Comparative Design and Analysis of PIDA Controller Using Kitti’s and Jung-Dorf Approach for Third Order Practical Systems

In this paper, PIDA controller is designed for third-order control systems using Novel Analytical methods. These approaches are based on Jung-Dorf method and Kittis method. This paper demonstrated the PIDA controller design for application of DC motor, Induction motor and AVR power system. The PIDA controller is an extension to the PID controller. The additional term A stands for acceleration, with this new term, a closed-loop system can respond faster with less overshoot. Originally, the PIDA controller design utilizes the Dominant pole concept proceeded in the s-plan.


Introduction
In the past decades, the Design of an effective and economic controller is always a non-intuitive and difficult task to control engineers. The PID (proportional-integral-derivative) controller, is widely used in industrial control systems [1]. The most popular design technique is Ziegler-Nichols method [2], which relies solely on parameters obtained from the plant step response. The PID controllers tuned in accordance with the Ziegler-Nichols method have generally a step response with a high percent overshoot. In many control applications, the systems are modeled as a third order. PID controllers are unsuitable, especially for third-order systems. This is the reason that a new structure of the controller becomes the necessity of such systems.
In 1996, Jung and Dorf have proposed a new structure of controller and termed as the proportionalintegral-derivative and acceleration (PIDA) controller [3]. This controller has less settling time and over-shoot compared to PID controller for third-order systems. The idea behind the PIDA controller design is to add an extra zero in standard PID controller. Thus, the effect of non-dominant roots is reduced [4]. A New Analytical approach of PIDA design was proposed by Kitti's [5] and extended to discrete system [6].
The application of PIDA was successfully carried out for torsional resonance suppression [4]. Dal-Young et.al, 2001, has used a pre-compensator to PIDA in ac motor [7]. The optimal designs of PIDA controller were presented using Genetic algorithm [8], Harmony search algorithm [9], Firefly algorithm [10] and Bat algorithm [11].
In this paper, the Analytical methods of PIDA controller design are compared. In the beginning, Jung-Dorf approach and Kitti's approach are reviewed briefly. The methods were applied to position control of DC motor, induction motor and AVR system.
The article is organized into four sections. The problem formulation is described in section 2. It includes the design methods and the systems under consideration. The design of PIDA controller for the systems with different approaches is included in section 3. The simulation results with analysis is included in this section. The article is concluded in section 4 and followed by references.

Position control of DC motor
As a Reference Armature controlled DC Servo motor can be considered as linear SISO plant model having third-order transfer function. The DC servomotors are found to have an excellent speed and position control [3,12]. A simple mathematical Relationship between the shaft angular position θ and voltage input Va to the DC motor may be derived from physical laws.
The dynamic behavior of the armature current-controlled DC servomotor is given by the following equations. The air gap flux ϕ of the motor is proportional to the field current so that The torque developed by motor is assumed to be related linearly to air gap flux and the armature current as follows: Where K1 and K f are constants. When a constant field current is established in a field coil .The motor torque is The armature current is related to the input voltage applied to the armature by Where ω(s) = sθ(s) the transform of the angular speed and the armature is current is The motor torque is equal to the torque delivered to the load which may be expressed as Where T l is the load torque and T d is the disturbance torque which is often negligible, so Therefore, the transfer function of the motor load combination with T d = 0 is Here the Angular displacement θ(s) is considered the output, and the armature voltage Va(s) is considered the input. The schematic diagram and block diagram representation of DC Motor is shown in Fig. 1 and Fig. 2, respectively. The transfer function parameters given in Table 1.

Induction motor
Here PIDA controller is designed for the simplified induction motor position control model that has been implemented in paper [12]. The induction motor model used here is the three phase, star connected, two poles 800 W, 60 Hz, 120 V / 5.4 A. The linearized control structure of induction motor is shown in Fig. 3.

Fig. 3. Block diagram representation of Induction motor
The transfer function of induction motor is given by Eq. 2.14.
Where KP , KI are PI controller gains and Kt is motor constant. On putting the parameter values

Automatic voltage regulator
An automatic voltage regulator (AVR) is commonly used in the generator excitation system of hydro and thermal power plants to regulate generator voltage and control the reactive power flow [13,9,10,14]. The main role of the AVR is to hold the terminal voltage of a synchronous generator at a specified level [11]. The schematic diagram is presented in Fig. 4.
The transfer function representation of the components for AVR power system as followings: • Exciter is represented by A simple AVR consists of four main components, i.e. amplifier, exciter, generator, and sensor, respectively. A simplified AVR system controlled by the PIDA controller is represented by the block diagram in Fig. 5

PIDA design methods
For the control system, the design procedure is as follows: 1. Obtain the mathematical model of plant G(s) as shown in Fig. 6.
2. Obtain the controller Gc(s), such that the desired specifications are acceptable.

Fig. 4. Schematic diagram of AVR system
The desired specifications necessary for the design of PIDA controller may as following: • Settling time (Ts) for tolerance of ±2% is given by − ln

Jung-Dorf approach
The PIDA controller suggested by Jung and Dorf is expressed as mentioned in Eq. 2.17.
Where a, b, c << d, e and two poles (d, e) of PIDA controller are far from the Zeros and excluded in the design for stability consideration. For the closed loop transfer function T (s), the performance is often decided by transient response which includes parameter such as settling time (Ts), percent overshoot (P O) and Peak time (Tp) [3]. Here the set desired specifications are as follows: (viii) Plot the responses for unity step input.

Kitti's approach
The design of PIDA controller using Kitti's approach follows the following steps: (iv) Compute the magnitude of zero(c) as shown in Fig. 7

Design using Jung-Dorf approach
The transfer function of DC servo motor is shown in Eq. 2.13. The procedure for designing the PIDA controller already mentioned in previous section. • The step responses of the plant with PIDA controller is shown in Fig. 8 and the step response information such as settling time, peak time, etc. are shown in Table 4.

Design using Kitti's approach
The procedural steps are already mentioned in section 2.2.2. In this section the PIDA controller is designed using Kitti's method as following: • The open-loop transfer function of the system is given by Eq. 3.8.

Gc(s)G(s)
• Set the location of zeros such as a = 10.0975 and b = 2.1025. The remaining Zeros associated to (s + c) and K must be solved using Eq. 3.9 and Eq. 3.10.
∠GC (s)G(s) = ±(2k + 1)π (3.9) |GC (s)G(s)| = 1 (3.10) • The location of zero of (s + c) can be find from the angle condition at the desired dominant closed loop poles s d = −ξωn ± jω n √ 1 − ξ 2 = −2.118 + i2.221. The angle condition at the desired closed loop poles (S d ) as follows: • Open loop Gain K can be find from magnitude condition. • The step responses of the plant with PIDA controller is shown in Fig. 8 and the step response information such as settling time, peak time, etc. are shown in Table 4.

Simulation results
The circuit for simulation in MATLAB is Considered as shown in Fig. 6, where, G(s) stands for the transfer function of DC motor as in Eq. 2.13 and Gc(s) stands for the transfer function of PIDA controller determined using Jung-Dorf method and Kitti's method as presented in Eq. 3.7 and Eq. 3.20, respectively. The combination of DC motor with PIDA controller is subjected to simulation on application of step input. The response of the system (DC Motor) without controller, with PIDA controller using Jung-Dorf approach and with PIDA using Kitti's approach are compared and shown in Fig. 8. The step response data are enlisted in Table 4.

Design using Jung-Dorf approach
The transfer function of induction motor have been presented in Eq.

Design using Kitti's approach
The design aspects of the PIDA controller for induction motor using Kitti's approach is considered in this section.

Simulation results
Considering the arrangement as shown in Fig. 6, where, G(s) stands for the transfer function of Induction motor (as in Eq. 2.15) and Gc(s) stands for the transfer function of PIDA controller determined Jung-Dorf method as presented in Eq. 3.21. The combination of Induction motor with PIDA controller is subjected to simulation on application of step input. The response of the system (Induction motor) without controller and with PIDA controller (based on both approaches) are compared and shown in Fig. 9.

Design using Jung-Dorf approach
The transfer function of AVR

Design using Kitti's approach
The design aspects of the PIDA controller for AVR power system using Kitti's approach is considered in this section.

PID controller
In this section, PID controller using Ziegler Nichols method is designed and mentioned in Eq. 3.25 [15].

Simulation results
The G(s) stands for the transfer function of AVR power system as in Eq. 2.16 and Gc(s) stands for the transfer function of PIDA controller determined using Jung-Dorf method and Kitti's method as presented in Eq. 3.23 and Eq. 3.24, respectively. The combination of AVR power system with PIDA controller is subjected to simulation on application of step input. The response of the system (AVR power system) without controller, with PIDA controller using Jung-Dorf approach and with PIDA using Kitti's approach are compared and shown in Fig. 10. The step response data are enlisted in Table 6. The step response of the system without controller, with PIDA controller using both techniques and with PID controller [15] are comapred in Fig. 11 and the information is included in the Table 6. The response of AVR power system outperforms that of with PIDA controller as compared to that of with PID controller.

Conclusion
The Analytical methods of PIDA controller design are compared. In this paper, three practical systems such as DC Motor, Induction motor and automatic voltage regulator (AVR), are considered for the optimal design of PIDA controller using Kitti's Approach and Jung-Dorf approach. The Kitti's Approach of PIDA design has appeared with more over-shoots as compared to Jung-Dorf approach. Both approaches have better steady-state characteristics than the system with PID controller [15].