Control reconfiguration

Control reconfiguration is an active approach to achieve fault-tolerant control for dynamic systems (Blanke et al. 2006). It is used when severe faults, such as actuator or sensor outages, cause a break-up of the control loop, which must be restructured to prevent failure at the system level. In addition to loop restructuring, the controller parameters must be adjusted to accommodate changed plant dynamics. Control reconfiguration is a building block towards increasing the dependability of systems under feedback control (Patton 1997).

Reconfiguration Problem

Control loop with supervisory level. In the nominal, i. e. fault-free situation, the lower control loop operates to meet the control goals. The fault detection (FDI) module monitors the loop to detect and isolate faults. The fault estimate is passed to the reconfiguration block, which modifies the control loop to reach the control goals in spite of the fault.

Fault Modelling

The figure to the right shows a plant controlled by a controller in a standard control loop. The plant is subject to a fault indicated by a red arrow and modelled by

${\begin{cases}{\dot {\mathbf {x} }}_{f}&=\mathbf {A} _{f}\mathbf {x} _{f}+\mathbf {B} _{f}\mathbf {u} ,\\\mathbf {y} _{f}&=\mathbf {C} _{f}\mathbf {x} _{f},\end{cases}}$

where the index $f$ indicates that the system is faulty. This approach models multiplicative faults by modified system matrices. Not all matrices need to change after every fault. In particular, actuator faults are represented by the input matrix $\mathbf {B} _{f}$ , sensor faults are represented by the output map $\mathbf {C} _{f}$ , and internal plant faults are represented by the system matrix $\mathbf {A} _{f}$ . Alternative scenarios model faults as an additive external signal $\mathbf {f}$ ,

${\begin{cases}{\dot {\mathbf {x} }}_{f}&=\mathbf {A} \mathbf {x} _{f}+\mathbf {B} \mathbf {u} +\mathbf {E} \mathbf {f} ,\\\mathbf {y} _{f}&=\mathbf {C} _{f}\mathbf {x} _{f}+\mathbf {F} \mathbf {f} .\end{cases}}$ .

The upper part of the figure shows a supervisory loop consisting of fault detection and isolation (FDI) and a reconfiguration which changes the loop by

choosing new input and output signals from the available inputs $\mathbf {u} ,\mathbf {y}$ to reach the control goal,
changing the controller internals (including dynamic structure and parameters),
adjusting the reference input $\mathbf {w}$ .

To this end, the vectors of inputs and outputs contain all available signals, not just those used by the controller in fault-free operation.

Reconfiguration Goals

The goal of reconfiguration is to keep the reconfigured control loop performance sufficient for preventing plant shutdown. The following goals are distinguished:

Stabilisation goal,
Equilibrium recovery goal,
Output trajectory recovery goal,
State trajectory recovery goal.

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 equlibrium 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 reference tracking for step-shape reference signals 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.

Reconfiguration Approaches

Fault hiding

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 (Steffen 2005).

Linear model following

In linear model following, a formal feature of the nominal closed loop is attempted to be recovered. In the classical pseudo-inverse method, the closed loop system matrix ${\bar {\mathbf {A} }}=\mathbf {A} -\mathbf {B} \mathbf {K}$ of a state-feedback control structure is used. The new controller $\mathbf {K} _{f}$ is found to approxmate ${\bar {\mathbf {A} }}$ in the sense of an induced matrix norm (Gao & Antsaklis 1991). (Staroswiecki 2005)

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.

Optimisation

Linear-quadratic regulator design (LQR), model predictive control (MPC) (Looze et al. 1985),(Lunze, Rowe-Serrano & Steffen 2003),(Maciejowski & Jones 2003)

Stochastic analysis

(Mahmoud, Zhang & Jiang 2003)

Learning control

Learning automata, neural networks etc. (Rauch 1994).

Mathematical Tools

The methods by which reconfiguration is achieved differ considerably. The following list gives an overview of mathematical approaches that are commonly used (Zhang & Jiang 2003).

Adaptive control (AC)
Disturbance decoupling (DD)
Eigenstructure assignment (EA)
Gain scheduling (GS)/linear parameter varying (LPV)
Generalised internal model control (GIMC)
Intelligent control (IC)
Linear matrix inequality (LMI)
Linear-quadratic regulator (LQR)
Model following (MF)
Model predictive control (MPC)
Pseudo-inverse method (PIM)
Robust control techniques

References