Design New PID like Fuzzy CTC Controller: Applied to Spherical Motor

Where COG(x _k ,y _k ) and COA(x_k,y_k) illustrates the crisp value of defuzzification output, U [ E U is discrete element of an output of the fuzzy set, g _u . (x _k ,y _k , U ^ ) is the fuzzy set membership function, and r is the number of fuzzy rules.

Design PID Controller: Design of a linear methodology to control of continuum robot manipulator was very straight forward. Since there was an output from the torque model, this means that there would be two inputs into the PID controller. Similarly, the outputs of the controller result from the two control inputs of the torque signal. In a typical PID method, the controller corrects the error between the desired input value and the measured value. Since the actual position is the measured signal. Figure 3 shows linear PID methodology, applied to spherical motor [21-34].

e(t) = 9_a(t)-9 d (t)                  (21)

UP[D=KPae + KVae + K[^e

Fig 3: Block diagram of linear PID method

The model-free control strategy is based on the assumption that the joints of the motors are all independent and the system can be decoupled into a group of single-axis control systems [18-23]. Therefore, the kinematic control method always results in a group of individual controllers, each for an active joint of the motor. With the independent joint assumption, no a priori knowledge of spherical motor dynamics is needed in the kinematic controller design, so the complex computation of its dynamics can be avoided and the controller design can be greatly simplified. This is suitable for real-time control applications when powerful processors, which can execute complex algorithms rapidly, are not accessible. However, since joints coupling is neglected, control performance degrades as operating speed increases and a motor controlled in this way is only appropriate for relatively slow motion [44, 46]. The fast motion requirement results in even higher dynamic coupling between the various spherical motor joints, which cannot be compensated for by a standard motor controller such as PID [50], and hence model-based control becomes the alternative.

• III.  Methodology

Conversely pure computed torque controller is a high-quality nonlinear controller; it has an important problem; nonlinear equivalent dynamic formulation in uncertain dynamic parameter. Computed torque controller is a nonlinear controller but it has a challenge in stability and robustness especially in presence of uncertainty and disturbance. Based on literature CTC formulation is written by;

т = H(q)(q d + K _y e + К р в) + N⁽q, q)      ⁽23)

The main challenge in this research is the role of nonlinearity term in presence of uncertainty. To solve this main challenge artificial intelligence based controller is introduce. This type of controller is intelligent therefore design a dynamic of system based on experience knowledge is done by this method. One of the main artificial intelligence techniques is fuzzy logic theory. In this theory the behavior and dynamic of controller is defined by rule base. However defined and number of rule base play important role to design high quality controller but system has limitation to the number of rule base to implementation and the speed of response. Based on literature PID controller can reduce or eliminate the steady state error and design stable controller. But this type of controller has three types of inputs; proportional part, integral part and derivative part. To design PID like fuzzy controller and if any input is described with seven linguistic values, and any rule has three conditions we will need 7 × 7 × 7 = 343 rules. It is too much work to write 343 rules, the speed of system is too low and design embedded controller based on FPGA or CPLD is very difficult. In PD like fuzzy controller error and change of error are the inputs and if any input is described with seven linguistic values, and any rule has two conditions we will need 7 × 7= 49 rules. Table 1 shows the rule table of PD like fuzzy controller based on seven linguistic variables for any inputs and totally 49 rules.

Table 1: Rule Table of PD like Fuzzy Controller

This table includes 49 rules. We are taking into account now not just the error but the change-of-error as well. It allows describing the dynamics of the controller.

In conventional PI controller error and integral of error are the inputs. Based on above discussion the PID-like fuzzy controller can be constructed as a parallel structure of a PD-like fuzzy controller and a conventional PI controller with the output approximated as:

U p[ _D = (^e + K v _ae) + (^e + K [ 2 e) ⁽²⁴⁾

In this type of design, we have 49 rule bases for PD like fuzzy controller. This PID like fuzzy controller applied to pure computed torque controller to remove the challenge in this conventional nonlinear controller. Figure 4 shows the block diagram of PID like fuzzy computed torque controller.

Fig 4: Block diagram of PID like Fuzzy Computed Torque Controller

• IV.    Results and discussion

PID like fuzzy computed torque controller was tested to Step response trajectory. This simulation is used to control position of three degrees of freedom spherical motor without and with external disturbance. The simulation was implemented in MATLAB/SIMULINK environment. These systems are tested by band limited white noise with a predefined 40% of relative to the input signal amplitude. This type of noise is used to external disturbance in continuous and hybrid systems and applied to nonlinear dynamic of these controllers.

Tracking performances: In proposed controller; the performance is depended on two important parameters; nonlinear equivalent part and PID like fuzzy controller. According to above discussion PID like fuzzy computed torque controller and pure computed torque controller have the same performance in certain system. Based on Fig 5, pure computed torque controller has a slight transient oscillation, to solve this challenge the output gain updating factor of PID like fuzzy computed torque controller is decreased. In this design pure computed torque controller has about 5% overshoot but both of two controllers have the same rise time.

Fig 5: Computed torque controller and proposed method

Disturbance rejection: Figure 6 shows the power disturbance elimination in proposed method and pure computed torque controller in presence of external disturbance and uncertainty parameters. The disturbance rejection is used to test and analyzed the robustness comparisons of these controllers for step trajectory. A band limited white noise with predefined of 40% the power of input signal value is applied to the step trajectory. According to the following graph, pure computed torque controller has moderate fluctuation in presence of external disturbance and uncertainty.

Fig 6: Computed torque controller and proposed method in presence of external disturbance

Based on above graph, pure computed torque controller has many challenges in presence of external disturbance. To eliminate above challenge, this research is used the following methodology; decrease the output scaling factor of the PD-part, increase the scaling factor for an integral input compared to other inputs, apply the centre of gravity defuzzification method, reduce the width of the membership function for the zero class of the error signal and Redistribute the membership functions, increasing their concentration around the zero point.

• V.    Conclusion

The central issues and challenges of non linear control and estimation problems are to satisfy the desired performance objectives in the presence of noises, disturbances, parameter perturbations, un-modeled dynamics, sensor failures, actuator failures and time delays. Evaluation algorithm PID like fuzzy computed torque controller has shown growing popularity in both industry and academia. To improve the optimality and robustness, we have proposed PD like fuzzy controller parallel with conventional PI controller based on 49 rule base. Computed torque controller provides us an effective tool to control nonlinear systems through the dynamic formulation of nonlinear system. Fuzzy logic controller is used to estimate highly nonlinear dynamic parameters. Mixed performance criteria have been used to design the controller and the relative weighting matrices of these criteria can be achieved by choosing different coefficient matrices. The simulation studies show that the proposed method provides a satisfactory alternative to the existing nonlinear control approaches.

ACKNOWLEDGMENT

The authors would like to thank the anonymous reviewers for their careful reading of this paper and for their helpful comments. This work was supported by the SSP Institute of Advance Science and Technology Program of Iran under grant no. 2013-Persian Gulf-2A.