Identification for Control with Application to Ill-Conditioned Systems. Jari Böling

Transcription

1 Identification for Control with Application to Ill-Conditioned Systems Jari Böling Process Control Laboratory Faculty of Chemical Engineering Åbo Akademi University Åbo 2001

2 2 ISBN Painotalo Gillot Oy 2001

3 3 Preface The work reported in this thesis has been done at the Process Control Laboratory, the Faculty of Chemical Engineering at Åbo Akademi University, during the years I wish to thank my supervisor, Professor Kurt-Erik Häggblom for most his valuable guidance and help during this work. I would also like to thank Professor Pertti Mäkilä for introducing me to system identification, and for supervising me during the preparation of my first paper. I wish to thank my co-worker Rasmus Nyström for interesting and fruitful cooperation. I am also grateful to professor Hannu Toivonen, for coordinating the postgraduate education at our Laboratory, and also for the many interesting and valuable discussions regarding my work. I also wish to thank Pekka Lehtiö, Sten Gustafsson and Stefan Rönnblad for their help with the experimental runs with the distillation column. Thanks also go to the other members of the staff at the laboratory. I would like to thank the financial supporters of this work, Academy of Finland, Graduate School of Chemical Engineering, and Stiftelsens för Åbo Akademi Forskningsinstitut. Finally, I would like to thank family and friends for being there for me. Most special thanks go to my wife Sanna and daughter Sofia for being the joys of my life. I also wish to thank my former spouse Vivi-Anne for her support during the first years of my PhDstudies. Åbo, June Jari Böling

4 4 Abstract In this thesis different aspects of identification of dynamic systems, for the purpose of designing feedback controllers, are investigated. Design of experiments, the use of feedback during identification experiments, formulation of identification criteria and estimation of model uncertainty are studied. Identification of ill-conditioned systems receives special attention in most of the papers included. Identification and robust control designs are experimentally tested on an ill-conditioned distillation column or on linear simulation models. An open-loop criterion that gives the same weighting as a certain closed-loop criterion is formulated. The weighting is iteratively determined using open-loop data only, and the need of repeated closed-loop experiments is eliminated. Identification of an ill-conditioned distillation column is experimentally investigated. It is shown that identification experiments, which do not excite the low-gain input direction of ill-conditioned systems explicitly, can result in models useless for feedback controller design. A method for estimating the low-gain input direction of an ill-conditioned distillation column is tested and verified, and this direction is used for design of identification experiments that give sufficient excitation of the low-gain input direction. A multimodel description is identified based on these experiments. A method for estimation of uncertainty is presented. In the method, experimental data is divided into parts, and different models are fitted to each data set. These models, together with the inputs and a nominal model, are used for separation of the experimental data into noise and uncertain dynamics. Expressions for obtaining uncertainty descriptions, minimal in norm or space, are also given. The method is applied to distillation column data, and the estimated uncertainty is used for design of controllers. The models and the uncertainty obtained are used for different controller designs. The controllers are tested experimentally on the distillation column. The design variables for bias distribution are reviewed, and the impact on ill-conditioned systems is studied. Recommendations on identification of multivariable, and thus potentially ill-conditioned systems are given. An iterative closed-loop scheme is suggested and tested using simulations.

5 5 List of publications This thesis is based on the following six papers: I. J.M. Böling and P.M. Mäkilä. On control relevant criteria in H identification, IEEE Trans. Automat. Contr., 1998, II. J.M. Böling and K.E. Häggblom. Control-relevant identification of an ill-conditioned distillation column. Proc. IEEE Conf. on Control Applications, 1996, , Dearborn, MI, USA. III. K.E. Häggblom and J.M. Böling. Multimodel identification for control of an illconditioned distillation column, J. Proc. Cont., 1998, IV. J.M. Böling, R.H. Nyström and K.E. Häggblom. Uncertainty estimation based on multiple-model identification with application to distillation. Submitted to Automatica, V. R.H. Nyström, J.M. Böling, J.M. Ramstedt, H.T. Toivonen and K.E. Häggblom. Application of robust and multimodel control methods to an ill-conditioned distillation column. Accepted for publication in J. Proc. Cont., VI. J.M. Böling. On closed-loop identification of ill-conditioned systems. Unpublished, 2001.

6 6 Introduction When designing a feedback controller some information about the dynamic behavior of the system to be controlled is needed. The minimum amount of information needed is often the sign of the gain of the system, that is, if the input of the system is increased, does the output increase or decrease? For example, if the temperature in a room (the output) is too low, should the heating power (the input) be increased or decreased in order to increase the temperature? A more difficult question is how much the heating power should be changed. The room temperature will not change instantaneously after the heating is changed, and the temperature will rise for a while after the heating is stopped. If the heating is increased too much, the temperature might rise uncomfortably high. If the heating is increased too little, it might take too long to reach the desired temperature. More information about the dynamic behavior of the room-temperature/heater system, that is, how the temperature will evolve in time after a change of the heating power is needed for successful control of the temperature. In practice, the heating power is dimensioned in such a way that simple on-off control can be sufficient. This will result in oscillations in the temperature, but the oscillations will be small enough to be satisfactory, if the heating power is suitable. If a more accurate temperature control is desired, a more accurate model describing the dynamic behavior of the system would be needed. Such a model could be obtained in basically three ways: 1. First-principles modeling, that is formulating mathematical equations describing laws of nature, such as conservation of mass and energy. In the above example, the heat transfer from the heating power source to the radiator, heat transfer from the radiator to the room, and the heat loss to the surroundings, would have to be modeled. 2. Black-box identification, that is experimental testing of how changes in the input will affect the output. The experimental input-output data is used for numerical fitting of parameters in a model, usually a linear differential or difference equation. The used model order is normally determined from data using statistical tests, without using knowledge of the system. That is, the system is treated as a black box. 3. Grey-box modeling, a combination of the other two alternatives. Experimental data is used for fitting uncertain parameters in a first-principles model. In the above example, the heat transfer parameters are most likely to some extent uncertain, and could be experimentally determined.

7 This thesis is about black-box identification of linear models, although system specific information is used in papers II and III. The benefits of using black-box identification is that it is relatively simple, and results in models which are quite easy to use for control design. Furthermore, the need of first-principles modeling is eliminated, which is also sometimes difficult or even impossible. When a system is identified in a black-box manner, the standard way is to use a least-squares fit of parameters in a transfer function model or a state-space model. This approach is treated in several textbooks, for example Ljung (1987), Söderström and Stoica (1989), and Goodwin and Payne (1977). The use of linear models means that the model is generally an approximation of the true system. This does not have to be a big disadvantage, because feedback control is often used to keep the system at a state, around which a linear model is normally accurate enough. The largest drawback with black-box identification is that it is necessary to apply excitation to the system, which can result in economic or other types of risks. If the system is a production unit, the excitation can result in poor product quality during the experiments. This is one of the initial motivations to the use of feedback during identification experiments, identification of open-loop unstable systems being the other one. There are, however, also some risks concerning model accuracy involved in closed-loop identification, and it has been a subject of intensive research. Closed-loop identification was initially studied during the 70 s, from which results are best summarized in the survey by Gustavsson et al. (1977). Interest for closed-loop identification grew in the late 80 s and 90 s, when closed-loop identification was used in several iterative schemes for control-relevant identification. More about this genre below, at this stage it should be mentioned that the fundamentals of closed-loop identification were revisited by Forsell and Ljung (1999). Closed-loop identification is also studied in this thesis, in papers I and VI. 7 A problem with all models, linear or nonlinear, is that the real world is too complex to be handled exactly; mathematical models are always approximations of the true system. Exact models are not necessary for design of feedback controllers, because the purpose of feedback is to correct the mistakes made by the controller, which possibly are induced by an erroneous model. However, at some point the feedback design will fail due to the model error. When this happens depends on the true system, which is generally unknown. There are some degrees of freedom in the modeling stage, due to the fact that there always will be a difference between the true system and the model. This difference, the bias, can be affected by a number of factors. These factors are treated in several papers in this thesis, the most general review is given in paper VI. Controllers designed using different models can and will result in different control performance. The purpose of identification for control is to try to model accurately what is important for the controller design. Consider the PID tuning rules by Ziegler and Nichols (1942), where the ultimate frequency and ultimate gain of a system is used for design of PID-controllers. The use of Ziegler-Nichols tuning rules does not always result in satisfactory control designs. The information used is not always enough, and might not always be the most important information about the system dynamics. What is the most important information about a system for design of a specific type of controller, and how this information can be obtained from identification

8 8 experiments, are the main questions addressed in the field of identification for control. The first study in this field was by Gevers and Ljung (1986), who claimed that models identified in closed loop are best suited for model based design. Another statement presented in this paper was, that the system should be controlled with an as good as possible controller during the identification experiments. This study was restricted to minimum variance control, further analysis and generalizations are given by Hjalmarsson et al. (1996). The model structure, including model order, is assumed to be known in this study, and the results do not hold when the model order is lower than the system order. The work by Gevers and Ljung probably initiated the iterative identfication-for-control schemes, for example the ones by Schrama (1992a, 1992b), Zhang et al. (1995), Lee et al. (1993), Rivera et al. (1992). In these studies, an identification step and a successive controller design step are iteratively repeated, which hopefully results in models suitable for highperformance control design. Paper I in this thesis is a comment on these studies, and it is shown that it is possible to use open-loop data and to iteratively calculate a weight, similar to the weighting obtained from closing the feedback loop. The impact of noise is not studied, however, it is assumed that the noise level is insignificant. The introduction so far was formulated with singlevariable systems, systems with one input and one output, in mind. Identification of multivariable systems pose special problems not present in the singlevariable case. In the multivariable case, the sign of the determinant of the gain matrix of the system is as important as the sign of the gain of a singlevariable system. If a multivariable system is ill-conditioned, that is the condition number of the gain matrix is high, the sign of the determinant can be changed by small model errors. Another characteristic of an ill-conditioned system is that it amplifies input vectors differently depending on the direction of the vector. The directions of the input vector which are least and most amplified, are denoted low-gain and high-gain input directions respectively. If an ill-conditioned system is identified using excitation of equal magnitude in all input directions, the response of the low-gain input will be very small compared with the response of the inputs in other directions. This can lead to incorrect estimates of the determinant of the steady-state matrix, and models which are useless for feedback control design. Ill-conditioned systems have been studied a lot, both from a control design and an identification perspective. The implications of ill-conditioning to control design was initially studied by Morari and Skogestad (1987), Yu and Luyben (1987) and McDonald et al. (1988). It was found that certain types of model inaccuracies were very harmful for a robust control design. This, on the other hand, has implications on identification of ill-conditioned systems, which was first studied by Luyben (1987) and Andersen et al (1988). In these studies it was recognized that the requirement on identification accuracy of individual elements in a ill-conditioned transfer function matrix is very high, thus making standard identification of ill-conditioned systems quite difficult. The need of additional excitation of the low-gain input direction was found necessary, although the low-gain input direction in general is unknown prior to identification. In Andersen and Kümmel (1992) it was suggested that closed-loop identification resulted in automatic emphasis of the low-gain input direction. The benefits of closed-loop identifi-

9 cation of ill-conditioned systems are also analyzed in paper VI of this thesis. Koung and McGregor (1993) suggested an iterative scheme, where the low-gain input direction of an identified model is used for design of a balanced input excitation. Further analysis of such iterative schemes, and variations with guaranteed convergence properties, are given by Lee et al. (1998). Häggblom (1995) used flow-gain information for estimation of the low-gain input direction of distillation columns. This estimate can be used for design of experiments, which is tested in practice in papers II and III of this thesis. Another important issue in identification and in control design is the robustness aspect. As mentioned earlier, a model can never be an exact representation of a real world system. This will mean that a controller designed based on a model may not behave as expected when applied on the system. However, robust control theory provides methods for designing controllers which guarantee stability and/or control performance, if the difference between the model and the system is smaller than some known limit. The estimation of uncertainty has also been given a lot of attention by the research community. Initially, the uncertainty of an identified model was determined on the basis of confidence intervals, given by statistical considerations. This approach does not take into account the possibility of unmodeled dynamics, it is assumed that the difference between experimental data and the response of the model is stochastic noise. A different approach for estimating uncertainty was studied by Helmicki et al. (1991) who introduced the first H identification method. This and other similar methods were not totally satisfactory from a practical point of view, which lead to the model validation concept introduced by Smith and Doyle (1992). A new method for doing such a model validation, which is quite simple and requires little a priori information compared with other similar methods, is presented in paper IV of this thesis. The contents of the six papers are summarized below. The contribution of the author of this thesis is also outlined for papers I-V, which are by several authors. I. On control relevant criteria in H identification The work is a comment on a suggested iterative closed-loop identification method, which is aimed to give models suitable for a special type of H control design. Models as well suited for such a design are obtained by using a weighted identification criterion, using open-loop data only. A similar iterative procedure is needed for design of a suitable weight as in the original work, but the need of repeated identification experiments is eliminated. The initial idea to use a weighted open-loop identification criterion, and the robust stability test in section IV-B is by Professor Mäkilä, the rest is mainly by the author of this thesis. 9

10 10 II. Control-relevant identification of an ill-conditioned distillation column Identification of ill-conditioned systems is investigated. It is experimentally shown that a black-box type open-loop excitation can result in models useless for control design. The low-gain input direction was estimated, and used for design of experiments which give a balanced excitation of the low- and the high-gain directions of the system. The data from these experiments resulted in models which satisfy integral controllability requirements. The design of experiments is by Dr. Häggblom, the identification is by the author of this thesis. Some of the contents in this paper are repeated in paper III, which is a journal version of this and another conference paper. III. Multimodel identification for control of an ill-conditioned distillation column The experimental data from the balanced excitation experiments used in paper II are also used here for identification of a multimodel set. The identification experiment consisted of a sequence of steps, each step being in either the low-gain or in the high-gain input direction. One model was fitted to the whole experiment, and furthermore a model was estimated from data from each separate step change. The models consist of two different parts, one describing the low-gain dynamics and one part describing the highgain dynamics of the plant. Due to this it was possible to fit only low-gain parameters to a low-gain step, and high-gain parameters to a high-gain step. The idea behind the identification of several models was to capture variations and non-linearities in the plant dynamics, and to use this for design of robust controllers. The used model structure with separate high-and low-gain dynamics was mainly by Dr. Häggblom. The identification is mainly by the author of this thesis. IV. Uncertainty estimation based on multiple-model identification with application to distillation A method for estimation of the difference between a nominal model and the true system, that is, the uncertainty of the nominal model, is developed. The method consists of basically two steps. An initial multimodel identification step, where different data are used for fitting of each model. In a second step, the difference between the responses of the multimodels and the responses of the nominal model is translated into a norm-bounded uncertainty. The difference between the multimodel responses and the experimental data is assumed to be noise. The experimental data can then be explained by the nominal model, the uncertainty model and the noise. Thus, the nominal model and the uncertainty bound are considered unfalsified and validated. This is a relatively simple way of obtaining a separation of dynamic uncertainty and noise, which is a crucial step in such a model validation. Furthermore, different structures and shapes of uncertainty models which all describe the data at hand are investigated. Uncertainty descriptions which minimize the

11 norm of or the space covered by the uncertainty are given. The different uncertainty types are used for µ-optimal controller designs, which are experimentally tested. The initial idea, which was the diagonal uncertainty case, and the connection to model validation was by the author of this thesis. A variation of this, which later led to weight 1, was by Mr. Nyström. This variation was used by the author of this thesis as initial weight W 0 in weight 4. The idea to use an arbitrary initial weight W 0 and the formulation of Equation (25) was by Dr. Häggblom. The good properties of weight 4 inspired Dr. Häggblom to further investigation, which led to the weight denoted optimal. The controller designs are by the author of this thesis. V. Application of robust and multimodel control methods to an ill-conditioned distillation column The models identified in paper III and the uncertainty estimated in paper IV are used for different control designs, and the controllers are experimentally tested. The µ-optimal design, the IMC-design, and the PID design are by the author of this thesis. The use of an LQ controller as a prefilter in the Glover-McFarlane robustification was suggested by the author of this thesis. VI. On closed-loop identification of ill-conditioned systems Design variables for bias distribution in identification of dynamic systems are reviewed, and the impacts on identification of multivariable and thus potentially ill-conditioned systems are analyzed. It is suggested that ill-conditioned systems are identified using closedloop experiments with a controller with integral action, which results in smooth closedloop transfer functions. Furthermore, the excitation signal should preferably be added to the setpoint, and an output-error criterion should be preferred to other prediction-error criteria. The design of such a controller requires a suitable model, so an iterative identification scheme is suggested to solve this conflict. The iterative scheme is tested using an ill-conditioned distillation column simulation model. Summary and conclusions Different aspects of identification of dynamic models for control design purposes have been studied in this thesis. The main point is that a model of a real-world system will not be an exact representation of the system. This means that when identification of the system is done in different ways, different models are most likely obtained. If the models are used for control design, the controllers will perform differently when applied to the true system. There are a number of steps in an identification procedure, the experimental design, the selection of model structure and the formulation of an identification criterion, which all affect the obtained model. Furthermore, information about the accuracy of the model, that is the model uncertainty, can be used in a control design. All these aspects 11

12 12 affect the obtained model and the resulting control design, and have been studied in this thesis. What has not been studied at all in this thesis, is the optimization methods used for finding optima for the used identification criteria. In the first paper of this thesis the formulation of an open-loop identification criterion, which gives the same weighting as certain closed-loop identification experiments, was studied. The purpose of the weighting was to give models accurate at frequencies which were important for the used control design method. Although a specific design method was studied, the use of weighted identification criteria can be useful in other similar situations. The impact of noise in this case was not clarified, and could be a subject of further research. The design of experiments and identification of models suited for control design for an ill-conditioned distillation column are the topics of papers II and III. An ill-conditioned system requires good excitation of the low-gain input direction, which is here estimated using certain flow-gains. The flow gains are somewhat simpler to estimate than the composition gains, mainly due to that the responses are much faster in the external flows than in the compositions. The low-gain input direction estimate was used for design of experiments which excite the low gain input direction sufficiently. These experiments gave models better suited for control design than black-box experiments. In paper IV a new method for estimating the model uncertainty was presented. Experimental data can be explained by different uncertainties, and expressions for uncertainty descriptions minimal in norm or space covered were given. The method was tested on distillation column data, where step excitation was used. The use of experimental data where other types of excitation are used was not tested. There is, however, no principal reason for the method not to work with other types of excitation. An uncertain model consists of two components, the nominal model and an uncertainty bound. Only the latter was studied, although the former is also important. However, the nominal model can be selected as it was done in paper IV, using a standard least-squares fit. The nominal model could also in some sense be centered in the uncertain model space, thus making the uncertainty minimal. However, this approach might not result in better control performance in practice, although it in principle should allow better robust performance. A robust performance control design is aimed to give a certain level of performance for all models included in an uncertain model space, but better performance is generally obtained for the nominal model. So it might be good from a practical point of view to have a nominal model which primarily describes the most likely behavior of the plant. That is, a model obtained from standard identification, fitted to as much relevant data as possible, should be used as a nominal model. If a smaller uncertainty is wanted, it could be obtained by minimization of it by means of multiplication of the nominal model with a scalar, or possibly by means of multiplication with a diagonal matrix at the outputs of the nominal model. This would not destroy the basic dynamic properties of the nominal model, which is possible if the nominal model is determined from an uncertainty minimization only. Another possible approach to reduce the uncertainty, estimated using the presented

13 method, is to penalize the size of the uncertainty in the multimodel identification stage. This could be viewed as explaining experimental data more as noise and less as uncertain dynamics. These arguments concerning the size of the uncertainty should be viewed as opinions, it would be a good topic for further research. The models and the estimated uncertainty were also tested for different controller designs in paper V. Most of the methods performed well and in a surprisingly similar way, which indicates that the used models are quite good. The multiloop SISO design was the only one which can be considered unsuitable, in this case. That this design was unsuitable was expected, due to the ill-conditioned nature of the plant. Identification of ill-conditioned systems in closed-loop is the topic of paper VI. Any multivariable system with unknown dynamics can be ill-conditioned, and the recommendations given in this paper can be of importance when a multivariable system is identified. The recommendations are here only tested in simulations, and an experimental test would be very interesting. The study of indirect identification in closed-loop identification of ill-conditioned systems would also be a natural continuation on this study. The fact that an iterative closed-loop identification scheme is suggested in paper VI, contrary to paper I, where it is suggested that it might not be needed, needs a comment. Consider the simple ill-conditioned system studied in paper IV: [ If both inputs are excited with a signal of magnitude of 1, this will result in output variations of magnitude 1 and for the first and second outputs, respectively. If both outputs are contaminated with noise with magnitudes of 0.1, the signal to noise ratio will be 10 and 0.01 respectively. The identification of the transfer functions associated with the first output should not be a problem, while the dynamics behind the second output is drowned in noise. No control-relevant weighting will change this signal to noise ratio, the identification experiments must be repeated using larger excitation of the second input. Paper I, on the other hand, was restricted to single-variable systems, and noise was not at all considered. Literature cited Forssell, U. and Ljung, L. (1999). Closed-loop identification revisited. Automatica, 35, Gevers M. and Ljung L. (1986). Optimal experiment designs with respect to the intended model application. Automatica, 22, Goodwin, G.C. and Payne, R.L. (1977). Dynamic system identification: Experiment design and data analysis. Academic Press, New York. ] 13

14 14 Gustavsson I., Ljung L. and Söderstrom T. (1977). Identification of processes in closed loop Identifiability and accuracy aspects. Automatica, 13, Häggblom, K.E. (1995). Proof of the relation between singular directions and external flows in distillation. Proc. IEEE Conf. on Decision and Control, , New Orleans, LA, USA. Helmicki, A.J., Jacobson, C.A., and Nett, C.N. (1991). Control oriented system identification : A worst-case/deterministic approach in H. IEEE Trans. Automat. Control, 36, Hjalmarsson H., Gevers M. and de Bruyne F. (1996). For model-based control design, closed-loop identification gives better performance. Automatica, 32, Koung, C.-W., and MacGregor, J.F. (1993). Design of identification experiments for robust control. A geometric approach for bivariate processes. Ind. Eng. Chem. Res., 32, Lee J., Cho, W. and Edgar T.F. (1998). Iterative identification methods for ill-conditioned processes. Ind. Eng. Chem. Res., 37, Ljung, L (1987). System identification: Theory for the user. Prentice Hall, New Jersey. Luyben, W.L. (1987). Sensitivity of distillation relative gain arrays to steady-state gains. Ind. Eng. Chem. Res., 26, McDonald, K.A., Palazoglu, A., and Bequette, B.W. (1988). Impact of model uncertainty descriptions for high-purity distillation control. AIChE J., 34, Rivera, D.E., Pollard, J.F. and Garcia, C.E. (1992). Control-relevant prefiltering : A systematic design approach and case study. IEEE Trans. Automat. Control, 37, Schrama, R.J.P. (1992a). Approximate identification and control design with application to a mechanical system. Ph.D. thesis, Delft Univ. Techn. Schrama, R.J.P. (1992b). Accurate identification for control : The necessity of a iterative scheme. IEEE Trans. Automat. Control, 37, Skogestad, S., and Morari, M. (1987). Implications of large RGA elements on control performance. Ind. Eng. Chem. Res., 26, Skogestad, S., Morari, M., and Doyle, J.C. (1988). Robust control of ill-conditioned plants: High-purity distillation. IEEE Trans. Autom. Control, 33, Smith, R.S., and Doyle, J.C. (1992). Model validation: A connection between robust control and identification. IEEE Trans. Automat. Control, 37, Södersström T. and Stoica, P. (1989). System identification. Prentice Hall, UK. Zang Z., Bitmead R.R. and M. Gevers (1995). Iterative weighted least-squares identification and weighted LQG control design. Automatica, 31, Ziegler J.G. and Nichols, N.B. (1942). Optimum settings for automatic controllers. Trans. ASME,