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Analysis of Achievable PID Performances Versus Optimal LQR Control
Copyright @ IFAC Intelligent Ccmponcnts md InltJUments for Control Applications, Budapest, Hungary, 1994
ANALYSIS OF ACHIEVABLE PID PERFORMANCES VERSUS OPTIMAL LQR CONTROL R. ARGELAGUET
.#, M. PONS $, J. QUEVEDO .,
J. AGUILAR MARTIN • +
* Dept. ESAII. Unive.Politecnica de Catalunya. ETSEIT. Cl Colom. 11. 08222 Terrassa. Spain + lAAS-CNRS. 7. Av. Colonel Roche. 31077 Toulouse Cedex. France $ Dept. Matematica Aplicada W. Universitat Politecnica de Catalunya EUPM. Avda. Bases de Manresa. 61-73. 08240 Manresa. Spain # Dept. Enginyeria Electronica. Universitat Politecnica de Catalunya EUPM. Avda. Bases de Manresa. 61-73. 08240 Manresa. Spain
Abstract: Supervision of process control techniques can be based either on the observation of the real behavior or on the theoretical achievable performances of the closed loop system. In this paper, the PID regulator is taken as an optimal LQR controller applied to an extended plant, in order to weight the integral of the output. By this statement of the problem, some properties of the closed loop system are studied as a function of the integral weight. As expected, the cost function of the PID increases with the weight, and we analyze its asymptotic behavior. The robustness loop margin decreases, and it is possible to determine which values of the parameters in the cost function define a PID with an acceptable margin. Moreover, the existence of an asymptotic PID is proved. Keywords: Supervision, Three term-controller, LQR
1. INTRODUCTION PID tuning methodology is oriented by practical desired performances (Ziegler Nichols, etc, ... ). Nevertheless, none of those methods give a basis for a comparison neither they open a way to evaluate what are the achievable performances of the closed loop system. On the contrary optimization methods (LQR, LQG, minimum time, ...), depend explicitly on a valuated criterion, therefore the performance can be located with respect to particular values. When imbedding PID design problem in linear quadratic optimization theory, it appears that some robustness properties may diminish, so it is useful to be able to link robustness with the quadratic cost parameters. Conversely the components of the criterion may be analyzed in terms of the regulator design parameters.
(1)
which approximates, at low frequencies, a first order plant with a pure delay T. The state space realization of the plant is:
x=Ax+Bu y=Cx where x =
(Xl X ) T 2
is the vector
of state variables
A= [OIl ~ ~
B
=1
c= [1
2. PID AS AN OPTIMIZAnON LQR
-hl
1
L bo+~blJ 0]
and the parameters bl,bQ,al,ao, are given by
PROBLEM
We consider a linear second order plant with input u(t), output yet) and transfer function: 373
b
o
= 2K 'tT
Fixed p, any value of A. > 0 defines a PID controller and, in order to know which is the best value of this parameter to obtain different design specifications, it is interesting to study the controller performance for different values of A..
T+ 2't
bI =K't-
~=
'tT
If we assume that all states are measured and available for feedback, it is well known that there exist a unique full state gain matrix F = (FI F2) such that the control law u*=-Fx minimizes the cost function
In order to determine the expression of the
regulator parameters in terms of A. and p we have to solve the Riccati equation: T T A· P+PA· -p-IpB·B· P+Q=O where
00
10 (u) =
f (y2 + pu2) dt
(2)
o
Q;[ c:c ~ ]
for any positive value of the parameter p. For the plant we are considering (1), this control law can be expressed as
u· (t) = -K Y- K dy o P 0 dt
In [1], the analytical solution of this equation, that is to say, the expression of the matrix P as a function of the parameters p, A. is obtained. From P we can explicitly obtain the parameters FI, F2, F3 and therefore Kp, KD, KI of the controller in terms of p and A..
(3)
that is to say, the linear quadratic optimal regulator is in this case a PD controller. It is known that an integrator is needed so as to transform the system from "type 0" to "type 1", to achieve a zero steady-state error for a constant reference input
3. ROBUSTNESS LOOP MARGIN In order to measure the robustness of the
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.
system 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 and the critical point -1:
The augmented plant is:
. =[A ~ ] C
A
The loop margin is connected to the sensitivity transfer function S, which relates the external input d to the output y
and the new cost function 00
l A(u) =
f (y + pu2 + 'A:z2) dt
r;O
o
where z =
f y dt o
~ can write M=
is the new state variable, and A. ~ O. The control law = - Flx l - F2Xz - F3 z
ISdn)
11... '
\\herell .1100
indicates the infmity norm
u;,
It is not difficult to see that for the system composed of the plant (1) and the PID regulator defmed from p, A., the expression of the loop margin is:
which minimizes JA. can be expressed now by t
u~(t) = - Kp Y - Ko y - KI
~r.;;:J~t d
~~L!T+t
t
f Y dt o
M=
The controller F that we obtain depend obviously of the two parameters p, A. which appear in the expression of JA. .
1 1 + bl F2
(4)
If P is fixed and A. increases, we know that F2 also increases and, therefore, M decreases as
374
expected. In the limit, when A,
~
00 we have
1 (Q) lim JHOo
F2~00 andM~O.
On the. other hand, if A is fixed and p increases, then F2 decreases, and when p ~ 00 we have F2 ~ 0 and M ~ 1. In this way, we can compensate the loss of robustness caused by the introduction of the parameter A by augmenting p. Specifically, given any loop margin Ml, it is necessary to take p> Pt to obtain M>Mt. Once p is selected, there exist Ap such that M>M t if and only if A< Ap For example, to obtain a loop margin M>0.5 we can take any
_0"_A=
JP
r::;-
~(~ x1(0) +"2(0» 2<10 bo
2 ="(
and it is not difficult to see that lim (1 (u*) - "( p-¥<> 0 A.
JP ) is fmite,
what proves that 10(u~) tends asymptotically to a parabola.
1 (u*) Finally, lim
p~
"r;:/~ = 2"( and 1A.(u~ .,p
tends also to a parabola. 5. ASYMPTOTIC PID
2
p>
K
(1+~)( 3+~)
For any positive parameters A, p the closed loop transfer function for the corresponding PID controller is:
and, once p is selected, choose any A such that
O~.r;:<
4fP
r1~_lJ[ 1~[J H~\K1 JPJ
KfL
T2
where A= 4. PID REGULATOR AND LQR COST
In this section we study the increase of the cost function when an integral weight A:;tO is introduced. Specifically, for any value of the parameter p, we compare 10(u~), 10(u~) and
It is not difficult to analyze the location of the poles as A changes, for a fixed value of the parameter p.
lA. (u~)
. If A=A", wtrre
bIP 2 (
ho2 )
A,,=-4 ~+p , 4bo
It is clear that 10(u~) ~ 10(u~) ~ lA. (u~)
There is a two order pole at All tree values are increasing functions of p, but their asymptotic behavior is very different
bI -s P =b-
While 10(u~) tends to a finite limit as p~oo, 10(u~) and lA.(u~)
tend to infinity. To be precise, it can be proved that lim 10(t\,) JH""
=--L(a1
2"o~
that is to say 10(u~ tends
~(O)+"2(O» 2+ CIox1(ol)
~ymptotically to a straight
line which depends on the initial conditions of the state variables. On the other hand,
o
For A <
~ ~+ ~ )
Ap there are two different negative real
poles which tend respectively to -2sp and 0 when A~ O. In this way, it is clear that the closed loop transfer function for the PID obtained for A.=O is F (b -b s) F (p, 0) = 2 o2 I s + sp Finally for A> Ap there are two complex conjugates poles which tend to infmity as A increases. In this case, the transfer function tends to
375
5. ACKNOLEDGEMENT when A ~
This work has been supported by the Spanish Board for Research and Technology (CICYT) under contract T AP93-0596-C04, and the paper has been partiall y supported by the group for the Study and Research in Automatic Control (CERCA).
00.
N=O
6. REFERENCES Argelaguet, R., M. Pons, J. Quevedo, J. Aguilar Martin. Analysis of the PID controllers designed from optimal linear regulator theory. To appear. Astrom, K.J. (1991). Assessment of Performance of simple feedback loops. Int. 1. Adapt. Control & Sig. Proc. No.5, pp. 3-19.
A~
It can be proved that F(p,oo) is the closed loop transfer function of a PID defined by the parameters: : K·= -t>
lirnK 1..-+00 -t>
bo+~bl b2
Athans, M. (1971). On the design of PID controllers using optimal linear regulator theory. Automatica, vol. 7, pp. 643-647. Pergamon Press.
=1..(1+4;) K
Boyd, S., C. Barrat, S. Norman, (1990). Linear Controller Design: Limits of Performance Via Convex Optimization. Proc. IEEE, vol. 78, No.3.
1
Ko- =
&,
K-=
lim K = _1_ = ..1. 0 b K
1..-+00
&, ~bo b
im K =
1..-+00
1
2
Parker,K.T., B.A., M. Sc. (1972). Design of proportional integral derivative controllers by the use of optimal linear regulator theory. Proc. IEEE, vo1.119, No.7.
=..1...(1+2;) Kf
1
It is interesting to notice that this PID limit is, in fact, independent of p. Thus , any PID obtained from any particular values of the parameters p, Alies between the optimal LQR regulator obtained for A = 0 (PD controller) and a PID limit obtained by increasing to infinity the integral weight
376
ANALYSIS OF ACHIEVABLE PID PERFORMANCES VERSUS OPTIMAL LQR CONTROL R. ARGELAGUET
.#, M. PONS $, J. QUEVEDO .,
J. AGUILAR MARTIN • +
* Dept. ESAII. Unive.Politecnica de Catalunya. ETSEIT. Cl Colom. 11. 08222 Terrassa. Spain + lAAS-CNRS. 7. Av. Colonel Roche. 31077 Toulouse Cedex. France $ Dept. Matematica Aplicada W. Universitat Politecnica de Catalunya EUPM. Avda. Bases de Manresa. 61-73. 08240 Manresa. Spain # Dept. Enginyeria Electronica. Universitat Politecnica de Catalunya EUPM. Avda. Bases de Manresa. 61-73. 08240 Manresa. Spain
Abstract: Supervision of process control techniques can be based either on the observation of the real behavior or on the theoretical achievable performances of the closed loop system. In this paper, the PID regulator is taken as an optimal LQR controller applied to an extended plant, in order to weight the integral of the output. By this statement of the problem, some properties of the closed loop system are studied as a function of the integral weight. As expected, the cost function of the PID increases with the weight, and we analyze its asymptotic behavior. The robustness loop margin decreases, and it is possible to determine which values of the parameters in the cost function define a PID with an acceptable margin. Moreover, the existence of an asymptotic PID is proved. Keywords: Supervision, Three term-controller, LQR
1. INTRODUCTION PID tuning methodology is oriented by practical desired performances (Ziegler Nichols, etc, ... ). Nevertheless, none of those methods give a basis for a comparison neither they open a way to evaluate what are the achievable performances of the closed loop system. On the contrary optimization methods (LQR, LQG, minimum time, ...), depend explicitly on a valuated criterion, therefore the performance can be located with respect to particular values. When imbedding PID design problem in linear quadratic optimization theory, it appears that some robustness properties may diminish, so it is useful to be able to link robustness with the quadratic cost parameters. Conversely the components of the criterion may be analyzed in terms of the regulator design parameters.
(1)
which approximates, at low frequencies, a first order plant with a pure delay T. The state space realization of the plant is:
x=Ax+Bu y=Cx where x =
(Xl X ) T 2
is the vector
of state variables
A= [OIl ~ ~
B
=1
c= [1
2. PID AS AN OPTIMIZAnON LQR
-hl
1
L bo+~blJ 0]
and the parameters bl,bQ,al,ao, are given by
PROBLEM
We consider a linear second order plant with input u(t), output yet) and transfer function: 373
b
o
= 2K 'tT
Fixed p, any value of A. > 0 defines a PID controller and, in order to know which is the best value of this parameter to obtain different design specifications, it is interesting to study the controller performance for different values of A..
T+ 2't
bI =K't-
~=
'tT
If we assume that all states are measured and available for feedback, it is well known that there exist a unique full state gain matrix F = (FI F2) such that the control law u*=-Fx minimizes the cost function
In order to determine the expression of the
regulator parameters in terms of A. and p we have to solve the Riccati equation: T T A· P+PA· -p-IpB·B· P+Q=O where
00
10 (u) =
f (y2 + pu2) dt
(2)
o
Q;[ c:c ~ ]
for any positive value of the parameter p. For the plant we are considering (1), this control law can be expressed as
u· (t) = -K Y- K dy o P 0 dt
In [1], the analytical solution of this equation, that is to say, the expression of the matrix P as a function of the parameters p, A. is obtained. From P we can explicitly obtain the parameters FI, F2, F3 and therefore Kp, KD, KI of the controller in terms of p and A..
(3)
that is to say, the linear quadratic optimal regulator is in this case a PD controller. It is known that an integrator is needed so as to transform the system from "type 0" to "type 1", to achieve a zero steady-state error for a constant reference input
3. ROBUSTNESS LOOP MARGIN In order to measure the robustness of the
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.
system 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 and the critical point -1:
The augmented plant is:
. =[A ~ ] C
A
The loop margin is connected to the sensitivity transfer function S, which relates the external input d to the output y
and the new cost function 00
l A(u) =
f (y + pu2 + 'A:z2) dt
r;O
o
where z =
f y dt o
~ can write M=
is the new state variable, and A. ~ O. The control law = - Flx l - F2Xz - F3 z
ISdn)
11... '
\\herell .1100
indicates the infmity norm
u;,
It is not difficult to see that for the system composed of the plant (1) and the PID regulator defmed from p, A., the expression of the loop margin is:
which minimizes JA. can be expressed now by t
u~(t) = - Kp Y - Ko y - KI
~r.;;:J~t d
~~L!T+t
t
f Y dt o
M=
The controller F that we obtain depend obviously of the two parameters p, A. which appear in the expression of JA. .
1 1 + bl F2
(4)
If P is fixed and A. increases, we know that F2 also increases and, therefore, M decreases as
374
expected. In the limit, when A,
~
00 we have
1 (Q) lim JHOo
F2~00 andM~O.
On the. other hand, if A is fixed and p increases, then F2 decreases, and when p ~ 00 we have F2 ~ 0 and M ~ 1. In this way, we can compensate the loss of robustness caused by the introduction of the parameter A by augmenting p. Specifically, given any loop margin Ml, it is necessary to take p> Pt to obtain M>Mt. Once p is selected, there exist Ap such that M>M t if and only if A< Ap For example, to obtain a loop margin M>0.5 we can take any
_0"_A=
JP
r::;-
~(~ x1(0) +"2(0» 2<10 bo
2 ="(
and it is not difficult to see that lim (1 (u*) - "( p-¥<> 0 A.
JP ) is fmite,
what proves that 10(u~) tends asymptotically to a parabola.
1 (u*) Finally, lim
p~
"r;:/~ = 2"( and 1A.(u~ .,p
tends also to a parabola. 5. ASYMPTOTIC PID
2
p>
K
(1+~)( 3+~)
For any positive parameters A, p the closed loop transfer function for the corresponding PID controller is:
and, once p is selected, choose any A such that
O~.r;:<
4fP
r1~_lJ[ 1~[J H~\K1 JPJ
KfL
T2
where A= 4. PID REGULATOR AND LQR COST
In this section we study the increase of the cost function when an integral weight A:;tO is introduced. Specifically, for any value of the parameter p, we compare 10(u~), 10(u~) and
It is not difficult to analyze the location of the poles as A changes, for a fixed value of the parameter p.
lA. (u~)
. If A=A", wtrre
bIP 2 (
ho2 )
A,,=-4 ~+p , 4bo
It is clear that 10(u~) ~ 10(u~) ~ lA. (u~)
There is a two order pole at All tree values are increasing functions of p, but their asymptotic behavior is very different
bI -s P =b-
While 10(u~) tends to a finite limit as p~oo, 10(u~) and lA.(u~)
tend to infinity. To be precise, it can be proved that lim 10(t\,) JH""
=--L(a1
2"o~
that is to say 10(u~ tends
~(O)+"2(O» 2+ CIox1(ol)
~ymptotically to a straight
line which depends on the initial conditions of the state variables. On the other hand,
o
For A <
~ ~+ ~ )
Ap there are two different negative real
poles which tend respectively to -2sp and 0 when A~ O. In this way, it is clear that the closed loop transfer function for the PID obtained for A.=O is F (b -b s) F (p, 0) = 2 o2 I s + sp Finally for A> Ap there are two complex conjugates poles which tend to infmity as A increases. In this case, the transfer function tends to
375
5. ACKNOLEDGEMENT when A ~
This work has been supported by the Spanish Board for Research and Technology (CICYT) under contract T AP93-0596-C04, and the paper has been partiall y supported by the group for the Study and Research in Automatic Control (CERCA).
00.
N=O
6. REFERENCES Argelaguet, R., M. Pons, J. Quevedo, J. Aguilar Martin. Analysis of the PID controllers designed from optimal linear regulator theory. To appear. Astrom, K.J. (1991). Assessment of Performance of simple feedback loops. Int. 1. Adapt. Control & Sig. Proc. No.5, pp. 3-19.
A~
It can be proved that F(p,oo) is the closed loop transfer function of a PID defined by the parameters: : K·= -t>
lirnK 1..-+00 -t>
bo+~bl b2
Athans, M. (1971). On the design of PID controllers using optimal linear regulator theory. Automatica, vol. 7, pp. 643-647. Pergamon Press.
=1..(1+4;) K
Boyd, S., C. Barrat, S. Norman, (1990). Linear Controller Design: Limits of Performance Via Convex Optimization. Proc. IEEE, vol. 78, No.3.
1
Ko- =
&,
K-=
lim K = _1_ = ..1. 0 b K
1..-+00
&, ~bo b
im K =
1..-+00
1
2
Parker,K.T., B.A., M. Sc. (1972). Design of proportional integral derivative controllers by the use of optimal linear regulator theory. Proc. IEEE, vo1.119, No.7.
=..1...(1+2;) Kf
1
It is interesting to notice that this PID limit is, in fact, independent of p. Thus , any PID obtained from any particular values of the parameters p, Alies between the optimal LQR regulator obtained for A = 0 (PD controller) and a PID limit obtained by increasing to infinity the integral weight
376