ALGORITHMS FOR INDUSTRIAL MODEL PREDICTIVE CONTROL.

D.J Sandoz, M Desforges, B Lennox and P GouldingControl Technology Centre Ltd.

School of EngineeringUniversity of Manchester

July 1999

SynopsisThis paper is concerned with control methods that have been embedded in an industrial

Model Predictive Control software package and which have been applied to a wide variety of industrial processes. Three methods are described and the various features

are evaluated by application to a constrained multivariable simulation. The relative attributes are contrasted by assessing the ability of the controllers to recover effectively

from the impact of a large unmeasured disturbance. One particular method, that employs Quadratic Programming to manage Cost Function minimisation and

Manipulated Variable constraints together with degrees of freedom prioritisation to manage Controlled Variable constraints, is suggested as the most appropriate for

general purpose application.

ABBREVIATIONS

ARX Auto-Regressive-eXogenousCV Controlled VariableDMC Dynamic Matrix ControlFIR Finite Impulse ResponseFSR Finite Step ResponseFV Feedforward VariableGPC Generalised Predictive ControlLP Linear ProgrammingLQ Linear QuadraticLR Long RangeLRQP Long Range QPMPC Model Predictive ControlMV Manipulated VariableNARX Nonlinear ARXQDMC Quadratic DMCQP Quadratic ProgrammingRBF Radial Basis FunctionRFC Residual Feedforward

CompensationRGA Relative Gain ArraySQP Sequential Quadratic

ProgrammingSVD Singular Value Decomposition

INTRODUCTION

This paper describes and compares algorithms that have been employed for

the exploitation of Model Predictive Control( MPC) in industry.

The most well known algorithm for industrial MPC is that of Dynamic Matrix Control( DMC). This algorithm was developed in the late 1970s(Cutler and Ramaker, 1980) in association with the Shell Oil Company in the USA. Today DMC is so standard that it is taught in many under-graduate control engineering courses and yet it retains its position as market leader in industry. There are various competitors to DMC, RMPCT, SMOC, Perfector, …, (Qin and Badgwell, 1996) and each carries its own style and idiosyncrasy. The only UK developed offering that is internationally competitive is the control engineering associated with the software product Connoisseur(Sandoz, 1996). Another very strong UK influence in this area of technology has been Generalised Predictive Control or GPC( Clarke D.W. et al, 1987). However, although GPC has become a standard in academic circles, it has not been engineered into any industrial product of current day significance.

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The initial exploitation of the control engineering of Connoisseur was in 1984, with application to Cement Kilns, Spray Drying Towers and Engine Test Beds( Hesketh and Sandoz, 1987). More recent applications have related to Petro-chemical processes, Grinding Mills, Power Generation Plant and Steel Annealing Furnaces(Warren 1992, Norberg 1997, Sandoz et al 1999), Since that time there has been a slow evolution in the capability of this control engineering, to some extent moving to keep pace with developments in numerical procedures and computing power. There has not been any detailed description of the progression of this engineering within the literature. This paper seeks to redress this omission and details three particular methods that are now available as options for use with MPC schemes.

1. The LR Method. This employs the Linear Quadratic( LQ) method from State-space Optimal Control Theory(Jacobs, 1974) in association with prioritisation of CVs( Controlled Variables) to provide a basis for pragmatic management of degrees of freedom in consequence of current and anticipated constraints( Sandoz 1984). This method has been the backbone of most industrial applications associated with Connoisseur to date.

2. The Quadratic Programming( QP) Method. This utilises QP to minimise the LQ cost function and simultaneously to resolve constraints associated with both CVs and MVs( Manipulated Variables), i.e. the soft and hard constraints respectively. QP has also been employed to produce an enhanced version of DMC known as QDMC( Garcia and Morari, 1986, Prett and Garcia, 1988). QDMC has been exploited widely by the Shell Oil Company in the USA although

application has been restricted to within this company.

3. The LRQP Method. This is a combination of the above two approaches. QP is used to solve the cost function minimisation and to simultaneously resolve constraints associated with MVs. Soft constraints are resolved using the same prioritisation approach as with the LR Method.

Comparative details of these methods are presented and simulation studies are used to highlight the various attributes. The LRQP method is argued to be the most favoured approach for industrial exploitation.

All three algorithms employ the same cause to effect model structure to describe process dynamics. This is a multivariable time series format that may take on both an ARX or an FIR structure( Sandoz and Wong, 1979). Many industrial MPC approaches, DMC being one, restrict consideration to FIR or FSR structures. An ARX structure can be a much more compact representation of process dynamics and hence the identification of ARX model coefficients is more efficient, particularly when being done in real-time adaptive mode( Hesketh and Sandoz, 1997). However, the ARX form is less accurate in representing the steady-state relationships of the process and is therefore not best suited for use with Linear Programming( LP) optimisation. The ARX form gives rise to superior control performance in rejecting the influence of unmeasured disturbances. Example of this largely unappreciated factor is illustrated below.

All industrial MPC algorithms employ the “Moving Horizon” principle. At each control instant a set of moves is computed to cover the complete design horizon( i.e. the future period of time across which the cost function is

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minimised) although only the first move is implemented, which is the one that relates to current time. The future moves are discarded. In practice this approach is essential to compensate for modelling error and for the influence of disturbances. If an infinite horizon is considered, then for the linear and unconstrained case the Moving Horizon approach gives identical results to those achieved by successively implementing the full set of computed moves. However, if the horizon is not infinite( in the sense that the solution has not converged) or if responses are moderated because of the need to honour CV constraint boundaries, then the results will not be the same. Of particular concern is the management of CV constraints that are predicted to arise in the future. The accuracy of such prediction when only the first move of a design is applied is questionable. Some methods of MPC focus upon response profiling across the future horizon but then compromise such analysis by employing “Moving Horizon”. Some of the implications of such compromise are considered.

The QP approaches are based upon a parametric analysis in that the Hessian and Gradient coefficients that are required by the QP are computed using the dynamic time-series model. An alternative, that is now becoming available in the product, is to use SQP( Sequential QP) which calculates approximate coefficients from model predictions and which effects a gradient based search mechanism( Bazaraa et al, 1993). SQP is computationally less efficient but does have the merit of being much simpler in code and also of being tractable with non-linear process relationships( Qin and Badgwell, 1998). Some of the practicalities of exploiting SQP are reviewed.

LINEAR MODELS FOR MPC

Industrial Process Dynamics may be characterised by a variety of different model types. Signals for such models are usually grouped into three categories, CVs , plant outputs that can be

monitored, MVs , plant inputs that can be

manipulated and Feed-forward Variables ( FVs), plant

inputs that cannot be manipulated and which disturb the process.

For MPC, consideration normally restricts to models of two basic categories; Parametric State-Space (ARX) or Non-parametric Finite-Step/Finite-Impulse Response (FSR/FIR). In fact both of these categories may be subsumed by one general purpose multivariable time-series format subject to the presence and scale of various dimensions. Thus, consider the general purpose linear model equation.

1.with

k the sampling instant, and k->k+1 the sampling interval, y a vector of p CVs that the model predicts,Y a vector of R samples of y i.e. Y y y yk k

TkT

k RT T | , , . . . . , |1 1

U a vector of S samples of u, with u a vector of m MVs andV a vector of S samples of v, with v a vector of q FVs.

The dimensions of the transition matrix A and the sampled vector Y are set by the number of CVs and the orders of process dynamics that prevail for each cause to effect path of the model. The transition matrix may be considered as a composite of all of the transfer function

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denominators associated with these paths. The offset vector d is not required if differences in the samples of y, u and v are taken across each sampling interval (e.g. use yk - yk-1 rather than yk etc.). The dimensions of the driving matrices B and C are set by the number of MVs and FVs and by however many samples S are necessary in U and V to establish a model that accurately reflects the process. This vague statement firms up very easily if the equation is representative of an FIR or FSR structure. The transition matrix is removed (i.e. the order of dynamics may be considered to be zero on all paths so that dynamic behaviour is represented entirely by the transfer function numerators). The number of samples for any path is then as many as required to encompass the time span to settle to steady state following a step change in any input (i.e. the sum total of any pure time delay and dynamic transients). The overall dimensions of B and C are thereby defined by the paths with the maximum sampling requirements and the matrices are padded with zeros to maintain validity. It can be that the matrices are quite sparse. A mechanism of cross-referencing pointers is then desirable and this can dramatically reduce workspace storage demands (which can be very large, particularly when the model involves a mix of fast and slow response issues).

If a transition matrix is declared, then in principle the dimensions of B and C should reduce significantly.

For any input, there needs to be at least as many samples as are needed to span the largest time delay into any CV, otherwise the equation would not be causal. A few extra samples may be necessary in order to validate the description of dynamics, should this be complex. Further, if there are multiple delay phenomena, for example such as

arise with processes that involve the recycling of material streams, then the sampling must extend to cover their successive impacts.

Many industrial technologies restrict to the FIR/FSR form, with the implication of very large model structures for sometimes very simple situations. Large structures impose significant computational burden in solving for control moves, but this is of little consequence except for very large systems, given the state of today’s low cost computer power. However, such structures do create problems for statistical identification methods because of the large number of coefficients that have to be determined. The identification of large amounts of coefficients requires large amounts of data that imposes a time constraint on delivering good results, irrespective of computer power (e.g. by requiring extended exposure to plant for data collection and experimentation). This is particularly the case for real-time identification which runs in parallel with MPC and which may, for example, be associated with a scheme for adaptive modelling and control.

The State-space parametric or ARX forms can be much more compact and therefore more suited for efficient identification of their parameters. However there are downsides. Accurate prediction for such a model is dependent upon good reflection of dynamics within the sampled history of the CVs. Should these signals be subject to significant levels of noise then, subject to sampling intervals and the ability to employ suitable filtering, the ability of the model to accurately predict steady state from a dynamic state reference can be compromised. In addition, multivariable structures that involve both fast and slow dynamics can present difficulties. For example it is very common for a controller to involve CVs that are liquid

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levels, which respond in seconds and other CVs which are analysers monitoring some chemical composition, which respond in tens of minutes. Proper description of the dynamic behaviour of the levels requires a very short sampling interval which then has to be imposed upon the model segment that describes the analysers. Accurate representation of analyser behaviour then relies upon a high degree of precision in the transition matrix coefficients which is not practicable, particularly with noise present.

The form of model used also has practical implications on controller operation, particularly with respect to initialising a controller in the first place. The very first move that a controller makes can only be correct if the sampled history in the model vectors properly reflects earlier plant behaviour. If the vectors are large, waiting time before a controller can go on line can be significant, with as much as one or two hours being quite common. Of course, this aspect is irrelevant if the controller is switched on with the process in a steady-state condition.

The proper description of processes that involve CV integration or which are open loop unstable, necessitates the involvement of a transition matrix A. Coefficients within A then reflect properties that are equivalent to poles being on or outside the unit circle (in z domain parlance). The problem with such systems is to obtain representative test data. Pulse testing, rather than step testing is often a good option in such cases, allowing rich data to be generated without the process going out of bounds. From the perspective of the model, issues of integration can be avoided by incorporating rate of change of CV although this may not be consistent with control requirements. From the perspective of control the inclusion of both a CV and its rate of

change can be beneficial for the management of aspects with very slow rates of integration, such as often prevail, for example, with temperatures in heating systems.

The transition matrix A can be presented in two forms. The first, termed homogeneous, restricts the prediction relationship for any CV to include only sampled history for that particular CV and not for any other. The second, termed heterogeneous, bases the prediction of each CV upon reference to the sampled history of all of the other CVs. The heterogeneous form is consistent with the standard state space equation representations of dynamic systems. The usual form adopted with chemical plants is homogeneous, because CVs are often distributed quite widely around the process and dynamic interdependence amongst the CVs does not necessarily make practical sense. Another advantage to the homogeneous form is that if a CV fails, it does not affect the ability of the model to predict behaviour in the other CVs. With the heterogeneous form, the failure of any CV will invalidate the ability to predict for every CV of the model.

CONTROL SYSTEM DESIGN

There are various techniques that can be employed to achieve satisfactory control once a representative cause to effect dynamic model is available. The choice of Industrial Model Predictive Controllers in the marketplace is quite limited and each one adopts a distinct and somewhat idiosyncratic approach to the development and application of the multivariable controller. Arguments fly as to their relative nuances and capabilities. In reality there is probably little, in performance terms, to choose between them. Most control methods may be engineered to work well, given a model that satisfactorily represents the process. The approach described here

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is straightforward as a direct variation of the standard LQ method of Optimal Control. A cost function of the general form:

2. is minimised, where E = y - s, with s a vector of set-points and e = u -t, with t a vector of MV targets. U is constructed from MV samples that are differenced. N is chosen large enough to ensure convergence so that the solution corresponds to that for an infinite horizon. The approach generates the optimal controller which has the general form:

3.

K is a matrix of Controller Gains that give rise to the optimal control response and u is an incremental move vector (i.e. a set of changes to be made in the MVs, the sampled data equivalent of integration with the controller so that unmeasured disturbances will be rejected). The vector E1 is given by

B1 and U1 correspond to B and U, with the term in u absent. B2 is the first m rows of B. V is comprised of FV samples that are differenced. A1 is derived from A in order to ensure certain properties reviewed below.

Equation 3 is derived by solving the Riccati equation that arises from the above expression of the optimal control problem. Very efficient procedures that involve orthogonal transformations (Householder) can be employed to calculate the matrix K( Silverman, 1976).

There is a feature of set-point handling that is distinctive in the approach described here and which has very advantageous properties for the controlled recovery from the impact of unmeasured disturbances. Consider the way in which the vector E1 is formed. The set-point vector s is a constant throughout all samples of E (i.e. it is not subject to the sampling index k). It is the origin of the space corresponding to the condition with U=0 and V=0. If s changes, so too do all of the E terms within E1. This has important and powerful implications in the performance of the controller. The controller does not employ change of set-point information in generating a response to error. This is different from the situation with conventional error feedback control systems and is also different from mechanisms adopted with alternative MPC procedures. In these cases, the change of set-point is a key contribution in generating the controlled response to the new set-point values. This has the drawback that such a controller lacks this vital contribution when responding to errors induced by unmeasured disturbances. The worst situation that can then arise is that recovery from the impact of an unmeasured disturbance is at a rate consistent with the open loop time constants of the process. A controller of the form indicated in equation 3 does not have this deficiency and the effort applied by the controller to minimise error is consistent however that error is induced.

The shaping of the response of CVs to set-points is by choice of the elements of the weighting matrices Q, P and q. In fact, without any significant loss in capability, this choice can be reduced to a single weight per signal, with the matrices being diagonal and with each weight duplicating on the diagonal as appropriate. The exercise of choice is made even more amenable by normalising the model matrices prior to

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application of the design algorithms. Normalisation of data is a necessary and standard feature that is used with the model identification procedures in order to prevent numerical problems with the computations. The model that arises is itself normalised and the relative scales between different variables are standardised to be equivalent. If this model is used to develop the control system, and if the weighting matrices are set as unit matrices, the resulting controller has very attractive properties, with effort and urgency being balanced evenly within the multivariable infrastructure. It is then usually quite straightforward for the design engineer to modify the weightings up or down from this standard in order to establish control responses that are appropriate for real application. This mechanism of course necessitates facilities for process simulation and interactive CAD.

The elements in e in equation 3 provide the basis for MVs to be also driven to targets (or set-points). This is of benefit when the controller is operating with more MVs than CVs so there is slack in the available degrees of freedom. This slack provides opportunity for an optimiser to position the “spare” MVs to best economic benefit.

The elements in V in equation 3, i.e. a sampled history of differences in the FVs, provide for feed-forward control. This leads to discussion of a basic weakness that is present in all sampled data control systems. The rate at which model and control system update (i.e. the interval k to k+1) is determined by a balance of factors; The need to properly regulate the

fastest time constants of the system (the rule of thumb for this is to chose the interval to be less than one third of the smallest time constant).

The desire to keep the model and controller dimension down to a

manageable number of coefficients. The smaller the interval, the greater the number of coefficients needed to describe the longer term dynamics and the less viable is the ARX description because of inability to detect dynamics across successive CV samples. This issue is particularly problematical when the multivariable system encapsulates a broad range of fast and slow dynamics.

The desire to keep the computational burden of control to a minimum and therefore to have the update interval as large as practicable.

The need to react promptly to incoming changes in the FVs or to the impact of unmeasured disturbances. If an FV changes half way through the control update interval then there is a delay before corrective action is taken. The faster the controller updates, the smaller this delay is.

A compromise has to be struck between these issues. A pragmatic address to the FV issue is to short cut the update of the controller when a large disturbance is detected, i.e. implement control action directly. This can be effective, most particularly when the process is close to steady state when the disturbance arises. It does, however, disrupt the sampling process which will give rise to a temporary deterioration in prediction and control accuracy. This mechanism should therefore only be used on an infrequent basis and only in response to disturbances of significance.

Consideration of the robustness of multivariable controllers has become topical, largely because academics on an international wave-front have been making significant contributions to this topic rather than because any particular variant of MPC is not robust. Robustness has two contexts;

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The ability of a controller to operate with an inaccurate model of the process.

The ability of a controller to achieve its objectives with the available degrees of freedom.

With the design of classical regulatory control systems, the first aspect is dealt with by gain and phase margin considerations. For multivariable control systems, consideration of the “infinity norm” rather than the squared cost (or two norm) criterion of equation 2 provides a basis for determination of a control system that will be stable for all model inaccuracies of a declared scale (Maciejowski 1989). Unfortunately, such consideration tunes the controller for the worst case, with the consequent risk of inducing mediocrity for normal situations in order to guarantee acceptable behaviour in the abnormal situation. A more pragmatic address is straightforward when working with a controller that is based upon the cost criterion of equation 2. A controller is more tolerant to model inaccuracy when the moves it is able to make are more constrained. This is achieved by increasing the weightings within P that are associated with the incremental MV moves. The pragmatic address to the robustness of an operating controller is therefore quite simple. If the controller begins to exhibit hyperactivity, increase the P weights as necessary in order to make things settle down (this exercise can even be made automatic by using on board control rules). A good rule, if control performance deteriorates, is to first check the process to find out if anything has changed or is faulty. It may be that the only way to bring the controller back to par is to obtain a new model. Adaptive real-time modelling can then contribute to avoid the need for repetition of open loop plant tests.

The management of degrees of freedom within a multivariable infrastructure can be a very problematical issue. Certain

combinations of MV to CV cause to effect paths may demonstrate high levels of dependence between parameters, in the extreme to the extent that required combinations of CV set-points cannot be achieved. Alternatively, very large combinations of MV moves may be required to produce small degrees of discrimination between the CVs. This issue has nothing to do with dynamics but rather, reflects in the steady state gain matrix of the multivariable process. If this matrix is close to being singular (or in the non square case, if a singular value is close to zero), then this suggests that the particular cause to effect description is not viable for control. The issue will be highlighted by a large Condition Number for the matrix (i.e. the ratio of the largest to the smallest singular value). With appropriate scaling, it is possible to analyse patterns of the singular values of the gain matrix to highlight which parameters exhibit dependence. Given such knowledge, it is then possible to select weightings to avoid contradictory objectives.

There is one more twist to the issue of the rejection of unmeasured disturbances. If a process is distributed, with some responses to input changes arising well before others, then these early responses can be used as a means for early detection of unmeasured disturbances. A mechanism which is termed “Residual Feed-forward Compensation” (RFC) operates with a separate model that is used to predict the early parameters. This model runs continuously against the process so that residuals are generated as continuous signals to reflect the error between plant and model. These residuals can then be treated as process signals in their own right and to be representative of the hitherto unmeasured disturbances. They may therefore be employed in the main multivariable controller as FVs to offer

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considerable anticipatory benefit in the management of the longer responding parameters within the multivariable system.

Given a control system that arises in consideration of the various design issues described above, it is essential to then provide the simulation environment that allows the engineer to thoroughly assess the implications of the design selections( i.e. to visualise step and disturbance rejection responses in a simulation context that is as close as possible to reality). It is also important to make it easy for the engineer to quickly modify design aspects and to assess the relative implications of such modification.

Note that in the industrial context it is most important that comprehensive facilities for Gain and/or Model scheduling are available in order to deal with nonlinearity in the piecewise linear sense or to accommodate situations of variability because of changes in plant or product.

INDUSTRIAL MODEL PREDICTIVE CONTROL

MPC is concerned with the operation of multivariable controllers in the face of process constraints. Such constraints come into three categories; MV minimum and maximum limits, MV incremental move limits and CV minimum and maximum limits;The MV constraints are termed “hard” since they can be rigorously enforced. The CV constraints are termed “soft” in that it is desirable that the constraints are honoured but process conditions might not always allow this.

It is unusual for a multivariable controller to be viable for industrial process application without having to cater for constraint issues. The problem is simple. Each MV move computed via

equation 3 is derived in the consideration that all of the other MV moves can be implemented. If any MV is constrained so that the computed move is not applied, then the calculated moves for the other MVs are not appropriate and control will break down (even if only one MV saturates, there will be consequent offset of all CVs from their set-points). If an MV saturates, a degree of freedom is lost and the number of CVs that can be driven to set-point is reduced by one. Practical management of constraints is a key area often neglected in the control literature. The effects of poor constraint management can nullify any benefits gained through advanced multivariable control.

Three approaches to MPC are discussed below. The first( The LR Method) is the

method that has been most widely exploited by Connoisseur users and is a pragmatic approach that involves prioritised and common sense control engineering to manage MPC.

The second( The QP Method) is the most sophisticated, with the complete problem of optimisation in the face of both hard and soft constraints being resolved by the Quadratic Programming( QP). It replaces the Riccati solving approach with a QP numerical optimiser that is able to minimise the cost J of equation 2 whilst also accounting for these signal constraints.

In the third approach( The LRQP Method) the constraints that are processed by the QP are restricted to the MV hard constraints and the CV soft constraints are dealt with in similar fashion to the LR Method.

The pros and cons for each of these methods are emphasised below.

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The LR Method

A fundamental approach adopted is to maintain the situation whereby the number of MVs in play is greater than or equal to the number of CVs. If an MV saturates, then that MV is locked at the saturation limit and a reduced controller is computed to appropriately take up the remaining degrees of freedom. To maintain balance this may require a CV to be dropped from the multivariable controller.

To deal with this in a sensibly managed way, the MVs and the CVs can be prioritised so that the CVs of least importance are eliminated first and so that the MVs that are in play are consistent with an acceptable condition number for the steady state gain matrix. The design engineer defines such priorities in consideration of the needs of the process and in consideration of the relative steady state sensitivities of the cause to effect paths. For the latter considerations, simulation of the various options quickly provides an appropriate perspective for design, particularly when used in association with RGA and SVD tools. Given definition of the priorities, mechanisms for selection of the signals that are to be involved in the controller can then be automatic and straightforward, altering as required in the face of variations in the encountered hard constraints.

Note that whether or not an MV or CV signal is present within a controller is not just a matter of saturation. It is also subject to the health of transducers and actuators. A multivariable controller can often continue with effective operations despite the failure of some of its signals. In fact there are further refinements possible here. An MV might be switched off for the purposes of automatic manipulation but may still be required to be involved in a feed-forward context (i.e. the MV becomes an FV). If a CV is

switched off then it can be desirable to substitute with an inferred value so that direct control of that CV can continue. Such inference is straightforward given the presence of the multivariable model that associates with the controller (inference of this nature should only be used for short periods since the inferred CVs can quickly drift away from reality if models are not precise). The mechanism can be very useful for providing interpolated values for analysers that only provide readings on an occasional basis.

The priority based approach has the advantage that it is simple to set up on the basis of sensible judgement by the process engineer. It does not individually address, however, all of the permutations of cause and effect that might arise. It is possible that certain selections might be inappropriate, for example it may be necessary to drop the highest priority CV rather than the lowest, in order to maintain an effective condition number. Fortunately, operating experience does suggest that exceptions of this nature are not common. It is important, however, that the engineer be given the facility to trap such exceptions and to override the standard procedure with more appropriate structure selections. This mechanism has been provided for the algorithms described here by an interpretative command language known as Director. Director subroutines, which are supplied with condition number and saturation status, may be called at critical stages in order to analyse and perhaps override the standard decision making procedures. Alternative cause and effect signals may be selected for the controller. The introduction of such Director usually evolves with the experience of controller operations.

For control purposes, CVs divide into two categories, those for control to set-points and those which are to be

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maintained within soft constraint boundaries. Constrained CVs normally float free. At each instant for execution of the controller, a closed loop simulation is carried out across a defined horizon into the future (the Long Range or LR horizon). It is closed loop so that the simulation reflects as closely as possible circumstances that are really going to happen. Should a constrained CV be predicted to exceed any constraint boundary at any stage, then that CV is introduced into the controller and the whole process is repeated. This simulation procedure is multi-pass, with soft constraints being brought in successively in consideration of priorities and available degrees of freedom. It would be usual to allocate constrained CVs a higher priority than set-point CVs so that in a crisis the controller diverts its attention to the maintaining of the process within bounds rather than maintaining required quality targets. When a constrained CV comes under control, there is the issue of selection of an appropriate set-point to control to. In principle this should be the constraint boundary itself (or inside it by a small comfort zone), on the presumption that an optimiser is pushing the controller to the boundaries. However, it is not wise to drive the process towards a boundary if a violation is predicted (it’s going to get there anyway and extra impetus might lead to unwelcome overshoot). Therefore the set-point is manipulated so that the CV will move gently towards the boundary.

The closed loop simulation for the detection of soft constraint violations can employ either a linear or a non-linear model( e.g. a neural network RBF model[ Haykin, 1994]) whereas the control design mechanism requires a linear model. A non-linear model provides the opportunity to predict potential constraint violations with greater accuracy with the consequent

possibility of holding the process closer to constraint boundaries for improved economic benefit. This ability to employ non-linear prediction within the MPC procedure is an important strength of this LR approach.

Note that every MV and CV combination that is encountered requires an individual solution of the Riccati equation that gives rise to a specific set of gains K for that situation. Certain combinations are encountered very frequently and it is more efficient, computationally, to save these gains in a table as they are computed. Then, when they are required again, the controller simply needs to point to the appropriate reference within the table. However, if any weighting is changed or if any signal is switched on or off, it is necessary to throw away all stored gains and re-compute from scratch. For large systems, computation in the initial stages as the tables are being created, can be heavy but demand reduces significantly as the look up mechanism phases in.

It is possible to dramatically improve the time required to solve the control design equations by Blocking together steps in the design horizon. The most appropriate form of Blocking is for MV moves to be compacted further out on the design horizon, on the basis that such adjustments should be fairly gentle. The earlier adjustments should processed without Blocking. However it unfortunately arises that such interference with the very efficient Householder solution algorithm means that the answer with Blocking takes longer to compute!(There is probably scope for some enlightened algebra here that could be devised to overcome this difficulty). An alternative form of Blocking that does give rise to computational efficiencies is therefore proposed. Thus, for example, suppose horizon steps are to be blocked in five’s

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and that the MVs are constrained to make adjustments of equal increment for each of the five steps within each block. Within the design process, an adjustment is then computed for every fifth step on the horizon, with one fifth of this adjustment being applied for each of the next successive five steps. The process is iterated for as many cycles as necessary to achieve convergence, in the same fashion as for the single step case, in order to generate the gains K. In actuality, only the very first adjustment is applied to the process (strict obedience for optimal control would require that such increments be recalculated only every fifth step, but this is unacceptable, for example because of the feed-forward compensation issues discussed above). The effect of updating the MV moves more quickly is interesting. It produces a dampening influence upon the controlled responses, similar to that achieved by increasing the MV weightings and is in consequence possibly also a major contribution to robustness. This is a variation on observations made when employing designs based upon a sampled data approach in a mode with continuous feedback( Sandoz and Appleby, 1972). There is the potential, still to be thoroughly investigated, that this approach obviates the need for the designer to be so heavily concerned with MV weightings in order to secure robust control, thereby simplifying design selections for the engineer.

The capability to set up a comprehensive real time simulation of the complete mechanism for this form of MPC is essential. Such simulation would normally be able to run faster than real time and provide the engineer with the capability to feel for and to tune and refine the modes of operation of the control system, prior to any encounter with the real process.

The QP Method

The numerical procedure of Quadratic Programming provides an alternative for solution of the control issue as expressed above. The QP algorithm requires the presentation of information that encapsulates predictive behaviour across the complete design horizon N, together with the cost weightings and constraints that are to apply (i.e. a Hessian matrix, a gradient vector and constraint inequality matrices). At each control instant k, the complete problem may be resolved by just a single call to the QP procedure. In contrast to the Ricatti approach, which produces a set of controller gains, the QP gives rise to a profile of current and future MV moves and only the current move is employed. With reference to equation 3, it is only when N is large enough to ensure convergence that a succession of first moves( as k iterates) produces the same responses as would arise by the implementation of the complete profile. In fact, this “Principle of Optimality” is only true without the involvement of soft CV constraints( see below). The QP method is computationally demanding, particularly if N is large, so the choice of N becomes an issue for design. Blocking can be employed in a fashion similar to that described for the LR Method to improve computational efficiency by compressing the horizon. Blocking for QP can be more sophisticated in that it is possible to efficiently compact MV moves further out on the design horizon. Such Blocking is extremely effective in reducing matrix dimensions and solution times. It does, however, need to be employed with caution. Blocking too early in the response horizon can degrade controlled performance significantly.

Consider the hard constraints. The QP delivers MV moves that account for the hard constraints and which in

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consequence are valid to employ on plant irrespective of whether these moves are constrained. There is no need to prioritise and reduce structure simply because of MV constraint and in particular because of incremental MV move constraints. In this respect, the QP approach is more “intelligent” and delivers better performance. However, even with QP, if an MV locks at a constraint boundary there is still the loss of a degree of freedom and continued effective management of set-points still requires re-computation with a reduced structure subject to priorities. One interpretation of “lock” is currently that the QP generates an MV move that is bounded for the complete design horizon, which appears satisfactory for resolution of the degrees of freedom issue.

A QP solver may also be presented with a matrix of constraint inequalities that relate to the CVs in addition to the aspects discussed above. This provides an elegant mechanism for solving within soft constraint bounds but which, from the Control Engineering perspective, has certain weaknesses. This form of QP implementation has

two objectives, first to satisfy soft constraints and then to minimise cost. Thus if the QP can satisfy a soft constraint issue by taking a sledgehammer to the process, it will do so. This can give rise to violent MV moves irrespective of cost function weightings. These exaggerated moves are simple to reduce by the use of the hard incremental move constraints. However, this perspective makes the proper selection of these constraints a different and a far more important design issue with the QP method than with the LR method. Too much constraint of the MV incremental moves can also give rise to unwelcome unfeasibility and soft constraint relaxation( see below).

It is possible that the QP will not be able to establish a solution that satisfies the soft constraint requirements. In such a case it is necessary to relax the soft constraint boundaries to a degree that allows a solution to prevail. Such relaxation is achieved by introducing an auxiliary cost function that involves dummy MVs( sometimes known as slack variables). A dummy MV adds to the soft constraint values that are within the inequality matrices that are presented to the QP, in order to broaden the range of validity. A very high cost penalty is imposed upon the dummy MVs so that they are only exploited if there is no alternative, i.e. if the QP could not otherwise obtain a solution. The dummy MVs may also be assigned relative cost weightings so that certain constraints are relaxed in preference to others. However, if such relaxation arises it is at the expense of the control engineering that is expressed in the primary cost function and unwelcome overall effects can arise. An example of such a situation is presented below.

Soft constraint considerations are on the basis that the complete horizon of MV moves is to be implemented on the process, rather than a sequence of first MV moves. The solving for soft constraints is not associated with the primary cost function and so the “Principle of Optimality” does not apply. It is possible that the MV moves at the first stage are influenced by boundary considerations further out. In such a circumstance the moving horizon approach will not generate the same responses as application of the full set of MV moves from a single QP iteration. Most activists in the arena of MPC appear to ignore the implications of this fundamental point, i.e. that sophisticated analysis

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is employed to generate a solution that is then ignored.

The LRQP Method

The LRQP method restricts the constraints handled by the QP to the MV hard constraints( both incremental and absolute) whilst the soft constraints are managed in similar fashion to the LR method( i.e. by simulation and prioritisation). It seeks to combine the strengths of the above two methods to provide a more effective overall address for control engineering application. This approach avoids the need to

have to address the issue of unfeasibility.

The dimension of the QP problem is significantly reduced to give enhanced computational efficiency.

The CV constraints are managed by feedback. This is more tolerant to model inaccuracies than the QP approach for which the constraint avoidance mechanisms are essentially open-loop( i.e. are not associated with the error based minimisation of the cost function).

The assessment of whether CVs will violate constraint boundaries is most accurately carried out by a simulation that reflects what is actually going to be done in the future, and that is with only the first move of the QP design being implemented at each step into the future. This argument justifies the incorporation of the QP approach to replace the control design engine within the LR prioritised management procedure. Note that one benefit of this is that it leaves open the opportunity to operate a non-linear simulation for detection of the soft constraint violations. The downside of this method of utilisation of QP is the computational effort to achieve solution. Each control step will involve multiple passes of the QP so that the optimisation calculations have to be repeatedly implemented (the

luxury of tables of Gains for fast reference is not available). This approach is therefore expensive computationally.

Each execution of the QP implicitly involves a simulation across the design horizon, since the QP delivers a complete set of MV moves for that horizon. Therefore, embedded within the QP is a basis for directly calculating future CV behaviour, without need for any further control calculations. This is the approach may be employed within the LRQP method as a compromise for the sake of computational efficiency, although there is a price to pay in terms of control effectiveness, as shown below.

SQP( Sequential QP)

Sequential (or Successive) Quadratic programming (SQP) is a constrained non-linear optimisation technique which minimises a specified function using a series of quadratic approximations. SQP is an iterative process, with two distinct stages at each iteration: firstly a QP problem is solved to yield a direction in which the solution will move and then a step length is estimated which reduces the objective function in this direction in some optimal way.

In terms of Model Predictive Control, use of an SQP method enables either linear or non-linear models to be used, with the SQP minimising the resulting quadratic or general non-linear MPC cost functions.

The advantage of using an SQP solution, rather than the QP methods discussed above, is that a range of model structures may be used without the need to re-define the QP problem specifically for each model type. Instead, it is only necessary to be able to evaluate the cost function for any given set of MVs. As a result, non-linear

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models based on, for example, Neural Networks or NARX structures, may be incorporated into the MPC problem for the control of general non-linear processes. For such models, the Hessian matrix required for the QP problem will not be known in advance; indeed it will vary with the solution as the SQP proceeds towards a minimum. Therefore, most SQP algorithms start with a generic initial value of the Hessian matrix, typically an identity matrix multiplied by a constant. The Hessian matrix is then updated with successive iterations of the SQP using, for example, the BFGS updating algorithm, converging to the true Hessian as the solution itself converges.

Clearly, the flexibility of the SQP approach comes at a cost. For linear models, where the Hessian is calculable, solution of the MPC problem using an SQP is massively inefficient. Typically, between 10 and 50 iterations of the SQP (each consisting of QP solution, Hessian update and step length calculation) would be performed at each control step for a reasonably sized MPC problem. The large number of iterations required, even for minimisation of a quadratic cost function, results largely from the initially unknown Hessian matrix. Use of a ‘warm start’ approach, where the Hessian is primed using the final value of the previous iteration, may alleviate the computational requirement somewhat.

As a general replacement for QP, SQP may be implemented in any of the ways described in the previous section i.e. with the soft constraints either included in the SQP solution or dealt with externally using the LR methodology. The latter is in fact the most advantageous. If only hard constraints are accommodated within the SQP, the coding is greatly simplified and errors

associated with linear constraint approximation are eliminated.

SIMULATED CASE STUDIES

The Simulated Process

The “Shell Heavy Oil Fractionator Problem” is used as the basis for simulated comparisons and examples( Prett, 1987). This problem was posed by Shell in 1987 to provide a benchmark for the assessment of multivariable control procedures. The model consists of 35 “first order plus time delay” transfer functions. These transfer functions are normalised so all process parameters have a nominal standard range of between –0.5 and +0.5. These transfer functions appear to have been chosen deliberately so that the column is ill conditioned, i.e. column interactions make it difficult to simultaneously control multiple issues because of heavy interaction and a mix of fast and slow time constants and short and long time delays. A simplified structure of Cause and Effect is chosen, as shown in fig. 1. There are three MVs, Top Draw( signal A1), Side Draw( signal A2) and Bottom Duty( signal A3). There are seven CVs, Top End Point Analysis( signal M1), Side End Point Analysis, and five column Temperatures( signals M3 to M7 from column top to bottom). There is a

Figure 1. Cause and Effect Diagram for Simulation Study.

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Causes and Effects

MVs

A1 Top DrawA2 Side DrawA3 Bottoms Duty

FV

A4 Intermediate Duty

CVs

M1 Top AnalysisM2 Side Analysis

M3 Top TemperatureM4M5M6M7 Bottom Temperature

single FV, signal A4, the Intermediate Duty.

For the purpose of comparison between the various methods of MPC that are reviewed above, an example is contrived which drives the process to an extreme constraint situation, but for which there is a final acceptable solution that can satisfy set-point demands within both hard and soft constraint boundaries. This involves the process being initially at a steady-state at the centre of the bounded region. A large and unmeasured disturbance is then applied to the Intermediate Duty( signal A4) and the comparison discussions are concerned with the manner in which the control engineering manages recovery.

Fig. 2 presents four sets of responses: Fig. 2a for the situation with the LR method; Figs. 2b and 2c with the LRQP and 2d with the QP. In each situation, the configuration and tuning selections are identical and the controller updates every 5 s. The controller is required to hold the two analysers( M1 and M2) at setpoint and to maintain the other 5 CVs within minimum and maximum temperature bounds. In fact, only two of these temperatures, M3( top) and M7( bottom) prove to have relevance. The CVs are prioritised in order of importance with M7 highest, then the other temperatures including M3, then M2 and then M1. Thus the management of CV constraints takes priority over set-points, which would be the normal application situation.

Consider Fig. 2a for the LR method. The figure shows, for a time span of 10 minutes: Analyser signals M1( top) and

M2( side) and their associated set-points which are constant at 0;

Temperatures M3( top) and M7( bottom) and their associated

set-points which are manoeuvred to accommodate anticipated soft constraint violations. The bounds are –0.5 to 0.5 for both M3 and M7;

The three MV signals(A1, A2 and A3) which are bounded to the range –0.5 to 0.5 and to a move constraint of no more than 0.1 per update; and

The “unmeasured” disturbance A4, which is seen to undergo a large change from 0 to 2 to the left side of the trends.

Fig. 2a is reviewed in detail to highlight the characteristics of the contrived problem that the various approaches are required to resolve. Progressing from left to right along the trends, detail may be observed as follows:Following the disturbance impact M1 peaks at 0.44 after 40s. and

recovers to set-point after 90s. following some undershoot.

M2 peaks at 1.208 after 35s. but takes more than 250s. to recover to set-point.

M3 peaks at 1.386 after 25s. It is quickly brought back within bounds but stays active as a soft constraint until about 190s. after the impact. It is at this point that the degree of freedom that is directed to hold M3 is released and redirected to bring M2 back to set-point.

M7 peaks at 0.458 after just 10s. it then moves down to the lower constraint boundary and it thereafter remains at that boundary.

A1 takes 5 steps to reach its minimum of –0.5 and then stays locked at this minimum for a further 4 steps. It then freely manoeuvres for about 260s. before finally locking at the minimum of –0.5.

A2 manipulates up and down with maximum moves of +-0.1 for a period of 60s. and thereafter moves around with less aggression at no

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Figure 2a) The LR Method

Figure 2b) The LRQP method( single pass)

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Figure 2c) The LRQP method( multi-pass)

Figure 2d) The QP method

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time moving outside the range –0.3 to 0.2.

A3 drives in sympathy with A1 to the minimum of –0.5 but then steps off immediately. It returns to this minimum after 150s, stays there for a further 60s. and then floats away freely.

It takes about 480s. for the process to recover to a steady-state. There are five main phases to this recovery. The first, for a period of about 60s.,

is crisis management to attempt to hold the process within soft constraint bounds and the MVs move as aggressively as they can for this purpose. Set-points for M1 and M2 are completely abandoned.

There is then a period of some 120s. during which M2 is not controlled because both M3 and M7 are under control to prevent soft constraint violation. The MVs can be seen to be slowly moving during this phase as the longer time constants of the process prevail.

Eventually, the constraint on M3 ceases to be predicted to be violated and the third phase is evident. The MVs then manoeuvre to bring M2 down to set-point, which takes about 70s.

Thereafter, for the fourth phase, the MVs move in a gentle fashion compensating for the long time constants, maintaining both M1 and M2 at set-point at the same time as holding M7 within soft constraint bounds.

Eventually A1 reaches the lower limit of –0.5 and the controller is just able to hold the two setpoints and the soft constraint associated with M7 despite having only two degrees of freedom left. This is because in the final steady-state, M7 settles just inside the lower soft constraint boundary.

Now consider Fig. 2b for the LRQP fast method that bases soft constraint evaluation from a single pass of the QP procedure. The responses follow the same general pattern as for the LR method but with the following major differences: The reaction to the initial crisis is

less fierce. The two temperatures are brought back within bounds more slowly and the MVs do not drive to their limits. Initial excursions of these two CVs are in general slightly larger and are outside bounds for about twice the period.

The time for recovery to set-point is faster at 180s rather than 250s.

Fig. 2c illustrates the LRQP method but this time with horizon behaviour being computed by multiple use of the QP to more accurately reflect the reality of the moving horizon mechanism. In this case the response timing closely follows the pattern of the LR method which also re-computes the MVs for each step of the simulation horizon. The time for recovery to set-point is back to 250s., a price to pay for the improved management of the CV constraints during the initial crisis. A major difference from the LR Method is much smoother control action during the recovery period following the initial crisis management phase and reduced amplitude in these manipulations and in the CV responses. This smoother action is as a result of better management of the MV constraints that arises from the application of QP.

Now consider Fig. 2d for the QP method. Following the disturbance impact, for the first 30s, the pattern of responses is very similar to the LR method, however, from then onwards things are quite different. A1 strikes the lower boundary for the second time after 90s. and then essentially stays locked on for the duration. A steady-state is

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Figure 3 Succession of Responses

resolved after 240s. however there is considerable offset. M1 settles at 0.141, M2 at –0.015, M3 at 0.515 and M7 at –0.521. Thus all requirements for control are dishonoured. Neither M1 or M2 are at set-point and both M3 and M7 are beyond their soft constraint boundaries. The reason for this is that the final resting-place is an unfeasible operating point for the QP. The slack variables are invoked to allow the constraint boundaries to relax. In consequence all attention is given to the auxiliary cost function and the set-point objectives suffer because of consequential lack of degrees of freedom.

The contrast between the four sets of responses may also be assessed by reference to Fig. 3, which incorporates the Fig. 2 items in succession on a single trend display. It is clear that the

LRQP multi-pass algorithm provides the most effective management of recovery

Fig 4. shows the performance of the SQP algorithm with LR soft constraint management on the same problem. The responses are very similar to those seen in Fig. 2b), as would be expected, as the SQP solution is simply replacing the QP solution used in the LRQP approach. Slight variations are seen due to the approximate nature of the SQP algorithm, and the use of a slightly different set of weights within the SQP. Unfortunately, whereas the previous methods performed real-time control successfully for 5s sample period on a standard desktop PC, the SQP solution tended to take around 30 seconds for each control step. Whilst a factor of 6 is recoverable through the use of a faster computer, or more efficient coding, the

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Figure 4 SQP control of linear system

relative inefficiency of the SQP with respect to the QP is highlighted.

Blocking with the LR method

Fig. 5 presents responses that indicate the implication of the use of Blocking with the LR Method. Responses to set-point changes from 0 to 0.05 and back again are presented. The first pair of transitions is for the situation without Blocking. The second is for the situation with a Blocking Width of 3 steps. The

Design Horizon N is 24 for this case. The solution is therefore obtained in 8 iterations for the case with Blocking, with the requirement that the MVs move the same amount for each step within the Width( approximating a ramp adjustment). In fact the MVs are recomputed and adjusted at every step. The effect is to dampen the responses. The manipulations are significantly reduced in amplitude and response to set-point change is slowed down.

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Figure 5 Effect of Blocking

Unmeasured Disturbance Rejection( comparison between FIR and ARX solutions).

Fig.6 presents a comparison between the behaviour of a controller that uses a compact ARX model representation and one that uses an FIR format. The left hand portion repeats the responses of fig. 2a for the LR Method. The right hand portion presents the equivalent with an FIR model being employed. The deviations and duration from set-point and constraint boundaries are much larger for the FIR case. The ARX model employs just 8 samples( i.e. S=8) within the U vector in contrast with over 30 for

the FIR case( 30 being the minimum to catch the complete impulse response profile). The reason for the better behaviour with the ARX model is simply that it will pick up and start tracking plant accurately just 8 steps following the disturbance impact. This interval is rather like a window of blindness that confuses the controller. The window is much larger for the FIR case since it takes over 30 steps before accurate prediction is once more in play. Under situations where all disturbances are measured, the performance of the two types of controller is essentially the same.

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Figure 6 Unmeasured disturbance rejection

CONCLUSIONS

This paper describes MPC algorithms that are in use to address a broad range of industrial process control situations. The LR method has, to date, been the most popular for exploitation because, in spite of pragmatism in dealing with constraint issues, it has proven robust, reliable and computationally efficient. The progress with the efficiency of computation( both hardware and software) now makes it practicable to consider the use of QP for medium to large scale industrial problems. QP provides a more elegant address for the management of constraints. However, it is argued that the straightforward application of QP in a control engineering context has its drawbacks, particularly in the management of constraints associated with Control Variables( i.e. soft constraints). The

LRQP method, which is a combination of the best attributes of the LR and QP approaches has therefore been introduced. LRQP uses prioritised control engineering to manage the soft constraints and QP to deal with the Manipulated Variable constraints. LRQP is now considered the favoured method for most industrial applications.

It must be said that the contrived example described in this paper represents a very extreme situation which would rarely be encountered in day to day situations. The example serves to emphasise the draw backs of being comprehensively elegant in dealing with both control engineering and constraint issues within a single QP address. Under normal operating conditions that do not require simultaneous relaxation of multiple constraints, the QP method provides a

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powerful control engine that can out perform the LRQP approach because of its uncompromising treatment of the soft constraint boundaries. However, it is unacceptable that a control method should fail because process operations are in crisis and for this reason the QP method is advised to be employed with extreme caution.

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ACKNOWLEDGEMENTS

The author’s gratefully acknowledge the support of Invensys PLC in funding aspects of this work.

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