Integrator-Backstepping Control Design for Nonlinear ...

AIAA 2015-1321 AIAA SciTech 5-9 January 2015, Kissimmee, Florida AIAA Guidance, Navigation, and Control Conference

Integrator-Backstepping Control Design for Nonlinear Flight System Dynamics 1

2

Downloaded by OLD DOMINION UNIVERSITY on March 10, 2015 | http://arc.aiaa.org | DOI: 10.2514/6.2015-1321

Thanh T. Tran and Brett Newman Department of Mechanical and Aerospace Engineering, Old Dominion University, Norfolk, VA 23529 This paper investigates an integrator-backstepping control (IBSC) methodology for a strict-feedback form of nonlinear dynamic systems in the presence of model parameter errors. A systematic procedure will be addressed firstly for formulating the IBSC law for the strict-feedback model. Formulation starts with a definition of modified tracking error by adding an integral term to normal tracking error and then a recursive sequence of coordinate transformations and Lyapunov function based feedback selections results in an IBSC law to make the system well-behaved in tracking and asymptotically stable. To show the applicability, flight path angle control corresponding to the nonlinear longitudinal dynamics of F-16 aircraft model will be considered. Assumption on lift force of aircraft as a sinusoidal function of attack angle supports for simplifying the longitudinal dynamics of aircraft to the standard strict-feedback form. The control design is applied for the standard model to achieve the IBSC law for nonlinear longitudinal dynamics of aircraft. Numerical simulation will be implemented to validate and evaluate the proposed algorithm via nonlinear closed-loop system of a high performance aircraft model.

Nomenclature a m

q

D g

 α L

 h

S t

= speed of sound, ft/s = aircraft velocity, ft/s = total mass of aircraft, slug = lift coefficient = throttle, % = drag coefficient = pitch rate, deg/s = inertial moment about y axis of aircraft, slug= thrust coefficient = positive gains = drag force, lbf = gravity, ft/ = pitch angle, deg = angle of attack, deg = lift force, lbf = control pitch moment coefficient, 1/ = thrust force, lbf = flight path angle, deg = altitude, ft = pitch control (elevator or horizontal vane), deg = density of air, slugs/ = thrust point offset, ft = pitching moment, lbf-ft = reference area, = time variable, s

1

Doctoral Candidate, Dept. of Mechanical and Aerospace Eng., Old Dominion Univ., 238 Kaufman Hall, Norfolk, VA 23529, Member AIAA. 2 Professor, Dept. of Mechanical and Aerospace Eng., Old Dominion Univ., 238 Kaufman Hall, Norfolk, VA 23529, Associate Fellow AIAA. American Institute of Aeronautics and Astronautics

Copyright © 2015 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

I.

Introduction


I

n recent years, the investigation and development of flight control algorithms for nonlinear aircraft dynamics has been addressed by many researchers. The standard approach for designing control systems for nonlinear aircraft is gain-scheduling. In this strategy, linear approximation of dynamic equations at several important operation points within the flight envelope is achieved. Depending on these points, linear controllers are designed and then combined continuously as the vehicle flies from one operating point to another. Due to linearization, the actual system performance and stability can be significantly different from the design results due the approximated nonlinearities. Instead of the need of interpolation or of gain-fitting technique of several operation points, the application of a variable-gain optimal output feedback control design methodology is proposed in Ref. 1 in which the feedback gains are calculated and scheduled as a function of the angle of attack. In this approach, the feedback gains are calculated and scheduled by minimizing the cost function that is a function of state and control vectors. The proposed approach is not fully effective and robust for short period mode control due to the computational cost and convergence of the associated constrained optimization problem. Nonlinear inverse dynamics (NDI) 2, 3 for flight control system is proposed to eliminate the drawbacks of gainscheduling based design. Reference 2 uses assumptions in which aerodynamic force coefficients and moment coefficients to be nonlinear functions of the angle of attack, sideslip angle, and thrust coefficient but linear functions of the elevator, aileron, and rudder, the motion equations can be rewritten as a triangular system of general form and then a nonlinear inverse dynamic controller is generated and proven valid over the entire flight envelope. The limitation of proposed strategy is that aerodynamic moments must be linearly represented in terms of stability derivatives and control variables. A better approach of NDI design for full nonlinear flight control is presented in Ref. 3 which uses a fact that control surface deflections do not affected directly to the slow dynamics. Therefore, full nonlinear flight dynamics are designed separately for slow-state variable dynamics and fast-state variable dynamics. With the designed fast-state controller, a separate and approximate inversion procedure is carried out to design slowstate controller for slow-state variable dynamics and the achieved slow-state controllers are used as commands for fast-state variable dynamics. A justification of reliability of proposed algorithm is confirmed analytically using the longitudinal dynamics. As general disadvantages of NDI approach that prevent the popularity of applying the method for nonlinear flight systems are the robustness of NDI based control design, i.e., system parameters of the aircraft dynamics are included in control law. Therefore, aircraft model used for control design needs to be accurate in order to achieve good performance and stability of the system. In recent years, many researchers have addressed backstepping control design which are introduced for the first time in Ref. 4 and has been a motivated basis in exploring a new direction in control design for nonlinear dynamic systems. Backstepping control design is seen as a recursive design process which breaks a design problem on the full system down to a sequence of sub-problems on lower order systems. Considering each lower order system with a CLF and paying attention to the interaction between two subsystems makes it modular and easy to design the stabilizing controller. The advantages of backstepping control are to stress on robustness, avoid the cancellation of dynamic nonlinearity, relax for a class of nonlinear system, and eliminate the requirement for the designed system to appear linear.5 Applications of backstepping design approach for nonlinear flight control have been paid an attention to many researchers.6-9 It is shown in the Ref. 6 how the equations of motion for aircraft are restructured in linear strict-feedback form, and then backstepping control design and adaptive gain scheduling are employed to achieve full envelope flight control. Ref. 7 assumes aerodynamic forces and moments as a linear function of attack angle, pitch angle and elevator. An aircraft model of strict-feedback form and backstepping control design are applied. The new contribution is that the aerodynamic parameters of aircraft are approximated by nominal ones and their model errors and then parameter adaptive scheme using the multilayer neural network is employed to improve the performance and stability of the aircraft. Limitations of these approach6, 7 are to ignore many considerable properties in which design model for generating the control law is used to be appeared like linear model. These disadvantages have been eliminated in Ref. 8 and Ref. 9. In these approaches, it is assumed that flight path angle is not significantly affected by the gravitational term which is fixed at the reference value. Also, the product of angle of attack and time derivative of flight path angle is assumed to be positive with nonzero attack angles. Model of nonlinear longitudinal dynamics of the aircraft is rewritten in strict-feedback form and then the backstepping based control algorithm is then used. The question arises concerning how to eliminate or improve the assumptions of linear strict-feedback form,6 neglection of the effect of flight path angle in the gravitational term, 7 and lift force as a linear function of angle of attack.8, 9 The paper10 is come up with the idea in which lift force can be represented as a sinusoidal function of attack angle. An F-16 longitudinal nonlinear dynamic model can be restructured to generalized strict-feedback form and an analysis and design schedule is presented systematically to formulate the American Institute of Aeronautics and Astronautics


backstepping controller for flight system dynamics. However, works1-10 has not addressed the robustness issues in control design. Approaches11, 12 are provided to improve the performance of aircraft under the presences of variation parameters of aircraft dynamics and disturbances. In these studies, the combination of stochastic robustness and dynamic inversion is proposed to minimize the probability of instability and probabilities of violations by using the genetic algorithm for searching the design parameter space. The question is whether or not the genetic algorithm is valid to be used to address the robustness issues in flight system dynamics. In this paper, the integrator-backstepping control (IBSC) design will be presented to eliminate the disadvantages of the mentioned works. In this approach, a modified tracking error is defined by adding an integral term to normal tracking error and control input is then generated by using the Lyapunov based design. A control design algorithm is provided to improve the stability, well-tracking commands, and robustness of a high-performance aircraft in the presence of unmodeled dynamics. . The remainder of this paper is organized as follows. In the Section II, a mathematical basis for modeling and control of for strict-feedback model of nonlinear dynamic system is provided firstly. Then an integratorbackstepping control law formulation is addressed for the F-16 model flight system dynamics in section III. Section IV shows the simulation results and discussions, and Section V draws conclusions to support the proposed control algorithm.

II.

Theoretical Background

In this section, a mathematical basis is provided to support for modeling and control of a strict-feedback system in the presence of model errors. A systematic formulation for generating the IBSC law control is then presented and then block diagram of the proposed control algorithm for the system is also provided for control design scheme. Some comments and discussions are provided in the end of the section. Consider a single-input single- output (SISO) strict-feedback model as follows (̂ ̇

(̂ ̇

)

(1) )

(2)

̂ ̇

(3) (4)

where parameters

̂ are known parameters of model and the unknown in the system are approximated by known constant parameters ̂ .

Definition 1 (Relative Degree) The system (1)-(4) is said to have relative degree at if is equal to the number of times in which the output y must be differentiated before the input u appears explicitly. Definition 2 (Control Lyapunov Function) is said to be Control Lyapunov function with respect to the system ̇ derivative of

with respect to

if the directional

such that ̇

Theorem 1 (Lyapunov Asymptotical Stability) Let V(x): be a radially unbounded function and ̇ with such that   when ̇  when then the system is globally asymptotically stable at x = 0.

be an equilibrium point of the system

American Institute of Aeronautics and Astronautics


Theorem 2 (State Decoupled Linear System) If the system (1)-(4) has relative degree and the variables are solvable explicitly in terms of other variables, then the strict-feedback system (1)-(4) can be transformed into a state decoupled linear system through a series of coordinate transformations and feedback selections. Proof The objective is to design a control law for the parameterized strict-feedback system (1)-(4) such that asymptotically where is a constant, and global asymptotic stability is achieved with zero or acceptably small overshoots in the system in the presence of the model parameter errors. The following steps are used for formulating the IBSC law for the system (1)-(4). Step 1: The state variable is regarded as a control input in Eq. (1) which is considered as the first subsystem. Thus, is chosen to make the first subsystem globally asymptotically stable. The chosen function is called a virtual control law. By introducing a modified tracking error as (5) where ∫ and

,

is defined as normal tracking error and

is a positive gain.

Differentiating in time of Eq. (5) and combining with Eq. (5) result in ̇ ̇

(̂ ̇

)

(6)

For the system in Eq. (6), a CLF in term of Definition 2 can be chosen such that when the virtual control law is applied, its time derivative becomes negative definite. The positive definite function is chosen as (7) Taking the derivative in time of Eq. (7) and combining with Eq. (6) achieves as ̇ ̇

(̂

)

(8)

By satisfying the asymptotically stable condition in term of Theorem 1 for Eq. (8), a virtual control law denoted as for can be chosen as ̂

(

) ̂

where

(9)

is the positive gain. By doing so, the CLF derivative is negative definite. ̇ ̇

(10)

Step 2: By choosing the state feedback in Eq. (9) and a change of coordinate indicated below ̂

(11)

the second subsystem can be rewritten as follows ̇ ̇

(12) (

̂

)

̇

̂


where ̇ is determined as a functions of gains, states, known parameters ̂ and command. The state variable is regarded as a control input in Eq. (12) and a CLF in term of Definition 2 can be chosen such that it makes the subsystem in Eq. (12) asymptotically stable with the virtual control law, i.e. (13) Taking the derivative of Eq. (13) in time and combining with Eq. (12) results in


̇

̂

(

)

̂ ̇

(14)

To meet the asymptotically stable condition in term of Theorem 1 for Eq. (14), a virtual control law denoted as for can be chosen such that ̂

(

)

̂ ̇

̂ where

(15)

is the positive gain. By doing so, the CLF derivative is again negative definite. ̇

(16)

Step 3: By choosing the state feedback in Eqs. (9), (15) and coordinate transformations as in Eqs. (5), (11) and the transformation below ̂

(17)

the complete system can be rewritten as follows ̇ ̇ ̇

̂

(

)

(18)

̂ ̇

where ̇ is determined as a functions of gains, states, known parameters ̂ and command A CLF in term of Definition 2 can be chosen such that it makes the system in Eq. (18) asymptotically stable with the associated control law. The CLF function in term of Definition 2 is (19) Taking the derivative of Eq. (19) in time and combining with Eq. (18) results in ̇

(

̂

)

̇

̂

(20)

To meet the asymptotically stable condition in term of Theorem 1 for Eq. (20), a control law u called the integratorbackstepping controller can be chosen such that ̂ ̂ ( ) ̇ ̂ where

are the positive gains. By doing so, the required sign condition on ̇


(21) is achieved.

̇

(22)

Thus, there exists a CLF in term of Definition 2 in Eq. (19), state feedback laws in Eqs. (9), (15), and (21), and changes of state transformations in Eqs. (5), (11), and (17), such that its derivative with respect to the system of Eqs. (1)- (4) is negative definite and the system is transformed into a state decoupled linear system as ̇ ̇


̇

(23)

Discussion By examining the system in Eq. (23), the state decoupling problem for the nonlinear system in Eqs. (1)-(4) can be obtained by coordinate transformations and feedback control laws and it is also easy to figure out that the state variable responses in new coordinates of the system (1)-(4) is exponentially globally stable at zeros with positive gains. Therefore, the output response in original coordinate of the system (1)-(4) is exponentially global stable and converges to the tracking command with no overshoots. The desired settling time and rising time of the system are obtained by optimization algorithm.14 Thus, the stability and performance specifications on the system in Eqs. (1)(4) are achieved with the proposed IBSC. By investigation, the output response in original coordinate of the system (1)-(4) is asymptotically rather than exponentially global stable. The exponential stability is only achieved if the initial condition for integrator term in the IBSC is selected properly. Therefore, a question concerning how to select a suitable initial condition for integrator term is still an open topic for study.

Figure 1. Block Diagram of Integrator-Backstepping Control Algorithm An integrator-backstepping control strategy is proposed in Fig. 1. In this approach, a virtual control law for the first subsystem is designed to enforce the output to asymptotically track to the command. For the second subsystem, the virtual control law is considered as command and second virtual control law for the second subsystem is designed to enforce the state to asymptotically track to the virtual command For the complete system, the virtual control law is considered as command and real control law for the complete system is designed to enforce the state to asymptotically track to the virtual command By doing so, the output is logically controlled by the real control input via virtual control laws. Also, an integrator is introduced to implement the proposed algorithm. American Institute of Aeronautics and Astronautics

III.

Application to Flight Path Angle Control

In this section, the nonlinear longitudinal dynamics model is introduced and then some assumptions are given to achieve the standard strict-feedback form for design suitability. The main point of this demonstration is to show the formulation for the IBSC law to nonlinear longitudinal flight systems. Also, a closed-loop system for flight path angle control is provided in the end of the section. Consider the aircraft longitudinal dynamics depicted in Fig. 2. The mathematical model of the longitudinal motion of an aircraft is of the form below. ̇ ̇ (24)


̇ ̇

(

)

where L=

,

,

Variables and parameters appearing in Eq. (32) include : aircraft velocity, : total mass of aircraft, : drag force, : thrust force, : angle of attack, : flight path angle, : gravity, : lift force, : pitch angle, : pitch rate, : inertial moment of aircraft, : pitch moment, : thrust point offset, : throttle, : pitch control (elevator or horizontal vane), : density of air, S: reference area, lift coefficient, drag coefficient, thrust coefficient.

xB

u

 V T



 xi w Figure 2. Aircraft Model of Longitudinal Dynamics Some assumptions are considered to assist in transforming the aircraft model in Eq. (24) to the structure in Eqs. (1)-(4).  Airspeed of aircraft is constant, i.e. ̇ . ̃  Lift force is a sinusoidal function of the angle of attack or , where ̃ is constant for a designated flight condition.  Thrust is constant for the controller design purpose.  Neglection of wind forces.


With the above assumptions, the mathematical model of longitudinal motion of the aircraft is rewritten in strictfeedback form in as follows ̃ ̇

(25) ̇

̇ where (26)


By using the relationship (27) the mathematical model of the aircraft can be further changed to ̃ ̇

(28) ̇ ̇

Note that the system in Eq. (28) possesses the standard strict-feedback form similar to Eqs. (1)-(3) with the above assumptions. The next step is to design the integrator-backstepping control law for nonlinear longitudinal dynamics of the aircraft by using the control design in Section II. The control design goal is to formulate an IBSC law for nonlinear flight dynamics which is required to keep flight path angle of the aircraft at a prescribed reference value or to follow a command value in the presence of the model parameter errors. The control design should satisfy the performance specifications with zero or small overshoot and short settling time, and achieve high precision and stability. By using the design model and control design for strictfeedback form in Section II developed theory, the complete system in Eq. (28) is divided into three subsystems. The first one consists of the first equation of Eq. (28), the second one consists of the first two equations of Eq. (28), and the last one consists of the whole system in Eq. (28). After applying the backstepping method with each subsystem as shown above, the resulting IBSC is proven to possess globally asymptotical stability using the CLF10. The following formulation is the procedure to get the IBSC law with a desired flight path angle of . Step 1: Consider the first subsystem and introduce the modified flight path error signal ̃ where

∫

, and

(

)

(29)

is defined as normal tracking error and

is a positive gains.

Taking the derivative in time of Eq. (29) and combining with the Eq. (28) result in ̃̇ ̃̇

{( ̃

)

(

̇

(

̃

)

)

(̃

)}

(

)

(30)

For the system in Eq. (30), a CLF ̃ in term of Definition 2 can be chosen such that when the virtual control law for is applied, its time derivative becomes negative definite, or mathematically ̃

̃


(31)

By taking the derivative in time of Eq. (31) and combining with the Eq. (30), one finds ̇ ̃

̃ ̃̇

̃{

{( ̃

)

(

̃

)

(̃

)}

(

(32)

)}

By satisfying the asymptotically stable condition in the sense of Lyapunov in for Eq. (32), the virtual control can be chosen with the following logic. ̃

{( ̃

)


{

(

̃

)

(̃

)} (

̃

(

) )

}

(33)

where is the positive gain By doing so, the correct definiteness condition is satisfied. ̇ ̃

̃ ̃̇

̃

̃

(34)

Step 2: By choosing the state feedback to be from Eq. (33) and using the change of state transformation (29), the second subsystem can be rewritten as follows. ̃̇ ̇ The state feedback angle is introduced as

̃

(35)

is considered as command tor the system (35) and the modified tracking error for pitch ̃

(36)

the second subsystem can be rewritten as ̃̇ ̃̇

̃

(37)

̇

is regarded as a control input in the second subsystem (37). So, can be chosen to make the subsystem (37) globally asymptotically stable. A CLF ̃ ̃ can be chosen such that it makes the (37) subsystem asymptotically stable with the virtual control law, i.e. ̃ ̃

(38)

̃

̃

Taking the derivative of Eq. (38) in time and combining with Eq. (37) results in ̇ ( ̃ ̃)

̃ ̃̇

̃

̃

̃

(39) ̇

By satisfying the asymptotically stable condition in the sense of Lyapunov for Eq. (39), a virtual control law be chosen. This control law is ̃ ̇ ̇ where

can

is the positive gain and ̇ is determined below.


(40)

(41) ̇

̃

√

{

̃

(

)

(42)

}

(43)

̃ ̃ ̃ then shows the necessary condition.

The time derivative of


̇ ( ̃ ̃)

̃

̃

̃ ̃

(44)

Step 3: By choosing the state feedbacks (33) and (40) and a change of state transformations (29) and (36), the complete system can be rewritten as ̃̇ ̃̇

̃ ̃

(45)

̇

The state feedback is considered as command tor the system (45) and the modified tracking error for pitch rate is introduced in Eq. (46), i.e. ̃

(46)

the complete system can be rewritten as ̃̇ ̃̇

̃ ̃

̃̇ ̇

(47)

where ̇ is determined below ̇

̇

̇ ̇ ̈

{ ̃ {

̃ ̃

̈ ̇

̈

̈

(48) ̇

(49)

⁄

( (

) )

(51)

} ̇

{

(50)

}

̇

}

Thus, the real control u can be chosen to make the system (47) globally asymptotically stable. A CLF be chosen such that it makes the complete subsystem (47) asymptotically stable with the control law. ( ̃ ̃ ̃)

( ̃ ̃)

̃

Taking the derivative of Eq. (53) in time and combining with the Eq. (47) results in


(52) ( ̃ ̃ ̃) can

(53)

̇ ( ̃ ̃ ̃)

̃

̃

̃ ̃̇

̃

̃

̃

̇

By satisfying the asymptotically stable condition in the sense of Lyapunov for Eq. (54), an IBSC law chosen as ̃

(55)

is the positive gain and ̇ is determined by the Eqs. (48)-(52). This selection leads to ̇ ( ̃ ̃ ̃)


can be

̇ ̇

where

(54)

̃

̃

̃

̃ ̃ ̃

(56)

Thus, there exist a CLF ( ̃ ̃ ̃) in Eq. (53) and the state feedbacks (33), (40), and (55) and a change of state transformations (29), (36), and (46) where the complete system can be transformed as ̃̇ ̃̇ ̃̇

̃ ̃

(57)

̃

By examining the system (57), it is clear that the time response of ̃ of the system is exponentially stable and tends to zero and no overshoot with positive gains. This implies that the flight path angle of the aircraft is with a well-tracking command. Also, the state response ̃ of the system tends exponentially to the origin or the output response of the system is with no overshoot with positive gains but this is only correct if the initial condition for integrator term is selected properly. The desired settling time and rise time of the system are achieved by optimally tuning the gains.10 Thus, the stability and performance specifications of the system (28) are obtained with the IBSC law (55). A block diagram is similar to the one shown in Fig. 1.

IV.

Flight Path Simulation Study

To explore feasibility of the proposed design method, a nonlinear simulation model of the F-16 is selected. The aerodynamic data of the F-16 aircraft model used for numerical simulation is provided in Ref. 13. This data is derived from low-speed static and dynamic wind-tunnel tests at the NASA Langley Research Center. In this research, it is assumed that the aircraft is in level fight at Mach 0.5 and at a height of 25000 ft. Also actuator and sensor dynamics and thrust point offset are not considered in this research. To evaluate the validity of the assumption that lift force is approximately sinusoidal with angle of attack for this model, an analysis of the aerodynamic data is considered here. By curve fitting the aerodynamic data at the indicated flight condition, lift is approximated as ̃ (58) A graph of the actual data and this approximation in Eq. (59) are shown in Fig. 3 in which the solid line shows the approximated lift force and the dashed line shows the real lift force, confirming the validity of the assumption. For other flight conditions, the data can be fit to trim speed and altitude, or ̃

(59)

In order to evaluate the effectiveness of the proposed control algorithm, the closed-loop simulator is tested with three different flight path command profiles away from the trim condition. In the first case, a small command of 5 degree for flight path angle is applied for design model and true model without the effectiveness of disturbances. In the second profile, the reference flight path angle will be put at 5 degree in the presence of wind effectiveness ( ) which has a magnitude less than 5 degree in Fig. 4. Also, the stability robustness of the proposed algorithm is examined in the last case by implementing the simulation via wind effectiveness in Fig. 5 in which wind direction created to aircraft speed is with an angle and wind speed acted on aircraft are with constant. The validation of assumptions for achieving the standard model is verified by comparing the simulation results in both design model and true model. Also, these results were generated with gains c1=1.32 s-1, c2=1.23 s-1, and c3=1.42 s-1 similar the American Institute of Aeronautics and Astronautics

ones in Ref. 10. These specific gain values were computed from an optimization process described further in Ref. 14. 4

8

True Lift vs. Aproximated Lift

x 10

6

Lift (lbf)

True Lift Approximated Lift

2

0

-2

-4 -20

-10

0

10 20 Angle of Attack (deg)

30

40

50

Figure 3 True Lift and Approximated Lift in Attack Angle

The Variation of  wind in time 5 4 3 2

 wind (deg)


4

1 0 -1 -2 -3 -4 -5

0

5

10

15

20

25

Time(s)

Figure 4. Wind Effectiveness on Attack Angle with Random Direction and Magnitude


Variation of  wind in Time 6

5

3

2

1

0

0

5

10

15

20

25

Time(s)

Figure 5. Wind Effectiveness on Attack Angle with Constant Direction and Magnitude

A. Case 1: Step Excitation without Wind Effectiveness Flight Path Angle Response for Step Command of 5 degree 6

5

4 Design Model True Model 3

 (deg)


 wind (deg)

4

2

1

0

-1

0

5

10

15

20

Time (s)

Figure 6. Flight Path Angle Response for Step Command of 5 degree American Institute of Aeronautics and Astronautics

25

Elevator Response for Step Command of 5 degree -1.2

-1.4

-1.6 Design Model True Model

 E (deg)


-1.8

-2

-2.2

-2.4

-2.6

-2.8

-3

0

5

10

15

20

25

Time(s)

Figure 7. Elevator Response for Step Command of 5 degree Figure 6 shows the time responses of flight path angle for an applied command of 5 degree without the presence of wind effective on flight condition. The red line shows time response of flight path angle of the aircraft for standard model. The blue line or dash line represents the ones for true model. Time responses in two models are with no overshoot and 6 second settling time. The small difference in simulation results for two models validates the given assumptions. The outcomes are verified again the prediction by the above theoretical development in Section II. Overall the flight path angle time response of the aircraft is well-behaved in tracking and the performance specifications are achieved with high reliability. However, initial condition on integrator term is computed by try and error method. A small change in initial condition on integrator term may lead to the small overshoot for time response of flight path angle. Thus, the problem for determining exactly with initial conditions should be addressed deeper. Figure 7 shows the time responses of elevator in which the red line shows time response of elevator of the aircraft for design model. The blue line or dash line represents the ones for true model. The response of the elevator simulated with design model has a larger magnitude the ones simulated with the true model. This is due to the approximated lift of the aircraft. B.

Case 2: Step Excitation with Random Wind Effectiveness Figure 8 shows the time responses of flight path angle for an applied command of 5 degree for the true model in two cases. The solid line shows time response of flight path angle of the aircraft with the presence of wind effect with random magnitude shown in Fig. 4. The dash line represents the ones without any disturbance. The results show that the response of flight path angle of the aircraft has small variation around command but it is still acceptable.


Flight Path Angle Response in the Wind Effectiveness 6

5

4 True Model with Wind Effectiveness True Model without Wind Effectiveness

2

1

0

-1

0

5

10

15

20

25

Time (s)

Figure 8. Flight Path Angle Response for Step Command with Wind Effectiveness

Elevator Response for Step Command of 5 degree with Wind Effectiveness -1.2

-1.4

-1.6

True Model without Wind Effectiveness True Model without Wind Effectiveness

-1.8

 E (deg)


 (deg)

3

-2

-2.2

-2.4

-2.6

-2.8

-3

0

5

10

15

20

Time(s)

Figure 9 Elevator Response for Multi-Step Command American Institute of Aeronautics and Astronautics

25

C. Case 3: Step Excitation with Wind Effectiveness from Constant Magnitude and Directional Speed Figure 10 and Figure 11 show the time responses of flight path angle and elevator of the aircraft with single excitation of 5 degree for the true model. In this simulation, the disturbance on attack angle is applied at with constant magnitude of 5 degree as shown in Fig. 5. Flight Path Angle Response in the Constant Speed Wind Effectiveness 6

5

True Model with Wind Effectiveness True Model without Wind Effectiveness

 (deg)

3

2

1

0

-1

0

5

10

15

20

25

Time (s)

Figure 10. Flight Path Angle Response for Multi-Step Command Elevator Response for Constant Speed Wind Effectiveness -1.2

-1.4 True Model without Wind Effectiveness True Model with Wind Effectiveness

-1.6

-1.8

 E (deg)


4

-2

-2.2

-2.4

-2.6

-2.8

-3

0

5

10

15

20

Time(s)

Figure 11. Elevator Response for Multi-Step Command


25

The solid line in Fig. 10 shows time response of flight path angle of the aircraft with the presence of wind effect on angle of attack shown in Fig. 5 and the dash line represents the one without any disturbance. The result shows the time response of flight path angle is recovered in the presence of the disturbance on attack angle with 5 second time. Figure 9 and figure 11 show that the elevator of the aircraft needs to be adjusted in order to track the command. Also, results in Fig. 9 and Fig. 11 show the elevator response lies within the actuator capabilities.13


V.

Conclusions

In this paper, a specific class of nonlinear strict-feedback systems in the presence of model parameter errors is introduced and developed. A systematic formulation for IBSC law for the strict-feedback systems is generated using Lyapunov based design approach. Also, some advantages and issues also discussed to validate and improve the proposed algorithm and then flight path angle control for nonlinear longitudinal dynamics is applied. Some assumptions are made to transform the original model into the standard model for design goal. A control strategy is provided for flight path angle control for longitudinal motion by using the proposed algorithm. From the primary results, some conclusions are made as follows  The proposed algorithm is employed in numerical nonlinear closed-loop simulation. The IBSC law provides with the stability and well-tracking command of for a class of nonlinear strict-feedback systems.  The paper shows also the applicability of the proposed theory for engineering systems with high accuracy and reliability.  The simulation results represent the robust stability of the proposed control strategy in the presence of the model parameter error and disturbance. The limit of the proposed algorithm is to determine the suitable initial condition for integrator in IBSC law. The try and error technique is used to find the initial condition for integrator in the paper. However, the values of initial condition for integrator should be a function of command by the simulation investigation. This issue needs to be a deeper understanding to improve the performances of the algorithm in the future works.

Acknowledgement This work was supported through a grant provided by the Vietnam Education Foundation and the Department of Mechanical and Aerospace Engineering, Old Dominion University.

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