- Email: [email protected]
SUBSPACE IDENTIFICATION IN INDUSTRIAL APC APPLICATIONS - A REVIEW OF RECENT PROGRESS AND INDUSTRIAL EXPERIENCE
14th IFAC Symposium on System Identification, Newcastle, Australia, 2006
SUBSPACE IDENTIFICATION IN INDUSTRIAL APC APPLICATIONS - A REVIEW OF RECENT PROGRESS AND INDUSTRIAL EXPERIENCE
Hong Zhao*, Michael Harmse, John Guiver, William M. Canney Aspen Technology, Inc. 2500 City West Blvd., Houston, Texas 77042, USA [email protected] Abstract: The subspace identification has been available in industrial advanced process control (APC) applications in a commercial APC package since 2000. After five years of industrial use, several issues were identified and addressed; the subspace identification technology has been widely accepted and is now used in numerous industrial APC projects. In this paper, a review of the 5-year application history with industrial experience and recent progress in the plant test and identification is given. Questions like why the parametric model identification (ID) methods, which have been theoretically proved optimal, such as prediction error approach, are not routinely used by industrial APC practitioners and why the APC practitioners are now in favour of subspace ID are answered from a point of view of industrial practice. Several important practical ID issues that are more concerned by industrial practitioners are discussed. More specifically, a new view on the issue of open-loop vs. closed-loop ID is provided with the recent progress in multi-variable constrained plant testing technology. It has been found that the subspace identification works very well with the innovative plant testing approach in a synergistic way, and able to substantially reduce plant test duration. As a result, improved identification efficiency from shorter but richer data sets results in better model accuracy, and consequently project costs have been reduced significantly over the past 4 years. With more applications of the subspace ID in industrial MPC projects, some known issues and the future needs are also provided to invite academic researchers to help address. Copyright © 2006 IFAC Keywords: system identification, subspace methods, model predictive control, industrial control, closed-loop identification, process control.
1. INTRODUCTION variable constrained plant testing technology and subspace identification technology work together in a synergistic way to reduce plant test duration substantially. Improved identification efficiency from shorter but richer data sets results in better model accuracy, and consequently project cost have been reduced significantly over the past 4 years, making it possible to deploy APC on process units that would not have yielded a suitable return on investment before. For example, plant testing and model identification had been completed in 48 hours instead of 2 weeks on a recent project. In addition, an APC project in a complex Fluid Catalytic Cracking (FCC) unit with 35 manipulated variables (MVs) and
The subspace identification approach was introduced into industrial advanced process control (APC) applications around 5 years ago. Although the method has received many attentions from academia for a considerable period, the subspace identification algorithm only became available as part of a commercially available industrial APC package in 2000. After five years of industrial use, several issues were identified that required algorithmic improvements, the subspace identification technology has been widely accepted and is now used in numerous industrial APC projects. We have found that multi*
To whom all correspondence should be addressed
1074
94 controlled variables (CVs) have been reportedly completed in only 4 days. Using the traditional manual step testing and FIR model identification approach, it would have taken 3-4 weeks to finish the project.
alternative model identification algorithms, especially those based on transfer function models, did not work well enough for typical process applications. Low order transfer functions have the benefit of always resulting in smooth step-response models, since only a small number of parameters are used to explain an entire data set. However, for processes with significant heat integration, recycles and interacting PID controllers, these model structures yield poor results. Given the state of the art in 1986, it was simply not possible to fit complicated responses using low order transfer functions, and the more flexible FIR model structure was selected. In contrast, FIR models are easy to identify using a Least Squares method.
2. A REVIEW OF THE HISTORY Academic scholars have questioned why the parametric model identification (ID) methods, which have been theoretically proved optimal, such as prediction error approach, are not routinely used by industrial APC practitioners. From a point of view of the practicing process control engineer, the following limitations may explain some of the major reasons for the slow adoption of parametric methods:
In contrast to the limitations of low-order transfer function methods, FIR ID method has the ability to fit very complicated process responses, since a large number of model parameters are available, typically 90 to 120 coefficients per MV~CV step-response curve. A typical model matrix may have thousands or even tens of thousands of coefficients. Over the last 20 years, the FIR ID method has been accepted by industrial APC practitioners with its following advantages: • The ID algorithm is simple and fast to solve large size model ID problem because the least square fitting and a MISO model structure; • FIR models are able to fit both fast and slow dynamics because of their non-parametric model structure; especially they are superior in identifying process models with inverse responses (non-minimum phase process). • The FIR identification is intuitive and easy to be understood by APC practitioners, because the traditional MPC controller uses a similar framework for model ID and model predictive control.
Problem size: for example, the FCC unit mentioned before needs a model matrix size of 35 (MVs) x 94 (CVs). Though the problem may be divided into a few smaller identification problems in practice, it will still be a major challenge for a traditional parametric identification algorithm to identify a large model efficiently and accurately. Model accuracy: given the typical model size, another challenge for traditional identification algorithms is to ensure that the same model is able to capture both fast dynamics (in minutes) and slow dynamics (in tens of hours) from limited plant test data. For example, in an ethylene plant, it is often seen that the reflex influences on product quality in a few minutes, while the re-boiler’s level’s change may affect the product quality in 3~10 hours. True MIMO problem: with the increase of number of MVs and CVs, some of the CVs are often highly correlated (i.e., they share dynamic states) in an identification problem, and this may easily cause numerical problems when the traditional parametric ID methods are applied to a large true MIMO model identification.
On the other hand, the non-parametric FIR identification method has its native limitations. The disadvantage of a large number of model parameters is that the identified models can be noisy, especially if the data set is small or the signal to noise ratio is poor. In some respects, FIR model identification is closer to a data interpolation technique than a true model identification technique. Another disadvantage of the FIR model is the truncation error. If the user selects a model with too low a Time to Steady State (TTSS) value, the model curves will be truncated (cut off) too early, and the gains will be inaccurate due to significant truncation errors.
2.1 Non-parametric FIR and low-order parametric model identification methods Because of the challenges listed above, the LeastSquare based non-parametric Finite Impulse Response (FIR) identification algorithm with smoothing functions and simple low order parametric model identification algorithms are widely used in the process industry (Qin and Badgwell, 2003), often relying on first order plus dead time models (or at most second order + dead time models). The nonparametric FIR model identification approach is able to work around the practical difficulties mentioned above to some extent, although FIR has its own limitations.
2.2 Subspace identification methods Subspace identification algorithms appeared in the 90’s. It has attracted lots of attentions in recent years. Among many algorithms the Canonical Variate Analysis (CVA) (Larimore, 1983, 1990), N4SID (Overschee and Moor, 1994), and MOESP (Verhaegen and Dewilde, 1992) are major representing ones. An internal review and comparative evaluations of several subspace
FIR model identification has been available for several decades now, and has been used in thousands of successful industrial MPC applications. When the original Dynamic Matrix Control [DMC] controller technology was built around two decades ago, those
1075
algorithms were conducted at Aspen Technology, Inc. in 1997. It was concluded that the Canonical Variate Analysis (CVA) method (Larimore, 1983, 1990) is one of the most appropriate and efficient algorithms for industrial MPC applications, because of the following features: • The CVA algorithm is able to well handle relatively large size MIMO ID problems; • The resultant model is well balanced in modelling both fast and slow dynamics; • As all other subspace ID algorithms, it does not require iterative computation and therefore there is no local minima problem that are often seen in traditional parametric ID algorithms; • The CVA algorithm is relatively robust in the presence of auto-correlated inputs and feedback correlation; • Its framework is similar to the MPC concept, i.e. using measured data from a past window to optimally predict the future responses over a future window. Therefore, it is somewhat easier for practicing engineers to understand.
capture the initial fast dynamics, e.g. inverse responses. • Subspace ID is able to offer an auto-balanced MIMO model that captures both slow overdamped and fast potentially under-damped dynamics in one parametric state space model of relatively modest order. Such morels are more appropriate for MPC applications. They have better control relevance since the MIMO state space model well reflects those shared dynamic states by multiple CVs. 3.2 Steady state gains and gain ratio If a dynamic model is to be used for constrained multivariable control of non-square systems, then the accuracy of the steady state gains and specifically the accuracy of the gain ratios of the MIMO model are far more important than the accuracy of the dynamics. MPC controllers are sometime considered as Steady-State, Multivariable, Constrained controllers. Therefore, an accurate steady-state gain matrix and accurate gain ratios that properly capture the true degrees of freedom of the process are critical to the performance of the controller.
Based on the CVA theory, a modified CVA identification algorithm with many data preprocessing capabilities was introduced into Aspen Tech’s industrial MPC package in 2000. Since then, the subspace model ID algorithm has been steadily improved based on feedback from both internal users and external customers. Combined with new multivariable constrained step test technology, the benefits have been very significant: The combined plant testing and model identification task, which is the most expensive activity in an APC project, has been reduced by more than half in several instances. The identified models proved to be more accurate, and performed on-line better than ever, because the gain ratios were more accurate when compared to the FIR identification method.
Specifically, the co-linear gains of a process model have to be accurate enough. For example, if the major MV handles of a MPC controller are either turned off or saturated, or the CVs included in the controller do not have strong MV handles, it will be very difficult to control these CVs, and weaker MV handles will have to be employed to move these CVs to their targets. Usually it requires much larger MV movement than before. This is effectively a completely new challenging control problem that is now consisting of a very different constraint set. If we had very accurately identified the model gains and then analyzed the sub-matrices, we will find that multiple 2x2, 3x3 and larger sub-matrices are close to co-linearity as indicated by an RGA number well above 10, or very large condition numbers. This would point to two separate but related characteristics of this particular control problem:
3. ADVANTAGES AND BENEFITS CAPTURED BY SUBSPACE IDENTIFICATION Industrial MPC practitioners are more concerned with the practical issues, rather than the theoretical properties of a model identification algorithm. These practical issues are essential for the success of an industrial MPC project.
Large MV movement: Controlling these CVs given only these weak (inappropriate) MVs is much more difficult, and large MV movement will be required.
3.1 The prediction capability
Excessive MV movement: The accuracy of the calculated MV Linear Programming (LP) targets (the most important driver of MPC controller performance) is dependent on the accuracy of the LP calculation, which depends on the gains. Effectively, the LP is inverting a square gain matrix with a poor condition number, meaning small errors in the gain matrix may result in large changes in the calculated delta MV values required to drive the process to the CV targets that the LP selected. Depending on the errors present in the gains, the MV movement may be far larger than required (potentially leading to instability). Excessive MV movement results when the real RGA numbers of the process are far smaller than we identified from the estimated gain values.
It is very important to MPC applications that an identified model must be able to predict for both fast, potentially complex (high order) dynamic effects caused by interacting PID loops, or extensive heat integration, and slow (lower order) dynamics caused by the larger hold-ups in the process. • FIR models are able to model both fast and slow dynamics, but may also over-fit the data and some undesirable effects may appear in the models due to the impact of unmeasured disturbances. • Low order parametric models can easily capture slow over-damped dynamics, but may fail to
1076
This can lead to large MV cycles or even closed loop instability.
significantly less (e.g. typically 40 data points rather than 180 data points for FIR).
Lazy control: The opposite is also possible. The MV movement may be way too small if the identified RGA numbers are low but the true process mechanism is far weaker than we think (much higher RGA numbers). This will lead to very poor disturbance rejection and potentially shutting down the process due to loss of levels or massive violation of safety critical process constraints. The purpose of the controller commissioning activities is to confirm that the controller has appropriate MV movement for the expected constraint sets.
Auto-correlated MVs and the presence of strong feedback correlation are supposed to affect the model identification negatively, and result in bias in the model estimation. In both traditional manual step testing, as well as the new automated multi-variable pulse testing, the MV moves are designed to minimize their cross-correlation. In addition, care needs to be taken to minimize the correlation between the MVs and the unmeasured disturbances (often referred to as feedback correlation). • The traditional FIR ID algorithm proved to be very sensitive to the MV cross-correlation, and is even more sensitive to feedback correlation. • The improved CVA based Subspace ID algorithm proved to be far more robust for correlated MVs, as well as data containing feedback correlation.
Small gain errors sometimes have a big impact: If the gains of nearly collinear sub-matrices are inaccurate by just a little bit, MVs may actually ramp in the wrong direction during controller commissioning. This is a clear indication that the gain ratios are close, but have flipped (the RGA has changed sign) sign and the model matrix has a directionality error. Under these conditions, we either have to do specific testing to discover the correct process direction or generate more data by simultaneous multivariable perturbation, so that the gain ratio difference will have the right sign, or we need to apply our process knowledge and confirm this process mechanism is real or not, so we can decide to fix the gain ratio and remove a degree of freedom from the controller.
What real benefit can be achieved from true MIMO model identification? It is not so obvious for traditional model ID algorithms. However, it is easy to see the advantages from the workflow of a subspace model identification activity: • As the subspace ID algorithm identifies state variables first, more relevant CVs that are included in the model will help to determine the true states with higher confidence or require less data. • For any given amount of plant test data, it has been proved that model accuracy can be improved by adding more correlated measurements as additional model outputs (Pseudo CVs). These dynamically related measurements will often not be included in the MPC system as they do not need to stay within specified constraints. These additional output variables are added to the subspace identification cases purely to improve the accuracy of the MIMO model. • Alternatively, by using more correlated measurements as outputs in a MIMO case, the model can converge faster, and the required testing time can be reduced then.
3.3 The different identification methods The traditional FIR model ID method places less emphasis on the model gains and gain ratios because the use of a MISO model structure and the use of differencing filter on both MV and CV data. The subspace ID method is proved to be able to identify the gain matrix and gain ratios more accurately by using the MIMO structure, and by using de-trending filters on the MVs and CVs data pre-processing, rather than differencing, which suppresses the low frequency (steady state gain) information. The amount of test data required to identify a sufficient model for MPC applications is also important as it affects project cost: • More test data will usually ensure a more reliable model, leading to improved controller performance. However, the duration of plant test is very costly and should be minimized. • If the test data have sections of bad data that requires multiple bad data slices to be defined, then initialization of the ID algorithm will “waste” some valuable test data. For a slow model, the FIR model ID method is relatively expensive in term of the data length to be used for initialization. • Subspace ID is superior in both model convergence rate and the amount of data used for initialization. A bench mark test shows that the subspace ID algorithm converges three times faster than the traditional FIR ID method, and the amount of data required for initialization is also
4. OPEN-LOOP VS. CLOSED-LOOP IDENTIFICATION Model identification from open-loop test data has been studied for over 40 years now. In recent years, lots of attentions have been paid to closed-loop identification. As a result, many new algorithms for closed-loop identification have been proposed, and their theoretical properties have been analyzed. Closed-loop tests provide for a moderate degree of constraint protection and cost less than a manual step test, but the data is more difficult to fit a model. It is true that some of the process properties, like colinearity, nonlinearity, and process operational stability, may need the model to be identified under closed-loop. On the other hand, the available closedloop identification algorithms have their own limitations for industrial MPC applications:
1077
The practical problem size is often too large for most of the published closed-loop ID algorithms. For example, a typical MIMO model for a MPC controller may contain 20~50 MVs and 30~90 CVs, while most published case studies uses simulation data with a model size often smaller than 10 x 5. Many reported closed-loop ID algorithms are either not robust enough for industrial process data (may work in some cases, may not in other cases), or show poor performance when there are strong process drift disturbances.
loop model identification problem then becomes very easy to solve and there are no longer special requirements on the closed-loop ID algorithm. • More significantly, the project implementation time can be shortened by more than half and the cost can then be reduced significantly. Many MPC applications where the multi-variable test methods have been applied in combination with the subspace method have clearly demonstrated that the plant testing and model identification cost can be reduced by more than 50%, while the identified model accuracy and controller performance is superior. As mentioned earlier, a recently completed MPC application on a FCC unit with 35 MVs and 95 CVs took only 4 days in total from plant testing to controller commissioned, a new world record (McIntosh, A. et al., 2003; Kalafatis, A. et al., 2005).
4.1 Recent progress in closed-loop identification In recent years, major progress in industrial MPC application has been made, namely, a “smart” constraint aware plant testing approach was developed to perform “pseudo” closed-loop identification when the process is under a form of discrete output MPC control. The “smart” tester consists of a MPC controller and a test signal generator. The tester starts with an initial model that can be easily obtained from pre-test data or from previously built process models (e.g. MPC revamp project). Practical applications have demonstrated that the requirements for the initial model are often easily satisfied. For example, for each CV, usually only one MV-CV approximate model is sufficient and the mismatch on the model’s gain can be up to 500% while the MPC tester can still work. During the test, multiple (usually all) MVs are perturbed automatically based on a balanced optimization, that is, to maximize the MVs’ perturbation while controlling the CVs strictly within their pre-set constraints. In such a way, the signal to noise ratio of the test data can be 4 times higher than that of those generated from a traditional test. As the test continues, model can be re-identified and updated into the MPC tester periodically from the collected new test data. This new technology brings significant benefits: • Under MPC control, all the important operational constraints can be satisfied, i.e. the process safety and product specification are ensured during the plant test period. • During plant test, the APC engineer no longer needs 24 hours supervision around the clock by using this new plant test tool. It is also a big cost reduction. • A “smart” algorithm can vary the amplitude of the MV pulses to drive all of the MVs at the same time to generate sufficiently independent perturbations to make the model identification problem far easier, while the cross-correlation among MVs are minimized. • More importantly, in those cases where a CV constraint may possibly be violated by several MV superimposing and driving the CVs way beyond their limits are predicted and prevented based on a Model Predictive Control algorithm that takes the worst case model uncertainty into account. • As a result, the plant test data has minimum crosscorrelation among the MVs and very low feedback correlation between the CVs and MVs. The close-
4.2 The new industrial state-space MPC controller needs accurate state-space model Another recent important progress in industrial MPC application is the development of a new state-space MPC controller (Froisy, J. B., 2006) that uses most of the latest MPC research results from academia and the new developed real-time environments for industrial MPC applications. This new industrial State-Space Controller (SSC) is referred to a MPC product bridging between the control theory and industrial practice. The major technical features of this SSC include: • State-space model • Real-time state estimation • Infinite horizon, constrained dynamic move plan These enabling technologies will help address an increasingly challenging set of industrial control problems. As seen above, the new SSC requires state-space models to describe and predict the process behaviours so that it can timely detect disturbances entering process with state estimator and optimally reject them by an infinite horizon MPC controller. Because the SSC’s performance will depend on the models’ quality, the subspace identification offers a perfect alternative to obtain accurate state-space models from plant testing data. 5. KNOWN ISSUES AND FUTURE NEEDS As the subspace algorithm has been accepted and is now widely used in industrial MPC projects, some of its limitations are better known, and work-around tricks have been developed. Industrial MPC practitioners would welcome academic scholars to study these limitations and help to address these issues. Different dead-times for different channels through the MIMO model. As well known, a state space model uses states to describe both pure dead-time and dynamics. In industrial process it is often seen that the pure dead-times are quite different among the
1078
process variables in the same unit, they are in a range from a few minutes to a few hours. In such a case, the model order needs to be set very high and it is difficult to identify a true MIMO state space model.
MV correlations and feedbacks if the MPC controller (tester) operates in a “correction” mode to avoid CV constraints violation or intervened by process operator. As a result, the plant test data contains “intermittent” feedbacks. Usually it is not a problem for the CVA-based subspace ID algorithm. But it may be an interesting topic for academic researchers to investigate how the currently available closed-loop identification algorithms work under such conditions.
Large unknown disturbances (e.g. strong drift disturbances, and periodic disturbances caused by sticky control valves) can impact the subspace ID results negatively. Data pre-processing has been used for removing some of the disturbances, a systematic approach, such as building explicit disturbance models would be very useful for industrial practitioners.
6. CONCLUSIONS After 5 years of practice and improvements, the CVA-based subspace identification technology has been accepted and widely used in industrial MPC applications because of its many special features. Specifically, with the major progress in multivariable plant testing technology, subspace ID offers more efficient ID tool and results more accurate process models. Although there are some known issues to be further addressed, the new way to do the “pseudo” closed-loop plant testing and model identification with subspace ID has demonstrated significant cost reduction in APC project implementation and significant improvements on the MPC controllers’ performance.
Very different dynamics within the same state space model are required sometime in order to satisfy the optimal process operation. The ratio between slowest time constant and the fast time constant can be in the range 20 to 100. In subspace identification, it is a challenging task to identify a model with detailed fast dynamics and accurate slow steady state gains. Currently a compromise between the two ends often has to be made in an expense of the model accuracy. More effective model uncertainty evaluation and validation methods are required to help industrial practitioners to know how good the identified models are. Currently available methods such as timedomain and frequency-domain model uncertainties are useful, but found not intuitive and effective enough in practice. For state-space model uncertainty, it seems there is no well known method available yet.
REFERENCES Froisy, J.B. (2006). Model predictive control – Building a bridge between theory and practice. Chemical Process Control (CPC-7), Lake Louise, Alberta, Canada, January 8-13, 2006. Kalafatis, A., Kalpesh, P. Harmse, M. and Zhang, Q. (2005). A unique multivariable test method for MPC projects dramatically reduces testing time of a Crude Unit. Hydrocarbon Processing, (in print). Larimore, W.E. (1983). System identification, reduced-order filtering and modelling via canonical variate analysis. In: Proceedings of the 1983 American control Conference, 445-451. Larimore, W.E. (1990). Canonical variate analysis in identification, filtering and adaptive control. In: Proceedings of the 29th Conference on Decision and Control, 596-604. McIntosh, A., Cooke, J. and Harmse, M. (2003). Techniques for implementing large scale DMCplus controllers, 7th Annual IEEE Advanced Process Control Applications for Industry Workshop, Vancouver, Canada, Arial 28-30, 2003. Overschee, P. Van and B. De. Moor (1994). N4SID: Subspace algorithms for the identification of combined deterministic-stochastic systems. Automatica, 31(12), 1853-1864. Qin, S.J. and T.A. Badgwell (2003). A survey of industrial model predictive control technology, Control Engineering Practice, 11(7), 733-764. Verhaegen, M. and P. Dewilde (1992). Subspace model identification. Part 1: the output error state-space model identification class of algorithms. International Journal of Control, 56, 1187-1210.
The speed of subspace ID algorithms for large models still needs to improve. An industrial benchmark test has demonstrated an exponential relationship between the ID speed and the model size. With the increase in number of MVs and CVs put into a MIMO model, the computation time required by the subspace ID algorithm grows exponentially. Very recently, a significant improvement on the speed has been made in AspenTech’s MPC product and this speed issue has been well addressed. Improvements in identification with highly correlated MVs and ill-condition system are still needed. Identifying accurate steady-state gains and gain ratios is far more important than those dynamics for an industrial MPC application with collinear MVs or ill-conditioned process. Though some academic studies have been reported in 1990s, industrial practitioners hope academic researchers can re-visit this issue and help to further address it. Identification from intermittent closed-loop data is a new requirement raised with the new multivariable plant testing applications. In most cases, the multivariable plant test generates information rich, un-correlated, good-quality and open-loop equivalent data from a ‘pseudo’ closed-loop environment. Some time a small portion of the testing data may contain
1079
SUBSPACE IDENTIFICATION IN INDUSTRIAL APC APPLICATIONS - A REVIEW OF RECENT PROGRESS AND INDUSTRIAL EXPERIENCE
Hong Zhao*, Michael Harmse, John Guiver, William M. Canney Aspen Technology, Inc. 2500 City West Blvd., Houston, Texas 77042, USA [email protected] Abstract: The subspace identification has been available in industrial advanced process control (APC) applications in a commercial APC package since 2000. After five years of industrial use, several issues were identified and addressed; the subspace identification technology has been widely accepted and is now used in numerous industrial APC projects. In this paper, a review of the 5-year application history with industrial experience and recent progress in the plant test and identification is given. Questions like why the parametric model identification (ID) methods, which have been theoretically proved optimal, such as prediction error approach, are not routinely used by industrial APC practitioners and why the APC practitioners are now in favour of subspace ID are answered from a point of view of industrial practice. Several important practical ID issues that are more concerned by industrial practitioners are discussed. More specifically, a new view on the issue of open-loop vs. closed-loop ID is provided with the recent progress in multi-variable constrained plant testing technology. It has been found that the subspace identification works very well with the innovative plant testing approach in a synergistic way, and able to substantially reduce plant test duration. As a result, improved identification efficiency from shorter but richer data sets results in better model accuracy, and consequently project costs have been reduced significantly over the past 4 years. With more applications of the subspace ID in industrial MPC projects, some known issues and the future needs are also provided to invite academic researchers to help address. Copyright © 2006 IFAC Keywords: system identification, subspace methods, model predictive control, industrial control, closed-loop identification, process control.
1. INTRODUCTION variable constrained plant testing technology and subspace identification technology work together in a synergistic way to reduce plant test duration substantially. Improved identification efficiency from shorter but richer data sets results in better model accuracy, and consequently project cost have been reduced significantly over the past 4 years, making it possible to deploy APC on process units that would not have yielded a suitable return on investment before. For example, plant testing and model identification had been completed in 48 hours instead of 2 weeks on a recent project. In addition, an APC project in a complex Fluid Catalytic Cracking (FCC) unit with 35 manipulated variables (MVs) and
The subspace identification approach was introduced into industrial advanced process control (APC) applications around 5 years ago. Although the method has received many attentions from academia for a considerable period, the subspace identification algorithm only became available as part of a commercially available industrial APC package in 2000. After five years of industrial use, several issues were identified that required algorithmic improvements, the subspace identification technology has been widely accepted and is now used in numerous industrial APC projects. We have found that multi*
To whom all correspondence should be addressed
1074
94 controlled variables (CVs) have been reportedly completed in only 4 days. Using the traditional manual step testing and FIR model identification approach, it would have taken 3-4 weeks to finish the project.
alternative model identification algorithms, especially those based on transfer function models, did not work well enough for typical process applications. Low order transfer functions have the benefit of always resulting in smooth step-response models, since only a small number of parameters are used to explain an entire data set. However, for processes with significant heat integration, recycles and interacting PID controllers, these model structures yield poor results. Given the state of the art in 1986, it was simply not possible to fit complicated responses using low order transfer functions, and the more flexible FIR model structure was selected. In contrast, FIR models are easy to identify using a Least Squares method.
2. A REVIEW OF THE HISTORY Academic scholars have questioned why the parametric model identification (ID) methods, which have been theoretically proved optimal, such as prediction error approach, are not routinely used by industrial APC practitioners. From a point of view of the practicing process control engineer, the following limitations may explain some of the major reasons for the slow adoption of parametric methods:
In contrast to the limitations of low-order transfer function methods, FIR ID method has the ability to fit very complicated process responses, since a large number of model parameters are available, typically 90 to 120 coefficients per MV~CV step-response curve. A typical model matrix may have thousands or even tens of thousands of coefficients. Over the last 20 years, the FIR ID method has been accepted by industrial APC practitioners with its following advantages: • The ID algorithm is simple and fast to solve large size model ID problem because the least square fitting and a MISO model structure; • FIR models are able to fit both fast and slow dynamics because of their non-parametric model structure; especially they are superior in identifying process models with inverse responses (non-minimum phase process). • The FIR identification is intuitive and easy to be understood by APC practitioners, because the traditional MPC controller uses a similar framework for model ID and model predictive control.
Problem size: for example, the FCC unit mentioned before needs a model matrix size of 35 (MVs) x 94 (CVs). Though the problem may be divided into a few smaller identification problems in practice, it will still be a major challenge for a traditional parametric identification algorithm to identify a large model efficiently and accurately. Model accuracy: given the typical model size, another challenge for traditional identification algorithms is to ensure that the same model is able to capture both fast dynamics (in minutes) and slow dynamics (in tens of hours) from limited plant test data. For example, in an ethylene plant, it is often seen that the reflex influences on product quality in a few minutes, while the re-boiler’s level’s change may affect the product quality in 3~10 hours. True MIMO problem: with the increase of number of MVs and CVs, some of the CVs are often highly correlated (i.e., they share dynamic states) in an identification problem, and this may easily cause numerical problems when the traditional parametric ID methods are applied to a large true MIMO model identification.
On the other hand, the non-parametric FIR identification method has its native limitations. The disadvantage of a large number of model parameters is that the identified models can be noisy, especially if the data set is small or the signal to noise ratio is poor. In some respects, FIR model identification is closer to a data interpolation technique than a true model identification technique. Another disadvantage of the FIR model is the truncation error. If the user selects a model with too low a Time to Steady State (TTSS) value, the model curves will be truncated (cut off) too early, and the gains will be inaccurate due to significant truncation errors.
2.1 Non-parametric FIR and low-order parametric model identification methods Because of the challenges listed above, the LeastSquare based non-parametric Finite Impulse Response (FIR) identification algorithm with smoothing functions and simple low order parametric model identification algorithms are widely used in the process industry (Qin and Badgwell, 2003), often relying on first order plus dead time models (or at most second order + dead time models). The nonparametric FIR model identification approach is able to work around the practical difficulties mentioned above to some extent, although FIR has its own limitations.
2.2 Subspace identification methods Subspace identification algorithms appeared in the 90’s. It has attracted lots of attentions in recent years. Among many algorithms the Canonical Variate Analysis (CVA) (Larimore, 1983, 1990), N4SID (Overschee and Moor, 1994), and MOESP (Verhaegen and Dewilde, 1992) are major representing ones. An internal review and comparative evaluations of several subspace
FIR model identification has been available for several decades now, and has been used in thousands of successful industrial MPC applications. When the original Dynamic Matrix Control [DMC] controller technology was built around two decades ago, those
1075
algorithms were conducted at Aspen Technology, Inc. in 1997. It was concluded that the Canonical Variate Analysis (CVA) method (Larimore, 1983, 1990) is one of the most appropriate and efficient algorithms for industrial MPC applications, because of the following features: • The CVA algorithm is able to well handle relatively large size MIMO ID problems; • The resultant model is well balanced in modelling both fast and slow dynamics; • As all other subspace ID algorithms, it does not require iterative computation and therefore there is no local minima problem that are often seen in traditional parametric ID algorithms; • The CVA algorithm is relatively robust in the presence of auto-correlated inputs and feedback correlation; • Its framework is similar to the MPC concept, i.e. using measured data from a past window to optimally predict the future responses over a future window. Therefore, it is somewhat easier for practicing engineers to understand.
capture the initial fast dynamics, e.g. inverse responses. • Subspace ID is able to offer an auto-balanced MIMO model that captures both slow overdamped and fast potentially under-damped dynamics in one parametric state space model of relatively modest order. Such morels are more appropriate for MPC applications. They have better control relevance since the MIMO state space model well reflects those shared dynamic states by multiple CVs. 3.2 Steady state gains and gain ratio If a dynamic model is to be used for constrained multivariable control of non-square systems, then the accuracy of the steady state gains and specifically the accuracy of the gain ratios of the MIMO model are far more important than the accuracy of the dynamics. MPC controllers are sometime considered as Steady-State, Multivariable, Constrained controllers. Therefore, an accurate steady-state gain matrix and accurate gain ratios that properly capture the true degrees of freedom of the process are critical to the performance of the controller.
Based on the CVA theory, a modified CVA identification algorithm with many data preprocessing capabilities was introduced into Aspen Tech’s industrial MPC package in 2000. Since then, the subspace model ID algorithm has been steadily improved based on feedback from both internal users and external customers. Combined with new multivariable constrained step test technology, the benefits have been very significant: The combined plant testing and model identification task, which is the most expensive activity in an APC project, has been reduced by more than half in several instances. The identified models proved to be more accurate, and performed on-line better than ever, because the gain ratios were more accurate when compared to the FIR identification method.
Specifically, the co-linear gains of a process model have to be accurate enough. For example, if the major MV handles of a MPC controller are either turned off or saturated, or the CVs included in the controller do not have strong MV handles, it will be very difficult to control these CVs, and weaker MV handles will have to be employed to move these CVs to their targets. Usually it requires much larger MV movement than before. This is effectively a completely new challenging control problem that is now consisting of a very different constraint set. If we had very accurately identified the model gains and then analyzed the sub-matrices, we will find that multiple 2x2, 3x3 and larger sub-matrices are close to co-linearity as indicated by an RGA number well above 10, or very large condition numbers. This would point to two separate but related characteristics of this particular control problem:
3. ADVANTAGES AND BENEFITS CAPTURED BY SUBSPACE IDENTIFICATION Industrial MPC practitioners are more concerned with the practical issues, rather than the theoretical properties of a model identification algorithm. These practical issues are essential for the success of an industrial MPC project.
Large MV movement: Controlling these CVs given only these weak (inappropriate) MVs is much more difficult, and large MV movement will be required.
3.1 The prediction capability
Excessive MV movement: The accuracy of the calculated MV Linear Programming (LP) targets (the most important driver of MPC controller performance) is dependent on the accuracy of the LP calculation, which depends on the gains. Effectively, the LP is inverting a square gain matrix with a poor condition number, meaning small errors in the gain matrix may result in large changes in the calculated delta MV values required to drive the process to the CV targets that the LP selected. Depending on the errors present in the gains, the MV movement may be far larger than required (potentially leading to instability). Excessive MV movement results when the real RGA numbers of the process are far smaller than we identified from the estimated gain values.
It is very important to MPC applications that an identified model must be able to predict for both fast, potentially complex (high order) dynamic effects caused by interacting PID loops, or extensive heat integration, and slow (lower order) dynamics caused by the larger hold-ups in the process. • FIR models are able to model both fast and slow dynamics, but may also over-fit the data and some undesirable effects may appear in the models due to the impact of unmeasured disturbances. • Low order parametric models can easily capture slow over-damped dynamics, but may fail to
1076
This can lead to large MV cycles or even closed loop instability.
significantly less (e.g. typically 40 data points rather than 180 data points for FIR).
Lazy control: The opposite is also possible. The MV movement may be way too small if the identified RGA numbers are low but the true process mechanism is far weaker than we think (much higher RGA numbers). This will lead to very poor disturbance rejection and potentially shutting down the process due to loss of levels or massive violation of safety critical process constraints. The purpose of the controller commissioning activities is to confirm that the controller has appropriate MV movement for the expected constraint sets.
Auto-correlated MVs and the presence of strong feedback correlation are supposed to affect the model identification negatively, and result in bias in the model estimation. In both traditional manual step testing, as well as the new automated multi-variable pulse testing, the MV moves are designed to minimize their cross-correlation. In addition, care needs to be taken to minimize the correlation between the MVs and the unmeasured disturbances (often referred to as feedback correlation). • The traditional FIR ID algorithm proved to be very sensitive to the MV cross-correlation, and is even more sensitive to feedback correlation. • The improved CVA based Subspace ID algorithm proved to be far more robust for correlated MVs, as well as data containing feedback correlation.
Small gain errors sometimes have a big impact: If the gains of nearly collinear sub-matrices are inaccurate by just a little bit, MVs may actually ramp in the wrong direction during controller commissioning. This is a clear indication that the gain ratios are close, but have flipped (the RGA has changed sign) sign and the model matrix has a directionality error. Under these conditions, we either have to do specific testing to discover the correct process direction or generate more data by simultaneous multivariable perturbation, so that the gain ratio difference will have the right sign, or we need to apply our process knowledge and confirm this process mechanism is real or not, so we can decide to fix the gain ratio and remove a degree of freedom from the controller.
What real benefit can be achieved from true MIMO model identification? It is not so obvious for traditional model ID algorithms. However, it is easy to see the advantages from the workflow of a subspace model identification activity: • As the subspace ID algorithm identifies state variables first, more relevant CVs that are included in the model will help to determine the true states with higher confidence or require less data. • For any given amount of plant test data, it has been proved that model accuracy can be improved by adding more correlated measurements as additional model outputs (Pseudo CVs). These dynamically related measurements will often not be included in the MPC system as they do not need to stay within specified constraints. These additional output variables are added to the subspace identification cases purely to improve the accuracy of the MIMO model. • Alternatively, by using more correlated measurements as outputs in a MIMO case, the model can converge faster, and the required testing time can be reduced then.
3.3 The different identification methods The traditional FIR model ID method places less emphasis on the model gains and gain ratios because the use of a MISO model structure and the use of differencing filter on both MV and CV data. The subspace ID method is proved to be able to identify the gain matrix and gain ratios more accurately by using the MIMO structure, and by using de-trending filters on the MVs and CVs data pre-processing, rather than differencing, which suppresses the low frequency (steady state gain) information. The amount of test data required to identify a sufficient model for MPC applications is also important as it affects project cost: • More test data will usually ensure a more reliable model, leading to improved controller performance. However, the duration of plant test is very costly and should be minimized. • If the test data have sections of bad data that requires multiple bad data slices to be defined, then initialization of the ID algorithm will “waste” some valuable test data. For a slow model, the FIR model ID method is relatively expensive in term of the data length to be used for initialization. • Subspace ID is superior in both model convergence rate and the amount of data used for initialization. A bench mark test shows that the subspace ID algorithm converges three times faster than the traditional FIR ID method, and the amount of data required for initialization is also
4. OPEN-LOOP VS. CLOSED-LOOP IDENTIFICATION Model identification from open-loop test data has been studied for over 40 years now. In recent years, lots of attentions have been paid to closed-loop identification. As a result, many new algorithms for closed-loop identification have been proposed, and their theoretical properties have been analyzed. Closed-loop tests provide for a moderate degree of constraint protection and cost less than a manual step test, but the data is more difficult to fit a model. It is true that some of the process properties, like colinearity, nonlinearity, and process operational stability, may need the model to be identified under closed-loop. On the other hand, the available closedloop identification algorithms have their own limitations for industrial MPC applications:
1077
The practical problem size is often too large for most of the published closed-loop ID algorithms. For example, a typical MIMO model for a MPC controller may contain 20~50 MVs and 30~90 CVs, while most published case studies uses simulation data with a model size often smaller than 10 x 5. Many reported closed-loop ID algorithms are either not robust enough for industrial process data (may work in some cases, may not in other cases), or show poor performance when there are strong process drift disturbances.
loop model identification problem then becomes very easy to solve and there are no longer special requirements on the closed-loop ID algorithm. • More significantly, the project implementation time can be shortened by more than half and the cost can then be reduced significantly. Many MPC applications where the multi-variable test methods have been applied in combination with the subspace method have clearly demonstrated that the plant testing and model identification cost can be reduced by more than 50%, while the identified model accuracy and controller performance is superior. As mentioned earlier, a recently completed MPC application on a FCC unit with 35 MVs and 95 CVs took only 4 days in total from plant testing to controller commissioned, a new world record (McIntosh, A. et al., 2003; Kalafatis, A. et al., 2005).
4.1 Recent progress in closed-loop identification In recent years, major progress in industrial MPC application has been made, namely, a “smart” constraint aware plant testing approach was developed to perform “pseudo” closed-loop identification when the process is under a form of discrete output MPC control. The “smart” tester consists of a MPC controller and a test signal generator. The tester starts with an initial model that can be easily obtained from pre-test data or from previously built process models (e.g. MPC revamp project). Practical applications have demonstrated that the requirements for the initial model are often easily satisfied. For example, for each CV, usually only one MV-CV approximate model is sufficient and the mismatch on the model’s gain can be up to 500% while the MPC tester can still work. During the test, multiple (usually all) MVs are perturbed automatically based on a balanced optimization, that is, to maximize the MVs’ perturbation while controlling the CVs strictly within their pre-set constraints. In such a way, the signal to noise ratio of the test data can be 4 times higher than that of those generated from a traditional test. As the test continues, model can be re-identified and updated into the MPC tester periodically from the collected new test data. This new technology brings significant benefits: • Under MPC control, all the important operational constraints can be satisfied, i.e. the process safety and product specification are ensured during the plant test period. • During plant test, the APC engineer no longer needs 24 hours supervision around the clock by using this new plant test tool. It is also a big cost reduction. • A “smart” algorithm can vary the amplitude of the MV pulses to drive all of the MVs at the same time to generate sufficiently independent perturbations to make the model identification problem far easier, while the cross-correlation among MVs are minimized. • More importantly, in those cases where a CV constraint may possibly be violated by several MV superimposing and driving the CVs way beyond their limits are predicted and prevented based on a Model Predictive Control algorithm that takes the worst case model uncertainty into account. • As a result, the plant test data has minimum crosscorrelation among the MVs and very low feedback correlation between the CVs and MVs. The close-
4.2 The new industrial state-space MPC controller needs accurate state-space model Another recent important progress in industrial MPC application is the development of a new state-space MPC controller (Froisy, J. B., 2006) that uses most of the latest MPC research results from academia and the new developed real-time environments for industrial MPC applications. This new industrial State-Space Controller (SSC) is referred to a MPC product bridging between the control theory and industrial practice. The major technical features of this SSC include: • State-space model • Real-time state estimation • Infinite horizon, constrained dynamic move plan These enabling technologies will help address an increasingly challenging set of industrial control problems. As seen above, the new SSC requires state-space models to describe and predict the process behaviours so that it can timely detect disturbances entering process with state estimator and optimally reject them by an infinite horizon MPC controller. Because the SSC’s performance will depend on the models’ quality, the subspace identification offers a perfect alternative to obtain accurate state-space models from plant testing data. 5. KNOWN ISSUES AND FUTURE NEEDS As the subspace algorithm has been accepted and is now widely used in industrial MPC projects, some of its limitations are better known, and work-around tricks have been developed. Industrial MPC practitioners would welcome academic scholars to study these limitations and help to address these issues. Different dead-times for different channels through the MIMO model. As well known, a state space model uses states to describe both pure dead-time and dynamics. In industrial process it is often seen that the pure dead-times are quite different among the
1078
process variables in the same unit, they are in a range from a few minutes to a few hours. In such a case, the model order needs to be set very high and it is difficult to identify a true MIMO state space model.
MV correlations and feedbacks if the MPC controller (tester) operates in a “correction” mode to avoid CV constraints violation or intervened by process operator. As a result, the plant test data contains “intermittent” feedbacks. Usually it is not a problem for the CVA-based subspace ID algorithm. But it may be an interesting topic for academic researchers to investigate how the currently available closed-loop identification algorithms work under such conditions.
Large unknown disturbances (e.g. strong drift disturbances, and periodic disturbances caused by sticky control valves) can impact the subspace ID results negatively. Data pre-processing has been used for removing some of the disturbances, a systematic approach, such as building explicit disturbance models would be very useful for industrial practitioners.
6. CONCLUSIONS After 5 years of practice and improvements, the CVA-based subspace identification technology has been accepted and widely used in industrial MPC applications because of its many special features. Specifically, with the major progress in multivariable plant testing technology, subspace ID offers more efficient ID tool and results more accurate process models. Although there are some known issues to be further addressed, the new way to do the “pseudo” closed-loop plant testing and model identification with subspace ID has demonstrated significant cost reduction in APC project implementation and significant improvements on the MPC controllers’ performance.
Very different dynamics within the same state space model are required sometime in order to satisfy the optimal process operation. The ratio between slowest time constant and the fast time constant can be in the range 20 to 100. In subspace identification, it is a challenging task to identify a model with detailed fast dynamics and accurate slow steady state gains. Currently a compromise between the two ends often has to be made in an expense of the model accuracy. More effective model uncertainty evaluation and validation methods are required to help industrial practitioners to know how good the identified models are. Currently available methods such as timedomain and frequency-domain model uncertainties are useful, but found not intuitive and effective enough in practice. For state-space model uncertainty, it seems there is no well known method available yet.
REFERENCES Froisy, J.B. (2006). Model predictive control – Building a bridge between theory and practice. Chemical Process Control (CPC-7), Lake Louise, Alberta, Canada, January 8-13, 2006. Kalafatis, A., Kalpesh, P. Harmse, M. and Zhang, Q. (2005). A unique multivariable test method for MPC projects dramatically reduces testing time of a Crude Unit. Hydrocarbon Processing, (in print). Larimore, W.E. (1983). System identification, reduced-order filtering and modelling via canonical variate analysis. In: Proceedings of the 1983 American control Conference, 445-451. Larimore, W.E. (1990). Canonical variate analysis in identification, filtering and adaptive control. In: Proceedings of the 29th Conference on Decision and Control, 596-604. McIntosh, A., Cooke, J. and Harmse, M. (2003). Techniques for implementing large scale DMCplus controllers, 7th Annual IEEE Advanced Process Control Applications for Industry Workshop, Vancouver, Canada, Arial 28-30, 2003. Overschee, P. Van and B. De. Moor (1994). N4SID: Subspace algorithms for the identification of combined deterministic-stochastic systems. Automatica, 31(12), 1853-1864. Qin, S.J. and T.A. Badgwell (2003). A survey of industrial model predictive control technology, Control Engineering Practice, 11(7), 733-764. Verhaegen, M. and P. Dewilde (1992). Subspace model identification. Part 1: the output error state-space model identification class of algorithms. International Journal of Control, 56, 1187-1210.
The speed of subspace ID algorithms for large models still needs to improve. An industrial benchmark test has demonstrated an exponential relationship between the ID speed and the model size. With the increase in number of MVs and CVs put into a MIMO model, the computation time required by the subspace ID algorithm grows exponentially. Very recently, a significant improvement on the speed has been made in AspenTech’s MPC product and this speed issue has been well addressed. Improvements in identification with highly correlated MVs and ill-condition system are still needed. Identifying accurate steady-state gains and gain ratios is far more important than those dynamics for an industrial MPC application with collinear MVs or ill-conditioned process. Though some academic studies have been reported in 1990s, industrial practitioners hope academic researchers can re-visit this issue and help to further address it. Identification from intermittent closed-loop data is a new requirement raised with the new multivariable plant testing applications. In most cases, the multivariable plant test generates information rich, un-correlated, good-quality and open-loop equivalent data from a ‘pseudo’ closed-loop environment. Some time a small portion of the testing data may contain
1079