18th IEEE International Conference on Control Applications Part of 2009 IEEE Multi-conference on Systems and Control Saint Petersburg, Russia, July 8-10, 2009
Robust Optimal Controller Design for a Non-minimum Phase Boiler System with a Saturable Actuator *
H. Zargarzadeh, **M. R. Jahed Motlagh
Abstract— Dealing with a multivariable non-minimum phase model of a boiler system, a LQG/LTR controller is designed which is well-known for its optimality and robustness. Firstly, the loop transfer function is shaped using LQG method; then, an observer is proposed with the aim of loop transfer recovery. Consequently, the designed controller is reduced to the lowest possible order for the sake of implementation easiness. Moreover, LQG design features are used to analyze stability margins when the system has a saturable actuator. Finally, numerical results demonstrate robustness of the system against the output disturbance, sensor response delay, and actuator saturability.
II. THE SYSTEM MODEL
I. INTRODUCTION LQG/Loop Transfer Recovery (LQG/LTR) which is introduced by Kwakernaak [1] and Doyle and Stein [2], is an optimal robust technique. This method deals with left invertible and minimum phase plants, and uses observer based state feedback controllers to recover a given (desired) Target Loop Transfer Function (TLTF). This procedure relies on cancellation of some dynamics (including zeros) of the plant by the observer dynamics. Therefore, the LQG/LTR technique, based on Kalman filter as observer, has some difficulties in facing with non-minimum phase systems. The main problem is that the observer poles asymptotically converge to the plant zeros when the compensator recovers the TLTF [3]. Saberi et al have shown that for any left invertible and minimum phase plants, any TLTF, designed via any state feedback control approach, could be recovered asymptotically using an observer based controller and employing an asymptotically infinite observer gain [2]. Nevertheless, it is not guaranteed that any TLTF for a particular non-minimum phase system, be recoverable. Coping with this problem, necessary and sufficient conditions, which a general TLTF has to satisfy for recoverability, are studied in [4]. In addition, necessary and sufficient conditions for a non-minimum phase system in order to have at least one recoverable target loop are proposed in [4] analytically. In order to have a stable observer, [5-6] propose the theory and design of LTR for non-minimum phase systems. The design algorithm is based on asymptotic and time-scale structure assignment (ATEA) procedure which is presented as a Matlab toolbox (LYNSYSKIT) in [7]. Controlling boiler systems, as typically non-minimum phase multivariable plants, have recently attracted much attention. Various controllers have been proposed for boiler or boilerturbine systems, e.g., inverse Nyquist array method [8], LQG method [9], LQG/LTR method [10], ,mixed-sensitivity approach [11], loop-shaping approach [12], internal-modelcontrol (IMC) method [13], predicative control method [14] and fuzzy auto-regressive moving average [15]. In addition, [16] provides a gain-scheduled ℓ -optimal controller for boilerturbine dynamics with actuator saturation. Authors are with Complex Systems Laboratory, Iran University of Science and Technology, Tehran, Iran (email: *
[email protected], **
[email protected]).
978-1-4244-4602-5/09/$25.00 ©2009 IEEE
In the following sections, using LQG method, an optimal robust TLTF is designed for the boiler system. Then, applying ATEA algorithm, a stable observer (which asymptotically recovers the designed LQG target loop) is introduced. After that, the total designed controller is reduced to a lower order controller without loss of performance in general. The last section analyzes the effect of saturability of the actuators using describing function method. Finally, the system robustness is numerically checked in the presence of the output disturbance, sensor response delay, and actuator saturability. W. Tan et al [12] have proposed a linear approximated model for a utility boiler system as: .
.
.
.
.
.
.
22.0459
. .
. .
0
.
.
.
. .
.
. .
. . .
.
.
.
.
(1)
.
Where the system inputs and outputs are defined as follows: . Feedwater flow rate (kg/s); . Fuel flow rate (kg/s); . Attemperator spray flow rate (kg/s); . Drum level (m); . Drum pressure (MPa); . Steam temperature ( ). Physical limitations of the system impose that the following 120, 0 7, 0 10 and constraints hold: 0 0.017 0.017. The operating point at which the system has been linearized is as follows: 40.62 2.274 , 2.876
1.0 6.45 466.7
.
(2)
By a laboratorial proof, it is shown that the linear model has a negligible deviation from the original nonlinear system in a sufficiently extended neighborhood of the operating point [12].
III. PRELIMINARIES In this section, a number of theoretical prerequisites, which are needed in the process of the controller design, are presented. Fig. 1 demonstrates a global observer based controller structure; where , , and are the plant transfer function, the controller(compensator) frequency response and the observer designing parameter respectively. Also, ,as a constant matrix, is the LQG controller state feedback gain, is a preis the designed observer gain matrix, controller used for the plant augmentation, and is the output , is applied to improve the final designed disturbance. loop transfer function; however, in the sequel, we assume that 1,1,1 until the LQG/LTR controller design is accomplished. Subsequently, in the last step of the controller design, will be modified in order to improve the time
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responses. Considering that the plant is a strictly proper LTI system, the plant is described as Σ , , ; which denotes , as a minimal realization for the plant model. Presume that is the TLTF. We say that exact loop transfer recovery (ELTR) is achieved, if , for all ; and it is asymptotically achieved (ALTR), if , pointwise in as ∞. Definition 1. The Geometric Subspace , , for the is the maximal subspace of which is system Σ , , invariant and contained in such that the eigenvalue of are in right half plane for some [4]. In Fig. 1, assume that the total augmented system has a minimal realization as Σ , , , and let have a state feedback . The TLTF, when the loop is control input as Φ where Φ . broken at the input, is Using any loop-shaping technique such as LQG, the desired and also the TLTF could be designed. Since there are a variety of choices for , selecting the most appropriate one is the art of control engineering. Recovering , one has to seek for a , such that the recovery error, , , is either asymptotically or exactly zero in the desired frequency region while guaranteeing the stability of the resulting closedloop system. Considering Fig. 1, the controller transfer function can be written as , . then Φ is called Definition 2. Let contains left half -plane admissible target loop. Where, denotes the set of eigenvalues of . points, and Theorem 1. Let be an admissible target loop for the system Σ , , that is not necessarily minimum phase or left , , invertible. is recoverable if and only if ν [4]. Theorem 2. Consider the system in Theorem.1 and the same TLTF then is recoverable by both a full and a reduced order observer based controller [5]. Theorem 3. Consider a stabilizable and detectable system Σ , , , which is not necessarily of minimum phase and which is not necessarily left invertible. Let be the dimension of , , . Also, let be any full rank matrix of dimension such that , , . Then the given system Σ has at least one recoverable target loop, if and only if an auxiliary system Σ , , is stabilizable by a static output feedback controller [5].
IV. LQG/LTR CONTROLLER DESIGN Evidently, from (1), the plant is a non-minimum phase system; and its right-half plane invariant zeros are 4.7521, 0.7087 and 0.0466. As we mentioned in the earlier section, such plants have some admissible target loops that are not recoverable [4]. This section pursues four steps to design a controller with the structure of Fig. 1. make a minimal realization for (1), , , Step 1. Let Σ the principal gains of the system are depicted in Fig. 2. This figure persuades us to augment the plant for three reasons: 1) The low frequency gains are very low. 2) The principal gains of loop transfer function , should have same high frequency rate. 3) The cut-off frequency and roll-off rate of principal gains of TLTF return ratio have to have close values in order to minimize the interactions in operating frequencies [3]. Choosing a pre-compensator such as
,
,
,
(3)
where all s and s have positive real values. The augmented system will have a realization as Σ , , Σ , , which means that the inputs of the plant are pre-filtered by . The entries of will be chosen in the next step.
Fig. 1. The structure of the proposed LQG/LTR controller.
Step 2. In this stage after system augmentation, we select a proper stabilizing state space feedback gain , for Σ , , such that the target loop transfer function return ratio, Φ , provide satisfactory principal gains at the plant input. Fig. 2 implies that one of the principal gains has a 20 / rate at high frequencies; therefore, one of diagonal entries of does not require a high frequency rate improvement. In other words, this entry is a PI-like compensator. The other two principal gains have a 0 / rate at high frequencies. Hence, should be as same as the other diagonal entries of integrators. By at most three trial and errors, we find that we have to have: 0, 0. Notice that in this case we are adding a negative real zero and three zero poles to the system. Furthermore, , the sole invariant zero of , could be deliberately located out of the desired operating frequencies. Choosing 0.5, one can select as 1
.
;
(4)
where, s are selected using Bryson loop shaping approach [17]. Fig. 3 illustrates principal gains of the compensated plant. Undoubtedly, there are infinite choices for . This might lead us to think that we will have a variety of final controllers as , , nonetheless choosing , in order to provide the TLTF, compensates the variety of . Hence, the final achieved TLTFs have similar frequency responses with a negligible deviation. Theorem 1 implies that we have to choose such that: ν , , . (5) Therefore, checking the recoverability of the desired TLTF, we ν , , is approximately null. can ensure if From Theorem 3 it could be easily verified that our compensated plant (Σ , , ) has at least one recoverable TLTF. This assures us that there is an such that ν 0. , , Here, we use LQG approach to design . Fig. 4 shows the principal gains of Φ , where minimizes the following cost , where , function 1 10 . Fig. 4 implies that makes a nearly reasonable TLTF. Thus, and need to be modified with the aim of
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Principal gains(dB)
shaping the loop reasonably. There are several loop-shaping techniques which use LQG methods for designing controllers. We will use the singular decomposition approach [3] and Bryson's inverse square method [17] for the sake of their simplicity. 50
-50 -100 1e-8
0.05 1e-6
1e-4 1e-2 1e0 Frequency(rad/s)
,
,
1e2
1e4
.
Using singular decomposition approach [3], modified as 285.37 440.49 0
440.49 0 711.78 0 0 9
can be
,
where , and 0.01 / is chosen as the cutoff frequency. The procedure of modifying is omitted for the sake of brevity. Here, the desired TLTF is attained. 200
Principal(dB)
150 100 50
20921 5660 80343 20085 44342 29672 9882 21065 32302 24810 4434 29668 4173 15233 14343 8270
200 18 294 58 155 285 69 152 265 192 47 254 36 109 89 112
.
(6)
It is shown that the LQG multivariable controller design approach, for each control loop, provides a phase margin greater than 60°, and a gain margin among 6 to ∞ should not get the system out of its [18]. Therefore, choosing safe margins. The frequency response of a specific control loop could not be distinguished from the principal gain graph of the system. Hence, experience shows that the slower loop of Fig. 5 belongs to the water level channel. Speeding up the this channel, we choose 3,1,1 . Here, the controller design is accomplished. Next section proposes a controller reduction approach which makes the controller more practical.
V. THE CONTROLLER REDUCTION
0 -50 -100 1e-8
1e-6
1e-4 1e-2 Frequency(rad/s)
Fig. 3. (Solid) principal gains of Σ
, ,
Fig. 4. (Dashed) principal gains of
Φ .
1e0
1e2
.
One should consider that the achieved value for is not uniquely the best choice, and there may be much better options, especially using other approaches. Nevertheless, since this value satisfies the system requirements, we content with it. Checking the condition (5) we have: ν
11457 1809 21650 7491 12914 15699 1854 4642 12442 9397 2346 10883 1996 5322 4566 2350
0
Fig. 2. Principal gains of Σ
10
Up to here, we have approximately recovered a diagonalized TLTF, therefore, the system is almost decoupled into three single loops. Reaching a desired rise or settling times, each loop could be compensated as a linear SISO system. The only condition that should be considered is the phase and gain margin of each loop.
,
,
1.62 0.51 1.49
5.31 0.27 0.48
0.71 0.53 3.37
10
.
This means that the target loop ( Φ ) is at least asymptotically recoverable. Step 3. In this stage, the observer gain should be designed such that , holds pointwise in ∞. , when Designing the observer gain , we utilize an observer design method which is based on ATEA algorithm [6]. The most noticeable advantage of this algorithm is that the resulted is a continuous function of . This advantageous property makes the algorithm exceptional where the designer achieves all possible observer gains in a matrix function of . Using LINSYSKIT toolbox which is developed for the above algorithm [7], for 0.05, is computed as given in (6). Fig. 5 depicts the principal gains of , , comparing with principal gains of the chosen TLTF in previous section. Obviously, the TLTF is asymptotically recovered. Step 4. Finally, we improve the closed loop system on its time responses by modifying as a pre-compensator.
Simple lower order controllers, such as PID controllers, are more preferable, if they could maintain the system in a reasonable stability margin. In fact, reliability, economic points of view and simplicity (especially for operators) are a number of reasons which make engineers prefer a lower order controller to a higher order or computer based one. The designed controller for the current system is reduced to a PID controller in [12,19] with an almost same approach. Furthermore, [19] showed that ignoring high frequency harmonics, the reduced controller can reasonably recover the TLTF. As another approach, we aim to reduce the controller order such that high and low frequency properties of the controller are more preserved. Fig. 6 depicts principal gains of the designed total controller , . The figure shows that the controller properties are as same as neither a PI nor PID controller. Therefore, checking the principal gains of the controller ( , ) in Fig. 7 may lead us to reduce , as:
, where , determined. Assume Σ where Estimating
,
,
and should be be a realization of , ,
, ,
,
.
means that
by
.
7
Fig. 7 shows that 2.7 could be a proper selection. We such that both sides of (7) have same low frequency choose gains: . Therefore, could be
approximated as where should be chosen such that
,
is achieved as an approximately real matrix. Applying trial and error method,
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made an improvement in frequency response properties. In other words, it was possible that this divergence make the worse. 40 20 Principal gains(dB)
0.03 is achieved as a reasonable frequency. However, one can write a simple algorithm to achieve this result by checking different operating frequencies. Generally, if the search for does not have a reasonable outcome, then we can choose the one which makes the smallest possible error; otherwise, the selected reduced model does not match , and should be modified.
Principal gains(dB)
100
50
0 -20 -40 -60 -80 1e-4
0
Fig. 7. The principal gains of
-50
-100 1e-6
1e-4
1e-2 1e0 Frequency(rad/s)
1e2
1e4
Fig. 5. Solid: the TLTF principal gains; dashed: Loop transfer recovery for 0.05 (Solid: desired loop principal gains, and dashed: recovered loop principal gains by , ).
Principal gains(dB)
1e-2
50
0
-50 1e-6
1e-4
1e-2 1e0 Frequency (rad/s)
Fig. 6. The principal gains of the designed controller,
Using the selected follows
and
0.4831 1.1824 0.0007 0.2677 1.7705 0.0003 , 0.0117 1.0133 0.0032 150.1402 190.4337 0.0125 48.9993 74.7872 0.060 17.8925 21.9436 0.2499
,
and
1e2
,
.
are obtained as
.
Fig. 7 shows how the principal gains of and , match; and Fig. 8 depicts the recovery that is attained by the reduced controller. VI. SIMULATION RESULTS In this section, the results that have obtained in the two preceding sections are evaluated by their frequency and time domain responses. Fig. 9 depicts the maximum singular values of sensitivity , and complementary sensitivity function function , for the three designed controllers in the previous sections (original target loop, recovered original target loop, and reduced controller). Obviously, sensitivity functions have a 20 / slope at low frequencies. It could be recognized that of the three controllers is less than 0.9 for the frequencies ). Therefore, a less than the cut-off frequency (0.01 reasonable disturbance rejection is achieved. Similarly, Fig. 9 shows that the complementary sensitivity function has realistic characteristics. Obviously, 1 for frequencies . This implies that sensor noise greater than 0.011 attenuation is guaranteed above this frequency. Besides, robust stability will be assured at all frequencies greater than 0.01 against multiplicative disturbances that have considerable amplitudes. It is noticeable in Fig.12 that the maximum singular value of the original controller shows a peak over 0 , while it does not happen for the reduced controller. This is a deviation which is a result of the controller reduction, although it has
1e0 Frequency(rad/s)
1e2
,
(dashed).
(solid) and
1e4
Completing the robustness analysis the designed controllers, characteristic loci at output of compensated plant, using the three controllers, are shown in Fig. 10. One could confirm that the above of 0 cross over frequency, the phase is at least 90°, and hence the gain 'roll off ' rate is at least 20 / [17]. Besides, the minimum gain and phase margin of the LQG controller (a gain margin of greater than 1 2 and a phase margin between 60° and 60°) can be approximately verified. This ensures us that our recovered controller and its reduced estimation provide sufficient gain and phase margins. Another object which should be considered is the practical constraint that the fuel actuator has. Fig. 11 illustrates an approach for simulation of such actuators which have a is constraint on the derivative of their outputs. In this figure, is the real fuel amount which is the fuel actuator input and injected by the actuator. Applying the describing function approach [20], it could be shown that the system of Fig. 11 has a gain ( ⁄ ) which is between 6 and 0 for frequencies less than 0.1 . Where the input is chosen as 2.5 0.5sin . In this case, phase lag is about 30°. Notice that the selection of relies on the operating point of the system and the maximum variation of the fuel flow rate at this point. Since is ten times bigger than the system operating frequency, we can guarantee that the system remains in a reasonable gain margin when the actuator has a realistic behavior. In the sequel, we assume that a constraint with a schematic as Fig. 11 is implemented to the fuel channel. Finally, we check the time domain responses of the system in the presence of the fuel actuator constraint. Using the reduced controller, Fig. 12 shows the step responses of the system to a 10% drum level increase, 5% drum pressure rise, and a 10% steam temperature fall separately. Fig. 13 compares the recovered and reduced controller in their step responses. In this figure, the level, pressure, and temperature changes are shown when the water level setpoint increases by 10%, the pressure setpoint rises by 5% and the temperature setpoint decreases by 10% . It could be verified that the reduced controller follows the original controller convincingly. Fig. 14 shows the controller outputs for the cases that were described for the Fig. 13; and Fig.15 shows the derivative of the fuel flow rate. Fig. 15 shows that the higher frequency harmonies are filtered by the reduced controller. This has beneficial effects on the systems which have such sluggish actuators. With the aim of numerical assessing of the system’s disturbance rejection, and phase margin, we turn the feedback
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signal into a delayed noise contaminated plant outputs as follows 1
,
1
Principal gains(dB)
where s, s, s and respectively are sensor outputs, actual outputs, injected white noises, and sensors’ time response delay. The variances of the injected noises are chosen as 2, and their sampling time are 600s (selected in the worst case according to Fig. 9). The sensor time delay is selected as 33s (about 20° phase delay in operating frequency). 50 0 -50 -100 1e-4
1e-2 1e0 Frequency(rad/s)
1e2
2 Imaginary
1
observer which is designed by ATEA algorithm. The most advantageous property of LQG approach is that it yields to controllers with guaranteed phase and gain margins. Finally, the 16 order controller is reduced to a 6 order controller. We mentioned that reducing the controller to simpler orders, has the cost of recoverability loss. For example, the estimated PI controller in [19] is not as reasonable as the reduced controller of this paper. Additionally, the controller robustness against an actuators constraint is analytically and numerically checked. Consequently, output disturbance rejection and the effect of sensors time delay response are examined simultaneously as the final numerical simulation result. We conclude that LQG/LTR method is a powerful classical controller design which provides remarkable margins of stability (the phase and gain margins) for the system. These margins make the system robust against the controller order reduction, sensors response delay, actuator saturation, and output disturbance.
Fig. 8. Solid: the principal gains of the TLTF, and dashed: the principal . gains of recovered loop by 10
-2 -4 -3.5
0
Imaginary
-20
2
-80 1e-4
4 2 0 -2 -4 -6 -8 1e-2.4
Imaginary
-40
-70
-2
-1.5
-1
-0.5
0
-2 -4 -2
-60
-2.5
0
-30
-50
-3
2
-10 Maximum principal gains(dB)
0
-1.5
-1
-0.5
0
0.5
-0.5
0
0.5
-2
Real
1e-1.6
1e-2
-1
0
-4 -2
1e-2
-1.5
1e0
1e2
Frequency(rad/s)
Fig. 9. Maximum singular values of S(s) and T(s). (Solid: target (original) controller, Dashed: recovered controller, Dotted: reduced controller).
Finally, we check the time domain responses of the system in the presence of the fuel actuator constraint. Using the reduced controller, Fig. 12 shows the step responses of the system to a 10% drum level increase, 5% drum pressure rise, and a 10% steam temperature fall separately. Finally, we check the time domain responses of the system in the presence of the fuel actuator constraint. Using the reduced controller, Fig. 12 shows the step responses of the system to a 10% drum level increase, 5% drum pressure rise, and a 10% steam temperature fall separately. Apparently, is the worst theoretical possible case which might not practically happen to the system. Such disturbances rarely happen except a serious fault takes place. Although is completely far from the reality, Fig. 17 shows that the system has peak errors less than 5% for pressure and temperature outputs, while this percentage riches to 20% for the water level. Since water level is an internal state of the system, 20% deviation is tolerable in the worst case.
VII. CONCLUSION This paper is involved with LQG/LTR controller design for a non-minimum phase plant. After designing a proper state feedback gain, the TLTF is asymptotically recovered by an
Fig. 10. Characteristic loci at output of compensated plant using original controller (upper), recovered controller (middle), reduced estimated (lower) and unit circle with – 1 as center (dashed).
Fig. 11. A schematic realization for an actuator with output rate constraint.
REFERENCES [1] [2] [3] [4] [5] [6] [7]
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H. Kwakernaak, “Optimal low-sensitivity linear feedback systems,” Automatica, vol. 5, pp. 276-285, 1969. J. Doyle and G. Stein, “Robustness with observers,” IEEE Transaction Automatic Control, vol. 24, pp. 607-611, 1979. J. M. Maciejowski, Multivariable feedback design, Addison-Wesley, chapter. 5, 1989. B. M. Chen, A. Saberi and P. Sannuti, “Necessary and sufficient conditions for a non-minimum phase plant to have Recoverable Target Loop – A stable design,” Automatica, vol. 28, no. 3, pp. 493-507, 1992. A. Saberi, B. M. Chen and P. Sannuti, “Theory of LTR for non-minimum phase systems, Recoverable target loops, and recovery in a subspace Part 1. Analysis,” Int. J. Control, vol. 53, no. 5, pp. 1067-1115, 1991. A. Saberi, B. M. Chen and P. Sannuti, “Theory of LTR for non-minimum phase systems, Recoverable target loops, and recovery in a subspace Part 2. Design,” Int. J. Control, vol. 53, no. 5, pp. 1117-1160, 1991. Z. Lin, A. Saberi and B. M. Chen, “Linear systems toolbox: System analysis and control design in the Matlab environment,” Proceedings of the First IEEE Conference on Control Applications, Dayton, Ohio, USA, pp. 659-664, 1992. Downloadable at: http://vlab.ee.nus.edu.sg/~bmchen/.
0.2 Pressure
Level(m)
1 Disturbance
Pressure(MPa)
0.95
6.8 6.6
-0.2
Temperator( oC)
500
0
5
10 Time(min)
15
20
Level
400
25
350
400
5% Pressure increase
10% temperator decrease 1.01
1 0.95 0.9
6.46
465
0
5
10 Time(min)
15
20
Temperature
6.4
470
6.44
490
480
480
460
470
440
460
420
450
400
0
5
10% level increase
10 15 Time(min)
20
25
5% pressure increase 44
42
42
41
40
0
5 10 Time(min)
15
2.27
2.3
2.26
2.26
2.2
2.25 3
2.91
3.05
2.9
3
2.89
2.95
2.28
2.88
2.9
2.87
2.85
10
2.95 2.9
0
5
10 Time(min)
15
20
2.85
0
5 Time(min)
10
10% temperature decrease
0 0 0
-0.01 -0.01 0.4
350
400
50
100
150
200 Time(min)
250
300
350
400
5 0 -5
0
0.2 Time(min)
0.4
-10
20
40 60 Time(min)
80
100
Fig. 18. Derivative of fuel flow rate for the situation of Fig. 16-17.
0.01
0.2 Time(min)
300
0
0.02
0
250
-15
0.01
-0.01
200
-5
5% pressure increase
10% level increase
0.01
150
Derivative of fuel flow rate
Fig. 14. The plant inputs. (Solid: original controller, Dashed: reduced controller). 0.02
400
100
40.5
2.4
5 Time(min)
350
50
10% temperator decrease
40 2.28
0
0
41
2.5
2.3
300
Fig. 17. Closed loop system outputs error (in percent) in the presence of the injected output disturbance of Fig. 16, and an imposed 20° (in operating frequency) phase lag to the sensor responses.
38
2.32
250
0
g/s2
43
2.34
200
-2
0
Fig. 13. The boiler step time response. (Solid: original controller, Dashed: reduced controller).
40
150
6.45
6.6
475
100
10
6.46
6.42
50
2
-4
0.99
6.8
6.44
0
4 Pressure
Level(m)
300
0
-20
1
1
Pressure(MPa)
250
-10
6.48
Temperator( oC)
200
10
0.95
feedwater flow(kg/s)
150
Error in percent
1.05
fuel flow rate (kg/s)
100
20
1.05
attemperator spray flow(Kg/s)
50
450
10% Level increase
Derivative of fuel flow rate(Kg/s 2)
0
Fig. 16. The injected disturbances to the plant outputs.
1.1
-0.02
0
0.05 Time(min)
0.1
Fig. 15. Derivative of fuel flow rates for the situation of Fig.16-17. (Solid: original controller, Dashed: reduced estimated controller). [8]
0
-0.1
Fig. 12. Plant step response for reduced controller when 10% drum level setpoint increases (Solid), 5% pressure setpoint increases (Dashed) and 10% temperature setpoint decreases (Dotted).
460
Level Tem perator
0.1
L. Johanssson and H. N. Koivo, “Inverse Nyquist array technique in the of a multivariable controller for solid-fuel boiler,” Int. J. Control, vol. 40, pp. 1077-1115, 1984. [9] R. Cori, C. Maffezzoni, “Practical optimal control of a drum boiler power plant,” Automatica, vol. 20, pp. 163-173, 1984. [10] W. H. Kwon, S. W. Kim and P. G. Park, “On the multivariable robust control of a boiler-turbine system.” in Proc. IFAC Symp. Power Syst. Power Plant Contr., Seoul, Korea, pp. 219–223, 1989.
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