Abstract-Modern aircraft include a variety of automatic control systems that aid the flight in navigation, flight management, and to augmenting the stability characteristics of airplane. In order to reduce cost and design cycling time and improve performance, Liu proposed closed-loop combination approaches in [1]. In this paper, we discuss the method again and apply the close-loop combination technique to the longitudinal control channels of the nonlinear F-16 fighter. We conclude that the closed-loop combination method cannot be applied in nonlinear F-16 system since the design of closed-loop combination controller does fully depend on the closed-loop transfer function and its minimal pole-zero cancellation form of the system, and in nonlinear system there are some unstable states that cannot be accurately cancelled due to the differences between linear model and nonlinear model. The simulation results are presented to show this limitation of the closed-loop combination method.

I. INTRODUCTION

The development of control and computing technology makes it possible for advanced flight control strategy. At the same time, the advance control techniques tend to make the control design and implementation more complicated with more control loops or channels, e.g. the autopilot of modern aircraft include a variety of automatic control systems that aid the flight in navigation, flight management, and to augmenting the stability characteristics of airplane [2]. Therefore, how to integrate the control loops with common functions or sharing similar control effectors so as to reduce design cost and cycle time becomes a very significant and interesting question in the design of integrate flight control system. Liu proposed closed-loop combination approach[1]

Y.K. Wong, is with the Department of Electrical Engineering, The Hong Kong Polytechnic Univercity, Hung Hom, Hong Kong. (E-mail: eeykwong@inet.polyu.edu.hk)Eric H.K. Fung, is with the Department of Mechanical Engineering, The Hong Kong Polytechnic Univercity, Hung Hom, Hong Kong. (E-mail: mmhkfung@inet.polyu.edu.hk)Hugh H.T. Liu is with the Institute for Aerospace Studies,University of Toronto, Toronto, Ontario, Canada, M3H 5T6. (E-mail: liu@utias.utoronto.ca)Y.C.Li is with the Department of Mechanical Engineering, The Hong Kong Polytechnic Univercity, Hung Hom, Hong Kong. (E-mail: mmlyc@inet.polyu.edu.hk)

to design the proper integrated controller and solve the above question.

In the longitudinal control channels of F-16 fighter, the pitch control loop and speed control loop are considered for the flight control integration [2, 3]. In this paper, we apply the closed-loop combination method to the 6DoF nonlinear F-16 fighter longitudinal control channels to discuss the method in the nonlinear system. We conclude that the closed-loop combination controller cannot be applied in nonlinear F-16 system though it satisfied all the specifications in the linear F-16 longitudinal control system. An explanation of the limitation of the method is that the closed-loop combination controller does fully depend on the closed-loop transfer function and its minimal pole-zero cancellation form. The unstable state derived from the controller can be cancelled in the linear system and the minimal stable form of the linear system will be achieved. But it not true for the nonlinear system due to the differences between the linear model and nonlinear model. The rest of the paper is organized as follows. In Section 2 we review the necessary theoretical background of the closed-loop combination method. And in Section 3, we give the 6DoF F-16 fighter nonlinear model and the linear model at trimmed operation point. In Section 4, we show and explain the limitation of the closed-loop combination method by the F-16 fighter simulation. The conclusions are in Section 5.

II. REVIEW OF THE CLOSED-LOOPED COMBINATION

We consider a integrated control system, shown in Figure 1.

ue

e

c ee

G

uK ucu

K

euG

Figure.1. Integrated control system

According to the closed-loop combination method [1], we should design the proper controller **, uKKK [Eq. (1)]:

Y.K. Wong, Eric H.K. Fung, Hugh H.T. Liu and Y.C. Li

The Limitation of The Closed-loop Combination Method

Proceedings of the2005 IEEE Conference on Control ApplicationsToronto, Canada, August 28-31, 2005

TA4.5

0-7803-9354-6/05/$20.00 ©2005 IEEE 693

JGJG

JGJK

JGJG

JGJK

ee

e

ee

e

uu

uu

uu

uu

1

)1(

1

)1(

*

*

(1)

where uJJ , denote the proper controllers for the two individual loops (see in Figure 2).

Je

Gec e

uJeuG uecu ue

Figure.2. Two individual loops

The above method is closed-loop combination method [1], it means that if we can design individual controller uJJ , to satisfy the separate specifications, then the derived integrated controller **, uKKK [Eq. (1)] will hold these specifications unexceptionally.

But when we apply it in the nonlinear F-16 model simulation, we find that it cannot feasible as its validity mainly depends on the closed-loop transfer function and the minimal pole-zero cancellation form of the model. Even if there is little difference between the theoretical model and real model, the controller cannot work properly due to the failure of unstable pole-zero cancellation.

III. NONLINEAR F-16 MODEL AND LINEARIZATION

In the following section, we show the limitation of the closed-loop combination method by the simulation example of the F-16 fighter. We give the non-linear Ordinary Differential Equations (ODE) describing the motion of F-16 fighter [4]:

qHCIsbq

qrpI

Ipq

III

r

rHCI

csqpr

II

prI

IIq

CIsbq

pqrII

qrI

IIp

Cmsq

gpvquw

Cmsq

grupwv

mT

Cmsq

gqwrvu

etnZZ

XZ

Z

YX

etmYY

XZ

Y

XZ

tlXX

XZ

X

ZY

tZ

tY

tX

,

,22

,

,

,

,

)(

)(

)(

coscos

sincos

sin

(2)

where wvu ,, and rqp ,, are the body-axes components of linear velocities and rotational velocities, respectively; yaw

angle , pitch angle , and roll angle ,that is, the Euler angles denote the attitudes of the aircraft with respect to the Earth. And other symbols are defined as follows: g isacceleration due to gravity, m airplane mass; q is the free-stream dynamic pressure; s denotes wing area, b is wing span, c is wing mean aerodynamic chord. T is the engine thrust, He is the engine angular momentum;

YZXZXYZYX IIIIII ,,,,, are inertia tensor ; the coefficient

tntmtltZtYtX CCCCCC ,,,,,, ,,,,, , are the total aerodynamic coefficient, which were derived from low-speed static and dynamic wind-tunnel tests conducted with subscale models of the F-16 in wind-tunnel facilities at the NASA Ames and Langley Research Centers [4].

The above motion equations and the kinematics equations together make up 12 independent ODEs, which is the F-16 nonlinear ODE model.

cos)cossin(sincos)sincos(

sin)}cossin(cos{sin)sincos(

cos)}cossin(cos{sin

sincoscos

cossin

wvuzwv

wvuy

wv

wvuxp

rq

rq

e

e

e (3)

where the Euler angles , , and denote the attitudes of the aircraft with respect to the Earth; the position of the aircraft with respect to the Earth-fixed reference frame is given by the coordinates eee zyx ,, .

In order to design the proper controller, we need to obtain the linearized model and synthesize the linear controller of the model based upon linear system theory. At the beginning, we need to find steady-state flight conditions which can be applied as ‘operating points’ for the linearization, and as initial conditions for simulations.

In this paper, we consider the steady wing-level flight of F-16 fighter at an altitude of 5000m and Mach number is 0.6. Then we compute the steady-state flight conditions and linearize the ODE nonlinear model by small-perturbation methods.

The linearized F-16 system described by the state-space matricies DCBA ,,, , can be denoted by the standard Matlab LTI system for study convenience. We consider one input two outputs subsystem, whose input is “deflection of elevator e ”, the outputs are “pitch attitude ”, “airspeed u along x-axes” and the state vector x is

],,,,,,,,,,,[ eee zyxrqpwvu . The transfer functions of the linearized F-16 fighter model can be derived from the state-space matricies DCBA ,,, as follows:

694

15-234

56789

23

45678

101.943+s0.01513+s1.867+s6.218+s269.8+

s196.5+s93.01+s31.3+s5.859+s0.0006163-s0.2525+s40.25+s75.16+

s85.27+s35.28+s9.25+s2.1+s0.03988

)(sGeu

15-234

56789

17-23

4567

101.943+s0.01513+s1.867+s6.218+s269.8+

s196.5+s93.01+s31.3+s5.859+s106.937-s0.0004771-s0.09165-s4.163-

s7.46-s3.155-s0.8351-s0.2012-

)(sGe

The following figure show the comparison of the open-loop linear model and nonlinear model when input step signal is 310 .

Figure.3. Comparison of the open-loop linear model and nonlinear model.

Because the matrix A is a 12-row-12-column matrix; it seems that the denominator of the transfer functions is complicated. It is true that under the conditions of small perturbations from steady-state, wings-level, non-side slipping flight, the rigid-aircraft equations of motion could be split into two uncoupled sets [3]. These are the longitudinal equations that involve ee zxqwu ,,,,, and the lateral-directional equations that involve eyrpv ,,,,, . It is possible to extract simplified sub-matrix LoA from A by

specifying a vector with the element number of the required state variables. The sub-matrices LoLoLo DCB ,, can be obtained from DCB ,, , respectively.

Similarly, the transfer functions of the simplified model are Given by:

15-2345

234

_108.666+s0.06756+s0.1149+s9.819+s2.454+s

0.002751-s1.462+s2.038+s1.964+s0.03988)(sG su e

15-2345

-1623

_108.666+s0.06756+s0.1149+s9.819+s2.454+s

103.094-s0.00213-s0.1501-s0.2012-)(sG se

By the simulation, we compare the step response of the opened original model and the simplified model. There exists so little difference between the two models that we can just find the controller for the simplified system model instead of the original linear model.

Figure.4. Comparison of the simplified linear model and original linear model

IV. SIMULATION AND THE LIMITATION OF THE CLOSED-LOOP COMBINATION

Now we consider the F-16 fighter longitudinal control system as the same structure as shown in Figure 1. Assume the overall multiple performance requirements are: the pitch attitude and speed control both have good design criteria in

695

term of tracking (small steady state error and fast setting time) and safety (acceptable overshoot).

;1)()(

;)1)((max)(

22

1)(1,01

uH

tuH

uess

tutuovershoot

c

;1)()(

;)1)((max)(

44

3)(1,03

ess

ttovershoot

H

tHc (4)

The desired specification values of the F-16 fighter longitudinal control system are defined by:

1 = 0.25; 2 = 0.02; 3 = 0.2; 4 = 0.02;

In order to apply the closed-loop combination to design the integrated controller, we need to find the proper controller for the two separate loops at first, and then derive the integrated controller .

Based on the linear system theory [5, 6], we design the proper PID controllers [Eq. (5)] of the two loops:

3+s0.5;s

10-s20-5s- 2

uJJ (5)

The simulation results with the PID controller of the two loops are shown in Figure 5. It can be seen that the separate specifications of the two loops are satisfied by the above controller.

Figure 5(a) Pitch control simulation

Figure 5(b) Speed control simulation According to the closed-loop combination method [1], we can derive the closed-loop combination controller [Eq. (6)]:

14.1)+s3.11+6.233)(s+s2.18+(s

0.1991)+s0.4191+0.01448)(s+0.3204)(s+1.154)(s-(s14.88)+3.088s+9.788)(s+2.444s+0.2942)(s+0.3995s+(s

0.006903)+0.01002s+0.5858)(s+(s3.414)+(s5.0997-

22

2

222

2

K

14.1)+3.11s+(s6.233)+2.18s+(s

0.1991)+0.4191s+(s0.001877)-0.01448)(s+(s0.3204)+(s1.154)-(s13.37)+3.104s+(s9.788)+2.444s+(s

0.1154)+0.3427s+(s0.006903)+0.01002s+(s0.0138)+(s6)+(ss0.5

22

2

22

22

uK

(6)

The simulation results of the simplified model with the closed-loop combination controller are given in Figure 6. We can see that the closed-loop combination controller satisfies the original individual performance successfully.

(a) u and of c ;

(b) u and of cuFigure 6. Step response of the simplified F-16 linear model

696

Then we apply the closed-loop combination controller to the original system model and get the simulation results as shown in Figure 7.

Due to the little difference of the simplified linear model and the original linear model, it can be seen that the step response of the original system is very close to the simplified linear system, i.e. the integrated system holds the original specifications as required.

But when we apply the closed-combination controller to the non-linear F-16 fighter system, the simulation curve changes unstable so quickly that we cannot continue the simulation in the Simulink .

It seems that the closed-loop combination method is not feasible in the F-16 nonlinear model. Let us analysis why such a simulation results appear.

(a) u and of c ;

(b) u and of cuFigure 7. Step response of the F-16 linear model

In the closed-loop combination controller [Eq. (6)], there exist unstable poles. And with the controller, we can derive the 22 non-minimal closed-loop transfer function matrix of the F-16 linear model. Without the loss of generality, we look at the one of the transfer function 11H .

15.62)+4.069s+(s10.86)+0.8958s+(s

9.793)+2.439s+(s9.794)+2.452s+9.773)(s+2.445s+(s

22.31)+7.87s+(s1.976)+1.296s+(s1.976)+1.296s+(s

0.454)+0.7861s+(s0.2558)+0.6998s+(s0.2558)+0.6998s+(s

0.09879)+0.3846s+(s0.006903)+0.01002s+(s

6.25)+5s+)(s103.522+0.003754s-(s

0.001869)-(s0.008248)+(s0.008252)+(s0.008261)+(s0.01379)+0.01447)(s+(s0.01447)+(s6.087)-(s

0.3144)+(s1.705)+(s2.497)+(s13.59)+(ss10.86)+0.8949s+(s9.786)+2.447s+(s

22.31)+7.87s+(s1.976)+1.296s+(s

1.976)+1.296s+(s0.4541)+0.7862s+(s

0.2558)+0.6998s+(s0.2558)+0.6997s+(s

0.006904)+0.01002s+(s6.257)+5.003s+(s

0.09882)+0.6287s+(s0.001868)-(s0.001877)-(s

0.001877)-(s0.008252)+(s0.01447)+(s6.087)-(s0.382)+(s

0.7317)+(s1.706)+(s2.493)+(s2.618)+(s13.59)+(ss2.012

232

222

2222

222

232

26-2

2

23

3232

222

22

22

322

2

332

3

11H

It can be seen that there still exist unstable poles in the non-minimal closed-loop transfer function 11H . It is obvious that the step responses of the non-minimal closed-loop transfer function 11H will change unstable quickly in the simulation. But we can get the minimal stable form min_11H by canceling the unstable and close zero-pole pairs, and the simulation results are satisfactory and have been shown in Figure 7.

222

22

23

min_11

15.62)+4.069s+0.09879)(s+0.3846s+(s

0.0002095)+0.02895s+0.01379)(s+(s0.3144)+(s0.09882)+0.6287s+(s0.01447)+(s

0.382)+(s0.7317)+(s2.618)+(s2.012

H

So we can deduce that the closed-loop combination method can be applied in the linear system just because in the linear system we can cancel the unstable and close pole-zero pairs (or unstable states) and get the stable minimal system, while in the nonlinear F-16 system, due to the difference of the nonlinear model and linear model, the unstable poles in the controller cannot be cancelled. So in the nonlinear simulation, the system is unstable. It means that the closed-loop combination method fully depends on the closed-loop transfer function and its minimal realization or pole-zero cancellation form, its application in nonlinear real-time F-16 system is not feasible.

In fact, the closed-loop combination controllers are derived from Eq.(7), see details in [1]

697

11 uu

uu

uu

uu

JG

JG

KGKG

KG

e

e

ee

e

11 JG

JG

KGKG

KG

e

e

ee

e

uu (7)

It implies that with the closed-loop combination contoller **, uKKK , though the integrated transfer function (see the

left side of the Eq.(7)) may produce the unstable poles (or states), after the cancellation, the minimal form of it will equal to the individual loop transfer function (see the right side of the Eq.(7)). While the non-minimal transfer functions of the both sides may not be identical.

So the closed-loop combination method requires the cancellation of the unstable states and achievement of the minimal stable form. And in the nonlinear system this requirement can not be satisfied due to the differences between the linear model and nonlinear model. When there are some unstable states that cannot accurately cancel, the system will be unstable.

In this paper the closed-loop combination controller is unstable, which leads to unstable states in the linear system. If we manage to synthesize the stable proper closed-loop combination controller, it will not have the unstable states in the system theoretically and thus improve the control effect in the nonlinear system. But how to design the proper stable controller is still a challenging job since in some cases it is not easy to avoid the unstable closed-loop combination controller with the method. We will continue the research of the closed-loop combination method and the application in the nonlinear flight control system in our future work.

V. CONCLUSION

In this paper, we manage to apply closed-loop combination method to the nonlinear F-16 fighter system and show the limitation of the method. We conclude that the closed-loop combination method cannot be applied in nonlinear F-16 system since the method depends on the closed-loop transfer function and its minimal pole-zero cancellation form. And in the nonlinear system some unstable states cannot be cancelled since there are some differences between the linear model and nonlinear model.

ACKNOWLEDGMENT

The authors would like to thank The Hong Kong Polytechnic University for the financial support (Project No. A-PE77) towards this work.

REFERENCES

[1] H.H.T Liu, “Design combination in integrated flight control”, InProceedings of American Control Conference Vol.1, pages 494 – 499, 25-27 June 2001. [2] B. Etkin, Dynamics of Flight – Stability and Control, Wiley, New York, USA, 2nd edition, 1982. [3] B. L. Stevens and F. L. Lewis, Aircraft Control and Simulation. John Wiley & Sons Inc., 1992. [4] T. N. Luat, M. E. Ogburn, and W. P. Gilbert, Simulator Study of Stall/Post-Stall Characteristics of a Fighter Airplane With Relaxed Longitudinal Static Stability, NASA Technique paper 1538, Dec.1979. [5] G. Graham, S. Graebe and M. Salgado, Control System Design, Prentice Hall, 2001. [6] F. Gene, J. D. Powell, and A. Emami-Naeini, Feedback Control of Dynamic Systems. Prentice-Hall, fourth edition, 2002.

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