PAPER FOR CONTROL 2006

MODELLING AND CONTROL OF A HIGHLY REDUNDANT ACTUATOR

X. Du*, R. Dixon, R.M. Goodall and A.C. Zolotas

Loughborough University, Loughborough, Leicestershire, LE11 3TU, UK Tel. +44 (0) 1509 227015 Email: x.du@lboro.ac.uk

Abstract: The paper focuses on control of a novel type of actuator for high integrity, safety critical applications. The highly redundant actuator comprising a relatively large number of actuation elements is controlled in such a way that faults in individual elements are inherently accommodated. Results from the modelling and control design show that the actuator can continue to operate safely after sub-actuator failures, without the need for reconfiguration. The paper also shows that the degree of performance lost when faults occur can be affected by the control algorithm employed. Copyright © 2006 LOUGHBOROUGH UNIVERSITY Keywords: fault tolerance, high redundancy

1. INTRODUCTION

High levels of availability and reliability are important objectives for the design of most modern engineering systems. This is particularly true in the case of safety critical applications. Hence engineers aim to produce fault tolerant designs (this applies to both the hardware/mechanisms and the control design). The result should be fault tolerant systems, which have the capability of tolerating component malfunctions whilst still maintaining desirable and robust performance and stability properties (Patton, 1997). In control terms this fault tolerance is often, but not exclusively, attained by some degree of re-configuration: i.e., detect the fault, reconfigure the system. Practical examples of fault tolerant systems can be found in aerospace systems, e.g. Airbus fly-by-wire system Briere, et al., 1995), and Boeing 737 trailing edge flap drive system (Ford, 1998). Here, low levels of functional redundancy in sensors and actuators (e.g. triplex and quadruplex) and even in the control computers can be used to provide the system with the capability of fault tolerance, thereby ensuring the safety and stability of the whole system. In the above approaches, redundancy is used for fault detection and isolation (FDI) based on the continuous comparison of the system under analysis with another system characterized by the same behaviour

and working in the same conditions (Isermann, 1984). Once a fault is detected the redundancy is then used to provide the extra control channels for the reconfiguration. A more general FDI with reconfiguration (R) scheme (as proposed by Patton, 1997) is illustrated in Fig. 1. The solid lines represent signal flow, and the dashed lines represent adaptation. The supervision system will reconfigure the actuator and/or sensor sets, and adapt the controller, based on the faults information collected by the FDI unit together with the inputs and outputs of the system, to accommodate the fault effects

Fig. 1. Scheme of fault-tolerant control system with

supervision subsystem (Patton, 1997)

Fig. 2. Several possible configurations of the high

redundancy actuator It could be argued that a disadvantage of such approaches is the possibility of faults occurring in the FDI unit and the supervision system. One might ask the question, “Who/what monitors the monitors?”. This paper proposes an alternative route to fault tolerant actuation. The highly redundant actuator suggested comprises a relatively large number of actuation elements in a matrix like structure and is controlled in such a way that faults in individual actuation elements are inherently accommodated. The FDI unit may still be used for monitoring, but it is no longer strictly necessary and no reconfiguration of the controller or hardware will be needed. Instead the actuation elements work together to complete the system’s objective by using the redundancy which is inherent in the structure. If this method can be proven to work, the main advantage would arise through removing the possibility of faults occurring within the FDI unit and supervision system. Several possible structures are shown above in Fig. 2. At this stage no attempt has been made to find the optimal structure for the actuators. The research has focused instead on modelling and control of a relatively simple (2 by 2) actuator with a view to discovering whether controlling such structures without reconfiguration is viable. Hence, the paper is structured as follows: work on modelling is presented in section 2; the control design using a classical approach is covered in section 3; then in section 4 a selection of simulation results is presented to give an indication of performance in the presence of faults alongside the fault-free case; the paper concludes in section 5, including comments on the future direction of this research.

2. MODELLING THE SYSTEM In this work, an electro-mechanical actuator is chosen as the sub-actuator of the high redundancy actuator because it is used in many industry applications, including manufacturing, process plant, railway vehicles and aerospace systems. A model for each individual electro-mechanical actuator will be introduced. Then, based on the individual actuator model, a two by two structure of the high redundancy actuator will be presented.

Fig. 3. Equivalent electromechanical actuator model 2.1 Modelling of individual actuator The electro-mechanical actuator can be considered as two parts. One is the D.C. motor, which generates the torque, and the other is the mechanical structure (screw), which translates the torque generated by the motor to the load. A schematic representation of the actuator with a load is given in Fig. 3., and the linear model, as implemented in Matlab/Simulink, is shown in Fig. 4. On the mechanical structure, n is the screw pitch,

mθ is the angular rotation of the motor shaft, mx and

actx are the screw and the end-of-actuator position respectively. Although the screw thread will normally be very stiff, it does possess finite stiffness and damping so these are included in the model. mK represents the motor stiffness, and sK and sC are the screw stiffness and damping, respectively. sM and M are the masses of the screw and the load. For the D.C. servo motor, armR is the resistance and

armL is the inductance of the windings. The field flux is assumed constant so that the electromotive force

re , (i.e. the back e.m.f.), is directly proportional to the speed rω and the electrical torque is proportional to the armature current ai . Hence

rer Ke ω= (1)

ate iKt = (2) where eK is the voltage constant depending on the number of armature turns and tK is the torque constant. The generated electromagnetic torque will be balanced by the load torque lt , the inertial torque at , and the damping torque dt . i.e.

dale tttt ++= (3) ( )dtdJt rma ω= (4)

rmd Ct ω= (5) where mJ accounts for motor inertia and mC accounts for the damping coefficient.

Fig. 4. Linear model of the individual electro-mechanical actuator At the output of the motor model, motor rotation is given. It is transformed by a gearing factor which provides a linear motion to the screw. The output of the actuator is a force and this force is generated by compression of the screw as the motor rotates. The actuator (and load) velocity and position are obtained based on Newton’s second law, and they are fed back to the actuator model to calculate the output force. The Simulink model is used in the time domain simulation and allows fault injection. However, to facilitate control design (in Matlab) it is also possible to obtain the linear model in the state-space formulation:

BuAxx +=& Cxy = (6) The state vector is chosen as

[ ]Tactactmmmma xxxxix &&& θθ= (7) the input u and output y are chosen as

avu = , [ ]Tactacta xxiy &= (8)

where av is the applied voltage. The state transition, input and observation matrices for the model are:

,

0100000

0000001000

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⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

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⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−

+−−

−−

−−

=

MK

MC

MK

MC

MK

MC

MKK

MC

MnK

JnK

JKn

JC

JK

LK

LR

A

ssss

s

s

s

s

s

ms

s

s

s

m

m

m

m

m

m

m

m

t

arm

e

arm

arm

(9)

[ ]1100001,0000001 =⎥⎦

⎤⎢⎣

⎡= C

LB

T

arm

(10)

2.2 Modelling of two by two structure Having developed the basic (sub-actuator) model, it is relatively straightforward to build multi-actuator systems by combining the individual sub-actuators. There are two basic connecting structures, one is the series structure and the other is the parallel structure. Taking advantage of both these connecting structures, two separate actuators have been developed and studied from a control perspective. In the initial study both were two by two networks, the first, “parallel in series”, is shown in Fig. 5, and the other, denoted “series in parallel”, is shown in Fig. 6. In the parallel in series structure, the two bottom actuators are connected in parallel, then a second pair of parallel actuators are connected in series (with the first pair). In the series in parallel structure, the first two sub-actuators are connected in series, and a second pair of series sub-actuators is connected in parallel connection to the load. In this present paper, the authors will focus on the series in parallel structure. The Simulink model for which can be seen in Fig. 7.

LoadBottom1

Bottom2

Top1

Top2 Fig. 5. Parallel in series two by two structure

LoadBottom1

Bottom2

Top1

Top2 Fig. 6. Series in parallel two by two structure

Fig. 7. Linear model of series in parallel two by two structure

3. CONTROL DESIGN The aim is to compare two different control algorithms for position control of the 2x2 actuator in the presence of faults. The control design starts with an individual electro-mechanical actuator (this will provide a benchmark). Then, the position controller for the high redundancy actuator will be designed. 3.1 Control of individual actuator Firstly, using frequency response methods, a classical PI feedback controller is designed for the individual actuator based on the linear model. Secondly, a position feedback controller with a current feedback inner loop is designed. The position sensor for both structures will be put on the end of the high redundancy actuator or on the load. The open loop Nichols chart for both uncompensated systems is shown in Fig. 8. The open loop Nichols chart for both compensated systems, i.e. without and with current control are shown in Fig. 9. For the system without current control inner loop, the phase margin (PM) and gain margin (GM) are 89.8deg and 20.5dB respectively. The closed loop bandwidth can be measured from the Nichols chart to be approximately 1Hz. For the system with current control inner loop, the system PM and GM are 62.3deg and 29.5dB, respectively. The bandwidth of the closed loop system is again 1 Hz. Note: for the current-control inner loop, the PM and GM are 73deg and ∞dB respectively and the bandwidth of the closed loop system is about 75 Hz. Simulation results with a 0.03m step input for both control systems are given in Fig. 10. Both the closed loop schemes are stable and have the same closed loop bandwidth. The control structure with only position controller has a faster response and has no overshoot. Note that this is due to a focus on achieving a similar closed-loop bandwidth during the design process.

Fig. 8. The open loop Nichols chart for the

uncompensated position control system

Fig. 9. The open loop Nichols chart for the

compensated position control systems

Fig. 10. Simulation results for the individual actuator

position feedback control

3.2 Control of two by two structure The control design for the series in parallel two-by-two structure is based on the same approach as used in section 3.1. Now, the position feedback control is based on the distance output of the high redundancy actuator, which will be referred to as the global controller. The current feedback control is based on the current output of each sub-actuator, which will be referred to as the local controller(s). For the first design a single global position controller is designed for the high redundancy actuator. The control signal from the controller is connected to the four sub-actuators directly. In the second design, four local current feedback controllers are applied to each sub-actuator, and subsequently a global position feedback controller is designed for the outer loop. The control structure with the current control of the two by two network is shown in Fig. 11. The open loop Nichols chart for the uncompensated system is similar to the individual actuator which was shown in Fig. 8., and for compensated systems is shown in Fig. 12. The simulation results with a 0.03m step input are given in Fig. 13.

LoadTop1C

GlobalController

C

Bottom1

C

Top2

C

Bottom2

C

Local Controller

Local Controller

Local Controller

Local Controller

Fig. 11. Position control structure with current

control of series in parallel two by two structure

Fig. 12. The open loop Nichols chart for the

compensated position control systems

Fig. 13. Simulation results for the control of series in

parallel two by two high redundancy actuator

For the system without current control, the PM and GM are 83.7 deg and 11.4 dB respectively. The closed loop bandwidth is approximately 2 Hz. For the system with current control, the system PM and GM are 63.5 deg and 20.7 dB respectively. The bandwidth of the closed loop system is again 2 Hz.

4. SIMULATION RESULTS UNDER FAULTS The primary aim of the high redundancy actuator concept is its ability to tolerate faults without restructuring. In this section, simulation results will be introduced for the high redundancy actuator under fault conditions. 4.1 Modelling of actuator faults The actuator faults can be divided into two types: motor faults which occur in the D.C. motor as electrical faults, and the mechanical faults which happen in the components where the torque is transferred to the load. The faults considered are summarised below. Open circuit. The effect is that the resistance of the rotor will be very large so that almost no current will flow. Hence no torque will be generated by the motor. Lockup: This fault occurs due to a jam in the gearbox or screw. It is generally a serious fault as it can not be solved (by reconfiguration) at the signal level. 4.2 Simulation results Tables showing performance of both the individual actuator and the high redundancy actuator with the controller under the healthy and faulty situations will be given in this section. The simulation results are for step responses (as shown for both actuators under the healthy situation in the previous section: Fig. 10. and 13.). No restructuring of the controllers is applied in any of the cases. Individual actuator: The individual actuator without redundancy has only a limited ability to tolerate faults. In general, with a closed-loop controller, the robustness of the actuator is increased and some kinds of faults are solved on the signal level. However with serious faults the actuator will lose the ability to complete its function. Table 1 and Table 2 list some key parameters of the individual actuator’s performance in both time and frequency domain without and with current control respectively. Table 1 Performance of individual actuator only with

position control Situations FV(m) RT(s) ST(s) OS Healthy 0.03 0.36 0.46 0%

Open circuit 0 -- -- -- Lockup 0 -- -- --

PM(deg) GM(dB) BW(Hz) Healthy 89.8 20.5 1

Open circuit -- -- -- Lockup -- -- --

KEY: FV is Final value of position, RT is Rise time, ST is Settling time, and OT is overshoot, GM is Gain margin, PM is Phase margin, BW is Band width. The same meaning will be used in the following tables.

Table 2 Performance of individual actuator with position control and current control

Situations FV(m) RT(s) ST(s) OS Healthy 0.03 0.35 1.37 11.3%

Open circuit 0 -- -- -- Lockup 0 -- -- --

PM(deg) GM(dB) BW(Hz) Healthy 62.3 29.5 1

Open circuit -- -- -- Lockup -- -- --

High redundancy actuator. The performance of the high redundancy actuator two-by-two structures will be listed in the tables following. The Bottom 1 sub-actuator is chosen as the faulty sub-actuator. Table 3 and Table 4 will list the performance of the series in parallel structure. Table 3 Performance of two by two series in parallel

structure with global position control Situations FV(m) RT(s) ST(s) OS Healthy 0.03 0.18 0.22 0%

Open circuit 0.03 0.18 0.27 0% Lockup 0.03 0.26 0.35 0%

PM(deg) GM(dB) BW(Hz) Healthy 83.7 11.4 2

Open circuit 82.9 6.27 2 Lockup 86 18.1 1.5

Table 4 Performance of two by two series in parallel

structure with global position control and local current control

Situations FV(m) RT(s) ST(s) OS Healthy 0.03 0.2 0.96 11%

Open circuit 0.03 0.2 1.02 11% Lockup 0.03 0.29 1.22 11%

PM(deg) GM(dB) BW(Hz) Healthy 63.5 20.7 2

Open circuit 85.5 5.82 2 Lockup 72.9 27.1 1.4

Evaluating the tables, a number of conclusions can be reached: Firstly, although the controller can increase the robustness of an individual actuator, some serious faults such as the open circuit and lockup cannot be solved on the signal level and thus cause a failure. When comparing the individual actuator with the high redundancy actuator it is clear that the latter can continue operating after both faults. Though, clearly, the performance is not as good as in the healthy situation because of the relatively low level of redundancy in the two by two configuration.

For the high redundancy actuator, the performance of the two different control structures is different in the first place as the simpler structure with only position controller has a faster response with bigger applied voltage. It is perhaps surprising that the degree of performance lost in the two fault cases is relatively independent of the controller structure. This will be analysed further in future work.

5. CONCLUSION AND FUTURE WORK In this paper, the individual electro-mechanical actuator and the high redundancy actuator have been modelled. Controllers have been designed for both actuators. The simulation results under healthy and faulty situations have been compared. The results show the capability of fault tolerance in the case of high redundancy actuator without the need to reconfigure. Other on-going work is investigating the design of alternative actuator system structures. Future studies will extend the range of faults considered, extend concepts to 3-by-3 structures and consider other controller designs based on modern robust control frameworks, i.e. 2H and ∞H . The work is continuing with funding from the UK’s research council (reference EP/D078350), and a lab scale actuator will also be developed as a demonstration. Through this practical exercise, the authors hope to gain an insight into the practicality and rough cost implications of such a system.

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Terry. Ford (1998). Actuation systems development. Aircraft Engineering and Aerospace Technology, Volume 70, Number 4, pp. 265-270