that no parametric faults happen during system mode identification and that an initial system mode is known. System mode identification in the presence of faults is more difficult. In [4], Arogeti et al. present an advanced method for this more general case that categorises ARRs into different types and provides a refined set of fault candidates to the fault parameter estimation procedure. Multiple fault detection, isolation and identification for hybrid systems with no available information on the nature of faults (abrupt or incipient) and on system mode changes has been recently addressed in [5].

Once a discrete system mode change takes places at a time instant $t_1$, the values of ‘measured’ inputs into the ARRs change and the discrete switch state variables used in the set of ARRs are no longer valid. As a result, the residuals of some ARRs will no longer be within bounds close to zero. Accordingly, some entries in the coherence vector will be equal to one. The coherence vector is compared with the rows of that part of the all-mode FSM that constitutes the MCSM to identify candidates for the switch state variable that may have changed and has caused the system mode change. Let $[m_s^0 \cdots m_2^0 \; m_1^0 ]$ denote the initial system mode. If the comparison of the coherence vector with the MCSM part of the all-mode FSM reveals say $m_2$ as candidate for the mode change then the next possible system mode is $[m_s^0 \cdots {m_2}\; m_1^0 ]$. This is checked by evaluating the ARRs with this set of discrete switch state variables. If the residuals of the adapted ARRs are not close to zero and if there is another switch state that may have changed then the ARRs are also evaluated with a second set of switch state variables. If a new mode cannot be identified, it is assumed that the change in the coherence vector is due to a parametric fault. In that case, the all-mode FSM is inspected for fault isolation by setting the discrete switch state variables to their values of the current system mode. If the inspection indicates that faults cannot be isolated, fault parameter estimation will have to be performed.

Example

For illustration, the method for identification of mode changes is applied to a modification of the simple network in Fig. 4.1 with two more switches displayed in Fig. 7.1.

Fig. 7.1

Network with three semiconductor switches

The network is easily converted into the DBG in Fig. 7.2.

Fig. 7.2

Diagnostic bond graph of the network in Fig. 7.1

The sum of flows at the two sensor junctions $0_1, 0_2$ yields two independent ARRs.

$$\begin{aligned} \mathrm{ARR}_1\!\!: 0&= \frac{1}{R_1} ( E - e_1 ) - C_1 \dot{e}_1 - \frac{m_1}{R_\mathrm{on}+ R_2} e_1 - \frac{m_2}{R_\mathrm{on}+ R_3} ( e_1 - e_2 ) \; = \; r_1 \end{aligned}$$

(7.2)

$$\begin{aligned} \mathrm{ARR}_2\!\!: 0&= \frac{m_2}{R_\mathrm{on}+ R_3} ( e_1 - e_2 ) - C_2 \dot{e}_2 - \frac{m_3}{R_\mathrm{on}+ R_4} e_2 \; = \; r_2 \end{aligned}$$

(7.3)

The structural information contained in the two ARRs (7.2)–(7.3) is given by the all-mode FSM in Table 7.1.

Table 7.1 All-mode FSM for the switched network in Fig. 7.1

The first three rows of the all-mode FSM constitute what has been called a Mode Change Signature Matrix (MCSM) in [3]. These rows indicate that $\mathrm{ARR}_1$ is sensitive to system mode changes caused by a change of the discrete state of switches $\mathrm{Sw}_1$ or $\mathrm{Sw}_2$, while ARR residual $r_2$ is affected when either switch $\mathrm{Sw}_2$ or $\mathrm{Sw}_3$ changes its state. Moreover, it can be seen that all three system mode changes caused by one of the three switches can be detected and can be isolated. There is an entry equal to one in each of the first three rows and their mode change signatures are unique.

An inspection of the lower part of the all-mode FSM reveals that a fault in resistance $R_3$ can be isolated in all system modes in which $\mathrm{Sw}_2$ is on ($m_2 = 1$). This is indicated by the entry $m_2$ in the last column. For instance, for mode $[m_3 \; m_2 \; m_1] = [ 1 \; 1 \; 1 ]$ (mode 7), the FSM is displayed in Table 7.2.

Table 7.2 FSM in mode 7 ($[m_3 \; m_2 \; m_1] = [ 1 \; 1 \; 1 ]$) for the switched network in Fig. 7.1

Identification of a Sequence of System Mode Changes

Now, let $[m_3 \; m_2 \; m_1] = [ 1 \; 1 \; 1 ]$ (mode 7) be the initial system mode. The initial system mode is detected by evaluating the two ARRs for all eight switch state combinations. As the system is healthy, all residuals must be close to zero. Let all switches be closed for some time interval $0 \le t < t_1$ so that the capacitors get charged to some extent. At $t = t_1$ a mode change may happen. An evaluation of the ARRs shows that residual $r_1(t_1)$ is no longer close to zero so that the coherence vector becomes $\varvec{c} = [ 1 \; 0]$. In other words, $\mathrm{ARR}_1$ is no longer consistent with the system mode. Comparison of the coherence vector with the first three rows of the all-mode FSM indicates that $\mathrm{Sw}_1$ must have changed its state so that the new system mode is likely to be $[ 1 \; 1 \; 0 ]$ (mode 6). As $\mathrm{ARR}_2$ is consistent with the new system mode, only $\mathrm{ARR}_1$ with the new set of discrete switch states is re-evaluated. If its residual is again close to zero, the new system mode is identified.

Let there be another mode change at $t = t_2$ and the coherence be $\varvec{c} = [ 1 \; 1]$. Comparison of its pattern with the rows of the MCSM part of the all-mode FSM indicates that $m_2$ has likely changed its value. As it is assumed that any system mode change is due to a single switch state change, a possible transition from mode $[ 1 \; 1 \; 0 ]$ (mode 6) to mode $[ 1 \; 0 \; 0 ]$ (mode 4) has taken place. As $m_2$ affects both residuals, the two ARRs must be evaluated again to validate the system mode hypothesis.

Likewise, a subsequent mode change at $t = t_3$ with the coherence vector $\varvec{c} = [ 0 \; 1]$ leads to mode $[ 0 \; 0 \; 0 ]$ (mode 0) and if a final mode change takes place at $t = t_4$ with the coherence vector $\varvec{c} = [ 1 \; 1]$ then mode $[ 0 \; 1 \; 0 ]$ (mode 2) is identified.

Table 7.3 FSM in mode 2 ($[ 0 \; 1 \; 0 ]$) for the switched network in Fig. 7.1

If $\varvec{c} = [ 1 \; 1]$ for a time instant $t_5 > t_4$, no new system mode is identified. It is then concluded that a parametric fault has happened. For this simple network, the parametric fault can be isolated by inspection of the FSM for system mode $[ 0 \; 1 \; 0 ]$ (mode 2) (Table 7.3) and identified as a fault in resistance $R_3$. No fault parameter estimation is needed. In addition, the FSM for mode 2 shows that in this mode, faults in resistances $R_2$ and $R_4$ cannot be detected.

Note that the re-evaluation of ARRs for validating a mode hypothesis is not necessary in the case of the considered simple network because the system is assumed to be healthy, the initial system mode is known, i.e. initial conditions for the discrete switch state variables are known and all system mode changes can be isolated by inspection of the MCSM part of the all-mode FSM. If the signature of a mode change is not unique so that it cannot be isolated by inspection of the MCSM part of the all-mode FSM, a subset of ARRs must be re-evaluated in order to identify the new system mode among a small set of potential mode candidates. If two or more mode changes take place in rapid sequence in real-time system mode identification while mode identification based on the evaluation of a subset of ARRs is still in progress, the last known system mode as a starting point for the decision about the next possible system mode and for the evaluation of subsets of ARRs is no longer valid. In such a case, the entire set of ARRs must be evaluated for identification of the next system mode.

The identification of the above sequence of mode changes is summarised in Table 7.4.

Table 7.4 System mode identification and FDI

Figure 7.3 depicts the time evolution of the signals $m_1(t)$, $m_2(t)$, and $m_3(t)$ controlling the discrete state of the switches $\mathrm{Sw}_1$, $\mathrm{Sw}_2$, and $\mathrm{Sw}_3$. Accordingly, the time history of the system mode is displayed in Fig. 7.4. Table 7.5 lists the parameters used for the simulation of the network’s dynamic behaviour. Figure 7.5 displays the time evolution of the capacitor voltages $u_{C_1}$ and $u_{C_2}$.

Fig. 7.3

Time evolution of the signals $m_1(t)$, $m_2(t)$, and $m_3(t)$ controlling the switches $\mathrm{Sw}_1$, $\mathrm{Sw}_2$, and $\mathrm{Sw}_3$

Fig. 7.4

Time history of the system mode

Table 7.5 Parameters of the switched network in Fig. 7.1

Fig. 7.5

Time evolution of the voltages across the capacitors of the network in Fig. 7.1

When switch $\mathrm{Sw}_1$ opens at $t = 10$ s, the current charging the two capacitors is increased and thus the rise of the voltages across the capacitors increases. This is less distinct for $u_{C_1}$. When switch $\mathrm{Sw}_2$ opens at $t = 20$ s, the rise of the voltage $u_{C_1}$ increases again, while capacitor $\mathrm{C}$ : $C_2$ discharges via the resistor $\mathrm{R}$ : $R_4$. After all, when $\mathrm{Sw}_3$ opens at $t = 30$ s, capacitor $\mathrm{C}$ : $C_2$ is neither charged nor can it discharge. It is charged again as of $t = 40$ s when switch $\mathrm{Sw}_2$ is closed. Finally, a parametric fault is introduced by abruptly increasing resistance $R_3$ at $t = 45$ s. As a result, the recharging of capacitor $\mathrm{C}$: $C_2$ is slowed down.

The mode changes and the abrupt fault in resistance $R_3$ are captured by the time evolution of the residuals $r_1(t)$ and $r_2(t)$ as depicted in Fig. 7.6.

Fig. 7.6

Time evolution of residuals $r_1(t)$ and $r_2(t)$. a Residual $r_1(t)$ b Residual $r_2(t)$

In real-time FDI and in real-time mode identification, ARRs are evaluated with sampled data. Residuals are used by a decision procedure that provides an update of the coherence vector. A system mode change is indicated by a coherence vector ($\varvec{c} \ne \varvec{0}$). Comparison of the coherence vector with the rows of the MCSM part of the all-mode FSM either isolates the new system mode or gives possible candidates, i.e. switch state combinations. For these mode candidates, a subset of ARRs has to be re-evaluated to isolate the new mode and that ends the mode identification.

To visualise mode changes by the time history of ARRs, their evaluation uses the last known, i.e. old and thus wrong values of the discrete switch state variables for a small time interval of length $\varDelta t$ once a system mode change has taken place at time $t = t_s$. This gives a peak in the affected residual at $t = t_s$ indicating the mode change. This use of wrong switch state values for a short while beyond the switching event is illustrated in Fig. 7.7 for the discrete state change of switch $\mathrm{Sw}$: $m_1$.

Fig. 7.7

Use of the last known state $m_1(t)$ of switch $\mathrm{Sw}$: $m_1$ for a small time $\varDelta t$ beyond the switching event at $t = 10$ s

The values of the signal $m_1(t)$ are used in the evaluation of the behavioural bond graph model replacing the real network, while the evaluation of the ARRs uses the signal $m_{11}(t)$.

7.2 Detection and Isolation of Switch Faults

Mode identification for systems represented by a hybrid model means identification of the current discrete state of all switches in the system and can therefore be used to detect switch faults. Under normal faultless conditions, the signal controlling a semiconductor switch toggles between two discrete values with a certain frequency and the switch opens and closes accordingly. This is no longer the case when an open or a short circuit fault happens in a switch. In a case where no other parametric faults have happened, system mode identification then reveals that a discrete switch state does not toggle any more as of some time instant. Clearly, a high switching frequency leaves little time for real-time ARR-based system mode identification. However, as has already been mentioned, evaluation of ARRs for all switch state combinations can be performed in parallel on multiple processors. Once discrete switch states have been identified, rule-based reasoning can detect and isolate switch faults. For instance, the two switches in the DC-DC buck converter in Fig. 2.19 have opposite states in normal operation, i.e. if one switch is open then the other one is closed. The two of them cannot be open or closed at the same time unless an open or a short circuit fault has happened in one of them.

Identification of System Mode Changes in the Presence of Parametric Faults

In the previous section, each discrete switch state has been taken into account by a row in the FSM. System mode changes can be viewed as faults in discrete switch states. This means that there are far more fault candidates than sensors. Accordingly, discrete switch states will share the same component fault signature so that a switch fault cannot be isolated. If multiple simultaneous faults have happened only in parameters that change continuously with time then parameter estimation can be used to isolate them. However, if there are discrete switch state faults among the multiple simultaneous faults then parameter estimation may result in meaningless real values for the discrete switch states. Therefore, in [6], Alavi and her co-authors propose to chose a combination of switch states, to insert them into the functional to be minimised and to perform a parameter estimation. This task is repeated for all feasible switch state combinations. The combination of switch states that gives the minimum of all optimal values of the functional identifies the system mode. At the same time, the minimisation of the functional provides estimated values for the continuous parameters. A comparison of estimated parameter values with their nominal ones then isolates parametric faults.

7.3 Summary

ARRs derived from hybrid system models are mode-dependent. For ARR-based FDI it is therefore necessary to know the current system mode so that ARRs with the correct set of discrete switch state values are evaluated. When ARR residuals are outside admissible parameter uncertainty bounds it is not clear whether a parametric fault has occurred or a mode change has happened. If a mode change has happened then an evaluation of ARRs with the discrete switch states of the last known mode yields wrong and misleading results. This chapter considers system mode identification for healthy systems only. For the more complex case of system mode changes in the presence of faults, it is suggested to see latest publications, e.g. [4].

The current system mode of a healthy system can be identified by evaluating all ARRs for all feasible switch state combinations. This may require considerable computational effort. The computational time, however, can be reduced by distributing the task on multiple parallel processors. Moreover, this task is only necessary if an initial system mode is not known or if the last known system mode is no longer valid because rapid system modes have taken place while system mode identification is still in progress. Once the current system mode is known, the all-mode FSM can be consulted to identify a subset of ARRs that is to be evaluated to identify the new current system. This has been illustrated by application to a simple circuit with three semiconductor switches.

Finally, the knowledge of the current values of discrete switch states can be used to detect and to isolate switch faults by rule-based reasoning. If switches toggle their state at a high frequency little time is left for mode identification. However, again, parallel processing can help to cope with the time constraints.