Commters them. Emm Vol. 19. Suoal.. DD. S369-S374. 1995

Pergamon 0098-1354(95)00043-7 - bpyrigh; Q’i9~5’klsevier S&&e Ltd

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Controllability analysis and modelling requirements: an industrial example

Stephen Walsh r Tahir Malik

Centre for Process Systems Engineering ICI Engineering Technology

Imperial College of Science and Technology PO Box 13

Prince Consort Road The Heath

London SW7 2BY, UK Runcorn WA7 4QF, UK

‘Author to whom correspondence should be addressed

ABSTRACT

This paper presents a controllability analysis procedure and modelling approach developed to address problems raised by industrial applications of controllability research. Most work on con-

trollability analysis assumes a nonlinear dynamic model as its starting point using linearisation of the model to give a model for frequency domain analysis. Such models may be expensive and

difficult to develop. This paper looks at the use of low order empirical dynamic models for control-

lability analysis. A heuristic controllability analysis procedure is presented which lends itself to

the use of simple models and addresses the use of cascade controllers, which are a feature of many

industrial problems, in a systematic way. An industrial example is presented and the procedure is applied successfully using simple gain-delay-lag models,

KEYWORDS

controllability, procedure, cascade, modelling, industrial, application

INTRODUCTION

The controllability of a system is, in its most general sense, the “best” dynamic performance

achievable for a system under closed loop control. It is often used in the more restricted sense

of the best dynamic performance achievable for a system under a specific form of control, e.g.

using a jixed linear controller. Most work has been based on the following approach: develop

a physically based nonlinear dynamic model; linearise that model; apply controllability analysis.

From an industrial perspective this focus on full dynamic models has several disadvantages:

l it pushes controllability analysis away from front-end design where it has maximum potential

impact ;

l it limits the use of controllability analysis in trouble-shooting and commissioning as the expense of developing a dynamic model may not be acceptable for many problems.

It is therefore important to consider what is the minimum information required to get useful results

from controllability analysis. Steady-state analysis can give some useful information, but issues

related to transient performance require the use of some sort of dynamic model. The contention

of this paper, supported by the example presented below, is that very limited information about

the dynamics (comparable to process reaction curves) may be adequate to answer useful questions about the controllability of quite complex processes, providing a means of making progress when

a full dynamic model is not available. To achieve widespread industrial use, controllability indicators must relate in a direct and intuitive way to commonly used performance requirements. Le., the criteria for “best” dynamic performance should be those of the controller designer rather than those most convenient mathematically. Typical performance requirements for an industrial problem would be: constraints on the maximum

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deviation of individual outputs; a specified set of disturbances which more usually takes the form

of steps rather than sinusoids.

Skogestad et al (1992aJ992b) present methods which go a considerable way to meeting these re- quirements, in that they focus on meeting constraints on outputs and inputs in the face of a set of

disturbances (a vector of sinusoids of bounded amplitudes and common, unknown frequency). The central idea of their work seems to be to use open loop frequency response to define a bandwidth

for which control is required (some output constraints are violated) and then to assess control structures based on inherent bandwidth limitations and ability to satisfy constraints under per-

fect control. Under perfect control the selected inputs are chosen to exactly cancel the effect of

the disturbances on the selected outputs. Control structures which satisfy all the constraints un-

der perfect control and which exhibit no characteristics preventing tight control, at least up to

the required control bandwidth, are considered as candidates for further analysis. The approach

presented by Skogestad et al. does however have a number of drawbacks.

l The disturbance model used is not related in an obvious way to typical step disturbances.

l There is ambiguity in the definition of achievable bandwidth.

- Multiple criteria are presented for estimating achievable bandwidth in the presence of time delays, non-minimum phase transmission zeros and phase lags - it is not clear

how to unify these.

- A single bandwidth is used for a multivariable system. This is quite acceptable if all the

controlled outputs are of a comparable nature, but may be quite misleading otherwise.

l The analysis focuses on square control systems, while most industrial control systems are

not square, but use more measurements than actuators (cascade controllers).

Before addressing modelling requirements we present a heuristic controllability procedure, incor- porating some novel elements to provide a partial solution to problems noted.

A HEURISTIC PROCEDURE FOR CONTROLLABILITY ANALYSIS

The procedure will be built around a number of heuristics which are presented below.

Heuristic 1: If a control system is inadequate under perfect control, i.e. with the inputs chosen to ensure no deviation of the selected outputs from their specified value, then it will be inadequate with an implementable controller.

It is not usually necessary to control any variable perfectly and doing so may disturb other variables

more than is desirable. The idea of perfect control is however a useful device to avoid the need to

either design a controller or to pose and solve an optimisation problem based on the true objectives.

When high loop gain is possible, i.e. within the achievable bandwidth, perfect control performance

can be approximated well by implementable controllers.

Heuristic 2: The transient response of a stable single-input single-output linear time invariant

dynamic system to a step is bounded by the maximum value (infinity norm) of the frequency response.

If no assumptions are made about the relationship between gain and phase or the variation of gain with frequency then there is no finite bound on the ratio of these two quantities. On the

other hand, experimentation with typical dynamic systems supports the approximate validity of

this bound. For systems showing a sharp peak in the frequency response the bound becomes quite

conservative, while for overdamped systems it is tight. This heuristic allows step disturbances to be related to a frequency domain analysis. The need for heuristics such as this is an Achilles’ heel of frequency domain controllability analysis. While many aspects of frequency domain controllability analysis can be transferred readily to the time domain, the concept of perfect control sits more readily in the frequency domain, where it can be analysed by matrix manipulation, than in the

time domain where its analysis involves high index dynamic models, or optimal control problems. For this reason, we utilise the frequency domain analysis in this paper.

European Symposium on Coaqhter Aided Process Engineering-S s371

Heuristic 3: The achievable bandwidth for a stable SISO system is bounded by the frequency at

which the open loop phase shift first reaches 180 degrees, wlso.

PID control makes no attempt to improve on this limit. While more complex controllers can

in principle do so they are in practice likely to be vulnerable to model errors and measurement noise. In preliminary controllability analysis it seems sensible to treat this bound as absolute. This measure aggregates the effect of time delays, right half plane zeros and other dynamic elements to

give a single value.

Heuristic 4: In a multivariable controller no individual feedback path should imply a bandwidth

beyond wlso, Corollary: perfect control is only realistic if the transfer function matrix between

the selected inputs and outputs is non-singular after paths whose bandwidth has been exceeded

are deleted.

This heuristic may be challenged on the basis that it is possible to have a multivariable controller

which is stable even if a particular path would be unstable in the absence of the other connections. This is indisputable, but it does not seem desirable. Indeed the basis of the widely used RGA and

Niederlinski index controllability tests is that this situation is undesirable as it implies a lack of

robustness to failures of actuators or measurements (poor integrity) and may cause commissioning

difficulties.

Building on these heuristics we propose the following procedure for controllability analysis of stable

systems.

1. Define control objectives and disturbances. The control objectives should be expressed as

bounds on variable values. The disturbances should be expressed in a form suitable for

frequency domain analysis. This may require the use of heuristic 2.

2. Compute the set of frequencies, W~ontro, - all frequencies for which the open-loop effect of

disturbances cause violation of the ith constraint. For multiple disturbances combine the

effects on a given constraint by summing the magnitudes of the individual effects at each

frequency.

3. Identify the possible measurements and actuators and hence the set of possible control struc- tures.

4. Compute the bandwidth limit (wlso) for each signal path between actuators and measure-

ments. If the largest value of wlso does not encompass W~Ontro, for all constraints then all

possible control structures fail.

5. Generate and evaluate square control structure options. For each structure (j) considered

determine the set of frequencies, Wi, for which control performance in respect is potentially

adequate in relation to each output constraint (i). Control is considered potentially adequate

at a particular frequency if:

(a) the transfer function between u and y is non-singular when paths for which wlso has

been exceeded have been deleted - if this is not the case perfect control of all outputs becomes unimplementable;

(b) deviations of the inputs are not excessive under perfect control;

(c) deviations of the constrained variables are not excessive.

6. If no structure has a IV’; encompassing W&ntro, for each output constraint, then it is necessary

to consider non-square controllers. If no combination of square controllers can be found such that the union of the individual Wi is adequate, then all structures fail the controllability test.

7. If multiple sets of structures satisfying the requirements above exist then favour those which

involve only a few, simple structures and which have sets Wi which are ordered in frequency and involve significant overlap. These characteristics lend themselves to straightforward

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implementation of a cascade control arrangement, e.g., with the control structure effective at

the higher frequency band being the secondary controller and the control structure effective

at the lower frequency band being the primary controller.

If at any point in the procedure no control scheme satisfying the tests can be found then control is likely to be difficult. Due to the underlying heuristics, failing the tests does not show that no

controller can meet the performance requirements. If further analysis is carried out it should focus

on structures for which the maximum constraint violation over frequency is small and occurs over

a narrow frequency range.

MODELLING REQUIREMENTS FOR CONTROLLABILITY ANALYSIS

Based on the above discussion we need a model to predict the phase crossover frequencies, 20~80,

of potential signal paths within a controller and to predict frequency responses of the system to

potential controlled inputs and disturbances.

A simple model meeting these requirements is a matrix of gain-delay-lag elements G linking

disturbances and inputs to the outputs. Complex responses may require summing several such

elements for a good representation. Such a model could be identified from simple plant tests

assuming that disturbance inputs can be perturbed or that suitable disturbances occur and can

be measured. The plant experimentation required for such a model is comparable to that for controller tuning using process reaction curves. If the starting point for the analysis is a steady- state model then the gains can be generated by linearisation and the delays and lags estimated

based on flowrates/ holdups and established correlations such as those given by McMillan (1983).

In the example below, a nonlinear dynamic model is available, so we will use the frequency response

of the full order linearised model to examine the accuracy of the analysis using this simple model.

AN INDUSTRIAL EXAMPLE

The example examined is an exothermic plug flow reactor configured as shown in Fig. 1. Target

steady-state values and constraints have been identified for two variables:

variable steady-state value value upper limit lower limit

Tin 400°C

I I

none 300°C

Tout 600°C 650°C none

There are only two potential manipulated variables, the split between the product and recycle,

Sl, and the split between the heat exchanger and the bypass S2. A split value of 1 corresponds to

100% of product recycled or bypassed. There are five directly measured variables for consideration

(see figure 1). It should be noted that Fi, is not itself available for control as it is determined by

the requirements of another plant. Ai, is a delayed composition measurement of the key reactant. The plant has been modelled in SPEEDUP using 421 states and 2574 equations. The key distur- bance is a well defined change between two operating regimes, A and B. This involves three feed component flowrates, but as the disturbance only occurs in a single direction with respect to these flowrates it is appropriate to represent it as a single disturbance. The objective is to make the

change between the two operating levels as quickly as possible, ideally as a step. The discussion below will be based on the disturbance occurring while at operating regime A as this was found

to be the limiting case.

The first step was to develop a matrix transfer function comprising delay-gain-lag elements. The manipulated variable deviations are required to lie within f.7, corresponding to the margin between the initial operating point and the maximum split value of 1. The temperature measurements, the disturbance and the manipulated variables are all normalised by their maximum acceptable deviation. Obtaining these models by step response testing was complicated by the presence of

a pronounced oscillatory mode in the response to the control inputs. The oscillations showed a steady amplitude and a frequency of about 11 rad/hr. Figure 2 shows the response of the reactor

European Symposium on Computer Aided Process Engineering-S s373

outlet temperature to a step change in the recycle split, Sl. The model was fitted to the mean value

of the oscillation, with the delay encompassing the initial “inverse response”. The results obtained

are shown below, with time in hours. For comparison steady-state gains from a linearisation of the model are shown in brackets.

I Ain

F recycle

T out Tin I[ d

Sl 1 (1) s2

The first point to note is that the model predicts that the full disturbance cannot be rejected at steady-state due to actuator limits (S1=2.2 (2.2) S2=.035 (.016)). However, solution of the

nonlinear model for the new steady state gives S1=.65, S2=.08 . This reflects the limitations of linearisation for controllability analysis when the disturbances are “large”. Rather than aborting

the analysis, at this point, we relaxed the actuator constraint on Sl from 1 to 3 to give a similar margin for actuator action at steady state to that observed using the full nonlinear model.

201s0 was determined from the above model as below (linearisation results in brackets).

Paths with no gain are assigned a value of 0 for WlsO. Open loop analysis indicated control was required for frequencies from 0 to 27 (30) rad/hr for Tout and from 0 to 7 (15) rad/hr for

Ti,. The discrepancies between the simple model and the full linearisation were attributable to the oscillatory mode which is captured by the linearisation as a resonant peak in the frequency

response. The maximum wlsO comfortably includes wi for each constraint.

Using the procedure presented, the next step is to evaluate the possible square control structures

to see if any scheme satisfies the performance requirements. As zero offset at steady state is

required for T&, and TO,, the control scheme must apply integral feedback from both the measured

temperatures, so the structure [Ti,, T,,$; Sl S2] must be part of any solution. Perfect control

of both temperatures fails above a frequency of 5 rad/hr as only one signal path (S2 to Ti,) is suitable for high gain (approximately perfect) control. Adequate actuator power is available at all

frequencies. Above this frequency perfect control can only be applied from Ti, to S2. This scheme

shows excessive deviation of Tour up to 28 rad/hr (26 rad/hr with linearised model). Based on these

observations no square feedback controller exists capable of delivering the required performance.

Two secondary measurements ( Frecycle, A) are available to augment the temperature feedback controller between 5 rad/hr and 30 rad/hr where improved performance is desired. Ti,, can also

be used in this frequency range. Only one square controller based on these measurements gives

significant improvement over the open-loop deviation of Tout - [Tin A; 5’1 S2]. Both the simple

model and the linearised model predict that this reduces the maximum deviation in this frequency range ( 5 goes to 3.2 with the simple model; 30 goes to 4.3 with the full linear model). Bandwidth

considerations suggest that the simplest implementation of the combined (non-square) controller

would be to pair Tin with S2 and cascade Tout through A onto Sl. RGA analysis supports the integrity of this multi-loop structure. The structure independently developed by the ICI engineers

was as above with the addition of a cascade from A to the ratio between the inlet and recycle Aow

which then determined Sl. The convergence between the two designs is quite encouraging.

The final aspect to consider is whether the analysis does provide a reasonable bound on the achievable performance. The disturbance considered in the nonlinear model is a smoothed step

with r = .04 hours. To,, is observed to peak at 680 “C. The smoothed step can be analysed as a step passed through a linear system with an associated frequency response. This frequency response

s374 European Symposium on Computer Aided Process Engineering-5

can then be multiplied at each frequency with the disturbance transfer function determined above,

and the response to the smoothed step estimated as for a step applied to this combined system.

Both the simple and the full linear model then predict a maximum transient value of 750 “C.

In this case the bound is almost a factor of 2 above the achieved performance supporting use of heuristic 2 to bound performance with “step-like” disturbances, but emphasising the point that it is only an approximate estimate of the performance actually achievable.

For this example, the simple model gives the same conclusions as the more complex model, despite

the fact that the original model exhibited complex dynamic response and was therefore not an

ideal choice to illustrate this approach. The results therefore support the use of simple empirical

dynamic models for controllability analysis when a full dynamic model

CONCLUSIONS

is not available.

The example illustrates that quite crude dynamic models can be used as a basis for preliminary

controllability analysis. The procedure proposed suggests the same control scheme as that chosen

in the control study carried out by ICI. While some of the analysis is quite involved, we believe the

techniques could be packaged so that process engineers with a moderate control background are

comfortable with the interpretation of the results in terms of required and achievable frequency

ranges. We hope that this example will encourage further use of controllability analysis industri- ally and stimulate further work on the problem of matching controllability analysis to industrial

practice. The work described has been carried out as part of an ongoing collaboration between ICI and

the CPSE at Imperial College aimed at overcoming problems with the industrial exploitation of controllability analysis and improving the tools available.

References

G. K. McMillan. Tuning and Control loop performance. Instrument Society of America Publica-

tions, 1983.

S. Skogestad and E. A. Wolff. Controllability measures for disturbance rejection. In Intemctions

Between Process Control and Process Design, pages 23-29, 1992.

E. A. Wolff, S. Skogestad, M. Hovd, and K. W. Mathisen. A procedure for controllability analysis. In Interactions Between Process Control and Process Design, pages 127-142, 1992.

,,----. : i,FRC$ .__,.* Sl to Tout

LLFJ re&zta

,,‘---‘\’

_______; \ Tout!

C__.’

Figure 1:Process schematic Oil0 2.00 mo (ho=4

Figure 2: Typical step response