- Email: [email protected]
A New Tuning of PID Controllers Based on LQR Optimization
Copyright Cl IFAC Digital Control: Past. Present and Future of PlO Control. Terrassa. Spain. 2000
A NEW TUNING OF PID CONTROLLERS BASED ON LQR OPTIMIZATION
Rosa Argelaguet .1
Montserrat Pons .. I Josep Aguilar ••• 1
Joseba Quevedo ••• 1
• Dept Engin. de Sistemes, Automatica i Informatica Industrial Universitat Politecnica de Catalunya. Av. de les Bases de Manresa 61-73, Manresa 08240, Spain. Phone: (34-3) 8777254; Fax: (34-3) 8777202 [email protected] •• Dept Matematica Aplicada III Universitat Politecnica de Catalunya. Av. de les Bases de Manresa 61-73, Manresa 08240, Spain. Phone: (34-3) 8777250; Fax: (34-3) 8777202 [email protected] ••• Dept Engin. de Sistemes, Automatica i Informatica Industrial Universitat Politecnica de Catalunya. Rambla Sant Nebridi, 10. Terrassa 08222, Spain. Phone: (34-3) 7398627; Fax: (34-3) 7398628 [email protected] / [email protected]
Abstract: This paper presents a new analytical tuning of Proportional-IntegralDerivative controllers based on the well known Linear Quadratic Regulator optimization problem for a wide class of processes whose behaviour can be modelled with with a constant time and a dead time delay. When the dead time delay is approximated by a first order Pade proposal, an optimal control problem gives an analytical solution for the tuning of the PlO parameters, and the performances obtained on the original model, without approximation, are very satisfactory. Also, the positive results obtained by simulation with this new PlO control design respect to the conventional empirical formulas to tune PlO controllers are very promising to be applied in the future to industrial processes. Copyright © 2000 IFAC.
Keywords: Optimization, PlO regulator.
I.
dead time) can be easily estimated by conventional techniques. For this class of processes, classical contributions (Ziegler and Nichols, Cohen and Coon, Lopez, Rovira...) give useful empirical formulas to tune PlO controllers
INTRODUCTION
A large number of industrial processes can be modelled by a dead time plus a first order system and their parameters (gain, constant time and
1 The
authors are members of LEA-SICA
271
in order to minimize a certain performance ( lSE, lAB, ITAE..).
(4)
Our objective is to give a new formula to tune PlO controllers for this class of processes in order to minimize the integral square error (ISE) of the controlled process for a abrupt change in the setpoint. The difference with respect to the precedent techniques is that the solution is obtained straightway when the delay is approximated by a first order Pade proposal. Generally, this approximation is acceptable when the constant time is much bigger than the pure delay, and the results can be applied to real processes.
If we assume that all states are measured and available for feedback, is well known that there exists a unique full state gain matrix F = (fl f2) such that the control law
minimizes
., JO(u)=f(yl+ PU 1 )dt o
(5)
the cost function: For the plant we are considering, which verifies the above conditions, this control law can be expressed as:
In the next section, a formulation of the PID as an optimization LQR problem will be presented and the ISE criteria minimization will be solved in the section Ill. The results obtained with this new tuning of PlO controllers has been compared with the conventional empirical formulas for the same simulated systems will be shown in the section IV and finally the conclusions of this paper will be presented in the section V. 2.
u· = - Fx
U o• ( t ) =-Kpy-K D -dy dt
(6)
The linear quadratic optimal regulator is in this case a PD controller. It is well known that to achieve a zero steady state (unbiased) error, for a constant set-point, the controller must incorporate an integrator in its structure. This can be accomplished by augmenting the plant with the integral of the output and considering a new LQR problem, weighting the integral output in the cost function [4]-[5].
PlO AS AN OPTIMIZATION LQR PROBLEM
We consider a linear second order plant with input u(t), output y(t) and transfer function: which approximates (first order Pade) a first order plant with a pure delay T, time constant 't and gain K. The strictly positive parameters bl> bo , al , ao are given by :
The augmented plant, whose states are still measurable and available for feedback, is:
. (A
A =
C
and we can consider a new cost function:
K
b,=r
00=-
rT
rT
(8)
o
(2)
T +2r
2K
bo=-
., J~(U)=f(yl+PU1+A.zl)dt
2
Q/
""'--;:r-
. x = Ax + Bu
= Jydt I
A state space realization of the plant is:
where z
is the new state variable, and
o
(3)
A ~ O.
y = ex
The control law u1..· = -fIX, - f2x2 - f3z which minimizes h can be expressed now by :
where x = ( XI X2 ) T is state variables vector, and the matrix A, Band C:
and the three PID constants are given as follows:
272
Kp
12
(10 )
1 + b , 12 =
13 1+ b,/
00
ISE
2
00
+00
o
-00
JI e(t) rdt = 2~ JIE (jw) rdw
(16)
The Laplace transform expression for the error signal is in our case: (17;
E (s)
~)
(12 )
A
where D is:
The analytical solution of the Riccati equation is obtained I and therefore the controller (K p , Ko , K( ) in terms of A. and p. The f l , f2 and f3 expressions are in the equation rT I,o bt bt I,·
ya
b'V-p·aD+
+
-
p.
(18)
In this way we obtain a expression of this criteria
rr
l~Vp
ISE as function of the parameters
(13)
Ji and p:
a, bo
, 1
bo
I,·
l+ b
,{T
,VL
bob,
I
J •
IF
These expressions (13) offer the possibility to study differents specifications for the system (plant - regulator) in terms of A. and p.
In order to fmd the values of A. and p which minimize ISE it is necessary to solve the equations system (20):
ISE CRITERIA MINIMIZAnON
a (ISE)
a:r;:
Criteria like ISE can be used to characterize set point responses if the error is interpreted as the error due to a unit step change of the set point. The block diagram of the PlO regulator (10)-(13) and the plant r
(15)
We can solve the square error integral by using the Parseval theorem:
where
3.
= Je 2 (t)dt o
The gain matrix F that we obtain depends obviously of the two parameters A. and p which appear in the expression of h. In this way any value of A. and p positives, defmes a PlO controller. In order to determine the expression of the regulator parameters in terms of A. and p we have to solve the Riccati equation:
(13).
(14)
We will found the values of the parameters A, p that minimize the integral square error (ISE):
1+ bl / 2
KD K,
e(t) = r(t) - y(t)
I,
Po is shown in figure
e
~
H
0 (20)
=
0
ap
1. After very tedious calculus we can find analytically expressions [1] for the values of A. and p that solve the equations system (20) and they are the unique solution of this equations system (21). We can prove that this solution is in fact a minimum.
T y
u
Re..I.,,,,
a (ISE)
=
Plan'
Fig. 1. Block diagram ofthe plant and the regulator
Where the error signal is:
273
(21)
A = a~ b; 2 bo
=..!t'
o.
/ / . . -K rovira /
06
2
/
+---Kemin
Replacing these values of A and p in the equations (10), (13) we can find the tuning of the optimal PlO controller:
02
o
o
15
'0
20
25
JO
Fig. 2 Unit closed loop response ofthe plant Po and the regulator K...;•.
The results show a faster response of our PID without overshoot in the response. These results has been obtained in both cases with a 2-degree of freedom structure of the PlO controller (see figure 3), that is mean that the proportional and derivative terms act only on the process variable and not on the control error in order to avoid undiserable impulse, due to the tep change of the set-point.
The proposed PlO controller based in LQR problem has an Internal Model Control (IMC) structure and tends to cancel the two model poles of the process. A comparative study of our proposed PlO controller and the derived from the IMC structure [3] applied to the same model has been done in [2].
4.
COMPARATIVE RESULTS
In order to analyze the performances of the new tuning PlO controller, different scenarios has been studied by simulation and a graphically comparison has been done with the conventional empirical tuning PlO controllers for the same simulated process.
Fig. 3 Two degree offreedom structure ofthe PID controller
4.2. Simulated second order process with dead time We are interested in to apply our PlO controller
4.1. Simulated process with dead time P (s) = I
In order to check the performance of this PlO controller ( ~min ) (22) with the original model, without approximations, some tests has been done with satisfactory results. In the figure 2 we show the results obtained for the plant Po _
10 5s+1
Po---e
1 e -0.16 s s1 +4 s+ 1
(23)
in high order plants with dead time (23). In the figure 4 we show a unit step response for the plant PI(S) (23) and the PlO controller ~in , according to the block diagram shown in figure 1. For to obtain the PID controller ~in , we approximate the plant (23) by a first order plant with a dead time and the parameters are :
-1s
and we compare the step response with one of the best empirical tuning of PlO controllers, proposed by Rovira to minimize the IAE [6].
K=l
274
r=1ns~
T=aBs~
1.2
r5. 0.8
SENSITIVITY TO THE UNCERTAINTY OFPROCESSP~ETERS
0.6
As a measure of robustness we will use the "loop margin". The loop margin M is simply the minimum distance in the complex plane between the Nyquist plot of the loop gain L=PoK and the critical point -1 :
0.4 0.2
Of-' -0.2 0
10
5
15
25
20
M
30
= minll + L(jw)! w
It can be proved that for the system composed of
the plant Po described in equation (I) and its corresponding ~in controller (22), the loop margin is always Mo=O.5 for any value of the plant parameters. In this section we study the changes in the loop margin M when the Kernin controller corresponding to the plant Po is applied to a new plant Po whose gain K , or whose dead
Fig. 4 Unit closed loop step response ofthe plant PI and the regulator K......
4.3. Flow control in a real process We have a real plant with two interconnected tanks, and we are interested now in the flow control. See figure 5.
time i are slightly different from those of the nominal plant Po. Especifically, we assume variations of the type:
-dJ-£h
K = (1 + a)K i = (1 + fJ)T where K and T are the nominal values of the gain < < 1 . Let's analyze and dead time, with different situations:
la! l.IfiI
!
Fig. 5 Two interconnected tanks for flow control
fJ = 0 the plant Po is given by:
5.1. When
The transfer function which approximate this real process is P(s): The figure 6 show us the unit step response for
A
(l+a)K(I-
~ s)
Po = ------0->---....,..
1
e -0.23s
(I+~{I+ ~ s)
(24)
0.2775 s+l and it can be proved that the loop margin M is: the real plant P and the regulator ~in according to the structure shown in the figure 3.
1
M = -(I-a) 2 Mo-M
K-K
a Mo K As expected, for positive values of a, MM o. The changes in M depend solely of a, and it is worth to notice that the relative change in M is the same as the relative change in K, that is to say: ----:~-
08 0.6
04
= --=
0.2
00
1.5
2.5
3
5.2. When a=O the plant
3.5
Fig. 6 Unit closed loop step response ofthe real plant P and the regulator K...i •.
275
Po is given by:
It can be proved that there exist a range of values of P for which the loop margin M remains equal
The comparative results obtained by simulation promise very good performances of this tuning of PID controller for simulated and real industrial processes.
to M=O.5. Especifically, if -1 < ps 0.17 there is no change in the loop margin, so that the system appears to be robust in front of changes on T which are inferior to 17%. For greater values of jJ < I the value of the loop margin decreases and it is given by:
5.3. When a by:
* 0 and p * 0 the plant
ACKNOWLEDGEMENTS This work has been partly supported by Research Commission of Generalitat de Catalunya (Group SAC), ref. 1999SGR 00134 and Comisi6n Interministerial de Ciencia y Tecnologia, ref. TAP96-1114-C03-0 1
Po is given
REFERENCES Argelaguet R., Pons M., Quevedo 1., Aguilar Martin J.,"Analysis of achievable PID performances versus optimal linear regulator theory". SICICA'94, Budapest. 1994. Argelaguet R. " Estudio de limites de prestaciones de controladores lineales con estructura fija". Tesis doctoral. UPC. 1996.
It can be proved that the loop margin depends only of a either if p < 0 or if p ~ 0 and.
Astrom K.J., "Assessement of Achievable Performance of simple feedback loops". Int. J. Adapt. Control & Sig. Pro.,No.5,pp.319,1991. In both cases, its value is:
M. Athans. The status of optimal control theory and applications ho deterministic systems.
IEEE. Transactions on Automatic Control,
I M =-(l-a)
AC-Il:580-596,1966.
2
The same that was obtained when
p = O.
Parker K.T., "Design of proportional integral derivative controllers by the use of optimal linear regulator theory". Proc. IEEE, vol. 119,No.7,1972.
It is
worth to notice that the second case is accomplished in particular for any a>O and any positive p < 0.17 . This allows us to ensure that if the change on T is less than 17% and the gain is increased by a % the loop margin M will decrease also by a %.
6.
Rovira A. A.PhD. Dissertation, Dept of Chemical Engineering Louisiana State University, 1981.
CONCLUSIONS
A new analytical tuning of PID controller has been established, based on an augmented LQR optimization problem, for a wide class of industrial processes, represented by a model with one constant time and a dead time. The proposed PID cancel the dominated poles of the model and the gain of the controller gives a very fast response of the system to a step change of the set-point, avoiding the overshoot of the response.
276
A NEW TUNING OF PID CONTROLLERS BASED ON LQR OPTIMIZATION
Rosa Argelaguet .1
Montserrat Pons .. I Josep Aguilar ••• 1
Joseba Quevedo ••• 1
• Dept Engin. de Sistemes, Automatica i Informatica Industrial Universitat Politecnica de Catalunya. Av. de les Bases de Manresa 61-73, Manresa 08240, Spain. Phone: (34-3) 8777254; Fax: (34-3) 8777202 [email protected] •• Dept Matematica Aplicada III Universitat Politecnica de Catalunya. Av. de les Bases de Manresa 61-73, Manresa 08240, Spain. Phone: (34-3) 8777250; Fax: (34-3) 8777202 [email protected] ••• Dept Engin. de Sistemes, Automatica i Informatica Industrial Universitat Politecnica de Catalunya. Rambla Sant Nebridi, 10. Terrassa 08222, Spain. Phone: (34-3) 7398627; Fax: (34-3) 7398628 [email protected] / [email protected]
Abstract: This paper presents a new analytical tuning of Proportional-IntegralDerivative controllers based on the well known Linear Quadratic Regulator optimization problem for a wide class of processes whose behaviour can be modelled with with a constant time and a dead time delay. When the dead time delay is approximated by a first order Pade proposal, an optimal control problem gives an analytical solution for the tuning of the PlO parameters, and the performances obtained on the original model, without approximation, are very satisfactory. Also, the positive results obtained by simulation with this new PlO control design respect to the conventional empirical formulas to tune PlO controllers are very promising to be applied in the future to industrial processes. Copyright © 2000 IFAC.
Keywords: Optimization, PlO regulator.
I.
dead time) can be easily estimated by conventional techniques. For this class of processes, classical contributions (Ziegler and Nichols, Cohen and Coon, Lopez, Rovira...) give useful empirical formulas to tune PlO controllers
INTRODUCTION
A large number of industrial processes can be modelled by a dead time plus a first order system and their parameters (gain, constant time and
1 The
authors are members of LEA-SICA
271
in order to minimize a certain performance ( lSE, lAB, ITAE..).
(4)
Our objective is to give a new formula to tune PlO controllers for this class of processes in order to minimize the integral square error (ISE) of the controlled process for a abrupt change in the setpoint. The difference with respect to the precedent techniques is that the solution is obtained straightway when the delay is approximated by a first order Pade proposal. Generally, this approximation is acceptable when the constant time is much bigger than the pure delay, and the results can be applied to real processes.
If we assume that all states are measured and available for feedback, is well known that there exists a unique full state gain matrix F = (fl f2) such that the control law
minimizes
., JO(u)=f(yl+ PU 1 )dt o
(5)
the cost function: For the plant we are considering, which verifies the above conditions, this control law can be expressed as:
In the next section, a formulation of the PID as an optimization LQR problem will be presented and the ISE criteria minimization will be solved in the section Ill. The results obtained with this new tuning of PlO controllers has been compared with the conventional empirical formulas for the same simulated systems will be shown in the section IV and finally the conclusions of this paper will be presented in the section V. 2.
u· = - Fx
U o• ( t ) =-Kpy-K D -dy dt
(6)
The linear quadratic optimal regulator is in this case a PD controller. It is well known that to achieve a zero steady state (unbiased) error, for a constant set-point, the controller must incorporate an integrator in its structure. This can be accomplished by augmenting the plant with the integral of the output and considering a new LQR problem, weighting the integral output in the cost function [4]-[5].
PlO AS AN OPTIMIZATION LQR PROBLEM
We consider a linear second order plant with input u(t), output y(t) and transfer function: which approximates (first order Pade) a first order plant with a pure delay T, time constant 't and gain K. The strictly positive parameters bl> bo , al , ao are given by :
The augmented plant, whose states are still measurable and available for feedback, is:
. (A
A =
C
and we can consider a new cost function:
K
b,=r
00=-
rT
rT
(8)
o
(2)
T +2r
2K
bo=-
., J~(U)=f(yl+PU1+A.zl)dt
2
Q/
""'--;:r-
. x = Ax + Bu
= Jydt I
A state space realization of the plant is:
where z
is the new state variable, and
o
(3)
A ~ O.
y = ex
The control law u1..· = -fIX, - f2x2 - f3z which minimizes h can be expressed now by :
where x = ( XI X2 ) T is state variables vector, and the matrix A, Band C:
and the three PID constants are given as follows:
272
Kp
12
(10 )
1 + b , 12 =
13 1+ b,/
00
ISE
2
00
+00
o
-00
JI e(t) rdt = 2~ JIE (jw) rdw
(16)
The Laplace transform expression for the error signal is in our case: (17;
E (s)
~)
(12 )
A
where D is:
The analytical solution of the Riccati equation is obtained I and therefore the controller (K p , Ko , K( ) in terms of A. and p. The f l , f2 and f3 expressions are in the equation rT I,o bt bt I,·
ya
b'V-p·aD+
+
-
p.
(18)
In this way we obtain a expression of this criteria
rr
l~Vp
ISE as function of the parameters
(13)
Ji and p:
a, bo
, 1
bo
I,·
l+ b
,{T
,VL
bob,
I
J •
IF
These expressions (13) offer the possibility to study differents specifications for the system (plant - regulator) in terms of A. and p.
In order to fmd the values of A. and p which minimize ISE it is necessary to solve the equations system (20):
ISE CRITERIA MINIMIZAnON
a (ISE)
a:r;:
Criteria like ISE can be used to characterize set point responses if the error is interpreted as the error due to a unit step change of the set point. The block diagram of the PlO regulator (10)-(13) and the plant r
(15)
We can solve the square error integral by using the Parseval theorem:
where
3.
= Je 2 (t)dt o
The gain matrix F that we obtain depends obviously of the two parameters A. and p which appear in the expression of h. In this way any value of A. and p positives, defmes a PlO controller. In order to determine the expression of the regulator parameters in terms of A. and p we have to solve the Riccati equation:
(13).
(14)
We will found the values of the parameters A, p that minimize the integral square error (ISE):
1+ bl / 2
KD K,
e(t) = r(t) - y(t)
I,
Po is shown in figure
e
~
H
0 (20)
=
0
ap
1. After very tedious calculus we can find analytically expressions [1] for the values of A. and p that solve the equations system (20) and they are the unique solution of this equations system (21). We can prove that this solution is in fact a minimum.
T y
u
Re..I.,,,,
a (ISE)
=
Plan'
Fig. 1. Block diagram ofthe plant and the regulator
Where the error signal is:
273
(21)
A = a~ b; 2 bo
=..!t'
o.
/ / . . -K rovira /
06
2
/
+---Kemin
Replacing these values of A and p in the equations (10), (13) we can find the tuning of the optimal PlO controller:
02
o
o
15
'0
20
25
JO
Fig. 2 Unit closed loop response ofthe plant Po and the regulator K...;•.
The results show a faster response of our PID without overshoot in the response. These results has been obtained in both cases with a 2-degree of freedom structure of the PlO controller (see figure 3), that is mean that the proportional and derivative terms act only on the process variable and not on the control error in order to avoid undiserable impulse, due to the tep change of the set-point.
The proposed PlO controller based in LQR problem has an Internal Model Control (IMC) structure and tends to cancel the two model poles of the process. A comparative study of our proposed PlO controller and the derived from the IMC structure [3] applied to the same model has been done in [2].
4.
COMPARATIVE RESULTS
In order to analyze the performances of the new tuning PlO controller, different scenarios has been studied by simulation and a graphically comparison has been done with the conventional empirical tuning PlO controllers for the same simulated process.
Fig. 3 Two degree offreedom structure ofthe PID controller
4.2. Simulated second order process with dead time We are interested in to apply our PlO controller
4.1. Simulated process with dead time P (s) = I
In order to check the performance of this PlO controller ( ~min ) (22) with the original model, without approximations, some tests has been done with satisfactory results. In the figure 2 we show the results obtained for the plant Po _
10 5s+1
Po---e
1 e -0.16 s s1 +4 s+ 1
(23)
in high order plants with dead time (23). In the figure 4 we show a unit step response for the plant PI(S) (23) and the PlO controller ~in , according to the block diagram shown in figure 1. For to obtain the PID controller ~in , we approximate the plant (23) by a first order plant with a dead time and the parameters are :
-1s
and we compare the step response with one of the best empirical tuning of PlO controllers, proposed by Rovira to minimize the IAE [6].
K=l
274
r=1ns~
T=aBs~
1.2
r5. 0.8
SENSITIVITY TO THE UNCERTAINTY OFPROCESSP~ETERS
0.6
As a measure of robustness we will use the "loop margin". The loop margin M is simply the minimum distance in the complex plane between the Nyquist plot of the loop gain L=PoK and the critical point -1 :
0.4 0.2
Of-' -0.2 0
10
5
15
25
20
M
30
= minll + L(jw)! w
It can be proved that for the system composed of
the plant Po described in equation (I) and its corresponding ~in controller (22), the loop margin is always Mo=O.5 for any value of the plant parameters. In this section we study the changes in the loop margin M when the Kernin controller corresponding to the plant Po is applied to a new plant Po whose gain K , or whose dead
Fig. 4 Unit closed loop step response ofthe plant PI and the regulator K......
4.3. Flow control in a real process We have a real plant with two interconnected tanks, and we are interested now in the flow control. See figure 5.
time i are slightly different from those of the nominal plant Po. Especifically, we assume variations of the type:
-dJ-£h
K = (1 + a)K i = (1 + fJ)T where K and T are the nominal values of the gain < < 1 . Let's analyze and dead time, with different situations:
la! l.IfiI
!
Fig. 5 Two interconnected tanks for flow control
fJ = 0 the plant Po is given by:
5.1. When
The transfer function which approximate this real process is P(s): The figure 6 show us the unit step response for
A
(l+a)K(I-
~ s)
Po = ------0->---....,..
1
e -0.23s
(I+~{I+ ~ s)
(24)
0.2775 s+l and it can be proved that the loop margin M is: the real plant P and the regulator ~in according to the structure shown in the figure 3.
1
M = -(I-a) 2 Mo-M
K-K
a Mo K As expected, for positive values of a, M
08 0.6
04
= --=
0.2
00
1.5
2.5
3
5.2. When a=O the plant
3.5
Fig. 6 Unit closed loop step response ofthe real plant P and the regulator K...i •.
275
Po is given by:
It can be proved that there exist a range of values of P for which the loop margin M remains equal
The comparative results obtained by simulation promise very good performances of this tuning of PID controller for simulated and real industrial processes.
to M=O.5. Especifically, if -1 < ps 0.17 there is no change in the loop margin, so that the system appears to be robust in front of changes on T which are inferior to 17%. For greater values of jJ < I the value of the loop margin decreases and it is given by:
5.3. When a by:
* 0 and p * 0 the plant
ACKNOWLEDGEMENTS This work has been partly supported by Research Commission of Generalitat de Catalunya (Group SAC), ref. 1999SGR 00134 and Comisi6n Interministerial de Ciencia y Tecnologia, ref. TAP96-1114-C03-0 1
Po is given
REFERENCES Argelaguet R., Pons M., Quevedo 1., Aguilar Martin J.,"Analysis of achievable PID performances versus optimal linear regulator theory". SICICA'94, Budapest. 1994. Argelaguet R. " Estudio de limites de prestaciones de controladores lineales con estructura fija". Tesis doctoral. UPC. 1996.
It can be proved that the loop margin depends only of a either if p < 0 or if p ~ 0 and.
Astrom K.J., "Assessement of Achievable Performance of simple feedback loops". Int. J. Adapt. Control & Sig. Pro.,No.5,pp.319,1991. In both cases, its value is:
M. Athans. The status of optimal control theory and applications ho deterministic systems.
IEEE. Transactions on Automatic Control,
I M =-(l-a)
AC-Il:580-596,1966.
2
The same that was obtained when
p = O.
Parker K.T., "Design of proportional integral derivative controllers by the use of optimal linear regulator theory". Proc. IEEE, vol. 119,No.7,1972.
It is
worth to notice that the second case is accomplished in particular for any a>O and any positive p < 0.17 . This allows us to ensure that if the change on T is less than 17% and the gain is increased by a % the loop margin M will decrease also by a %.
6.
Rovira A. A.PhD. Dissertation, Dept of Chemical Engineering Louisiana State University, 1981.
CONCLUSIONS
A new analytical tuning of PID controller has been established, based on an augmented LQR optimization problem, for a wide class of industrial processes, represented by a model with one constant time and a dead time. The proposed PID cancel the dominated poles of the model and the gain of the controller gives a very fast response of the system to a step change of the set-point, avoiding the overshoot of the response.
276