Particle swarm optimization algorithm reinforced fuzzy proportional–integral–derivative for a quadrotor attitude control

Abstract

In this article, the particle swarm optimization algorithm is presented using an improved stochastic variant strategy to optimize the gains of the fuzzy proportional–integral–derivative controller. The particle swarm optimization algorithm aims to renew the elite parameters for the schemes of fuzzy proportional–integral–derivative controls. The integral of time multiplied by absolute error criterion is used to estimate the system performance due to saving the setting and operation time. Facing nonlinear quadrotor attitude control models, the results demonstrate the proposed scheme with better performance.

Introduction

Particle swarm optimization (PSO), based on stochastic and population-based adaptive optimization and swarm intelligence technique, was presented by Kennedy and Eberhart, in 1995. Originally, since the PSO is only with few parameters able to be adjusted, it seems extremely effortless. However, its computing speed becomes better and with high accuracy. It also costs smaller memory size by comparison with other learning algorithms such as neural network and genetic algorithms. Several approaches^1–6 indicate that PSO performance will heavily depend on every PSO parameter. PSO has applied and achieved satisfied results in many fields such as in biped robot, ⁶ in electrical power energy systems, ⁴ in milling machine, ⁷ and in gun position control. ³

The unmanned aerial vehicles (UAVs), in the last decade have received a strong interest and exponential growth in military surveillance, civilian information search, and rescue operations. The current trends are to design the small flying vehicles capable of hovering maneuvers without or less oscillation as well as forward flight.^8–10 In such type, the significant represent is quadrotor that has four symmetric rotorcrafts in each couple.^8,9,11

The proportional–integral–derivative (PID) controllers^{3,8,9,12–15} are of low cost, uncomplicated structure, and easy design, which are applied to many hi-tech fields, from industrial machine to biped robot, and especially in UAV systems. However, the controller methods such as PID, backstepping, and fuzzy PID approaches for the UAVs^10,11,16 have problems in practical applications for better performance, robustness, and less oscillation.

Following growing tendency, this article copes with the fuzzy PID controller gains design. The parameter control gains is evaluated by the PSO algorithm and related to the indices of the system performance—the fitness function, the integral of time multiplied by absolute error (ITAE). ¹² This algorithm is exploited to optimize fuzzy PID controller gains and aims to solve the complex quadrotor attitude control systems.

System description

PSO algorithm

The fundamental idea of the PSO algorithm is originally generated by imitating the social behavior from animals just like the fish shoal, the bird flock, and the swarm.^1–6 PSO represents an optimization process that searching and updating the population based on best moving from an initial random. There are two vectors of position and velocity belonging to every particle. The particle velocity is computed from all particles to the best position by iterations, in which the best position is determined by the associated particles over the path of multiple generations. It assumes the searching space as Dth dimension and with m particles.

The ith particle in the searching space is located at X_i(t) = [x_i1(t), x_i2(t), …, x_iD(t)], and the position in each particle is a potential result. The velocity of the ith particle is given by V_i(t) = [v_i1(t), v_i2(t), …, v_iD(t)].

For each generation, the velocity vector of each particle needs to be adjusted by the best position P_i(t) which is the best particle from the neighbors P_g(t) at time t, where P_g(t) = [p_g1(t), p_g2(t), …, p_gD(t)]. From the previous works,^1–6 the PSO algorithm is verified as equation (1)

$$\{\begin{array}{c}{v}_{\mathrm{iD}}(t+1)=\omega \times v_{\mathrm{iD}}(t)\hfill \\ \multicolumn{1}{c}{+c_1\times \mathrm{ran}{d}_1(·)\times (p_{\mathrm{iD}}(t)-x_{\mathrm{iD}}(t))}\\ \multicolumn{1}{c}{+c_2\times \mathrm{ran}{d}_2(·)\times (p_{\mathrm{gD}}(t)-x_{\mathrm{iD}}(t))}\\ \multicolumn{1}{c}{x_{\mathrm{iD}}(t+1)=x_{\mathrm{iD}}(t)+v_{\mathrm{iD}}(t+1)}\end{array}$$

(1)

where v_iD indicates the velocity of the ith particle; p_iD is the best historical position of the ith particle, while the global best position is p_gD; and x_iD is the position of the ith particle with objective fitness value. The weighting ω is used to manage the values of v_iD related to the current value in learning rates c₁ and c₂.

In equation (1), the items rand₁(.) and rand₂(.) are random numbers between 0 and 1. The term c₁ × rand₁(.) × (p_iD(t) − x_iD(t)) mentions the cognitive component which returns the distance at the location where the solution P_i(t) is best. In contrast, the term of c₂ × rand₂(.) × (p_gD(t) − x_iD(t)) indicates the social component which reflects the distance between a particle and the best solution found by its neighbors P_g(t). Based on its earlier velocity, equation (1) is used to compute the particle’s new velocity, the distance between its current and the own best previous position, then the collaborative interaction between the particles.^1,2,17 This stage features cooperation among all particles by sharing information. The particle updates its position in final.

A set of PID parameter gains [k_P, k_I, k_D] and fuzzy scaling factor gains [k_e(t), k_de(t)] are represented in each individual, which starts by randomly generating an initial population of pop individuals. The optimization problem is found out by a group of $[k_P,k_I,k_D,k_{e(t)},k_{\mathrm{de}(t)}]\in f(E)$ such that the performance index of f(E) is the minima system. After the parameters of f(E) for all particles are determined, a group of solutions of each PSO particle is got.

Controllers design

The simplified PID controllers, whose key functions are to improve the dynamics and to reduce and/or eliminate the steady-state error, are extensively being used in industry, nowadays. By combining a fast learning scheme with PID parameters for high-grade solutions, an enhancement by fuzzy PID controllers is proposed.^3,13,17

As shown in Figure 1, the membership functions of the tracking error e(t) and the differential tracking error de(t) are introduced in the fuzzy inference system. The membership function of the output $e_{\mathrm{Fuzzy}}(t)$ is also adopting the triangular form for the computation convenience.

Figure 1.

Fuzzification interface: input e and de, output CI.

The PSO algorithm is adopted to discover the PID controller parameters, the outputs of the proposed fuzzy logic controllers (FLC), $k_P,k_I,k_D\in [0,50]$ . The reference input values for $e(t)$ and $\mathrm{de}(t)$ of $f(E)$ fitness functions made by PID gains are verified along with $f(E)\in [f_{min}(E),f_{max}(E)]\Rightarrow f(E)\in [0,100]$ .

The rules of FLC are able to be recognized by the mapping of the inputs e(t) and de(t) onto the output e_Fuzzy(t).^10,14–18 The strategy aims to bound the error into a stable sliding range; $k_{e(t)}$ and $k_{\mathrm{de}(t)}$ are sent to the fuzzy universe of discourse with the subsets ${\mu }_{e(t)}:U_{e(t)}\to [-k_{e(t)},k_{e(t)}]$ and ${\mu }_{\mathrm{de}(t)}:U_{\mathrm{de}(t)}\to [-k_{\mathrm{de}(t)},k_{\mathrm{de}(t)}]$ .

The FLC dynamic behavior, which is based on expert knowledge, is described by a set of linguistic rules. Mandani’s MIN–MAX inference engine type and center of area (COA) method defuzzification are employed. The error range is limited by sliding range zero error (ZE), and the PID controller gains are regulated by optimal system condition.

The basic implementation parameters of the fuzzy control algorithm, which have seven partitions, are decomposed: negative big (NB), negative medium (NM), negative small (NS), ZE, positive small (PS), positive medium (PM), and positive big (PB). The fuzzy control rules are then designed as shown in Table 1.

Table 1.

The fuzzy rule table.

CI(t)		e(t)
		NB	NM	NS	ZE	PS	PM	PB
de(t)	NB	ZE	NS	NM	NB	NB	NB	NB
	NM	PS	ZE	NS	NM	NB	NB	NB
	NS	PM	PS	ZE	NS	NM	NB	NB
	ZE	PB	PM	PS	ZE	NS	NM	NB
	PS	PB	PB	PM	PS	ZE	NS	NM
	PM	PB	PB	PB	PM	PS	ZE	NS
	PB	PB	PB	PB	PB	PM	PS	ZE

NB: negative big; NM: negative medium; NS: negative small; ZE: zero error; PS: positive small; PM: positive medium; PB: positive big.

In this article, there are four items of the system response as the rise time, settling time, peak time, and minimum overshoot were integrated to the fitness function of ITAE

$$\begin{array}{c}\multicolumn{1}{c}{f(E)={\lambda }_1\times \mathrm{Rise}\mathrm{time}}\\ +{\lambda }_2\times \mathrm{Settling}\mathrm{time}\hfill \\ +{\lambda }_3\times \mathrm{Peak}\mathrm{time}\hfill \\ +{\lambda }_4\times (|r-\mathrm{Overshoot}|)\hfill \end{array}$$

(2)

In equation (2), λ_i indicates the ith weighting. The output performance indices such as the rising time, the settling time, the peak time, and the maximum overshoot are weighed to define the fitness. For experiences, the weights could be chosen by [λ₁, λ₂, λ₃, λ₄] = [35, 25, 10, 5]. It raises output error performance, which is indicated by $f(E)$ . It will also increase the rate of decision of the PSO system to reject this particular particle group. Then, the particle groups, which contain a large concealed error, can be eliminated. Thus, the convergence speed is also accelerated.

To adapt the PSO algorithm, optimal control gains in quadrotor attitude system are to achieve an optimal compromise between the system stability and the easier flight pilot. Figure 2 shows the proposed controller. The system performance utilizes the fitness function ITAE as below

$$\mathrm{ITAE}=\underset{0}{\overset{\infty }{\int }}t·|e(t)|·\mathrm{dt}$$

(3)

Figure 2.

The diagram of PSO reinforced PID quadrotor attitude control.

Numerical simulation results

The transfer functions attitude model of quadrotor, which referred to Bouabdallah et al.,^8,9 is applied to proposed controllers:

• Roll channel (lateral control): $\varphi (s)=\frac{0.522}{0.004s^4+0.039s^3+0.009s^2}$

• Pitch channel (longitudinal control): $\theta (s)=\frac{0.522}{0.004s^4+0.039s^3+0.009s^2}$

• Yaw channel (pedal control): $\psi (s)=\frac{21.78}{0.008s^4+0.077s^3+0.18s^2}$

In the simulation results, the numbers of iterations for each control pilot channel are chosen as 30, 50, and 20, respectively. The parameters for tuning fuzzy PID controller gains are chosen as follows: the weighting factor [λ₁, λ₂, λ₃, λ₄] = [35, 25, 10, 5], fuzzy scaling factor gains are $k_{e(t)}=1$ and $k_{\mathrm{de}(t)}=1$ , and PID gains are in range k_PID∈ [0, 50]. The attitude control performances are displayed in each channel: roll angle is set as 1 rad (57.3°), pitch angle is set as 0.5 rad (28.65°), and yaw angle is set as 1 rad (57.3°) (Figures 3–5).

Figure 3.

Roll channel control: (a) roll angle response, (b) PID parameters in PSO convergence, (c) criterion of fuzzy gains, and (d) the fitness function.

Figure 4.

Pitch channel control: (a) pitch angle response, (b) PID parameters in PSO convergence, (c) criterion of fuzzy gains, and (d) the fitness function.

Figure 5.

Yaw channel control: (a) yaw angle response, (b) PID parameters in PSO convergence, (c) criterion of fuzzy gains, and (d) the fitness function.

The tracking results make obviously that the proposed controllers have the preeminent pilot as its performance reduced and/or eliminate the damping, overshoot and oscillation at all while the response is especially fast. All results have the maximum response time as just 0.6 s. It is the high effectiveness when compared to the reference results of Bouabdallah et al.,^8,9 which just obtain the response time after the third second.

Conclusion

In this article, the proposed PSO algorithm, which rapidly generates and updates the new elite control parameter gains, is used in reinforcement of fuzzy PID controllers. It is then implemented to control quadrotor attitude: roll–pitch–yaw channels. The simulation results demonstrate that the proposed controller really dominates the best performance in saving the settling time, reliability, stability, and less oscillation. Besides, the benefit of the proposed controller design scheme for quadrotor UAV pilot control is achieved.