Discrete PID Í ( n-2 ) Stage PD Cascade Controllers Proposed by Kitti

A Proportional-Integral-Derivative (PID) Controller has only two zeros and not enough for most industrial plants that are third or higher order plant. To achieve the desired specifications by PID controls, Associate Professor Dr. Kitti Tirasesth proposed the way to add more zeros to the PID controller according to the order of the plant. By this technique, the designer find only two unknown parameters based on the root locus magnitude and angle conditions. By increasing only the controller gain, the faster response with smaller overshoot can be obtained. Now the discrete-time proposed controllers are ready for real time implementation.


Introduction
According to the type and plant's order, most industrial plants are type 0 and consist of three to five first order lags or dead time plus one first order lag (1) .Then, the n th order plant to be controlled here, its transfer function is assumed to be given as where, the order of the plant is n = N + p.However, the Proportional-Integral-Derivative (PID) controller is properly applied to the typical second order plant only.The difficulty of satisfying the desired specifications with PID controller for a third order plant can easily show by the following example.In this example, the plant is type 0, 3rd order.Note that if one (3-2=1) more zero is available, the total angle for the zeros can be shared to each by smaller degrees such that they are able to arbitrarily place in LHP with the desired specifications.By this reason, the "PIDÍ(n-2) stage PD cascade controller for SISO systems" (2) had been proposed by Kitti ; Associate Professor Dr. at KMITL.

Methodology
There are 2 steps as shown in Fig. 5, for the design procedure of control system as follows: 1) Plant modeling, 2) Controller design.

Continuous-Time System
The open loop transfer function () () KsGs in Fig. 1, between the PID by (n-2) PD Controller and the nth order plant can be written as follows: The PID by (n-2) PD Controller designed by using Root Locus Technique for the 3rd order plant has the transfer function as written in Eqn.6, and the corresponding Root Locus Plot is shown in Fig. 6.The solutions for the case of using the method as proposed by Kitti, the PID by (n-2) PD controller's transfer function is given in Eqn. 7, the corresponding Root Locus Plot in Fig. 7, and a unit step response, in Fig. 8.Note that, the Root Locus Technique find 12 ,, pd zzz and K , but the Proposed by Kitti find only pd z and K .

Discrete-Time System
Recently there were two discretization method have been proposed for designing the Discrete-Time PID by (n-2) PD Controllers.The first is use 'zoh' (3) ; Zero Order Hold discretization method as shown in Fig. 9. While, the second use 'tustin' (4) ; the "Tustin's Method", discretize the plant first to obtain a sampled-data system or discrete-time system.
T h e D i s c r e t e -T i m e P I D b y ( n -2) PD Controllers transfer functions by the 'zoh' and 'tustin' methods can be written by Eqn. 8 and Eqn. 9, respectively.Last time, the Proposed by Kitti has been applied to find () Kz in Eqn. 10 in z-plane directly with the difficulty so much when the sampling time T is very small.

The Proposed by Kitti
This time, the Continuous-Time controller () Ks in s-domain will be discretized to Discrete-Time controller () Kz in z-domain as shown in Fig. 10.
Transforms the coefficients to obtain () Kz, Finally, the solutions to () Kz in Eqn. 10 be } [10 ]( 0.9399)( 0.8850) ( 0.8803) ( ). ( The state space model for the discrete-time controller block diagram in Fig. 12 is given by The unit step responses shown in Fig. 13 are for the designed controller gain K and for 10K with faster response and smaller in percentage of the overshoot.So, the designer should use the controller's coefficients that corresponding to this value of the controller gain K .

Conclusions
Since, the PID controller is suitable for a second order plant.But, for a third or higher order plant, the zeros provided by the PID controller is not enough.In order to give more zeros, the PIDÍ(n-2) stage PD cascade controll e r i s t h e n p r o p o s e d b y A s s o c i a t e P r o f e s s o r D r .K i t t i Tirasesth.Starting from the continuous-time controller, follow by two more of the discrete-time controllers.This paper is the third version of the discrete-time controller that directly transformed from the continuous-time controller with all features is maintained and ready for real time implementation.

Fig. 2 .F
Fig. 2. Find the angle and location for double c z .

Fig. 8 .
Fig. 8. CT -Unit step response for the Proposed by Kitti.
Fig. 11.DT -Root Locus Plot for the Proposed by Kitti.
Fig. 13.DT -Unit step responses for the Proposed by Kitti.