5
Dynamic Bayesian Networks: Integrating Reliability Computation in the Control System
5.1. Introduction
As discussed in the previous chapters, graphical probabilistic models are applied in risk management, maintenance and diagnosis. Nevertheless, these probabilistic models can also be used in control theory applications. The graphical probabilistic model is implemented online in the closed loop before interruption of system functioning for normal or maintenance actions. The purpose of this chapter is to integrate the probabilistic models into control theory to optimize the control strategy according to failures and their impacts on system reliability.
The control strategy has an impact on the system and its performance during the operation. For instance, modifying a control law according to the faults and failure can warrant the system functioning. Nevertheless, overcharging an actuator to compensate for decreased performance can accelerate the degradation of system components. From a long-term perspective, performance cannot be infinitely compensated for. The greater the compensation, the greater the necessary components’ overcharge and the faster the degradation rate.
The models necessary to estimate reliability under operational conditions have to be used online during the operational phase. To do this, it is necessary to use formalism well suited to the estimation and prognosis of the functioning modes induced by events occurring throughout the lifetime of the system. Moreover, control methods are not designed to integrate probabilistic information coming from reliability models. Thus, it is necessary to design methods that are able to integrate the probabilistic knowledge into the control algorithms.
This is an open question and this chapter is dedicated to showing our approach to this paradigm. Obviously, the proposed approach is based on DBN presented previously to estimate the fault and failure impact or context variations on fault-tolerant systems.
5.2. Integrating reliability information into the control
Component reliability and system reliability have been less closely examined in the literature on control theory. [GOK 05] proposed integrating the parameters to increase the life of actuators to reduce the maintenance costs. The method is based on the estimation of time before failure according to the past component use and the modification of the component’s functioning state if the estimated remaining lifetime is less than expected. [GOK 06] presented some algorithms for adaptive control. The first algorithm maintains the expected actuator lifetime by adjusting its performance level. The other algorithm offers a compromise between the actuator performance and the expected lifetime according to mission requirements.
[PER 10] proposed a solution based on model predictive control (MPC) strategy that is used to allocate the effort among the redundant actuators by fixing constraints on the actuator degradation. This degradation is computed by cumulating the control inputs. This constraint is integrated in the MPC strategy to protect against the dangerous degradation levels of some critical actuators. This method is not based on reliability computation but integrates the co-variables (control input) that have an impact on the component reliability. The principle is to focus on increasing the component reliability without considering the system reliability.
[GUE 07, GUE 11] focused on defining a structure that combines components to elaborate a system with higher reliability level after a component failure. It is based on a fault-tolerant control whose fundamental principle is to keep the performance levels closer to the performance level defined before the occurrence of failure. Fault tolerance is a control reconfiguration or a restructuring strategy integrating reliability analysis and component costs [GUE 04a, GUE 04b]. From the fault detection and isolation process, the reconfiguration task consists of determining the possible structures that ensure the initial system performance or accepted degraded performances by isolating the faulty components or switching to operating subsystems. For this purpose, an optimal structure is searched for from among all the possible structures [GUE 04a, GUE 05, GUE 06].
[KHE 11] proposed a fault-tolerant control strategy to warrant the system reliability. This new methodology requires adaptation of several reliability models or parameters to integrate them as constraints or conditioning criteria of the control law. The integration of the impact of reliability on the end of mission is a key point of this work.
5.3. Control integrating reliability modeled by DBN
The goal is to define a control strategy for over-actuated systems that allows us to optimally allocate the effort on actuators under the constraint of preserving the system reliability in the normal case or when component failure occurs. To optimize the actuator inputs, it is necessary to have sufficient free degrees in the control law. Clearly, this is the case in over-actuated systems. An over-actuated system is not necessarily a system with redundant components but a system where the control goals can be attained in a different manner.
Figure 5.1. Control structure of an over-actuated system integrating a reliability model
5.3.1. Modeling and controlling an over-actuated system
From a general point of view, an over-actuated system can be considered as a linear system with m actuators and described by the following discrete equation:
with A ∈ Rn×n, Bu ∈ Rn×m and C ∈ Rp×n being the state, control and output matrices respectively. 
is the state vector of the system, ũ ∈ Rm is the input control vector and ỹ ∈ Rp is the system output vector. The condition rank (Bu) = r < m characterizes over-actuated systems. Figure 5.1 shows the control principle of an over-actuated system integrating reliability information. The reliability model is used to allocate the control efforts on the actuators.
Matrix Bu can be factorized:
with Bv ∈ Rn×r and B ∈ Rr×m all of rank r. The system is then modeled by:
with 
representing the whole controlling effort required for the system to function is also called the virtual input vector. Control allocation aims to define the real control inputs of the system ũ(k) from the expected virtual control input, such as:
where 
is computed by an algorithm that should satisfy the control and target goals and ũmin ≤ ũ ≤ ũmax represents the physical limits of the actuators (saturations).
A solution to the allocation problem is given by the resolution of an optimization problem. If no solution exists, an optimal solution is defined with the limits ũ(k), such that Bũ(k) gives the best approximation of . The optimal control input can be obtained by minimizing the following criteria:
with Ψ all the possible solutions for the control input ũ(k), according to the controlling goals and 
being the expected control input. Matrix 
allows priority levels to be set for the actuators. W(k) is usually defined as a diagonal matrix:
5.3.2. Integrating reliability
The weighing matrix W(k) is considered the key to integrating the actuators’ reliability into the control input allocation problem of over-actuated systems. The control problem can be solved in several steps, as shown in Figure 5.2. To maximize system reliability, the weighing matrix W(k) is set from the actuator contributions 
to the system operation:
Figure 5.2. Control framework of an over-actuated system integrating a DBN reliability model
This contribution depends on the structure function φ(ek), where 
allows the calculation of system reliability according to the actuator 
. The actuators are taken into account in the control strategy proportional to their contribution to the system operation. The system state Sk is defined from the structure function φ(ek):
From the control point of view, the over-actuated system is hypothesised to be in a working state even if some actuators are in a failed state, i.e. 
. To satisfy the system goals, the actuators to be used depend on their availability and the structure function φ.
The unavailable actuators are isolated by the Maintenance function, as shown in Figure 5.2, from the Diagnostic function. An available actuator is used by the control law if at least one operating scenario exists, i.e. φ(ek) = 0, using . The probability of using an actuator and satisfying the system objectives is defined by the following conditional probability:
Table 5.1. Failure rates of the actuators
| Variable | Actuator name | Failure rates (10−4) |
| e1 | iEstrella12 | 1.2 |
| e2 | iEstrella3456 | 3.456 |
| e3 | iSJDSub | 6.3 |
| e4 | iSJDSpf | 9.5 |
| e5 | iRelleu | 1 |
| e6 | iCornella100 | 10 |
| e7 | iCornella50 | 5 |
| e8 | iSJD50 | 5 |
| e9 | iSJD10 | 1 |
To integrate the actuators’ reliability in the control strategy, the weighing matrix W(k) is estimated online according to the actuators’ state given by the Diagnostic function. Consequently, if an actuator 
is unavailable, i.e. , the system can work in the degraded mode because it is over-actuated and the scalar wi(k) of each actuator is defined by the following probability:
The weight wi(k) corresponds to the contribution probability of the actuator when the system is working, given the unavailability of some failed actuators. This probability assessment is not only based on the actuator’s state of health but also considers the system structure and the availability of other actuators. By using usual reliability modeling, assessing this probability is complex or impossible, given the structure function φ. However, this assessment can easily be realized by the inference mechanism of DBN.
5.4. Application to a drinking water network
The proposed method has been applied to a drinking water network (DWN) in Barcelona, Spain (Figure 5.3). A DWN is typically an over-actuated system because several paths can provide water to customers from various sources. The flow of water on these paths is controlled by actuators such as valves and pumps. The Barcelona DWN is supervised by a two-level monitoring system. At the top level, a control and monitoring system is installed at the control center which is in charge of the whole network, considering operational constraints and customer requirements.
A DWN is a flow network that connects sources of water to the consumers (sink sectors) through pipes. All the DWN parts can fail, i.e. sources, pipes, tanks, connectors, valves and pumps. Thanks to a DBN, all these elements’ reliability can be modeled, but the present work focuses on pumps and valves. The other components are considered as perfectly reliable and without any saturation.
The method has been applied in simulation with five sources and one sink, as shown in Figure 5.3. To satisfy the requested demand for each day d = yref(k) (C100CFE), at least three sources are required from among the five available sources (AportLL1, AportLL2, ApousE12, ApousE3456, and Aporta).
Figure 5.3. Part of the Barcelona DWN studied
Table 5.2. Paths linking the sources to the demand point
| Variable | Paths |
| P1 | {AportLL2, iSJDSpf, iSJD10, iCornella100} |
| P2 | {AportLL1, iSJDSub, iSJD10, iCornella100} |
| P3 | {AportLL2, iSJDSpf, iSJD10, iCornella50, iRelleu} |
| P4 | {AportLL1, iSJDSub, iSJD10, iCornella50, iRelleu} |
| P5 | {AportLL2, iSJDSpf, iSJD50, iRelleu} |
| P6 | {AportLL1, iSJDSub, iSJD50, iRelleu} |
| P7 | {ApousE12, iEstrella12, iRelleu} |
| P8 | {ApousE3456, iEstrella3456, iRelleu} |
| P9 | {Aporta, iRelleu} |
5.4.1. DBN modeling
All sources are considered to be available. The failure time distributions of all actuators are considered exponential and the failure rates are given in Table 5.1. All paths linking the sources (AportLL1, AportLL2, ApousE12, ApousE3456 and Aporta) to the demand point (C100CFE) are detailed in Table 5.2.
Table 5.3. Variables Computing the actuator Weighting
| Variable | Actuator weighing |
| αe1 | WiEstrella12 |
| αe2 | WiEstrella3456 |
| αe3 | WiSJDSub |
| αe4 | WiSJDSpf |
| αe5 | WiRelleu |
| αe6 | WiCornella100 |
| αe7 | WiCornella50 |
| αe8 | WiSJD50 |
| αe9 | WiSJD10 |
The DBN-based reliability model is shown in Figure 5.4. The Pi variables model the availability of paths linking the sources to the demand. The 
variables define the availability of sources through the DWN, according to the availability of the paths. The system availability is defined by the variable SC100CFE. The variables computing the weights wi(k) are defined in Table 5.3.
5.4.2. Results and discussion
A simulation is done over 3,600 hours by using Matlab to apply the methodology based on control input optimization and DWN reliability computation. The DBN is based on the Bayes Net Toolbox (BNT), as proposed by Murphy in 2002 [MUR 02].
The results for a particular simulation are illustrated in Figure 5.5. The figure shows the actuator reliability and the DWN reliability. The actuator reliability follows a decreasing exponential curve, whereas the DWN reliability is a bit more complex because it is the combination of the actuator reliability according to the system structure function φ.
Figure 5.6 shows the control inputs applied to the actuators and the weights used for the optimization. The curves αi(k) show the weight values 
according to the component reliability, the path availability linking sources to demand and the potential component failures or repairs. Each weight wi(k) is integrated in the trace of the matrix W(k) for the optimization process and the computation of ũ(k).
Figure 5.4. DBN model of the Barcelona DWN
Figure 5.5. Actuators and DWN reliability
Figure 5.6. Simulation of control inputs and weights of the Barcelona DWN
A simulation scenario is divided into three steps:
- – Step 1. No failure of the pumping station is considered but source (Aporta) unavailability is simulated by (k = 0) and (k = 1, 000). In this case, ApousE12 is easily available through path P7, with components e5 (iRelleu) and e1 (iEstrella12) that are the most reliable actuators. Paths P7 and P8 are used to satisfy consumer demand. The control inputs ũ1(k) and ũ2(k) are then used at maximum capacity. However, actuator saturations e1 and e2 do not meet the daily peak demand value. It is thus ũ3(k) and ũ8(k) that are considered to compensate for the saturations of ũ1(k) and ũ2(k). Component e5 is mainly considered because this component is more reliable than e6. The DWN functioning is thus based on links P6, P7 and P8.
- – Step 2. A critical failure occurs in e8 (iSJD50) at (k = 1, 000). A Diagnostic function is supposed to detect and isolate the failures correctly. The DBN computes the actuator weight wi(k) by integrating the unavailability of component e8. In Figure 5.6, the curve of α8(k) goes to 0. The path P6 cannot be used to provide water to the consumer. The control strategy switches to path P4. Actuators e7 and e9 that were previously unused are activated. Between (k = 1000) and (k = 2000), the components used are then e1, e2, e3, e5, e7 and e9. The corresponding control inputs are then different from 0.
- – Step 3. At (k = 2000), a second critical failure occurs and is simulated on pump e5 (iRelleu). The control strategy gives priority to e5 that is not valid and paths P4, P7 and P8 cannot be used. Consequently, the DBN computes the weights α1(k), α2(k), α5(k) and α7(k) as being equal to 0 and the weight α8(k). Only the paths containing e6 (iCornella100), i.e. P1 and P2, cannot be used. The system is then reconfigured to work with components e3, e4, e6 and e9. The DBN computes the weights α3(k), α4(k), α6(k) and α9(k) as equal to 1 because there is no other valid working scenario.
5.5. Conclusion
A modeling method based on DBN has been proposed in this chapter. It is dedicated to reliability modeling of the system and its actuators to integrate the component states into the control allocation of the system being modeled.
Each step of the system has been formalized. The control process combines a load allocation for all the actuators given the actuators’ reliability and their contribution to the system’s working state. It is not only the goal to preserve the components’ health but also to ensure system availability. This original strategy is applied both in normal conditions and in failure scenarios.
5.6. Acknowledgments
I want to warmly thank my scientific colleagues Prof. Vicenc Puig, Prof. Didier Theilliol, Prof. Fatiha Nejjari and Prof. Ramon Sarrate Estruch, from the Universitat Politécnica de Catalunya (UPC) (Spain) for their contribitions to this chapter.