a new perspective on the tuning, stability, and benefits ... · cascade control loops. the tuning...
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A New Perspective on the Tuning, Stability, and Benefits of
Cascade Control
AUTHOR
Jacques F. Smuts - OptiControls Inc, League City, Texas
KEYWORDS
Cascade Control, Stability, Disturbance Rejection, Controller Tuning
ABSTRACT
It is often said that for a cascade control system to be stable, the outer loop should be tuned for a
significantly slower response than the inner loop. However, the minimum ratio of outer to inner loop
response time required for stability is subject to some dispute. Guidance given in this regard has
ranged from ratios as low as 3:1 to as much as 20:1. This paper explores the minimum response-time
ratio required for cascade control systems to be stable, and the effect that process characteristics and
tuning methods have on this minimum. It also makes recommendations about the type of tuning
methods that should be used to obtain stable, responsive cascade control systems.
It has also been recommended that for cascade control to be beneficial, the inner loop in a cascade
control system must respond at least five times faster than the outer loop. This paper analyzes a few
typical, but distinctly different cascade control applications, and evaluates the benefits of cascade
control in each regard. It shows that in practice, cascade control is not always applied for improving
control performance – and explains how cascade control is sometimes applied solely to simplify a
control strategy.
INTRODUCTION
Cascade control is a form of feedback control that uses a specific arrangement of multiple control
loops to control a process. The most basic cascade control arrangement contains two feedback control
loops of which one, the inner loop, is nested inside another, the outer loop (Figure 1). With cascade
control, the primary process (PVP) variable is controlled by the primary controller through the outer
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control loop. The output of the primary controller (COP) drives the setpoint of the secondary controller
(SPS). The secondary controller controls the secondary process variable (PVS) through the inner loop.
Figure 1. Block diagram of a simple cascade control system.
The physical process being controlled by the cascade control system can be considered as having two
parts. The first part of the process (Process A) might be a flow control valve throttling the flow of
liquid into a tank and the remainder of the process (Process B) might be a tank in which the level is
being controlled (Figure 2).
Figure 2. Cascaded level control on a liquid-gas separator.
The secondary controller controls only Process A, forming an inner control loop. The primary
controller controls a pseudo-process consisting of the inner loop and Process B. It should be evident
from Figure 1 that the tuning of the secondary controller affects the dynamics of this pseudo-process
being controlled by the primary controller. For this reason, cascade control loops should be tuned
starting with the inner loop, then putting the secondary controller in cascade control mode (or remote
setpoint mode), and then tuning the outer loop [1, 2, 3].
TUNING AND STABILITY OF CASCADE CONTROL SYSTEMS
Since most industrial processes should be kept as close as possible to theirs setpoints, their feedback
control loops are tuned to respond quickly to demand changes and disturbances. However, a control
Primary
Controller
Secondary
ControllerProcess A Process B
Inner Loop
Outer Loop
SPP
PVP
SPS
PVS
COP COS
Aggregate Process being Controlled
PVS PVP
Pseudo-Process in Outer Loop
Separator
FT FC
LCLT
Gas
Liquid
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loop’s speed is in tradeoff with its stability. If a control loop is tuned for a faster response, its stability
is reduced. Stability problems can be compounded in complex, interactive control designs such as
cascade control loops.
The tuning and stability of cascade control systems have often been examined in academic literature.
Analysis is normally done from a theoretical perspective, supported with complex mathematics [e.g.
4]. The concept of cascade control is treated as a complex structure warranting very special tuning
considerations [5, 6]. Technically correct, but practically questionable conclusions have been drawn
for improving stability, such as a) not using integral action in the secondary controller, b) increasing
integral time in the primary controller, and c) using derivative action in the primary controller [7].
In contrast to the academic approach, tuning and stability of cascade controls are also covered in
controls training classes and practical textbooks. Because cascaded control loops are connected
through process design and control signals, the constituent loops interact on each other. This can
potentially cause an unstable system if the controllers are not tuned properly [8]. The minimum ratio of
the outer loop’s speed of response compared to that of the inner loop is an important factor to consider
for stability. A discussion on LinkedIn’s Control Engineering Group [9] revealed that practitioners
believe the minimum value for this ratio to be anything from 20:1 to 3:1, but no reasons have been
given for choosing a certain ratio.
The primary objective of this paper is to explore tuning and stability of cascade controls from a
practical perspective, considering three distinctly different processes and three tuning methods. These
have been chosen to cover the majority of processes and tuning rules in use today. The paper then
presents guidance on the use of certain tuning rules and the minimum ratio between the speed of
response of the outer and inner loops.
BENEFITS OF CASCADE CONTROL
Control system designs should be kept as simple as possible. Cascade control requires additional
instrumentation, wiring, inputs, configuration and tuning. It should therefore be implemented only
when needed, i.e., only when its benefits outweigh the cost of implementing, configuring, tuning, and
maintaining the additional controls.
The benefits of cascade control are mainly the ability of the secondary control loop to quickly react to
process disturbances and final control element nonlinearities to partially shield the primary control
loop from their negative effects. For example, if a change in pressure in a liquid-gas separator affects
the discharge flow rate, this will affect the level. A flow controller can quickly detect and compensate
for the change in flow rate, lessening its effect on the separator level (Figure 2). Or if the flow
characteristic of the control valve is somewhat nonlinear, deviations in flow rate can be detected and
corrected by the flow controller before the separator level is noticeably affected.
The secondary objective of this paper is to provide clarity on the benefit of cascade control. Benefits
depend on the distribution of process dynamics between Process A and Process B [2, 10]. This paper
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also presents guidance on how much benefit one can expect from cascade control arrangements on
different types of processes.
ANALYSES AND TEST SETUP
The stability and disturbance-rejection capability of cascade control loops were analyzed and the
results documented in this paper. Three distinctive types of processes under cascade control, and three
different tuning rules, were used during the analysis. These processes and tuning methods, as well as
the test setup used in the analyses are described below.
PROCESS TYPES
Since there is an infinite range of possible process characteristics, only certain relevant process
characteristics could be considered in this paper. However, the process characteristics were picked to
capture the extremes and midpoints of the possible range of processes that could be considered for
cascade control. These are described below.
Reference [1] recommends that for cascade control to be beneficial for disturbance-rejection, the
process in the inner loop should ideally respond a minimum of five times faster than the pseudo-
process in the outer control loop. This is a very common case in practice, since most cascade control
systems have a fast flow loop as the inner loop, and a much slower pressure, temperature, or level
process in the outer loop. An example is the outlet temperature control of a heat exchanger, cascaded
to a steam flow controller (Figure 3). The steam flow will have dynamics in the order of a few seconds,
while the temperature will respond much slower. Because it is typical for cascade control, a
configuration in which most of the process dynamics are contained in Process B will be used as the
first test case.
Figure 3. Heat exchanger with fast inner loop (flow) and slow outer loop (temperature).
Steam
Process flow
Condensate
TT
TC
Heat exchanger
Temperature
controller
FT FCCascaded
flow controller
Controller output
Set point
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Reference [3] describes total output control in which the only dynamics in the process are contained in
Process A, and the outer loop exists entirely in the control system. This control strategy is often used in
boilers with multiple mills/burners to control the total flow of fuel produced by all the mills/burners
according to a common boiler fuel demand signal [11]. Another example is that of export oil flow
control on an oil/gas platform, as shown in Figure 4. Because this scenario is the complete opposite of
that in the previous test case, a configuration in which all the process dynamics are contained in the
inner loop will be used as the second test case.
Figure 4. Total flow control strategy in which all process dynamics are contained in
the inner flow loops.
Reference 12 describes the dynamics of desuperheater and superheater outlet temperatures (Figure 5)
and shows that in some cases the superheater outlet temperature responds only a factor of two slower
than the desuperheater outlet temperature. Because of this significance, but also because it provides a
reasonable midpoint between the two test cases described already, the third test case was chosen to
contain a Process A and Process B with similar dynamics.
Total Flow
Controller
FC
FT
FT
FT
Pumps w. Individual
Flow Controllers
Total Discharge Flow RateΣ
Total Flow
Setpoint
FC
FC
FC
Common Flow
Setpoints
Product Flow Line
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Figure 5. Cascaded steam temperature control.
Table 1 provides a summary of the process types chosen for analysis. Since different process gains can
be perfectly cancelled with controller gains, all processes were assigned a gain of 1.
Table 1. Dynamics of processes analyzed in cascade control setups. td is dead time and τ is time constant.
Test
Case Condition Example Process Process A Process B
I Process A < Process B Temperature-flow cascade td = 3 sec; τ = 5 sec td = 15 sec; τ = 45 sec
II Process B = 0 Total output control td = 3 sec; τ = 5 sec td = 0 sec; τ = 0 sec
III Process A = Process B Steam temperature control td = 15 sec; τ = 45 sec td = 15 sec; τ = 45 sec
CONTROL LOOP TUNING
Much has been written about controller tuning [1, 2, 3, 8, 10] and it is assumed that the reader is
familiar with tuning techniques based on doing step tests, determining process characteristics, and
calculating tuning settings using proven tuning rules. Since control loop stability and speed of response
are in tradeoff with each other, an array of possible tuning objectives has to be considered. For
simplicity, the three tuning objectives shown in Table 2 reasonably cover the spectrum of loop
performance objectives from a stability perspective.
Table 2. Tuning objectives used in stability analysis of cascade control systems.
Tuning Objective Tuning Rule
Fast response / QAD Cohen-Coon [13] Semi-fast response Modified Cohen-Coon [14]
Very stable response Lambda / IMC [15]
Desuperheater
Spraywater
Control Valve
Desuperheater
Outlet Temperature
TC2
Desuperheater Outlet
Temperature
Controller
TT2
Spraywater
High-pressure
Turbine
Boiler
Drum
1st Stage
Superheater
2nd
Stage
Superheater
TC1
TT1
Main Steam
Temperature
Main Steam
Temperature
Controller
Furnace
Steam
Main Steam
Temperature
Set PointDesuperheater
Outlet Temperature
Set Point
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The fast-response tuning aims for a quarter-amplitude damping (QAD) response. This tuning objective
was included not because it is a good tuning method (quite the contrary), but because many tuners still
tune for a quarter-amplitude damping response. IMC tuning was included because of the very stable
response it provides. For this tuning method, the closed-loop time constant was set to the open loop
time constant. The semi-fast response objective was included as a midpoint. This type of tuning is
achieved by using the Cohen-Coon tuning rules, and reducing the calculated controller gain by a factor
of 2. In all cases, PI control only was used, since the derivative control mode is not used in most
industrial control loops.
Figure 6 and Figure 7 show the response of simulated temperature and flow control loops tuned with
the three methods described above.
Figure 6. Response curves of a simulated temperature control loop tuned with the three methods described in Table 2. Leftmost
trend is Cohen-Coon, middle is Modified Cohen-Coon, and rightmost trend is Lambda tuning.
Figure 7. Response curves of a simulated flow control loop tuned with the three methods described in Table 2. Leftmost trend is
Cohen-Coon, middle is Modified Cohen-Coon, and rightmost trend is Lambda tuning.
TEST SETUP
Previously developed single-loop simulation software was modified for analyzing the performance of
various cascade control setups under various tuning regimes (Figure 8). Simulated control loop
response of the new software was checked against other simulation software to ensure its accuracy.
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Figure 8. Control loop simulation software used in analysis.
TEST RESULTS
The stability and disturbance-rejection capability of cascade control loops were analyzed for the three
types of processes and three different tuning rules described above. The results of the analyses are
presented below. During the stability analysis, attention was paid to the closed-loop time constants of
the outer loop versus inner loop. These numbers are presented in the tables below and listed as t63.
STABILITY OF CASCADSE CONTROL WITH FAST INNER PROCESS, SLOW OUTER PROCESS
Test Case I represented the most common cascade control application in which the dynamics of the
inner loop’s process are much faster than the dynamics of the outer loop’s process, as found in a
temperature-to-flow cascade system. The dynamics that were assigned to Process A represent a first-
order + dead time model of the process in a typical flow control loop, i.e. a dead time of 3 seconds and
a time constant of 5 seconds. The dynamics that were assigned to Process B represent a first-order +
dead time model for a moderately fast-responding temperature process, i.e. a dead time of 15 seconds
and a time constant of 45 seconds.
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The results are shown in Table 3 and Figure 9. The cascade control system was stable in all cases, with
outer loop to inner loop response time ratios of 6:1, 7:1, and 9:1 respectively for the three tuning
methods.
Table 3. Stability of a typical cascade control system tuned using different tuning methods.
Cohen-Coon Tuning Modified Cohen-Coon IMC / Lambda Tuning
Process A td = 3 sec; τ = 5 sec td = 3 sec; τ = 5 sec td = 3 sec; τ = 5 sec
Inner loop tuning KC = 1.37; TI = 0.08 KC = 0.68; TI = 0.08 KC = 0.59; TI = 0.083
Inner loop t63 5 sec 7 sec 8 sec Process B td = 15 sec; τ = 45 sec td = 15 sec; τ = 45 sec td = 15 sec; τ = 45 sec
Aggregate process model td = 20 sec; τ = 42 sec td = 21 sec; τ = 45 sec td = 22 sec; τ = 46 sec
Outer loop tuning KC = 1.94; TI = 0.57 KC = 1.0; TI = 0.6 KC = 0.68; TI = 0.78 Outer loop t63 31 sec 48 sec 69 sec
t63 ratio 6.2:1 6.9:1 8.6:1
Stability assessment Stable, QAD Stable, slight overshoot Stable, no overshoot
Figure 9. Test Case I simulation results using Cohen-Coon (left), modified (middle), and IMC/Lambda (right) tuning rules.
STABILITY OF CASCADE CONTROL WITH DYNAMICS IN INNER PROCESS ONLY
Test Case II represented a less common cascade control application called total output control, in
which the only dynamics in the process are contained in Process A, and the outer loop exists entirely in
the control system. The dynamics that were assigned to Process A represent a first-order + dead time
model of the process in a typical flow control loop, i.e. a dead time of 3 seconds and a time constant of
5 seconds. No dynamics were assigned to Process B.
The results are shown in Table 4 and Figure 10. The cascade control system was unstable when very
fast tuning was used, but stable in the other cases. When the gain of the unstable controller in the outer
loop was decreased by a factor of three, the loop could be made stable. The outer to inner loop
response time ratios for stable response were 2:1, 1.1:1, and 1.1:1 respectively for the three tuning
methods.
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Table 4. Stability of a cascade control system with dynamics in inner process only, tuned using different tuning methods.
Cohen-Coon Tuning Modified Cohen-Coon IMC / Lambda Tuning
Process A td = 3 sec; τ = 5 sec td = 3 sec; τ = 5 sec td = 3 sec; τ = 5 sec Inner loop tuning KC = 1.37; TI = 0.08 KC = 0.68; TI = 0.08 KC = 0.59; TI = 0.083
Inner loop t63 5 sec 7 sec 8 sec
Process B td = 0 sec; τ = 0 sec td = 0 sec; τ = 0 sec td = 0 sec; τ = 0 sec Aggregate process model td = 3 sec; τ = 2 sec td = 3 sec; τ = 4 sec td = 3 sec; τ = 5 sec
Outer loop tuning KC = 0.63; TI = 0.048 KC = 0.6; TI = 0.075 KC = 0.6; TI = 0.088
Outer loop t63 5 sec 8 sec 9 sec
t63 ratio 1:1 1.1:1 1.1:1 Stability assessment Unstable Stable, some cycling Stable
Outer loop tuning for stability KC = 0.21; TI = 0.048
Outer loop t63 when stable 10 sec t63 ratio for stability 2:1
Figure 10. Test Case II simulation results using Cohen-Coon (left), modified (middle), and IMC/Lambda (right) tuning rules.
STABILITY OF CASCADE CONTROL WITH EQUIVALENT DYNAMICS IN PROCESSES A AND B
This test case represented a cascade control application in which the dynamics in the constituent
processes are equal. This could represent some processes in which both inner and outer loops control
temperature. A dead time of 15 seconds and a time constant of 45 seconds were assigned to Process A
and Process B.
The results are shown in Table 5 and Figure 11. The cascade control system was stable in all cases.
When the gain of the unstable controller in the outer loop was decreased by a factor of three, the loop
could be made stable. The outer to inner loop response time ratios were 2:7, 2.8:1, and 2.2:1
respectively for the three tuning methods.
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Table 5. Stability of a cascade control system with equal dynamics in Process A and B, tuned using different tuning methods.
Cohen-Coon Tuning Modified Cohen-Coon IMC / Lambda Tuning
Process A td = 15 sec; τ = 45 sec td = 15 sec; τ = 45 sec td = 15 sec; τ = 45 sec
Inner loop tuning KC = 2.7; TI = 0.5 KC = 1.35; TI = 0.5 KC = 0.74; TI = 0.75
Inner loop t63 25 sec 34 sec 60 sec Process B td = 15 sec; τ = 45 sec td = 15 sec; τ = 45 sec td = 15 sec; τ = 45 sec
Aggregate process model td = 40 sec; τ = 25 sec td = 43 sec; τ = 42 sec td = 44 sec; τ = 81 sec
Outer loop tuning KC = 0.64; TI = 0.56 KC = 0.49; TI = 0.8 KC = 0.65; TI = 1.36 Outer loop t63 67 sec 95 sec 130 sec
t63 ratio 2.7:1 2.8:1 2.2:1
Stability assessment Stable Stable Stable
Figure 11. Test Case III simulation results using Cohen-Coon (left), modified (middle), and IMC/Lambda (right) tuning rules.
ASSESSMENT OF CASCADE CONTROL BENEFITS
The disturbance-rejection benefits of using cascade control in each of the three test cases were
assessed. Each of the three tuning methods and resultant disturbance rejection capability was evaluated
on each test case. In every case, a disturbance was injected at the output of the secondary controller. As
a result, Process A was affected, which subsequently affected Process B. The deviation of the primary
control loop’s process variable under cascade control was compared to its deviation under single-loop
control.
The benefit of cascade control was calculated as the percentage reduction in the deviation of the
primary process variable achieved through cascade control, as shown in Equation (1). A benefit of
100% means that the disturbance was completely eliminated by the cascade control inner loop, and
50% means it was reduced by half, etc.
(1)
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The results of the disturbance-rejection evaluation are presented in Table 6. The benefits of cascade
control were more pronounced when the inner loop was substantially faster than the outer loop.
However, even in the case of ratios around 3:1, the benefit of using cascade control was still around
50%. Cascade control presented little benefit in the case where Process 2 had no dynamics. But in this
application, cascade control is typically used for controlling total flow rate from multiple loops, and
not necessarily for improved disturbance rejection.
Between using Cohen-Coon and Modified Cohen-Coon tuning, a relatively small amount of benefit
was sacrificed for a substantial improvement is loop stability. The use of IMC/Lambda tuning on loops
with long time constants resulted in poor disturbance rejection, as evident in the results from Test Case
III.
Table 6. Benefits of cascade versus single-loop control.
Cohen-Coon Tuning Modified Cohen-Coon IMC / Lambda Tuning
Test Case I 83% 77% 76%
Test Case II Unstable 6% 7%
Test Case III 66% 58% 46%
CONCLUSIONS
The stability and disturbance rejection capability of cascade control loops were analyzed using an
encompassing selection of different applications and tuning methods. The main conclusions drawn
from the analyses are noted below.
STABILITY OF CASCADE LOOPS
The requirement for an outer loop having to respond three times slower than the inner loop is needed
only when using aggressive tuning methods on cascade control applications that have most of their
dynamics in the inner loop, such as total flow control. If tuning is done by using step-tests to determine
process characteristics, the combined dynamics of the inner loop and outer process are contained in the
process model of the outer loop. If nonaggressive tuning rules are used for calculating controller
settings, stable cascade control can be achieved without any consideration of inner loop versus outer
loop dynamics or response times. Nonaggressive tuning can be obtained from quarter-amplitude
damping tuning rules by dividing the controller gain by a factor of two or more.
DISTURBANCE REJECTION OF CASCADE LOOPS
The disturbance-rejection capabilities of cascade control acting on disturbances entering the inner loop
are more effective when the inner loop is substantially faster than the outer loop. However, even in
cases where the closed-loop time constant ratios are around 3:1, cascade control can still reject 50%
more of the disturbance than single-loop control. Detuning the controllers from quarter-amplitude
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damping by dividing the controller gain by two results in a relatively small loss in disturbance rejection
capability for a substantial improvement in loop stability.
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