AN ALGEBRAIC SCHEME FOR THE DESIGN OF PID CONTROLLERS AND COMPENSATORS FOR LINEAR TIME INVARIANT CONTINUOUS SYSTEMS

Academic Open Internet Journal

www.acadjournal.com

Volume 13, 2004

AN ALGEBRAIC SCHEME FOR THE DESIGN OF PID CONTROLLERS AND COMPENSATORS FOR LINEAR TIME INVARIANT CONTINUOUS SYSTEMS

K. Porkumaran¹, S. N.Sivanandam²

1. Lecturer 2. Professor and Head

Department of Computer Science and Engineering, PSG College of Technology

Coimbatore, India. Email: kporkumaran@myway.com

Abstract:

The design of a control system is concerned with the arrangement or the plan of the system structure and selection of suitable components and parameters. In classical control systems, the process of output response stabilization is attained by the selection of either Proportional - Integral- Derivative (PID) controllers or Phase compensators (Lead, Lag and Lead-Lag networks). This paper proposes an algebraic procedure for the design of controllers or compensators for a given Linear Time Invariant Continuous Systems (LTICS). The steps involved in this scheme are illustrated with the examples.

Key Words: performance indices-PID controllers-phase compensators- LTICS - new algebraic criterion - bilinear transformation

1. Introduction

Stabilization is a process by which the output of a given LTIS is controlled for a specified signal under certain performance indices; these indices may be either in time domain or in frequency domain. Based on output response, the selection of the PID controllers or compensators for a given LTICS can be carried out employing graphical procedures due to Nyquist, Bode, Evans, Nichol, Neimark and Mitrovic [1,2]. For process control systems, the selection of PID controllers can be done using Zeigler- Nichols thumb rules [3]. This chapter proposes a novel technique for the design of controllers or compensators for a given Linear Time Invariant Continuous Systems (LTICS). With the help of bilinear transformation, design of these controllers or compensators for a Linear Time Invariant Discrete Systems (LTIDS) can be carried out by this scheme.

2. LTICS Description

A unity feed back control system having a plant transfer function G(s) is considered. The general closed loop transfer function of such a system is represented by

---- (1)

For unit step input, the time response of the output is obtained to analyze the nature and performance of the system in the time domain. If the system output response does not satisfy the designer’s criteria in time domain, a PID controller or compensator with a transfer function is added into the forward path of. This is one of the possible configurations for compensation. The feedback compensation, feed forward compensation and state feedback control are other possible compensations [4].

In this case, the closed loop transfer function is written as:

---- (2)

The characteristic polynomial is obtained as:

---- (3)

3. Selection of controller and compensator

To obtain an optimum transient response of the system, a PID controller is chosen with transfer function [5,6]

---- (4)

where - Proportional gain

- Integral gain

- Derivative gain

In the selection of phase compensator, the transfer function is assumed to be one of the following:

For Lead compensator,

---- (5)

For Lag compensator,

---- (6)

where the parameters K, A, and B are to be suitably chosen.

In the process of selection of controller (or) compensator, the different sets of () (or) () or (A, B) are chosen and the corresponding output response is compared for optimum response on the basis of specified overshoot, quick settling time and steady state error.

4. Fuller’s Scheme for Aperiodic stability [7- 10]

Formulate the transformed characteristic equation T(s) for Linear Time Invariant Continuous Systems, suggested by Fuller [9] as:

---- (7)

where and ---- (8)

This concept is suitably extended for formulating a criterion for the design of Controllers or Compensators for aperiodic response as given below.

5. Proposed Procedure

Without loss of generality and simplicity, the results of Hurwitz determinants in the Hurwitz criterion[11-14] suitably extended to deduce the newly proposed procedure for the design of controllers and compensators of Linear Time Invariant Systems.

The necessary condition for absolute stability is formulated as:

I. No missing term in the characteristic polynomial F (S).

II. All the coefficients in the characteristic polynomial F(S) must have same sign.

III. There should not be multiplicity of roots on the imaginary axis in the ‘s’ plane.

In addition to the above necessary conditions, the sufficient conditions are obtained from the following procedure. In this procedure, individual are not obtained, but by doing certain mathematical operations, the results of Hurwitz determinants are arrived as given below:

The Hurwitz determinant is written as:

(r =1, 2, 3,...n) ---- (9)

With suitable mathematical operations, the pseudo determinants are formulated:

For instance,

---- (10)

Where are obtained from first two rows of equation (9) as given below:

---- (10a)

---- (10b)

---- (10c)

---- (11)

whereare formed from the first two rows of equation (10): ----- (11a)

---- (11b)

---- (12)

The values of are formed from the first two rows of equation (11):

---- (12a)

---- (12b)

Similar steps can be extended for further evaluation of the pseudo determinants.

The above procedure is used to analyze the stability as well design of linear systems.

6. New Algebraic Criterion for Stabilization

The new algebraic criterion for the design of parameters of controllers or compensators can be stated as follows:

“In order for the roots of the characteristic polynomial to have distinct negative real values and to lie on the real axis of ‘s’ plane, it is necessary and sufficient that the first row, first column elementsof the corresponding pseudo determinants should be positive”.

In other words,

with . ---- (13)

7. Proof

According to the Hurwitz Criterion, the principal minors are obtained from the Hurwitz determinant in equation (9) as

In the new algebraic criterion, the first row, first column elements of the pseudo determinants are

It can be inferred that,

where are scaling factors.

From Hurwitz Criterion, the necessary and sufficient conditions for the roots to have negative real parts with are given by the following determinants:

In the new algebraic criterion, the scaling factors are all positive and it is easily observed that are also positive.

Conversely, if, then

Q. E. D.

8. Bilinear Transformation

In the case of Linear Time Invariant Discrete Systems (LTIDS), the design is followed by first applying bilinear transformation to transform LTIDS [15], to an equivalent LTICS and then applying newly proposed procedure.

9. Design specifications

The system is tested with unit step input and the design procedure is followed based on the following design specifications:

Maximum peak overshoot: less than 3%

Settling time : less than 3 seconds

Steady state error : 2% (assumed for optimum response)

Before proceeding on to the simulation, the starting values of the parameters of compensator are deduced using newly proposed procedures.

10. ALGORITHM FOR THE PROPOSED SCHEME

The design steps of the controller or compensator is iterative and given by the following algorithms for computer use.

10.1 Algorithm A: General Algorithm for Design of Controller or Compensator

STEPS

1. Read the open-loop transfer function of the system.

2. Form closed loop transfer function for the above system.

3. Obtain the time response using MATLAB-Simulink for unit step input.

4. Check for the time domain specifications.

5. If the specifications are not met with, proceed to design of controller or compensator using the following algorithms B, C and D respectively, else stop.

6. Incorporate the controller or compensator into the system.

7. Obtain output response for unit step input.

10.2 Algorithm B: Design of PID Controller

STEPS

1. Obtain the characteristic polynomial T(s) in terms of as given in equation (7).

2. Assume Obtain a limiting value for to be by applying

the new algebraic criterion for stabilization.

3. With , deduce the limiting value for to be for Repeat this step with and and get the limiting value of to be

4. With, test the system response for overshoot less than 3%, settling time less than 3 seconds and 2% steady state error. Else get the stability conditions for step 2.

5. Take any one of the stability conditions possessing all. If the degree of the characteristic equation is small, choose the last but one of the stability conditions.

6. Partially differentiate the chosen stability condition to get the step changes in the parameters. It is possible to develop a set of equations in terms of and with the following situations: (i) with , (ii) with , (iii) with , (iv) in terms of (i)
and (ii) , (v) in terms of (ii) and (iii), (vi) in terms of (iii) and (i).