- Email: [email protected]
Active FTC for Non-linear Aircraft based on Feedback Linearization and Robust Estimation
8th IFAC Symposium on Fault Detection, Supervision and Safety of Technical Processes (SAFEPROCESS) August 29-31, 2012. Mexico City, Mexico
Active FTC for Non-linear Aircraft based on Feedback Linearization and Robust Estimation Yimeng Tang Ron J Patton
Department of Engineering, University of Hull, Hull HU6 7RX, (e-mail: yimeng.tang@ 2010.hull.ac.uk, [email protected]). Abstract: The current work concerns the development of a novel approach to robust fault estimation to achieve active Fault Tolerant Control (FTC) for a nonlinear aircraft system. The novelty lies in the combination of a feedback linearization controller, with a state/fault estimator. The active FTC system combines these to compensate for the faults acting in the control system. The paper focuses on the development of an active FTC scheme applied to a nonlinear unmanned airborne vehicle (UAV) system with different faults acting on three actuators and with wind turbulence affecting the vertical force. The aircraft dynamics are approximately linearized on-line using a dynamic inversion controller based on differential geometry theory. For the linearized system with bounded input disturbance, a robust statespace observer is designed to simultaneously estimate system states and actuator faults by solving a Lyapunov equation. The FTC scheme is achieved via a linear matrix inequality (LMI) approach, based on the estimated states/faults. The simulation results demonstrate the efficiency and robustness of the proposed design.
Aircraft systems have non–linear dynamics and the joint model-based fault estimation and FTC problem is thus dependent on robustness strategies.
1. INTRODUCTION One of the most important problems in flight control is the achievement of high reliability through the use of redundancy methods of potential application to flight control systems (Patton, 1997). The model-based approach to fault detection and diagnosis (FDD) for dynamic systems (based on analytical redundancy) has long been emphasized for achieving robust and prompt detection of faults in the presence of modelling uncertainty (Chen and Patton, 1999); (Patton, Frank and Clarke, 2000); (Simani, Fantuzzi and Patton, 2003); (Blanke, Kinnaert, Lunze and Staroswiecki, 2003); (Chen and Speyer, 2004); (Ding, 2008) and (Edwards, Lombaerts, and Smaili, 2010). Some approaches are based on fault estimation as an alternative to the use residual-based FDD methods (Edwards, Spurgeon and Patton, 2000). Zhang and Li (2002) consider the fault estimation problem for aircraft actuators and sensors. The advantage of robust fault estimation is that the estimated fault can be used in a fault compensation/hiding approach to active FTC (Bin, Staroswiecki and Cocquempot, 2006); (Patton, Putra and Klinkhieo, 2010) and (Richter, Heemels, Wouw and Lunze, 2011). Here we consider the fault accommodation approach for FTC in flight control based on fault estimation. Estimation schemes can be used with fault accommodation control when severe subsystem failures and when structural damage occurs (Bing and Chowdhury, 2005); (Richter and Lunze, 2010). Considering flight control applications the response speed and accuracy in fault compensation/accommodation, i.e. a form of active FTC, is of paramount importance for increasing the aircraft survivability in the presence of severe subsystem failures and structural damage (Boskovic and Mehra, 1999); (Kim, Lee and Kim, 2003); (Maki, Jiang, Hagino, 2004); (Goupil, 2009). 978-3-902823-09-0/12/$20.00 © 2012 IFAC
210
Advanced high-performance aircraft, not only have the characteristics of high nonlinearity and are multi-input and multi-output from a control standpoint, but also require high manoeuvrability with static instability (Donald, 1990). For the purpose of efficiency and simplification, the feedback linearization technique is well-proven and has been developed to be one of the feasible control strategies in the study of nonlinear system, especially for aircraft (Lane et al, 1988); (Ochi and Kanai, 1991). Feedback linearization can remove nonlinear features from the system and provide a linearized and decoupled closed-loop form. In addition to these features, dynamic inversion control has advantages such as insensitivity to parameter changes and disturbances, and simplicity in physical realization (Snell, Enns and Garrard, 1992); (Wu and Zou, 2009). By using results from feedback linearization combined with the dynamic inversion strategy, an inner-loop controller design may ensure the stability of the nonlinear aircraft system. This strategy not only takes on the advantage of nonlinear transformation to simplify the control system design, but also brings into full play time-varying inversion modes against system perturbation and disturbance (Hovakimyan, Cao and Lavretsky, 2006). As another aspect to this work, the literature of the development of fault estimation within active FTC systems is well summarized in the work by Gao and Ho (2006). By using a linear matrix inequality (LMI) approach, a descriptor estimator was presented to simultaneously estimate system states, output noise and sensor faults for a class of nonlinear systems. Gao and Ding (2007) proposed a novel actuator 10.3182/20120829-3-MX-2028.00092
SAFEPROCESS 2012 August 29-31, 2012. Mexico City, Mexico
fault estimation approach for nonlinear descriptor systems. They described a robust FTC strategy incorporating both state and fault estimation and fault compensation. Bounds on the fault magnitudes are determined for guaranteeing closedloop system performance and robustness using the estimated states and faults together with a LMI strategy.
is invertible, then the system can be linearized by decoupling the non-linear terms in (3) by choosing as follows:
The contribution of this paper lies in the application of the combined dynamic inversion controller with a robust estimator and active fault compensation FTC approach to a nonlinear aircraft system. The aircraft example includes full force and moment longitudinal and lateral dynamics, together with wind turbulence and actuator faults. The proposed estimator and FTC scheme is efficient and reliable for realtime application.
,
( ),
/
(
( )
( )
)
(
[
[
)
(
( )
⋮
)-
( ) ⋯ ⋱ ⋯
⋮
]
-
(7)
( )
(8)
-
( )
(9)
(10)
3. ROBUST ESTIMATOR AND FTC SYSTEM After feedback linearization by the dynamic inversion controller, the system model can be simplified to a classical state-space system as: ̇
}
(11)
where refers to the input vector for the linearized system derived from the outer-loop FTC controller. A computational error is inevitable in solving the dynamic inversion matrix, thus the linearized system also needs online compensation for achieving high-accuracy. By including a model description of the action of both actuator faults and random disturbances, the system described in (11) can be broadened into a descriptor system (Gao and Ding, 2007) as: ̇
(2)
( )
,
Once linearization has been achieved, any further control objectives may be easily met.
with at least one of the , and is the row of . The input-output relation can then be defined as: ,
)-
̇
(1)
.
(
As a result, the output of the closed-loop system is given by the solution to the following linear system
Define to be the smallest integer such that at least one of the inputs appears in ( ) using Lie derivatives as: ∑
)
( ),
( ) is an n-D (dimensional) states vector where of the system, is the input vectors comes from the inner feedback linearization controller and is innerloop system output vectors, is a sufficiently smooth ( ) ( )- , ( vector field. ( ) , ( ) ) is an m-D sufficiently smooth vector field ， ( ) ( )- , ( ( ) , ( ) ) is a sufficiently smooth scalar function.
( )
(
which means ̇ is determined by (4) and (5) in the case that ( ) ( ) ( ) ( ). If ( ) is invertible, the control input can be obtained as:
For the multi-input, multi-output affine nonlinear system as: }
)
( ) ̇
The concept of feedback linearization makes use of the principle of transforming a smooth non-linear dynamical system into linear input-output form via feedback control (Isidori, 1985); (Descuss and Moog, 1985); (Sastry, 1989). ( ) ( )
(
By assuming that all the states of the system are measured, which is a special case of (1) as and it then follows that:
2. NONLINEAR FEEDBACK LINEARIZATION
( )
(6)
So, if the feedback signal is substituted into (3) the result is a closed-loop decoupled linear system. Hence, (3) becomes:
Section 2 introduces the theoretical foundation of the dynamic inversion controller design. The robust estimation procedure incorporated within the FTC system design is illustrated in Section 3. The mathematical model and simulation results for a nonlinear UAV, the Machan, are given in Section 4 to illustrate the proposed approach. The concluding discussion is given in Section 5.
̇
-
( )
( )
(3)
( )]
(4)
(5)
The new system function ( ) and input ( ) are introduced in (3) to achieve linear dependence between outputs and inputs are desirable. If the matrix ( ) 211
(
)
}
(12)
where is the unknown bounded process disturbance vector; is the unknown actuator fault vector; may be singular; , are constant real matrices of ( ) appropriate dimensions; is the real nonlinear vector function for compensating the inversion error. The actuator fault vector has unknown fault components, but the derivative ( ) , is assumed bounded. From the work of Gao and Ding (2007), the first step is to develop a robust state-space observer to estimate both the system states and the faults signal simultaneously by using the known input and the measurement output . The second step is to develop an efficient FTC scheme by using the estimated states and faults.
SAFEPROCESS 2012 August 29-31, 2012. Mexico City, Mexico
The complete design strategy is achieved by combining the above estimator with the dynamic inversion linearization procedure as illustrated in Fig. 1.
̅
( )
[
( )
̅
( ) ]
(18)
and is a positive scalar to be designed. ̅ [ ] includes the estimated states ̂ , the ) and the fault fault derivative estimates ̂ ( ̂ , which enable the observer (14) to be a estimates ̂ simultaneous state and fault estimator. Then there exists a robust observer in the form of (14) for the plant (12) to make the steady estimator error dynamics as small as any desired accuracy, if the derivative gain ̅ is well selected such that ̅ ̅ ̅ ̅ is non-singular. The proportional gain ̅ is computed as: ̅̅
̅
(19)
where ̅ can be solved from the Lyapunov equation: ̅
(
3.1 Robust State And Fault Estimator
(
)
(
)
(13)
and by using (12), an augmented descriptor process can be constructed as follows (Gao and Ding, 2007): ̅ ̅
̅ ̇̅
̅
̅
̅
̅(
( )
) }
̅ ̅
̅(
[
(14)
,̅ (
)
̅
(
̅
, ̅
, ̅
,
̅
̅
-
⋮
⋱ ⋯
(̅
)
(22)
}
(23) Then it is clear that:
⋯ [
(21)
̅
-
̅
̅+; the
where it is assumed that the observer gains ̅ , ̅ and the scalar parameter are designed according to the procedure given in Section 3.1. Let ̅ ( ) , with and ( ). By choosing:
̅
-
* as:
(20)
‖
) ̅ ̅ ] ̂̅ ̅ ̅ ̅( ̂ ) ̅ ̅) ( ̅ ̅ ) ̅ ̂̅
(
̅ ̅
)
,
̅
[ ̅ ( ̂̅
̅
-
)
̅
Consider system (12) and its augmented system (14). An observer-based controller can be constructed as (Gao and Ding, 2007):
̅
]
̅
3.2 Fault-Tolerant Control System
̇̅ , ̅
̅)
With a reasonably large , the designed observer can reduce the effects of the disturbance and the fault model error ( ) , as desired. The proof is given in Gao and Ding (2007).
where ̅
̅
‖̅
To simplify the notation in the determination of the robust estimation scheme let:
̅
̅(
̅)satisfying , ( ̅ is chosen with a positive scalar
with scalar
Fig. 1 Fault Estimation and FTC system
̅) ̅
̅ ̅
⋮
(24)
(15) Furthermore, choose:
]
(25)
Consider a state-space dynamic system as follows: ̇̅
̅ ̂̅
̅ ̂̅
( (̅
)̅ ( ̅ ̅) ( ̅
̅ ̂̅ ) ̅
̅ ( ̂̅ )
) } (16)
where ̂̅ ̅ is the estimate of the augmented state ̅ ̅ ; proportional gains ̅ and ̅ are to be designed the following forms: ̅
,
-
̅
The effect of the fault f on the plant can thus be removed. If there exist a positive definite matrix ( ) matrices , such that: (
)
(
, and
) (26)
(17)
212
[
]
SAFEPROCESS 2012 August 29-31, 2012. Mexico City, Mexico
(
where
̇
)
,
,
, ( ) is a matrix such that and rank ( ) ( ) . Furthermore, if a feasible solution ( )exists in the LMI (26), the statefeedback gain can be computed as: (
)
(27)
In this case, the normal dynamical output feedback controller (22) can perform a fault-tolerant operation by ensuring the closed-loop plant to be internally proper stable and attenuate the bounded input disturbance with prescribed performance index (Gao and Ding, 2005). The UAV (or remotely piloted vehicle) system, the Machan, was used as a development vehicle by Marconi Avionics, UK for research on high incidence flight and non-linear control laws, with RAE Farnborough UK and NASA Dryden in the 1980’s. The Machan Euler equations relating the forces , , and moments , , in the aircraft body axes to the angular and linear velocities in the inertial axes are given as: )
( ̇
)
( ̇
)
̇ ̇ ̇
(
)
(
)
(
)
(30)
where, , , , and represent the maximum engine power, the throttle demand, the propeller efficiency, the engine rises rate and the air flow rate respectively (Aslin, 1985). The open-loop Machan UAV is unstable, thus to achieve a stable fault estimator and FTC system, a closedloop “base-line” control system must be configured before the FTC system can be developed.
The nonlinear model system state and input vectors are chosen as: ,
,
,
-
-
(31)
where , , are the control surface elevator, aileron and rudder, respectively. (28)
where, is the mass of the aircraft. , , are the moments of inertia about the axes through the centre of gravity parallel to the aircraft body axes. , and are the forward, side and vertical velocity of the aircraft respectively. , and are the roll, pitch and yaw rates, respectively. The aerodynamic force and moment equations are: ( (
)
To achieve an inner-loop controller using the dynamic inversion control strategy, the system variables should be chosen carefully to form a square and invertible matrix. The nonlinear Machan model system has 14 state variables, which can be grouped according to the various response speeds. For instance, the fast variables: the angular rates , , with bandwidth between 5-10rad/s and the slow variables: the angles , , have a bandwidth between 1-2rad/s, etc (McLean, 1990). To simplify the system and just for illustration the methodology, only the fast variables are involved in this paper. The slow variables could also be controlled by the stable fast variables in another closed-loop.
4. NONLINEAR MACHAN AIRCRAFT STUDY
( ̇
(
To express the aircraft model in affine nonlinear form in (1), combine (28) to (30), the simplified (reduced order) nonlinear Machan equations can be expressed in terms of: ( )
, ( )
( )
( )-
(32)
where:
)
( )
(
)
( )
(
)
( )
(
)
(33)
and the parameters , , calculated from the aerodynamic coefficients have no relation to the inputs .
)
The control distribution matrix of (1) is determined as: (
) ̅
(
) (29)
where (degree) is the incidence, and (degree) are pitch angle and roll angle respectively, (N) is the side force, (%) is the position of the aircraft centre of gravity, (N) is the thrust force due to the engine, (N/m) is the tail moment, (N) is the force acting on the airframe, , and (N) represent the wing lift, total tail lift and tail lift due to the pitch rate respectively, , and (N/m) are the pitching moment, yawing moment and rolling moment component respectively, ̅ (m) is the mean aerodynamic chord and (N/m) is the rolling moment due to the engine.
( )
[
]
(34)
The parameters , , are calculated from the aerodynamic coefficients related to control input . Some ( ) exists of the model parameters are non-zero, so that and hence the dynamic inversion controller can be achieved. ( ), where Commonly, can be chosen as is the desired output refers to the outer-loop control law , is the bandwidth. In this study, the bandwidth can be chosen as rad/s (McLean, 1990). The desired fast variable states can be defined as:
The first order non-linear engine dynamic is given as: 213
SAFEPROCESS 2012 August 29-31, 2012. Mexico City, Mexico
̇ ̇
(
)
Therefore, the FTC gain matrix can be obtained as:
(
)
̅
( ̇
)
(35)
where the suffices and represent the command signals and the desired values, respectively. Thus the inner-loop control law is derived from the dynamic inversion strategy above as: ̇ ( ) {[ ̇ ] ̇
( ) [ ( )]} ( )
(36)
[
]
Fault test signals acting on the elevator, aileron and rudder are depicted as a multi-step up-step down signal (rad), which are useful for exciting non-linearity and testing for fault detectability properties. The input command signals are - , - , and the disturbance chosen as , model acting on the vertical force is chosen according to a Dryden wind turbulence spectrum approximated by the output of a colouring filter with zero-mean Gaussian distributed random noise with variance 80. With 3 faults acting on the actuators and wind disturbance affecting the vertical force, the estimation results corresponding to these faults are shown in Fig.3. The errors between estimated states and real states are shown in Fig.4.
Aeliron Fault Estimation
Elevator Fault Estimation
For this Machan study the engine thrust is not chosen to be a control variable, and is thus a constant value. The resulting inner-loop dynamic inversion controller, which represented in Fig.1 as “linearized system” block is shown in Fig.2.
Fig.2 Inner Loop Dynamic Inversion Controller
Rudder Fault Estimation
To achieve the robust observer (12), construct an augmented model in the form of (14) with derivative order . Choose ̅ , , . From (21) one can compute . By solving the Lyapunov function (20) and using (19), the observer gain can be determined as:
1
Elevator Fault Estimated Fault
0.5 0 -0.5 -1
0
10
20
30
40
50
1 Aeliron Fault Estimated Fault
0.5 0 -0.5 -1
0
10
20
30
40
0.1
50
Rudder Fault Estimated Fault
0.05 0 -0.05 -0.1
0
10
20
30
40
50
Time Sec
Fig.3 Faults and their Estimation ̅
-4
[
error of p
x 10
]
2 0 -2 0
error of q
1
Choose
-1
15
20
25
30
35
40
45
50
0
5
10
15
20
25
30
35
40
45
50
5
10
15
20
25 30 Time Sec
35
40
45
50
-6
error of r
2
] as in (25),
10
x 10
0
x 10
0 -2
[
5 -4
This Gao and Ding (2007) estimator has a high gain, which may lead to an unrealizable range of gain magnitudes. However, the condition number of the gain matrix ̅ is calculated as (̅ ) , which is acceptable from a practical standpoint and is suitable for eliminating the effect of nonlinearity. To achieve the robust FTC system, the choise of , and the solution of (26) & (27) leads to the following:
0
Fig.4 Error between Estimated States and Real States
is then defined as (23). 214
SAFEPROCESS 2012 August 29-31, 2012. Mexico City, Mexico
Descusse, J., and Moog, C.H. (1985). Decoupling with dynamic compensation for strongly invertible affine nonlinear systems. Int. J. Contr., 42, 1387-1398. Donald, M. (1990). Automatic Flight Control Systems. Prentice Hall. Edwards, C., Spurgeon, S. K. and Patton, R.J. (2000). Sliding mode observers for fault detection. Autom., 36, 541–553. Edwards, C. Lombaerts, T. & Smaili, H. (2010). Fault Tolerant Flight Control. Springer. Gao, Z. and Ho, D.(2006).State/noise estimator for descriptor systems with application to sensor fault diagnosis. IEEE Trans. on Signal Process., 54, 1316–1326. Gao, Z. and Steve X. D. (2007). Actuator fault robust estimation and fault-tolerant control for a class of nonlinear descriptor systems. Autom., 43, 912-920 Goupil, P. (2009). Oscillatory failure case detection in the A380 electrical flight control system by analytical redundancy. Contr. Engin. Pract., 18, 1110-1119. Hovakimyan, N., Lavretsky, E. and Cao C. (2006). Dynamic inversion of multi-input non-affine systems via time-scale separation, American Control Conference, 14, 6. Isidori, A. (1985). Nonlinear control systems: An introduction, Springer-Verlag, Berlin and New York. Kim, K. S., Lee, K. J. and Kim, Y. D. (2003). Reconfigurable flight control system design using direct adaptive method. J. Guid., Contr., Dyn., 26, 543. Lane, S. H. and Stengel, R. F. (1988). Flight control design using non-linear inverse dynamics. Autom., 24, 471-483. Maki, M., Jiang, J. and Hagino, K. (2004). A stability guaranteed active fault-tolerant control system against actuator failures. Int. J. Robust. & Nonli. Contr., 14, 10611077. McLean, D. (1990). Automatic flight control systems. Prentice Hall. Ochi, Y. and Kanai, K. (1991). Design of restructurable flight control systems using feedback linearization. J. of Guid., Contr. & Dyn., 14, 903-911. Patton, R. (1997). Fault-tolerant control systems: The 1997 situation. IFAC Symp. Safeprocess’97, Hull, UK, 1029-1052. Patton, R. J., Putra, D. & Klinkhieo, S. (2010). Friction compensation as a fault tolerant control problem, Int. J. Sys. Sci., 41, 987-1001. Richter, J. H. and Lunze, J. (2010). Reconfigurable control of Hammerstein systems after actuator failures: stability, tracking, and performance. Int. J. Contr., 83, 1612. Richter, J. H., Heemels, W. P. M. H., Wouw, N. and Lunze, J. (2011). Reconfigurable control of piecewise affine systems with actuator and sensor faults: Stability and tracking. Autom., 47, 678-691. Sastry, S. (1989). Adaptive Control of Linearizable Systems, IEEE Transaction on Automatic Control. 34, 1123-1131. Simani, S. and Patton, R. J. (2003). Fault diagnosis of non–linear dynamic processes using identified hybrid models. IEEE Conference on Decision and Control, 1, 445–450. Snell, S., Enns, D. and Garrard, W. (1992). Nonlinear inversion flight control for a super-manoeuvrable aircraft. J. of Guid. Contr. & Dyn, 15, 976-984. Zhang, Y. and Li, X. R. (2002). Detection and diagnosis of sensor and actuator failures using IMM estimator. IEEE Trans. Aero. & Electr. Syst., 34, 1293-1313. Wu, Y. and Zou, Q. (2009). Robust Inversion-Based 2-DOF Control Design for Output Tracking: Piezoelectric-Actuator Example. IEEE Trans.Contr.Syst.Technol., 17,1069– 1082.
p(rad/sec)
5 0 -5
0
5
10
15
20
25
30
35
40
45
50
0
5
10
15
20
25
30
35
40
45
50
0
5
10
15
20
25 30 Time Sec
35
40
45
50
q(rad/sec)
1 0 -1
r(rad/sec)
0.5 0 -0.5 -1
Fig.5 FTC System Responses with Faults and Disturbance Fig.5 shows the output responses resulting from the FTC system with disturbance on vertical force. 5. CONCLUSIONS The principle of dynamic inversion control design for nonlinear systems has been outlined through the use of the feedback linearization. The robust simultaneous fault and state estimator is designed for the on-line linearized model by solving a Lyapunov equation. Based on the estimated vectors, the robust FTC loop is accomplished by using LMI technique to ensure that the flight system satisfies robust performance. Both the estimator and the FTC are expressed in state-space form and are based on the original system matrices making them preferable for real control application. The full force and moment nonlinear UAV Machan has been used as an example of a system with highly non-linear dynamics and which is difficult to control. The simulation results demonstrate the robustness of the FTC with fault estimation, corresponding to disturbances and faults. REFERENCES Aslin, P. (1985). Aircraft simulation and robust flight control system design. DPhil thesis, Department of Electronics, University of York, UK. Bin, J. and Chowdhury, F. N. (2005). Fault estimation and accommodation for linear MIMO discrete-time system. IEEE Trans. Contr. Syst. Technol., 13, 493-499. Bin, J., Staroswiecki, M. and Cocquempot, V. (2006). Fault Accommodation for Nonlinear Dynamic Systems. IEEE Trans. Automatic Control, 51, 1578-1583. Patton, R. J., Frank, P. M. and Clarke, R. (2000). Issues of fault diagnosis for dynamic systems. Springer, Berlin. Blanke, M., Kinnaert, M., Lunze, J. and Staroswiecki, M. (2003). Diagnosis and fault-tolerant control. Springer. Boskovic, J. D. and Mehra, R. K. (1999). Stable multiple model adaptive flight control for accommodation of a large class of control effector failures. Proc. ACC ’99, 1920-1924. Chen, J. and Patton, R. J. (1999). Robust model-based fault diagnosis for dynamic systems. Kluwer Academic. Chen, R. and Speyer, J. (2004). Sensor and actuator fault reconstruction. J. of Guid. Contr. & Dyn., 27, 186–196. Ding, X. (2008). Model-based fault diagnosis techniques: design schemes, algorithms, and tools. Springer, Berlin.
215
Active FTC for Non-linear Aircraft based on Feedback Linearization and Robust Estimation Yimeng Tang Ron J Patton
Department of Engineering, University of Hull, Hull HU6 7RX, (e-mail: yimeng.tang@ 2010.hull.ac.uk, [email protected]). Abstract: The current work concerns the development of a novel approach to robust fault estimation to achieve active Fault Tolerant Control (FTC) for a nonlinear aircraft system. The novelty lies in the combination of a feedback linearization controller, with a state/fault estimator. The active FTC system combines these to compensate for the faults acting in the control system. The paper focuses on the development of an active FTC scheme applied to a nonlinear unmanned airborne vehicle (UAV) system with different faults acting on three actuators and with wind turbulence affecting the vertical force. The aircraft dynamics are approximately linearized on-line using a dynamic inversion controller based on differential geometry theory. For the linearized system with bounded input disturbance, a robust statespace observer is designed to simultaneously estimate system states and actuator faults by solving a Lyapunov equation. The FTC scheme is achieved via a linear matrix inequality (LMI) approach, based on the estimated states/faults. The simulation results demonstrate the efficiency and robustness of the proposed design.
Aircraft systems have non–linear dynamics and the joint model-based fault estimation and FTC problem is thus dependent on robustness strategies.
1. INTRODUCTION One of the most important problems in flight control is the achievement of high reliability through the use of redundancy methods of potential application to flight control systems (Patton, 1997). The model-based approach to fault detection and diagnosis (FDD) for dynamic systems (based on analytical redundancy) has long been emphasized for achieving robust and prompt detection of faults in the presence of modelling uncertainty (Chen and Patton, 1999); (Patton, Frank and Clarke, 2000); (Simani, Fantuzzi and Patton, 2003); (Blanke, Kinnaert, Lunze and Staroswiecki, 2003); (Chen and Speyer, 2004); (Ding, 2008) and (Edwards, Lombaerts, and Smaili, 2010). Some approaches are based on fault estimation as an alternative to the use residual-based FDD methods (Edwards, Spurgeon and Patton, 2000). Zhang and Li (2002) consider the fault estimation problem for aircraft actuators and sensors. The advantage of robust fault estimation is that the estimated fault can be used in a fault compensation/hiding approach to active FTC (Bin, Staroswiecki and Cocquempot, 2006); (Patton, Putra and Klinkhieo, 2010) and (Richter, Heemels, Wouw and Lunze, 2011). Here we consider the fault accommodation approach for FTC in flight control based on fault estimation. Estimation schemes can be used with fault accommodation control when severe subsystem failures and when structural damage occurs (Bing and Chowdhury, 2005); (Richter and Lunze, 2010). Considering flight control applications the response speed and accuracy in fault compensation/accommodation, i.e. a form of active FTC, is of paramount importance for increasing the aircraft survivability in the presence of severe subsystem failures and structural damage (Boskovic and Mehra, 1999); (Kim, Lee and Kim, 2003); (Maki, Jiang, Hagino, 2004); (Goupil, 2009). 978-3-902823-09-0/12/$20.00 © 2012 IFAC
210
Advanced high-performance aircraft, not only have the characteristics of high nonlinearity and are multi-input and multi-output from a control standpoint, but also require high manoeuvrability with static instability (Donald, 1990). For the purpose of efficiency and simplification, the feedback linearization technique is well-proven and has been developed to be one of the feasible control strategies in the study of nonlinear system, especially for aircraft (Lane et al, 1988); (Ochi and Kanai, 1991). Feedback linearization can remove nonlinear features from the system and provide a linearized and decoupled closed-loop form. In addition to these features, dynamic inversion control has advantages such as insensitivity to parameter changes and disturbances, and simplicity in physical realization (Snell, Enns and Garrard, 1992); (Wu and Zou, 2009). By using results from feedback linearization combined with the dynamic inversion strategy, an inner-loop controller design may ensure the stability of the nonlinear aircraft system. This strategy not only takes on the advantage of nonlinear transformation to simplify the control system design, but also brings into full play time-varying inversion modes against system perturbation and disturbance (Hovakimyan, Cao and Lavretsky, 2006). As another aspect to this work, the literature of the development of fault estimation within active FTC systems is well summarized in the work by Gao and Ho (2006). By using a linear matrix inequality (LMI) approach, a descriptor estimator was presented to simultaneously estimate system states, output noise and sensor faults for a class of nonlinear systems. Gao and Ding (2007) proposed a novel actuator 10.3182/20120829-3-MX-2028.00092
SAFEPROCESS 2012 August 29-31, 2012. Mexico City, Mexico
fault estimation approach for nonlinear descriptor systems. They described a robust FTC strategy incorporating both state and fault estimation and fault compensation. Bounds on the fault magnitudes are determined for guaranteeing closedloop system performance and robustness using the estimated states and faults together with a LMI strategy.
is invertible, then the system can be linearized by decoupling the non-linear terms in (3) by choosing as follows:
The contribution of this paper lies in the application of the combined dynamic inversion controller with a robust estimator and active fault compensation FTC approach to a nonlinear aircraft system. The aircraft example includes full force and moment longitudinal and lateral dynamics, together with wind turbulence and actuator faults. The proposed estimator and FTC scheme is efficient and reliable for realtime application.
,
( ),
/
(
( )
( )
)
(
[
[
)
(
( )
⋮
)-
( ) ⋯ ⋱ ⋯
⋮
]
-
(7)
( )
(8)
-
( )
(9)
(10)
3. ROBUST ESTIMATOR AND FTC SYSTEM After feedback linearization by the dynamic inversion controller, the system model can be simplified to a classical state-space system as: ̇
}
(11)
where refers to the input vector for the linearized system derived from the outer-loop FTC controller. A computational error is inevitable in solving the dynamic inversion matrix, thus the linearized system also needs online compensation for achieving high-accuracy. By including a model description of the action of both actuator faults and random disturbances, the system described in (11) can be broadened into a descriptor system (Gao and Ding, 2007) as: ̇
(2)
( )
,
Once linearization has been achieved, any further control objectives may be easily met.
with at least one of the , and is the row of . The input-output relation can then be defined as: ,
)-
̇
(1)
.
(
As a result, the output of the closed-loop system is given by the solution to the following linear system
Define to be the smallest integer such that at least one of the inputs appears in ( ) using Lie derivatives as: ∑
)
( ),
( ) is an n-D (dimensional) states vector where of the system, is the input vectors comes from the inner feedback linearization controller and is innerloop system output vectors, is a sufficiently smooth ( ) ( )- , ( vector field. ( ) , ( ) ) is an m-D sufficiently smooth vector field ， ( ) ( )- , ( ( ) , ( ) ) is a sufficiently smooth scalar function.
( )
(
which means ̇ is determined by (4) and (5) in the case that ( ) ( ) ( ) ( ). If ( ) is invertible, the control input can be obtained as:
For the multi-input, multi-output affine nonlinear system as: }
)
( ) ̇
The concept of feedback linearization makes use of the principle of transforming a smooth non-linear dynamical system into linear input-output form via feedback control (Isidori, 1985); (Descuss and Moog, 1985); (Sastry, 1989). ( ) ( )
(
By assuming that all the states of the system are measured, which is a special case of (1) as and it then follows that:
2. NONLINEAR FEEDBACK LINEARIZATION
( )
(6)
So, if the feedback signal is substituted into (3) the result is a closed-loop decoupled linear system. Hence, (3) becomes:
Section 2 introduces the theoretical foundation of the dynamic inversion controller design. The robust estimation procedure incorporated within the FTC system design is illustrated in Section 3. The mathematical model and simulation results for a nonlinear UAV, the Machan, are given in Section 4 to illustrate the proposed approach. The concluding discussion is given in Section 5.
̇
-
( )
( )
(3)
( )]
(4)
(5)
The new system function ( ) and input ( ) are introduced in (3) to achieve linear dependence between outputs and inputs are desirable. If the matrix ( ) 211
(
)
}
(12)
where is the unknown bounded process disturbance vector; is the unknown actuator fault vector; may be singular; , are constant real matrices of ( ) appropriate dimensions; is the real nonlinear vector function for compensating the inversion error. The actuator fault vector has unknown fault components, but the derivative ( ) , is assumed bounded. From the work of Gao and Ding (2007), the first step is to develop a robust state-space observer to estimate both the system states and the faults signal simultaneously by using the known input and the measurement output . The second step is to develop an efficient FTC scheme by using the estimated states and faults.
SAFEPROCESS 2012 August 29-31, 2012. Mexico City, Mexico
The complete design strategy is achieved by combining the above estimator with the dynamic inversion linearization procedure as illustrated in Fig. 1.
̅
( )
[
( )
̅
( ) ]
(18)
and is a positive scalar to be designed. ̅ [ ] includes the estimated states ̂ , the ) and the fault fault derivative estimates ̂ ( ̂ , which enable the observer (14) to be a estimates ̂ simultaneous state and fault estimator. Then there exists a robust observer in the form of (14) for the plant (12) to make the steady estimator error dynamics as small as any desired accuracy, if the derivative gain ̅ is well selected such that ̅ ̅ ̅ ̅ is non-singular. The proportional gain ̅ is computed as: ̅̅
̅
(19)
where ̅ can be solved from the Lyapunov equation: ̅
(
3.1 Robust State And Fault Estimator
(
)
(
)
(13)
and by using (12), an augmented descriptor process can be constructed as follows (Gao and Ding, 2007): ̅ ̅
̅ ̇̅
̅
̅
̅
̅(
( )
) }
̅ ̅
̅(
[
(14)
,̅ (
)
̅
(
̅
, ̅
, ̅
,
̅
̅
-
⋮
⋱ ⋯
(̅
)
(22)
}
(23) Then it is clear that:
⋯ [
(21)
̅
-
̅
̅+; the
where it is assumed that the observer gains ̅ , ̅ and the scalar parameter are designed according to the procedure given in Section 3.1. Let ̅ ( ) , with and ( ). By choosing:
̅
-
* as:
(20)
‖
) ̅ ̅ ] ̂̅ ̅ ̅ ̅( ̂ ) ̅ ̅) ( ̅ ̅ ) ̅ ̂̅
(
̅ ̅
)
,
̅
[ ̅ ( ̂̅
̅
-
)
̅
Consider system (12) and its augmented system (14). An observer-based controller can be constructed as (Gao and Ding, 2007):
̅
]
̅
3.2 Fault-Tolerant Control System
̇̅ , ̅
̅)
With a reasonably large , the designed observer can reduce the effects of the disturbance and the fault model error ( ) , as desired. The proof is given in Gao and Ding (2007).
where ̅
̅
‖̅
To simplify the notation in the determination of the robust estimation scheme let:
̅
̅(
̅)satisfying , ( ̅ is chosen with a positive scalar
with scalar
Fig. 1 Fault Estimation and FTC system
̅) ̅
̅ ̅
⋮
(24)
(15) Furthermore, choose:
]
(25)
Consider a state-space dynamic system as follows: ̇̅
̅ ̂̅
̅ ̂̅
( (̅
)̅ ( ̅ ̅) ( ̅
̅ ̂̅ ) ̅
̅ ( ̂̅ )
) } (16)
where ̂̅ ̅ is the estimate of the augmented state ̅ ̅ ; proportional gains ̅ and ̅ are to be designed the following forms: ̅
,
-
̅
The effect of the fault f on the plant can thus be removed. If there exist a positive definite matrix ( ) matrices , such that: (
)
(
, and
) (26)
(17)
212
[
]
SAFEPROCESS 2012 August 29-31, 2012. Mexico City, Mexico
(
where
̇
)
,
,
, ( ) is a matrix such that and rank ( ) ( ) . Furthermore, if a feasible solution ( )exists in the LMI (26), the statefeedback gain can be computed as: (
)
(27)
In this case, the normal dynamical output feedback controller (22) can perform a fault-tolerant operation by ensuring the closed-loop plant to be internally proper stable and attenuate the bounded input disturbance with prescribed performance index (Gao and Ding, 2005). The UAV (or remotely piloted vehicle) system, the Machan, was used as a development vehicle by Marconi Avionics, UK for research on high incidence flight and non-linear control laws, with RAE Farnborough UK and NASA Dryden in the 1980’s. The Machan Euler equations relating the forces , , and moments , , in the aircraft body axes to the angular and linear velocities in the inertial axes are given as: )
( ̇
)
( ̇
)
̇ ̇ ̇
(
)
(
)
(
)
(30)
where, , , , and represent the maximum engine power, the throttle demand, the propeller efficiency, the engine rises rate and the air flow rate respectively (Aslin, 1985). The open-loop Machan UAV is unstable, thus to achieve a stable fault estimator and FTC system, a closedloop “base-line” control system must be configured before the FTC system can be developed.
The nonlinear model system state and input vectors are chosen as: ,
,
,
-
-
(31)
where , , are the control surface elevator, aileron and rudder, respectively. (28)
where, is the mass of the aircraft. , , are the moments of inertia about the axes through the centre of gravity parallel to the aircraft body axes. , and are the forward, side and vertical velocity of the aircraft respectively. , and are the roll, pitch and yaw rates, respectively. The aerodynamic force and moment equations are: ( (
)
To achieve an inner-loop controller using the dynamic inversion control strategy, the system variables should be chosen carefully to form a square and invertible matrix. The nonlinear Machan model system has 14 state variables, which can be grouped according to the various response speeds. For instance, the fast variables: the angular rates , , with bandwidth between 5-10rad/s and the slow variables: the angles , , have a bandwidth between 1-2rad/s, etc (McLean, 1990). To simplify the system and just for illustration the methodology, only the fast variables are involved in this paper. The slow variables could also be controlled by the stable fast variables in another closed-loop.
4. NONLINEAR MACHAN AIRCRAFT STUDY
( ̇
(
To express the aircraft model in affine nonlinear form in (1), combine (28) to (30), the simplified (reduced order) nonlinear Machan equations can be expressed in terms of: ( )
, ( )
( )
( )-
(32)
where:
)
( )
(
)
( )
(
)
( )
(
)
(33)
and the parameters , , calculated from the aerodynamic coefficients have no relation to the inputs .
)
The control distribution matrix of (1) is determined as: (
) ̅
(
) (29)
where (degree) is the incidence, and (degree) are pitch angle and roll angle respectively, (N) is the side force, (%) is the position of the aircraft centre of gravity, (N) is the thrust force due to the engine, (N/m) is the tail moment, (N) is the force acting on the airframe, , and (N) represent the wing lift, total tail lift and tail lift due to the pitch rate respectively, , and (N/m) are the pitching moment, yawing moment and rolling moment component respectively, ̅ (m) is the mean aerodynamic chord and (N/m) is the rolling moment due to the engine.
( )
[
]
(34)
The parameters , , are calculated from the aerodynamic coefficients related to control input . Some ( ) exists of the model parameters are non-zero, so that and hence the dynamic inversion controller can be achieved. ( ), where Commonly, can be chosen as is the desired output refers to the outer-loop control law , is the bandwidth. In this study, the bandwidth can be chosen as rad/s (McLean, 1990). The desired fast variable states can be defined as:
The first order non-linear engine dynamic is given as: 213
SAFEPROCESS 2012 August 29-31, 2012. Mexico City, Mexico
̇ ̇
(
)
Therefore, the FTC gain matrix can be obtained as:
(
)
̅
( ̇
)
(35)
where the suffices and represent the command signals and the desired values, respectively. Thus the inner-loop control law is derived from the dynamic inversion strategy above as: ̇ ( ) {[ ̇ ] ̇
( ) [ ( )]} ( )
(36)
[
]
Fault test signals acting on the elevator, aileron and rudder are depicted as a multi-step up-step down signal (rad), which are useful for exciting non-linearity and testing for fault detectability properties. The input command signals are - , - , and the disturbance chosen as , model acting on the vertical force is chosen according to a Dryden wind turbulence spectrum approximated by the output of a colouring filter with zero-mean Gaussian distributed random noise with variance 80. With 3 faults acting on the actuators and wind disturbance affecting the vertical force, the estimation results corresponding to these faults are shown in Fig.3. The errors between estimated states and real states are shown in Fig.4.
Aeliron Fault Estimation
Elevator Fault Estimation
For this Machan study the engine thrust is not chosen to be a control variable, and is thus a constant value. The resulting inner-loop dynamic inversion controller, which represented in Fig.1 as “linearized system” block is shown in Fig.2.
Fig.2 Inner Loop Dynamic Inversion Controller
Rudder Fault Estimation
To achieve the robust observer (12), construct an augmented model in the form of (14) with derivative order . Choose ̅ , , . From (21) one can compute . By solving the Lyapunov function (20) and using (19), the observer gain can be determined as:
1
Elevator Fault Estimated Fault
0.5 0 -0.5 -1
0
10
20
30
40
50
1 Aeliron Fault Estimated Fault
0.5 0 -0.5 -1
0
10
20
30
40
0.1
50
Rudder Fault Estimated Fault
0.05 0 -0.05 -0.1
0
10
20
30
40
50
Time Sec
Fig.3 Faults and their Estimation ̅
-4
[
error of p
x 10
]
2 0 -2 0
error of q
1
Choose
-1
15
20
25
30
35
40
45
50
0
5
10
15
20
25
30
35
40
45
50
5
10
15
20
25 30 Time Sec
35
40
45
50
-6
error of r
2
] as in (25),
10
x 10
0
x 10
0 -2
[
5 -4
This Gao and Ding (2007) estimator has a high gain, which may lead to an unrealizable range of gain magnitudes. However, the condition number of the gain matrix ̅ is calculated as (̅ ) , which is acceptable from a practical standpoint and is suitable for eliminating the effect of nonlinearity. To achieve the robust FTC system, the choise of , and the solution of (26) & (27) leads to the following:
0
Fig.4 Error between Estimated States and Real States
is then defined as (23). 214
SAFEPROCESS 2012 August 29-31, 2012. Mexico City, Mexico
Descusse, J., and Moog, C.H. (1985). Decoupling with dynamic compensation for strongly invertible affine nonlinear systems. Int. J. Contr., 42, 1387-1398. Donald, M. (1990). Automatic Flight Control Systems. Prentice Hall. Edwards, C., Spurgeon, S. K. and Patton, R.J. (2000). Sliding mode observers for fault detection. Autom., 36, 541–553. Edwards, C. Lombaerts, T. & Smaili, H. (2010). Fault Tolerant Flight Control. Springer. Gao, Z. and Ho, D.(2006).State/noise estimator for descriptor systems with application to sensor fault diagnosis. IEEE Trans. on Signal Process., 54, 1316–1326. Gao, Z. and Steve X. D. (2007). Actuator fault robust estimation and fault-tolerant control for a class of nonlinear descriptor systems. Autom., 43, 912-920 Goupil, P. (2009). Oscillatory failure case detection in the A380 electrical flight control system by analytical redundancy. Contr. Engin. Pract., 18, 1110-1119. Hovakimyan, N., Lavretsky, E. and Cao C. (2006). Dynamic inversion of multi-input non-affine systems via time-scale separation, American Control Conference, 14, 6. Isidori, A. (1985). Nonlinear control systems: An introduction, Springer-Verlag, Berlin and New York. Kim, K. S., Lee, K. J. and Kim, Y. D. (2003). Reconfigurable flight control system design using direct adaptive method. J. Guid., Contr., Dyn., 26, 543. Lane, S. H. and Stengel, R. F. (1988). Flight control design using non-linear inverse dynamics. Autom., 24, 471-483. Maki, M., Jiang, J. and Hagino, K. (2004). A stability guaranteed active fault-tolerant control system against actuator failures. Int. J. Robust. & Nonli. Contr., 14, 10611077. McLean, D. (1990). Automatic flight control systems. Prentice Hall. Ochi, Y. and Kanai, K. (1991). Design of restructurable flight control systems using feedback linearization. J. of Guid., Contr. & Dyn., 14, 903-911. Patton, R. (1997). Fault-tolerant control systems: The 1997 situation. IFAC Symp. Safeprocess’97, Hull, UK, 1029-1052. Patton, R. J., Putra, D. & Klinkhieo, S. (2010). Friction compensation as a fault tolerant control problem, Int. J. Sys. Sci., 41, 987-1001. Richter, J. H. and Lunze, J. (2010). Reconfigurable control of Hammerstein systems after actuator failures: stability, tracking, and performance. Int. J. Contr., 83, 1612. Richter, J. H., Heemels, W. P. M. H., Wouw, N. and Lunze, J. (2011). Reconfigurable control of piecewise affine systems with actuator and sensor faults: Stability and tracking. Autom., 47, 678-691. Sastry, S. (1989). Adaptive Control of Linearizable Systems, IEEE Transaction on Automatic Control. 34, 1123-1131. Simani, S. and Patton, R. J. (2003). Fault diagnosis of non–linear dynamic processes using identified hybrid models. IEEE Conference on Decision and Control, 1, 445–450. Snell, S., Enns, D. and Garrard, W. (1992). Nonlinear inversion flight control for a super-manoeuvrable aircraft. J. of Guid. Contr. & Dyn, 15, 976-984. Zhang, Y. and Li, X. R. (2002). Detection and diagnosis of sensor and actuator failures using IMM estimator. IEEE Trans. Aero. & Electr. Syst., 34, 1293-1313. Wu, Y. and Zou, Q. (2009). Robust Inversion-Based 2-DOF Control Design for Output Tracking: Piezoelectric-Actuator Example. IEEE Trans.Contr.Syst.Technol., 17,1069– 1082.
p(rad/sec)
5 0 -5
0
5
10
15
20
25
30
35
40
45
50
0
5
10
15
20
25
30
35
40
45
50
0
5
10
15
20
25 30 Time Sec
35
40
45
50
q(rad/sec)
1 0 -1
r(rad/sec)
0.5 0 -0.5 -1
Fig.5 FTC System Responses with Faults and Disturbance Fig.5 shows the output responses resulting from the FTC system with disturbance on vertical force. 5. CONCLUSIONS The principle of dynamic inversion control design for nonlinear systems has been outlined through the use of the feedback linearization. The robust simultaneous fault and state estimator is designed for the on-line linearized model by solving a Lyapunov equation. Based on the estimated vectors, the robust FTC loop is accomplished by using LMI technique to ensure that the flight system satisfies robust performance. Both the estimator and the FTC are expressed in state-space form and are based on the original system matrices making them preferable for real control application. The full force and moment nonlinear UAV Machan has been used as an example of a system with highly non-linear dynamics and which is difficult to control. The simulation results demonstrate the robustness of the FTC with fault estimation, corresponding to disturbances and faults. REFERENCES Aslin, P. (1985). Aircraft simulation and robust flight control system design. DPhil thesis, Department of Electronics, University of York, UK. Bin, J. and Chowdhury, F. N. (2005). Fault estimation and accommodation for linear MIMO discrete-time system. IEEE Trans. Contr. Syst. Technol., 13, 493-499. Bin, J., Staroswiecki, M. and Cocquempot, V. (2006). Fault Accommodation for Nonlinear Dynamic Systems. IEEE Trans. Automatic Control, 51, 1578-1583. Patton, R. J., Frank, P. M. and Clarke, R. (2000). Issues of fault diagnosis for dynamic systems. Springer, Berlin. Blanke, M., Kinnaert, M., Lunze, J. and Staroswiecki, M. (2003). Diagnosis and fault-tolerant control. Springer. Boskovic, J. D. and Mehra, R. K. (1999). Stable multiple model adaptive flight control for accommodation of a large class of control effector failures. Proc. ACC ’99, 1920-1924. Chen, J. and Patton, R. J. (1999). Robust model-based fault diagnosis for dynamic systems. Kluwer Academic. Chen, R. and Speyer, J. (2004). Sensor and actuator fault reconstruction. J. of Guid. Contr. & Dyn., 27, 186–196. Ding, X. (2008). Model-based fault diagnosis techniques: design schemes, algorithms, and tools. Springer, Berlin.
215