a new perspective on the tuning, stability, and benefits ... · cascade control loops. the tuning...

Distributed with permission of author by ISA, 2013

Presented at ISA Automation Week 2013; http://www.isa.org

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



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



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



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



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



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



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.



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.



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.



Table 4. Stability of a cascade control system with dynamics in inner process only, tuned using different tuning methods.


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.



Table 5. Stability of a cascade control system with equal dynamics in Process A and B, tuned using different tuning methods.


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)



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.


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



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.

REFERENCES

1. K.J. Åström and T. Hägglund, Advanced PID Control, ISA, 2006.

2. A.B. Corripio, Tuning of Industrial Control Systems, ISA, 1990.

3. F.G. Shinskey, Process Control Systems: Application, Design, and Tuning, 4th Edition, McGraw-

Hill, 1996.

4. W. Tan et al, Robust Analysis and PID Tuning of Cascade Control Systems, Chemical Engineering

Communications, Volume 192, pp. 1204-1220, 2005.

5. O. Arrieta, R. Vilanova, P. Balaguer, Procedure for Cascade Control Systems Design: Choice of

Suitable PID Tunings, International Journal of Computers, Communications & Control, Vol. III,

No. 3, pp. 235-248, 2008.

6. V.M. Alfaro, R. Vilanova, O. Arrieta, Two-Degree-of-Freedom PI/PID Tuning Approach for

smooth Control on Cascade Control Systems, Proceedings of the 47th IEEE Conference on

Decision and Control, Cancun, Mexico, Dec. 9-11, 2008.

7. C.C. Yu and W.L. Luyben, Conditional Stability in Cascade Control, Industrial Engineering and

Chemical Fundamentals, Volume 25, pp. 171-174, 1986.

8. B.G. Liptak, Instrument Engineers’ Handbook: Vol. 2: Process Control, Third Edition, CRC Press,

1995.

9. http://www.linkedin.com/groups/cascade-loop-tuning-1967039.S.74565221

10. G.K. McMillan, Tuning and Control Loop Performance, Third Edition, ISA, 1994.

11. G.F. Gilman, Boiler Control Systems Engineering, ISA, 2005.

12. J.N. Sorge, C. Taft, et al, Advanced Steam Temperature Control Using Controller-Resident MPC

Algorithm, 52nd ISA POWID Symposium, 12-14 May, 2009.

13. G.H. Cohen and G.A. Coon, Theoretical Consideration of Retarded Control, Trans. ASME, 75, pp.

827-834, 1953.

14. J.F. Smuts, Process Control for Practitioners, OptiControls, 2011.

15. Rivera, D.E., M. Morari, and S. Skogestad, Internal Model Control 4. PID Controller Design,

Industrial Engineering and Chemical Process Design and Development, 25, p. 252, 1986.

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a new perspective on the tuning, stability, and benefits ... · cascade control loops. the tuning...

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