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New approach to milling circuit control—robust inverse Nyquist array design
Int. J. Miner. Process. 70 (2003) 171 – 182 www.elsevier.com/locate/ijminpro
New approach to milling circuit control—robust inverse Nyquist array design Dejan D. Ivezic´ a,*, Trajko B. Petrovic´ b a
Department of Mechanics and Thermodynamics, Faculty of Mining and Geology, University of Belgrade, Djusˇina 7, Belgrade 11000, Yugoslavia b Department of Control Engineering, Faculty of Electrical Engineering, University of Belgrade, Bulevar Revolucije 73, P.O. Box 35-54, Belgrade 11120, Yugoslavia Received 16 October 2002; received in revised form 10 December 2002; accepted 17 December 2002
Abstract The aim of this paper is to present an application of robust inverse Nyquist array control design method to a milling circuit in Majdanpek Copper Mine. An existent model of a milling circuit was adopted to respond to robust inverse Nyquist array methodology demand by introducing model uncertainty, and optimal robust controller of milling circuit is designed. The proposed controller is robust for reference signals. The robustness of multivariable feedback systems with designed controller is tested and compared with different, previously designed multivariable controllers. Simulation results, in time domain, demonstrate the preferable performance of the designed robust controller in setpoint tracking. Implementation of the new controller without any investments in new equipment significantly improves performance of the milling process. D 2003 Elsevier Science B.V. All rights reserved. Keywords: milling process; robust control; inverse Nyquist array
1. Introduction The ultimate goal of a control system designer is to build a system that will work in the real environment. As the real environment may change with time, and because operating conditions may vary due to disturbance changes, the control system must be able to withstand these changes. Even if the environment does not change, model uncertainty is
* Corresponding author. Fax: +381-11-324-3457. E-mail address: [email protected] (D.D. Ivezic´). 0301-7516/03/$ - see front matter D 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0301-7516(03)00022-X
172
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another issue. The uncertainties of a designed plant are caused by the linearization of a non-linear system around a non-linear operating point, neglected dynamics, component tolerances, and sensor noise. Conventional controllers, tuned for nominal systems obtained from a particular operating condition, give adequate performance for a certain range around the nominal operating point. The major disadvantage of such controllers is that they cannot guarantee overall system performance with varying operating conditions. Accordingly, the designer’s aim is to obtain a robust controller that will work with the actual plant in a real situation in order to achieve the design objectives. The conventional Inverse Nyquist Array (INA) method was a favourite engineering tool for development of controllers for milling processes in last decades, Koudstaal et al. (1981); Grujic´ (1995). Such controllers seem perfectly acceptable from an INA design point of view while the closed-loop performance is shown to be close to instability when modelling errors are present. The lack of robustness, which is not detected by INA, can be attributed to the fact that the method does not account for model uncertainty, which can adversely affect the diagonal dominance and drastically alter the stability and performance properties of multivariable control systems. Incorporating model uncertainty into the INA method enables the design of robust INA controllers, which improves the overall performance of the system. Robust INA control methodology was presented in detail in Arkun et al. (1984). The milling plant in Majdanpek Copper Mine was considered. Description of such system was given in detail in Grujic´ (1995). The model of milling circuit was obtained from its experimental identification around a particular operating point. An application of the inverse Nyquist array technique was presented for design of the INA controller (Rosenbrock, 1974). The output of this, classical design method, was a controller that had a predetermined structure and fixed parameters. For the same multivariable model, in this work, robust INA (RINA) controller is designed. The main task of the control law is to provide the perfect setpoint tracking. The final goal is to compare the robustness of closedloop systems with new RINA and previously designed Grujic´ (1995) INA controllers using transient analysis. This analysis also includes classical PI and decentralized controllers, designed in this work, for comparison and conclusion of the quality of RINA controller.
2. The plant The milling circuit at Majdanpek Concentrator is shown in Fig. 1. This circuit consists of two primary ball mills and two secondary ball mills closed by hydrocyclones. Also, in the secondary loop are two unit flotation cells that prevent the overgrinding of coarse, floatable material of high density. Both of the primary mill feeds have nuclear weight meters and feed-rate controllers. On the lines carrying the dilution water to the primary mills, unit-feed sump and cyclone-feed sump, are flow meters and control valves, so all the streams entering the plant could be monitored and controlled. On the product line that goes to the flotation plant is a flow meter and a nuclear density gauge. The concentrated stream from the unit is not monitored on-line, but is only a very small portion of the total product. In this
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Fig. 1. Milling circuit.
way, the input to the circuit and most of the output from the circuit can be measured, and so balances for water, solids and total slurry flow can be determined on-line for the whole plant. The levels in each of the sumps and the flow rates and density of the streams leaving the sumps are measured. Thus, the discharge stream from the secondary mills can be characterized from balances around the unit-feed sump and the primary mills, and the feed stream to the secondary mills can be characterized from balances around the cyclones and flotation-feed sump. The level in each sump was controlled by separate systems, which allowed them to act as a buffer against fluctuations in the incoming flow while the outgoing flow was kept under tight control. On the overflow line from the cyclone a particle size monitor (PSM) measures the particle size distribution and the mass percentage of the solids in this product stream. Only one point on the particle size distribution is measured, namely the percentage of the solids smaller than 75 Am, and no indication of the particle size spread is given.
3. Model of the plant Considering sump-level as a separate and well-solved control system, and according to a fixed ratio between solid feed rate and primary dilution, and too slow response of system to
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the unit sump dilution, solid feed (together with its dilution) and the cyclone dilution are taken for input variables, u1 and u2, respectively. As the fundamental object in milling is grinding of the ore down to a specific size, the particle size of the product, measured by PSM, was considered as first output variable—y1. One of the most critical points in a milling circuit is the cyclone underflow, since overloading at that point can lead to sandingup of the secondary mills and to discharge of very coarse material to the product. Cyclone underflow is not measured directly at Majdanpek Concentrator, but it is affected directly by flow of its feed, so flow rate of the feed to the cyclone was taken for the second output variable—y2. Using so defined variables, the dynamics of the system are determined in Grujic´ (1995) by performing step-response experiments. Their result was the transfer function matrix: 0 1 2 3 0:9362 e350s ð10:252s þ 2:819 103 Þ e200s P˜ 11 P˜ 12 B C 80218s2 þ 652s þ 1 C; 5 ¼ B 1164s þ 1 P˜ ¼ 4 @ A 36:49 1:1405 P˜ 21 P˜ 22 792s þ 1 179s þ 1 ˜ y ¼ Pu;
y ¼ ½y1 y2 T ;
u ¼ ½u1 u2 T
ð1Þ
4. The model uncertainty The assumption that the model (nominal state) P˜ of the plant matches the true plant P is only hypothetical because of the inherent existence of the model uncertainty. Let us assume that the transfer matrix of real process is in the form of 2 3 P11 P12 5 P¼4 P21 P22 2 3 k1 0:9362 ek2 350s ðk4 2:819 103 þ k5 10:252 sÞ ek6 200s 6 7 1 þ k3 1164 s 1 þ k7 652 s þ k8 80218 s2 7 ¼6 4 5 k 36:49 k 1:1405 10
1 þ k9 792 s ki a½0:9; 1:1; i ¼ 1; . . . ; 12
12
1 þ k11 179 s ð2Þ
i.e. with F 10% uncertainty in each parameter of Eq. (1). Instead of single-transfer matrix as an approximation of the system, a set of transfer matrixes were obtained as approximations of the uncontrolled plant. Therefore, introducing such a model uncertainty, formally, all other transfer matrixes which could be used as a model of the system according to step-response experiments are covered. However, the more important reason is that most other real cases, which could arise due to unknown changes in any factor that can affect the milling process, are covered. In this way, expensive, and often impossible, location of the exact influence of all those factors to system response is avoided, but we are able to account for their influence in future robust controller design.
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Fig. 2. Nyqist array of the nominal P˜ and perturbed P transfer matrix elements with uncertainty disks superimposed.
As the model of process was obtained from experimental identification of the system, uncertainty is most easily described in terms of uncertainty of the individual transfer matrix elements. Upper boundary eij(x) of additive uncertainty for individual transfer matrix elements is defined to satisfy the relation ð3Þ APij P˜ ij AVAeij A; bi; j for any change of parameter ki in Pij. It means that each element Pij(x) in plant P is independent, but confined to a disk with radius Aeij(x)A centered at P˜ij(x) in the Nyquist plane. So obtained upper boundary of uncertainty are (Fig. 2): s þ 0:05 s þ 0:00015 e12 ¼ 0:004 e11 ¼ 0:01 s þ 0:005 s þ 0:00145 ; ð4Þ s þ 0:25 s þ 0:5 e21 ¼ 0:05 e22 ¼ 0:0015 s þ 0:0025 s þ 0:005 i.e.
2
s þ 0:05 6 0:01 s þ 0:005 E ¼ P P˜ ¼ 6 4 s þ 0:25 0:05 s þ 0:0025
3 s þ 0:00015 s þ 0:00145 7 7 s þ 0:5 5 0:0015 s þ 0:005
0:004
ð5Þ
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As INA methodology demands inverse values of transfer matrixes defined above, the following notation will be used (here and in all following text symbol ^ denotes inversion): Pˆ˜ ¼ P˜ 1 Pˆ ¼ P1 Eˆ ¼ Pˆ Pˆ˜ ¼ ðP˜ þ EÞ1 P˜ 1
ð6Þ
5. Robust INA methodology We will study standard feedback system illustrated in Fig. 3. Input compensator transfer function matrix K consists of two matrixes: K ¼ K1 K2
ð7Þ
where such K1 must be selected to assure robust dominance feature of system: ˆ 1 Eˆ ˆ 1 Pˆ˜ þ K ˆ 1 Pˆ ¼ Q˜ þ QE ¼ K Qˆ ¼ K
ð8Þ
Treatment of dominance concept is the main difference between classical and robust INA methodology. The classical INA concept considers dominance regardless of model ˆ 1P˜ˆ part of Eq. (8). Such ˜ =K uncertainty, i.e. dominance is defined only with regard to the Q consideration excludes uncertainty of system and can lead to the design of a practically unacceptable controller when modelling errors are presented. Including QE = K1E˜ part of Eq. (8) in dominance consideration model uncertainty became an inherent part of design. However, this is the most difficult part of the design. Contrary to the classical INA method, such a compensator is not always guaranteed to exist. For a given nominal system, there is always error big enough to violate the robust dominance condition. If the ˆ cannot be made robustly dominant than dominance of Q ˆ + F is required system Q (dominance of closed-loop system) which can be achieved by using sufficiently low
Fig. 3. The standard multivariable feedback system.
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gains fi ( F = diag ( fi)). A direct way of synthesizing a compensator is to maximize the dominance ratios ˜ ii A AQ n X
ð9Þ
˜ ij A þ AQE AÞ þ AQE A ðAQ ij ii j¼1 j p i Graphical interpretation of robust row dominance condition P is represented by Gershgorin ˜ ˜ ii and radius ri ¼ n bands (set of circles with centre on Q j ¼ 1; jpi ðAQij A þ AQEij AÞþ ˆ is AQEii A). If each of the Gershgorin bands so produced excludes the origin, then Q ˆ robustly row-dominant. If dominance of Q + F is required, Ostrowski bands are used (Arkun et al., 1984; Rosenbrock, 1974). The rest of the robust INA design consists of single input single output (SISO) controllers design (K2) for the diagonal nominal eleˆ and adjusting the feedback gains ( fi) for the desired degree of robustness and ments of Q performance.
6. Robust INA controller design In Grujic´ (1995), Gershgorin row bands were showed and it was revealed that the uncompensated system is not nominally row-dominant. In the first design step, a ˆ robustly dominant is to be found. Individual entries of K were compensator that makes Q described as complex numbers dependent on frequency and dominance ratios which Eq. (9) tried to maximize. But the result was not fitted to stable proper transfer function matrix. Authors also tried to achieve robust column dominance by using post-compensators and solving multi-objective optimization problems given in Arkun et al. (1984) but with the same result. Thereby, it was necessary to investigate the dominance of the closed-loop ˆ + F. Transfer function matrixes, which lead to dominance of system, i.e. dominance of Q closed system, are: 3 2 7219 792s þ 1 0 7 6 79 179s þ 1 7 6 7 6 4287 7 ð10Þ K1 ¼ 6 s þ 1 7 6 204657 36156 14 7 6 5 4 70 53 8145 sþ1 109 2 3 2 3 f1 0 0:003 0 5¼4 5 F¼4 ð11Þ 0 f2 0 0:1 Robust Ostrowski row bands of closed-loop system are presented in Fig. 4 as appropriate verification of achieved dominance.
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Fig. 4. INA (only diagonal elements displayed) with robust Ostrowski bands for compensated closed system.
In the next design step, SISO controllers are designed for the diagonal transfer function obtained above. These controllers were designed to achieve certain dynamic performances (in this case, no steady-state offset and no overshoot) and are given by: 2 3 1 0 6 1 þ 150 s 7 7 K2 ¼ 6 ð12Þ 4 1 5 0 0:3 þ 500 s
7. PI and decentralized controller design For quality evaluation of proposed robust INA controller, it is interesting to compare it with industrially widespread classical PI and decentralized controllers. The structure of closed-loop systems with these controllers is covered by standard feedback system scheme (Fig. 3), where matrix F is the identity matrix and matrix K has the structure of a selected controller. More than half of the industrial controllers in use today are PI controllers. Their structure and tuning rules (Hang et al., 1991) are well known. In the discussed problem,
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parameters are tuned according to diagonal elements of nominal plant (Eq. (1)), i.e. parameters are tuning as in two SISO systems, and matrix K in that case, according to the above defined performances is: 3 2 1 1 0 6 kp1 þ Ti1 s 7 6 0:637 þ 1250s 7¼6 K¼6 4 1 5 4 0 kp2 þ 0 Ti2 s
3
2
0
7 7 1 5 100 þ 10s
ð13Þ
The concept of decentralized control (Morari and Zafirou, 1989) involves generating the ith control signal using only the ith output signal, while the influence of other output signals is neglected, i.e. controller is diagonal. The advantage is that they involve fewer tuning parameters than the full multivariable controller. It is also important that at least stability, but also performance, is preserved to a certain degree when individual sensors or actuators fail. The failure tolerance is generally easier to achieve with decentralized control systems, where parts can be turned off without significantly affecting the rest of the system. According to the desired dynamic performances matrix K, for decentralized controller, is: 2 3 2 3 ˜ 1 0:0011ð1164s þ 1Þ 0 P 0 c 1 11 1 5 5 ¼ 14 K¼ 4 ð14Þ s s 1 ˜ 0 0:4384ð179s þ 1Þ 0 c2 P22
8. The transient analysis Computer simulations are performed to evaluate the performance of the proposed robust controller. Time responses of closed-loop system, using RINA controller, classical PI controller, decentralized (DC) controller and previously designed Grujic´ (1995) INA controller are compared in order to observe how the system tracks setpoints changes. The
Fig. 5. Time responses of nominal plant to unity step signal in input u1.
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Fig. 6. Time responses of nominal plant to unity step signal in input u2.
most typical time responses of particle size of the product and flow rate of the feed to the cyclone on solid feed (together with its dilution) and the cyclone dilution with nominal and perturbed plant are shown in Figs. 5 –8. Time responses of nominal plant (Eq. (1)) are shown in Figs. 5 and 6. Responses of the closed-loop system to unity step signal in input u1 with both, RINA and INA, controllers show better performances compared with PI and DC controllers (zero y2 response), i.e. effect of including dominance concept in design procedure is obvious. The existence of time-delay in transfer matrix diagonal element precludes realization of y1 zero time responses to unity step signal in input u2 with both RINA and INA controllers, but the amplitude of that response is insignificant. RINA controller shows extremely better performances than INA controller (almost 100 times shorter settling time and no peak value) and similar performances as PI and DC controllers in output y2 to input u2.
Fig. 7. Time responses of perturbed plant (Eq. (15)) to unity step signal in input u1.
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Fig. 8. Time responses of perturbed plant (Eq. (15)) to unity step signal in input u2.
For satisfactory confirmation of robust performances, time responses for the worst-case assumption, i.e. for chosen combinations of maximal model parameters from Eq. (2), which give perturbation plant model in the form of:
2
1:1 0:9362 e1:1350s 6 1 þ 0:9 1164 s P¼6 4 1:1 36:49 1 þ 0:9 792 s
3 ð0:9 2:819 103 þ 1:1 10:252 sÞ e1:1200s 7 1 þ 1:1 652 s þ 0:9 80218 s2 7 5 0:9 1:1405 1 þ 1:1 179 s ð15Þ
are computed and shown in Figs. 7 and 8. Such time domain simulations are the best tools for extremely good confirmation of robust performances. In real cases, uncertainties will be smaller, so that corresponding time responses will have similar features, though less distinctive. Figs. 7 and 8 show that responses of closed-loop system with RINA controller and with perturbed plant have similar features as with nominal plant. It is quite clear according to fact that uncertainty of plant model was included in design procedure. Again, the disadvantage of the classical INA design, the fact that uncertainty is not taken into account, is visible. Dominance of the closed-loop system with the INA controller is significantly violated and responses are slower with greater peak values compared with RINA controller. Response is especially bad to step signal in input u1. This is because INA concept deals only with the nominal plant in achievement of dominance, therefore changes in the plant model seriously damage system dominance. Responses of closed-loop system with PI and DC controllers and with perturbed plant are worse compared with closed-loop system with RINA controller. Also, although their responses are worse than in the nominal case, their responses are better than the responses of INA controller.
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9. Conclusions The main purpose of this paper is to present the robust INA methodology in robust controller design of milling process to deal with model uncertainty. Such an approach guarantees the stable closed-loop system and certain performance requirements, which satisfy all possible operating conditions as defined by the system uncertainty descriptions. The used process has previously been modelled. The structure and the upper bound of process uncertainty are adopted in accordance with robust INA framework. These premises served as the basis for designing optimal robust controller of nominal process. In order to verify results, the most illustrative time responses are showed. Compared with previously designed INA controllers; and here, by classical methods, designed PI and decentralized controllers, the robust controller shows significantly better responses for both nominal and perturbed processes. Implementation of robust controller doesn’t demand any new investment. All existing equipment, measuring instruments and hardware remain in use. Robust INA requires the same control structure as INA controller (Fig. 3), and the only change is in the implemented software, according to new controller parameters. This will provide more quality, cheaper and more reliable work in the Majdanpek Copper mine milling circuit.
References Arkun, Y., Manousiothakis, B., Putz, P., 1984. Robust Nyquist array methodology: a new theoretical framework for analysis and design of robust multivariable feedback design. International Journal of Control 40 (4), 603 – 629. Grujic´, M., 1995. Intelligent control of Majdanpek flotation plant. Minerals Engineering 7 (14), 689 – 704. ˚ stro¨m, K.J., Ho, W.K., 1991. Refinements of the Zieger – Nichols tuning formula. IEE Proceedings. Hang, C.C., A Part D 138 (2), 111 – 118. Koudstaal, J., Hulbert, D.G., Braae, M., Gossman, G.I., 1981. The application of a multivariable controller to an industrial grinding circuit. Proceedings of the 8th Triennial World Congress. Control Science and Technology for the Progress of Society, vol. 22 (104 – 110). IFAC, Kyoto, pp. 204 – 215. Morari, M., Zafirou, E., 1989. Robust Process Control. Prentice Hall, Englewood Cliffs, NJ. Rosenbrock, H.H., 1974. Computer-Aided Control System Design. Academic Press, London.
New approach to milling circuit control—robust inverse Nyquist array design Dejan D. Ivezic´ a,*, Trajko B. Petrovic´ b a
Department of Mechanics and Thermodynamics, Faculty of Mining and Geology, University of Belgrade, Djusˇina 7, Belgrade 11000, Yugoslavia b Department of Control Engineering, Faculty of Electrical Engineering, University of Belgrade, Bulevar Revolucije 73, P.O. Box 35-54, Belgrade 11120, Yugoslavia Received 16 October 2002; received in revised form 10 December 2002; accepted 17 December 2002
Abstract The aim of this paper is to present an application of robust inverse Nyquist array control design method to a milling circuit in Majdanpek Copper Mine. An existent model of a milling circuit was adopted to respond to robust inverse Nyquist array methodology demand by introducing model uncertainty, and optimal robust controller of milling circuit is designed. The proposed controller is robust for reference signals. The robustness of multivariable feedback systems with designed controller is tested and compared with different, previously designed multivariable controllers. Simulation results, in time domain, demonstrate the preferable performance of the designed robust controller in setpoint tracking. Implementation of the new controller without any investments in new equipment significantly improves performance of the milling process. D 2003 Elsevier Science B.V. All rights reserved. Keywords: milling process; robust control; inverse Nyquist array
1. Introduction The ultimate goal of a control system designer is to build a system that will work in the real environment. As the real environment may change with time, and because operating conditions may vary due to disturbance changes, the control system must be able to withstand these changes. Even if the environment does not change, model uncertainty is
* Corresponding author. Fax: +381-11-324-3457. E-mail address: [email protected] (D.D. Ivezic´). 0301-7516/03/$ - see front matter D 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0301-7516(03)00022-X
172
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another issue. The uncertainties of a designed plant are caused by the linearization of a non-linear system around a non-linear operating point, neglected dynamics, component tolerances, and sensor noise. Conventional controllers, tuned for nominal systems obtained from a particular operating condition, give adequate performance for a certain range around the nominal operating point. The major disadvantage of such controllers is that they cannot guarantee overall system performance with varying operating conditions. Accordingly, the designer’s aim is to obtain a robust controller that will work with the actual plant in a real situation in order to achieve the design objectives. The conventional Inverse Nyquist Array (INA) method was a favourite engineering tool for development of controllers for milling processes in last decades, Koudstaal et al. (1981); Grujic´ (1995). Such controllers seem perfectly acceptable from an INA design point of view while the closed-loop performance is shown to be close to instability when modelling errors are present. The lack of robustness, which is not detected by INA, can be attributed to the fact that the method does not account for model uncertainty, which can adversely affect the diagonal dominance and drastically alter the stability and performance properties of multivariable control systems. Incorporating model uncertainty into the INA method enables the design of robust INA controllers, which improves the overall performance of the system. Robust INA control methodology was presented in detail in Arkun et al. (1984). The milling plant in Majdanpek Copper Mine was considered. Description of such system was given in detail in Grujic´ (1995). The model of milling circuit was obtained from its experimental identification around a particular operating point. An application of the inverse Nyquist array technique was presented for design of the INA controller (Rosenbrock, 1974). The output of this, classical design method, was a controller that had a predetermined structure and fixed parameters. For the same multivariable model, in this work, robust INA (RINA) controller is designed. The main task of the control law is to provide the perfect setpoint tracking. The final goal is to compare the robustness of closedloop systems with new RINA and previously designed Grujic´ (1995) INA controllers using transient analysis. This analysis also includes classical PI and decentralized controllers, designed in this work, for comparison and conclusion of the quality of RINA controller.
2. The plant The milling circuit at Majdanpek Concentrator is shown in Fig. 1. This circuit consists of two primary ball mills and two secondary ball mills closed by hydrocyclones. Also, in the secondary loop are two unit flotation cells that prevent the overgrinding of coarse, floatable material of high density. Both of the primary mill feeds have nuclear weight meters and feed-rate controllers. On the lines carrying the dilution water to the primary mills, unit-feed sump and cyclone-feed sump, are flow meters and control valves, so all the streams entering the plant could be monitored and controlled. On the product line that goes to the flotation plant is a flow meter and a nuclear density gauge. The concentrated stream from the unit is not monitored on-line, but is only a very small portion of the total product. In this
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Fig. 1. Milling circuit.
way, the input to the circuit and most of the output from the circuit can be measured, and so balances for water, solids and total slurry flow can be determined on-line for the whole plant. The levels in each of the sumps and the flow rates and density of the streams leaving the sumps are measured. Thus, the discharge stream from the secondary mills can be characterized from balances around the unit-feed sump and the primary mills, and the feed stream to the secondary mills can be characterized from balances around the cyclones and flotation-feed sump. The level in each sump was controlled by separate systems, which allowed them to act as a buffer against fluctuations in the incoming flow while the outgoing flow was kept under tight control. On the overflow line from the cyclone a particle size monitor (PSM) measures the particle size distribution and the mass percentage of the solids in this product stream. Only one point on the particle size distribution is measured, namely the percentage of the solids smaller than 75 Am, and no indication of the particle size spread is given.
3. Model of the plant Considering sump-level as a separate and well-solved control system, and according to a fixed ratio between solid feed rate and primary dilution, and too slow response of system to
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174
the unit sump dilution, solid feed (together with its dilution) and the cyclone dilution are taken for input variables, u1 and u2, respectively. As the fundamental object in milling is grinding of the ore down to a specific size, the particle size of the product, measured by PSM, was considered as first output variable—y1. One of the most critical points in a milling circuit is the cyclone underflow, since overloading at that point can lead to sandingup of the secondary mills and to discharge of very coarse material to the product. Cyclone underflow is not measured directly at Majdanpek Concentrator, but it is affected directly by flow of its feed, so flow rate of the feed to the cyclone was taken for the second output variable—y2. Using so defined variables, the dynamics of the system are determined in Grujic´ (1995) by performing step-response experiments. Their result was the transfer function matrix: 0 1 2 3 0:9362 e350s ð10:252s þ 2:819 103 Þ e200s P˜ 11 P˜ 12 B C 80218s2 þ 652s þ 1 C; 5 ¼ B 1164s þ 1 P˜ ¼ 4 @ A 36:49 1:1405 P˜ 21 P˜ 22 792s þ 1 179s þ 1 ˜ y ¼ Pu;
y ¼ ½y1 y2 T ;
u ¼ ½u1 u2 T
ð1Þ
4. The model uncertainty The assumption that the model (nominal state) P˜ of the plant matches the true plant P is only hypothetical because of the inherent existence of the model uncertainty. Let us assume that the transfer matrix of real process is in the form of 2 3 P11 P12 5 P¼4 P21 P22 2 3 k1 0:9362 ek2 350s ðk4 2:819 103 þ k5 10:252 sÞ ek6 200s 6 7 1 þ k3 1164 s 1 þ k7 652 s þ k8 80218 s2 7 ¼6 4 5 k 36:49 k 1:1405 10
1 þ k9 792 s ki a½0:9; 1:1; i ¼ 1; . . . ; 12
12
1 þ k11 179 s ð2Þ
i.e. with F 10% uncertainty in each parameter of Eq. (1). Instead of single-transfer matrix as an approximation of the system, a set of transfer matrixes were obtained as approximations of the uncontrolled plant. Therefore, introducing such a model uncertainty, formally, all other transfer matrixes which could be used as a model of the system according to step-response experiments are covered. However, the more important reason is that most other real cases, which could arise due to unknown changes in any factor that can affect the milling process, are covered. In this way, expensive, and often impossible, location of the exact influence of all those factors to system response is avoided, but we are able to account for their influence in future robust controller design.
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Fig. 2. Nyqist array of the nominal P˜ and perturbed P transfer matrix elements with uncertainty disks superimposed.
As the model of process was obtained from experimental identification of the system, uncertainty is most easily described in terms of uncertainty of the individual transfer matrix elements. Upper boundary eij(x) of additive uncertainty for individual transfer matrix elements is defined to satisfy the relation ð3Þ APij P˜ ij AVAeij A; bi; j for any change of parameter ki in Pij. It means that each element Pij(x) in plant P is independent, but confined to a disk with radius Aeij(x)A centered at P˜ij(x) in the Nyquist plane. So obtained upper boundary of uncertainty are (Fig. 2): s þ 0:05 s þ 0:00015 e12 ¼ 0:004 e11 ¼ 0:01 s þ 0:005 s þ 0:00145 ; ð4Þ s þ 0:25 s þ 0:5 e21 ¼ 0:05 e22 ¼ 0:0015 s þ 0:0025 s þ 0:005 i.e.
2
s þ 0:05 6 0:01 s þ 0:005 E ¼ P P˜ ¼ 6 4 s þ 0:25 0:05 s þ 0:0025
3 s þ 0:00015 s þ 0:00145 7 7 s þ 0:5 5 0:0015 s þ 0:005
0:004
ð5Þ
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As INA methodology demands inverse values of transfer matrixes defined above, the following notation will be used (here and in all following text symbol ^ denotes inversion): Pˆ˜ ¼ P˜ 1 Pˆ ¼ P1 Eˆ ¼ Pˆ Pˆ˜ ¼ ðP˜ þ EÞ1 P˜ 1
ð6Þ
5. Robust INA methodology We will study standard feedback system illustrated in Fig. 3. Input compensator transfer function matrix K consists of two matrixes: K ¼ K1 K2
ð7Þ
where such K1 must be selected to assure robust dominance feature of system: ˆ 1 Eˆ ˆ 1 Pˆ˜ þ K ˆ 1 Pˆ ¼ Q˜ þ QE ¼ K Qˆ ¼ K
ð8Þ
Treatment of dominance concept is the main difference between classical and robust INA methodology. The classical INA concept considers dominance regardless of model ˆ 1P˜ˆ part of Eq. (8). Such ˜ =K uncertainty, i.e. dominance is defined only with regard to the Q consideration excludes uncertainty of system and can lead to the design of a practically unacceptable controller when modelling errors are presented. Including QE = K1E˜ part of Eq. (8) in dominance consideration model uncertainty became an inherent part of design. However, this is the most difficult part of the design. Contrary to the classical INA method, such a compensator is not always guaranteed to exist. For a given nominal system, there is always error big enough to violate the robust dominance condition. If the ˆ cannot be made robustly dominant than dominance of Q ˆ + F is required system Q (dominance of closed-loop system) which can be achieved by using sufficiently low
Fig. 3. The standard multivariable feedback system.
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gains fi ( F = diag ( fi)). A direct way of synthesizing a compensator is to maximize the dominance ratios ˜ ii A AQ n X
ð9Þ
˜ ij A þ AQE AÞ þ AQE A ðAQ ij ii j¼1 j p i Graphical interpretation of robust row dominance condition P is represented by Gershgorin ˜ ˜ ii and radius ri ¼ n bands (set of circles with centre on Q j ¼ 1; jpi ðAQij A þ AQEij AÞþ ˆ is AQEii A). If each of the Gershgorin bands so produced excludes the origin, then Q ˆ robustly row-dominant. If dominance of Q + F is required, Ostrowski bands are used (Arkun et al., 1984; Rosenbrock, 1974). The rest of the robust INA design consists of single input single output (SISO) controllers design (K2) for the diagonal nominal eleˆ and adjusting the feedback gains ( fi) for the desired degree of robustness and ments of Q performance.
6. Robust INA controller design In Grujic´ (1995), Gershgorin row bands were showed and it was revealed that the uncompensated system is not nominally row-dominant. In the first design step, a ˆ robustly dominant is to be found. Individual entries of K were compensator that makes Q described as complex numbers dependent on frequency and dominance ratios which Eq. (9) tried to maximize. But the result was not fitted to stable proper transfer function matrix. Authors also tried to achieve robust column dominance by using post-compensators and solving multi-objective optimization problems given in Arkun et al. (1984) but with the same result. Thereby, it was necessary to investigate the dominance of the closed-loop ˆ + F. Transfer function matrixes, which lead to dominance of system, i.e. dominance of Q closed system, are: 3 2 7219 792s þ 1 0 7 6 79 179s þ 1 7 6 7 6 4287 7 ð10Þ K1 ¼ 6 s þ 1 7 6 204657 36156 14 7 6 5 4 70 53 8145 sþ1 109 2 3 2 3 f1 0 0:003 0 5¼4 5 F¼4 ð11Þ 0 f2 0 0:1 Robust Ostrowski row bands of closed-loop system are presented in Fig. 4 as appropriate verification of achieved dominance.
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Fig. 4. INA (only diagonal elements displayed) with robust Ostrowski bands for compensated closed system.
In the next design step, SISO controllers are designed for the diagonal transfer function obtained above. These controllers were designed to achieve certain dynamic performances (in this case, no steady-state offset and no overshoot) and are given by: 2 3 1 0 6 1 þ 150 s 7 7 K2 ¼ 6 ð12Þ 4 1 5 0 0:3 þ 500 s
7. PI and decentralized controller design For quality evaluation of proposed robust INA controller, it is interesting to compare it with industrially widespread classical PI and decentralized controllers. The structure of closed-loop systems with these controllers is covered by standard feedback system scheme (Fig. 3), where matrix F is the identity matrix and matrix K has the structure of a selected controller. More than half of the industrial controllers in use today are PI controllers. Their structure and tuning rules (Hang et al., 1991) are well known. In the discussed problem,
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parameters are tuned according to diagonal elements of nominal plant (Eq. (1)), i.e. parameters are tuning as in two SISO systems, and matrix K in that case, according to the above defined performances is: 3 2 1 1 0 6 kp1 þ Ti1 s 7 6 0:637 þ 1250s 7¼6 K¼6 4 1 5 4 0 kp2 þ 0 Ti2 s
3
2
0
7 7 1 5 100 þ 10s
ð13Þ
The concept of decentralized control (Morari and Zafirou, 1989) involves generating the ith control signal using only the ith output signal, while the influence of other output signals is neglected, i.e. controller is diagonal. The advantage is that they involve fewer tuning parameters than the full multivariable controller. It is also important that at least stability, but also performance, is preserved to a certain degree when individual sensors or actuators fail. The failure tolerance is generally easier to achieve with decentralized control systems, where parts can be turned off without significantly affecting the rest of the system. According to the desired dynamic performances matrix K, for decentralized controller, is: 2 3 2 3 ˜ 1 0:0011ð1164s þ 1Þ 0 P 0 c 1 11 1 5 5 ¼ 14 K¼ 4 ð14Þ s s 1 ˜ 0 0:4384ð179s þ 1Þ 0 c2 P22
8. The transient analysis Computer simulations are performed to evaluate the performance of the proposed robust controller. Time responses of closed-loop system, using RINA controller, classical PI controller, decentralized (DC) controller and previously designed Grujic´ (1995) INA controller are compared in order to observe how the system tracks setpoints changes. The
Fig. 5. Time responses of nominal plant to unity step signal in input u1.
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Fig. 6. Time responses of nominal plant to unity step signal in input u2.
most typical time responses of particle size of the product and flow rate of the feed to the cyclone on solid feed (together with its dilution) and the cyclone dilution with nominal and perturbed plant are shown in Figs. 5 –8. Time responses of nominal plant (Eq. (1)) are shown in Figs. 5 and 6. Responses of the closed-loop system to unity step signal in input u1 with both, RINA and INA, controllers show better performances compared with PI and DC controllers (zero y2 response), i.e. effect of including dominance concept in design procedure is obvious. The existence of time-delay in transfer matrix diagonal element precludes realization of y1 zero time responses to unity step signal in input u2 with both RINA and INA controllers, but the amplitude of that response is insignificant. RINA controller shows extremely better performances than INA controller (almost 100 times shorter settling time and no peak value) and similar performances as PI and DC controllers in output y2 to input u2.
Fig. 7. Time responses of perturbed plant (Eq. (15)) to unity step signal in input u1.
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Fig. 8. Time responses of perturbed plant (Eq. (15)) to unity step signal in input u2.
For satisfactory confirmation of robust performances, time responses for the worst-case assumption, i.e. for chosen combinations of maximal model parameters from Eq. (2), which give perturbation plant model in the form of:
2
1:1 0:9362 e1:1350s 6 1 þ 0:9 1164 s P¼6 4 1:1 36:49 1 þ 0:9 792 s
3 ð0:9 2:819 103 þ 1:1 10:252 sÞ e1:1200s 7 1 þ 1:1 652 s þ 0:9 80218 s2 7 5 0:9 1:1405 1 þ 1:1 179 s ð15Þ
are computed and shown in Figs. 7 and 8. Such time domain simulations are the best tools for extremely good confirmation of robust performances. In real cases, uncertainties will be smaller, so that corresponding time responses will have similar features, though less distinctive. Figs. 7 and 8 show that responses of closed-loop system with RINA controller and with perturbed plant have similar features as with nominal plant. It is quite clear according to fact that uncertainty of plant model was included in design procedure. Again, the disadvantage of the classical INA design, the fact that uncertainty is not taken into account, is visible. Dominance of the closed-loop system with the INA controller is significantly violated and responses are slower with greater peak values compared with RINA controller. Response is especially bad to step signal in input u1. This is because INA concept deals only with the nominal plant in achievement of dominance, therefore changes in the plant model seriously damage system dominance. Responses of closed-loop system with PI and DC controllers and with perturbed plant are worse compared with closed-loop system with RINA controller. Also, although their responses are worse than in the nominal case, their responses are better than the responses of INA controller.
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9. Conclusions The main purpose of this paper is to present the robust INA methodology in robust controller design of milling process to deal with model uncertainty. Such an approach guarantees the stable closed-loop system and certain performance requirements, which satisfy all possible operating conditions as defined by the system uncertainty descriptions. The used process has previously been modelled. The structure and the upper bound of process uncertainty are adopted in accordance with robust INA framework. These premises served as the basis for designing optimal robust controller of nominal process. In order to verify results, the most illustrative time responses are showed. Compared with previously designed INA controllers; and here, by classical methods, designed PI and decentralized controllers, the robust controller shows significantly better responses for both nominal and perturbed processes. Implementation of robust controller doesn’t demand any new investment. All existing equipment, measuring instruments and hardware remain in use. Robust INA requires the same control structure as INA controller (Fig. 3), and the only change is in the implemented software, according to new controller parameters. This will provide more quality, cheaper and more reliable work in the Majdanpek Copper mine milling circuit.
References Arkun, Y., Manousiothakis, B., Putz, P., 1984. Robust Nyquist array methodology: a new theoretical framework for analysis and design of robust multivariable feedback design. International Journal of Control 40 (4), 603 – 629. Grujic´, M., 1995. Intelligent control of Majdanpek flotation plant. Minerals Engineering 7 (14), 689 – 704. ˚ stro¨m, K.J., Ho, W.K., 1991. Refinements of the Zieger – Nichols tuning formula. IEE Proceedings. Hang, C.C., A Part D 138 (2), 111 – 118. Koudstaal, J., Hulbert, D.G., Braae, M., Gossman, G.I., 1981. The application of a multivariable controller to an industrial grinding circuit. Proceedings of the 8th Triennial World Congress. Control Science and Technology for the Progress of Society, vol. 22 (104 – 110). IFAC, Kyoto, pp. 204 – 215. Morari, M., Zafirou, E., 1989. Robust Process Control. Prentice Hall, Englewood Cliffs, NJ. Rosenbrock, H.H., 1974. Computer-Aided Control System Design. Academic Press, London.