- Email: [email protected]
Design and Implementation of µ Controllers for Real Industrial Plants
Copyright @ IF AC Robust Control Design, Prague, Czech Republic, 2000
DESIGN AND IMPLEMENTATION OF ft CONTROLLERS FOR REAL INDUSTRIAL PLANTS Marcio Fantini Miranda' Fabio Gont;alves Jota"
• Federal University of Minas Gerais - COLTEC - Brazil [email protected] •• Federal University of Minas Gerais - Elect. Eng. Dept - Brazil fabio@cpdee. 'Ilfmg. br'
Abstract: This paper considers the design and implementation of robust multi variable controllers for a real process control plant. The plant used for experimental tests is intended to control output flow rate and level of a product tank. First, a detailed description of a method for finding and representing uncertainties is presented. Next, a robust control is designed using the J1. synthesis. Comparisons of uncertainties (structured) parametric and non-parametric are also presented. They show how conservative the J1. analysis could be and, in some cases, how to reduce the conservatism. Copyright© 2000IFAC
Keywords: H-infinity Control; Robust Control; Uncertain Linear Systems: Process Control
l. INTRODUCTION
the output flow rate and level, the most common controlled variables (together with temperature) in process control.
The question of controlling uncertain systems has received considerable attention in the last ten years (Dorato et al., 1993). Depending on the structure and the type of the uncertainties present in the system , various frameworks have been considered such as, robust 11.00 control, e.g. (Doyle et al., 1989),(Skogestad and Postlethwaite, 199~) ; LMI, e.g. (Boyd et al., 1994), gain-scheduling, e.g. (Shamma and Athans, 1992) and nonlinear 11.= control, ego (James, 1995).
2. NOTATION The notation adopted here is standard in the 11.00 framework. A family or a set of plants , G(s) , has its nominal value expressed as Go(s). A (suboptimal) 11.00 controller, K(s), is the one that gives a closed loop matrix that satisfies:
The necessity of robustness may arise due to the existence of varying parameters, unmodelled dynamics, or a need to increase safety against exogenous inputs. In this work, we consider the ft-synthesis and analysis for an Interacting Tank System. This pilot scale plant presents characteristics similar to real industrial systems. The controller design is made so as to have a robust response against plant variations.
IIFt(Po, K)lloo ::; ,,(,
(1)
where 11 .1100 is the 11.00 norm, Ft represents the lower FLT of Po and K, and "( is some positive real number. Po(s) is the generalized matrix, which corresponds to Go plus the associated weighting matrices. J1. is the structured singular value which is used when the uncertain system has structured uncertainties (Packard and Doyle, 1993).
The use of such a system is directly related to the relevance of the variables to be controlled, that is,
585
3. THE INTERACTING TANKS SYSTEM
Rho and Rhi, respectively. We have considered h2 as being the state Xl and h3 as being the state X2 . FUnctions h(ut} and !2(U2) represent the steady state characteristics of valves FCV-Ol and FCV02, respectively. FUnction f4(P) represents the relation between output flow-rate and the relative aperture of the manual valve FV-04.
The experimental runs have been performed in a pilot-scale Interacting Tank System, ITS (Jota et al., 1995) (see figure 1). The ITS is composed of a 700 I reservoir (TQ-Ol) and two 300 1 passively interconnected tanks (TQ-02 and TQ-03). The coupling between these two tanks can be manually controlled by means of flow valves FV-03 and FV04. The basic operation of the system consists of pumping the liquid fluid from the reservoir (TQ-Ol) directly into the second tank (TQ-02). From TQ-02, the fluid flows naturally to the product tank (TQ-03) . The liquid is then pumped back to the reservoir by BA-02. The simultaneous control of level and flow-rate in the third tank is accomplished by equal-percentage pneumatic valves, FCV-Ol and FCV-02. A PLC is used to interface the plant to a microcomputer, where the control algorithms actually run. All the signal transmissions are accomplished via current loops of 4 to 20 mA.
The corresponding linear model is given by:
[;~] = [~~I
~~l] [:~] + [~ ~3] [:~]
(2)
[~~] = [~ t~]
[:~] + [~ -~k3] [:~] .
(3)
where k l , k2 and k3 vary with the chosen operating point and are given by: kI
=_
3.1 System Modeling
!4(P) 2AJXlO
-
(4)
X20
k2
=.!. Oh(UIO) JhlO + hbl -
ka
= __1_ 0!2(U20) J
A
he
(5)
+ hb2
(6)
OUI
RhoA
OU2
X20
k4 - _~ !2(U20) 1 . - 2 RhoA JX20 + h b2 ' k12 = kI - k4 tl
= !2(U20) Rho
Fig. 1. Schematic Diagram of the Interacting Tanks System, ITS
1 JX20
+ hb2
(7) (8)
(9)
From the previous state space model, a family of plants is derived and expressed as:
The modeling of the ITS is based on the equations of the mass balance between the tanks. The model can be expressed by the system of equations: dh2 dt h2 dh3 dt
h3 =
+ q23 A A +h +f
_ qo
= _ !2 (U2) J ARho
h
b2
3
4
(P) J h2 - h3 A 4. DEFINITION AND ANALYSIS OF THE ITS UNCERTAINTIES
where qi is the input flow-rate (in TQ-02), qo is the output flow- rate (in TQ-03), q23 is the flow-rate between TQ-02 and TQ-03, h2 is the level of TQ-02 and h3 is the level of TQ-03. The cross sections of both TQ-02 and TQ-03 is A. hI is the (average) level of the liquid in the tank TQ-Ol and he corresponds to the height of the water column that exerts pressure on BA-Ol (see figure 1). hbl and hb2 represent the pumping capacity of BA-Ol and BA-02, respectively. The hydraulic resistances of the output and input are
From equations (4) to (9) and from experiments performed on the real plant, it is possible to draw some conclusions about the values of the variables k i , (i = 1, ... ,4), k12 and tl defined for the linear model. (1) The variation of kI is a function of the difference of level between tanks TQ-02 and TQ-03 and the aperture of valve FV-04. Figure 2 (a)
586
Fig. 2. Variation of A (a) and control signal
U2
kl ( )
for different
(3) Define the operational (nominal) point. In this case , we have chosen, the sixth point for FV-04 100% open as the nominal operational point. (4 ) From experimental data and physical equations of the system , find the range of variation of the variables of the equations (10) to (15) . (5) From the 12 equilibrium points define different realizations (Ai , B; , C i , Di) , i == 1 .. . 12.
as function of the
Rh23 ·
shows the (experimental) relation kl with the output flow, defined by 1.12. It is noteworthy the fact that kl can vary up to 700 o/c if both out-flow and FV-04 aperture are changed. (2) The variation of k2 is a function of the gain of FCV-01 valve. So, k2 is important on the determination of the gll (s) transfer function . (3) The variation of ka is function of the gain of the FCV-02 valve. The importance of k3 variation is due to: i) k3 is the gain of g)2 (which can be considered as the perturbation on gll and ii) k3 varies about 800 o/c.
4.1.1. Range oJ Variation oJ The Linellr Constants. Considering the "quasi-steady state" points given in figure 3 the possible variations of the linear equation are given by (10) to (15) : -0.036::; k)
::; -3.2 X IQ-3
0.0062::; k2 ::; 0.0133
-2.6685 x
IQ-2
(11)
k12 -2.66627
X
IQ-2
0.0013::; k3 ::; 0.0082
-3.42 x 10- 5 2.17 x
::;
k4 ::; -1.07
IQ - 3::; t) ::;
6.78
(10) (12) (13)
X
IQ-5
X IQ-3 .
(14 ) (15 )
4.1 Uncertainty Representation
The uncertainties on the ITS model are mainly derived from the variations of valve FV-04, the output flow rate and the level difference between TQ-02 and TQ-03. The variation on the control valves , FCV-01 and FCV-02 , can also vary the operational point, resulting in a change of the nominal model. The uncertainty model is derived in the following steps: (1) From the curves of valves FCV-01 and FCV02, we get the equilibrium points, where i; == 0, that is points where q; == q23 == qo (see figure 3 (b));
5. PROBLEM SETUP As shown in section 3, the ITS plant has some interesting characteristics t.hat motivated the use of robust control. As it is possible to build a uncertainty model that fits well on the 1-£00 and J1. framework , we can look for a controller K(s) that results on a closed loop with robust stability and performance, defined by the J1. criteria. Definition 1 gives the formulation of the control objective. Definition 1. Find a controller K(s) that satisfies:
••
(1) Closed loop robust performance and stability (by the J1. synthesis we must have J1. ::; 1). It is important to note that we have a normalized system and we wrote the uncertainties in order to get 1161100 ::; 1; (2) good regulation response. Wich mean that we are interested in to keep the level and output flow rate fixed on specific values , despite perturbations; (3) steady state error less than 5%. It is a common design specification (Dorf and Bishop, 1998). In fact we require a zero steady state error; (4) the faster controller pole is limited by the sampling frequency. As the faster sampling frequency is 1 Hz (21l" rad/s) , the faster pole can not be much bigger than 1l" rad/ s.
~,.
!
6'·'
o. 02
~
~
~~~~a>~~'~A--~,.7.--~,,~~ Ib)
V ....... SilgNlf%)
Fig. 3. Characteristic curves of FCV-Ol and FCV-02 Valves. (a) Original curves; ( ) normalized curves with t e polynomial fit and t e equili rium points
(2) The equilibrium points which will define the "different plants" can be found from the condition "output flow == input flow , which is a very difficult condition to obtain in the open loop real plant.
587
6. THE DESIGN SETUP The designs are based on the well-known mixed sensitivity problem (Kwakernaak, 1993),(Skogestad and Postlethwaite, 1996) which uses the sensitivity functions for the performance specifications. Together, it is added the uncertainty description in order to fit the problem on the JL framework. According the uncertainty representation, two designs are considered: (1) The first one is based on the structured parametric uncertainty model. (2) The second design is the one in which the structured parametric uncertainty is represented as a non-parametric one, in order to reduce the conservativeness of the JL design. We also consider two kinds of experiments: one for the FV-04 valve 100% open and for 20% open. With these experiments we are testing the control for the larger possible variations. Both the designs are based on figure 4, which presents a standard mixed sensitivity problem with structured uncertainty. The weighting function W 1 is used to specify the desired sensitivity function S(s) and the weighting function W 2 is used to specify the control signal.
The design by the DK iteration resulted into 5 iterations with a JL peak value of about 1.5. The final controller was discretized and applied to the plant. Sections 9 and 10 present experimental results, in which FV-04 is 100% open, and a profile for level and flow rate with large variations. The JL peak value about 1.5 means that the robust performance has not been achieved. In order to improve this result we can: i) reduce the performance specification (W1 ) , by changing or the bandwidth or the sensitivity peak ;ii) reduce the W 2 , which implies in increasing the control signal; iii) reduce the uncertainties by a factor about 1.5. The experience has shown that the first option results on a very slow control and the second gives a control with a very high gain and very fast pole, which cause problems on the implementation. The third consideration can be a good solution, if we can guarantee that the real system can be represented by a less conservative uncertainty. We believe that we can not only reduce the uncertainties (and the performance specifications) by a factor of 1.5, because we can loose some information about the system variation. That is why we propose change the uncertainty representation. 8. SECOND DESIGN - JL SYNTHESIS WITH NON-PARAMETRIC UNCERTAINTIES
This design aimed to reduce the uncertainties and to produce a less conservative control. For such, the parametric uncertainties were manipulated to get a dynamic representation, similar to a "unmoldeled dynamic" . In this way, the six parametric uncertainties could be transformed into two dynamic uncertainties that represent the overall variations of the 4 entries of the nominal matrix, Go(s). As a result , an uncertainty system of the type: multiplicative input dynamic uncertainty system with a 2 x 2 diagonal matrix such as 1I~lloo ~ 1 was obtained. This uncertainty system can be represented by:
Fig. 4. Diagram of the S/KS Scheme Design with Structured Uncertainty
7. FIRST DESIGN - JL SYNTHESIS WITH PARAMETRIC UNCERTAINTIES This design has considered the parametric (structured) uncertainties on the model of the ITS plant. The six constants varying on the range given by equations (10) to (15) were taken into account. The uncertainty block ~ is a diagonal 6 x 6 matrix with 11~lloo ~ 1. We have also considered a fictitious uncertainty (Balas et al., 1992) for the performance specification defined by W 1 and W 2 , that is, a full 2 x 4 matrix. Final weighting functions were obtained after a few attempts, which results:
G(s)
= Go(s)(l + ~(S)Wd(S))
(16)
where ~(s) is any matrix such that 11~lloo ~ 1 and Wd(S) is a diagonal matrix that represents the weights of the uncertainties as a function of the frequency. The transformation from parametric to non-parametric uncertainty is done by the following procedure. Let a 2 x 2 uncertainty system be defined as:
0.5s + 2 x W1 =
S
[
o
10- 2
~
0.58 + x 10-'
1
=95 1(1 +Wll(S)~l)Ul + 95 2 (1 + W12(S)~2)U2 Y2 = 95 1 (1 + w21(S)~3)U1 +
(17)
+95 2 (1 + w22(s)~4)U2.
(18)
Y1
and
W _ [0.01 0 ] 2 0 0.1 .
588
Since 161 ~ 1, the maximum and minimum gain values of each (nominal) transfer function, gij (s), are determined from the relations:
g~az
= g~i (1 + w ij )
(19)
g:!.in
= g~j (1 - w
(20)
ij
).
On solving the system equations (19) and (20) for the functions w ij , we get:
Fig. 6. Time Response of the Flow Control (it is almost the same for all experiments)
(21)
= 1,2) we define the
With the functions w ij (i,j matrix W(s):
W(s)
9. EXPERIMENTAL RESULTS FOR FV-04 100% OPEN
wll(s) W12 (S)] w 22 (s) ,
As can be seen in figures 7 and 8 the time response of level and output flow rate presents good tracking response and good rejection perturbation (if we consider that both level and flow are perturbations for each other) . It is also worth of note that the time response due the two designs (with parametric and non-parametric) are very similar, despite the difference on the J.l peak value.
(22)
= [ w21(s)
Equation (21) gives the entries of W(s), whose frequency responses are depicted in figure 5.
:~ :: '''D''O] .. ::0 ':Qj .,
O'~O"
Tim. R•• pon •• tor Le.".1 Loop
u•
'0'
D, ~...
'0'
u.
•.•
o•
•n
u
~
~
or
~
~
Fig. 5. Frequency response of entries of W(s) Fig. 7. Time Response of the Level of the ITS Plant With
In order to represent the uncertainty system as in equation (16), we define a diagonal matrix,
Parametric Uncertainties and FV-04 100 % Open
(23) that is, the entries of Wd(s) are determined from the frequency response of W (s) so as to encompass the worst-case dynamics (see figure 5). It is possible to do this approximation because the functions 22 are similar, and the same for funcW 21 and W tions w 12 and W 22 . For performance specification it was used the same weighting functions as the first design. It can be seen on figures that the time response is almost the same. So the main advantage was the J.l peak reduction.
Fig. 8. Time Response of the Level of the ITS Plant With Non-Parametric Uncertainties and FV-04 100
% Open
With these uncertainty representation, the design by the DK iteration results into 3 iteration with a J.l peak value equal 1.0.
10. EXPERIMENTAL RESULTS FOR FV-04 20% OPEN
In the sequel it is presented the time response of experimental runs on the level and output flow rate. As the flow loop is not so difficult to control it will present only the time response of the level loop. For all experiments, the flow varies from 30% to 85%, as shown on figure 6.
The consideration of valve FV-04 20% open is a hard task for the control, since gain of the system can vary up to 700 %. So the time response of the level is not so good when the output flow changes
589
About the uncertainties representation, it was shown (by the comparison between of the nonparametric and parametric controllers) that, with a changing on the representation, the robustness test could be improved with almost the same controller. We have also shown that, within the p. framework it is possible to get a robust control even for a very large system variation.
from 30 to 85 %. Nevertheless the system keeps stable, despite this large perturbation. Figures 9 and 10 present the time response for the condition of FV04 20% opened only, for the parametric controller and the non-parametric, respectively.
For the experiments presented here, we can conclude that, for the overall variations considering the p. analysis results on conservatives indications, and when we reduce the uncertainty, we can find controller that satisfy the performance and stability tests with good time response.
12. REFERENCES Fig. 9. Time Response of the Level of the ITS Plant with
Balas, Gary J., Jonh C. Doyle, Keith Glover, Andy Packard and Roy Smith (1992). p.Analysis and Synthesis Toolbox - User's Guide. The Math Works Inc. Boyd, Stepen, Laurent El Ghaoui, Eric Feron and Venkataramanan Balakrishnan (1994). Linear Matrix Inequalities in System and Control Theory. SIAM. Dorato, P., R. Tempo and G. Muscato (1993). Bibliography on robust control. Automatica 29(1),201-213. Dorf, Richard C. and Robert H. Bishop (1998). Modern Control Systems. eighth ed .. Addison - Wesley. Doyle, John C., Keith Glover, Pramod P. Khargonekar and Bruce A. Francis (1989). State space solutions to standard h2 and 11.00 control problems. IEEE 1hms. Aut. Control 34(8),831-847. James, Mattew R. (1995). Recent developments in nonlinear 11.00 control. In: IFAC NOLCOS. IFAC. Jota, F. G., A. R. Braga, C. M. Polito and R. T. Pen a (1995). Development of an interacting tank system for the study of advanced process control strategies. In: 38th IEEE Midwest Symposium on Circuits and Systems (IEEE, Ed.). Vol. 1. pp. 441-445. Kwakernaak, H. (1993). Robust control and 11.00 optimiztion - tutorial paper. Automatica 29(2), 255-273. Packard, A. and J. C. Doyle (1993) . The complex structured singular value. Automatica 29(1),71-109. Shamma, Jeff S. and Michael Athans (1992). Gain scheduling: Potential hazards and possible remedies. IEEE Control Systems 12(3), 101107. Skogestad, Sigurd and lan Postlethwaite (1996). Multivariable Feedback Control - Analysis and Design. John Wiley and Sons.
Parametric Uncertainties and FV-04 20 % Open
tE-;:;g !20
°0
.
1000
' J.
2000
V~ 300CI
4000
sooo
Ir52'~'9~ o
1000
2000
3000
400C
1000
TirN (Iecondl)
Fig. 10. Time Response of the Level ofthe ITS Plant with Non-Parametric Uncertainties and FV-04 20 % Open
11. RESULTS ANALYSIS AND CONCLUSIONS All the designs, as performed in this paper, are based on the choice of the weighting functions which results on the best controller characteristics. This kind of design involves important tradeoffs between the weighting functions, the uncertainties and the controller performance. The selection of the weighting functions involves some competition between large close loop bandwidth (specified by W 1 ), the controller signal (specified by W2 ) and the robustness (specified by p.). In our case, the limitations imposed by the digital controller, that is, the controller can not have a pole faster than (about) 7r (in the s-plane) restricts the choice of the W 2 weighting function. This, in turn, limits the 11.00 norm of the closed loop system and affects the p. measure. Besides, it is not possible to increase the desired bandwidth because it can make the control signal very oscillatory. On the other hand, using very conservative weighting functions can result on very sluggish control.
590
DESIGN AND IMPLEMENTATION OF ft CONTROLLERS FOR REAL INDUSTRIAL PLANTS Marcio Fantini Miranda' Fabio Gont;alves Jota"
• Federal University of Minas Gerais - COLTEC - Brazil [email protected] •• Federal University of Minas Gerais - Elect. Eng. Dept - Brazil fabio@cpdee. 'Ilfmg. br'
Abstract: This paper considers the design and implementation of robust multi variable controllers for a real process control plant. The plant used for experimental tests is intended to control output flow rate and level of a product tank. First, a detailed description of a method for finding and representing uncertainties is presented. Next, a robust control is designed using the J1. synthesis. Comparisons of uncertainties (structured) parametric and non-parametric are also presented. They show how conservative the J1. analysis could be and, in some cases, how to reduce the conservatism. Copyright© 2000IFAC
Keywords: H-infinity Control; Robust Control; Uncertain Linear Systems: Process Control
l. INTRODUCTION
the output flow rate and level, the most common controlled variables (together with temperature) in process control.
The question of controlling uncertain systems has received considerable attention in the last ten years (Dorato et al., 1993). Depending on the structure and the type of the uncertainties present in the system , various frameworks have been considered such as, robust 11.00 control, e.g. (Doyle et al., 1989),(Skogestad and Postlethwaite, 199~) ; LMI, e.g. (Boyd et al., 1994), gain-scheduling, e.g. (Shamma and Athans, 1992) and nonlinear 11.= control, ego (James, 1995).
2. NOTATION The notation adopted here is standard in the 11.00 framework. A family or a set of plants , G(s) , has its nominal value expressed as Go(s). A (suboptimal) 11.00 controller, K(s), is the one that gives a closed loop matrix that satisfies:
The necessity of robustness may arise due to the existence of varying parameters, unmodelled dynamics, or a need to increase safety against exogenous inputs. In this work, we consider the ft-synthesis and analysis for an Interacting Tank System. This pilot scale plant presents characteristics similar to real industrial systems. The controller design is made so as to have a robust response against plant variations.
IIFt(Po, K)lloo ::; ,,(,
(1)
where 11 .1100 is the 11.00 norm, Ft represents the lower FLT of Po and K, and "( is some positive real number. Po(s) is the generalized matrix, which corresponds to Go plus the associated weighting matrices. J1. is the structured singular value which is used when the uncertain system has structured uncertainties (Packard and Doyle, 1993).
The use of such a system is directly related to the relevance of the variables to be controlled, that is,
585
3. THE INTERACTING TANKS SYSTEM
Rho and Rhi, respectively. We have considered h2 as being the state Xl and h3 as being the state X2 . FUnctions h(ut} and !2(U2) represent the steady state characteristics of valves FCV-Ol and FCV02, respectively. FUnction f4(P) represents the relation between output flow-rate and the relative aperture of the manual valve FV-04.
The experimental runs have been performed in a pilot-scale Interacting Tank System, ITS (Jota et al., 1995) (see figure 1). The ITS is composed of a 700 I reservoir (TQ-Ol) and two 300 1 passively interconnected tanks (TQ-02 and TQ-03). The coupling between these two tanks can be manually controlled by means of flow valves FV-03 and FV04. The basic operation of the system consists of pumping the liquid fluid from the reservoir (TQ-Ol) directly into the second tank (TQ-02). From TQ-02, the fluid flows naturally to the product tank (TQ-03) . The liquid is then pumped back to the reservoir by BA-02. The simultaneous control of level and flow-rate in the third tank is accomplished by equal-percentage pneumatic valves, FCV-Ol and FCV-02. A PLC is used to interface the plant to a microcomputer, where the control algorithms actually run. All the signal transmissions are accomplished via current loops of 4 to 20 mA.
The corresponding linear model is given by:
[;~] = [~~I
~~l] [:~] + [~ ~3] [:~]
(2)
[~~] = [~ t~]
[:~] + [~ -~k3] [:~] .
(3)
where k l , k2 and k3 vary with the chosen operating point and are given by: kI
=_
3.1 System Modeling
!4(P) 2AJXlO
-
(4)
X20
k2
=.!. Oh(UIO) JhlO + hbl -
ka
= __1_ 0!2(U20) J
A
he
(5)
+ hb2
(6)
OUI
RhoA
OU2
X20
k4 - _~ !2(U20) 1 . - 2 RhoA JX20 + h b2 ' k12 = kI - k4 tl
= !2(U20) Rho
Fig. 1. Schematic Diagram of the Interacting Tanks System, ITS
1 JX20
+ hb2
(7) (8)
(9)
From the previous state space model, a family of plants is derived and expressed as:
The modeling of the ITS is based on the equations of the mass balance between the tanks. The model can be expressed by the system of equations: dh2 dt h2 dh3 dt
h3 =
+ q23 A A +h +f
_ qo
= _ !2 (U2) J ARho
h
b2
3
4
(P) J h2 - h3 A 4. DEFINITION AND ANALYSIS OF THE ITS UNCERTAINTIES
where qi is the input flow-rate (in TQ-02), qo is the output flow- rate (in TQ-03), q23 is the flow-rate between TQ-02 and TQ-03, h2 is the level of TQ-02 and h3 is the level of TQ-03. The cross sections of both TQ-02 and TQ-03 is A. hI is the (average) level of the liquid in the tank TQ-Ol and he corresponds to the height of the water column that exerts pressure on BA-Ol (see figure 1). hbl and hb2 represent the pumping capacity of BA-Ol and BA-02, respectively. The hydraulic resistances of the output and input are
From equations (4) to (9) and from experiments performed on the real plant, it is possible to draw some conclusions about the values of the variables k i , (i = 1, ... ,4), k12 and tl defined for the linear model. (1) The variation of kI is a function of the difference of level between tanks TQ-02 and TQ-03 and the aperture of valve FV-04. Figure 2 (a)
586
Fig. 2. Variation of A (a) and control signal
U2
kl ( )
for different
(3) Define the operational (nominal) point. In this case , we have chosen, the sixth point for FV-04 100% open as the nominal operational point. (4 ) From experimental data and physical equations of the system , find the range of variation of the variables of the equations (10) to (15) . (5) From the 12 equilibrium points define different realizations (Ai , B; , C i , Di) , i == 1 .. . 12.
as function of the
Rh23 ·
shows the (experimental) relation kl with the output flow, defined by 1.12. It is noteworthy the fact that kl can vary up to 700 o/c if both out-flow and FV-04 aperture are changed. (2) The variation of k2 is a function of the gain of FCV-01 valve. So, k2 is important on the determination of the gll (s) transfer function . (3) The variation of ka is function of the gain of the FCV-02 valve. The importance of k3 variation is due to: i) k3 is the gain of g)2 (which can be considered as the perturbation on gll and ii) k3 varies about 800 o/c.
4.1.1. Range oJ Variation oJ The Linellr Constants. Considering the "quasi-steady state" points given in figure 3 the possible variations of the linear equation are given by (10) to (15) : -0.036::; k)
::; -3.2 X IQ-3
0.0062::; k2 ::; 0.0133
-2.6685 x
IQ-2
(11)
k12 -2.66627
X
IQ-2
0.0013::; k3 ::; 0.0082
-3.42 x 10- 5 2.17 x
::;
k4 ::; -1.07
IQ - 3::; t) ::;
6.78
(10) (12) (13)
X
IQ-5
X IQ-3 .
(14 ) (15 )
4.1 Uncertainty Representation
The uncertainties on the ITS model are mainly derived from the variations of valve FV-04, the output flow rate and the level difference between TQ-02 and TQ-03. The variation on the control valves , FCV-01 and FCV-02 , can also vary the operational point, resulting in a change of the nominal model. The uncertainty model is derived in the following steps: (1) From the curves of valves FCV-01 and FCV02, we get the equilibrium points, where i; == 0, that is points where q; == q23 == qo (see figure 3 (b));
5. PROBLEM SETUP As shown in section 3, the ITS plant has some interesting characteristics t.hat motivated the use of robust control. As it is possible to build a uncertainty model that fits well on the 1-£00 and J1. framework , we can look for a controller K(s) that results on a closed loop with robust stability and performance, defined by the J1. criteria. Definition 1 gives the formulation of the control objective. Definition 1. Find a controller K(s) that satisfies:
••
(1) Closed loop robust performance and stability (by the J1. synthesis we must have J1. ::; 1). It is important to note that we have a normalized system and we wrote the uncertainties in order to get 1161100 ::; 1; (2) good regulation response. Wich mean that we are interested in to keep the level and output flow rate fixed on specific values , despite perturbations; (3) steady state error less than 5%. It is a common design specification (Dorf and Bishop, 1998). In fact we require a zero steady state error; (4) the faster controller pole is limited by the sampling frequency. As the faster sampling frequency is 1 Hz (21l" rad/s) , the faster pole can not be much bigger than 1l" rad/ s.
~,.
!
6'·'
o. 02
~
~
~~~~a>~~'~A--~,.7.--~,,~~ Ib)
V ....... SilgNlf%)
Fig. 3. Characteristic curves of FCV-Ol and FCV-02 Valves. (a) Original curves; ( ) normalized curves with t e polynomial fit and t e equili rium points
(2) The equilibrium points which will define the "different plants" can be found from the condition "output flow == input flow , which is a very difficult condition to obtain in the open loop real plant.
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6. THE DESIGN SETUP The designs are based on the well-known mixed sensitivity problem (Kwakernaak, 1993),(Skogestad and Postlethwaite, 1996) which uses the sensitivity functions for the performance specifications. Together, it is added the uncertainty description in order to fit the problem on the JL framework. According the uncertainty representation, two designs are considered: (1) The first one is based on the structured parametric uncertainty model. (2) The second design is the one in which the structured parametric uncertainty is represented as a non-parametric one, in order to reduce the conservativeness of the JL design. We also consider two kinds of experiments: one for the FV-04 valve 100% open and for 20% open. With these experiments we are testing the control for the larger possible variations. Both the designs are based on figure 4, which presents a standard mixed sensitivity problem with structured uncertainty. The weighting function W 1 is used to specify the desired sensitivity function S(s) and the weighting function W 2 is used to specify the control signal.
The design by the DK iteration resulted into 5 iterations with a JL peak value of about 1.5. The final controller was discretized and applied to the plant. Sections 9 and 10 present experimental results, in which FV-04 is 100% open, and a profile for level and flow rate with large variations. The JL peak value about 1.5 means that the robust performance has not been achieved. In order to improve this result we can: i) reduce the performance specification (W1 ) , by changing or the bandwidth or the sensitivity peak ;ii) reduce the W 2 , which implies in increasing the control signal; iii) reduce the uncertainties by a factor about 1.5. The experience has shown that the first option results on a very slow control and the second gives a control with a very high gain and very fast pole, which cause problems on the implementation. The third consideration can be a good solution, if we can guarantee that the real system can be represented by a less conservative uncertainty. We believe that we can not only reduce the uncertainties (and the performance specifications) by a factor of 1.5, because we can loose some information about the system variation. That is why we propose change the uncertainty representation. 8. SECOND DESIGN - JL SYNTHESIS WITH NON-PARAMETRIC UNCERTAINTIES
This design aimed to reduce the uncertainties and to produce a less conservative control. For such, the parametric uncertainties were manipulated to get a dynamic representation, similar to a "unmoldeled dynamic" . In this way, the six parametric uncertainties could be transformed into two dynamic uncertainties that represent the overall variations of the 4 entries of the nominal matrix, Go(s). As a result , an uncertainty system of the type: multiplicative input dynamic uncertainty system with a 2 x 2 diagonal matrix such as 1I~lloo ~ 1 was obtained. This uncertainty system can be represented by:
Fig. 4. Diagram of the S/KS Scheme Design with Structured Uncertainty
7. FIRST DESIGN - JL SYNTHESIS WITH PARAMETRIC UNCERTAINTIES This design has considered the parametric (structured) uncertainties on the model of the ITS plant. The six constants varying on the range given by equations (10) to (15) were taken into account. The uncertainty block ~ is a diagonal 6 x 6 matrix with 11~lloo ~ 1. We have also considered a fictitious uncertainty (Balas et al., 1992) for the performance specification defined by W 1 and W 2 , that is, a full 2 x 4 matrix. Final weighting functions were obtained after a few attempts, which results:
G(s)
= Go(s)(l + ~(S)Wd(S))
(16)
where ~(s) is any matrix such that 11~lloo ~ 1 and Wd(S) is a diagonal matrix that represents the weights of the uncertainties as a function of the frequency. The transformation from parametric to non-parametric uncertainty is done by the following procedure. Let a 2 x 2 uncertainty system be defined as:
0.5s + 2 x W1 =
S
[
o
10- 2
~
0.58 + x 10-'
1
=95 1(1 +Wll(S)~l)Ul + 95 2 (1 + W12(S)~2)U2 Y2 = 95 1 (1 + w21(S)~3)U1 +
(17)
+95 2 (1 + w22(s)~4)U2.
(18)
Y1
and
W _ [0.01 0 ] 2 0 0.1 .
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Since 161 ~ 1, the maximum and minimum gain values of each (nominal) transfer function, gij (s), are determined from the relations:
g~az
= g~i (1 + w ij )
(19)
g:!.in
= g~j (1 - w
(20)
ij
).
On solving the system equations (19) and (20) for the functions w ij , we get:
Fig. 6. Time Response of the Flow Control (it is almost the same for all experiments)
(21)
= 1,2) we define the
With the functions w ij (i,j matrix W(s):
W(s)
9. EXPERIMENTAL RESULTS FOR FV-04 100% OPEN
wll(s) W12 (S)] w 22 (s) ,
As can be seen in figures 7 and 8 the time response of level and output flow rate presents good tracking response and good rejection perturbation (if we consider that both level and flow are perturbations for each other) . It is also worth of note that the time response due the two designs (with parametric and non-parametric) are very similar, despite the difference on the J.l peak value.
(22)
= [ w21(s)
Equation (21) gives the entries of W(s), whose frequency responses are depicted in figure 5.
:~ :: '''D''O] .. ::0 ':Qj .,
O'~O"
Tim. R•• pon •• tor Le.".1 Loop
u•
'0'
D, ~...
'0'
u.
•.•
o•
•n
u
~
~
or
~
~
Fig. 5. Frequency response of entries of W(s) Fig. 7. Time Response of the Level of the ITS Plant With
In order to represent the uncertainty system as in equation (16), we define a diagonal matrix,
Parametric Uncertainties and FV-04 100 % Open
(23) that is, the entries of Wd(s) are determined from the frequency response of W (s) so as to encompass the worst-case dynamics (see figure 5). It is possible to do this approximation because the functions 22 are similar, and the same for funcW 21 and W tions w 12 and W 22 . For performance specification it was used the same weighting functions as the first design. It can be seen on figures that the time response is almost the same. So the main advantage was the J.l peak reduction.
Fig. 8. Time Response of the Level of the ITS Plant With Non-Parametric Uncertainties and FV-04 100
% Open
With these uncertainty representation, the design by the DK iteration results into 3 iteration with a J.l peak value equal 1.0.
10. EXPERIMENTAL RESULTS FOR FV-04 20% OPEN
In the sequel it is presented the time response of experimental runs on the level and output flow rate. As the flow loop is not so difficult to control it will present only the time response of the level loop. For all experiments, the flow varies from 30% to 85%, as shown on figure 6.
The consideration of valve FV-04 20% open is a hard task for the control, since gain of the system can vary up to 700 %. So the time response of the level is not so good when the output flow changes
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About the uncertainties representation, it was shown (by the comparison between of the nonparametric and parametric controllers) that, with a changing on the representation, the robustness test could be improved with almost the same controller. We have also shown that, within the p. framework it is possible to get a robust control even for a very large system variation.
from 30 to 85 %. Nevertheless the system keeps stable, despite this large perturbation. Figures 9 and 10 present the time response for the condition of FV04 20% opened only, for the parametric controller and the non-parametric, respectively.
For the experiments presented here, we can conclude that, for the overall variations considering the p. analysis results on conservatives indications, and when we reduce the uncertainty, we can find controller that satisfy the performance and stability tests with good time response.
12. REFERENCES Fig. 9. Time Response of the Level of the ITS Plant with
Balas, Gary J., Jonh C. Doyle, Keith Glover, Andy Packard and Roy Smith (1992). p.Analysis and Synthesis Toolbox - User's Guide. The Math Works Inc. Boyd, Stepen, Laurent El Ghaoui, Eric Feron and Venkataramanan Balakrishnan (1994). Linear Matrix Inequalities in System and Control Theory. SIAM. Dorato, P., R. Tempo and G. Muscato (1993). Bibliography on robust control. Automatica 29(1),201-213. Dorf, Richard C. and Robert H. Bishop (1998). Modern Control Systems. eighth ed .. Addison - Wesley. Doyle, John C., Keith Glover, Pramod P. Khargonekar and Bruce A. Francis (1989). State space solutions to standard h2 and 11.00 control problems. IEEE 1hms. Aut. Control 34(8),831-847. James, Mattew R. (1995). Recent developments in nonlinear 11.00 control. In: IFAC NOLCOS. IFAC. Jota, F. G., A. R. Braga, C. M. Polito and R. T. Pen a (1995). Development of an interacting tank system for the study of advanced process control strategies. In: 38th IEEE Midwest Symposium on Circuits and Systems (IEEE, Ed.). Vol. 1. pp. 441-445. Kwakernaak, H. (1993). Robust control and 11.00 optimiztion - tutorial paper. Automatica 29(2), 255-273. Packard, A. and J. C. Doyle (1993) . The complex structured singular value. Automatica 29(1),71-109. Shamma, Jeff S. and Michael Athans (1992). Gain scheduling: Potential hazards and possible remedies. IEEE Control Systems 12(3), 101107. Skogestad, Sigurd and lan Postlethwaite (1996). Multivariable Feedback Control - Analysis and Design. John Wiley and Sons.
Parametric Uncertainties and FV-04 20 % Open
tE-;:;g !20
°0
.
1000
' J.
2000
V~ 300CI
4000
sooo
Ir52'~'9~ o
1000
2000
3000
400C
1000
TirN (Iecondl)
Fig. 10. Time Response of the Level ofthe ITS Plant with Non-Parametric Uncertainties and FV-04 20 % Open
11. RESULTS ANALYSIS AND CONCLUSIONS All the designs, as performed in this paper, are based on the choice of the weighting functions which results on the best controller characteristics. This kind of design involves important tradeoffs between the weighting functions, the uncertainties and the controller performance. The selection of the weighting functions involves some competition between large close loop bandwidth (specified by W 1 ), the controller signal (specified by W2 ) and the robustness (specified by p.). In our case, the limitations imposed by the digital controller, that is, the controller can not have a pole faster than (about) 7r (in the s-plane) restricts the choice of the W 2 weighting function. This, in turn, limits the 11.00 norm of the closed loop system and affects the p. measure. Besides, it is not possible to increase the desired bandwidth because it can make the control signal very oscillatory. On the other hand, using very conservative weighting functions can result on very sluggish control.
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