On the Performance of an Intelligently Tuned Fractional Order PID Roll Controller
S. Zaker1,* - S. Seyedtabaii 2
1 Elec. Eng. Dept., Shahed University, Tehran, Iran
2 Elec. Eng. Dept., Shahed University, Tehran, Iran ABSTRACT
Due to economic reasons, mostly just a rough model for Unmanned Aerial Vehicle (UAV) systems is available for control design. Besides, its low weight makes it more vulnerable to wind gust and environmental disturbances.
Hence, design of robust controls for UAV’s has gained more attention. In this respect, fractional order PID (FOPID) controllers as a robust control for roll stabilization of UAV is studied. Fractional order controllers give more flexibility in design than what traditional integer order controllers provide. For optimizing the controller performance, the parameters of FOPID are tuned using evolutionary algorithms. The proposed control method is simulated using a fixed- wing UAV mathematical model on MATLAB platform. There is drastic improvement in the performance of the system in facing 20% parameters tolerance, wind gust disturbance, payload variation and command following. The simulation results confirm the superiority of the design versus the performance of the well-tuned basic PID.
Keywords: UAV, Fractional Order Control, Intelligent Tuning, Wind Disturbance, GA, PSO.
1. INTRODUCTION
In recent years, there has been growing demand for autonomous unmanned aircraft equipped with autonomous control devices called unmanned aerial vehicles (UAVs) and micro aerial vehicles (MAVs) [1]. UAV has several basic advantages over manned systems including increased maneuverability, reduced cost, reduced radar signatures, longer endurance, and less risk to human life [2], so they are good candidates for replacement in both military and civilian applications.
One of the most crucial problems with UAVs for doing their tasks is to be stable under disturbances and tolerate dynamical uncertainties. Often, control is divided into two different relatively uncoupled scenarios, namely, horizontal lateral and longitudinal control [3]. Most commercial autopilots use traditional PID controllers because of its simplicity and effectiveness [4]. An application of PID in flight control has been reported in [5] that shows PID has the capacity of containing wind gusts. In [6] an indirect adaptive fuzzy controller is applied. A robust H∞ controller designed for controlling the lateral motion of UAV has been addressed in [7]. Adaptive neuro-fuzzy inference system (ANFIS) for UAV flight control has been investigated in [8] which it is resulted in instability in some flight conditions.
In comparison with integer order controllers, fractional order controllers provide more flexibilities in adjusting the gain and phase margins. Hence, these capabilities make fractional order control a powerful tool in designing robust control system with less controller parameters to be tuned [9]. In [10] a PIλ flight controller is designed and implemented on a roll channel of a fixed-wing UAV and it is shown that the controller outperform the PID. In [11]
preference of (PI)λ to PIλ is demonstrated.
In this paper, the design of a FOPID controller for a fixed-wing UAV roll control is discussed. The controller parameters are tuned using genetic algorithm (GA) and particle swarm optimization (PSO). The controller is tested against disturbances like winds, payload variation and parameter uncertainties. The quality of the system behavior is compared to the system equipped with well-tuned integer order PID controllers.
The paper is organized as follows. The preliminaries of UAV flight control basics are discussed in Section 2. In Section 3, the basics of genetic algorithm and particle swarm optimization are elaborated. In Section 4, design of controllers among them the proposed FOPID controller are discussed. Simulation results are portrayed in Section 5 and lastly conclusions come in Section 6.
2. UAV DYNAMICAL EQUATIONS
A typical UAV has been depicted in Fig. 1. The position control of UAV is governed by the control of the angle of roll (ϕ), pitch (θ) and yaw (ψ). The control surfaces of a fixed-wing UAV may include some or all of the followings:
Ailerons: to control the roll angle.
Elevator: to control the pitch angle (up and down).
Rudder: to control the yaw angle (left and right).
The state variable of a UAV include:
pn, pe and pd: the inertial (north, east) position and the altitude or the height.
vn, ve and vd: the speed with respect to the ground coordinate frame.
u, v and w: the velocities measured along body x, y and z axes.
ax, ay and az: the accelerations measured along body x, y and z axes.
ϕ, θ and ψ: the roll, pitch, and yaw angles.
p, q and r: the angular rates measured along body x, y and z axes.
Va, α and β: the airspeed, the angle of attack, and the sideslip angle.
Inertial coordinate
system
y x
z
Body-fixed coordinate system
xB
yB
zB
R Q
P T
U
V
W
Fig. 1. Body and earth frames coordinate of an Unmanned Aerial Vehicle (UAV)
Actually, small fixed-wing UAV’s motion are highly dynamical and nonlinear because of uncertainties caused by speed, altitude, weights, winds, and turbulences [12]. The dynamic model of three attitude channels, roll, pitch and yaw of UAV are given as follows [13]:
tan cos sin
cos sin
cos sin
cos
p r q
q r
r q
2 2 2 1
2 2
2 2
2
z xz x z y z xz y z x z xz
x z xz
y
xz x x xz y x xz y x z
x z xz
p I L I N I I I pq qr I I I I I I I
M pr I I I p r
q I
I L I N pq I I I I qrI I I I
r
I I I
where L, M and N represent the resultant moment components on body axis:
2
2
0
2
1 2 1 2 1 2
l l l a a l r r lr lp
m m m e e mq
n n n a a n r r nr np
L V S b C C C C C r C p
M V S c C C C C C q
N V S b C C C C C r C p
Where ρ is the atmosphere density (height dependent); V is airspeed, Sω, b, c represent wing area, span, and mean aerodynamic chord, respectively. It is obvious that the attitude dynamical model of UAV is nonlinear and there are strong coupling among three channels.
The state variables and controls involved in roll control are,
, , , , ,
T lat
T
lat a r
x v p r u
where its linearized (around the trim points) state space equation is as below:
* *
* * * * * * * *
* * * * * * * *
cos cos 0
0 0
0 0
0 1 cos tan cos tan sin tan 0 0 0
0 0 cos sec cos sec sin sec 0 0 0
a r
a r
a r
v p r
v p r
v p r
Y Y
v Y Y Y g v
L L
p
p L L L
r N N N r N N
q r
p r
a r
(1)
3. INTELLIGENT SEARCH METHODS
Metaheuristic optimization algorithms have become a popular choice for solving complex problems which are otherwise difficult to solve by traditional methods [14]. In this section we briefly describe the methods discussed in this paper. In this case, the methods of study are, genetic algorithm (GA) and particle swarm optimization (PSO). The following subsections briefly describe the basic theory of each method in its original form.
3.1 Genetic Algorithm (GA)
The GA is a population based nondeterministic optimization method that was developed by John Holland in the 1960s and first published in 1975. Based on the genetic theory of Darwin evolution, the GA simulates the evolution of a population of solutions to optimize a problem. Similarly to living organisms adapting to their environment over the generations, the solutions in the GA adapt to a fitness function over an iterative process using biology-like operators such as the crossovers of chromosomes, the mutations of genes and the inversions of genes. In recent years, the GA has been used for a wide range of applications, and in this paper we use this algorithm for tuning the fractional order parameters. Outline of basic genetic algorithm is given in the following steps:
1. Generate random population of n chromosomes (suitable solutions for the problem), 2. Evaluate the fitness f(x) of each chromosome x in the population,
3. Create a new population by repeating following steps until the new population is complete,
[Selection] Select two parent chromosomes from a population according to their fitness (the better fitness, the bigger chance to be selected),
[Crossover] With a crossover probability cross over the parents to form a new offspring (children). If no crossover was performed, offspring is an exact copy of parents,
[Mutation] With a mutation probability mutate new offspring at each locus (position in chromosome).
[Accepting] Place new offspring in a new population, 4. Use new generated population for a further run of algorithm,
5. If the end condition is satisfied, stop, and return the best solution in current population, 6. Go to step 2.
3.2 Particle Swarm Optimization (PSO)
A special approach of swarm intelligence based on simplified simulations of animals' social behaviors, such as fish schooling and bird flocking, is the particle swarm optimization (PSO) algorithm. PSO is a self-adaptive search optimization. The goal of particle swarm optimization is to solve the computationally hard optimization problems, where it is a robust optimization technique based on the movement and intelligence of swarms and applied successfully to a wide variety of search and optimization problems. It was inspired from the swarms in nature such as swarms of birds, fish, etc. The PSO developed in 1995 by James Kennedy and Russ Eberhart. The algorithm adopted uses a set of particles flying over a search space to locate a global optimum, where a swarm of n particles communicate either directly or indirectly with one another using search directions. In each iteration of PSO, each particle updates its position based on three components, by determines its velocity using, previous velocity, best previous position, and the best previous position of its neighborhood. The outline of basic particle swarm optimizer is as follows:
1. Initialize the population,
2. Evaluate fitness of the individual particle (update pbest), 3. Keep track of the individual's highest fitness (gbest), 4. Modify velocities based on pbest and gbest,
5. Update the particle position, 6. Terminate if the condition is met, 7.
The basic concept of PSO lies in accelerating each particle toward the best position found by it so far (pbest) and the global best position (gbest) obtained so far by any particle, with a random weighted acceleration at each time step, this is done by following equations:
1 1 2
1 1
0,1 0,1
t t t t
t t t
v w v C rand pbest x C rand gbest x
x x v
where: (gbest) = Global Best Position, (pbest) = Self Best Position,
C1 and C2 = Acceleration Coefficients, w = Inertial Weight.
Once the particle computes the new xt it then evaluates its new location. If fitness (xt) is better than fitness (pbest), then pbest = xt and fitness (pbest) = fitness (xt), in the end of iteration the fitness (gbest) = the better fitness (pbest), and gbest
= pbest.
4. INTELLIGENT CONTROLLER DESIGN
The understudy system is the Aerosonde UAV detailed in [15] with the aerodynamic coefficients given in [16].
Based on (1), the aileron-roll system transfer function is calculated as below,
5 3 4 2 3 4 2 14( ) 131.7 1784 3.057 *10 1.093*10
( ) 26.78 360.6 2981 43.16
a
s s s s
G s s s s s s s
For this system, a robust fractional order PID (FOPID) controller is designed and its parameters are intelligently tuned by the described methods. Besides, for the sake of comparison, a conventional PID (IOPID) controller is also worked out.
4.1 CHR PID Design
One of the design procedure for PID controller is based on the Chien, Hrones and Reswick (CHR) method which is developed for first order plus time delay (FOPTD) systems,
1K Ls
P s e
Ts
Considering the following FOPID approximation for the UAV system,
708.29 0.03471 68.948
P s e s
s
The controller parameters are given by,
1 1
( ) 1 3.36689 1 0.01388
0.08328
CCHR c d
i
G s K T s s
T s s
4.2 Intelligent PID Design
On the other hand, the parameters of the PID controller may be tuned using optimization methods. The PID control with the following expression,
PID( )
i
C p d
G s K K K s
s
has Kp, Ki and Kd parameters. For optimal design, the following objective function is introduced,
0
JITAE t e t dt
which is an integral-time absolute error (ITAE) minimum time criteria. By using ITAE as a cost function, GA and PSO are implemented to find appropriate parameters for PID those minimizing the cost function. To do this, the GA and PSO MATLAB programs are linked to the Simulink program, simulating the roll channel control. Upon the execution of the optimizations, the controller parameters depicted in Table 2 are derived. The algorithms initial settings have been:
npop = 30, Pcr = 0.7, Pm = 0.2 and maximum number of iteration 30 for GA, and for PSO we use C1 = C2 = 1.4962, w = 0.7298 for inertial weight, npop = 30 and maximum number of iteration 30.
Table 2. Parameters of the PID controller obtained by GA and PSO algorithms Minimum cost
Kd
Ki
Kp
Algorithm
0.0345 0.169
19.906 6.538
GA
0.0344 0.111
15.597 5.568
PSO 4.3 Intelligent FOPID Controller Design
The FOPID or PIλDμ controller has the following form of transfer function:
( ) i
c p d
G K
s K K s
s
(2)
It has five parameters: the proportional gain Kp, the integral gain Ki, the differential gain Kd, the integral order λ and the differential order μ. As Fig. 2 indicates, the integer type of PID control families are just special cases of the fractional order PID controller with λ=1 and μ=1. Hence, FOPID has two more degrees of freedom to be used in control design or in other word, the FOPID controller is just the generalized integer order PID controller.
λ μ
PID controller
PI controller PD
controller
λ=1 μ=1
P controller
0
Fig. 2. Fractional order PID controller
The 5 parameters of the controller (2) make the design as a multidimensional function optimization problem. This can be conducted using GA and/or PSO and so on. The tuning is performed similar to what pursued with the integer order PID controller. Fig. 3 shows the block diagram of the system used for the optimization.
UAV DYNAMICS FOPID
CONTROLLER ITAE CRITERIA
INTELLIGENT OPTIMIZATION
y(t)
r(t) e(t)
+ -
u(t)
Kp Ki Kd λ µ
Fig. 3. Block diagram representation of tuning controller
The optimization initials are npop = 30, Pcr = 0.7, Pm = 0.2 and maximum number of iteration 30 for GA, and for PSO parameters we use C1 = C2 = 1.4962, w = 0.7298, npop = 30 and maximum number of iteration 30. The result of optimization has been depicted in Table 3.
Table 3. Parameters of the FOPID controller obtained by GA and PSO algorithms Minimum cost μ
λ Kd
Ki
Kp
Algorithm
0.0188 0.4648
0.166 3.329
3.402 14.822
GA
0.0157 0.8781
0.0609 0.716
3.244 18.458
PSO 5. SIMULATION RESULTS
In this section the designs are evaluated using several test such as: reference following, disturbance rejection, payload variation and uncertainty.
5.1 Reference Following
The reference following behavior of the system using the various controllers has been shown in Fig. 4. The assigned command is 15° roll.
Fig. 4. Performance of the designed controllers
As the graphs indicate the best performance belongs to the FOPID controller designed using PSO. The GA tuned FOPID resides in the second place higher than the three version of the conventional PID controllers. Due to the unacceptable performance of CHR, it is not considered for other tests.
5.2 Disturbance Rejection
Second experiment is concerned with the robustness issues. Wind gust is a pretty common and nontrivial disturbance to the flight control systems. Especially for small or micro UAVs, the wind gust can even cause crashes if the controller is not well designed. So the designed controllers are tested under extreme conditions when the wind gust arrives 20 m/s for 0.25 second, while the UAV is in reference following condition in 30° roll angle. The results are shown in Fig. 5:
It is concluded from Fig. 5 that the intelligently tuned FOPID designed controllers has better performance than PID ones in extreme wind gust conditions and so the FOPID controllers have better robustness whenever a disturbance occurs.
5.3 Payload Variation
Payload variation is also a major issue for small and micro UAVs since the payload can have a big impact on the flight performance. A controller robust to the payload variations could save the UAV end users a lot of time while changing different payloads. So in the simulation we increase the mass by 30%. The results are shown in Fig. 6:
1 1.5 2 2.5 3
0 5 10 15 20 25
Time (sec.)
Roll (deg.)
Reference PIDCHR PID (GA) PID (PSO) FOPID (GA) FOPID (PSO)
Fig. 5. Wind gust disturbance effect
Fig. 6. Payload variation effect
5.4 Parameter Uncertainty
Another experiment that is done, is the impact of parameter uncertainties on the designed controllers. To do so, a 20 % change in the parameters of the dynamical equations is applied. A sample of behavior has been shown in Fig. 7.
There are changes in the response, however, still FOPID has secured its position at top.
1 2 3 4 5 6 7 8
0 10 20 30 40
Time (sec.)
Roll (deg.)
Reference PID (GA) PID (PSO) FOPID (GA) FOPID (PSO)
1 2 3 4
0 10 20 30 40 50
Time (sec.)
Roll (deg.)
Reference PID (GA) PID (PSO) FOPID (GA) FOPID (PSO)
1 1.5 2 2.5 3
0 10 20 30 40 50 60 70 80 90
Time (sec.)
Roll (deg.)
Reference PID (GA) PID (PSO) FOPID (GA) FOPID (PSO)
As it is expected, the intelligent FOPID controllers have better response in the presence of parameter uncertainties condition than PID ones.
Finally we consider the effect of wind disturbance that the gust portion is modeled by the standard Dryden gust model [15] and simultaneously random parameter uncertainty in the range ±20% was applied to the simulation model and the results of roll performance with the designed controllers are plotted in Fig. 8.
Fig. 8. Random parameter uncertainty and Dryden wind gust
It can be seen from the simulation results that the main advantages of the fractional order controller are robustness of the system whenever a disturbance occurs and in case of the uncertainty in the parameters.
6. CONCLSION
In this paper, an intelligently tuned fractional order PID controller is developed and applied to improve the roll control performance of a small fixed-wing UAV. The design of the parameters is conducted using GA and PSO methods of intelligent optimization. Besides, for the matters of comparison, three PID controllers are designed. One use CHR method and the other two again by using GA and PSO as optimization tools for acquiring best performances. The flight performances are compared in cases of wind gust disturbance and parameter uncertainties conditions. Simulation results show the effectiveness and superiority of the proposed design versus the competing algorithms in containing the unwanted phenomena.
ACKNOWLEDGEMENT
This work has been partially supported by the research department of Shahed University.
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