INTELLIGENT TRACKING CONTROL USING PSO-BASED INTERVAL TYPE-2 FUZZY LOGIC FOR A MIMO MANEUVERING SYSTEM

Abstract: Air vehicle modeling like the helicopter is very challenging assignment because of the highly nonlinear effects, effective cross-coupling between its axes, and the uncertainties and complexity in its aerodynamics. The Twin Rotor Mutli-Input Multi-Output System (TRMS) represents in its behavior a helicopter. TRMS has been widely used as an apparatus in Laboratories for experiments of control applications. The system consists of two degrees of freedom (DOF) model; that is yawing and pitching. This paper discusses the design of Four Interval Type-2 fuzzy logic controllers (IT2FLC) for yaw and pitch axes and their cross-couplings of a twin rotor MIMO system. The objectives of the designed controllers are to maintain the TRMS position within the predefined desired trajectories when exposed to changes during its maneuver. This must be achieved under uncertain or unknown dynamics of the system and due to external disturbances applied on the yaw and pitch angles. The coupling effects are determined as the uncertainties in the nonlinear TRMS. A PSO algorithm is used to tune the Inputs and output gains of the four Proportional-Derivative (PD) Like IT2FLCs to enhance the tracking characteristics of the TRMS model. Simulation results show the substantial enhancement in the performance using PSOBased Interval Type-2 fuzzy logic controllers compared with that of using IT2FLCs only. The maximum percentage of enhancements reaches about 33% and the average percentage of enhancements is about 17.1%. They also show the proposed controller effectiveness improving time domain characteristics and the simplicity of the controllers.


INTRODUCTION
TRMS, Twin Rotor MIMO System, has been widely used as an apparatus in Laboratories for experiments of control applications [1]. Since the model is of nonlinear type with significant coupling between the two axes (yaw and pitch) and complex aerodynamics, the controlling design using conventional, hybrid and intelligent methods is researchers challenge [2][3][4][5]. Fuzzy Logic Control (FLC) is a technique to control through the investigation and description of model behavior in terms of linguistic variables formalizing the rule base [6]. Different control method strategies combining FLC with conventional controllers (Like PD and PID), Neural Networks, sliding mode control and Self-Tuning algorithm have been used widely to control the axes of TRMS and track the desired trajectories efficiently [7][8][9]. Furthermore, Evolution algorithms like Differential Evolution ( Page 24 2018 Al-Qadisiyah Journal For Engineering Sciences . All rights reserved. (GA), Particle Swarm Optimization (PSO) and Artificial Bee Colony (ABC) have been used to tune the FLC parameters in order to enhance the response of tracking and minimize the steady state error [10][11][12][13]. To eliminate the effect of uncertainty, achieve robustness, and enhance the performance of the controlled system in recent time, an Interval Type 2 FLC (IT2FLC) was introduced as a new generation of Type 1 FLC. The difference in structure, mainly in the defuzzifier block, is the addition of the type reduction block during defuzzification [14]. Different researches have dealt with the use of IT2FLC and adaptive IT2FLC to control the TRMS [14][15][16][17].
In this paper, the design of Four Type-2 fuzzy logic controllers for the yaw and pitch axes with their couplings of a twin rotor MIMO system is discussed. The objectives of the designed controllers are to reduce overshoot and chattering, exist by the effect of external disturbances, in the yaw and pitch angles during when the TRMS system is exposed to changes during its maneuver. A PSO algorithm is used to tune the inputs and output gains of the Proportional-Derivative (PD) Like IT2FLCs to improve the tracking characteristics of the TRMS model. The remainder sections of this paper are as follows: section 1 describes the detailed TRMS model. Section 2 illustrates the structure of the Type-2 and Interval Type-2 Fuzzy Logic Control (IT2FLC). The detailed steps for the design of the four PD-Like Interval Type-2 FLCs and tuning the inputs and output gains of the mentioned controllers using PSO algorithm are explained in section 3 and section 4 respectively. Simulation Results are presented in section 5. Finally, concluding remarks are provided in the section of conclusion.

TWIN ROTOR MIMO SYSTEM MODEL
The helicopter as one of flight vehicles consists of many elastic parts like rotor, control surfaces and engine. This vehicle is acted by nonlinear aerodynamics forces and gravity, and complexity increases because of flexible surfaces structures which make a realistic analysis difficult [16]. To study the control of this aerodynamics model, the TRMS, a Lab. Setup, is designed by Feedback Company for control experiments [1]. The main parts of TRMS are the beam pivoted on its base which rotates in horizontal and vertical planes freely. Two rotors driven by two Direct Current (DC) motors located at such end of the beam. Aerodynamic force through the blades and coupling effect are produced by both motors. This produces non-linear and high order system with cross coupling [18]. However, there are many differences between the TRMS and helicopter. The pivot point location in the helicopter is located in the main rotor head while it is located in midway between two rotors of TRMS. Moreover, the lift generation of vertical axis in helicopter is by collective pitch control while it is generated in TRMS by speed control of the main rotor. Finally, the yaw is controlled in helicopter by pitch angle of tail rotor blades while is controlled in TRMS by tail rotor speed [18]. The setup of TRMS is shown in Figure 1 [9]. The mathematical model of the TRMS consists of electrical and mechanical parts where the electromechanical diagram is depicted in Figure 2 and the TRMS schematic diagram is shown in Figure 3. Figure 2. TRMS electro-mechanical model [1]. model (TRMS) [9]. The horizontal motion of the beam is described with the following equation:

Figure1.Twin rotor MIMO system
where 2 is the tail propeller thrust which is a nonlinear static function of the DC motor momentum and described by: ∅ is the friction forces momentum represented by: is the momentum of cross reaction approximated by: = .
( . +1) . +1 . 1 (4) The electrical circuit with the DC motor is approximated by a transfer function of first or and given in Laplace transform by: where the input voltage of the DC motor is 2 , 2 is the static gain of DC motor and 21 is the main rotor time constant. Moreover, the momentum equations for the vertical movement are described by: where 1 is the main propeller thrust which is a nonlinear static function of the DC motor momentum and described by: is the gravity momentum represented by: = .
( ) (8) The friction forces momentum is described by: ( ) (9) and the gyroscopic momentum is given by: = . 1 .̇. ( ) (10) The electrical circuit with the DC motor is approximated by a first order transfer function and the motor momentum is given in Laplace transform by: . 1 (11) where the input voltage of the DC motor is 1 , 1 is the static gain of DC motor and 11 is the time constant of the main rotor. In this paper, the physical parameters of the TRMS model are listed in table 1 [1].

INTERVAL TYPE-2 FUZZY LOGIC CONTROL (IT2FLC)
Fuzzy sets theory was introduced by Lotfi A. Zadeh in 1965 as a method to describe nonprobabilistic uncertainties. In 1975, the idea of Type-2 FLC (T2FLC) as an expansion of Type-1 FLC (T1FLC) was proposed by Zadeh too. The uncertainties in Fuzzy sets of membership functions (MFs) of T2FLC are in three dimensions while the ones in T1FLC are in two dimensions, that is the typical memberships of Type-2 consists of two Type-1 MFs. Fuzzy memberships in Type-2 have the Footprint Of Uncertainty (FOU) which is a bounded region of a fuzzy set (Ã) that can handle the uncertainties, nonlinearities and linguistics related with inputs and outputs of FLC and reducing them [19]. It represents the union of all primary membership functions, where: FOU(Ã) = U x ∈ J X (12) [21] where Ã is characterized by Type-2 MF u Ã (x, u), where x ⊂ X, X is the universe of discourse and u ∈ J x ⊆ [0, 1], then: (14) where ∬ denotes union over all admissible x and u. The upper and lower membership functions are defined by μ Ã ̅̅̅ (x) x ∈ X and μ Ã (x) x ∈ X respectively, as follows: and The secondary memberships functions (MFs) domain is within [0, 1]. Moreover, the two dimension plane whose axes are u and ̃( , ) is known as the vertical slice of ̃( , ) and represented as follows: where 0 ≤ f x1 (u) ≤ 1 and the secondary membership function is represented by μ Ã (X 1 ). It is the Type-1 fuzzy set where the primary membership function of x 1 is J x1 , It is secondary membership domain where J x1 ⊆ [0, 1] for all x 1 in X. Now, the interval set is defined when the secondary membership function is 1 ( ) = 1 ∈ 1 ⊆ [0, 1]. An Interval Type-2 (IT2) membership function is obtained when this it is true for 1 ∈ . The uniform uncertainty at the primary membership of is represented by secondary MF of Type-2. The membership function A of Type-2 with its secondary memberships is shown in Figure 4 [20].  Type-2 FLC is divided into two types; that is Mamdani type where the output membership functions are fuzzy sets and the Takagi-Sugeno-Kang (TSK) type where the output membership functions are either linear or constants. Figure 5 illustrates the structure of the T2FLC [19]. The difference between Type-1 and Type-2 FLC is in the nature of the membership functions used. The main blocks of T2FLC are [1]: a. Fuzzifier: It makes the inference engine works by crisp inputs into type-2 fuzzy sets mapping. b. Rule base: The difference between the rules in T2FLC and the rules in T1FLC is in the antecedents and consequents that are represented by the interval Type-2 fuzzy sets. c. Inference engine: The fuzzy inputs to fuzzy outputs are assigned in the inference engine block using the operators such as the intersection and union operators and the rule base. d. Type-reduction: Type-reduced sets are the outputs of Type-2 fuzzy sets for the inference engine when converted into fuzzy sets of Type-1. In Interval Type-2 FLC three methods for typereduction operation. That are, Karnik-Mendel (KM) iteration method, Enhanced Karnik-Mendel (EKM) iteration method, and Wu-Mendel Uncertainty Bounds method. In this paper, Modified Karnick Mendel is used to design the controller. It is an enhancement of the original KM algorithm with three improvements. First, reducing the number of iterations, better initialization is used. Second, one unnecessary iteration is removed by changing the termination condition. Third, reducing the cost of computation for each iteration, a subtle computing technique is used. The detailed algorithm is given in table 2 [22]. e. Defuzzification: The input to the defuzzification block is the type-reduction output block. This is done through two steps: first, by transforming the fuzzy sets of Type-2 into the fuzzy sets of Type-1. The left and right end points are used to calculate the type reduction sets. Second, by calculating the average of the points. Crisp value (Type-0) is produced by defuzzify the Type-1 fuzzy generated set using the fuzzy logic control known techniques. The calculations of type-reduction operations are very complex.

Symbol
To simplify calculations, the Interval Type-2 Fuzzy set is used [20]. In this paper, the Centroid method is used to calculate the defuzzified values as follows: y out = y l +y r 2 (18) . nd compute IF `= . stop and 1 = and = ; otherwise. continue.

DESIGN OF PD-LIKE IT2FLC FOR TRMS MODEL
The objective of Fuzzy controllers is to maintain the TRMS position within the pre-defined desired trajectory. This must be achieved under uncertain or unknown dynamics of the system. The MATLAB\Simulink of the PD Like IT2FLCs controlled TRMS system is shown in Figure 6 where the TRMS model explained in section 1 is simulated using MATLAB\Simulink. In order to take the effect of cross coupling between the pitch and yaw channels into consideration, four controllers are designed to control the Pitch (P), Pitch-Yaw (PW), Yaw-Pitch (YP) and Yaw (Y).

Figure 6. Simulink of TRMS controlled by IT2FLCs
Two controlled signals are generated from the outputs of the above controllers to control the pitch and yaw angles. The Pitch channel is controlled signal is generated by summing the outputs of the (P) and (YP) controllers. While the controlled signal of the yaw channel is generated by summing the outputs of the (Y) and (PY) controllers. The inputs to the (P) and (PY) controllers are the error (e(k)) and rate of error which are calculated in discrete time domain as follows: e(k)=ref- (19) e(k)= e(k)-e(k-1) (20) where k is the sampling instant. Moreover, the inputs to the (Y) and (YP) controllers are the error (e(k)) and rate of error in discrete time domain is calculated as follows: e(k)=ref- (21) e(k)= e (k)-e (k-1) (22) The inputs and output scaling factors of the four PD Like IT2FLCs are KP, KDP, KOP, KPY, KDPY, KOPY, KYP, KDYP, KOYP, KY, KDY, and KOY where K is the proportional gain, KD is the derivative gain and KO is the output gain for each controller. These gains will be tuned manually to reach the TRMS position within the pre-defined desired trajectory in the pitch and yaw axes when exposed to changes during its maneuver. This must be achieved under uncertainty due to the axes coupling effects and due to external disturbances that are represented by noise signal. Each controller is of Mamdani type where each input and output has two Trapezoid shaped Type-2 membership functions within the range of (-1.5,1.5) for the inputs and (-1, 1) for the outputs, see

 ANALYSIS OF SATBILITY
The guarantee of robustness and stability of IT2FLC is very big challenge because of the complexity in its structure. A Bounded Input Bounded Output (BIBO) is one of the approaches to realize the stability of IT2FLC [17]. Assume G1 and G2 are representing T2FLC and the controlled plant model respectively, see Figure 8. compared with other evolution algorithms, like Genetic Algorithm. Each population member in PSO algorithm is named as "Particle". Each particle (xi(k)) "files" around the multidimensional search space with a velocity (vi(k)) of that is updated by the own experience of the neighbors of particle in the swarm [24]. In this paper, the PSO algorithm with the constriction coefficient formula instead of weight is used; it's a good method that gives faster convergence ability with minimum number of iterations to reach a goal [24].The inputs and output gains (12 Gains) of the four PD-Like IT2FLCs are tuned to reach the best values depending on minimizing the following objective functions for pitch and yaw angles: (27) It is the overall performance index (PI) of Integral Square of Error (ISE) for the Pitch and Yaw motions, as follows: The minimization of this (PI) means that the TRMS model will follow the desired trajectories in both yaw and pitch motions in spite of the appearances of uncertainties in the model or the disturbances affecting them. The velocity of i th particle will be calculated as: vi(k+1)=w(vi(k)+c1r1(Xpbest i(k) -xi(k)) +c2r2 (Xgbest -xi(k))) (29) [25] where for the i th particle in the k th iteration, (xi) is the position, (Xpbesti) is the previous best position, (Xgbest) is the previous global best position of particles, (c1) and (c2) are the acceleration coefficients namely the cognitive and social scaling parameters, (r1) and (r2) are two random numbers in the range of [0 1] and (w) is a constriction coefficient given by: (30) [25] Where (ϕ =c1+c2, ϕ>4). The convergence of the particle is controlling the constriction coefficient. As a result, it prevents explosion and ensures convergence. A new position of the i th particle is then calculated as: xi(k+1)= xi(k) + vi(k+1) (31) [25] The PSO algorithm is repeated until the goal is achieved.

SIMULATION RESULTS
The physical parameters of the TRMS model simulated in this section are listed in table 1. The PD-Like IT2FLC controlled system has been simulated for 100 seconds with zero initial conditions for both; pitch and yaw angles. In this simulation, the reference signals of Sinusoidal wave and Saw tooth with amplitude of 0.2 rad and frequency of 0.02Hz and step input of 0.2 rad are applied to both angles [16]. To investigate the robustness of both controllers with respect to the measurement noise and parametric variations, a signal noise with is added to the measured variables. The measured signals from sensors are in general subject to noise in spite of the output of systems are measured using adequate sensors [16,17]. In the following simulations, a uniformly distributed random signal with amplitude of (0.01) is added to the measured pitch and yaw signals, see Figure 9. Page 32

AL-QADISIYAH JOURNAL FOR ENGINEERING SCIENCES
2018 Al-Qadisiyah Journal For Engineering Sciences . All rights reserved.  The time responses for the above three reference signals and the actual signals for the pitch and yaw angles without and with applying noise are shown in Figures 10-15. The control signals for controlling the pitch and yaw motors are also shown on the same previous figures.     As illustrated from results and from table 6 that the PSO-Based PD-Like IT2FLCs provide better performance than using PD-Like IT2FLCs. The proposed PSO based controller presents better performance and good convergence in both pitch and yaw channels with smaller oscillations. The control signals of the pitch and yaw channels for the PD-Like IT2FLCs contain higher oscillations which causes significant error in angles than using PSO Based PD-Like IT2FLCs. The less control efforts cause less consumption in power. Simulation results also show the effectiveness of the proposed controller in terms of the simplicity of the controller and improving time domain characteristics. The proposed controller uses two input membership function which reduces the rules into 4 as compared with the designed ones in references (9,13,16,17) which uses Type-1and Typ-2 FLC. Both controllers are BIBO stable for all reference trajectories applied without and with the application of noise, see equations (24 and 25). Table 5. Gains of controllers obtained by PSO Table 6. ISE Performance Index of TRMS motion CONCLUSIONS Type-2 FLC is a highly sensitive and robust controller through perturbations and uncertainties in the controlled system as compared with Type-1 FLC for the same class of systems. Type-1 FLC has higher tracking errors especially when disturbances exist. In this paper, Four PSO-Based IT2FLCs were designed for trajectory tracking for yaw and pitch axes and their cross-couplings of the 2DOF TRMS nonlinear model using MATLAB/Simulink. The PSO algorithm is used to tune the Inputs and output gains of the four Proportional-Derivative (PD) Like IT2FLCs to cancel high nonlinearities and to solve high the effect of coupling. Simulation results show that the PSO-Based IT2FLCs produce better stable tracking than IT2FLCs in terms of maintaining the TRMS position within the pre-defined desired trajectory, when exposed to changes during its maneuver without and with the presence of noise. The maximum percentage of enhancements reaches about 33% and the average percentage of enhancements is about 17.1%. They also show the effectiveness of the proposed controller in terms of improving time domain characteristics and the simplicity of the controllers compared with the designed ones proposed in previous published works.