The figure to the right shows a plant controlled by a controller in a standard control loop.
The nominal linear model of the plant is
The plant subject to a fault (indicated by a red arrow in the figure) is modelled in general by
where the subscript indicates that the system is faulty. This approach models multiplicative faults by modified system matrices. Specifically, actuator faults are represented by the new input matrix , sensor faults are represented by the output map , and internal plant faults are represented by the system matrix .
The upper part of the figure shows a supervisory loop consisting of fault detection and isolation (FDI) and reconfiguration which changes the loop by
To this end, the vectors of inputs and outputs contain all available signals, not just those used by the controller in fault-free operation.
Alternative scenarios model faults as an additive external signal influencing the state derivatives and outputs as follows:
The goal of reconfiguration is to keep the reconfigured control loop performance sufficient for preventing plant shutdown. The following goals are distinguished:
Internal stability of the reconfigured closed loop is usually the minimum requirement. The equilibrium recovery goal (also referred to as weak goal) refers to the steady-state output equilibrium which the reconfigured loop reaches after a given constant input. This equilibrium must equal the nominal equilibrium under the same input (as time tends to infinity). This goal ensures steady-state reference tracking after reconfiguration. The output trajectory recovery goal (also referred to as strong goal) is even stricter. It requires that the dynamic response to an input must equal the nominal response at all times. Further restrictions are imposed by the state trajectory recovery goal, which requires that the state trajectory be restored to the nominal case by the reconfiguration under any input.
Usually a combination of goals is pursued in practice, such as the equilibrium recovery goal with stability.
The question whether or not these or similar goals can be reached for specific faults is addressed by reconfigurability analysis.
This paradigm aims at keeping the nominal controller in the loop. To this end, a reconfiguration block is placed between the faulty plant and the nominal controller. Together with the faulty plant, it forms the reconfigured plant. The reconfiguration block has to fulfill the requirement that the behaviour of the reconfigured plant matches the behaviour of the nominal, that is fault-free plant .
In perfect model following, a dynamic compensator is introduced to allow for the exact recovery of the complete loop behaviour under certain conditions.
In eigenstructure assignment, the nominal closed loop eigenvalues and eigenvectors (the eigenstructure) is recovered to the nominal case after a fault.
Fault accommodation is another common approach to achieve fault tolerance. In contrast to control reconfiguration, accommodation is limited to internal controller changes. The sets of signals manipulated and measured by the controller are fixed, which means that the loop cannot be restructured .