Simultaneous adaptive decoupling and model matching control of a ...
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 11, NO. 4, JULY 2003
571
Simultaneous Adaptive Decoupling and Model Matching Control of a Fluidized Bed Combustor for Sewage Sludge Yingmin Jia, Hideki Kokame, and Jan Lunze
Abstract—The major difficulties in designing a controller for the fluidized bed combustor for sewage sludge result from the following reasons: 1) The dynamical behavior of the plant is highly unknown and the experimental data only render a coarse model; 2) the combustion process shows large time-delay; and 3) the plant model is a stiff differential equation because the parameter values among the subsystems differ severely. To circumvent these difficulties, this paper proposes to describe each channel of the multiple-input–multiple-output system by a first-order transfer function with variable coefficients and a fixed time delay. A model reference adaptive control scheme, which can simultaneously achieve adaptive decoupling and model matching, is proposed. The resulting adaptive control law guarantees closed-loop stability. The newly obtained controller has a structure that is similar to the well-known Smith predictor. This explains why the present scheme can cope with the time delay effects on the system performance. Due to its adaptation abilities, it can lower the limitations of the Smith scheme to accurate plant descriptions. The application study demonstrates that the proposed design can reach the desired performance requirements. Index Terms—Adaptive decoupling, fluidized bed combustors, model matching, process control, sewage sludge, time delay, uncertainty.
I. INTRODUCTION Sewage sludge is incinerated to form stabilized ash and to reduce the volume of the sludge. Sewage sludge combustors can only be industrially used if the gaseous emissions are kept within the strict legal limits. Currently, sewage sludge combustion plants with a bubbling fluidized bed are essentially controlled by hand and the combustion quality depends heavily on the operators’ experience. Hence, there is an increasing requirement for the automatic control of such plants. Fluidized bed combustors are characterized simultaneously by large model uncertainties, time delays, and a large variety of time constants which make the differential equations stiff [1]–[7]. As a consequence, the controller of such plants cannot be designed by standard methods. This paper proposes a controller whose structure is similar to the well-known Smith predictor. By using an internal model of the time-delay system, the controller can react before the control error is visible so that the Manuscript received July 23, 2002. Manuscript received in final form January 6, 2003. Recommended by Associate Editor F. Doyle. This work was supported by the JSPS, the AvH, the APEC, and the NSFC under Grant 69625506. Y. Jia is with the Seventh Research Division, Beijing University of Aeronautics and Astronautics, Beijing 100083, P.R. China (e-mail: [email protected]). H. Kokame is with the Department of Electrical and Electronic Systems, Osaka Prefecture University, Osaka 599-8531, Japan (e-mail: [email protected]). J. Lunze is with the Institute of Automation and Computer Control, RuhrUniversity Bochum, D-44780 Bochum, Germany (e-mail: [email protected]). Digital Object Identifier 10.1109/TCST.2003.813398
closed-loop system has the best performance under the performance limitations brought about by the time delays. This idea is combined here with the adaptation principle described in Section IV that makes it possible to cope with severe plant uncertainties. In more detail, this paper uses a model reference adaptive control (MRAC) scheme which simultaneously achieves adaptive decoupling and model-matching. An ideal model that satisfies all performance specifications is selected and a controller is designed such that the output of the closed-loop system tracks the model output. The main result is a new model-matching controller for multiple-input–multiple-output (MIMO) time-delay plants. Due to the proposed adaptation scheme the desired control objectives are met in spite of model uncertainties. The properties of the control scheme are shown by its application to a sewage sludge combustor. Relevant literature: A standard method for the control of delay-time systems is the Smith predictor [8]. However, the delay cancellation in this scheme depends strongly on the model accuracy of the plant which is hard to be satisfied for the present control has been studied for a variety of time-delay plant. systems [9]–[11]. For the multidelay case, the finite spectrum assignment can be reached by the method described in [12]. Quite a few existing results [4], [13] turn time delay into a multiplicative uncertainty and use robust control methods [14]. These methods are based on optimization principles but they are lacking definite rules to select the objective functions. For stiff systems these methods suffer from the large conservatism of the underlying stability tests. The MRAC scheme of single-input–single-output (SISO) systems with a single delay was proposed in [15]. The results cannot be used to deal with the multioutput delay case. Further, as pointed out in [16], the adaptive control law used there cannot guarantee the convergence of the system errors and boundedness of the control signal. In contrast to this, this paper obtains a new matching controller for multi-output delay systems by employing the decoupling principle of MIMO delay-free systems [17] which provides a globally stable adaptive control scheme and, therefore, can be viewed as a significant extension of the results in [15], [16] to general MIMO delay systems. Finally, the reader is referred to [18]–[20] for a description of the MRAC principle. Structure of the paper: In Section II, the plant model is given and the corresponding design objectives are formulated. Section III is devoted to developing the simultaneous decoupling and model-matching controller for the nominal plant. Extensions to the plant with parameter variations are made in Section IV where the adaptive control law to insure the global stability is given. Section V shows the applicability of the proposed method to the fluidized bed combustor.
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vide a firm basis for modeling the combustor from first principles. Therefore, a lot of experiments have been made to set up a dynamical model which describes the relationship between the input and the output . Here is propane mass is total air mass flow, is the bed temperature and flow, is O concentration. In this paper, we use the same model structure as in [3] with the following nominal parameter values obtained from the experimental data:
(1)
Fig. 1. Pilot fluidized bed combustor of the Technical University HamburgHarburg.
II. COMBUSTOR MODEL AND DESIGN OBJECTIVES A pilot fluidized bed combustor is shown in Fig. 1. The unit combusts wet sewage sludge (with 70% water contents) and propane in a bubbling fluidized bed. The combustor consists of a tube with a total height of 900 cm and a diameter in the bed area of 15 cm. The bed area of the combustor is filled with sand. The sand is fluidized with preheated air. The unit is equipped with the necessary facilities to feed propane, wet sludge, and primary air to the bed and secondary air into the riser. Gas samples are taken at the top of the riser, where the concentrations of O , CO , CO, NO , and SO are measured. The combustor is heated up by an electrical heating system in the combustor walls to compensate for the heat loss, which occurs due to the extreme height/diameter ratio of the combustor. Fluidized bed combustors have been studied intensively from a process-oriented viewpoint. For the combustor depicted in Fig. 1 the relevant results are given in [1]. However, all these results do not result in a dynamical model nor do they pro-
Seemingly, the model (1) is simple, but it captures almost all typical characteristics of the combustor which lead to the major difficulties of the controller design: • Uncertainty. The highly unknown dynamical behavior of the plant makes it impossible to model exactly the plant. • Time delay. The time delay occurring in the oxygen subsystem is up to 36 s, while in the temperature subsystem it is nearly zero. Hence, the system is a multioutput delay system. • Stiffness. The time constant of the temperature subsystem is over 50 times larger than that of the oxygen subsystem. The control objectives of the plant are determined by allowed emission limits stipulated by law [2]. Similar laws hold in all industrialized countries. Accordingly, the average concentration of NO or CO over a day should be less than 200 mg m or 50 mg m , respectively. The average value of CO over half an hour should be less than 100 mg/m and 90% of the measured CO-peaks within 24 h should be less than 150 mg/m . Combining the system structure with the chemical reaction principle, these objectives may be transformed into the following design specifications. 1) The closed-loop system should be robustly stable against the severe parameter variations. 2) The system should asymptotically track stepwise setpoint changes. 3) For set-point changes the settling time of the O -concentration should be under 5 min, the settling time of the bed temperature below 30 min, and the overshoot of both below 10%. 4) The coupling between the oxygen and temperature subsystems should be as small as possible. III. MODEL MATCHING CONTROL FOR THE NOMINAL PLANT For convenience, here we rewrite the model (1) of the fluidized bed combustor for sewage sludge in the following form:
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(2)
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 11, NO. 4, JULY 2003
where , ,
If
, ,
573
is chosen to satisfy
,
, and , . In the sequel, the symbol is used to denote , and the polynomial degree. Thus, we have , which is called the th row degree of the plant. A. Model-Matching Problem For the present plant whose design specifications are described in time domain by settling time, overshoot, etc., there is no effective control method which can achieve the desired performance. A possible choice is model-matching is chosen so that control, where first a reference model it can satisfy the expected performance and then a controller is designed such that the corresponding closed-loop transfer . function is equal to as a diagonal The following reasons suggest to choose transfer function matrix of the form:
(7) and , then, each is proper and stable which may be similar to the SISO case taken as the th input dynamics be the plant reference input. [17]. Let by For convenience, we denote , and thus, becomes just the new reference model. Here it should also be pointed out that the above model-matching problem for multioutput delay systems has not been solved so far, and the following discussion shows that it can not also be implied by any existing results.
where
B. Model-Matching Controller of Multidelay Systems In order to describe a model-matching controller, two groups of polynomial equations have to be defined. The first group is in (2) which are related to the elements in the first row of described as follows:
(3) First, controlling each output independently is the ideal feature of controlling multivariable systems and is especially emof the form (3) phasized in process industry. Therefore, can make the decoupling performance naturally be satisfied, in turn, the design specification 4) be satisfied. Then, it had been proved in [18] that due to the limitation of prior information, the practical adaptive control is feasible only when the Hermite normal form of the plant is diagonal. Thus, it becomes natural as the reference model. The adaptive to select a diagonal control will be discussed in the next section for the plant with parameter variations. it is easy to determine For the diagonal matrix and satisfying the desired performance specifications because only single transfer functions have to be dealt with. In addition, if there exists a controller making the closed-loop system match a diagonal reference model, then the plant should satisfy the following decoupling condition [17]:
(8) are unknown polynomials. where Correspondingly, the second group of polynomial equations are . However, related to the elements in the second row of due to the existence of time delays, this group of equations with , depend on unknown polynomials and as follows: additional polynomials
(9) where
(4) (10) , and the integer is nonsingular where is chosen such that tends to a nonzero finite vector and , here is the th row of . as -degree polynomials and Furthermore, introduce the -degree polynomials , each being monic and stable and let (5) Then,
can be rewritten as (6)
(11) for , and the zero of and are the zeros of for ( ). The following theorem shows that the polynomial equations (8) and (9) determine the required controller. Theorem 1: If (8) has the solutions with their degrees equal to or less than , and (9) has the with their degrees equal to solutions
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or less than
, there must exist a feedback control with each defined by
(16) Let
with
and
(12)
such that the closed-loop transfer function of the plant (2) is in (3). equal to with , and ” to Here, we have used “ by (12), we not only need represent that when computing , but also take the delay time . This regulation to take is also valid for the ensuing discussions. Proof: For simplification, we only give the proof for . The delay-free case of can similarly be dealt with. Let , . or , Then, a multiplication of the two equations in (9) by respectively, yields
(17) By substituting (15) and (16) into (17), we have
i.e.,
(13) Further
and
(14) Summing up the above equations leads to
(15) On the other hand, based on (10) and (11), we calculate the following integral:
(18) which is just the second desired input–output relationship. It satisfying (12) can achieve the expected model shows that matching. Thus, the proof is completed. Theorem 1 shows that for the reference model (3), there exists indeed a model-matching controller (12). However, this controller cannot satisfy our requirement because the second term of its right-hand side suffers from the following problems: 1) ’s coefficient includes the unknown as a factor of its denominator and, therefore, it is sensitive to model uncertainties, which in turn, results in poor closed-loop robustness; 2) it does not allow a parametric representation so that the adaptive scheme is hard to be applied to tolerate parameter variations in the present systems; and 3) the signal is itself difficult to be obtained. A direct observation, however, shows that if all can be taken as the same polynomial , then the input–output relationship of the open-loop plant results in (19)
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Now the right-hand side of (19) can be parameterized and disappears from ’s coefficient. At the the unknown same time, the signal in the right-hand side of (19) becomes obtainable. The following theorem provides an ideal solution to the problem because it confirms the existence of the required in the polynomial equations (8) and (9). polynomial Theorem 2: The polynomial equations (8) and (9) are solvwith their degrees equal to able for . Further, and can or less than , and and be taken as the same polynomial can be taken as the same polynomial . Proof: Similar to Theorem 1, here we only prove that (9) is solvable. The solvability of the delay-free equation (8) can be , reached by the similar way. Let , , and , as before, are roots of for where and roots of for . Note that and are monic, (10) and (11) can be rewritten as
(20)
575
write (9) as
(27) According to (25) and (26), the right-hand side of (27) van[roots of ], which guarantees the ishes at . Thus, the conclusion is established. solvability of Remark 1: It is easy to observe that the application of Theorem 2 to the present plant (2) leads to 0 polynomials and and 1 polynomials , , , and . In addition, it is also an interesting fact that Theorem 2 does not and . In this need the coprimeness of sense the model (2) is of generality because any MIMO system with output delays, if its each transfer function entry is not limited to be coprime, can be put into this form. Now, by combining Theorems 1 with 2, the following result is immediate. Theorem 3 (Model Matching Controller): For the given plant (2), there exists a realizable feedback control with each defined by
and
(21) For and
(please note the writing difference of ), (20) and (21), respectively, can be reduced to (22)
and (23) Equivalently, we have the following vector representation for :
with
and
(28)
such that the closed-loop transfer function of the plant (2) is in (3). equal to are 1 polynomials Remark 2: Note that is always utilizable. Therefore, Reand mark 1 implies the realizability of (28). On the other hand, as discussed in [16], the controller (28) has a similar structure to the Smith predictor which just cancels the effects of the time delays on the plant performance.
(24) . On the other hand, a necessary condition with is that (9) has solution of the form : that the following equation holds at (25) By comparing (24) with (25), it is known that if we take degree polynomial to satisfy (26) can then (25) holds. Thus, the problem becomes whether is solved from (9). For the be solved from (26), and then, former, it is obvious because the coefficient matrix of the unknown variables in (26) is a Vandermode matrix which is nonare distinct. For the latter, resingular when all
IV. ADAPTIVE CONTROL FOR THE PLANT WITH UNCERTAINTY From the MRAC point of view, the model-matching controller (28) in Theorem 3 provides an excellent controller structure that allows us to deal with the parameter variations of the plant with output delays by using an adaptive control scheme. To do this, we need to parameterize first (28), and then give the error dynamics of the plant and the augmented error. By employing the standard procedure of [18], the controller (28) has the following parametric representation:
with
and
(29)
are obtainable signal vectors, and where are unknown parameter vectors which will be replaced by
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their estimates or , respectively. The system error is defined by , where is the th real output of the plant and is the th desired output. can be represented by Thus, the augmented error
with
TABLE I REPRESENTATIVE OPERATING POINTS OF THE FLUIDIZED BED COMBUSTOR
and (30) ,
where
, ,
, and
with
and
(31)
denotes a time-delay operator, i.e., and . Write the signals and as
(32) The following adaptive law is chosen: Fig. 2.
(33) and . If the plant parameter variations do not change the and , then the adaptive controller (28) signs of the gains with the adaptive law (33) results in the bounded closed-loop . Due to space limitation, system signals and the proof is omitted. Remark 3: As we have seen, the time delay is considered to be fixed in the above discussion. However, it is not difficult to extend the present results to deal with the uncertainty in time delay. A possible way is to transform the delay uncertainty into unstructured multiplicative uncertainty as in [13] and then modify the adaptation law (33) according to [20] so that it tolerates the unstructured multiplicative uncertainty in the model.
where
V. NUMERICAL VERIFICATION The controller (28) with the adaptation law (33) is applied to the fluidized bed combustor. The simulation uses the foland lowing data: . Two arbitrary, but stable minimal-phase transfer functions and are chosen. The diagonal matching model has the diagonal elements
Step responses of the representative plants.
and which satisfy the required performance specifications. The initial conditions of the adaptation loop result from the nominal values of the plant parameters. For several representative plants listed in Table I the command step responses are shown in Fig. 2. Obviously, the model-matching performance of the temperature subsystem is quite satisfactory even for the plants with large uncertainties. Although the displayed responses vary with different plants, the oxygen subsystem satisfies the desired settling times and overshoot specifications. More precisely, the maximum overreaches shoot is less than 5.1%. After 300 s, the output of s 0.9883, s 0.9887 and s 1.0296, and 0.9642, their deviations from the expected final value are within 5%. Fig. 3 shows the step responses of two off-diagonal transfer functions. The displayed decoupling performance is quite acceptable. The system errors are shown in Figs. 4 and 5 for two and or and different scenarios where hold, respectively. In all cases, the presented adaptive controller meets the desired design objectives. VI. CONCLUSION In this paper, the control problem of a fluidized bed combustor for sewage sludge has been discussed. The main characteristics of the plant are captured by the model uncertainty, the time-delays and the stiffness. An MRAC scheme, which can simultaneously achieve the adaptive decoupling
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Fig. 3.
577
the closed-loop system can track the given model as closely as possible. The most important questions answered are whether there exists such controller and which control structure can achieve the desired tracking performance. The main results were stated in Theorems 1 to 3 which describe an appropriate controller structure for the present plant. The concept of the augmented errors has been used to guarantee that the system output errors converge to zero. The resulting closed-loop system is globally stable. The simulation results show that for the representative plant models the settling times of both the oxygen subsystem and the temperature subsystem are kept within the desired values of 5 and 30 min, respectively, with their overshoots under 10%. The coupling between the two subsystems and the effect of the parameter variations on the output responses have been reduced to a satisfactory level. Hence, the proposed method can be used to control the fluidized bed combustor for sewage sludge.
Decoupling characteristics of the representative plants.
REFERENCES
t
Fig. 4. System errors e ( ) and
e (t) for the case of m_
m
= 1 and _
= 0.
[1] T. Ogada, “Emissions and combustion characteristics of wet sewage sludge in a bubbling fluidized bed,” Ph.D. dissertation, Tech. Univ. Hamburg-Harburg, Germany, 1995. [2] K. Hansmann, Bundes-Immissionschutzgesetz und ergaenzende Vorschriften Baden-Baden, Germany, Mar. 1991, vol. 8. (German law concerning the restriction of pollutant emissions; similar laws hold in all industrialized countries). [3] J. Lunze and A. Wolff, “Robuste regelung einer wirbelschichtverbrennungsanlage fuer klaerschlamm,” presented at the Automatisierungstechnik 11, Muenchen, Germany, 1997. [4] Y. Jia, J. Lunze, and A. Wolff, “Application of robust multivariable control to a fluidized bed combustor for sewage sludge,” in Proc. 14th IFAC World Congr., vol. N, Beijing, China, July 1999, pp. 301–306. [5] , “Modeling and robust PI control of a fluidized bed combustor for sewage sludge,” Asian J. Contr., vol. 4, no. 4, pp. 482–493, 2002. [6] Y. Jia, H. Kokame, and J. Lunze, “Adaptive control of MIMO delay systems with application to a fluidized bed combustor for sewage sludge,” in Proc. SICE, session 215A-3, Nagoya, Japan, July 2001. [7] , “Direct model reference adaptive control of a fluidized bed combustor for sewage sludge,” in Proc. IEEE Conf. Control Applications, Mexico City, Mexico, Sept. 2001, pp. 13–17. [8] E. J. Adam, H. A. Latchman, and O. D. Crisalle, “Robustness of the Smith predictor with respect to uncertainty in the time-delay parameter,” in Proc. American Control Conf., Chicago, IL, June 2000, pp. 1452–1457. [9] G. Meinsma and H. Zwart, “On control for dead-time systems,” IEEE Trans. Automat. Contr., vol. 45, pp. 272–285, Feb. 2000. [10] G. Tadmor,, “The standard problems with a single input delay,” IEEE Trans. Automat. Contr., vol. 45, pp. 382–397, Mar. 2000. [11] K. M. Nagpal and R. Ravi, “ control and estimation problems with delayed measurements: State-space solutions,” SIAM J. Contr. Optimiz., vol. 35, no. 4, pp. 1217–1243, 1997. [12] Q. G. Wang, T. H. Lee, and K. K. Tan, Finite Spectrum Assignment for Time-Delay Systems, ser. Lecture Notes in Control and Information Sciences. Berlin, Germany: Springer-Verlag, 1999, vol. 239. [13] J. C. Doyle, B. A. Francis, and A. Tannenbaum, Feedback Control Theory. New York: Macmillan, 1991. [14] M. Morari and E. Zafiriou, Robust Process Control. London, U.K.: Prentice-Hall, 1989. [15] K. Ichikawa, “Adaptive control of delay system,” Int. J. Contr., vol. 43, no. 6, pp. 1653–1659, 1986. [16] R. Ortega and R. Lozano, “Globally stable adaptive controller for systems with delay,” Int. J. Contr., vol. 47, no. 1, pp. 17–23, 1988. [17] K. Ichikawa, Control System Design Based on Exact Model Matching Techniques, ser. Lecture Notes in Control and Information Sciences. Berlin, Germany: Springer-Verlag, 1985, vol. 74. [18] K. S. Narendra and A. M. Annaswamy, Stable Adaptive Systems. Englewood Cliffs, NJ: Prentice-Hall, 1989. [19] S. Sastry and M. Bodson, Adaptive Control: Stability, Convergence and Robustness. Englewood Cliffs, NJ: Prentice-Hall, 1989. [20] A. Datta, Adaptive Internal Model Control. Berlin, Germany: Springer-Verlag, 1998.
H H
Fig. 5. System errors
e (t) and e (t) for the case of m_
m
= 0 and _
= 1.
and model matching, has been proposed, in which the desired performance specifications are imposed on the reference model and the design task becomes how to find a controller such that
H
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571
Simultaneous Adaptive Decoupling and Model Matching Control of a Fluidized Bed Combustor for Sewage Sludge Yingmin Jia, Hideki Kokame, and Jan Lunze
Abstract—The major difficulties in designing a controller for the fluidized bed combustor for sewage sludge result from the following reasons: 1) The dynamical behavior of the plant is highly unknown and the experimental data only render a coarse model; 2) the combustion process shows large time-delay; and 3) the plant model is a stiff differential equation because the parameter values among the subsystems differ severely. To circumvent these difficulties, this paper proposes to describe each channel of the multiple-input–multiple-output system by a first-order transfer function with variable coefficients and a fixed time delay. A model reference adaptive control scheme, which can simultaneously achieve adaptive decoupling and model matching, is proposed. The resulting adaptive control law guarantees closed-loop stability. The newly obtained controller has a structure that is similar to the well-known Smith predictor. This explains why the present scheme can cope with the time delay effects on the system performance. Due to its adaptation abilities, it can lower the limitations of the Smith scheme to accurate plant descriptions. The application study demonstrates that the proposed design can reach the desired performance requirements. Index Terms—Adaptive decoupling, fluidized bed combustors, model matching, process control, sewage sludge, time delay, uncertainty.
I. INTRODUCTION Sewage sludge is incinerated to form stabilized ash and to reduce the volume of the sludge. Sewage sludge combustors can only be industrially used if the gaseous emissions are kept within the strict legal limits. Currently, sewage sludge combustion plants with a bubbling fluidized bed are essentially controlled by hand and the combustion quality depends heavily on the operators’ experience. Hence, there is an increasing requirement for the automatic control of such plants. Fluidized bed combustors are characterized simultaneously by large model uncertainties, time delays, and a large variety of time constants which make the differential equations stiff [1]–[7]. As a consequence, the controller of such plants cannot be designed by standard methods. This paper proposes a controller whose structure is similar to the well-known Smith predictor. By using an internal model of the time-delay system, the controller can react before the control error is visible so that the Manuscript received July 23, 2002. Manuscript received in final form January 6, 2003. Recommended by Associate Editor F. Doyle. This work was supported by the JSPS, the AvH, the APEC, and the NSFC under Grant 69625506. Y. Jia is with the Seventh Research Division, Beijing University of Aeronautics and Astronautics, Beijing 100083, P.R. China (e-mail: [email protected]). H. Kokame is with the Department of Electrical and Electronic Systems, Osaka Prefecture University, Osaka 599-8531, Japan (e-mail: [email protected]). J. Lunze is with the Institute of Automation and Computer Control, RuhrUniversity Bochum, D-44780 Bochum, Germany (e-mail: [email protected]). Digital Object Identifier 10.1109/TCST.2003.813398
closed-loop system has the best performance under the performance limitations brought about by the time delays. This idea is combined here with the adaptation principle described in Section IV that makes it possible to cope with severe plant uncertainties. In more detail, this paper uses a model reference adaptive control (MRAC) scheme which simultaneously achieves adaptive decoupling and model-matching. An ideal model that satisfies all performance specifications is selected and a controller is designed such that the output of the closed-loop system tracks the model output. The main result is a new model-matching controller for multiple-input–multiple-output (MIMO) time-delay plants. Due to the proposed adaptation scheme the desired control objectives are met in spite of model uncertainties. The properties of the control scheme are shown by its application to a sewage sludge combustor. Relevant literature: A standard method for the control of delay-time systems is the Smith predictor [8]. However, the delay cancellation in this scheme depends strongly on the model accuracy of the plant which is hard to be satisfied for the present control has been studied for a variety of time-delay plant. systems [9]–[11]. For the multidelay case, the finite spectrum assignment can be reached by the method described in [12]. Quite a few existing results [4], [13] turn time delay into a multiplicative uncertainty and use robust control methods [14]. These methods are based on optimization principles but they are lacking definite rules to select the objective functions. For stiff systems these methods suffer from the large conservatism of the underlying stability tests. The MRAC scheme of single-input–single-output (SISO) systems with a single delay was proposed in [15]. The results cannot be used to deal with the multioutput delay case. Further, as pointed out in [16], the adaptive control law used there cannot guarantee the convergence of the system errors and boundedness of the control signal. In contrast to this, this paper obtains a new matching controller for multi-output delay systems by employing the decoupling principle of MIMO delay-free systems [17] which provides a globally stable adaptive control scheme and, therefore, can be viewed as a significant extension of the results in [15], [16] to general MIMO delay systems. Finally, the reader is referred to [18]–[20] for a description of the MRAC principle. Structure of the paper: In Section II, the plant model is given and the corresponding design objectives are formulated. Section III is devoted to developing the simultaneous decoupling and model-matching controller for the nominal plant. Extensions to the plant with parameter variations are made in Section IV where the adaptive control law to insure the global stability is given. Section V shows the applicability of the proposed method to the fluidized bed combustor.
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vide a firm basis for modeling the combustor from first principles. Therefore, a lot of experiments have been made to set up a dynamical model which describes the relationship between the input and the output . Here is propane mass is total air mass flow, is the bed temperature and flow, is O concentration. In this paper, we use the same model structure as in [3] with the following nominal parameter values obtained from the experimental data:
(1)
Fig. 1. Pilot fluidized bed combustor of the Technical University HamburgHarburg.
II. COMBUSTOR MODEL AND DESIGN OBJECTIVES A pilot fluidized bed combustor is shown in Fig. 1. The unit combusts wet sewage sludge (with 70% water contents) and propane in a bubbling fluidized bed. The combustor consists of a tube with a total height of 900 cm and a diameter in the bed area of 15 cm. The bed area of the combustor is filled with sand. The sand is fluidized with preheated air. The unit is equipped with the necessary facilities to feed propane, wet sludge, and primary air to the bed and secondary air into the riser. Gas samples are taken at the top of the riser, where the concentrations of O , CO , CO, NO , and SO are measured. The combustor is heated up by an electrical heating system in the combustor walls to compensate for the heat loss, which occurs due to the extreme height/diameter ratio of the combustor. Fluidized bed combustors have been studied intensively from a process-oriented viewpoint. For the combustor depicted in Fig. 1 the relevant results are given in [1]. However, all these results do not result in a dynamical model nor do they pro-
Seemingly, the model (1) is simple, but it captures almost all typical characteristics of the combustor which lead to the major difficulties of the controller design: • Uncertainty. The highly unknown dynamical behavior of the plant makes it impossible to model exactly the plant. • Time delay. The time delay occurring in the oxygen subsystem is up to 36 s, while in the temperature subsystem it is nearly zero. Hence, the system is a multioutput delay system. • Stiffness. The time constant of the temperature subsystem is over 50 times larger than that of the oxygen subsystem. The control objectives of the plant are determined by allowed emission limits stipulated by law [2]. Similar laws hold in all industrialized countries. Accordingly, the average concentration of NO or CO over a day should be less than 200 mg m or 50 mg m , respectively. The average value of CO over half an hour should be less than 100 mg/m and 90% of the measured CO-peaks within 24 h should be less than 150 mg/m . Combining the system structure with the chemical reaction principle, these objectives may be transformed into the following design specifications. 1) The closed-loop system should be robustly stable against the severe parameter variations. 2) The system should asymptotically track stepwise setpoint changes. 3) For set-point changes the settling time of the O -concentration should be under 5 min, the settling time of the bed temperature below 30 min, and the overshoot of both below 10%. 4) The coupling between the oxygen and temperature subsystems should be as small as possible. III. MODEL MATCHING CONTROL FOR THE NOMINAL PLANT For convenience, here we rewrite the model (1) of the fluidized bed combustor for sewage sludge in the following form:
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(2)
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where , ,
If
, ,
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is chosen to satisfy
,
, and , . In the sequel, the symbol is used to denote , and the polynomial degree. Thus, we have , which is called the th row degree of the plant. A. Model-Matching Problem For the present plant whose design specifications are described in time domain by settling time, overshoot, etc., there is no effective control method which can achieve the desired performance. A possible choice is model-matching is chosen so that control, where first a reference model it can satisfy the expected performance and then a controller is designed such that the corresponding closed-loop transfer . function is equal to as a diagonal The following reasons suggest to choose transfer function matrix of the form:
(7) and , then, each is proper and stable which may be similar to the SISO case taken as the th input dynamics be the plant reference input. [17]. Let by For convenience, we denote , and thus, becomes just the new reference model. Here it should also be pointed out that the above model-matching problem for multioutput delay systems has not been solved so far, and the following discussion shows that it can not also be implied by any existing results.
where
B. Model-Matching Controller of Multidelay Systems In order to describe a model-matching controller, two groups of polynomial equations have to be defined. The first group is in (2) which are related to the elements in the first row of described as follows:
(3) First, controlling each output independently is the ideal feature of controlling multivariable systems and is especially emof the form (3) phasized in process industry. Therefore, can make the decoupling performance naturally be satisfied, in turn, the design specification 4) be satisfied. Then, it had been proved in [18] that due to the limitation of prior information, the practical adaptive control is feasible only when the Hermite normal form of the plant is diagonal. Thus, it becomes natural as the reference model. The adaptive to select a diagonal control will be discussed in the next section for the plant with parameter variations. it is easy to determine For the diagonal matrix and satisfying the desired performance specifications because only single transfer functions have to be dealt with. In addition, if there exists a controller making the closed-loop system match a diagonal reference model, then the plant should satisfy the following decoupling condition [17]:
(8) are unknown polynomials. where Correspondingly, the second group of polynomial equations are . However, related to the elements in the second row of due to the existence of time delays, this group of equations with , depend on unknown polynomials and as follows: additional polynomials
(9) where
(4) (10) , and the integer is nonsingular where is chosen such that tends to a nonzero finite vector and , here is the th row of . as -degree polynomials and Furthermore, introduce the -degree polynomials , each being monic and stable and let (5) Then,
can be rewritten as (6)
(11) for , and the zero of and are the zeros of for ( ). The following theorem shows that the polynomial equations (8) and (9) determine the required controller. Theorem 1: If (8) has the solutions with their degrees equal to or less than , and (9) has the with their degrees equal to solutions
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or less than
, there must exist a feedback control with each defined by
(16) Let
with
and
(12)
such that the closed-loop transfer function of the plant (2) is in (3). equal to with , and ” to Here, we have used “ by (12), we not only need represent that when computing , but also take the delay time . This regulation to take is also valid for the ensuing discussions. Proof: For simplification, we only give the proof for . The delay-free case of can similarly be dealt with. Let , . or , Then, a multiplication of the two equations in (9) by respectively, yields
(17) By substituting (15) and (16) into (17), we have
i.e.,
(13) Further
and
(14) Summing up the above equations leads to
(15) On the other hand, based on (10) and (11), we calculate the following integral:
(18) which is just the second desired input–output relationship. It satisfying (12) can achieve the expected model shows that matching. Thus, the proof is completed. Theorem 1 shows that for the reference model (3), there exists indeed a model-matching controller (12). However, this controller cannot satisfy our requirement because the second term of its right-hand side suffers from the following problems: 1) ’s coefficient includes the unknown as a factor of its denominator and, therefore, it is sensitive to model uncertainties, which in turn, results in poor closed-loop robustness; 2) it does not allow a parametric representation so that the adaptive scheme is hard to be applied to tolerate parameter variations in the present systems; and 3) the signal is itself difficult to be obtained. A direct observation, however, shows that if all can be taken as the same polynomial , then the input–output relationship of the open-loop plant results in (19)
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Now the right-hand side of (19) can be parameterized and disappears from ’s coefficient. At the the unknown same time, the signal in the right-hand side of (19) becomes obtainable. The following theorem provides an ideal solution to the problem because it confirms the existence of the required in the polynomial equations (8) and (9). polynomial Theorem 2: The polynomial equations (8) and (9) are solvwith their degrees equal to able for . Further, and can or less than , and and be taken as the same polynomial can be taken as the same polynomial . Proof: Similar to Theorem 1, here we only prove that (9) is solvable. The solvability of the delay-free equation (8) can be , reached by the similar way. Let , , and , as before, are roots of for where and roots of for . Note that and are monic, (10) and (11) can be rewritten as
(20)
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write (9) as
(27) According to (25) and (26), the right-hand side of (27) van[roots of ], which guarantees the ishes at . Thus, the conclusion is established. solvability of Remark 1: It is easy to observe that the application of Theorem 2 to the present plant (2) leads to 0 polynomials and and 1 polynomials , , , and . In addition, it is also an interesting fact that Theorem 2 does not and . In this need the coprimeness of sense the model (2) is of generality because any MIMO system with output delays, if its each transfer function entry is not limited to be coprime, can be put into this form. Now, by combining Theorems 1 with 2, the following result is immediate. Theorem 3 (Model Matching Controller): For the given plant (2), there exists a realizable feedback control with each defined by
and
(21) For and
(please note the writing difference of ), (20) and (21), respectively, can be reduced to (22)
and (23) Equivalently, we have the following vector representation for :
with
and
(28)
such that the closed-loop transfer function of the plant (2) is in (3). equal to are 1 polynomials Remark 2: Note that is always utilizable. Therefore, Reand mark 1 implies the realizability of (28). On the other hand, as discussed in [16], the controller (28) has a similar structure to the Smith predictor which just cancels the effects of the time delays on the plant performance.
(24) . On the other hand, a necessary condition with is that (9) has solution of the form : that the following equation holds at (25) By comparing (24) with (25), it is known that if we take degree polynomial to satisfy (26) can then (25) holds. Thus, the problem becomes whether is solved from (9). For the be solved from (26), and then, former, it is obvious because the coefficient matrix of the unknown variables in (26) is a Vandermode matrix which is nonare distinct. For the latter, resingular when all
IV. ADAPTIVE CONTROL FOR THE PLANT WITH UNCERTAINTY From the MRAC point of view, the model-matching controller (28) in Theorem 3 provides an excellent controller structure that allows us to deal with the parameter variations of the plant with output delays by using an adaptive control scheme. To do this, we need to parameterize first (28), and then give the error dynamics of the plant and the augmented error. By employing the standard procedure of [18], the controller (28) has the following parametric representation:
with
and
(29)
are obtainable signal vectors, and where are unknown parameter vectors which will be replaced by
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their estimates or , respectively. The system error is defined by , where is the th real output of the plant and is the th desired output. can be represented by Thus, the augmented error
with
TABLE I REPRESENTATIVE OPERATING POINTS OF THE FLUIDIZED BED COMBUSTOR
and (30) ,
where
, ,
, and
with
and
(31)
denotes a time-delay operator, i.e., and . Write the signals and as
(32) The following adaptive law is chosen: Fig. 2.
(33) and . If the plant parameter variations do not change the and , then the adaptive controller (28) signs of the gains with the adaptive law (33) results in the bounded closed-loop . Due to space limitation, system signals and the proof is omitted. Remark 3: As we have seen, the time delay is considered to be fixed in the above discussion. However, it is not difficult to extend the present results to deal with the uncertainty in time delay. A possible way is to transform the delay uncertainty into unstructured multiplicative uncertainty as in [13] and then modify the adaptation law (33) according to [20] so that it tolerates the unstructured multiplicative uncertainty in the model.
where
V. NUMERICAL VERIFICATION The controller (28) with the adaptation law (33) is applied to the fluidized bed combustor. The simulation uses the foland lowing data: . Two arbitrary, but stable minimal-phase transfer functions and are chosen. The diagonal matching model has the diagonal elements
Step responses of the representative plants.
and which satisfy the required performance specifications. The initial conditions of the adaptation loop result from the nominal values of the plant parameters. For several representative plants listed in Table I the command step responses are shown in Fig. 2. Obviously, the model-matching performance of the temperature subsystem is quite satisfactory even for the plants with large uncertainties. Although the displayed responses vary with different plants, the oxygen subsystem satisfies the desired settling times and overshoot specifications. More precisely, the maximum overreaches shoot is less than 5.1%. After 300 s, the output of s 0.9883, s 0.9887 and s 1.0296, and 0.9642, their deviations from the expected final value are within 5%. Fig. 3 shows the step responses of two off-diagonal transfer functions. The displayed decoupling performance is quite acceptable. The system errors are shown in Figs. 4 and 5 for two and or and different scenarios where hold, respectively. In all cases, the presented adaptive controller meets the desired design objectives. VI. CONCLUSION In this paper, the control problem of a fluidized bed combustor for sewage sludge has been discussed. The main characteristics of the plant are captured by the model uncertainty, the time-delays and the stiffness. An MRAC scheme, which can simultaneously achieve the adaptive decoupling
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Fig. 3.
577
the closed-loop system can track the given model as closely as possible. The most important questions answered are whether there exists such controller and which control structure can achieve the desired tracking performance. The main results were stated in Theorems 1 to 3 which describe an appropriate controller structure for the present plant. The concept of the augmented errors has been used to guarantee that the system output errors converge to zero. The resulting closed-loop system is globally stable. The simulation results show that for the representative plant models the settling times of both the oxygen subsystem and the temperature subsystem are kept within the desired values of 5 and 30 min, respectively, with their overshoots under 10%. The coupling between the two subsystems and the effect of the parameter variations on the output responses have been reduced to a satisfactory level. Hence, the proposed method can be used to control the fluidized bed combustor for sewage sludge.
Decoupling characteristics of the representative plants.
REFERENCES
t
Fig. 4. System errors e ( ) and
e (t) for the case of m_
m
= 1 and _
= 0.
[1] T. Ogada, “Emissions and combustion characteristics of wet sewage sludge in a bubbling fluidized bed,” Ph.D. dissertation, Tech. Univ. Hamburg-Harburg, Germany, 1995. [2] K. Hansmann, Bundes-Immissionschutzgesetz und ergaenzende Vorschriften Baden-Baden, Germany, Mar. 1991, vol. 8. (German law concerning the restriction of pollutant emissions; similar laws hold in all industrialized countries). [3] J. Lunze and A. Wolff, “Robuste regelung einer wirbelschichtverbrennungsanlage fuer klaerschlamm,” presented at the Automatisierungstechnik 11, Muenchen, Germany, 1997. [4] Y. Jia, J. Lunze, and A. Wolff, “Application of robust multivariable control to a fluidized bed combustor for sewage sludge,” in Proc. 14th IFAC World Congr., vol. N, Beijing, China, July 1999, pp. 301–306. [5] , “Modeling and robust PI control of a fluidized bed combustor for sewage sludge,” Asian J. Contr., vol. 4, no. 4, pp. 482–493, 2002. [6] Y. Jia, H. Kokame, and J. Lunze, “Adaptive control of MIMO delay systems with application to a fluidized bed combustor for sewage sludge,” in Proc. SICE, session 215A-3, Nagoya, Japan, July 2001. [7] , “Direct model reference adaptive control of a fluidized bed combustor for sewage sludge,” in Proc. IEEE Conf. Control Applications, Mexico City, Mexico, Sept. 2001, pp. 13–17. [8] E. J. Adam, H. A. Latchman, and O. D. Crisalle, “Robustness of the Smith predictor with respect to uncertainty in the time-delay parameter,” in Proc. American Control Conf., Chicago, IL, June 2000, pp. 1452–1457. [9] G. Meinsma and H. Zwart, “On control for dead-time systems,” IEEE Trans. Automat. Contr., vol. 45, pp. 272–285, Feb. 2000. [10] G. Tadmor,, “The standard problems with a single input delay,” IEEE Trans. Automat. Contr., vol. 45, pp. 382–397, Mar. 2000. [11] K. M. Nagpal and R. Ravi, “ control and estimation problems with delayed measurements: State-space solutions,” SIAM J. Contr. Optimiz., vol. 35, no. 4, pp. 1217–1243, 1997. [12] Q. G. Wang, T. H. Lee, and K. K. Tan, Finite Spectrum Assignment for Time-Delay Systems, ser. Lecture Notes in Control and Information Sciences. Berlin, Germany: Springer-Verlag, 1999, vol. 239. [13] J. C. Doyle, B. A. Francis, and A. Tannenbaum, Feedback Control Theory. New York: Macmillan, 1991. [14] M. Morari and E. Zafiriou, Robust Process Control. London, U.K.: Prentice-Hall, 1989. [15] K. Ichikawa, “Adaptive control of delay system,” Int. J. Contr., vol. 43, no. 6, pp. 1653–1659, 1986. [16] R. Ortega and R. Lozano, “Globally stable adaptive controller for systems with delay,” Int. J. Contr., vol. 47, no. 1, pp. 17–23, 1988. [17] K. Ichikawa, Control System Design Based on Exact Model Matching Techniques, ser. Lecture Notes in Control and Information Sciences. Berlin, Germany: Springer-Verlag, 1985, vol. 74. [18] K. S. Narendra and A. M. Annaswamy, Stable Adaptive Systems. Englewood Cliffs, NJ: Prentice-Hall, 1989. [19] S. Sastry and M. Bodson, Adaptive Control: Stability, Convergence and Robustness. Englewood Cliffs, NJ: Prentice-Hall, 1989. [20] A. Datta, Adaptive Internal Model Control. Berlin, Germany: Springer-Verlag, 1998.
H H
Fig. 5. System errors
e (t) and e (t) for the case of m_
m
= 0 and _
= 1.
and model matching, has been proposed, in which the desired performance specifications are imposed on the reference model and the design task becomes how to find a controller such that
H
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