lecture 10: cascade and ratio control 1. objectives recognize the cascade control configuration....
TRANSCRIPT
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Lecture 10:
Cascade and Ratio Control
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Objectives
• Recognize the cascade control configuration.
• Identify the situations in which cascade control is effective.
• List the procedure of tuning cascade controllers.
• Identify two configurations for ratio control and the advantages and disadvantages of each of them.
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Cascade Control
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Motivation for Cascade Control
Consider the following gas furnace where the purpose is to control the temperature of the outlet stream.
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The obvious idea is to control the temperature by manipulating the fuel gas flow valve.
Motivation for Cascade Control
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Problem: fuel flow may subject to large fluctuation due to the upstream pressure variation, P.
Motivation for Cascade Control
P
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• Let’s think about the process behavior.
• The change in the pressure disturbance P will quickly cause a change in the flow rate F. This, in turn, will cause a change in outlet temperature, T (in the long run because the process is slow).
• The idea of cascade control: why we do not try to measure and control the flow F (easily measurable) so that we can maintain it approximately constant and hence reduce the effect of the pressure disturbance on the temperature T.
Motivation for Cascade Control
v (valve) F Q T
P(heating oil)
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Cascade control solution: try to keep the flow constant. This will, hopefully, forbid pressure fluctuations from propagating to the outlet temperature, T.
Motivation for Cascade Control
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Cascade Control: Block Diagram
Although cascade control requires an additional measurement (usually flow rate) and an additional controller Gc2, large improvement in performance is obtained when the secondary (inner) loop is much faster than primary (outer) loop.
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Cascade Control Analogy to Management Hierarchy
• A cascade is a hierarchy, with decisions transmitted from upper to lower levels.
• What are the advantages of a hierarchy?
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When to use Cascade Control?
If you have a process with relatively slow dynamics (like level, temperature, composition, humidity) and a liquid or gas flow, or some other relatively-fast process, has to be manipulated to control the slow process.
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How to tune Cascade Controllers?
A cascade arrangement should be tuned starting with the innermost loop.
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Numerical ExampleConsider the following cascade system with a PI controller in the primary loop and a proportional controller in the slave loop.
For simplicity, consider first-order functions:
.1
75.0,
1
5.0,
12
8.0
sG
sG
sG Lvp
.,1
1 221 ccI
cc KGs
KG
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(a) How can the proper choice of Kc2 improves the actuator performance and eliminate disturbance in the manipulated variable (fuel-gas flow)?
To answer this question, we need to find the effect of adding the inner-loop on the relationship between
- the fuel flow and the primary control signal (point p),- the fuel flow and the disturbance (L).
These relationships can be found by manipulating the block diagram as follows:
.1
,1
2
*
2
2*
vc
LL
vc
vcv
GG
GG
L
f
GG
GGG
p
f
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Let us first consider Gv*. After substituting for Gv and Gc2, we
obtain
Comparing this to the case without cascade control
we can see that as the proportional gain Kc2 becomes larger, the steady state gain of Gv
* approaches unity. This means that the primary controller becomes more effective on the fuel-gas flow. Also, the effective actuator time constant of Gv
* become smaller, meaning a faster response.
.1
5.011
5.01
5.0
5.01
5.0
1
2
2
2
2
2
2
2*
sK
K
K
Ks
K
GG
GGG
c
c
c
c
c
vc
vcv
,1
5.0
sGv
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Similarly, if we consider GL*, after substituting for GL we
obtain
As the proportional gain Kc2 becomes larger, the steady state gain from the pressure disturbance to the flow approaches zero, meaning that the manipulated variable is becoming less sensitive to changes in the load compared to the case without cascade control:
.75.01
75.0
1 220
*
cvc
L
sL KGG
GG
.1
75.0
sGL
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(b) What is the proportional gain Kc2 if we want the inner-loop time constant to be only one tenth of the original valve time constant?
The transfer function of the valve is
Hence the time constant of the valve is 1.
From part (a), we already found the effective time constant of the inner loop Gv
* to be
181.05.01
12
2
c
c
KK
.1
5.0
sGv
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(c) For the value of the proportional gain Kc2 found in (b), find the steady state error in the inner loop.
• The steady state gain in the inner loop is
• Thus, the inner loop will have a 10% steady state error or offset. This is expected because we have used only a proportional controller in the inner loop.
• In general, this 10% offset is acceptable as long as we have integral action in the outer loop. The primary controller can make necessary adjustments in its output and ensure that there is no steady-state error in the controlled variable (the outlet stream temperature).
9.05.01
5.0
182
2
0
*
2
cKc
c
sv K
KG
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(d) Now we tackle the primary PI controller. Suppose that we want to choose the proper integral time constant among the given values of 0.05, 0.5, and 5s. Select and explain your choice of the integral time constant.
Among other methods, root locus is the most instructive in this case. With a PI primary controller and numerical values, the characteristic equation becomes
With MATLAB, we can easily prepare the root-locus plots of this equation for the cases of τI = 0.05, 0.5, and 5s.
.012
8.0
11.0
9.011
,01 *
sss
sK
GGG
I
Ic
pvc
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From the root-locus plots, it is clear that:
• The system may become unstable when τI = 0.05s.
• The system is always stable when τI = 5s, but the speed of the system response is limited by the dominant pole between the origin and − 0.2.
• The proper choice is τI = 0.5s, in which case the system is always stable but the closed-loop poles can move farther away from the origin.
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(e) Take the case without cascade control (that is without the inner controller and the secondary loop). Now we have only one controller (the PI controller). Using the integral time constant that you have selected in part (c), determine the range of proportional gain that you can use to maintain a stable system. How is this different from the cascade control case?
With cascade control:
(we already found that it is
always stable)
Without cascade control:
?
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With the choice of τI = 0.5s, but without the inner loop or the secondary controller, the closed-loop equation is
which can be expanded to
With Routh–Hurwitz analysis, we should find that, to have a stable system, we must keep Kc < 7.5. Compare this to cascade control, in which the system is always stable.
With cascade control, the system becomes more stable and allows us to use a larger proportional gain in the primary controller. The main reason is the much faster response (smaller time constant) of the actuator in the inner loop.
,012
8.0
1
5.0
5.0
15.011
sss
sKGGG cpvc
.04.0)2.05.0(5.1 23 cc KsKss
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CLASS EXERCISE
• Why do we retain the primary controller?
• Which modes are required for zero steady-state offset?
• Which modes are recommended for primary and secondary controllers?
• What is the additional cost for cascade control?
• What procedure is used for tuning cascade control?
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Quiz
In cascade control design, the sensor for the secondary variable should provide good
– accuracy – reproducibility
A constant bias in the secondary measurement will not seriously degrade the control performance. The primary controller will adjust the secondary set point to correct for a small bias. Remember, a sensor with good reproducibility is often less expensive than a highly accurate sensor.
√
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Quiz
In cascade control design, the sensor for the primary variable should provide good
– accuracy – reproducibility
Nothing can correct errors in the primary sensor. Therefore, the primary sensor must achieve the accuracy needed for the process application.
√
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Ratio Control
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Ratio Control
• Ratio control is used when the control objective is to keep the ratio between two variables, often flows, at a certain value, a.
• For example, in combustion, it is desired to control the fuel to air supply ratio, in order for the combustion to be as e cient as possible. ffi
The material on Ratio control is based on the following paper:Tore Hagglund. The Blend station: a new ratio control structure, Control Engineering Practice, 9, 1215–1220, (2001).
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Ratio Control: Traditional Approach
• Ratio control is normally solved as shown.
• There are two control loops: the main loop consists of process P1 and controller C1 and the second loop, consisting of process P2 and controller C2.
• It is attempted to control the flow y2
so that the ratio y2/y1 = a. This is obtained using a Ratio station where set-point r2 is determined by
r2 (t) = a y1(t)
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Ratio Control: Traditional Approach
• Provided that the controllers have an integral action, the traditional ratio control approach will work in a steady-state, i.e. y1 = r1 and y2 = ay1.
• However, during transients, the second flow y2 will always be delayed compared to the desired flow ay1. The length of this delay is determined by the dynamics in the second loop. This is a serious problem, since ratio control is normally applied to problems where the flows are supposed to vary and where steady-state conditions are uncommon.
• When set-point r1 is increasing, the delay causes an under-supply of the media corresponding to flow y2; and vice versa. There are cases when it is important never to get any under-supply of one of the two media.
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Ratio Control: Second Approach• A second approach is to apply the Ratio
station to set-point r1 instead of measurement signal y1. That is, set-point r2 is determined by
r2(t) = a r1(t).
• Now, the second flow is not necessarily delayed compared to the main flow as in the traditional approach. Instead, the transient behavior is determined by the dynamics in both loops. By tuning the controllers so that the closed-loop dynamics are the same, the ratio y2 = ay1 may be kept even during transients.
• The drawback with this method is that it is a kind of open-loop approach. If the dynamics in one of the loops change, so may the ratio y2 = ay1. Process dynamics often change mostly due to nonlinearities.