Modeling of Actuator Faults through Control Effectiveness
Function of Actuators in an Aircraft
Actuators that link control commands to physical actions on an aircraft are essential elements in any flight control system. In an aircraft, hydraulic mechanisms are widely used for actuation purposes due to their high force to inertia ratio. For instance in an F16 fighter, an F18 fighter, a JAS39 Gripen aircraft, and a Boeing 737, the actuators such as ailerons and elevators are all in the form of hydraulic driven control surfaces. The function of a control surface is to produce the required torque and moment to maneuver the aircraft. Different maneuvering commands are realized through deflecting appropriate control surfaces in various parts of the aircraft. When all components function normally, the desired control actions can be carried out exactly as the controller has demanded. However, when some failures occur in the actuators, the desired control commands cannot be completed as expected. The stability and the performance of the aircraft can suffer as a consequence.
Analysis of Faults in Hydraulic Driven Control Surfaces
A hydraulic driven actuator consists of three main parts: a hydraulic power supply, a servovalve, and control surface. The power supply delivers hydraulic fluid to the highpressure port of the servovalve at a constant pressure. As shown in Fig. 2.1, the torque motor converts the corresponding voltage from the controller into the angular displacement on the baffle, which generates a differential pressure in the servovalve. Subsequently, the servovalve regulates the motion of the actuator by directing the fluid flow to and from the actuator chamber. When the valve spool is pushed or pulled by the torque motor that is driven by the control signal, the amount of fluid delivered to the actuator chamber will change. The fluid will then move the piston in the actuator chamber. Since the control surface is physically connected to the piston through the piston rod, the displacement of the piston leads to movement in the control surface. The torque generated by the force through a hinge to deflect the control surface also has to overcome the aerodynamic force during a flight.
The nonlinear relationship that describes the motion, D_{spoo}i of the valve spool in response to the displacement of the piston pod, D_{po}d, as well as the corresponding linearization procedures are elaborated in Appendix A. The physical meanings of the symbols used in the derivations are summarized in Table 2.1. In the current work, modeling uncertainties of the actuator have not been explicitly considered.
FIGURE 2.1: Schematic diagram of a hydraulic driven control surface.
A common failure in a hydraulic powered actuator is loss of pressure in the supply pump due to leaks [70]. Reduction in pressure could cause stalling of the actuator when it can no longer balance the aerodynamic load imposed from the control surface during a flight. In an aircraft whose control surfaces are manipulated through hydraulic actuators, stalling of an actuator may have disastrous consequences, because control actions would not get executed as expected. In this chapter, an incorrect supply pressure is considered as the fault condition to illustrate the hybrid FTCS design concept and process. Other failure scenarios can be treated similarly, but this is beyond the scope of the current chapter.
From Appendix A, the relationship between the control command u_{ct} and the actual deflection of the control surface щ can be represented as
where
Since the gain к у is proportional to the gain k)_{u}, and the reduct ion of supply pressure can lead to the change of к у, к у is used to analyze the control
TABLE 2.1: Nomenclature of hydraulic actuator.
Symbol 
Physical Meaning 
Valve spool displacement 

Rod displacement 

Pressure sensitivity gain of the valve 

Effective discharge coefficient of the leakage orifice 

Piston effective area 

Length of rocker 

Hinge moment coefficient 

Hinge moment (load of control surface) 

Actuator cylinder column 

Elastic modulus of oil 

Flow sensitivity gains of the valve 

Control surface deflection gain 

Pipe coefficient of elasticity 

Piston damping coefficient 

Valve spool position gain 

Time constant of the transfer function of the controller output voltage and the valve spool 

Actual deflection 

Controller command 

Load pressure 

Supply pressure 

Density of the hydraulic fluid 

Valve coefficient of discharge 

Valve orifice area gradient 
effectiveness loss. From Ref. [72], the flow sensitivity gain of the valve is defined as: _
where C_{v}, w, P_{s}, P, and p represent the valve coefficient of discharge, valve orifice area gradient, supply pressure, load pressure at an operating condition, and the density of the hydraulic fluid, respectively. The nominal values of these parameters are provided in Ref. [72] as 0.6, 20.75 mm, 17.2 MPa, 11.5 MPa, and 847kg/m^{3}. Based on Eq. (2.2), the nominal value of k can he determined as 1.02.
When the supply pressure drops from nominal value, it means that a fault has occurred. As observed in Eq. (2.2), reduction in P_{s} induces loss in the steadystate gain of the actuator channel. For example, assuming the supply pressure is reduced from 17.2 MPa to 12 MPa, the postfault value of k will be reduced from 1.02 to 0.3025 accordingly. Since the steadystate gain ki_{la}k_{v }of the actuator is proportional to ki, this particular fault can be modeled as 29.66% loss in the effectiveness of the actuator.
It should be mentioned that when the actuator and aircraft are considered together as an integrated unit, the time constant of the actuator is much smaller than that of the aircraft [73]. Hence, the dynamics of the actuator can be ignored in the control system design process without causing significant error. As a result, only the gain of the transfer function of the actuator ki_{la}k_{v }is used to characterize the effectiveness of the control action. However, if one wants to examine the inner workings of the hydraulic actuators or to study the actuator under different failure modes, the full dynamic model of the actuator should be used.
Since the control signal is щ (t) = kh_{a}kvU_{cl} (t) under no fault condition, the relationship in the event of an actuator fault can be described as ^{u}i (0 = ^{k}L^{k} vu_{C}i (t). Thus, the ratio between u, (t) and u{ (t) is essentially the effectiveness of one actuator. Define a variable l, as
where uj (t) = li ■ щ (t) and 0 < l, < is known as the actuator effectiveness. For a healthy actuator, l, = 1. If the actuator suffers a complete failure (outage), then li = 0. A partial loss in the actuator effectiveness can be represented by ()
For simplicity, only a single control surface is considered in the above analysis. In practice, there are several control surfaces in an aircraft. These independently controlled surfaces form essential redundancies needed for fault tolerance, and together they generate the required moments to control the motion of the aircraft.
Modeling of Faults in Multiple Actuators
By extending the analysis of a single actuator, fault models for multiple independent actuators can be derived. In case of m actuators, the effectiveness factor li can be denoted as a diagonal element in the diagonal matrix L = diag{li,... ,l_{m}} for i^{th} actuator. When potential failures in all m actuators are considered, the matrix L can be used to describe the effectiveness in any of them. In fact, such modeling techniques have been used in the literature [9, 20, 21]. With the diagonal matrix L = diag{li,... ,l_{m}}, the effectiveness of m independent actuators can be described as:
where the diagonal element 0 < < 1, (i = 1,... ,m) represents the effectiveness for i^{th} control channel, и = [м i, 112,..., «_{т}]^{Г} is the desired control input
~lT
vector, and «/ = ,..., uf_{n} denotes the actual control actions exerted
on the aircraft.
Consider the model of an aircraft that has been linearized at a desired operating condition described by:
where x (t) € SR" is the state vector, u> (t) € 5R^{r} models a bounded external disturbance, и (t) 6 SR"^{1} and у (t) € SR^{P} denote the control surface deflection and the measurement output vector, respectively. A, В. G, and C are known matrices with appropriate dimensions.
The system with actuator faults modeled in terms of the control effectiveness matrix can then be written as:
where the postfault control input matrix can be represented by Bf = BL.
To ensure acceptable control performance in the presence of actuator faults, an FTCS has to counteract the effects of the faults by utilizing the existing physical redundancies in the system. The following definition of actuator redundancy is restated.
Definition 2.1. For a dynamic system described as Eq. (2.5), it is said to have {m — p) degrees of actuator redundancy if the pair (A,bi) is completely controllable Vi (1 < i < m), where bi is the i^{th} column of control input matrix B, and the number of independent control inputs is more than the number of system outputs being controlled.