| Academic Open Internet Journal |
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)
where
are 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 elements
of 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).
7. With suitable
increments, get up-dated sets of 
Choose the realizable sets and test for the output response.
8. The set that
gives good output response is again sharpened to get the optimum response, if
needed.
10.3 Algorithm C: Design of Lead
Compensator
The
transfer function for a Lead compensator is assumed to be
with
B > A, K > 0, A > 0 and B > 0
The
steps given in Algorithm B are adopted for this situation.
10.4 Algorithm D: Design of Lag
Compensator
The
transfer function for a Lag compensator is assumed to be
where
B > A, A > 0 and B > 0
STEPS
1. Obtain the characteristic polynomial T(s) as given in
equation (7).
2. Obtain the new
algebraic criterion’s stability conditions from first row, first column
elements 
of the pseudo determinants 
.
3. Assume B=0 and
find the limiting value of A and let it be
.
4. With 
, find the limiting value of B using new algebraic criterion
for aperiodic stability. Let it be
.
5. With 
, test the system response for an overshoot less than 3 %,
settling time 3 seconds and 2% steady state error. Else, get the stability
conditions from step 2.
6. Take any one
of the stability conditions possessing A and B. If the degree of the
characteristic equation is small, choose last but one stability condition.
7. Partially
differentiate the chosen stability condition to get the step changes in the
parameters. In this, there will be only one equation in terms of
and 
8. With suitable
increment, get up-dated values of A and B from new algebraic stabilization
conditions and test for the output response.
9. The set giving
good response is again sharpened to get the optimum response, if needed.
The following
examples illustrate the application of the newly proposed scheme.
11.
ILLUSTRATIONS
Example 1: [5]
It is given
that 
----
(14)
Let 
---- (15)
The
characteristic polynomial is obtained as:
---- (16)
The
transformed characteristic equation as given in equation (7) is obtained as:
---- (17)
As per new
algebraic criterion, it is found that 
are positive. Hence as per Algorithm A and B, an arbitrary
value of 
is assumed. With 
get the transformed characteristic polynomial T(s) is written
as:
---- (18)
Step 1:
---(19)
Step 2:
The first
pseudo determinant is given as:
---- (20)
Step 3:
The second
pseudo determinant is given as:
---- (21)
Step 4:
The third pseudo
determinant is given as:
---- (22)
Using the first row, first column
element of 
and stability
condition, it is estimated that 
. For the first trial, it is consider to be 
. With
and 
and using
, it is found that
. Since this value is too large, it is arbitrarily assumed
that 
(This chosen with regard to 
.
Thus, for this
initial set (48.5, 50, 0.12) and from the simulation, the output response is
found to be unsatisfactory. Further values of 
and 
are obtained using the updation procedure as given below.
To up-date the values, the first row
first column element of last but one 
is used:.
For 
and 
, the partial changes are obtained as shown below:
---- (23)
---- (24)
---- (25)
Simplifying and arranging equations
(23) – (25), we get
---- (26)
---- (27)
and
----
(28)
Let 
and 
This leads to the two sets:
(i) (48.5, 25, 0.12128) and (ii) (48.5, 25, and 0.11872).
For these two sets, the system output
responses are found to be unsatisfactory.
Considering the first set with 
and 
, the following two sets are obtained:
(iii)
(48.5, 12.5 0.12192) and (iv)
(48.5, 12.5, 0.120635)
In the above, the set (48.5, 12.5
0.12192) is considered and modified with 
and 
Thus (v) (14.01,
12.5, 1.08192) and (vi) (82.99, 12.5, 1.08192) are obtained. Using the set
(14.01, 12.5, 1.08192), the simulation was performed and found that the output
response is not satisfactory. With 
and
, the set with less value is formed as (9.51, 12.5, 0.655).
After few simulations, a satisfactory response is obtained for the set (9.47,
12.5, and 0.19). The output responses are shown in the Fig 1 and Fig. 2.
Example
2: [6]
It is given that 
----
(29)
It is proposed to design the Lead
compensator
----
(30)
The characteristic polynomial is
obtained as:
F(s)=1+G(s) 
=0
---- (31)
The new algebraic Criterion for is
applied to the equation (31),
Step
1:
---- (32)
Step
2:
The first pseudo determinant is
obtained as:
----
(33)
Step
3:
The second pseudo determinant with B=0,
---- (34)
From the element
of 
and using the stability condition, it is found that A < 2.
The value of B is assumed (twice the value of A) for initial simulation.
Assuming A = 1.95 and B = 4, the transformed characteristic equation T(S) as
given in equation (7) is obtained as:
---- (35)
The algebraic
criterion for stabilization is applied to the equation T(s) in the equation
(35).
Step
1:
---- (36)
Step
2:
The first pseudo determinant is
obtained as:
----(37)
Step
3:
The second pseudo determinant is
obtained as:
----(38)
Step
4:
The third pseudo determinant is
obtained as:
----(39)
From the first
row, first column element of 
and stability condition, it is assumed that the value of K as
0.92.Hence the initial set (1.95, 4 and 0.92) is considered and the output
response is found to be unsatisfactory.
To up-date
this set, the step sizes for (A, B, K) are deduced using first row, first column
element of last but one modified Hurwitz determinant with partial derivatives:
For K is constant
---- (40)
For A is constant:
---- (41)
For B is constant:
---- (42).
Arranging the above equations suitably
the following results is formed:
---- (43)
---- (44)
---- (45)
Let 
and 
For this situation 
The corresponding up-dated sets are (i) (1.95, 4.5, and
0.241728), (ii) (1.95, 4.5, and 1.718272). These two sets are used for
simulation and it is found that the set (1.95, 4.5, and 1.718272) satisfies the
design specifications giving the optimum response and hence further simulation
is terminated saving computational time. The Fig.3 and Fig 4 show the output
responses of the system with and without compensator.
12. SIMULATION
RESULTS
The
effectiveness of the newly proposed scheme for the design of PID controllers
and Phase compensators are demonstrated using computer simulations. The system
is simulated for a unit step input using MATLAB – SIMULINK software with and
without controllers or phase compensators [16]. The output responses of the
above simulation studies are given in the following Figures.
Fig
1. Response of a System without PID controller for a unit step input
Fig 2. Response of a System with PID controller for a unit step input
Fig 3. Response of a System without Lead compensator for a unit step
input
Fig 4. Response of a System with Lead compensator for a unit step input
13.Summary
In this
chapter, the newly proposed algebraic scheme presented to design PID
controllers and Phase compensators for the given Linear Time Invariant Systems
(LTIS). In this scheme, the initial sets obtained using new algebraic criterion
with Fuller’s scheme, are updated employing the relations deduced from any one
of the new algebraic criterion’s stability conditions. It is found that the
newly deduced sets are good enough to get the optimal response as per design
specifications. Further, the algorithms given are found to be effective in the
design process leading to efficient simulation run. The newly proposed scheme
can be extended to design LTIDS with bilinear transformation. The proposed
schemes are applied to certain illustrations.
14. References
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Technical College - Bourgas,