Design of gain schedule fractional PID control for nonlinear thrust vector control missile with uncertainty

ABSTRACT The purpose of this paper is to control the trajectory of the nonlinear missile model in the pitch channel by using Fractional PID controller (FPID) and Gain Schedule Fractional PID controller (GSFPID). FPID and GSFPID with nonlinear missile model are designed where their parameters are tuned by Simulink design optimization in the Matlab toolbox. This optimization method gives the optimal parameters that achieve the best tracking with step unit reference signal. The GSFPID controller compensates the restrictions that represent physical limits of actuators in the pitch channel. The GSFPID with nonlinear missile model is designed in two phases. The first phase is the boost phase where the thrust force is maximized and the second phase is sustain phase where the thrust force is minimized. The equations of motion for nonlinear missile model with FPID and GSFPID are modelled mathematically in the Matlab-Simulink environment. The results of FPID and GSFPID controllers with the nonlinear missile model are presented and compared. The wind effect and the dynamic uncertainties effects are researched and the results are compared. The closed-loop nonlinear system is linearized by the Simulink linear analysis tool at critical operating point t = 5.8 sec and the stability is studied.


Introduction and literature review
The quality requirements for automatic control have increased due to the great complication for plants and the increasingly complicated properties for production. The controllers with a six degree of freedom (6-DOF) missile model in pitch channel will be addressed in this paper. The 6-DOF missile model gives true information about the missile path. The purpose of this paper: (1) To develop a complicated mathematical model of flying path simulation for a missile in pitch channel. This mathematical model is utilized as an algorithm for designing, analysis, and development of the framework to present the missile motion by the Simulink environment to ease the design of the control structure. (2) According to the system features and application environment required, GSFPID control strategy is proposed to solve the missile attitude control problem. GSFPID controller compensates the restrictions that represent physical limits of actuators in the pitch channel. The dynamic uncertainties are studied where the aerodynamic coefficients of the missile are changed to study the robustness [1,2]. The closed-loop nonlinear system is linearized by the Simulink linear analysis tool at critical operating point t = 5.8 sec to study the stability system by the Nyquist diagram and Bode diagram.
In this paper, the autopilot is equipped with thrust vector control that does not depend on the dynamic pressure of the atmosphere. Hence, the thrust force and thrust moment change the motion behaviour of the missile, [3]. The missile system operates in two phases; the first phase is the boost phase where the thrust force is maximized. The second phase is the sustain phase where the thrust force is minimized [1,2]. FPID and GSFPID controllers' gains with nonlinear missile model are tuned by Simulink design optimization.
According to MacKenzie, guidance is defined as the procedure for guiding the path of an object toward a given point [1,4]. Moreover, Charles Stark Draper stated that Guidance depends on main basics and includes vehicles steering in different environments beyond the atmosphere with the gravitational field and in space [1,5]. The most wealthy and ripe for guidance is established with the steering missile. A guided missile is defined as a space-navigating unmanned vehicle that carries within itself the means for controlling its flying route [1,6]. Guided missiles are operated since World War II [1,4]. Today, missile control method involves a wide field of steering laws such as traditional control, optimal control, fuzzy logic and neural network control and differential geometric control rules. The steering missile evolution through and after World War II are found in the following literature [1,[7][8][9]. Moreover, Locke and Westrum placed the evolution of steering missile methods into a great view [1,10,11].
At the end of the nineteeth century, the initial definition of the fractional derivative was introduced by Liouville and Riemann. However, this idea started to be enjoyable for engineers in the late 1960s, especially when it was seen that the characteristic of some systems is more precise when the fractional derivative is utilized. There are increasing numbers of implementation of the fractional calculus such as the use of fractional order controllers, like the FPID (PI λ D δ ) controller [12]. The PI λ D δ controller has both the differentiation and integration of fractional order, which awards the ability to adjust control systems. The FPID controller that is utilized in the path-tracking trouble of the vehicle industry is given in [13]. Fractional order algorithms that are used in the hybrid control of robotic manipulators are observed in [14].
The novelty of the contribution of this paper can be summarized as follows (1) The main contribution is using the gain schedule design with fractional order calculus controller for the first time.
(2) The Simulink design optimization method is used to tune the controller parameters with nonlinear missile model to improve the tracking performance. (3) GSFPID control can achieve superior performance and effectiveness by comparing it with FPID control. (4) The robustness of the proposed control (GSFPID) is validated by simulation results where the wind disturbance effect and dynamic uncertainties will be studied. (5) The stability of the proposed control (GSFPID) is researched by using the Nyquist stability diagram and Bode diagram.
The rest of this paper is organized as follows. Section 2 represents the mathematical model of 6-DOF for the missile. Section 3 displays Gain Schedule FPID controller design for pitch channel of nonlinear missile model. Section 4 presents control implementation and outcomes. Finally, conclusions and future work are discussed in section 5.

Mathematical model of 6-DOF for missile
The 6-DOF equations for nonlinear missile model are divided into depicting kinematics, dynamics (thrust, gravity and aerodynamic), leading guidance descent model, and autopilot (electronics, actuators, and instruments). The inputs of the model are the initial situations, target movement, and target path description. The outputs of the model are the missile flying description (missile velocity and attitude angles).
This paper discusses the autopilot of a missile equipped with thrust vector control (TVC). This method of control does not rely on the dynamic pressure of the atmosphere but it is based on motor thrust. The variation of the thrust vector depends on the control demand. Hence, thrust force and thrust moment change the motion behaviour of the missile. It allows for the variation of the direction of the thrust vector with respect to the symmetry axes of the missile. Thrust vector control has been seen for many missile engines using solid or liquid propellants. The missile can fly in the desired attitude utilizing the thrust vector control [3,15,16]. The essential framing required for mathematical evolutions are the body, velocity and ground coordinates. The origin of these coordinates is the center of gravity (c.g) for the missile. In the ground coordinate system, the X g −Z g is the horizontal plane and the Y g axis complements a standard right-handed system and goes up perpendicularly. In the body coordinate, the positive X b axis corresponds to the center line of the missile and presents the roll-axis. The positive Z b axis is vertical to the X b axis in the horizontal surface and presents the pitch axis. The positive Y b axis goes up and presents the yaw axis. The body axis system is constant with the missile and gets about with it. In the velocity coordinate, X V corresponds to the direction of missile speed (V m ) that is linked to the direction for missile flying. The axis Z V supplements a standard right-handed system [1,2,[17][18][19].
The pitch plane is X-Y surface, the yaw plane is X-Z surface, and the roll plane is Y-Z surface. The ground coordinate and body coordinate are linked to each other through attitude angles ( , , ϒ). The ground coordinate and velocity coordinate are linked to each other through the angles(θ , σ ). The velocity coordinate system is linked to the body frame through the angle of attack (α) in the pitch plane and sideslip angle (β) in the yaw plane. The angles between various coordinate systems are represented in Figure 1 [1,2,17,19,20]. The relationship between the velocity and the body coordinate systems can be obtained as follows: The velocity and body axes system, as well as moments, forces and other quantities are shown in Figure 2.
There are six kinematic equations (three for rotational movement and three for translational movement) and six dynamic equations (three for rotational movement and three for translation movement) for 6-DOF missile. The missile 6-DOF equations are obtained as shown, [1,2,[19][20][21].
= ω y sin(ϒ) + ω z cos(ϒ) (12) . The moments and the forces present on the missile due to gravity, aerodynamic and thrust. These moments and forces are obtained as follows [1,2,17,19,20,22]. In this paper, the attitude control structure is designed in the pitch channel (X-Y plane), the yaw and roll channels are neglected. The nonlinear equations of motion in the pitch channel are presented as follows [23][24][25] The block diagram of pitch angle attitude control is depicted in Figure 3

Gain schedule FPID controller design for pitch channel of nonlinear missile model
The researchers stated that controllers utilize fractional order derivatives and integrals to realize performance and robustness outputs those got with classic (integer order) controllers. The Fractional-order PID controller (FPID) controller is the expansion of the classic PID controller depends on fractional calculus. The theories of fractional calculus are explained in [1].

Basic concepts of FPID controller PI λ D δ
The differential equation of the PI λ D δ controller is depicted in the time domain by: The continuous transfer function of the PIλ D δ controller is given through Laplace transform The FPID controller not only wants to design 3 parameters k p ,k i and k d , but also design 2 fractional orders λ, δ of integral and derivative controllers. The orders λ and δ are not necessarily integers but any real numbers, [1,26].

Optimal tuning FPID control parameters by Simulink design optimization
Tuned FPID controller with actuator restrictions is implemented in the Matlab using Simulink design optimization software that is named the nonlinear control design block set (NCD) [27,28]. This software has characteristics to optimize the design standard in any Simulink model by tuning required parameters that have natural actuation boundaries. Rise time, settling time, overshoot, and saturation limits are design requirements in Simulink response optimization. The gradient descent optimization method is selected to get the optimal FPID parameters. The gradient descent is the simplest method for optimization. The gradient descent method is explained in [3,26,29].
Simulink Design Optimization software gets gains that permit a proper solution with the given restrictions. Once the appropriate signals are constrained with signal limits the adjusted gains are set, and optimization tuning is given, the optimization is ready to run. Simulink Design Optimization software initials by drawing the starting response in blue in the Signal Constraint window. Through the optimization,  Figure 4 displays Simulink Design Optimization procedures [3,28,30].
FPID controller is designed with actuator restrictions so deviation response of the actuator and closedloop system meets the given constraints for tracking. FPID parameters, that are tuned, are kd (derivative gain), ki (integral gain), kp (proportional gain), λ (fractional order of integral gain) and δ (fractional order of derivative gain) [3,26,28,30].

Gain scheduling controller
In many situations, the dynamics of plants are varied with the operating conditions of the process. It is possible to vary the parameters of the controller by seeing the operating conditions of the process. This technique is called gain scheduling. Gain scheduling is simple to process in computer controlled systems. Gain scheduling depends on measurements of procedures of the process and it is the best way to compensate for varying process parameters or unknown nonlinearities. If the familiar definition of the adaptive controller is utilized, Gain schedule is a very helpful procedure for decreasing the effects of parameter variations. There are several commercial process control systems that utilize gain schedule to compensate for dynamic and static nonlinearities. It is possible to decrease the effects of parameter variations by varying the parameters of the controller as functions of the additional variables. Gain scheduling can be got as a feedback control system where the feedback gains are tuned by utilizing feedforward compensation. The essential problem in the design of systems with gain scheduling is to obtain appropriate scheduling variables. This is done by the information basis of the system. When scheduling variables are obtained, the controller parameters are calculated at a number of operating points. The controller is adjusted for each operating point. The stability and performance of the system are obtained by simulation [31][32][33][34][35]. The general block diagram of the gain schedule controller is shown in Figure 5

Control implementation and outcomes
In this section, the path of 6-DOF flight body model in pitch channel is controlled by GSFPID controller. GSF-PID controller is designed where their parameters are tuned by Simulink Design Optimization. The equation of motion for the nonlinear flying body with GSFPID controller is modelled mathematically in the Matlab-Simulink environment. The results of nonlinear flying body model with GSFPID will be given and compared with the results of the nonlinear flying body with FPID [1,11,20,36].

Nonlinear missile model description
Missile solid propellant thrust will be branched into 2 main phases: (1) Boost phase: At time 0-5.8 sec and thrust force is maximized. (2) Sustain phase: At time 5.8-25 sec and thrust force is minimized.
The deflection angle of the pitch actuator (δ α ) is limited with ±22.9°(±0.4 rad). The thrust force curve is represented in Figure 6.
Missile mass will be divided into two types according to variation with time: (1) Constant with time: That includes missile frame, the actuation system, gyroscope and electrical package.  [kg]; t sustain is the sustain time ( (32) where: m 0 is the initial missile body mass and the initial propellant mass [kg]; m is the missile mass and variable propellant mass [kg]. Figure 7 depicts the missile body and variable propellant mass curve.
The aerodynamic force and moment coefficients are usually allocated in the form of graphs obtained by experiments in wind tunnel. Aerodynamic coefficients are represented as a function of Mach number that is varied with missile velocity (V m ). All aerodynamic coefficients are dimensionless. Figure 8 shows the drag force coefficient at zero angle of attack (C x0 ). Drag force coefficient due to the angle of attack (C x ) is shown in Figure 9.
Lateral force coefficient (C y ) is displayed in Figure 10. The aerodynamic damping moment coefficient (m z0 ) is depicted in Figure 11. Figure 12 demonstrates the aerodynamic moment coefficient.

Controller design
The FPID controller has 5 tuning parameter k p , k i , k d , λ and δ which are tuned by Simulink Design Optimization through signal constraint that was explained in     section 3. The two FPID controllers are used in the GSFPID control. The first is tuned with boost phase (0 ≤ t < 5.8 sec) but the second is tuned with sustain phase (5.8 ≤ t < 25 sec). The optimized parameters of the FPID controller are shown in Table 1.
The optimized parameters of GSFPID controller at boost phase (0 ≤ t < 5.8 sec) are seen in Table 2.
The optimized parameters of gain schedule FPID controller at sustain phase (5.8 ≤ t < 25 sec) are seen in Table 3.

Simulation results
In this section, FPID and GSFPID controllers are designed in pitch channel with step unit reference signal where its initial value is 40°(0.698 rad) and its final value at first second is 41°(0.716 rad). Figure 13 gives simulink diagram of nonlinear dynamic system with GSFPID in pitch channel. Figure 14 displays a block diagram of nonlinear dynamic system with GSFPID in pitch Channel. Figure 15 represents pitch angle (plant output) comparison between FPID and GSFPID for nonlinear model. Figure 16 depicts the pitch error (the difference between the step unit reference signal and pitch angle) comparison between the FPID and GSFPID. The error for FPID has high overshoot at time 5.8 sec but the GSFPID does not. Figure 17 displays the pitch error comparison between FPID and GSFPID in boost phase. There is no steady state error for GSFPID. Figure 18 depicts the pitch actuator (plant input) comparison between FPID and GSFPID. Figure 19 displays the pitch actuator comparison between FPID and GSFPID responses at first second in the boost phase.     The down overshoot in GSFPID controller is less than that in FPID controller.
The missile velocity for FPID and GSFPID is given in Figure 20. The angle of attack for FPID and GS FPID controllers are shown in Figure 21

Wind effect
The wind effect is studied where wind velocity (that is presented in Figure 22) is summated to missile velocity (V m ) in the opposite direction and the results are compared. The profile of wind velocity was chosen from the horizontal wind model in Simulink library [28].     Figure 24. Figure 25 displays the pitch actuator action due to the wind effect for FPID and GSFPID controller.

Dynamic uncertainty
In this section, the aerodynamic coefficients uncertainties are studied. The responses of FPID and GSFPID   are compared during the change of the aerodynamic coefficient with 20% and 50%. Figures 26-28 show the pitch angle, pitch error and pitch actuator comparison between FPID and GSFPID up to 20% dynamic uncertainties.
From the above figures, we can conclude that FPID and GSFPID cope with the change in aerodynamic coefficient up to 20%. Figures 29 and 30 show the pitch angle and pitch actuator comparison between FPID and GSFPID up to 50% dynamic uncertainties. GSFPID cope with the change in aerodynamic coefficient up to 50% but FPID cannot because the pitch angle is out of control and   the actuator deflection is saturated during the sustain phase.

Performance and stability
Many issues have to be considered in the analysis and design of control systems. The basic requirements are: (1) Stability (2) Ability to follow reference signals (performance) Figure 30. Pitch actuator comparison with 50% dynamic uncertain.
(3) Reduction of effects of load disturbances (performance) (4) Reduction of effects of model uncertainties (robustness) Instability is the major drawback of feedback. Avoiding instability is thus a primary goal. It is also desirable that the process variable follows the reference signal. The system should also be able to reduce the effects of load disturbances. It must also be considered that the models used to design the control systems are inaccurate. The properties of the process may also change. The control system should be able to cope with moderate changes. In process control, the major emphasis is often on attenuation of load disturbances, while the ability to follow reference signals is the primary concern in motion control systems. In other cases, robustness may be the main requirement [37,38]. Figure 31 illustrates a stable system and an unstable system in Bode diagrams and polar plots [39].
The system is stable in the following cases: • All poles of a linear system with negative real parts (i.e. all poles in the left-hand side in S-plane). • The root locus in the Nyquist diagram of a linear system is not encirclement the point −1 + j0. • The gain and phase margins in the Bode diagram of the linear system are positive.
In this section, the nonlinear closed-loop missile system is linearized by the Simulink linear analysis tool. The step response; Bode diagram; poles and zeros in S-plane; and Nyquist diagram are obtained by the Simulink linear analysis tool to study the performance and stability of the closed-loop linear system. The linearization occurs at the critical operating point t = 5.8 sec where this point separates between boost and sustain phases. Figure 32 depicts the step unit response for the closed-loop linear system.
FPID and GSPID controllers with the linear system can follow step unit response so they give a good  Step unit response for closed-loop linear system. performance but GSPID control gives the best tracking and performance. Figure 33 represents the Bode diagram for the closed-loop linear system.
The Nyquist diagram of the closed-loop linear system is displayed in Figure 34. The poles and zeroes for the closed-loop linear system in S-plane are given in Figure 35.   The closed-loop linear system with FPID and GSF-PID are stable due to the following reasons: • All poles on the left-hand side in S-plane as shown in Figure 35.

Conclusion
In this paper, the flying of 6-DOF missile is simulated in the Simulink environment. FPID and GSFPID gains with nonlinear missile model simulation are tuned by Simulink design optimization. Simulink Design Optimization gives the best tuning parameters for FPID and GSFPID. The responses of FPID and GSFPID are compared. Pitch angle response with FPID and GSF-PID tracks step unit reference signal. GSFPID gives the best tracking where there is no steady state pitch error during boost phase and there is no overshoot at t = 5.8 sec (starting of sustain phase). The pitch actuator with GSFPID is the better response than FPID because the down overshoot at first second in boost phase is smaller than that in FPID controller. GSF-PID gives the best performance because pitch angle tracks pitch step unit response signal with nonlinear missile model and with linear missile model at critical operating point t = 5.8 sec.
The responses with GSFPID are slightly changed due to wind effect. FPID can overcome plant dynamic uncertainties up to 20%. The plant with FPID at 50% dynamic uncertainty is out of control and actuator deflection is saturated during the sustain phase. GSF-PID can cope with plant dynamic uncertainties up to 50%. GSFPID with nonlinear missile model is a robust control because it can overcome the wind effect and plant dynamic uncertainties up to 50%. FPID and GSF-PID with the closed-loop linear system at critical operating point t = 5.8 sec are stable because of all their poles are on the left-hand side in S-plane and it has positive phase and gain margins. Finally, GSFPID controller gives the best performance, stability and the best deflection actuator. GSFPID controller is not affected by wind and it copes with dynamic uncertainties so it is more robustness.

Future work
In the future work, the control can be improved to make a further research. This includes designing a new controller such as Two-Dimensional Dissipative Control for Roesser Model that is presented in appendix A. other controllers can be studied such as model predictive control and L1 adaptive control. GSFPID control can also be applied to other nonlinear processes such as robotic arm control to obtain better controller performance characteristics. The containment control of fractional order with linear time-varying (LTV) system will be investigated in future work. The control parameter can be tuned by using other optimization algorithms, like genetic algorithm, ant colony algorithm or immune algorithm to improve the control performance.
The proposed controller can be implemented in the real system such as the system-on-a-programmablechip (SoPC) based hardware to achieve the online real-time control. SoPC is embedded with many logic gates, flip-flops, on-chip memory, and silicon intellectual property (SIP) for faster development of systems and lower cost of devices. Typically, this is realized in field programmable gate array (FPGA). FPGA is popular in industrial applications for its low cost, high speed, small size, and faster execution speed [40].